{"text": "[STATEMENT]\nlemma inner_mult_diag_real':\n fixes B::\"complex Matrix.mat\"\n assumes \"diagonal_mat B\"\n and \"B \\ carrier_mat n n\"\n and \"\\i < n. B$$(i, i) \\ Reals\"\n and \"v \\ carrier_vec n\"\nshows \"inner_prod v (B *\\<^sub>v v) \\ Reals\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod v (B *\\<^sub>v v) \\ \\\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod v (B *\\<^sub>v v) \\ \\\n[PROOF STEP]\nhave \"inner_prod v (B *\\<^sub>v v) = \n (\\ i \\ {0 ..< n}. B $$ (i,i) * (vec_index v i * \n (conjugate (vec_index v i))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod v (B *\\<^sub>v v) = (\\i = 0..diagonal_mat ?B; ?B \\ carrier_mat ?n ?n; ?v \\ carrier_vec ?n\\ \\ Complex_Matrix.inner_prod ?v (?B *\\<^sub>v ?v) = (\\i = 0.. carrier_mat n n\n\\i \\\nv \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod v (B *\\<^sub>v v) = (\\i = 0..v v) = (\\i = 0..v v) \\ \\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod v (B *\\<^sub>v v) = (\\i = 0..v v) \\ \\\n[PROOF STEP]\nhave \"... \\ Reals\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0.. \\\n[PROOF STEP]\nproof (rule real_sum_real)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i. i < n \\ B $$ (i, i) * (v $ i * conjugate (v $ i)) \\ \\\n[PROOF STEP]\nshow \"\\i. i < n \\ \n B $$ (i, i) * ((vec_index v i) * conjugate (vec_index v i)) \\ \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i < n \\ B $$ (i, i) * (v $ i * conjugate (v $ i)) \\ \\\n[PROOF STEP]\nusing assms mult_conj_real\n[PROOF STATE]\nproof (prove)\nusing this:\ndiagonal_mat B\nB \\ carrier_mat n n\n\\i \\\nv \\ carrier_vec n\n?v * conjugate ?v \\ \\\n\ngoal (1 subgoal):\n 1. \\i. i < n \\ B $$ (i, i) * (v $ i * conjugate (v $ i)) \\ \\\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n?i < n \\ B $$ (?i, ?i) * (v $ ?i * conjugate (v $ ?i)) \\ \\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0.. \\\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod v (B *\\<^sub>v v) \\ \\\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nComplex_Matrix.inner_prod v (B *\\<^sub>v v) \\ \\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex_Matrix.inner_prod v (B *\\<^sub>v v) \\ \\\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod v (B *\\<^sub>v v) \\ \\\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod v (B *\\<^sub>v v) \\ \\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1639, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473713594991, "lm_q2_score": 0.8596637469145053, "lm_q1q2_score": 0.7499254098739264}} {"text": "[STATEMENT]\nlemma length_around_zero:\n assumes \"i >= 0\" \n shows \"length (around_zero i) = 2 * nat i + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (around_zero i) = 2 * nat i + 1\n[PROOF STEP]\nproof (induct rule: int_ge_induct [OF assms])\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. length (around_zero 0) = 2 * nat 0 + 1\n 2. \\i. \\0 \\ i; length (around_zero i) = 2 * nat i + 1\\ \\ length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\ncase 1\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. length (around_zero 0) = 2 * nat 0 + 1\n 2. \\i. \\0 \\ i; length (around_zero i) = 2 * nat i + 1\\ \\ length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\nfrom 1\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (around_zero 0) = 2 * nat 0 + 1\n[PROOF STEP]\nby (simp add: around_zero.simps)\n[PROOF STATE]\nproof (state)\nthis:\nlength (around_zero 0) = 2 * nat 0 + 1\n\ngoal (1 subgoal):\n 1. \\i. \\0 \\ i; length (around_zero i) = 2 * nat i + 1\\ \\ length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i. \\0 \\ i; length (around_zero i) = 2 * nat i + 1\\ \\ length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\ncase (2 i)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ i\nlength (around_zero i) = 2 * nat i + 1\n\ngoal (1 subgoal):\n 1. \\i. \\0 \\ i; length (around_zero i) = 2 * nat i + 1\\ \\ length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\nfrom 2\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ i\nlength (around_zero i) = 2 * nat i + 1\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ i\nlength (around_zero i) = 2 * nat i + 1\n\ngoal (1 subgoal):\n 1. length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\nby (simp add: around_zero.simps [of \"i + 1\"])\n[PROOF STATE]\nproof (state)\nthis:\nlength (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1021, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772417253256, "lm_q2_score": 0.8577681049901037, "lm_q1q2_score": 0.7497555792497094}} {"text": "[STATEMENT]\nlemma tensor_mat_id:\n assumes \"0 < d1\"\n and \"0 < d2\"\nshows \"tensor_mat (1\\<^sub>m d1) (1\\<^sub>m d2) = 1\\<^sub>m (d1 * d2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1\\<^sub>m d1 \\ 1\\<^sub>m d2 = 1\\<^sub>m (d1 * d2)\n[PROOF STEP]\nproof (rule eq_matI, auto)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\j. j < d1 * d2 \\ (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (j, j) = 1\n 2. \\i j. \\i < d1 * d2; j < d1 * d2; i \\ j\\ \\ (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (i, j) = 0\n[PROOF STEP]\nshow \"tensor_mat (1\\<^sub>m d1) (1\\<^sub>m d2) $$ (i, i) = 1\" if \"i < (d1 * d2)\" for i\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (i, i) = 1\n[PROOF STEP]\nusing that index_tensor_mat'[of \"1\\<^sub>m d1\" \"1\\<^sub>m d2\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ni < d1 * d2\n\\0 < dim_col (1\\<^sub>m d1); 0 < dim_col (1\\<^sub>m d2); ?i < dim_row (1\\<^sub>m d1) * dim_row (1\\<^sub>m d2); ?j < dim_col (1\\<^sub>m d1) * dim_col (1\\<^sub>m d2)\\ \\ (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (?i, ?j) = 1\\<^sub>m d1 $$ (?i div dim_row (1\\<^sub>m d2), ?j div dim_col (1\\<^sub>m d2)) * 1\\<^sub>m d2 $$ (?i mod dim_row (1\\<^sub>m d2), ?j mod dim_col (1\\<^sub>m d2))\n\ngoal (1 subgoal):\n 1. (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (i, i) = 1\n[PROOF STEP]\nby (simp add: assms less_mult_imp_div_less)\n[PROOF STATE]\nproof (state)\nthis:\n?i < d1 * d2 \\ (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (?i, ?i) = 1\n\ngoal (1 subgoal):\n 1. \\i j. \\i < d1 * d2; j < d1 * d2; i \\ j\\ \\ (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (i, j) = 0\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i j. \\i < d1 * d2; j < d1 * d2; i \\ j\\ \\ (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (i, j) = 0\n[PROOF STEP]\nshow \"tensor_mat (1\\<^sub>m d1) (1\\<^sub>m d2) $$ (i, j) = 0\" if \"i < d1 * d2\" \"j < d1 * d2\" \"i \\ j\" for i j\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (i, j) = 0\n[PROOF STEP]\nusing that index_tensor_mat[of \"1\\<^sub>m d1\" d1 d1 \"1\\<^sub>m d2\" d2 d2 i j]\n[PROOF STATE]\nproof (prove)\nusing this:\ni < d1 * d2\nj < d1 * d2\ni \\ j\n\\dim_row (1\\<^sub>m d1) = d1; dim_col (1\\<^sub>m d1) = d1; dim_row (1\\<^sub>m d2) = d2; dim_col (1\\<^sub>m d2) = d2; i < d1 * d2; j < d1 * d2; 0 < d1; 0 < d2\\ \\ (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (i, j) = 1\\<^sub>m d1 $$ (i div d2, j div d2) * 1\\<^sub>m d2 $$ (i mod d2, j mod d2)\n\ngoal (1 subgoal):\n 1. (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (i, j) = 0\n[PROOF STEP]\nby (metis assms(1) assms(2) index_one_mat(1) index_one_mat(2) index_one_mat(3) \n less_mult_imp_div_less mod_less_divisor mult_div_mod_eq mult_not_zero)\n[PROOF STATE]\nproof (state)\nthis:\n\\?i < d1 * d2; ?j < d1 * d2; ?i \\ ?j\\ \\ (1\\<^sub>m d1 \\ 1\\<^sub>m d2) $$ (?i, ?j) = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1596, "file": "Projective_Measurements_Linear_Algebra_Complements", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942290328345, "lm_q2_score": 0.8418256532040708, "lm_q1q2_score": 0.7494725208993805}} {"text": "[STATEMENT]\ntheorem euler_maclaurin_strong_nat':\n assumes \"a \\ b\"\n shows \"(\\k=a..b. f (real k)) = \n F (real b) + C + (1 / 2) *\\<^sub>R f (real b) +\n (\\i=1..N. (bernoulli (2*i) / fact (2*i)) *\\<^sub>R fs (2*i-1) (real b)) - \n EM_remainder (2*N+1) (fs (2*N+1)) b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = a..b. f (real k)) = F (real b) + C + (1 / 2) *\\<^sub>R f (real b) + (\\i = 1..N. (bernoulli (2 * i) / fact (2 * i)) *\\<^sub>R fs (2 * i - 1) (real b)) - EM_remainder (2 * N + 1) (fs (2 * N + 1)) (int b)\n[PROOF STEP]\nusing euler_maclaurin_strong_int'[of b]\n[PROOF STATE]\nproof (prove)\nusing this:\nint a \\ int b \\ (\\k = int a..int b. f (real_of_int k)) = F (real_of_int (int b)) + C + (1 / 2) *\\<^sub>R f (real_of_int (int b)) + (\\i = 1..N. (bernoulli (2 * i) / fact (2 * i)) *\\<^sub>R fs (2 * i - 1) (real_of_int (int b))) - EM_remainder (2 * N + 1) (fs (2 * N + 1)) (int b)\n\ngoal (1 subgoal):\n 1. (\\k = a..b. f (real k)) = F (real b) + C + (1 / 2) *\\<^sub>R f (real b) + (\\i = 1..N. (bernoulli (2 * i) / fact (2 * i)) *\\<^sub>R fs (2 * i - 1) (real b)) - EM_remainder (2 * N + 1) (fs (2 * N + 1)) (int b)\n[PROOF STEP]\nby (simp add: assms sum_atLeastAtMost_of_int_nat_transfer)", "meta": {"llama_tokens": 632, "file": "Euler_MacLaurin_Euler_MacLaurin", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9219218305645894, "lm_q2_score": 0.8128673269042767, "lm_q1q2_score": 0.7494001340257352}} {"text": "[STATEMENT]\nlemma sum_incr2: \"sum f (incr2 ` A) = sum (\\xy. f(fst xy,snd xy+1)) A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f (incr2 ` A) = (\\xy\\A. f (fst xy, snd xy + 1))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum f (incr2 ` A) = (\\xy\\A. f (fst xy, snd xy + 1))\n[PROOF STEP]\nhave \"sum f (incr2 ` A) = sum (f o incr2) A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f (incr2 ` A) = sum (f \\ incr2) A\n[PROOF STEP]\nby(subst sum.reindex)(auto simp: inj_on_def)\n[PROOF STATE]\nproof (state)\nthis:\nsum f (incr2 ` A) = sum (f \\ incr2) A\n\ngoal (1 subgoal):\n 1. sum f (incr2 ` A) = (\\xy\\A. f (fst xy, snd xy + 1))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum f (incr2 ` A) = sum (f \\ incr2) A\n\ngoal (1 subgoal):\n 1. sum f (incr2 ` A) = (\\xy\\A. f (fst xy, snd xy + 1))\n[PROOF STEP]\nhave \"f o incr2 = (\\xy. f(fst xy,snd xy+1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f \\ incr2 = (\\xy. f (fst xy, snd xy + 1))\n[PROOF STEP]\nby(auto simp: fun_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\nf \\ incr2 = (\\xy. f (fst xy, snd xy + 1))\n\ngoal (1 subgoal):\n 1. sum f (incr2 ` A) = (\\xy\\A. f (fst xy, snd xy + 1))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum f (incr2 ` A) = (\\a\\A. f (fst a, snd a + 1))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsum f (incr2 ` A) = (\\a\\A. f (fst a, snd a + 1))\n\ngoal (1 subgoal):\n 1. sum f (incr2 ` A) = (\\xy\\A. f (fst xy, snd xy + 1))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum f (incr2 ` A) = (\\xy\\A. f (fst xy, snd xy + 1))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 899, "file": "Optimal_BST_Weighted_Path_Length", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767906859264, "lm_q2_score": 0.8539127566694178, "lm_q1q2_score": 0.7492886252480532}} {"text": "[STATEMENT]\nlemma exp_not_zero_imp_exp_diff_not_zero:\n \\2 ^ (n - m) \\ 0\\ if \\2 ^ n \\ 0\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (2::'a) ^ (n - m) \\ (0::'a)\n[PROOF STEP]\nproof (cases \\m \\ n\\)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n 2. \\ m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nm \\ n\n\ngoal (2 subgoals):\n 1. m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n 2. \\ m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nm \\ n\n\ngoal (2 subgoals):\n 1. m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n 2. \\ m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n[PROOF STEP]\ndefine q where \\q = n - m\\\n[PROOF STATE]\nproof (state)\nthis:\nq = n - m\n\ngoal (2 subgoals):\n 1. m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n 2. \\ m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nm \\ n\nq = n - m\n[PROOF STEP]\nhave \\n = m + q\\\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ n\nq = n - m\n\ngoal (1 subgoal):\n 1. n = m + q\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn = m + q\n\ngoal (2 subgoals):\n 1. m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n 2. \\ m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n[PROOF STEP]\nwith that\n[PROOF STATE]\nproof (chain)\npicking this:\n(2::'a) ^ n \\ (0::'a)\nn = m + q\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(2::'a) ^ n \\ (0::'a)\nn = m + q\n\ngoal (1 subgoal):\n 1. (2::'a) ^ (n - m) \\ (0::'a)\n[PROOF STEP]\nby (simp add: exp_add_not_zero_imp_right)\n[PROOF STATE]\nproof (state)\nthis:\n(2::'a) ^ (n - m) \\ (0::'a)\n\ngoal (1 subgoal):\n 1. \\ m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ m \\ n\n\ngoal (1 subgoal):\n 1. \\ m \\ n \\ (2::'a) ^ (n - m) \\ (0::'a)\n[PROOF STEP]\nwith that\n[PROOF STATE]\nproof (chain)\npicking this:\n(2::'a) ^ n \\ (0::'a)\n\\ m \\ n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(2::'a) ^ n \\ (0::'a)\n\\ m \\ n\n\ngoal (1 subgoal):\n 1. (2::'a) ^ (n - m) \\ (0::'a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(2::'a) ^ (n - m) \\ (0::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1399, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213799730775, "lm_q2_score": 0.8333246015211008, "lm_q1q2_score": 0.7492599656851671}} {"text": "[STATEMENT]\nlemma invfun_surj:\"\\f \\ A \\ B; inj_on f A; surj_to f A B\\\n \\ surj_to (invfun A B f) B A \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ A \\ B; inj_on f A; surj_to f A B\\ \\ surj_to (f\\\\<^bsub>B,A\\<^esub>) B A\n[PROOF STEP]\napply (simp add:surj_to_def [of \"invfun A B f\" \"B\" \"A\"] image_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ A \\ B; inj_on f A; surj_to f A B\\ \\ {y. \\x\\B. y = (f\\\\<^bsub>B,A\\<^esub>) x} = A\n[PROOF STEP]\napply (rule equalityI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\f \\ A \\ B; inj_on f A; surj_to f A B\\ \\ {y. \\x\\B. y = (f\\\\<^bsub>B,A\\<^esub>) x} \\ A\n 2. \\f \\ A \\ B; inj_on f A; surj_to f A B\\ \\ A \\ {y. \\x\\B. y = (f\\\\<^bsub>B,A\\<^esub>) x}\n[PROOF STEP]\napply (rule subsetI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. \\f \\ A \\ B; inj_on f A; surj_to f A B; x \\ {y. \\x\\B. y = (f\\\\<^bsub>B,A\\<^esub>) x}\\ \\ x \\ A\n 2. \\f \\ A \\ B; inj_on f A; surj_to f A B\\ \\ A \\ {y. \\x\\B. y = (f\\\\<^bsub>B,A\\<^esub>) x}\n[PROOF STEP]\napply (simp add:CollectI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. \\f \\ A \\ B; inj_on f A; surj_to f A B; \\xa\\B. x = (f\\\\<^bsub>B,A\\<^esub>) xa\\ \\ x \\ A\n 2. \\f \\ A \\ B; inj_on f A; surj_to f A B\\ \\ A \\ {y. \\x\\B. y = (f\\\\<^bsub>B,A\\<^esub>) x}\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\xa. \\f \\ A \\ B; inj_on f A; surj_to f A B; xa \\ B\\ \\ (f\\\\<^bsub>B,A\\<^esub>) xa \\ A\n 2. \\x. \\f \\ A \\ B; inj_on f A; surj_to f A B; x \\ A\\ \\ \\xa\\B. x = (f\\\\<^bsub>B,A\\<^esub>) xa\n[PROOF STEP]\napply (simp add:invfun_mem)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\f \\ A \\ B; inj_on f A; surj_to f A B; x \\ A\\ \\ \\xa\\B. x = (f\\\\<^bsub>B,A\\<^esub>) xa\n[PROOF STEP]\napply (frule funcset_mem [of \"f\" \"A\" \"B\"], assumption+)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\f \\ A \\ B; inj_on f A; surj_to f A B; x \\ A; f x \\ B\\ \\ \\xa\\B. x = (f\\\\<^bsub>B,A\\<^esub>) xa\n[PROOF STEP]\napply (frule_tac t = x in invfun_l [of \"f\" \"A\" \"B\", THEN sym], assumption+)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\f \\ A \\ B; inj_on f A; surj_to f A B; x \\ A; f x \\ B; x = (f\\\\<^bsub>B,A\\<^esub>) (f x)\\ \\ \\xa\\B. x = (f\\\\<^bsub>B,A\\<^esub>) xa\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1516, "file": "Group-Ring-Module_Algebra1", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314828740729, "lm_q2_score": 0.8459424295406088, "lm_q1q2_score": 0.7491932483001453}} {"text": "[STATEMENT]\ntheorem integer_composition_enum_correct: \n \"set (integer_composition_enum n) = integer_compositions n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (integer_composition_enum n) = integer_compositions n\n[PROOF STEP]\nproof standard\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. set (integer_composition_enum n) \\ integer_compositions n\n 2. integer_compositions n \\ set (integer_composition_enum n)\n[PROOF STEP]\nshow \"set (integer_composition_enum n) \\ integer_compositions n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (integer_composition_enum n) \\ integer_compositions n\n[PROOF STEP]\nunfolding integer_compositions_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (integer_composition_enum n) \\ {xs. sum_list xs = n \\ 0 \\ set xs}\n[PROOF STEP]\nusing integer_composition_enum_not_null integer_composition_enum_sum\n[PROOF STATE]\nproof (prove)\nusing this:\n?xs \\ set (integer_composition_enum ?n) \\ 0 \\ set ?xs\n?xs \\ set (integer_composition_enum ?n) \\ sum_list ?xs = ?n\n\ngoal (1 subgoal):\n 1. set (integer_composition_enum n) \\ {xs. sum_list xs = n \\ 0 \\ set xs}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nset (integer_composition_enum n) \\ integer_compositions n\n\ngoal (1 subgoal):\n 1. integer_compositions n \\ set (integer_composition_enum n)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. integer_compositions n \\ set (integer_composition_enum n)\n[PROOF STEP]\nshow \"integer_compositions n \\ set (integer_composition_enum n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integer_compositions n \\ set (integer_composition_enum n)\n[PROOF STEP]\nunfolding integer_compositions_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {xs. sum_list xs = n \\ 0 \\ set xs} \\ set (integer_composition_enum n)\n[PROOF STEP]\nusing integer_composition_enum_correct_aux\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ set ?xs \\ ?xs \\ set (integer_composition_enum (sum_list ?xs))\n\ngoal (1 subgoal):\n 1. {xs. sum_list xs = n \\ 0 \\ set xs} \\ set (integer_composition_enum n)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninteger_compositions n \\ set (integer_composition_enum n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 895, "file": "Combinatorial_Enumeration_Algorithms_Integer_Compositions", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314738181875, "lm_q2_score": 0.8459424334245618, "lm_q1q2_score": 0.7491932440791386}} {"text": "[STATEMENT]\nlemma add_eq_inf_sup: \"a + b = sup a b + inf a b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a + b = sup a b + inf a b\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. a + b = sup a b + inf a b\n[PROOF STEP]\nhave \"0 = - inf 0 (a - b) + inf (a - b) 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) = - inf (0::'a) (a - b) + inf (a - b) (0::'a)\n[PROOF STEP]\nby (simp add: inf_commute)\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) = - inf (0::'a) (a - b) + inf (a - b) (0::'a)\n\ngoal (1 subgoal):\n 1. a + b = sup a b + inf a b\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(0::'a) = - inf (0::'a) (a - b) + inf (a - b) (0::'a)\n[PROOF STEP]\nhave \"0 = sup 0 (b - a) + inf (a - b) 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) = - inf (0::'a) (a - b) + inf (a - b) (0::'a)\n\ngoal (1 subgoal):\n 1. (0::'a) = sup (0::'a) (b - a) + inf (a - b) (0::'a)\n[PROOF STEP]\nby (simp add: inf_eq_neg_sup)\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) = sup (0::'a) (b - a) + inf (a - b) (0::'a)\n\ngoal (1 subgoal):\n 1. a + b = sup a b + inf a b\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(0::'a) = sup (0::'a) (b - a) + inf (a - b) (0::'a)\n[PROOF STEP]\nhave \"0 = (- a + sup a b) + (inf a b + (- b))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) = sup (0::'a) (b - a) + inf (a - b) (0::'a)\n\ngoal (1 subgoal):\n 1. (0::'a) = - a + sup a b + (inf a b + - b)\n[PROOF STEP]\nby (simp only: add_sup_distrib_left add_inf_distrib_right) simp\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) = - a + sup a b + (inf a b + - b)\n\ngoal (1 subgoal):\n 1. a + b = sup a b + inf a b\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(0::'a) = - a + sup a b + (inf a b + - b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) = - a + sup a b + (inf a b + - b)\n\ngoal (1 subgoal):\n 1. a + b = sup a b + inf a b\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\na + b = sup a b + inf a b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1026, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9111797003640646, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7491620495615238}} {"text": "[STATEMENT]\nlemma homeomorphic_path_connected_space_imp:\n \"\\path_connected_space X; X homeomorphic_space Y\\ \\ path_connected_space Y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\path_connected_space X; X homeomorphic_space Y\\ \\ path_connected_space Y\n[PROOF STEP]\nunfolding homeomorphic_space_def homeomorphic_maps_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\path_connected_space X; \\f g. continuous_map X Y f \\ continuous_map Y X g \\ (\\x\\topspace X. g (f x) = x) \\ (\\y\\topspace Y. f (g y) = y)\\ \\ path_connected_space Y\n[PROOF STEP]\nby (metis (no_types, opaque_lifting) continuous_map_closedin continuous_map_image_subset_topspace imageI order_class.order.antisym path_connectedin_continuous_map_image path_connectedin_topspace subsetI)", "meta": {"llama_tokens": 319, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9241418241572634, "lm_q2_score": 0.8104789155369047, "lm_q1q2_score": 0.7489974634452757}} {"text": "[STATEMENT]\nlemma (in prob_space) exponential_distributed_expectation:\n \"0 < l \\ distributed M lborel X (exponential_density l) \\ expectation X = 1 / l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < l; distributed M lborel X (\\x. ennreal (exponential_density l x))\\ \\ expectation X = 1 / l\n[PROOF STEP]\nusing erlang_ith_moment[of l X 0 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < l; distributed M lborel X (\\x. ennreal (exponential_density l x))\\ \\ expectation (\\x. X x ^ 1) = fact (0 + 1) / (fact 0 * l ^ 1)\n\ngoal (1 subgoal):\n 1. \\0 < l; distributed M lborel X (\\x. ennreal (exponential_density l x))\\ \\ expectation X = 1 / l\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 314, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9241418178895028, "lm_q2_score": 0.8104789178257653, "lm_q1q2_score": 0.7489974604806197}} {"text": "[STATEMENT]\nlemma min_plus_max:\n shows \"(min a b) + (max a b) = a + b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. min a b + max a b = a + b\n[PROOF STEP]\nproof (cases \"a \\ b\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. a \\ b \\ min a b + max a b = a + b\n 2. \\ a \\ b \\ min a b + max a b = a + b\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\na \\ b\n\ngoal (2 subgoals):\n 1. a \\ b \\ min a b + max a b = a + b\n 2. \\ a \\ b \\ min a b + max a b = a + b\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ b\n\ngoal (1 subgoal):\n 1. min a b + max a b = a + b\n[PROOF STEP]\nunfolding min_def max_def\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ b\n\ngoal (1 subgoal):\n 1. (if a \\ b then a else b) + (if a \\ b then b else a) = a + b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmin a b + max a b = a + b\n\ngoal (1 subgoal):\n 1. \\ a \\ b \\ min a b + max a b = a + b\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ a \\ b \\ min a b + max a b = a + b\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ a \\ b\n\ngoal (1 subgoal):\n 1. \\ a \\ b \\ min a b + max a b = a + b\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ a \\ b\n\ngoal (1 subgoal):\n 1. min a b + max a b = a + b\n[PROOF STEP]\nunfolding min_def max_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ a \\ b\n\ngoal (1 subgoal):\n 1. (if a \\ b then a else b) + (if a \\ b then b else a) = a + b\n[PROOF STEP]\nby (simp add: ac_simps)\n[PROOF STATE]\nproof (state)\nthis:\nmin a b + max a b = a + b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 795, "file": "Polynomials_Power_Products", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.879146761176671, "lm_q2_score": 0.8519528000888386, "lm_q1q2_score": 0.7489915448734983}} {"text": "[STATEMENT]\nlemma even_square_cong_4_int:\n fixes x :: int\n assumes \"even x\"\n shows \"[x ^ 2 = 0] (mod 4)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 0] (mod 4)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 0] (mod 4)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\neven x\n[PROOF STEP]\nhave \"even \\x\\\"\n[PROOF STATE]\nproof (prove)\nusing this:\neven x\n\ngoal (1 subgoal):\n 1. even \\x\\\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\neven \\x\\\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 0] (mod 4)\n[PROOF STEP]\nhence [simp]: \"\\x\\ mod 2 = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\neven \\x\\\n\ngoal (1 subgoal):\n 1. \\x\\ mod 2 = 0\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ mod 2 = 0\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 0] (mod 4)\n[PROOF STEP]\nhave \"(\\x\\ ^ 2) mod 4 = ((\\x\\ mod 4) ^ 2) mod 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\\\<^sup>2 mod 4 = (\\x\\ mod 4)\\<^sup>2 mod 4\n[PROOF STEP]\nby (simp add: power_mod)\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\\\<^sup>2 mod 4 = (\\x\\ mod 4)\\<^sup>2 mod 4\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 0] (mod 4)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\\\<^sup>2 mod 4 = (\\x\\ mod 4)\\<^sup>2 mod 4\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 0] (mod 4)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\neven x\n[PROOF STEP]\nhave \"\\x\\ mod 4 = 0 \\ \\x\\ mod 4 = 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\neven x\n\ngoal (1 subgoal):\n 1. \\x\\ mod 4 = 0 \\ \\x\\ mod 4 = 2\n[PROOF STEP]\nusing mod_double_modulus[of 2 \"\\x\\\"]\n[PROOF STATE]\nproof (prove)\nusing this:\neven x\n\\0 < 2; 0 \\ \\x\\\\ \\ \\x\\ mod (2 * 2) = \\x\\ mod 2 \\ \\x\\ mod (2 * 2) = \\x\\ mod 2 + 2\n\ngoal (1 subgoal):\n 1. \\x\\ mod 4 = 0 \\ \\x\\ mod 4 = 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ mod 4 = 0 \\ \\x\\ mod 4 = 2\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 0] (mod 4)\n[PROOF STEP]\nhence \"((\\x\\ mod 4) ^ 2) mod 4 = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\ mod 4 = 0 \\ \\x\\ mod 4 = 2\n\ngoal (1 subgoal):\n 1. (\\x\\ mod 4)\\<^sup>2 mod 4 = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\ mod 4)\\<^sup>2 mod 4 = 0\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 0] (mod 4)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x\\\\<^sup>2 mod 4 = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\\\<^sup>2 mod 4 = 0\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 0] (mod 4)\n[PROOF STEP]\nby (simp add: cong_def)\n[PROOF STATE]\nproof (state)\nthis:\n[x\\<^sup>2 = 0] (mod 4)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1493, "file": "Gaussian_Integers_Gaussian_Integers", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.89181104831338, "lm_q2_score": 0.8397339656668287, "lm_q1q2_score": 0.7488840282256863}} {"text": "[STATEMENT]\nlemma matrix_inv_matrix_change_of_basis:\n fixes X::\"'a::{field}^'n^'n\" and Y::\"'a^'n^'n\"\n assumes basis_X: \"is_basis (set_of_vector X)\" and basis_Y: \"is_basis (set_of_vector Y)\"\n shows\"matrix_change_of_basis Y X = matrix_inv (matrix_change_of_basis X Y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_change_of_basis Y X = matrix_inv (matrix_change_of_basis X Y)\n[PROOF STEP]\nproof (rule matrix_inv_unique[symmetric])\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n 2. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n[PROOF STEP]\nhave linear_id: \"linear ((*s)) ((*s)) id\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. linear (*s) (*s) id\n[PROOF STEP]\nby (metis vec.linear_id)\n[PROOF STATE]\nproof (state)\nthis:\nlinear (*s) (*s) id\n\ngoal (2 subgoals):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n 2. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n[PROOF STEP]\nhave \"(matrix_change_of_basis Y X) ** (matrix_change_of_basis X Y) = (matrix' Y X id) ** (matrix' X Y id)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = matrix' Y X id ** matrix' X Y id\n[PROOF STEP]\nunfolding matrix'_id_eq_matrix_change_of_basis[OF basis_X basis_Y]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = matrix' Y X id ** matrix_change_of_basis X Y\n[PROOF STEP]\nunfolding matrix'_id_eq_matrix_change_of_basis[OF basis_Y basis_X]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = matrix_change_of_basis Y X ** matrix_change_of_basis X Y\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_change_of_basis Y X ** matrix_change_of_basis X Y = matrix' Y X id ** matrix' X Y id\n\ngoal (2 subgoals):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n 2. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_change_of_basis Y X ** matrix_change_of_basis X Y = matrix' Y X id ** matrix' X Y id\n\ngoal (2 subgoals):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n 2. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n[PROOF STEP]\nhave \"... = matrix' X X (id \\ id)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix' Y X id ** matrix' X Y id = matrix' X X (id \\ id)\n[PROOF STEP]\nusing matrix'_compose[OF basis_X basis_Y basis_X linear_id linear_id]\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix' X X (id \\ id) = matrix' Y X id ** matrix' X Y id\n\ngoal (1 subgoal):\n 1. matrix' Y X id ** matrix' X Y id = matrix' X X (id \\ id)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nmatrix' Y X id ** matrix' X Y id = matrix' X X (id \\ id)\n\ngoal (2 subgoals):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n 2. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmatrix' Y X id ** matrix' X Y id = matrix' X X (id \\ id)\n\ngoal (2 subgoals):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n 2. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n[PROOF STEP]\nhave \"... = matrix_change_of_basis X X\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix' X X (id \\ id) = matrix_change_of_basis X X\n[PROOF STEP]\nusing matrix'_id_eq_matrix_change_of_basis[OF basis_X basis_X]\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix' X X id = matrix_change_of_basis X X\n\ngoal (1 subgoal):\n 1. matrix' X X (id \\ id) = matrix_change_of_basis X X\n[PROOF STEP]\nunfolding o_def id_def\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix' X X (\\x. x) = matrix_change_of_basis X X\n\ngoal (1 subgoal):\n 1. matrix' X X (\\x. x) = matrix_change_of_basis X X\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nmatrix' X X (id \\ id) = matrix_change_of_basis X X\n\ngoal (2 subgoals):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n 2. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmatrix' X X (id \\ id) = matrix_change_of_basis X X\n\ngoal (2 subgoals):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n 2. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n[PROOF STEP]\nhave \"... = mat 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_change_of_basis X X = mat (1::'a)\n[PROOF STEP]\nusing matrix_change_of_basis_mat_1[OF basis_X]\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix_change_of_basis X X = mat (1::'a)\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis X X = mat (1::'a)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_change_of_basis X X = mat (1::'a)\n\ngoal (2 subgoals):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n 2. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nmatrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n[PROOF STEP]\nshow \"matrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_change_of_basis Y X ** matrix_change_of_basis X Y = mat (1::'a)\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n[PROOF STEP]\nhave \"(matrix_change_of_basis X Y) ** (matrix_change_of_basis Y X) = (matrix' X Y id) ** (matrix' Y X id)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = matrix' X Y id ** matrix' Y X id\n[PROOF STEP]\nunfolding matrix'_id_eq_matrix_change_of_basis[OF basis_X basis_Y]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = matrix_change_of_basis X Y ** matrix' Y X id\n[PROOF STEP]\nunfolding matrix'_id_eq_matrix_change_of_basis[OF basis_Y basis_X]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = matrix_change_of_basis X Y ** matrix_change_of_basis Y X\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_change_of_basis X Y ** matrix_change_of_basis Y X = matrix' X Y id ** matrix' Y X id\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_change_of_basis X Y ** matrix_change_of_basis Y X = matrix' X Y id ** matrix' Y X id\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n[PROOF STEP]\nhave \"... = matrix' Y Y (id \\ id)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix' X Y id ** matrix' Y X id = matrix' Y Y (id \\ id)\n[PROOF STEP]\nusing matrix'_compose[OF basis_Y basis_X basis_Y linear_id linear_id]\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix' Y Y (id \\ id) = matrix' X Y id ** matrix' Y X id\n\ngoal (1 subgoal):\n 1. matrix' X Y id ** matrix' Y X id = matrix' Y Y (id \\ id)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nmatrix' X Y id ** matrix' Y X id = matrix' Y Y (id \\ id)\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmatrix' X Y id ** matrix' Y X id = matrix' Y Y (id \\ id)\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n[PROOF STEP]\nhave \"... = matrix_change_of_basis Y Y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix' Y Y (id \\ id) = matrix_change_of_basis Y Y\n[PROOF STEP]\nusing matrix'_id_eq_matrix_change_of_basis[OF basis_Y basis_Y]\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix' Y Y id = matrix_change_of_basis Y Y\n\ngoal (1 subgoal):\n 1. matrix' Y Y (id \\ id) = matrix_change_of_basis Y Y\n[PROOF STEP]\nunfolding o_def id_def\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix' Y Y (\\x. x) = matrix_change_of_basis Y Y\n\ngoal (1 subgoal):\n 1. matrix' Y Y (\\x. x) = matrix_change_of_basis Y Y\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nmatrix' Y Y (id \\ id) = matrix_change_of_basis Y Y\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmatrix' Y Y (id \\ id) = matrix_change_of_basis Y Y\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n[PROOF STEP]\nhave \"... = mat 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_change_of_basis Y Y = mat (1::'a)\n[PROOF STEP]\nusing matrix_change_of_basis_mat_1[OF basis_Y]\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix_change_of_basis Y Y = mat (1::'a)\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis Y Y = mat (1::'a)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_change_of_basis Y Y = mat (1::'a)\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nmatrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n[PROOF STEP]\nshow \"matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n\ngoal (1 subgoal):\n 1. matrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_change_of_basis X Y ** matrix_change_of_basis Y X = mat (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4311, "file": "Gauss_Jordan_Linear_Maps", "length": 44, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513786759491, "lm_q2_score": 0.8354835391516133, "lm_q1q2_score": 0.7488032738256947}} {"text": "[STATEMENT]\nlemma least_powerI:\n assumes \"(f ^^ n) x = x\" and \"n > 0\"\n shows \"(f ^^ (least_power f x)) x = x\" and \"least_power f x > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f ^^ least_power f x) x = x &&& 0 < least_power f x\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(f ^^ n) x = x\n0 < n\n\ngoal (1 subgoal):\n 1. (f ^^ least_power f x) x = x &&& 0 < least_power f x\n[PROOF STEP]\nunfolding least_power_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(f ^^ n) x = x\n0 < n\n\ngoal (1 subgoal):\n 1. (f ^^ (LEAST n. (f ^^ n) x = x \\ 0 < n)) x = x &&& 0 < (LEAST n. (f ^^ n) x = x \\ 0 < n)\n[PROOF STEP]\nby (metis (mono_tags, lifting) LeastI)+", "meta": {"llama_tokens": 329, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942014971871, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.7487756189527229}} {"text": "[STATEMENT]\nlemma rank_1_proj_trace_inner:\n fixes A :: \"'a::conjugatable_field Matrix.mat\" and v :: \"'a Matrix.vec\"\n assumes A: \"A \\ carrier_mat n n\"\n and v: \"v \\ carrier_vec n\"\n shows \"Complex_Matrix.trace (A * (rank_1_proj v)) = Complex_Matrix.inner_prod v (A *\\<^sub>v v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (A * rank_1_proj v) = Complex_Matrix.inner_prod v (A *\\<^sub>v v)\n[PROOF STEP]\nusing assms trace_outer_prod_right[of A]\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n n\nv \\ carrier_vec n\n\\A \\ carrier_mat ?n ?n; ?v \\ carrier_vec ?n; ?w \\ carrier_vec ?n\\ \\ Complex_Matrix.trace (A * outer_prod ?v ?w) = Complex_Matrix.inner_prod ?w (A *\\<^sub>v ?v)\n\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (A * rank_1_proj v) = Complex_Matrix.inner_prod v (A *\\<^sub>v v)\n[PROOF STEP]\nunfolding rank_1_proj_def\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n n\nv \\ carrier_vec n\n\\A \\ carrier_mat ?n ?n; ?v \\ carrier_vec ?n; ?w \\ carrier_vec ?n\\ \\ Complex_Matrix.trace (A * outer_prod ?v ?w) = Complex_Matrix.inner_prod ?w (A *\\<^sub>v ?v)\n\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (A * outer_prod v v) = Complex_Matrix.inner_prod v (A *\\<^sub>v v)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 562, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297941266014, "lm_q2_score": 0.831143054132195, "lm_q1q2_score": 0.7484690834274204}} {"text": "[STATEMENT]\ntheorem (in linear_map) rank_nullity_main: \n assumes fd: \"V.fin_dim\"\n shows \"(vectorspace.dim K (W.vs imT)) + (vectorspace.dim K (V.vs kerT)) = V.dim\" \n \"T ` (carrier V) = carrier W \\ W.fin_dim\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vectorspace.dim K (W.vs imT) + vectorspace.dim K (V.vs kerT) = V.dim &&& (T ` carrier V = carrier W \\ W.fin_dim)\n[PROOF STEP]\nproof - \n \\ \\First interpret kerT, imT as vectorspaces\\\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. vectorspace.dim K (W.vs imT) + vectorspace.dim K (V.vs kerT) = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave subs_ker: \"subspace K kerT V\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. subspace K kerT V\n[PROOF STEP]\nby (intro kerT_is_subspace)\n[PROOF STATE]\nproof (state)\nthis:\nsubspace K kerT V\n\ngoal (2 subgoals):\n 1. vectorspace.dim K (W.vs imT) + vectorspace.dim K (V.vs kerT) = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom subs_ker\n[PROOF STATE]\nproof (chain)\npicking this:\nsubspace K kerT V\n[PROOF STEP]\nhave vs_ker: \"vectorspace K (V.vs kerT)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsubspace K kerT V\n\ngoal (1 subgoal):\n 1. vectorspace K (V.vs kerT)\n[PROOF STEP]\nby (rule V.subspace_is_vs)\n[PROOF STATE]\nproof (state)\nthis:\nvectorspace K (V.vs kerT)\n\ngoal (2 subgoals):\n 1. vectorspace.dim K (W.vs imT) + vectorspace.dim K (V.vs kerT) = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom vs_ker\n[PROOF STATE]\nproof (chain)\npicking this:\nvectorspace K (V.vs kerT)\n[PROOF STEP]\ninterpret ker: vectorspace K \"(V.vs kerT)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nvectorspace K (V.vs kerT)\n\ngoal (1 subgoal):\n 1. vectorspace K (V.vs kerT)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. vectorspace.dim K (W.vs imT) + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave kerInC: \"kerT\\carrier V\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. kerT \\ carrier V\n[PROOF STEP]\nby (unfold ker_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nkerT \\ carrier V\n\ngoal (2 subgoals):\n 1. vectorspace.dim K (W.vs imT) + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave subs_im: \"subspace K imT W\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. subspace K imT W\n[PROOF STEP]\nby (intro imT_is_subspace)\n[PROOF STATE]\nproof (state)\nthis:\nsubspace K imT W\n\ngoal (2 subgoals):\n 1. vectorspace.dim K (W.vs imT) + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom subs_im\n[PROOF STATE]\nproof (chain)\npicking this:\nsubspace K imT W\n[PROOF STEP]\nhave vs_im: \"vectorspace K (W.vs imT)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsubspace K imT W\n\ngoal (1 subgoal):\n 1. vectorspace K (W.vs imT)\n[PROOF STEP]\nby (rule W.subspace_is_vs)\n[PROOF STATE]\nproof (state)\nthis:\nvectorspace K (W.vs imT)\n\ngoal (2 subgoals):\n 1. vectorspace.dim K (W.vs imT) + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom vs_im\n[PROOF STATE]\nproof (chain)\npicking this:\nvectorspace K (W.vs imT)\n[PROOF STEP]\ninterpret im: vectorspace K \"(W.vs imT)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nvectorspace K (W.vs imT)\n\ngoal (1 subgoal):\n 1. vectorspace K (W.vs imT)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave imInC: \"imT\\carrier W\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. imT \\ carrier W\n[PROOF STEP]\nby (unfold im_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nimT \\ carrier W\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\n(* obvious fact *)\n[PROOF STATE]\nproof (state)\nthis:\nimT \\ carrier W\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave zero_same[simp]: \"\\\\<^bsub>V.vs kerT\\<^esub> = \\\\<^bsub>V\\<^esub>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^bsub>V.vs kerT\\<^esub> = \\\\<^bsub>V\\<^esub>\n[PROOF STEP]\napply (unfold ker_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^bsub>V.vs {v \\ carrier V. T v = \\\\<^bsub>W\\<^esub>}\\<^esub> = \\\\<^bsub>V\\<^esub>\n[PROOF STEP]\nby auto\n \\ \\Show ker T has a finite basis. This is not obvious. Show that any linearly independent set \nhas size at most that of V. There exists a maximal linearly independent set, which is the basis.\\\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^bsub>V.vs kerT\\<^esub> = \\\\<^bsub>V\\<^esub>\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave every_li_small: \"\\A. (A \\ kerT)\\ ker.lin_indpt A \\ \n finite A \\ card A \\ V.dim\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A. A \\ kerT \\ ker.lin_indpt A \\ finite A \\ card A \\ V.dim\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\A. A \\ kerT \\ ker.lin_indpt A \\ finite A \\ card A \\ V.dim\n[PROOF STEP]\nfix A\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\A. A \\ kerT \\ ker.lin_indpt A \\ finite A \\ card A \\ V.dim\n[PROOF STEP]\nassume eli_asm: \"(A \\ kerT)\\ ker.lin_indpt A\"\n[PROOF STATE]\nproof (state)\nthis:\nA \\ kerT \\ ker.lin_indpt A\n\ngoal (1 subgoal):\n 1. \\A. A \\ kerT \\ ker.lin_indpt A \\ finite A \\ card A \\ V.dim\n[PROOF STEP]\n(*annoying: I can't just use subst V.module.span_li_not_depend(2) in the show ?thesis \n statement because it doesn't appear in the conclusion.*)\n[PROOF STATE]\nproof (state)\nthis:\nA \\ kerT \\ ker.lin_indpt A\n\ngoal (1 subgoal):\n 1. \\A. A \\ kerT \\ ker.lin_indpt A \\ finite A \\ card A \\ V.dim\n[PROOF STEP]\nnote V.module.span_li_not_depend(2)[where ?N=\"kerT\" and ?S=\"A\"]\n[PROOF STATE]\nproof (state)\nthis:\n\\A \\ kerT; LinearCombinations.submodule K kerT V\\ \\ ker.lin_dep A = V.module.lin_dep A\n\ngoal (1 subgoal):\n 1. \\A. A \\ kerT \\ ker.lin_indpt A \\ finite A \\ card A \\ V.dim\n[PROOF STEP]\nfrom this subs_ker fd eli_asm kerInC\n[PROOF STATE]\nproof (chain)\npicking this:\n\\A \\ kerT; LinearCombinations.submodule K kerT V\\ \\ ker.lin_dep A = V.module.lin_dep A\nsubspace K kerT V\nV.fin_dim\nA \\ kerT \\ ker.lin_indpt A\nkerT \\ carrier V\n[PROOF STEP]\nshow \"?thesis A\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A \\ kerT; LinearCombinations.submodule K kerT V\\ \\ ker.lin_dep A = V.module.lin_dep A\nsubspace K kerT V\nV.fin_dim\nA \\ kerT \\ ker.lin_indpt A\nkerT \\ carrier V\n\ngoal (1 subgoal):\n 1. finite A \\ card A \\ V.dim\n[PROOF STEP]\napply (intro conjI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\\\A \\ kerT; LinearCombinations.submodule K kerT V\\ \\ ker.lin_dep A = V.module.lin_dep A; subspace K kerT V; V.fin_dim; A \\ kerT \\ ker.lin_indpt A; kerT \\ carrier V\\ \\ finite A\n 2. \\\\A \\ kerT; LinearCombinations.submodule K kerT V\\ \\ ker.lin_dep A = V.module.lin_dep A; subspace K kerT V; V.fin_dim; A \\ kerT \\ ker.lin_indpt A; kerT \\ carrier V\\ \\ card A \\ V.dim\n[PROOF STEP]\nby (auto intro!: V.li_le_dim)\n[PROOF STATE]\nproof (state)\nthis:\nfinite A \\ card A \\ V.dim\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n?A1 \\ kerT \\ ker.lin_indpt ?A1 \\ finite ?A1 \\ card ?A1 \\ V.dim\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom every_li_small\n[PROOF STATE]\nproof (chain)\npicking this:\n?A1 \\ kerT \\ ker.lin_indpt ?A1 \\ finite ?A1 \\ card ?A1 \\ V.dim\n[PROOF STEP]\nhave exA: \n \"\\A. finite A \\ maximal A (\\S. S\\carrier (V.vs kerT) \\ ker.lin_indpt S)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?A1 \\ kerT \\ ker.lin_indpt ?A1 \\ finite ?A1 \\ card ?A1 \\ V.dim\n\ngoal (1 subgoal):\n 1. \\A. finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S)\n[PROOF STEP]\napply (intro maximal_exists[where ?N=\"V.dim\" and ?B=\"{}\"])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\A. \\\\A. A \\ kerT \\ ker.lin_indpt A \\ finite A \\ card A \\ V.dim; A \\ carrier (V.vs kerT) \\ ker.lin_indpt A\\ \\ finite A \\ card A \\ V.dim\n 2. (\\A. A \\ kerT \\ ker.lin_indpt A \\ finite A \\ card A \\ V.dim) \\ {} \\ carrier (V.vs kerT) \\ ker.lin_indpt {}\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\A. A \\ kerT \\ ker.lin_indpt A \\ finite A \\ card A \\ V.dim; ker.lin_dep {}\\ \\ False\n[PROOF STEP]\nby (unfold ker.lin_dep_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\n\\A. finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S)\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom exA\n[PROOF STATE]\nproof (chain)\npicking this:\n\\A. finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S)\n[PROOF STEP]\nobtain A where A:\" finite A \\ maximal A (\\S. S\\carrier (V.vs kerT) \\ ker.lin_indpt S)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A. finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S)\n\ngoal (1 subgoal):\n 1. (\\A. finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S) \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfinite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S)\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhence finA: \"finite A\" and Ainker: \"A\\carrier (V.vs kerT)\" and AinC: \"A\\carrier V\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S)\n\ngoal (1 subgoal):\n 1. finite A &&& A \\ carrier (V.vs kerT) &&& A \\ carrier V\n[PROOF STEP]\nby (unfold maximal_def ker_def, auto)\n \\ \\We obtain the basis A of kerT. It is also linearly independent when considered in V rather\nthan kerT\\\n[PROOF STATE]\nproof (state)\nthis:\nfinite A\nA \\ carrier (V.vs kerT)\nA \\ carrier V\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom A\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S)\n[PROOF STEP]\nhave Abasis: \"ker.basis A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S)\n\ngoal (1 subgoal):\n 1. ker.basis A\n[PROOF STEP]\nby (intro ker.max_li_is_basis, auto)\n[PROOF STATE]\nproof (state)\nthis:\nker.basis A\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom subs_ker Abasis\n[PROOF STATE]\nproof (chain)\npicking this:\nsubspace K kerT V\nker.basis A\n[PROOF STEP]\nhave spanA: \"V.module.span A = kerT\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsubspace K kerT V\nker.basis A\n\ngoal (1 subgoal):\n 1. V.module.span A = kerT\n[PROOF STEP]\napply (unfold ker.basis_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\subspace K kerT V; ker.lin_indpt A \\ ker.gen_set A \\ A \\ carrier (V.vs kerT)\\ \\ V.module.span A = kerT\n[PROOF STEP]\nby (subst sym[OF V.module.span_li_not_depend(1)[where ?N=\"kerT\"]], auto)\n[PROOF STATE]\nproof (state)\nthis:\nV.module.span A = kerT\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom Abasis\n[PROOF STATE]\nproof (chain)\npicking this:\nker.basis A\n[PROOF STEP]\nhave Akerli: \"ker.lin_indpt A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nker.basis A\n\ngoal (1 subgoal):\n 1. ker.lin_indpt A\n[PROOF STEP]\napply (unfold ker.basis_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ker.lin_indpt A \\ ker.gen_set A \\ A \\ carrier (V.vs kerT) \\ ker.lin_indpt A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nker.lin_indpt A\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom subs_ker Ainker Akerli\n[PROOF STATE]\nproof (chain)\npicking this:\nsubspace K kerT V\nA \\ carrier (V.vs kerT)\nker.lin_indpt A\n[PROOF STEP]\nhave Ali: \"V.module.lin_indpt A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsubspace K kerT V\nA \\ carrier (V.vs kerT)\nker.lin_indpt A\n\ngoal (1 subgoal):\n 1. V.module.lin_indpt A\n[PROOF STEP]\nby (auto simp add: V.module.span_li_not_depend(2))\n[PROOF STATE]\nproof (state)\nthis:\nV.module.lin_indpt A\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\ntxt\\Use the replacement theorem to find C such that $A\\cup C$ is a basis of V.\\\n[PROOF STATE]\nproof (state)\nthis:\nV.module.lin_indpt A\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom fd\n[PROOF STATE]\nproof (chain)\npicking this:\nV.fin_dim\n[PROOF STEP]\nobtain B where B: \"finite B\\ V.basis B\"\n[PROOF STATE]\nproof (prove)\nusing this:\nV.fin_dim\n\ngoal (1 subgoal):\n 1. (\\B. finite B \\ V.basis B \\ thesis) \\ thesis\n[PROOF STEP]\nby (metis V.finite_basis_exists)\n[PROOF STATE]\nproof (state)\nthis:\nfinite B \\ V.basis B\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom B\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite B \\ V.basis B\n[PROOF STEP]\nhave Bfin: \"finite B\" and Bbasis:\"V.basis B\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite B \\ V.basis B\n\ngoal (1 subgoal):\n 1. finite B &&& V.basis B\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite B\nV.basis B\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom B\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite B \\ V.basis B\n[PROOF STEP]\nhave Bcard: \"V.dim = card B\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite B \\ V.basis B\n\ngoal (1 subgoal):\n 1. V.dim = card B\n[PROOF STEP]\nby (intro V.dim_basis, auto)\n[PROOF STATE]\nproof (state)\nthis:\nV.dim = card B\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom Bbasis\n[PROOF STATE]\nproof (chain)\npicking this:\nV.basis B\n[PROOF STEP]\nhave 62: \"V.module.span B = carrier V\"\n[PROOF STATE]\nproof (prove)\nusing this:\nV.basis B\n\ngoal (1 subgoal):\n 1. V.module.gen_set B\n[PROOF STEP]\nby (unfold V.basis_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nV.module.gen_set B\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom A Abasis Ali B vs_ker\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S)\nker.basis A\nV.module.lin_indpt A\nfinite B \\ V.basis B\nvectorspace K (V.vs kerT)\n[PROOF STEP]\nhave \"\\C. finite C \\ C\\carrier V \\ C\\ V.module.span B \\ C\\A={} \n \\ int (card C) \\ (int (card B)) - (int (card A)) \\ (V.module.span (A \\ C) = V.module.span B)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S)\nker.basis A\nV.module.lin_indpt A\nfinite B \\ V.basis B\nvectorspace K (V.vs kerT)\n\ngoal (1 subgoal):\n 1. \\C. finite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n[PROOF STEP]\napply (intro V.replacement)\n[PROOF STATE]\nproof (prove)\ngoal (5 subgoals):\n 1. \\finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S); ker.basis A; V.module.lin_indpt A; finite B \\ V.basis B; vectorspace K (V.vs kerT)\\ \\ finite A\n 2. \\finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S); ker.basis A; V.module.lin_indpt A; finite B \\ V.basis B; vectorspace K (V.vs kerT)\\ \\ finite B\n 3. \\finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S); ker.basis A; V.module.lin_indpt A; finite B \\ V.basis B; vectorspace K (V.vs kerT)\\ \\ B \\ carrier V\n 4. \\finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S); ker.basis A; V.module.lin_indpt A; finite B \\ V.basis B; vectorspace K (V.vs kerT)\\ \\ V.module.lin_indpt A\n 5. \\finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S); ker.basis A; V.module.lin_indpt A; finite B \\ V.basis B; vectorspace K (V.vs kerT)\\ \\ A \\ V.module.span B\n[PROOF STEP]\napply (unfold vectorspace.basis_def V.basis_def)\n[PROOF STATE]\nproof (prove)\ngoal (5 subgoals):\n 1. \\finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S); ker.lin_indpt A \\ ker.gen_set A \\ A \\ carrier (V.vs kerT); V.module.lin_indpt A; finite B \\ V.module.lin_indpt B \\ V.module.gen_set B \\ B \\ carrier V; vectorspace K (V.vs kerT)\\ \\ finite A\n 2. \\finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S); ker.lin_indpt A \\ ker.gen_set A \\ A \\ carrier (V.vs kerT); V.module.lin_indpt A; finite B \\ V.module.lin_indpt B \\ V.module.gen_set B \\ B \\ carrier V; vectorspace K (V.vs kerT)\\ \\ finite B\n 3. \\finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S); ker.lin_indpt A \\ ker.gen_set A \\ A \\ carrier (V.vs kerT); V.module.lin_indpt A; finite B \\ V.module.lin_indpt B \\ V.module.gen_set B \\ B \\ carrier V; vectorspace K (V.vs kerT)\\ \\ B \\ carrier V\n 4. \\finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S); ker.lin_indpt A \\ ker.gen_set A \\ A \\ carrier (V.vs kerT); V.module.lin_indpt A; finite B \\ V.module.lin_indpt B \\ V.module.gen_set B \\ B \\ carrier V; vectorspace K (V.vs kerT)\\ \\ V.module.lin_indpt A\n 5. \\finite A \\ maximal A (\\S. S \\ carrier (V.vs kerT) \\ ker.lin_indpt S); ker.lin_indpt A \\ ker.gen_set A \\ A \\ carrier (V.vs kerT); V.module.lin_indpt A; finite B \\ V.module.lin_indpt B \\ V.module.gen_set B \\ B \\ carrier V; vectorspace K (V.vs kerT)\\ \\ A \\ V.module.span B\n[PROOF STEP]\nby (unfold ker_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\n\\C. finite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\ntxt \\From replacement we got $|C|\\leq |B|-|A|$. Equality must actually hold, because no generating set\ncan be smaller than $B$. Now $A\\cup C$ is a maximal generating set, hence a basis; its cardinality\nequals the dimension.\\\n[PROOF STATE]\nproof (state)\nthis:\n\\C. finite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\ntxt \\We claim that $T(C)$ is basis for $\\text{im}(T)$.\\\n[PROOF STATE]\nproof (state)\nthis:\n\\C. finite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\C. finite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n[PROOF STEP]\nobtain C where C: \"finite C \\ C\\carrier V \\ C\\ V.module.span B \\ C\\A={} \n \\ int (card C) \\ (int (card B)) - (int (card A)) \\ (V.module.span (A \\ C) = V.module.span B)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\C. finite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n\ngoal (1 subgoal):\n 1. (\\C. finite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhence Cfin: \"finite C\" and CinC: \"C\\carrier V\" and CinspanB: \" C\\V.module.span B\" and CAdis: \"C\\A={}\" \n and Ccard: \"int (card C) \\ (int (card B)) - (int (card A))\"\n and ACspanB: \"(V.module.span (A \\ C) = V.module.span B)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n\ngoal (1 subgoal):\n 1. (finite C &&& C \\ carrier V &&& C \\ V.module.span B) &&& C \\ A = {} &&& int (card C) \\ int (card B) - int (card A) &&& V.module.span (A \\ C) = V.module.span B\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite C\nC \\ carrier V\nC \\ V.module.span B\nC \\ A = {}\nint (card C) \\ int (card B) - int (card A)\nV.module.span (A \\ C) = V.module.span B\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom C\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n[PROOF STEP]\nhave cardLe: \"card A + card C \\ card B\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n\ngoal (1 subgoal):\n 1. card A + card C \\ card B\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard A + card C \\ card B\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom B C\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite B \\ V.basis B\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n[PROOF STEP]\nhave ACgen: \"V.module.gen_set (A\\C)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite B \\ V.basis B\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n\ngoal (1 subgoal):\n 1. V.module.gen_set (A \\ C)\n[PROOF STEP]\napply (unfold V.basis_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite B \\ V.module.lin_indpt B \\ V.module.gen_set B \\ B \\ carrier V; finite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\\ \\ V.module.gen_set (A \\ C)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nV.module.gen_set (A \\ C)\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom finA C ACgen AinC B\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\nV.module.gen_set (A \\ C)\nA \\ carrier V\nfinite B \\ V.basis B\n[PROOF STEP]\nhave cardGe: \"card (A\\C) \\ card B\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\nV.module.gen_set (A \\ C)\nA \\ carrier V\nfinite B \\ V.basis B\n\ngoal (1 subgoal):\n 1. card B \\ card (A \\ C)\n[PROOF STEP]\nby (intro V.li_smaller_than_gen, unfold V.basis_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\ncard B \\ card (A \\ C)\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom finA C\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n[PROOF STEP]\nhave cardUn: \"card (A\\C)\\ card A + card C\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n\ngoal (1 subgoal):\n 1. card (A \\ C) \\ card A + card C\n[PROOF STEP]\nby (metis Int_commute card_Un_disjoint le_refl)\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\ C) \\ card A + card C\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom cardLe cardUn cardGe Bcard\n[PROOF STATE]\nproof (chain)\npicking this:\ncard A + card C \\ card B\ncard (A \\ C) \\ card A + card C\ncard B \\ card (A \\ C)\nV.dim = card B\n[PROOF STEP]\nhave cardEq: \n \"card (A\\C) = card A + card C\" \n \"card (A\\C) = card B\" \n \"card (A\\C) = V.dim\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard A + card C \\ card B\ncard (A \\ C) \\ card A + card C\ncard B \\ card (A \\ C)\nV.dim = card B\n\ngoal (1 subgoal):\n 1. card (A \\ C) = card A + card C &&& card (A \\ C) = card B &&& card (A \\ C) = V.dim\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\ C) = card A + card C\ncard (A \\ C) = card B\ncard (A \\ C) = V.dim\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom Abasis C cardEq\n[PROOF STATE]\nproof (chain)\npicking this:\nker.basis A\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\ncard (A \\ C) = card A + card C\ncard (A \\ C) = card B\ncard (A \\ C) = V.dim\n[PROOF STEP]\nhave disj: \"A\\C={}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nker.basis A\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\ncard (A \\ C) = card A + card C\ncard (A \\ C) = card B\ncard (A \\ C) = V.dim\n\ngoal (1 subgoal):\n 1. A \\ C = {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA \\ C = {}\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom finA AinC C cardEq 62\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nA \\ carrier V\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\ncard (A \\ C) = card A + card C\ncard (A \\ C) = card B\ncard (A \\ C) = V.dim\nV.module.gen_set B\n[PROOF STEP]\nhave ACfin: \"finite (A\\C)\" and ACbasis: \"V.basis (A\\C)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nA \\ carrier V\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\ncard (A \\ C) = card A + card C\ncard (A \\ C) = card B\ncard (A \\ C) = V.dim\nV.module.gen_set B\n\ngoal (1 subgoal):\n 1. finite (A \\ C) &&& V.basis (A \\ C)\n[PROOF STEP]\nby (auto intro!: V.dim_gen_is_basis)\n[PROOF STATE]\nproof (state)\nthis:\nfinite (A \\ C)\nV.basis (A \\ C)\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave lm: \"linear_map K V W T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. linear_map K V W T\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nlinear_map K V W T\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\ntxt \\Let $C'$ be the image of $C$ under $T$. We will show $C'$ is a basis for $\\text{im}(T)$.\\\n[PROOF STATE]\nproof (state)\nthis:\nlinear_map K V W T\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nlet ?C' = \"T`C\"\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom Cfin\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite C\n[PROOF STEP]\nhave C'fin: \"finite ?C'\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite C\n\ngoal (1 subgoal):\n 1. finite (T ` C)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite (T ` C)\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom AinC C\n[PROOF STATE]\nproof (chain)\npicking this:\nA \\ carrier V\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n[PROOF STEP]\nhave cim: \"?C'\\imT\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier V\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\n\ngoal (1 subgoal):\n 1. T ` C \\ imT\n[PROOF STEP]\nby (unfold im_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nT ` C \\ imT\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\ntxt \\\"There is a subtle detail: we first have to show $T$ is injective on $C$.\\\n[PROOF STATE]\nproof (state)\nthis:\nT ` C \\ imT\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\ntxt \\We establish that no nontrivial linear combination of $C$ can have image 0 under $T$, \nbecause that would mean it is a linear combination of $A$, giving that $A\\cup C$ is linearly dependent, \ncontradiction. We use this result in 2 ways: (1) if $T$ is not injective on $C$, then we obtain $v$, $w\\in C$ \nsuch that $v-w$ is in the kernel, contradiction, (2) if $T(C)$ is linearly dependent, \ntaking the inverse image of that linear combination gives a linear combination of $C$ in the kernel, \ncontradiction. Hence $T$ is injective on $C$ and $T(C)$ is linearly independent.\\\n[PROOF STATE]\nproof (state)\nthis:\nT ` C \\ imT\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave lc_in_ker: \"\\d D v. \\D\\C; d\\D\\carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; \n v\\D; d v \\\\\\<^bsub>K\\<^esub>\\\\False\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfix d D v\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nassume D: \"D\\C\" and d: \"d\\D\\carrier K\" and T0: \"T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>\" \n and v: \"v\\D\" and dvnz: \"d v \\\\\\<^bsub>K\\<^esub>\"\n[PROOF STATE]\nproof (state)\nthis:\nD \\ C\nd \\ D \\ carrier K\nT (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>\nv \\ D\nd v \\ \\\\<^bsub>K\\<^esub>\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom D Cfin\n[PROOF STATE]\nproof (chain)\npicking this:\nD \\ C\nfinite C\n[PROOF STEP]\nhave Dfin: \"finite D\"\n[PROOF STATE]\nproof (prove)\nusing this:\nD \\ C\nfinite C\n\ngoal (1 subgoal):\n 1. finite D\n[PROOF STEP]\nby (auto intro: finite_subset)\n[PROOF STATE]\nproof (state)\nthis:\nfinite D\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom D CinC\n[PROOF STATE]\nproof (chain)\npicking this:\nD \\ C\nC \\ carrier V\n[PROOF STEP]\nhave DinC: \"D\\carrier V\"\n[PROOF STATE]\nproof (prove)\nusing this:\nD \\ C\nC \\ carrier V\n\ngoal (1 subgoal):\n 1. D \\ carrier V\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nD \\ carrier V\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom T0 d Dfin DinC\n[PROOF STATE]\nproof (chain)\npicking this:\nT (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>\nd \\ D \\ carrier K\nfinite D\nD \\ carrier V\n[PROOF STEP]\nhave lc_d: \"V.module.lincomb d D\\kerT\"\n[PROOF STATE]\nproof (prove)\nusing this:\nT (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>\nd \\ D \\ carrier K\nfinite D\nD \\ carrier V\n\ngoal (1 subgoal):\n 1. V.module.lincomb d D \\ kerT\n[PROOF STEP]\nby (unfold ker_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nV.module.lincomb d D \\ kerT\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom lc_d spanA AinC\n[PROOF STATE]\nproof (chain)\npicking this:\nV.module.lincomb d D \\ kerT\nV.module.span A = kerT\nA \\ carrier V\n[PROOF STEP]\nhave \"\\a' A'. A'\\A \\ a'\\A'\\carrier K \\\n V.module.lincomb a' A'= V.module.lincomb d D\"\n[PROOF STATE]\nproof (prove)\nusing this:\nV.module.lincomb d D \\ kerT\nV.module.span A = kerT\nA \\ carrier V\n\ngoal (1 subgoal):\n 1. \\a' A'. A' \\ A \\ a' \\ A' \\ carrier K \\ V.module.lincomb a' A' = V.module.lincomb d D\n[PROOF STEP]\nby (intro V.module.in_span, auto)\n[PROOF STATE]\nproof (state)\nthis:\n\\a' A'. A' \\ A \\ a' \\ A' \\ carrier K \\ V.module.lincomb a' A' = V.module.lincomb d D\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\a' A'. A' \\ A \\ a' \\ A' \\ carrier K \\ V.module.lincomb a' A' = V.module.lincomb d D\n[PROOF STEP]\nobtain a' A' where a': \"A'\\A \\ a'\\A'\\carrier K \\\n V.module.lincomb d D = V.module.lincomb a' A'\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a' A'. A' \\ A \\ a' \\ A' \\ carrier K \\ V.module.lincomb a' A' = V.module.lincomb d D\n\ngoal (1 subgoal):\n 1. (\\A' a'. A' \\ A \\ a' \\ A' \\ carrier K \\ V.module.lincomb d D = V.module.lincomb a' A' \\ thesis) \\ thesis\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\nA' \\ A \\ a' \\ A' \\ carrier K \\ V.module.lincomb d D = V.module.lincomb a' A'\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nhence A'sub: \"A'\\A\" and a'fun: \"a'\\A'\\carrier K\" \n and a'_lc:\"V.module.lincomb d D = V.module.lincomb a' A'\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA' \\ A \\ a' \\ A' \\ carrier K \\ V.module.lincomb d D = V.module.lincomb a' A'\n\ngoal (1 subgoal):\n 1. A' \\ A &&& a' \\ A' \\ carrier K &&& V.module.lincomb d D = V.module.lincomb a' A'\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA' \\ A\na' \\ A' \\ carrier K\nV.module.lincomb d D = V.module.lincomb a' A'\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom a' finA Dfin\n[PROOF STATE]\nproof (chain)\npicking this:\nA' \\ A \\ a' \\ A' \\ carrier K \\ V.module.lincomb d D = V.module.lincomb a' A'\nfinite A\nfinite D\n[PROOF STEP]\nhave A'fin: \"finite (A')\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA' \\ A \\ a' \\ A' \\ carrier K \\ V.module.lincomb d D = V.module.lincomb a' A'\nfinite A\nfinite D\n\ngoal (1 subgoal):\n 1. finite A'\n[PROOF STEP]\nby (auto intro: finite_subset)\n[PROOF STATE]\nproof (state)\nthis:\nfinite A'\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom AinC A'sub\n[PROOF STATE]\nproof (chain)\npicking this:\nA \\ carrier V\nA' \\ A\n[PROOF STEP]\nhave A'inC: \"A'\\carrier V\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier V\nA' \\ A\n\ngoal (1 subgoal):\n 1. A' \\ carrier V\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA' \\ carrier V\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nlet ?e = \"(\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub>\\\\<^bsub>K\\<^esub>\\\\<^bsub>K\\<^esub> d v)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom a'fun d\n[PROOF STATE]\nproof (chain)\npicking this:\na' \\ A' \\ carrier K\nd \\ D \\ carrier K\n[PROOF STEP]\nhave e_fun: \"?e \\ A' \\ D \\ carrier K\"\n[PROOF STATE]\nproof (prove)\nusing this:\na' \\ A' \\ carrier K\nd \\ D \\ carrier K\n\ngoal (1 subgoal):\n 1. (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K\n[PROOF STEP]\napply (unfold Pi_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a' \\ {f. \\x. x \\ A' \\ f x \\ carrier K}; d \\ {f. \\x. x \\ D \\ f x \\ carrier K}\\ \\ (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ {f. \\x. x \\ A' \\ D \\ f x \\ carrier K}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom\n A'fin Dfin (*finiteness*)\n A'inC DinC (*in carrier*)\n a'fun d e_fun (*coefficients valid*)\n disj D A'sub\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A'\nfinite D\nA' \\ carrier V\nD \\ carrier V\na' \\ A' \\ carrier K\nd \\ D \\ carrier K\n(\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K\nA \\ C = {}\nD \\ C\nA' \\ A\n[PROOF STEP]\n(*A and C disjoint*)\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A'\nfinite D\nA' \\ carrier V\nD \\ carrier V\na' \\ A' \\ carrier K\nd \\ D \\ carrier K\n(\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K\nA \\ C = {}\nD \\ C\nA' \\ A\n[PROOF STEP]\nhave lccomp1:\n \"V.module.lincomb a' A' \\\\<^bsub>V\\<^esub> \\\\<^bsub>K\\<^esub>\\\\<^bsub>K\\<^esub>\\\\<^bsub>V\\<^esub> V.module.lincomb d D = \n V.module.lincomb (\\v. if v\\A' then a' v else \\\\<^bsub>K\\<^esub>\\\\<^bsub>K\\<^esub>\\\\<^bsub>K\\<^esub> d v) (A'\\D)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A'\nfinite D\nA' \\ carrier V\nD \\ carrier V\na' \\ A' \\ carrier K\nd \\ D \\ carrier K\n(\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K\nA \\ C = {}\nD \\ C\nA' \\ A\n\ngoal (1 subgoal):\n 1. V.module.lincomb a' A' \\\\<^bsub>V\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>V\\<^esub> V.module.lincomb d D = V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D)\n[PROOF STEP]\napply (subst sym[OF V.module.lincomb_smult])\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; a' \\ A' \\ carrier K; d \\ D \\ carrier K; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K; A \\ C = {}; D \\ C; A' \\ A\\ \\ D \\ carrier V\n 2. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; a' \\ A' \\ carrier K; d \\ D \\ carrier K; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K; A \\ C = {}; D \\ C; A' \\ A\\ \\ d \\ D \\ carrier K\n 3. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; a' \\ A' \\ carrier K; d \\ D \\ carrier K; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K; A \\ C = {}; D \\ C; A' \\ A\\ \\ \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\ carrier K\n 4. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; a' \\ A' \\ carrier K; d \\ D \\ carrier K; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K; A \\ C = {}; D \\ C; A' \\ A\\ \\ V.module.lincomb a' A' \\\\<^bsub>V\\<^esub> V.module.lincomb (\\w. \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d w) D = V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D)\n[PROOF STEP]\napply (simp_all)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; a' \\ A' \\ carrier K; d \\ D \\ carrier K; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ carrier K \\ (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ D \\ carrier K; A \\ C = {}; D \\ C; A' \\ A\\ \\ V.module.lincomb a' A' \\\\<^bsub>V\\<^esub> V.module.lincomb (\\w. \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d w) D = V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D)\n[PROOF STEP]\napply (subst V.module.lincomb_union2)\n[PROOF STATE]\nproof (prove)\ngoal (6 subgoals):\n 1. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; a' \\ A' \\ carrier K; d \\ D \\ carrier K; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ carrier K \\ (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ D \\ carrier K; A \\ C = {}; D \\ C; A' \\ A\\ \\ finite (A' \\ D)\n 2. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; a' \\ A' \\ carrier K; d \\ D \\ carrier K; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ carrier K \\ (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ D \\ carrier K; A \\ C = {}; D \\ C; A' \\ A\\ \\ A' \\ D \\ carrier V\n 3. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; a' \\ A' \\ carrier K; d \\ D \\ carrier K; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ carrier K \\ (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ D \\ carrier K; A \\ C = {}; D \\ C; A' \\ A\\ \\ A' \\ D = {}\n 4. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; a' \\ A' \\ carrier K; d \\ D \\ carrier K; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ carrier K \\ (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ D \\ carrier K; A \\ C = {}; D \\ C; A' \\ A\\ \\ a' \\ A' \\ carrier K\n 5. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; a' \\ A' \\ carrier K; d \\ D \\ carrier K; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ carrier K \\ (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ D \\ carrier K; A \\ C = {}; D \\ C; A' \\ A\\ \\ (\\w. \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d w) \\ D \\ carrier K\n 6. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; a' \\ A' \\ carrier K; d \\ D \\ carrier K; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ carrier K \\ (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ D \\ carrier K; A \\ C = {}; D \\ C; A' \\ A\\ \\ V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D)\n[PROOF STEP]\nby (auto)\n[PROOF STATE]\nproof (state)\nthis:\nV.module.lincomb a' A' \\\\<^bsub>V\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>V\\<^esub> V.module.lincomb d D = V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D)\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom\n A'fin (*finiteness*)\n A'inC (*in carrier*)\n a'fun\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A'\nA' \\ carrier V\na' \\ A' \\ carrier K\n[PROOF STEP]\n(*coefficients valid*)\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A'\nA' \\ carrier V\na' \\ A' \\ carrier K\n[PROOF STEP]\nhave lccomp2: \n \"V.module.lincomb a' A' \\\\<^bsub>V\\<^esub> \\\\<^bsub>K\\<^esub>\\\\<^bsub>K\\<^esub>\\\\<^bsub>V\\<^esub> V.module.lincomb d D = \n \\\\<^bsub>V\\<^esub>\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A'\nA' \\ carrier V\na' \\ A' \\ carrier K\n\ngoal (1 subgoal):\n 1. V.module.lincomb a' A' \\\\<^bsub>V\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>V\\<^esub> V.module.lincomb d D = \\\\<^bsub>V\\<^esub>\n[PROOF STEP]\nby (simp add: a'_lc \n V.module.smult_minus_1 V.module.M.r_neg)\n[PROOF STATE]\nproof (state)\nthis:\nV.module.lincomb a' A' \\\\<^bsub>V\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>V\\<^esub> V.module.lincomb d D = \\\\<^bsub>V\\<^esub>\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom lccomp1 lccomp2\n[PROOF STATE]\nproof (chain)\npicking this:\nV.module.lincomb a' A' \\\\<^bsub>V\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>V\\<^esub> V.module.lincomb d D = V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D)\nV.module.lincomb a' A' \\\\<^bsub>V\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>V\\<^esub> V.module.lincomb d D = \\\\<^bsub>V\\<^esub>\n[PROOF STEP]\nhave lc0: \"V.module.lincomb (\\v. if v\\A' then a' v else \\\\<^bsub>K\\<^esub>\\\\<^bsub>K\\<^esub>\\\\<^bsub>K\\<^esub> d v) (A'\\D)\n =\\\\<^bsub>V\\<^esub>\"\n[PROOF STATE]\nproof (prove)\nusing this:\nV.module.lincomb a' A' \\\\<^bsub>V\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>V\\<^esub> V.module.lincomb d D = V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D)\nV.module.lincomb a' A' \\\\<^bsub>V\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>V\\<^esub> V.module.lincomb d D = \\\\<^bsub>V\\<^esub>\n\ngoal (1 subgoal):\n 1. V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = \\\\<^bsub>V\\<^esub>\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nV.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = \\\\<^bsub>V\\<^esub>\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom disj a' v D\n[PROOF STATE]\nproof (chain)\npicking this:\nA \\ C = {}\nA' \\ A \\ a' \\ A' \\ carrier K \\ V.module.lincomb d D = V.module.lincomb a' A'\nv \\ D\nD \\ C\n[PROOF STEP]\nhave v_nin: \"v\\A'\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ C = {}\nA' \\ A \\ a' \\ A' \\ carrier K \\ V.module.lincomb d D = V.module.lincomb a' A'\nv \\ D\nD \\ C\n\ngoal (1 subgoal):\n 1. v \\ A'\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv \\ A'\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom A'fin Dfin (*finiteness*)\n A'inC DinC (*in carrier*)\n e_fun d (*coefficients valid*)\n A'sub D disj (*A' D are disjoint subsets*)\n v dvnz (*d v is nonzero coefficient*)\n lc0\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A'\nfinite D\nA' \\ carrier V\nD \\ carrier V\n(\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K\nd \\ D \\ carrier K\nA' \\ A\nD \\ C\nA \\ C = {}\nv \\ D\nd v \\ \\\\<^bsub>K\\<^esub>\nV.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = \\\\<^bsub>V\\<^esub>\n[PROOF STEP]\nhave AC_ld: \"V.module.lin_dep (A\\C)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A'\nfinite D\nA' \\ carrier V\nD \\ carrier V\n(\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K\nd \\ D \\ carrier K\nA' \\ A\nD \\ C\nA \\ C = {}\nv \\ D\nd v \\ \\\\<^bsub>K\\<^esub>\nV.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = \\\\<^bsub>V\\<^esub>\n\ngoal (1 subgoal):\n 1. V.module.lin_dep (A \\ C)\n[PROOF STEP]\napply (intro V.module.lin_dep_crit[where ?A=\"A'\\D\" and \n ?S=\"A\\C\" and ?a=\"\\v. if v\\A' then a' v else \\\\<^bsub>K\\<^esub>\\\\<^bsub>K\\<^esub>\\\\<^bsub>K\\<^esub> d v\" and ?v=\"v\"])\n[PROOF STATE]\nproof (prove)\ngoal (6 subgoals):\n 1. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K; d \\ D \\ carrier K; A' \\ A; D \\ C; A \\ C = {}; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>; V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = \\\\<^bsub>V\\<^esub>\\ \\ finite (A' \\ D)\n 2. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K; d \\ D \\ carrier K; A' \\ A; D \\ C; A \\ C = {}; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>; V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = \\\\<^bsub>V\\<^esub>\\ \\ A' \\ D \\ A \\ C\n 3. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K; d \\ D \\ carrier K; A' \\ A; D \\ C; A \\ C = {}; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>; V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = \\\\<^bsub>V\\<^esub>\\ \\ (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K\n 4. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K; d \\ D \\ carrier K; A' \\ A; D \\ C; A \\ C = {}; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>; V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = \\\\<^bsub>V\\<^esub>\\ \\ v \\ A' \\ D\n 5. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K; d \\ D \\ carrier K; A' \\ A; D \\ C; A \\ C = {}; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>; V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = \\\\<^bsub>V\\<^esub>\\ \\ (if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ \\\\<^bsub>K\\<^esub>\n 6. \\finite A'; finite D; A' \\ carrier V; D \\ carrier V; (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) \\ A' \\ D \\ carrier K; d \\ D \\ carrier K; A' \\ A; D \\ C; A \\ C = {}; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>; V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = \\\\<^bsub>V\\<^esub>\\ \\ V.module.lincomb (\\v. if v \\ A' then a' v else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub> d v) (A' \\ D) = \\\\<^bsub>V\\<^esub>\n[PROOF STEP]\nby (auto dest: integral)\n[PROOF STATE]\nproof (state)\nthis:\nV.module.lin_dep (A \\ C)\n\ngoal (1 subgoal):\n 1. \\d D v. \\D \\ C; d \\ D \\ carrier K; T (V.module.lincomb d D) = \\\\<^bsub>W\\<^esub>; v \\ D; d v \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n[PROOF STEP]\nfrom AC_ld ACbasis\n[PROOF STATE]\nproof (chain)\npicking this:\nV.module.lin_dep (A \\ C)\nV.basis (A \\ C)\n[PROOF STEP]\nshow False\n[PROOF STATE]\nproof (prove)\nusing this:\nV.module.lin_dep (A \\ C)\nV.basis (A \\ C)\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby (unfold V.basis_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\?D1 \\ C; ?d1 \\ ?D1 \\ carrier K; T (V.module.lincomb ?d1 ?D1) = \\\\<^bsub>W\\<^esub>; ?v1 \\ ?D1; ?d1 ?v1 \\ \\\\<^bsub>K\\<^esub>\\ \\ False\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave C'_card: \"inj_on T C\" \"card C = card ?C'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on T C &&& card C = card (T ` C)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. inj_on T C\n 2. card C = card (T ` C)\n[PROOF STEP]\nshow \"inj_on T C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on T C\n[PROOF STEP]\nproof (rule ccontr)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ inj_on T C \\ False\n[PROOF STEP]\nassume \"\\inj_on T C\"\n[PROOF STATE]\nproof (state)\nthis:\n\\ inj_on T C\n\ngoal (1 subgoal):\n 1. \\ inj_on T C \\ False\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ inj_on T C\n[PROOF STEP]\nobtain v w where \"v\\C\" \"w\\C\" \"v\\w\" \"T v = T w\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ inj_on T C\n\ngoal (1 subgoal):\n 1. (\\v w. \\v \\ C; w \\ C; v \\ w; T v = T w\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (unfold inj_on_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nv \\ C\nw \\ C\nv \\ w\nT v = T w\n\ngoal (1 subgoal):\n 1. \\ inj_on T C \\ False\n[PROOF STEP]\nfrom this CinC\n[PROOF STATE]\nproof (chain)\npicking this:\nv \\ C\nw \\ C\nv \\ w\nT v = T w\nC \\ carrier V\n[PROOF STEP]\nshow False\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ C\nw \\ C\nv \\ w\nT v = T w\nC \\ carrier V\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\napply (intro lc_in_ker[where ?D1=\"{v,w}\" and ?d1=\"\\x. if x=v then \\\\<^bsub>K\\<^esub> else \\\\<^bsub>K\\<^esub>\\\\<^bsub>K\\<^esub>\"\n and ?v1=\"v\"])\n[PROOF STATE]\nproof (prove)\ngoal (5 subgoals):\n 1. \\v \\ C; w \\ C; v \\ w; T v = T w; C \\ carrier V\\ \\ {v, w} \\ C\n 2. \\v \\ C; w \\ C; v \\ w; T v = T w; C \\ carrier V\\ \\ (\\x. if x = v then \\\\<^bsub>K\\<^esub> else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub>) \\ {v, w} \\ carrier K\n 3. \\v \\ C; w \\ C; v \\ w; T v = T w; C \\ carrier V\\ \\ T (V.module.lincomb (\\x. if x = v then \\\\<^bsub>K\\<^esub> else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub>) {v, w}) = \\\\<^bsub>W\\<^esub>\n 4. \\v \\ C; w \\ C; v \\ w; T v = T w; C \\ carrier V\\ \\ v \\ {v, w}\n 5. \\v \\ C; w \\ C; v \\ w; T v = T w; C \\ carrier V\\ \\ (if v = v then \\\\<^bsub>K\\<^esub> else \\\\<^bsub>K\\<^esub> \\\\<^bsub>K\\<^esub>) \\ \\\\<^bsub>K\\<^esub>\n[PROOF STEP]\nby (auto simp add: V.module.lincomb_def hom_sum ring_subset_carrier \n W.module.smult_minus_1 r_neg T_im)\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ninj_on T C\n\ngoal (1 subgoal):\n 1. card C = card (T ` C)\n[PROOF STEP]\nfrom this Cfin\n[PROOF STATE]\nproof (chain)\npicking this:\ninj_on T C\nfinite C\n[PROOF STEP]\nshow \"card C = card ?C'\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on T C\nfinite C\n\ngoal (1 subgoal):\n 1. card C = card (T ` C)\n[PROOF STEP]\nby (metis card_image)\n[PROOF STATE]\nproof (state)\nthis:\ncard C = card (T ` C)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ninj_on T C\ncard C = card (T ` C)\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nlet ?f=\"the_inv_into C T\"\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave f: \"\\x. x\\C \\ ?f (T x) = x\" \"\\y. y\\?C' \\ T (?f y) = y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. x \\ C \\ the_inv_into C T (T x) = x) &&& (\\y. y \\ T ` C \\ T (the_inv_into C T y) = y)\n[PROOF STEP]\napply (insert C'_card(1))\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. \\x \\ C; inj_on T C\\ \\ the_inv_into C T (T x) = x\n 2. \\y. \\y \\ T ` C; inj_on T C\\ \\ T (the_inv_into C T y) = y\n[PROOF STEP]\napply (metis the_inv_into_f_f)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\y. \\y \\ T ` C; inj_on T C\\ \\ T (the_inv_into C T y) = y\n[PROOF STEP]\nby (metis f_the_inv_into_f)\n[PROOF STATE]\nproof (state)\nthis:\n?x1 \\ C \\ the_inv_into C T (T ?x1) = ?x1\n?y1 \\ T ` C \\ T (the_inv_into C T ?y1) = ?y1\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\n(*We show C' is a basis for the image. First we show it is linearly independent.*)\n[PROOF STATE]\nproof (state)\nthis:\n?x1 \\ C \\ the_inv_into C T (T ?x1) = ?x1\n?y1 \\ T ` C \\ T (the_inv_into C T ?y1) = ?y1\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave C'_li: \"im.lin_indpt ?C'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. im.lin_indpt (T ` C)\n[PROOF STEP]\nproof (rule ccontr)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ im.lin_indpt (T ` C) \\ False\n[PROOF STEP]\nassume Cld: \"\\im.lin_indpt ?C'\"\n[PROOF STATE]\nproof (state)\nthis:\n\\ im.lin_indpt (T ` C)\n\ngoal (1 subgoal):\n 1. \\ im.lin_indpt (T ` C) \\ False\n[PROOF STEP]\nfrom Cld cim subs_im\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ im.lin_indpt (T ` C)\nT ` C \\ imT\nsubspace K imT W\n[PROOF STEP]\nhave CldW: \"W.module.lin_dep ?C'\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ im.lin_indpt (T ` C)\nT ` C \\ imT\nsubspace K imT W\n\ngoal (1 subgoal):\n 1. W.module.lin_dep (T ` C)\n[PROOF STEP]\napply (subst sym[OF W.module.span_li_not_depend(2)[where ?S=\"T`C\" and ?N=\"imT\"]])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\\\ im.lin_indpt (T ` C); T ` C \\ imT; subspace K imT W\\ \\ T ` C \\ imT\n 2. \\\\ im.lin_indpt (T ` C); T ` C \\ imT; subspace K imT W\\ \\ LinearCombinations.submodule K imT W\n 3. \\\\ im.lin_indpt (T ` C); T ` C \\ imT; subspace K imT W\\ \\ im.lin_dep (T ` C)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nW.module.lin_dep (T ` C)\n\ngoal (1 subgoal):\n 1. \\ im.lin_indpt (T ` C) \\ False\n[PROOF STEP]\nfrom C CldW\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\nW.module.lin_dep (T ` C)\n[PROOF STEP]\nhave \"\\c' v'. (c'\\ (?C'\\carrier K)) \\ (W.module.lincomb c' ?C' = \\\\<^bsub>W\\<^esub>) \n \\ (v'\\?C') \\ (c' v'\\ \\\\<^bsub>K\\<^esub>)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\nW.module.lin_dep (T ` C)\n\ngoal (1 subgoal):\n 1. \\c' v'. c' \\ T ` C \\ carrier K \\ W.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub> \\ v' \\ T ` C \\ c' v' \\ \\\\<^bsub>K\\<^esub>\n[PROOF STEP]\nby (intro W.module.finite_lin_dep, auto)\n[PROOF STATE]\nproof (state)\nthis:\n\\c' v'. c' \\ T ` C \\ carrier K \\ W.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub> \\ v' \\ T ` C \\ c' v' \\ \\\\<^bsub>K\\<^esub>\n\ngoal (1 subgoal):\n 1. \\ im.lin_indpt (T ` C) \\ False\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\c' v'. c' \\ T ` C \\ carrier K \\ W.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub> \\ v' \\ T ` C \\ c' v' \\ \\\\<^bsub>K\\<^esub>\n[PROOF STEP]\nobtain c' v' where c': \"(c'\\ (?C'\\carrier K)) \\ (W.module.lincomb c' ?C' = \\\\<^bsub>W\\<^esub>) \n \\ (v'\\?C') \\ (c' v'\\ \\\\<^bsub>K\\<^esub>)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\c' v'. c' \\ T ` C \\ carrier K \\ W.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub> \\ v' \\ T ` C \\ c' v' \\ \\\\<^bsub>K\\<^esub>\n\ngoal (1 subgoal):\n 1. (\\c' v'. c' \\ T ` C \\ carrier K \\ W.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub> \\ v' \\ T ` C \\ c' v' \\ \\\\<^bsub>K\\<^esub> \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nc' \\ T ` C \\ carrier K \\ W.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub> \\ v' \\ T ` C \\ c' v' \\ \\\\<^bsub>K\\<^esub>\n\ngoal (1 subgoal):\n 1. \\ im.lin_indpt (T ` C) \\ False\n[PROOF STEP]\nhence c'fun: \"(c'\\ (?C'\\carrier K))\" and c'lc: \"(W.module.lincomb c' ?C' = \\\\<^bsub>W\\<^esub>)\" and \n v':\"(v'\\?C')\" and cvnz: \"(c' v'\\ \\\\<^bsub>K\\<^esub>)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc' \\ T ` C \\ carrier K \\ W.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub> \\ v' \\ T ` C \\ c' v' \\ \\\\<^bsub>K\\<^esub>\n\ngoal (1 subgoal):\n 1. (c' \\ T ` C \\ carrier K &&& W.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub>) &&& v' \\ T ` C &&& c' v' \\ \\\\<^bsub>K\\<^esub>\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nc' \\ T ` C \\ carrier K\nW.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub>\nv' \\ T ` C\nc' v' \\ \\\\<^bsub>K\\<^esub>\n\ngoal (1 subgoal):\n 1. \\ im.lin_indpt (T ` C) \\ False\n[PROOF STEP]\n(*can't get c' directly with metis/auto with W.module.finite_lin_dep*)\n[PROOF STATE]\nproof (state)\nthis:\nc' \\ T ` C \\ carrier K\nW.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub>\nv' \\ T ` C\nc' v' \\ \\\\<^bsub>K\\<^esub>\n\ngoal (1 subgoal):\n 1. \\ im.lin_indpt (T ` C) \\ False\n[PROOF STEP]\ntxt \\We take the inverse image of $C'$ under $T$ to get a linear combination of $C$ that is \nin the kernel and hence a linear combination of $A$. This contradicts $A\\cup C$ being linearly\nindependent.\\\n[PROOF STATE]\nproof (state)\nthis:\nc' \\ T ` C \\ carrier K\nW.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub>\nv' \\ T ` C\nc' v' \\ \\\\<^bsub>K\\<^esub>\n\ngoal (1 subgoal):\n 1. \\ im.lin_indpt (T ` C) \\ False\n[PROOF STEP]\nlet ?c=\"\\v. c' (T v)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ im.lin_indpt (T ` C) \\ False\n[PROOF STEP]\nfrom c'fun\n[PROOF STATE]\nproof (chain)\npicking this:\nc' \\ T ` C \\ carrier K\n[PROOF STEP]\nhave c_fun: \"?c\\ C\\carrier K\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc' \\ T ` C \\ carrier K\n\ngoal (1 subgoal):\n 1. (\\v. c' (T v)) \\ C \\ carrier K\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\v. c' (T v)) \\ C \\ carrier K\n\ngoal (1 subgoal):\n 1. \\ im.lin_indpt (T ` C) \\ False\n[PROOF STEP]\nfrom Cfin (*C finite*)\n c_fun c'fun (*coefficients valid*)\n C'_card (*bijective*)\n CinC (*C in carrier*)\n f (*inverse to T*) \n c'lc\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite C\n(\\v. c' (T v)) \\ C \\ carrier K\nc' \\ T ` C \\ carrier K\ninj_on T C\ncard C = card (T ` C)\nC \\ carrier V\n?x1 \\ C \\ the_inv_into C T (T ?x1) = ?x1\n?y1 \\ T ` C \\ T (the_inv_into C T ?y1) = ?y1\nW.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub>\n[PROOF STEP]\n(*lc c' = 0*)\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite C\n(\\v. c' (T v)) \\ C \\ carrier K\nc' \\ T ` C \\ carrier K\ninj_on T C\ncard C = card (T ` C)\nC \\ carrier V\n?x1 \\ C \\ the_inv_into C T (T ?x1) = ?x1\n?y1 \\ T ` C \\ T (the_inv_into C T ?y1) = ?y1\nW.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub>\n[PROOF STEP]\nhave \"T (V.module.lincomb ?c C) = \\\\<^bsub>W\\<^esub>\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite C\n(\\v. c' (T v)) \\ C \\ carrier K\nc' \\ T ` C \\ carrier K\ninj_on T C\ncard C = card (T ` C)\nC \\ carrier V\n?x1 \\ C \\ the_inv_into C T (T ?x1) = ?x1\n?y1 \\ T ` C \\ T (the_inv_into C T ?y1) = ?y1\nW.module.lincomb c' (T ` C) = \\\\<^bsub>W\\<^esub>\n\ngoal (1 subgoal):\n 1. T (V.module.lincomb (\\v. c' (T v)) C) = \\\\<^bsub>W\\<^esub>\n[PROOF STEP]\napply (unfold V.module.lincomb_def W.module.lincomb_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite C; (\\v. c' (T v)) \\ C \\ carrier K; c' \\ T ` C \\ carrier K; inj_on T C; card C = card (T ` C); C \\ carrier V; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; (\\\\<^bsub>W\\<^esub>v\\T ` C. c' v \\\\<^bsub>W\\<^esub> v) = \\\\<^bsub>W\\<^esub>\\ \\ T (\\\\<^bsub>V\\<^esub>v\\C. c' (T v) \\\\<^bsub>V\\<^esub> v) = \\\\<^bsub>W\\<^esub>\n[PROOF STEP]\napply (subst hom_sum, auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite C; (\\v. c' (T v)) \\ C \\ carrier K; c' \\ T ` C \\ carrier K; inj_on T C; card C = card (T ` C); C \\ carrier V; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; (\\\\<^bsub>W\\<^esub>v\\T ` C. c' v \\\\<^bsub>W\\<^esub> v) = \\\\<^bsub>W\\<^esub>\\ \\ (\\\\<^bsub>W\\<^esub>a\\C. T (c' (T a) \\\\<^bsub>V\\<^esub> a)) = \\\\<^bsub>W\\<^esub>\n[PROOF STEP]\napply (simp cong: finsum_cong add: ring_subset_carrier coeff_in_ring)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite C; c' \\ T ` C \\ carrier K; inj_on T C; card C = card (T ` C); C \\ carrier V; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; (\\\\<^bsub>W\\<^esub>v\\T ` C. c' v \\\\<^bsub>W\\<^esub> v) = \\\\<^bsub>W\\<^esub>\\ \\ (\\\\<^bsub>W\\<^esub>i\\C. c' (T i) \\\\<^bsub>W\\<^esub> T i) = \\\\<^bsub>W\\<^esub>\n[PROOF STEP]\napply (subst finsum_reindex[where ?f=\"\\w. c' w \\\\<^bsub>W\\<^esub> w\" and ?h=\"T\" and ?A=\"C\", THEN sym])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\finite C; c' \\ T ` C \\ carrier K; inj_on T C; card C = card (T ` C); C \\ carrier V; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; (\\\\<^bsub>W\\<^esub>v\\T ` C. c' v \\\\<^bsub>W\\<^esub> v) = \\\\<^bsub>W\\<^esub>\\ \\ (\\w. c' w \\\\<^bsub>W\\<^esub> w) \\ T ` C \\ carrier W\n 2. \\finite C; c' \\ T ` C \\ carrier K; inj_on T C; card C = card (T ` C); C \\ carrier V; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; (\\\\<^bsub>W\\<^esub>v\\T ` C. c' v \\\\<^bsub>W\\<^esub> v) = \\\\<^bsub>W\\<^esub>\\ \\ inj_on T C\n 3. \\finite C; c' \\ T ` C \\ carrier K; inj_on T C; card C = card (T ` C); C \\ carrier V; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; (\\\\<^bsub>W\\<^esub>v\\T ` C. c' v \\\\<^bsub>W\\<^esub> v) = \\\\<^bsub>W\\<^esub>\\ \\ (\\\\<^bsub>W\\<^esub>w\\T ` C. c' w \\\\<^bsub>W\\<^esub> w) = \\\\<^bsub>W\\<^esub>\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nT (V.module.lincomb (\\v. c' (T v)) C) = \\\\<^bsub>W\\<^esub>\n\ngoal (1 subgoal):\n 1. \\ im.lin_indpt (T ` C) \\ False\n[PROOF STEP]\nwith f c'fun cvnz v'\n[PROOF STATE]\nproof (chain)\npicking this:\n?x1 \\ C \\ the_inv_into C T (T ?x1) = ?x1\n?y1 \\ T ` C \\ T (the_inv_into C T ?y1) = ?y1\nc' \\ T ` C \\ carrier K\nc' v' \\ \\\\<^bsub>K\\<^esub>\nv' \\ T ` C\nT (V.module.lincomb (\\v. c' (T v)) C) = \\\\<^bsub>W\\<^esub>\n[PROOF STEP]\nshow False\n[PROOF STATE]\nproof (prove)\nusing this:\n?x1 \\ C \\ the_inv_into C T (T ?x1) = ?x1\n?y1 \\ T ` C \\ T (the_inv_into C T ?y1) = ?y1\nc' \\ T ` C \\ carrier K\nc' v' \\ \\\\<^bsub>K\\<^esub>\nv' \\ T ` C\nT (V.module.lincomb (\\v. c' (T v)) C) = \\\\<^bsub>W\\<^esub>\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby (intro lc_in_ker[where ?D1=\"C\" and ?d1=\"?c\" and ?v1=\"?f v'\"], auto)\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nim.lin_indpt (T ` C)\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave C'_gen: \"im.gen_set ?C'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. im.gen_set (T ` C)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. im.gen_set (T ` C)\n[PROOF STEP]\nhave C'_span: \"span ?C' = imT\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. W.module.span (T ` C) = imT\n[PROOF STEP]\nproof (rule equalityI)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. W.module.span (T ` C) \\ imT\n 2. imT \\ W.module.span (T ` C)\n[PROOF STEP]\nfrom cim subs_im\n[PROOF STATE]\nproof (chain)\npicking this:\nT ` C \\ imT\nsubspace K imT W\n[PROOF STEP]\nshow \"W.module.span ?C' \\ imT\"\n[PROOF STATE]\nproof (prove)\nusing this:\nT ` C \\ imT\nsubspace K imT W\n\ngoal (1 subgoal):\n 1. W.module.span (T ` C) \\ imT\n[PROOF STEP]\nby (intro span_is_subset, unfold subspace_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nW.module.span (T ` C) \\ imT\n\ngoal (1 subgoal):\n 1. imT \\ W.module.span (T ` C)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. imT \\ W.module.span (T ` C)\n[PROOF STEP]\nshow \"imT\\W.module.span ?C'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. imT \\ W.module.span (T ` C)\n[PROOF STEP]\nproof (auto)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ imT \\ x \\ W.module.span (T ` C)\n[PROOF STEP]\nfix w\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ imT \\ x \\ W.module.span (T ` C)\n[PROOF STEP]\nassume w: \"w\\imT\"\n[PROOF STATE]\nproof (state)\nthis:\nw \\ imT\n\ngoal (1 subgoal):\n 1. \\x. x \\ imT \\ x \\ W.module.span (T ` C)\n[PROOF STEP]\nfrom this finA Cfin AinC CinC\n[PROOF STATE]\nproof (chain)\npicking this:\nw \\ imT\nfinite A\nfinite C\nA \\ carrier V\nC \\ carrier V\n[PROOF STEP]\nobtain v where v_inC: \"v\\carrier V\" and w_eq_T_v: \"w= T v\"\n[PROOF STATE]\nproof (prove)\nusing this:\nw \\ imT\nfinite A\nfinite C\nA \\ carrier V\nC \\ carrier V\n\ngoal (1 subgoal):\n 1. (\\v. \\v \\ carrier V; w = T v\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (unfold im_def image_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nv \\ carrier V\nw = T v\n\ngoal (1 subgoal):\n 1. \\x. x \\ imT \\ x \\ W.module.span (T ` C)\n[PROOF STEP]\nfrom finA Cfin AinC CinC v_inC ACgen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nfinite C\nA \\ carrier V\nC \\ carrier V\nv \\ carrier V\nV.module.gen_set (A \\ C)\n[PROOF STEP]\nhave \"\\a. a \\ A\\C \\ carrier K\\ V.module.lincomb a (A\\C) = v\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite C\nA \\ carrier V\nC \\ carrier V\nv \\ carrier V\nV.module.gen_set (A \\ C)\n\ngoal (1 subgoal):\n 1. \\a. a \\ A \\ C \\ carrier K \\ V.module.lincomb a (A \\ C) = v\n[PROOF STEP]\nby (intro V.module.finite_in_span, auto)\n[PROOF STATE]\nproof (state)\nthis:\n\\a. a \\ A \\ C \\ carrier K \\ V.module.lincomb a (A \\ C) = v\n\ngoal (1 subgoal):\n 1. \\x. x \\ imT \\ x \\ W.module.span (T ` C)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\a. a \\ A \\ C \\ carrier K \\ V.module.lincomb a (A \\ C) = v\n[PROOF STEP]\nobtain a where \n a_fun: \"a \\ A\\C \\ carrier K\" and\n lc_a_v: \"v= V.module.lincomb a (A\\C)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a. a \\ A \\ C \\ carrier K \\ V.module.lincomb a (A \\ C) = v\n\ngoal (1 subgoal):\n 1. (\\a. \\a \\ A \\ C \\ carrier K; v = V.module.lincomb a (A \\ C)\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na \\ A \\ C \\ carrier K\nv = V.module.lincomb a (A \\ C)\n\ngoal (1 subgoal):\n 1. \\x. x \\ imT \\ x \\ W.module.span (T ` C)\n[PROOF STEP]\nlet ?a'=\"\\v. a (?f v)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ imT \\ x \\ W.module.span (T ` C)\n[PROOF STEP]\nfrom finA Cfin AinC CinC a_fun disj Ainker f C'_card\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nfinite C\nA \\ carrier V\nC \\ carrier V\na \\ A \\ C \\ carrier K\nA \\ C = {}\nA \\ carrier (V.vs kerT)\n?x1 \\ C \\ the_inv_into C T (T ?x1) = ?x1\n?y1 \\ T ` C \\ T (the_inv_into C T ?y1) = ?y1\ninj_on T C\ncard C = card (T ` C)\n[PROOF STEP]\nhave Tv: \"T v = W.module.lincomb ?a' ?C'\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite C\nA \\ carrier V\nC \\ carrier V\na \\ A \\ C \\ carrier K\nA \\ C = {}\nA \\ carrier (V.vs kerT)\n?x1 \\ C \\ the_inv_into C T (T ?x1) = ?x1\n?y1 \\ T ` C \\ T (the_inv_into C T ?y1) = ?y1\ninj_on T C\ncard C = card (T ` C)\n\ngoal (1 subgoal):\n 1. T v = W.module.lincomb (\\v. a (the_inv_into C T v)) (T ` C)\n[PROOF STEP]\napply (subst lc_a_v)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite C; A \\ carrier V; C \\ carrier V; a \\ A \\ C \\ carrier K; A \\ C = {}; A \\ carrier (V.vs kerT); \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; inj_on T C; card C = card (T ` C)\\ \\ T (V.module.lincomb a (A \\ C)) = W.module.lincomb (\\v. a (the_inv_into C T v)) (T ` C)\n[PROOF STEP]\napply (subst V.module.lincomb_union, simp_all)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite C; A \\ carrier V; C \\ carrier V; a \\ A \\ carrier K \\ a \\ C \\ carrier K; A \\ C = {}; A \\ kerT; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; inj_on T C; card C = card (T ` C)\\ \\ T (V.module.lincomb a A) \\\\<^bsub>W\\<^esub> T (V.module.lincomb a C) = W.module.lincomb (\\v. a (the_inv_into C T v)) (T ` C)\n[PROOF STEP]\n(*Break up the union A\\C*)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite C; A \\ carrier V; C \\ carrier V; a \\ A \\ carrier K \\ a \\ C \\ carrier K; A \\ C = {}; A \\ kerT; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; inj_on T C; card C = card (T ` C)\\ \\ T (V.module.lincomb a A) \\\\<^bsub>W\\<^esub> T (V.module.lincomb a C) = W.module.lincomb (\\v. a (the_inv_into C T v)) (T ` C)\n[PROOF STEP]\napply (unfold lincomb_def V.module.lincomb_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite C; A \\ carrier V; C \\ carrier V; a \\ A \\ carrier K \\ a \\ C \\ carrier K; A \\ C = {}; A \\ kerT; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; inj_on T C; card C = card (T ` C)\\ \\ T (\\\\<^bsub>V\\<^esub>v\\A. a v \\\\<^bsub>V\\<^esub> v) \\\\<^bsub>W\\<^esub> T (\\\\<^bsub>V\\<^esub>v\\C. a v \\\\<^bsub>V\\<^esub> v) = (\\\\<^bsub>W\\<^esub>v\\T ` C. a (the_inv_into C T v) \\\\<^bsub>W\\<^esub> v)\n[PROOF STEP]\napply (subst hom_sum, auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite C; A \\ carrier V; C \\ carrier V; A \\ C = {}; A \\ kerT; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; inj_on T C; card C = card (T ` C); a \\ A \\ carrier K; a \\ C \\ carrier K\\ \\ (\\\\<^bsub>W\\<^esub>aa\\A. T (a aa \\\\<^bsub>V\\<^esub> aa)) \\\\<^bsub>W\\<^esub> T (\\\\<^bsub>V\\<^esub>v\\C. a v \\\\<^bsub>V\\<^esub> v) = (\\\\<^bsub>W\\<^esub>v\\T ` C. a (the_inv_into C T v) \\\\<^bsub>W\\<^esub> v)\n[PROOF STEP]\n(*Take T inside the sum over A*)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite C; A \\ carrier V; C \\ carrier V; A \\ C = {}; A \\ kerT; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; inj_on T C; card C = card (T ` C); a \\ A \\ carrier K; a \\ C \\ carrier K\\ \\ (\\\\<^bsub>W\\<^esub>aa\\A. T (a aa \\\\<^bsub>V\\<^esub> aa)) \\\\<^bsub>W\\<^esub> T (\\\\<^bsub>V\\<^esub>v\\C. a v \\\\<^bsub>V\\<^esub> v) = (\\\\<^bsub>W\\<^esub>v\\T ` C. a (the_inv_into C T v) \\\\<^bsub>W\\<^esub> v)\n[PROOF STEP]\napply (simp add: subsetD coeff_in_ring\n hom_sum (*Take T inside the sum over C*)\n T_ker (*all terms become 0 because the vectors are in the kernel.*)\n )\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite C; A \\ carrier V; C \\ carrier V; A \\ C = {}; A \\ kerT; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; inj_on T C; card C = card (T ` C); a \\ A \\ carrier K; a \\ C \\ carrier K\\ \\ (\\\\<^bsub>W\\<^esub>v\\C. T (a v \\\\<^bsub>V\\<^esub> v)) = (\\\\<^bsub>W\\<^esub>v\\T ` C. a (the_inv_into C T v) \\\\<^bsub>W\\<^esub> v)\n[PROOF STEP]\napply (subst finsum_reindex[where ?h=\"T\" and ?f=\"\\v. ?a' v\\\\<^bsub>W\\<^esub> v\"], auto)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\xa. \\finite A; finite C; A \\ carrier V; C \\ carrier V; A \\ C = {}; A \\ kerT; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; inj_on T C; card C = card (T ` C); a \\ A \\ carrier K; a \\ C \\ carrier K; xa \\ C\\ \\ a xa \\\\<^bsub>W\\<^esub> T xa \\ carrier W\n 2. \\finite A; finite C; A \\ carrier V; C \\ carrier V; A \\ C = {}; A \\ kerT; \\x. x \\ C \\ the_inv_into C T (T x) = x; \\y. y \\ T ` C \\ T (the_inv_into C T y) = y; inj_on T C; card C = card (T ` C); a \\ A \\ carrier K; a \\ C \\ carrier K\\ \\ (\\\\<^bsub>W\\<^esub>v\\C. T (a v \\\\<^bsub>V\\<^esub> v)) = (\\\\<^bsub>W\\<^esub>x\\C. a (the_inv_into C T (T x)) \\\\<^bsub>W\\<^esub> T x)\n[PROOF STEP]\nby (auto cong: finsum_cong simp add: coeff_in_ring ring_subset_carrier)\n[PROOF STATE]\nproof (state)\nthis:\nT v = W.module.lincomb (\\v. a (the_inv_into C T v)) (T ` C)\n\ngoal (1 subgoal):\n 1. \\x. x \\ imT \\ x \\ W.module.span (T ` C)\n[PROOF STEP]\nfrom a_fun f\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ A \\ C \\ carrier K\n?x1 \\ C \\ the_inv_into C T (T ?x1) = ?x1\n?y1 \\ T ` C \\ T (the_inv_into C T ?y1) = ?y1\n[PROOF STEP]\nhave a'_fun: \"?a'\\?C' \\ carrier K\"\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ A \\ C \\ carrier K\n?x1 \\ C \\ the_inv_into C T (T ?x1) = ?x1\n?y1 \\ T ` C \\ T (the_inv_into C T ?y1) = ?y1\n\ngoal (1 subgoal):\n 1. (\\v. a (the_inv_into C T v)) \\ T ` C \\ carrier K\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\v. a (the_inv_into C T v)) \\ T ` C \\ carrier K\n\ngoal (1 subgoal):\n 1. \\x. x \\ imT \\ x \\ W.module.span (T ` C)\n[PROOF STEP]\nfrom C'fin CinC this w_eq_T_v a'_fun Tv\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (T ` C)\nC \\ carrier V\n(\\v. a (the_inv_into C T v)) \\ T ` C \\ carrier K\nw = T v\n(\\v. a (the_inv_into C T v)) \\ T ` C \\ carrier K\nT v = W.module.lincomb (\\v. a (the_inv_into C T v)) (T ` C)\n[PROOF STEP]\nshow \"w \\ LinearCombinations.module.span K W (T ` C)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (T ` C)\nC \\ carrier V\n(\\v. a (the_inv_into C T v)) \\ T ` C \\ carrier K\nw = T v\n(\\v. a (the_inv_into C T v)) \\ T ` C \\ carrier K\nT v = W.module.lincomb (\\v. a (the_inv_into C T v)) (T ` C)\n\ngoal (1 subgoal):\n 1. w \\ W.module.span (T ` C)\n[PROOF STEP]\nby (subst finite_span, auto)\n[PROOF STATE]\nproof (state)\nthis:\nw \\ W.module.span (T ` C)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nimT \\ W.module.span (T ` C)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nW.module.span (T ` C) = imT\n\ngoal (1 subgoal):\n 1. im.gen_set (T ` C)\n[PROOF STEP]\nfrom this subs_im CinC\n[PROOF STATE]\nproof (chain)\npicking this:\nW.module.span (T ` C) = imT\nsubspace K imT W\nC \\ carrier V\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nW.module.span (T ` C) = imT\nsubspace K imT W\nC \\ carrier V\n\ngoal (1 subgoal):\n 1. im.gen_set (T ` C)\n[PROOF STEP]\napply (subst span_li_not_depend(1))\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\W.module.span (T ` C) = imT; subspace K imT W; C \\ carrier V\\ \\ T ` C \\ imT\n 2. \\W.module.span (T ` C) = imT; subspace K imT W; C \\ carrier V\\ \\ LinearCombinations.submodule K imT W\n 3. \\W.module.span (T ` C) = imT; subspace K imT W; C \\ carrier V\\ \\ W.module.span (T ` C) = carrier (W.vs imT)\n[PROOF STEP]\nby (unfold im_def subspace_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nim.gen_set (T ` C)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nim.gen_set (T ` C)\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom C'_li C'_gen C cim\n[PROOF STATE]\nproof (chain)\npicking this:\nim.lin_indpt (T ` C)\nim.gen_set (T ` C)\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\nT ` C \\ imT\n[PROOF STEP]\nhave C'_basis: \"im.basis (T`C)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nim.lin_indpt (T ` C)\nim.gen_set (T ` C)\nfinite C \\ C \\ carrier V \\ C \\ V.module.span B \\ C \\ A = {} \\ int (card C) \\ int (card B) - int (card A) \\ V.module.span (A \\ C) = V.module.span B\nT ` C \\ imT\n\ngoal (1 subgoal):\n 1. im.basis (T ` C)\n[PROOF STEP]\nby (unfold im.basis_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nim.basis (T ` C)\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nhave C_card_im: \"card C = (vectorspace.dim K (W.vs imT))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card C = im.dim\n[PROOF STEP]\nusing C'_basis C'_card(2) C'fin im.dim_basis\n[PROOF STATE]\nproof (prove)\nusing this:\nim.basis (T ` C)\ncard C = card (T ` C)\nfinite (T ` C)\n\\finite ?A; im.basis ?A\\ \\ im.dim = card ?A\n\ngoal (1 subgoal):\n 1. card C = im.dim\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard C = im.dim\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom finA Abasis\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nker.basis A\n[PROOF STEP]\nhave \"ker.dim = card A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nker.basis A\n\ngoal (1 subgoal):\n 1. ker.dim = card A\n[PROOF STEP]\nby (rule ker.dim_basis)\n[PROOF STATE]\nproof (state)\nthis:\nker.dim = card A\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nnote * = this C_card_im cardEq\n[PROOF STATE]\nproof (state)\nthis:\nker.dim = card A\ncard C = im.dim\ncard (A \\ C) = card A + card C\ncard (A \\ C) = card B\ncard (A \\ C) = V.dim\n\ngoal (2 subgoals):\n 1. im.dim + ker.dim = V.dim\n 2. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nshow \"(vectorspace.dim K (W.vs imT)) + (vectorspace.dim K (V.vs kerT)) = V.dim\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. im.dim + ker.dim = V.dim\n[PROOF STEP]\nusing *\n[PROOF STATE]\nproof (prove)\nusing this:\nker.dim = card A\ncard C = im.dim\ncard (A \\ C) = card A + card C\ncard (A \\ C) = card B\ncard (A \\ C) = V.dim\n\ngoal (1 subgoal):\n 1. im.dim + ker.dim = V.dim\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nim.dim + ker.dim = V.dim\n\ngoal (1 subgoal):\n 1. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nassume \"T` (carrier V) = carrier W\"\n[PROOF STATE]\nproof (state)\nthis:\nT ` carrier V = carrier W\n\ngoal (1 subgoal):\n 1. T ` carrier V = carrier W \\ W.fin_dim\n[PROOF STEP]\nfrom * surj_imp_imT_carrier[OF this]\n[PROOF STATE]\nproof (chain)\npicking this:\nker.dim = card A\ncard C = im.dim\ncard (A \\ C) = card A + card C\ncard (A \\ C) = card B\ncard (A \\ C) = V.dim\nimT = carrier W\n[PROOF STEP]\nshow \"W.fin_dim\"\n[PROOF STATE]\nproof (prove)\nusing this:\nker.dim = card A\ncard C = im.dim\ncard (A \\ C) = card A + card C\ncard (A \\ C) = card B\ncard (A \\ C) = V.dim\nimT = carrier W\n\ngoal (1 subgoal):\n 1. W.fin_dim\n[PROOF STEP]\nusing C'_basis C'fin\n[PROOF STATE]\nproof (prove)\nusing this:\nker.dim = card A\ncard C = im.dim\ncard (A \\ C) = card A + card C\ncard (A \\ C) = card B\ncard (A \\ C) = V.dim\nimT = carrier W\nim.basis (T ` C)\nfinite (T ` C)\n\ngoal (1 subgoal):\n 1. W.fin_dim\n[PROOF STEP]\nunfolding W.fin_dim_def im.basis_def\n[PROOF STATE]\nproof (prove)\nusing this:\nker.dim = card A\ncard C = im.dim\ncard (A \\ C) = card A + card C\ncard (A \\ C) = card B\ncard (A \\ C) = V.dim\nimT = carrier W\nim.lin_indpt (T ` C) \\ im.gen_set (T ` C) \\ T ` C \\ carrier (W.vs imT)\nfinite (T ` C)\n\ngoal (1 subgoal):\n 1. \\A. finite A \\ A \\ carrier W \\ W.module.gen_set A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nW.fin_dim\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 43162, "file": "VectorSpace_VectorSpace", "length": 298, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297887874624, "lm_q2_score": 0.8311430520409023, "lm_q1q2_score": 0.7484690771065606}} {"text": "[STATEMENT]\nlemma ord_4_3 [simp]: \"ord 4 (3::nat) = 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ord 4 3 = 2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ord 4 3 = 2\n[PROOF STEP]\nhave \"[3 ^ 2 = (1 :: nat)] (mod 4)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [3\\<^sup>2 = 1] (mod 4)\n[PROOF STEP]\nby (simp add: cong_def)\n[PROOF STATE]\nproof (state)\nthis:\n[3\\<^sup>2 = 1] (mod 4)\n\ngoal (1 subgoal):\n 1. ord 4 3 = 2\n[PROOF STEP]\nhence \"ord 4 (3::nat) dvd 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n[3\\<^sup>2 = 1] (mod 4)\n\ngoal (1 subgoal):\n 1. ord 4 3 dvd 2\n[PROOF STEP]\nby (subst (asm) ord_divides) auto\n[PROOF STATE]\nproof (state)\nthis:\nord 4 3 dvd 2\n\ngoal (1 subgoal):\n 1. ord 4 3 = 2\n[PROOF STEP]\nhence \"ord 4 (3::nat) \\ 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nord 4 3 dvd 2\n\ngoal (1 subgoal):\n 1. ord 4 3 \\ 2\n[PROOF STEP]\nby (intro dvd_imp_le) auto\n[PROOF STATE]\nproof (state)\nthis:\nord 4 3 \\ 2\n\ngoal (1 subgoal):\n 1. ord 4 3 = 2\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nord 4 3 \\ 2\n\ngoal (1 subgoal):\n 1. ord 4 3 = 2\n[PROOF STEP]\nhave \"ord 4 (3::nat) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ord 4 3 \\ 1\n[PROOF STEP]\nby (auto simp: ord_eq_Suc_0_iff cong_def)\n[PROOF STATE]\nproof (state)\nthis:\nord 4 3 \\ 1\n\ngoal (1 subgoal):\n 1. ord 4 3 = 2\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nord 4 3 \\ 1\n\ngoal (1 subgoal):\n 1. ord 4 3 = 2\n[PROOF STEP]\nhave \"ord 4 (3::nat) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ord 4 3 \\ 0\n[PROOF STEP]\nby (auto simp: gcd_non_0_nat coprime_iff_gcd_eq_1)\n[PROOF STATE]\nproof (state)\nthis:\nord 4 3 \\ 0\n\ngoal (1 subgoal):\n 1. ord 4 3 = 2\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nord 4 3 \\ 2\nord 4 3 \\ 1\nord 4 3 \\ 0\n[PROOF STEP]\nshow \"ord 4 (3 :: nat) = 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nord 4 3 \\ 2\nord 4 3 \\ 1\nord 4 3 \\ 0\n\ngoal (1 subgoal):\n 1. ord 4 3 = 2\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\nord 4 3 = 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1132, "file": null, "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297834483234, "lm_q2_score": 0.8311430457670241, "lm_q1q2_score": 0.7484690670191582}} {"text": "[STATEMENT]\ntheorem IFNTT_INTT_gen_eq: \n \"length numbers = 2^l \\ 2^l \\ n \\ IFNTT numbers = INTT_gen (length numbers) numbers\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nproof(induction l arbitrary: numbers)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\numbers. \\length numbers = 2 ^ 0; 2 ^ 0 \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n 2. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nlength numbers = 2 ^ 0\n2 ^ 0 \\ n\n\ngoal (2 subgoals):\n 1. \\numbers. \\length numbers = 2 ^ 0; 2 ^ 0 \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n 2. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nhence \"local.IFNTT numbers = [numbers ! 0]\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ 0\n2 ^ 0 \\ n\n\ngoal (1 subgoal):\n 1. IFNTT numbers = [numbers ! 0]\n[PROOF STEP]\nby (metis IFNTT.simps(2) One_nat_def Suc_length_conv length_0_conv nth_Cons_0 power_0)\n[PROOF STATE]\nproof (state)\nthis:\nIFNTT numbers = [numbers ! 0]\n\ngoal (2 subgoals):\n 1. \\numbers. \\length numbers = 2 ^ 0; 2 ^ 0 \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n 2. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nIFNTT numbers = [numbers ! 0]\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nIFNTT numbers = [numbers ! 0]\n\ngoal (1 subgoal):\n 1. IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nunfolding INTT_gen_def intt_gen_def\n[PROOF STATE]\nproof (prove)\nusing this:\nIFNTT numbers = [numbers ! 0]\n\ngoal (1 subgoal):\n 1. IFNTT numbers = map (\\i. \\j = 0.. ^ (n div length numbers * i * j)) [0.. n\n\ngoal (1 subgoal):\n 1. IFNTT numbers = map (\\i. \\j = 0.. ^ (n div length numbers * i * j)) [0..l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ncase (Suc l)\n[PROOF STATE]\nproof (state)\nthis:\n\\length ?numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT ?numbers = INTT_gen (length ?numbers) ?numbers\nlength numbers = 2 ^ Suc l\n2 ^ Suc l \\ n\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ntext \\We define some lists that are used during the recursive call.\\\n[PROOF STATE]\nproof (state)\nthis:\n\\length ?numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT ?numbers = INTT_gen (length ?numbers) ?numbers\nlength numbers = 2 ^ Suc l\n2 ^ Suc l \\ n\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ndefine numbers1 where \"numbers1 = [numbers!i . i <- (filter even [0..l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ndefine numbers2 where \"numbers2 = [numbers!i . i <- (filter odd [0..l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ndefine ifntt1 where \"ifntt1 = IFNTT numbers1\"\n[PROOF STATE]\nproof (state)\nthis:\nifntt1 = IFNTT numbers1\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ndefine ifntt2 where \"ifntt2 = IFNTT numbers2\"\n[PROOF STATE]\nproof (state)\nthis:\nifntt2 = IFNTT numbers2\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ndefine sum1 where \n \"sum1 = map2 (+) ifntt1 (map2 ( \\ x k. x*(\\^( (n div (length numbers)) * k))) \n ifntt2 [0..<((length numbers) div 2)])\"\n[PROOF STATE]\nproof (state)\nthis:\nsum1 = map2 (+) ifntt1 (map2 (\\x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ndefine sum2 where \n \"sum2 = map2 (-) ifntt1 (map2 ( \\ x k. x*(\\^( (n div (length numbers)) * k))) \n ifntt2 [0..<((length numbers) div 2)])\"\n[PROOF STATE]\nproof (state)\nthis:\nsum2 = map2 (-) ifntt1 (map2 (\\x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ndefine l1 where \"l1 = length numbers1\"\n[PROOF STATE]\nproof (state)\nthis:\nl1 = length numbers1\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ndefine l2 where \"l2 = length numbers2\"\n[PROOF STATE]\nproof (state)\nthis:\nl2 = length numbers2\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ndefine llen where \"llen = length numbers\"\n[PROOF STATE]\nproof (state)\nthis:\nllen = length numbers\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ntext \\Properties of those lists\\\n[PROOF STATE]\nproof (state)\nthis:\nllen = length numbers\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nhave numbers1_even: \"length numbers1 = 2^l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length numbers1 = 2 ^ l\n[PROOF STEP]\nusing numbers1_def length_even_filter Suc\n[PROOF STATE]\nproof (prove)\nusing this:\nnumbers1 = map ((!) numbers) (filter even [0..length ?numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT ?numbers = INTT_gen (length ?numbers) ?numbers\nlength numbers = 2 ^ Suc l\n2 ^ Suc l \\ n\n\ngoal (1 subgoal):\n 1. length numbers1 = 2 ^ l\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlength numbers1 = 2 ^ l\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nhave numbers2_even: \"length numbers2 = 2^l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length numbers2 = 2 ^ l\n[PROOF STEP]\nusing numbers2_def length_odd_filter Suc\n[PROOF STATE]\nproof (prove)\nusing this:\nnumbers2 = map ((!) numbers) (filter odd [0..length ?numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT ?numbers = INTT_gen (length ?numbers) ?numbers\nlength numbers = 2 ^ Suc l\n2 ^ Suc l \\ n\n\ngoal (1 subgoal):\n 1. length numbers2 = 2 ^ l\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlength numbers2 = 2 ^ l\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nhave numbers1_ifntt: \"ifntt1 = INTT_gen (2^l) numbers1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt1 = INTT_gen (2 ^ l) numbers1\n[PROOF STEP]\nusing ifntt1_def Suc.IH[of numbers1] numbers1_even Suc(3)\n[PROOF STATE]\nproof (prove)\nusing this:\nifntt1 = IFNTT numbers1\n\\length numbers1 = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers1 = INTT_gen (length numbers1) numbers1\nlength numbers1 = 2 ^ l\n2 ^ Suc l \\ n\n\ngoal (1 subgoal):\n 1. ifntt1 = INTT_gen (2 ^ l) numbers1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nifntt1 = INTT_gen (2 ^ l) numbers1\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nhence ifntt1_by_index: \"ifntt1 ! i = intt_gen numbers1 (2^l) i\" if \"i < 2^l\" for i\n[PROOF STATE]\nproof (prove)\nusing this:\nifntt1 = INTT_gen (2 ^ l) numbers1\n\ngoal (1 subgoal):\n 1. ifntt1 ! i = intt_gen numbers1 (2 ^ l) i\n[PROOF STEP]\nunfolding INTT_gen_def\n[PROOF STATE]\nproof (prove)\nusing this:\nifntt1 = map (intt_gen numbers1 (2 ^ l)) [0.. ifntt1 ! ?i = intt_gen numbers1 (2 ^ l) ?i\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nhave numbers2_ifntt: \"ifntt2 = INTT_gen (2^l) numbers2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt2 = INTT_gen (2 ^ l) numbers2\n[PROOF STEP]\nusing ifntt2_def Suc.IH[of numbers2] numbers2_even Suc(3)\n[PROOF STATE]\nproof (prove)\nusing this:\nifntt2 = IFNTT numbers2\n\\length numbers2 = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers2 = INTT_gen (length numbers2) numbers2\nlength numbers2 = 2 ^ l\n2 ^ Suc l \\ n\n\ngoal (1 subgoal):\n 1. ifntt2 = INTT_gen (2 ^ l) numbers2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nifntt2 = INTT_gen (2 ^ l) numbers2\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nhence ifntt2_by_index: \"ifntt2 ! i = intt_gen numbers2 (2^l) i\" if \"i < 2^l\" for i\n[PROOF STATE]\nproof (prove)\nusing this:\nifntt2 = INTT_gen (2 ^ l) numbers2\n\ngoal (1 subgoal):\n 1. ifntt2 ! i = intt_gen numbers2 (2 ^ l) i\n[PROOF STEP]\nunfolding INTT_gen_def\n[PROOF STATE]\nproof (prove)\nusing this:\nifntt2 = map (intt_gen numbers2 (2 ^ l)) [0.. ifntt2 ! ?i = intt_gen numbers2 (2 ^ l) ?i\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nhave ifntt1_length: \"length ifntt1 = 2^l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length ifntt1 = 2 ^ l\n[PROOF STEP]\nunfolding numbers1_ifntt INTT_gen_def numbers1_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (map (intt_gen (map ((!) numbers) (filter even [0..l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nhave ifntt2_length: \"length ifntt2 = 2^l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length ifntt2 = 2 ^ l\n[PROOF STEP]\nunfolding numbers2_ifntt INTT_gen_def numbers2_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (map (intt_gen (map ((!) numbers) (filter odd [0..l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ntext \\Same proof structure as for the \\textit{FNTT} proof.\n $\\omega$s are just replaced by $\\mu$s.\\\n[PROOF STATE]\nproof (state)\nthis:\nlength ifntt2 = 2 ^ l\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nhave before_half: \"map (intt_gen numbers llen) [0..<(llen div 2)] = sum1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0..Length is important, since we want to use list lemmas later on.\\\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..We show equality by extensionality on indices.\\\n[PROOF STATE]\nproof (state)\nthis:\nlength sum1 = 2 ^ l\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0..First simplify this term.\\\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0..Expand the definition of $sum1$ and massage the result.\\\n[PROOF STATE]\nproof (state)\nthis:\nmap (intt_gen numbers llen) [0..^((n div llen) * i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum1 ! i = ifntt1 ! i + ifntt2 ! i * \\ ^ (n div llen * i)\n[PROOF STEP]\nunfolding sum1_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. map2 (+) ifntt1 (map2 (\\x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0.. ^ (n div llen * i)\n[PROOF STEP]\nusing map2_index\n \"00\" \"01\" INTT_gen_def add.left_neutral diff_zero ifntt1_length length_map length_upt map2_map_map map_nth nth_upt numbers2_even numbers2_ifntt that llen_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?i < length ?xs; ?i < length ?ys\\ \\ map2 ?f ?xs ?ys ! ?i = ?f (?xs ! ?i) (?ys ! ?i)\nlength (map (intt_gen numbers llen) [0..x. ?h (?f x) (?g x)) ?xs\nmap ((!) ?xs) [0.. [?i..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0.. ^ (n div llen * i)\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\nsum1 ! i = ifntt1 ! i + ifntt2 ! i * \\ ^ (n div llen * i)\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0.. ^ (n div llen * i)\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0..j=0..^((n div (2^l))*i*j))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n[PROOF STEP]\nunfolding l1_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n[PROOF STEP]\nusing ifntt1_by_index[of i] that\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt1 ! i = intt_gen numbers1 (2 ^ l) i\ni < 2 ^ l\n\ngoal (1 subgoal):\n 1. ifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n[PROOF STEP]\nunfolding intt_gen_def\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\ni < 2 ^ l\n\ngoal (1 subgoal):\n 1. ifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0..j=0..^((n div llen)*i*(2*j)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0.. ^ (n div 2 ^ l * i * j)) = (\\j = 0.. ^ (n div llen * i * (2 * j)))\n[PROOF STEP]\napply (rule sum_rules(2))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j < l1 \\ numbers1 ! j * \\ ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nsubgoal for j\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l1 \\ numbers1 ! j * \\ ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nunfolding numbers1_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l1 \\ map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\napply(subst llen_def[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l1 \\ map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. j < l1 \\ map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nassume ass: \"j < l1 \"\n[PROOF STATE]\nproof (state)\nthis:\nj < l1\n\ngoal (1 subgoal):\n 1. j < l1 \\ map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nhence \"map ((!) numbers) (filter even [0.. map ((!) numbers) (filter even [0.. map ((!) numbers) (filter even [0.. map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nmap ((!) numbers) (filter even [0.. map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nhave \"filter even [0..2 * j < llen; 2 * 2 ^ l = llen\\ \\ filter even [0..2 * j < length numbers; 2 * 2 ^ l = length numbers\\ \\ filter even [0.. map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfilter even [0.. map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nhave \"n div llen * (2 * j) = ((n div (2 ^ l)) * j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n div llen * (2 * j) = n div 2 ^ l * j\n[PROOF STEP]\nusing Suc(2) two_powrs_div[of l N] n_two_pot two_powr_div Suc(3) llen_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ Suc l\nl < N \\ 2 ^ N div 2 ^ Suc l * 2 = 2 ^ N div 2 ^ l\nn = 2 ^ N\n?j < ?i \\ 2 ^ ?i div 2 ^ ?j = 2 ^ (?i - ?j)\n2 ^ Suc l \\ n\nllen = length numbers\n\ngoal (1 subgoal):\n 1. n div llen * (2 * j) = n div 2 ^ l * j\n[PROOF STEP]\nby (metis One_nat_def div_if mult.assoc nat_less_le not_less_eq numeral_2_eq_2 power_eq_0_iff power_inject_exp zero_neq_numeral)\n[PROOF STATE]\nproof (state)\nthis:\nn div llen * (2 * j) = n div 2 ^ l * j\n\ngoal (1 subgoal):\n 1. j < l1 \\ map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nmap ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) =\n numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmap ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nunfolding llen_def l1_def l2_def\n[PROOF STATE]\nproof (prove)\nusing this:\nmap ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div length numbers * i * (2 * j))\n[PROOF STEP]\nby (metis (mono_tags, lifting) mult.assoc mult.left_commute)\n[PROOF STATE]\nproof (state)\nthis:\nmap ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0.. ^ (n div 2 ^ l * i * j)) = (\\j = 0.. ^ (n div llen * i * (2 * j)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0..j = 0.. ^ (n div 2 ^ l * i * j)) = (\\j = 0.. ^ (n div llen * i * (2 * j)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0..^((n div llen) * i)) = \n (\\j=0..^((n div (2^l))*i*j+ (n div llen) * i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i))\n[PROOF STEP]\napply(rule trans[where s = \"(\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i))\"])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i))\n 2. (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i)) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i))\n[PROOF STEP]\nsubgoal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i))\n[PROOF STEP]\nunfolding l2_def llen_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt2 ! i * \\ ^ (n div length numbers * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div length numbers * i))\n[PROOF STEP]\nusing ifntt2_by_index[of i] that sum_in[of _ \"(\\^((n div llen) * i))\" \"l2\"] comm_semiring_1_class.semiring_normalization_rules(26)[of \\]\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt2 ! i = intt_gen numbers2 (2 ^ l) i\ni < 2 ^ l\n(\\ia = 0.. ^ (n div llen * i)) = sum ?f {0.. ^ (n div llen * i)\n\\ ^ ?p * \\ ^ ?q = \\ ^ (?p + ?q)\n\ngoal (1 subgoal):\n 1. ifntt2 ! i * \\ ^ (n div length numbers * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div length numbers * i))\n[PROOF STEP]\nunfolding intt_gen_def\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt2 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\ni < 2 ^ l\n(\\ia = 0.. ^ (n div llen * i)) = sum ?f {0.. ^ (n div llen * i)\n\\ ^ ?p * \\ ^ ?q = \\ ^ (?p + ?q)\n\ngoal (1 subgoal):\n 1. ifntt2 ! i * \\ ^ (n div length numbers * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div length numbers * i))\n[PROOF STEP]\nusing sum_rules\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt2 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\ni < 2 ^ l\n(\\ia = 0.. ^ (n div llen * i)) = sum ?f {0.. ^ (n div llen * i)\n\\ ^ ?p * \\ ^ ?q = \\ ^ (?p + ?q)\n(\\i = 0..i. i < ?x \\ ?f i = ?g i) \\ sum ?f {0..i = 0..i = 0..j = 0..i = 0..i = 0.. sum ?f {0..l = ?a.. ^ (n div length numbers * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div length numbers * i))\n[PROOF STEP]\napply presburger\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i)) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i))\n[PROOF STEP]\napply (rule sum_rules(2))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i) = numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i)\n[PROOF STEP]\nsubgoal for j\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i) = numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i)\n[PROOF STEP]\nusing ifntt2_by_index[of i] that sum_in[of _ \"(\\^((n div llen) * i))\" \"l2\"] comm_semiring_1_class.semiring_normalization_rules(26)[of \\]\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt2 ! i = intt_gen numbers2 (2 ^ l) i\ni < 2 ^ l\n(\\ia = 0.. ^ (n div llen * i)) = sum ?f {0.. ^ (n div llen * i)\n\\ ^ ?p * \\ ^ ?q = \\ ^ (?p + ?q)\n\ngoal (1 subgoal):\n 1. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i) = numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i)\n[PROOF STEP]\nunfolding intt_gen_def\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt2 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\ni < 2 ^ l\n(\\ia = 0.. ^ (n div llen * i)) = sum ?f {0.. ^ (n div llen * i)\n\\ ^ ?p * \\ ^ ?q = \\ ^ (?p + ?q)\n\ngoal (1 subgoal):\n 1. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i) = numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i)\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\nifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0.. = (\\j=0..^((n div llen)*i*(2*j+1))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n[PROOF STEP]\napply (rule sum_rules(2))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nsubgoal for j\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nunfolding numbers2_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l2 \\ map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\napply(subst llen_def[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l2 \\ map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. j < l2 \\ map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nassume ass: \"j < l2 \"\n[PROOF STATE]\nproof (state)\nthis:\nj < l2\n\ngoal (1 subgoal):\n 1. j < l2 \\ map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nhence \"map ((!) numbers) (filter odd [0.. map ?f ?xs ! ?n = ?f (?xs ! ?n)\n\ngoal (1 subgoal):\n 1. map ((!) numbers) (filter odd [0.. map ?f ?xs ! ?n = ?f (?xs ! ?n)\n\ngoal (1 subgoal):\n 1. map ((!) numbers) (filter odd [0.. map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nmap ((!) numbers) (filter odd [0.. map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nhave \"filter odd [0..2 * j < length numbers; 2 * 2 ^ l = length numbers\\ \\ filter odd [0..2 * j < length numbers; 2 * 2 ^ l = length numbers\\ \\ filter odd [0.. map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfilter odd [0.. map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nhave \"n div llen * (2 * j) = ((n div (2 ^ l)) * j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n div llen * (2 * j) = n div 2 ^ l * j\n[PROOF STEP]\nusing Suc(2) two_powrs_div[of l N] n_two_pot two_powr_div Suc(3) llen_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ Suc l\nl < N \\ 2 ^ N div 2 ^ Suc l * 2 = 2 ^ N div 2 ^ l\nn = 2 ^ N\n?j < ?i \\ 2 ^ ?i div 2 ^ ?j = 2 ^ (?i - ?j)\n2 ^ Suc l \\ n\nllen = length numbers\n\ngoal (1 subgoal):\n 1. n div llen * (2 * j) = n div 2 ^ l * j\n[PROOF STEP]\nby (metis One_nat_def div_if mult.assoc nat_less_le not_less_eq numeral_2_eq_2 power_eq_0_iff power_inject_exp zero_neq_numeral)\n[PROOF STATE]\nproof (state)\nthis:\nn div llen * (2 * j) = n div 2 ^ l * j\n\ngoal (1 subgoal):\n 1. j < l2 \\ map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nmap ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) \n = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmap ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nunfolding llen_def\n[PROOF STATE]\nproof (prove)\nusing this:\nmap ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div length numbers * i) = numbers ! (2 * j + 1) * \\ ^ (n div length numbers * i * (2 * j + 1))\n[PROOF STEP]\nby (smt (z3) Groups.mult_ac(2) distrib_left mult.right_neutral mult_2 mult_cancel_left)\n[PROOF STATE]\nproof (state)\nthis:\nmap ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0..j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0..j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\nmap (intt_gen numbers llen) [0.. ^ (n div llen * i)\nifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n(\\j = 0.. ^ (n div 2 ^ l * i * j)) = (\\j = 0.. ^ (n div llen * i * (2 * j)))\nifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i))\n(\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [0..j = 0.. ^ (n div 2 ^ l * i * j + n div length numbers * i)) = (\\j = 0.. ^ (n div length numbers * i * (2 * j + 1)))\nmap (intt_gen numbers (length numbers)) [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0.. ^ (n div length numbers * i)\nifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n(\\j = 0.. ^ (n div 2 ^ l * i * j)) = (\\j = 0.. ^ (n div length numbers * i * (2 * j)))\nifntt2 ! i * \\ ^ (n div length numbers * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div length numbers * i))\n(\\j = 0.. ^ (n div 2 ^ l * i * j + n div length numbers * i)) = (\\j = 0.. ^ (n div length numbers * i * (2 * j + 1)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers (length numbers)) [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0.. d. numbers ! d * \\ ^ (n div length numbers * i * d)\" \"2^l\"] Suc(2)\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = 0.. ^ (n div 2 ^ l * i * j + n div length numbers * i)) = (\\j = 0.. ^ (n div length numbers * i * (2 * j + 1)))\nmap (intt_gen numbers (length numbers)) [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0.. ^ (n div length numbers * i)\nifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n(\\j = 0.. ^ (n div 2 ^ l * i * j)) = (\\j = 0.. ^ (n div length numbers * i * (2 * j)))\nifntt2 ! i * \\ ^ (n div length numbers * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div length numbers * i))\n(\\j = 0.. ^ (n div 2 ^ l * i * j + n div length numbers * i)) = (\\j = 0.. ^ (n div length numbers * i * (2 * j + 1)))\n(\\j = 0..<2 ^ l. numbers ! (2 * j) * \\ ^ (n div length numbers * i * (2 * j))) + (\\j = 0..<2 ^ l. numbers ! (2 * j + 1) * \\ ^ (n div length numbers * i * (2 * j + 1))) = (\\j = 0..<2 * 2 ^ l. numbers ! j * \\ ^ (n div length numbers * i * j))\nlength numbers = 2 ^ Suc l\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers (length numbers)) [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..j = 0.. ^ (n div 2 ^ l * i * j + n div length numbers * i)) = (\\j = 0.. ^ (n div length numbers * i * (2 * j + 1)))\nmap (\\i. \\j = 0.. ^ (n div length numbers * i * j)) [0..j = 0.. ^ (n div length numbers * i * j))\nmap2 (+) ifntt1 (map2 (\\x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0.. ^ (n div length numbers * i)\nifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n(\\j = 0.. ^ (n div 2 ^ l * i * j)) = (\\j = 0.. ^ (n div length numbers * i * (2 * j)))\nifntt2 ! i * \\ ^ (n div length numbers * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div length numbers * i))\n(\\j = 0.. ^ (n div 2 ^ l * i * j + n div length numbers * i)) = (\\j = 0.. ^ (n div length numbers * i * (2 * j + 1)))\n(\\j = 0..<2 ^ l. numbers ! (2 * j) * \\ ^ (n div length numbers * i * (2 * j))) + (\\j = 0..<2 ^ l. numbers ! (2 * j + 1) * \\ ^ (n div length numbers * i * (2 * j + 1))) = (\\j = 0..<2 * 2 ^ l. numbers ! j * \\ ^ (n div length numbers * i * j))\nlength numbers = 2 ^ Suc l\n\ngoal (1 subgoal):\n 1. map (\\i. \\j = 0.. ^ (n div length numbers * i * j)) [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0.. map (intt_gen numbers llen) [0.. map (intt_gen numbers llen) [0.. map (intt_gen numbers llen) [0..l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\ntext \\We show index-wise equality for the second halves\\\n[PROOF STATE]\nproof (state)\nthis:\nmap (intt_gen numbers llen) [0..l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nhave after_half: \"map (intt_gen numbers llen) [(llen div 2)..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0..Equality for every index\\\n[PROOF STATE]\nproof (state)\nthis:\nlength sum2 = 2 ^ l\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2..x y. x * \\ ^ (n div llen * y)) ifntt2 [0.. ^ (n div llen * i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. map2 (\\x y. x * \\ ^ (n div llen * y)) ifntt2 [0.. ^ (n div llen * i)\n[PROOF STEP]\nusing Suc(2) that\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ Suc l\ni < 2 ^ l\n\ngoal (1 subgoal):\n 1. map2 (\\x y. x * \\ ^ (n div llen * y)) ifntt2 [0.. ^ (n div llen * i)\n[PROOF STEP]\nby (simp add: ifntt2_length llen_def)\n[PROOF STATE]\nproof (state)\nthis:\nmap2 (\\x y. x * \\ ^ (n div llen * y)) ifntt2 [0.. ^ (n div llen * i)\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2.. ^ (n div llen * i) = ifntt2 ! i * \\ ^ (n div llen * (i+ llen div 2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - ifntt2 ! i * \\ ^ (n div llen * i) = ifntt2 ! i * \\ ^ (n div llen * (i + llen div 2))\n[PROOF STEP]\nusing Suc(2) my_div_exp_min1[of l]\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ Suc l\n2 ^ Suc l \\ n \\ (\\ ^ (n div 2 ^ Suc l)) ^ 2 ^ l = - 1\n\ngoal (1 subgoal):\n 1. - ifntt2 ! i * \\ ^ (n div llen * i) = ifntt2 ! i * \\ ^ (n div llen * (i + llen div 2))\n[PROOF STEP]\nunfolding llen_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ Suc l\n2 ^ Suc l \\ n \\ (\\ ^ (n div 2 ^ Suc l)) ^ 2 ^ l = - 1\n\ngoal (1 subgoal):\n 1. - ifntt2 ! i * \\ ^ (n div length numbers * i) = ifntt2 ! i * \\ ^ (n div length numbers * (i + length numbers div 2))\n[PROOF STEP]\nby (smt (z3) Suc.prems(2) mult.commute mult.left_commute mult_1s_ring_1(2) neq0_conv nonzero_mult_div_cancel_left numeral_One pos2 power_Suc power_add power_mult)\n[PROOF STATE]\nproof (state)\nthis:\n- ifntt2 ! i * \\ ^ (n div llen * i) = ifntt2 ! i * \\ ^ (n div llen * (i + llen div 2))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2..^((n div llen) * i))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n- ifntt2 ! i * \\ ^ (n div llen * i) = ifntt2 ! i * \\ ^ (n div llen * (i + llen div 2))\n\ngoal (1 subgoal):\n 1. sum2 ! i = ifntt1 ! i - ifntt2 ! i * \\ ^ (n div llen * i)\n[PROOF STEP]\nunfolding sum2_def llen_def\n[PROOF STATE]\nproof (prove)\nusing this:\n- ifntt2 ! i * \\ ^ (n div length numbers * i) = ifntt2 ! i * \\ ^ (n div length numbers * (i + length numbers div 2))\n\ngoal (1 subgoal):\n 1. map2 (-) ifntt1 (map2 (\\x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0.. ^ (n div length numbers * i)\n[PROOF STEP]\nby (simp add: Suc.prems(1) ifntt1_length ifntt2_length that)\n[PROOF STATE]\nproof (state)\nthis:\nsum2 ! i = ifntt1 ! i - ifntt2 ! i * \\ ^ (n div llen * i)\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2..j=0..^((n div (2^l))*i*j))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n[PROOF STEP]\nusing ifntt1_by_index that\n[PROOF STATE]\nproof (prove)\nusing this:\n?i < 2 ^ l \\ ifntt1 ! ?i = intt_gen numbers1 (2 ^ l) ?i\ni < 2 ^ l\n\ngoal (1 subgoal):\n 1. ifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n[PROOF STEP]\nunfolding intt_gen_def l1_def\n[PROOF STATE]\nproof (prove)\nusing this:\n?i < 2 ^ l \\ ifntt1 ! ?i = (\\j = 0.. ^ (n div 2 ^ l * ?i * j))\ni < 2 ^ l\n\ngoal (1 subgoal):\n 1. ifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2.. =(\\j=0..^((n div llen)*i*(2*j)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0.. ^ (n div 2 ^ l * i * j)) = (\\j = 0.. ^ (n div llen * i * (2 * j)))\n[PROOF STEP]\napply (rule sum_rules(2))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j < l1 \\ numbers1 ! j * \\ ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nsubgoal for j\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l1 \\ numbers1 ! j * \\ ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nunfolding numbers1_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l1 \\ map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\napply(subst llen_def[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l1 \\ map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. j < l1 \\ map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nassume ass: \"j < l1 \"\n[PROOF STATE]\nproof (state)\nthis:\nj < l1\n\ngoal (1 subgoal):\n 1. j < l1 \\ map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nhence \"map ((!) numbers) (filter even [0.. map ?f ?xs ! ?n = ?f (?xs ! ?n)\n\ngoal (1 subgoal):\n 1. map ((!) numbers) (filter even [0.. map ?f ?xs ! ?n = ?f (?xs ! ?n)\n\ngoal (1 subgoal):\n 1. map ((!) numbers) (filter even [0.. map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nmap ((!) numbers) (filter even [0.. map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nhave \"filter even [0..2 * ?j < ?l; 2 * ?x = ?l\\ \\ filter even [0.. map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfilter even [0.. map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nhave \"n div llen * (2 * j) = ((n div (2 ^ l)) * j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n div llen * (2 * j) = n div 2 ^ l * j\n[PROOF STEP]\nusing Suc(2) two_powrs_div[of l N] n_two_pot two_powr_div Suc(3) llen_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ Suc l\nl < N \\ 2 ^ N div 2 ^ Suc l * 2 = 2 ^ N div 2 ^ l\nn = 2 ^ N\n?j < ?i \\ 2 ^ ?i div 2 ^ ?j = 2 ^ (?i - ?j)\n2 ^ Suc l \\ n\nllen = length numbers\n\ngoal (1 subgoal):\n 1. n div llen * (2 * j) = n div 2 ^ l * j\n[PROOF STEP]\nby (metis One_nat_def div_if mult.assoc nat_less_le not_less_eq numeral_2_eq_2 power_eq_0_iff power_inject_exp zero_neq_numeral)\n[PROOF STATE]\nproof (state)\nthis:\nn div llen * (2 * j) = n div 2 ^ l * j\n\ngoal (1 subgoal):\n 1. j < l1 \\ map ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nmap ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) =\n numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmap ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n[PROOF STEP]\nby (metis (mono_tags, lifting) mult.assoc mult.left_commute)\n[PROOF STATE]\nproof (state)\nthis:\nmap ((!) numbers) (filter even [0.. ^ (n div 2 ^ l * i * j) = numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0.. ^ (n div 2 ^ l * i * j)) = (\\j = 0.. ^ (n div llen * i * (2 * j)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2.. = (\\j=0.. ^((n div llen)*(2^l + i)*(2*j))) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0.. ^ (n div llen * i * (2 * j))) = (\\j = 0.. ^ (n div llen * (2 ^ l + i) * (2 * j)))\n[PROOF STEP]\napply (rule sum_rules(2))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j < l1 \\ numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j)) = numbers ! (2 * j) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j))\n[PROOF STEP]\nsubgoal for j\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l1 \\ numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j)) = numbers ! (2 * j) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j))\n[PROOF STEP]\nusing Suc(2) Suc(3) my_div_exp_min1[of l] llen_def l1_def numbers1_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ Suc l\n2 ^ Suc l \\ n\n2 ^ Suc l \\ n \\ (\\ ^ (n div 2 ^ Suc l)) ^ 2 ^ l = - 1\nllen = length numbers\nl1 = length numbers1\nnumbers1 = map ((!) numbers) (filter even [0.. numbers ! (2 * j) * \\ ^ (n div llen * i * (2 * j)) = numbers ! (2 * j) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j))\n[PROOF STEP]\napply(smt (verit, del_insts) add.commute minus_power_mult_self mult_2 mult_minus1_right power_add power_mult)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0.. ^ (n div llen * i * (2 * j))) = (\\j = 0.. ^ (n div llen * (2 ^ l + i) * (2 * j)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2..j = 0.. ^ (n div llen * i * (2 * j))) = (\\j = 0.. ^ (n div llen * (2 ^ l + i) * (2 * j)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2..^((n div llen) * i)) =\n (\\j=0..^((n div (2^l))*i*j+ (n div llen) * i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i))\n[PROOF STEP]\napply(rule trans[where s = \"(\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i))\"])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i))\n 2. (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i)) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i))\n[PROOF STEP]\nsubgoal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i))\n[PROOF STEP]\nusing ifntt2_by_index[of i] that sum_in comm_semiring_1_class.semiring_normalization_rules(26)[of \\]\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt2 ! i = intt_gen numbers2 (2 ^ l) i\ni < 2 ^ l\n(\\i = 0.. ^ ?p * \\ ^ ?q = \\ ^ (?p + ?q)\n\ngoal (1 subgoal):\n 1. ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i))\n[PROOF STEP]\nunfolding intt_gen_def\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt2 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\ni < 2 ^ l\n(\\i = 0.. ^ ?p * \\ ^ ?q = \\ ^ (?p + ?q)\n\ngoal (1 subgoal):\n 1. ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i))\n[PROOF STEP]\nusing sum_rules l2_def\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt2 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\ni < 2 ^ l\n(\\i = 0.. ^ ?p * \\ ^ ?q = \\ ^ (?p + ?q)\n(\\i = 0..i. i < ?x \\ ?f i = ?g i) \\ sum ?f {0..i = 0..i = 0..j = 0..i = 0..i = 0.. sum ?f {0..l = ?a.. ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i))\n[PROOF STEP]\napply presburger\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0.. ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i)) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i))\n[PROOF STEP]\napply (rule sum_rules(2))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i) = numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i)\n[PROOF STEP]\nsubgoal for j\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i) = numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i)\n[PROOF STEP]\nusing ifntt2_by_index[of i] that sum_in comm_semiring_1_class.semiring_normalization_rules(26)[of \\]\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt2 ! i = intt_gen numbers2 (2 ^ l) i\ni < 2 ^ l\n(\\i = 0.. ^ ?p * \\ ^ ?q = \\ ^ (?p + ?q)\n\ngoal (1 subgoal):\n 1. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i) = numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i)\n[PROOF STEP]\nunfolding intt_gen_def\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ l \\ ifntt2 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\ni < 2 ^ l\n(\\i = 0.. ^ ?p * \\ ^ ?q = \\ ^ (?p + ?q)\n\ngoal (1 subgoal):\n 1. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j) * \\ ^ (n div llen * i) = numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i)\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\nifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2.. = (\\j=0..^((n div llen)*i*(2*j+1))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n[PROOF STEP]\napply (rule sum_rules(2))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nsubgoal for j\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l2 \\ numbers2 ! j * \\ ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nunfolding numbers2_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l2 \\ map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\napply(subst llen_def[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l2 \\ map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. j < l2 \\ map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nassume ass: \"j < l2 \"\n[PROOF STATE]\nproof (state)\nthis:\nj < l2\n\ngoal (1 subgoal):\n 1. j < l2 \\ map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nhence \"map ((!) numbers) (filter odd [0.. map ?f ?xs ! ?n = ?f (?xs ! ?n)\nllen = length numbers\nl2 = length numbers2\nnumbers2 = map ((!) numbers) (filter odd [0.. map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nmap ((!) numbers) (filter odd [0.. map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nhave \"filter odd [0..2 * ?j < ?l; 2 * ?x = ?l\\ \\ filter odd [0.. map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfilter odd [0.. map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nhave \"n div llen * (2 * j) = ((n div (2 ^ l)) * j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n div llen * (2 * j) = n div 2 ^ l * j\n[PROOF STEP]\nusing Suc(2) two_powrs_div[of l N] n_two_pot two_powr_div Suc(3) llen_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ Suc l\nl < N \\ 2 ^ N div 2 ^ Suc l * 2 = 2 ^ N div 2 ^ l\nn = 2 ^ N\n?j < ?i \\ 2 ^ ?i div 2 ^ ?j = 2 ^ (?i - ?j)\n2 ^ Suc l \\ n\nllen = length numbers\n\ngoal (1 subgoal):\n 1. n div llen * (2 * j) = n div 2 ^ l * j\n[PROOF STEP]\nby (metis One_nat_def div_if mult.assoc nat_less_le not_less_eq numeral_2_eq_2 power_eq_0_iff power_inject_exp zero_neq_numeral)\n[PROOF STATE]\nproof (state)\nthis:\nn div llen * (2 * j) = n div 2 ^ l * j\n\ngoal (1 subgoal):\n 1. j < l2 \\ map ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nmap ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i)\n = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmap ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n[PROOF STEP]\nby (smt (z3) Groups.mult_ac(2) distrib_left mult.right_neutral mult_2 mult_cancel_left)\n[PROOF STATE]\nproof (state)\nthis:\nmap ((!) numbers) (filter odd [0.. ^ (n div 2 ^ l * i * j + n div llen * i) = numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2..^((n div llen) * i)) = (\\j=0..^((n div llen)*i*(2*j+1)))) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n[PROOF STEP]\nusing 008 009\n[PROOF STATE]\nproof (prove)\nusing this:\nifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i))\n(\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n\ngoal (1 subgoal):\n 1. ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\nifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2..^((n div llen) * i)) =\n (\\j=0..^((n div llen)*i*(2*j+1)))) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1))))\n[PROOF STEP]\napply(rule neg_cong)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (- ifntt2 ! i * \\ ^ (n div llen * i)) = - (\\j = 0.. ^ (n div llen * i * (2 * j + 1))))\n[PROOF STEP]\napply(rule trans[where s=\"(\\j=0..^((n div llen)*i*(2*j+1))))\"])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. - (- ifntt2 ! i * \\ ^ (n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n 2. (\\j = 0.. ^ (n div llen * i * (2 * j + 1))) = - (\\j = 0.. ^ (n div llen * i * (2 * j + 1))))\n[PROOF STEP]\nsubgoal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (- ifntt2 ! i * \\ ^ (n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n[PROOF STEP]\nusing 008 009\n[PROOF STATE]\nproof (prove)\nusing this:\nifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i))\n(\\j = 0.. ^ (n div 2 ^ l * i * j + n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n\ngoal (1 subgoal):\n 1. - (- ifntt2 ! i * \\ ^ (n div llen * i)) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0.. ^ (n div llen * i * (2 * j + 1))) = - (\\j = 0.. ^ (n div llen * i * (2 * j + 1))))\n[PROOF STEP]\napply(rule sym)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n[PROOF STEP]\nusing sum_neg_in[of _ \"l2\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n- sum ?f {0..j = 0..j = 0.. ^ (n div llen * i * (2 * j + 1)))) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n- ifntt2 ! i * \\ ^ (n div llen * i) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1))))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2.. = (\\j=0..^((n div llen)*(2^l+i)*(2*j+1))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0.. ^ (n div llen * i * (2 * j + 1)))) = (\\j = 0.. ^ (n div llen * (2 ^ l + i) * (2 * j + 1)))\n[PROOF STEP]\napply(rule sum_rules(2))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j < l2 \\ - (numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))) = numbers ! (2 * j + 1) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j + 1))\n[PROOF STEP]\nsubgoal for j\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j < l2 \\ - (numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))) = numbers ! (2 * j + 1) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j + 1))\n[PROOF STEP]\nusing Suc(2) Suc(3) my_div_exp_min1[of l] llen_def l2_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ Suc l\n2 ^ Suc l \\ n\n2 ^ Suc l \\ n \\ (\\ ^ (n div 2 ^ Suc l)) ^ 2 ^ l = - 1\nllen = length numbers\nl2 = length numbers2\n\ngoal (1 subgoal):\n 1. j < l2 \\ - (numbers ! (2 * j + 1) * \\ ^ (n div llen * i * (2 * j + 1))) = numbers ! (2 * j + 1) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j + 1))\n[PROOF STEP]\napply (smt (z3) add.commute exp_rule mult.assoc mult_minus1_right plus_1_eq_Suc power_add power_minus1_odd power_mult)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0.. ^ (n div llen * i * (2 * j + 1)))) = (\\j = 0.. ^ (n div llen * (2 ^ l + i) * (2 * j + 1)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2..j = 0..<2 ^ l. numbers!(2*j) * \\ ^ (n div llen * (2^l + i) * (2*j)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ifntt1 ! i = (\\j = 0..<2 ^ l. numbers ! (2 * j) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j)))\n[PROOF STEP]\nusing 005 006 007 numbers1_even llen_def l1_def\n[PROOF STATE]\nproof (prove)\nusing this:\nifntt1 ! i = (\\j = 0.. ^ (n div 2 ^ l * i * j))\n(\\j = 0.. ^ (n div 2 ^ l * i * j)) = (\\j = 0.. ^ (n div llen * i * (2 * j)))\n(\\j = 0.. ^ (n div llen * i * (2 * j))) = (\\j = 0.. ^ (n div llen * (2 ^ l + i) * (2 * j)))\nlength numbers1 = 2 ^ l\nllen = length numbers\nl1 = length numbers1\n\ngoal (1 subgoal):\n 1. ifntt1 ! i = (\\j = 0..<2 ^ l. numbers ! (2 * j) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nifntt1 ! i = (\\j = 0..<2 ^ l. numbers ! (2 * j) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j)))\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2..j = 0..<2 ^ l. numbers ! (2*j + 1) * \\ ^ (n div llen* (2^l + i) * (2*j + 1))) =\n - ifntt2 ! i * \\ ^ (n div llen * i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..<2 ^ l. numbers ! (2 * j + 1) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j + 1))) = - ifntt2 ! i * \\ ^ (n div llen * i)\n[PROOF STEP]\nusing trans[OF l2_def numbers2_even] sym[OF 012] sym[OF 011]\n[PROOF STATE]\nproof (prove)\nusing this:\nl2 = 2 ^ l\n(\\j = 0.. ^ (n div llen * (2 ^ l + i) * (2 * j + 1))) = (\\j = 0.. ^ (n div llen * i * (2 * j + 1))))\n(\\j = 0.. ^ (n div llen * i * (2 * j + 1)))) = - ifntt2 ! i * \\ ^ (n div llen * i)\n\ngoal (1 subgoal):\n 1. (\\j = 0..<2 ^ l. numbers ! (2 * j + 1) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j + 1))) = - ifntt2 ! i * \\ ^ (n div llen * i)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..<2 ^ l. numbers ! (2 * j + 1) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j + 1))) = - ifntt2 ! i * \\ ^ (n div llen * i)\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2..^((n div llen) * i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. intt_gen numbers llen (2 ^ l + i) = ifntt1 ! i - ifntt2 ! i * \\ ^ (n div llen * i)\n[PROOF STEP]\nunfolding intt_gen_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0.. ^ (n div llen * (2 ^ l + i) * j)) = ifntt1 ! i - ifntt2 ! i * \\ ^ (n div llen * i)\n[PROOF STEP]\napply(subst Suc(2))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..<2 ^ Suc l. numbers ! j * \\ ^ (n div llen * (2 ^ l + i) * j)) = ifntt1 ! i - ifntt2 ! i * \\ ^ (n div llen * i)\n[PROOF STEP]\nusing sum_splice[of \"\\ d. numbers ! d * \\ ^ (n div llen * (2^l+i) * d)\" \"2^l\"] sym[OF 013] 014 Suc(2)\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\ia = 0..<2 * 2 ^ l. numbers ! ia * \\ ^ (n div llen * (2 ^ l + i) * ia)) = (\\ia = 0..<2 ^ l. numbers ! (2 * ia) * \\ ^ (n div llen * (2 ^ l + i) * (2 * ia))) + (\\ia = 0..<2 ^ l. numbers ! (2 * ia + 1) * \\ ^ (n div llen * (2 ^ l + i) * (2 * ia + 1)))\n(\\j = 0..<2 ^ l. numbers ! (2 * j) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j))) = ifntt1 ! i\n(\\j = 0..<2 ^ l. numbers ! (2 * j + 1) * \\ ^ (n div llen * (2 ^ l + i) * (2 * j + 1))) = - ifntt2 ! i * \\ ^ (n div llen * i)\nlength numbers = 2 ^ Suc l\n\ngoal (1 subgoal):\n 1. (\\j = 0..<2 ^ Suc l. numbers ! j * \\ ^ (n div llen * (2 ^ l + i) * j)) = ifntt1 ! i - ifntt2 ! i * \\ ^ (n div llen * i)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nintt_gen numbers llen (2 ^ l + i) = ifntt1 ! i - ifntt2 ! i * \\ ^ (n div llen * i)\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2.. ^ (n div llen * i)\n\ngoal (1 subgoal):\n 1. map (intt_gen numbers llen) [llen div 2.. ^ (n div llen * i)\nmap (intt_gen numbers llen) [llen div 2.. ^ (n div llen * i) = map2 (\\x y. x * \\ ^ (n div llen * y)) ifntt2 [0.. ^ (n div llen * i)\nsum2 = map2 (-) ifntt1 (map2 (\\x y. x * \\ ^ (n div length numbers * y)) ifntt2 [0.. map (intt_gen numbers llen) [llen div 2.. map (intt_gen numbers llen) [llen div 2.. map (intt_gen numbers llen) [llen div 2..l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nobtain x y xs where xyxs: \"numbers = x#y#xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x y xs. numbers = x # y # xs \\ thesis) \\ thesis\n[PROOF STEP]\nusing Suc(2)\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ Suc l\n\ngoal (1 subgoal):\n 1. (\\x y xs. numbers = x # y # xs \\ thesis) \\ thesis\n[PROOF STEP]\nby (metis FNTT.cases add.left_neutral even_Suc even_add length_Cons list.size(3) mult_2 power_Suc power_eq_0_iff zero_neq_numeral)\n[PROOF STATE]\nproof (state)\nthis:\nnumbers = x # y # xs\n\ngoal (1 subgoal):\n 1. \\l numbers. \\\\numbers. \\length numbers = 2 ^ l; 2 ^ l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers; length numbers = 2 ^ Suc l; 2 ^ Suc l \\ n\\ \\ IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. IFNTT numbers = INTT_gen (length numbers) numbers\n[PROOF STEP]\napply(subst xyxs)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. IFNTT (x # y # xs) = INTT_gen (length numbers) numbers\n[PROOF STEP]\napply(subst IFNTT.simps(3))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (let nn = length (x # y # xs); nums1 = map ((!) (x # y # xs)) (filter even [0..x y. x * \\ ^ (n div nn * y)) ifntt2 [0..x y. x * \\ ^ (n div nn * y)) ifntt2 [0..x y. x * \\ ^ (n div nn * y)) ifntt2 [0..x y. x * \\ ^ (n div nn * y)) ifntt2 [0..x y. x * \\ ^ (n div length numbers * y)) (IFNTT (map ((!) numbers) (filter odd [0..x y. x * \\ ^ (n div length numbers * y)) (IFNTT (map ((!) numbers) (filter odd [0..x y. x * \\ ^ (n div length numbers * y)) (IFNTT (map ((!) numbers) (filter odd [0..x y. x * \\ ^ (n div length numbers * y)) (IFNTT (map ((!) numbers) (filter odd [0..x y. x * \\ ^ (n div length numbers * y)) (IFNTT numbers2) [0..x y. x * \\ ^ (n div length numbers * y)) (IFNTT numbers2) [0..x y. x * \\ ^ (n div length numbers * y)) (IFNTT (map ((!) numbers) (filter odd [0..x y. x * \\ ^ (n div length numbers * y)) (IFNTT (map ((!) numbers) (filter odd [0.. nat\"\n assumes \"\\i \\ n. f i \\ 1\"\n shows \"(\\i \\ n. f i) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod f {..n} \\ 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i\\n. f i \\ 1\n\ngoal (1 subgoal):\n 1. prod f {..n} \\ 1\n[PROOF STEP]\nby (induct n) (auto intro: nat_mult_le_one)", "meta": {"llama_tokens": 200, "file": "Universal_Turing_Machine_Recs_alt_Def", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588023318196, "lm_q2_score": 0.8418256532040708, "lm_q1q2_score": 0.7481799593138516}} {"text": "[STATEMENT]\nlemma length_around_zero:\n assumes \"i >= 0\" \n shows \"length (around_zero i) = 2 * nat i + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (around_zero i) = 2 * nat i + 1\n[PROOF STEP]\nproof (induct rule: int_ge_induct [OF assms])\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. length (around_zero 0) = 2 * nat 0 + 1\n 2. \\i. \\0 \\ i; length (around_zero i) = 2 * nat i + 1\\ \\ length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\ncase 1\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. length (around_zero 0) = 2 * nat 0 + 1\n 2. \\i. \\0 \\ i; length (around_zero i) = 2 * nat i + 1\\ \\ length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\nfrom 1\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (around_zero 0) = 2 * nat 0 + 1\n[PROOF STEP]\nby (simp add: around_zero.simps)\n[PROOF STATE]\nproof (state)\nthis:\nlength (around_zero 0) = 2 * nat 0 + 1\n\ngoal (1 subgoal):\n 1. \\i. \\0 \\ i; length (around_zero i) = 2 * nat i + 1\\ \\ length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i. \\0 \\ i; length (around_zero i) = 2 * nat i + 1\\ \\ length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\ncase (2 i)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ i\nlength (around_zero i) = 2 * nat i + 1\n\ngoal (1 subgoal):\n 1. \\i. \\0 \\ i; length (around_zero i) = 2 * nat i + 1\\ \\ length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\nfrom 2\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ i\nlength (around_zero i) = 2 * nat i + 1\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ i\nlength (around_zero i) = 2 * nat i + 1\n\ngoal (1 subgoal):\n 1. length (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n[PROOF STEP]\nby (simp add: around_zero.simps [of \"i + 1\"])\n[PROOF STATE]\nproof (state)\nthis:\nlength (around_zero (i + 1)) = 2 * nat (i + 1) + 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1021, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772417253256, "lm_q2_score": 0.855851143290548, "lm_q1q2_score": 0.7480800066548686}} {"text": "[STATEMENT]\nlemma iso_paired2:\n assumes \"group G\" \"group H\"\n shows \"(\\(x,y). (f x,g y)) \\ iso (DirProd G H) (DirProd G' H') \\ f \\ iso G G' \\ g \\ iso H H'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\(x, y). (f x, g y)) \\ Group.iso (G \\\\ H) (G' \\\\ H')) = (f \\ Group.iso G G' \\ g \\ Group.iso H H')\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nGroup.group G\nGroup.group H\n\ngoal (1 subgoal):\n 1. ((\\(x, y). (f x, g y)) \\ Group.iso (G \\\\ H) (G' \\\\ H')) = (f \\ Group.iso G G' \\ g \\ Group.iso H H')\n[PROOF STEP]\nby (fastforce simp add: iso_def inj_on_def bij_betw_def hom_paired2 image_paired_Times\n times_eq_iff group_def monoid.carrier_not_empty)", "meta": {"llama_tokens": 358, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9073122163480667, "lm_q2_score": 0.8244619199068831, "lm_q1q2_score": 0.7480443718452964}} {"text": "[STATEMENT]\nlemma euclid:\n \"PRE (\\s::nat store. s ''x'' = x \\ s ''y'' = y)\n (WHILE (\\s. s ''y'' \\ 0) INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) \n DO\n (''z'' ::= (\\s. s ''y''));\n (''y'' ::= (\\s. s ''x'' mod s ''y''));\n (''x'' ::= (\\s. s ''z''))\n OD)\n POST (\\s. s ''x'' = gcd x y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rel_aka.dom_op \\\\s. s ''x'' = x \\ s ''y'' = y\\ \\ rel_aka.wp (rel_aka.while_inv \\\\s. s ''y'' \\ 0\\ \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\ ((''z'' ::= (\\s. s ''y'')) ; (''y'' ::= (\\s. s ''x'' mod s ''y'')) ; (''x'' ::= (\\s. s ''z'')))) \\\\s. s ''x'' = gcd x y\\\n[PROOF STEP]\napply (rule rel_aka.wp_while_inv, simp_all)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\\\s. s ''x'' = x \\ s ''y'' = y\\ \\ \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\\n 2. \\\\s. gcd (s ''x'') (s ''y'') = gcd x y \\ s ''y'' = 0\\ \\ \\\\s. s ''x'' = gcd x y\\\n 3. \\\\s. gcd (s ''x'') (s ''y'') = gcd x y \\ 0 < s ''y''\\ \\ \\\\s. gcd (s ''y'') (s ''x'' mod s ''y'') = gcd x y\\\n[PROOF STEP]\nusing gcd_red_nat\n[PROOF STATE]\nproof (prove)\nusing this:\ngcd ?x ?y = gcd ?y (?x mod ?y)\n\ngoal (3 subgoals):\n 1. \\\\s. s ''x'' = x \\ s ''y'' = y\\ \\ \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\\n 2. \\\\s. gcd (s ''x'') (s ''y'') = gcd x y \\ s ''y'' = 0\\ \\ \\\\s. s ''x'' = gcd x y\\\n 3. \\\\s. gcd (s ''x'') (s ''y'') = gcd x y \\ 0 < s ''y''\\ \\ \\\\s. gcd (s ''y'') (s ''x'' mod s ''y'') = gcd x y\\\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 931, "file": "Algebraic_VCs_AVC_KAD_VC_KAD_scratch", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.914900957313305, "lm_q2_score": 0.8175744806385542, "lm_q1q2_score": 0.7479996750111414}} {"text": "[STATEMENT]\nlemma infsetsum_add:\n assumes \"f abs_summable_on A\" and \"g abs_summable_on A\"\n shows \"infsetsum (\\x. f x + g x) A = infsetsum f A + infsetsum g A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. f x + g x) = infsetsum f A + infsetsum g A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nf abs_summable_on A\ng abs_summable_on A\n\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. f x + g x) = infsetsum f A + infsetsum g A\n[PROOF STEP]\nunfolding infsetsum_def abs_summable_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable (count_space A) f\nintegrable (count_space A) g\n\ngoal (1 subgoal):\n 1. LINT x|count_space A. f x + g x = integral\\<^sup>L (count_space A) f + integral\\<^sup>L (count_space A) g\n[PROOF STEP]\nby (rule Bochner_Integration.integral_add)", "meta": {"llama_tokens": 355, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9149009549929797, "lm_q2_score": 0.8175744695262775, "lm_q1q2_score": 0.7479996629474701}} {"text": "[STATEMENT]\nlemma Un_card1: \"\\ finite A; finite B \\ \\ card A \\ card (A \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite B\\ \\ card A \\ card (A \\ B)\n[PROOF STEP]\nby (rule card_mono, simp_all)", "meta": {"llama_tokens": 118, "file": "List-Infinite_CommonSet_SetInterval2", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505248181418, "lm_q2_score": 0.8267117919359419, "lm_q1q2_score": 0.7478852564481963}} {"text": "[STATEMENT]\nlemma L2_set_mult_ineq_lemma:\n fixes a b c d :: real\n shows \"2 * (a * c) * (b * d) \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * (a * c) * (b * d) \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 * (a * c) * (b * d) \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2\n[PROOF STEP]\nhave \"0 \\ (a * d - b * c)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ (a * d - b * c)\\<^sup>2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ (a * d - b * c)\\<^sup>2\n\ngoal (1 subgoal):\n 1. 2 * (a * c) * (b * d) \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ (a * d - b * c)\\<^sup>2\n\ngoal (1 subgoal):\n 1. 2 * (a * c) * (b * d) \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2\n[PROOF STEP]\nhave \"\\ = a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2 - 2 * (a * d) * (b * c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a * d - b * c)\\<^sup>2 = a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2 - 2 * (a * d) * (b * c)\n[PROOF STEP]\nby (simp only: power2_diff power_mult_distrib)\n[PROOF STATE]\nproof (state)\nthis:\n(a * d - b * c)\\<^sup>2 = a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2 - 2 * (a * d) * (b * c)\n\ngoal (1 subgoal):\n 1. 2 * (a * c) * (b * d) \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(a * d - b * c)\\<^sup>2 = a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2 - 2 * (a * d) * (b * c)\n\ngoal (1 subgoal):\n 1. 2 * (a * c) * (b * d) \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2\n[PROOF STEP]\nhave \"\\ = a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2 - 2 * (a * c) * (b * d)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2 - 2 * (a * d) * (b * c) = a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2 - 2 * (a * c) * (b * d)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2 - 2 * (a * d) * (b * c) = a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2 - 2 * (a * c) * (b * d)\n\ngoal (1 subgoal):\n 1. 2 * (a * c) * (b * d) \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2 - 2 * (a * c) * (b * d)\n[PROOF STEP]\nshow \"2 * (a * c) * (b * d) \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2 - 2 * (a * c) * (b * d)\n\ngoal (1 subgoal):\n 1. 2 * (a * c) * (b * d) \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * (a * c) * (b * d) \\ a\\<^sup>2 * d\\<^sup>2 + b\\<^sup>2 * c\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1511, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869786798664, "lm_q2_score": 0.8539127585282744, "lm_q1q2_score": 0.7478456748476678}} {"text": "[STATEMENT]\nlemma euclid2:\n \"PRE (\\s::nat store. s ''x'' = x \\ s ''y'' = y)\n (WHILE (\\s. s ''y'' \\ 0) INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) \n DO\n (''z'' ::= (\\s. s ''y''));\n (''y'' ::= (\\s. s ''x'' mod s ''y''));\n (''x'' ::= (\\s. s ''z''))\n OD)\n POST (\\s. s ''x'' = gcd x y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rdom \\\\s. s ''x'' = x \\ s ''y'' = y\\ \\ wp (WHILE (\\s. s ''y'' \\ zero_class.zero) INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) DO (''z'' ::= (\\s. s ''y'')) ; (''y'' ::= (\\s. s ''x'' mod s ''y'')) ; (''x'' ::= (\\s. s ''z'')) OD) \\\\s. s ''x'' = gcd x y\\\n[PROOF STEP]\napply hoare\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. rdom \\\\s. s ''x'' = x \\ s ''y'' = y\\ \\ rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\\n 2. rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\ ; pp_a \\\\s. s ''y'' \\ zero_class.zero\\ \\ rdom \\\\s. s ''x'' = gcd x y\\\n 3. rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\ ; rdom \\\\s. s ''y'' \\ zero_class.zero\\ \\ wp (''z'' ::= (\\s. s ''y'')) (wp (''y'' ::= (\\s. s ''x'' mod s ''y'')) (wp (''x'' ::= (\\s. s ''z'')) \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\))\n[PROOF STEP]\nusing gcd_red_nat\n[PROOF STATE]\nproof (prove)\nusing this:\ngcd ?x ?y = gcd ?y (?x mod ?y)\n\ngoal (3 subgoals):\n 1. rdom \\\\s. s ''x'' = x \\ s ''y'' = y\\ \\ rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\\n 2. rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\ ; pp_a \\\\s. s ''y'' \\ zero_class.zero\\ \\ rdom \\\\s. s ''x'' = gcd x y\\\n 3. rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\ ; rdom \\\\s. s ''y'' \\ zero_class.zero\\ \\ wp (''z'' ::= (\\s. s ''y'')) (wp (''y'' ::= (\\s. s ''x'' mod s ''y'')) (wp (''x'' ::= (\\s. s ''z'')) \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 1103, "file": "Algebraic_VCs_AVC_KAD_Path_Model_Example", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9173026618464795, "lm_q2_score": 0.8152324983301567, "lm_q1q2_score": 0.7478149407420083}} {"text": "[STATEMENT]\nlemma card_Collect_nat:\n assumes \"(j::nat)>i\"\n shows \"card {i..j} = j-i+1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {i..j} = j - i + 1\n[PROOF STEP]\nusing card_atLeastAtMost\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {?l..?u} = Suc ?u - ?l\n\ngoal (1 subgoal):\n 1. card {i..j} = j - i + 1\n[PROOF STEP]\nusing Suc_diff_le assms le_eq_less_or_eq\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {?l..?u} = Suc ?u - ?l\n?n \\ ?m \\ Suc ?m - ?n = Suc (?m - ?n)\ni < j\n(?m \\ ?n) = (?m < ?n \\ ?m = ?n)\n\ngoal (1 subgoal):\n 1. card {i..j} = j - i + 1\n[PROOF STEP]\nby presburger", "meta": {"llama_tokens": 306, "file": "Schutz_Spacetime_Util", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026595857203, "lm_q2_score": 0.8152324826183822, "lm_q1q2_score": 0.7478149244865115}} {"text": "[STATEMENT]\nlemma nat_minus_mod: \"(n - n mod m) mod m = 0\"\n for m n :: nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (n - n mod m) mod m = 0\n[PROOF STEP]\nby (induct n) (simp_all add: mod_Suc)", "meta": {"llama_tokens": 94, "file": "Word_Lib_More_Divides", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9489172673767974, "lm_q2_score": 0.787931188173138, "lm_q1q2_score": 0.7476815099622073}} {"text": "[STATEMENT]\nlemma M_cong_7_mod_12: \"[M = 7] (mod 12)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [M = 7] (mod 12)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. [M = 7] (mod 12)\n[PROOF STEP]\nhave \"[M = 8 - 1] (mod 12)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [M = 8 - 1] (mod 12)\n[PROOF STEP]\nusing p_gt_2 p_odd\n[PROOF STATE]\nproof (prove)\nusing this:\n2 < p\nodd p\n\ngoal (1 subgoal):\n 1. [M = 8 - 1] (mod 12)\n[PROOF STEP]\nunfolding M_def\n[PROOF STATE]\nproof (prove)\nusing this:\n2 < p\nodd p\n\ngoal (1 subgoal):\n 1. [2 ^ p - 1 = 8 - 1] (mod 12)\n[PROOF STEP]\nby (intro cong_diff_nat two_power_odd_mod_12) auto\n[PROOF STATE]\nproof (state)\nthis:\n[M = 8 - 1] (mod 12)\n\ngoal (1 subgoal):\n 1. [M = 7] (mod 12)\n[PROOF STEP]\nthus \"[M = 7] (mod 12)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n[M = 8 - 1] (mod 12)\n\ngoal (1 subgoal):\n 1. [M = 7] (mod 12)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n[M = 7] (mod 12)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 540, "file": "Mersenne_Primes_Lucas_Lehmer", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9019206659843132, "lm_q2_score": 0.8289388019824946, "lm_q1q2_score": 0.7476370363442902}} {"text": "[STATEMENT]\ntheorem equiv_msgrel: \"equivp msgrel\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. equivp (\\)\n[PROOF STEP]\nproof (rule equivpI)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. reflp (\\)\n 2. symp (\\)\n 3. transp (\\)\n[PROOF STEP]\nshow \"reflp msgrel\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. reflp (\\)\n[PROOF STEP]\nby (rule reflpI) (simp add: msgrel_refl)\n[PROOF STATE]\nproof (state)\nthis:\nreflp (\\)\n\ngoal (2 subgoals):\n 1. symp (\\)\n 2. transp (\\)\n[PROOF STEP]\nshow \"symp msgrel\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. symp (\\)\n[PROOF STEP]\nby (rule sympI) (blast intro: msgrel.SYM)\n[PROOF STATE]\nproof (state)\nthis:\nsymp (\\)\n\ngoal (1 subgoal):\n 1. transp (\\)\n[PROOF STEP]\nshow \"transp msgrel\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. transp (\\)\n[PROOF STEP]\nby (rule transpI) (blast intro: msgrel.TRANS)\n[PROOF STATE]\nproof (state)\nthis:\ntransp (\\)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 456, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894689081711, "lm_q2_score": 0.8354835371034368, "lm_q1q2_score": 0.7475818704463045}} {"text": "[STATEMENT]\nlemma norm_power_diff:\n fixes z w :: \"'a::{real_normed_algebra_1, comm_monoid_mult}\"\n assumes \"norm z \\ 1\" \"norm w \\ 1\"\n shows \"norm (z^m - w^m) \\ m * norm (z - w)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (z ^ m - w ^ m) \\ real m * norm (z - w)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm (z ^ m - w ^ m) \\ real m * norm (z - w)\n[PROOF STEP]\nhave \"norm (z^m - w^m) = norm ((\\ i < m. z) - (\\ i < m. w))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (z ^ m - w ^ m) = norm ((\\iiii real m * norm (z - w)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnorm (z ^ m - w ^ m) = norm ((\\ii real m * norm (z - w)\n[PROOF STEP]\nhave \"\\ \\ (\\iii (\\iii (\\i real m * norm (z - w)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnorm ((\\ii (\\i real m * norm (z - w)\n[PROOF STEP]\nhave \"\\ = m * norm (z - w)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ii real m * norm (z - w)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (z ^ m - w ^ m) \\ real m * norm (z - w)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (z ^ m - w ^ m) \\ real m * norm (z - w)\n\ngoal (1 subgoal):\n 1. norm (z ^ m - w ^ m) \\ real m * norm (z - w)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nnorm (z ^ m - w ^ m) \\ real m * norm (z - w)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1153, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894632969136, "lm_q2_score": 0.8354835309589074, "lm_q1q2_score": 0.747581860260131}} {"text": "[STATEMENT]\nlemma trigonometric_set_even:\n \"trigonometric_set(2*k) = (if k = 0 then (\\x. 1 / sqrt(2 * pi)) else (\\x. cos(k * x) / sqrt pi))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trigonometric_set (2 * k) = (if k = 0 then \\x. 1 / sqrt (2 * pi) else (\\x. cos (real k * x) / sqrt pi))\n[PROOF STEP]\nby (induction k) (auto simp: trigonometric_set_def add_divide_distrib split: if_split_asm)", "meta": {"llama_tokens": 185, "file": "Fourier_Fourier", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.899121388082479, "lm_q2_score": 0.8311430394931456, "lm_q1q2_score": 0.7472984833641677}} {"text": "[STATEMENT]\nlemma deriv_mult [simp]:\n \"\\f field_differentiable at z; g field_differentiable at z\\\n \\ deriv (\\w. f w * g w) z = f z * deriv g z + deriv f z * g z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f field_differentiable at z; g field_differentiable at z\\ \\ deriv (\\w. f w * g w) z = f z * deriv g z + deriv f z * g z\n[PROOF STEP]\nunfolding DERIV_deriv_iff_field_differentiable[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative deriv f z) (at z); (g has_field_derivative deriv g z) (at z)\\ \\ deriv (\\w. f w * g w) z = f z * deriv g z + deriv f z * g z\n[PROOF STEP]\nby (auto intro!: DERIV_imp_deriv derivative_eq_intros)", "meta": {"llama_tokens": 317, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213718636754, "lm_q2_score": 0.8311430457670241, "lm_q1q2_score": 0.7472984755250003}} {"text": "[STATEMENT]\nlemma multipl_div: \n fixes m k d1 d2 :: nat and f :: \"nat \\ complex\"\n assumes \"multiplicative_function f\" \"d1 dvd m\" \"d2 dvd k\" \"coprime m k\"\n shows \"f ((m*k) div (d1*d2)) = f(m div d1) * f(k div d2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (m * k div (d1 * d2)) = f (m div d1) * f (k div d2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nmultiplicative_function f\nd1 dvd m\nd2 dvd k\ncoprime m k\n\ngoal (1 subgoal):\n 1. f (m * k div (d1 * d2)) = f (m div d1) * f (k div d2)\n[PROOF STEP]\nunfolding multiplicative_function_def\n[PROOF STATE]\nproof (prove)\nusing this:\nf 0 = 0 \\ f 1 = 1 \\ (\\a b. 1 < a \\ 1 < b \\ coprime a b \\ f (a * b) = f a * f b)\nd1 dvd m\nd2 dvd k\ncoprime m k\n\ngoal (1 subgoal):\n 1. f (m * k div (d1 * d2)) = f (m div d1) * f (k div d2)\n[PROOF STEP]\nusing assms(1) multiplicative_function.mult_coprime\n[PROOF STATE]\nproof (prove)\nusing this:\nf 0 = 0 \\ f 1 = 1 \\ (\\a b. 1 < a \\ 1 < b \\ coprime a b \\ f (a * b) = f a * f b)\nd1 dvd m\nd2 dvd k\ncoprime m k\nmultiplicative_function f\n\\multiplicative_function ?f; coprime ?a ?b\\ \\ ?f (?a * ?b) = ?f ?a * ?f ?b\n\ngoal (1 subgoal):\n 1. f (m * k div (d1 * d2)) = f (m div d1) * f (k div d2)\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 630, "file": "Gauss_Sums_Gauss_Sums_Auxiliary", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213718636754, "lm_q2_score": 0.8311430394931456, "lm_q1q2_score": 0.7472984698840219}} {"text": "[STATEMENT]\nlemma linear_enclosure:\n fixes x::real\n assumes \"\\x. x \\ {a .. b} \\ (f has_field_derivative f' x) (at x within {a .. b})\"\n assumes \"\\x. x \\ {a .. b} \\ f' x \\ u\"\n assumes \"x \\ {a .. b}\"\n shows \"f x \\ {f b + u * (x - b) .. f a + u * (x - a)}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f x \\ {f b + u * (x - b)..f a + u * (x - a)}\n[PROOF STEP]\nusing linear_lower[OF assms] linear_upper[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\x. x \\ {a..b} \\ x \\ {a..b}; \\x. x \\ {a..b} \\ x \\ {a..b}\\ \\ f b + u * (x - b) \\ f x\n\\\\x. x \\ {a..b} \\ x \\ {a..b}; \\x. x \\ {a..b} \\ x \\ {a..b}\\ \\ f x \\ f a + u * (x - a)\n\ngoal (1 subgoal):\n 1. f x \\ {f b + u * (x - b)..f a + u * (x - a)}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 435, "file": "Affine_Arithmetic_Affine_Approximation", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9219218391455085, "lm_q2_score": 0.8104788995148792, "lm_q1q2_score": 0.7471981976293852}} {"text": "[STATEMENT]\ntheorem card_trees_of_size: \"card (trees_of_size n) = catalan n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (trees_of_size n) = catalan n\n[PROOF STEP]\nby (induction n rule: catalan.induct)\n (simp_all add: catalan_Suc trees_of_size_Suc card_image inj_on_def\n trees_of_size_disjoint Times_Int_Times catalan_Suc card_UN_disjoint)", "meta": {"llama_tokens": 144, "file": "Catalan_Numbers_Catalan_Numbers", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111797172476384, "lm_q2_score": 0.8198933293122507, "lm_q1q2_score": 0.7470701719759614}} {"text": "[STATEMENT]\nlemma find_zeros_from_vec_prop:\n fixes input_vec :: \"rat vec\"\n shows \"\\n < (dim_vec input_vec). ((input_vec $ n \\ 0) \\\n List.member (find_nonzeros_from_input_vec input_vec) n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n 0) = List.member (find_nonzeros_from_input_vec input_vec) n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n 0) = List.member (find_nonzeros_from_input_vec input_vec) n\n[PROOF STEP]\nhave h1: \"\\n < (dim_vec input_vec). ((input_vec $ n \\ 0) \\ List.member (find_nonzeros_from_input_vec input_vec) n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n 0 \\ List.member (find_nonzeros_from_input_vec input_vec) n\n[PROOF STEP]\nunfolding find_nonzeros_from_input_vec_def List.member_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n 0 \\ n \\ set (filter (\\i. input_vec $ i \\ 0) [0..n 0 \\ List.member (find_nonzeros_from_input_vec input_vec) n\n\ngoal (1 subgoal):\n 1. \\n 0) = List.member (find_nonzeros_from_input_vec input_vec) n\n[PROOF STEP]\nhave h2: \"\\n < (dim_vec input_vec). (List.member (find_nonzeros_from_input_vec input_vec) n \\ (input_vec $ n \\ 0) )\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n input_vec $ n \\ 0\n[PROOF STEP]\nunfolding find_nonzeros_from_input_vec_def List.member_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n set (filter (\\i. input_vec $ i \\ 0) [0.. input_vec $ n \\ 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\n input_vec $ n \\ 0\n\ngoal (1 subgoal):\n 1. \\n 0) = List.member (find_nonzeros_from_input_vec input_vec) n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n 0) = List.member (find_nonzeros_from_input_vec input_vec) n\n[PROOF STEP]\nusing h1 h2\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n 0 \\ List.member (find_nonzeros_from_input_vec input_vec) n\n\\n input_vec $ n \\ 0\n\ngoal (1 subgoal):\n 1. \\n 0) = List.member (find_nonzeros_from_input_vec input_vec) n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\n 0) = List.member (find_nonzeros_from_input_vec input_vec) n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1355, "file": "BenOr_Kozen_Reif_BKR_Proofs", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240895276223, "lm_q2_score": 0.8633915976709976, "lm_q1q2_score": 0.7470272090006881}} {"text": "[STATEMENT]\nlemma integral_norm_bound_integral:\n fixes f :: \"'a::euclidean_space \\ 'b::{banach,second_countable_topology}\"\n assumes \"integrable M f\" \"integrable M g\" \"\\x. x \\ space M \\ norm(f x) \\ g x\"\n shows \"norm (\\x. f x \\M) \\ (\\x. g x \\M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (integral\\<^sup>L M f) \\ integral\\<^sup>L M g\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm (integral\\<^sup>L M f) \\ integral\\<^sup>L M g\n[PROOF STEP]\nhave \"norm (\\x. f x \\M) \\ (\\x. norm (f x) \\M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (integral\\<^sup>L M f) \\ LINT x|M. norm (f x)\n[PROOF STEP]\nby (rule integral_norm_bound)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (integral\\<^sup>L M f) \\ LINT x|M. norm (f x)\n\ngoal (1 subgoal):\n 1. norm (integral\\<^sup>L M f) \\ integral\\<^sup>L M g\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnorm (integral\\<^sup>L M f) \\ LINT x|M. norm (f x)\n\ngoal (1 subgoal):\n 1. norm (integral\\<^sup>L M f) \\ integral\\<^sup>L M g\n[PROOF STEP]\nhave \"... \\ (\\x. g x \\M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LINT x|M. norm (f x) \\ integral\\<^sup>L M g\n[PROOF STEP]\nusing assms integrable_norm integral_mono\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable M f\nintegrable M g\n?x \\ space M \\ norm (f ?x) \\ g ?x\nintegrable ?M ?f \\ integrable ?M (\\x. norm (?f x))\n\\integrable ?M ?f; integrable ?M ?g; \\x. x \\ space ?M \\ ?f x \\ ?g x\\ \\ integral\\<^sup>L ?M ?f \\ integral\\<^sup>L ?M ?g\n\ngoal (1 subgoal):\n 1. LINT x|M. norm (f x) \\ integral\\<^sup>L M g\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nLINT x|M. norm (f x) \\ integral\\<^sup>L M g\n\ngoal (1 subgoal):\n 1. norm (integral\\<^sup>L M f) \\ integral\\<^sup>L M g\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (integral\\<^sup>L M f) \\ integral\\<^sup>L M g\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (integral\\<^sup>L M f) \\ integral\\<^sup>L M g\n\ngoal (1 subgoal):\n 1. norm (integral\\<^sup>L M f) \\ integral\\<^sup>L M g\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nnorm (integral\\<^sup>L M f) \\ integral\\<^sup>L M g\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1050, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9136765210631689, "lm_q2_score": 0.817574478416099, "lm_q1q2_score": 0.7469986051492562}} {"text": "[STATEMENT]\ntheorem M8: \"\\ \\[F \\ G]_v = (\\[F]_v \\ \\[G]_v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ \\[F \\ G]_v = (\\[F]_v \\ \\[G]_v)\n[PROOF STEP]\nby (rule int_iffI[OF M6 M7])", "meta": {"llama_tokens": 120, "file": "TLA_Rules", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9059898203834278, "lm_q2_score": 0.8244619306896956, "lm_q1q2_score": 0.7469541164985315}} {"text": "[STATEMENT]\nlemma op_norm_le_max_norm:\n fixes A :: \"real^('n::finite)^('m::finite)\"\n shows \"\\A\\\\<^sub>o\\<^sub>p \\ real CARD('m) * real CARD('n) * (\\A\\\\<^sub>m\\<^sub>a\\<^sub>x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ real CARD('m) * real CARD('n) * (\\A\\\\<^sub>m\\<^sub>a\\<^sub>x)\n[PROOF STEP]\napply(rule onorm_le_matrix_component)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. \\A $ i $ j\\ \\ \\A\\\\<^sub>m\\<^sub>a\\<^sub>x\n[PROOF STEP]\nunfolding max_norm_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. \\A $ i $ j\\ \\ Max {\\A $ i $ j\\ |i j. i \\ UNIV \\ j \\ UNIV}\n[PROOF STEP]\nby(rule Max_ge[OF max_norm_set_proptys]) force", "meta": {"llama_tokens": 366, "file": "Matrices_for_ODEs_MTX_Norms", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9059898153067649, "lm_q2_score": 0.8244619220634456, "lm_q1q2_score": 0.7469541044977215}} {"text": "[STATEMENT]\nlemma onorm_scaleC_left_lemma:\n fixes f :: \"'a::complex_normed_vector\"\n assumes r: \"bounded_clinear r\"\n shows \"onorm (\\x. r x *\\<^sub>C f) \\ onorm r * norm f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. onorm (\\x. r x *\\<^sub>C f) \\ onorm r * norm f\n[PROOF STEP]\nproof (rule onorm_bound)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 0 \\ onorm r * norm f\n 2. \\x. norm (r x *\\<^sub>C f) \\ onorm r * norm f * norm x\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 0 \\ onorm r * norm f\n 2. \\x. norm (r x *\\<^sub>C f) \\ onorm r * norm f * norm x\n[PROOF STEP]\nhave \"norm (r x *\\<^sub>C f) = norm (r x) * norm f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (r x *\\<^sub>C f) = cmod (r x) * norm f\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nnorm (r x *\\<^sub>C f) = cmod (r x) * norm f\n\ngoal (2 subgoals):\n 1. 0 \\ onorm r * norm f\n 2. \\x. norm (r x *\\<^sub>C f) \\ onorm r * norm f * norm x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnorm (r x *\\<^sub>C f) = cmod (r x) * norm f\n\ngoal (2 subgoals):\n 1. 0 \\ onorm r * norm f\n 2. \\x. norm (r x *\\<^sub>C f) \\ onorm r * norm f * norm x\n[PROOF STEP]\nhave \"\\ \\ onorm r * norm x * norm f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (r x) * norm f \\ onorm r * norm x * norm f\n[PROOF STEP]\nby (simp add: bounded_clinear.bounded_linear mult.commute mult_left_mono onorm r)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (r x) * norm f \\ onorm r * norm x * norm f\n\ngoal (2 subgoals):\n 1. 0 \\ onorm r * norm f\n 2. \\x. norm (r x *\\<^sub>C f) \\ onorm r * norm f * norm x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (r x *\\<^sub>C f) \\ onorm r * norm x * norm f\n[PROOF STEP]\nshow \"norm (r x *\\<^sub>C f) \\ onorm r * norm f * norm x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (r x *\\<^sub>C f) \\ onorm r * norm x * norm f\n\ngoal (1 subgoal):\n 1. norm (r x *\\<^sub>C f) \\ onorm r * norm f * norm x\n[PROOF STEP]\nby (simp add: ac_simps)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (r x *\\<^sub>C f) \\ onorm r * norm f * norm x\n\ngoal (1 subgoal):\n 1. 0 \\ onorm r * norm f\n[PROOF STEP]\nshow \"0 \\ onorm r * norm f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ onorm r * norm f\n[PROOF STEP]\nby (simp add: bounded_clinear.bounded_linear onorm_pos_le r)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ onorm r * norm f\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1149, "file": "Complex_Bounded_Operators_Complex_Vector_Spaces", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872046056466901, "lm_q2_score": 0.8418256412990657, "lm_q1q2_score": 0.7468715861120095}} {"text": "[STATEMENT]\nlemma gcd_lcm_distrib:\n \"gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n[PROOF STEP]\nproof (cases \"x = 0 \\ y = 0 \\ z = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x = (0::'a) \\ y = (0::'a) \\ z = (0::'a) \\ gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n 2. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nx = (0::'a) \\ y = (0::'a) \\ z = (0::'a)\n\ngoal (2 subgoals):\n 1. x = (0::'a) \\ y = (0::'a) \\ z = (0::'a) \\ gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n 2. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx = (0::'a) \\ y = (0::'a) \\ z = (0::'a)\n\ngoal (1 subgoal):\n 1. gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n[PROOF STEP]\nby (auto simp: lcm_proj1_if_dvd lcm_proj2_if_dvd)\n[PROOF STATE]\nproof (state)\nthis:\ngcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n\ngoal (1 subgoal):\n 1. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a))\n\ngoal (1 subgoal):\n 1. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n[PROOF STEP]\nhence \"normalize (gcd x (lcm y z)) = normalize (lcm (gcd x y) (gcd x z))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a))\n\ngoal (1 subgoal):\n 1. normalize (gcd x (lcm y z)) = normalize (lcm (gcd x y) (gcd x z))\n[PROOF STEP]\nby (intro associatedI prime_factorization_subset_imp_dvd)\n (auto simp: lcm_eq_0_iff prime_factorization_gcd prime_factorization_lcm\n subset_mset.inf_sup_distrib1)\n[PROOF STATE]\nproof (state)\nthis:\nnormalize (gcd x (lcm y z)) = normalize (lcm (gcd x y) (gcd x z))\n\ngoal (1 subgoal):\n 1. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnormalize (gcd x (lcm y z)) = normalize (lcm (gcd x y) (gcd x z))\n\ngoal (1 subgoal):\n 1. gcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ngcd x (lcm y z) = lcm (gcd x y) (gcd x z)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1427, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942041005328, "lm_q2_score": 0.8267118026095991, "lm_q1q2_score": 0.7467639797587546}} {"text": "[STATEMENT]\nlemma imirror_bounds_Int: \"\n \\ A \\ {..l + r}; B \\ {..l + r} \\ \\\n imirror_bounds (A \\ B) l r =\n imirror_bounds A l r \\ imirror_bounds B l r\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ {..l + r}; B \\ {..l + r}\\ \\ imirror_bounds (A \\ B) l r = imirror_bounds A l r \\ imirror_bounds B l r\n[PROOF STEP]\napply (unfold imirror_bounds_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ {..l + r}; B \\ {..l + r}\\ \\ (\\x. nat_mirror x l r) ` (A \\ B) = (\\x. nat_mirror x l r) ` A \\ (\\x. nat_mirror x l r) ` B\n[PROOF STEP]\napply (rule inj_on_image_Int[OF _ Un_upper1 Un_upper2])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ {..l + r}; B \\ {..l + r}\\ \\ inj_on (\\x. nat_mirror x l r) (A \\ B)\n[PROOF STEP]\napply (rule subset_inj_on[OF nat_mirror_inj_on])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ {..l + r}; B \\ {..l + r}\\ \\ A \\ B \\ {..l + r}\n[PROOF STEP]\napply (rule Un_least[of A _ B], assumption+)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 581, "file": "List-Infinite_CommonSet_SetIntervalCut", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942014971871, "lm_q2_score": 0.8267117962054049, "lm_q1q2_score": 0.7467639718216665}} {"text": "[STATEMENT]\nlemma matrix_match_condn_2:\n\"matrix_match A1 A2 B1 B2 \n \\((i<(row_length A1)*(row_length B1))\n \\(j<(length A2)*(length B2)))\n \\ ((A1 \\ B1)\\(A2 \\ B2))!j!i\n = scalar_product \n (vec_vec_Tensor \n (row A1 (i div row_length B1)) \n (row B1 (i mod row_length B1))) \n (vec_vec_Tensor \n (col A2 (j div length B2)) \n (col B2 (j mod length B2)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_match A1 A2 B1 B2 \\ i < row_length A1 * row_length B1 \\ j < length A2 * length B2 \\ ((A1 \\ B1) \\ (A2 \\ B2)) ! j ! i = scalar_product (vec_vec_Tensor (row A1 (i div row_length B1)) (row B1 (i mod row_length B1))) (vec_vec_Tensor (col A2 (j div length B2)) (col B2 (j mod length B2)))\n[PROOF STEP]\nusing elements_matrix_distribution2\n[PROOF STATE]\nproof (prove)\nusing this:\n\\mat (row_length ?A1.0) (length ?A1.0) ?A1.0; mat (row_length ?A2.0) (length ?A2.0) ?A2.0; mat (row_length ?B1.0) (length ?B1.0) ?B1.0; mat (row_length ?B2.0) (length ?B2.0) ?B2.0; length ?A1.0 = row_length ?A2.0; length ?B1.0 = row_length ?B2.0; ?A1.0 \\ [] \\ ?A2.0 \\ [] \\ ?B1.0 \\ [] \\ ?B2.0 \\ []; ?i < row_length ?A1.0 * row_length ?B1.0; ?j < length ?A2.0 * length ?B2.0\\ \\ ((?A1.0 \\ ?B1.0) \\ (?A2.0 \\ ?B2.0)) ! ?j ! ?i = scalar_product (vec_vec_Tensor (row ?A1.0 (?i div row_length ?B1.0)) (row ?B1.0 (?i mod row_length ?B1.0))) (vec_vec_Tensor (col ?A2.0 (?j div length ?B2.0)) (col ?B2.0 (?j mod length ?B2.0)))\n\ngoal (1 subgoal):\n 1. matrix_match A1 A2 B1 B2 \\ i < row_length A1 * row_length B1 \\ j < length A2 * length B2 \\ ((A1 \\ B1) \\ (A2 \\ B2)) ! j ! i = scalar_product (vec_vec_Tensor (row A1 (i div row_length B1)) (row B1 (i mod row_length B1))) (vec_vec_Tensor (col A2 (j div length B2)) (col B2 (j mod length B2)))\n[PROOF STEP]\nunfolding matrix_match_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\mat (row_length ?A1.0) (length ?A1.0) ?A1.0; mat (row_length ?A2.0) (length ?A2.0) ?A2.0; mat (row_length ?B1.0) (length ?B1.0) ?B1.0; mat (row_length ?B2.0) (length ?B2.0) ?B2.0; length ?A1.0 = row_length ?A2.0; length ?B1.0 = row_length ?B2.0; ?A1.0 \\ [] \\ ?A2.0 \\ [] \\ ?B1.0 \\ [] \\ ?B2.0 \\ []; ?i < row_length ?A1.0 * row_length ?B1.0; ?j < length ?A2.0 * length ?B2.0\\ \\ ((?A1.0 \\ ?B1.0) \\ (?A2.0 \\ ?B2.0)) ! ?j ! ?i = scalar_product (vec_vec_Tensor (row ?A1.0 (?i div row_length ?B1.0)) (row ?B1.0 (?i mod row_length ?B1.0))) (vec_vec_Tensor (col ?A2.0 (?j div length ?B2.0)) (col ?B2.0 (?j mod length ?B2.0)))\n\ngoal (1 subgoal):\n 1. (mat (row_length A1) (length A1) A1 \\ mat (row_length A2) (length A2) A2 \\ mat (row_length B1) (length B1) B1 \\ mat (row_length B2) (length B2) B2 \\ length A1 = row_length A2 \\ length B1 = row_length B2 \\ A1 \\ [] \\ A2 \\ [] \\ B1 \\ [] \\ B2 \\ []) \\ i < row_length A1 * row_length B1 \\ j < length A2 * length B2 \\ ((A1 \\ B1) \\ (A2 \\ B2)) ! j ! i = scalar_product (vec_vec_Tensor (row A1 (i div row_length B1)) (row B1 (i mod row_length B1))) (vec_vec_Tensor (col A2 (j div length B2)) (col B2 (j mod length B2)))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 1556, "file": "Matrix_Tensor_Matrix_Tensor", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9184802440252811, "lm_q2_score": 0.8128673087708699, "lm_q1q2_score": 0.746602564120042}} {"text": "[STATEMENT]\nlemma triangle_lemma:\n fixes x y z :: real\n assumes x: \"0 \\ x\"\n and y: \"0 \\ y\"\n and z: \"0 \\ z\"\n and xy: \"x\\<^sup>2 \\ y\\<^sup>2 + z\\<^sup>2\"\n shows \"x \\ y + z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y + z\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x \\ y + z\n[PROOF STEP]\nhave \"y\\<^sup>2 + z\\<^sup>2 \\ y\\<^sup>2 + 2 * y * z + z\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. y\\<^sup>2 + z\\<^sup>2 \\ y\\<^sup>2 + 2 * y * z + z\\<^sup>2\n[PROOF STEP]\nusing z y\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ z\n0 \\ y\n\ngoal (1 subgoal):\n 1. y\\<^sup>2 + z\\<^sup>2 \\ y\\<^sup>2 + 2 * y * z + z\\<^sup>2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ny\\<^sup>2 + z\\<^sup>2 \\ y\\<^sup>2 + 2 * y * z + z\\<^sup>2\n\ngoal (1 subgoal):\n 1. x \\ y + z\n[PROOF STEP]\nwith xy\n[PROOF STATE]\nproof (chain)\npicking this:\nx\\<^sup>2 \\ y\\<^sup>2 + z\\<^sup>2\ny\\<^sup>2 + z\\<^sup>2 \\ y\\<^sup>2 + 2 * y * z + z\\<^sup>2\n[PROOF STEP]\nhave th: \"x\\<^sup>2 \\ (y + z)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx\\<^sup>2 \\ y\\<^sup>2 + z\\<^sup>2\ny\\<^sup>2 + z\\<^sup>2 \\ y\\<^sup>2 + 2 * y * z + z\\<^sup>2\n\ngoal (1 subgoal):\n 1. x\\<^sup>2 \\ (y + z)\\<^sup>2\n[PROOF STEP]\nby (simp add: power2_eq_square field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nx\\<^sup>2 \\ (y + z)\\<^sup>2\n\ngoal (1 subgoal):\n 1. x \\ y + z\n[PROOF STEP]\nfrom y z\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ y\n0 \\ z\n[PROOF STEP]\nhave yz: \"y + z \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ y\n0 \\ z\n\ngoal (1 subgoal):\n 1. 0 \\ y + z\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ y + z\n\ngoal (1 subgoal):\n 1. x \\ y + z\n[PROOF STEP]\nfrom power2_le_imp_le[OF th yz]\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ y + z\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ y + z\n\ngoal (1 subgoal):\n 1. x \\ y + z\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nx \\ y + z\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1046, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473813156294, "lm_q2_score": 0.8558511396138366, "lm_q1q2_score": 0.7465995004381275}} {"text": "[STATEMENT]\nlemma in_set_least_squares_approximation_eq_full_rank_QR:\n fixes A::\"real^'cols::{mod_type}^'rows::{mod_type}\"\n assumes r: \"rank A = ncols A\"\n shows \"(x \\ set_least_squares_approximation A b) = ((snd (QR_decomposition A)) *v x = transpose (fst (QR_decomposition A)) *v b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nlet ?Q = \"fst (QR_decomposition A)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nlet ?R = \"snd (QR_decomposition A)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nhave inv_tR: \"invertible (transpose ?R)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. invertible (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)))\n[PROOF STEP]\nby (metis invertible_snd_QR_decomposition invertible_transpose r)\n[PROOF STATE]\nproof (state)\nthis:\ninvertible (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)))\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nhave inv_inv_tR: \"invertible (matrix_inv (transpose ?R))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. invertible (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))))\n[PROOF STEP]\nby (metis inv_tR invertible_fst_Gauss_Jordan_PA matrix_inv_Gauss_Jordan_PA)\n[PROOF STATE]\nproof (state)\nthis:\ninvertible (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))))\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nhave \"(x \\ set_least_squares_approximation A b) = (transpose A ** A *v x = transpose A *v b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (Finite_Cartesian_Product.transpose A ** A *v x = Finite_Cartesian_Product.transpose A *v b)\n[PROOF STEP]\nusing in_set_least_squares_approximation_eq\n[PROOF STATE]\nproof (prove)\nusing this:\n(?x \\ set_least_squares_approximation ?A ?b) = (Finite_Cartesian_Product.transpose ?A ** ?A *v ?x = Finite_Cartesian_Product.transpose ?A *v ?b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (Finite_Cartesian_Product.transpose A ** A *v x = Finite_Cartesian_Product.transpose A *v b)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(x \\ set_least_squares_approximation A b) = (Finite_Cartesian_Product.transpose A ** A *v x = Finite_Cartesian_Product.transpose A *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(x \\ set_least_squares_approximation A b) = (Finite_Cartesian_Product.transpose A ** A *v x = Finite_Cartesian_Product.transpose A *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nhave \"... = (transpose (?Q ** ?R) ** (?Q ** ?R) *v x = transpose (?Q ** ?R) *v b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose A ** A *v x = Finite_Cartesian_Product.transpose A *v b) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v b)\n[PROOF STEP]\nusing QR_decomposition_mult[OF r]\n[PROOF STATE]\nproof (prove)\nusing this:\nA = fst (QR_decomposition A) ** snd (QR_decomposition A)\n\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose A ** A *v x = Finite_Cartesian_Product.transpose A *v b) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v b)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose A ** A *v x = Finite_Cartesian_Product.transpose A *v b) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose A ** A *v x = Finite_Cartesian_Product.transpose A *v b) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nhave \"... = (transpose ?R ** transpose ?Q ** (?Q ** ?R) *v x = transpose ?R ** transpose ?Q *v b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v b) = (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nby (metis (opaque_lifting, no_types) matrix_transpose_mul)\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v b) = (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v b) = (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nhave \"... = (transpose ?R *v (transpose ?Q ** (?Q ** ?R) *v x) = transpose ?R *v (transpose ?Q *v b))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b) = (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b))\n[PROOF STEP]\nby (metis (opaque_lifting, no_types) matrix_vector_mul_assoc)\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b) = (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b))\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b) = (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b))\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nhave \"... = (matrix_inv (transpose ?R) *v (transpose ?R *v (transpose ?Q ** (?Q ** ?R) *v x)) \n = matrix_inv (transpose ?R) *v (transpose ?R *v (transpose ?Q *v b)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)) = (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x)) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)))\n[PROOF STEP]\nusing inv_matrix_vector_mul_left[OF inv_inv_tR]\n[PROOF STATE]\nproof (prove)\nusing this:\n(matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v ?x = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v ?y) = (?x = ?y)\n\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)) = (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x)) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)) = (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x)) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)))\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)) = (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x)) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)))\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nhave \"... = ((matrix_inv (transpose ?R) ** transpose ?R) *v (transpose ?Q ** (?Q ** ?R) *v x) \n = (matrix_inv (transpose ?R) ** transpose ?R) *v (transpose ?Q *v b))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x)) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b))) = (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b))\n[PROOF STEP]\nby (metis (opaque_lifting, no_types) matrix_vector_mul_assoc)\n[PROOF STATE]\nproof (state)\nthis:\n(matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x)) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b))) = (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b))\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x)) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) *v (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b))) = (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b))\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nhave \"... = (transpose ?Q ** (?Q ** ?R) *v x = transpose ?Q *v b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nunfolding matrix_inv_left[OF inv_tR]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (mat 1 *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = mat 1 *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nunfolding matrix_vector_mul_lid\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n(matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x) = matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))) ** Finite_Cartesian_Product.transpose (snd (QR_decomposition A)) *v (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nhave \"... = ((transpose ?Q ** ?Q) ** ?R *v x = transpose ?Q *v b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A) ** snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nby (metis (opaque_lifting, no_types) matrix_mul_assoc)\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A) ** snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** (fst (QR_decomposition A) ** snd (QR_decomposition A)) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b) = (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A) ** snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nhave \"... = (?R *v x = transpose ?Q *v b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A) ** snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nunfolding orthogonal_matrix_fst_QR_decomposition[OF r]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (mat 1 ** snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nunfolding matrix_mul_lid\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A) ** snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\nshow \"(x \\ set_least_squares_approximation A b) = (?R *v x = (transpose ?Q) *v b)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n\ngoal (1 subgoal):\n 1. (x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(x \\ set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 10928, "file": "QR_Decomposition_Least_Squares_Approximation", "length": 44, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297861178929, "lm_q2_score": 0.8289388040954683, "lm_q1q2_score": 0.746484083956914}} {"text": "[STATEMENT]\ntheorem eval_negation:\n \"eval i (Neg' p) =\n (\n if eval i p = Det False then Det True else\n if eval i p = Det True then Det False else\n eval i p\n )\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eval i (Neg' p) = (if eval i p = Det False then Det True else if eval i p = Det True then Det False else eval i p)\n[PROOF STEP]\nby (cases \"eval i p\") simp_all", "meta": {"llama_tokens": 141, "file": "Paraconsistency_Paraconsistency", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297754396142, "lm_q2_score": 0.828938799869521, "lm_q1q2_score": 0.7464840712996831}} {"text": "[STATEMENT]\nlemma linear_singular_into_hyperplane:\n fixes f :: \"'n::euclidean_space \\ 'n\"\n assumes \"linear f\"\n shows \"\\ inj f \\ (\\a. a \\ 0 \\ (\\x. a \\ f x = 0))\" (is \"_ = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ inj f) = (\\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0))\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\ inj f \\ \\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0)\n 2. \\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0) \\ \\ inj f\n[PROOF STEP]\nassume \"\\inj f\"\n[PROOF STATE]\nproof (state)\nthis:\n\\ inj f\n\ngoal (2 subgoals):\n 1. \\ inj f \\ \\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0)\n 2. \\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0) \\ \\ inj f\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ inj f\n[PROOF STEP]\nshow ?rhs\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ inj f\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0)\n[PROOF STEP]\nusing all_zero_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ inj f\n(\\u. ?x \\ u = 0) = (?x = (0::?'a))\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0)\n[PROOF STEP]\nby (metis (no_types, opaque_lifting) adjoint_clauses(2) adjoint_linear assms\n linear_injective_0 linear_injective_imp_surjective linear_surj_adj_imp_inj)\n[PROOF STATE]\nproof (state)\nthis:\n\\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0)\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0) \\ \\ inj f\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0) \\ \\ inj f\n[PROOF STEP]\nassume ?rhs\n[PROOF STATE]\nproof (state)\nthis:\n\\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0)\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0) \\ \\ inj f\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0)\n[PROOF STEP]\nshow \"\\inj f\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a. a \\ (0::'n) \\ (\\x. a \\ f x = 0)\n\ngoal (1 subgoal):\n 1. \\ inj f\n[PROOF STEP]\nby (metis assms linear_injective_isomorphism all_zero_iff)\n[PROOF STATE]\nproof (state)\nthis:\n\\ inj f\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1228, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.882427872638409, "lm_q2_score": 0.8459424334245618, "lm_q1q2_score": 0.7464831819013951}} {"text": "[STATEMENT]\nlemma setprod_one_le:\n fixes f::\"nat \\ nat\"\n assumes \"\\i \\ n. f i \\ 1\"\n shows \"(\\i \\ n. f i) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod f {..n} \\ 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i\\n. f i \\ 1\n\ngoal (1 subgoal):\n 1. prod f {..n} \\ 1\n[PROOF STEP]\nby (induct n) (auto intro: nat_mult_le_one)", "meta": {"llama_tokens": 200, "file": "Universal_Turing_Machine_Recs", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588023318196, "lm_q2_score": 0.8397339676722393, "lm_q1q2_score": 0.7463209553857263}} {"text": "[STATEMENT]\nlemma sqr_of_cmod_of_prod:\n shows \"(cmod (z1 * z2))\\<^sup>2 = (cmod z1)\\<^sup>2 * (cmod z2)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cmod (z1 * z2))\\<^sup>2 = (cmod z1)\\<^sup>2 * (cmod z2)\\<^sup>2\n[PROOF STEP]\nby (simp add: norm_mult power_mult_distrib)", "meta": {"llama_tokens": 146, "file": "Isabelle_Marries_Dirac_Basics", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9207896758909756, "lm_q2_score": 0.8104789063814616, "lm_q1q2_score": 0.7462806095234583}} {"text": "[STATEMENT]\ntheorem weak_integer_compositions_cardinality: \"card (weak_integer_compositions n k) = k multichoose n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (weak_integer_compositions n k) = k multichoose n\n[PROOF STEP]\nusing weak_integer_composition_enum_correct weak_integer_composition_enum_distinct composition_enum_length\n distinct_card\n[PROOF STATE]\nproof (prove)\nusing this:\nset (weak_integer_composition_enum ?i ?l) = weak_integer_compositions ?i ?l\ndistinct (weak_integer_composition_enum ?i ?l)\nlength (weak_integer_composition_enum ?i ?n) = ?n multichoose ?i\ndistinct ?xs \\ card (set ?xs) = length ?xs\n\ngoal (1 subgoal):\n 1. card (weak_integer_compositions n k) = k multichoose n\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 266, "file": "Combinatorial_Enumeration_Algorithms_Weak_Integer_Compositions", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9263037384317887, "lm_q2_score": 0.805632181981183, "lm_q1q2_score": 0.7462601019701289}} {"text": "[STATEMENT]\nlemma vec_norm_triangle_sq:\n fixes u::\"complex Matrix.vec\"\n assumes \"dim_vec u = dim_vec v\"\n shows \"(vec_norm (u+ v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nhave \"(vec_norm (u+v))\\<^sup>2 = inner_prod (u+v) (u+v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 = Complex_Matrix.inner_prod (u + v) (u + v)\n[PROOF STEP]\nby (simp add: inner_prod_vec_norm_pow2)\n[PROOF STATE]\nproof (state)\nthis:\n(vec_norm (u + v))\\<^sup>2 = Complex_Matrix.inner_prod (u + v) (u + v)\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(vec_norm (u + v))\\<^sup>2 = Complex_Matrix.inner_prod (u + v) (u + v)\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nhave \"... = inner_prod u u + inner_prod u v + inner_prod v u + \n inner_prod v v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (u + v) (u + v) = Complex_Matrix.inner_prod u u + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + Complex_Matrix.inner_prod v v\n[PROOF STEP]\nusing assms add_scalar_prod_distrib conjugate_add_vec\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec u = dim_vec v\n\\?v\\<^sub>1 \\ carrier_vec ?n; ?v\\<^sub>2 \\ carrier_vec ?n; ?v\\<^sub>3 \\ carrier_vec ?n\\ \\ (?v\\<^sub>1 + ?v\\<^sub>2) \\ ?v\\<^sub>3 = ?v\\<^sub>1 \\ ?v\\<^sub>3 + ?v\\<^sub>2 \\ ?v\\<^sub>3\n\\?v \\ carrier_vec ?n; ?w \\ carrier_vec ?n\\ \\ conjugate (?v + ?w) = conjugate ?v + conjugate ?w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (u + v) (u + v) = Complex_Matrix.inner_prod u u + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + Complex_Matrix.inner_prod v v\n[PROOF STEP]\nby (smt (z3) ab_semigroup_add_class.add_ac(1) carrier_vec_dim_vec \n dim_vec_conjugate index_add_vec(2) scalar_prod_add_distrib)\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (u + v) (u + v) = Complex_Matrix.inner_prod u u + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + Complex_Matrix.inner_prod v v\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (u + v) (u + v) = Complex_Matrix.inner_prod u u + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + Complex_Matrix.inner_prod v v\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nhave \"... = (vec_norm u)^2 + inner_prod u v + inner_prod v u + \n (vec_norm v)^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod u u + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + Complex_Matrix.inner_prod v v = (vec_norm u)\\<^sup>2 + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + (vec_norm v)\\<^sup>2\n[PROOF STEP]\nby (simp add: inner_prod_vec_norm_pow2)\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod u u + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + Complex_Matrix.inner_prod v v = (vec_norm u)\\<^sup>2 + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + (vec_norm v)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod u u + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + Complex_Matrix.inner_prod v v = (vec_norm u)\\<^sup>2 + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + (vec_norm v)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nhave \"... \\ (vec_norm u)^2 + 2 * cmod (inner_prod u v) + (vec_norm v)^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (vec_norm u)\\<^sup>2 + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + (vec_norm v)\\<^sup>2 \\ (vec_norm u)\\<^sup>2 + complex_of_real (2 * cmod (Complex_Matrix.inner_prod u v)) + (vec_norm v)\\<^sup>2\n[PROOF STEP]\nby (metis add_conj_le add_left_mono add_right_mono assms \n carrier_vec_dim_vec conjugate_complex_def inner_prod_swap \n is_num_normalize(1))\n[PROOF STATE]\nproof (state)\nthis:\n(vec_norm u)\\<^sup>2 + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + (vec_norm v)\\<^sup>2 \\ (vec_norm u)\\<^sup>2 + complex_of_real (2 * cmod (Complex_Matrix.inner_prod u v)) + (vec_norm v)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(vec_norm u)\\<^sup>2 + Complex_Matrix.inner_prod u v + Complex_Matrix.inner_prod v u + (vec_norm v)\\<^sup>2 \\ (vec_norm u)\\<^sup>2 + complex_of_real (2 * cmod (Complex_Matrix.inner_prod u v)) + (vec_norm v)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nhave \"... \\ (vec_norm u)^2 + 2 * ((vec_norm u)*(vec_norm v)) + \n (vec_norm v)^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (vec_norm u)\\<^sup>2 + complex_of_real (2 * cmod (Complex_Matrix.inner_prod u v)) + (vec_norm v)\\<^sup>2 \\ (vec_norm u)\\<^sup>2 + 2 * (vec_norm u * vec_norm v) + (vec_norm v)\\<^sup>2\n[PROOF STEP]\nusing Cauchy_Schwarz_complex_vec_norm[of u v] assms less_eq_complex_def\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec u = dim_vec v \\ complex_of_real (cmod (Complex_Matrix.inner_prod u v)) \\ vec_norm u * vec_norm v\ndim_vec u = dim_vec v\n(?x \\ ?y) = (Re ?x \\ Re ?y \\ Im ?x = Im ?y)\n\ngoal (1 subgoal):\n 1. (vec_norm u)\\<^sup>2 + complex_of_real (2 * cmod (Complex_Matrix.inner_prod u v)) + (vec_norm v)\\<^sup>2 \\ (vec_norm u)\\<^sup>2 + 2 * (vec_norm u * vec_norm v) + (vec_norm v)\\<^sup>2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(vec_norm u)\\<^sup>2 + complex_of_real (2 * cmod (Complex_Matrix.inner_prod u v)) + (vec_norm v)\\<^sup>2 \\ (vec_norm u)\\<^sup>2 + 2 * (vec_norm u * vec_norm v) + (vec_norm v)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(vec_norm u)\\<^sup>2 + complex_of_real (2 * cmod (Complex_Matrix.inner_prod u v)) + (vec_norm v)\\<^sup>2 \\ (vec_norm u)\\<^sup>2 + 2 * (vec_norm u * vec_norm v) + (vec_norm v)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nhave \"... = (vec_norm u + vec_norm v)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (vec_norm u)\\<^sup>2 + 2 * (vec_norm u * vec_norm v) + (vec_norm v)\\<^sup>2 = (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nby (simp add: power2_sum)\n[PROOF STATE]\nproof (state)\nthis:\n(vec_norm u)\\<^sup>2 + 2 * (vec_norm u * vec_norm v) + (vec_norm v)\\<^sup>2 = (vec_norm u + vec_norm v)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(vec_norm (u + v))\\<^sup>2 \\ (vec_norm u + vec_norm v)\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3451, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 24, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976953030553434, "lm_q2_score": 0.8311430457670241, "lm_q1q2_score": 0.7461132083521699}} {"text": "[STATEMENT]\nlemma has_integral_implies_lebesgue_measurable:\n fixes f :: \"'a :: euclidean_space \\ 'b :: euclidean_space\"\n assumes f: \"(f has_integral I) \\\"\n shows \"(\\x. indicator \\ x *\\<^sub>R f x) \\ lebesgue \\\\<^sub>M borel\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. indicat_real \\ x *\\<^sub>R f x) \\ borel_measurable lebesgue\n[PROOF STEP]\nproof (intro borel_measurable_euclidean_space[where 'c='b, THEN iffD2] ballI)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i. i \\ Basis \\ (\\x. indicat_real \\ x *\\<^sub>R f x \\ i) \\ borel_measurable lebesgue\n[PROOF STEP]\nfix i :: \"'b\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i. i \\ Basis \\ (\\x. indicat_real \\ x *\\<^sub>R f x \\ i) \\ borel_measurable lebesgue\n[PROOF STEP]\nassume \"i \\ Basis\"\n[PROOF STATE]\nproof (state)\nthis:\ni \\ Basis\n\ngoal (1 subgoal):\n 1. \\i. i \\ Basis \\ (\\x. indicat_real \\ x *\\<^sub>R f x \\ i) \\ borel_measurable lebesgue\n[PROOF STEP]\nhave \"(\\x. (f x \\ i) * indicator \\ x) \\ borel_measurable (completion lborel)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. f x \\ i * indicat_real \\ x) \\ borel_measurable lebesgue\n[PROOF STEP]\nusing has_integral_linear[OF f bounded_linear_inner_left, of i]\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\a. a \\ i) \\ f has_integral I \\ i) \\\n\ngoal (1 subgoal):\n 1. (\\x. f x \\ i * indicat_real \\ x) \\ borel_measurable lebesgue\n[PROOF STEP]\nby (intro has_integral_implies_lebesgue_measurable_real) (auto simp: comp_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. f x \\ i * indicat_real \\ x) \\ borel_measurable lebesgue\n\ngoal (1 subgoal):\n 1. \\i. i \\ Basis \\ (\\x. indicat_real \\ x *\\<^sub>R f x \\ i) \\ borel_measurable lebesgue\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x. f x \\ i * indicat_real \\ x) \\ borel_measurable lebesgue\n[PROOF STEP]\nshow \"(\\x. indicator \\ x *\\<^sub>R f x \\ i) \\ borel_measurable (completion lborel)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. f x \\ i * indicat_real \\ x) \\ borel_measurable lebesgue\n\ngoal (1 subgoal):\n 1. (\\x. indicat_real \\ x *\\<^sub>R f x \\ i) \\ borel_measurable lebesgue\n[PROOF STEP]\nby (simp add: ac_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. indicat_real \\ x *\\<^sub>R f x \\ i) \\ borel_measurable lebesgue\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1146, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.897695292107347, "lm_q2_score": 0.831143045767024, "lm_q1q2_score": 0.7461131992528187}} {"text": "[STATEMENT]\ntheorem smallest_dist_unique:\n \\ \\Theorem 2.5 in @{cite conway2013course}\\\n fixes M::\\'a::complex_inner set\\ and h\n assumes a1: \\convex M\\\n assumes \\is_arg_min (\\ x. dist x h) (\\ x. x \\ M) r\\\n assumes \\is_arg_min (\\ x. dist x h) (\\ x. x \\ M) s\\\n shows \\r = s\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r = s\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. r = s\n[PROOF STEP]\nhave *: \"is_arg_min (\\x. dist x h) (\\x. x\\M) k \\ is_arg_min (\\x. norm x) (\\x. x\\(\\x. x-h) ` M) (k-h)\" for k\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_arg_min (\\x. dist x h) (\\x. x \\ M) k = is_arg_min norm (\\x. x \\ (\\x. x - h) ` M) (k - h)\n[PROOF STEP]\nunfolding dist_norm is_arg_min_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (k \\ M \\ (\\y. y \\ M \\ norm (y - h) < norm (k - h))) = (k - h \\ (\\x. x - h) ` M \\ (\\y. y \\ (\\x. x - h) ` M \\ norm y < norm (k - h)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nis_arg_min (\\x. dist x h) (\\x. x \\ M) ?k = is_arg_min norm (\\x. x \\ (\\x. x - h) ` M) (?k - h)\n\ngoal (1 subgoal):\n 1. r = s\n[PROOF STEP]\nhave \\r - h = s - h\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r - h = s - h\n[PROOF STEP]\nusing _ assms(2,3)[unfolded *]\n[PROOF STATE]\nproof (prove)\nusing this:\nPROP ?psi \\ PROP ?psi\nis_arg_min norm (\\x. x \\ (\\x. x - h) ` M) (r - h)\nis_arg_min norm (\\x. x \\ (\\x. x - h) ` M) (s - h)\n\ngoal (1 subgoal):\n 1. r - h = s - h\n[PROOF STEP]\napply (rule smallest_norm_unique)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex ((\\x. x - h) ` M)\n[PROOF STEP]\nby (simp add: a1)\n[PROOF STATE]\nproof (state)\nthis:\nr - h = s - h\n\ngoal (1 subgoal):\n 1. r = s\n[PROOF STEP]\nthus \\r = s\\\n[PROOF STATE]\nproof (prove)\nusing this:\nr - h = s - h\n\ngoal (1 subgoal):\n 1. r = s\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nr = s\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n\n\n\\ \\Theorem 2.6 in @{cite conway2013course}\\", "meta": {"llama_tokens": 1072, "file": "Complex_Bounded_Operators_Complex_Inner_Product", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392725805823, "lm_q2_score": 0.8438951005915208, "lm_q1q2_score": 0.7460364108612455}} {"text": "[STATEMENT]\nlemma infsetsum_nonneg: \"(\\x. x \\ A \\ f x \\ (0::real)) \\ infsetsum f A \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. x \\ A \\ 0 \\ f x) \\ 0 \\ infsetsum f A\n[PROOF STEP]\nunfolding infsetsum_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. x \\ A \\ 0 \\ f x) \\ 0 \\ integral\\<^sup>L (count_space A) f\n[PROOF STEP]\nby (rule Bochner_Integration.integral_nonneg) auto", "meta": {"llama_tokens": 220, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392695254318, "lm_q2_score": 0.8438951025545425, "lm_q1q2_score": 0.746036410018407}} {"text": "[STATEMENT]\nlemma frechet_derivative_divide: \"frechet_derivative (\\x. f x / g x) (at x) =\n (\\h. frechet_derivative f (at x) h / (g x) -frechet_derivative g (at x) h * f x / (g x)\\<^sup>2)\"\n if \"f differentiable at x\" \"g differentiable at x\" \"g x \\ 0\" for f::\"_\\_::real_normed_field\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. frechet_derivative (\\x. f x / g x) (at x) = (\\h. frechet_derivative f (at x) h / g x - frechet_derivative g (at x) h * f x / (g x)\\<^sup>2)\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\nf differentiable at x\ng differentiable at x\ng x \\ (0::'d)\n\ngoal (1 subgoal):\n 1. frechet_derivative (\\x. f x / g x) (at x) = (\\h. frechet_derivative f (at x) h / g x - frechet_derivative g (at x) h * f x / (g x)\\<^sup>2)\n[PROOF STEP]\nby (auto simp: divide_inverse_commute bounded_bilinear.frechet_derivative[OF bounded_bilinear_mult]\n frechet_derivative_inverse)", "meta": {"llama_tokens": 428, "file": "Smooth_Manifolds_Analysis_More", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122238669025, "lm_q2_score": 0.8221891261650248, "lm_q1q2_score": 0.745982244499974}} {"text": "[STATEMENT]\nlemma general_digit_base:\n assumes \"t1 > t2\" and \"b>1\"\n shows \"nth_digit (a * b^t1) t2 b = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. nth_digit (a * b ^ t1) t2 b = 0\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. nth_digit (a * b ^ t1) t2 b = 0\n[PROOF STEP]\nhave 1: \"b^t1 div b^t2 = b^(t1-t2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. b ^ t1 div b ^ t2 = b ^ (t1 - t2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nt2 < t1\n1 < b\n\ngoal (1 subgoal):\n 1. b ^ t1 div b ^ t2 = b ^ (t1 - t2)\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\t2 < t1; Suc 0 < b\\ \\ b ^ t1 div b ^ t2 = b ^ (t1 - t2)\n[PROOF STEP]\nby (metis Suc_lessD less_imp_le less_numeral_extra(3) power_diff)\n[PROOF STATE]\nproof (state)\nthis:\nb ^ t1 div b ^ t2 = b ^ (t1 - t2)\n\ngoal (1 subgoal):\n 1. nth_digit (a * b ^ t1) t2 b = 0\n[PROOF STEP]\nhave \"b^t2 dvd b^t1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. b ^ t2 dvd b ^ t1\n[PROOF STEP]\nusing `t1 > t2`\n[PROOF STATE]\nproof (prove)\nusing this:\nt2 < t1\n\ngoal (1 subgoal):\n 1. b ^ t2 dvd b ^ t1\n[PROOF STEP]\nby (simp add: le_imp_power_dvd)\n[PROOF STATE]\nproof (state)\nthis:\nb ^ t2 dvd b ^ t1\n\ngoal (1 subgoal):\n 1. nth_digit (a * b ^ t1) t2 b = 0\n[PROOF STEP]\nhence \"(a * b^t1) div b^t2 = a * b^(t1-t2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nb ^ t2 dvd b ^ t1\n\ngoal (1 subgoal):\n 1. a * b ^ t1 div b ^ t2 = a * b ^ (t1 - t2)\n[PROOF STEP]\nusing div_mult_swap[of \"b^t2\" \"b^t1\" \"a\"] 1\n[PROOF STATE]\nproof (prove)\nusing this:\nb ^ t2 dvd b ^ t1\nb ^ t2 dvd b ^ t1 \\ a * (b ^ t1 div b ^ t2) = a * b ^ t1 div b ^ t2\nb ^ t1 div b ^ t2 = b ^ (t1 - t2)\n\ngoal (1 subgoal):\n 1. a * b ^ t1 div b ^ t2 = a * b ^ (t1 - t2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na * b ^ t1 div b ^ t2 = a * b ^ (t1 - t2)\n\ngoal (1 subgoal):\n 1. nth_digit (a * b ^ t1) t2 b = 0\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na * b ^ t1 div b ^ t2 = a * b ^ (t1 - t2)\n\ngoal (1 subgoal):\n 1. nth_digit (a * b ^ t1) t2 b = 0\n[PROOF STEP]\nusing nth_digit_def assms\n[PROOF STATE]\nproof (prove)\nusing this:\na * b ^ t1 div b ^ t2 = a * b ^ (t1 - t2)\nnth_digit ?num ?k ?base = ?num div ?base ^ ?k mod ?base\nt2 < t1\n1 < b\n\ngoal (1 subgoal):\n 1. nth_digit (a * b ^ t1) t2 b = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nnth_digit (a * b ^ t1) t2 b = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1270, "file": "Digit_Expansions_Bits_Digits", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.877476800298183, "lm_q2_score": 0.8499711699569787, "lm_q1q2_score": 0.7458299825595528}} {"text": "[STATEMENT]\nlemma satisfiableEquivalent: \n fixes formula1 :: Formula and formula2 :: Formula\n assumes \"equivalentFormulae formula1 formula2\"\n shows \"satisfiable formula1 = satisfiable formula2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. satisfiable formula1 = satisfiable formula2\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nequivalentFormulae formula1 formula2\n\ngoal (1 subgoal):\n 1. satisfiable formula1 = satisfiable formula2\n[PROOF STEP]\nunfolding equivalentFormulae_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\valuation. model valuation formula1 = model valuation formula2\n\ngoal (1 subgoal):\n 1. satisfiable formula1 = satisfiable formula2\n[PROOF STEP]\nunfolding satisfiable_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\valuation. model valuation formula1 = model valuation formula2\n\ngoal (1 subgoal):\n 1. (\\valuation. model valuation formula1) = (\\valuation. model valuation formula2)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 329, "file": "SATSolverVerification_CNF", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894661025424, "lm_q2_score": 0.8333245953120233, "lm_q1q2_score": 0.7456500697293625}} {"text": "[STATEMENT]\nlemma absdiff_norm_less:\n assumes \"sum (\\x. norm (f x - g x)) S < e\"\n shows \"\\sum (\\x. norm(f x)) S - sum (\\x. norm(g x)) S\\ < e\" (is \"?lhs < e\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ < e\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ < e\n[PROOF STEP]\nhave \"?lhs \\ (\\i\\S. \\norm (f i) - norm (g i)\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ \\ (\\i\\S. \\norm (f i) - norm (g i)\\)\n[PROOF STEP]\nby (metis (no_types) sum_abs sum_subtractf)\n[PROOF STATE]\nproof (state)\nthis:\n\\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ \\ (\\i\\S. \\norm (f i) - norm (g i)\\)\n\ngoal (1 subgoal):\n 1. \\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ < e\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ \\ (\\i\\S. \\norm (f i) - norm (g i)\\)\n\ngoal (1 subgoal):\n 1. \\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ < e\n[PROOF STEP]\nhave \"... \\ (\\x\\S. norm (f x - g x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\S. \\norm (f i) - norm (g i)\\) \\ (\\x\\S. norm (f x - g x))\n[PROOF STEP]\nby (simp add: norm_triangle_ineq3 sum_mono)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\S. \\norm (f i) - norm (g i)\\) \\ (\\x\\S. norm (f x - g x))\n\ngoal (1 subgoal):\n 1. \\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ < e\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\S. \\norm (f i) - norm (g i)\\) \\ (\\x\\S. norm (f x - g x))\n\ngoal (1 subgoal):\n 1. \\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ < e\n[PROOF STEP]\nhave \"... < e\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\S. norm (f x - g x)) < e\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x\\S. norm (f x - g x)) < e\n\ngoal (1 subgoal):\n 1. (\\x\\S. norm (f x - g x)) < e\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\S. norm (f x - g x)) < e\n\ngoal (1 subgoal):\n 1. \\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ < e\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ < e\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ < e\n\ngoal (1 subgoal):\n 1. \\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ < e\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\(\\x\\S. norm (f x)) - (\\x\\S. norm (g x))\\ < e\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1522, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.8333245994514084, "lm_q1q2_score": 0.7456500664192421}} {"text": "[STATEMENT]\nlemma sum_splice:\n \"(\\i::nat = 0..<2*n. f i) = (\\i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i::nat = 0..<2*n. f i) =\n sum f ((*) 2 ` {0..i. 2 * i + 1) ` {0..i. 2 * i + 1) ` {0..i = 0..i = 0..i. 2 * i + 1) ` {0..i = 0..i = 0..i = 0..i = 0..i. 2 * i + 1) ` {0..i = 0..i = 0..i. 2 * i + 1) ` {0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..{0..\\q-(q/(2*2^d))\\-1}\" \n \"of_nat d < \\(log 2 q)::real\\\"\n \"d>0\"\n shows \"abs ((decompress d (compress d x) - x) mod+- q) \\ \n round ( real_of_int q / real_of_int (2^(d+1)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(decompress d (compress d x) - x) mod+- q\\ \\ round (real_of_int q / real_of_int (2 ^ (d + 1)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\(decompress d (compress d x) - x) mod+- q\\ \\ round (real_of_int q / real_of_int (2 ^ (d + 1)))\n[PROOF STEP]\nhave \"abs ((decompress d (compress d x) - x) mod+- q) =\n abs ((decompress d (compress d x) - x)) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(decompress d (compress d x) - x) mod+- q\\ = \\decompress d (compress d x) - x\\\n[PROOF STEP]\nusing no_mod_plus_minus[OF decompress_compress_no_mod\n [OF assms(1) assms(2)] assms(3)]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\decompress d (compress d x) - x\\ = \\(decompress d (compress d x) - x) mod+- q\\\n\ngoal (1 subgoal):\n 1. \\(decompress d (compress d x) - x) mod+- q\\ = \\decompress d (compress d x) - x\\\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\(decompress d (compress d x) - x) mod+- q\\ = \\decompress d (compress d x) - x\\\n\ngoal (1 subgoal):\n 1. \\(decompress d (compress d x) - x) mod+- q\\ \\ round (real_of_int q / real_of_int (2 ^ (d + 1)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\(decompress d (compress d x) - x) mod+- q\\ = \\decompress d (compress d x) - x\\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\(decompress d (compress d x) - x) mod+- q\\ = \\decompress d (compress d x) - x\\\n\ngoal (1 subgoal):\n 1. \\(decompress d (compress d x) - x) mod+- q\\ \\ round (real_of_int q / real_of_int (2 ^ (d + 1)))\n[PROOF STEP]\nusing decompress_compress_no_mod\n [OF assms(1) assms(2)]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\(decompress d (compress d x) - x) mod+- q\\ = \\decompress d (compress d x) - x\\\n\\decompress d (compress d x) - x\\ \\ round (real_of_int q / real_of_int (2 ^ (d + 1)))\n\ngoal (1 subgoal):\n 1. \\(decompress d (compress d x) - x) mod+- q\\ \\ round (real_of_int q / real_of_int (2 ^ (d + 1)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\(decompress d (compress d x) - x) mod+- q\\ \\ round (real_of_int q / real_of_int (2 ^ (d + 1)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1197, "file": "CRYSTALS-Kyber_Compress", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9019206791658465, "lm_q2_score": 0.8267117962054049, "lm_q1q2_score": 0.7456284647079957}} {"text": "[STATEMENT]\nlemma Schwartz_inequality_abs:\n assumes \"f square_integrable S\" \"g square_integrable S\"\n shows \"\\l2product S f g\\ \\ l2norm S f * l2norm S g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\l2product S f g\\ \\ l2norm S f * l2norm S g\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\l2product S f g\\ \\ l2norm S f * l2norm S g\n[PROOF STEP]\nhave \"\\l2product S f g\\ \\ l2product S (\\x. \\f x\\) (\\x. \\g x\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\l2product S f g\\ \\ l2product S (\\x. \\f x\\) (\\x. \\g x\\)\n[PROOF STEP]\nunfolding l2product_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\LINT x|lebesgue_on S. f x * g x\\ \\ LINT x|lebesgue_on S. \\f x\\ * \\g x\\\n[PROOF STEP]\nproof (rule integral_abs_bound_integral)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. integrable (lebesgue_on S) (\\x. f x * g x)\n 2. integrable (lebesgue_on S) (\\x. \\f x\\ * \\g x\\)\n 3. \\x. x \\ space (lebesgue_on S) \\ \\f x * g x\\ \\ \\f x\\ * \\g x\\\n[PROOF STEP]\nshow \"integrable (lebesgue_on S) (\\x. f x * g x)\" \"integrable (lebesgue_on S) (\\x. \\f x\\ * \\g x\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integrable (lebesgue_on S) (\\x. f x * g x) &&& integrable (lebesgue_on S) (\\x. \\f x\\ * \\g x\\)\n[PROOF STEP]\nby (simp_all add: assms square_integrable_imp_integrable_product)\n[PROOF STATE]\nproof (state)\nthis:\nintegrable (lebesgue_on S) (\\x. f x * g x)\nintegrable (lebesgue_on S) (\\x. \\f x\\ * \\g x\\)\n\ngoal (1 subgoal):\n 1. \\x. x \\ space (lebesgue_on S) \\ \\f x * g x\\ \\ \\f x\\ * \\g x\\\n[PROOF STEP]\nqed (simp add: abs_mult)\n[PROOF STATE]\nproof (state)\nthis:\n\\l2product S f g\\ \\ l2product S (\\x. \\f x\\) (\\x. \\g x\\)\n\ngoal (1 subgoal):\n 1. \\l2product S f g\\ \\ l2norm S f * l2norm S g\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\l2product S f g\\ \\ l2product S (\\x. \\f x\\) (\\x. \\g x\\)\n\ngoal (1 subgoal):\n 1. \\l2product S f g\\ \\ l2norm S f * l2norm S g\n[PROOF STEP]\nhave \"\\ \\ l2norm S f * l2norm S g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. l2product S (\\x. \\f x\\) (\\x. \\g x\\) \\ l2norm S f * l2norm S g\n[PROOF STEP]\nby (simp add: Schwartz_inequality_strong assms)\n[PROOF STATE]\nproof (state)\nthis:\nl2product S (\\x. \\f x\\) (\\x. \\g x\\) \\ l2norm S f * l2norm S g\n\ngoal (1 subgoal):\n 1. \\l2product S f g\\ \\ l2norm S f * l2norm S g\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\l2product S f g\\ \\ l2norm S f * l2norm S g\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\l2product S f g\\ \\ l2norm S f * l2norm S g\n\ngoal (1 subgoal):\n 1. \\l2product S f g\\ \\ l2norm S f * l2norm S g\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\l2product S f g\\ \\ l2norm S f * l2norm S g\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1536, "file": "Fourier_Square_Integrable", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9019206686206199, "lm_q2_score": 0.8267117855317474, "lm_q1q2_score": 0.7456284463633401}} {"text": "[STATEMENT]\nlemma prime_factorization_subset_iff_dvd:\n assumes [simp]: \"x \\ 0\" \"y \\ 0\"\n shows \"prime_factorization x \\# prime_factorization y \\ x dvd y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (prime_factorization x \\# prime_factorization y) = (x dvd y)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (prime_factorization x \\# prime_factorization y) = (x dvd y)\n[PROOF STEP]\nhave \"x dvd y \\\n normalize (prod_mset (prime_factorization x)) dvd normalize (prod_mset (prime_factorization y))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x dvd y) = (normalize (\\\\<^sub># (prime_factorization x)) dvd normalize (\\\\<^sub># (prime_factorization y)))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ (0::'a)\ny \\ (0::'a)\n\ngoal (1 subgoal):\n 1. (x dvd y) = (normalize (\\\\<^sub># (prime_factorization x)) dvd normalize (\\\\<^sub># (prime_factorization y)))\n[PROOF STEP]\nby (subst (1 2) prod_mset_prime_factorization_weak) auto\n[PROOF STATE]\nproof (state)\nthis:\n(x dvd y) = (normalize (\\\\<^sub># (prime_factorization x)) dvd normalize (\\\\<^sub># (prime_factorization y)))\n\ngoal (1 subgoal):\n 1. (prime_factorization x \\# prime_factorization y) = (x dvd y)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(x dvd y) = (normalize (\\\\<^sub># (prime_factorization x)) dvd normalize (\\\\<^sub># (prime_factorization y)))\n\ngoal (1 subgoal):\n 1. (prime_factorization x \\# prime_factorization y) = (x dvd y)\n[PROOF STEP]\nhave \"\\ \\ prime_factorization x \\# prime_factorization y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (normalize (\\\\<^sub># (prime_factorization x)) dvd normalize (\\\\<^sub># (prime_factorization y))) = (prime_factorization x \\# prime_factorization y)\n[PROOF STEP]\nby (auto intro!: prod_mset_primes_dvd_imp_subset prod_mset_subset_imp_dvd)\n[PROOF STATE]\nproof (state)\nthis:\n(normalize (\\\\<^sub># (prime_factorization x)) dvd normalize (\\\\<^sub># (prime_factorization y))) = (prime_factorization x \\# prime_factorization y)\n\ngoal (1 subgoal):\n 1. (prime_factorization x \\# prime_factorization y) = (x dvd y)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(x dvd y) = (prime_factorization x \\# prime_factorization y)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(x dvd y) = (prime_factorization x \\# prime_factorization y)\n\ngoal (1 subgoal):\n 1. (prime_factorization x \\# prime_factorization y) = (x dvd y)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n(prime_factorization x \\# prime_factorization y) = (x dvd y)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1090, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314738181875, "lm_q2_score": 0.8418256492357358, "lm_q1q2_score": 0.7455472904305972}} {"text": "[STATEMENT]\nlemma orthogonal_complement_antimono_iff[simp]:\n fixes A B :: \\('a::chilbert_space) set\\\n assumes \\closed_csubspace A\\ and \\closed_csubspace B\\\n shows \\orthogonal_complement A \\ orthogonal_complement B \\ A \\ B\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (orthogonal_complement A \\ orthogonal_complement B) = (B \\ A)\n[PROOF STEP]\nproof (rule iffI)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. orthogonal_complement A \\ orthogonal_complement B \\ B \\ A\n 2. B \\ A \\ orthogonal_complement A \\ orthogonal_complement B\n[PROOF STEP]\nshow \\orthogonal_complement A \\ orthogonal_complement B\\ if \\A \\ B\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal_complement A \\ orthogonal_complement B\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\nB \\ A\n\ngoal (1 subgoal):\n 1. orthogonal_complement A \\ orthogonal_complement B\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nB \\ A \\ orthogonal_complement A \\ orthogonal_complement B\n\ngoal (1 subgoal):\n 1. orthogonal_complement A \\ orthogonal_complement B \\ B \\ A\n[PROOF STEP]\nassume \\orthogonal_complement A \\ orthogonal_complement B\\\n[PROOF STATE]\nproof (state)\nthis:\northogonal_complement A \\ orthogonal_complement B\n\ngoal (1 subgoal):\n 1. orthogonal_complement A \\ orthogonal_complement B \\ B \\ A\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\northogonal_complement A \\ orthogonal_complement B\n[PROOF STEP]\nhave \\orthogonal_complement (orthogonal_complement A) \\ orthogonal_complement (orthogonal_complement B)\\\n[PROOF STATE]\nproof (prove)\nusing this:\northogonal_complement A \\ orthogonal_complement B\n\ngoal (1 subgoal):\n 1. orthogonal_complement (orthogonal_complement B) \\ orthogonal_complement (orthogonal_complement A)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\northogonal_complement (orthogonal_complement B) \\ orthogonal_complement (orthogonal_complement A)\n\ngoal (1 subgoal):\n 1. orthogonal_complement A \\ orthogonal_complement B \\ B \\ A\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\northogonal_complement (orthogonal_complement B) \\ orthogonal_complement (orthogonal_complement A)\n[PROOF STEP]\nshow \\A \\ B\\\n[PROOF STATE]\nproof (prove)\nusing this:\northogonal_complement (orthogonal_complement B) \\ orthogonal_complement (orthogonal_complement A)\n\ngoal (1 subgoal):\n 1. B \\ A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\northogonal_complement (orthogonal_complement B) \\ orthogonal_complement (orthogonal_complement A)\nclosed_csubspace A\nclosed_csubspace B\n\ngoal (1 subgoal):\n 1. B \\ A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nB \\ A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1090, "file": "Complex_Bounded_Operators_Complex_Inner_Product", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213718636752, "lm_q2_score": 0.8289388104343892, "lm_q1q2_score": 0.7453166004288111}} {"text": "[STATEMENT]\nlemma length_of_bin_rep:\n fixes n m:: nat\n assumes \"m < 2^n\"\n shows \"length (bin_rep n m) = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (bin_rep n m) = n\n[PROOF STEP]\nusing assms length_of_bin_rep_aux bin_rep_def\n[PROOF STATE]\nproof (prove)\nusing this:\nm < 2 ^ n\n?m < 2 ^ ?n \\ length (bin_rep_aux ?n ?m) = ?n + 1\nbin_rep ?n ?m = butlast (bin_rep_aux ?n ?m)\n\ngoal (1 subgoal):\n 1. length (bin_rep n m) = n\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 216, "file": "Isabelle_Marries_Dirac_Binary_Nat", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110569397307, "lm_q2_score": 0.8354835391516132, "lm_q1q2_score": 0.745093458106547}} {"text": "[STATEMENT]\nlemma trace_1: \n shows \"Complex_Matrix.trace ((1\\<^sub>m n)::complex Matrix.mat) =(n::complex)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (1\\<^sub>m n) = of_nat n\n[PROOF STEP]\nusing one_mat_def\n[PROOF STATE]\nproof (prove)\nusing this:\n1\\<^sub>m ?n \\ Matrix.mat ?n ?n (\\(i, j). if i = j then 1::?'a else (0::?'a))\n\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (1\\<^sub>m n) = of_nat n\n[PROOF STEP]\nby (simp add: Complex_Matrix.trace_def Matrix.mat_def)", "meta": {"llama_tokens": 215, "file": "Projective_Measurements_Linear_Algebra_Complements", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110454379297, "lm_q2_score": 0.8354835330070839, "lm_q1q2_score": 0.7450934430172225}} {"text": "[STATEMENT]\nlemma le_Suc_eq: \"(\\a. (a < Suc n) = (a < Suc m)) = (\\a. (a < n) = (a < m))\"\n (is \"(\\a. ?A a) = (\\ a. ?B a)\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a. (a < Suc n) = (a < Suc m)) = (\\a. (a < n) = (a < m))\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\a. (a < Suc n) = (a < Suc m) \\ \\a. (a < n) = (a < m)\n 2. \\a. (a < n) = (a < m) \\ \\a. (a < Suc n) = (a < Suc m)\n[PROOF STEP]\nassume \"\\a. ?A a\"\n[PROOF STATE]\nproof (state)\nthis:\n\\a. (a < Suc n) = (a < Suc m)\n\ngoal (2 subgoals):\n 1. \\a. (a < Suc n) = (a < Suc m) \\ \\a. (a < n) = (a < m)\n 2. \\a. (a < n) = (a < m) \\ \\a. (a < Suc n) = (a < Suc m)\n[PROOF STEP]\nthus \"\\ a. ?B a\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a. (a < Suc n) = (a < Suc m)\n\ngoal (1 subgoal):\n 1. \\a. (a < n) = (a < m)\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n\\a. (a < n) = (a < m)\n\ngoal (1 subgoal):\n 1. \\a. (a < n) = (a < m) \\ \\a. (a < Suc n) = (a < Suc m)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a. (a < n) = (a < m) \\ \\a. (a < Suc n) = (a < Suc m)\n[PROOF STEP]\nassume B: \"\\ a. ?B a\"\n[PROOF STATE]\nproof (state)\nthis:\n\\a. (a < n) = (a < m)\n\ngoal (1 subgoal):\n 1. \\a. (a < n) = (a < m) \\ \\a. (a < Suc n) = (a < Suc m)\n[PROOF STEP]\nshow \"\\a. ?A a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a. (a < Suc n) = (a < Suc m)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a. (a < Suc n) = (a < Suc m)\n[PROOF STEP]\nfix a\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a. (a < Suc n) = (a < Suc m)\n[PROOF STEP]\nfrom B\n[PROOF STATE]\nproof (chain)\npicking this:\n\\a. (a < n) = (a < m)\n[PROOF STEP]\nshow \"?A a\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a. (a < n) = (a < m)\n\ngoal (1 subgoal):\n 1. (a < Suc n) = (a < Suc m)\n[PROOF STEP]\nby (cases a) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(a < Suc n) = (a < Suc m)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\a. (a < Suc n) = (a < Suc m)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1180, "file": "JiveDataStoreModel_Isabelle_Store_Store", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110454379297, "lm_q2_score": 0.8354835330070839, "lm_q1q2_score": 0.7450934430172225}} {"text": "[STATEMENT]\nlemma homotopic_with_linear:\n fixes f g :: \"_ \\ 'b::real_normed_vector\"\n assumes contf: \"continuous_on S f\"\n and contg:\"continuous_on S g\"\n and sub: \"\\x. x \\ S \\ closed_segment (f x) (g x) \\ t\"\n shows \"homotopic_with_canon (\\z. True) S t f g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. homotopic_with_canon (\\z. True) S t f g\n[PROOF STEP]\nunfolding homotopic_with_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\h. continuous_map (prod_topology (top_of_set {0..1}) (top_of_set S)) (top_of_set t) h \\ (\\x. h (0, x) = f x) \\ (\\x. h (1, x) = g x) \\ (\\t\\{0..1}. True)\n[PROOF STEP]\napply (rule_tac x=\"\\y. ((1 - (fst y)) *\\<^sub>R f(snd y) + (fst y) *\\<^sub>R g(snd y))\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous_map (prod_topology (top_of_set {0..1}) (top_of_set S)) (top_of_set t) (\\y. (1 - fst y) *\\<^sub>R f (snd y) + fst y *\\<^sub>R g (snd y)) \\ (\\x. (1 - fst (0, x)) *\\<^sub>R f (snd (0, x)) + fst (0, x) *\\<^sub>R g (snd (0, x)) = f x) \\ (\\x. (1 - fst (1, x)) *\\<^sub>R f (snd (1, x)) + fst (1, x) *\\<^sub>R g (snd (1, x)) = g x) \\ (\\t\\{0..1}. True)\n[PROOF STEP]\nusing sub closed_segment_def\n[PROOF STATE]\nproof (prove)\nusing this:\n?x \\ S \\ closed_segment (f ?x) (g ?x) \\ t\nclosed_segment ?a ?b = {(1 - u) *\\<^sub>R ?a + u *\\<^sub>R ?b |u. 0 \\ u \\ u \\ 1}\n\ngoal (1 subgoal):\n 1. continuous_map (prod_topology (top_of_set {0..1}) (top_of_set S)) (top_of_set t) (\\y. (1 - fst y) *\\<^sub>R f (snd y) + fst y *\\<^sub>R g (snd y)) \\ (\\x. (1 - fst (0, x)) *\\<^sub>R f (snd (0, x)) + fst (0, x) *\\<^sub>R g (snd (0, x)) = f x) \\ (\\x. (1 - fst (1, x)) *\\<^sub>R f (snd (1, x)) + fst (1, x) *\\<^sub>R g (snd (1, x)) = g x) \\ (\\t\\{0..1}. True)\n[PROOF STEP]\nby (fastforce intro: continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]\n continuous_on_subset [OF contg] continuous_on_compose2 [where g=g])", "meta": {"llama_tokens": 985, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110396870288, "lm_q2_score": 0.8354835330070838, "lm_q1q2_score": 0.7450934382124395}} {"text": "[STATEMENT]\nlemma index_tensor_vec[simp]:\n assumes \"0 < dim_vec v\" \n and \"i < dim_vec u * dim_vec v\"\nshows \"vec_index (tensor_vec u v) i = \n vec_index u (i div (dim_vec v)) * vec_index v (i mod dim_vec v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (u \\ v) $ i = u $ (i div dim_vec v) * v $ (i mod dim_vec v)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (u \\ v) $ i = u $ (i div dim_vec v) * v $ (i mod dim_vec v)\n[PROOF STEP]\nhave m: \"Matrix_Tensor.mult (1::complex) (*)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix_Tensor.mult 1 (*)\n[PROOF STEP]\nby (simp add: Matrix_Tensor.mult_def)\n[PROOF STATE]\nproof (state)\nthis:\nMatrix_Tensor.mult 1 (*)\n\ngoal (1 subgoal):\n 1. (u \\ v) $ i = u $ (i div dim_vec v) * v $ (i mod dim_vec v)\n[PROOF STEP]\nhave \"length (list_of_vec v) = dim_vec v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (list_of_vec v) = dim_vec v\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < dim_vec v\ni < dim_vec u * dim_vec v\n\ngoal (1 subgoal):\n 1. length (list_of_vec v) = dim_vec v\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlength (list_of_vec v) = dim_vec v\n\ngoal (1 subgoal):\n 1. (u \\ v) $ i = u $ (i div dim_vec v) * v $ (i mod dim_vec v)\n[PROOF STEP]\nhence \"vec_index (tensor_vec u v) i = (*) (vec_index u (i div dim_vec v)) (vec_index v (i mod dim_vec v))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (list_of_vec v) = dim_vec v\n\ngoal (1 subgoal):\n 1. (u \\ v) $ i = u $ (i div dim_vec v) * v $ (i mod dim_vec v)\n[PROOF STEP]\nunfolding tensor_vec_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (list_of_vec v) = dim_vec v\n\ngoal (1 subgoal):\n 1. vec_of_list (mult.vec_vec_Tensor (*) (list_of_vec u) (list_of_vec v)) $ i = u $ (i div dim_vec v) * v $ (i mod dim_vec v)\n[PROOF STEP]\nusing mult.vec_vec_Tensor_elements assms m\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (list_of_vec v) = dim_vec v\n\\Matrix_Tensor.mult ?id ?f; ?y \\ []\\ \\ \\i v) $ i = u $ (i div dim_vec v) * v $ (i mod dim_vec v)\n\ngoal (1 subgoal):\n 1. (u \\ v) $ i = u $ (i div dim_vec v) * v $ (i mod dim_vec v)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(u \\ v) $ i = u $ (i div dim_vec v) * v $ (i mod dim_vec v)\n\ngoal (1 subgoal):\n 1. (u \\ v) $ i = u $ (i div dim_vec v) * v $ (i mod dim_vec v)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(u \\ v) $ i = u $ (i div dim_vec v) * v $ (i mod dim_vec v)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1402, "file": "Projective_Measurements_Linear_Algebra_Complements", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110368115781, "lm_q2_score": 0.8354835309589073, "lm_q1q2_score": 0.7450934339834613}} {"text": "[STATEMENT]\nlemma tensor_mat_unitary:\n assumes \"m1 \\ carrier_mat d1 d1\"\n and \"m2 \\ carrier_mat d2 d2\"\n and \"unitary m1\"\n and \"unitary m2\"\n shows \"unitary (tensor_mat m1 m2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. unitary (tensor_mat m1 m2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nm1 \\ carrier_mat d1 d1\nm2 \\ carrier_mat d2 d2\nunitary m1\nunitary m2\n\ngoal (1 subgoal):\n 1. unitary (tensor_mat m1 m2)\n[PROOF STEP]\napply (auto simp add: unitary_def tensor_mat_adjoint)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m1 \\ carrier_mat d1 d1; m2 \\ carrier_mat d2 d2; inverts_mat m1 (adjoint m1); inverts_mat m2 (adjoint m2)\\ \\ inverts_mat (tensor_mat m1 m2) (tensor_mat (adjoint m1) (adjoint m2))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nm1 \\ carrier_mat d1 d1\nm2 \\ carrier_mat d2 d2\nunitary m1\nunitary m2\n\ngoal (1 subgoal):\n 1. \\m1 \\ carrier_mat d1 d1; m2 \\ carrier_mat d2 d2; inverts_mat m1 (adjoint m1); inverts_mat m2 (adjoint m2)\\ \\ inverts_mat (tensor_mat m1 m2) (tensor_mat (adjoint m1) (adjoint m2))\n[PROOF STEP]\nunfolding inverts_mat_def\n[PROOF STATE]\nproof (prove)\nusing this:\nm1 \\ carrier_mat d1 d1\nm2 \\ carrier_mat d2 d2\nunitary m1\nunitary m2\n\ngoal (1 subgoal):\n 1. \\m1 \\ carrier_mat d1 d1; m2 \\ carrier_mat d2 d2; m1 * adjoint m1 = 1\\<^sub>m (dim_row m1); m2 * adjoint m2 = 1\\<^sub>m (dim_row m2)\\ \\ tensor_mat m1 m2 * tensor_mat (adjoint m1) (adjoint m2) = 1\\<^sub>m (dim_row (tensor_mat m1 m2))\n[PROOF STEP]\napply (subst tensor_mat_mult[symmetric], auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m1 \\ carrier_mat d1 d1; m2 \\ carrier_mat d2 d2; unitary m1; unitary m2\\ \\ tensor_mat (1\\<^sub>m d1) (1\\<^sub>m d2) = 1\\<^sub>m d\n[PROOF STEP]\nby (rule tensor_mat_id)", "meta": {"llama_tokens": 886, "file": "QHLProver_Partial_State", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045996818985, "lm_q2_score": 0.839733963661418, "lm_q1q2_score": 0.7450158350695222}} {"text": "[STATEMENT]\nlemma set_incr_disjnt: \n assumes \"disjnt A B\" \n shows \"disjnt (set_incr n A) (set_incr n B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. disjnt (set_incr n A) (set_incr n B)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndisjnt A B\n\ngoal (1 subgoal):\n 1. disjnt (set_incr n A) (set_incr n B)\n[PROOF STEP]\nunfolding disjnt_def set_incr_def\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ B = {}\n\ngoal (1 subgoal):\n 1. (\\a. a + n) ` A \\ (\\a. a + n) ` B = {}\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 257, "file": "Hales_Jewett_Hales_Jewett", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9086178994073577, "lm_q2_score": 0.8198933425148213, "lm_q1q2_score": 0.7449697666138941}} {"text": "[STATEMENT]\nlemma set_of_distrib_right_left:\n \"set_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)\"\n for A1 :: \"'a::{linordered_idom, real_normed_algebra, linear_continuum_topology} interval\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. set_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)\n[PROOF STEP]\nhave \"set_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * (B1 + B2) + A2 * (B1 + B2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * (B1 + B2) + A2 * (B1 + B2))\n[PROOF STEP]\nby (rule set_of_distrib_right)\n[PROOF STATE]\nproof (state)\nthis:\nset_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * (B1 + B2) + A2 * (B1 + B2))\n\ngoal (1 subgoal):\n 1. set_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nset_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * (B1 + B2) + A2 * (B1 + B2))\n\ngoal (1 subgoal):\n 1. set_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)\n[PROOF STEP]\nhave \"... \\ set_of ((A1 * B1 + A1 * B2) + A2 * (B1 + B2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_of (A1 * (B1 + B2) + A2 * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * (B1 + B2))\n[PROOF STEP]\nby (rule set_of_add_inc_left[OF set_of_distrib_left])\n[PROOF STATE]\nproof (state)\nthis:\nset_of (A1 * (B1 + B2) + A2 * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * (B1 + B2))\n\ngoal (1 subgoal):\n 1. set_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nset_of (A1 * (B1 + B2) + A2 * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * (B1 + B2))\n\ngoal (1 subgoal):\n 1. set_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)\n[PROOF STEP]\nhave \"... \\ set_of ((A1 * B1 + A1 * B2) + (A2 * B1 + A2 * B2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_of (A1 * B1 + A1 * B2 + A2 * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + (A2 * B1 + A2 * B2))\n[PROOF STEP]\nby (rule set_of_add_inc_right[OF set_of_distrib_left])\n[PROOF STATE]\nproof (state)\nthis:\nset_of (A1 * B1 + A1 * B2 + A2 * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + (A2 * B1 + A2 * B2))\n\ngoal (1 subgoal):\n 1. set_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nset_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + (A2 * B1 + A2 * B2))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nset_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + (A2 * B1 + A2 * B2))\n\ngoal (1 subgoal):\n 1. set_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)\n[PROOF STEP]\nby (simp add: add.assoc)\n[PROOF STATE]\nproof (state)\nthis:\nset_of ((A1 + A2) * (B1 + B2)) \\ set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1705, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473846343394, "lm_q2_score": 0.8539127510928476, "lm_q1q2_score": 0.7449085551217592}} {"text": "[STATEMENT]\nlemma eucl_of_list_append_zeroes[simp]: \"eucl_of_list (xs @ replicate n 0) = eucl_of_list xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eucl_of_list (xs @ replicate n 0) = eucl_of_list xs\n[PROOF STEP]\nunfolding eucl_of_list_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list (map2 (*\\<^sub>R) (xs @ replicate n 0) Basis_list) = sum_list (map2 (*\\<^sub>R) xs Basis_list)\n[PROOF STEP]\napply (auto simp: sum_list_sum_nth)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..R Basis_list ! i) = (\\i = 0..R Basis_list ! i)\n[PROOF STEP]\napply (rule sum.mono_neutral_cong_right)\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. finite {0.. {0..i\\{0..R Basis_list ! i = (0::'a)\n 4. \\x. x \\ {0.. (xs @ replicate n 0) ! x *\\<^sub>R Basis_list ! x = xs ! x *\\<^sub>R Basis_list ! x\n[PROOF STEP]\nby (auto simp: nth_append)", "meta": {"llama_tokens": 565, "file": "Affine_Arithmetic_Executable_Euclidean_Space", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942067038785, "lm_q2_score": 0.824461928533133, "lm_q1q2_score": 0.7447316836918861}} {"text": "[STATEMENT]\nlemma arcsin_le_iff:\n assumes \"x \\ -1\" \"x \\ 1\" \"y \\ -pi/2\" \"y \\ pi/2\"\n shows \"arcsin x \\ y \\ x \\ sin y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (arcsin x \\ y) = (x \\ sin y)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (arcsin x \\ y) = (x \\ sin y)\n[PROOF STEP]\nhave \"arcsin x \\ y \\ sin (arcsin x) \\ sin y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (arcsin x \\ y) = (sin (arcsin x) \\ sin y)\n[PROOF STEP]\nusing arcsin_bounded[of x] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\- 1 \\ x; x \\ 1\\ \\ - (pi / 2) \\ arcsin x \\ arcsin x \\ pi / 2\n- 1 \\ x\nx \\ 1\n- pi / 2 \\ y\ny \\ pi / 2\n\ngoal (1 subgoal):\n 1. (arcsin x \\ y) = (sin (arcsin x) \\ sin y)\n[PROOF STEP]\nby (subst sin_mono_le_eq) auto\n[PROOF STATE]\nproof (state)\nthis:\n(arcsin x \\ y) = (sin (arcsin x) \\ sin y)\n\ngoal (1 subgoal):\n 1. (arcsin x \\ y) = (x \\ sin y)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(arcsin x \\ y) = (sin (arcsin x) \\ sin y)\n\ngoal (1 subgoal):\n 1. (arcsin x \\ y) = (x \\ sin y)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n- 1 \\ x\nx \\ 1\n- pi / 2 \\ y\ny \\ pi / 2\n[PROOF STEP]\nhave \"sin (arcsin x) = x\"\n[PROOF STATE]\nproof (prove)\nusing this:\n- 1 \\ x\nx \\ 1\n- pi / 2 \\ y\ny \\ pi / 2\n\ngoal (1 subgoal):\n 1. sin (arcsin x) = x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsin (arcsin x) = x\n\ngoal (1 subgoal):\n 1. (arcsin x \\ y) = (x \\ sin y)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(arcsin x \\ y) = (x \\ sin y)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(arcsin x \\ y) = (x \\ sin y)\n\ngoal (1 subgoal):\n 1. (arcsin x \\ y) = (x \\ sin y)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(arcsin x \\ y) = (x \\ sin y)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1017, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278695464501, "lm_q2_score": 0.843895106480586, "lm_q1q2_score": 0.7446765609323381}} {"text": "[STATEMENT]\nlemma CARD_INJ_IMAGE_2: \n fixes f s\n assumes \"finite s\" \"(\\x y. ((x \\ s) \\ (y \\ s)) \\ ((f x = f y) \\ (x = y)))\"\n shows \"(card (f ` s) = card s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (f ` s) = card s\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (f ` s) = card s\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (f ` s) = card s\n[PROOF STEP]\nfix x y\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (f ` s) = card s\n[PROOF STEP]\nassume \"x \\ s\" \"y \\ s\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ s\ny \\ s\n\ngoal (1 subgoal):\n 1. card (f ` s) = card s\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ s\ny \\ s\n[PROOF STEP]\nhave \"f x = f y \\ x = y\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ s\ny \\ s\n\ngoal (1 subgoal):\n 1. f x = f y \\ x = y\n[PROOF STEP]\nusing assms(2)\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ s\ny \\ s\n\\x y. x \\ s \\ y \\ s \\ (f x = f y) = (x = y)\n\ngoal (1 subgoal):\n 1. f x = f y \\ x = y\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nf x = f y \\ x = y\n\ngoal (1 subgoal):\n 1. card (f ` s) = card s\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n\\?x2 \\ s; ?y2 \\ s\\ \\ f ?x2 = f ?y2 \\ ?x2 = ?y2\n\ngoal (1 subgoal):\n 1. card (f ` s) = card s\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\?x2 \\ s; ?y2 \\ s\\ \\ f ?x2 = f ?y2 \\ ?x2 = ?y2\n[PROOF STEP]\nhave \"inj_on f s\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x2 \\ s; ?y2 \\ s\\ \\ f ?x2 = f ?y2 \\ ?x2 = ?y2\n\ngoal (1 subgoal):\n 1. inj_on f s\n[PROOF STEP]\nby (simp add: inj_onI)\n[PROOF STATE]\nproof (state)\nthis:\ninj_on f s\n\ngoal (1 subgoal):\n 1. card (f ` s) = card s\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ninj_on f s\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on f s\n\ngoal (1 subgoal):\n 1. card (f ` s) = card s\n[PROOF STEP]\nusing assms(1) inj_on_iff_eq_card\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on f s\nfinite s\nfinite ?A \\ inj_on ?f ?A = (card (?f ` ?A) = card ?A)\n\ngoal (1 subgoal):\n 1. card (f ` s) = card s\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ncard (f ` s) = card s\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1181, "file": "Factored_Transition_System_Bounding_SetUtils", "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772351648677, "lm_q2_score": 0.8519528057272543, "lm_q1q2_score": 0.7446725529210301}} {"text": "[STATEMENT]\nlemma tendsto_rabs_zero_iff: \"((\\x. \\f x\\) \\ (0::real)) F \\ (f \\ 0) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. \\f x\\) \\ 0) F = (f \\ 0) F\n[PROOF STEP]\nby (fold real_norm_def) (rule tendsto_norm_zero_iff)", "meta": {"llama_tokens": 144, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587993853655, "lm_q2_score": 0.8376199633332891, "lm_q1q2_score": 0.744442112953308}} {"text": "[STATEMENT]\nlemma A_inv_aux: \"b>2 \\ n>0 \\ \\ b n * \\ b n - \\ b (Suc n) * \\ b (n - Suc 0) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\2 < b; 0 < n\\ \\ \\ b n * \\ b n - \\ b (Suc n) * \\ b (n - Suc 0) = 1\n[PROOF STEP]\napply (induction n, auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. \\0 < n \\ \\ b n * \\ b n - \\ b (Suc n) * \\ b (n - Suc 0) = 1; 2 < b\\ \\ \\ b (Suc n) * \\ b (Suc n) - (int b * \\ b (Suc n) - \\ b n) * \\ b n = 1\n[PROOF STEP]\nsubgoal for n\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < n \\ \\ b n * \\ b n - \\ b (Suc n) * \\ b (n - Suc 0) = 1; 2 < b\\ \\ \\ b (Suc n) * \\ b (Suc n) - (int b * \\ b (Suc n) - \\ b n) * \\ b n = 1\n[PROOF STEP]\nusing alpha_det1[of b n]\n[PROOF STATE]\nproof (prove)\nusing this:\n2 < b \\ (\\ b (Suc n))\\<^sup>2 - int b * \\ b (Suc n) * \\ b n + (\\ b n)\\<^sup>2 = 1\n\ngoal (1 subgoal):\n 1. \\0 < n \\ \\ b n * \\ b n - \\ b (Suc n) * \\ b (n - Suc 0) = 1; 2 < b\\ \\ \\ b (Suc n) * \\ b (Suc n) - (int b * \\ b (Suc n) - \\ b n) * \\ b n = 1\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < n \\ \\ b n * \\ b n - \\ b (Suc n) * \\ b (n - Suc 0) = 1; 2 < b; (\\ b (Suc n))\\<^sup>2 - int b * \\ b (Suc n) * \\ b n + (\\ b n)\\<^sup>2 = 1\\ \\ \\ b (Suc n) * \\ b (Suc n) - (int b * \\ b (Suc n) - \\ b n) * \\ b n = 1\n[PROOF STEP]\nby algebra\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 896, "file": "DPRM_Theorem_Diophantine_Exponentiation", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.888758793492457, "lm_q2_score": 0.8376199653600372, "lm_q1q2_score": 0.7444421098185804}} {"text": "[STATEMENT]\ntheorem sum_of_two_squares_prime: assumes \"prime p\"\n shows \"is_sum2sq_nat p = [p\\3] (mod 4)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_sum2sq_nat p = [p \\ 3] (mod 4)\n[PROOF STEP]\nproof (cases \"[p=3] (mod 4)\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. [p = 3] (mod 4) \\ is_sum2sq_nat p = [p \\ 3] (mod 4)\n 2. [p \\ 3] (mod 4) \\ is_sum2sq_nat p = [p \\ 3] (mod 4)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\n[p = 3] (mod 4)\n\ngoal (2 subgoals):\n 1. [p = 3] (mod 4) \\ is_sum2sq_nat p = [p \\ 3] (mod 4)\n 2. [p \\ 3] (mod 4) \\ is_sum2sq_nat p = [p \\ 3] (mod 4)\n[PROOF STEP]\nhave \"odd (multiplicity p p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. odd (multiplicity p p)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. odd (multiplicity p p)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nodd (multiplicity p p)\n\ngoal (2 subgoals):\n 1. [p = 3] (mod 4) \\ is_sum2sq_nat p = [p \\ 3] (mod 4)\n 2. [p \\ 3] (mod 4) \\ is_sum2sq_nat p = [p \\ 3] (mod 4)\n[PROOF STEP]\nhence \"\\ (is_sum2sq_nat p)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nodd (multiplicity p p)\n\ngoal (1 subgoal):\n 1. \\ is_sum2sq_nat p\n[PROOF STEP]\nusing assms True sum_of_two_squares'\n[PROOF STATE]\nproof (prove)\nusing this:\nodd (multiplicity p p)\nprime p\n[p = 3] (mod 4)\nis_sum2sq_nat ?n = (\\p. prime p \\ [p = 3] (mod 4) \\ even (multiplicity p ?n))\n\ngoal (1 subgoal):\n 1. \\ is_sum2sq_nat p\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\ is_sum2sq_nat p\n\ngoal (2 subgoals):\n 1. [p = 3] (mod 4) \\ is_sum2sq_nat p = [p \\ 3] (mod 4)\n 2. [p \\ 3] (mod 4) \\ is_sum2sq_nat p = [p \\ 3] (mod 4)\n[PROOF STEP]\nwith True\n[PROOF STATE]\nproof (chain)\npicking this:\n[p = 3] (mod 4)\n\\ is_sum2sq_nat p\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n[p = 3] (mod 4)\n\\ is_sum2sq_nat p\n\ngoal (1 subgoal):\n 1. is_sum2sq_nat p = [p \\ 3] (mod 4)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nis_sum2sq_nat p = [p \\ 3] (mod 4)\n\ngoal (1 subgoal):\n 1. [p \\ 3] (mod 4) \\ is_sum2sq_nat p = [p \\ 3] (mod 4)\n[PROOF STEP]\nqed (simp add: fermat_two_squares assms)", "meta": {"llama_tokens": 1182, "file": "SumSquares_TwoSquares", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587875995482, "lm_q2_score": 0.8376199653600372, "lm_q1q2_score": 0.7444421048825622}} {"text": "[STATEMENT]\nlemma tendsto_rabs_zero_iff: \"((\\x. \\f x\\) \\ (0::real)) F \\ (f \\ 0) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. \\f x\\) \\ 0) F = (f \\ 0) F\n[PROOF STEP]\nby (fold real_norm_def) (rule tendsto_norm_zero_iff)", "meta": {"llama_tokens": 144, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587817066392, "lm_q2_score": 0.8376199592797929, "lm_q1q2_score": 0.7444420945426735}} {"text": "[STATEMENT]\nlemma \"filterlim (\\n. (1 + (2 / 3 :: real) ^ (n + 1)) ^ 2 ^ n / 2 powr (4 / 3) ^ (n - 1))\n at_top at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LIM n sequentially. (1 + (2 / 3) ^ (n + 1)) ^ 2 ^ n / 2 powr (4 / 3) ^ (n - 1) :> at_top\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. LIM n sequentially. (1 + (2 / 3) ^ (n + 1)) ^ 2 ^ n / 2 powr (4 / 3) ^ (n - 1) :> at_top\n[PROOF STEP]\nhave [simp]: \"ln 4 = 2 * ln (2 :: real)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ln 4 = 2 * ln 2\n[PROOF STEP]\nusing ln_realpow[of 2 2]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 2 \\ ln (2\\<^sup>2) = real 2 * ln 2\n\ngoal (1 subgoal):\n 1. ln 4 = 2 * ln 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nln 4 = 2 * ln 2\n\ngoal (1 subgoal):\n 1. LIM n sequentially. (1 + (2 / 3) ^ (n + 1)) ^ 2 ^ n / 2 powr (4 / 3) ^ (n - 1) :> at_top\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LIM n sequentially. (1 + (2 / 3) ^ (n + 1)) ^ 2 ^ n / 2 powr (4 / 3) ^ (n - 1) :> at_top\n[PROOF STEP]\nby (real_asymp simp: field_simps ln_div)\n[PROOF STATE]\nproof (state)\nthis:\nLIM n sequentially. (1 + (2 / 3) ^ (n + 1)) ^ 2 ^ n / 2 powr (4 / 3) ^ (n - 1) :> at_top\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 671, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933093975331751, "lm_q2_score": 0.8333246015211008, "lm_q1q2_score": 0.7444166977343877}} {"text": "[STATEMENT]\nlemma base_case_invertible_mat: \n shows \"invertible_mat (matrix_A [[1], [- 1]] [[], [0]])\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. invertible_mat (M_mat [[1], [- 1]] [[], [0]])\n[PROOF STEP]\nunfolding invertible_mat_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. square_mat (M_mat [[1], [- 1]] [[], [0]]) \\ (\\B. inverts_mat (M_mat [[1], [- 1]] [[], [0]]) B \\ inverts_mat B (M_mat [[1], [- 1]] [[], [0]]))\n[PROOF STEP]\nusing inverse_mat_base_case inverse_mat_base_case_2\n[PROOF STATE]\nproof (prove)\nusing this:\ninverts_mat (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]]) (mat_of_rows_list 2 [[1, 1], [1, - 1]])\ninverts_mat (mat_of_rows_list 2 [[1, 1], [1, - 1]]) (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]])\n\ngoal (1 subgoal):\n 1. square_mat (M_mat [[1], [- 1]] [[], [0]]) \\ (\\B. inverts_mat (M_mat [[1], [- 1]] [[], [0]]) B \\ inverts_mat B (M_mat [[1], [- 1]] [[], [0]]))\n[PROOF STEP]\napply (auto)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\inverts_mat (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]]) (mat_of_rows_list 2 [[1, 1], [1, - 1]]); inverts_mat (mat_of_rows_list 2 [[1, 1], [1, - 1]]) (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]])\\ \\ dim_col (M_mat [[1], [- 1]] [[], [0]]) = dim_row (M_mat [[1], [- 1]] [[], [0]])\n 2. \\inverts_mat (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]]) (mat_of_rows_list 2 [[1, 1], [1, - 1]]); inverts_mat (mat_of_rows_list 2 [[1, 1], [1, - 1]]) (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]])\\ \\ \\B. inverts_mat (M_mat [[1], [- 1]] [[], [0]]) B \\ inverts_mat B (M_mat [[1], [- 1]] [[], [0]])\n[PROOF STEP]\napply (simp add: mat_base_case mat_of_rows_list_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\inverts_mat (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]]) (mat_of_rows_list 2 [[1, 1], [1, - 1]]); inverts_mat (mat_of_rows_list 2 [[1, 1], [1, - 1]]) (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]])\\ \\ \\B. inverts_mat (M_mat [[1], [- 1]] [[], [0]]) B \\ inverts_mat B (M_mat [[1], [- 1]] [[], [0]])\n[PROOF STEP]\nusing mat_base_case\n[PROOF STATE]\nproof (prove)\nusing this:\nM_mat [[1], [- 1]] [[], [0]] = mat_of_rows_list 2 [[1, 1], [1, - 1]]\n\ngoal (1 subgoal):\n 1. \\inverts_mat (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]]) (mat_of_rows_list 2 [[1, 1], [1, - 1]]); inverts_mat (mat_of_rows_list 2 [[1, 1], [1, - 1]]) (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]])\\ \\ \\B. inverts_mat (M_mat [[1], [- 1]] [[], [0]]) B \\ inverts_mat B (M_mat [[1], [- 1]] [[], [0]])\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 1420, "file": "BenOr_Kozen_Reif_BKR_Proofs", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094032139576, "lm_q2_score": 0.8333245953120234, "lm_q1q2_score": 0.7444166969216963}} {"text": "[STATEMENT]\nlemma card_permutations_of_multiset_aux:\n \"card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nproof (induction A rule: multiset_remove_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\ncase (remove A)\n[PROOF STATE]\nproof (state)\nthis:\nA \\ {#}\n?x \\# A \\ card (permutations_of_multiset (A - {#?x#})) * (\\x\\set_mset (A - {#?x#}). fact (count (A - {#?x#}) x)) = fact (size (A - {#?x#}))\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nhave \"card (permutations_of_multiset A) = \n card (\\x\\set_mset A. (#) x ` permutations_of_multiset (A - {#x#}))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (permutations_of_multiset A) = card (\\x\\set_mset A. (#) x ` permutations_of_multiset (A - {#x#}))\n[PROOF STEP]\nby (simp add: permutations_of_multiset_nonempty remove.hyps)\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset A) = card (\\x\\set_mset A. (#) x ` permutations_of_multiset (A - {#x#}))\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset A) = card (\\x\\set_mset A. (#) x ` permutations_of_multiset (A - {#x#}))\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nhave \"\\ = (\\x\\set_mset A. card (permutations_of_multiset (A - {#x#})))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (\\x\\set_mset A. (#) x ` permutations_of_multiset (A - {#x#})) = (\\x\\set_mset A. card (permutations_of_multiset (A - {#x#})))\n[PROOF STEP]\nby (subst card_UN_disjoint) (auto simp: card_image)\n[PROOF STATE]\nproof (state)\nthis:\ncard (\\x\\set_mset A. (#) x ` permutations_of_multiset (A - {#x#})) = (\\x\\set_mset A. card (permutations_of_multiset (A - {#x#})))\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (\\x\\set_mset A. (#) x ` permutations_of_multiset (A - {#x#})) = (\\x\\set_mset A. card (permutations_of_multiset (A - {#x#})))\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nhave \"\\ * (\\x\\set_mset A. fact (count A x)) = \n (\\x\\set_mset A. card (permutations_of_multiset (A - {#x#})) * \n (\\y\\set_mset A. fact (count A y)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\set_mset A. card (permutations_of_multiset (A - {#x#}))) * (\\x\\set_mset A. fact (count A x)) = (\\x\\set_mset A. card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)))\n[PROOF STEP]\nby (subst sum_distrib_right) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\set_mset A. card (permutations_of_multiset (A - {#x#}))) * (\\x\\set_mset A. fact (count A x)) = (\\x\\set_mset A. card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)))\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\set_mset A. card (permutations_of_multiset (A - {#x#}))) * (\\x\\set_mset A. fact (count A x)) = (\\x\\set_mset A. card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)))\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nhave \"\\ = (\\x\\set_mset A. count A x * fact (size A - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\set_mset A. card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y))) = (\\x\\set_mset A. count A x * fact (size A - 1))\n[PROOF STEP]\nproof (intro sum.cong refl)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n[PROOF STEP]\nassume x: \"x \\# A\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\# A\n\ngoal (1 subgoal):\n 1. \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n[PROOF STEP]\nhave \"card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = \n count A x * (card (permutations_of_multiset (A - {#x#})) * \n (\\y\\set_mset (A - {#x#}). fact (count (A - {#x#}) y)))\" (is \"?lhs = _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * (card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset (A - {#x#}). fact (count (A - {#x#}) y)))\n[PROOF STEP]\nby (subst multiset_prod_fact_remove[OF x]) simp_all\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * (card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset (A - {#x#}). fact (count (A - {#x#}) y)))\n\ngoal (1 subgoal):\n 1. \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * (card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset (A - {#x#}). fact (count (A - {#x#}) y)))\n\ngoal (1 subgoal):\n 1. \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n[PROOF STEP]\nnote remove.IH[OF x]\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\n\ngoal (1 subgoal):\n 1. \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\n\ngoal (1 subgoal):\n 1. \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n[PROOF STEP]\nfrom x\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\# A\n[PROOF STEP]\nhave \"size (A - {#x#}) = size A - 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\# A\n\ngoal (1 subgoal):\n 1. size (A - {#x#}) = size A - 1\n[PROOF STEP]\nby (simp add: size_Diff_submset)\n[PROOF STATE]\nproof (state)\nthis:\nsize (A - {#x#}) = size A - 1\n\ngoal (1 subgoal):\n 1. \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n[PROOF STEP]\nshow \"?lhs = count A x * fact (size A - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y)) = count A x * fact (size A - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\set_mset A. card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y))) = (\\x\\set_mset A. count A x * fact (size A - 1))\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\set_mset A. card (permutations_of_multiset (A - {#x#})) * (\\y\\set_mset A. fact (count A y))) = (\\x\\set_mset A. count A x * fact (size A - 1))\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nhave \"(\\x\\set_mset A. count A x * fact (size A - 1)) =\n size A * fact (size A - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\set_mset A. count A x * fact (size A - 1)) = size A * fact (size A - 1)\n[PROOF STEP]\nby (simp add: sum_distrib_right size_multiset_overloaded_eq)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\set_mset A. count A x * fact (size A - 1)) = size A * fact (size A - 1)\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\set_mset A. count A x * fact (size A - 1)) = size A * fact (size A - 1)\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nfrom remove.hyps\n[PROOF STATE]\nproof (chain)\npicking this:\nA \\ {#}\n[PROOF STEP]\nhave \"\\ = fact (size A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ {#}\n\ngoal (1 subgoal):\n 1. size A * fact (size A - 1) = fact (size A)\n[PROOF STEP]\nby (cases \"size A\") auto\n[PROOF STATE]\nproof (state)\nthis:\nsize A * fact (size A - 1) = fact (size A)\n\ngoal (2 subgoals):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n 2. \\A. \\A \\ {#}; \\x. x \\# A \\ card (permutations_of_multiset (A - {#x#})) * (\\xa\\set_mset (A - {#x#}). fact (count (A - {#x#}) xa)) = fact (size (A - {#x#}))\\ \\ card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)) = fact (size A)\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset {#}) * (\\x\\set_mset {#}. fact (count {#} x)) = fact (size {#})\n[PROOF STEP]\nqed simp_all", "meta": {"llama_tokens": 7193, "file": null, "length": 38, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9184802440252811, "lm_q2_score": 0.81047890180374, "lm_q1q2_score": 0.744408859506041}} {"text": "[STATEMENT]\nlemma sum_splice_other_way_round: \" (\\j=(0::nat)..j=0..j=(0::nat)..<2*i. f j )\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..j = 0..x\\{column i (fst (QR_decomposition A))|i. i < k}. (x \\ (column k A) / (x \\ x)) *\\<^sub>R x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. column k A = column k (Gram_Schmidt_matrix A) + (\\x\\{column i (fst (QR_decomposition A)) |i. i < k}. (x \\ column k A / (x \\ x)) *\\<^sub>R x)\n[PROOF STEP]\nusing column_QR_decomposition[OF r]\n[PROOF STATE]\nproof (prove)\nusing this:\ncolumn ?k (Gram_Schmidt_matrix A) = column ?k A - (\\x\\{column i (fst (QR_decomposition A)) |i. i < ?k}. (x \\ column ?k A / (x \\ x)) *\\<^sub>R x)\n\ngoal (1 subgoal):\n 1. column k A = column k (Gram_Schmidt_matrix A) + (\\x\\{column i (fst (QR_decomposition A)) |i. i < k}. (x \\ column k A / (x \\ x)) *\\<^sub>R x)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 425, "file": "QR_Decomposition_QR_Decomposition", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9324533144915912, "lm_q2_score": 0.7981867825403176, "lm_q1q2_score": 0.7442719109630981}} {"text": "[STATEMENT]\nlemma polar_identity_minus:\n includes notation_norm\n shows \\\\x - y\\^2 = \\x\\^2 + \\y\\^2 - 2 * Re (x \\\\<^sub>C y)\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 - 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 - 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nhave \\\\x + (-y)\\^2 = \\x\\^2 + \\-y\\^2 + 2 * Re (x \\\\<^sub>C -y)\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x + - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\- y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C - y)\n[PROOF STEP]\nusing polar_identity\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x + ?y\\\\<^sup>2 = \\?x\\\\<^sup>2 + \\?y\\\\<^sup>2 + 2 * Re (?x \\\\<^sub>C ?y)\n\ngoal (1 subgoal):\n 1. \\x + - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\- y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C - y)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\x + - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\- y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C - y)\n\ngoal (1 subgoal):\n 1. \\x - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 - 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nhence \\\\x - y\\^2 = \\x\\^2 + \\y\\^2 - 2*Re (x \\\\<^sub>C y)\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x + - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\- y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C - y)\n\ngoal (1 subgoal):\n 1. \\x - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 - 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\x - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 - 2 * Re (x \\\\<^sub>C y)\n\ngoal (1 subgoal):\n 1. \\x - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 - 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 - 2 * Re (x \\\\<^sub>C y)\n\ngoal (1 subgoal):\n 1. \\x - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 - 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\x - y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 - 2 * Re (x \\\\<^sub>C y)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1228, "file": "Complex_Bounded_Operators_Complex_Inner_Product", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392909114836, "lm_q2_score": 0.8418256551882382, "lm_q1q2_score": 0.7442069552837052}} {"text": "[STATEMENT]\nlemma all_Suc_split: \"(\\i. P i) \\ (P 0 \\ (\\i. P (Suc i)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. P i) = (P 0 \\ (\\i. P (Suc i)))\n[PROOF STEP]\nusing nat_induct\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?P 0; \\n. ?P n \\ ?P (Suc n)\\ \\ ?P ?n\n\ngoal (1 subgoal):\n 1. (\\i. P i) = (P 0 \\ (\\i. P (Suc i)))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 220, "file": "Markov_Models_Markov_Models_Auxiliary", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392848011834, "lm_q2_score": 0.8418256512199033, "lm_q1q2_score": 0.7442069466317338}} {"text": "[STATEMENT]\nlemma two_powr_div:\n assumes \"j < (i::nat) \"\n shows \"((2^i) div ((2::nat)^j)) = 2^(i-j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ i div 2 ^ j = 2 ^ (i - j)\n[PROOF STEP]\nby (simp add: assms less_or_eq_imp_le power_diff)", "meta": {"llama_tokens": 121, "file": "Number_Theoretic_Transform_Preliminary_Lemmas", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9294404077216355, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.7441955008875164}} {"text": "[STATEMENT]\nlemma submatrix_carrier_first:\n assumes H: \"H \\ carrier_mat m n\"\n and k1: \"k1 \\ m\" and k2: \"k2 \\ n\"\n shows\"submatrix H {0.. carrier_mat k1 k2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. submatrix H {0.. carrier_mat k1 k2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. submatrix H {0.. carrier_mat k1 k2\n[PROOF STEP]\nhave km: \"k1\\m\" and kn: \"k2\\n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k1 \\ m &&& k2 \\ n\n[PROOF STEP]\nusing k1 k2\n[PROOF STATE]\nproof (prove)\nusing this:\nk1 \\ m\nk2 \\ n\n\ngoal (1 subgoal):\n 1. k1 \\ m &&& k2 \\ n\n[PROOF STEP]\nby simp+\n[PROOF STATE]\nproof (state)\nthis:\nk1 \\ m\nk2 \\ n\n\ngoal (1 subgoal):\n 1. submatrix H {0.. carrier_mat k1 k2\n[PROOF STEP]\nhave card_mk: \"card {i. i < m \\ i < k1} = k1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {i. i < m \\ i < k1} = k1\n[PROOF STEP]\nusing km\n[PROOF STATE]\nproof (prove)\nusing this:\nk1 \\ m\n\ngoal (1 subgoal):\n 1. card {i. i < m \\ i < k1} = k1\n[PROOF STEP]\nby (smt Collect_cong card_Collect_less_nat le_eq_less_or_eq nat_less_induct nat_neq_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncard {i. i < m \\ i < k1} = k1\n\ngoal (1 subgoal):\n 1. submatrix H {0.. carrier_mat k1 k2\n[PROOF STEP]\nhave card_nk: \"card {i. i < n \\ i < k2} = k2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {i. i < n \\ i < k2} = k2\n[PROOF STEP]\nusing kn\n[PROOF STATE]\nproof (prove)\nusing this:\nk2 \\ n\n\ngoal (1 subgoal):\n 1. card {i. i < n \\ i < k2} = k2\n[PROOF STEP]\nby (smt Collect_cong card_Collect_less_nat le_eq_less_or_eq nat_less_induct nat_neq_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncard {i. i < n \\ i < k2} = k2\n\ngoal (1 subgoal):\n 1. submatrix H {0.. carrier_mat k1 k2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. submatrix H {0.. carrier_mat k1 k2\n[PROOF STEP]\nby (smt Collect_cong H atLeastLessThan_iff card_mk card_nk carrier_matD \n carrier_matI dim_submatrix zero_order(1))\n[PROOF STATE]\nproof (state)\nthis:\nsubmatrix H {0.. carrier_mat k1 k2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1129, "file": "Modular_arithmetic_LLL_and_HNF_algorithms_HNF_Mod_Det_Soundness", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976953003183444, "lm_q2_score": 0.8289388125473628, "lm_q1q2_score": 0.7441344762752367}} {"text": "[STATEMENT]\nlemma powr_less_mono':\n assumes \"(x::real) > 0\" \"x < 1\" \"a < b\"\n shows \"x powr b < x powr a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x powr b < x powr a\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x powr b < x powr a\n[PROOF STEP]\nhave \"inverse x powr a < inverse x powr b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse x powr a < inverse x powr b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\nx < 1\na < b\n\ngoal (1 subgoal):\n 1. inverse x powr a < inverse x powr b\n[PROOF STEP]\nby (intro powr_less_mono) (simp_all add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\ninverse x powr a < inverse x powr b\n\ngoal (1 subgoal):\n 1. x powr b < x powr a\n[PROOF STEP]\nhence \"inverse (x powr a) < inverse (x powr b)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse x powr a < inverse x powr b\n\ngoal (1 subgoal):\n 1. inverse (x powr a) < inverse (x powr b)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse x powr a < inverse x powr b\n0 < x\nx < 1\na < b\n\ngoal (1 subgoal):\n 1. inverse (x powr a) < inverse (x powr b)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ninverse (x powr a) < inverse (x powr b)\n\ngoal (1 subgoal):\n 1. x powr b < x powr a\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < x\nx < 1\na < b\ninverse (x powr a) < inverse (x powr b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\nx < 1\na < b\ninverse (x powr a) < inverse (x powr b)\n\ngoal (1 subgoal):\n 1. x powr b < x powr a\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nx powr b < x powr a\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 767, "file": "Akra_Bazzi_Akra_Bazzi_Library", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8479677564567913, "lm_q1q2_score": 0.7440720173899664}} {"text": "[STATEMENT]\nlemma mat_det_left_def: assumes A: \"A \\ carrier_mat n n\"\n shows \"det A = (\\p\\{p. p permutes {0..i = 0 ..< dim_row A. A $$ (p i, i)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A = (\\p | p permutes {0..i = 0..p | p permutes {0..i = 0.. a b c. b = c \\ a * b = a * c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a b c. b = c \\ a * b = a * c\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n?b = ?c \\ ?a * ?b = ?a * ?c\n\ngoal (1 subgoal):\n 1. det A = (\\p | p permutes {0..i = 0..p | p permutes {0..i = 0..T = (\\p | p permutes {0..i = 0..p | p permutes {0..i = 0..T $$ (i, p i)) else (0::'a)) = (\\p | p permutes {0..i = 0.. carrier_mat n n\n\ngoal (1 subgoal):\n 1. (if dim_col A = dim_row A then \\p | p permutes {0..i = 0..T $$ (i, p i)) else (0::'a)) = (\\p | p permutes {0..i = 0..p | p permutes {0..i = 0.. carrier_mat d1 d1\"\n and \"m2 \\ carrier_mat d2 d2\"\n shows \"trace (tensor_mat m1 m2) = trace m1 * trace m2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (tensor_mat m1 m2) = trace m1 * trace m2\n[PROOF STEP]\napply (auto simp add: tensor_mat_carrier trace_def tensor_mat_eval)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..i = 0..i = 0..i = 0..i = 0..j = 0..i = 0..j = 0..i = 0..j = 0.. carrier_mat d1 d1\nm2 \\ carrier_mat d2 d2\n\ngoal (1 subgoal):\n 1. (\\i = 0..j = 0..i = 0..j = 0..homotopic_with p X1 X2 f g; continuous_map X2 X3 h; \\j. p j \\ q(h \\ j)\\\n \\ homotopic_with q X1 X3 (h \\ f) (h \\ g)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\homotopic_with p X1 X2 f g; continuous_map X2 X3 h; \\j. p j \\ q (h \\ j)\\ \\ homotopic_with q X1 X3 (h \\ f) (h \\ g)\n[PROOF STEP]\nunfolding homotopic_with_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\h. continuous_map (prod_topology (top_of_set {0..1}) X1) X2 h \\ (\\x. h (0, x) = f x) \\ (\\x. h (1, x) = g x) \\ (\\t\\{0..1}. p (\\x. h (t, x))); continuous_map X2 X3 h; \\j. p j \\ q (h \\ j)\\ \\ \\ha. continuous_map (prod_topology (top_of_set {0..1}) X1) X3 ha \\ (\\x. ha (0, x) = (h \\ f) x) \\ (\\x. ha (1, x) = (h \\ g) x) \\ (\\t\\{0..1}. q (\\x. ha (t, x)))\n[PROOF STEP]\napply clarify\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ha. \\continuous_map X2 X3 h; \\j. p j \\ q (h \\ j); continuous_map (prod_topology (top_of_set {0..1}) X1) X2 ha; \\x. ha (0, x) = f x; \\x. ha (1, x) = g x; \\t\\{0..1}. p (\\x. ha (t, x))\\ \\ \\ha. continuous_map (prod_topology (top_of_set {0..1}) X1) X3 ha \\ (\\x. ha (0, x) = (h \\ f) x) \\ (\\x. ha (1, x) = (h \\ g) x) \\ (\\t\\{0..1}. q (\\x. ha (t, x)))\n[PROOF STEP]\nsubgoal for k\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\continuous_map X2 X3 h; \\j. p j \\ q (h \\ j); continuous_map (prod_topology (top_of_set {0..1}) X1) X2 k; \\x. k (0, x) = f x; \\x. k (1, x) = g x; \\t\\{0..1}. p (\\x. k (t, x))\\ \\ \\ha. continuous_map (prod_topology (top_of_set {0..1}) X1) X3 ha \\ (\\x. ha (0, x) = (h \\ f) x) \\ (\\x. ha (1, x) = (h \\ g) x) \\ (\\t\\{0..1}. q (\\x. ha (t, x)))\n[PROOF STEP]\nby (rule_tac x=\"h \\ k\" in exI) (rule conjI continuous_map_compose | simp add: o_def)+\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1087, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942173896131, "lm_q2_score": 0.8354835411997897, "lm_q1q2_score": 0.7438261654543694}} {"text": "[STATEMENT]\nlemma prodinf_exp: \n assumes \"summable f\"\n shows \"prodinf (\\i. exp (f i)) = exp (suminf f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. exp (f i)) = exp (suminf f)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\i. exp (f i)) = exp (suminf f)\n[PROOF STEP]\nhave \"f sums suminf f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f sums suminf f\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable f\n\ngoal (1 subgoal):\n 1. f sums suminf f\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nf sums suminf f\n\ngoal (1 subgoal):\n 1. (\\i. exp (f i)) = exp (suminf f)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nf sums suminf f\n[PROOF STEP]\nhave \"(\\i. exp (f i)) has_prod exp (suminf f)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nf sums suminf f\n\ngoal (1 subgoal):\n 1. (\\i. exp (f i)) has_prod exp (suminf f)\n[PROOF STEP]\nby (simp add: has_prod_def sums_imp_has_prod_exp)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i. exp (f i)) has_prod exp (suminf f)\n\ngoal (1 subgoal):\n 1. (\\i. exp (f i)) = exp (suminf f)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i. exp (f i)) has_prod exp (suminf f)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i. exp (f i)) has_prod exp (suminf f)\n\ngoal (1 subgoal):\n 1. (\\i. exp (f i)) = exp (suminf f)\n[PROOF STEP]\nby (rule has_prod_unique [symmetric])\n[PROOF STATE]\nproof (state)\nthis:\n(\\i. exp (f i)) = exp (suminf f)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 716, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240930029118, "lm_q2_score": 0.8596637451167997, "lm_q1q2_score": 0.7438017841561694}} {"text": "[STATEMENT]\nlemma docking_station_arith:\n assumes \"(d::real) > x\" and \"v > 0\"\n shows \"(v = v\\<^sup>2 * t / (2 * d - 2 * x)) \\ (v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (v = v\\<^sup>2 * t / (2 * d - 2 * x)) = (v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. v = v\\<^sup>2 * t / (2 * d - 2 * x) \\ v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n 2. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nassume \"v = v\\<^sup>2 * t / (2 * d - 2 * x)\"\n[PROOF STATE]\nproof (state)\nthis:\nv = v\\<^sup>2 * t / (2 * d - 2 * x)\n\ngoal (2 subgoals):\n 1. v = v\\<^sup>2 * t / (2 * d - 2 * x) \\ v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n 2. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nhence \"v * t = 2 * (d - x)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv = v\\<^sup>2 * t / (2 * d - 2 * x)\n\ngoal (1 subgoal):\n 1. v * t = 2 * (d - x)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nv = v\\<^sup>2 * t / (2 * d - 2 * x)\nx < d\n0 < v\n\ngoal (1 subgoal):\n 1. v * t = 2 * (d - x)\n[PROOF STEP]\nby (simp add: eq_divide_eq power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\nv * t = 2 * (d - x)\n\ngoal (2 subgoals):\n 1. v = v\\<^sup>2 * t / (2 * d - 2 * x) \\ v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n 2. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nhence \"v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = 2 * (d - x) - 4 * (d - x)\\<^sup>2 / (4 * (d - x)) + x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv * t = 2 * (d - x)\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = 2 * (d - x) - 4 * (d - x)\\<^sup>2 / (4 * (d - x)) + x\n[PROOF STEP]\napply(subst power_mult_distrib[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v * t = 2 * (d - x) \\ v * t - (v * t)\\<^sup>2 / (4 * d - 4 * x) + x = 2 * (d - x) - 4 * (d - x)\\<^sup>2 / (4 * (d - x)) + x\n[PROOF STEP]\nby (erule ssubst, subst power_mult_distrib, simp)\n[PROOF STATE]\nproof (state)\nthis:\nv * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = 2 * (d - x) - 4 * (d - x)\\<^sup>2 / (4 * (d - x)) + x\n\ngoal (2 subgoals):\n 1. v = v\\<^sup>2 * t / (2 * d - 2 * x) \\ v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n 2. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nv * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = 2 * (d - x) - 4 * (d - x)\\<^sup>2 / (4 * (d - x)) + x\n\ngoal (2 subgoals):\n 1. v = v\\<^sup>2 * t / (2 * d - 2 * x) \\ v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n 2. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nhave \"... = d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * (d - x) - 4 * (d - x)\\<^sup>2 / (4 * (d - x)) + x = d\n[PROOF STEP]\napply(simp only: mult_divide_mult_cancel_left_if)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * (d - x) - (if 4 = 0 then 0 else (d - x)\\<^sup>2 / (d - x)) + x = d\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx < d\n0 < v\n\ngoal (1 subgoal):\n 1. 2 * (d - x) - (if 4 = 0 then 0 else (d - x)\\<^sup>2 / (d - x)) + x = d\n[PROOF STEP]\nby (auto simp: power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n2 * (d - x) - 4 * (d - x)\\<^sup>2 / (4 * (d - x)) + x = d\n\ngoal (2 subgoals):\n 1. v = v\\<^sup>2 * t / (2 * d - 2 * x) \\ v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n 2. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nv * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n[PROOF STEP]\nshow \"v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nv * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nassume \"v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\"\n[PROOF STATE]\nproof (state)\nthis:\nv * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nhence \"0 = v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d\n\ngoal (1 subgoal):\n 1. 0 = v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 = v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nhence \"0 = (4 * (d - x)) * (v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 = v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t\n\ngoal (1 subgoal):\n 1. 0 = 4 * (d - x) * (v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 = 4 * (d - x) * (v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t)\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n0 = 4 * (d - x) * (v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t)\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nhave \"... = v\\<^sup>2 * t\\<^sup>2 + 4 * (d - x)\\<^sup>2 - (4 * (d - x)) * (v * t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 4 * (d - x) * (v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t) = v\\<^sup>2 * t\\<^sup>2 + 4 * (d - x)\\<^sup>2 - 4 * (d - x) * (v * t)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx < d\n0 < v\n\ngoal (1 subgoal):\n 1. 4 * (d - x) * (v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t) = v\\<^sup>2 * t\\<^sup>2 + 4 * (d - x)\\<^sup>2 - 4 * (d - x) * (v * t)\n[PROOF STEP]\napply(simp add: distrib_left right_diff_distrib)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x < d; 0 < v\\ \\ (4 * d - 4 * x) * d - (4 * d - 4 * x) * x = 4 * (d - x)\\<^sup>2\n[PROOF STEP]\napply(subst right_diff_distrib[symmetric])+\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x < d; 0 < v\\ \\ 4 * (d - x) * (d - x) = 4 * (d - x)\\<^sup>2\n[PROOF STEP]\nby (simp add: power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n4 * (d - x) * (v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t) = v\\<^sup>2 * t\\<^sup>2 + 4 * (d - x)\\<^sup>2 - 4 * (d - x) * (v * t)\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n4 * (d - x) * (v\\<^sup>2 * t\\<^sup>2 / (4 * (d - x)) + (d - x) - v * t) = v\\<^sup>2 * t\\<^sup>2 + 4 * (d - x)\\<^sup>2 - 4 * (d - x) * (v * t)\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nhave \"... = (v * t - 2 * (d - x))\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v\\<^sup>2 * t\\<^sup>2 + 4 * (d - x)\\<^sup>2 - 4 * (d - x) * (v * t) = (v * t - 2 * (d - x))\\<^sup>2\n[PROOF STEP]\nby (simp only: power2_diff, auto simp: field_simps power2_diff)\n[PROOF STATE]\nproof (state)\nthis:\nv\\<^sup>2 * t\\<^sup>2 + 4 * (d - x)\\<^sup>2 - 4 * (d - x) * (v * t) = (v * t - 2 * (d - x))\\<^sup>2\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n0 = (v * t - 2 * (d - x))\\<^sup>2\n[PROOF STEP]\nhave \"0 = (v * t - 2 * (d - x))\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 = (v * t - 2 * (d - x))\\<^sup>2\n\ngoal (1 subgoal):\n 1. 0 = (v * t - 2 * (d - x))\\<^sup>2\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n0 = (v * t - 2 * (d - x))\\<^sup>2\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nhence \"v * t = 2 * (d - x)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 = (v * t - 2 * (d - x))\\<^sup>2\n\ngoal (1 subgoal):\n 1. v * t = 2 * (d - x)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv * t = 2 * (d - x)\n\ngoal (1 subgoal):\n 1. v * t - v\\<^sup>2 * t\\<^sup>2 / (4 * d - 4 * x) + x = d \\ v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\nthus \"v = v\\<^sup>2 * t / (2 * d - 2 * x)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv * t = 2 * (d - x)\n\ngoal (1 subgoal):\n 1. v = v\\<^sup>2 * t / (2 * d - 2 * x)\n[PROOF STEP]\napply(subst power2_eq_square, subst mult.assoc)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\v * t = 2 * (d - x); v * t = 2 * (d - x)\\ \\ v = v * (v * t) / (2 * d - 2 * x)\n[PROOF STEP]\napply(erule ssubst, subst right_diff_distrib[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v * t = 2 * (d - x) \\ v = v * (2 * (d - x)) / (2 * (d - x))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx < d\n0 < v\n\ngoal (1 subgoal):\n 1. v * t = 2 * (d - x) \\ v = v * (2 * (d - x)) / (2 * (d - x))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv = v\\<^sup>2 * t / (2 * d - 2 * x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5298, "file": "Matrices_for_ODEs_MTX_Examples", "length": 42, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505453836382, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7437938512474244}} {"text": "[STATEMENT]\nlemma image_affinity_interval:\n fixes c :: \"'a::ordered_real_vector\"\n shows \"((\\x. m *\\<^sub>R x + c) ` {a..b}) = \n (if {a..b}={} then {}\n else if 0 \\ m then {m *\\<^sub>R a + c .. m *\\<^sub>R b + c}\n else {m *\\<^sub>R b + c .. m *\\<^sub>R a + c})\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n[PROOF STEP]\nproof (cases \"m=0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. m = 0 \\ (\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n 2. m \\ 0 \\ (\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nm = 0\n\ngoal (2 subgoals):\n 1. m = 0 \\ (\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n 2. m \\ 0 \\ (\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nm = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nm = 0\n\ngoal (1 subgoal):\n 1. (\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n\ngoal (1 subgoal):\n 1. m \\ 0 \\ (\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. m \\ 0 \\ (\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nm \\ 0\n\ngoal (1 subgoal):\n 1. m \\ 0 \\ (\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\x. m *\\<^sub>R x + c) ` {a..b} \\ (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n 2. (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c}) \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n[PROOF STEP]\nshow \"?lhs \\ ?rhs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. m *\\<^sub>R x + c) ` {a..b} \\ (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n[PROOF STEP]\nby (auto simp: scaleR_left_mono scaleR_left_mono_neg)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. m *\\<^sub>R x + c) ` {a..b} \\ (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n\ngoal (1 subgoal):\n 1. (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c}) \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n[PROOF STEP]\nshow \"?rhs \\ ?lhs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c}) \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n[PROOF STEP]\nproof (clarsimp, intro conjI impI subsetI)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x. \\0 \\ m; a \\ b; x \\ {m *\\<^sub>R a + c..m *\\<^sub>R b + c}\\ \\ x \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n 2. \\x. \\\\ 0 \\ m; a \\ b; x \\ {m *\\<^sub>R b + c..m *\\<^sub>R a + c}\\ \\ x \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n[PROOF STEP]\nshow \"\\0 \\ m; a \\ b; x \\ {m *\\<^sub>R a + c..m *\\<^sub>R b + c}\\\n \\ x \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\" for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 \\ m; a \\ b; x \\ {m *\\<^sub>R a + c..m *\\<^sub>R b + c}\\ \\ x \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n[PROOF STEP]\nusing False\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ 0\n\ngoal (1 subgoal):\n 1. \\0 \\ m; a \\ b; x \\ {m *\\<^sub>R a + c..m *\\<^sub>R b + c}\\ \\ x \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n[PROOF STEP]\nby (rule_tac x=\"inverse m *\\<^sub>R (x-c)\" in image_eqI)\n (auto simp: pos_le_divideR_eq pos_divideR_le_eq le_diff_eq diff_le_eq)\n[PROOF STATE]\nproof (state)\nthis:\n\\0 \\ m; a \\ b; ?x \\ {m *\\<^sub>R a + c..m *\\<^sub>R b + c}\\ \\ ?x \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n\ngoal (1 subgoal):\n 1. \\x. \\\\ 0 \\ m; a \\ b; x \\ {m *\\<^sub>R b + c..m *\\<^sub>R a + c}\\ \\ x \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n[PROOF STEP]\nshow \"\\\\ 0 \\ m; a \\ b; x \\ {m *\\<^sub>R b + c..m *\\<^sub>R a + c}\\\n \\ x \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\" for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\ 0 \\ m; a \\ b; x \\ {m *\\<^sub>R b + c..m *\\<^sub>R a + c}\\ \\ x \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n[PROOF STEP]\nby (rule_tac x=\"inverse m *\\<^sub>R (x-c)\" in image_eqI)\n (auto simp add: neg_le_divideR_eq neg_divideR_le_eq le_diff_eq diff_le_eq)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\ 0 \\ m; a \\ b; ?x \\ {m *\\<^sub>R b + c..m *\\<^sub>R a + c}\\ \\ ?x \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c}) \\ (\\x. m *\\<^sub>R x + c) ` {a..b}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. m *\\<^sub>R x + c) ` {a..b} = (if {a..b} = {} then {} else if 0 \\ m then {m *\\<^sub>R a + c..m *\\<^sub>R b + c} else {m *\\<^sub>R b + c..m *\\<^sub>R a + c})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3611, "file": null, "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505402422645, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7437938450494953}} {"text": "[STATEMENT]\nlemma index_outer_prod:\n fixes v w :: \"'a::conjugatable_field vec\"\n assumes v: \"v \\ carrier_vec n\" and w: \"w \\ carrier_vec m\"\n and ij: \"i < n\" \"j < m\"\n shows \"(outer_prod v w)$$(i, j) = v $ i * conjugate (w $ j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. outer_prod v w $$ (i, j) = v $ i * conjugate (w $ j)\n[PROOF STEP]\nunfolding outer_prod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (mat (dim_vec v) 1 (\\(i, j). v $ i) * mat 1 (dim_vec w) (\\(i, y). conjugate w $ y)) $$ (i, j) = v $ i * conjugate (w $ j)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ carrier_vec n\nw \\ carrier_vec m\ni < n\nj < m\n\ngoal (1 subgoal):\n 1. (mat (dim_vec v) 1 (\\(i, j). v $ i) * mat 1 (dim_vec w) (\\(i, y). conjugate w $ y)) $$ (i, j) = v $ i * conjugate (w $ j)\n[PROOF STEP]\nby (simp add: scalar_prod_def)", "meta": {"llama_tokens": 403, "file": "QHLProver_Complex_Matrix", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505428129514, "lm_q2_score": 0.8221891283434877, "lm_q1q2_score": 0.7437938412508435}} {"text": "[STATEMENT]\nlemma list_sum_mean:\n fixes xs::\"real list\"\n shows \"\\:xs = ((mean xs) * (real (length xs)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list xs = mean xs * real (length xs)\n[PROOF STEP]\napply (induct_tac xs)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. sum_list [] = mean [] * real (length [])\n 2. \\a list. sum_list list = mean list * real (length list) \\ sum_list (a # list) = mean (a # list) * real (length (a # list))\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a list. sum_list list = mean list * real (length list) \\ sum_list (a # list) = mean (a # list) * real (length (a # list))\n[PROOF STEP]\napply clarsimp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a list. sum_list list = mean list * real (length list) \\ a + mean list * real (length list) = mean (a # list) * (1 + real (length list))\n[PROOF STEP]\napply (unfold mean_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a list. sum_list list = sum_list list / real (length list) * real (length list) \\ a + sum_list list / real (length list) * real (length list) = sum_list (a # list) / real (length (a # list)) * (1 + real (length list))\n[PROOF STEP]\napply clarsimp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 510, "file": "Cauchy_CauchysMeanTheorem", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505299595162, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7437938365951313}} {"text": "[STATEMENT]\nlemma norm_diff_triangle_less: \"norm (x - z) < e1 + e2\"\n if \"norm (x - y) < e1\" \"norm (y - z) < e2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (x - z) < e1 + e2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm (x - z) < e1 + e2\n[PROOF STEP]\nhave \"norm (x - z) \\ norm (x - y) + norm (y - z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (x - z) \\ norm (x - y) + norm (y - z)\n[PROOF STEP]\nby (metis norm_diff_triangle_ineq add_diff_cancel_left' diff_diff_eq2)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (x - z) \\ norm (x - y) + norm (y - z)\n\ngoal (1 subgoal):\n 1. norm (x - z) < e1 + e2\n[PROOF STEP]\nwith that\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (x - y) < e1\nnorm (y - z) < e2\nnorm (x - z) \\ norm (x - y) + norm (y - z)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (x - y) < e1\nnorm (y - z) < e2\nnorm (x - z) \\ norm (x - y) + norm (y - z)\n\ngoal (1 subgoal):\n 1. norm (x - z) < e1 + e2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nnorm (x - z) < e1 + e2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 540, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.879146761176671, "lm_q2_score": 0.845942439250491, "lm_q1q2_score": 0.7437075556089618}} {"text": "[STATEMENT]\nlemma finite_range_plus: \n assumes \"finite (range f)\"\n \"finite (range g)\"\n shows \"finite (range (\\x. f x + g x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (range (\\x. f x + g x))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. finite (range (\\x. f x + g x))\n[PROOF STEP]\nhave subs: \"range (\\x. (f x, g x)) \\ range f \\ range g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. range (\\x. (f x, g x)) \\ range f \\ range g\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrange (\\x. (f x, g x)) \\ range f \\ range g\n\ngoal (1 subgoal):\n 1. finite (range (\\x. f x + g x))\n[PROOF STEP]\nhave cart: \"finite (range f \\ range g)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (range f \\ range g)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (range f)\nfinite (range g)\n\ngoal (1 subgoal):\n 1. finite (range f \\ range g)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite (range f \\ range g)\n\ngoal (1 subgoal):\n 1. finite (range (\\x. f x + g x))\n[PROOF STEP]\nhave finite: \"finite (range (\\x. (f x, g x)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (range (\\x. (f x, g x)))\n[PROOF STEP]\nusing rev_finite_subset[OF cart subs]\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (range (\\x. (f x, g x)))\n\ngoal (1 subgoal):\n 1. finite (range (\\x. (f x, g x)))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nfinite (range (\\x. (f x, g x)))\n\ngoal (1 subgoal):\n 1. finite (range (\\x. f x + g x))\n[PROOF STEP]\nhave \"range (\\x. f x + g x) = (\\(a,b). a+b) ` range (\\x. (f x, g x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. range (\\x. f x + g x) = (\\(a, b). a + b) ` range (\\x. (f x, g x))\n[PROOF STEP]\nusing range_composition[of \"(\\(a,b). a+b)\" \"(\\x. (f x, g x))\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nrange (\\x. case (f x, g x) of (a, b) \\ a + b) = (\\(a, b). a + b) ` range (\\x. (f x, g x))\n\ngoal (1 subgoal):\n 1. range (\\x. f x + g x) = (\\(a, b). a + b) ` range (\\x. (f x, g x))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrange (\\x. f x + g x) = (\\(a, b). a + b) ` range (\\x. (f x, g x))\n\ngoal (1 subgoal):\n 1. finite (range (\\x. f x + g x))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nrange (\\x. f x + g x) = (\\(a, b). a + b) ` range (\\x. (f x, g x))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nrange (\\x. f x + g x) = (\\(a, b). a + b) ` range (\\x. (f x, g x))\n\ngoal (1 subgoal):\n 1. finite (range (\\x. f x + g x))\n[PROOF STEP]\nusing finite finite_image_set[where f = \"(\\(a,b). a+b)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nrange (\\x. f x + g x) = (\\(a, b). a + b) ` range (\\x. (f x, g x))\nfinite (range (\\x. (f x, g x)))\nfinite {x. ?P x} \\ finite {case x of (a, b) \\ a + b |x. ?P x}\n\ngoal (1 subgoal):\n 1. finite (range (\\x. f x + g x))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite (range (\\x. f x + g x))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1450, "file": "CRYSTALS-Kyber_Abs_Qr", "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467643431002, "lm_q2_score": 0.8459424334245618, "lm_q1q2_score": 0.743707553165732}} {"text": "[STATEMENT]\nlemma card_lucas_lehmer_Units:\n assumes \"m > 1\"\n shows \"card (Units (lucas_lehmer_ring_mod m)) < m ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (Units (lucas_lehmer_ring_mod m)) < m\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (Units (lucas_lehmer_ring_mod m)) < m\\<^sup>2\n[PROOF STEP]\ninterpret cring \"lucas_lehmer_ring_mod m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cring (lucas_lehmer_ring_mod m)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\n\ngoal (1 subgoal):\n 1. cring (lucas_lehmer_ring_mod m)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (Units (lucas_lehmer_ring_mod m)) < m\\<^sup>2\n[PROOF STEP]\nhave \"m ^ 2 > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < m\\<^sup>2\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\n\ngoal (1 subgoal):\n 1. 0 < m\\<^sup>2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < m\\<^sup>2\n\ngoal (1 subgoal):\n 1. card (Units (lucas_lehmer_ring_mod m)) < m\\<^sup>2\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < m\n[PROOF STEP]\nhave \"card (Units (lucas_lehmer_ring_mod m)) \\ card ({.. {.. card ({.. {.. card ({.. {..2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (Units (lucas_lehmer_ring_mod m)) \\ card ({.. {..2\n[PROOF STEP]\nhave \"\\ = m ^ 2 - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({.. {..2 - 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\n\ngoal (1 subgoal):\n 1. card ({.. {..2 - 1\n[PROOF STEP]\nby (subst card_Diff_subset) (auto simp: power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\ncard ({.. {..2 - 1\n\ngoal (1 subgoal):\n 1. card (Units (lucas_lehmer_ring_mod m)) < m\\<^sup>2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (Units (lucas_lehmer_ring_mod m)) \\ m\\<^sup>2 - 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (Units (lucas_lehmer_ring_mod m)) \\ m\\<^sup>2 - 1\n\ngoal (1 subgoal):\n 1. card (Units (lucas_lehmer_ring_mod m)) < m\\<^sup>2\n[PROOF STEP]\nusing \\m ^ 2 > 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (Units (lucas_lehmer_ring_mod m)) \\ m\\<^sup>2 - 1\n0 < m\\<^sup>2\n\ngoal (1 subgoal):\n 1. card (Units (lucas_lehmer_ring_mod m)) < m\\<^sup>2\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\ncard (Units (lucas_lehmer_ring_mod m)) < m\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1539, "file": "Mersenne_Primes_Lucas_Lehmer", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894745194281, "lm_q2_score": 0.8311430436757312, "lm_q1q2_score": 0.7436980473010857}} {"text": "[STATEMENT]\nlemma has_field_derivative_iff:\n \"(f has_field_derivative D) (at x within S) \\\n ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nhave \"((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) \n = ((\\y. (f y - f x) / (y - x) - D) \\ 0) (at x within S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n[PROOF STEP]\nby (smt (verit, best) Lim_cong_within divide_diff_eq_iff norm_divide right_minus_eq tendsto_norm_zero_iff)\n[PROOF STATE]\nproof (state)\nthis:\n((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nby (simp add: has_field_derivative_def has_derivative_iff_norm bounded_linear_mult_right LIM_zero_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1056, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314798554445, "lm_q2_score": 0.8397339616560072, "lm_q1q2_score": 0.7436948311462848}} {"text": "[STATEMENT]\nlemma sum_split_even_odd:\n fixes f :: \"nat \\ real\"\n shows \"(\\i<2 * n. if even i then f i else g i) = (\\iii<2 * n. if even i then f i else g i) = (\\iii<2 * 0. if even i then f i else g i) = (\\i<0. f (2 * i)) + (\\i<0. g (2 * i + 1))\n 2. \\n. (\\i<2 * n. if even i then f i else g i) = (\\ii (\\i<2 * Suc n. if even i then f i else g i) = (\\iii<2 * 0. if even i then f i else g i) = (\\i<0. f (2 * i)) + (\\i<0. g (2 * i + 1))\n 2. \\n. (\\i<2 * n. if even i then f i else g i) = (\\ii (\\i<2 * Suc n. if even i then f i else g i) = (\\iii<2 * 0. if even i then f i else g i) = (\\i<0. f (2 * i)) + (\\i<0. g (2 * i + 1))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\i<2 * 0. if even i then f i else g i) = (\\i<0. f (2 * i)) + (\\i<0. g (2 * i + 1))\n\ngoal (1 subgoal):\n 1. \\n. (\\i<2 * n. if even i then f i else g i) = (\\ii (\\i<2 * Suc n. if even i then f i else g i) = (\\iin. (\\i<2 * n. if even i then f i else g i) = (\\ii (\\i<2 * Suc n. if even i then f i else g i) = (\\iii<2 * n. if even i then f i else g i) = (\\iin. (\\i<2 * n. if even i then f i else g i) = (\\ii (\\i<2 * Suc n. if even i then f i else g i) = (\\iii<2 * Suc n. if even i then f i else g i) =\n (\\iii<2 * Suc n. if even i then f i else g i) = (\\iii<2 * n. if even i then f i else g i) = (\\iii<2 * Suc n. if even i then f i else g i) = (\\iii<2 * n. if even i then f i else g i) = (\\iii<2 * Suc n. if even i then f i else g i) = (\\iii<2 * Suc n. if even i then f i else g i) = (\\iin. (\\i<2 * n. if even i then f i else g i) = (\\ii (\\i<2 * Suc n. if even i then f i else g i) = (\\iii<2 * Suc n. if even i then f i else g i) = (\\iin. (\\i<2 * n. if even i then f i else g i) = (\\ii (\\i<2 * Suc n. if even i then f i else g i) = (\\ii = (\\iiiiiiiiiin. (\\i<2 * n. if even i then f i else g i) = (\\ii (\\i<2 * Suc n. if even i then f i else g i) = (\\iii<2 * Suc n. if even i then f i else g i) = (\\iii<2 * Suc n. if even i then f i else g i) = (\\iii<2 * Suc n. if even i then f i else g i) = (\\iii<2 * Suc n. if even i then f i else g i) = (\\ii\\<^sub>m A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. invertible_mat (k \\\\<^sub>m A)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. invertible_mat (k \\\\<^sub>m A)\n[PROOF STEP]\nobtain n where A: \"A \\ carrier_mat n n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. A \\ carrier_mat n n \\ thesis) \\ thesis\n[PROOF STEP]\nusing inv_A\n[PROOF STATE]\nproof (prove)\nusing this:\ninvertible_mat A\n\ngoal (1 subgoal):\n 1. (\\n. A \\ carrier_mat n n \\ thesis) \\ thesis\n[PROOF STEP]\nunfolding invertible_mat_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsquare_mat A \\ (\\B. inverts_mat A B \\ inverts_mat B A)\n\ngoal (1 subgoal):\n 1. (\\n. A \\ carrier_mat n n \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. invertible_mat (k \\\\<^sub>m A)\n[PROOF STEP]\nhave det_dvd_1: \"Determinant.det A dvd 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Determinant.det A dvd (1::'a)\n[PROOF STEP]\nusing inv_A invertible_iff_is_unit_JNF[OF A]\n[PROOF STATE]\nproof (prove)\nusing this:\ninvertible_mat A\ninvertible_mat A = (Determinant.det A dvd (1::'a))\n\ngoal (1 subgoal):\n 1. Determinant.det A dvd (1::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nDeterminant.det A dvd (1::'a)\n\ngoal (1 subgoal):\n 1. invertible_mat (k \\\\<^sub>m A)\n[PROOF STEP]\nhave \"Determinant.det (k \\\\<^sub>m A) = k ^ dim_col A * Determinant.det A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Determinant.det (k \\\\<^sub>m A) = k ^ dim_col A * Determinant.det A\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nDeterminant.det (k \\\\<^sub>m A) = k ^ dim_col A * Determinant.det A\n\ngoal (1 subgoal):\n 1. invertible_mat (k \\\\<^sub>m A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nDeterminant.det (k \\\\<^sub>m A) = k ^ dim_col A * Determinant.det A\n\ngoal (1 subgoal):\n 1. invertible_mat (k \\\\<^sub>m A)\n[PROOF STEP]\nhave \"... dvd 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k ^ dim_col A * Determinant.det A dvd (1::'a)\n[PROOF STEP]\nby (rule unit_prod, insert k det_dvd_1 dvd_power_same, force+)\n[PROOF STATE]\nproof (state)\nthis:\nk ^ dim_col A * Determinant.det A dvd (1::'a)\n\ngoal (1 subgoal):\n 1. invertible_mat (k \\\\<^sub>m A)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nDeterminant.det (k \\\\<^sub>m A) dvd (1::'a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nDeterminant.det (k \\\\<^sub>m A) dvd (1::'a)\n\ngoal (1 subgoal):\n 1. invertible_mat (k \\\\<^sub>m A)\n[PROOF STEP]\nusing invertible_iff_is_unit_JNF\n[PROOF STATE]\nproof (prove)\nusing this:\nDeterminant.det (k \\\\<^sub>m A) dvd (1::'a)\n?A \\ carrier_mat ?n ?n \\ invertible_mat ?A = (Determinant.det ?A dvd (1::?'a))\n\ngoal (1 subgoal):\n 1. invertible_mat (k \\\\<^sub>m A)\n[PROOF STEP]\nby (metis A smult_carrier_mat)\n[PROOF STATE]\nproof (state)\nthis:\ninvertible_mat (k \\\\<^sub>m A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1481, "file": "Smith_Normal_Form_SNF_Missing_Lemmas", "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206870747658, "lm_q2_score": 0.8244619285331332, "lm_q1q2_score": 0.74359926904959}} {"text": "[STATEMENT]\nlemma quasinorm_intersection_zero_space:\n \"zero_space\\<^sub>N (N1 \\\\<^sub>N N2) = zero_space\\<^sub>N N1 \\ zero_space\\<^sub>N N2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. zero_space\\<^sub>N (N1 \\\\<^sub>N N2) = zero_space\\<^sub>N N1 \\ zero_space\\<^sub>N N2\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x. x \\ zero_space\\<^sub>N (N1 \\\\<^sub>N N2) \\ x \\ zero_space\\<^sub>N N1\n 2. \\x. x \\ zero_space\\<^sub>N (N1 \\\\<^sub>N N2) \\ x \\ zero_space\\<^sub>N N2\n 3. \\x. \\x \\ zero_space\\<^sub>N N1; x \\ zero_space\\<^sub>N N2\\ \\ x \\ zero_space\\<^sub>N (N1 \\\\<^sub>N N2)\n[PROOF STEP]\nunfolding quasinorm_intersection(1) zero_spaceN_iff\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x. eNorm N1 x + eNorm N2 x = 0 \\ eNorm N1 x = 0\n 2. \\x. eNorm N1 x + eNorm N2 x = 0 \\ eNorm N2 x = 0\n 3. \\x. \\eNorm N1 x = 0; eNorm N2 x = 0\\ \\ eNorm N1 x + eNorm N2 x = 0\n[PROOF STEP]\nby (auto simp add: add_eq_0_iff_both_eq_0)", "meta": {"llama_tokens": 537, "file": "Lp_Functional_Spaces", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.899121388082479, "lm_q2_score": 0.8267117855317474, "lm_q1q2_score": 0.7433142481514494}} {"text": "[STATEMENT]\nlemma norm_cos_squared:\n \"norm(cos z) ^ 2 = cos(Re z) ^ 2 + (exp(Im z) - inverse(exp(Im z))) ^ 2 / 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cmod (cos z))\\<^sup>2 = (cos (Re z))\\<^sup>2 + (exp (Im z) - inverse (exp (Im z)))\\<^sup>2 / 4\n[PROOF STEP]\nproof (cases z)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x1 x2. z = Complex x1 x2 \\ (cmod (cos z))\\<^sup>2 = (cos (Re z))\\<^sup>2 + (exp (Im z) - inverse (exp (Im z)))\\<^sup>2 / 4\n[PROOF STEP]\ncase (Complex x1 x2)\n[PROOF STATE]\nproof (state)\nthis:\nz = Complex x1 x2\n\ngoal (1 subgoal):\n 1. \\x1 x2. z = Complex x1 x2 \\ (cmod (cos z))\\<^sup>2 = (cos (Re z))\\<^sup>2 + (exp (Im z) - inverse (exp (Im z)))\\<^sup>2 / 4\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nz = Complex x1 x2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nz = Complex x1 x2\n\ngoal (1 subgoal):\n 1. (cmod (cos z))\\<^sup>2 = (cos (Re z))\\<^sup>2 + (exp (Im z) - inverse (exp (Im z)))\\<^sup>2 / 4\n[PROOF STEP]\napply (simp only: cos_add cmod_power2 cos_of_real sin_of_real Complex_eq)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z = complex_of_real x1 + \\ * complex_of_real x2 \\ (Re (complex_of_real (cos x1) * cos (\\ * complex_of_real x2) - complex_of_real (sin x1) * sin (\\ * complex_of_real x2)))\\<^sup>2 + (Im (complex_of_real (cos x1) * cos (\\ * complex_of_real x2) - complex_of_real (sin x1) * sin (\\ * complex_of_real x2)))\\<^sup>2 = (cos (Re (complex_of_real x1 + \\ * complex_of_real x2)))\\<^sup>2 + (exp (Im (complex_of_real x1 + \\ * complex_of_real x2)) - inverse (exp (Im (complex_of_real x1 + \\ * complex_of_real x2))))\\<^sup>2 / 4\n[PROOF STEP]\napply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide power_divide)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z = complex_of_real x1 + \\ * complex_of_real x2 \\ (cos x1 * (exp x2 / (exp x2)\\<^sup>2 + exp x2))\\<^sup>2 / 4 + (sin x1 * (exp x2 / (exp x2)\\<^sup>2 - exp x2))\\<^sup>2 / 4 = (cos x1)\\<^sup>2 + (exp x2 - inverse (exp x2))\\<^sup>2 / 4\n[PROOF STEP]\napply (simp only: left_diff_distrib [symmetric] power_mult_distrib sin_squared_eq)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z = complex_of_real x1 + \\ * complex_of_real x2 \\ (cos x1)\\<^sup>2 * (exp x2 / (exp x2)\\<^sup>2 + exp x2)\\<^sup>2 / 4 + (1 - (cos x1)\\<^sup>2) * (exp x2 / (exp x2)\\<^sup>2 - exp x2)\\<^sup>2 / 4 = (cos x1)\\<^sup>2 + (exp x2 - inverse (exp x2))\\<^sup>2 / 4\n[PROOF STEP]\napply (simp add: power2_eq_square field_split_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (cos z))\\<^sup>2 = (cos (Re z))\\<^sup>2 + (exp (Im z) - inverse (exp (Im z)))\\<^sup>2 / 4\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1294, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045877523148, "lm_q2_score": 0.8376199673867852, "lm_q1q2_score": 0.7431402778585001}} {"text": "[STATEMENT]\nlemma adjoint_add:\n fixes A B :: \"'a::conjugatable_field mat\"\n assumes \"A \\ carrier_mat n m\" \"B \\ carrier_mat n m\"\n shows \"adjoint (A + B) = adjoint A + adjoint B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. adjoint (A + B) = adjoint A + adjoint B\n[PROOF STEP]\napply (rule eq_matI)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (adjoint A + adjoint B); j < dim_col (adjoint A + adjoint B)\\ \\ adjoint (A + B) $$ (i, j) = (adjoint A + adjoint B) $$ (i, j)\n 2. dim_row (adjoint (A + B)) = dim_row (adjoint A + adjoint B)\n 3. dim_col (adjoint (A + B)) = dim_col (adjoint A + adjoint B)\n[PROOF STEP]\nusing assms conjugatable_ring_class.conjugate_dist_add\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n m\nB \\ carrier_mat n m\nconjugate (?a + ?b) = conjugate ?a + conjugate ?b\n\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (adjoint A + adjoint B); j < dim_col (adjoint A + adjoint B)\\ \\ adjoint (A + B) $$ (i, j) = (adjoint A + adjoint B) $$ (i, j)\n 2. dim_row (adjoint (A + B)) = dim_row (adjoint A + adjoint B)\n 3. dim_col (adjoint (A + B)) = dim_col (adjoint A + adjoint B)\n[PROOF STEP]\nby( auto simp add: adjoint_eval)", "meta": {"llama_tokens": 540, "file": "QHLProver_Complex_Matrix", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045907347107, "lm_q2_score": 0.8376199633332891, "lm_q1q2_score": 0.7431402767603341}} {"text": "[STATEMENT]\nlemma Inf_closed_res_Fix: \n fixes X :: \"'a::quantale set\"\n assumes \"Inf_closed_set X\"\n and \"\\x. \\y \\ X. x \\ y \\ X\"\n and \"\\x. \\y \\ X. y \\ x \\ X\"\nshows \"X = Fix (\\y. \\{x \\ X. y \\ x})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. X = Fix (\\y. \\ {x \\ X. y \\ x})\n[PROOF STEP]\nunfolding Fix_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. X = {x. \\ {xa \\ X. x \\ xa} = x}\n[PROOF STEP]\napply safe\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. x \\ X \\ \\ {xa \\ X. x \\ xa} = x\n 2. \\x. \\ {xa \\ X. x \\ xa} = x \\ x \\ X\n[PROOF STEP]\napply (rule antisym, simp_all add: Inf_lower le_Inf_iff)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\ {xa \\ X. x \\ xa} = x \\ x \\ X\n[PROOF STEP]\nby (metis (no_types, lifting) Inf_closed_set_def assms(1) mem_Collect_eq subsetI)", "meta": {"llama_tokens": 478, "file": "Quantales_Quantic_Nuclei_Conuclei", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.919642531177793, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.743132984009578}} {"text": "[STATEMENT]\nlemma point_dist [simp]:\n \"dist (Point xA yA) (Point xB yB) = sqrt ((xA - xB)^2 + (yA - yB)^2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist (Point xA yA) (Point xB yB) = sqrt ((xA - xB)\\<^sup>2 + (yA - yB)\\<^sup>2)\n[PROOF STEP]\nunfolding point_dist_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt ((abscissa (Point xA yA - Point xB yB))\\<^sup>2 + (ordinate (Point xA yA - Point xB yB))\\<^sup>2) = sqrt ((xA - xB)\\<^sup>2 + (yA - yB)\\<^sup>2)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 245, "file": "Impossible_Geometry_Impossible_Geometry", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9196425311777929, "lm_q2_score": 0.8080672066194946, "lm_q1q2_score": 0.7431329712573206}} {"text": "[STATEMENT]\nlemma is_solution_eq_in_span_solve:\n fixes A::\"'a::{field}^'cols::{mod_type}^'rows::{mod_type}\"\n assumes con: \"consistent A b\"\n shows \"(is_solution x A b) = (x \\ {fst (the (solve A b))} + vec.span (snd (the (solve A b))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_solution x A b = (x \\ {fst (the (solve A b))} + vec.span (snd (the (solve A b))))\n[PROOF STEP]\nusing solution_set_rel_solve[OF con]\n[PROOF STATE]\nproof (prove)\nusing this:\nsolution_set A b = {fst (the (solve A b))} + vec.span (snd (the (solve A b)))\n\ngoal (1 subgoal):\n 1. is_solution x A b = (x \\ {fst (the (solve A b))} + vec.span (snd (the (solve A b))))\n[PROOF STEP]\nunfolding solution_set_def\n[PROOF STATE]\nproof (prove)\nusing this:\n{x. is_solution x A b} = {fst (the (solve A b))} + vec.span (snd (the (solve A b)))\n\ngoal (1 subgoal):\n 1. is_solution x A b = (x \\ {fst (the (solve A b))} + vec.span (snd (the (solve A b))))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 432, "file": "Gauss_Jordan_System_Of_Equations", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.92522995296862, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7431204560084509}} {"text": "[STATEMENT]\ntheorem (in prime_number_theorem)\n liminf_totient: \"liminf (\\n. totient n * ln (ln n) / n) = third_mertens_const\"\n (is \"_ = ereal ?c\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) = ereal third_mertens_const\n[PROOF STEP]\nproof (intro antisym)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n 2. ereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n[PROOF STEP]\nhave \"(\\k. totient (primorial' k) / (?c * primorial' k / ln (ln (primorial' k)))) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. real (totient (primorial' k)) / (third_mertens_const * real (primorial' k) / ln (ln (real (primorial' k))))) \\ 1\n[PROOF STEP]\nusing totient_primorial'_asymp_equiv'\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. real (totient (primorial' x))) \\[sequentially] (\\k. third_mertens_const * real (primorial' k) / ln (ln (real (primorial' k))))\n\ngoal (1 subgoal):\n 1. (\\k. real (totient (primorial' k)) / (third_mertens_const * real (primorial' k) / ln (ln (real (primorial' k))))) \\ 1\n[PROOF STEP]\nby (intro asymp_equivD_strong eventually_mono[OF eventually_gt_at_top[of 1]]) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. real (totient (primorial' k)) / (third_mertens_const * real (primorial' k) / ln (ln (real (primorial' k))))) \\ 1\n\ngoal (2 subgoals):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n 2. ereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n[PROOF STEP]\nhence \"(\\k. totient (primorial' k) / (?c * primorial' k / ln (ln (primorial' k))) * ?c)\n \\ 1 * ?c\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. real (totient (primorial' k)) / (third_mertens_const * real (primorial' k) / ln (ln (real (primorial' k))))) \\ 1\n\ngoal (1 subgoal):\n 1. (\\k. real (totient (primorial' k)) / (third_mertens_const * real (primorial' k) / ln (ln (real (primorial' k)))) * third_mertens_const) \\ 1 * third_mertens_const\n[PROOF STEP]\nby (intro tendsto_mult) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. real (totient (primorial' k)) / (third_mertens_const * real (primorial' k) / ln (ln (real (primorial' k)))) * third_mertens_const) \\ 1 * third_mertens_const\n\ngoal (2 subgoals):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n 2. ereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n[PROOF STEP]\nhence \"(\\k. totient (primorial' k) / (primorial' k / ln (ln (primorial' k)))) \\ ?c\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. real (totient (primorial' k)) / (third_mertens_const * real (primorial' k) / ln (ln (real (primorial' k)))) * third_mertens_const) \\ 1 * third_mertens_const\n\ngoal (1 subgoal):\n 1. (\\k. real (totient (primorial' k)) / (real (primorial' k) / ln (ln (real (primorial' k))))) \\ third_mertens_const\n[PROOF STEP]\nusing third_mertens_const_pos\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. real (totient (primorial' k)) / (third_mertens_const * real (primorial' k) / ln (ln (real (primorial' k)))) * third_mertens_const) \\ 1 * third_mertens_const\n0 < third_mertens_const\n\ngoal (1 subgoal):\n 1. (\\k. real (totient (primorial' k)) / (real (primorial' k) / ln (ln (real (primorial' k))))) \\ third_mertens_const\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. real (totient (primorial' k)) / (real (primorial' k) / ln (ln (real (primorial' k))))) \\ third_mertens_const\n\ngoal (2 subgoals):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n 2. ereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n[PROOF STEP]\nhence \"liminf ((\\n. totient n * ln (ln n) / n) \\ primorial') = ereal ?c\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. real (totient (primorial' k)) / (real (primorial' k) / ln (ln (real (primorial' k))))) \\ third_mertens_const\n\ngoal (1 subgoal):\n 1. liminf (\\x. ereal (((\\n. real (totient n) * ln (ln (real n)) / real n) \\ primorial') x)) = ereal third_mertens_const\n[PROOF STEP]\nby (intro lim_imp_Liminf tendsto_ereal) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nliminf (\\x. ereal (((\\n. real (totient n) * ln (ln (real n)) / real n) \\ primorial') x)) = ereal third_mertens_const\n\ngoal (2 subgoals):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n 2. ereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n[PROOF STEP]\nhence \"?c = liminf ((\\n. ereal (totient n * ln (ln n) / n)) \\ primorial')\"\n[PROOF STATE]\nproof (prove)\nusing this:\nliminf (\\x. ereal (((\\n. real (totient n) * ln (ln (real n)) / real n) \\ primorial') x)) = ereal third_mertens_const\n\ngoal (1 subgoal):\n 1. ereal third_mertens_const = liminf ((\\n. ereal (real (totient n) * ln (ln (real n)) / real n)) \\ primorial')\n[PROOF STEP]\nby (simp add: o_def)\n[PROOF STATE]\nproof (state)\nthis:\nereal third_mertens_const = liminf ((\\n. ereal (real (totient n) * ln (ln (real n)) / real n)) \\ primorial')\n\ngoal (2 subgoals):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n 2. ereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nereal third_mertens_const = liminf ((\\n. ereal (real (totient n) * ln (ln (real n)) / real n)) \\ primorial')\n\ngoal (2 subgoals):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n 2. ereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n[PROOF STEP]\nhave \"\\ \\ liminf (\\n. totient n * ln (ln n) / n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ liminf ((\\n. ereal (real (totient n) * ln (ln (real n)) / real n)) \\ primorial')\n[PROOF STEP]\nusing strict_mono_primorial'\n[PROOF STATE]\nproof (prove)\nusing this:\nstrict_mono primorial'\n\ngoal (1 subgoal):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ liminf ((\\n. ereal (real (totient n) * ln (ln (real n)) / real n)) \\ primorial')\n[PROOF STEP]\nby (rule liminf_subseq_mono)\n[PROOF STATE]\nproof (state)\nthis:\nliminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ liminf ((\\n. ereal (real (totient n) * ln (ln (real n)) / real n)) \\ primorial')\n\ngoal (2 subgoals):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n 2. ereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nliminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n[PROOF STEP]\nshow \"liminf (\\n. totient n * ln (ln n) / n) \\ ?c\"\n[PROOF STATE]\nproof (prove)\nusing this:\nliminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n\ngoal (1 subgoal):\n 1. liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nliminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x)) \\ ereal third_mertens_const\n\ngoal (1 subgoal):\n 1. ereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n[PROOF STEP]\nshow \"liminf (\\n. totient n * ln (ln n) / n) \\ ?c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n[PROOF STEP]\nunfolding le_Liminf_iff\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\y\\<^sub>F x in sequentially. y < ereal (real (totient x) * ln (ln (real x)) / real x)\n[PROOF STEP]\nproof safe\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\y. y < ereal third_mertens_const \\ \\\\<^sub>F x in sequentially. y < ereal (real (totient x) * ln (ln (real x)) / real x)\n[PROOF STEP]\nfix C'\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\y. y < ereal third_mertens_const \\ \\\\<^sub>F x in sequentially. y < ereal (real (totient x) * ln (ln (real x)) / real x)\n[PROOF STEP]\nassume \"C' < ereal ?c\"\n[PROOF STATE]\nproof (state)\nthis:\nC' < ereal third_mertens_const\n\ngoal (1 subgoal):\n 1. \\y. y < ereal third_mertens_const \\ \\\\<^sub>F x in sequentially. y < ereal (real (totient x) * ln (ln (real x)) / real x)\n[PROOF STEP]\nfrom ereal_dense2[OF this]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\z. C' < ereal z \\ ereal z < ereal third_mertens_const\n[PROOF STEP]\nobtain C where C: \"C < ?c\" \"ereal C > C'\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\z. C' < ereal z \\ ereal z < ereal third_mertens_const\n\ngoal (1 subgoal):\n 1. (\\C. \\C < third_mertens_const; C' < ereal C\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nC < third_mertens_const\nC' < ereal C\n\ngoal (1 subgoal):\n 1. \\y. y < ereal third_mertens_const \\ \\\\<^sub>F x in sequentially. y < ereal (real (totient x) * ln (ln (real x)) / real x)\n[PROOF STEP]\ndefine \\ where \"\\ = 1 - C / ?c\"\n[PROOF STATE]\nproof (state)\nthis:\n\\ = 1 - C / third_mertens_const\n\ngoal (1 subgoal):\n 1. \\y. y < ereal third_mertens_const \\ \\\\<^sub>F x in sequentially. y < ereal (real (totient x) * ln (ln (real x)) / real x)\n[PROOF STEP]\nfrom C\n[PROOF STATE]\nproof (chain)\npicking this:\nC < third_mertens_const\nC' < ereal C\n[PROOF STEP]\nhave \"\\ > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nC < third_mertens_const\nC' < ereal C\n\ngoal (1 subgoal):\n 1. 0 < \\\n[PROOF STEP]\nusing third_mertens_const_pos\n[PROOF STATE]\nproof (prove)\nusing this:\nC < third_mertens_const\nC' < ereal C\n0 < third_mertens_const\n\ngoal (1 subgoal):\n 1. 0 < \\\n[PROOF STEP]\nby (simp add: \\_def)\n[PROOF STATE]\nproof (state)\nthis:\n0 < \\\n\ngoal (1 subgoal):\n 1. \\y. y < ereal third_mertens_const \\ \\\\<^sub>F x in sequentially. y < ereal (real (totient x) * ln (ln (real x)) / real x)\n[PROOF STEP]\nhave \"eventually (\\n::nat. ln (ln n) > 0) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. 0 < ln (ln (real n))\n[PROOF STEP]\nby real_asymp\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F n in sequentially. 0 < ln (ln (real n))\n\ngoal (1 subgoal):\n 1. \\y. y < ereal third_mertens_const \\ \\\\<^sub>F x in sequentially. y < ereal (real (totient x) * ln (ln (real x)) / real x)\n[PROOF STEP]\nhence \"eventually (\\n. totient n * ln (ln n) / n > C) at_top\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F n in sequentially. 0 < ln (ln (real n))\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. C < real (totient n) * ln (ln (real n)) / real n\n[PROOF STEP]\nusing totient_lower_bound[OF \\\\ > 0\\] eventually_gt_at_top[of 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F n in sequentially. 0 < ln (ln (real n))\n\\\\<^sub>F n in sequentially. (1 - \\) * third_mertens_const * real n / ln (ln (real n)) < real (totient n)\neventually ((<) (1::?'a1)) at_top\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. C < real (totient n) * ln (ln (real n)) / real n\n[PROOF STEP]\nproof eventually_elim\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\0 < ln (ln (real n)); (1 - \\) * third_mertens_const * real n / ln (ln (real n)) < real (totient n); 1 < n\\ \\ C < real (totient n) * ln (ln (real n)) / real n\n[PROOF STEP]\ncase (elim n)\n[PROOF STATE]\nproof (state)\nthis:\n0 < ln (ln (real n))\n(1 - \\) * third_mertens_const * real n / ln (ln (real n)) < real (totient n)\n1 < n\n\ngoal (1 subgoal):\n 1. \\n. \\0 < ln (ln (real n)); (1 - \\) * third_mertens_const * real n / ln (ln (real n)) < real (totient n); 1 < n\\ \\ C < real (totient n) * ln (ln (real n)) / real n\n[PROOF STEP]\nhence \"totient n * ln (ln n) / n > (1 - \\) * ?c\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < ln (ln (real n))\n(1 - \\) * third_mertens_const * real n / ln (ln (real n)) < real (totient n)\n1 < n\n\ngoal (1 subgoal):\n 1. (1 - \\) * third_mertens_const < real (totient n) * ln (ln (real n)) / real n\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(1 - \\) * third_mertens_const < real (totient n) * ln (ln (real n)) / real n\n\ngoal (1 subgoal):\n 1. \\n. \\0 < ln (ln (real n)); (1 - \\) * third_mertens_const * real n / ln (ln (real n)) < real (totient n); 1 < n\\ \\ C < real (totient n) * ln (ln (real n)) / real n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(1 - \\) * third_mertens_const < real (totient n) * ln (ln (real n)) / real n\n\ngoal (1 subgoal):\n 1. \\n. \\0 < ln (ln (real n)); (1 - \\) * third_mertens_const * real n / ln (ln (real n)) < real (totient n); 1 < n\\ \\ C < real (totient n) * ln (ln (real n)) / real n\n[PROOF STEP]\nhave \"(1 - \\) * ?c = C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1 - \\) * third_mertens_const = C\n[PROOF STEP]\nusing third_mertens_const_pos\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < third_mertens_const\n\ngoal (1 subgoal):\n 1. (1 - \\) * third_mertens_const = C\n[PROOF STEP]\nby (simp add: field_simps \\_def)\n[PROOF STATE]\nproof (state)\nthis:\n(1 - \\) * third_mertens_const = C\n\ngoal (1 subgoal):\n 1. \\n. \\0 < ln (ln (real n)); (1 - \\) * third_mertens_const * real n / ln (ln (real n)) < real (totient n); 1 < n\\ \\ C < real (totient n) * ln (ln (real n)) / real n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nC < real (totient n) * ln (ln (real n)) / real n\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nC < real (totient n) * ln (ln (real n)) / real n\n\ngoal (1 subgoal):\n 1. C < real (totient n) * ln (ln (real n)) / real n\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nC < real (totient n) * ln (ln (real n)) / real n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F n in sequentially. C < real (totient n) * ln (ln (real n)) / real n\n\ngoal (1 subgoal):\n 1. \\y. y < ereal third_mertens_const \\ \\\\<^sub>F x in sequentially. y < ereal (real (totient x) * ln (ln (real x)) / real x)\n[PROOF STEP]\nthus \"eventually (\\n. ereal (totient n * ln (ln n) / n) > C') at_top\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F n in sequentially. C < real (totient n) * ln (ln (real n)) / real n\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. C' < ereal (real (totient n) * ln (ln (real n)) / real n)\n[PROOF STEP]\nby eventually_elim (rule less_trans[OF C(2)], auto)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F n in sequentially. C' < ereal (real (totient n) * ln (ln (real n)) / real n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nereal third_mertens_const \\ liminf (\\x. ereal (real (totient x) * ln (ln (real x)) / real x))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 6902, "file": "Prime_Distribution_Elementary_PNT_Consequences", "length": 54, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513786759491, "lm_q2_score": 0.82893881677331, "lm_q1q2_score": 0.742937557371089}} {"text": "[STATEMENT]\nlemma continuous_on_closure_norm_le:\n fixes f :: \"'a::metric_space \\ 'b::real_normed_vector\"\n assumes \"continuous_on (closure s) f\"\n and \"\\y \\ s. norm(f y) \\ b\"\n and \"x \\ (closure s)\"\n shows \"norm (f x) \\ b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (f x) \\ b\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm (f x) \\ b\n[PROOF STEP]\nhave *: \"f ` s \\ cball 0 b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f ` s \\ cball (0::'b) b\n[PROOF STEP]\nusing assms(2)[unfolded mem_cball_0[symmetric]]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\y\\s. f y \\ cball (0::'b) b\n\ngoal (1 subgoal):\n 1. f ` s \\ cball (0::'b) b\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nf ` s \\ cball (0::'b) b\n\ngoal (1 subgoal):\n 1. norm (f x) \\ b\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (f x) \\ b\n[PROOF STEP]\nby (meson \"*\" assms(1) assms(3) closed_cball image_closure_subset image_subset_iff mem_cball_0)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (f x) \\ b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 538, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513814471134, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.7429375520931927}} {"text": "[STATEMENT]\nlemma std_normal_distribution_odd_moments_abs:\n fixes k :: nat\n shows \"(LINT x|std_normal_distribution. \\x\\^(2 * k + 1)) = sqrt (2 / pi) * 2 ^ k * fact k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LINT x|std_normal_distribution. \\x\\ ^ (2 * k + 1) = sqrt (2 / pi) * 2 ^ k * fact k\n[PROOF STEP]\nusing integral_std_normal_moment_abs_odd[of k]\n[PROOF STATE]\nproof (prove)\nusing this:\nLBINT x. std_normal_density x * \\x\\ ^ (2 * k + 1) = sqrt (2 / pi) * 2 ^ k * fact k\n\ngoal (1 subgoal):\n 1. LINT x|std_normal_distribution. \\x\\ ^ (2 * k + 1) = sqrt (2 / pi) * 2 ^ k * fact k\n[PROOF STEP]\nby (subst integral_density) (auto simp: normal_density_nonneg)", "meta": {"llama_tokens": 311, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9086178919837705, "lm_q2_score": 0.817574478416099, "lm_q1q2_score": 0.7428627991181666}} {"text": "[STATEMENT]\nlemma sum_of_index_diff [simp]:\n fixes f:: \"nat \\ 'a::comm_monoid_add\"\n shows \"(\\i\\{a..i\\{..i = a..i = a..b. (\\i = a.. (\\i = a..i = a..b. (\\i = a.. (\\i = a..i = a..i = a..b. (\\i = a.. (\\i = a..b. (\\i = a.. (\\i = a..i = a..b. (\\i = a.. (\\i = a..i = a..i = a..i = a..i = a.. carrier_vec d\" \n shows \"mat d 1 (\\(i, j). a $ i) *\\<^sub>v vec 1 (\\k. c) = c \\\\<^sub>v a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat d 1 (\\(i, j). a $ i) *\\<^sub>v vec 1 (\\k. c) = c \\\\<^sub>v a\n[PROOF STEP]\napply (rule eq_vecI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\i. i < dim_vec (c \\\\<^sub>v a) \\ (mat d 1 (\\(i, j). a $ i) *\\<^sub>v vec 1 (\\k. c)) $ i = (c \\\\<^sub>v a) $ i\n 2. dim_vec (mat d 1 (\\(i, j). a $ i) *\\<^sub>v vec 1 (\\k. c)) = dim_vec (c \\\\<^sub>v a)\n[PROOF STEP]\nby (auto simp add: scalar_prod_def carrier_vecD[OF a])", "meta": {"llama_tokens": 353, "file": "QHLProver_Complex_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.905989822921759, "lm_q2_score": 0.8198933271118221, "lm_q1q2_score": 0.7428150102447715}} {"text": "[STATEMENT]\nlemma card_fPow: \"card (Pow (fset A)) = 2 ^ card (fset A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (Pow (fset A)) = 2 ^ card (fset A)\n[PROOF STEP]\nusing card_Pow[of \"fset A\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (fset A) \\ card (Pow (fset A)) = 2 ^ card (fset A)\n\ngoal (1 subgoal):\n 1. card (Pow (fset A)) = 2 ^ card (fset A)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 184, "file": "FSM_Tests_Test_Suite_Generator_Code_Export", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9219218305645894, "lm_q2_score": 0.8056321936479701, "lm_q1q2_score": 0.7427299067297024}} {"text": "[STATEMENT]\nlemma has_derivative_at':\n \"(f has_derivative f') (at x) \n \\ bounded_linear f' \\\n (\\e>0. \\d>0. \\x'. 0 < norm (x' - x) \\ norm (x' - x) < d \\\n norm (f x' - f x - f'(x' - x)) / norm (x' - x) < e)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f has_derivative f') (at x) = (bounded_linear f' \\ (\\e>0. \\d>0. \\x'. 0 < norm (x' - x) \\ norm (x' - x) < d \\ norm (f x' - f x - f' (x' - x)) / norm (x' - x) < e))\n[PROOF STEP]\nusing has_derivative_within' [of f f' x UNIV]\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_derivative f') (at x) = (bounded_linear f' \\ (\\e>0. \\d>0. \\x'\\UNIV. 0 < norm (x' - x) \\ norm (x' - x) < d \\ norm (f x' - f x - f' (x' - x)) / norm (x' - x) < e))\n\ngoal (1 subgoal):\n 1. (f has_derivative f') (at x) = (bounded_linear f' \\ (\\e>0. \\d>0. \\x'. 0 < norm (x' - x) \\ norm (x' - x) < d \\ norm (f x' - f x - f' (x' - x)) / norm (x' - x) < e))\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 504, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9136765328159726, "lm_q2_score": 0.8128673269042767, "lm_q1q2_score": 0.7426978008852873}} {"text": "[STATEMENT]\nlemma powr_less_mono':\n assumes \"(x::real) > 0\" \"x < 1\" \"a < b\"\n shows \"x powr b < x powr a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x powr b < x powr a\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x powr b < x powr a\n[PROOF STEP]\nhave \"inverse x powr a < inverse x powr b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse x powr a < inverse x powr b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\nx < 1\na < b\n\ngoal (1 subgoal):\n 1. inverse x powr a < inverse x powr b\n[PROOF STEP]\nby (intro powr_less_mono) (simp_all add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\ninverse x powr a < inverse x powr b\n\ngoal (1 subgoal):\n 1. x powr b < x powr a\n[PROOF STEP]\nhence \"inverse (x powr a) < inverse (x powr b)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse x powr a < inverse x powr b\n\ngoal (1 subgoal):\n 1. inverse (x powr a) < inverse (x powr b)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse x powr a < inverse x powr b\n0 < x\nx < 1\na < b\n\ngoal (1 subgoal):\n 1. inverse (x powr a) < inverse (x powr b)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ninverse (x powr a) < inverse (x powr b)\n\ngoal (1 subgoal):\n 1. x powr b < x powr a\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < x\nx < 1\na < b\ninverse (x powr a) < inverse (x powr b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\nx < 1\na < b\ninverse (x powr a) < inverse (x powr b)\n\ngoal (1 subgoal):\n 1. x powr b < x powr a\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nx powr b < x powr a\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 767, "file": "Landau_Symbols_Landau_Library", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757870046160258, "lm_q2_score": 0.8479677660619633, "lm_q1q2_score": 0.7426391498503497}} {"text": "[STATEMENT]\nlemma primorial_ge: \"primorial x \\ 2 powr primes_pi x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 powr primes_pi x \\ real (primorial x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 powr primes_pi x \\ real (primorial x)\n[PROOF STEP]\nhave \"2 powr primes_pi x = real (\\p | prime p \\ real p \\ x. 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 powr primes_pi x = real (\\p\\{p. prime p \\ real p \\ x}. 2)\n[PROOF STEP]\nby (simp add: primes_pi_def prime_sum_upto_def powr_realpow)\n[PROOF STATE]\nproof (state)\nthis:\n2 powr primes_pi x = real (\\p\\{p. prime p \\ real p \\ x}. 2)\n\ngoal (1 subgoal):\n 1. 2 powr primes_pi x \\ real (primorial x)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 powr primes_pi x = real (\\p\\{p. prime p \\ real p \\ x}. 2)\n\ngoal (1 subgoal):\n 1. 2 powr primes_pi x \\ real (primorial x)\n[PROOF STEP]\nhave \"(\\p | prime p \\ real p \\ x. 2) \\ (\\p | prime p \\ real p \\ x. p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\p\\{p. prime p \\ real p \\ x}. 2) \\ \\{p. prime p \\ real p \\ x}\n[PROOF STEP]\nby (intro prod_mono) (auto dest: prime_gt_1_nat)\n[PROOF STATE]\nproof (state)\nthis:\n(\\p\\{p. prime p \\ real p \\ x}. 2) \\ \\{p. prime p \\ real p \\ x}\n\ngoal (1 subgoal):\n 1. 2 powr primes_pi x \\ real (primorial x)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\p\\{p. prime p \\ real p \\ x}. 2) \\ \\{p. prime p \\ real p \\ x}\n\ngoal (1 subgoal):\n 1. 2 powr primes_pi x \\ real (primorial x)\n[PROOF STEP]\nhave \"\\ = primorial x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\{p. prime p \\ real p \\ x} = primorial x\n[PROOF STEP]\nby (simp add: primorial_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\{p. prime p \\ real p \\ x} = primorial x\n\ngoal (1 subgoal):\n 1. 2 powr primes_pi x \\ real (primorial x)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x y. x \\ y \\ real x \\ real y) \\ 2 powr primes_pi x \\ real (primorial x)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x y. x \\ y \\ real x \\ real y) \\ 2 powr primes_pi x \\ real (primorial x)\n\ngoal (1 subgoal):\n 1. 2 powr primes_pi x \\ real (primorial x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 powr primes_pi x \\ real (primorial x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1174, "file": "Prime_Distribution_Elementary_Primorial", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869851639065, "lm_q2_score": 0.847967764140929, "lm_q1q2_score": 0.7426391316731628}} {"text": "[STATEMENT]\nlemma gbinomial_factors: \"((a + 1) gchoose (Suc k)) = (a + 1) / of_nat (Suc k) * (a gchoose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nproof (cases k)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. k = 0 \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n 2. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nk = 0\n\ngoal (2 subgoals):\n 1. k = 0 \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n 2. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nk = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nk = 0\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\ncase (Suc b)\n[PROOF STATE]\nproof (state)\nthis:\nk = Suc b\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nk = Suc b\n[PROOF STEP]\nhave \"((a + 1) gchoose (Suc (Suc b))) = (\\i = 0 .. Suc b. a + (1 - of_nat i)) / fact (b + 2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nk = Suc b\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n[PROOF STEP]\nby (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nhave \"(\\i = 0 .. Suc b. a + (1 - of_nat i)) = (a + 1) * (\\i = 0..b. a - of_nat i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n[PROOF STEP]\nby (simp add: prod.atLeast0_atMost_Suc_shift del: prod.cl_ivl_Suc)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nhave \"\\ / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a + (1::'a)) * (\\i = 0..b. a - of_nat i) / fact (b + 2) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n[PROOF STEP]\nby (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)\n[PROOF STATE]\nproof (state)\nthis:\n(a + (1::'a)) * (\\i = 0..b. a - of_nat i) / fact (b + 2) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\na + (1::'a) gchoose Suc (Suc b) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na + (1::'a) gchoose Suc (Suc b) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nby (simp add: Suc)\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2429, "file": null, "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.877476800298183, "lm_q2_score": 0.8459424411924673, "lm_q1q2_score": 0.742294866534}} {"text": "[STATEMENT]\nlemma gbinomial_factors: \"((a + 1) gchoose (Suc k)) = (a + 1) / of_nat (Suc k) * (a gchoose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nproof (cases k)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. k = 0 \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n 2. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nk = 0\n\ngoal (2 subgoals):\n 1. k = 0 \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n 2. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nk = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nk = 0\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\ncase (Suc b)\n[PROOF STATE]\nproof (state)\nthis:\nk = Suc b\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nk = Suc b\n[PROOF STEP]\nhave \"((a + 1) gchoose (Suc (Suc b))) = (\\i = 0 .. Suc b. a + (1 - of_nat i)) / fact (b + 2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nk = Suc b\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n[PROOF STEP]\nby (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nhave \"(\\i = 0 .. Suc b. a + (1 - of_nat i)) = (a + 1) * (\\i = 0..b. a - of_nat i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n[PROOF STEP]\nby (simp add: prod.atLeast0_atMost_Suc_shift del: prod.cl_ivl_Suc)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nhave \"\\ / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a + (1::'a)) * (\\i = 0..b. a - of_nat i) / fact (b + 2) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n[PROOF STEP]\nby (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)\n[PROOF STATE]\nproof (state)\nthis:\n(a + (1::'a)) * (\\i = 0..b. a - of_nat i) / fact (b + 2) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\na + (1::'a) gchoose Suc (Suc b) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na + (1::'a) gchoose Suc (Suc b) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nby (simp add: Suc)\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2429, "file": null, "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.877476800298183, "lm_q2_score": 0.8459424353665381, "lm_q1q2_score": 0.7422948614218824}} {"text": "[STATEMENT]\nlemma gbinomial_factors: \"((a + 1) gchoose (Suc k)) = (a + 1) / of_nat (Suc k) * (a gchoose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nproof (cases k)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. k = 0 \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n 2. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nk = 0\n\ngoal (2 subgoals):\n 1. k = 0 \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n 2. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nk = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nk = 0\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\ncase (Suc b)\n[PROOF STATE]\nproof (state)\nthis:\nk = Suc b\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nk = Suc b\n[PROOF STEP]\nhave \"((a + 1) gchoose (Suc (Suc b))) = (\\i = 0 .. Suc b. a + (1 - of_nat i)) / fact (b + 2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nk = Suc b\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n[PROOF STEP]\nby (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nhave \"(\\i = 0 .. Suc b. a + (1 - of_nat i)) = (a + 1) * (\\i = 0..b. a - of_nat i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n[PROOF STEP]\nby (simp add: prod.atLeast0_atMost_Suc_shift del: prod.cl_ivl_Suc)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nhave \"\\ / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a + (1::'a)) * (\\i = 0..b. a - of_nat i) / fact (b + 2) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n[PROOF STEP]\nby (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)\n[PROOF STATE]\nproof (state)\nthis:\n(a + (1::'a)) * (\\i = 0..b. a - of_nat i) / fact (b + 2) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\na + (1::'a) gchoose Suc (Suc b) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na + (1::'a) gchoose Suc (Suc b) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nby (simp add: Suc)\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2429, "file": null, "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.877476800298183, "lm_q2_score": 0.8459424334245618, "lm_q1q2_score": 0.7422948597178431}} {"text": "[STATEMENT]\nlemma mods_with_nats:\n assumes \"(v::nat) > w\"\n and \"(v * b) mod a = (w * b) mod a\"\n shows \"((v - w) * b) mod a = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (v - w) * b mod a = 0\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nw < v\nv * b mod a = w * b mod a\n\ngoal (1 subgoal):\n 1. (v - w) * b mod a = 0\n[PROOF STEP]\nby (simp add: mod_eq_dvd_iff_nat algebra_simps)\n\n\\ \\The 0-vector of length \\n\\.\\", "meta": {"llama_tokens": 220, "file": "Diophantine_Eqns_Lin_Hom_List_Vector", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8633916099737805, "lm_q2_score": 0.8596637505099168, "lm_q1q2_score": 0.7422264695888555}} {"text": "[STATEMENT]\nlemma Re_less_pihalf_lemma:\n assumes \"\\Re z\\ < pi / 2\"\n shows \"0 < Re ((exp (\\*z) + inverse (exp (\\*z))) / 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < Re ((exp (\\ * z) + inverse (exp (\\ * z))) / 2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 < Re ((exp (\\ * z) + inverse (exp (\\ * z))) / 2)\n[PROOF STEP]\nhave \"0 < cos (Re z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < cos (Re z)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\Re z\\ < pi / 2\n\ngoal (1 subgoal):\n 1. 0 < cos (Re z)\n[PROOF STEP]\nusing cos_gt_zero_pi\n[PROOF STATE]\nproof (prove)\nusing this:\n\\Re z\\ < pi / 2\n\\- (pi / 2) < ?x; ?x < pi / 2\\ \\ 0 < cos ?x\n\ngoal (1 subgoal):\n 1. 0 < cos (Re z)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < cos (Re z)\n\ngoal (1 subgoal):\n 1. 0 < Re ((exp (\\ * z) + inverse (exp (\\ * z))) / 2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < cos (Re z)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < cos (Re z)\n\ngoal (1 subgoal):\n 1. 0 < Re ((exp (\\ * z) + inverse (exp (\\ * z))) / 2)\n[PROOF STEP]\nby (simp add: cos_i_times [symmetric] Re_cos Im_cos add_pos_pos)\n[PROOF STATE]\nproof (state)\nthis:\n0 < Re ((exp (\\ * z) + inverse (exp (\\ * z))) / 2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 669, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577681086260461, "lm_q2_score": 0.8652240704135291, "lm_q1q2_score": 0.7421616144163419}} {"text": "[STATEMENT]\nlemma nonzero_inverse_mult_distrib:\n assumes \"a \\ carrier R\" \"b \\ carrier R\"\n and \"a \\ \\\" \"b \\ \\\"\n shows \"inv (a \\ b) = inv b \\ inv a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inv (a \\ b) = inv b \\ inv a\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. inv (a \\ b) = inv b \\ inv a\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ carrier R\nb \\ carrier R\na \\ \\\nb \\ \\\n[PROOF STEP]\nhave \"a \\ (b \\ inv b) \\ inv a = \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ carrier R\nb \\ carrier R\na \\ \\\nb \\ \\\n\ngoal (1 subgoal):\n 1. a \\ (b \\ inv b) \\ inv a = \\\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na \\ (b \\ inv b) \\ inv a = \\\n\ngoal (1 subgoal):\n 1. inv (a \\ b) = inv b \\ inv a\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ carrier R\nb \\ carrier R\na \\ \\\nb \\ \\\na \\ (b \\ inv b) \\ inv a = \\\n[PROOF STEP]\nhave eq: \"a \\ b \\ (inv b \\ inv a) = \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ carrier R\nb \\ carrier R\na \\ \\\nb \\ \\\na \\ (b \\ inv b) \\ inv a = \\\n\ngoal (1 subgoal):\n 1. a \\ b \\ (inv b \\ inv a) = \\\n[PROOF STEP]\nby (simp only: m_assoc m_closed inv_closed assms)\n[PROOF STATE]\nproof (state)\nthis:\na \\ b \\ (inv b \\ inv a) = \\\n\ngoal (1 subgoal):\n 1. inv (a \\ b) = inv b \\ inv a\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ carrier R\nb \\ carrier R\na \\ \\\nb \\ \\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ carrier R\nb \\ carrier R\na \\ \\\nb \\ \\\n\ngoal (1 subgoal):\n 1. inv (a \\ b) = inv b \\ inv a\n[PROOF STEP]\nusing inverse_unique [OF _ _ eq]\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ carrier R\nb \\ carrier R\na \\ \\\nb \\ \\\n\\a \\ b \\ carrier R; inv b \\ inv a \\ carrier R\\ \\ inv (a \\ b) = inv b \\ inv a\n\ngoal (1 subgoal):\n 1. inv (a \\ b) = inv b \\ inv a\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ninv (a \\ b) = inv b \\ inv a\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1112, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267762381844, "lm_q2_score": 0.8539127455162773, "lm_q1q2_score": 0.7419022578756043}} {"text": "[STATEMENT]\nlemma has_field_derivative_iff:\n \"(f has_field_derivative D) (at x within S) \\\n ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nhave \"((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) \n = ((\\y. (f y - f x) / (y - x) - D) \\ 0) (at x within S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n[PROOF STEP]\nby (smt (verit, best) Lim_cong_within divide_diff_eq_iff norm_divide right_minus_eq tendsto_norm_zero_iff)\n[PROOF STATE]\nproof (state)\nthis:\n((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nby (simp add: has_field_derivative_def has_derivative_iff_norm bounded_linear_mult_right LIM_zero_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1056, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314798554445, "lm_q2_score": 0.8376199673867852, "lm_q1q2_score": 0.7418226112732278}} {"text": "[STATEMENT]\nlemma has_field_derivative_iff:\n \"(f has_field_derivative D) (at x within S) \\\n ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nhave \"((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) \n = ((\\y. (f y - f x) / (y - x) - D) \\ 0) (at x within S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n[PROOF STEP]\nby (smt (verit, best) Lim_cong_within divide_diff_eq_iff norm_divide right_minus_eq tendsto_norm_zero_iff)\n[PROOF STATE]\nproof (state)\nthis:\n((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\y. norm (f y - f x - D * (y - x)) / norm (y - x)) \\ 0) (at x within S) = ((\\y. (f y - f x) / (y - x) - D) \\ (0::'a)) (at x within S)\n\ngoal (1 subgoal):\n 1. (f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n[PROOF STEP]\nby (simp add: has_field_derivative_def has_derivative_iff_norm bounded_linear_mult_right LIM_zero_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_field_derivative D) (at x within S) = ((\\y. (f y - f x) / (y - x)) \\ D) (at x within S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1056, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314798554445, "lm_q2_score": 0.8376199673867852, "lm_q1q2_score": 0.7418226112732278}} {"text": "[STATEMENT]\nlemma eventually_ceiling_eq:\n fixes f::\"'a \\ 'b::{order_topology,floor_ceiling}\"\n assumes f: \"(f \\ l) F\"\n and l: \"l \\ \\\"\n shows \"\\\\<^sub>F x in F. ceiling (f x) = ceiling l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. \\f x\\ = \\l\\\n[PROOF STEP]\nusing eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in F. of_int \\l\\ < f x\n\\\\<^sub>F x in F. f x < of_int \\l\\\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. \\f x\\ = \\l\\\n[PROOF STEP]\nby eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)", "meta": {"llama_tokens": 326, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099070011518829, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7417857537125065}} {"text": "[STATEMENT]\nlemma effective_scalar_product_append:\nassumes \"length zs = length ws\" and \"(length xs = length ys)\" \n shows \"(scalar_product (xs@zs) (ys@ws)) = (scalar_product xs ys)+(scalar_product zs ws)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nusing scalar_product_append assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = ?n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\nlength zs = length ws\nlength xs = length ys\n\ngoal (1 subgoal):\n 1. scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 289, "file": "Matrix_Tensor_Matrix_Tensor", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894689081711, "lm_q2_score": 0.8289388104343892, "lm_q1q2_score": 0.7417257179459583}} {"text": "[STATEMENT]\nlemma Example2_lemma2_aux: \"!!b. j\n (\\i=0..i=0..i=0..b. j < n \\ sum b {0..i = 0..b. j < 0 \\ sum b {0..<0} = sum b {0..i = 0..<0 - Suc j. b (Suc j + i))\n 2. \\n b. \\\\b. j < n \\ sum b {0..i = 0.. \\ sum b {0..i = 0..n b. \\\\b. j < n \\ sum b {0..i = 0.. \\ sum b {0..i = 0..n b. \\\\b. j < n \\ sum b {0..i = 0.. j = n\\ \\ sum b {0..i = 0..n b. \\\\b. sum b {0..i = 0.. \\ (\\i = 0..i = 0..n b. \\\\b. sum b {0..i = 0.. \\ (\\i = 0..i = 0..n b. \\\\b. sum b {0..i = 0.. \\ n - j = Suc (n - Suc j)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n b. \\\\b. sum b {0..i = 0.. \\ n - j = Suc (n - Suc j)\n[PROOF STEP]\napply arith\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1323, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7417257048618469}} {"text": "[STATEMENT]\ntheorem euler_product_zeta:\n assumes \"Re s > 1\"\n shows \"(\\n. \\p\\n. if prime p then inverse (1 - 1 / of_nat p powr s) else 1) \\ zeta s\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. \\p\\n. if prime p then inverse (1 - 1 / of_nat p powr s) else 1) \\ zeta s\n[PROOF STEP]\nusing euler_product_fds_zeta[of s] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < s \\ 1 \\ (\\n. \\p\\n. if prime p then inverse (1 - 1 / nat_power p s) else 1) \\ eval_fds fds_zeta s\n1 < Re s\n\ngoal (1 subgoal):\n 1. (\\n. \\p\\n. if prime p then inverse (1 - 1 / of_nat p powr s) else 1) \\ zeta s\n[PROOF STEP]\nunfolding nat_power_complex_def\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < s \\ 1 \\ (\\n. \\p\\n. if prime p then inverse (1 - 1 / of_nat p powr s) else 1) \\ eval_fds fds_zeta s\n1 < Re s\n\ngoal (1 subgoal):\n 1. (\\n. \\p\\n. if prime p then inverse (1 - 1 / of_nat p powr s) else 1) \\ zeta s\n[PROOF STEP]\nby (simp add: eval_fds_zeta)", "meta": {"llama_tokens": 497, "file": "Zeta_Function_Zeta_Function", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505402422645, "lm_q2_score": 0.819893340314393, "lm_q1q2_score": 0.7417169532564505}} {"text": "[STATEMENT]\nlemma orthogonal_matrix_orthonormal_columns:\n fixes A :: \"real^'n^'n\"\n shows \"orthogonal_matrix A \\\n (\\i. norm(column i A) = 1) \\\n (\\i j. i \\ j \\ orthogonal (column i A) (column j A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal_matrix A = ((\\i. norm (column i A) = 1) \\ (\\i j. i \\ j \\ orthogonal (column i A) (column j A)))\n[PROOF STEP]\nby (auto simp: orthogonal_matrix matrix_mult_transpose_dot_column vec_eq_iff mat_def norm_eq_1 orthogonal_def)", "meta": {"llama_tokens": 213, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206870747658, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7415493914001637}} {"text": "[STATEMENT]\nlemma norm_sin_squared:\n \"norm(sin z) ^ 2 = (exp(2 * Im z) + inverse(exp(2 * Im z)) - 2 * cos(2 * Re z)) / 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cmod (sin z))\\<^sup>2 = (exp (2 * Im z) + inverse (exp (2 * Im z)) - 2 * cos (2 * Re z)) / 4\n[PROOF STEP]\nproof (cases z)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x1 x2. z = Complex x1 x2 \\ (cmod (sin z))\\<^sup>2 = (exp (2 * Im z) + inverse (exp (2 * Im z)) - 2 * cos (2 * Re z)) / 4\n[PROOF STEP]\ncase (Complex x1 x2)\n[PROOF STATE]\nproof (state)\nthis:\nz = Complex x1 x2\n\ngoal (1 subgoal):\n 1. \\x1 x2. z = Complex x1 x2 \\ (cmod (sin z))\\<^sup>2 = (exp (2 * Im z) + inverse (exp (2 * Im z)) - 2 * cos (2 * Re z)) / 4\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nz = Complex x1 x2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nz = Complex x1 x2\n\ngoal (1 subgoal):\n 1. (cmod (sin z))\\<^sup>2 = (exp (2 * Im z) + inverse (exp (2 * Im z)) - 2 * cos (2 * Re z)) / 4\n[PROOF STEP]\napply (simp only: sin_add cmod_power2 cos_of_real sin_of_real cos_double_cos exp_double Complex_eq)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z = complex_of_real x1 + \\ * complex_of_real x2 \\ (Re (complex_of_real (sin x1) * cos (\\ * complex_of_real x2) + complex_of_real (cos x1) * sin (\\ * complex_of_real x2)))\\<^sup>2 + (Im (complex_of_real (sin x1) * cos (\\ * complex_of_real x2) + complex_of_real (cos x1) * sin (\\ * complex_of_real x2)))\\<^sup>2 = ((exp (Im (complex_of_real x1 + \\ * complex_of_real x2)))\\<^sup>2 + inverse ((exp (Im (complex_of_real x1 + \\ * complex_of_real x2)))\\<^sup>2) - 2 * (2 * (cos (Re (complex_of_real x1 + \\ * complex_of_real x2)))\\<^sup>2 - 1)) / 4\n[PROOF STEP]\napply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide power_divide)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z = complex_of_real x1 + \\ * complex_of_real x2 \\ (sin x1 * (exp x2 / (exp x2)\\<^sup>2 + exp x2))\\<^sup>2 + (cos x1 * (exp x2 / (exp x2)\\<^sup>2 - exp x2))\\<^sup>2 = (exp x2)\\<^sup>2 + inverse ((exp x2)\\<^sup>2) - (4 * (cos x1)\\<^sup>2 - 2)\n[PROOF STEP]\napply (simp only: left_diff_distrib [symmetric] power_mult_distrib cos_squared_eq)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z = complex_of_real x1 + \\ * complex_of_real x2 \\ (sin x1)\\<^sup>2 * (exp x2 / (exp x2)\\<^sup>2 + exp x2)\\<^sup>2 + (1 - (sin x1)\\<^sup>2) * (exp x2 / (exp x2)\\<^sup>2 - exp x2)\\<^sup>2 = (exp x2)\\<^sup>2 + inverse ((exp x2)\\<^sup>2) - (4 * (1 - (sin x1)\\<^sup>2) - 2)\n[PROOF STEP]\napply (simp add: power2_eq_square field_split_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (sin z))\\<^sup>2 = (exp (2 * Im z) + inverse (exp (2 * Im z)) - 2 * cos (2 * Re z)) / 4\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1330, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213691605411, "lm_q2_score": 0.8244619199068831, "lm_q1q2_score": 0.7412913302474051}} {"text": "[STATEMENT]\nlemma law_of_cosines:\n shows \"(cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2*(cdist A C)*(cdist A B)*(cos (\\ (C-A) (B-A)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n[PROOF STEP]\nlet ?a = \"C-B\" and ?b = \"C-A\" and ?c = \"B-A\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n[PROOF STEP]\nhave \"?a = ?b - ?c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. C - B = C - A - (B - A)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nC - B = C - A - (B - A)\n\ngoal (1 subgoal):\n 1. (cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n[PROOF STEP]\nhence \"(cmod ?a)\\<^sup>2 = (cmod (?b - ?c))\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nC - B = C - A - (B - A)\n\ngoal (1 subgoal):\n 1. (cmod (C - B))\\<^sup>2 = (cmod (C - A - (B - A)))\\<^sup>2\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (C - B))\\<^sup>2 = (cmod (C - A - (B - A)))\\<^sup>2\n\ngoal (1 subgoal):\n 1. (cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (C - B))\\<^sup>2 = (cmod (C - A - (B - A)))\\<^sup>2\n\ngoal (1 subgoal):\n 1. (cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n[PROOF STEP]\nhave \"... = Re (scalprod (?b-?c) (?b-?c))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cmod (C - A - (B - A)))\\<^sup>2 = Re (scalprod (C - A - (B - A)) (C - A - (B - A)))\n[PROOF STEP]\nby (simp add: cmod_square)\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (C - A - (B - A)))\\<^sup>2 = Re (scalprod (C - A - (B - A)) (C - A - (B - A)))\n\ngoal (1 subgoal):\n 1. (cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (C - A - (B - A)))\\<^sup>2 = Re (scalprod (C - A - (B - A)) (C - A - (B - A)))\n\ngoal (1 subgoal):\n 1. (cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n[PROOF STEP]\nhave \"... = (cmod ?b)\\<^sup>2 + (cmod ?c)\\<^sup>2 - 2*Re (scalprod ?b ?c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Re (scalprod (C - A - (B - A)) (C - A - (B - A))) = (cmod (C - A))\\<^sup>2 + (cmod (B - A))\\<^sup>2 - 2 * Re (scalprod (C - A) (B - A))\n[PROOF STEP]\nby (simp add: cmod_square field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nRe (scalprod (C - A - (B - A)) (C - A - (B - A))) = (cmod (C - A))\\<^sup>2 + (cmod (B - A))\\<^sup>2 - 2 * Re (scalprod (C - A) (B - A))\n\ngoal (1 subgoal):\n 1. (cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(cmod (C - B))\\<^sup>2 = (cmod (C - A))\\<^sup>2 + (cmod (B - A))\\<^sup>2 - 2 * Re (scalprod (C - A) (B - A))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod (C - B))\\<^sup>2 = (cmod (C - A))\\<^sup>2 + (cmod (B - A))\\<^sup>2 - 2 * Re (scalprod (C - A) (B - A))\n\ngoal (1 subgoal):\n 1. (cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n[PROOF STEP]\nusing cos_cmod_scalprod[of ?b ?c]\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod (C - B))\\<^sup>2 = (cmod (C - A))\\<^sup>2 + (cmod (B - A))\\<^sup>2 - 2 * Re (scalprod (C - A) (B - A))\ncmod (C - A) * cmod (B - A) * cos (\\ (C - A) (B - A)) = Re (scalprod (C - A) (B - A))\n\ngoal (1 subgoal):\n 1. (cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(cdist B C)\\<^sup>2 = (cdist A C)\\<^sup>2 + (cdist A B)\\<^sup>2 - 2 * cdist A C * cdist A B * cos (\\ (C - A) (B - A))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2234, "file": "Complex_Geometry_Angles", "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045996818986, "lm_q2_score": 0.8354835452961425, "lm_q1q2_score": 0.7412448443452775}} {"text": "[STATEMENT]\nlemma Fourier_sum_offset_Dirichlet_kernel:\n assumes f: \"f absolutely_integrable_on {-pi..pi}\" and periodic: \"\\x. f(x + 2*pi) = f x\"\n shows\n \"(\\k\\2*n. Fourier_coefficient f k * trigonometric_set k t) =\n integral\\<^sup>L (lebesgue_on {-pi..pi}) (\\x. Dirichlet_kernel n x * f(x+t)) / pi\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nhave ft: \"(\\x. f(x+t)) absolutely_integrable_on {-pi..pi}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. f (x + t)) absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nusing absolutely_integrable_periodic_offset [OF f, of t] periodic\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. f (x + (pi - - pi)) = f x) \\ (\\x. f (x + t)) absolutely_integrable_on {- pi..pi}\n(\\x. f (x + (pi - - pi)) = f x) \\ (\\x. f (t + x)) absolutely_integrable_on {- pi..pi}\nf (?x + 2 * pi) = f ?x\n\ngoal (1 subgoal):\n 1. (\\x. f (x + t)) absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. f (x + t)) absolutely_integrable_on {- pi..pi}\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nhave \"?lhs = (\\k=0..n. Fourier_coefficient (\\x. f(x+t)) (2*k) * trigonometric_set (2*k) 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (\\k = 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) * trigonometric_set (2 * k) 0)\n[PROOF STEP]\nusing Fourier_sum_offset_unpaired assms atMost_atLeast0\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?f absolutely_integrable_on {- pi..pi}; \\x. ?f (x + 2 * pi) = ?f x\\ \\ (\\k\\2 * ?n. Fourier_coefficient ?f k * trigonometric_set k ?t) = (\\k\\?n. Fourier_coefficient (\\x. ?f (x + ?t)) (2 * k) * trigonometric_set (2 * k) 0)\nf absolutely_integrable_on {- pi..pi}\nf (?x + 2 * pi) = f ?x\n{..?n} = {0..?n}\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (\\k = 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) * trigonometric_set (2 * k) 0)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (\\k = 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) * trigonometric_set (2 * k) 0)\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (\\k = 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) * trigonometric_set (2 * k) 0)\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nhave \"\\ = Fourier_coefficient (\\x. f(x+t)) 0 / sqrt (2 * pi)\n + (\\k = Suc 0..n. Fourier_coefficient (\\x. f(x+t)) (2*k) / sqrt pi)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) * trigonometric_set (2 * k) 0) = Fourier_coefficient (\\x. f (x + t)) 0 / sqrt (2 * pi) + (\\k = Suc 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) / sqrt pi)\n[PROOF STEP]\nby (simp add: sum.atLeast_Suc_atMost trigonometric_set_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) * trigonometric_set (2 * k) 0) = Fourier_coefficient (\\x. f (x + t)) 0 / sqrt (2 * pi) + (\\k = Suc 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) / sqrt pi)\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) * trigonometric_set (2 * k) 0) = Fourier_coefficient (\\x. f (x + t)) 0 / sqrt (2 * pi) + (\\k = Suc 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) / sqrt pi)\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nhave \"\\ = (LINT x|lebesgue_on {-pi..pi}. f(x+t)) / (2 * pi) +\n (\\k = Suc 0..n. (LINT x|lebesgue_on {-pi..pi}. cos (real k * x) * f(x+t)) / pi)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Fourier_coefficient (\\x. f (x + t)) 0 / sqrt (2 * pi) + (\\k = Suc 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) / sqrt pi) = (LINT x|lebesgue_on {- pi..pi}. f (x + t)) / (2 * pi) + (\\k = Suc 0..n. (LINT x|lebesgue_on {- pi..pi}. cos (real k * x) * f (x + t)) / pi)\n[PROOF STEP]\nby (simp add: Fourier_coefficient_def orthonormal_coeff_def trigonometric_set_def l2product_def)\n[PROOF STATE]\nproof (state)\nthis:\nFourier_coefficient (\\x. f (x + t)) 0 / sqrt (2 * pi) + (\\k = Suc 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) / sqrt pi) = (LINT x|lebesgue_on {- pi..pi}. f (x + t)) / (2 * pi) + (\\k = Suc 0..n. (LINT x|lebesgue_on {- pi..pi}. cos (real k * x) * f (x + t)) / pi)\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nFourier_coefficient (\\x. f (x + t)) 0 / sqrt (2 * pi) + (\\k = Suc 0..n. Fourier_coefficient (\\x. f (x + t)) (2 * k) / sqrt pi) = (LINT x|lebesgue_on {- pi..pi}. f (x + t)) / (2 * pi) + (\\k = Suc 0..n. (LINT x|lebesgue_on {- pi..pi}. cos (real k * x) * f (x + t)) / pi)\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nhave \"\\ = LINT x|lebesgue_on {-pi..pi}.\n f(x+t) / (2 * pi) + (\\k = Suc 0..n. (cos (real k * x) * f(x+t)) / pi)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (LINT x|lebesgue_on {- pi..pi}. f (x + t)) / (2 * pi) + (\\k = Suc 0..n. (LINT x|lebesgue_on {- pi..pi}. cos (real k * x) * f (x + t)) / pi) = LINT x|lebesgue_on {- pi..pi}. f (x + t) / (2 * pi) + (\\k = Suc 0..n. cos (real k * x) * f (x + t) / pi)\n[PROOF STEP]\nusing Fourier_products_integrable_cos [OF ft] absolutely_integrable_imp_integrable [OF ft]\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable (lebesgue_on {- pi..pi}) (\\x. cos (?k * x) * f (x + t))\n{- pi..pi} \\ sets lebesgue \\ integrable (lebesgue_on {- pi..pi}) (\\x. f (x + t))\n\ngoal (1 subgoal):\n 1. (LINT x|lebesgue_on {- pi..pi}. f (x + t)) / (2 * pi) + (\\k = Suc 0..n. (LINT x|lebesgue_on {- pi..pi}. cos (real k * x) * f (x + t)) / pi) = LINT x|lebesgue_on {- pi..pi}. f (x + t) / (2 * pi) + (\\k = Suc 0..n. cos (real k * x) * f (x + t) / pi)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(LINT x|lebesgue_on {- pi..pi}. f (x + t)) / (2 * pi) + (\\k = Suc 0..n. (LINT x|lebesgue_on {- pi..pi}. cos (real k * x) * f (x + t)) / pi) = LINT x|lebesgue_on {- pi..pi}. f (x + t) / (2 * pi) + (\\k = Suc 0..n. cos (real k * x) * f (x + t) / pi)\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(LINT x|lebesgue_on {- pi..pi}. f (x + t)) / (2 * pi) + (\\k = Suc 0..n. (LINT x|lebesgue_on {- pi..pi}. cos (real k * x) * f (x + t)) / pi) = LINT x|lebesgue_on {- pi..pi}. f (x + t) / (2 * pi) + (\\k = Suc 0..n. cos (real k * x) * f (x + t) / pi)\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nhave \"\\ = (LINT x|lebesgue_on {-pi..pi}.\n f(x+t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f(x+t))) / pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LINT x|lebesgue_on {- pi..pi}. f (x + t) / (2 * pi) + (\\k = Suc 0..n. cos (real k * x) * f (x + t) / pi) = (LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t))) / pi\n[PROOF STEP]\nby (simp add: divide_simps sum_distrib_right mult.assoc)\n[PROOF STATE]\nproof (state)\nthis:\nLINT x|lebesgue_on {- pi..pi}. f (x + t) / (2 * pi) + (\\k = Suc 0..n. cos (real k * x) * f (x + t) / pi) = (LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t))) / pi\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nLINT x|lebesgue_on {- pi..pi}. f (x + t) / (2 * pi) + (\\k = Suc 0..n. cos (real k * x) * f (x + t) / pi) = (LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t))) / pi\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nhave \"\\ = ?rhs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t))) / pi = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t))) / pi = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nhave \"LINT x|lebesgue_on {-pi..pi}. f(x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f(x + t))\n = LINT x|lebesgue_on {-pi..pi}. Dirichlet_kernel n x * f(x + t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\nhave eq: \"f(x+t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f(x + t))\n = Dirichlet_kernel n x * f(x + t)\" if \"- pi \\ x\" \"x \\ pi\" for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\nproof (cases \"x = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x = 0 \\ f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n 2. x \\ 0 \\ f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nx \\ 0\n\ngoal (2 subgoals):\n 1. x = 0 \\ f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n 2. x \\ 0 \\ f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ 0\n[PROOF STEP]\nhave \"sin (x/2) \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ 0\n\ngoal (1 subgoal):\n 1. sin (x / 2) \\ 0\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ 0\n- pi \\ x\nx \\ pi\n\ngoal (1 subgoal):\n 1. sin (x / 2) \\ 0\n[PROOF STEP]\nby (auto simp: sin_zero_iff)\n[PROOF STATE]\nproof (state)\nthis:\nsin (x / 2) \\ 0\n\ngoal (2 subgoals):\n 1. x = 0 \\ f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n 2. x \\ 0 \\ f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nsin (x / 2) \\ 0\n[PROOF STEP]\nhave \"f(x + t) * (1/2 + (\\k = Suc 0..n. cos(real k * x))) = f(x + t) * sin((real n + 1/2) * x) / 2 / sin(x/2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsin (x / 2) \\ 0\n\ngoal (1 subgoal):\n 1. f (x + t) * (1 / 2 + (\\k = Suc 0..n. cos (real k * x))) = f (x + t) * sin ((real n + 1 / 2) * x) / 2 / sin (x / 2)\n[PROOF STEP]\nusing cosine_sum_lemma [of x n]\n[PROOF STATE]\nproof (prove)\nusing this:\nsin (x / 2) \\ 0\n(1 / 2 + (\\k = Suc 0..n. cos (real k * x))) * sin (x / 2) = sin ((real n + 1 / 2) * x) / 2\n\ngoal (1 subgoal):\n 1. f (x + t) * (1 / 2 + (\\k = Suc 0..n. cos (real k * x))) = f (x + t) * sin ((real n + 1 / 2) * x) / 2 / sin (x / 2)\n[PROOF STEP]\nby (simp add: divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\nf (x + t) * (1 / 2 + (\\k = Suc 0..n. cos (real k * x))) = f (x + t) * sin ((real n + 1 / 2) * x) / 2 / sin (x / 2)\n\ngoal (2 subgoals):\n 1. x = 0 \\ f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n 2. x \\ 0 \\ f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nf (x + t) * (1 / 2 + (\\k = Suc 0..n. cos (real k * x))) = f (x + t) * sin ((real n + 1 / 2) * x) / 2 / sin (x / 2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nf (x + t) * (1 / 2 + (\\k = Suc 0..n. cos (real k * x))) = f (x + t) * sin ((real n + 1 / 2) * x) / 2 / sin (x / 2)\n\ngoal (1 subgoal):\n 1. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\nby (simp add: Dirichlet_kernel_def False field_simps sum_distrib_left)\n[PROOF STATE]\nproof (state)\nthis:\nf (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n\ngoal (1 subgoal):\n 1. x = 0 \\ f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\nqed (simp add: Dirichlet_kernel_def algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\- pi \\ ?x; ?x \\ pi\\ \\ f (?x + t) / 2 + (\\k = Suc 0..n. cos (real k * ?x) * f (?x + t)) = Dirichlet_kernel n ?x * f (?x + t)\n\ngoal (1 subgoal):\n 1. LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\nby (rule Bochner_Integration.integral_cong [OF refl]) (simp add: eq)\n[PROOF STATE]\nproof (state)\nthis:\nLINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nLINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)\n\ngoal (1 subgoal):\n 1. (LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t))) / pi = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nLINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nLINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t)) = LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)\n\ngoal (1 subgoal):\n 1. (LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t))) / pi = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t))) / pi = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(LINT x|lebesgue_on {- pi..pi}. f (x + t) / 2 + (\\k = Suc 0..n. cos (real k * x) * f (x + t))) / pi = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n\ngoal (1 subgoal):\n 1. (\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 8586, "file": "Fourier_Fourier", "length": 51, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045996818986, "lm_q2_score": 0.8354835330070839, "lm_q1q2_score": 0.7412448334423681}} {"text": "[STATEMENT]\nlemma lcm_gcd_distrib:\n \"lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n[PROOF STEP]\nproof (cases \"x = 0 \\ y = 0 \\ z = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x = (0::'a) \\ y = (0::'a) \\ z = (0::'a) \\ lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n 2. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nx = (0::'a) \\ y = (0::'a) \\ z = (0::'a)\n\ngoal (2 subgoals):\n 1. x = (0::'a) \\ y = (0::'a) \\ z = (0::'a) \\ lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n 2. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx = (0::'a) \\ y = (0::'a) \\ z = (0::'a)\n\ngoal (1 subgoal):\n 1. lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n[PROOF STEP]\nby (auto simp: lcm_proj1_if_dvd lcm_proj2_if_dvd)\n[PROOF STATE]\nproof (state)\nthis:\nlcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n\ngoal (1 subgoal):\n 1. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a))\n\ngoal (1 subgoal):\n 1. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n[PROOF STEP]\nhence \"normalize (lcm x (gcd y z)) = normalize (gcd (lcm x y) (lcm x z))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a))\n\ngoal (1 subgoal):\n 1. normalize (lcm x (gcd y z)) = normalize (gcd (lcm x y) (lcm x z))\n[PROOF STEP]\nby (intro associatedI prime_factorization_subset_imp_dvd)\n (auto simp: lcm_eq_0_iff prime_factorization_gcd prime_factorization_lcm\n subset_mset.sup_inf_distrib1)\n[PROOF STATE]\nproof (state)\nthis:\nnormalize (lcm x (gcd y z)) = normalize (gcd (lcm x y) (lcm x z))\n\ngoal (1 subgoal):\n 1. \\ (x = (0::'a) \\ y = (0::'a) \\ z = (0::'a)) \\ lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnormalize (lcm x (gcd y z)) = normalize (gcd (lcm x y) (lcm x z))\n\ngoal (1 subgoal):\n 1. lcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlcm x (gcd y z) = gcd (lcm x y) (lcm x z)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1427, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045877523147, "lm_q2_score": 0.8354835391516133, "lm_q1q2_score": 0.741244828926852}} {"text": "[STATEMENT]\nlemma abs_impossible:\n \"\\y1\\ < x1 \\ \\y2\\ < x2 \\ x1 * x2 + y1 * y2 \\ 0\" for x1 x2::real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\y1\\ < x1; \\y2\\ < x2\\ \\ x1 * x2 + y1 * y2 \\ 0\n[PROOF STEP]\nproof goal_cases\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\y1\\ < x1; \\y2\\ < x2\\ \\ x1 * x2 + y1 * y2 \\ 0\n[PROOF STEP]\ncase 1\n[PROOF STATE]\nproof (state)\nthis:\n\\y1\\ < x1\n\\y2\\ < x2\n\ngoal (1 subgoal):\n 1. \\\\y1\\ < x1; \\y2\\ < x2\\ \\ x1 * x2 + y1 * y2 \\ 0\n[PROOF STEP]\nhave \"- y1 * y2 \\ abs y1 * abs y2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - y1 * y2 \\ \\y1\\ * \\y2\\\n[PROOF STEP]\nby (metis abs_ge_minus_self abs_mult mult.commute mult_minus_right)\n[PROOF STATE]\nproof (state)\nthis:\n- y1 * y2 \\ \\y1\\ * \\y2\\\n\ngoal (1 subgoal):\n 1. \\\\y1\\ < x1; \\y2\\ < x2\\ \\ x1 * x2 + y1 * y2 \\ 0\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n- y1 * y2 \\ \\y1\\ * \\y2\\\n\ngoal (1 subgoal):\n 1. \\\\y1\\ < x1; \\y2\\ < x2\\ \\ x1 * x2 + y1 * y2 \\ 0\n[PROOF STEP]\nhave \"\\ < x1 * x2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\y1\\ * \\y2\\ < x1 * x2\n[PROOF STEP]\napply (rule mult_strict_mono)\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. \\y1\\ < x1\n 2. \\y2\\ < x2\n 3. 0 < x1\n 4. 0 \\ \\y2\\\n[PROOF STEP]\nusing 1\n[PROOF STATE]\nproof (prove)\nusing this:\n\\y1\\ < x1\n\\y2\\ < x2\n\ngoal (4 subgoals):\n 1. \\y1\\ < x1\n 2. \\y2\\ < x2\n 3. 0 < x1\n 4. 0 \\ \\y2\\\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\y1\\ * \\y2\\ < x1 * x2\n\ngoal (1 subgoal):\n 1. \\\\y1\\ < x1; \\y2\\ < x2\\ \\ x1 * x2 + y1 * y2 \\ 0\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n- y1 * y2 < x1 * x2\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n- y1 * y2 < x1 * x2\n\ngoal (1 subgoal):\n 1. x1 * x2 + y1 * y2 \\ 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx1 * x2 + y1 * y2 \\ 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1250, "file": "Ordinary_Differential_Equations_IVP_Cones", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045847699185, "lm_q2_score": 0.8354835350552603, "lm_q1q2_score": 0.7412448228008058}} {"text": "[STATEMENT]\nlemma Log_divide: \n \"\\ 0 < a; a \\ 1; 0 < x; 0 < y \\\n \\ Log a (x/y) = Log a x - Log a y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < a; a \\ 1; 0 < x; 0 < y\\ \\ Log a (x / y) = Log a x - Log a y\n[PROOF STEP]\nby (metis Log_inverse Log_mult divide_real_def \n inverse_positive_iff_positive minus_real_def)", "meta": {"llama_tokens": 183, "file": "Real_Power_Log", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110569397306, "lm_q2_score": 0.831143054132195, "lm_q1q2_score": 0.7412225655737485}} {"text": "[STATEMENT]\nlemma power_diff_rev_if:\n assumes nz: \"(a::'a::field) ~= 0\"\n shows \"(a^m) / (a^n) = (if n <= m then a ^ (m-n) else (1/a) ^ (n-m))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n[PROOF STEP]\nproof (cases \"n <= m\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. n \\ m \\ a ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n 2. \\ n \\ m \\ a ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nn \\ m\n\ngoal (2 subgoals):\n 1. n \\ m \\ a ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n 2. \\ n \\ m \\ a ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n[PROOF STEP]\nwith nz\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ (0::'a)\nn \\ m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ (0::'a)\nn \\ m\n\ngoal (1 subgoal):\n 1. a ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n[PROOF STEP]\nby (simp add: power_diff)\n[PROOF STATE]\nproof (state)\nthis:\na ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n\ngoal (1 subgoal):\n 1. \\ n \\ m \\ a ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ n \\ m \\ a ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ n \\ m\n\ngoal (1 subgoal):\n 1. \\ n \\ m \\ a ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n[PROOF STEP]\nwith nz\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ (0::'a)\n\\ n \\ m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ (0::'a)\n\\ n \\ m\n\ngoal (1 subgoal):\n 1. a ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n[PROOF STEP]\nby (simp add: power_diff_inverse nonzero_inverse_eq_divide [THEN sym])\n[PROOF STATE]\nproof (state)\nthis:\na ^ m / a ^ n = (if n \\ m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1147, "file": "FFT_FFT", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772450055545, "lm_q2_score": 0.8479677622198947, "lm_q1q2_score": 0.7411893254546906}} {"text": "[STATEMENT]\nlemma powrfinitesum:\n fixes a::real and s::nat assumes \"s \\ n\"\n shows \" (\\j=s..(n::nat).(a powr (2^j))) = a powr (\\j=s..(n::nat).(2^j)) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}\n[PROOF STEP]\nusing \\s \\ n\\\n[PROOF STATE]\nproof (prove)\nusing this:\ns \\ n\n\ngoal (1 subgoal):\n 1. (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}\n[PROOF STEP]\nproof(induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. s \\ 0 \\ (\\j = s..0. a powr 2 ^ j) = a powr sum ((^) 2) {s..0}\n 2. \\n. \\s \\ n \\ (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}; s \\ Suc n\\ \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\ns \\ 0\n\ngoal (2 subgoals):\n 1. s \\ 0 \\ (\\j = s..0. a powr 2 ^ j) = a powr sum ((^) 2) {s..0}\n 2. \\n. \\s \\ n \\ (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}; s \\ Suc n\\ \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ns \\ 0\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ns \\ 0\n\ngoal (1 subgoal):\n 1. (\\j = s..0. a powr 2 ^ j) = a powr sum ((^) 2) {s..0}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = s..0. a powr 2 ^ j) = a powr sum ((^) 2) {s..0}\n\ngoal (1 subgoal):\n 1. \\n. \\s \\ n \\ (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}; s \\ Suc n\\ \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\s \\ n \\ (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}; s \\ Suc n\\ \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\ns \\ n \\ (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}\ns \\ Suc n\n\ngoal (1 subgoal):\n 1. \\n. \\s \\ n \\ (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}; s \\ Suc n\\ \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nhave ?case when \"s\\n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nusing Suc.hyps\n[PROOF STATE]\nproof (prove)\nusing this:\ns \\ n \\ (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}\n\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nby (metis Suc.prems add.commute linorder_not_le powr_add prod.nat_ivl_Suc' sum.cl_ivl_Suc that)\n[PROOF STATE]\nproof (state)\nthis:\ns \\ n \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n\ngoal (1 subgoal):\n 1. \\n. \\s \\ n \\ (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}; s \\ Suc n\\ \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ns \\ n \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n\ngoal (1 subgoal):\n 1. \\n. \\s \\ n \\ (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}; s \\ Suc n\\ \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nhave ?case when \"s=Suc n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nhave \"(\\j = s..Suc n. a powr 2 ^ j) =(a powr 2 ^(Suc n)) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr 2 ^ Suc n\n[PROOF STEP]\nusing \\s=Suc n\\\n[PROOF STATE]\nproof (prove)\nusing this:\ns = Suc n\n\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr 2 ^ Suc n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = s..Suc n. a powr 2 ^ j) = a powr 2 ^ Suc n\n\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = s..Suc n. a powr 2 ^ j) = a powr 2 ^ Suc n\n\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nhave \"a powr 2 ^ Suc n = a powr sum (power 2) {s..Suc n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a powr 2 ^ Suc n = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\ns = Suc n\n\ngoal (1 subgoal):\n 1. a powr 2 ^ Suc n = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na powr 2 ^ Suc n = a powr sum ((^) 2) {s..Suc n}\n\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\j = s..Suc n. a powr 2 ^ j) = a powr 2 ^ Suc n\na powr 2 ^ Suc n = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nshow \"(\\j = s..Suc n. a powr 2 ^ j) = a powr sum (power 2) {s..Suc n}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = s..Suc n. a powr 2 ^ j) = a powr 2 ^ Suc n\na powr 2 ^ Suc n = a powr sum ((^) 2) {s..Suc n}\n\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nusing \\s\\Suc n\\\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = s..Suc n. a powr 2 ^ j) = a powr 2 ^ Suc n\na powr 2 ^ Suc n = a powr sum ((^) 2) {s..Suc n}\ns \\ Suc n\n\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ns = Suc n \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n\ngoal (1 subgoal):\n 1. \\n. \\s \\ n \\ (\\j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}; s \\ Suc n\\ \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ns \\ n \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\ns = Suc n \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ns \\ n \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\ns = Suc n \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nusing \\s\\Suc n\\\n[PROOF STATE]\nproof (prove)\nusing this:\ns \\ n \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\ns = Suc n \\ (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\ns \\ Suc n\n\ngoal (1 subgoal):\n 1. (\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = s..Suc n. a powr 2 ^ j) = a powr sum ((^) 2) {s..Suc n}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3963, "file": "Irrationality_J_Hancl_Irrationality_J_Hancl", "length": 31, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278633625322, "lm_q2_score": 0.8397339656668287, "lm_q1q2_score": 0.7410046491163256}} {"text": "[STATEMENT]\nlemma eventually_powr_const_mono_nonneg:\n assumes \"p \\ (0 :: real)\" \"eventually (\\x. l x \\ 0) at_top\" \"eventually (\\x. l x \\ f x) at_top\"\n \"eventually (\\x. f x \\ g x) at_top\"\n shows \"eventually (\\x. f x powr p \\ g x powr p) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. f x powr p \\ g x powr p\n[PROOF STEP]\nusing assms(2-4)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in at_top. 0 \\ l x\n\\\\<^sub>F x in at_top. l x \\ f x\n\\\\<^sub>F x in at_top. f x \\ g x\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. f x powr p \\ g x powr p\n[PROOF STEP]\nby eventually_elim (auto simp: assms(1) intro!: powr_mono2)", "meta": {"llama_tokens": 338, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513731336204, "lm_q2_score": 0.8267117962054048, "lm_q1q2_score": 0.7409415825348558}} {"text": "[STATEMENT]\nlemma uint64_UNIV : \"(UNIV :: uint64 set) = uint64_of_nat ` {..<2 ^ 64}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. UNIV = uint64_of_nat ` {..<2 ^ 64}\n[PROOF STEP]\nusing nat_of_uint64_bij_betw\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw nat_of_uint64 UNIV {..<2 ^ 64}\n\ngoal (1 subgoal):\n 1. UNIV = uint64_of_nat ` {..<2 ^ 64}\n[PROOF STEP]\nby (metis UNIV_I UNIV_eq_I bij_betw_def card_UNIV_uint64 imageI image_eqI inj_on_contraD lessThan_iff rangeI uint64_nat_bij uint64_range)", "meta": {"llama_tokens": 247, "file": "FSM_Tests_Test_Suite_Generator_Code_Export", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898305367525, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7407141671784191}} {"text": "[STATEMENT]\nlemma cardOneImageCardOne: \n assumes \"card A = 1\" \n shows \"card (f`A) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (f ` A) = 1\n[PROOF STEP]\nusing assms card_image card_image_le\n[PROOF STATE]\nproof (prove)\nusing this:\ncard A = 1\ninj_on ?f ?A \\ card (?f ` ?A) = card ?A\nfinite ?A \\ card (?f ` ?A) \\ card ?A\n\ngoal (1 subgoal):\n 1. card (f ` A) = 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\card A = 1; \\f A. inj_on f A \\ card (f ` A) = card A; \\A f. finite A \\ card (f ` A) \\ card A\\ \\ card (f ` A) = 1\n[PROOF STEP]\nhave \"finite (f`A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (f ` A)\n[PROOF STEP]\nusing assms One_nat_def Suc_not_Zero card.infinite finite_imageI\n[PROOF STATE]\nproof (prove)\nusing this:\ncard A = 1\n1 = Suc 0\nSuc ?m \\ 0\ninfinite ?A \\ card ?A = 0\nfinite ?F \\ finite (?h ` ?F)\n\ngoal (1 subgoal):\n 1. finite (f ` A)\n[PROOF STEP]\nby (metis(no_types))\n[PROOF STATE]\nproof (state)\nthis:\nfinite (f ` A)\n\ngoal (1 subgoal):\n 1. \\card A = 1; \\f A. inj_on f A \\ card (f ` A) = card A; \\A f. finite A \\ card (f ` A) \\ card A\\ \\ card (f ` A) = 1\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite (f ` A)\n\ngoal (1 subgoal):\n 1. \\card A = 1; \\f A. inj_on f A \\ card (f ` A) = card A; \\A f. finite A \\ card (f ` A) \\ card A\\ \\ card (f ` A) = 1\n[PROOF STEP]\nhave \"f`A \\ {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f ` A \\ {}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncard A = 1\n\ngoal (1 subgoal):\n 1. f ` A \\ {}\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nf ` A \\ {}\n\ngoal (1 subgoal):\n 1. \\card A = 1; \\f A. inj_on f A \\ card (f ` A) = card A; \\A f. finite A \\ card (f ` A) \\ card A\\ \\ card (f ` A) = 1\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nf ` A \\ {}\n\ngoal (1 subgoal):\n 1. \\card A = 1; \\f A. inj_on f A \\ card (f ` A) = card A; \\A f. finite A \\ card (f ` A) \\ card A\\ \\ card (f ` A) = 1\n[PROOF STEP]\nhave \"card (f`A) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (f ` A) \\ 1\n[PROOF STEP]\nusing assms card_image_le One_nat_def Suc_not_Zero card.infinite\n[PROOF STATE]\nproof (prove)\nusing this:\ncard A = 1\nfinite ?A \\ card (?f ` ?A) \\ card ?A\n1 = Suc 0\nSuc ?m \\ 0\ninfinite ?A \\ card ?A = 0\n\ngoal (1 subgoal):\n 1. card (f ` A) \\ 1\n[PROOF STEP]\nby (metis)\n[PROOF STATE]\nproof (state)\nthis:\ncard (f ` A) \\ 1\n\ngoal (1 subgoal):\n 1. \\card A = 1; \\f A. inj_on f A \\ card (f ` A) = card A; \\A f. finite A \\ card (f ` A) \\ card A\\ \\ card (f ` A) = 1\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (f ` A)\nf ` A \\ {}\ncard (f ` A) \\ 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (f ` A)\nf ` A \\ {}\ncard (f ` A) \\ 1\n\ngoal (1 subgoal):\n 1. card (f ` A) = 1\n[PROOF STEP]\nby (metis assms image_empty image_insert \n cardinalityOneTheElemIdentity the_elem_eq)\n[PROOF STATE]\nproof (state)\nthis:\ncard (f ` A) = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1580, "file": "Vickrey_Clarke_Groves_MiscTools", "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.86153820232079, "lm_q2_score": 0.8596637451167997, "lm_q1q2_score": 0.7406331575682854}} {"text": "[STATEMENT]\nlemma lin_bound_arg_general_set: \n fixes A ::\"('a :: {field})vec set\"\n assumes \"A \\ carrier_vec nr\"\n assumes \"vec_space.lin_indpt_vs nr A\"\n shows \"card A \\ nr\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card A \\ nr\n[PROOF STEP]\nusing vec_space.lin_indpt_set_card_lt_dim[of \"A\" \"nr\"] vec_space.lin_indpt_vs_def[of nr A] \n vec_space.dim_is_n assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A \\ carrier_vec nr; \\ module.lin_dep class_ring (module_vec TYPE('a) nr) A\\ \\ card A \\ vectorspace.dim class_ring (module_vec TYPE('a) nr)\nvec_space.lin_indpt_vs nr A = (\\ module.lin_dep class_ring (module_vec TYPE('a) nr) A)\nvectorspace.dim class_ring (module_vec TYPE(?'a) ?n) = ?n\nA \\ carrier_vec nr\nvec_space.lin_indpt_vs nr A\n\ngoal (1 subgoal):\n 1. card A \\ nr\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 372, "file": "Fishers_Inequality_Linear_Bound_Argument", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032941988938414, "lm_q2_score": 0.8198933403143929, "lm_q1q2_score": 0.7406048980176853}} {"text": "[STATEMENT]\nlemma gcd_scm: \"VARS a b x y\n {0 0 < B \\ a = A \\ b = B \\ x = B \\ y = A}\nWHILE a \\ b INV {0 < a \\ 0 < b \\ Arith2.gcd A B = Arith2.gcd a b \\ 2 * A * B = a * x + b * y} VAR {0} \nDO IF a < b THEN b := b - a; x := x + y ELSE a := a - b; y := y + x FI OD\n{a = Arith2.gcd A B \\ 2 * A * B = a * (x + y)}\n[PROOF STEP]\napply vcg\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\a b x y. 0 < A \\ 0 < B \\ a = A \\ b = B \\ x = B \\ y = A \\ 0 < a \\ 0 < b \\ Arith2.gcd A B = Arith2.gcd a b \\ 2 * A * B = a * x + b * y\n 2. \\a b x y. (0 < a \\ 0 < b \\ Arith2.gcd A B = Arith2.gcd a b \\ 2 * A * B = a * x + b * y) \\ a \\ b \\ (a < b \\ 0 < a \\ 0 < b - a \\ Arith2.gcd A B = Arith2.gcd a (b - a) \\ 2 * A * B = a * (x + y) + (b - a) * y) \\ (\\ a < b \\ 0 < a - b \\ 0 < b \\ Arith2.gcd A B = Arith2.gcd (a - b) b \\ 2 * A * B = (a - b) * x + b * (y + x))\n 3. \\a b x y. (0 < a \\ 0 < b \\ Arith2.gcd A B = Arith2.gcd a b \\ 2 * A * B = a * x + b * y) \\ \\ a \\ b \\ a = Arith2.gcd A B \\ 2 * A * B = a * (x + y)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\a b x y. (0 < a \\ 0 < b \\ Arith2.gcd A B = Arith2.gcd a b \\ 2 * A * B = a * x + b * y) \\ a \\ b \\ (a < b \\ 0 < a \\ 0 < b - a \\ Arith2.gcd A B = Arith2.gcd a (b - a) \\ 2 * A * B = a * (x + y) + (b - a) * y) \\ (\\ a < b \\ 0 < a - b \\ 0 < b \\ Arith2.gcd A B = Arith2.gcd (a - b) b \\ 2 * A * B = (a - b) * x + b * (y + x))\n 2. \\a b x y. (0 < a \\ 0 < b \\ Arith2.gcd A B = Arith2.gcd a b \\ 2 * A * B = a * x + b * y) \\ \\ a \\ b \\ a = Arith2.gcd A B \\ 2 * A * B = a * (x + y)\n[PROOF STEP]\napply(simp add: distribs gcd_diff_r linorder_not_less gcd_diff_l)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a b x y. (0 < a \\ 0 < b \\ Arith2.gcd A B = Arith2.gcd a b \\ 2 * A * B = a * x + b * y) \\ \\ a \\ b \\ a = Arith2.gcd A B \\ 2 * A * B = a * (x + y)\n[PROOF STEP]\napply(simp add: distribs gcd_nnn)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1360, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942041005327, "lm_q2_score": 0.8198933337131076, "lm_q1q2_score": 0.740604896323714}} {"text": "[STATEMENT]\nlemma onorm_triangle:\n assumes f: \"bounded_linear f\"\n assumes g: \"bounded_linear g\"\n shows \"onorm (\\x. f x + g x) \\ onorm f + onorm g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. onorm (\\x. f x + g x) \\ onorm f + onorm g\n[PROOF STEP]\nproof (rule onorm_bound)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 0 \\ onorm f + onorm g\n 2. \\x. norm (f x + g x) \\ (onorm f + onorm g) * norm x\n[PROOF STEP]\nshow \"0 \\ onorm f + onorm g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ onorm f + onorm g\n[PROOF STEP]\nby (intro add_nonneg_nonneg onorm_pos_le f g)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ onorm f + onorm g\n\ngoal (1 subgoal):\n 1. \\x. norm (f x + g x) \\ (onorm f + onorm g) * norm x\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. norm (f x + g x) \\ (onorm f + onorm g) * norm x\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. norm (f x + g x) \\ (onorm f + onorm g) * norm x\n[PROOF STEP]\nhave \"norm (f x + g x) \\ norm (f x) + norm (g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (f x + g x) \\ norm (f x) + norm (g x)\n[PROOF STEP]\nby (rule norm_triangle_ineq)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (f x + g x) \\ norm (f x) + norm (g x)\n\ngoal (1 subgoal):\n 1. \\x. norm (f x + g x) \\ (onorm f + onorm g) * norm x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnorm (f x + g x) \\ norm (f x) + norm (g x)\n\ngoal (1 subgoal):\n 1. \\x. norm (f x + g x) \\ (onorm f + onorm g) * norm x\n[PROOF STEP]\nhave \"norm (f x) + norm (g x) \\ onorm f * norm x + onorm g * norm x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (f x) + norm (g x) \\ onorm f * norm x + onorm g * norm x\n[PROOF STEP]\nby (intro add_mono onorm f g)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (f x) + norm (g x) \\ onorm f * norm x + onorm g * norm x\n\ngoal (1 subgoal):\n 1. \\x. norm (f x + g x) \\ (onorm f + onorm g) * norm x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (f x + g x) \\ onorm f * norm x + onorm g * norm x\n[PROOF STEP]\nshow \"norm (f x + g x) \\ (onorm f + onorm g) * norm x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (f x + g x) \\ onorm f * norm x + onorm g * norm x\n\ngoal (1 subgoal):\n 1. norm (f x + g x) \\ (onorm f + onorm g) * norm x\n[PROOF STEP]\nby (simp only: distrib_right)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (f x + g x) \\ (onorm f + onorm g) * norm x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1173, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240930029117, "lm_q2_score": 0.8558511451289037, "lm_q1q2_score": 0.7405030307896591}} {"text": "[STATEMENT]\nlemma Log_less_cancel_iff [simp]:\n assumes \"1 < a\" \n and \"0 < x\"\n and \"0 < y\"\nshows \"(Log a x < Log a y) = (x < y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Log a x < Log a y) = (x < y)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. Log a x < Log a y \\ x < y\n 2. x < y \\ Log a x < Log a y\n[PROOF STEP]\nassume \"Log a x < Log a y\"\n[PROOF STATE]\nproof (state)\nthis:\nLog a x < Log a y\n\ngoal (2 subgoals):\n 1. Log a x < Log a y \\ x < y\n 2. x < y \\ Log a x < Log a y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nLog a x < Log a y\n[PROOF STEP]\nshow \"x < y\"\n[PROOF STATE]\nproof (prove)\nusing this:\nLog a x < Log a y\n\ngoal (1 subgoal):\n 1. x < y\n[PROOF STEP]\nusing powreal_Log_cancel assms powreal_less_cancel_iff\n[PROOF STATE]\nproof (prove)\nusing this:\nLog a x < Log a y\n\\0 < ?a; ?a \\ 1; 0 < ?x\\ \\ ?a pow\\<^sub>\\ Log ?a ?x = ?x\n1 < a\n0 < x\n0 < y\n1 < ?a \\ (?a pow\\<^sub>\\ ?r < ?a pow\\<^sub>\\ ?s) = (?r < ?s)\n\ngoal (1 subgoal):\n 1. x < y\n[PROOF STEP]\nby (metis less_irrefl real_inverse_bet_one_one_lemma \n inverse_positive_iff_positive)\n[PROOF STATE]\nproof (state)\nthis:\nx < y\n\ngoal (1 subgoal):\n 1. x < y \\ Log a x < Log a y\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x < y \\ Log a x < Log a y\n[PROOF STEP]\nassume \"x < y\"\n[PROOF STATE]\nproof (state)\nthis:\nx < y\n\ngoal (1 subgoal):\n 1. x < y \\ Log a x < Log a y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx < y\n[PROOF STEP]\nshow \"Log a x < Log a y\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx < y\n\ngoal (1 subgoal):\n 1. Log a x < Log a y\n[PROOF STEP]\nusing assms(1) assms(2) powreal_less_cancel_iff\n[PROOF STATE]\nproof (prove)\nusing this:\nx < y\n1 < a\n0 < x\n1 < ?a \\ (?a pow\\<^sub>\\ ?r < ?a pow\\<^sub>\\ ?s) = (?r < ?s)\n\ngoal (1 subgoal):\n 1. Log a x < Log a y\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nLog a x < Log a y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 954, "file": "Real_Power_Log", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.855851154320682, "lm_q2_score": 0.8652240808393984, "lm_q1q2_score": 0.7405030283324502}} {"text": "[STATEMENT]\nlemma linorder_rank_singleton: \n \"linorder_rank R {y} x = (if x \\ y \\ (y,x) \\ R then 1 else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. linorder_rank R {y} x = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. linorder_rank R {y} x = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n[PROOF STEP]\nhave \"linorder_rank R {y} x = card {z\\{y}-{x}. (z,x) \\ R}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. linorder_rank R {y} x = card {z \\ {y} - {x}. (z, x) \\ R}\n[PROOF STEP]\nby (simp add: linorder_rank_def)\n[PROOF STATE]\nproof (state)\nthis:\nlinorder_rank R {y} x = card {z \\ {y} - {x}. (z, x) \\ R}\n\ngoal (1 subgoal):\n 1. linorder_rank R {y} x = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlinorder_rank R {y} x = card {z \\ {y} - {x}. (z, x) \\ R}\n\ngoal (1 subgoal):\n 1. linorder_rank R {y} x = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n[PROOF STEP]\nhave \"{z\\{y}-{x}. (z,x) \\ R} = (if x \\ y \\ (y,x) \\ R then {y} else {})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {z \\ {y} - {x}. (z, x) \\ R} = (if x \\ y \\ (y, x) \\ R then {y} else {})\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{z \\ {y} - {x}. (z, x) \\ R} = (if x \\ y \\ (y, x) \\ R then {y} else {})\n\ngoal (1 subgoal):\n 1. linorder_rank R {y} x = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n{z \\ {y} - {x}. (z, x) \\ R} = (if x \\ y \\ (y, x) \\ R then {y} else {})\n\ngoal (1 subgoal):\n 1. linorder_rank R {y} x = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n[PROOF STEP]\nhave \"card \\ = (if x \\ y \\ (y,x) \\ R then 1 else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (if x \\ y \\ (y, x) \\ R then {y} else {}) = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (if x \\ y \\ (y, x) \\ R then {y} else {}) = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n\ngoal (1 subgoal):\n 1. linorder_rank R {y} x = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nlinorder_rank R {y} x = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlinorder_rank R {y} x = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n\ngoal (1 subgoal):\n 1. linorder_rank R {y} x = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nlinorder_rank R {y} x = (if x \\ y \\ (y, x) \\ R then 1 else 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1418, "file": "Comparison_Sort_Lower_Bound_Linorder_Relations", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.865224072151174, "lm_q2_score": 0.8558511506439708, "lm_q1q2_score": 0.7405030177154442}} {"text": "[STATEMENT]\nlemma card_face_cycle_sets_conv:\n shows \"card (pre_digraph_map.face_cycle_sets G M) = length (lists_fcs (remdups (snd G_list)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\ninterpret digraph_map G M\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. digraph_map (with_proj (list_digraph G_list)) (to_map' (snd G_list) xss)\n[PROOF STEP]\nby (rule digraph_map)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\nhave \"face_cycle_sets = {face_cycle_set a | a. a \\ parcs G}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. face_cycle_sets = {face_cycle_set a |a. a \\ parcs (list_digraph G_list)}\n[PROOF STEP]\nby (auto simp: face_cycle_sets_def)\n[PROOF STATE]\nproof (state)\nthis:\nface_cycle_sets = {face_cycle_set a |a. a \\ parcs (list_digraph G_list)}\n\ngoal (1 subgoal):\n 1. card face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nface_cycle_sets = {face_cycle_set a |a. a \\ parcs (list_digraph G_list)}\n\ngoal (1 subgoal):\n 1. card face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\nhave \"\\ = sset (lists_fcs (remdups (snd G_list)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {face_cycle_set a |a. a \\ parcs (list_digraph G_list)} = sset (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\nunfolding sset_lists_fcs\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {face_cycle_set a |a. a \\ parcs (list_digraph G_list)} = {face_cycle_set a |a. a \\ set (remdups (snd G_list))}\n[PROOF STEP]\nby (simp add: list_digraph_simps)\n[PROOF STATE]\nproof (state)\nthis:\n{face_cycle_set a |a. a \\ parcs (list_digraph G_list)} = sset (lists_fcs (remdups (snd G_list)))\n\ngoal (1 subgoal):\n 1. card face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n{face_cycle_set a |a. a \\ parcs (list_digraph G_list)} = sset (lists_fcs (remdups (snd G_list)))\n\ngoal (1 subgoal):\n 1. card face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\nhave \"card \\ = length (lists_fcs (remdups (snd G_list)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (sset (lists_fcs (remdups (snd G_list)))) = length (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\nby (simp add: card_image distincts_inj_on_set distinct_card distincts_distinct distincts_lists_fcs)\n[PROOF STATE]\nproof (state)\nthis:\ncard (sset (lists_fcs (remdups (snd G_list)))) = length (lists_fcs (remdups (snd G_list)))\n\ngoal (1 subgoal):\n 1. card face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n\ngoal (1 subgoal):\n 1. card face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard face_cycle_sets = length (lists_fcs (remdups (snd G_list)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1450, "file": "Planarity_Certificates_Planarity_Digraph_Map_Impl", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.893309411735131, "lm_q2_score": 0.8289388146603365, "lm_q1q2_score": 0.7404988448886419}} {"text": "[STATEMENT]\nlemma scalar_prod_ge_0: \"(x :: 'a :: linordered_idom vec) \\ x \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) \\ x \\ x\n[PROOF STEP]\nunfolding scalar_prod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) \\ (\\i = 0.. 'b::conditionally_complete_linorder\"\n assumes \"mono f\" \"bij f\" \"A \\ {}\" \"bdd_below A\"\n shows \"f (Inf A) = Inf (f`A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (\\ A) = \\ (f ` A)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. f (\\ A) = \\ (f ` A)\n[PROOF STEP]\nhave \"(inv f) (Inf (f`A)) \\ Inf ((inv f)`(f`A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inv f (\\ (f ` A)) \\ \\ (inv f ` f ` A)\n[PROOF STEP]\napply (rule cInf_greatest, auto simp add: assms(3))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xb. xb \\ A \\ inv f (\\ (f ` A)) \\ inv f (f xb)\n[PROOF STEP]\nusing mono_inv[OF assms(1) assms(2)] assms\n[PROOF STATE]\nproof (prove)\nusing this:\nmono (inv f)\nmono f\nbij f\nA \\ {}\nbdd_below A\n\ngoal (1 subgoal):\n 1. \\xb. xb \\ A \\ inv f (\\ (f ` A)) \\ inv f (f xb)\n[PROOF STEP]\nby (simp add: mono_def bdd_below_image_mono cInf_lower)\n[PROOF STATE]\nproof (state)\nthis:\ninv f (\\ (f ` A)) \\ \\ (inv f ` f ` A)\n\ngoal (1 subgoal):\n 1. f (\\ A) = \\ (f ` A)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ninv f (\\ (f ` A)) \\ \\ (inv f ` f ` A)\n[PROOF STEP]\nhave \"Inf (f`A) \\ f (Inf ((inv f)`(f`A)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninv f (\\ (f ` A)) \\ \\ (inv f ` f ` A)\n\ngoal (1 subgoal):\n 1. \\ (f ` A) \\ f (\\ (inv f ` f ` A))\n[PROOF STEP]\nby (metis (no_types, lifting) assms(1) assms(2) mono_def bij_inv_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\n\\ (f ` A) \\ f (\\ (inv f ` f ` A))\n\ngoal (1 subgoal):\n 1. f (\\ A) = \\ (f ` A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\ (f ` A) \\ f (\\ (inv f ` f ` A))\n\ngoal (1 subgoal):\n 1. f (\\ A) = \\ (f ` A)\n[PROOF STEP]\nhave \"... = f(Inf A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (\\ (inv f ` f ` A)) = f (\\ A)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nmono f\nbij f\nA \\ {}\nbdd_below A\n\ngoal (1 subgoal):\n 1. f (\\ (inv f ` f ` A)) = f (\\ A)\n[PROOF STEP]\nby (simp add: bij_is_inj)\n[PROOF STATE]\nproof (state)\nthis:\nf (\\ (inv f ` f ` A)) = f (\\ A)\n\ngoal (1 subgoal):\n 1. f (\\ A) = \\ (f ` A)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ (f ` A) \\ f (\\ A)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (f ` A) \\ f (\\ A)\n\ngoal (1 subgoal):\n 1. f (\\ A) = \\ (f ` A)\n[PROOF STEP]\nusing mono_cInf[OF assms(1) assms(3) assms(4)]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (f ` A) \\ f (\\ A)\nf (\\ A) \\ \\ (f ` A)\n\ngoal (1 subgoal):\n 1. f (\\ A) = \\ (f ` A)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nf (\\ A) = \\ (f ` A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1523, "file": "Ergodic_Theory_SG_Library_Complement", "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094032139577, "lm_q2_score": 0.8289388019824947, "lm_q1q2_score": 0.7404988264998754}} {"text": "[STATEMENT]\nlemma gbinomial_factors: \"((a + 1) gchoose (Suc k)) = (a + 1) / of_nat (Suc k) * (a gchoose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nproof (cases k)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. k = 0 \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n 2. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nk = 0\n\ngoal (2 subgoals):\n 1. k = 0 \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n 2. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nk = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nk = 0\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\ncase (Suc b)\n[PROOF STATE]\nproof (state)\nthis:\nk = Suc b\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nk = Suc b\n[PROOF STEP]\nhave \"((a + 1) gchoose (Suc (Suc b))) = (\\i = 0 .. Suc b. a + (1 - of_nat i)) / fact (b + 2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nk = Suc b\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n[PROOF STEP]\nby (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc (Suc b) = (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) / fact (b + 2)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nhave \"(\\i = 0 .. Suc b. a + (1 - of_nat i)) = (a + 1) * (\\i = 0..b. a - of_nat i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n[PROOF STEP]\nby (simp add: prod.atLeast0_atMost_Suc_shift del: prod.cl_ivl_Suc)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc b. a + ((1::'a) - of_nat i)) = (a + (1::'a)) * (\\i = 0..b. a - of_nat i)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nhave \"\\ / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a + (1::'a)) * (\\i = 0..b. a - of_nat i) / fact (b + 2) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n[PROOF STEP]\nby (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)\n[PROOF STATE]\nproof (state)\nthis:\n(a + (1::'a)) * (\\i = 0..b. a - of_nat i) / fact (b + 2) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n\ngoal (1 subgoal):\n 1. \\nat. k = Suc nat \\ a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\na + (1::'a) gchoose Suc (Suc b) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na + (1::'a) gchoose Suc (Suc b) = (a + (1::'a)) / of_nat (Suc (Suc b)) * (a gchoose Suc b)\n\ngoal (1 subgoal):\n 1. a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n[PROOF STEP]\nby (simp add: Suc)\n[PROOF STATE]\nproof (state)\nthis:\na + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2429, "file": null, "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.877476800298183, "lm_q2_score": 0.8438951005915208, "lm_q1q2_score": 0.740498372654361}} {"text": "[STATEMENT]\nlemma arctan_add_raw:\n assumes \"\\arctan x + arctan y\\ < pi/2\"\n shows \"arctan x + arctan y = arctan((x + y) / (1 - x * y))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arctan x + arctan y = arctan ((x + y) / (1 - x * y))\n[PROOF STEP]\nproof (rule arctan_unique [symmetric])\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. - (pi / 2) < arctan x + arctan y\n 2. arctan x + arctan y < pi / 2\n 3. tan (arctan x + arctan y) = (x + y) / (1 - x * y)\n[PROOF STEP]\nshow 12: \"- (pi / 2) < arctan x + arctan y\" \"arctan x + arctan y < pi / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (pi / 2) < arctan x + arctan y &&& arctan x + arctan y < pi / 2\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\arctan x + arctan y\\ < pi / 2\n\ngoal (1 subgoal):\n 1. - (pi / 2) < arctan x + arctan y &&& arctan x + arctan y < pi / 2\n[PROOF STEP]\nby linarith+\n[PROOF STATE]\nproof (state)\nthis:\n- (pi / 2) < arctan x + arctan y\narctan x + arctan y < pi / 2\n\ngoal (1 subgoal):\n 1. tan (arctan x + arctan y) = (x + y) / (1 - x * y)\n[PROOF STEP]\nshow \"tan (arctan x + arctan y) = (x + y) / (1 - x * y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tan (arctan x + arctan y) = (x + y) / (1 - x * y)\n[PROOF STEP]\nusing cos_gt_zero_pi [OF 12]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < cos (arctan x + arctan y)\n\ngoal (1 subgoal):\n 1. tan (arctan x + arctan y) = (x + y) / (1 - x * y)\n[PROOF STEP]\nby (simp add: arctan tan_add)\n[PROOF STATE]\nproof (state)\nthis:\ntan (arctan x + arctan y) = (x + y) / (1 - x * y)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 779, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.877476793890012, "lm_q2_score": 0.8438950986284991, "lm_q1q2_score": 0.7404983655240308}} {"text": "[STATEMENT]\nlemma sum_splice:\n \"(\\i::nat = 0..<2*nn. f i) = (\\i = 0..i = 0..i = 0..i = 0..i = 0..<0. f (2 * i)) + (\\i = 0..<0. f (2 * i + 1))\n 2. \\nn. sum f {0..<2 * nn} = (\\i = 0..i = 0.. sum f {0..<2 * Suc nn} = (\\i = 0..i = 0..i = 0..i = 0..i = 0..<0. f (2 * i)) + (\\i = 0..<0. f (2 * i + 1))\n 2. \\nn. sum f {0..<2 * nn} = (\\i = 0..i = 0.. sum f {0..<2 * Suc nn} = (\\i = 0..i = 0..i::nat = 0..<2*(n+1). f i) = (\\i::nat = 0..<(2*n). f i) + f(2*n+1) + f (2*n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f {0..<2 * (n + 1)} = sum f {0..<2 * n} + f (2 * n + 1) + f (2 * n)\n[PROOF STEP]\nby( simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nsum f {0..<2 * (n + 1)} = sum f {0..<2 * n} + f (2 * n + 1) + f (2 * n)\n\ngoal (2 subgoals):\n 1. sum f {0..<2 * 0} = (\\i = 0..<0. f (2 * i)) + (\\i = 0..<0. f (2 * i + 1))\n 2. \\nn. sum f {0..<2 * nn} = (\\i = 0..i = 0.. sum f {0..<2 * Suc nn} = (\\i = 0..i = 0..i = 0..<0. f (2 * i)) + (\\i = 0..<0. f (2 * i + 1))\n 2. \\nn. sum f {0..<2 * nn} = (\\i = 0..i = 0.. sum f {0..<2 * Suc nn} = (\\i = 0..i = 0.. = (\\i::nat = 0..i::nat = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..<0. f (2 * i)) + (\\i = 0..<0. f (2 * i + 1))\n 2. \\nn. sum f {0..<2 * nn} = (\\i = 0..i = 0.. sum f {0..<2 * Suc nn} = (\\i = 0..i = 0..i = 0..i = 0..i = 0..<0. f (2 * i)) + (\\i = 0..<0. f (2 * i + 1))\n 2. \\nn. sum f {0..<2 * nn} = (\\i = 0..i = 0.. sum f {0..<2 * Suc nn} = (\\i = 0..i = 0.. = (\\i::nat = 0..<(Suc n). f (2*i)) + (\\i::nat = 0..<(Suc n). f (2*i+1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..<0. f (2 * i)) + (\\i = 0..<0. f (2 * i + 1))\n 2. \\nn. sum f {0..<2 * nn} = (\\i = 0..i = 0.. sum f {0..<2 * Suc nn} = (\\i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..<0. f (2 * i)) + (\\i = 0..<0. f (2 * i + 1))\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 3099, "file": "Number_Theoretic_Transform_Preliminary_Lemmas", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392909114836, "lm_q2_score": 0.837619959279793, "lm_q1q2_score": 0.740488954855014}} {"text": "[STATEMENT]\ntheorem omega_lfp: \n \"x ^ \\ * y = lfp (\\ z . (x * z) \\ y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x ^ \\ * y = lfp (\\z. x * z \\ y)\n[PROOF STEP]\napply (rule order.antisym)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. x ^ \\ * y \\ lfp (\\z. x * z \\ y)\n 2. lfp (\\z. x * z \\ y) \\ x ^ \\ * y\n[PROOF STEP]\napply (rule lfp_greatest)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\u. x * u \\ y \\ u \\ x ^ \\ * y \\ u\n 2. lfp (\\z. x * z \\ y) \\ x ^ \\ * y\n[PROOF STEP]\napply (drule omega_least, simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lfp (\\z. x * z \\ y) \\ x ^ \\ * y\n[PROOF STEP]\napply (rule lfp_lowerbound)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * (x ^ \\ * y) \\ y \\ x ^ \\ * y\n[PROOF STEP]\napply (subst (2) omega_fix)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * (x ^ \\ * y) \\ y \\ x * x ^ \\ \\ (1::'a) * y\n[PROOF STEP]\nby (simp add: inf_comp mult.assoc)", "meta": {"llama_tokens": 536, "file": "MonoBoolTranAlgebra_Mono_Bool_Tran_Algebra", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392725805822, "lm_q2_score": 0.837619961306541, "lm_q1q2_score": 0.7404889412924099}} {"text": "[STATEMENT]\nlemma coeff_sum: \"MPoly_Type.coeff (sum f S) x = sum (\\i. MPoly_Type.coeff (f i) x) S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. MPoly_Type.coeff (sum f S) x = (\\i\\S. MPoly_Type.coeff (f i) x)\n[PROOF STEP]\napply (induction S rule: infinite_finite_induct)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\A. infinite A \\ MPoly_Type.coeff (sum f A) x = (\\i\\A. MPoly_Type.coeff (f i) x)\n 2. MPoly_Type.coeff (sum f {}) x = (\\i\\{}. MPoly_Type.coeff (f i) x)\n 3. \\xa F. \\finite F; xa \\ F; MPoly_Type.coeff (sum f F) x = (\\i\\F. MPoly_Type.coeff (f i) x)\\ \\ MPoly_Type.coeff (sum f (insert xa F)) x = (\\i\\insert xa F. MPoly_Type.coeff (f i) x)\n[PROOF STEP]\napply (auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xa F. \\finite F; xa \\ F; MPoly_Type.coeff (sum f F) x = (\\i\\F. MPoly_Type.coeff (f i) x)\\ \\ MPoly_Type.coeff (f xa + sum f F) x = MPoly_Type.coeff (f xa) x + (\\i\\F. MPoly_Type.coeff (f i) x)\n[PROOF STEP]\nby (metis More_MPoly_Type.coeff_add)", "meta": {"llama_tokens": 549, "file": "Virtual_Substitution_ExecutiblePolyProps", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297914570319, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7404058121324824}} {"text": "[STATEMENT]\nlemma matrix_construction_is_kronecker_product: \n fixes qs1 :: \"real poly list\"\n fixes subs1 subs2 :: \"nat list list\"\n fixes signs1 signs2 :: \"rat list list\"\n (* n1 is the number of polynomials in the \"1\" sets *)\n assumes \"\\l i. l \\ set subs1 \\ i \\ set l \\ i < n1\"\n assumes \"\\j. j \\ set signs1 \\ length j = n1\"\n shows \"\n (matrix_A (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2)) =\n kronecker_product (matrix_A signs1 subs1) (matrix_A signs2 subs2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2)\n[PROOF STEP]\nunfolding mat_eq_iff dim_row_matrix_A dim_col_matrix_A\n length_subsets_smash length_signs_smash dim_kronecker\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length subs1 * length subs2 = length subs1 * length subs2 \\ length signs1 * length signs2 = length signs1 * length signs2 \\ (\\i j. i < length subs1 * length subs2 \\ j < length signs1 * length signs2 \\ M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j))\n[PROOF STEP]\nproof safe\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nfix i j\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nassume i: \"i < length subs1 * length subs2\"\n and j: \"j < length signs1 * length signs2\"\n[PROOF STATE]\nproof (state)\nthis:\ni < length subs1 * length subs2\nj < length signs1 * length signs2\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave ld: \"i div length subs2 < length subs1\"\n \"j div length signs2 < length signs1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i div length subs2 < length subs1 &&& j div length signs2 < length signs1\n[PROOF STEP]\nusing i j less_mult_imp_div_less\n[PROOF STATE]\nproof (prove)\nusing this:\ni < length subs1 * length subs2\nj < length signs1 * length signs2\n?m < ?i * ?n \\ ?m div ?n < ?i\n\ngoal (1 subgoal):\n 1. i div length subs2 < length subs1 &&& j div length signs2 < length signs1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ni div length subs2 < length subs1\nj div length signs2 < length signs1\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave lm: \"i mod length subs2 < length subs2\"\n \"j mod length signs2 < length signs2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i mod length subs2 < length subs2 &&& j mod length signs2 < length signs2\n[PROOF STEP]\nusing i j less_mult_imp_mod_less\n[PROOF STATE]\nproof (prove)\nusing this:\ni < length subs1 * length subs2\nj < length signs1 * length signs2\n?m < ?n * ?i \\ ?m mod ?i < ?i\n\ngoal (1 subgoal):\n 1. i mod length subs2 < length subs2 &&& j mod length signs2 < length signs2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ni mod length subs2 < length subs2\nj mod length signs2 < length signs2\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave n1: \"n1 = length (signs1 ! (j div length signs2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n1 = length (signs1 ! (j div length signs2))\n[PROOF STEP]\nusing assms(2) ld(2) nth_mem\n[PROOF STATE]\nproof (prove)\nusing this:\n?j \\ set signs1 \\ length ?j = n1\nj div length signs2 < length signs1\n?n < length ?xs \\ ?xs ! ?n \\ set ?xs\n\ngoal (1 subgoal):\n 1. n1 = length (signs1 ! (j div length signs2))\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nn1 = length (signs1 ! (j div length signs2))\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave 1: \"matrix_A (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) =\n z (subsets_smash n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = z (subsets_smash n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\n[PROOF STEP]\nunfolding mat_of_rows_list_def matrix_A_def mtx_row_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (length (map (\\index_list. map (z index_list) (signs_smash signs1 signs2)) (subsets_smash n1 subs1 subs2))) (length (signs_smash signs1 signs2)) (\\(i, y). map (\\index_list. map (z index_list) (signs_smash signs1 signs2)) (subsets_smash n1 subs1 subs2) ! i ! y) $$ (i, j) = z (subsets_smash n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\n[PROOF STEP]\nusing i j\n[PROOF STATE]\nproof (prove)\nusing this:\ni < length subs1 * length subs2\nj < length signs1 * length signs2\n\ngoal (1 subgoal):\n 1. mat (length (map (\\index_list. map (z index_list) (signs_smash signs1 signs2)) (subsets_smash n1 subs1 subs2))) (length (signs_smash signs1 signs2)) (\\(i, y). map (\\index_list. map (z index_list) (signs_smash signs1 signs2)) (subsets_smash n1 subs1 subs2) ! i ! y) $$ (i, j) = z (subsets_smash n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\n[PROOF STEP]\nby (simp add: length_signs_smash length_subsets_smash)\n[PROOF STATE]\nproof (state)\nthis:\nM_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = z (subsets_smash n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave 2: \" ... = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2)))\n (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z (subsets_smash n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\nunfolding signs_smash_def subsets_smash_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z (concat (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1) ! i) (concat (map (\\l1. map ((@) l1) signs2) signs1) ! j) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (subst length_eq_concat)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x. x \\ set (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1) \\ length x = ?n\n 2. i < ?n * length (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1)\n 3. z (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1 ! (i div ?n) ! (i mod ?n)) (concat (map (\\l1. map ((@) l1) signs2) signs1) ! j) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\nusing i\n[PROOF STATE]\nproof (prove)\nusing this:\ni < length subs1 * length subs2\n\ngoal (3 subgoals):\n 1. \\x. x \\ set (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1) \\ length x = ?n\n 2. i < ?n * length (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1)\n 3. z (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1 ! (i div ?n) ! (i mod ?n)) (concat (map (\\l1. map ((@) l1) signs2) signs1) ! j) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (auto simp add: mult.commute)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i < length subs1 * length subs2 \\ z (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1 ! (i div length subs2) ! (i mod length subs2)) (concat (map (\\l1. map ((@) l1) signs2) signs1) ! j) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (subst length_eq_concat)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x. \\i < length subs1 * length subs2; x \\ set (map (\\l1. map ((@) l1) signs2) signs1)\\ \\ length x = ?n11\n 2. i < length subs1 * length subs2 \\ j < ?n11 * length (map (\\l1. map ((@) l1) signs2) signs1)\n 3. i < length subs1 * length subs2 \\ z (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1 ! (i div length subs2) ! (i mod length subs2)) (map (\\l1. map ((@) l1) signs2) signs1 ! (j div ?n11) ! (j mod ?n11)) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\nusing j\n[PROOF STATE]\nproof (prove)\nusing this:\nj < length signs1 * length signs2\n\ngoal (3 subgoals):\n 1. \\x. \\i < length subs1 * length subs2; x \\ set (map (\\l1. map ((@) l1) signs2) signs1)\\ \\ length x = ?n11\n 2. i < length subs1 * length subs2 \\ j < ?n11 * length (map (\\l1. map ((@) l1) signs2) signs1)\n 3. i < length subs1 * length subs2 \\ z (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1 ! (i div length subs2) ! (i mod length subs2)) (map (\\l1. map ((@) l1) signs2) signs1 ! (j div ?n11) ! (j mod ?n11)) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (auto simp add: mult.commute)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ z (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1 ! (i div length subs2) ! (i mod length subs2)) (map (\\l1. map ((@) l1) signs2) signs1 ! (j div length signs2) ! (j mod length signs2)) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\nusing ld lm\n[PROOF STATE]\nproof (prove)\nusing this:\ni div length subs2 < length subs1\nj div length signs2 < length signs1\ni mod length subs2 < length subs2\nj mod length signs2 < length signs2\n\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ z (map (\\l1. map (\\l2. l1 @ map ((+) n1) l2) subs2) subs1 ! (i div length subs2) ! (i mod length subs2)) (map (\\l1. map ((@) l1) signs2) signs1 ! (j div length signs2) ! (j mod length signs2)) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nz (subsets_smash n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave 3: \"... =\n z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) *\n z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2)) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\nunfolding n1\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z (subs1 ! (i div length subs2) @ map ((+) (length (signs1 ! (j div length signs2)))) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2)) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (subst z_append)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\ia. ia \\ set (subs1 ! (i div length subs2)) \\ ia < length (signs1 ! (j div length signs2))\n 2. z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2)) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (auto simp add: n1[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ia. ia \\ set (subs1 ! (i div length subs2)) \\ ia < n1\n[PROOF STEP]\nusing assms(1) ld(1) nth_mem\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?l \\ set subs1; ?i \\ set ?l\\ \\ ?i < n1\ni div length subs2 < length subs1\n?n < length ?xs \\ ?xs ! ?n \\ set ?xs\n\ngoal (1 subgoal):\n 1. \\ia. ia \\ set (subs1 ! (i div length subs2)) \\ ia < n1\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nz (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2)) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave 4: \"kronecker_product (matrix_A signs1 subs1) (matrix_A signs2 subs2) $$ (i,j) =\n z (subs1 ! (i div length subs2))\n (signs1 ! (j div length signs2)) *\n z (subs2 ! (i mod length subs2))\n (signs2 ! (j mod length signs2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\nunfolding kronecker_product_def matrix_A_def mat_of_rows_list_def mtx_row_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (let ra = dim_row (mat (length (map (\\index_list. map (z index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z index_list) signs1) subs1 ! i ! y)); ca = dim_col (mat (length (map (\\index_list. map (z index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z index_list) signs1) subs1 ! i ! y)); rb = dim_row (mat (length (map (\\index_list. map (z index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z index_list) signs2) subs2 ! i ! y)); cb = dim_col (mat (length (map (\\index_list. map (z index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z index_list) signs2) subs2 ! i ! y)) in mat (ra * rb) (ca * cb) (\\(i, j). mat (length (map (\\index_list. map (z index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z index_list) signs1) subs1 ! i ! y) $$ (i div rb, j div cb) * mat (length (map (\\index_list. map (z index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z index_list) signs2) subs2 ! i ! y) $$ (i mod rb, j mod cb))) $$ (i, j) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\nusing i j\n[PROOF STATE]\nproof (prove)\nusing this:\ni < length subs1 * length subs2\nj < length signs1 * length signs2\n\ngoal (1 subgoal):\n 1. (let ra = dim_row (mat (length (map (\\index_list. map (z index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z index_list) signs1) subs1 ! i ! y)); ca = dim_col (mat (length (map (\\index_list. map (z index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z index_list) signs1) subs1 ! i ! y)); rb = dim_row (mat (length (map (\\index_list. map (z index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z index_list) signs2) subs2 ! i ! y)); cb = dim_col (mat (length (map (\\index_list. map (z index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z index_list) signs2) subs2 ! i ! y)) in mat (ra * rb) (ca * cb) (\\(i, j). mat (length (map (\\index_list. map (z index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z index_list) signs1) subs1 ! i ! y) $$ (i div rb, j div cb) * mat (length (map (\\index_list. map (z index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z index_list) signs2) subs2 ! i ! y) $$ (i mod rb, j mod cb))) $$ (i, j) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (auto simp add: Let_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ mat (length subs1) (length signs1) (\\(i, y). map (\\index_list. map (z index_list) signs1) subs1 ! i ! y) $$ (i div length subs2, j div length signs2) * mat (length subs2) (length signs2) (\\(i, y). map (\\index_list. map (z index_list) signs2) subs2 ! i ! y) $$ (i mod length subs2, j mod length signs2) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (subst index_mat(1)[OF ld])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ (case (i div length subs2, j div length signs2) of (i, x) \\ map (\\index_list. map (z index_list) signs1) subs1 ! i ! x) * mat (length subs2) (length signs2) (\\(i, y). map (\\index_list. map (z index_list) signs2) subs2 ! i ! y) $$ (i mod length subs2, j mod length signs2) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (subst index_mat(1)[OF lm])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ (case (i div length subs2, j div length signs2) of (i, x) \\ map (\\index_list. map (z index_list) signs1) subs1 ! i ! x) * (case (i mod length subs2, j mod length signs2) of (i, x) \\ map (\\index_list. map (z index_list) signs2) subs2 ! i ! x) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\nusing ld lm\n[PROOF STATE]\nproof (prove)\nusing this:\ni div length subs2 < length subs1\nj div length signs2 < length signs1\ni mod length subs2 < length subs2\nj mod length signs2 < length signs2\n\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ (case (i div length subs2, j div length signs2) of (i, x) \\ map (\\index_list. map (z index_list) signs1) subs1 ! i ! x) * (case (i mod length subs2, j mod length signs2) of (i, x) \\ map (\\index_list. map (z index_list) signs2) subs2 ! i ! x) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nkronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nshow \"matrix_A (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) =\n kronecker_product (matrix_A signs1 subs1) (matrix_A signs2 subs2) $$ (i, j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nusing 1 2 3 4\n[PROOF STATE]\nproof (prove)\nusing this:\nM_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = z (subsets_smash n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\nz (subsets_smash n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\nz (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2)) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\nkronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n\ngoal (1 subgoal):\n 1. M_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nM_mat (signs_smash signs1 signs2) (subsets_smash n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 9451, "file": "BenOr_Kozen_Reif_BKR_Proofs", "length": 45, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297781091839, "lm_q2_score": 0.8221891392358015, "lm_q1q2_score": 0.7404058031197972}} {"text": "[STATEMENT]\nlemma split_sum_mid_less: assumes \"i<(n::nat)\"\n shows \"(\\jjj=i..jj\\{.. {i.. {i..i < n\\\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n\n\ngoal (1 subgoal):\n 1. sum f {.. {i.. {i.. {i.. = (\\jj=i.. {i.. {i..jjj=i..x::int. (x + n) gchoose k) = (\\x. (x + n - 1) gchoose (k - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bw_diff (\\x. x + n gchoose k) = (\\x. x + n - 1 gchoose (k - 1))\n[PROOF STEP]\nproof (rule ext)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. bw_diff (\\x. x + n gchoose k) x = x + n - 1 gchoose (k - 1)\n[PROOF STEP]\nfix x::int\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. bw_diff (\\x. x + n gchoose k) x = x + n - 1 gchoose (k - 1)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < k\n[PROOF STEP]\nhave eq: \"Suc (k - Suc 0) = k\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < k\n\ngoal (1 subgoal):\n 1. Suc (k - Suc 0) = k\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nSuc (k - Suc 0) = k\n\ngoal (1 subgoal):\n 1. \\x. bw_diff (\\x. x + n gchoose k) x = x + n - 1 gchoose (k - 1)\n[PROOF STEP]\nhave \"x + n gchoose k = (x + n - 1) + 1 gchoose (Suc (k - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x + n gchoose k = x + n - 1 + 1 gchoose Suc (k - 1)\n[PROOF STEP]\nby (simp add: eq)\n[PROOF STATE]\nproof (state)\nthis:\nx + n gchoose k = x + n - 1 + 1 gchoose Suc (k - 1)\n\ngoal (1 subgoal):\n 1. \\x. bw_diff (\\x. x + n gchoose k) x = x + n - 1 gchoose (k - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx + n gchoose k = x + n - 1 + 1 gchoose Suc (k - 1)\n\ngoal (1 subgoal):\n 1. \\x. bw_diff (\\x. x + n gchoose k) x = x + n - 1 gchoose (k - 1)\n[PROOF STEP]\nhave \"\\ = (x + n - 1) gchoose (k - 1) + ((x + n - 1) gchoose (Suc (k - 1)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x + n - 1 + 1 gchoose Suc (k - 1) = x + n - 1 gchoose (k - 1) + (x + n - 1 gchoose Suc (k - 1))\n[PROOF STEP]\nby (fact gbinomial_int_Suc_Suc)\n[PROOF STATE]\nproof (state)\nthis:\nx + n - 1 + 1 gchoose Suc (k - 1) = x + n - 1 gchoose (k - 1) + (x + n - 1 gchoose Suc (k - 1))\n\ngoal (1 subgoal):\n 1. \\x. bw_diff (\\x. x + n gchoose k) x = x + n - 1 gchoose (k - 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nx + n gchoose k = x + n - 1 gchoose (k - 1) + (x + n - 1 gchoose Suc (k - 1))\n[PROOF STEP]\nshow \"bw_diff (\\x. x + n gchoose k) x = x + n - 1 gchoose (k - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx + n gchoose k = x + n - 1 gchoose (k - 1) + (x + n - 1 gchoose Suc (k - 1))\n\ngoal (1 subgoal):\n 1. bw_diff (\\x. x + n gchoose k) x = x + n - 1 gchoose (k - 1)\n[PROOF STEP]\nby (simp add: eq bw_diff_def algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nbw_diff (\\x. x + n gchoose k) x = x + n - 1 gchoose (k - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1343, "file": "Groebner_Macaulay_Binomial_Int", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267728417087, "lm_q2_score": 0.8519527982093666, "lm_q1q2_score": 0.7401994002817075}} {"text": "[STATEMENT]\nlemma \" ((a::int) + b + c) ^ 2 = a ^ 2 + b ^ 2 + c ^ 2 + 2 * a * b + 2 * b * c + 2 * a * c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a + b + c)\\<^sup>2 = a\\<^sup>2 + b\\<^sup>2 + c\\<^sup>2 + 2 * a * b + 2 * b * c + 2 * a * c\n[PROOF STEP]\nby ring", "meta": {"llama_tokens": 143, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9362850004144265, "lm_q2_score": 0.7905303162021596, "lm_q1q2_score": 0.7401616774329558}} {"text": "[STATEMENT]\nlemma fix_ind_k:\n fixes k::nat\n assumes adm: \"adm P\"\n assumes base_k_steps: \"\\if\\\\)\"\n assumes step: \"\\x. (\\if\\x)) \\ P (iterate k\\f\\x)\"\n shows \"P (fix\\f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. P (fix\\f)\n[PROOF STEP]\nunfolding fix_def2\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. P (\\i. iterate i\\f\\\\)\n[PROOF STEP]\napply (rule admD [OF adm chain_iterate])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. P (iterate i\\f\\\\)\n[PROOF STEP]\napply(rule nat_k_induct[of k], simp add: base_k_steps)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. \\n\\<^sub>0. (\\i0 + i)\\f\\\\)) \\ P (iterate (n\\<^sub>0 + k)\\f\\\\)\n[PROOF STEP]\nusing step\n[PROOF STATE]\nproof (prove)\nusing this:\n\\if\\?x) \\ P (iterate k\\f\\?x)\n\ngoal (1 subgoal):\n 1. \\i. \\n\\<^sub>0. (\\i0 + i)\\f\\\\)) \\ P (iterate (n\\<^sub>0 + k)\\f\\\\)\n[PROOF STEP]\nby (subst (1 2) add.commute, unfold iterate_iterate[symmetric]) blast", "meta": {"llama_tokens": 584, "file": "CSP_RefTK_Fix_ind_ext", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467706759584, "lm_q2_score": 0.8418256432832333, "lm_q1q2_score": 0.7400882957646658}} {"text": "[STATEMENT]\ntheorem borel_measurable_det_Jacobian:\n fixes f :: \"real^'n::{finite,wellorder} \\ real^'n::_\"\n assumes S: \"S \\ sets lebesgue\" and f: \"\\x. x \\ S \\ (f has_derivative f' x) (at x within S)\"\n shows \"(\\x. det(matrix(f' x))) \\ borel_measurable (lebesgue_on S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. det (matrix (f' x))) \\ borel_measurable (lebesgue_on S)\n[PROOF STEP]\nunfolding det_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. \\p | p permutes UNIV. real_of_int (sign p) * (\\i\\UNIV. matrix (f' x) $ i $ p i)) \\ borel_measurable (lebesgue_on S)\n[PROOF STEP]\nby (intro measurable) (auto intro: f borel_measurable_partial_derivatives [OF S])", "meta": {"llama_tokens": 322, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9184802462567085, "lm_q2_score": 0.8056321959813275, "lm_q1q2_score": 0.7399572577572626}} {"text": "[STATEMENT]\nlemma cos_double_cos: \"cos (2 * w) = 2 * cos w ^ 2 - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos ((2::'a) * w) = (2::'a) * (cos w)\\<^sup>2 - (1::'a)\n[PROOF STEP]\nby (simp add: cos_double sin_squared_eq)", "meta": {"llama_tokens": 114, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9184802395624259, "lm_q2_score": 0.8056321866478979, "lm_q1q2_score": 0.7399572437915622}} {"text": "[STATEMENT]\ntheorem content_std_simplex:\n \"measure lborel (convex hull (insert 0 Basis :: 'a :: euclidean_space set)) =\n 1 / fact DIM('a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. content (convex hull insert (0::'a) Basis) = 1 / fact DIM('a)\n[PROOF STEP]\nby (simp add: measure_def emeasure_std_simplex)", "meta": {"llama_tokens": 128, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9241418178895029, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.739952962386541}} {"text": "[STATEMENT]\nlemma set_integral_add [simp, intro]:\n fixes f g :: \"_ \\ _ :: {banach, second_countable_topology}\"\n assumes \"set_integrable M A f\" \"set_integrable M A g\"\n shows \"set_integrable M A (\\x. f x + g x)\"\n and \"LINT x:A|M. f x + g x = (LINT x:A|M. f x) + (LINT x:A|M. g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_integrable M A (\\x. f x + g x) &&& LINT x:A|M. f x + g x = set_lebesgue_integral M A f + set_lebesgue_integral M A g\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nset_integrable M A f\nset_integrable M A g\n\ngoal (1 subgoal):\n 1. set_integrable M A (\\x. f x + g x) &&& LINT x:A|M. f x + g x = set_lebesgue_integral M A f + set_lebesgue_integral M A g\n[PROOF STEP]\nunfolding set_integrable_def set_lebesgue_integral_def\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable M (\\x. indicat_real A x *\\<^sub>R f x)\nintegrable M (\\x. indicat_real A x *\\<^sub>R g x)\n\ngoal (1 subgoal):\n 1. integrable M (\\x. indicat_real A x *\\<^sub>R (f x + g x)) &&& LINT x|M. indicat_real A x *\\<^sub>R (f x + g x) = (LINT x|M. indicat_real A x *\\<^sub>R f x) + (LINT x|M. indicat_real A x *\\<^sub>R g x)\n[PROOF STEP]\nby (simp_all add: scaleR_add_right)", "meta": {"llama_tokens": 551, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894632969137, "lm_q2_score": 0.8267117940706734, "lm_q1q2_score": 0.7397330025177264}} {"text": "[STATEMENT]\nlemma sum_setv_subset_eq: \n assumes \"finite A\"\n assumes \"set\\<^sub>v v \\ A\"\n shows \"(\\ i \\ set\\<^sub>v v. count_vec v i) = (\\ i \\ A. count_vec v i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum (count_vec v) (set\\<^sub>v v) = sum (count_vec v) A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum (count_vec v) (set\\<^sub>v v) = sum (count_vec v) A\n[PROOF STEP]\nhave ni: \"\\ x. x \\ set\\<^sub>v v \\ count_vec v x = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ set\\<^sub>v v \\ count_vec v x = 0\n[PROOF STEP]\nby (simp add: setv_not_in_count0_iff)\n[PROOF STATE]\nproof (state)\nthis:\n?x \\ set\\<^sub>v v \\ count_vec v ?x = 0\n\ngoal (1 subgoal):\n 1. sum (count_vec v) (set\\<^sub>v v) = sum (count_vec v) A\n[PROOF STEP]\nhave \"(\\ i \\ A. count_vec v i) = (\\ i \\ A - (set\\<^sub>v v). count_vec v i) + (\\ i \\ (set\\<^sub>v v). count_vec v i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum (count_vec v) A = sum (count_vec v) (A - set\\<^sub>v v) + sum (count_vec v) (set\\<^sub>v v)\n[PROOF STEP]\nusing sum.subset_diff assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?B \\ ?A; finite ?A\\ \\ sum ?g ?A = sum ?g (?A - ?B) + sum ?g ?B\nfinite A\nset\\<^sub>v v \\ A\n\ngoal (1 subgoal):\n 1. sum (count_vec v) A = sum (count_vec v) (A - set\\<^sub>v v) + sum (count_vec v) (set\\<^sub>v v)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsum (count_vec v) A = sum (count_vec v) (A - set\\<^sub>v v) + sum (count_vec v) (set\\<^sub>v v)\n\ngoal (1 subgoal):\n 1. sum (count_vec v) (set\\<^sub>v v) = sum (count_vec v) A\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nsum (count_vec v) A = sum (count_vec v) (A - set\\<^sub>v v) + sum (count_vec v) (set\\<^sub>v v)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsum (count_vec v) A = sum (count_vec v) (A - set\\<^sub>v v) + sum (count_vec v) (set\\<^sub>v v)\n\ngoal (1 subgoal):\n 1. sum (count_vec v) (set\\<^sub>v v) = sum (count_vec v) A\n[PROOF STEP]\nusing ni\n[PROOF STATE]\nproof (prove)\nusing this:\nsum (count_vec v) A = sum (count_vec v) (A - set\\<^sub>v v) + sum (count_vec v) (set\\<^sub>v v)\n?x \\ set\\<^sub>v v \\ count_vec v ?x = 0\n\ngoal (1 subgoal):\n 1. sum (count_vec v) (set\\<^sub>v v) = sum (count_vec v) A\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum (count_vec v) (set\\<^sub>v v) = sum (count_vec v) A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1148, "file": "Fishers_Inequality_Matrix_Vector_Extras", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894520743981, "lm_q2_score": 0.8267118004748677, "lm_q1q2_score": 0.739732998970346}} {"text": "[STATEMENT]\nlemma finite_imp_compact_convex_hull:\n fixes S :: \"'a::real_normed_vector set\"\n assumes \"finite S\"\n shows \"compact (convex hull S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. compact (convex hull S)\n[PROOF STEP]\nproof (cases \"S = {}\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. S = {} \\ compact (convex hull S)\n 2. S \\ {} \\ compact (convex hull S)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nS = {}\n\ngoal (2 subgoals):\n 1. S = {} \\ compact (convex hull S)\n 2. S \\ {} \\ compact (convex hull S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nS = {}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nS = {}\n\ngoal (1 subgoal):\n 1. compact (convex hull S)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncompact (convex hull S)\n\ngoal (1 subgoal):\n 1. S \\ {} \\ compact (convex hull S)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. S \\ {} \\ compact (convex hull S)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nS \\ {}\n\ngoal (1 subgoal):\n 1. S \\ {} \\ compact (convex hull S)\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite S\nS \\ {}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\nS \\ {}\n\ngoal (1 subgoal):\n 1. compact (convex hull S)\n[PROOF STEP]\nproof (induct rule: finite_ne_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x. compact (convex hull {x})\n 2. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\ncase (singleton x)\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\x. compact (convex hull {x})\n 2. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. compact (convex hull {x})\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncompact (convex hull {x})\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\ncase (insert x A)\n[PROOF STATE]\nproof (state)\nthis:\nfinite A\nA \\ {}\nx \\ A\ncompact (convex hull A)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\nlet ?f = \"\\(u, y::'a). u *\\<^sub>R x + (1 - u) *\\<^sub>R y\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\nlet ?T = \"{0..1::real} \\ (convex hull A)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\nhave \"continuous_on ?T ?f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous_on ({0..1} \\ (convex hull A)) (\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y)\n[PROOF STEP]\nunfolding split_def continuous_on\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xa\\{0..1} \\ (convex hull A). ((\\p. fst p *\\<^sub>R x + (1 - fst p) *\\<^sub>R snd p) \\ fst xa *\\<^sub>R x + (1 - fst xa) *\\<^sub>R snd xa) (at xa within {0..1} \\ (convex hull A))\n[PROOF STEP]\nby (intro ballI tendsto_intros)\n[PROOF STATE]\nproof (state)\nthis:\ncontinuous_on ({0..1} \\ (convex hull A)) (\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncontinuous_on ({0..1} \\ (convex hull A)) (\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\nhave \"compact ?T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. compact ({0..1} \\ (convex hull A))\n[PROOF STEP]\nby (intro compact_Times compact_Icc insert)\n[PROOF STATE]\nproof (state)\nthis:\ncompact ({0..1} \\ (convex hull A))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncontinuous_on ({0..1} \\ (convex hull A)) (\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y)\ncompact ({0..1} \\ (convex hull A))\n[PROOF STEP]\nhave \"compact (?f ` ?T)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous_on ({0..1} \\ (convex hull A)) (\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y)\ncompact ({0..1} \\ (convex hull A))\n\ngoal (1 subgoal):\n 1. compact ((\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y) ` ({0..1} \\ (convex hull A)))\n[PROOF STEP]\nby (rule compact_continuous_image)\n[PROOF STATE]\nproof (state)\nthis:\ncompact ((\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y) ` ({0..1} \\ (convex hull A)))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncompact ((\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y) ` ({0..1} \\ (convex hull A)))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\nhave \"?f ` ?T = convex hull (insert x A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y) ` ({0..1} \\ (convex hull A)) = convex hull insert x A\n[PROOF STEP]\nunfolding convex_hull_insert [OF \\A \\ {}\\]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y) ` ({0..1} \\ (convex hull A)) = {xa. \\u\\0. \\v\\0. \\b. u + v = 1 \\ b \\ convex hull A \\ xa = u *\\<^sub>R x + v *\\<^sub>R b}\n[PROOF STEP]\napply safe\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\xa a b. \\a \\ {0..1}; b \\ convex hull A\\ \\ \\u\\0. \\v\\0. \\ba. u + v = 1 \\ ba \\ convex hull A \\ a *\\<^sub>R x + (1 - a) *\\<^sub>R b = u *\\<^sub>R x + v *\\<^sub>R ba\n 2. \\xa u v b. \\0 \\ u; 0 \\ v; u + v = 1; b \\ convex hull A\\ \\ u *\\<^sub>R x + v *\\<^sub>R b \\ (\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y) ` ({0..1} \\ (convex hull A))\n[PROOF STEP]\napply (rule_tac x=a in exI, simp)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\a b. \\0 \\ a \\ a \\ 1; b \\ convex hull A\\ \\ \\v\\0. a + v = 1 \\ (\\ba. ba \\ convex hull A \\ (1 - a) *\\<^sub>R b = v *\\<^sub>R ba)\n 2. \\xa u v b. \\0 \\ u; 0 \\ v; u + v = 1; b \\ convex hull A\\ \\ u *\\<^sub>R x + v *\\<^sub>R b \\ (\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y) ` ({0..1} \\ (convex hull A))\n[PROOF STEP]\napply (rule_tac x=\"1 - a\" in exI, simp, fast)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xa u v b. \\0 \\ u; 0 \\ v; u + v = 1; b \\ convex hull A\\ \\ u *\\<^sub>R x + v *\\<^sub>R b \\ (\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y) ` ({0..1} \\ (convex hull A))\n[PROOF STEP]\napply (rule_tac x=\"(u, b)\" in image_eqI, simp_all)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(\\(u, y). u *\\<^sub>R x + (1 - u) *\\<^sub>R y) ` ({0..1} \\ (convex hull A)) = convex hull insert x A\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; F \\ {}; x \\ F; compact (convex hull F)\\ \\ compact (convex hull insert x F)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncompact (convex hull insert x A)\n[PROOF STEP]\nshow \"compact (convex hull (insert x A))\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncompact (convex hull insert x A)\n\ngoal (1 subgoal):\n 1. compact (convex hull insert x A)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncompact (convex hull insert x A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncompact (convex hull S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3998, "file": null, "length": 39, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473779969194, "lm_q2_score": 0.8479677583778258, "lm_q1q2_score": 0.7397224506468216}} {"text": "[STATEMENT]\nlemma lin_comb_imp_lin_dep_fin: \n fixes A :: \"'a vec set\"\n assumes \"finite A\"\n assumes \"A \\ carrier_vec n\"\n assumes \"lincomb c A = 0\\<^sub>v n\"\n assumes \"\\ a. a \\ A \\ c a \\ 0\"\n shows \"lin_dep A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lin_dep A\n[PROOF STEP]\nunfolding lin_dep_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Aa a v. finite Aa \\ Aa \\ A \\ True \\ lincomb a Aa = 0\\<^sub>v n \\ v \\ Aa \\ a v \\ (0::'a)\n[PROOF STEP]\nusing assms lincomb_as_lincomb_list_distinct sumlist_nth\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nA \\ carrier_vec n\nlincomb c A = 0\\<^sub>v n\n\\a. a \\ A \\ c a \\ (0::'a)\n\\set ?ws \\ carrier_vec n; distinct ?ws\\ \\ lincomb ?f (set ?ws) = lincomb_list (\\i. ?f (?ws ! i)) ?ws\n\\\\x\\set ?xs. dim_vec x = n; ?i < n\\ \\ M.sumlist ?xs $ ?i = (\\j = 0..Aa a v. finite Aa \\ Aa \\ A \\ True \\ lincomb a Aa = 0\\<^sub>v n \\ v \\ Aa \\ a v \\ (0::'a)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 541, "file": "Fishers_Inequality_Linear_Bound_Argument", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473614033683, "lm_q2_score": 0.8479677564567913, "lm_q1q2_score": 0.7397224349002159}} {"text": "[STATEMENT]\nlemma mignotte_coeff_helper:\n \"abs (coeff h i) \\ \n (degree h - 1 choose i) * mahler_measure h +\n (min i 1 * (degree h - 1 choose (i - 1)) * abs (lead_coeff h))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_of_int \\poly.coeff h i\\ \\ real (degree h - 1 choose i) * mahler_measure h + real_of_int (int (min i 1 * (degree h - 1 choose (i - 1))) * \\lead_coeff h\\)\n[PROOF STEP]\nunfolding mahler_measure_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_of_int \\poly.coeff h i\\ \\ real (degree h - 1 choose i) * mahler_measure_poly (of_int_poly h) + real_of_int (int (min i 1 * (degree h - 1 choose (i - 1))) * \\lead_coeff h\\)\n[PROOF STEP]\nusing mignotte_helper_coeff[of \"of_int_poly h\" i]\n[PROOF STATE]\nproof (prove)\nusing this:\ncmod (poly.coeff (of_int_poly h) i) \\ real (degree (of_int_poly h) - 1 choose i) * mahler_measure_poly (of_int_poly h) + real (min i 1 * (degree (of_int_poly h) - 1 choose (i - 1))) * cmod (lead_coeff (of_int_poly h))\n\ngoal (1 subgoal):\n 1. real_of_int \\poly.coeff h i\\ \\ real (degree h - 1 choose i) * mahler_measure_poly (of_int_poly h) + real_of_int (int (min i 1 * (degree h - 1 choose (i - 1))) * \\lead_coeff h\\)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 575, "file": "Berlekamp_Zassenhaus_Factor_Bound", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.907312213841788, "lm_q2_score": 0.8152324960856175, "lm_q1q2_score": 0.7396704008192084}} {"text": "[STATEMENT]\nlemma eventually_ceiling_eq:\n fixes f::\"'a \\ 'b::{order_topology,floor_ceiling}\"\n assumes f: \"(f \\ l) F\"\n and l: \"l \\ \\\"\n shows \"\\\\<^sub>F x in F. ceiling (f x) = ceiling l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. \\f x\\ = \\l\\\n[PROOF STEP]\nusing eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in F. of_int \\l\\ < f x\n\\\\<^sub>F x in F. f x < of_int \\l\\\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. \\f x\\ = \\l\\\n[PROOF STEP]\nby eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)", "meta": {"llama_tokens": 326, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099070060380482, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7396336716046934}} {"text": "[STATEMENT]\nlemma euclid_diff: \n \"PRE (\\s::nat store. s ''x'' = x \\ s ''y'' = y \\ x > 0 \\ y > 0)\n (WHILE (\\s. s ''x''\\ s ''y'') INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) \n DO\n (IF (\\s. s ''x'' > s ''y'')\n THEN (''x'' ::= (\\s. s ''x'' - s ''y''))\n ELSE (''y'' ::= (\\s. s ''y'' - s ''x''))\n FI)\n OD)\n POST (\\s. s ''x'' = gcd x y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rdom \\\\s. s ''x'' = x \\ s ''y'' = y \\ zero_class.zero < x \\ zero_class.zero < y\\ \\ wp (WHILE (\\s. s ''x'' \\ s ''y'') INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) DO (IF (\\s. s ''y'' < s ''x'') THEN (''x'' ::= (\\s. s ''x'' - s ''y'')) ELSE (''y'' ::= (\\s. s ''y'' - s ''x'')) FI) OD) \\\\s. s ''x'' = gcd x y\\\n[PROOF STEP]\napply (rule ppath_aka.fbox_whilei, simp_all)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\\\s. s ''x'' = x \\ s ''y'' = y \\ zero_class.zero < x \\ zero_class.zero < y\\ \\ \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\\n 2. \\(\\s. gcd (s ''x'') (s ''y'') = gcd x y) \\ - (\\s. s ''x'' \\ s ''y'')\\ \\ \\\\s. s ''x'' = gcd x y\\\n 3. \\(\\s. gcd (s ''x'') (s ''y'') = gcd x y) \\ (\\s. s ''x'' \\ s ''y'')\\ \\ \\(- (\\s. s ''y'' < s ''x'') \\ (\\s. gcd (s ''x'' - s ''y'') (s ''y'') = gcd x y)) \\ ((\\s. s ''y'' < s ''x'') \\ (\\s. gcd (s ''x'') (s ''y'' - s ''x'') = gcd x y))\\\n[PROOF STEP]\napply (simp_all add: p2pp_def)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. {Node s |s. s ''x'' = x \\ s ''y'' = y \\ zero_class.zero < x \\ zero_class.zero < y} \\ {Node s |s. gcd (s ''x'') (s ''y'') = gcd x y}\n 2. {Node s |s. gcd (s ''x'') (s ''y'') = gcd x y \\ s ''x'' = s ''y''} \\ {Node s |s. s ''x'' = gcd x y}\n 3. {Node s |s. gcd (s ''x'') (s ''y'') = gcd x y \\ s ''x'' \\ s ''y''} \\ {Node s |s. (s ''y'' < s ''x'' \\ gcd (s ''x'' - s ''y'') (s ''y'') = gcd x y) \\ (s ''y'' < s ''x'' \\ gcd (s ''x'') (s ''y'' - s ''x'') = gcd x y)}\n[PROOF STEP]\napply auto[2]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {Node s |s. gcd (s ''x'') (s ''y'') = gcd x y \\ s ''x'' \\ s ''y''} \\ {Node s |s. (s ''y'' < s ''x'' \\ gcd (s ''x'' - s ''y'') (s ''y'') = gcd x y) \\ (s ''y'' < s ''x'' \\ gcd (s ''x'') (s ''y'' - s ''x'') = gcd x y)}\n[PROOF STEP]\nby (safe, metis gcd.commute gcd_diff1_nat le_cases nat_less_le)", "meta": {"llama_tokens": 1330, "file": "Algebraic_VCs_AVC_KAD_Path_Model_Example", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505402422644, "lm_q2_score": 0.817574478416099, "lm_q1q2_score": 0.7396191935874115}} {"text": "[STATEMENT]\nlemma rank_1_proj_coord:\n assumes \"i < dim_vec v\"\n and \"j < dim_vec v\"\nshows \"(rank_1_proj v) $$ (i, j) = Matrix.vec_index v i * (cnj (Matrix.vec_index v j))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank_1_proj v $$ (i, j) = v $ i * cnj (v $ j)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_vec v\nj < dim_vec v\n\ngoal (1 subgoal):\n 1. rank_1_proj v $$ (i, j) = v $ i * cnj (v $ j)\n[PROOF STEP]\nunfolding rank_1_proj_def outer_prod_def\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_vec v\nj < dim_vec v\n\ngoal (1 subgoal):\n 1. (Matrix.mat (dim_vec v) 1 (\\(i, j). v $ i) * Matrix.mat 1 (dim_vec v) (\\(i, y). conjugate v $ y)) $$ (i, j) = v $ i * cnj (v $ j)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 350, "file": "Projective_Measurements_Linear_Algebra_Complements", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505299595162, "lm_q2_score": 0.8175744850834649, "lm_q1q2_score": 0.739619191212135}} {"text": "[STATEMENT]\nlemma unat_minus_one_word:\n \"unat (-1 :: 'a :: len word) = 2 ^ LENGTH('a) - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. unat (- 1) = 2 ^ LENGTH('a) - 1\n[PROOF STEP]\nby (simp add: mask_eq_exp_minus_1 unsigned_minus_1_eq_mask)", "meta": {"llama_tokens": 121, "file": "Word_Lib_More_Word", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.91243616285804, "lm_q2_score": 0.8104789040926008, "lm_q1q2_score": 0.7395102613276421}} {"text": "[STATEMENT]\nlemma transpose_rank:\n fixes A::\"'a::{conjugatable_ordered_field} mat\"\n shows \"vec_space.rank (dim_row A) A = vec_space.rank (dim_col A) (A\\<^sup>T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec_space.rank (dim_row A) A = vec_space.rank (dim_col A) A\\<^sup>T\n[PROOF STEP]\nby (metis carrier_mat_triv conjugatable_vec_space.rank_conjugate conjugate_transpose_rank index_transpose_mat(2))", "meta": {"llama_tokens": 169, "file": "BenOr_Kozen_Reif_More_Matrix", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361604769413, "lm_q2_score": 0.8104788995148792, "lm_q1q2_score": 0.739510255220933}} {"text": "[STATEMENT]\nlemma sum_add_split_nat_ivl:\n assumes le: \"m <= (k::nat)\" \"k <= n\"\n and g: \"!!i. [| m <= i; i < k |] ==> g i = f i\"\n and h: \"!!i. [| k <= i; i < n |] ==> h i = f i\"\n shows \"sum g {m.. k\nk \\ n\n\\m \\ ?i; ?i < k\\ \\ g ?i = f ?i\n\\k \\ ?i; ?i < n\\ \\ h ?i = f ?i\n\ngoal (1 subgoal):\n 1. sum g {m.. 0 \\ (Re z / cmod z)\\<^sup>2 + (Im z / cmod z)\\<^sup>2 = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z \\ 0 \\ (Re z / cmod z)\\<^sup>2 + (Im z / cmod z)\\<^sup>2 = 1\n[PROOF STEP]\nby (simp add: norm_complex_def complex_eq_iff power2_eq_square add_divide_distrib [symmetric])", "meta": {"llama_tokens": 159, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772351648677, "lm_q2_score": 0.8459424431344437, "lm_q1q2_score": 0.7394190318035678}} {"text": "[STATEMENT]\nlemma multiplicity_Lcm_atLeast1AtMost_Suc:\n fixes p n :: nat\n assumes p: \"prime p\" and n: \"n > 0\"\n shows \"multiplicity p (Lcm {1..Suc n}) =\n (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\ndefine k where \"k = Max (multiplicity p ` {1..n})\"\n[PROOF STATE]\nproof (state)\nthis:\nk = Max (multiplicity p ` {1..n})\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\ndefine l where \"l = multiplicity p (Suc n)\"\n[PROOF STATE]\nproof (state)\nthis:\nl = multiplicity p (Suc n)\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nhave eq: \"{1..Suc n} = insert (Suc n) {1..n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {1..Suc n} = insert (Suc n) {1..n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{1..Suc n} = insert (Suc n) {1..n}\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nfrom \\prime p\\\n[PROOF STATE]\nproof (chain)\npicking this:\nprime p\n[PROOF STEP]\nhave \"p > 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. 1 < p\n[PROOF STEP]\nby (auto dest: prime_gt_1_nat)\n[PROOF STATE]\nproof (state)\nthis:\n1 < p\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nhave \"multiplicity p (Lcm {1..Suc n}) = Max (multiplicity p ` {1..Suc n})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = Max (multiplicity p ` {1..Suc n})\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n0 < n\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = Max (multiplicity p ` {1..Suc n})\n[PROOF STEP]\nby (subst multiplicity_Lcm) auto\n[PROOF STATE]\nproof (state)\nthis:\nmultiplicity p (Lcm {1..Suc n}) = Max (multiplicity p ` {1..Suc n})\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmultiplicity p (Lcm {1..Suc n}) = Max (multiplicity p ` {1..Suc n})\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nhave \"multiplicity p ` {1..Suc n} =\n insert (multiplicity p (Suc n)) (multiplicity p ` {1..n})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. multiplicity p ` {1..Suc n} = insert (multiplicity p (Suc n)) (multiplicity p ` {1..n})\n[PROOF STEP]\nby (simp only: eq image_insert)\n[PROOF STATE]\nproof (state)\nthis:\nmultiplicity p ` {1..Suc n} = insert (multiplicity p (Suc n)) (multiplicity p ` {1..n})\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmultiplicity p ` {1..Suc n} = insert (multiplicity p (Suc n)) (multiplicity p ` {1..n})\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nhave \"Max \\ = max l k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Max (insert (multiplicity p (Suc n)) (multiplicity p ` {1..n})) = max l k\n[PROOF STEP]\nunfolding l_def k_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Max (insert (multiplicity p (Suc n)) (multiplicity p ` {1..n})) = max (multiplicity p (Suc n)) (Max (multiplicity p ` {1..n}))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n0 < n\n\ngoal (1 subgoal):\n 1. Max (insert (multiplicity p (Suc n)) (multiplicity p ` {1..n})) = max (multiplicity p (Suc n)) (Max (multiplicity p ` {1..n}))\n[PROOF STEP]\nby (subst Max.insert) auto\n[PROOF STATE]\nproof (state)\nthis:\nMax (insert (multiplicity p (Suc n)) (multiplicity p ` {1..n})) = max l k\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nMax (insert (multiplicity p (Suc n)) (multiplicity p ` {1..n})) = max l k\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nhave \"\\ = (if \\k. Suc n = p ^ k then 1 else 0) + k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nproof (cases \"\\k. Suc n = p ^ k\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n 2. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\k. Suc n = p ^ k\n\ngoal (2 subgoals):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n 2. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nhave \"p ^ l dvd Suc n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p ^ l dvd Suc n\n[PROOF STEP]\nunfolding l_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p ^ multiplicity p (Suc n) dvd Suc n\n[PROOF STEP]\nby (intro multiplicity_dvd)\n[PROOF STATE]\nproof (state)\nthis:\np ^ l dvd Suc n\n\ngoal (2 subgoals):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n 2. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nhence \"p ^ l \\ Suc n\"\n[PROOF STATE]\nproof (prove)\nusing this:\np ^ l dvd Suc n\n\ngoal (1 subgoal):\n 1. p ^ l \\ Suc n\n[PROOF STEP]\nunfolding l_def\n[PROOF STATE]\nproof (prove)\nusing this:\np ^ multiplicity p (Suc n) dvd Suc n\n\ngoal (1 subgoal):\n 1. p ^ multiplicity p (Suc n) \\ Suc n\n[PROOF STEP]\nby (intro dvd_imp_le multiplicity_dvd) auto\n[PROOF STATE]\nproof (state)\nthis:\np ^ l \\ Suc n\n\ngoal (2 subgoals):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n 2. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\np ^ l \\ Suc n\n\ngoal (2 subgoals):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n 2. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nhave \"Suc n \\ p ^ l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc n \\ p ^ l\n[PROOF STEP]\nusing False\n[PROOF STATE]\nproof (prove)\nusing this:\n\\k. Suc n = p ^ k\n\ngoal (1 subgoal):\n 1. Suc n \\ p ^ l\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nSuc n \\ p ^ l\n\ngoal (2 subgoals):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n 2. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\np ^ l \\ Suc n\nSuc n \\ p ^ l\n[PROOF STEP]\nhave \"p ^ l < Suc n\"\n[PROOF STATE]\nproof (prove)\nusing this:\np ^ l \\ Suc n\nSuc n \\ p ^ l\n\ngoal (1 subgoal):\n 1. p ^ l < Suc n\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\np ^ l < Suc n\n\ngoal (2 subgoals):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n 2. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\np ^ l < Suc n\n\ngoal (2 subgoals):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n 2. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nhave \"p ^ l > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < p ^ l\n[PROOF STEP]\nusing \\p > 1\\\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < p\n\ngoal (1 subgoal):\n 1. 0 < p ^ l\n[PROOF STEP]\nby (intro zero_less_power) auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < p ^ l\n\ngoal (2 subgoals):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n 2. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\np ^ l < Suc n\n0 < p ^ l\n[PROOF STEP]\nhave \"l = multiplicity p (p ^ l)\" and \"p ^ l \\ {1..n}\"\n[PROOF STATE]\nproof (prove)\nusing this:\np ^ l < Suc n\n0 < p ^ l\n\ngoal (1 subgoal):\n 1. l = multiplicity p (p ^ l) &&& p ^ l \\ {1..n}\n[PROOF STEP]\nusing \\prime p\\\n[PROOF STATE]\nproof (prove)\nusing this:\np ^ l < Suc n\n0 < p ^ l\nprime p\n\ngoal (1 subgoal):\n 1. l = multiplicity p (p ^ l) &&& p ^ l \\ {1..n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nl = multiplicity p (p ^ l)\np ^ l \\ {1..n}\n\ngoal (2 subgoals):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n 2. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nhence \"l \\ k\"\n[PROOF STATE]\nproof (prove)\nusing this:\nl = multiplicity p (p ^ l)\np ^ l \\ {1..n}\n\ngoal (1 subgoal):\n 1. l \\ k\n[PROOF STEP]\nunfolding k_def\n[PROOF STATE]\nproof (prove)\nusing this:\nl = multiplicity p (p ^ l)\np ^ l \\ {1..n}\n\ngoal (1 subgoal):\n 1. l \\ Max (multiplicity p ` {1..n})\n[PROOF STEP]\nby (intro Max.coboundedI) auto\n[PROOF STATE]\nproof (state)\nthis:\nl \\ k\n\ngoal (2 subgoals):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n 2. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nwith False\n[PROOF STATE]\nproof (chain)\npicking this:\n\\k. Suc n = p ^ k\nl \\ k\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\k. Suc n = p ^ k\nl \\ k\n\ngoal (1 subgoal):\n 1. max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nby (simp add: l_def k_def)\n[PROOF STATE]\nproof (state)\nthis:\nmax l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n\ngoal (1 subgoal):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\n\\k. Suc n = p ^ k\n\ngoal (1 subgoal):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\k. Suc n = p ^ k\n[PROOF STEP]\nobtain x where x: \"Suc n = p ^ x\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\k. Suc n = p ^ k\n\ngoal (1 subgoal):\n 1. (\\x. Suc n = p ^ x \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nSuc n = p ^ x\n\ngoal (1 subgoal):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nfrom x and \\n > 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\nSuc n = p ^ x\n0 < n\n[PROOF STEP]\nhave \"x > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc n = p ^ x\n0 < n\n\ngoal (1 subgoal):\n 1. 0 < x\n[PROOF STEP]\nby (intro Nat.gr0I) auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < x\n\ngoal (1 subgoal):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nfrom x\n[PROOF STATE]\nproof (chain)\npicking this:\nSuc n = p ^ x\n[PROOF STEP]\nhave [simp]: \"l = x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc n = p ^ x\n\ngoal (1 subgoal):\n 1. l = x\n[PROOF STEP]\nusing \\prime p\\\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc n = p ^ x\nprime p\n\ngoal (1 subgoal):\n 1. l = x\n[PROOF STEP]\nby (simp add: l_def)\n[PROOF STATE]\nproof (state)\nthis:\nl = x\n\ngoal (1 subgoal):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nhave \"x = k + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x = k + 1\n[PROOF STEP]\nproof (intro antisym)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x \\ k + 1\n 2. k + 1 \\ x\n[PROOF STEP]\nhave \"p ^ (x - 1) < Suc n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p ^ (x - 1) < Suc n\n[PROOF STEP]\nusing \\x > 0\\ \\p > 1\\\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n1 < p\n\ngoal (1 subgoal):\n 1. p ^ (x - 1) < Suc n\n[PROOF STEP]\nunfolding x\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n1 < p\n\ngoal (1 subgoal):\n 1. p ^ (x - 1) < p ^ x\n[PROOF STEP]\nby (intro power_strict_increasing) auto\n[PROOF STATE]\nproof (state)\nthis:\np ^ (x - 1) < Suc n\n\ngoal (2 subgoals):\n 1. x \\ k + 1\n 2. k + 1 \\ x\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\np ^ (x - 1) < Suc n\n\ngoal (2 subgoals):\n 1. x \\ k + 1\n 2. k + 1 \\ x\n[PROOF STEP]\nhave \"p ^ (x - 1) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < p ^ (x - 1)\n[PROOF STEP]\nusing \\p > 1\\\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < p\n\ngoal (1 subgoal):\n 1. 0 < p ^ (x - 1)\n[PROOF STEP]\nby (intro zero_less_power) auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < p ^ (x - 1)\n\ngoal (2 subgoals):\n 1. x \\ k + 1\n 2. k + 1 \\ x\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\np ^ (x - 1) < Suc n\n0 < p ^ (x - 1)\n[PROOF STEP]\nhave \"multiplicity p (p ^ (x - 1)) = x - 1\" and \"p ^ (x - 1) \\ {1..n}\"\n[PROOF STATE]\nproof (prove)\nusing this:\np ^ (x - 1) < Suc n\n0 < p ^ (x - 1)\n\ngoal (1 subgoal):\n 1. multiplicity p (p ^ (x - 1)) = x - 1 &&& p ^ (x - 1) \\ {1..n}\n[PROOF STEP]\nusing \\prime p\\\n[PROOF STATE]\nproof (prove)\nusing this:\np ^ (x - 1) < Suc n\n0 < p ^ (x - 1)\nprime p\n\ngoal (1 subgoal):\n 1. multiplicity p (p ^ (x - 1)) = x - 1 &&& p ^ (x - 1) \\ {1..n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmultiplicity p (p ^ (x - 1)) = x - 1\np ^ (x - 1) \\ {1..n}\n\ngoal (2 subgoals):\n 1. x \\ k + 1\n 2. k + 1 \\ x\n[PROOF STEP]\nhence \"x - 1 \\ k\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmultiplicity p (p ^ (x - 1)) = x - 1\np ^ (x - 1) \\ {1..n}\n\ngoal (1 subgoal):\n 1. x - 1 \\ k\n[PROOF STEP]\nunfolding k_def\n[PROOF STATE]\nproof (prove)\nusing this:\nmultiplicity p (p ^ (x - 1)) = x - 1\np ^ (x - 1) \\ {1..n}\n\ngoal (1 subgoal):\n 1. x - 1 \\ Max (multiplicity p ` {1..n})\n[PROOF STEP]\nby (intro Max.coboundedI) force+\n[PROOF STATE]\nproof (state)\nthis:\nx - 1 \\ k\n\ngoal (2 subgoals):\n 1. x \\ k + 1\n 2. k + 1 \\ x\n[PROOF STEP]\nthus \"x \\ k + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx - 1 \\ k\n\ngoal (1 subgoal):\n 1. x \\ k + 1\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\nx \\ k + 1\n\ngoal (1 subgoal):\n 1. k + 1 \\ x\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. k + 1 \\ x\n[PROOF STEP]\nhave \"multiplicity p y < x\" if \"y \\ {1..n}\" for y\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. multiplicity p y < x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. multiplicity p y < x\n[PROOF STEP]\nhave \"p ^ multiplicity p y \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p ^ multiplicity p y \\ y\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\ny \\ {1..n}\n\ngoal (1 subgoal):\n 1. p ^ multiplicity p y \\ y\n[PROOF STEP]\nby (intro dvd_imp_le multiplicity_dvd) auto\n[PROOF STATE]\nproof (state)\nthis:\np ^ multiplicity p y \\ y\n\ngoal (1 subgoal):\n 1. multiplicity p y < x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\np ^ multiplicity p y \\ y\n\ngoal (1 subgoal):\n 1. multiplicity p y < x\n[PROOF STEP]\nhave \"\\ < Suc n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. y < Suc n\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\ny \\ {1..n}\n\ngoal (1 subgoal):\n 1. y < Suc n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ny < Suc n\n\ngoal (1 subgoal):\n 1. multiplicity p y < x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ny < Suc n\n\ngoal (1 subgoal):\n 1. multiplicity p y < x\n[PROOF STEP]\nhave \"\\ = p ^ x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc n = p ^ x\n[PROOF STEP]\nby (fact x)\n[PROOF STATE]\nproof (state)\nthis:\nSuc n = p ^ x\n\ngoal (1 subgoal):\n 1. multiplicity p y < x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\np ^ multiplicity p y < p ^ x\n[PROOF STEP]\nshow \"multiplicity p y < x\"\n[PROOF STATE]\nproof (prove)\nusing this:\np ^ multiplicity p y < p ^ x\n\ngoal (1 subgoal):\n 1. multiplicity p y < x\n[PROOF STEP]\nusing \\p > 1\\\n[PROOF STATE]\nproof (prove)\nusing this:\np ^ multiplicity p y < p ^ x\n1 < p\n\ngoal (1 subgoal):\n 1. multiplicity p y < x\n[PROOF STEP]\nby (subst (asm) power_strict_increasing_iff)\n[PROOF STATE]\nproof (state)\nthis:\nmultiplicity p y < x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n?y \\ {1..n} \\ multiplicity p ?y < x\n\ngoal (1 subgoal):\n 1. k + 1 \\ x\n[PROOF STEP]\nhence \"k < x\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?y \\ {1..n} \\ multiplicity p ?y < x\n\ngoal (1 subgoal):\n 1. k < x\n[PROOF STEP]\nusing \\n > 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\n?y \\ {1..n} \\ multiplicity p ?y < x\n0 < n\n\ngoal (1 subgoal):\n 1. k < x\n[PROOF STEP]\nunfolding k_def\n[PROOF STATE]\nproof (prove)\nusing this:\n?y \\ {1..n} \\ multiplicity p ?y < x\n0 < n\n\ngoal (1 subgoal):\n 1. Max (multiplicity p ` {1..n}) < x\n[PROOF STEP]\nby (subst Max_less_iff) auto\n[PROOF STATE]\nproof (state)\nthis:\nk < x\n\ngoal (1 subgoal):\n 1. k + 1 \\ x\n[PROOF STEP]\nthus \"k + 1 \\ x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nk < x\n\ngoal (1 subgoal):\n 1. k + 1 \\ x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nk + 1 \\ x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nx = k + 1\n\ngoal (1 subgoal):\n 1. \\k. Suc n = p ^ k \\ max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx = k + 1\n\ngoal (1 subgoal):\n 1. max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nusing True\n[PROOF STATE]\nproof (prove)\nusing this:\nx = k + 1\n\\k. Suc n = p ^ k\n\ngoal (1 subgoal):\n 1. max l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmax l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nmax l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmax l k = (if \\k. Suc n = p ^ k then 1 else 0) + k\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nhave \"k = multiplicity p (Lcm {1..n})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k = multiplicity p (Lcm {1..n})\n[PROOF STEP]\nunfolding k_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Max (multiplicity p ` {1..n}) = multiplicity p (Lcm {1..n})\n[PROOF STEP]\nusing \\n > 0\\ and \\prime p\\\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\nprime p\n\ngoal (1 subgoal):\n 1. Max (multiplicity p ` {1..n}) = multiplicity p (Lcm {1..n})\n[PROOF STEP]\nby (subst multiplicity_Lcm) auto\n[PROOF STATE]\nproof (state)\nthis:\nk = multiplicity p (Lcm {1..n})\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nmultiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nmultiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n\ngoal (1 subgoal):\n 1. multiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nmultiplicity p (Lcm {1..Suc n}) = (if \\k. Suc n = p ^ k then 1 else 0) + multiplicity p (Lcm {1..n})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 9785, "file": "Prime_Distribution_Elementary_Lcm_Nat_Upto", "length": 119, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213853793452, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7392478339961402}} {"text": "[STATEMENT]\ntheorem catalan_integral_form:\n \"((\\x. x powr (real n - 1 / 2) * sqrt (4 - x) / (2*pi)) \n has_integral real (catalan n)) {0..4}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. x powr (real n - 1 / 2) * sqrt (4 - x) / (2 * pi)) has_integral real (catalan n)) {0..4}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\x. x powr (real n - 1 / 2) * sqrt (4 - x) / (2 * pi)) has_integral real (catalan n)) {0..4}\n[PROOF STEP]\nhave \"((\\x. x powr (real n - 1 / 2) * sqrt (4 - x) * inverse (2*pi)) has_integral \n I n * inverse (2 * pi)) {0..4}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. x powr (real n - 1 / 2) * sqrt (4 - x) * inverse (2 * pi)) has_integral I n * inverse (2 * pi)) {0..4}\n[PROOF STEP]\nunfolding I_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. x powr (real n - 1 / 2) * sqrt (4 - x) * inverse (2 * pi)) has_integral integral {0..4} (\\x. x powr (real n - 1 / 2) * sqrt (4 - x)) * inverse (2 * pi)) {0..4}\n[PROOF STEP]\nby (intro has_integral_mult_left integrable_integral integrable_I)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. x powr (real n - 1 / 2) * sqrt (4 - x) * inverse (2 * pi)) has_integral I n * inverse (2 * pi)) {0..4}\n\ngoal (1 subgoal):\n 1. ((\\x. x powr (real n - 1 / 2) * sqrt (4 - x) / (2 * pi)) has_integral real (catalan n)) {0..4}\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. x powr (real n - 1 / 2) * sqrt (4 - x) * inverse (2 * pi)) has_integral I n * inverse (2 * pi)) {0..4}\n\ngoal (1 subgoal):\n 1. ((\\x. x powr (real n - 1 / 2) * sqrt (4 - x) / (2 * pi)) has_integral real (catalan n)) {0..4}\n[PROOF STEP]\nby (simp add: catalan_eq_I field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. x powr (real n - 1 / 2) * sqrt (4 - x) / (2 * pi)) has_integral real (catalan n)) {0..4}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 905, "file": "Catalan_Numbers_Catalan_Numbers", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213853793452, "lm_q2_score": 0.8221891327004133, "lm_q1q2_score": 0.7392478320374378}} {"text": "[STATEMENT]\nlemma op_norm_def: \"\\A\\\\<^sub>o\\<^sub>p = Sup {\\A *v x\\ | x. \\x\\ = 1}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p = Sup {\\A *v x\\ |x. \\x\\ = 1}\n[PROOF STEP]\napply(rule antisym[OF onorm_le cSup_least[OF op_norm_set_proptys(3)]])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. \\A *v x\\ \\ Sup {\\A *v x\\ |x. \\x\\ = 1} * \\x\\\n 2. \\x. x \\ {\\A *v x\\ |x. \\x\\ = 1} \\ x \\ \\A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\napply(case_tac \"x = 0\", simp)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. x \\ 0 \\ \\A *v x\\ \\ Sup {\\A *v x\\ |x. \\x\\ = 1} * \\x\\\n 2. \\x. x \\ {\\A *v x\\ |x. \\x\\ = 1} \\ x \\ \\A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\napply(subst mult_norm_matrix_sgn_eq[symmetric], simp)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. x \\ 0 \\ \\A *v sgn x\\ \\ Sup {\\A *v x\\ |x. \\x\\ = 1}\n 2. \\x. x \\ {\\A *v x\\ |x. \\x\\ = 1} \\ x \\ \\A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\napply(rule cSup_upper[OF _ op_norm_set_proptys(2)])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. x \\ 0 \\ \\A *v sgn x\\ \\ {\\A *v x\\ |x. \\x\\ = 1}\n 2. \\x. x \\ {\\A *v x\\ |x. \\x\\ = 1} \\ x \\ \\A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\napply(force simp: norm_sgn)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ {\\A *v x\\ |x. \\x\\ = 1} \\ x \\ \\A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nunfolding onorm_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ {\\A *v x\\ |x. \\x\\ = 1} \\ x \\ (SUP x. \\A *v x\\ / \\x\\)\n[PROOF STEP]\napply(rule cSup_upper[OF _ onorm_set_proptys(2)])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ {\\A *v x\\ |x. \\x\\ = 1} \\ x \\ range (\\x. \\A *v x\\ / \\x\\)\n[PROOF STEP]\nby (simp add: image_def, clarsimp) (metis div_by_1)", "meta": {"llama_tokens": 1158, "file": "Matrices_for_ODEs_MTX_Norms", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213772699435, "lm_q2_score": 0.8221891392358014, "lm_q1q2_score": 0.7392478312460831}} {"text": "[STATEMENT]\nlemma norm_sum_Pythagorean:\n assumes \"finite I\" \"pairwise (\\i j. orthogonal (f i) (f j)) I\"\n shows \"(norm (sum f I))\\<^sup>2 = (\\i\\I. (norm (f i))\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (norm (sum f I))\\<^sup>2 = (\\i\\I. (norm (f i))\\<^sup>2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite I\npairwise (\\i j. orthogonal (f i) (f j)) I\n\ngoal (1 subgoal):\n 1. (norm (sum f I))\\<^sup>2 = (\\i\\I. (norm (f i))\\<^sup>2)\n[PROOF STEP]\nproof (induction I rule: finite_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. pairwise (\\i j. orthogonal (f i) (f j)) {} \\ (norm (sum f {}))\\<^sup>2 = (\\i\\{}. (norm (f i))\\<^sup>2)\n 2. \\x F. \\finite F; x \\ F; pairwise (\\i j. orthogonal (f i) (f j)) F \\ (norm (sum f F))\\<^sup>2 = (\\i\\F. (norm (f i))\\<^sup>2); pairwise (\\i j. orthogonal (f i) (f j)) (insert x F)\\ \\ (norm (sum f (insert x F)))\\<^sup>2 = (\\i\\insert x F. (norm (f i))\\<^sup>2)\n[PROOF STEP]\ncase empty\n[PROOF STATE]\nproof (state)\nthis:\npairwise (\\i j. orthogonal (f i) (f j)) {}\n\ngoal (2 subgoals):\n 1. pairwise (\\i j. orthogonal (f i) (f j)) {} \\ (norm (sum f {}))\\<^sup>2 = (\\i\\{}. (norm (f i))\\<^sup>2)\n 2. \\x F. \\finite F; x \\ F; pairwise (\\i j. orthogonal (f i) (f j)) F \\ (norm (sum f F))\\<^sup>2 = (\\i\\F. (norm (f i))\\<^sup>2); pairwise (\\i j. orthogonal (f i) (f j)) (insert x F)\\ \\ (norm (sum f (insert x F)))\\<^sup>2 = (\\i\\insert x F. (norm (f i))\\<^sup>2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\npairwise (\\i j. orthogonal (f i) (f j)) {}\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\npairwise (\\i j. orthogonal (f i) (f j)) {}\n\ngoal (1 subgoal):\n 1. (norm (sum f {}))\\<^sup>2 = (\\i\\{}. (norm (f i))\\<^sup>2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(norm (sum f {}))\\<^sup>2 = (\\i\\{}. (norm (f i))\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; pairwise (\\i j. orthogonal (f i) (f j)) F \\ (norm (sum f F))\\<^sup>2 = (\\i\\F. (norm (f i))\\<^sup>2); pairwise (\\i j. orthogonal (f i) (f j)) (insert x F)\\ \\ (norm (sum f (insert x F)))\\<^sup>2 = (\\i\\insert x F. (norm (f i))\\<^sup>2)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; pairwise (\\i j. orthogonal (f i) (f j)) F \\ (norm (sum f F))\\<^sup>2 = (\\i\\F. (norm (f i))\\<^sup>2); pairwise (\\i j. orthogonal (f i) (f j)) (insert x F)\\ \\ (norm (sum f (insert x F)))\\<^sup>2 = (\\i\\insert x F. (norm (f i))\\<^sup>2)\n[PROOF STEP]\ncase (insert x I)\n[PROOF STATE]\nproof (state)\nthis:\nfinite I\nx \\ I\npairwise (\\i j. orthogonal (f i) (f j)) I \\ (norm (sum f I))\\<^sup>2 = (\\i\\I. (norm (f i))\\<^sup>2)\npairwise (\\i j. orthogonal (f i) (f j)) (insert x I)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; pairwise (\\i j. orthogonal (f i) (f j)) F \\ (norm (sum f F))\\<^sup>2 = (\\i\\F. (norm (f i))\\<^sup>2); pairwise (\\i j. orthogonal (f i) (f j)) (insert x F)\\ \\ (norm (sum f (insert x F)))\\<^sup>2 = (\\i\\insert x F. (norm (f i))\\<^sup>2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite I\nx \\ I\npairwise (\\i j. orthogonal (f i) (f j)) I \\ (norm (sum f I))\\<^sup>2 = (\\i\\I. (norm (f i))\\<^sup>2)\npairwise (\\i j. orthogonal (f i) (f j)) (insert x I)\n[PROOF STEP]\nhave \"orthogonal (f x) (sum f I)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite I\nx \\ I\npairwise (\\i j. orthogonal (f i) (f j)) I \\ (norm (sum f I))\\<^sup>2 = (\\i\\I. (norm (f i))\\<^sup>2)\npairwise (\\i j. orthogonal (f i) (f j)) (insert x I)\n\ngoal (1 subgoal):\n 1. orthogonal (f x) (sum f I)\n[PROOF STEP]\nby (metis pairwise_insert orthogonal_rvsum)\n[PROOF STATE]\nproof (state)\nthis:\northogonal (f x) (sum f I)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; pairwise (\\i j. orthogonal (f i) (f j)) F \\ (norm (sum f F))\\<^sup>2 = (\\i\\F. (norm (f i))\\<^sup>2); pairwise (\\i j. orthogonal (f i) (f j)) (insert x F)\\ \\ (norm (sum f (insert x F)))\\<^sup>2 = (\\i\\insert x F. (norm (f i))\\<^sup>2)\n[PROOF STEP]\nwith insert\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite I\nx \\ I\npairwise (\\i j. orthogonal (f i) (f j)) I \\ (norm (sum f I))\\<^sup>2 = (\\i\\I. (norm (f i))\\<^sup>2)\npairwise (\\i j. orthogonal (f i) (f j)) (insert x I)\northogonal (f x) (sum f I)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite I\nx \\ I\npairwise (\\i j. orthogonal (f i) (f j)) I \\ (norm (sum f I))\\<^sup>2 = (\\i\\I. (norm (f i))\\<^sup>2)\npairwise (\\i j. orthogonal (f i) (f j)) (insert x I)\northogonal (f x) (sum f I)\n\ngoal (1 subgoal):\n 1. (norm (sum f (insert x I)))\\<^sup>2 = (\\i\\insert x I. (norm (f i))\\<^sup>2)\n[PROOF STEP]\nby (simp add: pairwise_insert norm_add_Pythagorean)\n[PROOF STATE]\nproof (state)\nthis:\n(norm (sum f (insert x I)))\\<^sup>2 = (\\i\\insert x I. (norm (f i))\\<^sup>2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2393, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213745668095, "lm_q2_score": 0.8221891327004133, "lm_q1q2_score": 0.7392478231474885}} {"text": "[STATEMENT]\nlemma card_permutations_of_multiset_remove_aux:\n assumes \"x \\# A\"\n shows \"card (permutations_of_multiset A) * count A x = \n size A * card (permutations_of_multiset (A - {#x#}))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (permutations_of_multiset A) * count A x = size A * card (permutations_of_multiset (A - {#x#}))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (permutations_of_multiset A) * count A x = size A * card (permutations_of_multiset (A - {#x#}))\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\# A\n[PROOF STEP]\nhave A: \"A - {#x#} + {#x#} = A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\# A\n\ngoal (1 subgoal):\n 1. A - {#x#} + {#x#} = A\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nA - {#x#} + {#x#} = A\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset A) * count A x = size A * card (permutations_of_multiset (A - {#x#}))\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\# A\n[PROOF STEP]\nhave B: \"size A = size (A - {#x#}) + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\# A\n\ngoal (1 subgoal):\n 1. size A = size (A - {#x#}) + 1\n[PROOF STEP]\nby (subst A [symmetric], subst size_union) simp\n[PROOF STATE]\nproof (state)\nthis:\nsize A = size (A - {#x#}) + 1\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset A) * count A x = size A * card (permutations_of_multiset (A - {#x#}))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (permutations_of_multiset A) * count A x = size A * card (permutations_of_multiset (A - {#x#}))\n[PROOF STEP]\nusing card_permutations_of_multiset_insert_aux[of \"A - {#x#}\" x, unfolded A] assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (permutations_of_multiset A) * (count (A - {#x#}) x + 1) = (size (A - {#x#}) + 1) * card (permutations_of_multiset (A - {#x#}))\nx \\# A\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset A) * count A x = size A * card (permutations_of_multiset (A - {#x#}))\n[PROOF STEP]\nby (simp add: B)\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset A) * count A x = size A * card (permutations_of_multiset (A - {#x#}))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1005, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869916479466, "lm_q2_score": 0.8438951084436077, "lm_q1q2_score": 0.7390723582902449}} {"text": "[STATEMENT]\nlemma Im_Arccos_bound: \"\\Im (Arccos w)\\ \\ cmod w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Im (Arccos w)\\ \\ cmod w\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\Im (Arccos w)\\ \\ cmod w\n[PROOF STEP]\nhave \"(Im (Arccos w))\\<^sup>2 \\ (cmod (cos (Arccos w)))\\<^sup>2 - (cos (Re (Arccos w)))\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Im (Arccos w))\\<^sup>2 \\ (cmod (cos (Arccos w)))\\<^sup>2 - (cos (Re (Arccos w)))\\<^sup>2\n[PROOF STEP]\nusing norm_cos_squared [of \"Arccos w\"] real_le_abs_sinh [of \"Im (Arccos w)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod (cos (Arccos w)))\\<^sup>2 = (cos (Re (Arccos w)))\\<^sup>2 + (exp (Im (Arccos w)) - inverse (exp (Im (Arccos w))))\\<^sup>2 / 4\n\\Im (Arccos w)\\ \\ \\(exp (Im (Arccos w)) - inverse (exp (Im (Arccos w)))) / 2\\\n\ngoal (1 subgoal):\n 1. (Im (Arccos w))\\<^sup>2 \\ (cmod (cos (Arccos w)))\\<^sup>2 - (cos (Re (Arccos w)))\\<^sup>2\n[PROOF STEP]\nby (simp only: abs_le_square_iff) (simp add: field_split_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(Im (Arccos w))\\<^sup>2 \\ (cmod (cos (Arccos w)))\\<^sup>2 - (cos (Re (Arccos w)))\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\Im (Arccos w)\\ \\ cmod w\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(Im (Arccos w))\\<^sup>2 \\ (cmod (cos (Arccos w)))\\<^sup>2 - (cos (Re (Arccos w)))\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\Im (Arccos w)\\ \\ cmod w\n[PROOF STEP]\nhave \"\\ \\ (cmod w)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cmod (cos (Arccos w)))\\<^sup>2 - (cos (Re (Arccos w)))\\<^sup>2 \\ (cmod w)\\<^sup>2\n[PROOF STEP]\nby (auto simp: cmod_power2)\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (cos (Arccos w)))\\<^sup>2 - (cos (Re (Arccos w)))\\<^sup>2 \\ (cmod w)\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\Im (Arccos w)\\ \\ cmod w\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(Im (Arccos w))\\<^sup>2 \\ (cmod w)\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(Im (Arccos w))\\<^sup>2 \\ (cmod w)\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\Im (Arccos w)\\ \\ cmod w\n[PROOF STEP]\nusing abs_le_square_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n(Im (Arccos w))\\<^sup>2 \\ (cmod w)\\<^sup>2\n(\\?x\\ \\ \\?y\\) = (?x\\<^sup>2 \\ ?y\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\Im (Arccos w)\\ \\ cmod w\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n\\Im (Arccos w)\\ \\ cmod w\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1279, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869981319863, "lm_q2_score": 0.8438950966654772, "lm_q1q2_score": 0.7390723534469608}} {"text": "[STATEMENT]\nlemma dim_kronecker[simp]:\n \"dim_row (kronecker_product A B) = dim_row A * dim_row B\"\n \"dim_col (kronecker_product A B) = dim_col A * dim_col B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_row (kronecker_product A B) = dim_row A * dim_row B &&& dim_col (kronecker_product A B) = dim_col A * dim_col B\n[PROOF STEP]\nunfolding kronecker_product_def Let_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_row (mat (dim_row A * dim_row B) (dim_col A * dim_col B) (\\(i, j). A $$ (i div dim_row B, j div dim_col B) * B $$ (i mod dim_row B, j mod dim_col B))) = dim_row A * dim_row B &&& dim_col (mat (dim_row A * dim_row B) (dim_col A * dim_col B) (\\(i, j). A $$ (i div dim_row B, j div dim_col B) * B $$ (i mod dim_row B, j mod dim_col B))) = dim_col A * dim_col B\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 353, "file": "BenOr_Kozen_Reif_More_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9230391643039738, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.7390700764795254}} {"text": "[STATEMENT]\nlemma orthogonal_iff_vangle: \"orthogonal u v \\ vangle u v = pi / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal u v = (vangle u v = pi / 2)\n[PROOF STEP]\nusing arccos_eq_iff[of \"u \\ v / (norm u * norm v)\" 0] Cauchy_Schwarz_ineq2[of u v]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\u \\ v / (norm u * norm v)\\ \\ 1 \\ \\0\\ \\ 1 \\ (arccos (u \\ v / (norm u * norm v)) = arccos 0) = (u \\ v / (norm u * norm v) = 0)\n\\u \\ v\\ \\ norm u * norm v\n\ngoal (1 subgoal):\n 1. orthogonal u v = (vangle u v = pi / 2)\n[PROOF STEP]\nby (auto simp: vangle_def orthogonal_def)", "meta": {"llama_tokens": 297, "file": "Triangle_Angles", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9173026573249612, "lm_q2_score": 0.8056321936479701, "lm_q1q2_score": 0.7390085520598207}} {"text": "[STATEMENT]\nlemma powrat_power_eq: \n \"0 < a \\ a pow\\<^sub>\\ rat_of_nat n = a ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < a \\ a pow\\<^sub>\\ rat_of_nat n = a ^ n\n[PROOF STEP]\nproof (induction n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 0 < a \\ a pow\\<^sub>\\ rat_of_nat 0 = a ^ 0\n 2. \\n. \\0 < a \\ a pow\\<^sub>\\ rat_of_nat n = a ^ n; 0 < a\\ \\ a pow\\<^sub>\\ rat_of_nat (Suc n) = a ^ Suc n\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n0 < a\n\ngoal (2 subgoals):\n 1. 0 < a \\ a pow\\<^sub>\\ rat_of_nat 0 = a ^ 0\n 2. \\n. \\0 < a \\ a pow\\<^sub>\\ rat_of_nat n = a ^ n; 0 < a\\ \\ a pow\\<^sub>\\ rat_of_nat (Suc n) = a ^ Suc n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < a\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n\ngoal (1 subgoal):\n 1. a pow\\<^sub>\\ rat_of_nat 0 = a ^ 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na pow\\<^sub>\\ rat_of_nat 0 = a ^ 0\n\ngoal (1 subgoal):\n 1. \\n. \\0 < a \\ a pow\\<^sub>\\ rat_of_nat n = a ^ n; 0 < a\\ \\ a pow\\<^sub>\\ rat_of_nat (Suc n) = a ^ Suc n\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\0 < a \\ a pow\\<^sub>\\ rat_of_nat n = a ^ n; 0 < a\\ \\ a pow\\<^sub>\\ rat_of_nat (Suc n) = a ^ Suc n\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n0 < a \\ a pow\\<^sub>\\ rat_of_nat n = a ^ n\n0 < a\n\ngoal (1 subgoal):\n 1. \\n. \\0 < a \\ a pow\\<^sub>\\ rat_of_nat n = a ^ n; 0 < a\\ \\ a pow\\<^sub>\\ rat_of_nat (Suc n) = a ^ Suc n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < a \\ a pow\\<^sub>\\ rat_of_nat n = a ^ n\n0 < a\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a \\ a pow\\<^sub>\\ rat_of_nat n = a ^ n\n0 < a\n\ngoal (1 subgoal):\n 1. a pow\\<^sub>\\ rat_of_nat (Suc n) = a ^ Suc n\n[PROOF STEP]\nusing powrat_add\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a \\ a pow\\<^sub>\\ rat_of_nat n = a ^ n\n0 < a\n0 < ?x \\ ?x pow\\<^sub>\\ (?r + ?s) = ?x pow\\<^sub>\\ ?r * ?x pow\\<^sub>\\ ?s\n\ngoal (1 subgoal):\n 1. a pow\\<^sub>\\ rat_of_nat (Suc n) = a ^ Suc n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na pow\\<^sub>\\ rat_of_nat (Suc n) = a ^ Suc n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1189, "file": "Real_Power_RatPower", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8633916099737806, "lm_q2_score": 0.8558511414521923, "lm_q1q2_score": 0.7389346949163061}} {"text": "[STATEMENT]\nlemma card_permutations_of_multiset_insert_aux:\n \"card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = \n (size A + 1) * card (permutations_of_multiset A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n[PROOF STEP]\nnote card_permutations_of_multiset_aux[of \"A + {#x#}\"]\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset (A + {#x#})) * (\\xa\\set_mset (A + {#x#}). fact (count (A + {#x#}) xa)) = fact (size (A + {#x#}))\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset (A + {#x#})) * (\\xa\\set_mset (A + {#x#}). fact (count (A + {#x#}) xa)) = fact (size (A + {#x#}))\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n[PROOF STEP]\nhave \"fact (size (A + {#x#})) = (size A + 1) * fact (size A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fact (size (A + {#x#})) = (size A + 1) * fact (size A)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfact (size (A + {#x#})) = (size A + 1) * fact (size A)\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfact (size (A + {#x#})) = (size A + 1) * fact (size A)\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n[PROOF STEP]\nnote multiset_prod_fact_insert[of A x]\n[PROOF STATE]\nproof (state)\nthis:\n(\\y\\set_mset (A + {#x#}). fact (count (A + {#x#}) y)) = (count A x + 1) * (\\y\\set_mset A. fact (count A y))\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\y\\set_mset (A + {#x#}). fact (count (A + {#x#}) y)) = (count A x + 1) * (\\y\\set_mset A. fact (count A y))\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n[PROOF STEP]\nnote card_permutations_of_multiset_aux[of A, symmetric]\n[PROOF STATE]\nproof (state)\nthis:\nfact (size A) = card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x))\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (permutations_of_multiset (A + {#x#})) * ((count A x + 1) * (\\y\\set_mset A. fact (count A y))) = (size A + 1) * (card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)))\n[PROOF STEP]\nhave \"card (permutations_of_multiset (A + {#x#})) * (count A x + 1) *\n (\\y\\set_mset A. fact (count A y)) =\n (size A + 1) * card (permutations_of_multiset A) *\n (\\x\\set_mset A. fact (count A x))\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (permutations_of_multiset (A + {#x#})) * ((count A x + 1) * (\\y\\set_mset A. fact (count A y))) = (size A + 1) * (card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x)))\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) * (\\y\\set_mset A. fact (count A y)) = (size A + 1) * card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x))\n[PROOF STEP]\nby (simp only: mult_ac)\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset (A + {#x#})) * (count A x + 1) * (\\y\\set_mset A. fact (count A y)) = (size A + 1) * card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x))\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (permutations_of_multiset (A + {#x#})) * (count A x + 1) * (\\y\\set_mset A. fact (count A y)) = (size A + 1) * card (permutations_of_multiset A) * (\\x\\set_mset A. fact (count A x))\n\ngoal (1 subgoal):\n 1. card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n[PROOF STEP]\nby (subst (asm) mult_right_cancel) simp_all\n[PROOF STATE]\nproof (state)\nthis:\ncard (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2278, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513814471134, "lm_q2_score": 0.8244619306896955, "lm_q1q2_score": 0.7389251443311938}} {"text": "[STATEMENT]\nlemma conjugate_exponent_ennrealI:\n assumes \"1/p + 1/q = (1::ennreal)\"\n shows \"p \\ 1\" \"q \\ 1\" \"p = conjugate_exponent q\" \"q = conjugate_exponent p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1 \\ p &&& 1 \\ q) &&& p = conjugate_exponent q &&& q = conjugate_exponent p\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. 1 \\ p\n 2. 1 \\ q\n 3. p = conjugate_exponent q\n 4. q = conjugate_exponent p\n[PROOF STEP]\nhave \"1/p \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 / p \\ 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 / p + 1 / q = 1\n\ngoal (1 subgoal):\n 1. 1 / p \\ 1\n[PROOF STEP]\nusing le_iff_add\n[PROOF STATE]\nproof (prove)\nusing this:\n1 / p + 1 / q = 1\n(?a \\ ?b) = (\\c. ?b = ?a + c)\n\ngoal (1 subgoal):\n 1. 1 / p \\ 1\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n1 / p \\ 1\n\ngoal (4 subgoals):\n 1. 1 \\ p\n 2. 1 \\ q\n 3. p = conjugate_exponent q\n 4. q = conjugate_exponent p\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n1 / p \\ 1\n[PROOF STEP]\nshow \"p \\ 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 / p \\ 1\n\ngoal (1 subgoal):\n 1. 1 \\ p\n[PROOF STEP]\nby (metis assms divide_ennreal_def ennreal_add_eq_top ennreal_divide_self ennreal_divide_zero ennreal_le_epsilon ennreal_one_neq_top mult.left_neutral mult_left_le zero_le)\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ p\n\ngoal (3 subgoals):\n 1. 1 \\ q\n 2. p = conjugate_exponent q\n 3. q = conjugate_exponent p\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n1 \\ p\n[PROOF STEP]\nshow \"q = conjugate_exponent p\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ p\n\ngoal (1 subgoal):\n 1. q = conjugate_exponent p\n[PROOF STEP]\nusing conjugate_exponent_ennreal_iff assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ p\n1 \\ ?p \\ (?q = conjugate_exponent ?p) = (1 / ?p + 1 / ?q = 1)\n1 / p + 1 / q = 1\n\ngoal (1 subgoal):\n 1. q = conjugate_exponent p\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nq = conjugate_exponent p\n\ngoal (2 subgoals):\n 1. 1 \\ q\n 2. p = conjugate_exponent q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nq = conjugate_exponent p\n[PROOF STEP]\nshow \"q \\ 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nq = conjugate_exponent p\n\ngoal (1 subgoal):\n 1. 1 \\ q\n[PROOF STEP]\nusing conjugate_exponent_ennreal[OF \\p \\ 1\\]\n[PROOF STATE]\nproof (prove)\nusing this:\nq = conjugate_exponent p\n1 / p + 1 / conjugate_exponent p = 1\n1 \\ conjugate_exponent p\nconjugate_exponent (conjugate_exponent p) = p\n\ngoal (1 subgoal):\n 1. 1 \\ q\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ q\n\ngoal (1 subgoal):\n 1. p = conjugate_exponent q\n[PROOF STEP]\nshow \"p = conjugate_exponent q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p = conjugate_exponent q\n[PROOF STEP]\nusing conjugate_exponent_ennreal_iff[OF \\q\\1\\, of p] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(p = conjugate_exponent q) = (1 / q + 1 / p = 1)\n1 / p + 1 / q = 1\n\ngoal (1 subgoal):\n 1. p = conjugate_exponent q\n[PROOF STEP]\nby (simp add: add.commute)\n[PROOF STATE]\nproof (state)\nthis:\np = conjugate_exponent q\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1518, "file": "Lp_Lp", "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588052782737, "lm_q2_score": 0.8311430499496096, "lm_q1q2_score": 0.7386857040885556}} {"text": "[STATEMENT]\nlemma prod_list_dvd_prod_list_subset:\nfixes A::\"'b::comm_monoid_mult list\"\nassumes dA: \"distinct A\"\n and dB: \"distinct B\" (*Maybe this condition could be avoided*)\n and s: \"set A \\ set B\"\nshows \"prod_list A dvd prod_list B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\nhave \"prod_list A = prod_list (map id A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod_list A = prod_list (map id A)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nprod_list A = prod_list (map id A)\n\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nprod_list A = prod_list (map id A)\n\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\nhave \"... = prod id (set A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod_list (map id A) = prod id (set A)\n[PROOF STEP]\nby (rule prod.distinct_set_conv_list[symmetric, OF dA])\n[PROOF STATE]\nproof (state)\nthis:\nprod_list (map id A) = prod id (set A)\n\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nprod_list (map id A) = prod id (set A)\n\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\nhave \"... dvd prod id (set B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod id (set A) dvd prod id (set B)\n[PROOF STEP]\nby (rule prod_dvd_prod_subset[OF _ s], auto)\n[PROOF STATE]\nproof (state)\nthis:\nprod id (set A) dvd prod id (set B)\n\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nprod id (set A) dvd prod id (set B)\n\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\nhave \"... = prod_list (map id B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod id (set B) = prod_list (map id B)\n[PROOF STEP]\nby (rule prod.distinct_set_conv_list[OF dB])\n[PROOF STATE]\nproof (state)\nthis:\nprod id (set B) = prod_list (map id B)\n\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nprod id (set B) = prod_list (map id B)\n\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\nhave \"... = prod_list B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod_list (map id B) = prod_list B\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nprod_list (map id B) = prod_list B\n\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nprod_list A dvd prod_list B\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nprod_list A dvd prod_list B\n\ngoal (1 subgoal):\n 1. prod_list A dvd prod_list B\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nprod_list A dvd prod_list B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1235, "file": "Berlekamp_Zassenhaus_Berlekamp_Type_Based", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587846530938, "lm_q2_score": 0.8311430541321951, "lm_q1q2_score": 0.7386856906633903}} {"text": "[STATEMENT]\nlemma kronecker_distr_left:\n assumes \"dim_row B = dim_row C\" \"dim_col B = dim_col C\"\n shows \"kronecker_product A (B+C) = kronecker_product A B + kronecker_product A C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. kronecker_product A (B + C) = kronecker_product A B + kronecker_product A C\n[PROOF STEP]\nunfolding kronecker_product_def Let_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (dim_row A * dim_row (B + C)) (dim_col A * dim_col (B + C)) (\\(i, j). A $$ (i div dim_row (B + C), j div dim_col (B + C)) * (B + C) $$ (i mod dim_row (B + C), j mod dim_col (B + C))) = mat (dim_row A * dim_row B) (dim_col A * dim_col B) (\\(i, j). A $$ (i div dim_row B, j div dim_col B) * B $$ (i mod dim_row B, j mod dim_col B)) + mat (dim_row A * dim_row C) (dim_col A * dim_col C) (\\(i, j). A $$ (i div dim_row C, j div dim_col C) * C $$ (i mod dim_row C, j mod dim_col C))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_row B = dim_row C\ndim_col B = dim_col C\n\ngoal (1 subgoal):\n 1. mat (dim_row A * dim_row (B + C)) (dim_col A * dim_col (B + C)) (\\(i, j). A $$ (i div dim_row (B + C), j div dim_col (B + C)) * (B + C) $$ (i mod dim_row (B + C), j mod dim_col (B + C))) = mat (dim_row A * dim_row B) (dim_col A * dim_col B) (\\(i, j). A $$ (i div dim_row B, j div dim_col B) * B $$ (i mod dim_row B, j mod dim_col B)) + mat (dim_row A * dim_row C) (dim_col A * dim_col C) (\\(i, j). A $$ (i div dim_row C, j div dim_col C) * C $$ (i mod dim_row C, j mod dim_col C))\n[PROOF STEP]\napply (auto simp add: mat_eq_iff)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. \\dim_row B = dim_row C; dim_col B = dim_col C; i < dim_row A * dim_row C; j < dim_col A * dim_col C\\ \\ A $$ (i div dim_row C, j div dim_col C) * (B + C) $$ (i mod dim_row C, j mod dim_col C) = A $$ (i div dim_row C, j div dim_col C) * B $$ (i mod dim_row C, j mod dim_col C) + A $$ (i div dim_row C, j div dim_col C) * C $$ (i mod dim_row C, j mod dim_col C)\n[PROOF STEP]\nby (metis (no_types, lifting) distrib_left index_add_mat(1) mod_less_divisor mult_eq_0_iff neq0_conv not_less_zero)", "meta": {"llama_tokens": 943, "file": "BenOr_Kozen_Reif_More_Matrix", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767970940975, "lm_q2_score": 0.8418256393148982, "lm_q1q2_score": 0.7386824656977279}} {"text": "[STATEMENT]\nlemma integral_norm_eq_0_iff:\n fixes f :: \"'a \\ 'b::{banach, second_countable_topology}\"\n assumes f[measurable]: \"integrable M f\"\n shows \"(\\x. norm (f x) \\M) = 0 \\ emeasure M {x\\space M. f x \\ 0} = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (LINT x|M. norm (f x) = 0) = (emeasure M {x \\ space M. f x \\ (0::'b)} = 0)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (LINT x|M. norm (f x) = 0) = (emeasure M {x \\ space M. f x \\ (0::'b)} = 0)\n[PROOF STEP]\nhave \"(\\\\<^sup>+x. norm (f x) \\M) = (\\x. norm (f x) \\M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (norm (f x)) \\M = ennreal (LINT x|M. norm (f x))\n[PROOF STEP]\nusing f\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable M f\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (norm (f x)) \\M = ennreal (LINT x|M. norm (f x))\n[PROOF STEP]\nby (intro nn_integral_eq_integral integrable_norm) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal (norm (f x)) \\M = ennreal (LINT x|M. norm (f x))\n\ngoal (1 subgoal):\n 1. (LINT x|M. norm (f x) = 0) = (emeasure M {x \\ space M. f x \\ (0::'b)} = 0)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sup>+ x. ennreal (norm (f x)) \\M = ennreal (LINT x|M. norm (f x))\n[PROOF STEP]\nhave \"(\\x. norm (f x) \\M) = 0 \\ (\\\\<^sup>+x. norm (f x) \\M) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>+ x. ennreal (norm (f x)) \\M = ennreal (LINT x|M. norm (f x))\n\ngoal (1 subgoal):\n 1. (LINT x|M. norm (f x) = 0) = (\\\\<^sup>+ x. ennreal (norm (f x)) \\M = 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(LINT x|M. norm (f x) = 0) = (\\\\<^sup>+ x. ennreal (norm (f x)) \\M = 0)\n\ngoal (1 subgoal):\n 1. (LINT x|M. norm (f x) = 0) = (emeasure M {x \\ space M. f x \\ (0::'b)} = 0)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(LINT x|M. norm (f x) = 0) = (\\\\<^sup>+ x. ennreal (norm (f x)) \\M = 0)\n\ngoal (1 subgoal):\n 1. (LINT x|M. norm (f x) = 0) = (emeasure M {x \\ space M. f x \\ (0::'b)} = 0)\n[PROOF STEP]\nhave \"\\ \\ emeasure M {x\\space M. ennreal (norm (f x)) \\ 0} = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sup>+ x. ennreal (norm (f x)) \\M = 0) = (emeasure M {x \\ space M. ennreal (norm (f x)) \\ 0} = 0)\n[PROOF STEP]\nby (intro nn_integral_0_iff) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\\\<^sup>+ x. ennreal (norm (f x)) \\M = 0) = (emeasure M {x \\ space M. ennreal (norm (f x)) \\ 0} = 0)\n\ngoal (1 subgoal):\n 1. (LINT x|M. norm (f x) = 0) = (emeasure M {x \\ space M. f x \\ (0::'b)} = 0)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(LINT x|M. norm (f x) = 0) = (emeasure M {x \\ space M. ennreal (norm (f x)) \\ 0} = 0)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(LINT x|M. norm (f x) = 0) = (emeasure M {x \\ space M. ennreal (norm (f x)) \\ 0} = 0)\n\ngoal (1 subgoal):\n 1. (LINT x|M. norm (f x) = 0) = (emeasure M {x \\ space M. f x \\ (0::'b)} = 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(LINT x|M. norm (f x) = 0) = (emeasure M {x \\ space M. f x \\ (0::'b)} = 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1677, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8418256432832333, "lm_q1q2_score": 0.7386824556934426}} {"text": "[STATEMENT]\nlemma poly_mult_lead_coeff:\n assumes \"subring K R\" \"polynomial K p1\" \"polynomial K p2\" and \"p1 \\ []\" and \"p2 \\ []\"\n shows \"lead_coeff (poly_mult p1 p2) = (lead_coeff p1) \\ (lead_coeff p2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lead_coeff (poly_mult p1 p2) = lead_coeff p1 \\ lead_coeff p2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. lead_coeff (poly_mult p1 p2) = lead_coeff p1 \\ lead_coeff p2\n[PROOF STEP]\nhave \"poly_mult p1 p2 \\ []\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly_mult p1 p2 \\ []\n[PROOF STEP]\nusing poly_mult_integral[OF assms(1-3)] assms(4-5)\n[PROOF STATE]\nproof (prove)\nusing this:\npoly_mult p1 p2 = [] \\ p1 = [] \\ p2 = []\np1 \\ []\np2 \\ []\n\ngoal (1 subgoal):\n 1. poly_mult p1 p2 \\ []\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\npoly_mult p1 p2 \\ []\n\ngoal (1 subgoal):\n 1. lead_coeff (poly_mult p1 p2) = lead_coeff p1 \\ lead_coeff p2\n[PROOF STEP]\nhence \"lead_coeff (poly_mult p1 p2) = (coeff (poly_mult p1 p2)) (degree p1 + degree p2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\npoly_mult p1 p2 \\ []\n\ngoal (1 subgoal):\n 1. lead_coeff (poly_mult p1 p2) = local.coeff (poly_mult p1 p2) (degree p1 + degree p2)\n[PROOF STEP]\nusing poly_mult_degree_eq[OF assms(1-3)] assms(4-5)\n[PROOF STATE]\nproof (prove)\nusing this:\npoly_mult p1 p2 \\ []\ndegree (poly_mult p1 p2) = (if p1 = [] \\ p2 = [] then 0 else degree p1 + degree p2)\np1 \\ []\np2 \\ []\n\ngoal (1 subgoal):\n 1. lead_coeff (poly_mult p1 p2) = local.coeff (poly_mult p1 p2) (degree p1 + degree p2)\n[PROOF STEP]\nby (metis coeff.simps(2) list.collapse)\n[PROOF STATE]\nproof (state)\nthis:\nlead_coeff (poly_mult p1 p2) = local.coeff (poly_mult p1 p2) (degree p1 + degree p2)\n\ngoal (1 subgoal):\n 1. lead_coeff (poly_mult p1 p2) = lead_coeff p1 \\ lead_coeff p2\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlead_coeff (poly_mult p1 p2) = local.coeff (poly_mult p1 p2) (degree p1 + degree p2)\n\ngoal (1 subgoal):\n 1. lead_coeff (poly_mult p1 p2) = lead_coeff p1 \\ lead_coeff p2\n[PROOF STEP]\nusing poly_mult_lead_coeff_aux[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nlead_coeff (poly_mult p1 p2) = local.coeff (poly_mult p1 p2) (degree p1 + degree p2)\nlocal.coeff (poly_mult p1 p2) (degree p1 + degree p2) = lead_coeff p1 \\ lead_coeff p2\n\ngoal (1 subgoal):\n 1. lead_coeff (poly_mult p1 p2) = lead_coeff p1 \\ lead_coeff p2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlead_coeff (poly_mult p1 p2) = lead_coeff p1 \\ lead_coeff p2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1234, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357666736772, "lm_q2_score": 0.8519528038477824, "lm_q1q2_score": 0.7386735524539509}} {"text": "[STATEMENT]\nlemma powr_inverse_numeral:\n \"0 < x \\ x powr (1 / numeral n) = root (numeral n) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ x powr (1 / numeral n) = root (numeral n) x\n[PROOF STEP]\nby (simp add: root_powr_inverse)", "meta": {"llama_tokens": 114, "file": "Amortized_Complexity_Splay_Tree_Analysis_Optimal", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9196425355825848, "lm_q2_score": 0.8031738010682209, "lm_q1q2_score": 0.7386327909278813}} {"text": "[STATEMENT]\nlemma card_prop_mono:\n assumes \\m \\ n\\\n shows \\card {i::nat. i \\ m \\ P i} \\ card {i. i \\ n \\ P i}\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {i. i \\ m \\ P i} \\ card {i. i \\ n \\ P i}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {i. i \\ m \\ P i} \\ card {i. i \\ n \\ P i}\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nm \\ n\n[PROOF STEP]\nhave \\{i. i \\ m \\ P i} \\ {i. i \\ n \\ P i}\\\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ n\n\ngoal (1 subgoal):\n 1. {i. i \\ m \\ P i} \\ {i. i \\ n \\ P i}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{i. i \\ m \\ P i} \\ {i. i \\ n \\ P i}\n\ngoal (1 subgoal):\n 1. card {i. i \\ m \\ P i} \\ card {i. i \\ n \\ P i}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n{i. i \\ m \\ P i} \\ {i. i \\ n \\ P i}\n\ngoal (1 subgoal):\n 1. card {i. i \\ m \\ P i} \\ card {i. i \\ n \\ P i}\n[PROOF STEP]\nhave \\finite {i. i \\ n \\ P i}\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {i. i \\ n \\ P i}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfinite {i. i \\ n \\ P i}\n\ngoal (1 subgoal):\n 1. card {i. i \\ m \\ P i} \\ card {i. i \\ n \\ P i}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n{i. i \\ m \\ P i} \\ {i. i \\ n \\ P i}\nfinite {i. i \\ n \\ P i}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n{i. i \\ m \\ P i} \\ {i. i \\ n \\ P i}\nfinite {i. i \\ n \\ P i}\n\ngoal (1 subgoal):\n 1. card {i. i \\ m \\ P i} \\ card {i. i \\ n \\ P i}\n[PROOF STEP]\nby (simp add: card_mono)\n[PROOF STATE]\nproof (state)\nthis:\ncard {i. i \\ m \\ P i} \\ card {i. i \\ n \\ P i}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 992, "file": "TESL_Language_StutteringLemmas", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392909114836, "lm_q2_score": 0.8354835289107307, "lm_q1q2_score": 0.7386002664664664}} {"text": "[STATEMENT]\nlemma continuous_UNIV_iff_lsc_usc:\n fixes f :: \"'a::metric_space \\ ereal\"\n shows \"(\\x. continuous (at x) f) \\ (lsc f) \\ (usc f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. isCont f x) = (lsc f \\ usc f)\n[PROOF STEP]\nby (metis continuous_iff_lsc_usc lsc_def usc_def)", "meta": {"llama_tokens": 145, "file": "Lower_Semicontinuous_Lower_Semicontinuous", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9086178969328287, "lm_q2_score": 0.8128673155708975, "lm_q1q2_score": 0.7385857907594628}} {"text": "[STATEMENT]\nlemma mod_0_div_mult_cancel: \"(n mod (m::nat) = 0) = (n div m * m = n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (n mod m = 0) = (n div m * m = n)\n[PROOF STEP]\napply (insert eq_diff_left_iff[OF mod_le_dividend le0, of n m])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (n - n mod m = n - 0) = (n mod m = 0) \\ (n mod m = 0) = (n div m * m = n)\n[PROOF STEP]\napply (simp add: mult.commute minus_mod_eq_mult_div [symmetric])\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 244, "file": "List-Infinite_CommonArith_Util_Div", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9086178870347122, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.7385857806540653}} {"text": "[STATEMENT]\nlemma measure_Union_le:\n \"finite F \\ (\\S. S \\ F \\ S \\ sets M) \\ measure M (\\F) \\ (\\S\\F. measure M S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite F; \\S. S \\ F \\ S \\ sets M\\ \\ Sigma_Algebra.measure M (\\ F) \\ sum (Sigma_Algebra.measure M) F\n[PROOF STEP]\nusing measure_UNION_le[of F \"\\x. x\" M]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite F; \\i. i \\ F \\ i \\ sets M\\ \\ Sigma_Algebra.measure M (\\i\\F. i) \\ sum (Sigma_Algebra.measure M) F\n\ngoal (1 subgoal):\n 1. \\finite F; \\S. S \\ F \\ S \\ sets M\\ \\ Sigma_Algebra.measure M (\\ F) \\ sum (Sigma_Algebra.measure M) F\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 351, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942014971871, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7385102896604113}} {"text": "[STATEMENT]\nlemma (in group) int_pow_mult:\n assumes \"x \\ carrier G\" shows \"x [^] (i + j::int) = x [^] i \\ x [^] j\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x [^] (i + j) = x [^] i \\ x [^] j\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x [^] (i + j) = x [^] i \\ x [^] j\n[PROOF STEP]\nhave [simp]: \"-i - j = -j - i\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - i - j = - j - i\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n- i - j = - j - i\n\ngoal (1 subgoal):\n 1. x [^] (i + j) = x [^] i \\ x [^] j\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x [^] (i + j) = x [^] i \\ x [^] j\n[PROOF STEP]\nby (auto simp: assms int_pow_def2 inv_solve_left inv_solve_right nat_add_distrib [symmetric] nat_pow_mult)\n[PROOF STATE]\nproof (state)\nthis:\nx [^] (i + j) = x [^] i \\ x [^] j\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 456, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094117351309, "lm_q2_score": 0.8267117940706734, "lm_q1q2_score": 0.7385094264357679}} {"text": "[STATEMENT]\nlemma vector_scalar_matrix_ac:\n fixes k :: \"'a::{field}\" and x :: \"'a::{field}^'n\" and A :: \"'a^'m^'n\"\n shows \"x v* (k *k A) = k *s (x v* A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x v* (k *k A) = k *s (x v* A)\n[PROOF STEP]\nusing scalar_vector_matrix_assoc\n[PROOF STATE]\nproof (prove)\nusing this:\n?k *s ?x v* ?A = ?k *s (?x v* ?A)\n\ngoal (1 subgoal):\n 1. x v* (k *k A) = k *s (x v* A)\n[PROOF STEP]\nunfolding vector_matrix_mult_def matrix_scalar_mult_def vec_eq_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i. (\\j. \\i\\UNIV. ?A $ i $ j * (?k *s ?x) $ i) $ i = (?k *s (\\j. \\i\\UNIV. ?A $ i $ j * ?x $ i)) $ i\n\ngoal (1 subgoal):\n 1. \\i. (\\j. \\i\\UNIV. (\\i j. k * A $ i $ j) $ i $ j * x $ i) $ i = (k *s (\\j. \\i\\UNIV. A $ i $ j * x $ i)) $ i\n[PROOF STEP]\nby (auto simp add: sum_distrib_left vector_space_over_itself.scale_scale)", "meta": {"llama_tokens": 450, "file": "Rank_Nullity_Theorem_Miscellaneous", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094032139577, "lm_q2_score": 0.8267118004748677, "lm_q1q2_score": 0.7385094251121406}} {"text": "[STATEMENT]\ntheorem zeta_even_nat: \n \"zeta (2 * of_nat n) = \n of_real ((-1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. zeta (2 * of_nat n) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\n[PROOF STEP]\nproof (cases \"n = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. n = 0 \\ zeta (2 * of_nat n) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\n 2. n \\ 0 \\ zeta (2 * of_nat n) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nn \\ 0\n\ngoal (2 subgoals):\n 1. n = 0 \\ zeta (2 * of_nat n) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\n 2. n \\ 0 \\ zeta (2 * of_nat n) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\n[PROOF STEP]\nhence \"(\\k. 1 / of_nat (Suc k) ^ (2 * n)) sums zeta (of_nat (2 * n))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\ 0\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n)) sums zeta (of_nat (2 * n))\n[PROOF STEP]\nby (intro sums_zeta_nat) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. 1 / of_nat (Suc k) ^ (2 * n)) sums zeta (of_nat (2 * n))\n\ngoal (2 subgoals):\n 1. n = 0 \\ zeta (2 * of_nat n) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\n 2. n \\ 0 \\ zeta (2 * of_nat n) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\n[PROOF STEP]\nfrom sums_unique2 [OF this nat_even_power_sums_complex] False\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < n \\ zeta (of_nat (2 * n)) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\nn \\ 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n \\ zeta (of_nat (2 * n)) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\nn \\ 0\n\ngoal (1 subgoal):\n 1. zeta (2 * of_nat n) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nzeta (2 * of_nat n) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\n\ngoal (1 subgoal):\n 1. n = 0 \\ zeta (2 * of_nat n) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))\n[PROOF STEP]\nqed (insert zeta_neg_of_nat[of 0], simp_all)", "meta": {"llama_tokens": 1330, "file": "Zeta_Function_Zeta_Function", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9219218412907381, "lm_q2_score": 0.8006920092299293, "lm_q1q2_score": 0.7381754514560371}} {"text": "[STATEMENT]\nlemma lm147: \n assumes \"card (Pow A) = 2\" \n shows \"card A = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card A = 1\n[PROOF STEP]\nusing assms lm146 [of A]\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (Pow A) = 2\ncard (Pow A) \\ 0 \\ card A = Discrete.log (card (Pow A))\n\ngoal (1 subgoal):\n 1. card A = 1\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 173, "file": "Vickrey_Clarke_Groves_MiscTools", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952975813453, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7380753220588372}} {"text": "[STATEMENT]\nlemma overdamped_door_arith:\n assumes \"b\\<^sup>2 + a * 4 > 0\" and \"a < 0\" and \"b \\ 0\" and \"t \\ 0\" and \"s1 > 0\"\n shows \"0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - \n(b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2) * s1 / sqrt (b\\<^sup>2 + a * 4)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2) * s1 / sqrt (b\\<^sup>2 + a * 4)\n[PROOF STEP]\nproof(subst diff_divide_distrib[symmetric], simp)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nhave f0: \"s1 / (2 * sqrt (b\\<^sup>2 + a * 4)) > 0\" (is \"s1/?c3 > 0\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nusing assms(1,5)\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < b\\<^sup>2 + a * 4\n0 < s1\n\ngoal (1 subgoal):\n 1. 0 < s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nhave f1: \"(b - sqrt (b\\<^sup>2 + 4 * a)) < (b + sqrt (b\\<^sup>2 + 4 * a))\" (is \"?c2 < ?c1\") \n and f2: \"(b + sqrt (b\\<^sup>2 + 4 * a)) < 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. b - sqrt (b\\<^sup>2 + 4 * a) < b + sqrt (b\\<^sup>2 + 4 * a) &&& b + sqrt (b\\<^sup>2 + 4 * a) < 0\n[PROOF STEP]\nusing sqrt_ge_absD[of b \"b\\<^sup>2 + 4 * a\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\b\\ \\ sqrt (b\\<^sup>2 + 4 * a) \\ b\\<^sup>2 \\ b\\<^sup>2 + 4 * a\n0 < b\\<^sup>2 + a * 4\na < 0\nb \\ 0\n0 \\ t\n0 < s1\n\ngoal (1 subgoal):\n 1. b - sqrt (b\\<^sup>2 + 4 * a) < b + sqrt (b\\<^sup>2 + 4 * a) &&& b + sqrt (b\\<^sup>2 + 4 * a) < 0\n[PROOF STEP]\nby (force, linarith)\n[PROOF STATE]\nproof (state)\nthis:\nb - sqrt (b\\<^sup>2 + 4 * a) < b + sqrt (b\\<^sup>2 + 4 * a)\nb + sqrt (b\\<^sup>2 + 4 * a) < 0\n\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nhence f3: \"exp (t * ?c2 / 2) \\ exp (t * ?c1 / 2)\" (is \"exp ?t1 \\ exp ?t2\")\n[PROOF STATE]\nproof (prove)\nusing this:\nb - sqrt (b\\<^sup>2 + 4 * a) < b + sqrt (b\\<^sup>2 + 4 * a)\nb + sqrt (b\\<^sup>2 + 4 * a) < 0\n\ngoal (1 subgoal):\n 1. exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)\n[PROOF STEP]\nunfolding exp_le_cancel_iff\n[PROOF STATE]\nproof (prove)\nusing this:\nb - sqrt (b\\<^sup>2 + 4 * a) < b + sqrt (b\\<^sup>2 + 4 * a)\nb + sqrt (b\\<^sup>2 + 4 * a) < 0\n\ngoal (1 subgoal):\n 1. t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 \\ t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2\n[PROOF STEP]\nusing assms(4)\n[PROOF STATE]\nproof (prove)\nusing this:\nb - sqrt (b\\<^sup>2 + 4 * a) < b + sqrt (b\\<^sup>2 + 4 * a)\nb + sqrt (b\\<^sup>2 + 4 * a) < 0\n0 \\ t\n\ngoal (1 subgoal):\n 1. t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 \\ t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2\n[PROOF STEP]\nby (case_tac \"t=0\", simp_all)\n[PROOF STATE]\nproof (state)\nthis:\nexp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)\n\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nhence \"?c2 * exp ?t2 \\ ?c2 * exp ?t1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nexp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)\n\ngoal (1 subgoal):\n 1. (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n[PROOF STEP]\nusing f1 f2 mult_le_cancel_left_pos[of \"-?c2\" \"exp ?t1\" \"exp ?t2\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nexp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)\nb - sqrt (b\\<^sup>2 + 4 * a) < b + sqrt (b\\<^sup>2 + 4 * a)\nb + sqrt (b\\<^sup>2 + 4 * a) < 0\n0 < - (b - sqrt (b\\<^sup>2 + 4 * a)) \\ (- (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) = (exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2))\n\ngoal (1 subgoal):\n 1. (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\n(b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nhave \"... < ?c1 * exp ?t1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) < (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n[PROOF STEP]\nusing f1\n[PROOF STATE]\nproof (prove)\nusing this:\nb - sqrt (b\\<^sup>2 + 4 * a) < b + sqrt (b\\<^sup>2 + 4 * a)\n\ngoal (1 subgoal):\n 1. (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) < (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) < (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) < (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nhave\"... \\ ?c1 * exp ?t1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n[PROOF STEP]\nusing f1 f2\n[PROOF STATE]\nproof (prove)\nusing this:\nb - sqrt (b\\<^sup>2 + 4 * a) < b + sqrt (b\\<^sup>2 + 4 * a)\nb + sqrt (b\\<^sup>2 + 4 * a) < 0\n\ngoal (1 subgoal):\n 1. (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) < (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n(b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n[PROOF STEP]\nshow \"0 \\ (?c1 * exp ?t1 - ?c2 * exp ?t2) * s1 / ?c3\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) < (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n(b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nusing f0 f1 assms(5)\n[PROOF STATE]\nproof (prove)\nusing this:\n(b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) < (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n(b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) \\ (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)\n0 < s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\nb - sqrt (b\\<^sup>2 + 4 * a) < b + sqrt (b\\<^sup>2 + 4 * a)\n0 < s1\n\ngoal (1 subgoal):\n 1. 0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) * s1 / (2 * sqrt (b\\<^sup>2 + a * 4))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5719, "file": "Matrices_for_ODEs_MTX_Examples", "length": 27, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.885631470799559, "lm_q2_score": 0.8333245932423308, "lm_q1q2_score": 0.7380184851666496}} {"text": "[STATEMENT]\nlemma upper_asymptotic_density_zero_union:\n assumes \"upper_asymptotic_density A = 0\" \"upper_asymptotic_density B = 0\"\n shows \"upper_asymptotic_density (A \\ B) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. upper_asymptotic_density (A \\ B) = 0\n[PROOF STEP]\nusing upper_asymptotic_density_in_01(3)[of \"A \\ B\"] upper_asymptotic_density_union[of A B]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ upper_asymptotic_density (A \\ B)\nupper_asymptotic_density (A \\ B) \\ upper_asymptotic_density A + upper_asymptotic_density B\n\ngoal (1 subgoal):\n 1. upper_asymptotic_density (A \\ B) = 0\n[PROOF STEP]\nunfolding assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ upper_asymptotic_density (A \\ B)\nupper_asymptotic_density (A \\ B) \\ 0 + 0\n\ngoal (1 subgoal):\n 1. upper_asymptotic_density (A \\ B) = 0\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 400, "file": "Ergodic_Theory_Asymptotic_Density", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8289388104343893, "lm_q1q2_score": 0.7379994319124411}} {"text": "[STATEMENT]\nlemma facet_of_convex_hull_affine_independent:\n fixes S :: \"'a::euclidean_space set\"\n assumes \"\\ affine_dependent S\"\n shows \"T facet_of (convex hull S) \\\n T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (T facet_of convex hull S) = (T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})))\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nassume ?lhs\n[PROOF STATE]\nproof (state)\nthis:\nT facet_of convex hull S\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nT facet_of convex hull S\n[PROOF STEP]\nhave \"T face_of (convex hull S)\" \"T \\ {}\"\n and afft: \"aff_dim T = aff_dim (convex hull S) - 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nT facet_of convex hull S\n\ngoal (1 subgoal):\n 1. (T face_of convex hull S &&& T \\ {}) &&& aff_dim T = aff_dim (convex hull S) - 1\n[PROOF STEP]\nby (auto simp: facet_of_def)\n[PROOF STATE]\nproof (state)\nthis:\nT face_of convex hull S\nT \\ {}\naff_dim T = aff_dim (convex hull S) - 1\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nT face_of convex hull S\nT \\ {}\naff_dim T = aff_dim (convex hull S) - 1\n[PROOF STEP]\nobtain c where \"c \\ S\" and c: \"T = convex hull c\"\n[PROOF STATE]\nproof (prove)\nusing this:\nT face_of convex hull S\nT \\ {}\naff_dim T = aff_dim (convex hull S) - 1\n\ngoal (1 subgoal):\n 1. (\\c. \\c \\ S; T = convex hull c\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: face_of_convex_hull_affine_independent [OF assms])\n[PROOF STATE]\nproof (state)\nthis:\nc \\ S\nT = convex hull c\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc \\ S\nT = convex hull c\n[PROOF STEP]\nhave affs: \"aff_dim S = aff_dim c + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ S\nT = convex hull c\n\ngoal (1 subgoal):\n 1. aff_dim S = aff_dim c + 1\n[PROOF STEP]\nby (metis aff_dim_convex_hull afft eq_diff_eq)\n[PROOF STATE]\nproof (state)\nthis:\naff_dim S = aff_dim c + 1\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nhave \"\\ affine_dependent c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ affine_dependent c\n[PROOF STEP]\nusing \\c \\ S\\ affine_dependent_subset assms\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ S\n\\affine_dependent ?s; ?s \\ ?t\\ \\ affine_dependent ?t\n\\ affine_dependent S\n\ngoal (1 subgoal):\n 1. \\ affine_dependent c\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\ affine_dependent c\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nwith affs\n[PROOF STATE]\nproof (chain)\npicking this:\naff_dim S = aff_dim c + 1\n\\ affine_dependent c\n[PROOF STEP]\nhave \"card (S - c) = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\naff_dim S = aff_dim c + 1\n\\ affine_dependent c\n\ngoal (1 subgoal):\n 1. card (S - c) = 1\n[PROOF STEP]\napply (simp add: aff_dim_affine_independent [symmetric] aff_dim_convex_hull)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\aff_dim S = int (card c); \\ affine_dependent c\\ \\ card (S - c) = Suc 0\n[PROOF STEP]\nby (metis aff_dim_affine_independent aff_independent_finite One_nat_def \\c \\ S\\ add.commute\n add_diff_cancel_right' assms card_Diff_subset card_mono of_nat_1 of_nat_diff of_nat_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncard (S - c) = 1\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (S - c) = 1\n[PROOF STEP]\nobtain u where u: \"u \\ S - c\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (S - c) = 1\n\ngoal (1 subgoal):\n 1. (\\u. u \\ S - c \\ thesis) \\ thesis\n[PROOF STEP]\nby (metis DiffI \\c \\ S\\ aff_independent_finite assms cancel_comm_monoid_add_class.diff_cancel\n card_Diff_subset subsetI subset_antisym zero_neq_one)\n[PROOF STATE]\nproof (state)\nthis:\nu \\ S - c\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nu \\ S - c\n[PROOF STEP]\nhave u: \"S = insert u c\"\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ S - c\n\ngoal (1 subgoal):\n 1. S = insert u c\n[PROOF STEP]\nby (metis Diff_subset \\c \\ S\\ \\card (S - c) = 1\\ card_1_singletonE double_diff insert_Diff insert_subset singletonD)\n[PROOF STATE]\nproof (state)\nthis:\nS = insert u c\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nhave \"T = convex hull (c - {u})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. T = convex hull (c - {u})\n[PROOF STEP]\nby (metis Diff_empty Diff_insert0 \\T facet_of convex hull S\\ c facet_of_irrefl insert_absorb u)\n[PROOF STATE]\nproof (state)\nthis:\nT = convex hull (c - {u})\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nwith \\T \\ {}\\\n[PROOF STATE]\nproof (chain)\npicking this:\nT \\ {}\nT = convex hull (c - {u})\n[PROOF STEP]\nshow ?rhs\n[PROOF STATE]\nproof (prove)\nusing this:\nT \\ {}\nT = convex hull (c - {u})\n\ngoal (1 subgoal):\n 1. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n[PROOF STEP]\nusing c u\n[PROOF STATE]\nproof (prove)\nusing this:\nT \\ {}\nT = convex hull (c - {u})\nT = convex hull c\nS = insert u c\n\ngoal (1 subgoal):\n 1. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nT \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n\ngoal (1 subgoal):\n 1. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nassume ?rhs\n[PROOF STATE]\nproof (state)\nthis:\nT \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n\ngoal (1 subgoal):\n 1. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nT \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n[PROOF STEP]\nobtain u where \"T \\ {}\" \"u \\ S\" and u: \"T = convex hull (S - {u})\"\n[PROOF STATE]\nproof (prove)\nusing this:\nT \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n\ngoal (1 subgoal):\n 1. (\\u. \\T \\ {}; u \\ S; T = convex hull (S - {u})\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (force simp: facet_of_def)\n[PROOF STATE]\nproof (state)\nthis:\nT \\ {}\nu \\ S\nT = convex hull (S - {u})\n\ngoal (1 subgoal):\n 1. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nT \\ {}\nu \\ S\nT = convex hull (S - {u})\n[PROOF STEP]\nhave \"\\ S \\ {u}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nT \\ {}\nu \\ S\nT = convex hull (S - {u})\n\ngoal (1 subgoal):\n 1. \\ S \\ {u}\n[PROOF STEP]\nusing \\T \\ {}\\ u\n[PROOF STATE]\nproof (prove)\nusing this:\nT \\ {}\nu \\ S\nT = convex hull (S - {u})\nT \\ {}\nT = convex hull (S - {u})\n\ngoal (1 subgoal):\n 1. \\ S \\ {u}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\ S \\ {u}\n\ngoal (1 subgoal):\n 1. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nhave \"aff_dim (S - {u}) = aff_dim S - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. aff_dim (S - {u}) = aff_dim S - 1\n[PROOF STEP]\nusing assms \\u \\ S\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ affine_dependent S\nu \\ S\n\ngoal (1 subgoal):\n 1. aff_dim (S - {u}) = aff_dim S - 1\n[PROOF STEP]\nunfolding affine_dependent_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (\\x\\S. x \\ affine hull (S - {x}))\nu \\ S\n\ngoal (1 subgoal):\n 1. aff_dim (S - {u}) = aff_dim S - 1\n[PROOF STEP]\nby (metis add_diff_cancel_right' aff_dim_insert insert_Diff [of u S])\n[PROOF STATE]\nproof (state)\nthis:\naff_dim (S - {u}) = aff_dim S - 1\n\ngoal (1 subgoal):\n 1. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\naff_dim (S - {u}) = aff_dim S - 1\n[PROOF STEP]\nhave \"aff_dim (convex hull (S - {u})) = aff_dim (convex hull S) - 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\naff_dim (S - {u}) = aff_dim S - 1\n\ngoal (1 subgoal):\n 1. aff_dim (convex hull (S - {u})) = aff_dim (convex hull S) - 1\n[PROOF STEP]\nby (simp add: aff_dim_convex_hull)\n[PROOF STATE]\nproof (state)\nthis:\naff_dim (convex hull (S - {u})) = aff_dim (convex hull S) - 1\n\ngoal (1 subgoal):\n 1. T \\ {} \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\naff_dim (convex hull (S - {u})) = aff_dim (convex hull S) - 1\n[PROOF STEP]\nshow ?lhs\n[PROOF STATE]\nproof (prove)\nusing this:\naff_dim (convex hull (S - {u})) = aff_dim (convex hull S) - 1\n\ngoal (1 subgoal):\n 1. T facet_of convex hull S\n[PROOF STEP]\nby (metis Diff_subset \\T \\ {}\\ assms face_of_convex_hull_affine_independent facet_of_def u)\n[PROOF STATE]\nproof (state)\nthis:\nT facet_of convex hull S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5221, "file": null, "length": 50, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.890294223211224, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7379994305629841}} {"text": "[STATEMENT]\nlemma power_subdist: \"x\\<^bsup>n\\<^esup> \\ (x + y)\\<^bsup>n\\<^esup>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x\\<^bsup>n\\<^esup> \\ (x + y)\\<^bsup>n\\<^esup>\n[PROOF STEP]\napply (induct n)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. x\\<^bsup>0\\<^esup> \\ (x + y)\\<^bsup>0\\<^esup>\n 2. \\n. x\\<^bsup>n\\<^esup> \\ (x + y)\\<^bsup>n\\<^esup> \\ x\\<^bsup>Suc n\\<^esup> \\ (x + y)\\<^bsup>Suc n\\<^esup>\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. x\\<^bsup>n\\<^esup> \\ (x + y)\\<^bsup>n\\<^esup> \\ x\\<^bsup>Suc n\\<^esup> \\ (x + y)\\<^bsup>Suc n\\<^esup>\n[PROOF STEP]\nusing local.mult_isol_var local.power_Suc2\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?u \\ ?x; ?v \\ ?y\\ \\ ?u \\ ?v \\ ?x \\ ?y\n?a\\<^bsup>Suc ?n\\<^esup> = ?a\\<^bsup>?n\\<^esup> \\ ?a\n\ngoal (1 subgoal):\n 1. \\n. x\\<^bsup>n\\<^esup> \\ (x + y)\\<^bsup>n\\<^esup> \\ x\\<^bsup>Suc n\\<^esup> \\ (x + y)\\<^bsup>Suc n\\<^esup>\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 547, "file": "Regular_Algebras_Dioid_Power_Sum", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942173896131, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.737999425737225}} {"text": "[STATEMENT]\nlemma sq_mtx_inv_mult:\n assumes \"mtx_invertible A\" and \"mtx_invertible B\"\n shows \"(A * B)\\<^sup>-\\<^sup>1 = B\\<^sup>-\\<^sup>1 * A\\<^sup>-\\<^sup>1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (A * B)\\<^sup>-\\<^sup>1 = B\\<^sup>-\\<^sup>1 * A\\<^sup>-\\<^sup>1\n[PROOF STEP]\nby (simp add: assms matrix_inv_matrix_mul sq_mtx_inv_def times_sq_mtx_def)", "meta": {"llama_tokens": 179, "file": "Matrices_for_ODEs_SQ_MTX", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942203004185, "lm_q2_score": 0.8289387998695209, "lm_q1q2_score": 0.7379994225065998}} {"text": "[STATEMENT]\ntheorem F1map_cong: \"\\\\z. z \\ F1set1 x \\ f1 z = g1 z; \\z. z \\ F1set2 x \\ f2 z = g2 z\\\n \\ F1map f1 f2 x = F1map g1 g2 x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\z. z \\ F1set1 x \\ f1 z = g1 z; \\z. z \\ F2set x \\ f2 z = g2 z\\ \\ F1map f1 f2 x = F1map g1 g2 x\n[PROOF STEP]\napply (rule F.map_cong0)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\z1. \\\\z. z \\ F1set1 x \\ f1 z = g1 z; \\z. z \\ F2set x \\ f2 z = g2 z; z1 \\ Fset1 x\\ \\ id z1 = id z1\n 2. \\z2. \\\\z. z \\ F1set1 x \\ f1 z = g1 z; \\z. z \\ F2set x \\ f2 z = g2 z; z2 \\ F1set1 x\\ \\ f1 z2 = g1 z2\n 3. \\z3. \\\\z. z \\ F1set1 x \\ f1 z = g1 z; \\z. z \\ F2set x \\ f2 z = g2 z; z3 \\ F2set x\\ \\ f2 z3 = g2 z3\n[PROOF STEP]\napply (rule refl)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\z2. \\\\z. z \\ F1set1 x \\ f1 z = g1 z; \\z. z \\ F2set x \\ f2 z = g2 z; z2 \\ F1set1 x\\ \\ f1 z2 = g1 z2\n 2. \\z3. \\\\z. z \\ F1set1 x \\ f1 z = g1 z; \\z. z \\ F2set x \\ f2 z = g2 z; z3 \\ F2set x\\ \\ f2 z3 = g2 z3\n[PROOF STEP]\napply assumption\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z3. \\\\z. z \\ F1set1 x \\ f1 z = g1 z; \\z. z \\ F2set x \\ f2 z = g2 z; z3 \\ F2set x\\ \\ f2 z3 = g2 z3\n[PROOF STEP]\napply assumption\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 878, "file": "BNF_Operations_Kill", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473779969194, "lm_q2_score": 0.8459424431344437, "lm_q1q2_score": 0.7379556722046401}} {"text": "[STATEMENT]\nlemma fls_deriv_X_power:\n \"fls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n-1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n - 1)\n[PROOF STEP]\nproof (cases n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. n = 0 \\ fls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n - 1)\n 2. \\nat. n = Suc nat \\ fls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n - 1)\n[PROOF STEP]\ncase (Suc m)\n[PROOF STATE]\nproof (state)\nthis:\nn = Suc m\n\ngoal (2 subgoals):\n 1. n = 0 \\ fls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n - 1)\n 2. \\nat. n = Suc nat \\ fls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n - 1)\n[PROOF STEP]\nhave \"fls_deriv (fls_X^Suc m) = of_nat (Suc m) * fls_X^m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fls_deriv (fls_X ^ Suc m) = of_nat (Suc m) * fls_X ^ m\n[PROOF STEP]\nby (induct m) (simp_all add: mult_of_nat_commute algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nfls_deriv (fls_X ^ Suc m) = of_nat (Suc m) * fls_X ^ m\n\ngoal (2 subgoals):\n 1. n = 0 \\ fls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n - 1)\n 2. \\nat. n = Suc nat \\ fls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n - 1)\n[PROOF STEP]\nwith Suc\n[PROOF STATE]\nproof (chain)\npicking this:\nn = Suc m\nfls_deriv (fls_X ^ Suc m) = of_nat (Suc m) * fls_X ^ m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn = Suc m\nfls_deriv (fls_X ^ Suc m) = of_nat (Suc m) * fls_X ^ m\n\ngoal (1 subgoal):\n 1. fls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n - 1)\n\ngoal (1 subgoal):\n 1. n = 0 \\ fls_deriv (fls_X ^ n) = of_nat n * fls_X ^ (n - 1)\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 927, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894604912848, "lm_q2_score": 0.8244619350028204, "lm_q1q2_score": 0.7377198500167744}} {"text": "[STATEMENT]\nlemma three_subset: \"\\x \\ y; x \\ z; y \\ z; {x,y,z} \\ X\\ \\ card X \\ 3 \\ infinite X\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x \\ y; x \\ z; y \\ z; {x, y, z} \\ X\\ \\ 3 \\ card X \\ infinite X\n[PROOF STEP]\napply (case_tac \"finite X\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x \\ y; x \\ z; y \\ z; {x, y, z} \\ X; finite X\\ \\ 3 \\ card X \\ infinite X\n 2. \\x \\ y; x \\ z; y \\ z; {x, y, z} \\ X; infinite X\\ \\ 3 \\ card X \\ infinite X\n[PROOF STEP]\napply (auto simp : card_mono)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x \\ y; x \\ z; y \\ z; finite X; x \\ X; y \\ X; z \\ X\\ \\ 3 \\ card X\n[PROOF STEP]\napply (erule_tac Y = \"{x,y,z}\" in card_subset_finite)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x \\ y; x \\ z; y \\ z; x \\ X; y \\ X; z \\ X\\ \\ card {x, y, z} = 3\n 2. \\x \\ y; x \\ z; y \\ z; x \\ X; y \\ X; z \\ X\\ \\ {x, y, z} \\ X\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 615, "file": "Schutz_Spacetime_Util", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894689081711, "lm_q2_score": 0.8244619199068831, "lm_q1q2_score": 0.7377198434484911}} {"text": "[STATEMENT]\nlemma cosine_law_triangle':\n \"2 * dist a b * dist a c * cos (angle b a c) = (dist a b ^ 2 + dist a c ^ 2 - dist b c ^ 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * dist a b * dist a c * cos (angle b a c) = (dist a b)\\<^sup>2 + (dist a c)\\<^sup>2 - (dist b c)\\<^sup>2\n[PROOF STEP]\nusing cosine_law_triangle[of b c a]\n[PROOF STATE]\nproof (prove)\nusing this:\n(dist b c)\\<^sup>2 = (dist a b)\\<^sup>2 + (dist a c)\\<^sup>2 - 2 * dist a b * dist a c * cos (angle b a c)\n\ngoal (1 subgoal):\n 1. 2 * dist a b * dist a c * cos (angle b a c) = (dist a b)\\<^sup>2 + (dist a c)\\<^sup>2 - (dist b c)\\<^sup>2\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 296, "file": "Triangle_Triangle", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9241418283357702, "lm_q2_score": 0.7981867681382279, "lm_q1q2_score": 0.7376377792606814}} {"text": "[STATEMENT]\nlemma L2_set_mono:\n assumes \"\\i. i \\ K \\ f i \\ g i\"\n assumes \"\\i. i \\ K \\ 0 \\ f i\"\n shows \"L2_set f K \\ L2_set g K\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. L2_set f K \\ L2_set g K\n[PROOF STEP]\nunfolding L2_set_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (\\i\\K. (f i)\\<^sup>2) \\ sqrt (\\i\\K. (g i)\\<^sup>2)\n[PROOF STEP]\nby (simp add: sum_nonneg sum_mono power_mono assms)", "meta": {"llama_tokens": 238, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772417253256, "lm_q2_score": 0.8438951045175643, "lm_q1q2_score": 0.737629505262218}} {"text": "[STATEMENT]\nlemma geometric_pmf_prob_atMost:\n assumes \"p \\ {0<..1}\"\n shows \"measure_pmf.prob (geometric_pmf p) {..n} = (1 - (1 - p)^(n + 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. measure_pmf.prob (geometric_pmf p) {..n} = 1 - (1 - p) ^ (n + 1)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. measure_pmf.prob (geometric_pmf p) {..n} = 1 - (1 - p) ^ (n + 1)\n[PROOF STEP]\nhave \"(\\x\\n. (1 - p) ^ x * p) = 1 - (1 - p) * (1 - p) ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\n. (1 - p) ^ x * p) = 1 - (1 - p) * (1 - p) ^ n\n[PROOF STEP]\nby (induction n) (auto simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\n. (1 - p) ^ x * p) = 1 - (1 - p) * (1 - p) ^ n\n\ngoal (1 subgoal):\n 1. measure_pmf.prob (geometric_pmf p) {..n} = 1 - (1 - p) ^ (n + 1)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x\\n. (1 - p) ^ x * p) = 1 - (1 - p) * (1 - p) ^ n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x\\n. (1 - p) ^ x * p) = 1 - (1 - p) * (1 - p) ^ n\n\ngoal (1 subgoal):\n 1. measure_pmf.prob (geometric_pmf p) {..n} = 1 - (1 - p) ^ (n + 1)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x\\n. (1 - p) ^ x * p) = 1 - (1 - p) * (1 - p) ^ n\np \\ {0<..1}\n\ngoal (1 subgoal):\n 1. measure_pmf.prob (geometric_pmf p) {..n} = 1 - (1 - p) ^ (n + 1)\n[PROOF STEP]\nby (auto simp add: measure_pmf_conv_infsetsum)\n[PROOF STATE]\nproof (state)\nthis:\nmeasure_pmf.prob (geometric_pmf p) {..n} = 1 - (1 - p) ^ (n + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 828, "file": "Skip_Lists_Geometric_PMF", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772318846386, "lm_q2_score": 0.8438951104066295, "lm_q1q2_score": 0.7376295021052082}} {"text": "[STATEMENT]\nlemma not_equal_vector7 : \n fixes x::\"real^7\" and y::\"real^7\" \n assumes \"x = vector[x1,x2,x3,x4,x5,x6,x7] \" and \"y= vector [y1,y2,y3,y4,y5,y6,y7]\"\nand \"x$1 \\ y$1 \\ x$2 \\ y$2 \\ x$3 \\ y$3 \\ x$4 \\ y$4 \\ x$5 \\ y$5 \\ x$6 \\ y$6 \\ x$7 \\ y$7 \"\nshows \"x \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx = vector [x1, x2, x3, x4, x5, x6, x7]\ny = vector [y1, y2, y3, y4, y5, y6, y7]\nx $ 1 \\ y $ 1 \\ x $ 2 \\ y $ 2 \\ x $ 3 \\ y $ 3 \\ x $ 4 \\ y $ 4 \\ x $ 5 \\ y $ 5 \\ x $ 6 \\ y $ 6 \\ x $ 7 \\ y $ 7\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 441, "file": "Octonions_Cross_Product_7", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122213606241, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7375244559318265}} {"text": "[STATEMENT]\ntheorem expected_lookup_cost:\n assumes \"bst t\" \"t \\ Leaf\"\n shows \"measure_pmf.expectation (pmf_of_set (set_tree t)) (\\x. lookup_cost x t) =\n ipl t / size t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. measure_pmf.expectation (pmf_of_set (set_tree t)) (\\x. real (lookup_cost x t)) = real (ipl t) / real (size t)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nbst t\nt \\ \\\\\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (pmf_of_set (set_tree t)) (\\x. real (lookup_cost x t)) = real (ipl t) / real (size t)\n[PROOF STEP]\nby (subst integral_pmf_of_set)\n (simp_all add: sum_lookup_costs of_nat_sum [symmetric] card_set_tree_bst)", "meta": {"llama_tokens": 297, "file": "Random_BSTs_Random_BSTs", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122163480667, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.7375244436309517}} {"text": "[STATEMENT]\nlemma measure_pmf_of_list:\n assumes \"pmf_of_list_wf xs\"\n shows \"measure (pmf_of_list xs) A = sum_list (map snd (filter (\\x. fst x \\ A) xs))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. measure_pmf.prob (pmf_of_list xs) A = sum_list (map snd (filter (\\x. fst x \\ A) xs))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\npmf_of_list_wf xs\n\ngoal (1 subgoal):\n 1. measure_pmf.prob (pmf_of_list xs) A = sum_list (map snd (filter (\\x. fst x \\ A) xs))\n[PROOF STEP]\nunfolding pmf_of_list_wf_def Sigma_Algebra.measure_def\n[PROOF STATE]\nproof (prove)\nusing this:\nBall (set (map snd xs)) ((\\) 0) \\ sum_list (map snd xs) = 1\n\ngoal (1 subgoal):\n 1. enn2real (emeasure (measure_pmf (pmf_of_list xs)) A) = sum_list (map snd (filter (\\x. fst x \\ A) xs))\n[PROOF STEP]\nby (subst emeasure_pmf_of_list [OF assms], subst enn2real_ennreal) (auto intro!: sum_list_nonneg)", "meta": {"llama_tokens": 422, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505299595162, "lm_q2_score": 0.8152324826183822, "lm_q1q2_score": 0.7375004974409316}} {"text": "[STATEMENT]\nlemma sin_diff_sin: \"sin w - sin z = 2 * sin ((w - z) / 2) * cos ((w + z) / 2)\"\n for w :: \"'a::{real_normed_field,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin w - sin z = (2::'a) * sin ((w - z) / (2::'a)) * cos ((w + z) / (2::'a))\n[PROOF STEP]\napply (simp add: mult.assoc sin_times_cos)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin w * (2::'a) - sin z * (2::'a) = (2::'a) * sin ((w - z) / (2::'a) + (w + z) / (2::'a)) + (2::'a) * sin ((w - z) / (2::'a) - (w + z) / (2::'a))\n[PROOF STEP]\napply (simp add: field_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 311, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099070109242131, "lm_q2_score": 0.8104788995148791, "lm_q1q2_score": 0.7374604328747293}} {"text": "[STATEMENT]\nlemma higher_deriv_power:\n shows \"(deriv ^^ j) (\\w. (w - z) ^ n) w =\n pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)\n[PROOF STEP]\nproof (induction j arbitrary: w)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\w. (deriv ^^ 0) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - 0)) 0 * (w - z) ^ (n - 0)\n 2. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\w. (deriv ^^ 0) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - 0)) 0 * (w - z) ^ (n - 0)\n 2. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (deriv ^^ 0) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - 0)) 0 * (w - z) ^ (n - 0)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(deriv ^^ 0) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - 0)) 0 * (w - z) ^ (n - 0)\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\ncase (Suc j w)\n[PROOF STATE]\nproof (state)\nthis:\n(deriv ^^ j) (\\w. (w - z) ^ n) ?w = pochhammer (of_nat (Suc n - j)) j * (?w - z) ^ (n - j)\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nhave \"(deriv ^^ Suc j) (\\w. (w - z) ^ n) w = deriv ((deriv ^^ j) (\\w. (w - z) ^ n)) w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = deriv ((deriv ^^ j) (\\w. (w - z) ^ n)) w\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(deriv ^^ Suc j) (\\w. (w - z) ^ n) w = deriv ((deriv ^^ j) (\\w. (w - z) ^ n)) w\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(deriv ^^ Suc j) (\\w. (w - z) ^ n) w = deriv ((deriv ^^ j) (\\w. (w - z) ^ n)) w\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nhave \"(deriv ^^ j) (\\w. (w - z) ^ n) =\n (\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (deriv ^^ j) (\\w. (w - z) ^ n) = (\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\n(deriv ^^ j) (\\w. (w - z) ^ n) ?w = pochhammer (of_nat (Suc n - j)) j * (?w - z) ^ (n - j)\n\ngoal (1 subgoal):\n 1. (deriv ^^ j) (\\w. (w - z) ^ n) = (\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))\n[PROOF STEP]\nby (intro Suc.IH ext)\n[PROOF STATE]\nproof (state)\nthis:\n(deriv ^^ j) (\\w. (w - z) ^ n) = (\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(deriv ^^ j) (\\w. (w - z) ^ n) = (\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\n(deriv ^^ j) (\\w. (w - z) ^ n) = (\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nhave \"(\\ has_field_derivative of_nat (n - j) *\n pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) has_field_derivative of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)\n[PROOF STEP]\nusing Suc.prems\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) has_field_derivative of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)\n[PROOF STEP]\nby (auto intro!: derivative_eq_intros)\n[PROOF STATE]\nproof (state)\nthis:\n((\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) has_field_derivative of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n((\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) has_field_derivative of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nhave \"of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j =\n pochhammer (of_nat (Suc n - Suc j)) (Suc j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j = pochhammer (of_nat (Suc n - Suc j)) (Suc j)\n[PROOF STEP]\nby (cases \"Suc j \\ n\", subst pochhammer_rec)\n (insert Suc.prems, simp_all add: algebra_simps Suc_diff_le pochhammer_0_left)\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (n - j) * pochhammer (of_nat (Suc n - j)) j = pochhammer (of_nat (Suc n - Suc j)) (Suc j)\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n((\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) has_field_derivative pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)) (at w)\n[PROOF STEP]\nhave \"deriv (\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w =\n \\ * (w - z) ^ (n - Suc j)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) has_field_derivative pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)) (at w)\n\ngoal (1 subgoal):\n 1. deriv (\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nby (rule DERIV_imp_deriv)\n[PROOF STATE]\nproof (state)\nthis:\nderiv (\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\nderiv (\\w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n\ngoal (1 subgoal):\n 1. \\j w. (\\w. (deriv ^^ j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) \\ (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n\ngoal (1 subgoal):\n 1. (deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(deriv ^^ Suc j) (\\w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - Suc j)) (Suc j) * (w - z) ^ (n - Suc j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5181, "file": null, "length": 28, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045907347108, "lm_q2_score": 0.831143054132195, "lm_q1q2_score": 0.7373939331833517}} {"text": "[STATEMENT]\nlemma dirichlet_prod_prime_power:\n assumes \"prime p\"\n shows \"dirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\n[PROOF STEP]\nhave \"dirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ k div p ^ i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ k div p ^ i))\n[PROOF STEP]\nunfolding dirichlet_prod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\d | d dvd p ^ k. f d * g (p ^ k div d)) = (\\i\\k. f (p ^ i) * g (p ^ k div p ^ i))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. (\\d | d dvd p ^ k. f d * g (p ^ k div d)) = (\\i\\k. f (p ^ i) * g (p ^ k div p ^ i))\n[PROOF STEP]\nby (intro sum.reindex_bij_betw [symmetric] bij_betw_prime_power_divisors)\n[PROOF STATE]\nproof (state)\nthis:\ndirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ k div p ^ i))\n\ngoal (1 subgoal):\n 1. dirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ k div p ^ i))\n\ngoal (1 subgoal):\n 1. dirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nprime p\n[PROOF STEP]\nhave \"\\ = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. (\\i\\k. f (p ^ i) * g (p ^ k div p ^ i)) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\n[PROOF STEP]\nby (intro sum.cong refl) (auto simp: power_diff')\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\k. f (p ^ i) * g (p ^ k div p ^ i)) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\n\ngoal (1 subgoal):\n 1. dirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\n\ngoal (1 subgoal):\n 1. dirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ndirichlet_prod f g (p ^ k) = (\\i\\k. f (p ^ i) * g (p ^ (k - i)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1331, "file": "Dirichlet_Series_Dirichlet_Product", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045847699186, "lm_q2_score": 0.8311430436757312, "lm_q1q2_score": 0.7373939189487334}} {"text": "[STATEMENT]\nlemma arctan_series:\n assumes \"\\x\\ \\ 1\"\n shows \"arctan x = (\\k. (-1)^k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\"\n (is \"_ = suminf (\\ n. ?c x n)\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nlet ?c' = \"\\x n. (-1)^n * x^(n*2)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nhave DERIV_arctan_suminf: \"DERIV (\\ x. suminf (?c x)) x :> (suminf (?c' x))\"\n if \"0 < r\" and \"r < 1\" and \"\\x\\ < r\" for r x :: real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. \\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) has_real_derivative (\\n. (- 1) ^ n * x ^ (n * 2))) (at x)\n[PROOF STEP]\nproof (rule DERIV_arctan_series)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x\\ < 1\n[PROOF STEP]\nfrom that\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < r\nr < 1\n\\x\\ < r\n[PROOF STEP]\nshow \"\\x\\ < 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < r\nr < 1\n\\x\\ < r\n\ngoal (1 subgoal):\n 1. \\x\\ < 1\n[PROOF STEP]\nusing \\r < 1\\ and \\\\x\\ < r\\\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < r\nr < 1\n\\x\\ < r\nr < 1\n\\x\\ < r\n\ngoal (1 subgoal):\n 1. \\x\\ < 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ < 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\0 < ?r1; ?r1 < 1; \\?x1\\ < ?r1\\ \\ ((\\x. \\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) has_real_derivative (\\n. (- 1) ^ n * ?x1 ^ (n * 2))) (at ?x1)\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\n\\0 < ?r1; ?r1 < 1; \\?x1\\ < ?r1\\ \\ ((\\x. \\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) has_real_derivative (\\n. (- 1) ^ n * ?x1 ^ (n * 2))) (at ?x1)\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nfix x :: real\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nassume \"\\x\\ \\ 1\"\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ \\ 1\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nnote summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]\n[PROOF STATE]\nproof (state)\nthis:\nsummable (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n0 < 1 / real (0 * 2 + 1) * x ^ (0 * 2 + 1) \\ (\\n. (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1))) \\ {\\i<2 * n. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1))..\\i<2 * n + 1. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1))})\n1 / real (0 * 2 + 1) * x ^ (0 * 2 + 1) < 0 \\ (\\n. (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1))) \\ {\\i<2 * n + 1. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1))..\\i<2 * n. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1))})\n(\\n. \\i<2 * n. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1))) \\ (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1)))\n(\\n. \\i<2 * n + 1. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1))) \\ (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1)))\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n\\?xa3\\ \\ 1 \\ summable (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * ?xa3 ^ (n * 2 + 1)))\n\\?xa3\\ \\ 1 \\ 0 < 1 / real (0 * 2 + 1) * ?xa3 ^ (0 * 2 + 1) \\ (\\n. (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))) \\ {\\i<2 * n. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))..\\i<2 * n + 1. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))})\n\\?xa3\\ \\ 1 \\ 1 / real (0 * 2 + 1) * ?xa3 ^ (0 * 2 + 1) < 0 \\ (\\n. (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))) \\ {\\i<2 * n + 1. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))..\\i<2 * n. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))})\n\\?xa3\\ \\ 1 \\ (\\n. \\i<2 * n. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))) \\ (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1)))\n\\?xa3\\ \\ 1 \\ (\\n. \\i<2 * n + 1. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))) \\ (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1)))\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nnote arctan_series_borders = this\n[PROOF STATE]\nproof (state)\nthis:\n\\?xa3\\ \\ 1 \\ summable (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * ?xa3 ^ (n * 2 + 1)))\n\\?xa3\\ \\ 1 \\ 0 < 1 / real (0 * 2 + 1) * ?xa3 ^ (0 * 2 + 1) \\ (\\n. (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))) \\ {\\i<2 * n. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))..\\i<2 * n + 1. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))})\n\\?xa3\\ \\ 1 \\ 1 / real (0 * 2 + 1) * ?xa3 ^ (0 * 2 + 1) < 0 \\ (\\n. (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))) \\ {\\i<2 * n + 1. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))..\\i<2 * n. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))})\n\\?xa3\\ \\ 1 \\ (\\n. \\i<2 * n. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))) \\ (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1)))\n\\?xa3\\ \\ 1 \\ (\\n. \\i<2 * n + 1. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1))) \\ (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * ?xa3 ^ (i * 2 + 1)))\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nhave when_less_one: \"arctan x = (\\k. ?c x k)\" if \"\\x\\ < 1\" for x :: real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nobtain r where \"\\x\\ < r\" and \"r < 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\r. \\\\x\\ < r; r < 1\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing dense[OF \\\\x\\ < 1\\]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\z>\\x\\. z < 1\n\ngoal (1 subgoal):\n 1. (\\r. \\\\x\\ < r; r < 1\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ < r\nr < 1\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x\\ < r\nr < 1\n[PROOF STEP]\nhave \"0 < r\" and \"- r < x\" and \"x < r\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\ < r\nr < 1\n\ngoal (1 subgoal):\n 1. 0 < r &&& - r < x &&& x < r\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < r\n- r < x\nx < r\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nhave suminf_eq_arctan_bounded: \"suminf (?c x) - arctan x = suminf (?c a) - arctan a\"\n if \"-r < a\" and \"b < r\" and \"a < b\" and \"a \\ x\" and \"x \\ b\" for x a b\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a\n[PROOF STEP]\nfrom that\n[PROOF STATE]\nproof (chain)\npicking this:\n- r < a\nb < r\na < b\na \\ x\nx \\ b\n[PROOF STEP]\nhave \"\\x\\ < r\"\n[PROOF STATE]\nproof (prove)\nusing this:\n- r < a\nb < r\na < b\na \\ x\nx \\ b\n\ngoal (1 subgoal):\n 1. \\x\\ < r\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ < r\n\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a\n[PROOF STEP]\nshow \"suminf (?c x) - arctan x = suminf (?c a) - arctan a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a\n[PROOF STEP]\nproof (rule DERIV_isconst2[of \"a\" \"b\"])\n[PROOF STATE]\nproof (state)\ngoal (5 subgoals):\n 1. a < b\n 2. continuous_on {a..b} (\\a. (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a)\n 3. \\x. \\a < x; x < b\\ \\ ((\\a. (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a) has_real_derivative 0) (at x)\n 4. a \\ x\n 5. x \\ b\n[PROOF STEP]\nshow \"a < b\" and \"a \\ x\" and \"x \\ b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a < b &&& a \\ x &&& x \\ b\n[PROOF STEP]\nusing \\a < b\\ \\a \\ x\\ \\x \\ b\\\n[PROOF STATE]\nproof (prove)\nusing this:\na < b\na \\ x\nx \\ b\n\ngoal (1 subgoal):\n 1. a < b &&& a \\ x &&& x \\ b\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na < b\na \\ x\nx \\ b\n\ngoal (2 subgoals):\n 1. continuous_on {a..b} (\\a. (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a)\n 2. \\x. \\a < x; x < b\\ \\ ((\\a. (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a) has_real_derivative 0) (at x)\n[PROOF STEP]\nhave \"\\x. - r < x \\ x < r \\ DERIV (\\ x. suminf (?c x) - arctan x) x :> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nproof (rule allI, rule impI)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nassume \"-r < x \\ x < r\"\n[PROOF STATE]\nproof (state)\nthis:\n- r < x \\ x < r\n\ngoal (1 subgoal):\n 1. \\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n- r < x \\ x < r\n[PROOF STEP]\nhave \"\\x\\ < r\"\n[PROOF STATE]\nproof (prove)\nusing this:\n- r < x \\ x < r\n\ngoal (1 subgoal):\n 1. \\x\\ < r\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ < r\n\ngoal (1 subgoal):\n 1. \\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nwith \\r < 1\\\n[PROOF STATE]\nproof (chain)\npicking this:\nr < 1\n\\x\\ < r\n[PROOF STEP]\nhave \"\\x\\ < 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nr < 1\n\\x\\ < r\n\ngoal (1 subgoal):\n 1. \\x\\ < 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ < 1\n\ngoal (1 subgoal):\n 1. \\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nhave \"\\- (x\\<^sup>2)\\ < 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\- x\\<^sup>2\\ < 1\n[PROOF STEP]\nusing abs_square_less_1 \\\\x\\ < 1\\\n[PROOF STATE]\nproof (prove)\nusing this:\n(?x\\<^sup>2 < (1::?'a)) = (\\?x\\ < (1::?'a))\n\\x\\ < 1\n\ngoal (1 subgoal):\n 1. \\- x\\<^sup>2\\ < 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\- x\\<^sup>2\\ < 1\n\ngoal (1 subgoal):\n 1. \\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\- x\\<^sup>2\\ < 1\n[PROOF STEP]\nhave \"(\\n. (- (x\\<^sup>2)) ^ n) sums (1 / (1 - (- (x\\<^sup>2))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\- x\\<^sup>2\\ < 1\n\ngoal (1 subgoal):\n 1. (^) (- x\\<^sup>2) sums (1 / (1 - - x\\<^sup>2))\n[PROOF STEP]\nunfolding real_norm_def[symmetric]\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (- x\\<^sup>2) < 1\n\ngoal (1 subgoal):\n 1. (^) (- x\\<^sup>2) sums (1 / (1 - - x\\<^sup>2))\n[PROOF STEP]\nby (rule geometric_sums)\n[PROOF STATE]\nproof (state)\nthis:\n(^) (- x\\<^sup>2) sums (1 / (1 - - x\\<^sup>2))\n\ngoal (1 subgoal):\n 1. \\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(^) (- x\\<^sup>2) sums (1 / (1 - - x\\<^sup>2))\n[PROOF STEP]\nhave \"(?c' x) sums (1 / (1 - (- (x\\<^sup>2))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(^) (- x\\<^sup>2) sums (1 / (1 - - x\\<^sup>2))\n\ngoal (1 subgoal):\n 1. (\\n. (- 1) ^ n * x ^ (n * 2)) sums (1 / (1 - - x\\<^sup>2))\n[PROOF STEP]\nunfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2]\n[PROOF STATE]\nproof (prove)\nusing this:\n(^) (- x\\<^sup>2) sums (1 / (1 - - x\\<^sup>2))\n\ngoal (1 subgoal):\n 1. (^) (- 1 * x\\<^sup>2) sums (1 / (1 - - x\\<^sup>2))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. (- 1) ^ n * x ^ (n * 2)) sums (1 / (1 - - x\\<^sup>2))\n\ngoal (1 subgoal):\n 1. \\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n. (- 1) ^ n * x ^ (n * 2)) sums (1 / (1 - - x\\<^sup>2))\n[PROOF STEP]\nhave suminf_c'_eq_geom: \"inverse (1 + x\\<^sup>2) = suminf (?c' x)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. (- 1) ^ n * x ^ (n * 2)) sums (1 / (1 - - x\\<^sup>2))\n\ngoal (1 subgoal):\n 1. inverse (1 + x\\<^sup>2) = (\\n. (- 1) ^ n * x ^ (n * 2))\n[PROOF STEP]\nusing sums_unique\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. (- 1) ^ n * x ^ (n * 2)) sums (1 / (1 - - x\\<^sup>2))\n?f sums ?s \\ ?s = suminf ?f\n\ngoal (1 subgoal):\n 1. inverse (1 + x\\<^sup>2) = (\\n. (- 1) ^ n * x ^ (n * 2))\n[PROOF STEP]\nunfolding inverse_eq_divide\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. (- 1) ^ n * x ^ (n * 2)) sums (1 / (1 - - x\\<^sup>2))\n?f sums ?s \\ ?s = suminf ?f\n\ngoal (1 subgoal):\n 1. 1 / (1 + x\\<^sup>2) = (\\n. (- 1) ^ n * x ^ (n * 2))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninverse (1 + x\\<^sup>2) = (\\n. (- 1) ^ n * x ^ (n * 2))\n\ngoal (1 subgoal):\n 1. \\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nhave \"DERIV (\\ x. suminf (?c x)) x :> (inverse (1 + x\\<^sup>2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. \\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) has_real_derivative inverse (1 + x\\<^sup>2)) (at x)\n[PROOF STEP]\nunfolding suminf_c'_eq_geom\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. \\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) has_real_derivative (\\n. (- 1) ^ n * x ^ (n * 2))) (at x)\n[PROOF STEP]\nby (rule DERIV_arctan_suminf[OF \\0 < r\\ \\r < 1\\ \\\\x\\ < r\\])\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. \\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) has_real_derivative inverse (1 + x\\<^sup>2)) (at x)\n\ngoal (1 subgoal):\n 1. \\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nfrom DERIV_diff [OF this DERIV_arctan]\n[PROOF STATE]\nproof (chain)\npicking this:\n((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative inverse (1 + x\\<^sup>2) - inverse (1 + x\\<^sup>2)) (at x)\n[PROOF STEP]\nshow \"DERIV (\\x. suminf (?c x) - arctan x) x :> 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative inverse (1 + x\\<^sup>2) - inverse (1 + x\\<^sup>2)) (at x)\n\ngoal (1 subgoal):\n 1. ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n\ngoal (2 subgoals):\n 1. continuous_on {a..b} (\\a. (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a)\n 2. \\x. \\a < x; x < b\\ \\ ((\\a. (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a) has_real_derivative 0) (at x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n[PROOF STEP]\nhave DERIV_in_rball: \"\\y. a \\ y \\ y \\ b \\ DERIV (\\x. suminf (?c x) - arctan x) y :> 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n\ngoal (1 subgoal):\n 1. \\y. a \\ y \\ y \\ b \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at y)\n[PROOF STEP]\nusing \\-r < a\\ \\b < r\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. - r < x \\ x < r \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at x)\n- r < a\nb < r\n\ngoal (1 subgoal):\n 1. \\y. a \\ y \\ y \\ b \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at y)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\y. a \\ y \\ y \\ b \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at y)\n\ngoal (2 subgoals):\n 1. continuous_on {a..b} (\\a. (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a)\n 2. \\x. \\a < x; x < b\\ \\ ((\\a. (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a) has_real_derivative 0) (at x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\y. a \\ y \\ y \\ b \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at y)\n[PROOF STEP]\nshow \"\\y. \\a < y; y < b\\ \\ DERIV (\\x. suminf (?c x) - arctan x) y :> 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\y. a \\ y \\ y \\ b \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at y)\n\ngoal (1 subgoal):\n 1. \\y. \\a < y; y < b\\ \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at y)\n[PROOF STEP]\nusing \\\\x\\ < r\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\y. a \\ y \\ y \\ b \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at y)\n\\x\\ < r\n\ngoal (1 subgoal):\n 1. \\y. \\a < y; y < b\\ \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at y)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\a < ?y1; ?y1 < b\\ \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at ?y1)\n\ngoal (1 subgoal):\n 1. continuous_on {a..b} (\\a. (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a)\n[PROOF STEP]\nshow \"continuous_on {a..b} (\\x. suminf (?c x) - arctan x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous_on {a..b} (\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x)\n[PROOF STEP]\nusing DERIV_in_rball DERIV_atLeastAtMost_imp_continuous_on\n[PROOF STATE]\nproof (prove)\nusing this:\n\\y. a \\ y \\ y \\ b \\ ((\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x) has_real_derivative 0) (at y)\n(\\x. \\?a \\ x; x \\ ?b\\ \\ \\y. (?f has_field_derivative y) (at x)) \\ continuous_on {?a..?b} ?f\n\ngoal (1 subgoal):\n 1. continuous_on {a..b} (\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ncontinuous_on {a..b} (\\x. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = (\\b. (- 1) ^ b * (1 / real (b * 2 + 1) * a ^ (b * 2 + 1))) - arctan a\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\- r < ?a1; ?b1 < r; ?a1 < ?b1; ?a1 \\ ?x1; ?x1 \\ ?b1\\ \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * ?x1 ^ (a * 2 + 1))) - arctan ?x1 = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * ?a1 ^ (a * 2 + 1))) - arctan ?a1\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nhave suminf_arctan_zero: \"suminf (?c 0) - arctan 0 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 0 ^ (a * 2 + 1))) - arctan 0 = 0\n[PROOF STEP]\nunfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 - 0 = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 0 ^ (a * 2 + 1))) - arctan 0 = 0\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nhave \"suminf (?c x) - arctan x = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nproof (cases \"x = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x = 0 \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n 2. x \\ 0 \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nx = 0\n\ngoal (2 subgoals):\n 1. x = 0 \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n 2. x \\ 0 \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx = 0\n\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nusing suminf_arctan_zero\n[PROOF STATE]\nproof (prove)\nusing this:\nx = 0\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 0 ^ (a * 2 + 1))) - arctan 0 = 0\n\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n\ngoal (1 subgoal):\n 1. x \\ 0 \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x \\ 0 \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nx \\ 0\n\ngoal (1 subgoal):\n 1. x \\ 0 \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ 0\n[PROOF STEP]\nhave \"0 < \\x\\\" and \"- \\x\\ < \\x\\\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ 0\n\ngoal (1 subgoal):\n 1. 0 < \\x\\ &&& - \\x\\ < \\x\\\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < \\x\\\n- \\x\\ < \\x\\\n\ngoal (1 subgoal):\n 1. x \\ 0 \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nhave \"suminf (?c (- \\x\\)) - arctan (- \\x\\) = suminf (?c 0) - arctan 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * (- \\x\\) ^ (a * 2 + 1))) - arctan (- \\x\\) = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 0 ^ (a * 2 + 1))) - arctan 0\n[PROOF STEP]\nby (rule suminf_eq_arctan_bounded[where x1=0 and a1=\"-\\x\\\" and b1=\"\\x\\\", symmetric])\n (simp_all only: \\\\x\\ < r\\ \\-\\x\\ < \\x\\\\ neg_less_iff_less)\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * (- \\x\\) ^ (a * 2 + 1))) - arctan (- \\x\\) = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 0 ^ (a * 2 + 1))) - arctan 0\n\ngoal (1 subgoal):\n 1. x \\ 0 \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * (- \\x\\) ^ (a * 2 + 1))) - arctan (- \\x\\) = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 0 ^ (a * 2 + 1))) - arctan 0\n\ngoal (1 subgoal):\n 1. x \\ 0 \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nhave \"suminf (?c x) - arctan x = suminf (?c (- \\x\\)) - arctan (- \\x\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * (- \\x\\) ^ (a * 2 + 1))) - arctan (- \\x\\)\n[PROOF STEP]\nby (rule suminf_eq_arctan_bounded[where x1=x and a1=\"- \\x\\\" and b1=\"\\x\\\"])\n (simp_all only: \\\\x\\ < r\\ \\- \\x\\ < \\x\\\\ neg_less_iff_less)\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * (- \\x\\) ^ (a * 2 + 1))) - arctan (- \\x\\)\n\ngoal (1 subgoal):\n 1. x \\ 0 \\ (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * (- \\x\\) ^ (a * 2 + 1))) - arctan (- \\x\\) = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 0 ^ (a * 2 + 1))) - arctan 0\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * (- \\x\\) ^ (a * 2 + 1))) - arctan (- \\x\\)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * (- \\x\\) ^ (a * 2 + 1))) - arctan (- \\x\\) = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 0 ^ (a * 2 + 1))) - arctan 0\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * (- \\x\\) ^ (a * 2 + 1))) - arctan (- \\x\\)\n\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nusing suminf_arctan_zero\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * (- \\x\\) ^ (a * 2 + 1))) - arctan (- \\x\\) = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 0 ^ (a * 2 + 1))) - arctan 0\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * (- \\x\\) ^ (a * 2 + 1))) - arctan (- \\x\\)\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 0 ^ (a * 2 + 1))) - arctan 0 = 0\n\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) - arctan x = 0\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\narctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\?x1\\ < 1 \\ arctan ?x1 = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * ?x1 ^ (k * 2 + 1)))\n\ngoal (1 subgoal):\n 1. arctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n[PROOF STEP]\nshow \"arctan x = suminf (\\n. ?c x n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nproof (cases \"\\x\\ < 1\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n 2. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ < 1\n\ngoal (2 subgoals):\n 1. \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n 2. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x\\ < 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\ < 1\n\ngoal (1 subgoal):\n 1. arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nby (rule when_less_one)\n[PROOF STATE]\nproof (state)\nthis:\narctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n\ngoal (1 subgoal):\n 1. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ \\x\\ < 1\n\ngoal (1 subgoal):\n 1. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ \\x\\ < 1\n[PROOF STEP]\nhave \"\\x\\ = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ \\x\\ < 1\n\ngoal (1 subgoal):\n 1. \\x\\ = 1\n[PROOF STEP]\nusing \\\\x\\ \\ 1\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ \\x\\ < 1\n\\x\\ \\ 1\n\ngoal (1 subgoal):\n 1. \\x\\ = 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ = 1\n\ngoal (1 subgoal):\n 1. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nlet ?a = \"\\x n. \\1 / real (n * 2 + 1) * x^(n * 2 + 1)\\\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nlet ?diff = \"\\x n. \\arctan x - (\\i\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave \"?diff 1 n \\ ?a 1 n\" for n :: nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n[PROOF STEP]\nhave \"0 < (1 :: real)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < 1\n\ngoal (1 subgoal):\n 1. \\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n0 < 1\n\ngoal (1 subgoal):\n 1. \\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n[PROOF STEP]\nhave \"?diff x n \\ ?a x n\" if \"0 < x\" and \"x < 1\" for x :: real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nfrom that\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < x\nx < 1\n[PROOF STEP]\nhave \"\\x\\ \\ 1\" and \"\\x\\ < 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\nx < 1\n\ngoal (1 subgoal):\n 1. \\x\\ \\ 1 &&& \\x\\ < 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ \\ 1\n\\x\\ < 1\n\ngoal (1 subgoal):\n 1. \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nfrom \\0 < x\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < x\n[PROOF STEP]\nhave \"0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n\ngoal (1 subgoal):\n 1. 0 < 1 / real (0 * 2 + 1) * x ^ (0 * 2 + 1)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < 1 / real (0 * 2 + 1) * x ^ (0 * 2 + 1)\n\ngoal (1 subgoal):\n 1. \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nnote bounds = mp[OF arctan_series_borders(2)[OF \\\\x\\ \\ 1\\] this, unfolded when_less_one[OF \\\\x\\ < 1\\, symmetric], THEN spec]\n[PROOF STATE]\nproof (state)\nthis:\narctan x \\ {\\i<2 * ?x. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1))..\\i<2 * ?x + 1. (- 1) ^ i * (1 / real (i * 2 + 1) * x ^ (i * 2 + 1))}\n\ngoal (1 subgoal):\n 1. \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nhave \"0 < 1 / real (n*2+1) * x^(n*2+1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < 1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\n[PROOF STEP]\nby (rule mult_pos_pos) (simp_all only: zero_less_power[OF \\0 < x\\], auto)\n[PROOF STATE]\nproof (state)\nthis:\n0 < 1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\n\ngoal (1 subgoal):\n 1. \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < 1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\n[PROOF STEP]\nhave a_pos: \"?a x n = 1 / real (n*2+1) * x^(n*2+1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\n\ngoal (1 subgoal):\n 1. \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\ = 1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\n[PROOF STEP]\nby (rule abs_of_pos)\n[PROOF STATE]\nproof (state)\nthis:\n\\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\ = 1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\n\ngoal (1 subgoal):\n 1. \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nproof (cases \"even n\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. even n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n 2. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\neven n\n\ngoal (2 subgoals):\n 1. even n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n 2. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\neven n\n[PROOF STEP]\nhave sgn_pos: \"(-1)^n = (1::real)\"\n[PROOF STATE]\nproof (prove)\nusing this:\neven n\n\ngoal (1 subgoal):\n 1. (- 1) ^ n = 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(- 1) ^ n = 1\n\ngoal (2 subgoals):\n 1. even n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n 2. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nfrom \\even n\\\n[PROOF STATE]\nproof (chain)\npicking this:\neven n\n[PROOF STEP]\nobtain m where \"n = 2 * m\"\n[PROOF STATE]\nproof (prove)\nusing this:\neven n\n\ngoal (1 subgoal):\n 1. (\\m. n = 2 * m \\ thesis) \\ thesis\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nn = 2 * m\n\ngoal (2 subgoals):\n 1. even n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n 2. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = 2 * m\n[PROOF STEP]\nhave \"2 * m = n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 2 * m\n\ngoal (1 subgoal):\n 1. 2 * m = n\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n2 * m = n\n\ngoal (2 subgoals):\n 1. even n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n 2. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nfrom bounds[of m, unfolded this atLeastAtMost_iff]\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i arctan x \\ arctan x \\ (\\iarctan x - (\\i \\ (\\iii arctan x \\ arctan x \\ (\\iarctan x - (\\i \\ (\\iiarctan x - (\\i \\ (\\ii \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n 2. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\arctan x - (\\i \\ (\\ii \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n 2. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nhave \"\\ = ?c x n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\iiii \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n 2. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\ii \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n 2. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nhave \"\\ = ?a x n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)) = \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nunfolding sgn_pos a_pos\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)) = 1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)) = \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n\ngoal (2 subgoals):\n 1. even n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n 2. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n\ngoal (1 subgoal):\n 1. \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n\ngoal (1 subgoal):\n 1. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nodd n\n\ngoal (1 subgoal):\n 1. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nodd n\n[PROOF STEP]\nhave sgn_neg: \"(-1)^n = (-1::real)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nodd n\n\ngoal (1 subgoal):\n 1. (- 1) ^ n = - 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(- 1) ^ n = - 1\n\ngoal (1 subgoal):\n 1. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nfrom \\odd n\\\n[PROOF STATE]\nproof (chain)\npicking this:\nodd n\n[PROOF STEP]\nobtain m where \"n = 2 * m + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nodd n\n\ngoal (1 subgoal):\n 1. (\\m. n = 2 * m + 1 \\ thesis) \\ thesis\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nn = 2 * m + 1\n\ngoal (1 subgoal):\n 1. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = 2 * m + 1\n[PROOF STEP]\nhave m_def: \"2 * m + 1 = n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 2 * m + 1\n\ngoal (1 subgoal):\n 1. 2 * m + 1 = n\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n2 * m + 1 = n\n\ngoal (1 subgoal):\n 1. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n2 * m + 1 = n\n[PROOF STEP]\nhave m_plus: \"2 * (m + 1) = n + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * m + 1 = n\n\ngoal (1 subgoal):\n 1. 2 * (m + 1) = n + 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n2 * (m + 1) = n + 1\n\ngoal (1 subgoal):\n 1. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nfrom bounds[of \"m + 1\", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i arctan x\narctan x \\ (\\iarctan x - (\\i \\ (\\iii arctan x\narctan x \\ (\\iarctan x - (\\i \\ (\\iiarctan x - (\\i \\ (\\ii \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\arctan x - (\\i \\ (\\ii \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nhave \"\\ = - ?c x n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\iiii \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\ii \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nhave \"\\ = ?a x n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - ((- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1))) = \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nunfolding sgn_neg a_pos\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (- 1 * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1))) = 1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n- ((- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1))) = \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n\ngoal (1 subgoal):\n 1. odd n \\ \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n\ngoal (1 subgoal):\n 1. \\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\arctan x - (\\i \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\0 < ?x1; ?x1 < 1\\ \\ \\arctan ?x1 - (\\i \\ \\1 / real (n * 2 + 1) * ?x1 ^ (n * 2 + 1)\\\n\ngoal (1 subgoal):\n 1. \\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n[PROOF STEP]\nhence \"\\x \\ { 0 <..< 1 }. 0 \\ ?a x n - ?diff x n\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < ?x1; ?x1 < 1\\ \\ \\arctan ?x1 - (\\i \\ \\1 / real (n * 2 + 1) * ?x1 ^ (n * 2 + 1)\\\n\ngoal (1 subgoal):\n 1. \\x\\{0<..<1}. 0 \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\ - \\arctan x - (\\i\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\{0<..<1}. 0 \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\ - \\arctan x - (\\i\n\ngoal (1 subgoal):\n 1. \\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\{0<..<1}. 0 \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\ - \\arctan x - (\\i\n\ngoal (1 subgoal):\n 1. \\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n[PROOF STEP]\nhave \"isCont (\\ x. ?a x n - ?diff x n) x\" for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. isCont (\\x. \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\ - \\arctan x - (\\i) x\n[PROOF STEP]\nunfolding diff_conv_add_uminus divide_inverse\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. isCont (\\x. \\1 * inverse (real (n * 2 + 1)) * x ^ (n * 2 + 1)\\ + - \\arctan x + - (\\i) x\n[PROOF STEP]\nby (auto intro!: isCont_add isCont_rabs continuous_ident isCont_minus isCont_arctan\n continuous_at_within_inverse isCont_mult isCont_power continuous_const isCont_sum\n simp del: add_uminus_conv_diff)\n[PROOF STATE]\nproof (state)\nthis:\nisCont (\\x. \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\ - \\arctan x - (\\i) ?x1\n\ngoal (1 subgoal):\n 1. \\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < 1\n\\x\\{0<..<1}. 0 \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\ - \\arctan x - (\\i\nisCont (\\x. \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\ - \\arctan x - (\\i) ?x1\n[PROOF STEP]\nhave \"0 \\ ?a 1 n - ?diff 1 n\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1\n\\x\\{0<..<1}. 0 \\ \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\ - \\arctan x - (\\i\nisCont (\\x. \\1 / real (n * 2 + 1) * x ^ (n * 2 + 1)\\ - \\arctan x - (\\i) ?x1\n\ngoal (1 subgoal):\n 1. 0 \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\ - \\arctan 1 - (\\i\n[PROOF STEP]\nby (rule LIM_less_bound)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\ - \\arctan 1 - (\\i\n\ngoal (1 subgoal):\n 1. \\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\ - \\arctan 1 - (\\i\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\ - \\arctan 1 - (\\i\n\ngoal (1 subgoal):\n 1. \\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\arctan 1 - (\\i \\ \\1 / real (?n1 * 2 + 1) * 1 ^ (?n1 * 2 + 1)\\\n\ngoal (1 subgoal):\n 1. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave \"?a 1 \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\) \\ 0\n[PROOF STEP]\nunfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. 1 * inverse (real (n * 2 + Suc 0)) * 1) \\ 0\n[PROOF STEP]\nby (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat simp del: of_nat_Suc)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\) \\ 0\n\ngoal (1 subgoal):\n 1. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave \"?diff 1 \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. \\arctan 1 - (\\i) \\ 0\n[PROOF STEP]\nproof (rule LIMSEQ_I)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\r. 0 < r \\ \\no. \\n\\no. norm (\\arctan 1 - (\\i - 0) < r\n[PROOF STEP]\nfix r :: real\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\r. 0 < r \\ \\no. \\n\\no. norm (\\arctan 1 - (\\i - 0) < r\n[PROOF STEP]\nassume \"0 < r\"\n[PROOF STATE]\nproof (state)\nthis:\n0 < r\n\ngoal (1 subgoal):\n 1. \\r. 0 < r \\ \\no. \\n\\no. norm (\\arctan 1 - (\\i - 0) < r\n[PROOF STEP]\nobtain N :: nat where N_I: \"N \\ n \\ ?a 1 n < r\" for n\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\N. (\\n. N \\ n \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\ < r) \\ thesis) \\ thesis\n[PROOF STEP]\nusing LIMSEQ_D[OF \\?a 1 \\ 0\\ \\0 < r\\]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\no. \\n\\no. norm (\\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\ - 0) < r\n\ngoal (1 subgoal):\n 1. (\\N. (\\n. N \\ n \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\ < r) \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nN \\ ?n1 \\ \\1 / real (?n1 * 2 + 1) * 1 ^ (?n1 * 2 + 1)\\ < r\n\ngoal (1 subgoal):\n 1. \\r. 0 < r \\ \\no. \\n\\no. norm (\\arctan 1 - (\\i - 0) < r\n[PROOF STEP]\nhave \"norm (?diff 1 n - 0) < r\" if \"N \\ n\" for n\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (\\arctan 1 - (\\i - 0) < r\n[PROOF STEP]\nusing \\?diff 1 n \\ ?a 1 n\\ N_I[OF that]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\arctan 1 - (\\i \\ \\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\\n\\1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)\\ < r\n\ngoal (1 subgoal):\n 1. norm (\\arctan 1 - (\\i - 0) < r\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nN \\ ?n1 \\ norm (\\arctan 1 - (\\i - 0) < r\n\ngoal (1 subgoal):\n 1. \\r. 0 < r \\ \\no. \\n\\no. norm (\\arctan 1 - (\\i - 0) < r\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nN \\ ?n1 \\ norm (\\arctan 1 - (\\i - 0) < r\n[PROOF STEP]\nshow \"\\N. \\ n \\ N. norm (?diff 1 n - 0) < r\"\n[PROOF STATE]\nproof (prove)\nusing this:\nN \\ ?n1 \\ norm (\\arctan 1 - (\\i - 0) < r\n\ngoal (1 subgoal):\n 1. \\N. \\n\\N. norm (\\arctan 1 - (\\i - 0) < r\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\N. \\n\\N. norm (\\arctan 1 - (\\i - 0) < r\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. \\arctan 1 - (\\i) \\ 0\n\ngoal (1 subgoal):\n 1. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nfrom this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of \"- arctan 1\", THEN tendsto_minus]\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x. - (arctan 1 - (\\i - (0 + - arctan 1)\n[PROOF STEP]\nhave \"(?c 1) sums (arctan 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. - (arctan 1 - (\\i - (0 + - arctan 1)\n\ngoal (1 subgoal):\n 1. (\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 1 ^ (a * 2 + 1))) sums arctan 1\n[PROOF STEP]\nunfolding sums_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. - (arctan 1 - (\\i - (0 + - arctan 1)\n\ngoal (1 subgoal):\n 1. (\\n. \\a arctan 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 1 ^ (a * 2 + 1))) sums arctan 1\n\ngoal (1 subgoal):\n 1. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 1 ^ (a * 2 + 1))) sums arctan 1\n[PROOF STEP]\nhave \"arctan 1 = (\\i. ?c 1 i)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\a. (- 1) ^ a * (1 / real (a * 2 + 1) * 1 ^ (a * 2 + 1))) sums arctan 1\n\ngoal (1 subgoal):\n 1. arctan 1 = (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1)))\n[PROOF STEP]\nby (rule sums_unique)\n[PROOF STATE]\nproof (state)\nthis:\narctan 1 = (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1)))\n\ngoal (1 subgoal):\n 1. \\ \\x\\ < 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nproof (cases \"x = 1\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x = 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n 2. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nx = 1\n\ngoal (2 subgoals):\n 1. x = 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n 2. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx = 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx = 1\n\ngoal (1 subgoal):\n 1. arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nby (simp add: \\arctan 1 = (\\ i. ?c 1 i)\\)\n[PROOF STATE]\nproof (state)\nthis:\narctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nx \\ 1\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ 1\n[PROOF STEP]\nhave \"x = -1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ 1\n\ngoal (1 subgoal):\n 1. x = - 1\n[PROOF STEP]\nusing \\\\x\\ = 1\\\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ 1\n\\x\\ = 1\n\ngoal (1 subgoal):\n 1. x = - 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx = - 1\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave \"- (pi/2) < 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (pi / 2) < 0\n[PROOF STEP]\nusing pi_gt_zero\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < pi\n\ngoal (1 subgoal):\n 1. - (pi / 2) < 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n- (pi / 2) < 0\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave \"- (2 * pi) < 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (2 * pi) < 0\n[PROOF STEP]\nusing pi_gt_zero\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < pi\n\ngoal (1 subgoal):\n 1. - (2 * pi) < 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n- (2 * pi) < 0\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave c_minus_minus: \"?c (- 1) i = - ?c 1 i\" for i\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (- 1) ^ i * (1 / real (i * 2 + 1) * (- 1) ^ (i * 2 + 1)) = - ((- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(- 1) ^ ?i1 * (1 / real (?i1 * 2 + 1) * (- 1) ^ (?i1 * 2 + 1)) = - ((- 1) ^ ?i1 * (1 / real (?i1 * 2 + 1) * 1 ^ (?i1 * 2 + 1)))\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave \"arctan (- 1) = arctan (tan (-(pi / 4)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arctan (- 1) = arctan (tan (- (pi / 4)))\n[PROOF STEP]\nunfolding tan_45 tan_minus\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arctan (- 1) = arctan (- 1)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\narctan (- 1) = arctan (tan (- (pi / 4)))\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\narctan (- 1) = arctan (tan (- (pi / 4)))\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave \"\\ = - (pi / 4)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arctan (tan (- (pi / 4))) = - (pi / 4)\n[PROOF STEP]\nby (rule arctan_tan) (auto simp: order_less_trans[OF \\- (pi/2) < 0\\ pi_gt_zero])\n[PROOF STATE]\nproof (state)\nthis:\narctan (tan (- (pi / 4))) = - (pi / 4)\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\narctan (tan (- (pi / 4))) = - (pi / 4)\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave \"\\ = - (arctan (tan (pi / 4)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (pi / 4) = - arctan (tan (pi / 4))\n[PROOF STEP]\nunfolding neg_equal_iff_equal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pi / 4 = arctan (tan (pi / 4))\n[PROOF STEP]\nby (rule arctan_tan[symmetric]) (auto simp: order_less_trans[OF \\- (2 * pi) < 0\\ pi_gt_zero])\n[PROOF STATE]\nproof (state)\nthis:\n- (pi / 4) = - arctan (tan (pi / 4))\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n- (pi / 4) = - arctan (tan (pi / 4))\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave \"\\ = - (arctan 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - arctan (tan (pi / 4)) = - arctan 1\n[PROOF STEP]\nunfolding tan_45\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - arctan 1 = - arctan 1\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n- arctan (tan (pi / 4)) = - arctan 1\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n- arctan (tan (pi / 4)) = - arctan 1\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave \"\\ = - (\\ i. ?c 1 i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - arctan 1 = - (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1)))\n[PROOF STEP]\nusing \\arctan 1 = (\\ i. ?c 1 i)\\\n[PROOF STATE]\nproof (prove)\nusing this:\narctan 1 = (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1)))\n\ngoal (1 subgoal):\n 1. - arctan 1 = - (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n- arctan 1 = - (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1)))\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n- arctan 1 = - (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1)))\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nhave \"\\ = (\\ i. ?c (- 1) i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1))) = (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * (- 1) ^ (i * 2 + 1)))\n[PROOF STEP]\nusing suminf_minus[OF sums_summable[OF \\(?c 1) sums (arctan 1)\\]]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. - ((- 1) ^ n * (1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)))) = - (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)))\n\ngoal (1 subgoal):\n 1. - (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1))) = (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * (- 1) ^ (i * 2 + 1)))\n[PROOF STEP]\nunfolding c_minus_minus\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. - ((- 1) ^ n * (1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)))) = - (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * 1 ^ (n * 2 + 1)))\n\ngoal (1 subgoal):\n 1. - (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1))) = (\\i. - ((- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1))))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n- (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * 1 ^ (i * 2 + 1))) = (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * (- 1) ^ (i * 2 + 1)))\n\ngoal (1 subgoal):\n 1. x \\ 1 \\ arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\narctan (- 1) = (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * (- 1) ^ (i * 2 + 1)))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\narctan (- 1) = (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * (- 1) ^ (i * 2 + 1)))\n\ngoal (1 subgoal):\n 1. arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nusing \\x = -1\\\n[PROOF STATE]\nproof (prove)\nusing this:\narctan (- 1) = (\\i. (- 1) ^ i * (1 / real (i * 2 + 1) * (- 1) ^ (i * 2 + 1)))\nx = - 1\n\ngoal (1 subgoal):\n 1. arctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\narctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\narctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\narctan x = (\\n. (- 1) ^ n * (1 / real (n * 2 + 1) * x ^ (n * 2 + 1)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 40084, "file": null, "length": 282, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637505099168, "lm_q2_score": 0.8577681068080749, "lm_q1q2_score": 0.7373921477664206}} {"text": "[STATEMENT]\nlemma orthogonal_unit_vec [simp]:\n assumes \"i < n\" and \"j < n\" and \"i \\ j\"\n shows \"\\unit_vec n i|unit_vec n j\\ = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\unit_vec n i|unit_vec n j\\ = 0\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\unit_vec n i|unit_vec n j\\ = 0\n[PROOF STEP]\nhave \"\\unit_vec n i|unit_vec n j\\ = unit_vec n i \\ unit_vec n j\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\unit_vec n i|unit_vec n j\\ = unit_vec n i \\ unit_vec n j\n[PROOF STEP]\nusing assms unit_vec_def inner_prod_def scalar_prod_def\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n\nj < n\ni \\ j\nunit_vec ?n ?i = vec ?n (\\j. if j = ?i then 1::?'a else (0::?'a))\n\\?u|?v\\ \\ \\i = 0.. ?w \\ \\i = 0..unit_vec n i|unit_vec n j\\ = unit_vec n i \\ unit_vec n j\n[PROOF STEP]\nby (smt complex_cnj_zero index_unit_vec(3) index_vec inner_prod_with_row_bra_vec row_bra_vec \n scalar_prod_right_unit)\n[PROOF STATE]\nproof (state)\nthis:\n\\unit_vec n i|unit_vec n j\\ = unit_vec n i \\ unit_vec n j\n\ngoal (1 subgoal):\n 1. \\unit_vec n i|unit_vec n j\\ = 0\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\unit_vec n i|unit_vec n j\\ = unit_vec n i \\ unit_vec n j\n\ngoal (1 subgoal):\n 1. \\unit_vec n i|unit_vec n j\\ = 0\n[PROOF STEP]\nusing assms scalar_prod_def unit_vec_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\unit_vec n i|unit_vec n j\\ = unit_vec n i \\ unit_vec n j\ni < n\nj < n\ni \\ j\n?v \\ ?w \\ \\i = 0..j. if j = ?i then 1::?'a else (0::?'a))\n\ngoal (1 subgoal):\n 1. \\unit_vec n i|unit_vec n j\\ = 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\unit_vec n i|unit_vec n j\\ = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 957, "file": "Isabelle_Marries_Dirac_Quantum", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577680904463333, "lm_q2_score": 0.8596637505099168, "lm_q1q2_score": 0.7373921337008245}} {"text": "[STATEMENT]\ntheorem log_lin_time_explicitly:\n assumes \"length numbers = 2^l\"\n shows \"T\\<^sub>F\\<^sub>N\\<^sub>T\\<^sub>T numbers \\ 30 * Discrete.log (length numbers) * length numbers + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. T\\<^sub>F\\<^sub>N\\<^sub>T\\<^sub>T numbers \\ 30 * Discrete.log (length numbers) * length numbers + 1\n[PROOF STEP]\nusing log_lin_time[of numbers l] assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength numbers = 2 ^ l \\ T\\<^sub>F\\<^sub>N\\<^sub>T\\<^sub>T numbers \\ 30 * l * length numbers + 1\nlength numbers = 2 ^ l\n\ngoal (1 subgoal):\n 1. T\\<^sub>F\\<^sub>N\\<^sub>T\\<^sub>T numbers \\ 30 * Discrete.log (length numbers) * length numbers + 1\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 288, "file": "Number_Theoretic_Transform_Butterfly", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361652391385, "lm_q2_score": 0.8080672112416737, "lm_q1q2_score": 0.7373097474808377}} {"text": "[STATEMENT]\ntheorem \"\\\\ \\True\\\n \\S :== 0;; \\I :== 1;;\n WHILE \\I \\ n\n INV \\\\S = (SUMM j<\\I. j)\\\n DO\n \\S :== \\S + \\I;;\n \\I :== \\I + 1\n OD\n \\\\S = (SUMM j\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\ \\True\\ \\S :== 0;; \\I :== 1;; WHILE \\I \\ n INV \\\\S = (SUMM j<\\I. j)\\ DO \\S :== \\S + \\I;; \\I :== \\I + 1 OD \\\\S = (SUMM j\n[PROOF STEP]\napply vcg\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. True \\ 0 = (SUMM j<1. j)\n 2. \\I S. \\S = (SUMM j n\\ \\ S + I = (SUMM jI S. \\S = (SUMM j I \\ n\\ \\ S = (SUMM jz. z \\ S \\ (f has_field_derivative f' z) (at z within S)\"\n and dn: \"\\z. z \\ S \\ norm (f' z) \\ B\"\n and \"x \\ S\" \"y \\ S\"\n shows \"norm(f x - f y) \\ B * norm(x - y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (f x - f y) \\ B * norm (x - y)\n[PROOF STEP]\napply (rule differentiable_bound [OF cvs])\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. \\x. x \\ S \\ (f has_derivative ?f' x) (at x within S)\n 2. \\x. x \\ S \\ onorm (?f' x) \\ B\n 3. x \\ S\n 4. y \\ S\n[PROOF STEP]\napply (erule df [unfolded has_field_derivative_def])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x. x \\ S \\ onorm ((*) (f' x)) \\ B\n 2. x \\ S\n 3. y \\ S\n[PROOF STEP]\napply (rule onorm_le, simp_all add: norm_mult mult_right_mono assms)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 481, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110396870287, "lm_q2_score": 0.8267118026095991, "lm_q1q2_score": 0.7372707122068042}} {"text": "[STATEMENT]\nlemma sin_cos_squared_add [simp]: \"(sin x)\\<^sup>2 + (cos x)\\<^sup>2 = 1\"\n for x :: \"'a::{real_normed_field,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (sin x)\\<^sup>2 + (cos x)\\<^sup>2 = (1::'a)\n[PROOF STEP]\nusing cos_add [of x \"-x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ncos (x + - x) = cos x * cos (- x) - sin x * sin (- x)\n\ngoal (1 subgoal):\n 1. (sin x)\\<^sup>2 + (cos x)\\<^sup>2 = (1::'a)\n[PROOF STEP]\nby (simp add: power2_eq_square algebra_simps)", "meta": {"llama_tokens": 232, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110454379297, "lm_q2_score": 0.8267117876664789, "lm_q1q2_score": 0.7372707036347023}} {"text": "[STATEMENT]\ntheorem sigma_primepower:\n assumes \"prime p\"\n shows \"(p - 1) * sigma(p^e) = p^(e+1) - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (p - 1) * sigma (p ^ e) = p ^ (e + 1) - 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (p - 1) * sigma (p ^ e) = p ^ (e + 1) - 1\n[PROOF STEP]\nhave \"sigma(p^e) = (\\i=0..e . p^i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sigma (p ^ e) = sum ((^) p) {0..e}\n[PROOF STEP]\nusing assms divisors_def dvd_prime_power_iff prime_nat_iff rewrite_sum_of_powers\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\ndivisors ?m \\ {n. n dvd ?m}\nprime ?p \\ {d. d dvd ?p ^ ?n} = (^) ?p ` {0..?n}\nprime ?n = (1 < ?n \\ (\\m. m dvd ?n \\ m = 1 \\ m = ?n))\n1 < ?p \\ \\ ((^) ?p ` {0..?n}) = sum ((^) ?p) {0..?n}\n\ngoal (1 subgoal):\n 1. sigma (p ^ e) = sum ((^) p) {0..e}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsigma (p ^ e) = sum ((^) p) {0..e}\n\ngoal (1 subgoal):\n 1. (p - 1) * sigma (p ^ e) = p ^ (e + 1) - 1\n[PROOF STEP]\nthus \"(p - 1)*sigma(p^e) = p^(e+1) - 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsigma (p ^ e) = sum ((^) p) {0..e}\n\ngoal (1 subgoal):\n 1. (p - 1) * sigma (p ^ e) = p ^ (e + 1) - 1\n[PROOF STEP]\nusing sum_of_powers_nat\n[PROOF STATE]\nproof (prove)\nusing this:\nsigma (p ^ e) = sum ((^) p) {0..e}\n(?x - 1) * sum ((^) ?x) {0..?n} = ?x ^ Suc ?n - 1\n\ngoal (1 subgoal):\n 1. (p - 1) * sigma (p ^ e) = p ^ (e + 1) - 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(p - 1) * sigma (p ^ e) = p ^ (e + 1) - 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 833, "file": "Perfect-Number-Thm_Sigma", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916029436189, "lm_q2_score": 0.8539127492339909, "lm_q1q2_score": 0.7372610973351279}} {"text": "[STATEMENT]\nlemma card_set_of_intersecting_sets_by_card:\n assumes \"A \\ I\" \"finite I\" \"k \\ n\" \"n \\ card I\" \"k \\ card A\"\n shows \"card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * ((card I - card A) choose (n - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nnote finite_A = finite_subset[OF assms(1,2)]\n[PROOF STATE]\nproof (state)\nthis:\nfinite A\n\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nhave \"card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = card ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\" (is \"card ?lhs = card ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = card ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nproof (rule bij_betw_same_card[symmetric])\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. bij_betw ?f ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n[PROOF STEP]\nlet ?f = \"\\(K, B'). K \\ B'\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. bij_betw ?f ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n[PROOF STEP]\nhave \"inj_on ?f ?rhs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on (\\(x, y). x \\ y) ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nby (blast intro: inj_onI)\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\(x, y). x \\ y) ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n\ngoal (1 subgoal):\n 1. bij_betw ?f ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\(x, y). x \\ y) ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n\ngoal (1 subgoal):\n 1. bij_betw ?f ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n[PROOF STEP]\nhave \"?f ` ?rhs = ?lhs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) = {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n[PROOF STEP]\nproof (rule set_eqI, rule iffI)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x. x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) \\ x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n 2. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nfix B\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x. x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) \\ x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n 2. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nassume \"B \\ ?f ` ?rhs\"\n[PROOF STATE]\nproof (state)\nthis:\nB \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n\ngoal (2 subgoals):\n 1. \\x. x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) \\ x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n 2. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nB \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nobtain K B' where K: \"K \\ A\" \"card K = k\" \"B' \\ I - A\" \"card B' = n - k\" \"K \\ B' = B\"\n[PROOF STATE]\nproof (prove)\nusing this:\nB \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n\ngoal (1 subgoal):\n 1. (\\K B'. \\K \\ A; card K = k; B' \\ I - A; card B' = n - k; K \\ B' = B\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nK \\ A\ncard K = k\nB' \\ I - A\ncard B' = n - k\nK \\ B' = B\n\ngoal (2 subgoals):\n 1. \\x. x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) \\ x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n 2. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nshow \"B \\ ?lhs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. B \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n[PROOF STEP]\nproof safe\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\x. x \\ B \\ x \\ I\n 2. card B = n\n 3. card (A \\ B) = k\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\x. x \\ B \\ x \\ I\n 2. card B = n\n 3. card (A \\ B) = k\n[PROOF STEP]\nassume \"x \\ B\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ B\n\ngoal (3 subgoals):\n 1. \\x. x \\ B \\ x \\ I\n 2. card B = n\n 3. card (A \\ B) = k\n[PROOF STEP]\nthus \"x \\ I\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ B\n\ngoal (1 subgoal):\n 1. x \\ I\n[PROOF STEP]\nusing K \\A \\ I\\\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ B\nK \\ A\ncard K = k\nB' \\ I - A\ncard B' = n - k\nK \\ B' = B\nA \\ I\n\ngoal (1 subgoal):\n 1. x \\ I\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nx \\ I\n\ngoal (2 subgoals):\n 1. card B = n\n 2. card (A \\ B) = k\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. card B = n\n 2. card (A \\ B) = k\n[PROOF STEP]\nhave \"card B = card K + card B' - card (K \\ B')\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card B = card K + card B' - card (K \\ B')\n[PROOF STEP]\nusing K assms\n[PROOF STATE]\nproof (prove)\nusing this:\nK \\ A\ncard K = k\nB' \\ I - A\ncard B' = n - k\nK \\ B' = B\nA \\ I\nfinite I\nk \\ n\nn \\ card I\nk \\ card A\n\ngoal (1 subgoal):\n 1. card B = card K + card B' - card (K \\ B')\n[PROOF STEP]\nby (metis card_union finite_A finite_subset finite_Diff)\n[PROOF STATE]\nproof (state)\nthis:\ncard B = card K + card B' - card (K \\ B')\n\ngoal (2 subgoals):\n 1. card B = n\n 2. card (A \\ B) = k\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncard B = card K + card B' - card (K \\ B')\n\ngoal (2 subgoals):\n 1. card B = n\n 2. card (A \\ B) = k\n[PROOF STEP]\nhave \"K \\ B' = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. K \\ B' = {}\n[PROOF STEP]\nusing K assms\n[PROOF STATE]\nproof (prove)\nusing this:\nK \\ A\ncard K = k\nB' \\ I - A\ncard B' = n - k\nK \\ B' = B\nA \\ I\nfinite I\nk \\ n\nn \\ card I\nk \\ card A\n\ngoal (1 subgoal):\n 1. K \\ B' = {}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nK \\ B' = {}\n\ngoal (2 subgoals):\n 1. card B = n\n 2. card (A \\ B) = k\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncard B = card K + card B' - card (K \\ B')\nK \\ B' = {}\n[PROOF STEP]\nshow \"card B = n\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard B = card K + card B' - card (K \\ B')\nK \\ B' = {}\n\ngoal (1 subgoal):\n 1. card B = n\n[PROOF STEP]\nusing K assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncard B = card K + card B' - card (K \\ B')\nK \\ B' = {}\nK \\ A\ncard K = k\nB' \\ I - A\ncard B' = n - k\nK \\ B' = B\nA \\ I\nfinite I\nk \\ n\nn \\ card I\nk \\ card A\n\ngoal (1 subgoal):\n 1. card B = n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard B = n\n\ngoal (1 subgoal):\n 1. card (A \\ B) = k\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (A \\ B) = k\n[PROOF STEP]\nhave \"A \\ B = K\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A \\ B = K\n[PROOF STEP]\nusing K assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nK \\ A\ncard K = k\nB' \\ I - A\ncard B' = n - k\nK \\ B' = B\nA \\ I\n\ngoal (1 subgoal):\n 1. A \\ B = K\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nA \\ B = K\n\ngoal (1 subgoal):\n 1. card (A \\ B) = k\n[PROOF STEP]\nthus \"card (A \\ B) = k\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ B = K\n\ngoal (1 subgoal):\n 1. card (A \\ B) = k\n[PROOF STEP]\nusing K\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ B = K\nK \\ A\ncard K = k\nB' \\ I - A\ncard B' = n - k\nK \\ B' = B\n\ngoal (1 subgoal):\n 1. card (A \\ B) = k\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\ B) = k\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nB \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nfix B\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nassume \"B \\ ?lhs\"\n[PROOF STATE]\nproof (state)\nthis:\nB \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nhence B: \"B \\ I\" \"card B = n\" \"card (A \\ B) = k\"\n[PROOF STATE]\nproof (prove)\nusing this:\nB \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n\ngoal (1 subgoal):\n 1. B \\ I &&& card B = n &&& card (A \\ B) = k\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nB \\ I\ncard B = n\ncard (A \\ B) = k\n\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nlet ?K = \"A \\ B\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nlet ?B' = \"B - A\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nhave \"?K \\ A\" \"card ?K = k\" \"?B' \\ I - A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A \\ B \\ A &&& card (A \\ B) = k &&& B - A \\ I - A\n[PROOF STEP]\nusing B\n[PROOF STATE]\nproof (prove)\nusing this:\nB \\ I\ncard B = n\ncard (A \\ B) = k\n\ngoal (1 subgoal):\n 1. A \\ B \\ A &&& card (A \\ B) = k &&& B - A \\ I - A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA \\ B \\ A\ncard (A \\ B) = k\nB - A \\ I - A\n\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nA \\ B \\ A\ncard (A \\ B) = k\nB - A \\ I - A\n\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nhave \"card ?B' = n - k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (B - A) = n - k\n[PROOF STEP]\nusing B finite_A assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nB \\ I\ncard B = n\ncard (A \\ B) = k\nfinite A\nA \\ I\n\ngoal (1 subgoal):\n 1. card (B - A) = n - k\n[PROOF STEP]\nby (metis Int_commute card_Diff_subset_Int finite_Un inf.left_idem le_iff_inf sup_absorb2)\n[PROOF STATE]\nproof (state)\nthis:\ncard (B - A) = n - k\n\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nA \\ B \\ A\ncard (A \\ B) = k\nB - A \\ I - A\ncard (B - A) = n - k\n[PROOF STEP]\nhave \"(?K, ?B') \\ ?rhs\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ B \\ A\ncard (A \\ B) = k\nB - A \\ I - A\ncard (B - A) = n - k\n\ngoal (1 subgoal):\n 1. (A \\ B, B - A) \\ {K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(A \\ B, B - A) \\ {K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}\n\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(A \\ B, B - A) \\ {K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}\n\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nhave \"B = ?f (?K, ?B')\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. B = (case (A \\ B, B - A) of (x, xa) \\ x \\ xa)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nB = (case (A \\ B, B - A) of (x, xa) \\ x \\ xa)\n\ngoal (1 subgoal):\n 1. \\x. x \\ {B. B \\ I \\ card B = n \\ card (A \\ B) = k} \\ x \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(A \\ B, B - A) \\ {K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}\nB = (case (A \\ B, B - A) of (x, xa) \\ x \\ xa)\n[PROOF STEP]\nshow \"B \\ ?f ` ?rhs\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(A \\ B, B - A) \\ {K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}\nB = (case (A \\ B, B - A) of (x, xa) \\ x \\ xa)\n\ngoal (1 subgoal):\n 1. B \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nB \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) = {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n\ngoal (1 subgoal):\n 1. bij_betw ?f ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ninj_on (\\(x, y). x \\ y) ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n(\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) = {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n[PROOF STEP]\nshow \"bij_betw ?f ?rhs ?lhs\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on (\\(x, y). x \\ y) ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n(\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) = {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n\ngoal (1 subgoal):\n 1. bij_betw (\\(x, y). x \\ y) ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n[PROOF STEP]\nunfolding bij_betw_def\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on (\\(x, y). x \\ y) ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n(\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) = {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n\ngoal (1 subgoal):\n 1. inj_on (\\(x, y). x \\ y) ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) \\ (\\(x, y). x \\ y) ` ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) = {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw (\\(x, y). x \\ y) ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) {B. B \\ I \\ card B = n \\ card (A \\ B) = k}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncard {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = card ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = card ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k})\n\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nhave \"\\ = (\\K | K \\ A \\ card K = k. card {B'. B' \\ I - A \\ card B' = n - k})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) = (\\K | K \\ A \\ card K = k. card {B'. B' \\ I - A \\ card B' = n - k})\n[PROOF STEP]\nproof (rule card_SigmaI, safe)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. finite {K. K \\ A \\ card K = k}\n 2. \\K. \\K \\ A; k = card K\\ \\ finite {B'. B' \\ I - A \\ card B' = n - card K}\n[PROOF STEP]\nshow \"finite {K. K \\ A \\ card K = k}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {K. K \\ A \\ card K = k}\n[PROOF STEP]\nby (blast intro: finite_subset[where B = \"Pow A\"] finite_A)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {K. K \\ A \\ card K = k}\n\ngoal (1 subgoal):\n 1. \\K. \\K \\ A; k = card K\\ \\ finite {B'. B' \\ I - A \\ card B' = n - card K}\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\K. \\K \\ A; k = card K\\ \\ finite {B'. B' \\ I - A \\ card B' = n - card K}\n[PROOF STEP]\nfix K\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\K. \\K \\ A; k = card K\\ \\ finite {B'. B' \\ I - A \\ card B' = n - card K}\n[PROOF STEP]\nassume \"K \\ A\"\n[PROOF STATE]\nproof (state)\nthis:\nK \\ A\n\ngoal (1 subgoal):\n 1. \\K. \\K \\ A; k = card K\\ \\ finite {B'. B' \\ I - A \\ card B' = n - card K}\n[PROOF STEP]\nthus \"finite {B'. B' \\ I - A \\ card B' = n - card K}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nK \\ A\n\ngoal (1 subgoal):\n 1. finite {B'. B' \\ I - A \\ card B' = n - card K}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nK \\ A\nA \\ I\nfinite I\nk \\ n\nn \\ card I\nk \\ card A\n\ngoal (1 subgoal):\n 1. finite {B'. B' \\ I - A \\ card B' = n - card K}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite {B'. B' \\ I - A \\ card B' = n - card K}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncard ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) = (\\K | K \\ A \\ card K = k. card {B'. B' \\ I - A \\ card B' = n - k})\n\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ({K. K \\ A \\ card K = k} \\ {B'. B' \\ I - A \\ card B' = n - k}) = (\\K | K \\ A \\ card K = k. card {B'. B' \\ I - A \\ card B' = n - k})\n\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nhave \"\\ = card {K. K \\ A \\ card K = k} * card {B'. B' \\ I - A \\ card B' = n - k}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\K | K \\ A \\ card K = k. card {B'. B' \\ I - A \\ card B' = n - k}) = card {K. K \\ A \\ card K = k} * card {B'. B' \\ I - A \\ card B' = n - k}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\K | K \\ A \\ card K = k. card {B'. B' \\ I - A \\ card B' = n - k}) = card {K. K \\ A \\ card K = k} * card {B'. B' \\ I - A \\ card B' = n - k}\n\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\K | K \\ A \\ card K = k. card {B'. B' \\ I - A \\ card B' = n - k}) = card {K. K \\ A \\ card K = k} * card {B'. B' \\ I - A \\ card B' = n - k}\n\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nhave \"\\ = (card A choose k) * (card (I - A) choose (n - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {K. K \\ A \\ card K = k} * card {B'. B' \\ I - A \\ card B' = n - k} = (card A choose k) * (card (I - A) choose (n - k))\n[PROOF STEP]\nby (simp only: n_subsets[OF finite_A] n_subsets[OF finite_Diff[OF assms(2)]])\n[PROOF STATE]\nproof (state)\nthis:\ncard {K. K \\ A \\ card K = k} * card {B'. B' \\ I - A \\ card B' = n - k} = (card A choose k) * (card (I - A) choose (n - k))\n\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {K. K \\ A \\ card K = k} * card {B'. B' \\ I - A \\ card B' = n - k} = (card A choose k) * (card (I - A) choose (n - k))\n\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nhave \"\\ = (card A choose k) * ((card I - card A) choose (n - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (card A choose k) * (card (I - A) choose (n - k)) = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nby (simp only: card_Diff_subset[OF finite_A assms(1)])\n[PROOF STATE]\nproof (state)\nthis:\n(card A choose k) * (card (I - A) choose (n - k)) = (card A choose k) * (card I - card A choose (n - k))\n\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n\ngoal (1 subgoal):\n 1. card {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {B. B \\ I \\ card B = n \\ card (A \\ B) = k} = (card A choose k) * (card I - card A choose (n - k))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 12401, "file": "Random_Graph_Subgraph_Threshold_Ugraph_Misc", "length": 96, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213799730774, "lm_q2_score": 0.8198933425148214, "lm_q1q2_score": 0.7371836335526653}} {"text": "[STATEMENT]\nlemma inner_prod_minus_distrib_right:\n fixes u v w :: \"('a::conjugatable_field) vec\"\n assumes dimu: \"u \\ carrier_vec n\" and dimv:\"v \\ carrier_vec n\" and dimw: \"w \\ carrier_vec n\" \n shows \"inner_prod u (v - w) = inner_prod u v - inner_prod u w\" (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inner_prod u (v - w) = inner_prod u v - inner_prod u w\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. inner_prod u (v - w) = inner_prod u v - inner_prod u w\n[PROOF STEP]\nhave dimvw: \"v - w \\ carrier_vec n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v - w \\ carrier_vec n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ carrier_vec n\nv \\ carrier_vec n\nw \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. v - w \\ carrier_vec n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv - w \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. inner_prod u (v - w) = inner_prod u v - inner_prod u w\n[PROOF STEP]\nhave dimcu: \"conjugate u \\ carrier_vec n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. conjugate u \\ carrier_vec n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ carrier_vec n\nv \\ carrier_vec n\nw \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. conjugate u \\ carrier_vec n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nconjugate u \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. inner_prod u (v - w) = inner_prod u v - inner_prod u w\n[PROOF STEP]\nhave \"(v - w) \\ (conjugate u) = v \\ conjugate u - w \\ conjugate u\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inner_prod u (v - w) = inner_prod u v - inner_prod u w\n[PROOF STEP]\napply (simp add: comm_scalar_prod[OF dimvw dimcu])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. conjugate u \\ (v - w) = inner_prod u v - inner_prod u w\n[PROOF STEP]\napply (simp add: scalar_prod_minus_distrib[OF dimcu dimv dimw])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. conjugate u \\ v - conjugate u \\ w = inner_prod u v - inner_prod u w\n[PROOF STEP]\napply (insert dimv dimw dimcu, simp add: comm_scalar_prod[of _ n])\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\ninner_prod u (v - w) = inner_prod u v - inner_prod u w\n\ngoal (1 subgoal):\n 1. inner_prod u (v - w) = inner_prod u v - inner_prod u w\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ninner_prod u (v - w) = inner_prod u v - inner_prod u w\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ninner_prod u (v - w) = inner_prod u v - inner_prod u w\n\ngoal (1 subgoal):\n 1. inner_prod u (v - w) = inner_prod u v - inner_prod u w\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninner_prod u (v - w) = inner_prod u v - inner_prod u w\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1235, "file": "QHLProver_Complex_Matrix", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240825770433, "lm_q2_score": 0.8519527963298946, "lm_q1q2_score": 0.7371300766034797}} {"text": "[STATEMENT]\nlemma inverse_matrix_code_rings[code_unfold]: \n fixes A::\"'a::{euclidean_ring}^'n::{mod_type}^'n::{mod_type}\"\n shows \"inverse_matrix A = (let d=det A in if is_unit d then Some ((1 div d) *k adjugate A) else None)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse_matrix A = (let d = det A in if is_unit d then Some ((1::'a) div d *k adjugate A) else None)\n[PROOF STEP]\nusing invertible_imp_matrix_inv[of A]\n[PROOF STATE]\nproof (prove)\nusing this:\ninvertible A \\ matrix_inv A = (1::'a) div det A *k adjugate A\n\ngoal (1 subgoal):\n 1. inverse_matrix A = (let d = det A in if is_unit d then Some ((1::'a) div d *k adjugate A) else None)\n[PROOF STEP]\nunfolding inverse_matrix_def invertible_iff_is_unit\n[PROOF STATE]\nproof (prove)\nusing this:\nis_unit (det A) \\ matrix_inv A = (1::'a) div det A *k adjugate A\n\ngoal (1 subgoal):\n 1. (if is_unit (det A) then Some (matrix_inv A) else None) = (let d = det A in if is_unit d then Some ((1::'a) div d *k adjugate A) else None)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 417, "file": "Echelon_Form_Echelon_Form_Inverse", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9149009503523291, "lm_q2_score": 0.8056321843145404, "lm_q1q2_score": 0.7370736510637957}} {"text": "[STATEMENT]\nlemma bounded_hyperplane_eq_trivial_0:\n fixes a :: \"'a::euclidean_space\"\n assumes \"a \\ 0\"\n shows \"bounded {x. a \\ x = 0} \\ DIM('a) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bounded {x. a \\ x = 0} = (DIM('a) = 1)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. bounded {x. a \\ x = 0} \\ DIM('a) = 1\n 2. DIM('a) = 1 \\ bounded {x. a \\ x = 0}\n[PROOF STEP]\nassume \"bounded {x. a \\ x = 0}\"\n[PROOF STATE]\nproof (state)\nthis:\nbounded {x. a \\ x = 0}\n\ngoal (2 subgoals):\n 1. bounded {x. a \\ x = 0} \\ DIM('a) = 1\n 2. DIM('a) = 1 \\ bounded {x. a \\ x = 0}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nbounded {x. a \\ x = 0}\n[PROOF STEP]\nhave \"aff_dim {x. a \\ x = 0} \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbounded {x. a \\ x = 0}\n\ngoal (1 subgoal):\n 1. aff_dim {x. a \\ x = 0} \\ 0\n[PROOF STEP]\nby (simp add: affine_bounded_eq_lowdim affine_hyperplane)\n[PROOF STATE]\nproof (state)\nthis:\naff_dim {x. a \\ x = 0} \\ 0\n\ngoal (2 subgoals):\n 1. bounded {x. a \\ x = 0} \\ DIM('a) = 1\n 2. DIM('a) = 1 \\ bounded {x. a \\ x = 0}\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ (0::'a)\naff_dim {x. a \\ x = 0} \\ 0\n[PROOF STEP]\nshow \"DIM('a) = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ (0::'a)\naff_dim {x. a \\ x = 0} \\ 0\n\ngoal (1 subgoal):\n 1. DIM('a) = 1\n[PROOF STEP]\nby (simp add: le_Suc_eq)\n[PROOF STATE]\nproof (state)\nthis:\nDIM('a) = 1\n\ngoal (1 subgoal):\n 1. DIM('a) = 1 \\ bounded {x. a \\ x = 0}\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. DIM('a) = 1 \\ bounded {x. a \\ x = 0}\n[PROOF STEP]\nassume \"DIM('a) = 1\"\n[PROOF STATE]\nproof (state)\nthis:\nDIM('a) = 1\n\ngoal (1 subgoal):\n 1. DIM('a) = 1 \\ bounded {x. a \\ x = 0}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nDIM('a) = 1\n[PROOF STEP]\nshow \"bounded {x. a \\ x = 0}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nDIM('a) = 1\n\ngoal (1 subgoal):\n 1. bounded {x. a \\ x = 0}\n[PROOF STEP]\nby (simp add: affine_bounded_eq_lowdim affine_hyperplane assms)\n[PROOF STATE]\nproof (state)\nthis:\nbounded {x. a \\ x = 0}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1156, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357563664174, "lm_q2_score": 0.8499711699569787, "lm_q1q2_score": 0.7369553962332978}} {"text": "[STATEMENT]\nlemma fermat_theorem:\n fixes p a :: nat\n assumes \"prime p\" and \"\\ p dvd a\"\n shows \"[a ^ (p - 1) = 1] (mod p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [a ^ (p - 1) = 1] (mod p)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. [a ^ (p - 1) = 1] (mod p)\n[PROOF STEP]\nfrom assms prime_imp_coprime [of p a]\n[PROOF STATE]\nproof (chain)\npicking this:\nprime p\n\\ p dvd a\n\\prime p; \\ p dvd a\\ \\ coprime p a\n[PROOF STEP]\nhave \"coprime a p\"\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\\ p dvd a\n\\prime p; \\ p dvd a\\ \\ coprime p a\n\ngoal (1 subgoal):\n 1. coprime a p\n[PROOF STEP]\nby (auto simp add: ac_simps)\n[PROOF STATE]\nproof (state)\nthis:\ncoprime a p\n\ngoal (1 subgoal):\n 1. [a ^ (p - 1) = 1] (mod p)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncoprime a p\n[PROOF STEP]\nhave \"[a ^ totient p = 1] (mod p)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime a p\n\ngoal (1 subgoal):\n 1. [a ^ totient p = 1] (mod p)\n[PROOF STEP]\nby (rule euler_theorem)\n[PROOF STATE]\nproof (state)\nthis:\n[a ^ totient p = 1] (mod p)\n\ngoal (1 subgoal):\n 1. [a ^ (p - 1) = 1] (mod p)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n[a ^ totient p = 1] (mod p)\n\ngoal (1 subgoal):\n 1. [a ^ (p - 1) = 1] (mod p)\n[PROOF STEP]\nhave \"totient p = p - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. totient p = p - 1\n[PROOF STEP]\nby (rule totient_prime) (rule assms)\n[PROOF STATE]\nproof (state)\nthis:\ntotient p = p - 1\n\ngoal (1 subgoal):\n 1. [a ^ (p - 1) = 1] (mod p)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n[a ^ (p - 1) = 1] (mod p)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n[a ^ (p - 1) = 1] (mod p)\n\ngoal (1 subgoal):\n 1. [a ^ (p - 1) = 1] (mod p)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n[a ^ (p - 1) = 1] (mod p)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 952, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.849971175657575, "lm_q2_score": 0.8670357494949104, "lm_q1q2_score": 0.7369553953353356}} {"text": "[STATEMENT]\nlemma ivl_integral_bound:\n fixes f::\"real \\ 'a::banach\"\n assumes \"continuous_on (closed_segment a b) f\"\n assumes \"\\t. t \\ (closed_segment a b) \\ norm (f t) \\ B\"\n shows \"norm (ivl_integral a b f) \\ B * abs (b - a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (ivl_integral a b f) \\ B * \\b - a\\\n[PROOF STEP]\nusing integral_bound[of a b f B]\n integral_bound[of b a f B]\n assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a \\ b; continuous_on {a..b} f; \\t. t \\ {a..b} \\ norm (f t) \\ B\\ \\ norm (integral {a..b} f) \\ B * (b - a)\n\\b \\ a; continuous_on {b..a} f; \\t. t \\ {b..a} \\ norm (f t) \\ B\\ \\ norm (integral {b..a} f) \\ B * (a - b)\ncontinuous_on (closed_segment a b) f\n?t \\ closed_segment a b \\ norm (f ?t) \\ B\n\ngoal (1 subgoal):\n 1. norm (ivl_integral a b f) \\ B * \\b - a\\\n[PROOF STEP]\nby (auto simp: closed_segment_eq_real_ivl has_ivl_integral_def ivl_integral_def split: if_splits)", "meta": {"llama_tokens": 482, "file": "Ordinary_Differential_Equations_Library_Interval_Integral_HK", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513842182775, "lm_q2_score": 0.8221891370573386, "lm_q1q2_score": 0.7368881521768709}} {"text": "[STATEMENT]\nlemma lemma2: \n \"[| P A n ; P B m ; ! A n. P A n --> P A (Suc n) |]\n ==> ? n . P A n & P B n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\P A n; P B m; \\A n. P A n \\ P A (Suc n)\\ \\ \\n. P A n \\ P B n\n[PROOF STEP]\napply (rule exI[of _ \"n+m\"], rule)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\P A n; P B m; \\A n. P A n \\ P A (Suc n)\\ \\ P A (n + m)\n 2. \\P A n; P B m; \\A n. P A n \\ P A (Suc n)\\ \\ P B (n + m)\n[PROOF STEP]\napply(blast intro!: lemma1)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\P A n; P B m; \\A n. P A n \\ P A (Suc n)\\ \\ P B (n + m)\n[PROOF STEP]\napply(rule subst[OF add.commute])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\P A n; P B m; \\A n. P A n \\ P A (Suc n)\\ \\ P B (m + n)\n[PROOF STEP]\napply(blast intro!: lemma1)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 508, "file": "Completeness_Completeness", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513675912912, "lm_q2_score": 0.8221891305219504, "lm_q1q2_score": 0.7368881326489927}} {"text": "[STATEMENT]\nlemma nn_integral_0_1_ln_times_power:\n \"(\\\\<^sup>+y\\{0<..<1}. ennreal (-ln y * y ^ n) \\lborel) = ennreal (1 / (n + 1)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = ennreal (1 / (real n + 1)\\<^sup>2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = ennreal (1 / (real n + 1)\\<^sup>2)\n[PROOF STEP]\nhave \"(\\\\<^sup>+y\\{0<..<1}. ennreal (-ln y * y ^ n) \\lborel) =\n (\\\\<^sup>+y\\{0<..<1}. ennreal (-ln y * y powr real n) \\lborel)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = \\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x powr real n)\\lborel\n[PROOF STEP]\nby (intro set_nn_integral_cong) (auto simp: powr_realpow)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = \\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x powr real n)\\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = ennreal (1 / (real n + 1)\\<^sup>2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = \\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x powr real n)\\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = ennreal (1 / (real n + 1)\\<^sup>2)\n[PROOF STEP]\nhave \"\\ = ennreal (1 / (n + 1)^2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x powr real n)\\lborel = ennreal (1 / real ((n + 1)\\<^sup>2))\n[PROOF STEP]\nby (subst nn_integral_0_1_ln_times_powr) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x powr real n)\\lborel = ennreal (1 / real ((n + 1)\\<^sup>2))\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = ennreal (1 / (real n + 1)\\<^sup>2)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = ennreal (1 / real ((n + 1)\\<^sup>2))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = ennreal (1 / real ((n + 1)\\<^sup>2))\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = ennreal (1 / (real n + 1)\\<^sup>2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+x\\{0<..<1}. ennreal (- ln x * x ^ n)\\lborel = ennreal (1 / (real n + 1)\\<^sup>2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1403, "file": "Zeta_3_Irrational_Zeta_3_Irrational", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513675912913, "lm_q2_score": 0.8221891261650248, "lm_q1q2_score": 0.7368881287440923}} {"text": "[STATEMENT]\nlemma x_times_x_minus_1_nonneg: \"x \\ 0 \\ x \\ 1 \\ (x::_::linordered_idom) * (x - 1) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ (0::'a) \\ (1::'a) \\ x \\ (0::'a) \\ x * (x - (1::'a))\n[PROOF STEP]\nproof (elim disjE)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x \\ (0::'a) \\ (0::'a) \\ x * (x - (1::'a))\n 2. (1::'a) \\ x \\ (0::'a) \\ x * (x - (1::'a))\n[PROOF STEP]\nassume x: \"x \\ 0\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ (0::'a)\n\ngoal (2 subgoals):\n 1. x \\ (0::'a) \\ (0::'a) \\ x * (x - (1::'a))\n 2. (1::'a) \\ x \\ (0::'a) \\ x * (x - (1::'a))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx \\ (0::'a)\n\ngoal (2 subgoals):\n 1. x \\ (0::'a) \\ (0::'a) \\ x * (x - (1::'a))\n 2. (1::'a) \\ x \\ (0::'a) \\ x * (x - (1::'a))\n[PROOF STEP]\nhave \"0 \\ x^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) \\ x\\<^sup>2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) \\ x\\<^sup>2\n\ngoal (2 subgoals):\n 1. x \\ (0::'a) \\ (0::'a) \\ x * (x - (1::'a))\n 2. (1::'a) \\ x \\ (0::'a) \\ x * (x - (1::'a))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ x\\<^sup>2\n[PROOF STEP]\nshow \"x * (x - 1) \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ x\\<^sup>2\n\ngoal (1 subgoal):\n 1. (0::'a) \\ x * (x - (1::'a))\n[PROOF STEP]\nby (simp add: power2_eq_square algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) \\ x * (x - (1::'a))\n\ngoal (1 subgoal):\n 1. (1::'a) \\ x \\ (0::'a) \\ x * (x - (1::'a))\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 914, "file": "Akra_Bazzi_Akra_Bazzi_Library", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767970940974, "lm_q2_score": 0.8397339716830606, "lm_q1q2_score": 0.7368470758835576}} {"text": "[STATEMENT]\nlemma crb_lem_pos: \n fixes x:: \"real\"\n fixes p:: \"real poly\"\n assumes x: \"poly p x = 0\" \n assumes p: \"p \\ 0\" \n shows \"x < crb p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x < real_of_int (crb p)\n[PROOF STEP]\nusing cauchy_root_bound[of p x]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\poly p x = 0; p \\ 0\\ \\ norm x \\ 1 + max_list_non_empty (map (\\i. norm (coeff p i)) [0..poly p x = 0; p \\ 0\\ \\ \\x\\ \\ 1 + max_list_non_empty (map (\\i. \\coeff p i\\) [0..lead_coeff p\\) \\ x < real_of_int (crb p)\n[PROOF STEP]\nunfolding crb_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\poly p x = 0; p \\ 0\\ \\ \\x\\ \\ 1 + max_list_non_empty (map (\\i. \\coeff p i\\) [0..lead_coeff p\\) \\ x < real_of_int \\2 + max_list_non_empty (map (\\i. norm (coeff p i)) [0..\n[PROOF STEP]\napply (auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\poly p x = 0; p \\ 0\\ \\ \\x\\ \\ 1 + max_list_non_empty (map (\\i. \\coeff p i\\) [0..lead_coeff p\\) \\ x < real_of_int \\2 + max_list_non_empty (map (\\i. \\coeff p i\\) [0..lead_coeff p\\\\\n[PROOF STEP]\nusing p x\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ 0\npoly p x = 0\n\ngoal (1 subgoal):\n 1. (\\poly p x = 0; p \\ 0\\ \\ \\x\\ \\ 1 + max_list_non_empty (map (\\i. \\coeff p i\\) [0..lead_coeff p\\) \\ x < real_of_int \\2 + max_list_non_empty (map (\\i. \\coeff p i\\) [0..lead_coeff p\\\\\n[PROOF STEP]\nby linarith", "meta": {"llama_tokens": 968, "file": "BenOr_Kozen_Reif_BKR_Decision", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.877476784277755, "lm_q2_score": 0.8397339616560073, "lm_q1q2_score": 0.736847056322733}} {"text": "[STATEMENT]\nlemma tendsto_of_int_floor:\n fixes f::\"'a \\ 'b::{order_topology,floor_ceiling}\"\n assumes \"(f \\ l) F\"\n and \"l \\ \\\"\n shows \"((\\x. of_int (floor (f x)) :: 'c::{ring_1,topological_space}) \\ of_int (floor l)) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. of_int \\f x\\) \\ of_int \\l\\) F\n[PROOF STEP]\nusing eventually_floor_eq[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in F. \\f x\\ = \\l\\\n\ngoal (1 subgoal):\n 1. ((\\x. of_int \\f x\\) \\ of_int \\l\\) F\n[PROOF STEP]\nby (simp add: eventually_mono topological_tendstoI)", "meta": {"llama_tokens": 313, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026641072387, "lm_q2_score": 0.803173801068221, "lm_q1q2_score": 0.7367534674610164}} {"text": "[STATEMENT]\nlemma tendsto_of_int_floor:\n fixes f::\"'a \\ 'b::{order_topology,floor_ceiling}\"\n assumes \"(f \\ l) F\"\n and \"l \\ \\\"\n shows \"((\\x. of_int (floor (f x)) :: 'c::{ring_1,topological_space}) \\ of_int (floor l)) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. of_int \\f x\\) \\ of_int \\l\\) F\n[PROOF STEP]\nusing eventually_floor_eq[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in F. \\f x\\ = \\l\\\n\ngoal (1 subgoal):\n 1. ((\\x. of_int \\f x\\) \\ of_int \\l\\) F\n[PROOF STEP]\nby (simp add: eventually_mono topological_tendstoI)", "meta": {"llama_tokens": 313, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026482819238, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7367534525896852}} {"text": "[STATEMENT]\nlemma ceiling_log_nat_eq_powr_iff: fixes b n k :: nat\n shows \"\\ b \\ 2; k > 0 \\ \\\n ceiling (log b (real k)) = int n + 1 \\ (b^n < k \\ k \\ b^(n+1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\2 \\ b; 0 < k\\ \\ (\\log (real b) (real k)\\ = int n + 1) = (b ^ n < k \\ k \\ b ^ (n + 1))\n[PROOF STEP]\nusing ceiling_log_eq_powr_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < ?x; 1 < ?b\\ \\ (\\log ?b ?x\\ = int ?k + 1) = (?b powr real ?k < ?x \\ ?x \\ ?b powr real (?k + 1))\n\ngoal (1 subgoal):\n 1. \\2 \\ b; 0 < k\\ \\ (\\log (real b) (real k)\\ = int n + 1) = (b ^ n < k \\ k \\ b ^ (n + 1))\n[PROOF STEP]\nby (auto simp: powr_add powr_realpow of_nat_power[symmetric] of_nat_mult[symmetric] ac_simps\n simp del: of_nat_power of_nat_mult)", "meta": {"llama_tokens": 443, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026550642018, "lm_q2_score": 0.8031737892899221, "lm_q1q2_score": 0.7367534493936213}} {"text": "[STATEMENT]\ntheorem FreeGroup_universal_property:\n fixes f :: \"'a \\ 'b::group_add\"\n shows \"\\!T::'a freeword\\'b. (\\s\\S. T (Abs_freeletter s) = f s) \\\n GroupHom (FreeGroup S) T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\!T. (\\s\\S. T (Abs_freeletter s) = f s) \\ GroupHom (FreeGroup S) T\n[PROOF STEP]\nproof (rule ex1I, rule conjI)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\s\\S. ?a (Abs_freeletter s) = f s\n 2. GroupHom (FreeGroup S) ?a\n 3. \\T. (\\s\\S. T (Abs_freeletter s) = f s) \\ GroupHom (FreeGroup S) T \\ T = ?a\n[PROOF STEP]\nshow \"\\s\\S. res_freeword_funlift f S (Abs_freeletter s) = f s\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\s\\S. res_freeword_funlift f S (Abs_freeletter s) = f s\n[PROOF STEP]\nusing Abs_freeletter_in_FreeGroup_iff[of _ S] freeword_funlift_Abs_freeletter\n[PROOF STATE]\nproof (prove)\nusing this:\n(Abs_freeletter ?s \\ FreeGroup S) = (?s \\ S)\nfreeword_funlift ?f (Abs_freeletter ?s) = ?f ?s\n\ngoal (1 subgoal):\n 1. \\s\\S. res_freeword_funlift f S (Abs_freeletter s) = f s\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\s\\S. res_freeword_funlift f S (Abs_freeletter s) = f s\n\ngoal (2 subgoals):\n 1. GroupHom (FreeGroup S) (res_freeword_funlift f S)\n 2. \\T. (\\s\\S. T (Abs_freeletter s) = f s) \\ GroupHom (FreeGroup S) T \\ T = res_freeword_funlift f S\n[PROOF STEP]\nshow \"\\T. (\\s\\S. T (Abs_freeletter s) = f s) \\\n GroupHom (FreeGroup S) T \\\n T = restrict0 (freeword_funlift f) (FreeGroup S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\T. (\\s\\S. T (Abs_freeletter s) = f s) \\ GroupHom (FreeGroup S) T \\ T = res_freeword_funlift f S\n[PROOF STEP]\nusing uniqueness_of_restricted_lift\n[PROOF STATE]\nproof (prove)\nusing this:\n\\GroupHom (FreeGroup ?S) ?T; \\s\\?S. ?T (Abs_freeletter s) = ?f s\\ \\ ?T = res_freeword_funlift ?f ?S\n\ngoal (1 subgoal):\n 1. \\T. (\\s\\S. T (Abs_freeletter s) = f s) \\ GroupHom (FreeGroup S) T \\ T = res_freeword_funlift f S\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\s\\S. ?T (Abs_freeletter s) = f s) \\ GroupHom (FreeGroup S) ?T \\ ?T = res_freeword_funlift f S\n\ngoal (1 subgoal):\n 1. GroupHom (FreeGroup S) (res_freeword_funlift f S)\n[PROOF STEP]\nqed (rule hom_restrict0_freeword_funlift)", "meta": {"llama_tokens": 1131, "file": "Buildings_Algebra", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267864276107, "lm_q2_score": 0.8479677545357569, "lm_q1q2_score": 0.7367370991675387}} {"text": "[STATEMENT]\nlemma has_vector_derivative_scaleR[derivative_intros]:\n \"(f has_field_derivative f') (at x within s) \\ (g has_vector_derivative g') (at x within s) \\\n ((\\x. f x *\\<^sub>R g x) has_vector_derivative (f x *\\<^sub>R g' + f' *\\<^sub>R g x)) (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_real_derivative f') (at x within s); (g has_vector_derivative g') (at x within s)\\ \\ ((\\x. f x *\\<^sub>R g x) has_vector_derivative f x *\\<^sub>R g' + f' *\\<^sub>R g x) (at x within s)\n[PROOF STEP]\nunfolding has_real_derivative_iff_has_vector_derivative\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_vector_derivative f') (at x within s); (g has_vector_derivative g') (at x within s)\\ \\ ((\\x. f x *\\<^sub>R g x) has_vector_derivative f x *\\<^sub>R g' + f' *\\<^sub>R g x) (at x within s)\n[PROOF STEP]\nby (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_scaleR])", "meta": {"llama_tokens": 424, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887588023318195, "lm_q2_score": 0.82893881677331, "lm_q1q2_score": 0.7367266700018026}} {"text": "[STATEMENT]\nlemma has_vector_derivative_scaleR[derivative_intros]:\n \"(f has_field_derivative f') (at x within s) \\ (g has_vector_derivative g') (at x within s) \\\n ((\\x. f x *\\<^sub>R g x) has_vector_derivative (f x *\\<^sub>R g' + f' *\\<^sub>R g x)) (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_real_derivative f') (at x within s); (g has_vector_derivative g') (at x within s)\\ \\ ((\\x. f x *\\<^sub>R g x) has_vector_derivative f x *\\<^sub>R g' + f' *\\<^sub>R g x) (at x within s)\n[PROOF STEP]\nunfolding has_real_derivative_iff_has_vector_derivative\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_vector_derivative f') (at x within s); (g has_vector_derivative g') (at x within s)\\ \\ ((\\x. f x *\\<^sub>R g x) has_vector_derivative f x *\\<^sub>R g' + f' *\\<^sub>R g x) (at x within s)\n[PROOF STEP]\nby (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_scaleR])", "meta": {"llama_tokens": 424, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887588023318195, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.736726662490107}} {"text": "[STATEMENT]\nlemma has_vector_derivative_scaleR[derivative_intros]:\n \"(f has_field_derivative f') (at x within s) \\ (g has_vector_derivative g') (at x within s) \\\n ((\\x. f x *\\<^sub>R g x) has_vector_derivative (f x *\\<^sub>R g' + f' *\\<^sub>R g x)) (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_real_derivative f') (at x within s); (g has_vector_derivative g') (at x within s)\\ \\ ((\\x. f x *\\<^sub>R g x) has_vector_derivative f x *\\<^sub>R g' + f' *\\<^sub>R g x) (at x within s)\n[PROOF STEP]\nunfolding has_real_derivative_iff_has_vector_derivative\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_vector_derivative f') (at x within s); (g has_vector_derivative g') (at x within s)\\ \\ ((\\x. f x *\\<^sub>R g x) has_vector_derivative f x *\\<^sub>R g' + f' *\\<^sub>R g x) (at x within s)\n[PROOF STEP]\nby (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_scaleR])", "meta": {"llama_tokens": 424, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887588023318195, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.736726662490107}} {"text": "[STATEMENT]\nlemma card_eq_1_singleton:\n assumes \"card A = 1\"\n obtains x where \"A = {x}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. A = {x} \\ thesis) \\ thesis\n[PROOF STEP]\nusing assms[simplified]\n[PROOF STATE]\nproof (prove)\nusing this:\ncard A = Suc 0\n\ngoal (1 subgoal):\n 1. (\\x. A = {x} \\ thesis) \\ thesis\n[PROOF STEP]\nby - (drule card_eq_SucD, auto)", "meta": {"llama_tokens": 181, "file": "Formal_SSA_FormalSSA_Misc", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.8289388104343892, "lm_q1q2_score": 0.7367266570407403}} {"text": "[STATEMENT]\ntheorem effective_matrix_Tensor_elements2: \n fixes M1 M2 \n assumes \"mat (row_length M1) (length M1) M1\"\n and \"mat (row_length M2) (length M2) M2\"\n shows\n \"(\\i <((row_length M1)*(row_length M2)).\n \\j < ((length M1)*(length M2))\n .((M1 \\ M2)!j!i) = f (M1!(j div (length M2))!(i div (row_length M2))) \n (M2!(j mod length M2)!(i mod (row_length M2))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ij M2) ! j ! i = M1 ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)\n[PROOF STEP]\nusing matrix_Tensor_elements assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. (i < row_length ?M1.0 * row_length ?M2.0 \\ j < length ?M1.0 * length ?M2.0) \\ mat (row_length ?M1.0) (length ?M1.0) ?M1.0 \\ mat (row_length ?M2.0) (length ?M2.0) ?M2.0 \\ (?M1.0 \\ ?M2.0) ! j ! i = ?M1.0 ! (j div length ?M2.0) ! (i div row_length ?M2.0) * ?M2.0 ! (j mod length ?M2.0) ! (i mod row_length ?M2.0)\nmat (row_length M1) (length M1) M1\nmat (row_length M2) (length M2) M2\n\ngoal (1 subgoal):\n 1. \\ij M2) ! j ! i = M1 ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 634, "file": "Matrix_Tensor_Matrix_Tensor", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392878563336, "lm_q2_score": 0.8333245953120233, "lm_q1q2_score": 0.7366916817928084}} {"text": "[STATEMENT]\nlemma SA_nat_pow_closed:\n assumes \"is_semialg_function n f\"\n shows \"f [^]\\<^bsub>SA n\\<^esub> (k::nat) \\ carrier (SA n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f [^]\\<^bsub>SA n\\<^esub> k \\ carrier (SA n)\n[PROOF STEP]\napply(induction k)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. f [^]\\<^bsub>SA n\\<^esub> 0 \\ carrier (SA n)\n 2. \\k. f [^]\\<^bsub>SA n\\<^esub> k \\ carrier (SA n) \\ f [^]\\<^bsub>SA n\\<^esub> Suc k \\ carrier (SA n)\n[PROOF STEP]\nusing nat_pow_def[of \"SA n\" f ]\n[PROOF STATE]\nproof (prove)\nusing this:\nf [^]\\<^bsub>SA n\\<^esub> ?n = rec_nat \\\\<^bsub>SA n\\<^esub> (\\u b. b \\\\<^bsub>SA n\\<^esub> f) ?n\n\ngoal (2 subgoals):\n 1. f [^]\\<^bsub>SA n\\<^esub> 0 \\ carrier (SA n)\n 2. \\k. f [^]\\<^bsub>SA n\\<^esub> k \\ carrier (SA n) \\ f [^]\\<^bsub>SA n\\<^esub> Suc k \\ carrier (SA n)\n[PROOF STEP]\napply (metis Group.nat_pow_0 monoid.one_closed SA_is_monoid)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\k. f [^]\\<^bsub>SA n\\<^esub> k \\ carrier (SA n) \\ f [^]\\<^bsub>SA n\\<^esub> Suc k \\ carrier (SA n)\n[PROOF STEP]\nby (metis Group.nat_pow_Suc SA_car assms(1) assms SA_mult_closed semialg_functions_memE(1))", "meta": {"llama_tokens": 595, "file": "Padic_Field_Padic_Semialgebraic_Function_Ring", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392909114835, "lm_q2_score": 0.8333245911726382, "lm_q1q2_score": 0.736691680679361}} {"text": "[STATEMENT]\nlemma card_lists_distinct_length_eq:\n assumes \"finite A\" \"k \\ card A\"\n shows \"card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1 .. card A}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nk \\ card A\n\ngoal (1 subgoal):\n 1. card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}\n[PROOF STEP]\nproof (induct k)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\finite A; 0 \\ card A\\ \\ card {xs. length xs = 0 \\ distinct xs \\ set xs \\ A} = \\{card A - 0 + 1..card A}\n 2. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nfinite A\n0 \\ card A\n\ngoal (2 subgoals):\n 1. \\finite A; 0 \\ card A\\ \\ card {xs. length xs = 0 \\ distinct xs \\ set xs \\ A} = \\{card A - 0 + 1..card A}\n 2. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\n0 \\ card A\n[PROOF STEP]\nhave \"{xs. length xs = 0 \\ distinct xs \\ set xs \\ A} = {[]}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n0 \\ card A\n\ngoal (1 subgoal):\n 1. {xs. length xs = 0 \\ distinct xs \\ set xs \\ A} = {[]}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{xs. length xs = 0 \\ distinct xs \\ set xs \\ A} = {[]}\n\ngoal (2 subgoals):\n 1. \\finite A; 0 \\ card A\\ \\ card {xs. length xs = 0 \\ distinct xs \\ set xs \\ A} = \\{card A - 0 + 1..card A}\n 2. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{xs. length xs = 0 \\ distinct xs \\ set xs \\ A} = {[]}\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n{xs. length xs = 0 \\ distinct xs \\ set xs \\ A} = {[]}\n\ngoal (1 subgoal):\n 1. card {xs. length xs = 0 \\ distinct xs \\ set xs \\ A} = \\{card A - 0 + 1..card A}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard {xs. length xs = 0 \\ distinct xs \\ set xs \\ A} = \\{card A - 0 + 1..card A}\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\ncase (Suc k)\n[PROOF STATE]\nproof (state)\nthis:\n\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}\nfinite A\nSuc k \\ card A\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nlet \"?k_list\" = \"\\k xs. length xs = k \\ distinct xs \\ set xs \\ A\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nhave inj_Cons: \"\\A. inj_on (\\(xs, n). n # xs) A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A. inj_on (\\(xs, n). n # xs) A\n[PROOF STEP]\nby (rule inj_onI) auto\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\(xs, n). n # xs) ?A\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nfrom Suc\n[PROOF STATE]\nproof (chain)\npicking this:\n\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}\nfinite A\nSuc k \\ card A\n[PROOF STEP]\nhave \"k \\ card A\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}\nfinite A\nSuc k \\ card A\n\ngoal (1 subgoal):\n 1. k \\ card A\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nk \\ card A\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nk \\ card A\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nnote \\finite A\\\n[PROOF STATE]\nproof (state)\nthis:\nfinite A\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite A\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nhave \"finite {xs. ?k_list k xs}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {xs. length xs = k \\ distinct xs \\ set xs \\ A}\n[PROOF STEP]\nby (rule finite_subset) (use finite_lists_length_eq[OF \\finite A\\, of k] in auto)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {xs. length xs = k \\ distinct xs \\ set xs \\ A}\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite {xs. length xs = k \\ distinct xs \\ set xs \\ A}\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nhave \"\\i j. i \\ j \\ {i} \\ (A - set i) \\ {j} \\ (A - set j) = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. i \\ j \\ {i} \\ (A - set i) \\ {j} \\ (A - set j) = {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n?i1 \\ ?j1 \\ {?i1} \\ (A - set ?i1) \\ {?j1} \\ (A - set ?j1) = {}\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n?i1 \\ ?j1 \\ {?i1} \\ (A - set ?i1) \\ {?j1} \\ (A - set ?j1) = {}\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nhave \"\\i. i \\ {xs. ?k_list k xs} \\ card (A - set i) = card A - k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i \\ {xs. length xs = k \\ distinct xs \\ set xs \\ A} \\ card (A - set i) = card A - k\n[PROOF STEP]\nby (simp add: card_Diff_subset distinct_card)\n[PROOF STATE]\nproof (state)\nthis:\n?i1 \\ {xs. length xs = k \\ distinct xs \\ set xs \\ A} \\ card (A - set ?i1) = card A - k\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n?i1 \\ {xs. length xs = k \\ distinct xs \\ set xs \\ A} \\ card (A - set ?i1) = card A - k\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nhave \"{xs. ?k_list (Suc k) xs} =\n (\\(xs, n). n#xs) ` \\((\\xs. {xs} \\ (A - set xs)) ` {xs. ?k_list k xs})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = (\\(xs, n). n # xs) ` (\\xs\\{xs. length xs = k \\ distinct xs \\ set xs \\ A}. {xs} \\ (A - set xs))\n[PROOF STEP]\nby (auto simp: length_Suc_conv)\n[PROOF STATE]\nproof (state)\nthis:\n{xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = (\\(xs, n). n # xs) ` (\\xs\\{xs. length xs = k \\ distinct xs \\ set xs \\ A}. {xs} \\ (A - set xs))\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n{xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = (\\(xs, n). n # xs) ` (\\xs\\{xs. length xs = k \\ distinct xs \\ set xs \\ A}. {xs} \\ (A - set xs))\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nhave \"Suc (card A - Suc k) = card A - k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc (card A - Suc k) = card A - k\n[PROOF STEP]\nusing Suc.prems\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nSuc k \\ card A\n\ngoal (1 subgoal):\n 1. Suc (card A - Suc k) = card A - k\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nSuc (card A - Suc k) = card A - k\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nSuc (card A - Suc k) = card A - k\n[PROOF STEP]\nhave \"(card A - k) * \\{Suc (card A - k)..card A} = \\{Suc (card A - Suc k)..card A}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc (card A - Suc k) = card A - k\n\ngoal (1 subgoal):\n 1. (card A - k) * \\{Suc (card A - k)..card A} = \\{Suc (card A - Suc k)..card A}\n[PROOF STEP]\nby (subst prod.insert[symmetric]) (simp add: atLeastAtMost_insertL)+\n[PROOF STATE]\nproof (state)\nthis:\n(card A - k) * \\{Suc (card A - k)..card A} = \\{Suc (card A - Suc k)..card A}\n\ngoal (1 subgoal):\n 1. \\k. \\\\finite A; k \\ card A\\ \\ card {xs. length xs = k \\ distinct xs \\ set xs \\ A} = \\{card A - k + 1..card A}; finite A; Suc k \\ card A\\ \\ card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nk \\ card A\nfinite A\nfinite {xs. length xs = k \\ distinct xs \\ set xs \\ A}\n?i1 \\ ?j1 \\ {?i1} \\ (A - set ?i1) \\ {?j1} \\ (A - set ?j1) = {}\n?i1 \\ {xs. length xs = k \\ distinct xs \\ set xs \\ A} \\ card (A - set ?i1) = card A - k\n{xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = (\\(xs, n). n # xs) ` (\\xs\\{xs. length xs = k \\ distinct xs \\ set xs \\ A}. {xs} \\ (A - set xs))\n(card A - k) * \\{Suc (card A - k)..card A} = \\{Suc (card A - Suc k)..card A}\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ card A\nfinite A\nfinite {xs. length xs = k \\ distinct xs \\ set xs \\ A}\n?i1 \\ ?j1 \\ {?i1} \\ (A - set ?i1) \\ {?j1} \\ (A - set ?j1) = {}\n?i1 \\ {xs. length xs = k \\ distinct xs \\ set xs \\ A} \\ card (A - set ?i1) = card A - k\n{xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = (\\(xs, n). n # xs) ` (\\xs\\{xs. length xs = k \\ distinct xs \\ set xs \\ A}. {xs} \\ (A - set xs))\n(card A - k) * \\{Suc (card A - k)..card A} = \\{Suc (card A - Suc k)..card A}\n\ngoal (1 subgoal):\n 1. card {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n[PROOF STEP]\nby (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\ncard {xs. length xs = Suc k \\ distinct xs \\ set xs \\ A} = \\{card A - Suc k + 1..card A}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 7074, "file": null, "length": 42, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094003735663, "lm_q2_score": 0.8244619285331332, "lm_q1q2_score": 0.7364995910087673}} {"text": "[STATEMENT]\nlemma set_rel_alt: \"\\R\\set_rel = {(A,B). A \\ R\\``B \\ B \\ R``A}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\R\\set_rel = {(A, B). A \\ R\\ `` B \\ B \\ R `` A}\n[PROOF STEP]\nunfolding set_rel_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {(A, B). (\\x\\A. \\y\\B. (x, y) \\ R) \\ (\\y\\B. \\x\\A. (x, y) \\ R)} = {(A, B). A \\ R\\ `` B \\ B \\ R `` A}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 260, "file": "Automatic_Refinement_Parametricity_Relators", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9086179068309441, "lm_q2_score": 0.8104789063814617, "lm_q1q2_score": 0.7364156474469564}} {"text": "[STATEMENT]\nlemma index_component_mult:\nassumes \"i < dim_vec v\" \"i < dim_vec w\"\nshows \"component_mult v w $ i = v $ i * w $ i\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. component_mult v w $ i = v $ i * w $ i\n[PROOF STEP]\nunfolding component_mult_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec (min (dim_vec v) (dim_vec w)) (\\i. v $ i * w $ i) $ i = v $ i * w $ i\n[PROOF STEP]\nusing assms index_vec\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_vec v\ni < dim_vec w\n?i < ?n \\ vec ?n ?f $ ?i = ?f ?i\n\ngoal (1 subgoal):\n 1. vec (min (dim_vec v) (dim_vec w)) (\\i. v $ i * w $ i) $ i = v $ i * w $ i\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 298, "file": "Jordan_Normal_Form_Matrix", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942171172603, "lm_q2_score": 0.815232489352, "lm_q1q2_score": 0.7363947932377701}} {"text": "[STATEMENT]\nlemma filterlim_at_bot_linear_iff:\n fixes f::\"'a::linordered_field \\ 'b\"\n assumes \"c\\0\"\n shows \"(LIM x at_bot. f (x * c + b) :> F2) \\ (if c>0 then (LIM x at_bot. f x :> F2) \n else (LIM x at_top. f x :> F2)) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (LIM x at_bot. f (x * c + b) :> F2) = (if (0::'a) < c then filterlim f F2 at_bot else filterlim f F2 at_top)\n[PROOF STEP]\nunfolding filterlim_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (filtermap (\\x. f (x * c + b)) at_bot \\ F2) = (if (0::'a) < c then filtermap f at_bot \\ F2 else filtermap f at_top \\ F2)\n[PROOF STEP]\napply (subst filtermap_filtermap[of f \"\\x. x * c + b\",symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (filtermap f (filtermap (\\x. x * c + b) at_bot) \\ F2) = (if (0::'a) < c then filtermap f at_bot \\ F2 else filtermap f at_top \\ F2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'a)\n\ngoal (1 subgoal):\n 1. (filtermap f (filtermap (\\x. x * c + b) at_bot) \\ F2) = (if (0::'a) < c then filtermap f at_bot \\ F2 else filtermap f at_top \\ F2)\n[PROOF STEP]\nby (auto simp add:filtermap_at_bot_linear_eq)", "meta": {"llama_tokens": 558, "file": "Winding_Number_Eval_Missing_Topology", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942067038785, "lm_q2_score": 0.8152324960856175, "lm_q1q2_score": 0.7363947908308807}} {"text": "[STATEMENT]\nlemma matrix_mult_index: \n assumes \"m1 \\ []\"\n and wf1: \"mat nr n m1\"\n and wf2: \"mat n nc m2\"\n and i: \"i < nr\"\n and j: \"j < nc\"\n shows \"matrix_mult m1 m2 ! j ! i = scalar_product (row m1 i) (col m2 j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (m1 \\ m2) ! j ! i = scalar_product (row m1 i) (col m2 j)\n[PROOF STEP]\nusing matrix_index unique_row_col assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\mat (row_length ?m1.0) ?n ?m1.0; mat ?n ?nc ?m2.0; ?i < row_length ?m1.0; ?j < ?nc\\ \\ (?m1.0 \\ ?m2.0) ! ?j ! ?i = scalar_product (row ?m1.0 ?i) (col ?m2.0 ?j)\n\\mat ?nr1.0 ?nc1.0 ?M; mat ?nr2.0 ?nc2.0 ?M; ?M \\ []\\ \\ ?nr1.0 = ?nr2.0\n\\mat ?nr1.0 ?nc1.0 ?M; mat ?nr2.0 ?nc2.0 ?M; ?M \\ []\\ \\ ?nc1.0 = ?nc2.0\nm1 \\ []\nmat nr n m1\nmat n nc m2\ni < nr\nj < nc\n\ngoal (1 subgoal):\n 1. (m1 \\ m2) ! j ! i = scalar_product (row m1 i) (col m2 j)\n[PROOF STEP]\nby (metis matrix_row_length)", "meta": {"llama_tokens": 513, "file": "Matrix_Tensor_Matrix_Tensor", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942041005328, "lm_q2_score": 0.8152324871074607, "lm_q1q2_score": 0.7363947805986316}} {"text": "[STATEMENT]\nlemma subgroup_inter:\n assumes \"subgroup A\" and \"subgroup B\"\n shows \"subgroup (A \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. subgroup (A \\ B)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsubgroup A\nsubgroup B\n\ngoal (1 subgoal):\n 1. subgroup (A \\ B)\n[PROOF STEP]\nunfolding subgroup_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) \\ A \\ (\\a\\A. \\b\\A. a + b \\ A) \\ (\\a\\A. - a \\ A)\n(0::'a) \\ B \\ (\\a\\B. \\b\\B. a + b \\ B) \\ (\\a\\B. - a \\ B)\n\ngoal (1 subgoal):\n 1. (0::'a) \\ A \\ B \\ (\\a\\A \\ B. \\b\\A \\ B. a + b \\ A \\ B) \\ (\\a\\A \\ B. - a \\ A \\ B)\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 382, "file": "Echelon_Form_Rings2", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467675095292, "lm_q2_score": 0.8376199694135333, "lm_q1q2_score": 0.7363908885113386}} {"text": "[STATEMENT]\nlemma rank_argument: \n fixes M :: \"('c :: {conjugatable_ordered_field}) mat\"\n assumes \"M \\ carrier_mat x y\"\n assumes \"vec_space.rank x (M* M\\<^sup>T) = x\"\n shows \"x \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nlet ?B = \"(M * M\\<^sup>T)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nhave Mt_car: \"M\\<^sup>T \\ carrier_mat y x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M\\<^sup>T \\ carrier_mat y x\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nM \\ carrier_mat x y\nrank x (M * M\\<^sup>T) = x\n\ngoal (1 subgoal):\n 1. M\\<^sup>T \\ carrier_mat y x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nM\\<^sup>T \\ carrier_mat y x\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nhave b_car: \"?B \\ carrier_mat x x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M * M\\<^sup>T \\ carrier_mat x x\n[PROOF STEP]\nusing transpose_carrier_mat assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(?A\\<^sup>T \\ carrier_mat ?nc ?nr) = (?A \\ carrier_mat ?nr ?nc)\nM \\ carrier_mat x y\nrank x (M * M\\<^sup>T) = x\n\ngoal (1 subgoal):\n 1. M * M\\<^sup>T \\ carrier_mat x x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nM * M\\<^sup>T \\ carrier_mat x x\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nM * M\\<^sup>T \\ carrier_mat x x\n[PROOF STEP]\nhave \"rank x ?B \\ min (rank x M) (rank y M\\<^sup>T)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nM * M\\<^sup>T \\ carrier_mat x x\n\ngoal (1 subgoal):\n 1. rank x (M * M\\<^sup>T) \\ min (rank x M) (rank y M\\<^sup>T)\n[PROOF STEP]\nusing rank_mat_mult_lt_min_rank_factor Mt_car b_car assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nM * M\\<^sup>T \\ carrier_mat x x\n\\?A \\ carrier_mat ?n ?m; ?B \\ carrier_mat ?m ?nc\\ \\ rank ?n (?A * ?B) \\ min (rank ?n ?A) (rank ?m ?B)\nM\\<^sup>T \\ carrier_mat y x\nM * M\\<^sup>T \\ carrier_mat x x\nM \\ carrier_mat x y\n\ngoal (1 subgoal):\n 1. rank x (M * M\\<^sup>T) \\ min (rank x M) (rank y M\\<^sup>T)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nrank x (M * M\\<^sup>T) \\ min (rank x M) (rank y M\\<^sup>T)\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nrank x (M * M\\<^sup>T) \\ min (rank x M) (rank y M\\<^sup>T)\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nusing le_trans vec_space.rank_le_nc\n[PROOF STATE]\nproof (prove)\nusing this:\nrank x (M * M\\<^sup>T) \\ min (rank x M) (rank y M\\<^sup>T)\n\\?i \\ ?j; ?j \\ ?k\\ \\ ?i \\ ?k\n?A \\ carrier_mat ?n ?nc \\ rank ?n ?A \\ ?nc\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nby (metis assms(1) assms(2) min.bounded_iff)\n[PROOF STATE]\nproof (state)\nthis:\nx \\ y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1398, "file": "Fishers_Inequality_Rank_Argument_General", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467611766711, "lm_q2_score": 0.8376199552262966, "lm_q1q2_score": 0.736390870734147}} {"text": "[STATEMENT]\nlemma metric_bound_lemma: \"cmod (x - y) \\ \\Re x - Re y\\ + \\Im x - Im y\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (x - y) \\ \\Re x - Re y\\ + \\Im x - Im y\\\n[PROOF STEP]\nusing real_sqrt_sum_squares_triangle_ineq[of \"Re x - Re y\" 0 0 \"Im x - Im y\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt ((Re x - Re y + 0)\\<^sup>2 + (0 + (Im x - Im y))\\<^sup>2) \\ sqrt ((Re x - Re y)\\<^sup>2 + 0\\<^sup>2) + sqrt (0\\<^sup>2 + (Im x - Im y)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. cmod (x - y) \\ \\Re x - Re y\\ + \\Im x - Im y\\\n[PROOF STEP]\nunfolding cmod_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt ((Re x - Re y + 0)\\<^sup>2 + (0 + (Im x - Im y))\\<^sup>2) \\ sqrt ((Re x - Re y)\\<^sup>2 + 0\\<^sup>2) + sqrt (0\\<^sup>2 + (Im x - Im y)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. sqrt ((Re (x - y))\\<^sup>2 + (Im (x - y))\\<^sup>2) \\ \\Re x - Re y\\ + \\Im x - Im y\\\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 478, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9111797172476384, "lm_q2_score": 0.8080672089305841, "lm_q1q2_score": 0.736294450950458}} {"text": "[STATEMENT]\nlemma Ilsm_Ilam: \"$(I^{l}s)^{m}_n = (1+i)^n * $(I^{l}a)^{m}_n\"\n if \"l \\ 0\" \"m \\ 0\" for l n m :: nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. $(I^{l}s)^{m}_n = (1 + i) ^ n * $(I^{l}a)^{m}_n\n[PROOF STEP]\nunfolding acc_incr_def ann_incr_def v_pres_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\kreal (l * (k + 1)) / real m\\ / (real l * real m)) = (1 + i) ^ n * (\\kreal (l * (k + 1)) / real m\\ / (real l * real m))\n[PROOF STEP]\nusing v_futr_pos powr_realpow\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 + i\n0 < ?x \\ ?x .^ real ?n = ?x ^ ?n\n\ngoal (1 subgoal):\n 1. (\\kreal (l * (k + 1)) / real m\\ / (real l * real m)) = (1 + i) ^ n * (\\kreal (l * (k + 1)) / real m\\ / (real l * real m))\n[PROOF STEP]\napply (subst inverse_powr, simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < 1 + i; \\x n. 0 < x \\ x .^ real n = x ^ n\\ \\ (\\kreal (l * (k + 1)) / real m\\ / (real l * real m)) = (1 + i) ^ n * (\\kreal (l * (k + 1)) / real m\\ / (real l * real m))\n[PROOF STEP]\napply (subst sum_distrib_left)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < 1 + i; \\x n. 0 < x \\ x .^ real n = x ^ n\\ \\ (\\kreal (l * (k + 1)) / real m\\ / (real l * real m)) = (\\nareal (l * (na + 1)) / real m\\ / (real l * real m)))\n[PROOF STEP]\nby (subst minus_real_def, subst powr_add, subst times_divide_eq_right, subst mult.assoc, simp)", "meta": {"llama_tokens": 1025, "file": "Actuarial_Mathematics_Interest", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111797124237604, "lm_q2_score": 0.8080672089305841, "lm_q1q2_score": 0.7362944470524404}} {"text": "[STATEMENT]\nlemma measure_Union_AE:\n \"finite F \\ (\\S. S \\ F \\ S \\ fmeasurable M) \\ pairwise (\\S T. AE x in M. x \\ S \\ x \\ T) F \\\n measure M (\\F) = (\\S\\F. measure M S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite F; \\S. S \\ F \\ S \\ fmeasurable M; pairwise (\\S T. AE x in M. x \\ S \\ x \\ T) F\\ \\ Sigma_Algebra.measure M (\\ F) = sum (Sigma_Algebra.measure M) F\n[PROOF STEP]\nusing measure_UNION_AE[of F \"\\x. x\" M]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite F; \\i. i \\ F \\ i \\ fmeasurable M; pairwise (\\i j. AE x in M. x \\ i \\ x \\ j) F\\ \\ Sigma_Algebra.measure M (\\i\\F. i) = sum (Sigma_Algebra.measure M) F\n\ngoal (1 subgoal):\n 1. \\finite F; \\S. S \\ F \\ S \\ fmeasurable M; pairwise (\\S T. AE x in M. x \\ S \\ x \\ T) F\\ \\ Sigma_Algebra.measure M (\\ F) = sum (Sigma_Algebra.measure M) F\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 479, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297914570318, "lm_q2_score": 0.8175744850834648, "lm_q1q2_score": 0.7362501805528027}} {"text": "[STATEMENT]\nlemma rank_matrix_dim_range:\nassumes lf: \"linear ((*s)) ((*s)) f\"\nshows \"rank (matrix f::'a::{field}^'cols::{mod_type}^'rows::{mod_type}) = vec.dim (range f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank (matrix f) = vec.dim (range f)\n[PROOF STEP]\nunfolding rank_col_rank[of \"matrix f\"] col_rank_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (col_space (matrix f)) = vec.dim (range f)\n[PROOF STEP]\nunfolding col_space_eq'\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (range ((*v) (matrix f))) = vec.dim (range f)\n[PROOF STEP]\nusing matrix_works[OF lf]\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix f *v ?x = f ?x\n\ngoal (1 subgoal):\n 1. vec.dim (range ((*v) (matrix f))) = vec.dim (range f)\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 333, "file": "Gauss_Jordan_Linear_Maps", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297754396142, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.7362501554490477}} {"text": "[STATEMENT]\nlemma lowdim_subset_hyperplane:\n fixes S :: \"'a::euclidean_space set\"\n assumes d: \"dim S < DIM('a)\"\n shows \"\\a::'a. a \\ 0 \\ span S \\ {x. a \\ x = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n[PROOF STEP]\nassume \"span S = UNIV\"\n[PROOF STATE]\nproof (state)\nthis:\nspan S = UNIV\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nspan S = UNIV\n[PROOF STEP]\nhave \"dim (span S) = dim (UNIV :: ('a) set)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nspan S = UNIV\n\ngoal (1 subgoal):\n 1. dim (span S) = dim UNIV\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim (span S) = dim UNIV\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndim (span S) = dim UNIV\n[PROOF STEP]\nhave \"dim S = DIM('a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndim (span S) = dim UNIV\n\ngoal (1 subgoal):\n 1. dim S = DIM('a)\n[PROOF STEP]\nby (metis Euclidean_Space.dim_UNIV dim_span)\n[PROOF STATE]\nproof (state)\nthis:\ndim S = DIM('a)\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n[PROOF STEP]\nwith d\n[PROOF STATE]\nproof (chain)\npicking this:\ndim S < DIM('a)\ndim S = DIM('a)\n[PROOF STEP]\nhave False\n[PROOF STATE]\nproof (prove)\nusing this:\ndim S < DIM('a)\ndim S = DIM('a)\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\nspan S = UNIV \\ False\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nspan S = UNIV \\ False\n[PROOF STEP]\nhave th: \"span S \\ UNIV\"\n[PROOF STATE]\nproof (prove)\nusing this:\nspan S = UNIV \\ False\n\ngoal (1 subgoal):\n 1. span S \\ UNIV\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nspan S \\ UNIV\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n[PROOF STEP]\nfrom span_not_univ_subset_hyperplane[OF th]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\a. a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1486, "file": null, "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473846343394, "lm_q2_score": 0.8438950966654772, "lm_q1q2_score": 0.736169680481872}} {"text": "[STATEMENT]\nlemma select_index_suc_odd:\n fixes n k i:: nat\n assumes \"k \\ 2^n -1\" and \"select_index n i k\"\n shows \"select_index (Suc n) i (2*k+1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nhave \"((2*k+1) mod 2^(Suc n - i) \\ 2^(n - i)) = \n(((2*k+1) div 2) mod 2^(n - i) \\ 2^(n-1-i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n[PROOF STEP]\nhave \"2*k+1 < 2^(n + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * k + 1 < 2 ^ (n + 1)\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ 2 ^ n - 1\n\ngoal (1 subgoal):\n 1. 2 * k + 1 < 2 ^ (n + 1)\n[PROOF STEP]\nby (smt Suc_1 Suc_eq_plus1 Suc_le_lessD Suc_le_mono add_Suc_right distrib_left_numeral le_add_diff_inverse mult_le_mono2 nat_mult_1_right one_le_numeral one_le_power plus_1_eq_Suc power_add power_one_right)\n[PROOF STATE]\nproof (state)\nthis:\n2 * k + 1 < 2 ^ (n + 1)\n\ngoal (1 subgoal):\n 1. (2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n2 * k + 1 < 2 ^ (n + 1)\n\ngoal (1 subgoal):\n 1. (2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n[PROOF STEP]\nhave \"i < n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i < n\n[PROOF STEP]\nusing assms(2) select_index_def\n[PROOF STATE]\nproof (prove)\nusing this:\nselect_index n i k\nselect_index ?n ?i ?j \\ ?i \\ ?n - 1 \\ ?j \\ 2 ^ ?n - 1 \\ 2 ^ (?n - 1 - ?i) \\ ?j mod 2 ^ (?n - ?i)\n\ngoal (1 subgoal):\n 1. i < n\n[PROOF STEP]\nby (metis (no_types, opaque_lifting) add_cancel_left_left add_diff_inverse_nat diff_le_self div_by_1 le_antisym less_le_trans less_one mod_div_trivial not_le power_0)\n[PROOF STATE]\nproof (state)\nthis:\ni < n\n\ngoal (1 subgoal):\n 1. (2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n2 * k + 1 < 2 ^ (n + 1)\ni < n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * k + 1 < 2 ^ (n + 1)\ni < n\n\ngoal (1 subgoal):\n 1. (2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n[PROOF STEP]\nusing select_index_div_2[of \"2*k+1\" \"n\" i] select_index_def\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * k + 1 < 2 ^ (n + 1)\ni < n\n\\2 * k + 1 < 2 ^ (n + 1); i < n\\ \\ select_index n i ((2 * k + 1) div 2) = select_index (n + 1) i (2 * k + 1)\nselect_index ?n ?i ?j \\ ?i \\ ?n - 1 \\ ?j \\ 2 ^ ?n - 1 \\ 2 ^ (?n - 1 - ?i) \\ ?j mod 2 ^ (?n - ?i)\n\ngoal (1 subgoal):\n 1. (2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n[PROOF STEP]\nby (metis Nat.le_diff_conv2 Suc_eq_plus1 Suc_leI assms(2) diff_Suc_1 less_imp_le less_power_add_imp_div_less one_le_numeral one_le_power power_one_right)\n[PROOF STATE]\nproof (state)\nthis:\n(2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n\ngoal (1 subgoal):\n 1. select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n\ngoal (1 subgoal):\n 1. select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nhave \"\\ = (k mod 2^(n - i) \\ 2^(n-1-i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)) = (2 ^ (n - 1 - i) \\ k mod 2 ^ (n - i))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)) = (2 ^ (n - 1 - i) \\ k mod 2 ^ (n - i))\n\ngoal (1 subgoal):\n 1. select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n(2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)) = (2 ^ (n - 1 - i) \\ k mod 2 ^ (n - i))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\n(2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)) = (2 ^ (n - 1 - i) \\ k mod 2 ^ (n - i))\n\ngoal (1 subgoal):\n 1. select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\(2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)); (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)) = (2 ^ (n - 1 - i) \\ k mod 2 ^ (n - i))\\ \\ select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nhave \"i \\ Suc n -1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i \\ Suc n - 1\n[PROOF STEP]\nusing assms(2) select_index_def\n[PROOF STATE]\nproof (prove)\nusing this:\nselect_index n i k\nselect_index ?n ?i ?j \\ ?i \\ ?n - 1 \\ ?j \\ 2 ^ ?n - 1 \\ 2 ^ (?n - 1 - ?i) \\ ?j mod 2 ^ (?n - ?i)\n\ngoal (1 subgoal):\n 1. i \\ Suc n - 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ni \\ Suc n - 1\n\ngoal (1 subgoal):\n 1. \\(2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)); (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)) = (2 ^ (n - 1 - i) \\ k mod 2 ^ (n - i))\\ \\ select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ni \\ Suc n - 1\n\ngoal (1 subgoal):\n 1. \\(2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)); (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)) = (2 ^ (n - 1 - i) \\ k mod 2 ^ (n - i))\\ \\ select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nhave \"2*k+1 \\ 2^(Suc n)-1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * k + 1 \\ 2 ^ Suc n - 1\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ 2 ^ n - 1\n\ngoal (1 subgoal):\n 1. 2 * k + 1 \\ 2 ^ Suc n - 1\n[PROOF STEP]\nby (smt Suc_diff_1 Suc_eq_plus1 add_diff_cancel_right' diff_Suc_diff_eq2 diff_diff_left diff_is_0_eq diff_mult_distrib2 le_add2 mult_2 mult_Suc_right plus_1_eq_Suc pos2 power_Suc zero_less_power)\n[PROOF STATE]\nproof (state)\nthis:\n2 * k + 1 \\ 2 ^ Suc n - 1\n\ngoal (1 subgoal):\n 1. \\(2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)); (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)) = (2 ^ (n - 1 - i) \\ k mod 2 ^ (n - i))\\ \\ select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ni \\ Suc n - 1\n2 * k + 1 \\ 2 ^ Suc n - 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ni \\ Suc n - 1\n2 * k + 1 \\ 2 ^ Suc n - 1\n\ngoal (1 subgoal):\n 1. select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nusing select_index_def\n[PROOF STATE]\nproof (prove)\nusing this:\ni \\ Suc n - 1\n2 * k + 1 \\ 2 ^ Suc n - 1\nselect_index ?n ?i ?j \\ ?i \\ ?n - 1 \\ ?j \\ 2 ^ ?n - 1 \\ 2 ^ (?n - 1 - ?i) \\ ?j mod 2 ^ (?n - ?i)\n\ngoal (1 subgoal):\n 1. select_index (Suc n) i (2 * k + 1)\n[PROOF STEP]\nby (metis \\(2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i)) = (2 ^ (n - 1 - i) \\ k mod 2 ^ (n - i))\\ \\(2 ^ (n - i) \\ (2 * k + 1) mod 2 ^ (Suc n - i)) = (2 ^ (n - 1 - i) \\ (2 * k + 1) div 2 mod 2 ^ (n - i))\\ assms(2) diff_Suc_1)\n[PROOF STATE]\nproof (state)\nthis:\nselect_index (Suc n) i (2 * k + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nselect_index (Suc n) i (2 * k + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4472, "file": "Isabelle_Marries_Dirac_Measurement", "length": 34, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.890294223211224, "lm_q2_score": 0.8267118026095992, "lm_q1q2_score": 0.7360167421238638}} {"text": "[STATEMENT]\nlemma adder_helper:\n assumes lw: \"0 < max (length w1) (length w2)\"\n shows \"((2::int) ^ (length w1 - 1)) + (2 ^ (length w2 - 1)) \\ 2 ^ max (length w1) (length w2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ max (length w1) (length w2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ max (length w1) (length w2)\n[PROOF STEP]\nhave \"((2::int) ^ (length w1 - 1)) + (2 ^ (length w2 - 1)) \\\n 2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1)\n[PROOF STEP]\nby (auto simp:max_def)\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1)\n\ngoal (1 subgoal):\n 1. 2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ max (length w1) (length w2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1)\n\ngoal (1 subgoal):\n 1. 2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ max (length w1) (length w2)\n[PROOF STEP]\nhave \"... = 2 ^ max (length w1) (length w2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1) = 2 ^ max (length w1) (length w2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1) = 2 ^ max (length w1) (length w2)\n[PROOF STEP]\nfrom lw\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < max (length w1) (length w2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < max (length w1) (length w2)\n\ngoal (1 subgoal):\n 1. 2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1) = 2 ^ max (length w1) (length w2)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < max (length w1) (length w2) \\ 2 * 2 ^ (max (length w1) (length w2) - Suc 0) = 2 ^ max (length w1) (length w2)\n[PROOF STEP]\napply (subst power_Suc [symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < max (length w1) (length w2) \\ 2 ^ Suc (max (length w1) (length w2) - Suc 0) = 2 ^ max (length w1) (length w2)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1) = 2 ^ max (length w1) (length w2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1) = 2 ^ max (length w1) (length w2)\n\ngoal (1 subgoal):\n 1. 2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ max (length w1) (length w2)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ max (length w1) (length w2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ max (length w1) (length w2)\n\ngoal (1 subgoal):\n 1. 2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ max (length w1) (length w2)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \\ 2 ^ max (length w1) (length w2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1741, "file": "RSAPSS_Word", "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8267117940706734, "lm_q1q2_score": 0.7360167321153104}} {"text": "[STATEMENT]\nlemma not_icard_seteq: \"\\(A::nat set) B. (A \\ B \\ icard B \\ icard A \\ \\ A = B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A B. A \\ B \\ icard B \\ icard A \\ A \\ B\n[PROOF STEP]\napply (rule_tac x=\"{1..}\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\B. {1..} \\ B \\ icard B \\ icard {1..} \\ {1..} \\ B\n[PROOF STEP]\napply (rule_tac x=\"{0..}\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {1..} \\ {0..} \\ icard {0..} \\ icard {1..} \\ {1..} \\ {0..}\n[PROOF STEP]\napply (fastforce simp add: infinite_atLeast)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 340, "file": "List-Infinite_CommonSet_InfiniteSet2", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8267117940706734, "lm_q1q2_score": 0.7360167321153104}} {"text": "[STATEMENT]\nlemma ennreal_SUP:\n \"\\ (SUP a\\A. ennreal (f a)) \\ \\; A \\ {} \\ \\ ennreal (SUP a\\A. f a) = (SUP a\\A. ennreal (f a))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(\\a\\A. ennreal (f a)) \\ \\; A \\ {}\\ \\ ennreal (\\ (f ` A)) = (\\a\\A. ennreal (f a))\n[PROOF STEP]\nusing ennreal_Sup[of \"f ` A\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\ (ennreal ` f ` A) \\ \\; f ` A \\ {}\\ \\ ennreal (\\ (f ` A)) = \\ (ennreal ` f ` A)\n\ngoal (1 subgoal):\n 1. \\(\\a\\A. ennreal (f a)) \\ \\; A \\ {}\\ \\ ennreal (\\ (f ` A)) = (\\a\\A. ennreal (f a))\n[PROOF STEP]\nby (auto simp add: image_comp)", "meta": {"llama_tokens": 419, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942290328345, "lm_q2_score": 0.8267117855317474, "lm_q1q2_score": 0.736016731732345}} {"text": "[STATEMENT]\nlemma card_bijections_range_permutation:\n assumes \"finite A\" \"finite B\"\n shows \"card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // range_permutation A B) = iverson (card A = card B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // range_permutation A B) = iverson (card A = card B)\n[PROOF STEP]\nusing assms card_bijections_range_permutation_eq_0 card_bijections_range_permutation_eq_1\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\ncard ?A \\ card ?B \\ card ({f \\ ?A \\\\<^sub>E ?B. bij_betw f ?A ?B} // range_permutation ?A ?B) = 0\n\\finite ?A; finite ?B; card ?A = card ?B\\ \\ card ({f \\ ?A \\\\<^sub>E ?B. bij_betw f ?A ?B} // range_permutation ?A ?B) = 1\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // range_permutation A B) = iverson (card A = card B)\n[PROOF STEP]\nunfolding iverson_def\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\ncard ?A \\ card ?B \\ card ({f \\ ?A \\\\<^sub>E ?B. bij_betw f ?A ?B} // range_permutation ?A ?B) = 0\n\\finite ?A; finite ?B; card ?A = card ?B\\ \\ card ({f \\ ?A \\\\<^sub>E ?B. bij_betw f ?A ?B} // range_permutation ?A ?B) = 1\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // range_permutation A B) = (if card A = card B then 1 else 0)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 623, "file": "Twelvefold_Way_Card_Bijections", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952838963489, "lm_q2_score": 0.8198933359135361, "lm_q1q2_score": 0.7360143809476263}} {"text": "[STATEMENT]\nlemma fInf_unfold: \"(f::nat \\ 'a) 0 \\ (\\n. f (Suc n)) = (\\n. f n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f 0 \\ (\\n. f (Suc n)) = \\ range f\n[PROOF STEP]\napply (intro order.antisym inf_greatest)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. f 0 \\ (\\n. f (Suc n)) \\ \\ range f\n 2. \\ range f \\ f 0\n 3. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\napply (rule Inf_greatest, safe)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x n. n \\ UNIV \\ f 0 \\ (\\n. f (Suc n)) \\ f n\n 2. \\ range f \\ f 0\n 3. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\napply (case_tac n)\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. \\x n. \\n \\ UNIV; n = 0\\ \\ f 0 \\ (\\n. f (Suc n)) \\ f n\n 2. \\x n nat. \\n \\ UNIV; n = Suc nat\\ \\ f 0 \\ (\\n. f (Suc n)) \\ f n\n 3. \\ range f \\ f 0\n 4. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\napply simp_all\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\n nat. n = Suc nat \\ f 0 \\ (\\n. f (Suc n)) \\ f (Suc nat)\n 2. \\ range f \\ f 0\n 3. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\nusing Inf_lower inf.coboundedI2\n[PROOF STATE]\nproof (prove)\nusing this:\n?x \\ ?A \\ \\ ?A \\ ?x\n?b \\ ?c \\ ?a \\ ?b \\ ?c\n\ngoal (3 subgoals):\n 1. \\n nat. n = Suc nat \\ f 0 \\ (\\n. f (Suc n)) \\ f (Suc nat)\n 2. \\ range f \\ f 0\n 3. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\napply force\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\ range f \\ f 0\n 2. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\napply (simp add: Inf_lower)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\nby (auto intro: Inf_mono)", "meta": {"llama_tokens": 1025, "file": "Order_Lattice_Props_Order_Lattice_Props", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.880797081106935, "lm_q2_score": 0.8354835330070838, "lm_q1q2_score": 0.7358914571855489}} {"text": "[STATEMENT]\nlemma atLeastLessThan_less_eq3:\n\"(\\f. inj_on f {0..<(m::nat)} \\ f ` {0.. {0.. n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\f. inj_on f {0.. f ` {0.. {0.. n)\n[PROOF STEP]\nusing atLeastLessThan_less_eq2\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on ?f {0.. ?f ` {0.. {0.. ?m \\ ?n\n\ngoal (1 subgoal):\n 1. (\\f. inj_on f {0.. f ` {0.. {0.. n)\n[PROOF STEP]\nproof(auto)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\f m n. inj_on f {0.. f ` {0.. {0.. m \\ n; m \\ n\\ \\ \\f. inj_on f {0.. f ` {0.. {0.. n\"\n[PROOF STATE]\nproof (state)\nthis:\nm \\ n\n\ngoal (1 subgoal):\n 1. \\\\f m n. inj_on f {0.. f ` {0.. {0.. m \\ n; m \\ n\\ \\ \\f. inj_on f {0.. f ` {0.. {0.. id ` {0.. {0.. n\n\ngoal (1 subgoal):\n 1. inj_on id {0.. id ` {0.. {0.. n\n\ngoal (1 subgoal):\n 1. (\\x\\{0..y\\{0.. x = y) \\ id ` {0.. {0.. id ` {0.. {0..\\f m n. inj_on f {0.. f ` {0.. {0.. m \\ n; m \\ n\\ \\ \\f. inj_on f {0.. f ` {0.. {0..f. inj_on f {0.. f ` {0.. {0.. id ` {0.. {0..f. inj_on f {0.. f ` {0.. {0..f. inj_on f {0.. f ` {0.. {0..d. 0 < d \\ (\\y. \\x - y\\ < d \\ a < y \\ y < b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nusing linorder_linear [of \"x - a\" \"b - x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nx - a \\ b - x \\ b - x \\ x - a\n\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x - a \\ b - x \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n 2. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nassume \"x - a \\ b - x\"\n[PROOF STATE]\nproof (state)\nthis:\nx - a \\ b - x\n\ngoal (2 subgoals):\n 1. x - a \\ b - x \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n 2. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\na < x\nx < b\nx - a \\ b - x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na < x\nx < b\nx - a \\ b - x\n\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nby (rule_tac x = \"x - a\" in exI) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n\ngoal (1 subgoal):\n 1. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nassume \"b - x \\ x - a\"\n[PROOF STATE]\nproof (state)\nthis:\nb - x \\ x - a\n\ngoal (1 subgoal):\n 1. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\na < x\nx < b\nb - x \\ x - a\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na < x\nx < b\nb - x \\ x - a\n\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nby (rule_tac x = \"b - x\" in exI) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1241, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8558511396138365, "lm_q2_score": 0.8596637541053281, "lm_q1q2_score": 0.735744203635754}} {"text": "[STATEMENT]\nlemma lemma_interval_lt: \n fixes a b x :: real\n assumes \"a < x\" \"x < b\"\n shows \"\\d. 0 < d \\ (\\y. \\x - y\\ < d \\ a < y \\ y < b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nusing linorder_linear [of \"x - a\" \"b - x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nx - a \\ b - x \\ b - x \\ x - a\n\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x - a \\ b - x \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n 2. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nassume \"x - a \\ b - x\"\n[PROOF STATE]\nproof (state)\nthis:\nx - a \\ b - x\n\ngoal (2 subgoals):\n 1. x - a \\ b - x \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n 2. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\na < x\nx < b\nx - a \\ b - x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na < x\nx < b\nx - a \\ b - x\n\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nby (rule_tac x = \"x - a\" in exI) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n\ngoal (1 subgoal):\n 1. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nassume \"b - x \\ x - a\"\n[PROOF STATE]\nproof (state)\nthis:\nb - x \\ x - a\n\ngoal (1 subgoal):\n 1. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\na < x\nx < b\nb - x \\ x - a\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na < x\nx < b\nb - x \\ x - a\n\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nby (rule_tac x = \"b - x\" in exI) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1241, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8558511396138365, "lm_q2_score": 0.8596637541053281, "lm_q1q2_score": 0.735744203635754}} {"text": "[STATEMENT]\nlemma norm_bfun_le_norm_vec: \"norm (bfun.Bfun (($) (x :: real^'c :: finite))) \\ norm x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (bfun.Bfun (($) x)) \\ norm x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm (bfun.Bfun (($) x)) \\ norm x\n[PROOF STEP]\nhave \"norm (bfun.Bfun (($) (x :: real^'c :: finite))) \\ (\\xa. \\x $ xa\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (bfun.Bfun (($) x)) \\ (\\xa. \\x $ xa\\)\n[PROOF STEP]\nunfolding norm_bfun_def dist_bfun_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. map_fun apply_bfun (map_fun apply_bfun id) (\\f g. \\x. dist (f x) (g x)) (bfun.Bfun (($) x)) 0 \\ (\\xa. \\x $ xa\\)\n[PROOF STEP]\nby (auto simp: Bfun_inverse)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (bfun.Bfun (($) x)) \\ (\\xa. \\x $ xa\\)\n\ngoal (1 subgoal):\n 1. norm (bfun.Bfun (($) x)) \\ norm x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnorm (bfun.Bfun (($) x)) \\ (\\xa. \\x $ xa\\)\n\ngoal (1 subgoal):\n 1. norm (bfun.Bfun (($) x)) \\ norm x\n[PROOF STEP]\nhave \"\\ \\ norm x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\xa. \\x $ xa\\) \\ norm x\n[PROOF STEP]\nusing component_le_norm_cart\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x $ ?i\\ \\ norm ?x\n\ngoal (1 subgoal):\n 1. (\\xa. \\x $ xa\\) \\ norm x\n[PROOF STEP]\nby (auto intro: cSUP_least)\n[PROOF STATE]\nproof (state)\nthis:\n(\\xa. \\x $ xa\\) \\ norm x\n\ngoal (1 subgoal):\n 1. norm (bfun.Bfun (($) x)) \\ norm x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (bfun.Bfun (($) x)) \\ norm x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (bfun.Bfun (($) x)) \\ norm x\n\ngoal (1 subgoal):\n 1. norm (bfun.Bfun (($) x)) \\ norm x\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nnorm (bfun.Bfun (($) x)) \\ norm x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 984, "file": "MDP-Rewards_Bounded_Functions", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637361282706, "lm_q2_score": 0.8558511488056151, "lm_q1q2_score": 0.7357441961519076}} {"text": "[STATEMENT]\nlemma lemma_interval_lt: \n fixes a b x :: real\n assumes \"a < x\" \"x < b\"\n shows \"\\d. 0 < d \\ (\\y. \\x - y\\ < d \\ a < y \\ y < b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nusing linorder_linear [of \"x - a\" \"b - x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nx - a \\ b - x \\ b - x \\ x - a\n\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x - a \\ b - x \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n 2. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nassume \"x - a \\ b - x\"\n[PROOF STATE]\nproof (state)\nthis:\nx - a \\ b - x\n\ngoal (2 subgoals):\n 1. x - a \\ b - x \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n 2. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\na < x\nx < b\nx - a \\ b - x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na < x\nx < b\nx - a \\ b - x\n\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nby (rule_tac x = \"x - a\" in exI) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n\ngoal (1 subgoal):\n 1. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nassume \"b - x \\ x - a\"\n[PROOF STATE]\nproof (state)\nthis:\nb - x \\ x - a\n\ngoal (1 subgoal):\n 1. b - x \\ x - a \\ \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\na < x\nx < b\nb - x \\ x - a\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na < x\nx < b\nb - x \\ x - a\n\ngoal (1 subgoal):\n 1. \\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n[PROOF STEP]\nby (rule_tac x = \"b - x\" in exI) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\d>0. \\y. \\x - y\\ < d \\ a < y \\ y < b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1241, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8558511396138365, "lm_q2_score": 0.8596637433190938, "lm_q1q2_score": 0.735744194404343}} {"text": "[STATEMENT]\nlemma mult_powr_eq_mult_powr_iff:\n \"\\ 2 dvd m1 \\ \\ 2 dvd m2 \\ m1 * 2 powr e1 = m2 * 2 powr e2 \\ m1 = m2 \\ e1 = e2\"\n for m1 m2 e1 e2 :: int\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\odd m1; odd m2\\ \\ (real_of_int m1 * 2 powr real_of_int e1 = real_of_int m2 * 2 powr real_of_int e2) = (m1 = m2 \\ e1 = e2)\n[PROOF STEP]\nusing mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\odd m1; e1 \\ e2\\ \\ (real_of_int m1 * 2 powr real_of_int e1 = real_of_int m2 * 2 powr real_of_int e2) = (m1 = m2 \\ e1 = e2)\n\ngoal (1 subgoal):\n 1. \\odd m1; odd m2\\ \\ (real_of_int m1 * 2 powr real_of_int e1 = real_of_int m2 * 2 powr real_of_int e2) = (m1 = m2 \\ e1 = e2)\n[PROOF STEP]\nusing mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\odd m1; e1 \\ e2\\ \\ (real_of_int m1 * 2 powr real_of_int e1 = real_of_int m2 * 2 powr real_of_int e2) = (m1 = m2 \\ e1 = e2)\n\\odd m2; e2 \\ e1\\ \\ (real_of_int m2 * 2 powr real_of_int e2 = real_of_int m1 * 2 powr real_of_int e1) = (m2 = m1 \\ e2 = e1)\n\ngoal (1 subgoal):\n 1. \\odd m1; odd m2\\ \\ (real_of_int m1 * 2 powr real_of_int e1 = real_of_int m2 * 2 powr real_of_int e2) = (m1 = m2 \\ e1 = e2)\n[PROOF STEP]\nby (cases e1 e2 rule: linorder_le_cases) auto", "meta": {"llama_tokens": 756, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894661025423, "lm_q2_score": 0.8221891283434876, "lm_q1q2_score": 0.7356861711857838}} {"text": "[STATEMENT]\nlemma absolutely_integrable_reflect[simp]:\n fixes f :: \"'a::euclidean_space \\ 'b::euclidean_space\"\n shows \"(\\x. f(-x)) absolutely_integrable_on cbox (-b) (-a) \\ f absolutely_integrable_on cbox a b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. f (- x)) absolutely_integrable_on cbox (- b) (- a)) = (f absolutely_integrable_on cbox a b)\n[PROOF STEP]\nunfolding absolutely_integrable_on_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. f (- x)) integrable_on cbox (- b) (- a) \\ (\\x. norm (f (- x))) integrable_on cbox (- b) (- a)) = (f integrable_on cbox a b \\ (\\x. norm (f x)) integrable_on cbox a b)\n[PROOF STEP]\nby (metis (mono_tags, lifting) integrable_eq integrable_reflect)", "meta": {"llama_tokens": 309, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894661025424, "lm_q2_score": 0.8221891239865619, "lm_q1q2_score": 0.7356861672872527}} {"text": "[STATEMENT]\nlemma measure_finite_Union:\n \"finite S \\ A`S \\ sets M \\ disjoint_family_on A S \\ (\\i. i \\ S \\ emeasure M (A i) \\ \\) \\\n measure M (\\i\\S. A i) = (\\i\\S. measure M (A i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite S; A ` S \\ sets M; disjoint_family_on A S; \\i. i \\ S \\ emeasure M (A i) \\ \\\\ \\ Sigma_Algebra.measure M (\\ (A ` S)) = (\\i\\S. Sigma_Algebra.measure M (A i))\n[PROOF STEP]\nby (induction S rule: finite_induct)\n (auto simp: disjoint_family_on_insert measure_Union sum_emeasure[symmetric] sets.countable_UN'[OF countable_finite])", "meta": {"llama_tokens": 293, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894632969136, "lm_q2_score": 0.822189121808099, "lm_q1q2_score": 0.7356861630312296}} {"text": "[STATEMENT]\nlemma convex_sums:\n assumes \"convex S\"\n and \"convex T\"\n shows \"convex (\\x\\ S. \\y \\ T. {x + y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex (\\x\\S. \\y\\T. {x + y})\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. convex (\\x\\S. \\y\\T. {x + y})\n[PROOF STEP]\nhave \"linear (\\(x, y). x + y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. linear (\\(x, y). x + y)\n[PROOF STEP]\nby (auto intro: linearI simp: scaleR_add_right)\n[PROOF STATE]\nproof (state)\nthis:\nlinear (\\(x, y). x + y)\n\ngoal (1 subgoal):\n 1. convex (\\x\\S. \\y\\T. {x + y})\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\nconvex S\nconvex T\nlinear (\\(x, y). x + y)\n[PROOF STEP]\nhave \"convex ((\\(x, y). x + y) ` (S \\ T))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nconvex S\nconvex T\nlinear (\\(x, y). x + y)\n\ngoal (1 subgoal):\n 1. convex ((\\(x, y). x + y) ` (S \\ T))\n[PROOF STEP]\nby (intro convex_linear_image convex_Times)\n[PROOF STATE]\nproof (state)\nthis:\nconvex ((\\(x, y). x + y) ` (S \\ T))\n\ngoal (1 subgoal):\n 1. convex (\\x\\S. \\y\\T. {x + y})\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nconvex ((\\(x, y). x + y) ` (S \\ T))\n\ngoal (1 subgoal):\n 1. convex (\\x\\S. \\y\\T. {x + y})\n[PROOF STEP]\nhave \"((\\(x, y). x + y) ` (S \\ T)) = (\\x\\ S. \\y \\ T. {x + y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(x, y). x + y) ` (S \\ T) = (\\x\\S. \\y\\T. {x + y})\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\(x, y). x + y) ` (S \\ T) = (\\x\\S. \\y\\T. {x + y})\n\ngoal (1 subgoal):\n 1. convex (\\x\\S. \\y\\T. {x + y})\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nconvex (\\x\\S. \\y\\T. {x + y})\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nconvex (\\x\\S. \\y\\T. {x + y})\n\ngoal (1 subgoal):\n 1. convex (\\x\\S. \\y\\T. {x + y})\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nconvex (\\x\\S. \\y\\T. {x + y})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1096, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872046026642945, "lm_q2_score": 0.8289387998695209, "lm_q1q2_score": 0.7354383185712554}} {"text": "[STATEMENT]\nlemma sumset_stabilizer_eq:\n shows \"sumset (stabilizer A) (stabilizer A) = stabilizer A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sumset (stabilizer A) (stabilizer A) = stabilizer A\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. sumset (stabilizer A) (stabilizer A) \\ stabilizer A\n 2. stabilizer A \\ sumset (stabilizer A) (stabilizer A)\n[PROOF STEP]\nshow \"sumset (stabilizer A) (stabilizer A) \\ stabilizer A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sumset (stabilizer A) (stabilizer A) \\ stabilizer A\n[PROOF STEP]\nusing stabilizer_is_subgroup subgroup.axioms(1) subsetI\n[PROOF STATE]\nproof (prove)\nusing this:\nsubgroup (stabilizer ?S) G (\\) \\\nsubgroup ?G ?M ?composition ?unit \\ submonoid ?G ?M ?composition ?unit\n(\\x. x \\ ?A \\ x \\ ?B) \\ ?A \\ ?B\n\ngoal (1 subgoal):\n 1. sumset (stabilizer A) (stabilizer A) \\ stabilizer A\n[PROOF STEP]\nby (metis (mono_tags, lifting) additive_abelian_group.sumset.simps additive_abelian_group_axioms \n submonoid.sub_composition_closed)\n[PROOF STATE]\nproof (state)\nthis:\nsumset (stabilizer A) (stabilizer A) \\ stabilizer A\n\ngoal (1 subgoal):\n 1. stabilizer A \\ sumset (stabilizer A) (stabilizer A)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. stabilizer A \\ sumset (stabilizer A) (stabilizer A)\n[PROOF STEP]\nshow \"stabilizer A \\ sumset (stabilizer A) (stabilizer A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. stabilizer A \\ sumset (stabilizer A) (stabilizer A)\n[PROOF STEP]\nusing Left_Coset_eq_sumset stabilizer_is_nonempty \n stabilizer_subset_group sub_unit_closed additive_abelian_group_axioms right_unit \n subset_iff sumsetI\n[PROOF STATE]\nproof (prove)\nusing this:\n?x \\ G \\ sumset {?x} (stabilizer ?A) = ?x \\| stabilizer ?A\nstabilizer ?S \\ {}\nstabilizer ?S \\ G\n\\ \\ stabilizer ?A\nadditive_abelian_group G (\\) \\\n?a \\ G \\ ?a \\ \\ = ?a\n(?A \\ ?B) = (\\t. t \\ ?A \\ t \\ ?B)\n\\?a \\ ?A; ?a \\ G; ?b \\ ?B; ?b \\ G\\ \\ ?a \\ ?b \\ sumset ?A ?B\n\ngoal (1 subgoal):\n 1. stabilizer A \\ sumset (stabilizer A) (stabilizer A)\n[PROOF STEP]\nby (smt (verit, best))\n[PROOF STATE]\nproof (state)\nthis:\nstabilizer A \\ sumset (stabilizer A) (stabilizer A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1023, "file": "Kneser_Cauchy_Davenport_Kneser_Cauchy_Davenport_preliminaries", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869981319862, "lm_q2_score": 0.8397339616560072, "lm_q1q2_score": 0.735428085508195}} {"text": "[STATEMENT]\nlemma quadratic_form_mat:\n fixes A :: \"complex mat\" and v :: \"complex vec\" and m :: nat\n assumes dimv: \"dim_vec v = m\" and dimA: \"A \\ carrier_mat m m\"\n shows \"inner_prod v (A *\\<^sub>v v) = (\\i=0..j=0..v v) = (\\i = 0..j = 0..v v) = (\\i = 0..j = 0..v v) = (\\i=0..j=0..v v) = (\\i = 0..j = 0..i = 0..v v) $ i * conjugate v $ i) = (\\i = 0..j = 0.. carrier_mat m m\n\ngoal (1 subgoal):\n 1. (\\i = 0..v v) $ i * conjugate v $ i) = (\\i = 0..j = 0..dim_vec v = m; A \\ carrier_mat m m\\ \\ (\\i = 0..n = 0..i = 0..j = 0..v v) = (\\i = 0..j = 0..v v) = (\\i = 0..j = 0..v v) = (\\i = 0..j = 0..v v) = (\\i = 0..j = 0..v v) = (\\i = 0..j = 0..v v) = (\\i = 0..j = 0.. a \\ 1 \\ 0 < x \\ log a (inverse x) = - log a x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < a; a \\ 1; 0 < x\\ \\ log a (inverse x) = - log a x\n[PROOF STEP]\nusing ln_inverse log_def\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < ?x \\ ln (inverse ?x) = - ln ?x\nlog ?a ?x = ln ?x / ln ?a\n\ngoal (1 subgoal):\n 1. \\0 < a; a \\ 1; 0 < x\\ \\ log a (inverse x) = - log a x\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 250, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505299595163, "lm_q2_score": 0.8128673246376008, "lm_q1q2_score": 0.7353608560201798}} {"text": "[STATEMENT]\nlemma connected_space_iff_components_eq:\n \"connected_space X \\ (\\C \\ connected_components_of X. \\C' \\ connected_components_of X. C = C')\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. connected_space X = (\\C\\connected_components_of X. \\C'\\connected_components_of X. C = C')\n[PROOF STEP]\napply (rule iffI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. connected_space X \\ \\C\\connected_components_of X. \\C'\\connected_components_of X. C = C'\n 2. \\C\\connected_components_of X. \\C'\\connected_components_of X. C = C' \\ connected_space X\n[PROOF STEP]\napply (force simp: connected_components_of_def connected_space_connected_component_set image_iff)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\C\\connected_components_of X. \\C'\\connected_components_of X. C = C' \\ connected_space X\n[PROOF STEP]\nby (metis connected_component_in_connected_components_of connected_component_of_refl connected_space_iff_connected_component mem_Collect_eq)", "meta": {"llama_tokens": 394, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122288794594, "lm_q2_score": 0.8104789155369047, "lm_q1q2_score": 0.735357431315596}} {"text": "[STATEMENT]\nlemma orthonormal_Fourier_partial_sum_diff_squared:\n assumes os: \"orthonormal_system S w\" and w: \"\\i. (w i) square_integrable S\"\n and f: \"f square_integrable S\" and \"finite I\"\n shows \"(l2norm S (\\x. f x -(\\i\\I. orthonormal_coeff S w f i * w i x)))\\<^sup>2 =\n (l2norm S f)\\<^sup>2 - (\\i\\I. (orthonormal_coeff S w f i)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (l2norm S (\\x. f x - (\\i\\I. orthonormal_coeff S w f i * w i x)))\\<^sup>2 = (l2norm S f)\\<^sup>2 - (\\i\\I. (orthonormal_coeff S w f i)\\<^sup>2)\n[PROOF STEP]\nusing orthonormal_partial_sum_diff [OF assms, where a = \"orthonormal_coeff S w f\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(l2norm S (\\x. f x - (\\i\\I. orthonormal_coeff S w f i * w i x)))\\<^sup>2 = (l2norm S f)\\<^sup>2 + (\\i\\I. (orthonormal_coeff S w f i)\\<^sup>2) - 2 * (\\i\\I. orthonormal_coeff S w f i * orthonormal_coeff S w f i)\n\ngoal (1 subgoal):\n 1. (l2norm S (\\x. f x - (\\i\\I. orthonormal_coeff S w f i * w i x)))\\<^sup>2 = (l2norm S f)\\<^sup>2 - (\\i\\I. (orthonormal_coeff S w f i)\\<^sup>2)\n[PROOF STEP]\nby (simp add: power2_eq_square)", "meta": {"llama_tokens": 565, "file": "Fourier_Fourier", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122263731811, "lm_q2_score": 0.8104789109591832, "lm_q1q2_score": 0.7353574251308878}} {"text": "[STATEMENT]\nlemma bezout_ring_imp_diagonal_2x2_admits_SNF_JNF:\n assumes c: \"OFCLASS('a::comm_ring_1, bezout_ring_class)\"\n shows \"\\A. (A::'a mat) \\ carrier_mat 2 2 \\ isDiagonal_mat A\n \\ (\\P Q. P \\ carrier_mat 2 2 \\ Q \\ carrier_mat 2 2 \n \\ invertible_mat P \\ invertible_mat Q \\ Smith_normal_form_mat (P*A*Q))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A. A \\ carrier_mat 2 2 \\ isDiagonal_mat A \\ (\\P Q. P \\ carrier_mat 2 2 \\ Q \\ carrier_mat 2 2 \\ invertible_mat P \\ invertible_mat Q \\ Smith_normal_form_mat (P * A * Q))\n[PROOF STEP]\nusing bezout_ring_imp_diagonal_admits_SNF_JNF\n [OF OFCLASS_bezout_ring_imp_class_bezout_ring[where ?'a='a, OF c]]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A. admits_SNF_JNF A\n\ngoal (1 subgoal):\n 1. \\A. A \\ carrier_mat 2 2 \\ isDiagonal_mat A \\ (\\P Q. P \\ carrier_mat 2 2 \\ Q \\ carrier_mat 2 2 \\ invertible_mat P \\ invertible_mat Q \\ Smith_normal_form_mat (P * A * Q))\n[PROOF STEP]\nunfolding admits_SNF_JNF_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A. square_mat A \\ isDiagonal_mat A \\ (\\P Q. P \\ carrier_mat (dim_row A) (dim_row A) \\ Q \\ carrier_mat (dim_row A) (dim_row A) \\ invertible_mat P \\ invertible_mat Q \\ Smith_normal_form_mat (P * A * Q))\n\ngoal (1 subgoal):\n 1. \\A. A \\ carrier_mat 2 2 \\ isDiagonal_mat A \\ (\\P Q. P \\ carrier_mat 2 2 \\ Q \\ carrier_mat 2 2 \\ invertible_mat P \\ invertible_mat Q \\ Smith_normal_form_mat (P * A * Q))\n[PROOF STEP]\nusing \\\\A. admits_SNF_JNF A\\ admits_SNF_JNF_alt_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A. square_mat A \\ isDiagonal_mat A \\ (\\P Q. P \\ carrier_mat (dim_row A) (dim_row A) \\ Q \\ carrier_mat (dim_row A) (dim_row A) \\ invertible_mat P \\ invertible_mat Q \\ Smith_normal_form_mat (P * A * Q))\n\\A. admits_SNF_JNF A\n(\\A. admits_SNF_JNF A) = (\\A n. A \\ carrier_mat n n \\ isDiagonal_mat A \\ (\\P Q. P \\ carrier_mat n n \\ Q \\ carrier_mat n n \\ invertible_mat P \\ invertible_mat Q \\ Smith_normal_form_mat (P * A * Q)))\n\ngoal (1 subgoal):\n 1. \\A. A \\ carrier_mat 2 2 \\ isDiagonal_mat A \\ (\\P Q. P \\ carrier_mat 2 2 \\ Q \\ carrier_mat 2 2 \\ invertible_mat P \\ invertible_mat Q \\ Smith_normal_form_mat (P * A * Q))\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 1149, "file": "Smith_Normal_Form_Alternative_Proofs", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.907312213841788, "lm_q2_score": 0.8104789155369047, "lm_q1q2_score": 0.7353574191278804}} {"text": "[STATEMENT]\nlemma effective_row_formula:\n fixes M1 and M2\n assumes \"i < (row_length M1)*(row_length M2)\" \n and \"(mat (row_length M1) (length M1) M1)\"\n and \"(mat (row_length M2) (length M2) M2)\"\n shows \"row (M1 \\ M2) i \n = vec_vec_Tensor \n (row M1 (i div row_length M2)) \n (row M2 (i mod row_length M2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row (M1 \\ M2) i = vec_vec_Tensor (row M1 (i div row_length M2)) (row M2 (i mod row_length M2))\n[PROOF STEP]\nusing assms row_formula\n[PROOF STATE]\nproof (prove)\nusing this:\ni < row_length M1 * row_length M2\nmat (row_length M1) (length M1) M1\nmat (row_length M2) (length M2) M2\n\\i. i < row_length ?M1.0 * row_length ?M2.0 \\ mat (row_length ?M1.0) (length ?M1.0) ?M1.0 \\ mat (row_length ?M2.0) (length ?M2.0) ?M2.0 \\ row (?M1.0 \\ ?M2.0) i = vec_vec_Tensor (row ?M1.0 (i div row_length ?M2.0)) (row ?M2.0 (i mod row_length ?M2.0))\n\ngoal (1 subgoal):\n 1. row (M1 \\ M2) i = vec_vec_Tensor (row M1 (i div row_length M2)) (row M2 (i mod row_length M2))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 498, "file": "Matrix_Tensor_Matrix_Tensor", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122213606241, "lm_q2_score": 0.8104789063814617, "lm_q1q2_score": 0.7353574169148933}} {"text": "[STATEMENT]\nlemma poly_mult_in_carrier:\n \"\\ set p1 \\ carrier R; set p2 \\ carrier R \\ \\ set (poly_mult p1 p2) \\ carrier R\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\set p1 \\ carrier R; set p2 \\ carrier R\\ \\ set (poly_mult p1 p2) \\ carrier R\n[PROOF STEP]\nusing poly_mult_is_polynomial polynomial_in_carrier carrier_is_subring\n[PROOF STATE]\nproof (prove)\nusing this:\n\\subring ?K R; set ?p1.0 \\ ?K; set ?p2.0 \\ ?K\\ \\ polynomial ?K (poly_mult ?p1.0 ?p2.0)\n\\subring ?K R; polynomial ?K ?p\\ \\ set ?p \\ carrier R\nsubring (carrier R) R\n\ngoal (1 subgoal):\n 1. \\set p1 \\ carrier R; set p2 \\ carrier R\\ \\ set (poly_mult p1 p2) \\ carrier R\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 361, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9241418199787564, "lm_q2_score": 0.7956580952177051, "lm_q1q2_score": 0.7353009201953207}} {"text": "[STATEMENT]\nlemma onorm_sum:\n assumes \"finite S\"\n assumes \"\\s. s \\ S \\ bounded_linear (f s)\"\n shows \"onorm (\\x. sum (\\s. f s x) S) \\ sum (\\s. onorm (f s)) S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. onorm (\\x. \\s\\S. f s x) \\ (\\s\\S. onorm (f s))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\n?s \\ S \\ bounded_linear (f ?s)\n\ngoal (1 subgoal):\n 1. onorm (\\x. \\s\\S. f s x) \\ (\\s\\S. onorm (f s))\n[PROOF STEP]\nby (induction) (auto simp: onorm_zero intro!: onorm_triangle_le bounded_linear_sum)", "meta": {"llama_tokens": 280, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.901920681802153, "lm_q2_score": 0.8152324848629214, "lm_q1q2_score": 0.7352750385748295}} {"text": "[STATEMENT]\nlemma measure_preservingE:\n assumes \"f \\ measure_preserving M N\"\n shows \"f \\ measurable M N\"\n \"\\A. A \\ sets N \\ emeasure M (f-`A \\ space M) = emeasure N A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f \\ M \\\\<^sub>M N &&& (\\A. A \\ sets N \\ emeasure M (f -` A \\ space M) = emeasure N A)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nf \\ measure_preserving M N\n\ngoal (1 subgoal):\n 1. f \\ M \\\\<^sub>M N &&& (\\A. A \\ sets N \\ emeasure M (f -` A \\ space M) = emeasure N A)\n[PROOF STEP]\nunfolding measure_preserving_def\n[PROOF STATE]\nproof (prove)\nusing this:\nf \\ {f \\ M \\\\<^sub>M N. \\A\\sets N. emeasure M (f -` A \\ space M) = emeasure N A}\n\ngoal (1 subgoal):\n 1. f \\ M \\\\<^sub>M N &&& (\\A. A \\ sets N \\ emeasure M (f -` A \\ space M) = emeasure N A)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 413, "file": "Ergodic_Theory_Measure_Preserving_Transformations", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206659843131, "lm_q2_score": 0.8152324960856175, "lm_q1q2_score": 0.735275035801594}} {"text": "[STATEMENT]\nlemma eln_mult_add:\n assumes \"\\((a = \\ \\ b = 0) \\ (a = 0 \\ b = \\))\"\n shows \"eln(a * b) = eln a + eln b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eln (a * b) = eln a + eln b\n[PROOF STEP]\nby (smt assms ennexp.simps(2) ennexp.simps(3) ennexp_add_mult ennexp_eln eln_ennexp)", "meta": {"llama_tokens": 164, "file": "Gromov_Hyperbolicity_Eexp_Eln", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099070060380482, "lm_q2_score": 0.8080672181749422, "lm_q1q2_score": 0.7352660231670559}} {"text": "[STATEMENT]\nlemma hom_paired2:\n assumes \"group G\" \"group H\"\n shows \"(\\(x,y). (f x,g y)) \\ hom (DirProd G H) (DirProd G' H') \\ f \\ hom G G' \\ g \\ hom H H'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\(x, y). (f x, g y)) \\ hom (G \\\\ H) (G' \\\\ H')) = (f \\ hom G G' \\ g \\ hom H H')\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nGroup.group G\nGroup.group H\n\ngoal (1 subgoal):\n 1. ((\\(x, y). (f x, g y)) \\ hom (G \\\\ H) (G' \\\\ H')) = (f \\ hom G G' \\ g \\ hom H H')\n[PROOF STEP]\nby (fastforce simp: hom_def Pi_def dest!: group.is_monoid)", "meta": {"llama_tokens": 312, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099070084811306, "lm_q2_score": 0.8080672135527632, "lm_q1q2_score": 0.7352660209354777}} {"text": "[STATEMENT]\nlemma norm_sq_mtx_diag: \"\\sq_mtx_diag f\\ = Max {\\f i\\ |i. i \\ UNIV}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\sq_mtx_diag f\\ = Max {\\f i\\ |i. i \\ UNIV}\n[PROOF STEP]\nunfolding norm_sq_mtx_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\to_vec (sq_mtx_diag f)\\\\<^sub>o\\<^sub>p = Max {\\f i\\ |i. i \\ UNIV}\n[PROOF STEP]\napply transfer\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f. \\diag_mat f\\\\<^sub>o\\<^sub>p = Max {\\f i\\ |i. i \\ UNIV}\n[PROOF STEP]\nby (rule op_norm_diag_mat_eq)", "meta": {"llama_tokens": 303, "file": "Matrices_for_ODEs_SQ_MTX", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110569397306, "lm_q2_score": 0.8244619220634456, "lm_q1q2_score": 0.7352642581219633}} {"text": "[STATEMENT]\nlemma fermat_little_theorem:\n assumes \"prime (P :: nat)\" \n shows \"[x^P = x] (mod P)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [x ^ P = x] (mod P)\n[PROOF STEP]\nproof(cases \"P dvd x\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. P dvd x \\ [x ^ P = x] (mod P)\n 2. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nP dvd x\n\ngoal (2 subgoals):\n 1. P dvd x \\ [x ^ P = x] (mod P)\n 2. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nhence \"x mod P = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nP dvd x\n\ngoal (1 subgoal):\n 1. x mod P = 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx mod P = 0\n\ngoal (2 subgoals):\n 1. P dvd x \\ [x ^ P = x] (mod P)\n 2. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nx mod P = 0\n\ngoal (2 subgoals):\n 1. P dvd x \\ [x ^ P = x] (mod P)\n 2. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nhave \"x ^ P mod P = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x ^ P mod P = 0\n[PROOF STEP]\nby (simp add: True assms prime_dvd_power_nat_iff prime_gt_0_nat)\n[PROOF STATE]\nproof (state)\nthis:\nx ^ P mod P = 0\n\ngoal (2 subgoals):\n 1. P dvd x \\ [x ^ P = x] (mod P)\n 2. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nx mod P = 0\nx ^ P mod P = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx mod P = 0\nx ^ P mod P = 0\n\ngoal (1 subgoal):\n 1. [x ^ P = x] (mod P)\n[PROOF STEP]\nby (simp add: cong_def)\n[PROOF STATE]\nproof (state)\nthis:\n[x ^ P = x] (mod P)\n\ngoal (1 subgoal):\n 1. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ P dvd x\n\ngoal (1 subgoal):\n 1. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nhence \"[x ^ (P - 1) = 1] (mod P)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ P dvd x\n\ngoal (1 subgoal):\n 1. [x ^ (P - 1) = 1] (mod P)\n[PROOF STEP]\nusing fermat_theorem assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ P dvd x\n\\prime ?p; \\ ?p dvd ?a\\ \\ [?a ^ (?p - 1) = 1] (mod ?p)\nprime P\n\ngoal (1 subgoal):\n 1. [x ^ (P - 1) = 1] (mod P)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n[x ^ (P - 1) = 1] (mod P)\n\ngoal (1 subgoal):\n 1. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[x ^ (P - 1) = 1] (mod P)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n[x ^ (P - 1) = 1] (mod P)\n\ngoal (1 subgoal):\n 1. [x ^ P = x] (mod P)\n[PROOF STEP]\nby (metis Suc_diff_1 assms cong_scalar_left nat_mult_1_right not_gr_zero not_prime_0 power_Suc)\n[PROOF STATE]\nproof (state)\nthis:\n[x ^ P = x] (mod P)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1454, "file": "Sigma_Commit_Crypto_Number_Theory_Aux", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.89181104831338, "lm_q2_score": 0.824461928533133, "lm_q1q2_score": 0.7352642567796044}} {"text": "[STATEMENT]\nlemma NatMultiples_product_multiple: \"\n \\ a \\ NatMultiples F; b \\ NatMultiples F \\ \\ a * b \\ NatMultiples F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a \\ NatMultiples F; b \\ NatMultiples F\\ \\ a * b \\ NatMultiples F\n[PROOF STEP]\napply (induct a rule: NatMultiples.induct)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\k. \\k \\ F; b \\ NatMultiples F\\ \\ k * b \\ NatMultiples F\n 2. \\k m. \\k \\ F; m \\ NatMultiples F; b \\ NatMultiples F \\ m * b \\ NatMultiples F; b \\ NatMultiples F\\ \\ k * m * b \\ NatMultiples F\n[PROOF STEP]\napply (simp add: NatMultiples_Product)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\k m. \\k \\ F; m \\ NatMultiples F; b \\ NatMultiples F \\ m * b \\ NatMultiples F; b \\ NatMultiples F\\ \\ k * m * b \\ NatMultiples F\n[PROOF STEP]\napply (simp add: mult.assoc[of _ _ b] NatMultiples_Product)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 473, "file": "Nat-Interval-Logic_IL_IntervalOperators", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213745668095, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7350986928407394}} {"text": "[STATEMENT]\nlemma ivl_integral_nonneg:\n fixes f :: \"real \\ real\"\n assumes \"f integrable_on (closed_segment a b)\"\n and \"\\x. a \\ x \\ x \\ b \\ 0 \\ f x\"\n and \"\\x. b \\ x \\ x \\ a \\ f x \\ 0\"\n shows \"0 \\ ivl_integral a b f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ ivl_integral a b f\n[PROOF STEP]\nby (rule has_ivl_integral_nonneg[OF assms(1)[unfolded has_ivl_integral_ivl_integral] assms(2-3)])", "meta": {"llama_tokens": 217, "file": "Ordinary_Differential_Equations_Library_Interval_Integral_HK", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213772699435, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.7350986850594673}} {"text": "[STATEMENT]\nlemma matrix_vector_mult_diff_distrib [algebra_simps]:\n fixes A :: \"'a::ring_1^'n^'m\"\n shows \"A *v (x - y) = A *v x - A *v y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *v (x - y) = A *v x - A *v y\n[PROOF STEP]\nby (vector matrix_vector_mult_def sum_subtractf right_diff_distrib)", "meta": {"llama_tokens": 137, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213772699435, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.7350986850594673}} {"text": "[STATEMENT]\nlemma has_integral_add_cbox:\n fixes f :: \"'n::euclidean_space \\ 'a::real_normed_vector\"\n assumes \"(f has_integral k) (cbox a b)\" \"(g has_integral l) (cbox a b)\"\n shows \"((\\x. f x + g x) has_integral (k + l)) (cbox a b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. f x + g x) has_integral k + l) (cbox a b)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_integral k) (cbox a b)\n(g has_integral l) (cbox a b)\n\ngoal (1 subgoal):\n 1. ((\\x. f x + g x) has_integral k + l) (cbox a b)\n[PROOF STEP]\nunfolding has_integral_cbox\n[PROOF STATE]\nproof (prove)\nusing this:\n(sum (\\(x, k). content k *\\<^sub>R f x) \\ k) (division_filter (cbox a b))\n(sum (\\(x, k). content k *\\<^sub>R g x) \\ l) (division_filter (cbox a b))\n\ngoal (1 subgoal):\n 1. (sum (\\(x, k). content k *\\<^sub>R (f x + g x)) \\ k + l) (division_filter (cbox a b))\n[PROOF STEP]\nby (simp add: split_beta' scaleR_add_right sum.distrib[abs_def] tendsto_add)", "meta": {"llama_tokens": 454, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.912436153333645, "lm_q2_score": 0.8056321959813274, "lm_q1q2_score": 0.7350879419029396}} {"text": "[STATEMENT]\nlemma integral_sin_and_cos_N [simp]:\n fixes m n::real\n assumes \"m \\ \\\" \"n \\ \\\"\n shows \"integral\\<^sup>L (lebesgue_on {-pi..pi}) (\\x. sin (m * x) * sin (n * x)) = (if m = n \\ n \\ 0 then pi else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LINT x|lebesgue_on {- pi..pi}. sin (m * x) * sin (n * x) = (if m = n \\ n \\ 0 then pi else 0)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ \\\nn \\ \\\n\ngoal (1 subgoal):\n 1. LINT x|lebesgue_on {- pi..pi}. sin (m * x) * sin (n * x) = (if m = n \\ n \\ 0 then pi else 0)\n[PROOF STEP]\nunfolding Nats_altdef1\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ {real_of_int n |n. 0 \\ n}\nn \\ {real_of_int n |n. 0 \\ n}\n\ngoal (1 subgoal):\n 1. LINT x|lebesgue_on {- pi..pi}. sin (m * x) * sin (n * x) = (if m = n \\ n \\ 0 then pi else 0)\n[PROOF STEP]\nby (auto simp: integral_sin_and_cos)", "meta": {"llama_tokens": 446, "file": "Fourier_Fourier", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361533336451, "lm_q2_score": 0.8056321843145404, "lm_q1q2_score": 0.7350879312577415}} {"text": "[STATEMENT]\nlemma disj_family_sum:\n shows \"finite I \\ disjoint_family_on A I \\ (\\i. i \\ I \\ finite (A i)) \\ \n (\\ i \\ (\\n \\ I. A n). f i) = (\\ n\\ I. (\\ i \\ A n. f i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite I; disjoint_family_on A I; \\i. i \\ I \\ finite (A i)\\ \\ sum f (\\ (A ` I)) = (\\n\\I. sum f (A n))\n[PROOF STEP]\nproof (induct rule:finite_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\disjoint_family_on A {}; \\i. i \\ {} \\ finite (A i)\\ \\ sum f (\\ (A ` {})) = (\\n\\{}. sum f (A n))\n 2. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\ncase empty\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on A {}\n?i \\ {} \\ finite (A ?i)\n\ngoal (2 subgoals):\n 1. \\disjoint_family_on A {}; \\i. i \\ {} \\ finite (A i)\\ \\ sum f (\\ (A ` {})) = (\\n\\{}. sum f (A n))\n 2. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndisjoint_family_on A {}\n?i \\ {} \\ finite (A ?i)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ndisjoint_family_on A {}\n?i \\ {} \\ finite (A ?i)\n\ngoal (1 subgoal):\n 1. sum f (\\ (A ` {})) = (\\n\\{}. sum f (A n))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum f (\\ (A ` {})) = (\\n\\{}. sum f (A n))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\ncase (insert x F)\n[PROOF STATE]\nproof (state)\nthis:\nfinite F\nx \\ F\n\\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n))\ndisjoint_family_on A (insert x F)\n?i \\ insert x F \\ finite (A ?i)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nhence \"disjoint_family_on A F\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite F\nx \\ F\n\\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n))\ndisjoint_family_on A (insert x F)\n?i \\ insert x F \\ finite (A ?i)\n\ngoal (1 subgoal):\n 1. disjoint_family_on A F\n[PROOF STEP]\nby (meson disjoint_family_on_mono subset_insertI)\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on A F\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nhave \"(\\n \\ (insert x F). A n) = A x \\ (\\n \\ F. A n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (A ` insert x F) = A x \\ \\ (A ` F)\n[PROOF STEP]\nusing insert\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite F\nx \\ F\n\\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n))\ndisjoint_family_on A (insert x F)\n?i \\ insert x F \\ finite (A ?i)\n\ngoal (1 subgoal):\n 1. \\ (A ` insert x F) = A x \\ \\ (A ` F)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\ (A ` insert x F) = A x \\ \\ (A ` F)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nhence \"(\\ i \\ (\\n \\ (insert x F). A n). f i) = (\\ i \\ (A x \\ (\\n \\ F. A n)). f i)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (A ` insert x F) = A x \\ \\ (A ` F)\n\ngoal (1 subgoal):\n 1. sum f (\\ (A ` insert x F)) = sum f (A x \\ \\ (A ` F))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum f (\\ (A ` insert x F)) = sum f (A x \\ \\ (A ` F))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum f (\\ (A ` insert x F)) = sum f (A x \\ \\ (A ` F))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nhave \"... = (\\ i \\ A x. f i) + (\\ i \\ (\\n \\ F. A n). f i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f (A x \\ \\ (A ` F)) = sum f (A x) + sum f (\\ (A ` F))\n[PROOF STEP]\nby (rule sum.union_disjoint, (simp add: insert disjoint_Un)+)\n[PROOF STATE]\nproof (state)\nthis:\nsum f (A x \\ \\ (A ` F)) = sum f (A x) + sum f (\\ (A ` F))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum f (A x \\ \\ (A ` F)) = sum f (A x) + sum f (\\ (A ` F))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nhave \"... = (\\ i \\ A x. f i) + (\\n\\F. sum f (A n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f (A x) + sum f (\\ (A ` F)) = sum f (A x) + (\\n\\F. sum f (A n))\n[PROOF STEP]\nusing \\disjoint_family_on A F\\\n[PROOF STATE]\nproof (prove)\nusing this:\ndisjoint_family_on A F\n\ngoal (1 subgoal):\n 1. sum f (A x) + sum f (\\ (A ` F)) = sum f (A x) + (\\n\\F. sum f (A n))\n[PROOF STEP]\nby (simp add: insert)\n[PROOF STATE]\nproof (state)\nthis:\nsum f (A x) + sum f (\\ (A ` F)) = sum f (A x) + (\\n\\F. sum f (A n))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum f (A x) + sum f (\\ (A ` F)) = sum f (A x) + (\\n\\F. sum f (A n))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nhave \"... = (\\n\\(insert x F). sum f (A n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f (A x) + (\\n\\F. sum f (A n)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nusing insert\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite F\nx \\ F\n\\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n))\ndisjoint_family_on A (insert x F)\n?i \\ insert x F \\ finite (A ?i)\n\ngoal (1 subgoal):\n 1. sum f (A x) + (\\n\\F. sum f (A n)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum f (A x) + (\\n\\F. sum f (A n)) = (\\n\\insert x F. sum f (A n))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\disjoint_family_on A F; \\i. i \\ F \\ finite (A i)\\ \\ sum f (\\ (A ` F)) = (\\n\\F. sum f (A n)); disjoint_family_on A (insert x F); \\i. i \\ insert x F \\ finite (A i)\\ \\ sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nsum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n\ngoal (1 subgoal):\n 1. sum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nsum f (\\ (A ` insert x F)) = (\\n\\insert x F. sum f (A n))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5095, "file": "Projective_Measurements_Linear_Algebra_Complements", "length": 29, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767970940975, "lm_q2_score": 0.8376199653600372, "lm_q1q2_score": 0.7349920843861945}} {"text": "[STATEMENT]\nlemma RSpan_append : \"RSpan (ms @ ns) = RSpan ms + RSpan ns\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. RSpan (ms @ ns) = RSpan ms + RSpan ns\n[PROOF STEP]\nproof (induct ms)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. RSpan ([] @ ns) = RSpan [] + RSpan ns\n 2. \\a ms. RSpan (ms @ ns) = RSpan ms + RSpan ns \\ RSpan ((a # ms) @ ns) = RSpan (a # ms) + RSpan ns\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. RSpan ([] @ ns) = RSpan [] + RSpan ns\n 2. \\a ms. RSpan (ms @ ns) = RSpan ms + RSpan ns \\ RSpan ((a # ms) @ ns) = RSpan (a # ms) + RSpan ns\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. RSpan ([] @ ns) = RSpan [] + RSpan ns\n[PROOF STEP]\nusing add_0_left[of \"RSpan ns\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 + RSpan ns = RSpan ns\n\ngoal (1 subgoal):\n 1. RSpan ([] @ ns) = RSpan [] + RSpan ns\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nRSpan ([] @ ns) = RSpan [] + RSpan ns\n\ngoal (1 subgoal):\n 1. \\a ms. RSpan (ms @ ns) = RSpan ms + RSpan ns \\ RSpan ((a # ms) @ ns) = RSpan (a # ms) + RSpan ns\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a ms. RSpan (ms @ ns) = RSpan ms + RSpan ns \\ RSpan ((a # ms) @ ns) = RSpan (a # ms) + RSpan ns\n[PROOF STEP]\ncase (Cons m ms)\n[PROOF STATE]\nproof (state)\nthis:\nRSpan (ms @ ns) = RSpan ms + RSpan ns\n\ngoal (1 subgoal):\n 1. \\a ms. RSpan (ms @ ns) = RSpan ms + RSpan ns \\ RSpan ((a # ms) @ ns) = RSpan (a # ms) + RSpan ns\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nRSpan (ms @ ns) = RSpan ms + RSpan ns\n\ngoal (1 subgoal):\n 1. RSpan ((m # ms) @ ns) = RSpan (m # ms) + RSpan ns\n[PROOF STEP]\nusing RSpan_Cons[of m \"ms@ns\"] add.assoc\n[PROOF STATE]\nproof (prove)\nusing this:\nRSpan (ms @ ns) = RSpan ms + RSpan ns\nRSpan (m # ms @ ns) = RSpan [m] + RSpan (ms @ ns)\n?a + ?b + ?c = ?a + (?b + ?c)\n\ngoal (1 subgoal):\n 1. RSpan ((m # ms) @ ns) = RSpan (m # ms) + RSpan ns\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nRSpan ((m # ms) @ ns) = RSpan (m # ms) + RSpan ns\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 972, "file": "Rep_Fin_Groups_Rep_Fin_Groups", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267626522814, "lm_q2_score": 0.8459424373085146, "lm_q1q2_score": 0.7349774291969373}} {"text": "[STATEMENT]\nlemma UN_Int_Compl_subset:\n \"(\\i \\ lessThan n. A i) \\ (- A n) \\ \n (\\i \\ lessThan n. (A i) \\ (- A (Suc i)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (A ` {.. - A n \\ (\\i - A (Suc i))\n[PROOF STEP]\nby (induct n) (auto simp: lessThan_Suc)", "meta": {"llama_tokens": 163, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513786759492, "lm_q2_score": 0.8198933447152497, "lm_q1q2_score": 0.7348305405682778}} {"text": "[STATEMENT]\nlemma mult_adjugate_det: \"A ** adjugate A = mat (det A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A ** adjugate A = mat (det A)\n[PROOF STEP]\nusing mult_adjugate_det[of \"from_vec A\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nfrom_vec A * Square_Matrix.adjugate (from_vec A) = diag (Square_Matrix.det (from_vec A))\n\ngoal (1 subgoal):\n 1. A ** adjugate A = mat (det A)\n[PROOF STEP]\nunfolding det_sq_matrix_eq adjugate_eq to_vec_eq_iff[symmetric] to_vec_matrix_matrix_mult to_vec_from_vec\n[PROOF STATE]\nproof (prove)\nusing this:\nA ** to_vec (Square_Matrix.adjugate (from_vec A)) = to_vec (diag (det A))\n\ngoal (1 subgoal):\n 1. A ** (to_vec \\ Square_Matrix.adjugate \\ from_vec) A = mat (det A)\n[PROOF STEP]\nby (simp add: to_vec_diag)", "meta": {"llama_tokens": 326, "file": "Echelon_Form_Echelon_Form_Inverse", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.914900945711678, "lm_q2_score": 0.8031738057795403, "lm_q1q2_score": 0.734824474478549}} {"text": "[STATEMENT]\nlemma covar_indep_eq_zero:\n fixes f g :: \"'a \\ real\"\n assumes \"integrable M f\"\n assumes \"integrable M g\"\n assumes \"indep_var borel f borel g\"\n shows \"covariance f g = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. covariance f g = 0\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. covariance f g = 0\n[PROOF STEP]\nhave a:\"indep_var borel ((\\t. t - expectation f) \\ f) borel ((\\t. t - expectation g) \\ g)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. indep_var borel ((\\t. t - expectation f) \\ f) borel ((\\t. t - expectation g) \\ g)\n[PROOF STEP]\nby (rule indep_var_compose[OF assms(3)], auto)\n[PROOF STATE]\nproof (state)\nthis:\nindep_var borel ((\\t. t - expectation f) \\ f) borel ((\\t. t - expectation g) \\ g)\n\ngoal (1 subgoal):\n 1. covariance f g = 0\n[PROOF STEP]\nhave b:\"expectation (\\\\. (f \\ - expectation f) * (g \\ - expectation g)) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. expectation (\\\\. (f \\ - expectation f) * (g \\ - expectation g)) = 0\n[PROOF STEP]\nusing a assms\n[PROOF STATE]\nproof (prove)\nusing this:\nindep_var borel ((\\t. t - expectation f) \\ f) borel ((\\t. t - expectation g) \\ g)\nintegrable M f\nintegrable M g\nindep_var borel f borel g\n\ngoal (1 subgoal):\n 1. expectation (\\\\. (f \\ - expectation f) * (g \\ - expectation g)) = 0\n[PROOF STEP]\nby (subst indep_var_lebesgue_integral, auto simp add:comp_def prob_space)\n[PROOF STATE]\nproof (state)\nthis:\nexpectation (\\\\. (f \\ - expectation f) * (g \\ - expectation g)) = 0\n\ngoal (1 subgoal):\n 1. covariance f g = 0\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nexpectation (\\\\. (f \\ - expectation f) * (g \\ - expectation g)) = 0\n\ngoal (1 subgoal):\n 1. covariance f g = 0\n[PROOF STEP]\nby (simp add:covariance_def)\n[PROOF STATE]\nproof (state)\nthis:\ncovariance f g = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 849, "file": "Frequency_Moments_Probability_Ext", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392878563335, "lm_q2_score": 0.831143054132195, "lm_q1q2_score": 0.7347631136817637}} {"text": "[STATEMENT]\nlemma Dirichlet_kernel_continuous: \"continuous_on {-pi..pi} (Dirichlet_kernel n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous_on {- pi..pi} (Dirichlet_kernel n)\n[PROOF STEP]\napply (rule continuous_on_subset [OF Dirichlet_kernel_continuous_strong], clarsimp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\- pi \\ x; x \\ pi\\ \\ - (2 * pi) < x \\ x < 2 * pi\n[PROOF STEP]\nusing pi_gt_zero\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < pi\n\ngoal (1 subgoal):\n 1. \\x. \\- pi \\ x; x \\ pi\\ \\ - (2 * pi) < x \\ x < 2 * pi\n[PROOF STEP]\nby linarith", "meta": {"llama_tokens": 285, "file": "Fourier_Fourier", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392848011834, "lm_q2_score": 0.8311430520409023, "lm_q1q2_score": 0.734763109293712}} {"text": "[STATEMENT]\nlemma infnorm_2:\n fixes x :: \"real^2\"\n shows \"infnorm x = max \\x$1\\ \\x$2\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. infnorm x = max \\x $ 1\\ \\x $ 2\\\n[PROOF STEP]\nunfolding infnorm_cart UNIV_2\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sup {\\x $ i\\ |i. i \\ {1, 2}} = max \\x $ 1\\ \\x $ 2\\\n[PROOF STEP]\nby (rule cSup_eq) auto", "meta": {"llama_tokens": 206, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392817460333, "lm_q2_score": 0.8311430520409023, "lm_q1q2_score": 0.7347631067544452}} {"text": "[STATEMENT]\nlemma double_sum_split_square_diff: \"finite {0.. \n (\\ i \\ {0.. j \\ ({0..< x} - {i}) . c i * c j)) = \n (\\ i \\ {0.. i \\ {0.. (\\i = 0::'b..j\\{0::'b..2 - (\\i = 0::'b.. i. c i\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {0::'b.. (sum c {0::'b..2 = (\\i = 0::'b..i = 0::'b..j\\{0::'b.. (\\i = 0::'b..j\\{0::'b..2 - (\\i = 0::'b..Cn(\\{Cn(B)|B. P B}) = Cn(\\{B. P B})\\ (is \\?A = ?B\\)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Cn (\\ {Cn B |B. P B}) = Cn (\\ {B. P B})\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. Cn (\\ {Cn B |B. P B}) \\ Cn (\\ {B. P B})\n 2. Cn (\\ {B. P B}) \\ Cn (\\ {Cn B |B. P B})\n[PROOF STEP]\nhave \\?A \\ Cn ?B\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Cn (\\ {Cn B |B. P B}) \\ Cn (Cn (\\ {B. P B}))\n[PROOF STEP]\napply(rule monotonicity_L, rule Union_least, auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x B. \\x \\ Cn B; P B\\ \\ x \\ Cn (\\ (Collect P))\n[PROOF STEP]\nby (metis Sup_upper in_mono mem_Collect_eq monotonicity_L)\n[PROOF STATE]\nproof (state)\nthis:\nCn (\\ {Cn B |B. P B}) \\ Cn (Cn (\\ {B. P B}))\n\ngoal (2 subgoals):\n 1. Cn (\\ {Cn B |B. P B}) \\ Cn (\\ {B. P B})\n 2. Cn (\\ {B. P B}) \\ Cn (\\ {Cn B |B. P B})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nCn (\\ {Cn B |B. P B}) \\ Cn (Cn (\\ {B. P B}))\n[PROOF STEP]\nshow \\?A \\ ?B\\\n[PROOF STATE]\nproof (prove)\nusing this:\nCn (\\ {Cn B |B. P B}) \\ Cn (Cn (\\ {B. P B}))\n\ngoal (1 subgoal):\n 1. Cn (\\ {Cn B |B. P B}) \\ Cn (\\ {B. P B})\n[PROOF STEP]\nby (simp add: idempotency_L)\n[PROOF STATE]\nproof (state)\nthis:\nCn (\\ {Cn B |B. P B}) \\ Cn (\\ {B. P B})\n\ngoal (1 subgoal):\n 1. Cn (\\ {B. P B}) \\ Cn (\\ {Cn B |B. P B})\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Cn (\\ {B. P B}) \\ Cn (\\ {Cn B |B. P B})\n[PROOF STEP]\nshow \\?B \\ ?A\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Cn (\\ {B. P B}) \\ Cn (\\ {Cn B |B. P B})\n[PROOF STEP]\nby (metis (mono_tags, lifting) Union_subsetI inclusion_L mem_Collect_eq monotonicity_L)\n[PROOF STATE]\nproof (state)\nthis:\nCn (\\ {B. P B}) \\ Cn (\\ {Cn B |B. P B})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n\n\\ \\\nThe intersection of two closures is closed.\n\\", "meta": {"llama_tokens": 1079, "file": "Belief_Revision_AGM_Logic", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438950868503682, "lm_q2_score": 0.8705972583359805, "lm_q1q2_score": 0.7346927489351347}} {"text": "[STATEMENT]\nlemma sum_list_replicate: \"sum_list (replicate n c) = n*c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list (replicate n c) = n * c\n[PROOF STEP]\napply(induction n)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. sum_list (replicate 0 c) = 0 * c\n 2. \\n. sum_list (replicate n c) = n * c \\ sum_list (replicate (Suc n) c) = Suc n * c\n[PROOF STEP]\napply(auto simp add: ring_class.ring_distribs(2))\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 223, "file": "BTree_BTree_Height", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.879146761176671, "lm_q2_score": 0.8354835432479663, "lm_q1q2_score": 0.7345126510628587}} {"text": "[STATEMENT]\nlemma dim_vec_lincomb [simp]:\n assumes \"finite F\" and \"f: F \\ UNIV\" and \"F \\ carrier_vec n\"\n shows \"dim_vec (module.lincomb (module_cpx_vec n) f F) = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_vec (module.lincomb (module_cpx_vec n) f F) = n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite F\nf \\ F \\ UNIV\nF \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. dim_vec (module.lincomb (module_cpx_vec n) f F) = n\n[PROOF STEP]\nproof(induct F)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\f \\ {} \\ UNIV; {} \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f {}) = n\n 2. \\x F. \\finite F; x \\ F; \\f \\ F \\ UNIV; F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f F) = n; f \\ insert x F \\ UNIV; insert x F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = n\n[PROOF STEP]\ncase empty\n[PROOF STATE]\nproof (state)\nthis:\nf \\ {} \\ UNIV\n{} \\ carrier_vec n\n\ngoal (2 subgoals):\n 1. \\f \\ {} \\ UNIV; {} \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f {}) = n\n 2. \\x F. \\finite F; x \\ F; \\f \\ F \\ UNIV; F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f F) = n; f \\ insert x F \\ UNIV; insert x F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = n\n[PROOF STEP]\nshow \"dim_vec (module.lincomb (module_cpx_vec n) f {}) = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_vec (module.lincomb (module_cpx_vec n) f {}) = n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dim_vec (module.lincomb (module_cpx_vec n) f {}) = n\n[PROOF STEP]\nhave \"module.lincomb (module_cpx_vec n) f {} = 0\\<^sub>v n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. module.lincomb (module_cpx_vec n) f {} = 0\\<^sub>v n\n[PROOF STEP]\nusing module.lincomb_def abelian_monoid.finsum_empty module_cpx_vec_def vecspace_cpx_vec vectorspace_def\n[PROOF STATE]\nproof (prove)\nusing this:\nModule.module ?R ?M \\ module.lincomb ?M ?a ?A = (\\\\<^bsub>?M\\<^esub>v\\?A. ?a v \\\\<^bsub>?M\\<^esub> v)\nabelian_monoid ?G \\ finsum ?G ?f {} = \\\\<^bsub>?G\\<^esub>\nmodule_cpx_vec ?n \\ module_vec TYPE(complex) ?n\nvectorspace cpx_rng (module_cpx_vec ?n)\nvectorspace ?K ?V \\ Module.module ?K ?V \\ field ?K\n\ngoal (1 subgoal):\n 1. module.lincomb (module_cpx_vec n) f {} = 0\\<^sub>v n\n[PROOF STEP]\nby (smt abelian_group_def Module.module_def module_vec_simps(2))\n[PROOF STATE]\nproof (state)\nthis:\nmodule.lincomb (module_cpx_vec n) f {} = 0\\<^sub>v n\n\ngoal (1 subgoal):\n 1. dim_vec (module.lincomb (module_cpx_vec n) f {}) = n\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nmodule.lincomb (module_cpx_vec n) f {} = 0\\<^sub>v n\n\ngoal (1 subgoal):\n 1. dim_vec (module.lincomb (module_cpx_vec n) f {}) = n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec (module.lincomb (module_cpx_vec n) f {}) = n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec (module.lincomb (module_cpx_vec n) f {}) = n\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\f \\ F \\ UNIV; F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f F) = n; f \\ insert x F \\ UNIV; insert x F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = n\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\f \\ F \\ UNIV; F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f F) = n; f \\ insert x F \\ UNIV; insert x F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = n\n[PROOF STEP]\ncase (insert x F)\n[PROOF STATE]\nproof (state)\nthis:\nfinite F\nx \\ F\n\\f \\ F \\ UNIV; F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f F) = n\nf \\ insert x F \\ UNIV\ninsert x F \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\f \\ F \\ UNIV; F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f F) = n; f \\ insert x F \\ UNIV; insert x F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = n\n[PROOF STEP]\nhence \"module.lincomb (module_cpx_vec n) f (insert x F) = \n (f x \\\\<^sub>v x) \\\\<^bsub>module_cpx_vec n\\<^esub> module.lincomb (module_cpx_vec n) f F\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite F\nx \\ F\n\\f \\ F \\ UNIV; F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f F) = n\nf \\ insert x F \\ UNIV\ninsert x F \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. module.lincomb (module_cpx_vec n) f (insert x F) = f x \\\\<^sub>v x \\\\<^bsub>module_cpx_vec n\\<^esub> module.lincomb (module_cpx_vec n) f F\n[PROOF STEP]\nusing module_cpx_vec_def module_vec_def module_cpx_vec module.lincomb_insert cpx_rng_def insert_subset\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite F\nx \\ F\n\\f \\ F \\ UNIV; F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f F) = n\nf \\ insert x F \\ UNIV\ninsert x F \\ carrier_vec n\nmodule_cpx_vec ?n \\ module_vec TYPE(complex) ?n\nmodule_vec ?ty ?n \\ \\carrier = carrier_vec ?n, monoid.mult = undefined, one = undefined, zero = 0\\<^sub>v ?n, add = (+), module.smult = (\\\\<^sub>v)\\\nModule.module cpx_rng (module_cpx_vec ?n)\n\\Module.module ?R ?M; finite ?S; ?S \\ carrier ?M; ?a \\ ?S \\ {?v} \\ carrier ?R; ?v \\ ?S; ?v \\ carrier ?M\\ \\ module.lincomb ?M ?a (?S \\ {?v}) = ?a ?v \\\\<^bsub>?M\\<^esub> ?v \\\\<^bsub>?M\\<^esub> module.lincomb ?M ?a ?S\ncpx_rng \\ \\carrier = UNIV, monoid.mult = (*), one = 1, zero = 0, add = (+)\\\n(insert ?x ?A \\ ?B) = (?x \\ ?B \\ ?A \\ ?B)\n\ngoal (1 subgoal):\n 1. module.lincomb (module_cpx_vec n) f (insert x F) = f x \\\\<^sub>v x \\\\<^bsub>module_cpx_vec n\\<^esub> module.lincomb (module_cpx_vec n) f F\n[PROOF STEP]\nby (smt Pi_I' UNIV_I Un_insert_right module_vec_simps(4) partial_object.select_convs(1) sup_bot.comm_neutral)\n[PROOF STATE]\nproof (state)\nthis:\nmodule.lincomb (module_cpx_vec n) f (insert x F) = f x \\\\<^sub>v x \\\\<^bsub>module_cpx_vec n\\<^esub> module.lincomb (module_cpx_vec n) f F\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\f \\ F \\ UNIV; F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f F) = n; f \\ insert x F \\ UNIV; insert x F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = n\n[PROOF STEP]\nhence \"dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = \n dim_vec (module.lincomb (module_cpx_vec n) f F)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmodule.lincomb (module_cpx_vec n) f (insert x F) = f x \\\\<^sub>v x \\\\<^bsub>module_cpx_vec n\\<^esub> module.lincomb (module_cpx_vec n) f F\n\ngoal (1 subgoal):\n 1. dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = dim_vec (module.lincomb (module_cpx_vec n) f F)\n[PROOF STEP]\nusing index_add_vec\n[PROOF STATE]\nproof (prove)\nusing this:\nmodule.lincomb (module_cpx_vec n) f (insert x F) = f x \\\\<^sub>v x \\\\<^bsub>module_cpx_vec n\\<^esub> module.lincomb (module_cpx_vec n) f F\n?i < dim_vec ?v\\<^sub>2 \\ (?v\\<^sub>1 + ?v\\<^sub>2) $ ?i = ?v\\<^sub>1 $ ?i + ?v\\<^sub>2 $ ?i\ndim_vec (?v\\<^sub>1 + ?v\\<^sub>2) = dim_vec ?v\\<^sub>2\n\ngoal (1 subgoal):\n 1. dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = dim_vec (module.lincomb (module_cpx_vec n) f F)\n[PROOF STEP]\nby (simp add: module_cpx_vec_def module_vec_simps(1))\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = dim_vec (module.lincomb (module_cpx_vec n) f F)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\f \\ F \\ UNIV; F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f F) = n; f \\ insert x F \\ UNIV; insert x F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = n\n[PROOF STEP]\nthus \"dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = n\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = dim_vec (module.lincomb (module_cpx_vec n) f F)\n\ngoal (1 subgoal):\n 1. dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = n\n[PROOF STEP]\nusing insert.hyps(3) insert.prems(2)\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = dim_vec (module.lincomb (module_cpx_vec n) f F)\n\\f \\ F \\ UNIV; F \\ carrier_vec n\\ \\ dim_vec (module.lincomb (module_cpx_vec n) f F) = n\ninsert x F \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. dim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec (module.lincomb (module_cpx_vec n) f (insert x F)) = n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4183, "file": "Isabelle_Marries_Dirac_Complex_Vectors", "length": 23, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467580102419, "lm_q2_score": 0.8354835452961425, "lm_q1q2_score": 0.7345126502180068}} {"text": "[STATEMENT]\nlemma local_lipschitz_temp_dyn:\n assumes \"0 < (a::real)\"\n shows \"local_lipschitz UNIV UNIV (\\t::real. f a L)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local_lipschitz UNIV UNIV (\\t. f a L)\n[PROOF STEP]\napply(unfold local_lipschitz_def lipschitz_on_def dist_norm)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\UNIV. \\t\\UNIV. \\u>0. \\La. \\t\\cball t u \\ UNIV. 0 \\ La \\ (\\xa\\cball x u \\ UNIV. \\y\\cball x u \\ UNIV. \\f a L xa - f a L y\\ \\ La \\ \\xa - y\\)\n[PROOF STEP]\napply(clarsimp, rule_tac x=1 in exI, clarsimp, rule_tac x=a in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a \\ \\xa - y\\)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a \\ \\xa - y\\)\n[PROOF STEP]\napply(simp_all add: norm_diff_temp_dyn)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. \\dist t ta \\ 1; 0 < a\\ \\ \\xa\\cball x 1. \\y\\cball x 1. \\xa $ 1 - y $ 1\\ \\ \\xa - y\\\n[PROOF STEP]\napply(simp add: norm_vec_def L2_set_def, unfold UNIV_4, clarsimp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ \\xa $ 1 - y $ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nunfolding real_sqrt_abs[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2) \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nby (rule real_le_lsqrt) auto", "meta": {"llama_tokens": 1105, "file": "Hybrid_Systems_VCs_PredicateTransformers_HS_VC_PT_Examples", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467580102418, "lm_q2_score": 0.8354835432479661, "lm_q1q2_score": 0.7345126484173591}} {"text": "[STATEMENT]\nlemma local_lipschitz_temp_dyn:\n assumes \"0 < (a::real)\"\n shows \"local_lipschitz UNIV UNIV (\\t::real. f a L)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local_lipschitz UNIV UNIV (\\t. f a L)\n[PROOF STEP]\napply(unfold local_lipschitz_def lipschitz_on_def dist_norm)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\UNIV. \\t\\UNIV. \\u>0. \\La. \\t\\cball t u \\ UNIV. 0 \\ La \\ (\\xa\\cball x u \\ UNIV. \\y\\cball x u \\ UNIV. \\f a L xa - f a L y\\ \\ La \\ \\xa - y\\)\n[PROOF STEP]\napply(clarsimp, rule_tac x=1 in exI, clarsimp, rule_tac x=a in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a \\ \\xa - y\\)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a \\ \\xa - y\\)\n[PROOF STEP]\napply(simp_all add: norm_diff_temp_dyn)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. \\dist t ta \\ 1; 0 < a\\ \\ \\xa\\cball x 1. \\y\\cball x 1. \\xa $ 1 - y $ 1\\ \\ \\xa - y\\\n[PROOF STEP]\napply(simp add: norm_vec_def L2_set_def, unfold UNIV_4, clarsimp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ \\xa $ 1 - y $ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nunfolding real_sqrt_abs[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2) \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nby (rule real_le_lsqrt) auto", "meta": {"llama_tokens": 1105, "file": "Hybrid_Systems_VCs_ModalKleeneAlgebra_HS_VC_MKA_Examples_ndfun", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467580102418, "lm_q2_score": 0.8354835432479661, "lm_q1q2_score": 0.7345126484173591}} {"text": "[STATEMENT]\nlemma per_col_adjoint_row:\n assumes \"A \\ carrier_mat n n\"\n and \"i < n\" \n and \"f i < n\"\nshows \"Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = \n Matrix.row (Complex_Matrix.adjoint A) (f i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\nhave \"per_col A f \\ carrier_mat n n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. per_col A f \\ carrier_mat n n\n[PROOF STEP]\nusing assms per_col_carrier[of A]\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n n\ni < n\nf i < n\nA \\ carrier_mat ?n ?m \\ per_col A ?f \\ carrier_mat ?n ?m\n\ngoal (1 subgoal):\n 1. per_col A f \\ carrier_mat n n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nper_col A f \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\nhence \"Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = \n conjugate (Matrix.col (per_col A f) i)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nper_col A f \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = conjugate (Matrix.col (per_col A f) i)\n[PROOF STEP]\nusing assms adjoint_row[of i \"per_col A f\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nper_col A f \\ carrier_mat n n\nA \\ carrier_mat n n\ni < n\nf i < n\ni < dim_col (per_col A f) \\ Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = conjugate (Matrix.col (per_col A f) i)\n\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = conjugate (Matrix.col (per_col A f) i)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nMatrix.row (Complex_Matrix.adjoint (per_col A f)) i = conjugate (Matrix.col (per_col A f) i)\n\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nMatrix.row (Complex_Matrix.adjoint (per_col A f)) i = conjugate (Matrix.col (per_col A f) i)\n\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\nhave \"... = conjugate (Matrix.col A (f i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. conjugate (Matrix.col (per_col A f) i) = conjugate (Matrix.col A (f i))\n[PROOF STEP]\nusing assms per_col_col\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n n\ni < n\nf i < n\n\\?A \\ carrier_mat ?n ?m; ?j < ?m\\ \\ Matrix.col (per_col ?A ?f) ?j = Matrix.col ?A (?f ?j)\n\ngoal (1 subgoal):\n 1. conjugate (Matrix.col (per_col A f) i) = conjugate (Matrix.col A (f i))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nconjugate (Matrix.col (per_col A f) i) = conjugate (Matrix.col A (f i))\n\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nconjugate (Matrix.col (per_col A f) i) = conjugate (Matrix.col A (f i))\n\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\nhave \"... = Matrix.row (Complex_Matrix.adjoint A) (f i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. conjugate (Matrix.col A (f i)) = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\nusing assms \n adjoint_row[of \"f i\" A]\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n n\ni < n\nf i < n\nf i < dim_col A \\ Matrix.row (Complex_Matrix.adjoint A) (f i) = conjugate (Matrix.col A (f i))\n\ngoal (1 subgoal):\n 1. conjugate (Matrix.col A (f i)) = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nconjugate (Matrix.col A (f i)) = Matrix.row (Complex_Matrix.adjoint A) (f i)\n\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nMatrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nMatrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nMatrix.row (Complex_Matrix.adjoint (per_col A f)) i = Matrix.row (Complex_Matrix.adjoint A) (f i)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2057, "file": "Commuting_Hermitian_Commuting_Hermitian", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467580102419, "lm_q2_score": 0.8354835309589074, "lm_q1q2_score": 0.734512637613473}} {"text": "[STATEMENT]\nlemma half_1_minus_sum:\n \"1 - (\\iii<(2::nat). P i) \\ P 0 \\ P 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i<2. P i) = (P 0 \\ P 1)\n[PROOF STEP]\nusing less_2_cases\n[PROOF STATE]\nproof (prove)\nusing this:\n?n < 2 \\ ?n = 0 \\ ?n = Suc 0\n\ngoal (1 subgoal):\n 1. (\\i<2. P i) = (P 0 \\ P 1)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 188, "file": "Poincare_Bendixson_Poincare_Bendixson", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094117351309, "lm_q2_score": 0.8221891305219504, "lm_q1q2_score": 0.7344692885215823}} {"text": "[STATEMENT]\nlemma mult_mat_vec_smult_vec_assoc:\n fixes A :: \"'a::comm_ring_1 mat\"\n assumes A: \"A \\ carrier_mat n m\" and w: \"w \\ carrier_vec m\"\n shows \"A *\\<^sub>v (a \\\\<^sub>v w) = a \\\\<^sub>v (A *\\<^sub>v w)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *\\<^sub>v (a \\\\<^sub>v w) = a \\\\<^sub>v (A *\\<^sub>v w)\n[PROOF STEP]\napply (rule eq_vecI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\i. i < dim_vec (a \\\\<^sub>v (A *\\<^sub>v w)) \\ (A *\\<^sub>v (a \\\\<^sub>v w)) $ i = (a \\\\<^sub>v (A *\\<^sub>v w)) $ i\n 2. dim_vec (A *\\<^sub>v (a \\\\<^sub>v w)) = dim_vec (a \\\\<^sub>v (A *\\<^sub>v w))\n[PROOF STEP]\napply (simp add: scalar_prod_def carrier_matD[OF A] carrier_vecD[OF w])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\i. i < n \\ (\\ia = 0..ia = 0..v (a \\\\<^sub>v w)) = dim_vec (a \\\\<^sub>v (A *\\<^sub>v w))\n[PROOF STEP]\napply (subst sum_distrib_left)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\i. i < n \\ (\\ia = 0..n = 0..v (a \\\\<^sub>v w)) = dim_vec (a \\\\<^sub>v (A *\\<^sub>v w))\n[PROOF STEP]\napply (rule sum.cong, simp)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\i x. \\i < n; x \\ {0.. \\ A $$ (i, x) * (a * w $ x) = a * (A $$ (i, x) * w $ x)\n 2. dim_vec (A *\\<^sub>v (a \\\\<^sub>v w)) = dim_vec (a \\\\<^sub>v (A *\\<^sub>v w))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 807, "file": "QHLProver_Complex_Matrix", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094174159129, "lm_q2_score": 0.8221891239865619, "lm_q1q2_score": 0.7344692873541353}} {"text": "[STATEMENT]\nlemma hom_component_plus: \"hom_component (p + q) n = hom_component p n + hom_component q n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. hom_component (p + q) n = hom_component p n + hom_component q n\n[PROOF STEP]\nby (rule poly_mapping_eqI) (simp add: hom_component_def lookup_except lookup_add)", "meta": {"llama_tokens": 115, "file": "Polynomials_MPoly_PM", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933093946927837, "lm_q2_score": 0.8221891283434876, "lm_q1q2_score": 0.7344692725635084}} {"text": "[STATEMENT]\nlemma polyfun_Prod:\nassumes \"finite I\"\nassumes \"\\i. i\\I \\ polyfun N (f i)\"\nshows \"polyfun N (\\x. \\i\\I. f i x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. polyfun N (\\x. \\i\\I. f i x)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite I\n?i \\ I \\ polyfun N (f ?i)\n\ngoal (1 subgoal):\n 1. polyfun N (\\x. \\i\\I. f i x)\n[PROOF STEP]\napply (induction I rule:finite_induct)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (\\i. i \\ {} \\ polyfun N (f i)) \\ polyfun N (\\x. \\i\\{}. f i x)\n 2. \\x F. \\finite F; x \\ F; (\\i. i \\ F \\ polyfun N (f i)) \\ polyfun N (\\x. \\i\\F. f i x); \\i. i \\ insert x F \\ polyfun N (f i)\\ \\ polyfun N (\\xa. \\i\\insert x F. f i xa)\n[PROOF STEP]\napply (simp add: polyfun_const)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; (\\i. i \\ F \\ polyfun N (f i)) \\ polyfun N (\\x. \\i\\F. f i x); \\i. i \\ insert x F \\ polyfun N (f i)\\ \\ polyfun N (\\xa. \\i\\insert x F. f i xa)\n[PROOF STEP]\nusing comm_monoid_add_class.sum.insert polyfun_mult\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite ?A; ?x \\ ?A\\ \\ sum ?g (insert ?x ?A) = ?g ?x + sum ?g ?A\n\\polyfun ?N ?f; polyfun ?N ?g\\ \\ polyfun ?N (\\x. ?f x * ?g x)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; (\\i. i \\ F \\ polyfun N (f i)) \\ polyfun N (\\x. \\i\\F. f i x); \\i. i \\ insert x F \\ polyfun N (f i)\\ \\ polyfun N (\\xa. \\i\\insert x F. f i xa)\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 853, "file": "Polynomials_More_MPoly_Type", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473746782093, "lm_q2_score": 0.8418256452674008, "lm_q1q2_score": 0.7343643915858066}} {"text": "[STATEMENT]\nlemma integrable_Beta':\n assumes \"a > 0\" \"b > (0::real)\"\n shows \"(\\t. t powr (a - 1) * (1 - t) powr (b - 1)) integrable_on {0..1}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\t. t powr (a - 1) * (1 - t) powr (b - 1)) integrable_on {0..1}\n[PROOF STEP]\nusing integrable_Beta[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nset_integrable lborel {0..1} (\\t. t powr (a - 1) * (1 - t) powr (b - 1))\n\ngoal (1 subgoal):\n 1. (\\t. t powr (a - 1) * (1 - t) powr (b - 1)) integrable_on {0..1}\n[PROOF STEP]\nby (rule set_borel_integral_eq_integral)", "meta": {"llama_tokens": 286, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942014971871, "lm_q2_score": 0.8128673201042492, "lm_q1q2_score": 0.7342583368367261}} {"text": "[STATEMENT]\ntheorem harry_sum_closed_form: \"harry_sum n = n * (n + 1) * 2 ^ n div 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. harry_sum n = n * (n + 1) * 2 ^ n div 4\n[PROOF STEP]\nusing harry_sum_closed_form_aux[of n]\n[PROOF STATE]\nproof (prove)\nusing this:\n4 * harry_sum n = n * (n + 1) * 2 ^ n\n\ngoal (1 subgoal):\n 1. harry_sum n = n * (n + 1) * 2 ^ n div 4\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 193, "file": "IMO2019_IMO2019_Q5", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942119105695, "lm_q2_score": 0.8128673087708699, "lm_q1q2_score": 0.7342583350640485}} {"text": "[STATEMENT]\nlemma cos_tan_half: \"cos x \\0 \\ cos (2*x) = (1 - (tan x)^2) / (1+ (tan x)^2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos x \\ (0::'a) \\ cos ((2::'a) * x) = ((1::'a) - (tan x)\\<^sup>2) / ((1::'a) + (tan x)\\<^sup>2)\n[PROOF STEP]\nunfolding cos_double tan_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos x \\ (0::'a) \\ (cos x)\\<^sup>2 - (sin x)\\<^sup>2 = ((1::'a) - (sin x / cos x)\\<^sup>2) / ((1::'a) + (sin x / cos x)\\<^sup>2)\n[PROOF STEP]\nby (auto simp add:field_simps )", "meta": {"llama_tokens": 273, "file": "Winding_Number_Eval_Missing_Transcendental", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942014971871, "lm_q2_score": 0.8128673178375734, "lm_q1q2_score": 0.7342583347892511}} {"text": "[STATEMENT]\ntheorem exp_times_self_eq_iff:\n \"w * exp w = x \\ x \\ -exp (-1) \\ (w = Lambert_W x \\ x < 0 \\ w = Lambert_W' x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (w * exp w = x) = (- exp (- 1) \\ x \\ (w = Lambert_W x \\ x < 0 \\ w = Lambert_W' x))\n[PROOF STEP]\nusing exp_times_self_eqD[of w x]\n[PROOF STATE]\nproof (prove)\nusing this:\nw * exp w = x \\ - exp (- 1) \\ x\nw * exp w = x \\ w = Lambert_W x \\ x < 0 \\ w = Lambert_W' x\n\ngoal (1 subgoal):\n 1. (w * exp w = x) = (- exp (- 1) \\ x \\ (w = Lambert_W x \\ x < 0 \\ w = Lambert_W' x))\n[PROOF STEP]\nby (auto simp: Lambert_W_times_exp_self Lambert_W'_times_exp_self)", "meta": {"llama_tokens": 333, "file": "Lambert_W_Lambert_W", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9086179018818865, "lm_q2_score": 0.8080672181749422, "lm_q1q2_score": 0.7342243403576486}} {"text": "[STATEMENT]\nlemma mat_kernel: assumes \"A \\ carrier_mat nr nc\" \n shows \"mat_kernel A = {v. v \\ carrier_vec nc \\ A *\\<^sub>v v = 0\\<^sub>v nr}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat_kernel A = {v \\ carrier_vec nc. A *\\<^sub>v v = 0\\<^sub>v nr}\n[PROOF STEP]\nunfolding mat_kernel_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {v \\ carrier_vec (dim_col A). A *\\<^sub>v v = 0\\<^sub>v (dim_row A)} = {v \\ carrier_vec nc. A *\\<^sub>v v = 0\\<^sub>v nr}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat nr nc\n\ngoal (1 subgoal):\n 1. {v \\ carrier_vec (dim_col A). A *\\<^sub>v v = 0\\<^sub>v (dim_row A)} = {v \\ carrier_vec nc. A *\\<^sub>v v = 0\\<^sub>v nr}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 356, "file": "Jordan_Normal_Form_Matrix_Kernel", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297807787537, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7341411389624231}} {"text": "[STATEMENT]\nlemma arsinh_minus_real [simp]: \"arsinh (-x::real) = -arsinh x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arsinh (- x) = - arsinh x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. arsinh (- x) = - arsinh x\n[PROOF STEP]\nhave \"arsinh (-x) = ln (sqrt (x\\<^sup>2 + 1) - x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arsinh (- x) = ln (sqrt (x\\<^sup>2 + 1) - x)\n[PROOF STEP]\nby (simp add: arsinh_real_def)\n[PROOF STATE]\nproof (state)\nthis:\narsinh (- x) = ln (sqrt (x\\<^sup>2 + 1) - x)\n\ngoal (1 subgoal):\n 1. arsinh (- x) = - arsinh x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\narsinh (- x) = ln (sqrt (x\\<^sup>2 + 1) - x)\n\ngoal (1 subgoal):\n 1. arsinh (- x) = - arsinh x\n[PROOF STEP]\nhave \"sqrt (x^2 + 1) - x = inverse (sqrt (x^2 + 1) + x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (x\\<^sup>2 + 1) - x = inverse (sqrt (x\\<^sup>2 + 1) + x)\n[PROOF STEP]\nusing arsinh_real_aux[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x + sqrt (x\\<^sup>2 + 1)\n\ngoal (1 subgoal):\n 1. sqrt (x\\<^sup>2 + 1) - x = inverse (sqrt (x\\<^sup>2 + 1) + x)\n[PROOF STEP]\nby (simp add: field_split_simps algebra_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (x\\<^sup>2 + 1) - x = inverse (sqrt (x\\<^sup>2 + 1) + x)\n\ngoal (1 subgoal):\n 1. arsinh (- x) = - arsinh x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (x\\<^sup>2 + 1) - x = inverse (sqrt (x\\<^sup>2 + 1) + x)\n\ngoal (1 subgoal):\n 1. arsinh (- x) = - arsinh x\n[PROOF STEP]\nhave \"ln \\ = -arsinh x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ln (inverse (sqrt (x\\<^sup>2 + 1) + x)) = - arsinh x\n[PROOF STEP]\nusing arsinh_real_aux[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x + sqrt (x\\<^sup>2 + 1)\n\ngoal (1 subgoal):\n 1. ln (inverse (sqrt (x\\<^sup>2 + 1) + x)) = - arsinh x\n[PROOF STEP]\nby (simp add: arsinh_real_def ln_inverse)\n[PROOF STATE]\nproof (state)\nthis:\nln (inverse (sqrt (x\\<^sup>2 + 1) + x)) = - arsinh x\n\ngoal (1 subgoal):\n 1. arsinh (- x) = - arsinh x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\narsinh (- x) = - arsinh x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\narsinh (- x) = - arsinh x\n\ngoal (1 subgoal):\n 1. arsinh (- x) = - arsinh x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\narsinh (- x) = - arsinh x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1151, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637577007394, "lm_q2_score": 0.8539127585282744, "lm_q1q2_score": 0.7340778507450205}} {"text": "[STATEMENT]\nlemma diag_mat_mul_diag_ele:\n fixes A B :: \"complex mat\"\n assumes dimA: \"A \\ carrier_mat n n\" and dimB: \"B \\ carrier_mat n n\"\n and dA: \"diagonal_mat A\" and dB: \"diagonal_mat B\"\n shows \"\\iii(i,j). if i = j then (A$$(i, i)) * (B$$(i, i)) else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A * B = mat n n (\\(i, j). if i = j then A $$ (i, i) * B $$ (i, i) else 0)\n[PROOF STEP]\nusing diag_mat_mult_diag_mat[of A n B] dimA dimB dA dB\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A \\ carrier_mat n n; B \\ carrier_mat n n; diagonal_mat A; diagonal_mat B\\ \\ A * B = mat n n (\\(i, j). if i = j then A $$ (i, i) * B $$ (i, i) else 0)\nA \\ carrier_mat n n\nB \\ carrier_mat n n\ndiagonal_mat A\ndiagonal_mat B\n\ngoal (1 subgoal):\n 1. A * B = mat n n (\\(i, j). if i = j then A $$ (i, i) * B $$ (i, i) else 0)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA * B = mat n n (\\(i, j). if i = j then A $$ (i, i) * B $$ (i, i) else 0)\n\ngoal (1 subgoal):\n 1. \\i(i, j). if i = j then A $$ (i, i) * B $$ (i, i) else 0)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nA * B = mat n n (\\(i, j). if i = j then A $$ (i, i) * B $$ (i, i) else 0)\n\ngoal (1 subgoal):\n 1. \\i(i, j). if i = j then A $$ (i, i) * B $$ (i, i) else 0)\nA * B = mat n n (\\(i, j). if i = j then A $$ (i, i) * B $$ (i, i) else 0)\n\ngoal (1 subgoal):\n 1. \\iiia. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ ia $ ia = 1\n 2. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\ndefine Q where \"Q = fst (QR_decomposition A)\"\n[PROOF STATE]\nproof (state)\nthis:\nQ = fst (QR_decomposition A)\n\ngoal (2 subgoals):\n 1. \\ia. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ ia $ ia = 1\n 2. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nhave n: \"\\i. norm (column i Q) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. norm (column i Q) = 1\n[PROOF STEP]\nunfolding Q_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. norm (column i (fst (QR_decomposition A))) = 1\n[PROOF STEP]\nusing norm_columns_fst_QR_decomposition[OF r]\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (column ?i (fst (QR_decomposition A))) = 1\n\ngoal (1 subgoal):\n 1. \\i. norm (column i (fst (QR_decomposition A))) = 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i. norm (column i Q) = 1\n\ngoal (2 subgoals):\n 1. \\ia. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ ia $ ia = 1\n 2. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nhave c: \"card (columns Q) = ncols A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (columns Q) = ncols A\n[PROOF STEP]\nunfolding Q_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (columns (fst (QR_decomposition A))) = ncols A\n[PROOF STEP]\nby (metis full_rank_imp_is_basis2 independent_columns_fst_QR_decomposition r)\n[PROOF STATE]\nproof (state)\nthis:\ncard (columns Q) = ncols A\n\ngoal (2 subgoals):\n 1. \\ia. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ ia $ ia = 1\n 2. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nhave p: \"pairwise orthogonal (columns Q)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pairwise orthogonal (columns Q)\n[PROOF STEP]\nby (metis Q_def orthogonal_fst_QR_decomposition)\n[PROOF STATE]\nproof (state)\nthis:\npairwise orthogonal (columns Q)\n\ngoal (2 subgoals):\n 1. \\ia. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ ia $ ia = 1\n 2. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nfix ia\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\ia. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ ia $ ia = 1\n 2. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nhave \"(transpose Q ** Q) $ ia $ ia = column ia Q \\ column ia Q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose Q ** Q) $ ia $ ia = column ia Q \\ column ia Q\n[PROOF STEP]\nunfolding matrix_matrix_mult_inner_mult\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row ia (Finite_Cartesian_Product.transpose Q) \\ column ia Q = column ia Q \\ column ia Q\n[PROOF STEP]\nunfolding row_transpose\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. column ia Q \\ column ia Q = column ia Q \\ column ia Q\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose Q ** Q) $ ia $ ia = column ia Q \\ column ia Q\n\ngoal (2 subgoals):\n 1. \\ia. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ ia $ ia = 1\n 2. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose Q ** Q) $ ia $ ia = column ia Q \\ column ia Q\n\ngoal (2 subgoals):\n 1. \\ia. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ ia $ ia = 1\n 2. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nhave \"... = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. column ia Q \\ column ia Q = 1\n[PROOF STEP]\nusing n norm_eq_1\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i. norm (column i Q) = 1\n(norm ?x = 1) = (?x \\ ?x = 1)\n\ngoal (1 subgoal):\n 1. column ia Q \\ column ia Q = 1\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ncolumn ia Q \\ column ia Q = 1\n\ngoal (2 subgoals):\n 1. \\ia. (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ ia $ ia = 1\n 2. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(Finite_Cartesian_Product.transpose Q ** Q) $ ia $ ia = 1\n[PROOF STEP]\nshow \"(transpose Q ** Q) $ ia $ ia = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(Finite_Cartesian_Product.transpose Q ** Q) $ ia $ ia = 1\n\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose Q ** Q) $ ia $ ia = 1\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose Q ** Q) $ ia $ ia = 1\n\ngoal (1 subgoal):\n 1. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nfix i\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nassume i_not_ia: \"i \\ ia\"\n[PROOF STATE]\nproof (state)\nthis:\ni \\ ia\n\ngoal (1 subgoal):\n 1. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nhave column_i_not_ia: \"column i Q \\ column ia Q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. column i Q \\ column ia Q\n[PROOF STEP]\nproof (rule ccontr, simp)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. column i Q = column ia Q \\ False\n[PROOF STEP]\nassume col_i_ia: \"column i Q = column ia Q\"\n[PROOF STATE]\nproof (state)\nthis:\ncolumn i Q = column ia Q\n\ngoal (1 subgoal):\n 1. column i Q = column ia Q \\ False\n[PROOF STEP]\nhave rw: \"(\\i. column i Q)` (UNIV-{ia}) = {column i Q|i. i\\ia}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. column i Q) ` (UNIV - {ia}) = {column i Q |i. i \\ ia}\n[PROOF STEP]\nunfolding columns_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. column i Q) ` (UNIV - {ia}) = {column i Q |i. i \\ ia}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\i. column i Q) ` (UNIV - {ia}) = {column i Q |i. i \\ ia}\n\ngoal (1 subgoal):\n 1. column i Q = column ia Q \\ False\n[PROOF STEP]\nhave \"card (columns Q) = card ({column i Q|i. i\\ia})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (columns Q) = card {column i Q |i. i \\ ia}\n[PROOF STEP]\nby (rule bij_betw_same_card[of id], unfold bij_betw_def columns_def, auto, metis col_i_ia i_not_ia)\n[PROOF STATE]\nproof (state)\nthis:\ncard (columns Q) = card {column i Q |i. i \\ ia}\n\ngoal (1 subgoal):\n 1. column i Q = column ia Q \\ False\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (columns Q) = card {column i Q |i. i \\ ia}\n\ngoal (1 subgoal):\n 1. column i Q = column ia Q \\ False\n[PROOF STEP]\nhave \"... = card ((\\i. column i Q)` (UNIV-{ia}))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {column i Q |i. i \\ ia} = card ((\\i. column i Q) ` (UNIV - {ia}))\n[PROOF STEP]\nunfolding rw\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {column i Q |i. i \\ ia} = card {column i Q |i. i \\ ia}\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\ncard {column i Q |i. i \\ ia} = card ((\\i. column i Q) ` (UNIV - {ia}))\n\ngoal (1 subgoal):\n 1. column i Q = column ia Q \\ False\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {column i Q |i. i \\ ia} = card ((\\i. column i Q) ` (UNIV - {ia}))\n\ngoal (1 subgoal):\n 1. column i Q = column ia Q \\ False\n[PROOF STEP]\nhave \"... \\ card (UNIV - {ia})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((\\i. column i Q) ` (UNIV - {ia})) \\ card (UNIV - {ia})\n[PROOF STEP]\nby (metis card_image_le finite_code)\n[PROOF STATE]\nproof (state)\nthis:\ncard ((\\i. column i Q) ` (UNIV - {ia})) \\ card (UNIV - {ia})\n\ngoal (1 subgoal):\n 1. column i Q = column ia Q \\ False\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((\\i. column i Q) ` (UNIV - {ia})) \\ card (UNIV - {ia})\n\ngoal (1 subgoal):\n 1. column i Q = column ia Q \\ False\n[PROOF STEP]\nhave \"... < CARD ('n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (UNIV - {ia}) < CARD('n)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (UNIV - {ia}) < CARD('n)\n\ngoal (1 subgoal):\n 1. column i Q = column ia Q \\ False\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (columns Q) < CARD('n)\n[PROOF STEP]\nshow False\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (columns Q) < CARD('n)\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nusing c\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (columns Q) < CARD('n)\ncard (columns Q) = ncols A\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nunfolding ncols_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (columns Q) < CARD('n)\ncard (columns Q) = CARD('n)\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncolumn i Q \\ column ia Q\n\ngoal (1 subgoal):\n 1. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nhence oia: \"orthogonal (column i Q) (column ia Q)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncolumn i Q \\ column ia Q\n\ngoal (1 subgoal):\n 1. orthogonal (column i Q) (column ia Q)\n[PROOF STEP]\nusing p\n[PROOF STATE]\nproof (prove)\nusing this:\ncolumn i Q \\ column ia Q\npairwise orthogonal (columns Q)\n\ngoal (1 subgoal):\n 1. orthogonal (column i Q) (column ia Q)\n[PROOF STEP]\nunfolding pairwise_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncolumn i Q \\ column ia Q\n\\x\\columns Q. \\y\\columns Q. x \\ y \\ orthogonal x y\n\ngoal (1 subgoal):\n 1. orthogonal (column i Q) (column ia Q)\n[PROOF STEP]\nunfolding columns_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncolumn i Q \\ column ia Q\n\\x\\{column i Q |i. i \\ UNIV}. \\y\\{column i Q |i. i \\ UNIV}. x \\ y \\ orthogonal x y\n\ngoal (1 subgoal):\n 1. orthogonal (column i Q) (column ia Q)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\northogonal (column i Q) (column ia Q)\n\ngoal (1 subgoal):\n 1. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nhave \"(transpose Q ** Q) $ i $ ia = column i Q \\ column ia Q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose Q ** Q) $ i $ ia = column i Q \\ column ia Q\n[PROOF STEP]\nunfolding matrix_matrix_mult_inner_mult\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row i (Finite_Cartesian_Product.transpose Q) \\ column ia Q = column i Q \\ column ia Q\n[PROOF STEP]\nunfolding row_transpose\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. column i Q \\ column ia Q = column i Q \\ column ia Q\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose Q ** Q) $ i $ ia = column i Q \\ column ia Q\n\ngoal (1 subgoal):\n 1. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose Q ** Q) $ i $ ia = column i Q \\ column ia Q\n\ngoal (1 subgoal):\n 1. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nhave \"... = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. column i Q \\ column ia Q = 0\n[PROOF STEP]\nusing oia\n[PROOF STATE]\nproof (prove)\nusing this:\northogonal (column i Q) (column ia Q)\n\ngoal (1 subgoal):\n 1. column i Q \\ column ia Q = 0\n[PROOF STEP]\nunfolding orthogonal_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncolumn i Q \\ column ia Q = 0\n\ngoal (1 subgoal):\n 1. column i Q \\ column ia Q = 0\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncolumn i Q \\ column ia Q = 0\n\ngoal (1 subgoal):\n 1. \\i ia. i \\ ia \\ (Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A)) $ i $ ia = 0\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(Finite_Cartesian_Product.transpose Q ** Q) $ i $ ia = 0\n[PROOF STEP]\nshow \"(transpose Q ** Q) $ i $ ia = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(Finite_Cartesian_Product.transpose Q ** Q) $ i $ ia = 0\n\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose Q ** Q) $ i $ ia = 0\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose Q ** Q) $ i $ ia = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 6297, "file": "QR_Decomposition_QR_Decomposition", "length": 67, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942377652497, "lm_q2_score": 0.8244619242200082, "lm_q1q2_score": 0.7340137003899233}} {"text": "[STATEMENT]\nlemma scalar_left_mono: assumes \n \"u \\ carrier_vec n\" \"v \\ carrier_vec n\" \"w \\ carrier_vec n\" \n and \"\\ i. i < n \\ u $ i \\ v $ i\"\n and \"\\ i. i < n \\ w $ i \\ (0 :: 'a :: ordered_semiring_0)\"\n shows \"u \\ w \\ v \\ w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v \\ w \\ u \\ w\n[PROOF STEP]\nunfolding scalar_prod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0.. (\\i = 0..(i, j). if i = j then a else if Suc i = j then 1::'b else (0::'b)) + - b \\\\<^sub>m 1\\<^sub>m (dim_row (mat n n (\\(i, j). if i = j then a else if Suc i = j then 1::'b else (0::'b)))) = mat n n (\\(i, j). if i = j then a - b else if Suc i = j then 1::'b else (0::'b))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 283, "file": "Jordan_Normal_Form_Jordan_Normal_Form", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942203004185, "lm_q2_score": 0.8244619306896955, "lm_q1q2_score": 0.7340136917507601}} {"text": "[STATEMENT]\ntheorem \"\\\\ \\True\\\n \\S :== 0;; \\I :== 1;;\n WHILE \\I \\ n\n INV \\\\S = (SUMM j<\\I. j)\\\n DO\n \\S :== \\S + \\I;;\n \\I :== \\I + 1\n OD\n \\\\S = (SUMM j\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\ \\True\\ \\S :== 0;; \\I :== 1;; WHILE \\I \\ n INV \\\\S = (SUMM j<\\I. j)\\ DO \\S :== \\S + \\I;; \\I :== \\I + 1 OD \\\\S = (SUMM j\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\ \\True\\ \\S :== 0;; \\I :== 1;; WHILE \\I \\ n INV \\\\S = (SUMM j<\\I. j)\\ DO \\S :== \\S + \\I;; \\I :== \\I + 1 OD \\\\S = (SUMM j\n[PROOF STEP]\nlet ?sum = \"\\k. SUMM j\\ \\True\\ \\S :== 0;; \\I :== 1;; WHILE \\I \\ n INV \\\\S = (SUMM j<\\I. j)\\ DO \\S :== \\S + \\I;; \\I :== \\I + 1 OD \\\\S = (SUMM j\n[PROOF STEP]\nlet ?inv = \"\\s i. s = ?sum i\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\ \\True\\ \\S :== 0;; \\I :== 1;; WHILE \\I \\ n INV \\\\S = (SUMM j<\\I. j)\\ DO \\S :== \\S + \\I;; \\I :== \\I + 1 OD \\\\S = (SUMM j\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\ \\True\\ \\S :== 0;; \\I :== 1;; WHILE \\I \\ n INV \\\\S = (SUMM j<\\I. j)\\ DO \\S :== \\S + \\I;; \\I :== \\I + 1 OD \\\\S = (SUMM j\n[PROOF STEP]\nproof vcg\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. True \\ 0 = (SUMM j<1. j)\n 2. \\I S. \\S = (SUMM j n\\ \\ S + I = (SUMM jI S. \\S = (SUMM j I \\ n\\ \\ S = (SUMM jI S. \\S = (SUMM j n\\ \\ S + I = (SUMM jI S. \\S = (SUMM j I \\ n\\ \\ S = (SUMM jI S. \\S = (SUMM j n\\ \\ S + I = (SUMM jI S. \\S = (SUMM j I \\ n\\ \\ S = (SUMM jI S. \\S = (SUMM j n\\ \\ S + I = (SUMM jI S. \\S = (SUMM j I \\ n\\ \\ S = (SUMM j n\"\n[PROOF STATE]\nproof (state)\nthis:\ns = (SUMM j n\n\ngoal (2 subgoals):\n 1. \\I S. \\S = (SUMM j n\\ \\ S + I = (SUMM jI S. \\S = (SUMM j I \\ n\\ \\ S = (SUMM j n\n\ngoal (1 subgoal):\n 1. s + i = (SUMM jI S. \\S = (SUMM j I \\ n\\ \\ S = (SUMM jI S. \\S = (SUMM j I \\ n\\ \\ S = (SUMM jI S. \\S = (SUMM j I \\ n\\ \\ S = (SUMM j i \\ n\"\n[PROOF STATE]\nproof (state)\nthis:\ns = (SUMM j i \\ n\n\ngoal (1 subgoal):\n 1. \\I S. \\S = (SUMM j I \\ n\\ \\ S = (SUMM j i \\ n\n\ngoal (1 subgoal):\n 1. s = (SUMM j\\ \\True\\ \\S :== 0;; \\I :== 1;; WHILE \\I \\ n INV \\\\S = (SUMM j<\\I. j)\\ DO \\S :== \\S + \\I;; \\I :== \\I + 1 OD \\\\S = (SUMM j\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2665, "file": "Simpl_ex_VcgEx", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772318846386, "lm_q2_score": 0.8397339716830606, "lm_q1q2_score": 0.7339923454882231}} {"text": "[STATEMENT]\nlemma trace_matrix_sum_linear:\n fixes f :: \"nat \\ complex mat\"\n shows \"(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = sum (\\k. trace (f k)) {0..k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < 0 \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f 0) = (\\k = 0..<0. trace (f k))\n 2. \\n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0.. f ?k \\ carrier_mat d d\n\ngoal (2 subgoals):\n 1. (\\k. k < 0 \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f 0) = (\\k = 0..<0. trace (f k))\n 2. \\n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0..k = 0..<0. trace (f k))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace (matrix_sum d f 0) = (\\k = 0..<0. trace (f k))\n\ngoal (1 subgoal):\n 1. \\n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0..n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0..k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0.. f ?k \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. \\n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0..k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0.. f ?k \\ carrier_mat d d\n[PROOF STEP]\nhave \"\\k. k < n \\ f k \\ carrier_mat d d\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0.. f ?k \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. \\k. k < n \\ f k \\ carrier_mat d d\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n?k < n \\ f ?k \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. \\n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0.. f ?k \\ carrier_mat d d\n[PROOF STEP]\nhave ds: \"matrix_sum d f n \\ carrier_mat d d\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?k < n \\ f ?k \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. matrix_sum d f n \\ carrier_mat d d\n[PROOF STEP]\nusing matrix_sum_dim\n[PROOF STATE]\nproof (prove)\nusing this:\n?k < n \\ f ?k \\ carrier_mat d d\n(\\k. k < ?n \\ ?f k \\ carrier_mat ?d ?d) \\ matrix_sum ?d ?f ?n \\ carrier_mat ?d ?d\n\ngoal (1 subgoal):\n 1. matrix_sum d f n \\ carrier_mat d d\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_sum d f n \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. \\n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0.. carrier_mat d d\n 2. f n \\ carrier_mat d d\n[PROOF STEP]\nusing ds Suc\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix_sum d f n \\ carrier_mat d d\n(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0.. f ?k \\ carrier_mat d d\n\ngoal (2 subgoals):\n 1. matrix_sum d f n \\ carrier_mat d d\n 2. f n \\ carrier_mat d d\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace (matrix_sum d f (Suc n)) = trace (f n) + trace (matrix_sum d f n)\n\ngoal (1 subgoal):\n 1. \\n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0..n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0.. = sum (trace \\ f) {0.. f) n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (f n) + trace (matrix_sum d f n) = sum (trace \\ f) {0.. f) n\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0.. f ?k \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. trace (f n) + trace (matrix_sum d f n) = sum (trace \\ f) {0.. f) n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace (f n) + trace (matrix_sum d f n) = sum (trace \\ f) {0.. f) n\n\ngoal (1 subgoal):\n 1. \\n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0.. f) {0.. f) n\n\ngoal (1 subgoal):\n 1. \\n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0.. = sum (trace \\ f) {0.. f) {0.. f) n = sum (trace \\ f) {0.. f) {0.. f) n = sum (trace \\ f) {0..n. \\(\\k. k < n \\ f k \\ carrier_mat d d) \\ trace (matrix_sum d f n) = (\\k = 0..k. k < Suc n \\ f k \\ carrier_mat d d\\ \\ trace (matrix_sum d f (Suc n)) = (\\k = 0.. f) {0.. f) {0..k = 0..k = 0..) = - pi / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Arg (cor k * \\) = - pi / 2\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Arg (cor k * \\) = - pi / 2\n[PROOF STEP]\nhave \"sin (Arg (Complex 0 k)) < 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin (Arg (Complex 0 k)) < 0\n[PROOF STEP]\nusing \\k < 0\\ rcis_cmod_Arg[of \"Complex 0 k\"] Im_rcis[of \"cmod (Complex 0 k)\" \"Arg (Complex 0 k)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nk < 0\nrcis (cmod (Complex 0 k)) (Arg (Complex 0 k)) = Complex 0 k\nIm (rcis (cmod (Complex 0 k)) (Arg (Complex 0 k))) = cmod (Complex 0 k) * sin (Arg (Complex 0 k))\n\ngoal (1 subgoal):\n 1. sin (Arg (Complex 0 k)) < 0\n[PROOF STEP]\nby (metis complex.sel(2) mult_less_0_iff norm_not_less_zero)\n[PROOF STATE]\nproof (state)\nthis:\nsin (Arg (Complex 0 k)) < 0\n\ngoal (1 subgoal):\n 1. Arg (cor k * \\) = - pi / 2\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsin (Arg (Complex 0 k)) < 0\n\ngoal (1 subgoal):\n 1. Arg (cor k * \\) = - pi / 2\n[PROOF STEP]\nusing assms is_imag_arg2[of \"cor k * \\\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nsin (Arg (Complex 0 k)) < 0\nk < 0\n\\is_imag (cor k * \\); cor k * \\ \\ 0\\ \\ Arg (cor k * \\) = pi / 2 \\ Arg (cor k * \\) = - pi / 2\n\ngoal (1 subgoal):\n 1. Arg (cor k * \\) = - pi / 2\n[PROOF STEP]\nusing Arg_zero complex_of_real_i[of k]\n[PROOF STATE]\nproof (prove)\nusing this:\nsin (Arg (Complex 0 k)) < 0\nk < 0\n\\is_imag (cor k * \\); cor k * \\ \\ 0\\ \\ Arg (cor k * \\) = pi / 2 \\ Arg (cor k * \\) = - pi / 2\nArg 0 = 0\ncor k * \\ = Complex 0 k\n\ngoal (1 subgoal):\n 1. Arg (cor k * \\) = - pi / 2\n[PROOF STEP]\nby (smt complex.sel(1) sin_pi_half sin_zero)\n[PROOF STATE]\nproof (state)\nthis:\nArg (cor k * \\) = - pi / 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 949, "file": "Complex_Geometry_More_Complex", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519528019683105, "lm_q2_score": 0.8615382094310357, "lm_q1q2_score": 0.7339898915275319}} {"text": "[STATEMENT]\nlemma suminf_ln_real: \n fixes f :: \"nat \\ real\"\n assumes f: \"convergent_prod f\" and 0: \"\\x. f x > 0\"\n shows \"suminf (\\i. ln (f i)) = ln (prodinf f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. ln (f i)) = ln (prodinf f)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\i. ln (f i)) = ln (prodinf f)\n[PROOF STEP]\nhave \"f has_prod prodinf f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f has_prod prodinf f\n[PROOF STEP]\nby (simp add: f has_prod_iff)\n[PROOF STATE]\nproof (state)\nthis:\nf has_prod prodinf f\n\ngoal (1 subgoal):\n 1. (\\i. ln (f i)) = ln (prodinf f)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nf has_prod prodinf f\n[PROOF STEP]\nhave \"raw_has_prod f 0 (prodinf f)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nf has_prod prodinf f\n\ngoal (1 subgoal):\n 1. raw_has_prod f 0 (prodinf f)\n[PROOF STEP]\nby (metis \"0\" has_prod_def less_irrefl)\n[PROOF STATE]\nproof (state)\nthis:\nraw_has_prod f 0 (prodinf f)\n\ngoal (1 subgoal):\n 1. (\\i. ln (f i)) = ln (prodinf f)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nraw_has_prod f 0 (prodinf f)\n[PROOF STEP]\nhave \"(\\i. ln (f i)) sums ln (prodinf f)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nraw_has_prod f 0 (prodinf f)\n\ngoal (1 subgoal):\n 1. (\\i. ln (f i)) sums ln (prodinf f)\n[PROOF STEP]\nusing \"0\" has_prod_imp_sums_ln_real\n[PROOF STATE]\nproof (prove)\nusing this:\nraw_has_prod f 0 (prodinf f)\n0 < f ?x\n\\raw_has_prod ?f 0 ?p; \\x. 0 < ?f x\\ \\ (\\i. ln (?f i)) sums ln ?p\n\ngoal (1 subgoal):\n 1. (\\i. ln (f i)) sums ln (prodinf f)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(\\i. ln (f i)) sums ln (prodinf f)\n\ngoal (1 subgoal):\n 1. (\\i. ln (f i)) = ln (prodinf f)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i. ln (f i)) sums ln (prodinf f)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i. ln (f i)) sums ln (prodinf f)\n\ngoal (1 subgoal):\n 1. (\\i. ln (f i)) = ln (prodinf f)\n[PROOF STEP]\nby (rule sums_unique [symmetric])\n[PROOF STATE]\nproof (state)\nthis:\n(\\i. ln (f i)) = ln (prodinf f)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1060, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615382094310357, "lm_q2_score": 0.8519527963298946, "lm_q1q2_score": 0.7339898866698212}} {"text": "[STATEMENT]\nlemma cindex_polyE_rec:\n fixes p q::\"real poly\"\n assumes \"a < b\" \"coprime p q\"\n shows \"cindex_polyE a b q p = cross_alt q p a b/2 + cindex_polyE a b (- (p mod q)) q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cindex_polyE a b q p = real_of_int (cross_alt q p a b) / 2 + cindex_polyE a b (- (p mod q)) q\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. cindex_polyE a b q p = real_of_int (cross_alt q p a b) / 2 + cindex_polyE a b (- (p mod q)) q\n[PROOF STEP]\nnote cindex_polyE_inverse_add_cross[OF assms]\n[PROOF STATE]\nproof (state)\nthis:\ncindex_polyE a b q p + cindex_polyE a b p q = real_of_int (cross_alt p q a b) / 2\n\ngoal (1 subgoal):\n 1. cindex_polyE a b q p = real_of_int (cross_alt q p a b) / 2 + cindex_polyE a b (- (p mod q)) q\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncindex_polyE a b q p + cindex_polyE a b p q = real_of_int (cross_alt p q a b) / 2\n\ngoal (1 subgoal):\n 1. cindex_polyE a b q p = real_of_int (cross_alt q p a b) / 2 + cindex_polyE a b (- (p mod q)) q\n[PROOF STEP]\nhave \"cindex_polyE a b (- (p mod q)) q = - cindex_polyE a b p q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cindex_polyE a b (- (p mod q)) q = - cindex_polyE a b p q\n[PROOF STEP]\nusing cindex_polyE_mod cindex_polyE_smult_1[of a b \"-1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ncindex_polyE ?a ?b ?q ?p = cindex_polyE ?a ?b (?q mod ?p) ?p\ncindex_polyE a b (smult (- 1) ?q) ?p = sgn (- 1) * cindex_polyE a b ?q ?p\n\ngoal (1 subgoal):\n 1. cindex_polyE a b (- (p mod q)) q = - cindex_polyE a b p q\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncindex_polyE a b (- (p mod q)) q = - cindex_polyE a b p q\n\ngoal (1 subgoal):\n 1. cindex_polyE a b q p = real_of_int (cross_alt q p a b) / 2 + cindex_polyE a b (- (p mod q)) q\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncindex_polyE a b q p + cindex_polyE a b p q = real_of_int (cross_alt p q a b) / 2\ncindex_polyE a b (- (p mod q)) q = - cindex_polyE a b p q\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncindex_polyE a b q p + cindex_polyE a b p q = real_of_int (cross_alt p q a b) / 2\ncindex_polyE a b (- (p mod q)) q = - cindex_polyE a b p q\n\ngoal (1 subgoal):\n 1. cindex_polyE a b q p = real_of_int (cross_alt q p a b) / 2 + cindex_polyE a b (- (p mod q)) q\n[PROOF STEP]\nby (auto simp add:field_simps cross_alt_poly_commute)\n[PROOF STATE]\nproof (state)\nthis:\ncindex_polyE a b q p = real_of_int (cross_alt q p a b) / 2 + cindex_polyE a b (- (p mod q)) q\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1172, "file": "Count_Complex_Roots_Extended_Sturm", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970842359877, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.7339898755959436}} {"text": "[STATEMENT]\nlemma fInf_unfold: \"(f::nat \\ 'a) 0 \\ (\\n. f (Suc n)) = (\\n. f n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f 0 \\ (\\n. f (Suc n)) = \\ range f\n[PROOF STEP]\napply (intro order.antisym inf_greatest)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. f 0 \\ (\\n. f (Suc n)) \\ \\ range f\n 2. \\ range f \\ f 0\n 3. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\napply (rule Inf_greatest, safe)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x n. n \\ UNIV \\ f 0 \\ (\\n. f (Suc n)) \\ f n\n 2. \\ range f \\ f 0\n 3. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\napply (case_tac n)\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. \\x n. \\n \\ UNIV; n = 0\\ \\ f 0 \\ (\\n. f (Suc n)) \\ f n\n 2. \\x n nat. \\n \\ UNIV; n = Suc nat\\ \\ f 0 \\ (\\n. f (Suc n)) \\ f n\n 3. \\ range f \\ f 0\n 4. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\napply simp_all\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\n nat. n = Suc nat \\ f 0 \\ (\\n. f (Suc n)) \\ f (Suc nat)\n 2. \\ range f \\ f 0\n 3. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\nusing Inf_lower inf.coboundedI2\n[PROOF STATE]\nproof (prove)\nusing this:\n?x \\ ?A \\ \\ ?A \\ ?x\n?b \\ ?c \\ ?a \\ ?b \\ ?c\n\ngoal (3 subgoals):\n 1. \\n nat. n = Suc nat \\ f 0 \\ (\\n. f (Suc n)) \\ f (Suc nat)\n 2. \\ range f \\ f 0\n 3. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\napply force\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\ range f \\ f 0\n 2. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\napply (simp add: Inf_lower)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ range f \\ (\\n. f (Suc n))\n[PROOF STEP]\nby (auto intro: Inf_mono)", "meta": {"llama_tokens": 1025, "file": "Order_Lattice_Props_Order_Lattice_Props_Loc", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.880797081106935, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.7339898729884271}} {"text": "[STATEMENT]\nlemma det_dim_2: assumes A: \"A \\ carrier_mat n n\"\n and n: \"n = 2\"\nshows \"Determinant.det A = A $$ (0,0) * A $$ (1,1) - A $$ (0,1) * A $$ (1,0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Determinant.det A = A $$ (0, 0) * A $$ (1, 1) - A $$ (0, 1) * A $$ (1, 0)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Determinant.det A = A $$ (0, 0) * A $$ (1, 1) - A $$ (0, 1) * A $$ (1, 0)\n[PROOF STEP]\nhave set: \"(\\i<(2 :: nat). f i) = f 0 + f 1\" for f\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f {..<2} = f 0 + f 1\n[PROOF STEP]\nby (subst sum.cong[of _ \"{0,1}\" f f], auto)\n[PROOF STATE]\nproof (state)\nthis:\nsum ?f1 {..<2} = ?f1 0 + ?f1 1\n\ngoal (1 subgoal):\n 1. Determinant.det A = A $$ (0, 0) * A $$ (1, 1) - A $$ (0, 1) * A $$ (1, 0)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Determinant.det A = A $$ (0, 0) * A $$ (1, 1) - A $$ (0, 1) * A $$ (1, 0)\n[PROOF STEP]\napply (subst laplace_expansion_column[OF A[unfolded n], of 0], insert A n,\n auto simp: cofactor_def mat_delete_def set)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ carrier_mat 2 2; n = 2\\ \\ A $$ (0, 0) * Determinant.det (Matrix.mat (Suc 0) (Suc 0) (\\(i', j'). A $$ (Suc i', Suc j'))) - A $$ (Suc 0, 0) * Determinant.det (Matrix.mat (Suc 0) (Suc 0) (\\(i', j'). A $$ (if i' = 0 then i' else Suc i', Suc j'))) = A $$ (0, 0) * A $$ (Suc 0, Suc 0) - A $$ (0, Suc 0) * A $$ (Suc 0, 0)\n[PROOF STEP]\napply (subst (1 2) det_dim_1, auto)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\nDeterminant.det A = A $$ (0, 0) * A $$ (1, 1) - A $$ (0, 1) * A $$ (1, 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 905, "file": "Perron_Frobenius_Perron_Frobenius_Aux", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970779778824, "lm_q2_score": 0.8333245953120234, "lm_q1q2_score": 0.7339898685579316}} {"text": "[STATEMENT]\nlemma vec_of_list_dot_rewrite:\n assumes \"length xs = length ys\"\n shows \"vec_of_list xs \\ vec_of_list ys =\n sum_list (map2 (*) xs ys)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec_of_list xs \\ vec_of_list ys = sum_list (map2 (*) xs ys)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength xs = length ys\n\ngoal (1 subgoal):\n 1. vec_of_list xs \\ vec_of_list ys = sum_list (map2 (*) xs ys)\n[PROOF STEP]\nproof (induction xs arbitrary:ys)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\ys. length [] = length ys \\ vec_of_list [] \\ vec_of_list ys = sum_list (map2 (*) [] ys)\n 2. \\a xs ys. \\\\ys. length xs = length ys \\ vec_of_list xs \\ vec_of_list ys = sum_list (map2 (*) xs ys); length (a # xs) = length ys\\ \\ vec_of_list (a # xs) \\ vec_of_list ys = sum_list (map2 (*) (a # xs) ys)\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\nlength [] = length ys\n\ngoal (2 subgoals):\n 1. \\ys. length [] = length ys \\ vec_of_list [] \\ vec_of_list ys = sum_list (map2 (*) [] ys)\n 2. \\a xs ys. \\\\ys. length xs = length ys \\ vec_of_list xs \\ vec_of_list ys = sum_list (map2 (*) xs ys); length (a # xs) = length ys\\ \\ vec_of_list (a # xs) \\ vec_of_list ys = sum_list (map2 (*) (a # xs) ys)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength [] = length ys\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nlength [] = length ys\n\ngoal (1 subgoal):\n 1. vec_of_list [] \\ vec_of_list ys = sum_list (map2 (*) [] ys)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nvec_of_list [] \\ vec_of_list ys = sum_list (map2 (*) [] ys)\n\ngoal (1 subgoal):\n 1. \\a xs ys. \\\\ys. length xs = length ys \\ vec_of_list xs \\ vec_of_list ys = sum_list (map2 (*) xs ys); length (a # xs) = length ys\\ \\ vec_of_list (a # xs) \\ vec_of_list ys = sum_list (map2 (*) (a # xs) ys)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a xs ys. \\\\ys. length xs = length ys \\ vec_of_list xs \\ vec_of_list ys = sum_list (map2 (*) xs ys); length (a # xs) = length ys\\ \\ vec_of_list (a # xs) \\ vec_of_list ys = sum_list (map2 (*) (a # xs) ys)\n[PROOF STEP]\ncase (Cons a xs)\n[PROOF STATE]\nproof (state)\nthis:\nlength xs = length ?ys \\ vec_of_list xs \\ vec_of_list ?ys = sum_list (map2 (*) xs ?ys)\nlength (a # xs) = length ys\n\ngoal (1 subgoal):\n 1. \\a xs ys. \\\\ys. length xs = length ys \\ vec_of_list xs \\ vec_of_list ys = sum_list (map2 (*) xs ys); length (a # xs) = length ys\\ \\ vec_of_list (a # xs) \\ vec_of_list ys = sum_list (map2 (*) (a # xs) ys)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength xs = length ?ys \\ vec_of_list xs \\ vec_of_list ?ys = sum_list (map2 (*) xs ?ys)\nlength (a # xs) = length ys\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nlength xs = length ?ys \\ vec_of_list xs \\ vec_of_list ?ys = sum_list (map2 (*) xs ?ys)\nlength (a # xs) = length ys\n\ngoal (1 subgoal):\n 1. vec_of_list (a # xs) \\ vec_of_list ys = sum_list (map2 (*) (a # xs) ys)\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\ys. length xs = length ys \\ vec_of_list xs \\ vec_of_list ys = sum_list (map2 (*) xs ys); Suc (length xs) = length ys\\ \\ vCons a (vec_of_list xs) \\ vec_of_list ys = sum_list (map2 (*) (a # xs) ys)\n[PROOF STEP]\nby (smt (verit, best) Suc_length_conv list.simps(9) old.prod.case scalar_prod_vCons sum_list.Cons vec_of_list_Cons zip_Cons_Cons)\n[PROOF STATE]\nproof (state)\nthis:\nvec_of_list (a # xs) \\ vec_of_list ys = sum_list (map2 (*) (a # xs) ys)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1706, "file": "BenOr_Kozen_Reif_Matrix_Equation_Construction", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.880797081106935, "lm_q2_score": 0.8333245911726382, "lm_q1q2_score": 0.7339898675194897}} {"text": "[STATEMENT]\ntheorem Join_Un: \"\\(A \\ B) = \\A \\ \\B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(A \\ B) = \\A \\ \\B\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\(A \\ B) = \\A \\ \\B\n[PROOF STEP]\nhave \"dual (\\(A \\ B)) = \\(dual ` A \\ dual ` B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dual (\\(A \\ B)) = \\(dual ` A \\ dual ` B)\n[PROOF STEP]\nby (simp only: dual_Join image_Un)\n[PROOF STATE]\nproof (state)\nthis:\ndual (\\(A \\ B)) = \\(dual ` A \\ dual ` B)\n\ngoal (1 subgoal):\n 1. \\(A \\ B) = \\A \\ \\B\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndual (\\(A \\ B)) = \\(dual ` A \\ dual ` B)\n\ngoal (1 subgoal):\n 1. \\(A \\ B) = \\A \\ \\B\n[PROOF STEP]\nhave \"\\ = \\(dual ` A) \\ \\(dual ` B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(dual ` A \\ dual ` B) = \\dual ` A \\ \\dual ` B\n[PROOF STEP]\nby (rule Meet_Un)\n[PROOF STATE]\nproof (state)\nthis:\n\\(dual ` A \\ dual ` B) = \\dual ` A \\ \\dual ` B\n\ngoal (1 subgoal):\n 1. \\(A \\ B) = \\A \\ \\B\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\(dual ` A \\ dual ` B) = \\dual ` A \\ \\dual ` B\n\ngoal (1 subgoal):\n 1. \\(A \\ B) = \\A \\ \\B\n[PROOF STEP]\nhave \"\\ = dual (\\A \\ \\B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\dual ` A \\ \\dual ` B = dual (\\A \\ \\B)\n[PROOF STEP]\nby (simp only: dual_join dual_Join)\n[PROOF STATE]\nproof (state)\nthis:\n\\dual ` A \\ \\dual ` B = dual (\\A \\ \\B)\n\ngoal (1 subgoal):\n 1. \\(A \\ B) = \\A \\ \\B\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndual (\\(A \\ B)) = dual (\\A \\ \\B)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndual (\\(A \\ B)) = dual (\\A \\ \\B)\n\ngoal (1 subgoal):\n 1. \\(A \\ B) = \\A \\ \\B\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n\\(A \\ B) = \\A \\ \\B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1283, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970685907242, "lm_q2_score": 0.8333245932423308, "lm_q1q2_score": 0.7339898589124025}} {"text": "[STATEMENT]\nlemma jordan_block_poly_bound: \n assumes i: \"i < n\" and j: \"j < n\" and a: \"norm a = 1\"\n shows \"norm ((jordan_block n a ^\\<^sub>m k) $$ (i,j)) \\ max 1 (of_nat k ^ (n - 1))\"\n (is \"?lhs \\ ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm ((jordan_block n a ^\\<^sub>m k) $$ (i, j)) \\ max 1 (real k ^ (n - 1))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm ((jordan_block n a ^\\<^sub>m k) $$ (i, j)) \\ max 1 (real k ^ (n - 1))\n[PROOF STEP]\nfrom jordan_block_bound[OF i j, of k, unfolded a]\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm ((jordan_block n a ^\\<^sub>m k) $$ (i, j)) \\ 1 ^ (k + i - j) * max 1 (real k ^ (n - 1))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm ((jordan_block n a ^\\<^sub>m k) $$ (i, j)) \\ 1 ^ (k + i - j) * max 1 (real k ^ (n - 1))\n\ngoal (1 subgoal):\n 1. norm ((jordan_block n a ^\\<^sub>m k) $$ (i, j)) \\ max 1 (real k ^ (n - 1))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nnorm ((jordan_block n a ^\\<^sub>m k) $$ (i, j)) \\ max 1 (real k ^ (n - 1))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 564, "file": "Jordan_Normal_Form_Jordan_Normal_Form", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9136765163620469, "lm_q2_score": 0.8031737963569014, "lm_q1q2_score": 0.7338410362886538}} {"text": "[STATEMENT]\nlemma integral_cos_nx:\n \"integral {-pi..pi} (\\x. cos(x * real_of_int n)) = (if n = 0 then 2 * pi else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral {- pi..pi} (\\x. cos (x * real_of_int n)) = (if n = 0 then 2 * pi else 0)\n[PROOF STEP]\nusing has_integral_cos_nx [of n]\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. cos (real_of_int n * x)) has_integral (if n = 0 then 2 * pi else 0)) {- pi..pi}\n\ngoal (1 subgoal):\n 1. integral {- pi..pi} (\\x. cos (x * real_of_int n)) = (if n = 0 then 2 * pi else 0)\n[PROOF STEP]\nby (force simp: mult.commute)", "meta": {"llama_tokens": 267, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894745194283, "lm_q2_score": 0.8198933293122507, "lm_q1q2_score": 0.7336319212972933}} {"text": "[STATEMENT]\nlemma collinear_det:\n assumes \"\\ collinear z1 z2 z3\"\n shows \"det2 (z3 - z1) (cnj (z3 - z1)) (z1 - z2) (cnj (z1 - z2)) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det2 (z3 - z1) (cnj (z3 - z1)) (z1 - z2) (cnj (z1 - z2)) \\ 0\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. det2 (z3 - z1) (cnj (z3 - z1)) (z1 - z2) (cnj (z1 - z2)) \\ 0\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ collinear z1 z2 z3\n[PROOF STEP]\nhave \"((z3 - z1) / (z2 - z1)) - cnj ((z3 - z1) / (z2 - z1)) \\ 0\" \"z2 \\ z1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ collinear z1 z2 z3\n\ngoal (1 subgoal):\n 1. (z3 - z1) / (z2 - z1) - cnj ((z3 - z1) / (z2 - z1)) \\ 0 &&& z2 \\ z1\n[PROOF STEP]\nunfolding collinear_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (z1 = z2 \\ is_real ((z3 - z1) / (z2 - z1)))\n\ngoal (1 subgoal):\n 1. (z3 - z1) / (z2 - z1) - cnj ((z3 - z1) / (z2 - z1)) \\ 0 &&& z2 \\ z1\n[PROOF STEP]\nusing Complex_Im_express_cnj[of \"(z3 - z1) / (z2 - z1)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (z1 = z2 \\ is_real ((z3 - z1) / (z2 - z1)))\nComplex 0 (Im ((z3 - z1) / (z2 - z1))) = ((z3 - z1) / (z2 - z1) - cnj ((z3 - z1) / (z2 - z1))) / 2\n\ngoal (1 subgoal):\n 1. (z3 - z1) / (z2 - z1) - cnj ((z3 - z1) / (z2 - z1)) \\ 0 &&& z2 \\ z1\n[PROOF STEP]\nby (auto simp add: Complex_eq)\n[PROOF STATE]\nproof (state)\nthis:\n(z3 - z1) / (z2 - z1) - cnj ((z3 - z1) / (z2 - z1)) \\ 0\nz2 \\ z1\n\ngoal (1 subgoal):\n 1. det2 (z3 - z1) (cnj (z3 - z1)) (z1 - z2) (cnj (z1 - z2)) \\ 0\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(z3 - z1) / (z2 - z1) - cnj ((z3 - z1) / (z2 - z1)) \\ 0\nz2 \\ z1\n\ngoal (1 subgoal):\n 1. det2 (z3 - z1) (cnj (z3 - z1)) (z1 - z2) (cnj (z1 - z2)) \\ 0\n[PROOF STEP]\nby (auto simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\ndet2 (z3 - z1) (cnj (z3 - z1)) (z1 - z2) (cnj (z1 - z2)) \\ 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1112, "file": "Complex_Geometry_Elementary_Complex_Geometry", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869948899665, "lm_q2_score": 0.8376199694135333, "lm_q1q2_score": 0.733576675872504}} {"text": "[STATEMENT]\nlemma trace_smult: \n assumes \"A \\ carrier_mat n n\"\n shows \"trace (c \\\\<^sub>m A) = c * trace A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (c \\\\<^sub>m A) = c * trace A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. trace (c \\\\<^sub>m A) = c * trace A\n[PROOF STEP]\nhave \"trace (c \\\\<^sub>m A) = (\\i = 0..\\<^sub>m A) = (\\i = 0..i = 0..\\<^sub>m A). (c \\\\<^sub>m A) $$ (i, i)) = (\\i = 0.. carrier_mat n n\n\ngoal (1 subgoal):\n 1. (\\i = 0..\\<^sub>m A). (c \\\\<^sub>m A) $$ (i, i)) = (\\i = 0..\\<^sub>m A) = (\\i = 0..\\<^sub>m A) = c * trace A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ntrace (c \\\\<^sub>m A) = (\\i = 0..\\<^sub>m A) = c * trace A\n[PROOF STEP]\nhave \"\\ = c * (\\i = 0..i = 0..i = 0..i = 0..i = 0..\\<^sub>m A) = c * trace A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..i = 0..\\<^sub>m A) = c * trace A\n[PROOF STEP]\nhave \"\\ = c * trace A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c * (\\i = 0..i = 0..i = 0..i = 0..\\<^sub>m A) = c * trace A\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ntrace (c \\\\<^sub>m A) = c * (\\i = 0..i = 0..\\<^sub>m A) = c * (\\i = 0..i = 0..\\<^sub>m A) = c * trace A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace (c \\\\<^sub>m A) = c * trace A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1481, "file": "QHLProver_Complex_Matrix", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869786798663, "lm_q2_score": 0.837619961306541, "lm_q1q2_score": 0.7335766551946021}} {"text": "[STATEMENT]\ntheorem fib_mult_eq_sum_nat: \"fib (Suc n) * fib n = (\\k \\ {..n}. fib k * fib k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fib (Suc n) * fib n = (\\k\\n. fib k * fib k)\n[PROOF STEP]\nby (induct n rule: nat.induct) (auto simp add: field_simps)", "meta": {"llama_tokens": 126, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9219218305645894, "lm_q2_score": 0.7956581000631541, "lm_q1q2_score": 0.7335345721137663}} {"text": "[STATEMENT]\nlemma real_sqrt_one [simp]: \"sqrt 1 = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt 1 = 1\n[PROOF STEP]\nunfolding sqrt_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. root 2 1 = 1\n[PROOF STEP]\nby (rule real_root_one [OF pos2])", "meta": {"llama_tokens": 123, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9161096090086367, "lm_q2_score": 0.800691997339971, "lm_q1q2_score": 0.7335216326194652}} {"text": "[STATEMENT]\nlemma jumpF_tan_comp:\n \"jumpF (f o tan) (at_right x) = (if cos x = 0 \n then jumpF f at_bot else jumpF f (at_right (tan x)))\"\n \"jumpF (f o tan) (at_left x) = (if cos x =0 \n then jumpF f at_top else jumpF f (at_left (tan x)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. jumpF (f \\ tan) (at_right x) = (if cos x = 0 then jumpF f at_bot else jumpF f (at_right (tan x))) &&& jumpF (f \\ tan) (at_left x) = (if cos x = 0 then jumpF f at_top else jumpF f (at_left (tan x)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. jumpF (f \\ tan) (at_right x) = (if cos x = 0 then jumpF f at_bot else jumpF f (at_right (tan x)))\n 2. jumpF (f \\ tan) (at_left x) = (if cos x = 0 then jumpF f at_top else jumpF f (at_left (tan x)))\n[PROOF STEP]\nhave \"filtermap (f \\ tan) (at_right x) = \n (if cos x = 0 then filtermap f at_bot else filtermap f (at_right (tan x)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. filtermap (f \\ tan) (at_right x) = (if cos x = 0 then filtermap f at_bot else filtermap f (at_right (tan x)))\n[PROOF STEP]\nunfolding comp_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. filtermap (\\x. f (tan x)) (at_right x) = (if cos x = 0 then filtermap f at_bot else filtermap f (at_right (tan x)))\n[PROOF STEP]\napply (subst filtermap_filtermap[of f tan,symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. filtermap f (filtermap tan (at_right x)) = (if cos x = 0 then filtermap f at_bot else filtermap f (at_right (tan x)))\n[PROOF STEP]\nusing filtermap_tan_at_right_inf filtermap_tan_at_right\n[PROOF STATE]\nproof (prove)\nusing this:\ncos ?a = 0 \\ filtermap tan (at_right ?a) = at_bot\ncos ?a \\ 0 \\ filtermap tan (at_right ?a) = at_right (tan ?a)\n\ngoal (1 subgoal):\n 1. filtermap f (filtermap tan (at_right x)) = (if cos x = 0 then filtermap f at_bot else filtermap f (at_right (tan x)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfiltermap (f \\ tan) (at_right x) = (if cos x = 0 then filtermap f at_bot else filtermap f (at_right (tan x)))\n\ngoal (2 subgoals):\n 1. jumpF (f \\ tan) (at_right x) = (if cos x = 0 then jumpF f at_bot else jumpF f (at_right (tan x)))\n 2. jumpF (f \\ tan) (at_left x) = (if cos x = 0 then jumpF f at_top else jumpF f (at_left (tan x)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfiltermap (f \\ tan) (at_right x) = (if cos x = 0 then filtermap f at_bot else filtermap f (at_right (tan x)))\n[PROOF STEP]\nshow \"jumpF (f o tan) (at_right x) = (if cos x = 0 \n then jumpF f at_bot else jumpF f (at_right (tan x)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfiltermap (f \\ tan) (at_right x) = (if cos x = 0 then filtermap f at_bot else filtermap f (at_right (tan x)))\n\ngoal (1 subgoal):\n 1. jumpF (f \\ tan) (at_right x) = (if cos x = 0 then jumpF f at_bot else jumpF f (at_right (tan x)))\n[PROOF STEP]\nunfolding jumpF_def filterlim_def\n[PROOF STATE]\nproof (prove)\nusing this:\nfiltermap (f \\ tan) (at_right x) = (if cos x = 0 then filtermap f at_bot else filtermap f (at_right (tan x)))\n\ngoal (1 subgoal):\n 1. (if filtermap (f \\ tan) (at_right x) \\ at_top then 1 / 2 else if filtermap (f \\ tan) (at_right x) \\ at_bot then - 1 / 2 else 0) = (if cos x = 0 then if filtermap f at_bot \\ at_top then 1 / 2 else if filtermap f at_bot \\ at_bot then - 1 / 2 else 0 else if filtermap f (at_right (tan x)) \\ at_top then 1 / 2 else if filtermap f (at_right (tan x)) \\ at_bot then - 1 / 2 else 0)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\njumpF (f \\ tan) (at_right x) = (if cos x = 0 then jumpF f at_bot else jumpF f (at_right (tan x)))\n\ngoal (1 subgoal):\n 1. jumpF (f \\ tan) (at_left x) = (if cos x = 0 then jumpF f at_top else jumpF f (at_left (tan x)))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. jumpF (f \\ tan) (at_left x) = (if cos x = 0 then jumpF f at_top else jumpF f (at_left (tan x)))\n[PROOF STEP]\nhave \"filtermap (f \\ tan) (at_left x) = \n (if cos x = 0 then filtermap f at_top else filtermap f (at_left (tan x)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. filtermap (f \\ tan) (at_left x) = (if cos x = 0 then filtermap f at_top else filtermap f (at_left (tan x)))\n[PROOF STEP]\nunfolding comp_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. filtermap (\\x. f (tan x)) (at_left x) = (if cos x = 0 then filtermap f at_top else filtermap f (at_left (tan x)))\n[PROOF STEP]\napply (subst filtermap_filtermap[of f tan,symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. filtermap f (filtermap tan (at_left x)) = (if cos x = 0 then filtermap f at_top else filtermap f (at_left (tan x)))\n[PROOF STEP]\nusing filtermap_tan_at_left_inf filtermap_tan_at_left\n[PROOF STATE]\nproof (prove)\nusing this:\ncos ?a = 0 \\ filtermap tan (at_left ?a) = at_top\ncos ?a \\ 0 \\ filtermap tan (at_left ?a) = at_left (tan ?a)\n\ngoal (1 subgoal):\n 1. filtermap f (filtermap tan (at_left x)) = (if cos x = 0 then filtermap f at_top else filtermap f (at_left (tan x)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfiltermap (f \\ tan) (at_left x) = (if cos x = 0 then filtermap f at_top else filtermap f (at_left (tan x)))\n\ngoal (1 subgoal):\n 1. jumpF (f \\ tan) (at_left x) = (if cos x = 0 then jumpF f at_top else jumpF f (at_left (tan x)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfiltermap (f \\ tan) (at_left x) = (if cos x = 0 then filtermap f at_top else filtermap f (at_left (tan x)))\n[PROOF STEP]\nshow \"jumpF (f o tan) (at_left x) = (if cos x = 0 \n then jumpF f at_top else jumpF f (at_left (tan x)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfiltermap (f \\ tan) (at_left x) = (if cos x = 0 then filtermap f at_top else filtermap f (at_left (tan x)))\n\ngoal (1 subgoal):\n 1. jumpF (f \\ tan) (at_left x) = (if cos x = 0 then jumpF f at_top else jumpF f (at_left (tan x)))\n[PROOF STEP]\nunfolding jumpF_def filterlim_def\n[PROOF STATE]\nproof (prove)\nusing this:\nfiltermap (f \\ tan) (at_left x) = (if cos x = 0 then filtermap f at_top else filtermap f (at_left (tan x)))\n\ngoal (1 subgoal):\n 1. (if filtermap (f \\ tan) (at_left x) \\ at_top then 1 / 2 else if filtermap (f \\ tan) (at_left x) \\ at_bot then - 1 / 2 else 0) = (if cos x = 0 then if filtermap f at_top \\ at_top then 1 / 2 else if filtermap f at_top \\ at_bot then - 1 / 2 else 0 else if filtermap f (at_left (tan x)) \\ at_top then 1 / 2 else if filtermap f (at_left (tan x)) \\ at_bot then - 1 / 2 else 0)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\njumpF (f \\ tan) (at_left x) = (if cos x = 0 then jumpF f at_top else jumpF f (at_left (tan x)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2826, "file": "Winding_Number_Eval_Cauchy_Index_Theorem", "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872046026642944, "lm_q2_score": 0.8267117983401363, "lm_q1q2_score": 0.7334625125642449}} {"text": "[STATEMENT]\nlemma smallo_imp_eventually_sgn:\n fixes f g :: \"real \\ real\"\n assumes \"g \\ o(f)\"\n shows \"eventually (\\x. sgn (f x + g x) = sgn (f x)) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. sgn (f x + g x) = sgn (f x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. sgn (f x + g x) = sgn (f x)\n[PROOF STEP]\nhave \"0 < (1/2 :: real)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < 1 / 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < 1 / 2\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. sgn (f x + g x) = sgn (f x)\n[PROOF STEP]\nfrom landau_o.smallD[OF assms, OF this]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F x in at_top. norm (g x) \\ 1 / 2 * norm (f x)\n[PROOF STEP]\nhave \"eventually (\\x. \\g x\\ \\ 1/2 * \\f x\\) at_top\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in at_top. norm (g x) \\ 1 / 2 * norm (f x)\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. \\g x\\ \\ 1 / 2 * \\f x\\\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in at_top. \\g x\\ \\ 1 / 2 * \\f x\\\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. sgn (f x + g x) = sgn (f x)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in at_top. \\g x\\ \\ 1 / 2 * \\f x\\\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. sgn (f x + g x) = sgn (f x)\n[PROOF STEP]\nproof eventually_elim\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. \\g x\\ \\ 1 / 2 * \\f x\\ \\ sgn (f x + g x) = sgn (f x)\n[PROOF STEP]\ncase (elim x)\n[PROOF STATE]\nproof (state)\nthis:\n\\g x\\ \\ 1 / 2 * \\f x\\\n\ngoal (1 subgoal):\n 1. \\x. \\g x\\ \\ 1 / 2 * \\f x\\ \\ sgn (f x + g x) = sgn (f x)\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n\\g x\\ \\ 1 / 2 * \\f x\\\n\ngoal (1 subgoal):\n 1. sgn (f x + g x) = sgn (f x)\n[PROOF STEP]\nby (cases \"f x\" \"0::real\" rule: linorder_cases;\n cases \"f x + g x\" \"0::real\" rule: linorder_cases) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nsgn (f x + g x) = sgn (f x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in at_top. sgn (f x + g x) = sgn (f x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1202, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357529306639, "lm_q2_score": 0.8459424431344437, "lm_q1q2_score": 0.7334623431190778}} {"text": "[STATEMENT]\nlemma fermat_little:\n assumes \"prime (P :: nat)\" \n shows \"[x^P = x] (mod P)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [x ^ P = x] (mod P)\n[PROOF STEP]\nproof(cases \"P dvd x\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. P dvd x \\ [x ^ P = x] (mod P)\n 2. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nP dvd x\n\ngoal (2 subgoals):\n 1. P dvd x \\ [x ^ P = x] (mod P)\n 2. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nhence \"x mod P = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nP dvd x\n\ngoal (1 subgoal):\n 1. x mod P = 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx mod P = 0\n\ngoal (2 subgoals):\n 1. P dvd x \\ [x ^ P = x] (mod P)\n 2. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nx mod P = 0\n\ngoal (2 subgoals):\n 1. P dvd x \\ [x ^ P = x] (mod P)\n 2. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nhave \"x ^ P mod P = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x ^ P mod P = 0\n[PROOF STEP]\nby (simp add: True assms prime_dvd_power_nat_iff prime_gt_0_nat)\n[PROOF STATE]\nproof (state)\nthis:\nx ^ P mod P = 0\n\ngoal (2 subgoals):\n 1. P dvd x \\ [x ^ P = x] (mod P)\n 2. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nx mod P = 0\nx ^ P mod P = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx mod P = 0\nx ^ P mod P = 0\n\ngoal (1 subgoal):\n 1. [x ^ P = x] (mod P)\n[PROOF STEP]\nby (simp add: cong_def)\n[PROOF STATE]\nproof (state)\nthis:\n[x ^ P = x] (mod P)\n\ngoal (1 subgoal):\n 1. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ P dvd x\n\ngoal (1 subgoal):\n 1. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nhence \"[x ^ (P - 1) = 1] (mod P)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ P dvd x\n\ngoal (1 subgoal):\n 1. [x ^ (P - 1) = 1] (mod P)\n[PROOF STEP]\nusing fermat_theorem assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ P dvd x\n\\prime ?p; \\ ?p dvd ?a\\ \\ [?a ^ (?p - 1) = 1] (mod ?p)\nprime P\n\ngoal (1 subgoal):\n 1. [x ^ (P - 1) = 1] (mod P)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n[x ^ (P - 1) = 1] (mod P)\n\ngoal (1 subgoal):\n 1. \\ P dvd x \\ [x ^ P = x] (mod P)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[x ^ (P - 1) = 1] (mod P)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n[x ^ (P - 1) = 1] (mod P)\n\ngoal (1 subgoal):\n 1. [x ^ P = x] (mod P)\n[PROOF STEP]\nby (metis assms cong_def diff_diff_cancel diff_is_0_eq' diff_zero mod_mult_right_eq power_eq_if power_one_right prime_ge_1_nat zero_le_one)\n[PROOF STATE]\nproof (state)\nthis:\n[x ^ P = x] (mod P)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1467, "file": "Multi_Party_Computation_Number_Theory_Aux", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.88242786954645, "lm_q2_score": 0.8311430436757313, "lm_q1q2_score": 0.7334237853191277}} {"text": "[STATEMENT]\nlemma infsum_Re:\n assumes \"f summable_on M\"\n shows \"infsum (\\x. Re (f x)) M = Re (infsum f M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sub>\\x\\M. Re (f x)) = Re (infsum f M)\n[PROOF STEP]\napply (rule infsum_comm_additive[where f=Re, unfolded o_def])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. additive Re\n 2. isCont Re (infsum f M)\n 3. f summable_on M\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nf summable_on M\n\ngoal (3 subgoals):\n 1. additive Re\n 2. isCont Re (infsum f M)\n 3. f summable_on M\n[PROOF STEP]\nby (auto intro!: additive.intro)", "meta": {"llama_tokens": 276, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278664544911, "lm_q2_score": 0.8311430436757312, "lm_q1q2_score": 0.7334237827492674}} {"text": "[STATEMENT]\nlemma emeasure_lborel_cbox_cart_eq:\n fixes a b :: \"real ^ ('n :: finite)\"\n shows \"emeasure lborel (cbox a b) = ennreal (\\i \\ UNIV. max 0 ((b - a) $ i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. emeasure lborel (cbox a b) = ennreal (\\i\\UNIV. max 0 ((b - a) $ i))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. emeasure lborel (cbox a b) = ennreal (\\i\\UNIV. max 0 ((b - a) $ i))\n[PROOF STEP]\nhave \"emeasure lborel (cbox a b) = ennreal (\\e\\Basis. max 0 ((b - a) \\ e))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. emeasure lborel (cbox a b) = ennreal (\\e\\Basis. max 0 ((b - a) \\ e))\n[PROOF STEP]\nunfolding emeasure_lborel_cbox_eq'\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ennreal (\\e\\Basis. max 0 ((b - a) \\ e)) = ennreal (\\e\\Basis. max 0 ((b - a) \\ e))\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nemeasure lborel (cbox a b) = ennreal (\\e\\Basis. max 0 ((b - a) \\ e))\n\ngoal (1 subgoal):\n 1. emeasure lborel (cbox a b) = ennreal (\\i\\UNIV. max 0 ((b - a) $ i))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nemeasure lborel (cbox a b) = ennreal (\\e\\Basis. max 0 ((b - a) \\ e))\n\ngoal (1 subgoal):\n 1. emeasure lborel (cbox a b) = ennreal (\\i\\UNIV. max 0 ((b - a) $ i))\n[PROOF STEP]\nhave \"(Basis :: (real ^ 'n) set) = range (\\k. axis k 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Basis = range (\\k. axis k 1)\n[PROOF STEP]\nunfolding Basis_vec_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. \\u\\Basis. {axis i u}) = range (\\k. axis k 1)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nBasis = range (\\k. axis k 1)\n\ngoal (1 subgoal):\n 1. emeasure lborel (cbox a b) = ennreal (\\i\\UNIV. max 0 ((b - a) $ i))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nBasis = range (\\k. axis k 1)\n\ngoal (1 subgoal):\n 1. emeasure lborel (cbox a b) = ennreal (\\i\\UNIV. max 0 ((b - a) $ i))\n[PROOF STEP]\nhave \"(\\e\\\\. max 0 ((b - a) \\ e)) = (\\ i \\ UNIV . max 0 ((b - a) $ i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\e\\range (\\k. axis k 1). max 0 ((b - a) \\ e)) = (\\i\\UNIV. max 0 ((b - a) $ i))\n[PROOF STEP]\nby (subst prod.reindex) (auto intro!: inj_axis simp: algebra_simps inner_axis)\n[PROOF STATE]\nproof (state)\nthis:\n(\\e\\range (\\k. axis k 1). max 0 ((b - a) \\ e)) = (\\i\\UNIV. max 0 ((b - a) $ i))\n\ngoal (1 subgoal):\n 1. emeasure lborel (cbox a b) = ennreal (\\i\\UNIV. max 0 ((b - a) $ i))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nemeasure lborel (cbox a b) = ennreal (\\i\\UNIV. max 0 ((b - a) $ i))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nemeasure lborel (cbox a b) = ennreal (\\i\\UNIV. max 0 ((b - a) $ i))\n\ngoal (1 subgoal):\n 1. emeasure lborel (cbox a b) = ennreal (\\i\\UNIV. max 0 ((b - a) $ i))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nemeasure lborel (cbox a b) = ennreal (\\i\\UNIV. max 0 ((b - a) $ i))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1549, "file": "Minkowskis_Theorem_Minkowskis_Theorem", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110339361275, "lm_q2_score": 0.8221891327004132, "lm_q1q2_score": 0.7332373405246034}} {"text": "[STATEMENT]\nlemma null_space_eq_solution_set: \nshows \"null_space A = solution_set A 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. null_space A = solution_set A 0\n[PROOF STEP]\nunfolding null_space_def solution_set_def is_solution_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x. A *v x = 0} = {x. A *v x = 0}\n[PROOF STEP]\n..", "meta": {"llama_tokens": 154, "file": "Gauss_Jordan_System_Of_Equations", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.901920681802153, "lm_q2_score": 0.8128673269042767, "lm_q1q2_score": 0.7331418536961988}} {"text": "[STATEMENT]\nlemma eval_fds_mult:\n fixes s :: \"'a :: {nat_power, real_normed_field, banach, second_countable_topology}\"\n assumes \"fds_abs_converges f s\" \"fds_abs_converges g s\"\n shows \"eval_fds (f * g) s = eval_fds f s * eval_fds g s\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eval_fds (f * g) s = eval_fds f s * eval_fds g s\n[PROOF STEP]\nusing suminf_dirichlet_prod[OF _ _ assms[unfolded fds_abs_converges_def]]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\fds_nth f 0 = (0::'a); fds_nth g 0 = (0::'a)\\ \\ (\\n. dirichlet_prod (fds_nth f) (fds_nth g) n / nat_power n s) = (\\n. fds_nth f n / nat_power n s) * (\\n. fds_nth g n / nat_power n s)\n\ngoal (1 subgoal):\n 1. eval_fds (f * g) s = eval_fds f s * eval_fds g s\n[PROOF STEP]\nby (simp_all add: eval_fds_def fds_nth_mult)", "meta": {"llama_tokens": 382, "file": "Dirichlet_Series_Dirichlet_Series_Analysis", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206844384594, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7331418517504426}} {"text": "[STATEMENT]\nlemma max_all_edges_between: \n assumes \"finite X\" \"finite Y\"\n shows \"card (all_edges_between X Y) \\ card X * card Y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (all_edges_between X Y) \\ card X * card Y\n[PROOF STEP]\nby (metis assms card_mono finite_SigmaI all_edges_between_subset card_cartesian_product)", "meta": {"llama_tokens": 132, "file": "Undirected_Graph_Theory_Undirected_Graph_Basics", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.90192067652954, "lm_q2_score": 0.8128673269042767, "lm_q1q2_score": 0.733141849410264}} {"text": "[STATEMENT]\nlemma sum_list_map_filter_sum: fixes f :: \"'a \\ 'b :: comm_monoid_add\" \n shows \"sum_list (map f (filter g xs)) + sum_list (map f (filter (Not o g) xs)) = sum_list (map f xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list (map f (filter g xs)) + sum_list (map f (filter (Not \\ g) xs)) = sum_list (map f xs)\n[PROOF STEP]\nby (induct xs, auto simp: ac_simps)", "meta": {"llama_tokens": 161, "file": "Farkas_Farkas", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206791658465, "lm_q2_score": 0.8128673178375735, "lm_q1q2_score": 0.7331418433757843}} {"text": "[STATEMENT]\nlemma concat_length_le:\n fixes g :: \"nat \\ real\"\n assumes \"\\ x \\ set xs . real (length (f x)) \\ g x\"\n shows \"length (concat (map f xs)) \\ (\\x\\xs. g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (length (concat (map f xs))) \\ sum_list (map g xs)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\set xs. real (length (f x)) \\ g x\n\ngoal (1 subgoal):\n 1. real (length (concat (map f xs))) \\ sum_list (map g xs)\n[PROOF STEP]\nby (induction xs) force+", "meta": {"llama_tokens": 226, "file": "Pratt_Certificate_Pratt_Certificate", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206712569267, "lm_q2_score": 0.8128673201042492, "lm_q1q2_score": 0.7331418389912435}} {"text": "[STATEMENT]\nlemma norm_square_vec_eq: \"norm x ^ 2 = (\\i\\UNIV. x $ i ^ 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (norm x)\\<^sup>2 = (\\i\\UNIV. (x $ i)\\<^sup>2)\n[PROOF STEP]\nby (auto simp: norm_vec_def L2_set_def intro!: sum_nonneg)", "meta": {"llama_tokens": 127, "file": "Octonions_Cross_Product_7", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9184802529509909, "lm_q2_score": 0.7981867729389246, "lm_q1q2_score": 0.7331187891110786}} {"text": "[STATEMENT]\nlemma not_list_ex_equals_list_all_not: \"(\\list_ex P xs) = list_all (\\x. \\P x) xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ list_ex P xs) = list_all (\\x. \\ P x) xs\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\ list_ex P xs) = list_all (\\x. \\ P x) xs\n[PROOF STEP]\nhave \"(\\list_ex P xs) = (\\Bex (set xs) P)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ list_ex P xs) = (\\ Bex (set xs) P)\n[PROOF STEP]\nusing list_ex_iff\n[PROOF STATE]\nproof (prove)\nusing this:\nlist_ex ?P ?xs = Bex (set ?xs) ?P\n\ngoal (1 subgoal):\n 1. (\\ list_ex P xs) = (\\ Bex (set xs) P)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(\\ list_ex P xs) = (\\ Bex (set xs) P)\n\ngoal (1 subgoal):\n 1. (\\ list_ex P xs) = list_all (\\x. \\ P x) xs\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\ list_ex P xs) = (\\ Bex (set xs) P)\n\ngoal (1 subgoal):\n 1. (\\ list_ex P xs) = list_all (\\x. \\ P x) xs\n[PROOF STEP]\nhave \"\\ = Ball (set xs) (\\x. \\P x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ Bex (set xs) P) = (\\x\\set xs. \\ P x)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(\\ Bex (set xs) P) = (\\x\\set xs. \\ P x)\n\ngoal (1 subgoal):\n 1. (\\ list_ex P xs) = list_all (\\x. \\ P x) xs\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\ list_ex P xs) = (\\x\\set xs. \\ P x)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\ list_ex P xs) = (\\x\\set xs. \\ P x)\n\ngoal (1 subgoal):\n 1. (\\ list_ex P xs) = list_all (\\x. \\ P x) xs\n[PROOF STEP]\nby (simp add: Ball_set_list_all)\n[PROOF STATE]\nproof (state)\nthis:\n(\\ list_ex P xs) = list_all (\\x. \\ P x) xs\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 900, "file": "Verified_SAT_Based_AI_Planning_List_Supplement", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767970940975, "lm_q2_score": 0.8354835411997897, "lm_q1q2_score": 0.7331174217568259}} {"text": "[STATEMENT]\nlemma Polygamma_real_strict_mono:\n assumes \"x > 0\" \"x < (y::real)\" \"even n\"\n shows \"Polygamma n x < Polygamma n y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Polygamma n x < Polygamma n y\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Polygamma n x < Polygamma n y\n[PROOF STEP]\nhave \"\\\\. x < \\ \\ \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\>x. \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\nx < y\neven n\n\ngoal (1 subgoal):\n 1. \\\\>x. \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n[PROOF STEP]\nby (intro MVT2 derivative_intros impI allI) (auto elim!: nonpos_Ints_cases)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\>x. \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n\ngoal (1 subgoal):\n 1. Polygamma n x < Polygamma n y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\>x. \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n[PROOF STEP]\nobtain \\\n where \\: \"x < \\\" \"\\ < y\"\n and Polygamma: \"Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\>x. \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n\ngoal (1 subgoal):\n 1. (\\\\. \\x < \\; \\ < y; Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx < \\\n\\ < y\nPolygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n\ngoal (1 subgoal):\n 1. Polygamma n x < Polygamma n y\n[PROOF STEP]\nnote Polygamma\n[PROOF STATE]\nproof (state)\nthis:\nPolygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n\ngoal (1 subgoal):\n 1. Polygamma n x < Polygamma n y\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nPolygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n\ngoal (1 subgoal):\n 1. Polygamma n x < Polygamma n y\n[PROOF STEP]\nfrom \\ assms\n[PROOF STATE]\nproof (chain)\npicking this:\nx < \\\n\\ < y\n0 < x\nx < y\neven n\n[PROOF STEP]\nhave \"(y - x) * Polygamma (Suc n) \\ > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx < \\\n\\ < y\n0 < x\nx < y\neven n\n\ngoal (1 subgoal):\n 1. 0 < (y - x) * Polygamma (Suc n) \\\n[PROOF STEP]\nby (intro mult_pos_pos Polygamma_real_odd_pos) (auto elim!: nonpos_Ints_cases)\n[PROOF STATE]\nproof (state)\nthis:\n0 < (y - x) * Polygamma (Suc n) \\\n\ngoal (1 subgoal):\n 1. Polygamma n x < Polygamma n y\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < Polygamma n y - Polygamma n x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < Polygamma n y - Polygamma n x\n\ngoal (1 subgoal):\n 1. Polygamma n x < Polygamma n y\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nPolygamma n x < Polygamma n y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1376, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767970940975, "lm_q2_score": 0.8354835350552603, "lm_q1q2_score": 0.7331174163651439}} {"text": "[STATEMENT]\nlemma ring_hom_uminus:\n assumes \"ring S\"\n assumes \"f \\ (ring_hom S R)\"\n assumes \"a \\ carrier S\"\n shows \"f (\\\\<^bsub>S\\<^esub> a) = \\ (f a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (\\\\<^bsub>S\\<^esub> a) = \\ f a\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. f (\\\\<^bsub>S\\<^esub> a) = \\ f a\n[PROOF STEP]\nhave \"f (a \\\\<^bsub>S\\<^esub> a) = (f a) \\ f (\\\\<^bsub>S\\<^esub> a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (a \\\\<^bsub>S\\<^esub> a) = f a \\ f (\\\\<^bsub>S\\<^esub> a)\n[PROOF STEP]\nunfolding a_minus_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (a \\\\<^bsub>S\\<^esub> \\\\<^bsub>S\\<^esub> a) = f a \\ f (\\\\<^bsub>S\\<^esub> a)\n[PROOF STEP]\nby (simp add: assms(1) assms(2) assms(3) ring.ring_simprules(3) ring_hom_add)\n[PROOF STATE]\nproof (state)\nthis:\nf (a \\\\<^bsub>S\\<^esub> a) = f a \\ f (\\\\<^bsub>S\\<^esub> a)\n\ngoal (1 subgoal):\n 1. f (\\\\<^bsub>S\\<^esub> a) = \\ f a\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nf (a \\\\<^bsub>S\\<^esub> a) = f a \\ f (\\\\<^bsub>S\\<^esub> a)\n[PROOF STEP]\nhave \"(f a) \\ f (\\\\<^bsub>S\\<^esub> a) = \\ \"\n[PROOF STATE]\nproof (prove)\nusing this:\nf (a \\\\<^bsub>S\\<^esub> a) = f a \\ f (\\\\<^bsub>S\\<^esub> a)\n\ngoal (1 subgoal):\n 1. f a \\ f (\\\\<^bsub>S\\<^esub> a) = \\\n[PROOF STEP]\nby (metis R.ring_axioms a_minus_def assms(1) assms(2) assms(3) \n ring.ring_simprules(16) ring_hom_zero)\n[PROOF STATE]\nproof (state)\nthis:\nf a \\ f (\\\\<^bsub>S\\<^esub> a) = \\\n\ngoal (1 subgoal):\n 1. f (\\\\<^bsub>S\\<^esub> a) = \\ f a\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nf a \\ f (\\\\<^bsub>S\\<^esub> a) = \\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nf a \\ f (\\\\<^bsub>S\\<^esub> a) = \\\n\ngoal (1 subgoal):\n 1. f (\\\\<^bsub>S\\<^esub> a) = \\ f a\n[PROOF STEP]\nby (metis (no_types, lifting) R.add.m_comm R.minus_equality assms(1)\n assms(2) assms(3) ring.ring_simprules(3) ring_hom_closed)\n[PROOF STATE]\nproof (state)\nthis:\nf (\\\\<^bsub>S\\<^esub> a) = \\ f a\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1106, "file": "Padic_Ints_Cring_Poly", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.8354835371034369, "lm_q1q2_score": 0.7331174101314891}} {"text": "[STATEMENT]\nlemma powr_real_of_int:\n \"x > 0 \\ x powr real_of_int n = (if n \\ 0 then x ^ nat n else inverse (x ^ nat (- n)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ x powr real_of_int n = (if 0 \\ n then x ^ nat n else inverse (x ^ nat (- n)))\n[PROOF STEP]\nusing powr_realpow[of x \"nat n\"] powr_realpow[of x \"nat (-n)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x \\ x powr real (nat n) = x ^ nat n\n0 < x \\ x powr real (nat (- n)) = x ^ nat (- n)\n\ngoal (1 subgoal):\n 1. 0 < x \\ x powr real_of_int n = (if 0 \\ n then x ^ nat n else inverse (x ^ nat (- n)))\n[PROOF STEP]\nby (auto simp: field_simps powr_minus)", "meta": {"llama_tokens": 291, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099070035949656, "lm_q2_score": 0.8056321936479701, "lm_q1q2_score": 0.7330503753218636}} {"text": "[STATEMENT]\nlemma pderiv_power_Suc: \"pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p\n[PROOF STEP]\nproof (induction n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. pderiv (p ^ Suc 0) = smult (of_nat (Suc 0)) (p ^ 0) * pderiv p\n 2. \\n. pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p \\ pderiv (p ^ Suc (Suc n)) = smult (of_nat (Suc (Suc n))) (p ^ Suc n) * pderiv p\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\npderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p\n\ngoal (2 subgoals):\n 1. pderiv (p ^ Suc 0) = smult (of_nat (Suc 0)) (p ^ 0) * pderiv p\n 2. \\n. pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p \\ pderiv (p ^ Suc (Suc n)) = smult (of_nat (Suc (Suc n))) (p ^ Suc n) * pderiv p\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\npderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\npderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p\n\ngoal (1 subgoal):\n 1. pderiv (p ^ Suc (Suc n)) = smult (of_nat (Suc (Suc n))) (p ^ Suc n) * pderiv p\n[PROOF STEP]\nby (simp add: pderiv_mult smult_add_left algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\npderiv (p ^ Suc (Suc n)) = smult (of_nat (Suc (Suc n))) (p ^ Suc n) * pderiv p\n\ngoal (1 subgoal):\n 1. pderiv (p ^ Suc 0) = smult (of_nat (Suc 0)) (p ^ 0) * pderiv p\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 777, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213664574069, "lm_q2_score": 0.815232489352, "lm_q1q2_score": 0.7329929498066436}} {"text": "[STATEMENT]\nlemma lnorm_triangle_fun:\n assumes f: \"f \\ lspace M p\" and g: \"g \\ lspace M p\" and \"p \\ 1\"\n shows \"lnorm M p (f + g) \\ lnorm M p f + lnorm M p g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lnorm M p (f + g) \\ lnorm M p f + lnorm M p g\n[PROOF STEP]\nusing lnorm_triangle [OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nlnorm M p (\\x. f x + g x) \\ lnorm M p f + lnorm M p g\n\ngoal (1 subgoal):\n 1. lnorm M p (f + g) \\ lnorm M p f + lnorm M p g\n[PROOF STEP]\nby (simp add: plus_fun_def)", "meta": {"llama_tokens": 246, "file": "Fourier_Square_Integrable", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.870597268408361, "lm_q2_score": 0.8418256551882382, "lm_q1q2_score": 0.732891115882959}} {"text": "[STATEMENT]\nlemma homeomorphic_space_prod_topology:\n \"\\X homeomorphic_space X''; Y homeomorphic_space Y'\\\n \\ prod_topology X Y homeomorphic_space prod_topology X'' Y'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\X homeomorphic_space X''; Y homeomorphic_space Y'\\ \\ prod_topology X Y homeomorphic_space prod_topology X'' Y'\n[PROOF STEP]\nusing homeomorphic_maps_prod\n[PROOF STATE]\nproof (prove)\nusing this:\nhomeomorphic_maps (prod_topology ?X ?Y) (prod_topology ?X' ?Y') (\\(x, y). (?f x, ?g y)) (\\(x, y). (?f' x, ?g' y)) = (topspace (prod_topology ?X ?Y) = {} \\ topspace (prod_topology ?X' ?Y') = {} \\ homeomorphic_maps ?X ?X' ?f ?f' \\ homeomorphic_maps ?Y ?Y' ?g ?g')\n\ngoal (1 subgoal):\n 1. \\X homeomorphic_space X''; Y homeomorphic_space Y'\\ \\ prod_topology X Y homeomorphic_space prod_topology X'' Y'\n[PROOF STEP]\nunfolding homeomorphic_space_def\n[PROOF STATE]\nproof (prove)\nusing this:\nhomeomorphic_maps (prod_topology ?X ?Y) (prod_topology ?X' ?Y') (\\(x, y). (?f x, ?g y)) (\\(x, y). (?f' x, ?g' y)) = (topspace (prod_topology ?X ?Y) = {} \\ topspace (prod_topology ?X' ?Y') = {} \\ homeomorphic_maps ?X ?X' ?f ?f' \\ homeomorphic_maps ?Y ?Y' ?g ?g')\n\ngoal (1 subgoal):\n 1. \\\\f g. homeomorphic_maps X X'' f g; \\f g. homeomorphic_maps Y Y' f g\\ \\ \\f g. homeomorphic_maps (prod_topology X Y) (prod_topology X'' Y') f g\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 612, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972684083609, "lm_q2_score": 0.8418256412990658, "lm_q1q2_score": 0.7328911037910834}} {"text": "[STATEMENT]\nlemma mod_2_is_both_even_or_odd:\n \"((even i \\ even j) \\ (odd i \\ odd j)) \\ (i mod 2 = j mod 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (even i \\ even j \\ odd i \\ odd j) = (i mod (2::'a) = j mod (2::'a))\n[PROOF STEP]\nby (metis even_iff_mod_2_eq_zero odd_iff_mod_2_eq_one)", "meta": {"llama_tokens": 158, "file": "Isabelle_Marries_Dirac_Deutsch_Jozsa", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361509525463, "lm_q2_score": 0.8031737940012418, "lm_q1q2_score": 0.7328448051444464}} {"text": "[STATEMENT]\nlemma rewrite_sum_of_powers:\n assumes p: \"(p::nat)>1\"\n shows \"\\ ((^) p ` {0..n}) = (\\ i = 0 .. n . p^i)\" (is \"?l = ?r\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ ((^) p ` {0..n}) = sum ((^) p) {0..n}\n[PROOF STEP]\nby (metis inj_on_def p power_inject_exp sum.reindex_cong)", "meta": {"llama_tokens": 147, "file": "Perfect-Number-Thm_Sigma", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513842182777, "lm_q2_score": 0.8175744695262775, "lm_q1q2_score": 0.7327522500144503}} {"text": "[STATEMENT]\nlemma sum_pair_less_iff:\n \"sum (\\((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m \\ n} =\n sum (\\s. sum (\\i. a i * b (s - i) * c s) {0..s}) {0..n}\"\n (is \"?l = ?r\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(k, m)\\{(k, m). k + m \\ n}. a k * b m * c (k + m)) = (\\s = 0..n. \\i = 0..s. a i * b (s - i) * c s)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\(k, m)\\{(k, m). k + m \\ n}. a k * b m * c (k + m)) = (\\s = 0..n. \\i = 0..s. a i * b (s - i) * c s)\n[PROOF STEP]\nhave th0: \"{(k, m). k + m \\ n} = (\\s\\{0..n}. \\i\\{0..s}. {(i, s - i)})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {(k, m). k + m \\ n} = (\\s\\{0..n}. \\i\\{0..s}. {(i, s - i)})\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{(k, m). k + m \\ n} = (\\s\\{0..n}. \\i\\{0..s}. {(i, s - i)})\n\ngoal (1 subgoal):\n 1. (\\(k, m)\\{(k, m). k + m \\ n}. a k * b m * c (k + m)) = (\\s = 0..n. \\i = 0..s. a i * b (s - i) * c s)\n[PROOF STEP]\nshow \"?l = ?r\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(k, m)\\{(k, m). k + m \\ n}. a k * b m * c (k + m)) = (\\s = 0..n. \\i = 0..s. a i * b (s - i) * c s)\n[PROOF STEP]\nunfolding th0\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(k, m)\\(\\s\\{0..n}. \\i\\{0..s}. {(i, s - i)}). a k * b m * c (k + m)) = (\\s = 0..n. \\i = 0..s. a i * b (s - i) * c s)\n[PROOF STEP]\nby (simp add: sum.UNION_disjoint eq_diff_iff disjoint_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(\\(k, m)\\{(k, m). k + m \\ n}. a k * b m * c (k + m)) = (\\s = 0..n. \\i = 0..s. a i * b (s - i) * c s)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 953, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513675912913, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.7327522384125295}} {"text": "[STATEMENT]\nlemma tendsto_ln_over_ln:\n assumes \"(a::real) > 0\" \"c > 0\"\n shows \"((\\x. ln (a*x) / ln (c*x)) \\ 1) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. ln (a * x) / ln (c * x)) \\ 1) at_top\n[PROOF STEP]\nproof (rule lhospital_at_top_at_top)\n[PROOF STATE]\nproof (state)\ngoal (5 subgoals):\n 1. LIM x at_top. ln (c * x) :> at_top\n 2. \\\\<^sub>F x in at_top. ?g' x \\ 0\n 3. \\\\<^sub>F x in at_top. ((\\x. ln (a * x)) has_real_derivative ?f' x) (at x)\n 4. \\\\<^sub>F x in at_top. ((\\x. ln (c * x)) has_real_derivative ?g' x) (at x)\n 5. ((\\x. ?f' x / ?g' x) \\ 1) at_top\n[PROOF STEP]\nshow \"LIM x at_top. ln (c*x) :> at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LIM x at_top. ln (c * x) :> at_top\n[PROOF STEP]\nby (intro filterlim_compose[OF ln_at_top] filterlim_tendsto_pos_mult_at_top[OF tendsto_const] \n filterlim_ident assms(2))\n[PROOF STATE]\nproof (state)\nthis:\nLIM x at_top. ln (c * x) :> at_top\n\ngoal (4 subgoals):\n 1. \\\\<^sub>F x in at_top. ?g' x \\ 0\n 2. \\\\<^sub>F x in at_top. ((\\x. ln (a * x)) has_real_derivative ?f' x) (at x)\n 3. \\\\<^sub>F x in at_top. ((\\x. ln (c * x)) has_real_derivative ?g' x) (at x)\n 4. ((\\x. ?f' x / ?g' x) \\ 1) at_top\n[PROOF STEP]\nshow \"eventually (\\x. ((\\x. ln (a*x)) has_real_derivative (inverse x)) (at x)) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. ((\\x. ln (a * x)) has_real_derivative inverse x) (at x)\n[PROOF STEP]\nusing eventually_gt_at_top[of \"inverse a\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\neventually ((<) (inverse a)) at_top\n0 < a\n0 < c\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. ((\\x. ln (a * x)) has_real_derivative inverse x) (at x)\n[PROOF STEP]\nby (auto elim!: eventually_mono intro!: derivative_eq_intros simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in at_top. ((\\x. ln (a * x)) has_real_derivative inverse x) (at x)\n\ngoal (3 subgoals):\n 1. \\\\<^sub>F x in at_top. ?g' x \\ 0\n 2. \\\\<^sub>F x in at_top. ((\\x. ln (c * x)) has_real_derivative ?g' x) (at x)\n 3. ((\\x. inverse x / ?g' x) \\ 1) at_top\n[PROOF STEP]\nshow \"eventually (\\x. ((\\x. ln (c*x)) has_real_derivative (inverse x)) (at x)) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. ((\\x. ln (c * x)) has_real_derivative inverse x) (at x)\n[PROOF STEP]\nusing eventually_gt_at_top[of \"inverse c\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\neventually ((<) (inverse c)) at_top\n0 < a\n0 < c\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. ((\\x. ln (c * x)) has_real_derivative inverse x) (at x)\n[PROOF STEP]\nby (auto elim!: eventually_mono intro!: derivative_eq_intros simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in at_top. ((\\x. ln (c * x)) has_real_derivative inverse x) (at x)\n\ngoal (2 subgoals):\n 1. \\\\<^sub>F x in at_top. inverse x \\ 0\n 2. ((\\x. inverse x / inverse x) \\ 1) at_top\n[PROOF STEP]\nshow \"((\\x::real. inverse x / inverse x) \\ 1) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. inverse x / inverse x) \\ 1) at_top\n[PROOF STEP]\nby (subst tendsto_cong[of _ \"\\_. 1\"]) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. inverse x / inverse x) \\ 1) at_top\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. inverse x \\ 0\n[PROOF STEP]\nqed simp_all", "meta": {"llama_tokens": 1680, "file": "Landau_Symbols_Landau_Real_Products", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588023318196, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.7327477942546258}} {"text": "[STATEMENT]\nlemma local_lipschitz_temp_dyn:\n assumes \"0 < (a::real)\"\n shows \"local_lipschitz UNIV UNIV (\\t::real. f a L)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local_lipschitz UNIV UNIV (\\t. f a L)\n[PROOF STEP]\napply(unfold local_lipschitz_def lipschitz_on_def dist_norm)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\UNIV. \\t\\UNIV. \\u>0. \\La. \\t\\cball t u \\ UNIV. 0 \\ La \\ (\\xa\\cball x u \\ UNIV. \\y\\cball x u \\ UNIV. \\f a L xa - f a L y\\ \\ La * \\xa - y\\)\n[PROOF STEP]\napply(clarsimp, rule_tac x=1 in exI, clarsimp, rule_tac x=a in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a * \\xa - y\\)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a * \\xa - y\\)\n[PROOF STEP]\napply(simp add: norm_diff_temp_dyn)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. \\dist t ta \\ 1; 0 < a\\ \\ \\xa\\cball x 1. \\y\\cball x 1. \\xa $ 1 - y $ 1\\ \\ \\xa - y\\\n[PROOF STEP]\napply(simp add: norm_vec_def L2_set_def, unfold UNIV_4, clarsimp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ \\xa $ 1 - y $ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nunfolding real_sqrt_abs[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2) \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nby (rule real_le_lsqrt) auto", "meta": {"llama_tokens": 1097, "file": "Hybrid_Systems_VCs_HS_VC_Examples", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587905460026, "lm_q2_score": 0.8244619285331332, "lm_q1q2_score": 0.7327477864543323}} {"text": "[STATEMENT]\nlemma summable_Cauchy_product:\n fixes a b :: \"nat \\ 'a::{real_normed_algebra,banach}\"\n assumes \"summable (\\k. norm (a k))\"\n and \"summable (\\k. norm (b k))\"\n shows \"summable (\\k. \\i\\k. a i * b (k - i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. summable (\\k. \\i\\k. a i * b (k - i))\n[PROOF STEP]\nusing Cauchy_product_sums[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. \\i\\k. a i * b (k - i)) sums ((\\k. a k) * (\\k. b k))\n\ngoal (1 subgoal):\n 1. summable (\\k. \\i\\k. a i * b (k - i))\n[PROOF STEP]\nby (simp add: sums_iff)", "meta": {"llama_tokens": 296, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467675095294, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.7326146260743155}} {"text": "[STATEMENT]\nlemma summable_Cauchy_product:\n fixes a b :: \"nat \\ 'a::{real_normed_algebra,banach}\"\n assumes \"summable (\\k. norm (a k))\"\n and \"summable (\\k. norm (b k))\"\n shows \"summable (\\k. \\i\\k. a i * b (k - i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. summable (\\k. \\i\\k. a i * b (k - i))\n[PROOF STEP]\nusing Cauchy_product_sums[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. \\i\\k. a i * b (k - i)) sums ((\\k. a k) * (\\k. b k))\n\ngoal (1 subgoal):\n 1. summable (\\k. \\i\\k. a i * b (k - i))\n[PROOF STEP]\nby (simp add: sums_iff)", "meta": {"llama_tokens": 296, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467675095294, "lm_q2_score": 0.8333245932423308, "lm_q1q2_score": 0.7326146224351885}} {"text": "[STATEMENT]\ntheorem supplement1_Legendre:\n \"prime p \\ 2 < p \\ Legendre (-1) p = (-1)^((p-1) div 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\prime p; 2 < p\\ \\ Legendre (- 1) (int p) = (- 1) ^ ((p - 1) div 2)\n[PROOF STEP]\nusing euler_criterion[of p \"-1\"] Legendre_in_cong_eq[symmetric, of p]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\prime p; 2 < p\\ \\ [Legendre (- 1) (int p) = (- 1) ^ ((p - 1) div 2)] (mod int p)\n\\2 < int p; ?b \\ {- 1, 0, 1}\\ \\ (Legendre ?a ?m = ?b) = [Legendre ?a ?m = ?b] (mod int p)\n\ngoal (1 subgoal):\n 1. \\prime p; 2 < p\\ \\ Legendre (- 1) (int p) = (- 1) ^ ((p - 1) div 2)\n[PROOF STEP]\nby (simp add: minus_one_power_iff)", "meta": {"llama_tokens": 366, "file": "Probabilistic_Prime_Tests_Legendre_Symbol", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467611766711, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.7326146207969889}} {"text": "[STATEMENT]\nlemma summable_Cauchy_product:\n fixes a b :: \"nat \\ 'a::{real_normed_algebra,banach}\"\n assumes \"summable (\\k. norm (a k))\"\n and \"summable (\\k. norm (b k))\"\n shows \"summable (\\k. \\i\\k. a i * b (k - i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. summable (\\k. \\i\\k. a i * b (k - i))\n[PROOF STEP]\nusing Cauchy_product_sums[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. \\i\\k. a i * b (k - i)) sums ((\\k. a k) * (\\k. b k))\n\ngoal (1 subgoal):\n 1. summable (\\k. \\i\\k. a i * b (k - i))\n[PROOF STEP]\nby (simp add: sums_iff)", "meta": {"llama_tokens": 296, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467675095294, "lm_q2_score": 0.8333245911726382, "lm_q1q2_score": 0.732614620615625}} {"text": "[STATEMENT]\nlemma local_lipschitz_temp_dyn:\n assumes \"0 < (a::real)\"\n shows \"local_lipschitz UNIV UNIV (\\t::real. f a L)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local_lipschitz UNIV UNIV (\\t. f a L)\n[PROOF STEP]\napply(unfold local_lipschitz_def lipschitz_on_def dist_norm)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\UNIV. \\t\\UNIV. \\u>0. \\La. \\t\\cball t u \\ UNIV. 0 \\ La \\ (\\xa\\cball x u \\ UNIV. \\y\\cball x u \\ UNIV. \\f a L xa - f a L y\\ \\ La \\ \\xa - y\\)\n[PROOF STEP]\napply(clarsimp, rule_tac x=1 in exI, clarsimp, rule_tac x=a in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a \\ \\xa - y\\)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a \\ \\xa - y\\)\n[PROOF STEP]\napply(simp_all add: norm_diff_temp_dyn)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. \\dist t ta \\ 1; 0 < a\\ \\ \\xa\\cball x 1. \\y\\cball x 1. \\xa $ 1 - y $ 1\\ \\ \\xa - y\\\n[PROOF STEP]\napply(simp add: norm_vec_def L2_set_def, unfold UNIV_4, clarsimp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ \\xa $ 1 - y $ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nunfolding real_sqrt_abs[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2) \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nby (rule real_le_lsqrt) auto", "meta": {"llama_tokens": 1105, "file": "Hybrid_Systems_VCs_ModalKleeneAlgebra_HS_VC_MKA_Examples_rel", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467580102418, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.7326146181583255}} {"text": "[STATEMENT]\nlemma tan_60: \"tan (pi / 3) = sqrt 3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tan (pi / 3) = sqrt 3\n[PROOF STEP]\nunfolding tan_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin (pi / 3) / cos (pi / 3) = sqrt 3\n[PROOF STEP]\nby (simp add: sin_60 cos_60)", "meta": {"llama_tokens": 143, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9149009596336303, "lm_q2_score": 0.800691997339971, "lm_q1q2_score": 0.7325538767373075}} {"text": "[STATEMENT]\nlemma foldl1_map:\n assumes \"associative f\"\n and \"xs \\ []\"\n and \"ys \\ []\"\n shows \"foldl1 f (map h (xs @ ys)) \n = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. foldl1 f (map h (xs @ ys)) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. foldl1 f (map h (xs @ ys)) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n[PROOF STEP]\nhave \n \"foldl1 f (map h (xs @ ys)) \n = foldl1 f (map h xs @ map h ys)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. foldl1 f (map h (xs @ ys)) = foldl1 f (map h xs @ map h ys)\n[PROOF STEP]\nusing L5\n[PROOF STATE]\nproof (prove)\nusing this:\nmap ?f (?xs @ ?ys) = map ?f ?xs @ map ?f ?ys\n\ngoal (1 subgoal):\n 1. foldl1 f (map h (xs @ ys)) = foldl1 f (map h xs @ map h ys)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfoldl1 f (map h (xs @ ys)) = foldl1 f (map h xs @ map h ys)\n\ngoal (1 subgoal):\n 1. foldl1 f (map h (xs @ ys)) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfoldl1 f (map h (xs @ ys)) = foldl1 f (map h xs @ map h ys)\n\ngoal (1 subgoal):\n 1. foldl1 f (map h (xs @ ys)) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n[PROOF STEP]\nhave \n \"\\ = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. foldl1 f (map h xs @ map h ys) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n[PROOF STEP]\nusing assms and L8 [where f=f]\n[PROOF STATE]\nproof (prove)\nusing this:\nassociative f\nxs \\ []\nys \\ []\n\\associative f; ?xs \\ []; ?ys \\ []\\ \\ foldl1 f (?xs @ ?ys) = f (foldl1 f ?xs) (foldl1 f ?ys)\n\ngoal (1 subgoal):\n 1. foldl1 f (map h xs @ map h ys) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfoldl1 f (map h xs @ map h ys) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n\ngoal (1 subgoal):\n 1. foldl1 f (map h (xs @ ys)) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nfoldl1 f (map h (xs @ ys)) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfoldl1 f (map h (xs @ ys)) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n\ngoal (1 subgoal):\n 1. foldl1 f (map h (xs @ ys)) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nfoldl1 f (map h (xs @ ys)) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1249, "file": "MuchAdoAboutTwo_MuchAdoAboutTwo", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473779969194, "lm_q2_score": 0.8397339736884711, "lm_q1q2_score": 0.732539730162072}} {"text": "[STATEMENT]\nlemma sum_Un_ge:\n fixes f :: \"_ \\ real\"\n assumes \"finite M\" \"finite N\" \"\\B \\ M \\ N. 0 < f B\"\n shows \"sum f M \\ sum f (M \\ N)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nhave \"0 \\ sum f N - sum f (M \\ N)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ sum f N - sum f (M \\ N)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite M\nfinite N\n\\B\\M \\ N. 0 < f B\n\ngoal (1 subgoal):\n 1. 0 \\ sum f N - sum f (M \\ N)\n[PROOF STEP]\nby (smt DiffD1 inf.cobounded2 UnCI sum_mono2)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ sum f N - sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ sum f N - sum f (M \\ N)\n[PROOF STEP]\nhave \"sum f M \\ sum f M + sum f N - sum f (M \\ N)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ sum f N - sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f M + sum f N - sum f (M \\ N)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum f M \\ sum f M + sum f N - sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum f M \\ sum f M + sum f N - sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nhave \"... = sum f (M \\ N)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f M + sum f N - sum f (M \\ N) = sum f (M \\ N)\n[PROOF STEP]\nusing sum_Un[OF assms(1,2), symmetric]\n[PROOF STATE]\nproof (prove)\nusing this:\nsum ?f M + sum ?f N - sum ?f (M \\ N) = sum ?f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M + sum f N - sum f (M \\ N) = sum f (M \\ N)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nsum f M + sum f N - sum f (M \\ N) = sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum f M \\ sum f (M \\ N)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsum f M \\ sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nsum f M \\ sum f (M \\ N)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1149, "file": "Approximation_Algorithms_Approx_BP_Hoare", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511543206819, "lm_q2_score": 0.8558511451289037, "lm_q1q2_score": 0.7324811904852497}} {"text": "[STATEMENT]\nlemma sum_Un_ge:\n fixes f :: \"_ \\ real\"\n assumes \"finite M\" \"finite N\" \"\\B \\ M \\ N. 0 < f B\"\n shows \"sum f M \\ sum f (M \\ N)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nhave \"0 \\ sum f N - sum f (M \\ N)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ sum f N - sum f (M \\ N)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite M\nfinite N\n\\B\\M \\ N. 0 < f B\n\ngoal (1 subgoal):\n 1. 0 \\ sum f N - sum f (M \\ N)\n[PROOF STEP]\nby (smt DiffD1 inf.cobounded2 UnCI sum_mono2)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ sum f N - sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ sum f N - sum f (M \\ N)\n[PROOF STEP]\nhave \"sum f M \\ sum f M + sum f N - sum f (M \\ N)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ sum f N - sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f M + sum f N - sum f (M \\ N)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum f M \\ sum f M + sum f N - sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum f M \\ sum f M + sum f N - sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nhave \"... = sum f (M \\ N)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f M + sum f N - sum f (M \\ N) = sum f (M \\ N)\n[PROOF STEP]\nusing sum_Un[OF assms(1,2), symmetric]\n[PROOF STATE]\nproof (prove)\nusing this:\nsum ?f M + sum ?f N - sum ?f (M \\ N) = sum ?f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M + sum f N - sum f (M \\ N) = sum f (M \\ N)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nsum f M + sum f N - sum f (M \\ N) = sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum f M \\ sum f (M \\ N)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsum f M \\ sum f (M \\ N)\n\ngoal (1 subgoal):\n 1. sum f M \\ sum f (M \\ N)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nsum f M \\ sum f (M \\ N)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1149, "file": "Approximation_Algorithms_Approx_BP_Hoare", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511543206819, "lm_q2_score": 0.855851143290548, "lm_q1q2_score": 0.7324811889118908}} {"text": "[STATEMENT]\nlemma leq_antisym: \"\\ (x, y) \\ leq; (y, x) \\ leq \\ \\ x = y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(x, y) \\ leq; (y, x) \\ leq\\ \\ x = y\n[PROOF STEP]\nusing po\n[PROOF STATE]\nproof (prove)\nusing this:\nPartial_order leq\n\ngoal (1 subgoal):\n 1. \\(x, y) \\ leq; (y, x) \\ leq\\ \\ x = y\n[PROOF STEP]\nby(simp add: order_on_defs antisym_def)", "meta": {"llama_tokens": 220, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8519528170040852, "lm_q2_score": 0.8596637505099167, "lm_q1q2_score": 0.7323929539232207}} {"text": "[STATEMENT]\nlemma sum_squares_bound: \"2 * x * y \\ x\\<^sup>2 + y\\<^sup>2\"\n for x y :: \"'a::linordered_field\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (2::'a) * x * y \\ x\\<^sup>2 + y\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (2::'a) * x * y \\ x\\<^sup>2 + y\\<^sup>2\n[PROOF STEP]\nhave \"(x - y)\\<^sup>2 = x * x - 2 * x * y + y * y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x - y)\\<^sup>2 = x * x - (2::'a) * x * y + y * y\n[PROOF STEP]\nby algebra\n[PROOF STATE]\nproof (state)\nthis:\n(x - y)\\<^sup>2 = x * x - (2::'a) * x * y + y * y\n\ngoal (1 subgoal):\n 1. (2::'a) * x * y \\ x\\<^sup>2 + y\\<^sup>2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(x - y)\\<^sup>2 = x * x - (2::'a) * x * y + y * y\n[PROOF STEP]\nhave \"0 \\ x\\<^sup>2 - 2 * x * y + y\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(x - y)\\<^sup>2 = x * x - (2::'a) * x * y + y * y\n\ngoal (1 subgoal):\n 1. (0::'a) \\ x\\<^sup>2 - (2::'a) * x * y + y\\<^sup>2\n[PROOF STEP]\nby (metis sum_power2_ge_zero zero_le_double_add_iff_zero_le_single_add power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) \\ x\\<^sup>2 - (2::'a) * x * y + y\\<^sup>2\n\ngoal (1 subgoal):\n 1. (2::'a) * x * y \\ x\\<^sup>2 + y\\<^sup>2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(0::'a) \\ x\\<^sup>2 - (2::'a) * x * y + y\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) \\ x\\<^sup>2 - (2::'a) * x * y + y\\<^sup>2\n\ngoal (1 subgoal):\n 1. (2::'a) * x * y \\ x\\<^sup>2 + y\\<^sup>2\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\n(2::'a) * x * y \\ x\\<^sup>2 + y\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 873, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637361282707, "lm_q2_score": 0.8519528000888387, "lm_q1q2_score": 0.7323929271293128}} {"text": "[STATEMENT]\nlemma finprod_Suc2:\n \"f \\ {..Suc n} \\ carrier G \\\n finprod G f {..Suc n} = (finprod G (%i. f (Suc i)) {..n} \\ f 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f \\ {..Suc n} \\ carrier G \\ finprod G f {..Suc n} = (\\i\\{..n}. f (Suc i)) \\ f 0\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. f \\ {..Suc 0} \\ carrier G \\ finprod G f {..Suc 0} = (\\i\\{..0}. f (Suc i)) \\ f 0\n 2. \\n. \\f \\ {..Suc n} \\ carrier G \\ finprod G f {..Suc n} = (\\i\\{..n}. f (Suc i)) \\ f 0; f \\ {..Suc (Suc n)} \\ carrier G\\ \\ finprod G f {..Suc (Suc n)} = (\\i\\{..Suc n}. f (Suc i)) \\ f 0\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nf \\ {..Suc 0} \\ carrier G\n\ngoal (2 subgoals):\n 1. f \\ {..Suc 0} \\ carrier G \\ finprod G f {..Suc 0} = (\\i\\{..0}. f (Suc i)) \\ f 0\n 2. \\n. \\f \\ {..Suc n} \\ carrier G \\ finprod G f {..Suc n} = (\\i\\{..n}. f (Suc i)) \\ f 0; f \\ {..Suc (Suc n)} \\ carrier G\\ \\ finprod G f {..Suc (Suc n)} = (\\i\\{..Suc n}. f (Suc i)) \\ f 0\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nf \\ {..Suc 0} \\ carrier G\n\ngoal (1 subgoal):\n 1. finprod G f {..Suc 0} = (\\i\\{..0}. f (Suc i)) \\ f 0\n[PROOF STEP]\nby (simp add: Pi_def)\n[PROOF STATE]\nproof (state)\nthis:\nfinprod G f {..Suc 0} = (\\i\\{..0}. f (Suc i)) \\ f 0\n\ngoal (1 subgoal):\n 1. \\n. \\f \\ {..Suc n} \\ carrier G \\ finprod G f {..Suc n} = (\\i\\{..n}. f (Suc i)) \\ f 0; f \\ {..Suc (Suc n)} \\ carrier G\\ \\ finprod G f {..Suc (Suc n)} = (\\i\\{..Suc n}. f (Suc i)) \\ f 0\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\f \\ {..Suc n} \\ carrier G \\ finprod G f {..Suc n} = (\\i\\{..n}. f (Suc i)) \\ f 0; f \\ {..Suc (Suc n)} \\ carrier G\\ \\ finprod G f {..Suc (Suc n)} = (\\i\\{..Suc n}. f (Suc i)) \\ f 0\n[PROOF STEP]\ncase Suc\n[PROOF STATE]\nproof (state)\nthis:\nf \\ {..Suc n_} \\ carrier G \\ finprod G f {..Suc n_} = (\\i\\{..n_}. f (Suc i)) \\ f 0\nf \\ {..Suc (Suc n_)} \\ carrier G\n\ngoal (1 subgoal):\n 1. \\n. \\f \\ {..Suc n} \\ carrier G \\ finprod G f {..Suc n} = (\\i\\{..n}. f (Suc i)) \\ f 0; f \\ {..Suc (Suc n)} \\ carrier G\\ \\ finprod G f {..Suc (Suc n)} = (\\i\\{..Suc n}. f (Suc i)) \\ f 0\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nf \\ {..Suc n_} \\ carrier G \\ finprod G f {..Suc n_} = (\\i\\{..n_}. f (Suc i)) \\ f 0\nf \\ {..Suc (Suc n_)} \\ carrier G\n\ngoal (1 subgoal):\n 1. finprod G f {..Suc (Suc n_)} = (\\i\\{..Suc n_}. f (Suc i)) \\ f 0\n[PROOF STEP]\nby (simp add: m_assoc Pi_def)\n[PROOF STATE]\nproof (state)\nthis:\nfinprod G f {..Suc (Suc n_)} = (\\i\\{..Suc n_}. f (Suc i)) \\ f 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1601, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382094310357, "lm_q2_score": 0.849971175657575, "lm_q1q2_score": 0.7322826447440194}} {"text": "[STATEMENT]\nlemma (in group) ord_conv_Least: (* Manuel Eberl *)\n assumes \"x \\ carrier G\" \"\\n::nat > 0. x [^] n = \\\"\n shows \"ord x = (LEAST n::nat. 0 < n \\ x [^] n = \\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ord x = (LEAST n. 0 < n \\ x [^] n = \\)\n[PROOF STEP]\nproof (rule antisym)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ord x \\ (LEAST n. 0 < n \\ x [^] n = \\)\n 2. (LEAST n. 0 < n \\ x [^] n = \\) \\ ord x\n[PROOF STEP]\nshow \"ord x \\ (LEAST n::nat. 0 < n \\ x [^] n = \\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ord x \\ (LEAST n. 0 < n \\ x [^] n = \\)\n[PROOF STEP]\nusing assms LeastI_ex[OF assms(2)]\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ carrier G\n\\n>0. x [^] n = \\\n0 < (LEAST n. 0 < n \\ x [^] n = \\) \\ x [^] (LEAST n. 0 < n \\ x [^] n = \\) = \\\n\ngoal (1 subgoal):\n 1. ord x \\ (LEAST n. 0 < n \\ x [^] n = \\)\n[PROOF STEP]\nby (intro ord_le) auto\n[PROOF STATE]\nproof (state)\nthis:\nord x \\ (LEAST n. 0 < n \\ x [^] n = \\)\n\ngoal (1 subgoal):\n 1. (LEAST n. 0 < n \\ x [^] n = \\) \\ ord x\n[PROOF STEP]\nshow \"ord x \\ (LEAST n::nat. 0 < n \\ x [^] n = \\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (LEAST n. 0 < n \\ x [^] n = \\) \\ ord x\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ carrier G\n\\n>0. x [^] n = \\\n\ngoal (1 subgoal):\n 1. (LEAST n. 0 < n \\ x [^] n = \\) \\ ord x\n[PROOF STEP]\nby (intro Least_le) (auto intro: pow_ord_eq_1 ord_pos)\n[PROOF STATE]\nproof (state)\nthis:\n(LEAST n. 0 < n \\ x [^] n = \\) \\ ord x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 854, "file": "Finitely_Generated_Abelian_Groups_Miscellaneous_Groups", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8499711604559848, "lm_q2_score": 0.8615382058759129, "lm_q1q2_score": 0.7322826286255167}} {"text": "[STATEMENT]\nlemma tensor_vec_dim[simp]:\n shows \"dim_vec (tensor_vec u v) = dim_vec u * (dim_vec v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_vec (u \\ v) = dim_vec u * dim_vec v\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dim_vec (u \\ v) = dim_vec u * dim_vec v\n[PROOF STEP]\nhave \"length (mult.vec_vec_Tensor (*) (list_of_vec u) (list_of_vec v)) = \n length (list_of_vec u) * length (list_of_vec v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (mult.vec_vec_Tensor (*) (list_of_vec u) (list_of_vec v)) = length (list_of_vec u) * length (list_of_vec v)\n[PROOF STEP]\nusing mult.vec_vec_Tensor_length[of \"1::real\" \"(*)\" \"list_of_vec u\" \"list_of_vec v\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nMatrix_Tensor.mult (complex_of_real 1) (*) \\ length (mult.vec_vec_Tensor (*) (list_of_vec u) (list_of_vec v)) = length (list_of_vec u) * length (list_of_vec v)\n\ngoal (1 subgoal):\n 1. length (mult.vec_vec_Tensor (*) (list_of_vec u) (list_of_vec v)) = length (list_of_vec u) * length (list_of_vec v)\n[PROOF STEP]\nby (simp add: Matrix_Tensor.mult_def)\n[PROOF STATE]\nproof (state)\nthis:\nlength (mult.vec_vec_Tensor (*) (list_of_vec u) (list_of_vec v)) = length (list_of_vec u) * length (list_of_vec v)\n\ngoal (1 subgoal):\n 1. dim_vec (u \\ v) = dim_vec u * dim_vec v\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (mult.vec_vec_Tensor (*) (list_of_vec u) (list_of_vec v)) = length (list_of_vec u) * length (list_of_vec v)\n\ngoal (1 subgoal):\n 1. dim_vec (u \\ v) = dim_vec u * dim_vec v\n[PROOF STEP]\nunfolding tensor_vec_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (mult.vec_vec_Tensor (*) (list_of_vec u) (list_of_vec v)) = length (list_of_vec u) * length (list_of_vec v)\n\ngoal (1 subgoal):\n 1. dim_vec (vec_of_list (mult.vec_vec_Tensor (*) (list_of_vec u) (list_of_vec v))) = dim_vec u * dim_vec v\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec (u \\ v) = dim_vec u * dim_vec v\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 904, "file": "Projective_Measurements_Linear_Algebra_Complements", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772482857833, "lm_q2_score": 0.8376199613065411, "lm_q1q2_score": 0.7321445508880657}} {"text": "[STATEMENT]\nlemma sum_of_2_squares_nat_mult [intro]:\n assumes \"sum_of_2_squares_nat x\" \"sum_of_2_squares_nat y\"\n shows \"sum_of_2_squares_nat (x * y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_of_2_squares_nat (x * y)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum_of_2_squares_nat (x * y)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nsum_of_2_squares_nat x\nsum_of_2_squares_nat y\n[PROOF STEP]\nobtain z1 z2 where \"x = gauss_int_norm z1\" \"y = gauss_int_norm z2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsum_of_2_squares_nat x\nsum_of_2_squares_nat y\n\ngoal (1 subgoal):\n 1. (\\z1 z2. \\x = gauss_int_norm z1; y = gauss_int_norm z2\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: sum_of_2_squares_nat_altdef)\n[PROOF STATE]\nproof (state)\nthis:\nx = gauss_int_norm z1\ny = gauss_int_norm z2\n\ngoal (1 subgoal):\n 1. sum_of_2_squares_nat (x * y)\n[PROOF STEP]\nhence \"x * y = gauss_int_norm (z1 * z2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx = gauss_int_norm z1\ny = gauss_int_norm z2\n\ngoal (1 subgoal):\n 1. x * y = gauss_int_norm (z1 * z2)\n[PROOF STEP]\nby (simp add: gauss_int_norm_mult)\n[PROOF STATE]\nproof (state)\nthis:\nx * y = gauss_int_norm (z1 * z2)\n\ngoal (1 subgoal):\n 1. sum_of_2_squares_nat (x * y)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx * y = gauss_int_norm (z1 * z2)\n\ngoal (1 subgoal):\n 1. sum_of_2_squares_nat (x * y)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsum_of_2_squares_nat (x * y)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 764, "file": "Gaussian_Integers_Gaussian_Integers_Sums_Of_Two_Squares", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916205190225, "lm_q2_score": 0.847967764140929, "lm_q1q2_score": 0.732128262029529}} {"text": "[STATEMENT]\nlemma card_bijections_domain_and_range_permutation_eq_1:\n assumes \"finite A\" \"finite B\"\n assumes \"card A = card B\"\n shows \"card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 1\n[PROOF STEP]\nusing bij_betw_quotient_domain_and_range_permutation_eq[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n{f \\ A \\\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B = {{f \\ A \\\\<^sub>E B. bij_betw f A B}}\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 1\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 325, "file": "Twelvefold_Way_Card_Bijections_Direct", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942067038785, "lm_q2_score": 0.8104789086703224, "lm_q1q2_score": 0.732100902857584}} {"text": "[STATEMENT]\ntheorem prime_gauss_int_of_nat:\n fixes n :: nat\n assumes prime: \"prime n\" and \"[n = 3] (mod 4)\"\n shows \"prime (of_nat n :: gauss_int)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prime (of_nat n)\n[PROOF STEP]\nusing prime_elem_gauss_int_of_nat[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nprime_elem (of_nat n)\n\ngoal (1 subgoal):\n 1. prime (of_nat n)\n[PROOF STEP]\nunfolding prime_def\n[PROOF STATE]\nproof (prove)\nusing this:\nprime_elem (of_nat n)\n\ngoal (1 subgoal):\n 1. prime_elem (of_nat n) \\ normalize (of_nat n) = of_nat n\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 255, "file": "Gaussian_Integers_Gaussian_Integers", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9059898203834277, "lm_q2_score": 0.8080672135527632, "lm_q1q2_score": 0.7321006696644049}} {"text": "[STATEMENT]\ntheorem wens_set_single_eq:\n \"[|F = mk_program (init, {act}, allowed); single_valued act|]\n ==> wens_set F B =\n insert (wens_single act B) (range (wens_single_finite act B))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\F = mk_program (init, {act}, allowed); single_valued act\\ \\ wens_set F B = insert (wens_single act B) (range (wens_single_finite act B))\n[PROOF STEP]\napply (rule equalityI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\F = mk_program (init, {act}, allowed); single_valued act\\ \\ wens_set F B \\ insert (wens_single act B) (range (wens_single_finite act B))\n 2. \\F = mk_program (init, {act}, allowed); single_valued act\\ \\ insert (wens_single act B) (range (wens_single_finite act B)) \\ wens_set F B\n[PROOF STEP]\napply (simp add: wens_set_subset_single)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\F = mk_program (init, {act}, allowed); single_valued act\\ \\ insert (wens_single act B) (range (wens_single_finite act B)) \\ wens_set F B\n[PROOF STEP]\napply (erule ssubst, erule single_subset_wens_set)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 489, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898153067649, "lm_q2_score": 0.8080672066194946, "lm_q1q2_score": 0.7321006592806494}} {"text": "[STATEMENT]\nlemma mat_inv_mult_mm:\n assumes \"mat_det A \\ 0\" and \"mat_det B \\ 0\"\n shows \"mat_inv (A *\\<^sub>m\\<^sub>m B) = mat_inv B *\\<^sub>m\\<^sub>m mat_inv A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat_inv (A *\\<^sub>m\\<^sub>m B) = mat_inv B *\\<^sub>m\\<^sub>m mat_inv A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nmat_det A \\ 0\nmat_det B \\ 0\n\ngoal (1 subgoal):\n 1. mat_inv (A *\\<^sub>m\\<^sub>m B) = mat_inv B *\\<^sub>m\\<^sub>m mat_inv A\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\mat_det A \\ 0; mat_det B \\ 0\\ \\ mat_inv (A *\\<^sub>m\\<^sub>m B) = mat_inv B *\\<^sub>m\\<^sub>m mat_inv A\n[PROOF STEP]\nhave \"(A *\\<^sub>m\\<^sub>m B) *\\<^sub>m\\<^sub>m (mat_inv B *\\<^sub>m\\<^sub>m mat_inv A) = eye\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *\\<^sub>m\\<^sub>m B *\\<^sub>m\\<^sub>m (mat_inv B *\\<^sub>m\\<^sub>m mat_inv A) = eye\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nmat_det A \\ 0\nmat_det B \\ 0\n\ngoal (1 subgoal):\n 1. A *\\<^sub>m\\<^sub>m B *\\<^sub>m\\<^sub>m (mat_inv B *\\<^sub>m\\<^sub>m mat_inv A) = eye\n[PROOF STEP]\nby (metis mat_inv_r mult_mm_assoc mult_mm_inv_r)\n[PROOF STATE]\nproof (state)\nthis:\nA *\\<^sub>m\\<^sub>m B *\\<^sub>m\\<^sub>m (mat_inv B *\\<^sub>m\\<^sub>m mat_inv A) = eye\n\ngoal (1 subgoal):\n 1. \\mat_det A \\ 0; mat_det B \\ 0\\ \\ mat_inv (A *\\<^sub>m\\<^sub>m B) = mat_inv B *\\<^sub>m\\<^sub>m mat_inv A\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nA *\\<^sub>m\\<^sub>m B *\\<^sub>m\\<^sub>m (mat_inv B *\\<^sub>m\\<^sub>m mat_inv A) = eye\n\ngoal (1 subgoal):\n 1. mat_inv (A *\\<^sub>m\\<^sub>m B) = mat_inv B *\\<^sub>m\\<^sub>m mat_inv A\n[PROOF STEP]\nusing mult_mm_inv_l[of \"A *\\<^sub>m\\<^sub>m B\" \"mat_inv B *\\<^sub>m\\<^sub>m mat_inv A\" eye] assms mat_eye_r\n[PROOF STATE]\nproof (prove)\nusing this:\nA *\\<^sub>m\\<^sub>m B *\\<^sub>m\\<^sub>m (mat_inv B *\\<^sub>m\\<^sub>m mat_inv A) = eye\n\\mat_det (A *\\<^sub>m\\<^sub>m B) \\ 0; A *\\<^sub>m\\<^sub>m B *\\<^sub>m\\<^sub>m (mat_inv B *\\<^sub>m\\<^sub>m mat_inv A) = eye\\ \\ mat_inv B *\\<^sub>m\\<^sub>m mat_inv A = mat_inv (A *\\<^sub>m\\<^sub>m B) *\\<^sub>m\\<^sub>m eye\nmat_det A \\ 0\nmat_det B \\ 0\n?A *\\<^sub>m\\<^sub>m eye = ?A\n\ngoal (1 subgoal):\n 1. mat_inv (A *\\<^sub>m\\<^sub>m B) = mat_inv B *\\<^sub>m\\<^sub>m mat_inv A\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmat_inv (A *\\<^sub>m\\<^sub>m B) = mat_inv B *\\<^sub>m\\<^sub>m mat_inv A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1255, "file": "Complex_Geometry_Matrices", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970842359877, "lm_q2_score": 0.8311430436757313, "lm_q1q2_score": 0.7320683694526083}} {"text": "[STATEMENT]\ntheorem nat_even_power_sums_complex:\n assumes n': \"n' > 0\"\n shows \"(\\k. 1 / of_nat (Suc k) ^ (2*n') :: complex) sums\n of_real ((-1) ^ Suc n' * bernoulli (2*n') * (2 * pi) ^ (2 * n') / (2 * fact (2*n')))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\ndefine n where \"n = 2 * n'\"\n[PROOF STATE]\nproof (state)\nthis:\nn = 2 * n'\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nfrom n'\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < n'\n[PROOF STEP]\nhave n: \"n \\ 2\" \"even n\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n'\n\ngoal (1 subgoal):\n 1. 2 \\ n &&& even n\n[PROOF STEP]\nby (auto simp: n_def)\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ n\neven n\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\ndefine zeta :: complex where \"zeta = (\\k. 1 / of_nat (Suc k) ^ n)\"\n[PROOF STATE]\nproof (state)\nthis:\nzeta = (\\k. 1 / of_nat (Suc k) ^ n)\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave \"summable (\\k. 1 / of_nat (Suc k) ^ n :: complex)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. summable (\\k. 1 / of_nat (Suc k) ^ n)\n[PROOF STEP]\nusing inverse_power_summable[of n] n\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n \\ summable (\\na. inverse (of_nat na ^ n))\n2 \\ n\neven n\n\ngoal (1 subgoal):\n 1. summable (\\k. 1 / of_nat (Suc k) ^ n)\n[PROOF STEP]\nby (subst summable_Suc_iff) (simp add: divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\nsummable (\\k. 1 / of_nat (Suc k) ^ n)\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhence \"(\\k. \\i zeta\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\k. 1 / of_nat (Suc k) ^ n)\n\ngoal (1 subgoal):\n 1. (\\k. \\i zeta\n[PROOF STEP]\nby (subst (asm) summable_sums_iff) (simp add: sums_def zeta_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. \\i zeta\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. \\i zeta\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave \"(\\k. \\ik. \\i\\{0<..k}. 1 / of_nat i ^ n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. \\ik. \\i\\{0<..k}. (1::'a) / of_nat i ^ n)\n[PROOF STEP]\nby (intro ext sum.reindex_bij_witness[of _ \"\\n. n - 1\" Suc]) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. \\ik. \\i\\{0<..k}. (1::?'a1) / of_nat i ^ n)\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\k. \\i\\{0<..k}. 1 / of_nat i ^ n) \\ zeta\n[PROOF STEP]\nhave zeta_limit: \"(\\k. \\i\\{0<..k}. 1 / of_nat i ^ n) \\ zeta\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. \\i\\{0<..k}. 1 / of_nat i ^ n) \\ zeta\n\ngoal (1 subgoal):\n 1. (\\k. \\i\\{0<..k}. 1 / of_nat i ^ n) \\ zeta\n[PROOF STEP]\n.\n\n \\ \\This is the exponential generating function of the Bernoulli numbers.\\\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. \\i\\{0<..k}. 1 / of_nat i ^ n) \\ zeta\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\ndefine f where \"f = (\\z::complex. if z = 0 then 1 else z / (exp z - 1))\"\n\n \\ \\We will integrate over this function, since its residue at the origin\n is the $n$-th coefficient of @{term f}. Note that it has singularities\n at all points $2ik\\pi$ for $k\\in\\mathbb{Z}$.\\\n[PROOF STATE]\nproof (state)\nthis:\nf = (\\z. if z = 0 then 1 else z / (exp z - 1))\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\ndefine g where \"g = (\\z::complex. 1 / (z ^ n * (exp z - 1)))\"\n\n \\ \\We integrate along a rectangle of width $2m$ and height $2(2m+1)\\pi$\n with its centre at the origin. The benefit of the rectangular path is that\n it is easier to bound the value of the exponential appearing in the integrand.\n The horizontal lines of the rectangle are always right in the middle between \n two adjacent singularities.\\\n[PROOF STATE]\nproof (state)\nthis:\ng = (\\z. 1 / (z ^ n * (exp z - 1)))\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\ndefine \\ :: \"nat \\ real \\ complex\" \n where \"\\ = (\\m. rectpath (-real m - real (2*m+1)*pi*\\) (real m + real (2*m+1)*pi*\\))\"\n\n \\ \\This set is a convex open enclosing set the contains our path.\\\n[PROOF STATE]\nproof (state)\nthis:\n\\ = (\\m. rectpath (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\))\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\ndefine A where \"A = (\\m::nat. box (-(real m+1) - (2*m+2)*pi*\\) (real m+1 + (2*m+2)*pi*\\))\"\n\n \\ \\These are all the singularities in the enclosing inside the path\n (and also inside @{term A}).\\\n[PROOF STATE]\nproof (state)\nthis:\nA = (\\m. box (complex_of_real (- (real m + 1)) - complex_of_real (real (2 * m + 2) * pi) * \\) (complex_of_real (real m + 1) + complex_of_real (real (2 * m + 2) * pi) * \\))\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\ndefine S where \"S = (\\m::nat. (\\n. 2 * pi * of_int n * \\) ` {-m..m})\"\n\n \\ \\Any singularity in @{term A} is of the form $2ki\\pi$ where $|k| \\leq m$.\\\n[PROOF STATE]\nproof (state)\nthis:\nS = (\\m. (\\n. complex_of_real (2 * pi * real_of_int n) * \\) ` {- int m..int m})\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave int_bound: \"k \\ {-int m..int m}\" if \"2 * pi * k * \\ \\ A m\" for k m\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k \\ {- int m..int m}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. k \\ {- int m..int m}\n[PROOF STEP]\nfrom that\n[PROOF STATE]\nproof (chain)\npicking this:\ncomplex_of_real (2 * pi * real_of_int k) * \\ \\ A m\n[PROOF STEP]\nhave \"(-real (Suc m)) * (2 * pi) < real_of_int k * (2 * pi) \\ \n real (Suc m) * (2 * pi) > real_of_int k * (2 * pi)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (2 * pi * real_of_int k) * \\ \\ A m\n\ngoal (1 subgoal):\n 1. - real (Suc m) * (2 * pi) < real_of_int k * (2 * pi) \\ real_of_int k * (2 * pi) < real (Suc m) * (2 * pi)\n[PROOF STEP]\nby (auto simp: A_def in_box_complex_iff algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n- real (Suc m) * (2 * pi) < real_of_int k * (2 * pi) \\ real_of_int k * (2 * pi) < real (Suc m) * (2 * pi)\n\ngoal (1 subgoal):\n 1. k \\ {- int m..int m}\n[PROOF STEP]\nhence \"-real (Suc m) < real_of_int k \\ real_of_int k < real (Suc m)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n- real (Suc m) * (2 * pi) < real_of_int k * (2 * pi) \\ real_of_int k * (2 * pi) < real (Suc m) * (2 * pi)\n\ngoal (1 subgoal):\n 1. - real (Suc m) < real_of_int k \\ real_of_int k < real (Suc m)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n- real (Suc m) < real_of_int k \\ real_of_int k < real (Suc m)\n\ngoal (1 subgoal):\n 1. k \\ {- int m..int m}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n- real (Suc m) < real_of_int k \\ real_of_int k < real (Suc m)\n\ngoal (1 subgoal):\n 1. k \\ {- int m..int m}\n[PROOF STEP]\nhave \"-real (Suc m) = real_of_int (-int (Suc m))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - real (Suc m) = real_of_int (- int (Suc m))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n- real (Suc m) = real_of_int (- int (Suc m))\n\ngoal (1 subgoal):\n 1. k \\ {- int m..int m}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n- real (Suc m) = real_of_int (- int (Suc m))\n\ngoal (1 subgoal):\n 1. k \\ {- int m..int m}\n[PROOF STEP]\nhave \"real (Suc m) = real_of_int (int (Suc m))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (Suc m) = real_of_int (int (Suc m))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal (Suc m) = real_of_int (int (Suc m))\n\ngoal (1 subgoal):\n 1. k \\ {- int m..int m}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (Suc m) = real_of_int (int (Suc m))\n\ngoal (1 subgoal):\n 1. k \\ {- int m..int m}\n[PROOF STEP]\nhave \"real_of_int (- int (Suc m)) < real_of_int k \\ \n real_of_int k < real_of_int (int (Suc m)) \\ k \\ {-int m..int m}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (real_of_int (- int (Suc m)) < real_of_int k \\ real_of_int k < real_of_int (int (Suc m))) = (k \\ {- int m..int m})\n[PROOF STEP]\nby (subst of_int_less_iff) auto\n[PROOF STATE]\nproof (state)\nthis:\n(real_of_int (- int (Suc m)) < real_of_int k \\ real_of_int k < real_of_int (int (Suc m))) = (k \\ {- int m..int m})\n\ngoal (1 subgoal):\n 1. k \\ {- int m..int m}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nk \\ {- int m..int m}\n[PROOF STEP]\nshow \"k \\ {-int m..int m}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ {- int m..int m}\n\ngoal (1 subgoal):\n 1. k \\ {- int m..int m}\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nk \\ {- int m..int m}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (2 * pi * real_of_int ?k1) * \\ \\ A ?m1 \\ ?k1 \\ {- int ?m1..int ?m1}\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave zeros: \"\\k\\{-int m..int m}. z = 2 * pi * of_int k * \\\" if \"z \\ A m\" \"exp z = 1\" for z m\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\k\\{- int m..int m}. z = complex_of_real (2 * pi * real_of_int k) * \\\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\k\\{- int m..int m}. z = complex_of_real (2 * pi * real_of_int k) * \\\n[PROOF STEP]\nfrom that(2)\n[PROOF STATE]\nproof (chain)\npicking this:\nexp z = 1\n[PROOF STEP]\nobtain k where z_eq: \"z = 2 * pi * of_int k * \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\nexp z = 1\n\ngoal (1 subgoal):\n 1. (\\k. z = complex_of_real (2 * pi * real_of_int k) * \\ \\ thesis) \\ thesis\n[PROOF STEP]\nunfolding exp_eq_1\n[PROOF STATE]\nproof (prove)\nusing this:\nRe z = 0 \\ (\\n. Im z = real_of_int (2 * n) * pi)\n\ngoal (1 subgoal):\n 1. (\\k. z = complex_of_real (2 * pi * real_of_int k) * \\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: complex_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\nz = complex_of_real (2 * pi * real_of_int k) * \\\n\ngoal (1 subgoal):\n 1. \\k\\{- int m..int m}. z = complex_of_real (2 * pi * real_of_int k) * \\\n[PROOF STEP]\nwith int_bound[of k] and that(1)\n[PROOF STATE]\nproof (chain)\npicking this:\ncomplex_of_real (2 * pi * real_of_int k) * \\ \\ A ?m1 \\ k \\ {- int ?m1..int ?m1}\nz \\ A m\nz = complex_of_real (2 * pi * real_of_int k) * \\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (2 * pi * real_of_int k) * \\ \\ A ?m1 \\ k \\ {- int ?m1..int ?m1}\nz \\ A m\nz = complex_of_real (2 * pi * real_of_int k) * \\\n\ngoal (1 subgoal):\n 1. \\k\\{- int m..int m}. z = complex_of_real (2 * pi * real_of_int k) * \\\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\k\\{- int m..int m}. z = complex_of_real (2 * pi * real_of_int k) * \\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\?z1 \\ A ?m1; exp ?z1 = 1\\ \\ \\k\\{- int ?m1..int ?m1}. ?z1 = complex_of_real (2 * pi * real_of_int k) * \\\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave zeros': \"z ^ n * (exp z - 1) \\ 0\" if \"z \\ A m - S m\" for z m\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z ^ n * (exp z - 1) \\ 0\n[PROOF STEP]\nusing zeros[of z] that\n[PROOF STATE]\nproof (prove)\nusing this:\n\\z \\ A ?m1; exp z = 1\\ \\ \\k\\{- int ?m1..int ?m1}. z = complex_of_real (2 * pi * real_of_int k) * \\\nz \\ A m - S m\n\ngoal (1 subgoal):\n 1. z ^ n * (exp z - 1) \\ 0\n[PROOF STEP]\nby (auto simp: S_def)\n\n \\ \\The singularities all lie strictly inside the integration path.\\\n[PROOF STATE]\nproof (state)\nthis:\n?z1 \\ A ?m1 - S ?m1 \\ ?z1 ^ n * (exp ?z1 - 1) \\ 0\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave subset: \"S m \\ box (-real m - real(2*m+1)*pi*\\) (real m + real(2*m+1)*pi*\\)\" if \"m > 0\" for m\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. S m \\ box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n[PROOF STEP]\nproof (rule, goal_cases)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ S m \\ x \\ box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n[PROOF STEP]\ncase (1 z)\n[PROOF STATE]\nproof (state)\nthis:\nz \\ S m\n\ngoal (1 subgoal):\n 1. \\x. x \\ S m \\ x \\ box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nz \\ S m\n[PROOF STEP]\nobtain k :: int where k: \"k \\ {-int m..int m}\" \"z = 2 * pi * k * \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ S m\n\ngoal (1 subgoal):\n 1. (\\k. \\k \\ {- int m..int m}; z = complex_of_real (2 * pi * real_of_int k) * \\\\ \\ thesis) \\ thesis\n[PROOF STEP]\nunfolding S_def\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ (\\n. complex_of_real (2 * pi * real_of_int n) * \\) ` {- int m..int m}\n\ngoal (1 subgoal):\n 1. (\\k. \\k \\ {- int m..int m}; z = complex_of_real (2 * pi * real_of_int k) * \\\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nk \\ {- int m..int m}\nz = complex_of_real (2 * pi * real_of_int k) * \\\n\ngoal (1 subgoal):\n 1. \\x. x \\ S m \\ x \\ box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n[PROOF STEP]\nhave \"2 * pi * -m + -pi < 2 * pi * k + 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * pi * real_of_int (- int m) + - pi < 2 * pi * real_of_int k + 0\n[PROOF STEP]\nusing k\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ {- int m..int m}\nz = complex_of_real (2 * pi * real_of_int k) * \\\n\ngoal (1 subgoal):\n 1. 2 * pi * real_of_int (- int m) + - pi < 2 * pi * real_of_int k + 0\n[PROOF STEP]\nby (intro add_le_less_mono mult_left_mono) auto\n[PROOF STATE]\nproof (state)\nthis:\n2 * pi * real_of_int (- int m) + - pi < 2 * pi * real_of_int k + 0\n\ngoal (1 subgoal):\n 1. \\x. x \\ S m \\ x \\ box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n2 * pi * real_of_int (- int m) + - pi < 2 * pi * real_of_int k + 0\n\ngoal (1 subgoal):\n 1. \\x. x \\ S m \\ x \\ box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n[PROOF STEP]\nhave \"2 * pi * k + 0 < 2 * pi * m + pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * pi * real_of_int k + 0 < 2 * pi * real m + pi\n[PROOF STEP]\nusing k\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ {- int m..int m}\nz = complex_of_real (2 * pi * real_of_int k) * \\\n\ngoal (1 subgoal):\n 1. 2 * pi * real_of_int k + 0 < 2 * pi * real m + pi\n[PROOF STEP]\nby (intro add_le_less_mono mult_left_mono) auto\n[PROOF STATE]\nproof (state)\nthis:\n2 * pi * real_of_int k + 0 < 2 * pi * real m + pi\n\ngoal (1 subgoal):\n 1. \\x. x \\ S m \\ x \\ box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n2 * pi * real_of_int (- int m) + - pi < 2 * pi * real_of_int k + 0\n2 * pi * real_of_int k + 0 < 2 * pi * real m + pi\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * pi * real_of_int (- int m) + - pi < 2 * pi * real_of_int k + 0\n2 * pi * real_of_int k + 0 < 2 * pi * real m + pi\n\ngoal (1 subgoal):\n 1. z \\ box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n[PROOF STEP]\nusing k \\m > 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * pi * real_of_int (- int m) + - pi < 2 * pi * real_of_int k + 0\n2 * pi * real_of_int k + 0 < 2 * pi * real m + pi\nk \\ {- int m..int m}\nz = complex_of_real (2 * pi * real_of_int k) * \\\n0 < m\n\ngoal (1 subgoal):\n 1. z \\ box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n[PROOF STEP]\nby (auto simp: A_def in_box_complex_iff algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nz \\ box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n0 < ?m1 \\ S ?m1 \\ box (complex_of_real (- real ?m1) - complex_of_real (real (2 * ?m1 + 1) * pi) * \\) (complex_of_real (real ?m1) + complex_of_real (real (2 * ?m1 + 1) * pi) * \\)\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nfrom n and zeros'\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ n\neven n\n?z1 \\ A ?m1 - S ?m1 \\ ?z1 ^ n * (exp ?z1 - 1) \\ 0\n[PROOF STEP]\nhave holo: \"g holomorphic_on A m - S m\" for m\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\neven n\n?z1 \\ A ?m1 - S ?m1 \\ ?z1 ^ n * (exp ?z1 - 1) \\ 0\n\ngoal (1 subgoal):\n 1. g holomorphic_on A m - S m\n[PROOF STEP]\nunfolding g_def\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\neven n\n?z1 \\ A ?m1 - S ?m1 \\ ?z1 ^ n * (exp ?z1 - 1) \\ 0\n\ngoal (1 subgoal):\n 1. (\\z. 1 / (z ^ n * (exp z - 1))) holomorphic_on A m - S m\n[PROOF STEP]\nby (intro holomorphic_intros) auto\n\n \\ \\The integration path lies completely inside $A$ and does not cross\n any singularities.\\\n[PROOF STATE]\nproof (state)\nthis:\ng holomorphic_on A ?m1 - S ?m1\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave path_subset: \"path_image (\\ m) \\ A m - S m\" if \"m > 0\" for m\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ A m - S m\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ A m - S m\n[PROOF STEP]\nhave \"path_image (\\ m) \\ cbox (-real m - (2 * m + 1) * pi * \\) (real m + (2 * m + 1) * pi * \\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ cbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n[PROOF STEP]\nunfolding \\_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. path_image (rectpath (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)) \\ cbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n[PROOF STEP]\nby (rule path_image_rectpath_subset_cbox) auto\n[PROOF STATE]\nproof (state)\nthis:\npath_image (\\ m) \\ cbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ A m - S m\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\npath_image (\\ m) \\ cbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\n\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ A m - S m\n[PROOF STEP]\nhave \"\\ \\ A m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\) \\ A m\n[PROOF STEP]\nunfolding A_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\) \\ box (complex_of_real (- (real m + 1)) - complex_of_real (real (2 * m + 2) * pi) * \\) (complex_of_real (real m + 1) + complex_of_real (real (2 * m + 2) * pi) * \\)\n[PROOF STEP]\nby (subst subset_box_complex) auto\n[PROOF STATE]\nproof (state)\nthis:\ncbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\) \\ A m\n\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ A m - S m\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\npath_image (\\ m) \\ A m\n[PROOF STEP]\nhave \"path_image (\\ m) \\ A m\"\n[PROOF STATE]\nproof (prove)\nusing this:\npath_image (\\ m) \\ A m\n\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ A m\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\npath_image (\\ m) \\ A m\n\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ A m - S m\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\npath_image (\\ m) \\ A m\n\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ A m - S m\n[PROOF STEP]\nhave \"path_image (\\ m) \\ S m = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ S m = {}\n[PROOF STEP]\nproof safe\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. \\x \\ path_image (\\ m); x \\ S m\\ \\ x \\ {}\n[PROOF STEP]\nfix z\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. \\x \\ path_image (\\ m); x \\ S m\\ \\ x \\ {}\n[PROOF STEP]\nassume z: \"z \\ path_image (\\ m)\" \"z \\ S m\"\n[PROOF STATE]\nproof (state)\nthis:\nz \\ path_image (\\ m)\nz \\ S m\n\ngoal (1 subgoal):\n 1. \\x. \\x \\ path_image (\\ m); x \\ S m\\ \\ x \\ {}\n[PROOF STEP]\nfrom this(2)\n[PROOF STATE]\nproof (chain)\npicking this:\nz \\ S m\n[PROOF STEP]\nobtain k :: int where k: \"z = 2 * pi * k * \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ S m\n\ngoal (1 subgoal):\n 1. (\\k. z = complex_of_real (2 * pi * real_of_int k) * \\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: S_def)\n[PROOF STATE]\nproof (state)\nthis:\nz = complex_of_real (2 * pi * real_of_int k) * \\\n\ngoal (1 subgoal):\n 1. \\x. \\x \\ path_image (\\ m); x \\ S m\\ \\ x \\ {}\n[PROOF STEP]\nhence [simp]: \"Re z = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz = complex_of_real (2 * pi * real_of_int k) * \\\n\ngoal (1 subgoal):\n 1. Re z = 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nRe z = 0\n\ngoal (1 subgoal):\n 1. \\x. \\x \\ path_image (\\ m); x \\ S m\\ \\ x \\ {}\n[PROOF STEP]\nfrom z(1)\n[PROOF STATE]\nproof (chain)\npicking this:\nz \\ path_image (\\ m)\n[PROOF STEP]\nhave \"\\Im z\\ = of_int (2*m+1) * pi\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ path_image (\\ m)\n\ngoal (1 subgoal):\n 1. \\Im z\\ = real_of_int (int (2 * m + 1)) * pi\n[PROOF STEP]\nusing \\m > 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ path_image (\\ m)\n0 < m\n\ngoal (1 subgoal):\n 1. \\Im z\\ = real_of_int (int (2 * m + 1)) * pi\n[PROOF STEP]\nby (auto simp: \\_def path_image_rectpath)\n[PROOF STATE]\nproof (state)\nthis:\n\\Im z\\ = real_of_int (int (2 * m + 1)) * pi\n\ngoal (1 subgoal):\n 1. \\x. \\x \\ path_image (\\ m); x \\ S m\\ \\ x \\ {}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\Im z\\ = real_of_int (int (2 * m + 1)) * pi\n\ngoal (1 subgoal):\n 1. \\x. \\x \\ path_image (\\ m); x \\ S m\\ \\ x \\ {}\n[PROOF STEP]\nhave \"\\Im z\\ = of_int (2 * \\k\\) * pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Im z\\ = real_of_int (2 * \\k\\) * pi\n[PROOF STEP]\nby (simp add: k abs_mult)\n[PROOF STATE]\nproof (state)\nthis:\n\\Im z\\ = real_of_int (2 * \\k\\) * pi\n\ngoal (1 subgoal):\n 1. \\x. \\x \\ path_image (\\ m); x \\ S m\\ \\ x \\ {}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal_of_int (2 * \\k\\) * pi = real_of_int (int (2 * m + 1)) * pi\n[PROOF STEP]\nhave \"2 * \\k\\ = 2 * m + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal_of_int (2 * \\k\\) * pi = real_of_int (int (2 * m + 1)) * pi\n\ngoal (1 subgoal):\n 1. 2 * \\k\\ = int (2 * m + 1)\n[PROOF STEP]\nby (subst (asm) mult_cancel_right, subst (asm) of_int_eq_iff) simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * \\k\\ = int (2 * m + 1)\n\ngoal (1 subgoal):\n 1. \\x. \\x \\ path_image (\\ m); x \\ S m\\ \\ x \\ {}\n[PROOF STEP]\nhence False\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * \\k\\ = int (2 * m + 1)\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal (1 subgoal):\n 1. \\x. \\x \\ path_image (\\ m); x \\ S m\\ \\ x \\ {}\n[PROOF STEP]\nthus \"z \\ {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nFalse\n\ngoal (1 subgoal):\n 1. z \\ {}\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nz \\ {}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\npath_image (\\ m) \\ S m = {}\n\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ A m - S m\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\npath_image (\\ m) \\ A m\npath_image (\\ m) \\ S m = {}\n[PROOF STEP]\nshow \"path_image (\\ m) \\ A m - S m\"\n[PROOF STATE]\nproof (prove)\nusing this:\npath_image (\\ m) \\ A m\npath_image (\\ m) \\ S m = {}\n\ngoal (1 subgoal):\n 1. path_image (\\ m) \\ A m - S m\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\npath_image (\\ m) \\ A m - S m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n\n \\ \\We now obtain a closed form for the Bernoulli numbers using the integral.\\\n[PROOF STATE]\nproof (state)\nthis:\n0 < ?m1 \\ path_image (\\ ?m1) \\ A ?m1 - S ?m1\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave eq: \"(\\x\\{0<..m}. 1 / of_nat x ^ n) =\n contour_integral (\\ m) g * (2 * pi * \\) ^ n / (4 * pi * \\) -\n complex_of_real (bernoulli n / fact n) * (2 * pi * \\) ^ n / 2\" \n if m: \"m > 0\" for m\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nproof -\n \\ \\We relate the formal power series of the Bernoulli numbers to the\n corresponding complex function.\\\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"subdegree (fps_exp 1 - 1 :: complex fps) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. subdegree (fps_exp 1 - 1) = 1\n[PROOF STEP]\nby (intro subdegreeI) auto\n[PROOF STATE]\nproof (state)\nthis:\nsubdegree (fps_exp 1 - 1) = 1\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhence expansion: \"f has_fps_expansion bernoulli_fps\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsubdegree (fps_exp 1 - 1) = 1\n\ngoal (1 subgoal):\n 1. f has_fps_expansion bernoulli_fps\n[PROOF STEP]\nunfolding f_def bernoulli_fps_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsubdegree (fps_exp 1 - 1) = 1\n\ngoal (1 subgoal):\n 1. (\\z. if z = 0 then 1 else z / (exp z - 1)) has_fps_expansion fps_X / (fps_exp 1 - 1)\n[PROOF STEP]\nby (auto intro!: fps_expansion_intros)\n\n \\ \\We use the Residue Theorem to explicitly compute the integral.\\\n[PROOF STATE]\nproof (state)\nthis:\nf has_fps_expansion bernoulli_fps\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"contour_integral (\\ m) g =\n 2 * pi * \\ * (\\z\\S m. winding_number (\\ m) z * residue g z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. contour_integral (\\ m) g = complex_of_real (2 * pi) * \\ * (\\z\\S m. winding_number (\\ m) z * residue g z)\n[PROOF STEP]\nproof (rule Residue_theorem)\n[PROOF STATE]\nproof (state)\ngoal (8 subgoals):\n 1. open ?s\n 2. connected ?s\n 3. finite (S m)\n 4. g holomorphic_on ?s - S m\n 5. valid_path (\\ m)\n 6. pathfinish (\\ m) = pathstart (\\ m)\n 7. path_image (\\ m) \\ ?s - S m\n 8. \\z. z \\ ?s \\ winding_number (\\ m) z = 0\n[PROOF STEP]\nhave \"cbox (-real m - (2 * m + 1) * pi * \\) (real m + (2 * m + 1) * pi * \\) \\ A m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\) \\ A m\n[PROOF STEP]\nunfolding A_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\) \\ box (complex_of_real (- (real m + 1)) - complex_of_real (real (2 * m + 2) * pi) * \\) (complex_of_real (real m + 1) + complex_of_real (real (2 * m + 2) * pi) * \\)\n[PROOF STEP]\nby (subst subset_box_complex) simp_all\n[PROOF STATE]\nproof (state)\nthis:\ncbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\) \\ A m\n\ngoal (8 subgoals):\n 1. open ?s\n 2. connected ?s\n 3. finite (S m)\n 4. g holomorphic_on ?s - S m\n 5. valid_path (\\ m)\n 6. pathfinish (\\ m) = pathstart (\\ m)\n 7. path_image (\\ m) \\ ?s - S m\n 8. \\z. z \\ ?s \\ winding_number (\\ m) z = 0\n[PROOF STEP]\nthus \"\\z. z \\ A m \\ winding_number (\\ m) z = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\) \\ A m\n\ngoal (1 subgoal):\n 1. \\z. z \\ A m \\ winding_number (\\ m) z = 0\n[PROOF STEP]\nunfolding \\_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncbox (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\) \\ A m\n\ngoal (1 subgoal):\n 1. \\z. z \\ A m \\ winding_number (rectpath (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)) z = 0\n[PROOF STEP]\nby (intro winding_number_rectpath_outside allI impI) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\z. z \\ A m \\ winding_number (\\ m) z = 0\n\ngoal (7 subgoals):\n 1. open (A m)\n 2. connected (A m)\n 3. finite (S m)\n 4. g holomorphic_on A m - S m\n 5. valid_path (\\ m)\n 6. pathfinish (\\ m) = pathstart (\\ m)\n 7. path_image (\\ m) \\ A m - S m\n[PROOF STEP]\nqed (insert holo path_subset m, auto simp: \\_def A_def S_def intro: convex_connected)\n \\ \\Clearly, all the winding numbers are 1\\\n[PROOF STATE]\nproof (state)\nthis:\ncontour_integral (\\ m) g = complex_of_real (2 * pi) * \\ * (\\z\\S m. winding_number (\\ m) z * residue g z)\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncontour_integral (\\ m) g = complex_of_real (2 * pi) * \\ * (\\z\\S m. winding_number (\\ m) z * residue g z)\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"winding_number (\\ m) z = 1\" if \"z \\ S m\" for z\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. winding_number (\\ m) z = 1\n[PROOF STEP]\nunfolding \\_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. winding_number (rectpath (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)) z = 1\n[PROOF STEP]\nusing subset[of m] that m\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < m \\ S m \\ box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)\nz \\ S m\n0 < m\n\ngoal (1 subgoal):\n 1. winding_number (rectpath (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)) z = 1\n[PROOF STEP]\nby (subst winding_number_rectpath) blast+\n[PROOF STATE]\nproof (state)\nthis:\n?z1 \\ S m \\ winding_number (\\ m) ?z1 = 1\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhence \"(\\z\\S m. winding_number (\\ m) z * residue g z) = (\\z\\S m. residue g z)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?z1 \\ S m \\ winding_number (\\ m) ?z1 = 1\n\ngoal (1 subgoal):\n 1. (\\z\\S m. winding_number (\\ m) z * residue g z) = sum (residue g) (S m)\n[PROOF STEP]\nby (intro sum.cong) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(\\z\\S m. winding_number (\\ m) z * residue g z) = sum (residue g) (S m)\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\z\\S m. winding_number (\\ m) z * residue g z) = sum (residue g) (S m)\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"\\ = (\\k=-int m..int m. residue g (2 * pi * of_int k * \\))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum (residue g) (S m) = (\\k = - int m..int m. residue g (complex_of_real (2 * pi * real_of_int k) * \\))\n[PROOF STEP]\nunfolding S_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum (residue g) ((\\n. complex_of_real (2 * pi * real_of_int n) * \\) ` {- int m..int m}) = (\\k = - int m..int m. residue g (complex_of_real (2 * pi * real_of_int k) * \\))\n[PROOF STEP]\nby (subst sum.reindex) (auto simp: inj_on_def o_def)\n[PROOF STATE]\nproof (state)\nthis:\nsum (residue g) (S m) = (\\k = - int m..int m. residue g (complex_of_real (2 * pi * real_of_int k) * \\))\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum (residue g) (S m) = (\\k = - int m..int m. residue g (complex_of_real (2 * pi * real_of_int k) * \\))\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"{-int m..int m} = insert 0 ({-int m..int m}-{0})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {- int m..int m} = insert 0 ({- int m..int m} - {0})\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{- int m..int m} = insert 0 ({- int m..int m} - {0})\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n{- int m..int m} = insert 0 ({- int m..int m} - {0})\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"(\\k\\\\. residue g (2 * pi * of_int k * \\)) = \n residue g 0 + (\\k\\{-int m..m}-{0}. residue g (2 * pi * of_int k * \\))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\insert 0 ({- int m..int m} - {0}). residue g (complex_of_real (2 * pi * real_of_int k) * \\)) = residue g 0 + (\\k\\{- int m..int m} - {0}. residue g (complex_of_real (2 * pi * real_of_int k) * \\))\n[PROOF STEP]\nby (subst sum.insert) auto\n \\ \\The residue at the origin is just the $n$-th coefficient of $f$.\\\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\insert 0 ({- int m..int m} - {0}). residue g (complex_of_real (2 * pi * real_of_int k) * \\)) = residue g 0 + (\\k\\{- int m..int m} - {0}. residue g (complex_of_real (2 * pi * real_of_int k) * \\))\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\insert 0 ({- int m..int m} - {0}). residue g (complex_of_real (2 * pi * real_of_int k) * \\)) = residue g 0 + (\\k\\{- int m..int m} - {0}. residue g (complex_of_real (2 * pi * real_of_int k) * \\))\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"residue g 0 = residue (\\z. f z / z ^ Suc n) 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. residue g 0 = residue (\\z. f z / z ^ Suc n) 0\n[PROOF STEP]\nunfolding f_def g_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. residue (\\z. 1 / (z ^ n * (exp z - 1))) 0 = residue (\\z. (if z = 0 then 1 else z / (exp z - 1)) / z ^ Suc n) 0\n[PROOF STEP]\nby (intro residue_cong eventually_mono[OF eventually_at_ball[of 1]]) auto\n[PROOF STATE]\nproof (state)\nthis:\nresidue g 0 = residue (\\z. f z / z ^ Suc n) 0\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nresidue g 0 = residue (\\z. f z / z ^ Suc n) 0\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"\\ = fps_nth bernoulli_fps n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. residue (\\z. f z / z ^ Suc n) 0 = bernoulli_fps $ n\n[PROOF STEP]\nby (rule residue_fps_expansion_over_power_at_0 [OF expansion])\n[PROOF STATE]\nproof (state)\nthis:\nresidue (\\z. f z / z ^ Suc n) 0 = bernoulli_fps $ n\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nresidue (\\z. f z / z ^ Suc n) 0 = bernoulli_fps $ n\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"\\ = of_real (bernoulli n / fact n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bernoulli_fps $ n = complex_of_real (bernoulli n / fact n)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nbernoulli_fps $ n = complex_of_real (bernoulli n / fact n)\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nbernoulli_fps $ n = complex_of_real (bernoulli n / fact n)\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"(\\k\\{-int m..m}-{0}. residue g (2 * pi * of_int k * \\)) = \n (\\k\\{-int m..m}-{0}. 1 / of_int k ^ n) / (2 * pi * \\) ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\{- int m..int m} - {0}. residue g (complex_of_real (2 * pi * real_of_int k) * \\)) = (\\k\\{- int m..int m} - {0}. 1 / of_int k ^ n) / (complex_of_real (2 * pi) * \\) ^ n\n[PROOF STEP]\nproof (subst sum_divide_distrib, intro refl sum.cong, goal_cases)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ {- int m..int m} - {0} \\ residue g (complex_of_real (2 * pi * real_of_int x) * \\) = 1 / of_int x ^ n / (complex_of_real (2 * pi) * \\) ^ n\n[PROOF STEP]\ncase (1 k)\n[PROOF STATE]\nproof (state)\nthis:\nk \\ {- int m..int m} - {0}\n\ngoal (1 subgoal):\n 1. \\x. x \\ {- int m..int m} - {0} \\ residue g (complex_of_real (2 * pi * real_of_int x) * \\) = 1 / of_int x ^ n / (complex_of_real (2 * pi) * \\) ^ n\n[PROOF STEP]\nhence *: \"residue g (2 * pi * of_int k * \\) = 1 / (2 * complex_of_real pi * of_int k * \\) ^ n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ {- int m..int m} - {0}\n\ngoal (1 subgoal):\n 1. residue g (complex_of_real (2 * pi * real_of_int k) * \\) = 1 / (2 * complex_of_real pi * of_int k * \\) ^ n\n[PROOF STEP]\nunfolding g_def\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ {- int m..int m} - {0}\n\ngoal (1 subgoal):\n 1. residue (\\z. 1 / (z ^ n * (exp z - 1))) (complex_of_real (2 * pi * real_of_int k) * \\) = 1 / (2 * complex_of_real pi * of_int k * \\) ^ n\n[PROOF STEP]\nby (subst residue_bernoulli) auto\n[PROOF STATE]\nproof (state)\nthis:\nresidue g (complex_of_real (2 * pi * real_of_int k) * \\) = 1 / (2 * complex_of_real pi * of_int k * \\) ^ n\n\ngoal (1 subgoal):\n 1. \\x. x \\ {- int m..int m} - {0} \\ residue g (complex_of_real (2 * pi * real_of_int x) * \\) = 1 / of_int x ^ n / (complex_of_real (2 * pi) * \\) ^ n\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nresidue g (complex_of_real (2 * pi * real_of_int k) * \\) = 1 / (2 * complex_of_real pi * of_int k * \\) ^ n\n\ngoal (1 subgoal):\n 1. residue g (complex_of_real (2 * pi * real_of_int k) * \\) = 1 / of_int k ^ n / (complex_of_real (2 * pi) * \\) ^ n\n[PROOF STEP]\nusing 1\n[PROOF STATE]\nproof (prove)\nusing this:\nresidue g (complex_of_real (2 * pi * real_of_int k) * \\) = 1 / (2 * complex_of_real pi * of_int k * \\) ^ n\nk \\ {- int m..int m} - {0}\n\ngoal (1 subgoal):\n 1. residue g (complex_of_real (2 * pi * real_of_int k) * \\) = 1 / of_int k ^ n / (complex_of_real (2 * pi) * \\) ^ n\n[PROOF STEP]\nby (subst *) (simp add: divide_simps power_mult_distrib)\n[PROOF STATE]\nproof (state)\nthis:\nresidue g (complex_of_real (2 * pi * real_of_int k) * \\) = 1 / of_int k ^ n / (complex_of_real (2 * pi) * \\) ^ n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\{- int m..int m} - {0}. residue g (complex_of_real (2 * pi * real_of_int k) * \\)) = (\\k\\{- int m..int m} - {0}. 1 / of_int k ^ n) / (complex_of_real (2 * pi) * \\) ^ n\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\{- int m..int m} - {0}. residue g (complex_of_real (2 * pi * real_of_int k) * \\)) = (\\k\\{- int m..int m} - {0}. 1 / of_int k ^ n) / (complex_of_real (2 * pi) * \\) ^ n\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"(\\k\\{-int m..m}-{0}. 1 / of_int k ^ n) =\n (\\(a,b)\\{0<..m}\\{-1,1::int}. 1 / of_int (int a) ^ n :: complex)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\{- int m..int m} - {0}. 1 / of_int k ^ n) = (\\(a, b)\\{0<..m} \\ {- 1, 1}. 1 / of_int (int a) ^ n)\n[PROOF STEP]\nusing n\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\neven n\n\ngoal (1 subgoal):\n 1. (\\k\\{- int m..int m} - {0}. 1 / of_int k ^ n) = (\\(a, b)\\{0<..m} \\ {- 1, 1}. 1 / of_int (int a) ^ n)\n[PROOF STEP]\nby (intro sum.reindex_bij_witness[of _ \"\\k. snd k * int (fst k)\" \"\\k. (nat \\k\\,sgn k)\"])\n (auto split: if_splits simp: abs_if)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\{- int m..int m} - {0}. 1 / of_int k ^ n) = (\\(a, b)\\{0<..m} \\ {- 1, 1}. 1 / of_int (int a) ^ n)\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\{- int m..int m} - {0}. 1 / of_int k ^ n) = (\\(a, b)\\{0<..m} \\ {- 1, 1}. 1 / of_int (int a) ^ n)\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"\\ = (\\x\\{0<..m}. 2 / of_nat x ^ n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(a, b)\\{0<..m} \\ {- 1, 1}. 1 / of_int (int a) ^ n) = (\\x\\{0<..m}. 2 / of_nat x ^ n)\n[PROOF STEP]\nusing n\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\neven n\n\ngoal (1 subgoal):\n 1. (\\(a, b)\\{0<..m} \\ {- 1, 1}. 1 / of_int (int a) ^ n) = (\\x\\{0<..m}. 2 / of_nat x ^ n)\n[PROOF STEP]\nby (subst sum.Sigma [symmetric]) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\(a, b)\\{0<..m} \\ {- 1, 1}. 1 / of_int (int a) ^ n) = (\\x\\{0<..m}. 2 / of_nat x ^ n)\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\(a, b)\\{0<..m} \\ {- 1, 1}. 1 / of_int (int a) ^ n) = (\\x\\{0<..m}. 2 / of_nat x ^ n)\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"\\ = (\\x\\{0<..m}. 1 / of_nat x ^ n) * 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 2 / of_nat x ^ n) = (\\x\\{0<..m}. 1 / of_nat x ^ n) * 2\n[PROOF STEP]\nby (simp add: sum_distrib_right)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{0<..m}. 2 / of_nat x ^ n) = (\\x\\{0<..m}. 1 / of_nat x ^ n) * 2\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncontour_integral (\\ m) g = complex_of_real (2 * pi) * \\ * (complex_of_real (bernoulli n / fact n) + (\\x\\{0<..m}. 1 / of_nat x ^ n) * 2 / (complex_of_real (2 * pi) * \\) ^ n)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncontour_integral (\\ m) g = complex_of_real (2 * pi) * \\ * (complex_of_real (bernoulli n / fact n) + (\\x\\{0<..m}. 1 / of_nat x ^ n) * 2 / (complex_of_real (2 * pi) * \\) ^ n)\n\ngoal (1 subgoal):\n 1. (\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{0<..m}. 1 / of_nat x ^ n) = contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n\n \\ \\The ugly part: We have to prove a bound on the integral by splitting\n it into four integrals over lines and bounding each part separately.\\\n[PROOF STATE]\nproof (state)\nthis:\n0 < ?m1 \\ (\\x\\{0<..?m1}. 1 / of_nat x ^ n) = contour_integral (\\ ?m1) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave \"eventually (\\m. norm (contour_integral (\\ m) g) \\ \n ((4 + 12 * pi) + 6 * pi / m) / real m ^ (n - 1)) sequentially\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F m in sequentially. cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nusing eventually_gt_at_top[of \"1::nat\"]\n[PROOF STATE]\nproof (prove)\nusing this:\neventually ((<) 1) sequentially\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F m in sequentially. cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nproof eventually_elim\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\ncase (elim m)\n[PROOF STATE]\nproof (state)\nthis:\n1 < m\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nlet ?c = \"(2*m+1) * pi * \\\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\ndefine I where \"I = (\\p1 p2. contour_integral (linepath p1 p2) g)\"\n[PROOF STATE]\nproof (state)\nthis:\nI = (\\p1 p2. contour_integral (linepath p1 p2) g)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\ndefine p1 p2 p3 p4 where \"p1 = -real m - ?c\" and \"p2 = real m - ?c\" \n and \"p3 = real m + ?c\" and \"p4 = -real m + ?c\"\n[PROOF STATE]\nproof (state)\nthis:\np1 = complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\\np2 = complex_of_real (real m) - complex_of_real (real (2 * m + 1) * pi) * \\\np3 = complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\\np4 = complex_of_real (- real m) + complex_of_real (real (2 * m + 1) * pi) * \\\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave eq: \"\\ m = linepath p1 p2 +++ linepath p2 p3 +++ linepath p3 p4 +++ linepath p4 p1\"\n (is \"\\ m = ?\\'\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ m = linepath p1 p2 +++ linepath p2 p3 +++ linepath p3 p4 +++ linepath p4 p1\n[PROOF STEP]\nunfolding \\_def rectpath_def Let_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. linepath (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) (Complex (Re (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)) (Im (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\))) +++ linepath (Complex (Re (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\)) (Im (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\))) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\) +++ linepath (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\) (Complex (Re (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\)) (Im (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\))) +++ linepath (Complex (Re (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\)) (Im (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \\))) (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \\) = linepath p1 p2 +++ linepath p2 p3 +++ linepath p3 p4 +++ linepath p4 p1\n[PROOF STEP]\nby (intro joinpaths_cong linepath_cong) \n (simp_all add: p1_def p2_def p3_def p4_def complex_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\n\\ m = linepath p1 p2 +++ linepath p2 p3 +++ linepath p3 p4 +++ linepath p4 p1\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave integrable: \"g contour_integrable_on \\ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. g contour_integrable_on \\ m\n[PROOF STEP]\nusing elim\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\n\ngoal (1 subgoal):\n 1. g contour_integrable_on \\ m\n[PROOF STEP]\nby (intro contour_integrable_holomorphic_simple[OF holo _ _ path_subset])\n (auto simp: \\_def A_def S_def intro!: finite_imp_closed)\n[PROOF STATE]\nproof (state)\nthis:\ng contour_integrable_on \\ m\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"norm (contour_integral (\\ m) g) = norm (I p1 p2 + I p2 p3 + I p3 p4 + I p4 p1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (contour_integral (\\ m) g) = cmod (I p1 p2 + I p2 p3 + I p3 p4 + I p4 p1)\n[PROOF STEP]\nunfolding I_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (contour_integral (\\ m) g) = cmod (contour_integral (linepath p1 p2) g + contour_integral (linepath p2 p3) g + contour_integral (linepath p3 p4) g + contour_integral (linepath p4 p1) g)\n[PROOF STEP]\nby (insert integrable, unfold eq)\n (subst contour_integral_join; (force simp: add_ac)?)+\n[PROOF STATE]\nproof (state)\nthis:\ncmod (contour_integral (\\ m) g) = cmod (I p1 p2 + I p2 p3 + I p3 p4 + I p4 p1)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncmod (contour_integral (\\ m) g) = cmod (I p1 p2 + I p2 p3 + I p3 p4 + I p4 p1)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"\\ \\ norm (I p1 p2) + norm (I p2 p3) + norm (I p3 p4) + norm (I p4 p1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (I p1 p2 + I p2 p3 + I p3 p4 + I p4 p1) \\ cmod (I p1 p2) + cmod (I p2 p3) + cmod (I p3 p4) + cmod (I p4 p1)\n[PROOF STEP]\nby (intro norm_triangle_mono order.refl)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (I p1 p2 + I p2 p3 + I p3 p4 + I p4 p1) \\ cmod (I p1 p2) + cmod (I p2 p3) + cmod (I p3 p4) + cmod (I p4 p1)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncmod (I p1 p2 + I p2 p3 + I p3 p4 + I p4 p1) \\ cmod (I p1 p2) + cmod (I p2 p3) + cmod (I p3 p4) + cmod (I p4 p1)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"norm (I p1 p2) \\ 1 / real m ^ n * norm (p2 - p1)\" (is \"_ \\ ?B1 * _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (I p1 p2) \\ 1 / real m ^ n * cmod (p2 - p1)\n[PROOF STEP]\nunfolding I_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (contour_integral (linepath p1 p2) g) \\ 1 / real m ^ n * cmod (p2 - p1)\n[PROOF STEP]\nproof (intro contour_integral_bound_linepath)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfix z\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nassume z: \"z \\ closed_segment p1 p2\"\n[PROOF STATE]\nproof (state)\nthis:\nz \\ closed_segment p1 p2\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\ndefine a where \"a = Re z\"\n[PROOF STATE]\nproof (state)\nthis:\na = Re z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfrom z\n[PROOF STATE]\nproof (chain)\npicking this:\nz \\ closed_segment p1 p2\n[PROOF STEP]\nhave z: \"z = a - (2*m+1) * pi * \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ closed_segment p1 p2\n\ngoal (1 subgoal):\n 1. z = complex_of_real a - complex_of_real (real (2 * m + 1) * pi) * \\\n[PROOF STEP]\nby (subst (asm) closed_segment_same_Im)\n (auto simp: p1_def p2_def complex_eq_iff a_def)\n[PROOF STATE]\nproof (state)\nthis:\nz = complex_of_real a - complex_of_real (real (2 * m + 1) * pi) * \\\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"real m * 1 \\ (2*m+1) * pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real m * 1 \\ real (2 * m + 1) * pi\n[PROOF STEP]\nusing pi_ge_two\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ pi\n\ngoal (1 subgoal):\n 1. real m * 1 \\ real (2 * m + 1) * pi\n[PROOF STEP]\nby (intro mult_mono) auto\n[PROOF STATE]\nproof (state)\nthis:\nreal m * 1 \\ real (2 * m + 1) * pi\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal m * 1 \\ real (2 * m + 1) * pi\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"(2*m+1) * pi = \\Im z\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (2 * m + 1) * pi = \\Im z\\\n[PROOF STEP]\nby (simp add: z)\n[PROOF STATE]\nproof (state)\nthis:\nreal (2 * m + 1) * pi = \\Im z\\\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (2 * m + 1) * pi = \\Im z\\\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"\\Im z\\ \\ norm z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Im z\\ \\ cmod z\n[PROOF STEP]\nby (rule abs_Im_le_cmod)\n[PROOF STATE]\nproof (state)\nthis:\n\\Im z\\ \\ cmod z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal m * 1 \\ cmod z\n[PROOF STEP]\nhave \"norm z \\ m\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m * 1 \\ cmod z\n\ngoal (1 subgoal):\n 1. real m \\ cmod z\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal m \\ cmod z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nreal m \\ cmod z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\nreal m \\ cmod z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"exp z - 1 = -of_real (exp a + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp z - 1 = - complex_of_real (exp a + 1)\n[PROOF STEP]\nusing exp_integer_2pi_plus1[of m]\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m \\ \\ \\ exp (complex_of_real ((2 * real m + 1) * pi) * \\) = - 1\n\ngoal (1 subgoal):\n 1. exp z - 1 = - complex_of_real (exp a + 1)\n[PROOF STEP]\nby (simp add: z exp_diff algebra_simps exp_of_real)\n[PROOF STATE]\nproof (state)\nthis:\nexp z - 1 = - complex_of_real (exp a + 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nexp z - 1 = - complex_of_real (exp a + 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"norm \\ \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 \\ cmod (- complex_of_real (exp a + 1))\n[PROOF STEP]\nunfolding norm_minus_cancel norm_of_real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 \\ \\exp a + 1\\\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ cmod (- complex_of_real (exp a + 1))\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n1 \\ cmod (exp z - 1)\n[PROOF STEP]\nhave \"norm (exp z - 1) \\ 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ cmod (exp z - 1)\n\ngoal (1 subgoal):\n 1. 1 \\ cmod (exp z - 1)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nreal m \\ cmod z\n1 \\ cmod (exp z - 1)\n[PROOF STEP]\nhave \"norm z ^ n * norm (exp z - 1) \\ real m ^ n * 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m \\ cmod z\n1 \\ cmod (exp z - 1)\n\ngoal (1 subgoal):\n 1. real m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n[PROOF STEP]\nby (intro mult_mono power_mono) auto\n[PROOF STATE]\nproof (state)\nthis:\nreal m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p1 p2 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nthus \"norm (g z) \\ 1 / real m ^ n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n\ngoal (1 subgoal):\n 1. cmod (g z) \\ 1 / real m ^ n\n[PROOF STEP]\nusing elim\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n1 < m\n\ngoal (1 subgoal):\n 1. cmod (g z) \\ 1 / real m ^ n\n[PROOF STEP]\nby (simp add: g_def divide_simps norm_divide norm_mult norm_power mult_less_0_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (g z) \\ 1 / real m ^ n\n\ngoal (2 subgoals):\n 1. g contour_integrable_on linepath p1 p2\n 2. 0 \\ 1 / real m ^ n\n[PROOF STEP]\nqed (insert integrable, auto simp: eq)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (I p1 p2) \\ 1 / real m ^ n * cmod (p2 - p1)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncmod (I p1 p2) \\ 1 / real m ^ n * cmod (p2 - p1)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"norm (p2 - p1) = 2 * m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (p2 - p1) = real (2 * m)\n[PROOF STEP]\nby (simp add: p2_def p1_def)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (p2 - p1) = real (2 * m)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncmod (p2 - p1) = real (2 * m)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"norm (I p3 p4) \\ 1 / real m ^ n * norm (p4 - p3)\" (is \"_ \\ ?B3 * _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (I p3 p4) \\ 1 / real m ^ n * cmod (p4 - p3)\n[PROOF STEP]\nunfolding I_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (contour_integral (linepath p3 p4) g) \\ 1 / real m ^ n * cmod (p4 - p3)\n[PROOF STEP]\nproof (intro contour_integral_bound_linepath)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfix z\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nassume z: \"z \\ closed_segment p3 p4\"\n[PROOF STATE]\nproof (state)\nthis:\nz \\ closed_segment p3 p4\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\ndefine a where \"a = Re z\"\n[PROOF STATE]\nproof (state)\nthis:\na = Re z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfrom z\n[PROOF STATE]\nproof (chain)\npicking this:\nz \\ closed_segment p3 p4\n[PROOF STEP]\nhave z: \"z = a + (2*m+1) * pi * \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ closed_segment p3 p4\n\ngoal (1 subgoal):\n 1. z = complex_of_real a + complex_of_real (real (2 * m + 1) * pi) * \\\n[PROOF STEP]\nby (subst (asm) closed_segment_same_Im)\n (auto simp: p3_def p4_def complex_eq_iff a_def)\n[PROOF STATE]\nproof (state)\nthis:\nz = complex_of_real a + complex_of_real (real (2 * m + 1) * pi) * \\\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"real m * 1 \\ (2*m+1) * pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real m * 1 \\ real (2 * m + 1) * pi\n[PROOF STEP]\nusing pi_ge_two\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ pi\n\ngoal (1 subgoal):\n 1. real m * 1 \\ real (2 * m + 1) * pi\n[PROOF STEP]\nby (intro mult_mono) auto\n[PROOF STATE]\nproof (state)\nthis:\nreal m * 1 \\ real (2 * m + 1) * pi\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal m * 1 \\ real (2 * m + 1) * pi\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"(2*m+1) * pi = \\Im z\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (2 * m + 1) * pi = \\Im z\\\n[PROOF STEP]\nby (simp add: z)\n[PROOF STATE]\nproof (state)\nthis:\nreal (2 * m + 1) * pi = \\Im z\\\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (2 * m + 1) * pi = \\Im z\\\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"\\Im z\\ \\ norm z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Im z\\ \\ cmod z\n[PROOF STEP]\nby (rule abs_Im_le_cmod)\n[PROOF STATE]\nproof (state)\nthis:\n\\Im z\\ \\ cmod z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal m * 1 \\ cmod z\n[PROOF STEP]\nhave \"norm z \\ m\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m * 1 \\ cmod z\n\ngoal (1 subgoal):\n 1. real m \\ cmod z\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal m \\ cmod z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nreal m \\ cmod z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\nreal m \\ cmod z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"exp z - 1 = -of_real (exp a + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp z - 1 = - complex_of_real (exp a + 1)\n[PROOF STEP]\nusing exp_integer_2pi_plus1[of m]\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m \\ \\ \\ exp (complex_of_real ((2 * real m + 1) * pi) * \\) = - 1\n\ngoal (1 subgoal):\n 1. exp z - 1 = - complex_of_real (exp a + 1)\n[PROOF STEP]\nby (simp add: z exp_add algebra_simps exp_of_real)\n[PROOF STATE]\nproof (state)\nthis:\nexp z - 1 = - complex_of_real (exp a + 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nexp z - 1 = - complex_of_real (exp a + 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"norm \\ \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 \\ cmod (- complex_of_real (exp a + 1))\n[PROOF STEP]\nunfolding norm_minus_cancel norm_of_real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 \\ \\exp a + 1\\\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ cmod (- complex_of_real (exp a + 1))\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n1 \\ cmod (exp z - 1)\n[PROOF STEP]\nhave \"norm (exp z - 1) \\ 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ cmod (exp z - 1)\n\ngoal (1 subgoal):\n 1. 1 \\ cmod (exp z - 1)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nreal m \\ cmod z\n1 \\ cmod (exp z - 1)\n[PROOF STEP]\nhave \"norm z ^ n * norm (exp z - 1) \\ real m ^ n * 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m \\ cmod z\n1 \\ cmod (exp z - 1)\n\ngoal (1 subgoal):\n 1. real m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n[PROOF STEP]\nby (intro mult_mono power_mono) auto\n[PROOF STATE]\nproof (state)\nthis:\nreal m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p3 p4 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nthus \"norm (g z) \\ 1 / real m ^ n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n\ngoal (1 subgoal):\n 1. cmod (g z) \\ 1 / real m ^ n\n[PROOF STEP]\nusing elim\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n1 < m\n\ngoal (1 subgoal):\n 1. cmod (g z) \\ 1 / real m ^ n\n[PROOF STEP]\nby (simp add: g_def divide_simps norm_divide norm_mult norm_power mult_less_0_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (g z) \\ 1 / real m ^ n\n\ngoal (2 subgoals):\n 1. g contour_integrable_on linepath p3 p4\n 2. 0 \\ 1 / real m ^ n\n[PROOF STEP]\nqed (insert integrable, auto simp: eq)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (I p3 p4) \\ 1 / real m ^ n * cmod (p4 - p3)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncmod (I p3 p4) \\ 1 / real m ^ n * cmod (p4 - p3)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"norm (p4 - p3) = 2 * m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (p4 - p3) = real (2 * m)\n[PROOF STEP]\nby (simp add: p4_def p3_def)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (p4 - p3) = real (2 * m)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncmod (p4 - p3) = real (2 * m)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"norm (I p2 p3) \\ (1 / real m ^ n) * norm (p3 - p2)\" (is \"_ \\ ?B2 * _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (I p2 p3) \\ 1 / real m ^ n * cmod (p3 - p2)\n[PROOF STEP]\nunfolding I_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (contour_integral (linepath p2 p3) g) \\ 1 / real m ^ n * cmod (p3 - p2)\n[PROOF STEP]\nproof (rule contour_integral_bound_linepath)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfix z\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nassume z: \"z \\ closed_segment p2 p3\"\n[PROOF STATE]\nproof (state)\nthis:\nz \\ closed_segment p2 p3\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\ndefine b where \"b = Im z\"\n[PROOF STATE]\nproof (state)\nthis:\nb = Im z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfrom z\n[PROOF STATE]\nproof (chain)\npicking this:\nz \\ closed_segment p2 p3\n[PROOF STEP]\nhave z: \"z = m + b * \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ closed_segment p2 p3\n\ngoal (1 subgoal):\n 1. z = of_nat m + complex_of_real b * \\\n[PROOF STEP]\nby (subst (asm) closed_segment_same_Re)\n (auto simp: p2_def p3_def algebra_simps complex_eq_iff b_def)\n[PROOF STATE]\nproof (state)\nthis:\nz = of_nat m + complex_of_real b * \\\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfrom elim\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < m\n[PROOF STEP]\nhave \"2 \\ 1 + real m\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\n\ngoal (1 subgoal):\n 1. 2 \\ 1 + real m\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ 1 + real m\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ 1 + real m\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"\\ \\ exp (real m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + real m \\ exp (real m)\n[PROOF STEP]\nby (rule exp_ge_add_one_self)\n[PROOF STATE]\nproof (state)\nthis:\n1 + real m \\ exp (real m)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 + real m \\ exp (real m)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"exp (real m) - 1 = norm (exp z) - norm (1::complex)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp (real m) - 1 = cmod (exp z) - cmod 1\n[PROOF STEP]\nby (simp add: z)\n[PROOF STATE]\nproof (state)\nthis:\nexp (real m) - 1 = cmod (exp z) - cmod 1\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nexp (real m) - 1 = cmod (exp z) - cmod 1\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"\\ \\ norm (exp z - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (exp z) - cmod 1 \\ cmod (exp z - 1)\n[PROOF STEP]\nby (rule norm_triangle_ineq2)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (exp z) - cmod 1 \\ cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x y. x \\ y \\ x - 1 \\ y - 1) \\ 2 - 1 \\ cmod (exp z - 1)\n[PROOF STEP]\nhave \"norm (exp z - 1) \\ 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x y. x \\ y \\ x - 1 \\ y - 1) \\ 2 - 1 \\ cmod (exp z - 1)\n\ngoal (1 subgoal):\n 1. 1 \\ cmod (exp z - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nhave \"norm z \\ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real m \\ cmod z\n[PROOF STEP]\nusing z and abs_Re_le_cmod[of z]\n[PROOF STATE]\nproof (prove)\nusing this:\nz = of_nat m + complex_of_real b * \\\n\\Re z\\ \\ cmod z\n\ngoal (1 subgoal):\n 1. real m \\ cmod z\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal m \\ cmod z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n1 \\ cmod (exp z - 1)\nreal m \\ cmod z\n[PROOF STEP]\nhave \"norm z ^ n * norm (exp z - 1) \\ real m ^ n * 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ cmod (exp z - 1)\nreal m \\ cmod z\n\ngoal (1 subgoal):\n 1. real m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n[PROOF STEP]\nusing elim\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ cmod (exp z - 1)\nreal m \\ cmod z\n1 < m\n\ngoal (1 subgoal):\n 1. real m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n[PROOF STEP]\nby (intro mult_mono power_mono) (auto simp: z)\n[PROOF STATE]\nproof (state)\nthis:\nreal m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n 3. \\x. x \\ closed_segment p2 p3 \\ cmod (g x) \\ 1 / real m ^ n\n[PROOF STEP]\nthus \"norm (g z) \\ 1 / real m ^ n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n\ngoal (1 subgoal):\n 1. cmod (g z) \\ 1 / real m ^ n\n[PROOF STEP]\nusing n and elim\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m ^ n * 1 \\ cmod z ^ n * cmod (exp z - 1)\n2 \\ n\neven n\n1 < m\n\ngoal (1 subgoal):\n 1. cmod (g z) \\ 1 / real m ^ n\n[PROOF STEP]\nby (simp add: g_def norm_mult norm_divide norm_power divide_simps mult_less_0_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (g z) \\ 1 / real m ^ n\n\ngoal (2 subgoals):\n 1. g contour_integrable_on linepath p2 p3\n 2. 0 \\ 1 / real m ^ n\n[PROOF STEP]\nqed (insert integrable, auto simp: eq)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (I p2 p3) \\ 1 / real m ^ n * cmod (p3 - p2)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncmod (I p2 p3) \\ 1 / real m ^ n * cmod (p3 - p2)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"p3 - p2 = of_real (2*(2*real m+1)*pi) * \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p3 - p2 = complex_of_real (2 * (2 * real m + 1) * pi) * \\\n[PROOF STEP]\nby (simp add: p2_def p3_def)\n[PROOF STATE]\nproof (state)\nthis:\np3 - p2 = complex_of_real (2 * (2 * real m + 1) * pi) * \\\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\np3 - p2 = complex_of_real (2 * (2 * real m + 1) * pi) * \\\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"norm \\ = 2 * (2 * real m + 1) * pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (complex_of_real (2 * (2 * real m + 1) * pi) * \\) = 2 * (2 * real m + 1) * pi\n[PROOF STEP]\nunfolding norm_mult norm_of_real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\2 * (2 * real m + 1) * pi\\ * cmod \\ = 2 * (2 * real m + 1) * pi\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncmod (complex_of_real (2 * (2 * real m + 1) * pi) * \\) = 2 * (2 * real m + 1) * pi\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncmod (complex_of_real (2 * (2 * real m + 1) * pi) * \\) = 2 * (2 * real m + 1) * pi\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"norm (I p4 p1) \\ (2 / real m ^ n) * norm (p1 - p4)\" (is \"_ \\ ?B4 * _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (I p4 p1) \\ 2 / real m ^ n * cmod (p1 - p4)\n[PROOF STEP]\nunfolding I_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (contour_integral (linepath p4 p1) g) \\ 2 / real m ^ n * cmod (p1 - p4)\n[PROOF STEP]\nproof (rule contour_integral_bound_linepath)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nfix z\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nassume z: \"z \\ closed_segment p4 p1\"\n[PROOF STATE]\nproof (state)\nthis:\nz \\ closed_segment p4 p1\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\ndefine b where \"b = Im z\"\n[PROOF STATE]\nproof (state)\nthis:\nb = Im z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nfrom z\n[PROOF STATE]\nproof (chain)\npicking this:\nz \\ closed_segment p4 p1\n[PROOF STEP]\nhave z: \"z = -real m + b * \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ closed_segment p4 p1\n\ngoal (1 subgoal):\n 1. z = complex_of_real (- real m) + complex_of_real b * \\\n[PROOF STEP]\nby (subst (asm) closed_segment_same_Re)\n (auto simp: p1_def p4_def algebra_simps b_def complex_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\nz = complex_of_real (- real m) + complex_of_real b * \\\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nfrom elim\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < m\n[PROOF STEP]\nhave \"2 \\ 1 + real m\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\n\ngoal (1 subgoal):\n 1. 2 \\ 1 + real m\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ 1 + real m\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ 1 + real m\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nhave \"\\ \\ exp (real m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + real m \\ exp (real m)\n[PROOF STEP]\nby (rule exp_ge_add_one_self)\n[PROOF STATE]\nproof (state)\nthis:\n1 + real m \\ exp (real m)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ exp (real m)\n[PROOF STEP]\nhave \"1 / 2 \\ 1 - exp (-real m)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ exp (real m)\n\ngoal (1 subgoal):\n 1. 1 / 2 \\ 1 - exp (- real m)\n[PROOF STEP]\nby (subst exp_minus) (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n1 / 2 \\ 1 - exp (- real m)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 / 2 \\ 1 - exp (- real m)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nhave \"1 - exp (-real m) = norm (1::complex) - norm (exp z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 - exp (- real m) = cmod 1 - cmod (exp z)\n[PROOF STEP]\nby (simp add: z)\n[PROOF STATE]\nproof (state)\nthis:\n1 - exp (- real m) = cmod 1 - cmod (exp z)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 - exp (- real m) = cmod 1 - cmod (exp z)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nhave \"\\ \\ norm (exp z - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod 1 - cmod (exp z) \\ cmod (exp z - 1)\n[PROOF STEP]\nby (subst norm_minus_commute, rule norm_triangle_ineq2)\n[PROOF STATE]\nproof (state)\nthis:\ncmod 1 - cmod (exp z) \\ cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n1 / 2 \\ cmod (exp z - 1)\n[PROOF STEP]\nhave \"norm (exp z - 1) \\ 1 / 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 / 2 \\ cmod (exp z - 1)\n\ngoal (1 subgoal):\n 1. 1 / 2 \\ cmod (exp z - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 / 2 \\ cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n1 / 2 \\ cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nhave \"norm z \\ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real m \\ cmod z\n[PROOF STEP]\nusing z and abs_Re_le_cmod[of z]\n[PROOF STATE]\nproof (prove)\nusing this:\nz = complex_of_real (- real m) + complex_of_real b * \\\n\\Re z\\ \\ cmod z\n\ngoal (1 subgoal):\n 1. real m \\ cmod z\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal m \\ cmod z\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n1 / 2 \\ cmod (exp z - 1)\nreal m \\ cmod z\n[PROOF STEP]\nhave \"norm z ^ n * norm (exp z - 1) \\ real m ^ n * (1 / 2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 / 2 \\ cmod (exp z - 1)\nreal m \\ cmod z\n\ngoal (1 subgoal):\n 1. real m ^ n * (1 / 2) \\ cmod z ^ n * cmod (exp z - 1)\n[PROOF STEP]\nusing elim\n[PROOF STATE]\nproof (prove)\nusing this:\n1 / 2 \\ cmod (exp z - 1)\nreal m \\ cmod z\n1 < m\n\ngoal (1 subgoal):\n 1. real m ^ n * (1 / 2) \\ cmod z ^ n * cmod (exp z - 1)\n[PROOF STEP]\nby (intro mult_mono power_mono) (auto simp: z)\n[PROOF STATE]\nproof (state)\nthis:\nreal m ^ n * (1 / 2) \\ cmod z ^ n * cmod (exp z - 1)\n\ngoal (3 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n 3. \\x. x \\ closed_segment p4 p1 \\ cmod (g x) \\ 2 / real m ^ n\n[PROOF STEP]\nthus \"norm (g z) \\ 2 / real m ^ n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m ^ n * (1 / 2) \\ cmod z ^ n * cmod (exp z - 1)\n\ngoal (1 subgoal):\n 1. cmod (g z) \\ 2 / real m ^ n\n[PROOF STEP]\nusing n and elim\n[PROOF STATE]\nproof (prove)\nusing this:\nreal m ^ n * (1 / 2) \\ cmod z ^ n * cmod (exp z - 1)\n2 \\ n\neven n\n1 < m\n\ngoal (1 subgoal):\n 1. cmod (g z) \\ 2 / real m ^ n\n[PROOF STEP]\nby (simp add: g_def norm_mult norm_divide norm_power divide_simps mult_less_0_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (g z) \\ 2 / real m ^ n\n\ngoal (2 subgoals):\n 1. g contour_integrable_on linepath p4 p1\n 2. 0 \\ 2 / real m ^ n\n[PROOF STEP]\nqed (insert integrable, auto simp: eq)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (I p4 p1) \\ 2 / real m ^ n * cmod (p1 - p4)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncmod (I p4 p1) \\ 2 / real m ^ n * cmod (p1 - p4)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"p1 - p4 = -of_real (2*(2*real m+1)*pi) * \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p1 - p4 = - complex_of_real (2 * (2 * real m + 1) * pi) * \\\n[PROOF STEP]\nby (simp add: p1_def p4_def algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\np1 - p4 = - complex_of_real (2 * (2 * real m + 1) * pi) * \\\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\np1 - p4 = - complex_of_real (2 * (2 * real m + 1) * pi) * \\\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"norm \\ = 2 * (2 * real m + 1) * pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (- complex_of_real (2 * (2 * real m + 1) * pi) * \\) = 2 * (2 * real m + 1) * pi\n[PROOF STEP]\nunfolding norm_mult norm_of_real norm_minus_cancel\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\2 * (2 * real m + 1) * pi\\ * cmod \\ = 2 * (2 * real m + 1) * pi\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncmod (- complex_of_real (2 * (2 * real m + 1) * pi) * \\) = 2 * (2 * real m + 1) * pi\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncmod (- complex_of_real (2 * (2 * real m + 1) * pi) * \\) = 2 * (2 * real m + 1) * pi\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"?B1 * (2*m) + ?B2 * (2*(2*real m+1)*pi) + ?B3 * (2*m) + ?B4 * (2*(2*real m+1)*pi) =\n (4 * m + 6 * (2 * m + 1) * pi) / real m ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 / real m ^ n * real (2 * m) + 1 / real m ^ n * (2 * (2 * real m + 1) * pi) + 1 / real m ^ n * real (2 * m) + 2 / real m ^ n * (2 * (2 * real m + 1) * pi) = (real (4 * m) + real (6 * (2 * m + 1)) * pi) / real m ^ n\n[PROOF STEP]\nby (simp add: divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\n1 / real m ^ n * real (2 * m) + 1 / real m ^ n * (2 * (2 * real m + 1) * pi) + 1 / real m ^ n * real (2 * m) + 2 / real m ^ n * (2 * (2 * real m + 1) * pi) = (real (4 * m) + real (6 * (2 * m + 1)) * pi) / real m ^ n\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 / real m ^ n * real (2 * m) + 1 / real m ^ n * (2 * (2 * real m + 1) * pi) + 1 / real m ^ n * real (2 * m) + 2 / real m ^ n * (2 * (2 * real m + 1) * pi) = (real (4 * m) + real (6 * (2 * m + 1)) * pi) / real m ^ n\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"(4 * m + 6 * (2 * m + 1) * pi) = (4 + 12 * pi) * m + 6 * pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (4 * m) + real (6 * (2 * m + 1)) * pi = (4 + 12 * pi) * real m + 6 * pi\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nreal (4 * m) + real (6 * (2 * m + 1)) * pi = (4 + 12 * pi) * real m + 6 * pi\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (4 * m) + real (6 * (2 * m + 1)) * pi = (4 + 12 * pi) * real m + 6 * pi\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nhave \"\\ / real m ^ n = ((4 + 12 * pi) + 6 * pi / m) / real m ^ (n - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((4 + 12 * pi) * real m + 6 * pi) / real m ^ n = (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nusing n\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\neven n\n\ngoal (1 subgoal):\n 1. ((4 + 12 * pi) * real m + 6 * pi) / real m ^ n = (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nby (cases n) (simp_all add: divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\n((4 + 12 * pi) * real m + 6 * pi) / real m ^ n = (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n\ngoal (1 subgoal):\n 1. \\m. 1 < m \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\x y. x \\ y \\ x + cmod (I p2 p3) + cmod (I p3 p4) + cmod (I p4 p1) \\ y + cmod (I p2 p3) + cmod (I p3 p4) + cmod (I p4 p1); \\x y. x \\ y \\ 1 / real m ^ n * real (2 * m) + cmod (I p2 p3) + x + cmod (I p4 p1) \\ 1 / real m ^ n * real (2 * m) + cmod (I p2 p3) + y + cmod (I p4 p1); \\x y. x \\ y \\ 1 / real m ^ n * real (2 * m) + x + 1 / real m ^ n * real (2 * m) + cmod (I p4 p1) \\ 1 / real m ^ n * real (2 * m) + y + 1 / real m ^ n * real (2 * m) + cmod (I p4 p1); \\x y. x \\ y \\ 1 / real m ^ n * real (2 * m) + 1 / real m ^ n * (2 * (2 * real m + 1) * pi) + 1 / real m ^ n * real (2 * m) + x \\ 1 / real m ^ n * real (2 * m) + 1 / real m ^ n * (2 * (2 * real m + 1) * pi) + 1 / real m ^ n * real (2 * m) + y\\ \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nshow \"cmod (contour_integral (\\ m) g) \\ \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\x y. x \\ y \\ x + cmod (I p2 p3) + cmod (I p3 p4) + cmod (I p4 p1) \\ y + cmod (I p2 p3) + cmod (I p3 p4) + cmod (I p4 p1); \\x y. x \\ y \\ 1 / real m ^ n * real (2 * m) + cmod (I p2 p3) + x + cmod (I p4 p1) \\ 1 / real m ^ n * real (2 * m) + cmod (I p2 p3) + y + cmod (I p4 p1); \\x y. x \\ y \\ 1 / real m ^ n * real (2 * m) + x + 1 / real m ^ n * real (2 * m) + cmod (I p4 p1) \\ 1 / real m ^ n * real (2 * m) + y + 1 / real m ^ n * real (2 * m) + cmod (I p4 p1); \\x y. x \\ y \\ 1 / real m ^ n * real (2 * m) + 1 / real m ^ n * (2 * (2 * real m + 1) * pi) + 1 / real m ^ n * real (2 * m) + x \\ 1 / real m ^ n * real (2 * m) + 1 / real m ^ n * (2 * (2 * real m + 1) * pi) + 1 / real m ^ n * real (2 * m) + y\\ \\ cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n\ngoal (1 subgoal):\n 1. cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n\n \\ \\It is clear that this bound goes to 0 since @{prop \"n \\ 2\"}.\\\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F m in sequentially. cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F m in sequentially. cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave \"(\\m. (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\m. (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)) \\ 0\n[PROOF STEP]\nby (rule real_tendsto_divide_at_top tendsto_add tendsto_const \n filterlim_real_sequentially filterlim_pow_at_top | use n in simp)+\n[PROOF STATE]\nproof (state)\nthis:\n(\\m. (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)) \\ 0\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F m in sequentially. cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n(\\m. (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)) \\ 0\n[PROOF STEP]\nhave *: \"(\\m. contour_integral (\\ m) g) \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F m in sequentially. cmod (contour_integral (\\ m) g) \\ (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)\n(\\m. (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)) \\ 0\n\ngoal (1 subgoal):\n 1. (\\m. contour_integral (\\ m) g) \\ 0\n[PROOF STEP]\nby (rule Lim_null_comparison)\n\n \\ \\Since the infinite sum over the residues can expressed using the\n zeta function, we have now related the Bernoulli numbers at even\n positive integers to the zeta function.\\\n[PROOF STATE]\nproof (state)\nthis:\n(\\m. contour_integral (\\ m) g) \\ 0\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave \"(\\m. contour_integral (\\ m) g * (2 * pi * \\) ^ n / (4 * pi * \\) -\n of_real (bernoulli n / fact n) * (2 * pi * \\) ^ n / 2) \\\n 0 * (2 * pi * \\) ^ n / (4 * pi * \\) - \n of_real (bernoulli n / fact n) * (2 * pi * \\) ^ n / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\m. contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2) \\ 0 * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nusing n\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\neven n\n\ngoal (1 subgoal):\n 1. (\\m. contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2) \\ 0 * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nby (intro tendsto_intros * zeta_limit) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\m. contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2) \\ 0 * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\m. contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2) \\ 0 * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave \"?this \\ (\\m. \\k\\{0<..m}. 1 / of_nat k ^ n) \\ \n - of_real (bernoulli n / fact n) * (2 * pi * \\) ^ n / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\m. contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2) \\ 0 * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2 = (\\m. \\k\\{0<..m}. 1 / of_nat k ^ n) \\ - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nby (intro filterlim_cong eventually_mono [OF eventually_gt_at_top[of \"0::nat\"]])\n (use eq in simp_all)\n[PROOF STATE]\nproof (state)\nthis:\n(\\m. contour_integral (\\ m) g * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2) \\ 0 * (complex_of_real (2 * pi) * \\) ^ n / (complex_of_real (4 * pi) * \\) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2 = (\\m. \\k\\{0<..m}. 1 / of_nat k ^ n) \\ - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\m. \\k\\{0<..m}. 1 / of_nat k ^ n) \\ - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\nhave \"(\\m. \\k\\{0<..m}. 1 / of_nat k ^ n)\n \\ - of_real (bernoulli n / fact n) * (of_real (2 * pi) * \\) ^ n / 2\" \n (is \"_ \\ ?L\")\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\m. \\k\\{0<..m}. 1 / of_nat k ^ n) \\ - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n\ngoal (1 subgoal):\n 1. (\\m. \\k\\{0<..m}. 1 / of_nat k ^ n) \\ - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\m. \\k\\{0<..m}. 1 / of_nat k ^ n) \\ - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\m. \\k\\{0<..m}. 1 / of_nat k ^ n) \\ - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave \"(\\m. \\k\\{0<..m}. 1 / of_nat k ^ n) = (\\m. \\k\\{..m. \\k\\{0<..m}. (1::'a) / of_nat k ^ n) = (\\m. \\kn. n - 1\"]) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\m. \\k\\{0<..m}. (1::?'a2) / of_nat k ^ n) = (\\m. \\kk. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\m. \\k\\{0<..m}. (1::?'a2) / of_nat k ^ n) = (\\m. \\kk. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave \"\\ \\ ?L \\ (\\k. 1 / of_nat (Suc k) ^ n) sums ?L\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\m. \\k - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2 = (\\k. 1 / of_nat (Suc k) ^ n) sums (- complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2)\n[PROOF STEP]\nby (simp add: sums_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\m. \\k - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2 = (\\k. 1 / of_nat (Suc k) ^ n) sums (- complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2)\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\m. \\k - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2 = (\\k. 1 / of_nat (Suc k) ^ n) sums (- complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \\) ^ n / 2)\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave \"(2 * pi * \\) ^ n = (2 * pi) ^ n * (-1) ^ n'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (complex_of_real (2 * pi) * \\) ^ n = complex_of_real ((2 * pi) ^ n * (- 1) ^ n')\n[PROOF STEP]\nby (simp add: n_def divide_simps power_mult_distrib power_mult power_minus')\n[PROOF STATE]\nproof (state)\nthis:\n(complex_of_real (2 * pi) * \\) ^ n = complex_of_real ((2 * pi) ^ n * (- 1) ^ n')\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(complex_of_real (2 * pi) * \\) ^ n = complex_of_real ((2 * pi) ^ n * (- 1) ^ n')\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nhave \"- of_real (bernoulli n / fact n) * \\ / 2 =\n of_real ((-1) ^ Suc n' * bernoulli (2*n') * (2*pi)^(2*n') / (2 * fact (2*n')))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - complex_of_real (bernoulli n / fact n) * complex_of_real ((2 * pi) ^ n * (- 1) ^ n') / 2 = complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nby (simp add: n_def divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\n- complex_of_real (bernoulli n / fact n) * complex_of_real ((2 * pi) ^ n * (- 1) ^ n') / 2 = complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\k. 1 / of_nat (Suc k) ^ n) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. 1 / of_nat (Suc k) ^ n) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\nunfolding n_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n\ngoal (1 subgoal):\n 1. (\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 56081, "file": "Bernoulli_Bernoulli_Zeta", "length": 434, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970811069351, "lm_q2_score": 0.8311430415844384, "lm_q1q2_score": 0.7320683650099133}} {"text": "[STATEMENT]\nlemma transpose_mat_mult_entries: \"i < dim_row A \\ j < dim_row A \\ \n (A * A\\<^sup>T) $$ (i, j) = (\\k\\ {0..<(dim_col A)}. (A $$ (i, k)) * (A $$ (j, k)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i < dim_row A; j < dim_row A\\ \\ (A * A\\<^sup>T) $$ (i, j) = (\\k = 0..x. f x + g x) (A \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f A + sum g B = sum f (A - B) + sum g (B - A) + (\\x\\A \\ B. f x + g x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum f A + sum g B = sum f (A - B) + sum g (B - A) + (\\x\\A \\ B. f x + g x)\n[PROOF STEP]\nhave 1: \"sum f A = sum f (A - B) + sum f (A \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f A = sum f (A - B) + sum f (A \\ B)\n[PROOF STEP]\nby (metis A Int_Diff_disjoint Un_Diff_Int finite_Diff finite_Int inf_sup_aci(1) local.sum.union_disjoint)\n[PROOF STATE]\nproof (state)\nthis:\nsum f A = sum f (A - B) + sum f (A \\ B)\n\ngoal (1 subgoal):\n 1. sum f A + sum g B = sum f (A - B) + sum g (B - A) + (\\x\\A \\ B. f x + g x)\n[PROOF STEP]\nhave 2: \"sum g B = sum g (B - A) + sum g (A \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum g B = sum g (B - A) + sum g (A \\ B)\n[PROOF STEP]\nby (metis B Int_Diff_disjoint Int_commute Un_Diff_Int finite_Diff finite_Int local.sum.union_disjoint)\n[PROOF STATE]\nproof (state)\nthis:\nsum g B = sum g (B - A) + sum g (A \\ B)\n\ngoal (1 subgoal):\n 1. sum f A + sum g B = sum f (A - B) + sum g (B - A) + (\\x\\A \\ B. f x + g x)\n[PROOF STEP]\nhave 3: \"sum f (A \\ B) + sum g (A \\ B) = sum (\\x. f x + g x) (A \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f (A \\ B) + sum g (A \\ B) = (\\x\\A \\ B. f x + g x)\n[PROOF STEP]\nby (simp add: sum.distrib)\n[PROOF STATE]\nproof (state)\nthis:\nsum f (A \\ B) + sum g (A \\ B) = (\\x\\A \\ B. f x + g x)\n\ngoal (1 subgoal):\n 1. sum f A + sum g B = sum f (A - B) + sum g (B - A) + (\\x\\A \\ B. f x + g x)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f A + sum g B = sum f (A - B) + sum g (B - A) + (\\x\\A \\ B. f x + g x)\n[PROOF STEP]\nby (simp add: \"1\" \"2\" \"3\" add.assoc add.left_commute)\n[PROOF STATE]\nproof (state)\nthis:\nsum f A + sum g B = sum f (A - B) + sum g (B - A) + (\\x\\A \\ B. f x + g x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1092, "file": "Echelon_Form_Rings2", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942290328345, "lm_q2_score": 0.8221891261650248, "lm_q1q2_score": 0.7319902341982707}} {"text": "[STATEMENT]\nlemma path_component_imp_homotopic_points:\n assumes \"path_component S a b\"\n shows \"homotopic_loops S (linepath a a) (linepath b b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. homotopic_loops S (linepath a a) (linepath b b)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. homotopic_loops S (linepath a a) (linepath b b)\n[PROOF STEP]\nobtain g :: \"real \\ 'a\" where g: \"continuous_on {0..1} g\" \"g ` {0..1} \\ S\" \"g 0 = a\" \"g 1 = b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\g. \\continuous_on {0..1} g; g ` {0..1} \\ S; g 0 = a; g 1 = b\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\npath_component S a b\n\ngoal (1 subgoal):\n 1. (\\g. \\continuous_on {0..1} g; g ` {0..1} \\ S; g 0 = a; g 1 = b\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: path_defs)\n[PROOF STATE]\nproof (state)\nthis:\ncontinuous_on {0..1} g\ng ` {0..1} \\ S\ng 0 = a\ng 1 = b\n\ngoal (1 subgoal):\n 1. homotopic_loops S (linepath a a) (linepath b b)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncontinuous_on {0..1} g\ng ` {0..1} \\ S\ng 0 = a\ng 1 = b\n[PROOF STEP]\nhave \"continuous_on ({0..1} \\ {0..1}) (g \\ fst)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous_on {0..1} g\ng ` {0..1} \\ S\ng 0 = a\ng 1 = b\n\ngoal (1 subgoal):\n 1. continuous_on ({0..1} \\ {0::'b..1::'b}) (g \\ fst)\n[PROOF STEP]\nby (fastforce intro!: continuous_intros)+\n[PROOF STATE]\nproof (state)\nthis:\ncontinuous_on ({0..1} \\ {0::?'b1..1::?'b1}) (g \\ fst)\n\ngoal (1 subgoal):\n 1. homotopic_loops S (linepath a a) (linepath b b)\n[PROOF STEP]\nwith g\n[PROOF STATE]\nproof (chain)\npicking this:\ncontinuous_on {0..1} g\ng ` {0..1} \\ S\ng 0 = a\ng 1 = b\ncontinuous_on ({0..1} \\ {0::?'b1..1::?'b1}) (g \\ fst)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous_on {0..1} g\ng ` {0..1} \\ S\ng 0 = a\ng 1 = b\ncontinuous_on ({0..1} \\ {0::?'b1..1::?'b1}) (g \\ fst)\n\ngoal (1 subgoal):\n 1. homotopic_loops S (linepath a a) (linepath b b)\n[PROOF STEP]\nby (auto simp add: homotopic_loops_def homotopic_with_def path_defs image_subset_iff)\n[PROOF STATE]\nproof (state)\nthis:\nhomotopic_loops S (linepath a a) (linepath b b)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1128, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869851639066, "lm_q2_score": 0.8354835350552604, "lm_q1q2_score": 0.7317056063201296}} {"text": "[STATEMENT]\nlemma unit_cpx_vec_length [simp]:\n assumes \"i < n\"\n shows \"\\unit_vec n i\\ = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\unit_vec n i\\ = 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\unit_vec n i\\ = 1\n[PROOF STEP]\nhave \"(\\j2) = (\\jj2) = (\\j?j < ?n; ?j \\ ?i\\ \\ cmod (unit_vec ?n ?i $ ?j) = 0\n\\?j < ?n; ?j = ?i\\ \\ cmod (unit_vec ?n ?i $ ?j) = 1\n\ngoal (1 subgoal):\n 1. (\\j2) = (\\jj2) = (\\junit_vec n i\\ = 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\j2) = (\\junit_vec n i\\ = 1\n[PROOF STEP]\nhave \"\\ = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\jjjunit_vec n i\\ = 1\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\j2) = 1\n[PROOF STEP]\nhave \"sqrt (\\j2) = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j2) = 1\n\ngoal (1 subgoal):\n 1. sqrt (\\j2) = 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (\\j2) = 1\n\ngoal (1 subgoal):\n 1. \\unit_vec n i\\ = 1\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (\\j2) = 1\n\ngoal (1 subgoal):\n 1. \\unit_vec n i\\ = 1\n[PROOF STEP]\nusing cpx_vec_length_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (\\j2) = 1\n\\?v\\ \\ sqrt (\\i2)\n\ngoal (1 subgoal):\n 1. \\unit_vec n i\\ = 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\unit_vec n i\\ = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1403, "file": "Isabelle_Marries_Dirac_Quantum", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869819218865, "lm_q2_score": 0.8354835371034369, "lm_q1q2_score": 0.7317056054052415}} {"text": "[STATEMENT]\nlemma r01_to_r01_r01_snd_def':\n \"r01_to_r01_r01_snd r = (\\n. real (r01_binary_expansion' r (2*n + 1)) * (1/2)^(n+1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r01_to_r01_r01_snd r = (\\n. real (r01_binary_expansion' r (2 * n + 1)) * (1 / 2) ^ (n + 1))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. r01_to_r01_r01_snd r = (\\n. real (r01_binary_expansion' r (2 * n + 1)) * (1 / 2) ^ (n + 1))\n[PROOF STEP]\nhave \"r01_to_r01_r01_snd_sum r = (\\n. \\i=0..n. real (r01_binary_expansion' r (2*i + 1)) * (1/2)^(i+1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r01_to_r01_r01_snd_sum r = (\\n. \\i = 0..n. real (r01_binary_expansion' r (2 * i + 1)) * (1 / 2) ^ (i + 1))\n[PROOF STEP]\nby(auto simp add: r01_to_r01_r01_snd_sum_def r01_binary_sum_def r01_to_r01_r01_snd'_def)\n[PROOF STATE]\nproof (state)\nthis:\nr01_to_r01_r01_snd_sum r = (\\n. \\i = 0..n. real (r01_binary_expansion' r (2 * i + 1)) * (1 / 2) ^ (i + 1))\n\ngoal (1 subgoal):\n 1. r01_to_r01_r01_snd r = (\\n. real (r01_binary_expansion' r (2 * n + 1)) * (1 / 2) ^ (n + 1))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nr01_to_r01_r01_snd_sum r = (\\n. \\i = 0..n. real (r01_binary_expansion' r (2 * i + 1)) * (1 / 2) ^ (i + 1))\n\ngoal (1 subgoal):\n 1. r01_to_r01_r01_snd r = (\\n. real (r01_binary_expansion' r (2 * n + 1)) * (1 / 2) ^ (n + 1))\n[PROOF STEP]\nusing lim_sum_ai real01_binary_expansion'_0or1\n[PROOF STATE]\nproof (prove)\nusing this:\nr01_to_r01_r01_snd_sum r = (\\n. \\i = 0..n. real (r01_binary_expansion' r (2 * i + 1)) * (1 / 2) ^ (i + 1))\n(\\n. ?a n \\ {0, 1}) \\ lim (\\n. \\i = 0..n. real (?a i) * (1 / 2) ^ Suc i) = (\\n. real (?a n) * (1 / 2) ^ Suc n)\nr01_binary_expansion' ?r ?n \\ {0, 1}\n\ngoal (1 subgoal):\n 1. r01_to_r01_r01_snd r = (\\n. real (r01_binary_expansion' r (2 * n + 1)) * (1 / 2) ^ (n + 1))\n[PROOF STEP]\nby(simp add: r01_to_r01_r01_snd_def)\n[PROOF STATE]\nproof (state)\nthis:\nr01_to_r01_r01_snd r = (\\n. real (r01_binary_expansion' r (2 * n + 1)) * (1 / 2) ^ (n + 1))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1222, "file": "Quasi_Borel_Spaces_StandardBorel", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869786798664, "lm_q2_score": 0.8354835330070838, "lm_q1q2_score": 0.7317055991090544}} {"text": "[STATEMENT]\nlemma compact_scaling:\n fixes s :: \"'a::real_normed_vector set\"\n assumes \"compact s\"\n shows \"compact ((\\x. c *\\<^sub>R x) ` s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. compact ((*\\<^sub>R) c ` s)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. compact ((*\\<^sub>R) c ` s)\n[PROOF STEP]\nlet ?f = \"\\x. scaleR c x\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. compact ((*\\<^sub>R) c ` s)\n[PROOF STEP]\nhave *: \"bounded_linear ?f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bounded_linear ((*\\<^sub>R) c)\n[PROOF STEP]\nby (rule bounded_linear_scaleR_right)\n[PROOF STATE]\nproof (state)\nthis:\nbounded_linear ((*\\<^sub>R) c)\n\ngoal (1 subgoal):\n 1. compact ((*\\<^sub>R) c ` s)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. compact ((*\\<^sub>R) c ` s)\n[PROOF STEP]\nusing compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\continuous_on s ((*\\<^sub>R) c); compact s\\ \\ compact ((*\\<^sub>R) c ` s)\n\\x\\s. isCont ((*\\<^sub>R) c) x \\ continuous_on s ((*\\<^sub>R) c)\n\ngoal (1 subgoal):\n 1. compact ((*\\<^sub>R) c ` s)\n[PROOF STEP]\nusing linear_continuous_at[OF *] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\continuous_on s ((*\\<^sub>R) c); compact s\\ \\ compact ((*\\<^sub>R) c ` s)\n\\x\\s. isCont ((*\\<^sub>R) c) x \\ continuous_on s ((*\\<^sub>R) c)\nisCont ((*\\<^sub>R) c) ?a\ncompact s\n\ngoal (1 subgoal):\n 1. compact ((*\\<^sub>R) c ` s)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncompact ((*\\<^sub>R) c ` s)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 756, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357701094303, "lm_q2_score": 0.8438950947024555, "lm_q1q2_score": 0.7316872333269141}} {"text": "[STATEMENT]\nlemma card_lists_length_le:\n assumes \"finite A\" shows \"card {xs. set xs \\ A \\ length xs \\ n} = (\\i\\n. card A^i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {xs. set xs \\ A \\ length xs \\ n} = sum ((^) (card A)) {..n}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {xs. set xs \\ A \\ length xs \\ n} = sum ((^) (card A)) {..n}\n[PROOF STEP]\nhave \"(\\i\\n. card A^i) = card (\\i\\n. {xs. set xs \\ A \\ length xs = i})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((^) (card A)) {..n} = card (\\i\\n. {xs. set xs \\ A \\ length xs = i})\n[PROOF STEP]\nusing \\finite A\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. sum ((^) (card A)) {..n} = card (\\i\\n. {xs. set xs \\ A \\ length xs = i})\n[PROOF STEP]\nby (subst card_UN_disjoint)\n (auto simp add: card_lists_length_eq finite_lists_length_eq)\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) (card A)) {..n} = card (\\i\\n. {xs. set xs \\ A \\ length xs = i})\n\ngoal (1 subgoal):\n 1. card {xs. set xs \\ A \\ length xs \\ n} = sum ((^) (card A)) {..n}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) (card A)) {..n} = card (\\i\\n. {xs. set xs \\ A \\ length xs = i})\n\ngoal (1 subgoal):\n 1. card {xs. set xs \\ A \\ length xs \\ n} = sum ((^) (card A)) {..n}\n[PROOF STEP]\nhave \"(\\i\\n. {xs. set xs \\ A \\ length xs = i}) = {xs. set xs \\ A \\ length xs \\ n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\n. {xs. set xs \\ A \\ length xs = i}) = {xs. set xs \\ A \\ length xs \\ n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. {xs. set xs \\ A \\ length xs = i}) = {xs. set xs \\ A \\ length xs \\ n}\n\ngoal (1 subgoal):\n 1. card {xs. set xs \\ A \\ length xs \\ n} = sum ((^) (card A)) {..n}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum ((^) (card A)) {..n} = card {xs. set xs \\ A \\ length xs \\ n}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsum ((^) (card A)) {..n} = card {xs. set xs \\ A \\ length xs \\ n}\n\ngoal (1 subgoal):\n 1. card {xs. set xs \\ A \\ length xs \\ n} = sum ((^) (card A)) {..n}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard {xs. set xs \\ A \\ length xs \\ n} = sum ((^) (card A)) {..n}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1142, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357598021708, "lm_q2_score": 0.8438951045175643, "lm_q1q2_score": 0.7316872331387188}} {"text": "[STATEMENT]\nlemma ld_bounded: \"Max {i. 2 ^ i \\ Suc n} \\ Suc n\" (is \"?m \\ Suc n\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Max {i. 2 ^ i \\ Suc n} \\ Suc n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Max {i. 2 ^ i \\ Suc n} \\ Suc n\n[PROOF STEP]\nhave \"?m \\ 2 ^ ?m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Max {i. 2 ^ i \\ Suc n} \\ 2 ^ Max {i. 2 ^ i \\ Suc n}\n[PROOF STEP]\nby (rule le_pow2) simp\n[PROOF STATE]\nproof (state)\nthis:\nMax {i. 2 ^ i \\ Suc n} \\ 2 ^ Max {i. 2 ^ i \\ Suc n}\n\ngoal (1 subgoal):\n 1. Max {i. 2 ^ i \\ Suc n} \\ Suc n\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nMax {i. 2 ^ i \\ Suc n} \\ 2 ^ Max {i. 2 ^ i \\ Suc n}\n\ngoal (1 subgoal):\n 1. Max {i. 2 ^ i \\ Suc n} \\ Suc n\n[PROOF STEP]\nhave \"?m \\ {i. 2 ^ i \\ Suc n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Max {i. 2 ^ i \\ Suc n} \\ {i. 2 ^ i \\ Suc n}\n[PROOF STEP]\nby (rule Max_in) (auto intro: exI[of _ 0])\n[PROOF STATE]\nproof (state)\nthis:\nMax {i. 2 ^ i \\ Suc n} \\ {i. 2 ^ i \\ Suc n}\n\ngoal (1 subgoal):\n 1. Max {i. 2 ^ i \\ Suc n} \\ Suc n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nMax {i. 2 ^ i \\ Suc n} \\ {i. 2 ^ i \\ Suc n}\n[PROOF STEP]\nhave \"2 ^ ?m \\ Suc n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nMax {i. 2 ^ i \\ Suc n} \\ {i. 2 ^ i \\ Suc n}\n\ngoal (1 subgoal):\n 1. 2 ^ Max {i. 2 ^ i \\ Suc n} \\ Suc n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ Max {i. 2 ^ i \\ Suc n} \\ Suc n\n\ngoal (1 subgoal):\n 1. Max {i. 2 ^ i \\ Suc n} \\ Suc n\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nMax {i. 2 ^ i \\ Suc n} \\ 2 ^ Max {i. 2 ^ i \\ Suc n}\n2 ^ Max {i. 2 ^ i \\ Suc n} \\ Suc n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nMax {i. 2 ^ i \\ Suc n} \\ 2 ^ Max {i. 2 ^ i \\ Suc n}\n2 ^ Max {i. 2 ^ i \\ Suc n} \\ Suc n\n\ngoal (1 subgoal):\n 1. Max {i. 2 ^ i \\ Suc n} \\ Suc n\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\nMax {i. 2 ^ i \\ Suc n} \\ Suc n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1155, "file": "Formula_Derivatives_WS1S_Formula", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357666736773, "lm_q2_score": 0.8438950966654774, "lm_q1q2_score": 0.7316872321295091}} {"text": "[STATEMENT]\nlemma all_le_eq_imp_eq: \"\\ c::nat. (\\a. (a < d) = (a < c)) \\ (d = c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\c. (\\a. (a < d) = (a < c)) \\ d = c\n[PROOF STEP]\nproof (induct d)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\c. (\\a. (a < 0) = (a < c)) \\ 0 = c\n 2. \\d c. (\\c. (\\a. (a < d) = (a < c)) \\ d = c) \\ (\\a. (a < Suc d) = (a < c)) \\ Suc d = c\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\c. (\\a. (a < 0) = (a < c)) \\ 0 = c\n 2. \\d c. (\\c. (\\a. (a < d) = (a < c)) \\ d = c) \\ (\\a. (a < Suc d) = (a < c)) \\ Suc d = c\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a. (a < 0) = (a < c)) \\ 0 = c\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (a < 0) = (a < c)) \\ 0 = c\n\ngoal (1 subgoal):\n 1. \\d c. (\\c. (\\a. (a < d) = (a < c)) \\ d = c) \\ (\\a. (a < Suc d) = (a < c)) \\ Suc d = c\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\d c. (\\c. (\\a. (a < d) = (a < c)) \\ d = c) \\ (\\a. (a < Suc d) = (a < c)) \\ Suc d = c\n[PROOF STEP]\ncase (Suc n c)\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (a < n) = (a < ?c)) \\ n = ?c\n\ngoal (1 subgoal):\n 1. \\d c. (\\c. (\\a. (a < d) = (a < c)) \\ d = c) \\ (\\a. (a < Suc d) = (a < c)) \\ Suc d = c\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\a. (a < n) = (a < ?c)) \\ n = ?c\n\ngoal (1 subgoal):\n 1. (\\a. (a < Suc n) = (a < c)) \\ Suc n = c\n[PROOF STEP]\nby (cases c) (auto simp add: le_Suc_eq)\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. (a < Suc n) = (a < c)) \\ Suc n = c\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 971, "file": "JiveDataStoreModel_Isabelle_Store_Store", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357563664174, "lm_q2_score": 0.8438951005915208, "lm_q1q2_score": 0.7316872268352832}} {"text": "[STATEMENT]\ntheorem derangements_formula:\n assumes \"n \\ 0\" \"finite S\" \"card S = n\"\n shows \"int (card (derangements S)) = round (fact n / exp 1 :: real)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. int (card (derangements S)) = round (fact n / exp 1)\n[PROOF STEP]\nusing count_derangements_approximation[of n] assms\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\ 0 \\ \\real (count_derangements n) - fact n / exp 1\\ < 1 / 2\nn \\ 0\nfinite S\ncard S = n\n\ngoal (1 subgoal):\n 1. int (card (derangements S)) = round (fact n / exp 1)\n[PROOF STEP]\nby (intro round_unique' [symmetric]) (auto simp: card_derangements abs_minus_commute)", "meta": {"llama_tokens": 267, "file": "Derangements_Derangements", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9136765163620469, "lm_q2_score": 0.8006920044739461, "lm_q1q2_score": 0.7315734813266995}} {"text": "[STATEMENT]\nlemma \"Carmichael 122200 = 1380\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Carmichael 122200 = 1380\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Carmichael 122200 = 1380\n[PROOF STEP]\nhave \"prime_factorization (2^3 * 5^2 * 13 * 47) = {#2, 2, 2, 5, 5, 13, 47::nat#}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prime_factorization (2 ^ 3 * 5\\<^sup>2 * 13 * 47) = {#2, 2, 2, 5, 5, 13, 47#}\n[PROOF STEP]\nby (intro prime_factorization_eqI) auto\n[PROOF STATE]\nproof (state)\nthis:\nprime_factorization (2 ^ 3 * 5\\<^sup>2 * 13 * 47) = {#2, 2, 2, 5, 5, 13, 47#}\n\ngoal (1 subgoal):\n 1. Carmichael 122200 = 1380\n[PROOF STEP]\nfrom eval_Carmichael[OF this]\n[PROOF STATE]\nproof (chain)\npicking this:\nCarmichael (2 ^ 3 * 5\\<^sup>2 * 13 * 47) = (LCM p\\set_mset {#2, 2, 2, 5, 5, 13, 47#}. let k = count {#2, 2, 2, 5, 5, 13, 47#} p in if p = 2 \\ 2 < k then 2 ^ (k - 2) else p ^ (k - 1) * (p - 1))\n[PROOF STEP]\nshow \"Carmichael 122200 = 1380\"\n[PROOF STATE]\nproof (prove)\nusing this:\nCarmichael (2 ^ 3 * 5\\<^sup>2 * 13 * 47) = (LCM p\\set_mset {#2, 2, 2, 5, 5, 13, 47#}. let k = count {#2, 2, 2, 5, 5, 13, 47#} p in if p = 2 \\ 2 < k then 2 ^ (k - 2) else p ^ (k - 1) * (p - 1))\n\ngoal (1 subgoal):\n 1. Carmichael 122200 = 1380\n[PROOF STEP]\nby (simp add: lcm_nat_def gcd_non_0_nat)\n[PROOF STATE]\nproof (state)\nthis:\nCarmichael 122200 = 1380\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 829, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9136765281148512, "lm_q2_score": 0.8006919925839875, "lm_q1q2_score": 0.7315734798734999}} {"text": "[STATEMENT]\nlemma icard_Un_Int: \"icard A + icard B = icard (A \\ B) + icard (A \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. icard A + icard B = icard (A \\ B) + icard (A \\ B)\n[PROOF STEP]\napply (case_tac \"finite A\", case_tac \"finite B\")\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\finite A; finite B\\ \\ icard A + icard B = icard (A \\ B) + icard (A \\ B)\n 2. \\finite A; infinite B\\ \\ icard A + icard B = icard (A \\ B) + icard (A \\ B)\n 3. infinite A \\ icard A + icard B = icard (A \\ B) + icard (A \\ B)\n[PROOF STEP]\napply (simp add: icard_finite card_Un_Int[of A])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\finite A; infinite B\\ \\ icard A + icard B = icard (A \\ B) + icard (A \\ B)\n 2. infinite A \\ icard A + icard B = icard (A \\ B) + icard (A \\ B)\n[PROOF STEP]\napply simp_all\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 450, "file": "List-Infinite_CommonSet_InfiniteSet2", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9136765140114859, "lm_q2_score": 0.8006919949619793, "lm_q1q2_score": 0.7315734707537634}} {"text": "[STATEMENT]\nlemma T'_mom_select_le': \"\\C\\<^sub>1 C\\<^sub>2. \\n. T'_mom_select n \\ C\\<^sub>1 * n + C\\<^sub>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\C\\<^sub>1 C\\<^sub>2. \\n. T'_mom_select n \\ C\\<^sub>1 * n + C\\<^sub>2\n[PROOF STEP]\nproof (rule akra_bazzi_light_nat)\n[PROOF STATE]\nproof (state)\ngoal (5 subgoals):\n 1. 0 < ?a\n 2. 0 < ?c\n 3. ?a + ?c < 1\n 4. 0 \\ ?C\\<^sub>1\n 5. \\n>?n\\<^sub>0. T'_mom_select n = T'_mom_select (nat \\?a * real n + ?b\\) + T'_mom_select (nat \\?c * real n + ?d\\) + ?C\\<^sub>1 * n + ?C\\<^sub>2\n[PROOF STEP]\nshow \"\\n>20. T'_mom_select n = T'_mom_select (nat \\0.2 * n + 0\\) +\n T'_mom_select (nat \\0.7 * n + 3\\) + 17 * n + 50\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n>20. T'_mom_select n = T'_mom_select (nat \\2 / 10 * real n + 0\\) + T'_mom_select (nat \\7 / 10 * real n + 3\\) + 17 * n + 50\n[PROOF STEP]\nusing T'_mom_select.simps\n[PROOF STATE]\nproof (prove)\nusing this:\nT'_mom_select ?n = (if ?n \\ 20 then 463 else T'_mom_select (nat \\2 / 10 * real ?n\\) + T'_mom_select (nat \\7 / 10 * real ?n + 3\\) + 17 * ?n + 50)\n\ngoal (1 subgoal):\n 1. \\n>20. T'_mom_select n = T'_mom_select (nat \\2 / 10 * real n + 0\\) + T'_mom_select (nat \\7 / 10 * real n + 3\\) + 17 * n + 50\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\n>20. T'_mom_select n = T'_mom_select (nat \\2 / 10 * real n + 0\\) + T'_mom_select (nat \\7 / 10 * real n + 3\\) + 17 * n + 50\n\ngoal (4 subgoals):\n 1. 0 < 2 / 10\n 2. 0 < 7 / 10\n 3. 2 / 10 + 7 / 10 < 1\n 4. 0 \\ 17\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 901, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.894789468908171, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7315570373120753}} {"text": "[STATEMENT]\nlemma convex_contains_segment:\n \"convex S \\ (\\a\\S. \\b\\S. closed_segment a b \\ S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex S = (\\a\\S. \\b\\S. closed_segment a b \\ S)\n[PROOF STEP]\nunfolding convex_alt closed_segment_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\S. \\y\\S. \\u. 0 \\ u \\ u \\ 1 \\ (1 - u) *\\<^sub>R x + u *\\<^sub>R y \\ S) = (\\a\\S. \\b\\S. {(1 - u) *\\<^sub>R a + u *\\<^sub>R b |u. 0 \\ u \\ u \\ 1} \\ S)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 289, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894661025424, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.731557035018265}} {"text": "[STATEMENT]\ntheorem squarefree_asymptotics'':\n \"(\\x. card {n. real n \\ x \\ squarefree n}) \\[at_top] (\\x. 6 / pi\\<^sup>2 * x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. real (card {n. real n \\ x \\ squarefree n})) \\[at_top] (*) (6 / pi\\<^sup>2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x. real (card {n. real n \\ x \\ squarefree n})) \\[at_top] (*) (6 / pi\\<^sup>2)\n[PROOF STEP]\nhave \"(\\x. card {n. real n \\ x \\ squarefree n} - 6 / pi\\<^sup>2 * x) \\ O(\\x. sqrt x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. real (card {n. real n \\ x \\ squarefree n}) - 6 / pi\\<^sup>2 * x) \\ O(sqrt)\n[PROOF STEP]\nby (rule squarefree_asymptotics)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. real (card {n. real n \\ x \\ squarefree n}) - 6 / pi\\<^sup>2 * x) \\ O(sqrt)\n\ngoal (1 subgoal):\n 1. (\\x. real (card {n. real n \\ x \\ squarefree n})) \\[at_top] (*) (6 / pi\\<^sup>2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. real (card {n. real n \\ x \\ squarefree n}) - 6 / pi\\<^sup>2 * x) \\ O(sqrt)\n\ngoal (1 subgoal):\n 1. (\\x. real (card {n. real n \\ x \\ squarefree n})) \\[at_top] (*) (6 / pi\\<^sup>2)\n[PROOF STEP]\nhave \"(sqrt :: real \\ real) \\ \\(\\x. x powr (1/2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt \\ \\(\\x. x powr (1 / 2))\n[PROOF STEP]\nby (intro bigthetaI_cong eventually_mono[OF eventually_ge_at_top[of \"0::real\"]]) \n (auto simp: powr_half_sqrt)\n[PROOF STATE]\nproof (state)\nthis:\nsqrt \\ \\(\\x. x powr (1 / 2))\n\ngoal (1 subgoal):\n 1. (\\x. real (card {n. real n \\ x \\ squarefree n})) \\[at_top] (*) (6 / pi\\<^sup>2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsqrt \\ \\(\\x. x powr (1 / 2))\n\ngoal (1 subgoal):\n 1. (\\x. real (card {n. real n \\ x \\ squarefree n})) \\[at_top] (*) (6 / pi\\<^sup>2)\n[PROOF STEP]\nhave \"(\\x::real. x powr (1/2)) \\ o(\\x. 6 / pi ^ 2 * x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. x powr (1 / 2)) \\ o((*) (6 / pi\\<^sup>2))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. x powr (1 / 2)) \\ o((*) (6 / pi\\<^sup>2))\n\ngoal (1 subgoal):\n 1. (\\x. real (card {n. real n \\ x \\ squarefree n})) \\[at_top] (*) (6 / pi\\<^sup>2)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x. real (card {n. real n \\ x \\ squarefree n}) - 6 / pi\\<^sup>2 * x) \\ o((*) (6 / pi\\<^sup>2))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. real (card {n. real n \\ x \\ squarefree n}) - 6 / pi\\<^sup>2 * x) \\ o((*) (6 / pi\\<^sup>2))\n\ngoal (1 subgoal):\n 1. (\\x. real (card {n. real n \\ x \\ squarefree n})) \\[at_top] (*) (6 / pi\\<^sup>2)\n[PROOF STEP]\nby (simp add: asymp_equiv_altdef)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. real (card {n. real n \\ x \\ squarefree n})) \\[at_top] (*) (6 / pi\\<^sup>2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1456, "file": "Dirichlet_Series_Arithmetic_Summatory_Asymptotics", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.894789468908171, "lm_q2_score": 0.8175744761936437, "lm_q1q2_score": 0.7315570313461865}} {"text": "[STATEMENT]\nlemma dim_affine_hull:\n fixes S :: \"'n::euclidean_space set\"\n shows \"dim (affine hull S) = dim S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim (affine hull S) = dim S\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dim (affine hull S) = dim S\n[PROOF STEP]\nhave \"dim (affine hull S) \\ dim S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim S \\ dim (affine hull S)\n[PROOF STEP]\nusing dim_subset\n[PROOF STATE]\nproof (prove)\nusing this:\n?S \\ ?T \\ dim ?S \\ dim ?T\n\ngoal (1 subgoal):\n 1. dim S \\ dim (affine hull S)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndim S \\ dim (affine hull S)\n\ngoal (1 subgoal):\n 1. dim (affine hull S) = dim S\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndim S \\ dim (affine hull S)\n\ngoal (1 subgoal):\n 1. dim (affine hull S) = dim S\n[PROOF STEP]\nhave \"dim (span S) \\ dim (affine hull S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim (affine hull S) \\ dim (span S)\n[PROOF STEP]\nusing dim_subset affine_hull_subset_span\n[PROOF STATE]\nproof (prove)\nusing this:\n?S \\ ?T \\ dim ?S \\ dim ?T\naffine hull ?s \\ span ?s\n\ngoal (1 subgoal):\n 1. dim (affine hull S) \\ dim (span S)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ndim (affine hull S) \\ dim (span S)\n\ngoal (1 subgoal):\n 1. dim (affine hull S) = dim S\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndim (affine hull S) \\ dim (span S)\n\ngoal (1 subgoal):\n 1. dim (affine hull S) = dim S\n[PROOF STEP]\nhave \"dim (span S) = dim S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim (span S) = dim S\n[PROOF STEP]\nusing dim_span\n[PROOF STATE]\nproof (prove)\nusing this:\ndim (span ?S) = dim ?S\n\ngoal (1 subgoal):\n 1. dim (span S) = dim S\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndim (span S) = dim S\n\ngoal (1 subgoal):\n 1. dim (affine hull S) = dim S\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ndim S \\ dim (affine hull S)\ndim (affine hull S) \\ dim (span S)\ndim (span S) = dim S\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndim S \\ dim (affine hull S)\ndim (affine hull S) \\ dim (span S)\ndim (span S) = dim S\n\ngoal (1 subgoal):\n 1. dim (affine hull S) = dim S\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndim (affine hull S) = dim S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1077, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267694452331, "lm_q2_score": 0.8418256393148982, "lm_q1q2_score": 0.731400650642131}} {"text": "[STATEMENT]\nlemma \"(a::int) ^ 3 - b ^ 3 = (a - b) * (a ^ 2 + a * b + b ^ 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a ^ 3 - b ^ 3 = (a - b) * (a\\<^sup>2 + a * b + b\\<^sup>2)\n[PROOF STEP]\nby ring", "meta": {"llama_tokens": 109, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9343951625409307, "lm_q2_score": 0.7826624738835051, "lm_q1q2_score": 0.7313160294990646}} {"text": "[STATEMENT]\nlemma tan_45: \"tan (pi / 4) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tan (pi / 4) = 1\n[PROOF STEP]\nunfolding tan_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin (pi / 4) / cos (pi / 4) = 1\n[PROOF STEP]\nby (simp add: sin_45 cos_45)", "meta": {"llama_tokens": 137, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9161096090086367, "lm_q2_score": 0.7981867729389246, "lm_q1q2_score": 0.7312265724729436}} {"text": "[STATEMENT]\nlemma multTensor2 [simp]:\n assumes a1:\"A = Matrix.mat 2 1 (\\(i,j). if i = 0 then a0 else a1)\" and \n a2:\"B = Matrix.mat 2 1 (\\(i,j). if i = 0 then b0 else b1)\"\n shows \"mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = [[a0*b0, a0*b1, a1*b0, a1*b1]]\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = [[a0 * b0, a0 * b1, a1 * b0, a1 * b1]]\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = [[a0 * b0, a0 * b1, a1 * b0, a1 * b1]]\n[PROOF STEP]\nhave \"mat_to_cols_list A = [[a0, a1]]\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat_to_cols_list A = [[a0, a1]]\n[PROOF STEP]\nby (auto simp: a1 mat_to_cols_list_def) (simp add: numeral_2_eq_2)\n[PROOF STATE]\nproof (state)\nthis:\nmat_to_cols_list A = [[a0, a1]]\n\ngoal (1 subgoal):\n 1. mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = [[a0 * b0, a0 * b1, a1 * b0, a1 * b1]]\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nmat_to_cols_list A = [[a0, a1]]\n\ngoal (1 subgoal):\n 1. mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = [[a0 * b0, a0 * b1, a1 * b0, a1 * b1]]\n[PROOF STEP]\nhave f2:\"mat_to_cols_list B = [[b0, b1]]\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat_to_cols_list B = [[b0, b1]]\n[PROOF STEP]\nby (auto simp: a2 mat_to_cols_list_def) (simp add: numeral_2_eq_2)\n[PROOF STATE]\nproof (state)\nthis:\nmat_to_cols_list B = [[b0, b1]]\n\ngoal (1 subgoal):\n 1. mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = [[a0 * b0, a0 * b1, a1 * b0, a1 * b1]]\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nmat_to_cols_list A = [[a0, a1]]\nmat_to_cols_list B = [[b0, b1]]\n[PROOF STEP]\nhave \"mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = \n mult.Tensor (*) [[a0,a1]] [[b0,b1]]\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmat_to_cols_list A = [[a0, a1]]\nmat_to_cols_list B = [[b0, b1]]\n\ngoal (1 subgoal):\n 1. mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = mult.Tensor (*) [[a0, a1]] [[b0, b1]]\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = mult.Tensor (*) [[a0, a1]] [[b0, b1]]\n\ngoal (1 subgoal):\n 1. mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = [[a0 * b0, a0 * b1, a1 * b0, a1 * b1]]\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nmult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = mult.Tensor (*) [[a0, a1]] [[b0, b1]]\n\ngoal (1 subgoal):\n 1. mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = [[a0 * b0, a0 * b1, a1 * b0, a1 * b1]]\n[PROOF STEP]\nusing mult.Tensor_def[of \"(1::complex)\" \"(*)\"] mult.times_def[of \"(1::complex)\" \"(*)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nmult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = mult.Tensor (*) [[a0, a1]] [[b0, b1]]\nMatrix_Tensor.mult 1 (*) \\ mult.Tensor (*) \\ rec_list (\\xs. []) (\\x xs xsa ys. mult.vec_mat_Tensor (*) x ys @ xsa ys)\nMatrix_Tensor.mult 1 (*) \\ mult.times (*) \\ \\uu uua. rec_list (\\n. []) (\\y ys ysa n. n * y # ysa n) uua uu\n\ngoal (1 subgoal):\n 1. mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = [[a0 * b0, a0 * b1, a1 * b0, a1 * b1]]\n[PROOF STEP]\nby (metis (mono_tags, lifting) append_self_conv list.simps(6) mult.Tensor.simps(2) mult.vec_mat_Tensor.simps(1) \nmult.vec_mat_Tensor.simps(2) plus_mult_cpx plus_mult_def tensor_prod_2)\n[PROOF STATE]\nproof (state)\nthis:\nmult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = [[a0 * b0, a0 * b1, a1 * b0, a1 * b1]]\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1805, "file": "Isabelle_Marries_Dirac_More_Tensor", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767938900121, "lm_q2_score": 0.8333245911726382, "lm_q1q2_score": 0.7312229905318717}} {"text": "[STATEMENT]\nlemma natUnion_subset: \"(\\ n. \\ m. f n \\ g m) \\ natUnion f \\ natUnion g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. \\m. f n \\ g m) \\ natUnion f \\ natUnion g\n[PROOF STEP]\nby (meson natUnion_elem natUnion_upperbound subset_iff)", "meta": {"llama_tokens": 124, "file": "LocalLexing_Limit", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110569397307, "lm_q2_score": 0.8198933403143929, "lm_q1q2_score": 0.731189946403625}} {"text": "[STATEMENT]\nlemma power_mult_cong:\n assumes \"[x^n = a](mod m)\" \"[y^n = b](mod m)\"\n shows \"[(x*y)^n = a*b](mod m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [(x * y) ^ n = a * b] (mod m)\n[PROOF STEP]\nusing assms cong_mult[of \"x^n\" a m \"y^n\" b] power_mult_distrib\n[PROOF STATE]\nproof (prove)\nusing this:\n[x ^ n = a] (mod m)\n[y ^ n = b] (mod m)\n\\[x ^ n = a] (mod m); [y ^ n = b] (mod m)\\ \\ [x ^ n * y ^ n = a * b] (mod m)\n(?a * ?b) ^ ?n = ?a ^ ?n * ?b ^ ?n\n\ngoal (1 subgoal):\n 1. [(x * y) ^ n = a * b] (mod m)\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 286, "file": "Probabilistic_Prime_Tests_Algebraic_Auxiliaries", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110511888303, "lm_q2_score": 0.8198933271118221, "lm_q1q2_score": 0.7311899299143015}} {"text": "[STATEMENT]\nlemma erfc_aux_unroll: \n \"x > 0 \\ \n erfc_aux x = (\\i erfc_aux x = (\\i erfc x = exp (- x\\<^sup>2) / sqrt pi * (\\i erfc_aux x = (\\im\\<^sub>m M2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. unitary (M1 *\\<^sub>m\\<^sub>m M2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nunitary M1\nunitary M2\n\ngoal (1 subgoal):\n 1. unitary (M1 *\\<^sub>m\\<^sub>m M2)\n[PROOF STEP]\nunfolding unitary_def\n[PROOF STATE]\nproof (prove)\nusing this:\nmat_adj M1 *\\<^sub>m\\<^sub>m M1 = eye\nmat_adj M2 *\\<^sub>m\\<^sub>m M2 = eye\n\ngoal (1 subgoal):\n 1. mat_adj (M1 *\\<^sub>m\\<^sub>m M2) *\\<^sub>m\\<^sub>m (M1 *\\<^sub>m\\<^sub>m M2) = eye\n[PROOF STEP]\nby (metis mat_adj_mult_mm mat_eye_l mult_mm_assoc)", "meta": {"llama_tokens": 316, "file": "Complex_Geometry_Unitary_Matrices", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9073122163480667, "lm_q2_score": 0.8056321983146849, "lm_q1q2_score": 0.730959935414262}} {"text": "[STATEMENT]\nlemma (in group) int_pow_eq_id:\n assumes \"x \\ carrier G\"\n shows \"(pow G x i = one G \\ int (ord x) dvd i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x [^] i = \\) = (int (ord x) dvd i)\n[PROOF STEP]\nproof (cases i rule: int_cases2)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\n. i = int n \\ (x [^] i = \\) = (int (ord x) dvd i)\n 2. \\n. i = - int n \\ (x [^] i = \\) = (int (ord x) dvd i)\n[PROOF STEP]\ncase (nonneg n)\n[PROOF STATE]\nproof (state)\nthis:\ni = int n\n\ngoal (2 subgoals):\n 1. \\n. i = int n \\ (x [^] i = \\) = (int (ord x) dvd i)\n 2. \\n. i = - int n \\ (x [^] i = \\) = (int (ord x) dvd i)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni = int n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ni = int n\n\ngoal (1 subgoal):\n 1. (x [^] i = \\) = (int (ord x) dvd i)\n[PROOF STEP]\nby (simp add: int_pow_int pow_eq_id assms)\n[PROOF STATE]\nproof (state)\nthis:\n(x [^] i = \\) = (int (ord x) dvd i)\n\ngoal (1 subgoal):\n 1. \\n. i = - int n \\ (x [^] i = \\) = (int (ord x) dvd i)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. i = - int n \\ (x [^] i = \\) = (int (ord x) dvd i)\n[PROOF STEP]\ncase (nonpos n)\n[PROOF STATE]\nproof (state)\nthis:\ni = - int n\n\ngoal (1 subgoal):\n 1. \\n. i = - int n \\ (x [^] i = \\) = (int (ord x) dvd i)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni = - int n\n[PROOF STEP]\nhave \"x [^] i = inv (x [^] n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni = - int n\n\ngoal (1 subgoal):\n 1. x [^] i = inv (x [^] n)\n[PROOF STEP]\nby (simp add: assms int_pow_int int_pow_neg)\n[PROOF STATE]\nproof (state)\nthis:\nx [^] i = inv (x [^] n)\n\ngoal (1 subgoal):\n 1. \\n. i = - int n \\ (x [^] i = \\) = (int (ord x) dvd i)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx [^] i = inv (x [^] n)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx [^] i = inv (x [^] n)\n\ngoal (1 subgoal):\n 1. (x [^] i = \\) = (int (ord x) dvd i)\n[PROOF STEP]\nby (simp add: assms pow_eq_id nonpos)\n[PROOF STATE]\nproof (state)\nthis:\n(x [^] i = \\) = (int (ord x) dvd i)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1104, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392756357326, "lm_q2_score": 0.8267117876664789, "lm_q1q2_score": 0.7308456899281955}} {"text": "[STATEMENT]\nlemma q64_upto_def: \"q64 = (\\k. k\\<^sup>2 mod 64) ` {..<64}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. q64 = (\\k. k\\<^sup>2 mod 64) ` {..<64}\n[PROOF STEP]\nby (simp add: q64_def lessThan_nat_numeral lessThan_Suc insert_commute)", "meta": {"llama_tokens": 130, "file": "Pell_Efficient_Discrete_Sqrt", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127380808499, "lm_q2_score": 0.8558511524823263, "lm_q1q2_score": 0.7308222010058343}} {"text": "[STATEMENT]\nlemma ai_exists0_less_than1:\n assumes \"\\n. a n \\ {0,1}\"\n and \"\\i. a i = 0\"\n shows \"(\\n::nat. real (a n) * (1/2)^(Suc n)) < 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. real (a n) * (1 / 2) ^ Suc n) < 1\n[PROOF STEP]\nusing ai_exists0_less_than_sum[of a 0] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\n. a n \\ {0, 1}; 0 \\ ?i; a ?i = 0\\ \\ (\\n. real (a (n + 0)) * (1 / 2) ^ Suc (n + 0)) < (1 / 2) ^ 0\na ?n \\ {0, 1}\n\\i. a i = 0\n\ngoal (1 subgoal):\n 1. (\\n. real (a n) * (1 / 2) ^ Suc n) < 1\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 329, "file": "Quasi_Borel_Spaces_StandardBorel", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099070060380482, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7308134665147525}} {"text": "[STATEMENT]\nlemma bij_betw_image_mset_set:\n assumes \"bij_betw f A B\"\n shows \"image_mset f (mset_set A) = mset_set B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. image_mset f (mset_set A) = mset_set B\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw f A B\n\ngoal (1 subgoal):\n 1. image_mset f (mset_set A) = mset_set B\n[PROOF STEP]\nby (simp add: bij_betw_def image_mset_mset_set)", "meta": {"llama_tokens": 193, "file": "Hermite_Lindemann_Misc_HLW", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099070011518829, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7308134540165875}} {"text": "[STATEMENT]\nlemma crk_is_preserved:\nfixes A::\"'a::{field}^'cols::{finite, wellorder}^'rows\"\n and P::\"'a::{field}^'rows^'rows\"\nassumes inv_P: \"invertible P\"\nshows \"col_rank A = col_rank (P**A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. col_rank A = col_rank (P ** A)\n[PROOF STEP]\nusing rank_nullity_theorem_matrices\n[PROOF STATE]\nproof (prove)\nusing this:\nncols ?A = vec.dim (null_space ?A) + vec.dim (col_space ?A)\n\ngoal (1 subgoal):\n 1. col_rank A = col_rank (P ** A)\n[PROOF STEP]\nunfolding ncols_def\n[PROOF STATE]\nproof (prove)\nusing this:\nCARD(?'cols) = vec.dim (null_space ?A) + vec.dim (col_space ?A)\n\ngoal (1 subgoal):\n 1. col_rank A = col_rank (P ** A)\n[PROOF STEP]\nby (metis col_rank_def inv_P add_left_cancel null_space_is_preserved)", "meta": {"llama_tokens": 333, "file": "Gauss_Jordan_Rank", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.918480237330998, "lm_q2_score": 0.795658104908603, "lm_q1q2_score": 0.7307962450307858}} {"text": "[STATEMENT]\nlemma take_map: \"take n (map f xs) = map f (take n xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. take n (map f xs) = map f (take n xs)\n[PROOF STEP]\nproof (induct n arbitrary: xs)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\xs. take 0 (map f xs) = map f (take 0 xs)\n 2. \\n xs. (\\xs. take n (map f xs) = map f (take n xs)) \\ take (Suc n) (map f xs) = map f (take (Suc n) xs)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\xs. take 0 (map f xs) = map f (take 0 xs)\n 2. \\n xs. (\\xs. take n (map f xs) = map f (take n xs)) \\ take (Suc n) (map f xs) = map f (take (Suc n) xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. take 0 (map f xs) = map f (take 0 xs)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ntake 0 (map f xs) = map f (take 0 xs)\n\ngoal (1 subgoal):\n 1. \\n xs. (\\xs. take n (map f xs) = map f (take n xs)) \\ take (Suc n) (map f xs) = map f (take (Suc n) xs)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n xs. (\\xs. take n (map f xs) = map f (take n xs)) \\ take (Suc n) (map f xs) = map f (take (Suc n) xs)\n[PROOF STEP]\ncase Suc\n[PROOF STATE]\nproof (state)\nthis:\ntake n_ (map f ?xs) = map f (take n_ ?xs)\n\ngoal (1 subgoal):\n 1. \\n xs. (\\xs. take n (map f xs) = map f (take n xs)) \\ take (Suc n) (map f xs) = map f (take (Suc n) xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ntake n_ (map f ?xs) = map f (take n_ ?xs)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ntake n_ (map f ?xs) = map f (take n_ ?xs)\n\ngoal (1 subgoal):\n 1. take (Suc n_) (map f xs) = map f (take (Suc n_) xs)\n[PROOF STEP]\nby (cases xs) simp_all\n[PROOF STATE]\nproof (state)\nthis:\ntake (Suc n_) (map f xs) = map f (take (Suc n_) xs)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 896, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527982093666, "lm_q2_score": 0.8577681068080748, "lm_q1q2_score": 0.7307779388098902}} {"text": "[STATEMENT]\nlemma bij_betw_totatives:\n assumes \"m1 > 1\" \"m2 > 1\" \"coprime m1 m2\"\n shows \"bij_betw (\\x. (x mod m1, x mod m2)) (totatives (m1 * m2)) \n (totatives m1 \\ totatives m2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw (\\x. (x mod m1, x mod m2)) (totatives (m1 * m2)) (totatives m1 \\ totatives m2)\n[PROOF STEP]\nunfolding bij_betw_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on (\\x. (x mod m1, x mod m2)) (totatives (m1 * m2)) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2) = totatives m1 \\ totatives m2\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. inj_on (\\x. (x mod m1, x mod m2)) (totatives (m1 * m2))\n 2. (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2) = totatives m1 \\ totatives m2\n[PROOF STEP]\nshow \"inj_on (\\x. (x mod m1, x mod m2)) (totatives (m1 * m2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on (\\x. (x mod m1, x mod m2)) (totatives (m1 * m2))\n[PROOF STEP]\nproof (intro inj_onI, clarify)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x y. \\x \\ totatives (m1 * m2); y \\ totatives (m1 * m2); x mod m1 = y mod m1; x mod m2 = y mod m2\\ \\ x = y\n[PROOF STEP]\nfix x y\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x y. \\x \\ totatives (m1 * m2); y \\ totatives (m1 * m2); x mod m1 = y mod m1; x mod m2 = y mod m2\\ \\ x = y\n[PROOF STEP]\nassume xy: \"x \\ totatives (m1 * m2)\" \"y \\ totatives (m1 * m2)\"\n \"x mod m1 = y mod m1\" \"x mod m2 = y mod m2\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ totatives (m1 * m2)\ny \\ totatives (m1 * m2)\nx mod m1 = y mod m1\nx mod m2 = y mod m2\n\ngoal (1 subgoal):\n 1. \\x y. \\x \\ totatives (m1 * m2); y \\ totatives (m1 * m2); x mod m1 = y mod m1; x mod m2 = y mod m2\\ \\ x = y\n[PROOF STEP]\nhave ex: \"\\!z. z < m1 * m2 \\ [z = x] (mod m1) \\ [z = x] (mod m2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\!z. z < m1 * m2 \\ [z = x] (mod m1) \\ [z = x] (mod m2)\n[PROOF STEP]\nby (rule binary_chinese_remainder_unique_nat) (insert assms, simp_all)\n[PROOF STATE]\nproof (state)\nthis:\n\\!z. z < m1 * m2 \\ [z = x] (mod m1) \\ [z = x] (mod m2)\n\ngoal (1 subgoal):\n 1. \\x y. \\x \\ totatives (m1 * m2); y \\ totatives (m1 * m2); x mod m1 = y mod m1; x mod m2 = y mod m2\\ \\ x = y\n[PROOF STEP]\nhave \"x < m1 * m2 \\ [x = x] (mod m1) \\ [x = x] (mod m2)\"\n \"y < m1 * m2 \\ [y = x] (mod m1) \\ [y = x] (mod m2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x < m1 * m2 \\ [x = x] (mod m1) \\ [x = x] (mod m2) &&& y < m1 * m2 \\ [y = x] (mod m1) \\ [y = x] (mod m2)\n[PROOF STEP]\nusing xy assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ totatives (m1 * m2)\ny \\ totatives (m1 * m2)\nx mod m1 = y mod m1\nx mod m2 = y mod m2\n1 < m1\n1 < m2\ncoprime m1 m2\n\ngoal (1 subgoal):\n 1. x < m1 * m2 \\ [x = x] (mod m1) \\ [x = x] (mod m2) &&& y < m1 * m2 \\ [y = x] (mod m1) \\ [y = x] (mod m2)\n[PROOF STEP]\nby (simp_all add: totatives_less one_less_mult cong_def)\n[PROOF STATE]\nproof (state)\nthis:\nx < m1 * m2 \\ [x = x] (mod m1) \\ [x = x] (mod m2)\ny < m1 * m2 \\ [y = x] (mod m1) \\ [y = x] (mod m2)\n\ngoal (1 subgoal):\n 1. \\x y. \\x \\ totatives (m1 * m2); y \\ totatives (m1 * m2); x mod m1 = y mod m1; x mod m2 = y mod m2\\ \\ x = y\n[PROOF STEP]\nfrom this[THEN the1_equality[OF ex]]\n[PROOF STATE]\nproof (chain)\npicking this:\n(THE xa. xa < m1 * m2 \\ [xa = x] (mod m1) \\ [xa = x] (mod m2)) = x\n(THE xa. xa < m1 * m2 \\ [xa = x] (mod m1) \\ [xa = x] (mod m2)) = y\n[PROOF STEP]\nshow \"x = y\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(THE xa. xa < m1 * m2 \\ [xa = x] (mod m1) \\ [xa = x] (mod m2)) = x\n(THE xa. xa < m1 * m2 \\ [xa = x] (mod m1) \\ [xa = x] (mod m2)) = y\n\ngoal (1 subgoal):\n 1. x = y\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx = y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\x. (x mod m1, x mod m2)) (totatives (m1 * m2))\n\ngoal (1 subgoal):\n 1. (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2) = totatives m1 \\ totatives m2\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2) = totatives m1 \\ totatives m2\n[PROOF STEP]\nshow \"(\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2) = totatives m1 \\ totatives m2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2) = totatives m1 \\ totatives m2\n[PROOF STEP]\nproof safe\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\a b x. x \\ totatives (m1 * m2) \\ x mod m1 \\ totatives m1\n 2. \\a b x. x \\ totatives (m1 * m2) \\ x mod m2 \\ totatives m2\n 3. \\a b. \\a \\ totatives m1; b \\ totatives m2\\ \\ (a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\a b x. x \\ totatives (m1 * m2) \\ x mod m1 \\ totatives m1\n 2. \\a b x. x \\ totatives (m1 * m2) \\ x mod m2 \\ totatives m2\n 3. \\a b. \\a \\ totatives m1; b \\ totatives m2\\ \\ (a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n[PROOF STEP]\nassume \"x \\ totatives (m1 * m2)\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ totatives (m1 * m2)\n\ngoal (3 subgoals):\n 1. \\a b x. x \\ totatives (m1 * m2) \\ x mod m1 \\ totatives m1\n 2. \\a b x. x \\ totatives (m1 * m2) \\ x mod m2 \\ totatives m2\n 3. \\a b. \\a \\ totatives m1; b \\ totatives m2\\ \\ (a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < m1\n1 < m2\ncoprime m1 m2\nx \\ totatives (m1 * m2)\n[PROOF STEP]\nshow \"x mod m1 \\ totatives m1\" \"x mod m2 \\ totatives m2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m1\n1 < m2\ncoprime m1 m2\nx \\ totatives (m1 * m2)\n\ngoal (1 subgoal):\n 1. x mod m1 \\ totatives m1 &&& x mod m2 \\ totatives m2\n[PROOF STEP]\nusing coprime_common_divisor [of x m1 m1] coprime_common_divisor [of x m2 m2]\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m1\n1 < m2\ncoprime m1 m2\nx \\ totatives (m1 * m2)\n\\coprime x m1; m1 dvd x; m1 dvd m1\\ \\ is_unit m1\n\\coprime x m2; m2 dvd x; m2 dvd m2\\ \\ is_unit m2\n\ngoal (1 subgoal):\n 1. x mod m1 \\ totatives m1 &&& x mod m2 \\ totatives m2\n[PROOF STEP]\nby (auto simp add: in_totatives_iff mod_greater_zero_iff_not_dvd)\n[PROOF STATE]\nproof (state)\nthis:\nx mod m1 \\ totatives m1\nx mod m2 \\ totatives m2\n\ngoal (1 subgoal):\n 1. \\a b. \\a \\ totatives m1; b \\ totatives m2\\ \\ (a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a b. \\a \\ totatives m1; b \\ totatives m2\\ \\ (a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n[PROOF STEP]\nfix a b\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a b. \\a \\ totatives m1; b \\ totatives m2\\ \\ (a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n[PROOF STEP]\nassume ab: \"a \\ totatives m1\" \"b \\ totatives m2\"\n[PROOF STATE]\nproof (state)\nthis:\na \\ totatives m1\nb \\ totatives m2\n\ngoal (1 subgoal):\n 1. \\a b. \\a \\ totatives m1; b \\ totatives m2\\ \\ (a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < m1\n1 < m2\ncoprime m1 m2\na \\ totatives m1\nb \\ totatives m2\n[PROOF STEP]\nhave ab': \"a < m1\" \"b < m2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m1\n1 < m2\ncoprime m1 m2\na \\ totatives m1\nb \\ totatives m2\n\ngoal (1 subgoal):\n 1. a < m1 &&& b < m2\n[PROOF STEP]\nby (auto simp: totatives_less)\n[PROOF STATE]\nproof (state)\nthis:\na < m1\nb < m2\n\ngoal (1 subgoal):\n 1. \\a b. \\a \\ totatives m1; b \\ totatives m2\\ \\ (a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n[PROOF STEP]\nwith binary_chinese_remainder_unique_nat[OF assms(3), of a b]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\m1 \\ 0; m2 \\ 0\\ \\ \\!x. x < m1 * m2 \\ [x = a] (mod m1) \\ [x = b] (mod m2)\na < m1\nb < m2\n[PROOF STEP]\nobtain x\n where x: \"x < m1 * m2\" \"x mod m1 = a\" \"x mod m2 = b\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\m1 \\ 0; m2 \\ 0\\ \\ \\!x. x < m1 * m2 \\ [x = a] (mod m1) \\ [x = b] (mod m2)\na < m1\nb < m2\n\ngoal (1 subgoal):\n 1. (\\x. \\x < m1 * m2; x mod m1 = a; x mod m2 = b\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: cong_def)\n[PROOF STATE]\nproof (state)\nthis:\nx < m1 * m2\nx mod m1 = a\nx mod m2 = b\n\ngoal (1 subgoal):\n 1. \\a b. \\a \\ totatives m1; b \\ totatives m2\\ \\ (a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n[PROOF STEP]\nfrom x ab assms(3)\n[PROOF STATE]\nproof (chain)\npicking this:\nx < m1 * m2\nx mod m1 = a\nx mod m2 = b\na \\ totatives m1\nb \\ totatives m2\ncoprime m1 m2\n[PROOF STEP]\nhave \"x \\ totatives (m1 * m2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx < m1 * m2\nx mod m1 = a\nx mod m2 = b\na \\ totatives m1\nb \\ totatives m2\ncoprime m1 m2\n\ngoal (1 subgoal):\n 1. x \\ totatives (m1 * m2)\n[PROOF STEP]\nby (auto intro: ccontr simp add: in_totatives_iff)\n[PROOF STATE]\nproof (state)\nthis:\nx \\ totatives (m1 * m2)\n\ngoal (1 subgoal):\n 1. \\a b. \\a \\ totatives m1; b \\ totatives m2\\ \\ (a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n[PROOF STEP]\nwith x\n[PROOF STATE]\nproof (chain)\npicking this:\nx < m1 * m2\nx mod m1 = a\nx mod m2 = b\nx \\ totatives (m1 * m2)\n[PROOF STEP]\nshow \"(a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1*m2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx < m1 * m2\nx mod m1 = a\nx mod m2 = b\nx \\ totatives (m1 * m2)\n\ngoal (1 subgoal):\n 1. (a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(a, b) \\ (\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. (x mod m1, x mod m2)) ` totatives (m1 * m2) = totatives m1 \\ totatives m2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5227, "file": null, "length": 41, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587993853654, "lm_q2_score": 0.8221891305219504, "lm_q1q2_score": 0.7307278245103861}} {"text": "[STATEMENT]\nlemma pi_approx: \"3.141592653588 \\ pi\" \"pi \\ 3.1415926535899\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 3141592653588 / 10 ^ 12 \\ pi &&& pi \\ 31415926535899 / 10 ^ 13\n[PROOF STEP]\nunfolding pi_machin\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 3141592653588 / 10 ^ 12 \\ 16 * arctan (1 / 5) - 4 * arctan (1 / 239) &&& 16 * arctan (1 / 5) - 4 * arctan (1 / 239) \\ 31415926535899 / 10 ^ 13\n[PROOF STEP]\nusing arctan_bounds[of \"1/5\" 4]\n arctan_bounds[of \"1/239\" 4]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 \\ 1 / 5; 1 / 5 < 1\\ \\ (\\k<2 * 4. (- 1) ^ k * (1 / real (k * 2 + 1) * (1 / 5) ^ (k * 2 + 1))) \\ arctan (1 / 5)\n\\0 \\ 1 / 5; 1 / 5 < 1\\ \\ arctan (1 / 5) \\ (\\k<2 * 4 + 1. (- 1) ^ k * (1 / real (k * 2 + 1) * (1 / 5) ^ (k * 2 + 1)))\n\\0 \\ 1 / 239; 1 / 239 < 1\\ \\ (\\k<2 * 4. (- 1) ^ k * (1 / real (k * 2 + 1) * (1 / 239) ^ (k * 2 + 1))) \\ arctan (1 / 239)\n\\0 \\ 1 / 239; 1 / 239 < 1\\ \\ arctan (1 / 239) \\ (\\k<2 * 4 + 1. (- 1) ^ k * (1 / real (k * 2 + 1) * (1 / 239) ^ (k * 2 + 1)))\n\ngoal (1 subgoal):\n 1. 3141592653588 / 10 ^ 12 \\ 16 * arctan (1 / 5) - 4 * arctan (1 / 239) &&& 16 * arctan (1 / 5) - 4 * arctan (1 / 239) \\ 31415926535899 / 10 ^ 13\n[PROOF STEP]\nby (simp_all add: eval_nat_numeral)", "meta": {"llama_tokens": 872, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587993853654, "lm_q2_score": 0.8221891239865619, "lm_q1q2_score": 0.7307278187020021}} {"text": "[STATEMENT]\nlemma summable_Cauchy_product:\n fixes a b :: \"nat \\ 'a::{real_normed_algebra,banach}\"\n assumes \"summable (\\k. norm (a k))\"\n and \"summable (\\k. norm (b k))\"\n shows \"summable (\\k. \\i\\k. a i * b (k - i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. summable (\\k. \\i\\k. a i * b (k - i))\n[PROOF STEP]\nusing Cauchy_product_sums[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. \\i\\k. a i * b (k - i)) sums ((\\k. a k) * (\\k. b k))\n\ngoal (1 subgoal):\n 1. summable (\\k. \\i\\k. a i * b (k - i))\n[PROOF STEP]\nby (simp add: sums_iff)", "meta": {"llama_tokens": 296, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467675095294, "lm_q2_score": 0.8311430499496096, "lm_q1q2_score": 0.7306967257012106}} {"text": "[STATEMENT]\nlemma convex_hull_scaling:\n \"convex hull ((\\x. c *\\<^sub>R x) ` S) = (\\x. c *\\<^sub>R x) ` (convex hull S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex hull (*\\<^sub>R) c ` S = (*\\<^sub>R) c ` (convex hull S)\n[PROOF STEP]\nusing linear_scaleR\n[PROOF STATE]\nproof (prove)\nusing this:\nlinear ((*\\<^sub>R) ?c)\n\ngoal (1 subgoal):\n 1. convex hull (*\\<^sub>R) c ` S = (*\\<^sub>R) c ` (convex hull S)\n[PROOF STEP]\nby (rule convex_hull_linear_image [symmetric])", "meta": {"llama_tokens": 220, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467548438124, "lm_q2_score": 0.8311430520409023, "lm_q1q2_score": 0.7306967170127412}} {"text": "[STATEMENT]\nlemma snd_safe_distance_iff_nonneg_D2: \n assumes \"a\\<^sub>e < a\\<^sub>o\"\n shows \"s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance \\ 0 \\ D2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = (0 \\ D2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = (0 \\ D2)\n[PROOF STEP]\nfrom snd_safe_distance_def assms pos_le_divide_eq[of \"2 * (a\\<^sub>o - a\\<^sub>e)\"]\n[PROOF STATE]\nproof (chain)\npicking this:\nsnd_safe_distance = (v\\<^sub>o - v\\<^sub>e)\\<^sup>2 / (2 * (a\\<^sub>o - a\\<^sub>e))\na\\<^sub>e < a\\<^sub>o\n0 < 2 * (a\\<^sub>o - a\\<^sub>e) \\ (?a \\ ?b / (2 * (a\\<^sub>o - a\\<^sub>e))) = (?a * (2 * (a\\<^sub>o - a\\<^sub>e)) \\ ?b)\n[PROOF STEP]\nhave \"s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance \\ (s\\<^sub>o - s\\<^sub>e) * (2 * (a\\<^sub>o - a\\<^sub>e)) \\ (v\\<^sub>o - v\\<^sub>e)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsnd_safe_distance = (v\\<^sub>o - v\\<^sub>e)\\<^sup>2 / (2 * (a\\<^sub>o - a\\<^sub>e))\na\\<^sub>e < a\\<^sub>o\n0 < 2 * (a\\<^sub>o - a\\<^sub>e) \\ (?a \\ ?b / (2 * (a\\<^sub>o - a\\<^sub>e))) = (?a * (2 * (a\\<^sub>o - a\\<^sub>e)) \\ ?b)\n\ngoal (1 subgoal):\n 1. (s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = ((s\\<^sub>o - s\\<^sub>e) * (2 * (a\\<^sub>o - a\\<^sub>e)) \\ (v\\<^sub>o - v\\<^sub>e)\\<^sup>2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = ((s\\<^sub>o - s\\<^sub>e) * (2 * (a\\<^sub>o - a\\<^sub>e)) \\ (v\\<^sub>o - v\\<^sub>e)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = (0 \\ D2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = ((s\\<^sub>o - s\\<^sub>e) * (2 * (a\\<^sub>o - a\\<^sub>e)) \\ (v\\<^sub>o - v\\<^sub>e)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = (0 \\ D2)\n[PROOF STEP]\nhave \"... \\ 0 \\ D2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((s\\<^sub>o - s\\<^sub>e) * (2 * (a\\<^sub>o - a\\<^sub>e)) \\ (v\\<^sub>o - v\\<^sub>e)\\<^sup>2) = (0 \\ D2)\n[PROOF STEP]\nby (auto simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n((s\\<^sub>o - s\\<^sub>e) * (2 * (a\\<^sub>o - a\\<^sub>e)) \\ (v\\<^sub>o - v\\<^sub>e)\\<^sup>2) = (0 \\ D2)\n\ngoal (1 subgoal):\n 1. (s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = (0 \\ D2)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = (0 \\ D2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = (0 \\ D2)\n\ngoal (1 subgoal):\n 1. (s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = (0 \\ D2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(s\\<^sub>o - s\\<^sub>e \\ snd_safe_distance) = (0 \\ D2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1487, "file": "Safe_Distance_Safe_Distance", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467548438124, "lm_q2_score": 0.8311430415844384, "lm_q1q2_score": 0.7306967078199749}} {"text": "[STATEMENT]\nlemma invertible_imp_matrix_inv:\n assumes i: \"invertible (A :: ('a :: {comm_ring_1,euclidean_semiring}) ^ 'b ^ 'b)\"\n shows \"matrix_inv A = (1 div (det A)) *k adjugate A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_inv A = (1::'a) div det A *k adjugate A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. matrix_inv A = (1::'a) div det A *k adjugate A\n[PROOF STEP]\nlet ?A = \"adjugate A\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. matrix_inv A = (1::'a) div det A *k adjugate A\n[PROOF STEP]\nhave \"A ** ?A = det A *k mat 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A ** adjugate A = det A *k mat (1::'a)\n[PROOF STEP]\nunfolding mult_adjugate_det\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (det A) = det A *k mat (1::'a)\n[PROOF STEP]\nby (simp add: scalar_mult_mat)\n[PROOF STATE]\nproof (state)\nthis:\nA ** adjugate A = det A *k mat (1::'a)\n\ngoal (1 subgoal):\n 1. matrix_inv A = (1::'a) div det A *k adjugate A\n[PROOF STEP]\nhence \"matrix_inv A ** (A ** ?A) = matrix_inv A ** (det A *k mat 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA ** adjugate A = det A *k mat (1::'a)\n\ngoal (1 subgoal):\n 1. matrix_inv A ** (A ** adjugate A) = matrix_inv A ** (det A *k mat (1::'a))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_inv A ** (A ** adjugate A) = matrix_inv A ** (det A *k mat (1::'a))\n\ngoal (1 subgoal):\n 1. matrix_inv A = (1::'a) div det A *k adjugate A\n[PROOF STEP]\nhence \"?A = det A *k matrix_inv A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix_inv A ** (A ** adjugate A) = matrix_inv A ** (det A *k mat (1::'a))\n\ngoal (1 subgoal):\n 1. adjugate A = det A *k matrix_inv A\n[PROOF STEP]\nunfolding matrix_mul_assoc matrix_inv_left[OF i] matrix_mul_lid scalar_mult_mat matrix_mul_mat\n[PROOF STATE]\nproof (prove)\nusing this:\nadjugate A = det A * (1::'a) *k matrix_inv A\n\ngoal (1 subgoal):\n 1. adjugate A = det A *k matrix_inv A\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nadjugate A = det A *k matrix_inv A\n\ngoal (1 subgoal):\n 1. matrix_inv A = (1::'a) div det A *k adjugate A\n[PROOF STEP]\nwith i\n[PROOF STATE]\nproof (chain)\npicking this:\ninvertible A\nadjugate A = det A *k matrix_inv A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ninvertible A\nadjugate A = det A *k matrix_inv A\n\ngoal (1 subgoal):\n 1. matrix_inv A = (1::'a) div det A *k adjugate A\n[PROOF STEP]\nby (metis (no_types, lifting) dvd_mult_div_cancel invertible_iff_is_unit\n matrix_mul_assoc matrix_mul_mat matrix_mul_rid scalar_mult_mat mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_inv A = (1::'a) div det A *k adjugate A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1175, "file": "Echelon_Form_Echelon_Form_Inverse", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473879530491, "lm_q2_score": 0.8376199714402812, "lm_q1q2_score": 0.7306955941832369}} {"text": "[STATEMENT]\nlemma card_update:\n fixes X v b\n assumes \"finite (X :: 'a set)\" \"v \\ X\"\n shows \"\n card ((\\s. fmupd v b s) ` {s :: 'a state. fmdom' s = X})\n = card {s :: 'a state. fmdom' s = X}\n \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (fmupd v b ` {s. fmdom' s = X}) = card {s. fmdom' s = X}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (fmupd v b ` {s. fmdom' s = X}) = card {s. fmdom' s = X}\n[PROOF STEP]\nhave \"inj_on (\\s. fmupd v b s) {s :: 'a state. fmdom' s = X}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on (fmupd v b) {s. fmdom' s = X}\n[PROOF STEP]\nusing assms bool_update_inj\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite X\nv \\ X\n\\finite ?X; ?v \\ ?X\\ \\ inj_on (fmupd ?v ?b) {s. fmdom' s = ?X}\n\ngoal (1 subgoal):\n 1. inj_on (fmupd v b) {s. fmdom' s = X}\n[PROOF STEP]\nby fast\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (fmupd v b) {s. fmdom' s = X}\n\ngoal (1 subgoal):\n 1. card (fmupd v b ` {s. fmdom' s = X}) = card {s. fmdom' s = X}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ninj_on (fmupd v b) {s. fmdom' s = X}\n[PROOF STEP]\nshow\n \"card ((\\s. fmupd v b s) ` {s :: 'a state. fmdom' s = X}) = card {s :: 'a state. fmdom' s = X}\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on (fmupd v b) {s. fmdom' s = X}\n\ngoal (1 subgoal):\n 1. card (fmupd v b ` {s. fmdom' s = X}) = card {s. fmdom' s = X}\n[PROOF STEP]\nusing card_image\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on (fmupd v b) {s. fmdom' s = X}\ninj_on ?f ?A \\ card (?f ` ?A) = card ?A\n\ngoal (1 subgoal):\n 1. card (fmupd v b ` {s. fmdom' s = X}) = card {s. fmdom' s = X}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ncard (fmupd v b ` {s. fmdom' s = X}) = card {s. fmdom' s = X}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n\n\\ \\NOTE added lemma.\\", "meta": {"llama_tokens": 937, "file": "Factored_Transition_System_Bounding_FactoredSystem", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473680407889, "lm_q2_score": 0.8376199714402812, "lm_q1q2_score": 0.7306955775043301}} {"text": "[STATEMENT]\nlemma uniformly_continuous_on_add[continuous_intros]:\n fixes f g :: \"'a::metric_space \\ 'b::real_normed_vector\"\n assumes \"uniformly_continuous_on s f\"\n and \"uniformly_continuous_on s g\"\n shows \"uniformly_continuous_on s (\\x. f x + g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. uniformly_continuous_on s (\\x. f x + g x)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nuniformly_continuous_on s f\nuniformly_continuous_on s g\n\ngoal (1 subgoal):\n 1. uniformly_continuous_on s (\\x. f x + g x)\n[PROOF STEP]\nunfolding uniformly_continuous_on_sequentially\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x y. (\\n. x n \\ s) \\ (\\n. y n \\ s) \\ (\\n. dist (x n) (y n)) \\ 0 \\ (\\n. dist (f (x n)) (f (y n))) \\ 0\n\\x y. (\\n. x n \\ s) \\ (\\n. y n \\ s) \\ (\\n. dist (x n) (y n)) \\ 0 \\ (\\n. dist (g (x n)) (g (y n))) \\ 0\n\ngoal (1 subgoal):\n 1. \\x y. (\\n. x n \\ s) \\ (\\n. y n \\ s) \\ (\\n. dist (x n) (y n)) \\ 0 \\ (\\n. dist (f (x n) + g (x n)) (f (y n) + g (y n))) \\ 0\n[PROOF STEP]\nunfolding dist_norm tendsto_norm_zero_iff add_diff_add\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x y. (\\n. x n \\ s) \\ (\\n. y n \\ s) \\ (\\n. dist (x n) (y n)) \\ 0 \\ (\\n. f (x n) - f (y n)) \\ (0::'b)\n\\x y. (\\n. x n \\ s) \\ (\\n. y n \\ s) \\ (\\n. dist (x n) (y n)) \\ 0 \\ (\\n. g (x n) - g (y n)) \\ (0::'b)\n\ngoal (1 subgoal):\n 1. \\x y. (\\n. x n \\ s) \\ (\\n. y n \\ s) \\ (\\n. dist (x n) (y n)) \\ 0 \\ (\\n. f (x n) - f (y n) + (g (x n) - g (y n))) \\ (0::'b)\n[PROOF STEP]\nby (auto intro: tendsto_add_zero)", "meta": {"llama_tokens": 891, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.849971190859164, "lm_q2_score": 0.8596637523076225, "lm_q1q2_score": 0.7306894232873672}} {"text": "[STATEMENT]\nlemma lower_asymptotic_density_one_lim:\n assumes \"lower_asymptotic_density A = 1\"\n shows \"(\\n. card(A \\ {.. 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. real (card (A \\ {.. 1\n[PROOF STEP]\napply (rule lower_asymptotic_density_eq_upper)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. lower_asymptotic_density A = 1\n 2. upper_asymptotic_density A = 1\n[PROOF STEP]\nusing assms lower_asymptotic_density_le_upper[of A] upper_asymptotic_density_in_01(2)[of A]\n[PROOF STATE]\nproof (prove)\nusing this:\nlower_asymptotic_density A = 1\nlower_asymptotic_density A \\ upper_asymptotic_density A\nupper_asymptotic_density A \\ 1\n\ngoal (2 subgoals):\n 1. lower_asymptotic_density A = 1\n 2. upper_asymptotic_density A = 1\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 378, "file": "Ergodic_Theory_Asymptotic_Density", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513786759491, "lm_q2_score": 0.8152324960856175, "lm_q1q2_score": 0.7306532485581699}} {"text": "[STATEMENT]\ntheorem FNTT_inv_IFNTT: \n assumes \"length numbers = n\"\n shows \"FNTT (IFNTT numbers) = map ((*) (of_int_mod_ring (int n))) numbers\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. FNTT (IFNTT numbers) = map ((*) (of_int_mod_ring (int n))) numbers\n[PROOF STEP]\nby (simp add: FNTT_correct IFNTT_correct assms inv_ntt_correct length_INTT)", "meta": {"llama_tokens": 144, "file": "Number_Theoretic_Transform_Butterfly", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513759047847, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.7306532382523413}} {"text": "[STATEMENT]\nlemma tanh_real_gt_neg1: \"tanh (x :: real) > -1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - 1 < tanh x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. - 1 < tanh x\n[PROOF STEP]\nhave \"- cosh x < sinh x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - cosh x < sinh x\n[PROOF STEP]\nby (simp add: sinh_def cosh_def field_split_simps)\n[PROOF STATE]\nproof (state)\nthis:\n- cosh x < sinh x\n\ngoal (1 subgoal):\n 1. - 1 < tanh x\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n- cosh x < sinh x\n\ngoal (1 subgoal):\n 1. - 1 < tanh x\n[PROOF STEP]\nby (simp add: tanh_def field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n- 1 < tanh x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 348, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513675912912, "lm_q2_score": 0.8152324826183822, "lm_q1q2_score": 0.7306532274515686}} {"text": "[STATEMENT]\ntheorem polyfun_eq_const:\n fixes c :: \"nat \\ 'a::{comm_ring,real_normed_div_algebra}\"\n shows \"(\\z. (\\i\\n. c i * z^i) = k) \\ c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\nfix z\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\nhave \"(\\i\\n. c i * z^i) = (\\i\\n. (if i = 0 then c 0 - k else c i) * z^i) + k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\n. c i * z ^ i) = (\\i\\n. (if i = 0 then c 0 - k else c i) * z ^ i) + k\n[PROOF STEP]\nby (induct n) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. c i * z ^ i) = (\\i\\n. (if i = 0 then c 0 - k else c i) * z ^ i) + k\n\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. c i * ?z2 ^ i) = (\\i\\n. (if i = 0 then c 0 - k else c i) * ?z2 ^ i) + k\n\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i\\n. c i * ?z2 ^ i) = (\\i\\n. (if i = 0 then c 0 - k else c i) * ?z2 ^ i) + k\n[PROOF STEP]\nhave \"(\\z. (\\i\\n. c i * z^i) = k) \\ (\\z. (\\i\\n. (if i = 0 then c 0 - k else c i) * z^i) = 0)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i\\n. c i * ?z2 ^ i) = (\\i\\n. (if i = 0 then c 0 - k else c i) * ?z2 ^ i) + k\n\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. c i * z ^ i) = k) = (\\z. (\\i\\n. (if i = 0 then c 0 - k else c i) * z ^ i) = (0::'a))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\z. (\\i\\n. c i * z ^ i) = k) = (\\z. (\\i\\n. (if i = 0 then c 0 - k else c i) * z ^ i) = (0::'a))\n\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\z. (\\i\\n. c i * z ^ i) = k) = (\\z. (\\i\\n. (if i = 0 then c 0 - k else c i) * z ^ i) = (0::'a))\n\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\nhave \"... \\ c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. (if i = 0 then c 0 - k else c i) * z ^ i) = (0::'a)) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\nby (auto simp: polyfun_eq_0)\n[PROOF STATE]\nproof (state)\nthis:\n(\\z. (\\i\\n. (if i = 0 then c 0 - k else c i) * z ^ i) = (0::'a)) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n\ngoal (1 subgoal):\n 1. (\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\z. (\\i\\n. c i * z ^ i) = k) = (c 0 = k \\ (\\k. k \\ 0 \\ k \\ n \\ c k = (0::'a)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2359, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916099737806, "lm_q2_score": 0.8459424373085146, "lm_q1q2_score": 0.7303796028929423}} {"text": "[STATEMENT]\nlemma hermitian_decomp_eigenvalues: \n assumes \"hermitian_decomp A B U\"\n shows \"diag_mat B = (eigvals A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. diag_mat B = eigvals A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nhermitian_decomp A B U\n\ngoal (1 subgoal):\n 1. diag_mat B = eigvals A\n[PROOF STEP]\nunfolding hermitian_decomp_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsimilar_mat_wit A B U (Complex_Matrix.adjoint U) \\ diagonal_mat B \\ diag_mat B = eigvals A \\ Complex_Matrix.unitary U \\ (\\i \\)\n\ngoal (1 subgoal):\n 1. diag_mat B = eigvals A\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 288, "file": "Projective_Measurements_Projective_Measurements", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094117351309, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7303469803342223}} {"text": "[STATEMENT]\nlemma rref_condition3:\n assumes r: \"reduced_row_echelon_form A\"\n shows \"(\\i. i \\ (is_zero_row i A) \\ \\ (is_zero_row (i+1) A) \\ ((LEAST n. A $ i $ n \\ 0) < (LEAST n. A $ (i+1) $ n \\ 0)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i < i + (1::'c) \\ \\ is_zero_row i A \\ \\ is_zero_row (i + (1::'c)) A \\ (LEAST n. A $ i $ n \\ (0::'a)) < (LEAST n. A $ (i + (1::'c)) $ n \\ (0::'a))\n[PROOF STEP]\nusing r\n[PROOF STATE]\nproof (prove)\nusing this:\nreduced_row_echelon_form A\n\ngoal (1 subgoal):\n 1. \\i. i < i + (1::'c) \\ \\ is_zero_row i A \\ \\ is_zero_row (i + (1::'c)) A \\ (LEAST n. A $ i $ n \\ (0::'a)) < (LEAST n. A $ (i + (1::'c)) $ n \\ (0::'a))\n[PROOF STEP]\nunfolding reduced_row_echelon_form_def'\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i. is_zero_row i A \\ \\ (\\j>i. \\ is_zero_row j A)) \\ (\\i. \\ is_zero_row i A \\ A $ i $ (LEAST k. A $ i $ k \\ (0::'a)) = (1::'a)) \\ (\\i. i < i + (1::'c) \\ \\ is_zero_row i A \\ \\ is_zero_row (i + (1::'c)) A \\ (LEAST k. A $ i $ k \\ (0::'a)) < (LEAST k. A $ (i + (1::'c)) $ k \\ (0::'a))) \\ (\\i. \\ is_zero_row i A \\ (\\j. i \\ j \\ A $ j $ (LEAST k. A $ i $ k \\ (0::'a)) = (0::'a)))\n\ngoal (1 subgoal):\n 1. \\i. i < i + (1::'c) \\ \\ is_zero_row i A \\ \\ is_zero_row (i + (1::'c)) A \\ (LEAST n. A $ i $ n \\ (0::'a)) < (LEAST n. A $ (i + (1::'c)) $ n \\ (0::'a))\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 825, "file": "Gauss_Jordan_Rref", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933093975331751, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.7303469587963646}} {"text": "[STATEMENT]\nlemma harm_altdef: \"harm n = (\\kkk = 1..n. inverse (of_nat k)) = (\\kf : a \\ b\\ \\ equivalence_map f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\f. \\f : a \\ b\\ \\ equivalence_map f \\ thesis) \\ thesis\n[PROOF STEP]\nusing assms equivalent_objects_def\n[PROOF STATE]\nproof (prove)\nusing this:\nequivalent_objects a b\nequivalent_objects b c\nequivalent_objects ?a ?b \\ \\f. \\f : ?a \\ ?b\\ \\ equivalence_map f\n\ngoal (1 subgoal):\n 1. (\\f. \\f : a \\ b\\ \\ equivalence_map f \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\f : a \\ b\\ \\ equivalence_map f\n\ngoal (1 subgoal):\n 1. equivalent_objects a c\n[PROOF STEP]\nobtain g where g: \"\\g : b \\ c\\ \\ equivalence_map g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\g. \\g : b \\ c\\ \\ equivalence_map g \\ thesis) \\ thesis\n[PROOF STEP]\nusing assms equivalent_objects_def\n[PROOF STATE]\nproof (prove)\nusing this:\nequivalent_objects a b\nequivalent_objects b c\nequivalent_objects ?a ?b \\ \\f. \\f : ?a \\ ?b\\ \\ equivalence_map f\n\ngoal (1 subgoal):\n 1. (\\g. \\g : b \\ c\\ \\ equivalence_map g \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\g : b \\ c\\ \\ equivalence_map g\n\ngoal (1 subgoal):\n 1. equivalent_objects a c\n[PROOF STEP]\nhave \"\\g \\ f : a \\ c\\ \\ equivalence_map (g \\ f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\g \\ f : a \\ c\\ \\ equivalence_map (g \\ f)\n[PROOF STEP]\nusing f g equivalence_maps_compose\n[PROOF STATE]\nproof (prove)\nusing this:\n\\f : a \\ b\\ \\ equivalence_map f\n\\g : b \\ c\\ \\ equivalence_map g\n\\equivalence_map ?f; equivalence_map ?f'; src ?f' = trg ?f\\ \\ equivalence_map (?f' \\ ?f)\n\ngoal (1 subgoal):\n 1. \\g \\ f : a \\ c\\ \\ equivalence_map (g \\ f)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\g \\ f : a \\ c\\ \\ equivalence_map (g \\ f)\n\ngoal (1 subgoal):\n 1. equivalent_objects a c\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\g \\ f : a \\ c\\ \\ equivalence_map (g \\ f)\n\ngoal (1 subgoal):\n 1. equivalent_objects a c\n[PROOF STEP]\nusing equivalent_objects_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\g \\ f : a \\ c\\ \\ equivalence_map (g \\ f)\nequivalent_objects ?a ?b \\ \\f. \\f : ?a \\ ?b\\ \\ equivalence_map f\n\ngoal (1 subgoal):\n 1. equivalent_objects a c\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nequivalent_objects a c\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1404, "file": "Bicategory_InternalEquivalence", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942348544448, "lm_q2_score": 0.8198933293122506, "lm_q1q2_score": 0.7299463042823134}} {"text": "[STATEMENT]\nlemma Bseq_mult:\n fixes f g :: \"nat \\ 'a::real_normed_field\"\n assumes \"Bseq f\" and \"Bseq g\"\n shows \"Bseq (\\x. f x * g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Bseq (\\x. f x * g x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Bseq (\\x. f x * g x)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nBseq f\nBseq g\n[PROOF STEP]\nobtain K1 K2 where K: \"norm (f x) \\ K1\" \"K1 > 0\" \"norm (g x) \\ K2\" \"K2 > 0\"\n for x\n[PROOF STATE]\nproof (prove)\nusing this:\nBseq f\nBseq g\n\ngoal (1 subgoal):\n 1. (\\K1 K2. \\\\x. norm (f x) \\ K1; 0 < K1; \\x. norm (g x) \\ K2; 0 < K2\\ \\ thesis) \\ thesis\n[PROOF STEP]\nunfolding Bseq_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\K>0. \\n. norm (f n) \\ K\n\\K>0. \\n. norm (g n) \\ K\n\ngoal (1 subgoal):\n 1. (\\K1 K2. \\\\x. norm (f x) \\ K1; 0 < K1; \\x. norm (g x) \\ K2; 0 < K2\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nnorm (f ?x) \\ K1\n0 < K1\nnorm (g ?x) \\ K2\n0 < K2\n\ngoal (1 subgoal):\n 1. Bseq (\\x. f x * g x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (f ?x) \\ K1\n0 < K1\nnorm (g ?x) \\ K2\n0 < K2\n[PROOF STEP]\nhave \"norm (f x * g x) \\ K1 * K2\" for x\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (f ?x) \\ K1\n0 < K1\nnorm (g ?x) \\ K2\n0 < K2\n\ngoal (1 subgoal):\n 1. norm (f x * g x) \\ K1 * K2\n[PROOF STEP]\nby (auto simp: norm_mult intro!: mult_mono)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (f ?x * g ?x) \\ K1 * K2\n\ngoal (1 subgoal):\n 1. Bseq (\\x. f x * g x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (f ?x * g ?x) \\ K1 * K2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (f ?x * g ?x) \\ K1 * K2\n\ngoal (1 subgoal):\n 1. Bseq (\\x. f x * g x)\n[PROOF STEP]\nby (rule BseqI')\n[PROOF STATE]\nproof (state)\nthis:\nBseq (\\x. f x * g x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1037, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357735451835, "lm_q2_score": 0.8418256512199033, "lm_q1q2_score": 0.7298929546956267}} {"text": "[STATEMENT]\nlemma transpose_matrix_add: \"transpose_matrix ((A::('a::monoid_add) matrix)+B) = transpose_matrix A + transpose_matrix B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. transpose_matrix (A + B) = transpose_matrix A + transpose_matrix B\n[PROOF STEP]\nby (simp add: plus_matrix_def transpose_combine_matrix)", "meta": {"llama_tokens": 119, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9263037302939515, "lm_q2_score": 0.7879311856832191, "lm_q1q2_score": 0.729863596513302}} {"text": "[STATEMENT]\nlemma tendsto_ln_powr_over_powr: \n assumes \"(a::real) > 0\" \"b > 0\"\n shows \"((\\x. ln x powr a / x powr b) \\ 0) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. ln x powr a / x powr b) \\ 0) at_top\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\x. ln x powr a / x powr b) \\ 0) at_top\n[PROOF STEP]\nhave \"eventually (\\x. ln x powr a / x powr b = (ln x / x powr (b/a)) powr a) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. ln x powr a / x powr b = (ln x / x powr (b / a)) powr a\n[PROOF STEP]\nusing assms eventually_gt_at_top[of \"1::real\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n0 < b\neventually ((<) 1) at_top\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. ln x powr a / x powr b = (ln x / x powr (b / a)) powr a\n[PROOF STEP]\nby (elim eventually_mono) (simp add: powr_divide powr_powr)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in at_top. ln x powr a / x powr b = (ln x / x powr (b / a)) powr a\n\ngoal (1 subgoal):\n 1. ((\\x. ln x powr a / x powr b) \\ 0) at_top\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in at_top. ln x powr a / x powr b = (ln x / x powr (b / a)) powr a\n\ngoal (1 subgoal):\n 1. ((\\x. ln x powr a / x powr b) \\ 0) at_top\n[PROOF STEP]\nhave \"eventually (\\x. 0 < ln x / x powr (b / a)) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. 0 < ln x / x powr (b / a)\n[PROOF STEP]\nusing eventually_gt_at_top[of \"1::real\"]\n[PROOF STATE]\nproof (prove)\nusing this:\neventually ((<) 1) at_top\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. 0 < ln x / x powr (b / a)\n[PROOF STEP]\nby (elim eventually_mono) simp\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in at_top. 0 < ln x / x powr (b / a)\n\ngoal (1 subgoal):\n 1. ((\\x. ln x powr a / x powr b) \\ 0) at_top\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < a\n0 < b\n\\\\<^sub>F x in at_top. 0 < ln x / x powr (b / a)\n[PROOF STEP]\nhave \"((\\x. (ln x / x powr (b/a)) powr a) \\ 0) at_top\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n0 < b\n\\\\<^sub>F x in at_top. 0 < ln x / x powr (b / a)\n\ngoal (1 subgoal):\n 1. ((\\x. (ln x / x powr (b / a)) powr a) \\ 0) at_top\n[PROOF STEP]\nby (intro tendsto_zero_powrI tendsto_ln_over_powr) (simp_all add: eventually_mono)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. (ln x / x powr (b / a)) powr a) \\ 0) at_top\n\ngoal (1 subgoal):\n 1. ((\\x. ln x powr a / x powr b) \\ 0) at_top\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F x in at_top. ln x powr a / x powr b = (ln x / x powr (b / a)) powr a\n((\\x. (ln x / x powr (b / a)) powr a) \\ 0) at_top\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in at_top. ln x powr a / x powr b = (ln x / x powr (b / a)) powr a\n((\\x. (ln x / x powr (b / a)) powr a) \\ 0) at_top\n\ngoal (1 subgoal):\n 1. ((\\x. ln x powr a / x powr b) \\ 0) at_top\n[PROOF STEP]\nby (subst tendsto_cong) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. ln x powr a / x powr b) \\ 0) at_top\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1604, "file": "Landau_Symbols_Landau_Real_Products", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.875787001374006, "lm_q2_score": 0.8333246015211009, "lm_q1q2_score": 0.7298148539373533}} {"text": "[STATEMENT]\nlemma Ints_scalar_prod: \n \"v \\ carrier_vec n \\ w \\ carrier_vec n\n \\ (\\ i. i < n \\ v $ i \\ \\) \\ (\\ i. i < n \\ w $ i \\ \\) \\ v \\ w \\ \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\v \\ carrier_vec n; w \\ carrier_vec n; \\i. i < n \\ v $ i \\ \\; \\i. i < n \\ w $ i \\ \\\\ \\ v \\ w \\ \\\n[PROOF STEP]\nunfolding scalar_prod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\v \\ carrier_vec n; w \\ carrier_vec n; \\i. i < n \\ v $ i \\ \\; \\i. i < n \\ w $ i \\ \\\\ \\ (\\i = 0.. \\\n[PROOF STEP]\nby (intro Ints_sum Ints_mult, auto)", "meta": {"llama_tokens": 372, "file": "LLL_Basis_Reduction_Gram_Schmidt_2", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869948899665, "lm_q2_score": 0.8333245994514082, "lm_q1q2_score": 0.7298148467214338}} {"text": "[STATEMENT]\nlemma dist_riemann_sphere_sym: \n shows \"dist_riemann_sphere' M1 M2 = dist_riemann_sphere' M2 M1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = dist_riemann_sphere' M2 M1\n[PROOF STEP]\nproof transfer\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\M1 M2. \\M1 \\ unit_sphere; M2 \\ unit_sphere\\ \\ dist_riemann_sphere_r3 M1 M2 = dist_riemann_sphere_r3 M2 M1\n[PROOF STEP]\nfix M1 M2 :: R3\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\M1 M2. \\M1 \\ unit_sphere; M2 \\ unit_sphere\\ \\ dist_riemann_sphere_r3 M1 M2 = dist_riemann_sphere_r3 M2 M1\n[PROOF STEP]\nobtain x1 y1 z1 x2 y2 z2 where MM: \"(x1, y1, z1) = M1\" \"(x2, y2, z2) = M2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x1 y1 z1 x2 y2 z2. \\(x1, y1, z1) = M1; (x2, y2, z2) = M2\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (cases \"M1\", cases \"M2\", auto)\n[PROOF STATE]\nproof (state)\nthis:\n(x1, y1, z1) = M1\n(x2, y2, z2) = M2\n\ngoal (1 subgoal):\n 1. \\M1 M2. \\M1 \\ unit_sphere; M2 \\ unit_sphere\\ \\ dist_riemann_sphere_r3 M1 M2 = dist_riemann_sphere_r3 M2 M1\n[PROOF STEP]\nshow \"dist_riemann_sphere_r3 M1 M2 = dist_riemann_sphere_r3 M2 M1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist_riemann_sphere_r3 M1 M2 = dist_riemann_sphere_r3 M2 M1\n[PROOF STEP]\nusing norm_minus_cancel[of \"(x1 - x2, y1 - y2, z1 - z2)\"] MM[symmetric]\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (- (x1 - x2, y1 - y2, z1 - z2)) = norm (x1 - x2, y1 - y2, z1 - z2)\nM1 = (x1, y1, z1)\nM2 = (x2, y2, z2)\n\ngoal (1 subgoal):\n 1. dist_riemann_sphere_r3 M1 M2 = dist_riemann_sphere_r3 M2 M1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndist_riemann_sphere_r3 M1 M2 = dist_riemann_sphere_r3 M2 M1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 938, "file": "Complex_Geometry_Chordal_Metric", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869819218865, "lm_q2_score": 0.8333246015211008, "lm_q1q2_score": 0.7298148377274236}} {"text": "[STATEMENT]\ntheorem ln_lower_9_eq: \"0 < x \\\n ln_lower_9 x = (1/30)*(6 + 481*x + 1881*x^2 + 1281*x^3 + 131*x^4)*(x - 1) /\n (x*(5 + 40*x + 60*x^2 + 20*x^3 + x^4))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ ln_lower_9 x = 1 / 30 * (6 + 481 * x + 1881 * x\\<^sup>2 + 1281 * x ^ 3 + 131 * x ^ 4) * (x - 1) / (x * (5 + 40 * x + 60 * x\\<^sup>2 + 20 * x ^ 3 + x ^ 4))\n[PROOF STEP]\nunfolding ln_lower_9_def ln_upper_9_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ - ((6 * inverse x ^ 4 + 481 * inverse x ^ 3 + 1881 * (inverse x)\\<^sup>2 + 1281 * inverse x + 131) * (inverse x - 1) / (30 * (5 * inverse x ^ 4 + 40 * inverse x ^ 3 + 60 * (inverse x)\\<^sup>2 + 20 * inverse x + 1))) = 1 / 30 * (6 + 481 * x + 1881 * x\\<^sup>2 + 1281 * x ^ 3 + 131 * x ^ 4) * (x - 1) / (x * (5 + 40 * x + 60 * x\\<^sup>2 + 20 * x ^ 3 + x ^ 4))\n[PROOF STEP]\nby (simp add: zero_less_mult_iff add_pos_pos dual_order.strict_implies_not_eq divide_simps)\n algebra", "meta": {"llama_tokens": 562, "file": "Special_Function_Bounds_Log_CF_Bounds", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976953003183443, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.729707166935599}} {"text": "[STATEMENT]\nlemma pochhammer_absorb_comp: \"(r - of_nat k) * pochhammer (- r) k = r * pochhammer (-r + 1) k\"\n (is \"?lhs = ?rhs\")\n for r :: \"'a::comm_ring_1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (r - of_nat k) * pochhammer (- r) k = r * pochhammer (- r + (1::'a)) k\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (r - of_nat k) * pochhammer (- r) k = r * pochhammer (- r + (1::'a)) k\n[PROOF STEP]\nhave \"?lhs = - pochhammer (- r) (Suc k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (r - of_nat k) * pochhammer (- r) k = - pochhammer (- r) (Suc k)\n[PROOF STEP]\nby (subst pochhammer_rec') (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(r - of_nat k) * pochhammer (- r) k = - pochhammer (- r) (Suc k)\n\ngoal (1 subgoal):\n 1. (r - of_nat k) * pochhammer (- r) k = r * pochhammer (- r + (1::'a)) k\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(r - of_nat k) * pochhammer (- r) k = - pochhammer (- r) (Suc k)\n\ngoal (1 subgoal):\n 1. (r - of_nat k) * pochhammer (- r) k = r * pochhammer (- r + (1::'a)) k\n[PROOF STEP]\nhave \"\\ = ?rhs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - pochhammer (- r) (Suc k) = r * pochhammer (- r + (1::'a)) k\n[PROOF STEP]\nby (subst pochhammer_rec) simp\n[PROOF STATE]\nproof (state)\nthis:\n- pochhammer (- r) (Suc k) = r * pochhammer (- r + (1::'a)) k\n\ngoal (1 subgoal):\n 1. (r - of_nat k) * pochhammer (- r) k = r * pochhammer (- r + (1::'a)) k\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(r - of_nat k) * pochhammer (- r) k = r * pochhammer (- r + (1::'a)) k\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(r - of_nat k) * pochhammer (- r) k = r * pochhammer (- r + (1::'a)) k\n\ngoal (1 subgoal):\n 1. (r - of_nat k) * pochhammer (- r) k = r * pochhammer (- r + (1::'a)) k\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(r - of_nat k) * pochhammer (- r) k = r * pochhammer (- r + (1::'a)) k\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 946, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267864276108, "lm_q2_score": 0.8397339756938819, "lm_q1q2_score": 0.7295833715561968}} {"text": "[STATEMENT]\nlemma prime_factors_gcd [simp]: \n \"a \\ 0 \\ b \\ 0 \\ prime_factors (gcd a b) = \n prime_factors a \\ prime_factors b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a \\ (0::'a); b \\ (0::'a)\\ \\ prime_factors (gcd a b) = prime_factors a \\ prime_factors b\n[PROOF STEP]\nby (subst prime_factorization_gcd) auto", "meta": {"llama_tokens": 172, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278788223265, "lm_q2_score": 0.8267117855317474, "lm_q1q2_score": 0.729513527304198}} {"text": "[STATEMENT]\nlemma exp_integer_2pi_plus1:\n assumes \"n \\ \\\"\n shows \"exp(((2 * n + 1) * pi) * \\) = - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp (complex_of_real ((2 * n + 1) * pi) * \\) = - 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. exp (complex_of_real ((2 * n + 1) * pi) * \\) = - 1\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nn \\ \\\n[PROOF STEP]\nobtain n' where [simp]: \"n = of_int n'\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\ \\\n\ngoal (1 subgoal):\n 1. (\\n'. n = real_of_int n' \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: Ints_def)\n[PROOF STATE]\nproof (state)\nthis:\nn = real_of_int n'\n\ngoal (1 subgoal):\n 1. exp (complex_of_real ((2 * n + 1) * pi) * \\) = - 1\n[PROOF STEP]\nhave \"exp(((2 * n + 1) * pi) * \\) = exp (pi * \\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp (complex_of_real ((2 * n + 1) * pi) * \\) = exp (complex_of_real pi * \\)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\ \\\n\ngoal (1 subgoal):\n 1. exp (complex_of_real ((2 * n + 1) * pi) * \\) = exp (complex_of_real pi * \\)\n[PROOF STEP]\nby (subst exp_eq) (auto intro!: exI[of _ n'] simp: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nexp (complex_of_real ((2 * n + 1) * pi) * \\) = exp (complex_of_real pi * \\)\n\ngoal (1 subgoal):\n 1. exp (complex_of_real ((2 * n + 1) * pi) * \\) = - 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nexp (complex_of_real ((2 * n + 1) * pi) * \\) = exp (complex_of_real pi * \\)\n\ngoal (1 subgoal):\n 1. exp (complex_of_real ((2 * n + 1) * pi) * \\) = - 1\n[PROOF STEP]\nhave \"\\ = - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp (complex_of_real pi * \\) = - 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nexp (complex_of_real pi * \\) = - 1\n\ngoal (1 subgoal):\n 1. exp (complex_of_real ((2 * n + 1) * pi) * \\) = - 1\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nexp (complex_of_real ((2 * n + 1) * pi) * \\) = - 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nexp (complex_of_real ((2 * n + 1) * pi) * \\) = - 1\n\ngoal (1 subgoal):\n 1. exp (complex_of_real ((2 * n + 1) * pi) * \\) = - 1\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nexp (complex_of_real ((2 * n + 1) * pi) * \\) = - 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1133, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.882427872638409, "lm_q2_score": 0.8267117855317474, "lm_q1q2_score": 0.7295135221918805}} {"text": "[STATEMENT]\ntheorem c_card_is_pr: \"c_card \\ PrimRec1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c_card \\ PrimRec1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. c_card \\ PrimRec1\n[PROOF STEP]\ndefine g :: \"nat \\ nat\" where \"g x = 1\" for x\n[PROOF STATE]\nproof (state)\nthis:\ng ?x = 1\n\ngoal (1 subgoal):\n 1. c_card \\ PrimRec1\n[PROOF STEP]\nhave g_is_pr: \"g \\ PrimRec1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. g \\ PrimRec1\n[PROOF STEP]\nby (unfold g_def, rule const_is_pr)\n[PROOF STATE]\nproof (state)\nthis:\ng \\ PrimRec1\n\ngoal (1 subgoal):\n 1. c_card \\ PrimRec1\n[PROOF STEP]\nhave \"c_card = (\\ u. sum g (nat_to_set u))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c_card = (\\u. sum g (nat_to_set u))\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\u. c_card u = sum g (nat_to_set u)\n[PROOF STEP]\nfix u\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\u. c_card u = sum g (nat_to_set u)\n[PROOF STEP]\nshow \"c_card u = sum g (nat_to_set u)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c_card u = sum g (nat_to_set u)\n[PROOF STEP]\nby (unfold c_card_def, unfold g_def, rule card_eq_sum)\n[PROOF STATE]\nproof (state)\nthis:\nc_card u = sum g (nat_to_set u)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nc_card = (\\u. sum g (nat_to_set u))\n\ngoal (1 subgoal):\n 1. c_card \\ PrimRec1\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nc_card = (\\u. sum g (nat_to_set u))\n\ngoal (1 subgoal):\n 1. c_card \\ PrimRec1\n[PROOF STEP]\nfrom g_is_pr\n[PROOF STATE]\nproof (chain)\npicking this:\ng \\ PrimRec1\n[PROOF STEP]\nhave \"(\\ u. sum g (nat_to_set u)) \\ PrimRec1\"\n[PROOF STATE]\nproof (prove)\nusing this:\ng \\ PrimRec1\n\ngoal (1 subgoal):\n 1. (\\u. sum g (nat_to_set u)) \\ PrimRec1\n[PROOF STEP]\nby (rule sum_is_pr)\n[PROOF STATE]\nproof (state)\nthis:\n(\\u. sum g (nat_to_set u)) \\ PrimRec1\n\ngoal (1 subgoal):\n 1. c_card \\ PrimRec1\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nc_card = (\\u. sum g (nat_to_set u))\n(\\u. sum g (nat_to_set u)) \\ PrimRec1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nc_card = (\\u. sum g (nat_to_set u))\n(\\u. sum g (nat_to_set u)) \\ PrimRec1\n\ngoal (1 subgoal):\n 1. c_card \\ PrimRec1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nc_card \\ PrimRec1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1179, "file": "Recursion-Theory-I_PRecFinSet", "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278664544911, "lm_q2_score": 0.8267117898012105, "lm_q1q2_score": 0.7295135208470559}} {"text": "[STATEMENT]\nlemma degree_polyadd:\n fixes p::\"'a::{plus,zero} poly\"\n assumes np: \"isnpolyh p n0\"\n and nq: \"isnpolyh q n1\"\n shows \"degree (p +\\<^sub>p q) \\ max (degree p) (degree q)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Polynomial_Expression.degree (p +\\<^sub>p q) \\ max (Polynomial_Expression.degree p) (Polynomial_Expression.degree q)\n[PROOF STEP]\nusing degreen_polyadd[OF np nq, where m= \"0\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ max n0 n1 \\ degreen (p +\\<^sub>p q) 0 \\ max (degreen p 0) (degreen q 0)\n\ngoal (1 subgoal):\n 1. Polynomial_Expression.degree (p +\\<^sub>p q) \\ max (Polynomial_Expression.degree p) (Polynomial_Expression.degree q)\n[PROOF STEP]\nby (simp add: degree_eq_degreen0)", "meta": {"llama_tokens": 311, "file": "Taylor_Models_Polynomial_Expression", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894632969136, "lm_q2_score": 0.8152324871074607, "lm_q1q2_score": 0.7294614396010928}} {"text": "[STATEMENT]\nlemma euclid_diff2: \n \"PRE (\\s::nat store. s ''x'' = x \\ s ''y'' = y \\ x > 0 \\ y > 0)\n (WHILE (\\s. s ''x''\\ s ''y'') INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) \n DO\n (IF (\\s. s ''x'' > s ''y'')\n THEN (''x'' ::= (\\s. s ''x'' - s ''y''))\n ELSE (''y'' ::= (\\s. s ''y'' - s ''x''))\n FI)\n OD)\n POST (\\s. s ''x'' = gcd x y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rdom \\\\s. s ''x'' = x \\ s ''y'' = y \\ 0 < x \\ 0 < y\\ \\ wp (WHILE (\\s. s ''x'' \\ s ''y'') INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) DO (IF (\\s. s ''y'' < s ''x'') THEN (''x'' ::= (\\s. s ''x'' - s ''y'')) ELSE (''y'' ::= (\\s. s ''y'' - s ''x'')) FI) OD) \\\\s. s ''x'' = gcd x y\\\n[PROOF STEP]\napply (hoare, simp_all)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\s. gcd (s ''x'') (s ''y'') = gcd x y \\ s ''x'' = s ''y'' \\ s ''y'' = gcd x y\n 2. \\s. gcd (s ''x'') (s ''y'') = gcd x y \\ s ''x'' \\ s ''y'' \\ (s ''y'' < s ''x'' \\ gcd (s ''x'' - s ''y'') (s ''y'') = gcd x y) \\ (s ''y'' < s ''x'' \\ gcd (s ''x'') (s ''y'' - s ''x'') = gcd x y)\n[PROOF STEP]\napply auto[1]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\s. gcd (s ''x'') (s ''y'') = gcd x y \\ s ''x'' \\ s ''y'' \\ (s ''y'' < s ''x'' \\ gcd (s ''x'' - s ''y'') (s ''y'') = gcd x y) \\ (s ''y'' < s ''x'' \\ gcd (s ''x'') (s ''y'' - s ''x'') = gcd x y)\n[PROOF STEP]\nby (metis gcd.commute gcd_diff1_nat le_cases nat_less_le)", "meta": {"llama_tokens": 823, "file": "Algebraic_VCs_AVC_KAD_VC_KAD_Examples2", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894632969136, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.7294614396010928}} {"text": "[STATEMENT]\nlemma contour_integral_subpath_combine_less:\n assumes \"f contour_integrable_on g\" \"valid_path g\" \"u \\ {0..1}\" \"v \\ {0..1}\" \"w \\ {0..1}\"\n \"ux. f (g x) * vector_derivative g (at x)) integrable_on {u..w}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. f (g x) * vector_derivative g (at x)) integrable_on {u..w}\n[PROOF STEP]\nusing integrable_on_subcbox [where a=u and b=w and S = \"{0..1}\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?f integrable_on {0..1}; cbox u w \\ {0..1}\\ \\ ?f integrable_on cbox u w\nf contour_integrable_on g\nvalid_path g\nu \\ {0..1}\nv \\ {0..1}\nw \\ {0..1}\nu < v\nv < w\n\ngoal (1 subgoal):\n 1. (\\x. f (g x) * vector_derivative g (at x)) integrable_on {u..w}\n[PROOF STEP]\nby (auto simp: contour_integrable_on)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. f (g x) * vector_derivative g (at x)) integrable_on {u..w}\n\ngoal (1 subgoal):\n 1. contour_integral (subpath u v g) f + contour_integral (subpath v w g) f = contour_integral (subpath u w g) f\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\nf contour_integrable_on g\nvalid_path g\nu \\ {0..1}\nv \\ {0..1}\nw \\ {0..1}\nu < v\nv < w\n(\\x. f (g x) * vector_derivative g (at x)) integrable_on {u..w}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nf contour_integrable_on g\nvalid_path g\nu \\ {0..1}\nv \\ {0..1}\nw \\ {0..1}\nu < v\nv < w\n(\\x. f (g x) * vector_derivative g (at x)) integrable_on {u..w}\n\ngoal (1 subgoal):\n 1. contour_integral (subpath u v g) f + contour_integral (subpath v w g) f = contour_integral (subpath u w g) f\n[PROOF STEP]\nby (auto simp: contour_integral_subcontour_integral Henstock_Kurzweil_Integration.integral_combine)\n[PROOF STATE]\nproof (state)\nthis:\ncontour_integral (subpath u v g) f + contour_integral (subpath v w g) f = contour_integral (subpath u w g) f\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1092, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894661025424, "lm_q2_score": 0.8152324803738429, "lm_q1q2_score": 0.7294614358631623}} {"text": "[STATEMENT]\nlemma vector_derivative_part_circlepath01:\n \"\\0 \\ x; x \\ 1\\\n \\ vector_derivative (part_circlepath z r s t) (at x within {0..1}) =\n \\ * r * (of_real t - of_real s) * exp(\\ * linepath s t x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 \\ x; x \\ 1\\ \\ vector_derivative (part_circlepath z r s t) (at x within {0..1}) = \\ * complex_of_real r * (complex_of_real t - complex_of_real s) * exp (\\ * complex_of_real (linepath s t x))\n[PROOF STEP]\nusing has_vector_derivative_part_circlepath\n[PROOF STATE]\nproof (prove)\nusing this:\n(part_circlepath ?z ?r ?s ?t has_vector_derivative \\ * complex_of_real ?r * (complex_of_real ?t - complex_of_real ?s) * exp (\\ * complex_of_real (linepath ?s ?t ?x))) (at ?x within ?X)\n\ngoal (1 subgoal):\n 1. \\0 \\ x; x \\ 1\\ \\ vector_derivative (part_circlepath z r s t) (at x within {0..1}) = \\ * complex_of_real r * (complex_of_real t - complex_of_real s) * exp (\\ * complex_of_real (linepath s t x))\n[PROOF STEP]\nby (auto simp: vector_derivative_at_within_ivl)", "meta": {"llama_tokens": 475, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9196425377849806, "lm_q2_score": 0.7931059609645724, "lm_q1q2_score": 0.7293739786738551}} {"text": "[STATEMENT]\nlemma sqrt_two_squared_cpx: \"complex_of_real (sqrt 2) * complex_of_real (sqrt 2) = 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (sqrt 2) * complex_of_real (sqrt 2) = 2\n[PROOF STEP]\nby (metis mult_2_right numeral_Bit0 of_real_mult of_real_numeral real_sqrt_four real_sqrt_mult)", "meta": {"llama_tokens": 136, "file": "Isabelle_Marries_Dirac_Quantum_Prisoners_Dilemma", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9196425267730007, "lm_q2_score": 0.7931059560743422, "lm_q1q2_score": 0.7293739654429245}} {"text": "[STATEMENT]\nlemma mtx_idx_unique_conv[simp]: \n fixes M :: nat\n assumes \"j (i=i' \\ j=j')\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (i * M + j = i' * M + j') = (i = i' \\ j = j')\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nj < M\nj' < M\n\ngoal (1 subgoal):\n 1. (i * M + j = i' * M + j') = (i = i' \\ j = j')\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\j < M; j' < M; i * M + j = i' * M + j'\\ \\ i = i'\n 2. \\j < M; j' < M; i * M + j = i' * M + j'\\ \\ j = j'\n[PROOF STEP]\nsubgoal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j < M; j' < M; i * M + j = i' * M + j'\\ \\ i = i'\n[PROOF STEP]\nby (metis add_right_cancel div_if div_mult_self3 linorder_neqE_nat not_less0)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j < M; j' < M; i * M + j = i' * M + j'\\ \\ j = j'\n[PROOF STEP]\nsubgoal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j < M; j' < M; i * M + j = i' * M + j'\\ \\ j = j'\n[PROOF STEP]\nusing \\\\j < M; j' < M; i * M + j = i' * M + j'\\ \\ i = i'\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\j < M; j' < M; i * M + j = i' * M + j'\\ \\ i = i'\n\ngoal (1 subgoal):\n 1. \\j < M; j' < M; i * M + j = i' * M + j'\\ \\ j = j'\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 774, "file": "Refine_Imperative_HOL_IICF_Impl_IICF_Array_Matrix", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.831143054132195, "lm_q1q2_score": 0.729308731751657}} {"text": "[STATEMENT]\ntheorem pr_index_enumerator_inj1: \"pr_index_enumerator n1 m = pr_index_enumerator n2 m \\ n1 = n2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pr_index_enumerator n1 m = pr_index_enumerator n2 m \\ n1 = n2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. pr_index_enumerator n1 m = pr_index_enumerator n2 m \\ n1 = n2\n[PROOF STEP]\nassume A: \"pr_index_enumerator n1 m = pr_index_enumerator n2 m\"\n[PROOF STATE]\nproof (state)\nthis:\npr_index_enumerator n1 m = pr_index_enumerator n2 m\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n1 m = pr_index_enumerator n2 m \\ n1 = n2\n[PROOF STEP]\ndefine f where \"f x = pr_index_enumerator x m\" for x\n[PROOF STATE]\nproof (state)\nthis:\nf ?x = pr_index_enumerator ?x m\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n1 m = pr_index_enumerator n2 m \\ n1 = n2\n[PROOF STEP]\nhave f_mono: \"\\ x y. (x < y \\ f x < f y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x y. x < y \\ f x < f y\n[PROOF STEP]\nproof (rule allI, rule allI)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x y. x < y \\ f x < f y\n[PROOF STEP]\nfix x y\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x y. x < y \\ f x < f y\n[PROOF STEP]\nshow \"x < y \\ f x < f y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x < y \\ f x < f y\n[PROOF STEP]\nby (unfold f_def, simp add: pr_index_enumerator_mono1)\n[PROOF STATE]\nproof (state)\nthis:\nx < y \\ f x < f y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\x y. x < y \\ f x < f y\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n1 m = pr_index_enumerator n2 m \\ n1 = n2\n[PROOF STEP]\nfrom f_mono\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x y. x < y \\ f x < f y\n[PROOF STEP]\nhave \"\\ x y. (f x = f y \\ x = y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x y. x < y \\ f x < f y\n\ngoal (1 subgoal):\n 1. \\x y. f x = f y \\ x = y\n[PROOF STEP]\nby (rule f_mono_inj)\n[PROOF STATE]\nproof (state)\nthis:\n\\x y. f x = f y \\ x = y\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n1 m = pr_index_enumerator n2 m \\ n1 = n2\n[PROOF STEP]\nwith A f_def\n[PROOF STATE]\nproof (chain)\npicking this:\npr_index_enumerator n1 m = pr_index_enumerator n2 m\nf ?x = pr_index_enumerator ?x m\n\\x y. f x = f y \\ x = y\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\npr_index_enumerator n1 m = pr_index_enumerator n2 m\nf ?x = pr_index_enumerator ?x m\n\\x y. f x = f y \\ x = y\n\ngoal (1 subgoal):\n 1. n1 = n2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nn1 = n2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1252, "file": "Recursion-Theory-I_PRecFun2", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.853912747375134, "lm_q1q2_score": 0.7291669785424479}} {"text": "[STATEMENT]\nlemma eventually_mono_sequentially:\n assumes \"eventually P sequentially\"\n assumes \"\\x. P (x+k) \\ Q (x+k)\"\n shows \"eventually Q sequentially\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eventually Q sequentially\n[PROOF STEP]\nusing sequentially_offset[OF assms(1),of k]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F i in sequentially. P (i + k)\n\ngoal (1 subgoal):\n 1. eventually Q sequentially\n[PROOF STEP]\napply (subst eventually_sequentially_seg[symmetric,of _ k])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F i in sequentially. P (i + k) \\ \\\\<^sub>F n in sequentially. Q (n + k)\n[PROOF STEP]\napply (elim eventually_mono)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. P (x + k) \\ Q (x + k)\n[PROOF STEP]\nby fact", "meta": {"llama_tokens": 320, "file": "Irrational_Series_Erdos_Straus_Irrational_Series_Erdos_Straus", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.85391273808085, "lm_q2_score": 0.8539127548105611, "lm_q1q2_score": 0.7291669785424477}} {"text": "[STATEMENT]\nlemma decseq_harm_diff_ln: \"decseq (\\n. harm (Suc n) - ln (Suc n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. decseq (\\n. harm (Suc n) - ln (real (Suc n)))\n[PROOF STEP]\nproof (rule decseq_SucI)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. harm (Suc (Suc n)) - ln (real (Suc (Suc n))) \\ harm (Suc n) - ln (real (Suc n))\n[PROOF STEP]\nfix m :: nat\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. harm (Suc (Suc n)) - ln (real (Suc (Suc n))) \\ harm (Suc n) - ln (real (Suc n))\n[PROOF STEP]\ndefine n where \"n = Suc m\"\n[PROOF STATE]\nproof (state)\nthis:\nn = Suc m\n\ngoal (1 subgoal):\n 1. \\n. harm (Suc (Suc n)) - ln (real (Suc (Suc n))) \\ harm (Suc n) - ln (real (Suc n))\n[PROOF STEP]\nhave \"n > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < n\n[PROOF STEP]\nby (simp add: n_def)\n[PROOF STATE]\nproof (state)\nthis:\n0 < n\n\ngoal (1 subgoal):\n 1. \\n. harm (Suc (Suc n)) - ln (real (Suc (Suc n))) \\ harm (Suc n) - ln (real (Suc n))\n[PROOF STEP]\nhave \"convex_on {0<..} (\\x :: real. -ln x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex_on {0<..} (\\x. - ln x)\n[PROOF STEP]\nby (rule convex_on_realI[where f' = \"\\x. -1/x\"])\n (auto intro!: derivative_eq_intros simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nconvex_on {0<..} (\\x. - ln x)\n\ngoal (1 subgoal):\n 1. \\n. harm (Suc (Suc n)) - ln (real (Suc (Suc n))) \\ harm (Suc n) - ln (real (Suc n))\n[PROOF STEP]\nhence \"(-1 / (n + 1)) * (real n - real (n + 1)) \\ (- ln (real n)) - (-ln (real (n + 1)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nconvex_on {0<..} (\\x. - ln x)\n\ngoal (1 subgoal):\n 1. - 1 / real (n + 1) * (real n - real (n + 1)) \\ - ln (real n) - - ln (real (n + 1))\n[PROOF STEP]\nusing \\n > 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\nconvex_on {0<..} (\\x. - ln x)\n0 < n\n\ngoal (1 subgoal):\n 1. - 1 / real (n + 1) * (real n - real (n + 1)) \\ - ln (real n) - - ln (real (n + 1))\n[PROOF STEP]\nby (intro convex_on_imp_above_tangent[where A = \"{0<..}\"])\n (auto intro!: derivative_eq_intros simp: interior_open)\n[PROOF STATE]\nproof (state)\nthis:\n- 1 / real (n + 1) * (real n - real (n + 1)) \\ - ln (real n) - - ln (real (n + 1))\n\ngoal (1 subgoal):\n 1. \\n. harm (Suc (Suc n)) - ln (real (Suc (Suc n))) \\ harm (Suc n) - ln (real (Suc n))\n[PROOF STEP]\nthus \"harm (Suc n) - ln (Suc n) \\ harm n - ln n\"\n[PROOF STATE]\nproof (prove)\nusing this:\n- 1 / real (n + 1) * (real n - real (n + 1)) \\ - ln (real n) - - ln (real (n + 1))\n\ngoal (1 subgoal):\n 1. harm (Suc n) - ln (real (Suc n)) \\ harm n - ln (real n)\n[PROOF STEP]\nby (auto simp: harm_Suc field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nharm (Suc n) - ln (real (Suc n)) \\ harm n - ln (real n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1377, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.855851143290548, "lm_q2_score": 0.8519527982093666, "lm_q1q2_score": 0.729144776377068}} {"text": "[STATEMENT]\nlemma minus_q_mult_cancel: \n assumes \"[a = e + b - q * c - d] (mod q)\" \n and \"e + b - d > 0\" \n and \"e + b - q * c - d > 0\"\n shows \"[a = e + b - d] (mod q)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [a = e + b - d] (mod q)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. [a = e + b - d] (mod q)\n[PROOF STEP]\nhave \"a mod q = (e + b - q * c - d) mod q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a mod q = (e + b - q * c - d) mod q\n[PROOF STEP]\nusing assms(1) cong_def\n[PROOF STATE]\nproof (prove)\nusing this:\n[a = e + b - q * c - d] (mod q)\n[?b = ?c] (mod ?a) = (?b mod ?a = ?c mod ?a)\n\ngoal (1 subgoal):\n 1. a mod q = (e + b - q * c - d) mod q\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\na mod q = (e + b - q * c - d) mod q\n\ngoal (1 subgoal):\n 1. [a = e + b - d] (mod q)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na mod q = (e + b - q * c - d) mod q\n[PROOF STEP]\nhave \"a mod q = (e + b - d) mod q\"\n[PROOF STATE]\nproof (prove)\nusing this:\na mod q = (e + b - q * c - d) mod q\n\ngoal (1 subgoal):\n 1. a mod q = (e + b - d) mod q\n[PROOF STEP]\nby (metis (no_types) add_cancel_left_left assms(3) diff_commute diff_is_0_eq' linordered_semidom_class.add_diff_inverse mod_add_left_eq mod_mult_self1_is_0 nat_less_le)\n[PROOF STATE]\nproof (state)\nthis:\na mod q = (e + b - d) mod q\n\ngoal (1 subgoal):\n 1. [a = e + b - d] (mod q)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na mod q = (e + b - d) mod q\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na mod q = (e + b - d) mod q\n\ngoal (1 subgoal):\n 1. [a = e + b - d] (mod q)\n[PROOF STEP]\nusing cong_def\n[PROOF STATE]\nproof (prove)\nusing this:\na mod q = (e + b - d) mod q\n[?b = ?c] (mod ?a) = (?b mod ?a = ?c mod ?a)\n\ngoal (1 subgoal):\n 1. [a = e + b - d] (mod q)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n[a = e + b - d] (mod q)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 949, "file": "Multi_Party_Computation_Secure_Multiplication", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527982093668, "lm_q2_score": 0.8558511414521923, "lm_q1q2_score": 0.7291447748108758}} {"text": "[STATEMENT]\nlemma power_diff_inverse:\n assumes nz: \"(a::'a::field) ~= 0\"\n shows \"m <= n ==> (inverse a) ^ (n-m) = (a^m) / (a^n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. m \\ n \\ inverse a ^ (n - m) = a ^ m / a ^ n\n[PROOF STEP]\napply (induct n m rule: diff_induct)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x. 0 \\ x \\ inverse a ^ (x - 0) = a ^ 0 / a ^ x\n 2. \\y. Suc y \\ 0 \\ inverse a ^ (0 - Suc y) = a ^ Suc y / a ^ 0\n 3. \\x y. \\y \\ x \\ inverse a ^ (x - y) = a ^ y / a ^ x; Suc y \\ Suc x\\ \\ inverse a ^ (Suc x - Suc y) = a ^ Suc y / a ^ Suc x\n[PROOF STEP]\napply (simp add: power_inverse\n nonzero_inverse_eq_divide [THEN sym] nz)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\y. Suc y \\ 0 \\ inverse a ^ (0 - Suc y) = a ^ Suc y / a ^ 0\n 2. \\x y. \\y \\ x \\ inverse a ^ (x - y) = a ^ y / a ^ x; Suc y \\ Suc x\\ \\ inverse a ^ (Suc x - Suc y) = a ^ Suc y / a ^ Suc x\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x y. \\y \\ x \\ inverse a ^ (x - y) = a ^ y / a ^ x; Suc y \\ Suc x\\ \\ inverse a ^ (Suc x - Suc y) = a ^ Suc y / a ^ Suc x\n[PROOF STEP]\napply (simp add: nz)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 654, "file": "FFT_FFT", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.855851143290548, "lm_q2_score": 0.8519527963298946, "lm_q1q2_score": 0.7291447747685197}} {"text": "[STATEMENT]\nlemma linepath_of_real: \"(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. linepath (of_real a) (of_real b) x = of_real ((1 - x) * a + x * b)\n[PROOF STEP]\nby (simp add: scaleR_conv_of_real linepath_def)", "meta": {"llama_tokens": 127, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110454379297, "lm_q2_score": 0.8175744695262777, "lm_q1q2_score": 0.7291219423915904}} {"text": "[STATEMENT]\nlemma real_sqrt_zero [simp]: \"sqrt 0 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt 0 = 0\n[PROOF STEP]\nunfolding sqrt_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. root 2 0 = 0\n[PROOF STEP]\nby (rule real_root_zero)", "meta": {"llama_tokens": 119, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110339361275, "lm_q2_score": 0.8175744673038222, "lm_q1q2_score": 0.7291219310060003}} {"text": "[STATEMENT]\nlemma dvd_power_minus_1_conv2:\n fixes x::nat\n assumes x: \"x > 1\" \n and a: \"a > 0\" \n and a_dvd_b: \"a dvd b\" \n and b0: \"b > 0\"\n shows \"x ^ a - 1 dvd x^b - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x ^ a - 1 dvd x ^ b - 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x ^ a - 1 dvd x ^ b - 1\n[PROOF STEP]\ndefine q where q[simp]: \"q = b div a\"\n[PROOF STATE]\nproof (state)\nthis:\nq = b div a\n\ngoal (1 subgoal):\n 1. x ^ a - 1 dvd x ^ b - 1\n[PROOF STEP]\nhave b: \"b = a * q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. b = a * q\n[PROOF STEP]\nusing a_dvd_b\n[PROOF STATE]\nproof (prove)\nusing this:\na dvd b\n\ngoal (1 subgoal):\n 1. b = a * q\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nb = a * q\n\ngoal (1 subgoal):\n 1. x ^ a - 1 dvd x ^ b - 1\n[PROOF STEP]\nhave \"x^b - 1 = ((x ^ a) - 1) * sum ((^) (x ^ a)) {..k. x ^ b - 1 = (x ^ a - 1) * k\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx ^ a - 1 dvd x ^ b - 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 930, "file": "Berlekamp_Zassenhaus_Distinct_Degree_Factorization", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637648915617, "lm_q2_score": 0.8479677622198946, "lm_q1q2_score": 0.7289671589766271}} {"text": "[STATEMENT]\ntheorem (in group) group_inverse_equality:\n assumes eq: \"x' * x = 1\"\n shows \"inverse x = x'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\nhave \"inverse x = 1 * inverse x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse x = (1::'a) * inverse x\n[PROOF STEP]\nby (simp only: group_left_one)\n[PROOF STATE]\nproof (state)\nthis:\ninverse x = (1::'a) * inverse x\n\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ninverse x = (1::'a) * inverse x\n\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\nhave \"\\ = (x' * x) * inverse x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1::'a) * inverse x = x' * x * inverse x\n[PROOF STEP]\nby (simp only: eq)\n[PROOF STATE]\nproof (state)\nthis:\n(1::'a) * inverse x = x' * x * inverse x\n\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(1::'a) * inverse x = x' * x * inverse x\n\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\nhave \"\\ = x' * (x * inverse x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x' * x * inverse x = x' * (x * inverse x)\n[PROOF STEP]\nby (simp only: group_assoc)\n[PROOF STATE]\nproof (state)\nthis:\nx' * x * inverse x = x' * (x * inverse x)\n\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx' * x * inverse x = x' * (x * inverse x)\n\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\nhave \"\\ = x' * 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x' * (x * inverse x) = x' * (1::'a)\n[PROOF STEP]\nby (simp only: group_right_inverse)\n[PROOF STATE]\nproof (state)\nthis:\nx' * (x * inverse x) = x' * (1::'a)\n\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx' * (x * inverse x) = x' * (1::'a)\n\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\nhave \"\\ = x'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x' * (1::'a) = x'\n[PROOF STEP]\nby (simp only: group_right_one)\n[PROOF STATE]\nproof (state)\nthis:\nx' * (1::'a) = x'\n\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ninverse x = x'\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse x = x'\n\ngoal (1 subgoal):\n 1. inverse x = x'\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ninverse x = x'\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1127, "file": null, "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677430095496, "lm_q2_score": 0.8596637451167997, "lm_q1q2_score": 0.7289671256938293}} {"text": "[STATEMENT]\nlemma sin_exp_eq: \"sin z = (exp(\\ * z) - exp(-(\\ * z))) / (2*\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin z = (exp (\\ * z) - exp (- (\\ * z))) / (2 * \\)\n[PROOF STEP]\nby (simp add: exp_Euler exp_minus_Euler)", "meta": {"llama_tokens": 119, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9161096044278532, "lm_q2_score": 0.7956580976404297, "lm_q1q2_score": 0.7289100250891922}} {"text": "[STATEMENT]\nlemma prime_gr4 :\n assumes \"is_prime p\"\n and \"p mod 4 = 1\"\n shows \"p > 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 4 < p\n[PROOF STEP]\nproof(rule ccontr, drule leI)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. p \\ 4 \\ False\n[PROOF STEP]\nassume \"p \\ 4\"\n[PROOF STATE]\nproof (state)\nthis:\np \\ 4\n\ngoal (1 subgoal):\n 1. p \\ 4 \\ False\n[PROOF STEP]\nthus False\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ 4\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby (metis assms dvd_imp_mod_0 dvd_triv_left is_prime_def mod_less mult.right_neutral \n order_less_le zero_neq_one)\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 330, "file": "Involutions2Squares_Involutions2Squares", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392817460332, "lm_q2_score": 0.824461932846258, "lm_q1q2_score": 0.7288567349403522}} {"text": "[STATEMENT]\nlemma m_s_anti_mono_rep: assumes \"\\x. S1 x \\ S2 x\"\nshows \"(\\x\\X. m (S2 x)) \\ (\\x\\X. m (S1 x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\X. m (S2 x)) \\ (\\x\\X. m (S1 x))\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x\\X. m (S2 x)) \\ (\\x\\X. m (S1 x))\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x. S1 x \\ S2 x\n[PROOF STEP]\nhave \"\\x. m(S1 x) \\ m(S2 x)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. S1 x \\ S2 x\n\ngoal (1 subgoal):\n 1. \\x. m (S2 x) \\ m (S1 x)\n[PROOF STEP]\nby (metis m_anti_mono)\n[PROOF STATE]\nproof (state)\nthis:\n\\x. m (S2 x) \\ m (S1 x)\n\ngoal (1 subgoal):\n 1. (\\x\\X. m (S2 x)) \\ (\\x\\X. m (S1 x))\n[PROOF STEP]\nthus \"(\\x\\X. m (S2 x)) \\ (\\x\\X. m (S1 x))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. m (S2 x) \\ m (S1 x)\n\ngoal (1 subgoal):\n 1. (\\x\\X. m (S2 x)) \\ (\\x\\X. m (S1 x))\n[PROOF STEP]\nby (metis sum_mono)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\X. m (S2 x)) \\ (\\x\\X. m (S1 x))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 654, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473647220786, "lm_q2_score": 0.8354835432479661, "lm_q1q2_score": 0.7288318672210281}} {"text": "[STATEMENT]\nlemma equal_card_inter_fin_eq_sets: \"finite A \\ finite B \\ card A = card B \\ \n card (A \\ B) = card A \\ A = B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite B; card A = card B; card (A \\ B) = card A\\ \\ A = B\n[PROOF STEP]\nby (metis Int_lower1 Int_lower2 card_subset_eq)", "meta": {"llama_tokens": 148, "file": "Fishers_Inequality_Set_Multiset_Extras", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.901920681802153, "lm_q2_score": 0.8080672227971211, "lm_q1q2_score": 0.7288125405271517}} {"text": "[STATEMENT]\nlemma sin_cos_squared_add2 [simp]: \"(cos x)\\<^sup>2 + (sin x)\\<^sup>2 = 1\"\n for x :: \"'a::{real_normed_field,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cos x)\\<^sup>2 + (sin x)\\<^sup>2 = (1::'a)\n[PROOF STEP]\nby (subst add.commute, rule sin_cos_squared_add)", "meta": {"llama_tokens": 137, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206712569267, "lm_q2_score": 0.8080672181749422, "lm_q1q2_score": 0.7288125278370613}} {"text": "[STATEMENT]\nlemma \\_incr_upper:\n \"\\ f j (l + 1) = \\ f j l + f (of_int l)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ f j (l + 1) = \\ f j l + f (of_int l)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ f j (l + 1) = \\ f j l + f (of_int l)\n[PROOF STEP]\nhave \"{l.. f j (l + 1) = \\ f j l + f (of_int l)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{l.. f l (l + 1) = f (of_int l)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{l.. f l (l + 1) = f (of_int l)\n[PROOF STEP]\nby (simp add: \\_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\ f l (l + 1) = f (of_int l)\n\ngoal (1 subgoal):\n 1. \\ f j (l + 1) = \\ f j l + f (of_int l)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\ f l (l + 1) = f (of_int l)\n\ngoal (1 subgoal):\n 1. \\ f j (l + 1) = \\ f j l + f (of_int l)\n[PROOF STEP]\nhave \"\\ f j (l + 1) = \\ f j l + \\ f l (l + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ f j (l + 1) = \\ f j l + \\ f l (l + 1)\n[PROOF STEP]\nby (simp add: \\_concat)\n[PROOF STATE]\nproof (state)\nthis:\n\\ f j (l + 1) = \\ f j l + \\ f l (l + 1)\n\ngoal (1 subgoal):\n 1. \\ f j (l + 1) = \\ f j l + f (of_int l)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ f l (l + 1) = f (of_int l)\n\\ f j (l + 1) = \\ f j l + \\ f l (l + 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ f l (l + 1) = f (of_int l)\n\\ f j (l + 1) = \\ f j l + \\ f l (l + 1)\n\ngoal (1 subgoal):\n 1. \\ f j (l + 1) = \\ f j l + f (of_int l)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\ f j (l + 1) = \\ f j l + f (of_int l)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1068, "file": "Discrete_Summation_Discrete_Summation", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382236515258, "lm_q2_score": 0.8459424431344437, "lm_q1q2_score": 0.7288117497694805}} {"text": "[STATEMENT]\nlemma rank_nullity_theorem_matrices:\n fixes A::\"'a::{field}^'cols::{finite, wellorder}^'rows\"\n shows \"ncols A = vec.dim (null_space A) + vec.dim (col_space A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ncols A = vec.dim (null_space A) + vec.dim (col_space A)\n[PROOF STEP]\nusing vec.rank_nullity_theorem[OF matrix_vector_mul_linear_gen, of A]\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite_dimensional_vector_space.dimension cart_basis = vec.dim {x. A *v x = 0} + vec.dim (range ((*v) A))\n\ngoal (1 subgoal):\n 1. ncols A = vec.dim (null_space A) + vec.dim (col_space A)\n[PROOF STEP]\napply (subst (2 3) matrix_of_matrix_vector_mul [of A, symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite_dimensional_vector_space.dimension cart_basis = vec.dim {x. A *v x = 0} + vec.dim (range ((*v) A)) \\ ncols A = vec.dim (null_space (matrix ((*v) A))) + vec.dim (col_space (matrix ((*v) A)))\n[PROOF STEP]\nunfolding null_space_eq_ker[OF matrix_vector_mul_linear_gen]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite_dimensional_vector_space.dimension cart_basis = vec.dim {x. A *v x = 0} + vec.dim (range ((*v) A)) \\ ncols A = vec.dim {x. A *v x = 0} + vec.dim (col_space (matrix ((*v) A)))\n[PROOF STEP]\nunfolding col_space_eq_range [OF matrix_vector_mul_linear_gen]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite_dimensional_vector_space.dimension cart_basis = vec.dim {x. A *v x = 0} + vec.dim (range ((*v) A)) \\ ncols A = vec.dim {x. A *v x = 0} + vec.dim (range ((*v) A))\n[PROOF STEP]\nunfolding vec.dimension_def ncols_def card_cart_basis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. CARD('cols) = vec.dim {x. A *v x = 0} + vec.dim (range ((*v) A)) \\ CARD('cols) = vec.dim {x. A *v x = 0} + vec.dim (range ((*v) A))\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 768, "file": "Rank_Nullity_Theorem_Dim_Formula", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382058759128, "lm_q2_score": 0.8459424314825853, "lm_q1q2_score": 0.7288117246938138}} {"text": "[STATEMENT]\nlemma real_sqrt_diff_squares_triangle_ineq:\n fixes a b c d :: real\n shows \"sqrt ((a - c)^2 + (b - d)^2) \\ sqrt (a^2 + b^2) + sqrt (c^2 + d^2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nhave \"sqrt ((a - c)^2 + (b - d)^2) \\ sqrt (a^2 + b^2) + sqrt ((-c)^2 + (-d)^2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt ((- c)\\<^sup>2 + (- d)\\<^sup>2)\n[PROOF STEP]\nby (metis diff_conv_add_uminus real_sqrt_sum_squares_triangle_ineq)\n[PROOF STATE]\nproof (state)\nthis:\nsqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt ((- c)\\<^sup>2 + (- d)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. sqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt ((- c)\\<^sup>2 + (- d)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. sqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nhave \"... = sqrt (a^2 + b^2) + sqrt (c^2 + d^2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt ((- c)\\<^sup>2 + (- d)\\<^sup>2) = sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt ((- c)\\<^sup>2 + (- d)\\<^sup>2) = sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n\ngoal (1 subgoal):\n 1. sqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n\ngoal (1 subgoal):\n 1. sqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nsqrt ((a - c)\\<^sup>2 + (b - d)\\<^sup>2) \\ sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1373, "file": "Impossible_Geometry_Impossible_Geometry", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213853793453, "lm_q2_score": 0.8104788995148792, "lm_q1q2_score": 0.7287189109525454}} {"text": "[STATEMENT]\nlemma abs_power_eq_0 [simp]: \"\\a\\^n = 0 \\ a = 0 \\ n \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a\\ ^ n = (0::'a)) = (a = (0::'a) \\ n \\ 0)\n[PROOF STEP]\napply (induct n, force)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. (\\a\\ ^ n = (0::'a)) = (a = (0::'a) \\ n \\ 0) \\ (\\a\\ ^ Suc n = (0::'a)) = (a = (0::'a) \\ Suc n \\ 0)\n[PROOF STEP]\napply (unfold power_Suc)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. (\\a\\ ^ n = (0::'a)) = (a = (0::'a) \\ n \\ 0) \\ (\\a\\ * \\a\\ ^ n = (0::'a)) = (a = (0::'a) \\ Suc n \\ 0)\n[PROOF STEP]\napply (subst real_mult_eq_0_iff, auto)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\n.", "meta": {"llama_tokens": 426, "file": "LLL_Basis_Reduction_Norms", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213664574069, "lm_q2_score": 0.810478913248044, "lm_q1q2_score": 0.7287189079644955}} {"text": "[STATEMENT]\nlemma equivalentFormulaeSymmetry: \n fixes formula1 :: Formula and formula2 :: Formula\n shows \"equivalentFormulae formula1 formula2 = equivalentFormulae formula2 formula1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. equivalentFormulae formula1 formula2 = equivalentFormulae formula2 formula1\n[PROOF STEP]\nunfolding equivalentFormulae_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\valuation. model valuation formula1 = model valuation formula2) = (\\valuation. model valuation formula2 = model valuation formula1)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 181, "file": "SATSolverVerification_CNF", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588052782737, "lm_q2_score": 0.8198933293122507, "lm_q1q2_score": 0.7286874158151821}} {"text": "[STATEMENT]\ntheorem singleton_intersection:\n assumes A:\"card A = 1\"\n assumes B:\"card B = 1\"\n assumes noteq:\"A \\ B\"\n shows \"A \\ B = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A \\ B = {}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncard A = 1\ncard B = 1\nA \\ B\n\ngoal (1 subgoal):\n 1. A \\ B = {}\n[PROOF STEP]\nby(auto simp:card_Suc_eq)", "meta": {"llama_tokens": 177, "file": "Secondary_Sylow_GroupAction", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.888758793492457, "lm_q2_score": 0.8198933381139645, "lm_q1q2_score": 0.7286874139746702}} {"text": "[STATEMENT]\nlemma perzeta_conv_hurwitz_zeta_multiplication':\n fixes k :: nat and a :: int and s :: complex\n assumes \"k > 0\" \"s \\ 1\"\n shows \"perzeta (a / k) s = k powr -s *\n (\\n=1..k. exp (2 * pi * n * a / k * \\) * hurwitz_zeta (n / k) s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. perzeta (real_of_int a / real k) s = of_nat k powr - s * (\\n = 1..k. exp (complex_of_real (2 * pi * real n * real_of_int a / real k) * \\) * hurwitz_zeta (real n / real k) s)\n[PROOF STEP]\nusing perzeta_conv_hurwitz_zeta_multiplication[of k s a] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < k; s \\ 1\\ \\ of_nat k powr s * perzeta (real_of_int a / real k) s = (\\n = 1..k. exp (complex_of_real (2 * pi * real n * real_of_int a / real k) * \\) * hurwitz_zeta (real n / real k) s)\n0 < k\ns \\ 1\n\ngoal (1 subgoal):\n 1. perzeta (real_of_int a / real k) s = of_nat k powr - s * (\\n = 1..k. exp (complex_of_real (2 * pi * real n * real_of_int a / real k) * \\) * hurwitz_zeta (real n / real k) s)\n[PROOF STEP]\nby (simp add: powr_minus field_simps)", "meta": {"llama_tokens": 484, "file": "Zeta_Function_Zeta_Function", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099070060380481, "lm_q2_score": 0.8006920044739461, "lm_q1q2_score": 0.7285552645494917}} {"text": "[STATEMENT]\nlemma euclid2:\n \"PRE (\\s::nat store. s ''x'' = x \\ s ''y'' = y)\n (WHILE (\\s. s ''y'' \\ 0) INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) \n DO\n (''z'' ::= (\\s. s ''y''));\n (''y'' ::= (\\s. s ''x'' mod s ''y''));\n (''x'' ::= (\\s. s ''z''))\n OD)\n POST (\\s. s ''x'' = gcd x y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rdom \\\\s. s ''x'' = x \\ s ''y'' = y\\ \\ wp (WHILE (\\s. s ''y'' \\ 0) INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) DO (''z'' ::= (\\s. s ''y'')) ; (''y'' ::= (\\s. s ''x'' mod s ''y'')) ; (''x'' ::= (\\s. s ''z'')) OD) \\\\s. s ''x'' = gcd x y\\\n[PROOF STEP]\napply hoare\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. rdom \\\\s. s ''x'' = x \\ s ''y'' = y\\ \\ rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\\n 2. rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\ ; rel_ad \\\\s. s ''y'' \\ 0\\ \\ rdom \\\\s. s ''x'' = gcd x y\\\n 3. rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\ ; rdom \\\\s. s ''y'' \\ 0\\ \\ wp (''z'' ::= (\\s. s ''y'')) (wp (''y'' ::= (\\s. s ''x'' mod s ''y'')) (wp (''x'' ::= (\\s. s ''z'')) \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\))\n[PROOF STEP]\nusing gcd_red_nat\n[PROOF STATE]\nproof (prove)\nusing this:\ngcd ?x ?y = gcd ?y (?x mod ?y)\n\ngoal (3 subgoals):\n 1. rdom \\\\s. s ''x'' = x \\ s ''y'' = y\\ \\ rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\\n 2. rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\ ; rel_ad \\\\s. s ''y'' \\ 0\\ \\ rdom \\\\s. s ''x'' = gcd x y\\\n 3. rdom \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\ ; rdom \\\\s. s ''y'' \\ 0\\ \\ wp (''z'' ::= (\\s. s ''y'')) (wp (''y'' ::= (\\s. s ''x'' mod s ''y'')) (wp (''x'' ::= (\\s. s ''z'')) \\\\s. gcd (s ''x'') (s ''y'') = gcd x y\\))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 1088, "file": "Algebraic_VCs_AVC_KAD_VC_KAD_Examples2", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099069962657176, "lm_q2_score": 0.8006919997179627, "lm_q1q2_score": 0.7285552523973622}} {"text": "[STATEMENT]\nlemma cos_double: \"cos(2*x) = (cos x)\\<^sup>2 - (sin x)\\<^sup>2\"\n for x :: \"'a::{real_normed_field,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos ((2::'a) * x) = (cos x)\\<^sup>2 - (sin x)\\<^sup>2\n[PROOF STEP]\nusing cos_add [where x=x and y=x]\n[PROOF STATE]\nproof (prove)\nusing this:\ncos (x + x) = cos x * cos x - sin x * sin x\n\ngoal (1 subgoal):\n 1. cos ((2::'a) * x) = (cos x)\\<^sup>2 - (sin x)\\<^sup>2\n[PROOF STEP]\nby (simp add: power2_eq_square)", "meta": {"llama_tokens": 233, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772286044095, "lm_q2_score": 0.8333245953120233, "lm_q1q2_score": 0.7283900527982244}} {"text": "[STATEMENT]\nlemma cis_mult: \"cis a * cis b = cis (a + b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cis a * cis b = cis (a + b)\n[PROOF STEP]\nby (simp add: complex_eq_iff cos_add sin_add)", "meta": {"llama_tokens": 91, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361676202372, "lm_q2_score": 0.7981867849406659, "lm_q1q2_score": 0.7282944910963796}} {"text": "[STATEMENT]\nlemma dist_midpoint:\n fixes a b :: \"'a::real_normed_vector\" shows\n \"dist a (midpoint a b) = (dist a b) / 2\" (is ?t1)\n \"dist b (midpoint a b) = (dist a b) / 2\" (is ?t2)\n \"dist (midpoint a b) a = (dist a b) / 2\" (is ?t3)\n \"dist (midpoint a b) b = (dist a b) / 2\" (is ?t4)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist a (midpoint a b) = dist a b / 2 &&& dist b (midpoint a b) = dist a b / 2) &&& dist (midpoint a b) a = dist a b / 2 &&& dist (midpoint a b) b = dist a b / 2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. dist a (midpoint a b) = dist a b / 2\n 2. dist b (midpoint a b) = dist a b / 2\n 3. dist (midpoint a b) a = dist a b / 2\n 4. dist (midpoint a b) b = dist a b / 2\n[PROOF STEP]\nhave *: \"\\x y::'a. 2 *\\<^sub>R x = - y \\ norm x = (norm y) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x y. 2 *\\<^sub>R x = - y \\ norm x = norm y / 2\n[PROOF STEP]\nunfolding equation_minus_iff\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x y. y = - (2 *\\<^sub>R x) \\ norm x = norm y / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n2 *\\<^sub>R ?x = - ?y \\ norm ?x = norm ?y / 2\n\ngoal (4 subgoals):\n 1. dist a (midpoint a b) = dist a b / 2\n 2. dist b (midpoint a b) = dist a b / 2\n 3. dist (midpoint a b) a = dist a b / 2\n 4. dist (midpoint a b) b = dist a b / 2\n[PROOF STEP]\nhave **: \"\\x y::'a. 2 *\\<^sub>R x = y \\ norm x = (norm y) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x y. 2 *\\<^sub>R x = y \\ norm x = norm y / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n2 *\\<^sub>R ?x = ?y \\ norm ?x = norm ?y / 2\n\ngoal (4 subgoals):\n 1. dist a (midpoint a b) = dist a b / 2\n 2. dist b (midpoint a b) = dist a b / 2\n 3. dist (midpoint a b) a = dist a b / 2\n 4. dist (midpoint a b) b = dist a b / 2\n[PROOF STEP]\nnote scaleR_right_distrib [simp]\n[PROOF STATE]\nproof (state)\nthis:\n?a *\\<^sub>R (?x + ?y) = ?a *\\<^sub>R ?x + ?a *\\<^sub>R ?y\n\ngoal (4 subgoals):\n 1. dist a (midpoint a b) = dist a b / 2\n 2. dist b (midpoint a b) = dist a b / 2\n 3. dist (midpoint a b) a = dist a b / 2\n 4. dist (midpoint a b) b = dist a b / 2\n[PROOF STEP]\nshow ?t1\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist a (midpoint a b) = dist a b / 2\n[PROOF STEP]\nunfolding midpoint_def dist_norm\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (a - (a + b) /\\<^sub>R 2) = norm (a - b) / 2\n[PROOF STEP]\napply (rule **)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 *\\<^sub>R (a - (a + b) /\\<^sub>R 2) = a - b\n[PROOF STEP]\napply (simp add: scaleR_right_diff_distrib)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 *\\<^sub>R a - a = a\n[PROOF STEP]\napply (simp add: scaleR_2)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\ndist a (midpoint a b) = dist a b / 2\n\ngoal (3 subgoals):\n 1. dist b (midpoint a b) = dist a b / 2\n 2. dist (midpoint a b) a = dist a b / 2\n 3. dist (midpoint a b) b = dist a b / 2\n[PROOF STEP]\nshow ?t2\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist b (midpoint a b) = dist a b / 2\n[PROOF STEP]\nunfolding midpoint_def dist_norm\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (b - (a + b) /\\<^sub>R 2) = norm (a - b) / 2\n[PROOF STEP]\napply (rule *)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 *\\<^sub>R (b - (a + b) /\\<^sub>R 2) = - (a - b)\n[PROOF STEP]\napply (simp add: scaleR_right_diff_distrib)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 *\\<^sub>R b - b = b\n[PROOF STEP]\napply (simp add: scaleR_2)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\ndist b (midpoint a b) = dist a b / 2\n\ngoal (2 subgoals):\n 1. dist (midpoint a b) a = dist a b / 2\n 2. dist (midpoint a b) b = dist a b / 2\n[PROOF STEP]\nshow ?t3\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist (midpoint a b) a = dist a b / 2\n[PROOF STEP]\nunfolding midpoint_def dist_norm\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm ((a + b) /\\<^sub>R 2 - a) = norm (a - b) / 2\n[PROOF STEP]\napply (rule *)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 *\\<^sub>R ((a + b) /\\<^sub>R 2 - a) = - (a - b)\n[PROOF STEP]\napply (simp add: scaleR_right_diff_distrib)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a - 2 *\\<^sub>R a = - a\n[PROOF STEP]\napply (simp add: scaleR_2)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\ndist (midpoint a b) a = dist a b / 2\n\ngoal (1 subgoal):\n 1. dist (midpoint a b) b = dist a b / 2\n[PROOF STEP]\nshow ?t4\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist (midpoint a b) b = dist a b / 2\n[PROOF STEP]\nunfolding midpoint_def dist_norm\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm ((a + b) /\\<^sub>R 2 - b) = norm (a - b) / 2\n[PROOF STEP]\napply (rule **)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 *\\<^sub>R ((a + b) /\\<^sub>R 2 - b) = a - b\n[PROOF STEP]\napply (simp add: scaleR_right_diff_distrib)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. b - 2 *\\<^sub>R b = - b\n[PROOF STEP]\napply (simp add: scaleR_2)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\ndist (midpoint a b) b = dist a b / 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2506, "file": null, "length": 32, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.893309411735131, "lm_q2_score": 0.8152324848629214, "lm_q1q2_score": 0.7282548514802655}} {"text": "[STATEMENT]\nlemma (in encoding) indRelR_iff_exists_source_target_relation:\n fixes Pred :: \"(('procS, 'procT) Proc \\ ('procS, 'procT) Proc) \\ bool\"\n shows \"(\\(P, Q) \\ indRelR. Pred (P, Q))\n = (\\Rel. (\\S. (SourceTerm S, TargetTerm (\\S\\)) \\ Rel) \\ (\\(P, Q) \\ Rel. Pred (P, Q)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(P, Q)\\indRelR. Pred (P, Q)) = (\\Rel. (\\S. (SourceTerm S, TargetTerm (\\S\\)) \\ Rel) \\ (\\(P, Q)\\Rel. Pred (P, Q)))\n[PROOF STEP]\nusing indRelR_impl_exists_source_target_relation(2)[where PredB=\"Pred\"]\n source_target_relation_impl_indRelR[where Pred=\"Pred\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\(P, Q)\\indRelR. Pred (P, Q) \\ \\Rel. (\\S. (SourceTerm S, TargetTerm (\\S\\)) \\ Rel) \\ (\\(P, Q)\\Rel. Pred (P, Q))\n(\\S. (SourceTerm S, TargetTerm (\\S\\)) \\ ?Rel) \\ (\\(P, Q)\\?Rel. Pred (P, Q)) \\ \\(P, Q)\\indRelR. Pred (P, Q)\n\ngoal (1 subgoal):\n 1. (\\(P, Q)\\indRelR. Pred (P, Q)) = (\\Rel. (\\S. (SourceTerm S, TargetTerm (\\S\\)) \\ Rel) \\ (\\(P, Q)\\Rel. Pred (P, Q)))\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 567, "file": "Encodability_Process_Calculi_SourceTargetRelation", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094032139577, "lm_q2_score": 0.8152324803738429, "lm_q1q2_score": 0.7282548405233921}} {"text": "[STATEMENT]\nlemma transpose_matrix_diff: \"transpose_matrix ((A::('a::group_add) matrix)-B) = transpose_matrix A - transpose_matrix B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. transpose_matrix (A - B) = transpose_matrix A - transpose_matrix B\n[PROOF STEP]\nby (simp add: diff_matrix_def transpose_combine_matrix)", "meta": {"llama_tokens": 117, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9304582516374121, "lm_q2_score": 0.782662489091802, "lm_q1q2_score": 0.7282347712225432}} {"text": "[STATEMENT]\nlemma matrix_vector_mult_zero_eq:\n assumes P: \"invertible P\"\n shows \"((P**A)*v x = 0) = (A *v x = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (P ** A *v x = 0) = (A *v x = 0)\n[PROOF STEP]\nproof (rule iffI)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. P ** A *v x = 0 \\ A *v x = 0\n 2. A *v x = 0 \\ P ** A *v x = 0\n[PROOF STEP]\nassume \"P ** A *v x = 0\"\n[PROOF STATE]\nproof (state)\nthis:\nP ** A *v x = 0\n\ngoal (2 subgoals):\n 1. P ** A *v x = 0 \\ A *v x = 0\n 2. A *v x = 0 \\ P ** A *v x = 0\n[PROOF STEP]\nhence \"matrix_inv P *v (P ** A *v x) = matrix_inv P *v 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nP ** A *v x = 0\n\ngoal (1 subgoal):\n 1. matrix_inv P *v (P ** A *v x) = matrix_inv P *v 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_inv P *v (P ** A *v x) = matrix_inv P *v 0\n\ngoal (2 subgoals):\n 1. P ** A *v x = 0 \\ A *v x = 0\n 2. A *v x = 0 \\ P ** A *v x = 0\n[PROOF STEP]\nhence \"matrix_inv P *v (P ** A *v x) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix_inv P *v (P ** A *v x) = matrix_inv P *v 0\n\ngoal (1 subgoal):\n 1. matrix_inv P *v (P ** A *v x) = 0\n[PROOF STEP]\nby (metis matrix_vector_mult_0_right)\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_inv P *v (P ** A *v x) = 0\n\ngoal (2 subgoals):\n 1. P ** A *v x = 0 \\ A *v x = 0\n 2. A *v x = 0 \\ P ** A *v x = 0\n[PROOF STEP]\nhence \"(matrix_inv P ** P ** A) *v x = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix_inv P *v (P ** A *v x) = 0\n\ngoal (1 subgoal):\n 1. matrix_inv P ** P ** A *v x = 0\n[PROOF STEP]\nby (metis matrix_vector_mul_assoc)\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_inv P ** P ** A *v x = 0\n\ngoal (2 subgoals):\n 1. P ** A *v x = 0 \\ A *v x = 0\n 2. A *v x = 0 \\ P ** A *v x = 0\n[PROOF STEP]\nthus \"A *v x = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix_inv P ** P ** A *v x = 0\n\ngoal (1 subgoal):\n 1. A *v x = 0\n[PROOF STEP]\nby (metis assms matrix_inv_left matrix_mul_lid)\n[PROOF STATE]\nproof (state)\nthis:\nA *v x = 0\n\ngoal (1 subgoal):\n 1. A *v x = 0 \\ P ** A *v x = 0\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. A *v x = 0 \\ P ** A *v x = 0\n[PROOF STEP]\nassume \"A *v x = 0\"\n[PROOF STATE]\nproof (state)\nthis:\nA *v x = 0\n\ngoal (1 subgoal):\n 1. A *v x = 0 \\ P ** A *v x = 0\n[PROOF STEP]\nthus \"P ** A *v x = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA *v x = 0\n\ngoal (1 subgoal):\n 1. P ** A *v x = 0\n[PROOF STEP]\nby (metis matrix_vector_mul_assoc matrix_vector_mult_0_right)\n[PROOF STATE]\nproof (state)\nthis:\nP ** A *v x = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1312, "file": "Rank_Nullity_Theorem_Miscellaneous", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970748488297, "lm_q2_score": 0.8267117983401363, "lm_q1q2_score": 0.7281653337210077}} {"text": "[STATEMENT]\nlemma convolution_of_aezfun_is_aezfun :\n fixes f g :: \"'a::group_add \\ 'b::{comm_monoid_add,times}\"\n shows \"f \\ aezfun_set \\ g \\ aezfun_set \\ convolution f g \\ aezfun_set\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ aezfun_set; g \\ aezfun_set\\ \\ convolution f g \\ aezfun_set\n[PROOF STEP]\nusing supp_convolution_subset_sum_supp[of f g]\n finite_set_plus[of \"supp f\" \"supp g\"] finite_subset\n[PROOF STATE]\nproof (prove)\nusing this:\nsupp (convolution f g) \\ supp f + supp g\n\\finite (supp f); finite (supp g)\\ \\ finite (supp f + supp g)\n\\?A \\ ?B; finite ?B\\ \\ finite ?A\n\ngoal (1 subgoal):\n 1. \\f \\ aezfun_set; g \\ aezfun_set\\ \\ convolution f g \\ aezfun_set\n[PROOF STEP]\nunfolding aezfun_set_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsupp (convolution f g) \\ supp f + supp g\n\\finite (supp f); finite (supp g)\\ \\ finite (supp f + supp g)\n\\?A \\ ?B; finite ?B\\ \\ finite ?A\n\ngoal (1 subgoal):\n 1. \\f \\ {f. finite (supp f)}; g \\ {f. finite (supp f)}\\ \\ convolution f g \\ {f. finite (supp f)}\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 541, "file": "Rep_Fin_Groups_Rep_Fin_Groups", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314768368161, "lm_q2_score": 0.822189121808099, "lm_q1q2_score": 0.7281565661860716}} {"text": "[STATEMENT]\nlemma Bseq_mult:\n fixes f g :: \"nat \\ 'a::real_normed_field\"\n assumes \"Bseq f\" and \"Bseq g\"\n shows \"Bseq (\\x. f x * g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Bseq (\\x. f x * g x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Bseq (\\x. f x * g x)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nBseq f\nBseq g\n[PROOF STEP]\nobtain K1 K2 where K: \"norm (f x) \\ K1\" \"K1 > 0\" \"norm (g x) \\ K2\" \"K2 > 0\"\n for x\n[PROOF STATE]\nproof (prove)\nusing this:\nBseq f\nBseq g\n\ngoal (1 subgoal):\n 1. (\\K1 K2. \\\\x. norm (f x) \\ K1; 0 < K1; \\x. norm (g x) \\ K2; 0 < K2\\ \\ thesis) \\ thesis\n[PROOF STEP]\nunfolding Bseq_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\K>0. \\n. norm (f n) \\ K\n\\K>0. \\n. norm (g n) \\ K\n\ngoal (1 subgoal):\n 1. (\\K1 K2. \\\\x. norm (f x) \\ K1; 0 < K1; \\x. norm (g x) \\ K2; 0 < K2\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nnorm (f ?x) \\ K1\n0 < K1\nnorm (g ?x) \\ K2\n0 < K2\n\ngoal (1 subgoal):\n 1. Bseq (\\x. f x * g x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (f ?x) \\ K1\n0 < K1\nnorm (g ?x) \\ K2\n0 < K2\n[PROOF STEP]\nhave \"norm (f x * g x) \\ K1 * K2\" for x\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (f ?x) \\ K1\n0 < K1\nnorm (g ?x) \\ K2\n0 < K2\n\ngoal (1 subgoal):\n 1. norm (f x * g x) \\ K1 * K2\n[PROOF STEP]\nby (auto simp: norm_mult intro!: mult_mono)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (f ?x * g ?x) \\ K1 * K2\n\ngoal (1 subgoal):\n 1. Bseq (\\x. f x * g x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (f ?x * g ?x) \\ K1 * K2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (f ?x * g ?x) \\ K1 * K2\n\ngoal (1 subgoal):\n 1. Bseq (\\x. f x * g x)\n[PROOF STEP]\nby (rule BseqI')\n[PROOF STATE]\nproof (state)\nthis:\nBseq (\\x. f x * g x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1037, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357598021707, "lm_q2_score": 0.8397339656668287, "lm_q1q2_score": 0.7280793769536288}} {"text": "[STATEMENT]\nlemma L2_set_mono2:\n assumes a1: \"finite L\" and a2: \"K \\ L\"\n shows \"L2_set f K \\ L2_set f L\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. L2_set f K \\ L2_set f L\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. L2_set f K \\ L2_set f L\n[PROOF STEP]\nhave \"(\\i\\K. (f i)\\<^sup>2) \\ (\\i\\L. (f i)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\K. (f i)\\<^sup>2) \\ (\\i\\L. (f i)\\<^sup>2)\n[PROOF STEP]\napply (rule sum_mono2)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. finite L\n 2. K \\ L\n 3. \\b. b \\ L - K \\ 0 \\ (f b)\\<^sup>2\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite L\nK \\ L\n\ngoal (3 subgoals):\n 1. finite L\n 2. K \\ L\n 3. \\b. b \\ L - K \\ 0 \\ (f b)\\<^sup>2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\K. (f i)\\<^sup>2) \\ (\\i\\L. (f i)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. L2_set f K \\ L2_set f L\n[PROOF STEP]\nhence \"sqrt (\\i\\K. (f i)\\<^sup>2) \\ sqrt (\\i\\L. (f i)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i\\K. (f i)\\<^sup>2) \\ (\\i\\L. (f i)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. sqrt (\\i\\K. (f i)\\<^sup>2) \\ sqrt (\\i\\L. (f i)\\<^sup>2)\n[PROOF STEP]\nby (rule real_sqrt_le_mono)\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (\\i\\K. (f i)\\<^sup>2) \\ sqrt (\\i\\L. (f i)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. L2_set f K \\ L2_set f L\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (\\i\\K. (f i)\\<^sup>2) \\ sqrt (\\i\\L. (f i)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. L2_set f K \\ L2_set f L\n[PROOF STEP]\nunfolding L2_set_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (\\i\\K. (f i)\\<^sup>2) \\ sqrt (\\i\\L. (f i)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. sqrt (\\i\\K. (f i)\\<^sup>2) \\ sqrt (\\i\\L. (f i)\\<^sup>2)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nL2_set f K \\ L2_set f L\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1122, "file": "Complex_Bounded_Operators_extra_Extra_General", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869916479466, "lm_q2_score": 0.8311430520409023, "lm_q1q2_score": 0.7279042731759946}} {"text": "[STATEMENT]\nlemma p_LIMSEQ:\n assumes x: \"x > 0\"\n shows \"(\\n. p n x) \\ P x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. p n x) \\ P x\n[PROOF STEP]\nproof (rule Lim_transform_eventually)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ?f \\ P x\n 2. \\\\<^sub>F xa in sequentially. ?f xa = p xa x\n[PROOF STEP]\nfrom D_summable[OF x]\n[PROOF STATE]\nproof (chain)\npicking this:\nsummable (\\n. D (real n + x))\n[PROOF STEP]\nhave \"(\\n. D (real n + x)) sums P x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\n. D (real n + x))\n\ngoal (1 subgoal):\n 1. (\\n. D (real n + x)) sums P x\n[PROOF STEP]\nunfolding P_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\n. D (real n + x))\n\ngoal (1 subgoal):\n 1. (\\n. D (real n + x)) sums (\\n. D (real n + x))\n[PROOF STEP]\nby (simp add: sums_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. D (real n + x)) sums P x\n\ngoal (2 subgoals):\n 1. ?f \\ P x\n 2. \\\\<^sub>F xa in sequentially. ?f xa = p xa x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n. D (real n + x)) sums P x\n[PROOF STEP]\nshow \"(\\n. \\r P x\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. D (real n + x)) sums P x\n\ngoal (1 subgoal):\n 1. (\\n. \\r P x\n[PROOF STEP]\nby (simp add: sums_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. \\r P x\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F xa in sequentially. (\\rn. \\r P x\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F xa in sequentially. (\\rn. (\\r\\<^sub>F n in sequentially. (\\r\\<^sub>F n in sequentially. (\\r lotteries_on outcomes\" \n and \"q \\ lotteries_on outcomes\"\n shows \"p \\[relation] q \\ \n (\\z\\outcomes. (pmf p z) * (u z)) > (\\z\\outcomes. (pmf q z) * (u z))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (p \\ q \\ (q, p) \\ relation) = ((\\z\\outcomes. pmf q z * u z) < (\\z\\outcomes. pmf p z * u z))\n[PROOF STEP]\nusing sum_equals_pmf_expectation[of p, OF assms(1)] assms(1) assms(2)\n sum_equals_pmf_expectation[of q, OF assms(2)] strict_prefernce_iff_strict_utility\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\z\\outcomes. pmf p z * u z) = measure_pmf.expectation p u\np \\ \\

outcomes\nq \\ \\

outcomes\n(\\z\\outcomes. pmf q z * u z) = measure_pmf.expectation q u\n\\?x \\ \\

outcomes; ?y \\ \\

. \\y \\ \\

. x \\[\\] y \\ \n measure_pmf.expectation x ocU \\ measure_pmf.expectation y ocU\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\local.\\

. \\y\\local.\\

. x \\[\\] y = (measure_pmf.expectation y ocU \\ measure_pmf.expectation x ocU)\n[PROOF STEP]\nusing ordinal_utility_expected_value\n[PROOF STATE]\nproof (prove)\nusing this:\nordinal_utility local.\\

\\ (\\x. measure_pmf.expectation x ocU)\n\ngoal (1 subgoal):\n 1. \\x\\local.\\

. \\y\\local.\\

. x \\[\\] y = (measure_pmf.expectation y ocU \\ measure_pmf.expectation x ocU)\n[PROOF STEP]\nby (meson ordinal_utility.util_def)", "meta": {"llama_tokens": 336, "file": "Neumann_Morgenstern_Utility_Neumann_Morgenstern_Utility_Theorem", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099069962657176, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7193090632272857}} {"text": "[STATEMENT]\nlemma Leaf_in_set_random_bst_iff:\n assumes \"finite A\"\n shows \"Leaf \\ set_pmf (random_bst A) \\ A = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\ \\ set_pmf (random_bst A)) = (A = {})\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\\\ \\ set_pmf (random_bst A) \\ A = {}\n 2. A = {} \\ \\\\ \\ set_pmf (random_bst A)\n[PROOF STEP]\nassume \"Leaf \\ set_pmf (random_bst A)\"\n[PROOF STATE]\nproof (state)\nthis:\n\\\\ \\ set_pmf (random_bst A)\n\ngoal (2 subgoals):\n 1. \\\\ \\ set_pmf (random_bst A) \\ A = {}\n 2. A = {} \\ \\\\ \\ set_pmf (random_bst A)\n[PROOF STEP]\nfrom size_random_bst[OF this] and assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A \\ size \\\\ = card A\nfinite A\n[PROOF STEP]\nshow \"A = {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A \\ size \\\\ = card A\nfinite A\n\ngoal (1 subgoal):\n 1. A = {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA = {}\n\ngoal (1 subgoal):\n 1. A = {} \\ \\\\ \\ set_pmf (random_bst A)\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 515, "file": "Random_BSTs_Random_BSTs", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473813156294, "lm_q2_score": 0.8244619285331332, "lm_q1q2_score": 0.7192172043503123}} {"text": "[STATEMENT]\nlemma lead_coeff_p: \"lead_coeff (p s) = (\\i = 0..i = 0..x = 0..i = 0.. 0\"\n shows \"summable (\\n. 1 / (real (Suc n) + d) ^ 2)\"\n \"(\\n. 1 / (real (Suc n) + d) ^ 2) \\ 1 / d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. summable (\\n. 1 / (real (Suc n) + d)\\<^sup>2) &&& (\\n. 1 / (real (Suc n) + d)\\<^sup>2) \\ 1 / d\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. summable (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n 2. (\\n. 1 / (real (Suc n) + d)\\<^sup>2) \\ 1 / d\n[PROOF STEP]\nshow *: \"summable (\\n. 1 / (real (Suc n) + d) ^ 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. summable (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n[PROOF STEP]\nproof (rule summable_comparison_test, intro allI exI impI)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\n. ?N3 \\ n \\ norm (1 / (real (Suc n) + d)\\<^sup>2) \\ ?g n\n 2. summable ?g\n[PROOF STEP]\nfix n :: nat\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\n. ?N3 \\ n \\ norm (1 / (real (Suc n) + d)\\<^sup>2) \\ ?g n\n 2. summable ?g\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < d\n[PROOF STEP]\nshow \"norm (1 / (real (Suc n) + d) ^ 2) \\ 1 / real (Suc n) ^ 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n\ngoal (1 subgoal):\n 1. norm (1 / (real (Suc n) + d)\\<^sup>2) \\ 1 / (real (Suc n))\\<^sup>2\n[PROOF STEP]\nunfolding norm_divide norm_one norm_power\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n\ngoal (1 subgoal):\n 1. 1 / (norm (real (Suc n) + d))\\<^sup>2 \\ 1 / (real (Suc n))\\<^sup>2\n[PROOF STEP]\nby (intro divide_left_mono power_mono) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nnorm (1 / (real (Suc n) + d)\\<^sup>2) \\ 1 / (real (Suc n))\\<^sup>2\n\ngoal (1 subgoal):\n 1. summable (\\n. 1 / (real (Suc n))\\<^sup>2)\n[PROOF STEP]\nqed (insert inverse_squares_sums, simp add: sums_iff)\n[PROOF STATE]\nproof (state)\nthis:\nsummable (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (\\n. 1 / (real (Suc n) + d)\\<^sup>2) \\ 1 / d\n[PROOF STEP]\nshow \"(\\n. 1 / (real (Suc n) + d) ^ 2) \\ 1 / d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. 1 / (real (Suc n) + d)\\<^sup>2) \\ 1 / d\n[PROOF STEP]\nproof (rule sums_le)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\n. ?f n \\ ?g n\n 2. ?f sums (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n 3. ?g sums (1 / d)\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\n. ?f n \\ ?g n\n 2. ?f sums (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n 3. ?g sums (1 / d)\n[PROOF STEP]\nhave \"1 / (real (Suc n) + d) ^ 2 \\ 1 / ((real n + d) * (real (Suc n) + d))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 / (real (Suc n) + d)\\<^sup>2 \\ 1 / ((real n + d) * (real (Suc n) + d))\n[PROOF STEP]\nunfolding power2_eq_square\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 / ((real (Suc n) + d) * (real (Suc n) + d)) \\ 1 / ((real n + d) * (real (Suc n) + d))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n\ngoal (1 subgoal):\n 1. 1 / ((real (Suc n) + d) * (real (Suc n) + d)) \\ 1 / ((real n + d) * (real (Suc n) + d))\n[PROOF STEP]\nby (intro divide_left_mono mult_mono mult_pos_pos add_nonneg_pos) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n1 / (real (Suc n) + d)\\<^sup>2 \\ 1 / ((real n + d) * (real (Suc n) + d))\n\ngoal (3 subgoals):\n 1. \\n. ?f n \\ ?g n\n 2. ?f sums (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n 3. ?g sums (1 / d)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 / (real (Suc n) + d)\\<^sup>2 \\ 1 / ((real n + d) * (real (Suc n) + d))\n\ngoal (3 subgoals):\n 1. \\n. ?f n \\ ?g n\n 2. ?f sums (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n 3. ?g sums (1 / d)\n[PROOF STEP]\nhave \"\\ = 1 / (real n + d) - 1 / (real (Suc n) + d)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 / ((real n + d) * (real (Suc n) + d)) = 1 / (real n + d) - 1 / (real (Suc n) + d)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n\ngoal (1 subgoal):\n 1. 1 / ((real n + d) * (real (Suc n) + d)) = 1 / (real n + d) - 1 / (real (Suc n) + d)\n[PROOF STEP]\nby (simp add: divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\n1 / ((real n + d) * (real (Suc n) + d)) = 1 / (real n + d) - 1 / (real (Suc n) + d)\n\ngoal (3 subgoals):\n 1. \\n. ?f n \\ ?g n\n 2. ?f sums (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n 3. ?g sums (1 / d)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n1 / (real (Suc n) + d)\\<^sup>2 \\ 1 / (real n + d) - 1 / (real (Suc n) + d)\n[PROOF STEP]\nshow \"1 / (real (Suc n) + d)\\<^sup>2 \\ 1 / (real n + d) - 1 / (real (Suc n) + d)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 / (real (Suc n) + d)\\<^sup>2 \\ 1 / (real n + d) - 1 / (real (Suc n) + d)\n\ngoal (1 subgoal):\n 1. 1 / (real (Suc n) + d)\\<^sup>2 \\ 1 / (real n + d) - 1 / (real (Suc n) + d)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n1 / (real (Suc n) + d)\\<^sup>2 \\ 1 / (real n + d) - 1 / (real (Suc n) + d)\n\ngoal (2 subgoals):\n 1. (\\n. 1 / (real (Suc (?n7 n)) + d)\\<^sup>2) sums (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n 2. (\\n. 1 / (real (?n7 n) + d) - 1 / (real (Suc (?n7 n)) + d)) sums (1 / d)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\n. 1 / (real (Suc (?n7 n)) + d)\\<^sup>2) sums (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n 2. (\\n. 1 / (real (?n7 n) + d) - 1 / (real (Suc (?n7 n)) + d)) sums (1 / d)\n[PROOF STEP]\nshow \"(\\n. 1 / (real (Suc n) + d)\\<^sup>2) sums (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. 1 / (real (Suc n) + d)\\<^sup>2) sums (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n[PROOF STEP]\nusing *\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (\\n. 1 / (real (Suc n) + d)\\<^sup>2) sums (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n[PROOF STEP]\nby (simp add: sums_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. 1 / (real (Suc n) + d)\\<^sup>2) sums (\\n. 1 / (real (Suc n) + d)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (\\n. 1 / (real n + d) - 1 / (real (Suc n) + d)) sums (1 / d)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\n. 1 / (real n + d) - 1 / (real (Suc n) + d)) sums (1 / d)\n[PROOF STEP]\nhave \"(\\n. 1 / (real n + d) - 1 / (real (Suc n) + d)) sums (1 / (real 0 + d) - 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. 1 / (real n + d) - 1 / (real (Suc n) + d)) sums (1 / (real 0 + d) - 0)\n[PROOF STEP]\nby (intro telescope_sums' real_tendsto_divide_at_top[OF tendsto_const],\n subst add.commute, rule filterlim_tendsto_add_at_top[OF tendsto_const \n filterlim_real_sequentially])\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. 1 / (real n + d) - 1 / (real (Suc n) + d)) sums (1 / (real 0 + d) - 0)\n\ngoal (1 subgoal):\n 1. (\\n. 1 / (real n + d) - 1 / (real (Suc n) + d)) sums (1 / d)\n[PROOF STEP]\nthus \"(\\n. 1 / (real n + d) - 1 / (real (Suc n) + d)) sums (1 / d)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. 1 / (real n + d) - 1 / (real (Suc n) + d)) sums (1 / (real 0 + d) - 0)\n\ngoal (1 subgoal):\n 1. (\\n. 1 / (real n + d) - 1 / (real (Suc n) + d)) sums (1 / d)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. 1 / (real n + d) - 1 / (real (Suc n) + d)) sums (1 / d)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. 1 / (real (Suc n) + d)\\<^sup>2) \\ 1 / d\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3846, "file": "Dirichlet_Series_Arithmetic_Summatory_Asymptotics", "length": 34, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473680407889, "lm_q2_score": 0.8244619199068831, "lm_q1q2_score": 0.7192171858806252}} {"text": "[STATEMENT]\nlemma valid_pratt_tree_imp_prime:\n assumes \"valid_pratt_tree t\"\n shows \"prime (pratt_tree_number t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prime (pratt_tree_number t)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nvalid_pratt_tree t\n\ngoal (1 subgoal):\n 1. prime (pratt_tree_number t)\n[PROOF STEP]\nproof (induction t rule: valid_pratt_tree.induct)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n a ts. \\\\x. \\x \\ set ts; valid_pratt_tree x\\ \\ prime (pratt_tree_number x); valid_pratt_tree (Pratt_Node (n, a, ts))\\ \\ prime (pratt_tree_number (Pratt_Node (n, a, ts)))\n[PROOF STEP]\ncase (1 n a ts)\n[PROOF STATE]\nproof (state)\nthis:\n\\?x \\ set ts; valid_pratt_tree ?x\\ \\ prime (pratt_tree_number ?x)\nvalid_pratt_tree (Pratt_Node (n, a, ts))\n\ngoal (1 subgoal):\n 1. \\n a ts. \\\\x. \\x \\ set ts; valid_pratt_tree x\\ \\ prime (pratt_tree_number x); valid_pratt_tree (Pratt_Node (n, a, ts))\\ \\ prime (pratt_tree_number (Pratt_Node (n, a, ts)))\n[PROOF STEP]\nfrom 1\n[PROOF STATE]\nproof (chain)\npicking this:\n\\?x \\ set ts; valid_pratt_tree ?x\\ \\ prime (pratt_tree_number ?x)\nvalid_pratt_tree (Pratt_Node (n, a, ts))\n[PROOF STEP]\nhave \"prime_factors (n - 1) \\ set (map pratt_tree_number ts)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x \\ set ts; valid_pratt_tree ?x\\ \\ prime (pratt_tree_number ?x)\nvalid_pratt_tree (Pratt_Node (n, a, ts))\n\ngoal (1 subgoal):\n 1. prime_factors (n - 1) \\ set (map pratt_tree_number ts)\n[PROOF STEP]\nby (intro check_prime_factors_subset_correct) (auto simp: list.pred_set)\n[PROOF STATE]\nproof (state)\nthis:\nprime_factors (n - 1) \\ set (map pratt_tree_number ts)\n\ngoal (1 subgoal):\n 1. \\n a ts. \\\\x. \\x \\ set ts; valid_pratt_tree x\\ \\ prime (pratt_tree_number x); valid_pratt_tree (Pratt_Node (n, a, ts))\\ \\ prime (pratt_tree_number (Pratt_Node (n, a, ts)))\n[PROOF STEP]\nwith 1\n[PROOF STATE]\nproof (chain)\npicking this:\n\\?x \\ set ts; valid_pratt_tree ?x\\ \\ prime (pratt_tree_number ?x)\nvalid_pratt_tree (Pratt_Node (n, a, ts))\nprime_factors (n - 1) \\ set (map pratt_tree_number ts)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x \\ set ts; valid_pratt_tree ?x\\ \\ prime (pratt_tree_number ?x)\nvalid_pratt_tree (Pratt_Node (n, a, ts))\nprime_factors (n - 1) \\ set (map pratt_tree_number ts)\n\ngoal (1 subgoal):\n 1. prime (pratt_tree_number (Pratt_Node (n, a, ts)))\n[PROOF STEP]\nby (intro lehmers_theorem[where a = a]) auto\n[PROOF STATE]\nproof (state)\nthis:\nprime (pratt_tree_number (Pratt_Node (n, a, ts)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1249, "file": "Pratt_Certificate_Pratt_Certificate", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240930029117, "lm_q2_score": 0.8311430415844384, "lm_q1q2_score": 0.719124984310577}} {"text": "[STATEMENT]\nlemma (in equivalence) set_eq_trans_aux:\n assumes \"A \\ carrier S\" \"B \\ carrier S\" \"C \\ carrier S\"\n and \"A {.=} B\" \"B {.=} C\"\n shows \"\\a. a \\ A \\ a .\\ C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a. a \\ A \\ a .\\ C\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier S\nB \\ carrier S\nC \\ carrier S\nA {.=} B\nB {.=} C\n\ngoal (1 subgoal):\n 1. \\a. a \\ A \\ a .\\ C\n[PROOF STEP]\nby (simp add: eq_elem_cong_r subset_iff)", "meta": {"llama_tokens": 252, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240756264639, "lm_q2_score": 0.8311430499496096, "lm_q1q2_score": 0.7191249771060109}} {"text": "[STATEMENT]\nlemma finite_nonempty_carrier_has_maximum:\n assumes \"carrier \\ {}\"\n shows \"\\e \\ carrier. \\m \\ carrier. e \\[relation] m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\e\\carrier. \\m\\carrier. e \\ m\n[PROOF STEP]\nusing finite_nempty_preorder_has_max[of carrier relation] assms\n \\finite carrier\\ reflexivity total transitivity\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite carrier; carrier \\ {}; refl_on carrier relation; trans relation; total_on carrier relation\\ \\ \\x\\carrier. \\y\\carrier. x \\ y\ncarrier \\ {}\nfinite carrier\nrefl_on carrier relation\ntotal_on carrier relation\ntrans relation\n\ngoal (1 subgoal):\n 1. \\e\\carrier. \\m\\carrier. e \\ m\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 332, "file": "First_Welfare_Theorem_Preferences", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872046026642944, "lm_q2_score": 0.8104789132480439, "lm_q1q2_score": 0.7190606221960198}} {"text": "[STATEMENT]\nlemma Least_imp_disj_eq: \"\n \\ \\x. P x; \\x. P x \\ Q x \\ \\\n (LEAST (x::'a::wellorder). P x \\ Q x) = (LEAST x. Q x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x. P x; \\x. P x \\ Q x\\ \\ (LEAST x. P x \\ Q x) = (LEAST x. Q x)\n[PROOF STEP]\napply (subst Least_disj, assumption, blast)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x. P x; \\x. P x \\ Q x\\ \\ min (Least P) (Least Q) = Least Q\n[PROOF STEP]\napply (subst min.commute)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x. P x; \\x. P x \\ Q x\\ \\ min (Least Q) (Least P) = Least Q\n[PROOF STEP]\napply (rule min.absorb_iff1[THEN iffD1])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x. P x; \\x. P x \\ Q x\\ \\ Least Q \\ Least P\n[PROOF STEP]\napply (rule Least_imp_le, assumption, blast)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 491, "file": "List-Infinite_CommonSet_SetInterval2", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.88720460564669, "lm_q2_score": 0.8104789063814616, "lm_q1q2_score": 0.7190606185211252}} {"text": "[STATEMENT]\nlemma conefield_right_vangleI:\n assumes ij: \"i \\ Basis\" \"j \\ Basis\" and ij_neq: \"i \\ j\"\n assumes \"y \\ {y1 .. y2}\" \"y1 < y2\"\n shows \"(i + y *\\<^sub>R j) \\ conefield (i + y1 *\\<^sub>R j) (i + y2 *\\<^sub>R j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i + y *\\<^sub>R j \\ conefield (i + y1 *\\<^sub>R j) (i + y2 *\\<^sub>R j)\n[PROOF STEP]\nunfolding conefield_alt_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i + y *\\<^sub>R j \\ cone hull {i + y1 *\\<^sub>R j--i + y2 *\\<^sub>R j}\n[PROOF STEP]\napply (rule hull_inc)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i + y *\\<^sub>R j \\ {i + y1 *\\<^sub>R j--i + y2 *\\<^sub>R j}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ni \\ Basis\nj \\ Basis\ni \\ j\ny \\ {y1..y2}\ny1 < y2\n\ngoal (1 subgoal):\n 1. i + y *\\<^sub>R j \\ {i + y1 *\\<^sub>R j--i + y2 *\\<^sub>R j}\n[PROOF STEP]\nby (auto simp: in_segment divide_simps inner_Basis algebra_simps\n intro!: exI[where x=\"(y - y1) / (y2 - y1)\"] euclidean_eqI[where 'a='a] )", "meta": {"llama_tokens": 521, "file": "Ordinary_Differential_Equations_IVP_Cones", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045937171068, "lm_q2_score": 0.8104789155369047, "lm_q1q2_score": 0.7190606169752009}} {"text": "[STATEMENT]\nlemma \"finite (A:: nat set) \\ A \\ {}\\ Max A \\ A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; A \\ {}\\ \\ Max A \\ A\n[PROOF STEP]\nproof(induction A rule: finite.induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. {} \\ {} \\ Max {} \\ {}\n 2. \\A a. \\finite A; A \\ {} \\ Max A \\ A; insert a A \\ {}\\ \\ Max (insert a A) \\ insert a A\n[PROOF STEP]\ncase emptyI\n[PROOF STATE]\nproof (state)\nthis:\n{} \\ {}\n\ngoal (2 subgoals):\n 1. {} \\ {} \\ Max {} \\ {}\n 2. \\A a. \\finite A; A \\ {} \\ Max A \\ A; insert a A \\ {}\\ \\ Max (insert a A) \\ insert a A\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{} \\ {}\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n{} \\ {}\n\ngoal (1 subgoal):\n 1. Max {} \\ {}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nMax {} \\ {}\n\ngoal (1 subgoal):\n 1. \\A a. \\finite A; A \\ {} \\ Max A \\ A; insert a A \\ {}\\ \\ Max (insert a A) \\ insert a A\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\A a. \\finite A; A \\ {} \\ Max A \\ A; insert a A \\ {}\\ \\ Max (insert a A) \\ insert a A\n[PROOF STEP]\ncase (insertI A a)\n[PROOF STATE]\nproof (state)\nthis:\nfinite A\nA \\ {} \\ Max A \\ A\ninsert a A \\ {}\n\ngoal (1 subgoal):\n 1. \\A a. \\finite A; A \\ {} \\ Max A \\ A; insert a A \\ {}\\ \\ Max (insert a A) \\ insert a A\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nA \\ {} \\ Max A \\ A\ninsert a A \\ {}\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nA \\ {} \\ Max A \\ A\ninsert a A \\ {}\n\ngoal (1 subgoal):\n 1. Max (insert a A) \\ insert a A\n[PROOF STEP]\nby (meson Max_in finite_insert)\n[PROOF STATE]\nproof (state)\nthis:\nMax (insert a A) \\ insert a A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 955, "file": "Van_Emde_Boas_Trees_VEBT_Pred", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104788995148791, "lm_q2_score": 0.8872045892435128, "lm_q1q2_score": 0.7190605991346325}} {"text": "[STATEMENT]\nlemma gamma_inverse:\n assumes \"\\ \\ {0<..<1}\"\n and \"\\ \\ {0<..<1}\"\n shows \"(1::real) - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\n[PROOF STEP]\nhave \"1 - (\\ - \\) / (1 - \\) = (1 - \\)/(1 - \\) - (\\ - \\) / (1 - \\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\) - (\\ - \\) / (1 - \\)\n[PROOF STEP]\nusing assms(2)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ \\ {0<..<1}\n\ngoal (1 subgoal):\n 1. 1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\) - (\\ - \\) / (1 - \\)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\) - (\\ - \\) / (1 - \\)\n\ngoal (1 subgoal):\n 1. 1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\) - (\\ - \\) / (1 - \\)\n\ngoal (1 subgoal):\n 1. 1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\n[PROOF STEP]\nhave \"... = (1 - \\ - (\\ - \\)) / (1 - \\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1 - \\) / (1 - \\) - (\\ - \\) / (1 - \\) = (1 - \\ - (\\ - \\)) / (1 - \\)\n[PROOF STEP]\nby (metis diff_divide_distrib)\n[PROOF STATE]\nproof (state)\nthis:\n(1 - \\) / (1 - \\) - (\\ - \\) / (1 - \\) = (1 - \\ - (\\ - \\)) / (1 - \\)\n\ngoal (1 subgoal):\n 1. 1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(1 - \\) / (1 - \\) - (\\ - \\) / (1 - \\) = (1 - \\ - (\\ - \\)) / (1 - \\)\n\ngoal (1 subgoal):\n 1. 1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\n[PROOF STEP]\nhave \"... = (1 - \\) / (1 - \\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1 - \\ - (\\ - \\)) / (1 - \\) = (1 - \\) / (1 - \\)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(1 - \\ - (\\ - \\)) / (1 - \\) = (1 - \\) / (1 - \\)\n\ngoal (1 subgoal):\n 1. 1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\n\ngoal (1 subgoal):\n 1. 1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n1 - (\\ - \\) / (1 - \\) = (1 - \\) / (1 - \\)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1455, "file": "Neumann_Morgenstern_Utility_Neumann_Morgenstern_Utility_Theorem", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8459424334245617, "lm_q2_score": 0.8499711775577736, "lm_q1q2_score": 0.7190266862839632}} {"text": "[STATEMENT]\nlemma infsum_cmult_left:\n fixes f :: \"'a \\ 'b :: {t2_space, topological_semigroup_mult, semiring_0}\"\n assumes \\c \\ 0 \\ f summable_on A\\\n shows \"infsum (\\x. f x * c) A = infsum f A * c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n[PROOF STEP]\nproof (cases \\c=0\\)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. c = (0::'b) \\ (\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n 2. c \\ (0::'b) \\ (\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nc = (0::'b)\n\ngoal (2 subgoals):\n 1. c = (0::'b) \\ (\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n 2. c \\ (0::'b) \\ (\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc = (0::'b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nc = (0::'b)\n\ngoal (1 subgoal):\n 1. (\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n\ngoal (1 subgoal):\n 1. c \\ (0::'b) \\ (\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. c \\ (0::'b) \\ (\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nc \\ (0::'b)\n\ngoal (1 subgoal):\n 1. c \\ (0::'b) \\ (\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc \\ (0::'b)\n[PROOF STEP]\nhave \\has_sum f A (infsum f A)\\\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'b)\n\ngoal (1 subgoal):\n 1. has_sum f A (infsum f A)\n[PROOF STEP]\nby (simp add: assms)\n[PROOF STATE]\nproof (state)\nthis:\nhas_sum f A (infsum f A)\n\ngoal (1 subgoal):\n 1. c \\ (0::'b) \\ (\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nhas_sum f A (infsum f A)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nhas_sum f A (infsum f A)\n\ngoal (1 subgoal):\n 1. (\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n[PROOF STEP]\nby (auto intro!: infsumI has_sum_cmult_left)\n[PROOF STATE]\nproof (state)\nthis:\n(\\\\<^sub>\\x\\A. f x * c) = infsum f A * c\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1218, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.851952809486198, "lm_q2_score": 0.8438950986284991, "lm_q1q2_score": 0.7189588001881819}} {"text": "[STATEMENT]\ntheorem Stewart':\n fixes A B C D :: \"'a::euclidean_space\"\n assumes \"between (B, C) D\"\n shows \"(dist A C)\\<^sup>2 * dist B D + (dist B A)\\<^sup>2 * dist C D = dist B C * ((dist A D)\\<^sup>2 + dist B D * dist C D)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist A C)\\<^sup>2 * dist B D + (dist B A)\\<^sup>2 * dist C D = dist B C * ((dist A D)\\<^sup>2 + dist B D * dist C D)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nbetween (B, C) D\n\ngoal (1 subgoal):\n 1. (dist A C)\\<^sup>2 * dist B D + (dist B A)\\<^sup>2 * dist C D = dist B C * ((dist A D)\\<^sup>2 + dist B D * dist C D)\n[PROOF STEP]\nby (auto intro: Stewart)", "meta": {"llama_tokens": 287, "file": "Stewart_Apollonius_Stewart_Apollonius", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9314625031628428, "lm_q2_score": 0.7718434873426302, "lm_q1q2_score": 0.7189432667701043}} {"text": "[STATEMENT]\nlemma totient_double: \"totient (2 * n) = (if even n then 2 * totient n else totient n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. totient (2 * n) = (if even n then 2 * totient n else totient n)\n[PROOF STEP]\nby (simp add: totient_mult ac_simps odd_imp_coprime_nat)", "meta": {"llama_tokens": 116, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361604769413, "lm_q2_score": 0.7879312056025699, "lm_q1q2_score": 0.7189369239599763}} {"text": "[STATEMENT]\nlemma mat_mult [code abstract]: \"vec_nth (m ** m') = mat_mult_row m m'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ($) (m ** m') = mat_mult_row m m'\n[PROOF STEP]\nunfolding matrix_matrix_mult_def mat_mult_row_def[abs_def]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ($) (\\i j. \\k\\UNIV. m $ i $ k * m' $ k $ j) = (\\f. \\c. \\k\\UNIV. m $ f $ k * m' $ k $ c)\n[PROOF STEP]\nusing vec_lambda_beta\n[PROOF STATE]\nproof (prove)\nusing this:\nvec_lambda ?g $ ?i = ?g ?i\n\ngoal (1 subgoal):\n 1. ($) (\\i j. \\k\\UNIV. m $ i $ k * m' $ k $ j) = (\\f. \\c. \\k\\UNIV. m $ f $ k * m' $ k $ c)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 325, "file": "Gauss_Jordan_Code_Matrix", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361652391386, "lm_q2_score": 0.7879311956428947, "lm_q1q2_score": 0.7189369186246923}} {"text": "[STATEMENT]\nlemma exp_first_two_terms: \"exp x = 1 + x + (\\n. inverse (fact (n + 2)) *\\<^sub>R (x ^ (n + 2)))\"\n for x :: \"'a::{real_normed_algebra_1,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp x = (1::'a) + x + (\\n. x ^ (n + 2) /\\<^sub>R fact (n + 2))\n[PROOF STEP]\nusing exp_first_terms[of x 2]\n[PROOF STATE]\nproof (prove)\nusing this:\nexp x = (\\n<2. x ^ n /\\<^sub>R fact n) + (\\n. x ^ (n + 2) /\\<^sub>R fact (n + 2))\n\ngoal (1 subgoal):\n 1. exp x = (1::'a) + x + (\\n. x ^ (n + 2) /\\<^sub>R fact (n + 2))\n[PROOF STEP]\nby (simp add: eval_nat_numeral)", "meta": {"llama_tokens": 292, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9124361604769414, "lm_q2_score": 0.7879311906630568, "lm_q1q2_score": 0.7189369103286245}} {"text": "[STATEMENT]\nlemma sum_of_2_powers: \"(\\i=0.. Suc (Suc n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc (Suc n) \\ nth_prime n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Suc (Suc n) \\ nth_prime n\n[PROOF STEP]\nhave \"n = card {q. prime q \\ q < nth_prime n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n = card {q. prime q \\ q < nth_prime n}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn = card {q. prime q \\ q < nth_prime n}\n\ngoal (1 subgoal):\n 1. Suc (Suc n) \\ nth_prime n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nn = card {q. prime q \\ q < nth_prime n}\n\ngoal (1 subgoal):\n 1. Suc (Suc n) \\ nth_prime n\n[PROOF STEP]\nhave \"\\ \\ card {2.. q < nth_prime n} \\ card {2.. q < nth_prime n} \\ card {2.. nth_prime n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {q. prime q \\ q < nth_prime n} \\ card {2.. nth_prime n\n[PROOF STEP]\nhave \"\\ = nth_prime n - 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {2.. nth_prime n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nn \\ nth_prime n - 2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\ nth_prime n - 2\n\ngoal (1 subgoal):\n 1. Suc (Suc n) \\ nth_prime n\n[PROOF STEP]\nusing nth_prime_ge_2[of n]\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\ nth_prime n - 2\n2 \\ nth_prime n\n\ngoal (1 subgoal):\n 1. Suc (Suc n) \\ nth_prime n\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\nSuc (Suc n) \\ nth_prime n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1003, "file": "Prime_Number_Theorem_Prime_Counting_Functions", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127641048443, "lm_q2_score": 0.8418256512199033, "lm_q1q2_score": 0.7188456687275482}} {"text": "[STATEMENT]\nlemma card_of_Func_squared:\n fixes A :: \"'a set\"\n shows \"|Func (UNIV :: bool set) A| =o |A \\ A|\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. |Func UNIV A| =o |A \\ A|\n[PROOF STEP]\nproof (rule ordIso_symmetric)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. |A \\ A| =o |Func UNIV A|\n[PROOF STEP]\ndefine f where \"f = (\\(x::'a,y) b. if A = {} then undefined else if b then x else y)\"\n[PROOF STATE]\nproof (state)\nthis:\nf = (\\(x, y) b. if A = {} then undefined else if b then x else y)\n\ngoal (1 subgoal):\n 1. |A \\ A| =o |Func UNIV A|\n[PROOF STEP]\nhave \"Func (UNIV :: bool set) A \\ f ` (A \\ A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Func UNIV A \\ f ` (A \\ A)\n[PROOF STEP]\nunfolding f_def Func_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {f. (\\a\\UNIV. f a \\ A) \\ (\\a. a \\ UNIV \\ f a = undefined)} \\ (\\(x, y) b. if A = {} then undefined else if b then x else y) ` (A \\ A)\n[PROOF STEP]\nby (auto simp: image_iff fun_eq_iff split: option.splits if_split_asm) blast\n[PROOF STATE]\nproof (state)\nthis:\nFunc UNIV A \\ f ` (A \\ A)\n\ngoal (1 subgoal):\n 1. |A \\ A| =o |Func UNIV A|\n[PROOF STEP]\nhence \"bij_betw f (A \\ A) (Func (UNIV :: bool set) A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nFunc UNIV A \\ f ` (A \\ A)\n\ngoal (1 subgoal):\n 1. bij_betw f (A \\ A) (Func UNIV A)\n[PROOF STEP]\nunfolding bij_betw_def inj_on_def f_def Func_def\n[PROOF STATE]\nproof (prove)\nusing this:\n{f. (\\a\\UNIV. f a \\ A) \\ (\\a. a \\ UNIV \\ f a = undefined)} \\ (\\(x, y) b. if A = {} then undefined else if b then x else y) ` (A \\ A)\n\ngoal (1 subgoal):\n 1. (\\x\\A \\ A. \\y\\A \\ A. (case x of (x, y) \\ \\b. if A = {} then undefined else if b then x else y) = (case y of (x, y) \\ \\b. if A = {} then undefined else if b then x else y) \\ x = y) \\ (\\(x, y) b. if A = {} then undefined else if b then x else y) ` (A \\ A) = {f. (\\a\\UNIV. f a \\ A) \\ (\\a. a \\ UNIV \\ f a = undefined)}\n[PROOF STEP]\nby (auto simp: fun_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw f (A \\ A) (Func UNIV A)\n\ngoal (1 subgoal):\n 1. |A \\ A| =o |Func UNIV A|\n[PROOF STEP]\nthus \"|A \\ A| =o |Func (UNIV :: bool set) A|\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw f (A \\ A) (Func UNIV A)\n\ngoal (1 subgoal):\n 1. |A \\ A| =o |Func UNIV A|\n[PROOF STEP]\nusing card_of_ordIso\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw f (A \\ A) (Func UNIV A)\n(\\f. bij_betw f ?A ?B) = (|?A| =o |?B|)\n\ngoal (1 subgoal):\n 1. |A \\ A| =o |Func UNIV A|\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n|A \\ A| =o |Func UNIV A|\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1300, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127529517043, "lm_q2_score": 0.8418256412990657, "lm_q1q2_score": 0.7188456508670191}} {"text": "[STATEMENT]\nlemma bitwise_inner_prod_with_zero:\n assumes \"m < 2^n\"\n shows \"(0 \\\\<^bsub>n\\<^esub> m) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\\\<^bsub>n\\<^esub> m = 0\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 \\\\<^bsub>n\\<^esub> m = 0\n[PROOF STEP]\nhave \"(0 \\\\<^bsub>n\\<^esub> m) = (\\j\\{0..\\<^bsub>n\\<^esub> m = (\\j = 0..\\<^bsub>?n\\<^esub> ?j = (\\k = 0..\\<^bsub>n\\<^esub> m = (\\j = 0..\\<^bsub>n\\<^esub> m = (\\j = 0..\\<^bsub>n\\<^esub> m = 0\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n0 \\\\<^bsub>n\\<^esub> m = (\\j = 0..\\<^bsub>n\\<^esub> m = 0\n[PROOF STEP]\nhave \"(\\j\\{0..j\\{0..j = 0..j = 0..j = 0..j = 0..\\<^bsub>n\\<^esub> m = 0\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\\\<^bsub>n\\<^esub> m = (\\j = 0..j = 0..j = 0..\\<^bsub>n\\<^esub> m = (\\j = 0..j = 0..j = 0..\\<^bsub>n\\<^esub> m = 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 \\\\<^bsub>n\\<^esub> m = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1221, "file": "Isabelle_Marries_Dirac_Quantum", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127417985636, "lm_q2_score": 0.8418256472515683, "lm_q1q2_score": 0.7188456465609371}} {"text": "[STATEMENT]\nlemma (in lower_semilattice) meet_assoc_lemma:\n assumes L: \"x \\ carrier L\" \"y \\ carrier L\" \"z \\ carrier L\"\n shows \"x \\ (y \\ z) = \\{x, y, z}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ (y \\ z) = \\{x, y, z}\n[PROOF STEP]\nusing weak_meet_assoc_lemma L\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x \\ carrier L; ?y \\ carrier L; ?z \\ carrier L\\ \\ ?x \\ (?y \\ ?z) .= \\{?x, ?y, ?z}\nx \\ carrier L\ny \\ carrier L\nz \\ carrier L\n\ngoal (1 subgoal):\n 1. x \\ (y \\ z) = \\{x, y, z}\n[PROOF STEP]\nunfolding eq_is_equal\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x \\ carrier L; ?y \\ carrier L; ?z \\ carrier L\\ \\ ?x \\ (?y \\ ?z) = \\{?x, ?y, ?z}\nx \\ carrier L\ny \\ carrier L\nz \\ carrier L\n\ngoal (1 subgoal):\n 1. x \\ (y \\ z) = \\{x, y, z}\n[PROOF STEP]\n.", "meta": {"llama_tokens": 474, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297914570319, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7187909725016214}} {"text": "[STATEMENT]\nlemma ell2_norm_finite: \n \"ell2_norm (x::'a::finite\\complex) = sqrt (sum (\\i. (norm(x i))^2) UNIV)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ell2_norm x = sqrt (\\i\\UNIV. (cmod (x i))\\<^sup>2)\n[PROOF STEP]\nby (simp add: ell2_norm_def)", "meta": {"llama_tokens": 130, "file": "Complex_Bounded_Operators_Complex_L2", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297807787538, "lm_q2_score": 0.798186775339273, "lm_q1q2_score": 0.7187909618167759}} {"text": "[STATEMENT]\ntheorem ln_lower_5_eq: \"0\n ln_lower_5 x = (1/3)*(10*x^2 + 19*x + 1)*(x - 1) / (x*(x^2 + 6*x + 3))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ ln_lower_5 x = 1 / 3 * (10 * x\\<^sup>2 + 19 * x + 1) * (x - 1) / (x * (x\\<^sup>2 + 6 * x + 3))\n[PROOF STEP]\nunfolding ln_lower_5_def ln_upper_5_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ - (((inverse x)\\<^sup>2 + 19 * inverse x + 10) * (inverse x - 1) / (3 * (3 * (inverse x)\\<^sup>2 + 6 * inverse x + 1))) = 1 / 3 * (10 * x\\<^sup>2 + 19 * x + 1) * (x - 1) / (x * (x\\<^sup>2 + 6 * x + 3))\n[PROOF STEP]\nby (simp add: zero_less_mult_iff add_pos_pos dual_order.strict_implies_not_eq divide_simps)\n algebra", "meta": {"llama_tokens": 384, "file": "Special_Function_Bounds_Log_CF_Bounds", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976952948443462, "lm_q2_score": 0.8006920068519378, "lm_q1q2_score": 0.7187774471704615}} {"text": "[STATEMENT]\nlemma totient_4 [simp]: \"totient 4 = 2\"\n and totient_8 [simp]: \"totient 8 = 4\"\n and totient_9 [simp]: \"totient 9 = 6\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. totient 4 = 2 &&& totient 8 = 4 &&& totient 9 = 6\n[PROOF STEP]\nusing totient_prime_power[of 2 2] totient_prime_power[of 2 3] totient_prime_power[of 3 2]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\prime 2; 0 < 2\\ \\ totient (2\\<^sup>2) = 2 ^ (2 - 1) * (2 - 1)\n\\prime 2; 0 < 3\\ \\ totient (2 ^ 3) = 2 ^ (3 - 1) * (2 - 1)\n\\prime 3; 0 < 2\\ \\ totient (3\\<^sup>2) = 3 ^ (2 - 1) * (3 - 1)\n\ngoal (1 subgoal):\n 1. totient 4 = 2 &&& totient 8 = 4 &&& totient 9 = 6\n[PROOF STEP]\nby simp_all", "meta": {"llama_tokens": 371, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.897695292107347, "lm_q2_score": 0.8006920044739461, "lm_q1q2_score": 0.7187774428442562}} {"text": "[STATEMENT]\nlemma smult_diff_right: \"smult a (p - q) = smult a p - smult a q\"\n for a :: \"'a::comm_ring\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. smult a (p - q) = smult a p - smult a q\n[PROOF STEP]\nby (rule poly_eqI) (simp add: algebra_simps)", "meta": {"llama_tokens": 113, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952866333484, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.7187774363265573}} {"text": "[STATEMENT]\nlemma image_mset_mono_pos: \n \"\\b. 0 \\ zcount A b \\ \\b. 0 \\ zcount B b \\ A \\#\\<^sub>z B \\ image_zmset f A \\#\\<^sub>z image_zmset f B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\b. 0 \\ zcount A b; \\b. 0 \\ zcount B b; A \\#\\<^sub>z B\\ \\ Auxiliary.image_zmset f A \\#\\<^sub>z Auxiliary.image_zmset f B\n[PROOF STEP]\nunfolding subseteq_zmset_def zcount_image_zmset\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\b. 0 \\ zcount A b; \\b. 0 \\ zcount B b; \\a. zcount A a \\ zcount B a\\ \\ \\a. sum (zcount A) (f -` {a} \\ set_zmset A) \\ sum (zcount B) (f -` {a} \\ set_zmset B)\n[PROOF STEP]\napply (intro allI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a. \\\\b. 0 \\ zcount A b; \\b. 0 \\ zcount B b; \\a. zcount A a \\ zcount B a\\ \\ sum (zcount A) (f -` {a} \\ set_zmset A) \\ sum (zcount B) (f -` {a} \\ set_zmset B)\n[PROOF STEP]\napply (rule order_trans[OF sum_mono sum_mono2])\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. \\a i. \\\\b. 0 \\ zcount A b; \\b. 0 \\ zcount B b; \\a. zcount A a \\ zcount B a; i \\ f -` {a} \\ set_zmset A\\ \\ zcount A i \\ zcount B i\n 2. \\a. \\\\b. 0 \\ zcount A b; \\b. 0 \\ zcount B b; \\a. zcount A a \\ zcount B a\\ \\ finite (f -` {a} \\ set_zmset B)\n 3. \\a. \\\\b. 0 \\ zcount A b; \\b. 0 \\ zcount B b; \\a. zcount A a \\ zcount B a\\ \\ f -` {a} \\ set_zmset A \\ f -` {a} \\ set_zmset B\n 4. \\a b. \\\\b. 0 \\ zcount A b; \\b. 0 \\ zcount B b; \\a. zcount A a \\ zcount B a; b \\ f -` {a} \\ set_zmset B - f -` {a} \\ set_zmset A\\ \\ 0 \\ zcount B b\n[PROOF STEP]\napply simp_all\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a. \\\\b. 0 \\ zcount A b; \\b. 0 \\ zcount B b; \\a. zcount A a \\ zcount B a\\ \\ f -` {a} \\ set_zmset A \\ set_zmset B\n[PROOF STEP]\nby (metis Int_subset_iff antisym subsetI zcount_ne_zero_iff)", "meta": {"llama_tokens": 1095, "file": "Progress_Tracking_Propagate", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467706759584, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7187679663943232}} {"text": "[STATEMENT]\nlemma equivalentFormulaeTransitivity: \n fixes formula1 :: Formula and formula2 :: Formula and formula3 :: Formula\n assumes \"equivalentFormulae formula1 formula2\" and \"equivalentFormulae formula2 formula3\"\n shows \"equivalentFormulae formula1 formula3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. equivalentFormulae formula1 formula3\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nequivalentFormulae formula1 formula2\nequivalentFormulae formula2 formula3\n\ngoal (1 subgoal):\n 1. equivalentFormulae formula1 formula3\n[PROOF STEP]\nunfolding equivalentFormulae_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\valuation. model valuation formula1 = model valuation formula2\n\\valuation. model valuation formula2 = model valuation formula3\n\ngoal (1 subgoal):\n 1. \\valuation. model valuation formula1 = model valuation formula3\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 283, "file": "SATSolverVerification_CNF", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467675095292, "lm_q2_score": 0.8175744695262777, "lm_q1q2_score": 0.7187679520823451}} {"text": "[STATEMENT]\nlemma epi_compose:\n \"\\f \\ epi G H; g \\ epi H K\\ \\ (g \\ f) \\ epi G K\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ epi G H; g \\ epi H K\\ \\ g \\ f \\ epi G K\n[PROOF STEP]\nusing hom_compose\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?f \\ hom ?G ?H; ?g \\ hom ?H ?I\\ \\ ?g \\ ?f \\ hom ?G ?I\n\ngoal (1 subgoal):\n 1. \\f \\ epi G H; g \\ epi H K\\ \\ g \\ f \\ epi G K\n[PROOF STEP]\nby (force simp: epi_def hom_compose simp flip: image_image)", "meta": {"llama_tokens": 287, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.879146761176671, "lm_q2_score": 0.8175744739711883, "lm_q1q2_score": 0.7187679508124907}} {"text": "[STATEMENT]\nlemma det_swaprows: \n assumes *: \"k < n\" \"l < n\" and k: \"k \\ l\" and A: \"A \\ carrier_mat n n\"\n shows \"det (swaprows k l A) = - det A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (swaprows k l A) = - det A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. det (swaprows k l A) = - det A\n[PROOF STEP]\nhave \"swaprows k l A = swaprows_mat n k l * A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. swaprows k l A = swaprows_mat n k l * A\n[PROOF STEP]\nby (rule swaprows_mat[OF A *])\n[PROOF STATE]\nproof (state)\nthis:\nswaprows k l A = swaprows_mat n k l * A\n\ngoal (1 subgoal):\n 1. det (swaprows k l A) = - det A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nswaprows k l A = swaprows_mat n k l * A\n\ngoal (1 subgoal):\n 1. det (swaprows k l A) = - det A\n[PROOF STEP]\nhave \"det (swaprows_mat n k l * A) = det (swaprows_mat n k l) * det A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (swaprows_mat n k l * A) = det (swaprows_mat n k l) * det A\n[PROOF STEP]\nby (rule det_mult[OF _ A], insert A, auto)\n[PROOF STATE]\nproof (state)\nthis:\ndet (swaprows_mat n k l * A) = det (swaprows_mat n k l) * det A\n\ngoal (1 subgoal):\n 1. det (swaprows k l A) = - det A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndet (swaprows_mat n k l * A) = det (swaprows_mat n k l) * det A\n\ngoal (1 subgoal):\n 1. det (swaprows k l A) = - det A\n[PROOF STEP]\nhave \"det (swaprows_mat n k l) = - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (swaprows_mat n k l) = - (1::'b)\n[PROOF STEP]\nby (rule det_swaprows_mat[OF * k])\n[PROOF STATE]\nproof (state)\nthis:\ndet (swaprows_mat n k l) = - (1::?'b1)\n\ngoal (1 subgoal):\n 1. det (swaprows k l A) = - det A\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndet (swaprows k l A) = - (1::'a) * det A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndet (swaprows k l A) = - (1::'a) * det A\n\ngoal (1 subgoal):\n 1. det (swaprows k l A) = - det A\n[PROOF STEP]\nusing A\n[PROOF STATE]\nproof (prove)\nusing this:\ndet (swaprows k l A) = - (1::'a) * det A\nA \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. det (swaprows k l A) = - det A\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndet (swaprows k l A) = - det A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1028, "file": "Jordan_Normal_Form_Determinant", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357666736772, "lm_q2_score": 0.8289388019824946, "lm_q1q2_score": 0.7187195897024516}} {"text": "[STATEMENT]\nlemma Read_Show_nat_id: \\Read\\<^sub>n\\<^sub>a\\<^sub>t(Show\\<^sub>n\\<^sub>a\\<^sub>t n) = n\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Read\\<^sub>n\\<^sub>a\\<^sub>t (Show\\<^sub>n\\<^sub>a\\<^sub>t n) = n\n[PROOF STEP]\napply(unfold Read\\<^sub>n\\<^sub>a\\<^sub>t_def Show\\<^sub>n\\<^sub>a\\<^sub>t_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. nat_implode (map nat_of_digit (map digit_of_nat (nat_explode n))) = n\n[PROOF STEP]\nusing bar' nat_of_digit_digit_of_nat_id nat_explode_digits\n[PROOF STATE]\nproof (prove)\nusing this:\n\\d\\set ?l. d < 10 \\ map nat_of_digit (map digit_of_nat ?l) = ?l\n?x < 10 \\ nat_of_digit (digit_of_nat ?x) = ?x\n\\d\\set (nat_explode ?n). d < 10\n\ngoal (1 subgoal):\n 1. nat_implode (map nat_of_digit (map digit_of_nat (nat_explode n))) = n\n[PROOF STEP]\nusing nat_implode_explode_id\n[PROOF STATE]\nproof (prove)\nusing this:\n\\d\\set ?l. d < 10 \\ map nat_of_digit (map digit_of_nat ?l) = ?l\n?x < 10 \\ nat_of_digit (digit_of_nat ?x) = ?x\n\\d\\set (nat_explode ?n). d < 10\nnat_implode (nat_explode ?n) = ?n\n\ngoal (1 subgoal):\n 1. nat_implode (map nat_of_digit (map digit_of_nat (nat_explode n))) = n\n[PROOF STEP]\nby presburger", "meta": {"llama_tokens": 568, "file": "Solidity_ReadShow", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357598021707, "lm_q2_score": 0.8289388040954683, "lm_q1q2_score": 0.7187195858384171}} {"text": "[STATEMENT]\nlemma complex_roots_unity:\n assumes \"1 \\ n\"\n shows \"{z::complex. z^n = 1} = {exp(2 * of_real pi * \\ * of_nat j / of_nat n) | j. j < n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {z. z ^ n = 1} = {exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) |j. j < n}\n[PROOF STEP]\napply (rule Finite_Set.card_seteq [symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. finite {z. z ^ n = 1}\n 2. {exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) |j. j < n} \\ {z. z ^ n = 1}\n 3. card {z. z ^ n = 1} \\ card {exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) |j. j < n}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ n\n\ngoal (3 subgoals):\n 1. finite {z. z ^ n = 1}\n 2. {exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) |j. j < n} \\ {z. z ^ n = 1}\n 3. card {z. z ^ n = 1} \\ card {exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) |j. j < n}\n[PROOF STEP]\napply (auto simp: card_complex_roots_unity_explicit finite_roots_unity complex_root_unity card_roots_unity)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 537, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942119105696, "lm_q2_score": 0.7956580952177051, "lm_q1q2_score": 0.7187133520699419}} {"text": "[STATEMENT]\nlemma distinct_seteq_set_Cons':\n \"\\Finite_Set.card xs = Suc n; x \\ {ys. set ys = xs \\ distinct ys}\\\n \\ \\y ys zs. y # ys = x \\ Finite_Set.card zs = n \\ distinct ys \\ set ys = zs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Finite_Set.card xs = Suc n; x \\ {ys. set ys = xs \\ distinct ys}\\ \\ \\y ys zs. y # ys = x \\ Finite_Set.card zs = n \\ distinct ys \\ set ys = zs\n[PROOF STEP]\nusing distinct_seteq_set_length_eq[of x] Suc_length_conv[of n x]\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ {ys. set ys = ?xs \\ distinct ys} \\ length x = Finite_Set.card ?xs\n(Suc n = length x) = (\\y ys. x = y # ys \\ length ys = n)\n\ngoal (1 subgoal):\n 1. \\Finite_Set.card xs = Suc n; x \\ {ys. set ys = xs \\ distinct ys}\\ \\ \\y ys zs. y # ys = x \\ Finite_Set.card zs = n \\ distinct ys \\ set ys = zs\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 447, "file": "Query_Optimization_IKKBZ_Optimality", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942093072239, "lm_q2_score": 0.7956580927949806, "lm_q1q2_score": 0.7187133478101357}} {"text": "[STATEMENT]\nlemma upper_triangular_invertible:\n fixes A :: \"real^'n::{finite,wellorder}^'n::{finite,wellorder}\"\n assumes u: \"upper_triangular A\"\n and d: \"\\i. A $ i $ i \\ 0\" \n shows \"invertible A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. invertible A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. invertible A\n[PROOF STEP]\nhave det_R: \"det A = (prod (\\i. A$i$i) (UNIV::'n set))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A = (\\i\\UNIV. A $ i $ i)\n[PROOF STEP]\nusing det_upperdiagonal u\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i j. j < i \\ ?A $ i $ j = (0::?'a)) \\ det ?A = (\\i\\UNIV. ?A $ i $ i)\nupper_triangular A\n\ngoal (1 subgoal):\n 1. det A = (\\i\\UNIV. A $ i $ i)\n[PROOF STEP]\nunfolding upper_triangular_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i j. j < i \\ ?A $ i $ j = (0::?'a)) \\ det ?A = (\\i\\UNIV. ?A $ i $ i)\n\\i j. j < i \\ A $ i $ j = 0\n\ngoal (1 subgoal):\n 1. det A = (\\i\\UNIV. A $ i $ i)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ndet A = (\\i\\UNIV. A $ i $ i)\n\ngoal (1 subgoal):\n 1. invertible A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndet A = (\\i\\UNIV. A $ i $ i)\n\ngoal (1 subgoal):\n 1. invertible A\n[PROOF STEP]\nhave \"... \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\UNIV. A $ i $ i) \\ 0\n[PROOF STEP]\nusing d\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i. A $ i $ i \\ 0\n\ngoal (1 subgoal):\n 1. (\\i\\UNIV. A $ i $ i) \\ 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\UNIV. A $ i $ i) \\ 0\n\ngoal (1 subgoal):\n 1. invertible A\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndet A \\ 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndet A \\ 0\n\ngoal (1 subgoal):\n 1. invertible A\n[PROOF STEP]\nby (metis invertible_det_nz)\n[PROOF STATE]\nproof (state)\nthis:\ninvertible A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 985, "file": "QR_Decomposition_QR_Decomposition", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511616741042, "lm_q2_score": 0.8397339676722393, "lm_q1q2_score": 0.7186872917294906}} {"text": "[STATEMENT]\nlemma periodic_arithmetic_homothecy:\n assumes \"periodic_arithmetic f k\"\n shows \"periodic_arithmetic (\\l. f (l*a)) k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. periodic_arithmetic (\\l. f (l * a)) k\n[PROOF STEP]\nunfolding periodic_arithmetic_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. f ((n + k) * a) = f (n * a)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. f ((n + k) * a) = f (n * a)\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. f ((n + k) * a) = f (n * a)\n[PROOF STEP]\nhave \"f ((n + k) * a) = f(n*a+k*a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f ((n + k) * a) = f (n * a + k * a)\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nf ((n + k) * a) = f (n * a + k * a)\n\ngoal (1 subgoal):\n 1. \\n. f ((n + k) * a) = f (n * a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nf ((n + k) * a) = f (n * a + k * a)\n\ngoal (1 subgoal):\n 1. \\n. f ((n + k) * a) = f (n * a)\n[PROOF STEP]\nhave \"\\ = f(n*a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (n * a + k * a) = f (n * a)\n[PROOF STEP]\nusing mult_period[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nperiodic_arithmetic f (k * ?q)\n\ngoal (1 subgoal):\n 1. f (n * a + k * a) = f (n * a)\n[PROOF STEP]\nunfolding periodic_arithmetic_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n. f (n + k * ?q) = f n\n\ngoal (1 subgoal):\n 1. f (n * a + k * a) = f (n * a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nf (n * a + k * a) = f (n * a)\n\ngoal (1 subgoal):\n 1. \\n. f ((n + k) * a) = f (n * a)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nf ((n + k) * a) = f (n * a)\n[PROOF STEP]\nshow \"f ((n + k) * a) = f (n * a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nf ((n + k) * a) = f (n * a)\n\ngoal (1 subgoal):\n 1. f ((n + k) * a) = f (n * a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nf ((n + k) * a) = f (n * a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 999, "file": "Gauss_Sums_Periodic_Arithmetic", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.855851154320682, "lm_q2_score": 0.8397339716830605, "lm_q1q2_score": 0.7186872889872383}} {"text": "[STATEMENT]\nlemma proots_sphere_pos_interval:\n fixes a b::real\n defines \"q1\\[:a,b:]\" and \"q2\\[:1,1:]\"\n assumes \"p\\0\" \"a x < b} = proots_count (fcompose p q1 q2) {x. 0 < x}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. proots_count p {x. a < x \\ x < b} = proots_count (fcompose p q1 q2) {x. 0 < x}\n[PROOF STEP]\napply (rule proots_fcompose_bij_eq[OF _ \\p\\0\\])\n[PROOF STATE]\nproof (prove)\ngoal (5 subgoals):\n 1. bij_betw (\\x. poly q1 x / poly q2 x) {x. 0 < x} {x. a < x \\ x < b}\n 2. \\x\\{x. 0 < x}. poly q2 x \\ 0\n 3. max (degree q1) (degree q2) \\ 1\n 4. \\c. q1 \\ smult c q2\n 5. infinite UNIV\n[PROOF STEP]\nunfolding q1_def q2_def\n[PROOF STATE]\nproof (prove)\ngoal (5 subgoals):\n 1. bij_betw (\\x. poly [:a, b:] x / poly [:1, 1:] x) {x. 0 < x} {x. a < x \\ x < b}\n 2. \\x\\{x. 0 < x}. poly [:1, 1:] x \\ 0\n 3. max (degree [:a, b:]) (degree [:1, 1:]) \\ 1\n 4. \\c. [:a, b:] \\ smult c [:1, 1:]\n 5. infinite UNIV\n[PROOF STEP]\nusing bij_betw_pos_interval[OF \\a] \\a\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw (\\x. (a + b * x) / (1 + x)) {x. 0 < x} {x. a < x \\ x < b}\na < b\n\ngoal (5 subgoals):\n 1. bij_betw (\\x. poly [:a, b:] x / poly [:1, 1:] x) {x. 0 < x} {x. a < x \\ x < b}\n 2. \\x\\{x. 0 < x}. poly [:1, 1:] x \\ 0\n 3. max (degree [:a, b:]) (degree [:1, 1:]) \\ 1\n 4. \\c. [:a, b:] \\ smult c [:1, 1:]\n 5. infinite UNIV\n[PROOF STEP]\nby (auto simp add:algebra_simps infinite_UNIV_char_0)", "meta": {"llama_tokens": 828, "file": "Budan_Fourier_Descartes_Roots_Test", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511543206819, "lm_q2_score": 0.8397339676722393, "lm_q1q2_score": 0.7186872855545722}} {"text": "[STATEMENT]\nlemma ceiling_log_nat_eq_if: fixes b n k :: nat\n assumes \"b^n < k\" \"k \\ b^(n+1)\" \"b \\ 2\"\n shows \"ceiling (log b (real k)) = int n + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\log (real b) (real k)\\ = int n + 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\log (real b) (real k)\\ = int n + 1\n[PROOF STEP]\nhave \"k \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 \\ k\n[PROOF STEP]\nusing assms(1,3) one_le_power[of b n]\n[PROOF STATE]\nproof (prove)\nusing this:\nb ^ n < k\n2 \\ b\n1 \\ b \\ 1 \\ b ^ n\n\ngoal (1 subgoal):\n 1. 1 \\ k\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ k\n\ngoal (1 subgoal):\n 1. \\log (real b) (real k)\\ = int n + 1\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\nb ^ n < k\nk \\ b ^ (n + 1)\n2 \\ b\n1 \\ k\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nb ^ n < k\nk \\ b ^ (n + 1)\n2 \\ b\n1 \\ k\n\ngoal (1 subgoal):\n 1. \\log (real b) (real k)\\ = int n + 1\n[PROOF STEP]\nby(simp add: ceiling_log_nat_eq_powr_iff)\n[PROOF STATE]\nproof (state)\nthis:\n\\log (real b) (real k)\\ = int n + 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 626, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511396138366, "lm_q2_score": 0.8397339616560072, "lm_q1q2_score": 0.7186872680557356}} {"text": "[STATEMENT]\nlemma INF_of_enat_nat_conv2: \n assumes \"finite A\" \n shows \"enat d = (INF x \\ A. enat (f x)) \\ (\\x\\A. d = f x \\ (\\y\\A. f x \\ f y))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (enat d = (INF x\\A. enat (f x))) = (\\x\\A. d = f x \\ (\\y\\A. f x \\ f y))\n[PROOF STEP]\nusing INF_of_enat_nat_conv1[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n((INF x\\A. enat (?f x)) = enat ?d) = (\\x\\A. ?d = ?f x \\ (\\y\\A. ?f x \\ ?f y))\n\ngoal (1 subgoal):\n 1. (enat d = (INF x\\A. enat (f x))) = (\\x\\A. d = f x \\ (\\y\\A. f x \\ f y))\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 346, "file": "Prim_Dijkstra_Simple_Common", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894632969136, "lm_q2_score": 0.803173801068221, "lm_q1q2_score": 0.7186714543919754}} {"text": "[STATEMENT]\nlemma differentiable_inverse [simp, derivative_intros]:\n fixes f :: \"'a::real_normed_vector \\ 'b::real_normed_field\"\n shows \"f differentiable (at x within s) \\ f x \\ 0 \\\n (\\x. inverse (f x)) differentiable (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f differentiable at x within s; f x \\ (0::'b)\\ \\ (\\x. inverse (f x)) differentiable at x within s\n[PROOF STEP]\nunfolding differentiable_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\D. (f has_derivative D) (at x within s); f x \\ (0::'b)\\ \\ \\D. ((\\x. inverse (f x)) has_derivative D) (at x within s)\n[PROOF STEP]\nby (blast intro: has_derivative_inverse)", "meta": {"llama_tokens": 303, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.8031738057795403, "lm_q1q2_score": 0.7186714541007992}} {"text": "[STATEMENT]\nlemma INF_eq_zero_iff_ennreal: \"((\\i\\A. f i) = (0::ennreal)) = (\\x>0. \\i\\A. f i < x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ (f ` A) = 0) = (\\x>0. \\i\\A. f i < x)\n[PROOF STEP]\nusing INF_eq_bot_iff[where 'a=ennreal]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\ (?f ` ?A) = \\) = (\\x>\\. \\i\\?A. ?f i < x)\n\ngoal (1 subgoal):\n 1. (\\ (f ` A) = 0) = (\\x>0. \\i\\A. f i < x)\n[PROOF STEP]\nunfolding bot_ennreal_def zero_ennreal_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\ (?f ` ?A) = e2ennreal 0) = (\\x>e2ennreal 0. \\i\\?A. ?f i < x)\n\ngoal (1 subgoal):\n 1. (\\ (f ` A) = e2ennreal 0) = (\\x>e2ennreal 0. \\i\\A. f i < x)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 414, "file": "Markov_Models_Discrete_Time_Markov_Chain", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894689081711, "lm_q2_score": 0.8031737940012418, "lm_q1q2_score": 0.7186714525753319}} {"text": "[STATEMENT]\nlemma cosh_complex:\n fixes z :: complex\n shows \"(exp z + inverse (exp z)) / 2 = cos(\\ * z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (exp z + inverse (exp z)) / 2 = cos (\\ * z)\n[PROOF STEP]\nby (simp add: cos_exp_eq field_split_simps exp_minus exp_of_real)", "meta": {"llama_tokens": 119, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894632969137, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7186714522841561}} {"text": "[STATEMENT]\nlemma ind_in_set_less:\n fixes x :: nat and A :: \"nat set\"\n assumes \"finite A\" \"x \\ A\"\n shows \"ind_in_set A x < card A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ind_in_set A x < card A\n[PROOF STEP]\nunfolding ind_in_set_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {i \\ A. i < x} < card A\n[PROOF STEP]\napply (rule psubset_card_mono)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. finite A\n 2. {i \\ A. i < x} \\ A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nx \\ A\n\ngoal (2 subgoals):\n 1. finite A\n 2. {i \\ A. i < x} \\ A\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 297, "file": "QHLProver_Partial_State", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.8031738034238807, "lm_q1q2_score": 0.7186714519929799}} {"text": "[STATEMENT]\nlemma fds_totient_times_zeta: \n \"fds (\\n. of_nat (totient n) :: 'a :: comm_semiring_1) * fds_zeta = fds of_nat\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fds (\\n. of_nat (totient n)) * fds_zeta = fds of_nat\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. 0 < n \\ fds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = fds_nth (fds of_nat) n\n[PROOF STEP]\nfix n :: nat\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. 0 < n \\ fds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = fds_nth (fds of_nat) n\n[PROOF STEP]\nassume n: \"n > 0\"\n[PROOF STATE]\nproof (state)\nthis:\n0 < n\n\ngoal (1 subgoal):\n 1. \\n. 0 < n \\ fds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = fds_nth (fds of_nat) n\n[PROOF STEP]\nhave \"fds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = \n dirichlet_prod (\\n. of_nat (totient n)) (\\_. 1) n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = dirichlet_prod (\\n. of_nat (totient n)) (\\_. 1::'b) n\n[PROOF STEP]\nby (simp add: fds_nth_mult)\n[PROOF STATE]\nproof (state)\nthis:\nfds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = dirichlet_prod (\\n. of_nat (totient n)) (\\_. 1::?'b1) n\n\ngoal (1 subgoal):\n 1. \\n. 0 < n \\ fds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = fds_nth (fds of_nat) n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = dirichlet_prod (\\n. of_nat (totient n)) (\\_. 1::?'b1) n\n\ngoal (1 subgoal):\n 1. \\n. 0 < n \\ fds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = fds_nth (fds of_nat) n\n[PROOF STEP]\nfrom n\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < n\n[PROOF STEP]\nhave \"\\ = fds_nth (fds of_nat) n\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\n\ngoal (1 subgoal):\n 1. dirichlet_prod (\\n. of_nat (totient n)) (\\_. 1::'b) n = fds_nth (fds of_nat) n\n[PROOF STEP]\nby (simp add: fds_nth_fds dirichlet_prod_def totient_divisor_sum of_nat_sum [symmetric]\n del: of_nat_sum)\n[PROOF STATE]\nproof (state)\nthis:\ndirichlet_prod (\\n. of_nat (totient n)) (\\_. 1::?'b2) n = fds_nth (fds of_nat) n\n\ngoal (1 subgoal):\n 1. \\n. 0 < n \\ fds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = fds_nth (fds of_nat) n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nfds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = fds_nth (fds of_nat) n\n[PROOF STEP]\nshow \"fds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = fds_nth (fds of_nat) n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = fds_nth (fds of_nat) n\n\ngoal (1 subgoal):\n 1. fds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = fds_nth (fds of_nat) n\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nfds_nth (fds (\\n. of_nat (totient n)) * fds_zeta) n = fds_nth (fds of_nat) n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1470, "file": "Dirichlet_Series_More_Totient", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894661025425, "lm_q2_score": 0.8031737940012417, "lm_q1q2_score": 0.7186714503219245}} {"text": "[STATEMENT]\nlemma sin_zero_norm_cos_one:\n fixes x :: \"'a::{real_normed_field,banach}\"\n assumes \"sin x = 0\"\n shows \"norm (cos x) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (cos x) = 1\n[PROOF STEP]\nusing sin_cos_squared_add [of x, unfolded assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a)\\<^sup>2 + (cos x)\\<^sup>2 = (1::'a)\n\ngoal (1 subgoal):\n 1. norm (cos x) = 1\n[PROOF STEP]\nby (simp add: square_norm_one)", "meta": {"llama_tokens": 203, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894632969137, "lm_q2_score": 0.803173791645582, "lm_q1q2_score": 0.7186714459606974}} {"text": "[STATEMENT]\nlemma differentiable_inverse [simp, derivative_intros]:\n fixes f :: \"'a::real_normed_vector \\ 'b::real_normed_field\"\n shows \"f differentiable (at x within s) \\ f x \\ 0 \\\n (\\x. inverse (f x)) differentiable (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f differentiable at x within s; f x \\ (0::'b)\\ \\ (\\x. inverse (f x)) differentiable at x within s\n[PROOF STEP]\nunfolding differentiable_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\D. (f has_derivative D) (at x within s); f x \\ (0::'b)\\ \\ \\D. ((\\x. inverse (f x)) has_derivative D) (at x within s)\n[PROOF STEP]\nby (blast intro: has_derivative_inverse)", "meta": {"llama_tokens": 303, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7186714393460628}} {"text": "[STATEMENT]\nlemma adjuster_in_span:\n assumes w: \"(w :: 'a vec) : carrier_vec n\"\n and us: \"set (us :: 'a vec list) \\ carrier_vec n\"\n and dist: \"distinct us\"\n shows \"adjuster n w us : span (set us)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. adjuster n w us \\ local.span (set us)\n[PROOF STEP]\nusing adjuster_lincomb[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nadjuster n w us = lincomb (\\u. - (w \\c u) / (u \\c u)) (set us)\n\ngoal (1 subgoal):\n 1. adjuster n w us \\ local.span (set us)\n[PROOF STEP]\nunfolding finite_span[OF finite_set us]\n[PROOF STATE]\nproof (prove)\nusing this:\nadjuster n w us = lincomb (\\u. - (w \\c u) / (u \\c u)) (set us)\n\ngoal (1 subgoal):\n 1. adjuster n w us \\ {uu_. \\a. uu_ = lincomb a (set us)}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 349, "file": "Jordan_Normal_Form_Gram_Schmidt", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789454880027, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7186714370926552}} {"text": "[STATEMENT]\nlemma ladder_stepdown_length:\n assumes \"length L > 1\"\n shows \"length (ladder_stepdown L) = length L - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (ladder_stepdown L) = length L - 1\n[PROOF STEP]\napply (subst ladder_stepdown_prepare[OF assms(1)])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (ladder_stepdown ((ladder_n L 0, ladder_j L 0, ladder_i L 0) # (ladder_n L 1, ladder_j L 1, ladder_ix L 1) # drop 2 L)) = length L - 1\n[PROOF STEP]\napply (simp add: ladder_shift_n_length)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc (length L - 2) = length L - Suc 0\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < length L\n\ngoal (1 subgoal):\n 1. Suc (length L - 2) = length L - Suc 0\n[PROOF STEP]\napply arith\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 375, "file": "LocalLexing_Ladder", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772482857833, "lm_q2_score": 0.8221891305219504, "lm_q1q2_score": 0.7186568127771071}} {"text": "[STATEMENT]\nlemma Im_Arccos_of_real:\n assumes \"\\x\\ \\ 1\"\n shows \"Im (Arccos (of_real x)) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Im (Arccos (complex_of_real x)) = 0\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Im (Arccos (complex_of_real x)) = 0\n[PROOF STEP]\nhave \"csqrt (1 - (of_real x)\\<^sup>2) = (if x^2 \\ 1 then sqrt (1 - x^2) else \\ * sqrt (x^2 - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. csqrt (1 - (complex_of_real x)\\<^sup>2) = (if x\\<^sup>2 \\ 1 then complex_of_real (sqrt (1 - x\\<^sup>2)) else \\ * complex_of_real (sqrt (x\\<^sup>2 - 1)))\n[PROOF STEP]\nby (simp add: of_real_sqrt del: csqrt_of_real_nonneg)\n[PROOF STATE]\nproof (state)\nthis:\ncsqrt (1 - (complex_of_real x)\\<^sup>2) = (if x\\<^sup>2 \\ 1 then complex_of_real (sqrt (1 - x\\<^sup>2)) else \\ * complex_of_real (sqrt (x\\<^sup>2 - 1)))\n\ngoal (1 subgoal):\n 1. Im (Arccos (complex_of_real x)) = 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncsqrt (1 - (complex_of_real x)\\<^sup>2) = (if x\\<^sup>2 \\ 1 then complex_of_real (sqrt (1 - x\\<^sup>2)) else \\ * complex_of_real (sqrt (x\\<^sup>2 - 1)))\n[PROOF STEP]\nhave \"cmod (of_real x + \\ * csqrt (1 - (of_real x)\\<^sup>2))^2 = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncsqrt (1 - (complex_of_real x)\\<^sup>2) = (if x\\<^sup>2 \\ 1 then complex_of_real (sqrt (1 - x\\<^sup>2)) else \\ * complex_of_real (sqrt (x\\<^sup>2 - 1)))\n\ngoal (1 subgoal):\n 1. (cmod (complex_of_real x + \\ * csqrt (1 - (complex_of_real x)\\<^sup>2)))\\<^sup>2 = 1\n[PROOF STEP]\nusing assms abs_square_le_1\n[PROOF STATE]\nproof (prove)\nusing this:\ncsqrt (1 - (complex_of_real x)\\<^sup>2) = (if x\\<^sup>2 \\ 1 then complex_of_real (sqrt (1 - x\\<^sup>2)) else \\ * complex_of_real (sqrt (x\\<^sup>2 - 1)))\n\\x\\ \\ 1\n(?x\\<^sup>2 \\ (1::?'a)) = (\\?x\\ \\ (1::?'a))\n\ngoal (1 subgoal):\n 1. (cmod (complex_of_real x + \\ * csqrt (1 - (complex_of_real x)\\<^sup>2)))\\<^sup>2 = 1\n[PROOF STEP]\nby (force simp add: Complex.cmod_power2)\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (complex_of_real x + \\ * csqrt (1 - (complex_of_real x)\\<^sup>2)))\\<^sup>2 = 1\n\ngoal (1 subgoal):\n 1. Im (Arccos (complex_of_real x)) = 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(cmod (complex_of_real x + \\ * csqrt (1 - (complex_of_real x)\\<^sup>2)))\\<^sup>2 = 1\n[PROOF STEP]\nhave \"cmod (of_real x + \\ * csqrt (1 - (of_real x)\\<^sup>2)) = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod (complex_of_real x + \\ * csqrt (1 - (complex_of_real x)\\<^sup>2)))\\<^sup>2 = 1\n\ngoal (1 subgoal):\n 1. cmod (complex_of_real x + \\ * csqrt (1 - (complex_of_real x)\\<^sup>2)) = 1\n[PROOF STEP]\nby (simp add: norm_complex_def)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (complex_of_real x + \\ * csqrt (1 - (complex_of_real x)\\<^sup>2)) = 1\n\ngoal (1 subgoal):\n 1. Im (Arccos (complex_of_real x)) = 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncmod (complex_of_real x + \\ * csqrt (1 - (complex_of_real x)\\<^sup>2)) = 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncmod (complex_of_real x + \\ * csqrt (1 - (complex_of_real x)\\<^sup>2)) = 1\n\ngoal (1 subgoal):\n 1. Im (Arccos (complex_of_real x)) = 0\n[PROOF STEP]\nby (simp add: Im_Arccos exp_minus)\n[PROOF STATE]\nproof (state)\nthis:\nIm (Arccos (complex_of_real x)) = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1620, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772318846386, "lm_q2_score": 0.8221891305219503, "lm_q1q2_score": 0.7186567992922641}} {"text": "[STATEMENT]\nlemma matrix_vect_scaleR: \"(A :: 'a :: real_normed_algebra_1 ^ 'n ^ 'k) *v (a *\\<^sub>R v) = a *\\<^sub>R (A *v v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *v a *\\<^sub>R v = a *\\<^sub>R (A *v v)\n[PROOF STEP]\nunfolding vec_eq_iff\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. (A *v a *\\<^sub>R v) $h i = (a *\\<^sub>R (A *v v)) $h i\n[PROOF STEP]\nby (auto simp: matrix_vector_mult_def scaleR_vec_def scaleR_sum_right\n intro!: sum.cong)", "meta": {"llama_tokens": 226, "file": "Perron_Frobenius_Perron_Frobenius_Aux", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.874077222043951, "lm_q2_score": 0.8221891327004133, "lm_q1q2_score": 0.7186567931055027}} {"text": "[STATEMENT]\nlemma card_less_if_surj_not_inj:\n \"\\ finite A; f ` A = B; \\ inj_on f A \\ \\ card B < card A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; f ` A = B; \\ inj_on f A\\ \\ card B < card A\n[PROOF STEP]\nby (metis card_image_le inj_on_iff_eq_card order_le_neq_trans)", "meta": {"llama_tokens": 152, "file": "Bondy_Bondy", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392695254319, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.7186066258745645}} {"text": "[STATEMENT]\nlemma is_norm_1[simp]: \"is_norm_1 x = (norm x = 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_norm_1 x = (cmod x = 1)\n[PROOF STEP]\nunfolding is_norm_1_def norm_complex_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((Re x)\\<^sup>2 + (Im x)\\<^sup>2 = 1) = (sqrt ((Re x)\\<^sup>2 + (Im x)\\<^sup>2) = 1)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 177, "file": "Algebraic_Numbers_Compare_Complex", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9059898203834277, "lm_q2_score": 0.7931059560743422, "lm_q1q2_score": 0.71854592268882}} {"text": "[STATEMENT]\nlemma QR_decomposition_mult:\n fixes A::\"real^'n::{mod_type}^'m::{mod_type}\"\n assumes r: \"rank A = ncols A\"\n shows \"A = (fst (QR_decomposition A)) ** (snd (QR_decomposition A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A = fst (QR_decomposition A) ** snd (QR_decomposition A)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. A = fst (QR_decomposition A) ** snd (QR_decomposition A)\n[PROOF STEP]\nhave \"\\b. column b A = column b ((fst (QR_decomposition A)) ** (snd (QR_decomposition A)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\b. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n[PROOF STEP]\nproof (clarify)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\b. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n[PROOF STEP]\nfix b\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\b. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n[PROOF STEP]\nhave \"(fst (QR_decomposition A) ** snd (QR_decomposition A)) \n = (\\ i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fst (QR_decomposition A) ** snd (QR_decomposition A) = (\\i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A))\n[PROOF STEP]\nunfolding matrix_matrix_mult_def R_Qi_Aj\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A)) = (\\i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfst (QR_decomposition A) ** snd (QR_decomposition A) = (\\i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A))\n\ngoal (1 subgoal):\n 1. \\b. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n[PROOF STEP]\nhence \"column b ((fst (QR_decomposition A) ** snd (QR_decomposition A))) = \n column b ((\\ i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfst (QR_decomposition A) ** snd (QR_decomposition A) = (\\i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A))\n\ngoal (1 subgoal):\n 1. column b (fst (QR_decomposition A) ** snd (QR_decomposition A)) = column b (\\i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncolumn b (fst (QR_decomposition A) ** snd (QR_decomposition A)) = column b (\\i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A))\n\ngoal (1 subgoal):\n 1. \\b. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncolumn b (fst (QR_decomposition A) ** snd (QR_decomposition A)) = column b (\\i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A))\n\ngoal (1 subgoal):\n 1. \\b. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n[PROOF STEP]\nhave \"... = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. (x \\ column b A) *\\<^sub>R x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. column b (\\i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. (x \\ column b A) *\\<^sub>R x)\n[PROOF STEP]\nproof (subst column_def, subst vec_eq_iff, auto)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i. (\\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ i)\n[PROOF STEP]\nfix a\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i. (\\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ i)\n[PROOF STEP]\ndefine f where \"f i = column i (fst (QR_decomposition A))\" for i\n[PROOF STATE]\nproof (state)\nthis:\nf ?i = column ?i (fst (QR_decomposition A))\n\ngoal (1 subgoal):\n 1. \\i. (\\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ i)\n[PROOF STEP]\ndefine g where \"g x = (THE i. x = column i (fst (QR_decomposition A)))\" for x\n[PROOF STATE]\nproof (state)\nthis:\ng ?x = (THE i. ?x = column i (fst (QR_decomposition A)))\n\ngoal (1 subgoal):\n 1. \\i. (\\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ i)\n[PROOF STEP]\nhave f_eq: \"f`UNIV = {column i (fst (QR_decomposition A)) |i. i\\UNIV}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. range f = {column i (fst (QR_decomposition A)) |i. i \\ UNIV}\n[PROOF STEP]\nunfolding f_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. range (\\i. column i (fst (QR_decomposition A))) = {column i (fst (QR_decomposition A)) |i. i \\ UNIV}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrange f = {column i (fst (QR_decomposition A)) |i. i \\ UNIV}\n\ngoal (1 subgoal):\n 1. \\i. (\\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ i)\n[PROOF STEP]\nhave inj_f: \"inj f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj f\n[PROOF STEP]\nby (metis inj_on_def f_def column_eq_fst_QR_decomposition r)\n[PROOF STATE]\nproof (state)\nthis:\ninj f\n\ngoal (1 subgoal):\n 1. \\i. (\\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ i)\n[PROOF STEP]\nhave \"(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) \n = (\\x\\{column i (fst (QR_decomposition A)) |i. i\\UNIV}. x \\ column b A * x $ a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\nlet ?c= \"{column i (fst (QR_decomposition A)) |i. i\\UNIV}\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\nlet ?d= \"{column i (fst (QR_decomposition A)) |i. i\\b}\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\nlet ?f = \"{column i (fst (QR_decomposition A)) |i. i>b}\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\nhave set_rw: \"?c = ?d \\ ?f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {column i (fst (QR_decomposition A)) |i. i \\ UNIV} = {column i (fst (QR_decomposition A)) |i. i \\ b} \\ {column i (fst (QR_decomposition A)) |i. b < i}\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n{column i (fst (QR_decomposition A)) |i. i \\ UNIV} = {column i (fst (QR_decomposition A)) |i. i \\ b} \\ {column i (fst (QR_decomposition A)) |i. b < i}\n\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\nhave \"(\\x\\?c. x \\ column b A * x $ a) \n = (\\x\\(?d \\ ?f). x \\ column b A * x $ a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b} \\ {column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a)\n[PROOF STEP]\nusing set_rw\n[PROOF STATE]\nproof (prove)\nusing this:\n{column i (fst (QR_decomposition A)) |i. i \\ UNIV} = {column i (fst (QR_decomposition A)) |i. i \\ b} \\ {column i (fst (QR_decomposition A)) |i. b < i}\n\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b} \\ {column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b} \\ {column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a)\n\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b} \\ {column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a)\n\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\nhave \"... = (\\x\\?d. x \\ column b A * x $ a) + (\\x\\?f. x \\ column b A * x $ a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b} \\ {column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) + (\\x\\{column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a)\n[PROOF STEP]\nby (rule sum.union_disjoint, auto, metis f_def inj_eq inj_f not_le)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b} \\ {column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) + (\\x\\{column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a)\n\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b} \\ {column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) + (\\x\\{column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a)\n\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\nhave \"... = (\\x\\?d. x \\ column b A * x $ a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) + (\\x\\{column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a)\n[PROOF STEP]\nusing sums_columns_Q_0[OF r]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x\\{column i (fst (QR_decomposition A)) |i. ?b < i}. x \\ column ?b A * x $ ?a) = 0\n\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) + (\\x\\{column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) + (\\x\\{column i (fst (QR_decomposition A)) |i. b < i}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a)\n\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a)\n\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n\ngoal (1 subgoal):\n 1. \\i. (\\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ i)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a)\n\ngoal (1 subgoal):\n 1. \\i. (\\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ i)\n[PROOF STEP]\nhave \"... = (\\x\\f`UNIV. x \\ column b A * x $ a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a) = (\\x\\range f. x \\ column b A * x $ a)\n[PROOF STEP]\nusing f_eq\n[PROOF STATE]\nproof (prove)\nusing this:\nrange f = {column i (fst (QR_decomposition A)) |i. i \\ UNIV}\n\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a) = (\\x\\range f. x \\ column b A * x $ a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a) = (\\x\\range f. x \\ column b A * x $ a)\n\ngoal (1 subgoal):\n 1. \\i. (\\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ i)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ UNIV}. x \\ column b A * x $ a) = (\\x\\range f. x \\ column b A * x $ a)\n\ngoal (1 subgoal):\n 1. \\i. (\\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ i)\n[PROOF STEP]\nhave \"... = (\\k\\UNIV. fst (QR_decomposition A) $ a $ k * (column k (fst (QR_decomposition A)) \\ column b A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\range f. x \\ column b A * x $ a) = (\\k\\UNIV. fst (QR_decomposition A) $ a $ k * (column k (fst (QR_decomposition A)) \\ column b A))\n[PROOF STEP]\nunfolding sum.reindex[OF inj_f]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((\\x. x \\ column b A * x $ a) \\ f) UNIV = (\\k\\UNIV. fst (QR_decomposition A) $ a $ k * (column k (fst (QR_decomposition A)) \\ column b A))\n[PROOF STEP]\nunfolding f_def column_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((\\x. x \\ (\\i. A $ i $ b) * x $ a) \\ (\\i. \\ia. fst (QR_decomposition A) $ ia $ i)) UNIV = (\\k\\UNIV. fst (QR_decomposition A) $ a $ k * ((\\i. fst (QR_decomposition A) $ i $ k) \\ (\\i. A $ i $ b)))\n[PROOF STEP]\nby (rule sum.cong, simp_all)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\range f. x \\ column b A * x $ a) = (\\k\\UNIV. fst (QR_decomposition A) $ a $ k * (column k (fst (QR_decomposition A)) \\ column b A))\n\ngoal (1 subgoal):\n 1. \\i. (\\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ i)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\k\\UNIV. fst (QR_decomposition A) $ a $ k * (column k (fst (QR_decomposition A)) \\ column b A))\n[PROOF STEP]\nshow \" (\\k\\UNIV. fst (QR_decomposition A) $ a $ k * (column k (fst (QR_decomposition A)) \\ column b A)) =\n (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a) = (\\k\\UNIV. fst (QR_decomposition A) $ a $ k * (column k (fst (QR_decomposition A)) \\ column b A))\n\ngoal (1 subgoal):\n 1. (\\k\\UNIV. fst (QR_decomposition A) $ a $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\UNIV. fst (QR_decomposition A) $ a $ k * (column k (fst (QR_decomposition A)) \\ column b A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. x \\ column b A * x $ a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncolumn b (\\i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. (x \\ column b A) *\\<^sub>R x)\n\ngoal (1 subgoal):\n 1. \\b. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncolumn b (\\i j. \\k\\UNIV. fst (QR_decomposition A) $ i $ k * (column k (fst (QR_decomposition A)) \\ column j A)) = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. (x \\ column b A) *\\<^sub>R x)\n\ngoal (1 subgoal):\n 1. \\b. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n[PROOF STEP]\nhave \"... = column b A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. (x \\ column b A) *\\<^sub>R x) = column b A\n[PROOF STEP]\nusing column_QR_decomposition2[OF r]\n[PROOF STATE]\nproof (prove)\nusing this:\ncolumn ?k A = (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ ?k}. (x \\ column ?k A) *\\<^sub>R x)\n\ngoal (1 subgoal):\n 1. (\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. (x \\ column b A) *\\<^sub>R x) = column b A\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\{column i (fst (QR_decomposition A)) |i. i \\ b}. (x \\ column b A) *\\<^sub>R x) = column b A\n\ngoal (1 subgoal):\n 1. \\b. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncolumn b (fst (QR_decomposition A) ** snd (QR_decomposition A)) = column b A\n[PROOF STEP]\nshow \"column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncolumn b (fst (QR_decomposition A) ** snd (QR_decomposition A)) = column b A\n\ngoal (1 subgoal):\n 1. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\ncolumn b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\b. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n\ngoal (1 subgoal):\n 1. A = fst (QR_decomposition A) ** snd (QR_decomposition A)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\b. column b A = column b (fst (QR_decomposition A) ** snd (QR_decomposition A))\n\ngoal (1 subgoal):\n 1. A = fst (QR_decomposition A) ** snd (QR_decomposition A)\n[PROOF STEP]\nunfolding column_def vec_eq_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\b i. (\\i. A $ i $ b) $ i = (\\i. (fst (QR_decomposition A) ** snd (QR_decomposition A)) $ i $ b) $ i\n\ngoal (1 subgoal):\n 1. \\i ia. A $ i $ ia = (fst (QR_decomposition A) ** snd (QR_decomposition A)) $ i $ ia\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA = fst (QR_decomposition A) ** snd (QR_decomposition A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 10586, "file": "QR_Decomposition_QR_Decomposition", "length": 66, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681195338728, "lm_q2_score": 0.8376199694135332, "lm_q1q2_score": 0.7184837060478664}} {"text": "[STATEMENT]\nlemma absolutely_integrable_cos_product:\n assumes \"f absolutely_integrable_on {-pi..pi}\"\n shows \"(\\x. cos(k * x) * f x) absolutely_integrable_on {-pi..pi}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. cos (k * x) * f x) absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nproof (rule absolutely_integrable_bounded_measurable_product_real)\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. (\\x. cos (k * x)) \\ borel_measurable (lebesgue_on {- pi..pi})\n 2. {- pi..pi} \\ sets lebesgue\n 3. bounded ((\\x. cos (k * x)) ` {- pi..pi})\n 4. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nshow \"(\\x. cos (k * x)) \\ borel_measurable (lebesgue_on {-pi..pi})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. cos (k * x)) \\ borel_measurable (lebesgue_on {- pi..pi})\n[PROOF STEP]\nby (metis borel_measurable_integrable integrable_cos_cx mult_commute_abs)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. cos (k * x)) \\ borel_measurable (lebesgue_on {- pi..pi})\n\ngoal (3 subgoals):\n 1. {- pi..pi} \\ sets lebesgue\n 2. bounded ((\\x. cos (k * x)) ` {- pi..pi})\n 3. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nshow \"bounded ((\\x. cos (k * x)) ` {-pi..pi})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bounded ((\\x. cos (k * x)) ` {- pi..pi})\n[PROOF STEP]\nby (metis (mono_tags, lifting) abs_cos_le_one bounded_iff imageE real_norm_def)\n[PROOF STATE]\nproof (state)\nthis:\nbounded ((\\x. cos (k * x)) ` {- pi..pi})\n\ngoal (2 subgoals):\n 1. {- pi..pi} \\ sets lebesgue\n 2. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nqed (auto simp: assms)", "meta": {"llama_tokens": 717, "file": "Fourier_Fourier", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577680904463333, "lm_q2_score": 0.837619959279793, "lm_q1q2_score": 0.7184836729911636}} {"text": "[STATEMENT]\nlemma matrix_mult_sum:\n \"(A::'a::comm_semiring_1^'n^'m) *v x = sum (\\i. (x$i) *s column i A) (UNIV:: 'n set)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *v x = (\\i\\UNIV. x $ i *s column i A)\n[PROOF STEP]\nby (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)", "meta": {"llama_tokens": 144, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9149009457116781, "lm_q2_score": 0.7853085808877581, "lm_q1q2_score": 0.7184795633297058}} {"text": "[STATEMENT]\nlemma arcsin_arccos_sqrt_neg: \"-1 \\ x \\ x \\ 0 \\ arcsin x = -arccos(sqrt(1 - x\\<^sup>2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\- 1 \\ x; x \\ 0\\ \\ arcsin x = - arccos (sqrt (1 - x\\<^sup>2))\n[PROOF STEP]\nusing arcsin_arccos_sqrt_pos [of \"-x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 \\ - x; - x \\ 1\\ \\ arcsin (- x) = arccos (sqrt (1 - (- x)\\<^sup>2))\n\ngoal (1 subgoal):\n 1. \\- 1 \\ x; x \\ 0\\ \\ arcsin x = - arccos (sqrt (1 - x\\<^sup>2))\n[PROOF STEP]\nby (simp add: arcsin_minus)", "meta": {"llama_tokens": 302, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110569397306, "lm_q2_score": 0.8056321843145405, "lm_q1q2_score": 0.7184716897982143}} {"text": "[STATEMENT]\ntheorem thales:\n fixes A B C :: \"'a :: real_inner\"\n assumes \"dist B (midpoint A C) = dist A C / 2\"\n shows \"orthogonal (A - B) (C - B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal (A - B) (C - B)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. orthogonal (A - B) (C - B)\n[PROOF STEP]\nhave \"dist A C ^ 2 = dist B (midpoint A C) ^ 2 * 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist A C)\\<^sup>2 = (dist B (midpoint A C))\\<^sup>2 * 4\n[PROOF STEP]\nby (subst assms) (simp add: field_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n(dist A C)\\<^sup>2 = (dist B (midpoint A C))\\<^sup>2 * 4\n\ngoal (1 subgoal):\n 1. orthogonal (A - B) (C - B)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(dist A C)\\<^sup>2 = (dist B (midpoint A C))\\<^sup>2 * 4\n\ngoal (1 subgoal):\n 1. orthogonal (A - B) (C - B)\n[PROOF STEP]\nby (auto simp: orthogonal_def dist_norm power2_norm_eq_inner midpoint_def\n algebra_simps inner_commute)\n[PROOF STATE]\nproof (state)\nthis:\northogonal (A - B) (C - B)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 494, "file": "Triangle_Triangle", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110425624792, "lm_q2_score": 0.805632181981183, "lm_q1q2_score": 0.7184716761345238}} {"text": "[STATEMENT]\nlemma linear_homeomorphism_image:\n fixes f :: \"'a::euclidean_space \\ 'b::euclidean_space\"\n assumes \"linear f\" \"inj f\"\n obtains g where \"homeomorphism (f ` S) S g f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\g. homeomorphism (f ` S) S g f \\ thesis) \\ thesis\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\g. homeomorphism (f ` S) S g f \\ thesis) \\ thesis\n[PROOF STEP]\nobtain g where \"linear g\" \"g \\ f = id\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\g. \\linear g; g \\ f = id\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing assms linear_injective_left_inverse\n[PROOF STATE]\nproof (prove)\nusing this:\nlinear f\ninj f\n\\linear ?f; inj ?f\\ \\ \\g. linear g \\ g \\ ?f = id\n\ngoal (1 subgoal):\n 1. (\\g. \\linear g; g \\ f = id\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nlinear g\ng \\ f = id\n\ngoal (1 subgoal):\n 1. (\\g. homeomorphism (f ` S) S g f \\ thesis) \\ thesis\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlinear g\ng \\ f = id\n[PROOF STEP]\nhave \"homeomorphism (f ` S) S g f\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlinear g\ng \\ f = id\n\ngoal (1 subgoal):\n 1. homeomorphism (f ` S) S g f\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlinear g\ng \\ f = id\nlinear f\ninj f\n\ngoal (1 subgoal):\n 1. homeomorphism (f ` S) S g f\n[PROOF STEP]\nunfolding homeomorphism_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlinear g\ng \\ f = id\nlinear f\ninj f\n\ngoal (1 subgoal):\n 1. (\\x\\f ` S. f (g x) = x) \\ g ` f ` S = S \\ continuous_on (f ` S) g \\ (\\y\\S. g (f y) = y) \\ f ` S = f ` S \\ continuous_on S f\n[PROOF STEP]\nby (auto simp: eq_id_iff [symmetric] image_comp linear_conv_bounded_linear linear_continuous_on)\n[PROOF STATE]\nproof (state)\nthis:\nhomeomorphism (f ` S) S g f\n\ngoal (1 subgoal):\n 1. (\\g. homeomorphism (f ` S) S g f \\ thesis) \\ thesis\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nhomeomorphism (f ` S) S g f\n[PROOF STEP]\nshow thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nhomeomorphism (f ` S) S g f\n\ngoal (1 subgoal):\n 1. thesis\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nthesis\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1035, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267830311354, "lm_q2_score": 0.8267117983401363, "lm_q1q2_score": 0.7182693522457453}} {"text": "[STATEMENT]\nlemma minus_zero_vec_eq:\n fixes v::\"'a::{ab_group_add} Matrix.vec\"\n assumes \"dim_vec v = n\"\n and \"dim_vec w = n\" \n and \"v - w = 0\\<^sub>v n\"\nshows \"v = w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v = w\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. v = w\n[PROOF STEP]\nhave \"v = v - w + w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v = v - w + w\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec v = n\ndim_vec w = n\nv - w = 0\\<^sub>v n\n\ngoal (1 subgoal):\n 1. v = v - w + w\n[PROOF STEP]\nby (metis carrier_vec_dim_vec comm_add_vec left_zero_vec \n minus_add_minus_vec minus_cancel_vec uminus_eq_vec \n zero_minus_vec)\n[PROOF STATE]\nproof (state)\nthis:\nv = v - w + w\n\ngoal (1 subgoal):\n 1. v = w\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nv = v - w + w\n\ngoal (1 subgoal):\n 1. v = w\n[PROOF STEP]\nhave \"... = 0\\<^sub>v n + w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v - w + w = 0\\<^sub>v n + w\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec v = n\ndim_vec w = n\nv - w = 0\\<^sub>v n\n\ngoal (1 subgoal):\n 1. v - w + w = 0\\<^sub>v n + w\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nv - w + w = 0\\<^sub>v n + w\n\ngoal (1 subgoal):\n 1. v = w\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nv - w + w = 0\\<^sub>v n + w\n\ngoal (1 subgoal):\n 1. v = w\n[PROOF STEP]\nhave \"... = w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0\\<^sub>v n + w = w\n[PROOF STEP]\nusing assms left_zero_vec[of w n]\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec v = n\ndim_vec w = n\nv - w = 0\\<^sub>v n\nw \\ carrier_vec n \\ 0\\<^sub>v n + w = w\n\ngoal (1 subgoal):\n 1. 0\\<^sub>v n + w = w\n[PROOF STEP]\nby (metis carrier_vec_dim_vec)\n[PROOF STATE]\nproof (state)\nthis:\n0\\<^sub>v n + w = w\n\ngoal (1 subgoal):\n 1. v = w\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nv = w\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nv = w\n\ngoal (1 subgoal):\n 1. v = w\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nv = w\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1026, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267118026095992, "lm_q2_score": 0.8688267779364222, "lm_q1q2_score": 0.7182693517433095}} {"text": "[STATEMENT]\nlemma ordered_map_list_all:\n assumes \"finite S\" \"is_map S\"\n shows \"list_all P (ordered_map S) = Ball S P\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. list_all P (ordered_map S) = Ball S P\n[PROOF STEP]\nunfolding list_all_iff\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Ball (set (ordered_map S)) P = Ball S P\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\nis_map S\n\ngoal (1 subgoal):\n 1. Ball (set (ordered_map S)) P = Ball S P\n[PROOF STEP]\nby (simp add: ordered_map_set_eq)", "meta": {"llama_tokens": 224, "file": "CakeML_Codegen_Utils_Compiler_Utils", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267898240861, "lm_q2_score": 0.8267117898012104, "lm_q1q2_score": 0.7182693504427103}} {"text": "[STATEMENT]\nlemma take_drop: \"take n (drop m xs) = drop m (take (n + m) xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. take n (drop m xs) = drop m (take (n + m) xs)\n[PROOF STEP]\nproof (induct m arbitrary: xs n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\xs n. take n (drop 0 xs) = drop 0 (take (n + 0) xs)\n 2. \\m xs n. (\\xs n. take n (drop m xs) = drop m (take (n + m) xs)) \\ take n (drop (Suc m) xs) = drop (Suc m) (take (n + Suc m) xs)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\xs n. take n (drop 0 xs) = drop 0 (take (n + 0) xs)\n 2. \\m xs n. (\\xs n. take n (drop m xs) = drop m (take (n + m) xs)) \\ take n (drop (Suc m) xs) = drop (Suc m) (take (n + Suc m) xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. take n (drop 0 xs) = drop 0 (take (n + 0) xs)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ntake n (drop 0 xs) = drop 0 (take (n + 0) xs)\n\ngoal (1 subgoal):\n 1. \\m xs n. (\\xs n. take n (drop m xs) = drop m (take (n + m) xs)) \\ take n (drop (Suc m) xs) = drop (Suc m) (take (n + Suc m) xs)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m xs n. (\\xs n. take n (drop m xs) = drop m (take (n + m) xs)) \\ take n (drop (Suc m) xs) = drop (Suc m) (take (n + Suc m) xs)\n[PROOF STEP]\ncase Suc\n[PROOF STATE]\nproof (state)\nthis:\ntake ?n (drop m_ ?xs) = drop m_ (take (?n + m_) ?xs)\n\ngoal (1 subgoal):\n 1. \\m xs n. (\\xs n. take n (drop m xs) = drop m (take (n + m) xs)) \\ take n (drop (Suc m) xs) = drop (Suc m) (take (n + Suc m) xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ntake ?n (drop m_ ?xs) = drop m_ (take (?n + m_) ?xs)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ntake ?n (drop m_ ?xs) = drop m_ (take (?n + m_) ?xs)\n\ngoal (1 subgoal):\n 1. take n (drop (Suc m_) xs) = drop (Suc m_) (take (n + Suc m_) xs)\n[PROOF STEP]\nby (cases xs; cases n) simp_all\n[PROOF STATE]\nproof (state)\nthis:\ntake n (drop (Suc m_) xs) = drop (Suc m_) (take (n + Suc m_) xs)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1017, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267830311354, "lm_q2_score": 0.8267117876664789, "lm_q1q2_score": 0.7182693429721859}} {"text": "[STATEMENT]\nlemma op_norm_le_sum_column: \"\\A\\\\<^sub>o\\<^sub>p \\ (\\i\\UNIV. \\column i A\\)\" for A :: \"real^'n^'m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nproof(unfold op_norm_def, rule cSup_least[OF op_norm_set_proptys(3)], clarsimp)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\xa. \\xa\\ = 1 \\ \\A *v xa\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nfix x :: \"real^'n\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\xa. \\xa\\ = 1 \\ \\A *v xa\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nassume x_def:\"\\x\\ = 1\"\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ = 1\n\ngoal (1 subgoal):\n 1. \\xa. \\xa\\ = 1 \\ \\A *v xa\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nhence x_hyp:\"\\i. \\x $ i\\ \\ 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\ = 1\n\ngoal (1 subgoal):\n 1. \\i. \\x $ i\\ \\ 1\n[PROOF STEP]\nby (simp add: norm_bound_component_le_cart)\n[PROOF STATE]\nproof (state)\nthis:\n\\x $ ?i\\ \\ 1\n\ngoal (1 subgoal):\n 1. \\xa. \\xa\\ = 1 \\ \\A *v xa\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nhave \"(\\A *v x\\) = \\(\\i\\UNIV. x $ i *s column i A)\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A *v x\\ = \\\\i\\UNIV. x $ i *s column i A\\\n[PROOF STEP]\nby(subst matrix_mult_sum[of A], simp)\n[PROOF STATE]\nproof (state)\nthis:\n\\A *v x\\ = \\\\i\\UNIV. x $ i *s column i A\\\n\ngoal (1 subgoal):\n 1. \\xa. \\xa\\ = 1 \\ \\A *v xa\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\A *v x\\ = \\\\i\\UNIV. x $ i *s column i A\\\n\ngoal (1 subgoal):\n 1. \\xa. \\xa\\ = 1 \\ \\A *v xa\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nhave \"... \\ (\\i\\UNIV. \\x $ i *s column i A\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\i\\UNIV. x $ i *s column i A\\ \\ (\\i\\UNIV. \\x $ i *s column i A\\)\n[PROOF STEP]\nby (simp add: sum_norm_le)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\i\\UNIV. x $ i *s column i A\\ \\ (\\i\\UNIV. \\x $ i *s column i A\\)\n\ngoal (1 subgoal):\n 1. \\xa. \\xa\\ = 1 \\ \\A *v xa\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\i\\UNIV. x $ i *s column i A\\ \\ (\\i\\UNIV. \\x $ i *s column i A\\)\n\ngoal (1 subgoal):\n 1. \\xa. \\xa\\ = 1 \\ \\A *v xa\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nhave \"... = (\\i\\UNIV. (\\x $ i\\) * (\\column i A\\))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\UNIV. \\x $ i *s column i A\\) = (\\i\\UNIV. \\x $ i\\ * \\column i A\\)\n[PROOF STEP]\nby (simp add: mult_norm_matrix_sgn_eq)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\UNIV. \\x $ i *s column i A\\) = (\\i\\UNIV. \\x $ i\\ * \\column i A\\)\n\ngoal (1 subgoal):\n 1. \\xa. \\xa\\ = 1 \\ \\A *v xa\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\UNIV. \\x $ i *s column i A\\) = (\\i\\UNIV. \\x $ i\\ * \\column i A\\)\n\ngoal (1 subgoal):\n 1. \\xa. \\xa\\ = 1 \\ \\A *v xa\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nhave \"... \\ (\\i\\UNIV. \\column i A\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\UNIV. \\x $ i\\ * \\column i A\\) \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nusing x_hyp\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x $ ?i\\ \\ 1\n\ngoal (1 subgoal):\n 1. (\\i\\UNIV. \\x $ i\\ * \\column i A\\) \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nby (simp add: mult_left_le_one_le sum_mono)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\UNIV. \\x $ i\\ * \\column i A\\) \\ (\\i\\UNIV. \\column i A\\)\n\ngoal (1 subgoal):\n 1. \\xa. \\xa\\ = 1 \\ \\A *v xa\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\A *v x\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\nshow \"\\A *v x\\ \\ (\\i\\UNIV. \\column i A\\)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A *v x\\ \\ (\\i\\UNIV. \\column i A\\)\n\ngoal (1 subgoal):\n 1. \\A *v x\\ \\ (\\i\\UNIV. \\column i A\\)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\A *v x\\ \\ (\\i\\UNIV. \\column i A\\)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2510, "file": "Matrices_for_ODEs_MTX_Norms", "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267762381843, "lm_q2_score": 0.8267117919359419, "lm_q1q2_score": 0.718269341065797}} {"text": "[STATEMENT]\nlemma double_sum_split_case: \n assumes \"finite A\"\n shows \"(\\ i \\ A . (\\ j \\ A . f i j)) = (\\ i \\ A . (f i i)) + (\\ i \\ A . (\\ j \\ (A - {i}) . f i j))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\A. sum (f i) A) = (\\i\\A. f i i) + (\\i\\A. sum (f i) (A - {i}))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\i\\A. sum (f i) A) = (\\i\\A. f i i) + (\\i\\A. sum (f i) (A - {i}))\n[PROOF STEP]\nhave \"\\ i. i \\ A \\ (\\ j \\ A . f i j) = f i i + (\\ j \\ (A - {i}) . f i j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i \\ A \\ sum (f i) A = f i i + sum (f i) (A - {i})\n[PROOF STEP]\nusing sum.remove assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite ?A; ?x \\ ?A\\ \\ sum ?g ?A = ?g ?x + sum ?g (?A - {?x})\nfinite A\n\ngoal (1 subgoal):\n 1. \\i. i \\ A \\ sum (f i) A = f i i + sum (f i) (A - {i})\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\n?i1 \\ A \\ sum (f ?i1) A = f ?i1 ?i1 + sum (f ?i1) (A - {?i1})\n\ngoal (1 subgoal):\n 1. (\\i\\A. sum (f i) A) = (\\i\\A. f i i) + (\\i\\A. sum (f i) (A - {i}))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n?i1 \\ A \\ sum (f ?i1) A = f ?i1 ?i1 + sum (f ?i1) (A - {?i1})\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n?i1 \\ A \\ sum (f ?i1) A = f ?i1 ?i1 + sum (f ?i1) (A - {?i1})\n\ngoal (1 subgoal):\n 1. (\\i\\A. sum (f i) A) = (\\i\\A. f i i) + (\\i\\A. sum (f i) (A - {i}))\n[PROOF STEP]\nby (simp add: sum.distrib)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\A. sum (f i) A) = (\\i\\A. f i i) + (\\i\\A. sum (f i) (A - {i}))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 945, "file": "Fishers_Inequality_Set_Multiset_Extras", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.868826769445233, "lm_q2_score": 0.8267117962054049, "lm_q1q2_score": 0.7182693391594077}} {"text": "[STATEMENT]\nlemma det_four_block_mat_lower_left_zero: fixes A1 :: \"'a :: idom mat\" \n assumes A1: \"A1 \\ carrier_mat n n\"\n and A2: \"A2 \\ carrier_mat n m\" and A30: \"A3 = 0\\<^sub>m m n\"\n and A4: \"A4 \\ carrier_mat m m\" \nshows \"det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nhave A3: \"A3 \\ carrier_mat m n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A3 \\ carrier_mat m n\n[PROOF STEP]\nusing A30\n[PROOF STATE]\nproof (prove)\nusing this:\nA3 = 0\\<^sub>m m n\n\ngoal (1 subgoal):\n 1. A3 \\ carrier_mat m n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA3 \\ carrier_mat m n\n\ngoal (1 subgoal):\n 1. det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\napply (subst det_transpose[OF four_block_carrier_mat[OF A1 A4], symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (four_block_mat A1 A2 A3 A4)\\<^sup>T = det A1 * det A4\n[PROOF STEP]\napply (subst transpose_four_block_mat[OF A1 A2 A3 A4])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (four_block_mat A1\\<^sup>T A3\\<^sup>T A2\\<^sup>T A4\\<^sup>T) = det A1 * det A4\n[PROOF STEP]\napply (subst det_four_block_mat_upper_right_zero[of _ n _ m], \n insert A1 A2 A30 A4, auto simp: det_transpose)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\ndet (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 833, "file": "Jordan_Normal_Form_Determinant", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637648915618, "lm_q2_score": 0.8354835350552603, "lm_q1q2_score": 0.7182349212505162}} {"text": "[STATEMENT]\nlemma absolutely_integrable_mult_Fejer_kernel_reflected_part5:\n assumes f: \"f absolutely_integrable_on {-pi..pi}\"\n and periodic: \"\\x. f(x + 2*pi) = f x\" and \"d \\ pi\"\n shows \"(\\x. Fejer_kernel n x * ((f(t + x) + f(t - x)) - c)) absolutely_integrable_on {0..d}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. Fejer_kernel n x * (f (t + x) + f (t - x) - c)) absolutely_integrable_on {0..d}\n[PROOF STEP]\nunfolding distrib_left right_diff_distrib\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. Fejer_kernel n x * f (t + x) + Fejer_kernel n x * f (t - x) - Fejer_kernel n x * c) absolutely_integrable_on {0..d}\n[PROOF STEP]\nby (intro set_integral_add set_integral_diff absolutely_integrable_on_const\n absolutely_integrable_mult_Fejer_kernel_reflected_part1 absolutely_integrable_mult_Fejer_kernel_reflected_part2 assms, auto)", "meta": {"llama_tokens": 361, "file": "Fourier_Fourier", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587993853655, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.718176852701835}} {"text": "[STATEMENT]\nlemma continuous_sgn:\n fixes f :: \"'a::t2_space \\ 'b::real_normed_vector\"\n assumes \"continuous F f\"\n and \"f (Lim F (\\x. x)) \\ 0\"\n shows \"continuous F (\\x. sgn (f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous F (\\x. sgn (f x))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous F f\nf (Lim F (\\x. x)) \\ (0::'b)\n\ngoal (1 subgoal):\n 1. continuous F (\\x. sgn (f x))\n[PROOF STEP]\nunfolding continuous_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ f (Lim F (\\x. x))) F\nf (Lim F (\\x. x)) \\ (0::'b)\n\ngoal (1 subgoal):\n 1. ((\\x. sgn (f x)) \\ sgn (f (Lim F (\\x. x)))) F\n[PROOF STEP]\nby (rule tendsto_sgn)", "meta": {"llama_tokens": 344, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588052782736, "lm_q2_score": 0.8080672112416737, "lm_q1q2_score": 0.7181768492476962}} {"text": "[STATEMENT]\nlemma ringhom1:\"\\ Ring A; Ring R; x \\ carrier A; y \\ carrier A;\n f \\ rHom A R \\ \\ f (x \\\\<^bsub>A\\<^esub> y) = (f x) \\\\<^bsub>R\\<^esub> (f y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Ring A; Ring R; x \\ carrier A; y \\ carrier A; f \\ rHom A R\\ \\ f (x \\\\<^bsub>A\\<^esub> y) = f x \\\\<^bsub>R\\<^esub> f y\n[PROOF STEP]\napply (simp add:rHom_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Ring A; Ring R; x \\ carrier A; y \\ carrier A; f \\ aHom A R \\ (\\x\\carrier A. \\y\\carrier A. f (x \\\\<^sub>r\\<^bsub>A\\<^esub> y) = f x \\\\<^sub>r\\<^bsub>R\\<^esub> f y) \\ f 1\\<^sub>r\\<^bsub>A\\<^esub> = 1\\<^sub>r\\<^bsub>R\\<^esub>\\ \\ f (x \\\\<^bsub>A\\<^esub> y) = f x \\\\<^bsub>R\\<^esub> f y\n[PROOF STEP]\napply (erule conjE)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Ring A; Ring R; x \\ carrier A; y \\ carrier A; f \\ aHom A R; (\\x\\carrier A. \\y\\carrier A. f (x \\\\<^sub>r\\<^bsub>A\\<^esub> y) = f x \\\\<^sub>r\\<^bsub>R\\<^esub> f y) \\ f 1\\<^sub>r\\<^bsub>A\\<^esub> = 1\\<^sub>r\\<^bsub>R\\<^esub>\\ \\ f (x \\\\<^bsub>A\\<^esub> y) = f x \\\\<^bsub>R\\<^esub> f y\n[PROOF STEP]\napply (frule Ring.ring_is_ag [of \"A\"])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Ring A; Ring R; x \\ carrier A; y \\ carrier A; f \\ aHom A R; (\\x\\carrier A. \\y\\carrier A. f (x \\\\<^sub>r\\<^bsub>A\\<^esub> y) = f x \\\\<^sub>r\\<^bsub>R\\<^esub> f y) \\ f 1\\<^sub>r\\<^bsub>A\\<^esub> = 1\\<^sub>r\\<^bsub>R\\<^esub>; aGroup A\\ \\ f (x \\\\<^bsub>A\\<^esub> y) = f x \\\\<^bsub>R\\<^esub> f y\n[PROOF STEP]\napply (frule Ring.ring_is_ag [of \"R\"])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Ring A; Ring R; x \\ carrier A; y \\ carrier A; f \\ aHom A R; (\\x\\carrier A. \\y\\carrier A. f (x \\\\<^sub>r\\<^bsub>A\\<^esub> y) = f x \\\\<^sub>r\\<^bsub>R\\<^esub> f y) \\ f 1\\<^sub>r\\<^bsub>A\\<^esub> = 1\\<^sub>r\\<^bsub>R\\<^esub>; aGroup A; aGroup R\\ \\ f (x \\\\<^bsub>A\\<^esub> y) = f x \\\\<^bsub>R\\<^esub> f y\n[PROOF STEP]\napply (rule aHom_add, assumption+)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1146, "file": "Group-Ring-Module_Algebra4", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587964389112, "lm_q2_score": 0.8080672135527631, "lm_q1q2_score": 0.7181768441588984}} {"text": "[STATEMENT]\nlemma nth_mset_commute_mono:\n assumes \"mono f\"\n assumes \"k < size M\"\n shows \"f (nth_mset k M) = nth_mset k (image_mset f M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (nth_mset k M) = nth_mset k (image_mset f M)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. f (nth_mset k M) = nth_mset k (image_mset f M)\n[PROOF STEP]\nhave a:\"k < length (sorted_list_of_multiset M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k < length (sorted_list_of_multiset M)\n[PROOF STEP]\nby (metis assms(2) mset_sorted_list_of_multiset size_mset)\n[PROOF STATE]\nproof (state)\nthis:\nk < length (sorted_list_of_multiset M)\n\ngoal (1 subgoal):\n 1. f (nth_mset k M) = nth_mset k (image_mset f M)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (nth_mset k M) = nth_mset k (image_mset f M)\n[PROOF STEP]\nusing a\n[PROOF STATE]\nproof (prove)\nusing this:\nk < length (sorted_list_of_multiset M)\n\ngoal (1 subgoal):\n 1. f (nth_mset k M) = nth_mset k (image_mset f M)\n[PROOF STEP]\nby (simp add:nth_mset_def sorted_list_of_multiset_image_commute[OF assms(1)])\n[PROOF STATE]\nproof (state)\nthis:\nf (nth_mset k M) = nth_mset k (image_mset f M)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 580, "file": "Frequency_Moments_K_Smallest", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587905460027, "lm_q2_score": 0.8080672181749422, "lm_q1q2_score": 0.7181768435050345}} {"text": "[STATEMENT]\nlemma continuous_sgn:\n fixes f :: \"'a::t2_space \\ 'b::real_normed_vector\"\n assumes \"continuous F f\"\n and \"f (Lim F (\\x. x)) \\ 0\"\n shows \"continuous F (\\x. sgn (f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous F (\\x. sgn (f x))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous F f\nf (Lim F (\\x. x)) \\ (0::'b)\n\ngoal (1 subgoal):\n 1. continuous F (\\x. sgn (f x))\n[PROOF STEP]\nunfolding continuous_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ f (Lim F (\\x. x))) F\nf (Lim F (\\x. x)) \\ (0::'b)\n\ngoal (1 subgoal):\n 1. ((\\x. sgn (f x)) \\ sgn (f (Lim F (\\x. x)))) F\n[PROOF STEP]\nby (rule tendsto_sgn)", "meta": {"llama_tokens": 344, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588052782736, "lm_q2_score": 0.808067204308405, "lm_q1q2_score": 0.7181768430856925}} {"text": "[STATEMENT]\nlemma integral_combine_tagged_division_topdown:\n fixes f :: \"'n::euclidean_space \\ 'a::banach\"\n assumes \"f integrable_on cbox a b\"\n and \"p tagged_division_of (cbox a b)\"\n shows \"integral (cbox a b) f = sum (\\(x,k). integral k f) p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral (cbox a b) f = (\\(x, k)\\p. integral k f)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nf integrable_on cbox a b\np tagged_division_of cbox a b\n\ngoal (1 subgoal):\n 1. integral (cbox a b) f = (\\(x, k)\\p. integral k f)\n[PROOF STEP]\nby (auto intro: integral_unique [OF has_integral_combine_tagged_division_topdown])", "meta": {"llama_tokens": 275, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587875995482, "lm_q2_score": 0.8080672135527632, "lm_q1q2_score": 0.718176837016099}} {"text": "[STATEMENT]\nlemma spectrum_char_poly_root: \n fixes A::\"complex Matrix.mat\"\n assumes \"A\\ carrier_mat n n\"\nand \"k \\ spectrum A\"\nshows \"poly (char_poly A) k = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly (char_poly A) k = 0\n[PROOF STEP]\nusing eigvals_poly_length[of A n] assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n n \\ char_poly A = (\\a\\eigvals A. [:- a, 1:]) \\ length (eigvals A) = dim_row A\nA \\ carrier_mat n n\nk \\ spectrum A\n\ngoal (1 subgoal):\n 1. poly (char_poly A) k = 0\n[PROOF STEP]\nunfolding spectrum_def eigenvalue_root_char_poly\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n n \\ char_poly A = (\\a\\eigvals A. [:- a, 1:]) \\ length (eigvals A) = dim_row A\nA \\ carrier_mat n n\nk \\ set (eigvals A)\n\ngoal (1 subgoal):\n 1. poly (char_poly A) k = 0\n[PROOF STEP]\nby (simp add: linear_poly_root)", "meta": {"llama_tokens": 408, "file": "Projective_Measurements_Projective_Measurements", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.888758793492457, "lm_q2_score": 0.8080672066194945, "lm_q1q2_score": 0.7181768356159619}} {"text": "[STATEMENT]\nlemma absolutely_integrable_Gamma_integral':\n assumes \"Re z > 0\"\n shows \"(\\t. complex_of_real t powr (z - 1) / of_real (exp t)) absolutely_integrable_on {0<..}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_set_integrable lebesgue {0<..} (\\t. complex_of_real t powr (z - 1) / complex_of_real (exp t))\n[PROOF STEP]\nusing absolutely_integrable_Gamma_integral [OF assms zero_less_one]\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_set_integrable lebesgue {0<..} (\\t. complex_of_real t powr (z - 1) / complex_of_real (exp (1 * t)))\n\ngoal (1 subgoal):\n 1. complex_set_integrable lebesgue {0<..} (\\t. complex_of_real t powr (z - 1) / complex_of_real (exp t))\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 302, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.8080672043084051, "lm_q1q2_score": 0.7181768335619607}} {"text": "[STATEMENT]\nlemma \"((\\x::real. (ln(ln x + ln (ln x)) - ln (ln x)) / \n (ln (ln x + ln (ln (ln x)))) * ln x) \\ 1) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. (ln (ln x + ln (ln x)) - ln (ln x)) / ln (ln x + ln (ln (ln x))) * ln x) \\ 1) at_top\n[PROOF STEP]\nby real_asymp", "meta": {"llama_tokens": 164, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9304582593509315, "lm_q2_score": 0.7718434978390747, "lm_q1q2_score": 0.7181681574906799}} {"text": "[STATEMENT]\nlemma lfp_while_lattice:\n fixes f :: \"'a::complete_lattice \\ 'a\"\n assumes \"mono f\" and \"finite (UNIV :: 'a set)\"\n shows \"lfp f = while (\\A. f A \\ A) f bot\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lfp f = while (\\A. f A \\ A) f bot\n[PROOF STEP]\nunfolding while_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lfp f = the (while_option (\\A. f A \\ A) f bot)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nmono f\nfinite UNIV\n\ngoal (1 subgoal):\n 1. lfp f = the (while_option (\\A. f A \\ A) f bot)\n[PROOF STEP]\nby (rule lfp_the_while_option_lattice)", "meta": {"llama_tokens": 282, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970779778824, "lm_q2_score": 0.8152324826183822, "lm_q1q2_score": 0.7180543885629259}} {"text": "[STATEMENT]\nlemma csorted_list_of_set_set [simp]:\n \"\\ ID CCOMPARE('a :: ccompare) = Some c; linorder.sorted (le_of_comp c) xs; distinct xs \\ \n \\ linorder.sorted_list_of_set (le_of_comp c) (set xs) = xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ID ccompare = Some c; sorted_wrt (le_of_comp c) xs; distinct xs\\ \\ linorder.sorted_list_of_set (le_of_comp c) (set xs) = xs\n[PROOF STEP]\nby(simp add: distinct_remdups_id linorder.sorted_list_of_set_sort_remdups[OF ID_ccompare] linorder.sorted_sort_id[OF ID_ccompare])", "meta": {"llama_tokens": 236, "file": "Containers_Set_Impl", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869916479466, "lm_q2_score": 0.8198933447152497, "lm_q1q2_score": 0.7180519258403414}} {"text": "[STATEMENT]\nlemma (in comm_monoid_add) sum_list_plus :\n \"length xs = length ys\n \\ sum_list xs + sum_list ys = sum_list [a+b. (a,b)\\zip xs ys]\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length xs = length ys \\ sum_list xs + sum_list ys = sum_list (map2 (+) xs ys)\n[PROOF STEP]\nproof (induct xs ys rule: list_induct2)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. sum_list [] + sum_list [] = sum_list (map2 (+) [] [])\n 2. \\x xs y ys. \\length xs = length ys; sum_list xs + sum_list ys = sum_list (map2 (+) xs ys)\\ \\ sum_list (x # xs) + sum_list (y # ys) = sum_list (map2 (+) (x # xs) (y # ys))\n[PROOF STEP]\ncase Cons\n[PROOF STATE]\nproof (state)\nthis:\nlength xs_ = length ys_\nsum_list xs_ + sum_list ys_ = sum_list (map2 (+) xs_ ys_)\n\ngoal (2 subgoals):\n 1. sum_list [] + sum_list [] = sum_list (map2 (+) [] [])\n 2. \\x xs y ys. \\length xs = length ys; sum_list xs + sum_list ys = sum_list (map2 (+) xs ys)\\ \\ sum_list (x # xs) + sum_list (y # ys) = sum_list (map2 (+) (x # xs) (y # ys))\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nlength xs_ = length ys_\nsum_list xs_ + sum_list ys_ = sum_list (map2 (+) xs_ ys_)\n\ngoal (1 subgoal):\n 1. sum_list (x_ # xs_) + sum_list (y_ # ys_) = sum_list (map2 (+) (x_ # xs_) (y_ # ys_))\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nsum_list (x_ # xs_) + sum_list (y_ # ys_) = sum_list (map2 (+) (x_ # xs_) (y_ # ys_))\n\ngoal (1 subgoal):\n 1. sum_list [] + sum_list [] = sum_list (map2 (+) [] [])\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 722, "file": "Rep_Fin_Groups_Rep_Fin_Groups", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869948899665, "lm_q2_score": 0.8198933359135361, "lm_q1q2_score": 0.7180519207900256}} {"text": "[STATEMENT]\nlemma insertion_sum_var : \"((insertion f (\\i\\{0..<(n::nat)}.g(i))) = (\\i\\{0..i = 0..i = 0..<0. insertion f (g i))\n 2. \\n. insertion f (sum g {0..i = 0.. insertion f (sum g {0..i = 0..i = 0..<0. insertion f (g i))\n 2. \\n. insertion f (sum g {0..i = 0.. insertion f (sum g {0..i = 0..i = 0..<0. insertion f (g i))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninsertion f (sum g {0..<0}) = (\\i = 0..<0. insertion f (g i))\n\ngoal (1 subgoal):\n 1. \\n. insertion f (sum g {0..i = 0.. insertion f (sum g {0..i = 0..n. insertion f (sum g {0..i = 0.. insertion f (sum g {0..i = 0..i = 0..n. insertion f (sum g {0..i = 0.. insertion f (sum g {0..i = 0..i = 0..i = 0..i = 0..i = 0..A \\ B; B \\ C; B \\ {}\\ \\ A \\ C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ B; B \\ C; B \\ {}\\ \\ A \\ C\n[PROOF STEP]\nunfolding less_sets_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x\\A. \\y\\B. x < y; \\x\\B. \\y\\C. x < y; B \\ {}\\ \\ \\x\\A. \\y\\C. x < y\n[PROOF STEP]\nusing less_trans\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x < ?y; ?y < ?z\\ \\ ?x < ?z\n\ngoal (1 subgoal):\n 1. \\\\x\\A. \\y\\B. x < y; \\x\\B. \\y\\C. x < y; B \\ {}\\ \\ \\x\\A. \\y\\C. x < y\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 410, "file": "Nash_Williams_Nash_Extras", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869851639066, "lm_q2_score": 0.8198933337131077, "lm_q1q2_score": 0.7180519108885874}} {"text": "[STATEMENT]\nlemma prob0_fst_qubits_eq:\n fixes n:: nat\n shows \"prob0_fst_qubits n v = (cmod(v $$ (0,0)))\\<^sup>2 + (cmod(v $$ (1,0)))\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prob0_fst_qubits n v = (cmod (v $$ (0, 0)))\\<^sup>2 + (cmod (v $$ (1, 0)))\\<^sup>2\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. prob0_fst_qubits n v = (cmod (v $$ (0, 0)))\\<^sup>2 + (cmod (v $$ (1, 0)))\\<^sup>2\n[PROOF STEP]\nhave \"prob0_fst_qubits n v = (\\j\\{k| k::nat. (k<2^(n+1)) \\ (\\i\\{0.. select_index (n+1) i k)}. (cmod(v $$ (j,0)))\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prob0_fst_qubits n v = (\\j\\{k |k. k < 2 ^ (n + 1) \\ (\\i\\{0.. select_index (n + 1) i k)}. (cmod (v $$ (j, 0)))\\<^sup>2)\n[PROOF STEP]\nusing prob0_fst_qubits_def\n[PROOF STATE]\nproof (prove)\nusing this:\nprob0_fst_qubits ?n ?v \\ \\j\\{k |k. k < 2 ^ (?n + 1) \\ (\\i\\{0.. select_index (?n + 1) i k)}. (cmod (?v $$ (j, 0)))\\<^sup>2\n\ngoal (1 subgoal):\n 1. prob0_fst_qubits n v = (\\j\\{k |k. k < 2 ^ (n + 1) \\ (\\i\\{0.. select_index (n + 1) i k)}. (cmod (v $$ (j, 0)))\\<^sup>2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nprob0_fst_qubits n v = (\\j\\{k |k. k < 2 ^ (n + 1) \\ (\\i\\{0.. select_index (n + 1) i k)}. (cmod (v $$ (j, 0)))\\<^sup>2)\n\ngoal (1 subgoal):\n 1. prob0_fst_qubits n v = (cmod (v $$ (0, 0)))\\<^sup>2 + (cmod (v $$ (1, 0)))\\<^sup>2\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nprob0_fst_qubits n v = (\\j\\{k |k. k < 2 ^ (n + 1) \\ (\\i\\{0.. select_index (n + 1) i k)}. (cmod (v $$ (j, 0)))\\<^sup>2)\n\ngoal (1 subgoal):\n 1. prob0_fst_qubits n v = (cmod (v $$ (0, 0)))\\<^sup>2 + (cmod (v $$ (1, 0)))\\<^sup>2\n[PROOF STEP]\nhave \"\\ = (\\j\\{0,1}. (cmod(v $$ (j,0)))\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j\\{k |k. k < 2 ^ (n + 1) \\ (\\i\\{0.. select_index (n + 1) i k)}. (cmod (v $$ (j, 0)))\\<^sup>2) = (\\j\\{0, 1}. (cmod (v $$ (j, 0)))\\<^sup>2)\n[PROOF STEP]\nusing prob0_fst_qubits_index\n[PROOF STATE]\nproof (prove)\nusing this:\n{k |k. k < 2 ^ (?n + 1) \\ (\\i\\{0.. select_index (?n + 1) i k)} = {0, 1}\n\ngoal (1 subgoal):\n 1. (\\j\\{k |k. k < 2 ^ (n + 1) \\ (\\i\\{0.. select_index (n + 1) i k)}. (cmod (v $$ (j, 0)))\\<^sup>2) = (\\j\\{0, 1}. (cmod (v $$ (j, 0)))\\<^sup>2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\{k |k. k < 2 ^ (n + 1) \\ (\\i\\{0.. select_index (n + 1) i k)}. (cmod (v $$ (j, 0)))\\<^sup>2) = (\\j\\{0, 1}. (cmod (v $$ (j, 0)))\\<^sup>2)\n\ngoal (1 subgoal):\n 1. prob0_fst_qubits n v = (cmod (v $$ (0, 0)))\\<^sup>2 + (cmod (v $$ (1, 0)))\\<^sup>2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nprob0_fst_qubits n v = (\\j\\{0, 1}. (cmod (v $$ (j, 0)))\\<^sup>2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nprob0_fst_qubits n v = (\\j\\{0, 1}. (cmod (v $$ (j, 0)))\\<^sup>2)\n\ngoal (1 subgoal):\n 1. prob0_fst_qubits n v = (cmod (v $$ (0, 0)))\\<^sup>2 + (cmod (v $$ (1, 0)))\\<^sup>2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nprob0_fst_qubits n v = (cmod (v $$ (0, 0)))\\<^sup>2 + (cmod (v $$ (1, 0)))\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1846, "file": "Isabelle_Marries_Dirac_Measurement", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382129861583, "lm_q2_score": 0.8333245891029457, "lm_q1q2_score": 0.7179409773331765}} {"text": "[STATEMENT]\ntheorem insert_list_set: \n assumes \"sorted_less xs\"\n shows \"insert_list x xs = ins_list x xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. insert_list x xs = ins_list x xs\n[PROOF STEP]\nusing assms split_list_conc\n[PROOF STATE]\nproof (prove)\nusing this:\nsorted_less xs\nsplit_list ?ks ?p = (?kls, ?krs) \\ ?ks = ?kls @ ?krs\n\ngoal (1 subgoal):\n 1. insert_list x xs = ins_list x xs\n[PROOF STEP]\nusing insert_sorted_split_list[of xs x]\n[PROOF STATE]\nproof (prove)\nusing this:\nsorted_less xs\nsplit_list ?ks ?p = (?kls, ?krs) \\ ?ks = ?kls @ ?krs\n\\sorted_less xs; split_list xs x = (?ls, ?rs)\\ \\ ins_list x xs = ?ls @ ins_list x ?rs\n\ngoal (1 subgoal):\n 1. insert_list x xs = ins_list x xs\n[PROOF STEP]\nusing insert_sorted_split_list_right[of xs x]\n[PROOF STATE]\nproof (prove)\nusing this:\nsorted_less xs\nsplit_list ?ks ?p = (?kls, ?krs) \\ ?ks = ?kls @ ?krs\n\\sorted_less xs; split_list xs x = (?ls, ?rs)\\ \\ ins_list x xs = ?ls @ ins_list x ?rs\n\\split_list xs x = (?ls, ?sep # ?rs); sorted_less xs; x \\ ?sep\\ \\ ins_list x (?sep # ?rs) = x # ?sep # ?rs\n\ngoal (1 subgoal):\n 1. insert_list x xs = ins_list x xs\n[PROOF STEP]\nby (auto split!: list.splits prod.splits)", "meta": {"llama_tokens": 537, "file": "BTree_BPlusTree_Set", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382129861584, "lm_q2_score": 0.8333245870332531, "lm_q1q2_score": 0.7179409755500573}} {"text": "[STATEMENT]\nlemma sum_of_odd_squares:\n \"3 * (\\i=0..i = 0..i. \\j\\UNIV. to_fun A i j *s row B j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A * B = upd_rows 0 UNIV (\\i. \\j\\UNIV. to_fun A i j *s Square_Matrix.row B j)\n[PROOF STEP]\nby transfer simp\n[PROOF STATE]\nproof (state)\nthis:\nA * B = upd_rows 0 UNIV (\\i. \\j\\UNIV. to_fun A i j *s Square_Matrix.row B j)\n\ngoal (1 subgoal):\n 1. Square_Matrix.det (A * B) = Square_Matrix.det A * Square_Matrix.det B\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nA * B = upd_rows 0 UNIV (\\i. \\j\\UNIV. to_fun A i j *s Square_Matrix.row B j)\n\ngoal (1 subgoal):\n 1. Square_Matrix.det (A * B) = Square_Matrix.det A * Square_Matrix.det B\n[PROOF STEP]\nhave \"\\f. upd_rows 0 UNIV (\\i. Square_Matrix.row B (f i)) = perm_rows B f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f. upd_rows 0 UNIV (\\i. Square_Matrix.row B (f i)) = perm_rows B f\n[PROOF STEP]\nby transfer simp\n[PROOF STATE]\nproof (state)\nthis:\nupd_rows 0 UNIV (\\i. Square_Matrix.row B (?f i)) = perm_rows B ?f\n\ngoal (1 subgoal):\n 1. Square_Matrix.det (A * B) = Square_Matrix.det A * Square_Matrix.det B\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nupd_rows 0 UNIV (\\i. Square_Matrix.row B (?f i)) = perm_rows B ?f\n\ngoal (1 subgoal):\n 1. Square_Matrix.det (A * B) = Square_Matrix.det A * Square_Matrix.det B\n[PROOF STEP]\nhave \"det A = (\\p | p permutes UNIV. of_int (sign p) * (\\i\\UNIV. to_fun A i (p i)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Square_Matrix.det A = (\\p | p permutes UNIV. of_int (sign p) * (\\i\\UNIV. to_fun A i (p i)))\n[PROOF STEP]\nby transfer rule\n[PROOF STATE]\nproof (state)\nthis:\nSquare_Matrix.det A = (\\p | p permutes UNIV. of_int (sign p) * (\\i\\UNIV. to_fun A i (p i)))\n\ngoal (1 subgoal):\n 1. Square_Matrix.det (A * B) = Square_Matrix.det A * Square_Matrix.det B\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nA * B = upd_rows 0 UNIV (\\i. \\j\\UNIV. to_fun A i j *s Square_Matrix.row B j)\nupd_rows 0 UNIV (\\i. Square_Matrix.row B (?f i)) = perm_rows B ?f\nSquare_Matrix.det A = (\\p | p permutes UNIV. of_int (sign p) * (\\i\\UNIV. to_fun A i (p i)))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nA * B = upd_rows 0 UNIV (\\i. \\j\\UNIV. to_fun A i j *s Square_Matrix.row B j)\nupd_rows 0 UNIV (\\i. Square_Matrix.row B (?f i)) = perm_rows B ?f\nSquare_Matrix.det A = (\\p | p permutes UNIV. of_int (sign p) * (\\i\\UNIV. to_fun A i (p i)))\n\ngoal (1 subgoal):\n 1. Square_Matrix.det (A * B) = Square_Matrix.det A * Square_Matrix.det B\n[PROOF STEP]\nby (auto simp add: det_rows_sum det_rows_mult sum_distrib_right det_perm_rows_If\n split: if_split_asm intro!: sum.mono_neutral_cong_right)\n[PROOF STATE]\nproof (state)\nthis:\nSquare_Matrix.det (A * B) = Square_Matrix.det A * Square_Matrix.det B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1440, "file": "Cayley_Hamilton_Square_Matrix", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314858927011, "lm_q2_score": 0.8104789086703225, "lm_q1q2_score": 0.7177856401703925}} {"text": "[STATEMENT]\nlemma extreme_point_of_convex_hull_eq:\n fixes S :: \"'a::euclidean_space set\"\n shows\n \"\\compact S; \\T. T \\ S \\ convex hull T \\ convex hull S\\\n \\ (x extreme_point_of (convex hull S) \\ x \\ S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\compact S; \\T. T \\ S \\ convex hull T \\ convex hull S\\ \\ (x extreme_point_of convex hull S) = (x \\ S)\n[PROOF STEP]\nusing extreme_points_of_convex_hull_eq\n[PROOF STATE]\nproof (prove)\nusing this:\n\\compact ?S; \\T. T \\ ?S \\ convex hull T \\ convex hull ?S\\ \\ {x. x extreme_point_of convex hull ?S} = ?S\n\ngoal (1 subgoal):\n 1. \\compact S; \\T. T \\ S \\ convex hull T \\ convex hull S\\ \\ (x extreme_point_of convex hull S) = (x \\ S)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 383, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314738181875, "lm_q2_score": 0.8104789155369048, "lm_q1q2_score": 0.7177856364655153}} {"text": "[STATEMENT]\nlemma minus_mtx3: \"mtx \n ([a\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3] # \n [a\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3] # \n [a\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3] # []) - mtx \n ([b\\<^sub>1\\<^sub>1, b\\<^sub>1\\<^sub>2, b\\<^sub>1\\<^sub>3] # \n [b\\<^sub>2\\<^sub>1, b\\<^sub>2\\<^sub>2, b\\<^sub>2\\<^sub>3] # \n [b\\<^sub>3\\<^sub>1, b\\<^sub>3\\<^sub>2, b\\<^sub>3\\<^sub>3] # []) = (mtx \n ([a\\<^sub>1\\<^sub>1-b\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2-b\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3-b\\<^sub>1\\<^sub>3] # \n [a\\<^sub>2\\<^sub>1-b\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2-b\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3-b\\<^sub>2\\<^sub>3] # \n [a\\<^sub>3\\<^sub>1-b\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2-b\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3-b\\<^sub>3\\<^sub>3] # [])::3 sq_mtx)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mtx [[a\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3], [a\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3], [a\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3]] - mtx [[b\\<^sub>1\\<^sub>1, b\\<^sub>1\\<^sub>2, b\\<^sub>1\\<^sub>3], [b\\<^sub>2\\<^sub>1, b\\<^sub>2\\<^sub>2, b\\<^sub>2\\<^sub>3], [b\\<^sub>3\\<^sub>1, b\\<^sub>3\\<^sub>2, b\\<^sub>3\\<^sub>3]] = mtx [[a\\<^sub>1\\<^sub>1 - b\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2 - b\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3 - b\\<^sub>1\\<^sub>3], [a\\<^sub>2\\<^sub>1 - b\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2 - b\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3 - b\\<^sub>2\\<^sub>3], [a\\<^sub>3\\<^sub>1 - b\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2 - b\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3 - b\\<^sub>3\\<^sub>3]]\n[PROOF STEP]\nby (simp add: sq_mtx_eq_iff)", "meta": {"llama_tokens": 864, "file": "Matrices_for_ODEs_SQ_MTX", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314828740729, "lm_q2_score": 0.8104789040926008, "lm_q1q2_score": 0.7177856336696836}} {"text": "[STATEMENT]\nlemma local_lipschitz_therm_dyn:\n assumes \"0 < (a::real)\"\n shows \"local_lipschitz UNIV UNIV (\\t::real. f a L)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local_lipschitz UNIV UNIV (\\t. f a L)\n[PROOF STEP]\napply(unfold local_lipschitz_def lipschitz_on_def dist_norm)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\UNIV. \\t\\UNIV. \\u>0. \\La. \\t\\cball t u \\ UNIV. 0 \\ La \\ (\\xa\\cball x u \\ UNIV. \\y\\cball x u \\ UNIV. \\f a L xa - f a L y\\ \\ La \\ \\xa - y\\)\n[PROOF STEP]\napply(clarsimp, rule_tac x=1 in exI, clarsimp, rule_tac x=a in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a \\ \\xa - y\\)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a \\ \\xa - y\\)\n[PROOF STEP]\napply(simp_all add: norm_diff_therm_dyn)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. \\dist t ta \\ 1; 0 < a\\ \\ \\xa\\cball x 1. \\y\\cball x 1. \\xa $ 1 - y $ 1\\ \\ \\xa - y\\\n[PROOF STEP]\napply(simp add: norm_vec_def L2_set_def, unfold UNIV_4, clarsimp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ \\xa $ 1 - y $ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nunfolding real_sqrt_abs[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2) \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nby (rule real_le_lsqrt) auto", "meta": {"llama_tokens": 1107, "file": "Hybrid_Systems_VCs_KleeneAlgebraTests_HS_VC_KAT_Examples_ndfun", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972616934406, "lm_q2_score": 0.8244619306896956, "lm_q1q2_score": 0.7177742992289363}} {"text": "[STATEMENT]\ntheorem integral_has_vector_derivative':\n fixes f :: \"real \\ 'b::banach\"\n assumes \"continuous_on {a..b} f\"\n and \"x \\ {a..b}\"\n shows \"((\\u. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\u. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\u. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})\n[PROOF STEP]\nhave *: \"integral {x..b} f = integral {a .. b} f - integral {a .. x} f\" if \"a \\ x\" \"x \\ b\" for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral {x..b} f = integral {a..b} f - integral {a..x} f\n[PROOF STEP]\nusing integral_combine[of a x b for x, OF that integrable_continuous_real[OF assms(1)]]\n[PROOF STATE]\nproof (prove)\nusing this:\nintegral {a..x} f + integral {x..b} f = integral {a..b} f\n\ngoal (1 subgoal):\n 1. integral {x..b} f = integral {a..b} f - integral {a..x} f\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\a \\ ?x; ?x \\ b\\ \\ integral {?x..b} f = integral {a..b} f - integral {a..?x} f\n\ngoal (1 subgoal):\n 1. ((\\u. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\u. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})\n[PROOF STEP]\nusing \\x \\ _\\ *\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ {a..b}\n\\a \\ ?x; ?x \\ b\\ \\ integral {?x..b} f = integral {a..b} f - integral {a..?x} f\n\ngoal (1 subgoal):\n 1. ((\\u. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})\n[PROOF STEP]\nby (rule has_vector_derivative_transform)\n (auto intro!: derivative_eq_intros assms integral_has_vector_derivative)\n[PROOF STATE]\nproof (state)\nthis:\n((\\u. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 916, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972684083609, "lm_q2_score": 0.8244619242200081, "lm_q1q2_score": 0.7177742991326401}} {"text": "[STATEMENT]\nlemma ID_cEnum: \n \"ID cEnum = Some (enum, enum_all, enum_ex)\n \\ UNIV = set enum \\ enum_all = Ball UNIV \\ enum_ex = Bex UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ID cEnum = Some (enum, enum_all, enum_ex) \\ UNIV = set enum \\ enum_all = Ball UNIV \\ enum_ex = Bex UNIV\n[PROOF STEP]\nunfolding ID_def id_apply fun_eq_iff\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cEnum = Some (enum, enum_all, enum_ex) \\ UNIV = set enum \\ (\\x. enum_all x = Ball UNIV x) \\ (\\x. enum_ex x = Bex UNIV x)\n[PROOF STEP]\nby(intro conjI allI UNIV_cenum cenum_all_UNIV cenum_ex_UNIV fun_eq_iff)", "meta": {"llama_tokens": 280, "file": "Containers_Collection_Enum", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972684083609, "lm_q2_score": 0.8244619199068831, "lm_q1q2_score": 0.7177742953776453}} {"text": "[STATEMENT]\nlemma closure_approachable_le:\n fixes S :: \"'a::metric_space set\"\n shows \"x \\ closure S \\ (\\e>0. \\y\\S. dist y x \\ e)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x \\ closure S) = (\\e>0. \\y\\S. dist y x \\ e)\n[PROOF STEP]\nunfolding closure_approachable\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\e>0. \\y\\S. dist y x < e) = (\\e>0. \\y\\S. dist y x \\ e)\n[PROOF STEP]\nusing dense\n[PROOF STATE]\nproof (prove)\nusing this:\n?x < ?y \\ \\z>?x. z < ?y\n\ngoal (1 subgoal):\n 1. (\\e>0. \\y\\S. dist y x < e) = (\\e>0. \\y\\S. dist y x \\ e)\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 330, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213826762113, "lm_q2_score": 0.7981867849406659, "lm_q1q2_score": 0.7176668057097312}} {"text": "[STATEMENT]\nlemma Carmichael_prod_coprime:\n assumes \"finite A\" \"\\i j. i \\ A \\ j \\ A \\ i \\ j \\ coprime (f i) (f j)\"\n shows \"Carmichael (\\i\\A. f i) = (LCM i\\A. Carmichael (f i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Carmichael (prod f A) = (LCM i\\A. Carmichael (f i))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\\?i \\ A; ?j \\ A; ?i \\ ?j\\ \\ coprime (f ?i) (f ?j)\n\ngoal (1 subgoal):\n 1. Carmichael (prod f A) = (LCM i\\A. Carmichael (f i))\n[PROOF STEP]\nby (induction A rule: finite_induct)\n (simp, simp, subst Carmichael_mult_coprime[OF prod_coprime_right], auto)", "meta": {"llama_tokens": 329, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213826762113, "lm_q2_score": 0.7981867825403176, "lm_q1q2_score": 0.7176668035515268}} {"text": "[STATEMENT]\nlemma nth_prime_numeral:\n \"nth_prime (numeral n) = smallest_prime_beyond (Suc (nth_prime (pred_numeral n)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. nth_prime (numeral n) = smallest_prime_beyond (Suc (nth_prime (pred_numeral n)))\n[PROOF STEP]\nby (subst nth_prime_Suc[symmetric]) auto", "meta": {"llama_tokens": 125, "file": "Prime_Distribution_Elementary_Prime_Distribution_Elementary_Library", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213826762114, "lm_q2_score": 0.7981867801399695, "lm_q1q2_score": 0.7176668013933225}} {"text": "[STATEMENT]\nlemma convex_comb_dets:\n assumes \"det3 p q r > 0\"\n shows \"s = (det3 s q r / det3 p q r) *\\<^sub>R p + (det3 p s r / det3 p q r) *\\<^sub>R q +\n (det3 p q s / det3 p q r) *\\<^sub>R r\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. s = (det3 s q r / det3 p q r) *\\<^sub>R p + (det3 p s r / det3 p q r) *\\<^sub>R q + (det3 p q s / det3 p q r) *\\<^sub>R r\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. s = (det3 s q r / det3 p q r) *\\<^sub>R p + (det3 p s r / det3 p q r) *\\<^sub>R q + (det3 p q s / det3 p q r) *\\<^sub>R r\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < det3 p q r\n[PROOF STEP]\nhave \"det3 p q r *\\<^sub>R ?lhs = det3 p q r *\\<^sub>R ?rhs\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < det3 p q r\n\ngoal (1 subgoal):\n 1. det3 p q r *\\<^sub>R s = det3 p q r *\\<^sub>R ((det3 s q r / det3 p q r) *\\<^sub>R p + (det3 p s r / det3 p q r) *\\<^sub>R q + (det3 p q s / det3 p q r) *\\<^sub>R r)\n[PROOF STEP]\nby (simp add: field_simps prod_eq_iff scaleR_add_right) (simp add: algebra_simps det3_def')\n[PROOF STATE]\nproof (state)\nthis:\ndet3 p q r *\\<^sub>R s = det3 p q r *\\<^sub>R ((det3 s q r / det3 p q r) *\\<^sub>R p + (det3 p s r / det3 p q r) *\\<^sub>R q + (det3 p q s / det3 p q r) *\\<^sub>R r)\n\ngoal (1 subgoal):\n 1. s = (det3 s q r / det3 p q r) *\\<^sub>R p + (det3 p s r / det3 p q r) *\\<^sub>R q + (det3 p q s / det3 p q r) *\\<^sub>R r\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndet3 p q r *\\<^sub>R s = det3 p q r *\\<^sub>R ((det3 s q r / det3 p q r) *\\<^sub>R p + (det3 p s r / det3 p q r) *\\<^sub>R q + (det3 p q s / det3 p q r) *\\<^sub>R r)\n\ngoal (1 subgoal):\n 1. s = (det3 s q r / det3 p q r) *\\<^sub>R p + (det3 p s r / det3 p q r) *\\<^sub>R q + (det3 p q s / det3 p q r) *\\<^sub>R r\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndet3 p q r *\\<^sub>R s = det3 p q r *\\<^sub>R ((det3 s q r / det3 p q r) *\\<^sub>R p + (det3 p s r / det3 p q r) *\\<^sub>R q + (det3 p q s / det3 p q r) *\\<^sub>R r)\n0 < det3 p q r\n\ngoal (1 subgoal):\n 1. s = (det3 s q r / det3 p q r) *\\<^sub>R p + (det3 p s r / det3 p q r) *\\<^sub>R q + (det3 p q s / det3 p q r) *\\<^sub>R r\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ns = (det3 s q r / det3 p q r) *\\<^sub>R p + (det3 p s r / det3 p q r) *\\<^sub>R q + (det3 p q s / det3 p q r) *\\<^sub>R r\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1185, "file": "Affine_Arithmetic_Counterclockwise_2D_Strict", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213772699436, "lm_q2_score": 0.7981867681382279, "lm_q1q2_score": 0.7176667862870886}} {"text": "[STATEMENT]\nlemma upper_asymptotic_density_zero_lim:\n assumes \"upper_asymptotic_density A = 0\"\n shows \"(\\n. card(A \\ {.. 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. real (card (A \\ {.. 0\n[PROOF STEP]\napply (rule lower_asymptotic_density_eq_upper)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. lower_asymptotic_density A = 0\n 2. upper_asymptotic_density A = 0\n[PROOF STEP]\nusing assms lower_asymptotic_density_le_upper[of A] lower_asymptotic_density_in_01(3)[of A]\n[PROOF STATE]\nproof (prove)\nusing this:\nupper_asymptotic_density A = 0\nlower_asymptotic_density A \\ upper_asymptotic_density A\n0 \\ lower_asymptotic_density A\n\ngoal (2 subgoals):\n 1. lower_asymptotic_density A = 0\n 2. upper_asymptotic_density A = 0\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 377, "file": "Ergodic_Theory_Asymptotic_Density", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513842182777, "lm_q2_score": 0.8006920044739461, "lm_q1q2_score": 0.7176213173422815}} {"text": "[STATEMENT]\nlemma matrix_left_invertible_span_rows_gen:\n \"(\\(B::'a^'m^'n). B ** (A::'a::field^'n^'m) = mat 1) \\ vec.span (rows A) = UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\B. B ** A = mat (1::'a)) = (vec.span (rows A) = UNIV)\n[PROOF STEP]\nunfolding right_invertible_transpose[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\B. Finite_Cartesian_Product.transpose A ** B = mat (1::'a)) = (vec.span (rows A) = UNIV)\n[PROOF STEP]\nunfolding columns_transpose[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\B. Finite_Cartesian_Product.transpose A ** B = mat (1::'a)) = (vec.span (columns (Finite_Cartesian_Product.transpose A)) = UNIV)\n[PROOF STEP]\nunfolding matrix_right_invertible_span_columns\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (vec.span (columns (Finite_Cartesian_Product.transpose A)) = UNIV) = (vec.span (columns (Finite_Cartesian_Product.transpose A)) = UNIV)\n[PROOF STEP]\n..", "meta": {"llama_tokens": 413, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513675912912, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.7176213018979081}} {"text": "[STATEMENT]\nlemma uminus_Inf: \"- (\\A) = \\(uminus ` A)\" for A :: \\'a::complete_orthocomplemented_lattice set\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - \\ A = \\ (uminus ` A)\n[PROOF STEP]\nproof (rule order.antisym)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. - \\ A \\ \\ (uminus ` A)\n 2. \\ (uminus ` A) \\ - \\ A\n[PROOF STEP]\nshow \"- \\A \\ \\(uminus ` A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - \\ A \\ \\ (uminus ` A)\n[PROOF STEP]\nby (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp\n[PROOF STATE]\nproof (state)\nthis:\n- \\ A \\ \\ (uminus ` A)\n\ngoal (1 subgoal):\n 1. \\ (uminus ` A) \\ - \\ A\n[PROOF STEP]\nshow \"\\(uminus ` A) \\ - \\A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (uminus ` A) \\ - \\ A\n[PROOF STEP]\nby (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\ (uminus ` A) \\ - \\ A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 544, "file": "Complex_Bounded_Operators_extra_Extra_General", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513703624558, "lm_q2_score": 0.8006919973399709, "lm_q1q2_score": 0.7176212998542008}} {"text": "[STATEMENT]\nlemma sqrt_sum_squares_le_sum:\n \"\\0 \\ x; 0 \\ y\\ \\ sqrt (x\\<^sup>2 + y\\<^sup>2) \\ x + y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 \\ x; 0 \\ y\\ \\ sqrt (x\\<^sup>2 + y\\<^sup>2) \\ x + y\n[PROOF STEP]\nby (rule power2_le_imp_le) (simp_all add: power2_sum)", "meta": {"llama_tokens": 169, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513675912912, "lm_q2_score": 0.800691997339971, "lm_q1q2_score": 0.7176212976353515}} {"text": "[STATEMENT]\nlemma ceiling_zero [simp]: \"\\0\\ = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nusing ceiling_of_int [of 0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\of_int 0\\ = 0\n\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 158, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391624034103, "lm_q2_score": 0.8311430415844384, "lm_q1q2_score": 0.7176019404782322}} {"text": "[STATEMENT]\nlemma ceiling_zero [simp]: \"\\0\\ = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nusing ceiling_of_int [of 0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\of_int 0\\ = 0\n\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 158, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391624034103, "lm_q2_score": 0.8311430415844384, "lm_q1q2_score": 0.7176019404782322}} {"text": "[STATEMENT]\nlemma lagrange_exists:\n assumes d: \"distinct (map fst zs_ws)\"\n defines e: \"(p :: complex poly) \\ lagrange_interpolation_poly zs_ws\"\n shows \"degree p \\ (length zs_ws)-1\"\n \"(\\x y. (x,y) \\ set zs_ws \\ poly p x = y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. degree p \\ length zs_ws - 1 &&& \\x y. (x, y) \\ set zs_ws \\ poly p x = y\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. degree p \\ length zs_ws - 1\n 2. \\x y. (x, y) \\ set zs_ws \\ poly p x = y\n[PROOF STEP]\nfrom e\n[PROOF STATE]\nproof (chain)\npicking this:\np \\ lagrange_interpolation_poly zs_ws\n[PROOF STEP]\nshow \"degree p \\ (length zs_ws - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ lagrange_interpolation_poly zs_ws\n\ngoal (1 subgoal):\n 1. degree p \\ length zs_ws - 1\n[PROOF STEP]\nusing degree_lagrange_interpolation_poly\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ lagrange_interpolation_poly zs_ws\ndegree (lagrange_interpolation_poly ?xs_ys) \\ length ?xs_ys - 1\n\ngoal (1 subgoal):\n 1. degree p \\ length zs_ws - 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndegree p \\ length zs_ws - 1\n\ngoal (1 subgoal):\n 1. \\x y. (x, y) \\ set zs_ws \\ poly p x = y\n[PROOF STEP]\nfrom e d\n[PROOF STATE]\nproof (chain)\npicking this:\np \\ lagrange_interpolation_poly zs_ws\ndistinct (map fst zs_ws)\n[PROOF STEP]\nhave \n \"poly p x = y\" if \"(x,y) \\ set zs_ws\" for x y\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ lagrange_interpolation_poly zs_ws\ndistinct (map fst zs_ws)\n\ngoal (1 subgoal):\n 1. poly p x = y\n[PROOF STEP]\nusing that lagrange_interpolation_poly\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ lagrange_interpolation_poly zs_ws\ndistinct (map fst zs_ws)\n(x, y) \\ set zs_ws\n\\distinct (map fst ?xs_ys); ?p = lagrange_interpolation_poly ?xs_ys; (?x, ?y) \\ set ?xs_ys\\ \\ poly ?p ?x = ?y\n\ngoal (1 subgoal):\n 1. poly p x = y\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(?x, ?y) \\ set zs_ws \\ poly p ?x = ?y\n\ngoal (1 subgoal):\n 1. \\x y. (x, y) \\ set zs_ws \\ poly p x = y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(?x, ?y) \\ set zs_ws \\ poly p ?x = ?y\n[PROOF STEP]\nshow \"(\\x y. (x,y) \\ set zs_ws \\ poly p x = y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(?x, ?y) \\ set zs_ws \\ poly p ?x = ?y\n\ngoal (1 subgoal):\n 1. \\x y. (x, y) \\ set zs_ws \\ poly p x = y\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\x y. (x, y) \\ set zs_ws \\ poly p x = y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1225, "file": "Gauss_Sums_Finite_Fourier_Series", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8633916134888613, "lm_q2_score": 0.8311430394931456, "lm_q1q2_score": 0.7176019299080233}} {"text": "[STATEMENT]\nlemma bin_lcs_rev: \"bin_lcs x y = rev (bin_lcp (rev x) (rev y))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bin_lcs x y = rev (bin_lcp (rev x) (rev y))\n[PROOF STEP]\nunfolding bin_lcp_def bin_lcs_def longest_common_suffix_def rev_append\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rev (rev y \\ rev x \\\\<^sub>p rev x \\ rev y) = rev (rev x \\ rev y \\\\<^sub>p rev y \\ rev x)\n[PROOF STEP]\nusing lcp_sym\n[PROOF STATE]\nproof (prove)\nusing this:\n?u \\\\<^sub>p ?v = ?v \\\\<^sub>p ?u\n\ngoal (1 subgoal):\n 1. rev (rev y \\ rev x \\\\<^sub>p rev x \\ rev y) = rev (rev x \\ rev y \\\\<^sub>p rev y \\ rev x)\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 313, "file": "Combinatorics_Words_Submonoids", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505299595162, "lm_q2_score": 0.7931059438487663, "lm_q1q2_score": 0.7174837124168287}} {"text": "[STATEMENT]\nlemma legendre_aux_set_eq:\n assumes \"prime p\" \"x \\ 1\"\n shows \"{m. m > 0 \\ real (p ^ m) \\ x} = {0<..nat \\log (real p) x\\}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {m. 0 < m \\ real (p ^ m) \\ x} = {0<..nat \\log (real p) x\\}\n[PROOF STEP]\nusing prime_gt_1_nat[OF assms(1)] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < p\nprime p\n1 \\ x\n\ngoal (1 subgoal):\n 1. {m. 0 < m \\ real (p ^ m) \\ x} = {0<..nat \\log (real p) x\\}\n[PROOF STEP]\nby (auto simp: le_nat_iff le_log_iff le_floor_iff powr_realpow)", "meta": {"llama_tokens": 287, "file": "Prime_Distribution_Elementary_More_Dirichlet_Misc", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.89330940889474, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.717482711367661}} {"text": "[STATEMENT]\nlemma coeff_linear_poly_power:\n fixes c :: \"'a :: semiring_1\"\n assumes \"i \\ n\"\n shows \"coeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. coeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)\n[PROOF STEP]\nhave \"[:a, b:] = monom b 1 + [:a:]\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [:a, b:] = monom b 1 + [:a:]\n[PROOF STEP]\nby (simp add: monom_altdef)\n[PROOF STATE]\nproof (state)\nthis:\n[:a, b:] = monom b 1 + [:a:]\n\ngoal (1 subgoal):\n 1. coeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n[:a, b:] = monom b 1 + [:a:]\n\ngoal (1 subgoal):\n 1. coeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)\n[PROOF STEP]\nhave \"coeff (\\ ^ n) i = (\\k\\n. a^(n-k) * of_nat (n choose k) * (if k = i then b ^ k else 0))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coeff ((monom b 1 + [:a:]) ^ n) i = (\\k\\n. a ^ (n - k) * of_nat (n choose k) * (if k = i then b ^ k else (0::'b)))\n[PROOF STEP]\nby (subst binomial_ring) (simp add: coeff_sum of_nat_poly monom_power poly_const_pow mult_ac)\n[PROOF STATE]\nproof (state)\nthis:\ncoeff ((monom b 1 + [:a:]) ^ n) i = (\\k\\n. a ^ (n - k) * of_nat (n choose k) * (if k = i then b ^ k else (0::'b)))\n\ngoal (1 subgoal):\n 1. coeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncoeff ((monom b 1 + [:a:]) ^ n) i = (\\k\\n. a ^ (n - k) * of_nat (n choose k) * (if k = i then b ^ k else (0::'b)))\n\ngoal (1 subgoal):\n 1. coeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)\n[PROOF STEP]\nhave \"\\ = (\\k\\{i}. a ^ (n - i) * b ^ i * of_nat (n choose k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\n. a ^ (n - k) * of_nat (n choose k) * (if k = i then b ^ k else (0::'b))) = (\\k\\{i}. a ^ (n - i) * b ^ i * of_nat (n choose k))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ni \\ n\n\ngoal (1 subgoal):\n 1. (\\k\\n. a ^ (n - k) * of_nat (n choose k) * (if k = i then b ^ k else (0::'b))) = (\\k\\{i}. a ^ (n - i) * b ^ i * of_nat (n choose k))\n[PROOF STEP]\nby (intro sum.mono_neutral_cong_right) (auto simp: mult_ac)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\n. a ^ (n - k) * of_nat (n choose k) * (if k = i then b ^ k else (0::'b))) = (\\k\\{i}. a ^ (n - i) * b ^ i * of_nat (n choose k))\n\ngoal (1 subgoal):\n 1. coeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncoeff ([:a, b:] ^ n) i = (\\k\\{i}. a ^ (n - i) * b ^ i * of_nat (n choose k))\n[PROOF STEP]\nshow *: ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncoeff ([:a, b:] ^ n) i = (\\k\\{i}. a ^ (n - i) * b ^ i * of_nat (n choose k))\n\ngoal (1 subgoal):\n 1. coeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)\n[PROOF STEP]\nby (simp add: mult_ac)\n[PROOF STATE]\nproof (state)\nthis:\ncoeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1590, "file": "Formal_Puiseux_Series_Puiseux_Polynomial_Library", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767970940975, "lm_q2_score": 0.8175744806385542, "lm_q1q2_score": 0.7174026366565888}} {"text": "[STATEMENT]\nlemma divisor_count_squarefree:\n assumes \"squarefree n\" \"n > 0\"\n shows \"divisor_count n = 2 ^ primes_omega n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. divisor_count n = 2 ^ primes_omega n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. divisor_count n = 2 ^ primes_omega n\n[PROOF STEP]\nhave \"divisor_count n = (\\p\\prime_factors n. Suc (multiplicity p n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. divisor_count n = (\\p\\prime_factors n. Suc (multiplicity p n))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsquarefree n\n0 < n\n\ngoal (1 subgoal):\n 1. divisor_count n = (\\p\\prime_factors n. Suc (multiplicity p n))\n[PROOF STEP]\nby (subst divisor_count.prod_prime_factors') auto\n[PROOF STATE]\nproof (state)\nthis:\ndivisor_count n = (\\p\\prime_factors n. Suc (multiplicity p n))\n\ngoal (1 subgoal):\n 1. divisor_count n = 2 ^ primes_omega n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndivisor_count n = (\\p\\prime_factors n. Suc (multiplicity p n))\n\ngoal (1 subgoal):\n 1. divisor_count n = 2 ^ primes_omega n\n[PROOF STEP]\nhave \"\\ = (\\p\\prime_factors n. 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\p\\prime_factors n. Suc (multiplicity p n)) = (\\p\\prime_factors n. 2)\n[PROOF STEP]\nusing assms assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsquarefree n\n0 < n\nsquarefree n\n0 < n\n\ngoal (1 subgoal):\n 1. (\\p\\prime_factors n. Suc (multiplicity p n)) = (\\p\\prime_factors n. 2)\n[PROOF STEP]\nby (intro prod.cong) (auto simp: squarefree_factorial_semiring')\n[PROOF STATE]\nproof (state)\nthis:\n(\\p\\prime_factors n. Suc (multiplicity p n)) = (\\p\\prime_factors n. 2)\n\ngoal (1 subgoal):\n 1. divisor_count n = 2 ^ primes_omega n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndivisor_count n = (\\p\\prime_factors n. 2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndivisor_count n = (\\p\\prime_factors n. 2)\n\ngoal (1 subgoal):\n 1. divisor_count n = 2 ^ primes_omega n\n[PROOF STEP]\nby (simp add: primes_omega_def)\n[PROOF STATE]\nproof (state)\nthis:\ndivisor_count n = 2 ^ primes_omega n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 994, "file": "Prime_Distribution_Elementary_Primes_Omega", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767970940975, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.7174026288559769}} {"text": "[STATEMENT]\nlemma complex_div_gt_0: \"(Re (a / b) > 0 \\ Re (a * cnj b) > 0) \\ (Im (a / b) > 0 \\ Im (a * cnj b) > 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n[PROOF STEP]\nproof (cases \"b = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. b = 0 \\ (0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n 2. b \\ 0 \\ (0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nb = 0\n\ngoal (2 subgoals):\n 1. b = 0 \\ (0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n 2. b \\ 0 \\ (0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nb = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nb = 0\n\ngoal (1 subgoal):\n 1. (0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n\ngoal (1 subgoal):\n 1. b \\ 0 \\ (0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. b \\ 0 \\ (0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nb \\ 0\n\ngoal (1 subgoal):\n 1. b \\ 0 \\ (0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nb \\ 0\n[PROOF STEP]\nhave \"0 < (Re b)\\<^sup>2 + (Im b)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nb \\ 0\n\ngoal (1 subgoal):\n 1. 0 < (Re b)\\<^sup>2 + (Im b)\\<^sup>2\n[PROOF STEP]\nby (simp add: complex_eq_iff sum_power2_gt_zero_iff)\n[PROOF STATE]\nproof (state)\nthis:\n0 < (Re b)\\<^sup>2 + (Im b)\\<^sup>2\n\ngoal (1 subgoal):\n 1. b \\ 0 \\ (0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < (Re b)\\<^sup>2 + (Im b)\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < (Re b)\\<^sup>2 + (Im b)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n[PROOF STEP]\nby (simp add: Re_divide Im_divide zero_less_divide_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(0 < Re (a / b)) = (0 < Re (a * cnj b)) \\ (0 < Im (a / b)) = (0 < Im (a * cnj b))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1458, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767842777551, "lm_q2_score": 0.8175744806385543, "lm_q1q2_score": 0.7174026261782744}} {"text": "[STATEMENT]\nlemma suminf_half_series_ereal: \"(\\n. (1/2 :: ereal) ^ Suc n) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. (1 / 2) ^ Suc n) = 1\n[PROOF STEP]\nusing sums_ereal[THEN iffD2, OF power_half_series, THEN sums_unique, symmetric]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. ereal ((1 / 2) ^ Suc x)) = ereal 1\n\ngoal (1 subgoal):\n 1. (\\n. (1 / 2) ^ Suc n) = 1\n[PROOF STEP]\nby (simp add: one_ereal_def)", "meta": {"llama_tokens": 213, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767938900121, "lm_q2_score": 0.8175744695262777, "lm_q1q2_score": 0.7174026242862456}} {"text": "[STATEMENT]\nlemma odd_half_floor:\nassumes \"odd x\"\nshows \"\\real_of_int x / 2\\ = (x-1) div 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\real_of_int x / 2\\ = (x - 1) div 2\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nodd x\n\ngoal (1 subgoal):\n 1. \\real_of_int x / 2\\ = (x - 1) div 2\n[PROOF STEP]\nby (metis add.commute diff_add_cancel even_add \n even_succ_div_2 floor_divide_of_int_eq odd_one of_int_numeral)", "meta": {"llama_tokens": 226, "file": "CRYSTALS-Kyber_Abs_Qr", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767746654974, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7174026202696907}} {"text": "[STATEMENT]\nlemma erf_remainder_conv_Suc:\n assumes \"x > 0\"\n shows \"erf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) *\n exp (- x\\<^sup>2) / (x ^ (2 * n + 1)) + erf_remainder (Suc n) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. erf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. erf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x\n[PROOF STEP]\nhave \"erf_remainder n x =\n (- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) *\n exp (- x\\<^sup>2) / (2 * x ^ (2 * n + 1)) + -(\n (- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) *\n real (2 * n + 1) / 2 * erf_remainder_integral (Suc n) x)\" (is \"_ = ?A + ?B\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. erf_remainder n x = (- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / (2 * x ^ (2 * n + 1)) + - ((- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * real (2 * n + 1) / 2 * erf_remainder_integral (Suc n) x)\n[PROOF STEP]\nunfolding erf_remainder_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * erf_remainder_integral n x = (- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / (2 * x ^ (2 * n + 1)) + - ((- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * real (2 * n + 1) / 2 * erf_remainder_integral (Suc n) x)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n\ngoal (1 subgoal):\n 1. (- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * erf_remainder_integral n x = (- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / (2 * x ^ (2 * n + 1)) + - ((- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * real (2 * n + 1) / 2 * erf_remainder_integral (Suc n) x)\n[PROOF STEP]\nby (subst erf_remainder_integral_conv_Suc)\n (auto simp: assms algebra_simps simp del: power_Suc)\n[PROOF STATE]\nproof (state)\nthis:\nerf_remainder n x = (- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / (2 * x ^ (2 * n + 1)) + - ((- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * real (2 * n + 1) / 2 * erf_remainder_integral (Suc n) x)\n\ngoal (1 subgoal):\n 1. erf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nerf_remainder n x = (- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / (2 * x ^ (2 * n + 1)) + - ((- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * real (2 * n + 1) / 2 * erf_remainder_integral (Suc n) x)\n\ngoal (1 subgoal):\n 1. erf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x\n[PROOF STEP]\nhave \"?B = erf_remainder (Suc n) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - ((- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * real (2 * n + 1) / 2 * erf_remainder_integral (Suc n) x) = erf_remainder (Suc n) x\n[PROOF STEP]\nby (simp add: divide_simps erf_remainder_def)\n[PROOF STATE]\nproof (state)\nthis:\n- ((- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * real (2 * n + 1) / 2 * erf_remainder_integral (Suc n) x) = erf_remainder (Suc n) x\n\ngoal (1 subgoal):\n 1. erf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n- ((- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * real (2 * n + 1) / 2 * erf_remainder_integral (Suc n) x) = erf_remainder (Suc n) x\n\ngoal (1 subgoal):\n 1. erf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x\n[PROOF STEP]\nhave \"?A = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) *\n exp (- x\\<^sup>2) / (x ^ (2 * n + 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / (2 * x ^ (2 * n + 1)) = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1)\n[PROOF STEP]\nby (simp add: divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / (2 * x ^ (2 * n + 1)) = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1)\n\ngoal (1 subgoal):\n 1. erf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nerf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nerf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x\n\ngoal (1 subgoal):\n 1. erf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nerf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2747, "file": "Error_Function_Error_Function_Asymptotics", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8175744761936437, "lm_q1q2_score": 0.7174026196583897}} {"text": "[STATEMENT]\nlemma nat_less_power_trans2:\n fixes n :: nat\n shows \"\\n < 2 ^ (m - k); k \\ m\\ \\ n * 2 ^ k < 2 ^ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n < 2 ^ (m - k); k \\ m\\ \\ n * 2 ^ k < 2 ^ m\n[PROOF STEP]\nby (subst mult.commute, erule (1) nat_less_power_trans)", "meta": {"llama_tokens": 158, "file": "Word_Lib_Many_More", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767746654976, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.7174026105189262}} {"text": "[STATEMENT]\nlemma tendsto_powr:\n fixes a b :: real\n assumes f: \"(f \\ a) F\"\n and g: \"(g \\ b) F\"\n and a: \"a \\ 0\"\n shows \"((\\x. f x powr g x) \\ a powr b) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. f x powr g x) \\ a powr b) F\n[PROOF STEP]\nunfolding powr_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. if f x = 0 then 0 else exp (g x * ln (f x))) \\ (if a = 0 then 0 else exp (b * ln a))) F\n[PROOF STEP]\nproof (rule filterlim_If)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ((\\x. 0) \\ (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))\n 2. ((\\x. exp (g x * ln (f x))) \\ (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x \\ 0}))\n[PROOF STEP]\nfrom f\n[PROOF STATE]\nproof (chain)\npicking this:\n(f \\ a) F\n[PROOF STEP]\nshow \"((\\x. 0) \\ (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ a) F\n\ngoal (1 subgoal):\n 1. ((\\x. 0) \\ (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))\n[PROOF STEP]\nby simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_mono dest: t1_space_nhds)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. 0) \\ (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))\n\ngoal (1 subgoal):\n 1. ((\\x. exp (g x * ln (f x))) \\ (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x \\ 0}))\n[PROOF STEP]\nfrom f g a\n[PROOF STATE]\nproof (chain)\npicking this:\n(f \\ a) F\n(g \\ b) F\na \\ 0\n[PROOF STEP]\nshow \"((\\x. exp (g x * ln (f x))) \\ (if a = 0 then 0 else exp (b * ln a)))\n (inf F (principal {x. f x \\ 0}))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ a) F\n(g \\ b) F\na \\ 0\n\ngoal (1 subgoal):\n 1. ((\\x. exp (g x * ln (f x))) \\ (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x \\ 0}))\n[PROOF STEP]\nby (auto intro!: tendsto_intros intro: tendsto_mono inf_le1)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. exp (g x * ln (f x))) \\ (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x \\ 0}))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1108, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278664544912, "lm_q2_score": 0.8128673178375734, "lm_q1q2_score": 0.7172967729899947}} {"text": "[STATEMENT]\nlemma filtermap_times_pos_at_right:\n fixes c::\"'a::{linordered_field, linorder_topology}\"\n assumes \"c > 0\"\n shows \"filtermap (times c) (at_right p) = at_right (c * p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. filtermap ((*) c) (at_right p) = at_right (c * p)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) < c\n\ngoal (1 subgoal):\n 1. filtermap ((*) c) (at_right p) = at_right (c * p)\n[PROOF STEP]\nby (intro filtermap_fun_inverse[where g=\"\\x. inverse c * x\"])\n (auto intro!: filterlim_ident filterlim_times_pos)", "meta": {"llama_tokens": 239, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.882427872638409, "lm_q2_score": 0.8128673087708699, "lm_q1q2_score": 0.7172967700159875}} {"text": "[STATEMENT]\nlemma set_card_diff_ge_zero: \"finite A \\ finite B \\ A \\ B \\ card A = card B \\ \n card (A - B) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite B; A \\ B; card A = card B\\ \\ 0 < card (A - B)\n[PROOF STEP]\nby (meson Diff_eq_empty_iff card_0_eq card_subset_eq finite_Diff neq0_conv)", "meta": {"llama_tokens": 161, "file": "Design_Theory_Multisets_Extras", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278602705732, "lm_q2_score": 0.8128673201042493, "lm_q1q2_score": 0.7172967699634678}} {"text": "[STATEMENT]\nlemma aux_sum_formula: \"(\\i 3 * n^2 + 7 * (n :: nat)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i 3 * n\\<^sup>2 + 7 * n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\i 3 * n\\<^sup>2 + 7 * n\n[PROOF STEP]\nhave \"(\\iii {..i {..i 3 * n\\<^sup>2 + 7 * n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i {..i 3 * n\\<^sup>2 + 7 * n\n[PROOF STEP]\nhave \"\\ \\ 10 * n + 3 * ((n - 1) * n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 10 * n + 5 * \\ {.. 10 * n + 3 * ((n - 1) * n)\n[PROOF STEP]\nby (unfold gauss_sum_nat2, rule add_left_mono, cases n, auto)\n[PROOF STATE]\nproof (state)\nthis:\n10 * n + 5 * \\ {.. 10 * n + 3 * ((n - 1) * n)\n\ngoal (1 subgoal):\n 1. (\\i 3 * n\\<^sup>2 + 7 * n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n10 * n + 5 * \\ {.. 10 * n + 3 * ((n - 1) * n)\n\ngoal (1 subgoal):\n 1. (\\i 3 * n\\<^sup>2 + 7 * n\n[PROOF STEP]\nhave \"\\ = 3 * n^2 + 7 * n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 10 * n + 3 * ((n - 1) * n) = 3 * n\\<^sup>2 + 7 * n\n[PROOF STEP]\nunfolding power2_eq_square\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 10 * n + 3 * ((n - 1) * n) = 3 * (n * n) + 7 * n\n[PROOF STEP]\nby (cases n, auto)\n[PROOF STATE]\nproof (state)\nthis:\n10 * n + 3 * ((n - 1) * n) = 3 * n\\<^sup>2 + 7 * n\n\ngoal (1 subgoal):\n 1. (\\i 3 * n\\<^sup>2 + 7 * n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i 3 * n\\<^sup>2 + 7 * n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i 3 * n\\<^sup>2 + 7 * n\n\ngoal (1 subgoal):\n 1. (\\i 3 * n\\<^sup>2 + 7 * n\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\i 3 * n\\<^sup>2 + 7 * n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1349, "file": "Multitape_To_Singletape_TM_TM_Common", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278633625322, "lm_q2_score": 0.8128673155708975, "lm_q1q2_score": 0.7172967684764643}} {"text": "[STATEMENT]\nlemma lucas_lehmer_mult_assoc:\n assumes m: \"m > 0\"\n shows \"lucas_lehmer_mult m x (lucas_lehmer_mult m y z) =\n lucas_lehmer_mult m (lucas_lehmer_mult m x y) z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lucas_lehmer_mult m x (lucas_lehmer_mult m y z) = lucas_lehmer_mult m (lucas_lehmer_mult m x y) z\n[PROOF STEP]\nproof (rule prod_eqI)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n 2. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nlet ?mul = \"lucas_lehmer_mult m\"\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n 2. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nhave \"[fst (?mul x (?mul y z)) = fst x * (fst y * fst z + 3 * snd y * snd z) +\n 3 * snd x * (fst y * snd z + snd y * fst z)] (mod m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst x * (fst y * fst z + 3 * snd y * snd z) + 3 * snd x * (fst y * snd z + snd y * fst z)] (mod m)\n[PROOF STEP]\nby (rule lucas_lehmer_mult_cong[THEN cong_trans] cong_add cong_mult cong_refl)+\n[PROOF STATE]\nproof (state)\nthis:\n[fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst x * (fst y * fst z + 3 * snd y * snd z) + 3 * snd x * (fst y * snd z + snd y * fst z)] (mod m)\n\ngoal (2 subgoals):\n 1. fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n 2. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n[fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst x * (fst y * fst z + 3 * snd y * snd z) + 3 * snd x * (fst y * snd z + snd y * fst z)] (mod m)\n\ngoal (2 subgoals):\n 1. fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n 2. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nhave \"fst x * (fst y * fst z + 3 * snd y * snd z) +\n 3 * snd x * (fst y * snd z + snd y * fst z) =\n (fst x * fst y + 3 * snd x * snd y) * fst z +\n 3 * (fst x * snd y + snd x * fst y) * snd z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fst x * (fst y * fst z + 3 * snd y * snd z) + 3 * snd x * (fst y * snd z + snd y * fst z) = (fst x * fst y + 3 * snd x * snd y) * fst z + 3 * (fst x * snd y + snd x * fst y) * snd z\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nfst x * (fst y * fst z + 3 * snd y * snd z) + 3 * snd x * (fst y * snd z + snd y * fst z) = (fst x * fst y + 3 * snd x * snd y) * fst z + 3 * (fst x * snd y + snd x * fst y) * snd z\n\ngoal (2 subgoals):\n 1. fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n 2. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfst x * (fst y * fst z + 3 * snd y * snd z) + 3 * snd x * (fst y * snd z + snd y * fst z) = (fst x * fst y + 3 * snd x * snd y) * fst z + 3 * (fst x * snd y + snd x * fst y) * snd z\n\ngoal (2 subgoals):\n 1. fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n 2. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nhave \"[\\ = fst (?mul (?mul x y) z)] (mod m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [(fst x * fst y + 3 * snd x * snd y) * fst z + 3 * (fst x * snd y + snd x * fst y) * snd z = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)] (mod m)\n[PROOF STEP]\nby (rule cong_sym, (rule lucas_lehmer_mult_cong[THEN cong_trans] cong_add cong_mult cong_refl)+)\n[PROOF STATE]\nproof (state)\nthis:\n[(fst x * fst y + 3 * snd x * snd y) * fst z + 3 * (fst x * snd y + snd x * fst y) * snd z = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)] (mod m)\n\ngoal (2 subgoals):\n 1. fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n 2. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n[fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)] (mod m)\n[PROOF STEP]\nshow \"fst (?mul x (?mul y z)) = fst (?mul (?mul x y) z)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n[fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)] (mod m)\n\ngoal (1 subgoal):\n 1. fst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nby (rule cong_less_modulus_unique_nat)\n (use m in \\auto simp: lucas_lehmer_mult_def case_prod_unfold\\)\n[PROOF STATE]\nproof (state)\nthis:\nfst (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n\ngoal (1 subgoal):\n 1. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nhave \"[snd (?mul x (?mul y z)) = fst x * (fst y * snd z + snd y * fst z) +\n snd x * (fst y * fst z + 3 * snd y * snd z)] (mod m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst x * (fst y * snd z + snd y * fst z) + snd x * (fst y * fst z + 3 * snd y * snd z)] (mod m)\n[PROOF STEP]\nby (rule lucas_lehmer_mult_cong[THEN cong_trans] cong_add cong_mult cong_refl)+\n[PROOF STATE]\nproof (state)\nthis:\n[snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst x * (fst y * snd z + snd y * fst z) + snd x * (fst y * fst z + 3 * snd y * snd z)] (mod m)\n\ngoal (1 subgoal):\n 1. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n[snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = fst x * (fst y * snd z + snd y * fst z) + snd x * (fst y * fst z + 3 * snd y * snd z)] (mod m)\n\ngoal (1 subgoal):\n 1. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nhave \"fst x * (fst y * snd z + snd y * fst z) + snd x * (fst y * fst z + 3 * snd y * snd z) =\n (fst x * fst y + 3 * snd x * snd y) * snd z + (fst x * snd y + snd x * fst y) * fst z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fst x * (fst y * snd z + snd y * fst z) + snd x * (fst y * fst z + 3 * snd y * snd z) = (fst x * fst y + 3 * snd x * snd y) * snd z + (fst x * snd y + snd x * fst y) * fst z\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nfst x * (fst y * snd z + snd y * fst z) + snd x * (fst y * fst z + 3 * snd y * snd z) = (fst x * fst y + 3 * snd x * snd y) * snd z + (fst x * snd y + snd x * fst y) * fst z\n\ngoal (1 subgoal):\n 1. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfst x * (fst y * snd z + snd y * fst z) + snd x * (fst y * fst z + 3 * snd y * snd z) = (fst x * fst y + 3 * snd x * snd y) * snd z + (fst x * snd y + snd x * fst y) * fst z\n\ngoal (1 subgoal):\n 1. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nhave \"[\\ = snd (?mul (?mul x y) z)] (mod m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [(fst x * fst y + 3 * snd x * snd y) * snd z + (fst x * snd y + snd x * fst y) * fst z = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)] (mod m)\n[PROOF STEP]\nby (rule cong_sym, (rule lucas_lehmer_mult_cong[THEN cong_trans] cong_add cong_mult cong_refl)+)\n[PROOF STATE]\nproof (state)\nthis:\n[(fst x * fst y + 3 * snd x * snd y) * snd z + (fst x * snd y + snd x * fst y) * fst z = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)] (mod m)\n\ngoal (1 subgoal):\n 1. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n[snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)] (mod m)\n[PROOF STEP]\nshow \"snd (?mul x (?mul y z)) = snd (?mul (?mul x y) z)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n[snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)] (mod m)\n\ngoal (1 subgoal):\n 1. snd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n[PROOF STEP]\nby (rule cong_less_modulus_unique_nat)\n (use m in \\auto simp: lucas_lehmer_mult_def case_prod_unfold\\)\n[PROOF STATE]\nproof (state)\nthis:\nsnd (lucas_lehmer_mult m x (lucas_lehmer_mult m y z)) = snd (lucas_lehmer_mult m (lucas_lehmer_mult m x y) z)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4802, "file": "Mersenne_Primes_Lucas_Lehmer", "length": 25, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278571786139, "lm_q2_score": 0.8128673201042492, "lm_q1q2_score": 0.7172967674501151}} {"text": "[STATEMENT]\nlemma sin_z_over_z_series:\n fixes z :: \"'a :: {real_normed_field,banach}\"\n assumes \"z \\ 0\"\n shows \"(\\n. (-1)^n / fact (2*n+1) * z^(2*n)) sums (sin z / z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. (- (1::'a)) ^ n / fact (2 * n + 1) * z ^ (2 * n)) sums (sin z / z)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\n. (- (1::'a)) ^ n / fact (2 * n + 1) * z ^ (2 * n)) sums (sin z / z)\n[PROOF STEP]\nfrom sin_series[of z]\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n. ((- 1) ^ n / fact (2 * n + 1)) *\\<^sub>R z ^ (2 * n + 1)) sums sin z\n[PROOF STEP]\nhave \"(\\n. z * ((-1)^n / fact (2*n+1)) * z^(2*n)) sums sin z\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. ((- 1) ^ n / fact (2 * n + 1)) *\\<^sub>R z ^ (2 * n + 1)) sums sin z\n\ngoal (1 subgoal):\n 1. (\\n. z * ((- (1::'a)) ^ n / fact (2 * n + 1)) * z ^ (2 * n)) sums sin z\n[PROOF STEP]\nby (simp add: field_simps scaleR_conv_of_real)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. z * ((- (1::'a)) ^ n / fact (2 * n + 1)) * z ^ (2 * n)) sums sin z\n\ngoal (1 subgoal):\n 1. (\\n. (- (1::'a)) ^ n / fact (2 * n + 1) * z ^ (2 * n)) sums (sin z / z)\n[PROOF STEP]\nfrom sums_mult[OF this, of \"inverse z\"] and assms\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n. inverse z * (z * ((- (1::'a)) ^ n / fact (2 * n + 1)) * z ^ (2 * n))) sums (inverse z * sin z)\nz \\ (0::'a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. inverse z * (z * ((- (1::'a)) ^ n / fact (2 * n + 1)) * z ^ (2 * n))) sums (inverse z * sin z)\nz \\ (0::'a)\n\ngoal (1 subgoal):\n 1. (\\n. (- (1::'a)) ^ n / fact (2 * n + 1) * z ^ (2 * n)) sums (sin z / z)\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. (- (1::'a)) ^ n / fact (2 * n + 1) * z ^ (2 * n)) sums (sin z / z)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 926, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711832583695, "lm_q2_score": 0.8438950986284991, "lm_q1q2_score": 0.7172865155272038}} {"text": "[STATEMENT]\nlemma expands_to_sqrt:\n assumes \"trimmed_pos F\" \"basis_wf basis\" \"(f expands_to F) basis\"\n shows \"((\\x. sqrt (f x)) expands_to powr_expansion False F (1/2)) basis\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. sqrt (f x)) expands_to powr_expansion False F (1 / 2)) basis\n[PROOF STEP]\nusing expands_to_root[of 2 F basis f] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < 2; trimmed_pos F; basis_wf basis; (f expands_to F) basis\\ \\ ((\\x. root 2 (f x)) expands_to powr_expansion False F (inverse (real 2))) basis\ntrimmed_pos F\nbasis_wf basis\n(f expands_to F) basis\n\ngoal (1 subgoal):\n 1. ((\\x. sqrt (f x)) expands_to powr_expansion False F (1 / 2)) basis\n[PROOF STEP]\nby (simp add: sqrt_def)", "meta": {"llama_tokens": 333, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.907312213841788, "lm_q2_score": 0.7905303285397349, "lm_q1q2_score": 0.7172578224964629}} {"text": "[STATEMENT]\nlemma foldl_prod_prod:\n \"foldl (\\(r::'b::comm_ring_1) (x::'a::comm_ring_1). r * f x) v as * foldl (\\r x. r * g x) w as =\n foldl (\\r x. r * f x * g x) (v * w) as\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. foldl (\\r x. r * f x) v as * foldl (\\r x. r * g x) w as = foldl (\\r x. r * f x * g x) (v * w) as\n[PROOF STEP]\nby (induct as arbitrary: v w) (simp_all add: algebra_simps)", "meta": {"llama_tokens": 201, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942377652497, "lm_q2_score": 0.8056321866478979, "lm_q1q2_score": 0.7172496935308417}} {"text": "[STATEMENT]\nlemma suminf_geometric: \"norm c < 1 \\ suminf (\\n. c^n) = 1 / (1 - c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm c < 1 \\ (\\n. c ^ n) = (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nby (rule sums_unique[symmetric]) (rule geometric_sums)", "meta": {"llama_tokens": 125, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8056321913146128, "lm_q1q2_score": 0.7172496836153609}} {"text": "[STATEMENT]\nlemma suminf_geometric: \"norm c < 1 \\ suminf (\\n. c^n) = 1 / (1 - c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm c < 1 \\ (\\n. c ^ n) = (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nby (rule sums_unique[symmetric]) (rule geometric_sums)", "meta": {"llama_tokens": 125, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8056321866478979, "lm_q1q2_score": 0.7172496794606116}} {"text": "[STATEMENT]\nlemma simple_integral_add[simp]:\n assumes f: \"simple_function M f\" and \"\\x. 0 \\ f x\" and g: \"simple_function M g\" and \"\\x. 0 \\ g x\"\n shows \"(\\\\<^sup>Sx. f x + g x \\M) = integral\\<^sup>S M f + integral\\<^sup>S M g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n[PROOF STEP]\nhave \"(\\\\<^sup>Sx. f x + g x \\M) =\n (\\y\\(\\x. (f x, g x))`space M. (fst y + snd y) * emeasure M {x\\space M. (f x, g x) = y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>S x. f x + g x \\M = (\\y\\(\\x. (f x, g x)) ` space M. (fst y + snd y) * emeasure M {x \\ space M. (f x, g x) = y})\n[PROOF STEP]\nby (intro simple_function_partition) (auto intro: f g)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>S x. f x + g x \\M = (\\y\\(\\x. (f x, g x)) ` space M. (fst y + snd y) * emeasure M {x \\ space M. (f x, g x) = y})\n\ngoal (1 subgoal):\n 1. \\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>S x. f x + g x \\M = (\\y\\(\\x. (f x, g x)) ` space M. (fst y + snd y) * emeasure M {x \\ space M. (f x, g x) = y})\n\ngoal (1 subgoal):\n 1. \\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n[PROOF STEP]\nhave \"\\ = (\\y\\(\\x. (f x, g x))`space M. fst y * emeasure M {x\\space M. (f x, g x) = y}) +\n (\\y\\(\\x. (f x, g x))`space M. snd y * emeasure M {x\\space M. (f x, g x) = y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\y\\(\\x. (f x, g x)) ` space M. (fst y + snd y) * emeasure M {x \\ space M. (f x, g x) = y}) = (\\y\\(\\x. (f x, g x)) ` space M. fst y * emeasure M {x \\ space M. (f x, g x) = y}) + (\\y\\(\\x. (f x, g x)) ` space M. snd y * emeasure M {x \\ space M. (f x, g x) = y})\n[PROOF STEP]\nusing assms(2,4)\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ f ?x\n0 \\ g ?x\n\ngoal (1 subgoal):\n 1. (\\y\\(\\x. (f x, g x)) ` space M. (fst y + snd y) * emeasure M {x \\ space M. (f x, g x) = y}) = (\\y\\(\\x. (f x, g x)) ` space M. fst y * emeasure M {x \\ space M. (f x, g x) = y}) + (\\y\\(\\x. (f x, g x)) ` space M. snd y * emeasure M {x \\ space M. (f x, g x) = y})\n[PROOF STEP]\nby (auto intro!: sum.cong distrib_right simp: sum.distrib[symmetric])\n[PROOF STATE]\nproof (state)\nthis:\n(\\y\\(\\x. (f x, g x)) ` space M. (fst y + snd y) * emeasure M {x \\ space M. (f x, g x) = y}) = (\\y\\(\\x. (f x, g x)) ` space M. fst y * emeasure M {x \\ space M. (f x, g x) = y}) + (\\y\\(\\x. (f x, g x)) ` space M. snd y * emeasure M {x \\ space M. (f x, g x) = y})\n\ngoal (1 subgoal):\n 1. \\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\y\\(\\x. (f x, g x)) ` space M. (fst y + snd y) * emeasure M {x \\ space M. (f x, g x) = y}) = (\\y\\(\\x. (f x, g x)) ` space M. fst y * emeasure M {x \\ space M. (f x, g x) = y}) + (\\y\\(\\x. (f x, g x)) ` space M. snd y * emeasure M {x \\ space M. (f x, g x) = y})\n\ngoal (1 subgoal):\n 1. \\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n[PROOF STEP]\nhave \"(\\y\\(\\x. (f x, g x))`space M. fst y * emeasure M {x\\space M. (f x, g x) = y}) = (\\\\<^sup>Sx. f x \\M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\y\\(\\x. (f x, g x)) ` space M. fst y * emeasure M {x \\ space M. (f x, g x) = y}) = integral\\<^sup>S M f\n[PROOF STEP]\nby (intro simple_function_partition[symmetric]) (auto intro: f g)\n[PROOF STATE]\nproof (state)\nthis:\n(\\y\\(\\x. (f x, g x)) ` space M. fst y * emeasure M {x \\ space M. (f x, g x) = y}) = integral\\<^sup>S M f\n\ngoal (1 subgoal):\n 1. \\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\y\\(\\x. (f x, g x)) ` space M. fst y * emeasure M {x \\ space M. (f x, g x) = y}) = integral\\<^sup>S M f\n\ngoal (1 subgoal):\n 1. \\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n[PROOF STEP]\nhave \"(\\y\\(\\x. (f x, g x))`space M. snd y * emeasure M {x\\space M. (f x, g x) = y}) = (\\\\<^sup>Sx. g x \\M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\y\\(\\x. (f x, g x)) ` space M. snd y * emeasure M {x \\ space M. (f x, g x) = y}) = integral\\<^sup>S M g\n[PROOF STEP]\nby (intro simple_function_partition[symmetric]) (auto intro: f g)\n[PROOF STATE]\nproof (state)\nthis:\n(\\y\\(\\x. (f x, g x)) ` space M. snd y * emeasure M {x \\ space M. (f x, g x) = y}) = integral\\<^sup>S M g\n\ngoal (1 subgoal):\n 1. \\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n\ngoal (1 subgoal):\n 1. \\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>S x. f x + g x \\M = integral\\<^sup>S M f + integral\\<^sup>S M g\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2734, "file": null, "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8056321796478255, "lm_q1q2_score": 0.7172496732284875}} {"text": "[STATEMENT]\ntheorem card_equiv_k_classes_eq_card_partitions_k_parts:\n \"card {R. equiv A R \\ card (A // R) = k} = card {P. partition_on A P \\ card P = k}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {R. equiv A R \\ card (A // R) = k} = card {P. partition_on A P \\ card P = k}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {R. equiv A R \\ card (A // R) = k} = card {P. partition_on A P \\ card P = k}\n[PROOF STEP]\nhave \"bij_betw (\\R. A // R) {R. equiv A R \\ card (A // R) = k} {P. partition_on A P \\ card P = k}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw ((//) A) {R. equiv A R \\ card (A // R) = k} {P. partition_on A P \\ card P = k}\n[PROOF STEP]\nby (rule bij_betw_partition_of_equiv_with_k_classes)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw ((//) A) {R. equiv A R \\ card (A // R) = k} {P. partition_on A P \\ card P = k}\n\ngoal (1 subgoal):\n 1. card {R. equiv A R \\ card (A // R) = k} = card {P. partition_on A P \\ card P = k}\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nbij_betw ((//) A) {R. equiv A R \\ card (A // R) = k} {P. partition_on A P \\ card P = k}\n[PROOF STEP]\nshow \"card {R. equiv A R \\ card (A // R) = k} = card {P. partition_on A P \\ card P = k}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw ((//) A) {R. equiv A R \\ card (A // R) = k} {P. partition_on A P \\ card P = k}\n\ngoal (1 subgoal):\n 1. card {R. equiv A R \\ card (A // R) = k} = card {P. partition_on A P \\ card P = k}\n[PROOF STEP]\nby (rule bij_betw_same_card)\n[PROOF STATE]\nproof (state)\nthis:\ncard {R. equiv A R \\ card (A // R) = k} = card {P. partition_on A P \\ card P = k}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 796, "file": "Card_Equiv_Relations_Card_Equiv_Relations", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473647220787, "lm_q2_score": 0.8221891327004133, "lm_q1q2_score": 0.717234523214337}} {"text": "[STATEMENT]\nlemma proj2_set_Col_coeff:\n assumes \"proj2_set_Col S\" and \"{p,q,r} \\ S\" and \"p \\ q\" and \"r \\ p\"\n shows \"r = proj2_abs (proj2_Col_coeff p q r *\\<^sub>R proj2_rep p + proj2_rep q)\"\n (is \"r = proj2_abs (?i *\\<^sub>R ?u + ?v)\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r = proj2_abs (proj2_Col_coeff p q r *\\<^sub>R proj2_rep p + proj2_rep q)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. r = proj2_abs (proj2_Col_coeff p q r *\\<^sub>R proj2_rep p + proj2_rep q)\n[PROOF STEP]\nfrom \\{p,q,r} \\ S\\ and \\proj2_set_Col S\\\n[PROOF STATE]\nproof (chain)\npicking this:\n{p, q, r} \\ S\nproj2_set_Col S\n[PROOF STEP]\nhave \"proj2_set_Col {p,q,r}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{p, q, r} \\ S\nproj2_set_Col S\n\ngoal (1 subgoal):\n 1. proj2_set_Col {p, q, r}\n[PROOF STEP]\nby (rule proj2_subset_Col)\n[PROOF STATE]\nproof (state)\nthis:\nproj2_set_Col {p, q, r}\n\ngoal (1 subgoal):\n 1. r = proj2_abs (proj2_Col_coeff p q r *\\<^sub>R proj2_rep p + proj2_rep q)\n[PROOF STEP]\nhence \"proj2_Col p q r\"\n[PROOF STATE]\nproof (prove)\nusing this:\nproj2_set_Col {p, q, r}\n\ngoal (1 subgoal):\n 1. proj2_Col p q r\n[PROOF STEP]\nby (subst proj2_Col_iff_set_Col)\n[PROOF STATE]\nproof (state)\nthis:\nproj2_Col p q r\n\ngoal (1 subgoal):\n 1. r = proj2_abs (proj2_Col_coeff p q r *\\<^sub>R proj2_rep p + proj2_rep q)\n[PROOF STEP]\nwith \\p \\ q\\ and \\r \\ p\\ and proj2_Col_coeff\n[PROOF STATE]\nproof (chain)\npicking this:\np \\ q\nr \\ p\n\\proj2_Col ?a ?r ?t; ?a \\ ?r; ?t \\ ?a\\ \\ ?t = proj2_abs (proj2_Col_coeff ?a ?r ?t *\\<^sub>R proj2_rep ?a + proj2_rep ?r)\nproj2_Col p q r\n[PROOF STEP]\nshow \"r = proj2_abs (?i *\\<^sub>R ?u + ?v)\"\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ q\nr \\ p\n\\proj2_Col ?a ?r ?t; ?a \\ ?r; ?t \\ ?a\\ \\ ?t = proj2_abs (proj2_Col_coeff ?a ?r ?t *\\<^sub>R proj2_rep ?a + proj2_rep ?r)\nproj2_Col p q r\n\ngoal (1 subgoal):\n 1. r = proj2_abs (proj2_Col_coeff p q r *\\<^sub>R proj2_rep p + proj2_rep q)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nr = proj2_abs (proj2_Col_coeff p q r *\\<^sub>R proj2_rep p + proj2_rep q)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1144, "file": "Tarskis_Geometry_Projective", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240964782012, "lm_q2_score": 0.8289388125473629, "lm_q1q2_score": 0.7172178351220051}} {"text": "[STATEMENT]\nlemma neg_prod_sum_le:\n fixes c :: \"'a::linordered_field\"\n assumes \"c < 0\"\n shows \"c * x + t \\ 0 \\ x \\ (- 1 / c) * t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c * x + t \\ (0::'a) \\ - (1::'a) / c * t \\ x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. c * x + t \\ (0::'a) \\ - (1::'a) / c * t \\ x\n[PROOF STEP]\nhave \"c * x + t \\ 0 \\ c * x \\ -t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (c * x + t \\ (0::'a)) = (c * x \\ - t)\n[PROOF STEP]\nby (subst le_iff_diff_le_0 [of \"c*x\" \"-t\"]) simp\n[PROOF STATE]\nproof (state)\nthis:\n(c * x + t \\ (0::'a)) = (c * x \\ - t)\n\ngoal (1 subgoal):\n 1. c * x + t \\ (0::'a) \\ - (1::'a) / c * t \\ x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(c * x + t \\ (0::'a)) = (c * x \\ - t)\n\ngoal (1 subgoal):\n 1. c * x + t \\ (0::'a) \\ - (1::'a) / c * t \\ x\n[PROOF STEP]\nhave \"\\ \\ - t / c \\ x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (c * x \\ - t) = (- t / c \\ x)\n[PROOF STEP]\nby (simp only: neg_divide_le_eq[OF \\c < 0\\] algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(c * x \\ - t) = (- t / c \\ x)\n\ngoal (1 subgoal):\n 1. c * x + t \\ (0::'a) \\ - (1::'a) / c * t \\ x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(c * x \\ - t) = (- t / c \\ x)\n\ngoal (1 subgoal):\n 1. c * x + t \\ (0::'a) \\ - (1::'a) / c * t \\ x\n[PROOF STEP]\nhave \"\\ \\ (- 1 / c) * t \\ x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (- t / c \\ x) = (- (1::'a) / c * t \\ x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(- t / c \\ x) = (- (1::'a) / c * t \\ x)\n\ngoal (1 subgoal):\n 1. c * x + t \\ (0::'a) \\ - (1::'a) / c * t \\ x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(c * x + t \\ (0::'a)) = (- (1::'a) / c * t \\ x)\n[PROOF STEP]\nshow \"PROP ?thesis\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(c * x + t \\ (0::'a)) = (- (1::'a) / c * t \\ x)\n\ngoal (1 subgoal):\n 1. c * x + t \\ (0::'a) \\ - (1::'a) / c * t \\ x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nc * x + t \\ (0::'a) \\ - (1::'a) / c * t \\ x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1197, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240825770432, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.7172178199424043}} {"text": "[STATEMENT]\nlemma invertible_mat_first_column_not0:\n fixes A::\"'a :: comm_ring_1 mat\"\n assumes A: \"A \\ carrier_mat n n\" and inv_A: \"invertible_mat A\" and n0: \"0 (0\\<^sub>v n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. col A 0 \\ 0\\<^sub>v n\n[PROOF STEP]\nproof (rule ccontr)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ col A 0 \\ 0\\<^sub>v n \\ False\n[PROOF STEP]\nassume \" \\ col A 0 \\ 0\\<^sub>v n\"\n[PROOF STATE]\nproof (state)\nthis:\n\\ col A 0 \\ 0\\<^sub>v n\n\ngoal (1 subgoal):\n 1. \\ col A 0 \\ 0\\<^sub>v n \\ False\n[PROOF STEP]\nhence col_A0: \"col A 0 = 0\\<^sub>v n\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ col A 0 \\ 0\\<^sub>v n\n\ngoal (1 subgoal):\n 1. col A 0 = 0\\<^sub>v n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncol A 0 = 0\\<^sub>v n\n\ngoal (1 subgoal):\n 1. \\ col A 0 \\ 0\\<^sub>v n \\ False\n[PROOF STEP]\nhave \"(det A dvd 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A dvd (1::'a)\n[PROOF STEP]\nusing inv_A invertible_iff_is_unit_JNF[OF A]\n[PROOF STATE]\nproof (prove)\nusing this:\ninvertible_mat A\ninvertible_mat A = (det A dvd (1::'a))\n\ngoal (1 subgoal):\n 1. det A dvd (1::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndet A dvd (1::'a)\n\ngoal (1 subgoal):\n 1. \\ col A 0 \\ 0\\<^sub>v n \\ False\n[PROOF STEP]\nhence 1: \"det A \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndet A dvd (1::'a)\n\ngoal (1 subgoal):\n 1. det A \\ (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndet A \\ (0::'a)\n\ngoal (1 subgoal):\n 1. \\ col A 0 \\ 0\\<^sub>v n \\ False\n[PROOF STEP]\nhave \"det A = (\\iii col A 0 \\ 0\\<^sub>v n \\ False\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndet A = (\\i col A 0 \\ 0\\<^sub>v n \\ False\n[PROOF STEP]\nhave \"... = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ii col A 0 \\ 0\\<^sub>v n \\ False\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndet A = (0::'a)\n[PROOF STEP]\nshow False\n[PROOF STATE]\nproof (prove)\nusing this:\ndet A = (0::'a)\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nusing 1\n[PROOF STATE]\nproof (prove)\nusing this:\ndet A = (0::'a)\ndet A \\ (0::'a)\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby contradiction\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1536, "file": "Modular_arithmetic_LLL_and_HNF_algorithms_HNF_Mod_Det_Soundness", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240895276223, "lm_q2_score": 0.8289387998695209, "lm_q1q2_score": 0.7172178183912261}} {"text": "[STATEMENT]\nlemma jordan_nf_root_char_poly: fixes A :: \"'a :: {semiring_no_zero_divisors, idom} mat\"\n assumes \"jordan_nf A n_as\" \n and \"(m, lam) \\ set n_as\" \nshows \"poly (char_poly A) lam = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly (char_poly A) lam = (0::'a)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. poly (char_poly A) lam = (0::'a)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\njordan_nf A n_as\n(m, lam) \\ set n_as\n[PROOF STEP]\nhave m0: \"m \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\njordan_nf A n_as\n(m, lam) \\ set n_as\n\ngoal (1 subgoal):\n 1. m \\ 0\n[PROOF STEP]\nunfolding jordan_nf_def\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ fst ` set n_as \\ similar_mat A (jordan_matrix n_as)\n(m, lam) \\ set n_as\n\ngoal (1 subgoal):\n 1. m \\ 0\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\nm \\ 0\n\ngoal (1 subgoal):\n 1. poly (char_poly A) lam = (0::'a)\n[PROOF STEP]\nfrom split_list[OF assms(2)]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ys zs. n_as = ys @ (m, lam) # zs\n[PROOF STEP]\nobtain as bs where nas: \"n_as = as @ (m, lam) # bs\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ys zs. n_as = ys @ (m, lam) # zs\n\ngoal (1 subgoal):\n 1. (\\as bs. n_as = as @ (m, lam) # bs \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nn_as = as @ (m, lam) # bs\n\ngoal (1 subgoal):\n 1. poly (char_poly A) lam = (0::'a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly (char_poly A) lam = (0::'a)\n[PROOF STEP]\nusing m0\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ 0\n\ngoal (1 subgoal):\n 1. poly (char_poly A) lam = (0::'a)\n[PROOF STEP]\nunfolding jordan_nf_char_poly[OF assms(1)] nas poly_prod_list prod_list_zero_iff\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ 0\n\ngoal (1 subgoal):\n 1. (0::'a) \\ set (map (\\p. poly p lam) (map (\\(n, a). [:- a, 1::'a:] ^ n) (as @ (m, lam) # bs)))\n[PROOF STEP]\nby (auto simp: o_def)\n[PROOF STATE]\nproof (state)\nthis:\npoly (char_poly A) lam = (0::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1050, "file": "Perron_Frobenius_Perron_Frobenius_Aux", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240756264639, "lm_q2_score": 0.8289388019824946, "lm_q1q2_score": 0.7172178086962122}} {"text": "[STATEMENT]\nlemma iter_tensor_of_H_on_zero_tensor: \n assumes \"n \\ 1\"\n shows \"(H \\\\<^bsup>n\\<^esup>) * ( |zero\\ \\\\<^bsup>n\\<^esup>) = \\\\<^sub>1\\<^sub>0 n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ n\n\ngoal (1 subgoal):\n 1. H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n[PROOF STEP]\nproof(rule nat_induct_at_least)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. H \\\\<^bsup>1\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>1\\<^esup> = Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1))\n 2. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nshow \"(H \\\\<^bsup>1\\<^esup>) * ( |zero\\ \\\\<^bsup>1\\<^esup>) = \\\\<^sub>1\\<^sub>0 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. H \\\\<^bsup>1\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>1\\<^esup> = Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1))\n[PROOF STEP]\nusing H_on_ket_zero\n[PROOF STATE]\nproof (prove)\nusing this:\nH * |Deutsch.zero\\ = Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1))\n\ngoal (1 subgoal):\n 1. H \\\\<^bsup>1\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>1\\<^esup> = Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nH \\\\<^bsup>1\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>1\\<^esup> = Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1))\n\ngoal (1 subgoal):\n 1. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nfix n:: nat\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nassume a0: \"n \\ 1\" and IH: \"(H \\\\<^bsup>n\\<^esup>) * ( |zero\\ \\\\<^bsup>n\\<^esup>) = \\\\<^sub>1\\<^sub>0 n\"\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ n\nH \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n\ngoal (1 subgoal):\n 1. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n1 \\ n\nH \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n[PROOF STEP]\nhave \"(H \\\\<^bsup>(Suc n)\\<^esup>) * ( |zero\\ \\\\<^bsup>(Suc n)\\<^esup>) = (H * |zero\\) \\ ((H \\\\<^bsup>n\\<^esup>) * ( |zero\\ \\\\<^bsup>n\\<^esup>))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ n\nH \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n\ngoal (1 subgoal):\n 1. H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = H * |Deutsch.zero\\ \\ H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup>\n[PROOF STEP]\nusing iter_tensor_mult_distr[of \"n\" \"H\" \"|zero\\\"] a0 ket_vec_def H_def\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ n\nH \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n\\1 \\ n; dim_col H = dim_row |Deutsch.zero\\; 0 < dim_col H; 0 < dim_col |Deutsch.zero\\\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = H * |Deutsch.zero\\ \\ H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup>\n1 \\ n\n|?v\\ \\ Matrix.mat (dim_vec ?v) 1 (\\(i, j). ?v $ i)\nH \\ complex_of_real (1 / sqrt 2) \\\\<^sub>m Matrix.mat 2 2 (\\(i, j). if i \\ j then 1 else if i = 0 then 1 else - 1)\n\ngoal (1 subgoal):\n 1. H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = H * |Deutsch.zero\\ \\ H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup>\n[PROOF STEP]\nby(simp add: H_def)\n[PROOF STATE]\nproof (state)\nthis:\nH \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = H * |Deutsch.zero\\ \\ H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup>\n\ngoal (1 subgoal):\n 1. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nH \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = H * |Deutsch.zero\\ \\ H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup>\n\ngoal (1 subgoal):\n 1. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nhave \"... = (H * |zero\\) \\ (\\\\<^sub>1\\<^sub>0 n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. H * |Deutsch.zero\\ \\ H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = H * |Deutsch.zero\\ \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n[PROOF STEP]\nusing IH\n[PROOF STATE]\nproof (prove)\nusing this:\nH \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n\ngoal (1 subgoal):\n 1. H * |Deutsch.zero\\ \\ H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = H * |Deutsch.zero\\ \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nH * |Deutsch.zero\\ \\ H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = H * |Deutsch.zero\\ \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n\ngoal (1 subgoal):\n 1. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nH * |Deutsch.zero\\ \\ H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = H * |Deutsch.zero\\ \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n\ngoal (1 subgoal):\n 1. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nhave \"... = (\\\\<^sub>1\\<^sub>0 1) \\ (\\\\<^sub>1\\<^sub>0 n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. H * |Deutsch.zero\\ \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n)) = Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1)) \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n[PROOF STEP]\nusing H_on_ket_zero\n[PROOF STATE]\nproof (prove)\nusing this:\nH * |Deutsch.zero\\ = Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1))\n\ngoal (1 subgoal):\n 1. H * |Deutsch.zero\\ \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n)) = Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1)) \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nH * |Deutsch.zero\\ \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n)) = Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1)) \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n\ngoal (1 subgoal):\n 1. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nH * |Deutsch.zero\\ \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n)) = Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1)) \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\n\ngoal (1 subgoal):\n 1. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nhave \"... = (\\\\<^sub>1\\<^sub>0 (Suc n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1)) \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n)) = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nusing \\\\<^sub>1\\<^sub>0_tensor a0\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ ?n \\ Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1)) \\ Matrix.mat (2 ^ ?n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ ?n)) = Matrix.mat (2 ^ Suc ?n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc ?n))\n1 \\ n\n\ngoal (1 subgoal):\n 1. Matrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1)) \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n)) = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nMatrix.mat (2 ^ 1) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ 1)) \\ Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n)) = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n\ngoal (1 subgoal):\n 1. \\n. \\1 \\ n; H \\\\<^bsup>n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>n\\<^esup> = Matrix.mat (2 ^ n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ n))\\ \\ H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nH \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nshow \"(H \\\\<^bsup>(Suc n)\\<^esup>) * ( |zero\\ \\\\<^bsup>(Suc n)\\<^esup>) = (\\\\<^sub>1\\<^sub>0 (Suc n))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nH \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n\ngoal (1 subgoal):\n 1. H \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nH \\\\<^bsup>Suc n\\<^esup> * |Deutsch.zero\\ \\\\<^bsup>Suc n\\<^esup> = Matrix.mat (2 ^ Suc n) 1 (\\x. complex_of_real (case x of (i, j) \\ 1 / sqrt 2 ^ Suc n))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 7154, "file": "Isabelle_Marries_Dirac_Deutsch_Jozsa", "length": 28, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.851952809486198, "lm_q2_score": 0.8418256452674008, "lm_q1q2_score": 0.7171957235830936}} {"text": "[STATEMENT]\nlemma vec_plus_assoc: assumes vec: \"vec nr u\" \"vec nr v\" \"vec nr w\"\n shows \"vec_plus u (vec_plus v w) = vec_plus (vec_plus u v) w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec_plus u (vec_plus v w) = vec_plus (vec_plus u v) w\n[PROOF STEP]\nproof (rule vec_eqI)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. vec ?n (vec_plus u (vec_plus v w))\n 2. vec ?n (vec_plus (vec_plus u v) w)\n 3. \\i. i < ?n \\ vec_plus u (vec_plus v w) ! i = vec_plus (vec_plus u v) w ! i\n[PROOF STEP]\nfix i\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. vec ?n (vec_plus u (vec_plus v w))\n 2. vec ?n (vec_plus (vec_plus u v) w)\n 3. \\i. i < ?n \\ vec_plus u (vec_plus v w) ! i = vec_plus (vec_plus u v) w ! i\n[PROOF STEP]\nassume i: \"i < nr\"\n[PROOF STATE]\nproof (state)\nthis:\ni < nr\n\ngoal (3 subgoals):\n 1. vec ?n (vec_plus u (vec_plus v w))\n 2. vec ?n (vec_plus (vec_plus u v) w)\n 3. \\i. i < ?n \\ vec_plus u (vec_plus v w) ! i = vec_plus (vec_plus u v) w ! i\n[PROOF STEP]\nnote [simp] = vec_plus_index[OF _ _ i]\n[PROOF STATE]\nproof (state)\nthis:\n\\vec nr ?v1.0; vec nr ?v2.0\\ \\ vec_plusI ?pl ?v1.0 ?v2.0 ! i = ?pl (?v1.0 ! i) (?v2.0 ! i)\n\ngoal (3 subgoals):\n 1. vec ?n (vec_plus u (vec_plus v w))\n 2. vec ?n (vec_plus (vec_plus u v) w)\n 3. \\i. i < ?n \\ vec_plus u (vec_plus v w) ! i = vec_plus (vec_plus u v) w ! i\n[PROOF STEP]\nfrom vec\n[PROOF STATE]\nproof (chain)\npicking this:\nvec nr u\nvec nr v\nvec nr w\n[PROOF STEP]\nshow \"vec_plus u (vec_plus v w) ! i = vec_plus (vec_plus u v) w ! i\"\n[PROOF STATE]\nproof (prove)\nusing this:\nvec nr u\nvec nr v\nvec nr w\n\ngoal (1 subgoal):\n 1. vec_plus u (vec_plus v w) ! i = vec_plus (vec_plus u v) w ! i\n[PROOF STEP]\nby (auto simp: add.assoc)\n[PROOF STATE]\nproof (state)\nthis:\nvec_plus u (vec_plus v w) ! i = vec_plus (vec_plus u v) w ! i\n\ngoal (2 subgoals):\n 1. vec nr (vec_plus u (vec_plus v w))\n 2. vec nr (vec_plus (vec_plus u v) w)\n[PROOF STEP]\nqed (auto intro: vec)", "meta": {"llama_tokens": 932, "file": "Matrix_Matrix_Legacy", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527944504227, "lm_q2_score": 0.8418256492357358, "lm_q1q2_score": 0.7171957143064265}} {"text": "[STATEMENT]\nlemma mersenne_mod_less':\n assumes \"k \\ 5 * 2 ^ n\"\n shows \"mersenne_mod k n < 2 ^ n + 5\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nhave \"mersenne_mod k n = k mod 2 ^ n + k div 2 ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mersenne_mod k n = k mod 2 ^ n + k div 2 ^ n\n[PROOF STEP]\nby (simp add: mersenne_mod_def)\n[PROOF STATE]\nproof (state)\nthis:\nmersenne_mod k n = k mod 2 ^ n + k div 2 ^ n\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmersenne_mod k n = k mod 2 ^ n + k div 2 ^ n\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nhave \"k mod 2 ^ n < 2 ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k mod 2 ^ n < 2 ^ n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nk mod 2 ^ n < 2 ^ n\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nk mod 2 ^ n < 2 ^ n\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\nk mod 2 ^ n < 2 ^ n\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nhave \"k div 2 ^ n * 2 ^ n + 0 \\ k div 2 ^ n * 2 ^ n + k mod (2 ^ n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k div 2 ^ n * 2 ^ n + 0 \\ k div 2 ^ n * 2 ^ n + k mod 2 ^ n\n[PROOF STEP]\nby (intro add_mono) auto\n[PROOF STATE]\nproof (state)\nthis:\nk div 2 ^ n * 2 ^ n + 0 \\ k div 2 ^ n * 2 ^ n + k mod 2 ^ n\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nk div 2 ^ n * 2 ^ n + 0 \\ k div 2 ^ n * 2 ^ n + k mod 2 ^ n\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nhave \"\\ = k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k div 2 ^ n * 2 ^ n + k mod 2 ^ n = k\n[PROOF STEP]\nby (subst mult.commute) auto\n[PROOF STATE]\nproof (state)\nthis:\nk div 2 ^ n * 2 ^ n + k mod 2 ^ n = k\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nk div 2 ^ n * 2 ^ n + k mod 2 ^ n = k\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nhave \"\\ \\ 5 * 2 ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k \\ 5 * 2 ^ n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ 5 * 2 ^ n\n\ngoal (1 subgoal):\n 1. k \\ 5 * 2 ^ n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nk \\ 5 * 2 ^ n\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\k div 2 ^ n * 2 ^ n + 0 \\ UNIV; k \\ UNIV; 5 * 2 ^ n \\ UNIV\\ \\ k div 2 ^ n * 2 ^ n + 0 \\ 5 * 2 ^ n\n[PROOF STEP]\nhave \"k div 2 ^ n \\ 5\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\k div 2 ^ n * 2 ^ n + 0 \\ UNIV; k \\ UNIV; 5 * 2 ^ n \\ UNIV\\ \\ k div 2 ^ n * 2 ^ n + 0 \\ 5 * 2 ^ n\n\ngoal (1 subgoal):\n 1. k div 2 ^ n \\ 5\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nk div 2 ^ n \\ 5\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\nk div 2 ^ n \\ 5\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\x y. x < y \\ x + k div 2 ^ n < y + k div 2 ^ n; \\x y. x \\ y \\ 2 ^ n + x \\ 2 ^ n + y\\ \\ mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\x y. x < y \\ x + k div 2 ^ n < y + k div 2 ^ n; \\x y. x \\ y \\ 2 ^ n + x \\ 2 ^ n + y\\ \\ mersenne_mod k n < 2 ^ n + 5\n\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 5\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmersenne_mod k n < 2 ^ n + 5\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2010, "file": "Mersenne_Primes_Lucas_Lehmer_Code", "length": 25, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339676722393, "lm_q2_score": 0.853912747375134, "lm_q1q2_score": 0.7170595393992238}} {"text": "[STATEMENT]\nlemma minus_add_minus_mat: fixes u :: \"'a :: ab_group_add mat\"\n assumes \"u \\ carrier_mat nr nc\" \"v \\ carrier_mat nr nc\" \"w \\ carrier_mat nr nc\"\n shows \"u - (v + w) = u - v - w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. u - (v + w) = u - v - w\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ carrier_mat nr nc\nv \\ carrier_mat nr nc\nw \\ carrier_mat nr nc\n\ngoal (1 subgoal):\n 1. u - (v + w) = u - v - w\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 225, "file": "Jordan_Normal_Form_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.8397339616560072, "lm_q1q2_score": 0.7170595327009415}} {"text": "[STATEMENT]\nlemma echelon_form_def': \n\"echelon_form A = (\n (\\i. is_zero_row i A \\ \\ (\\j. j>i \\ \\ is_zero_row j A)) \n \\ \n (\\i j. i \\ (is_zero_row i A) \\ \\ (is_zero_row j A) \n \\ ((LEAST n. A $ i $ n \\ 0) < (LEAST n. A $ j $ n \\ 0))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. echelon_form A = ((\\i. is_zero_row i A \\ \\ (\\j>i. \\ is_zero_row j A)) \\ (\\i j. i < j \\ \\ is_zero_row i A \\ \\ is_zero_row j A \\ (LEAST n. A $h i $h n \\ (0::'a)) < (LEAST n. A $h j $h n \\ (0::'a))))\n[PROOF STEP]\nunfolding echelon_form_def echelon_form_upt_k_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\i. is_zero_row_upt_k i (ncols A) A \\ \\ (\\j>i. \\ is_zero_row_upt_k j (ncols A) A)) \\ (\\i j. i < j \\ \\ is_zero_row_upt_k i (ncols A) A \\ \\ is_zero_row_upt_k j (ncols A) A \\ (LEAST n. A $h i $h n \\ (0::'a)) < (LEAST n. A $h j $h n \\ (0::'a)))) = ((\\i. is_zero_row i A \\ \\ (\\j>i. \\ is_zero_row j A)) \\ (\\i j. i < j \\ \\ is_zero_row i A \\ \\ is_zero_row j A \\ (LEAST n. A $h i $h n \\ (0::'a)) < (LEAST n. A $h j $h n \\ (0::'a))))\n[PROOF STEP]\nunfolding is_zero_row_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\i. is_zero_row_upt_k i (ncols A) A \\ \\ (\\j>i. \\ is_zero_row_upt_k j (ncols A) A)) \\ (\\i j. i < j \\ \\ is_zero_row_upt_k i (ncols A) A \\ \\ is_zero_row_upt_k j (ncols A) A \\ (LEAST n. A $h i $h n \\ (0::'a)) < (LEAST n. A $h j $h n \\ (0::'a)))) = ((\\i. is_zero_row_upt_k i (ncols A) A \\ \\ (\\j>i. \\ is_zero_row_upt_k j (ncols A) A)) \\ (\\i j. i < j \\ \\ is_zero_row_upt_k i (ncols A) A \\ \\ is_zero_row_upt_k j (ncols A) A \\ (LEAST n. A $h i $h n \\ (0::'a)) < (LEAST n. A $h j $h n \\ (0::'a))))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 1030, "file": "Modular_arithmetic_LLL_and_HNF_algorithms_Uniqueness_Hermite_JNF", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099069962657176, "lm_q2_score": 0.7879311856832191, "lm_q1q2_score": 0.7169440984291032}} {"text": "[STATEMENT]\nlemma poly_add_in_carrier:\n \"\\ set p1 \\ carrier R; set p2 \\ carrier R \\ \\ set (poly_add p1 p2) \\ carrier R\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\set p1 \\ carrier R; set p2 \\ carrier R\\ \\ set (poly_add p1 p2) \\ carrier R\n[PROOF STEP]\nusing polynomial_incl[OF poly_add_is_polynomial[OF carrier_is_subring]]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\set ?p1.1 \\ carrier R; set ?p2.1 \\ carrier R\\ \\ set (poly_add ?p1.1 ?p2.1) \\ carrier R\n\ngoal (1 subgoal):\n 1. \\set p1 \\ carrier R; set p2 \\ carrier R\\ \\ set (poly_add p1 p2) \\ carrier R\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 321, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872046056466901, "lm_q2_score": 0.8080672089305841, "lm_q1q2_score": 0.7169209494352805}} {"text": "[STATEMENT]\nlemma arccos_arcsin_sqrt_neg: \"-1 \\ x \\ x \\ 0 \\ arccos x = pi - arcsin(sqrt(1 - x\\<^sup>2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\- 1 \\ x; x \\ 0\\ \\ arccos x = pi - arcsin (sqrt (1 - x\\<^sup>2))\n[PROOF STEP]\nusing arccos_arcsin_sqrt_pos [of \"-x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 \\ - x; - x \\ 1\\ \\ arccos (- x) = arcsin (sqrt (1 - (- x)\\<^sup>2))\n\ngoal (1 subgoal):\n 1. \\- 1 \\ x; x \\ 0\\ \\ arccos x = pi - arcsin (sqrt (1 - x\\<^sup>2))\n[PROOF STEP]\nby (simp add: arccos_minus)", "meta": {"llama_tokens": 305, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045817875224, "lm_q2_score": 0.8080672181749422, "lm_q1q2_score": 0.7169209383571062}} {"text": "[STATEMENT]\nlemma exp_1_lt_3: \"exp (1::real) < 3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp 1 < 3\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. exp 1 < 3\n[PROOF STEP]\nfrom Taylor_up[of 3 \"\\_. exp\" exp 0 1 0]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\0 < 3; exp = exp; \\m t. m < 3 \\ 0 \\ t \\ t \\ 1 \\ (exp has_real_derivative exp t) (at t); 0 \\ 0; 0 < 1\\ \\ \\t>0. t < 1 \\ exp 1 = (\\m<3. exp 0 / fact m * (1 - 0) ^ m) + exp t / fact 3 * (1 - 0) ^ 3\n[PROOF STEP]\nobtain t :: real where \"t > 0\" \"t < 1\" \"exp 1 = 5/2 + exp t / 6\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < 3; exp = exp; \\m t. m < 3 \\ 0 \\ t \\ t \\ 1 \\ (exp has_real_derivative exp t) (at t); 0 \\ 0; 0 < 1\\ \\ \\t>0. t < 1 \\ exp 1 = (\\m<3. exp 0 / fact m * (1 - 0) ^ m) + exp t / fact 3 * (1 - 0) ^ 3\n\ngoal (1 subgoal):\n 1. (\\t. \\0 < t; t < 1; exp 1 = 5 / 2 + exp t / 6\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: eval_nat_numeral)\n[PROOF STATE]\nproof (state)\nthis:\n0 < t\nt < 1\nexp 1 = 5 / 2 + exp t / 6\n\ngoal (1 subgoal):\n 1. exp 1 < 3\n[PROOF STEP]\nnote this(3)\n[PROOF STATE]\nproof (state)\nthis:\nexp 1 = 5 / 2 + exp t / 6\n\ngoal (1 subgoal):\n 1. exp 1 < 3\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nexp 1 = 5 / 2 + exp t / 6\n\ngoal (1 subgoal):\n 1. exp 1 < 3\n[PROOF STEP]\nfrom \\t < 1\\\n[PROOF STATE]\nproof (chain)\npicking this:\nt < 1\n[PROOF STEP]\nhave \"exp t < exp 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nt < 1\n\ngoal (1 subgoal):\n 1. exp t < exp 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nexp t < exp 1\n\ngoal (1 subgoal):\n 1. exp 1 < 3\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x y. x < y \\ 5 / 2 + x / 6 < 5 / 2 + y / 6) \\ exp 1 < 5 / 2 + exp 1 / 6\n[PROOF STEP]\nshow \"exp (1::real) < 3\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x y. x < y \\ 5 / 2 + x / 6 < 5 / 2 + y / 6) \\ exp 1 < 5 / 2 + exp 1 / 6\n\ngoal (1 subgoal):\n 1. exp 1 < 3\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nexp 1 < 3\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1155, "file": "Akra_Bazzi_Akra_Bazzi_Method", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8376199552262967, "lm_q2_score": 0.8558511524823263, "lm_q1q2_score": 0.7168780040226207}} {"text": "[STATEMENT]\nlemma ordLeq_Func:\nassumes \"{b1,b2} \\ B\" \"b1 \\ b2\"\nshows \"|A| \\o |Func A B|\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. |A| \\o |Func A B|\n[PROOF STEP]\nunfolding card_of_ordLeq[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f. inj_on f A \\ f ` A \\ Func A B\n[PROOF STEP]\nproof(intro exI conjI)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. inj_on ?f A\n 2. ?f ` A \\ Func A B\n[PROOF STEP]\nlet ?F = \"\\ aa a. if a \\ A then (if a = aa then b1 else b2) else undefined\"\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. inj_on ?f A\n 2. ?f ` A \\ Func A B\n[PROOF STEP]\nshow \"inj_on ?F A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on (\\aa a. if a \\ A then if a = aa then b1 else b2 else undefined) A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n{b1, b2} \\ B\nb1 \\ b2\n\ngoal (1 subgoal):\n 1. inj_on (\\aa a. if a \\ A then if a = aa then b1 else b2 else undefined) A\n[PROOF STEP]\nunfolding inj_on_def fun_eq_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n{b1, b2} \\ B\nb1 \\ b2\n\ngoal (1 subgoal):\n 1. \\x\\A. \\y\\A. (\\xa. (if xa \\ A then if xa = x then b1 else b2 else undefined) = (if xa \\ A then if xa = y then b1 else b2 else undefined)) \\ x = y\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\aa a. if a \\ A then if a = aa then b1 else b2 else undefined) A\n\ngoal (1 subgoal):\n 1. (\\aa a. if a \\ A then if a = aa then b1 else b2 else undefined) ` A \\ Func A B\n[PROOF STEP]\nshow \"?F ` A \\ Func A B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\aa a. if a \\ A then if a = aa then b1 else b2 else undefined) ` A \\ Func A B\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n{b1, b2} \\ B\nb1 \\ b2\n\ngoal (1 subgoal):\n 1. (\\aa a. if a \\ A then if a = aa then b1 else b2 else undefined) ` A \\ Func A B\n[PROOF STEP]\nunfolding Func_def\n[PROOF STATE]\nproof (prove)\nusing this:\n{b1, b2} \\ B\nb1 \\ b2\n\ngoal (1 subgoal):\n 1. (\\aa a. if a \\ A then if a = aa then b1 else b2 else undefined) ` A \\ {f. (\\a\\A. f a \\ B) \\ (\\a. a \\ A \\ f a = undefined)}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\aa a. if a \\ A then if a = aa then b1 else b2 else undefined) ` A \\ Func A B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1126, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511396138365, "lm_q2_score": 0.8376199653600372, "lm_q1q2_score": 0.7168780019166902}} {"text": "[STATEMENT]\nlemma triangle_num_eq_iff: \"triangle_num m = triangle_num n \\ m = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (triangle_num m = triangle_num n) = (m = n)\n[PROOF STEP]\nusing strict_mono_eq[OF strict_mono_triangle_num]\n[PROOF STATE]\nproof (prove)\nusing this:\n(triangle_num ?x = triangle_num ?y) = (?x = ?y)\n\ngoal (1 subgoal):\n 1. (triangle_num m = triangle_num n) = (m = n)\n[PROOF STEP]\n.", "meta": {"llama_tokens": 178, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357598021708, "lm_q2_score": 0.8267117876664789, "lm_q1q2_score": 0.7167886829568164}} {"text": "[STATEMENT]\nlemma double_csin_square:\n \"2 * csin (a::real) * csin a = 1 - ccos (2 * a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * complex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos (2 * a))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 * complex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos (2 * a))\n[PROOF STEP]\nhave eq: \"ccos (2 * a) = ccos a * ccos a - csin a * csin a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (cos (2 * a)) = complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a)\n[PROOF STEP]\nusing cos_add[of a a]\n[PROOF STATE]\nproof (prove)\nusing this:\ncos (a + a) = cos a * cos a - sin a * sin a\n\ngoal (1 subgoal):\n 1. complex_of_real (cos (2 * a)) = complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (cos (2 * a)) = complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a)\n\ngoal (1 subgoal):\n 1. 2 * complex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos (2 * a))\n[PROOF STEP]\nhave \"ccos a * ccos a = 1 - csin a * csin a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (cos a) * complex_of_real (cos a) = 1 - complex_of_real (sin a) * complex_of_real (sin a)\n[PROOF STEP]\nusing csin_ccos_squared_add[of a]\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (cos a) * complex_of_real (cos a) + complex_of_real (sin a) * complex_of_real (sin a) = 1\n\ngoal (1 subgoal):\n 1. complex_of_real (cos a) * complex_of_real (cos a) = 1 - complex_of_real (sin a) * complex_of_real (sin a)\n[PROOF STEP]\nby (auto intro: add_implies_diff)\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (cos a) * complex_of_real (cos a) = 1 - complex_of_real (sin a) * complex_of_real (sin a)\n\ngoal (1 subgoal):\n 1. 2 * complex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos (2 * a))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncomplex_of_real (cos a) * complex_of_real (cos a) = 1 - complex_of_real (sin a) * complex_of_real (sin a)\n[PROOF STEP]\nhave \"ccos a * ccos a - csin a * csin a = 1 - 2 * csin (a::real) * csin a\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (cos a) * complex_of_real (cos a) = 1 - complex_of_real (sin a) * complex_of_real (sin a)\n\ngoal (1 subgoal):\n 1. complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a) = 1 - 2 * complex_of_real (sin a) * complex_of_real (sin a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a) = 1 - 2 * complex_of_real (sin a) * complex_of_real (sin a)\n\ngoal (1 subgoal):\n 1. 2 * complex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos (2 * a))\n[PROOF STEP]\nwith eq\n[PROOF STATE]\nproof (chain)\npicking this:\ncomplex_of_real (cos (2 * a)) = complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a)\ncomplex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a) = 1 - 2 * complex_of_real (sin a) * complex_of_real (sin a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (cos (2 * a)) = complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a)\ncomplex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a) = 1 - 2 * complex_of_real (sin a) * complex_of_real (sin a)\n\ngoal (1 subgoal):\n 1. 2 * complex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos (2 * a))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * complex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos (2 * a))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1720, "file": "QHLProver_Grover", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357598021707, "lm_q2_score": 0.8267117855317473, "lm_q1q2_score": 0.7167886811059277}} {"text": "[STATEMENT]\ntheorem integral_Markov_inequality_measure:\n assumes [measurable]: \"integrable M u\" and \"A \\ sets M\" and \"AE x in M. 0 \\ u x\" \"0 < (c::real)\"\n shows \"measure M {x\\space M. u x \\ c} \\ (\\x. u x \\M) / c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M {x \\ space M. c \\ u x} \\ integral\\<^sup>L M u / c\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M {x \\ space M. c \\ u x} \\ integral\\<^sup>L M u / c\n[PROOF STEP]\nhave le: \"emeasure M {x\\space M. u x \\ c} \\ ennreal ((1/c) * (\\x. u x \\M))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. emeasure M {x \\ space M. c \\ u x} \\ ennreal (1 / c * integral\\<^sup>L M u)\n[PROOF STEP]\nby (rule integral_Markov_inequality) (use assms in auto)\n[PROOF STATE]\nproof (state)\nthis:\nemeasure M {x \\ space M. c \\ u x} \\ ennreal (1 / c * integral\\<^sup>L M u)\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M {x \\ space M. c \\ u x} \\ integral\\<^sup>L M u / c\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nemeasure M {x \\ space M. c \\ u x} \\ ennreal (1 / c * integral\\<^sup>L M u)\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M {x \\ space M. c \\ u x} \\ integral\\<^sup>L M u / c\n[PROOF STEP]\nhave \"\\ < top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ennreal (1 / c * integral\\<^sup>L M u) < top\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nennreal (1 / c * integral\\<^sup>L M u) < top\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M {x \\ space M. c \\ u x} \\ integral\\<^sup>L M u / c\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nemeasure M {x \\ space M. c \\ u x} < top\n[PROOF STEP]\nhave \"ennreal (measure M {x\\space M. u x \\ c}) = emeasure M {x\\space M. u x \\ c}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nemeasure M {x \\ space M. c \\ u x} < top\n\ngoal (1 subgoal):\n 1. ennreal (Sigma_Algebra.measure M {x \\ space M. c \\ u x}) = emeasure M {x \\ space M. c \\ u x}\n[PROOF STEP]\nby (intro emeasure_eq_ennreal_measure [symmetric]) auto\n[PROOF STATE]\nproof (state)\nthis:\nennreal (Sigma_Algebra.measure M {x \\ space M. c \\ u x}) = emeasure M {x \\ space M. c \\ u x}\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M {x \\ space M. c \\ u x} \\ integral\\<^sup>L M u / c\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nennreal (Sigma_Algebra.measure M {x \\ space M. c \\ u x}) = emeasure M {x \\ space M. c \\ u x}\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M {x \\ space M. c \\ u x} \\ integral\\<^sup>L M u / c\n[PROOF STEP]\nnote le\n[PROOF STATE]\nproof (state)\nthis:\nemeasure M {x \\ space M. c \\ u x} \\ ennreal (1 / c * integral\\<^sup>L M u)\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M {x \\ space M. c \\ u x} \\ integral\\<^sup>L M u / c\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nennreal (Sigma_Algebra.measure M {x \\ space M. c \\ u x}) \\ ennreal (1 / c * integral\\<^sup>L M u)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nennreal (Sigma_Algebra.measure M {x \\ space M. c \\ u x}) \\ ennreal (1 / c * integral\\<^sup>L M u)\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M {x \\ space M. c \\ u x} \\ integral\\<^sup>L M u / c\n[PROOF STEP]\nby (subst (asm) ennreal_le_iff)\n (auto intro!: divide_nonneg_pos Bochner_Integration.integral_nonneg_AE assms)\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M {x \\ space M. c \\ u x} \\ integral\\<^sup>L M u / c\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1616, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357494949105, "lm_q2_score": 0.8267117876664789, "lm_q1q2_score": 0.7167886744356828}} {"text": "[STATEMENT]\nlemma integral_exp_times:\n \"integral {t0 .. t} (\\t. exp (c * t)) = (if c = 0 then t - t0 else exp (c * t) / c - exp (c * t0) / c)\"\n if \"t0 \\ t\"\n for c t::real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral {t0..t} (\\t. exp (c * t)) = (if c = 0 then t - t0 else exp (c * t) / c - exp (c * t0) / c)\n[PROOF STEP]\nusing exp_times_has_integral[OF that, of c] that\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\t. exp (c * t)) has_integral (if c = 0 then t else exp (c * t) / c) - (if c = 0 then t0 else exp (c * t0) / c)) {t0..t}\nt0 \\ t\n\ngoal (1 subgoal):\n 1. integral {t0..t} (\\t. exp (c * t)) = (if c = 0 then t - t0 else exp (c * t) / c - exp (c * t0) / c)\n[PROOF STEP]\nby (auto split: if_splits)", "meta": {"llama_tokens": 352, "file": "Laplace_Transform_Existence", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467675095292, "lm_q2_score": 0.8152324960856177, "lm_q1q2_score": 0.7167090137023957}} {"text": "[STATEMENT]\nlemma real_sqrt_le_iff [simp]: \"sqrt x \\ sqrt y \\ x \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (sqrt x \\ sqrt y) = (x \\ y)\n[PROOF STEP]\nunfolding sqrt_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (root 2 x \\ root 2 y) = (x \\ y)\n[PROOF STEP]\nby (rule real_root_le_iff [OF pos2])", "meta": {"llama_tokens": 163, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467675095294, "lm_q2_score": 0.8152324848629214, "lm_q1q2_score": 0.7167090038359987}} {"text": "[STATEMENT]\nlemma card_sequences:\n assumes \"finite A\" \"finite B\" \"card A \\ card B\"\n shows \"card {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\n[PROOF STEP]\nobtain enum where \"bij_betw enum {0..enum. bij_betw enum {0.. thesis) \\ thesis\n[PROOF STEP]\nusing \\finite A\\ ex_bij_betw_nat_finite\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite ?M \\ \\h. bij_betw h {0..enum. bij_betw enum {0.. thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw enum {0.. B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\n[PROOF STEP]\nhave \"bij_betw (function_of A enum) {xs. set xs \\ B \\ length xs = card A \\ distinct xs} {f \\ A \\\\<^sub>E B. inj_on f A}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw (function_of A enum) {xs. set xs \\ B \\ length xs = card A \\ distinct xs} {f \\ A \\\\<^sub>E B. inj_on f A}\n[PROOF STEP]\nusing \\bij_betw enum {0..\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw enum {0.. B \\ length xs = card A \\ distinct xs} {f \\ A \\\\<^sub>E B. inj_on f A}\n[PROOF STEP]\nby (rule bij_betw_function_of)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw (function_of A enum) {xs. set xs \\ B \\ length xs = card A \\ distinct xs} {f \\ A \\\\<^sub>E B. inj_on f A}\n\ngoal (1 subgoal):\n 1. card {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nbij_betw (function_of A enum) {xs. set xs \\ B \\ length xs = card A \\ distinct xs} {f \\ A \\\\<^sub>E B. inj_on f A}\n[PROOF STEP]\nhave \"card {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = card {f \\ A \\\\<^sub>E B. inj_on f A}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw (function_of A enum) {xs. set xs \\ B \\ length xs = card A \\ distinct xs} {f \\ A \\\\<^sub>E B. inj_on f A}\n\ngoal (1 subgoal):\n 1. card {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = card {f \\ A \\\\<^sub>E B. inj_on f A}\n[PROOF STEP]\nby (rule bij_betw_same_card)\n[PROOF STATE]\nproof (state)\nthis:\ncard {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = card {f \\ A \\\\<^sub>E B. inj_on f A}\n\ngoal (1 subgoal):\n 1. card {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = card {f \\ A \\\\<^sub>E B. inj_on f A}\n\ngoal (1 subgoal):\n 1. card {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\n[PROOF STEP]\nhave \"card {f \\ A \\\\<^sub>E B. inj_on f A} = fact (card B) div fact (card B - card A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {f \\ A \\\\<^sub>E B. inj_on f A} = fact (card B) div fact (card B - card A)\n[PROOF STEP]\nusing \\finite A\\ \\finite B\\ \\card A \\ card B\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\ncard A \\ card B\n\ngoal (1 subgoal):\n 1. card {f \\ A \\\\<^sub>E B. inj_on f A} = fact (card B) div fact (card B - card A)\n[PROOF STEP]\nby (rule card_extensional_funcset_inj_on)\n[PROOF STATE]\nproof (state)\nthis:\ncard {f \\ A \\\\<^sub>E B. inj_on f A} = fact (card B) div fact (card B - card A)\n\ngoal (1 subgoal):\n 1. card {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\n\ngoal (1 subgoal):\n 1. card {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {xs. set xs \\ B \\ length xs = card A \\ distinct xs} = fact (card B) div fact (card B - card A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2064, "file": "Twelvefold_Way_Twelvefold_Way_Entry2", "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467643431002, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.7167090032279023}} {"text": "[STATEMENT]\nlemma (in group) ord_pow:\n assumes \"x \\ carrier G\" \"k dvd ord x\" \"k \\ 0\"\n shows \"ord (pow G x k) = ord x div k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ord (x [^] k) = ord x div k\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ord (x [^] k) = ord x div k\n[PROOF STEP]\nhave \"(x [^] k) [^] (ord x div k) = \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x [^] k) [^] (ord x div k) = \\\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ carrier G\nk dvd ord x\nk \\ 0\n\ngoal (1 subgoal):\n 1. (x [^] k) [^] (ord x div k) = \\\n[PROOF STEP]\nby (simp add: nat_pow_pow)\n[PROOF STATE]\nproof (state)\nthis:\n(x [^] k) [^] (ord x div k) = \\\n\ngoal (1 subgoal):\n 1. ord (x [^] k) = ord x div k\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(x [^] k) [^] (ord x div k) = \\\n\ngoal (1 subgoal):\n 1. ord (x [^] k) = ord x div k\n[PROOF STEP]\nhave \"ord x dvd k * ord (x [^] k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ord x dvd k * ord (x [^] k)\n[PROOF STEP]\nby (metis assms(1) pow_ord_eq_1 pow_eq_id nat_pow_closed nat_pow_pow)\n[PROOF STATE]\nproof (state)\nthis:\nord x dvd k * ord (x [^] k)\n\ngoal (1 subgoal):\n 1. ord (x [^] k) = ord x div k\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(x [^] k) [^] (ord x div k) = \\\nord x dvd k * ord (x [^] k)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(x [^] k) [^] (ord x div k) = \\\nord x dvd k * ord (x [^] k)\n\ngoal (1 subgoal):\n 1. ord (x [^] k) = ord x div k\n[PROOF STEP]\nby (metis assms div_dvd_div dvd_antisym dvd_triv_left pow_eq_id nat_pow_closed nonzero_mult_div_cancel_left)\n[PROOF STATE]\nproof (state)\nthis:\nord (x [^] k) = ord x div k\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 871, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681195338728, "lm_q2_score": 0.8354835391516133, "lm_q1q2_score": 0.7166511442795841}} {"text": "[STATEMENT]\nlemma fib_lowerbound:\n defines \"\\ \\ (1 + sqrt 5) / 2\"\n shows \"real (fib(n+2)) \\ \\ ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ ^ n \\ real (fib (n + 2))\n[PROOF STEP]\nproof (induction n rule: fib.induct)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\ ^ 0 \\ real (fib (0 + 2))\n 2. \\ ^ Suc 0 \\ real (fib (Suc 0 + 2))\n 3. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\ncase 1\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (3 subgoals):\n 1. \\ ^ 0 \\ real (fib (0 + 2))\n 2. \\ ^ Suc 0 \\ real (fib (Suc 0 + 2))\n 3. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ ^ 0 \\ real (fib (0 + 2))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\ ^ 0 \\ real (fib (0 + 2))\n\ngoal (2 subgoals):\n 1. \\ ^ Suc 0 \\ real (fib (Suc 0 + 2))\n 2. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\ ^ Suc 0 \\ real (fib (Suc 0 + 2))\n 2. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\ncase 2\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\ ^ Suc 0 \\ real (fib (Suc 0 + 2))\n 2. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ ^ Suc 0 \\ real (fib (Suc 0 + 2))\n[PROOF STEP]\nby (simp add: \\_def real_le_lsqrt)\n[PROOF STATE]\nproof (state)\nthis:\n\\ ^ Suc 0 \\ real (fib (Suc 0 + 2))\n\ngoal (1 subgoal):\n 1. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\ncase (3 n)\n[PROOF STATE]\nproof (state)\nthis:\n\\ ^ Suc n \\ real (fib (Suc n + 2))\n\\ ^ n \\ real (fib (n + 2))\n\ngoal (1 subgoal):\n 1. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nhave \"\\ ^ Suc (Suc n) = \\ ^ 2 * \\ ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ ^ Suc (Suc n) = \\\\<^sup>2 * \\ ^ n\n[PROOF STEP]\nby (simp add: field_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n\\ ^ Suc (Suc n) = \\\\<^sup>2 * \\ ^ n\n\ngoal (1 subgoal):\n 1. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\ ^ Suc (Suc n) = \\\\<^sup>2 * \\ ^ n\n\ngoal (1 subgoal):\n 1. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nhave \"\\ = (\\ + 1) * \\ ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>2 * \\ ^ n = (\\ + 1) * \\ ^ n\n[PROOF STEP]\nby (simp_all add: \\_def power2_eq_square field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>2 * \\ ^ n = (\\ + 1) * \\ ^ n\n\ngoal (1 subgoal):\n 1. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>2 * \\ ^ n = (\\ + 1) * \\ ^ n\n\ngoal (1 subgoal):\n 1. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nhave \"\\ = \\ ^ Suc n + \\ ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ + 1) * \\ ^ n = \\ ^ Suc n + \\ ^ n\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\ + 1) * \\ ^ n = \\ ^ Suc n + \\ ^ n\n\ngoal (1 subgoal):\n 1. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\ + 1) * \\ ^ n = \\ ^ Suc n + \\ ^ n\n\ngoal (1 subgoal):\n 1. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nhave \"\\ \\ real (fib (Suc n + 2)) + real (fib (n + 2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ ^ Suc n + \\ ^ n \\ real (fib (Suc n + 2)) + real (fib (n + 2))\n[PROOF STEP]\nby (intro add_mono \"3.IH\")\n[PROOF STATE]\nproof (state)\nthis:\n\\ ^ Suc n + \\ ^ n \\ real (fib (Suc n + 2)) + real (fib (n + 2))\n\ngoal (1 subgoal):\n 1. \\n. \\\\ ^ Suc n \\ real (fib (Suc n + 2)); \\ ^ n \\ real (fib (n + 2))\\ \\ \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ ^ Suc (Suc n) \\ real (fib (Suc n + 2)) + real (fib (n + 2))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ ^ Suc (Suc n) \\ real (fib (Suc n + 2)) + real (fib (n + 2))\n\ngoal (1 subgoal):\n 1. \\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\ ^ Suc (Suc n) \\ real (fib (Suc (Suc n) + 2))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3285, "file": null, "length": 27, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681122619885, "lm_q2_score": 0.8354835411997897, "lm_q1q2_score": 0.7166511399609049}} {"text": "[STATEMENT]\nlemma ldrop_iterates: \"ldrop (enat n) (iterates f x) = iterates f ((f ^^ n) x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ldrop (enat n) (iterates f x) = iterates f ((f ^^ n) x)\n[PROOF STEP]\nproof(induct n arbitrary: x)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x. ldrop (enat 0) (iterates f x) = iterates f ((f ^^ 0) x)\n 2. \\n x. (\\x. ldrop (enat n) (iterates f x) = iterates f ((f ^^ n) x)) \\ ldrop (enat (Suc n)) (iterates f x) = iterates f ((f ^^ Suc n) x)\n[PROOF STEP]\ncase Suc\n[PROOF STATE]\nproof (state)\nthis:\nldrop (enat n_) (iterates f ?x) = iterates f ((f ^^ n_) ?x)\n\ngoal (2 subgoals):\n 1. \\x. ldrop (enat 0) (iterates f x) = iterates f ((f ^^ 0) x)\n 2. \\n x. (\\x. ldrop (enat n) (iterates f x) = iterates f ((f ^^ n) x)) \\ ldrop (enat (Suc n)) (iterates f x) = iterates f ((f ^^ Suc n) x)\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nldrop (enat n_) (iterates f ?x) = iterates f ((f ^^ n_) ?x)\n\ngoal (1 subgoal):\n 1. ldrop (enat (Suc n_)) (iterates f x) = iterates f ((f ^^ Suc n_) x)\n[PROOF STEP]\nby(subst iterates)(simp add: eSuc_enat[symmetric] funpow_swap1)\n[PROOF STATE]\nproof (state)\nthis:\nldrop (enat (Suc n_)) (iterates f x) = iterates f ((f ^^ Suc n_) x)\n\ngoal (1 subgoal):\n 1. \\x. ldrop (enat 0) (iterates f x) = iterates f ((f ^^ 0) x)\n[PROOF STEP]\nqed(simp add: zero_enat_def[symmetric])", "meta": {"llama_tokens": 658, "file": "Coinductive_Coinductive_List", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772417253256, "lm_q2_score": 0.8198933337131076, "lm_q1q2_score": 0.716650103640935}} {"text": "[STATEMENT]\nlemma Lcm_eq_Max_nat:\n fixes M :: \"nat set\" \n assumes M: \"finite M\" \"M \\ {}\" \"0 \\ M\" and lcm: \"\\m n. \\m \\ M; n \\ M\\ \\ lcm m n \\ M\"\n shows \"Lcm M = Max M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Lcm M = Max M\n[PROOF STEP]\nproof (rule antisym)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. Lcm M \\ Max M\n 2. Max M \\ Lcm M\n[PROOF STEP]\nshow \"Lcm M \\ Max M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Lcm M \\ Max M\n[PROOF STEP]\nby (simp add: Lcm_in_lcm_closed_set_nat \\finite M\\ \\M \\ {}\\ lcm)\n[PROOF STATE]\nproof (state)\nthis:\nLcm M \\ Max M\n\ngoal (1 subgoal):\n 1. Max M \\ Lcm M\n[PROOF STEP]\nshow \"Max M \\ Lcm M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Max M \\ Lcm M\n[PROOF STEP]\nby (meson Lcm_0_iff Max_in M dvd_Lcm dvd_imp_le le_0_eq not_le)\n[PROOF STATE]\nproof (state)\nthis:\nMax M \\ Lcm M\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 480, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.874077222043951, "lm_q2_score": 0.8198933425148213, "lm_q1q2_score": 0.7166500951976846}} {"text": "[STATEMENT]\nlemma converse_power: fixes r :: \"'a rel\" shows \"(r\\)^^n = (r^^n)\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r\\ ^^ n = (r ^^ n)\\\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. r\\ ^^ 0 = (r ^^ 0)\\\n 2. \\n. r\\ ^^ n = (r ^^ n)\\ \\ r\\ ^^ Suc n = (r ^^ Suc n)\\\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nr\\ ^^ n = (r ^^ n)\\\n\ngoal (2 subgoals):\n 1. r\\ ^^ 0 = (r ^^ 0)\\\n 2. \\n. r\\ ^^ n = (r ^^ n)\\ \\ r\\ ^^ Suc n = (r ^^ Suc n)\\\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r\\ ^^ Suc n = (r ^^ Suc n)\\\n[PROOF STEP]\nunfolding relpow.simps(2)[of _ \"r\\\"] relpow_Suc[of _ r]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r\\ ^^ n O r\\ = (r O r ^^ n)\\\n[PROOF STEP]\nby (simp add: Suc converse_relcomp)\n[PROOF STATE]\nproof (state)\nthis:\nr\\ ^^ Suc n = (r ^^ Suc n)\\\n\ngoal (1 subgoal):\n 1. r\\ ^^ 0 = (r ^^ 0)\\\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 588, "file": "Abstract-Rewriting_Abstract_Rewriting", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772318846387, "lm_q2_score": 0.8198933271118221, "lm_q1q2_score": 0.716650089802588}} {"text": "[STATEMENT]\nlemma matrix_match_condn_1:\n\"matrix_match A1 A2 B1 B2 \n \\((i<(row_length A1)*(row_length B1))\n \\(j<(length A2)*(length B2)))\n \\ ((matrix_mult A1 A2)\\(matrix_mult B1 B2))!j!i\n = f\n (scalar_product \n (row A1 (i div (row_length B1))) \n (col A2 (j div (length B2))))\n (scalar_product \n (row B1 (i mod (row_length B1))) \n (col B2 (j mod (length B2))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_match A1 A2 B1 B2 \\ i < row_length A1 * row_length B1 \\ j < length A2 * length B2 \\ (A1 \\ A2 \\ B1 \\ B2) ! j ! i = scalar_product (row A1 (i div row_length B1)) (col A2 (j div length B2)) * scalar_product (row B1 (i mod row_length B1)) (col B2 (j mod length B2))\n[PROOF STEP]\nusing elements_matrix_distribution_1\n[PROOF STATE]\nproof (prove)\nusing this:\n\\mat (row_length ?A1.0) (length ?A1.0) ?A1.0; mat (row_length ?A2.0) (length ?A2.0) ?A2.0; mat (row_length ?B1.0) (length ?B1.0) ?B1.0; mat (row_length ?B2.0) (length ?B2.0) ?B2.0; length ?A1.0 = row_length ?A2.0; length ?B1.0 = row_length ?B2.0; ?A1.0 \\ [] \\ ?A2.0 \\ [] \\ ?B1.0 \\ [] \\ ?B2.0 \\ []; ?i < row_length ?A1.0 * row_length ?B1.0; ?j < length ?A2.0 * length ?B2.0\\ \\ (?A1.0 \\ ?A2.0 \\ ?B1.0 \\ ?B2.0) ! ?j ! ?i = scalar_product (row ?A1.0 (?i div row_length ?B1.0)) (col ?A2.0 (?j div length ?B2.0)) * scalar_product (row ?B1.0 (?i mod row_length ?B1.0)) (col ?B2.0 (?j mod length ?B2.0))\n\ngoal (1 subgoal):\n 1. matrix_match A1 A2 B1 B2 \\ i < row_length A1 * row_length B1 \\ j < length A2 * length B2 \\ (A1 \\ A2 \\ B1 \\ B2) ! j ! i = scalar_product (row A1 (i div row_length B1)) (col A2 (j div length B2)) * scalar_product (row B1 (i mod row_length B1)) (col B2 (j mod length B2))\n[PROOF STEP]\nunfolding matrix_match_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\mat (row_length ?A1.0) (length ?A1.0) ?A1.0; mat (row_length ?A2.0) (length ?A2.0) ?A2.0; mat (row_length ?B1.0) (length ?B1.0) ?B1.0; mat (row_length ?B2.0) (length ?B2.0) ?B2.0; length ?A1.0 = row_length ?A2.0; length ?B1.0 = row_length ?B2.0; ?A1.0 \\ [] \\ ?A2.0 \\ [] \\ ?B1.0 \\ [] \\ ?B2.0 \\ []; ?i < row_length ?A1.0 * row_length ?B1.0; ?j < length ?A2.0 * length ?B2.0\\ \\ (?A1.0 \\ ?A2.0 \\ ?B1.0 \\ ?B2.0) ! ?j ! ?i = scalar_product (row ?A1.0 (?i div row_length ?B1.0)) (col ?A2.0 (?j div length ?B2.0)) * scalar_product (row ?B1.0 (?i mod row_length ?B1.0)) (col ?B2.0 (?j mod length ?B2.0))\n\ngoal (1 subgoal):\n 1. (mat (row_length A1) (length A1) A1 \\ mat (row_length A2) (length A2) A2 \\ mat (row_length B1) (length B1) B1 \\ mat (row_length B2) (length B2) B2 \\ length A1 = row_length A2 \\ length B1 = row_length B2 \\ A1 \\ [] \\ A2 \\ [] \\ B1 \\ [] \\ B2 \\ []) \\ i < row_length A1 * row_length B1 \\ j < length A2 * length B2 \\ (A1 \\ A2 \\ B1 \\ B2) ! j ! i = scalar_product (row A1 (i div row_length B1)) (col A2 (j div length B2)) * scalar_product (row B1 (i mod row_length B1)) (col B2 (j mod length B2))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 1495, "file": "Matrix_Tensor_Matrix_Tensor", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361557147439, "lm_q2_score": 0.7853085708384735, "lm_q1q2_score": 0.7165439334256964}} {"text": "[STATEMENT]\nlemma cong_mersenne_mod2: \"[mersenne_mod2 k n = k] (mod (2 ^ n - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [mersenne_mod2 k n = k] (mod 2 ^ n - 1)\n[PROOF STEP]\nunfolding mersenne_mod2_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [mersenne_mod (mersenne_mod k n) n = k] (mod 2 ^ n - 1)\n[PROOF STEP]\nby (rule cong_trans) (rule cong_mersenne_mod)+", "meta": {"llama_tokens": 180, "file": "Mersenne_Primes_Lucas_Lehmer_Code", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976952975813453, "lm_q2_score": 0.7981867801399695, "lm_q1q2_score": 0.7165285191232457}} {"text": "[STATEMENT]\nlemma edge_density_Un:\n assumes \"disjnt X1 X2\" \"finite X1\" \"finite X2\"\n shows \"edge_density (X1 \\ X2) Y G = (edge_density X1 Y G * card X1 + edge_density X2 Y G * card X2) / (card X1 + card X2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. edge_density (X1 \\ X2) Y G = (edge_density X1 Y G * real (card X1) + edge_density X2 Y G * real (card X2)) / real (card X1 + card X2)\n[PROOF STEP]\nproof (cases \"finite Y\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. finite Y \\ edge_density (X1 \\ X2) Y G = (edge_density X1 Y G * real (card X1) + edge_density X2 Y G * real (card X2)) / real (card X1 + card X2)\n 2. infinite Y \\ edge_density (X1 \\ X2) Y G = (edge_density X1 Y G * real (card X1) + edge_density X2 Y G * real (card X2)) / real (card X1 + card X2)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nfinite Y\n\ngoal (2 subgoals):\n 1. finite Y \\ edge_density (X1 \\ X2) Y G = (edge_density X1 Y G * real (card X1) + edge_density X2 Y G * real (card X2)) / real (card X1 + card X2)\n 2. infinite Y \\ edge_density (X1 \\ X2) Y G = (edge_density X1 Y G * real (card X1) + edge_density X2 Y G * real (card X2)) / real (card X1 + card X2)\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\ndisjnt X1 X2\nfinite X1\nfinite X2\nfinite Y\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndisjnt X1 X2\nfinite X1\nfinite X2\nfinite Y\n\ngoal (1 subgoal):\n 1. edge_density (X1 \\ X2) Y G = (edge_density X1 Y G * real (card X1) + edge_density X2 Y G * real (card X2)) / real (card X1 + card X2)\n[PROOF STEP]\nby (simp add: edge_density_def all_edges_between_disjnt1 all_edges_between_Un1 finite_all_edges_between card_Un_disjnt card_ge_0_finite divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\nedge_density (X1 \\ X2) Y G = (edge_density X1 Y G * real (card X1) + edge_density X2 Y G * real (card X2)) / real (card X1 + card X2)\n\ngoal (1 subgoal):\n 1. infinite Y \\ edge_density (X1 \\ X2) Y G = (edge_density X1 Y G * real (card X1) + edge_density X2 Y G * real (card X2)) / real (card X1 + card X2)\n[PROOF STEP]\nqed (simp add: edge_density_def)", "meta": {"llama_tokens": 928, "file": "Szemeredi_Regularity_Szemeredi", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.897695292107347, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7165285125991914}} {"text": "[STATEMENT]\nlemma absolutely_integrable_change_of_variables_linear:\n fixes f :: \"real^'m::{finite,wellorder} \\ real^'n\" and g :: \"real^'m::_ \\ real^'m::_\"\n assumes \"linear g\"\n shows \"(\\x. \\det (matrix g)\\ *\\<^sub>R f(g x)) absolutely_integrable_on S\n \\ f absolutely_integrable_on (g ` S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. \\det (matrix g)\\ *\\<^sub>R f (g x)) absolutely_integrable_on S) = (f absolutely_integrable_on g ` S)\n[PROOF STEP]\nusing assms has_absolute_integral_change_of_variables_linear\n[PROOF STATE]\nproof (prove)\nusing this:\nlinear g\nlinear ?g \\ ((\\x. \\det (matrix ?g)\\ *\\<^sub>R ?f (?g x)) absolutely_integrable_on ?S \\ integral ?S (\\x. \\det (matrix ?g)\\ *\\<^sub>R ?f (?g x)) = ?b) = (?f absolutely_integrable_on ?g ` ?S \\ integral (?g ` ?S) ?f = ?b)\n\ngoal (1 subgoal):\n 1. ((\\x. \\det (matrix g)\\ *\\<^sub>R f (g x)) absolutely_integrable_on S) = (f absolutely_integrable_on g ` S)\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 441, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297967961707, "lm_q2_score": 0.7956581073313276, "lm_q1q2_score": 0.7165138337143062}} {"text": "[STATEMENT]\nlemma list_every_elemnt_bound_sum_bound_real:\"\\ x \\ set (xs::'a list). (f::'a\\real) x \\ (bound::real) \\ foldr (\\ a b. a+b) (map f xs) i \\ real(length xs) * bound + i\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\set xs. f x \\ bound \\ foldr (+) (map f xs) i \\ real (length xs) * bound + i\n[PROOF STEP]\napply(induction xs)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x\\set []. f x \\ bound \\ foldr (+) (map f []) i \\ real (length []) * bound + i\n 2. \\a xs. \\\\x\\set xs. f x \\ bound \\ foldr (+) (map f xs) i \\ real (length xs) * bound + i; \\x\\set (a # xs). f x \\ bound\\ \\ foldr (+) (map f (a # xs)) i \\ real (length (a # xs)) * bound + i\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a xs. \\\\x\\set xs. f x \\ bound \\ foldr (+) (map f xs) i \\ real (length xs) * bound + i; \\x\\set (a # xs). f x \\ bound\\ \\ foldr (+) (map f (a # xs)) i \\ real (length (a # xs)) * bound + i\n[PROOF STEP]\napply (simp add: algebra_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 540, "file": "Van_Emde_Boas_Trees_VEBT_Space", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297967961706, "lm_q2_score": 0.7956580952177051, "lm_q1q2_score": 0.7165138228056281}} {"text": "[STATEMENT]\nlemma gbinomial_altdef_of_nat: \"a gchoose k = (\\i = 0..i = 0.. \\S) \\ (\\B \\ S. card (A \\ B))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\n\ngoal (1 subgoal):\n 1. card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nproof (induction \"card S\" arbitrary: S)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\S. \\0 = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n 2. \\x S. \\\\S. \\x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B)); Suc x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n\\n = card ?S; finite ?S\\ \\ card (A \\ \\ ?S) \\ (\\B\\?S. card (A \\ B))\nSuc n = card S\nfinite S\n\ngoal (2 subgoals):\n 1. \\S. \\0 = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n 2. \\x S. \\\\S. \\x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B)); Suc x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\n = card ?S; finite ?S\\ \\ card (A \\ \\ ?S) \\ (\\B\\?S. card (A \\ B))\nSuc n = card S\nfinite S\n[PROOF STEP]\nobtain B T where *: \"S = { B } \\ T\" \"card T = n\" \"B \\ T\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n = card ?S; finite ?S\\ \\ card (A \\ \\ ?S) \\ (\\B\\?S. card (A \\ B))\nSuc n = card S\nfinite S\n\ngoal (1 subgoal):\n 1. (\\B T. \\S = {B} \\ T; card T = n; B \\ T\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (metis card_Suc_eq Suc_eq_plus1 insert_is_Un)\n[PROOF STATE]\nproof (state)\nthis:\nS = {B} \\ T\ncard T = n\nB \\ T\n\ngoal (2 subgoals):\n 1. \\S. \\0 = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n 2. \\x S. \\\\S. \\x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B)); Suc x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nhence \"card (A \\ \\S) = card (A \\ \\({ B } \\ T))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nS = {B} \\ T\ncard T = n\nB \\ T\n\ngoal (1 subgoal):\n 1. card (A \\ \\ S) = card (A \\ \\ ({B} \\ T))\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\ \\ S) = card (A \\ \\ ({B} \\ T))\n\ngoal (2 subgoals):\n 1. \\S. \\0 = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n 2. \\x S. \\\\S. \\x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B)); Suc x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\ \\ S) = card (A \\ \\ ({B} \\ T))\n\ngoal (2 subgoals):\n 1. \\S. \\0 = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n 2. \\x S. \\\\S. \\x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B)); Suc x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nhave \"... \\ card (A \\ B) + card (A \\ \\T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (A \\ \\ ({B} \\ T)) \\ card (A \\ B) + card (A \\ \\ T)\n[PROOF STEP]\nby (simp add: card_Un_le inf_sup_distrib1)\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\ \\ ({B} \\ T)) \\ card (A \\ B) + card (A \\ \\ T)\n\ngoal (2 subgoals):\n 1. \\S. \\0 = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n 2. \\x S. \\\\S. \\x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B)); Suc x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\ \\ ({B} \\ T)) \\ card (A \\ B) + card (A \\ \\ T)\n\ngoal (2 subgoals):\n 1. \\S. \\0 = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n 2. \\x S. \\\\S. \\x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B)); Suc x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nhave \"... \\ card (A \\ B) + (\\B \\ T. card (A \\ B))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (A \\ B) + card (A \\ \\ T) \\ card (A \\ B) + (\\B\\T. card (A \\ B))\n[PROOF STEP]\nusing Suc *\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n = card ?S; finite ?S\\ \\ card (A \\ \\ ?S) \\ (\\B\\?S. card (A \\ B))\nSuc n = card S\nfinite S\nS = {B} \\ T\ncard T = n\nB \\ T\n\ngoal (1 subgoal):\n 1. card (A \\ B) + card (A \\ \\ T) \\ card (A \\ B) + (\\B\\T. card (A \\ B))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\ B) + card (A \\ \\ T) \\ card (A \\ B) + (\\B\\T. card (A \\ B))\n\ngoal (2 subgoals):\n 1. \\S. \\0 = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n 2. \\x S. \\\\S. \\x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B)); Suc x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\ B) + card (A \\ \\ T) \\ card (A \\ B) + (\\B\\T. card (A \\ B))\n\ngoal (2 subgoals):\n 1. \\S. \\0 = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n 2. \\x S. \\\\S. \\x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B)); Suc x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nhave \"... \\ (\\B \\ S. card (A \\ B))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (A \\ B) + (\\B\\T. card (A \\ B)) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nusing Suc.prems *\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\nS = {B} \\ T\ncard T = n\nB \\ T\n\ngoal (1 subgoal):\n 1. card (A \\ B) + (\\B\\T. card (A \\ B)) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\ B) + (\\B\\T. card (A \\ B)) \\ (\\B\\S. card (A \\ B))\n\ngoal (2 subgoals):\n 1. \\S. \\0 = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n 2. \\x S. \\\\S. \\x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B)); Suc x = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n\ngoal (1 subgoal):\n 1. card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n\ngoal (1 subgoal):\n 1. \\S. \\0 = card S; finite S\\ \\ card (A \\ \\ S) \\ (\\B\\S. card (A \\ B))\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 4048, "file": "Closest_Pair_Points_Common", "length": 23, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392817460332, "lm_q2_score": 0.810478913248044, "lm_q1q2_score": 0.7164951963381063}} {"text": "[STATEMENT]\nlemma differentiable_inverse [simp, derivative_intros]:\n fixes f :: \"'a::real_normed_vector \\ 'b::real_normed_field\"\n shows \"f differentiable (at x within s) \\ f x \\ 0 \\\n (\\x. inverse (f x)) differentiable (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f differentiable at x within s; f x \\ (0::'b)\\ \\ (\\x. inverse (f x)) differentiable at x within s\n[PROOF STEP]\nunfolding differentiable_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\D. (f has_derivative D) (at x within s); f x \\ (0::'b)\\ \\ \\D. ((\\x. inverse (f x)) has_derivative D) (at x within s)\n[PROOF STEP]\nby (blast intro: has_derivative_inverse)", "meta": {"llama_tokens": 303, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.8006920116079209, "lm_q1q2_score": 0.7164507708398885}} {"text": "[STATEMENT]\nlemma binomial_code[code]:\n \"n choose k = (if k \\ n then fact n div (fact k * fact (n - k)) else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n choose k = (if k \\ n then fact n div (fact k * fact (n - k)) else 0)\n[PROOF STEP]\nusing binomial_eq_0[of n k] binomial_altdef_nat[of k n]\n[PROOF STATE]\nproof (prove)\nusing this:\nn < k \\ n choose k = 0\nk \\ n \\ n choose k = fact n div (fact k * fact (n - k))\n\ngoal (1 subgoal):\n 1. n choose k = (if k \\ n then fact n div (fact k * fact (n - k)) else 0)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 239, "file": "Polynomial_Interpolation_Improved_Code_Equations", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894632969136, "lm_q2_score": 0.800691997339971, "lm_q1q2_score": 0.7164507625659664}} {"text": "[STATEMENT]\nlemma convex_Inter: \"(\\s. s\\f \\ convex s) \\ convex(\\f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\s. s \\ f \\ convex s) \\ convex (\\ f)\n[PROOF STEP]\nunfolding convex_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\s. s \\ f \\ \\x\\s. \\y\\s. \\u\\0. \\v\\0. u + v = 1 \\ u *\\<^sub>R x + v *\\<^sub>R y \\ s) \\ \\x\\\\ f. \\y\\\\ f. \\u\\0. \\v\\0. u + v = 1 \\ u *\\<^sub>R x + v *\\<^sub>R y \\ \\ f\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 289, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9184802484881361, "lm_q2_score": 0.7799929104825006, "lm_q1q2_score": 0.7164080822389517}} {"text": "[STATEMENT]\nlemma [simp]: \"(a * b = 1) = (a = 1 \\ b = 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a * b = (1::'a)) = (a = (1::'a) \\ b = (1::'a))\n[PROOF STEP]\napply (rule iffI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. a * b = (1::'a) \\ a = (1::'a) \\ b = (1::'a)\n 2. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply (rule conjI)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. a * b = (1::'a) \\ a = (1::'a)\n 2. a * b = (1::'a) \\ b = (1::'a)\n 3. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply (rule order.antisym)\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. a * b = (1::'a) \\ a \\ (1::'a)\n 2. a * b = (1::'a) \\ (1::'a) \\ a\n 3. a * b = (1::'a) \\ b = (1::'a)\n 4. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. a * b = (1::'a) \\ (1::'a) \\ a\n 2. a * b = (1::'a) \\ b = (1::'a)\n 3. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply (rule_tac y = \"a*b\" in order_trans)\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. a * b = (1::'a) \\ (1::'a) \\ a * b\n 2. a * b = (1::'a) \\ a * b \\ a\n 3. a * b = (1::'a) \\ b = (1::'a)\n 4. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. a * b = (1::'a) \\ a * b \\ a\n 2. a * b = (1::'a) \\ b = (1::'a)\n 3. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply (drule drop_assumption)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. True \\ a * b \\ a\n 2. a * b = (1::'a) \\ b = (1::'a)\n 3. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. a * b = (1::'a) \\ b = (1::'a)\n 2. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply (rule order.antisym)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. a * b = (1::'a) \\ b \\ (1::'a)\n 2. a * b = (1::'a) \\ (1::'a) \\ b\n 3. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. a * b = (1::'a) \\ (1::'a) \\ b\n 2. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply (rule_tac y = \"a*b\" in order_trans)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. a * b = (1::'a) \\ (1::'a) \\ a * b\n 2. a * b = (1::'a) \\ a * b \\ b\n 3. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. a * b = (1::'a) \\ a * b \\ b\n 2. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply (drule drop_assumption)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. True \\ a * b \\ b\n 2. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a = (1::'a) \\ b = (1::'a) \\ a * b = (1::'a)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 1698, "file": "PseudoHoops_PseudoHoopFilters", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942119105696, "lm_q2_score": 0.7931059487389968, "lm_q1q2_score": 0.7164080129277767}} {"text": "[STATEMENT]\nlemma scalar_product_append:\n \"\\xs ys zs ws.(length zs = length ws)\n \\(length xs = length ys) \n \\(length xs = n) \\ \n (scalar_product (xs@zs) (ys@ws))\n = (scalar_product xs ys)\n +(scalar_product zs ws)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\napply(rule allI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xs. \\ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\napply(rule allI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xs ys. \\zs ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\napply(rule allI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xs ys zs. \\ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\napply(rule allI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nproof(induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n 2. \\n xs ys zs ws. (\\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws) \\ length zs = length ws \\ length xs = length ys \\ length xs = Suc n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n 2. \\n xs ys zs ws. (\\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws) \\ length zs = length ws \\ length xs = length ys \\ length xs = Suc n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nhave \"(length zs = length ws) \\(length xs = length ys) \\(length xs = 0)\n \\\n (scalar_product (xs@zs) (ys@ws))\n = (scalar_product xs ys)\n +(scalar_product zs ws)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nassume assms:\"(length zs = length ws)\\(length xs = length ys)\n \\(length xs = 0)\"\n[PROOF STATE]\nproof (state)\nthis:\nlength zs = length ws \\ length xs = length ys \\ length xs = 0\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nhave 1:\"xs = []\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. xs = []\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength zs = length ws \\ length xs = length ys \\ length xs = 0\n\ngoal (1 subgoal):\n 1. xs = []\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nxs = []\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nxs = []\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nhave 2:\"ys = []\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ys = []\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength zs = length ws \\ length xs = length ys \\ length xs = 0\n\ngoal (1 subgoal):\n 1. ys = []\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nys = []\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nxs = []\nys = []\n[PROOF STEP]\nhave \"scalar_product xs ys = zer\"\n[PROOF STATE]\nproof (prove)\nusing this:\nxs = []\nys = []\n\ngoal (1 subgoal):\n 1. scalar_product xs ys = zer\n[PROOF STEP]\nunfolding scalar_product_def scalar_prodI_def zip_def\n[PROOF STATE]\nproof (prove)\nusing this:\nxs = []\nys = []\n\ngoal (1 subgoal):\n 1. foldr (\\(x, y). (+) (x * y)) (rec_list (\\xs. []) (\\y ys ysa. case_list [] (\\z zs. (z, y) # ysa zs)) ys xs) zer = zer\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product xs ys = zer\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product xs ys = zer\n[PROOF STEP]\nhave \"(scalar_product xs ys)+(scalar_product zs ws) \n = (scalar_product zs ws)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product xs ys = zer\n\ngoal (1 subgoal):\n 1. scalar_product xs ys + scalar_product zs ws = scalar_product zs ws\n[PROOF STEP]\nusing plus_left_id\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product xs ys = zer\nzer + ?x = ?x\n\ngoal (1 subgoal):\n 1. scalar_product xs ys + scalar_product zs ws = scalar_product zs ws\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product xs ys + scalar_product zs ws = scalar_product zs ws\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product xs ys + scalar_product zs ws = scalar_product zs ws\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nhave \"(scalar_product (xs@zs) (ys@ws)) = (scalar_product zs ws)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. scalar_product (xs @ zs) (ys @ ws) = scalar_product zs ws\n[PROOF STEP]\nusing 1 2\n[PROOF STATE]\nproof (prove)\nusing this:\nxs = []\nys = []\n\ngoal (1 subgoal):\n 1. scalar_product (xs @ zs) (ys @ ws) = scalar_product zs ws\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (xs @ zs) (ys @ ws) = scalar_product zs ws\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product xs ys + scalar_product zs ws = scalar_product zs ws\nscalar_product (xs @ zs) (ys @ ws) = scalar_product zs ws\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product xs ys + scalar_product zs ws = scalar_product zs ws\nscalar_product (xs @ zs) (ys @ ws) = scalar_product zs ws\n\ngoal (1 subgoal):\n 1. scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nlength zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n\ngoal (2 subgoals):\n 1. \\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n 2. \\n xs ys zs ws. (\\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws) \\ length zs = length ws \\ length xs = length ys \\ length xs = Suc n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nlength zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength zs = length ws \\ length xs = length ys \\ length xs = 0 \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n\ngoal (1 subgoal):\n 1. \\n xs ys zs ws. (\\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws) \\ length zs = length ws \\ length xs = length ys \\ length xs = Suc n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n xs ys zs ws. (\\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws) \\ length zs = length ws \\ length xs = length ys \\ length xs = Suc n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\ncase (Suc k)\n[PROOF STATE]\nproof (state)\nthis:\nlength ?zs = length ?ws \\ length ?xs = length ?ys \\ length ?xs = k \\ scalar_product (?xs @ ?zs) (?ys @ ?ws) = scalar_product ?xs ?ys + scalar_product ?zs ?ws\n\ngoal (1 subgoal):\n 1. \\n xs ys zs ws. (\\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws) \\ length zs = length ws \\ length xs = length ys \\ length xs = Suc n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nhave \"(length zs = length ws)\\(length xs = length ys)\\(length xs = (Suc k)) \\\n (scalar_product (xs@zs) (ys@ws))\n = (scalar_product xs ys)\n +(scalar_product zs ws)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nassume assms:\"(length zs = length ws)\n \\(length xs = length ys)\n \\(length xs = (Suc k))\"\n[PROOF STATE]\nproof (state)\nthis:\nlength zs = length ws \\ length xs = length ys \\ length xs = Suc k\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nhave \"\\x xss.(xs = x#xss)\\(length xss = k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x xss. xs = x # xss \\ length xss = k\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength zs = length ws \\ length xs = length ys \\ length xs = Suc k\n\ngoal (1 subgoal):\n 1. \\x xss. xs = x # xss \\ length xss = k\n[PROOF STEP]\nby (metis Suc_length_conv)\n[PROOF STATE]\nproof (state)\nthis:\n\\x xss. xs = x # xss \\ length xss = k\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x xss. xs = x # xss \\ length xss = k\n[PROOF STEP]\nobtain x xss where \"(xs = x#xss)\\(length xss = k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x xss. xs = x # xss \\ length xss = k\n\ngoal (1 subgoal):\n 1. (\\x xss. xs = x # xss \\ length xss = k \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nxs = x # xss \\ length xss = k\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nxs = x # xss \\ length xss = k\n[PROOF STEP]\nhave 1:\"(xs = x#xss)\\(length xss = k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nxs = x # xss \\ length xss = k\n\ngoal (1 subgoal):\n 1. xs = x # xss \\ length xss = k\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nxs = x # xss \\ length xss = k\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nhave \"\\y yss.(ys = y#yss)\\(length yss = k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\y yss. ys = y # yss \\ length yss = k\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength zs = length ws \\ length xs = length ys \\ length xs = Suc k\n\ngoal (1 subgoal):\n 1. \\y yss. ys = y # yss \\ length yss = k\n[PROOF STEP]\nby (metis Suc_length_conv)\n[PROOF STATE]\nproof (state)\nthis:\n\\y yss. ys = y # yss \\ length yss = k\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\y yss. ys = y # yss \\ length yss = k\n[PROOF STEP]\nobtain y yss where \"(ys = y#yss)\\(length yss = k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\y yss. ys = y # yss \\ length yss = k\n\ngoal (1 subgoal):\n 1. (\\y yss. ys = y # yss \\ length yss = k \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nys = y # yss \\ length yss = k\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nys = y # yss \\ length yss = k\n[PROOF STEP]\nhave 2:\"(ys = y#yss)\\(length yss = k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nys = y # yss \\ length yss = k\n\ngoal (1 subgoal):\n 1. ys = y # yss \\ length yss = k\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nys = y # yss \\ length yss = k\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nwith 1\n[PROOF STATE]\nproof (chain)\npicking this:\nxs = x # xss \\ length xss = k\nys = y # yss \\ length yss = k\n[PROOF STEP]\nhave \"length xss = length yss \\ length xss = k\"\n[PROOF STATE]\nproof (prove)\nusing this:\nxs = x # xss \\ length xss = k\nys = y # yss \\ length yss = k\n\ngoal (1 subgoal):\n 1. length xss = length yss \\ length xss = k\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength xss = length yss \\ length xss = k\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength xss = length yss \\ length xss = k\n[PROOF STEP]\nhave 3:\"(scalar_product (xss@zs) (yss@ws))\n = (scalar_product xss yss)\n +(scalar_product zs ws)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength xss = length yss \\ length xss = k\n\ngoal (1 subgoal):\n 1. scalar_product (xss @ zs) (yss @ ws) = scalar_product xss yss + scalar_product zs ws\n[PROOF STEP]\nusing 1 2 assms Suc\n[PROOF STATE]\nproof (prove)\nusing this:\nlength xss = length yss \\ length xss = k\nxs = x # xss \\ length xss = k\nys = y # yss \\ length yss = k\nlength zs = length ws \\ length xs = length ys \\ length xs = Suc k\nlength ?zs = length ?ws \\ length ?xs = length ?ys \\ length ?xs = k \\ scalar_product (?xs @ ?zs) (?ys @ ?ws) = scalar_product ?xs ?ys + scalar_product ?zs ?ws\n\ngoal (1 subgoal):\n 1. scalar_product (xss @ zs) (yss @ ws) = scalar_product xss yss + scalar_product zs ws\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (xss @ zs) (yss @ ws) = scalar_product xss yss + scalar_product zs ws\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product (xss @ zs) (yss @ ws) = scalar_product xss yss + scalar_product zs ws\n[PROOF STEP]\nhave 4:\"(scalar_product ((x#xss)@zs) ((y#yss)@ws)) = \n (scalar_product (x#(xss@zs)) (y#(yss@ws)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product (xss @ zs) (yss @ ws) = scalar_product xss yss + scalar_product zs ws\n\ngoal (1 subgoal):\n 1. scalar_product ((x # xss) @ zs) ((y # yss) @ ws) = scalar_product (x # xss @ zs) (y # yss @ ws)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product ((x # xss) @ zs) ((y # yss) @ ws) = scalar_product (x # xss @ zs) (y # yss @ ws)\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product ((x # xss) @ zs) ((y # yss) @ ws) = scalar_product (x # xss @ zs) (y # yss @ ws)\n[PROOF STEP]\nhave \"... = (x*y) + (scalar_product (xss@zs) (yss@ws))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product ((x # xss) @ zs) ((y # yss) @ ws) = scalar_product (x # xss @ zs) (y # yss @ ws)\n\ngoal (1 subgoal):\n 1. scalar_product (x # xss @ zs) (y # yss @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\n[PROOF STEP]\nunfolding scalar_product_def scalar_prodI_def\n[PROOF STATE]\nproof (prove)\nusing this:\nfoldr (\\(x, y). (+) (x * y)) (zip ((x # xss) @ zs) ((y # yss) @ ws)) zer = foldr (\\(x, y). (+) (x * y)) (zip (x # xss @ zs) (y # yss @ ws)) zer\n\ngoal (1 subgoal):\n 1. foldr (\\(x, y). (+) (x * y)) (zip (x # xss @ zs) (y # yss @ ws)) zer = x * y + foldr (\\(x, y). (+) (x * y)) (zip (xss @ zs) (yss @ ws)) zer\n[PROOF STEP]\nusing zip_Cons scalar_prodI_def scalar_prod_cons\n[PROOF STATE]\nproof (prove)\nusing this:\nfoldr (\\(x, y). (+) (x * y)) (zip ((x # xss) @ zs) ((y # yss) @ ws)) zer = foldr (\\(x, y). (+) (x * y)) (zip (x # xss @ zs) (y # yss @ ws)) zer\nlength ?v = length ?w \\ zip (?a # ?v) (?b # ?w) = (?a, ?b) # zip ?v ?w\nscalar_prodI ?ze ?pl ?ti ?v ?w = foldr (\\(x, y). ?pl (?ti x y)) (zip ?v ?w) ?ze\nscalar_prodI ?ze ?pl ?ti (?a # ?as) (?b # ?bs) = ?pl (?ti ?a ?b) (scalar_prodI ?ze ?pl ?ti ?as ?bs)\n\ngoal (1 subgoal):\n 1. foldr (\\(x, y). (+) (x * y)) (zip (x # xss @ zs) (y # yss @ ws)) zer = x * y + foldr (\\(x, y). (+) (x * y)) (zip (xss @ zs) (yss @ ws)) zer\n[PROOF STEP]\nby (metis)\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (x # xss @ zs) (y # yss @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nwith 4\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product ((x # xss) @ zs) ((y # yss) @ ws) = scalar_product (x # xss @ zs) (y # yss @ ws)\nscalar_product (x # xss @ zs) (y # yss @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\n[PROOF STEP]\nhave 5:\"(scalar_product (xs@zs) ((ys)@ws))\n = (x*y) + (scalar_product (xss@zs) (yss@ws))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product ((x # xss) @ zs) ((y # yss) @ ws) = scalar_product (x # xss @ zs) (y # yss @ ws)\nscalar_product (x # xss @ zs) (y # yss @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\n\ngoal (1 subgoal):\n 1. scalar_product (xs @ zs) (ys @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\n[PROOF STEP]\nusing 1 2\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product ((x # xss) @ zs) ((y # yss) @ ws) = scalar_product (x # xss @ zs) (y # yss @ ws)\nscalar_product (x # xss @ zs) (y # yss @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\nxs = x # xss \\ length xss = k\nys = y # yss \\ length yss = k\n\ngoal (1 subgoal):\n 1. scalar_product (xs @ zs) (ys @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (xs @ zs) (ys @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (xs @ zs) (ys @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nhave \"(scalar_product xs ys) = (x*y) + (scalar_product xss yss)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. scalar_product xs ys = x * y + scalar_product xss yss\n[PROOF STEP]\nunfolding scalar_product_def scalar_prodI_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. foldr (\\(x, y). (+) (x * y)) (zip xs ys) zer = x * y + foldr (\\(x, y). (+) (x * y)) (zip xss yss) zer\n[PROOF STEP]\nusing zip_Cons\n[PROOF STATE]\nproof (prove)\nusing this:\nlength ?v = length ?w \\ zip (?a # ?v) (?b # ?w) = (?a, ?b) # zip ?v ?w\n\ngoal (1 subgoal):\n 1. foldr (\\(x, y). (+) (x * y)) (zip xs ys) zer = x * y + foldr (\\(x, y). (+) (x * y)) (zip xss yss) zer\n[PROOF STEP]\nby (metis \"1\" \"2\" scalar_prodI_def scalar_prod_cons)\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product xs ys = x * y + scalar_product xss yss\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product xs ys = x * y + scalar_product xss yss\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product xs ys = x * y + scalar_product xss yss\n[PROOF STEP]\nhave \"(scalar_product xs ys)+(scalar_product zs ws)\n = (x*y) \n + (scalar_product xss yss) \n + (scalar_product zs ws)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product xs ys = x * y + scalar_product xss yss\n\ngoal (1 subgoal):\n 1. scalar_product xs ys + scalar_product zs ws = x * y + scalar_product xss yss + scalar_product zs ws\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product xs ys + scalar_product zs ws = x * y + scalar_product xss yss + scalar_product zs ws\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product (xs @ zs) (ys @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\nscalar_product xs ys = x * y + scalar_product xss yss\nscalar_product xs ys + scalar_product zs ws = x * y + scalar_product xss yss + scalar_product zs ws\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product (xs @ zs) (ys @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\nscalar_product xs ys = x * y + scalar_product xss yss\nscalar_product xs ys + scalar_product zs ws = x * y + scalar_product xss yss + scalar_product zs ws\n\ngoal (1 subgoal):\n 1. scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nusing 3 plus_assoc\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product (xs @ zs) (ys @ ws) = x * y + scalar_product (xss @ zs) (yss @ ws)\nscalar_product xs ys = x * y + scalar_product xss yss\nscalar_product xs ys + scalar_product zs ws = x * y + scalar_product xss yss + scalar_product zs ws\nscalar_product (xss @ zs) (yss @ ws) = scalar_product xss yss + scalar_product zs ws\n?a + ?b + ?c = ?a + (?b + ?c)\n\ngoal (1 subgoal):\n 1. scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nlength zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n\ngoal (1 subgoal):\n 1. \\n xs ys zs ws. (\\xs ys zs ws. length zs = length ws \\ length xs = length ys \\ length xs = n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws) \\ length zs = length ws \\ length xs = length ys \\ length xs = Suc n \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nlength zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n\ngoal (1 subgoal):\n 1. length zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength zs = length ws \\ length xs = length ys \\ length xs = Suc k \\ scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 11334, "file": "Matrix_Tensor_Matrix_Tensor", "length": 95, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032941988938414, "lm_q2_score": 0.7931059487389968, "lm_q1q2_score": 0.7164080026041322}} {"text": "[STATEMENT]\nlemma (in finite_measure) finite_measure_dist:\n assumes AE: \"AE x in M. x \\ C \\ (x \\ A \\ x \\ B)\"\n assumes sets: \"A \\ sets M\" \"B \\ sets M\" \"C \\ sets M\"\n shows \"dist (measure M A) (measure M B) \\ measure M C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nhave \"measure M A \\ measure M (B \\ C)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M A \\ Sigma_Algebra.measure M (B \\ C)\n[PROOF STEP]\nusing AE sets\n[PROOF STATE]\nproof (prove)\nusing this:\nAE x in M. x \\ C \\ (x \\ A) = (x \\ B)\nA \\ sets M\nB \\ sets M\nC \\ sets M\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M A \\ Sigma_Algebra.measure M (B \\ C)\n[PROOF STEP]\nby (auto intro: finite_measure_mono_AE)\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M A \\ Sigma_Algebra.measure M (B \\ C)\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M A \\ Sigma_Algebra.measure M (B \\ C)\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nhave \"\\ \\ measure M B + measure M C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (B \\ C) \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\n[PROOF STEP]\nusing sets\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ sets M\nB \\ sets M\nC \\ sets M\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (B \\ C) \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\n[PROOF STEP]\nby (auto intro: finite_measure_subadditive)\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M (B \\ C) \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nSigma_Algebra.measure M A \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\n[PROOF STEP]\nhave A: \"measure M A \\ measure M B + measure M C\"\n[PROOF STATE]\nproof (prove)\nusing this:\nSigma_Algebra.measure M A \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M A \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M A \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M A \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M A \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M A \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nhave \"measure M B \\ measure M (A \\ C)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M B \\ Sigma_Algebra.measure M (A \\ C)\n[PROOF STEP]\nusing AE sets\n[PROOF STATE]\nproof (prove)\nusing this:\nAE x in M. x \\ C \\ (x \\ A) = (x \\ B)\nA \\ sets M\nB \\ sets M\nC \\ sets M\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M B \\ Sigma_Algebra.measure M (A \\ C)\n[PROOF STEP]\nby (auto intro: finite_measure_mono_AE)\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M B \\ Sigma_Algebra.measure M (A \\ C)\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M B \\ Sigma_Algebra.measure M (A \\ C)\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nhave \"\\ \\ measure M A + measure M C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (A \\ C) \\ Sigma_Algebra.measure M A + Sigma_Algebra.measure M C\n[PROOF STEP]\nusing sets\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ sets M\nB \\ sets M\nC \\ sets M\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (A \\ C) \\ Sigma_Algebra.measure M A + Sigma_Algebra.measure M C\n[PROOF STEP]\nby (auto intro: finite_measure_subadditive)\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M (A \\ C) \\ Sigma_Algebra.measure M A + Sigma_Algebra.measure M C\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nSigma_Algebra.measure M B \\ Sigma_Algebra.measure M A + Sigma_Algebra.measure M C\n[PROOF STEP]\nhave B: \"measure M B \\ measure M A + measure M C\"\n[PROOF STATE]\nproof (prove)\nusing this:\nSigma_Algebra.measure M B \\ Sigma_Algebra.measure M A + Sigma_Algebra.measure M C\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M B \\ Sigma_Algebra.measure M A + Sigma_Algebra.measure M C\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M B \\ Sigma_Algebra.measure M A + Sigma_Algebra.measure M C\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M B \\ Sigma_Algebra.measure M A + Sigma_Algebra.measure M C\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nSigma_Algebra.measure M A \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\nSigma_Algebra.measure M B \\ Sigma_Algebra.measure M A + Sigma_Algebra.measure M C\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nSigma_Algebra.measure M A \\ Sigma_Algebra.measure M B + Sigma_Algebra.measure M C\nSigma_Algebra.measure M B \\ Sigma_Algebra.measure M A + Sigma_Algebra.measure M C\n\ngoal (1 subgoal):\n 1. dist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n[PROOF STEP]\nby (simp add: dist_real_def)\n[PROOF STATE]\nproof (state)\nthis:\ndist (Sigma_Algebra.measure M A) (Sigma_Algebra.measure M B) \\ Sigma_Algebra.measure M C\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2983, "file": "Probabilistic_Noninterference_Trace_Based", "length": 30, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637648915617, "lm_q2_score": 0.8333245994514084, "lm_q1q2_score": 0.7163789625411503}} {"text": "[STATEMENT]\nlemma Polygamma_real_strict_antimono:\n assumes \"x > 0\" \"x < (y::real)\" \"odd n\"\n shows \"Polygamma n x > Polygamma n y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Polygamma n y < Polygamma n x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Polygamma n y < Polygamma n x\n[PROOF STEP]\nhave \"\\\\. x < \\ \\ \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\>x. \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\nx < y\nodd n\n\ngoal (1 subgoal):\n 1. \\\\>x. \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n[PROOF STEP]\nby (intro MVT2 derivative_intros impI allI) (auto elim!: nonpos_Ints_cases)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\>x. \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n\ngoal (1 subgoal):\n 1. Polygamma n y < Polygamma n x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\>x. \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n[PROOF STEP]\nobtain \\\n where \\: \"x < \\\" \"\\ < y\"\n and Polygamma: \"Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\>x. \\ < y \\ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n\ngoal (1 subgoal):\n 1. (\\\\. \\x < \\; \\ < y; Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx < \\\n\\ < y\nPolygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n\ngoal (1 subgoal):\n 1. Polygamma n y < Polygamma n x\n[PROOF STEP]\nnote Polygamma\n[PROOF STATE]\nproof (state)\nthis:\nPolygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n\ngoal (1 subgoal):\n 1. Polygamma n y < Polygamma n x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nPolygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \\\n\ngoal (1 subgoal):\n 1. Polygamma n y < Polygamma n x\n[PROOF STEP]\nfrom \\ assms\n[PROOF STATE]\nproof (chain)\npicking this:\nx < \\\n\\ < y\n0 < x\nx < y\nodd n\n[PROOF STEP]\nhave \"(y - x) * Polygamma (Suc n) \\ < 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx < \\\n\\ < y\n0 < x\nx < y\nodd n\n\ngoal (1 subgoal):\n 1. (y - x) * Polygamma (Suc n) \\ < 0\n[PROOF STEP]\nby (intro mult_pos_neg Polygamma_real_even_neg) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(y - x) * Polygamma (Suc n) \\ < 0\n\ngoal (1 subgoal):\n 1. Polygamma n y < Polygamma n x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nPolygamma n y - Polygamma n x < 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nPolygamma n y - Polygamma n x < 0\n\ngoal (1 subgoal):\n 1. Polygamma n y < Polygamma n x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nPolygamma n y < Polygamma n x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1372, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637433190938, "lm_q2_score": 0.8333246015211009, "lm_q1q2_score": 0.7163789463435217}} {"text": "[STATEMENT]\nlemma Majority_exQ: \n assumes asm1: \"\\D \\ MajoritySet. \\d\\D. P d\"\n shows \"\\D\\MajoritySet.\\d\\D. P d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\D\\MajoritySet. \\d\\D. P d\n[PROOF STEP]\nusing asm1\n[PROOF STATE]\nproof (prove)\nusing this:\n\\D\\MajoritySet. \\d\\D. P d\n\ngoal (1 subgoal):\n 1. \\D\\MajoritySet. \\d\\D. P d\n[PROOF STEP]\nproof(auto simp add: MajoritySet_def)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\D Da. \\IsMajority D; \\x\\D. P x; IsMajority Da\\ \\ \\x\\Da. P x\n[PROOF STEP]\nfix D1 D2\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\D Da. \\IsMajority D; \\x\\D. P x; IsMajority Da\\ \\ \\x\\Da. P x\n[PROOF STEP]\nassume D1: \"IsMajority D1\" and D2: \"IsMajority D2\"\n and Px: \"\\x\\D1. P x\"\n[PROOF STATE]\nproof (state)\nthis:\nIsMajority D1\nIsMajority D2\n\\x\\D1. P x\n\ngoal (1 subgoal):\n 1. \\D Da. \\IsMajority D; \\x\\D. P x; IsMajority Da\\ \\ \\x\\Da. P x\n[PROOF STEP]\nfrom D1 D2 majorities_intersect\n[PROOF STATE]\nproof (chain)\npicking this:\nIsMajority D1\nIsMajority D2\n\\S T. IsMajority S \\ IsMajority T \\ S \\ T \\ {}\n[PROOF STEP]\nhave \"\\d\\D1. d\\D2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nIsMajority D1\nIsMajority D2\n\\S T. IsMajority S \\ IsMajority T \\ S \\ T \\ {}\n\ngoal (1 subgoal):\n 1. \\d\\D1. d \\ D2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\d\\D1. d \\ D2\n\ngoal (1 subgoal):\n 1. \\D Da. \\IsMajority D; \\x\\D. P x; IsMajority Da\\ \\ \\x\\Da. P x\n[PROOF STEP]\nwith Px\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x\\D1. P x\n\\d\\D1. d \\ D2\n[PROOF STEP]\nshow \"\\x\\D2. P x\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\D1. P x\n\\d\\D1. d \\ D2\n\ngoal (1 subgoal):\n 1. \\x\\D2. P x\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\D2. P x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1100, "file": "DiskPaxos_DiskPaxos_Inv4", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267898240861, "lm_q2_score": 0.8244619242200082, "lm_q1q2_score": 0.7163146069522587}} {"text": "[STATEMENT]\nlemma (in complete_measure) null_sets_outer:\n \"S \\ null_sets M \\ (\\e>0. \\T\\fmeasurable M. S \\ T \\ measure M T < e)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (S \\ null_sets M) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (S \\ null_sets M) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n[PROOF STEP]\nhave \"S \\ null_sets M \\ (S \\ fmeasurable M \\ 0 = measure M S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (S \\ null_sets M) = (S \\ fmeasurable M \\ 0 = Sigma_Algebra.measure M S)\n[PROOF STEP]\nby (auto simp: null_sets_def emeasure_eq_measure2 intro: fmeasurableI) (simp add: measure_def)\n[PROOF STATE]\nproof (state)\nthis:\n(S \\ null_sets M) = (S \\ fmeasurable M \\ 0 = Sigma_Algebra.measure M S)\n\ngoal (1 subgoal):\n 1. (S \\ null_sets M) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(S \\ null_sets M) = (S \\ fmeasurable M \\ 0 = Sigma_Algebra.measure M S)\n\ngoal (1 subgoal):\n 1. (S \\ null_sets M) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n[PROOF STEP]\nhave \"\\ = (\\e>0. \\T\\fmeasurable M. S \\ T \\ measure M T < e)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (S \\ fmeasurable M \\ 0 = Sigma_Algebra.measure M S) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n[PROOF STEP]\nunfolding fmeasurable_measure_inner_outer\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\e>0. \\T\\fmeasurable M. T \\ S \\ 0 - e < Sigma_Algebra.measure M T) \\ (\\e>0. \\U\\fmeasurable M. S \\ U \\ Sigma_Algebra.measure M U < 0 + e)) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(S \\ fmeasurable M \\ 0 = Sigma_Algebra.measure M S) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n\ngoal (1 subgoal):\n 1. (S \\ null_sets M) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(S \\ null_sets M) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(S \\ null_sets M) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n\ngoal (1 subgoal):\n 1. (S \\ null_sets M) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(S \\ null_sets M) = (\\e>0. \\T\\fmeasurable M. S \\ T \\ Sigma_Algebra.measure M T < e)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1414, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267796346599, "lm_q2_score": 0.824461932846258, "lm_q1q2_score": 0.7163146060461816}} {"text": "[STATEMENT]\nlemma H_tensor_Id_1: \n defines d:\"v \\ mat_of_cols_list 4 [[1/sqrt(2), 0, 1/sqrt(2), 0],\n [0, 1/sqrt(2), 0, 1/sqrt(2)],\n [1/sqrt(2), 0, -1/sqrt(2), 0],\n [0, 1/sqrt(2), 0, -1/sqrt(2)]]\"\n shows \"(H \\ Id 1) = v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. H \\ Quantum.Id 1 = v\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row v; j < dim_col v\\ \\ (H \\ Quantum.Id 1) $$ (i, j) = v $$ (i, j)\n 2. dim_row (H \\ Quantum.Id 1) = dim_row v\n 3. dim_col (H \\ Quantum.Id 1) = dim_col v\n[PROOF STEP]\nshow \"dim_col (H \\ Id 1) = dim_col v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_col (H \\ Quantum.Id 1) = dim_col v\n[PROOF STEP]\nby (simp add: d H_def Id_def mat_of_cols_list_def)\n[PROOF STATE]\nproof (state)\nthis:\ndim_col (H \\ Quantum.Id 1) = dim_col v\n\ngoal (2 subgoals):\n 1. \\i j. \\i < dim_row v; j < dim_col v\\ \\ (H \\ Quantum.Id 1) $$ (i, j) = v $$ (i, j)\n 2. dim_row (H \\ Quantum.Id 1) = dim_row v\n[PROOF STEP]\nshow \"dim_row (H \\ Id 1) = dim_row v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_row (H \\ Quantum.Id 1) = dim_row v\n[PROOF STEP]\nby (simp add: d H_def Id_def mat_of_cols_list_def)\n[PROOF STATE]\nproof (state)\nthis:\ndim_row (H \\ Quantum.Id 1) = dim_row v\n\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_row v; j < dim_col v\\ \\ (H \\ Quantum.Id 1) $$ (i, j) = v $$ (i, j)\n[PROOF STEP]\nfix i j:: nat\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_row v; j < dim_col v\\ \\ (H \\ Quantum.Id 1) $$ (i, j) = v $$ (i, j)\n[PROOF STEP]\nassume \"i < dim_row v\" and \"j < dim_col v\"\n[PROOF STATE]\nproof (state)\nthis:\ni < dim_row v\nj < dim_col v\n\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_row v; j < dim_col v\\ \\ (H \\ Quantum.Id 1) $$ (i, j) = v $$ (i, j)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni < dim_row v\nj < dim_col v\n[PROOF STEP]\nhave \"i \\ {0..<4} \\ j \\ {0..<4}\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_row v\nj < dim_col v\n\ngoal (1 subgoal):\n 1. i \\ {0..<4} \\ j \\ {0..<4}\n[PROOF STEP]\nby (auto simp add: d mat_of_cols_list_def)\n[PROOF STATE]\nproof (state)\nthis:\ni \\ {0..<4} \\ j \\ {0..<4}\n\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_row v; j < dim_col v\\ \\ (H \\ Quantum.Id 1) $$ (i, j) = v $$ (i, j)\n[PROOF STEP]\nthus \"(H \\ Id 1) $$ (i, j) = v $$ (i, j)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni \\ {0..<4} \\ j \\ {0..<4}\n\ngoal (1 subgoal):\n 1. (H \\ Quantum.Id 1) $$ (i, j) = v $$ (i, j)\n[PROOF STEP]\nby (auto simp add: d Id_def H_def mat_of_cols_list_def)\n[PROOF STATE]\nproof (state)\nthis:\n(H \\ Quantum.Id 1) $$ (i, j) = v $$ (i, j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1490, "file": "Isabelle_Marries_Dirac_Deutsch", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267796346598, "lm_q2_score": 0.8244619199068831, "lm_q1q2_score": 0.7163145948041061}} {"text": "[STATEMENT]\nlemma (in prime_number_theorem) totient_primorial'_asymp_equiv:\n \"(\\k. totient (primorial' k)) \\[at_top] (\\k. third_mertens_const * primorial' k / ln k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. real (totient (primorial' x))) \\[sequentially] (\\k. third_mertens_const * real (primorial' k) / ln (real k))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x. real (totient (primorial' x))) \\[sequentially] (\\k. third_mertens_const * real (primorial' k) / ln (real k))\n[PROOF STEP]\nlet ?C = third_mertens_const\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x. real (totient (primorial' x))) \\[sequentially] (\\k. third_mertens_const * real (primorial' k) / ln (real k))\n[PROOF STEP]\nhave \"(\\k. totient (primorial' k)) \\[at_top] (\\k. primorial' k * (\\ix. real (totient (primorial' x))) \\[sequentially] (\\k. real (primorial' k) * (\\ix. real (totient (primorial' x))) \\[sequentially] (\\k. real (primorial' k) * (\\ix. real (totient (primorial' x))) \\[sequentially] (\\k. third_mertens_const * real (primorial' k) / ln (real k))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. real (totient (primorial' x))) \\[sequentially] (\\k. real (primorial' k) * (\\ix. real (totient (primorial' x))) \\[sequentially] (\\k. third_mertens_const * real (primorial' k) / ln (real k))\n[PROOF STEP]\nhave \"\\ \\[at_top] (\\k. primorial' k * (?C / ln k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. real (primorial' k) * (\\i[sequentially] (\\k. real (primorial' k) * (third_mertens_const / ln (real k)))\n[PROOF STEP]\nby (intro asymp_equiv_intros mertens_third_theorem_asymp_equiv')\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. real (primorial' k) * (\\i[sequentially] (\\k. real (primorial' k) * (third_mertens_const / ln (real k)))\n\ngoal (1 subgoal):\n 1. (\\x. real (totient (primorial' x))) \\[sequentially] (\\k. third_mertens_const * real (primorial' k) / ln (real k))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\c d. c \\[sequentially] d \\ c \\[sequentially] d) \\ (\\x. real (totient (primorial' x))) \\[sequentially] (\\a. real (primorial' a) * (third_mertens_const / ln (real a)))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\c d. c \\[sequentially] d \\ c \\[sequentially] d) \\ (\\x. real (totient (primorial' x))) \\[sequentially] (\\a. real (primorial' a) * (third_mertens_const / ln (real a)))\n\ngoal (1 subgoal):\n 1. (\\x. real (totient (primorial' x))) \\[sequentially] (\\k. third_mertens_const * real (primorial' k) / ln (real k))\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. real (totient (primorial' x))) \\[sequentially] (\\k. third_mertens_const * real (primorial' k) / ln (real k))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1442, "file": "Prime_Distribution_Elementary_PNT_Consequences", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110540642806, "lm_q2_score": 0.8031737963569014, "lm_q1q2_score": 0.7162792699258581}} {"text": "[STATEMENT]\nlemma nu_witness_properties:\n \"mu < nu_witness\"\n \"nu_witness \\ lambda + mu\"\n \"lambda dvd nu_witness\"\n \"mu = 0 \\ nu_witness = lambda\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (mu < nu_witness &&& nu_witness \\ lambda + mu) &&& lambda dvd nu_witness &&& (mu = 0 \\ nu_witness = lambda)\n[PROOF STEP]\nunfolding nu_witness_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (mu < mu + lambda - mu mod lambda &&& mu + lambda - mu mod lambda \\ lambda + mu) &&& lambda dvd mu + lambda - mu mod lambda &&& (mu = 0 \\ mu + lambda - mu mod lambda = lambda)\n[PROOF STEP]\nusing properties_lambda_gt_0\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < lambda\n\ngoal (1 subgoal):\n 1. (mu < mu + lambda - mu mod lambda &&& mu + lambda - mu mod lambda \\ lambda + mu) &&& lambda dvd mu + lambda - mu mod lambda &&& (mu = 0 \\ mu + lambda - mu mod lambda = lambda)\n[PROOF STEP]\napply (simp_all add: less_diff_conv divide_simps)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < lambda \\ lambda dvd mu + lambda - mu mod lambda\n[PROOF STEP]\napply (metis minus_mod_eq_div_mult [symmetric] dvd_def mod_add_self2 mult.commute)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 493, "file": "TortoiseHare_TortoiseHare", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110569397306, "lm_q2_score": 0.8031737916455819, "lm_q1q2_score": 0.7162792680337373}} {"text": "[STATEMENT]\nlemma card_extensional_funcset:\n assumes \"finite A\"\n shows \"card (A \\\\<^sub>E B) = card B ^ card A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (A \\\\<^sub>E B) = card B ^ card A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. card (A \\\\<^sub>E B) = card B ^ card A\n[PROOF STEP]\nby (simp add: card_PiE prod_constant)", "meta": {"llama_tokens": 167, "file": "Twelvefold_Way_Preliminaries", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110454379297, "lm_q2_score": 0.8031737940012417, "lm_q1q2_score": 0.7162792608965957}} {"text": "[STATEMENT]\nlemma lin_bound_arg_general_list: \n fixes xs ::\"('a :: {field})vec list\"\n assumes \"distinct xs\"\n assumes \"(set xs) \\ carrier_vec nr\"\n assumes \"vec_space.lin_indpt_vs nr (set xs)\"\n shows \"length (xs) \\ nr\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length xs \\ nr\n[PROOF STEP]\nusing lin_bound_arg_general_set[of \"set xs\" nr] distinct_card assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\set xs \\ carrier_vec nr; vec_space.lin_indpt_vs nr (set xs)\\ \\ card (set xs) \\ nr\ndistinct ?xs \\ card (set ?xs) = length ?xs\ndistinct xs\nset xs \\ carrier_vec nr\nvec_space.lin_indpt_vs nr (set xs)\n\ngoal (1 subgoal):\n 1. length xs \\ nr\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 294, "file": "Fishers_Inequality_Linear_Bound_Argument", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110339361275, "lm_q2_score": 0.803173791645582, "lm_q1q2_score": 0.7162792495578463}} {"text": "[STATEMENT]\nlemma arcsin_arccos_sqrt_pos: \"0 \\ x \\ x \\ 1 \\ arcsin x = arccos(sqrt(1 - x\\<^sup>2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 \\ x; x \\ 1\\ \\ arcsin x = arccos (sqrt (1 - x\\<^sup>2))\n[PROOF STEP]\napply (simp add: abs_square_le_1 arcsin_eq_Re_Arcsin arccos_eq_Re_Arccos)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 \\ x; x \\ 1\\ \\ Re (Arcsin (complex_of_real x)) = Re (Arccos (complex_of_real (sqrt (1 - x\\<^sup>2))))\n[PROOF STEP]\napply (subst Arcsin_Arccos_csqrt_pos)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\0 \\ x; x \\ 1\\ \\ 0 < Re (complex_of_real x) \\ Re (complex_of_real x) = 0 \\ 0 \\ Im (complex_of_real x)\n 2. \\0 \\ x; x \\ 1\\ \\ Re (Arccos (csqrt (1 - (complex_of_real x)\\<^sup>2))) = Re (Arccos (complex_of_real (sqrt (1 - x\\<^sup>2))))\n[PROOF STEP]\napply (auto simp: power_le_one csqrt_1_diff_eq)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 511, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898279984213, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7162124229679887}} {"text": "[STATEMENT]\ntheorem pappus_iff_pascal: \"is_pappus = pascal_prop\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_pappus = pascal_prop\n[PROOF STEP]\nusing pappus_pascal pascal_pappus\n[PROOF STATE]\nproof (prove)\nusing this:\nis_pappus \\ pascal_prop\npascal_prop \\ is_pappus\n\ngoal (1 subgoal):\n 1. is_pappus = pascal_prop\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 160, "file": "Projective_Geometry_Pascal_Property", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9059898203834278, "lm_q2_score": 0.7905303186696747, "lm_q1q2_score": 0.7162124214191925}} {"text": "[STATEMENT]\nlemma Z_XpZ_rho_psim[simp]:\n shows \"Complex_Matrix.trace (rho_psim * Z_XpZ) =1/ (sqrt 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (rho_psim * Z_XpZ) = complex_of_real (1 / sqrt 2)\n[PROOF STEP]\napply (auto simp add: rho_psim_def ket_psim_def ket_10_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (rank_1_proj (1 / complex_of_real (sqrt 2) \\\\<^sub>v (ket_01 - (ket_1 \\ ket_0))) * Z_XpZ) = 1 / complex_of_real (sqrt 2)\n[PROOF STEP]\napply (auto simp add: Z_XpZ_def XpZ_def X_def Z_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (rank_1_proj (1 / complex_of_real (sqrt 2) \\\\<^sub>v (ket_01 - (ket_1 \\ ket_0))) * (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) \\ - (1 / complex_of_real (sqrt 2)) \\\\<^sub>m (Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1) + Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1)))) = 1 / complex_of_real (sqrt 2)\n[PROOF STEP]\napply (auto simp add: rank_1_proj_def outer_prod_def ket_01_def ket_1_def ket_0_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (Matrix.mat 4 (Suc 0) (\\(i, j). (1 / complex_of_real (sqrt 2) \\\\<^sub>v ((unit_vec 2 0 \\ unit_vec 2 (Suc 0)) - (unit_vec 2 (Suc 0) \\ unit_vec 2 0))) $ i) * Matrix.mat (Suc 0) 4 (\\(i, j). conjugate (1 / complex_of_real (sqrt 2) \\\\<^sub>v ((unit_vec 2 0 \\ unit_vec 2 (Suc 0)) - (unit_vec 2 (Suc 0) \\ unit_vec 2 0))) $ j) * (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) \\ - (1 / complex_of_real (sqrt 2)) \\\\<^sub>m (Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1) + Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1)))) = 1 / complex_of_real (sqrt 2)\n[PROOF STEP]\napply (auto simp add: Complex_Matrix.trace_def sum_4_elems)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 / (complex_of_real (sqrt 2) * (complex_of_real (sqrt 2) * complex_of_real (sqrt 2))) = 1 / complex_of_real (sqrt 2)\n[PROOF STEP]\napply (simp add: csqrt_2_sq)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1036, "file": "Projective_Measurements_CHSH_Inequality", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9059898102301019, "lm_q2_score": 0.7905303236047049, "lm_q1q2_score": 0.7162124178637677}} {"text": "[STATEMENT]\nlemma sorted_list_of_set_Un:\n assumes AB: \"less_sets A B\" and fin: \"finite A\" \"finite B\"\n shows \"sorted_list_of_set (A \\ B) = sorted_list_of_set A @ sorted_list_of_set B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sorted_list_of_set (A \\ B) = sorted_list_of_set A @ sorted_list_of_set B\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sorted_list_of_set (A \\ B) = sorted_list_of_set A @ sorted_list_of_set B\n[PROOF STEP]\nhave \"strict_sorted (sorted_list_of_set A @ sorted_list_of_set B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. strict_sorted (sorted_list_of_set A @ sorted_list_of_set B)\n[PROOF STEP]\nusing AB\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ B\n\ngoal (1 subgoal):\n 1. strict_sorted (sorted_list_of_set A @ sorted_list_of_set B)\n[PROOF STEP]\nunfolding less_sets_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\A. \\y\\B. x < y\n\ngoal (1 subgoal):\n 1. strict_sorted (sorted_list_of_set A @ sorted_list_of_set B)\n[PROOF STEP]\nby (metis fin set_sorted_list_of_set sorted_wrt_append strict_sorted_list_of_set)\n[PROOF STATE]\nproof (state)\nthis:\nstrict_sorted (sorted_list_of_set A @ sorted_list_of_set B)\n\ngoal (1 subgoal):\n 1. sorted_list_of_set (A \\ B) = sorted_list_of_set A @ sorted_list_of_set B\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nstrict_sorted (sorted_list_of_set A @ sorted_list_of_set B)\n\ngoal (1 subgoal):\n 1. sorted_list_of_set (A \\ B) = sorted_list_of_set A @ sorted_list_of_set B\n[PROOF STEP]\nhave \"card A + card B = card (A \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card A + card B = card (A \\ B)\n[PROOF STEP]\nusing less_sets_imp_disjnt [OF AB]\n[PROOF STATE]\nproof (prove)\nusing this:\ndisjnt A B\n\ngoal (1 subgoal):\n 1. card A + card B = card (A \\ B)\n[PROOF STEP]\nby (simp add: assms card_Un_disjoint disjnt_def)\n[PROOF STATE]\nproof (state)\nthis:\ncard A + card B = card (A \\ B)\n\ngoal (1 subgoal):\n 1. sorted_list_of_set (A \\ B) = sorted_list_of_set A @ sorted_list_of_set B\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nstrict_sorted (sorted_list_of_set A @ sorted_list_of_set B)\ncard A + card B = card (A \\ B)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nstrict_sorted (sorted_list_of_set A @ sorted_list_of_set B)\ncard A + card B = card (A \\ B)\n\ngoal (1 subgoal):\n 1. sorted_list_of_set (A \\ B) = sorted_list_of_set A @ sorted_list_of_set B\n[PROOF STEP]\nby (simp add: assms strict_sorted_equal)\n[PROOF STATE]\nproof (state)\nthis:\nsorted_list_of_set (A \\ B) = sorted_list_of_set A @ sorted_list_of_set B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1160, "file": "Ordinal_Partitions_Library_Additions", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382236515258, "lm_q2_score": 0.8311430436757313, "lm_q1q2_score": 0.7160615014487121}} {"text": "[STATEMENT]\nlemma birthday_paradox:\n assumes \"card S = 23\" \"card T = 365\"\n shows \"2 * card {f \\ extensional_funcset S T. \\ inj_on f S} \\ card (extensional_funcset S T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (S \\\\<^sub>E T) \\ 2 * card {f \\ S \\\\<^sub>E T. \\ inj_on f S}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (S \\\\<^sub>E T) \\ 2 * card {f \\ S \\\\<^sub>E T. \\ inj_on f S}\n[PROOF STEP]\nfrom \\card S = 23\\ \\card T = 365\\\n[PROOF STATE]\nproof (chain)\npicking this:\ncard S = 23\ncard T = 365\n[PROOF STEP]\nhave \"finite S\" \"finite T\" \"card S <= card T\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard S = 23\ncard T = 365\n\ngoal (1 subgoal):\n 1. finite S &&& finite T &&& card S \\ card T\n[PROOF STEP]\nby (auto intro: card_ge_0_finite)\n[PROOF STATE]\nproof (state)\nthis:\nfinite S\nfinite T\ncard S \\ card T\n\ngoal (1 subgoal):\n 1. card (S \\\\<^sub>E T) \\ 2 * card {f \\ S \\\\<^sub>E T. \\ inj_on f S}\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\ncard S = 23\ncard T = 365\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard S = 23\ncard T = 365\n\ngoal (1 subgoal):\n 1. card (S \\\\<^sub>E T) \\ 2 * card {f \\ S \\\\<^sub>E T. \\ inj_on f S}\n[PROOF STEP]\nusing card_PiE[OF \\finite S\\, of \"\\i. T\"] \\finite S\\\n card_extensional_funcset_not_inj_on[OF \\finite S\\ \\finite T\\ \\card S <= card T\\]\n[PROOF STATE]\nproof (prove)\nusing this:\ncard S = 23\ncard T = 365\ncard (S \\\\<^sub>E T) = (\\i\\S. card T)\nfinite S\ncard {f \\ S \\\\<^sub>E T. \\ inj_on f S} = card T ^ card S - fact (card T) div fact (card T - card S)\n\ngoal (1 subgoal):\n 1. card (S \\\\<^sub>E T) \\ 2 * card {f \\ S \\\\<^sub>E T. \\ inj_on f S}\n[PROOF STEP]\nby (simp add: fact_div_fact prod_upto_nat_unfold prod_constant)\n[PROOF STATE]\nproof (state)\nthis:\ncard (S \\\\<^sub>E T) \\ 2 * card {f \\ S \\\\<^sub>E T. \\ inj_on f S}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 967, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615382129861583, "lm_q2_score": 0.8311430520409024, "lm_q1q2_score": 0.7160614997911806}} {"text": "[STATEMENT]\nlemma complex_quadratic_equation_two_roots:\n fixes \\ :: complex\n assumes \"a \\ 0\" and \"a*\\\\<^sup>2 + b * \\ + c = 0\"\n shows \"\\ = (-b + ccsqrt(b\\<^sup>2 - 4*a*c)) / (2*a) \\\n \\ = (-b - ccsqrt(b\\<^sup>2 - 4*a*c)) / (2*a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ 0\na * \\\\<^sup>2 + b * \\ + c = 0\n[PROOF STEP]\nhave \"\\\\<^sup>2 + (b/a) * \\ + (c/a) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ 0\na * \\\\<^sup>2 + b * \\ + c = 0\n\ngoal (1 subgoal):\n 1. \\\\<^sup>2 + b / a * \\ + c / a = 0\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>2 + b / a * \\ + c / a = 0\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nhence \"\\ = (-(b/a) + ccsqrt((b/a)\\<^sup>2 - 4*(c/a))) / 2 \\ \\ = (-(b/a) - ccsqrt((b/a)\\<^sup>2 - 4*(c/a))) / 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>2 + b / a * \\ + c / a = 0\n\ngoal (1 subgoal):\n 1. \\ = (- (b / a) + ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2 \\ \\ = (- (b / a) - ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\n[PROOF STEP]\nusing complex_quadratic_equation_monic_only_two_roots[of \\ \"b/a\" \"c/a\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>2 + b / a * \\ + c / a = 0\n\\\\<^sup>2 + b / a * \\ + c / a = 0 \\ \\ = (- (b / a) + ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2 \\ \\ = (- (b / a) - ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\n\ngoal (1 subgoal):\n 1. \\ = (- (b / a) + ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2 \\ \\ = (- (b / a) - ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\ = (- (b / a) + ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2 \\ \\ = (- (b / a) - ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nhence \"\\ k. \\ = (-(b/a) + (-1)^k * ccsqrt((b/a)\\<^sup>2 - 4*(c/a))) / 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ = (- (b / a) + ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2 \\ \\ = (- (b / a) - ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\n\ngoal (1 subgoal):\n 1. \\k. \\ = (- (b / a) + (- 1) ^ k * ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\n[PROOF STEP]\nby safe (rule_tac x=\"2\" in exI, simp, rule_tac x=\"1\" in exI, simp)\n[PROOF STATE]\nproof (state)\nthis:\n\\k. \\ = (- (b / a) + (- 1) ^ k * ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\k. \\ = (- (b / a) + (- 1) ^ k * ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\n[PROOF STEP]\nobtain k1 where \"\\ = (-(b/a) + (-1)^k1 * ccsqrt((b/a)\\<^sup>2 - 4*(c/a))) / 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\k. \\ = (- (b / a) + (- 1) ^ k * ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\n\ngoal (1 subgoal):\n 1. (\\k1. \\ = (- (b / a) + (- 1) ^ k1 * ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2 \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\ = (- (b / a) + (- 1) ^ k1 * ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\ = (- (b / a) + (- 1) ^ k1 * ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nhave \"(b / a)\\<^sup>2 - 4 * (c / a) = (b\\<^sup>2 - 4 * a * c) * (1 / a\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (b / a)\\<^sup>2 - 4 * (c / a) = (b\\<^sup>2 - 4 * a * c) * (1 / a\\<^sup>2)\n[PROOF STEP]\nusing \\a \\ 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ 0\n\ngoal (1 subgoal):\n 1. (b / a)\\<^sup>2 - 4 * (c / a) = (b\\<^sup>2 - 4 * a * c) * (1 / a\\<^sup>2)\n[PROOF STEP]\nby (simp add: field_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n(b / a)\\<^sup>2 - 4 * (c / a) = (b\\<^sup>2 - 4 * a * c) * (1 / a\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nhence \"ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1/a\\<^sup>2) \\\n ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = - ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1/a\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(b / a)\\<^sup>2 - 4 * (c / a) = (b\\<^sup>2 - 4 * a * c) * (1 / a\\<^sup>2)\n\ngoal (1 subgoal):\n 1. ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2) \\ ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = - ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\n[PROOF STEP]\nusing ccsqrt_mult[of \"b\\<^sup>2 - 4 * a * c\" \"1/a\\<^sup>2\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(b / a)\\<^sup>2 - 4 * (c / a) = (b\\<^sup>2 - 4 * a * c) * (1 / a\\<^sup>2)\nccsqrt ((b\\<^sup>2 - 4 * a * c) * (1 / a\\<^sup>2)) = ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2) \\ ccsqrt ((b\\<^sup>2 - 4 * a * c) * (1 / a\\<^sup>2)) = - ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\n\ngoal (1 subgoal):\n 1. ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2) \\ ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = - ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2) \\ ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = - ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nhence \"\\ k. ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = (-1)^k * ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2) \\ ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = - ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\k. ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = (- 1) ^ k * ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\n[PROOF STEP]\nby safe (rule_tac x=\"2\" in exI, simp, rule_tac x=\"1\" in exI, simp)\n[PROOF STATE]\nproof (state)\nthis:\n\\k. ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = (- 1) ^ k * ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\k. ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = (- 1) ^ k * ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\n[PROOF STEP]\nobtain k2 where \"ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = (-1)^k2 * ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\k. ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = (- 1) ^ k * ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (\\k2. ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = (- 1) ^ k2 * ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2) \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = (- 1) ^ k2 * ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = (- 1) ^ k2 * ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nhave \"ccsqrt (1 / a\\<^sup>2) = 1/a \\ ccsqrt (1 / a\\<^sup>2) = -1/a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ccsqrt (1 / a\\<^sup>2) = 1 / a \\ ccsqrt (1 / a\\<^sup>2) = - 1 / a\n[PROOF STEP]\nusing ccsqrt[of \"1/a\" \"1 / a\\<^sup>2\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n1 / a * (1 / a) = 1 / a\\<^sup>2 \\ 1 / a = ccsqrt (1 / a\\<^sup>2) \\ 1 / a = - ccsqrt (1 / a\\<^sup>2)\n\ngoal (1 subgoal):\n 1. ccsqrt (1 / a\\<^sup>2) = 1 / a \\ ccsqrt (1 / a\\<^sup>2) = - 1 / a\n[PROOF STEP]\nby (auto simp add: power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\nccsqrt (1 / a\\<^sup>2) = 1 / a \\ ccsqrt (1 / a\\<^sup>2) = - 1 / a\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nhence \"\\ k. ccsqrt (1 / a\\<^sup>2) = (-1)^k * 1/a\"\n[PROOF STATE]\nproof (prove)\nusing this:\nccsqrt (1 / a\\<^sup>2) = 1 / a \\ ccsqrt (1 / a\\<^sup>2) = - 1 / a\n\ngoal (1 subgoal):\n 1. \\k. ccsqrt (1 / a\\<^sup>2) = (- 1) ^ k * 1 / a\n[PROOF STEP]\nby safe (rule_tac x=\"2\" in exI, simp, rule_tac x=\"1\" in exI, simp)\n[PROOF STATE]\nproof (state)\nthis:\n\\k. ccsqrt (1 / a\\<^sup>2) = (- 1) ^ k * 1 / a\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\k. ccsqrt (1 / a\\<^sup>2) = (- 1) ^ k * 1 / a\n[PROOF STEP]\nobtain k3 where \"ccsqrt (1 / a\\<^sup>2) = (-1)^k3 * 1/a\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\k. ccsqrt (1 / a\\<^sup>2) = (- 1) ^ k * 1 / a\n\ngoal (1 subgoal):\n 1. (\\k3. ccsqrt (1 / a\\<^sup>2) = (- 1) ^ k3 * 1 / a \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nccsqrt (1 / a\\<^sup>2) = (- 1) ^ k3 * 1 / a\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ = (- (b / a) + (- 1) ^ k1 * ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\nccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = (- 1) ^ k2 * ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\nccsqrt (1 / a\\<^sup>2) = (- 1) ^ k3 * 1 / a\n[PROOF STEP]\nhave \"\\ = (- (b / a) + ((-1) ^ k1 * (-1) ^ k2 * (-1) ^ k3) * ccsqrt (b\\<^sup>2 - 4 * a * c) * 1/a) / 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ = (- (b / a) + (- 1) ^ k1 * ccsqrt ((b / a)\\<^sup>2 - 4 * (c / a))) / 2\nccsqrt ((b / a)\\<^sup>2 - 4 * (c / a)) = (- 1) ^ k2 * ccsqrt (b\\<^sup>2 - 4 * a * c) * ccsqrt (1 / a\\<^sup>2)\nccsqrt (1 / a\\<^sup>2) = (- 1) ^ k3 * 1 / a\n\ngoal (1 subgoal):\n 1. \\ = (- (b / a) + (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 * ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\ = (- (b / a) + (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 * ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\ = (- (b / a) + (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 * ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nhave \"(-(1::complex)) ^ k1 * (-1) ^ k2 * (-1) ^ k3 = 1 \\ (-(1::complex)) ^ k1 * (-1) ^ k2 * (-1) ^ k3 = -1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = 1 \\ (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = - 1\n[PROOF STEP]\nusing neg_one_even_power[of \"k1 + k2 + k3\"]\n[PROOF STATE]\nproof (prove)\nusing this:\neven (k1 + k2 + k3) \\ (- (1::?'a)) ^ (k1 + k2 + k3) = (1::?'a)\n\ngoal (1 subgoal):\n 1. (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = 1 \\ (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = - 1\n[PROOF STEP]\nusing neg_one_odd_power[of \"k1 + k2 + k3\"]\n[PROOF STATE]\nproof (prove)\nusing this:\neven (k1 + k2 + k3) \\ (- (1::?'a)) ^ (k1 + k2 + k3) = (1::?'a)\nodd (k1 + k2 + k3) \\ (- (1::?'a)) ^ (k1 + k2 + k3) = - (1::?'a)\n\ngoal (1 subgoal):\n 1. (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = 1 \\ (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = - 1\n[PROOF STEP]\nby (smt power_add)\n[PROOF STATE]\nproof (state)\nthis:\n(- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = 1 \\ (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = - 1\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ = (- (b / a) + (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 * ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2\n(- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = 1 \\ (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = - 1\n[PROOF STEP]\nhave \"\\ = (- (b / a) + ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2 \\ \\ = (- (b / a) - ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ = (- (b / a) + (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 * ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2\n(- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = 1 \\ (- 1) ^ k1 * (- 1) ^ k2 * (- 1) ^ k3 = - 1\n\ngoal (1 subgoal):\n 1. \\ = (- (b / a) + ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2 \\ \\ = (- (b / a) - ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\ = (- (b / a) + ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2 \\ \\ = (- (b / a) - ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ = (- (b / a) + ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2 \\ \\ = (- (b / a) - ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nusing \\a \\ 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ = (- (b / a) + ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2 \\ \\ = (- (b / a) - ccsqrt (b\\<^sup>2 - 4 * a * c) * 1 / a) / 2\na \\ 0\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\ = (- b + ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a) \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * a * c)) / (2 * a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 8377, "file": "Complex_Geometry_Quadratic", "length": 48, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382094310355, "lm_q2_score": 0.8311430499496096, "lm_q1q2_score": 0.7160614950346365}} {"text": "[STATEMENT]\nlemma card_eq_UNIV[simp]: \"card (S::'a::finite set) = card (UNIV::'a set) \\ S=UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (card S = card UNIV) = (S = UNIV)\n[PROOF STEP]\nproof (auto)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. card S = card UNIV \\ x \\ S\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. card S = card UNIV \\ x \\ S\n[PROOF STEP]\nassume A: \"card S = card (UNIV::'a set)\"\n[PROOF STATE]\nproof (state)\nthis:\ncard S = card UNIV\n\ngoal (1 subgoal):\n 1. \\x. card S = card UNIV \\ x \\ S\n[PROOF STEP]\nshow \"x\\S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ S\n[PROOF STEP]\nproof (rule ccontr)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x \\ S \\ False\n[PROOF STEP]\nassume \"x\\S\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ S\n\ngoal (1 subgoal):\n 1. x \\ S \\ False\n[PROOF STEP]\nhence \"S\\UNIV\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ S\n\ngoal (1 subgoal):\n 1. S \\ UNIV\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nS \\ UNIV\n\ngoal (1 subgoal):\n 1. x \\ S \\ False\n[PROOF STEP]\nwith psubset_card_mono[of UNIV S]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\finite UNIV; S \\ UNIV\\ \\ card S < card UNIV\nS \\ UNIV\n[PROOF STEP]\nhave \"card S < card (UNIV::'a set)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite UNIV; S \\ UNIV\\ \\ card S < card UNIV\nS \\ UNIV\n\ngoal (1 subgoal):\n 1. card S < card UNIV\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard S < card UNIV\n\ngoal (1 subgoal):\n 1. x \\ S \\ False\n[PROOF STEP]\nwith A\n[PROOF STATE]\nproof (chain)\npicking this:\ncard S = card UNIV\ncard S < card UNIV\n[PROOF STEP]\nshow False\n[PROOF STATE]\nproof (prove)\nusing this:\ncard S = card UNIV\ncard S < card UNIV\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nx \\ S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 953, "file": "Automatic_Refinement_Lib_Misc", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744850834648, "lm_q2_score": 0.8757869835428966, "lm_q1q2_score": 0.7160210921128846}} {"text": "[STATEMENT]\nlemma HComplex_diff [simp]:\n \"\\x1 y1 x2 y2. HComplex x1 y1 - HComplex x2 y2 = HComplex (x1 - x2) (y1 - y2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x1 y1 x2 y2. HComplex x1 y1 - HComplex x2 y2 = HComplex (x1 - x2) (y1 - y2)\n[PROOF STEP]\nby transfer (rule complex_diff)", "meta": {"llama_tokens": 147, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869786798664, "lm_q2_score": 0.8175744673038222, "lm_q1q2_score": 0.7160210725658157}} {"text": "[STATEMENT]\nlemma Log_inverse:\n \"\\ a > 0; a \\ 1; x > 0 \\ \\ Log a (inverse x) = - Log a x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < a; a \\ 1; 0 < x\\ \\ Log a (inverse x) = - Log a x\n[PROOF STEP]\nby (metis Log_powreal_cancel powreal_Log_cancel powreal_minus)", "meta": {"llama_tokens": 151, "file": "Real_Power_Log", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587993853654, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7160126990989762}} {"text": "[STATEMENT]\nlemma odd_odd_even_nat': \n fixes a b :: nat \n assumes \"odd a\" \"odd b\"\n shows \"even (((a * b)\\<^sup>2 - 1) div 8) \\ even (((a\\<^sup>2 - 1) div 8) + ((b\\<^sup>2 - 1) div 8))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. even (((a * b)\\<^sup>2 - 1) div 8) = even ((a\\<^sup>2 - 1) div 8 + (b\\<^sup>2 - 1) div 8)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. even (((a * b)\\<^sup>2 - 1) div 8) = even ((a\\<^sup>2 - 1) div 8 + (b\\<^sup>2 - 1) div 8)\n[PROOF STEP]\nobtain x where [simp]: \"a = 2*x + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. a = 2 * x + 1 \\ thesis) \\ thesis\n[PROOF STEP]\nusing \\odd a\\\n[PROOF STATE]\nproof (prove)\nusing this:\nodd a\n\ngoal (1 subgoal):\n 1. (\\x. a = 2 * x + 1 \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto elim: oddE)\n[PROOF STATE]\nproof (state)\nthis:\na = 2 * x + 1\n\ngoal (1 subgoal):\n 1. even (((a * b)\\<^sup>2 - 1) div 8) = even ((a\\<^sup>2 - 1) div 8 + (b\\<^sup>2 - 1) div 8)\n[PROOF STEP]\nobtain y where [simp]: \"b = 2*y + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\y. b = 2 * y + 1 \\ thesis) \\ thesis\n[PROOF STEP]\nusing \\odd b\\\n[PROOF STATE]\nproof (prove)\nusing this:\nodd b\n\ngoal (1 subgoal):\n 1. (\\y. b = 2 * y + 1 \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto elim: oddE)\n[PROOF STATE]\nproof (state)\nthis:\nb = 2 * y + 1\n\ngoal (1 subgoal):\n 1. even (((a * b)\\<^sup>2 - 1) div 8) = even ((a\\<^sup>2 - 1) div 8 + (b\\<^sup>2 - 1) div 8)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. even (((a * b)\\<^sup>2 - 1) div 8) = even ((a\\<^sup>2 - 1) div 8 + (b\\<^sup>2 - 1) div 8)\n[PROOF STEP]\nby (cases \"even x\"; cases \"even y\"; elim oddE evenE)\n (auto simp: power2_eq_square algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\neven (((a * b)\\<^sup>2 - 1) div 8) = even ((a\\<^sup>2 - 1) div 8 + (b\\<^sup>2 - 1) div 8)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 980, "file": "Probabilistic_Prime_Tests_Jacobi_Symbol", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588052782736, "lm_q2_score": 0.8056321843145404, "lm_q1q2_score": 0.7160126976251168}} {"text": "[STATEMENT]\nlemma log4_log2: \"log 4 x = log 2 x / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. log 4 x = log 2 x / 2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. log 4 x = log 2 x / 2\n[PROOF STEP]\nhave \"log 4 x = log (2^2) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. log 4 x = log (2\\<^sup>2) x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlog 4 x = log (2\\<^sup>2) x\n\ngoal (1 subgoal):\n 1. log 4 x = log 2 x / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlog 4 x = log (2\\<^sup>2) x\n\ngoal (1 subgoal):\n 1. log 4 x = log 2 x / 2\n[PROOF STEP]\nhave \"\\ = log 2 x / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. log (2\\<^sup>2) x = log 2 x / 2\n[PROOF STEP]\nby(simp only: log_base_pow)\n[PROOF STATE]\nproof (state)\nthis:\nlog (2\\<^sup>2) x = log 2 x / 2\n\ngoal (1 subgoal):\n 1. log 4 x = log 2 x / 2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nlog 4 x = log 2 x / 2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlog 4 x = log 2 x / 2\n\ngoal (1 subgoal):\n 1. log 4 x = log 2 x / 2\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nlog 4 x = log 2 x / 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 605, "file": "Amortized_Complexity_Splay_Tree_Analysis_Optimal", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587846530938, "lm_q2_score": 0.8056321983146848, "lm_q1q2_score": 0.7160126934515595}} {"text": "[STATEMENT]\nlemma gcd_unique_int:\n \"d \\ 0 \\ d dvd a \\ d dvd b \\ (\\e. e dvd a \\ e dvd b \\ e dvd d) \\ d = gcd a b\"\n for d a :: int\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0 \\ d \\ d dvd a \\ d dvd b \\ (\\e. e dvd a \\ e dvd b \\ e dvd d)) = (d = gcd a b)\n[PROOF STEP]\nusing zdvd_antisym_nonneg\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 \\ ?m; 0 \\ ?n; ?m dvd ?n; ?n dvd ?m\\ \\ ?m = ?n\n\ngoal (1 subgoal):\n 1. (0 \\ d \\ d dvd a \\ d dvd b \\ (\\e. e dvd a \\ e dvd b \\ e dvd d)) = (d = gcd a b)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 325, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587875995482, "lm_q2_score": 0.805632181981183, "lm_q1q2_score": 0.7160126813087748}} {"text": "[STATEMENT]\nlemma real_powr_at_top_neg: \n assumes \"(a::real) > 0\" \"a < 1\"\n shows \"((\\x. a powr x) \\ 0) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((powr) a \\ 0) at_top\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((powr) a \\ 0) at_top\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < a\na < 1\n[PROOF STEP]\nhave \"LIM x at_top. ln (inverse a) * x :> at_top\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\na < 1\n\ngoal (1 subgoal):\n 1. filterlim ((*) (ln (inverse a))) at_top at_top\n[PROOF STEP]\nby (intro filterlim_tendsto_pos_mult_at_top[OF tendsto_const])\n (simp_all add: filterlim_ident field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nfilterlim ((*) (ln (inverse a))) at_top at_top\n\ngoal (1 subgoal):\n 1. ((powr) a \\ 0) at_top\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < a\na < 1\nfilterlim ((*) (ln (inverse a))) at_top at_top\n[PROOF STEP]\nhave \"LIM x at_top. ln a * x :> at_bot\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\na < 1\nfilterlim ((*) (ln (inverse a))) at_top at_top\n\ngoal (1 subgoal):\n 1. filterlim ((*) (ln a)) at_bot at_top\n[PROOF STEP]\nby (subst filterlim_uminus_at_bot) (simp add: ln_inverse)\n[PROOF STATE]\nproof (state)\nthis:\nfilterlim ((*) (ln a)) at_bot at_top\n\ngoal (1 subgoal):\n 1. ((powr) a \\ 0) at_top\n[PROOF STEP]\nhence \"((\\x. exp (x * ln a)) \\ 0) at_top\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfilterlim ((*) (ln a)) at_bot at_top\n\ngoal (1 subgoal):\n 1. ((\\x. exp (x * ln a)) \\ 0) at_top\n[PROOF STEP]\nby (intro filterlim_compose[OF exp_at_bot]) (simp_all add: mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. exp (x * ln a)) \\ 0) at_top\n\ngoal (1 subgoal):\n 1. ((powr) a \\ 0) at_top\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < a\na < 1\n((\\x. exp (x * ln a)) \\ 0) at_top\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\na < 1\n((\\x. exp (x * ln a)) \\ 0) at_top\n\ngoal (1 subgoal):\n 1. ((powr) a \\ 0) at_top\n[PROOF STEP]\nunfolding powr_def\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\na < 1\n((\\x. exp (x * ln a)) \\ 0) at_top\n\ngoal (1 subgoal):\n 1. ((\\x. if a = 0 then 0 else exp (x * ln a)) \\ 0) at_top\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n((powr) a \\ 0) at_top\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1172, "file": "Akra_Bazzi_Akra_Bazzi_Library", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.880797071719777, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7159711572410067}} {"text": "[STATEMENT]\nlemma exponential_distributedI:\n assumes X[measurable]: \"X \\ borel_measurable M\" and [arith]: \"0 < l\"\n and X_distr: \"\\a. 0 \\ a \\ emeasure M {x\\space M. X x \\ a} = 1 - exp (- a * l)\"\n shows \"distributed M lborel X (exponential_density l)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. distributed M lborel X (\\x. ennreal (exponential_density l x))\n[PROOF STEP]\nproof (rule erlang_distributedI)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. X \\ borel_measurable M\n 2. 0 < l\n 3. \\a. 0 \\ a \\ emeasure M {x \\ space M. X x \\ a} = ennreal (erlang_CDF 0 l a)\n[PROOF STEP]\nfix a :: real\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. X \\ borel_measurable M\n 2. 0 < l\n 3. \\a. 0 \\ a \\ emeasure M {x \\ space M. X x \\ a} = ennreal (erlang_CDF 0 l a)\n[PROOF STEP]\nassume \"0 \\ a\"\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ a\n\ngoal (3 subgoals):\n 1. X \\ borel_measurable M\n 2. 0 < l\n 3. \\a. 0 \\ a \\ emeasure M {x \\ space M. X x \\ a} = ennreal (erlang_CDF 0 l a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ a\n[PROOF STEP]\nshow \"emeasure M {x \\ space M. X x \\ a} = ennreal (erlang_CDF 0 l a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ a\n\ngoal (1 subgoal):\n 1. emeasure M {x \\ space M. X x \\ a} = ennreal (erlang_CDF 0 l a)\n[PROOF STEP]\nusing X_distr[of a]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ a\n0 \\ a \\ emeasure M {x \\ space M. X x \\ a} = 1 - ennreal (exp (- a * l))\n\ngoal (1 subgoal):\n 1. emeasure M {x \\ space M. X x \\ a} = ennreal (erlang_CDF 0 l a)\n[PROOF STEP]\nby (simp add: erlang_CDF_def ennreal_minus ennreal_1[symmetric] del: ennreal_1)\n[PROOF STATE]\nproof (state)\nthis:\nemeasure M {x \\ space M. X x \\ a} = ennreal (erlang_CDF 0 l a)\n\ngoal (2 subgoals):\n 1. X \\ borel_measurable M\n 2. 0 < l\n[PROOF STEP]\nqed fact+", "meta": {"llama_tokens": 919, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.870597271765821, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7157956195976138}} {"text": "[STATEMENT]\nlemma round_down_shift: \"round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. round_down p (x * 2 powr real_of_int k) = 2 powr real_of_int k * round_down (p + k) x\n[PROOF STEP]\nunfolding round_down_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_of_int \\x * 2 powr real_of_int k * 2 powr real_of_int p\\ * 2 powr - real_of_int p = 2 powr real_of_int k * (real_of_int \\x * 2 powr real_of_int (p + k)\\ * 2 powr - real_of_int (p + k))\n[PROOF STEP]\nby (simp add: powr_add powr_mult field_simps powr_diff)\n (simp flip: powr_add)", "meta": {"llama_tokens": 296, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972717658209, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7157956195976137}} {"text": "[STATEMENT]\nlemma local_lipschitz_therm_dyn:\n assumes \"0 < (a::real)\"\n shows \"local_lipschitz UNIV UNIV (\\t::real. f a L)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local_lipschitz UNIV UNIV (\\t. f a L)\n[PROOF STEP]\napply(unfold local_lipschitz_def lipschitz_on_def dist_norm)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\UNIV. \\t\\UNIV. \\u>0. \\La. \\t\\cball t u \\ UNIV. 0 \\ La \\ (\\xa\\cball x u \\ UNIV. \\y\\cball x u \\ UNIV. \\f a L xa - f a L y\\ \\ La \\ \\xa - y\\)\n[PROOF STEP]\napply(clarsimp, rule_tac x=1 in exI, clarsimp, rule_tac x=a in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a \\ \\xa - y\\)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n\ngoal (1 subgoal):\n 1. \\x t ta. dist t ta \\ 1 \\ 0 \\ a \\ (\\xa\\cball x 1. \\y\\cball x 1. \\f a L xa - f a L y\\ \\ a \\ \\xa - y\\)\n[PROOF STEP]\napply(simp_all add: norm_diff_therm_dyn)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta. \\dist t ta \\ 1; 0 < a\\ \\ \\xa\\cball x 1. \\y\\cball x 1. \\xa $ 1 - y $ 1\\ \\ \\xa - y\\\n[PROOF STEP]\napply(simp add: norm_vec_def L2_set_def, unfold UNIV_4, clarsimp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ \\xa $ 1 - y $ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nunfolding real_sqrt_abs[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x t ta xa y. \\dist t ta \\ 1; 0 < a; dist x xa \\ 1; dist x y \\ 1\\ \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2) \\ sqrt ((xa $ 1 - y $ 1)\\<^sup>2 + ((xa $ 2 - y $ 2)\\<^sup>2 + ((xa $ 3 - y $ 3)\\<^sup>2 + (xa $ 4 - y $ 4)\\<^sup>2)))\n[PROOF STEP]\nby (rule real_le_lsqrt) auto", "meta": {"llama_tokens": 1107, "file": "Hybrid_Systems_VCs_KleeneAlgebraTests_HS_VC_KAT_Examples_rel", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972616934406, "lm_q2_score": 0.8221891283434876, "lm_q1q2_score": 0.7157956037299571}} {"text": "[STATEMENT]\nlemma mat_plus_left_mono: \"A \\\\<^sub>m (B :: 'a :: ordered_ab_semigroup mat) \n \\ A \\ carrier_mat nr nc \\ B \\ carrier_mat nr nc \\ C \\ carrier_mat nr nc \n \\ A + C \\\\<^sub>m B + C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\\\<^sub>m B; A \\ carrier_mat nr nc; B \\ carrier_mat nr nc; C \\ carrier_mat nr nc\\ \\ A + C \\\\<^sub>m B + C\n[PROOF STEP]\nby (intro mat_geI[of _ nr nc], auto simp: plus_left_mono)", "meta": {"llama_tokens": 228, "file": "Jordan_Normal_Form_Matrix_Comparison", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972549785201, "lm_q2_score": 0.8221891327004132, "lm_q1q2_score": 0.71579560200215}} {"text": "[STATEMENT]\nlemma ceiling_zero [simp]: \"\\0\\ = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nusing ceiling_of_int [of 0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\of_int 0\\ = 0\n\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 158, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391624034103, "lm_q2_score": 0.8289388125473629, "lm_q1q2_score": 0.7156988275901685}} {"text": "[STATEMENT]\nlemma ceiling_zero [simp]: \"\\0\\ = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nusing ceiling_of_int [of 0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\of_int 0\\ = 0\n\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 158, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391624034103, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7156988221171973}} {"text": "[STATEMENT]\nlemma neg_of_aezfun_is_aezfun :\n fixes f :: \"'a\\'b::group_add\"\n shows \"f \\ aezfun_set \\ - f \\ aezfun_set\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f \\ aezfun_set \\ - f \\ aezfun_set\n[PROOF STEP]\nusing supp_neg_eq_supp[of f]\n[PROOF STATE]\nproof (prove)\nusing this:\nsupp (- f) = supp f\n\ngoal (1 subgoal):\n 1. f \\ aezfun_set \\ - f \\ aezfun_set\n[PROOF STEP]\nunfolding aezfun_set_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsupp (- f) = supp f\n\ngoal (1 subgoal):\n 1. f \\ {f. finite (supp f)} \\ - f \\ {f. finite (supp f)}\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 288, "file": "Rep_Fin_Groups_Rep_Fin_Groups", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8633916029436189, "lm_q2_score": 0.82893881677331, "lm_q1q2_score": 0.715698813756095}} {"text": "[STATEMENT]\nlemma LIMSEQ_const_iff: \"(\\n. k) \\ l \\ k = l\"\n for k l :: \"'a::t2_space\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. k) \\ l = (k = l)\n[PROOF STEP]\nusing trivial_limit_sequentially\n[PROOF STATE]\nproof (prove)\nusing this:\nsequentially \\ bot\n\ngoal (1 subgoal):\n 1. (\\n. k) \\ l = (k = l)\n[PROOF STEP]\nby (rule tendsto_const_iff)", "meta": {"llama_tokens": 186, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391602943619, "lm_q2_score": 0.8289388146603365, "lm_q1q2_score": 0.7156988119317714}} {"text": "[STATEMENT]\nlemma LIMSEQ_const_iff: \"(\\n. k) \\ l \\ k = l\"\n for k l :: \"'a::t2_space\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. k) \\ l = (k = l)\n[PROOF STEP]\nusing trivial_limit_sequentially\n[PROOF STATE]\nproof (prove)\nusing this:\nsequentially \\ bot\n\ngoal (1 subgoal):\n 1. (\\n. k) \\ l = (k = l)\n[PROOF STEP]\nby (rule tendsto_const_iff)", "meta": {"llama_tokens": 186, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391602943619, "lm_q2_score": 0.8289388146603365, "lm_q1q2_score": 0.7156988119317714}} {"text": "[STATEMENT]\nlemma LIMSEQ_const_iff: \"(\\n. k) \\ l \\ k = l\"\n for k l :: \"'a::t2_space\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. k) \\ l = (k = l)\n[PROOF STEP]\nusing trivial_limit_sequentially\n[PROOF STATE]\nproof (prove)\nusing this:\nsequentially \\ bot\n\ngoal (1 subgoal):\n 1. (\\n. k) \\ l = (k = l)\n[PROOF STEP]\nby (rule tendsto_const_iff)", "meta": {"llama_tokens": 186, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391602943619, "lm_q2_score": 0.8289388146603365, "lm_q1q2_score": 0.7156988119317714}} {"text": "[STATEMENT]\nlemma LIMSEQ_const_iff: \"(\\n. k) \\ l \\ k = l\"\n for k l :: \"'a::t2_space\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. k) \\ l = (k = l)\n[PROOF STEP]\nusing trivial_limit_sequentially\n[PROOF STATE]\nproof (prove)\nusing this:\nsequentially \\ bot\n\ngoal (1 subgoal):\n 1. (\\n. k) \\ l = (k = l)\n[PROOF STEP]\nby (rule tendsto_const_iff)", "meta": {"llama_tokens": 186, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391602943619, "lm_q2_score": 0.8289388104343892, "lm_q1q2_score": 0.715698808283124}} {"text": "[STATEMENT]\ntheorem card_partial_equiv:\n assumes \"finite A\"\n shows \"card {R. partial_equiv A R} = Bell (card A + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nlet ?expr = \"do {\n k \\ {0..card A};\n A' \\ {A'. A' \\ A \\ card A' = k};\n {R. equiv A' R}\n }\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nhave \"card {R. partial_equiv A R} = card (all_partial_equivs_on A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = card (all_partial_equivs_on A)\n[PROOF STEP]\nusing \\finite A\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = card (all_partial_equivs_on A)\n[PROOF STEP]\nby (simp add: all_partial_equivs_on)\n[PROOF STATE]\nproof (state)\nthis:\ncard {R. partial_equiv A R} = card (all_partial_equivs_on A)\n\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {R. partial_equiv A R} = card (all_partial_equivs_on A)\n\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nhave \"card (all_partial_equivs_on A) = card ?expr\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (all_partial_equivs_on A) = card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})))\n[PROOF STEP]\nunfolding all_partial_equivs_on_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. Collect (equiv A')))) = card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})))\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\ncard (all_partial_equivs_on A) = card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})))\n\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (all_partial_equivs_on A) = card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})))\n\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nhave \"card ?expr = (\\k = 0..card A. (card A choose k) * Bell k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nlet \"?S \\ ?comp\" = ?expr\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nfix k\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nassume k: \"k \\ {..card A}\"\n[PROOF STATE]\nproof (state)\nthis:\nk \\ {..card A}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nlet ?expr = \"?comp k\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nlet \"?S \\ ?comp\" = ?expr\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nhave \"finite ?S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {A'. A' \\ A \\ card A' = k}\n[PROOF STEP]\nusing \\finite A\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. finite {A'. A' \\ A \\ card A' = k}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfinite {A'. A' \\ A \\ card A' = k}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite {A'. A' \\ A \\ card A' = k}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\nfinite {A'. A' \\ A \\ card A' = k}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nfix A'\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nassume A': \"A' \\ {A'. A' \\ A \\ card A' = k}\"\n[PROOF STATE]\nproof (state)\nthis:\nA' \\ {A'. A' \\ A \\ card A' = k}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nA' \\ {A'. A' \\ A \\ card A' = k}\n[PROOF STEP]\nhave \"A' \\ A\" and \"card A' = k\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA' \\ {A'. A' \\ A \\ card A' = k}\n\ngoal (1 subgoal):\n 1. A' \\ A &&& card A' = k\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA' \\ A\ncard A' = k\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nlet ?expr = \"?comp A'\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nhave \"finite A'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite A'\n[PROOF STEP]\nusing \\finite A\\ \\A' \\ A\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nA' \\ A\n\ngoal (1 subgoal):\n 1. finite A'\n[PROOF STEP]\nby (simp add: finite_subset)\n[PROOF STATE]\nproof (state)\nthis:\nfinite A'\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nhave \"card ?expr = Bell k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {R. equiv A' R} = Bell k\n[PROOF STEP]\nusing \\finite A\\ \\finite A'\\ \\A' \\ A\\ \\card A' = k\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite A'\nA' \\ A\ncard A' = k\n\ngoal (1 subgoal):\n 1. card {R. equiv A' R} = Bell k\n[PROOF STEP]\nby (simp add: card_equiv_rel_eq_Bell)\n[PROOF STATE]\nproof (state)\nthis:\ncard {R. equiv A' R} = Bell k\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncard {R. equiv A' R} = Bell k\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nhave \"finite ?expr\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {R. equiv A' R}\n[PROOF STEP]\nusing \\finite A'\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A'\n\ngoal (1 subgoal):\n 1. finite {R. equiv A' R}\n[PROOF STEP]\nby (simp add: finite_equiv)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {R. equiv A' R}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {R. equiv A' R} = Bell k\nfinite {R. equiv A' R}\n[PROOF STEP]\nhave \"finite ?expr \\ card ?expr = Bell k\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {R. equiv A' R} = Bell k\nfinite {R. equiv A' R}\n\ngoal (1 subgoal):\n 1. finite {R. equiv A' R} \\ card {R. equiv A' R} = Bell k\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfinite {R. equiv A' R} \\ card {R. equiv A' R} = Bell k\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n?A'2 \\ {A'. A' \\ A \\ card A' = k} \\ finite {R. equiv ?A'2 R} \\ card {R. equiv ?A'2 R} = Bell k\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nnote inner = this\n[PROOF STATE]\nproof (state)\nthis:\n?A'2 \\ {A'. A' \\ A \\ card A' = k} \\ finite {R. equiv ?A'2 R} \\ card {R. equiv ?A'2 R} = Bell k\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n?A'2 \\ {A'. A' \\ A \\ card A' = k} \\ finite {R. equiv ?A'2 R} \\ card {R. equiv ?A'2 R} = Bell k\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nhave \"disjoint_family_on ?comp ?S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. disjoint_family_on (\\A'. {R. equiv A' R}) {A'. A' \\ A \\ card A' = k}\n[PROOF STEP]\nby (injectivity_solver rule: injectivity(1))\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on (\\A'. {R. equiv A' R}) {A'. A' \\ A \\ card A' = k}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on (\\A'. {R. equiv A' R}) {A'. A' \\ A \\ card A' = k}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nhave \"card ?S = card A choose k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {A'. A' \\ A \\ card A' = k} = card A choose k\n[PROOF STEP]\nusing \\finite A\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. card {A'. A' \\ A \\ card A' = k} = card A choose k\n[PROOF STEP]\nby (simp add: n_subsets)\n[PROOF STATE]\nproof (state)\nthis:\ncard {A'. A' \\ A \\ card A' = k} = card A choose k\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite {A'. A' \\ A \\ card A' = k}\n?A'2 \\ {A'. A' \\ A \\ card A' = k} \\ finite {R. equiv ?A'2 R} \\ card {R. equiv ?A'2 R} = Bell k\ndisjoint_family_on (\\A'. {R. equiv A' R}) {A'. A' \\ A \\ card A' = k}\ncard {A'. A' \\ A \\ card A' = k} = card A choose k\n[PROOF STEP]\nhave \"card ?expr = (card A choose k) * Bell k\" (is \"_ = ?formula\")\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {A'. A' \\ A \\ card A' = k}\n?A'2 \\ {A'. A' \\ A \\ card A' = k} \\ finite {R. equiv ?A'2 R} \\ card {R. equiv ?A'2 R} = Bell k\ndisjoint_family_on (\\A'. {R. equiv A' R}) {A'. A' \\ A \\ card A' = k}\ncard {A'. A' \\ A \\ card A' = k} = card A choose k\n\ngoal (1 subgoal):\n 1. card ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) = (card A choose k) * Bell k\n[PROOF STEP]\nby (subst card_bind_constant) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) = (card A choose k) * Bell k\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncard ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) = (card A choose k) * Bell k\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nhave \"finite ?expr\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))\n[PROOF STEP]\nusing \\finite ?S\\ inner\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {A'. A' \\ A \\ card A' = k}\n?A'2 \\ {A'. A' \\ A \\ card A' = k} \\ finite {R. equiv ?A'2 R} \\ card {R. equiv ?A'2 R} = Bell k\n\ngoal (1 subgoal):\n 1. finite ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))\n[PROOF STEP]\nby (auto intro!: finite_bind)\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncard ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) = (card A choose k) * Bell k\nfinite ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))\n[PROOF STEP]\nhave \"finite ?expr \\ card ?expr = ?formula\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) = (card A choose k) * Bell k\nfinite ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))\n\ngoal (1 subgoal):\n 1. finite ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) \\ card ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) = (card A choose k) * Bell k\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) \\ card ({A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) = (card A choose k) * Bell k\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n?k2 \\ {..card A} \\ finite ({A'. A' \\ A \\ card A' = ?k2} \\ (\\A'. {R. equiv A' R})) \\ card ({A'. A' \\ A \\ card A' = ?k2} \\ (\\A'. {R. equiv A' R})) = (card A choose ?k2) * Bell ?k2\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n?k2 \\ {..card A} \\ finite ({A'. A' \\ A \\ card A' = ?k2} \\ (\\A'. {R. equiv A' R})) \\ card ({A'. A' \\ A \\ card A' = ?k2} \\ (\\A'. {R. equiv A' R})) = (card A choose ?k2) * Bell ?k2\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nhave \"finite ?S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {0..card A}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfinite {0..card A}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite {0..card A}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nhave \"disjoint_family_on ?comp ?S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. disjoint_family_on (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) {0..card A}\n[PROOF STEP]\nby (injectivity_solver rule: injectivity(2))\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) {0..card A}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n?k2 \\ {..card A} \\ finite ({A'. A' \\ A \\ card A' = ?k2} \\ (\\A'. {R. equiv A' R})) \\ card ({A'. A' \\ A \\ card A' = ?k2} \\ (\\A'. {R. equiv A' R})) = (card A choose ?k2) * Bell ?k2\nfinite {0..card A}\ndisjoint_family_on (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) {0..card A}\n[PROOF STEP]\nshow \"card ?expr = (\\k = 0..card A. (card A choose k) * Bell k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?k2 \\ {..card A} \\ finite ({A'. A' \\ A \\ card A' = ?k2} \\ (\\A'. {R. equiv A' R})) \\ card ({A'. A' \\ A \\ card A' = ?k2} \\ (\\A'. {R. equiv A' R})) = (card A choose ?k2) * Bell ?k2\nfinite {0..card A}\ndisjoint_family_on (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R})) {0..card A}\n\ngoal (1 subgoal):\n 1. card ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n[PROOF STEP]\nby (subst card_bind) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncard ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ({0..card A} \\ (\\k. {A'. A' \\ A \\ card A' = k} \\ (\\A'. {R. equiv A' R}))) = (\\k = 0..card A. (card A choose k) * Bell k)\n\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nhave \"\\ = (\\k\\card A. (card A choose k) * Bell k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..card A. (card A choose k) * Bell k) = (\\k\\card A. (card A choose k) * Bell k)\n[PROOF STEP]\nby (auto intro: sum.cong)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..card A. (card A choose k) * Bell k) = (\\k\\card A. (card A choose k) * Bell k)\n\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..card A. (card A choose k) * Bell k) = (\\k\\card A. (card A choose k) * Bell k)\n\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nhave \"\\ = Bell (card A + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\card A. (card A choose k) * Bell k) = Bell (card A + 1)\n[PROOF STEP]\nusing Bell_recursive_eq\n[PROOF STATE]\nproof (prove)\nusing this:\nBell (?n + 1) = (\\k\\?n. (?n choose k) * Bell k)\n\ngoal (1 subgoal):\n 1. (\\k\\card A. (card A choose k) * Bell k) = Bell (card A + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\card A. (card A choose k) * Bell k) = Bell (card A + 1)\n\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {R. partial_equiv A R} = Bell (card A + 1)\n\ngoal (1 subgoal):\n 1. card {R. partial_equiv A R} = Bell (card A + 1)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {R. partial_equiv A R} = Bell (card A + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 9899, "file": "Card_Equiv_Relations_Card_Partial_Equiv_Relations", "length": 83, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314828740729, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.7156497707409746}} {"text": "[STATEMENT]\nlemma countable_UN[intro, simp]:\n fixes I :: \"'i set\" and A :: \"'i => 'a set\"\n assumes I: \"countable I\"\n assumes A: \"\\i. i \\ I \\ countable (A i)\"\n shows \"countable (\\i\\I. A i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. countable (\\ (A ` I))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. countable (\\ (A ` I))\n[PROOF STEP]\nhave \"(\\i\\I. A i) = snd ` (SIGMA i : I. A i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (A ` I) = snd ` Sigma I A\n[PROOF STEP]\nby (auto simp: image_iff)\n[PROOF STATE]\nproof (state)\nthis:\n\\ (A ` I) = snd ` Sigma I A\n\ngoal (1 subgoal):\n 1. countable (\\ (A ` I))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ (A ` I) = snd ` Sigma I A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (A ` I) = snd ` Sigma I A\n\ngoal (1 subgoal):\n 1. countable (\\ (A ` I))\n[PROOF STEP]\nby (simp add: assms)\n[PROOF STATE]\nproof (state)\nthis:\ncountable (\\ (A ` I))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 491, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314798554444, "lm_q2_score": 0.8080672089305841, "lm_q1q2_score": 0.7156497580678518}} {"text": "[STATEMENT]\nlemma Youngs_inequality:\n fixes p::real\n assumes \"p>1\" \"q>1\" \"1/p + 1/q = 1\" \"a\\0\" \"b\\0\"\n shows \"a * b \\ a powr p / p + b powr q / q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a * b \\ a powr p / p + b powr q / q\n[PROOF STEP]\nproof (cases \"a=0 \\ b=0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. a = 0 \\ b = 0 \\ a * b \\ a powr p / p + b powr q / q\n 2. \\ (a = 0 \\ b = 0) \\ a * b \\ a powr p / p + b powr q / q\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ (a = 0 \\ b = 0)\n\ngoal (2 subgoals):\n 1. a = 0 \\ b = 0 \\ a * b \\ a powr p / p + b powr q / q\n 2. \\ (a = 0 \\ b = 0) \\ a * b \\ a powr p / p + b powr q / q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ (a = 0 \\ b = 0)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (a = 0 \\ b = 0)\n\ngoal (1 subgoal):\n 1. a * b \\ a powr p / p + b powr q / q\n[PROOF STEP]\nusing Youngs_inequality_0 [of \"1/p\" \"1/q\" \"a powr p\" \"b powr q\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (a = 0 \\ b = 0)\n\\0 \\ 1 / p; 0 \\ 1 / q; 1 / p + 1 / q = 1; 0 < a powr p; 0 < b powr q\\ \\ (a powr p) powr (1 / p) * (b powr q) powr (1 / q) \\ 1 / p * a powr p + 1 / q * b powr q\n1 < p\n1 < q\n1 / p + 1 / q = 1\n0 \\ a\n0 \\ b\n\ngoal (1 subgoal):\n 1. a * b \\ a powr p / p + b powr q / q\n[PROOF STEP]\nby (simp add: powr_powr)\n[PROOF STATE]\nproof (state)\nthis:\na * b \\ a powr p / p + b powr q / q\n\ngoal (1 subgoal):\n 1. a = 0 \\ b = 0 \\ a * b \\ a powr p / p + b powr q / q\n[PROOF STEP]\nqed (use assms in auto)", "meta": {"llama_tokens": 867, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.84594244507642, "lm_q2_score": 0.8459424334245618, "lm_q1q2_score": 0.7156186105250705}} {"text": "[STATEMENT]\nlemma unitary_operator_keep_trace:\n fixes U A :: \"complex mat\"\n assumes dU: \"U \\ carrier_mat n n\" and dA: \"A \\ carrier_mat n n\" and u: \"unitary U\"\n shows \"trace A = trace (adjoint U * A * U)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace A = trace (adjoint U * A * U)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. trace A = trace (adjoint U * A * U)\n[PROOF STEP]\nhave u': \"U * adjoint U = 1\\<^sub>m n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. U * adjoint U = 1\\<^sub>m n\n[PROOF STEP]\nusing u\n[PROOF STATE]\nproof (prove)\nusing this:\nunitary U\n\ngoal (1 subgoal):\n 1. U * adjoint U = 1\\<^sub>m n\n[PROOF STEP]\nunfolding unitary_def inverts_mat_def\n[PROOF STATE]\nproof (prove)\nusing this:\nU \\ carrier_mat (dim_row U) (dim_row U) \\ U * adjoint U = 1\\<^sub>m (dim_row U)\n\ngoal (1 subgoal):\n 1. U * adjoint U = 1\\<^sub>m n\n[PROOF STEP]\nusing dU\n[PROOF STATE]\nproof (prove)\nusing this:\nU \\ carrier_mat (dim_row U) (dim_row U) \\ U * adjoint U = 1\\<^sub>m (dim_row U)\nU \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. U * adjoint U = 1\\<^sub>m n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nU * adjoint U = 1\\<^sub>m n\n\ngoal (1 subgoal):\n 1. trace A = trace (adjoint U * A * U)\n[PROOF STEP]\nhave \"trace (adjoint U * A * U) = trace (U * adjoint U * A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (adjoint U * A * U) = trace (U * adjoint U * A)\n[PROOF STEP]\nusing dU dA\n[PROOF STATE]\nproof (prove)\nusing this:\nU \\ carrier_mat n n\nA \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. trace (adjoint U * A * U) = trace (U * adjoint U * A)\n[PROOF STEP]\nby (mat_assoc n)\n[PROOF STATE]\nproof (state)\nthis:\ntrace (adjoint U * A * U) = trace (U * adjoint U * A)\n\ngoal (1 subgoal):\n 1. trace A = trace (adjoint U * A * U)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ntrace (adjoint U * A * U) = trace (U * adjoint U * A)\n\ngoal (1 subgoal):\n 1. trace A = trace (adjoint U * A * U)\n[PROOF STEP]\nhave \"\\ = trace A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (U * adjoint U * A) = trace A\n[PROOF STEP]\nusing u' dA\n[PROOF STATE]\nproof (prove)\nusing this:\nU * adjoint U = 1\\<^sub>m n\nA \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. trace (U * adjoint U * A) = trace A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace (U * adjoint U * A) = trace A\n\ngoal (1 subgoal):\n 1. trace A = trace (adjoint U * A * U)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ntrace (adjoint U * A * U) = trace A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntrace (adjoint U * A * U) = trace A\n\ngoal (1 subgoal):\n 1. trace A = trace (adjoint U * A * U)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace A = trace (adjoint U * A * U)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1252, "file": "QHLProver_Complex_Matrix", "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8479677506936878, "lm_q2_score": 0.843895106480586, "lm_q1q2_score": 0.7155958352637526}} {"text": "[STATEMENT]\nlemma card_all_edges_loops: \n assumes \"finite S\"\n shows \"card (all_edges_loops S) = (card S) choose 2 + card S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (all_edges_loops S) = card S choose 2 + card S\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (all_edges_loops S) = card S choose 2 + card S\n[PROOF STEP]\nhave \"card (all_edges_loops S) = card (all_edges S \\ {{v} | v. v \\ S})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (all_edges_loops S) = card (all_edges S \\ {{v} |v. v \\ S})\n[PROOF STEP]\nby (simp add: all_edges_loops_def)\n[PROOF STATE]\nproof (state)\nthis:\ncard (all_edges_loops S) = card (all_edges S \\ {{v} |v. v \\ S})\n\ngoal (1 subgoal):\n 1. card (all_edges_loops S) = card S choose 2 + card S\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (all_edges_loops S) = card (all_edges S \\ {{v} |v. v \\ S})\n\ngoal (1 subgoal):\n 1. card (all_edges_loops S) = card S choose 2 + card S\n[PROOF STEP]\nhave \"... = card (all_edges S) + card {{v} | v. v \\ S}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (all_edges S \\ {{v} |v. v \\ S}) = card (all_edges S) + card {{v} |v. v \\ S}\n[PROOF STEP]\nusing loops_disjoint assms card_Un_disjoint[of \"all_edges S\" \"{{v} | v. v \\ S}\"] \n all_edges_loops_ss finite_all_edges_loops finite_subset\n[PROOF STATE]\nproof (prove)\nusing this:\nall_edges ?S \\ {{v} |v. v \\ ?S} = {}\nfinite S\n\\finite (all_edges S); finite {{v} |v. v \\ S}; all_edges S \\ {{v} |v. v \\ S} = {}\\ \\ card (all_edges S \\ {{v} |v. v \\ S}) = card (all_edges S) + card {{v} |v. v \\ S}\nall_edges ?S \\ all_edges_loops ?S\n{{v} |v. v \\ ?S} \\ all_edges_loops ?S\nfinite ?S \\ finite (all_edges_loops ?S)\n\\?A \\ ?B; finite ?B\\ \\ finite ?A\n\ngoal (1 subgoal):\n 1. card (all_edges S \\ {{v} |v. v \\ S}) = card (all_edges S) + card {{v} |v. v \\ S}\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\ncard (all_edges S \\ {{v} |v. v \\ S}) = card (all_edges S) + card {{v} |v. v \\ S}\n\ngoal (1 subgoal):\n 1. card (all_edges_loops S) = card S choose 2 + card S\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (all_edges S \\ {{v} |v. v \\ S}) = card (all_edges S) + card {{v} |v. v \\ S}\n\ngoal (1 subgoal):\n 1. card (all_edges_loops S) = card S choose 2 + card S\n[PROOF STEP]\nhave \"... = (card S) choose 2 + card {{v} | v. v \\ S}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (all_edges S) + card {{v} |v. v \\ S} = card S choose 2 + card {{v} |v. v \\ S}\n[PROOF STEP]\nby(simp add: card_all_edges assms)\n[PROOF STATE]\nproof (state)\nthis:\ncard (all_edges S) + card {{v} |v. v \\ S} = card S choose 2 + card {{v} |v. v \\ S}\n\ngoal (1 subgoal):\n 1. card (all_edges_loops S) = card S choose 2 + card S\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (all_edges_loops S) = card S choose 2 + card {{v} |v. v \\ S}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (all_edges_loops S) = card S choose 2 + card {{v} |v. v \\ S}\n\ngoal (1 subgoal):\n 1. card (all_edges_loops S) = card S choose 2 + card S\n[PROOF STEP]\nusing assms card_singletons\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (all_edges_loops S) = card S choose 2 + card {{v} |v. v \\ S}\nfinite S\nfinite ?S \\ card {{v} |v. v \\ ?S} = card ?S\n\ngoal (1 subgoal):\n 1. card (all_edges_loops S) = card S choose 2 + card S\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (all_edges_loops S) = card S choose 2 + card S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1687, "file": "Undirected_Graph_Theory_Undirected_Graph_Basics", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256512199033, "lm_q2_score": 0.8499711775577735, "lm_q1q2_score": 0.7155275400657207}} {"text": "[STATEMENT]\nlemma unique_cross_ratio:\n assumes \"u \\ v\" and \"v \\ w\" and \"u \\ w\"\n assumes \"cross_ratio z u v w = cross_ratio z' u v w\"\n shows \"z = z'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z = z'\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. z = z'\n[PROOF STEP]\nobtain M where \"(\\ z. cross_ratio z u v w) = moebius_pt M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\M. (\\z. cross_ratio z u v w) = moebius_pt M \\ thesis) \\ thesis\n[PROOF STEP]\nusing is_moebius_cross_ratio[OF assms(1-3)]\n[PROOF STATE]\nproof (prove)\nusing this:\nis_moebius (\\z. cross_ratio z u v w)\n\ngoal (1 subgoal):\n 1. (\\M. (\\z. cross_ratio z u v w) = moebius_pt M \\ thesis) \\ thesis\n[PROOF STEP]\nunfolding is_moebius_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\M. (\\z. cross_ratio z u v w) = moebius_pt M\n\ngoal (1 subgoal):\n 1. (\\M. (\\z. cross_ratio z u v w) = moebius_pt M \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\z. cross_ratio z u v w) = moebius_pt M\n\ngoal (1 subgoal):\n 1. z = z'\n[PROOF STEP]\nhence \"moebius_pt M z = moebius_pt M z'\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\z. cross_ratio z u v w) = moebius_pt M\n\ngoal (1 subgoal):\n 1. moebius_pt M z = moebius_pt M z'\n[PROOF STEP]\nusing assms(4)\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\z. cross_ratio z u v w) = moebius_pt M\ncross_ratio z u v w = cross_ratio z' u v w\n\ngoal (1 subgoal):\n 1. moebius_pt M z = moebius_pt M z'\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\nmoebius_pt M z = moebius_pt M z'\n\ngoal (1 subgoal):\n 1. z = z'\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nmoebius_pt M z = moebius_pt M z'\n\ngoal (1 subgoal):\n 1. z = z'\n[PROOF STEP]\nusing moebius_pt_eq_I\n[PROOF STATE]\nproof (prove)\nusing this:\nmoebius_pt M z = moebius_pt M z'\nmoebius_pt ?M ?z1.0 = moebius_pt ?M ?z2.0 \\ ?z1.0 = ?z2.0\n\ngoal (1 subgoal):\n 1. z = z'\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\nz = z'\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1012, "file": "Complex_Geometry_Moebius", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8499711832583695, "lm_q2_score": 0.8418256432832333, "lm_q1q2_score": 0.7155275381186879}} {"text": "[STATEMENT]\nlemma absolutely_integrable_mult_Dirichlet_kernel:\n assumes \"f absolutely_integrable_on {-pi..pi}\"\n shows \"(\\x. Dirichlet_kernel n x * f x) absolutely_integrable_on {-pi..pi}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. Dirichlet_kernel n x * f x) absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nproof (rule absolutely_integrable_bounded_measurable_product_real)\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. Dirichlet_kernel n \\ borel_measurable (lebesgue_on {- pi..pi})\n 2. {- pi..pi} \\ sets lebesgue\n 3. bounded (Dirichlet_kernel n ` {- pi..pi})\n 4. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nshow \"Dirichlet_kernel n \\ borel_measurable (lebesgue_on {-pi..pi})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Dirichlet_kernel n \\ borel_measurable (lebesgue_on {- pi..pi})\n[PROOF STEP]\nby (simp add: Dirichlet_kernel_continuous continuous_imp_measurable_on_sets_lebesgue)\n[PROOF STATE]\nproof (state)\nthis:\nDirichlet_kernel n \\ borel_measurable (lebesgue_on {- pi..pi})\n\ngoal (3 subgoals):\n 1. {- pi..pi} \\ sets lebesgue\n 2. bounded (Dirichlet_kernel n ` {- pi..pi})\n 3. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nhave \"compact (Dirichlet_kernel n ` {-pi..pi})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. compact (Dirichlet_kernel n ` {- pi..pi})\n[PROOF STEP]\nby (auto simp: compact_continuous_image [OF Dirichlet_kernel_continuous])\n[PROOF STATE]\nproof (state)\nthis:\ncompact (Dirichlet_kernel n ` {- pi..pi})\n\ngoal (3 subgoals):\n 1. {- pi..pi} \\ sets lebesgue\n 2. bounded (Dirichlet_kernel n ` {- pi..pi})\n 3. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncompact (Dirichlet_kernel n ` {- pi..pi})\n[PROOF STEP]\nshow \"bounded (Dirichlet_kernel n ` {-pi..pi})\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncompact (Dirichlet_kernel n ` {- pi..pi})\n\ngoal (1 subgoal):\n 1. bounded (Dirichlet_kernel n ` {- pi..pi})\n[PROOF STEP]\nusing compact_imp_bounded\n[PROOF STATE]\nproof (prove)\nusing this:\ncompact (Dirichlet_kernel n ` {- pi..pi})\ncompact ?U \\ bounded ?U\n\ngoal (1 subgoal):\n 1. bounded (Dirichlet_kernel n ` {- pi..pi})\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nbounded (Dirichlet_kernel n ` {- pi..pi})\n\ngoal (2 subgoals):\n 1. {- pi..pi} \\ sets lebesgue\n 2. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nqed (use assms in auto)", "meta": {"llama_tokens": 981, "file": "Fourier_Fourier", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.849971175657575, "lm_q2_score": 0.8418256492357358, "lm_q1q2_score": 0.7155275367795997}} {"text": "[STATEMENT]\nlemma abs_ln_one_plus_x_minus_x_bound:\n fixes x :: real\n assumes \"\\x\\ \\ 1 / 2\"\n shows \"\\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n[PROOF STEP]\nproof (cases \"0 \\ x\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 0 \\ x \\ \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n 2. \\ 0 \\ x \\ \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ x\n\ngoal (2 subgoals):\n 1. 0 \\ x \\ \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n 2. \\ 0 \\ x \\ \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ x\n\ngoal (1 subgoal):\n 1. \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n[PROOF STEP]\nusing abs_ln_one_plus_x_minus_x_bound_nonneg assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ x\n\\0 \\ ?x; ?x \\ 1\\ \\ \\ln (1 + ?x) - ?x\\ \\ ?x\\<^sup>2\n\\x\\ \\ 1 / 2\n\ngoal (1 subgoal):\n 1. \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n\\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\ 0 \\ x \\ \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ 0 \\ x \\ \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ 0 \\ x\n\ngoal (1 subgoal):\n 1. \\ 0 \\ x \\ \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ 0 \\ x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 0 \\ x\n\ngoal (1 subgoal):\n 1. \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n[PROOF STEP]\nusing abs_ln_one_plus_x_minus_x_bound_nonpos assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 0 \\ x\n\\- (1 / 2) \\ ?x; ?x \\ 0\\ \\ \\ln (1 + ?x) - ?x\\ \\ 2 * ?x\\<^sup>2\n\\x\\ \\ 1 / 2\n\ngoal (1 subgoal):\n 1. \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1243, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8499711642563823, "lm_q2_score": 0.8418256432832333, "lm_q1q2_score": 0.7155275221223277}} {"text": "[STATEMENT]\nlemma \"count_true (vec (2::nat) (\\i. True)) = 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. count_true (vec 2 (\\i. True)) = 2\n[PROOF STEP]\nunfolding count_true_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..i. True)). if vec 2 (\\i. True) $ i then 1 else 0) = 2\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 167, "file": "Simplicial_complexes_and_boolean_functions_Boolean_functions", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9173026528034425, "lm_q2_score": 0.7799929053683038, "lm_q1q2_score": 0.7154895612622095}} {"text": "[STATEMENT]\nlemma sin_tan_half: \"sin (2*x) = 2 * tan x / (1 + (tan x)^2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin ((2::'a) * x) = (2::'a) * tan x / ((1::'a) + (tan x)\\<^sup>2)\n[PROOF STEP]\nunfolding sin_double tan_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (2::'a) * sin x * cos x = (2::'a) * (sin x / cos x) / ((1::'a) + (sin x / cos x)\\<^sup>2)\n[PROOF STEP]\napply (cases \"cos x=0\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. cos x = (0::'a) \\ (2::'a) * sin x * cos x = (2::'a) * (sin x / cos x) / ((1::'a) + (sin x / cos x)\\<^sup>2)\n 2. cos x \\ (0::'a) \\ (2::'a) * sin x * cos x = (2::'a) * (sin x / cos x) / ((1::'a) + (sin x / cos x)\\<^sup>2)\n[PROOF STEP]\nby (auto simp add:field_simps power2_eq_square)", "meta": {"llama_tokens": 383, "file": "Winding_Number_Eval_Missing_Transcendental", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026595857203, "lm_q2_score": 0.7799928900257126, "lm_q1q2_score": 0.7154895524785384}} {"text": "[STATEMENT]\nlemma cos_periodic_nat[simp]:\n fixes n :: nat\n shows \"cos (x + n * (2 * pi)) = cos x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos (x + real n * (2 * pi)) = cos x\n[PROOF STEP]\nproof (induct n arbitrary: x)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x. cos (x + real 0 * (2 * pi)) = cos x\n 2. \\n x. (\\x. cos (x + real n * (2 * pi)) = cos x) \\ cos (x + real (Suc n) * (2 * pi)) = cos x\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\x. cos (x + real 0 * (2 * pi)) = cos x\n 2. \\n x. (\\x. cos (x + real n * (2 * pi)) = cos x) \\ cos (x + real (Suc n) * (2 * pi)) = cos x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos (x + real 0 * (2 * pi)) = cos x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncos (x + real 0 * (2 * pi)) = cos x\n\ngoal (1 subgoal):\n 1. \\n x. (\\x. cos (x + real n * (2 * pi)) = cos x) \\ cos (x + real (Suc n) * (2 * pi)) = cos x\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n x. (\\x. cos (x + real n * (2 * pi)) = cos x) \\ cos (x + real (Suc n) * (2 * pi)) = cos x\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\ncos (?x + real n * (2 * pi)) = cos ?x\n\ngoal (1 subgoal):\n 1. \\n x. (\\x. cos (x + real n * (2 * pi)) = cos x) \\ cos (x + real (Suc n) * (2 * pi)) = cos x\n[PROOF STEP]\nhave split_pi_off: \"x + (Suc n) * (2 * pi) = (x + n * (2 * pi)) + 2 * pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x + real (Suc n) * (2 * pi) = x + real n * (2 * pi) + 2 * pi\n[PROOF STEP]\nunfolding Suc_eq_plus1 of_nat_add of_int_1 distrib_right\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x + (real n * (2 * pi) + real 1 * (2 * pi)) = x + real n * (2 * pi) + 2 * pi\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx + real (Suc n) * (2 * pi) = x + real n * (2 * pi) + 2 * pi\n\ngoal (1 subgoal):\n 1. \\n x. (\\x. cos (x + real n * (2 * pi)) = cos x) \\ cos (x + real (Suc n) * (2 * pi)) = cos x\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos (x + real (Suc n) * (2 * pi)) = cos x\n[PROOF STEP]\nunfolding split_pi_off\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos (x + real n * (2 * pi) + 2 * pi) = cos x\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\ncos (?x + real n * (2 * pi)) = cos ?x\n\ngoal (1 subgoal):\n 1. cos (x + real n * (2 * pi) + 2 * pi) = cos x\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncos (x + real (Suc n) * (2 * pi)) = cos x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1258, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519528057272544, "lm_q2_score": 0.8397339596505965, "lm_q1q2_score": 0.7154137029887827}} {"text": "[STATEMENT]\nlemma norm_add_Pythagorean:\n assumes \"orthogonal a b\"\n shows \"norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (norm (a + b))\\<^sup>2 = (norm a)\\<^sup>2 + (norm b)\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (norm (a + b))\\<^sup>2 = (norm a)\\<^sup>2 + (norm b)\\<^sup>2\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\northogonal a b\n[PROOF STEP]\nhave \"(a - (0 - b)) \\ (a - (0 - b)) = a \\ a - (0 - b \\ b)\"\n[PROOF STATE]\nproof (prove)\nusing this:\northogonal a b\n\ngoal (1 subgoal):\n 1. (a - ((0::'a) - b)) \\ (a - ((0::'a) - b)) = a \\ a - (0 - b \\ b)\n[PROOF STEP]\nby (simp add: algebra_simps orthogonal_def inner_commute)\n[PROOF STATE]\nproof (state)\nthis:\n(a - ((0::'a) - b)) \\ (a - ((0::'a) - b)) = a \\ a - (0 - b \\ b)\n\ngoal (1 subgoal):\n 1. (norm (a + b))\\<^sup>2 = (norm a)\\<^sup>2 + (norm b)\\<^sup>2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(a - ((0::'a) - b)) \\ (a - ((0::'a) - b)) = a \\ a - (0 - b \\ b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(a - ((0::'a) - b)) \\ (a - ((0::'a) - b)) = a \\ a - (0 - b \\ b)\n\ngoal (1 subgoal):\n 1. (norm (a + b))\\<^sup>2 = (norm a)\\<^sup>2 + (norm b)\\<^sup>2\n[PROOF STEP]\nby (simp add: power2_norm_eq_inner)\n[PROOF STATE]\nproof (state)\nthis:\n(norm (a + b))\\<^sup>2 = (norm a)\\<^sup>2 + (norm b)\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 733, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527944504227, "lm_q2_score": 0.8397339616560072, "lm_q1q2_score": 0.7154136952277594}} {"text": "[STATEMENT]\nlemma outer_measure_of_mono: \"A \\ B \\ outer_measure_of M A \\ outer_measure_of M B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A \\ B \\ outer_measure_of M A \\ outer_measure_of M B\n[PROOF STEP]\nunfolding outer_measure_of_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A \\ B \\ Inf (emeasure M ` {B \\ sets M. A \\ B}) \\ Inf (emeasure M ` {Ba \\ sets M. B \\ Ba})\n[PROOF STEP]\nby (intro INF_superset_mono) auto", "meta": {"llama_tokens": 208, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213772699435, "lm_q2_score": 0.7956581049086031, "lm_q1q2_score": 0.7153932111214164}} {"text": "[STATEMENT]\nlemma cmod_real_prod_squared: \n fixes x y:: real\n shows \"(cmod (complex_of_real x * complex_of_real y))\\<^sup>2 = x\\<^sup>2 * y\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cmod (complex_of_real x * complex_of_real y))\\<^sup>2 = x\\<^sup>2 * y\\<^sup>2\n[PROOF STEP]\nby (simp add: norm_mult power_mult_distrib)", "meta": {"llama_tokens": 144, "file": "Isabelle_Marries_Dirac_Quantum_Prisoners_Dilemma", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213718636754, "lm_q2_score": 0.7956581024858786, "lm_q1q2_score": 0.715393204641552}} {"text": "[STATEMENT]\nlemma filterlim_at_top_linear_iff:\n fixes f::\"'a::linordered_field \\ 'b\"\n assumes \"c\\0\"\n shows \"(LIM x at_top. f (x * c + b) :> F2) \\ (if c>0 then (LIM x at_top. f x :> F2) \n else (LIM x at_bot. f x :> F2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (LIM x at_top. f (x * c + b) :> F2) = (if (0::'a) < c then filterlim f F2 at_top else filterlim f F2 at_bot)\n[PROOF STEP]\nunfolding filterlim_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (filtermap (\\x. f (x * c + b)) at_top \\ F2) = (if (0::'a) < c then filtermap f at_top \\ F2 else filtermap f at_bot \\ F2)\n[PROOF STEP]\napply (subst filtermap_filtermap[of f \"\\x. x * c + b\",symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (filtermap f (filtermap (\\x. x * c + b) at_top) \\ F2) = (if (0::'a) < c then filtermap f at_top \\ F2 else filtermap f at_bot \\ F2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'a)\n\ngoal (1 subgoal):\n 1. (filtermap f (filtermap (\\x. x * c + b) at_top) \\ F2) = (if (0::'a) < c then filtermap f at_top \\ F2 else filtermap f at_bot \\ F2)\n[PROOF STEP]\nby (auto simp add:filtermap_at_top_linear_eq)", "meta": {"llama_tokens": 558, "file": "Winding_Number_Eval_Missing_Topology", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513842182777, "lm_q2_score": 0.7981867777396211, "lm_q1q2_score": 0.7153760044138622}} {"text": "[STATEMENT]\nlemma lemma_tan_add1: \"cos x \\ 0 \\ cos y \\ 0 \\ 1 - tan x * tan y = cos (x + y)/(cos x * cos y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\cos x \\ (0::'a); cos y \\ (0::'a)\\ \\ (1::'a) - tan x * tan y = cos (x + y) / (cos x * cos y)\n[PROOF STEP]\nby (simp add: tan_def cos_add field_simps)", "meta": {"llama_tokens": 171, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513731336204, "lm_q2_score": 0.798186775339273, "lm_q1q2_score": 0.71537599341492}} {"text": "[STATEMENT]\nlemma less_two_pow_divD:\n \"\\ (x :: nat) < 2 ^ n div 2 ^ m \\\n \\ n \\ m \\ (x < 2 ^ (n - m))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x < 2 ^ n div 2 ^ m \\ m \\ n \\ x < 2 ^ (n - m)\n[PROOF STEP]\napply (rule context_conjI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. x < 2 ^ n div 2 ^ m \\ m \\ n\n 2. \\x < 2 ^ n div 2 ^ m; m \\ n\\ \\ x < 2 ^ (n - m)\n[PROOF STEP]\napply (rule ccontr)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x < 2 ^ n div 2 ^ m; \\ m \\ n\\ \\ False\n 2. \\x < 2 ^ n div 2 ^ m; m \\ n\\ \\ x < 2 ^ (n - m)\n[PROOF STEP]\napply (simp add: power_strict_increasing)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x < 2 ^ n div 2 ^ m; m \\ n\\ \\ x < 2 ^ (n - m)\n[PROOF STEP]\napply (simp add: power_sub)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 484, "file": "Word_Lib_More_Divides", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513703624557, "lm_q2_score": 0.7981867705385762, "lm_q1q2_score": 0.715375986900382}} {"text": "[STATEMENT]\nlemma Inf_le_iff_less:\n fixes x :: \"'a :: {complete_linorder, dense_linorder}\"\n shows \"(INF i\\A. f i) \\ x \\ (\\y>x. \\i\\A. f i \\ y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Inf (f ` A) \\ x) = (\\y>x. \\i\\A. f i \\ y)\n[PROOF STEP]\nunfolding INF_le_iff\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\y>x. \\i\\A. f i < y) = (\\y>x. \\i\\A. f i \\ y)\n[PROOF STEP]\nby (blast intro: less_imp_le less_trans le_less_trans dest: dense)", "meta": {"llama_tokens": 257, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8152324960856175, "lm_q1q2_score": 0.7153475864918604}} {"text": "[STATEMENT]\nlemma card_of_Times_ordLess_infinite_Field[simp]:\nassumes INF: \"\\finite (Field r)\" and r: \"Card_order r\" and\n LESS1: \"|A| B| B| B| B| |?C| =o r\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r =o |Field r| \\ |Field r| =o r\n[PROOF STEP]\nusing r card_of_Field_ordIso\n ordIso_symmetric\n[PROOF STATE]\nproof (prove)\nusing this:\nCard_order r\nCard_order ?r \\ |Field ?r| =o ?r\n?r =o ?r' \\ ?r' =o ?r\n\ngoal (1 subgoal):\n 1. r =o |Field r| \\ |Field r| =o r\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nr =o |Field r| \\ |Field r| =o r\n\ngoal (1 subgoal):\n 1. |A \\ B| |Field r| =o r\n\ngoal (1 subgoal):\n 1. |A| |Field r| =o r\n|A| ?r \\ ?r B| B| B| infinite ?C; |?A| \\ |?A \\ ?B| B| B| B| B| B| B| |Field r| =o r\n\\?r \\ ?r B| B| x xs. finite {xa. strict_prefix xa xs} \\ finite {xa. strict_prefix xa (xs @ [x])}\n[PROOF STEP]\napply (subgoal_tac \"{xa. strict_prefix xa (xs @ [x])} = {xa. strict_prefix xa xs} \\ {xs}\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x xs. \\finite {xa. strict_prefix xa xs}; {xa. strict_prefix xa (xs @ [x])} = {xa. strict_prefix xa xs} \\ {xs}\\ \\ finite {xa. strict_prefix xa (xs @ [x])}\n 2. \\x xs. finite {xa. strict_prefix xa xs} \\ {xa. strict_prefix xa (xs @ [x])} = {xa. strict_prefix xa xs} \\ {xs}\n[PROOF STEP]\nby (auto simp:strict_prefix_def)", "meta": {"llama_tokens": 382, "file": "Myhill-Nerode_Myhill_2", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736694, "lm_q2_score": 0.8152324848629215, "lm_q1q2_score": 0.7153475766442053}} {"text": "[STATEMENT]\nlemma norm_cblinfun_of_matrix:\n \"norm (cblinfun_of_matrix a) \\ (\\i\\CBasis. \\j\\CBasis. cmod (a i j))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (cblinfun_of_matrix a) \\ (\\i\\CBasis. \\j\\CBasis. cmod (a i j))\n[PROOF STEP]\napply (rule norm_cblinfun_bound)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. 0 \\ (\\i\\CBasis. \\j\\CBasis. cmod (a i j))\n 2. \\x. norm (cblinfun_apply (cblinfun_of_matrix a) x) \\ (\\i\\CBasis. \\j\\CBasis. cmod (a i j)) * norm x\n[PROOF STEP]\napply (simp add: sum_nonneg)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. norm (cblinfun_apply (cblinfun_of_matrix a) x) \\ (\\i\\CBasis. \\j\\CBasis. cmod (a i j)) * norm x\n[PROOF STEP]\napply (simp only: cblinfun_of_matrix_apply sum_distrib_right)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. norm (\\i\\CBasis. \\j\\CBasis. (j \\\\<^sub>C x * a i j) *\\<^sub>C i) \\ (\\n\\CBasis. \\na\\CBasis. cmod (a n na) * norm x)\n[PROOF STEP]\napply (rule order_trans[OF norm_sum sum_mono])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x n. n \\ CBasis \\ norm (\\j\\CBasis. (j \\\\<^sub>C x * a n j) *\\<^sub>C n) \\ (\\na\\CBasis. cmod (a n na) * norm x)\n[PROOF STEP]\napply (rule order_trans[OF norm_sum sum_mono])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x n na. \\n \\ CBasis; na \\ CBasis\\ \\ norm ((na \\\\<^sub>C x * a n na) *\\<^sub>C n) \\ cmod (a n na) * norm x\n[PROOF STEP]\napply (simp add: abs_mult mult_right_mono ac_simps Basis_le_norm)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x n na. \\n \\ CBasis; na \\ CBasis\\ \\ cmod (na \\\\<^sub>C x * a n na) \\ norm x * cmod (a n na)\n[PROOF STEP]\nby (metis complex_inner_class.Cauchy_Schwarz_ineq2 complex_scaleC_def mult.left_neutral mult_right_mono norm_CBasis norm_ge_zero norm_scaleC)", "meta": {"llama_tokens": 941, "file": "Complex_Bounded_Operators_Complex_Bounded_Linear_Function0", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8152324803738429, "lm_q1q2_score": 0.7153475727051429}} {"text": "[STATEMENT]\nlemma decr_grading_p_monom_mult:\n assumes \"hom_grading d\"\n shows \"decr_grading_p d n (monom_mult c t p) = monom_mult c (decr_grading d n t) (decr_grading_p d n p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. decr_grading_p d n (monom_mult c t p) = monom_mult c (decr_grading d n t) (decr_grading_p d n p)\n[PROOF STEP]\nproof (induct p rule: poly_mapping_plus_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. decr_grading_p d n (monom_mult c t 0) = monom_mult c (decr_grading d n t) (decr_grading_p d n 0)\n 2. \\p ca ta. \\ca \\ (0::'b); ta \\ keys p; decr_grading_p d n (monom_mult c t p) = monom_mult c (decr_grading d n t) (decr_grading_p d n p)\\ \\ decr_grading_p d n (monom_mult c t (monomial ca ta + p)) = monom_mult c (decr_grading d n t) (decr_grading_p d n (monomial ca ta + p))\n[PROOF STEP]\ncase 1\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. decr_grading_p d n (monom_mult c t 0) = monom_mult c (decr_grading d n t) (decr_grading_p d n 0)\n 2. \\p ca ta. \\ca \\ (0::'b); ta \\ keys p; decr_grading_p d n (monom_mult c t p) = monom_mult c (decr_grading d n t) (decr_grading_p d n p)\\ \\ decr_grading_p d n (monom_mult c t (monomial ca ta + p)) = monom_mult c (decr_grading d n t) (decr_grading_p d n (monomial ca ta + p))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. decr_grading_p d n (monom_mult c t 0) = monom_mult c (decr_grading d n t) (decr_grading_p d n 0)\n[PROOF STEP]\nby (simp add: decr_grading_p_zero)\n[PROOF STATE]\nproof (state)\nthis:\ndecr_grading_p d n (monom_mult c t 0) = monom_mult c (decr_grading d n t) (decr_grading_p d n 0)\n\ngoal (1 subgoal):\n 1. \\p ca ta. \\ca \\ (0::'b); ta \\ keys p; decr_grading_p d n (monom_mult c t p) = monom_mult c (decr_grading d n t) (decr_grading_p d n p)\\ \\ decr_grading_p d n (monom_mult c t (monomial ca ta + p)) = monom_mult c (decr_grading d n t) (decr_grading_p d n (monomial ca ta + p))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\p ca ta. \\ca \\ (0::'b); ta \\ keys p; decr_grading_p d n (monom_mult c t p) = monom_mult c (decr_grading d n t) (decr_grading_p d n p)\\ \\ decr_grading_p d n (monom_mult c t (monomial ca ta + p)) = monom_mult c (decr_grading d n t) (decr_grading_p d n (monomial ca ta + p))\n[PROOF STEP]\ncase (2 p a s)\n[PROOF STATE]\nproof (state)\nthis:\na \\ (0::'b)\ns \\ keys p\ndecr_grading_p d n (monom_mult c t p) = monom_mult c (decr_grading d n t) (decr_grading_p d n p)\n\ngoal (1 subgoal):\n 1. \\p ca ta. \\ca \\ (0::'b); ta \\ keys p; decr_grading_p d n (monom_mult c t p) = monom_mult c (decr_grading d n t) (decr_grading_p d n p)\\ \\ decr_grading_p d n (monom_mult c t (monomial ca ta + p)) = monom_mult c (decr_grading d n t) (decr_grading_p d n (monomial ca ta + p))\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nhom_grading d\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nhom_grading d\n\ngoal (1 subgoal):\n 1. decr_grading_p d n (monom_mult c t (monomial a s + p)) = monom_mult c (decr_grading d n t) (decr_grading_p d n (monomial a s + p))\n[PROOF STEP]\nby (simp add: monom_mult_dist_right decr_grading_p_plus 2(3) monom_mult_monomial\n decr_grading_p_monomial decr_grading_term_splus)\n[PROOF STATE]\nproof (state)\nthis:\ndecr_grading_p d n (monom_mult c t (monomial a s + p)) = monom_mult c (decr_grading d n t) (decr_grading_p d n (monomial a s + p))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1680, "file": "Polynomials_Quasi_PM_Power_Products", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240791017536, "lm_q2_score": 0.8267117855317474, "lm_q1q2_score": 0.7152909433192726}} {"text": "[STATEMENT]\nlemma \n defines \"f \\ (\\x::real. ln (ln (x * exp (x * exp x) + 1)) - exp (exp (ln (ln x) + 1 / x)))\"\n shows \"(f \\ 0) at_top\" (is ?thesis1)\n \"f \\ (\\x. -(ln x ^ 2) / (2*x))\" (is ?thesis2)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f \\ 0) at_top &&& f \\[at_top] (\\x. - (ln x)\\<^sup>2 / (2 * x))\n[PROOF STEP]\nunfolding f_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. ln (ln (x * exp (x * exp x) + 1)) - exp (exp (ln (ln x) + 1 / x))) \\ 0) at_top &&& (\\x. ln (ln (x * exp (x * exp x) + 1)) - exp (exp (ln (ln x) + 1 / x))) \\[at_top] (\\x. - (ln x)\\<^sup>2 / (2 * x))\n[PROOF STEP]\nby real_asymp+", "meta": {"llama_tokens": 354, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094117351309, "lm_q2_score": 0.8006920092299293, "lm_q1q2_score": 0.7152657077462081}} {"text": "[STATEMENT]\nlemma bt_map_append: \"bt_map f (append t u) = append (bt_map f t) (bt_map f u)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bt_map f (Binary_Tree.append t u) = Binary_Tree.append (bt_map f t) (bt_map f u)\n[PROOF STEP]\napply (induct t)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. bt_map f (Binary_Tree.append Lf u) = Binary_Tree.append (bt_map f Lf) (bt_map f u)\n 2. \\x1 t1 t2. \\bt_map f (Binary_Tree.append t1 u) = Binary_Tree.append (bt_map f t1) (bt_map f u); bt_map f (Binary_Tree.append t2 u) = Binary_Tree.append (bt_map f t2) (bt_map f u)\\ \\ bt_map f (Binary_Tree.append (Br x1 t1 t2) u) = Binary_Tree.append (bt_map f (Br x1 t1 t2)) (bt_map f u)\n[PROOF STEP]\napply (metis append.simps(1) bt_map.simps(1))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x1 t1 t2. \\bt_map f (Binary_Tree.append t1 u) = Binary_Tree.append (bt_map f t1) (bt_map f u); bt_map f (Binary_Tree.append t2 u) = Binary_Tree.append (bt_map f t2) (bt_map f u)\\ \\ bt_map f (Binary_Tree.append (Br x1 t1 t2) u) = Binary_Tree.append (bt_map f (Br x1 t1 t2)) (bt_map f u)\n[PROOF STEP]\nby (metis append.simps(2) bt_map.simps(2))", "meta": {"llama_tokens": 547, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094117351309, "lm_q2_score": 0.8006920068519376, "lm_q1q2_score": 0.7152657056219258}} {"text": "[STATEMENT]\nlemma mon_iff_hom_one:\n \"\\group G; group H\\ \\ f \\ mon G H \\ f \\ hom G H \\ (\\x. x \\ carrier G \\ f x = \\\\<^bsub>H\\<^esub> \\ x = \\\\<^bsub>G\\<^esub>)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Group.group G; Group.group H\\ \\ (f \\ mon G H) = (f \\ hom G H \\ (\\x. x \\ carrier G \\ f x = \\\\<^bsub>H\\<^esub> \\ x = \\\\<^bsub>G\\<^esub>))\n[PROOF STEP]\nby (auto simp: mon_def inj_on_one_iff')", "meta": {"llama_tokens": 240, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094060543489, "lm_q2_score": 0.8006920116079209, "lm_q1q2_score": 0.7152657053219337}} {"text": "[STATEMENT]\nlemma scalar_prod_inter_num_lift_01: \n assumes \"j1 < \\\" \"j2 < \\\"\n shows \"((col (lift_01_mat N) j1) \\ (col (lift_01_mat N) j2)) = (of_nat ((\\s ! j1) |\\| (\\s ! j2)) :: ('b :: {ring_1}))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. col (lift_01_mat N) j1 \\ col (lift_01_mat N) j2 = of_nat (\\s ! j1 |\\| \\s ! j2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. col (lift_01_mat N) j1 \\ col (lift_01_mat N) j2 = of_nat (\\s ! j1 |\\| \\s ! j2)\n[PROOF STEP]\ninterpret z1: zero_one_matrix_ring_1 \"(lift_01_mat N)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. zero_one_matrix_ring_1 (lift_01_mat N)\n[PROOF STEP]\nby (intro_locales) (simp add: lift_mat_is_0_1)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. col (lift_01_mat N) j1 \\ col (lift_01_mat N) j2 = of_nat (\\s ! j1 |\\| \\s ! j2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. col (lift_01_mat N) j1 \\ col (lift_01_mat N) j2 = of_nat (\\s ! j1 |\\| \\s ! j2)\n[PROOF STEP]\nusing assms z1.scalar_prod_inc_vec_mat_inter_num preserve_mat_inter_num \n mat_inter_num_conv lift_01_mat_def blocks_list_length inc_mat_dim_col\n[PROOF STATE]\nproof (prove)\nusing this:\nj1 < \\\nj2 < \\\n\\?j1.0 < dim_col (lift_01_mat N); ?j2.0 < dim_col (lift_01_mat N)\\ \\ col (lift_01_mat N) ?j1.0 \\ col (lift_01_mat N) ?j2.0 = of_nat (mat_inter_num (lift_01_mat N) ?j1.0 ?j2.0)\n\\inj_on_01_hom ?f; ?j1.0 < dim_col N; ?j2.0 < dim_col N\\ \\ mat_inter_num N ?j1.0 ?j2.0 = mat_inter_num (map_mat ?f N) ?j1.0 ?j2.0\n\\?j1.0 < dim_col N; ?j2.0 < dim_col N\\ \\ \\s ! ?j1.0 |\\| \\s ! ?j2.0 = mat_inter_num N ?j1.0 ?j2.0\nlift_01_mat ?M \\ map_mat of_zero_neq_one ?M\nlength \\s = \\\ndim_col (inc_mat_of ?Vs ?Bs) = length ?Bs\n\ngoal (1 subgoal):\n 1. col (lift_01_mat N) j1 \\ col (lift_01_mat N) j2 = of_nat (\\s ! j1 |\\| \\s ! j2)\n[PROOF STEP]\nby (metis lift_01_mat_simp(2) of_inj_on_01_hom.inj_on_01_hom_axioms)\n[PROOF STATE]\nproof (state)\nthis:\ncol (lift_01_mat N) j1 \\ col (lift_01_mat N) j2 = of_nat (\\s ! j1 |\\| \\s ! j2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1189, "file": "Fishers_Inequality_Incidence_Matrices", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094145755219, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.7152657036476395}} {"text": "[STATEMENT]\nlemma jordan_matrix_Cons: \"jordan_matrix (Cons (n,a) n_as) = four_block_mat \n (jordan_block n a) (0\\<^sub>m n (sum_list (map fst n_as))) \n (0\\<^sub>m (sum_list (map fst n_as)) n) (jordan_matrix n_as)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. jordan_matrix ((n, a) # n_as) = four_block_mat (jordan_block n a) (0\\<^sub>m n (sum_list (map fst n_as))) (0\\<^sub>m (sum_list (map fst n_as)) n) (jordan_matrix n_as)\n[PROOF STEP]\nunfolding jordan_matrix_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. diag_block_mat (map (\\(x, y). jordan_block x y) ((n, a) # n_as)) = four_block_mat (jordan_block n a) (0\\<^sub>m n (sum_list (map fst n_as))) (0\\<^sub>m (sum_list (map fst n_as)) n) (diag_block_mat (map (\\(x, y). jordan_block x y) n_as))\n[PROOF STEP]\nby (simp, simp add: jordan_matrix_def[symmetric])", "meta": {"llama_tokens": 391, "file": "Jordan_Normal_Form_Jordan_Normal_Form", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933093975331751, "lm_q2_score": 0.8006919997179627, "lm_q1q2_score": 0.7152656878776864}} {"text": "[STATEMENT]\nlemma affine_hull_2_alt:\n fixes a b :: \"'a::real_vector\"\n shows \"affine hull {a,b} = range (\\u. a + u *\\<^sub>R (b - a))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. affine hull {a, b} = range (\\u. a + u *\\<^sub>R (b - a))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. affine hull {a, b} = range (\\u. a + u *\\<^sub>R (b - a))\n[PROOF STEP]\nhave 1: \"u *\\<^sub>R a + v *\\<^sub>R b = a + v *\\<^sub>R (b - a)\" if \"u + v = 1\" for u v\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. u *\\<^sub>R a + v *\\<^sub>R b = a + v *\\<^sub>R (b - a)\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\nu + v = 1\n\ngoal (1 subgoal):\n 1. u *\\<^sub>R a + v *\\<^sub>R b = a + v *\\<^sub>R (b - a)\n[PROOF STEP]\nby (simp add: algebra_simps flip: scaleR_add_left)\n[PROOF STATE]\nproof (state)\nthis:\n?u + ?v = 1 \\ ?u *\\<^sub>R a + ?v *\\<^sub>R b = a + ?v *\\<^sub>R (b - a)\n\ngoal (1 subgoal):\n 1. affine hull {a, b} = range (\\u. a + u *\\<^sub>R (b - a))\n[PROOF STEP]\nhave 2: \"a + u *\\<^sub>R (b - a) = (1 - u) *\\<^sub>R a + u *\\<^sub>R b\" for u\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a + u *\\<^sub>R (b - a) = (1 - u) *\\<^sub>R a + u *\\<^sub>R b\n[PROOF STEP]\nby (auto simp: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\na + ?u *\\<^sub>R (b - a) = (1 - ?u) *\\<^sub>R a + ?u *\\<^sub>R b\n\ngoal (1 subgoal):\n 1. affine hull {a, b} = range (\\u. a + u *\\<^sub>R (b - a))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. affine hull {a, b} = range (\\u. a + u *\\<^sub>R (b - a))\n[PROOF STEP]\nby (force simp add: affine_hull_2 dest: 1 intro!: 2)\n[PROOF STATE]\nproof (state)\nthis:\naffine hull {a, b} = range (\\u. a + u *\\<^sub>R (b - a))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 873, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.853912760387131, "lm_q2_score": 0.837619961306541, "lm_q1q2_score": 0.7152543733146304}} {"text": "[STATEMENT]\nlemma l2norm_diff:\n assumes \"f square_integrable S\" \"g square_integrable S\"\n shows \"l2norm S (\\x. f x - g x) = l2norm S (\\x. g x - f x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. l2norm S (\\x. f x - g x) = l2norm S (\\x. g x - f x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. l2norm S (\\x. f x - g x) = l2norm S (\\x. g x - f x)\n[PROOF STEP]\nhave \"(\\x. f x - g x) square_integrable S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. f x - g x) square_integrable S\n[PROOF STEP]\nusing assms square_integrable_diff\n[PROOF STATE]\nproof (prove)\nusing this:\nf square_integrable S\ng square_integrable S\n\\?f square_integrable ?S; ?g square_integrable ?S\\ \\ (\\x. ?f x - ?g x) square_integrable ?S\n\ngoal (1 subgoal):\n 1. (\\x. f x - g x) square_integrable S\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. f x - g x) square_integrable S\n\ngoal (1 subgoal):\n 1. l2norm S (\\x. f x - g x) = l2norm S (\\x. g x - f x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x. f x - g x) square_integrable S\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. f x - g x) square_integrable S\n\ngoal (1 subgoal):\n 1. l2norm S (\\x. f x - g x) = l2norm S (\\x. g x - f x)\n[PROOF STEP]\nusing l2norm_neg [of \"\\x. f x - g x\" S]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. f x - g x) square_integrable S\n(\\x. f x - g x) square_integrable S \\ l2norm S (\\x. - (f x - g x)) = l2norm S (\\x. f x - g x)\n\ngoal (1 subgoal):\n 1. l2norm S (\\x. f x - g x) = l2norm S (\\x. g x - f x)\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nl2norm S (\\x. f x - g x) = l2norm S (\\x. g x - f x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 870, "file": "Fourier_Square_Integrable", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199754937772, "lm_q2_score": 0.8539127455162773, "lm_q1q2_score": 0.7152543729731682}} {"text": "[STATEMENT]\nlemma forall_nat_expansion:\n \\(\\n \\ (n\\<^sub>0::nat). P n) = (P n\\<^sub>0 \\ (\\n \\ Suc n\\<^sub>0. P n))\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n\\n\\<^sub>0. P n) = (P n\\<^sub>0 \\ (\\n\\Suc n\\<^sub>0. P n))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\n\\n\\<^sub>0. P n) = (P n\\<^sub>0 \\ (\\n\\Suc n\\<^sub>0. P n))\n[PROOF STEP]\nhave \\(\\n \\ (n\\<^sub>0::nat). P n) = (\\n. (n = n\\<^sub>0 \\ n > n\\<^sub>0) \\ P n)\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n\\n\\<^sub>0. P n) = (\\n. n = n\\<^sub>0 \\ n\\<^sub>0 < n \\ P n)\n[PROOF STEP]\nusing le_less\n[PROOF STATE]\nproof (prove)\nusing this:\n(?x \\ ?y) = (?x < ?y \\ ?x = ?y)\n\ngoal (1 subgoal):\n 1. (\\n\\n\\<^sub>0. P n) = (\\n. n = n\\<^sub>0 \\ n\\<^sub>0 < n \\ P n)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(\\n\\n\\<^sub>0. P n) = (\\n. n = n\\<^sub>0 \\ n\\<^sub>0 < n \\ P n)\n\ngoal (1 subgoal):\n 1. (\\n\\n\\<^sub>0. P n) = (P n\\<^sub>0 \\ (\\n\\Suc n\\<^sub>0. P n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\n\\n\\<^sub>0. P n) = (\\n. n = n\\<^sub>0 \\ n\\<^sub>0 < n \\ P n)\n\ngoal (1 subgoal):\n 1. (\\n\\n\\<^sub>0. P n) = (P n\\<^sub>0 \\ (\\n\\Suc n\\<^sub>0. P n))\n[PROOF STEP]\nhave \\... = (P n\\<^sub>0 \\ (\\n > n\\<^sub>0. P n))\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. n = n\\<^sub>0 \\ n\\<^sub>0 < n \\ P n) = (P n\\<^sub>0 \\ (\\n>n\\<^sub>0. P n))\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. n = n\\<^sub>0 \\ n\\<^sub>0 < n \\ P n) = (P n\\<^sub>0 \\ (\\n>n\\<^sub>0. P n))\n\ngoal (1 subgoal):\n 1. (\\n\\n\\<^sub>0. P n) = (P n\\<^sub>0 \\ (\\n\\Suc n\\<^sub>0. P n))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n\\n\\<^sub>0. P n) = (P n\\<^sub>0 \\ (\\n>n\\<^sub>0. P n))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n\\n\\<^sub>0. P n) = (P n\\<^sub>0 \\ (\\n>n\\<^sub>0. P n))\n\ngoal (1 subgoal):\n 1. (\\n\\n\\<^sub>0. P n) = (P n\\<^sub>0 \\ (\\n\\Suc n\\<^sub>0. P n))\n[PROOF STEP]\nusing Suc_le_eq\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n\\n\\<^sub>0. P n) = (P n\\<^sub>0 \\ (\\n>n\\<^sub>0. P n))\n(Suc ?m \\ ?n) = (?m < ?n)\n\ngoal (1 subgoal):\n 1. (\\n\\n\\<^sub>0. P n) = (P n\\<^sub>0 \\ (\\n\\Suc n\\<^sub>0. P n))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\n\\n\\<^sub>0. P n) = (P n\\<^sub>0 \\ (\\n\\Suc n\\<^sub>0. P n))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1429, "file": "TESL_Language_Corecursive_Prop", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127417985637, "lm_q2_score": 0.8376199714402813, "lm_q1q2_score": 0.7152543663978053}} {"text": "[STATEMENT]\nlemma prime_power_not_one:\n assumes \"prime p\" \"k > 0\"\n shows \"p ^ k \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p ^ k \\ (1::'a)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. p ^ k = (1::'a) \\ False\n[PROOF STEP]\nassume \"p ^ k = 1\"\n[PROOF STATE]\nproof (state)\nthis:\np ^ k = (1::'a)\n\ngoal (1 subgoal):\n 1. p ^ k = (1::'a) \\ False\n[PROOF STEP]\nhence \"is_unit (p ^ k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\np ^ k = (1::'a)\n\ngoal (1 subgoal):\n 1. is_unit (p ^ k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nis_unit (p ^ k)\n\ngoal (1 subgoal):\n 1. p ^ k = (1::'a) \\ False\n[PROOF STEP]\nthus False\n[PROOF STATE]\nproof (prove)\nusing this:\nis_unit (p ^ k)\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nis_unit (p ^ k)\nprime p\n0 < k\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby (simp add: is_unit_power_iff)\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 495, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473746782093, "lm_q2_score": 0.8198933425148213, "lm_q1q2_score": 0.7152318048589462}} {"text": "[STATEMENT]\nlemma trace_adjoint_eq_u:\n fixes A :: \"complex mat\"\n shows \"trace (A * adjoint A) = (\\ i \\ {0 ..< dim_row A}. \\ j \\ {0 ..< dim_col A}. (norm(A $$ (i,j)))\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (A * adjoint A) = complex_of_real (\\i = 0..j = 0..2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. trace (A * adjoint A) = complex_of_real (\\i = 0..j = 0..2)\n[PROOF STEP]\nhave \"trace (A * adjoint A) = (\\ i \\ {0 ..< dim_row A}. row A i \\ conjugate (row A i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (A * adjoint A) = (\\i = 0..i = 0..i = 0..j = 0..2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ntrace (A * adjoint A) = (\\i = 0..i = 0..j = 0..2)\n[PROOF STEP]\nhave \"\\ = (\\ i \\ {0 ..< dim_row A}. \\ j \\ {0 ..< dim_col A}. (norm(A $$ (i,j)))\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..i = 0..j = 0..2)\n[PROOF STEP]\nproof (simp add: scalar_prod_def cmod_def)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\i = 0..ia = 0..x = 0..xa = 0..2) + (\\x = 0..xa = 0..2)\n[PROOF STEP]\nhave cnjmul: \"\\ i ia. A $$ (i, ia) * cnj (A $$ (i, ia)) =\n ((complex_of_real (Re (A $$ (i, ia))))\\<^sup>2 + (complex_of_real (Im (A $$ (i, ia))))\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i ia. A $$ (i, ia) * cnj (A $$ (i, ia)) = (complex_of_real (Re (A $$ (i, ia))))\\<^sup>2 + (complex_of_real (Im (A $$ (i, ia))))\\<^sup>2\n[PROOF STEP]\nby (simp add: complex_mult_cnj)\n[PROOF STATE]\nproof (state)\nthis:\n\\i ia. A $$ (i, ia) * cnj (A $$ (i, ia)) = (complex_of_real (Re (A $$ (i, ia))))\\<^sup>2 + (complex_of_real (Im (A $$ (i, ia))))\\<^sup>2\n\ngoal (1 subgoal):\n 1. (\\i = 0..ia = 0..x = 0..xa = 0..2) + (\\x = 0..xa = 0..2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i ia. A $$ (i, ia) * cnj (A $$ (i, ia)) = (complex_of_real (Re (A $$ (i, ia))))\\<^sup>2 + (complex_of_real (Im (A $$ (i, ia))))\\<^sup>2\n[PROOF STEP]\nhave \"\\ i. (\\ia = 0..ia = 0..2 + (complex_of_real (Im (A $$ (i, ia))))\\<^sup>2))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i ia. A $$ (i, ia) * cnj (A $$ (i, ia)) = (complex_of_real (Re (A $$ (i, ia))))\\<^sup>2 + (complex_of_real (Im (A $$ (i, ia))))\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\i. (\\ia = 0..ia = 0..2 + (complex_of_real (Im (A $$ (i, ia))))\\<^sup>2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i. (\\ia = 0..ia = 0..2 + (complex_of_real (Im (A $$ (i, ia))))\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (\\i = 0..ia = 0..x = 0..xa = 0..2) + (\\x = 0..xa = 0..2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i. (\\ia = 0..ia = 0..2 + (complex_of_real (Im (A $$ (i, ia))))\\<^sup>2)\n[PROOF STEP]\nshow\"(\\i = 0..ia = 0..x = 0..xa = 0..2) +\n (\\x = 0..xa = 0..2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i. (\\ia = 0..ia = 0..2 + (complex_of_real (Im (A $$ (i, ia))))\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (\\i = 0..ia = 0..x = 0..xa = 0..2) + (\\x = 0..xa = 0..2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..ia = 0..x = 0..xa = 0..2) + (\\x = 0..xa = 0..2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..i = 0..j = 0..2)\n\ngoal (1 subgoal):\n 1. trace (A * adjoint A) = complex_of_real (\\i = 0..j = 0..2)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ntrace (A * adjoint A) = complex_of_real (\\i = 0..j = 0..2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntrace (A * adjoint A) = complex_of_real (\\i = 0..j = 0..2)\n\ngoal (1 subgoal):\n 1. trace (A * adjoint A) = complex_of_real (\\i = 0..j = 0..2)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ntrace (A * adjoint A) = complex_of_real (\\i = 0..j = 0..2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3490, "file": "QHLProver_Matrix_Limit", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473813156295, "lm_q2_score": 0.819893335913536, "lm_q1q2_score": 0.7152318045423088}} {"text": "[STATEMENT]\nlemma falling_fact_pochhammer: \"prod (\\i. a - int i) {0..i = 0..i = 0..x. z * f x) {0..n}\" for z::int and n f\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z ^ Suc n * prod f {0..n} = (\\x = 0..n. z * f x)\n[PROOF STEP]\nby (induct n) (simp_all add: ac_simps)\n[PROOF STATE]\nproof (state)\nthis:\n?z ^ Suc ?n * prod ?f {0..?n} = (\\x = 0..?n. ?z * ?f x)\n\ngoal (1 subgoal):\n 1. (\\i = 0..i = 0.. (\\i = 0..nat. k = Suc nat \\ (\\i = 0.. (\\i = 0..nat. k = Suc nat \\ (\\i = 0..i = 0..i = 0..nat. k = Suc nat \\ (\\i = 0..nat. k = Suc nat \\ (\\i = 0..nat. k = Suc nat \\ (\\i = 0..i = 0..i = 0..i = 0..infinite (- (\\p \\ psubst (\\n::nat. 2 * n) ` S. params p))\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. infinite (- \\ (params ` psubst ((*) 2) ` S))\n[PROOF STEP]\nproof (rule infinite_super)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ?S \\ - \\ (params ` psubst ((*) 2) ` S)\n 2. infinite ?S\n[PROOF STEP]\nshow \\infinite (range (\\n::nat. 2 * n + 1))\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. infinite (range (\\n. 2 * n + 1))\n[PROOF STEP]\nusing inj_onI Suc_1 Suc_mult_cancel1 add_right_imp_eq finite_imageD infinite_UNIV_char_0\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x y. \\x \\ ?A; y \\ ?A; ?f x = ?f y\\ \\ x = y) \\ inj_on ?f ?A\nSuc 1 = 2\n(Suc ?k * ?m = Suc ?k * ?n) = (?m = ?n)\n?b + ?a = ?c + ?a \\ ?b = ?c\n\\finite (?f ` ?A); inj_on ?f ?A\\ \\ finite ?A\ninfinite UNIV\n\ngoal (1 subgoal):\n 1. infinite (range (\\n. 2 * n + 1))\n[PROOF STEP]\nby (metis (no_types, lifting))\n[PROOF STATE]\nproof (state)\nthis:\ninfinite (range (\\n. 2 * n + 1))\n\ngoal (1 subgoal):\n 1. range (\\n. 2 * n + 1) \\ - \\ (params ` psubst ((*) 2) ` S)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. range (\\n. 2 * n + 1) \\ - \\ (params ` psubst ((*) 2) ` S)\n[PROOF STEP]\nhave \\\\m n. Suc (2 * m) \\ 2 * n\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m n. Suc (2 * m) \\ 2 * n\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\nSuc (2 * ?m) \\ 2 * ?n\n\ngoal (1 subgoal):\n 1. range (\\n. 2 * n + 1) \\ - \\ (params ` psubst ((*) 2) ` S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nSuc (2 * ?m) \\ 2 * ?n\n[PROOF STEP]\nshow \\range (\\n. 2 * n + 1)\n \\ - (\\p::(nat, 'a) form \\ psubst (\\n . 2 * n) ` S. params p)\\\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc (2 * ?m) \\ 2 * ?n\n\ngoal (1 subgoal):\n 1. range (\\n. 2 * n + 1) \\ - \\ (params ` psubst ((*) 2) ` S)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrange (\\n. 2 * n + 1) \\ - \\ (params ` psubst ((*) 2) ` S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1088, "file": "FOL-Fitting_FOL_Fitting", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278726384089, "lm_q2_score": 0.8104789063814616, "lm_q1q2_score": 0.7151891771764973}} {"text": "[STATEMENT]\nlemma ivl_integral_has_vderiv_on_compact_interval:\n fixes f :: \"real \\ 'a::banach\"\n assumes \"continuous_on A f\"\n and \"c \\ A\" \"is_interval A\" \"compact A\"\n shows \"((\\u. ivl_integral c u f) has_vderiv_on f) A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\u. ivl_integral c u f) has_vderiv_on f) A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\u. ivl_integral c u f) has_vderiv_on f) A\n[PROOF STEP]\nhave \"A = {Inf A .. Sup A}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A = {Inf A..Sup A}\n[PROOF STEP]\nby (rule compact_interval_eq_Inf_Sup) (use assms in auto)\n[PROOF STATE]\nproof (state)\nthis:\nA = {Inf A..Sup A}\n\ngoal (1 subgoal):\n 1. ((\\u. ivl_integral c u f) has_vderiv_on f) A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nA = {Inf A..Sup A}\n\ngoal (1 subgoal):\n 1. ((\\u. ivl_integral c u f) has_vderiv_on f) A\n[PROOF STEP]\nhave \"\\ = closed_segment (Inf A) (Sup A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {Inf A..Sup A} = closed_segment (Inf A) (Sup A)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous_on A f\nc \\ A\nis_interval A\ncompact A\n\ngoal (1 subgoal):\n 1. {Inf A..Sup A} = closed_segment (Inf A) (Sup A)\n[PROOF STEP]\nby (auto simp add: closed_segment_eq_real_ivl\n intro!: cInf_le_cSup bounded_imp_bdd_below bounded_imp_bdd_above compact_imp_bounded)\n[PROOF STATE]\nproof (state)\nthis:\n{Inf A..Sup A} = closed_segment (Inf A) (Sup A)\n\ngoal (1 subgoal):\n 1. ((\\u. ivl_integral c u f) has_vderiv_on f) A\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nA = closed_segment (Inf A) (Sup A)\n[PROOF STEP]\nhave *: \"A = closed_segment (Inf A) (Sup A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA = closed_segment (Inf A) (Sup A)\n\ngoal (1 subgoal):\n 1. A = closed_segment (Inf A) (Sup A)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nA = closed_segment (Inf A) (Sup A)\n\ngoal (1 subgoal):\n 1. ((\\u. ivl_integral c u f) has_vderiv_on f) A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\u. ivl_integral c u f) has_vderiv_on f) A\n[PROOF STEP]\napply (subst *)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\u. ivl_integral c u f) has_vderiv_on f) (closed_segment (Inf A) (Sup A))\n[PROOF STEP]\napply (rule ivl_integral_has_vderiv_on_subset_segment)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. continuous_on (closed_segment (Inf A) (Sup A)) f\n 2. c \\ closed_segment (Inf A) (Sup A)\n[PROOF STEP]\nunfolding *[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. continuous_on A f\n 2. c \\ A\n[PROOF STEP]\nby fact+\n[PROOF STATE]\nproof (state)\nthis:\n((\\u. ivl_integral c u f) has_vderiv_on f) A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1291, "file": "Ordinary_Differential_Equations_Library_Interval_Integral_HK", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278540866547, "lm_q2_score": 0.810478913248044, "lm_q1q2_score": 0.7151891681999555}} {"text": "[STATEMENT]\nlemma sin_cube_plus_cos_cube_gt_zero_iff':\n \"(sin x ^ 3 + cos x ^ 3 > 0) = (sin (x + pi/4) > 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0 < sin x ^ 3 + cos x ^ 3) = (0 < sin (x + pi / 4))\n[PROOF STEP]\nby (smt (verit, best) mult_pos_pos real_sqrt_gt_0_iff\n sin_cube_plus_cos_cube_gt_zero_iff sin_plus_cos_eq_45 zero_less_mult_pos)", "meta": {"llama_tokens": 178, "file": "Hyperdual_AnalyticTestFunction", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505325302033, "lm_q2_score": 0.7905303162021597, "lm_q1q2_score": 0.7151536715335538}} {"text": "[STATEMENT]\nlemma suminf_geometric: \"norm c < 1 \\ suminf (\\n. c^n) = 1 / (1 - c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm c < 1 \\ (\\n. c ^ n) = (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nby (rule sums_unique[symmetric]) (rule geometric_sums)", "meta": {"llama_tokens": 125, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8031737963569016, "lm_q1q2_score": 0.7150609887932948}} {"text": "[STATEMENT]\nlemma suminf_geometric: \"norm c < 1 \\ suminf (\\n. c^n) = 1 / (1 - c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm c < 1 \\ (\\n. c ^ n) = (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nby (rule sums_unique[symmetric]) (rule geometric_sums)", "meta": {"llama_tokens": 125, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8031737963569016, "lm_q1q2_score": 0.7150609887932948}} {"text": "[STATEMENT]\ntheorem machin: \"pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nhave \"\\1 / 5\\ < (1 :: real)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\1 / 5\\ < 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\1 / 5\\ < 1\n\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nfrom arctan_add[OF less_imp_le[OF this] this]\n[PROOF STATE]\nproof (chain)\npicking this:\narctan (1 / 5) + arctan (1 / 5) = arctan ((1 / 5 + 1 / 5) / (1 - 1 / 5 * (1 / 5)))\n[PROOF STEP]\nhave \"2 * arctan (1 / 5) = arctan (5 / 12)\"\n[PROOF STATE]\nproof (prove)\nusing this:\narctan (1 / 5) + arctan (1 / 5) = arctan ((1 / 5 + 1 / 5) / (1 - 1 / 5 * (1 / 5)))\n\ngoal (1 subgoal):\n 1. 2 * arctan (1 / 5) = arctan (5 / 12)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n2 * arctan (1 / 5) = arctan (5 / 12)\n\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n2 * arctan (1 / 5) = arctan (5 / 12)\n\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nhave \"\\5 / 12\\ < (1 :: real)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\5 / 12\\ < 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\5 / 12\\ < 1\n\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nfrom arctan_add[OF less_imp_le[OF this] this]\n[PROOF STATE]\nproof (chain)\npicking this:\narctan (5 / 12) + arctan (5 / 12) = arctan ((5 / 12 + 5 / 12) / (1 - 5 / 12 * (5 / 12)))\n[PROOF STEP]\nhave \"2 * arctan (5 / 12) = arctan (120 / 119)\"\n[PROOF STATE]\nproof (prove)\nusing this:\narctan (5 / 12) + arctan (5 / 12) = arctan ((5 / 12 + 5 / 12) / (1 - 5 / 12 * (5 / 12)))\n\ngoal (1 subgoal):\n 1. 2 * arctan (5 / 12) = arctan (120 / 119)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n2 * arctan (5 / 12) = arctan (120 / 119)\n\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n2 * arctan (5 / 12) = arctan (120 / 119)\n\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nhave \"\\1\\ \\ (1::real)\" and \"\\1 / 239\\ < (1::real)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\1\\ \\ 1 &&& \\1 / 239\\ < 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\1\\ \\ 1\n\\1 / 239\\ < 1\n\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nfrom arctan_add[OF this]\n[PROOF STATE]\nproof (chain)\npicking this:\narctan 1 + arctan (1 / 239) = arctan ((1 + 1 / 239) / (1 - 1 * (1 / 239)))\n[PROOF STEP]\nhave \"arctan 1 + arctan (1 / 239) = arctan (120 / 119)\"\n[PROOF STATE]\nproof (prove)\nusing this:\narctan 1 + arctan (1 / 239) = arctan ((1 + 1 / 239) / (1 - 1 * (1 / 239)))\n\ngoal (1 subgoal):\n 1. arctan 1 + arctan (1 / 239) = arctan (120 / 119)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\narctan 1 + arctan (1 / 239) = arctan (120 / 119)\n\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n2 * arctan (1 / 5) = arctan (5 / 12)\n2 * arctan (5 / 12) = arctan (120 / 119)\narctan 1 + arctan (1 / 239) = arctan (120 / 119)\n[PROOF STEP]\nhave \"arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * arctan (1 / 5) = arctan (5 / 12)\n2 * arctan (5 / 12) = arctan (120 / 119)\narctan 1 + arctan (1 / 239) = arctan (120 / 119)\n\ngoal (1 subgoal):\n 1. arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\narctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)\n\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\narctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\narctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)\n\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nunfolding arctan_one\n[PROOF STATE]\nproof (prove)\nusing this:\npi / 4 + arctan (1 / 239) = 4 * arctan (1 / 5)\n\ngoal (1 subgoal):\n 1. pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n[PROOF STEP]\nby algebra\n[PROOF STATE]\nproof (state)\nthis:\npi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2545, "file": null, "length": 26, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004187, "lm_q2_score": 0.8031737940012417, "lm_q1q2_score": 0.7150609866960645}} {"text": "[STATEMENT]\nlemma homotopic_paths_linear:\n fixes g h :: \"real \\ 'a::real_normed_vector\"\n assumes \"path g\" \"path h\" \"pathstart h = pathstart g\" \"pathfinish h = pathfinish g\"\n \"\\t. t \\ {0..1} \\ closed_segment (g t) (h t) \\ S\"\n shows \"homotopic_paths S g h\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. homotopic_paths S g h\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\npath g\npath h\npathstart h = pathstart g\npathfinish h = pathfinish g\n?t \\ {0..1} \\ closed_segment (g ?t) (h ?t) \\ S\n\ngoal (1 subgoal):\n 1. homotopic_paths S g h\n[PROOF STEP]\nunfolding path_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous_on {0..1} g\ncontinuous_on {0..1} h\npathstart h = pathstart g\npathfinish h = pathfinish g\n?t \\ {0..1} \\ closed_segment (g ?t) (h ?t) \\ S\n\ngoal (1 subgoal):\n 1. homotopic_paths S g h\n[PROOF STEP]\napply (simp add: closed_segment_def pathstart_def pathfinish_def homotopic_paths_def homotopic_with_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\continuous_on {0..1} g; continuous_on {0..1} h; h 0 = g 0; h 1 = g 1; \\t. 0 \\ t \\ t \\ 1 \\ {(1 - u) *\\<^sub>R g t + u *\\<^sub>R h t |u. 0 \\ u \\ u \\ 1} \\ S\\ \\ \\ha. continuous_on ({0..1} \\ {0..1}) ha \\ ha ` ({0..1} \\ {0..1}) \\ S \\ (\\x. ha (0, x) = g x) \\ (\\x. ha (1, x) = h x) \\ (\\t\\{0..1}. ha (t, 0) = g 0 \\ ha (t, 1) = g 1)\n[PROOF STEP]\napply (rule_tac x=\"\\y. ((1 - (fst y)) *\\<^sub>R (g \\ snd) y + (fst y) *\\<^sub>R (h \\ snd) y)\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\continuous_on {0..1} g; continuous_on {0..1} h; h 0 = g 0; h 1 = g 1; \\t. 0 \\ t \\ t \\ 1 \\ {(1 - u) *\\<^sub>R g t + u *\\<^sub>R h t |u. 0 \\ u \\ u \\ 1} \\ S\\ \\ continuous_on ({0..1} \\ {0..1}) (\\y. (1 - fst y) *\\<^sub>R (g \\ snd) y + fst y *\\<^sub>R (h \\ snd) y) \\ (\\y. (1 - fst y) *\\<^sub>R (g \\ snd) y + fst y *\\<^sub>R (h \\ snd) y) ` ({0..1} \\ {0..1}) \\ S \\ (\\x. (1 - fst (0, x)) *\\<^sub>R (g \\ snd) (0, x) + fst (0, x) *\\<^sub>R (h \\ snd) (0, x) = g x) \\ (\\x. (1 - fst (1, x)) *\\<^sub>R (g \\ snd) (1, x) + fst (1, x) *\\<^sub>R (h \\ snd) (1, x) = h x) \\ (\\t\\{0..1}. (1 - fst (t, 0)) *\\<^sub>R (g \\ snd) (t, 0) + fst (t, 0) *\\<^sub>R (h \\ snd) (t, 0) = g 0 \\ (1 - fst (t, 1)) *\\<^sub>R (g \\ snd) (t, 1) + fst (t, 1) *\\<^sub>R (h \\ snd) (t, 1) = g 1)\n[PROOF STEP]\napply (intro conjI subsetI continuous_intros; force)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1351, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942232112239, "lm_q2_score": 0.8031737892899221, "lm_q1q2_score": 0.7150609848394864}} {"text": "[STATEMENT]\nlemma complex_quadratic_equation_monic_only_two_roots:\n fixes \\ :: complex\n assumes \"\\\\<^sup>2 + b * \\ + c = 0\"\n shows \"\\ = (-b + ccsqrt(b\\<^sup>2 - 4*c)) / 2 \\ \\ = (-b - ccsqrt(b\\<^sup>2 - 4*c)) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>2 + b * \\ + c = 0\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\<^sup>2 + b * \\ + c = 0 \\ \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sup>2 + b * \\ + c = 0\n[PROOF STEP]\nhave \"(2 * (\\ + b/2))\\<^sup>2 = b\\<^sup>2 - 4*c\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>2 + b * \\ + c = 0\n\ngoal (1 subgoal):\n 1. (2 * (\\ + b / 2))\\<^sup>2 = b\\<^sup>2 - 4 * c\n[PROOF STEP]\nby (simp add: power2_eq_square field_simps)\n (metis (no_types, lifting) distrib_right_numeral mult.assoc mult_zero_left)\n[PROOF STATE]\nproof (state)\nthis:\n(2 * (\\ + b / 2))\\<^sup>2 = b\\<^sup>2 - 4 * c\n\ngoal (1 subgoal):\n 1. \\\\<^sup>2 + b * \\ + c = 0 \\ \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n[PROOF STEP]\nhence \"2 * (\\ + b/2) = ccsqrt (b\\<^sup>2 - 4*c) \\ 2 * (\\ + b/2) = - ccsqrt (b\\<^sup>2 - 4*c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(2 * (\\ + b / 2))\\<^sup>2 = b\\<^sup>2 - 4 * c\n\ngoal (1 subgoal):\n 1. 2 * (\\ + b / 2) = ccsqrt (b\\<^sup>2 - 4 * c) \\ 2 * (\\ + b / 2) = - ccsqrt (b\\<^sup>2 - 4 * c)\n[PROOF STEP]\nusing ccsqrt[of \"(2 * (\\ + b / 2))\" \"b\\<^sup>2 - 4 * c\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(2 * (\\ + b / 2))\\<^sup>2 = b\\<^sup>2 - 4 * c\n2 * (\\ + b / 2) * (2 * (\\ + b / 2)) = b\\<^sup>2 - 4 * c \\ 2 * (\\ + b / 2) = ccsqrt (b\\<^sup>2 - 4 * c) \\ 2 * (\\ + b / 2) = - ccsqrt (b\\<^sup>2 - 4 * c)\n\ngoal (1 subgoal):\n 1. 2 * (\\ + b / 2) = ccsqrt (b\\<^sup>2 - 4 * c) \\ 2 * (\\ + b / 2) = - ccsqrt (b\\<^sup>2 - 4 * c)\n[PROOF STEP]\nby (simp add: power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n2 * (\\ + b / 2) = ccsqrt (b\\<^sup>2 - 4 * c) \\ 2 * (\\ + b / 2) = - ccsqrt (b\\<^sup>2 - 4 * c)\n\ngoal (1 subgoal):\n 1. \\\\<^sup>2 + b * \\ + c = 0 \\ \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * (\\ + b / 2) = ccsqrt (b\\<^sup>2 - 4 * c) \\ 2 * (\\ + b / 2) = - ccsqrt (b\\<^sup>2 - 4 * c)\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n[PROOF STEP]\nusing mult_cancel_right[of \"b + \\ * 2\" 2 \"ccsqrt (b\\<^sup>2 - 4*c)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * (\\ + b / 2) = ccsqrt (b\\<^sup>2 - 4 * c) \\ 2 * (\\ + b / 2) = - ccsqrt (b\\<^sup>2 - 4 * c)\n((b + \\ * 2) * 2 = ccsqrt (b\\<^sup>2 - 4 * c) * 2) = (2 = 0 \\ b + \\ * 2 = ccsqrt (b\\<^sup>2 - 4 * c))\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n[PROOF STEP]\nusing mult_cancel_right[of \"b + \\ * 2\" 2 \"-ccsqrt (b\\<^sup>2 - 4*c)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * (\\ + b / 2) = ccsqrt (b\\<^sup>2 - 4 * c) \\ 2 * (\\ + b / 2) = - ccsqrt (b\\<^sup>2 - 4 * c)\n((b + \\ * 2) * 2 = ccsqrt (b\\<^sup>2 - 4 * c) * 2) = (2 = 0 \\ b + \\ * 2 = ccsqrt (b\\<^sup>2 - 4 * c))\n((b + \\ * 2) * 2 = - ccsqrt (b\\<^sup>2 - 4 * c) * 2) = (2 = 0 \\ b + \\ * 2 = - ccsqrt (b\\<^sup>2 - 4 * c))\n\ngoal (1 subgoal):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n[PROOF STEP]\nby (auto simp add: field_simps) (metis add_diff_cancel diff_minus_eq_add minus_diff_eq)\n[PROOF STATE]\nproof (state)\nthis:\n\\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2274, "file": "Complex_Geometry_Quadratic", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511616741041, "lm_q2_score": 0.8354835452961425, "lm_q1q2_score": 0.7150495628013026}} {"text": "[STATEMENT]\nlemma floor_zero [simp]: \"\\0\\ = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nusing floor_of_int [of 0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\of_int 0\\ = 0\n\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 156, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511616741042, "lm_q2_score": 0.8354835391516133, "lm_q1q2_score": 0.7150495575425002}} {"text": "[STATEMENT]\nlemma bag_of_nths_Un_Int:\n \"bag_of (nths l (A Un B)) + bag_of (nths l (A Int B)) = \n bag_of (nths l A) + bag_of (nths l B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mset (nths l (A \\ B)) + mset (nths l (A \\ B)) = mset (nths l A) + mset (nths l B)\n[PROOF STEP]\napply (subgoal_tac \"A Int B Int {.. B \\ {.. {.. (B \\ {.. mset (nths l (A \\ B)) + mset (nths l (A \\ B)) = mset (nths l A) + mset (nths l B)\n 2. A \\ B \\ {.. {.. (B \\ {.. {1..<1 + k}) = fact (k :: nat)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (\\{1..<1 + k}) = fact k\n[PROOF STEP]\nby (simp add: fact_prod atLeastLessThanSuc_atLeastAtMost)", "meta": {"llama_tokens": 104, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9196425245706048, "lm_q2_score": 0.7772998611746912, "lm_q1q2_score": 0.7148380066790736}} {"text": "[STATEMENT]\nlemma associativity:\n fixes M1 M2 M3\n shows\n \" (mat (row_length M1) (length M1) M1) \n \\ (mat (row_length M2) (length M2) M2)\n \\ (mat (row_length M3) (length M3) M3)\n \\\n M1 \\ (M2 \\ M3) = (M1 \\ M2) \\ M3\" (is \"?x \\?l = ?r\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nfix j\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nassume 0:\" (mat (row_length M1) (length M1) M1) \n \\ (mat (row_length M2) (length M2) M2)\n \\ (mat (row_length M3) (length M3) M3)\"\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave 1:\"length ((M1 \\ M2) \\ M3) \n = (length M1)*(length M2)* (length M3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. length (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\n[PROOF STEP]\nhave \"length (M2 \\ M3) = (length M2)* (length M3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (M2 \\ M3) = length M2 * length M3\n[PROOF STEP]\nby (metis length_Tensor)\n[PROOF STATE]\nproof (state)\nthis:\nlength (M2 \\ M3) = length M2 * length M3\n\ngoal (1 subgoal):\n 1. length (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength (M2 \\ M3) = length M2 * length M3\n[PROOF STEP]\nhave \"length (M1 \\ (M2 \\ M3)) \n = (length M1)*(length M2)* (length M3)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (M2 \\ M3) = length M2 * length M3\n\ngoal (1 subgoal):\n 1. length (M1 \\ (M2 \\ M3)) = length M1 * length M2 * length M3\n[PROOF STEP]\nusing mult.assoc length_Tensor\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (M2 \\ M3) = length M2 * length M3\nMatrix_Tensor.mult ?id ?f \\ ?f (?f ?a ?b) ?c = ?f ?a (?f ?b ?c)\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\n\ngoal (1 subgoal):\n 1. length (M1 \\ (M2 \\ M3)) = length M1 * length M2 * length M3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength (M1 \\ (M2 \\ M3)) = length M1 * length M2 * length M3\n\ngoal (1 subgoal):\n 1. length (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nlength (M1 \\ (M2 \\ M3)) = length M1 * length M2 * length M3\n\ngoal (1 subgoal):\n 1. length (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\n[PROOF STEP]\nhave \" length (M1 \\ M2) = (length M1)* (length M2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (M1 \\ M2) = length M1 * length M2\n[PROOF STEP]\nby (metis length_Tensor)\n[PROOF STATE]\nproof (state)\nthis:\nlength (M1 \\ M2) = length M1 * length M2\n\ngoal (1 subgoal):\n 1. length (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nlength (M1 \\ (M2 \\ M3)) = length M1 * length M2 * length M3\nlength (M1 \\ M2) = length M1 * length M2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (M1 \\ (M2 \\ M3)) = length M1 * length M2 * length M3\nlength (M1 \\ M2) = length M1 * length M2\n\ngoal (1 subgoal):\n 1. length (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\n[PROOF STEP]\nusing mult.assoc length_Tensor\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (M1 \\ (M2 \\ M3)) = length M1 * length M2 * length M3\nlength (M1 \\ M2) = length M1 * length M2\nMatrix_Tensor.mult ?id ?f \\ ?f (?f ?a ?b) ?c = ?f ?a (?f ?b ?c)\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\n\ngoal (1 subgoal):\n 1. length (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nlength (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave 2:\"row_length ((M1 \\ M2) \\ M3) \n = (row_length M1)*(row_length M2)* (row_length M3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. row_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n[PROOF STEP]\nhave \"row_length (M2 \\ M3) = (row_length M2)* (row_length M3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row_length (M2 \\ M3) = row_length M2 * row_length M3\n[PROOF STEP]\nusing row_length_mat assoc\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\n?a * ?b * ?c = ?a * (?b * ?c)\n\ngoal (1 subgoal):\n 1. row_length (M2 \\ M3) = row_length M2 * row_length M3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrow_length (M2 \\ M3) = row_length M2 * row_length M3\n\ngoal (1 subgoal):\n 1. row_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nrow_length (M2 \\ M3) = row_length M2 * row_length M3\n[PROOF STEP]\nhave \"row_length (M1 \\ (M2 \\ M3)) \n = (row_length M1)*(row_length M2)* (row_length M3)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length (M2 \\ M3) = row_length M2 * row_length M3\n\ngoal (1 subgoal):\n 1. row_length (M1 \\ (M2 \\ M3)) = row_length M1 * row_length M2 * row_length M3\n[PROOF STEP]\nusing row_length_mat assoc\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length (M2 \\ M3) = row_length M2 * row_length M3\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\n?a * ?b * ?c = ?a * (?b * ?c)\n\ngoal (1 subgoal):\n 1. row_length (M1 \\ (M2 \\ M3)) = row_length M1 * row_length M2 * row_length M3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrow_length (M1 \\ (M2 \\ M3)) = row_length M1 * row_length M2 * row_length M3\n\ngoal (1 subgoal):\n 1. row_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nrow_length (M1 \\ (M2 \\ M3)) = row_length M1 * row_length M2 * row_length M3\n\ngoal (1 subgoal):\n 1. row_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n[PROOF STEP]\nhave \" row_length (M1 \\ M2) \n = (row_length M1)* (row_length M2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row_length (M1 \\ M2) = row_length M1 * row_length M2\n[PROOF STEP]\nusing row_length_mat\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\n\ngoal (1 subgoal):\n 1. row_length (M1 \\ M2) = row_length M1 * row_length M2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrow_length (M1 \\ M2) = row_length M1 * row_length M2\n\ngoal (1 subgoal):\n 1. row_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nrow_length (M1 \\ (M2 \\ M3)) = row_length M1 * row_length M2 * row_length M3\nrow_length (M1 \\ M2) = row_length M1 * row_length M2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length (M1 \\ (M2 \\ M3)) = row_length M1 * row_length M2 * row_length M3\nrow_length (M1 \\ M2) = row_length M1 * row_length M2\n\ngoal (1 subgoal):\n 1. row_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n[PROOF STEP]\nusing row_length_mat assoc\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length (M1 \\ (M2 \\ M3)) = row_length M1 * row_length M2 * row_length M3\nrow_length (M1 \\ M2) = row_length M1 * row_length M2\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\n?a * ?b * ?c = ?a * (?b * ?c)\n\ngoal (1 subgoal):\n 1. row_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrow_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nrow_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave 3:\n \"\\i.\\j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n (((M1 \\ M2) \\ M3)!j!i) \n = f \n ((M1 \\ M2)!(j div (length M3))!(i div (row_length M3))) \n (M3!(j mod length M3)!(i mod (row_length M3))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nusing 0 matrix_Tensor_elements 1 2 effective_well_defined_Tensor \n length_Tensor row_length_mat\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\\i j. (i < row_length ?M1.0 * row_length ?M2.0 \\ j < length ?M1.0 * length ?M2.0) \\ mat (row_length ?M1.0) (length ?M1.0) ?M1.0 \\ mat (row_length ?M2.0) (length ?M2.0) ?M2.0 \\ (?M1.0 \\ ?M2.0) ! j ! i = ?M1.0 ! (j div length ?M2.0) ! (i div row_length ?M2.0) * ?M2.0 ! (j mod length ?M2.0) ! (i mod row_length ?M2.0)\nlength (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\nrow_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n\\mat (row_length ?M1.0) (length ?M1.0) ?M1.0; mat (row_length ?M2.0) (length ?M2.0) ?M2.0\\ \\ mat (row_length ?M1.0 * row_length ?M2.0) (length ?M1.0 * length ?M2.0) (?M1.0 \\ ?M2.0)\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave \n \"\\j.(j < (length M1)*(length M2)*(length M3)) \n \\ (j div (length M3)) < (length M1)*(length M2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\jj. j < length M1 * length M2 * length M3 \\ j div length M3 < length M1 * length M2\n[PROOF STEP]\napply(simp add:div_left_ineq)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n\\j mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\j mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave \"\\i.(i < (row_length M1)*(row_length M2)*(row_length M3)) \n \\ (i div (row_length M3)) \n < (row_length M1)*(row_length M2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ii. i < row_length M1 * row_length M2 * row_length M3 \\ i div row_length M3 < row_length M1 * row_length M2\n[PROOF STEP]\napply(simp add:div_left_ineq)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n\\i mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\\jii.\\j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n ((i div (row_length M3)) < (row_length M1)*(row_length M2))\n \\ ((j div (length M3)) < (length M1)*(length M2)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\\jii j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i div row_length M3 < row_length M1 * row_length M2 \\ j div length M3 < length M1 * length M2\n[PROOF STEP]\nusing allI 0\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\\jix. ?P x) \\ \\x. ?P x\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i div row_length M3 < row_length M1 * row_length M2 \\ j div length M3 < length M1 * length M2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i div row_length M3 < row_length M1 * row_length M2 \\ j div length M3 < length M1 * length M2\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave \" (mat (row_length M1) (length M1) M1) \n \\ (mat (row_length M2) (length M2) M2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2\n[PROOF STEP]\nusing 0\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2\n[PROOF STEP]\nhave \"\\i.\\j.(((i div (row_length M3)) < (row_length M1)*(row_length M2))\n \\ ((j div (length M3)) < (length M1)*(length M2))\n \\\n (((M1 \\ M2))!(j div (length M3))!(i div row_length M3)) \n = f \n ((M1)!((j div (length M3)) div (length M2))\n !((i div (row_length M3)) div (row_length M2))) \n (M2!((j div (length M3)) mod (length M2))\n !((i div (row_length M3)) mod (row_length M2))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2\n\ngoal (1 subgoal):\n 1. \\i j. i div row_length M3 < row_length M1 * row_length M2 \\ j div length M3 < length M1 * length M2 \\ (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2)\n[PROOF STEP]\nusing effective_matrix_tensor_elements\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2\n(?i < row_length ?M1.0 * row_length ?M2.0 \\ ?j < length ?M1.0 * length ?M2.0) \\ mat (row_length ?M1.0) (length ?M1.0) ?M1.0 \\ mat (row_length ?M2.0) (length ?M2.0) ?M2.0 \\ (?M1.0 \\ ?M2.0) ! ?j ! ?i = ?M1.0 ! (?j div length ?M2.0) ! (?i div row_length ?M2.0) * ?M2.0 ! (?j mod length ?M2.0) ! (?i mod row_length ?M2.0)\n\ngoal (1 subgoal):\n 1. \\i j. i div row_length M3 < row_length M1 * row_length M2 \\ j div length M3 < length M1 * length M2 \\ (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i div row_length M3 < row_length M1 * row_length M2 \\ j div length M3 < length M1 * length M2 \\ (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nwith 4\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i div row_length M3 < row_length M1 * row_length M2 \\ j div length M3 < length M1 * length M2\n\\i j. i div row_length M3 < row_length M1 * row_length M2 \\ j div length M3 < length M1 * length M2 \\ (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2)\n[PROOF STEP]\nhave 5:\"\\i j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ (((M1 \\ M2))!(j div (length M3))!(i div row_length M3)) \n = f \n ((M1)!((j div (length M3)) div (length M2))\n !((i div (row_length M3)) div (row_length M2))) \n (M2!((j div (length M3)) mod (length M2))\n !((i div (row_length M3)) mod (row_length M2))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i div row_length M3 < row_length M1 * row_length M2 \\ j div length M3 < length M1 * length M2\n\\i j. i div row_length M3 < row_length M1 * row_length M2 \\ j div length M3 < length M1 * length M2 \\ (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2)\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nwith 3\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2)\n[PROOF STEP]\nhave 6:\n \"\\i.\\j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n (((M1 \\ M2) \\ M3)!j!i) \n = f \n (f \n ((M1)!((j div (length M3)) div (length M2))\n !((i div (row_length M3)) div (row_length M2))) \n (M2!((j div (length M3)) mod (length M2))\n !((i div (row_length M3)) mod (row_length M2)))) \n (M3!(j mod length M3)!(i mod (row_length M3))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2) ! (j div length M3) ! (i div row_length M3) = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2)\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave \"(j div (length M3))div (length M2) = (j div ((length M3)*(length M2)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j div length M3 div length M2 = j div (length M3 * length M2)\n[PROOF STEP]\nusing div_mult2_eq\n[PROOF STATE]\nproof (prove)\nusing this:\n?m div (?n * ?q) = ?m div ?n div ?q\n\ngoal (1 subgoal):\n 1. j div length M3 div length M2 = j div (length M3 * length M2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nj div length M3 div length M2 = j div (length M3 * length M2)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nj div length M3 div length M2 = j div (length M3 * length M2)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave \"((i div (row_length M3)) div (row_length M2)) = (i div ((row_length M3)*(row_length M2)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i div row_length M3 div row_length M2 = i div (row_length M3 * row_length M2)\n[PROOF STEP]\nusing div_mult2_eq\n[PROOF STATE]\nproof (prove)\nusing this:\n?m div (?n * ?q) = ?m div ?n div ?q\n\ngoal (1 subgoal):\n 1. i div row_length M3 div row_length M2 = i div (row_length M3 * row_length M2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ni div row_length M3 div row_length M2 = i div (row_length M3 * row_length M2)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nj div length M3 div length M2 = j div (length M3 * length M2)\ni div row_length M3 div row_length M2 = i div (row_length M3 * row_length M2)\n[PROOF STEP]\nhave step1:\"\\i.\\j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n (((M1 \\ M2) \\ M3)!j!i) \n = f \n (f \n ((M1)!(j div ((length M3)*(length M2)))! (i div ((row_length M3)*(row_length M2)))) \n (M2!((j div (length M3)) mod (length M2))!((i div (row_length M3)) mod (row_length M2)))) \n (M3!(j mod length M3)!(i mod (row_length M3))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nj div length M3 div length M2 = j div (length M3 * length M2)\ni div row_length M3 div row_length M2 = i div (row_length M3 * row_length M2)\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div (length M3 * length M2)) ! (i div (row_length M3 * row_length M2)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nusing 6\n[PROOF STATE]\nproof (prove)\nusing this:\nj div length M3 div length M2 = j div (length M3 * length M2)\ni div row_length M3 div row_length M2 = i div (row_length M3 * row_length M2)\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div (length M3 * length M2)) ! (i div (row_length M3 * row_length M2)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nby (metis \"3\" \"5\" div_mult2_eq)\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div (length M3 * length M2)) ! (i div (row_length M3 * row_length M2)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div (length M3 * length M2)) ! (i div (row_length M3 * row_length M2)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nhave step1:\"\\i j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n (((M1 \\ M2) \\ M3)!j!i) \n = f \n (f \n ((M1)!(j div ((length M2)*(length M3)))! (i div ((row_length M2)*(row_length M3)))) \n (M2!((j div (length M3)) mod (length M2))!((i div (row_length M3)) mod (row_length M2)))) \n (M3!(j mod length M3)!(i mod (row_length M3))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div (length M3 * length M2)) ! (i div (row_length M3 * row_length M2)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nby (metis mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave 7:\n \"\\i.\\j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n ((M1 \\ (M2 \\ M3))!j!i) \n = f \n ((M1)!(j div (length (M2 \\ M3)))!(i div (row_length (M2 \\ M3)))) \n ((M2 \\ M3)!(j mod length (M2 \\M3))!(i mod (row_length (M2 \\ M3)))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div length (M2 \\ M3)) ! (i div row_length (M2 \\ M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n[PROOF STEP]\nusing 0 matrix_Tensor_elements 1 2 effective_well_defined_Tensor \n length_Tensor row_length_mat\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\\i j. (i < row_length ?M1.0 * row_length ?M2.0 \\ j < length ?M1.0 * length ?M2.0) \\ mat (row_length ?M1.0) (length ?M1.0) ?M1.0 \\ mat (row_length ?M2.0) (length ?M2.0) ?M2.0 \\ (?M1.0 \\ ?M2.0) ! j ! i = ?M1.0 ! (j div length ?M2.0) ! (i div row_length ?M2.0) * ?M2.0 ! (j mod length ?M2.0) ! (i mod row_length ?M2.0)\nlength (M1 \\ M2 \\ M3) = length M1 * length M2 * length M3\nrow_length (M1 \\ M2 \\ M3) = row_length M1 * row_length M2 * row_length M3\n\\mat (row_length ?M1.0) (length ?M1.0) ?M1.0; mat (row_length ?M2.0) (length ?M2.0) ?M2.0\\ \\ mat (row_length ?M1.0 * row_length ?M2.0) (length ?M1.0 * length ?M2.0) (?M1.0 \\ ?M2.0)\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div length (M2 \\ M3)) ! (i div row_length (M2 \\ M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div length (M2 \\ M3)) ! (i div row_length (M2 \\ M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div length (M2 \\ M3)) ! (i div row_length (M2 \\ M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n[PROOF STEP]\nhave \n \"\\i.\\j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n ((M1 \\ (M2 \\ M3))!j!i) \n = f \n ((M1)!(j div ((length M2)*(length M3)))!(i div ((row_length M2)*(row_length M3)))) \n ((M2 \\ M3)!(j mod length (M2 \\M3))!(i mod (row_length (M2 \\ M3)))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div length (M2 \\ M3)) ! (i div row_length (M2 \\ M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n[PROOF STEP]\nusing length_Tensor row_length_mat\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div length (M2 \\ M3)) ! (i div row_length (M2 \\ M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n[PROOF STEP]\nhave \n \"\\i.\\j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n ((M1 \\ (M2 \\ M3))!j!i) \n = f \n ((M1)!(j div ((length M3)*(length M2)))\n !(i div ((row_length M3)*(row_length M2)))) \n ((M2 \\ M3)!(j mod length (M2 \\M3))\n !(i mod (row_length (M2 \\ M3)))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M3 * length M2)) ! (i div (row_length M3 * row_length M2)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n[PROOF STEP]\nusing mult.commute\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n?a * ?b = ?b * ?a\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M3 * length M2)) ! (i div (row_length M3 * row_length M2)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n[PROOF STEP]\nby (metis)\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M3 * length M2)) ! (i div (row_length M3 * row_length M2)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave 8:\n \"\\j.((j < (length M1)*(length M2)*(length M3)))\n \\ (j mod (length (M2 \\ M3))) < (length (M2 \\ M3))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\nproof(cases \"length (M2 \\ M3) = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. length (M2 \\ M3) = 0 \\ \\j M3) < length (M2 \\ M3)\n 2. length (M2 \\ M3) \\ 0 \\ \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nlength (M2 \\ M3) = 0\n\ngoal (2 subgoals):\n 1. length (M2 \\ M3) = 0 \\ \\j M3) < length (M2 \\ M3)\n 2. length (M2 \\ M3) \\ 0 \\ \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\nhave \"(length M2)*(length M3) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length M2 * length M3 = 0\n[PROOF STEP]\nusing length_Tensor True\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\nlength (M2 \\ M3) = 0\n\ngoal (1 subgoal):\n 1. length M2 * length M3 = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength M2 * length M3 = 0\n\ngoal (2 subgoals):\n 1. length (M2 \\ M3) = 0 \\ \\j M3) < length (M2 \\ M3)\n 2. length (M2 \\ M3) \\ 0 \\ \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength M2 * length M3 = 0\n[PROOF STEP]\nhave \"(length M1)*(length M2)*(length M3) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength M2 * length M3 = 0\n\ngoal (1 subgoal):\n 1. length M1 * length M2 * length M3 = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength M1 * length M2 * length M3 = 0\n\ngoal (2 subgoals):\n 1. length (M2 \\ M3) = 0 \\ \\j M3) < length (M2 \\ M3)\n 2. length (M2 \\ M3) \\ 0 \\ \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength M1 * length M2 * length M3 = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlength M1 * length M2 * length M3 = 0\n\ngoal (1 subgoal):\n 1. \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\nby (metis less_nat_zero_code)\n[PROOF STATE]\nproof (state)\nthis:\n\\j M3) < length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. length (M2 \\ M3) \\ 0 \\ \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. length (M2 \\ M3) \\ 0 \\ \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nlength (M2 \\ M3) \\ 0\n\ngoal (1 subgoal):\n 1. length (M2 \\ M3) \\ 0 \\ \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\nhave \"length (M2 \\ M3) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < length (M2 \\ M3)\n[PROOF STEP]\nusing False\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (M2 \\ M3) \\ 0\n\ngoal (1 subgoal):\n 1. 0 < length (M2 \\ M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. length (M2 \\ M3) \\ 0 \\ \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < length (M2 \\ M3)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\nusing mod_less_divisor\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < length (M2 \\ M3)\n0 < ?n \\ ?m mod ?n < ?n\n\ngoal (1 subgoal):\n 1. \\j M3) < length (M2 \\ M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\j M3) < length (M2 \\ M3)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\j M3) < length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\j M3) < length (M2 \\ M3)\n[PROOF STEP]\nhave 9:\n \"\\i.((i < (row_length M1)*(row_length M2)*(row_length M3)))\n \\ (i mod (row_length (M2 \\ M3))) < (row_length (M2 \\ M3))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\j M3) < length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\nproof(cases \"row_length (M2 \\ M3) = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) = 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n 2. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) \\ 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nrow_length (M2 \\ M3) = 0\n\ngoal (2 subgoals):\n 1. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) = 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n 2. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) \\ 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\nhave \"(row_length M2)*(row_length M3) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row_length M2 * row_length M3 = 0\n[PROOF STEP]\nusing True\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length (M2 \\ M3) = 0\n\ngoal (1 subgoal):\n 1. row_length M2 * row_length M3 = 0\n[PROOF STEP]\nby (metis row_length_mat)\n[PROOF STATE]\nproof (state)\nthis:\nrow_length M2 * row_length M3 = 0\n\ngoal (2 subgoals):\n 1. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) = 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n 2. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) \\ 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nrow_length M2 * row_length M3 = 0\n[PROOF STEP]\nhave \"(row_length M1)*(row_length M2)*(row_length M3) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length M2 * row_length M3 = 0\n\ngoal (1 subgoal):\n 1. row_length M1 * row_length M2 * row_length M3 = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrow_length M1 * row_length M2 * row_length M3 = 0\n\ngoal (2 subgoals):\n 1. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) = 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n 2. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) \\ 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nrow_length M1 * row_length M2 * row_length M3 = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length M1 * row_length M2 * row_length M3 = 0\n\ngoal (1 subgoal):\n 1. \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\nby (metis less_nat_zero_code)\n[PROOF STATE]\nproof (state)\nthis:\n\\i M3) < row_length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) \\ 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) \\ 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nrow_length (M2 \\ M3) \\ 0\n\ngoal (1 subgoal):\n 1. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) \\ 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\nhave \"row_length (M2 \\ M3) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < row_length (M2 \\ M3)\n[PROOF STEP]\nusing False\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length (M2 \\ M3) \\ 0\n\ngoal (1 subgoal):\n 1. 0 < row_length (M2 \\ M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < row_length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. \\\\j M3) < length (M2 \\ M3); row_length (M2 \\ M3) \\ 0\\ \\ \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < row_length (M2 \\ M3)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < row_length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\nusing mod_less_divisor\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < row_length (M2 \\ M3)\n0 < ?n \\ ?m mod ?n < ?n\n\ngoal (1 subgoal):\n 1. \\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i M3) < row_length (M2 \\ M3)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\i M3) < row_length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nwith 8\n[PROOF STATE]\nproof (chain)\npicking this:\n\\j M3) < length (M2 \\ M3)\n\\i M3) < row_length (M2 \\ M3)\n[PROOF STEP]\nhave 10:\"\\i.\\j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n (i mod (row_length (M2 \\ M3))) < (row_length (M2 \\ M3))\n \\ (j mod (length (M2 \\ M3))) < (length (M2 \\ M3)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\j M3) < length (M2 \\ M3)\n\\i M3) < row_length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i mod row_length (M2 \\ M3) < row_length (M2 \\ M3) \\ j mod length (M2 \\ M3) < length (M2 \\ M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i mod row_length (M2 \\ M3) < row_length (M2 \\ M3) \\ j mod length (M2 \\ M3) < length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i mod row_length (M2 \\ M3) < row_length (M2 \\ M3) \\ j mod length (M2 \\ M3) < length (M2 \\ M3)\n[PROOF STEP]\nhave 11:\"\\ i j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n (i mod (row_length (M2 \\ M3))) \n < (row_length M2)*(row_length M3)\n \\(j mod (length (M2 \\ M3))) < (length M2)*(length M3))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i mod row_length (M2 \\ M3) < row_length (M2 \\ M3) \\ j mod length (M2 \\ M3) < length (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i mod row_length (M2 \\ M3) < row_length M2 * row_length M3 \\ j mod length (M2 \\ M3) < length M2 * length M3\n[PROOF STEP]\nusing length_Tensor row_length_mat\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i mod row_length (M2 \\ M3) < row_length (M2 \\ M3) \\ j mod length (M2 \\ M3) < length (M2 \\ M3)\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i mod row_length (M2 \\ M3) < row_length M2 * row_length M3 \\ j mod length (M2 \\ M3) < length M2 * length M3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i mod row_length (M2 \\ M3) < row_length M2 * row_length M3 \\ j mod length (M2 \\ M3) < length M2 * length M3\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave \"(mat (row_length M2) (length M2) M2) \n \\ (mat (row_length M3) (length M3) M3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n[PROOF STEP]\nusing 0\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\ngoal (1 subgoal):\n 1. mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nmat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n[PROOF STEP]\nhave \"\\ i j.(((i mod (row_length (M2 \\ M3))) \n < (row_length M2)*(row_length M3))\n \\((j mod (length (M2\\M3))) < (length M2)*(length M3))\n \\\n (((M2 \\ M3))!(j mod (length (M2 \\ M3)))!(i mod row_length (M2 \\ M3))) \n = f \n ((M2)!((j mod (length (M2 \\ M3))) div (length M3))\n !((i mod (row_length (M2 \\ M3))) div (row_length M3))) \n (M3!((j mod (length (M2 \\ M3))) mod (length M3))\n !((i mod (row_length (M2 \\ M3))) mod (row_length M3))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\ngoal (1 subgoal):\n 1. \\i j. i mod row_length (M2 \\ M3) < row_length M2 * row_length M3 \\ j mod length (M2 \\ M3) < length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n[PROOF STEP]\nusing matrix_Tensor_elements\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\\i j. (i < row_length ?M1.0 * row_length ?M2.0 \\ j < length ?M1.0 * length ?M2.0) \\ mat (row_length ?M1.0) (length ?M1.0) ?M1.0 \\ mat (row_length ?M2.0) (length ?M2.0) ?M2.0 \\ (?M1.0 \\ ?M2.0) ! j ! i = ?M1.0 ! (j div length ?M2.0) ! (i div row_length ?M2.0) * ?M2.0 ! (j mod length ?M2.0) ! (i mod row_length ?M2.0)\n\ngoal (1 subgoal):\n 1. \\i j. i mod row_length (M2 \\ M3) < row_length M2 * row_length M3 \\ j mod length (M2 \\ M3) < length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i mod row_length (M2 \\ M3) < row_length M2 * row_length M3 \\ j mod length (M2 \\ M3) < length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i mod row_length (M2 \\ M3) < row_length M2 * row_length M3 \\ j mod length (M2 \\ M3) < length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n[PROOF STEP]\nhave \"\\ i j.\n ((i < (row_length M1)*(row_length M2)*(row_length M3))\n \\(j < (length M1)*(length M2)*(length M3) )\n \\ \n (((M2 \\ M3))!(j mod (length (M2 \\ M3)))\n !(i mod row_length (M2 \\ M3))) \n = \n f \n ((M2)!((j mod (length (M2 \\ M3))) div (length M3))\n !((i mod (row_length (M2 \\ M3))) div (row_length M3))) \n (M3!((j mod (length (M2 \\ M3))) mod (length M3))\n !((i mod (row_length (M2 \\ M3))) mod (row_length M3))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i mod row_length (M2 \\ M3) < row_length M2 * row_length M3 \\ j mod length (M2 \\ M3) < length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n[PROOF STEP]\nusing 11\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i mod row_length (M2 \\ M3) < row_length M2 * row_length M3 \\ j mod length (M2 \\ M3) < length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ i mod row_length (M2 \\ M3) < row_length M2 * row_length M3 \\ j mod length (M2 \\ M3) < length M2 * length M3\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n[PROOF STEP]\nhave \"\\j.(j mod (length (M2 \\ M3))) mod (length M3)\n = j mod (length M3)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n\ngoal (1 subgoal):\n 1. \\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\j. ?x < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (?x mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (?x mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (?x mod row_length (M2 \\ M3) mod row_length M3) \\ \\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n[PROOF STEP]\nhave \"\\j.((j mod (length (M2 \\ M3))) \n = (j mod ((length M2) *(length M3))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n[PROOF STEP]\nusing length_Tensor\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\n\ngoal (1 subgoal):\n 1. \\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n\ngoal (1 subgoal):\n 1. \\j. ?x < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (?x mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (?x mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (?x mod row_length (M2 \\ M3) mod row_length M3) \\ \\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n\ngoal (1 subgoal):\n 1. \\j. ?x < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (?x mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (?x mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (?x mod row_length (M2 \\ M3) mod row_length M3) \\ \\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n[PROOF STEP]\nhave \n \"\\j.((j mod ((length M2) *(length M3))) mod (length M3)\n = (j mod (length M3)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j mod (length M2 * length M3) mod length M3 = j mod length M3\n[PROOF STEP]\nusing mod_prop1\n[PROOF STATE]\nproof (prove)\nusing this:\n?a mod (?b * ?c) mod ?c = ?a mod ?c\n\ngoal (1 subgoal):\n 1. \\j. j mod (length M2 * length M3) mod length M3 = j mod length M3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\j. j mod (length M2 * length M3) mod length M3 = j mod length M3\n\ngoal (1 subgoal):\n 1. \\j. ?x < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (?x mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (?x mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (?x mod row_length (M2 \\ M3) mod row_length M3) \\ \\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n\\j. j mod (length M2 * length M3) mod length M3 = j mod length M3\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n\\j. j mod (length M2 * length M3) mod length M3 = j mod length M3\n\ngoal (1 subgoal):\n 1. \\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n[PROOF STEP]\nhave \"\\i.(i mod (row_length (M2 \\ M3))) mod (row_length M3)\n = i mod (row_length M3)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n\ngoal (1 subgoal):\n 1. \\i. i mod row_length (M2 \\ M3) mod row_length M3 = i mod row_length M3\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ?x mod length (M2 \\ M3) mod length M3 = ?x mod length M3 \\ \\i. i mod row_length (M2 \\ M3) mod row_length M3 = i mod row_length M3\n[PROOF STEP]\nhave \"\\i.((i mod (row_length (M2 \\ M3))) \n = (i mod ((row_length M2) *(row_length M3))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n[PROOF STEP]\nusing row_length_mat\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\n\ngoal (1 subgoal):\n 1. \\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n\ngoal (1 subgoal):\n 1. ?x mod length (M2 \\ M3) mod length M3 = ?x mod length M3 \\ \\i. i mod row_length (M2 \\ M3) mod row_length M3 = i mod row_length M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n\ngoal (1 subgoal):\n 1. ?x mod length (M2 \\ M3) mod length M3 = ?x mod length M3 \\ \\i. i mod row_length (M2 \\ M3) mod row_length M3 = i mod row_length M3\n[PROOF STEP]\nhave \"\\i.((i mod ((row_length M2)*(row_length M3))) \n mod (row_length M3)\n = (i mod (row_length M3)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i mod (row_length M2 * row_length M3) mod row_length M3 = i mod row_length M3\n[PROOF STEP]\nusing mod_prop1\n[PROOF STATE]\nproof (prove)\nusing this:\n?a mod (?b * ?c) mod ?c = ?a mod ?c\n\ngoal (1 subgoal):\n 1. \\i. i mod (row_length M2 * row_length M3) mod row_length M3 = i mod row_length M3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i. i mod (row_length M2 * row_length M3) mod row_length M3 = i mod row_length M3\n\ngoal (1 subgoal):\n 1. ?x mod length (M2 \\ M3) mod length M3 = ?x mod length M3 \\ \\i. i mod row_length (M2 \\ M3) mod row_length M3 = i mod row_length M3\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n\\i. i mod (row_length M2 * row_length M3) mod row_length M3 = i mod row_length M3\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n\\i. i mod (row_length M2 * row_length M3) mod row_length M3 = i mod row_length M3\n\ngoal (1 subgoal):\n 1. \\i. i mod row_length (M2 \\ M3) mod row_length M3 = i mod row_length M3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i. i mod row_length (M2 \\ M3) mod row_length M3 = i mod row_length M3\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\i. i mod row_length (M2 \\ M3) mod row_length M3 = i mod row_length M3\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n\\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n\\i. i mod row_length (M2 \\ M3) mod row_length M3 = i mod row_length M3\n[PROOF STEP]\nhave 12:\"\\ i j.((i < (row_length M1)\n *(row_length M2)\n *(row_length M3))\n \\(j < (length M1)*(length M2)*(length M3) )\n \\ \n (((M2 \\ M3))!(j mod (length (M2 \\ M3)))\n !(i mod row_length (M2 \\ M3))) \n = f \n ((M2)!((j mod (length (M2 \\ M3))) div (length M3))\n !((i mod (row_length (M2 \\ M3))) div (row_length M3))) \n (M3!(j mod (length M3))!(i mod (row_length M3))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length (M2 \\ M3) mod length M3) ! (i mod row_length (M2 \\ M3) mod row_length M3)\n\\j. j mod length (M2 \\ M3) mod length M3 = j mod length M3\n\\i. i mod row_length (M2 \\ M3) mod row_length M3 = i mod row_length M3\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave \"\\j.(j mod (length (M2 \\ M3))) div (length M3)\n = (j div (length M3)) mod (length M2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j mod length (M2 \\ M3) div length M3 = j div length M3 mod length M2\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\j. j mod length (M2 \\ M3) div length M3 = j div length M3 mod length M2\n[PROOF STEP]\nhave \"\\j.((j mod (length (M2 \\ M3))) \n = (j mod ((length M2)*(length M3))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n[PROOF STEP]\nusing length_Tensor\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\n\ngoal (1 subgoal):\n 1. \\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n\ngoal (1 subgoal):\n 1. \\j. j mod length (M2 \\ M3) div length M3 = j div length M3 mod length M2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n\ngoal (1 subgoal):\n 1. \\j. j mod length (M2 \\ M3) div length M3 = j div length M3 mod length M2\n[PROOF STEP]\nusing mod_div_relation\n[PROOF STATE]\nproof (prove)\nusing this:\n\\j. j mod length (M2 \\ M3) = j mod (length M2 * length M3)\n?a mod (?b * ?c) div ?c = ?a div ?c mod ?b\n\ngoal (1 subgoal):\n 1. \\j. j mod length (M2 \\ M3) div length M3 = j div length M3 mod length M2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\j. j mod length (M2 \\ M3) div length M3 = j div length M3 mod length M2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\j. j mod length (M2 \\ M3) div length M3 = j div length M3 mod length M2\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\j. j mod length (M2 \\ M3) div length M3 = j div length M3 mod length M2\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave \"\\i.(i mod (row_length (M2 \\ M3))) div (row_length M3)\n = (i div (row_length M3)) mod (row_length M2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i mod row_length (M2 \\ M3) div row_length M3 = i div row_length M3 mod row_length M2\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i. i mod row_length (M2 \\ M3) div row_length M3 = i div row_length M3 mod row_length M2\n[PROOF STEP]\nhave \"\\i.((i mod (row_length (M2 \\ M3))) \n = (i mod ((row_length M2)*(row_length M3))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n[PROOF STEP]\nusing row_length_mat\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\n\ngoal (1 subgoal):\n 1. \\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n\ngoal (1 subgoal):\n 1. \\i. i mod row_length (M2 \\ M3) div row_length M3 = i div row_length M3 mod row_length M2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n\ngoal (1 subgoal):\n 1. \\i. i mod row_length (M2 \\ M3) div row_length M3 = i div row_length M3 mod row_length M2\n[PROOF STEP]\nusing mod_div_relation\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i. i mod row_length (M2 \\ M3) = i mod (row_length M2 * row_length M3)\n?a mod (?b * ?c) div ?c = ?a div ?c mod ?b\n\ngoal (1 subgoal):\n 1. \\i. i mod row_length (M2 \\ M3) div row_length M3 = i div row_length M3 mod row_length M2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i. i mod row_length (M2 \\ M3) div row_length M3 = i div row_length M3 mod row_length M2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\i. i mod row_length (M2 \\ M3) div row_length M3 = i div row_length M3 mod row_length M2\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\\j. j mod length (M2 \\ M3) div length M3 = j div length M3 mod length M2\n\\i. i mod row_length (M2 \\ M3) div row_length M3 = i div row_length M3 mod row_length M2\n[PROOF STEP]\nhave \"\\ i j.\n ((i < (row_length M1)*(row_length M2)*(row_length M3))\n \\(j < (length M1)*(length M2)*(length M3) )\n \\ \n (((M2 \\ M3))!(j mod (length (M2 \\ M3)))\n !(i mod row_length (M2 \\ M3))) \n = f \n ((M2)!((j div (length M3)) mod (length M2))\n !((i div (row_length M3)) mod (row_length M2)))\n (M3!(j mod (length M3))!(i mod (row_length M3))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j mod length (M2 \\ M3) div length M3) ! (i mod row_length (M2 \\ M3) div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\\j. j mod length (M2 \\ M3) div length M3 = j div length M3 mod length M2\n\\i. i mod row_length (M2 \\ M3) div row_length M3 = i div row_length M3 mod row_length M2\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nwith 7\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div length (M2 \\ M3)) ! (i div row_length (M2 \\ M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nhave 13:\"\\i j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n ((M1 \\ (M2 \\ M3))!j!i) \n = f \n ((M1)!(j div ((length M2)*(length M3)))\n !(i div ((row_length M2)*(row_length M3)))) \n (f \n ((M2)!((j div (length M3)) mod (length M2))!((i div (row_length M3)) mod (row_length M2)))\n (M3!(j mod (length M3))\n !(i mod (row_length M3)))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div length (M2 \\ M3)) ! (i div row_length (M2 \\ M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3))\n[PROOF STEP]\nusing length_Tensor row_length_mat\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div length (M2 \\ M3)) ! (i div row_length (M2 \\ M3)) * (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3))\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M2 \\ M3) ! (j mod length (M2 \\ M3)) ! (i mod row_length (M2 \\ M3)) = M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3))\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3))\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave \"\\ i j.( f \n ((M1)!(j div ((length M2)*(length M3)))\n !(i div ((row_length M2)*(row_length M3)))) \n (f \n ((M2)!((j div (length M3)) mod (length M2))!((i div (row_length M3)) mod (row_length M2)))\n (M3!(j mod (length M3))\n !(i mod (row_length M3)))))\n = f (f \n ((M1)!(j div ((length M2)*(length M3)))\n !(i div ((row_length M2)*(row_length M3)))) \n ((M2)!((j div (length M3)) mod (length M2))\n !((i div (row_length M3)) mod (row_length M2))))\n (M3!(j mod (length M3))\n !(i mod (row_length M3)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)) = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nusing assoc\n[PROOF STATE]\nproof (prove)\nusing this:\n?a * ?b * ?c = ?a * (?b * ?c)\n\ngoal (1 subgoal):\n 1. \\i j. M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)) = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)) = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nwith 13\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3))\n\\i j. M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)) = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nhave \"\\i j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n ((M1 \\ (M2 \\ M3))!j!i) \n = f (f \n ((M1)!(j div ((length M2)*(length M3)))\n !(i div ((row_length M2)*(row_length M3)))) \n ((M2)!((j div (length M3)) mod (length M2))\n !((i div (row_length M3)) mod (row_length M2))))\n (M3!(j mod (length M3))\n !(i mod (row_length M3))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3))\n\\i j. M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)) = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nwith step1\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n[PROOF STEP]\nhave step2: \n \"\\i j.(((i<((row_length M1)*(row_length M2)*(row_length M3)))\n \\(j < (length M1)*(length M2)*(length M3)))\n \\ \n ((M1 \\ (M2 \\ M3))!j!i) = (((M1 \\ M2) \\ M3)!j!i))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ M2 \\ M3) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)\n\ngoal (1 subgoal):\n 1. \\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = (M1 \\ M2 \\ M3) ! j ! i\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = (M1 \\ M2 \\ M3) ! j ! i\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = (M1 \\ M2 \\ M3) ! j ! i\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave \"mat ((row_length M1)*(row_length M2)*(row_length M3))\n ((length M1)*(length M2)*(length M3))\n (M1 \\ (M2 \\ M3))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\n[PROOF STEP]\nhave \"mat ((row_length M2)*(row_length M3)) ((length M2)*(length M3)) (M2 \\ M3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (row_length M2 * row_length M3) (length M2 * length M3) (M2 \\ M3)\n[PROOF STEP]\nusing 0 effective_well_defined_Tensor row_length_mat length_Tensor\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\\mat (row_length ?M1.0) (length ?M1.0) ?M1.0; mat (row_length ?M2.0) (length ?M2.0) ?M2.0\\ \\ mat (row_length ?M1.0 * row_length ?M2.0) (length ?M1.0 * length ?M2.0) (?M1.0 \\ ?M2.0)\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\n\ngoal (1 subgoal):\n 1. mat (row_length M2 * row_length M3) (length M2 * length M3) (M2 \\ M3)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M2 * row_length M3) (length M2 * length M3) (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M2 * row_length M3) (length M2 * length M3) (M2 \\ M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\n[PROOF STEP]\nhave \"mat ((row_length M1)*((row_length (M2 \\ M3))))\n ((length M1)*((length (M2 \\ M3))))\n (M1 \\ (M2 \\ M3))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length (M2 \\ M3)) (length M1 * length (M2 \\ M3)) (M1 \\ (M2 \\ M3))\n[PROOF STEP]\nusing 0 effective_well_defined_Tensor row_length_mat length_Tensor\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\\mat (row_length ?M1.0) (length ?M1.0) ?M1.0; mat (row_length ?M2.0) (length ?M2.0) ?M2.0\\ \\ mat (row_length ?M1.0 * row_length ?M2.0) (length ?M1.0 * length ?M2.0) (?M1.0 \\ ?M2.0)\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length (M2 \\ M3)) (length M1 * length (M2 \\ M3)) (M1 \\ (M2 \\ M3))\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M1 * row_length (M2 \\ M3)) (length M1 * length (M2 \\ M3)) (M1 \\ (M2 \\ M3))\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nmat (row_length M2 * row_length M3) (length M2 * length M3) (M2 \\ M3)\nmat (row_length M1 * row_length (M2 \\ M3)) (length M1 * length (M2 \\ M3)) (M1 \\ (M2 \\ M3))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M2 * row_length M3) (length M2 * length M3) (M2 \\ M3)\nmat (row_length M1 * row_length (M2 \\ M3)) (length M1 * length (M2 \\ M3)) (M1 \\ (M2 \\ M3))\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\n[PROOF STEP]\nusing row_length_mat length_Tensor mult.assoc\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M2 * row_length M3) (length M2 * length M3) (M2 \\ M3)\nmat (row_length M1 * row_length (M2 \\ M3)) (length M1 * length (M2 \\ M3)) (M1 \\ (M2 \\ M3))\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\nMatrix_Tensor.mult ?id ?f \\ ?f (?f ?a ?b) ?c = ?f ?a (?f ?b ?c)\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\n[PROOF STEP]\nby (simp add: length_Tensor row_length_mat semigroup_mult_class.mult.assoc)\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nhave \"mat ((row_length M1)*(row_length M2)*(row_length M3))\n ((length M1)*(length M2)*(length M3))\n ((M1 \\ M2) \\ M3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n[PROOF STEP]\nhave \"mat ((row_length M1)*(row_length M2)) ((length M1)*(length M2)) (M1 \\ M2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2) (length M1 * length M2) (M1 \\ M2)\n[PROOF STEP]\nusing 0 effective_well_defined_Tensor row_length_mat length_Tensor\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\\mat (row_length ?M1.0) (length ?M1.0) ?M1.0; mat (row_length ?M2.0) (length ?M2.0) ?M2.0\\ \\ mat (row_length ?M1.0 * row_length ?M2.0) (length ?M1.0 * length ?M2.0) (?M1.0 \\ ?M2.0)\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2) (length M1 * length M2) (M1 \\ M2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M1 * row_length M2) (length M1 * length M2) (M1 \\ M2)\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M1 * row_length M2) (length M1 * length M2) (M1 \\ M2)\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n[PROOF STEP]\nhave \"mat ((row_length (M1 \\ M2))*(row_length M3))\n ((length (M1 \\ M2))*(length M3))\n ((M1 \\ M2 )\\ M3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (row_length (M1 \\ M2) * row_length M3) (length (M1 \\ M2) * length M3) (M1 \\ M2 \\ M3)\n[PROOF STEP]\nusing 0 effective_well_defined_Tensor row_length_mat length_Tensor\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3\n\\mat (row_length ?M1.0) (length ?M1.0) ?M1.0; mat (row_length ?M2.0) (length ?M2.0) ?M2.0\\ \\ mat (row_length ?M1.0 * row_length ?M2.0) (length ?M1.0 * length ?M2.0) (?M1.0 \\ ?M2.0)\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\n\ngoal (1 subgoal):\n 1. mat (row_length (M1 \\ M2) * row_length M3) (length (M1 \\ M2) * length M3) (M1 \\ M2 \\ M3)\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length (M1 \\ M2) * row_length M3) (length (M1 \\ M2) * length M3) (M1 \\ M2 \\ M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nmat (row_length M1 * row_length M2) (length M1 * length M2) (M1 \\ M2)\nmat (row_length (M1 \\ M2) * row_length M3) (length (M1 \\ M2) * length M3) (M1 \\ M2 \\ M3)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1 * row_length M2) (length M1 * length M2) (M1 \\ M2)\nmat (row_length (M1 \\ M2) * row_length M3) (length (M1 \\ M2) * length M3) (M1 \\ M2 \\ M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n[PROOF STEP]\nusing row_length_mat length_Tensor\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length M1 * row_length M2) (length M1 * length M2) (M1 \\ M2)\nmat (row_length (M1 \\ M2) * row_length M3) (length (M1 \\ M2) * length M3) (M1 \\ M2 \\ M3)\nrow_length (?m1.0 \\ ?m2.0) = row_length ?m1.0 * row_length ?m2.0\nlength (?M1.0 \\ ?M2.0) = length ?M1.0 * length ?M2.0\n\ngoal (1 subgoal):\n 1. mat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n[PROOF STEP]\nby (metis mult.assoc)\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nmat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n\ngoal (1 subgoal):\n 1. mat (row_length M1) (length M1) M1 \\ mat (row_length M2) (length M2) M2 \\ mat (row_length M3) (length M3) M3 \\ M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3))\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = (M1 \\ M2 \\ M3) ! j ! i\nmat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\nmat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3))\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = (M1 \\ M2 \\ M3) ! j ! i\nmat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\nmat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n\ngoal (1 subgoal):\n 1. M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nusing mat_eqI\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = M1 ! (j div (length M2 * length M3)) ! (i div (row_length M2 * row_length M3)) * (M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3))\n\\i j. i < row_length M1 * row_length M2 * row_length M3 \\ j < length M1 * length M2 * length M3 \\ (M1 \\ (M2 \\ M3)) ! j ! i = (M1 \\ M2 \\ M3) ! j ! i\nmat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ (M2 \\ M3))\nmat (row_length M1 * row_length M2 * row_length M3) (length M1 * length M2 * length M3) (M1 \\ M2 \\ M3)\n\\mat ?nr ?nc ?m1.0; mat ?nr ?nc ?m2.0; \\i j. \\i < ?nc; j < ?nr\\ \\ ?m1.0 ! i ! j = ?m2.0 ! i ! j\\ \\ ?m1.0 = ?m2.0\n\ngoal (1 subgoal):\n 1. M1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nM1 \\ (M2 \\ M3) = M1 \\ M2 \\ M3\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 49956, "file": "Matrix_Tensor_Matrix_Tensor", "length": 262, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357735451834, "lm_q2_score": 0.824461932846258, "lm_q1q2_score": 0.7148379897039123}} {"text": "[STATEMENT]\nlemma cauchy_schwarz_ineq_var_uniform:\n fixes X :: \"'a \\ real\"\n assumes \"M = uniform_count_measure S\"\n assumes \"finite S\"\n shows \"expectation (\\ x. (X x)^2) \\ (expectation (\\ x . (X x)))^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (expectation X)\\<^sup>2 \\ expectation (\\x. (X x)\\<^sup>2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (expectation X)\\<^sup>2 \\ expectation (\\x. (X x)\\<^sup>2)\n[PROOF STEP]\nhave borel: \"X \\ borel_measurable M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. random_variable borel X\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nM = uniform_count_measure S\nfinite S\n\ngoal (1 subgoal):\n 1. random_variable borel X\n[PROOF STEP]\nby (simp)\n[PROOF STATE]\nproof (state)\nthis:\nrandom_variable borel X\n\ngoal (1 subgoal):\n 1. (expectation X)\\<^sup>2 \\ expectation (\\x. (X x)\\<^sup>2)\n[PROOF STEP]\nhave \"integrable M X\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integrable M X\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nM = uniform_count_measure S\nfinite S\n\ngoal (1 subgoal):\n 1. integrable M X\n[PROOF STEP]\nby (simp add: integrable_uniform_count_measure_finite)\n[PROOF STATE]\nproof (state)\nthis:\nintegrable M X\n\ngoal (1 subgoal):\n 1. (expectation X)\\<^sup>2 \\ expectation (\\x. (X x)\\<^sup>2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nintegrable M X\n[PROOF STEP]\nhave \"integrable M (\\ x. (X x)^2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable M X\n\ngoal (1 subgoal):\n 1. integrable M (\\x. (X x)\\<^sup>2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable M X\nM = uniform_count_measure S\nfinite S\n\ngoal (1 subgoal):\n 1. integrable M (\\x. (X x)\\<^sup>2)\n[PROOF STEP]\nby (simp add: integrable_uniform_count_measure_finite)\n[PROOF STATE]\nproof (state)\nthis:\nintegrable M (\\x. (X x)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (expectation X)\\<^sup>2 \\ expectation (\\x. (X x)\\<^sup>2)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable M (\\x. (X x)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (expectation X)\\<^sup>2 \\ expectation (\\x. (X x)\\<^sup>2)\n[PROOF STEP]\nusing cauchy_schwarz_ineq_var borel\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable M (\\x. (X x)\\<^sup>2)\n\\integrable M (\\x. (?X x)\\<^sup>2); random_variable borel ?X\\ \\ (expectation ?X)\\<^sup>2 \\ expectation (\\x. (?X x)\\<^sup>2)\nrandom_variable borel X\n\ngoal (1 subgoal):\n 1. (expectation X)\\<^sup>2 \\ expectation (\\x. (X x)\\<^sup>2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(expectation X)\\<^sup>2 \\ expectation (\\x. (X x)\\<^sup>2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1221, "file": "Balog_Szemeredi_Gowers_Prob_Space_Lemmas", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357701094303, "lm_q2_score": 0.8244619328462579, "lm_q1q2_score": 0.7148379868712647}} {"text": "[STATEMENT]\nlemma doubleton_eq_iff: \"\\a,b\\ = \\c,d\\ \\ (a=c \\ b=d) \\ (a=d \\ b=c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a, b\\ = \\c, d\\) = (a = c \\ b = d \\ a = d \\ b = c)\n[PROOF STEP]\nby auto (metis hmem_hempty hmem_hinsert)+", "meta": {"llama_tokens": 162, "file": "HereditarilyFinite_HF", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357666736772, "lm_q2_score": 0.8244619306896955, "lm_q1q2_score": 0.7148379821688002}} {"text": "[STATEMENT]\nlemma rel_interior_closure_convex_segment:\n fixes S :: \"_::euclidean_space set\"\n assumes \"convex S\" \"a \\ rel_interior S\" \"b \\ closure S\"\n shows \"open_segment a b \\ rel_interior S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. open_segment a b \\ rel_interior S\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ open_segment a b \\ x \\ rel_interior S\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ open_segment a b \\ x \\ rel_interior S\n[PROOF STEP]\nhave [simp]: \"(1 - u) *\\<^sub>R a + u *\\<^sub>R b = b - (1 - u) *\\<^sub>R (b - a)\" for u\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1 - u) *\\<^sub>R a + u *\\<^sub>R b = b - (1 - u) *\\<^sub>R (b - a)\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(1 - ?u) *\\<^sub>R a + ?u *\\<^sub>R b = b - (1 - ?u) *\\<^sub>R (b - a)\n\ngoal (1 subgoal):\n 1. \\x. x \\ open_segment a b \\ x \\ rel_interior S\n[PROOF STEP]\nassume \"x \\ open_segment a b\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ open_segment a b\n\ngoal (1 subgoal):\n 1. \\x. x \\ open_segment a b \\ x \\ rel_interior S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ open_segment a b\n[PROOF STEP]\nshow \"x \\ rel_interior S\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ open_segment a b\n\ngoal (1 subgoal):\n 1. x \\ rel_interior S\n[PROOF STEP]\nunfolding closed_segment_def open_segment_def\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ {(1 - u) *\\<^sub>R a + u *\\<^sub>R b |u. 0 \\ u \\ u \\ 1} - {a, b}\n\ngoal (1 subgoal):\n 1. x \\ rel_interior S\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ {(1 - u) *\\<^sub>R a + u *\\<^sub>R b |u. 0 \\ u \\ u \\ 1} - {a, b}\nconvex S\na \\ rel_interior S\nb \\ closure S\n\ngoal (1 subgoal):\n 1. x \\ rel_interior S\n[PROOF STEP]\nby (auto intro: rel_interior_closure_convex_shrink)\n[PROOF STATE]\nproof (state)\nthis:\nx \\ rel_interior S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 954, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681049901037, "lm_q2_score": 0.8333245911726382, "lm_q1q2_score": 0.7147992554118068}} {"text": "[STATEMENT]\nlemma ennexp_add_mult:\n assumes \"\\((a = \\ \\ b = -\\) \\ (a = -\\ \\ b = \\))\"\n shows \"ennexp(a+b) = ennexp a * ennexp b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ennexp (a + b) = ennexp a * ennexp b\n[PROOF STEP]\napply (cases a, cases b)\n[PROOF STATE]\nproof (prove)\ngoal (5 subgoals):\n 1. \\r ra. \\a = ereal r; b = ereal ra\\ \\ ennexp (a + b) = ennexp a * ennexp b\n 2. \\r. \\a = ereal r; b = \\\\ \\ ennexp (a + b) = ennexp a * ennexp b\n 3. \\r. \\a = ereal r; b = - \\\\ \\ ennexp (a + b) = ennexp a * ennexp b\n 4. a = \\ \\ ennexp (a + b) = ennexp a * ennexp b\n 5. a = - \\ \\ ennexp (a + b) = ennexp a * ennexp b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (a = \\ \\ b = - \\ \\ a = - \\ \\ b = \\)\n\ngoal (5 subgoals):\n 1. \\r ra. \\a = ereal r; b = ereal ra\\ \\ ennexp (a + b) = ennexp a * ennexp b\n 2. \\r. \\a = ereal r; b = \\\\ \\ ennexp (a + b) = ennexp a * ennexp b\n 3. \\r. \\a = ereal r; b = - \\\\ \\ ennexp (a + b) = ennexp a * ennexp b\n 4. a = \\ \\ ennexp (a + b) = ennexp a * ennexp b\n 5. a = - \\ \\ ennexp (a + b) = ennexp a * ennexp b\n[PROOF STEP]\nby (auto simp add: ennreal_mult'' exp_add ennreal_top_eq_mult_iff)", "meta": {"llama_tokens": 727, "file": "Gromov_Hyperbolicity_Eexp_Eln", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045937171068, "lm_q2_score": 0.8056321959813275, "lm_q1q2_score": 0.7147605851210342}} {"text": "[STATEMENT]\nlemma multiplicity_cong:\n \"(\\r. p ^ r dvd a \\ p ^ r dvd b) \\ multiplicity p a = multiplicity p b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\r. (p ^ r dvd a) = (p ^ r dvd b)) \\ multiplicity p a = multiplicity p b\n[PROOF STEP]\nby (simp add: multiplicity_def)", "meta": {"llama_tokens": 128, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045937171068, "lm_q2_score": 0.8056321959813274, "lm_q1q2_score": 0.7147605851210341}} {"text": "[STATEMENT]\nlemma matrix_mult_mono: \n assumes \"0 \\ E\" \"0 \\ C\" \"(E :: real^'c^'c) \\ B\" \"C \\ D\"\n shows \"E ** C \\ B ** D\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. E ** C \\ B ** D\n[PROOF STEP]\nusing order.trans[OF assms(1) assms(3)] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ B\n0 \\ E\n0 \\ C\nE \\ B\nC \\ D\n\ngoal (1 subgoal):\n 1. E ** C \\ B ** D\n[PROOF STEP]\nunfolding Finite_Cartesian_Product.less_eq_vec_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i ia. 0 $ i $ ia \\ B $ i $ ia\n\\i ia. 0 $ i $ ia \\ E $ i $ ia\n\\i ia. 0 $ i $ ia \\ C $ i $ ia\n\\i ia. E $ i $ ia \\ B $ i $ ia\n\\i ia. C $ i $ ia \\ D $ i $ ia\n\ngoal (1 subgoal):\n 1. \\i ia. (E ** C) $ i $ ia \\ (B ** D) $ i $ ia\n[PROOF STEP]\nby (auto intro!: sum_mono mult_mono simp: matrix_matrix_mult_def)", "meta": {"llama_tokens": 433, "file": "MDP-Algorithms_Matrix_Util", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872046056466901, "lm_q2_score": 0.805632181981183, "lm_q1q2_score": 0.714760582310898}} {"text": "[STATEMENT]\nlemma tensor_vec_add2:\n fixes v1 v2 v3 :: \"'a::comm_ring vec\"\n assumes \"v1 \\ carrier_vec d1\"\n and \"v2 \\ carrier_vec d2\"\n and \"v3 \\ carrier_vec d2\"\n shows \"tensor_vec v1 (v2 + v3) = tensor_vec v1 v2 + tensor_vec v1 v3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tensor_vec v1 (v2 + v3) = tensor_vec v1 v2 + tensor_vec v1 v3\n[PROOF STEP]\napply (rule eq_vecI, auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i < d \\ tensor_vec v1 (v2 + v3) $ i = tensor_vec v1 v2 $ i + tensor_vec v1 v3 $ i\n[PROOF STEP]\nunfolding tensor_vec_eval\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i < d \\ v1 $ encode1 i * (v2 + v3) $ encode2 i = v1 $ encode1 i * v2 $ encode2 i + v1 $ encode1 i * v3 $ encode2 i\n[PROOF STEP]\nusing assms(3) semiring_class.distrib_left\n[PROOF STATE]\nproof (prove)\nusing this:\nv3 \\ carrier_vec d2\n?a * (?b + ?c) = ?a * ?b + ?a * ?c\n\ngoal (1 subgoal):\n 1. \\i. i < d \\ v1 $ encode1 i * (v2 + v3) $ encode2 i = v1 $ encode1 i * v2 $ encode2 i + v1 $ encode1 i * v3 $ encode2 i\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 505, "file": "QHLProver_Partial_State", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045847699186, "lm_q2_score": 0.8056321889812553, "lm_q1q2_score": 0.7147605717023953}} {"text": "[STATEMENT]\nlemma part_equiv_trans_sym: \n \"\\ part_equiv R; (a,b)\\R; (c,b)\\R \\ \\ (a,c)\\R\"\n \"\\ part_equiv R; (a,b)\\R; (a,c)\\R \\ \\ (b,c)\\R\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\part_equiv R; (a, b) \\ R; (c, b) \\ R\\ \\ (a, c) \\ R) &&& (\\part_equiv R; (a, b) \\ R; (a, c) \\ R\\ \\ (b, c) \\ R)\n[PROOF STEP]\napply (metis part_equiv_sym part_equiv_trans)+\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 289, "file": "Collections_Lib_Partial_Equivalence_Relation", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467675095294, "lm_q2_score": 0.8128673178375735, "lm_q1q2_score": 0.7146296748910439}} {"text": "[STATEMENT]\nlemma L2_set_strict_mono:\n assumes \"finite K\" and \"K \\ {}\"\n assumes \"\\i. i \\ K \\ f i < g i\"\n assumes \"\\i. i \\ K \\ 0 \\ f i\"\n shows \"L2_set f K < L2_set g K\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. L2_set f K < L2_set g K\n[PROOF STEP]\nunfolding L2_set_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (\\i\\K. (f i)\\<^sup>2) < sqrt (\\i\\K. (g i)\\<^sup>2)\n[PROOF STEP]\nby (simp add: sum_strict_mono power_strict_mono assms)", "meta": {"llama_tokens": 250, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772351648678, "lm_q2_score": 0.8175744673038222, "lm_q1q2_score": 0.7146232299223145}} {"text": "[STATEMENT]\nlemma unitarily_equiv_eq:\n assumes \"unitarily_equiv A B U\"\n shows \"A = U * B * (Complex_Matrix.adjoint U)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A = U * B * Complex_Matrix.adjoint U\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nunitarily_equiv A B U\n\ngoal (1 subgoal):\n 1. A = U * B * Complex_Matrix.adjoint U\n[PROOF STEP]\nunfolding unitarily_equiv_def similar_mat_wit_def\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex_Matrix.unitary U \\ (let n = dim_row A in {A, B, U, Complex_Matrix.adjoint U} \\ carrier_mat n n \\ U * Complex_Matrix.adjoint U = 1\\<^sub>m n \\ Complex_Matrix.adjoint U * U = 1\\<^sub>m n \\ A = U * B * Complex_Matrix.adjoint U)\n\ngoal (1 subgoal):\n 1. A = U * B * Complex_Matrix.adjoint U\n[PROOF STEP]\nby meson", "meta": {"llama_tokens": 330, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099070109242131, "lm_q2_score": 0.7853085758631159, "lm_q1q2_score": 0.7145577789167584}} {"text": "[STATEMENT]\nlemma CDERIV_inverse_fun:\n \"DERIV f x :> d \\ f x \\ 0 \\ DERIV (\\x. inverse (f x)) x :> - (d * inverse ((f x)\\<^sup>2))\"\n for x :: complex\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative d) (at x); f x \\ 0\\ \\ ((\\x. inverse (f x)) has_field_derivative - (d * inverse ((f x)\\<^sup>2))) (at x)\n[PROOF STEP]\nunfolding numeral_2_eq_2\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative d) (at x); f x \\ 0\\ \\ ((\\x. inverse (f x)) has_field_derivative - (d * inverse (f x ^ Suc (Suc 0)))) (at x)\n[PROOF STEP]\nby (rule DERIV_inverse_fun)", "meta": {"llama_tokens": 314, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099069987088004, "lm_q2_score": 0.7853085859124002, "lm_q1q2_score": 0.7145577784678042}} {"text": "[STATEMENT]\nlemma deriv_deriv_real_sqrt [simp]:\n assumes \"x > 0\"\n shows \"deriv(deriv sqrt) x = - inverse ((sqrt x)^3)/4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. deriv (deriv sqrt) x = - inverse (sqrt x ^ 3) / 4\n[PROOF STEP]\nusing DERIV_imp_deriv assms has_real_derivative_deriv_sqrt\n[PROOF STATE]\nproof (prove)\nusing this:\n(?f has_field_derivative ?f') (at ?x) \\ deriv ?f ?x = ?f'\n0 < x\n0 < ?x \\ (deriv sqrt has_real_derivative - inverse (sqrt ?x ^ 3) / 4) (at ?x)\n\ngoal (1 subgoal):\n 1. deriv (deriv sqrt) x = - inverse (sqrt x ^ 3) / 4\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 262, "file": "Hyperdual_TwiceFieldDifferentiable", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.909907001151883, "lm_q2_score": 0.7853085758631159, "lm_q1q2_score": 0.7145577712424638}} {"text": "[STATEMENT]\nlemma continuous_injective_image_open_segment_1:\n fixes f :: \"'a::euclidean_space \\ real\"\n assumes contf: \"continuous_on (closed_segment a b) f\"\n and injf: \"inj_on f (closed_segment a b)\"\n shows \"f ` (open_segment a b) = open_segment (f a) (f b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f ` open_segment a b = open_segment (f a) (f b)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. f ` open_segment a b = open_segment (f a) (f b)\n[PROOF STEP]\nhave \"f ` (open_segment a b) = f ` (closed_segment a b) - {f a, f b}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f ` open_segment a b = f ` closed_segment a b - {f a, f b}\n[PROOF STEP]\nby (metis (no_types, opaque_lifting) empty_subsetI ends_in_segment image_insert image_is_empty inj_on_image_set_diff injf insert_subset open_segment_def segment_open_subset_closed)\n[PROOF STATE]\nproof (state)\nthis:\nf ` open_segment a b = f ` closed_segment a b - {f a, f b}\n\ngoal (1 subgoal):\n 1. f ` open_segment a b = open_segment (f a) (f b)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nf ` open_segment a b = f ` closed_segment a b - {f a, f b}\n\ngoal (1 subgoal):\n 1. f ` open_segment a b = open_segment (f a) (f b)\n[PROOF STEP]\nhave \"... = open_segment (f a) (f b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f ` closed_segment a b - {f a, f b} = open_segment (f a) (f b)\n[PROOF STEP]\nusing continuous_injective_image_segment_1 [OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nf ` closed_segment a b = closed_segment (f a) (f b)\n\ngoal (1 subgoal):\n 1. f ` closed_segment a b - {f a, f b} = open_segment (f a) (f b)\n[PROOF STEP]\nby (simp add: open_segment_def inj_on_image_set_diff [OF injf])\n[PROOF STATE]\nproof (state)\nthis:\nf ` closed_segment a b - {f a, f b} = open_segment (f a) (f b)\n\ngoal (1 subgoal):\n 1. f ` open_segment a b = open_segment (f a) (f b)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nf ` open_segment a b = open_segment (f a) (f b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nf ` open_segment a b = open_segment (f a) (f b)\n\ngoal (1 subgoal):\n 1. f ` open_segment a b = open_segment (f a) (f b)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nf ` open_segment a b = open_segment (f a) (f b)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 975, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637505099167, "lm_q2_score": 0.8311430415844385, "lm_q1q2_score": 0.7145035443386981}} {"text": "[STATEMENT]\nlemma swap_triple:\n assumes \"a \\ c\" and \"b \\ c\"\n shows \"Fun.swap a b (Fun.swap b c (Fun.swap a b f)) = Fun.swap a c f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f \\ Transposition.transpose a b \\ Transposition.transpose b c \\ Transposition.transpose a b = f \\ Transposition.transpose a c\n[PROOF STEP]\nusing assms transpose_comp_triple [of a c b]\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ c\nb \\ c\n\\a \\ c; b \\ c\\ \\ Transposition.transpose a b \\ Transposition.transpose b c \\ Transposition.transpose a b = Transposition.transpose a c\n\ngoal (1 subgoal):\n 1. f \\ Transposition.transpose a b \\ Transposition.transpose b c \\ Transposition.transpose a b = f \\ Transposition.transpose a c\n[PROOF STEP]\nby (simp add: comp_assoc)", "meta": {"llama_tokens": 320, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392939666336, "lm_q2_score": 0.8080672158638527, "lm_q1q2_score": 0.7143631709898636}} {"text": "[STATEMENT]\nlemma gbinomial_altdef_of_nat: \"a gchoose k = (\\i = 0..i = 0..i = 0..i = 0..i = 0..i = 0.. carrier_mat d1 d1\"\n and \"m2 \\ carrier_mat d2 d2\"\n and \"m3 \\ carrier_mat d2 d2\"\n shows \"tensor_mat m1 (m2 + m3) = tensor_mat m1 m2 + tensor_mat m1 m3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tensor_mat m1 (m2 + m3) = tensor_mat m1 m2 + tensor_mat m1 m3\n[PROOF STEP]\napply (rule eq_matI, auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. \\i < d; j < d\\ \\ tensor_mat m1 (m2 + m3) $$ (i, j) = tensor_mat m1 m2 $$ (i, j) + tensor_mat m1 m3 $$ (i, j)\n[PROOF STEP]\nunfolding tensor_mat_eval\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. \\i < d; j < d\\ \\ m1 $$ (encode1 i, encode1 j) * (m2 + m3) $$ (encode2 i, encode2 j) = m1 $$ (encode1 i, encode1 j) * m2 $$ (encode2 i, encode2 j) + m1 $$ (encode1 i, encode1 j) * m3 $$ (encode2 i, encode2 j)\n[PROOF STEP]\nusing assms(3) semiring_class.distrib_left\n[PROOF STATE]\nproof (prove)\nusing this:\nm3 \\ carrier_mat d2 d2\n?a * (?b + ?c) = ?a * ?b + ?a * ?c\n\ngoal (1 subgoal):\n 1. \\i j. \\i < d; j < d\\ \\ m1 $$ (encode1 i, encode1 j) * (m2 + m3) $$ (encode2 i, encode2 j) = m1 $$ (encode1 i, encode1 j) * m2 $$ (encode2 i, encode2 j) + m1 $$ (encode1 i, encode1 j) * m3 $$ (encode2 i, encode2 j)\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 624, "file": "QHLProver_Partial_State", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392878563336, "lm_q2_score": 0.8080672089305841, "lm_q1q2_score": 0.7143631599230487}} {"text": "[STATEMENT]\nlemma hermitian_decomp_diag_mat: \n assumes \"hermitian_decomp A B U\"\n shows \"diagonal_mat B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. diagonal_mat B\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nhermitian_decomp A B U\n\ngoal (1 subgoal):\n 1. diagonal_mat B\n[PROOF STEP]\nunfolding hermitian_decomp_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsimilar_mat_wit A B U (Complex_Matrix.adjoint U) \\ diagonal_mat B \\ diag_mat B = eigvals A \\ Complex_Matrix.unitary U \\ (\\i \\)\n\ngoal (1 subgoal):\n 1. diagonal_mat B\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 268, "file": "Projective_Measurements_Projective_Measurements", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392878563335, "lm_q2_score": 0.808067208930584, "lm_q1q2_score": 0.7143631599230486}} {"text": "[STATEMENT]\nlemma bounded_differences:\n fixes S :: \"'a::real_normed_vector set\"\n assumes \"bounded S\" and \"bounded T\"\n shows \"bounded (\\x\\ S. \\y \\ T. {x - y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bounded (\\x\\S. \\y\\T. {x - y})\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nbounded S\nbounded T\n\ngoal (1 subgoal):\n 1. bounded (\\x\\S. \\y\\T. {x - y})\n[PROOF STEP]\nby (simp add: bounded_iff) (meson add_mono norm_triangle_le_diff)", "meta": {"llama_tokens": 221, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267898240861, "lm_q2_score": 0.8221891261650247, "lm_q1q2_score": 0.714339939114229}} {"text": "[STATEMENT]\ntheorem gs1: \"\\ Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a \\ \\\n \\A. Gale_Shapley1 P\\<^sub>a P\\<^sub>b = Some A\n \\ Pref.matching P\\<^sub>a (list A) { Pref.stable P\\<^sub>a P\\<^sub>b (list A) { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b (list A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a\\ \\ \\A. Gale_Shapley2.Gale_Shapley1 P\\<^sub>a P\\<^sub>b = Some A \\ Pref.matching P\\<^sub>a (list A) { Pref.stable P\\<^sub>a P\\<^sub>b (list A) { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b (list A)\n[PROOF STEP]\nunfolding Gale_Shapley1_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a\\ \\ \\A. (if Pref P\\<^sub>a P\\<^sub>b then Some (fst (Gale_Shapley2.gs1 (length P\\<^sub>a) (pref_array P\\<^sub>a) (rank_array P\\<^sub>b))) else None) = Some A \\ Pref.matching P\\<^sub>a (list A) { Pref.stable P\\<^sub>a P\\<^sub>b (list A) { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b (list A)\n[PROOF STEP]\nusing Pref.gs1\n[PROOF STATE]\nproof (prove)\nusing this:\n\\Pref ?P\\<^sub>a ?P\\<^sub>b; ?R\\<^sub>b = rank_array ?P\\<^sub>b\\ \\ Gale_Shapley2.gs1 (length ?P\\<^sub>a) (pref_array ?P\\<^sub>a) ?R\\<^sub>b = (?A, ?B, ?M, ?ai, ?a) \\ Pref.matching ?P\\<^sub>a (list ?A) {a} \\ Pref.stable ?P\\<^sub>a ?P\\<^sub>b (list ?A) {a} \\ Pref.opti\\<^sub>a ?P\\<^sub>a ?P\\<^sub>b (list ?A)\n\ngoal (1 subgoal):\n 1. \\Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a\\ \\ \\A. (if Pref P\\<^sub>a P\\<^sub>b then Some (fst (Gale_Shapley2.gs1 (length P\\<^sub>a) (pref_array P\\<^sub>a) (rank_array P\\<^sub>b))) else None) = Some A \\ Pref.matching P\\<^sub>a (list A) { Pref.stable P\\<^sub>a P\\<^sub>b (list A) { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b (list A)\n[PROOF STEP]\nby (metis fst_conv surj_pair)", "meta": {"llama_tokens": 941, "file": "Gale_Shapley_Gale_Shapley2", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952975813454, "lm_q2_score": 0.7956581049086031, "lm_q1q2_score": 0.7142585392589378}} {"text": "[STATEMENT]\nlemma LPDs_Star: \"LPDs (MStar r) \\ {MStar r} \\ MTimesR (LPDs r) (MStar r)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LPDs (MStar r) \\ {MStar r} \\ MTimesR (LPDs r) (MStar r)\n[PROOF STEP]\nunfolding LPDs_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. LPDi i (MStar r)) \\ {MStar r} \\ MTimesR (\\i. LPDi i r) (MStar r)\n[PROOF STEP]\nusing LPDi_Star[OF order_refl, of _ r]\n[PROOF STATE]\nproof (prove)\nusing this:\nLPDi ?i (MStar r) \\ {MStar r} \\ MTimesR (\\j\\?i. LPDi j r) (MStar r)\n\ngoal (1 subgoal):\n 1. (\\i. LPDi i (MStar r)) \\ {MStar r} \\ MTimesR (\\i. LPDi i r) (MStar r)\n[PROOF STEP]\nby (force simp: MTimesR_def)", "meta": {"llama_tokens": 346, "file": "MFODL_Monitor_Optimized_Monitor", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952893703477, "lm_q2_score": 0.7956580927949806, "lm_q1q2_score": 0.714258521851449}} {"text": "[STATEMENT]\nlemma incseq_of_rat_interlaced_seq:\n \"\\ (\\n. of_rat (r n)) \\ (x::real); \n (\\n. of_rat (s n)) \\ (x::real);\n \\n. of_rat (r n) < x; \\n. of_rat (s n) < x \\ \n \\ incseq (\\n. real_of_rat (interlaced_seq r s n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(\\n. real_of_rat (r n)) \\ x; (\\n. real_of_rat (s n)) \\ x; \\n. real_of_rat (r n) < x; \\n. real_of_rat (s n) < x\\ \\ incseq (\\n. real_of_rat (interlaced_seq r s n))\n[PROOF STEP]\nusing incseq_interlaced_seq incseq_of_rat\n[PROOF STATE]\nproof (prove)\nusing this:\n\\(\\n. real_of_rat (?r n)) \\ ?x; (\\n. real_of_rat (?s n)) \\ ?x; \\n. real_of_rat (?r n) < ?x; \\n. real_of_rat (?s n) < ?x\\ \\ incseq (interlaced_seq ?r ?s)\nincseq ?s \\ incseq (\\n. of_rat (?s n))\n\ngoal (1 subgoal):\n 1. \\(\\n. real_of_rat (r n)) \\ x; (\\n. real_of_rat (s n)) \\ x; \\n. real_of_rat (r n) < x; \\n. real_of_rat (s n) < x\\ \\ incseq (\\n. real_of_rat (interlaced_seq r s n))\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 578, "file": "Real_Power_RealPower", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952838963489, "lm_q2_score": 0.7956580976404297, "lm_q1q2_score": 0.7142585218457544}} {"text": "[STATEMENT]\nlemma Polygamma_approx_1_real: \n \"x > (0 :: real) \\ \n Polygamma_approx (Suc 0) m x = ln x - 1 / (2*x) + stirling_sum (Suc 0) m x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ Polygamma_approx (Suc 0) m x = ln x - 1 / (2 * x) + stirling_sum (Suc 0) m x\n[PROOF STEP]\nunfolding Polygamma_approx_Suc Polygamma_approx_0\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ deriv (\\x. (x - 1 / 2) * ln x - x + of_real (ln (2 * pi)) / 2 + stirling_sum 0 m x) x = ln x - 1 / (2 * x) + stirling_sum (Suc 0) m x\n[PROOF STEP]\nby (intro DERIV_imp_deriv) \n (auto intro!: derivative_eq_intros elim!: nonpos_Reals_cases simp: field_simps)", "meta": {"llama_tokens": 320, "file": "Stirling_Formula_Gamma_Asymptotics", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297914570319, "lm_q2_score": 0.7931059560743422, "lm_q1q2_score": 0.7142155412269573}} {"text": "[STATEMENT]\nlemma all_onbs_same_card:\n fixes E F :: \\'a::chilbert_space set\\\n assumes \\is_onb E\\ \\is_onb F\\\n shows \\\\f. bij_betw f E F\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f. bij_betw f E F\n[PROOF STEP]\napply (rule all_ortho_bases_same_card)\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. is_ortho_set E\n 2. is_ortho_set F\n 3. ccspan E = \\\n 4. ccspan F = \\\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nis_onb E\nis_onb F\n\ngoal (4 subgoals):\n 1. is_ortho_set E\n 2. is_ortho_set F\n 3. ccspan E = \\\n 4. ccspan F = \\\n[PROOF STEP]\nby (auto simp: is_onb_def)", "meta": {"llama_tokens": 320, "file": "Complex_Bounded_Operators_Complex_Bounded_Linear_Function", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297807787538, "lm_q2_score": 0.793105951184112, "lm_q1q2_score": 0.7142155283541534}} {"text": "[STATEMENT]\nlemma Ln'_of_real_neg:\n assumes \"x < 0\"\n shows \"Ln' (of_real x) = of_real (ln (-x)) + \\ * pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Ln' (complex_of_real x) = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Ln' (complex_of_real x) = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n[PROOF STEP]\nhave \"Ln' (of_real x) = Ln (of_real (-x) * \\) + \\ * pi / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Ln' (complex_of_real x) = Ln (complex_of_real (- x) * \\) + \\ * complex_of_real pi / 2\n[PROOF STEP]\nby (simp add: Ln'_def mult_ac)\n[PROOF STATE]\nproof (state)\nthis:\nLn' (complex_of_real x) = Ln (complex_of_real (- x) * \\) + \\ * complex_of_real pi / 2\n\ngoal (1 subgoal):\n 1. Ln' (complex_of_real x) = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nLn' (complex_of_real x) = Ln (complex_of_real (- x) * \\) + \\ * complex_of_real pi / 2\n\ngoal (1 subgoal):\n 1. Ln' (complex_of_real x) = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n[PROOF STEP]\nhave \"\\ = of_real (ln (-x)) + \\ * pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Ln (complex_of_real (- x) * \\) + \\ * complex_of_real pi / 2 = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx < 0\n\ngoal (1 subgoal):\n 1. Ln (complex_of_real (- x) * \\) + \\ * complex_of_real pi / 2 = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n[PROOF STEP]\nby (subst Ln_times_of_real) (auto simp: Ln_Reals_eq)\n[PROOF STATE]\nproof (state)\nthis:\nLn (complex_of_real (- x) * \\) + \\ * complex_of_real pi / 2 = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n\ngoal (1 subgoal):\n 1. Ln' (complex_of_real x) = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nLn' (complex_of_real x) = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nLn' (complex_of_real x) = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n\ngoal (1 subgoal):\n 1. Ln' (complex_of_real x) = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nLn' (complex_of_real x) = complex_of_real (ln (- x)) + \\ * complex_of_real pi\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1119, "file": "Zeta_Function_Zeta_Function", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894717137996, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7142091251825756}} {"text": "[STATEMENT]\nlemma dist_metric_completion_limit':\n fixes x y::\"nat \\ 'a\"\n assumes \"Cauchy x\" \"Cauchy y\"\n shows \"(\\n. dist (x n) (y n)) \\ dist (abs_metric_completion x) (abs_metric_completion y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. dist (x n) (y n)) \\ dist (abs_metric_completion x) (abs_metric_completion y)\n[PROOF STEP]\napply (subst dist_metric_completion.abs_eq)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. Cauchy x \\ Cauchy x \\ (\\n. dist (x n) (x n)) \\ 0\n 2. Cauchy y \\ Cauchy y \\ (\\n. dist (y n) (y n)) \\ 0\n 3. (\\n. dist (x n) (y n)) \\ lim (\\n. dist (x n) (y n))\n[PROOF STEP]\nusing assms convergent_Cauchy_dist[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nCauchy x\nCauchy y\nconvergent (\\n. dist (x n) (y n))\n\ngoal (3 subgoals):\n 1. Cauchy x \\ Cauchy x \\ (\\n. dist (x n) (x n)) \\ 0\n 2. Cauchy y \\ Cauchy y \\ (\\n. dist (y n) (y n)) \\ 0\n 3. (\\n. dist (x n) (y n)) \\ lim (\\n. dist (x n) (y n))\n[PROOF STEP]\nby (auto simp add: convergent_LIMSEQ_iff)", "meta": {"llama_tokens": 529, "file": "Gromov_Hyperbolicity_Metric_Completion", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.7981867825403177, "lm_q1q2_score": 0.7142091182811096}} {"text": "[STATEMENT]\nlemma real_inverse_bet_one_one_lemma: \n \"1 < (a::real) \\ 0 < inverse a \\ inverse a < 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 < a \\ 0 < inverse a \\ inverse a < 1\n[PROOF STEP]\nby (metis inverse_less_1_iff inverse_positive_iff_positive \n le_less_trans zero_le_one)", "meta": {"llama_tokens": 134, "file": "Real_Power_RealPower", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894632969136, "lm_q2_score": 0.7981867753392728, "lm_q1q2_score": 0.7142091163165221}} {"text": "[STATEMENT]\nlemma card_subgrp_dvd:\n assumes \"subgroup H G\"\n shows \"card H dvd order G\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card H dvd order G\n[PROOF STEP]\nproof(cases \"finite (carrier G)\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. finite (carrier G) \\ card H dvd order G\n 2. infinite (carrier G) \\ card H dvd order G\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nfinite (carrier G)\n\ngoal (2 subgoals):\n 1. finite (carrier G) \\ card H dvd order G\n 2. infinite (carrier G) \\ card H dvd order G\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\nsubgroup H G\nfinite (carrier G)\n[PROOF STEP]\nhave \"card (rcosets H) * card H = order G\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsubgroup H G\nfinite (carrier G)\n\ngoal (1 subgoal):\n 1. card (rcosets H) * card H = order G\n[PROOF STEP]\nby (metis lagrange)\n[PROOF STATE]\nproof (state)\nthis:\ncard (rcosets H) * card H = order G\n\ngoal (2 subgoals):\n 1. finite (carrier G) \\ card H dvd order G\n 2. infinite (carrier G) \\ card H dvd order G\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (rcosets H) * card H = order G\n\ngoal (1 subgoal):\n 1. card H dvd order G\n[PROOF STEP]\nby (metis dvd_triv_left mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\ncard H dvd order G\n\ngoal (1 subgoal):\n 1. infinite (carrier G) \\ card H dvd order G\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. infinite (carrier G) \\ card H dvd order G\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\ninfinite (carrier G)\n\ngoal (1 subgoal):\n 1. infinite (carrier G) \\ card H dvd order G\n[PROOF STEP]\nhence \"order G = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninfinite (carrier G)\n\ngoal (1 subgoal):\n 1. order G = 0\n[PROOF STEP]\nunfolding order_def\n[PROOF STATE]\nproof (prove)\nusing this:\ninfinite (carrier G)\n\ngoal (1 subgoal):\n 1. card (carrier G) = 0\n[PROOF STEP]\nby (metis card.infinite)\n[PROOF STATE]\nproof (state)\nthis:\norder G = 0\n\ngoal (1 subgoal):\n 1. infinite (carrier G) \\ card H dvd order G\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\norder G = 0\n\ngoal (1 subgoal):\n 1. card H dvd order G\n[PROOF STEP]\nby (metis dvd_0_right)\n[PROOF STATE]\nproof (state)\nthis:\ncard H dvd order G\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 983, "file": "Secondary_Sylow_SndSylow", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289388083214156, "lm_q2_score": 0.8615382076534743, "lm_q1q2_score": 0.7141624551756394}} {"text": "[STATEMENT]\nlemma mtx_hOsc_solution_eq:\n fixes a b :: real\n defines \"\\\\<^sub>1 \\ (b - sqrt (b\\<^sup>2+4*a))/2\" and \"\\\\<^sub>2 \\ (b + sqrt (b\\<^sup>2+4*a))/2\"\n defines \"\\ t \\ mtx (\n [\\\\<^sub>2*exp(t*\\\\<^sub>1) - \\\\<^sub>1*exp(t*\\\\<^sub>2), exp(t*\\\\<^sub>2)-exp(t*\\\\<^sub>1)]#\n [a*exp(t*\\\\<^sub>2) - a*exp(t*\\\\<^sub>1), \\\\<^sub>2*exp(t*\\\\<^sub>2)-\\\\<^sub>1*exp(t*\\\\<^sub>1)]#[])\"\n assumes \"b\\<^sup>2 + a * 4 > 0\" and \"a \\ 0\"\n shows \"P (-\\\\<^sub>2/a) (-\\\\<^sub>1/a) * (\\\\\\\\ i. exp (t * (if i=1 then \\\\<^sub>1 else \\\\<^sub>2))) * (P (-\\\\<^sub>2/a) (-\\\\<^sub>1/a))\\<^sup>-\\<^sup>1 \n = (1/sqrt (b\\<^sup>2 + a * 4)) *\\<^sub>R (\\ t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. P (- \\\\<^sub>2 / a) (- \\\\<^sub>1 / a) * (\\\\\\\\ i. exp (t * (if i = 1 then \\\\<^sub>1 else \\\\<^sub>2))) * P (- \\\\<^sub>2 / a) (- \\\\<^sub>1 / a)\\<^sup>-\\<^sup>1 = (1 / sqrt (b\\<^sup>2 + a * 4)) *\\<^sub>R \\ t\n[PROOF STEP]\nunfolding assms\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. P (- ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / a) (- ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / a) * (\\\\\\\\ i. exp (t * (if i = 1 then (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 else (b + sqrt (b\\<^sup>2 + 4 * a)) / 2))) * P (- ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / a) (- ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / a)\\<^sup>-\\<^sup>1 = (1 / sqrt (b\\<^sup>2 + a * 4)) *\\<^sub>R mtx [[(b + sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2)) - (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)), exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2))], [a * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - a * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2)), (b + sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2))]]\n[PROOF STEP]\napply(subst inv_mtx_chB_hOsc)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. - ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / a \\ - ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / a\n 2. P (- ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / a) (- ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / a) * (\\\\\\\\ i. exp (t * (if i = 1 then (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 else (b + sqrt (b\\<^sup>2 + 4 * a)) / 2))) * (1 / (- ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / a - - ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / a)) *\\<^sub>R mtx [[1, - (- ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / a)], [- 1, - ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / a]] = (1 / sqrt (b\\<^sup>2 + a * 4)) *\\<^sub>R mtx [[(b + sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2)) - (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)), exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2))], [a * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - a * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2)), (b + sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2))]]\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>1 \\ (b - sqrt (b\\<^sup>2 + 4 * a)) / 2\n\\\\<^sub>2 \\ (b + sqrt (b\\<^sup>2 + 4 * a)) / 2\n\\ ?t \\ mtx [[\\\\<^sub>2 * exp (?t * \\\\<^sub>1) - \\\\<^sub>1 * exp (?t * \\\\<^sub>2), exp (?t * \\\\<^sub>2) - exp (?t * \\\\<^sub>1)], [a * exp (?t * \\\\<^sub>2) - a * exp (?t * \\\\<^sub>1), \\\\<^sub>2 * exp (?t * \\\\<^sub>2) - \\\\<^sub>1 * exp (?t * \\\\<^sub>1)]]\n0 < b\\<^sup>2 + a * 4\na \\ 0\n\ngoal (2 subgoals):\n 1. - ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / a \\ - ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / a\n 2. P (- ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / a) (- ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / a) * (\\\\\\\\ i. exp (t * (if i = 1 then (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 else (b + sqrt (b\\<^sup>2 + 4 * a)) / 2))) * (1 / (- ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / a - - ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / a)) *\\<^sub>R mtx [[1, - (- ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / a)], [- 1, - ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / a]] = (1 / sqrt (b\\<^sup>2 + a * 4)) *\\<^sub>R mtx [[(b + sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2)) - (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)), exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2))], [a * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - a * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2)), (b + sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2))]]\n[PROOF STEP]\napply(simp_all add: mtx_times_scaleR_commute, subst sq_mtx_eq_iff)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\\\<^sub>1 \\ (b - sqrt (b\\<^sup>2 + 4 * a)) / 2; \\\\<^sub>2 \\ (b + sqrt (b\\<^sup>2 + 4 * a)) / 2; \\t. \\ t \\ mtx [[(b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2, exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)], [a * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - a * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2), (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2]]; 0 < b\\<^sup>2 + a * 4; a \\ 0; \\\\<^sub>1 \\ (b - sqrt (b\\<^sup>2 + 4 * a)) / 2; \\\\<^sub>2 \\ (b + sqrt (b\\<^sup>2 + 4 * a)) / 2; \\t. \\ t \\ mtx [[\\\\<^sub>2 * exp (t * \\\\<^sub>1) - \\\\<^sub>1 * exp (t * \\\\<^sub>2), exp (t * \\\\<^sub>2) - exp (t * \\\\<^sub>1)], [a * exp (t * \\\\<^sub>2) - a * exp (t * \\\\<^sub>1), \\\\<^sub>2 * exp (t * \\\\<^sub>2) - \\\\<^sub>1 * exp (t * \\\\<^sub>1)]]; 0 < b\\<^sup>2 + a * 4; a \\ 0\\ \\ \\i j. ((1 / ((b - sqrt (b\\<^sup>2 + 4 * a)) / (2 * a) - (b + sqrt (b\\<^sup>2 + 4 * a)) / (2 * a))) *\\<^sub>R (P (- ((b + sqrt (b\\<^sup>2 + 4 * a)) / (2 * a))) (- ((b - sqrt (b\\<^sup>2 + 4 * a)) / (2 * a))) * (\\\\\\\\ i. exp (t * (if i = 1 then (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 else (b + sqrt (b\\<^sup>2 + 4 * a)) / 2))) * mtx [[1, (b - sqrt (b\\<^sup>2 + 4 * a)) / (2 * a)], [- 1, - ((b + sqrt (b\\<^sup>2 + 4 * a)) / (2 * a))]])) $$ i $ j = ((1 / sqrt (b\\<^sup>2 + a * 4)) *\\<^sub>R mtx [[(b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2, exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)], [a * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - a * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2), (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2]]) $$ i $ j\n[PROOF STEP]\nunfolding UNIV_2 sq_mtx_times_eq sq_mtx_scaleR_eq sq_mtx_uminus_eq\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(b - sqrt (b\\<^sup>2 + 4 * a)) / 2 \\ (b - sqrt (b\\<^sup>2 + 4 * a)) / 2; (b + sqrt (b\\<^sup>2 + 4 * a)) / 2 \\ (b + sqrt (b\\<^sup>2 + 4 * a)) / 2; \\t. \\ t \\ mtx [[(b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2, exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)], [a * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - a * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2), (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2]]; 0 < b\\<^sup>2 + a * 4; a \\ 0; \\\\<^sub>1 \\ (b - sqrt (b\\<^sup>2 + 4 * a)) / 2; \\\\<^sub>2 \\ (b + sqrt (b\\<^sup>2 + 4 * a)) / 2; \\t. \\ t \\ mtx [[(b + sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2)) - (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)), exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2))], [a * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - a * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2)), (b + sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) - (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 * exp (t * ((b - sqrt (b\\<^sup>2 + 4 * a)) / 2))]]; 0 < b\\<^sup>2 + a * 4; a \\ 0\\ \\ \\i j. to_mtx (\\i j. (1 / ((b - sqrt (b\\<^sup>2 + 4 * a)) / (2 * a) - (b + sqrt (b\\<^sup>2 + 4 * a)) / (2 * a))) *\\<^sub>R to_mtx (\\i j. \\k\\insert 1 (skip 2). to_mtx (\\i j. \\k\\insert 1 (skip 2). P (- ((b + sqrt (b\\<^sup>2 + 4 * a)) / (2 * a))) (- ((b - sqrt (b\\<^sup>2 + 4 * a)) / (2 * a))) $$ i $ k * (\\\\\\\\ i. exp (t * (if i = 1 then (b - sqrt (b\\<^sup>2 + 4 * a)) / 2 else (b + sqrt (b\\<^sup>2 + 4 * a)) / 2))) $$ k $ j) $$ i $ k * mtx [[1, (b - sqrt (b\\<^sup>2 + 4 * a)) / (2 * a)], [- 1, - ((b + sqrt (b\\<^sup>2 + 4 * a)) / (2 * a))]] $$ k $ j) $$ i $ j) $$ i $ j = to_mtx (\\i j. (1 / sqrt (b\\<^sup>2 + a * 4)) *\\<^sub>R mtx [[(b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2, exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)], [a * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - a * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2), (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2]] $$ i $ j) $$ i $ j\n[PROOF STEP]\napply(simp_all add: axis_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(b - sqrt (b\\<^sup>2 + 4 * a)) / 2 \\ (b - sqrt (b\\<^sup>2 + 4 * a)) / 2; (b + sqrt (b\\<^sup>2 + 4 * a)) / 2 \\ (b + sqrt (b\\<^sup>2 + 4 * a)) / 2; \\t. mtx [[(b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2, exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)], [a * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - a * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2), (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2]] \\ mtx [[(b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2, exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)], [a * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - a * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2), (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2]]; \\\\<^sub>1 \\ (b - sqrt (b\\<^sup>2 + 4 * a)) / 2; \\\\<^sub>2 \\ (b + sqrt (b\\<^sup>2 + 4 * a)) / 2; \\t. \\ t \\ mtx [[(b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2, exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)], [a * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - a * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2), (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2]]; 0 < b\\<^sup>2 + a * 4; a \\ 0\\ \\ \\i. (i = 2 \\ (\\j. (j = 2 \\ (exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) * (b - sqrt (b\\<^sup>2 + 4 * a)) / (2 * a) - exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) * (b + sqrt (b\\<^sup>2 + 4 * a)) / (2 * a)) / ((b - sqrt (b\\<^sup>2 + 4 * a)) / (2 * a) - (b + sqrt (b\\<^sup>2 + 4 * a)) / (2 * a)) = ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2) / sqrt (b\\<^sup>2 + a * 4)) \\ (j \\ 2 \\ j = 1 \\ (exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) - exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2)) / ((b - sqrt (b\\<^sup>2 + 4 * a)) / (2 * a) - (b + sqrt (b\\<^sup>2 + 4 * a)) / (2 * a)) = (a * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - a * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)) / sqrt (b\\<^sup>2 + a * 4)))) \\ (i \\ 2 \\ i = 1 \\ (\\j. (j = 2 \\ ((b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) * (b + sqrt (b\\<^sup>2 + 4 * a)) / (4 * (a * a)) - (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) * (b - sqrt (b\\<^sup>2 + 4 * a)) / (4 * (a * a))) / ((b - sqrt (b\\<^sup>2 + 4 * a)) / (2 * a) - (b + sqrt (b\\<^sup>2 + 4 * a)) / (2 * a)) = (exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) - exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2)) / sqrt (b\\<^sup>2 + a * 4)) \\ (j \\ 2 \\ j = 1 \\ ((b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / (2 * a) - (b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / (2 * a)) / ((b - sqrt (b\\<^sup>2 + 4 * a)) / (2 * a) - (b + sqrt (b\\<^sup>2 + 4 * a)) / (2 * a)) = ((b + sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b - sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2 - (b - sqrt (b\\<^sup>2 + 4 * a)) * exp (t * (b + sqrt (b\\<^sup>2 + 4 * a)) / 2) / 2) / sqrt (b\\<^sup>2 + a * 4))))\n[PROOF STEP]\nby (auto simp: field_simps, auto simp: field_power_simps)+", "meta": {"llama_tokens": 7685, "file": "Matrices_for_ODEs_MTX_Examples", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9252299570920386, "lm_q2_score": 0.7718435083355187, "lm_q1q2_score": 0.7141327360990405}} {"text": "[STATEMENT]\nlemma bernoulli'_upto_20 [simp]:\n \"bernoulli' 2 = 1 / 6\" \n \"bernoulli' 4 = -(1 / 30)\" \n \"bernoulli' 6 = 1 / 42\" \n \"bernoulli' 8 = - (1 / 30)\"\n \"bernoulli' 10 = 5 / 66\" \n \"bernoulli' 12 = - (691 / 2730)\" \n \"bernoulli' 14 = 7 / 6\"\n \"bernoulli' 16 = -(3617 / 510)\" \n \"bernoulli' 18 = 43867 / 798\" \n \"bernoulli' 20 = -(174611 / 330)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((bernoulli' 2 = 1 / 6 &&& bernoulli' 4 = - (1 / 30)) &&& bernoulli' 6 = 1 / 42 &&& bernoulli' 8 = - (1 / 30) &&& bernoulli' 10 = 5 / 66) &&& (bernoulli' 12 = - (691 / 2730) &&& bernoulli' 14 = 7 / 6) &&& bernoulli' 16 = - (3617 / 510) &&& bernoulli' 18 = 43867 / 798 &&& bernoulli' 20 = - (174611 / 330)\n[PROOF STEP]\nby (simp_all add: bernoulli'_def)", "meta": {"llama_tokens": 461, "file": "Bernoulli_Bernoulli_FPS", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9252299550303293, "lm_q2_score": 0.7718435030872968, "lm_q1q2_score": 0.7141327296519114}} {"text": "[STATEMENT]\nlemma \n \"((\\x. ((a powr x - x * ln a) / (b powr x - x * ln b)) powr (1 / x ^ 2)) \\\n exp ((ln a ^ 2 - ln b ^ 2) / 2)) (at 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. ((a powr x - x * ln a) / (b powr x - x * ln b)) powr (1 / x\\<^sup>2)) \\0\\ exp (((ln a)\\<^sup>2 - (ln b)\\<^sup>2) / 2)\n[PROOF STEP]\nusing ab\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n0 < b\n\ngoal (1 subgoal):\n 1. (\\x. ((a powr x - x * ln a) / (b powr x - x * ln b)) powr (1 / x\\<^sup>2)) \\0\\ exp (((ln a)\\<^sup>2 - (ln b)\\<^sup>2) / 2)\n[PROOF STEP]\nby (real_asymp simp: power2_eq_square)", "meta": {"llama_tokens": 328, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9124361628580401, "lm_q2_score": 0.7826624840223699, "lm_q1q2_score": 0.7141295537343133}} {"text": "[STATEMENT]\nlemma arcs_graph_G_ge_2vertsG:\n \"\\graph G; connected G; Suc n = card (verts G)\\ \\ card (arcs G) \\ 2 * (card (verts G) - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\graph G; Digraph_Component.connected G; Suc n = card (verts G)\\ \\ 2 * (card (verts G) - 1) \\ card (arcs G)\n[PROOF STEP]\nusing arcs_graph_G_ge_2vertsT connected_verts_G_eq_T\n[PROOF STATE]\nproof (prove)\nusing this:\ngraph G \\ 2 * (card (verts T) - 1) \\ card (arcs G)\n\\graph G; Digraph_Component.connected G; Suc n = card (verts G)\\ \\ verts T = verts G\n\ngoal (1 subgoal):\n 1. \\graph G; Digraph_Component.connected G; Suc n = card (verts G)\\ \\ 2 * (card (verts G) - 1) \\ card (arcs G)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 356, "file": "Query_Optimization_Directed_Tree_Additions", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361580958426, "lm_q2_score": 0.7826624789529375, "lm_q1q2_score": 0.7141295453815866}} {"text": "[STATEMENT]\nlemma exp_mult_2: \"exp (y * 2 :: real) = exp y * exp y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp (y * 2) = exp y * exp y\n[PROOF STEP]\nby (subst exp_add [symmetric]) simp", "meta": {"llama_tokens": 86, "file": "Stirling_Formula_Stirling_Formula", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9124361509525462, "lm_q2_score": 0.7826624789529375, "lm_q1q2_score": 0.7141295397907964}} {"text": "[STATEMENT]\nlemma DERIV_fun_powr:\n fixes r :: real\n assumes g: \"DERIV g x :> m\"\n and pos: \"g x > 0\"\n shows \"DERIV (\\x. (g x) powr r) x :> r * (g x) powr (r - of_nat 1) * m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. g x powr r) has_real_derivative r * g x powr (r - real 1) * m) (at x)\n[PROOF STEP]\nusing DERIV_powr[OF g pos DERIV_const, of r] pos\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. g x powr r) has_real_derivative g x powr r * (0 * ln (g x) + m * r / g x)) (at x)\n0 < g x\n\ngoal (1 subgoal):\n 1. ((\\x. g x powr r) has_real_derivative r * g x powr (r - real 1) * m) (at x)\n[PROOF STEP]\nby (simp add: powr_diff field_simps)", "meta": {"llama_tokens": 328, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942014971872, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7140814485042566}} {"text": "[STATEMENT]\nlemma ring_iso_set_trans:\n \"\\ f \\ ring_iso R S; g \\ ring_iso S Q \\ \\ (g \\ f) \\ ring_iso R Q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ ring_iso R S; g \\ ring_iso S Q\\ \\ g \\ f \\ ring_iso R Q\n[PROOF STEP]\nunfolding ring_iso_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ {h \\ ring_hom R S. bij_betw h (carrier R) (carrier S)}; g \\ {h \\ ring_hom S Q. bij_betw h (carrier S) (carrier Q)}\\ \\ g \\ f \\ {h \\ ring_hom R Q. bij_betw h (carrier R) (carrier Q)}\n[PROOF STEP]\nusing bij_betw_trans ring_hom_trans\n[PROOF STATE]\nproof (prove)\nusing this:\n\\bij_betw ?f ?A ?B; bij_betw ?g ?B ?C\\ \\ bij_betw (?g \\ ?f) ?A ?C\n\\?f \\ ring_hom ?R ?S; ?g \\ ring_hom ?S ?T\\ \\ ?g \\ ?f \\ ring_hom ?R ?T\n\ngoal (1 subgoal):\n 1. \\f \\ {h \\ ring_hom R S. bij_betw h (carrier R) (carrier S)}; g \\ {h \\ ring_hom S Q. bij_betw h (carrier S) (carrier Q)}\\ \\ g \\ f \\ {h \\ ring_hom R Q. bij_betw h (carrier R) (carrier Q)}\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 559, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032941988938414, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7140814464462328}} {"text": "[STATEMENT]\nlemma matpow_mono: \"0 \\ C \\ C \\ D \\ matpow (C :: real^_^_) n \\ matpow D n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 \\ C; C \\ D\\ \\ matpow C n \\ matpow D n\n[PROOF STEP]\nby (induction n) (auto intro!: matrix_mult_mono nonneg_matpow)", "meta": {"llama_tokens": 139, "file": "MDP-Algorithms_Matrix_Util", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110511888303, "lm_q2_score": 0.8006920092299293, "lm_q1q2_score": 0.7140659824298399}} {"text": "[STATEMENT]\nlemma infsetsum_cmult_right:\n fixes f :: \"'a \\ 'b :: {banach, real_normed_algebra, second_countable_topology}\"\n assumes \"c \\ 0 \\ f abs_summable_on A\"\n shows \"infsetsum (\\x. c * f x) A = c * infsetsum f A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. c * f x) = c * infsetsum f A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'b) \\ f abs_summable_on A\n\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. c * f x) = c * infsetsum f A\n[PROOF STEP]\nunfolding infsetsum_def abs_summable_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'b) \\ integrable (count_space A) f\n\ngoal (1 subgoal):\n 1. LINT x|count_space A. c * f x = c * integral\\<^sup>L (count_space A) f\n[PROOF STEP]\nby (rule Bochner_Integration.integral_mult_right)", "meta": {"llama_tokens": 369, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110454379297, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.714065971462982}} {"text": "[STATEMENT]\ntheorem inorder_ins:\n \"\\ avl t; sorted(inorder t) \\ \\ inorder(tree(ins x t)) = ins_list x (inorder t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\avl t; Sorted_Less.sorted (Tree2.inorder t)\\ \\ Tree2.inorder (tree (ins x t)) = ins_list x (Tree2.inorder t)\n[PROOF STEP]\napply(induction t)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\avl \\\\; Sorted_Less.sorted (Tree2.inorder \\\\)\\ \\ Tree2.inorder (tree (ins x \\\\)) = ins_list x (Tree2.inorder \\\\)\n 2. \\t1 x2 t2. \\\\avl t1; Sorted_Less.sorted (Tree2.inorder t1)\\ \\ Tree2.inorder (tree (ins x t1)) = ins_list x (Tree2.inorder t1); \\avl t2; Sorted_Less.sorted (Tree2.inorder t2)\\ \\ Tree2.inorder (tree (ins x t2)) = ins_list x (Tree2.inorder t2); avl \\t1, x2, t2\\; Sorted_Less.sorted (Tree2.inorder \\t1, x2, t2\\)\\ \\ Tree2.inorder (tree (ins x \\t1, x2, t2\\)) = ins_list x (Tree2.inorder \\t1, x2, t2\\)\n[PROOF STEP]\napply (auto simp: ins_list_simps split!: splits)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 549, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110396870287, "lm_q2_score": 0.800691997339971, "lm_q1q2_score": 0.7140659626168431}} {"text": "[STATEMENT]\nlemma has_derivative_at_within:\n \"(f has_derivative f') (at x within s) \\\n (bounded_linear f' \\ ((\\y. ((f y - f x) - f' (y - x)) /\\<^sub>R norm (y - x)) \\ 0) (at x within s))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nproof (cases \"at x within s = bot\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. at x within s = bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n 2. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nat x within s = bot\n\ngoal (2 subgoals):\n 1. at x within s = bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n 2. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nat x within s = bot\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nat x within s = bot\n\ngoal (1 subgoal):\n 1. (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nby (metis (no_types, lifting) has_derivative_within tendsto_bot)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n\ngoal (1 subgoal):\n 1. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nat x within s \\ bot\n\ngoal (1 subgoal):\n 1. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nat x within s \\ bot\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nat x within s \\ bot\n\ngoal (1 subgoal):\n 1. (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nby (simp add: Lim_ident_at has_derivative_def)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1550, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.875787001374006, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7139700212037309}} {"text": "[STATEMENT]\nlemma has_derivative_at_within:\n \"(f has_derivative f') (at x within s) \\\n (bounded_linear f' \\ ((\\y. ((f y - f x) - f' (y - x)) /\\<^sub>R norm (y - x)) \\ 0) (at x within s))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nproof (cases \"at x within s = bot\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. at x within s = bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n 2. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nat x within s = bot\n\ngoal (2 subgoals):\n 1. at x within s = bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n 2. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nat x within s = bot\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nat x within s = bot\n\ngoal (1 subgoal):\n 1. (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nby (metis (no_types, lifting) has_derivative_within tendsto_bot)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n\ngoal (1 subgoal):\n 1. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nat x within s \\ bot\n\ngoal (1 subgoal):\n 1. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nat x within s \\ bot\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nat x within s \\ bot\n\ngoal (1 subgoal):\n 1. (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nby (simp add: Lim_ident_at has_derivative_def)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1550, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.875787001374006, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7139700212037309}} {"text": "[STATEMENT]\nlemma has_derivative_at_within:\n \"(f has_derivative f') (at x within s) \\\n (bounded_linear f' \\ ((\\y. ((f y - f x) - f' (y - x)) /\\<^sub>R norm (y - x)) \\ 0) (at x within s))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nproof (cases \"at x within s = bot\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. at x within s = bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n 2. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nat x within s = bot\n\ngoal (2 subgoals):\n 1. at x within s = bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n 2. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nat x within s = bot\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nat x within s = bot\n\ngoal (1 subgoal):\n 1. (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nby (metis (no_types, lifting) has_derivative_within tendsto_bot)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n\ngoal (1 subgoal):\n 1. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nat x within s \\ bot\n\ngoal (1 subgoal):\n 1. at x within s \\ bot \\ (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nat x within s \\ bot\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nat x within s \\ bot\n\ngoal (1 subgoal):\n 1. (f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n[PROOF STEP]\nby (simp add: Lim_ident_at has_derivative_def)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_derivative f') (at x within s) = (bounded_linear f' \\ ((\\y. (f y - f x - f' (y - x)) /\\<^sub>R norm (y - x)) \\ (0::'b)) (at x within s))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1550, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.875787001374006, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.713970015306516}} {"text": "[STATEMENT]\nlemma enumerate_Suc'':\n fixes S :: \"'a::wellorder set\"\n assumes \"infinite S\"\n shows \"enumerate S (Suc n) = (LEAST s. s \\ S \\ enumerate S n < s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. wellorder_class.enumerate S (Suc n) = (LEAST s. s \\ S \\ wellorder_class.enumerate S n < s)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ninfinite S\n\ngoal (1 subgoal):\n 1. wellorder_class.enumerate S (Suc n) = (LEAST s. s \\ S \\ wellorder_class.enumerate S n < s)\n[PROOF STEP]\nproof (induct n arbitrary: S)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\S. infinite S \\ wellorder_class.enumerate S (Suc 0) = (LEAST s. s \\ S \\ wellorder_class.enumerate S 0 < s)\n 2. \\n S. \\\\S. infinite S \\ wellorder_class.enumerate S (Suc n) = (LEAST s. s \\ S \\ wellorder_class.enumerate S n < s); infinite S\\ \\ wellorder_class.enumerate S (Suc (Suc n)) = (LEAST s. s \\ S \\ wellorder_class.enumerate S (Suc n) < s)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\ninfinite S\n\ngoal (2 subgoals):\n 1. \\S. infinite S \\ wellorder_class.enumerate S (Suc 0) = (LEAST s. s \\ S \\ wellorder_class.enumerate S 0 < s)\n 2. \\n S. \\\\S. infinite S \\ wellorder_class.enumerate S (Suc n) = (LEAST s. s \\ S \\ wellorder_class.enumerate S n < s); infinite S\\ \\ wellorder_class.enumerate S (Suc (Suc n)) = (LEAST s. s \\ S \\ wellorder_class.enumerate S (Suc n) < s)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ninfinite S\n[PROOF STEP]\nhave \"\\s \\ S. enumerate S 0 \\ s\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninfinite S\n\ngoal (1 subgoal):\n 1. \\s\\S. wellorder_class.enumerate S 0 \\ s\n[PROOF STEP]\nby (auto simp: enumerate.simps intro: Least_le)\n[PROOF STATE]\nproof (state)\nthis:\n\\s\\S. wellorder_class.enumerate S 0 \\ s\n\ngoal (2 subgoals):\n 1. \\S. infinite S \\ wellorder_class.enumerate S (Suc 0) = (LEAST s. s \\ S \\ wellorder_class.enumerate S 0 < s)\n 2. \\n S. \\\\S. infinite S \\ wellorder_class.enumerate S (Suc n) = (LEAST s. s \\ S \\ wellorder_class.enumerate S n < s); infinite S\\ \\ wellorder_class.enumerate S (Suc (Suc n)) = (LEAST s. s \\ S \\ wellorder_class.enumerate S (Suc n) < s)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\s\\S. wellorder_class.enumerate S 0 \\ s\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n\\s\\S. wellorder_class.enumerate S 0 \\ s\n\ngoal (1 subgoal):\n 1. wellorder_class.enumerate S (Suc 0) = (LEAST s. s \\ S \\ wellorder_class.enumerate S 0 < s)\n[PROOF STEP]\nunfolding enumerate_Suc' enumerate_0[of \"S - {enumerate S 0}\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\s\\S. wellorder_class.enumerate S 0 \\ s\n\ngoal (1 subgoal):\n 1. (LEAST n. n \\ S - {wellorder_class.enumerate S 0}) = (LEAST s. s \\ S \\ wellorder_class.enumerate S 0 < s)\n[PROOF STEP]\nby (intro arg_cong[where f = Least] ext) auto\n[PROOF STATE]\nproof (state)\nthis:\nwellorder_class.enumerate S (Suc 0) = (LEAST s. s \\ S \\ wellorder_class.enumerate S 0 < s)\n\ngoal (1 subgoal):\n 1. \\n S. \\\\S. infinite S \\ wellorder_class.enumerate S (Suc n) = (LEAST s. s \\ S \\ wellorder_class.enumerate S n < s); infinite S\\ \\ wellorder_class.enumerate S (Suc (Suc n)) = (LEAST s. s \\ S \\ wellorder_class.enumerate S (Suc n) < s)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n S. \\\\S. infinite S \\ wellorder_class.enumerate S (Suc n) = (LEAST s. s \\ S \\ wellorder_class.enumerate S n < s); infinite S\\ \\ wellorder_class.enumerate S (Suc (Suc n)) = (LEAST s. s \\ S \\ wellorder_class.enumerate S (Suc n) < s)\n[PROOF STEP]\ncase (Suc n S)\n[PROOF STATE]\nproof (state)\nthis:\ninfinite ?S \\ wellorder_class.enumerate ?S (Suc n) = (LEAST s. s \\ ?S \\ wellorder_class.enumerate ?S n < s)\ninfinite S\n\ngoal (1 subgoal):\n 1. \\n S. \\\\S. infinite S \\ wellorder_class.enumerate S (Suc n) = (LEAST s. s \\ S \\ wellorder_class.enumerate S n < s); infinite S\\ \\ wellorder_class.enumerate S (Suc (Suc n)) = (LEAST s. s \\ S \\ wellorder_class.enumerate S (Suc n) < s)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. wellorder_class.enumerate S (Suc (Suc n)) = (LEAST s. s \\ S \\ wellorder_class.enumerate S (Suc n) < s)\n[PROOF STEP]\nusing enumerate_mono[OF zero_less_Suc \\infinite S\\, of n] \\infinite S\\\n[PROOF STATE]\nproof (prove)\nusing this:\nwellorder_class.enumerate S 0 < wellorder_class.enumerate S (Suc n)\ninfinite S\n\ngoal (1 subgoal):\n 1. wellorder_class.enumerate S (Suc (Suc n)) = (LEAST s. s \\ S \\ wellorder_class.enumerate S (Suc n) < s)\n[PROOF STEP]\napply (subst (1 2) enumerate_Suc')\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\wellorder_class.enumerate S 0 < wellorder_class.enumerate S (Suc n); infinite S\\ \\ wellorder_class.enumerate (S - {wellorder_class.enumerate S 0}) (Suc n) = (LEAST s. s \\ S \\ wellorder_class.enumerate (S - {wellorder_class.enumerate S 0}) n < s)\n[PROOF STEP]\napply (subst Suc)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\wellorder_class.enumerate S 0 < wellorder_class.enumerate S (Suc n); infinite S\\ \\ infinite (S - {wellorder_class.enumerate S 0})\n 2. \\wellorder_class.enumerate S 0 < wellorder_class.enumerate S (Suc n); infinite S\\ \\ (LEAST s. s \\ S - {wellorder_class.enumerate S 0} \\ wellorder_class.enumerate (S - {wellorder_class.enumerate S 0}) n < s) = (LEAST s. s \\ S \\ wellorder_class.enumerate (S - {wellorder_class.enumerate S 0}) n < s)\n[PROOF STEP]\napply (use \\infinite S\\ in simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\wellorder_class.enumerate S 0 < wellorder_class.enumerate S (Suc n); infinite S\\ \\ (LEAST s. s \\ S - {wellorder_class.enumerate S 0} \\ wellorder_class.enumerate (S - {wellorder_class.enumerate S 0}) n < s) = (LEAST s. s \\ S \\ wellorder_class.enumerate (S - {wellorder_class.enumerate S 0}) n < s)\n[PROOF STEP]\napply (intro arg_cong[where f = Least] ext)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\s. \\wellorder_class.enumerate S 0 < wellorder_class.enumerate S (Suc n); infinite S\\ \\ (s \\ S - {wellorder_class.enumerate S 0} \\ wellorder_class.enumerate (S - {wellorder_class.enumerate S 0}) n < s) = (s \\ S \\ wellorder_class.enumerate (S - {wellorder_class.enumerate S 0}) n < s)\n[PROOF STEP]\napply (auto simp flip: enumerate_Suc')\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\nwellorder_class.enumerate S (Suc (Suc n)) = (LEAST s. s \\ S \\ wellorder_class.enumerate S (Suc n) < s)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2876, "file": null, "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869948899666, "lm_q2_score": 0.815232489352, "lm_q1q2_score": 0.7139700119862548}} {"text": "[STATEMENT]\nlemma abs_power_le [simp]: \"\\a\\^n \\ \\b\\^n \\ n = 0 \\ \\a\\ \\ \\b\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a\\ ^ n \\ \\b\\ ^ n) = (n = 0 \\ \\a\\ \\ \\b\\)\n[PROOF STEP]\nby (subst nonneg_power_le, auto)", "meta": {"llama_tokens": 157, "file": "LLL_Basis_Reduction_Norms", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869916479467, "lm_q2_score": 0.8152324826183822, "lm_q1q2_score": 0.71397000344604}} {"text": "[STATEMENT]\nlemma cos_0_iff_canon:\n assumes \"cos \\ = 0\" and \"-pi < \\\" and \"\\ \\ pi\"\n shows \"\\ = pi/2 \\ \\ = -pi/2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ = pi / 2 \\ \\ = - pi / 2\n[PROOF STEP]\nby (smt (verit, best) arccos_0 arccos_cos assms cos_minus divide_minus_left)", "meta": {"llama_tokens": 149, "file": "Complex_Geometry_More_Transcendental", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869786798663, "lm_q2_score": 0.815232489352, "lm_q1q2_score": 0.7139699987712543}} {"text": "[STATEMENT]\nlemma degree_monic_charpoly: fixes A :: \"'a :: comm_ring_1 ^ 'n ^ 'n\" \n shows \"degree (charpoly A) = CARD('n) \\ monic (charpoly A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. degree (charpoly A) = CARD('n) \\ monic (charpoly A)\n[PROOF STEP]\nproof (transfer, goal_cases)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\A. A \\ carrier_mat CARD('n) CARD('n) \\ degree (char_poly A) = CARD('n) \\ monic (char_poly A)\n[PROOF STEP]\ncase 1\n[PROOF STATE]\nproof (state)\nthis:\nA_ \\ carrier_mat CARD('n) CARD('n)\n\ngoal (1 subgoal):\n 1. \\A. A \\ carrier_mat CARD('n) CARD('n) \\ degree (char_poly A) = CARD('n) \\ monic (char_poly A)\n[PROOF STEP]\nfrom degree_monic_char_poly[OF 1]\n[PROOF STATE]\nproof (chain)\npicking this:\ndegree (char_poly A_) = CARD('n) \\ coeff (char_poly A_) CARD('n) = (1::'a)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ndegree (char_poly A_) = CARD('n) \\ coeff (char_poly A_) CARD('n) = (1::'a)\n\ngoal (1 subgoal):\n 1. degree (char_poly A_) = CARD('n) \\ monic (char_poly A_)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndegree (char_poly A_) = CARD('n) \\ monic (char_poly A_)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 560, "file": "Perron_Frobenius_HMA_Connect", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970717197768, "lm_q2_score": 0.8104789155369047, "lm_q1q2_score": 0.713867455495526}} {"text": "[STATEMENT]\nlemma cofinite_closure_infinite:\n fixes X :: \"'a cofinite set\"\n assumes \"infinite X\"\n shows \"closure X = UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. closure X = UNIV\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ninfinite X\n\ngoal (1 subgoal):\n 1. closure X = UNIV\n[PROOF STEP]\nby (metis Compl_empty_eq closure_subset double_compl finite_subset interior_complement open_cofinite_def open_interior)", "meta": {"llama_tokens": 172, "file": "Kuratowski_Closure_Complement_KuratowskiClosureComplementTheorem", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970779778824, "lm_q2_score": 0.8104789063814617, "lm_q1q2_score": 0.7138674525035011}} {"text": "[STATEMENT]\nlemma nat_exists_least_iff: \"(\\(n::nat). P n) \\ (\\n. P n \\ (\\m < n. \\ P m))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. P n) = (\\n. P n \\ (\\m P m))\n[PROOF STEP]\nby (metis ex_least_nat_le not_less0)", "meta": {"llama_tokens": 138, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9059898127684335, "lm_q2_score": 0.7879311931529758, "lm_q1q2_score": 0.7138576341590729}} {"text": "[STATEMENT]\nlemma list_sel_aux_eq_foldl: \"list_sel_aux f x y = foldl (\\a b. a * f x b) 1 y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. list_sel_aux f x y = foldl (\\a b. a * f x b) 1 y\n[PROOF STEP]\napply(induction y)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. list_sel_aux f x [] = foldl (\\a b. a * f x b) 1 []\n 2. \\a y. list_sel_aux f x y = foldl (\\a b. a * f x b) 1 y \\ list_sel_aux f x (a # y) = foldl (\\a b. a * f x b) 1 (a # y)\n[PROOF STEP]\napply(auto)[2]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a y. list_sel_aux f x y = foldl (\\a b. a * f x b) 1 y \\ f x a * foldl (\\a b. a * f x b) 1 y = foldl (\\a b. a * f x b) (f x a) y\n[PROOF STEP]\nusing foldl_acc_extr\n[PROOF STATE]\nproof (prove)\nusing this:\nfoldl (\\a b. a * ?f ?x b) ?z ?y = ?z * foldl (\\a b. a * ?f ?x b) 1 ?y\n\ngoal (1 subgoal):\n 1. \\a y. list_sel_aux f x y = foldl (\\a b. a * f x b) 1 y \\ f x a * foldl (\\a b. a * f x b) 1 y = foldl (\\a b. a * f x b) (f x a) y\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 530, "file": "Query_Optimization_Selectivities", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.841825655188238, "lm_q2_score": 0.8479677583778257, "lm_q1q2_score": 0.7138410137749146}} {"text": "[STATEMENT]\nlemma sum_of_squares_nat_bound:\n fixes x y n :: nat\n assumes \"x ^ 2 + y ^ 2 = n\"\n shows \"x \\ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ n\n[PROOF STEP]\nproof (cases \"x = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x = 0 \\ x \\ n\n 2. x \\ 0 \\ x \\ n\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nx \\ 0\n\ngoal (2 subgoals):\n 1. x = 0 \\ x \\ n\n 2. x \\ 0 \\ x \\ n\n[PROOF STEP]\nhence \"x * 1 \\ x ^ 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ 0\n\ngoal (1 subgoal):\n 1. x * 1 \\ x\\<^sup>2\n[PROOF STEP]\nunfolding power2_eq_square\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ 0\n\ngoal (1 subgoal):\n 1. x * 1 \\ x * x\n[PROOF STEP]\nby (intro mult_mono) auto\n[PROOF STATE]\nproof (state)\nthis:\nx * 1 \\ x\\<^sup>2\n\ngoal (2 subgoals):\n 1. x = 0 \\ x \\ n\n 2. x \\ 0 \\ x \\ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx * 1 \\ x\\<^sup>2\n\ngoal (2 subgoals):\n 1. x = 0 \\ x \\ n\n 2. x \\ 0 \\ x \\ n\n[PROOF STEP]\nhave \"\\ \\ x ^ 2 + y ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x\\<^sup>2 \\ x\\<^sup>2 + y\\<^sup>2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx\\<^sup>2 \\ x\\<^sup>2 + y\\<^sup>2\n\ngoal (2 subgoals):\n 1. x = 0 \\ x \\ n\n 2. x \\ 0 \\ x \\ n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx\\<^sup>2 \\ x\\<^sup>2 + y\\<^sup>2\n\ngoal (2 subgoals):\n 1. x = 0 \\ x \\ n\n 2. x \\ 0 \\ x \\ n\n[PROOF STEP]\nhave \"\\ = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x\\<^sup>2 + y\\<^sup>2 = n\n[PROOF STEP]\nby fact\n[PROOF STATE]\nproof (state)\nthis:\nx\\<^sup>2 + y\\<^sup>2 = n\n\ngoal (2 subgoals):\n 1. x = 0 \\ x \\ n\n 2. x \\ 0 \\ x \\ n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nx * 1 \\ n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx * 1 \\ n\n\ngoal (1 subgoal):\n 1. x \\ n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx \\ n\n\ngoal (1 subgoal):\n 1. x = 0 \\ x \\ n\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 1087, "file": "Gaussian_Integers_Gaussian_Integers", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8479677506936878, "lm_q2_score": 0.8418256412990658, "lm_q1q2_score": 0.7138409955286401}} {"text": "[STATEMENT]\nlemma productdivides: \"\\x mod a = (0::nat); x mod b = 0; prime a; prime b; a \\ b\\ \\ x mod (a*b) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x mod a = 0; x mod b = 0; prime a; prime b; a \\ b\\ \\ x mod (a * b) = 0\n[PROOF STEP]\nby (simp add: mod_eq_0_iff_dvd primes_coprime divides_mult)", "meta": {"llama_tokens": 170, "file": "RSAPSS_Productdivides", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587993853654, "lm_q2_score": 0.8031737963569016, "lm_q1q2_score": 0.7138277789479458}} {"text": "[STATEMENT]\nlemma self_cscalar_prod_geq_0:\n fixes v :: \"'a::conjugatable_ordered_field vec\"\n shows \"v \\c v \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) \\ inner_prod v v\n[PROOF STEP]\nby (auto simp add: scalar_prod_def, rule sum_nonneg, rule conjugate_square_positive)", "meta": {"llama_tokens": 128, "file": "QHLProver_Complex_Matrix", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587875995483, "lm_q2_score": 0.8031738034238807, "lm_q1q2_score": 0.7138277757627262}} {"text": "[STATEMENT]\nlemma asymp_equiv_divide [asymp_equiv_intros]:\n assumes \"f1 \\[F] g1\" \"f2 \\[F] g2\"\n shows \"(\\x. f1 x / f2 x) \\[F] (\\x. g1 x / g2 x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. f1 x / f2 x) \\[F] (\\x. g1 x / g2 x)\n[PROOF STEP]\nusing asymp_equiv_mult[OF assms(1) asymp_equiv_inverse[OF assms(2)]]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. f1 x * inverse (f2 x)) \\[F] (\\x. g1 x * inverse (g2 x))\n\ngoal (1 subgoal):\n 1. (\\x. f1 x / f2 x) \\[F] (\\x. g1 x / g2 x)\n[PROOF STEP]\nby (simp add: field_simps)", "meta": {"llama_tokens": 294, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587905460026, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7138277739420145}} {"text": "[STATEMENT]\ntheorem subgroup_Inter:\n assumes subgr: \"(\\H. H \\ A \\ subgroup H G)\"\n and not_empty: \"A \\ {}\"\n shows \"subgroup (\\A) G\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. subgroup (\\ A) G\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. \\ A \\ carrier G\n 2. \\x y. \\x \\ \\ A; y \\ \\ A\\ \\ x \\\\<^bsub>G\\<^esub> y \\ \\ A\n 3. \\\\<^bsub>G\\<^esub> \\ \\ A\n 4. \\x. x \\ \\ A \\ inv\\<^bsub>G\\<^esub> x \\ \\ A\n[PROOF STEP]\nshow \"\\ A \\ carrier G\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ A \\ carrier G\n[PROOF STEP]\nby (simp add: Inf_less_eq not_empty subgr subgroup.subset)\n[PROOF STATE]\nproof (state)\nthis:\n\\ A \\ carrier G\n\ngoal (3 subgoals):\n 1. \\x y. \\x \\ \\ A; y \\ \\ A\\ \\ x \\\\<^bsub>G\\<^esub> y \\ \\ A\n 2. \\\\<^bsub>G\\<^esub> \\ \\ A\n 3. \\x. x \\ \\ A \\ inv\\<^bsub>G\\<^esub> x \\ \\ A\n[PROOF STEP]\nqed (auto simp: subgr subgroup.m_closed subgroup.one_closed subgroup.m_inv_closed)", "meta": {"llama_tokens": 531, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587905460026, "lm_q2_score": 0.8031737869342623, "lm_q1q2_score": 0.7138277634739478}} {"text": "[STATEMENT]\nlemma le_list_antisym:\n \"\\ order r; xs <=[r] ys; ys <=[r] xs \\ \\ xs = ys\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\order r; xs <=[r] ys; ys <=[r] xs\\ \\ xs = ys\n[PROOF STEP]\napply (unfold unfold_lesub_list)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\order r; Listn.le r xs ys; Listn.le r ys xs\\ \\ xs = ys\n[PROOF STEP]\napply (simp add: Listn.le_def list_all2_conv_all_nth)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\order r; length xs = length ys \\ (\\i\\<^bsub>r\\<^esub> ys ! i); \\i\\<^bsub>r\\<^esub> xs ! i\\ \\ xs = ys\n[PROOF STEP]\napply (rule nth_equalityI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\order r; length xs = length ys \\ (\\i\\<^bsub>r\\<^esub> ys ! i); \\i\\<^bsub>r\\<^esub> xs ! i\\ \\ length xs = length ys\n 2. \\i. \\order r; length xs = length ys \\ (\\i\\<^bsub>r\\<^esub> ys ! i); \\i\\<^bsub>r\\<^esub> xs ! i; i < length xs\\ \\ xs ! i = ys ! i\n[PROOF STEP]\napply blast\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. \\order r; length xs = length ys \\ (\\i\\<^bsub>r\\<^esub> ys ! i); \\i\\<^bsub>r\\<^esub> xs ! i; i < length xs\\ \\ xs ! i = ys ! i\n[PROOF STEP]\napply clarify\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. \\order r; \\i\\<^bsub>r\\<^esub> xs ! i; i < length xs; length xs = length ys; \\i\\<^bsub>r\\<^esub> ys ! i\\ \\ xs ! i = ys ! i\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. \\order r; \\i\\<^bsub>r\\<^esub> xs ! i; i < length ys; length xs = length ys; \\i\\<^bsub>r\\<^esub> ys ! i\\ \\ xs ! i = ys ! i\n[PROOF STEP]\napply (blast intro: order_antisym)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1018, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.870597271765821, "lm_q2_score": 0.8198933425148214, "lm_q1q2_score": 0.7137969071323633}} {"text": "[STATEMENT]\nlemma exp_to_cos:\n fixes x:: real\n shows \"exp (\\ * x) + exp (-(\\ * x)) = 2 * (cos x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp (\\ * complex_of_real x) + exp (- (\\ * complex_of_real x)) = complex_of_real (2 * cos x)\n[PROOF STEP]\nusing exp_of_real exp_of_real_inv\n[PROOF STATE]\nproof (prove)\nusing this:\nexp (\\ * complex_of_real ?x) = complex_of_real (cos ?x) + \\ * complex_of_real (sin ?x)\nexp (- (\\ * complex_of_real ?x)) = complex_of_real (cos ?x) - \\ * complex_of_real (sin ?x)\n\ngoal (1 subgoal):\n 1. exp (\\ * complex_of_real x) + exp (- (\\ * complex_of_real x)) = complex_of_real (2 * cos x)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 290, "file": "Isabelle_Marries_Dirac_Quantum_Prisoners_Dilemma", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972751232809, "lm_q2_score": 0.8198933381139645, "lm_q1q2_score": 0.7137969060537482}} {"text": "[STATEMENT]\nlemma gauss_int_norm_mult: \"gauss_int_norm (z * u) = gauss_int_norm z * gauss_int_norm u\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. gauss_int_norm (z * u) = gauss_int_norm z * gauss_int_norm u\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. gauss_int_norm (z * u) = gauss_int_norm z * gauss_int_norm u\n[PROOF STEP]\nhave \"real (gauss_int_norm (z * u)) = real (gauss_int_norm z * gauss_int_norm u)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (gauss_int_norm (z * u)) = real (gauss_int_norm z * gauss_int_norm u)\n[PROOF STEP]\nunfolding of_nat_mult\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (gauss_int_norm (z * u)) = real (gauss_int_norm z) * real (gauss_int_norm u)\n[PROOF STEP]\nby (simp add: real_gauss_int_norm norm_power norm_mult power_mult_distrib)\n[PROOF STATE]\nproof (state)\nthis:\nreal (gauss_int_norm (z * u)) = real (gauss_int_norm z * gauss_int_norm u)\n\ngoal (1 subgoal):\n 1. gauss_int_norm (z * u) = gauss_int_norm z * gauss_int_norm u\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (gauss_int_norm (z * u)) = real (gauss_int_norm z * gauss_int_norm u)\n\ngoal (1 subgoal):\n 1. gauss_int_norm (z * u) = gauss_int_norm z * gauss_int_norm u\n[PROOF STEP]\nby (subst (asm) of_nat_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\ngauss_int_norm (z * u) = gauss_int_norm z * gauss_int_norm u\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 642, "file": "Gaussian_Integers_Gaussian_Integers", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972751232809, "lm_q2_score": 0.8198933315126791, "lm_q1q2_score": 0.7137969003066872}} {"text": "[STATEMENT]\nlemma uint64_range : \"range nat_of_uint64 = {..<2 ^ 64}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. range nat_of_uint64 = {..<2 ^ 64}\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. range nat_of_uint64 \\ {..<2 ^ 64}\n 2. {..<2 ^ 64} \\ range nat_of_uint64\n[PROOF STEP]\nshow \"{..<2 ^ 64} \\ range nat_of_uint64\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {..<2 ^ 64} \\ range nat_of_uint64\n[PROOF STEP]\nusing uint64_nat_bij\n[PROOF STATE]\nproof (prove)\nusing this:\n?x < 2 ^ 64 \\ nat_of_uint64 (uint64_of_nat ?x) = ?x\n\ngoal (1 subgoal):\n 1. {..<2 ^ 64} \\ range nat_of_uint64\n[PROOF STEP]\nby (metis lessThan_iff range_eqI subsetI)\n[PROOF STATE]\nproof (state)\nthis:\n{..<2 ^ 64} \\ range nat_of_uint64\n\ngoal (1 subgoal):\n 1. range nat_of_uint64 \\ {..<2 ^ 64}\n[PROOF STEP]\nhave \"\\ x . nat_of_uint64 x < 2^64\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. nat_of_uint64 x < 2 ^ 64\n[PROOF STEP]\napply transfer\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. unat x < 2 ^ 64\n[PROOF STEP]\nusing take_bit_nat_eq_self\n[PROOF STATE]\nproof (prove)\nusing this:\n?m < 2 ^ ?n \\ take_bit ?n ?m = ?m\n\ngoal (1 subgoal):\n 1. \\x. unat x < 2 ^ 64\n[PROOF STEP]\nby (metis uint64.size_eq_length unsigned_less)\n[PROOF STATE]\nproof (state)\nthis:\nnat_of_uint64 ?x < 2 ^ 64\n\ngoal (1 subgoal):\n 1. range nat_of_uint64 \\ {..<2 ^ 64}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nnat_of_uint64 ?x < 2 ^ 64\n[PROOF STEP]\nshow \"range nat_of_uint64 \\ {..<2 ^ 64}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nnat_of_uint64 ?x < 2 ^ 64\n\ngoal (1 subgoal):\n 1. range nat_of_uint64 \\ {..<2 ^ 64}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrange nat_of_uint64 \\ {..<2 ^ 64}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 927, "file": "FSM_Tests_Test_Suite_Generator_Code_Export", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972650509008, "lm_q2_score": 0.8198933359135361, "lm_q1q2_score": 0.713796895879784}} {"text": "[STATEMENT]\nlemma matrix_inv_matrix_mul:\n assumes \"invertible A\" and \"invertible B\"\n shows \"(A ** B)\\<^sup>-\\<^sup>1 = B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (A ** B)\\<^sup>-\\<^sup>1 = B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1\n[PROOF STEP]\nproof(rule matrix_inv_unique)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. A ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = mat (1::'a)\n 2. B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nhave \"A ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = A ** (B ** B\\<^sup>-\\<^sup>1) ** A\\<^sup>-\\<^sup>1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = A ** (B ** B\\<^sup>-\\<^sup>1) ** A\\<^sup>-\\<^sup>1\n[PROOF STEP]\nby (simp add: matrix_mul_assoc)\n[PROOF STATE]\nproof (state)\nthis:\nA ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = A ** (B ** B\\<^sup>-\\<^sup>1) ** A\\<^sup>-\\<^sup>1\n\ngoal (2 subgoals):\n 1. A ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = mat (1::'a)\n 2. B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nA ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = A ** (B ** B\\<^sup>-\\<^sup>1) ** A\\<^sup>-\\<^sup>1\n\ngoal (2 subgoals):\n 1. A ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = mat (1::'a)\n 2. B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nhave \"... = mat 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A ** (B ** B\\<^sup>-\\<^sup>1) ** A\\<^sup>-\\<^sup>1 = mat (1::'a)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ninvertible A\ninvertible B\n\ngoal (1 subgoal):\n 1. A ** (B ** B\\<^sup>-\\<^sup>1) ** A\\<^sup>-\\<^sup>1 = mat (1::'a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nA ** (B ** B\\<^sup>-\\<^sup>1) ** A\\<^sup>-\\<^sup>1 = mat (1::'a)\n\ngoal (2 subgoals):\n 1. A ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = mat (1::'a)\n 2. B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nA ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = mat (1::'a)\n[PROOF STEP]\nshow \"A ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = mat 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = mat (1::'a)\n\ngoal (1 subgoal):\n 1. A ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = mat (1::'a)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nA ** B ** (B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1) = mat (1::'a)\n\ngoal (1 subgoal):\n 1. B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nhave \"B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = B\\<^sup>-\\<^sup>1 ** (A\\<^sup>-\\<^sup>1 ** A) ** B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = B\\<^sup>-\\<^sup>1 ** (A\\<^sup>-\\<^sup>1 ** A) ** B\n[PROOF STEP]\nby (simp add: matrix_mul_assoc)\n[PROOF STATE]\nproof (state)\nthis:\nB\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = B\\<^sup>-\\<^sup>1 ** (A\\<^sup>-\\<^sup>1 ** A) ** B\n\ngoal (1 subgoal):\n 1. B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nB\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = B\\<^sup>-\\<^sup>1 ** (A\\<^sup>-\\<^sup>1 ** A) ** B\n\ngoal (1 subgoal):\n 1. B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nhave \"... = mat 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. B\\<^sup>-\\<^sup>1 ** (A\\<^sup>-\\<^sup>1 ** A) ** B = mat (1::'a)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ninvertible A\ninvertible B\n\ngoal (1 subgoal):\n 1. B\\<^sup>-\\<^sup>1 ** (A\\<^sup>-\\<^sup>1 ** A) ** B = mat (1::'a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nB\\<^sup>-\\<^sup>1 ** (A\\<^sup>-\\<^sup>1 ** A) ** B = mat (1::'a)\n\ngoal (1 subgoal):\n 1. B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nB\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nshow \"B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nB\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n\ngoal (1 subgoal):\n 1. B\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nB\\<^sup>-\\<^sup>1 ** A\\<^sup>-\\<^sup>1 ** (A ** B) = mat (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2404, "file": "Matrices_for_ODEs_MTX_Preliminaries", "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.863391617003942, "lm_q2_score": 0.8267117898012104, "lm_q1q2_score": 0.71377602899269}} {"text": "[STATEMENT]\nlemma real_vector_affinity_eq:\n fixes x :: \"'a :: real_vector\"\n assumes m0: \"m \\ 0\"\n shows \"m *\\<^sub>R x + c = y \\ x = inverse m *\\<^sub>R y - (inverse m *\\<^sub>R c)\"\n (is \"?lhs \\ ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (m *\\<^sub>R x + c = y) = (x = y /\\<^sub>R m - c /\\<^sub>R m)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. m *\\<^sub>R x + c = y \\ x = y /\\<^sub>R m - c /\\<^sub>R m\n 2. x = y /\\<^sub>R m - c /\\<^sub>R m \\ m *\\<^sub>R x + c = y\n[PROOF STEP]\nassume ?lhs\n[PROOF STATE]\nproof (state)\nthis:\nm *\\<^sub>R x + c = y\n\ngoal (2 subgoals):\n 1. m *\\<^sub>R x + c = y \\ x = y /\\<^sub>R m - c /\\<^sub>R m\n 2. x = y /\\<^sub>R m - c /\\<^sub>R m \\ m *\\<^sub>R x + c = y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nm *\\<^sub>R x + c = y\n[PROOF STEP]\nhave \"m *\\<^sub>R x = y - c\"\n[PROOF STATE]\nproof (prove)\nusing this:\nm *\\<^sub>R x + c = y\n\ngoal (1 subgoal):\n 1. m *\\<^sub>R x = y - c\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nm *\\<^sub>R x = y - c\n\ngoal (2 subgoals):\n 1. m *\\<^sub>R x + c = y \\ x = y /\\<^sub>R m - c /\\<^sub>R m\n 2. x = y /\\<^sub>R m - c /\\<^sub>R m \\ m *\\<^sub>R x + c = y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nm *\\<^sub>R x = y - c\n[PROOF STEP]\nhave \"inverse m *\\<^sub>R (m *\\<^sub>R x) = inverse m *\\<^sub>R (y - c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nm *\\<^sub>R x = y - c\n\ngoal (1 subgoal):\n 1. m *\\<^sub>R x /\\<^sub>R m = (y - c) /\\<^sub>R m\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nm *\\<^sub>R x /\\<^sub>R m = (y - c) /\\<^sub>R m\n\ngoal (2 subgoals):\n 1. m *\\<^sub>R x + c = y \\ x = y /\\<^sub>R m - c /\\<^sub>R m\n 2. x = y /\\<^sub>R m - c /\\<^sub>R m \\ m *\\<^sub>R x + c = y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nm *\\<^sub>R x /\\<^sub>R m = (y - c) /\\<^sub>R m\n[PROOF STEP]\nshow \"x = inverse m *\\<^sub>R y - (inverse m *\\<^sub>R c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nm *\\<^sub>R x /\\<^sub>R m = (y - c) /\\<^sub>R m\n\ngoal (1 subgoal):\n 1. x = y /\\<^sub>R m - c /\\<^sub>R m\n[PROOF STEP]\nusing m0\n[PROOF STATE]\nproof (prove)\nusing this:\nm *\\<^sub>R x /\\<^sub>R m = (y - c) /\\<^sub>R m\nm \\ 0\n\ngoal (1 subgoal):\n 1. x = y /\\<^sub>R m - c /\\<^sub>R m\n[PROOF STEP]\nby (simp add: scaleR_diff_right)\n[PROOF STATE]\nproof (state)\nthis:\nx = y /\\<^sub>R m - c /\\<^sub>R m\n\ngoal (1 subgoal):\n 1. x = y /\\<^sub>R m - c /\\<^sub>R m \\ m *\\<^sub>R x + c = y\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x = y /\\<^sub>R m - c /\\<^sub>R m \\ m *\\<^sub>R x + c = y\n[PROOF STEP]\nassume ?rhs\n[PROOF STATE]\nproof (state)\nthis:\nx = y /\\<^sub>R m - c /\\<^sub>R m\n\ngoal (1 subgoal):\n 1. x = y /\\<^sub>R m - c /\\<^sub>R m \\ m *\\<^sub>R x + c = y\n[PROOF STEP]\nwith m0\n[PROOF STATE]\nproof (chain)\npicking this:\nm \\ 0\nx = y /\\<^sub>R m - c /\\<^sub>R m\n[PROOF STEP]\nshow \"m *\\<^sub>R x + c = y\"\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ 0\nx = y /\\<^sub>R m - c /\\<^sub>R m\n\ngoal (1 subgoal):\n 1. m *\\<^sub>R x + c = y\n[PROOF STEP]\nby (simp add: scaleR_diff_right)\n[PROOF STATE]\nproof (state)\nthis:\nm *\\<^sub>R x + c = y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1608, "file": null, "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391599428538, "lm_q2_score": 0.8267117983401363, "lm_q1q2_score": 0.7137760218353333}} {"text": "[STATEMENT]\nlemma LIMSEQ_const_iff: \"(\\n. k) \\ l \\ k = l\"\n for k l :: \"'a::t2_space\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. k) \\ l = (k = l)\n[PROOF STEP]\nusing trivial_limit_sequentially\n[PROOF STATE]\nproof (prove)\nusing this:\nsequentially \\ bot\n\ngoal (1 subgoal):\n 1. (\\n. k) \\ l = (k = l)\n[PROOF STEP]\nby (rule tendsto_const_iff)", "meta": {"llama_tokens": 186, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391602943619, "lm_q2_score": 0.826711791935942, "lm_q1q2_score": 0.7137760192119647}} {"text": "[STATEMENT]\nlemma connected_openin:\n \"connected S \\\n \\(\\E1 E2. openin (top_of_set S) E1 \\\n openin (top_of_set S) E2 \\\n S \\ E1 \\ E2 \\ E1 \\ E2 = {} \\ E1 \\ {} \\ E2 \\ {})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. connected S = (\\E1 E2. openin (top_of_set S) E1 \\ openin (top_of_set S) E2 \\ S \\ E1 \\ E2 \\ E1 \\ E2 = {} \\ E1 \\ {} \\ E2 \\ {})\n[PROOF STEP]\nunfolding connected_def openin_open disjoint_iff_not_equal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\A B. open A \\ open B \\ S \\ A \\ B \\ (\\x\\A \\ B. \\y\\S. x \\ y) \\ \\ (\\x\\A. \\y\\S. x \\ y) \\ \\ (\\x\\B. \\y\\S. x \\ y)) = (\\E1 E2. (\\T. open T \\ E1 = S \\ T) \\ (\\T. open T \\ E2 = S \\ T) \\ S \\ E1 \\ E2 \\ (\\x\\E1. \\y\\E2. x \\ y) \\ E1 \\ {} \\ E2 \\ {})\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 523, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391595913457, "lm_q2_score": 0.8267117855317474, "lm_q1q2_score": 0.713776007870719}} {"text": "[STATEMENT]\nlemma set_check_generate:\n \"set (check_generate a b) = {(x, y).\n (x, y) \\ (zeroes (length a), zeroes (length b)) \\\n length x = length a \\ length y = length b \\ a \\ x = b \\ y \\\n (\\i Max (set b)) \\ (\\j Max (set a))}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (check_generate a b) = {(x, y). (x, y) \\ (zeroes (length a), zeroes (length b)) \\ length x = length a \\ length y = length b \\ a \\ x = b \\ y \\ (\\i Max (set b)) \\ (\\j Max (set a))}\n[PROOF STEP]\nunfolding check_def and set_filter and set_generate and set_gen2\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x \\ {(x, y). (x, y) \\ (zeroes (length a), zeroes (length b)) \\ (x, y) \\ {(x, y). length x = length a \\ length y = length b \\ (\\i Max (set b)) \\ (\\j Max (set a))}}. case x of (x, y) \\ a \\ x = b \\ y} = {(x, y). (x, y) \\ (zeroes (length a), zeroes (length b)) \\ length x = length a \\ length y = length b \\ a \\ x = b \\ y \\ (\\i Max (set b)) \\ (\\j Max (set a))}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 567, "file": "Diophantine_Eqns_Lin_Hom_Simple_Algorithm", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.914900957313305, "lm_q2_score": 0.7799929053683038, "lm_q1q2_score": 0.7136162558190473}} {"text": "[STATEMENT]\nlemma lipschitz_on_normD:\n \"norm (f x - f y) \\ L * norm (x - y)\"\n if \"lipschitz_on L X f\" \"x \\ X\" \"y \\ X\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (f x - f y) \\ L * norm (x - y)\n[PROOF STEP]\nusing lipschitz_onD[OF that]\n[PROOF STATE]\nproof (prove)\nusing this:\ndist (f x) (f y) \\ L * dist x y\n\ngoal (1 subgoal):\n 1. norm (f x - f y) \\ L * norm (x - y)\n[PROOF STEP]\nby (simp add: dist_norm)", "meta": {"llama_tokens": 212, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8519528094861981, "lm_q2_score": 0.8376199592797929, "lm_q1q2_score": 0.7136126775901345}} {"text": "[STATEMENT]\nlemma DERIV_pos_inc_right:\n fixes f :: \"real \\ real\"\n assumes der: \"DERIV f x :> l\"\n and l: \"0 < l\"\n shows \"\\d > 0. \\h > 0. h < d \\ f x < f (x + h)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\d>0. \\h>0. h < d \\ f x < f (x + h)\n[PROOF STEP]\nusing has_real_derivative_pos_inc_right[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\d>0. \\h>0. x + h \\ UNIV \\ h < d \\ f x < f (x + h)\n\ngoal (1 subgoal):\n 1. \\d>0. \\h>0. h < d \\ f x < f (x + h)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 277, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314768368161, "lm_q2_score": 0.8056321959813275, "lm_q1q2_score": 0.7134932315142304}} {"text": "[STATEMENT]\nlemma DERIV_pos_inc_right:\n fixes f :: \"real \\ real\"\n assumes der: \"DERIV f x :> l\"\n and l: \"0 < l\"\n shows \"\\d > 0. \\h > 0. h < d \\ f x < f (x + h)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\d>0. \\h>0. h < d \\ f x < f (x + h)\n[PROOF STEP]\nusing has_real_derivative_pos_inc_right[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\d>0. \\h>0. x + h \\ UNIV \\ h < d \\ f x < f (x + h)\n\ngoal (1 subgoal):\n 1. \\d>0. \\h>0. h < d \\ f x < f (x + h)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 277, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314768368161, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7134932273812409}} {"text": "[STATEMENT]\nlemma d_IN_plus_flow:\n assumes wf: \"wf_residual_network\"\n and f: \"flow \\ f\"\n and g: \"flow (residual_network f) g\"\n shows \"d_IN (f \\ g) x \\ d_IN f x + d_IN g x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. d_IN (f \\ g) x \\ d_IN f x + d_IN g x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. d_IN (f \\ g) x \\ d_IN f x + d_IN g x\n[PROOF STEP]\nhave \"d_IN (f \\ g) x \\ (\\\\<^sup>+ y. f (y, x) + g (y, x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. d_IN (f \\ g) x \\ (\\\\<^sup>+ y. f (y, x) + g (y, x))\n[PROOF STEP]\nunfolding d_IN_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sup>+ xa. (f \\ g) (xa, x)) \\ (\\\\<^sup>+ y. f (y, x) + g (y, x))\n[PROOF STEP]\nby(rule nn_integral_mono)(auto intro: diff_le_self_ennreal)\n[PROOF STATE]\nproof (state)\nthis:\nd_IN (f \\ g) x \\ (\\\\<^sup>+ y. f (y, x) + g (y, x))\n\ngoal (1 subgoal):\n 1. d_IN (f \\ g) x \\ d_IN f x + d_IN g x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nd_IN (f \\ g) x \\ (\\\\<^sup>+ y. f (y, x) + g (y, x))\n\ngoal (1 subgoal):\n 1. d_IN (f \\ g) x \\ d_IN f x + d_IN g x\n[PROOF STEP]\nhave \"\\ = d_IN f x + d_IN g x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sup>+ y. f (y, x) + g (y, x)) = d_IN f x + d_IN g x\n[PROOF STEP]\nby(subst nn_integral_add)(simp_all add: d_IN_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\\\<^sup>+ y. f (y, x) + g (y, x)) = d_IN f x + d_IN g x\n\ngoal (1 subgoal):\n 1. d_IN (f \\ g) x \\ d_IN f x + d_IN g x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nd_IN (f \\ g) x \\ d_IN f x + d_IN g x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nd_IN (f \\ g) x \\ d_IN f x + d_IN g x\n\ngoal (1 subgoal):\n 1. d_IN (f \\ g) x \\ d_IN f x + d_IN g x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nd_IN (f \\ g) x \\ d_IN f x + d_IN g x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1027, "file": "MFMC_Countable_MFMC_Flow_Attainability", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.885631470799559, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7134932225174321}} {"text": "[STATEMENT]\nlemma polytope_imp_convex: \"polytope S \\ convex S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. polytope S \\ convex S\n[PROOF STEP]\nby (metis convex_convex_hull polytope_def)", "meta": {"llama_tokens": 85, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314828740728, "lm_q2_score": 0.8056321796478255, "lm_q1q2_score": 0.7134932219125751}} {"text": "[STATEMENT]\nlemma argmaxEquivalence: \n assumes \"\\x\\X. f x = g x\" \n shows \"argmax f X = argmax g X\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. argmax f X = argmax g X\n[PROOF STEP]\nusing assms argmaxLemma Collect_cong image_cong\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\X. f x = g x\nargmax ?f ?A = {x \\ ?A. ?f x = Max (?f ` ?A)}\n(\\x. ?P x = ?Q x) \\ {x. ?P x} = {x. ?Q x}\n\\?M = ?N; \\x. x \\ ?N \\ ?f x = ?g x\\ \\ ?f ` ?M = ?g ` ?N\n\ngoal (1 subgoal):\n 1. argmax f X = argmax g X\n[PROOF STEP]\nby (metis(no_types,lifting))", "meta": {"llama_tokens": 288, "file": "Vickrey_Clarke_Groves_Argmax", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314738181875, "lm_q2_score": 0.8056321819811829, "lm_q1q2_score": 0.7134932166833573}} {"text": "[STATEMENT]\nlemma sum_index_shift: \"(\\l = (a::nat)..< b. f(l+c)) = (\\l = (a+c)..< (b+c). f l )\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\l = a..vec_of_list l\\ = sqrt(\\i2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\vec_of_list l\\ = sqrt (\\i2)\n[PROOF STEP]\nby (auto simp: cpx_vec_length_def vec_of_list_def vec_of_list_index)\n (metis (no_types, lifting) dim_vec_of_list sum.cong vec_of_list.abs_eq vec_of_list_index)", "meta": {"llama_tokens": 197, "file": "Isabelle_Marries_Dirac_Quantum", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314677809303, "lm_q2_score": 0.805632181981183, "lm_q1q2_score": 0.7134932118195486}} {"text": "[STATEMENT]\nlemma aff_dim_subset:\n fixes S T :: \"'n::euclidean_space set\"\n assumes \"S \\ T\"\n shows \"aff_dim S \\ aff_dim T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. aff_dim S \\ aff_dim T\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. aff_dim S \\ aff_dim T\n[PROOF STEP]\nobtain B where B: \"\\ affine_dependent B\" \"B \\ S\" \"affine hull B = affine hull S\"\n \"of_nat (card B) = aff_dim S + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\B. \\\\ affine_dependent B; B \\ S; affine hull B = affine hull S; int (card B) = aff_dim S + 1\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing aff_dim_inner_basis_exists[of S]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\B\\S. affine hull B = affine hull S \\ \\ affine_dependent B \\ int (card B) = aff_dim S + 1\n\ngoal (1 subgoal):\n 1. (\\B. \\\\ affine_dependent B; B \\ S; affine hull B = affine hull S; int (card B) = aff_dim S + 1\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\ affine_dependent B\nB \\ S\naffine hull B = affine hull S\nint (card B) = aff_dim S + 1\n\ngoal (1 subgoal):\n 1. aff_dim S \\ aff_dim T\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ affine_dependent B\nB \\ S\naffine hull B = affine hull S\nint (card B) = aff_dim S + 1\n[PROOF STEP]\nhave \"int (card B) \\ aff_dim T + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ affine_dependent B\nB \\ S\naffine hull B = affine hull S\nint (card B) = aff_dim S + 1\n\ngoal (1 subgoal):\n 1. int (card B) \\ aff_dim T + 1\n[PROOF STEP]\nusing assms independent_card_le_aff_dim[of B T]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ affine_dependent B\nB \\ S\naffine hull B = affine hull S\nint (card B) = aff_dim S + 1\nS \\ T\n\\B \\ T; \\ affine_dependent B\\ \\ int (card B) \\ aff_dim T + 1\n\ngoal (1 subgoal):\n 1. int (card B) \\ aff_dim T + 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nint (card B) \\ aff_dim T + 1\n\ngoal (1 subgoal):\n 1. aff_dim S \\ aff_dim T\n[PROOF STEP]\nwith B\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ affine_dependent B\nB \\ S\naffine hull B = affine hull S\nint (card B) = aff_dim S + 1\nint (card B) \\ aff_dim T + 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ affine_dependent B\nB \\ S\naffine hull B = affine hull S\nint (card B) = aff_dim S + 1\nint (card B) \\ aff_dim T + 1\n\ngoal (1 subgoal):\n 1. aff_dim S \\ aff_dim T\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\naff_dim S \\ aff_dim T\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1199, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127678225575, "lm_q2_score": 0.8354835432479661, "lm_q1q2_score": 0.7134300648850682}} {"text": "[STATEMENT]\nlemma adjoint_mult:\n fixes A B :: \"'a::conjugatable_field mat\"\n assumes \"A \\ carrier_mat n m\" \"B \\ carrier_mat m l\"\n shows \"adjoint (A * B) = adjoint B * adjoint A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. adjoint (A * B) = adjoint B * adjoint A\n[PROOF STEP]\nproof (rule eq_matI, auto simp add: adjoint_eval adjoint_row adjoint_col)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_col B; j < dim_row A\\ \\ conjugate (row A j \\ col B i) = inner_prod (row A j) (conjugate (col B i))\n[PROOF STEP]\nfix i j\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_col B; j < dim_row A\\ \\ conjugate (row A j \\ col B i) = inner_prod (row A j) (conjugate (col B i))\n[PROOF STEP]\nassume \"i < dim_col B\" \"j < dim_row A\"\n[PROOF STATE]\nproof (state)\nthis:\ni < dim_col B\nj < dim_row A\n\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_col B; j < dim_row A\\ \\ conjugate (row A j \\ col B i) = inner_prod (row A j) (conjugate (col B i))\n[PROOF STEP]\nshow \"conjugate (row A j \\ col B i) = conjugate (col B i) \\ conjugate (row A j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. conjugate (row A j \\ col B i) = inner_prod (row A j) (conjugate (col B i))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n m\nB \\ carrier_mat m l\n\ngoal (1 subgoal):\n 1. conjugate (row A j \\ col B i) = inner_prod (row A j) (conjugate (col B i))\n[PROOF STEP]\napply (simp add: conjugate_scalar_prod)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ carrier_mat n m; B \\ carrier_mat m l\\ \\ inner_prod (col B i) (conjugate (row A j)) = inner_prod (row A j) (conjugate (col B i))\n[PROOF STEP]\napply (subst comm_scalar_prod[where n=\"dim_row B\"])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\A \\ carrier_mat n m; B \\ carrier_mat m l\\ \\ conjugate (row A j) \\ carrier_vec (dim_row B)\n 2. \\A \\ carrier_mat n m; B \\ carrier_mat m l\\ \\ conjugate (col B i) \\ carrier_vec (dim_row B)\n 3. \\A \\ carrier_mat n m; B \\ carrier_mat m l\\ \\ inner_prod (row A j) (conjugate (col B i)) = inner_prod (row A j) (conjugate (col B i))\n[PROOF STEP]\nby (auto simp add: carrier_vecI)\n[PROOF STATE]\nproof (state)\nthis:\nconjugate (row A j \\ col B i) = inner_prod (row A j) (conjugate (col B i))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1109, "file": "QHLProver_Complex_Matrix", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127529517043, "lm_q2_score": 0.8354835432479663, "lm_q1q2_score": 0.7134300524607152}} {"text": "[STATEMENT]\nlemma icard_empty_iff: \"(icard A = 0) = (A = {})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (icard A = 0) = (A = {})\n[PROOF STEP]\napply (unfold zero_enat_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (icard A = enat 0) = (A = {})\n[PROOF STEP]\napply (rule iffI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. icard A = enat 0 \\ A = {}\n 2. A = {} \\ icard A = enat 0\n[PROOF STEP]\napply (frule icard_eq_enat_imp)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\icard A = enat 0; finite A\\ \\ A = {}\n 2. A = {} \\ icard A = enat 0\n[PROOF STEP]\napply (simp add: icard_finite)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A = {} \\ icard A = enat 0\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 392, "file": "List-Infinite_CommonSet_InfiniteSet2", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127529517043, "lm_q2_score": 0.8354835350552604, "lm_q1q2_score": 0.7134300454648592}} {"text": "[STATEMENT]\nlemma fib_asymptotics:\n fixes \\ :: real\n defines \"\\ \\ (1 + sqrt 5) / 2\"\n shows \"(\\n. real (fib n) / (\\ ^ n / sqrt 5)) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. real (fib n) / (\\ ^ n / sqrt 5)) \\ 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\n. real (fib n) / (\\ ^ n / sqrt 5)) \\ 1\n[PROOF STEP]\ndefine \\ :: real where \"\\ \\ (1 - sqrt 5) / 2\"\n[PROOF STATE]\nproof (state)\nthis:\n\\ \\ (1 - sqrt 5) / 2\n\ngoal (1 subgoal):\n 1. (\\n. real (fib n) / (\\ ^ n / sqrt 5)) \\ 1\n[PROOF STEP]\nhave \"\\ > 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 < \\\n[PROOF STEP]\nby (simp add: \\_def)\n[PROOF STATE]\nproof (state)\nthis:\n1 < \\\n\ngoal (1 subgoal):\n 1. (\\n. real (fib n) / (\\ ^ n / sqrt 5)) \\ 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < \\\n[PROOF STEP]\nhave *: \"\\ \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < \\\n\ngoal (1 subgoal):\n 1. \\ \\ 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\ \\ 0\n\ngoal (1 subgoal):\n 1. (\\n. real (fib n) / (\\ ^ n / sqrt 5)) \\ 1\n[PROOF STEP]\nhave \"(\\n. (\\ / \\) ^ n) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (^) (\\ / \\) \\ 0\n[PROOF STEP]\nby (rule LIMSEQ_power_zero) (simp_all add: \\_def \\_def field_simps add_pos_pos)\n[PROOF STATE]\nproof (state)\nthis:\n(^) (\\ / \\) \\ 0\n\ngoal (1 subgoal):\n 1. (\\n. real (fib n) / (\\ ^ n / sqrt 5)) \\ 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(^) (\\ / \\) \\ 0\n[PROOF STEP]\nhave \"(\\n. 1 - (\\ / \\) ^ n) \\ 1 - 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(^) (\\ / \\) \\ 0\n\ngoal (1 subgoal):\n 1. (\\n. 1 - (\\ / \\) ^ n) \\ 1 - 0\n[PROOF STEP]\nby (intro tendsto_diff tendsto_const)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. 1 - (\\ / \\) ^ n) \\ 1 - 0\n\ngoal (1 subgoal):\n 1. (\\n. real (fib n) / (\\ ^ n / sqrt 5)) \\ 1\n[PROOF STEP]\nwith *\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ \\ 0\n(\\n. 1 - (\\ / \\) ^ n) \\ 1 - 0\n[PROOF STEP]\nhave \"(\\n. (\\ ^ n - \\ ^ n) / \\ ^ n) \\ 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ \\ 0\n(\\n. 1 - (\\ / \\) ^ n) \\ 1 - 0\n\ngoal (1 subgoal):\n 1. (\\n. (\\ ^ n - \\ ^ n) / \\ ^ n) \\ 1\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. (\\ ^ n - \\ ^ n) / \\ ^ n) \\ 1\n\ngoal (1 subgoal):\n 1. (\\n. real (fib n) / (\\ ^ n / sqrt 5)) \\ 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n. (\\ ^ n - \\ ^ n) / \\ ^ n) \\ 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. (\\ ^ n - \\ ^ n) / \\ ^ n) \\ 1\n\ngoal (1 subgoal):\n 1. (\\n. real (fib n) / (\\ ^ n / sqrt 5)) \\ 1\n[PROOF STEP]\nby (simp add: fib_closed_form \\_def \\_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. real (fib n) / (\\ ^ n / sqrt 5)) \\ 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1662, "file": null, "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127492339909, "lm_q2_score": 0.8354835330070838, "lm_q1q2_score": 0.7134300406098066}} {"text": "[STATEMENT]\nlemma arctan_series':\n assumes \"\\x\\ \\ 1\"\n shows \"(\\k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1))) sums arctan x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) sums arctan x\n[PROOF STEP]\nusing summable_arctan_series[OF assms] arctan_series[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\narctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n\ngoal (1 subgoal):\n 1. (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) sums arctan x\n[PROOF STEP]\nby (simp add: sums_iff)", "meta": {"llama_tokens": 340, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.8354835350552604, "lm_q1q2_score": 0.7134300392526823}} {"text": "[STATEMENT]\nlemma arctan_series':\n assumes \"\\x\\ \\ 1\"\n shows \"(\\k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1))) sums arctan x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) sums arctan x\n[PROOF STEP]\nusing summable_arctan_series[OF assms] arctan_series[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\narctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n\ngoal (1 subgoal):\n 1. (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) sums arctan x\n[PROOF STEP]\nby (simp add: sums_iff)", "meta": {"llama_tokens": 340, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.8354835309589074, "lm_q1q2_score": 0.7134300357547543}} {"text": "[STATEMENT]\nlemma cbox_complex_of_real: \"cbox (complex_of_real x) (complex_of_real y) = complex_of_real ` {x..y}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cbox (complex_of_real x) (complex_of_real y) = complex_of_real ` {x..y}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. cbox (complex_of_real x) (complex_of_real y) = complex_of_real ` {x..y}\n[PROOF STEP]\nhave \"(x \\ Re z \\ Re z \\ y \\ Im z = 0) = (z \\ complex_of_real ` {x..y})\" for z\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x \\ Re z \\ Re z \\ y \\ Im z = 0) = (z \\ complex_of_real ` {x..y})\n[PROOF STEP]\nby (cases z) (simp add: complex_eq_cancel_iff2 image_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(x \\ Re ?z \\ Re ?z \\ y \\ Im ?z = 0) = (?z \\ complex_of_real ` {x..y})\n\ngoal (1 subgoal):\n 1. cbox (complex_of_real x) (complex_of_real y) = complex_of_real ` {x..y}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(x \\ Re ?z \\ Re ?z \\ y \\ Im ?z = 0) = (?z \\ complex_of_real ` {x..y})\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(x \\ Re ?z \\ Re ?z \\ y \\ Im ?z = 0) = (?z \\ complex_of_real ` {x..y})\n\ngoal (1 subgoal):\n 1. cbox (complex_of_real x) (complex_of_real y) = complex_of_real ` {x..y}\n[PROOF STEP]\nby (auto simp: in_cbox_complex_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncbox (complex_of_real x) (complex_of_real y) = complex_of_real ` {x..y}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 698, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240895276223, "lm_q2_score": 0.8244619177503205, "lm_q1q2_score": 0.7133443121357185}} {"text": "[STATEMENT]\nlemma Max_mono_plus:\n assumes \"finite (range (f::_\\_::ordered_ab_semigroup_add))\" \n \"finite (range g)\"\n shows \"(MAX x. f x + g x) \\ (MAX x. f x) + (MAX x. g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (MAX x. f x + g x) \\ Max (range f) + Max (range g)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (MAX x. f x + g x) \\ Max (range f) + Max (range g)\n[PROOF STEP]\nobtain xmax where xmax_def: \"f xmax + g xmax = (MAX x. f x + g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\xmax. f xmax + g xmax = (MAX x. f x + g x) \\ thesis) \\ thesis\n[PROOF STEP]\nusing finite_range_plus[OF assms] Max_in\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (range (\\x. f x + g x))\n\\finite ?A; ?A \\ {}\\ \\ Max ?A \\ ?A\n\ngoal (1 subgoal):\n 1. (\\xmax. f xmax + g xmax = (MAX x. f x + g x) \\ thesis) \\ thesis\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nf xmax + g xmax = (MAX x. f x + g x)\n\ngoal (1 subgoal):\n 1. (MAX x. f x + g x) \\ Max (range f) + Max (range g)\n[PROOF STEP]\nhave \"(MAX x. f x + g x) = f xmax + g xmax\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (MAX x. f x + g x) = f xmax + g xmax\n[PROOF STEP]\nusing xmax_def\n[PROOF STATE]\nproof (prove)\nusing this:\nf xmax + g xmax = (MAX x. f x + g x)\n\ngoal (1 subgoal):\n 1. (MAX x. f x + g x) = f xmax + g xmax\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(MAX x. f x + g x) = f xmax + g xmax\n\ngoal (1 subgoal):\n 1. (MAX x. f x + g x) \\ Max (range f) + Max (range g)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(MAX x. f x + g x) = f xmax + g xmax\n\ngoal (1 subgoal):\n 1. (MAX x. f x + g x) \\ Max (range f) + Max (range g)\n[PROOF STEP]\nhave \"\\ \\ (MAX x. f x) + g xmax\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f xmax + g xmax \\ Max (range f) + g xmax\n[PROOF STEP]\nusing Max_ge[OF assms(1), of \"f xmax\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nf xmax \\ range f \\ f xmax \\ Max (range f)\n\ngoal (1 subgoal):\n 1. f xmax + g xmax \\ Max (range f) + g xmax\n[PROOF STEP]\nby (auto simp add: add_right_mono[of \"f xmax\"])\n[PROOF STATE]\nproof (state)\nthis:\nf xmax + g xmax \\ Max (range f) + g xmax\n\ngoal (1 subgoal):\n 1. (MAX x. f x + g x) \\ Max (range f) + Max (range g)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nf xmax + g xmax \\ Max (range f) + g xmax\n\ngoal (1 subgoal):\n 1. (MAX x. f x + g x) \\ Max (range f) + Max (range g)\n[PROOF STEP]\nhave \"\\ \\ (MAX x. f x) + (MAX x. g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Max (range f) + g xmax \\ Max (range f) + Max (range g)\n[PROOF STEP]\nusing Max_ge[OF assms(2), of \"g xmax\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ng xmax \\ range g \\ g xmax \\ Max (range g)\n\ngoal (1 subgoal):\n 1. Max (range f) + g xmax \\ Max (range f) + Max (range g)\n[PROOF STEP]\nby (auto simp add: add_left_mono[of \"g xmax\"])\n[PROOF STATE]\nproof (state)\nthis:\nMax (range f) + g xmax \\ Max (range f) + Max (range g)\n\ngoal (1 subgoal):\n 1. (MAX x. f x + g x) \\ Max (range f) + Max (range g)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(MAX x. f x + g x) \\ Max (range f) + Max (range g)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(MAX x. f x + g x) \\ Max (range f) + Max (range g)\n\ngoal (1 subgoal):\n 1. (MAX x. f x + g x) \\ Max (range f) + Max (range g)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(MAX x. f x + g x) \\ Max (range f) + Max (range g)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1658, "file": "CRYSTALS-Kyber_Abs_Qr", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240721511739, "lm_q2_score": 0.8244619306896955, "lm_q1q2_score": 0.7133443090049573}} {"text": "[STATEMENT]\ntheorem weak_conv_imp_integral_bdd_continuous_conv:\n fixes f :: \"real \\ 'a::{banach, second_countable_topology}\"\n assumes\n \"\\x. isCont f x\" and\n \"\\x. norm (f x) \\ B\"\n shows\n \"(\\ n. integral\\<^sup>L (\\ n) f) \\ integral\\<^sup>L M f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. integral\\<^sup>L (\\ n) f) \\ integral\\<^sup>L M f\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nisCont f ?x\nnorm (f ?x) \\ B\n\ngoal (1 subgoal):\n 1. (\\n. integral\\<^sup>L (\\ n) f) \\ integral\\<^sup>L M f\n[PROOF STEP]\nby (intro weak_conv_imp_bdd_ae_continuous_conv)\n (auto intro!: borel_measurable_continuous_onI continuous_at_imp_continuous_on)", "meta": {"llama_tokens": 316, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9241418137109956, "lm_q2_score": 0.7718435030872968, "lm_q1q2_score": 0.7132928548441428}} {"text": "[STATEMENT]\nlemma mat_kernelI: assumes \"A \\ carrier_mat nr nc\" \"v \\ carrier_vec nc\" \"A *\\<^sub>v v = 0\\<^sub>v nr\"\n shows \"v \\ mat_kernel A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v \\ mat_kernel A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat nr nc\nv \\ carrier_vec nc\nA *\\<^sub>v v = 0\\<^sub>v nr\n\ngoal (1 subgoal):\n 1. v \\ mat_kernel A\n[PROOF STEP]\nunfolding mat_kernel_def\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat nr nc\nv \\ carrier_vec nc\nA *\\<^sub>v v = 0\\<^sub>v nr\n\ngoal (1 subgoal):\n 1. v \\ {v \\ carrier_vec (dim_col A). A *\\<^sub>v v = 0\\<^sub>v (dim_row A)}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 322, "file": "Jordan_Normal_Form_Matrix_Kernel", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.877476793890012, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7132722118919982}} {"text": "[STATEMENT]\nlemma sums_stieltjes_gamma:\n \"(\\k. ln (k+1) ^ n / (k+1) - (ln (k+2) ^ (n+1) - ln (k+1) ^ (n + 1)) / (n + 1))\n sums stieltjes_gamma n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. ln (real (k + 1)) ^ n / real (k + 1) - (ln (real (k + 2)) ^ (n + 1) - ln (real (k + 1)) ^ (n + 1)) / (real n + 1)) sums stieltjes_gamma n\n[PROOF STEP]\nusing stieltjes_gamma_summable[of n]\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\k. ln (real (k + 1)) ^ n / real (k + 1) - (ln (real (k + 2)) ^ (n + 1) - ln (real (k + 1)) ^ (n + 1)) / (real n + 1))\n\ngoal (1 subgoal):\n 1. (\\k. ln (real (k + 1)) ^ n / real (k + 1) - (ln (real (k + 2)) ^ (n + 1) - ln (real (k + 1)) ^ (n + 1)) / (real n + 1)) sums stieltjes_gamma n\n[PROOF STEP]\nunfolding stieltjes_gamma_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\k. ln (real (k + 1)) ^ n / real (k + 1) - (ln (real (k + 2)) ^ (n + 1) - ln (real (k + 1)) ^ (n + 1)) / (real n + 1))\n\ngoal (1 subgoal):\n 1. (\\k. ln (real (k + 1)) ^ n / real (k + 1) - (ln (real (k + 2)) ^ (n + 1) - ln (real (k + 1)) ^ (n + 1)) / (real n + 1)) sums of_real (\\k. ln (real (k + 1)) ^ n / real (k + 1) - (ln (real (k + 2)) ^ (n + 1) - ln (real (k + 1)) ^ (n + 1)) / (real n + 1))\n[PROOF STEP]\nby (simp add: summable_sums)", "meta": {"llama_tokens": 668, "file": "Zeta_Function_Zeta_Laurent_Expansion", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767842777551, "lm_q2_score": 0.8128673201042493, "lm_q1q2_score": 0.7132722020895533}} {"text": "[STATEMENT]\nlemma disjointify_union:\n assumes \"finite As\"\n shows \"\\ (disjointify As) = \\ As\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (disjointify As) = \\ As\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite As\n\ngoal (1 subgoal):\n 1. \\ (disjointify As) = \\ As\n[PROOF STEP]\nby (simp add: disjointify_def enumerate_enumerates rec_disjointify_finite_set)", "meta": {"llama_tokens": 168, "file": "Padic_Field_Generated_Boolean_Algebra", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.713272201474012}} {"text": "[STATEMENT]\nlemma (in ccpo) Sup_image_mono:\n assumes ccpo: \"class.ccpo luba orda lessa\"\n and mono: \"monotone orda (\\) f\"\n and chain: \"Complete_Partial_Order.chain orda A\"\n and \"A \\ {}\"\n shows \"Sup (f ` A) \\ (f (luba A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (f ` A) \\ f (luba A)\n[PROOF STEP]\nproof(rule ccpo_Sup_least)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. Complete_Partial_Order.chain (\\) (f ` A)\n 2. \\x. x \\ f ` A \\ x \\ f (luba A)\n[PROOF STEP]\nfrom chain\n[PROOF STATE]\nproof (chain)\npicking this:\nComplete_Partial_Order.chain orda A\n[PROOF STEP]\nshow \"Complete_Partial_Order.chain (\\) (f ` A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nComplete_Partial_Order.chain orda A\n\ngoal (1 subgoal):\n 1. Complete_Partial_Order.chain (\\) (f ` A)\n[PROOF STEP]\nby(rule chain_imageI)(rule monotoneD[OF mono])\n[PROOF STATE]\nproof (state)\nthis:\nComplete_Partial_Order.chain (\\) (f ` A)\n\ngoal (1 subgoal):\n 1. \\x. x \\ f ` A \\ x \\ f (luba A)\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ f ` A \\ x \\ f (luba A)\n[PROOF STEP]\nassume \"x \\ f ` A\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ f ` A\n\ngoal (1 subgoal):\n 1. \\x. x \\ f ` A \\ x \\ f (luba A)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ f ` A\n[PROOF STEP]\nobtain y where \"x = f y\" \"y \\ A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ f ` A\n\ngoal (1 subgoal):\n 1. (\\y. \\x = f y; y \\ A\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nx = f y\ny \\ A\n\ngoal (1 subgoal):\n 1. \\x. x \\ f ` A \\ x \\ f (luba A)\n[PROOF STEP]\nfrom \\y \\ A\\\n[PROOF STATE]\nproof (chain)\npicking this:\ny \\ A\n[PROOF STEP]\nhave \"orda y (luba A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ny \\ A\n\ngoal (1 subgoal):\n 1. orda y (luba A)\n[PROOF STEP]\nby(rule ccpo.ccpo_Sup_upper[OF ccpo chain])\n[PROOF STATE]\nproof (state)\nthis:\norda y (luba A)\n\ngoal (1 subgoal):\n 1. \\x. x \\ f ` A \\ x \\ f (luba A)\n[PROOF STEP]\nhence \"f y \\ f (luba A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\norda y (luba A)\n\ngoal (1 subgoal):\n 1. f y \\ f (luba A)\n[PROOF STEP]\nby(rule monotoneD[OF mono])\n[PROOF STATE]\nproof (state)\nthis:\nf y \\ f (luba A)\n\ngoal (1 subgoal):\n 1. \\x. x \\ f ` A \\ x \\ f (luba A)\n[PROOF STEP]\nthus \"x \\ f (luba A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nf y \\ f (luba A)\n\ngoal (1 subgoal):\n 1. x \\ f (luba A)\n[PROOF STEP]\nusing \\x = f y\\\n[PROOF STATE]\nproof (prove)\nusing this:\nf y \\ f (luba A)\nx = f y\n\ngoal (1 subgoal):\n 1. x \\ f (luba A)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx \\ f (luba A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1363, "file": "CryptHOL_Misc_CryptHOL", "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.7132721987271834}} {"text": "[STATEMENT]\nlemma \"(a::int) ^ 4 - b ^ 4 = (a - b) * (a + b) * (a ^ 2 + b ^ 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a ^ 4 - b ^ 4 = (a - b) * (a + b) * (a\\<^sup>2 + b\\<^sup>2)\n[PROOF STEP]\nby ring", "meta": {"llama_tokens": 113, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9207896715436482, "lm_q2_score": 0.7745833893685269, "lm_q1q2_score": 0.7132283846798116}} {"text": "[STATEMENT]\nlemma fib_plus_2: \"fib (n + 2) = fib (n + 1) + fib n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fib (n + 2) = fib (n + 1) + fib n\n[PROOF STEP]\nby (metis Suc_eq_plus1 add_2_eq_Suc' fib.simps(3))", "meta": {"llama_tokens": 111, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9207896671963207, "lm_q2_score": 0.7745833789613196, "lm_q1q2_score": 0.713228371729595}} {"text": "[STATEMENT]\nlemma trace_measure2_id:\n assumes dM0: \"M0 \\ carrier_mat n n\" and dM1: \"M1 \\ carrier_mat n n\" \n and id: \"adjoint M0 * M0 + adjoint M1 * M1 = 1\\<^sub>m n\"\n and dA: \"A \\ carrier_mat n n\"\n shows \"trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\n[PROOF STEP]\nhave \"trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace ((adjoint M0 * M0 + adjoint M1 * M1) * A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace ((adjoint M0 * M0 + adjoint M1 * M1) * A)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nM0 \\ carrier_mat n n\nM1 \\ carrier_mat n n\nadjoint M0 * M0 + adjoint M1 * M1 = 1\\<^sub>m n\nA \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace ((adjoint M0 * M0 + adjoint M1 * M1) * A)\n[PROOF STEP]\nby (mat_assoc n)\n[PROOF STATE]\nproof (state)\nthis:\ntrace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace ((adjoint M0 * M0 + adjoint M1 * M1) * A)\n\ngoal (1 subgoal):\n 1. trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ntrace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace ((adjoint M0 * M0 + adjoint M1 * M1) * A)\n\ngoal (1 subgoal):\n 1. trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\n[PROOF STEP]\nhave \"\\ = trace (1\\<^sub>m n * A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace ((adjoint M0 * M0 + adjoint M1 * M1) * A) = trace (1\\<^sub>m n * A)\n[PROOF STEP]\nusing id\n[PROOF STATE]\nproof (prove)\nusing this:\nadjoint M0 * M0 + adjoint M1 * M1 = 1\\<^sub>m n\n\ngoal (1 subgoal):\n 1. trace ((adjoint M0 * M0 + adjoint M1 * M1) * A) = trace (1\\<^sub>m n * A)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace ((adjoint M0 * M0 + adjoint M1 * M1) * A) = trace (1\\<^sub>m n * A)\n\ngoal (1 subgoal):\n 1. trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ntrace ((adjoint M0 * M0 + adjoint M1 * M1) * A) = trace (1\\<^sub>m n * A)\n\ngoal (1 subgoal):\n 1. trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\n[PROOF STEP]\nhave \"\\ = trace A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (1\\<^sub>m n * A) = trace A\n[PROOF STEP]\nusing dA\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. trace (1\\<^sub>m n * A) = trace A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace (1\\<^sub>m n * A) = trace A\n\ngoal (1 subgoal):\n 1. trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ntrace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntrace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\n\ngoal (1 subgoal):\n 1. trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ntrace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1565, "file": "QHLProver_Quantum_Program", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473746782093, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7132089537276963}} {"text": "[STATEMENT]\nlemma Suc_0_mod_eq_Suc_0_iff:\n \"Suc 0 mod n = Suc 0 \\ n \\ Suc 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Suc 0 mod n = Suc 0) = (n \\ Suc 0)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (Suc 0 mod n = Suc 0) = (n \\ Suc 0)\n[PROOF STEP]\nconsider \"n = 0\" | \"n = Suc 0\" | \"n > 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n = 0 \\ thesis; n = Suc 0 \\ thesis; 1 < n \\ thesis\\ \\ thesis\n[PROOF STEP]\nby (cases n) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\n = 0 \\ ?thesis; n = Suc 0 \\ ?thesis; 1 < n \\ ?thesis\\ \\ ?thesis\n\ngoal (1 subgoal):\n 1. (Suc 0 mod n = Suc 0) = (n \\ Suc 0)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n = 0 \\ ?thesis; n = Suc 0 \\ ?thesis; 1 < n \\ ?thesis\\ \\ ?thesis\n\ngoal (1 subgoal):\n 1. (Suc 0 mod n = Suc 0) = (n \\ Suc 0)\n[PROOF STEP]\nby cases auto\n[PROOF STATE]\nproof (state)\nthis:\n(Suc 0 mod n = Suc 0) = (n \\ Suc 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 564, "file": "Pratt_Certificate_Pratt_Certificate", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473647220786, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7132089455878179}} {"text": "[STATEMENT]\nlemma floor_zero [simp]: \"\\0\\ = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nusing floor_of_int [of 0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\of_int 0\\ = 0\n\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 156, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8558511616741042, "lm_q2_score": 0.8333245932423308, "lm_q1q2_score": 0.7132018211780492}} {"text": "[STATEMENT]\nlemma floor_zero [simp]: \"\\0\\ = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nusing floor_of_int [of 0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\of_int 0\\ = 0\n\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 156, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8558511616741041, "lm_q2_score": 0.8333245911726382, "lm_q1q2_score": 0.7132018194067004}} {"text": "[STATEMENT]\nlemma dfa_accepts_negate: \n assumes \"wf_dfa A n\"\n and \"list_all (is_alph n) bss\"\n shows \"dfa_accepts (negate_dfa A) bss = (\\ dfa_accepts A bss)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dfa_accepts (negate_dfa A) bss = (\\ dfa_accepts A bss)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dfa_accepts (negate_dfa A) bss = (\\ dfa_accepts A bss)\n[PROOF STEP]\nhave \"dfa_steps (negate_dfa A) 0 bss = dfa_steps A 0 bss\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dfa_steps (negate_dfa A) 0 bss = dfa_steps A 0 bss\n[PROOF STEP]\nby (simp add: negate_dfa_def dfa_trans_def [abs_def] split_beta)\n[PROOF STATE]\nproof (state)\nthis:\ndfa_steps (negate_dfa A) 0 bss = dfa_steps A 0 bss\n\ngoal (1 subgoal):\n 1. dfa_accepts (negate_dfa A) bss = (\\ dfa_accepts A bss)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndfa_steps (negate_dfa A) 0 bss = dfa_steps A 0 bss\n\ngoal (1 subgoal):\n 1. dfa_accepts (negate_dfa A) bss = (\\ dfa_accepts A bss)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nwf_dfa A n\nlist_all (is_alph n) bss\n[PROOF STEP]\nhave \"dfa_is_node A (dfa_steps A 0 bss)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nwf_dfa A n\nlist_all (is_alph n) bss\n\ngoal (1 subgoal):\n 1. dfa_is_node A (dfa_steps A 0 bss)\n[PROOF STEP]\nby (simp add: dfa_steps_is_node dfa_startnode_is_node)\n[PROOF STATE]\nproof (state)\nthis:\ndfa_is_node A (dfa_steps A 0 bss)\n\ngoal (1 subgoal):\n 1. dfa_accepts (negate_dfa A) bss = (\\ dfa_accepts A bss)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ndfa_steps (negate_dfa A) 0 bss = dfa_steps A 0 bss\ndfa_is_node A (dfa_steps A 0 bss)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndfa_steps (negate_dfa A) 0 bss = dfa_steps A 0 bss\ndfa_is_node A (dfa_steps A 0 bss)\n\ngoal (1 subgoal):\n 1. dfa_accepts (negate_dfa A) bss = (\\ dfa_accepts A bss)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndfa_steps (negate_dfa A) 0 bss = dfa_steps A 0 bss\ndfa_is_node A (dfa_steps A 0 bss)\nwf_dfa A n\nlist_all (is_alph n) bss\n\ngoal (1 subgoal):\n 1. dfa_accepts (negate_dfa A) bss = (\\ dfa_accepts A bss)\n[PROOF STEP]\nby (simp add: accepts_def dfa_accepting_def wf_dfa_def dfa_is_node_def negate_dfa_def split_beta)\n[PROOF STATE]\nproof (state)\nthis:\ndfa_accepts (negate_dfa A) bss = (\\ dfa_accepts A bss)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1179, "file": "Presburger-Automata_Presburger_Automata", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511469672594, "lm_q2_score": 0.8333246015211008, "lm_q1q2_score": 0.7132018160078685}} {"text": "[STATEMENT]\nlemma det_identical_cols:\n assumes A: \"A \\ carrier_mat n n\"\n and ij: \"i \\ j\"\n and i: \"i < n\" and j: \"j < n\"\n and r: \"col A i = col A j\"\n shows \"det A = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A = (0::'a)\n[PROOF STEP]\nusing det_identical_rows det_transpose\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?A \\ carrier_mat ?n ?n; ?i \\ ?j; ?i < ?n; ?j < ?n; row ?A ?i = row ?A ?j\\ \\ det ?A = (0::?'a)\n?A \\ carrier_mat ?n ?n \\ det ?A\\<^sup>T = det ?A\n\ngoal (1 subgoal):\n 1. det A = (0::'a)\n[PROOF STEP]\nby (metis A i ij j carrier_matD(2) transpose_carrier_mat r row_transpose)", "meta": {"llama_tokens": 314, "file": "Jordan_Normal_Form_DL_Rank", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111797075998823, "lm_q2_score": 0.7826624789529375, "lm_q1q2_score": 0.7131461687217366}} {"text": "[STATEMENT]\nlemma DERIV_inverse_fun:\n \"(f has_field_derivative d) (at x within s) \\ f x \\ 0 \\\n ((\\x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative d) (at x within s); f x \\ (0::'a)\\ \\ ((\\x. inverse (f x)) has_field_derivative - (d * inverse (f x ^ Suc (Suc 0)))) (at x within s)\n[PROOF STEP]\nby (drule (1) DERIV_inverse') (simp add: ac_simps nonzero_inverse_mult_distrib)", "meta": {"llama_tokens": 242, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513675912913, "lm_q2_score": 0.7956581073313275, "lm_q1q2_score": 0.7131096668308007}} {"text": "[STATEMENT]\nlemma DERIV_inverse_fun:\n \"(f has_field_derivative d) (at x within s) \\ f x \\ 0 \\\n ((\\x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative d) (at x within s); f x \\ (0::'a)\\ \\ ((\\x. inverse (f x)) has_field_derivative - (d * inverse (f x ^ Suc (Suc 0)))) (at x within s)\n[PROOF STEP]\nby (drule (1) DERIV_inverse') (simp add: ac_simps nonzero_inverse_mult_distrib)", "meta": {"llama_tokens": 242, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513675912913, "lm_q2_score": 0.7956581073313275, "lm_q1q2_score": 0.7131096668308007}} {"text": "[STATEMENT]\nlemma DERIV_inverse_fun:\n \"(f has_field_derivative d) (at x within s) \\ f x \\ 0 \\\n ((\\x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative d) (at x within s); f x \\ (0::'a)\\ \\ ((\\x. inverse (f x)) has_field_derivative - (d * inverse (f x ^ Suc (Suc 0)))) (at x within s)\n[PROOF STEP]\nby (drule (1) DERIV_inverse') (simp add: ac_simps nonzero_inverse_mult_distrib)", "meta": {"llama_tokens": 242, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513675912913, "lm_q2_score": 0.7956581073313275, "lm_q1q2_score": 0.7131096668308007}} {"text": "[STATEMENT]\nlemma mono_gfp_eqI:\n assumes MONO: \"mono f\"\n assumes FIXP: \"a \\ f a\"\n assumes GREATEST: \"\\x. f x = x \\ x\\a\"\n shows \"gfp f = a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. gfp f = a\n[PROOF STEP]\napply (rule antisym)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. gfp f \\ a\n 2. a \\ gfp f\n[PROOF STEP]\napply (metis GREATEST MONO gfp_unfold)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ gfp f\n[PROOF STEP]\napply (rule gfp_upperbound)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ f a\n[PROOF STEP]\napply (rule FIXP)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 317, "file": "Refine_Monadic_Refine_Misc", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513675912912, "lm_q2_score": 0.7956580927949806, "lm_q1q2_score": 0.7131096538025798}} {"text": "[STATEMENT]\nlemma card_complex_roots_unity_explicit:\n \"card {exp(2 * of_real pi * \\ * of_nat j / of_nat n) | j::nat. j < n} = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) |j. j < n} = n\n[PROOF STEP]\nby (simp add: Finite_Set.bij_betw_same_card [OF bij_betw_roots_unity, symmetric])", "meta": {"llama_tokens": 160, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.88242786954645, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.7130610358238104}} {"text": "[STATEMENT]\nlemma sum_list_mono2: fixes xs :: \"'a ::ordered_comm_monoid_add list\"\nshows \"\\ length xs = length ys; \\i. i < length xs \\ xs!i \\ ys!i \\\n \\ sum_list xs \\ sum_list ys\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\length xs = length ys; \\i. i < length xs \\ xs ! i \\ ys ! i\\ \\ sum_list xs \\ sum_list ys\n[PROOF STEP]\napply(induction xs ys rule: list_induct2)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (\\i. i < length [] \\ [] ! i \\ [] ! i) \\ sum_list [] \\ sum_list []\n 2. \\x xs y ys. \\length xs = length ys; (\\i. i < length xs \\ xs ! i \\ ys ! i) \\ sum_list xs \\ sum_list ys; \\i. i < length (x # xs) \\ (x # xs) ! i \\ (y # ys) ! i\\ \\ sum_list (x # xs) \\ sum_list (y # ys)\n[PROOF STEP]\nby(auto simp: nth_Cons' less_Suc_eq_0_disj imp_ex add_mono)", "meta": {"llama_tokens": 418, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278664544912, "lm_q2_score": 0.8080672158638528, "lm_q1q2_score": 0.7130610292465605}} {"text": "[STATEMENT]\nlemma INT_Un_Compl_subset:\n \"(\\i \\ lessThan n. -A i \\ A (Suc i)) \\ \n (\\i \\ lessThan n. -A i) \\ A n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i A (Suc i)) \\ (\\i A n\n[PROOF STEP]\nby (induct n) (auto simp: lessThan_Suc)", "meta": {"llama_tokens": 160, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278602705732, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.7130610283282783}} {"text": "[STATEMENT]\nlemma vectorize_plus: \"vectorize_poly (p + q) = vectorize_poly p + vectorize_poly q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vectorize_poly (p + q) = vectorize_poly p + vectorize_poly q\n[PROOF STEP]\nby (rule poly_mapping_eqI, simp add: lookup_vectorize_poly lookup_add proj_plus)", "meta": {"llama_tokens": 115, "file": "Polynomials_MPoly_Type_Class", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278633625321, "lm_q2_score": 0.8080672158638527, "lm_q1q2_score": 0.7130610267480495}} {"text": "[STATEMENT]\nlemma exp_times_self_strict_mono:\n assumes \"x \\ -1\" \"x < (y :: real)\"\n shows \"x * exp x < y * exp y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * exp x < y * exp y\n[PROOF STEP]\nusing assms(2)\n[PROOF STATE]\nproof (prove)\nusing this:\nx < y\n\ngoal (1 subgoal):\n 1. x * exp x < y * exp y\n[PROOF STEP]\nproof (rule DERIV_pos_imp_increasing_open)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\xa. \\x < xa; xa < y\\ \\ \\y. ((\\a. a * exp a) has_real_derivative y) (at xa) \\ 0 < y\n 2. continuous_on {x..y} (\\a. a * exp a)\n[PROOF STEP]\nfix t\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\xa. \\x < xa; xa < y\\ \\ \\y. ((\\a. a * exp a) has_real_derivative y) (at xa) \\ 0 < y\n 2. continuous_on {x..y} (\\a. a * exp a)\n[PROOF STEP]\nassume t: \"x < t\" \"t < y\"\n[PROOF STATE]\nproof (state)\nthis:\nx < t\nt < y\n\ngoal (2 subgoals):\n 1. \\xa. \\x < xa; xa < y\\ \\ \\y. ((\\a. a * exp a) has_real_derivative y) (at xa) \\ 0 < y\n 2. continuous_on {x..y} (\\a. a * exp a)\n[PROOF STEP]\nhave \"((\\x. x * exp x) has_real_derivative (t + 1) * exp t) (at t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. x * exp x) has_real_derivative (t + 1) * exp t) (at t)\n[PROOF STEP]\nby (auto intro!: derivative_eq_intros simp: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. x * exp x) has_real_derivative (t + 1) * exp t) (at t)\n\ngoal (2 subgoals):\n 1. \\xa. \\x < xa; xa < y\\ \\ \\y. ((\\a. a * exp a) has_real_derivative y) (at xa) \\ 0 < y\n 2. continuous_on {x..y} (\\a. a * exp a)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. x * exp x) has_real_derivative (t + 1) * exp t) (at t)\n\ngoal (2 subgoals):\n 1. \\xa. \\x < xa; xa < y\\ \\ \\y. ((\\a. a * exp a) has_real_derivative y) (at xa) \\ 0 < y\n 2. continuous_on {x..y} (\\a. a * exp a)\n[PROOF STEP]\nhave \"(t + 1) * exp t > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < (t + 1) * exp t\n[PROOF STEP]\nusing t assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx < t\nt < y\n- 1 \\ x\nx < y\n\ngoal (1 subgoal):\n 1. 0 < (t + 1) * exp t\n[PROOF STEP]\nby (intro mult_pos_pos) auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < (t + 1) * exp t\n\ngoal (2 subgoals):\n 1. \\xa. \\x < xa; xa < y\\ \\ \\y. ((\\a. a * exp a) has_real_derivative y) (at xa) \\ 0 < y\n 2. continuous_on {x..y} (\\a. a * exp a)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n((\\x. x * exp x) has_real_derivative (t + 1) * exp t) (at t)\n0 < (t + 1) * exp t\n[PROOF STEP]\nshow \"\\y. ((\\a. a * exp a) has_real_derivative y) (at t) \\ 0 < y\"\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. x * exp x) has_real_derivative (t + 1) * exp t) (at t)\n0 < (t + 1) * exp t\n\ngoal (1 subgoal):\n 1. \\y. ((\\a. a * exp a) has_real_derivative y) (at t) \\ 0 < y\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\y. ((\\a. a * exp a) has_real_derivative y) (at t) \\ 0 < y\n\ngoal (1 subgoal):\n 1. continuous_on {x..y} (\\a. a * exp a)\n[PROOF STEP]\nqed (auto intro!: continuous_intros)", "meta": {"llama_tokens": 1613, "file": "Lambert_W_Lambert_W", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278540866547, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.7130610233312565}} {"text": "[STATEMENT]\nlemma mat_mult_invertible_lattice_eq: \n assumes fs: \"set fs \\ carrier_vec n\"\n and gs: \"set gs \\ carrier_vec n\" \n and P: \"P \\ carrier_mat m m\" and invertible_P: \"invertible_mat P\"\n and length_fs: \"length fs = m\" and length_gs: \"length gs = m\"\n and prod: \"mat_of_rows n fs = (map_mat of_int P) * mat_of_rows n gs\" \n shows \"lattice_of fs = lattice_of gs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lattice_of fs = lattice_of gs\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. lattice_of fs \\ lattice_of gs\n 2. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nthm mat_mult_sub_lattice\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. lattice_of fs \\ lattice_of gs\n 2. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nshow \"lattice_of fs \\ lattice_of gs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lattice_of fs \\ lattice_of gs\n[PROOF STEP]\nby (rule mat_mult_sub_lattice[OF fs gs _ prod],simp add: length_fs length_gs P)\n[PROOF STATE]\nproof (state)\nthis:\nlattice_of fs \\ lattice_of gs\n\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nobtain inv_P where P_inv_P: \"inverts_mat P inv_P\" and inv_P_P: \"inverts_mat inv_P P\"\n and inv_P: \"inv_P \\ carrier_mat m m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\inv_P. \\inverts_mat P inv_P; inverts_mat inv_P P; inv_P \\ carrier_mat m m\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing P invertible_P obtain_inverse_matrix\n[PROOF STATE]\nproof (prove)\nusing this:\nP \\ carrier_mat m m\ninvertible_mat P\n\\?A \\ carrier_mat ?n ?n; invertible_mat ?A; \\B. \\inverts_mat ?A B; inverts_mat B ?A; B \\ carrier_mat ?n ?n\\ \\ ?thesis\\ \\ ?thesis\n\ngoal (1 subgoal):\n 1. (\\inv_P. \\inverts_mat P inv_P; inverts_mat inv_P P; inv_P \\ carrier_mat m m\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ninverts_mat P inv_P\ninverts_mat inv_P P\ninv_P \\ carrier_mat m m\n\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nhave \"of_int_hom.mat_hom (inv_P) * mat_of_rows n fs \n = of_int_hom.mat_hom (inv_P) * ((map_mat of_int P) * mat_of_rows n gs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int_hom.mat_hom inv_P * mat_of_rows n fs = of_int_hom.mat_hom inv_P * (of_int_hom.mat_hom P * mat_of_rows n gs)\n[PROOF STEP]\nusing prod\n[PROOF STATE]\nproof (prove)\nusing this:\nmat_of_rows n fs = of_int_hom.mat_hom P * mat_of_rows n gs\n\ngoal (1 subgoal):\n 1. of_int_hom.mat_hom inv_P * mat_of_rows n fs = of_int_hom.mat_hom inv_P * (of_int_hom.mat_hom P * mat_of_rows n gs)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nof_int_hom.mat_hom inv_P * mat_of_rows n fs = of_int_hom.mat_hom inv_P * (of_int_hom.mat_hom P * mat_of_rows n gs)\n\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nof_int_hom.mat_hom inv_P * mat_of_rows n fs = of_int_hom.mat_hom inv_P * (of_int_hom.mat_hom P * mat_of_rows n gs)\n\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nhave \"... = of_int_hom.mat_hom (inv_P) * (map_mat of_int P) * mat_of_rows n gs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int_hom.mat_hom inv_P * (of_int_hom.mat_hom P * mat_of_rows n gs) = of_int_hom.mat_hom inv_P * of_int_hom.mat_hom P * mat_of_rows n gs\n[PROOF STEP]\nby (smt P assoc_mult_mat inv_P length_gs map_carrier_mat mat_of_rows_carrier(1))\n[PROOF STATE]\nproof (state)\nthis:\nof_int_hom.mat_hom inv_P * (of_int_hom.mat_hom P * mat_of_rows n gs) = of_int_hom.mat_hom inv_P * of_int_hom.mat_hom P * mat_of_rows n gs\n\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nof_int_hom.mat_hom inv_P * (of_int_hom.mat_hom P * mat_of_rows n gs) = of_int_hom.mat_hom inv_P * of_int_hom.mat_hom P * mat_of_rows n gs\n\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nhave \"... = of_int_hom.mat_hom (inv_P * P) * mat_of_rows n gs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int_hom.mat_hom inv_P * of_int_hom.mat_hom P * mat_of_rows n gs = of_int_hom.mat_hom (inv_P * P) * mat_of_rows n gs\n[PROOF STEP]\nby (metis P inv_P of_int_hom.mat_hom_mult)\n[PROOF STATE]\nproof (state)\nthis:\nof_int_hom.mat_hom inv_P * of_int_hom.mat_hom P * mat_of_rows n gs = of_int_hom.mat_hom (inv_P * P) * mat_of_rows n gs\n\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nof_int_hom.mat_hom inv_P * of_int_hom.mat_hom P * mat_of_rows n gs = of_int_hom.mat_hom (inv_P * P) * mat_of_rows n gs\n\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nhave \"... = mat_of_rows n gs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int_hom.mat_hom (inv_P * P) * mat_of_rows n gs = mat_of_rows n gs\n[PROOF STEP]\nby (metis carrier_matD(1) inv_P inv_P_P inverts_mat_def left_mult_one_mat' \n length_gs mat_of_rows_carrier(2) of_int_hom.mat_hom_one)\n[PROOF STATE]\nproof (state)\nthis:\nof_int_hom.mat_hom (inv_P * P) * mat_of_rows n gs = mat_of_rows n gs\n\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nof_int_hom.mat_hom inv_P * mat_of_rows n fs = mat_of_rows n gs\n[PROOF STEP]\nhave prod: \"mat_of_rows n gs = of_int_hom.mat_hom (inv_P) * mat_of_rows n fs\"\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int_hom.mat_hom inv_P * mat_of_rows n fs = mat_of_rows n gs\n\ngoal (1 subgoal):\n 1. mat_of_rows n gs = of_int_hom.mat_hom inv_P * mat_of_rows n fs\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nmat_of_rows n gs = of_int_hom.mat_hom inv_P * mat_of_rows n fs\n\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nshow \"lattice_of gs \\ lattice_of fs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lattice_of gs \\ lattice_of fs\n[PROOF STEP]\nby (rule mat_mult_sub_lattice[OF gs fs _ prod], simp add: length_fs length_gs inv_P)\n[PROOF STATE]\nproof (state)\nthis:\nlattice_of gs \\ lattice_of fs\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2872, "file": "Modular_arithmetic_LLL_and_HNF_algorithms_HNF_Mod_Det_Soundness", "length": 26, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278664544911, "lm_q2_score": 0.8080672089305841, "lm_q1q2_score": 0.7130610231284509}} {"text": "[STATEMENT]\ntheorem gs: \"\\ Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a \\ \\\n \\A. Gale_Shapley P\\<^sub>a P\\<^sub>b = Some(A) \\ Pref.matching P\\<^sub>a A {\n Pref.stable P\\<^sub>a P\\<^sub>b A { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a\\ \\ \\A. Gale_Shapley P\\<^sub>a P\\<^sub>b = Some A \\ Pref.matching P\\<^sub>a A { Pref.stable P\\<^sub>a P\\<^sub>b A { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b A\n[PROOF STEP]\nunfolding Gale_Shapley_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a\\ \\ \\A. (if Pref P\\<^sub>a P\\<^sub>b then Some (fst (gs (length P\\<^sub>a) P\\<^sub>a (map ranking P\\<^sub>b))) else None) = Some A \\ Pref.matching P\\<^sub>a A { Pref.stable P\\<^sub>a P\\<^sub>b A { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b A\n[PROOF STEP]\nusing Pref.gs\n[PROOF STATE]\nproof (prove)\nusing this:\n\\Pref ?P\\<^sub>a ?P\\<^sub>b; ?R\\<^sub>b = map ranking ?P\\<^sub>b\\ \\ gs (length ?P\\<^sub>a) ?P\\<^sub>a ?R\\<^sub>b = (?A, ?BMai) \\ Pref.matching ?P\\<^sub>a ?A {a} \\ Pref.stable ?P\\<^sub>a ?P\\<^sub>b ?A {a} \\ Pref.opti\\<^sub>a ?P\\<^sub>a ?P\\<^sub>b ?A\n\ngoal (1 subgoal):\n 1. \\Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a\\ \\ \\A. (if Pref P\\<^sub>a P\\<^sub>b then Some (fst (gs (length P\\<^sub>a) P\\<^sub>a (map ranking P\\<^sub>b))) else None) = Some A \\ Pref.matching P\\<^sub>a A { Pref.stable P\\<^sub>a P\\<^sub>b A { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b A\n[PROOF STEP]\nby (metis fst_conv surj_pair)", "meta": {"llama_tokens": 832, "file": "Gale_Shapley_Gale_Shapley1", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278571786139, "lm_q2_score": 0.8080672158638528, "lm_q1q2_score": 0.713061021751028}} {"text": "[STATEMENT]\nlemma mat_inverse_simps[simp]:\n fixes c :: \"'a::division_ring\"\n assumes \"c \\ 0\"\n shows \"mat (inverse c) ** mat c = mat 1\" \n and \"mat c ** mat (inverse c) = mat 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (inverse c) ** mat c = mat (1::'a) &&& mat c ** mat (inverse c) = mat (1::'a)\n[PROOF STEP]\nunfolding matrix_matrix_mult_def mat_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i j. \\k\\UNIV. (\\i j. if i = j then inverse c else (0::'a)) $ i $ k * (\\i j. if i = j then c else (0::'a)) $ k $ j) = (\\i j. if i = j then 1::'a else (0::'a)) &&& (\\i j. \\k\\UNIV. (\\i j. if i = j then c else (0::'a)) $ i $ k * (\\i j. if i = j then inverse c else (0::'a)) $ k $ j) = (\\i j. if i = j then 1::'a else (0::'a))\n[PROOF STEP]\nby (auto simp: vec_eq_iff assms)", "meta": {"llama_tokens": 386, "file": "Matrices_for_ODEs_MTX_Preliminaries", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278571786139, "lm_q2_score": 0.8080672158638527, "lm_q1q2_score": 0.7130610217510279}} {"text": "[STATEMENT]\nlemma lower_asymptotic_density_one_intersection:\n assumes \"lower_asymptotic_density A = 1\" \"lower_asymptotic_density B = 1\"\n shows \"lower_asymptotic_density (A \\ B) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lower_asymptotic_density (A \\ B) = 1\n[PROOF STEP]\nusing lower_asymptotic_density_in_01(2)[of \"A \\ B\"] lower_asymptotic_density_intersection[of A B]\n[PROOF STATE]\nproof (prove)\nusing this:\nlower_asymptotic_density (A \\ B) \\ 1\nlower_asymptotic_density A + lower_asymptotic_density B \\ lower_asymptotic_density (A \\ B) + 1\n\ngoal (1 subgoal):\n 1. lower_asymptotic_density (A \\ B) = 1\n[PROOF STEP]\nunfolding assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlower_asymptotic_density (A \\ B) \\ 1\n1 + 1 \\ lower_asymptotic_density (A \\ B) + 1\n\ngoal (1 subgoal):\n 1. lower_asymptotic_density (A \\ B) = 1\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 409, "file": "Ergodic_Theory_Asymptotic_Density", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094003735664, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7130277518086902}} {"text": "[STATEMENT]\nlemma inv\\<^sub>L_compose: \n assumes \"invertible\\<^sub>L f\" \"invertible\\<^sub>L g\" \n shows\"inv\\<^sub>L (f o\\<^sub>L g) = (inv\\<^sub>L g) o\\<^sub>L (inv\\<^sub>L f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inv\\<^sub>L (f o\\<^sub>L g) = inv\\<^sub>L g o\\<^sub>L inv\\<^sub>L f\n[PROOF STEP]\nusing assms inv\\<^sub>L_eq is_inverse\\<^sub>L_compose\n[PROOF STATE]\nproof (prove)\nusing this:\ninvertible\\<^sub>L f\ninvertible\\<^sub>L g\nis_inverse\\<^sub>L ?f ?g \\ inv\\<^sub>L ?f = ?g\n\\invertible\\<^sub>L ?f; invertible\\<^sub>L ?g\\ \\ is_inverse\\<^sub>L (?f o\\<^sub>L ?g) (inv\\<^sub>L ?g o\\<^sub>L inv\\<^sub>L ?f)\n\ngoal (1 subgoal):\n 1. inv\\<^sub>L (f o\\<^sub>L g) = inv\\<^sub>L g o\\<^sub>L inv\\<^sub>L f\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 352, "file": "MDP-Rewards_Blinfun_Util", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206712569267, "lm_q2_score": 0.7905303186696748, "lm_q1q2_score": 0.7129956356635052}} {"text": "[STATEMENT]\nlemma preordered_finite_set_has_maxima:\n assumes \"finite A\" \"A \\ {}\"\n shows \"\\a::'a::{preorder} \\ A. \\b \\ A. \\(a < b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a\\A. \\b\\A. \\ a < b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nA \\ {}\n\ngoal (1 subgoal):\n 1. \\a\\A. \\b\\A. \\ a < b\n[PROOF STEP]\nproof (induction A rule: finite_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. {} \\ {} \\ \\a\\{}. \\b\\{}. \\ a < b\n 2. \\x F. \\finite F; x \\ F; F \\ {} \\ \\a\\F. \\b\\F. \\ a < b; insert x F \\ {}\\ \\ \\a\\insert x F. \\b\\insert x F. \\ a < b\n[PROOF STEP]\ncase (insert a A)\n[PROOF STATE]\nproof (state)\nthis:\nfinite A\na \\ A\nA \\ {} \\ \\a\\A. \\b\\A. \\ a < b\ninsert a A \\ {}\n\ngoal (2 subgoals):\n 1. {} \\ {} \\ \\a\\{}. \\b\\{}. \\ a < b\n 2. \\x F. \\finite F; x \\ F; F \\ {} \\ \\a\\F. \\b\\F. \\ a < b; insert x F \\ {}\\ \\ \\a\\insert x F. \\b\\insert x F. \\ a < b\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\na \\ A\nA \\ {} \\ \\a\\A. \\b\\A. \\ a < b\ninsert a A \\ {}\n\ngoal (1 subgoal):\n 1. \\aa\\insert a A. \\b\\insert a A. \\ aa < b\n[PROOF STEP]\nby (cases \"A = {}\", simp, metis insert_iff order_trans less_le_not_le)\n[PROOF STATE]\nproof (state)\nthis:\n\\aa\\insert a A. \\b\\insert a A. \\ aa < b\n\ngoal (1 subgoal):\n 1. {} \\ {} \\ \\a\\{}. \\b\\{}. \\ a < b\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 856, "file": "Stateful_Protocol_Composition_and_Typing_Miscellaneous", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681158979307, "lm_q2_score": 0.8311430541321951, "lm_q1q2_score": 0.7129280115846248}} {"text": "[STATEMENT]\ntheorem (in group_action) burnside:\n assumes \"finite (carrier G)\" \"finite E\"\n shows \"card (orbits G E \\) * order G = (\\g \\ carrier G. card(invariants E \\ g))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nhave \"(\\g \\ carrier G. card(invariants E \\ g)) =\n (\\g \\ carrier G. \\x \\ E. (if (\\ g) x = x then 1 else 0))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\g\\carrier G. card (invariants E \\ g)) = (\\g\\carrier G. \\x\\E. if \\ g x = x then 1 else 0)\n[PROOF STEP]\nby (simp add: assms(2) card_as_sums invariants_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\g\\carrier G. card (invariants E \\ g)) = (\\g\\carrier G. \\x\\E. if \\ g x = x then 1 else 0)\n\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\g\\carrier G. card (invariants E \\ g)) = (\\g\\carrier G. \\x\\E. if \\ g x = x then 1 else 0)\n\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nhave \" ... = (\\x \\ E. \\g \\ carrier G. (if (\\ g) x = x then 1 else 0))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\g\\carrier G. \\x\\E. if \\ g x = x then 1 else 0) = (\\x\\E. \\g\\carrier G. if \\ g x = x then 1 else 0)\n[PROOF STEP]\nusing sum_invertion[where ?f = \"\\ g x. (if (\\ g) x = x then 1 else 0)\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite ?A; finite ?B\\ \\ (\\x\\?A. \\y\\?B. if \\ x y = y then 1::?'d1 else (0::?'d1)) = (\\y\\?B. \\x\\?A. if \\ x y = y then 1::?'d1 else (0::?'d1))\nfinite (carrier G)\nfinite E\n\ngoal (1 subgoal):\n 1. (\\g\\carrier G. \\x\\E. if \\ g x = x then 1 else 0) = (\\x\\E. \\g\\carrier G. if \\ g x = x then 1 else 0)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\g\\carrier G. \\x\\E. if \\ g x = x then 1 else 0) = (\\x\\E. \\g\\carrier G. if \\ g x = x then 1 else 0)\n\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\g\\carrier G. \\x\\E. if \\ g x = x then 1 else 0) = (\\x\\E. \\g\\carrier G. if \\ g x = x then 1 else 0)\n\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nhave \" ... = (\\x \\ E. card (stabilizer G \\ x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\E. \\g\\carrier G. if \\ g x = x then 1 else 0) = (\\x\\E. card (stabilizer G \\ x))\n[PROOF STEP]\nby (simp add: assms(1) card_as_sums stabilizer_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\E. \\g\\carrier G. if \\ g x = x then 1 else 0) = (\\x\\E. card (stabilizer G \\ x))\n\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\E. \\g\\carrier G. if \\ g x = x then 1 else 0) = (\\x\\E. card (stabilizer G \\ x))\n\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nhave \" ... = (\\orbit \\ (orbits G E \\). \\x \\ orbit. card (stabilizer G \\ x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\E. card (stabilizer G \\ x)) = sum (sum (\\x. card (stabilizer G \\ x))) (orbits G E \\)\n[PROOF STEP]\nusing disjoint_sum orbits_coverture assms(2)\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite E \\ sum (sum ?f) (orbits G E \\) = sum ?f E\n\\ (orbits G E \\) = E\nfinite E\n\ngoal (1 subgoal):\n 1. (\\x\\E. card (stabilizer G \\ x)) = sum (sum (\\x. card (stabilizer G \\ x))) (orbits G E \\)\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\E. card (stabilizer G \\ x)) = sum (sum (\\x. card (stabilizer G \\ x))) (orbits G E \\)\n\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\E. card (stabilizer G \\ x)) = sum (sum (\\x. card (stabilizer G \\ x))) (orbits G E \\)\n\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nhave \" ... = (\\orbit \\ (orbits G E \\). order G)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum (sum (\\x. card (stabilizer G \\ x))) (orbits G E \\) = (\\orbit\\orbits G E \\. order G)\n[PROOF STEP]\nby (simp add: assms(1) card_stablizer_sum)\n[PROOF STATE]\nproof (state)\nthis:\nsum (sum (\\x. card (stabilizer G \\ x))) (orbits G E \\) = (\\orbit\\orbits G E \\. order G)\n\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\g\\carrier G. card (invariants E \\ g)) = (\\orbit\\orbits G E \\. order G)\n[PROOF STEP]\nhave \"(\\g \\ carrier G. card(invariants E \\ g)) = card (orbits G E \\) * order G\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\g\\carrier G. card (invariants E \\ g)) = (\\orbit\\orbits G E \\. order G)\n\ngoal (1 subgoal):\n 1. (\\g\\carrier G. card (invariants E \\ g)) = card (orbits G E \\) * order G\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\g\\carrier G. card (invariants E \\ g)) = card (orbits G E \\) * order G\n\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\g\\carrier G. card (invariants E \\ g)) = card (orbits G E \\) * order G\n\ngoal (1 subgoal):\n 1. card (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (orbits G E \\) * order G = (\\g\\carrier G. card (invariants E \\ g))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3018, "file": null, "length": 23, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681158979306, "lm_q2_score": 0.8311430520409023, "lm_q1q2_score": 0.7129280097907804}} {"text": "[STATEMENT]\nlemma bij_betw_image_Fpow:\nassumes \"bij_betw f A B\"\nshows \"bij_betw (image f) (Fpow A) (Fpow B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw ((`) f) (Fpow A) (Fpow B)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw f A B\n\ngoal (1 subgoal):\n 1. bij_betw ((`) f) (Fpow A) (Fpow B)\n[PROOF STEP]\nunfolding bij_betw_def\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on f A \\ f ` A = B\n\ngoal (1 subgoal):\n 1. inj_on ((`) f) (Fpow A) \\ (`) f ` Fpow A = Fpow B\n[PROOF STEP]\nby (auto simp add: inj_on_image_Fpow image_Fpow_surjective)", "meta": {"llama_tokens": 278, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357666736772, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7128673887992788}} {"text": "[STATEMENT]\nlemma compute_ofsm_table_list_props:\n \"length (compute_ofsm_table_list M k) = Suc k\"\n \"\\ i q . i < Suc k \\ ofsm_lookup ((compute_ofsm_table_list M k) ! i) q = ofsm_table M (\\q . states M) i q\"\n \"\\ i . i < Suc k \\ Mapping.keys ((compute_ofsm_table_list M k) ! i) = states M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (compute_ofsm_table_list M k) = Suc k &&& (\\i q. i < Suc k \\ ofsm_lookup (compute_ofsm_table_list M k ! i) q = ofsm_table M (\\q. FSM.states M) i q) &&& (\\i. i < Suc k \\ Mapping.keys (compute_ofsm_table_list M k ! i) = FSM.states M)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. length (compute_ofsm_table_list M k) = Suc k\n 2. \\i q. i < Suc k \\ ofsm_lookup (compute_ofsm_table_list M k ! i) q = ofsm_table M (\\q. FSM.states M) i q\n 3. \\i. i < Suc k \\ Mapping.keys (compute_ofsm_table_list M k ! i) = FSM.states M\n[PROOF STEP]\ndefine t where \"t = (\\ k . (foldr (\\ _ prev . (next_ofsm_table M (hd prev)) # prev) (rev [0..k. foldr (\\_ prev. next_ofsm_table M (hd prev) # prev) (rev [0..i q. i < Suc k \\ ofsm_lookup (compute_ofsm_table_list M k ! i) q = ofsm_table M (\\q. FSM.states M) i q\n 3. \\i. i < Suc k \\ Mapping.keys (compute_ofsm_table_list M k ! i) = FSM.states M\n[PROOF STEP]\nhave t_props:\"length (t k) = Suc k\n \\ (\\ i q . i < Suc k \\ ofsm_lookup (t k ! (k-i)) q = ofsm_table M (\\q . states M) i q)\n \\ (\\ i . i < Suc k \\ Mapping.keys (t k ! i) = states M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i (\\i q. i < Suc 0 \\ ofsm_lookup (t 0 ! (0 - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ik. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i (\\i q. i < Suc 0 \\ ofsm_lookup (t 0 ! (0 - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ik. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i_ prev. next_ofsm_table M (hd prev) # prev) (rev [0..<0]) [initial_ofsm_table M] = [initial_ofsm_table M]\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nt 0 = [initial_ofsm_table M]\n\ngoal (2 subgoals):\n 1. length (t 0) = Suc 0 \\ (\\i q. i < Suc 0 \\ ofsm_lookup (t 0 ! (0 - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ik. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i (\\i q. i < Suc 0 \\ ofsm_lookup (t 0 ! (0 - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\it 0 = [initial_ofsm_table M]\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length [initial_ofsm_table M] = Suc 0 \\ (\\i q. i < Suc 0 \\ ofsm_lookup ([initial_ofsm_table M] ! (0 - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\iq. FSM.states M) 0 ?q\n\ngoal (1 subgoal):\n 1. length [initial_ofsm_table M] = Suc 0 \\ (\\i q. i < Suc 0 \\ ofsm_lookup ([initial_ofsm_table M] ! (0 - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\iq. FSM.states M) 0 ?q\nMapping.keys (initial_ofsm_table M) = FSM.states M\n\ngoal (1 subgoal):\n 1. length [initial_ofsm_table M] = Suc 0 \\ (\\i q. i < Suc 0 \\ ofsm_lookup ([initial_ofsm_table M] ! (0 - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i (\\i q. i < Suc 0 \\ ofsm_lookup (t 0 ! (0 - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ik. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ik. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ik. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ik. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\irev [0..\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. foldr (\\_ prev. next_ofsm_table M (hd prev) # prev) (k # rev [0.._ prev. next_ofsm_table M (hd prev) # prev) (rev [0.._ prev. next_ofsm_table M (hd prev) # prev) (rev [0..k. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ii q . i < Suc k \\ ofsm_lookup (t k ! (k-i)) q = ofsm_table M (\\q. FSM.states M) i q\"\n and IH3: \"\\i . i < Suc k \\ Mapping.keys (t k ! i) = FSM.states M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (t k) = Suc k &&& (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) &&& (\\i. i < Suc k \\ Mapping.keys (t k ! i) = FSM.states M)\n[PROOF STEP]\nusing Suc.IH\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ii q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) &&& (\\i. i < Suc k \\ Mapping.keys (t k ! i) = FSM.states M)\n[PROOF STEP]\nby blast+\n[PROOF STATE]\nproof (state)\nthis:\nlength (t k) = Suc k\n?i < Suc k \\ ofsm_lookup (t k ! (k - ?i)) ?q = ofsm_table M (\\q. FSM.states M) ?i ?q\n?i < Suc k \\ Mapping.keys (t k ! ?i) = FSM.states M\n\ngoal (1 subgoal):\n 1. \\k. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ik. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ik. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ii q . i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! ((Suc k)-i)) q = ofsm_table M (\\q. FSM.states M) i q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nfix i q\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nassume \"i < Suc (Suc k)\"\n[PROOF STATE]\nproof (state)\nthis:\ni < Suc (Suc k)\n\ngoal (1 subgoal):\n 1. \\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni < Suc (Suc k)\n[PROOF STEP]\nconsider \"i = Suc k\" | \"i < Suc k\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < Suc (Suc k)\n\ngoal (1 subgoal):\n 1. \\i = Suc k \\ thesis; i < Suc k \\ thesis\\ \\ thesis\n[PROOF STEP]\nusing less_Suc_eq\n[PROOF STATE]\nproof (prove)\nusing this:\ni < Suc (Suc k)\n(?m < Suc ?n) = (?m < ?n \\ ?m = ?n)\n\ngoal (1 subgoal):\n 1. \\i = Suc k \\ thesis; i < Suc k \\ thesis\\ \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\i = Suc k \\ ?thesis; i < Suc k \\ ?thesis\\ \\ ?thesis\n\ngoal (1 subgoal):\n 1. \\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i = Suc k \\ ?thesis; i < Suc k \\ ?thesis\\ \\ ?thesis\n[PROOF STEP]\nshow \"ofsm_lookup (t (Suc k) ! ((Suc k)-i)) q = ofsm_table M (\\q. FSM.states M) i q\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i = Suc k \\ ?thesis; i < Suc k \\ ?thesis\\ \\ ?thesis\n\ngoal (1 subgoal):\n 1. ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nproof cases\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. i = Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n 2. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\ncase 1\n[PROOF STATE]\nproof (state)\nthis:\ni = Suc k\n\ngoal (2 subgoals):\n 1. i = Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n 2. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni = Suc k\n[PROOF STEP]\nhave \"(t (Suc k) ! ((Suc k)-i)) = hd (t (Suc k))\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni = Suc k\n\ngoal (1 subgoal):\n 1. t (Suc k) ! (Suc k - i) = hd (t (Suc k))\n[PROOF STEP]\nby (metis \"*\" diff_self_eq_0 list.sel(1) nth_Cons_0)\n[PROOF STATE]\nproof (state)\nthis:\nt (Suc k) ! (Suc k - i) = hd (t (Suc k))\n\ngoal (2 subgoals):\n 1. i = Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n 2. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nt (Suc k) ! (Suc k - i) = hd (t (Suc k))\n[PROOF STEP]\nhave \"(t (Suc k) ! ((Suc k)-i)) = next_ofsm_table M (hd (t k))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nt (Suc k) ! (Suc k - i) = hd (t (Suc k))\n\ngoal (1 subgoal):\n 1. t (Suc k) ! (Suc k - i) = next_ofsm_table M (hd (t k))\n[PROOF STEP]\nunfolding *\n[PROOF STATE]\nproof (prove)\nusing this:\n(next_ofsm_table M (hd (t k)) # t k) ! (Suc k - i) = hd (next_ofsm_table M (hd (t k)) # t k)\n\ngoal (1 subgoal):\n 1. (next_ofsm_table M (hd (t k)) # t k) ! (Suc k - i) = next_ofsm_table M (hd (t k))\n[PROOF STEP]\nby (metis list.sel(1))\n[PROOF STATE]\nproof (state)\nthis:\nt (Suc k) ! (Suc k - i) = next_ofsm_table M (hd (t k))\n\ngoal (2 subgoals):\n 1. i = Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n 2. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nt (Suc k) ! (Suc k - i) = next_ofsm_table M (hd (t k))\n[PROOF STEP]\nhave \"ofsm_lookup (t (Suc k) ! ((Suc k)-i)) q = ofsm_lookup (next_ofsm_table M (hd (t k))) q\"\n[PROOF STATE]\nproof (prove)\nusing this:\nt (Suc k) ! (Suc k - i) = next_ofsm_table M (hd (t k))\n\ngoal (1 subgoal):\n 1. ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_lookup (next_ofsm_table M (hd (t k))) q\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_lookup (next_ofsm_table M (hd (t k))) q\n\ngoal (2 subgoals):\n 1. i = Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n 2. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nhave \"(hd (t k)) = (t k ! (k-k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. hd (t k) = t k ! (k - k)\n[PROOF STEP]\nby (metis IH1 diff_self_eq_0 hd_conv_nth list.size(3) nat.simps(3))\n[PROOF STATE]\nproof (state)\nthis:\nhd (t k) = t k ! (k - k)\n\ngoal (2 subgoals):\n 1. i = Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n 2. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nhd (t k) = t k ! (k - k)\n\ngoal (2 subgoals):\n 1. i = Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n 2. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nhave \"k < Suc k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k < Suc k\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nk < Suc k\n\ngoal (2 subgoals):\n 1. i = Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n 2. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nhd (t k) = t k ! (k - k)\nk < Suc k\n[PROOF STEP]\nhave \"ofsm_lookup (next_ofsm_table M (hd (t k))) q = ofsm_table M (\\q. FSM.states M) i q\"\n[PROOF STATE]\nproof (prove)\nusing this:\nhd (t k) = t k ! (k - k)\nk < Suc k\n\ngoal (1 subgoal):\n 1. ofsm_lookup (next_ofsm_table M (hd (t k))) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nby (metis \"1\" IH2 next_ofsm_table_lookup_invar)\n[PROOF STATE]\nproof (state)\nthis:\nofsm_lookup (next_ofsm_table M (hd (t k))) q = ofsm_table M (\\q. FSM.states M) i q\n\ngoal (2 subgoals):\n 1. i = Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n 2. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nofsm_lookup (next_ofsm_table M (hd (t k))) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nofsm_lookup (next_ofsm_table M (hd (t k))) q = ofsm_table M (\\q. FSM.states M) i q\n\ngoal (1 subgoal):\n 1. ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nunfolding \\ofsm_lookup (t (Suc k) ! ((Suc k)-i)) q = ofsm_lookup (next_ofsm_table M (hd (t k))) q\\\n[PROOF STATE]\nproof (prove)\nusing this:\nofsm_lookup (next_ofsm_table M (hd (t k))) q = ofsm_table M (\\q. FSM.states M) i q\n\ngoal (1 subgoal):\n 1. ofsm_lookup (next_ofsm_table M (hd (t k))) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n\ngoal (1 subgoal):\n 1. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\ncase 2\n[PROOF STATE]\nproof (state)\nthis:\ni < Suc k\n\ngoal (1 subgoal):\n 1. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni < Suc k\n[PROOF STEP]\nhave \"((Suc k)-i) > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < Suc k\n\ngoal (1 subgoal):\n 1. 0 < Suc k - i\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < Suc k - i\n\ngoal (1 subgoal):\n 1. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < Suc k - i\n[PROOF STEP]\nhave \"(t (Suc k) ! ((Suc k)-i)) = t k ! (((Suc k)-i) - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < Suc k - i\n\ngoal (1 subgoal):\n 1. t (Suc k) ! (Suc k - i) = t k ! (Suc k - i - 1)\n[PROOF STEP]\nunfolding *\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < Suc k - i\n\ngoal (1 subgoal):\n 1. (next_ofsm_table M (hd (t k)) # t k) ! (Suc k - i) = t k ! (Suc k - i - 1)\n[PROOF STEP]\nby (meson nth_Cons_pos)\n[PROOF STATE]\nproof (state)\nthis:\nt (Suc k) ! (Suc k - i) = t k ! (Suc k - i - 1)\n\ngoal (1 subgoal):\n 1. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nt (Suc k) ! (Suc k - i) = t k ! (Suc k - i - 1)\n[PROOF STEP]\nhave \"(t (Suc k) ! ((Suc k)-i)) = t k ! (k-i)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nt (Suc k) ! (Suc k - i) = t k ! (Suc k - i - 1)\n\ngoal (1 subgoal):\n 1. t (Suc k) ! (Suc k - i) = t k ! (k - i)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nt (Suc k) ! (Suc k - i) = t k ! (k - i)\n\ngoal (1 subgoal):\n 1. i < Suc k \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nshow \"ofsm_lookup (t (Suc k) ! ((Suc k)-i)) q = ofsm_table M (\\q. FSM.states M) i q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nusing IH2[OF 2]\n[PROOF STATE]\nproof (prove)\nusing this:\nofsm_lookup (t k ! (k - i)) ?q = ofsm_table M (\\q. FSM.states M) i ?q\n\ngoal (1 subgoal):\n 1. ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nunfolding \\(t (Suc k) ! ((Suc k)-i)) = t k ! (k-i)\\\n[PROOF STATE]\nproof (prove)\nusing this:\nofsm_lookup (t k ! (k - i)) ?q = ofsm_table M (\\q. FSM.states M) i ?q\n\ngoal (1 subgoal):\n 1. ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\nofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n?i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - ?i)) ?q = ofsm_table M (\\q. FSM.states M) ?i ?q\n\ngoal (1 subgoal):\n 1. \\k. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i ofsm_lookup (t (Suc k) ! (Suc k - ?i)) ?q = ofsm_table M (\\q. FSM.states M) ?i ?q\n\ngoal (1 subgoal):\n 1. \\k. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ii . i < Suc (Suc k) \\ Mapping.keys (t (Suc k) ! i) = FSM.states M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i < Suc (Suc k) \\ Mapping.keys (t (Suc k) ! i) = FSM.states M\n[PROOF STEP]\nby (metis \"*\" IH3 Suc_diff_1 Suc_less_eq less_Suc_eq_0_disj next_ofsm_table_keys_invar nth_Cons')\n[PROOF STATE]\nproof (state)\nthis:\n?i < Suc (Suc k) \\ Mapping.keys (t (Suc k) ! ?i) = FSM.states M\n\ngoal (1 subgoal):\n 1. \\k. length (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i ofsm_lookup (t (Suc k) ! (Suc k - ?i)) ?q = ofsm_table M (\\q. FSM.states M) ?i ?q\n?i < Suc (Suc k) \\ Mapping.keys (t (Suc k) ! ?i) = FSM.states M\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (t (Suc k)) = Suc (Suc k)\n?i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - ?i)) ?q = ofsm_table M (\\q. FSM.states M) ?i ?q\n?i < Suc (Suc k) \\ Mapping.keys (t (Suc k) ! ?i) = FSM.states M\n\ngoal (1 subgoal):\n 1. length (t (Suc k)) = Suc (Suc k) \\ (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i (\\i q. i < Suc (Suc k) \\ ofsm_lookup (t (Suc k) ! (Suc k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ii q. i < Suc k \\ ofsm_lookup (compute_ofsm_table_list M k ! i) q = ofsm_table M (\\q. FSM.states M) i q\n 3. \\i. i < Suc k \\ Mapping.keys (compute_ofsm_table_list M k ! i) = FSM.states M\n[PROOF STEP]\nhave *:\"(compute_ofsm_table_list M k) = rev (t k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. compute_ofsm_table_list M k = rev (t k)\n[PROOF STEP]\nunfolding compute_ofsm_table_list.simps t_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rev (foldr (\\_ prev. next_ofsm_table M (hd prev) # prev) [0.._ prev. next_ofsm_table M (hd prev) # prev) (rev [0.. prev . (next_ofsm_table M (hd prev)) # prev)\", OF length_rev]\n[PROOF STATE]\nproof (prove)\nusing this:\nfoldr (\\_ x. next_ofsm_table M (hd x) # x) (rev [0..a x. next_ofsm_table M (hd x) # x) [0.._ prev. next_ofsm_table M (hd prev) # prev) [0.._ prev. next_ofsm_table M (hd prev) # prev) (rev [0..i q. i < Suc k \\ ofsm_lookup (compute_ofsm_table_list M k ! i) q = ofsm_table M (\\q. FSM.states M) i q\n 3. \\i. i < Suc k \\ Mapping.keys (compute_ofsm_table_list M k ! i) = FSM.states M\n[PROOF STEP]\nshow \"length (compute_ofsm_table_list M k) = Suc k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (compute_ofsm_table_list M k) = Suc k\n[PROOF STEP]\nusing t_props\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ii q. i < Suc k \\ ofsm_lookup (compute_ofsm_table_list M k ! i) q = ofsm_table M (\\q. FSM.states M) i q\n 2. \\i. i < Suc k \\ Mapping.keys (compute_ofsm_table_list M k ! i) = FSM.states M\n[PROOF STEP]\nhave \"\\ i . i < Suc k \\ (rev (t k) ! i) = t k ! (k - i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i < Suc k \\ rev (t k) ! i = t k ! (k - i)\n[PROOF STEP]\nby (simp add: rev_nth t_props)\n[PROOF STATE]\nproof (state)\nthis:\n?i < Suc k \\ rev (t k) ! ?i = t k ! (k - ?i)\n\ngoal (2 subgoals):\n 1. \\i q. i < Suc k \\ ofsm_lookup (compute_ofsm_table_list M k ! i) q = ofsm_table M (\\q. FSM.states M) i q\n 2. \\i. i < Suc k \\ Mapping.keys (compute_ofsm_table_list M k ! i) = FSM.states M\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n?i < Suc k \\ rev (t k) ! ?i = t k ! (k - ?i)\n[PROOF STEP]\nshow \"\\i q. i < Suc k \\\n ofsm_lookup (compute_ofsm_table_list M k ! i) q = ofsm_table M (\\q. FSM.states M) i q\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?i < Suc k \\ rev (t k) ! ?i = t k ! (k - ?i)\n\ngoal (1 subgoal):\n 1. \\i q. i < Suc k \\ ofsm_lookup (compute_ofsm_table_list M k ! i) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nunfolding *\n[PROOF STATE]\nproof (prove)\nusing this:\n?i < Suc k \\ rev (t k) ! ?i = t k ! (k - ?i)\n\ngoal (1 subgoal):\n 1. \\i q. i < Suc k \\ ofsm_lookup (rev (t k) ! i) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nusing t_props\n[PROOF STATE]\nproof (prove)\nusing this:\n?i < Suc k \\ rev (t k) ! ?i = t k ! (k - ?i)\nlength (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\ii q. i < Suc k \\ ofsm_lookup (rev (t k) ! i) q = ofsm_table M (\\q. FSM.states M) i q\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\n?i < Suc k \\ ofsm_lookup (compute_ofsm_table_list M k ! ?i) ?q = ofsm_table M (\\q. FSM.states M) ?i ?q\n\ngoal (1 subgoal):\n 1. \\i. i < Suc k \\ Mapping.keys (compute_ofsm_table_list M k ! i) = FSM.states M\n[PROOF STEP]\nshow \"\\i. i < Suc k \\ Mapping.keys (compute_ofsm_table_list M k ! i) = FSM.states M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i < Suc k \\ Mapping.keys (compute_ofsm_table_list M k ! i) = FSM.states M\n[PROOF STEP]\nunfolding *\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i < Suc k \\ Mapping.keys (rev (t k) ! i) = FSM.states M\n[PROOF STEP]\nusing t_props \\\\ i . i < Suc k \\ (rev (t k) ! i) = t k ! (k - i)\\\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (t k) = Suc k \\ (\\i q. i < Suc k \\ ofsm_lookup (t k ! (k - i)) q = ofsm_table M (\\q. FSM.states M) i q) \\ (\\i rev (t k) ! ?i = t k ! (k - ?i)\n\ngoal (1 subgoal):\n 1. \\i. i < Suc k \\ Mapping.keys (rev (t k) ! i) = FSM.states M\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n?i < Suc k \\ Mapping.keys (compute_ofsm_table_list M k ! ?i) = FSM.states M\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 16041, "file": "FSM_Tests_OFSM_Tables_Refined", "length": 107, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357666736773, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7128673869104738}} {"text": "[STATEMENT]\nlemma scalar_prod_block_size_lift_01: \n assumes \"i < \\\"\n shows \"((col (lift_01_mat N) i) \\ (col (lift_01_mat N) i)) = (of_nat (card (\\s ! i)) :: ('b :: {ring_1}))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. col (lift_01_mat N) i \\ col (lift_01_mat N) i = of_nat (card (\\s ! i))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. col (lift_01_mat N) i \\ col (lift_01_mat N) i = of_nat (card (\\s ! i))\n[PROOF STEP]\ninterpret z1: zero_one_matrix_ring_1 \"(lift_01_mat N)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. zero_one_matrix_ring_1 (lift_01_mat N)\n[PROOF STEP]\nby (intro_locales) (simp add: lift_mat_is_0_1)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. col (lift_01_mat N) i \\ col (lift_01_mat N) i = of_nat (card (\\s ! i))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. col (lift_01_mat N) i \\ col (lift_01_mat N) i = of_nat (card (\\s ! i))\n[PROOF STEP]\nusing assms z1.scalar_prod_inc_vec_block_size_mat preserve_mat_block_size \n mat_block_size_N_col lift_01_mat_def\n[PROOF STATE]\nproof (prove)\nusing this:\ni < \\\n?j < dim_col (lift_01_mat N) \\ col (lift_01_mat N) ?j \\ col (lift_01_mat N) ?j = of_nat (mat_block_size (lift_01_mat N) ?j)\n\\inj_on_01_hom ?f; ?j < dim_col N\\ \\ mat_block_size N ?j = mat_block_size (map_mat ?f N) ?j\n?j < \\ \\ mat_block_size N ?j = card (\\s ! ?j)\nlift_01_mat ?M \\ map_mat of_zero_neq_one ?M\n\ngoal (1 subgoal):\n 1. col (lift_01_mat N) i \\ col (lift_01_mat N) i = of_nat (card (\\s ! i))\n[PROOF STEP]\nby (metis inc_mat_dim_col lift_01_mat_simp(2) of_inj_on_01_hom.inj_on_01_hom_axioms size_mset)\n[PROOF STATE]\nproof (state)\nthis:\ncol (lift_01_mat N) i \\ col (lift_01_mat N) i = of_nat (card (\\s ! i))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 922, "file": "Fishers_Inequality_Incidence_Matrices", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357735451834, "lm_q2_score": 0.8221891261650248, "lm_q1q2_score": 0.7128673850049306}} {"text": "[STATEMENT]\nlemma interval_split_cart:\n \"{a..b::real^'n} \\ {x. x$k \\ c} = {a .. (\\ i. if i = k then min (b$k) c else b$i)}\"\n \"cbox a b \\ {x. x$k \\ c} = {(\\ i. if i = k then max (a$k) c else a$i) .. b}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {a..b} \\ {x. x $ k \\ c} = {a..\\i. if i = k then min (b $ k) c else b $ i} &&& cbox a b \\ {x. c \\ x $ k} = {\\i. if i = k then max (a $ k) c else a $ i..b}\n[PROOF STEP]\napply (rule_tac[!] set_eqI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. (x \\ {a..b} \\ {x. x $ k \\ c}) = (x \\ {a..\\i. if i = k then min (b $ k) c else b $ i})\n 2. \\x. (x \\ cbox a b \\ {x. c \\ x $ k}) = (x \\ {\\i. if i = k then max (a $ k) c else a $ i..b})\n[PROOF STEP]\nunfolding Int_iff mem_box_cart mem_Collect_eq interval_cbox_cart\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. ((\\i. a $ i \\ x $ i \\ x $ i \\ b $ i) \\ x $ k \\ c) = (\\i. a $ i \\ x $ i \\ x $ i \\ (\\i. if i = k then min (b $ k) c else b $ i) $ i)\n 2. \\x. ((\\i. a $ i \\ x $ i \\ x $ i \\ b $ i) \\ c \\ x $ k) = (\\i. (\\i. if i = k then max (a $ k) c else a $ i) $ i \\ x $ i \\ x $ i \\ b $ i)\n[PROOF STEP]\nunfolding vec_lambda_beta\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. ((\\i. a $ i \\ x $ i \\ x $ i \\ b $ i) \\ x $ k \\ c) = (\\i. a $ i \\ x $ i \\ x $ i \\ (if i = k then min (b $ k) c else b $ i))\n 2. \\x. ((\\i. a $ i \\ x $ i \\ x $ i \\ b $ i) \\ c \\ x $ k) = (\\i. (if i = k then max (a $ k) c else a $ i) \\ x $ i \\ x $ i \\ b $ i)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 879, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357632379241, "lm_q2_score": 0.8221891305219505, "lm_q1q2_score": 0.7128673803080245}} {"text": "[STATEMENT]\nlemma subset_atMost_Max_le_conv: \"\n \\ finite A; A \\ {} \\ \\ (A \\ {..n}) = (Max A \\ n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; A \\ {}\\ \\ (A \\ {..n}) = (Max A \\ n)\n[PROOF STEP]\napply (rule iffI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\finite A; A \\ {}; A \\ {..n}\\ \\ Max A \\ n\n 2. \\finite A; A \\ {}; Max A \\ n\\ \\ A \\ {..n}\n[PROOF STEP]\napply (blast intro: subset_atMost_imp_Max_le)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; A \\ {}; Max A \\ n\\ \\ A \\ {..n}\n[PROOF STEP]\napply (rule Max_le_imp_subset_atMost, assumption+)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 387, "file": "List-Infinite_CommonSet_SetInterval2", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357529306639, "lm_q2_score": 0.8221891392358015, "lm_q1q2_score": 0.7128673793887277}} {"text": "[STATEMENT]\ntheorem effective_matrix_tensor_elements2:\n assumes \" i<(row_length M1)*(row_length M2)\"\n and \"j < (length M1)*(length M2)\"\n and \"mat (row_length M1) (length M1) M1\"\n and \"mat (row_length M2) (length M2) M2\"\n shows \"(M1 \\ M2)!j!i = \n (M1!(j div (length M2))!(i div (row_length M2)))\n * (M2!(j mod length M2)!(i mod (row_length M2)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (M1 \\ M2) ! j ! i = M1 ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)\n[PROOF STEP]\nusing assms matrix_Tensor_elements\n[PROOF STATE]\nproof (prove)\nusing this:\ni < row_length M1 * row_length M2\nj < length M1 * length M2\nmat (row_length M1) (length M1) M1\nmat (row_length M2) (length M2) M2\n\\i j. (i < row_length ?M1.0 * row_length ?M2.0 \\ j < length ?M1.0 * length ?M2.0) \\ mat (row_length ?M1.0) (length ?M1.0) ?M1.0 \\ mat (row_length ?M2.0) (length ?M2.0) ?M2.0 \\ (?M1.0 \\ ?M2.0) ! j ! i = ?M1.0 ! (j div length ?M2.0) ! (i div row_length ?M2.0) * ?M2.0 ! (j mod length ?M2.0) ! (i mod row_length ?M2.0)\n\ngoal (1 subgoal):\n 1. (M1 \\ M2) ! j ! i = M1 ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 580, "file": "Matrix_Tensor_Matrix_Tensor", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505299595163, "lm_q2_score": 0.7879311956428946, "lm_q1q2_score": 0.7128023737099799}} {"text": "[STATEMENT]\nlemma conjugate_exponent_ennreal_iff:\n assumes \"p \\ (1::ennreal)\"\n shows \"q = conjugate_exponent p \\ (1/p + 1/q = 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (q = conjugate_exponent p) = (1 / p + 1 / q = 1)\n[PROOF STEP]\nusing conjugate_exponent_ennreal[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n1 / p + 1 / conjugate_exponent p = 1\n1 \\ conjugate_exponent p\nconjugate_exponent (conjugate_exponent p) = p\n\ngoal (1 subgoal):\n 1. (q = conjugate_exponent p) = (1 / p + 1 / q = 1)\n[PROOF STEP]\nby (auto, metis ennreal_add_diff_cancel_left ennreal_add_eq_top ennreal_top_neq_one one_divide_one_divide_ennreal)", "meta": {"llama_tokens": 290, "file": "Lp_Lp", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505248181418, "lm_q2_score": 0.7879311956428946, "lm_q1q2_score": 0.7128023696589305}} {"text": "[STATEMENT]\nlemma mset_mult_add:\n assumes \"mset (a::'a::field list) = mset b + mset c\"\n shows \"prod_list a = prod_list b * prod_list c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod_list a = prod_list b * prod_list c\n[PROOF STEP]\nunfolding prod_mset_prod_list[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub># (mset a) = \\\\<^sub># (mset b) * \\\\<^sub># (mset c)\n[PROOF STEP]\nusing prod_mset_Un[of \"mset b\" \"mset c\",unfolded assms[symmetric]]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub># (mset a) = \\\\<^sub># (mset b) * \\\\<^sub># (mset c)\n\ngoal (1 subgoal):\n 1. \\\\<^sub># (mset a) = \\\\<^sub># (mset b) * \\\\<^sub># (mset c)\n[PROOF STEP]\n.", "meta": {"llama_tokens": 333, "file": "Berlekamp_Zassenhaus_Mahler_Measure", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9136765328159726, "lm_q2_score": 0.7799929053683038, "lm_q1q2_score": 0.7126612133979688}} {"text": "[STATEMENT]\ntheorem p51: \"\\z w. \\x y. F(x,y) \\ x = z \\ y = w \\\n \\z. \\x. (\\w. \\y. F(x,y) \\ y = w) \\ x = z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z w. \\x y. F (x, y) = (x = z \\ y = w) \\ \\z. \\x. (\\w. \\y. F (x, y) = (y = w)) = (x = z)\n[PROOF STEP]\n@proof\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z w. \\x y. F (x, y) = (x = z \\ y = w) \\ \\z. \\x. (\\w. \\y. F (x, y) = (y = w)) = (x = z)\n[PROOF STEP]\n@obtain z w where \"\\x y. F(x,y) \\ x = z \\ y = w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z w. \\x y. F (x, y) = (x = z \\ y = w) \\ \\z. \\x. (\\w. \\y. F (x, y) = (y = w)) = (x = z)\n[PROOF STEP]\n@have \"\\x. (\\w. \\y. F(x,y) \\ y = w) \\ x = z\" @with\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z w. \\x y. F (x, y) = (x = z \\ y = w) \\ \\z. \\x. (\\w. \\y. F (x, y) = (y = w)) = (x = z)\n[PROOF STEP]\n@case \"x = z\" @with\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z w. \\x y. F (x, y) = (x = z \\ y = w) \\ \\z. \\x. (\\w. \\y. F (x, y) = (y = w)) = (x = z)\n[PROOF STEP]\n@have \"\\y. F(x,y) \\ y = w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z w. \\x y. F (x, y) = (x = z \\ y = w) \\ \\z. \\x. (\\w. \\y. F (x, y) = (y = w)) = (x = z)\n[PROOF STEP]\n@end\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z w. \\x y. F (x, y) = (x = z \\ y = w) \\ \\z. \\x. (\\w. \\y. F (x, y) = (y = w)) = (x = z)\n[PROOF STEP]\n@end\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z w. \\x y. F (x, y) = (x = z \\ y = w) \\ \\z. \\x. (\\w. \\y. F (x, y) = (y = w)) = (x = z)\n[PROOF STEP]\n@qed", "meta": {"llama_tokens": 1039, "file": "Auto2_HOL_HOL_Pelletier", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.913676530465412, "lm_q2_score": 0.7799928900257126, "lm_q1q2_score": 0.7126611975463827}} {"text": "[STATEMENT]\nlemma normf_least: \"S \\ {} \\ (\\x. x \\ S \\ \\f x\\ \\ M) \\ normf f \\ M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\S \\ {}; \\x. x \\ S \\ \\f x\\ \\ M\\ \\ normf f \\ M\n[PROOF STEP]\nby (simp add: normf_def cSUP_least)", "meta": {"llama_tokens": 159, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637433190939, "lm_q2_score": 0.8289388146603365, "lm_q1q2_score": 0.7126086443933974}} {"text": "[STATEMENT]\nlemma freeIn_list_conj : \"(\\f\\set(F). freeIn var f) \\ freeIn var (list_conj F)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f\\set F. freeIn var f \\ freeIn var (list_conj F)\n[PROOF STEP]\nproof(induct F)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. Ball (set []) (freeIn var) \\ freeIn var (list_conj [])\n 2. \\a F. \\Ball (set F) (freeIn var) \\ freeIn var (list_conj F); Ball (set (a # F)) (freeIn var)\\ \\ freeIn var (list_conj (a # F))\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\n\\a\\set []. freeIn var a\n\ngoal (2 subgoals):\n 1. Ball (set []) (freeIn var) \\ freeIn var (list_conj [])\n 2. \\a F. \\Ball (set F) (freeIn var) \\ freeIn var (list_conj F); Ball (set (a # F)) (freeIn var)\\ \\ freeIn var (list_conj (a # F))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\a\\set []. freeIn var a\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a\\set []. freeIn var a\n\ngoal (1 subgoal):\n 1. freeIn var (list_conj [])\n[PROOF STEP]\nby(simp add: list_conj_def)\n[PROOF STATE]\nproof (state)\nthis:\nfreeIn var (list_conj [])\n\ngoal (1 subgoal):\n 1. \\a F. \\Ball (set F) (freeIn var) \\ freeIn var (list_conj F); Ball (set (a # F)) (freeIn var)\\ \\ freeIn var (list_conj (a # F))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a F. \\Ball (set F) (freeIn var) \\ freeIn var (list_conj F); Ball (set (a # F)) (freeIn var)\\ \\ freeIn var (list_conj (a # F))\n[PROOF STEP]\ncase (Cons a F)\n[PROOF STATE]\nproof (state)\nthis:\n\\a\\set F. freeIn var a \\ freeIn var (list_conj F)\n\\a\\set (a # F). freeIn var a\n\ngoal (1 subgoal):\n 1. \\a F. \\Ball (set F) (freeIn var) \\ freeIn var (list_conj F); Ball (set (a # F)) (freeIn var)\\ \\ freeIn var (list_conj (a # F))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\a\\set F. freeIn var a \\ freeIn var (list_conj F)\n\\a\\set (a # F). freeIn var a\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a\\set F. freeIn var a \\ freeIn var (list_conj F)\n\\a\\set (a # F). freeIn var a\n\ngoal (1 subgoal):\n 1. freeIn var (list_conj (a # F))\n[PROOF STEP]\nby (simp add: PolyAtoms.and_def list_conj_def)\n[PROOF STATE]\nproof (state)\nthis:\nfreeIn var (list_conj (a # F))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1127, "file": "Virtual_Substitution_PolyAtoms", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637433190938, "lm_q2_score": 0.8289388104343892, "lm_q1q2_score": 0.7126086407605037}} {"text": "[STATEMENT]\nlemma Lcm_fin_1_iff:\n \"Lcm\\<^sub>f\\<^sub>i\\<^sub>n A = 1 \\ (\\a\\A. is_unit a) \\ finite A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Lcm\\<^sub>f\\<^sub>i\\<^sub>n A = (1::'a)) = ((\\a\\A. is_unit a) \\ finite A)\n[PROOF STEP]\nby (induct A rule: infinite_finite_induct) simp_all", "meta": {"llama_tokens": 157, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045937171068, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7125794837710032}} {"text": "[STATEMENT]\nlemma prod_pow[simp]: \"(\\i = 0..i = 0..i j. i < j \\ X $ i $ j = (0::'a)\n\\i j. i < j \\ Y $ i $ j = (0::'a)\n\ngoal (1 subgoal):\n 1. \\i j. i < j \\ (\\i j. \\k\\UNIV. X $ i $ k * Y $ k $ j) $ i $ j = (0::'a)\n[PROOF STEP]\nby (auto intro!: sum.neutral) (metis mult_not_zero neqE less_trans)", "meta": {"llama_tokens": 344, "file": "MDP-Algorithms_Matrix_Util", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772482857833, "lm_q2_score": 0.8152324983301568, "lm_q1q2_score": 0.7125761788535678}} {"text": "[STATEMENT]\nlemma (in Group) tOp_assocTr45:\"\\a \\ carrier G; b \\ carrier G; c \\ carrier G; \nd \\ carrier G\\ \\ a \\ b \\ c \\ d = a \\ (b \\ (c \\ d))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a \\ carrier G; b \\ carrier G; c \\ carrier G; d \\ carrier G\\ \\ a \\ b \\ c \\ d = a \\ (b \\ (c \\ d))\n[PROOF STEP]\napply (frule mult_closed[of \"c\" \"d\"], assumption+)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a \\ carrier G; b \\ carrier G; c \\ carrier G; d \\ carrier G; c \\ d \\ carrier G\\ \\ a \\ b \\ c \\ d = a \\ (b \\ (c \\ d))\n[PROOF STEP]\napply (simp add:tassoc[of \"a\" \"b\" \"c \\ d\", THEN sym])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a \\ carrier G; b \\ carrier G; c \\ carrier G; d \\ carrier G; c \\ d \\ carrier G\\ \\ a \\ b \\ c \\ d = a \\ b \\ (c \\ d)\n[PROOF STEP]\napply (simp add:tOp_assocTr41)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 526, "file": "Group-Ring-Module_Algebra2", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772351648677, "lm_q2_score": 0.8152324960856175, "lm_q1q2_score": 0.7125761661950704}} {"text": "[STATEMENT]\nlemma below_val_zero:\n assumes \"prime p\"\n assumes \"x \\ (padic_set p)\"\n assumes \"x (Suc n) \\ \\\\<^bsub>residue_ring (p^(Suc n))\\<^esub>\"\n shows \"int n \\ (padic_val p x )\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. padic_val p x \\ int n\n[PROOF STEP]\nproof(cases \"x = padic_zero p\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x = padic_zero p \\ padic_val p x \\ int n\n 2. x \\ padic_zero p \\ padic_val p x \\ int n\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nx = padic_zero p\n\ngoal (2 subgoals):\n 1. x = padic_zero p \\ padic_val p x \\ int n\n 2. x \\ padic_zero p \\ padic_val p x \\ int n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx = padic_zero p\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx = padic_zero p\n\ngoal (1 subgoal):\n 1. padic_val p x \\ int n\n[PROOF STEP]\nusing assms(3) padic_zero_simp\n[PROOF STATE]\nproof (prove)\nusing this:\nx = padic_zero p\nx (Suc n) \\ \\\\<^bsub>residue_ring (p ^ Suc n)\\<^esub>\npadic_zero ?p ?n = \\\\<^bsub>residue_ring (?p ^ ?n)\\<^esub>\npadic_zero ?p ?n = 0\n\ngoal (1 subgoal):\n 1. padic_val p x \\ int n\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\npadic_val p x \\ int n\n\ngoal (1 subgoal):\n 1. x \\ padic_zero p \\ padic_val p x \\ int n\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x \\ padic_zero p \\ padic_val p x \\ int n\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nx \\ padic_zero p\n\ngoal (1 subgoal):\n 1. x \\ padic_zero p \\ padic_val p x \\ int n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ padic_zero p\n[PROOF STEP]\nhave \"padic_val p x = int (LEAST k::nat. x (Suc k) \\ \\\\<^bsub>residue_ring (p ^ Suc k)\\<^esub>)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ padic_zero p\n\ngoal (1 subgoal):\n 1. padic_val p x = int (LEAST k. x (Suc k) \\ \\\\<^bsub>residue_ring (p ^ Suc k)\\<^esub>)\n[PROOF STEP]\nusing padic_val_def\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ padic_zero p\npadic_val ?p ?f \\ if ?f = padic_zero ?p then - 1 else int (LEAST k. ?f (Suc k) \\ \\\\<^bsub>residue_ring (?p ^ Suc k)\\<^esub>)\n\ngoal (1 subgoal):\n 1. padic_val p x = int (LEAST k. x (Suc k) \\ \\\\<^bsub>residue_ring (p ^ Suc k)\\<^esub>)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\npadic_val p x = int (LEAST k. x (Suc k) \\ \\\\<^bsub>residue_ring (p ^ Suc k)\\<^esub>)\n\ngoal (1 subgoal):\n 1. x \\ padic_zero p \\ padic_val p x \\ int n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\npadic_val p x = int (LEAST k. x (Suc k) \\ \\\\<^bsub>residue_ring (p ^ Suc k)\\<^esub>)\n[PROOF STEP]\nshow \"of_nat n \\ (padic_val p x )\"\n[PROOF STATE]\nproof (prove)\nusing this:\npadic_val p x = int (LEAST k. x (Suc k) \\ \\\\<^bsub>residue_ring (p ^ Suc k)\\<^esub>)\n\ngoal (1 subgoal):\n 1. padic_val p x \\ int n\n[PROOF STEP]\nby (metis (mono_tags, lifting) Least_le assms(3) nat_int nat_le_iff)\n[PROOF STATE]\nproof (state)\nthis:\npadic_val p x \\ int n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1501, "file": "Padic_Ints_Padic_Construction", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772286044095, "lm_q2_score": 0.8152324983301567, "lm_q1q2_score": 0.7125761628086722}} {"text": "[STATEMENT]\nlemma euler_mascheroni_sum_real:\n \"(\\n. inverse (of_nat (n+1)) + ln (of_nat (n+1)) - ln (of_nat (n+2)) :: real)\n sums euler_mascheroni\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. inverse (real (n + 1)) + ln (real (n + 1)) - ln (real (n + 2))) sums euler_mascheroni\n[PROOF STEP]\nusing sums_add[OF telescope_sums[OF LIMSEQ_Suc[OF euler_mascheroni_LIMSEQ]]\n telescope_sums'[OF LIMSEQ_inverse_real_of_nat]]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. harm (Suc (Suc n)) - ln (real (Suc (Suc n))) - (harm (Suc n) - ln (real (Suc n))) + (inverse (real (Suc n)) - inverse (real (Suc (Suc n))))) sums (euler_mascheroni - (harm (Suc 0) - ln (real (Suc 0))) + (inverse (real (Suc 0)) - 0))\n\ngoal (1 subgoal):\n 1. (\\n. inverse (real (n + 1)) + ln (real (n + 1)) - ln (real (n + 2))) sums euler_mascheroni\n[PROOF STEP]\nby (simp_all add: harm_def algebra_simps)", "meta": {"llama_tokens": 417, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772286044095, "lm_q2_score": 0.815232489352, "lm_q1q2_score": 0.71257615496107}} {"text": "[STATEMENT]\nlemma has_integral_interval_reverse: fixes f :: \"real \\ real\" and a b :: real\n assumes \"a \\ b\"\n and \"continuous_on {a..b} f\"\n shows \"((\\x. f (a+b-x)) has_integral (integral {a..b} f)) {a..b}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. f (a + b - x)) has_integral integral {a..b} f) {a..b}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\x. f (a + b - x)) has_integral integral {a..b} f) {a..b}\n[PROOF STEP]\nlet ?g = \"\\x. a + b - x\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\x. f (a + b - x)) has_integral integral {a..b} f) {a..b}\n[PROOF STEP]\nlet ?g' = \"\\x. -1\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\x. f (a + b - x)) has_integral integral {a..b} f) {a..b}\n[PROOF STEP]\nhave g_C0: \"continuous_on {a..b} ?g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous_on {a..b} ((-) (a + b))\n[PROOF STEP]\nusing continuous_on_op_minus\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous_on ?s ((-) ?x)\n\ngoal (1 subgoal):\n 1. continuous_on {a..b} ((-) (a + b))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncontinuous_on {a..b} ((-) (a + b))\n\ngoal (1 subgoal):\n 1. ((\\x. f (a + b - x)) has_integral integral {a..b} f) {a..b}\n[PROOF STEP]\nhave Dg_g': \"\\x. x\\{a..b} \\ (?g has_field_derivative ?g' x) (at x within {a..b})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ {a..b} \\ ((-) (a + b) has_real_derivative - 1) (at x within {a..b})\n[PROOF STEP]\nby (auto intro!: derivative_eq_intros)\n[PROOF STATE]\nproof (state)\nthis:\n?x1 \\ {a..b} \\ ((-) (a + b) has_real_derivative - 1) (at ?x1 within {a..b})\n\ngoal (1 subgoal):\n 1. ((\\x. f (a + b - x)) has_integral integral {a..b} f) {a..b}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. f (a + b - x)) has_integral integral {a..b} f) {a..b}\n[PROOF STEP]\nusing has_integral_substitution_general\n [of \"{}\" a b ?g a b f, simplified, OF assms g_C0 Dg_g', simplified]\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. - f (a + b - x)) has_integral integral {b..a} f - integral {a..b} f) {a..b}\n\ngoal (1 subgoal):\n 1. ((\\x. f (a + b - x)) has_integral integral {a..b} f) {a..b}\n[PROOF STEP]\napply (simp add: has_integral_null_interval[OF assms(1), THEN integral_unique])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. - f (a + b - x)) has_integral - integral {a..b} f) {a..b} \\ ((\\x. f (a + b - x)) has_integral integral {a..b} f) {a..b}\n[PROOF STEP]\nby (simp add: has_integral_neg_iff)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. f (a + b - x)) has_integral integral {a..b} f) {a..b}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1284, "file": "Actuarial_Mathematics_Preliminaries", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467738423874, "lm_q2_score": 0.810478913248044, "lm_q1q2_score": 0.7125299218493021}} {"text": "[STATEMENT]\nlemma uints_mono_iff: \"uints l \\ uints m \\ l \\ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (uints l \\ uints m) = (l \\ m)\n[PROOF STEP]\nusing power_increasing_iff[of \"2::int\" l m]\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < 2 \\ (2 ^ l \\ 2 ^ m) = (l \\ m)\n\ngoal (1 subgoal):\n 1. (uints l \\ uints m) = (l \\ m)\n[PROOF STEP]\napply (auto simp: uints_num subset_iff simp del: power_increasing_iff)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\ 2 ^ l \\ 2 ^ m; \\ l \\ m; \\t. 0 \\ t \\ t < 2 ^ l \\ t < 2 ^ m\\ \\ False\n[PROOF STEP]\napply (meson less_irrefl not_le zero_le_numeral zero_le_power)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 368, "file": "Word_Lib_Typedef_Morphisms", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467738423874, "lm_q2_score": 0.8104789109591832, "lm_q1q2_score": 0.7125299198370575}} {"text": "[STATEMENT]\nlemma infsum_Sigma:\n fixes A :: \"'a set\" and B :: \"'a \\ 'b set\"\n and f :: \\'a \\ 'b \\ 'c::{comm_monoid_add, t2_space, uniform_space}\\\n assumes plus_cont: \\uniformly_continuous_on UNIV (\\(x::'c,y). x+y)\\\n assumes summableAB: \"f summable_on (Sigma A B)\"\n assumes summableB: \\\\x. x\\A \\ (\\y. f (x, y)) summable_on (B x)\\\n shows \"infsum f (Sigma A B) = infsum (\\x. infsum (\\y. f (x, y)) (B x)) A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. infsum f (Sigma A B) = (\\\\<^sub>\\x\\A. \\\\<^sub>\\y\\B x. f (x, y))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. infsum f (Sigma A B) = (\\\\<^sub>\\x\\A. \\\\<^sub>\\y\\B x. f (x, y))\n[PROOF STEP]\nfrom summableAB\n[PROOF STATE]\nproof (chain)\npicking this:\nf summable_on Sigma A B\n[PROOF STEP]\nhave a: \\has_sum f (Sigma A B) (infsum f (Sigma A B))\\\n[PROOF STATE]\nproof (prove)\nusing this:\nf summable_on Sigma A B\n\ngoal (1 subgoal):\n 1. has_sum f (Sigma A B) (infsum f (Sigma A B))\n[PROOF STEP]\nusing has_sum_infsum\n[PROOF STATE]\nproof (prove)\nusing this:\nf summable_on Sigma A B\n?f summable_on ?S \\ has_sum ?f ?S (infsum ?f ?S)\n\ngoal (1 subgoal):\n 1. has_sum f (Sigma A B) (infsum f (Sigma A B))\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nhas_sum f (Sigma A B) (infsum f (Sigma A B))\n\ngoal (1 subgoal):\n 1. infsum f (Sigma A B) = (\\\\<^sub>\\x\\A. \\\\<^sub>\\y\\B x. f (x, y))\n[PROOF STEP]\nfrom summableB\n[PROOF STATE]\nproof (chain)\npicking this:\n?x \\ A \\ (\\y. f (?x, y)) summable_on B ?x\n[PROOF STEP]\nhave b: \\\\x. x\\A \\ has_sum (\\y. f (x, y)) (B x) (infsum (\\y. f (x, y)) (B x))\\\n[PROOF STATE]\nproof (prove)\nusing this:\n?x \\ A \\ (\\y. f (?x, y)) summable_on B ?x\n\ngoal (1 subgoal):\n 1. \\x. x \\ A \\ has_sum (\\y. f (x, y)) (B x) (\\\\<^sub>\\y\\B x. f (x, y))\n[PROOF STEP]\nby (auto intro!: has_sum_infsum)\n[PROOF STATE]\nproof (state)\nthis:\n?x \\ A \\ has_sum (\\y. f (?x, y)) (B ?x) (\\\\<^sub>\\y\\B ?x. f (?x, y))\n\ngoal (1 subgoal):\n 1. infsum f (Sigma A B) = (\\\\<^sub>\\x\\A. \\\\<^sub>\\y\\B x. f (x, y))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. infsum f (Sigma A B) = (\\\\<^sub>\\x\\A. \\\\<^sub>\\y\\B x. f (x, y))\n[PROOF STEP]\nusing plus_cont a b\n[PROOF STATE]\nproof (prove)\nusing this:\nuniformly_continuous_on UNIV (\\(x, y). x + y)\nhas_sum f (Sigma A B) (infsum f (Sigma A B))\n?x \\ A \\ has_sum (\\y. f (?x, y)) (B ?x) (\\\\<^sub>\\y\\B ?x. f (?x, y))\n\ngoal (1 subgoal):\n 1. infsum f (Sigma A B) = (\\\\<^sub>\\x\\A. \\\\<^sub>\\y\\B x. f (x, y))\n[PROOF STEP]\nby (auto intro: infsumI[symmetric] has_sum_Sigma simp: summable_on_def)\n[PROOF STATE]\nproof (state)\nthis:\ninfsum f (Sigma A B) = (\\\\<^sub>\\x\\A. \\\\<^sub>\\y\\B x. f (x, y))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1460, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467675095292, "lm_q2_score": 0.8104789109591831, "lm_q1q2_score": 0.7125299147044094}} {"text": "[STATEMENT]\nlemma Max_le_iMin_imp_singleton: \"\\ finite I; I \\ {}; Max I \\ iMin I \\ \\ I = {iMin I}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite I; I \\ {}; Max I \\ iMin I\\ \\ I = {iMin I}\n[PROOF STEP]\nby (simp add: iMin_Min_conv Max_le_Min_imp_singleton)", "meta": {"llama_tokens": 147, "file": "List-Infinite_CommonSet_SetIntervalStep", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467675095294, "lm_q2_score": 0.8104789040926008, "lm_q1q2_score": 0.7125299086676758}} {"text": "[STATEMENT]\nlemma imirror_bounds_iMin: \"\n \\ finite I; I \\ {}; iMin I \\ l + r \\ \\\n iMin (imirror_bounds I l r) = l + r - Max I\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite I; I \\ {}; iMin I \\ l + r\\ \\ iMin (imirror_bounds I l r) = l + r - Max I\n[PROOF STEP]\napply (unfold imirror_bounds_def nat_mirror_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite I; I \\ {}; iMin I \\ l + r\\ \\ iMin ((-) (l + r) ` I) = l + r - Max I\n[PROOF STEP]\napply (rule iMin_equality)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\finite I; I \\ {}; iMin I \\ l + r\\ \\ l + r - Max I \\ (-) (l + r) ` I\n 2. \\x. \\finite I; I \\ {}; iMin I \\ l + r; x \\ (-) (l + r) ` I\\ \\ l + r - Max I \\ x\n[PROOF STEP]\napply (blast intro: Max_in)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\finite I; I \\ {}; iMin I \\ l + r; x \\ (-) (l + r) ` I\\ \\ l + r - Max I \\ x\n[PROOF STEP]\napply (blast intro: Max_ge diff_le_mono2)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 558, "file": "List-Infinite_CommonSet_SetIntervalCut", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.879146761176671, "lm_q2_score": 0.8104788995148792, "lm_q1q2_score": 0.7125298995105386}} {"text": "[STATEMENT]\nlemma card_injective_functions_domain_permutation:\n assumes \"finite A\" \"finite B\"\n shows \"card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A\n[PROOF STEP]\nhave \"bij_betw (subset_of A) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) {X. X \\ B \\ card X = card A}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw (subset_of A) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) {X. X \\ B \\ card X = card A}\n[PROOF STEP]\nusing \\finite A\\ \\finite B\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\n\ngoal (1 subgoal):\n 1. bij_betw (subset_of A) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) {X. X \\ B \\ card X = card A}\n[PROOF STEP]\nby (rule bij_betw_subset_of)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw (subset_of A) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) {X. X \\ B \\ card X = card A}\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nbij_betw (subset_of A) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) {X. X \\ B \\ card X = card A}\n[PROOF STEP]\nhave \"card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card {X. X \\ B \\ card X = card A}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw (subset_of A) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) {X. X \\ B \\ card X = card A}\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card {X. X \\ B \\ card X = card A}\n[PROOF STEP]\nby (rule bij_betw_same_card)\n[PROOF STATE]\nproof (state)\nthis:\ncard ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card {X. X \\ B \\ card X = card A}\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card {X. X \\ B \\ card X = card A}\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A\n[PROOF STEP]\nhave \"card {X. X \\ B \\ card X = card A} = card B choose card A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {X. X \\ B \\ card X = card A} = card B choose card A\n[PROOF STEP]\nusing \\finite B\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite B\n\ngoal (1 subgoal):\n 1. card {X. X \\ B \\ card X = card A} = card B choose card A\n[PROOF STEP]\nby (rule n_subsets)\n[PROOF STATE]\nproof (state)\nthis:\ncard {X. X \\ B \\ card X = card A} = card B choose card A\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1668, "file": "Twelvefold_Way_Twelvefold_Way_Entry5", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267898240861, "lm_q2_score": 0.819893340314393, "lm_q1q2_score": 0.7123452988635011}} {"text": "[STATEMENT]\nlemma matpow_scaleR: \"matpow (c *\\<^sub>R (X :: 'b :: real_algebra_1^_^_)) n = (c^n) *\\<^sub>R (matpow X) n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matpow (c *\\<^sub>R X) n = c ^ n *\\<^sub>R matpow X n\n[PROOF STEP]\nproof (induction n arbitrary: X c)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\X c. matpow (c *\\<^sub>R X) 0 = c ^ 0 *\\<^sub>R matpow X 0\n 2. \\n X c. (\\X c. matpow (c *\\<^sub>R X) n = c ^ n *\\<^sub>R matpow X n) \\ matpow (c *\\<^sub>R X) (Suc n) = c ^ Suc n *\\<^sub>R matpow X (Suc n)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nmatpow (?c *\\<^sub>R ?X) n = ?c ^ n *\\<^sub>R matpow ?X n\n\ngoal (2 subgoals):\n 1. \\X c. matpow (c *\\<^sub>R X) 0 = c ^ 0 *\\<^sub>R matpow X 0\n 2. \\n X c. (\\X c. matpow (c *\\<^sub>R X) n = c ^ n *\\<^sub>R matpow X n) \\ matpow (c *\\<^sub>R X) (Suc n) = c ^ Suc n *\\<^sub>R matpow X (Suc n)\n[PROOF STEP]\nhave \"matpow (c *\\<^sub>R X) (Suc n) = (c^n)*\\<^sub>R (matpow X) n ** c *\\<^sub>R X\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matpow (c *\\<^sub>R X) (Suc n) = c ^ n *\\<^sub>R matpow X n ** c *\\<^sub>R X\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\nmatpow (?c *\\<^sub>R ?X) n = ?c ^ n *\\<^sub>R matpow ?X n\n\ngoal (1 subgoal):\n 1. matpow (c *\\<^sub>R X) (Suc n) = c ^ n *\\<^sub>R matpow X n ** c *\\<^sub>R X\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmatpow (c *\\<^sub>R X) (Suc n) = c ^ n *\\<^sub>R matpow X n ** c *\\<^sub>R X\n\ngoal (2 subgoals):\n 1. \\X c. matpow (c *\\<^sub>R X) 0 = c ^ 0 *\\<^sub>R matpow X 0\n 2. \\n X c. (\\X c. matpow (c *\\<^sub>R X) n = c ^ n *\\<^sub>R matpow X n) \\ matpow (c *\\<^sub>R X) (Suc n) = c ^ Suc n *\\<^sub>R matpow X (Suc n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmatpow (c *\\<^sub>R X) (Suc n) = c ^ n *\\<^sub>R matpow X n ** c *\\<^sub>R X\n\ngoal (2 subgoals):\n 1. \\X c. matpow (c *\\<^sub>R X) 0 = c ^ 0 *\\<^sub>R matpow X 0\n 2. \\n X c. (\\X c. matpow (c *\\<^sub>R X) n = c ^ n *\\<^sub>R matpow X n) \\ matpow (c *\\<^sub>R X) (Suc n) = c ^ Suc n *\\<^sub>R matpow X (Suc n)\n[PROOF STEP]\nhave \"\\ = c *\\<^sub>R ((c^n) *\\<^sub>R (matpow X) n ** X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c ^ n *\\<^sub>R matpow X n ** c *\\<^sub>R X = c *\\<^sub>R (c ^ n *\\<^sub>R matpow X n ** X)\n[PROOF STEP]\nusing scalar_matrix_assoc'\n[PROOF STATE]\nproof (prove)\nusing this:\n?k *\\<^sub>R (?C ** ?D) = ?C ** ?k *\\<^sub>R ?D\n\ngoal (1 subgoal):\n 1. c ^ n *\\<^sub>R matpow X n ** c *\\<^sub>R X = c *\\<^sub>R (c ^ n *\\<^sub>R matpow X n ** X)\n[PROOF STEP]\nby (auto simp: scalar_matrix_assoc')\n[PROOF STATE]\nproof (state)\nthis:\nc ^ n *\\<^sub>R matpow X n ** c *\\<^sub>R X = c *\\<^sub>R (c ^ n *\\<^sub>R matpow X n ** X)\n\ngoal (2 subgoals):\n 1. \\X c. matpow (c *\\<^sub>R X) 0 = c ^ 0 *\\<^sub>R matpow X 0\n 2. \\n X c. (\\X c. matpow (c *\\<^sub>R X) n = c ^ n *\\<^sub>R matpow X n) \\ matpow (c *\\<^sub>R X) (Suc n) = c ^ Suc n *\\<^sub>R matpow X (Suc n)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nmatpow (c *\\<^sub>R X) (Suc n) = c *\\<^sub>R (c ^ n *\\<^sub>R matpow X n ** X)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nmatpow (c *\\<^sub>R X) (Suc n) = c *\\<^sub>R (c ^ n *\\<^sub>R matpow X n ** X)\n\ngoal (1 subgoal):\n 1. matpow (c *\\<^sub>R X) (Suc n) = c ^ Suc n *\\<^sub>R matpow X (Suc n)\n[PROOF STEP]\nby (simp add: scalar_matrix_assoc)\n[PROOF STATE]\nproof (state)\nthis:\nmatpow (c *\\<^sub>R X) (Suc n) = c ^ Suc n *\\<^sub>R matpow X (Suc n)\n\ngoal (1 subgoal):\n 1. \\X c. matpow (c *\\<^sub>R X) 0 = c ^ 0 *\\<^sub>R matpow X 0\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 1795, "file": "MDP-Algorithms_Matrix_Util", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267864276108, "lm_q2_score": 0.8198933425148214, "lm_q1q2_score": 0.7123452979905447}} {"text": "[STATEMENT]\nlemma sum_list_log:\n fixes b::real\n fixes xs::\"nat list\"\n assumes b: \"b > 0\" \"b \\ 1\"\n assumes xs:\"\\ x \\ set xs . x \\ b\"\n shows \"(\\x\\xs. log b x) = log b (prod_list xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list (map (log b) (map real xs)) = log b (real (prod_list xs))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < b\nb \\ 1\n\\x\\set xs. b \\ real x\n\ngoal (1 subgoal):\n 1. sum_list (map (log b) (map real xs)) = log b (real (prod_list xs))\n[PROOF STEP]\nproof (induction xs)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\0 < b; b \\ 1; \\x\\set []. b \\ real x\\ \\ sum_list (map (log b) (map real [])) = log b (real (prod_list []))\n 2. \\a xs. \\\\0 < b; b \\ 1; \\x\\set xs. b \\ real x\\ \\ sum_list (map (log b) (map real xs)) = log b (real (prod_list xs)); 0 < b; b \\ 1; \\x\\set (a # xs). b \\ real x\\ \\ sum_list (map (log b) (map real (a # xs))) = log b (real (prod_list (a # xs)))\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\n0 < b\nb \\ 1\n\\x\\set []. b \\ real x\n\ngoal (2 subgoals):\n 1. \\0 < b; b \\ 1; \\x\\set []. b \\ real x\\ \\ sum_list (map (log b) (map real [])) = log b (real (prod_list []))\n 2. \\a xs. \\\\0 < b; b \\ 1; \\x\\set xs. b \\ real x\\ \\ sum_list (map (log b) (map real xs)) = log b (real (prod_list xs)); 0 < b; b \\ 1; \\x\\set (a # xs). b \\ real x\\ \\ sum_list (map (log b) (map real (a # xs))) = log b (real (prod_list (a # xs)))\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < b\nb \\ 1\n\\x\\set []. b \\ real x\n\ngoal (1 subgoal):\n 1. sum_list (map (log b) (map real [])) = log b (real (prod_list []))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum_list (map (log b) (map real [])) = log b (real (prod_list []))\n\ngoal (1 subgoal):\n 1. \\a xs. \\\\0 < b; b \\ 1; \\x\\set xs. b \\ real x\\ \\ sum_list (map (log b) (map real xs)) = log b (real (prod_list xs)); 0 < b; b \\ 1; \\x\\set (a # xs). b \\ real x\\ \\ sum_list (map (log b) (map real (a # xs))) = log b (real (prod_list (a # xs)))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a xs. \\\\0 < b; b \\ 1; \\x\\set xs. b \\ real x\\ \\ sum_list (map (log b) (map real xs)) = log b (real (prod_list xs)); 0 < b; b \\ 1; \\x\\set (a # xs). b \\ real x\\ \\ sum_list (map (log b) (map real (a # xs))) = log b (real (prod_list (a # xs)))\n[PROOF STEP]\ncase (Cons y ys)\n[PROOF STATE]\nproof (state)\nthis:\n\\0 < b; b \\ 1; \\x\\set ys. b \\ real x\\ \\ sum_list (map (log b) (map real ys)) = log b (real (prod_list ys))\n0 < b\nb \\ 1\n\\x\\set (y # ys). b \\ real x\n\ngoal (1 subgoal):\n 1. \\a xs. \\\\0 < b; b \\ 1; \\x\\set xs. b \\ real x\\ \\ sum_list (map (log b) (map real xs)) = log b (real (prod_list xs)); 0 < b; b \\ 1; \\x\\set (a # xs). b \\ real x\\ \\ sum_list (map (log b) (map real (a # xs))) = log b (real (prod_list (a # xs)))\n[PROOF STEP]\nhave \"real (prod_list ys) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < real (prod_list ys)\n[PROOF STEP]\nusing prod_list_ge Cons.prems\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\set ?xs. 1 \\ x \\ 1 \\ prod_list ?xs\n0 < b\nb \\ 1\n\\x\\set (y # ys). b \\ real x\n\ngoal (1 subgoal):\n 1. 0 < real (prod_list ys)\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n0 < real (prod_list ys)\n\ngoal (1 subgoal):\n 1. \\a xs. \\\\0 < b; b \\ 1; \\x\\set xs. b \\ real x\\ \\ sum_list (map (log b) (map real xs)) = log b (real (prod_list xs)); 0 < b; b \\ 1; \\x\\set (a # xs). b \\ real x\\ \\ sum_list (map (log b) (map real (a # xs))) = log b (real (prod_list (a # xs)))\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < real (prod_list ys)\n\ngoal (1 subgoal):\n 1. sum_list (map (log b) (map real (y # ys))) = log b (real (prod_list (y # ys)))\n[PROOF STEP]\nusing log_mult[OF Cons.prems(1-2)] Cons\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < real (prod_list ys)\n\\0 < ?x; 0 < ?y\\ \\ log b (?x * ?y) = log b ?x + log b ?y\n\\0 < b; b \\ 1; \\x\\set ys. b \\ real x\\ \\ sum_list (map (log b) (map real ys)) = log b (real (prod_list ys))\n0 < b\nb \\ 1\n\\x\\set (y # ys). b \\ real x\n\ngoal (1 subgoal):\n 1. sum_list (map (log b) (map real (y # ys))) = log b (real (prod_list (y # ys)))\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\nsum_list (map (log b) (map real (y # ys))) = log b (real (prod_list (y # ys)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2349, "file": "Pratt_Certificate_Pratt_Certificate", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267864276108, "lm_q2_score": 0.8198933403143929, "lm_q1q2_score": 0.7123452960787534}} {"text": "[STATEMENT]\nlemma orthogonal_spaces_sym: \\orthogonal_spaces S T \\ orthogonal_spaces T S\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal_spaces S T \\ orthogonal_spaces T S\n[PROOF STEP]\nunfolding orthogonal_spaces_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\space_as_set S. \\y\\space_as_set T. is_orthogonal x y \\ \\x\\space_as_set T. \\y\\space_as_set S. is_orthogonal x y\n[PROOF STEP]\nusing is_orthogonal_sym\n[PROOF STATE]\nproof (prove)\nusing this:\nis_orthogonal ?\\ ?\\ = is_orthogonal ?\\ ?\\\n\ngoal (1 subgoal):\n 1. \\x\\space_as_set S. \\y\\space_as_set T. is_orthogonal x y \\ \\x\\space_as_set T. \\y\\space_as_set S. is_orthogonal x y\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 335, "file": "Complex_Bounded_Operators_Complex_Inner_Product", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267762381843, "lm_q2_score": 0.8198933425148214, "lm_q1q2_score": 0.7123452896363017}} {"text": "[STATEMENT]\nlemma cos_eq_1:\n fixes z::complex\n shows \"cos z = 1 \\ (\\n::int. z = complex_of_real(2 * n * pi))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cos z = 1) = (\\x. z = complex_of_real (2 * real_of_int x * pi))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (cos z = 1) = (\\x. z = complex_of_real (2 * real_of_int x * pi))\n[PROOF STEP]\nhave \"cos z = cos (2*(z/2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos z = cos (2 * (z / 2))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncos z = cos (2 * (z / 2))\n\ngoal (1 subgoal):\n 1. (cos z = 1) = (\\x. z = complex_of_real (2 * real_of_int x * pi))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncos z = cos (2 * (z / 2))\n\ngoal (1 subgoal):\n 1. (cos z = 1) = (\\x. z = complex_of_real (2 * real_of_int x * pi))\n[PROOF STEP]\nhave \"\\ = 1 - 2 * sin (z/2) ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos (2 * (z / 2)) = 1 - 2 * (sin (z / 2))\\<^sup>2\n[PROOF STEP]\nby (simp only: cos_double_sin)\n[PROOF STATE]\nproof (state)\nthis:\ncos (2 * (z / 2)) = 1 - 2 * (sin (z / 2))\\<^sup>2\n\ngoal (1 subgoal):\n 1. (cos z = 1) = (\\x. z = complex_of_real (2 * real_of_int x * pi))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncos z = 1 - 2 * (sin (z / 2))\\<^sup>2\n[PROOF STEP]\nhave [simp]: \"cos z = 1 \\ sin (z/2) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncos z = 1 - 2 * (sin (z / 2))\\<^sup>2\n\ngoal (1 subgoal):\n 1. (cos z = 1) = (sin (z / 2) = 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(cos z = 1) = (sin (z / 2) = 0)\n\ngoal (1 subgoal):\n 1. (cos z = 1) = (\\x. z = complex_of_real (2 * real_of_int x * pi))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cos z = 1) = (\\x. z = complex_of_real (2 * real_of_int x * pi))\n[PROOF STEP]\nby (auto simp: sin_eq_0)\n[PROOF STATE]\nproof (state)\nthis:\n(cos z = 1) = (\\x. z = complex_of_real (2 * real_of_int x * pi))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 998, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267694452331, "lm_q2_score": 0.819893340314393, "lm_q1q2_score": 0.7123452821550151}} {"text": "[STATEMENT]\nlemma mergesort_by_rel_merge_simps[simp] :\n \"mergesort_by_rel_merge R (x#xs) (y#ys) =\n (if R x y then x # mergesort_by_rel_merge R xs (y#ys) else y # mergesort_by_rel_merge R (x#xs) ys)\"\n \"mergesort_by_rel_merge R xs [] = xs\"\n \"mergesort_by_rel_merge R [] ys = ys\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mergesort_by_rel_merge R (x # xs) (y # ys) = (if R x y then x # mergesort_by_rel_merge R xs (y # ys) else y # mergesort_by_rel_merge R (x # xs) ys) &&& mergesort_by_rel_merge R xs [] = xs &&& mergesort_by_rel_merge R [] ys = ys\n[PROOF STEP]\napply (simp_all add: mergesort_by_rel_merge.simps)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mergesort_by_rel_merge R [] ys = ys\n[PROOF STEP]\napply (cases ys)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. ys = [] \\ mergesort_by_rel_merge R [] ys = ys\n 2. \\a list. ys = a # list \\ mergesort_by_rel_merge R [] ys = ys\n[PROOF STEP]\napply (simp_all add: mergesort_by_rel_merge.simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 485, "file": "Automatic_Refinement_Lib_Misc", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267728417087, "lm_q2_score": 0.8198933359135361, "lm_q1q2_score": 0.7123452811161806}} {"text": "[STATEMENT]\nlemma compact_closed_differences:\n fixes S T :: \"'a::real_normed_vector set\"\n assumes \"compact S\" \"closed T\"\n shows \"closed (\\x\\ S. \\y \\ T. {x - y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. closed (\\x\\S. \\y\\T. {x - y})\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. closed (\\x\\S. \\y\\T. {x - y})\n[PROOF STEP]\nhave \"(\\x\\ S. \\y \\ uminus ` T. {x + y}) = (\\x\\ S. \\y \\ T. {x - y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\S. \\y\\uminus ` T. {x + y}) = (\\x\\S. \\y\\T. {x - y})\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\S. \\y\\uminus ` T. {x + y}) = (\\x\\S. \\y\\T. {x - y})\n\ngoal (1 subgoal):\n 1. closed (\\x\\S. \\y\\T. {x - y})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x\\S. \\y\\uminus ` T. {x + y}) = (\\x\\S. \\y\\T. {x - y})\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x\\S. \\y\\uminus ` T. {x + y}) = (\\x\\S. \\y\\T. {x - y})\n\ngoal (1 subgoal):\n 1. closed (\\x\\S. \\y\\T. {x - y})\n[PROOF STEP]\nby (metis assms closed_negations compact_closed_sums)\n[PROOF STATE]\nproof (state)\nthis:\nclosed (\\x\\S. \\y\\T. {x - y})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 712, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267728417086, "lm_q2_score": 0.8198933315126792, "lm_q1q2_score": 0.7123452772925982}} {"text": "[STATEMENT]\nlemma vec_norm_geq_0:\n fixes v :: \"complex vec\"\n shows \"vec_norm v \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ vec_norm v\n[PROOF STEP]\nunfolding vec_norm_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ csqrt (inner_prod v v)\n[PROOF STEP]\nby (insert self_cscalar_prod_geq_0[of v], simp add: less_eq_complex_def)", "meta": {"llama_tokens": 163, "file": "QHLProver_Complex_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267694452331, "lm_q2_score": 0.8198933337131076, "lm_q1q2_score": 0.7123452764196416}} {"text": "[STATEMENT]\nlemma mat_plus_right_mono: \"B \\\\<^sub>m (C :: 'a :: ordered_ab_semigroup mat) \n \\ A \\ carrier_mat nr nc \\ B \\ carrier_mat nr nc \\ C \\ carrier_mat nr nc \n \\ A + B \\\\<^sub>m A + C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\B \\\\<^sub>m C; A \\ carrier_mat nr nc; B \\ carrier_mat nr nc; C \\ carrier_mat nr nc\\ \\ A + B \\\\<^sub>m A + C\n[PROOF STEP]\nby (intro mat_geI[of _ nr nc], auto simp: plus_right_mono)", "meta": {"llama_tokens": 228, "file": "Jordan_Normal_Form_Matrix_Comparison", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.868826769445233, "lm_q2_score": 0.8198933271118222, "lm_q1q2_score": 0.7123452706842681}} {"text": "[STATEMENT]\nlemma rtranclp_imp_eq_image: \"(\\x y. R x y \\ f x = f y) \\ R\\<^sup>*\\<^sup>* x y \\ f x = f y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x y. R x y \\ f x = f y; R\\<^sup>*\\<^sup>* x y\\ \\ f x = f y\n[PROOF STEP]\nby (erule rtranclp.induct) auto", "meta": {"llama_tokens": 154, "file": "Functional_Ordered_Resolution_Prover_Deterministic_FO_Ordered_Resolution_Prover", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267626522814, "lm_q2_score": 0.8198933315126791, "lm_q1q2_score": 0.7123452689383547}} {"text": "[STATEMENT]\ntheorem vcgaAlgDefinedness:\n assumes \"distinct \\\" \"set \\ \\ {}\" \"finite N\" \n shows \"card (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocationsAlg N \\ b)) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocationsAlg N \\ b)) = 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocationsAlg N \\ b)) = 1\n[PROOF STEP]\nhave \"card (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocations N (set \\) b)) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocations N (set \\) b)) = 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndistinct \\\nset \\ \\ {}\nfinite N\n\ngoal (1 subgoal):\n 1. card (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocations N (set \\) b)) = 1\n[PROOF STEP]\nby (rule tiebreakingGivesSingleton)\n[PROOF STATE]\nproof (state)\nthis:\ncard (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocations N (set \\) b)) = 1\n\ngoal (1 subgoal):\n 1. card (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocationsAlg N \\ b)) = 1\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncard (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocations N (set \\) b)) = 1\n\ngoal (1 subgoal):\n 1. card (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocationsAlg N \\ b)) = 1\n[PROOF STEP]\nhave \"maximalStrictAllocations N (set \\) b = maximalStrictAllocationsAlg N \\ b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. maximalStrictAllocations N (set \\) b = maximalStrictAllocationsAlg N \\ b\n[PROOF STEP]\nusing assms(3,1)\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite N\ndistinct \\\n\ngoal (1 subgoal):\n 1. maximalStrictAllocations N (set \\) b = maximalStrictAllocationsAlg N \\ b\n[PROOF STEP]\nby (rule maximalAllocationBridgingTheorem)\n[PROOF STATE]\nproof (state)\nthis:\nmaximalStrictAllocations N (set \\) b = maximalStrictAllocationsAlg N \\ b\n\ngoal (1 subgoal):\n 1. card (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocationsAlg N \\ b)) = 1\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocations N (set \\) b)) = 1\nmaximalStrictAllocations N (set \\) b = maximalStrictAllocationsAlg N \\ b\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocations N (set \\) b)) = 1\nmaximalStrictAllocations N (set \\) b = maximalStrictAllocationsAlg N \\ b\n\ngoal (1 subgoal):\n 1. card (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocationsAlg N \\ b)) = 1\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\ncard (argmax (sum (randomBidsAlg N \\ b r)) (maximalStrictAllocationsAlg N \\ b)) = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1253, "file": "Vickrey_Clarke_Groves_CombinatorialAuction", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.919642528975397, "lm_q2_score": 0.7745833893685269, "lm_q1q2_score": 0.7123398271012067}} {"text": "[STATEMENT]\nlemma cis_Arg_power[simp]: assumes \"x \\ 0\" \n shows \"cis (Arg (x ^ n)) = cis (Arg x * real n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cis (Arg (x ^ n)) = cis (Arg x * real n)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. cis (Arg (x ^ 0)) = cis (Arg x * real 0)\n 2. \\n. cis (Arg (x ^ n)) = cis (Arg x * real n) \\ cis (Arg (x ^ Suc n)) = cis (Arg x * real (Suc n))\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\ncis (Arg (x ^ n)) = cis (Arg x * real n)\n\ngoal (2 subgoals):\n 1. cis (Arg (x ^ 0)) = cis (Arg x * real 0)\n 2. \\n. cis (Arg (x ^ n)) = cis (Arg x * real n) \\ cis (Arg (x ^ Suc n)) = cis (Arg x * real (Suc n))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cis (Arg (x ^ Suc n)) = cis (Arg x * real (Suc n))\n[PROOF STEP]\nunfolding power.simps\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cis (Arg (x * x ^ n)) = cis (Arg x * real (Suc n))\n[PROOF STEP]\nproof (subst cis_arg_mult)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x * x ^ n \\ 0\n 2. cis (Arg x + Arg (x ^ n)) = cis (Arg x * real (Suc n))\n[PROOF STEP]\nshow \"cis (Arg x + Arg (x ^ n)) = cis (Arg x * real (Suc n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cis (Arg x + Arg (x ^ n)) = cis (Arg x * real (Suc n))\n[PROOF STEP]\nunfolding mult.commute[of \"Arg x\"] DeMoivre[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cis (Arg x + Arg (x ^ n)) = cis (Arg x) ^ Suc n\n[PROOF STEP]\nunfolding power.simps\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cis (Arg x + Arg (x ^ n)) = cis (Arg x) * cis (Arg x) ^ n\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\ncis (Arg (x ^ n)) = cis (Arg x * real n)\n\ngoal (1 subgoal):\n 1. cis (Arg x + Arg (x ^ n)) = cis (Arg x) * cis (Arg x) ^ n\n[PROOF STEP]\nby (metis DeMoivre cis_mult mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\ncis (Arg x + Arg (x ^ n)) = cis (Arg x * real (Suc n))\n\ngoal (1 subgoal):\n 1. x * x ^ n \\ 0\n[PROOF STEP]\nshow \"x * x ^ n \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * x ^ n \\ 0\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ 0\n\ngoal (1 subgoal):\n 1. x * x ^ n \\ 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx * x ^ n \\ 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncis (Arg (x ^ Suc n)) = cis (Arg x * real (Suc n))\n\ngoal (1 subgoal):\n 1. cis (Arg (x ^ 0)) = cis (Arg x * real 0)\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 1219, "file": "Cubic_Quartic_Equations_Complex_Roots", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382200964035, "lm_q2_score": 0.8267118026095991, "lm_q1q2_score": 0.7122438149529632}} {"text": "[STATEMENT]\nlemma card_lists_distinct_length_eq_union:\n assumes \"finite X\" \"finite Y\" \"X \\ Y = {}\"\n shows \"card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} =\n (\\k=0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\"\n (is \"card ?S = _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nlet ?expr = \"do {\n k \\ {0..n};\n xs \\ {xs. length xs = k \\ distinct xs \\ set xs \\ X};\n ys \\ {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y};\n {zs. interleavings xs ys zs}\n }\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nfrom \\X \\ Y = {}\\\n[PROOF STATE]\nproof (chain)\npicking this:\nX \\ Y = {}\n[PROOF STEP]\nhave \"card ?S = card ?expr\"\n[PROOF STATE]\nproof (prove)\nusing this:\nX \\ Y = {}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = card ({0..n} \\ (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))))\n[PROOF STEP]\nby (simp add: lists_distinct_union_by_interleavings)\n[PROOF STATE]\nproof (state)\nthis:\ncard {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = card ({0..n} \\ (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))))\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nlet \"?S \\ ?comp\" = \"?expr\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nfix k\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nassume \"k \\ ?S\"\n[PROOF STATE]\nproof (state)\nthis:\nk \\ {0..n}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nlet \"?expr\" = \"?comp k\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nlet \"?S \\ ?comp\" = \"?expr\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nfrom \\finite X\\\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite X\n[PROOF STEP]\nhave \"finite ?S\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite X\n\ngoal (1 subgoal):\n 1. finite {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\nfinite {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nfix xs\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nassume xs: \"xs \\ ?S\"\n[PROOF STATE]\nproof (state)\nthis:\nxs \\ {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nlet ?expr = \"?comp xs\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nlet \"?S \\ ?comp\" = ?expr\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nfrom \\finite Y\\\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite Y\n[PROOF STEP]\nhave \"finite ?S\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite Y\n\ngoal (1 subgoal):\n 1. finite {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\nfinite {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nfix ys\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nassume ys: \"ys \\ ?S\"\n[PROOF STATE]\nproof (state)\nthis:\nys \\ {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nlet ?expr = \"?comp ys\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nhave \"finite ?expr\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {zs. interleavings xs ys zs}\n[PROOF STEP]\nby (simp add: finite_interleavings)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {zs. interleavings xs ys zs}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite {zs. interleavings xs ys zs}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nhave \"card ?expr = (n choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {zs. interleavings xs ys zs} = n choose k\n[PROOF STEP]\nusing xs ys \\X \\ Y = {}\\ \\k \\ _\\\n[PROOF STATE]\nproof (prove)\nusing this:\nxs \\ {xs. length xs = k \\ distinct xs \\ set xs \\ X}\nys \\ {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\nX \\ Y = {}\nk \\ {0..n}\n\ngoal (1 subgoal):\n 1. card {zs. interleavings xs ys zs} = n choose k\n[PROOF STEP]\nby (subst card_interleavings) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard {zs. interleavings xs ys zs} = n choose k\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite {zs. interleavings xs ys zs}\ncard {zs. interleavings xs ys zs} = n choose k\n[PROOF STEP]\nhave \"finite ?expr \\ card ?expr = (n choose k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {zs. interleavings xs ys zs}\ncard {zs. interleavings xs ys zs} = n choose k\n\ngoal (1 subgoal):\n 1. finite {zs. interleavings xs ys zs} \\ card {zs. interleavings xs ys zs} = n choose k\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nfinite {zs. interleavings xs ys zs} \\ card {zs. interleavings xs ys zs} = n choose k\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n?ys2 \\ {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ finite {zs. interleavings xs ?ys2 zs} \\ card {zs. interleavings xs ?ys2 zs} = n choose k\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n?ys2 \\ {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ finite {zs. interleavings xs ?ys2 zs} \\ card {zs. interleavings xs ?ys2 zs} = n choose k\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nhave \"disjoint_family_on ?comp ?S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. disjoint_family_on (\\ys. {zs. interleavings xs ys zs}) {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n[PROOF STEP]\nusing \\k \\ {0..n}\\ \\xs \\ {xs. length xs = k \\ distinct xs \\ set xs \\ X}\\\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ {0..n}\nxs \\ {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n\ngoal (1 subgoal):\n 1. disjoint_family_on (\\ys. {zs. interleavings xs ys zs}) {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n[PROOF STEP]\nby (injectivity_solver rule: injectivity(3)[OF \\X \\ Y = {}\\])\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on (\\ys. {zs. interleavings xs ys zs}) {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on (\\ys. {zs. interleavings xs ys zs}) {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nhave \"card ?S = ffact (n - k) (card Y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} = ffact (n - k) (card Y)\n[PROOF STEP]\nusing \\finite Y\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite Y\n\ngoal (1 subgoal):\n 1. card {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} = ffact (n - k) (card Y)\n[PROOF STEP]\nby (simp add: card_lists_distinct_length_eq)\n[PROOF STATE]\nproof (state)\nthis:\ncard {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} = ffact (n - k) (card Y)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n?ys2 \\ {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ finite {zs. interleavings xs ?ys2 zs} \\ card {zs. interleavings xs ?ys2 zs} = n choose k\ndisjoint_family_on (\\ys. {zs. interleavings xs ys zs}) {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\ncard {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} = ffact (n - k) (card Y)\n[PROOF STEP]\nhave \"card ?expr = (n choose k) * ffact (n - k) (card Y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n?ys2 \\ {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ finite {zs. interleavings xs ?ys2 zs} \\ card {zs. interleavings xs ?ys2 zs} = n choose k\ndisjoint_family_on (\\ys. {zs. interleavings xs ys zs}) {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\ncard {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} = ffact (n - k) (card Y)\n\ngoal (1 subgoal):\n 1. card ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) = (n choose k) * ffact (n - k) (card Y)\n[PROOF STEP]\nby (subst card_bind_constant) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) = (n choose k) * ffact (n - k) (card Y)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncard ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) = (n choose k) * ffact (n - k) (card Y)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nhave \"finite ?expr\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))\n[PROOF STEP]\nusing \\finite ?S\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y}\n\ngoal (1 subgoal):\n 1. finite ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))\n[PROOF STEP]\nby (auto intro!: finite_bind finite_interleavings)\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncard ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) = (n choose k) * ffact (n - k) (card Y)\nfinite ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))\n[PROOF STEP]\nhave \"finite ?expr \\ card ?expr = (n choose k) * ffact (n - k) (card Y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) = (n choose k) * ffact (n - k) (card Y)\nfinite ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))\n\ngoal (1 subgoal):\n 1. finite ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) \\ card ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) = (n choose k) * ffact (n - k) (card Y)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) \\ card ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) = (n choose k) * ffact (n - k) (card Y)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n?xs2 \\ {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ finite ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings ?xs2 ys zs})) \\ card ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings ?xs2 ys zs})) = (n choose k) * ffact (n - k) (card Y)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n?xs2 \\ {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ finite ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings ?xs2 ys zs})) \\ card ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings ?xs2 ys zs})) = (n choose k) * ffact (n - k) (card Y)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nhave \"disjoint_family_on ?comp ?S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. disjoint_family_on (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n[PROOF STEP]\nusing \\k \\ {0..n}\\\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ {0..n}\n\ngoal (1 subgoal):\n 1. disjoint_family_on (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n[PROOF STEP]\nby (injectivity_solver rule: injectivity(2)[OF \\X \\ Y = {}\\])\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nhave \"card ?S = ffact k (card X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {xs. length xs = k \\ distinct xs \\ set xs \\ X} = ffact k (card X)\n[PROOF STEP]\nusing \\finite X\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite X\n\ngoal (1 subgoal):\n 1. card {xs. length xs = k \\ distinct xs \\ set xs \\ X} = ffact k (card X)\n[PROOF STEP]\nby (simp add: card_lists_distinct_length_eq)\n[PROOF STATE]\nproof (state)\nthis:\ncard {xs. length xs = k \\ distinct xs \\ set xs \\ X} = ffact k (card X)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n?xs2 \\ {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ finite ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings ?xs2 ys zs})) \\ card ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings ?xs2 ys zs})) = (n choose k) * ffact (n - k) (card Y)\ndisjoint_family_on (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) {xs. length xs = k \\ distinct xs \\ set xs \\ X}\ncard {xs. length xs = k \\ distinct xs \\ set xs \\ X} = ffact k (card X)\n[PROOF STEP]\nhave \"card ?expr = (n choose k) * ffact k (card X) * ffact (n - k) (card Y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {xs. length xs = k \\ distinct xs \\ set xs \\ X}\n?xs2 \\ {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ finite ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings ?xs2 ys zs})) \\ card ({ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings ?xs2 ys zs})) = (n choose k) * ffact (n - k) (card Y)\ndisjoint_family_on (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})) {xs. length xs = k \\ distinct xs \\ set xs \\ X}\ncard {xs. length xs = k \\ distinct xs \\ set xs \\ X} = ffact k (card X)\n\ngoal (1 subgoal):\n 1. card ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) = (n choose k) * ffact k (card X) * ffact (n - k) (card Y)\n[PROOF STEP]\nby (subst card_bind_constant) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) = (n choose k) * ffact k (card X) * ffact (n - k) (card Y)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncard ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) = (n choose k) * ffact k (card X) * ffact (n - k) (card Y)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nhave \"finite ?expr\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})))\n[PROOF STEP]\nusing \\finite ?S\\ \\finite Y\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {xs. length xs = k \\ distinct xs \\ set xs \\ X}\nfinite Y\n\ngoal (1 subgoal):\n 1. finite ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})))\n[PROOF STEP]\nby (auto intro!: finite_bind finite_interleavings)\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})))\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncard ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) = (n choose k) * ffact k (card X) * ffact (n - k) (card Y)\nfinite ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})))\n[PROOF STEP]\nhave \"finite ?expr \\ card ?expr = (n choose k) * ffact k (card X) * ffact (n - k) (card Y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) = (n choose k) * ffact k (card X) * ffact (n - k) (card Y)\nfinite ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})))\n\ngoal (1 subgoal):\n 1. finite ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) \\ card ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) = (n choose k) * ffact k (card X) * ffact (n - k) (card Y)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) \\ card ({xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) = (n choose k) * ffact k (card X) * ffact (n - k) (card Y)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n?k2 \\ {0..n} \\ finite ({xs. length xs = ?k2 \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - ?k2 \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) \\ card ({xs. length xs = ?k2 \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - ?k2 \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) = (n choose ?k2) * ffact ?k2 (card X) * ffact (n - ?k2) (card Y)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n?k2 \\ {0..n} \\ finite ({xs. length xs = ?k2 \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - ?k2 \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) \\ card ({xs. length xs = ?k2 \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - ?k2 \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) = (n choose ?k2) * ffact ?k2 (card X) * ffact (n - ?k2) (card Y)\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nhave \"disjoint_family_on ?comp ?S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. disjoint_family_on (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) {0..n}\n[PROOF STEP]\nby (injectivity_solver rule: injectivity(1)[OF \\X \\ Y = {}\\])\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) {0..n}\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n?k2 \\ {0..n} \\ finite ({xs. length xs = ?k2 \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - ?k2 \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) \\ card ({xs. length xs = ?k2 \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - ?k2 \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) = (n choose ?k2) * ffact ?k2 (card X) * ffact (n - ?k2) (card Y)\ndisjoint_family_on (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) {0..n}\n[PROOF STEP]\nhave \"card ?expr = (\\k=0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?k2 \\ {0..n} \\ finite ({xs. length xs = ?k2 \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - ?k2 \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) \\ card ({xs. length xs = ?k2 \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - ?k2 \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) = (n choose ?k2) * ffact ?k2 (card X) * ffact (n - ?k2) (card Y)\ndisjoint_family_on (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))) {0..n}\n\ngoal (1 subgoal):\n 1. card ({0..n} \\ (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})))) = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nby (auto simp add: card_bind)\n[PROOF STATE]\nproof (state)\nthis:\ncard ({0..n} \\ (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})))) = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nfrom \\card _ = card ?expr\\ this\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = card ({0..n} \\ (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))))\ncard ({0..n} \\ (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})))) = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = card ({0..n} \\ (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs}))))\ncard ({0..n} \\ (\\k. {xs. length xs = k \\ distinct xs \\ set xs \\ X} \\ (\\xs. {ys. length ys = n - k \\ distinct ys \\ set ys \\ Y} \\ (\\ys. {zs. interleavings xs ys zs})))) = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n\ngoal (1 subgoal):\n 1. card {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard {zs. length zs = n \\ distinct zs \\ set zs \\ X \\ Y} = (\\k = 0..n. (n choose k) * ffact k (card X) * ffact (n - k) (card Y))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 14753, "file": "Falling_Factorial_Sum_Falling_Factorial_Sum_Combinatorics", "length": 86, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382165412809, "lm_q2_score": 0.8267118004748678, "lm_q1q2_score": 0.7122438101747489}} {"text": "[STATEMENT]\nlemma lin_indpt_id:\n shows \"lin_indpt (set (cols (1\\<^sub>m n)::'a vec list))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lin_indpt (set (cols (1\\<^sub>m n)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. lin_indpt (set (cols (1\\<^sub>m n)))\n[PROOF STEP]\nhave *: \"set (cols (1\\<^sub>m n)) = set (rows (1\\<^sub>m n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (cols (1\\<^sub>m n)) = set (rows (1\\<^sub>m n))\n[PROOF STEP]\nby (metis cols_transpose transpose_one)\n[PROOF STATE]\nproof (state)\nthis:\nset (cols (1\\<^sub>m n)) = set (rows (1\\<^sub>m n))\n\ngoal (1 subgoal):\n 1. lin_indpt (set (cols (1\\<^sub>m n)))\n[PROOF STEP]\nhave \"det (1\\<^sub>m n) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (1\\<^sub>m n) \\ (0::'b)\n[PROOF STEP]\nusing det_one\n[PROOF STATE]\nproof (prove)\nusing this:\ndet (1\\<^sub>m ?n) = (1::?'a)\n\ngoal (1 subgoal):\n 1. det (1\\<^sub>m n) \\ (0::'b)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndet (1\\<^sub>m n) \\ (0::?'b3)\n\ngoal (1 subgoal):\n 1. lin_indpt (set (cols (1\\<^sub>m n)))\n[PROOF STEP]\nfrom det_not_0_imp_lin_indpt_rows[OF _ this]\n[PROOF STATE]\nproof (chain)\npicking this:\n1\\<^sub>m n \\ carrier_mat n n \\ lin_indpt (set (rows (1\\<^sub>m n)))\n[PROOF STEP]\nhave \"lin_indpt (set (rows (1\\<^sub>m n)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1\\<^sub>m n \\ carrier_mat n n \\ lin_indpt (set (rows (1\\<^sub>m n)))\n\ngoal (1 subgoal):\n 1. lin_indpt (set (rows (1\\<^sub>m n)))\n[PROOF STEP]\nusing one_carrier_mat\n[PROOF STATE]\nproof (prove)\nusing this:\n1\\<^sub>m n \\ carrier_mat n n \\ lin_indpt (set (rows (1\\<^sub>m n)))\n1\\<^sub>m ?n \\ carrier_mat ?n ?n\n\ngoal (1 subgoal):\n 1. lin_indpt (set (rows (1\\<^sub>m n)))\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nlin_indpt (set (rows (1\\<^sub>m n)))\n\ngoal (1 subgoal):\n 1. lin_indpt (set (cols (1\\<^sub>m n)))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlin_indpt (set (rows (1\\<^sub>m n)))\n\ngoal (1 subgoal):\n 1. lin_indpt (set (cols (1\\<^sub>m n)))\n[PROOF STEP]\nby (simp add: *)\n[PROOF STATE]\nproof (state)\nthis:\nlin_indpt (set (cols (1\\<^sub>m n)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1046, "file": "BenOr_Kozen_Reif_More_Matrix", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615382236515258, "lm_q2_score": 0.8267117919359419, "lm_q1q2_score": 0.7122438086962611}} {"text": "[STATEMENT]\nlemma cmod_conjugate_square_eq:\n fixes z::complex\n shows \"cmod (z * (conjugate z)) = z * (conjugate z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (cmod (z * conjugate z)) = z * conjugate z\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. complex_of_real (cmod (z * conjugate z)) = z * conjugate z\n[PROOF STEP]\nhave \"0 \\ z * (conjugate z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ z * conjugate z\n[PROOF STEP]\nusing conjugate_square_positive less_eq_complex_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::?'a) \\ ?a * conjugate ?a\n(?x \\ ?y) = (Re ?x \\ Re ?y \\ Im ?x = Im ?y)\n\ngoal (1 subgoal):\n 1. 0 \\ z * conjugate z\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ z * conjugate z\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (z * conjugate z)) = z * conjugate z\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ z * conjugate z\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (z * conjugate z)) = z * conjugate z\n[PROOF STEP]\nusing cmod_pos\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ z * conjugate z\n0 \\ ?z \\ complex_of_real (cmod ?z) = ?z\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (z * conjugate z)) = z * conjugate z\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (cmod (z * conjugate z)) = z * conjugate z\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 629, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382165412809, "lm_q2_score": 0.8267117898012105, "lm_q1q2_score": 0.7122438009789852}} {"text": "[STATEMENT]\nlemma rat_precision_pos:\n assumes \"0 \\ x\"\n and \"0 < y\"\n and \"2 * x < y\"\n shows \"rat_precision n (int x) (int y) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < rat_precision n (int x) (int y)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 < rat_precision n (int x) (int y)\n[PROOF STEP]\nhave \"0 < x \\ log 2 x + 1 = log 2 (2 * x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ log 2 (real x) + 1 = log 2 (real (2 * x))\n[PROOF STEP]\nby (simp add: log_mult)\n[PROOF STATE]\nproof (state)\nthis:\n0 < x \\ log 2 (real x) + 1 = log 2 (real (2 * x))\n\ngoal (1 subgoal):\n 1. 0 < rat_precision n (int x) (int y)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < x \\ log 2 (real x) + 1 = log 2 (real (2 * x))\n[PROOF STEP]\nhave \"bitlen (int x) < bitlen (int y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x \\ log 2 (real x) + 1 = log 2 (real (2 * x))\n\ngoal (1 subgoal):\n 1. bitlen (int x) < bitlen (int y)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x \\ log 2 (real x) + 1 = log 2 (real (2 * x))\n0 \\ x\n0 < y\n2 * x < y\n\ngoal (1 subgoal):\n 1. bitlen (int x) < bitlen (int y)\n[PROOF STEP]\nby (simp add: bitlen_alt_def)\n (auto intro!: floor_mono simp add: one_add_floor)\n[PROOF STATE]\nproof (state)\nthis:\nbitlen (int x) < bitlen (int y)\n\ngoal (1 subgoal):\n 1. 0 < rat_precision n (int x) (int y)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nbitlen (int x) < bitlen (int y)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nbitlen (int x) < bitlen (int y)\n\ngoal (1 subgoal):\n 1. 0 < rat_precision n (int x) (int y)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nbitlen (int x) < bitlen (int y)\n0 \\ x\n0 < y\n2 * x < y\n\ngoal (1 subgoal):\n 1. 0 < rat_precision n (int x) (int y)\n[PROOF STEP]\nby (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)\n[PROOF STATE]\nproof (state)\nthis:\n0 < rat_precision n (int x) (int y)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 957, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.86153820232079, "lm_q2_score": 0.8267118004748678, "lm_q1q2_score": 0.7122437984185013}} {"text": "[STATEMENT]\nlemma sq_mtx_norm_le_sum_col: \"\\A\\ \\ (\\i\\UNIV. \\\\\\\\ i A\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A\\ \\ (\\i\\UNIV. \\\\\\\\ i A\\)\n[PROOF STEP]\nusing op_norm_le_sum_column[of \"to_vec A\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\to_vec A\\\\<^sub>o\\<^sub>p \\ (\\i\\UNIV. \\column i (to_vec A)\\)\n\ngoal (1 subgoal):\n 1. \\A\\ \\ (\\i\\UNIV. \\\\\\\\ i A\\)\n[PROOF STEP]\napply(simp add: norm_sq_mtx_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\to_vec A\\\\<^sub>o\\<^sub>p \\ (\\i\\UNIV. \\column i (to_vec A)\\) \\ \\to_vec A\\\\<^sub>o\\<^sub>p \\ (\\i\\UNIV. \\\\\\\\ i A\\)\n[PROOF STEP]\nby(transfer, simp add: op_norm_le_sum_column)", "meta": {"llama_tokens": 412, "file": "Matrices_for_ODEs_SQ_MTX", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392817460332, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.712210503761253}} {"text": "[STATEMENT]\nlemma powr_minus_divide: \"x powr (- a) = 1/(x powr a)\"\n for a x :: \"'a::{ln,real_normed_field}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x powr - a = (1::'a) / x powr a\n[PROOF STEP]\nby (simp add: divide_inverse powr_minus)", "meta": {"llama_tokens": 113, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392817460333, "lm_q2_score": 0.8056321796478255, "lm_q1q2_score": 0.712210493447355}} {"text": "[STATEMENT]\nlemma has_derivative_zero_unique:\n fixes f :: \"'a::real_normed_vector \\ 'b::real_normed_vector\"\n assumes \"convex s\"\n and \"\\x. x \\ s \\ (f has_derivative (\\h. 0)) (at x within s)\"\n and \"x \\ s\" \"y \\ s\"\n shows \"f x = f y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f x = f y\n[PROOF STEP]\nusing has_derivative_zero_constant[OF assms(1,2)] assms(3-)\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. x \\ s \\ x \\ s) \\ \\c. \\x\\s. f x = c\nx \\ s\ny \\ s\n\ngoal (1 subgoal):\n 1. f x = f y\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 276, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392756357326, "lm_q2_score": 0.8056321843145405, "lm_q1q2_score": 0.7122104926502594}} {"text": "[STATEMENT]\nlemma poly_shift_monom: \"poly_shift n (monom c m) = (if m \\ n then monom c (m - n) else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly_shift n (monom c m) = (if n \\ m then monom c (m - n) else 0)\n[PROOF STEP]\nby (auto simp add: poly_eq_iff coeff_poly_shift)", "meta": {"llama_tokens": 127, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392695254318, "lm_q2_score": 0.8056321796478255, "lm_q1q2_score": 0.7122104836020451}} {"text": "[STATEMENT]\nlemma (in comm_semiring_0) poly_mult: \"poly (p1 *** p2) x = poly p1 x * poly p2 x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly (p1 *** p2) x = poly p1 x * poly p2 x\n[PROOF STEP]\nproof (induct p1 arbitrary: p2)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\p2. poly ([] *** p2) x = poly [] x * poly p2 x\n 2. \\a p1 p2. (\\p2. poly (p1 *** p2) x = poly p1 x * poly p2 x) \\ poly ((a # p1) *** p2) x = poly (a # p1) x * poly p2 x\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\p2. poly ([] *** p2) x = poly [] x * poly p2 x\n 2. \\a p1 p2. (\\p2. poly (p1 *** p2) x = poly p1 x * poly p2 x) \\ poly ((a # p1) *** p2) x = poly (a # p1) x * poly p2 x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly ([] *** p2) x = poly [] x * poly p2 x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\npoly ([] *** p2) x = poly [] x * poly p2 x\n\ngoal (1 subgoal):\n 1. \\a p1 p2. (\\p2. poly (p1 *** p2) x = poly p1 x * poly p2 x) \\ poly ((a # p1) *** p2) x = poly (a # p1) x * poly p2 x\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a p1 p2. (\\p2. poly (p1 *** p2) x = poly p1 x * poly p2 x) \\ poly ((a # p1) *** p2) x = poly (a # p1) x * poly p2 x\n[PROOF STEP]\ncase (Cons a as)\n[PROOF STATE]\nproof (state)\nthis:\npoly (as *** ?p2.1) x = poly as x * poly ?p2.1 x\n\ngoal (1 subgoal):\n 1. \\a p1 p2. (\\p2. poly (p1 *** p2) x = poly p1 x * poly p2 x) \\ poly ((a # p1) *** p2) x = poly (a # p1) x * poly p2 x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\npoly (as *** ?p2.1) x = poly as x * poly ?p2.1 x\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\npoly (as *** ?p2.1) x = poly as x * poly ?p2.1 x\n\ngoal (1 subgoal):\n 1. poly ((a # as) *** p2) x = poly (a # as) x * poly p2 x\n[PROOF STEP]\nby (cases as) (simp_all add: poly_cmult poly_add distrib_right distrib_left ac_simps)\n[PROOF STATE]\nproof (state)\nthis:\npoly ((a # as) *** p2) x = poly (a # as) x * poly p2 x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1015, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438950947024555, "lm_q2_score": 0.8438951005915208, "lm_q1q2_score": 0.7121589358326197}} {"text": "[STATEMENT]\nlemma vector_derivative_scaleR_at [simp]:\n \"\\f differentiable at a; g differentiable at a\\\n \\ vector_derivative (\\x. f x *\\<^sub>R g x) (at a) = f a *\\<^sub>R vector_derivative g (at a) + vector_derivative f (at a) *\\<^sub>R g a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f differentiable at a; g differentiable at a\\ \\ vector_derivative (\\x. f x *\\<^sub>R g x) (at a) = f a *\\<^sub>R vector_derivative g (at a) + vector_derivative f (at a) *\\<^sub>R g a\n[PROOF STEP]\napply (rule vector_derivative_at)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f differentiable at a; g differentiable at a\\ \\ ((\\x. f x *\\<^sub>R g x) has_vector_derivative f a *\\<^sub>R vector_derivative g (at a) + vector_derivative f (at a) *\\<^sub>R g a) (at a)\n[PROOF STEP]\napply (rule has_vector_derivative_scaleR)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\f differentiable at a; g differentiable at a\\ \\ (f has_real_derivative vector_derivative f (at a)) (at a)\n 2. \\f differentiable at a; g differentiable at a\\ \\ (g has_vector_derivative vector_derivative g (at a)) (at a)\n[PROOF STEP]\napply (auto simp: vector_derivative_works has_vector_derivative_def has_field_derivative_def mult_commute_abs)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 572, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.909907001151883, "lm_q2_score": 0.7826624688140726, "lm_q1q2_score": 0.712150059912742}} {"text": "[STATEMENT]\nlemma mult_conj_cmod_square:\n fixes z::complex\n shows \"z * conjugate z = (cmod z)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z * conjugate z = complex_of_real ((cmod z)\\<^sup>2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. z * conjugate z = complex_of_real ((cmod z)\\<^sup>2)\n[PROOF STEP]\nhave \"z * conjugate z = (Re z)\\<^sup>2 + (Im z)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z * conjugate z = complex_of_real ((Re z)\\<^sup>2 + (Im z)\\<^sup>2)\n[PROOF STEP]\nusing complex_mult_cnj\n[PROOF STATE]\nproof (prove)\nusing this:\n?z * cnj ?z = complex_of_real ((Re ?z)\\<^sup>2 + (Im ?z)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. z * conjugate z = complex_of_real ((Re z)\\<^sup>2 + (Im z)\\<^sup>2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nz * conjugate z = complex_of_real ((Re z)\\<^sup>2 + (Im z)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. z * conjugate z = complex_of_real ((cmod z)\\<^sup>2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nz * conjugate z = complex_of_real ((Re z)\\<^sup>2 + (Im z)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. z * conjugate z = complex_of_real ((cmod z)\\<^sup>2)\n[PROOF STEP]\nhave \"... = (cmod z)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real ((Re z)\\<^sup>2 + (Im z)\\<^sup>2) = complex_of_real ((cmod z)\\<^sup>2)\n[PROOF STEP]\nunfolding cmod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real ((Re z)\\<^sup>2 + (Im z)\\<^sup>2) = complex_of_real ((sqrt ((Re z)\\<^sup>2 + (Im z)\\<^sup>2))\\<^sup>2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real ((Re z)\\<^sup>2 + (Im z)\\<^sup>2) = complex_of_real ((cmod z)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. z * conjugate z = complex_of_real ((cmod z)\\<^sup>2)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nz * conjugate z = complex_of_real ((cmod z)\\<^sup>2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nz * conjugate z = complex_of_real ((cmod z)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. z * conjugate z = complex_of_real ((cmod z)\\<^sup>2)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nz * conjugate z = complex_of_real ((cmod z)\\<^sup>2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1030, "file": "Projective_Measurements_Linear_Algebra_Complements", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.849971190859164, "lm_q2_score": 0.8376199633332891, "lm_q1q2_score": 0.7119528377218051}} {"text": "[STATEMENT]\nlemma ereal_Inf_cmult:\n assumes \"c>(0::real)\"\n shows \"Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}\n[PROOF STEP]\nhave \"(\\x::ereal. c * x) (Inf {x::ereal. P x}) = Inf ((\\x::ereal. c * x)`{x::ereal. P x})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ereal c * Inf {x. P x} = Inf ((*) (ereal c) ` {x. P x})\n[PROOF STEP]\napply (rule mono_bij_Inf)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. mono ((*) (ereal c))\n 2. bij ((*) (ereal c))\n[PROOF STEP]\napply (simp add: assms ereal_mult_left_mono less_imp_le mono_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij ((*) (ereal c))\n[PROOF STEP]\napply (rule bij_betw_byWitness[of _ \"\\x. (x::ereal) / c\"], auto simp add: assms ereal_mult_divide)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a. ereal c * a / ereal c = a\n[PROOF STEP]\nusing assms ereal_divide_eq\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < c\n\\?b \\ 0; \\?b\\ \\ \\\\ \\ (?a / ?b = ?c) = (?a = ?b * ?c)\n\ngoal (1 subgoal):\n 1. \\a. ereal c * a / ereal c = a\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\nereal c * Inf {x. P x} = Inf ((*) (ereal c) ` {x. P x})\n\ngoal (1 subgoal):\n 1. Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nereal c * Inf {x. P x} = Inf ((*) (ereal c) ` {x. P x})\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nereal c * Inf {x. P x} = Inf ((*) (ereal c) ` {x. P x})\n\ngoal (1 subgoal):\n 1. Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}\n[PROOF STEP]\nby (simp only: setcompr_eq_image[symmetric])\n[PROOF STATE]\nproof (state)\nthis:\nInf {ereal c * x |x. P x} = ereal c * Inf {x. P x}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 982, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199673867852, "lm_q2_score": 0.8499711699569787, "lm_q1q2_score": 0.7119528236590722}} {"text": "[STATEMENT]\ntheorem three_divides_nat:\n shows \"(3 dvd n) = (3 dvd sumdig n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (3 dvd n) = (3 dvd sumdig n)\n[PROOF STEP]\nproof (unfold sumdig_def)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (3 dvd n) = (3 dvd (\\xxxxxxxxxxxxxxxxxxxxxxxR v) = s *\\<^sub>R (f v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f (a + b) = f a + f b &&& f (a - b) = f a - f b) &&& f (0::'a) = (0::'b) &&& f (- a) = - f a &&& f (s *\\<^sub>R v) = s *\\<^sub>R f v\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (5 subgoals):\n 1. f (a + b) = f a + f b\n 2. f (a - b) = f a - f b\n 3. f (0::'a) = (0::'b)\n 4. f (- a) = - f a\n 5. f (s *\\<^sub>R v) = s *\\<^sub>R f v\n[PROOF STEP]\ninterpret f: bounded_linear f\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bounded_linear f\n[PROOF STEP]\nby fact\n[PROOF STATE]\nproof (state)\ngoal (5 subgoals):\n 1. f (a + b) = f a + f b\n 2. f (a - b) = f a - f b\n 3. f (0::'a) = (0::'b)\n 4. f (- a) = - f a\n 5. f (s *\\<^sub>R v) = s *\\<^sub>R f v\n[PROOF STEP]\nshow \"f (a + b) = f a + f b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (a + b) = f a + f b\n[PROOF STEP]\nby (rule f.add)\n[PROOF STATE]\nproof (state)\nthis:\nf (a + b) = f a + f b\n\ngoal (4 subgoals):\n 1. f (a - b) = f a - f b\n 2. f (0::'a) = (0::'b)\n 3. f (- a) = - f a\n 4. f (s *\\<^sub>R v) = s *\\<^sub>R f v\n[PROOF STEP]\nshow \"f (a - b) = f a - f b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (a - b) = f a - f b\n[PROOF STEP]\nby (rule f.diff)\n[PROOF STATE]\nproof (state)\nthis:\nf (a - b) = f a - f b\n\ngoal (3 subgoals):\n 1. f (0::'a) = (0::'b)\n 2. f (- a) = - f a\n 3. f (s *\\<^sub>R v) = s *\\<^sub>R f v\n[PROOF STEP]\nshow \"f 0 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (0::'a) = (0::'b)\n[PROOF STEP]\nby (rule f.zero)\n[PROOF STATE]\nproof (state)\nthis:\nf (0::'a) = (0::'b)\n\ngoal (2 subgoals):\n 1. f (- a) = - f a\n 2. f (s *\\<^sub>R v) = s *\\<^sub>R f v\n[PROOF STEP]\nshow \"f (- a) = - f a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (- a) = - f a\n[PROOF STEP]\nby (rule f.neg)\n[PROOF STATE]\nproof (state)\nthis:\nf (- a) = - f a\n\ngoal (1 subgoal):\n 1. f (s *\\<^sub>R v) = s *\\<^sub>R f v\n[PROOF STEP]\nshow \"f (s *\\<^sub>R v) = s *\\<^sub>R (f v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (s *\\<^sub>R v) = s *\\<^sub>R f v\n[PROOF STEP]\nby (rule f.scale)\n[PROOF STATE]\nproof (state)\nthis:\nf (s *\\<^sub>R v) = s *\\<^sub>R f v\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1241, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894661025424, "lm_q2_score": 0.7956581073313275, "lm_q1q2_score": 0.7119464930591579}} {"text": "[STATEMENT]\nlemma order_statistics_measurable:\n fixes X :: \"nat \\ 'a \\ ('b :: {linorder_topology, second_countable_topology})\"\n assumes \"n \\ 1\" \n assumes \"j < n\"\n assumes \"\\i. i < n \\ X i \\ measurable M borel\"\n shows \"(\\x. (sort (map (\\i. X i x) [0.. measurable M borel\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. sort (map (\\i. X i x) [0.. borel_measurable M\n[PROOF STEP]\napply (subst sort_map_eq_sort[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. map (sort_map (\\i. X i x) n) [0.. borel_measurable M\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ n\nj < n\n?i < n \\ X ?i \\ borel_measurable M\n\ngoal (1 subgoal):\n 1. (\\x. map (sort_map (\\i. X i x) n) [0.. borel_measurable M\n[PROOF STEP]\nby (simp add:order_statistics_measurable_aux del:sort_map.simps)", "meta": {"llama_tokens": 427, "file": "Median_Method_Median", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894717137996, "lm_q2_score": 0.7956581000631542, "lm_q1q2_score": 0.7119464910203153}} {"text": "[STATEMENT]\nlemma has_derivative_inverse_basic_x:\n fixes f :: \"'a::real_normed_vector \\ 'b::real_normed_vector\"\n assumes \"(f has_derivative f') (at x)\"\n and \"bounded_linear g'\"\n and \"g' \\ f' = id\"\n and \"continuous (at (f x)) g\"\n and \"g (f x) = x\"\n and \"open T\"\n and \"f x \\ T\"\n and \"\\y. y \\ T \\ f (g y) = y\"\n shows \"(g has_derivative g') (at (f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (g has_derivative g') (at (f x))\n[PROOF STEP]\nby (rule has_derivative_inverse_basic) (use assms in auto)", "meta": {"llama_tokens": 241, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894632969136, "lm_q2_score": 0.7956581000631542, "lm_q1q2_score": 0.7119464843233517}} {"text": "[STATEMENT]\nlemma convex_lower:\n assumes \"convex_on S f\"\n and \"x \\ S\"\n and \"y \\ S\"\n and \"0 \\ u\"\n and \"0 \\ v\"\n and \"u + v = 1\"\n shows \"f (u *\\<^sub>R x + v *\\<^sub>R y) \\ max (f x) (f y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (u *\\<^sub>R x + v *\\<^sub>R y) \\ max (f x) (f y)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. f (u *\\<^sub>R x + v *\\<^sub>R y) \\ max (f x) (f y)\n[PROOF STEP]\nlet ?m = \"max (f x) (f y)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. f (u *\\<^sub>R x + v *\\<^sub>R y) \\ max (f x) (f y)\n[PROOF STEP]\nhave \"u * f x + v * f y \\ u * max (f x) (f y) + v * max (f x) (f y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. u * f x + v * f y \\ u * max (f x) (f y) + v * max (f x) (f y)\n[PROOF STEP]\nusing assms(4,5)\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ u\n0 \\ v\n\ngoal (1 subgoal):\n 1. u * f x + v * f y \\ u * max (f x) (f y) + v * max (f x) (f y)\n[PROOF STEP]\nby (auto simp: mult_left_mono add_mono)\n[PROOF STATE]\nproof (state)\nthis:\nu * f x + v * f y \\ u * max (f x) (f y) + v * max (f x) (f y)\n\ngoal (1 subgoal):\n 1. f (u *\\<^sub>R x + v *\\<^sub>R y) \\ max (f x) (f y)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nu * f x + v * f y \\ u * max (f x) (f y) + v * max (f x) (f y)\n\ngoal (1 subgoal):\n 1. f (u *\\<^sub>R x + v *\\<^sub>R y) \\ max (f x) (f y)\n[PROOF STEP]\nhave \"\\ = max (f x) (f y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. u * max (f x) (f y) + v * max (f x) (f y) = max (f x) (f y)\n[PROOF STEP]\nusing assms(6)\n[PROOF STATE]\nproof (prove)\nusing this:\nu + v = 1\n\ngoal (1 subgoal):\n 1. u * max (f x) (f y) + v * max (f x) (f y) = max (f x) (f y)\n[PROOF STEP]\nby (simp add: distrib_right [symmetric])\n[PROOF STATE]\nproof (state)\nthis:\nu * max (f x) (f y) + v * max (f x) (f y) = max (f x) (f y)\n\ngoal (1 subgoal):\n 1. f (u *\\<^sub>R x + v *\\<^sub>R y) \\ max (f x) (f y)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nu * f x + v * f y \\ max (f x) (f y)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nu * f x + v * f y \\ max (f x) (f y)\n\ngoal (1 subgoal):\n 1. f (u *\\<^sub>R x + v *\\<^sub>R y) \\ max (f x) (f y)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu * f x + v * f y \\ max (f x) (f y)\nconvex_on S f\nx \\ S\ny \\ S\n0 \\ u\n0 \\ v\nu + v = 1\n\ngoal (1 subgoal):\n 1. f (u *\\<^sub>R x + v *\\<^sub>R y) \\ max (f x) (f y)\n[PROOF STEP]\nunfolding convex_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\nu * f x + v * f y \\ max (f x) (f y)\n\\x\\S. \\y\\S. \\u\\0. \\v\\0. u + v = 1 \\ f (u *\\<^sub>R x + v *\\<^sub>R y) \\ u * f x + v * f y\nx \\ S\ny \\ S\n0 \\ u\n0 \\ v\nu + v = 1\n\ngoal (1 subgoal):\n 1. f (u *\\<^sub>R x + v *\\<^sub>R y) \\ max (f x) (f y)\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nf (u *\\<^sub>R x + v *\\<^sub>R y) \\ max (f x) (f y)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1553, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.875787001374006, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7118986347741499}} {"text": "[STATEMENT]\nlemma card_ran_le_dom: \"finite (dom m) \\ card (ran m) \\ card (dom m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (dom m) \\ card (ran m) \\ card (dom m)\n[PROOF STEP]\nby(simp add: ran_alt_def card_image_le)", "meta": {"llama_tokens": 106, "file": "CryptHOL_Misc_CryptHOL", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869851639066, "lm_q2_score": 0.8128673201042492, "lm_q1q2_score": 0.7118986196123647}} {"text": "[STATEMENT]\nlemma fold_random_permutation_foldr:\n assumes \"finite A\"\n shows \"fold_random_permutation f x A =\n map_pmf (\\xs. foldr f xs x) (pmf_of_set (permutations_of_set A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = map_pmf (\\xs. foldr f xs x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = map_pmf (\\xs. foldr f xs x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nhave \"fold_random_permutation f x A =\n map_pmf (foldl (\\x y. f y x) x \\ rev) (pmf_of_set (permutations_of_set A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x \\ rev) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x \\ rev) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nby (subst fold_random_permutation_foldl [OF assms])\n (simp_all add: pmf.map_comp [symmetric] map_pmf_of_set_inj)\n[PROOF STATE]\nproof (state)\nthis:\nfold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x \\ rev) (pmf_of_set (permutations_of_set A))\n\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = map_pmf (\\xs. foldr f xs x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x \\ rev) (pmf_of_set (permutations_of_set A))\n\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = map_pmf (\\xs. foldr f xs x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nhave \"foldl (\\x y. f y x) x \\ rev = (\\xs. foldr f xs x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. foldl (\\x y. f y x) x \\ rev = (\\xs. foldr f xs x)\n[PROOF STEP]\nby (intro ext) (simp add: foldl_conv_foldr)\n[PROOF STATE]\nproof (state)\nthis:\nfoldl (\\x y. f y x) x \\ rev = (\\xs. foldr f xs x)\n\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = map_pmf (\\xs. foldr f xs x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nfold_random_permutation f x A = map_pmf (\\a. foldr f a x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfold_random_permutation f x A = map_pmf (\\a. foldr f a x) (pmf_of_set (permutations_of_set A))\n\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = map_pmf (\\xs. foldr f xs x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nfold_random_permutation f x A = map_pmf (\\xs. foldr f xs x) (pmf_of_set (permutations_of_set A))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1285, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869916479466, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.7118986189276532}} {"text": "[STATEMENT]\nlemma (in group) normal_subgroup_intersect:\n assumes \"M \\ G\" and \"N \\ G\"\n shows \"M \\ N \\ G\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M \\ N \\ G\n[PROOF STEP]\nusing assms subgroup_intersect is_group normal_inv_iff\n[PROOF STATE]\nproof (prove)\nusing this:\nM \\ G\nN \\ G\n\\subgroup ?H G; subgroup ?H' G\\ \\ subgroup (?H \\ ?H') G\nGroup.group G\n?N \\ G = (subgroup ?N G \\ (\\x\\carrier G. \\h\\?N. x \\ h \\ inv x \\ ?N))\n\ngoal (1 subgoal):\n 1. M \\ N \\ G\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 270, "file": "Jordan_Hoelder_SubgroupsAndNormalSubgroups", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869851639066, "lm_q2_score": 0.8128673087708699, "lm_q1q2_score": 0.7118986096867386}} {"text": "[STATEMENT]\ntheorem ln_lower_7_eq: \"0 < x \\\n ln_lower_7 x = (1/12)*(47*x^3 + 239*x^2 + 131*x + 3)*(x - 1) / (x*(x^3 + 12*x^2 + 18*x + 4))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ ln_lower_7 x = 1 / 12 * (47 * x ^ 3 + 239 * x\\<^sup>2 + 131 * x + 3) * (x - 1) / (x * (x ^ 3 + 12 * x\\<^sup>2 + 18 * x + 4))\n[PROOF STEP]\nunfolding ln_lower_7_def ln_upper_7_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ - ((3 * inverse x ^ 3 + 131 * (inverse x)\\<^sup>2 + 239 * inverse x + 47) * (inverse x - 1) / (12 * (4 * inverse x ^ 3 + 18 * (inverse x)\\<^sup>2 + 12 * inverse x + 1))) = 1 / 12 * (47 * x ^ 3 + 239 * x\\<^sup>2 + 131 * x + 3) * (x - 1) / (x * (x ^ 3 + 12 * x\\<^sup>2 + 18 * x + 4))\n[PROOF STEP]\nby (simp add: zero_less_mult_iff add_pos_pos dual_order.strict_implies_not_eq divide_simps)\n algebra", "meta": {"llama_tokens": 474, "file": "Special_Function_Bounds_Log_CF_Bounds", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297967961706, "lm_q2_score": 0.7905303112671294, "lm_q1q2_score": 0.7118961005666015}} {"text": "[STATEMENT]\nlemma proots_card_ball_plane_eq:\n defines \"q1\\[:\\,-1:]\" and \"q2\\[:\\,1:]\"\n assumes \"p\\0\"\n shows \"card (proots_within p (ball 0 1)) = card (proots_within (fcompose p q1 q2) {x. 0 < Im x})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (proots_within p (ball 0 1)) = card (proots_within (fcompose p q1 q2) {x. 0 < Im x})\n[PROOF STEP]\nunfolding q1_def q2_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (proots_within p (ball 0 1)) = card (proots_within (fcompose p [:\\, - 1:] [:\\, 1:]) {x. 0 < Im x})\n[PROOF STEP]\nproof (rule proots_card_fcompose_bij_eq[OF _ \\p\\0\\])\n[PROOF STATE]\nproof (state)\ngoal (5 subgoals):\n 1. bij_betw (\\x. poly [:\\, - 1:] x / poly [:\\, 1:] x) {x. 0 < Im x} (ball 0 1)\n 2. \\x\\{x. 0 < Im x}. poly [:\\, 1:] x \\ 0\n 3. max (degree [:\\, - 1:]) (degree [:\\, 1:]) \\ 1\n 4. \\c. [:\\, - 1:] \\ smult c [:\\, 1:]\n 5. infinite UNIV\n[PROOF STEP]\nshow \"\\x\\{x. 0 < Im x}. poly [:\\, 1:] x \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\{x. 0 < Im x}. poly [:\\, 1:] x \\ 0\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. 0 < Im x \\ \\ + x \\ 0\n[PROOF STEP]\nby (metis add_less_same_cancel2 imaginary_unit.simps(2) not_one_less_zero \n plus_complex.simps(2) zero_complex.simps(2))\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\{x. 0 < Im x}. poly [:\\, 1:] x \\ 0\n\ngoal (4 subgoals):\n 1. bij_betw (\\x. poly [:\\, - 1:] x / poly [:\\, 1:] x) {x. 0 < Im x} (ball 0 1)\n 2. max (degree [:\\, - 1:]) (degree [:\\, 1:]) \\ 1\n 3. \\c. [:\\, - 1:] \\ smult c [:\\, 1:]\n 4. infinite UNIV\n[PROOF STEP]\nqed (use bij_betw_plane_ball infinite_UNIV_char_0 in auto)", "meta": {"llama_tokens": 919, "file": "Count_Complex_Roots_Count_Line", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.863391617003942, "lm_q2_score": 0.8244619285331332, "lm_q1q2_score": 0.7118335176344103}} {"text": "[STATEMENT]\ntheorem sigma_Inter:\n \"(\\i::nat. a i \\ sigma A) \\ (\\i. a i) \\ sigma A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. a i \\ sigma A) \\ \\ (range a) \\ sigma A\n[PROOF STEP]\n(*<*)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. a i \\ sigma A) \\ \\ (range a) \\ sigma A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\i. a i \\ sigma A) \\ \\ (range a) \\ sigma A\n[PROOF STEP]\nassume \"\\i::nat. a i \\ sigma A\"\n[PROOF STATE]\nproof (state)\nthis:\na ?i \\ sigma A\n\ngoal (1 subgoal):\n 1. (\\i. a i \\ sigma A) \\ \\ (range a) \\ sigma A\n[PROOF STEP]\nhence \"\\i::nat. -(a i) \\ sigma A\"\n[PROOF STATE]\nproof (prove)\nusing this:\na ?i \\ sigma A\n\ngoal (1 subgoal):\n 1. \\i. - a i \\ sigma A\n[PROOF STEP]\nby (rule sigma.complement)\n[PROOF STATE]\nproof (state)\nthis:\n- a ?i \\ sigma A\n\ngoal (1 subgoal):\n 1. (\\i. a i \\ sigma A) \\ \\ (range a) \\ sigma A\n[PROOF STEP]\nhence \"(\\i. -(a i)) \\ sigma A\"\n[PROOF STATE]\nproof (prove)\nusing this:\n- a ?i \\ sigma A\n\ngoal (1 subgoal):\n 1. (\\i. - a i) \\ sigma A\n[PROOF STEP]\nby (rule sigma.Union)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i. - a i) \\ sigma A\n\ngoal (1 subgoal):\n 1. (\\i. a i \\ sigma A) \\ \\ (range a) \\ sigma A\n[PROOF STEP]\nhence \"-(\\i. -(a i)) \\ sigma A\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i. - a i) \\ sigma A\n\ngoal (1 subgoal):\n 1. - (\\i. - a i) \\ sigma A\n[PROOF STEP]\nby (rule sigma.complement)\n[PROOF STATE]\nproof (state)\nthis:\n- (\\i. - a i) \\ sigma A\n\ngoal (1 subgoal):\n 1. (\\i. a i \\ sigma A) \\ \\ (range a) \\ sigma A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n- (\\i. - a i) \\ sigma A\n\ngoal (1 subgoal):\n 1. (\\i. a i \\ sigma A) \\ \\ (range a) \\ sigma A\n[PROOF STEP]\nhave \"-(\\i. -(a i)) = (\\i. a i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (\\i. - a i) = \\ (range a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n- (\\i. - a i) = \\ (range a)\n\ngoal (1 subgoal):\n 1. (\\i. a i \\ sigma A) \\ \\ (range a) \\ sigma A\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ (range a) \\ sigma A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (range a) \\ sigma A\n\ngoal (1 subgoal):\n 1. \\ (range a) \\ sigma A\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\ (range a) \\ sigma A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1266, "file": "Integration_Sigma_Algebra", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916099737806, "lm_q2_score": 0.8244619306896956, "lm_q1q2_score": 0.7118335137002678}} {"text": "[STATEMENT]\nlemma integer_winding_number_eq:\n assumes \\: \"path \\\" and z: \"z \\ path_image \\\"\n shows \"winding_number \\ z \\ \\ \\ pathfinish \\ = pathstart \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (pathfinish \\ = pathstart \\)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (pathfinish \\ = pathstart \\)\n[PROOF STEP]\nobtain p where p: \"valid_path p\" \"z \\ path_image p\"\n \"pathstart p = pathstart \\\" \"pathfinish p = pathfinish \\\"\n and eq: \"contour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\p. \\valid_path p; z \\ path_image p; pathstart p = pathstart \\; pathfinish p = pathfinish \\; contour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing winding_number [OF assms, of 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 \\ \\p. winding_number_prop \\ z 1 p (winding_number \\ z)\n\ngoal (1 subgoal):\n 1. (\\p. \\valid_path p; z \\ path_image p; pathstart p = pathstart \\; pathfinish p = pathfinish \\; contour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\\ \\ thesis) \\ thesis\n[PROOF STEP]\nunfolding winding_number_prop_def\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 \\ \\p. valid_path p \\ z \\ path_image p \\ pathstart p = pathstart \\ \\ pathfinish p = pathfinish \\ \\ (\\t\\{0..1}. cmod (\\ t - p t) < 1) \\ contour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\n\ngoal (1 subgoal):\n 1. (\\p. \\valid_path p; z \\ path_image p; pathstart p = pathstart \\; pathfinish p = pathfinish \\; contour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nvalid_path p\nz \\ path_image p\npathstart p = pathstart \\\npathfinish p = pathfinish \\\ncontour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\n\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (pathfinish \\ = pathstart \\)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nvalid_path p\nz \\ path_image p\npathstart p = pathstart \\\npathfinish p = pathfinish \\\ncontour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\n[PROOF STEP]\nhave wneq: \"winding_number \\ z = winding_number p z\"\n[PROOF STATE]\nproof (prove)\nusing this:\nvalid_path p\nz \\ path_image p\npathstart p = pathstart \\\npathfinish p = pathfinish \\\ncontour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\n\ngoal (1 subgoal):\n 1. winding_number \\ z = winding_number p z\n[PROOF STEP]\nusing eq winding_number_valid_path\n[PROOF STATE]\nproof (prove)\nusing this:\nvalid_path p\nz \\ path_image p\npathstart p = pathstart \\\npathfinish p = pathfinish \\\ncontour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\ncontour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\n\\valid_path ?\\; ?z \\ path_image ?\\\\ \\ winding_number ?\\ ?z = 1 / (complex_of_real (2 * pi) * \\) * contour_integral ?\\ (\\w. 1 / (w - ?z))\n\ngoal (1 subgoal):\n 1. winding_number \\ z = winding_number p z\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\nwinding_number \\ z = winding_number p z\n\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (pathfinish \\ = pathstart \\)\n[PROOF STEP]\nhave iff: \"(winding_number \\ z \\ \\) \\ (exp (contour_integral p (\\w. 1 / (w - z))) = 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (exp (contour_integral p (\\w. 1 / (w - z))) = 1)\n[PROOF STEP]\nusing eq\n[PROOF STATE]\nproof (prove)\nusing this:\ncontour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\n\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (exp (contour_integral p (\\w. 1 / (w - z))) = 1)\n[PROOF STEP]\nby (simp add: exp_eq_1 complex_is_Int_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(winding_number \\ z \\ \\) = (exp (contour_integral p (\\w. 1 / (w - z))) = 1)\n\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (pathfinish \\ = pathstart \\)\n[PROOF STEP]\nhave \"\\ 0 \\ z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ 0 \\ z\n[PROOF STEP]\nby (metis pathstart_def pathstart_in_path_image z)\n[PROOF STATE]\nproof (state)\nthis:\n\\ 0 \\ z\n\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (pathfinish \\ = pathstart \\)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ 0 \\ z\n[PROOF STEP]\nhave \"exp (contour_integral p (\\w. 1 / (w - z))) = (\\ 1 - z) / (\\ 0 - z)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 0 \\ z\n\ngoal (1 subgoal):\n 1. exp (contour_integral p (\\w. 1 / (w - z))) = (\\ 1 - z) / (\\ 0 - z)\n[PROOF STEP]\nusing p winding_number_exp_integral(2) [of p 0 1 z]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 0 \\ z\nvalid_path p\nz \\ path_image p\npathstart p = pathstart \\\npathfinish p = pathfinish \\\n\\p piecewise_C1_differentiable_on {0..1}; 0 \\ 1; z \\ p ` {0..1}\\ \\ exp (- integral {0..1} (\\x. vector_derivative p (at x) / (p x - z))) * (p 1 - z) = p 0 - z\n\ngoal (1 subgoal):\n 1. exp (contour_integral p (\\w. 1 / (w - z))) = (\\ 1 - z) / (\\ 0 - z)\n[PROOF STEP]\nby (simp add: valid_path_def path_defs contour_integral_integral exp_minus field_split_simps)\n[PROOF STATE]\nproof (state)\nthis:\nexp (contour_integral p (\\w. 1 / (w - z))) = (\\ 1 - z) / (\\ 0 - z)\n\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (pathfinish \\ = pathstart \\)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nexp (contour_integral p (\\w. 1 / (w - z))) = (\\ 1 - z) / (\\ 0 - z)\n[PROOF STEP]\nhave \"winding_number p z \\ \\ \\ pathfinish p = pathstart p\"\n[PROOF STATE]\nproof (prove)\nusing this:\nexp (contour_integral p (\\w. 1 / (w - z))) = (\\ 1 - z) / (\\ 0 - z)\n\ngoal (1 subgoal):\n 1. (winding_number p z \\ \\) = (pathfinish p = pathstart p)\n[PROOF STEP]\nusing p wneq iff\n[PROOF STATE]\nproof (prove)\nusing this:\nexp (contour_integral p (\\w. 1 / (w - z))) = (\\ 1 - z) / (\\ 0 - z)\nvalid_path p\nz \\ path_image p\npathstart p = pathstart \\\npathfinish p = pathfinish \\\nwinding_number \\ z = winding_number p z\n(winding_number \\ z \\ \\) = (exp (contour_integral p (\\w. 1 / (w - z))) = 1)\n\ngoal (1 subgoal):\n 1. (winding_number p z \\ \\) = (pathfinish p = pathstart p)\n[PROOF STEP]\nby (auto simp: path_defs)\n[PROOF STATE]\nproof (state)\nthis:\n(winding_number p z \\ \\) = (pathfinish p = pathstart p)\n\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (pathfinish \\ = pathstart \\)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(winding_number p z \\ \\) = (pathfinish p = pathstart p)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(winding_number p z \\ \\) = (pathfinish p = pathstart p)\n\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (pathfinish \\ = pathstart \\)\n[PROOF STEP]\nusing p eq\n[PROOF STATE]\nproof (prove)\nusing this:\n(winding_number p z \\ \\) = (pathfinish p = pathstart p)\nvalid_path p\nz \\ path_image p\npathstart p = pathstart \\\npathfinish p = pathfinish \\\ncontour_integral p (\\w. 1 / (w - z)) = complex_of_real (2 * pi) * \\ * winding_number \\ z\n\ngoal (1 subgoal):\n 1. (winding_number \\ z \\ \\) = (pathfinish \\ = pathstart \\)\n[PROOF STEP]\nby (auto simp: winding_number_valid_path)\n[PROOF STATE]\nproof (state)\nthis:\n(winding_number \\ z \\ \\) = (pathfinish \\ = pathstart \\)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3641, "file": null, "length": 27, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110454379296, "lm_q2_score": 0.7981867705385763, "lm_q1q2_score": 0.7118317782887326}} {"text": "[STATEMENT]\nlemma iter_tensor_of_H_fst_neg:\n fixes n i j:: nat\n assumes \"i \\ 2^n \\ j \\ 2^n\" and \"i < 2^(n+1) \\ j < 2^(n+1)\"\n shows \"(H^\\<^sub>\\ (Suc n)) $$ (i,j) = -1/sqrt(2) * (H^\\<^sub>\\ n) $$ (i mod 2^n, j mod 2^n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (- 1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>n\\<^esub> j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (- 1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>n\\<^esub> j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)\n[PROOF STEP]\nhave \"(H^\\<^sub>\\ (Suc n)) $$ (i,j) = (-1)^(bip i (n+1) j)/(sqrt 2)^(n+1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real ((- 1) ^ i \\\\<^bsub>n + 1\\<^esub> j / sqrt 2 ^ (n + 1))\n[PROOF STEP]\nusing assms(2)\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ (n + 1) \\ j < 2 ^ (n + 1)\n\ngoal (1 subgoal):\n 1. Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real ((- 1) ^ i \\\\<^bsub>n + 1\\<^esub> j / sqrt 2 ^ (n + 1))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nMatrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real ((- 1) ^ i \\\\<^bsub>n + 1\\<^esub> j / sqrt 2 ^ (n + 1))\n\ngoal (1 subgoal):\n 1. Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (- 1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>n\\<^esub> j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nMatrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real ((- 1) ^ i \\\\<^bsub>n + 1\\<^esub> j / sqrt 2 ^ (n + 1))\n\ngoal (1 subgoal):\n 1. Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (- 1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>n\\<^esub> j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)\n[PROOF STEP]\nhave \"bip i (n+1) j = 1 + bip (i mod 2^n) n (j mod 2^n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i \\\\<^bsub>n + 1\\<^esub> j = 1 + i mod 2 ^ n \\\\<^bsub>n\\<^esub> j mod 2 ^ n\n[PROOF STEP]\nusing bitwise_inner_prod_fst_el_is_1 assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\2 ^ ?n \\ ?i \\ 2 ^ ?n \\ ?j; ?i < 2 ^ (?n + 1) \\ ?j < 2 ^ (?n + 1)\\ \\ ?i \\\\<^bsub>?n + 1\\<^esub> ?j = 1 + ?i mod 2 ^ ?n \\\\<^bsub>?n\\<^esub> ?j mod 2 ^ ?n\n2 ^ n \\ i \\ 2 ^ n \\ j\ni < 2 ^ (n + 1) \\ j < 2 ^ (n + 1)\n\ngoal (1 subgoal):\n 1. i \\\\<^bsub>n + 1\\<^esub> j = 1 + i mod 2 ^ n \\\\<^bsub>n\\<^esub> j mod 2 ^ n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ni \\\\<^bsub>n + 1\\<^esub> j = 1 + i mod 2 ^ n \\\\<^bsub>n\\<^esub> j mod 2 ^ n\n\ngoal (1 subgoal):\n 1. Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (- 1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>n\\<^esub> j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nMatrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real ((- 1) ^ i \\\\<^bsub>n + 1\\<^esub> j / sqrt 2 ^ (n + 1))\ni \\\\<^bsub>n + 1\\<^esub> j = 1 + i mod 2 ^ n \\\\<^bsub>n\\<^esub> j mod 2 ^ n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nMatrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real ((- 1) ^ i \\\\<^bsub>n + 1\\<^esub> j / sqrt 2 ^ (n + 1))\ni \\\\<^bsub>n + 1\\<^esub> j = 1 + i mod 2 ^ n \\\\<^bsub>n\\<^esub> j mod 2 ^ n\n\ngoal (1 subgoal):\n 1. Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (- 1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>n\\<^esub> j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nMatrix.mat (2 ^ Suc n) (2 ^ Suc n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>Suc n\\<^esub> j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (- 1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (\\x. complex_of_real (case x of (i, j) \\ (- 1) ^ i \\\\<^bsub>n\\<^esub> j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2980, "file": "Isabelle_Marries_Dirac_Deutsch_Jozsa", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110339361275, "lm_q2_score": 0.798186775339273, "lm_q1q2_score": 0.7118317733894605}} {"text": "[STATEMENT]\nlemma map_poly_times:\n assumes \"f 0 = 0\" and \"\\a b. f (a + b) = f a + f b\" and \"\\a b. f (a * b) = f a * f b\"\n shows \"map_poly f (p * q) = map_poly f p * map_poly f q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. map_poly f (p * q) = map_poly f p * map_poly f q\n[PROOF STEP]\nproof (induct p)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. map_poly f (0 * q) = map_poly f 0 * map_poly f q\n 2. \\a p. \\a \\ (0::'b) \\ p \\ 0; map_poly f (p * q) = map_poly f p * map_poly f q\\ \\ map_poly f (pCons a p * q) = map_poly f (pCons a p) * map_poly f q\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. map_poly f (0 * q) = map_poly f 0 * map_poly f q\n 2. \\a p. \\a \\ (0::'b) \\ p \\ 0; map_poly f (p * q) = map_poly f p * map_poly f q\\ \\ map_poly f (pCons a p * q) = map_poly f (pCons a p) * map_poly f q\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. map_poly f (0 * q) = map_poly f 0 * map_poly f q\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmap_poly f (0 * q) = map_poly f 0 * map_poly f q\n\ngoal (1 subgoal):\n 1. \\a p. \\a \\ (0::'b) \\ p \\ 0; map_poly f (p * q) = map_poly f p * map_poly f q\\ \\ map_poly f (pCons a p * q) = map_poly f (pCons a p) * map_poly f q\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a p. \\a \\ (0::'b) \\ p \\ 0; map_poly f (p * q) = map_poly f p * map_poly f q\\ \\ map_poly f (pCons a p * q) = map_poly f (pCons a p) * map_poly f q\n[PROOF STEP]\ncase (pCons c p)\n[PROOF STATE]\nproof (state)\nthis:\nc \\ (0::'b) \\ p \\ 0\nmap_poly f (p * q) = map_poly f p * map_poly f q\n\ngoal (1 subgoal):\n 1. \\a p. \\a \\ (0::'b) \\ p \\ 0; map_poly f (p * q) = map_poly f p * map_poly f q\\ \\ map_poly f (pCons a p * q) = map_poly f (pCons a p) * map_poly f q\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. map_poly f (pCons c p * q) = map_poly f (pCons c p) * map_poly f q\n[PROOF STEP]\nby (simp add: assms map_poly_plus map_poly_smult map_poly_pCons pCons)\n[PROOF STATE]\nproof (state)\nthis:\nmap_poly f (pCons c p * q) = map_poly f (pCons c p) * map_poly f q\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1127, "file": "Nullstellensatz_Nullstellensatz", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527869325345, "lm_q2_score": 0.8354835371034368, "lm_q1q2_score": 0.7117925278715246}} {"text": "[STATEMENT]\ntheorem bf_ifex_rel_single: \"single_valued bf_ifex_rel\" \"single_valued (bf_ifex_rel\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. single_valued bf_ifex_rel &&& single_valued (bf_ifex_rel\\)\n[PROOF STEP]\nunfolding single_valued_def bf_ifex_rel_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x y. (x, y) \\ {(a, b). (\\ass. a ass = val_ifex b ass) \\ ro_ifex b} \\ (\\z. (x, z) \\ {(a, b). (\\ass. a ass = val_ifex b ass) \\ ro_ifex b} \\ y = z) &&& \\x y. (x, y) \\ {(a, b). (\\ass. a ass = val_ifex b ass) \\ ro_ifex b}\\ \\ (\\z. (x, z) \\ {(a, b). (\\ass. a ass = val_ifex b ass) \\ ro_ifex b}\\ \\ y = z)\n[PROOF STEP]\nusing ro_ifex_unique\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ro_ifex ?x; ro_ifex ?y; \\ass. val_ifex ?x ass = val_ifex ?y ass\\ \\ ?x = ?y\n\ngoal (1 subgoal):\n 1. \\x y. (x, y) \\ {(a, b). (\\ass. a ass = val_ifex b ass) \\ ro_ifex b} \\ (\\z. (x, z) \\ {(a, b). (\\ass. a ass = val_ifex b ass) \\ ro_ifex b} \\ y = z) &&& \\x y. (x, y) \\ {(a, b). (\\ass. a ass = val_ifex b ass) \\ ro_ifex b}\\ \\ (\\z. (x, z) \\ {(a, b). (\\ass. a ass = val_ifex b ass) \\ ro_ifex b}\\ \\ y = z)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 666, "file": "ROBDD_BDT", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972684083609, "lm_q2_score": 0.8175744828610096, "lm_q1q2_score": 0.7117781114991732}} {"text": "[STATEMENT]\nlemma Inf_pres_downset_var: \"(\\x \\ X. \\x) = \\(\\X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (\\ ` X) = \\ (\\ X)\n[PROOF STEP]\nunfolding downset_prop\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\X. {y. y \\ x}) = {y. y \\ \\ X}\n[PROOF STEP]\nby (safe, simp_all add: le_Inf_iff)", "meta": {"llama_tokens": 184, "file": "Order_Lattice_Props_Order_Lattice_Props_Loc", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972751232809, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.7117781073148027}} {"text": "[STATEMENT]\nlemma rsum_component_le:\n fixes f :: \"'a::euclidean_space \\ 'b::euclidean_space\"\n assumes p: \"p tagged_division_of (cbox a b)\"\n and \"\\x. x \\ cbox a b \\ (f x)\\i \\ (g x)\\i\"\n shows \"(\\(x, K)\\p. content K *\\<^sub>R f x) \\ i \\ (\\(x, K)\\p. content K *\\<^sub>R g x) \\ i\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(x, K)\\p. content K *\\<^sub>R f x) \\ i \\ (\\(x, K)\\p. content K *\\<^sub>R g x) \\ i\n[PROOF STEP]\nunfolding inner_sum_left\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\p. (case x of (x, K) \\ content K *\\<^sub>R f x) \\ i) \\ (\\x\\p. (case x of (x, K) \\ content K *\\<^sub>R g x) \\ i)\n[PROOF STEP]\nproof (rule sum_mono, clarify)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a b. (a, b) \\ p \\ content b *\\<^sub>R f a \\ i \\ content b *\\<^sub>R g a \\ i\n[PROOF STEP]\nfix x K\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a b. (a, b) \\ p \\ content b *\\<^sub>R f a \\ i \\ content b *\\<^sub>R g a \\ i\n[PROOF STEP]\nassume ab: \"(x, K) \\ p\"\n[PROOF STATE]\nproof (state)\nthis:\n(x, K) \\ p\n\ngoal (1 subgoal):\n 1. \\a b. (a, b) \\ p \\ content b *\\<^sub>R f a \\ i \\ content b *\\<^sub>R g a \\ i\n[PROOF STEP]\nwith p\n[PROOF STATE]\nproof (chain)\npicking this:\np tagged_division_of cbox a b\n(x, K) \\ p\n[PROOF STEP]\nobtain u v where K: \"K = cbox u v\"\n[PROOF STATE]\nproof (prove)\nusing this:\np tagged_division_of cbox a b\n(x, K) \\ p\n\ngoal (1 subgoal):\n 1. (\\u v. K = cbox u v \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nK = cbox u v\n\ngoal (1 subgoal):\n 1. \\a b. (a, b) \\ p \\ content b *\\<^sub>R f a \\ i \\ content b *\\<^sub>R g a \\ i\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nK = cbox u v\n[PROOF STEP]\nshow \"(content K *\\<^sub>R f x) \\ i \\ (content K *\\<^sub>R g x) \\ i\"\n[PROOF STATE]\nproof (prove)\nusing this:\nK = cbox u v\n\ngoal (1 subgoal):\n 1. content K *\\<^sub>R f x \\ i \\ content K *\\<^sub>R g x \\ i\n[PROOF STEP]\nby (metis ab assms inner_scaleR_left measure_nonneg mult_left_mono tag_in_interval)\n[PROOF STATE]\nproof (state)\nthis:\ncontent K *\\<^sub>R f x \\ i \\ content K *\\<^sub>R g x \\ i\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1120, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972650509008, "lm_q2_score": 0.8175744695262775, "lm_q1q2_score": 0.7117780971450183}} {"text": "[STATEMENT]\nlemma sums_logderiv_zeta:\n assumes \"Re s > 1\"\n shows \"(\\p. if prime p then of_real (ln (real p)) / (of_nat p powr s - 1) else 0) sums\n -(deriv zeta s / zeta s)\" (is \"?f sums _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\p. if prime p then complex_of_real (ln (real p)) / (of_nat p powr s - 1) else 0) sums - (deriv zeta s / zeta s)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\p. if prime p then complex_of_real (ln (real p)) / (of_nat p powr s - 1) else 0) sums - (deriv zeta s / zeta s)\n[PROOF STEP]\nnote * = eval_fds_logderiv_zeta[OF assms]\n[PROOF STATE]\nproof (state)\nthis:\n(\\p. complex_of_real (ln (real p)) / (of_nat p powr s - 1)) abs_summable_on {p. prime p}\nderiv zeta s / zeta s = - (\\\\<^sub>ap | prime p. complex_of_real (ln (real p)) / (of_nat p powr s - 1))\n\ngoal (1 subgoal):\n 1. (\\p. if prime p then complex_of_real (ln (real p)) / (of_nat p powr s - 1) else 0) sums - (deriv zeta s / zeta s)\n[PROOF STEP]\nfrom sums_infsetsum_nat[OF *(1)] and *(2)\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n. if n \\ {p. prime p} then complex_of_real (ln (real n)) / (of_nat n powr s - 1) else 0) sums (\\\\<^sub>ap | prime p. complex_of_real (ln (real p)) / (of_nat p powr s - 1))\nderiv zeta s / zeta s = - (\\\\<^sub>ap | prime p. complex_of_real (ln (real p)) / (of_nat p powr s - 1))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. if n \\ {p. prime p} then complex_of_real (ln (real n)) / (of_nat n powr s - 1) else 0) sums (\\\\<^sub>ap | prime p. complex_of_real (ln (real p)) / (of_nat p powr s - 1))\nderiv zeta s / zeta s = - (\\\\<^sub>ap | prime p. complex_of_real (ln (real p)) / (of_nat p powr s - 1))\n\ngoal (1 subgoal):\n 1. (\\p. if prime p then complex_of_real (ln (real p)) / (of_nat p powr s - 1) else 0) sums - (deriv zeta s / zeta s)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\p. if prime p then complex_of_real (ln (real p)) / (of_nat p powr s - 1) else 0) sums - (deriv zeta s / zeta s)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 930, "file": "Prime_Number_Theorem_Prime_Number_Theorem_Library", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970842359877, "lm_q2_score": 0.8080672135527631, "lm_q1q2_score": 0.711743245563973}} {"text": "[STATEMENT]\nlemma gen_fib_recurrence: \"gen_fib a b (Suc (Suc n)) = gen_fib a b n + gen_fib a b (Suc n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. gen_fib a b (Suc (Suc n)) = gen_fib a b n + gen_fib a b (Suc n)\n[PROOF STEP]\nby (induct a b n rule: gen_fib.induct) simp_all", "meta": {"llama_tokens": 136, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970842359876, "lm_q2_score": 0.8080672066194946, "lm_q1q2_score": 0.7117432394571701}} {"text": "[STATEMENT]\nlemma dist_vector_1:\n fixes x :: \"'a::real_normed_vector^1\"\n shows \"dist x y = dist (x$1) (y$1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist x y = dist (x $ 1) (y $ 1)\n[PROOF STEP]\nby (simp add: dist_norm norm_vector_1)", "meta": {"llama_tokens": 112, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.880797071719777, "lm_q2_score": 0.8080672112416737, "lm_q1q2_score": 0.7117432334144326}} {"text": "[STATEMENT]\nlemma diag_PAQ_dvd_Gcd_minors:\n fixes P A Q::\"'a::{semiring_Gcd,comm_ring_1} mat\"\n assumes A: \"A \\ carrier_mat m n\"\n and P: \"P \\ carrier_mat m m\"\n and Q: \"Q \\ carrier_mat n n\"\n and inv_P: \"invertible_mat P\"\n and inv_Q: \"invertible_mat Q\"\n and SNF: \"Smith_normal_form_mat (P*A*Q)\"\n shows \"(\\i=0..i = 0..i = 0..i=0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0.. n = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (m * n = 0) = (m = 0 \\ n = 0)\n[PROOF STEP]\nby (rule mult_eq_0_iff)", "meta": {"llama_tokens": 106, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032941962904955, "lm_q2_score": 0.7879311956428947, "lm_q1q2_score": 0.7117336761004577}} {"text": "[STATEMENT]\nlemma ld_ld_1_less:\n \"\\x > 0; y > 0 \\ \\ 1 + log 2 x + log 2 y < 2 * log 2 (x+y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < x; 0 < y\\ \\ 1 + log 2 x + log 2 y < 2 * log 2 (x + y)\n[PROOF STEP]\nusing ld_sum_inequality[of x y]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < x; 0 < y\\ \\ log 2 x + log 2 y + 2 \\ 2 * log 2 (x + y)\n\ngoal (1 subgoal):\n 1. \\0 < x; 0 < y\\ \\ 1 + log 2 x + log 2 y < 2 * log 2 (x + y)\n[PROOF STEP]\nby linarith", "meta": {"llama_tokens": 288, "file": "Amortized_Complexity_Lemmas_log", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.888758793492457, "lm_q2_score": 0.8006920092299292, "lm_q1q2_score": 0.7116220640822432}} {"text": "[STATEMENT]\nlemma NSDERIV_inverse_fun:\n \"NSDERIV f x :> d \\ f x \\ 0 \\\n NSDERIV (\\x. inverse (f x)) x :> (- (d * inverse (f x ^ Suc (Suc 0))))\"\n for x :: \"'a::{real_normed_field}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\NSDERIV f x :> d; f x \\ (0::'a)\\ \\ NSDERIV (\\x. inverse (f x)) x :> - (d * inverse (f x ^ Suc (Suc 0)))\n[PROOF STEP]\nby (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: power_Suc)", "meta": {"llama_tokens": 220, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.8006920068519376, "lm_q1q2_score": 0.7116220619687821}} {"text": "[STATEMENT]\nlemma homotopic_loops_linear:\n fixes g h :: \"real \\ 'a::real_normed_vector\"\n assumes \"path g\" \"path h\" \"pathfinish g = pathstart g\" \"pathfinish h = pathstart h\"\n \"\\t x. t \\ {0..1} \\ closed_segment (g t) (h t) \\ S\"\n shows \"homotopic_loops S g h\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. homotopic_loops S g h\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\npath g\npath h\npathfinish g = pathstart g\npathfinish h = pathstart h\n?t \\ {0..1} \\ closed_segment (g ?t) (h ?t) \\ S\n\ngoal (1 subgoal):\n 1. homotopic_loops S g h\n[PROOF STEP]\nunfolding path_defs homotopic_loops_def homotopic_with_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous_on {0..1} g\ncontinuous_on {0..1} h\ng 1 = g 0\nh 1 = h 0\n?t \\ {0..1} \\ closed_segment (g ?t) (h ?t) \\ S\n\ngoal (1 subgoal):\n 1. \\ha. continuous_map (prod_topology (top_of_set {0..1}) (top_of_set {0..1})) (top_of_set S) ha \\ (\\x. ha (0, x) = g x) \\ (\\x. ha (1, x) = h x) \\ (\\t\\{0..1}. ha (t, 1) = ha (t, 0))\n[PROOF STEP]\napply (rule_tac x=\"\\y. ((1 - (fst y)) *\\<^sub>R g(snd y) + (fst y) *\\<^sub>R h(snd y))\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\continuous_on {0..1} g; continuous_on {0..1} h; g 1 = g 0; h 1 = h 0; \\t. t \\ {0..1} \\ closed_segment (g t) (h t) \\ S\\ \\ continuous_map (prod_topology (top_of_set {0..1}) (top_of_set {0..1})) (top_of_set S) (\\y. (1 - fst y) *\\<^sub>R g (snd y) + fst y *\\<^sub>R h (snd y)) \\ (\\x. (1 - fst (0, x)) *\\<^sub>R g (snd (0, x)) + fst (0, x) *\\<^sub>R h (snd (0, x)) = g x) \\ (\\x. (1 - fst (1, x)) *\\<^sub>R g (snd (1, x)) + fst (1, x) *\\<^sub>R h (snd (1, x)) = h x) \\ (\\t\\{0..1}. (1 - fst (t, 1)) *\\<^sub>R g (snd (t, 1)) + fst (t, 1) *\\<^sub>R h (snd (t, 1)) = (1 - fst (t, 0)) *\\<^sub>R g (snd (t, 0)) + fst (t, 0) *\\<^sub>R h (snd (t, 0)))\n[PROOF STEP]\nby (force simp: closed_segment_def intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])", "meta": {"llama_tokens": 1026, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587993853654, "lm_q2_score": 0.8006919997179627, "lm_q1q2_score": 0.7116220603468039}} {"text": "[STATEMENT]\nlemma mset_bubblesort:\n \"mset (bubblesort xs) = mset xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mset (bubblesort xs) = mset xs\n[PROOF STEP]\napply(induction xs rule: bubblesort.induct)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. mset (bubblesort []) = mset []\n 2. \\x. mset (bubblesort [x]) = mset [x]\n 3. \\v vb vc. (\\x21 x22. bubble_min (v # vb # vc) = x21 # x22 \\ mset (bubblesort x22) = mset x22) \\ mset (bubblesort (v # vb # vc)) = mset (v # vb # vc)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. mset (bubblesort [x]) = mset [x]\n 2. \\v vb vc. (\\x21 x22. bubble_min (v # vb # vc) = x21 # x22 \\ mset (bubblesort x22) = mset x22) \\ mset (bubblesort (v # vb # vc)) = mset (v # vb # vc)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\v vb vc. (\\x21 x22. bubble_min (v # vb # vc) = x21 # x22 \\ mset (bubblesort x22) = mset x22) \\ mset (bubblesort (v # vb # vc)) = mset (v # vb # vc)\n[PROOF STEP]\nby(auto split: list.splits if_splits dest: bubble_minD_mset)", "meta": {"llama_tokens": 552, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587993853655, "lm_q2_score": 0.8006919973399709, "lm_q1q2_score": 0.7116220582333428}} {"text": "[STATEMENT]\nlemma inj_on_mult_units: \n assumes 1: \"coprime x (q::nat)\" shows \"inj_on (\\ b. x*b mod q) ({..b. x * b mod q) ({..xa y. \\xa < q; 0 < xa; y < q; x * xa mod q = x * y mod q; 0 < y\\ \\ xa = y\n[PROOF STEP]\nusing 1\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime x q\n\ngoal (1 subgoal):\n 1. \\xa y. \\xa < q; 0 < xa; y < q; x * xa mod q = x * y mod q; 0 < y\\ \\ xa = y\n[PROOF STEP]\nby(simp only: inj_mult)", "meta": {"llama_tokens": 324, "file": "Multi_Party_Computation_Uniform_Sampling", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.8006919925839875, "lm_q1q2_score": 0.711622049288016}} {"text": "[STATEMENT]\ntheorem interval_integral_substitution:\n assumes integrable: \"set_integrable lborel {g a..g b} f\"\n assumes derivg: \"\\x. x \\ {a..b} \\ (g has_real_derivative g' x) (at x)\"\n assumes contg': \"continuous_on {a..b} g'\"\n assumes derivg_nonneg: \"\\x. x \\ {a..b} \\ g' x \\ 0\"\n assumes \"a \\ b\"\n shows \"set_integrable lborel {a..b} (\\x. f (g x) * g' x)\"\n and \"(LBINT x=g a..g b. f x) = (LBINT x=a..b. f (g x) * g' x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_integrable lborel {a..b} (\\x. f (g x) * g' x) &&& interval_lebesgue_integral lborel (ereal (g a)) (ereal (g b)) f = LBINT x=ereal a..ereal b. f (g x) * g' x\n[PROOF STEP]\napply (rule integral_substitution[OF assms], simp, simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. interval_lebesgue_integral lborel (ereal (g a)) (ereal (g b)) f = LBINT x=ereal a..ereal b. f (g x) * g' x\n[PROOF STEP]\napply (subst (1 2) interval_integral_Icc, fact)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. g a \\ g b\n 2. set_lebesgue_integral lborel {g a..g b} f = LBINT x:{a..b}. f (g x) * g' x\n[PROOF STEP]\napply (rule deriv_nonneg_imp_mono[OF derivg derivg_nonneg], simp, simp, fact)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_lebesgue_integral lborel {g a..g b} f = LBINT x:{a..b}. f (g x) * g' x\n[PROOF STEP]\nusing integral_substitution(2)[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\x. x \\ {a..b} \\ x \\ {a..b}; \\x. x \\ {a..b} \\ x \\ {a..b}\\ \\ LBINT x. f x * indicat_real {g a..g b} x = LBINT x. f (g x) * g' x * indicat_real {a..b} x\n\ngoal (1 subgoal):\n 1. set_lebesgue_integral lborel {g a..g b} f = LBINT x:{a..b}. f (g x) * g' x\n[PROOF STEP]\napply (simp add: mult.commute set_lebesgue_integral_def)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 876, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587846530938, "lm_q2_score": 0.8006919997179627, "lm_q1q2_score": 0.7116220485507918}} {"text": "[STATEMENT]\nlemma (in prob_space) erlang_distributed_variance:\n assumes [arith]: \"0 < l\" and \"distributed M lborel X (erlang_density k l)\"\n shows \"variance X = (k + 1) / l\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. expectation (\\x. (X x - expectation X)\\<^sup>2) = real (k + 1) / l\\<^sup>2\n[PROOF STEP]\nproof (subst variance_eq)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. integrable M X\n 2. integrable M (\\x. (X x)\\<^sup>2)\n 3. expectation (\\x. (X x)\\<^sup>2) - (expectation X)\\<^sup>2 = real (k + 1) / l\\<^sup>2\n[PROOF STEP]\nshow \"integrable M X\" \"integrable M (\\x. (X x)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integrable M X &&& integrable M (\\x. (X x)\\<^sup>2)\n[PROOF STEP]\nusing erlang_ith_moment_integrable[OF assms, of 1] erlang_ith_moment_integrable[OF assms, of 2]\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable M (\\x. X x ^ 1)\nintegrable M (\\x. (X x)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. integrable M X &&& integrable M (\\x. (X x)\\<^sup>2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nintegrable M X\nintegrable M (\\x. (X x)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. expectation (\\x. (X x)\\<^sup>2) - (expectation X)\\<^sup>2 = real (k + 1) / l\\<^sup>2\n[PROOF STEP]\nshow \"expectation (\\x. (X x)\\<^sup>2) - (expectation X)\\<^sup>2 = real (k + 1) / l\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. expectation (\\x. (X x)\\<^sup>2) - (expectation X)\\<^sup>2 = real (k + 1) / l\\<^sup>2\n[PROOF STEP]\nusing erlang_ith_moment[OF assms, of 1] erlang_ith_moment[OF assms, of 2]\n[PROOF STATE]\nproof (prove)\nusing this:\nexpectation (\\x. X x ^ 1) = fact (k + 1) / (fact k * l ^ 1)\nexpectation (\\x. (X x)\\<^sup>2) = fact (k + 2) / (fact k * l\\<^sup>2)\n\ngoal (1 subgoal):\n 1. expectation (\\x. (X x)\\<^sup>2) - (expectation X)\\<^sup>2 = real (k + 1) / l\\<^sup>2\n[PROOF STEP]\nby simp (auto simp: power2_eq_square field_simps of_nat_Suc)\n[PROOF STATE]\nproof (state)\nthis:\nexpectation (\\x. (X x)\\<^sup>2) - (expectation X)\\<^sup>2 = real (k + 1) / l\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 979, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127566694178, "lm_q2_score": 0.8333245994514084, "lm_q1q2_score": 0.7115865059179906}} {"text": "[STATEMENT]\nlemma trace_measurement2:\n assumes m: \"measurement n 2 M\" and dA: \"A \\ carrier_mat n n\"\n shows \"trace ((M 0) * A * adjoint (M 0)) + trace ((M 1) * A * adjoint (M 1)) = trace A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. trace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A\n[PROOF STEP]\nfrom m\n[PROOF STATE]\nproof (chain)\npicking this:\nmeasurement n 2 M\n[PROOF STEP]\nhave dM0: \"M 0 \\ carrier_mat n n\" and dM1: \"M 1 \\ carrier_mat n n\" and id: \"adjoint (M 0) * (M 0) + adjoint (M 1) * (M 1) = 1\\<^sub>m n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmeasurement n 2 M\n\ngoal (1 subgoal):\n 1. M 0 \\ carrier_mat n n &&& M 1 \\ carrier_mat n n &&& adjoint (M 0) * M 0 + adjoint (M 1) * M 1 = 1\\<^sub>m n\n[PROOF STEP]\nusing measurement_def measurement_id2\n[PROOF STATE]\nproof (prove)\nusing this:\nmeasurement n 2 M\nmeasurement ?d ?n ?M = ((\\j carrier_mat ?d ?d) \\ matrix_sum ?d (\\j. adjoint (?M j) * ?M j) ?n = 1\\<^sub>m ?d)\nmeasurement ?d 2 ?M \\ adjoint (?M 0) * ?M 0 + adjoint (?M 1) * ?M 1 = 1\\<^sub>m ?d\n\ngoal (1 subgoal):\n 1. M 0 \\ carrier_mat n n &&& M 1 \\ carrier_mat n n &&& adjoint (M 0) * M 0 + adjoint (M 1) * M 1 = 1\\<^sub>m n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nM 0 \\ carrier_mat n n\nM 1 \\ carrier_mat n n\nadjoint (M 0) * M 0 + adjoint (M 1) * M 1 = 1\\<^sub>m n\n\ngoal (1 subgoal):\n 1. trace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A\n[PROOF STEP]\nhave \"trace (M 1 * A * adjoint (M 1)) + trace (M 0 * A * adjoint (M 0))\n = trace ((adjoint (M 0) * M 0 + adjoint (M 1) * M 1) * A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (M 1 * A * adjoint (M 1)) + trace (M 0 * A * adjoint (M 0)) = trace ((adjoint (M 0) * M 0 + adjoint (M 1) * M 1) * A)\n[PROOF STEP]\nusing dM0 dM1 dA\n[PROOF STATE]\nproof (prove)\nusing this:\nM 0 \\ carrier_mat n n\nM 1 \\ carrier_mat n n\nA \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. trace (M 1 * A * adjoint (M 1)) + trace (M 0 * A * adjoint (M 0)) = trace ((adjoint (M 0) * M 0 + adjoint (M 1) * M 1) * A)\n[PROOF STEP]\nby (mat_assoc n)\n[PROOF STATE]\nproof (state)\nthis:\ntrace (M 1 * A * adjoint (M 1)) + trace (M 0 * A * adjoint (M 0)) = trace ((adjoint (M 0) * M 0 + adjoint (M 1) * M 1) * A)\n\ngoal (1 subgoal):\n 1. trace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ntrace (M 1 * A * adjoint (M 1)) + trace (M 0 * A * adjoint (M 0)) = trace ((adjoint (M 0) * M 0 + adjoint (M 1) * M 1) * A)\n\ngoal (1 subgoal):\n 1. trace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A\n[PROOF STEP]\nhave \"\\ = trace (1\\<^sub>m n * A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace ((adjoint (M 0) * M 0 + adjoint (M 1) * M 1) * A) = trace (1\\<^sub>m n * A)\n[PROOF STEP]\nusing id\n[PROOF STATE]\nproof (prove)\nusing this:\nadjoint (M 0) * M 0 + adjoint (M 1) * M 1 = 1\\<^sub>m n\n\ngoal (1 subgoal):\n 1. trace ((adjoint (M 0) * M 0 + adjoint (M 1) * M 1) * A) = trace (1\\<^sub>m n * A)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace ((adjoint (M 0) * M 0 + adjoint (M 1) * M 1) * A) = trace (1\\<^sub>m n * A)\n\ngoal (1 subgoal):\n 1. trace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ntrace ((adjoint (M 0) * M 0 + adjoint (M 1) * M 1) * A) = trace (1\\<^sub>m n * A)\n\ngoal (1 subgoal):\n 1. trace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A\n[PROOF STEP]\nhave \"\\ = trace A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (1\\<^sub>m n * A) = trace A\n[PROOF STEP]\nusing dA\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. trace (1\\<^sub>m n * A) = trace A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace (1\\<^sub>m n * A) = trace A\n\ngoal (1 subgoal):\n 1. trace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ntrace (M 1 * A * adjoint (M 1)) + trace (M 0 * A * adjoint (M 0)) = trace A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntrace (M 1 * A * adjoint (M 1)) + trace (M 0 * A * adjoint (M 0)) = trace A\n\ngoal (1 subgoal):\n 1. trace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A\n[PROOF STEP]\nusing dA dM0 dM1 local.id state_sig.trace_measure2_id\n[PROOF STATE]\nproof (prove)\nusing this:\ntrace (M 1 * A * adjoint (M 1)) + trace (M 0 * A * adjoint (M 0)) = trace A\nA \\ carrier_mat n n\nM 0 \\ carrier_mat n n\nM 1 \\ carrier_mat n n\nadjoint (M 0) * M 0 + adjoint (M 1) * M 1 = 1\\<^sub>m n\n\\?M0.0 \\ carrier_mat ?n ?n; ?M1.0 \\ carrier_mat ?n ?n; adjoint ?M0.0 * ?M0.0 + adjoint ?M1.0 * ?M1.0 = 1\\<^sub>m ?n; ?A \\ carrier_mat ?n ?n\\ \\ trace (?M0.0 * ?A * adjoint ?M0.0) + trace (?M1.0 * ?A * adjoint ?M1.0) = trace ?A\n\ngoal (1 subgoal):\n 1. trace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ntrace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2590, "file": "QHLProver_Quantum_Hoare", "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333245911726382, "lm_q2_score": 0.8539127585282744, "lm_q1q2_score": 0.711586500397674}} {"text": "[STATEMENT]\nlemma arctan_series':\n assumes \"\\x\\ \\ 1\"\n shows \"(\\k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1))) sums arctan x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) sums arctan x\n[PROOF STEP]\nusing summable_arctan_series[OF assms] arctan_series[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\narctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n\ngoal (1 subgoal):\n 1. (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) sums arctan x\n[PROOF STEP]\nby (simp add: sums_iff)", "meta": {"llama_tokens": 340, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.8333246035907933, "lm_q1q2_score": 0.7115865001584777}} {"text": "[STATEMENT]\nlemma arctan_series':\n assumes \"\\x\\ \\ 1\"\n shows \"(\\k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1))) sums arctan x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) sums arctan x\n[PROOF STEP]\nusing summable_arctan_series[OF assms] arctan_series[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\narctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n\ngoal (1 subgoal):\n 1. (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) sums arctan x\n[PROOF STEP]\nby (simp add: sums_iff)", "meta": {"llama_tokens": 340, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.8333245932423308, "lm_q1q2_score": 0.7115864913217936}} {"text": "[STATEMENT]\nlemma arctan_series':\n assumes \"\\x\\ \\ 1\"\n shows \"(\\k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1))) sums arctan x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) sums arctan x\n[PROOF STEP]\nusing summable_arctan_series[OF assms] arctan_series[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\narctan x = (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))\n\ngoal (1 subgoal):\n 1. (\\k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) sums arctan x\n[PROOF STEP]\nby (simp add: sums_iff)", "meta": {"llama_tokens": 340, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.8333245932423308, "lm_q1q2_score": 0.7115864913217936}} {"text": "[STATEMENT]\nlemma floor_zero [simp]: \"\\0\\ = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nusing floor_of_int [of 0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\of_int 0\\ = 0\n\ngoal (1 subgoal):\n 1. \\0::'a\\ = 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 156, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127380808499, "lm_q2_score": 0.8333245870332531, "lm_q1q2_score": 0.7115864798236586}} {"text": "[STATEMENT]\nlemma (in strategic_space_2p) U\\<^sub>B_cnj_times_U\\<^sub>B:\n shows \"U\\<^sub>B\\<^sup>\\ * U\\<^sub>B = 1\\<^sub>m 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]] = 1\\<^sub>m 2\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (1\\<^sub>m 2); j < dim_col (1\\<^sub>m 2)\\ \\ (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) $$ (i, j) = 1\\<^sub>m 2 $$ (i, j)\n 2. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_row (1\\<^sub>m 2)\n 3. dim_col (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_col (1\\<^sub>m 2)\n[PROOF STEP]\nfix i j\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (1\\<^sub>m 2); j < dim_col (1\\<^sub>m 2)\\ \\ (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) $$ (i, j) = 1\\<^sub>m 2 $$ (i, j)\n 2. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_row (1\\<^sub>m 2)\n 3. dim_col (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_col (1\\<^sub>m 2)\n[PROOF STEP]\nassume a0:\"i < dim_row (1\\<^sub>m 2)\" and a1:\"j < dim_col (1\\<^sub>m 2)\"\n[PROOF STATE]\nproof (state)\nthis:\ni < dim_row (1\\<^sub>m 2)\nj < dim_col (1\\<^sub>m 2)\n\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (1\\<^sub>m 2); j < dim_col (1\\<^sub>m 2)\\ \\ (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) $$ (i, j) = 1\\<^sub>m 2 $$ (i, j)\n 2. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_row (1\\<^sub>m 2)\n 3. dim_col (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_col (1\\<^sub>m 2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni < dim_row (1\\<^sub>m 2)\nj < dim_col (1\\<^sub>m 2)\n[PROOF STEP]\nshow \"(U\\<^sub>B\\<^sup>\\ * U\\<^sub>B) $$ (i,j) = 1\\<^sub>m 2 $$ (i,j)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_row (1\\<^sub>m 2)\nj < dim_col (1\\<^sub>m 2)\n\ngoal (1 subgoal):\n 1. (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) $$ (i, j) = 1\\<^sub>m 2 $$ (i, j)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i < dim_row (1\\<^sub>m 2); j < dim_col (1\\<^sub>m 2)\\ \\ (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) $$ (i, j) = 1\\<^sub>m 2 $$ (i, j)\n[PROOF STEP]\nhave \"i\\{0,1} \\ j\\{0,1}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i \\ {0, 1} \\ j \\ {0, 1}\n[PROOF STEP]\nusing a0 a1 mat_of_cols_list_def\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_row (1\\<^sub>m 2)\nj < dim_col (1\\<^sub>m 2)\nTensor.mat_of_cols_list ?nr ?cs = Matrix.mat ?nr (length ?cs) (\\(i, j). ?cs ! j ! i)\n\ngoal (1 subgoal):\n 1. i \\ {0, 1} \\ j \\ {0, 1}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ni \\ {0, 1} \\ j \\ {0, 1}\n\ngoal (1 subgoal):\n 1. \\i < dim_row (1\\<^sub>m 2); j < dim_col (1\\<^sub>m 2)\\ \\ (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) $$ (i, j) = 1\\<^sub>m 2 $$ (i, j)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ni \\ {0, 1} \\ j \\ {0, 1}\n\ngoal (1 subgoal):\n 1. (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) $$ (i, j) = 1\\<^sub>m 2 $$ (i, j)\n[PROOF STEP]\nusing mat_of_cols_list_def cos_sin_squared_add_cpx hermite_cnj_of_U\\<^sub>B exp_sin_cos_squared_add[of \"\\\\<^sub>B\" \"\\\\<^sub>B / 2\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ni \\ {0, 1} \\ j \\ {0, 1}\nTensor.mat_of_cols_list ?nr ?cs = Matrix.mat ?nr (length ?cs) (\\(i, j). ?cs ! j ! i)\ncomplex_of_real (cos (?\\ / 2)) * complex_of_real (cos (?\\ / 2)) - \\ * complex_of_real (sin (?\\ / 2)) * (\\ * complex_of_real (sin (?\\ / 2))) = 1\nTensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ = Tensor.mat_of_cols_list 2 [exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [sin (\\\\<^sub>B / 2)], [complex_of_real (- sin (\\\\<^sub>B / 2)), exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\nexp (- (\\ * complex_of_real \\\\<^sub>B)) * complex_of_real (cos (\\\\<^sub>B / 2)) * (exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))) + complex_of_real (sin (\\\\<^sub>B / 2) * sin (\\\\<^sub>B / 2)) = 1\n\ngoal (1 subgoal):\n 1. (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) $$ (i, j) = 1\\<^sub>m 2 $$ (i, j)\n[PROOF STEP]\nby (auto simp add: set_2 algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) $$ (i, j) = 1\\<^sub>m 2 $$ (i, j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) $$ (i, j) = 1\\<^sub>m 2 $$ (i, j)\n\ngoal (2 subgoals):\n 1. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_row (1\\<^sub>m 2)\n 2. dim_col (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_col (1\\<^sub>m 2)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_row (1\\<^sub>m 2)\n 2. dim_col (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_col (1\\<^sub>m 2)\n[PROOF STEP]\nshow \"dim_row (U\\<^sub>B\\<^sup>\\ * U\\<^sub>B) = dim_row (1\\<^sub>m 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_row (1\\<^sub>m 2)\n[PROOF STEP]\nusing mat_of_cols_list_def\n[PROOF STATE]\nproof (prove)\nusing this:\nTensor.mat_of_cols_list ?nr ?cs = Matrix.mat ?nr (length ?cs) (\\(i, j). ?cs ! j ! i)\n\ngoal (1 subgoal):\n 1. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_row (1\\<^sub>m 2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_row (1\\<^sub>m 2)\n\ngoal (1 subgoal):\n 1. dim_col (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_col (1\\<^sub>m 2)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dim_col (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_col (1\\<^sub>m 2)\n[PROOF STEP]\nshow \"dim_col (U\\<^sub>B\\<^sup>\\ * U\\<^sub>B) = dim_col (1\\<^sub>m 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_col (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_col (1\\<^sub>m 2)\n[PROOF STEP]\nusing mat_of_cols_list_def\n[PROOF STATE]\nproof (prove)\nusing this:\nTensor.mat_of_cols_list ?nr ?cs = Matrix.mat ?nr (length ?cs) (\\(i, j). ?cs ! j ! i)\n\ngoal (1 subgoal):\n 1. dim_col (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_col (1\\<^sub>m 2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_col (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = dim_col (1\\<^sub>m 2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 10065, "file": "Isabelle_Marries_Dirac_Quantum_Prisoners_Dilemma", "length": 22, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9219218391455084, "lm_q2_score": 0.7718434978390747, "lm_q1q2_score": 0.7115793770603019}} {"text": "[STATEMENT]\nlemma locally_translation:\n fixes P :: \"'a :: real_normed_vector set \\ bool\"\n shows \"(\\S. P ((+) a ` S) = P S) \\ locally P ((+) a ` S) = locally P S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\S. P ((+) a ` S) = P S) \\ locally P ((+) a ` S) = locally P S\n[PROOF STEP]\nusing homeomorphism_locally [OF homeomorphism_translation]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\S T. homeomorphism S T ((+) (- ?a1)) ((+) ?a1) \\ ?P S = ?Q T) \\ locally ?P ((+) ?a1 ` ?T) = locally ?Q ?T\n\ngoal (1 subgoal):\n 1. (\\S. P ((+) a ` S) = P S) \\ locally P ((+) a ` S) = locally P S\n[PROOF STEP]\nby (metis (full_types) homeomorphism_image2)", "meta": {"llama_tokens": 287, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9059898254600902, "lm_q2_score": 0.7853085859124002, "lm_q1q2_score": 0.7114815886830856}} {"text": "[STATEMENT]\nlemma integral_0_iff:\n fixes f :: \"'a \\ real\"\n shows \"integrable M f \\ (\\x. \\f x\\ \\M) = 0 \\ emeasure M {x\\space M. f x \\ 0} = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integrable M f \\ (LINT x|M. \\f x\\ = 0) = (emeasure M {x \\ space M. f x \\ 0} = 0)\n[PROOF STEP]\nusing integral_norm_eq_0_iff[of M f]\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable M f \\ (LINT x|M. norm (f x) = 0) = (emeasure M {x \\ space M. f x \\ 0} = 0)\n\ngoal (1 subgoal):\n 1. integrable M f \\ (LINT x|M. \\f x\\ = 0) = (emeasure M {x \\ space M. f x \\ 0} = 0)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 331, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898254600902, "lm_q2_score": 0.7853085758631159, "lm_q1q2_score": 0.7114815795785363}} {"text": "[STATEMENT]\nlemma contour_integral_bound_linepath:\n shows\n \"\\f contour_integrable_on (linepath a b);\n 0 \\ B; \\x. x \\ closed_segment a b \\ norm(f x) \\ B\\\n \\ norm(contour_integral (linepath a b) f) \\ B*norm(b - a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f contour_integrable_on linepath a b; 0 \\ B; \\x. x \\ closed_segment a b \\ cmod (f x) \\ B\\ \\ cmod (contour_integral (linepath a b) f) \\ B * cmod (b - a)\n[PROOF STEP]\nusing has_contour_integral_bound_linepath [of f]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\(f has_contour_integral ?i) (linepath ?a ?b); 0 \\ ?B; \\x. x \\ closed_segment ?a ?b \\ cmod (f x) \\ ?B\\ \\ cmod ?i \\ ?B * cmod (?b - ?a)\n\ngoal (1 subgoal):\n 1. \\f contour_integrable_on linepath a b; 0 \\ B; \\x. x \\ closed_segment a b \\ cmod (f x) \\ B\\ \\ cmod (contour_integral (linepath a b) f) \\ B * cmod (b - a)\n[PROOF STEP]\nby (auto simp: has_contour_integral_integral)", "meta": {"llama_tokens": 483, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898153067649, "lm_q2_score": 0.785308580887758, "lm_q1q2_score": 0.7114815761573176}} {"text": "[STATEMENT]\nlemma exp_of_minus_half_pi: \n fixes x:: real\n assumes \"x = pi/2\"\n shows \"exp (-(\\ * complex_of_real x)) = -\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp (- (\\ * complex_of_real x)) = - \\\n[PROOF STEP]\nusing assms cis_conv_exp cis_minus_pi_half\n[PROOF STATE]\nproof (prove)\nusing this:\nx = pi / 2\ncis ?b = exp (\\ * complex_of_real ?b)\ncis (- (pi / 2)) = - \\\n\ngoal (1 subgoal):\n 1. exp (- (\\ * complex_of_real x)) = - \\\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 222, "file": "Isabelle_Marries_Dirac_Basics", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240825770433, "lm_q2_score": 0.8221891348788759, "lm_q1q2_score": 0.7113778399303883}} {"text": "[STATEMENT]\nlemma emeasure_point_measure:\n assumes A: \"finite {a\\X. 0 < f a}\" \"X \\ A\"\n shows \"emeasure (point_measure A f) X = (\\a|a\\X \\ 0 < f a. f a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. emeasure (point_measure A f) X = (\\a | a \\ X \\ 0 < f a. f a)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. emeasure (point_measure A f) X = (\\a | a \\ X \\ 0 < f a. f a)\n[PROOF STEP]\nhave \"{a. (a \\ X \\ a \\ A \\ 0 < f a) \\ a \\ X} = {a\\X. 0 < f a}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {a. (a \\ X \\ a \\ A \\ 0 < f a) \\ a \\ X} = {a \\ X. 0 < f a}\n[PROOF STEP]\nusing \\X \\ A\\\n[PROOF STATE]\nproof (prove)\nusing this:\nX \\ A\n\ngoal (1 subgoal):\n 1. {a. (a \\ X \\ a \\ A \\ 0 < f a) \\ a \\ X} = {a \\ X. 0 < f a}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{a. (a \\ X \\ a \\ A \\ 0 < f a) \\ a \\ X} = {a \\ X. 0 < f a}\n\ngoal (1 subgoal):\n 1. emeasure (point_measure A f) X = (\\a | a \\ X \\ 0 < f a. f a)\n[PROOF STEP]\nwith A\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite {a \\ X. 0 < f a}\nX \\ A\n{a. (a \\ X \\ a \\ A \\ 0 < f a) \\ a \\ X} = {a \\ X. 0 < f a}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {a \\ X. 0 < f a}\nX \\ A\n{a. (a \\ X \\ a \\ A \\ 0 < f a) \\ a \\ X} = {a \\ X. 0 < f a}\n\ngoal (1 subgoal):\n 1. emeasure (point_measure A f) X = (\\a | a \\ X \\ 0 < f a. f a)\n[PROOF STEP]\nby (simp add: emeasure_density nn_integral_count_space point_measure_def indicator_def of_bool_def)\n[PROOF STATE]\nproof (state)\nthis:\nemeasure (point_measure A f) X = (\\a | a \\ X \\ 0 < f a. f a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 921, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8558511469672594, "lm_q2_score": 0.8311430415844384, "lm_q1q2_score": 0.7113347254338982}} {"text": "[STATEMENT]\nlemma tendsto_null_sum:\n fixes f :: \"'a \\ 'b \\ 'c::topological_comm_monoid_add\"\n assumes \"\\i. i \\ I \\ ((\\x. f x i) \\ 0) F\"\n shows \"((\\i. sum (f i) I) \\ 0) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\i. sum (f i) I) \\ (0::'c)) F\n[PROOF STEP]\nusing tendsto_sum [of I \"\\x y. f y x\" \"\\x. 0\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i. i \\ I \\ ((\\y. f y i) \\ (0::'c)) ?F) \\ ((\\x. sum (f x) I) \\ (\\i\\I. (0::'c))) ?F\n?i \\ I \\ ((\\x. f x ?i) \\ (0::'c)) F\n\ngoal (1 subgoal):\n 1. ((\\i. sum (f i) I) \\ (0::'c)) F\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 365, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314798554444, "lm_q2_score": 0.8031738057795402, "lm_q1q2_score": 0.7113160061936635}} {"text": "[STATEMENT]\nlemma cross7_triple1: \"(x \\\\<^sub>7 y) \\ z = (y \\\\<^sub>7 z) \\ x\" \nand cross7_triple2: \"(x \\\\<^sub>7 y) \\ z = x \\ (y \\\\<^sub>7 z) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\\\<^sub>7 y \\ z = y \\\\<^sub>7 z \\ x &&& x \\\\<^sub>7 y \\ z = x \\ y \\\\<^sub>7 z\n[PROOF STEP]\nby (simp_all add: cross7_def inner_vec_def sum_7 vec_eq_iff algebra_simps)", "meta": {"llama_tokens": 199, "file": "Octonions_Cross_Product_7", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314858927011, "lm_q2_score": 0.8031737987125613, "lm_q1q2_score": 0.7113160047838909}} {"text": "[STATEMENT]\nlemma exp_inv_limit_0_right:\n \"((\\(t::real). exp(-inverse t)) \\ 0) (at_right 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\t. exp (- inverse t)) \\ 0) (at_right 0)\n[PROOF STEP]\napply (rule filterlim_compose[where g = exp])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (exp \\ 0) ?F2.0\n 2. LIM x at_right 0. - inverse x :> ?F2.0\n[PROOF STEP]\napply (rule exp_at_bot)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LIM x at_right 0. - inverse x :> at_bot\n[PROOF STEP]\napply (rule filterlim_compose[where g = uminus])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. filterlim uminus at_bot ?F2.4\n 2. filterlim inverse ?F2.4 (at_right 0)\n[PROOF STEP]\napply (rule filterlim_uminus_at_bot_at_top)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. filterlim inverse at_top (at_right 0)\n[PROOF STEP]\nby (rule filterlim_inverse_at_top_right)", "meta": {"llama_tokens": 404, "file": "Smooth_Manifolds_Bump_Function", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314798554445, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7113159999349246}} {"text": "[STATEMENT]\nlemma tendsto_null_sum:\n fixes f :: \"'a \\ 'b \\ 'c::topological_comm_monoid_add\"\n assumes \"\\i. i \\ I \\ ((\\x. f x i) \\ 0) F\"\n shows \"((\\i. sum (f i) I) \\ 0) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\i. sum (f i) I) \\ (0::'c)) F\n[PROOF STEP]\nusing tendsto_sum [of I \"\\x y. f y x\" \"\\x. 0\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i. i \\ I \\ ((\\y. f y i) \\ (0::'c)) ?F) \\ ((\\x. sum (f x) I) \\ (\\i\\I. (0::'c))) ?F\n?i \\ I \\ ((\\x. f x ?i) \\ (0::'c)) F\n\ngoal (1 subgoal):\n 1. ((\\i. sum (f i) I) \\ (0::'c)) F\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 365, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314798554444, "lm_q2_score": 0.803173791645582, "lm_q1q2_score": 0.7113159936761851}} {"text": "[STATEMENT]\nlemma sum_eq_0_iff:\n fixes f :: \"_ \\ 'a :: {comm_monoid_add,ordered_ab_group_add}\"\n shows \"finite A \\ (\\i. i \\ A \\ 0 \\ f i) \\ (\\i\\A. f i) = 0 \\ (\\i\\A. f i = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; \\i. i \\ A \\ (0::'a) \\ f i\\ \\ (sum f A = (0::'a)) = (\\i\\A. f i = (0::'a))\n[PROOF STEP]\napply rule\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\finite A; \\i. i \\ A \\ (0::'a) \\ f i; sum f A = (0::'a)\\ \\ \\i\\A. f i = (0::'a)\n 2. \\finite A; \\i. i \\ A \\ (0::'a) \\ f i; \\i\\A. f i = (0::'a)\\ \\ sum f A = (0::'a)\n[PROOF STEP]\napply (blast intro: sum_nonneg_0)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; \\i. i \\ A \\ (0::'a) \\ f i; \\i\\A. f i = (0::'a)\\ \\ sum f A = (0::'a)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 541, "file": "Probabilistic_Noninterference_Trace_Based", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314768368161, "lm_q2_score": 0.8031737940012418, "lm_q1q2_score": 0.7113159933379485}} {"text": "[STATEMENT]\nlemma power_inject_exp_less_one [simp]:\n \"\\0 < a; (a::'a::{linordered_field}) < 1 \\ \n \\ a ^ m = a ^ n \\ m = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(0::'a) < a; a < (1::'a)\\ \\ (a ^ m = a ^ n) = (m = n)\n[PROOF STEP]\nby (metis less_irrefl nat_neq_iff power_strict_decreasing)", "meta": {"llama_tokens": 171, "file": "Real_Power_RatPower", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314798554444, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7113159915899387}} {"text": "[STATEMENT]\nlemma complex_eq_0: \"z=0 \\ (Re z)\\<^sup>2 + (Im z)\\<^sup>2 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (z = 0) = ((Re z)\\<^sup>2 + (Im z)\\<^sup>2 = 0)\n[PROOF STEP]\nby (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)", "meta": {"llama_tokens": 131, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314677809303, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.711315990236991}} {"text": "[STATEMENT]\nlemma plus_mtx3: \"mtx \n ([a\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3] # \n [a\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3] # \n [a\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3] # []) + mtx \n ([b\\<^sub>1\\<^sub>1, b\\<^sub>1\\<^sub>2, b\\<^sub>1\\<^sub>3] # \n [b\\<^sub>2\\<^sub>1, b\\<^sub>2\\<^sub>2, b\\<^sub>2\\<^sub>3] # \n [b\\<^sub>3\\<^sub>1, b\\<^sub>3\\<^sub>2, b\\<^sub>3\\<^sub>3] # []) = (mtx \n ([a\\<^sub>1\\<^sub>1+b\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2+b\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3+b\\<^sub>1\\<^sub>3] # \n [a\\<^sub>2\\<^sub>1+b\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2+b\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3+b\\<^sub>2\\<^sub>3] # \n [a\\<^sub>3\\<^sub>1+b\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2+b\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3+b\\<^sub>3\\<^sub>3] # [])::3 sq_mtx)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mtx [[a\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3], [a\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3], [a\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3]] + mtx [[b\\<^sub>1\\<^sub>1, b\\<^sub>1\\<^sub>2, b\\<^sub>1\\<^sub>3], [b\\<^sub>2\\<^sub>1, b\\<^sub>2\\<^sub>2, b\\<^sub>2\\<^sub>3], [b\\<^sub>3\\<^sub>1, b\\<^sub>3\\<^sub>2, b\\<^sub>3\\<^sub>3]] = mtx [[a\\<^sub>1\\<^sub>1 + b\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2 + b\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3 + b\\<^sub>1\\<^sub>3], [a\\<^sub>2\\<^sub>1 + b\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2 + b\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3 + b\\<^sub>2\\<^sub>3], [a\\<^sub>3\\<^sub>1 + b\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2 + b\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3 + b\\<^sub>3\\<^sub>3]]\n[PROOF STEP]\nby (subst sq_mtx_eq_iff) simp", "meta": {"llama_tokens": 863, "file": "Matrices_for_ODEs_SQ_MTX", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314617436727, "lm_q2_score": 0.803173801068221, "lm_q1q2_score": 0.7113159874742704}} {"text": "[STATEMENT]\nlemma aff_dim_halfspace_le:\n fixes a :: \"'a::euclidean_space\"\n shows \"aff_dim {x. a \\ x \\ r} =\n (if a = 0 \\ r < 0 then -1 else DIM('a))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. aff_dim {x. a \\ x \\ r} = (if a = (0::'a) \\ r < 0 then - 1 else int DIM('a))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. aff_dim {x. a \\ x \\ r} = (if a = (0::'a) \\ r < 0 then - 1 else int DIM('a))\n[PROOF STEP]\nhave \"int (DIM('a)) = aff_dim (UNIV::'a set)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. int DIM('a) = aff_dim UNIV\n[PROOF STEP]\nby (simp)\n[PROOF STATE]\nproof (state)\nthis:\nint DIM('a) = aff_dim UNIV\n\ngoal (1 subgoal):\n 1. aff_dim {x. a \\ x \\ r} = (if a = (0::'a) \\ r < 0 then - 1 else int DIM('a))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nint DIM('a) = aff_dim UNIV\n[PROOF STEP]\nhave \"aff_dim (affine hull {x. a \\ x \\ r}) = DIM('a)\" if \"(a = 0 \\ r \\ 0)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nint DIM('a) = aff_dim UNIV\n\ngoal (1 subgoal):\n 1. aff_dim (affine hull {x. a \\ x \\ r}) = int DIM('a)\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\nint DIM('a) = aff_dim UNIV\na = (0::'a) \\ 0 \\ r\n\ngoal (1 subgoal):\n 1. aff_dim (affine hull {x. a \\ x \\ r}) = int DIM('a)\n[PROOF STEP]\nby (simp add: affine_hull_halfspace_le not_less)\n[PROOF STATE]\nproof (state)\nthis:\na = (0::'a) \\ 0 \\ r \\ aff_dim (affine hull {x. a \\ x \\ r}) = int DIM('a)\n\ngoal (1 subgoal):\n 1. aff_dim {x. a \\ x \\ r} = (if a = (0::'a) \\ r < 0 then - 1 else int DIM('a))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na = (0::'a) \\ 0 \\ r \\ aff_dim (affine hull {x. a \\ x \\ r}) = int DIM('a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na = (0::'a) \\ 0 \\ r \\ aff_dim (affine hull {x. a \\ x \\ r}) = int DIM('a)\n\ngoal (1 subgoal):\n 1. aff_dim {x. a \\ x \\ r} = (if a = (0::'a) \\ r < 0 then - 1 else int DIM('a))\n[PROOF STEP]\nby (force)\n[PROOF STATE]\nproof (state)\nthis:\naff_dim {x. a \\ x \\ r} = (if a = (0::'a) \\ r < 0 then - 1 else int DIM('a))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1104, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767906859264, "lm_q2_score": 0.8104789018037399, "lm_q1q2_score": 0.7111764256733998}} {"text": "[STATEMENT]\nlemma box_midpoint:\n fixes a :: \"'a::euclidean_space\"\n assumes \"box a b \\ {}\"\n shows \"((1/2) *\\<^sub>R (a + b)) \\ box a b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1 / 2) *\\<^sub>R (a + b) \\ box a b\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (1 / 2) *\\<^sub>R (a + b) \\ box a b\n[PROOF STEP]\nhave \"a \\ i < ((1 / 2) *\\<^sub>R (a + b)) \\ i \\ ((1 / 2) *\\<^sub>R (a + b)) \\ i < b \\ i\" if \"i \\ Basis\" for i\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ i < (1 / 2) *\\<^sub>R (a + b) \\ i \\ (1 / 2) *\\<^sub>R (a + b) \\ i < b \\ i\n[PROOF STEP]\nusing assms that\n[PROOF STATE]\nproof (prove)\nusing this:\nbox a b \\ {}\ni \\ Basis\n\ngoal (1 subgoal):\n 1. a \\ i < (1 / 2) *\\<^sub>R (a + b) \\ i \\ (1 / 2) *\\<^sub>R (a + b) \\ i < b \\ i\n[PROOF STEP]\nby (auto simp: inner_add_left box_ne_empty)\n[PROOF STATE]\nproof (state)\nthis:\n?i \\ Basis \\ a \\ ?i < (1 / 2) *\\<^sub>R (a + b) \\ ?i \\ (1 / 2) *\\<^sub>R (a + b) \\ ?i < b \\ ?i\n\ngoal (1 subgoal):\n 1. (1 / 2) *\\<^sub>R (a + b) \\ box a b\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n?i \\ Basis \\ a \\ ?i < (1 / 2) *\\<^sub>R (a + b) \\ ?i \\ (1 / 2) *\\<^sub>R (a + b) \\ ?i < b \\ ?i\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n?i \\ Basis \\ a \\ ?i < (1 / 2) *\\<^sub>R (a + b) \\ ?i \\ (1 / 2) *\\<^sub>R (a + b) \\ ?i < b \\ ?i\n\ngoal (1 subgoal):\n 1. (1 / 2) *\\<^sub>R (a + b) \\ box a b\n[PROOF STEP]\nunfolding mem_box\n[PROOF STATE]\nproof (prove)\nusing this:\n?i \\ Basis \\ a \\ ?i < (1 / 2) *\\<^sub>R (a + b) \\ ?i \\ (1 / 2) *\\<^sub>R (a + b) \\ ?i < b \\ ?i\n\ngoal (1 subgoal):\n 1. \\i\\Basis. a \\ i < (1 / 2) *\\<^sub>R (a + b) \\ i \\ (1 / 2) *\\<^sub>R (a + b) \\ i < b \\ i\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(1 / 2) *\\<^sub>R (a + b) \\ box a b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1058, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767746654974, "lm_q2_score": 0.8104789040926008, "lm_q1q2_score": 0.7111764146976024}} {"text": "[STATEMENT]\nlemma sum_of_bool_eq_card:\n assumes \"finite S\"\n shows \"(\\a \\ S. of_bool (P a)) = real (card {a \\ S . P a })\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a\\S. of_bool (P a)) = real (card {a \\ S. P a})\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\a\\S. of_bool (P a)) = real (card {a \\ S. P a})\n[PROOF STEP]\nhave \"(\\a \\ S. of_bool (P a) :: real) = (\\a \\ {x\\S. P x}. 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a\\S. of_bool (P a)) = (\\a\\{x \\ S. P x}. 1)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\n\ngoal (1 subgoal):\n 1. (\\a\\S. of_bool (P a)) = (\\a\\{x \\ S. P x}. 1)\n[PROOF STEP]\nby (intro sum.mono_neutral_cong_right) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\a\\S. of_bool (P a)) = (\\a\\{x \\ S. P x}. 1)\n\ngoal (1 subgoal):\n 1. (\\a\\S. of_bool (P a)) = real (card {a \\ S. P a})\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\a\\S. of_bool (P a)) = (\\a\\{x \\ S. P x}. 1)\n\ngoal (1 subgoal):\n 1. (\\a\\S. of_bool (P a)) = real (card {a \\ S. P a})\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\a\\S. of_bool (P a)) = real (card {a \\ S. P a})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 678, "file": "Probabilistic_Prime_Tests_Algebraic_Auxiliaries", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473779969194, "lm_q2_score": 0.8152324915965392, "lm_q1q2_score": 0.7111659265021366}} {"text": "[STATEMENT]\nlemma Sup_downset_adj_var: \"(Sup (X::'a::complete_lattice set) \\ y) = (X \\ \\y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ X \\ y) = (X \\ \\ y)\n[PROOF STEP]\nusing Sup_downset_adj adj_def\n[PROOF STATE]\nproof (prove)\nusing this:\nSup \\ \\\n?f \\ ?g = (\\x y. (?f x \\ y) = (x \\ ?g y))\n\ngoal (1 subgoal):\n 1. (\\ X \\ y) = (X \\ \\ y)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 221, "file": "Order_Lattice_Props_Galois_Connections", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473746782094, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7111659257546343}} {"text": "[STATEMENT]\nlemma hyperbola_method_sqrt:\n fixes f g :: \"nat \\ 'a :: comm_ring\"\n assumes \"x \\ 0\"\n shows \"sum_upto (dirichlet_prod f g) x = \n sum_upto (\\n. f n * sum_upto g (x / real n)) (sqrt x) +\n sum_upto (\\n. sum_upto f (x / real n) * g n) (sqrt x) -\n sum_upto f (sqrt x) * sum_upto g (sqrt x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_upto (dirichlet_prod f g) x = sum_upto (\\n. f n * sum_upto g (x / real n)) (sqrt x) + sum_upto (\\n. sum_upto f (x / real n) * g n) (sqrt x) - sum_upto f (sqrt x) * sum_upto g (sqrt x)\n[PROOF STEP]\nusing assms hyperbola_method[of \"sqrt x\" \"sqrt x\" x]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ x\n\\0 \\ sqrt x; 0 \\ sqrt x; sqrt x * sqrt x = x\\ \\ sum_upto (dirichlet_prod ?f ?g) x = sum_upto (\\n. ?f n * sum_upto ?g (x / real n)) (sqrt x) + sum_upto (\\n. sum_upto ?f (x / real n) * ?g n) (sqrt x) - sum_upto ?f (sqrt x) * sum_upto ?g (sqrt x)\n\ngoal (1 subgoal):\n 1. sum_upto (dirichlet_prod f g) x = sum_upto (\\n. f n * sum_upto g (x / real n)) (sqrt x) + sum_upto (\\n. sum_upto f (x / real n) * g n) (sqrt x) - sum_upto f (sqrt x) * sum_upto g (sqrt x)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 566, "file": "Dirichlet_Series_Arithmetic_Summatory", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9149009619539554, "lm_q2_score": 0.7772998560157665, "lm_q1q2_score": 0.7111523859954958}} {"text": "[STATEMENT]\nlemma connected_component_intermediate_subset:\n \"\\connected_component_set U a \\ T; T \\ U\\\n \\ connected_component_set T a = connected_component_set U a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\connected_component_set U a \\ T; T \\ U\\ \\ connected_component_set T a = connected_component_set U a\n[PROOF STEP]\napply (case_tac \"a \\ U\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\connected_component_set U a \\ T; T \\ U; a \\ U\\ \\ connected_component_set T a = connected_component_set U a\n 2. \\connected_component_set U a \\ T; T \\ U; a \\ U\\ \\ connected_component_set T a = connected_component_set U a\n[PROOF STEP]\napply (simp add: connected_component_maximal connected_component_mono subset_antisym)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\connected_component_set U a \\ T; T \\ U; a \\ U\\ \\ connected_component_set T a = connected_component_set U a\n[PROOF STEP]\nusing connected_component_eq_empty\n[PROOF STATE]\nproof (prove)\nusing this:\n(connected_component_set ?S ?x = {}) = (?x \\ ?S)\n\ngoal (1 subgoal):\n 1. \\connected_component_set U a \\ T; T \\ U; a \\ U\\ \\ connected_component_set T a = connected_component_set U a\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 525, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9086178969328286, "lm_q2_score": 0.782662489091802, "lm_q1q2_score": 0.711141144846806}} {"text": "[STATEMENT]\nlemma scalar_prod_minus_distrib: fixes v\\<^sub>1 :: \"'a :: ring vec\"\n assumes v: \"v\\<^sub>1 \\ carrier_vec n\" \"v\\<^sub>2 \\ carrier_vec n\" \"v\\<^sub>3 \\ carrier_vec n\"\n shows \"v\\<^sub>1 \\ (v\\<^sub>2 - v\\<^sub>3) = v\\<^sub>1 \\ v\\<^sub>2 - v\\<^sub>1 \\ v\\<^sub>3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v\\<^sub>1 \\ (v\\<^sub>2 - v\\<^sub>3) = v\\<^sub>1 \\ v\\<^sub>2 - v\\<^sub>1 \\ v\\<^sub>3\n[PROOF STEP]\nunfolding minus_add_uminus_vec[OF v(2-3)]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v\\<^sub>1 \\ (v\\<^sub>2 + - v\\<^sub>3) = v\\<^sub>1 \\ v\\<^sub>2 - v\\<^sub>1 \\ v\\<^sub>3\n[PROOF STEP]\nby (subst scalar_prod_add_distrib[OF v(1)], insert v, auto)", "meta": {"llama_tokens": 339, "file": "Jordan_Normal_Form_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9086178870347122, "lm_q2_score": 0.7826624738835052, "lm_q1q2_score": 0.7111411232813911}} {"text": "[STATEMENT]\nlemma permutation_mat_dim[simp]: \"permutation_mat n p \\ carrier_mat n n\" \n \"dim_row (permutation_mat n p) = n\"\n \"dim_col (permutation_mat n p) = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. permutation_mat n p \\ carrier_mat n n &&& dim_row (permutation_mat n p) = n &&& dim_col (permutation_mat n p) = n\n[PROOF STEP]\nunfolding permutation_mat_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix.mat n n (\\(i, j). if i = p j then 1::'a else (0::'a)) \\ carrier_mat n n &&& dim_row (Matrix.mat n n (\\(i, j). if i = p j then 1::'b else (0::'b))) = n &&& dim_col (Matrix.mat n n (\\(i, j). if i = p j then 1::'c else (0::'c))) = n\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 309, "file": "Perron_Frobenius_Perron_Frobenius_General", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9086178919837705, "lm_q2_score": 0.7826624688140726, "lm_q1q2_score": 0.7111411225486562}} {"text": "[STATEMENT]\nlemma card_remove_fset_less1: \n shows \"x |\\| xs \\ card_fset (remove_fset x xs) < card_fset xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x |\\| xs \\ card_fset (remove_fset x xs) < card_fset xs\n[PROOF STEP]\nunfolding card_fset in_fset remove_fset\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ fset xs \\ card (fset xs - {x}) < card (fset xs)\n[PROOF STEP]\nby (rule card_Diff1_less[OF finite_fset])", "meta": {"llama_tokens": 201, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577680977182187, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.7110372647386678}} {"text": "[STATEMENT]\nlemma star_power_slide: \n assumes \"k \\ n\" \n shows \"x\\<^bsup>k \\<^esup>\\ (x\\<^bsup>n\\<^esup>)\\<^sup>\\ = (x\\<^bsup>n\\<^esup>)\\<^sup>\\ \\ x\\<^bsup>k\\<^esup>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x\\<^bsup>k\\<^esup> \\ (x\\<^bsup>n\\<^esup>)\\<^sup>\\ = (x\\<^bsup>n\\<^esup>)\\<^sup>\\ \\ x\\<^bsup>k\\<^esup>\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x\\<^bsup>k\\<^esup> \\ (x\\<^bsup>n\\<^esup>)\\<^sup>\\ = (x\\<^bsup>n\\<^esup>)\\<^sup>\\ \\ x\\<^bsup>k\\<^esup>\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nk \\ n\n[PROOF STEP]\nhave \"x\\<^bsup>k \\<^esup>\\ (x\\<^bsup>n\\<^esup>)\\<^sup>\\ = (x\\<^bsup>k \\<^esup>\\ x\\<^bsup>n-k\\<^esup>)\\<^sup>\\ \\ x\\<^bsup>k\\<^esup>\"\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ n\n\ngoal (1 subgoal):\n 1. x\\<^bsup>k\\<^esup> \\ (x\\<^bsup>n\\<^esup>)\\<^sup>\\ = (x\\<^bsup>k\\<^esup> \\ x\\<^bsup>n - k\\<^esup>)\\<^sup>\\ \\ x\\<^bsup>k\\<^esup>\n[PROOF STEP]\nby (metis (full_types) le_add_diff_inverse2 power_add star_slide)\n[PROOF STATE]\nproof (state)\nthis:\nx\\<^bsup>k\\<^esup> \\ (x\\<^bsup>n\\<^esup>)\\<^sup>\\ = (x\\<^bsup>k\\<^esup> \\ x\\<^bsup>n - k\\<^esup>)\\<^sup>\\ \\ x\\<^bsup>k\\<^esup>\n\ngoal (1 subgoal):\n 1. x\\<^bsup>k\\<^esup> \\ (x\\<^bsup>n\\<^esup>)\\<^sup>\\ = (x\\<^bsup>n\\<^esup>)\\<^sup>\\ \\ x\\<^bsup>k\\<^esup>\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\nk \\ n\nx\\<^bsup>k\\<^esup> \\ (x\\<^bsup>n\\<^esup>)\\<^sup>\\ = (x\\<^bsup>k\\<^esup> \\ x\\<^bsup>n - k\\<^esup>)\\<^sup>\\ \\ x\\<^bsup>k\\<^esup>\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ n\nx\\<^bsup>k\\<^esup> \\ (x\\<^bsup>n\\<^esup>)\\<^sup>\\ = (x\\<^bsup>k\\<^esup> \\ x\\<^bsup>n - k\\<^esup>)\\<^sup>\\ \\ x\\<^bsup>k\\<^esup>\n\ngoal (1 subgoal):\n 1. x\\<^bsup>k\\<^esup> \\ (x\\<^bsup>n\\<^esup>)\\<^sup>\\ = (x\\<^bsup>n\\<^esup>)\\<^sup>\\ \\ x\\<^bsup>k\\<^esup>\n[PROOF STEP]\nby (metis (full_types) le_add_diff_inverse2 ab_semigroup_add_class.add.commute power_add)\n[PROOF STATE]\nproof (state)\nthis:\nx\\<^bsup>k\\<^esup> \\ (x\\<^bsup>n\\<^esup>)\\<^sup>\\ = (x\\<^bsup>n\\<^esup>)\\<^sup>\\ \\ x\\<^bsup>k\\<^esup>\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1163, "file": "Regular_Algebras_Regular_Algebras", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577680940822761, "lm_q2_score": 0.8289388083214155, "lm_q1q2_score": 0.7110372617246937}} {"text": "[STATEMENT]\nlemma complement_4_language: \"language (complement_3 A) = language (complement_4 A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. language (complement_3 A) = language (complement_4 A)\n[PROOF STEP]\nusing complement_4_language_1 complement_4_language_2\n[PROOF STATE]\nproof (prove)\nusing this:\nlanguage (complement_3 ?A) \\ language (complement_4 ?A)\nlanguage (complement_4 ?A) \\ language (complement_3 ?A)\n\ngoal (1 subgoal):\n 1. language (complement_3 A) = language (complement_4 A)\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 197, "file": "Buchi_Complementation_Complementation_Implement", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577680904463333, "lm_q2_score": 0.8289388104343892, "lm_q1q2_score": 0.7110372605231611}} {"text": "[STATEMENT]\nlemma strict_dominate_trans: \n assumes \"strict_dominate n1 n2\"\n and \"strict_dominate n2 n3\"\n shows \"strict_dominate n1 n3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. strict_dominate n1 n3\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nstrict_dominate n1 n2\nstrict_dominate n2 n3\n\ngoal (1 subgoal):\n 1. strict_dominate n1 n3\n[PROOF STEP]\napply(subgoal_tac \"dominate n2 n3\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\strict_dominate n1 n2; strict_dominate n2 n3; dominate n2 n3\\ \\ strict_dominate n1 n3\n 2. \\strict_dominate n1 n2; strict_dominate n2 n3\\ \\ dominate n2 n3\n[PROOF STEP]\napply(rule strict_dominate_trans1)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\strict_dominate n1 n2; strict_dominate n2 n3; dominate n2 n3\\ \\ strict_dominate n1 ?n2.3\n 2. \\strict_dominate n1 n2; strict_dominate n2 n3; dominate n2 n3\\ \\ dominate ?n2.3 n3\n 3. \\strict_dominate n1 n2; strict_dominate n2 n3\\ \\ dominate n2 n3\n[PROOF STEP]\napply (auto simp add: strict_dominate_def dominate_def)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 536, "file": "Dominance_CHK_Cfg", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278788223264, "lm_q2_score": 0.8056321866478979, "lm_q1q2_score": 0.7109123015746971}} {"text": "[STATEMENT]\nlemma euclid_diff: \n \"PRE (\\s::nat store. s ''x'' = x \\ s ''y'' = y \\ x > 0 \\ y > 0)\n (WHILE (\\s. s ''x''\\ s ''y'') INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) \n DO\n (IF (\\s. s ''x'' > s ''y'')\n THEN (''x'' ::= (\\s. s ''x'' - s ''y''))\n ELSE (''y'' ::= (\\s. s ''y'' - s ''x''))\n FI)\n OD)\n POST (\\s. s ''x'' = gcd x y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rdom \\\\s. s ''x'' = x \\ s ''y'' = y \\ 0 < x \\ 0 < y\\ \\ wp (WHILE (\\s. s ''x'' \\ s ''y'') INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) DO (IF (\\s. s ''y'' < s ''x'') THEN (''x'' ::= (\\s. s ''x'' - s ''y'')) ELSE (''y'' ::= (\\s. s ''y'' - s ''x'')) FI) OD) \\\\s. s ''x'' = gcd x y\\\n[PROOF STEP]\napply (rule rel_antidomain_kleene_algebra.fbox_whilei, simp_all)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\s. gcd (s ''x'') (s ''y'') = gcd x y \\ s ''x'' = s ''y'' \\ s ''y'' = gcd x y\n 2. \\s. gcd (s ''x'') (s ''y'') = gcd x y \\ s ''x'' \\ s ''y'' \\ (s ''y'' < s ''x'' \\ gcd (s ''x'' - s ''y'') (s ''y'') = gcd x y) \\ (s ''y'' < s ''x'' \\ gcd (s ''x'') (s ''y'' - s ''x'') = gcd x y)\n[PROOF STEP]\napply auto[1]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\s. gcd (s ''x'') (s ''y'') = gcd x y \\ s ''x'' \\ s ''y'' \\ (s ''y'' < s ''x'' \\ gcd (s ''x'' - s ''y'') (s ''y'') = gcd x y) \\ (s ''y'' < s ''x'' \\ gcd (s ''x'') (s ''y'' - s ''x'') = gcd x y)\n[PROOF STEP]\nby (metis gcd.commute gcd_diff1_nat le_cases nat_less_le)", "meta": {"llama_tokens": 839, "file": "Algebraic_VCs_AVC_KAD_VC_KAD_Examples", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278726384089, "lm_q2_score": 0.805632181981183, "lm_q1q2_score": 0.7109122924746949}} {"text": "[STATEMENT]\nlemma scaleC_cadjoint:\n fixes A::\"'a::chilbert_space \\ 'b::complex_inner\"\n assumes \"bounded_clinear A\"\n shows \\(\\t. a *\\<^sub>C A t)\\<^sup>\\ = (\\s. cnj a *\\<^sub>C (A\\<^sup>\\) s)\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\t. a *\\<^sub>C A t)\\<^sup>\\ = (\\s. cnj a *\\<^sub>C (A\\<^sup>\\) s)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\t. a *\\<^sub>C A t)\\<^sup>\\ = (\\s. cnj a *\\<^sub>C (A\\<^sup>\\) s)\n[PROOF STEP]\nhave b3: \\((\\ s. (cnj a) *\\<^sub>C ((A\\<^sup>\\) s)) x \\\\<^sub>C y) = (x \\\\<^sub>C (\\ t. a *\\<^sub>C (A t)) y)\\\n for x y\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cnj a *\\<^sub>C (A\\<^sup>\\) x \\\\<^sub>C y = x \\\\<^sub>C a *\\<^sub>C A y\n[PROOF STEP]\nby (simp add: assms cadjoint_univ_prop)\n[PROOF STATE]\nproof (state)\nthis:\ncnj a *\\<^sub>C (A\\<^sup>\\) ?x \\\\<^sub>C ?y = ?x \\\\<^sub>C a *\\<^sub>C A ?y\n\ngoal (1 subgoal):\n 1. (\\t. a *\\<^sub>C A t)\\<^sup>\\ = (\\s. cnj a *\\<^sub>C (A\\<^sup>\\) s)\n[PROOF STEP]\nhave \"((\\t. a *\\<^sub>C A t)\\<^sup>\\) b = cnj a *\\<^sub>C (A\\<^sup>\\) b\"\n for b::'b\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\t. a *\\<^sub>C A t)\\<^sup>\\) b = cnj a *\\<^sub>C (A\\<^sup>\\) b\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\t. a *\\<^sub>C A t)\\<^sup>\\) b = cnj a *\\<^sub>C (A\\<^sup>\\) b\n[PROOF STEP]\nhave \"bounded_clinear (\\t. a *\\<^sub>C A t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bounded_clinear (\\t. a *\\<^sub>C A t)\n[PROOF STEP]\nby (simp add: assms bounded_clinear_const_scaleC)\n[PROOF STATE]\nproof (state)\nthis:\nbounded_clinear (\\t. a *\\<^sub>C A t)\n\ngoal (1 subgoal):\n 1. ((\\t. a *\\<^sub>C A t)\\<^sup>\\) b = cnj a *\\<^sub>C (A\\<^sup>\\) b\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nbounded_clinear (\\t. a *\\<^sub>C A t)\n\ngoal (1 subgoal):\n 1. ((\\t. a *\\<^sub>C A t)\\<^sup>\\) b = cnj a *\\<^sub>C (A\\<^sup>\\) b\n[PROOF STEP]\nby (metis (no_types) cadjoint_eqI b3)\n[PROOF STATE]\nproof (state)\nthis:\n((\\t. a *\\<^sub>C A t)\\<^sup>\\) b = cnj a *\\<^sub>C (A\\<^sup>\\) b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n((\\t. a *\\<^sub>C A t)\\<^sup>\\) ?b = cnj a *\\<^sub>C (A\\<^sup>\\) ?b\n\ngoal (1 subgoal):\n 1. (\\t. a *\\<^sub>C A t)\\<^sup>\\ = (\\s. cnj a *\\<^sub>C (A\\<^sup>\\) s)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\t. a *\\<^sub>C A t)\\<^sup>\\) ?b = cnj a *\\<^sub>C (A\\<^sup>\\) ?b\n\ngoal (1 subgoal):\n 1. (\\t. a *\\<^sub>C A t)\\<^sup>\\ = (\\s. cnj a *\\<^sub>C (A\\<^sup>\\) s)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(\\t. a *\\<^sub>C A t)\\<^sup>\\ = (\\s. cnj a *\\<^sub>C (A\\<^sup>\\) s)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1468, "file": "Complex_Bounded_Operators_Complex_Inner_Product", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278633625322, "lm_q2_score": 0.8056321843145405, "lm_q1q2_score": 0.7109122870607697}} {"text": "[STATEMENT]\nlemma length_strict_mono: \"strict_mono (f::nat \\ 'a list) \\ length (f i) \\ i + length (f 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. strict_mono f \\ i + length (f 0) \\ length (f i)\n[PROOF STEP]\napply(induct i, simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. \\strict_mono f \\ i + length (f 0) \\ length (f i); strict_mono f\\ \\ Suc i + length (f 0) \\ length (f (Suc i))\n[PROOF STEP]\nby (metis dual_order.trans lessI less_length_mono not_less not_less_eq_eq plus_nat.simps(2) strict_mono_def)", "meta": {"llama_tokens": 260, "file": "HOL-CSP_CSP", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357735451835, "lm_q2_score": 0.8198933447152497, "lm_q1q2_score": 0.7108768603597343}} {"text": "[STATEMENT]\nlemma Inf_pres_downset_var: \"(\\x \\ X. \\x) = \\(\\X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (\\ ` X) = \\ (\\ X)\n[PROOF STEP]\nunfolding downset_prop\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\X. {y. y \\ x}) = {y. y \\ \\ X}\n[PROOF STEP]\nby (safe, simp_all add: le_Inf_iff)", "meta": {"llama_tokens": 184, "file": "Order_Lattice_Props_Order_Lattice_Props", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357598021707, "lm_q2_score": 0.8198933337131076, "lm_q1q2_score": 0.710876839552679}} {"text": "[STATEMENT]\nlemma line_equation:\n assumes \"z1 \\ z2\" and \"\\ = rot90 (z2 - z1)\"\n shows \"line z1 z2 = {z. cnj \\*z + \\*cnj z - (cnj \\ * z1 + \\ * cnj z1) = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nfix z\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nhave \"z \\ line z1 z2 \\ Im ((z - z1)/(z2 - z1)) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (z \\ line z1 z2) = is_real ((z - z1) / (z2 - z1))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nz1 \\ z2\n\\ = rot90 (z2 - z1)\n\ngoal (1 subgoal):\n 1. (z \\ line z1 z2) = is_real ((z - z1) / (z2 - z1))\n[PROOF STEP]\nby (simp add: line_def collinear_def)\n[PROOF STATE]\nproof (state)\nthis:\n(z \\ line z1 z2) = is_real ((z - z1) / (z2 - z1))\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(z \\ line z1 z2) = is_real ((z - z1) / (z2 - z1))\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nhave \"... \\ (z - z1)/(z2 - z1) = cnj ((z - z1)/(z2 - z1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_real ((z - z1) / (z2 - z1)) = ((z - z1) / (z2 - z1) = cnj ((z - z1) / (z2 - z1)))\n[PROOF STEP]\nusing complex_diff_cnj[of \"(z - z1)/(z2 - z1)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(z - z1) / (z2 - z1) - cnj ((z - z1) / (z2 - z1)) = cor (2 * Im ((z - z1) / (z2 - z1))) * \\\n\ngoal (1 subgoal):\n 1. is_real ((z - z1) / (z2 - z1)) = ((z - z1) / (z2 - z1) = cnj ((z - z1) / (z2 - z1)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nis_real ((z - z1) / (z2 - z1)) = ((z - z1) / (z2 - z1) = cnj ((z - z1) / (z2 - z1)))\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nis_real ((z - z1) / (z2 - z1)) = ((z - z1) / (z2 - z1) = cnj ((z - z1) / (z2 - z1)))\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nhave \"... \\ (z - z1)*(cnj z2 - cnj z1) = (cnj z - cnj z1)*(z2 - z1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((z - z1) / (z2 - z1) = cnj ((z - z1) / (z2 - z1))) = ((z - z1) * (cnj z2 - cnj z1) = (cnj z - cnj z1) * (z2 - z1))\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nz1 \\ z2\n\ngoal (1 subgoal):\n 1. ((z - z1) / (z2 - z1) = cnj ((z - z1) / (z2 - z1))) = ((z - z1) * (cnj z2 - cnj z1) = (cnj z - cnj z1) * (z2 - z1))\n[PROOF STEP]\nusing \\(z \\ line z1 z2) = is_real ((z - z1) / (z2 - z1))\\ calculation is_real_div\n[PROOF STATE]\nproof (prove)\nusing this:\nz1 \\ z2\n(z \\ line z1 z2) = is_real ((z - z1) / (z2 - z1))\n(z \\ line z1 z2) = ((z - z1) / (z2 - z1) = cnj ((z - z1) / (z2 - z1)))\n?b \\ 0 \\ is_real (?a / ?b) = (?a * cnj ?b = ?b * cnj ?a)\n\ngoal (1 subgoal):\n 1. ((z - z1) / (z2 - z1) = cnj ((z - z1) / (z2 - z1))) = ((z - z1) * (cnj z2 - cnj z1) = (cnj z - cnj z1) * (z2 - z1))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n((z - z1) / (z2 - z1) = cnj ((z - z1) / (z2 - z1))) = ((z - z1) * (cnj z2 - cnj z1) = (cnj z - cnj z1) * (z2 - z1))\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n((z - z1) / (z2 - z1) = cnj ((z - z1) / (z2 - z1))) = ((z - z1) * (cnj z2 - cnj z1) = (cnj z - cnj z1) * (z2 - z1))\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nhave \"... \\ cnj(z2 - z1)*z - (z2 - z1)*cnj z - (cnj(z2 - z1)*z1 - (z2 - z1)*cnj z1) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((z - z1) * (cnj z2 - cnj z1) = (cnj z - cnj z1) * (z2 - z1)) = (cnj (z2 - z1) * z - (z2 - z1) * cnj z - (cnj (z2 - z1) * z1 - (z2 - z1) * cnj z1) = 0)\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n((z - z1) * (cnj z2 - cnj z1) = (cnj z - cnj z1) * (z2 - z1)) = (cnj (z2 - z1) * z - (z2 - z1) * cnj z - (cnj (z2 - z1) * z1 - (z2 - z1) * cnj z1) = 0)\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n((z - z1) * (cnj z2 - cnj z1) = (cnj z - cnj z1) * (z2 - z1)) = (cnj (z2 - z1) * z - (z2 - z1) * cnj z - (cnj (z2 - z1) * z1 - (z2 - z1) * cnj z1) = 0)\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nhave \"... \\ cnj \\ * z + \\ * cnj z - (cnj \\ * z1 + \\ * cnj z1) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cnj (z2 - z1) * z - (z2 - z1) * cnj z - (cnj (z2 - z1) * z1 - (z2 - z1) * cnj z1) = 0) = (cnj_mix \\ z - cnj_mix \\ z1 = 0)\n[PROOF STEP]\napply (subst assms)+\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cnj (z2 - z1) * z - (z2 - z1) * cnj z - (cnj (z2 - z1) * z1 - (z2 - z1) * cnj z1) = 0) = (cnj_mix (rot90 (z2 - z1)) z - cnj_mix (rot90 (z2 - z1)) z1 = 0)\n[PROOF STEP]\napply (subst cnj_mix_minus)+\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ * cnj_mix (rot90 (z2 - z1)) z - \\ * cnj_mix (rot90 (z2 - z1)) z1 = 0) = (cnj_mix (rot90 (z2 - z1)) z - cnj_mix (rot90 (z2 - z1)) z1 = 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(cnj (z2 - z1) * z - (z2 - z1) * cnj z - (cnj (z2 - z1) * z1 - (z2 - z1) * cnj z1) = 0) = (cnj_mix \\ z - cnj_mix \\ z1 = 0)\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(z \\ line z1 z2) = (cnj_mix \\ z - cnj_mix \\ z1 = 0)\n[PROOF STEP]\nhave \"z \\ line z1 z2 \\ cnj \\ * z + \\ * cnj z - (cnj \\ * z1 + \\ * cnj z1) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(z \\ line z1 z2) = (cnj_mix \\ z - cnj_mix \\ z1 = 0)\n\ngoal (1 subgoal):\n 1. (z \\ line z1 z2) = (cnj_mix \\ z - cnj_mix \\ z1 = 0)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(z \\ line z1 z2) = (cnj_mix \\ z - cnj_mix \\ z1 = 0)\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n(?z2 \\ line z1 z2) = (cnj_mix \\ ?z2 - cnj_mix \\ z1 = 0)\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(?z2 \\ line z1 z2) = (cnj_mix \\ ?z2 - cnj_mix \\ z1 = 0)\n\ngoal (1 subgoal):\n 1. line z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nline z1 z2 = {z. cnj_mix \\ z - cnj_mix \\ z1 = 0}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3897, "file": "Complex_Geometry_Elementary_Complex_Geometry", "length": 30, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357598021707, "lm_q2_score": 0.8198933293122506, "lm_q1q2_score": 0.7108768357369786}} {"text": "[STATEMENT]\nlemma set_coeffs_map_poly:\n \"(\\x. f x = 0 \\ x = 0) \\ set (coeffs (map_poly f p)) = f ` set (coeffs p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. (f x = (0::'b)) = (x = (0::'a))) \\ set (coeffs (map_poly f p)) = f ` set (coeffs p)\n[PROOF STEP]\nby (simp add: coeffs_map_poly')", "meta": {"llama_tokens": 161, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357460591569, "lm_q2_score": 0.8198933381139645, "lm_q1q2_score": 0.7108768321005737}} {"text": "[STATEMENT]\nlemma heron_step_real: \"\\t > 0; n \\ 0\\ \\ (t + n/t) / 2 \\ sqrt n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < t; 0 \\ n\\ \\ sqrt n \\ (t + n / t) / 2\n[PROOF STEP]\nusing arith_geo_mean_sqrt[of t \"n/t\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 \\ t; 0 \\ n / t\\ \\ sqrt (t * (n / t)) \\ (t + n / t) / 2\n\ngoal (1 subgoal):\n 1. \\0 < t; 0 \\ n\\ \\ sqrt n \\ (t + n / t) / 2\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 273, "file": "Pell_Efficient_Discrete_Sqrt", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513842182777, "lm_q2_score": 0.7931059560743423, "lm_q1q2_score": 0.7108223109633898}} {"text": "[STATEMENT]\nlemma iMin_subset_idem: \"\\ B \\ {}; B \\ A \\ \\ min (iMin B) (iMin A) = iMin A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\B \\ {}; B \\ A\\ \\ min (iMin B) (iMin A) = iMin A\n[PROOF STEP]\nby (metis iMin_subset min.absorb2)", "meta": {"llama_tokens": 140, "file": "List-Infinite_CommonSet_SetInterval2", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513731336202, "lm_q2_score": 0.7931059560743422, "lm_q1q2_score": 0.7108223021720819}} {"text": "[STATEMENT]\nlemma poly_cpoly_of_real_iff:\n shows \"poly (cpoly_of pR pI) (of_real t) =0 \\ poly pR t = 0 \\ poly pI t=0 \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (poly (cpoly_of pR pI) (complex_of_real t) = 0) = (poly pR t = 0 \\ poly pI t = 0)\n[PROOF STEP]\nunfolding poly_cpoly_of_real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Complex (poly pR t) (poly pI t) = 0) = (poly pR t = 0 \\ poly pI t = 0)\n[PROOF STEP]\nusing Complex_eq_0\n[PROOF STATE]\nproof (prove)\nusing this:\n(Complex ?a ?b = 0) = (?a = 0 \\ ?b = 0)\n\ngoal (1 subgoal):\n 1. (Complex (poly pR t) (poly pI t) = 0) = (poly pR t = 0 \\ poly pI t = 0)\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 336, "file": "Count_Complex_Roots_CC_Polynomials_Extra", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513648201267, "lm_q2_score": 0.7931059609645724, "lm_q1q2_score": 0.7108222999614762}} {"text": "[STATEMENT]\nlemma polyfun_eq_0: \"(\\x. (\\i\\n. c i * x^i) = 0) \\ (\\i\\n. c i = 0)\"\n for c :: \"nat \\ 'a::{idom,real_normed_div_algebra}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. (\\i\\n. c i * x ^ i) = (0::'a)) = (\\i\\n. c i = (0::'a))\n[PROOF STEP]\nusing zero_polynom_imp_zero_coeffs\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\w. (\\i\\?n. ?c i * w ^ i) = (0::?'a); ?k \\ ?n\\ \\ ?c ?k = (0::?'a)\n\ngoal (1 subgoal):\n 1. (\\x. (\\i\\n. c i * x ^ i) = (0::'a)) = (\\i\\n. c i = (0::'a))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 328, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513786759491, "lm_q2_score": 0.7931059462938815, "lm_q1q2_score": 0.7108222978019846}} {"text": "[STATEMENT]\nlemma card_injective_functions_domain_and_range_permutation:\n assumes \"finite A\" \"finite B\"\n shows \"card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \\ card B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \\ card B)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \\ card B)\n[PROOF STEP]\nhave \"bij_betw (number_partition_of A B) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) {N. (\\n. n\\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw (number_partition_of A B) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) {N. (\\n. n \\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B}\n[PROOF STEP]\nusing \\finite A\\ \\finite B\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\n\ngoal (1 subgoal):\n 1. bij_betw (number_partition_of A B) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) {N. (\\n. n \\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B}\n[PROOF STEP]\nby (rule bij_betw_number_partition_of)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw (number_partition_of A B) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) {N. (\\n. n \\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B}\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \\ card B)\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nbij_betw (number_partition_of A B) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) {N. (\\n. n \\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B}\n[PROOF STEP]\nhave \"card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = card {N. (\\n. n\\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw (number_partition_of A B) ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) {N. (\\n. n \\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B}\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = card {N. (\\n. n \\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B}\n[PROOF STEP]\nby (rule bij_betw_same_card)\n[PROOF STATE]\nproof (state)\nthis:\ncard ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = card {N. (\\n. n \\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B}\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \\ card B)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = card {N. (\\n. n \\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B}\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \\ card B)\n[PROOF STEP]\nhave \"card {N. (\\n. n\\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B} = iverson (card A \\ card B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {N. (\\n. n \\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B} = iverson (card A \\ card B)\n[PROOF STEP]\nby (rule card_number_partitions_with_only_parts_1)\n[PROOF STATE]\nproof (state)\nthis:\ncard {N. (\\n. n \\# N \\ n = 1) \\ number_partition (card A) N \\ size N \\ card B} = iverson (card A \\ card B)\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \\ card B)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \\ card B)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \\ card B)\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \\ card B)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard ({f \\ A \\\\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \\ card B)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2159, "file": "Twelvefold_Way_Twelvefold_Way_Entry11", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513731336204, "lm_q2_score": 0.7931059462938815, "lm_q1q2_score": 0.7108222934063307}} {"text": "[STATEMENT]\nlemma prob_space_erlang_density:\n assumes l[arith]: \"0 < l\"\n shows \"prob_space (density lborel (erlang_density k l))\" (is \"prob_space ?D\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prob_space (density lborel (\\x. ennreal (erlang_density k l x)))\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. emeasure (density lborel (\\x. ennreal (erlang_density k l x))) (space (density lborel (\\x. ennreal (erlang_density k l x)))) = 1\n[PROOF STEP]\nshow \"emeasure ?D (space ?D) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. emeasure (density lborel (\\x. ennreal (erlang_density k l x))) (space (density lborel (\\x. ennreal (erlang_density k l x)))) = 1\n[PROOF STEP]\nusing nn_integral_erlang_ith_moment[OF l, where k=k and i=0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>+ x. ennreal (erlang_density k l x * x ^ 0) \\lborel = ennreal (fact (k + 0) / (fact k * l ^ 0))\n\ngoal (1 subgoal):\n 1. emeasure (density lborel (\\x. ennreal (erlang_density k l x))) (space (density lborel (\\x. ennreal (erlang_density k l x)))) = 1\n[PROOF STEP]\nby (simp add: emeasure_density)\n[PROOF STATE]\nproof (state)\nthis:\nemeasure (density lborel (\\x. ennreal (erlang_density k l x))) (space (density lborel (\\x. ennreal (erlang_density k l x)))) = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 594, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513648201266, "lm_q2_score": 0.7931059511841119, "lm_q1q2_score": 0.7108222911957249}} {"text": "[STATEMENT]\ntheorem Ball_def': \"(\\x\\S. P x) \\ (\\x. x \\ S \\ P x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\S. P x) = (\\x. x \\ S \\ P x)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 109, "file": "Auto2_HOL_HOL_HOL_Base", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213853793452, "lm_q2_score": 0.7905303186696747, "lm_q1q2_score": 0.7107827153066532}} {"text": "[STATEMENT]\ntheorem effective_matrix_Tensor_elements: \n fixes M1 M2 i j \n assumes \"i<((row_length M1)*(row_length M2))\"\n and \"j < (length M1)*(length M2)\"\n and \"mat (row_length M1) (length M1) M1\"\n and \"mat (row_length M2) (length M2) M2\"\n shows\n \"((M1 \\ M2)!j!i) = f (M1!(j div (length M2))!(i div (row_length M2))) \n(M2!(j mod length M2)!(i mod (row_length M2)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (M1 \\ M2) ! j ! i = M1 ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)\n[PROOF STEP]\nusing matrix_Tensor_elements assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. (i < row_length ?M1.0 * row_length ?M2.0 \\ j < length ?M1.0 * length ?M2.0) \\ mat (row_length ?M1.0) (length ?M1.0) ?M1.0 \\ mat (row_length ?M2.0) (length ?M2.0) ?M2.0 \\ (?M1.0 \\ ?M2.0) ! j ! i = ?M1.0 ! (j div length ?M2.0) ! (i div row_length ?M2.0) * ?M2.0 ! (j mod length ?M2.0) ! (i mod row_length ?M2.0)\ni < row_length M1 * row_length M2\nj < length M1 * length M2\nmat (row_length M1) (length M1) M1\nmat (row_length M2) (length M2) M2\n\ngoal (1 subgoal):\n 1. (M1 \\ M2) ! j ! i = M1 ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 593, "file": "Matrix_Tensor_Matrix_Tensor", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213745668094, "lm_q2_score": 0.7905303211371899, "lm_q1q2_score": 0.7107827089776114}} {"text": "[STATEMENT]\nlemma to_bl_plus_carry:\n \"to_bl (x + y) =\n rev (foldr (\\(x, y) res car. xor3 x y car # res (carry x y car))\n (rev (zip (to_bl x) (to_bl y))) (\\_. []) False)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. to_bl (x + y) = rev (foldr (\\(x, y) res car. xor3 x y car # res (carry x y car)) (rev (zip (to_bl x) (to_bl y))) (\\_. []) False)\n[PROOF STEP]\nusing rbl_add_suc_carry_fold[where xs=\"rev (to_bl x)\" and ys=\"rev (to_bl y)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (rev (to_bl x)) = length (rev (to_bl y)) \\ \\car. (if car then rbl_succ else id) (rbl_add (rev (to_bl x)) (rev (to_bl y))) = foldr (\\(x, y) res car. xor3 x y car # res (carry x y car)) (zip (rev (to_bl x)) (rev (to_bl y))) (\\_. []) car\n\ngoal (1 subgoal):\n 1. to_bl (x + y) = rev (foldr (\\(x, y) res car. xor3 x y car # res (carry x y car)) (rev (zip (to_bl x) (to_bl y))) (\\_. []) False)\n[PROOF STEP]\napply (simp add: word_add_rbl[OF refl refl])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\car. (if car then rbl_succ else id) (rbl_add (rev (to_bl x)) (rev (to_bl y))) = foldr (\\(x, y) res car. xor3 x y car # res (carry x y car)) (zip (rev (to_bl x)) (rev (to_bl y))) (\\_. []) car \\ rbl_add (rev (to_bl x)) (rev (to_bl y)) = foldr (\\(x, y) res car. xor3 x y car # res (carry x y car)) (rev (zip (to_bl x) (to_bl y))) (\\_. []) False\n[PROOF STEP]\napply (drule_tac x=False in spec)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if False then rbl_succ else id) (rbl_add (rev (to_bl x)) (rev (to_bl y))) = foldr (\\(x, y) res car. xor3 x y car # res (carry x y car)) (zip (rev (to_bl x)) (rev (to_bl y))) (\\_. []) False \\ rbl_add (rev (to_bl x)) (rev (to_bl y)) = foldr (\\(x, y) res car. xor3 x y car # res (carry x y car)) (rev (zip (to_bl x) (to_bl y))) (\\_. []) False\n[PROOF STEP]\napply (simp add: zip_rev)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 891, "file": "Word_Lib_Bitwise", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094117351309, "lm_q2_score": 0.795658104908603, "lm_q1q2_score": 0.7107688736381932}} {"text": "[STATEMENT]\nlemma has_integral_cmul_iff:\n assumes \"c \\ 0\"\n shows \"((\\x. c *\\<^sub>R f x) has_integral (c *\\<^sub>R I)) A \\ (f has_integral I) A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. c *\\<^sub>R f x) has_integral c *\\<^sub>R I) A = (f has_integral I) A\n[PROOF STEP]\nusing assms has_integral_cmul[of f I A c]\n has_integral_cmul[of \"\\x. c *\\<^sub>R f x\" \"c *\\<^sub>R I\" A \"inverse c\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ 0\n(f has_integral I) A \\ ((\\x. c *\\<^sub>R f x) has_integral c *\\<^sub>R I) A\n((\\x. c *\\<^sub>R f x) has_integral c *\\<^sub>R I) A \\ ((\\x. c *\\<^sub>R f x /\\<^sub>R c) has_integral c *\\<^sub>R I /\\<^sub>R c) A\n\ngoal (1 subgoal):\n 1. ((\\x. c *\\<^sub>R f x) has_integral c *\\<^sub>R I) A = (f has_integral I) A\n[PROOF STEP]\nby (auto simp: field_simps)", "meta": {"llama_tokens": 417, "file": "Prime_Number_Theorem_Prime_Number_Theorem_Library", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094060543488, "lm_q2_score": 0.795658104908603, "lm_q1q2_score": 0.7107688691182329}} {"text": "[STATEMENT]\nlemma rel_interior_mono:\n \"\\S \\ T; affine hull S = affine hull T\\\n \\ (rel_interior S) \\ (rel_interior T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\S \\ T; affine hull S = affine hull T\\ \\ rel_interior S \\ rel_interior T\n[PROOF STEP]\nby (auto simp: rel_interior_def)", "meta": {"llama_tokens": 153, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094003735664, "lm_q2_score": 0.7956581024858786, "lm_q1q2_score": 0.7107688624340299}} {"text": "[STATEMENT]\nlemma homeomorphicI [intro?]:\n \"\\f ` S = T; g ` T = S;\n continuous_on S f; continuous_on T g;\n \\x. x \\ S \\ g(f(x)) = x;\n \\y. y \\ T \\ f(g(y)) = y\\ \\ S homeomorphic T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f ` S = T; g ` T = S; continuous_on S f; continuous_on T g; \\x. x \\ S \\ g (f x) = x; \\y. y \\ T \\ f (g y) = y\\ \\ S homeomorphic T\n[PROOF STEP]\nunfolding homeomorphic_def homeomorphism_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f ` S = T; g ` T = S; continuous_on S f; continuous_on T g; \\x. x \\ S \\ g (f x) = x; \\y. y \\ T \\ f (g y) = y\\ \\ \\f g. (\\x\\S. g (f x) = x) \\ f ` S = T \\ continuous_on S f \\ (\\y\\T. f (g y) = y) \\ g ` T = S \\ continuous_on T g\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 421, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933093975331751, "lm_q2_score": 0.795658104908603, "lm_q1q2_score": 0.710768862338292}} {"text": "[STATEMENT]\nlemma integrable_stretch:\n fixes f :: \"'a::euclidean_space \\ 'b::real_normed_vector\"\n assumes \"f integrable_on cbox a b\"\n and \"\\k\\Basis. m k \\ 0\"\n shows \"(\\x::'a. f (\\k\\Basis. (m k * (x\\k))*\\<^sub>R k)) integrable_on\n ((\\x. \\k\\Basis. (1 / m k * (x\\k))*\\<^sub>R k) ` cbox a b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. f (\\k\\Basis. (m k * (x \\ k)) *\\<^sub>R k)) integrable_on (\\x. \\k\\Basis. (1 / m k * (x \\ k)) *\\<^sub>R k) ` cbox a b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nf integrable_on cbox a b\n\\k\\Basis. m k \\ 0\n\ngoal (1 subgoal):\n 1. (\\x. f (\\k\\Basis. (m k * (x \\ k)) *\\<^sub>R k)) integrable_on (\\x. \\k\\Basis. (1 / m k * (x \\ k)) *\\<^sub>R k) ` cbox a b\n[PROOF STEP]\nunfolding integrable_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\y. (f has_integral y) (cbox a b)\n\\k\\Basis. m k \\ 0\n\ngoal (1 subgoal):\n 1. \\y. ((\\x. f (\\k\\Basis. (m k * (x \\ k)) *\\<^sub>R k)) has_integral y) ((\\x. \\k\\Basis. (1 / m k * (x \\ k)) *\\<^sub>R k) ` cbox a b)\n[PROOF STEP]\nby (force dest: has_integral_stretch)", "meta": {"llama_tokens": 612, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094003735663, "lm_q2_score": 0.7956580927949806, "lm_q1q2_score": 0.7107688537770595}} {"text": "[STATEMENT]\nlemma cong_mersenne_mod: \"[mersenne_mod k n = k] (mod (2 ^ n - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [mersenne_mod k n = k] (mod 2 ^ n - 1)\n[PROOF STEP]\nunfolding mersenne_mod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [k mod 2 ^ n + k div 2 ^ n = k] (mod 2 ^ n - 1)\n[PROOF STEP]\nby (rule cong_mersenne_number_int)", "meta": {"llama_tokens": 171, "file": "Mersenne_Primes_Lucas_Lehmer_Code", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9111797172476384, "lm_q2_score": 0.7799929002541068, "lm_q1q2_score": 0.7107137103087025}} {"text": "[STATEMENT]\nlemma has_field_derivative_linepath_within [derivative_intros]:\n assumes [derivative_intros]: \n \"(f has_field_derivative f') (at x within S)\" \"(g has_field_derivative g') (at x within S)\"\n \"(h has_real_derivative h') (at x within S)\"\n shows \"((\\x. linepath (f x) (g x) (h x)) has_field_derivative \n (1 - h x) *\\<^sub>R f' + h x *\\<^sub>R g' - h' *\\<^sub>R (f x - g x)) (at x within S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. linepath (f x) (g x) (h x)) has_real_derivative (1 - h x) *\\<^sub>R f' + h x *\\<^sub>R g' - h' *\\<^sub>R (f x - g x)) (at x within S)\n[PROOF STEP]\nunfolding linepath_def [abs_def]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. (1 - h x) *\\<^sub>R f x + h x *\\<^sub>R g x) has_real_derivative (1 - h x) *\\<^sub>R f' + h x *\\<^sub>R g' - h' *\\<^sub>R (f x - g x)) (at x within S)\n[PROOF STEP]\nby (auto intro!: derivative_eq_intros simp: field_simps scaleR_diff_right)", "meta": {"llama_tokens": 429, "file": "Error_Function_Error_Function", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111797100118214, "lm_q2_score": 0.7799928951399098, "lm_q1q2_score": 0.710713700004864}} {"text": "[STATEMENT]\nlemma winding_number_split_linepath:\n assumes \"c \\ closed_segment a b\" \"z \\ closed_segment a b\"\n shows \"winding_number(linepath a b) z = winding_number(linepath a c) z + winding_number(linepath c b) z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. winding_number (linepath a b) z = winding_number (linepath a c) z + winding_number (linepath c b) z\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. winding_number (linepath a b) z = winding_number (linepath a c) z + winding_number (linepath c b) z\n[PROOF STEP]\nhave \"z \\ closed_segment a c\" \"z \\ closed_segment c b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z \\ closed_segment a c &&& z \\ closed_segment c b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ closed_segment a b\nz \\ closed_segment a b\n\ngoal (1 subgoal):\n 1. z \\ closed_segment a c &&& z \\ closed_segment c b\n[PROOF STEP]\nby (meson convex_contains_segment convex_segment ends_in_segment subsetCE)+\n[PROOF STATE]\nproof (state)\nthis:\nz \\ closed_segment a c\nz \\ closed_segment c b\n\ngoal (1 subgoal):\n 1. winding_number (linepath a b) z = winding_number (linepath a c) z + winding_number (linepath c b) z\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nz \\ closed_segment a c\nz \\ closed_segment c b\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ closed_segment a c\nz \\ closed_segment c b\n\ngoal (1 subgoal):\n 1. winding_number (linepath a b) z = winding_number (linepath a c) z + winding_number (linepath c b) z\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ closed_segment a c\nz \\ closed_segment c b\nc \\ closed_segment a b\nz \\ closed_segment a b\n\ngoal (1 subgoal):\n 1. winding_number (linepath a b) z = winding_number (linepath a c) z + winding_number (linepath c b) z\n[PROOF STEP]\nby (simp add: winding_number_valid_path contour_integral_split_linepath [symmetric] continuous_on_inversediff field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nwinding_number (linepath a b) z = winding_number (linepath a c) z + winding_number (linepath c b) z\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 857, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637577007393, "lm_q2_score": 0.8267117898012104, "lm_q1q2_score": 0.7106941637560122}} {"text": "[STATEMENT]\nlemma fib_rec_even: \"fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. fib (2 * 0) = (fib (0 - 1) + fib (0 + 1)) * fib 0\n 2. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. fib (2 * 0) = (fib (0 - 1) + fib (0 + 1)) * fib 0\n 2. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fib (2 * 0) = (fib (0 - 1) + fib (0 + 1)) * fib 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfib (2 * 0) = (fib (0 - 1) + fib (0 + 1)) * fib 0\n\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nfib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n\n\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nlet ?rfib = \"\\x. real (fib x)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nhave \"2 * (Suc n) = Suc (Suc (2 * n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * Suc n = Suc (Suc (2 * n))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * Suc n = Suc (Suc (2 * n))\n\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 * Suc n = Suc (Suc (2 * n))\n\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nhave \"real (fib \\) = ?rfib n^2 + ?rfib (Suc n)^2 + (?rfib (n - 1) + ?rfib (n + 1)) * ?rfib n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (fib (Suc (Suc (2 * n)))) = (real (fib n))\\<^sup>2 + (real (fib (Suc n)))\\<^sup>2 + (real (fib (n - 1)) + real (fib (n + 1))) * real (fib n)\n[PROOF STEP]\nby (simp add: fib_rec_odd Suc)\n[PROOF STATE]\nproof (state)\nthis:\nreal (fib (Suc (Suc (2 * n)))) = (real (fib n))\\<^sup>2 + (real (fib (Suc n)))\\<^sup>2 + (real (fib (n - 1)) + real (fib (n + 1))) * real (fib n)\n\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (fib (Suc (Suc (2 * n)))) = (real (fib n))\\<^sup>2 + (real (fib (Suc n)))\\<^sup>2 + (real (fib (n - 1)) + real (fib (n + 1))) * real (fib n)\n\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nhave \"(?rfib (n - 1) + ?rfib (n + 1)) * ?rfib n = (2 * ?rfib (n + 1) - ?rfib n) * ?rfib n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (real (fib (n - 1)) + real (fib (n + 1))) * real (fib n) = (2 * real (fib (n + 1)) - real (fib n)) * real (fib n)\n[PROOF STEP]\nby (cases n) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(real (fib (n - 1)) + real (fib (n + 1))) * real (fib n) = (2 * real (fib (n + 1)) - real (fib n)) * real (fib n)\n\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(real (fib (n - 1)) + real (fib (n + 1))) * real (fib n) = (2 * real (fib (n + 1)) - real (fib n)) * real (fib n)\n\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nhave \"?rfib n^2 + ?rfib (Suc n)^2 + \\ = (?rfib (Suc n) + 2 * ?rfib n) * ?rfib (Suc n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (real (fib n))\\<^sup>2 + (real (fib (Suc n)))\\<^sup>2 + (2 * real (fib (n + 1)) - real (fib n)) * real (fib n) = (real (fib (Suc n)) + 2 * real (fib n)) * real (fib (Suc n))\n[PROOF STEP]\nby (simp add: algebra_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n(real (fib n))\\<^sup>2 + (real (fib (Suc n)))\\<^sup>2 + (2 * real (fib (n + 1)) - real (fib n)) * real (fib n) = (real (fib (Suc n)) + 2 * real (fib n)) * real (fib (Suc n))\n\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(real (fib n))\\<^sup>2 + (real (fib (Suc n)))\\<^sup>2 + (2 * real (fib (n + 1)) - real (fib n)) * real (fib n) = (real (fib (Suc n)) + 2 * real (fib n)) * real (fib (Suc n))\n\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nhave \"\\ = real ((fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (real (fib (Suc n)) + 2 * real (fib n)) * real (fib (Suc n)) = real ((fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(real (fib (Suc n)) + 2 * real (fib n)) * real (fib (Suc n)) = real ((fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n))\n\ngoal (1 subgoal):\n 1. \\n. fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n \\ fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (fib (2 * Suc n)) = real ((fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (fib (2 * Suc n)) = real ((fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n))\n\ngoal (1 subgoal):\n 1. fib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n[PROOF STEP]\nby (simp only: of_nat_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\nfib (2 * Suc n) = (fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3493, "file": null, "length": 26, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637469145054, "lm_q2_score": 0.8267117962054049, "lm_q1q2_score": 0.7106941603443594}} {"text": "[STATEMENT]\ntheorem sum_of_cubes: \"real (\\k\\n::nat. k ^ 3) = real (n ^ 2 + n) ^ 2 / 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (\\k\\n. k ^ 3) = (real (n\\<^sup>2 + n))\\<^sup>2 / 4\n[PROOF STEP]\nby (simp only: of_nat_sum of_nat_power sum_of_powers)\n (simp add: bernoulli_unroll_all field_simps power2_eq_square power_numeral_reduce)", "meta": {"llama_tokens": 170, "file": "Bernoulli_Bernoulli", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206818021529, "lm_q2_score": 0.787931188173138, "lm_q1q2_score": 0.7106514344502971}} {"text": "[STATEMENT]\ntheorem nn_integral_liminf:\n fixes u :: \"nat \\ 'a \\ ennreal\"\n assumes u: \"\\i. u i \\ borel_measurable M\"\n shows \"(\\\\<^sup>+ x. liminf (\\n. u n x) \\M) \\ liminf (\\n. integral\\<^sup>N M (u n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. liminf (\\n. u n x) \\M \\ liminf (\\n. integral\\<^sup>N M (u n))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. liminf (\\n. u n x) \\M \\ liminf (\\n. integral\\<^sup>N M (u n))\n[PROOF STEP]\nhave \"(\\\\<^sup>+ x. liminf (\\n. u n x) \\M) = (SUP n. \\\\<^sup>+ x. (INF i\\{n..}. u i x) \\M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. liminf (\\n. u n x) \\M = (SUP n. \\\\<^sup>+ x. (INF i\\{n..}. u i x) \\M)\n[PROOF STEP]\nunfolding liminf_SUP_INF\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. (SUP n. INF n\\{n..}. u n x) \\M = (SUP n. \\\\<^sup>+ x. (INF i\\{n..}. u i x) \\M)\n[PROOF STEP]\nusing u\n[PROOF STATE]\nproof (prove)\nusing this:\nu ?i \\ borel_measurable M\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. (SUP n. INF n\\{n..}. u n x) \\M = (SUP n. \\\\<^sup>+ x. (INF i\\{n..}. u i x) \\M)\n[PROOF STEP]\nby (intro nn_integral_monotone_convergence_SUP_AE)\n (auto intro!: AE_I2 intro: INF_greatest INF_superset_mono)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. liminf (\\n. u n x) \\M = (SUP n. \\\\<^sup>+ x. (INF i\\{n..}. u i x) \\M)\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. liminf (\\n. u n x) \\M \\ liminf (\\n. integral\\<^sup>N M (u n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. liminf (\\n. u n x) \\M = (SUP n. \\\\<^sup>+ x. (INF i\\{n..}. u i x) \\M)\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. liminf (\\n. u n x) \\M \\ liminf (\\n. integral\\<^sup>N M (u n))\n[PROOF STEP]\nhave \"\\ \\ liminf (\\n. integral\\<^sup>N M (u n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (SUP n. \\\\<^sup>+ x. (INF i\\{n..}. u i x) \\M) \\ liminf (\\n. integral\\<^sup>N M (u n))\n[PROOF STEP]\nby (auto simp: liminf_SUP_INF intro!: SUP_mono INF_greatest nn_integral_mono INF_lower)\n[PROOF STATE]\nproof (state)\nthis:\n(SUP n. \\\\<^sup>+ x. (INF i\\{n..}. u i x) \\M) \\ liminf (\\n. integral\\<^sup>N M (u n))\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. liminf (\\n. u n x) \\M \\ liminf (\\n. integral\\<^sup>N M (u n))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sup>+ x. liminf (\\n. u n x) \\M \\ liminf (\\n. integral\\<^sup>N M (u n))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>+ x. liminf (\\n. u n x) \\M \\ liminf (\\n. integral\\<^sup>N M (u n))\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. liminf (\\n. u n x) \\M \\ liminf (\\n. integral\\<^sup>N M (u n))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. liminf (\\n. u n x) \\M \\ liminf (\\n. integral\\<^sup>N M (u n))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1534, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206844384594, "lm_q2_score": 0.787931185683219, "lm_q1q2_score": 0.7106514342818157}} {"text": "[STATEMENT]\nlemma card_Int_conv: \"\n \\ finite A; finite B \\ \\\n card (A \\ B) = card A + card B - card (A \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite B\\ \\ card (A \\ B) = card A + card B - card (A \\ B)\n[PROOF STEP]\nby (simp only: card_Un_Int diff_add_inverse)", "meta": {"llama_tokens": 149, "file": "List-Infinite_CommonSet_SetInterval2", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.90192067652954, "lm_q2_score": 0.787931190663057, "lm_q1q2_score": 0.7106514325415503}} {"text": "[STATEMENT]\ntheorem meld_correct:\n assumes I: \"invar q\" \"invar q'\"\n shows \n \"invar (meld q q')\"\n \"queue_to_multiset (meld q q') = queue_to_multiset q + queue_to_multiset q'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. invar (meld q q') &&& queue_to_multiset (meld q q') = queue_to_multiset q + queue_to_multiset q'\n[PROOF STEP]\nusing meld_invar[of q q'] meld_mset[of q q'] I\n[PROOF STATE]\nproof (prove)\nusing this:\n\\invar q; invar q'\\ \\ invar (meld q q')\n\\queue_invar q; queue_invar q'\\ \\ queue_to_multiset (meld q q') = queue_to_multiset q + queue_to_multiset q'\ninvar q\ninvar q'\n\ngoal (1 subgoal):\n 1. invar (meld q q') &&& queue_to_multiset (meld q q') = queue_to_multiset q + queue_to_multiset q'\n[PROOF STEP]\nunfolding invar_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\queue_invar q \\ rank_skew_invar q; queue_invar q' \\ rank_skew_invar q'\\ \\ queue_invar (meld q q') \\ rank_skew_invar (meld q q')\n\\queue_invar q; queue_invar q'\\ \\ queue_to_multiset (meld q q') = queue_to_multiset q + queue_to_multiset q'\nqueue_invar q \\ rank_skew_invar q\nqueue_invar q' \\ rank_skew_invar q'\n\ngoal (1 subgoal):\n 1. queue_invar (meld q q') \\ rank_skew_invar (meld q q') &&& queue_to_multiset (meld q q') = queue_to_multiset q + queue_to_multiset q'\n[PROOF STEP]\nby simp_all", "meta": {"llama_tokens": 629, "file": "Binomial-Heaps_SkewBinomialHeap", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942319436397, "lm_q2_score": 0.7981867825403177, "lm_q1q2_score": 0.7106210885092971}} {"text": "[STATEMENT]\nlemma lubmap_below_cong:\n assumes \"\\ k v. m k = Some v \\ f1 k v \\ (f2 k v :: 'a :: Join_cpo)\"\n shows \"(\\ k\\v\\m. f1 k v) \\ (\\ k\\v\\m. f2 k v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lub (mapCollect f1 m) \\ lub (mapCollect f2 m)\n[PROOF STEP]\napply (rule lubmap_belowI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\k v. m k = Some v \\ f1 k v \\ lub (mapCollect f2 m)\n[PROOF STEP]\napply (rule below_trans[OF assms], assumption)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\k v. m k = Some v \\ f2 k v \\ lub (mapCollect f2 m)\n[PROOF STEP]\napply (rule below_lubmapI, assumption)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 348, "file": "Call_Arity_AList-Utils-HOLCF", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.890294223211224, "lm_q2_score": 0.798186784940666, "lm_q1q2_score": 0.7106210836762145}} {"text": "[STATEMENT]\nlemma eventually_powr_const_mono_nonpos:\n assumes \"p \\ (0 :: real)\" \"eventually (\\x. l x > 0) at_top\" \"eventually (\\x. l x \\ f x) at_top\"\n \"eventually (\\x. f x \\ g x) at_top\"\n shows \"eventually (\\x. f x powr p \\ g x powr p) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. g x powr p \\ f x powr p\n[PROOF STEP]\nusing assms(2-4)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in at_top. 0 < l x\n\\\\<^sub>F x in at_top. l x \\ f x\n\\\\<^sub>F x in at_top. f x \\ g x\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. g x powr p \\ f x powr p\n[PROOF STEP]\nby eventually_elim (auto simp: assms(1) intro!: powr_mono2')", "meta": {"llama_tokens": 333, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.7981867729389246, "lm_q1q2_score": 0.7106210753144997}} {"text": "[STATEMENT]\nlemma multiplicity_power_nat: \n \"prime p \\ n > 0 \\ multiplicity p (n ^ k :: nat) = k * multiplicity p n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\prime p; 0 < n\\ \\ multiplicity p (n ^ k) = k * multiplicity p n\n[PROOF STEP]\nby (induction k) (simp_all add: prime_elem_multiplicity_mult_distrib)", "meta": {"llama_tokens": 139, "file": "Prime_Harmonic_Series_Prime_Harmonic_Misc", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.7981867705385762, "lm_q1q2_score": 0.7106210731774835}} {"text": "[STATEMENT]\nlemma matrix_invertible_gen:\n fixes f :: \"'a::field^'m \\ 'a::field^'n\"\n assumes \"Vector_Spaces.linear (*s) (*s) f\"\n shows \"invertible (matrix f) \\ (\\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id)\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. invertible (matrix f) = (\\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. invertible (matrix f) \\ \\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id\n 2. \\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id \\ invertible (matrix f)\n[PROOF STEP]\nassume ?lhs\n[PROOF STATE]\nproof (state)\nthis:\ninvertible (matrix f)\n\ngoal (2 subgoals):\n 1. invertible (matrix f) \\ \\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id\n 2. \\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id \\ invertible (matrix f)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ninvertible (matrix f)\n[PROOF STEP]\nshow ?rhs\n[PROOF STATE]\nproof (prove)\nusing this:\ninvertible (matrix f)\n\ngoal (1 subgoal):\n 1. \\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id\n[PROOF STEP]\nby (metis assms invertible_def left_right_inverse_eq matrix_left_invertible_gen matrix_right_invertible_gen)\n[PROOF STATE]\nproof (state)\nthis:\n\\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id\n\ngoal (1 subgoal):\n 1. \\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id \\ invertible (matrix f)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id \\ invertible (matrix f)\n[PROOF STEP]\nassume ?rhs\n[PROOF STATE]\nproof (state)\nthis:\n\\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id\n\ngoal (1 subgoal):\n 1. \\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id \\ invertible (matrix f)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id\n[PROOF STEP]\nshow ?lhs\n[PROOF STATE]\nproof (prove)\nusing this:\n\\g. Vector_Spaces.linear (*s) (*s) g \\ f \\ g = id \\ g \\ f = id\n\ngoal (1 subgoal):\n 1. invertible (matrix f)\n[PROOF STEP]\nby (metis assms invertible_def matrix_compose_gen matrix_id_mat_1)\n[PROOF STATE]\nproof (state)\nthis:\ninvertible (matrix f)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1193, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.7981867729389246, "lm_q1q2_score": 0.7106210706677671}} {"text": "[STATEMENT]\nlemma cos_sin_squared_add_cpx: \n \"complex_of_real (cos (\\/2)) * complex_of_real (cos (\\/2)) -\n \\*complex_of_real (sin (\\/2)) * (\\*complex_of_real (sin (\\/2))) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (cos (\\ / 2)) * complex_of_real (cos (\\ / 2)) - \\ * complex_of_real (sin (\\ / 2)) * (\\ * complex_of_real (sin (\\ / 2))) = 1\n[PROOF STEP]\napply (auto simp add: algebra_simps)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (cos (\\ / 2)) * complex_of_real (cos (\\ / 2)) + complex_of_real (sin (\\ / 2)) * complex_of_real (sin (\\ / 2)) = 1\n[PROOF STEP]\nby (metis of_real_add of_real_hom.hom_one of_real_mult sin_cos_squared_add3)", "meta": {"llama_tokens": 329, "file": "Isabelle_Marries_Dirac_Quantum_Prisoners_Dilemma", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026663679976, "lm_q2_score": 0.7745833789613196, "lm_q1q2_score": 0.7105273988455516}} {"text": "[STATEMENT]\nlemma ZmX_inv:\n \"ZmX * ZmX = 1\\<^sub>m 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ZmX * ZmX = 1\\<^sub>m 2\n[PROOF STEP]\nunfolding ZmX_def X_def Z_def times_mat_def one_mat_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix.mat (dim_row (complex_of_real (1 / sqrt 2) \\\\<^sub>m (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) - Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1)))) (dim_col (complex_of_real (1 / sqrt 2) \\\\<^sub>m (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) - Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1)))) (\\(i, j). Matrix.row (complex_of_real (1 / sqrt 2) \\\\<^sub>m (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) - Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1))) i \\ Matrix.col (complex_of_real (1 / sqrt 2) \\\\<^sub>m (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) - Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1))) j) = Matrix.mat 2 2 (\\(i, j). if i = j then 1 else 0)\n[PROOF STEP]\napply (rule cong_mat, simp+)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. \\i < 2; j < 2\\ \\ (i = j \\ Matrix.row (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) - Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1)) j \\ Matrix.col (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) - Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1)) j = complex_of_real (sqrt 2) * complex_of_real (sqrt 2)) \\ (i \\ j \\ Matrix.row (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) - Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1)) i \\ Matrix.col (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) - Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1)) j = 0)\n[PROOF STEP]\napply (auto simp add: Matrix.scalar_prod_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. j < 2 \\ complex_of_real (sqrt 2) * complex_of_real (sqrt 2) = 2\n[PROOF STEP]\napply (auto simp add: Gates.csqrt_2_sq)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1066, "file": "Projective_Measurements_CHSH_Inequality", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026595857203, "lm_q2_score": 0.7745833737577158, "lm_q1q2_score": 0.7105273888188327}} {"text": "[STATEMENT]\nlemma dyadics_levels: \"dyadics = (\\K. \\k m. {of_nat m / 2^k})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dyadics = (\\K. \\km. {of_nat m / (2::'a) ^ k})\n[PROOF STEP]\nunfolding dyadics_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k m. {of_nat m / (2::'a) ^ k}) = (\\K. \\km. {of_nat m / (2::'a) ^ k})\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 213, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772417253255, "lm_q2_score": 0.8128673178375735, "lm_q1q2_score": 0.7105088230641298}} {"text": "[STATEMENT]\nlemma contour_integral_circlepath:\n assumes \"r > 0\"\n shows \"contour_integral (circlepath z r) (\\w. 1 / (w - z)) = 2 * complex_of_real pi * \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. contour_integral (circlepath z r) (\\w. 1 / (w - z)) = 2 * complex_of_real pi * \\\n[PROOF STEP]\nproof (rule contour_integral_unique)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\w. 1 / (w - z)) has_contour_integral 2 * complex_of_real pi * \\) (circlepath z r)\n[PROOF STEP]\nshow \"((\\w. 1 / (w - z)) has_contour_integral 2 * complex_of_real pi * \\) (circlepath z r)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\w. 1 / (w - z)) has_contour_integral 2 * complex_of_real pi * \\) (circlepath z r)\n[PROOF STEP]\nunfolding has_contour_integral_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. 1 / (circlepath z r x - z) * vector_derivative (circlepath z r) (at x within {0..1})) has_integral 2 * complex_of_real pi * \\) {0..1}\n[PROOF STEP]\nusing assms has_integral_const_real [of _ 0 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < r\n((\\x. ?c) has_integral content {0..1} *\\<^sub>R ?c) {0..1}\n\ngoal (1 subgoal):\n 1. ((\\x. 1 / (circlepath z r x - z) * vector_derivative (circlepath z r) (at x within {0..1})) has_integral 2 * complex_of_real pi * \\) {0..1}\n[PROOF STEP]\napply (subst has_integral_cong)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. \\0 < r; \\c. ((\\x. c) has_integral content {0..1} *\\<^sub>R c) {0..1}; x \\ {0..1}\\ \\ 1 / (circlepath z r x - z) * vector_derivative (circlepath z r) (at x within {0..1}) = ?g3 x\n 2. \\0 < r; \\c. ((\\x. c) has_integral content {0..1} *\\<^sub>R c) {0..1}\\ \\ (?g3 has_integral 2 * complex_of_real pi * \\) {0..1}\n[PROOF STEP]\napply (simp add: vector_derivative_circlepath01)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < r; \\c. ((\\x. c) has_integral content {0..1} *\\<^sub>R c) {0..1}\\ \\ ((\\x. 2 * complex_of_real pi * \\ * complex_of_real r * exp (2 * complex_of_real pi * \\ * complex_of_real x) / (circlepath z r x - z)) has_integral 2 * complex_of_real pi * \\) {0..1}\n[PROOF STEP]\napply (force simp: circlepath)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n((\\w. 1 / (w - z)) has_contour_integral 2 * complex_of_real pi * \\) (circlepath z r)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1134, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772384450968, "lm_q2_score": 0.8128673201042493, "lm_q1q2_score": 0.7105088223789887}} {"text": "[STATEMENT]\nlemma left_invertible_iff_invertible:\n fixes M :: \"real^'n^'n\"\n shows \"(\\ N. N ** M = mat 1) \\ invertible M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\N. N ** M = mat 1) = invertible M\n[PROOF STEP]\nby (simp add: invertible_def matrix_left_right_inverse)", "meta": {"llama_tokens": 127, "file": "Tarskis_Geometry_Linear_Algebra2", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772351648677, "lm_q2_score": 0.8128673201042493, "lm_q1q2_score": 0.7105088197125977}} {"text": "[STATEMENT]\nlemma connectedin_Un:\n \"\\connectedin X S; connectedin X T; S \\ T \\ {}\\ \\ connectedin X (S \\ T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\connectedin X S; connectedin X T; S \\ T \\ {}\\ \\ connectedin X (S \\ T)\n[PROOF STEP]\nusing connectedin_Union [of \"{S,T}\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\Sa. Sa \\ {S, T} \\ connectedin ?X Sa; \\ {S, T} \\ {}\\ \\ connectedin ?X (\\ {S, T})\n\ngoal (1 subgoal):\n 1. \\connectedin X S; connectedin X T; S \\ T \\ {}\\ \\ connectedin X (S \\ T)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 293, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.874077222043951, "lm_q2_score": 0.8128673269042767, "lm_q1q2_score": 0.7105088149907823}} {"text": "[STATEMENT]\nlemma abs_non_negative_sum: assumes \" abs x + abs y = 0\"\nshows \"abs x= 0\" and \"abs y = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Preliminaries.abs x = 0 &&& Preliminaries.abs y = 0\n[PROOF STEP]\nusing abs_def diff_0 abs_non_negative neg_0_le_iff_le \n add_nonneg_eq_0_iff assms\n[PROOF STATE]\nproof (prove)\nusing this:\nPreliminaries.abs ?x \\ if 0 \\ ?x then ?x else 0 - ?x\n(0::?'a) - ?a = - ?a\n0 \\ Preliminaries.abs ?x\n((0::?'a) \\ - ?a) = (?a \\ (0::?'a))\n\\(0::?'a) \\ ?x; (0::?'a) \\ ?y\\ \\ (?x + ?y = (0::?'a)) = (?x = (0::?'a) \\ ?y = (0::?'a))\nPreliminaries.abs x + Preliminaries.abs y = 0\n\ngoal (1 subgoal):\n 1. Preliminaries.abs x = 0 &&& Preliminaries.abs y = 0\n[PROOF STEP]\napply (metis)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Preliminaries.abs y = 0\n[PROOF STEP]\nby (metis abs_non_negative add_nonneg_eq_0_iff assms)", "meta": {"llama_tokens": 438, "file": "Knot_Theory_Preliminaries", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772384450968, "lm_q2_score": 0.8128673110375457, "lm_q1q2_score": 0.7105088144539895}} {"text": "[STATEMENT]\nlemma split_sum_first_elt_less: assumes \"ni\\{n..i\\{Suc n .. sum ?g {?m..s::nat store. s ''x'' = x \\ s ''y'' = y)\n (WHILE (\\s. s ''y'' \\ 0) INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) \n DO\n (''z'' ::= (\\s. s ''y''));\n (''y'' ::= (\\s. s ''x'' mod s ''y''));\n (''x'' ::= (\\s. s ''z''))\n OD)\n POST (\\s. s ''x'' = gcd x y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. PRE (\\s. s ''x'' = x \\ s ''y'' = y) (WHILE (\\s. s ''y'' \\ 0) INV (\\s. gcd (s ''x'') (s ''y'') = gcd x y) DO (''z'' ::= (\\s. s ''y'')) ; (''y'' ::= (\\s. s ''x'' mod s ''y'')) ; (''x'' ::= (\\s. s ''z'')) OD) POST (\\s. s ''x'' = gcd x y)\n[PROOF STEP]\napply hoare\n[PROOF STATE]\nproof (prove)\ngoal (5 subgoals):\n 1. \\s. s ''x'' = x \\ s ''y'' = y \\ gcd (s ''x'') (s ''y'') = gcd x y\n 2. \\s. gcd (s ''x'') (s ''y'') = gcd x y \\ \\ s ''y'' \\ 0 \\ s ''x'' = gcd x y\n 3. \\s. ?Q28 s \\ gcd ((s(''x'' := s ''z'')) ''x'') ((s(''x'' := s ''z'')) ''y'') = gcd x y\n 4. \\s. ?Q20 s \\ ?Q28 (s(''y'' := s ''x'' mod s ''y''))\n 5. \\s. ((\\s. gcd (s ''x'') (s ''y'') = gcd x y) \\ (\\s. s ''y'' \\ 0)) s \\ ?Q20 (s(''z'' := s ''y''))\n[PROOF STEP]\nusing gcd_red_nat\n[PROOF STATE]\nproof (prove)\nusing this:\ngcd ?x ?y = gcd ?y (?x mod ?y)\n\ngoal (5 subgoals):\n 1. \\s. s ''x'' = x \\ s ''y'' = y \\ gcd (s ''x'') (s ''y'') = gcd x y\n 2. \\s. gcd (s ''x'') (s ''y'') = gcd x y \\ \\ s ''y'' \\ 0 \\ s ''x'' = gcd x y\n 3. \\s. ?Q28 s \\ gcd ((s(''x'' := s ''z'')) ''x'') ((s(''x'' := s ''z'')) ''y'') = gcd x y\n 4. \\s. ?Q20 s \\ ?Q28 (s(''y'' := s ''x'' mod s ''y''))\n 5. \\s. ((\\s. gcd (s ''x'') (s ''y'') = gcd x y) \\ (\\s. s ''y'' \\ 0)) s \\ ?Q20 (s(''z'' := s ''y''))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 990, "file": "Algebraic_VCs_AVC_KAT_VC_KAT_Examples2", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505428129514, "lm_q2_score": 0.7853085758631159, "lm_q1q2_score": 0.7104298294302336}} {"text": "[STATEMENT]\nlemma divide_option_mpoly_poly:\n fixes p q :: \"'a :: idom_divide mpoly poly\"\n shows \"p div? q =\n (if p = 0 then Some 0\n else if q = 0 then None\n else let dp = Polynomial.degree p; dq = Polynomial.degree q\n in if dp < dq then None\n else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of\n None \\ None\n | Some c \\ (\n case (p - Polynomial.monom c (dp - dq) * q) div? q of\n None \\ None\n | Some r \\ Some (Polynomial.monom c (dp - dq) + r)))\"\n (is \"_ = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nproof (cases \"p = 0\"; cases \"q = 0\")\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. \\p = 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\p = 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 3. \\p \\ 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 4. \\p \\ 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nassume [simp]: \"p \\ 0\" \"q \\ 0\"\n[PROOF STATE]\nproof (state)\nthis:\np \\ 0\nq \\ 0\n\ngoal (4 subgoals):\n 1. \\p = 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\p = 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 3. \\p \\ 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 4. \\p \\ 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\ndefine dp where \"dp = Polynomial.degree p\"\n[PROOF STATE]\nproof (state)\nthis:\ndp = Polynomial.degree p\n\ngoal (4 subgoals):\n 1. \\p = 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\p = 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 3. \\p \\ 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 4. \\p \\ 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\ndefine dq where \"dq = Polynomial.degree q\"\n[PROOF STATE]\nproof (state)\nthis:\ndq = Polynomial.degree q\n\ngoal (4 subgoals):\n 1. \\p = 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\p = 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 3. \\p \\ 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 4. \\p \\ 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\ndefine cp where \"cp = Polynomial.lead_coeff p\"\n[PROOF STATE]\nproof (state)\nthis:\ncp = Polynomial.lead_coeff p\n\ngoal (4 subgoals):\n 1. \\p = 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\p = 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 3. \\p \\ 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 4. \\p \\ 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\ndefine cq where \"cq = Polynomial.lead_coeff q\"\n[PROOF STATE]\nproof (state)\nthis:\ncq = Polynomial.lead_coeff q\n\ngoal (4 subgoals):\n 1. \\p = 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\p = 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 3. \\p \\ 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 4. \\p \\ 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\ndefine mon where \"mon = Polynomial.monom (cp div cq) (dp - dq)\"\n[PROOF STATE]\nproof (state)\nthis:\nmon = Polynomial.monom (cp div cq) (dp - dq)\n\ngoal (4 subgoals):\n 1. \\p = 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\p = 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 3. \\p \\ 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 4. \\p \\ 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nproof (cases \"dp < dq\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. dp < dq \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\ dp < dq \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\ndp < dq\n\ngoal (2 subgoals):\n 1. dp < dq \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\ dp < dq \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nhence \"\\q dvd p\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndp < dq\n\ngoal (1 subgoal):\n 1. \\ q dvd p\n[PROOF STEP]\nunfolding dp_def dq_def\n[PROOF STATE]\nproof (prove)\nusing this:\nPolynomial.degree p < Polynomial.degree q\n\ngoal (1 subgoal):\n 1. \\ q dvd p\n[PROOF STEP]\nby (meson \\p \\ 0\\ divides_degree leD)\n[PROOF STATE]\nproof (state)\nthis:\n\\ q dvd p\n\ngoal (2 subgoals):\n 1. dp < dq \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\ dp < dq \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ q dvd p\n\ngoal (1 subgoal):\n 1. p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nusing True\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ q dvd p\ndp < dq\n\ngoal (1 subgoal):\n 1. p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nby (simp add: divide_option_def dp_def dq_def)\n[PROOF STATE]\nproof (state)\nthis:\np div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n\ngoal (1 subgoal):\n 1. \\ dp < dq \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ dp < dq \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\ncase deg: False\n[PROOF STATE]\nproof (state)\nthis:\n\\ dp < dq\n\ngoal (1 subgoal):\n 1. \\ dp < dq \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nproof (cases \"cq dvd cp\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. cq dvd cp \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\ cq dvd cp \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ cq dvd cp\n\ngoal (2 subgoals):\n 1. cq dvd cp \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\ cq dvd cp \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nhence \"\\q dvd p\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ cq dvd cp\n\ngoal (1 subgoal):\n 1. \\ q dvd p\n[PROOF STEP]\nunfolding cq_def cp_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ Polynomial.lead_coeff q dvd Polynomial.lead_coeff p\n\ngoal (1 subgoal):\n 1. \\ q dvd p\n[PROOF STEP]\nusing poly_lead_coeff_dvd_lead_coeff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ Polynomial.lead_coeff q dvd Polynomial.lead_coeff p\n?p dvd ?q \\ Polynomial.lead_coeff ?p dvd Polynomial.lead_coeff ?q\n\ngoal (1 subgoal):\n 1. \\ q dvd p\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\ q dvd p\n\ngoal (2 subgoals):\n 1. cq dvd cp \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\ cq dvd cp \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ q dvd p\n\ngoal (1 subgoal):\n 1. p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nusing deg False\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ q dvd p\n\\ dp < dq\n\\ cq dvd cp\n\ngoal (1 subgoal):\n 1. p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nby (simp add: dp_def dq_def Let_def divide_option_def cp_def cq_def)\n[PROOF STATE]\nproof (state)\nthis:\np div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n\ngoal (1 subgoal):\n 1. cq dvd cp \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. cq dvd cp \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\ncase dvd1: True\n[PROOF STATE]\nproof (state)\nthis:\ncq dvd cp\n\ngoal (1 subgoal):\n 1. cq dvd cp \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nproof (cases \"q dvd (p - mon * q)\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\ q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ q dvd p - mon * q\n\ngoal (2 subgoals):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\ q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nhence \"\\q dvd p\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ q dvd p - mon * q\n\ngoal (1 subgoal):\n 1. \\ q dvd p\n[PROOF STEP]\nby (meson dvd_diff dvd_triv_right)\n[PROOF STATE]\nproof (state)\nthis:\n\\ q dvd p\n\ngoal (2 subgoals):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\ q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ q dvd p\n\ngoal (1 subgoal):\n 1. p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nusing deg dvd1 False\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ q dvd p\n\\ dp < dq\ncq dvd cp\n\\ q dvd p - mon * q\n\ngoal (1 subgoal):\n 1. p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nby (simp add: dp_def dq_def Let_def divide_option_def cp_def cq_def mon_def)\n[PROOF STATE]\nproof (state)\nthis:\np div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n\ngoal (1 subgoal):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\ncase dvd2: True\n[PROOF STATE]\nproof (state)\nthis:\nq dvd p - mon * q\n\ngoal (1 subgoal):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nhence \"q dvd p\"\n[PROOF STATE]\nproof (prove)\nusing this:\nq dvd p - mon * q\n\ngoal (1 subgoal):\n 1. q dvd p\n[PROOF STEP]\nby (metis diff_eq_eq dvd_add dvd_triv_right)\n[PROOF STATE]\nproof (state)\nthis:\nq dvd p\n\ngoal (1 subgoal):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nhave \"?rhs = Some (mon + (p - mon * q) div q)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r)) = Some (mon + (p - mon * q) div q)\n[PROOF STEP]\nusing deg dvd1 dvd2\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ dp < dq\ncq dvd cp\nq dvd p - mon * q\n\ngoal (1 subgoal):\n 1. (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r)) = Some (mon + (p - mon * q) div q)\n[PROOF STEP]\nby (simp add: dp_def dq_def Let_def divide_option_def cp_def cq_def mon_def)\n[PROOF STATE]\nproof (state)\nthis:\n(if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r)) = Some (mon + (p - mon * q) div q)\n\ngoal (1 subgoal):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r)) = Some (mon + (p - mon * q) div q)\n\ngoal (1 subgoal):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nhave \"mon + (p - mon * q) div q = p div q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mon + (p - mon * q) div q = p div q\n[PROOF STEP]\nusing dvd2\n[PROOF STATE]\nproof (prove)\nusing this:\nq dvd p - mon * q\n\ngoal (1 subgoal):\n 1. mon + (p - mon * q) div q = p div q\n[PROOF STEP]\nby (elim dvdE) (auto simp: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nmon + (p - mon * q) div q = p div q\n\ngoal (1 subgoal):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmon + (p - mon * q) div q = p div q\n\ngoal (1 subgoal):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nhave \"Some \\ = p div? q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Some (p div q) = p div? q\n[PROOF STEP]\nusing \\q dvd p\\\n[PROOF STATE]\nproof (prove)\nusing this:\nq dvd p\n\ngoal (1 subgoal):\n 1. Some (p div q) = p div? q\n[PROOF STEP]\nby (simp add: divide_option_def)\n[PROOF STATE]\nproof (state)\nthis:\nSome (p div q) = p div? q\n\ngoal (1 subgoal):\n 1. q dvd p - mon * q \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r)) = p div? q\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r)) = p div? q\n\ngoal (1 subgoal):\n 1. p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\np div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\np div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\np div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\np div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n\ngoal (3 subgoals):\n 1. \\p = 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 2. \\p = 0; q \\ 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n 3. \\p \\ 0; q = 0\\ \\ p div? q = (if p = 0 then Some 0 else if q = 0 then None else let dp = Polynomial.degree p; dq = Polynomial.degree q in if dp < dq then None else case Polynomial.lead_coeff p div? Polynomial.lead_coeff q of None \\ None | Some c \\ case (p - Polynomial.monom c (dp - dq) * q) div? q of None \\ None | Some r \\ Some (Polynomial.monom c (dp - dq) + r))\n[PROOF STEP]\nqed (auto simp: divide_option_def)", "meta": {"llama_tokens": 16961, "file": "Factor_Algebraic_Polynomial_MPoly_Divide", "length": 60, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505248181417, "lm_q2_score": 0.785308580887758, "lm_q1q2_score": 0.7104298198443003}} {"text": "[STATEMENT]\nlemma lim_null_mult_left_bounded:\n fixes f :: \"'a \\ 'b::real_normed_div_algebra\"\n assumes g: \"eventually (\\x. norm(g x) \\ B) F\" and f: \"(f \\ 0) F\"\n shows \"((\\z. g z * f z) \\ 0) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\z. g z * f z) \\ (0::'b)) F\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\z. g z * f z) \\ (0::'b)) F\n[PROOF STEP]\nhave \"((\\x. norm (g x) * norm (f x)) \\ 0) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. norm (g x) * norm (f x)) \\ 0) F\n[PROOF STEP]\nproof (rule Lim_null_comparison)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\\\<^sub>F x in F. norm (norm (g x) * norm (f x)) \\ ?g x\n 2. (?g \\ 0) F\n[PROOF STEP]\nshow \"\\\\<^sub>F x in F. norm (norm (g x) * norm (f x)) \\ B * norm (f x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. norm (norm (g x) * norm (f x)) \\ B * norm (f x)\n[PROOF STEP]\nby (simp add: eventually_mono [OF g] mult_right_mono)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in F. norm (norm (g x) * norm (f x)) \\ B * norm (f x)\n\ngoal (1 subgoal):\n 1. ((\\x. B * norm (f x)) \\ 0) F\n[PROOF STEP]\nshow \"((\\x. B * norm (f x)) \\ 0) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. B * norm (f x)) \\ 0) F\n[PROOF STEP]\nby (simp add: f tendsto_mult_right_zero tendsto_norm_zero)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. B * norm (f x)) \\ 0) F\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. norm (g x) * norm (f x)) \\ 0) F\n\ngoal (1 subgoal):\n 1. ((\\z. g z * f z) \\ (0::'b)) F\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n((\\x. norm (g x) * norm (f x)) \\ 0) F\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. norm (g x) * norm (f x)) \\ 0) F\n\ngoal (1 subgoal):\n 1. ((\\z. g z * f z) \\ (0::'b)) F\n[PROOF STEP]\nby (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)\n[PROOF STATE]\nproof (state)\nthis:\n((\\z. g z * f z) \\ (0::'b)) F\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1078, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8438951104066293, "lm_q2_score": 0.8418256532040708, "lm_q1q2_score": 0.7104125525537822}} {"text": "[STATEMENT]\nlemma inv_fun_plus [simp]: \n \"\\inv (f1 :: nat \\ 'a::monoid_add); inv f2\\ \\ inv (fun_plus f1 f2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Abstract_Linear_Poly.inv f1; Abstract_Linear_Poly.inv f2\\ \\ Abstract_Linear_Poly.inv (fun_plus f1 f2)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\Abstract_Linear_Poly.inv f1; Abstract_Linear_Poly.inv f2\\ \\ Abstract_Linear_Poly.inv (fun_plus f1 f2)\n[PROOF STEP]\nhave *: \"{v. f1 v + f2 v \\ (0 :: 'a)} \\ {v. f1 v \\ (0 :: 'a)} \\ {v. f2 v \\ (0 :: 'a)}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {v. f1 v + f2 v \\ (0::'a)} \\ {v. f1 v \\ (0::'a)} \\ {v. f2 v \\ (0::'a)}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{v. f1 v + f2 v \\ (0::'a)} \\ {v. f1 v \\ (0::'a)} \\ {v. f2 v \\ (0::'a)}\n\ngoal (1 subgoal):\n 1. \\Abstract_Linear_Poly.inv f1; Abstract_Linear_Poly.inv f2\\ \\ Abstract_Linear_Poly.inv (fun_plus f1 f2)\n[PROOF STEP]\nassume \"inv f1\" \"inv f2\"\n[PROOF STATE]\nproof (state)\nthis:\nAbstract_Linear_Poly.inv f1\nAbstract_Linear_Poly.inv f2\n\ngoal (1 subgoal):\n 1. \\Abstract_Linear_Poly.inv f1; Abstract_Linear_Poly.inv f2\\ \\ Abstract_Linear_Poly.inv (fun_plus f1 f2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nAbstract_Linear_Poly.inv f1\nAbstract_Linear_Poly.inv f2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nAbstract_Linear_Poly.inv f1\nAbstract_Linear_Poly.inv f2\n\ngoal (1 subgoal):\n 1. Abstract_Linear_Poly.inv (fun_plus f1 f2)\n[PROOF STEP]\nusing *\n[PROOF STATE]\nproof (prove)\nusing this:\nAbstract_Linear_Poly.inv f1\nAbstract_Linear_Poly.inv f2\n{v. f1 v + f2 v \\ (0::'a)} \\ {v. f1 v \\ (0::'a)} \\ {v. f2 v \\ (0::'a)}\n\ngoal (1 subgoal):\n 1. Abstract_Linear_Poly.inv (fun_plus f1 f2)\n[PROOF STEP]\nby (auto simp add: finite_subset)\n[PROOF STATE]\nproof (state)\nthis:\nAbstract_Linear_Poly.inv (fun_plus f1 f2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 983, "file": "Simplex_Abstract_Linear_Poly", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256512199033, "lm_q2_score": 0.8438950966654774, "lm_q1q2_score": 0.7104125393116987}} {"text": "[STATEMENT]\nlemma evenperm_comp:\n assumes \"permutation p\" \"permutation q\"\n shows \"evenperm (p \\ q) \\ evenperm p = evenperm q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. evenperm (p \\ q) = (evenperm p = evenperm q)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. evenperm (p \\ q) = (evenperm p = evenperm q)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\npermutation p\npermutation q\n[PROOF STEP]\nobtain n m where n: \"swapidseq n p\" and m: \"swapidseq m q\"\n[PROOF STATE]\nproof (prove)\nusing this:\npermutation p\npermutation q\n\ngoal (1 subgoal):\n 1. (\\n m. \\swapidseq n p; swapidseq m q\\ \\ thesis) \\ thesis\n[PROOF STEP]\nunfolding permutation_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n. swapidseq n p\n\\n. swapidseq n q\n\ngoal (1 subgoal):\n 1. (\\n m. \\swapidseq n p; swapidseq m q\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nswapidseq n p\nswapidseq m q\n\ngoal (1 subgoal):\n 1. evenperm (p \\ q) = (evenperm p = evenperm q)\n[PROOF STEP]\nhave \"even (n + m) \\ (even n \\ even m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. even (n + m) = (even n = even m)\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\neven (n + m) = (even n = even m)\n\ngoal (1 subgoal):\n 1. evenperm (p \\ q) = (evenperm p = evenperm q)\n[PROOF STEP]\nfrom evenperm_unique[OF n refl] evenperm_unique[OF m refl]\n and evenperm_unique[OF swapidseq_comp_add[OF n m] this]\n[PROOF STATE]\nproof (chain)\npicking this:\nevenperm p = even n\nevenperm q = even m\nevenperm (p \\ q) = (even n = even m)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nevenperm p = even n\nevenperm q = even m\nevenperm (p \\ q) = (even n = even m)\n\ngoal (1 subgoal):\n 1. evenperm (p \\ q) = (evenperm p = evenperm q)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nevenperm (p \\ q) = (evenperm p = evenperm q)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 878, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256432832333, "lm_q2_score": 0.8438951025545426, "lm_q1q2_score": 0.710412537571548}} {"text": "[STATEMENT]\nlemma path_connectedin_Un:\n \"\\path_connectedin X S; path_connectedin X T; S \\ T \\ {}\\\n \\ path_connectedin X (S \\ T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\path_connectedin X S; path_connectedin X T; S \\ T \\ {}\\ \\ path_connectedin X (S \\ T)\n[PROOF STEP]\nby (blast intro: path_connectedin_Union [of \"{S,T}\", simplified])", "meta": {"llama_tokens": 170, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467675095294, "lm_q2_score": 0.8080672135527632, "lm_q1q2_score": 0.7104096787253443}} {"text": "[STATEMENT]\nlemma aff_dim_sums_Int_0:\n assumes \"affine S\"\n and \"affine T\"\n and \"0 \\ S\" \"0 \\ T\"\n shows \"aff_dim {x + y| x y. x \\ S \\ y \\ T} = (aff_dim S + aff_dim T) - aff_dim(S \\ T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. aff_dim {x + y |x y. x \\ S \\ y \\ T} = aff_dim S + aff_dim T - aff_dim (S \\ T)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. aff_dim {x + y |x y. x \\ S \\ y \\ T} = aff_dim S + aff_dim T - aff_dim (S \\ T)\n[PROOF STEP]\nhave \"0 \\ {x + y |x y. x \\ S \\ y \\ T}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) \\ {x + y |x y. x \\ S \\ y \\ T}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\naffine S\naffine T\n(0::'a) \\ S\n(0::'a) \\ T\n\ngoal (1 subgoal):\n 1. (0::'a) \\ {x + y |x y. x \\ S \\ y \\ T}\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) \\ {x + y |x y. x \\ S \\ y \\ T}\n\ngoal (1 subgoal):\n 1. aff_dim {x + y |x y. x \\ S \\ y \\ T} = aff_dim S + aff_dim T - aff_dim (S \\ T)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(0::'a) \\ {x + y |x y. x \\ S \\ y \\ T}\n[PROOF STEP]\nhave 0: \"0 \\ affine hull {x + y |x y. x \\ S \\ y \\ T}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) \\ {x + y |x y. x \\ S \\ y \\ T}\n\ngoal (1 subgoal):\n 1. (0::'a) \\ affine hull {x + y |x y. x \\ S \\ y \\ T}\n[PROOF STEP]\nby (metis (lifting) hull_inc)\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) \\ affine hull {x + y |x y. x \\ S \\ y \\ T}\n\ngoal (1 subgoal):\n 1. aff_dim {x + y |x y. x \\ S \\ y \\ T} = aff_dim S + aff_dim T - aff_dim (S \\ T)\n[PROOF STEP]\nhave sub: \"subspace S\" \"subspace T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. subspace S &&& subspace T\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\naffine S\naffine T\n(0::'a) \\ S\n(0::'a) \\ T\n\ngoal (1 subgoal):\n 1. subspace S &&& subspace T\n[PROOF STEP]\nby (auto simp: subspace_affine)\n[PROOF STATE]\nproof (state)\nthis:\nsubspace S\nsubspace T\n\ngoal (1 subgoal):\n 1. aff_dim {x + y |x y. x \\ S \\ y \\ T} = aff_dim S + aff_dim T - aff_dim (S \\ T)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. aff_dim {x + y |x y. x \\ S \\ y \\ T} = aff_dim S + aff_dim T - aff_dim (S \\ T)\n[PROOF STEP]\nusing dim_sums_Int [OF sub]\n[PROOF STATE]\nproof (prove)\nusing this:\ndim {x + y |x y. x \\ S \\ y \\ T} + dim (S \\ T) = dim S + dim T\n\ngoal (1 subgoal):\n 1. aff_dim {x + y |x y. x \\ S \\ y \\ T} = aff_dim S + aff_dim T - aff_dim (S \\ T)\n[PROOF STEP]\nby (simp add: aff_dim_zero assms 0 hull_inc)\n[PROOF STATE]\nproof (state)\nthis:\naff_dim {x + y |x y. x \\ S \\ y \\ T} = aff_dim S + aff_dim T - aff_dim (S \\ T)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1454, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467643431002, "lm_q2_score": 0.8080672135527632, "lm_q1q2_score": 0.7104096761666568}} {"text": "[STATEMENT]\nlemma card_UN_disjoint':\n assumes \"disjoint_family_on A I\" \"\\i. i \\ I \\ finite (A i)\" \"finite I\"\n shows \"card (\\i\\I. A i) = (\\i\\I. card (A i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (\\ (A ` I)) = (\\i\\I. card (A i))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndisjoint_family_on A I\n?i \\ I \\ finite (A ?i)\nfinite I\n\ngoal (1 subgoal):\n 1. card (\\ (A ` I)) = (\\i\\I. card (A i))\n[PROOF STEP]\nby (simp add: card_UN_disjoint disjoint_family_on_def)", "meta": {"llama_tokens": 251, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467675095294, "lm_q2_score": 0.808067208930584, "lm_q1q2_score": 0.7104096746617704}} {"text": "[STATEMENT]\nlemma floor_log_nat_eq_powr_iff: fixes b n k :: nat\n shows \"\\ b \\ 2; k > 0 \\ \\\n floor (log b (real k)) = n \\ b^n \\ k \\ k < b^(n+1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\2 \\ b; 0 < k\\ \\ (\\log (real b) (real k)\\ = int n) = (b ^ n \\ k \\ k < b ^ (n + 1))\n[PROOF STEP]\nby (auto simp: floor_log_eq_powr_iff powr_add powr_realpow\n of_nat_power[symmetric] of_nat_mult[symmetric] ac_simps\n simp del: of_nat_power of_nat_mult)", "meta": {"llama_tokens": 250, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045937171068, "lm_q2_score": 0.8006920092299292, "lm_q1q2_score": 0.7103776287413733}} {"text": "[STATEMENT]\nlemma construct_NofI_prop_R:\n fixes p:: \"real poly\"\n fixes I1:: \"real poly list\"\n fixes I2:: \"real poly list\"\n assumes nonzero: \"p\\0\"\n shows \"construct_NofI_R p I1 I2 =\n rat_of_int (int (card {x. poly p x = 0 \\ (\\q \\ set I1. poly q x = 0) \\ poly (prod_list I2) x > 0}) - \n int (card {x. poly p x = 0 \\ (\\q \\ set I1. poly q x = 0) \\ poly (prod_list I2) x < 0}))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. construct_NofI_R p I1 I2 = rat_of_int (int (card {x. poly p x = 0 \\ (\\q\\set I1. poly q x = 0) \\ 0 < poly (prod_list I2) x}) - int (card {x. poly p x = 0 \\ (\\q\\set I1. poly q x = 0) \\ poly (prod_list I2) x < 0}))\n[PROOF STEP]\nunfolding construct_NofI_R_prop_helper[OF nonzero]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rat_of_int (int (card {x. poly (sum_list (map power2 (p # I1))) x = 0 \\ 0 < poly (prod_list I2) x}) - int (card {x. poly (sum_list (map power2 (p # I1))) x = 0 \\ poly (prod_list I2) x < 0})) = rat_of_int (int (card {x. poly p x = 0 \\ (\\q\\set I1. poly q x = 0) \\ 0 < poly (prod_list I2) x}) - int (card {x. poly p x = 0 \\ (\\q\\set I1. poly q x = 0) \\ poly (prod_list I2) x < 0}))\n[PROOF STEP]\nusing zer_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n(poly (sum_list (map power2 ?ls)) ?x = 0) = (\\i\\set ?ls. poly i ?x = 0)\n\ngoal (1 subgoal):\n 1. rat_of_int (int (card {x. poly (sum_list (map power2 (p # I1))) x = 0 \\ 0 < poly (prod_list I2) x}) - int (card {x. poly (sum_list (map power2 (p # I1))) x = 0 \\ poly (prod_list I2) x < 0})) = rat_of_int (int (card {x. poly p x = 0 \\ (\\q\\set I1. poly q x = 0) \\ 0 < poly (prod_list I2) x}) - int (card {x. poly p x = 0 \\ (\\q\\set I1. poly q x = 0) \\ poly (prod_list I2) x < 0}))\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ls x. (poly (sum_list (map power2 ls)) x = 0) = (\\i\\set ls. poly i x = 0)) \\ rat_of_nat (card {x. (poly p x)\\<^sup>2 + poly (sum_list (map power2 I1)) x = 0 \\ 0 < poly (prod_list I2) x}) - rat_of_nat (card {x. (poly p x)\\<^sup>2 + poly (sum_list (map power2 I1)) x = 0 \\ poly (prod_list I2) x < 0}) = rat_of_nat (card {x. poly p x = 0 \\ (\\q\\set I1. poly q x = 0) \\ 0 < poly (prod_list I2) x}) - rat_of_nat (card {x. poly p x = 0 \\ (\\q\\set I1. poly q x = 0) \\ poly (prod_list I2) x < 0})\n[PROOF STEP]\nby (smt (verit, del_insts) Collect_cong sum_list_map_power2_poly zero_le_power2 zero_less_power2)", "meta": {"llama_tokens": 1199, "file": "BenOr_Kozen_Reif_Renegar_Proofs", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045847699185, "lm_q2_score": 0.800691997339971, "lm_q1q2_score": 0.7103776110286056}} {"text": "[STATEMENT]\nlemma log2_at_1: \"log2 1 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. log2 1 = 0\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. log2 1 = 0\n[PROOF STEP]\nlet ?v = \"log2 1\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. log2 1 = 0\n[PROOF STEP]\nhave S1: \"0 \\ ?v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ log2 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ log2 1\n\ngoal (1 subgoal):\n 1. log2 1 = 0\n[PROOF STEP]\nhave S2: \"?v = Least(%(z::nat). (1::nat)<2^(z+1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. log2 1 = (LEAST z. 1 < 2 ^ (z + 1))\n[PROOF STEP]\nby (simp add: log2_def)\n[PROOF STATE]\nproof (state)\nthis:\nlog2 1 = (LEAST z. 1 < 2 ^ (z + 1))\n\ngoal (1 subgoal):\n 1. log2 1 = 0\n[PROOF STEP]\nhave S3: \"(1::nat)<2^(0+1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 < 2 ^ (0 + 1)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n1 < 2 ^ (0 + 1)\n\ngoal (1 subgoal):\n 1. log2 1 = 0\n[PROOF STEP]\nfrom S3\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < 2 ^ (0 + 1)\n[PROOF STEP]\nhave S4: \"Least(%(z::nat). (1::nat)<2^(z+1)) \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < 2 ^ (0 + 1)\n\ngoal (1 subgoal):\n 1. (LEAST z. 1 < 2 ^ (z + 1)) \\ 0\n[PROOF STEP]\nby (rule Least_le)\n[PROOF STATE]\nproof (state)\nthis:\n(LEAST z. 1 < 2 ^ (z + 1)) \\ 0\n\ngoal (1 subgoal):\n 1. log2 1 = 0\n[PROOF STEP]\nfrom S2 S4\n[PROOF STATE]\nproof (chain)\npicking this:\nlog2 1 = (LEAST z. 1 < 2 ^ (z + 1))\n(LEAST z. 1 < 2 ^ (z + 1)) \\ 0\n[PROOF STEP]\nhave S5: \"?v \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlog2 1 = (LEAST z. 1 < 2 ^ (z + 1))\n(LEAST z. 1 < 2 ^ (z + 1)) \\ 0\n\ngoal (1 subgoal):\n 1. log2 1 \\ 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlog2 1 \\ 0\n\ngoal (1 subgoal):\n 1. log2 1 = 0\n[PROOF STEP]\nfrom S1 S5\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ log2 1\nlog2 1 \\ 0\n[PROOF STEP]\nhave S6: \"?v = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ log2 1\nlog2 1 \\ 0\n\ngoal (1 subgoal):\n 1. log2 1 = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlog2 1 = 0\n\ngoal (1 subgoal):\n 1. log2 1 = 0\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlog2 1 = 0\n\ngoal (1 subgoal):\n 1. log2 1 = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlog2 1 = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1275, "file": "Recursion-Theory-I_PRecFinSet", "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424256566558, "lm_q2_score": 0.8397339716830606, "lm_q1q2_score": 0.7103665929118658}} {"text": "[STATEMENT]\nlemma equal_div_mod:assumes \"((j::nat) div a) = (i div a)\"\n and \"(j mod a) = (i mod a)\"\n shows \"j = i\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j = i\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. j = i\n[PROOF STEP]\nhave \"j = a*(j div a) + (j mod a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. j = a *n j div a + j mod a\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nj = a *n j div a + j mod a\n\ngoal (1 subgoal):\n 1. j = i\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nj = a *n j div a + j mod a\n[PROOF STEP]\nhave \"j = a*(i div a) + (i mod a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nj = a *n j div a + j mod a\n\ngoal (1 subgoal):\n 1. j = a *n i div a + i mod a\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nj = a *n j div a + j mod a\nj div a = i div a\nj mod a = i mod a\n\ngoal (1 subgoal):\n 1. j = a *n i div a + i mod a\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nj = a *n i div a + i mod a\n\ngoal (1 subgoal):\n 1. j = i\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nj = a *n i div a + i mod a\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nj = a *n i div a + i mod a\n\ngoal (1 subgoal):\n 1. j = i\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nj = i\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 633, "file": "Knot_Theory_Kauffman_Matrix", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424217727027, "lm_q2_score": 0.839733963661418, "lm_q1q2_score": 0.7103665828645307}} {"text": "[STATEMENT]\nlemma ket_1_norm:\n shows \"\\ket_1\\ = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ket_1\\ = 1\n[PROOF STEP]\nusing unit_cpx_vec_length\n[PROOF STATE]\nproof (prove)\nusing this:\n?i < ?n \\ \\unit_vec ?n ?i\\ = 1\n\ngoal (1 subgoal):\n 1. \\ket_1\\ = 1\n[PROOF STEP]\nunfolding ket_1_def\n[PROOF STATE]\nproof (prove)\nusing this:\n?i < ?n \\ \\unit_vec ?n ?i\\ = 1\n\ngoal (1 subgoal):\n 1. \\unit_vec 2 1\\ = 1\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 248, "file": "Projective_Measurements_CHSH_Inequality", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267762381843, "lm_q2_score": 0.8175744806385543, "lm_q1q2_score": 0.7103306003478029}} {"text": "[STATEMENT]\nlemma add_scale_img:\n assumes \"a < b\" shows \"(\\x::real. a + (b - a) * x) ` {0 .. 1} = {a .. b}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. a + (b - a) * x) ` {0..1} = {a..b}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\na < b\n\ngoal (1 subgoal):\n 1. (\\x. a + (b - a) * x) ` {0..1} = {a..b}\n[PROOF STEP]\napply (auto simp: algebra_simps affine_ineq image_iff)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\xa. \\a < b; 0 \\ xa; xa \\ 1\\ \\ xa * a \\ xa * b\n 2. \\x. \\a < b; a \\ x; x \\ b\\ \\ \\xa\\{0..1}. x + xa * a = a + xa * b\n[PROOF STEP]\nusing less_eq_real_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(?x \\ ?y) = (?x < ?y \\ ?x = ?y)\n\ngoal (2 subgoals):\n 1. \\xa. \\a < b; 0 \\ xa; xa \\ 1\\ \\ xa * a \\ xa * b\n 2. \\x. \\a < b; a \\ x; x \\ b\\ \\ \\xa\\{0..1}. x + xa * a = a + xa * b\n[PROOF STEP]\napply force\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\a < b; a \\ x; x \\ b\\ \\ \\xa\\{0..1}. x + xa * a = a + xa * b\n[PROOF STEP]\napply (rule_tac x=\"(x-a)/(b-a)\" in bexI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. \\a < b; a \\ x; x \\ b\\ \\ x + (x - a) / (b - a) * a = a + (x - a) / (b - a) * b\n 2. \\x. \\a < b; a \\ x; x \\ b\\ \\ (x - a) / (b - a) \\ {0..1}\n[PROOF STEP]\napply (auto simp: field_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 836, "file": "Green_Green", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267694452331, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.710330596724988}} {"text": "[STATEMENT]\nlemma filterlim_at_infinity_times:\n fixes f :: \"'a \\ 'b::real_normed_field\"\n assumes \"filterlim f at_infinity F\" \"filterlim g at_infinity F\"\n shows \"filterlim (\\x. f x * g x) at_infinity F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LIM x F. f x * g x :> at_infinity\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. LIM x F. f x * g x :> at_infinity\n[PROOF STEP]\nhave \"((\\x. inverse (f x) * inverse (g x)) \\ 0 * 0) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. inverse (f x) * inverse (g x)) \\ (0::'b) * (0::'b)) F\n[PROOF STEP]\nby (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. inverse (f x) * inverse (g x)) \\ (0::'b) * (0::'b)) F\n\ngoal (1 subgoal):\n 1. LIM x F. f x * g x :> at_infinity\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n((\\x. inverse (f x) * inverse (g x)) \\ (0::'b) * (0::'b)) F\n[PROOF STEP]\nhave \"filterlim (\\x. inverse (f x) * inverse (g x)) (at 0) F\"\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. inverse (f x) * inverse (g x)) \\ (0::'b) * (0::'b)) F\n\ngoal (1 subgoal):\n 1. LIM x F. inverse (f x) * inverse (g x) :> at (0::'b)\n[PROOF STEP]\nunfolding filterlim_at\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. inverse (f x) * inverse (g x)) \\ (0::'b) * (0::'b)) F\n\ngoal (1 subgoal):\n 1. (\\\\<^sub>F x in F. inverse (f x) * inverse (g x) \\ UNIV \\ inverse (f x) * inverse (g x) \\ (0::'b)) \\ ((\\x. inverse (f x) * inverse (g x)) \\ (0::'b)) F\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. inverse (f x) * inverse (g x)) \\ (0::'b) * (0::'b)) F\nfilterlim f at_infinity F\nfilterlim g at_infinity F\n\ngoal (1 subgoal):\n 1. (\\\\<^sub>F x in F. inverse (f x) * inverse (g x) \\ UNIV \\ inverse (f x) * inverse (g x) \\ (0::'b)) \\ ((\\x. inverse (f x) * inverse (g x)) \\ (0::'b)) F\n[PROOF STEP]\nby (auto intro: filterlim_at_infinity_imp_eventually_ne tendsto_imp_eventually_ne eventually_conj)\n[PROOF STATE]\nproof (state)\nthis:\nLIM x F. inverse (f x) * inverse (g x) :> at (0::'b)\n\ngoal (1 subgoal):\n 1. LIM x F. f x * g x :> at_infinity\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nLIM x F. inverse (f x) * inverse (g x) :> at (0::'b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nLIM x F. inverse (f x) * inverse (g x) :> at (0::'b)\n\ngoal (1 subgoal):\n 1. LIM x F. f x * g x :> at_infinity\n[PROOF STEP]\nby (subst filterlim_inverse_at_iff[symmetric]) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nLIM x F. f x * g x :> at_infinity\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1257, "file": "Winding_Number_Eval_Missing_Topology", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267626522813, "lm_q2_score": 0.817574478416099, "lm_q1q2_score": 0.7103305873093867}} {"text": "[STATEMENT]\nlemma scaleR_2:\n fixes x :: \"'a::real_vector\"\n shows \"scaleR 2 x = x + x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 *\\<^sub>R x = x + x\n[PROOF STEP]\nunfolding one_add_one [symmetric] scaleR_left_distrib\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 *\\<^sub>R x + 1 *\\<^sub>R x = x + x\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 157, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615382094310355, "lm_q2_score": 0.8244619350028205, "lm_q1q2_score": 0.7103054592263768}} {"text": "[STATEMENT]\nlemma prime_gt_3_impl_p_minus_one_not_prime:\n fixes p::nat\n assumes \"prime p\" \"p>3\"\n shows \"\\ prime (p - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ prime (p - 1)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. prime (p - 1) \\ False\n[PROOF STEP]\nassume \"prime (p - 1)\"\n[PROOF STATE]\nproof (state)\nthis:\nprime (p - 1)\n\ngoal (1 subgoal):\n 1. prime (p - 1) \\ False\n[PROOF STEP]\nhave \"\\ even p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. odd p\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n3 < p\n\ngoal (1 subgoal):\n 1. odd p\n[PROOF STEP]\nby (simp add: prime_odd_nat)\n[PROOF STATE]\nproof (state)\nthis:\nodd p\n\ngoal (1 subgoal):\n 1. prime (p - 1) \\ False\n[PROOF STEP]\nhence \"2 dvd (p - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nodd p\n\ngoal (1 subgoal):\n 1. even (p - 1)\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\neven (p - 1)\n\ngoal (1 subgoal):\n 1. prime (p - 1) \\ False\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\neven (p - 1)\n[PROOF STEP]\nobtain q where \"p - 1 = 2 * q\"\n[PROOF STATE]\nproof (prove)\nusing this:\neven (p - 1)\n\ngoal (1 subgoal):\n 1. (\\q. p - 1 = 2 * q \\ thesis) \\ thesis\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\np - 1 = 2 * q\n\ngoal (1 subgoal):\n 1. prime (p - 1) \\ False\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\np - 1 = 2 * q\n[PROOF STEP]\nhave \"2 \\ prime_factors (p - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\np - 1 = 2 * q\n\ngoal (1 subgoal):\n 1. 2 \\# prime_factorization (p - 1)\n[PROOF STEP]\nusing \\p>3\\\n[PROOF STATE]\nproof (prove)\nusing this:\np - 1 = 2 * q\n3 < p\n\ngoal (1 subgoal):\n 1. 2 \\# prime_factorization (p - 1)\n[PROOF STEP]\nby (auto simp: prime_factorization_times_prime)\n[PROOF STATE]\nproof (state)\nthis:\n2 \\# prime_factorization (p - 1)\n\ngoal (1 subgoal):\n 1. prime (p - 1) \\ False\n[PROOF STEP]\nthus False\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\# prime_factorization (p - 1)\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nusing prime_factors_of_prime \\p>3\\ \\prime (p - 1)\\\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\# prime_factorization (p - 1)\nprime ?p \\ prime_factors ?p = {?p}\n3 < p\nprime (p - 1)\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1143, "file": "Pratt_Certificate_Pratt_Certificate", "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382094310357, "lm_q2_score": 0.824461932846258, "lm_q1q2_score": 0.7103054573684159}} {"text": "[STATEMENT]\nlemma finite_Pow_card:\n assumes \"finite X\"\n shows \"card (Pow X) = 2 powr (card X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (card (Pow X)) = 2 powr real (card X)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite X\n\ngoal (1 subgoal):\n 1. real (card (Pow X)) = 2 powr real (card X)\n[PROOF STEP]\nproof (induct X rule: finite_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. real (card (Pow {})) = 2 powr real (card {})\n 2. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\ncase empty\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. real (card (Pow {})) = 2 powr real (card {})\n 2. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (card (Pow {})) = 2 powr real (card {})\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nreal (card (Pow {})) = 2 powr real (card {})\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\ncase (insert x X)\n[PROOF STATE]\nproof (state)\nthis:\nfinite X\nx \\ X\nreal (card (Pow X)) = 2 powr real (card X)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nhave \"0 \\ (2 :: real)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ 2\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nhence two_powr_one: \"(2 :: real) = 2 powr 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ 2\n\ngoal (1 subgoal):\n 1. 2 = 2 powr 1\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n2 = 2 powr 1\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nhave \"bij_betw (\\ x. fst x \\ snd x) ({{},{x}} \\ Pow X) (Pow (insert x X))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X) (Pow (insert x X))\n[PROOF STEP]\nunfolding bij_betw_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X) \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\n 2. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\n 2. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\nfix y z\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\n 2. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\nassume \n \"y \\ {{}, {x}} \\ Pow X\"\n \"z \\ {{}, {x}} \\ Pow X\"\n \"fst y \\ snd y = fst z \\ snd z\"\n (is \"?Uy = ?Uz\")\n[PROOF STATE]\nproof (state)\nthis:\ny \\ {{}, {x}} \\ Pow X\nz \\ {{}, {x}} \\ Pow X\nfst y \\ snd y = fst z \\ snd z\n\ngoal (2 subgoals):\n 1. inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\n 2. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\nhence \n \"x \\ snd y\"\n \"x \\ snd z\"\n \"fst y = {x} \\ fst y = {}\"\n \"fst z = {x} \\ fst z = {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\ny \\ {{}, {x}} \\ Pow X\nz \\ {{}, {x}} \\ Pow X\nfst y \\ snd y = fst z \\ snd z\n\ngoal (1 subgoal):\n 1. (x \\ snd y &&& x \\ snd z) &&& fst y = {x} \\ fst y = {} &&& fst z = {x} \\ fst z = {}\n[PROOF STEP]\nusing insert.hyps(2)\n[PROOF STATE]\nproof (prove)\nusing this:\ny \\ {{}, {x}} \\ Pow X\nz \\ {{}, {x}} \\ Pow X\nfst y \\ snd y = fst z \\ snd z\nx \\ X\n\ngoal (1 subgoal):\n 1. (x \\ snd y &&& x \\ snd z) &&& fst y = {x} \\ fst y = {} &&& fst z = {x} \\ fst z = {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx \\ snd y\nx \\ snd z\nfst y = {x} \\ fst y = {}\nfst z = {x} \\ fst z = {}\n\ngoal (2 subgoals):\n 1. inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\n 2. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\nhence \n \"x \\ ?Uy \\ fst y = {x}\"\n \"x \\ ?Uz \\ fst z = {x}\"\n \"x \\ ?Uy \\ fst y = {}\"\n \"x \\ ?Uz \\ fst z = {}\"\n \"snd y = ?Uy - {x}\"\n \"snd z = ?Uz - {x}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ snd y\nx \\ snd z\nfst y = {x} \\ fst y = {}\nfst z = {x} \\ fst z = {}\n\ngoal (1 subgoal):\n 1. ((x \\ fst y \\ snd y) = (fst y = {x}) &&& (x \\ fst z \\ snd z) = (fst z = {x}) &&& (x \\ fst y \\ snd y) = (fst y = {})) &&& (x \\ fst z \\ snd z) = (fst z = {}) &&& snd y = fst y \\ snd y - {x} &&& snd z = fst z \\ snd z - {x}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(x \\ fst y \\ snd y) = (fst y = {x})\n(x \\ fst z \\ snd z) = (fst z = {x})\n(x \\ fst y \\ snd y) = (fst y = {})\n(x \\ fst z \\ snd z) = (fst z = {})\nsnd y = fst y \\ snd y - {x}\nsnd z = fst z \\ snd z - {x}\n\ngoal (2 subgoals):\n 1. inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\n 2. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\nhence \n \"x \\ ?Uy \\ y = ({x}, ?Uy - {x})\"\n \"x \\ ?Uz \\ z = ({x}, ?Uz - {x})\"\n \"x \\ ?Uy \\ y = ({}, ?Uy - {x})\"\n \"x \\ ?Uz \\ z = ({}, ?Uz - {x})\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(x \\ fst y \\ snd y) = (fst y = {x})\n(x \\ fst z \\ snd z) = (fst z = {x})\n(x \\ fst y \\ snd y) = (fst y = {})\n(x \\ fst z \\ snd z) = (fst z = {})\nsnd y = fst y \\ snd y - {x}\nsnd z = fst z \\ snd z - {x}\n\ngoal (1 subgoal):\n 1. ((x \\ fst y \\ snd y) = (y = ({x}, fst y \\ snd y - {x})) &&& (x \\ fst z \\ snd z) = (z = ({x}, fst z \\ snd z - {x}))) &&& (x \\ fst y \\ snd y) = (y = ({}, fst y \\ snd y - {x})) &&& (x \\ fst z \\ snd z) = (z = ({}, fst z \\ snd z - {x}))\n[PROOF STEP]\nby (metis fst_conv prod.collapse)+\n[PROOF STATE]\nproof (state)\nthis:\n(x \\ fst y \\ snd y) = (y = ({x}, fst y \\ snd y - {x}))\n(x \\ fst z \\ snd z) = (z = ({x}, fst z \\ snd z - {x}))\n(x \\ fst y \\ snd y) = (y = ({}, fst y \\ snd y - {x}))\n(x \\ fst z \\ snd z) = (z = ({}, fst z \\ snd z - {x}))\n\ngoal (2 subgoals):\n 1. inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\n 2. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\nhence \"y = z\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(x \\ fst y \\ snd y) = (y = ({x}, fst y \\ snd y - {x}))\n(x \\ fst z \\ snd z) = (z = ({x}, fst z \\ snd z - {x}))\n(x \\ fst y \\ snd y) = (y = ({}, fst y \\ snd y - {x}))\n(x \\ fst z \\ snd z) = (z = ({}, fst z \\ snd z - {x}))\n\ngoal (1 subgoal):\n 1. y = z\n[PROOF STEP]\nusing \\?Uy = ?Uz\\\n[PROOF STATE]\nproof (prove)\nusing this:\n(x \\ fst y \\ snd y) = (y = ({x}, fst y \\ snd y - {x}))\n(x \\ fst z \\ snd z) = (z = ({x}, fst z \\ snd z - {x}))\n(x \\ fst y \\ snd y) = (y = ({}, fst y \\ snd y - {x}))\n(x \\ fst z \\ snd z) = (z = ({}, fst z \\ snd z - {x}))\nfst y \\ snd y = fst z \\ snd z\n\ngoal (1 subgoal):\n 1. y = z\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\ny = z\n\ngoal (2 subgoals):\n 1. inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\n 2. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n\\?y2 \\ {{}, {x}} \\ Pow X; ?z2 \\ {{}, {x}} \\ Pow X; fst ?y2 \\ snd ?y2 = fst ?z2 \\ snd ?z2\\ \\ ?y2 = ?z2\n\ngoal (2 subgoals):\n 1. inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\n 2. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\nthus \"inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?y2 \\ {{}, {x}} \\ Pow X; ?z2 \\ {{}, {x}} \\ Pow X; fst ?y2 \\ snd ?y2 = fst ?z2 \\ snd ?z2\\ \\ ?y2 = ?z2\n\ngoal (1 subgoal):\n 1. inj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nunfolding inj_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?y2 \\ {{}, {x}} \\ Pow X; ?z2 \\ {{}, {x}} \\ Pow X; fst ?y2 \\ snd ?y2 = fst ?z2 \\ snd ?z2\\ \\ ?y2 = ?z2\n\ngoal (1 subgoal):\n 1. \\xa\\{{}, {x}} \\ Pow X. \\y\\{{}, {x}} \\ Pow X. fst xa \\ snd xa = fst y \\ snd y \\ xa = y\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X)\n\ngoal (1 subgoal):\n 1. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\nshow \"(\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n[PROOF STEP]\nproof (intro equalityI subsetI)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\xa. xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) \\ xa \\ Pow (insert x X)\n 2. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nfix y\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\xa. xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) \\ xa \\ Pow (insert x X)\n 2. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nassume \"y \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\"\n[PROOF STATE]\nproof (state)\nthis:\ny \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n\ngoal (2 subgoals):\n 1. \\xa. xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) \\ xa \\ Pow (insert x X)\n 2. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\ny \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nobtain z where\n \"z \\ ({{}, {x}} \\ Pow X)\"\n \"y = fst z \\ snd z\"\n[PROOF STATE]\nproof (prove)\nusing this:\ny \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n\ngoal (1 subgoal):\n 1. (\\z. \\z \\ {{}, {x}} \\ Pow X; y = fst z \\ snd z\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nz \\ {{}, {x}} \\ Pow X\ny = fst z \\ snd z\n\ngoal (2 subgoals):\n 1. \\xa. xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) \\ xa \\ Pow (insert x X)\n 2. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nhence \n \"snd z \\ X\"\n \"fst z \\ insert x X\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ {{}, {x}} \\ Pow X\ny = fst z \\ snd z\n\ngoal (1 subgoal):\n 1. snd z \\ X &&& fst z \\ insert x X\n[PROOF STEP]\nusing SigmaE\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ {{}, {x}} \\ Pow X\ny = fst z \\ snd z\n\\?c \\ Sigma ?A ?B; \\x y. \\x \\ ?A; y \\ ?B x; ?c = (x, y)\\ \\ ?P\\ \\ ?P\n\ngoal (1 subgoal):\n 1. snd z \\ X &&& fst z \\ insert x X\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsnd z \\ X\nfst z \\ insert x X\n\ngoal (2 subgoals):\n 1. \\xa. xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) \\ xa \\ Pow (insert x X)\n 2. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nthus \"y \\ Pow (insert x X)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsnd z \\ X\nfst z \\ insert x X\n\ngoal (1 subgoal):\n 1. y \\ Pow (insert x X)\n[PROOF STEP]\nusing \\y = fst z \\ snd z\\\n[PROOF STATE]\nproof (prove)\nusing this:\nsnd z \\ X\nfst z \\ insert x X\ny = fst z \\ snd z\n\ngoal (1 subgoal):\n 1. y \\ Pow (insert x X)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ny \\ Pow (insert x X)\n\ngoal (1 subgoal):\n 1. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nfix y\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nassume \"y \\ Pow (insert x X)\"\n[PROOF STATE]\nproof (state)\nthis:\ny \\ Pow (insert x X)\n\ngoal (1 subgoal):\n 1. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nlet ?z = \"(if x \\ y then {x} else {}, y - {x})\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nhave \"?z \\ ({{}, {x}} \\ Pow X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if x \\ y then {x} else {}, y - {x}) \\ {{}, {x}} \\ Pow X\n[PROOF STEP]\nusing \\y \\ Pow (insert x X)\\\n[PROOF STATE]\nproof (prove)\nusing this:\ny \\ Pow (insert x X)\n\ngoal (1 subgoal):\n 1. (if x \\ y then {x} else {}, y - {x}) \\ {{}, {x}} \\ Pow X\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(if x \\ y then {x} else {}, y - {x}) \\ {{}, {x}} \\ Pow X\n\ngoal (1 subgoal):\n 1. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(if x \\ y then {x} else {}, y - {x}) \\ {{}, {x}} \\ Pow X\n\ngoal (1 subgoal):\n 1. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nhave \"(\\x. fst x \\ snd x) ?z = y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fst (if x \\ y then {x} else {}, y - {x}) \\ snd (if x \\ y then {x} else {}, y - {x}) = y\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfst (if x \\ y then {x} else {}, y - {x}) \\ snd (if x \\ y then {x} else {}, y - {x}) = y\n\ngoal (1 subgoal):\n 1. \\xa. xa \\ Pow (insert x X) \\ xa \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(if x \\ y then {x} else {}, y - {x}) \\ {{}, {x}} \\ Pow X\nfst (if x \\ y then {x} else {}, y - {x}) \\ snd (if x \\ y then {x} else {}, y - {x}) = y\n[PROOF STEP]\nshow \"y \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(if x \\ y then {x} else {}, y - {x}) \\ {{}, {x}} \\ Pow X\nfst (if x \\ y then {x} else {}, y - {x}) \\ snd (if x \\ y then {x} else {}, y - {x}) = y\n\ngoal (1 subgoal):\n 1. y \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ny \\ (\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. fst x \\ snd x) ` ({{}, {x}} \\ Pow X) = Pow (insert x X)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X) (Pow (insert x X))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nhence \"card (Pow (insert x X)) = card ({{},{x}} \\ Pow X)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X) (Pow (insert x X))\n\ngoal (1 subgoal):\n 1. card (Pow (insert x X)) = card ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nusing bij_betw_same_card\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw (\\x. fst x \\ snd x) ({{}, {x}} \\ Pow X) (Pow (insert x X))\nbij_betw ?f ?A ?B \\ card ?A = card ?B\n\ngoal (1 subgoal):\n 1. card (Pow (insert x X)) = card ({{}, {x}} \\ Pow X)\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\ncard (Pow (insert x X)) = card ({{}, {x}} \\ Pow X)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (Pow (insert x X)) = card ({{}, {x}} \\ Pow X)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nhave \"\\ = 2 * card (Pow X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({{}, {x}} \\ Pow X) = 2 * card (Pow X)\n[PROOF STEP]\nby (simp add: insert.hyps(1))\n[PROOF STATE]\nproof (state)\nthis:\ncard ({{}, {x}} \\ Pow X) = 2 * card (Pow X)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ({{}, {x}} \\ Pow X) = 2 * card (Pow X)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nhave \"\\ = 2 * (2 powr (card X))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (2 * card (Pow X)) = 2 * 2 powr real (card X)\n[PROOF STEP]\nby (simp add: insert.hyps(3))\n[PROOF STATE]\nproof (state)\nthis:\nreal (2 * card (Pow X)) = 2 * 2 powr real (card X)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (2 * card (Pow X)) = 2 * 2 powr real (card X)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nhave \"\\ = (2 powr 1) * 2 powr (card X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * 2 powr real (card X) = 2 powr 1 * 2 powr real (card X)\n[PROOF STEP]\nusing two_powr_one\n[PROOF STATE]\nproof (prove)\nusing this:\n2 = 2 powr 1\n\ngoal (1 subgoal):\n 1. 2 * 2 powr real (card X) = 2 powr 1 * 2 powr real (card X)\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n2 * 2 powr real (card X) = 2 powr 1 * 2 powr real (card X)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 * 2 powr real (card X) = 2 powr 1 * 2 powr real (card X)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nhave \"\\ = 2 powr (1 + card X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 powr 1 * 2 powr real (card X) = 2 powr real (1 + card X)\n[PROOF STEP]\nby (simp add: powr_add)\n[PROOF STATE]\nproof (state)\nthis:\n2 powr 1 * 2 powr real (card X) = 2 powr real (1 + card X)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 powr 1 * 2 powr real (card X) = 2 powr real (1 + card X)\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nhave \"\\ = 2 powr (card (insert x X))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 powr real (1 + card X) = 2 powr real (card (insert x X))\n[PROOF STEP]\nby (simp add: insert.hyps(1) insert.hyps(2))\n[PROOF STATE]\nproof (state)\nthis:\n2 powr real (1 + card X) = 2 powr real (card (insert x X))\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; real (card (Pow F)) = 2 powr real (card F)\\ \\ real (card (Pow (insert x F))) = 2 powr real (card (insert x F))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (card (Pow (insert x X))) = 2 powr real (card (insert x X))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (card (Pow (insert x X))) = 2 powr real (card (insert x X))\n\ngoal (1 subgoal):\n 1. real (card (Pow (insert x X))) = 2 powr real (card (insert x X))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nreal (card (Pow (insert x X))) = 2 powr real (card (insert x X))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 10823, "file": "Birkhoff_Finite_Distributive_Lattices_Birkhoff_Finite_Distributive_Lattices", "length": 84, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615381987656672, "lm_q2_score": 0.8244619306896955, "lm_q1q2_score": 0.7103054467172646}} {"text": "[STATEMENT]\nlemma permutations_complete: (* could generalize with multi-sets *)\n assumes \"distinct xs\" \"distinct ys\" \"set xs = set ys\"\n shows \"ys \\ set (permutations xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ys \\ set (permutations xs)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndistinct xs\ndistinct ys\nset xs = set ys\n\ngoal (1 subgoal):\n 1. ys \\ set (permutations xs)\n[PROOF STEP]\nproof (induct \"length xs\" arbitrary: xs ys)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\xs ys. \\0 = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs)\n 2. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs); Suc x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n0 = length xs\ndistinct xs\ndistinct ys\nset xs = set ys\n\ngoal (2 subgoals):\n 1. \\xs ys. \\0 = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs)\n 2. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs); Suc x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 = length xs\ndistinct xs\ndistinct ys\nset xs = set ys\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n0 = length xs\ndistinct xs\ndistinct ys\nset xs = set ys\n\ngoal (1 subgoal):\n 1. ys \\ set (permutations xs)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nys \\ set (permutations xs)\n\ngoal (1 subgoal):\n 1. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs); Suc x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs); Suc x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n\\n = length ?xs; distinct ?xs; distinct ?ys; set ?xs = set ?ys\\ \\ ?ys \\ set (permutations ?xs)\nSuc n = length xs\ndistinct xs\ndistinct ys\nset xs = set ys\n\ngoal (1 subgoal):\n 1. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs); Suc x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs)\n[PROOF STEP]\nfrom Suc.hyps\n[PROOF STATE]\nproof (chain)\npicking this:\n\\n = length ?xs; distinct ?xs; distinct ?ys; set ?xs = set ?ys\\ \\ ?ys \\ set (permutations ?xs)\nSuc n = length xs\n[PROOF STEP]\nhave \"xs \\ []\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n = length ?xs; distinct ?xs; distinct ?ys; set ?xs = set ?ys\\ \\ ?ys \\ set (permutations ?xs)\nSuc n = length xs\n\ngoal (1 subgoal):\n 1. xs \\ []\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nxs \\ []\n\ngoal (1 subgoal):\n 1. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs); Suc x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nxs \\ []\n[PROOF STEP]\nobtain y ys' where [simp]: \"ys = y # ys'\" \"y \\ set xs\"\n[PROOF STATE]\nproof (prove)\nusing this:\nxs \\ []\n\ngoal (1 subgoal):\n 1. (\\y ys'. \\ys = y # ys'; y \\ set xs\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing Suc.prems\n[PROOF STATE]\nproof (prove)\nusing this:\nxs \\ []\ndistinct xs\ndistinct ys\nset xs = set ys\n\ngoal (1 subgoal):\n 1. (\\y ys'. \\ys = y # ys'; y \\ set xs\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (cases ys) auto\n[PROOF STATE]\nproof (state)\nthis:\nys = y # ys'\ny \\ set xs\n\ngoal (1 subgoal):\n 1. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs); Suc x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs)\n[PROOF STEP]\nhave \"ys' \\ set (permutations (remove1 y xs))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ys' \\ set (permutations (remove1 y xs))\n[PROOF STEP]\nusing Suc.prems \\Suc n = _\\\n[PROOF STATE]\nproof (prove)\nusing this:\ndistinct xs\ndistinct ys\nset xs = set ys\nSuc n = length xs\n\ngoal (1 subgoal):\n 1. ys' \\ set (permutations (remove1 y xs))\n[PROOF STEP]\nby (intro Suc.hyps) (simp_all add: length_remove1 )\n[PROOF STATE]\nproof (state)\nthis:\nys' \\ set (permutations (remove1 y xs))\n\ngoal (1 subgoal):\n 1. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs); Suc x = length xs; distinct xs; distinct ys; set xs = set ys\\ \\ ys \\ set (permutations xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nys' \\ set (permutations (remove1 y xs))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nys' \\ set (permutations (remove1 y xs))\n\ngoal (1 subgoal):\n 1. ys \\ set (permutations xs)\n[PROOF STEP]\nusing \\xs \\ []\\\n[PROOF STATE]\nproof (prove)\nusing this:\nys' \\ set (permutations (remove1 y xs))\nxs \\ []\n\ngoal (1 subgoal):\n 1. ys \\ set (permutations xs)\n[PROOF STEP]\nby (auto simp: set_permutations_step)\n[PROOF STATE]\nproof (state)\nthis:\nys \\ set (permutations xs)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2572, "file": "Planarity_Certificates_Planarity_Executable_Permutations", "length": 23, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615381987656672, "lm_q2_score": 0.8244619220634457, "lm_q1q2_score": 0.7103054392854209}} {"text": "[STATEMENT]\nlemma sin_cos_eq_iff: \"sin y = sin x \\ cos y = cos x \\ (\\n::int. y = x + 2 * pi * n)\" (is \"?L=?R\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (sin y = sin x \\ cos y = cos x) = (\\xa. y = x + 2 * pi * real_of_int xa)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. sin y = sin x \\ cos y = cos x \\ \\xa. y = x + 2 * pi * real_of_int xa\n 2. \\xa. y = x + 2 * pi * real_of_int xa \\ sin y = sin x \\ cos y = cos x\n[PROOF STEP]\nassume ?L\n[PROOF STATE]\nproof (state)\nthis:\nsin y = sin x \\ cos y = cos x\n\ngoal (2 subgoals):\n 1. sin y = sin x \\ cos y = cos x \\ \\xa. y = x + 2 * pi * real_of_int xa\n 2. \\xa. y = x + 2 * pi * real_of_int xa \\ sin y = sin x \\ cos y = cos x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nsin y = sin x \\ cos y = cos x\n[PROOF STEP]\nhave \"cos (y-x) = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsin y = sin x \\ cos y = cos x\n\ngoal (1 subgoal):\n 1. cos (y - x) = 1\n[PROOF STEP]\nusing cos_add [of y \"-x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nsin y = sin x \\ cos y = cos x\ncos (y + - x) = cos y * cos (- x) - sin y * sin (- x)\n\ngoal (1 subgoal):\n 1. cos (y - x) = 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncos (y - x) = 1\n\ngoal (2 subgoals):\n 1. sin y = sin x \\ cos y = cos x \\ \\xa. y = x + 2 * pi * real_of_int xa\n 2. \\xa. y = x + 2 * pi * real_of_int xa \\ sin y = sin x \\ cos y = cos x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncos (y - x) = 1\n[PROOF STEP]\nshow ?R\n[PROOF STATE]\nproof (prove)\nusing this:\ncos (y - x) = 1\n\ngoal (1 subgoal):\n 1. \\xa. y = x + 2 * pi * real_of_int xa\n[PROOF STEP]\nby (metis cos_one_2pi_int add.commute diff_add_cancel mult.assoc mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\n\\xa. y = x + 2 * pi * real_of_int xa\n\ngoal (1 subgoal):\n 1. \\xa. y = x + 2 * pi * real_of_int xa \\ sin y = sin x \\ cos y = cos x\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\xa. y = x + 2 * pi * real_of_int xa \\ sin y = sin x \\ cos y = cos x\n[PROOF STEP]\nassume ?R\n[PROOF STATE]\nproof (state)\nthis:\n\\xa. y = x + 2 * pi * real_of_int xa\n\ngoal (1 subgoal):\n 1. \\xa. y = x + 2 * pi * real_of_int xa \\ sin y = sin x \\ cos y = cos x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\xa. y = x + 2 * pi * real_of_int xa\n[PROOF STEP]\nshow ?L\n[PROOF STATE]\nproof (prove)\nusing this:\n\\xa. y = x + 2 * pi * real_of_int xa\n\ngoal (1 subgoal):\n 1. sin y = sin x \\ cos y = cos x\n[PROOF STEP]\nby (auto simp: sin_add cos_add)\n[PROOF STATE]\nproof (state)\nthis:\nsin y = sin x \\ cos y = cos x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1301, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677737461007, "lm_q2_score": 0.8376199714402812, "lm_q1q2_score": 0.7102747424274877}} {"text": "[STATEMENT]\nlemma norm_ell2_vec_of_basis_enum: \"norm \\ =\n (let \\' = vec_of_basis_enum \\ in\n sqrt (\\ i \\ {0 ..< dim_vec \\'}. let z = vec_index \\' i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\"\n (is \"_ = ?rhs\") for \\ :: \"'a::onb_enum\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm \\ = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm \\ = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n[PROOF STEP]\nhave \"norm \\ = sqrt (cmod (\\i = 0..).\n vec_of_basis_enum \\ $ i * conjugate (vec_of_basis_enum \\) $ i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm \\ = sqrt (cmod (\\i = 0..). vec_of_basis_enum \\ $ i * conjugate (vec_of_basis_enum \\) $ i))\n[PROOF STEP]\nunfolding norm_eq_sqrt_cinner[where 'a='a] cscalar_prod_vec_of_basis_enum[symmetric] scalar_prod_def dim_vec_conjugate\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (cmod (\\i = 0..). vec_of_basis_enum \\ $ i * conjugate (vec_of_basis_enum \\) $ i)) = sqrt (cmod (\\i = 0..). vec_of_basis_enum \\ $ i * conjugate (vec_of_basis_enum \\) $ i))\n[PROOF STEP]\nby rule\n[PROOF STATE]\nproof (state)\nthis:\nnorm \\ = sqrt (cmod (\\i = 0..). vec_of_basis_enum \\ $ i * conjugate (vec_of_basis_enum \\) $ i))\n\ngoal (1 subgoal):\n 1. norm \\ = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnorm \\ = sqrt (cmod (\\i = 0..). vec_of_basis_enum \\ $ i * conjugate (vec_of_basis_enum \\) $ i))\n\ngoal (1 subgoal):\n 1. norm \\ = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n[PROOF STEP]\nhave \"\\ = sqrt (cmod (\\x = 0..).\n let z = vec_of_basis_enum \\ $ x in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (cmod (\\i = 0..). vec_of_basis_enum \\ $ i * conjugate (vec_of_basis_enum \\) $ i)) = sqrt (cmod (complex_of_real (\\x = 0..). let z = vec_of_basis_enum \\ $ x in (Re z)\\<^sup>2 + (Im z)\\<^sup>2)))\n[PROOF STEP]\napply (subst sum.cong, rule refl)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. x \\ {0..)} \\ vec_of_basis_enum \\ $ x * conjugate (vec_of_basis_enum \\) $ x = ?h x\n 2. sqrt (cmod (sum ?h {0..)})) = sqrt (cmod (complex_of_real (\\x = 0..). let z = vec_of_basis_enum \\ $ x in (Re z)\\<^sup>2 + (Im z)\\<^sup>2)))\n[PROOF STEP]\napply (subst vec_index_conjugate)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x. x \\ {0..)} \\ x < dim_vec (vec_of_basis_enum \\)\n 2. \\x. x \\ {0..)} \\ vec_of_basis_enum \\ $ x * conjugate (vec_of_basis_enum \\ $ x) = ?h3 x x\n 3. sqrt (cmod (\\x = 0..). ?h3 x x)) = sqrt (cmod (complex_of_real (\\x = 0..). let z = vec_of_basis_enum \\ $ x in (Re z)\\<^sup>2 + (Im z)\\<^sup>2)))\n[PROOF STEP]\nby (auto simp: Let_def complex_mult_cnj)\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (cmod (\\i = 0..). vec_of_basis_enum \\ $ i * conjugate (vec_of_basis_enum \\) $ i)) = sqrt (cmod (complex_of_real (\\x = 0..). let z = vec_of_basis_enum \\ $ x in (Re z)\\<^sup>2 + (Im z)\\<^sup>2)))\n\ngoal (1 subgoal):\n 1. norm \\ = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (cmod (\\i = 0..). vec_of_basis_enum \\ $ i * conjugate (vec_of_basis_enum \\) $ i)) = sqrt (cmod (complex_of_real (\\x = 0..). let z = vec_of_basis_enum \\ $ x in (Re z)\\<^sup>2 + (Im z)\\<^sup>2)))\n\ngoal (1 subgoal):\n 1. norm \\ = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n[PROOF STEP]\nhave \"\\ = ?rhs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (cmod (complex_of_real (\\x = 0..). let z = vec_of_basis_enum \\ $ x in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))) = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n[PROOF STEP]\nunfolding Let_def norm_of_real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt \\\\x = 0..). (Re (vec_of_basis_enum \\ $ x))\\<^sup>2 + (Im (vec_of_basis_enum \\ $ x))\\<^sup>2\\ = sqrt (\\i = 0..). (Re (vec_of_basis_enum \\ $ i))\\<^sup>2 + (Im (vec_of_basis_enum \\ $ i))\\<^sup>2)\n[PROOF STEP]\napply (subst abs_of_nonneg)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. 0 \\ (\\x = 0..). (Re (vec_of_basis_enum \\ $ x))\\<^sup>2 + (Im (vec_of_basis_enum \\ $ x))\\<^sup>2)\n 2. sqrt (\\x = 0..). (Re (vec_of_basis_enum \\ $ x))\\<^sup>2 + (Im (vec_of_basis_enum \\ $ x))\\<^sup>2) = sqrt (\\i = 0..). (Re (vec_of_basis_enum \\ $ i))\\<^sup>2 + (Im (vec_of_basis_enum \\ $ i))\\<^sup>2)\n[PROOF STEP]\napply (rule sum_nonneg)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. x \\ {0..)} \\ 0 \\ (Re (vec_of_basis_enum \\ $ x))\\<^sup>2 + (Im (vec_of_basis_enum \\ $ x))\\<^sup>2\n 2. sqrt (\\x = 0..). (Re (vec_of_basis_enum \\ $ x))\\<^sup>2 + (Im (vec_of_basis_enum \\ $ x))\\<^sup>2) = sqrt (\\i = 0..). (Re (vec_of_basis_enum \\ $ i))\\<^sup>2 + (Im (vec_of_basis_enum \\ $ i))\\<^sup>2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (cmod (complex_of_real (\\x = 0..). let z = vec_of_basis_enum \\ $ x in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))) = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n\ngoal (1 subgoal):\n 1. norm \\ = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm \\ = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm \\ = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n\ngoal (1 subgoal):\n 1. norm \\ = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n[PROOF STEP]\nby -\n[PROOF STATE]\nproof (state)\nthis:\nnorm \\ = (let \\' = vec_of_basis_enum \\ in sqrt (\\i = 0..'. let z = \\' $ i in (Re z)\\<^sup>2 + (Im z)\\<^sup>2))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3842, "file": "Complex_Bounded_Operators_Cblinfun_Matrix", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199552262967, "lm_q2_score": 0.8479677602988602, "lm_q1q2_score": 0.7102747174148745}} {"text": "[STATEMENT]\ntheorem set_times_rearrange4: \"C * (a *o D) = a *o (C * D)\"\n for a :: \"'a::comm_monoid_mult\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. C * a *o D = a *o (C * D)\n[PROOF STEP]\nby (metis mult.commute set_times_rearrange3)", "meta": {"llama_tokens": 114, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9136765328159726, "lm_q2_score": 0.7772998714925403, "lm_q1q2_score": 0.7102006515436053}} {"text": "[STATEMENT]\nlemma sum_of_3_powers: \"2 * (\\i=0.. s \\ y \\ t}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. compact {x - y |x y. x \\ s \\ y \\ t}\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. compact {x - y |x y. x \\ s \\ y \\ t}\n[PROOF STEP]\nhave \"{x - y | x y. x\\s \\ y \\ t} = {x + y | x y. x \\ s \\ y \\ (uminus ` t)}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x - y |x y. x \\ s \\ y \\ t} = {x + y |x y. x \\ s \\ y \\ uminus ` t}\n[PROOF STEP]\nusing diff_conv_add_uminus\n[PROOF STATE]\nproof (prove)\nusing this:\n?a - ?b = ?a + - ?b\n\ngoal (1 subgoal):\n 1. {x - y |x y. x \\ s \\ y \\ t} = {x + y |x y. x \\ s \\ y \\ uminus ` t}\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n{x - y |x y. x \\ s \\ y \\ t} = {x + y |x y. x \\ s \\ y \\ uminus ` t}\n\ngoal (1 subgoal):\n 1. compact {x - y |x y. x \\ s \\ y \\ t}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{x - y |x y. x \\ s \\ y \\ t} = {x + y |x y. x \\ s \\ y \\ uminus ` t}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n{x - y |x y. x \\ s \\ y \\ t} = {x + y |x y. x \\ s \\ y \\ uminus ` t}\n\ngoal (1 subgoal):\n 1. compact {x - y |x y. x \\ s \\ y \\ t}\n[PROOF STEP]\nusing compact_sums[OF assms(1) compact_negations[OF assms(2)]]\n[PROOF STATE]\nproof (prove)\nusing this:\n{x - y |x y. x \\ s \\ y \\ t} = {x + y |x y. x \\ s \\ y \\ uminus ` t}\ncompact {x + y |x y. x \\ s \\ y \\ uminus ` t}\n\ngoal (1 subgoal):\n 1. compact {x - y |x y. x \\ s \\ y \\ t}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncompact {x - y |x y. x \\ s \\ y \\ t}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 950, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711870587667, "lm_q2_score": 0.8354835391516132, "lm_q1q2_score": 0.7101369355407563}} {"text": "[STATEMENT]\nlemma exp_golomb_bit_count_est:\n assumes \"n \\ m \"\n shows \"bit_count (N\\<^sub>e n) \\ (2 * log 2 (real m+1) + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real m + 1) + 1)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. bit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real m + 1) + 1)\n[PROOF STEP]\nhave \"bit_count (N\\<^sub>e n) \\ (2 * log 2 (real n+1) + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real n + 1) + 1)\n[PROOF STEP]\nusing exp_golomb_bit_count\n[PROOF STATE]\nproof (prove)\nusing this:\nbit_count (N\\<^sub>e ?n) \\ ereal (2 * log 2 (real ?n + 1) + 1)\n\ngoal (1 subgoal):\n 1. bit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real n + 1) + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nbit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real n + 1) + 1)\n\ngoal (1 subgoal):\n 1. bit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real m + 1) + 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nbit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real n + 1) + 1)\n\ngoal (1 subgoal):\n 1. bit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real m + 1) + 1)\n[PROOF STEP]\nhave \"... \\ (2 * log 2 (real m+1) + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ereal (2 * log 2 (real n + 1) + 1) \\ ereal (2 * log 2 (real m + 1) + 1)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\ m\n\ngoal (1 subgoal):\n 1. ereal (2 * log 2 (real n + 1) + 1) \\ ereal (2 * log 2 (real m + 1) + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nereal (2 * log 2 (real n + 1) + 1) \\ ereal (2 * log 2 (real m + 1) + 1)\n\ngoal (1 subgoal):\n 1. bit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real m + 1) + 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nbit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real m + 1) + 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nbit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real m + 1) + 1)\n\ngoal (1 subgoal):\n 1. bit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real m + 1) + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nbit_count (N\\<^sub>e n) \\ ereal (2 * log 2 (real m + 1) + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1157, "file": "Prefix_Free_Code_Combinators_Prefix_Free_Code_Combinators", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711832583696, "lm_q2_score": 0.8354835411997897, "lm_q1q2_score": 0.710136934106478}} {"text": "[STATEMENT]\nlemma det_identical_columns:\n assumes A: \"A \\ carrier_mat n n\" \n and ij: \"i \\ j\"\n and i: \"i < n\" and j: \"j < n\"\n and r: \"col A i = col A j\"\n shows \"det A = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A = (0::'a)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. det A = (0::'a)\n[PROOF STEP]\nhave \"det A = det A\\<^sup>T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A = det A\\<^sup>T\n[PROOF STEP]\nusing det_transpose[OF A]\n[PROOF STATE]\nproof (prove)\nusing this:\ndet A\\<^sup>T = det A\n\ngoal (1 subgoal):\n 1. det A = det A\\<^sup>T\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\ndet A = det A\\<^sup>T\n\ngoal (1 subgoal):\n 1. det A = (0::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndet A = det A\\<^sup>T\n\ngoal (1 subgoal):\n 1. det A = (0::'a)\n[PROOF STEP]\nhave \"... = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A\\<^sup>T = (0::'a)\n[PROOF STEP]\nproof (rule det_identical_rows[of _ n i j])\n[PROOF STATE]\nproof (state)\ngoal (5 subgoals):\n 1. A\\<^sup>T \\ carrier_mat n n\n 2. i \\ j\n 3. i < n\n 4. j < n\n 5. row A\\<^sup>T i = row A\\<^sup>T j\n[PROOF STEP]\nshow \"row (transpose_mat A) i = row (transpose_mat A) j\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row A\\<^sup>T i = row A\\<^sup>T j\n[PROOF STEP]\nusing A i j r\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n n\ni < n\nj < n\ncol A i = col A j\n\ngoal (1 subgoal):\n 1. row A\\<^sup>T i = row A\\<^sup>T j\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrow A\\<^sup>T i = row A\\<^sup>T j\n\ngoal (4 subgoals):\n 1. A\\<^sup>T \\ carrier_mat n n\n 2. i \\ j\n 3. i < n\n 4. j < n\n[PROOF STEP]\nqed (auto simp add: assms)\n[PROOF STATE]\nproof (state)\nthis:\ndet A\\<^sup>T = (0::'a)\n\ngoal (1 subgoal):\n 1. det A = (0::'a)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndet A = (0::'a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndet A = (0::'a)\n\ngoal (1 subgoal):\n 1. det A = (0::'a)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ndet A = (0::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1024, "file": "Jordan_Normal_Form_Determinant", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8354835371034368, "lm_q2_score": 0.8499711794579722, "lm_q1q2_score": 0.7101369274495266}} {"text": "[STATEMENT]\nlemma max_all_edges_between: \n assumes \"finite X\" \"finite Y\"\n shows \"card (all_edges_between X Y G) \\ card X * card Y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (all_edges_between X Y G) \\ card X * card Y\n[PROOF STEP]\nby (metis assms card_mono finite_SigmaI all_edges_between_subset card_cartesian_product)", "meta": {"llama_tokens": 134, "file": "Szemeredi_Regularity_Szemeredi", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392939666336, "lm_q2_score": 0.8031737916455819, "lm_q1q2_score": 0.7100371916988643}} {"text": "[STATEMENT]\nlemma Arg_minus_ii [simp]: \"Arg (-\\) = -pi/2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Arg (- \\) = - pi / 2\n[PROOF STEP]\nproof (rule cis_Arg_unique)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. sgn (- \\) = cis (- pi / 2)\n 2. - pi < - pi / 2\n 3. - pi / 2 \\ pi\n[PROOF STEP]\nshow \"sgn (- \\) = cis (- pi / 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sgn (- \\) = cis (- pi / 2)\n[PROOF STEP]\nby (simp add: sgn_eq)\n[PROOF STATE]\nproof (state)\nthis:\nsgn (- \\) = cis (- pi / 2)\n\ngoal (2 subgoals):\n 1. - pi < - pi / 2\n 2. - pi / 2 \\ pi\n[PROOF STEP]\nshow \"- pi / 2 \\ pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - pi / 2 \\ pi\n[PROOF STEP]\nusing pi_not_less_zero\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ pi < 0\n\ngoal (1 subgoal):\n 1. - pi / 2 \\ pi\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\n- pi / 2 \\ pi\n\ngoal (1 subgoal):\n 1. - pi < - pi / 2\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 479, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392848011835, "lm_q2_score": 0.8031737963569014, "lm_q1q2_score": 0.7100371885024066}} {"text": "[STATEMENT]\nlemma Least_imp_le: \"\n \\ \\x. P x; \\x. P x \\ Q x \\ \\\n (LEAST (x::'a::wellorder). Q x) \\ (LEAST x. P x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x. P x; \\x. P x \\ Q x\\ \\ (LEAST x. Q x) \\ (LEAST x. P x)\n[PROOF STEP]\nby (blast intro: Least_le LeastI2_ex)", "meta": {"llama_tokens": 177, "file": "List-Infinite_CommonSet_SetInterval2", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392695254318, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7100371699858359}} {"text": "[STATEMENT]\nlemma length_chop:\n assumes \"n > 0\"\n shows \"length (chop n xs) = nat \\length xs / n\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (chop n xs) = nat \\real (length xs) / real n\\\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. length (chop n xs) = nat \\real (length xs) / real n\\\n[PROOF STEP]\nfrom \\n > 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < n\n[PROOF STEP]\nhave \"real n * length (chop n xs) \\ length xs\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\n\ngoal (1 subgoal):\n 1. real (length xs) \\ real n * real (length (chop n xs))\n[PROOF STEP]\nby (induction n xs rule: chop.induct; subst chop.simps) (auto simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nreal (length xs) \\ real n * real (length (chop n xs))\n\ngoal (1 subgoal):\n 1. length (chop n xs) = nat \\real (length xs) / real n\\\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nreal (length xs) \\ real n * real (length (chop n xs))\n\ngoal (1 subgoal):\n 1. length (chop n xs) = nat \\real (length xs) / real n\\\n[PROOF STEP]\nfrom \\n > 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < n\n[PROOF STEP]\nhave \"real n * length (chop n xs) < length xs + n\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\n\ngoal (1 subgoal):\n 1. real n * real (length (chop n xs)) < real (length xs + n)\n[PROOF STEP]\nby (induction n xs rule: chop.induct; subst chop.simps)\n (auto simp: field_simps split: nat_diff_split_asm)+\n[PROOF STATE]\nproof (state)\nthis:\nreal n * real (length (chop n xs)) < real (length xs + n)\n\ngoal (1 subgoal):\n 1. length (chop n xs) = nat \\real (length xs) / real n\\\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (length xs) \\ real n * real (length (chop n xs))\nreal n * real (length (chop n xs)) < real (length xs + n)\n[PROOF STEP]\nhave \"length (chop n xs) \\ length xs / n\" and \"length (chop n xs) < length xs / n + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (length xs) \\ real n * real (length (chop n xs))\nreal n * real (length (chop n xs)) < real (length xs + n)\n\ngoal (1 subgoal):\n 1. real (length xs) / real n \\ real (length (chop n xs)) &&& real (length (chop n xs)) < real (length xs) / real n + 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (length xs) \\ real n * real (length (chop n xs))\nreal n * real (length (chop n xs)) < real (length xs + n)\n0 < n\n\ngoal (1 subgoal):\n 1. real (length xs) / real n \\ real (length (chop n xs)) &&& real (length (chop n xs)) < real (length xs) / real n + 1\n[PROOF STEP]\nby (auto simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nreal (length xs) / real n \\ real (length (chop n xs))\nreal (length (chop n xs)) < real (length xs) / real n + 1\n\ngoal (1 subgoal):\n 1. length (chop n xs) = nat \\real (length xs) / real n\\\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (length xs) / real n \\ real (length (chop n xs))\nreal (length (chop n xs)) < real (length xs) / real n + 1\n\ngoal (1 subgoal):\n 1. length (chop n xs) = nat \\real (length xs) / real n\\\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\nlength (chop n xs) = nat \\real (length xs) / real n\\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1371, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8519528019683106, "lm_q2_score": 0.8333245994514084, "lm_q1q2_score": 0.7099532274517474}} {"text": "[STATEMENT]\nlemma convergent_prod_iff_nz_lim:\n fixes f :: \"nat \\ 'a :: {topological_semigroup_mult,t2_space,idom}\"\n assumes \"\\i. f i \\ 0\"\n shows \"convergent_prod f \\ (\\L. (\\n. \\i\\n. f i) \\ L \\ L \\ 0)\"\n (is \"?lhs \\ ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convergent_prod f = (\\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a))\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. convergent_prod f \\ \\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a)\n 2. \\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a) \\ convergent_prod f\n[PROOF STEP]\nassume ?lhs\n[PROOF STATE]\nproof (state)\nthis:\nconvergent_prod f\n\ngoal (2 subgoals):\n 1. convergent_prod f \\ \\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a)\n 2. \\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a) \\ convergent_prod f\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nconvergent_prod f\n[PROOF STEP]\nshow ?rhs\n[PROOF STATE]\nproof (prove)\nusing this:\nconvergent_prod f\n\ngoal (1 subgoal):\n 1. \\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a)\n[PROOF STEP]\nusing assms convergentD convergent_prod_imp_convergent convergent_prod_to_zero_iff\n[PROOF STATE]\nproof (prove)\nusing this:\nconvergent_prod f\nf ?i \\ (0::'a)\nconvergent ?X \\ \\L. ?X \\ L\nconvergent_prod ?f \\ convergent (\\n. prod ?f {..n})\nconvergent_prod ?f \\ (\\n. prod ?f {..n}) \\ (0::?'a) = (\\i. ?f i = (0::?'a))\n\ngoal (1 subgoal):\n 1. \\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a)\n\ngoal (1 subgoal):\n 1. \\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a) \\ convergent_prod f\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a) \\ convergent_prod f\n[PROOF STEP]\nassume ?rhs\n[PROOF STATE]\nproof (state)\nthis:\n\\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a)\n\ngoal (1 subgoal):\n 1. \\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a) \\ convergent_prod f\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a)\n[PROOF STEP]\nshow ?lhs\n[PROOF STATE]\nproof (prove)\nusing this:\n\\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a)\n\ngoal (1 subgoal):\n 1. convergent_prod f\n[PROOF STEP]\nunfolding prod_defs\n[PROOF STATE]\nproof (prove)\nusing this:\n\\L. (\\n. prod f {..n}) \\ L \\ L \\ (0::'a)\n\ngoal (1 subgoal):\n 1. \\M p. (\\n. \\i\\n. f (i + M)) \\ p \\ p \\ (0::'a)\n[PROOF STEP]\nby (rule_tac x=0 in exI) auto\n[PROOF STATE]\nproof (state)\nthis:\nconvergent_prod f\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1469, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.851952809486198, "lm_q2_score": 0.8333245891029456, "lm_q1q2_score": 0.7099532249001861}} {"text": "[STATEMENT]\nlemma decompose_init_prod:\n fixes n::nat\n shows \"(\\ i\\ {0..n}. f i) = f 0 * (\\ i\\ {1..n}. f i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod f {0..n} = f 0 * prod f {1..n}\n[PROOF STEP]\nproof (cases \"Suc 0 \\ n\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. Suc 0 \\ n \\ prod f {0..n} = f 0 * prod f {1..n}\n 2. \\ Suc 0 \\ n \\ prod f {0..n} = f 0 * prod f {1..n}\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nSuc 0 \\ n\n\ngoal (2 subgoals):\n 1. Suc 0 \\ n \\ prod f {0..n} = f 0 * prod f {1..n}\n 2. \\ Suc 0 \\ n \\ prod f {0..n} = f 0 * prod f {1..n}\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc 0 \\ n\n\ngoal (1 subgoal):\n 1. prod f {0..n} = f 0 * prod f {1..n}\n[PROOF STEP]\nby (metis One_nat_def Suc_le_D True prod.atLeast0_atMost_Suc_shift prod.atLeast_Suc_atMost_Suc_shift)\n[PROOF STATE]\nproof (state)\nthis:\nprod f {0..n} = f 0 * prod f {1..n}\n\ngoal (1 subgoal):\n 1. \\ Suc 0 \\ n \\ prod f {0..n} = f 0 * prod f {1..n}\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ Suc 0 \\ n \\ prod f {0..n} = f 0 * prod f {1..n}\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ Suc 0 \\ n\n\ngoal (1 subgoal):\n 1. \\ Suc 0 \\ n \\ prod f {0..n} = f 0 * prod f {1..n}\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ Suc 0 \\ n\n\ngoal (1 subgoal):\n 1. prod f {0..n} = f 0 * prod f {1..n}\n[PROOF STEP]\nby (metis One_nat_def atLeastLessThanSuc_atLeastAtMost prod.atLeast0_lessThan_Suc_shift\n prod.atLeast_Suc_lessThan_Suc_shift)\n[PROOF STATE]\nproof (state)\nthis:\nprod f {0..n} = f 0 * prod f {1..n}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 887, "file": "DiscretePricing_Infinite_Coin_Toss_Space", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527982093666, "lm_q2_score": 0.8333245911726382, "lm_q1q2_score": 0.7099532172662056}} {"text": "[STATEMENT]\nlemma fUnionM_funionM:\n \"fUnionM ((\\y. funionM (f y) (g y)) |`| A) = funionM (fUnionM (f |`| A)) (fUnionM (g |`| A))\" for f g A\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fUnionM ((\\y. funionM (f y) (g y)) |`| A) = funionM (fUnionM (f |`| A)) (fUnionM (g |`| A))\n[PROOF STEP]\nby(induction A)(simp_all add: funionM_assoc funionM_commute fun_left_comm)", "meta": {"llama_tokens": 186, "file": "Monomorphic_Monad_Monomorphic_Monad", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8633916240341031, "lm_q2_score": 0.8221891283434876, "lm_q1q2_score": 0.7098712067836673}} {"text": "[STATEMENT]\nlemma var_sum_1:\n fixes f :: \"'b \\ 'a \\ real\"\n assumes \"finite I\"\n assumes \"\\i. i \\ I \\ f i \\ borel_measurable M\"\n assumes \"\\i. i \\ I \\ integrable M (\\\\. f i \\^2)\"\n shows \n \"variance (\\\\. (\\i \\ I. f i \\)) = (\\i \\ I. (\\j \\ I. covariance (f i) (f j)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nhave a:\"\\i j. i \\ I \\ j \\ I \\ integrable M (\\\\. (f i \\ - expectation (f i)) * (f j \\ - expectation (f j)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. \\i \\ I; j \\ I\\ \\ integrable M (\\\\. (f i \\ - expectation (f i)) * (f j \\ - expectation (f j)))\n[PROOF STEP]\nusing assms covar_integrable\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite I\n?i12 \\ I \\ random_variable borel (f ?i12)\n?i12 \\ I \\ integrable M (\\\\. (f ?i12 \\)\\<^sup>2)\n\\random_variable borel ?f; random_variable borel ?g; integrable M (\\\\. (?f \\)\\<^sup>2); integrable M (\\\\. (?g \\)\\<^sup>2)\\ \\ integrable M (\\\\. (?f \\ - expectation ?f) * (?g \\ - expectation ?g))\n\ngoal (1 subgoal):\n 1. \\i j. \\i \\ I; j \\ I\\ \\ integrable M (\\\\. (f i \\ - expectation (f i)) * (f j \\ - expectation (f j)))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\?i12 \\ I; ?j12 \\ I\\ \\ integrable M (\\\\. (f ?i12 \\ - expectation (f ?i12)) * (f ?j12 \\ - expectation (f ?j12)))\n\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nhave \"variance (\\\\. (\\i \\ I. f i \\)) = expectation (\\\\. (\\i\\I. f i \\ - expectation (f i))\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = expectation (\\\\. (\\i\\I. f i \\ - expectation (f i))\\<^sup>2)\n[PROOF STEP]\nusing square_integrable_imp_integrable[OF assms(2,3)]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?i13 \\ I; ?i13 \\ I\\ \\ integrable M (f ?i13)\n\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = expectation (\\\\. (\\i\\I. f i \\ - expectation (f i))\\<^sup>2)\n[PROOF STEP]\nby (simp add: Bochner_Integration.integral_sum sum_subtractf)\n[PROOF STATE]\nproof (state)\nthis:\nexpectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = expectation (\\\\. (\\i\\I. f i \\ - expectation (f i))\\<^sup>2)\n\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nexpectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = expectation (\\\\. (\\i\\I. f i \\ - expectation (f i))\\<^sup>2)\n\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nhave \"... = expectation (\\\\. (\\i \\ I. (\\j \\ I. (f i \\ - expectation (f i)) * (f j \\ - expectation (f j)))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. expectation (\\\\. (\\i\\I. f i \\ - expectation (f i))\\<^sup>2) = expectation (\\\\. \\i\\I. \\j\\I. (f i \\ - expectation (f i)) * (f j \\ - expectation (f j)))\n[PROOF STEP]\nby (simp add: power2_eq_square sum_distrib_right sum_distrib_left mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\nexpectation (\\\\. (\\i\\I. f i \\ - expectation (f i))\\<^sup>2) = expectation (\\\\. \\i\\I. \\j\\I. (f i \\ - expectation (f i)) * (f j \\ - expectation (f j)))\n\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nexpectation (\\\\. (\\i\\I. f i \\ - expectation (f i))\\<^sup>2) = expectation (\\\\. \\i\\I. \\j\\I. (f i \\ - expectation (f i)) * (f j \\ - expectation (f j)))\n\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nhave \"... = (\\i \\ I. (\\j \\ I. covariance (f i) (f j)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. expectation (\\\\. \\i\\I. \\j\\I. (f i \\ - expectation (f i)) * (f j \\ - expectation (f j))) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nusing a\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?i12 \\ I; ?j12 \\ I\\ \\ integrable M (\\\\. (f ?i12 \\ - expectation (f ?i12)) * (f ?j12 \\ - expectation (f ?j12)))\n\ngoal (1 subgoal):\n 1. expectation (\\\\. \\i\\I. \\j\\I. (f i \\ - expectation (f i)) * (f j \\ - expectation (f j))) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nby (simp add: Bochner_Integration.integral_sum covariance_def)\n[PROOF STATE]\nproof (state)\nthis:\nexpectation (\\\\. \\i\\I. \\j\\I. (f i \\ - expectation (f i)) * (f j \\ - expectation (f j))) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nexpectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nexpectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nexpectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. \\j\\I. covariance (f i) (f j))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3339, "file": "Frequency_Moments_Probability_Ext", "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391617003942, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7098712066461449}} {"text": "[STATEMENT]\nlemma hnorm_add_less:\n \"\\(x::'a::real_normed_vector star) y r s. hnorm x < r \\ hnorm y < s \\ hnorm (x + y) < r + s\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x y r s. \\hnorm x < r; hnorm y < s\\ \\ hnorm (x + y) < r + s\n[PROOF STEP]\nby transfer (rule norm_add_less)", "meta": {"llama_tokens": 149, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8633916099737806, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7098712008660226}} {"text": "[STATEMENT]\nlemma shortest_path_Min_alt: \n assumes \"u \\ V\" \"v \\ V\"\n shows \"shortest_path u v = Min ((\\ p. enat (walk_length p)) ` (connecting_paths u v))\" (is \"shortest_path u v = Min ?A\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. shortest_path u v = (MIN p\\connecting_paths u v. enat (walk_length p))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. shortest_path u v = (MIN p\\connecting_paths u v. enat (walk_length p))\n[PROOF STEP]\nhave ne: \"?A \\ {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\p. enat (walk_length p)) ` connecting_paths u v \\ {}\n[PROOF STEP]\nusing connecting_paths_not_empty assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?u \\ V; ?v \\ V\\ \\ connecting_paths ?u ?v \\ {}\nu \\ V\nv \\ V\n\ngoal (1 subgoal):\n 1. (\\p. enat (walk_length p)) ` connecting_paths u v \\ {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\p. enat (walk_length p)) ` connecting_paths u v \\ {}\n\ngoal (1 subgoal):\n 1. shortest_path u v = (MIN p\\connecting_paths u v. enat (walk_length p))\n[PROOF STEP]\nhave \"finite (connecting_paths u v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (connecting_paths u v)\n[PROOF STEP]\nby (simp add: fin_connecting_paths)\n[PROOF STATE]\nproof (state)\nthis:\nfinite (connecting_paths u v)\n\ngoal (1 subgoal):\n 1. shortest_path u v = (MIN p\\connecting_paths u v. enat (walk_length p))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (connecting_paths u v)\n[PROOF STEP]\nhave fin: \"finite ?A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (connecting_paths u v)\n\ngoal (1 subgoal):\n 1. finite ((\\p. enat (walk_length p)) ` connecting_paths u v)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfinite ((\\p. enat (walk_length p)) ` connecting_paths u v)\n\ngoal (1 subgoal):\n 1. shortest_path u v = (MIN p\\connecting_paths u v. enat (walk_length p))\n[PROOF STEP]\nhave \"shortest_path u v = Inf ?A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. shortest_path u v = (INF p\\connecting_paths u v. enat (walk_length p))\n[PROOF STEP]\nunfolding shortest_path_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (INF p\\connecting_paths u v. enat (walk_length p)) = (INF p\\connecting_paths u v. enat (walk_length p))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nshortest_path u v = (INF p\\connecting_paths u v. enat (walk_length p))\n\ngoal (1 subgoal):\n 1. shortest_path u v = (MIN p\\connecting_paths u v. enat (walk_length p))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nshortest_path u v = (INF p\\connecting_paths u v. enat (walk_length p))\n\ngoal (1 subgoal):\n 1. shortest_path u v = (MIN p\\connecting_paths u v. enat (walk_length p))\n[PROOF STEP]\nusing Min_Inf ne\n[PROOF STATE]\nproof (prove)\nusing this:\nshortest_path u v = (INF p\\connecting_paths u v. enat (walk_length p))\n\\finite ?A; ?A \\ {}\\ \\ Min ?A = Inf ?A\n(\\p. enat (walk_length p)) ` connecting_paths u v \\ {}\n\ngoal (1 subgoal):\n 1. shortest_path u v = (MIN p\\connecting_paths u v. enat (walk_length p))\n[PROOF STEP]\nby (metis fin)\n[PROOF STATE]\nproof (state)\nthis:\nshortest_path u v = (MIN p\\connecting_paths u v. enat (walk_length p))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1419, "file": "Undirected_Graph_Theory_Connectivity", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8633916170039421, "lm_q2_score": 0.822189123986562, "lm_q1q2_score": 0.7098711972418124}} {"text": "[STATEMENT]\nlemma connected_iff_connected_component:\n \"connected S \\ (\\x \\ S. \\y \\ S. connected_component S x y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. connected S = (\\x\\S. \\y\\S. connected_component S x y)\n[PROOF STEP]\nusing connected_component_in\n[PROOF STATE]\nproof (prove)\nusing this:\nconnected_component ?S ?x ?y \\ ?x \\ ?S \\ ?y \\ ?S\n\ngoal (1 subgoal):\n 1. connected S = (\\x\\S. \\y\\S. connected_component S x y)\n[PROOF STEP]\nby (auto simp: connected_iff_eq_connected_component_set)", "meta": {"llama_tokens": 229, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8633916170039421, "lm_q2_score": 0.8221891239865619, "lm_q1q2_score": 0.7098711972418122}} {"text": "[STATEMENT]\nlemma max_decr:\n fixes X :: \"nat set\"\n assumes \"\\x \\ X. x \\ k\" \"finite X\"\n shows \"Max ((\\x. x - k) ` X) = Max X - k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (MAX x\\X. x - k) = Max X - k\n[PROOF STEP]\nproof (rule mono_Max_commute[symmetric])\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. mono (\\x. x - k)\n 2. finite X\n 3. X \\ {}\n[PROOF STEP]\nshow \"mono (\\x. x - k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mono (\\x. x - k)\n[PROOF STEP]\nby (rule monoI) linarith\n[PROOF STATE]\nproof (state)\nthis:\nmono (\\x. x - k)\n\ngoal (2 subgoals):\n 1. finite X\n 2. X \\ {}\n[PROOF STEP]\nshow \"finite X\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite X\n[PROOF STEP]\nby fact\n[PROOF STATE]\nproof (state)\nthis:\nfinite X\n\ngoal (1 subgoal):\n 1. X \\ {}\n[PROOF STEP]\nshow \"X \\ {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. X \\ {}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\X. k \\ x\nfinite X\n\ngoal (1 subgoal):\n 1. X \\ {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nX \\ {}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 563, "file": "CakeML_Codegen_Utils_Compiler_Utils", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916099737807, "lm_q2_score": 0.8221891261650248, "lm_q1q2_score": 0.7098711933425567}} {"text": "[STATEMENT]\nlemma has_real_derivative_inverse_sqrt:\n assumes \"x > 0\"\n shows \"((\\x. inverse (sqrt x) / 2) has_real_derivative - (inverse (sqrt x ^ 3) / 4)) (at x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. inverse (sqrt x) / 2) has_real_derivative - (inverse (sqrt x ^ 3) / 4)) (at x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\x. inverse (sqrt x) / 2) has_real_derivative - (inverse (sqrt x ^ 3) / 4)) (at x)\n[PROOF STEP]\nhave inv_sqrt_mult: \"(inverse (sqrt x)/2) * (sqrt x * 2) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse (sqrt x) / 2 * (sqrt x * 2) = 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n\ngoal (1 subgoal):\n 1. inverse (sqrt x) / 2 * (sqrt x * 2) = 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ninverse (sqrt x) / 2 * (sqrt x * 2) = 1\n\ngoal (1 subgoal):\n 1. ((\\x. inverse (sqrt x) / 2) has_real_derivative - (inverse (sqrt x ^ 3) / 4)) (at x)\n[PROOF STEP]\nhave inv_sqrt_mult2: \"(- inverse ((sqrt x)^3)/2)* x * (sqrt x * 2) = -1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - inverse (sqrt x ^ 3) / 2 * x * (sqrt x * 2) = - 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n\ngoal (1 subgoal):\n 1. - inverse (sqrt x ^ 3) / 2 * x * (sqrt x * 2) = - 1\n[PROOF STEP]\nby (simp add: field_simps power3_eq_cube)\n[PROOF STATE]\nproof (state)\nthis:\n- inverse (sqrt x ^ 3) / 2 * x * (sqrt x * 2) = - 1\n\ngoal (1 subgoal):\n 1. ((\\x. inverse (sqrt x) / 2) has_real_derivative - (inverse (sqrt x ^ 3) / 4)) (at x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n- inverse (sqrt x ^ 3) / 2 * x * (sqrt x * 2) = - 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n- inverse (sqrt x ^ 3) / 2 * x * (sqrt x * 2) = - 1\n\ngoal (1 subgoal):\n 1. ((\\x. inverse (sqrt x) / 2) has_real_derivative - (inverse (sqrt x ^ 3) / 4)) (at x)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n- inverse (sqrt x ^ 3) / 2 * x * (sqrt x * 2) = - 1\n0 < x\n\ngoal (1 subgoal):\n 1. ((\\x. inverse (sqrt x) / 2) has_real_derivative - (inverse (sqrt x ^ 3) / 4)) (at x)\n[PROOF STEP]\nby (safe intro!: DERIV_imp_deriv derivative_eq_intros)\n (auto intro: derivative_eq_intros inv_sqrt_mult [THEN ssubst] inv_sqrt_mult2 [THEN ssubst]\n simp add: divide_simps power3_eq_cube)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. inverse (sqrt x) / 2) has_real_derivative - (inverse (sqrt x ^ 3) / 4)) (at x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1183, "file": "Hyperdual_TwiceFieldDifferentiable", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916064586998, "lm_q2_score": 0.8221891283434876, "lm_q1q2_score": 0.7098711923333618}} {"text": "[STATEMENT]\nlemma homeomorphic_trans [trans]:\n assumes \"S homeomorphic T\"\n and \"T homeomorphic U\"\n shows \"S homeomorphic U\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. S homeomorphic U\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nS homeomorphic T\nT homeomorphic U\n\ngoal (1 subgoal):\n 1. S homeomorphic U\n[PROOF STEP]\nunfolding homeomorphic_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\f g. homeomorphism S T f g\n\\f g. homeomorphism T U f g\n\ngoal (1 subgoal):\n 1. \\f g. homeomorphism S U f g\n[PROOF STEP]\nby (metis homeomorphism_compose)", "meta": {"llama_tokens": 224, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633915959134572, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7098711893057774}} {"text": "[STATEMENT]\nlemma poly_roots_degree:\n fixes p :: \"'a::idom poly\"\n shows \"p \\ 0 \\ card {x. poly p x = 0} \\ degree p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p \\ 0 \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nproof (induct n \\ \"degree p\" arbitrary: p)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\p. \\0 = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\n p. \\\\p. \\n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p; Suc n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\ncase (0 p)\n[PROOF STATE]\nproof (state)\nthis:\n0 = degree p\np \\ 0\n\ngoal (2 subgoals):\n 1. \\p. \\0 = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\n p. \\\\p. \\n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p; Suc n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 = degree p\np \\ 0\n[PROOF STEP]\nobtain a where \"a \\ 0\" and \"p = [:a:]\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 = degree p\np \\ 0\n\ngoal (1 subgoal):\n 1. (\\a. \\a \\ (0::'a); p = [:a:]\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (cases p, simp split: if_splits)\n[PROOF STATE]\nproof (state)\nthis:\na \\ (0::'a)\np = [:a:]\n\ngoal (2 subgoals):\n 1. \\p. \\0 = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\n p. \\\\p. \\n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p; Suc n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ (0::'a)\np = [:a:]\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ (0::'a)\np = [:a:]\n\ngoal (1 subgoal):\n 1. card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard {x. poly p x = (0::'a)} \\ degree p\n\ngoal (1 subgoal):\n 1. \\n p. \\\\p. \\n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p; Suc n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n p. \\\\p. \\n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p; Suc n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\ncase (Suc n p)\n[PROOF STATE]\nproof (state)\nthis:\n\\n = degree ?p; ?p \\ 0\\ \\ card {x. poly ?p x = (0::'a)} \\ degree ?p\nSuc n = degree p\np \\ 0\n\ngoal (1 subgoal):\n 1. \\n p. \\\\p. \\n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p; Suc n = degree p; p \\ 0\\ \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nproof (cases \"\\x. poly p x = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\n\\x. poly p x = (0::'a)\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x. poly p x = (0::'a)\n[PROOF STEP]\nobtain a where a: \"poly p a = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. poly p x = (0::'a)\n\ngoal (1 subgoal):\n 1. (\\a. poly p a = (0::'a) \\ thesis) \\ thesis\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\npoly p a = (0::'a)\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\npoly p a = (0::'a)\n[PROOF STEP]\nhave \"[:-a, 1:] dvd p\"\n[PROOF STATE]\nproof (prove)\nusing this:\npoly p a = (0::'a)\n\ngoal (1 subgoal):\n 1. [:- a, 1::'a:] dvd p\n[PROOF STEP]\nby (simp only: poly_eq_0_iff_dvd)\n[PROOF STATE]\nproof (state)\nthis:\n[:- a, 1::'a:] dvd p\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[:- a, 1::'a:] dvd p\n[PROOF STEP]\nobtain k where k: \"p = [:-a, 1:] * k\"\n[PROOF STATE]\nproof (prove)\nusing this:\n[:- a, 1::'a:] dvd p\n\ngoal (1 subgoal):\n 1. (\\k. p = [:- a, 1::'a:] * k \\ thesis) \\ thesis\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\np = [:- a, 1::'a:] * k\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nwith \\p \\ 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\np \\ 0\np = [:- a, 1::'a:] * k\n[PROOF STEP]\nhave \"k \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ 0\np = [:- a, 1::'a:] * k\n\ngoal (1 subgoal):\n 1. k \\ 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nk \\ 0\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nwith k\n[PROOF STATE]\nproof (chain)\npicking this:\np = [:- a, 1::'a:] * k\nk \\ 0\n[PROOF STEP]\nhave \"degree p = Suc (degree k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\np = [:- a, 1::'a:] * k\nk \\ 0\n\ngoal (1 subgoal):\n 1. degree p = Suc (degree k)\n[PROOF STEP]\nby (simp add: degree_mult_eq del: mult_pCons_left)\n[PROOF STATE]\nproof (state)\nthis:\ndegree p = Suc (degree k)\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nwith \\Suc n = degree p\\\n[PROOF STATE]\nproof (chain)\npicking this:\nSuc n = degree p\ndegree p = Suc (degree k)\n[PROOF STEP]\nhave \"n = degree k\"\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc n = degree p\ndegree p = Suc (degree k)\n\ngoal (1 subgoal):\n 1. n = degree k\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn = degree k\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nfrom Suc.hyps(1)[OF this \\k \\ 0\\]\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {x. poly k x = (0::'a)} \\ degree k\n[PROOF STEP]\nhave le: \"card {x. poly k x = 0} \\ degree k\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {x. poly k x = (0::'a)} \\ degree k\n\ngoal (1 subgoal):\n 1. card {x. poly k x = (0::'a)} \\ degree k\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {x. poly k x = (0::'a)} \\ degree k\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nhave \"card {x. poly p x = 0} = card {x. poly ([:-a, 1:] * k) x = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {x. poly p x = (0::'a)} = card {x. poly ([:- a, 1::'a:] * k) x = (0::'a)}\n[PROOF STEP]\nunfolding k\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {x. poly ([:- a, 1::'a:] * k) x = (0::'a)} = card {x. poly ([:- a, 1::'a:] * k) x = (0::'a)}\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\ncard {x. poly p x = (0::'a)} = card {x. poly ([:- a, 1::'a:] * k) x = (0::'a)}\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {x. poly p x = (0::'a)} = card {x. poly ([:- a, 1::'a:] * k) x = (0::'a)}\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nhave \"{x. poly ([:-a, 1:] * k) x = 0} = insert a {x. poly k x = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x. poly ([:- a, 1::'a:] * k) x = (0::'a)} = insert a {x. poly k x = (0::'a)}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{x. poly ([:- a, 1::'a:] * k) x = (0::'a)} = insert a {x. poly k x = (0::'a)}\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n{x. poly ([:- a, 1::'a:] * k) x = (0::'a)} = insert a {x. poly k x = (0::'a)}\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nhave \"card \\ \\ Suc (card {x. poly k x = 0})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (insert a {x. poly k x = (0::'a)}) \\ Suc (card {x. poly k x = (0::'a)})\n[PROOF STEP]\nunfolding card_insert_if[OF poly_roots_finite[OF \\k \\ 0\\]]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if a \\ {x. poly k x = (0::'a)} then card {x. poly k x = (0::'a)} else Suc (card {x. poly k x = (0::'a)})) \\ Suc (card {x. poly k x = (0::'a)})\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (insert a {x. poly k x = (0::'a)}) \\ Suc (card {x. poly k x = (0::'a)})\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (insert a {x. poly k x = (0::'a)}) \\ Suc (card {x. poly k x = (0::'a)})\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nhave \"\\ \\ Suc (degree k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc (card {x. poly k x = (0::'a)}) \\ Suc (degree k)\n[PROOF STEP]\nusing le\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {x. poly k x = (0::'a)} \\ degree k\n\ngoal (1 subgoal):\n 1. Suc (card {x. poly k x = (0::'a)}) \\ Suc (degree k)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nSuc (card {x. poly k x = (0::'a)}) \\ Suc (degree k)\n\ngoal (2 subgoals):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n 2. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {x. poly p x = (0::'a)} \\ Suc (degree k)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {x. poly p x = (0::'a)} \\ Suc (degree k)\n\ngoal (1 subgoal):\n 1. card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nusing \\degree p = Suc (degree k)\\\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {x. poly p x = (0::'a)} \\ Suc (degree k)\ndegree p = Suc (degree k)\n\ngoal (1 subgoal):\n 1. card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard {x. poly p x = (0::'a)} \\ degree p\n\ngoal (1 subgoal):\n 1. \\x. poly p x = (0::'a) \\ card {x. poly p x = (0::'a)} \\ degree p\n[PROOF STEP]\nqed simp\n[PROOF STATE]\nproof (state)\nthis:\ncard {x. poly p x = (0::'a)} \\ degree p\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5957, "file": "Polynomial_Interpolation_Missing_Polynomial", "length": 54, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391595913457, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7098711893057773}} {"text": "[STATEMENT]\nlemma closed_segment_translation_eq [simp]:\n \"d + x \\ closed_segment (d + a) (d + b) \\ x \\ closed_segment a b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (d + x \\ closed_segment (d + a) (d + b)) = (x \\ closed_segment a b)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (d + x \\ closed_segment (d + a) (d + b)) = (x \\ closed_segment a b)\n[PROOF STEP]\nhave *: \"\\d x a b. x \\ closed_segment a b \\ d + x \\ closed_segment (d + a) (d + b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\d x a b. x \\ closed_segment a b \\ d + x \\ closed_segment (d + a) (d + b)\n[PROOF STEP]\napply (simp add: closed_segment_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\d x a b. \\u. x = (1 - u) *\\<^sub>R a + u *\\<^sub>R b \\ 0 \\ u \\ u \\ 1 \\ \\u. d + x = (1 - u) *\\<^sub>R (d + a) + u *\\<^sub>R (d + b) \\ 0 \\ u \\ u \\ 1\n[PROOF STEP]\napply (erule ex_forward)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\d x a b u. x = (1 - u) *\\<^sub>R a + u *\\<^sub>R b \\ 0 \\ u \\ u \\ 1 \\ d + x = (1 - u) *\\<^sub>R (d + a) + u *\\<^sub>R (d + b) \\ 0 \\ u \\ u \\ 1\n[PROOF STEP]\napply (simp add: algebra_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n?x \\ closed_segment ?a ?b \\ ?d + ?x \\ closed_segment (?d + ?a) (?d + ?b)\n\ngoal (1 subgoal):\n 1. (d + x \\ closed_segment (d + a) (d + b)) = (x \\ closed_segment a b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (d + x \\ closed_segment (d + a) (d + b)) = (x \\ closed_segment a b)\n[PROOF STEP]\nusing * [where d = \"-d\"] *\n[PROOF STATE]\nproof (prove)\nusing this:\n?x \\ closed_segment ?a ?b \\ - d + ?x \\ closed_segment (- d + ?a) (- d + ?b)\n?x \\ closed_segment ?a ?b \\ ?d + ?x \\ closed_segment (?d + ?a) (?d + ?b)\n\ngoal (1 subgoal):\n 1. (d + x \\ closed_segment (d + a) (d + b)) = (x \\ closed_segment a b)\n[PROOF STEP]\nby (fastforce simp add:)\n[PROOF STATE]\nproof (state)\nthis:\n(d + x \\ closed_segment (d + a) (d + b)) = (x \\ closed_segment a b)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1044, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869884059266, "lm_q2_score": 0.8104789086703225, "lm_q1q2_score": 0.7098068825909037}} {"text": "[STATEMENT]\nlemma d_OUT_h_plus_not_infty [simp]: \"d_OUT (h_plus i) x \\ top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. d_OUT (h_plus i) x \\ \\\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. d_OUT (h_plus i) x \\ \\\n[PROOF STEP]\nhave \"d_OUT (h_plus i) x \\ (\\\\<^sup>+ y\\UNIV. \\\\<^sup>+ x. h_plus i (x, y))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. d_OUT (h_plus i) x \\ (\\\\<^sup>+ y. \\\\<^sup>+ x. h_plus i (x, y))\n[PROOF STEP]\nunfolding d_OUT_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sup>+ y. h_plus i (x, y)) \\ (\\\\<^sup>+ y. \\\\<^sup>+ x. h_plus i (x, y))\n[PROOF STEP]\nby(rule nn_integral_mono nn_integral_ge_point)+ simp\n[PROOF STATE]\nproof (state)\nthis:\nd_OUT (h_plus i) x \\ (\\\\<^sup>+ y. \\\\<^sup>+ x. h_plus i (x, y))\n\ngoal (1 subgoal):\n 1. d_OUT (h_plus i) x \\ \\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nd_OUT (h_plus i) x \\ (\\\\<^sup>+ y. \\\\<^sup>+ x. h_plus i (x, y))\n\ngoal (1 subgoal):\n 1. d_OUT (h_plus i) x \\ \\\n[PROOF STEP]\nhave \"\\ < \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sup>+ y. \\\\<^sup>+ x. h_plus i (x, y)) < \\\n[PROOF STEP]\nusing h_plus_sum_finite\n[PROOF STATE]\nproof (prove)\nusing this:\nintegral\\<^sup>N (count_space UNIV) (h_plus ?i) \\ \\\n\ngoal (1 subgoal):\n 1. (\\\\<^sup>+ y. \\\\<^sup>+ x. h_plus i (x, y)) < \\\n[PROOF STEP]\nby(simp add: nn_integral_snd_count_space less_top)\n[PROOF STATE]\nproof (state)\nthis:\n(\\\\<^sup>+ y. \\\\<^sup>+ x. h_plus i (x, y)) < \\\n\ngoal (1 subgoal):\n 1. d_OUT (h_plus i) x \\ \\\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nd_OUT (h_plus i) x < \\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nd_OUT (h_plus i) x < \\\n\ngoal (1 subgoal):\n 1. d_OUT (h_plus i) x \\ \\\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nd_OUT (h_plus i) x \\ \\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 998, "file": "MFMC_Countable_MFMC_Flow_Attainability", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.870597271765821, "lm_q2_score": 0.8152324938410783, "lm_q1q2_score": 0.7097391849928892}} {"text": "[STATEMENT]\nlemma fmrel_converse_tranclp_induct:\n \"fmrel r\\<^sup>+\\<^sup>+ a b \\\n reflp r \\\n (\\y. fmrel r y b \\ P y) \\\n (\\y z. fmrel r y z \\ fmrel r\\<^sup>+\\<^sup>+ z b \\ P z \\ P y) \\ P a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\fmrel r\\<^sup>+\\<^sup>+ a b; reflp r; \\y. fmrel r y b \\ P y; \\y z. \\fmrel r y z; fmrel r\\<^sup>+\\<^sup>+ z b; P z\\ \\ P y\\ \\ P a\n[PROOF STEP]\napply (drule fmrel_to_trancl, simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\fmrel r\\<^sup>+\\<^sup>+ a b; \\y. fmrel r y b \\ P y; \\y z. \\fmrel r y z; fmrel r\\<^sup>+\\<^sup>+ z b; P z\\ \\ P y; (fmrel r)\\<^sup>+\\<^sup>+ a b\\ \\ P a\n[PROOF STEP]\nby (erule converse_tranclp_induct; simp add: trancl_to_fmrel)", "meta": {"llama_tokens": 436, "file": "Safe_OCL_Finite_Map_Ext", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972784807408, "lm_q2_score": 0.8152324826183822, "lm_q1q2_score": 0.7097391806966613}} {"text": "[STATEMENT]\ntheorem (in state_space)\n shows \"\\\\ \\True\\\n \\S :== 0;; \\I :== 1;;\n WHILE \\I \\ n\n INV \\\\S = (SUMM j<\\I. j)\\\n DO\n \\S :== \\S + \\I;;\n \\I :== \\I + 1\n OD\n \\\\S = (SUMM j\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\ \\True\\ \\S :== 0;; \\I :== 1;; WHILE \\I \\ n INV \\\\S = (SUMM j<\\I. j)\\ DO \\S :== \\S + \\I;; \\I :== \\I + 1 OD \\\\S = (SUMM j\n[PROOF STEP]\napply vcg\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. True \\ 0 = (SUMM j<1. j)\n 2. \\I S. \\S = (SUMM j n\\ \\ S + I = (SUMM jI S. \\S = (SUMM j I \\ n\\ \\ S = (SUMM j 'a :: comm_semiring_0\"\n assumes \"x \\ 0\"\n shows \"sum_upto (dirichlet_prod f g) x + sum_upto f (sqrt x) * sum_upto g (sqrt x) = \n sum_upto (\\n. f n * sum_upto g (x / real n)) (sqrt x) +\n sum_upto (\\n. sum_upto f (x / real n) * g n) (sqrt x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_upto (dirichlet_prod f g) x + sum_upto f (sqrt x) * sum_upto g (sqrt x) = sum_upto (\\n. f n * sum_upto g (x / real n)) (sqrt x) + sum_upto (\\n. sum_upto f (x / real n) * g n) (sqrt x)\n[PROOF STEP]\nusing assms hyperbola_method_semiring[of \"sqrt x\" \"sqrt x\" x]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ x\n\\0 \\ sqrt x; 0 \\ sqrt x; sqrt x * sqrt x = x\\ \\ sum_upto (dirichlet_prod ?f ?g) x + sum_upto ?f (sqrt x) * sum_upto ?g (sqrt x) = sum_upto (\\n. ?f n * sum_upto ?g (x / real n)) (sqrt x) + sum_upto (\\n. sum_upto ?f (x / real n) * ?g n) (sqrt x)\n\ngoal (1 subgoal):\n 1. sum_upto (dirichlet_prod f g) x + sum_upto f (sqrt x) * sum_upto g (sqrt x) = sum_upto (\\n. f n * sum_upto g (x / real n)) (sqrt x) + sum_upto (\\n. sum_upto f (x / real n) * g n) (sqrt x)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 573, "file": "Dirichlet_Series_Arithmetic_Summatory", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099070035949657, "lm_q2_score": 0.7799929104825007, "lm_q1q2_score": 0.7097210120024485}} {"text": "[STATEMENT]\nlemma fps_mult_assoc_lemma:\n fixes k :: nat\n and f :: \"nat \\ nat \\ nat \\ 'a::comm_monoid_add\"\n shows \"(\\j=0..k. \\i=0..j. f i (j - i) (n - j)) =\n (\\j=0..k. \\i=0..k - j. f j i (n - j - i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..k. \\i = 0..j. f i (j - i) (n - j)) = (\\j = 0..k. \\i = 0..k - j. f j i (n - j - i))\n[PROOF STEP]\nby (induct k) (simp_all add: Suc_diff_le sum.distrib add.assoc)", "meta": {"llama_tokens": 243, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099070158103778, "lm_q2_score": 0.7799928900257127, "lm_q1q2_score": 0.7097210029166084}} {"text": "[STATEMENT]\nlemma has_integral_cmul_iff:\n assumes \"c \\ 0\"\n shows \"((\\x. c *\\<^sub>R f x) has_integral (c *\\<^sub>R I)) A \\ (f has_integral I) A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. c *\\<^sub>R f x) has_integral c *\\<^sub>R I) A = (f has_integral I) A\n[PROOF STEP]\nusing assms has_integral_cmul[of f I A c]\n has_integral_cmul[of \"\\x. c *\\<^sub>R f x\" \"c *\\<^sub>R I\" A \"inverse c\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ 0\n(f has_integral I) A \\ ((\\x. c *\\<^sub>R f x) has_integral c *\\<^sub>R I) A\n((\\x. c *\\<^sub>R f x) has_integral c *\\<^sub>R I) A \\ ((\\x. c *\\<^sub>R f x /\\<^sub>R c) has_integral c *\\<^sub>R I /\\<^sub>R c) A\n\ngoal (1 subgoal):\n 1. ((\\x. c *\\<^sub>R f x) has_integral c *\\<^sub>R I) A = (f has_integral I) A\n[PROOF STEP]\nby (auto simp: field_simps)", "meta": {"llama_tokens": 417, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894745194283, "lm_q2_score": 0.7931059609645724, "lm_q1q2_score": 0.709662866049716}} {"text": "[STATEMENT]\nlemma UN_Ioc_eq_UNIV: \"(\\n. { -real n <.. real n}) = UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. {- real n<..real n}) = UNIV\n[PROOF STEP]\nby auto\n (metis le_less_trans minus_minus neg_less_iff_less not_le real_arch_simple\n of_nat_0_le_iff reals_Archimedean2)", "meta": {"llama_tokens": 135, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976953030553434, "lm_q2_score": 0.7905303186696747, "lm_q1q2_score": 0.7096553539926108}} {"text": "[STATEMENT]\nlemma ex2_3: \"\\integral {0 .. 1} (\\x::real. arctan (sqrt (x\\<^sup>2 + 2)) / (sqrt (x\\<^sup>2 + 2) * (x\\<^sup>2 + 1))) - 5*pi\\<^sup>2/96\\ \\ 1 / 10^3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\integral {0..1} (\\x. arctan (sqrt (x\\<^sup>2 + 2)) / (sqrt (x\\<^sup>2 + 2) * (x\\<^sup>2 + 1))) - 5 * pi\\<^sup>2 / 96\\ \\ 1 / 10 ^ 3\n[PROOF STEP]\nby (rule abs_minus_leI) (tactic \\integral_bnds_tac 1 30 10 6 12 \"\" @{thms pow_gt_zero} @{context} 1\\)", "meta": {"llama_tokens": 267, "file": "Ordinary_Differential_Equations_Ex_Examples_Integral", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.929440403812707, "lm_q2_score": 0.7634837581726991, "lm_q1q2_score": 0.7096126525004766}} {"text": "[STATEMENT]\nlemma inter_eq_blocks_eq_card: \"card b1 = card b2 \\ finite b1 \\ finite b2 \\ b1 |\\| b2 = card b1 \n \\ b1 = b2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\card b1 = card b2; finite b1; finite b2; b1 |\\| b2 = card b1\\ \\ b1 = b2\n[PROOF STEP]\nusing equal_card_inter_fin_eq_sets intersection_number_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite ?A; finite ?B; card ?A = card ?B; card (?A \\ ?B) = card ?A\\ \\ ?A = ?B\n?b1.0 |\\| ?b2.0 \\ card (?b1.0 \\ ?b2.0)\n\ngoal (1 subgoal):\n 1. \\card b1 = card b2; finite b1; finite b2; b1 |\\| b2 = card b1\\ \\ b1 = b2\n[PROOF STEP]\nby (metis)", "meta": {"llama_tokens": 334, "file": "Fishers_Inequality_Design_Extras", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970779778825, "lm_q2_score": 0.8056321936479701, "lm_q1q2_score": 0.7095984820900437}} {"text": "[STATEMENT]\nlemma ordLeq_Sigma_mono1:\nassumes \"\\i \\ I. p i \\o r i\"\nshows \"|SIGMA i : I. Field(p i)| \\o |SIGMA i : I. Field(r i)|\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. |SIGMA i:I. Field (p i)| \\o |SIGMA i:I. Field (r i)|\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i\\I. p i \\o r i\n\ngoal (1 subgoal):\n 1. |SIGMA i:I. Field (p i)| \\o |SIGMA i:I. Field (r i)|\n[PROOF STEP]\nby (auto simp add: card_of_Sigma_mono1)", "meta": {"llama_tokens": 240, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970842359877, "lm_q2_score": 0.8056321796478255, "lm_q1q2_score": 0.7095984748004881}} {"text": "[STATEMENT]\nlemma left_null_space_orthogonal_complement_col_space:\n fixes A::\"real^'cols::{finite,wellorder}^'rows\"\n shows \"left_null_space A = orthogonal_complement (col_space A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. left_null_space A = orthogonal_complement (col_space A)\n[PROOF STEP]\nusing null_space_orthogonal_complement_row_space[of \"transpose A\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nnull_space (Finite_Cartesian_Product.transpose A) = orthogonal_complement (row_space (Finite_Cartesian_Product.transpose A))\n\ngoal (1 subgoal):\n 1. left_null_space A = orthogonal_complement (col_space A)\n[PROOF STEP]\nunfolding left_null_space_eq_null_space_transpose\n[PROOF STATE]\nproof (prove)\nusing this:\nnull_space (Finite_Cartesian_Product.transpose A) = orthogonal_complement (row_space (Finite_Cartesian_Product.transpose A))\n\ngoal (1 subgoal):\n 1. null_space (Finite_Cartesian_Product.transpose A) = orthogonal_complement (col_space A)\n[PROOF STEP]\nunfolding col_space_eq_row_space_transpose\n[PROOF STATE]\nproof (prove)\nusing this:\nnull_space (Finite_Cartesian_Product.transpose A) = orthogonal_complement (row_space (Finite_Cartesian_Product.transpose A))\n\ngoal (1 subgoal):\n 1. null_space (Finite_Cartesian_Product.transpose A) = orthogonal_complement (row_space (Finite_Cartesian_Product.transpose A))\n[PROOF STEP]\n.", "meta": {"llama_tokens": 480, "file": "QR_Decomposition_Least_Squares_Approximation", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110368115781, "lm_q2_score": 0.7956581049086031, "lm_q1q2_score": 0.7095766794860767}} {"text": "[STATEMENT]\ntheorem height_expectation_bound:\n assumes \"finite A\" \"A \\ {}\"\n shows \"measure_pmf.expectation (random_bst A) height\n \\ height_exp_approx (card A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhave \"convex_on UNIV ((powr) 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex_on UNIV ((powr) 2)\n[PROOF STEP]\nby (intro convex_on_realI[where f' = \"\\x. ln 2 * 2 powr x\"])\n (auto intro!: derivative_eq_intros DERIV_powr simp: powr_def [abs_def])\n[PROOF STATE]\nproof (state)\nthis:\nconvex_on UNIV ((powr) 2)\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhence \"2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t - 1)) \\\n measure_pmf.expectation (random_bst A) (\\t. 2 powr real (height t - 1))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nconvex_on UNIV ((powr) 2)\n\ngoal (1 subgoal):\n 1. 2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t - 1)) \\ measure_pmf.expectation (random_bst A) (\\t. 2 powr real (height t - 1))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nconvex_on UNIV ((powr) 2)\nfinite A\nA \\ {}\n\ngoal (1 subgoal):\n 1. 2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t - 1)) \\ measure_pmf.expectation (random_bst A) (\\t. 2 powr real (height t - 1))\n[PROOF STEP]\nby (intro measure_pmf.jensens_inequality[where I = UNIV])\n (auto intro!: integrable_measure_pmf_finite)\n[PROOF STATE]\nproof (state)\nthis:\n2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t - 1)) \\ measure_pmf.expectation (random_bst A) (\\t. 2 powr real (height t - 1))\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t - 1)) \\ measure_pmf.expectation (random_bst A) (\\t. 2 powr real (height t - 1))\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhave \"(\\t. 2 powr real (height t - 1)) = (\\t. 2 ^ (height t - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\t. 2 powr real (height t - 1)) = (\\t. 2 ^ (height t - 1))\n[PROOF STEP]\nby (simp add: powr_realpow)\n[PROOF STATE]\nproof (state)\nthis:\n(\\t. 2 powr real (height t - 1)) = (\\t. 2 ^ (height t - 1))\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\t. 2 powr real (height t - 1)) = (\\t. 2 ^ (height t - 1))\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhave \"measure_pmf.expectation (random_bst A) (\\t. 2 ^ (height t - 1)) =\n measure_pmf.expectation (random_bst A) (\\t. real (eheight t))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\t. 2 ^ (height t - 1)) = measure_pmf.expectation (random_bst A) (\\t. real (eheight t))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nA \\ {}\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\t. 2 ^ (height t - 1)) = measure_pmf.expectation (random_bst A) (\\t. real (eheight t))\n[PROOF STEP]\nby (intro integral_cong_AE)\n (auto simp: AE_measure_pmf_iff random_bst_altdef eheight_def)\n[PROOF STATE]\nproof (state)\nthis:\nmeasure_pmf.expectation (random_bst A) (\\t. 2 ^ (height t - 1)) = measure_pmf.expectation (random_bst A) (\\t. real (eheight t))\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmeasure_pmf.expectation (random_bst A) (\\t. 2 ^ (height t - 1)) = measure_pmf.expectation (random_bst A) (\\t. real (eheight t))\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhave \"\\ = measure_pmf.expectation (map_pmf eheight (random_bst A)) real\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\t. real (eheight t)) = measure_pmf.expectation (map_pmf eheight (random_bst A)) real\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmeasure_pmf.expectation (random_bst A) (\\t. real (eheight t)) = measure_pmf.expectation (map_pmf eheight (random_bst A)) real\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmeasure_pmf.expectation (random_bst A) (\\t. real (eheight t)) = measure_pmf.expectation (map_pmf eheight (random_bst A)) real\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhave \"map_pmf eheight (random_bst A) = eheight_rbst (card A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. map_pmf eheight (random_bst A) = eheight_rbst (card A)\n[PROOF STEP]\nby (rule eheight_rbst [symmetric]) fact+\n[PROOF STATE]\nproof (state)\nthis:\nmap_pmf eheight (random_bst A) = eheight_rbst (card A)\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmap_pmf eheight (random_bst A) = eheight_rbst (card A)\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhave \"measure_pmf.expectation \\ real = eheight_exp (card A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. measure_pmf.expectation (eheight_rbst (card A)) real = eheight_exp (card A)\n[PROOF STEP]\nby (simp add: eheight_exp_def)\n[PROOF STATE]\nproof (state)\nthis:\nmeasure_pmf.expectation (eheight_rbst (card A)) real = eheight_exp (card A)\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmeasure_pmf.expectation (eheight_rbst (card A)) real = eheight_exp (card A)\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhave \"\\ \\ real ((card A + 3) choose 3) / 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eheight_exp (card A) \\ real (card A + 3 choose 3) / 4\n[PROOF STEP]\nby (rule eheight_exp_bound)\n[PROOF STATE]\nproof (state)\nthis:\neheight_exp (card A) \\ real (card A + 3 choose 3) / 4\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\neheight_exp (card A) \\ real (card A + 3 choose 3) / 4\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhave \"measure_pmf.expectation (random_bst A) (\\t. real (height t - 1)) =\n measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\t. real (height t - 1)) = measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)\n[PROOF STEP]\nproof (intro integral_cong_AE AE_pmfI, goal_cases)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. measure_pmf.random_variable (random_bst A) borel (\\t. real (height t - 1))\n 2. measure_pmf.random_variable (random_bst A) borel (\\t. real (height t) - 1)\n 3. \\y. y \\ set_pmf (random_bst A) \\ real (height y - 1) = real (height y) - 1\n[PROOF STEP]\ncase (3 t)\n[PROOF STATE]\nproof (state)\nthis:\nt \\ set_pmf (random_bst A)\n\ngoal (3 subgoals):\n 1. measure_pmf.random_variable (random_bst A) borel (\\t. real (height t - 1))\n 2. measure_pmf.random_variable (random_bst A) borel (\\t. real (height t) - 1)\n 3. \\y. y \\ set_pmf (random_bst A) \\ real (height y - 1) = real (height y) - 1\n[PROOF STEP]\nwith \\A \\ {}\\ and assms\n[PROOF STATE]\nproof (chain)\npicking this:\nA \\ {}\nfinite A\nA \\ {}\nt \\ set_pmf (random_bst A)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ {}\nfinite A\nA \\ {}\nt \\ set_pmf (random_bst A)\n\ngoal (1 subgoal):\n 1. real (height t - 1) = real (height t) - 1\n[PROOF STEP]\nby (subst of_nat_diff) (auto simp: Suc_le_eq random_bst_altdef)\n[PROOF STATE]\nproof (state)\nthis:\nreal (height t - 1) = real (height t) - 1\n\ngoal (2 subgoals):\n 1. measure_pmf.random_variable (random_bst A) borel (\\t. real (height t - 1))\n 2. measure_pmf.random_variable (random_bst A) borel (\\t. real (height t) - 1)\n[PROOF STEP]\nqed auto\n[PROOF STATE]\nproof (state)\nthis:\nmeasure_pmf.expectation (random_bst A) (\\t. real (height t - 1)) = measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1) \\ real (card A + 3 choose 3) / 4\n[PROOF STEP]\nhave \"2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)\n \\ real ((card A + 3) choose 3) / 4\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1) \\ real (card A + 3 choose 3) / 4\n\ngoal (1 subgoal):\n 1. 2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1) \\ real (card A + 3 choose 3) / 4\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1) \\ real (card A + 3 choose 3) / 4\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhence \"log 2 (2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)) \\\n log 2 (real ((card A + 3) choose 3) / 4)\" (is \"?lhs \\ ?rhs\")\n[PROOF STATE]\nproof (prove)\nusing this:\n2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1) \\ real (card A + 3 choose 3) / 4\n\ngoal (1 subgoal):\n 1. log 2 (2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)) \\ log 2 (real (card A + 3 choose 3) / 4)\n[PROOF STEP]\nby (subst log_le_cancel_iff) auto\n[PROOF STATE]\nproof (state)\nthis:\nlog 2 (2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)) \\ log 2 (real (card A + 3 choose 3) / 4)\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlog 2 (2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)) \\ log 2 (real (card A + 3 choose 3) / 4)\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhave \"?lhs = measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. log 2 (2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)) = measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlog 2 (2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)) = measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlog 2 (2 powr measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)) = measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1)\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nhave \"\\ = measure_pmf.expectation (random_bst A) (\\t. real (height t)) - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1) = measure_pmf.expectation (random_bst A) (\\t. real (height t)) - 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nA \\ {}\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\t. real (height t) - 1) = measure_pmf.expectation (random_bst A) (\\t. real (height t)) - 1\n[PROOF STEP]\nby (subst Bochner_Integration.integral_diff) (auto intro!: integrable_measure_pmf_finite)\n[PROOF STATE]\nproof (state)\nthis:\nmeasure_pmf.expectation (random_bst A) (\\t. real (height t) - 1) = measure_pmf.expectation (random_bst A) (\\t. real (height t)) - 1\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nmeasure_pmf.expectation (random_bst A) (\\t. real (height t)) - 1 \\ log 2 (real (card A + 3 choose 3) / 4)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nmeasure_pmf.expectation (random_bst A) (\\t. real (height t)) - 1 \\ log 2 (real (card A + 3 choose 3) / 4)\n\ngoal (1 subgoal):\n 1. measure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n[PROOF STEP]\nby (simp add: height_exp_approx_def)\n[PROOF STATE]\nproof (state)\nthis:\nmeasure_pmf.expectation (random_bst A) (\\x. real (height x)) \\ height_exp_approx (card A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5928, "file": "Random_BSTs_Random_BSTs", "length": 49, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110396870287, "lm_q2_score": 0.7956580952177051, "lm_q1q2_score": 0.7095766731315025}} {"text": "[STATEMENT]\nlemma cos_exp_eq: \"cos z = (exp(\\ * z) + exp(-(\\ * z))) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos z = (exp (\\ * z) + exp (- (\\ * z))) / 2\n[PROOF STEP]\nby (simp add: exp_Euler exp_minus_Euler)", "meta": {"llama_tokens": 112, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297807787538, "lm_q2_score": 0.78793120560257, "lm_q1q2_score": 0.7095555158500215}} {"text": "[STATEMENT]\nlemma complex_root_unity_eq_1:\n fixes j::nat and k::nat\n assumes \"1 \\ n\"\n shows \"exp(2 * of_real pi * \\ * of_nat j / of_nat n) = 1 \\ n dvd j\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) = 1) = (n dvd j)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) = 1) = (n dvd j)\n[PROOF STEP]\nhave \"1 = exp(2 * of_real pi * \\ * (of_nat n / of_nat n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 = exp (2 * complex_of_real pi * \\ * (of_nat n / of_nat n))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ n\n\ngoal (1 subgoal):\n 1. 1 = exp (2 * complex_of_real pi * \\ * (of_nat n / of_nat n))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 = exp (2 * complex_of_real pi * \\ * (of_nat n / of_nat n))\n\ngoal (1 subgoal):\n 1. (exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) = 1) = (n dvd j)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n1 = exp (2 * complex_of_real pi * \\ * (of_nat n / of_nat n))\n[PROOF STEP]\nhave \"exp(2 * of_real pi * \\ * (of_nat j / of_nat n)) = 1 \\ j mod n = n mod n\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 = exp (2 * complex_of_real pi * \\ * (of_nat n / of_nat n))\n\ngoal (1 subgoal):\n 1. (exp (2 * complex_of_real pi * \\ * (of_nat j / of_nat n)) = 1) = (j mod n = n mod n)\n[PROOF STEP]\nusing complex_root_unity_eq [of n j n] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 = exp (2 * complex_of_real pi * \\ * (of_nat n / of_nat n))\n1 \\ n \\ (exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) = exp (2 * complex_of_real pi * \\ * of_nat n / of_nat n)) = (j mod n = n mod n)\n1 \\ n\n\ngoal (1 subgoal):\n 1. (exp (2 * complex_of_real pi * \\ * (of_nat j / of_nat n)) = 1) = (j mod n = n mod n)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(exp (2 * complex_of_real pi * \\ * (of_nat j / of_nat n)) = 1) = (j mod n = n mod n)\n\ngoal (1 subgoal):\n 1. (exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) = 1) = (n dvd j)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(exp (2 * complex_of_real pi * \\ * (of_nat j / of_nat n)) = 1) = (j mod n = n mod n)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(exp (2 * complex_of_real pi * \\ * (of_nat j / of_nat n)) = 1) = (j mod n = n mod n)\n\ngoal (1 subgoal):\n 1. (exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) = 1) = (n dvd j)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(exp (2 * complex_of_real pi * \\ * of_nat j / of_nat n) = 1) = (n dvd j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1277, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511396138365, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7094482219636282}} {"text": "[STATEMENT]\nlemma bit_twiddle_min:\n \"(y::'a::len word) XOR (((x::'a::len word) XOR y) AND (if x < y then -1 else 0)) = min x y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. y XOR (x XOR y) AND (if x < y then - 1 else 0) = min x y\n[PROOF STEP]\nby (rule bit_eqI) (auto simp add: bit_simps)", "meta": {"llama_tokens": 136, "file": "Word_Lib_More_Word", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588023318196, "lm_q2_score": 0.7981867825403177, "lm_q1q2_score": 0.7093955288876213}} {"text": "[STATEMENT]\nlemma matrix_mul_sum_alt:\n fixes A B :: \"'a::comm_ring_1^'n^'n\"\n shows \"A ** B = (\\ i. sum (\\k. A$i$k *s B $ k) (UNIV :: 'n set))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A ** B = (\\i. \\k\\UNIV. A $ i $ k *s B $ k)\n[PROOF STEP]\nby (vector matrix_matrix_mult_def sum_component)", "meta": {"llama_tokens": 152, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587993853655, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7093955222691394}} {"text": "[STATEMENT]\nlemma (in upper_semilattice) join_assoc_lemma:\n assumes L: \"x \\ carrier L\" \"y \\ carrier L\" \"z \\ carrier L\"\n shows \"x \\ (y \\ z) = \\{x, y, z}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ (y \\ z) = \\{x, y, z}\n[PROOF STEP]\nusing weak_join_assoc_lemma L\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x \\ carrier L; ?y \\ carrier L; ?z \\ carrier L\\ \\ ?x \\ (?y \\ ?z) .= \\{?x, ?y, ?z}\nx \\ carrier L\ny \\ carrier L\nz \\ carrier L\n\ngoal (1 subgoal):\n 1. x \\ (y \\ z) = \\{x, y, z}\n[PROOF STEP]\nunfolding eq_is_equal\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x \\ carrier L; ?y \\ carrier L; ?z \\ carrier L\\ \\ ?x \\ (?y \\ ?z) = \\{?x, ?y, ?z}\nx \\ carrier L\ny \\ carrier L\nz \\ carrier L\n\ngoal (1 subgoal):\n 1. x \\ (y \\ z) = \\{x, y, z}\n[PROOF STEP]\n.", "meta": {"llama_tokens": 491, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587964389112, "lm_q2_score": 0.7981867801399695, "lm_q1q2_score": 0.709395522050649}} {"text": "[STATEMENT]\nlemma sorted_list_of_set_from_up_to:\n assumes \"(i::nat) < j\"\n assumes \"k < j - i\"\n shows \"sorted_list_of_set {i..i < j; 0 < j - i\\ \\ sorted_list_of_set {i..k. \\\\i < j; k < j - i\\ \\ sorted_list_of_set {i.. \\ sorted_list_of_set {i..k. \\\\i < j; k < j - i\\ \\ sorted_list_of_set {i.. \\ sorted_list_of_set {i.. 'b::{comm_monoid_add}\"\n assumes \"finite A\"\n assumes \"z \\ A\"\n assumes \"\\y. y \\ A \\ y \\ z \\ f y = 0\"\n shows \"sum f A = f z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f A = f z\n[PROOF STEP]\nusing sum.union_disjoint[where A=\"A-{z}\" and B=\"{z}\" and g=\"f\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite (A - {z}); finite {z}; (A - {z}) \\ {z} = {}\\ \\ sum f (A - {z} \\ {z}) = sum f (A - {z}) + sum f {z}\n\ngoal (1 subgoal):\n 1. sum f A = f z\n[PROOF STEP]\nby (simp add: assms sum.insert_if)", "meta": {"llama_tokens": 283, "file": "Equivalence_Relation_Enumeration_Equivalence_Relation_Enumeration", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240756264638, "lm_q2_score": 0.819893340314393, "lm_q1q2_score": 0.7093914574858143}} {"text": "[STATEMENT]\nlemma nat_less_power_trans:\n fixes n :: nat\n assumes nv: \"n < 2 ^ (m - k)\"\n and kv: \"k \\ m\"\n shows \"2 ^ k * n < 2 ^ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ k * n < 2 ^ m\n[PROOF STEP]\nproof (rule order_less_le_trans)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 2 ^ k * n < ?y\n 2. ?y \\ 2 ^ m\n[PROOF STEP]\nshow \"2 ^ k * n < 2 ^ k * 2 ^ (m - k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ k * n < 2 ^ k * 2 ^ (m - k)\n[PROOF STEP]\nby (rule mult_less_mono2 [OF nv zero_less_power]) simp\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ k * n < 2 ^ k * 2 ^ (m - k)\n\ngoal (1 subgoal):\n 1. 2 ^ k * 2 ^ (m - k) \\ 2 ^ m\n[PROOF STEP]\nshow \"(2::nat) ^ k * 2 ^ (m - k) \\ 2 ^ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ k * 2 ^ (m - k) \\ 2 ^ m\n[PROOF STEP]\nusing nv kv\n[PROOF STATE]\nproof (prove)\nusing this:\nn < 2 ^ (m - k)\nk \\ m\n\ngoal (1 subgoal):\n 1. 2 ^ k * 2 ^ (m - k) \\ 2 ^ m\n[PROOF STEP]\nby (subst power_add [symmetric]) simp\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ k * 2 ^ (m - k) \\ 2 ^ m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 570, "file": "Word_Lib_More_Arithmetic", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240686758841, "lm_q2_score": 0.8198933271118221, "lm_q1q2_score": 0.7093914403638982}} {"text": "[STATEMENT]\nlemma eventually_upper_bound_power_even_nonpos:\n assumes \"even n\" \"eventually (\\x. u x \\ (0::real)) at_top\"\n \"eventually (\\x. l x \\ f x) at_top\" \"eventually (\\x. f x \\ u x) at_top\"\n shows \"eventually (\\x. f x ^ n \\ l x ^ n) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. f x ^ n \\ l x ^ n\n[PROOF STEP]\nusing assms(2-)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in at_top. u x \\ 0\n\\\\<^sub>F x in at_top. l x \\ f x\n\\\\<^sub>F x in at_top. f x \\ u x\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. f x ^ n \\ l x ^ n\n[PROOF STEP]\nby eventually_elim (rule power_mono_even, auto simp: assms(1))", "meta": {"llama_tokens": 323, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942119105695, "lm_q2_score": 0.7853085758631159, "lm_q1q2_score": 0.7093646911408849}} {"text": "[STATEMENT]\ntheorem Fiber_equality:\n \"\\ a \\ S; b \\ S \\ \\ Class a = Class b \\ \\ a = \\ b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a \\ S; b \\ S\\ \\ (Class a = Class b) = (\\ a = \\ b)\n[PROOF STEP]\nunfolding Class_equivalence\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a \\ S; b \\ S\\ \\ ((a, b) \\ E(\\)) = (\\ a = \\ b)\n[PROOF STEP]\nunfolding Fiber_Relation_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a \\ S; b \\ S\\ \\ ((a, b) \\ {(x, y). x \\ S \\ y \\ S \\ \\ x = \\ y}) = (\\ a = \\ b)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 342, "file": "Jacobson_Basic_Algebra_Set_Theory", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.903294214513915, "lm_q2_score": 0.7853085708384736, "lm_q1q2_score": 0.7093646886465842}} {"text": "[STATEMENT]\nlemma ld_ld_less2: assumes \"x \\ 2\" \"y \\ 2\"\n shows \"1 + log 2 x + log 2 y \\ 2 * log 2 (x + y - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + log 2 x + log 2 y \\ 2 * log 2 (x + y - 1)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 1 + log 2 x + log 2 y \\ 2 * log 2 (x + y - 1)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ x\n2 \\ y\n[PROOF STEP]\nhave \"2*x \\ x*x\" \"2*y \\ y*y\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ x\n2 \\ y\n\ngoal (1 subgoal):\n 1. 2 * x \\ x * x &&& 2 * y \\ y * y\n[PROOF STEP]\nby simp_all\n[PROOF STATE]\nproof (state)\nthis:\n2 * x \\ x * x\n2 * y \\ y * y\n\ngoal (1 subgoal):\n 1. 1 + log 2 x + log 2 y \\ 2 * log 2 (x + y - 1)\n[PROOF STEP]\nhence 1: \"2 * x * y \\ (x + y - 1)^2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * x \\ x * x\n2 * y \\ y * y\n\ngoal (1 subgoal):\n 1. 2 * x * y \\ (x + y - 1)\\<^sup>2\n[PROOF STEP]\nby(simp add: numeral_eq_Suc algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n2 * x * y \\ (x + y - 1)\\<^sup>2\n\ngoal (1 subgoal):\n 1. 1 + log 2 x + log 2 y \\ 2 * log 2 (x + y - 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + log 2 x + log 2 y \\ 2 * log 2 (x + y - 1)\n[PROOF STEP]\napply(rule powr_le_cancel_iff[of 2, THEN iffD1])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. 1 < 2\n 2. 2 powr (1 + log 2 x + log 2 y) \\ 2 powr (2 * log 2 (x + y - 1))\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 powr (1 + log 2 x + log 2 y) \\ 2 powr (2 * log 2 (x + y - 1))\n[PROOF STEP]\nusing assms 1\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ x\n2 \\ y\n2 * x * y \\ (x + y - 1)\\<^sup>2\n\ngoal (1 subgoal):\n 1. 2 powr (1 + log 2 x + log 2 y) \\ 2 powr (2 * log 2 (x + y - 1))\n[PROOF STEP]\nby(simp add: powr_add log_powr[symmetric] powr_numeral)\n[PROOF STATE]\nproof (state)\nthis:\n1 + log 2 x + log 2 y \\ 2 * log 2 (x + y - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1065, "file": "Amortized_Complexity_Lemmas_log", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681195338728, "lm_q2_score": 0.8267118026095992, "lm_q1q2_score": 0.7091270283208941}} {"text": "[STATEMENT]\nlemma fact_prod_rev: \"fact n = of_nat (\\i = 0..{1..n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod Suc {0..{1..n}\n[PROOF STEP]\nby (simp add: atLeast0LessThan prod.atLeast1_atMost_eq)\n[PROOF STATE]\nproof (state)\nthis:\nprod Suc {0..{1..n}\n\ngoal (1 subgoal):\n 1. fact n = of_nat (prod ((-) n) {0..{1..n}\n[PROOF STEP]\nhave \"prod Suc {0..{1..n}\n\ngoal (1 subgoal):\n 1. prod Suc {0..i. i\" 1 n]\n[PROOF STATE]\nproof (prove)\nusing this:\nprod Suc {0..{1..n}\n\\{1..n} = prod ((-) (n + 1)) {1..n}\n\ngoal (1 subgoal):\n 1. prod Suc {0.. affine_dependent S\"\n shows \"(convex hull S) - rel_interior(convex hull S) = (\\a\\S. convex hull (S - {a}))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nhave \"finite S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite S\n[PROOF STEP]\nby (metis assms aff_independent_finite)\n[PROOF STATE]\nproof (state)\nthis:\nfinite S\n\ngoal (1 subgoal):\n 1. convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite S\n[PROOF STEP]\nconsider \"card S = 0\" | \"card S = 1\" | \"2 \\ card S\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\n\ngoal (1 subgoal):\n 1. \\card S = 0 \\ thesis; card S = 1 \\ thesis; 2 \\ card S \\ thesis\\ \\ thesis\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\n\\card S = 0 \\ ?thesis; card S = 1 \\ ?thesis; 2 \\ card S \\ ?thesis\\ \\ ?thesis\n\ngoal (1 subgoal):\n 1. convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\card S = 0 \\ ?thesis; card S = 1 \\ ?thesis; 2 \\ card S \\ ?thesis\\ \\ ?thesis\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\card S = 0 \\ ?thesis; card S = 1 \\ ?thesis; 2 \\ card S \\ ?thesis\\ \\ ?thesis\n\ngoal (1 subgoal):\n 1. convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nproof cases\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. card S = 0 \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n 2. card S = 1 \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n 3. 2 \\ card S \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\ncase 1\n[PROOF STATE]\nproof (state)\nthis:\ncard S = 0\n\ngoal (3 subgoals):\n 1. card S = 0 \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n 2. card S = 1 \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n 3. 2 \\ card S \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncard S = 0\n[PROOF STEP]\nhave \"S = {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard S = 0\n\ngoal (1 subgoal):\n 1. S = {}\n[PROOF STEP]\nby (simp add: \\finite S\\)\n[PROOF STATE]\nproof (state)\nthis:\nS = {}\n\ngoal (3 subgoals):\n 1. card S = 0 \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n 2. card S = 1 \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n 3. 2 \\ card S \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nS = {}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nS = {}\n\ngoal (1 subgoal):\n 1. convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nconvex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n\ngoal (2 subgoals):\n 1. card S = 1 \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n 2. 2 \\ card S \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. card S = 1 \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n 2. 2 \\ card S \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\ncase 2\n[PROOF STATE]\nproof (state)\nthis:\ncard S = 1\n\ngoal (2 subgoals):\n 1. card S = 1 \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n 2. 2 \\ card S \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nby (auto intro: card_1_singletonE [OF \\card S = 1\\])\n[PROOF STATE]\nproof (state)\nthis:\nconvex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n\ngoal (1 subgoal):\n 1. 2 \\ card S \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 \\ card S \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\ncase 3\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ card S\n\ngoal (1 subgoal):\n 1. 2 \\ card S \\ convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ affine_dependent S\n2 \\ card S\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ affine_dependent S\n2 \\ card S\n\ngoal (1 subgoal):\n 1. convex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n[PROOF STEP]\nby (auto simp: polyhedron_convex_hull rel_boundary_of_polyhedron facet_of_convex_hull_affine_independent_alt \\finite S\\)\n[PROOF STATE]\nproof (state)\nthis:\nconvex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nconvex hull S - rel_interior (convex hull S) = (\\a\\S. convex hull (S - {a}))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2722, "file": null, "length": 27, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681122619883, "lm_q2_score": 0.8267117855317474, "lm_q1q2_score": 0.7091270076603048}} {"text": "[STATEMENT]\nlemma emeasure_cball_aux_integral:\n \"(\\\\<^sup>+x. indicator {-1..1} x * sqrt (1 - x\\<^sup>2) ^ n \\lborel) =\n ennreal (Beta (1 / 2) (real n / 2 + 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nhave \"((\\t. t powr (-1 / 2) * (1 - t) powr (real n / 2)) has_integral\n Beta (1 / 2) (real n / 2 + 1)) {0..1}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\t. t powr (- 1 / 2) * (1 - t) powr (real n / 2)) has_integral Beta (1 / 2) (real n / 2 + 1)) {0..1}\n[PROOF STEP]\nusing has_integral_Beta_real[of \"1/2\" \"n / 2 + 1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < 1 / 2; 0 < real n / 2 + 1\\ \\ ((\\t. t powr (1 / 2 - 1) * (1 - t) powr (real n / 2 + 1 - 1)) has_integral Beta (1 / 2) (real n / 2 + 1)) {0..1}\n\ngoal (1 subgoal):\n 1. ((\\t. t powr (- 1 / 2) * (1 - t) powr (real n / 2)) has_integral Beta (1 / 2) (real n / 2 + 1)) {0..1}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n((\\t. t powr (- 1 / 2) * (1 - t) powr (real n / 2)) has_integral Beta (1 / 2) (real n / 2 + 1)) {0..1}\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nfrom nn_integral_has_integral_lebesgue[OF _ this]\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x. 0 \\ x powr (- 1 / 2) * (1 - x) powr (real n / 2)) \\ \\\\<^sup>+ x. ennreal (indicat_real {0..1} x * (x powr (- 1 / 2) * (1 - x) powr (real n / 2))) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nhave\n \"ennreal (Beta (1 / 2) (real n / 2 + 1)) =\n nn_integral lborel (\\t. ennreal (t powr (-1 / 2) * (1 - t) powr (real n / 2) *\n indicator {0^2..1^2} t))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. 0 \\ x powr (- 1 / 2) * (1 - x) powr (real n / 2)) \\ \\\\<^sup>+ x. ennreal (indicat_real {0..1} x * (x powr (- 1 / 2) * (1 - x) powr (real n / 2))) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n\ngoal (1 subgoal):\n 1. ennreal (Beta (1 / 2) (real n / 2 + 1)) = \\\\<^sup>+ t. ennreal (t powr (- 1 / 2) * (1 - t) powr (real n / 2) * indicat_real {0\\<^sup>2..1\\<^sup>2} t) \\lborel\n[PROOF STEP]\nby (simp add: mult_ac ennreal_mult' ennreal_indicator)\n[PROOF STATE]\nproof (state)\nthis:\nennreal (Beta (1 / 2) (real n / 2 + 1)) = \\\\<^sup>+ t. ennreal (t powr (- 1 / 2) * (1 - t) powr (real n / 2) * indicat_real {0\\<^sup>2..1\\<^sup>2} t) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nennreal (Beta (1 / 2) (real n / 2 + 1)) = \\\\<^sup>+ t. ennreal (t powr (- 1 / 2) * (1 - t) powr (real n / 2) * indicat_real {0\\<^sup>2..1\\<^sup>2} t) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nhave \"\\ = (\\\\<^sup>+ x. ennreal (x\\<^sup>2 powr - (1 / 2) * (1 - x\\<^sup>2) powr (real n / 2) * (2 * x) *\n indicator {0..1} x) \\lborel)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ t. ennreal (t powr (- 1 / 2) * (1 - t) powr (real n / 2) * indicat_real {0\\<^sup>2..1\\<^sup>2} t) \\lborel = \\\\<^sup>+ x. ennreal (x\\<^sup>2 powr - (1 / 2) * (1 - x\\<^sup>2) powr (real n / 2) * (2 * x) * indicat_real {0..1} x) \\lborel\n[PROOF STEP]\nby (subst nn_integral_substitution[where g = \"\\x. x ^ 2\" and g' = \"\\x. 2 * x\"])\n (auto intro!: derivative_eq_intros continuous_intros simp: set_borel_measurable_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ t. ennreal (t powr (- 1 / 2) * (1 - t) powr (real n / 2) * indicat_real {0\\<^sup>2..1\\<^sup>2} t) \\lborel = \\\\<^sup>+ x. ennreal (x\\<^sup>2 powr - (1 / 2) * (1 - x\\<^sup>2) powr (real n / 2) * (2 * x) * indicat_real {0..1} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ t. ennreal (t powr (- 1 / 2) * (1 - t) powr (real n / 2) * indicat_real {0\\<^sup>2..1\\<^sup>2} t) \\lborel = \\\\<^sup>+ x. ennreal (x\\<^sup>2 powr - (1 / 2) * (1 - x\\<^sup>2) powr (real n / 2) * (2 * x) * indicat_real {0..1} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nhave \"\\ = (\\\\<^sup>+ x. 2 * ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicator {0..1} x) \\lborel)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (x\\<^sup>2 powr - (1 / 2) * (1 - x\\<^sup>2) powr (real n / 2) * (2 * x) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. 2 * ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel\n[PROOF STEP]\nby (intro nn_integral_cong_AE AE_I[of _ _ \"{0}\"])\n (auto simp: indicator_def powr_minus powr_half_sqrt field_split_simps ennreal_mult')\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal (x\\<^sup>2 powr - (1 / 2) * (1 - x\\<^sup>2) powr (real n / 2) * (2 * x) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. 2 * ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal (x\\<^sup>2 powr - (1 / 2) * (1 - x\\<^sup>2) powr (real n / 2) * (2 * x) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. 2 * ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nhave \"\\ = (\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicator {0..1} x) \\lborel) +\n (\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicator {0..1} x) \\lborel)\"\n (is \"_ = ?I + _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. 2 * ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel\n[PROOF STEP]\nby (simp add: mult_2 nn_integral_add)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. 2 * ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. 2 * ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nhave \"?I = (\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicator {-1..0} x) \\lborel)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..0} x) \\lborel\n[PROOF STEP]\nby (subst nn_integral_real_affine[of _ \"-1\" 0])\n (auto simp: indicator_def intro!: nn_integral_cong)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..0} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nhence \"?I + ?I = \\ + ?I\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..0} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..0} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..0} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..0} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nhave \"\\ = (\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) *\n (indicator {-1..0} x + indicator{0..1} x)) \\lborel)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..0} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * (indicat_real {- 1..0} x + indicat_real {0..1} x)) \\lborel\n[PROOF STEP]\nby (subst nn_integral_add [symmetric]) (auto simp: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..0} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * (indicat_real {- 1..0} x + indicat_real {0..1} x)) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..0} x) \\lborel + \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {0..1} x) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * (indicat_real {- 1..0} x + indicat_real {0..1} x)) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nhave \"\\ = (\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicator {-1..1} x) \\lborel)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * (indicat_real {- 1..0} x + indicat_real {0..1} x)) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..1} x) \\lborel\n[PROOF STEP]\nby (intro nn_integral_cong_AE AE_I[of _ _ \"{0}\"]) (auto simp: indicator_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * (indicat_real {- 1..0} x + indicat_real {0..1} x)) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..1} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * (indicat_real {- 1..0} x + indicat_real {0..1} x)) \\lborel = \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..1} x) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nhave \"\\ = (\\\\<^sup>+ x. ennreal (indicator {-1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..1} x) \\lborel = \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel\n[PROOF STEP]\nby (intro nn_integral_cong_AE AE_I[of _ _ \"{1, -1}\"])\n (auto simp: powr_half_sqrt [symmetric] indicator_def abs_square_le_1\n abs_square_eq_1 powr_def exp_of_nat_mult [symmetric] emeasure_lborel_countable)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal ((1 - x\\<^sup>2) powr (real n / 2) * indicat_real {- 1..1} x) \\lborel = \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nennreal (Beta (1 / 2) (real n / 2 + 1)) = \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nennreal (Beta (1 / 2) (real n / 2 + 1)) = \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal (indicat_real {- 1..1} x * sqrt (1 - x\\<^sup>2) ^ n) \\lborel = ennreal (Beta (1 / 2) (real n / 2 + 1))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 7789, "file": null, "length": 34, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577680977182187, "lm_q2_score": 0.826711787666479, "lm_q1q2_score": 0.7091269974679036}} {"text": "[STATEMENT]\nlemma poly_all_0_iff_0: \"(\\x. poly p x = 0) \\ p = 0\"\n for p :: \"'a::{ring_char_0,comm_ring_1,ring_no_zero_divisors} poly\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. poly p x = (0::'a)) = (p = 0)\n[PROOF STEP]\nby (auto simp add: poly_eq_poly_eq_iff [symmetric])", "meta": {"llama_tokens": 147, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314617436728, "lm_q2_score": 0.8006920068519376, "lm_q1q2_score": 0.7091180324347565}} {"text": "[STATEMENT]\nlemma transpose_mat_plus: assumes wf: \"mat nr nc m1\" \"mat nr nc m2\"\n shows \"transpose nr (mat_plusI pl m1 m2) = mat_plusI pl (transpose nr m1) (transpose nr m2)\" (is \"?l = ?r\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix_Legacy.transpose nr (mat_plusI pl m1 m2) = mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2)\n[PROOF STEP]\nproof (rule mat_eqI)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. mat ?nr ?nc (Matrix_Legacy.transpose nr (mat_plusI pl m1 m2))\n 2. mat ?nr ?nc (mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2))\n 3. \\i j. \\i < ?nc; j < ?nr\\ \\ Matrix_Legacy.transpose nr (mat_plusI pl m1 m2) ! i ! j = mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2) ! i ! j\n[PROOF STEP]\nfix i j\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. mat ?nr ?nc (Matrix_Legacy.transpose nr (mat_plusI pl m1 m2))\n 2. mat ?nr ?nc (mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2))\n 3. \\i j. \\i < ?nc; j < ?nr\\ \\ Matrix_Legacy.transpose nr (mat_plusI pl m1 m2) ! i ! j = mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2) ! i ! j\n[PROOF STEP]\nassume i: \"i < nr\" and j: \"j < nc\"\n[PROOF STATE]\nproof (state)\nthis:\ni < nr\nj < nc\n\ngoal (3 subgoals):\n 1. mat ?nr ?nc (Matrix_Legacy.transpose nr (mat_plusI pl m1 m2))\n 2. mat ?nr ?nc (mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2))\n 3. \\i j. \\i < ?nc; j < ?nr\\ \\ Matrix_Legacy.transpose nr (mat_plusI pl m1 m2) ! i ! j = mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2) ! i ! j\n[PROOF STEP]\nnote [simp] = transpose_index[OF _ this] mat_plus_index[OF _ _ j i] mat_plus_index[OF _ _ this]\n[PROOF STATE]\nproof (state)\nthis:\nmat nr nc ?m \\ Matrix_Legacy.transpose nr ?m ! i ! j = ?m ! j ! i\n\\mat nr nc ?m1.0; mat nr nc ?m2.0\\ \\ mat_plusI ?pl ?m1.0 ?m2.0 ! j ! i = ?pl (?m1.0 ! j ! i) (?m2.0 ! j ! i)\n\\mat nc nr ?m1.0; mat nc nr ?m2.0\\ \\ mat_plusI ?pl ?m1.0 ?m2.0 ! i ! j = ?pl (?m1.0 ! i ! j) (?m2.0 ! i ! j)\n\ngoal (3 subgoals):\n 1. mat ?nr ?nc (Matrix_Legacy.transpose nr (mat_plusI pl m1 m2))\n 2. mat ?nr ?nc (mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2))\n 3. \\i j. \\i < ?nc; j < ?nr\\ \\ Matrix_Legacy.transpose nr (mat_plusI pl m1 m2) ! i ! j = mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2) ! i ! j\n[PROOF STEP]\nshow \"?l ! i ! j = ?r ! i ! j\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix_Legacy.transpose nr (mat_plusI pl m1 m2) ! i ! j = mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2) ! i ! j\n[PROOF STEP]\nusing wf\n[PROOF STATE]\nproof (prove)\nusing this:\nmat nr nc m1\nmat nr nc m2\n\ngoal (1 subgoal):\n 1. Matrix_Legacy.transpose nr (mat_plusI pl m1 m2) ! i ! j = mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2) ! i ! j\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nMatrix_Legacy.transpose nr (mat_plusI pl m1 m2) ! i ! j = mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2) ! i ! j\n\ngoal (2 subgoals):\n 1. mat nc nr (Matrix_Legacy.transpose nr (mat_plusI pl m1 m2))\n 2. mat nc nr (mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2))\n[PROOF STEP]\nqed (auto intro: wf)", "meta": {"llama_tokens": 1558, "file": "Matrix_Matrix_Legacy", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314707995588, "lm_q2_score": 0.8006919973399709, "lm_q1q2_score": 0.7091180312616349}} {"text": "[STATEMENT]\nlemma list_sum_distrib_aux:\n shows \"(\\:xs/(n :: 'a :: archimedean_field) + \\:xs) = (1 + (1/n)) * \\:xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list xs / n + sum_list xs = ((1::'a) + (1::'a) / n) * sum_list xs\n[PROOF STEP]\nproof (induct xs)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. sum_list [] / n + sum_list [] = ((1::'a) + (1::'a) / n) * sum_list []\n 2. \\a xs. sum_list xs / n + sum_list xs = ((1::'a) + (1::'a) / n) * sum_list xs \\ sum_list (a # xs) / n + sum_list (a # xs) = ((1::'a) + (1::'a) / n) * sum_list (a # xs)\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. sum_list [] / n + sum_list [] = ((1::'a) + (1::'a) / n) * sum_list []\n 2. \\a xs. sum_list xs / n + sum_list xs = ((1::'a) + (1::'a) / n) * sum_list xs \\ sum_list (a # xs) / n + sum_list (a # xs) = ((1::'a) + (1::'a) / n) * sum_list (a # xs)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list [] / n + sum_list [] = ((1::'a) + (1::'a) / n) * sum_list []\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum_list [] / n + sum_list [] = ((1::'a) + (1::'a) / n) * sum_list []\n\ngoal (1 subgoal):\n 1. \\a xs. sum_list xs / n + sum_list xs = ((1::'a) + (1::'a) / n) * sum_list xs \\ sum_list (a # xs) / n + sum_list (a # xs) = ((1::'a) + (1::'a) / n) * sum_list (a # xs)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a xs. sum_list xs / n + sum_list xs = ((1::'a) + (1::'a) / n) * sum_list xs \\ sum_list (a # xs) / n + sum_list (a # xs) = ((1::'a) + (1::'a) / n) * sum_list (a # xs)\n[PROOF STEP]\ncase (Cons x xs)\n[PROOF STATE]\nproof (state)\nthis:\nsum_list xs / n + sum_list xs = ((1::'a) + (1::'a) / n) * sum_list xs\n\ngoal (1 subgoal):\n 1. \\a xs. sum_list xs / n + sum_list xs = ((1::'a) + (1::'a) / n) * sum_list xs \\ sum_list (a # xs) / n + sum_list (a # xs) = ((1::'a) + (1::'a) / n) * sum_list (a # xs)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nhave\n \"\\:(x#xs)/n = x/n + \\:xs/n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n = x / n + sum_list xs / n\n[PROOF STEP]\nby (simp add: add_divide_distrib)\n[PROOF STATE]\nproof (state)\nthis:\nsum_list (x # xs) / n = x / n + sum_list xs / n\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum_list (x # xs) / n = x / n + sum_list xs / n\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nwith Cons\n[PROOF STATE]\nproof (chain)\npicking this:\nsum_list xs / n + sum_list xs = ((1::'a) + (1::'a) / n) * sum_list xs\nsum_list (x # xs) / n = x / n + sum_list xs / n\n[PROOF STEP]\nhave\n \"\\ = x/n + (1+1/n)*\\:xs - \\:xs\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsum_list xs / n + sum_list xs = ((1::'a) + (1::'a) / n) * sum_list xs\nsum_list (x # xs) / n = x / n + sum_list xs / n\n\ngoal (1 subgoal):\n 1. x / n + sum_list xs / n = x / n + ((1::'a) + (1::'a) / n) * sum_list xs - sum_list xs\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx / n + sum_list xs / n = x / n + ((1::'a) + (1::'a) / n) * sum_list xs - sum_list xs\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum_list (x # xs) / n = x / n + ((1::'a) + (1::'a) / n) * sum_list xs - sum_list xs\n[PROOF STEP]\nhave\n \"\\:(x#xs) / n + \\:(x#xs) = x/n + (1+1/n)*\\:xs - \\:xs + \\:(x#xs)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsum_list (x # xs) / n = x / n + ((1::'a) + (1::'a) / n) * sum_list xs - sum_list xs\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = x / n + ((1::'a) + (1::'a) / n) * sum_list xs - sum_list xs + sum_list (x # xs)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum_list (x # xs) / n + sum_list (x # xs) = x / n + ((1::'a) + (1::'a) / n) * sum_list xs - sum_list xs + sum_list (x # xs)\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum_list (x # xs) / n + sum_list (x # xs) = x / n + ((1::'a) + (1::'a) / n) * sum_list xs - sum_list xs + sum_list (x # xs)\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nhave\n \"\\ = x/n + (1+(1/n)- 1)*\\:xs + \\:(x#xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x / n + ((1::'a) + (1::'a) / n) * sum_list xs - sum_list xs + sum_list (x # xs) = x / n + ((1::'a) + (1::'a) / n - (1::'a)) * sum_list xs + sum_list (x # xs)\n[PROOF STEP]\nby (subst mult_1_left [symmetric, of \"\\:xs\"]) (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nx / n + ((1::'a) + (1::'a) / n) * sum_list xs - sum_list xs + sum_list (x # xs) = x / n + ((1::'a) + (1::'a) / n - (1::'a)) * sum_list xs + sum_list (x # xs)\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx / n + ((1::'a) + (1::'a) / n) * sum_list xs - sum_list xs + sum_list (x # xs) = x / n + ((1::'a) + (1::'a) / n - (1::'a)) * sum_list xs + sum_list (x # xs)\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nhave\n \"\\ = x/n + (1/n)*\\:xs + \\:(x#xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x / n + ((1::'a) + (1::'a) / n - (1::'a)) * sum_list xs + sum_list (x # xs) = x / n + (1::'a) / n * sum_list xs + sum_list (x # xs)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx / n + ((1::'a) + (1::'a) / n - (1::'a)) * sum_list xs + sum_list (x # xs) = x / n + (1::'a) / n * sum_list xs + sum_list (x # xs)\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx / n + ((1::'a) + (1::'a) / n - (1::'a)) * sum_list xs + sum_list (x # xs) = x / n + (1::'a) / n * sum_list xs + sum_list (x # xs)\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nhave\n \"\\ = (1/n)*\\:(x#xs) + 1*\\:(x#xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x / n + (1::'a) / n * sum_list xs + sum_list (x # xs) = (1::'a) / n * sum_list (x # xs) + (1::'a) * sum_list (x # xs)\n[PROOF STEP]\nby(simp add: divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\nx / n + (1::'a) / n * sum_list xs + sum_list (x # xs) = (1::'a) / n * sum_list (x # xs) + (1::'a) * sum_list (x # xs)\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum_list (x # xs) / n + sum_list (x # xs) = (1::'a) / n * sum_list (x # xs) + (1::'a) * sum_list (x # xs)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsum_list (x # xs) / n + sum_list (x # xs) = (1::'a) / n * sum_list (x # xs) + (1::'a) * sum_list (x # xs)\n\ngoal (1 subgoal):\n 1. sum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nsum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nsum_list (x # xs) / n + sum_list (x # xs) = ((1::'a) + (1::'a) / n) * sum_list (x # xs)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3940, "file": "Cauchy_CauchysMeanTheorem", "length": 31, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473746782093, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7091026746319822}} {"text": "[STATEMENT]\nlemma sum_disc_lim: \n assumes \"\\c :: real\\ < 1\" \n shows \"(\\x. c^x * B) = B /(1-c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. c ^ x * B) = B / (1 - c)\n[PROOF STEP]\nby (simp add: assms suminf_geometric summable_geometric suminf_mult2[symmetric])", "meta": {"llama_tokens": 133, "file": "MDP-Rewards_MDP_reward_Util", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473879530491, "lm_q2_score": 0.8128673087708698, "lm_q1q2_score": 0.7091026735586929}} {"text": "[STATEMENT]\nlemma CDIM_prod[simp]: \"CDIM('a \\ 'b) = CDIM('a) + CDIM('b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. CDIM('a \\ 'b) = CDIM('a) + CDIM('b)\n[PROOF STEP]\nunfolding CBasis_prod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((\\u. (u, 0::'b)) ` CBasis \\ Pair (0::'a) ` CBasis) = CDIM('a) + CDIM('b)\n[PROOF STEP]\nby (subst card_Un_disjoint) (auto intro!: card_image arg_cong2[where f=\"(+)\"] inj_onI)", "meta": {"llama_tokens": 226, "file": "Complex_Bounded_Operators_Complex_Euclidean_Space0", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473614033683, "lm_q2_score": 0.8128673201042492, "lm_q1q2_score": 0.709102661863969}} {"text": "[STATEMENT]\nlemma card_eq_1_iff: \"card A = Suc 0 \\ (\\x. A = {x})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (card A = Suc 0) = (\\x. A = {x})\n[PROOF STEP]\nusing card_eq_SucD\n[PROOF STATE]\nproof (prove)\nusing this:\ncard ?A = Suc ?k \\ \\b B. ?A = insert b B \\ b \\ B \\ card B = ?k \\ (?k = 0 \\ B = {})\n\ngoal (1 subgoal):\n 1. (card A = Suc 0) = (\\x. A = {x})\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 214, "file": "MFMC_Countable_MFMC_Misc", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.872347368040789, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.7091026613273252}} {"text": "[STATEMENT]\nlemma alpha_l_beta_l_add_norm:\n \"alpha_l l * alpha_l l + beta_l l * beta_l l = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. alpha_l l * alpha_l l + beta_l l * beta_l l = 1\n[PROOF STEP]\nusing alpha_l_def beta_l_def csin_ccos_squared_add\n[PROOF STATE]\nproof (prove)\nusing this:\nalpha_l ?l = complex_of_real (cos ((real ?l + 1 / 2) * \\))\nbeta_l ?l = complex_of_real (sin ((real ?l + 1 / 2) * \\))\ncomplex_of_real (cos ?a) * complex_of_real (cos ?a) + complex_of_real (sin ?a) * complex_of_real (sin ?a) = 1\n\ngoal (1 subgoal):\n 1. alpha_l l * alpha_l l + beta_l l * beta_l l = 1\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 289, "file": "QHLProver_Grover", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898203834278, "lm_q2_score": 0.7826624688140726, "lm_q1q2_score": 0.7090842295417118}} {"text": "[STATEMENT]\nlemma scaleR_vector_assoc: \"c *\\<^sub>R (A *v x) = (c *\\<^sub>R A) *v x\" for x :: \"('a::real_normed_algebra_1)^'n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c *\\<^sub>R (A *v x) = c *\\<^sub>R A *v x\n[PROOF STEP]\nunfolding matrix_vector_mult_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c *\\<^sub>R (\\i. \\j\\UNIV. A $ i $ j * x $ j) = (\\i. \\j\\UNIV. (c *\\<^sub>R A) $ i $ j * x $ j)\n[PROOF STEP]\nby(auto simp: vec_eq_iff scaleR_right.sum)", "meta": {"llama_tokens": 240, "file": "Matrices_for_ODEs_MTX_Preliminaries", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.877476800298183, "lm_q2_score": 0.8080672227971211, "lm_q1q2_score": 0.7090602410858567}} {"text": "[STATEMENT]\nlemma SUP_sigma_sigma:\n \"M \\ {} \\ (\\m. m \\ M \\ f m \\ Pow \\) \\ (SUP m\\M. sigma \\ (f m)) = sigma \\ (\\m\\M. f m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\M \\ {}; \\m. m \\ M \\ f m \\ Pow \\\\ \\ (SUP m\\M. sigma \\ (f m)) = sigma \\ (\\ (f ` M))\n[PROOF STEP]\nusing Sup_sigma[of \"f`M\" \\]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\f ` M \\ {}; \\m. m \\ f ` M \\ m \\ Pow \\\\ \\ Sup (sigma \\ ` f ` M) = sigma \\ (\\ (f ` M))\n\ngoal (1 subgoal):\n 1. \\M \\ {}; \\m. m \\ M \\ f m \\ Pow \\\\ \\ (SUP m\\M. sigma \\ (f m)) = sigma \\ (\\ (f ` M))\n[PROOF STEP]\nby (auto simp: image_comp)", "meta": {"llama_tokens": 394, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9184802350995703, "lm_q2_score": 0.7718434925908525, "lm_q1q2_score": 0.7089229925349196}} {"text": "[STATEMENT]\nlemma d_OUT_add: \"d_OUT (\\e. f e + g e) x = d_OUT f x + d_OUT g x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. d_OUT (\\e. f e + g e) x = d_OUT f x + d_OUT g x\n[PROOF STEP]\nunfolding d_OUT_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sup>+ y. f (x, y) + g (x, y)) = (\\\\<^sup>+ y. f (x, y)) + (\\\\<^sup>+ y. g (x, y))\n[PROOF STEP]\nby(simp add: nn_integral_add)", "meta": {"llama_tokens": 206, "file": "MFMC_Countable_MFMC_Network", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357666736772, "lm_q2_score": 0.817574478416099, "lm_q1q2_score": 0.7088663147063341}} {"text": "[STATEMENT]\nlemma collect_suc_eq_lt[simp]:\n \"{f i |i::nat. i < Suc n} = {f i |i. i = 0} \\ {f (Suc i) |i. i < n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {f i |i. i < Suc n} = {f i |i. i = 0} \\ {f (Suc i) |i. i < n}\n[PROOF STEP]\napply(induct n)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. {f i |i. i < Suc 0} = {f i |i. i = 0} \\ {f (Suc i) |i. i < 0}\n 2. \\n. {f i |i. i < Suc n} = {f i |i. i = 0} \\ {f (Suc i) |i. i < n} \\ {f i |i. i < Suc (Suc n)} = {f i |i. i = 0} \\ {f (Suc i) |i. i < Suc n}\n[PROOF STEP]\napply(simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. {f i |i. i < Suc n} = {f i |i. i = 0} \\ {f (Suc i) |i. i < n} \\ {f i |i. i < Suc (Suc n)} = {f i |i. i = 0} \\ {f (Suc i) |i. i < Suc n}\n[PROOF STEP]\napply(clarsimp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. {f i |i. i < Suc n} = insert (f 0) {f (Suc i) |i. i < n} \\ {f i |i. i < Suc (Suc n)} = insert (f 0) {f (Suc i) |i. i < Suc n}\n[PROOF STEP]\napply(simp add: less_Suc_eq)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. {f i |i. i < n \\ i = n} = insert (f 0) {f (Suc i) |i. i < n} \\ {f i |i. i < n \\ i = n \\ i = Suc n} = insert (f 0) {f (Suc i) |i. i < n \\ i = n}\n[PROOF STEP]\napply(simp add: image_Collect[THEN sym])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. f ` {i. i < n \\ i = n} = insert (f 0) ((\\x. f (Suc x)) ` {i. i < n}) \\ f ` {i. i < n \\ i = n \\ i = Suc n} = insert (f 0) ((\\x. f (Suc x)) ` {i. i < n \\ i = n})\n[PROOF STEP]\napply(simp add: Collect_disj_eq)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. insert (f n) (f ` {i. i < n}) = insert (f 0) ((\\x. f (Suc x)) ` {i. i < n}) \\ insert (f (Suc n)) (insert (f 0) ((\\x. f (Suc x)) ` {i. i < n})) = insert (f 0) (insert (f (Suc n)) ((\\x. f (Suc x)) ` {i. i < n}))\n[PROOF STEP]\napply(blast)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1068, "file": "LightweightJava_Lightweight_Java_Proof", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357598021708, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.7088663033075212}} {"text": "[STATEMENT]\nlemma sigma_sets_singletons:\n assumes \"countable S\"\n shows \"sigma_sets S ((\\s. {s})`S) = Pow S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sigma_sets S ((\\s. {s}) ` S) = Pow S\n[PROOF STEP]\nproof safe\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x xa. \\x \\ sigma_sets S ((\\s. {s}) ` S); xa \\ x\\ \\ xa \\ S\n 2. \\x. x \\ S \\ x \\ sigma_sets S ((\\s. {s}) ` S)\n[PROOF STEP]\ninterpret sigma_algebra S \"sigma_sets S ((\\s. {s})`S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sigma_algebra S (sigma_sets S ((\\s. {s}) ` S))\n[PROOF STEP]\nby (rule sigma_algebra_sigma_sets) auto\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x xa. \\x \\ sigma_sets S ((\\s. {s}) ` S); xa \\ x\\ \\ xa \\ S\n 2. \\x. x \\ S \\ x \\ sigma_sets S ((\\s. {s}) ` S)\n[PROOF STEP]\nfix A\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x xa. \\x \\ sigma_sets S ((\\s. {s}) ` S); xa \\ x\\ \\ xa \\ S\n 2. \\x. x \\ S \\ x \\ sigma_sets S ((\\s. {s}) ` S)\n[PROOF STEP]\nassume \"A \\ S\"\n[PROOF STATE]\nproof (state)\nthis:\nA \\ S\n\ngoal (2 subgoals):\n 1. \\x xa. \\x \\ sigma_sets S ((\\s. {s}) ` S); xa \\ x\\ \\ xa \\ S\n 2. \\x. x \\ S \\ x \\ sigma_sets S ((\\s. {s}) ` S)\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\ncountable S\nA \\ S\n[PROOF STEP]\nhave \"(\\a\\A. {a}) \\ sigma_sets S ((\\s. {s})`S)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncountable S\nA \\ S\n\ngoal (1 subgoal):\n 1. (\\a\\A. {a}) \\ sigma_sets S ((\\s. {s}) ` S)\n[PROOF STEP]\nby (intro countable_UN') (auto dest: countable_subset)\n[PROOF STATE]\nproof (state)\nthis:\n(\\a\\A. {a}) \\ sigma_sets S ((\\s. {s}) ` S)\n\ngoal (2 subgoals):\n 1. \\x xa. \\x \\ sigma_sets S ((\\s. {s}) ` S); xa \\ x\\ \\ xa \\ S\n 2. \\x. x \\ S \\ x \\ sigma_sets S ((\\s. {s}) ` S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\a\\A. {a}) \\ sigma_sets S ((\\s. {s}) ` S)\n[PROOF STEP]\nshow \"A \\ sigma_sets S ((\\s. {s})`S)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\a\\A. {a}) \\ sigma_sets S ((\\s. {s}) ` S)\n\ngoal (1 subgoal):\n 1. A \\ sigma_sets S ((\\s. {s}) ` S)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nA \\ sigma_sets S ((\\s. {s}) ` S)\n\ngoal (1 subgoal):\n 1. \\x xa. \\x \\ sigma_sets S ((\\s. {s}) ` S); xa \\ x\\ \\ xa \\ S\n[PROOF STEP]\nqed (auto dest: sigma_sets_into_sp[rotated])", "meta": {"llama_tokens": 1375, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357460591569, "lm_q2_score": 0.8175744695262775, "lm_q1q2_score": 0.7088662901446355}} {"text": "[STATEMENT]\nlemma coeff_mult: \"coeff (p * q) n = (\\i\\n. coeff p i * coeff q (n-i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coeff (p * q) n = (\\i\\n. coeff p i * coeff q (n - i))\n[PROOF STEP]\nproof (induct p arbitrary: n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\n. coeff (0 * q) n = (\\i\\n. coeff 0 i * coeff q (n - i))\n 2. \\a p n. \\a \\ (0::'a) \\ p \\ 0; \\n. coeff (p * q) n = (\\i\\n. coeff p i * coeff q (n - i))\\ \\ coeff (pCons a p * q) n = (\\i\\n. coeff (pCons a p) i * coeff q (n - i))\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\n. coeff (0 * q) n = (\\i\\n. coeff 0 i * coeff q (n - i))\n 2. \\a p n. \\a \\ (0::'a) \\ p \\ 0; \\n. coeff (p * q) n = (\\i\\n. coeff p i * coeff q (n - i))\\ \\ coeff (pCons a p * q) n = (\\i\\n. coeff (pCons a p) i * coeff q (n - i))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coeff (0 * q) n = (\\i\\n. coeff 0 i * coeff q (n - i))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncoeff (0 * q) n = (\\i\\n. coeff 0 i * coeff q (n - i))\n\ngoal (1 subgoal):\n 1. \\a p n. \\a \\ (0::'a) \\ p \\ 0; \\n. coeff (p * q) n = (\\i\\n. coeff p i * coeff q (n - i))\\ \\ coeff (pCons a p * q) n = (\\i\\n. coeff (pCons a p) i * coeff q (n - i))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a p n. \\a \\ (0::'a) \\ p \\ 0; \\n. coeff (p * q) n = (\\i\\n. coeff p i * coeff q (n - i))\\ \\ coeff (pCons a p * q) n = (\\i\\n. coeff (pCons a p) i * coeff q (n - i))\n[PROOF STEP]\ncase (pCons a p n)\n[PROOF STATE]\nproof (state)\nthis:\na \\ (0::'a) \\ p \\ 0\ncoeff (p * q) ?n = (\\i\\?n. coeff p i * coeff q (?n - i))\n\ngoal (1 subgoal):\n 1. \\a p n. \\a \\ (0::'a) \\ p \\ 0; \\n. coeff (p * q) n = (\\i\\n. coeff p i * coeff q (n - i))\\ \\ coeff (pCons a p * q) n = (\\i\\n. coeff (pCons a p) i * coeff q (n - i))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ (0::'a) \\ p \\ 0\ncoeff (p * q) ?n = (\\i\\?n. coeff p i * coeff q (?n - i))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ (0::'a) \\ p \\ 0\ncoeff (p * q) ?n = (\\i\\?n. coeff p i * coeff q (?n - i))\n\ngoal (1 subgoal):\n 1. coeff (pCons a p * q) n = (\\i\\n. coeff (pCons a p) i * coeff q (n - i))\n[PROOF STEP]\nby (cases n) (simp_all add: sum.atMost_Suc_shift del: sum.atMost_Suc)\n[PROOF STATE]\nproof (state)\nthis:\ncoeff (pCons a p * q) n = (\\i\\n. coeff (pCons a p) i * coeff q (n - i))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1457, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637648915617, "lm_q2_score": 0.8244619199068831, "lm_q1q2_score": 0.7087600380768764}} {"text": "[STATEMENT]\nlemma sum_union_disjoint':\n assumes \"finite A\"\n and \"finite B\"\n and \"A \\ B = {}\"\n and \"A \\ B = C\"\n shows \"sum g C = sum g A + sum g B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum g C = sum g A + sum g B\n[PROOF STEP]\nusing sum.union_disjoint[OF assms(1-3)] and assms(4)\n[PROOF STATE]\nproof (prove)\nusing this:\nsum ?g (A \\ B) = sum ?g A + sum ?g B\nA \\ B = C\n\ngoal (1 subgoal):\n 1. sum g C = sum g A + sum g B\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 213, "file": "DiscretePricing_Infinite_Coin_Toss_Space", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637577007393, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.708760037710073}} {"text": "[STATEMENT]\nlemma arcosh_mono:\n fixes x y::real\n assumes \"x \\ 1\" \"y \\ 1\"\n shows \"arcosh x \\ arcosh y \\ x \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (arcosh y \\ arcosh x) = (y \\ x)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ x\n1 \\ y\n\ngoal (1 subgoal):\n 1. (arcosh y \\ arcosh x) = (y \\ x)\n[PROOF STEP]\nby (smt arcosh_ge_0 cosh_arcosh cosh_real_nonneg_less_iff)", "meta": {"llama_tokens": 223, "file": "Poincare_Disc_Hyperbolic_Functions", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637577007393, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.708760037710073}} {"text": "[STATEMENT]\nlemma unitary_inner_prod:\n assumes \"v\\ carrier_vec n\"\n and \"w \\ carrier_vec n\"\n and \"U \\ carrier_mat n n\"\n and \"Complex_Matrix.unitary U\"\nshows \"inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = inner_prod v w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod v w\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod v w\n[PROOF STEP]\nhave \"inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = \n inner_prod (((Complex_Matrix.adjoint U) * U) *\\<^sub>v v) w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint U * U *\\<^sub>v v) w\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ carrier_vec n\nw \\ carrier_vec n\nU \\ carrier_mat n n\nComplex_Matrix.unitary U\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint U * U *\\<^sub>v v) w\n[PROOF STEP]\nby (simp add: inner_prod_mult_mat_vec_left)\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint U * U *\\<^sub>v v) w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod v w\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint U * U *\\<^sub>v v) w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod v w\n[PROOF STEP]\nhave \"... = inner_prod (1\\<^sub>m n *\\<^sub>v v) w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Complex_Matrix.adjoint U * U *\\<^sub>v v) w = Complex_Matrix.inner_prod (1\\<^sub>m n *\\<^sub>v v) w\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ carrier_vec n\nw \\ carrier_vec n\nU \\ carrier_mat n n\nComplex_Matrix.unitary U\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Complex_Matrix.adjoint U * U *\\<^sub>v v) w = Complex_Matrix.inner_prod (1\\<^sub>m n *\\<^sub>v v) w\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (Complex_Matrix.adjoint U * U *\\<^sub>v v) w = Complex_Matrix.inner_prod (1\\<^sub>m n *\\<^sub>v v) w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod v w\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (Complex_Matrix.adjoint U * U *\\<^sub>v v) w = Complex_Matrix.inner_prod (1\\<^sub>m n *\\<^sub>v v) w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod v w\n[PROOF STEP]\nhave \"... = inner_prod v w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (1\\<^sub>m n *\\<^sub>v v) w = Complex_Matrix.inner_prod v w\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ carrier_vec n\nw \\ carrier_vec n\nU \\ carrier_mat n n\nComplex_Matrix.unitary U\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (1\\<^sub>m n *\\<^sub>v v) w = Complex_Matrix.inner_prod v w\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (1\\<^sub>m n *\\<^sub>v v) w = Complex_Matrix.inner_prod v w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod v w\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nComplex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod v w\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod v w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod v w\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (U *\\<^sub>v v) (U *\\<^sub>v w) = Complex_Matrix.inner_prod v w\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1804, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637541053281, "lm_q2_score": 0.824461928533133, "lm_q1q2_score": 0.7087600365997119}} {"text": "[STATEMENT]\nlemma conv_radius_mult_ge:\n fixes f g :: \"nat \\ ('a :: {banach,real_normed_div_algebra})\"\n shows \"conv_radius (\\x. \\i\\x. f i * g (x - i)) \\ min (conv_radius f) (conv_radius g)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. min (conv_radius f) (conv_radius g) \\ conv_radius (\\x. \\i\\x. f i * g (x - i))\n[PROOF STEP]\nproof (rule conv_radius_geI_ex')\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\r. \\0 < r; ereal r < min (conv_radius f) (conv_radius g)\\ \\ summable (\\n. (\\i\\n. f i * g (n - i)) * of_real r ^ n)\n[PROOF STEP]\nfix r\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\r. \\0 < r; ereal r < min (conv_radius f) (conv_radius g)\\ \\ summable (\\n. (\\i\\n. f i * g (n - i)) * of_real r ^ n)\n[PROOF STEP]\nassume r: \"r > 0\" \"ereal r < min (conv_radius f) (conv_radius g)\"\n[PROOF STATE]\nproof (state)\nthis:\n0 < r\nereal r < min (conv_radius f) (conv_radius g)\n\ngoal (1 subgoal):\n 1. \\r. \\0 < r; ereal r < min (conv_radius f) (conv_radius g)\\ \\ summable (\\n. (\\i\\n. f i * g (n - i)) * of_real r ^ n)\n[PROOF STEP]\nfrom r\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < r\nereal r < min (conv_radius f) (conv_radius g)\n[PROOF STEP]\nhave \"summable (\\n. (\\i\\n. (f i * of_real r^i) * (g (n - i) * of_real r^(n - i))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < r\nereal r < min (conv_radius f) (conv_radius g)\n\ngoal (1 subgoal):\n 1. summable (\\n. \\i\\n. f i * of_real r ^ i * (g (n - i) * of_real r ^ (n - i)))\n[PROOF STEP]\nby (intro summable_Cauchy_product abs_summable_in_conv_radius) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nsummable (\\n. \\i\\n. f i * of_real r ^ i * (g (n - i) * of_real r ^ (n - i)))\n\ngoal (1 subgoal):\n 1. \\r. \\0 < r; ereal r < min (conv_radius f) (conv_radius g)\\ \\ summable (\\n. (\\i\\n. f i * g (n - i)) * of_real r ^ n)\n[PROOF STEP]\nthus \"summable (\\n. (\\i\\n. f i * g (n - i)) * of_real r ^ n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\n. \\i\\n. f i * of_real r ^ i * (g (n - i) * of_real r ^ (n - i)))\n\ngoal (1 subgoal):\n 1. summable (\\n. (\\i\\n. f i * g (n - i)) * of_real r ^ n)\n[PROOF STEP]\nby (simp add: algebra_simps of_real_def power_add [symmetric] scaleR_sum_right)\n[PROOF STATE]\nproof (state)\nthis:\nsummable (\\n. (\\i\\n. f i * g (n - i)) * of_real r ^ n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1179, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637505099168, "lm_q2_score": 0.8244619306896955, "lm_q1q2_score": 0.7087600354893507}} {"text": "[STATEMENT]\nlemma lm003: \n assumes \"\\ x1 \\ X. (x1 \\ {} & (\\ x2 \\ X-{x1}. x1 \\ x2 = {}))\" \n shows \"is_non_overlapping X\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_non_overlapping X\n[PROOF STEP]\nunfolding is_non_overlapping_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Xa\\X. \\Y\\X. (Xa \\ Y \\ {}) = (Xa = Y)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x1\\X. x1 \\ {} \\ (\\x2\\X - {x1}. x1 \\ x2 = {})\n\ngoal (1 subgoal):\n 1. \\Xa\\X. \\Y\\X. (Xa \\ Y \\ {}) = (Xa = Y)\n[PROOF STEP]\nby fast", "meta": {"llama_tokens": 328, "file": "Vickrey_Clarke_Groves_Universes", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278726384089, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7087429465568351}} {"text": "[STATEMENT]\nlemma not_less_Ord_Least: \"\\k < (LEAST x. Ord x \\ P x); Ord k\\ \\ \\ P k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\k < (LEAST x. Ord x \\ P x); Ord k\\ \\ \\ P k\n[PROOF STEP]\nusing Ord_Least_le less_le_not_le\n[PROOF STATE]\nproof (prove)\nusing this:\n\\Ord ?k; ?P ?k\\ \\ (LEAST i. Ord i \\ ?P i) \\ ?k\n(?x < ?y) = (?x \\ ?y \\ \\ ?y \\ ?x)\n\ngoal (1 subgoal):\n 1. \\k < (LEAST x. Ord x \\ P x); Ord k\\ \\ \\ P k\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 278, "file": "ZFC_in_HOL_ZFC_in_HOL", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278602705731, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7087429366233133}} {"text": "[STATEMENT]\nlemma det_cols_mult: \"det (upd_cols M T (\\i. c i *s a i)) = (\\i\\T. c i) * det (upd_cols M T a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Square_Matrix.det (upd_cols M T (\\i. c i *s a i)) = prod c T * Square_Matrix.det (upd_cols M T a)\n[PROOF STEP]\nusing det_rows_mult[of \"transpose M\" T c a]\n[PROOF STATE]\nproof (prove)\nusing this:\nSquare_Matrix.det (upd_rows (Square_Matrix.transpose M) T (\\i. c i *s a i)) = prod c T * Square_Matrix.det (upd_rows (Square_Matrix.transpose M) T a)\n\ngoal (1 subgoal):\n 1. Square_Matrix.det (upd_cols M T (\\i. c i *s a i)) = prod c T * Square_Matrix.det (upd_cols M T a)\n[PROOF STEP]\nby (simp add: det_transpose upd_rows_transpose)", "meta": {"llama_tokens": 311, "file": "Cayley_Hamilton_Square_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278540866548, "lm_q2_score": 0.8031737963569014, "lm_q1q2_score": 0.7087429295778525}} {"text": "[STATEMENT]\nlemma coplanar_translation_imp: \n assumes \"coplanar S\" shows \"coplanar ((\\x. a + x) ` S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coplanar ((+) a ` S)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. coplanar ((+) a ` S)\n[PROOF STEP]\nobtain u v w where \"S \\ affine hull {u,v,w}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\u v w. S \\ affine hull {u, v, w} \\ thesis) \\ thesis\n[PROOF STEP]\nby (meson assms coplanar_def)\n[PROOF STATE]\nproof (state)\nthis:\nS \\ affine hull {u, v, w}\n\ngoal (1 subgoal):\n 1. coplanar ((+) a ` S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nS \\ affine hull {u, v, w}\n[PROOF STEP]\nhave \"(+) a ` S \\ affine hull {u + a, v + a, w + a}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ affine hull {u, v, w}\n\ngoal (1 subgoal):\n 1. (+) a ` S \\ affine hull {u + a, v + a, w + a}\n[PROOF STEP]\nusing affine_hull_translation [of a \"{u,v,w}\" for u v w]\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ affine hull {u, v, w}\naffine hull (+) a ` {?ua2, ?va2, ?wa2} = (+) a ` (affine hull {?ua2, ?va2, ?wa2})\n\ngoal (1 subgoal):\n 1. (+) a ` S \\ affine hull {u + a, v + a, w + a}\n[PROOF STEP]\nby (force simp: add.commute)\n[PROOF STATE]\nproof (state)\nthis:\n(+) a ` S \\ affine hull {u + a, v + a, w + a}\n\ngoal (1 subgoal):\n 1. coplanar ((+) a ` S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(+) a ` S \\ affine hull {u + a, v + a, w + a}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(+) a ` S \\ affine hull {u + a, v + a, w + a}\n\ngoal (1 subgoal):\n 1. coplanar ((+) a ` S)\n[PROOF STEP]\nunfolding coplanar_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(+) a ` S \\ affine hull {u + a, v + a, w + a}\n\ngoal (1 subgoal):\n 1. \\u v w. (+) a ` S \\ affine hull {u, v, w}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ncoplanar ((+) a ` S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 943, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438950986284991, "lm_q2_score": 0.8397339696776499, "lm_q1q2_score": 0.7086473811628214}} {"text": "[STATEMENT]\ntheorem inrange: assumes \"invar_vebt t n\" shows \" set_vebt' t \\ {0..2^n-1}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_vebt' t \\ {0..2 ^ n - 1}\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ set_vebt' t \\ x \\ {0..2 ^ n - 1}\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ set_vebt' t \\ x \\ {0..2 ^ n - 1}\n[PROOF STEP]\nassume \"x \\ set_vebt' t\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ set_vebt' t\n\ngoal (1 subgoal):\n 1. \\x. x \\ set_vebt' t \\ x \\ {0..2 ^ n - 1}\n[PROOF STEP]\nhence \"vebt_member t x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ set_vebt' t\n\ngoal (1 subgoal):\n 1. vebt_member t x\n[PROOF STEP]\nusing set_vebt'_def\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ set_vebt' t\nset_vebt' ?t = {x. vebt_member ?t x}\n\ngoal (1 subgoal):\n 1. vebt_member t x\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nvebt_member t x\n\ngoal (1 subgoal):\n 1. \\x. x \\ set_vebt' t \\ x \\ {0..2 ^ n - 1}\n[PROOF STEP]\nhence \"x < 2^n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nvebt_member t x\n\ngoal (1 subgoal):\n 1. x < 2 ^ n\n[PROOF STEP]\nusing assms member_bound\n[PROOF STATE]\nproof (prove)\nusing this:\nvebt_member t x\ninvar_vebt t n\n\\vebt_member ?tree ?x; invar_vebt ?tree ?n\\ \\ ?x < 2 ^ ?n\n\ngoal (1 subgoal):\n 1. x < 2 ^ n\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nx < 2 ^ n\n\ngoal (1 subgoal):\n 1. \\x. x \\ set_vebt' t \\ x \\ {0..2 ^ n - 1}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx < 2 ^ n\n[PROOF STEP]\nshow \"x \\ {0..2^n-1}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx < 2 ^ n\n\ngoal (1 subgoal):\n 1. x \\ {0..2 ^ n - 1}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx \\ {0..2 ^ n - 1}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 947, "file": "Van_Emde_Boas_Trees_VEBT_Member", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199795472731, "lm_q2_score": 0.8459424295406088, "lm_q1q2_score": 0.7085782805299753}} {"text": "[STATEMENT]\nlemma fds_deriv_inverse [simp]:\n fixes f :: \"'a :: {real_algebra, field} fds\"\n assumes \"fds_nth f (Suc 0) \\ 0\"\n shows \"fds_deriv (inverse f) = -fds_deriv f / f ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nhave \"(0 :: 'a fds) = fds_deriv 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 = fds_deriv 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 = fds_deriv 1\n\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n0 = fds_deriv 1\n\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfds_nth f (Suc 0) \\ (0::'a)\n[PROOF STEP]\nhave \"(1 :: 'a fds) = inverse f * f\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfds_nth f (Suc 0) \\ (0::'a)\n\ngoal (1 subgoal):\n 1. 1 = inverse f * f\n[PROOF STEP]\nby (simp add: fds_left_inverse)\n[PROOF STATE]\nproof (state)\nthis:\n1 = inverse f * f\n\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 = inverse f * f\n\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nhave \"fds_deriv \\ = fds_deriv (inverse f) * f + inverse f * fds_deriv f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fds_deriv (inverse f * f) = fds_deriv (inverse f) * f + inverse f * fds_deriv f\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfds_deriv (inverse f * f) = fds_deriv (inverse f) * f + inverse f * fds_deriv f\n\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfds_deriv (inverse f * f) = fds_deriv (inverse f) * f + inverse f * fds_deriv f\n\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nhave \"\\ * inverse f = fds_deriv (inverse f) * (f * inverse f) + \n inverse f ^ 2 * fds_deriv f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (fds_deriv (inverse f) * f + inverse f * fds_deriv f) * inverse f = fds_deriv (inverse f) * (f * inverse f) + (inverse f)\\<^sup>2 * fds_deriv f\n[PROOF STEP]\nby (simp add: algebra_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n(fds_deriv (inverse f) * f + inverse f * fds_deriv f) * inverse f = fds_deriv (inverse f) * (f * inverse f) + (inverse f)\\<^sup>2 * fds_deriv f\n\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(fds_deriv (inverse f) * f + inverse f * fds_deriv f) * inverse f = fds_deriv (inverse f) * (f * inverse f) + (inverse f)\\<^sup>2 * fds_deriv f\n\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfds_nth f (Suc 0) \\ (0::'a)\n[PROOF STEP]\nhave \"f * inverse f = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfds_nth f (Suc 0) \\ (0::'a)\n\ngoal (1 subgoal):\n 1. f * inverse f = 1\n[PROOF STEP]\nby (simp add: fds_right_inverse)\n[PROOF STATE]\nproof (state)\nthis:\nf * inverse f = 1\n\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n0 * inverse f = fds_deriv (inverse f) * 1 + (inverse f)\\<^sup>2 * fds_deriv f\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 * inverse f = fds_deriv (inverse f) * 1 + (inverse f)\\<^sup>2 * fds_deriv f\n\ngoal (1 subgoal):\n 1. fds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n[PROOF STEP]\nby (simp add: algebra_simps power2_eq_square divide_fds_def inverse_mult_fds add_eq_0_iff)\n[PROOF STATE]\nproof (state)\nthis:\nfds_deriv (inverse f) = - fds_deriv f / f\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1859, "file": "Dirichlet_Series_Dirichlet_Series", "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424295406088, "lm_q2_score": 0.8376199653600372, "lm_q1q2_score": 0.7085782685283906}} {"text": "[STATEMENT]\nlemma newton_coefficients_main: \n \"k < n \\ newton_coefficients_main (rev (map f [0.. i. map (\\ j. xij_f j i) [0.. local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0..k. \\k < n \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0..k. \\k < n \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..k. \\k < n \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0..k. \\k < n \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0..k. \\k < n \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0..k. \\k < n \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..k. \\k < n \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. f. rev (map f [0.. f. f k # rev (map f [0..f. rev (map f [0..f. f k # rev (map f [0..k. \\k < n \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0.. \\ local.newton_coefficients_main (rev (map f [0..i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0.. map (\\j. xij_f j k) [0..j. xij_f j k) [0.. rev (map (\\i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..j. xij_f j k) [0.. rev (map (\\i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..j. xij_f j k) [0.. rev (map (\\i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..j. xij_f j k) [0.. rev (map (\\i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..j. xij_f j k) [0.. rev (map (\\i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..j. xij_f j k) [0.. rev (map (\\i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..j. xij_f j k) [0.. rev (map (\\i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..j. xij_f j k)) [nn..< Suc k]) (f (Suc k)) (x (Suc k)) (map x [nn ..< n]) =\n map ((\\j. xij_f j (Suc k))) [nn..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..j. xij_f j k) [0.. rev (map (\\i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..j. xij_f j k) [0.. rev (map (\\i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..nn. \\0 = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..m nn. \\\\nn. \\m = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn.. \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..nn. \\0 = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..m nn. \\\\nn. \\m = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn.. \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..nn. \\0 = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..m nn. \\\\nn. \\m = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn.. \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [Suc k..j. xij_f j (Suc k)) [Suc k..j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..m nn. \\\\nn. \\m = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn.. \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..m nn. \\\\nn. \\m = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn.. \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..m = Suc k - ?nn1; ?nn1 < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [?nn1..j. xij_f j (Suc k)) [?nn1..m nn. \\\\nn. \\m = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn.. \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..Suc k < n\\\n[PROOF STATE]\nproof (chain)\npicking this:\nSuc k < n\n\\m = Suc k - ?nn1; ?nn1 < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [?nn1..j. xij_f j (Suc k)) [?nn1..m = Suc k - ?nn1; ?nn1 < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [?nn1..j. xij_f j (Suc k)) [?nn1..m nn. \\\\nn. \\m = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn.. \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..m nn. \\\\nn. \\m = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn.. \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..m nn. \\\\nn. \\m = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn.. \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [Suc nn..j. xij_f j (Suc k)) [Suc nn..m nn. \\\\nn. \\m = Suc k - nn; nn < Suc (Suc k)\\ \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn.. \\ local.divided_differences_impl (map (\\j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j (Suc k)) [Suc nn..j. xij_f j (Suc k)) [Suc nn..j. xij_f j (Suc k)) [nn..j. xij_f j (Suc k)) [Suc (Suc nn).. xij_f (Suc nn) (Suc k) = xij_f (Suc nn) (Suc k) \\ map (\\j. xij_f j (Suc k)) [Suc (Suc nn)..j. xij_f j (Suc k)) [Suc (Suc nn)..j. xij_f j (Suc k)) [Suc (Suc nn).. xij_f (Suc nn) (Suc k) = xij_f (Suc nn) (Suc k) \\ map (\\j. xij_f j (Suc k)) [Suc (Suc nn)..j. xij_f j (Suc k)) [Suc (Suc nn)..j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..j. xij_f j k) [0.. rev (map (\\i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j (Suc k)) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j (Suc k)) [nn..j. xij_f j (Suc k)) [nn..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..i. map (\\j. xij_f j i) [0..j. xij_f j (Suc k)) [0..j. xij_f j k) [0..j. xij_f j k) [0.. rev (map (\\i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..i. map (\\j. xij_f j i) [0..x. f x + g x)\" and\n \"interval_lebesgue_integral M a b (\\x. f x + g x) =\n interval_lebesgue_integral M a b f + interval_lebesgue_integral M a b g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. interval_lebesgue_integrable M a b (\\x. f x + g x) &&& LINT x=a..b|M. (f x + g x) = interval_lebesgue_integral M a b f + interval_lebesgue_integral M a b g\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ninterval_lebesgue_integrable M a b f\ninterval_lebesgue_integrable M a b g\n\ngoal (1 subgoal):\n 1. interval_lebesgue_integrable M a b (\\x. f x + g x) &&& LINT x=a..b|M. (f x + g x) = interval_lebesgue_integral M a b f + interval_lebesgue_integral M a b g\n[PROOF STEP]\nby (auto simp: interval_lebesgue_integral_def interval_lebesgue_integrable_def\n field_simps)", "meta": {"llama_tokens": 414, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513648201267, "lm_q2_score": 0.7905303211371898, "lm_q1q2_score": 0.7085138792508994}} {"text": "[STATEMENT]\nlemma fref_nest: \"fref P1 R1 (fref P2 R2 S) \n \\ CURRY (fref (\\(a,b). P1 a \\ P2 b) (R1\\\\<^sub>rR2) S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [P1]\\<^sub>f R1 \\ [P2]\\<^sub>f R2 \\ S \\ CURRY ([\\(a, b). P1 a \\ P2 b]\\<^sub>f R1 \\\\<^sub>r R2 \\ S)\n[PROOF STEP]\napply (rule eq_reflection)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [P1]\\<^sub>f R1 \\ [P2]\\<^sub>f R2 \\ S = CURRY ([\\(a, b). P1 a \\ P2 b]\\<^sub>f R1 \\\\<^sub>r R2 \\ S)\n[PROOF STEP]\nby (auto simp: fref_def CURRY_def)", "meta": {"llama_tokens": 297, "file": "Refine_Imperative_HOL_Sepref_Rules", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513675912912, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7085138748070476}} {"text": "[STATEMENT]\nlemma has_integral_powr_from_0:\n assumes a: \"a > (-1::real)\" and c: \"c \\ 0\"\n shows \"((\\x. x powr a) has_integral (c powr (a+1) / (a+1))) {0..c}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nproof (cases \"c = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nc \\ 0\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\ndefine f where \"f = (\\k x. if x \\ {inverse (of_nat (Suc k))..c} then x powr a else 0)\"\n[PROOF STATE]\nproof (state)\nthis:\nf = (\\k x. if x \\ {inverse (real (Suc k))..c} then x powr a else 0)\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\ndefine F where \"F = (\\k. if inverse (of_nat (Suc k)) \\ c then\n c powr (a+1)/(a+1) - inverse (real (Suc k)) powr (a+1)/(a+1) else 0)\"\n[PROOF STATE]\nproof (state)\nthis:\nF = (\\k. if inverse (real (Suc k)) \\ c then c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1) else 0)\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\nF = (\\k. if inverse (real (Suc k)) \\ c then c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1) else 0)\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nfix k :: nat\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nhave \"(f k has_integral F k) {0..c}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f k has_integral F k) {0..c}\n[PROOF STEP]\nproof (cases \"inverse (of_nat (Suc k)) \\ c\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\ninverse (real (Suc k)) \\ c\n\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\ninverse (real (Suc k)) \\ c\n\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\nassume x: \"x \\ inverse (1 + real k)\"\n[PROOF STATE]\nproof (state)\nthis:\ninverse (1 + real k) \\ x\n\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\nhave \"0 < inverse (1 + real k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < inverse (1 + real k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < inverse (1 + real k)\n\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n0 < inverse (1 + real k)\n\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\nnote x\n[PROOF STATE]\nproof (state)\nthis:\ninverse (1 + real k) \\ x\n\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < x\n[PROOF STEP]\nhave \"x > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n\ngoal (1 subgoal):\n 1. 0 < x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n0 < x\n\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\ninverse (1 + real k) \\ ?x2 \\ 0 < ?x2\n\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\nnote x = this\n[PROOF STATE]\nproof (state)\nthis:\ninverse (1 + real k) \\ ?x2 \\ 0 < ?x2\n\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\nhence \"((\\x. x powr a) has_integral c powr (a + 1) / (a + 1) -\n inverse (real (Suc k)) powr (a + 1) / (a + 1)) {inverse (real (Suc k))..c}\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse (1 + real k) \\ ?x2 \\ 0 < ?x2\n\ngoal (1 subgoal):\n 1. ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) {inverse (real (Suc k))..c}\n[PROOF STEP]\nusing True a\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse (1 + real k) \\ ?x2 \\ 0 < ?x2\ninverse (real (Suc k)) \\ c\n- 1 < a\n\ngoal (1 subgoal):\n 1. ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) {inverse (real (Suc k))..c}\n[PROOF STEP]\nby (intro fundamental_theorem_of_calculus)\n (auto intro!: derivative_eq_intros continuous_on_powr' continuous_on_const\n simp: has_real_derivative_iff_has_vector_derivative [symmetric])\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. x powr a) has_integral c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) {inverse (real (Suc k))..c}\n\ngoal (2 subgoals):\n 1. inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n 2. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\nwith True\n[PROOF STATE]\nproof (chain)\npicking this:\ninverse (real (Suc k)) \\ c\n((\\x. x powr a) has_integral c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) {inverse (real (Suc k))..c}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse (real (Suc k)) \\ c\n((\\x. x powr a) has_integral c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) {inverse (real (Suc k))..c}\n\ngoal (1 subgoal):\n 1. (f k has_integral F k) {0..c}\n[PROOF STEP]\nunfolding f_def F_def\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse (real (Suc k)) \\ c\n((\\x. x powr a) has_integral c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) {inverse (real (Suc k))..c}\n\ngoal (1 subgoal):\n 1. ((\\x. if x \\ {inverse (real (Suc k))..c} then x powr a else 0) has_integral (if inverse (real (Suc k)) \\ c then c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1) else 0)) {0..c}\n[PROOF STEP]\nby (subst has_integral_restrict) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(f k has_integral F k) {0..c}\n\ngoal (1 subgoal):\n 1. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ inverse (real (Suc k)) \\ c\n\ngoal (1 subgoal):\n 1. \\ inverse (real (Suc k)) \\ c \\ (f k has_integral F k) {0..c}\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ inverse (real (Suc k)) \\ c\n\ngoal (1 subgoal):\n 1. (f k has_integral F k) {0..c}\n[PROOF STEP]\nunfolding f_def F_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ inverse (real (Suc k)) \\ c\n\ngoal (1 subgoal):\n 1. ((\\x. if x \\ {inverse (real (Suc k))..c} then x powr a else 0) has_integral (if inverse (real (Suc k)) \\ c then c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1) else 0)) {0..c}\n[PROOF STEP]\nby (subst has_integral_restrict) auto\n[PROOF STATE]\nproof (state)\nthis:\n(f k has_integral F k) {0..c}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(f k has_integral F k) {0..c}\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n(f ?k2 has_integral F ?k2) {0..c}\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nnote has_integral_f = this\n[PROOF STATE]\nproof (state)\nthis:\n(f ?k2 has_integral F ?k2) {0..c}\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nhave integral_f: \"integral {0..c} (f k) = F k\" for k\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral {0..c} (f k) = F k\n[PROOF STEP]\nusing has_integral_f[of k]\n[PROOF STATE]\nproof (prove)\nusing this:\n(f k has_integral F k) {0..c}\n\ngoal (1 subgoal):\n 1. integral {0..c} (f k) = F k\n[PROOF STEP]\nby (rule integral_unique)\n[PROOF STATE]\nproof (state)\nthis:\nintegral {0..c} (f ?k) = F ?k\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nhave A: \"(\\x. x powr a) integrable_on {0..c} \\\n (\\k. integral {0..c} (f k)) \\ integral {0..c} (\\x. x powr a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. x powr a) integrable_on {0..c} \\ (\\k. integral {0..c} (f k)) \\ integral {0..c} (\\x. x powr a)\n[PROOF STEP]\nproof (intro monotone_convergence_increasing ballI allI)\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. \\k. f k integrable_on {0..c}\n 2. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 3. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 4. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nfix k\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. \\k. f k integrable_on {0..c}\n 2. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 3. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 4. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nfrom has_integral_f[of k]\n[PROOF STATE]\nproof (chain)\npicking this:\n(f k has_integral F k) {0..c}\n[PROOF STEP]\nshow \"f k integrable_on {0..c}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(f k has_integral F k) {0..c}\n\ngoal (1 subgoal):\n 1. f k integrable_on {0..c}\n[PROOF STEP]\nby (auto simp: integrable_on_def)\n[PROOF STATE]\nproof (state)\nthis:\nf k integrable_on {0..c}\n\ngoal (3 subgoals):\n 1. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 2. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 3. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 2. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 3. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nfix k :: nat and x :: real\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 2. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 3. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 2. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 3. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nassume x: \"inverse (real (Suc k)) \\ x\"\n[PROOF STATE]\nproof (state)\nthis:\ninverse (real (Suc k)) \\ x\n\ngoal (3 subgoals):\n 1. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 2. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 3. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nhave \"inverse (real (Suc (Suc k))) \\ inverse (real (Suc k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse (real (Suc (Suc k))) \\ inverse (real (Suc k))\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\ninverse (real (Suc (Suc k))) \\ inverse (real (Suc k))\n\ngoal (3 subgoals):\n 1. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 2. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 3. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ninverse (real (Suc (Suc k))) \\ inverse (real (Suc k))\n\ngoal (3 subgoals):\n 1. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 2. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 3. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nnote x\n[PROOF STATE]\nproof (state)\nthis:\ninverse (real (Suc k)) \\ x\n\ngoal (3 subgoals):\n 1. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 2. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 3. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ninverse (real (Suc (Suc k))) \\ x\n[PROOF STEP]\nhave \"inverse (real (Suc (Suc k))) \\ x\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse (real (Suc (Suc k))) \\ x\n\ngoal (1 subgoal):\n 1. inverse (real (Suc (Suc k))) \\ x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ninverse (real (Suc (Suc k))) \\ x\n\ngoal (3 subgoals):\n 1. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 2. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 3. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\ninverse (real (Suc k)) \\ x \\ inverse (real (Suc (Suc k))) \\ x\n\ngoal (3 subgoals):\n 1. \\k x. x \\ {0..c} \\ f k x \\ f (Suc k) x\n 2. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 3. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nthus \"f k x \\ f (Suc k) x\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse (real (Suc k)) \\ x \\ inverse (real (Suc (Suc k))) \\ x\n\ngoal (1 subgoal):\n 1. f k x \\ f (Suc k) x\n[PROOF STEP]\nby (auto simp: f_def simp del: of_nat_Suc)\n[PROOF STATE]\nproof (state)\nthis:\nf k x \\ f (Suc k) x\n\ngoal (2 subgoals):\n 1. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 2. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 2. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 2. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nassume x: \"x \\ {0..c}\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ {0..c}\n\ngoal (2 subgoals):\n 1. \\x. x \\ {0..c} \\ (\\k. f k x) \\ x powr a\n 2. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nshow \"(\\k. f k x) \\ x powr a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. f k x) \\ x powr a\n[PROOF STEP]\nproof (cases \"x = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x = 0 \\ (\\k. f k x) \\ x powr a\n 2. x \\ 0 \\ (\\k. f k x) \\ x powr a\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nx \\ 0\n\ngoal (2 subgoals):\n 1. x = 0 \\ (\\k. f k x) \\ x powr a\n 2. x \\ 0 \\ (\\k. f k x) \\ x powr a\n[PROOF STEP]\nwith x\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ {0..c}\nx \\ 0\n[PROOF STEP]\nhave \"x > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ {0..c}\nx \\ 0\n\ngoal (1 subgoal):\n 1. 0 < x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < x\n\ngoal (2 subgoals):\n 1. x = 0 \\ (\\k. f k x) \\ x powr a\n 2. x \\ 0 \\ (\\k. f k x) \\ x powr a\n[PROOF STEP]\nfrom order_tendstoD(2)[OF LIMSEQ_inverse_real_of_nat this]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F xa in sequentially. inverse (real (Suc xa)) < x\n[PROOF STEP]\nhave \"eventually (\\k. x powr a = f k x) sequentially\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F xa in sequentially. inverse (real (Suc xa)) < x\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F k in sequentially. x powr a = f k x\n[PROOF STEP]\nby eventually_elim (insert x, simp add: f_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F k in sequentially. x powr a = f k x\n\ngoal (2 subgoals):\n 1. x = 0 \\ (\\k. f k x) \\ x powr a\n 2. x \\ 0 \\ (\\k. f k x) \\ x powr a\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F k in sequentially. x powr a = f k x\n\ngoal (2 subgoals):\n 1. x = 0 \\ (\\k. f k x) \\ x powr a\n 2. x \\ 0 \\ (\\k. f k x) \\ x powr a\n[PROOF STEP]\nhave \"(\\_. x powr a) \\ x powr a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\_. x powr a) \\ x powr a\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\_. x powr a) \\ x powr a\n\ngoal (2 subgoals):\n 1. x = 0 \\ (\\k. f k x) \\ x powr a\n 2. x \\ 0 \\ (\\k. f k x) \\ x powr a\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F k in sequentially. x powr a = f k x\n(\\_. x powr a) \\ x powr a\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F k in sequentially. x powr a = f k x\n(\\_. x powr a) \\ x powr a\n\ngoal (1 subgoal):\n 1. (\\k. f k x) \\ x powr a\n[PROOF STEP]\nby (blast intro: Lim_transform_eventually)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. f k x) \\ x powr a\n\ngoal (1 subgoal):\n 1. x = 0 \\ (\\k. f k x) \\ x powr a\n[PROOF STEP]\nqed (simp_all add: f_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. f k x) \\ x powr a\n\ngoal (1 subgoal):\n 1. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nfix k\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nfrom a\n[PROOF STATE]\nproof (chain)\npicking this:\n- 1 < a\n[PROOF STEP]\nhave \"F k \\ c powr (a + 1) / (a + 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n- 1 < a\n\ngoal (1 subgoal):\n 1. F k \\ c powr (a + 1) / (a + 1)\n[PROOF STEP]\nby (auto simp add: F_def divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\nF k \\ c powr (a + 1) / (a + 1)\n\ngoal (1 subgoal):\n 1. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nF k \\ c powr (a + 1) / (a + 1)\n\ngoal (1 subgoal):\n 1. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nfrom a\n[PROOF STATE]\nproof (chain)\npicking this:\n- 1 < a\n[PROOF STEP]\nhave \"F k \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n- 1 < a\n\ngoal (1 subgoal):\n 1. 0 \\ F k\n[PROOF STEP]\nby (auto simp: F_def divide_simps simp del: of_nat_Suc intro!: powr_mono2)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ F k\n\ngoal (1 subgoal):\n 1. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nhence \"F k = abs (F k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ F k\n\ngoal (1 subgoal):\n 1. F k = \\F k\\\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nF k = \\F k\\\n\ngoal (1 subgoal):\n 1. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\F k\\ \\ c powr (a + 1) / (a + 1)\n[PROOF STEP]\nhave \"abs (F k) \\ c powr (a + 1) / (a + 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\F k\\ \\ c powr (a + 1) / (a + 1)\n\ngoal (1 subgoal):\n 1. \\F k\\ \\ c powr (a + 1) / (a + 1)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\F k\\ \\ c powr (a + 1) / (a + 1)\n\ngoal (1 subgoal):\n 1. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n\\F ?k2\\ \\ c powr (a + 1) / (a + 1)\n\ngoal (1 subgoal):\n 1. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nthus \"bounded (range(\\k. integral {0..c} (f k)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\F ?k2\\ \\ c powr (a + 1) / (a + 1)\n\ngoal (1 subgoal):\n 1. bounded (range (\\k. integral {0..c} (f k)))\n[PROOF STEP]\nby (intro boundedI[of _ \"c powr (a+1) / (a+1)\"]) (auto simp: integral_f)\n[PROOF STATE]\nproof (state)\nthis:\nbounded (range (\\k. integral {0..c} (f k)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. x powr a) integrable_on {0..c} \\ (\\k. integral {0..c} (f k)) \\ integral {0..c} (\\x. x powr a)\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nfrom False c\n[PROOF STATE]\nproof (chain)\npicking this:\nc \\ 0\n0 \\ c\n[PROOF STEP]\nhave \"c > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ 0\n0 \\ c\n\ngoal (1 subgoal):\n 1. 0 < c\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < c\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nfrom order_tendstoD(2)[OF LIMSEQ_inverse_real_of_nat this]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F x in sequentially. inverse (real (Suc x)) < c\n[PROOF STEP]\nhave \"eventually (\\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a+1) / (a+1) =\n integral {0..c} (f k)) sequentially\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in sequentially. inverse (real (Suc x)) < c\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F k in sequentially. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1) = integral {0..c} (f k)\n[PROOF STEP]\nby eventually_elim (simp add: integral_f F_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F k in sequentially. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1) = integral {0..c} (f k)\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F k in sequentially. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1) = integral {0..c} (f k)\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nhave \"(\\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1))\n \\ c powr (a + 1) / (a + 1) - 0 powr (a + 1) / (a + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) \\ c powr (a + 1) / (a + 1) - 0 powr (a + 1) / (a + 1)\n[PROOF STEP]\nusing a\n[PROOF STATE]\nproof (prove)\nusing this:\n- 1 < a\n\ngoal (1 subgoal):\n 1. (\\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) \\ c powr (a + 1) / (a + 1) - 0 powr (a + 1) / (a + 1)\n[PROOF STEP]\nby (intro tendsto_intros LIMSEQ_inverse_real_of_nat) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) \\ c powr (a + 1) / (a + 1) - 0 powr (a + 1) / (a + 1)\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nhence \"(\\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1))\n \\ c powr (a + 1) / (a + 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) \\ c powr (a + 1) / (a + 1) - 0 powr (a + 1) / (a + 1)\n\ngoal (1 subgoal):\n 1. (\\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) \\ c powr (a + 1) / (a + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) \\ c powr (a + 1) / (a + 1)\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F k in sequentially. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1) = integral {0..c} (f k)\n(\\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) \\ c powr (a + 1) / (a + 1)\n[PROOF STEP]\nhave \"(\\k. integral {0..c} (f k)) \\ c powr (a+1) / (a+1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F k in sequentially. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1) = integral {0..c} (f k)\n(\\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) \\ c powr (a + 1) / (a + 1)\n\ngoal (1 subgoal):\n 1. (\\k. integral {0..c} (f k)) \\ c powr (a + 1) / (a + 1)\n[PROOF STEP]\nby (blast intro: Lim_transform_eventually)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k. integral {0..c} (f k)) \\ c powr (a + 1) / (a + 1)\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nwith A\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x. x powr a) integrable_on {0..c} \\ (\\k. integral {0..c} (f k)) \\ integral {0..c} (\\x. x powr a)\n(\\k. integral {0..c} (f k)) \\ c powr (a + 1) / (a + 1)\n[PROOF STEP]\nhave \"integral {0..c} (\\x. x powr a) = c powr (a+1) / (a+1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. x powr a) integrable_on {0..c} \\ (\\k. integral {0..c} (f k)) \\ integral {0..c} (\\x. x powr a)\n(\\k. integral {0..c} (f k)) \\ c powr (a + 1) / (a + 1)\n\ngoal (1 subgoal):\n 1. integral {0..c} (\\x. x powr a) = c powr (a + 1) / (a + 1)\n[PROOF STEP]\nby (blast intro: LIMSEQ_unique)\n[PROOF STATE]\nproof (state)\nthis:\nintegral {0..c} (\\x. x powr a) = c powr (a + 1) / (a + 1)\n\ngoal (2 subgoals):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n 2. c \\ 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nwith A\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x. x powr a) integrable_on {0..c} \\ (\\k. integral {0..c} (f k)) \\ integral {0..c} (\\x. x powr a)\nintegral {0..c} (\\x. x powr a) = c powr (a + 1) / (a + 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. x powr a) integrable_on {0..c} \\ (\\k. integral {0..c} (f k)) \\ integral {0..c} (\\x. x powr a)\nintegral {0..c} (\\x. x powr a) = c powr (a + 1) / (a + 1)\n\ngoal (1 subgoal):\n 1. ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nby (simp add: has_integral_integral)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n\ngoal (1 subgoal):\n 1. c = 0 \\ ((\\x. x powr a) has_integral c powr (a + 1) / (a + 1)) {0..c}\n[PROOF STEP]\nqed (simp_all add: has_integral_refl)", "meta": {"llama_tokens": 14390, "file": null, "length": 119, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094145755219, "lm_q2_score": 0.7931059560743422, "lm_q1q2_score": 0.7084890173171302}} {"text": "[STATEMENT]\nlemma diag_jordan_block_pow: \"diag_mat (jordan_block n (a :: 'a) ^\\<^sub>m k) = replicate n (a ^ k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. diag_mat (jordan_block n a ^\\<^sub>m k) = replicate n (a ^ k)\n[PROOF STEP]\nunfolding diag_mat_def jordan_block_pow\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. map (\\i. mat n n (\\(i, j). if i \\ j then of_nat (k choose (j - i)) * a ^ (k + i - j) else (0::'a)) $$ (i, i)) [0..(i, j). if i \\ j then of_nat (k choose (j - i)) * a ^ (k + i - j) else (0::'a)))] = replicate n (a ^ k)\n[PROOF STEP]\nby (intro nth_equalityI, auto)", "meta": {"llama_tokens": 293, "file": "Jordan_Normal_Form_Jordan_Normal_Form", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094060543487, "lm_q2_score": 0.7931059585194573, "lm_q1q2_score": 0.7084890127431813}} {"text": "[STATEMENT]\nlemma pos_succ:\n assumes \"x \\ last_el\"\n shows \"pos (succ x) = pos x + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pos (succ x) = pos x + 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. pos (succ x) = pos x + 1\n[PROOF STEP]\nhave \"x \\ last_el\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ last_el\n[PROOF STEP]\nby (rule last_el_greatest)\n[PROOF STATE]\nproof (state)\nthis:\nx \\ last_el\n\ngoal (1 subgoal):\n 1. pos (succ x) = pos x + 1\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ last_el\nx \\ last_el\n[PROOF STEP]\nhave \"x < last_el\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ last_el\nx \\ last_el\n\ngoal (1 subgoal):\n 1. x < last_el\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx < last_el\n\ngoal (1 subgoal):\n 1. pos (succ x) = pos x + 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx < last_el\n[PROOF STEP]\nhave \"pos x < pos last_el\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx < last_el\n\ngoal (1 subgoal):\n 1. pos x < pos last_el\n[PROOF STEP]\nby (simp add: less_pos)\n[PROOF STATE]\nproof (state)\nthis:\npos x < pos last_el\n\ngoal (1 subgoal):\n 1. pos (succ x) = pos x + 1\n[PROOF STEP]\nwith rangeI [of pos x]\n[PROOF STATE]\nproof (chain)\npicking this:\npos x \\ range pos\npos x < pos last_el\n[PROOF STEP]\nhave \"pos x + 1 \\ range pos\"\n[PROOF STATE]\nproof (prove)\nusing this:\npos x \\ range pos\npos x < pos last_el\n\ngoal (1 subgoal):\n 1. pos x + 1 \\ range pos\n[PROOF STEP]\nby (simp add: range_pos last_el_def pos_val)\n[PROOF STATE]\nproof (state)\nthis:\npos x + 1 \\ range pos\n\ngoal (1 subgoal):\n 1. pos (succ x) = pos x + 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\npos x + 1 \\ range pos\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\npos x + 1 \\ range pos\n\ngoal (1 subgoal):\n 1. pos (succ x) = pos x + 1\n[PROOF STEP]\nby (simp add: succ_def pos_val)\n[PROOF STATE]\nproof (state)\nthis:\npos (succ x) = pos x + 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 944, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677660619633, "lm_q2_score": 0.8354835350552604, "lm_q1q2_score": 0.7084631068023612}} {"text": "[STATEMENT]\ntheorem c_cons_is_pr: \"c_cons \\ PrimRec2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c_cons \\ PrimRec2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. c_cons \\ PrimRec2\n[PROOF STEP]\nhave \"c_cons = (\\ x u. (sgn2 u)*((c_pair 0 x)+1) + (sgn1 u)*((c_pair (c_len u) (c_pair x (c_snd (u-(1::nat))))) + 1))\" (is \"_ = ?R\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c_cons = (\\x u. sgn2 u * (c_pair 0 x + 1) + sgn1 u * (c_pair (c_len u) (c_pair x (c_snd (u - 1))) + 1))\n[PROOF STEP]\nby (simp add: c_cons_aux3)\n[PROOF STATE]\nproof (state)\nthis:\nc_cons = (\\x u. sgn2 u * (c_pair 0 x + 1) + sgn1 u * (c_pair (c_len u) (c_pair x (c_snd (u - 1))) + 1))\n\ngoal (1 subgoal):\n 1. c_cons \\ PrimRec2\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nc_cons = (\\x u. sgn2 u * (c_pair 0 x + 1) + sgn1 u * (c_pair (c_len u) (c_pair x (c_snd (u - 1))) + 1))\n\ngoal (1 subgoal):\n 1. c_cons \\ PrimRec2\n[PROOF STEP]\nfrom c_pair_is_pr c_len_is_pr\n[PROOF STATE]\nproof (chain)\npicking this:\nc_pair \\ PrimRec2\nc_len \\ PrimRec1\n[PROOF STEP]\nhave \"?R \\ PrimRec2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc_pair \\ PrimRec2\nc_len \\ PrimRec1\n\ngoal (1 subgoal):\n 1. (\\x u. sgn2 u * (c_pair 0 x + 1) + sgn1 u * (c_pair (c_len u) (c_pair x (c_snd (u - 1))) + 1)) \\ PrimRec2\n[PROOF STEP]\nby prec\n[PROOF STATE]\nproof (state)\nthis:\n(\\x u. sgn2 u * (c_pair 0 x + 1) + sgn1 u * (c_pair (c_len u) (c_pair x (c_snd (u - 1))) + 1)) \\ PrimRec2\n\ngoal (1 subgoal):\n 1. c_cons \\ PrimRec2\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nc_cons = (\\x u. sgn2 u * (c_pair 0 x + 1) + sgn1 u * (c_pair (c_len u) (c_pair x (c_snd (u - 1))) + 1))\n(\\x u. sgn2 u * (c_pair 0 x + 1) + sgn1 u * (c_pair (c_len u) (c_pair x (c_snd (u - 1))) + 1)) \\ PrimRec2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nc_cons = (\\x u. sgn2 u * (c_pair 0 x + 1) + sgn1 u * (c_pair (c_len u) (c_pair x (c_snd (u - 1))) + 1))\n(\\x u. sgn2 u * (c_pair 0 x + 1) + sgn1 u * (c_pair (c_len u) (c_pair x (c_snd (u - 1))) + 1)) \\ PrimRec2\n\ngoal (1 subgoal):\n 1. c_cons \\ PrimRec2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nc_cons \\ PrimRec2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1201, "file": "Recursion-Theory-I_PRecList", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677545357568, "lm_q2_score": 0.8354835289107307, "lm_q1q2_score": 0.7084630919620424}} {"text": "[STATEMENT]\ntheorem Right_Cosets_cardinality:\n \"\\ x \\ G; y \\ G \\ \\ card (H |\\ x) = card (H |\\ y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x \\ G; y \\ G\\ \\ card (H |\\ x) = card (H |\\ y)\n[PROOF STEP]\nby (fast intro: bij_betw_same_card Right_Coset_bij)", "meta": {"llama_tokens": 158, "file": "Jacobson_Basic_Algebra_Group_Theory", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.899121388082479, "lm_q2_score": 0.7879311981328135, "lm_q1q2_score": 0.7084457925786661}} {"text": "[STATEMENT]\nlemma finprod_reindex_bij_betw: \"bij_betw h S T \n \\ g \\ h ` S \\ carrier G \n \\ finprod G (\\x. g (h x)) S = finprod G g T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\bij_betw h S T; g \\ h ` S \\ carrier G\\ \\ (\\x\\S. g (h x)) = finprod G g T\n[PROOF STEP]\nusing finprod_reindex[of g h S]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\g \\ h ` S \\ carrier G; inj_on h S\\ \\ finprod G g (h ` S) = (\\x\\S. g (h x))\n\ngoal (1 subgoal):\n 1. \\bij_betw h S T; g \\ h ` S \\ carrier G\\ \\ (\\x\\S. g (h x)) = finprod G g T\n[PROOF STEP]\nunfolding bij_betw_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\g \\ h ` S \\ carrier G; inj_on h S\\ \\ finprod G g (h ` S) = (\\x\\S. g (h x))\n\ngoal (1 subgoal):\n 1. \\inj_on h S \\ h ` S = T; g \\ h ` S \\ carrier G\\ \\ (\\x\\S. g (h x)) = finprod G g T\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 491, "file": "Jordan_Normal_Form_Missing_Ring", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772384450967, "lm_q2_score": 0.8104789155369047, "lm_q1q2_score": 0.7084211723104744}} {"text": "[STATEMENT]\nlemma irreducible_monic_factor: fixes p :: \"'a :: field poly\" \n assumes \"degree p > 0\" \n shows \"\\ q r. irreducible q \\ p = q * r \\ monic q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\q r. irreducible q \\ p = q * r \\ monic q\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\q r. irreducible q \\ p = q * r \\ monic q\n[PROOF STEP]\nfrom irreducible\\<^sub>d_factorization_exists[OF assms]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\fs. fs \\ [] \\ (\\f\\set fs. irreducible\\<^sub>d f \\ degree f \\ degree p) \\ p = prod_list fs\n\\ irreducible\\<^sub>d p \\ \\fs. 1 < length fs \\ (\\f\\set fs. irreducible\\<^sub>d f \\ degree f < degree p) \\ p = prod_list fs\n[PROOF STEP]\nobtain fs where \"fs \\ []\" and \"set fs \\ Collect irreducible\" and \"p = prod_list fs\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\fs. fs \\ [] \\ (\\f\\set fs. irreducible\\<^sub>d f \\ degree f \\ degree p) \\ p = prod_list fs\n\\ irreducible\\<^sub>d p \\ \\fs. 1 < length fs \\ (\\f\\set fs. irreducible\\<^sub>d f \\ degree f < degree p) \\ p = prod_list fs\n\ngoal (1 subgoal):\n 1. (\\fs. \\fs \\ []; set fs \\ Collect irreducible; p = prod_list fs\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfs \\ []\nset fs \\ Collect irreducible\np = prod_list fs\n\ngoal (1 subgoal):\n 1. \\q r. irreducible q \\ p = q * r \\ monic q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfs \\ []\nset fs \\ Collect irreducible\np = prod_list fs\n[PROOF STEP]\nhave q: \"irreducible (hd fs)\" and p: \"p = hd fs * prod_list (tl fs)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfs \\ []\nset fs \\ Collect irreducible\np = prod_list fs\n\ngoal (1 subgoal):\n 1. irreducible (hd fs) &&& p = hd fs * prod_list (tl fs)\n[PROOF STEP]\nby (atomize(full), cases fs, auto)\n[PROOF STATE]\nproof (state)\nthis:\nirreducible (hd fs)\np = hd fs * prod_list (tl fs)\n\ngoal (1 subgoal):\n 1. \\q r. irreducible q \\ p = q * r \\ monic q\n[PROOF STEP]\ndefine c where \"c = coeff (hd fs) (degree (hd fs))\"\n[PROOF STATE]\nproof (state)\nthis:\nc = lead_coeff (hd fs)\n\ngoal (1 subgoal):\n 1. \\q r. irreducible q \\ p = q * r \\ monic q\n[PROOF STEP]\nfrom q\n[PROOF STATE]\nproof (chain)\npicking this:\nirreducible (hd fs)\n[PROOF STEP]\nhave c: \"c \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nirreducible (hd fs)\n\ngoal (1 subgoal):\n 1. c \\ (0::'a)\n[PROOF STEP]\nunfolding c_def irreducible\\<^sub>d_def\n[PROOF STATE]\nproof (prove)\nusing this:\nirreducible (hd fs)\n\ngoal (1 subgoal):\n 1. lead_coeff (hd fs) \\ (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nc \\ (0::'a)\n\ngoal (1 subgoal):\n 1. \\q r. irreducible q \\ p = q * r \\ monic q\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\q r. irreducible q \\ p = q * r \\ monic q\n[PROOF STEP]\nby (rule exI[of _ \"smult (1/c) (hd fs)\"], rule exI[of _ \"smult c (prod_list (tl fs))\"], unfold p,\n insert q c, auto simp: c_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\q r. irreducible q \\ p = q * r \\ monic q\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1417, "file": "Polynomial_Interpolation_Missing_Polynomial", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772450055545, "lm_q2_score": 0.8104789086703225, "lm_q1q2_score": 0.7084211716256639}} {"text": "[STATEMENT]\nlemma ln_gt_1:\n assumes \"x > (3 :: real)\"\n shows \"ln x > 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 < ln x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 1 < ln x\n[PROOF STEP]\nhave \"x > exp 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp 1 < x\n[PROOF STEP]\nusing exp_le assms\n[PROOF STATE]\nproof (prove)\nusing this:\nexp 1 \\ 3\n3 < x\n\ngoal (1 subgoal):\n 1. exp 1 < x\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\nexp 1 < x\n\ngoal (1 subgoal):\n 1. 1 < ln x\n[PROOF STEP]\nhence \"ln x > ln (exp 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nexp 1 < x\n\ngoal (1 subgoal):\n 1. ln (exp 1) < ln x\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nexp 1 < x\n3 < x\n\ngoal (1 subgoal):\n 1. ln (exp 1) < ln x\n[PROOF STEP]\nby (subst ln_less_cancel_iff) auto\n[PROOF STATE]\nproof (state)\nthis:\nln (exp 1) < ln x\n\ngoal (1 subgoal):\n 1. 1 < ln x\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nln (exp 1) < ln x\n\ngoal (1 subgoal):\n 1. 1 < ln x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 < ln x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 572, "file": "Prime_Distribution_Elementary_PNT_Consequences", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772417253256, "lm_q2_score": 0.8104789086703225, "lm_q1q2_score": 0.7084211689671076}} {"text": "[STATEMENT]\ntheorem integer_partitions_enum_correct:\n \"set (map mset (integer_partitions_enum n)) = integer_partitions n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (map mset (integer_partitions_enum n)) = integer_partitions n\n[PROOF STEP]\nproof(standard)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. set (map mset (integer_partitions_enum n)) \\ integer_partitions n\n 2. integer_partitions n \\ set (map mset (integer_partitions_enum n))\n[PROOF STEP]\nhave \"\\xs \\ set (integer_partitions_enum_aux n n)\\ \\ \\\\<^sub># (mset xs) = n\" for xs\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. xs \\ set (integer_partitions_enum_aux n n) \\ \\\\<^sub># (mset xs) = n\n[PROOF STEP]\nby (simp add: integer_partitions_enum_aux_sum sum_mset_sum_list)\n[PROOF STATE]\nproof (state)\nthis:\n?xs \\ set (integer_partitions_enum_aux n n) \\ \\\\<^sub># (mset ?xs) = n\n\ngoal (2 subgoals):\n 1. set (map mset (integer_partitions_enum n)) \\ integer_partitions n\n 2. integer_partitions n \\ set (map mset (integer_partitions_enum n))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n?xs \\ set (integer_partitions_enum_aux n n) \\ \\\\<^sub># (mset ?xs) = n\n\ngoal (2 subgoals):\n 1. set (map mset (integer_partitions_enum n)) \\ integer_partitions n\n 2. integer_partitions n \\ set (map mset (integer_partitions_enum n))\n[PROOF STEP]\nhave \"xs \\ set (integer_partitions_enum_aux n n) \\ 0 \\# mset xs\" for xs\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. xs \\ set (integer_partitions_enum_aux n n) \\ 0 \\# mset xs\n[PROOF STEP]\nusing integer_partitions_enum_aux_not_null\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?xs \\ set (integer_partitions_enum_aux ?n ?m); ?x \\ set ?xs\\ \\ ?x \\ 0\n\ngoal (1 subgoal):\n 1. xs \\ set (integer_partitions_enum_aux n n) \\ 0 \\# mset xs\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n?xs \\ set (integer_partitions_enum_aux n n) \\ 0 \\# mset ?xs\n\ngoal (2 subgoals):\n 1. set (map mset (integer_partitions_enum n)) \\ integer_partitions n\n 2. integer_partitions n \\ set (map mset (integer_partitions_enum n))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n?xs \\ set (integer_partitions_enum_aux n n) \\ \\\\<^sub># (mset ?xs) = n\n?xs \\ set (integer_partitions_enum_aux n n) \\ 0 \\# mset ?xs\n[PROOF STEP]\nshow \"set (map mset (integer_partitions_enum n)) \\ integer_partitions n\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?xs \\ set (integer_partitions_enum_aux n n) \\ \\\\<^sub># (mset ?xs) = n\n?xs \\ set (integer_partitions_enum_aux n n) \\ 0 \\# mset ?xs\n\ngoal (1 subgoal):\n 1. set (map mset (integer_partitions_enum n)) \\ integer_partitions n\n[PROOF STEP]\nunfolding integer_partitions_def\n[PROOF STATE]\nproof (prove)\nusing this:\n?xs \\ set (integer_partitions_enum_aux n n) \\ \\\\<^sub># (mset ?xs) = n\n?xs \\ set (integer_partitions_enum_aux n n) \\ 0 \\# mset ?xs\n\ngoal (1 subgoal):\n 1. set (map mset (integer_partitions_enum n)) \\ {A. \\\\<^sub># A = n \\ 0 \\# A}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nset (map mset (integer_partitions_enum n)) \\ integer_partitions n\n\ngoal (1 subgoal):\n 1. integer_partitions n \\ set (map mset (integer_partitions_enum n))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. integer_partitions n \\ set (map mset (integer_partitions_enum n))\n[PROOF STEP]\nhave \"0 \\# A \\ A \\ mset ` set (integer_partitions_enum_aux (\\\\<^sub># A) (\\\\<^sub># A))\" for A\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\# A \\ A \\ mset ` set (integer_partitions_enum_aux (\\\\<^sub># A) (\\\\<^sub># A))\n[PROOF STEP]\nunfolding image_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\# A \\ A \\ {y. \\x\\set (integer_partitions_enum_aux (\\\\<^sub># A) (\\\\<^sub># A)). y = mset x}\n[PROOF STEP]\nusing integer_partitions_enum_correct_aux1\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 \\# ?A; \\x\\#?A. x \\ ?m\\ \\ \\xs\\set (integer_partitions_enum_aux (\\\\<^sub># ?A) ?m). ?A = mset xs\n\ngoal (1 subgoal):\n 1. 0 \\# A \\ A \\ {y. \\x\\set (integer_partitions_enum_aux (\\\\<^sub># A) (\\\\<^sub># A)). y = mset x}\n[PROOF STEP]\nby (simp add: sum_mset.remove)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\# ?A \\ ?A \\ mset ` set (integer_partitions_enum_aux (\\\\<^sub># ?A) (\\\\<^sub># ?A))\n\ngoal (1 subgoal):\n 1. integer_partitions n \\ set (map mset (integer_partitions_enum n))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\# ?A \\ ?A \\ mset ` set (integer_partitions_enum_aux (\\\\<^sub># ?A) (\\\\<^sub># ?A))\n[PROOF STEP]\nshow \"integer_partitions n \\ set (map mset (integer_partitions_enum n))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\# ?A \\ ?A \\ mset ` set (integer_partitions_enum_aux (\\\\<^sub># ?A) (\\\\<^sub># ?A))\n\ngoal (1 subgoal):\n 1. integer_partitions n \\ set (map mset (integer_partitions_enum n))\n[PROOF STEP]\nunfolding integer_partitions_def\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\# ?A \\ ?A \\ mset ` set (integer_partitions_enum_aux (\\\\<^sub># ?A) (\\\\<^sub># ?A))\n\ngoal (1 subgoal):\n 1. {A. \\\\<^sub># A = n \\ 0 \\# A} \\ set (map mset (integer_partitions_enum n))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninteger_partitions n \\ set (map mset (integer_partitions_enum n))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2361, "file": "Combinatorial_Enumeration_Algorithms_Integer_Partitions", "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772482857831, "lm_q2_score": 0.8104788995148791, "lm_q1q2_score": 0.7084211662816552}} {"text": "[STATEMENT]\nlemma adjoint_mat_incr:\n \"adjoint (mat_incr n) = mat n n (\\(i,j). if j = 0 then (if i = n - 1 then 1 else 0) else (if j = i + 1 then 1 else 0))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. adjoint (mat_incr n) = mat n n (\\(i, j). if j = 0 then if i = n - 1 then 1 else 0 else if j = i + 1 then 1 else 0)\n[PROOF STEP]\napply (rule eq_matI)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (mat n n (\\(i, j). if j = 0 then if i = n - 1 then 1 else 0 else if j = i + 1 then 1 else 0)); j < dim_col (mat n n (\\(i, j). if j = 0 then if i = n - 1 then 1 else 0 else if j = i + 1 then 1 else 0))\\ \\ adjoint (mat_incr n) $$ (i, j) = mat n n (\\(i, j). if j = 0 then if i = n - 1 then 1 else 0 else if j = i + 1 then 1 else 0) $$ (i, j)\n 2. dim_row (adjoint (mat_incr n)) = dim_row (mat n n (\\(i, j). if j = 0 then if i = n - 1 then 1 else 0 else if j = i + 1 then 1 else 0))\n 3. dim_col (adjoint (mat_incr n)) = dim_col (mat n n (\\(i, j). if j = 0 then if i = n - 1 then 1 else 0 else if j = i + 1 then 1 else 0))\n[PROOF STEP]\nunfolding mat_incr_def\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (mat n n (\\(i, j). if j = 0 then if i = n - 1 then 1 else 0 else if j = i + 1 then 1 else 0)); j < dim_col (mat n n (\\(i, j). if j = 0 then if i = n - 1 then 1 else 0 else if j = i + 1 then 1 else 0))\\ \\ adjoint (mat n n (\\(i, j). if i = 0 then if j = n - 1 then 1 else 0 else if i = j + 1 then 1 else 0)) $$ (i, j) = mat n n (\\(i, j). if j = 0 then if i = n - 1 then 1 else 0 else if j = i + 1 then 1 else 0) $$ (i, j)\n 2. dim_row (adjoint (mat n n (\\(i, j). if i = 0 then if j = n - 1 then 1 else 0 else if i = j + 1 then 1 else 0))) = dim_row (mat n n (\\(i, j). if j = 0 then if i = n - 1 then 1 else 0 else if j = i + 1 then 1 else 0))\n 3. dim_col (adjoint (mat n n (\\(i, j). if i = 0 then if j = n - 1 then 1 else 0 else if i = j + 1 then 1 else 0))) = dim_col (mat n n (\\(i, j). if j = 0 then if i = n - 1 then 1 else 0 else if j = i + 1 then 1 else 0))\n[PROOF STEP]\nby (auto simp add: adjoint_eval)", "meta": {"llama_tokens": 1006, "file": "QHLProver_Gates", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772286044095, "lm_q2_score": 0.810478913248044, "lm_q1q2_score": 0.708421162334164}} {"text": "[STATEMENT]\nlemma coprime_crossproduct_nat:\n fixes a b c d :: nat\n assumes \"coprime a d\" and \"coprime b c\"\n shows \"a * c = b * d \\ a = b \\ c = d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a * c = b * d) = (a = b \\ c = d)\n[PROOF STEP]\nusing assms coprime_crossproduct [of a d b c]\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime a d\ncoprime b c\n\\coprime a d; coprime b c\\ \\ (normalize a * normalize c = normalize b * normalize d) = (normalize a = normalize b \\ normalize c = normalize d)\n\ngoal (1 subgoal):\n 1. (a * c = b * d) = (a = b \\ c = d)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 273, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772286044095, "lm_q2_score": 0.8104789063814617, "lm_q1q2_score": 0.7084211563322407}} {"text": "[STATEMENT]\nlemma nat_Least_mono: \"\n \\ A \\ {}; mono (f::(nat\\nat)) \\ \\\n (LEAST x. x \\ f ` A) = f (LEAST x. x \\ A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ {}; incseq f\\ \\ (LEAST x. x \\ f ` A) = f (LEAST x. x \\ A)\n[PROOF STEP]\nunfolding iMin_def[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ {}; incseq f\\ \\ iMin (f ` A) = f (iMin A)\n[PROOF STEP]\nby (blast intro: iMin_mono2)", "meta": {"llama_tokens": 249, "file": "List-Infinite_CommonSet_SetInterval2", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942377652497, "lm_q2_score": 0.7956581024858786, "lm_q1q2_score": 0.7083698238744103}} {"text": "[STATEMENT]\ntheorem card_permutations_of_multiset:\n \"card (permutations_of_multiset A) = fact (size A) div (\\x\\set_mset A. fact (count A x))\"\n \"(\\x\\set_mset A. fact (count A x) :: nat) dvd fact (size A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (permutations_of_multiset A) = fact (size A) div (\\x\\set_mset A. fact (count A x)) &&& (\\x\\set_mset A. fact (count A x)) dvd fact (size A)\n[PROOF STEP]\nby (simp_all flip: card_permutations_of_multiset_aux[of A])", "meta": {"llama_tokens": 223, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942203004187, "lm_q2_score": 0.7956581024858786, "lm_q1q2_score": 0.708369809978376}} {"text": "[STATEMENT]\nlemma polytope_imp_compact:\n fixes S :: \"'a::real_normed_vector set\"\n shows \"polytope S \\ compact S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. polytope S \\ compact S\n[PROOF STEP]\nby (metis finite_imp_compact_convex_hull polytope_def)", "meta": {"llama_tokens": 109, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.7956581000631542, "lm_q1q2_score": 0.7083698078214383}} {"text": "[STATEMENT]\nlemma csqrt_2_sq:\n \"complex_of_real (sqrt 2) * complex_of_real (sqrt 2) = 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (sqrt 2) * complex_of_real (sqrt 2) = 2\n[PROOF STEP]\nby (smt of_real_add of_real_hom.hom_one of_real_power one_add_one power2_eq_square real_sqrt_pow2)", "meta": {"llama_tokens": 139, "file": "QHLProver_Gates", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.7956581000631542, "lm_q1q2_score": 0.7083698078214383}} {"text": "[STATEMENT]\nlemma triangle_num_gt_0_iff [simp]: \"triangle_num n > 0 \\ n > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0 < triangle_num n) = (0 < n)\n[PROOF STEP]\nusing triangle_num_eq_0_iff[of n]\n[PROOF STATE]\nproof (prove)\nusing this:\n(triangle_num n = 0) = (n = 0)\n\ngoal (1 subgoal):\n 1. (0 < triangle_num n) = (0 < n)\n[PROOF STEP]\nby linarith", "meta": {"llama_tokens": 169, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615382165412809, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7083473609231655}} {"text": "[STATEMENT]\nlemma totient_formula1:\n assumes \"n > 0\"\n shows \"totient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. totient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. totient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < n\n[PROOF STEP]\nhave \"n = (\\p\\prime_factors n. p ^ multiplicity p n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\n\ngoal (1 subgoal):\n 1. n = (\\p\\prime_factors n. p ^ multiplicity p n)\n[PROOF STEP]\nby (rule prime_factorization_nat)\n[PROOF STATE]\nproof (state)\nthis:\nn = (\\p\\prime_factors n. p ^ multiplicity p n)\n\ngoal (1 subgoal):\n 1. totient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nn = (\\p\\prime_factors n. p ^ multiplicity p n)\n\ngoal (1 subgoal):\n 1. totient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n[PROOF STEP]\nhave \"totient \\ = (\\x\\prime_factors n. totient (x ^ multiplicity x n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. totient (\\p\\prime_factors n. p ^ multiplicity p n) = (\\x\\prime_factors n. totient (x ^ multiplicity x n))\n[PROOF STEP]\nproof (rule totient_prod_coprime)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. pairwise coprime ((\\p. p ^ multiplicity p n) ` prime_factors n)\n 2. inj_on (\\p. p ^ multiplicity p n) (prime_factors n)\n[PROOF STEP]\nshow \"pairwise coprime ((\\p. p ^ multiplicity p n) ` prime_factors n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pairwise coprime ((\\p. p ^ multiplicity p n) ` prime_factors n)\n[PROOF STEP]\nproof (rule pairwiseI, clarify)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x y p pa. \\p ^ multiplicity p n \\ pa ^ multiplicity pa n; p \\# prime_factorization n; pa \\# prime_factorization n\\ \\ coprime (p ^ multiplicity p n) (pa ^ multiplicity pa n)\n[PROOF STEP]\nfix p q\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x y p pa. \\p ^ multiplicity p n \\ pa ^ multiplicity pa n; p \\# prime_factorization n; pa \\# prime_factorization n\\ \\ coprime (p ^ multiplicity p n) (pa ^ multiplicity pa n)\n[PROOF STEP]\nassume *: \"p \\# prime_factorization n\" \"q \\# prime_factorization n\" \n \"p ^ multiplicity p n \\ q ^ multiplicity q n\"\n[PROOF STATE]\nproof (state)\nthis:\np \\# prime_factorization n\nq \\# prime_factorization n\np ^ multiplicity p n \\ q ^ multiplicity q n\n\ngoal (1 subgoal):\n 1. \\x y p pa. \\p ^ multiplicity p n \\ pa ^ multiplicity pa n; p \\# prime_factorization n; pa \\# prime_factorization n\\ \\ coprime (p ^ multiplicity p n) (pa ^ multiplicity pa n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\np \\# prime_factorization n\nq \\# prime_factorization n\np ^ multiplicity p n \\ q ^ multiplicity q n\n[PROOF STEP]\nhave \"multiplicity p n > 0\" \"multiplicity q n > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\np \\# prime_factorization n\nq \\# prime_factorization n\np ^ multiplicity p n \\ q ^ multiplicity q n\n\ngoal (1 subgoal):\n 1. 0 < multiplicity p n &&& 0 < multiplicity q n\n[PROOF STEP]\nby (simp_all add: prime_factors_multiplicity)\n[PROOF STATE]\nproof (state)\nthis:\n0 < multiplicity p n\n0 < multiplicity q n\n\ngoal (1 subgoal):\n 1. \\x y p pa. \\p ^ multiplicity p n \\ pa ^ multiplicity pa n; p \\# prime_factorization n; pa \\# prime_factorization n\\ \\ coprime (p ^ multiplicity p n) (pa ^ multiplicity pa n)\n[PROOF STEP]\nwith * primes_coprime [of p q]\n[PROOF STATE]\nproof (chain)\npicking this:\np \\# prime_factorization n\nq \\# prime_factorization n\np ^ multiplicity p n \\ q ^ multiplicity q n\n\\prime p; prime q; p \\ q\\ \\ coprime p q\n0 < multiplicity p n\n0 < multiplicity q n\n[PROOF STEP]\nshow \"coprime (p ^ multiplicity p n) (q ^ multiplicity q n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\np \\# prime_factorization n\nq \\# prime_factorization n\np ^ multiplicity p n \\ q ^ multiplicity q n\n\\prime p; prime q; p \\ q\\ \\ coprime p q\n0 < multiplicity p n\n0 < multiplicity q n\n\ngoal (1 subgoal):\n 1. coprime (p ^ multiplicity p n) (q ^ multiplicity q n)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncoprime (p ^ multiplicity p n) (q ^ multiplicity q n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\npairwise coprime ((\\p. p ^ multiplicity p n) ` prime_factors n)\n\ngoal (1 subgoal):\n 1. inj_on (\\p. p ^ multiplicity p n) (prime_factors n)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. inj_on (\\p. p ^ multiplicity p n) (prime_factors n)\n[PROOF STEP]\nshow \"inj_on (\\p. p ^ multiplicity p n) (prime_factors n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on (\\p. p ^ multiplicity p n) (prime_factors n)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x y. \\x \\# prime_factorization n; y \\# prime_factorization n; x ^ multiplicity x n = y ^ multiplicity y n\\ \\ x = y\n[PROOF STEP]\nfix p q\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x y. \\x \\# prime_factorization n; y \\# prime_factorization n; x ^ multiplicity x n = y ^ multiplicity y n\\ \\ x = y\n[PROOF STEP]\nassume pq: \"p \\# prime_factorization n\" \"q \\# prime_factorization n\" \n \"p ^ multiplicity p n = q ^ multiplicity q n\"\n[PROOF STATE]\nproof (state)\nthis:\np \\# prime_factorization n\nq \\# prime_factorization n\np ^ multiplicity p n = q ^ multiplicity q n\n\ngoal (1 subgoal):\n 1. \\x y. \\x \\# prime_factorization n; y \\# prime_factorization n; x ^ multiplicity x n = y ^ multiplicity y n\\ \\ x = y\n[PROOF STEP]\nfrom assms and pq\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < n\np \\# prime_factorization n\nq \\# prime_factorization n\np ^ multiplicity p n = q ^ multiplicity q n\n[PROOF STEP]\nhave \"prime p\" \"prime q\" \"multiplicity p n > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\np \\# prime_factorization n\nq \\# prime_factorization n\np ^ multiplicity p n = q ^ multiplicity q n\n\ngoal (1 subgoal):\n 1. prime p &&& prime q &&& 0 < multiplicity p n\n[PROOF STEP]\nby (simp_all add: prime_factors_multiplicity)\n[PROOF STATE]\nproof (state)\nthis:\nprime p\nprime q\n0 < multiplicity p n\n\ngoal (1 subgoal):\n 1. \\x y. \\x \\# prime_factorization n; y \\# prime_factorization n; x ^ multiplicity x n = y ^ multiplicity y n\\ \\ x = y\n[PROOF STEP]\nfrom prime_power_eq_imp_eq[OF this pq(3)]\n[PROOF STATE]\nproof (chain)\npicking this:\np = q\n[PROOF STEP]\nshow \"p = q\"\n[PROOF STATE]\nproof (prove)\nusing this:\np = q\n\ngoal (1 subgoal):\n 1. p = q\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\np = q\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\p. p ^ multiplicity p n) (prime_factors n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ntotient (\\p\\prime_factors n. p ^ multiplicity p n) = (\\x\\prime_factors n. totient (x ^ multiplicity x n))\n\ngoal (1 subgoal):\n 1. totient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ntotient (\\p\\prime_factors n. p ^ multiplicity p n) = (\\x\\prime_factors n. totient (x ^ multiplicity x n))\n\ngoal (1 subgoal):\n 1. totient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n[PROOF STEP]\nhave \"\\ = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\prime_factors n. totient (x ^ multiplicity x n)) = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n[PROOF STEP]\nby (intro prod.cong refl totient_prime_power) (auto simp: prime_factors_multiplicity)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\prime_factors n. totient (x ^ multiplicity x n)) = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n\ngoal (1 subgoal):\n 1. totient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ntotient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntotient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n\ngoal (1 subgoal):\n 1. totient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ntotient n = (\\p\\prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3791, "file": null, "length": 38, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382058759129, "lm_q2_score": 0.8221891392358015, "lm_q1q2_score": 0.7083473559078736}} {"text": "[STATEMENT]\nlemma max_reg_list_append:\n \"max_reg_list (as@bs) = max (max_reg_list as) (max_reg_list bs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. max_reg_list (as @ bs) = max (max_reg_list as) (max_reg_list bs)\n[PROOF STEP]\nproof(induct bs rule: rev_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. max_reg_list (as @ []) = max (max_reg_list as) (max_reg_list [])\n 2. \\x xs. max_reg_list (as @ xs) = max (max_reg_list as) (max_reg_list xs) \\ max_reg_list (as @ xs @ [x]) = max (max_reg_list as) (max_reg_list (xs @ [x]))\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. max_reg_list (as @ []) = max (max_reg_list as) (max_reg_list [])\n 2. \\x xs. max_reg_list (as @ xs) = max (max_reg_list as) (max_reg_list xs) \\ max_reg_list (as @ xs @ [x]) = max (max_reg_list as) (max_reg_list (xs @ [x]))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. max_reg_list (as @ []) = max (max_reg_list as) (max_reg_list [])\n[PROOF STEP]\nby (metis append_Nil2 fold_simps(1) list.simps(8) max_reg_list_def sup_None_2 sup_max)\n[PROOF STATE]\nproof (state)\nthis:\nmax_reg_list (as @ []) = max (max_reg_list as) (max_reg_list [])\n\ngoal (1 subgoal):\n 1. \\x xs. max_reg_list (as @ xs) = max (max_reg_list as) (max_reg_list xs) \\ max_reg_list (as @ xs @ [x]) = max (max_reg_list as) (max_reg_list (xs @ [x]))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x xs. max_reg_list (as @ xs) = max (max_reg_list as) (max_reg_list xs) \\ max_reg_list (as @ xs @ [x]) = max (max_reg_list as) (max_reg_list (xs @ [x]))\n[PROOF STEP]\ncase (snoc x xs)\n[PROOF STATE]\nproof (state)\nthis:\nmax_reg_list (as @ xs) = max (max_reg_list as) (max_reg_list xs)\n\ngoal (1 subgoal):\n 1. \\x xs. max_reg_list (as @ xs) = max (max_reg_list as) (max_reg_list xs) \\ max_reg_list (as @ xs @ [x]) = max (max_reg_list as) (max_reg_list (xs @ [x]))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nmax_reg_list (as @ xs) = max (max_reg_list as) (max_reg_list xs)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nmax_reg_list (as @ xs) = max (max_reg_list as) (max_reg_list xs)\n\ngoal (1 subgoal):\n 1. max_reg_list (as @ xs @ [x]) = max (max_reg_list as) (max_reg_list (xs @ [x]))\n[PROOF STEP]\nby (metis append_assoc max.assoc max_reg_list_append_singleton)\n[PROOF STATE]\nproof (state)\nthis:\nmax_reg_list (as @ xs @ [x]) = max (max_reg_list as) (max_reg_list (xs @ [x]))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1169, "file": "Extended_Finite_State_Machines_GExp", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615381952105441, "lm_q2_score": 0.8221891392358014, "lm_q1q2_score": 0.7083473471389231}} {"text": "[STATEMENT]\nlemma vangle_eq_zeroD: \"vangle u v = 0 \\ (\\k. v = k *\\<^sub>R u)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vangle u v = 0 \\ \\k. v = k *\\<^sub>R u\n[PROOF STEP]\napply (auto simp: vangle_def split: if_splits)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\arccos (u \\ v / (norm u * norm v)) = 0; u \\ (0::'a); v \\ (0::'a)\\ \\ \\k. v = k *\\<^sub>R u\n[PROOF STEP]\napply (subst (asm) arccos_eq_zero_iff)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\u \\ (0::'a); v \\ (0::'a)\\ \\ - 1 \\ u \\ v / (norm u * norm v)\n 2. \\u \\ (0::'a); v \\ (0::'a)\\ \\ u \\ v / (norm u * norm v) \\ 1\n 3. \\u \\ v / (norm u * norm v) = 1; u \\ (0::'a); v \\ (0::'a)\\ \\ \\k. v = k *\\<^sub>R u\n[PROOF STEP]\napply (auto simp: divide_simps mult_less_0_iff split: if_splits)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\u \\ (0::'a); v \\ (0::'a)\\ \\ - (norm u * norm v) \\ u \\ v\n 2. \\u \\ (0::'a); v \\ (0::'a)\\ \\ u \\ v \\ norm u * norm v\n 3. \\u \\ v = norm u * norm v; u \\ (0::'a); v \\ (0::'a)\\ \\ \\k. v = k *\\<^sub>R u\n[PROOF STEP]\napply (metis Real_Vector_Spaces.norm_minus_cancel inner_minus_left minus_le_iff norm_cauchy_schwarz)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\u \\ (0::'a); v \\ (0::'a)\\ \\ u \\ v \\ norm u * norm v\n 2. \\u \\ v = norm u * norm v; u \\ (0::'a); v \\ (0::'a)\\ \\ \\k. v = k *\\<^sub>R u\n[PROOF STEP]\napply (metis norm_cauchy_schwarz)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\u \\ v = norm u * norm v; u \\ (0::'a); v \\ (0::'a)\\ \\ \\k. v = k *\\<^sub>R u\n[PROOF STEP]\nby (metis Cauchy_Schwarz_eq2_iff abs_of_pos inner_commute mult.commute mult_sign_intros(5) zero_less_norm_iff)", "meta": {"llama_tokens": 1009, "file": "Ordinary_Differential_Equations_IVP_Cones", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711870587667, "lm_q2_score": 0.8333245953120233, "lm_q1q2_score": 0.7083018954826268}} {"text": "[STATEMENT]\nlemma card_of_Times_mono[simp]:\nassumes \"|A| \\o |B|\" and \"|C| \\o |D|\"\nshows \"|A \\ C| \\o |B \\ D|\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. |A \\ C| \\o |B \\ D|\n[PROOF STEP]\nusing assms card_of_Times_mono1[of A B C] card_of_Times_mono2[of C D B]\n ordLeq_transitive[of \"|A \\ C|\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n|A| \\o |B|\n|C| \\o |D|\n|A| \\o |B| \\ |A \\ C| \\o |B \\ C|\n|C| \\o |D| \\ |B \\ C| \\o |B \\ D|\n\\|A \\ C| \\o ?r'; ?r' \\o ?r''\\ \\ |A \\ C| \\o ?r''\n\ngoal (1 subgoal):\n 1. |A \\ C| \\o |B \\ D|\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 368, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8499711604559846, "lm_q2_score": 0.8333245911726382, "lm_q1q2_score": 0.7083018697955163}} {"text": "[STATEMENT]\nlemma bounded_clinear_equal_ket:\n fixes f g :: \\'a ell2 \\ _\\\n assumes \\bounded_clinear f\\\n assumes \\bounded_clinear g\\\n assumes \\\\i. f (ket i) = g (ket i)\\\n shows \\f = g\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f = g\n[PROOF STEP]\napply (rule ext)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. f x = g x\n[PROOF STEP]\napply (rule bounded_clinear_eq_on[of f g \\range ket\\])\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. \\x. bounded_clinear f\n 2. \\x. bounded_clinear g\n 3. \\x xa. xa \\ range ket \\ f xa = g xa\n 4. \\x. x \\ closure (cspan (range ket))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nbounded_clinear f\nbounded_clinear g\nf (ket ?i) = g (ket ?i)\n\ngoal (4 subgoals):\n 1. \\x. bounded_clinear f\n 2. \\x. bounded_clinear g\n 3. \\x xa. xa \\ range ket \\ f xa = g xa\n 4. \\x. x \\ closure (cspan (range ket))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 471, "file": "Complex_Bounded_Operators_Complex_L2", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267864276108, "lm_q2_score": 0.8152324915965392, "lm_q1q2_score": 0.7082958258651953}} {"text": "[STATEMENT]\nlemma transpose_minus: \"A \\ carrier_mat nr nc \\ B \\ carrier_mat nr nc\n \\ transpose_mat (A - B) = transpose_mat A - transpose_mat B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ carrier_mat nr nc; B \\ carrier_mat nr nc\\ \\ transpose_mat (A - B) = transpose_mat A - transpose_mat B\n[PROOF STEP]\nby (intro eq_matI, auto)", "meta": {"llama_tokens": 164, "file": "Jordan_Normal_Form_Matrix", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267762381843, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7082958195085596}} {"text": "[STATEMENT]\nlemma inner_prod_mult_mat_vec_left:\n assumes \"v \\ carrier_vec n\"\n and \"w \\ carrier_vec n'\"\n and \"A \\ carrier_mat m n\"\n and \"B \\ carrier_mat m n'\"\nshows \"inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = \n inner_prod (((Complex_Matrix.adjoint B) * A) *\\<^sub>v v) w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n[PROOF STEP]\nhave \"inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = \n inner_prod (Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v)) w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v)) w\n[PROOF STEP]\nusing adjoint_def_alter\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?v \\ carrier_vec ?n; ?w \\ carrier_vec ?m; ?A \\ carrier_mat ?n ?m\\ \\ Complex_Matrix.inner_prod ?v (?A *\\<^sub>v ?w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint ?A *\\<^sub>v ?v) ?w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v)) w\n[PROOF STEP]\nby (metis assms mult_mat_vec_carrier)\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v)) w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v)) w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n[PROOF STEP]\nhave \"... = inner_prod (((Complex_Matrix.adjoint B) * A) *\\<^sub>v v) w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v)) w = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v)) w = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n[PROOF STEP]\nhave \"Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v) = \n ((Complex_Matrix.adjoint B) * A) *\\<^sub>v v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v) = Complex_Matrix.adjoint B * A *\\<^sub>v v\n[PROOF STEP]\nproof (rule assoc_mult_mat_vec[symmetric], (auto simp add: assms))\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. A \\ carrier_mat (dim_row B) ?n\\<^sub>3\n 2. v \\ carrier_vec ?n\\<^sub>3\n[PROOF STEP]\nshow \"v \\ carrier_vec n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v \\ carrier_vec n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ carrier_vec n\nw \\ carrier_vec n'\nA \\ carrier_mat m n\nB \\ carrier_mat m n'\n\ngoal (1 subgoal):\n 1. v \\ carrier_vec n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nv \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. A \\ carrier_mat (dim_row B) n\n[PROOF STEP]\nshow \"A \\ carrier_mat (dim_row B) n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A \\ carrier_mat (dim_row B) n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ carrier_vec n\nw \\ carrier_vec n'\nA \\ carrier_mat m n\nB \\ carrier_mat m n'\n\ngoal (1 subgoal):\n 1. A \\ carrier_mat (dim_row B) n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA \\ carrier_mat (dim_row B) n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v) = Complex_Matrix.adjoint B * A *\\<^sub>v v\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v)) w = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v) = Complex_Matrix.adjoint B * A *\\<^sub>v v\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v)) w = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v)) w = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (Complex_Matrix.adjoint B *\\<^sub>v (A *\\<^sub>v v)) w = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nComplex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (A *\\<^sub>v v) (B *\\<^sub>v w) = Complex_Matrix.inner_prod (Complex_Matrix.adjoint B * A *\\<^sub>v v) w\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2650, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 23, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267762381844, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.7082958136582124}} {"text": "[STATEMENT]\nlemma tan_minus_45: \"tan (-(pi/4)) = -1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tan (- (pi / 4)) = - 1\n[PROOF STEP]\nunfolding tan_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin (- (pi / 4)) / cos (- (pi / 4)) = - 1\n[PROOF STEP]\nby (simp add: sin_45 cos_45)", "meta": {"llama_tokens": 144, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.901920681802153, "lm_q2_score": 0.7853085808877581, "lm_q1q2_score": 0.708286050699368}} {"text": "[STATEMENT]\nlemma cos0_scalprod0:\n assumes \"z1 \\ 0\" and \"z2 \\ 0\"\n shows \"cos (\\ z1 z2) = 0 \\ scalprod z1 z2 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cos (\\ z1 z2) = 0) = (scalprod z1 z2 = 0)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nz1 \\ 0\nz2 \\ 0\n\ngoal (1 subgoal):\n 1. (cos (\\ z1 z2) = 0) = (scalprod z1 z2 = 0)\n[PROOF STEP]\nusing cnj_mix_real[of z1 z2]\n[PROOF STATE]\nproof (prove)\nusing this:\nz1 \\ 0\nz2 \\ 0\nis_real (cnj_mix z1 z2)\n\ngoal (1 subgoal):\n 1. (cos (\\ z1 z2) = 0) = (scalprod z1 z2 = 0)\n[PROOF STEP]\nusing cos_cmod_scalprod[of z1 z2]\n[PROOF STATE]\nproof (prove)\nusing this:\nz1 \\ 0\nz2 \\ 0\nis_real (cnj_mix z1 z2)\ncmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n\ngoal (1 subgoal):\n 1. (cos (\\ z1 z2) = 0) = (scalprod z1 z2 = 0)\n[PROOF STEP]\nby (auto simp add: complex_eq_if_Re_eq)", "meta": {"llama_tokens": 480, "file": "Complex_Geometry_Angles", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206738932334, "lm_q2_score": 0.785308580887758, "lm_q1q2_score": 0.7082860444884255}} {"text": "[STATEMENT]\nlemma invertible_mat_four_block_mat_lower_right:\n assumes A: \"(A::'a::comm_ring_1 mat) \\ carrier_mat n n\" and inv_A: \"invertible_mat A\"\n shows \"invertible_mat (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. invertible_mat (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. invertible_mat (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A)\n[PROOF STEP]\nlet ?I = \"(four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. invertible_mat (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A)\n[PROOF STEP]\nhave \"Determinant.det ?I = Determinant.det (1\\<^sub>m 1) * Determinant.det A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Determinant.det (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A) = Determinant.det (1\\<^sub>m 1) * Determinant.det A\n[PROOF STEP]\nby (rule det_four_block_mat_lower_left_zero_col, insert assms, auto)\n[PROOF STATE]\nproof (state)\nthis:\nDeterminant.det (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A) = Determinant.det (1\\<^sub>m 1) * Determinant.det A\n\ngoal (1 subgoal):\n 1. invertible_mat (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nDeterminant.det (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A) = Determinant.det (1\\<^sub>m 1) * Determinant.det A\n\ngoal (1 subgoal):\n 1. invertible_mat (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A)\n[PROOF STEP]\nhave \"... = Determinant.det A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Determinant.det (1\\<^sub>m 1) * Determinant.det A = Determinant.det A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nDeterminant.det (1\\<^sub>m 1) * Determinant.det A = Determinant.det A\n\ngoal (1 subgoal):\n 1. invertible_mat (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nDeterminant.det (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A) = Determinant.det A\n[PROOF STEP]\nhave \"Determinant.det ?I = Determinant.det A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nDeterminant.det (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A) = Determinant.det A\n\ngoal (1 subgoal):\n 1. Determinant.det (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A) = Determinant.det A\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nDeterminant.det (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A) = Determinant.det A\n\ngoal (1 subgoal):\n 1. invertible_mat (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nDeterminant.det (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A) = Determinant.det A\n\ngoal (1 subgoal):\n 1. invertible_mat (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A)\n[PROOF STEP]\nby (metis (no_types, lifting) assms carrier_matD(1) carrier_matD(2) carrier_mat_triv\n index_mat_four_block(2) index_mat_four_block(3) index_one_mat(2) index_one_mat(3)\n invertible_iff_is_unit_JNF)\n[PROOF STATE]\nproof (state)\nthis:\ninvertible_mat (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 n) (0\\<^sub>m n 1) A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1681, "file": "Smith_Normal_Form_SNF_Missing_Lemmas", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467738423873, "lm_q2_score": 0.80563219364797, "lm_q1q2_score": 0.7082689439491783}} {"text": "[STATEMENT]\nlemma closed_orbit_period_nonneg:\n assumes \"closed_orbit x\"\n shows \"period x \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ period x\n[PROOF STEP]\nunfolding period_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ Inf {T \\ existence_ivl0 x. 0 < T \\ flow0 x T = x}\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nclosed_orbit x\n\ngoal (1 subgoal):\n 1. 0 \\ Inf {T \\ existence_ivl0 x. 0 < T \\ flow0 x T = x}\n[PROOF STEP]\nunfolding closed_orbit_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\T\\existence_ivl0 x. T \\ 0 \\ flow0 x T = x\n\ngoal (1 subgoal):\n 1. 0 \\ Inf {T \\ existence_ivl0 x. 0 < T \\ flow0 x T = x}\n[PROOF STEP]\napply (auto intro!:cInf_greatest)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\T. \\T \\ existence_ivl0 x; T \\ 0; flow0 x T = x; \\xa>0. xa \\ existence_ivl0 x \\ flow0 x xa \\ x\\ \\ False\n[PROOF STEP]\nby (smt recurrence_time_flip_sign)", "meta": {"llama_tokens": 479, "file": "Poincare_Bendixson_Periodic_Orbit", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467643431002, "lm_q2_score": 0.8056321959813275, "lm_q1q2_score": 0.7082689383476105}} {"text": "[STATEMENT]\nlemma nn_integral_erlang_density:\n assumes [arith]: \"0 < l\"\n shows \"(\\\\<^sup>+ x. ennreal (erlang_density k l x) * indicator {.. a} x \\lborel) = erlang_CDF k l a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nproof (cases \"0 \\ a\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\ncase [arith]: True\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ a\n\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nhave eq: \"\\x. indicator {0..a} (x / l) = indicator {0..a*l} x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. indicator {0..a} (x / l) = indicator {0..a * l} x\n[PROOF STEP]\nby (simp add: field_simps split: split_indicator)\n[PROOF STATE]\nproof (state)\nthis:\nindicator {0..a} (?x / l) = indicator {0..a * l} ?x\n\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nhave \"(\\\\<^sup>+x. ennreal (erlang_density k l x) * indicator {.. a} x \\lborel) =\n (\\\\<^sup>+x. (l/fact k) * (ennreal ((l*x)^k * exp (- (l*x))) * indicator {0 .. a} x) \\lborel)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = \\\\<^sup>+ x. ennreal (l / fact k) * (ennreal ((l * x) ^ k * exp (- (l * x))) * indicator {0..a} x) \\lborel\n[PROOF STEP]\nby (intro nn_integral_cong)\n (auto simp: erlang_density_def power_mult_distrib ennreal_mult[symmetric] split: split_indicator)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = \\\\<^sup>+ x. ennreal (l / fact k) * (ennreal ((l * x) ^ k * exp (- (l * x))) * indicator {0..a} x) \\lborel\n\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = \\\\<^sup>+ x. ennreal (l / fact k) * (ennreal ((l * x) ^ k * exp (- (l * x))) * indicator {0..a} x) \\lborel\n\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nhave \"\\ = (l/fact k) * (\\\\<^sup>+x. ennreal ((l*x)^k * exp (- (l*x))) * indicator {0 .. a} x \\lborel)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (l / fact k) * (ennreal ((l * x) ^ k * exp (- (l * x))) * indicator {0..a} x) \\lborel = ennreal (l / fact k) * (\\\\<^sup>+x\\{0..a}. ennreal ((l * x) ^ k * exp (- (l * x)))\\lborel)\n[PROOF STEP]\nby (intro nn_integral_cmult) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal (l / fact k) * (ennreal ((l * x) ^ k * exp (- (l * x))) * indicator {0..a} x) \\lborel = ennreal (l / fact k) * (\\\\<^sup>+x\\{0..a}. ennreal ((l * x) ^ k * exp (- (l * x)))\\lborel)\n\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. ennreal (l / fact k) * (ennreal ((l * x) ^ k * exp (- (l * x))) * indicator {0..a} x) \\lborel = ennreal (l / fact k) * (\\\\<^sup>+x\\{0..a}. ennreal ((l * x) ^ k * exp (- (l * x)))\\lborel)\n\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nhave \"\\ = ennreal (l/fact k) * ((1/l) * (\\\\<^sup>+x. ennreal (x^k * exp (- x)) * indicator {0 .. l * a} x \\lborel))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ennreal (l / fact k) * (\\\\<^sup>+x\\{0..a}. ennreal ((l * x) ^ k * exp (- (l * x)))\\lborel) = ennreal (l / fact k) * (ennreal (1 / l) * (\\\\<^sup>+x\\{0..l * a}. ennreal (x ^ k * exp (- x))\\lborel))\n[PROOF STEP]\nby (subst nn_integral_real_affine[where c=\"1 / l\" and t=0]) (auto simp: field_simps eq)\n[PROOF STATE]\nproof (state)\nthis:\nennreal (l / fact k) * (\\\\<^sup>+x\\{0..a}. ennreal ((l * x) ^ k * exp (- (l * x)))\\lborel) = ennreal (l / fact k) * (ennreal (1 / l) * (\\\\<^sup>+x\\{0..l * a}. ennreal (x ^ k * exp (- x))\\lborel))\n\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nennreal (l / fact k) * (\\\\<^sup>+x\\{0..a}. ennreal ((l * x) ^ k * exp (- (l * x)))\\lborel) = ennreal (l / fact k) * (ennreal (1 / l) * (\\\\<^sup>+x\\{0..l * a}. ennreal (x ^ k * exp (- x))\\lborel))\n\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nhave \"\\ = (1 - (\\n\\k. ((l * a)^n * exp (-(l * a))) / fact n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ennreal (l / fact k) * (ennreal (1 / l) * (\\\\<^sup>+x\\{0..l * a}. ennreal (x ^ k * exp (- x))\\lborel)) = ennreal (1 - (\\n\\k. (l * a) ^ n * exp (- (l * a)) / fact n))\n[PROOF STEP]\nby (subst nn_intergal_power_times_exp_Icc) (auto simp: ennreal_mult'[symmetric])\n[PROOF STATE]\nproof (state)\nthis:\nennreal (l / fact k) * (ennreal (1 / l) * (\\\\<^sup>+x\\{0..l * a}. ennreal (x ^ k * exp (- x))\\lborel)) = ennreal (1 - (\\n\\k. (l * a) ^ n * exp (- (l * a)) / fact n))\n\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nennreal (l / fact k) * (ennreal (1 / l) * (\\\\<^sup>+x\\{0..l * a}. ennreal (x ^ k * exp (- x))\\lborel)) = ennreal (1 - (\\n\\k. (l * a) ^ n * exp (- (l * a)) / fact n))\n\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nhave \"\\ = erlang_CDF k l a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ennreal (1 - (\\n\\k. (l * a) ^ n * exp (- (l * a)) / fact n)) = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nby (auto simp: erlang_CDF_def)\n[PROOF STATE]\nproof (state)\nthis:\nennreal (1 - (\\n\\k. (l * a) ^ n * exp (- (l * a)) / fact n)) = ennreal (erlang_CDF k l a)\n\ngoal (2 subgoals):\n 1. 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n 2. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n\ngoal (1 subgoal):\n 1. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ 0 \\ a\n\ngoal (1 subgoal):\n 1. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ 0 \\ a\n[PROOF STEP]\nhave \"(\\\\<^sup>+ x. ennreal (erlang_density k l x) * indicator {.. a} x \\lborel) = (\\\\<^sup>+x. 0 \\(lborel::real measure))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 0 \\ a\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = \\\\<^sup>+ x. 0 \\lborel\n[PROOF STEP]\nby (intro nn_integral_cong) (auto simp: erlang_density_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = \\\\<^sup>+ x. 0 \\lborel\n\ngoal (1 subgoal):\n 1. \\ 0 \\ a \\ \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nwith False\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ 0 \\ a\n\\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = \\\\<^sup>+ x. 0 \\lborel\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 0 \\ a\n\\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = \\\\<^sup>+ x. 0 \\lborel\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n[PROOF STEP]\nby (simp add: erlang_CDF_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+x\\{..a}. ennreal (erlang_density k l x)\\lborel = ennreal (erlang_CDF k l a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5262, "file": null, "length": 30, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872046026642944, "lm_q2_score": 0.7981867753392728, "lm_q1q2_score": 0.708154980866774}} {"text": "[STATEMENT]\nlemma condition_2_part_1:\n fixes A::\"'a::{field}^'columns::{mod_type}^'rows::{mod_type}\" and k::nat\n defines ia:\"ia\\(if \\m. is_zero_row_upt_k m k A then 0 else to_nat (GREATEST n. \\ is_zero_row_upt_k n k A) + 1)\"\n defines B:\"B\\(snd (Gauss_Jordan_column_k (ia,A) k))\"\n assumes not_zero_i_suc_k: \"\\ is_zero_row_upt_k i (Suc k) B\"\n and all_zero: \"\\m. is_zero_row_upt_k m k A\"\n and all_zero_k: \"\\m. A $ m $ from_nat k = 0\"\n shows \"A $ i $ (LEAST k. A $ i $ k \\ 0) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A $ i $ (LEAST k. A $ i $ k \\ (0::'a)) = (1::'a)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. A $ i $ (LEAST k. A $ i $ k \\ (0::'a)) = (1::'a)\n[PROOF STEP]\nhave ia2: \"ia = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ia = 0\n[PROOF STEP]\nusing ia all_zero\n[PROOF STATE]\nproof (prove)\nusing this:\nia \\ if \\m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \\ is_zero_row_upt_k n k A) + 1\n\\m. is_zero_row_upt_k m k A\n\ngoal (1 subgoal):\n 1. ia = 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nia = 0\n\ngoal (1 subgoal):\n 1. A $ i $ (LEAST k. A $ i $ k \\ (0::'a)) = (1::'a)\n[PROOF STEP]\nhave B_eq_A: \"B=A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. B = A\n[PROOF STEP]\nunfolding B Gauss_Jordan_column_k_def Let_def fst_conv snd_conv ia2\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. snd (if (\\m\\mod_type_class.from_nat 0. A $ m $ mod_type_class.from_nat k = (0::'a)) \\ 0 = nrows A then (0, A) else (0 + 1, Gauss_Jordan_in_ij A (mod_type_class.from_nat 0) (mod_type_class.from_nat k))) = A\n[PROOF STEP]\nusing all_zero_k\n[PROOF STATE]\nproof (prove)\nusing this:\n\\m. A $ m $ mod_type_class.from_nat k = (0::'a)\n\ngoal (1 subgoal):\n 1. snd (if (\\m\\mod_type_class.from_nat 0. A $ m $ mod_type_class.from_nat k = (0::'a)) \\ 0 = nrows A then (0, A) else (0 + 1, Gauss_Jordan_in_ij A (mod_type_class.from_nat 0) (mod_type_class.from_nat k))) = A\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nB = A\n\ngoal (1 subgoal):\n 1. A $ i $ (LEAST k. A $ i $ k \\ (0::'a)) = (1::'a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A $ i $ (LEAST k. A $ i $ k \\ (0::'a)) = (1::'a)\n[PROOF STEP]\nusing all_zero_k condition_1_part_1[OF _ all_zero] not_zero_i_suc_k\n[PROOF STATE]\nproof (prove)\nusing this:\n\\m. A $ m $ mod_type_class.from_nat k = (0::'a)\n\\m\\mod_type_class.from_nat 0. A $ m $ mod_type_class.from_nat k = (0::'a) \\ is_zero_row_upt_k ?j (Suc k) A\n\\ is_zero_row_upt_k i (Suc k) B\n\ngoal (1 subgoal):\n 1. A $ i $ (LEAST k. A $ i $ k \\ (0::'a)) = (1::'a)\n[PROOF STEP]\nunfolding B_eq_A\n[PROOF STATE]\nproof (prove)\nusing this:\n\\m. A $ m $ mod_type_class.from_nat k = (0::'a)\n\\m\\mod_type_class.from_nat 0. A $ m $ mod_type_class.from_nat k = (0::'a) \\ is_zero_row_upt_k ?j (Suc k) A\n\\ is_zero_row_upt_k i (Suc k) A\n\ngoal (1 subgoal):\n 1. A $ i $ (LEAST k. A $ i $ k \\ (0::'a)) = (1::'a)\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\nA $ i $ (LEAST k. A $ i $ k \\ (0::'a)) = (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1625, "file": "Gauss_Jordan_Gauss_Jordan", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045966995027, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7081549782353562}} {"text": "[STATEMENT]\nlemma isometry_on_compose:\n assumes \"isometry_on X f\"\n \"isometry_on (f`X) g\"\n shows \"isometry_on X (\\x. g(f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. isometry_on X (\\x. g (f x))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nisometry_on X f\nisometry_on (f ` X) g\n\ngoal (1 subgoal):\n 1. isometry_on X (\\x. g (f x))\n[PROOF STEP]\nunfolding isometry_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\X. \\y\\X. dist (f x) (f y) = dist x y\n\\x\\f ` X. \\y\\f ` X. dist (g x) (g y) = dist x y\n\ngoal (1 subgoal):\n 1. \\x\\X. \\y\\X. dist (g (f x)) (g (f y)) = dist x y\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 332, "file": "Gromov_Hyperbolicity_Isometries", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045877523148, "lm_q2_score": 0.798186784940666, "lm_q1q2_score": 0.7081549774826291}} {"text": "[STATEMENT]\nlemma list_eq_prod [simp]:\n fixes xs::\"real list\"\n shows \"\\:(list_eq xs m) = (m ^ (length (list_eq xs m)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod_list (list_eq xs m) = m ^ length (list_eq xs m)\n[PROOF STEP]\napply (induct_tac xs)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. prod_list (list_eq [] m) = m ^ length (list_eq [] m)\n 2. \\a list. prod_list (list_eq list m) = m ^ length (list_eq list m) \\ prod_list (list_eq (a # list) m) = m ^ length (list_eq (a # list) m)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a list. prod_list (list_eq list m) = m ^ length (list_eq list m) \\ prod_list (list_eq (a # list) m) = m ^ length (list_eq (a # list) m)\n[PROOF STEP]\napply clarsimp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 361, "file": "Cauchy_CauchysMeanTheorem", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045937171068, "lm_q2_score": 0.798186768138228, "lm_q1q2_score": 0.7081549673364471}} {"text": "[STATEMENT]\nlemma Int_interior_Union_intervals:\n \"\\finite \\; open S; \\T. T\\\\ \\ \\a b. T = cbox a b; \\T. T\\\\ \\ S \\ (interior T) = {}\\ \n \\ S \\ interior (\\\\) = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite \\; open S; \\T. T \\ \\ \\ \\a b. T = cbox a b; \\T. T \\ \\ \\ S \\ interior T = {}\\ \\ S \\ interior (\\ \\) = {}\n[PROOF STEP]\nusing interior_Union_subset_cbox[of \\ \"UNIV - S\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite \\; \\s. s \\ \\ \\ \\a b. s = cbox a b; \\s. s \\ \\ \\ interior s \\ UNIV - S; closed (UNIV - S)\\ \\ interior (\\ \\) \\ UNIV - S\n\ngoal (1 subgoal):\n 1. \\finite \\; open S; \\T. T \\ \\ \\ \\a b. T = cbox a b; \\T. T \\ \\ \\ S \\ interior T = {}\\ \\ S \\ interior (\\ \\) = {}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 462, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045877523147, "lm_q2_score": 0.7981867729389246, "lm_q1q2_score": 0.708154966834629}} {"text": "[STATEMENT]\nlemma cpx_vec_length_inner_prod [simp]:\n \"\\v\\\\<^sup>2 = \\v|v\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (\\v\\\\<^sup>2) = \\v|v\\\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. complex_of_real (\\v\\\\<^sup>2) = \\v|v\\\n[PROOF STEP]\nhave \"\\v\\\\<^sup>2 = (\\i2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\v\\\\<^sup>2 = (\\i2)\n[PROOF STEP]\nusing cpx_vec_length_def complex_of_real_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?v\\ \\ sqrt (\\i2)\ncomplex_of_real ?r = Complex ?r 0\n\ngoal (1 subgoal):\n 1. \\v\\\\<^sup>2 = (\\i2)\n[PROOF STEP]\nby (metis (no_types, lifting) real_sqrt_power real_sqrt_unique sum_nonneg zero_le_power2)\n[PROOF STATE]\nproof (state)\nthis:\n\\v\\\\<^sup>2 = (\\i2)\n\ngoal (1 subgoal):\n 1. complex_of_real (\\v\\\\<^sup>2) = \\v|v\\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\v\\\\<^sup>2 = (\\i2)\n\ngoal (1 subgoal):\n 1. complex_of_real (\\v\\\\<^sup>2) = \\v|v\\\n[PROOF STEP]\nhave \"\\ = (\\ii2) = (\\i2) = ?z * cnj ?z\n?a * ?b = ?b * ?a\n\ngoal (1 subgoal):\n 1. complex_of_real (\\i2) = (\\ii2) = (\\iv\\\\<^sup>2) = \\v|v\\\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncomplex_of_real (\\v\\\\<^sup>2) = (\\iv\\\\<^sup>2) = (\\iv\\\\<^sup>2) = \\v|v\\\n[PROOF STEP]\nusing inner_prod_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (\\v\\\\<^sup>2) = (\\i?u|?v\\ \\ \\i = 0..v\\\\<^sup>2) = \\v|v\\\n[PROOF STEP]\nby (simp add: lessThan_atLeast0)\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (\\v\\\\<^sup>2) = \\v|v\\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1421, "file": "Isabelle_Marries_Dirac_Quantum", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527982093666, "lm_q2_score": 0.831143045767024, "lm_q1q2_score": 0.7080946435534718}} {"text": "[STATEMENT]\nlemma irrelevant_set_bit[simp]:\n fixes p m n :: nat\n assumes \"n \\ m\"\n shows \"(p + 2 ^ m) mod 2 ^ n = p mod 2 ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (p + 2 ^ m) mod 2 ^ n = p mod 2 ^ n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (p + 2 ^ m) mod 2 ^ n = p mod 2 ^ n\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nn \\ m\n[PROOF STEP]\nobtain q :: nat where \"2 ^ m = q * 2 ^ n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\ m\n\ngoal (1 subgoal):\n 1. (\\q. 2 ^ m = q * 2 ^ n \\ thesis) \\ thesis\n[PROOF STEP]\nby (metis le_add_diff_inverse mult.commute power_add)\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ m = q * 2 ^ n\n\ngoal (1 subgoal):\n 1. (p + 2 ^ m) mod 2 ^ n = p mod 2 ^ n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n2 ^ m = q * 2 ^ n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 ^ m = q * 2 ^ n\n\ngoal (1 subgoal):\n 1. (p + 2 ^ m) mod 2 ^ n = p mod 2 ^ n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(p + 2 ^ m) mod 2 ^ n = p mod 2 ^ n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 543, "file": "Formula_Derivatives_Presburger_Formula", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527869325345, "lm_q2_score": 0.8311430541321951, "lm_q1q2_score": 0.7080946413075421}} {"text": "[STATEMENT]\nlemma L2_set_constant: \"L2_set (\\x. y) A = sqrt (of_nat (card A)) * \\y\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. L2_set (\\x. y) A = sqrt (real (card A)) * \\y\\\n[PROOF STEP]\nunfolding L2_set_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (\\i\\A. y\\<^sup>2) = sqrt (real (card A)) * \\y\\\n[PROOF STEP]\nby (simp add: real_sqrt_mult)", "meta": {"llama_tokens": 194, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505402422644, "lm_q2_score": 0.7826624789529375, "lm_q1q2_score": 0.7080360344121248}} {"text": "[STATEMENT]\nlemma all_less_Suc_eq: \"(\\x < Suc n. P x) \\ (\\x < n. P x) \\ P n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\xx P n)\n[PROOF STEP]\nby (auto elim: less_SucE)", "meta": {"llama_tokens": 125, "file": "Planarity_Certificates_Verification_Check_Planarity_Verification", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505402422645, "lm_q2_score": 0.7826624688140726, "lm_q1q2_score": 0.7080360252399953}} {"text": "[STATEMENT]\nlemma diff_leq_zero_weak_preference:\n assumes \"p \\ lotteries_on outcomes\"\n and \"q \\ lotteries_on outcomes\"\n shows \"p \\ q \\ ((\\a\\outcomes. pmf q a * u a) - (\\a\\outcomes. pmf p a * u a) \\ 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p \\ q = ((\\a\\outcomes. pmf q a * u a) - (\\a\\outcomes. pmf p a * u a) \\ 0)\n[PROOF STEP]\nusing assms(1) assms(2) diff_le_0_iff_le\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ \\

outcomes\nq \\ \\

outcomes\n(?a - ?b \\ (0::?'a)) = (?a \\ ?b)\n\ngoal (1 subgoal):\n 1. p \\ q = ((\\a\\outcomes. pmf q a * u a) - (\\a\\outcomes. pmf p a * u a) \\ 0)\n[PROOF STEP]\nby (metis (mono_tags, lifting) expected_utility_weak_preference)", "meta": {"llama_tokens": 377, "file": "Neumann_Morgenstern_Utility_Expected_Utility", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505299595163, "lm_q2_score": 0.7826624688140726, "lm_q1q2_score": 0.7080360171920742}} {"text": "[STATEMENT]\nlemma ln_fact: \"ln (fact n) = (\\d=1..n. ln d)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ln (fact n) = (\\d = 1..n. ln (real d))\n[PROOF STEP]\nby (induction n) (simp_all add: ln_mult)", "meta": {"llama_tokens": 104, "file": "Bertrands_Postulate_Bertrand", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9273632876167045, "lm_q2_score": 0.7634837527911057, "lm_q1q2_score": 0.7080268030302991}} {"text": "[STATEMENT]\nlemma proots_card_sphere_axis_eq:\n defines \"q1\\[:\\,-1:]\" and \"q2\\[:\\,1:]\"\n assumes \"p\\0\"\n shows \"card (proots_within p (sphere 0 1 - {- 1})) \n = card (proots_within (fcompose p q1 q2) {x. 0 = Im x})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (proots_within p (sphere 0 1 - {- 1})) = card (proots_within (fcompose p q1 q2) {x. 0 = Im x})\n[PROOF STEP]\nunfolding q1_def q2_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (proots_within p (sphere 0 1 - {- 1})) = card (proots_within (fcompose p [:\\, - 1:] [:\\, 1:]) {x. 0 = Im x})\n[PROOF STEP]\nproof (rule proots_card_fcompose_bij_eq[OF _ \\p\\0\\])\n[PROOF STATE]\nproof (state)\ngoal (5 subgoals):\n 1. bij_betw (\\x. poly [:\\, - 1:] x / poly [:\\, 1:] x) {x. 0 = Im x} (sphere 0 1 - {- 1})\n 2. \\x\\{x. 0 = Im x}. poly [:\\, 1:] x \\ 0\n 3. max (degree [:\\, - 1:]) (degree [:\\, 1:]) \\ 1\n 4. \\c. [:\\, - 1:] \\ smult c [:\\, 1:]\n 5. infinite UNIV\n[PROOF STEP]\nshow \"\\x\\{x. 0 = Im x}. poly [:\\, 1:] x \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\{x. 0 = Im x}. poly [:\\, 1:] x \\ 0\n[PROOF STEP]\nby (simp add: Complex_eq_0 plus_complex.code)\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\{x. 0 = Im x}. poly [:\\, 1:] x \\ 0\n\ngoal (4 subgoals):\n 1. bij_betw (\\x. poly [:\\, - 1:] x / poly [:\\, 1:] x) {x. 0 = Im x} (sphere 0 1 - {- 1})\n 2. max (degree [:\\, - 1:]) (degree [:\\, 1:]) \\ 1\n 3. \\c. [:\\, - 1:] \\ smult c [:\\, 1:]\n 4. infinite UNIV\n[PROOF STEP]\nqed (use bij_betw_axis_sphere infinite_UNIV_char_0 in auto)", "meta": {"llama_tokens": 856, "file": "Count_Complex_Roots_Count_Line", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916240341031, "lm_q2_score": 0.8198933359135361, "lm_q1q2_score": 0.7078890388291263}} {"text": "[STATEMENT]\nlemma Digamma_real_three_halves_pos: \"Digamma (3/2 :: real) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < Digamma (3 / 2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 < Digamma (3 / 2)\n[PROOF STEP]\nhave \"-Digamma (3/2 :: real) = -Digamma (of_nat 1 + 1/2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - Digamma (3 / 2) = - Digamma (real 1 + 1 / 2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n- Digamma (3 / 2) = - Digamma (real 1 + 1 / 2)\n\ngoal (1 subgoal):\n 1. 0 < Digamma (3 / 2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n- Digamma (3 / 2) = - Digamma (real 1 + 1 / 2)\n\ngoal (1 subgoal):\n 1. 0 < Digamma (3 / 2)\n[PROOF STEP]\nhave \"\\ = 2 * ln 2 + euler_mascheroni - 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - Digamma (real 1 + 1 / 2) = 2 * ln 2 + euler_mascheroni - 2\n[PROOF STEP]\nby (subst Digamma_half_integer) simp\n[PROOF STATE]\nproof (state)\nthis:\n- Digamma (real 1 + 1 / 2) = 2 * ln 2 + euler_mascheroni - 2\n\ngoal (1 subgoal):\n 1. 0 < Digamma (3 / 2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n- Digamma (real 1 + 1 / 2) = 2 * ln 2 + euler_mascheroni - 2\n\ngoal (1 subgoal):\n 1. 0 < Digamma (3 / 2)\n[PROOF STEP]\nnote euler_mascheroni_less_13_over_22\n[PROOF STATE]\nproof (state)\nthis:\neuler_mascheroni < 13 / 22\n\ngoal (1 subgoal):\n 1. 0 < Digamma (3 / 2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\neuler_mascheroni < 13 / 22\n\ngoal (1 subgoal):\n 1. 0 < Digamma (3 / 2)\n[PROOF STEP]\nnote ln2_le_25_over_36\n[PROOF STATE]\nproof (state)\nthis:\nln 2 \\ 25 / 36\n\ngoal (1 subgoal):\n 1. 0 < Digamma (3 / 2)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\x y. x < y \\ 2 * ln 2 + x - 2 < 2 * ln 2 + y - 2; \\x y. x \\ y \\ 2 * x + 13 / 22 - 2 \\ 2 * y + 13 / 22 - 2\\ \\ - Digamma (3 / 2) < 2 * (25 / 36) + 13 / 22 - 2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\x y. x < y \\ 2 * ln 2 + x - 2 < 2 * ln 2 + y - 2; \\x y. x \\ y \\ 2 * x + 13 / 22 - 2 \\ 2 * y + 13 / 22 - 2\\ \\ - Digamma (3 / 2) < 2 * (25 / 36) + 13 / 22 - 2\n\ngoal (1 subgoal):\n 1. 0 < Digamma (3 / 2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < Digamma (3 / 2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1218, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916134888613, "lm_q2_score": 0.819893340314393, "lm_q1q2_score": 0.7078890339828158}} {"text": "[STATEMENT]\nlemma coeff_poly_mult_rat_main: \"coeff (poly_mult_rat_main n d f) i = coeff f i * n ^ (degree f - i) * d ^ i\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coeff (poly_mult_rat_main n d f) i = coeff f i * n ^ (degree f - i) * d ^ i\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. coeff (poly_mult_rat_main n d f) i = coeff f i * n ^ (degree f - i) * d ^ i\n[PROOF STEP]\nhave id: \"coeff (poly_mult_rat_main n d f) i = (coeff f i * d ^ i) * n ^ (length (coeffs f) - Suc i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coeff (poly_mult_rat_main n d f) i = coeff f i * d ^ i * n ^ (length (coeffs f) - Suc i)\n[PROOF STEP]\nunfolding poly_mult_rat_main_def Let_def poly_of_list_def coeff_Poly\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. nth_default (0::'a) (map2 (\\fi i. fi * d ^ i * n ^ (length (coeffs f) - Suc i)) (coeffs f) [0..fi i. fi * d ^ i * n ^ (length (coeffs f) - Suc i)) (coeffs f) [0..fi i. fi * d ^ i * n ^ (length (coeffs f) - Suc i)) (coeffs f) [0..fi i. fi * d ^ i * n ^ (length (coeffs f) - Suc i)) (coeffs f) [0.. \\ list_ex pred tr\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Filtermap.filtermap pred func tr = []) = (\\ list_ex pred tr)\n[PROOF STEP]\nproof(induction tr)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (Filtermap.filtermap pred func [] = []) = (\\ list_ex pred [])\n 2. \\a tr. (Filtermap.filtermap pred func tr = []) = (\\ list_ex pred tr) \\ (Filtermap.filtermap pred func (a # tr) = []) = (\\ list_ex pred (a # tr))\n[PROOF STEP]\ncase (Cons trn tr)\n[PROOF STATE]\nproof (state)\nthis:\n(Filtermap.filtermap pred func tr = []) = (\\ list_ex pred tr)\n\ngoal (2 subgoals):\n 1. (Filtermap.filtermap pred func [] = []) = (\\ list_ex pred [])\n 2. \\a tr. (Filtermap.filtermap pred func tr = []) = (\\ list_ex pred tr) \\ (Filtermap.filtermap pred func (a # tr) = []) = (\\ list_ex pred (a # tr))\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(Filtermap.filtermap pred func tr = []) = (\\ list_ex pred tr)\n\ngoal (1 subgoal):\n 1. (Filtermap.filtermap pred func (trn # tr) = []) = (\\ list_ex pred (trn # tr))\n[PROOF STEP]\nby (cases \"pred trn\") auto\n[PROOF STATE]\nproof (state)\nthis:\n(Filtermap.filtermap pred func (trn # tr) = []) = (\\ list_ex pred (trn # tr))\n\ngoal (1 subgoal):\n 1. (Filtermap.filtermap pred func [] = []) = (\\ list_ex pred [])\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 568, "file": "Bounded_Deducibility_Security_Filtermap", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.863391602943619, "lm_q2_score": 0.8198933293122507, "lm_q1q2_score": 0.7078890158376846}} {"text": "[STATEMENT]\nlemma hom_into_subgroup_eq_gen:\n \"group G \\\n h \\ hom K (subgroup_generated G H)\n \\ h \\ hom K G \\ h ` (carrier K) \\ carrier(subgroup_generated G H)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Group.group G \\ (h \\ hom K (subgroup_generated G H)) = (h \\ hom K G \\ h ` carrier K \\ carrier (subgroup_generated G H))\n[PROOF STEP]\nusing group.carrier_subgroup_generated_subset [of G H]\n[PROOF STATE]\nproof (prove)\nusing this:\nGroup.group G \\ carrier (subgroup_generated G H) \\ carrier G\n\ngoal (1 subgoal):\n 1. Group.group G \\ (h \\ hom K (subgroup_generated G H)) = (h \\ hom K G \\ h ` carrier K \\ carrier (subgroup_generated G H))\n[PROOF STEP]\nby (auto simp: hom_def)", "meta": {"llama_tokens": 302, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392939666336, "lm_q2_score": 0.8006920092299293, "lm_q1q2_score": 0.707843198524352}} {"text": "[STATEMENT]\nlemma leadsTo_UN_UN:\n \"(!! i. i \\ I ==> F \\ (A i) leadsTo (A' i)) \n ==> F \\ (\\i \\ I. A i) leadsTo (\\i \\ I. A' i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. i \\ I \\ F \\ A i \\ A' i) \\ F \\ \\ (A ` I) \\ \\ (A' ` I)\n[PROOF STEP]\napply (blast intro: leadsTo_Union leadsTo_weaken_R)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 212, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392909114836, "lm_q2_score": 0.8006920092299292, "lm_q1q2_score": 0.7078431960781177}} {"text": "[STATEMENT]\nlemma sin_arctan: \"sin (arctan x) = x / sqrt (1 + x\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin (arctan x) = x / sqrt (1 + x\\<^sup>2)\n[PROOF STEP]\nusing add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 + x\\<^sup>2\n\ngoal (1 subgoal):\n 1. sin (arctan x) = x / sqrt (1 + x\\<^sup>2)\n[PROOF STEP]\nusing tan_arctan [of x]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 + x\\<^sup>2\ntan (arctan x) = x\n\ngoal (1 subgoal):\n 1. sin (arctan x) = x / sqrt (1 + x\\<^sup>2)\n[PROOF STEP]\nunfolding tan_def cos_arctan\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 + x\\<^sup>2\nsin (arctan x) / (1 / sqrt (1 + x\\<^sup>2)) = x\n\ngoal (1 subgoal):\n 1. sin (arctan x) = x / sqrt (1 + x\\<^sup>2)\n[PROOF STEP]\nby (simp add: eq_divide_eq)", "meta": {"llama_tokens": 381, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392878563335, "lm_q2_score": 0.8006920068519376, "lm_q1q2_score": 0.7078431915296455}} {"text": "[STATEMENT]\nlemma cpx_mat_cnj_prod:\n assumes \"dim_col M = dim_row N\"\n shows \"(M * N)\\<^sup>\\ = (M\\<^sup>\\) * (N\\<^sup>\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ = M\\<^sup>\\ * N\\<^sup>\\\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n 2. dim_row M * N\\<^sup>\\ = dim_row (M\\<^sup>\\ * N\\<^sup>\\)\n 3. dim_col M * N\\<^sup>\\ = dim_col (M\\<^sup>\\ * N\\<^sup>\\)\n[PROOF STEP]\nshow \"dim_row (M * N)\\<^sup>\\ = dim_row ((M\\<^sup>\\) * (N\\<^sup>\\))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_row M * N\\<^sup>\\ = dim_row (M\\<^sup>\\ * N\\<^sup>\\)\n[PROOF STEP]\nby (simp add: cpx_mat_cnj_def)\n[PROOF STATE]\nproof (state)\nthis:\ndim_row M * N\\<^sup>\\ = dim_row (M\\<^sup>\\ * N\\<^sup>\\)\n\ngoal (2 subgoals):\n 1. \\i j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n 2. dim_col M * N\\<^sup>\\ = dim_col (M\\<^sup>\\ * N\\<^sup>\\)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\i j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n 2. dim_col M * N\\<^sup>\\ = dim_col (M\\<^sup>\\ * N\\<^sup>\\)\n[PROOF STEP]\nshow \"dim_col ((M * N)\\<^sup>\\) = dim_col ((M\\<^sup>\\) * (N\\<^sup>\\))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_col M * N\\<^sup>\\ = dim_col (M\\<^sup>\\ * N\\<^sup>\\)\n[PROOF STEP]\nby (simp add: cpx_mat_cnj_def)\n[PROOF STATE]\nproof (state)\nthis:\ndim_col M * N\\<^sup>\\ = dim_col (M\\<^sup>\\ * N\\<^sup>\\)\n\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n[PROOF STEP]\nfix i j::nat\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n[PROOF STEP]\nassume a1:\"i < dim_row ((M\\<^sup>\\) * (N\\<^sup>\\))\" and a2:\"j < dim_col ((M\\<^sup>\\) * (N\\<^sup>\\))\"\n[PROOF STATE]\nproof (state)\nthis:\ni < dim_row (M\\<^sup>\\ * N\\<^sup>\\)\nj < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\n\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni < dim_row (M\\<^sup>\\ * N\\<^sup>\\)\nj < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\n[PROOF STEP]\nhave \"(M * N)\\<^sup>\\ $$ (i,j) = cnj (\\k<(dim_row N). M $$ (i,k) * N $$ (k,j))\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_row (M\\<^sup>\\ * N\\<^sup>\\)\nj < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\n\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ $$ (i, j) = cnj (\\k\\ * N\\<^sup>\\)\nj < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\ndim_col M = dim_row N\n?M\\<^sup>\\ \\ mat (dim_row ?M) (dim_col ?M) (\\(i, j). cnj (?M $$ (i, j)))\n\\?i < ?nr; ?j < ?nc\\ \\ mat ?nr ?nc ?f $$ (?i, ?j) = ?f (?i, ?j)\n\\?i < ?nr; ?j < ?nc\\ \\ mat\\<^sub>r ?nr ?nc ?g $$ (?i, ?j) = ?g ?i $ ?j\n?A * ?B \\ mat (dim_row ?A) (dim_col ?B) (\\(i, j). row ?A i \\ col ?B j)\n?v \\ ?w \\ \\i = 0..j. ?A $$ (?i, j))\ncol ?A ?j = vec (dim_row ?A) (\\i. ?A $$ (i, ?j))\ndim_row (?M\\<^sup>\\ * ?N\\<^sup>\\) = dim_row ?M\ndim_col (?M\\<^sup>\\ * ?N\\<^sup>\\) = dim_col ?N\n\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ $$ (i, j) = cnj (\\k\\ $$ (i, j) = cnj (\\ki j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nM * N\\<^sup>\\ $$ (i, j) = cnj (\\ki j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n[PROOF STEP]\nhave \"\\ = (\\k<(dim_row N). cnj(M $$ (i,k)) * cnj(N $$ (k,j)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cnj (\\kkkki j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncnj (\\kki j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n[PROOF STEP]\nhave \"((M\\<^sup>\\) * (N\\<^sup>\\)) $$ (i,j) = \n (\\k<(dim_row N). cnj(M $$ (i,k)) * cnj(N $$ (k,j)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j) = (\\k\\ * N\\<^sup>\\)\nj < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\n?M\\<^sup>\\ \\ mat (dim_row ?M) (dim_col ?M) (\\(i, j). cnj (?M $$ (i, j)))\n\\?i < ?nr; ?j < ?nc\\ \\ mat ?nr ?nc ?f $$ (?i, ?j) = ?f (?i, ?j)\n\\?i < ?nr; ?j < ?nc\\ \\ mat\\<^sub>r ?nr ?nc ?g $$ (?i, ?j) = ?g ?i $ ?j\n?A * ?B \\ mat (dim_row ?A) (dim_col ?B) (\\(i, j). row ?A i \\ col ?B j)\n?v \\ ?w \\ \\i = 0..j. ?A $$ (?i, j))\ncol ?A ?j = vec (dim_row ?A) (\\i. ?A $$ (i, ?j))\n\ngoal (1 subgoal):\n 1. (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j) = (\\k\\ * N\\<^sup>\\) $$ (i, j) = (\\ki j. \\i < dim_row (M\\<^sup>\\ * N\\<^sup>\\); j < dim_col (M\\<^sup>\\ * N\\<^sup>\\)\\ \\ M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j) = M * N\\<^sup>\\ $$ (i, j)\n[PROOF STEP]\nshow \"(M * N)\\<^sup>\\ $$ (i, j) = ((M\\<^sup>\\) * (N\\<^sup>\\)) $$ (i, j)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j) = M * N\\<^sup>\\ $$ (i, j)\n\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nM * N\\<^sup>\\ $$ (i, j) = (M\\<^sup>\\ * N\\<^sup>\\) $$ (i, j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4336, "file": "Isabelle_Marries_Dirac_Quantum", "length": 24, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392725805823, "lm_q2_score": 0.8006920068519376, "lm_q1q2_score": 0.7078431792984736}} {"text": "[STATEMENT]\nlemma arcs_undir_G_eq_2vertsG:\n \"\\undirected_tree G; Suc n = card (verts G)\\ \\ card (arcs G) = 2 * (card (verts G) - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\undirected_tree G; Suc n = card (verts G)\\ \\ card (arcs G) = 2 * (card (verts G) - 1)\n[PROOF STEP]\nusing arcs_graph_G_ge_2vertsG undirected_tree.acyclic undirected_tree.axioms(1)\n undirected_tree.connected\n[PROOF STATE]\nproof (prove)\nusing this:\n\\graph G; Digraph_Component.connected G; Suc n = card (verts G)\\ \\ 2 * (card (verts G) - 1) \\ card (arcs G)\nundirected_tree ?G \\ card (arcs ?G) \\ 2 * (card (verts ?G) - 1)\nundirected_tree ?G \\ graph ?G\nundirected_tree ?G \\ Digraph_Component.connected ?G\n\ngoal (1 subgoal):\n 1. \\undirected_tree G; Suc n = card (verts G)\\ \\ card (arcs G) = 2 * (card (verts G) - 1)\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 419, "file": "Query_Optimization_Directed_Tree_Additions", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392695254319, "lm_q2_score": 0.8006920068519376, "lm_q1q2_score": 0.7078431768522391}} {"text": "[STATEMENT]\nlemma fps_divide_deriv:\n assumes \"b dvd (a :: 'a :: field fps)\"\n shows \"fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b\\<^sup>2\n[PROOF STEP]\nhave eq_divide_imp: \"c \\ 0 \\ a * c = b \\ a = b div c\" for a b c :: \"'a :: field fps\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\c \\ 0; a * c = b\\ \\ a = b / c\n[PROOF STEP]\nby (drule sym) (simp add: mult.assoc)\n[PROOF STATE]\nproof (state)\nthis:\n\\?c \\ 0; ?a * ?c = ?b\\ \\ ?a = ?b / ?c\n\ngoal (1 subgoal):\n 1. fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b\\<^sup>2\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nb dvd a\n[PROOF STEP]\nhave \"a = a / b * b\"\n[PROOF STATE]\nproof (prove)\nusing this:\nb dvd a\n\ngoal (1 subgoal):\n 1. a = a / b * b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na = a / b * b\n\ngoal (1 subgoal):\n 1. fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\na = a / b * b\n\ngoal (1 subgoal):\n 1. fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b\\<^sup>2\n[PROOF STEP]\nhave \"fps_deriv (a / b * b) = fps_deriv (a / b) * b + a / b * fps_deriv b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fps_deriv (a / b * b) = fps_deriv (a / b) * b + a / b * fps_deriv b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfps_deriv (a / b * b) = fps_deriv (a / b) * b + a / b * fps_deriv b\n\ngoal (1 subgoal):\n 1. fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b\\<^sup>2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nfps_deriv a = fps_deriv (a / b) * b + a / b * fps_deriv b\n[PROOF STEP]\nhave \"fps_deriv (a / b) * b^2 = fps_deriv a * b - a * fps_deriv b\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfps_deriv a = fps_deriv (a / b) * b + a / b * fps_deriv b\n\ngoal (1 subgoal):\n 1. fps_deriv (a / b) * b\\<^sup>2 = fps_deriv a * b - a * fps_deriv b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfps_deriv a = fps_deriv (a / b) * b + a / b * fps_deriv b\nb dvd a\n\ngoal (1 subgoal):\n 1. fps_deriv (a / b) * b\\<^sup>2 = fps_deriv a * b - a * fps_deriv b\n[PROOF STEP]\nby (simp add: power2_eq_square algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nfps_deriv (a / b) * b\\<^sup>2 = fps_deriv a * b - a * fps_deriv b\n\ngoal (1 subgoal):\n 1. fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b\\<^sup>2\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfps_deriv (a / b) * b\\<^sup>2 = fps_deriv a * b - a * fps_deriv b\n\ngoal (1 subgoal):\n 1. fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b\\<^sup>2\n[PROOF STEP]\nby (cases \"b = 0\") (simp_all add: eq_divide_imp)\n[PROOF STATE]\nproof (state)\nthis:\nfps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1560, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392725805823, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.7078431750939976}} {"text": "[STATEMENT]\nlemma count_take_less: assumes \"n\\m\" \n shows \"count_list (take n cs) x \\ count_list (take m cs) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. count_list (take n cs) x \\ count_list (take m cs) x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. count_list (take n cs) x \\ count_list (take m cs) x\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nn \\ m\n[PROOF STEP]\nhave \"count_list (take n cs) x = count_list (take n (take m cs)) x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\ m\n\ngoal (1 subgoal):\n 1. count_list (take n cs) x = count_list (take n (take m cs)) x\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncount_list (take n cs) x = count_list (take n (take m cs)) x\n\ngoal (1 subgoal):\n 1. count_list (take n cs) x \\ count_list (take m cs) x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncount_list (take n cs) x = count_list (take n (take m cs)) x\n\ngoal (1 subgoal):\n 1. count_list (take n cs) x \\ count_list (take m cs) x\n[PROOF STEP]\nhave \"\\ \\ count_list (take n (take m cs) @ drop n (take m cs)) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. count_list (take n (take m cs)) x \\ count_list (take n (take m cs) @ drop n (take m cs)) x\n[PROOF STEP]\nby (simp)\n[PROOF STATE]\nproof (state)\nthis:\ncount_list (take n (take m cs)) x \\ count_list (take n (take m cs) @ drop n (take m cs)) x\n\ngoal (1 subgoal):\n 1. count_list (take n cs) x \\ count_list (take m cs) x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncount_list (take n (take m cs)) x \\ count_list (take n (take m cs) @ drop n (take m cs)) x\n\ngoal (1 subgoal):\n 1. count_list (take n cs) x \\ count_list (take m cs) x\n[PROOF STEP]\nhave \"\\ = count_list (take m cs) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. count_list (take n (take m cs) @ drop n (take m cs)) x = count_list (take m cs) x\n[PROOF STEP]\nby(simp only: append_take_drop_id)\n[PROOF STATE]\nproof (state)\nthis:\ncount_list (take n (take m cs) @ drop n (take m cs)) x = count_list (take m cs) x\n\ngoal (1 subgoal):\n 1. count_list (take n cs) x \\ count_list (take m cs) x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncount_list (take n cs) x \\ count_list (take m cs) x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncount_list (take n cs) x \\ count_list (take m cs) x\n\ngoal (1 subgoal):\n 1. count_list (take n cs) x \\ count_list (take m cs) x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncount_list (take n cs) x \\ count_list (take m cs) x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1104, "file": "List_Update_TS", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.853912760387131, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.707841426005759}} {"text": "[STATEMENT]\nlemma Arg_eq_Im_Ln:\n assumes \"z \\ 0\" shows \"Arg z = Im (Ln z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Arg z = Im (Ln z)\n[PROOF STEP]\nproof (rule cis_Arg_unique)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. sgn z = cis (Im (Ln z))\n 2. - pi < Im (Ln z)\n 3. Im (Ln z) \\ pi\n[PROOF STEP]\nshow \"sgn z = cis (Im (Ln z))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sgn z = cis (Im (Ln z))\n[PROOF STEP]\nby (metis assms exp_Ln exp_eq_polar nonzero_mult_div_cancel_left norm_eq_zero\n norm_exp_eq_Re of_real_eq_0_iff sgn_eq)\n[PROOF STATE]\nproof (state)\nthis:\nsgn z = cis (Im (Ln z))\n\ngoal (2 subgoals):\n 1. - pi < Im (Ln z)\n 2. Im (Ln z) \\ pi\n[PROOF STEP]\nshow \"- pi < Im (Ln z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - pi < Im (Ln z)\n[PROOF STEP]\nby (simp add: assms mpi_less_Im_Ln)\n[PROOF STATE]\nproof (state)\nthis:\n- pi < Im (Ln z)\n\ngoal (1 subgoal):\n 1. Im (Ln z) \\ pi\n[PROOF STEP]\nshow \"Im (Ln z) \\ pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Im (Ln z) \\ pi\n[PROOF STEP]\nby (simp add: Im_Ln_le_pi assms)\n[PROOF STATE]\nproof (state)\nthis:\nIm (Ln z) \\ pi\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 597, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.853912760387131, "lm_q2_score": 0.8289388019824946, "lm_q1q2_score": 0.7078414205928734}} {"text": "[STATEMENT]\nlemma nat_intv_frac_decomp:\n fixes c :: nat and d :: real\n assumes \"c < d\" \"d < c + 1\"\n shows \"d = c + frac d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. d = real c + frac d\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. d = real c + frac d\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nreal c < d\nd < real (c + 1)\n[PROOF STEP]\nhave \"int c = \\d\\\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal c < d\nd < real (c + 1)\n\ngoal (1 subgoal):\n 1. int c = \\d\\\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\nint c = \\d\\\n\ngoal (1 subgoal):\n 1. d = real c + frac d\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nint c = \\d\\\n\ngoal (1 subgoal):\n 1. d = real c + frac d\n[PROOF STEP]\nby (simp add: frac_def)\n[PROOF STATE]\nproof (state)\nthis:\nd = real c + frac d\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 438, "file": "Timed_Automata_Misc", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127455162772, "lm_q2_score": 0.8289388062084421, "lm_q1q2_score": 0.7078414118744359}} {"text": "[STATEMENT]\nlemma ceiling_one [simp]: \"\\1\\ = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\1::'a\\ = 1\n[PROOF STEP]\nusing ceiling_of_int [of 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\of_int 1\\ = 1\n\ngoal (1 subgoal):\n 1. \\1::'a\\ = 1\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 158, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.8289388019824946, "lm_q1q2_score": 0.7078414082658456}} {"text": "[STATEMENT]\nlemma sum_of_cubes: \"4 * (\\i=0..n. i * i * i) = n * n * Suc n * Suc n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 4 * (\\i = 0..n. i * i * i) = n * n * Suc n * Suc n\n[PROOF STEP]\nby (induct n) auto", "meta": {"llama_tokens": 116, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9136765210631689, "lm_q2_score": 0.7745833841649233, "lm_q1q2_score": 0.7077186517171432}} {"text": "[STATEMENT]\nlemma col_scalar_prod_as_sum:\nassumes \"dim_vec v = dim_row A\"\nshows \"col A j \\ v = (\\i = 0.. v = (\\i = 0.. v = (\\i = 0..i = 0..i. A $$ (i, j)) $ i * v $ i) = (\\i = 0.. gf_U' x y z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (gf_U x y has_real_derivative gf_U' x y z) (at z)\n[PROOF STEP]\nunfolding gf_U_def[abs_def] gf_U'_def real_scaleR_def u_def[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\z. \\n. u x y n * z ^ Suc n) has_real_derivative (\\n. u x y n * real (Suc n) * z ^ n)) (at z)\n[PROOF STEP]\nusing z\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < z\nz < 1\n\ngoal (1 subgoal):\n 1. ((\\z. \\n. u x y n * z ^ Suc n) has_real_derivative (\\n. u x y n * real (Suc n) * z ^ n)) (at z)\n[PROOF STEP]\nby (intro DERIV_power_series'[where R=1] summable_gf_U') auto", "meta": {"llama_tokens": 373, "file": "Markov_Models_Classifying_Markov_Chain_States", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869819218865, "lm_q2_score": 0.8080672158638527, "lm_q1q2_score": 0.7076947481714251}} {"text": "[STATEMENT]\nlemma minus_log_convex:\n fixes b :: real\n assumes \"b > 1\"\n shows \"convex_on {0 <..} (\\ x. - log b x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex_on {0<..} (\\x. - log b x)\n[PROOF STEP]\nusing assms concave_on_def log_concave\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < b\nconcave_on ?S ?f \\ convex_on ?S (\\x. - ?f x)\n1 < ?b \\ concave_on {0<..} (log ?b)\n\ngoal (1 subgoal):\n 1. convex_on {0<..} (\\x. - log b x)\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 231, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869851639066, "lm_q2_score": 0.8080672066194946, "lm_q1q2_score": 0.7076947426951068}} {"text": "[STATEMENT]\nlemma card_bijections_domain_permutation:\n assumes \"finite A\" \"finite B\"\n shows \"card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // domain_permutation A B) = iverson (card A = card B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // domain_permutation A B) = iverson (card A = card B)\n[PROOF STEP]\nusing assms card_bijections_domain_permutation_eq_0 card_bijections_domain_permutation_eq_1\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\ncard ?A \\ card ?B \\ card ({f \\ ?A \\\\<^sub>E ?B. bij_betw f ?A ?B} // domain_permutation ?A ?B) = 0\n\\finite ?A; finite ?B; card ?A = card ?B\\ \\ card ({f \\ ?A \\\\<^sub>E ?B. bij_betw f ?A ?B} // domain_permutation ?A ?B) = 1\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // domain_permutation A B) = iverson (card A = card B)\n[PROOF STEP]\nunfolding iverson_def\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\ncard ?A \\ card ?B \\ card ({f \\ ?A \\\\<^sub>E ?B. bij_betw f ?A ?B} // domain_permutation ?A ?B) = 0\n\\finite ?A; finite ?B; card ?A = card ?B\\ \\ card ({f \\ ?A \\\\<^sub>E ?B. bij_betw f ?A ?B} // domain_permutation ?A ?B) = 1\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // domain_permutation A B) = (if card A = card B then 1 else 0)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 623, "file": "Twelvefold_Way_Card_Bijections", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869786798663, "lm_q2_score": 0.8080672112416737, "lm_q1q2_score": 0.7076947415036107}} {"text": "[STATEMENT]\nlemma matrix_right_invertible_independent_rows:\n fixes A :: \"'a::{field}^'n^'m\"\n shows \"(\\(B::'a^'m^'n). A ** B = mat 1) \\\n (\\c. sum (\\i. c i *s row i A) (UNIV :: 'm set) = 0 \\ (\\i. c i = 0))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\B. A ** B = mat (1::'a)) = (\\c. (\\i\\UNIV. c i *s row i A) = 0 \\ (\\i. c i = (0::'a)))\n[PROOF STEP]\nunfolding left_invertible_transpose[symmetric]\n matrix_left_invertible_independent_columns\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\c. (\\i\\UNIV. c i *s column i (Finite_Cartesian_Product.transpose A)) = 0 \\ (\\i. c i = (0::'a))) = (\\c. (\\i\\UNIV. c i *s row i A) = 0 \\ (\\i. c i = (0::'a)))\n[PROOF STEP]\nby (simp add:)", "meta": {"llama_tokens": 373, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972583359805, "lm_q2_score": 0.8128673201042492, "lm_q1q2_score": 0.7076800602736751}} {"text": "[STATEMENT]\nlemma powr_nat_bounds_ge_neg:\n assumes \"powr_nat (abs x) y \\ u\"\n shows \"powr_nat x y \\ -u\" \"powr_nat x y \\ u\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - u \\ powr_nat x y &&& powr_nat x y \\ u\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. - u \\ powr_nat x y\n 2. powr_nat x y \\ u\n[PROOF STEP]\nhave \"abs (powr_nat x y) \\ powr_nat (abs x) y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\powr_nat x y\\ \\ powr_nat \\x\\ y\n[PROOF STEP]\nby (rule abs_powr_nat_le)\n[PROOF STATE]\nproof (state)\nthis:\n\\powr_nat x y\\ \\ powr_nat \\x\\ y\n\ngoal (2 subgoals):\n 1. - u \\ powr_nat x y\n 2. powr_nat x y \\ u\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\powr_nat x y\\ \\ powr_nat \\x\\ y\n\ngoal (2 subgoals):\n 1. - u \\ powr_nat x y\n 2. powr_nat x y \\ u\n[PROOF STEP]\nhave \"\\ \\ u\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. powr_nat \\x\\ y \\ u\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\npowr_nat \\x\\ y \\ u\n\ngoal (1 subgoal):\n 1. powr_nat \\x\\ y \\ u\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\npowr_nat \\x\\ y \\ u\n\ngoal (2 subgoals):\n 1. - u \\ powr_nat x y\n 2. powr_nat x y \\ u\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\powr_nat x y\\ \\ u\n[PROOF STEP]\nshow \"powr_nat x y \\ -u\" \"powr_nat x y \\ u\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\powr_nat x y\\ \\ u\n\ngoal (1 subgoal):\n 1. - u \\ powr_nat x y &&& powr_nat x y \\ u\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n- u \\ powr_nat x y\npowr_nat x y \\ u\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 880, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511469672594, "lm_q2_score": 0.8267117962054049, "lm_q1q2_score": 0.707542238993759}} {"text": "[STATEMENT]\nlemma not_zero_divisor_mult:\n fixes a b :: \"'a :: {ab_semigroup_mult, mult_zero}\"\n assumes \"\\ zero_divisor (a * b)\"\n shows \"\\ zero_divisor a\" and \"\\ zero_divisor b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ zero_divisor a &&& \\ zero_divisor b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ zero_divisor (a * b)\n\ngoal (1 subgoal):\n 1. \\ zero_divisor a &&& \\ zero_divisor b\n[PROOF STEP]\nby (auto dest: zero_divisor_mult_right zero_divisor_mult_left)", "meta": {"llama_tokens": 230, "file": "Polynomial_Interpolation_Missing_Polynomial", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117855317473, "lm_q2_score": 0.8558511451289037, "lm_q1q2_score": 0.7075422283389066}} {"text": "[STATEMENT]\nlemma DERIV_cot [simp]: \"sin x \\ 0 \\ DERIV cot x :> -inverse ((sin x)\\<^sup>2)\"\n for x :: \"'a::{real_normed_field,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin x \\ (0::'a) \\ (cot has_field_derivative - inverse ((sin x)\\<^sup>2)) (at x)\n[PROOF STEP]\nunfolding cot_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin x \\ (0::'a) \\ ((\\x. cos x / sin x) has_field_derivative - inverse ((sin x)\\<^sup>2)) (at x)\n[PROOF STEP]\nusing cos_squared_eq[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\n(cos x)\\<^sup>2 = (1::'a) - (sin x)\\<^sup>2\n\ngoal (1 subgoal):\n 1. sin x \\ (0::'a) \\ ((\\x. cos x / sin x) has_field_derivative - inverse ((sin x)\\<^sup>2)) (at x)\n[PROOF STEP]\nby (auto intro!: derivative_eq_intros) (simp add: divide_inverse power2_eq_square)", "meta": {"llama_tokens": 383, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970904940927, "lm_q2_score": 0.8031738034238806, "lm_q1q2_score": 0.7074331492168283}} {"text": "[STATEMENT]\nlemma prime_factorization_unique':\n assumes \"\\p \\# M. prime p\" \"\\p \\# N. prime p\" \"(\\i \\# M. i) = (\\i \\# N. i)\"\n shows \"M = N\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M = N\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. M = N\n[PROOF STEP]\nhave \"prime_factorization (\\i \\# M. i) = prime_factorization (\\i \\# N. i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prime_factorization (\\i\\#M. i) = prime_factorization (\\i\\#N. i)\n[PROOF STEP]\nby (simp only: assms)\n[PROOF STATE]\nproof (state)\nthis:\nprime_factorization (\\i\\#M. i) = prime_factorization (\\i\\#N. i)\n\ngoal (1 subgoal):\n 1. M = N\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nprime_factorization (\\i\\#M. i) = prime_factorization (\\i\\#N. i)\n\ngoal (1 subgoal):\n 1. M = N\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n\\p\\#M. prime p\n\\p\\#N. prime p\n(\\i\\#M. i) = (\\i\\#N. i)\n[PROOF STEP]\nhave \"prime_factorization (\\i \\# M. i) = M\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\p\\#M. prime p\n\\p\\#N. prime p\n(\\i\\#M. i) = (\\i\\#N. i)\n\ngoal (1 subgoal):\n 1. prime_factorization (\\i\\#M. i) = M\n[PROOF STEP]\nby (subst prime_factorization_prod_mset_primes) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nprime_factorization (\\i\\#M. i) = M\n\ngoal (1 subgoal):\n 1. M = N\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nprime_factorization (\\i\\#M. i) = M\n\ngoal (1 subgoal):\n 1. M = N\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n\\p\\#M. prime p\n\\p\\#N. prime p\n(\\i\\#M. i) = (\\i\\#N. i)\n[PROOF STEP]\nhave \"prime_factorization (\\i \\# N. i) = N\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\p\\#M. prime p\n\\p\\#N. prime p\n(\\i\\#M. i) = (\\i\\#N. i)\n\ngoal (1 subgoal):\n 1. prime_factorization (\\i\\#N. i) = N\n[PROOF STEP]\nby (subst prime_factorization_prod_mset_primes) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nprime_factorization (\\i\\#N. i) = N\n\ngoal (1 subgoal):\n 1. M = N\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nM = N\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nM = N\n\ngoal (1 subgoal):\n 1. M = N\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nM = N\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1190, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970904940926, "lm_q2_score": 0.8031738010682209, "lm_q1q2_score": 0.7074331471419701}} {"text": "[STATEMENT]\nlemma iT_Plus_mem_iff: \"x \\ (I \\ k) = (k \\ x \\ (x - k) \\ I)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x \\ I \\ k) = (k \\ x \\ x - k \\ I)\n[PROOF STEP]\napply (simp add: iT_Plus_def image_iff)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\xa\\I. x = xa + k) = (k \\ x \\ x - k \\ I)\n[PROOF STEP]\napply (rule iffI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\xa\\I. x = xa + k \\ k \\ x \\ x - k \\ I\n 2. k \\ x \\ x - k \\ I \\ \\xa\\I. x = xa + k\n[PROOF STEP]\napply fastforce\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k \\ x \\ x - k \\ I \\ \\xa\\I. x = xa + k\n[PROOF STEP]\napply (rule_tac x=\"x - k\" in bexI, simp+)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 421, "file": "Nat-Interval-Logic_IL_IntervalOperators", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970873650401, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7074331342545059}} {"text": "[STATEMENT]\ntheorem kleene_qfp_is_dual_extreme:\n assumes comp: \"omega_complete A (\\)\"\n and cont: \"omega_continuous A (\\) f\" and bA: \"b \\ A\" and bot: \"\\x \\ A. b \\ x\"\n shows \"extreme_bound A (\\) {f^n b |. n :: nat} = extreme {s \\ A. f s \\ s} (\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. extreme_bound A (\\) {f^n b |. n} = extreme {s \\ A. f s \\ s} (\\x y. y \\ x)\n[PROOF STEP]\napply (rule kleene_qfp_iff_least[OF bA bot cont comp])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\q\\A. \\x\\A. f q \\ q \\ x \\ f q \\ x \\ q\n 2. \\p\\A. \\q\\A. \\x\\A. p \\ q \\ q \\ x \\ p \\ x\n[PROOF STEP]\nusing cont[THEN omega_continuousDdom]\n[PROOF STATE]\nproof (prove)\nusing this:\nf ` A \\ A\n\ngoal (2 subgoals):\n 1. \\q\\A. \\x\\A. f q \\ q \\ x \\ f q \\ x \\ q\n 2. \\p\\A. \\q\\A. \\x\\A. p \\ q \\ q \\ x \\ p \\ x\n[PROOF STEP]\nby (auto dest: sym_order_trans order_sym_trans)", "meta": {"llama_tokens": 534, "file": "Complete_Non_Orders_Kleene_Fixed_Point", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970811069351, "lm_q2_score": 0.8031737940012418, "lm_q1q2_score": 0.7074331333778765}} {"text": "[STATEMENT]\nlemma measure_closure:\n assumes \"bounded S\" and neg: \"negligible (frontier S)\"\n shows \"measure lebesgue (closure S) = measure lebesgue S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure lebesgue (closure S) = Sigma_Algebra.measure lebesgue S\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure lebesgue (closure S) = Sigma_Algebra.measure lebesgue S\n[PROOF STEP]\nhave \"measure lebesgue (frontier S) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure lebesgue (frontier S) = 0\n[PROOF STEP]\nby (metis neg negligible_imp_measure0)\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure lebesgue (frontier S) = 0\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure lebesgue (closure S) = Sigma_Algebra.measure lebesgue S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nSigma_Algebra.measure lebesgue (frontier S) = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nSigma_Algebra.measure lebesgue (frontier S) = 0\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure lebesgue (closure S) = Sigma_Algebra.measure lebesgue S\n[PROOF STEP]\nby (metis assms lmeasurable_iff_integrable_on eq_iff_diff_eq_0 has_integral_interior integrable_on_def integral_unique lmeasurable_interior lmeasure_integral measure_frontier)\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure lebesgue (closure S) = Sigma_Algebra.measure lebesgue S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 579, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.880797068590724, "lm_q2_score": 0.803173801068221, "lm_q1q2_score": 0.7074331295497585}} {"text": "[STATEMENT]\nlemma or_dfa_accepts:\n assumes \"wf_dfa M n\" and \"wf_dfa N n\"\n and \"list_all (is_alph n) bs\"\n shows \"dfa_accepts (or_dfa M N) bs = (dfa_accepts M bs \\ dfa_accepts N bs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dfa_accepts (or_dfa M N) bs = (dfa_accepts M bs \\ dfa_accepts N bs)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nwf_dfa M n\nwf_dfa N n\nlist_all (is_alph n) bs\n\ngoal (1 subgoal):\n 1. dfa_accepts (or_dfa M N) bs = (dfa_accepts M bs \\ dfa_accepts N bs)\n[PROOF STEP]\nby (simp add: binop_dfa_accepts or_dfa_def)", "meta": {"llama_tokens": 278, "file": "Presburger-Automata_Presburger_Automata", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970779778825, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.707433126714987}} {"text": "[STATEMENT]\nlemma inj_on_mult: \n assumes coprime: \"coprime x (q::nat)\" \n shows \"inj_on (\\ b. x*b mod q) {..b. x * b mod q) {..xa y. \\xa < q; y < q; x * xa mod q = x * y mod q\\ \\ xa = y\n[PROOF STEP]\nusing coprime\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime x q\n\ngoal (1 subgoal):\n 1. \\xa y. \\xa < q; y < q; x * xa mod q = x * y mod q\\ \\ xa = y\n[PROOF STEP]\nby(simp only: inj_mult)", "meta": {"llama_tokens": 295, "file": "Multi_Party_Computation_Uniform_Sampling", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240825770431, "lm_q2_score": 0.8175744828610096, "lm_q1q2_score": 0.7073851318718175}} {"text": "[STATEMENT]\nlemma mset_set_Union:\n \"finite A \\ finite B \\ A \\ B = {} \\ mset_set (A \\ B) = mset_set A + mset_set B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite B; A \\ B = {}\\ \\ mset_set (A \\ B) = mset_set A + mset_set B\n[PROOF STEP]\nby (induction A rule: finite_induct) auto", "meta": {"llama_tokens": 156, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240825770432, "lm_q2_score": 0.8175744784160989, "lm_q1q2_score": 0.7073851280259738}} {"text": "[STATEMENT]\nlemma card_minus_fset:\n shows \"card_fset (A - B) = card_fset A - card_fset (A |\\| B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card_fset (A - B) = card_fset A - card_fset (A |\\| B)\n[PROOF STEP]\nby (simp add: card_fset card_Diff_subset_Int)", "meta": {"llama_tokens": 125, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789468908171, "lm_q2_score": 0.7905303236047049, "lm_q1q2_score": 0.7073582084140585}} {"text": "[STATEMENT]\nlemma differentiable_mult [simp, derivative_intros]:\n fixes f g :: \"'a::real_normed_vector \\ 'b::real_normed_algebra\"\n shows \"f differentiable (at x within s) \\ g differentiable (at x within s) \\\n (\\x. f x * g x) differentiable (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f differentiable at x within s; g differentiable at x within s\\ \\ (\\x. f x * g x) differentiable at x within s\n[PROOF STEP]\nunfolding differentiable_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\D. (f has_derivative D) (at x within s); \\D. (g has_derivative D) (at x within s)\\ \\ \\D. ((\\x. f x * g x) has_derivative D) (at x within s)\n[PROOF STEP]\nby (blast intro: has_derivative_mult)", "meta": {"llama_tokens": 309, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894661025424, "lm_q2_score": 0.7905303236047049, "lm_q1q2_score": 0.707358206196124}} {"text": "[STATEMENT]\nlemma open_Collect_less_Int:\n fixes f g :: \"'a::topological_space \\ real\"\n assumes f: \"continuous_on s f\"\n and g: \"continuous_on s g\"\n shows \"\\A. open A \\ A \\ s = {x\\s. f x < g x}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A. open A \\ A \\ s = {x \\ s. f x < g x}\n[PROOF STEP]\nusing open_Collect_positive[OF continuous_on_diff[OF g f]]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A. open A \\ A \\ s = {x \\ s. 0 < g x - f x}\n\ngoal (1 subgoal):\n 1. \\A. open A \\ A \\ s = {x \\ s. f x < g x}\n[PROOF STEP]\nby (simp add: field_simps)", "meta": {"llama_tokens": 281, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894745194281, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7073582040183013}} {"text": "[STATEMENT]\nlemma open_Collect_less_Int:\n fixes f g :: \"'a::topological_space \\ real\"\n assumes f: \"continuous_on s f\"\n and g: \"continuous_on s g\"\n shows \"\\A. open A \\ A \\ s = {x\\s. f x < g x}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A. open A \\ A \\ s = {x \\ s. f x < g x}\n[PROOF STEP]\nusing open_Collect_positive[OF continuous_on_diff[OF g f]]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A. open A \\ A \\ s = {x \\ s. 0 < g x - f x}\n\ngoal (1 subgoal):\n 1. \\A. open A \\ A \\ s = {x \\ s. f x < g x}\n[PROOF STEP]\nby (simp add: field_simps)", "meta": {"llama_tokens": 281, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894745194281, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7073582040183013}} {"text": "[STATEMENT]\nlemma differentiable_mult [simp, derivative_intros]:\n fixes f g :: \"'a::real_normed_vector \\ 'b::real_normed_algebra\"\n shows \"f differentiable (at x within s) \\ g differentiable (at x within s) \\\n (\\x. f x * g x) differentiable (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f differentiable at x within s; g differentiable at x within s\\ \\ (\\x. f x * g x) differentiable at x within s\n[PROOF STEP]\nunfolding differentiable_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\D. (f has_derivative D) (at x within s); \\D. (g has_derivative D) (at x within s)\\ \\ \\D. ((\\x. f x * g x) has_derivative D) (at x within s)\n[PROOF STEP]\nby (blast intro: has_derivative_mult)", "meta": {"llama_tokens": 309, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894661025424, "lm_q2_score": 0.7905303186696747, "lm_q1q2_score": 0.707358201780311}} {"text": "[STATEMENT]\nlemma differentiable_mult [simp, derivative_intros]:\n fixes f g :: \"'a::real_normed_vector \\ 'b::real_normed_algebra\"\n shows \"f differentiable (at x within s) \\ g differentiable (at x within s) \\\n (\\x. f x * g x) differentiable (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f differentiable at x within s; g differentiable at x within s\\ \\ (\\x. f x * g x) differentiable at x within s\n[PROOF STEP]\nunfolding differentiable_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\D. (f has_derivative D) (at x within s); \\D. (g has_derivative D) (at x within s)\\ \\ \\D. ((\\x. f x * g x) has_derivative D) (at x within s)\n[PROOF STEP]\nby (blast intro: has_derivative_mult)", "meta": {"llama_tokens": 309, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894661025424, "lm_q2_score": 0.7905303112671295, "lm_q1q2_score": 0.7073581951565915}} {"text": "[STATEMENT]\nlemma (in strategic_space_2p) U\\<^sub>B_is_gate:\n shows \"gate 1 U\\<^sub>B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. gate 1 (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = 2 ^ 1\n 2. square_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n 3. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n[PROOF STEP]\nshow \"dim_row U\\<^sub>B = 2^1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = 2 ^ 1\n[PROOF STEP]\nusing mat_of_cols_list_def\n[PROOF STATE]\nproof (prove)\nusing this:\nTensor.mat_of_cols_list ?nr ?cs = Matrix.mat ?nr (length ?cs) (\\(i, j). ?cs ! j ! i)\n\ngoal (1 subgoal):\n 1. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = 2 ^ 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = 2 ^ 1\n\ngoal (2 subgoals):\n 1. square_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n 2. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = 2 ^ 1\n\ngoal (2 subgoals):\n 1. square_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n 2. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n[PROOF STEP]\nshow \"square_mat U\\<^sub>B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. square_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n[PROOF STEP]\nusing mat_of_cols_list_def\n[PROOF STATE]\nproof (prove)\nusing this:\nTensor.mat_of_cols_list ?nr ?cs = Matrix.mat ?nr (length ?cs) (\\(i, j). ?cs ! j ! i)\n\ngoal (1 subgoal):\n 1. square_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsquare_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n\ngoal (1 subgoal):\n 1. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ndim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = 2 ^ 1\nsquare_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n[PROOF STEP]\nshow \"unitary U\\<^sub>B\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = 2 ^ 1\nsquare_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n\ngoal (1 subgoal):\n 1. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n[PROOF STEP]\nusing mat_of_cols_list_def unitary_def U\\<^sub>B_cnj_times_U\\<^sub>B U\\<^sub>B_times_U\\<^sub>B_cnj\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]) = 2 ^ 1\nsquare_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\nTensor.mat_of_cols_list ?nr ?cs = Matrix.mat ?nr (length ?cs) (\\(i, j). ?cs ! j ! i)\nunitary ?M \\ ?M\\<^sup>\\ * ?M = 1\\<^sub>m (dim_col ?M) \\ ?M * ?M\\<^sup>\\ = 1\\<^sub>m (dim_row ?M)\nTensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]] = 1\\<^sub>m 2\nTensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]] * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]]\\<^sup>\\ = 1\\<^sub>m 2\n\ngoal (1 subgoal):\n 1. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nunitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2)) # map complex_of_real [- sin (\\\\<^sub>B / 2)], [complex_of_real (sin (\\\\<^sub>B / 2)), exp (- \\ * complex_of_real \\\\<^sub>B) * complex_of_real (cos (\\\\<^sub>B / 2))]])\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4554, "file": "Isabelle_Marries_Dirac_Quantum_Prisoners_Dilemma", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976953003183444, "lm_q2_score": 0.7879311956428947, "lm_q1q2_score": 0.7073221313028406}} {"text": "[STATEMENT]\nlemma map_poly_sum_commute:\n assumes \"h 0 = 0\" \"\\p q. h (p + q) = h p + h q\"\n shows \"sum (\\i. map_poly h (f i)) S = map_poly h (sum f S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\S. map_poly h (f i)) = map_poly h (sum f S)\n[PROOF STEP]\napply(induct S rule: infinite_finite_induct)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\A. infinite A \\ (\\i\\A. map_poly h (f i)) = map_poly h (sum f A)\n 2. (\\i\\{}. map_poly h (f i)) = map_poly h (sum f {})\n 3. \\x F. \\finite F; x \\ F; (\\i\\F. map_poly h (f i)) = map_poly h (sum f F)\\ \\ (\\i\\insert x F. map_poly h (f i)) = map_poly h (sum f (insert x F))\n[PROOF STEP]\nusing map_poly_add[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nmap_poly h (?p + ?q) = map_poly h ?p + map_poly h ?q\n\ngoal (3 subgoals):\n 1. \\A. infinite A \\ (\\i\\A. map_poly h (f i)) = map_poly h (sum f A)\n 2. (\\i\\{}. map_poly h (f i)) = map_poly h (sum f {})\n 3. \\x F. \\finite F; x \\ F; (\\i\\F. map_poly h (f i)) = map_poly h (sum f F)\\ \\ (\\i\\insert x F. map_poly h (f i)) = map_poly h (sum f (insert x F))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 575, "file": "Algebraic_Numbers_Bivariate_Polynomials", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110454379297, "lm_q2_score": 0.7931059585194573, "lm_q1q2_score": 0.7073006540102885}} {"text": "[STATEMENT]\nlemma contour_integrable_linepath_same_Im_iff:\n fixes a b :: complex and f :: \"complex \\ complex\"\n assumes \"Im a = Im b\" \"Re a < Re b\"\n shows \"(f contour_integrable_on linepath a b) \\\n (\\x. f (of_real x + Im a * \\)) integrable_on {Re a..Re b}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f contour_integrable_on linepath a b) = ((\\x. f (complex_of_real x + complex_of_real (Im a) * \\)) integrable_on {Re a..Re b})\n[PROOF STEP]\nusing has_contour_integral_linepath_same_Im_iff[OF assms, of f]\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_contour_integral ?I) (linepath a b) = ((\\x. f (complex_of_real x + complex_of_real (Im a) * \\)) has_integral ?I) {Re a..Re b}\n\ngoal (1 subgoal):\n 1. (f contour_integrable_on linepath a b) = ((\\x. f (complex_of_real x + complex_of_real (Im a) * \\)) integrable_on {Re a..Re b})\n[PROOF STEP]\nby (auto simp: contour_integrable_on_def integrable_on_def)", "meta": {"llama_tokens": 408, "file": "Prime_Number_Theorem_Prime_Number_Theorem_Library", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110454379297, "lm_q2_score": 0.7931059585194573, "lm_q1q2_score": 0.7073006540102885}} {"text": "[STATEMENT]\nlemma lm04: \n assumes \"runiq f\" \"y1 \\ Range f\" \n shows \"(f^-1 `` {y1} \\ f^-1 `` {y2} \\ {}) = (f^-1``{y1}=f^-1``{y2})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f\\ `` {y1} \\ f\\ `` {y2} \\ {}) = (f\\ `` {y1} = f\\ `` {y2})\n[PROOF STEP]\nusing assms rightUniqueFunctionAfterInverse\n[PROOF STATE]\nproof (prove)\nusing this:\nruniq f\ny1 \\ Range f\nruniq ?f \\ ?f `` ?f\\ `` ?Y \\ ?Y\n\ngoal (1 subgoal):\n 1. (f\\ `` {y1} \\ f\\ `` {y2} \\ {}) = (f\\ `` {y1} = f\\ `` {y2})\n[PROOF STEP]\nby fast", "meta": {"llama_tokens": 307, "file": "Vickrey_Clarke_Groves_RelationProperties", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110425624792, "lm_q2_score": 0.7931059487389968, "lm_q1q2_score": 0.7073006430074289}} {"text": "[STATEMENT]\nlemma card_ceros_count_UNIV:\n shows \"card (ceros_of_boolean_input a) + count_true ((a::bool vec)) = dim_vec a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (ceros_of_boolean_input a) + count_true a = dim_vec a\n[PROOF STEP]\nusing card_complementary [of a]\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (ceros_of_boolean_input a) + card {x. x < dim_vec a \\ a $ x = True} = dim_vec a\n\ngoal (1 subgoal):\n 1. card (ceros_of_boolean_input a) + count_true a = dim_vec a\n[PROOF STEP]\nusing card_boolean_function\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (ceros_of_boolean_input a) + card {x. x < dim_vec a \\ a $ x = True} = dim_vec a\n?v \\ carrier_vec ?n \\ card {x. x < ?n \\ ?v $ x = True} = (\\i = 0.. a $ x = False} + card {x. x < dim_vec a \\ a $ x = True} = dim_vec a\n?v \\ carrier_vec ?n \\ card {x. x < ?n \\ ?v $ x = True} = (\\i = 0.. a $ x = False} + count_true a = dim_vec a\n[PROOF STEP]\nunfolding count_true_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {x. x < dim_vec a \\ a $ x = False} + card {x. x < dim_vec a \\ a $ x = True} = dim_vec a\n?v \\ carrier_vec ?n \\ card {x. x < ?n \\ ?v $ x = True} = (\\i = 0.. a $ x = False} + (\\i = 0..1 \\ carrier_vec n\" \"v\\<^sub>2 \\ carrier_vec n\" \"v\\<^sub>3 \\ carrier_vec n\"\n shows \"v\\<^sub>1 \\ (v\\<^sub>2 + v\\<^sub>3) = v\\<^sub>1 \\ v\\<^sub>2 + v\\<^sub>1 \\ v\\<^sub>3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v\\<^sub>1 \\ (v\\<^sub>2 + v\\<^sub>3) = v\\<^sub>1 \\ v\\<^sub>2 + v\\<^sub>1 \\ v\\<^sub>3\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. v\\<^sub>1 \\ (v\\<^sub>2 + v\\<^sub>3) = v\\<^sub>1 \\ v\\<^sub>2 + v\\<^sub>1 \\ v\\<^sub>3\n[PROOF STEP]\nhave \"(\\i\\{0..3}. v\\<^sub>1 $ i * (v\\<^sub>2 + v\\<^sub>3) $ i) = (\\i\\{0..3}. v\\<^sub>1 $ i * v\\<^sub>2 $ i + v\\<^sub>1 $ i * v\\<^sub>3 $ i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..3. v\\<^sub>1 $ i * (v\\<^sub>2 + v\\<^sub>3) $ i) = (\\i = 0..3. v\\<^sub>1 $ i * v\\<^sub>2 $ i + v\\<^sub>1 $ i * v\\<^sub>3 $ i)\n[PROOF STEP]\nby (rule sum.cong, insert v, auto simp: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..3. v\\<^sub>1 $ i * (v\\<^sub>2 + v\\<^sub>3) $ i) = (\\i = 0..3. v\\<^sub>1 $ i * v\\<^sub>2 $ i + v\\<^sub>1 $ i * v\\<^sub>3 $ i)\n\ngoal (1 subgoal):\n 1. v\\<^sub>1 \\ (v\\<^sub>2 + v\\<^sub>3) = v\\<^sub>1 \\ v\\<^sub>2 + v\\<^sub>1 \\ v\\<^sub>3\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i = 0..3. v\\<^sub>1 $ i * (v\\<^sub>2 + v\\<^sub>3) $ i) = (\\i = 0..3. v\\<^sub>1 $ i * v\\<^sub>2 $ i + v\\<^sub>1 $ i * v\\<^sub>3 $ i)\n\ngoal (1 subgoal):\n 1. v\\<^sub>1 \\ (v\\<^sub>2 + v\\<^sub>3) = v\\<^sub>1 \\ v\\<^sub>2 + v\\<^sub>1 \\ v\\<^sub>3\n[PROOF STEP]\nunfolding scalar_prod_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i = 0..3. v\\<^sub>1 $ i * (v\\<^sub>2 + v\\<^sub>3) $ i) = (\\i = 0..3. v\\<^sub>1 $ i * v\\<^sub>2 $ i + v\\<^sub>1 $ i * v\\<^sub>3 $ i)\n\ngoal (1 subgoal):\n 1. (\\i = 0..2 + v\\<^sub>3). v\\<^sub>1 $ i * (v\\<^sub>2 + v\\<^sub>3) $ i) = (\\i = 0..2. v\\<^sub>1 $ i * v\\<^sub>2 $ i) + (\\i = 0..3. v\\<^sub>1 $ i * v\\<^sub>3 $ i)\n[PROOF STEP]\nusing v\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i = 0..3. v\\<^sub>1 $ i * (v\\<^sub>2 + v\\<^sub>3) $ i) = (\\i = 0..3. v\\<^sub>1 $ i * v\\<^sub>2 $ i + v\\<^sub>1 $ i * v\\<^sub>3 $ i)\nv\\<^sub>1 \\ carrier_vec n\nv\\<^sub>2 \\ carrier_vec n\nv\\<^sub>3 \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. (\\i = 0..2 + v\\<^sub>3). v\\<^sub>1 $ i * (v\\<^sub>2 + v\\<^sub>3) $ i) = (\\i = 0..2. v\\<^sub>1 $ i * v\\<^sub>2 $ i) + (\\i = 0..3. v\\<^sub>1 $ i * v\\<^sub>3 $ i)\n[PROOF STEP]\nby (auto intro: sum.distrib)\n[PROOF STATE]\nproof (state)\nthis:\nv\\<^sub>1 \\ (v\\<^sub>2 + v\\<^sub>3) = v\\<^sub>1 \\ v\\<^sub>2 + v\\<^sub>1 \\ v\\<^sub>3\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1542, "file": "Jordan_Normal_Form_Matrix", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681195338728, "lm_q2_score": 0.8244619285331332, "lm_q1q2_score": 0.7071971580651358}} {"text": "[STATEMENT]\nlemma two_power_sum: \"sum (\\ x. (2::nat) ^ x) {i. i< Suc m} = (2 ^ Suc m) - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc m} = 2 ^ Suc m - 1\n[PROOF STEP]\nproof (induct m)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. sum ((^) 2) {i. i < Suc 0} = 2 ^ Suc 0 - 1\n 2. \\m. sum ((^) 2) {i. i < Suc m} = 2 ^ Suc m - 1 \\ sum ((^) 2) {i. i < Suc (Suc m)} = 2 ^ Suc (Suc m) - 1\n[PROOF STEP]\nshow \"sum (\\ x. (2::nat) ^ x) {i. i< Suc 0} = (2 ^ Suc 0) - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc 0} = 2 ^ Suc 0 - 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) 2) {i. i < Suc 0} = 2 ^ Suc 0 - 1\n\ngoal (1 subgoal):\n 1. \\m. sum ((^) 2) {i. i < Suc m} = 2 ^ Suc m - 1 \\ sum ((^) 2) {i. i < Suc (Suc m)} = 2 ^ Suc (Suc m) - 1\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m. sum ((^) 2) {i. i < Suc m} = 2 ^ Suc m - 1 \\ sum ((^) 2) {i. i < Suc (Suc m)} = 2 ^ Suc (Suc m) - 1\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m. sum ((^) 2) {i. i < Suc m} = 2 ^ Suc m - 1 \\ sum ((^) 2) {i. i < Suc (Suc m)} = 2 ^ Suc (Suc m) - 1\n[PROOF STEP]\nassume A: \"sum (\\ x. (2::nat) ^ x) {i. i< Suc n} = (2 ^ Suc n) - 1\"\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) 2) {i. i < Suc n} = 2 ^ Suc n - 1\n\ngoal (1 subgoal):\n 1. \\m. sum ((^) 2) {i. i < Suc m} = 2 ^ Suc m - 1 \\ sum ((^) 2) {i. i < Suc (Suc m)} = 2 ^ Suc (Suc m) - 1\n[PROOF STEP]\nshow \"sum (\\ x. (2::nat) ^ x) {i. i< Suc (Suc n)} = (2 ^ Suc (Suc n)) - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nlet ?f = \"\\ x. (2::nat) ^ x\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nhave S1: \"{i. i< Suc (Suc n)} = {i. i \\ Suc n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {i. i < Suc (Suc n)} = {i. i \\ Suc n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{i. i < Suc (Suc n)} = {i. i \\ Suc n}\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nhave S2: \"{i. i \\ Suc n} = {i. i < Suc n} \\ { Suc n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {i. i \\ Suc n} = {i. i < Suc n} \\ {Suc n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{i. i \\ Suc n} = {i. i < Suc n} \\ {Suc n}\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nfrom S1 S2\n[PROOF STATE]\nproof (chain)\npicking this:\n{i. i < Suc (Suc n)} = {i. i \\ Suc n}\n{i. i \\ Suc n} = {i. i < Suc n} \\ {Suc n}\n[PROOF STEP]\nhave S3: \"{i. i< Suc (Suc n)} = {i. i < Suc n} \\ { Suc n}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{i. i < Suc (Suc n)} = {i. i \\ Suc n}\n{i. i \\ Suc n} = {i. i < Suc n} \\ {Suc n}\n\ngoal (1 subgoal):\n 1. {i. i < Suc (Suc n)} = {i. i < Suc n} \\ {Suc n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{i. i < Suc (Suc n)} = {i. i < Suc n} \\ {Suc n}\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nhave S4: \"{i. i < Suc n} = (\\ x. x) ` {i. i < Suc n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {i. i < Suc n} = (\\x. x) ` {i. i < Suc n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{i. i < Suc n} = (\\x. x) ` {i. i < Suc n}\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{i. i < Suc n} = (\\x. x) ` {i. i < Suc n}\n[PROOF STEP]\nhave S5: \"finite {i. i < Suc n}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{i. i < Suc n} = (\\x. x) ` {i. i < Suc n}\n\ngoal (1 subgoal):\n 1. finite {i. i < Suc n}\n[PROOF STEP]\nby (rule nat_seg_image_imp_finite)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {i. i < Suc n}\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nhave S6: \"Suc n \\ {i. i < Suc n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc n \\ {i. i < Suc n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nSuc n \\ {i. i < Suc n}\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nfrom S5 S6 sum.insert\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite {i. i < Suc n}\nSuc n \\ {i. i < Suc n}\n\\finite ?A; ?x \\ ?A\\ \\ sum ?g (insert ?x ?A) = ?g ?x + sum ?g ?A\n[PROOF STEP]\nhave S7: \"sum ?f ({i. i< Suc n} \\ {Suc n}) = 2 ^ Suc n + sum ?f {i. i< Suc n}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {i. i < Suc n}\nSuc n \\ {i. i < Suc n}\n\\finite ?A; ?x \\ ?A\\ \\ sum ?g (insert ?x ?A) = ?g ?x + sum ?g ?A\n\ngoal (1 subgoal):\n 1. sum ((^) 2) ({i. i < Suc n} \\ {Suc n}) = 2 ^ Suc n + sum ((^) 2) {i. i < Suc n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) 2) ({i. i < Suc n} \\ {Suc n}) = 2 ^ Suc n + sum ((^) 2) {i. i < Suc n}\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nfrom S3\n[PROOF STATE]\nproof (chain)\npicking this:\n{i. i < Suc (Suc n)} = {i. i < Suc n} \\ {Suc n}\n[PROOF STEP]\nhave \"sum ?f {i. i< Suc (Suc n)} = sum ?f ({i. i< Suc n} \\ {Suc n})\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{i. i < Suc (Suc n)} = {i. i < Suc n} \\ {Suc n}\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = sum ((^) 2) ({i. i < Suc n} \\ {Suc n})\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) 2) {i. i < Suc (Suc n)} = sum ((^) 2) ({i. i < Suc n} \\ {Suc n})\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) 2) {i. i < Suc (Suc n)} = sum ((^) 2) ({i. i < Suc n} \\ {Suc n})\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nfrom S7\n[PROOF STATE]\nproof (chain)\npicking this:\nsum ((^) 2) ({i. i < Suc n} \\ {Suc n}) = 2 ^ Suc n + sum ((^) 2) {i. i < Suc n}\n[PROOF STEP]\nhave \"\\ = 2 ^ Suc n + sum ?f {i. i< Suc n}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsum ((^) 2) ({i. i < Suc n} \\ {Suc n}) = 2 ^ Suc n + sum ((^) 2) {i. i < Suc n}\n\ngoal (1 subgoal):\n 1. sum ((^) 2) ({i. i < Suc n} \\ {Suc n}) = 2 ^ Suc n + sum ((^) 2) {i. i < Suc n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) 2) ({i. i < Suc n} \\ {Suc n}) = 2 ^ Suc n + sum ((^) 2) {i. i < Suc n}\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) 2) ({i. i < Suc n} \\ {Suc n}) = 2 ^ Suc n + sum ((^) 2) {i. i < Suc n}\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nfrom A\n[PROOF STATE]\nproof (chain)\npicking this:\nsum ((^) 2) {i. i < Suc n} = 2 ^ Suc n - 1\n[PROOF STEP]\nhave \"\\ = 2 ^ Suc n + (((2::nat) ^ Suc n)-(1::nat))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsum ((^) 2) {i. i < Suc n} = 2 ^ Suc n - 1\n\ngoal (1 subgoal):\n 1. 2 ^ Suc n + sum ((^) 2) {i. i < Suc n} = 2 ^ Suc n + (2 ^ Suc n - 1)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ Suc n + sum ((^) 2) {i. i < Suc n} = 2 ^ Suc n + (2 ^ Suc n - 1)\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ Suc n + sum ((^) 2) {i. i < Suc n} = 2 ^ Suc n + (2 ^ Suc n - 1)\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nhave \"\\ = (2 ^ Suc (Suc n)) - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ Suc n + (2 ^ Suc n - 1) = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ Suc n + (2 ^ Suc n - 1) = 2 ^ Suc (Suc n) - 1\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n\ngoal (1 subgoal):\n 1. sum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) 2) {i. i < Suc (Suc n)} = 2 ^ Suc (Suc n) - 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4674, "file": "Recursion-Theory-I_PRecFinSet", "length": 45, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577680977182187, "lm_q2_score": 0.824461928533133, "lm_q1q2_score": 0.7071971400789595}} {"text": "[STATEMENT]\nlemma le_floor_add: \"\\x\\ + \\y\\ \\ \\x + y\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\ + \\y\\ \\ \\x + y\\\n[PROOF STEP]\nby (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)", "meta": {"llama_tokens": 138, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297914570319, "lm_q2_score": 0.7853085909370422, "lm_q1q2_score": 0.7071937816259503}} {"text": "[STATEMENT]\nlemma matrix_vector_mul_injective_on_rowspace:\n fixes A :: \"real^'n^'m\"\n shows \"\\A *v x = A *v y; x \\ span(rows A); y \\ span(rows A)\\ \\ x = y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A *v x = A *v y; x \\ span (rows A); y \\ span (rows A)\\ \\ x = y\n[PROOF STEP]\nusing nullspace_inter_rowspace [of A \"x-y\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(A *v (x - y) = 0 \\ x - y \\ span (rows A)) = (x - y = 0)\n\ngoal (1 subgoal):\n 1. \\A *v x = A *v y; x \\ span (rows A); y \\ span (rows A)\\ \\ x = y\n[PROOF STEP]\nby (metis diff_eq_diff_eq diff_self matrix_vector_mult_diff_distrib span_diff)", "meta": {"llama_tokens": 321, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.900529778109184, "lm_q2_score": 0.7853085708384736, "lm_q1q2_score": 0.707193753044411}} {"text": "[STATEMENT]\nlemma Im_exp: \"Im (exp z) = exp (Re z) * sin (Im z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Im (exp z) = exp (Re z) * sin (Im z)\n[PROOF STEP]\nunfolding exp_eq_polar\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Im (complex_of_real (exp (Re z)) * cis (Im z)) = exp (Re z) * sin (Im z)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 154, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588052782736, "lm_q2_score": 0.7956581073313276, "lm_q1q2_score": 0.707148148881763}} {"text": "[STATEMENT]\nlemma mset_size_ne0_set_card: \"size A > 0 \\ card (set_mset A) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < size A \\ 0 < card (set_mset A)\n[PROOF STEP]\nusing mset_nempty_set_nempty\n[PROOF STATE]\nproof (prove)\nusing this:\n(?A \\ {#}) = (set_mset ?A \\ {})\n\ngoal (1 subgoal):\n 1. 0 < size A \\ 0 < card (set_mset A)\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 186, "file": "Design_Theory_Multisets_Extras", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587993853655, "lm_q2_score": 0.7956581073313275, "lm_q1q2_score": 0.7071481441930229}} {"text": "[STATEMENT]\nlemma isCont_sgn[continuous_intros]:\n fixes f :: \"'a::t2_space \\ 'b::real_normed_vector\"\n assumes \"isCont f a\"\n and \"f a \\ 0\"\n shows \"isCont (\\x. sgn (f x)) a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. isCont (\\x. sgn (f x)) a\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nisCont f a\nf a \\ (0::'b)\n\ngoal (1 subgoal):\n 1. isCont (\\x. sgn (f x)) a\n[PROOF STEP]\nunfolding continuous_at\n[PROOF STATE]\nproof (prove)\nusing this:\nf \\a\\ f a\nf a \\ (0::'b)\n\ngoal (1 subgoal):\n 1. (\\x. sgn (f x)) \\a\\ sgn (f a)\n[PROOF STEP]\nby (rule tendsto_sgn)", "meta": {"llama_tokens": 307, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.7956581000631542, "lm_q1q2_score": 0.7071481330446294}} {"text": "[STATEMENT]\nlemma det_perm_rows:\n fixes A :: \"'a::comm_ring_1^^'n\"\n assumes p: \"p permutes UNIV\"\n shows \"det (perm_rows A p) = of_int (sign p) * det A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Square_Matrix.det (perm_rows A p) = of_int (sign p) * Square_Matrix.det A\n[PROOF STEP]\nusing det_perm_cols[OF p, of \"transpose A\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nSquare_Matrix.det (perm_cols (Square_Matrix.transpose A) p) = of_int (sign p) * Square_Matrix.det (Square_Matrix.transpose A)\n\ngoal (1 subgoal):\n 1. Square_Matrix.det (perm_rows A p) = of_int (sign p) * Square_Matrix.det A\n[PROOF STEP]\nby (simp add: det_transpose perm_cols_transpose)", "meta": {"llama_tokens": 269, "file": "Cayley_Hamilton_Square_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.7956581000631541, "lm_q1q2_score": 0.7071481330446293}} {"text": "[STATEMENT]\nlemma frechet_derivative_inverse: \"frechet_derivative (\\x. inverse (f x)) (at x) =\n (\\h. - 1 / (f x)\\<^sup>2 * frechet_derivative f (at x) h)\"\n if \"f differentiable at x\" \"f x \\ 0\" for f::\"_\\_::real_normed_field\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. frechet_derivative (\\x. inverse (f x)) (at x) = (\\h. - (1::'d) / (f x)\\<^sup>2 * frechet_derivative f (at x) h)\n[PROOF STEP]\napply (rule frechet_derivative_at')\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. inverse (f x)) has_derivative (\\h. - (1::'d) / (f x)\\<^sup>2 * frechet_derivative f (at x) h)) (at x)\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\nf differentiable at x\nf x \\ (0::'d)\n\ngoal (1 subgoal):\n 1. ((\\x. inverse (f x)) has_derivative (\\h. - (1::'d) / (f x)\\<^sup>2 * frechet_derivative f (at x) h)) (at x)\n[PROOF STEP]\nby (auto intro!: derivative_eq_intros frechet_derivative_worksI\n simp: divide_simps algebra_simps power2_eq_square)", "meta": {"llama_tokens": 466, "file": "Smooth_Manifolds_Analysis_More", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587875995482, "lm_q2_score": 0.7956581024858786, "lm_q1q2_score": 0.7071481305091065}} {"text": "[STATEMENT]\nlemma isCont_sgn[continuous_intros]:\n fixes f :: \"'a::t2_space \\ 'b::real_normed_vector\"\n assumes \"isCont f a\"\n and \"f a \\ 0\"\n shows \"isCont (\\x. sgn (f x)) a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. isCont (\\x. sgn (f x)) a\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nisCont f a\nf a \\ (0::'b)\n\ngoal (1 subgoal):\n 1. isCont (\\x. sgn (f x)) a\n[PROOF STEP]\nunfolding continuous_at\n[PROOF STATE]\nproof (prove)\nusing this:\nf \\a\\ f a\nf a \\ (0::'b)\n\ngoal (1 subgoal):\n 1. (\\x. sgn (f x)) \\a\\ sgn (f a)\n[PROOF STEP]\nby (rule tendsto_sgn)", "meta": {"llama_tokens": 307, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.7956580952177051, "lm_q1q2_score": 0.707148128738194}} {"text": "[STATEMENT]\nlemma exp1: \"exp 3 (A) + exp 3 (B) < 3 * ((exp 3 A) * (exp 3 B))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp 3 A + exp 3 B < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\nusing exp3Min[of A] exp3Min[of B]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < exp 3 A\n0 < exp 3 B\n\ngoal (1 subgoal):\n 1. exp 3 A + exp 3 B < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\napply(case_tac \"exp 3 A\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\0 < exp 3 A; 0 < exp 3 B; exp 3 A = 0\\ \\ exp 3 A + exp 3 B < 3 * (exp 3 A * exp 3 B)\n 2. \\nat. \\0 < exp 3 A; 0 < exp 3 B; exp 3 A = Suc nat\\ \\ exp 3 A + exp 3 B < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\napply(simp add: exp3Min)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\nat. \\0 < exp 3 A; 0 < exp 3 B; exp 3 A = Suc nat\\ \\ exp 3 A + exp 3 B < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\napply(case_tac \"exp 3 B\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\nat. \\0 < exp 3 A; 0 < exp 3 B; exp 3 A = Suc nat; exp 3 B = 0\\ \\ exp 3 A + exp 3 B < 3 * (exp 3 A * exp 3 B)\n 2. \\nat nata. \\0 < exp 3 A; 0 < exp 3 B; exp 3 A = Suc nat; exp 3 B = Suc nata\\ \\ exp 3 A + exp 3 B < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\napply (simp add: exp3Min, simp)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 711, "file": "Completeness_Completeness", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587875995482, "lm_q2_score": 0.7956580927949806, "lm_q1q2_score": 0.7071481218962358}} {"text": "[STATEMENT]\nlemma log_divide: \"0 < a \\ a \\ 1 \\ 0 < x \\ 0 < y \\ log a (x/y) = log a x - log a y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < a; a \\ 1; 0 < x; 0 < y\\ \\ log a (x / y) = log a x - log a y\n[PROOF STEP]\nby (simp add: log_mult divide_inverse log_inverse)", "meta": {"llama_tokens": 157, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473879530491, "lm_q2_score": 0.8104789155369047, "lm_q1q2_score": 0.7070191649596387}} {"text": "[STATEMENT]\nlemma groups_of_size_zero: \"groups_of_size 0 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. groups_of_size 0 = 0\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. groups_of_size 0 = 0\n[PROOF STEP]\nhave empty: \"{G \\ \\ . card G = 0} = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {G \\ \\. card G = 0} = {}\n[PROOF STEP]\nusing min_group_size\n[PROOF STATE]\nproof (prove)\nusing this:\n?G \\ \\ \\ 1 \\ card ?G\n\ngoal (1 subgoal):\n 1. {G \\ \\. card G = 0} = {}\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n{G \\ \\. card G = 0} = {}\n\ngoal (1 subgoal):\n 1. groups_of_size 0 = 0\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n{G \\ \\. card G = 0} = {}\n\ngoal (1 subgoal):\n 1. groups_of_size 0 = 0\n[PROOF STEP]\nunfolding groups_of_size_def\n[PROOF STATE]\nproof (prove)\nusing this:\n{G \\ \\. card G = 0} = {}\n\ngoal (1 subgoal):\n 1. card {G \\ \\. card G = 0} = 0\n[PROOF STEP]\nby (simp add: empty)\n[PROOF STATE]\nproof (state)\nthis:\ngroups_of_size 0 = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 535, "file": "Design_Theory_Group_Divisible_Designs", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473813156294, "lm_q2_score": 0.8104789086703225, "lm_q1q2_score": 0.7070191535901049}} {"text": "[STATEMENT]\nlemma content_cbox_plus:\n fixes x :: \"'a::euclidean_space\"\n shows \"content(cbox x (x + h *\\<^sub>R One)) = (if h \\ 0 then h ^ DIM('a) else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. content (cbox x (x + h *\\<^sub>R One)) = (if 0 \\ h then h ^ DIM('a) else 0)\n[PROOF STEP]\nby (simp add: algebra_simps content_cbox_if box_eq_empty)", "meta": {"llama_tokens": 158, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.877476793890012, "lm_q2_score": 0.8056321959813275, "lm_q1q2_score": 0.7069235563842651}} {"text": "[STATEMENT]\nlemma card_remove_fset_less2: \n shows \"x |\\| xs \\ y |\\| xs \\ card_fset (remove_fset y (remove_fset x xs)) < card_fset xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x |\\| xs; y |\\| xs\\ \\ card_fset (remove_fset y (remove_fset x xs)) < card_fset xs\n[PROOF STEP]\nunfolding card_fset remove_fset in_fset\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x \\ fset xs; y \\ fset xs\\ \\ card (fset xs - {x} - {y}) < card (fset xs)\n[PROOF STEP]\nby (rule card_Diff2_less[OF finite_fset])", "meta": {"llama_tokens": 263, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.8056321936479701, "lm_q1q2_score": 0.7069235491741691}} {"text": "[STATEMENT]\nlemma subresultant_product: fixes F :: \"'a :: idom poly\"\n assumes \"F = B * G\"\n and FG: \"degree F \\ degree G\"\nshows \"subresultant J F G = (if J < degree G then 0 else\n if J < degree F then smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nproof (cases \"J < degree G\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n 2. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\ncase J: True\n[PROOF STATE]\nproof (state)\nthis:\nJ < degree G\n\ngoal (2 subgoals):\n 1. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n 2. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nF = B * G\ndegree G \\ degree F\n[PROOF STEP]\nhave eq: \"F + (-B) * G = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nF = B * G\ndegree G \\ degree F\n\ngoal (1 subgoal):\n 1. F + - B * G = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nF + - B * G = 0\n\ngoal (2 subgoals):\n 1. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n 2. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nfrom J\n[PROOF STATE]\nproof (chain)\npicking this:\nJ < degree G\n[PROOF STEP]\nhave lt: \"degree 0 < degree G \\ b\" for b\n[PROOF STATE]\nproof (prove)\nusing this:\nJ < degree G\n\ngoal (1 subgoal):\n 1. degree 0 < degree G \\ b\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndegree 0 < degree G \\ ?b1\n\ngoal (2 subgoals):\n 1. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n 2. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nfrom BT_lemma_1_13[OF eq FG lt lt]\n[PROOF STATE]\nproof (chain)\npicking this:\nsubresultant (degree 0) F G = Polynomial.smult ((- (1::'a)) ^ ((degree F - degree 0) * (degree G - degree 0)) * lead_coeff G ^ (degree F - degree 0) * lead_coeff 0 ^ (degree G - degree 0 - 1)) 0\n[PROOF STEP]\nhave \"subresultant 0 F G = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsubresultant (degree 0) F G = Polynomial.smult ((- (1::'a)) ^ ((degree F - degree 0) * (degree G - degree 0)) * lead_coeff G ^ (degree F - degree 0) * lead_coeff 0 ^ (degree G - degree 0 - 1)) 0\n\ngoal (1 subgoal):\n 1. subresultant 0 F G = 0\n[PROOF STEP]\nusing J\n[PROOF STATE]\nproof (prove)\nusing this:\nsubresultant (degree 0) F G = Polynomial.smult ((- (1::'a)) ^ ((degree F - degree 0) * (degree G - degree 0)) * lead_coeff G ^ (degree F - degree 0) * lead_coeff 0 ^ (degree G - degree 0 - 1)) 0\nJ < degree G\n\ngoal (1 subgoal):\n 1. subresultant 0 F G = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsubresultant 0 F G = 0\n\ngoal (2 subgoals):\n 1. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n 2. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nwith BT_lemma_1_14[OF eq FG lt, of J]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\degree 0 < J; J < degree G - 1\\ \\ subresultant J F G = 0\nsubresultant 0 F G = 0\n[PROOF STEP]\nhave 00: \"J = 0 \\ J < degree G - 1 \\ subresultant J F G = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\degree 0 < J; J < degree G - 1\\ \\ subresultant J F G = 0\nsubresultant 0 F G = 0\n\ngoal (1 subgoal):\n 1. J = 0 \\ J < degree G - 1 \\ subresultant J F G = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nJ = 0 \\ J < degree G - 1 \\ subresultant J F G = 0\n\ngoal (2 subgoals):\n 1. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n 2. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nfrom BT_lemma_1_15[OF eq FG lt lt] J\n[PROOF STATE]\nproof (chain)\npicking this:\nsubresultant (degree G - 1) F G = Polynomial.smult ((- (1::'a)) ^ (degree F - degree G + 1) * lead_coeff G ^ (degree F - degree G + 1)) 0\nJ < degree G\n[PROOF STEP]\nhave 01: \"subresultant (degree G - 1) F G = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsubresultant (degree G - 1) F G = Polynomial.smult ((- (1::'a)) ^ (degree F - degree G + 1) * lead_coeff G ^ (degree F - degree G + 1)) 0\nJ < degree G\n\ngoal (1 subgoal):\n 1. subresultant (degree G - 1) F G = 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsubresultant (degree G - 1) F G = 0\n\ngoal (2 subgoals):\n 1. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n 2. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nfrom J\n[PROOF STATE]\nproof (chain)\npicking this:\nJ < degree G\n[PROOF STEP]\nhave \"(J = 0 \\ J < degree G - 1) \\ J = degree G - 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nJ < degree G\n\ngoal (1 subgoal):\n 1. (J = 0 \\ J < degree G - 1) \\ J = degree G - 1\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\n(J = 0 \\ J < degree G - 1) \\ J = degree G - 1\n\ngoal (2 subgoals):\n 1. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n 2. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nwith 00 01\n[PROOF STATE]\nproof (chain)\npicking this:\nJ = 0 \\ J < degree G - 1 \\ subresultant J F G = 0\nsubresultant (degree G - 1) F G = 0\n(J = 0 \\ J < degree G - 1) \\ J = degree G - 1\n[PROOF STEP]\nhave \"subresultant J F G = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nJ = 0 \\ J < degree G - 1 \\ subresultant J F G = 0\nsubresultant (degree G - 1) F G = 0\n(J = 0 \\ J < degree G - 1) \\ J = degree G - 1\n\ngoal (1 subgoal):\n 1. subresultant J F G = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsubresultant J F G = 0\n\ngoal (2 subgoals):\n 1. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n 2. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsubresultant J F G = 0\n\ngoal (1 subgoal):\n 1. subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nusing J\n[PROOF STATE]\nproof (prove)\nusing this:\nsubresultant J F G = 0\nJ < degree G\n\ngoal (1 subgoal):\n 1. subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsubresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\ncase J: False\n[PROOF STATE]\nproof (state)\nthis:\n\\ J < degree G\n\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nhence dg: \"degree G - J = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ J < degree G\n\ngoal (1 subgoal):\n 1. degree G - J = 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndegree G - J = 0\n\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nlet ?n = \"degree F - J\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nhave *: \"(j :: nat) < 0 \\ False\" \"j - 0 = j\" for j\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (j < 0) = False &&& j - 0 = j\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(?j < 0) = False\n?j - 0 = ?j\n\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nlet ?M = \"mat ?n ?n\n (\\(i, j).\n if i = ?n - 1 then monom 1 (?n - 1 - j) * G\n else [:coeff_int G (int (degree G) - int i + int j):])\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nhave \"subresultant J F G = det ?M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. subresultant J F G = det (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]))\n[PROOF STEP]\nunfolding subresultant_def subresultant_mat_def Let_def dg *\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (mat (degree F - J + 0) (degree F - J + 0) (\\(i, j). if False then if i = degree F - J + 0 - 1 then monom (1::'a) (0 - 1 - j) * F else [:coeff_int F (int (degree F) - int i + int j):] else if i = degree F - J + 0 - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):])) = det (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsubresultant J F G = det (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]))\n\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsubresultant J F G = det (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]))\n\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nhave \"det ?M = prod_list (diag_mat ?M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):])) = prod_list (diag_mat (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):])))\n[PROOF STEP]\nby (rule det_lower_triangular[of ?n], auto intro: coeff_int_eq_0)\n[PROOF STATE]\nproof (state)\nthis:\ndet (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):])) = prod_list (diag_mat (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):])))\n\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndet (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):])) = prod_list (diag_mat (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):])))\n\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nhave \"\\ = (\\i = 0..< ?n. ?M $$ (i,i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod_list (diag_mat (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]))) = (\\i = 0..(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]) $$ (i, i))\n[PROOF STEP]\nunfolding prod_list_diag_prod\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):])). mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]) $$ (i, i)) = (\\i = 0..(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]) $$ (i, i))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nprod_list (diag_mat (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]))) = (\\i = 0..(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]) $$ (i, i))\n\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nprod_list (diag_mat (mat (degree F - J) (degree F - J) (\\(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]))) = (\\i = 0..(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]) $$ (i, i))\n\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nhave \"\\ = (\\i = 0..< ?n. if i = ?n - 1 then G else [: lead_coeff G :])\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]) $$ (i, i)) = (\\i = 0..i = 0..(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]) $$ (i, i)) = (\\i = 0.. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..(i, j). if i = degree F - J - 1 then monom (1::'a) (degree F - J - 1 - j) * G else [:coeff_int G (int (degree G) - int i + int j):]) $$ (i, i)) = (\\i = 0.. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nhave \"\\ = (if J < degree F then smult (lead_coeff G ^ (?n - 1)) G else 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0.. (\\i = 0.. J < degree F \\ (\\i = 0.. (\\i = 0.. J < degree F \\ (\\i = 0.. {?n - 1}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nJ < degree F\n\ngoal (1 subgoal):\n 1. {0.. {degree F - J - 1}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{0.. {degree F - J - 1}\n\ngoal (2 subgoals):\n 1. J < degree F \\ (\\i = 0.. J < degree F \\ (\\i = 0..i = 0..< ?n. if i = ?n - 1 then G else [: lead_coeff G :])\n = (\\i = 0 ..< ?n - 1. if i = ?n - 1 then G else [: lead_coeff G :]) * G\" (is \"_ = ?P * G\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..i = 0..i\\{0.. {degree F - J - 1}. if i = degree F - J - 1 then G else [:lead_coeff G:]) = (\\i = 0..i = 0..i = 0.. (\\i = 0.. J < degree F \\ (\\i = 0..i = 0..i = 0.. (\\i = 0.. J < degree F \\ (\\i = 0..i = 0 ..< ?n - 1. [: lead_coeff G :])\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..i = 0..i = 0..i = 0.. (\\i = 0.. J < degree F \\ (\\i = 0..i = 0..i = 0.. (\\i = 0.. J < degree F \\ (\\i = 0.. = [: lead_coeff G ^ (?n - 1) :]\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..i = 0.. (\\i = 0.. J < degree F \\ (\\i = 0..i = 0..i = 0..i = 0..i = 0.. J < degree F \\ (\\i = 0..i = 0.. J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsubresultant J F G = (if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nhave \"subresultant J F G =\n (if J < degree F then smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsubresultant J F G = (if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n\ngoal (1 subgoal):\n 1. subresultant J F G = (if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nsubresultant J F G = (if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n\ngoal (1 subgoal):\n 1. \\ J < degree G \\ subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsubresultant J F G = (if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n\ngoal (1 subgoal):\n 1. subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nusing J\n[PROOF STATE]\nproof (prove)\nusing this:\nsubresultant J F G = (if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n\\ J < degree G\n\ngoal (1 subgoal):\n 1. subresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsubresultant J F G = (if J < degree G then 0 else if J < degree F then Polynomial.smult (lead_coeff G ^ (degree F - J - 1)) G else 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 12176, "file": "Subresultants_Subresultant", "length": 74, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.877476793890012, "lm_q2_score": 0.8056321796478255, "lm_q1q2_score": 0.7069235420519961}} {"text": "[STATEMENT]\nlemma ordinality_of_utility_function :\n fixes f :: \"real \\ real\"\n assumes monot: \"monotone (>) (>) f\"\n shows \"(f \\ u) x > (f \\ u) y \\ u x > u y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((f \\ u) y < (f \\ u) x) = (u y < u x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((f \\ u) y < (f \\ u) x) = (u y < u x)\n[PROOF STEP]\nlet ?func = \"(\\x. f(u x))\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((f \\ u) y < (f \\ u) x) = (u y < u x)\n[PROOF STEP]\nhave \"\\m n . u m \\ u n \\ ?func m \\ ?func n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m n. (u n \\ u m) = (f (u n) \\ f (u m))\n[PROOF STEP]\nby (metis le_less monot monotone_def not_less)\n[PROOF STATE]\nproof (state)\nthis:\n\\m n. (u n \\ u m) = (f (u n) \\ f (u m))\n\ngoal (1 subgoal):\n 1. ((f \\ u) y < (f \\ u) x) = (u y < u x)\n[PROOF STEP]\nhence \"u x > u y \\ ?func x > ?func y\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\m n. (u n \\ u m) = (f (u n) \\ f (u m))\n\ngoal (1 subgoal):\n 1. (u y < u x) = (f (u y) < f (u x))\n[PROOF STEP]\nusing not_le\n[PROOF STATE]\nproof (prove)\nusing this:\n\\m n. (u n \\ u m) = (f (u n) \\ f (u m))\n(\\ ?x \\ ?y) = (?y < ?x)\n\ngoal (1 subgoal):\n 1. (u y < u x) = (f (u y) < f (u x))\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(u y < u x) = (f (u y) < f (u x))\n\ngoal (1 subgoal):\n 1. ((f \\ u) y < (f \\ u) x) = (u y < u x)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(u y < u x) = (f (u y) < f (u x))\n\ngoal (1 subgoal):\n 1. ((f \\ u) y < (f \\ u) x) = (u y < u x)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n((f \\ u) y < (f \\ u) x) = (u y < u x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 933, "file": "First_Welfare_Theorem_Utility_Functions", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256551882382, "lm_q2_score": 0.8397339716830606, "lm_q1q2_score": 0.7069096008959139}} {"text": "[STATEMENT]\nlemma consistent_imp_nonneg_constant:\n assumes \"consistent x t T B p incr\"\n assumes \"t < T\"\n shows \"B \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ B\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 \\ B\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nconsistent x t T B p incr\nt < T\n[PROOF STEP]\nhave \"T - t > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nconsistent x t T B p incr\nt < T\n\ngoal (1 subgoal):\n 1. 0 < T - t\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < T - t\n\ngoal (1 subgoal):\n 1. 0 \\ B\n[PROOF STEP]\nhave \"0 \\ dist (x T) (discrete_evolution incr T t (x t))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ dist (x T) (discrete_evolution incr T t (x t))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ dist (x T) (discrete_evolution incr T t (x t))\n\ngoal (1 subgoal):\n 1. 0 \\ B\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ dist (x T) (discrete_evolution incr T t (x t))\n\ngoal (1 subgoal):\n 1. 0 \\ B\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nconsistent x t T B p incr\nt < T\n[PROOF STEP]\nhave \"... \\ B * (T - t) ^ (p + 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nconsistent x t T B p incr\nt < T\n\ngoal (1 subgoal):\n 1. dist (x T) (discrete_evolution incr T t (x t)) \\ B * (T - t) ^ (p + 1)\n[PROOF STEP]\nunfolding consistent_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\h\\0. t + h \\ T \\ dist (x (t + h)) (discrete_evolution incr (t + h) t (x t)) \\ B * h ^ (p + 1)\nt < T\n\ngoal (1 subgoal):\n 1. dist (x T) (discrete_evolution incr T t (x t)) \\ B * (T - t) ^ (p + 1)\n[PROOF STEP]\nby (auto dest: spec[where x=\"T - t\"])\n[PROOF STATE]\nproof (state)\nthis:\ndist (x T) (discrete_evolution incr T t (x t)) \\ B * (T - t) ^ (p + 1)\n\ngoal (1 subgoal):\n 1. 0 \\ B\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ B * (T - t) ^ (p + 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ B * (T - t) ^ (p + 1)\n\ngoal (1 subgoal):\n 1. 0 \\ B\n[PROOF STEP]\nusing zero_less_power[OF \\T - t > 0\\, of \"p+1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ B * (T - t) ^ (p + 1)\n0 < (T - t) ^ (p + 1)\n\ngoal (1 subgoal):\n 1. 0 \\ B\n[PROOF STEP]\nby (simp add: zero_le_mult_iff)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1164, "file": "Ordinary_Differential_Equations_Numerics_One_Step_Method", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256512199033, "lm_q2_score": 0.8397339716830606, "lm_q1q2_score": 0.7069095975635683}} {"text": "[STATEMENT]\nlemma between_nth_primes_imp_nonprime:\n assumes \"n > nth_prime k\" \"n < nth_prime (Suc k)\"\n shows \"\\prime n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ prime n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nnth_prime k < n\nn < nth_prime (Suc k)\n\ngoal (1 subgoal):\n 1. \\ prime n\n[PROOF STEP]\nby (metis Suc_leI not_le nth_prime_Suc smallest_prime_beyond_smallest)", "meta": {"llama_tokens": 184, "file": "Prime_Distribution_Elementary_Prime_Distribution_Elementary_Library", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314677809303, "lm_q2_score": 0.7981867801399695, "lm_q1q2_score": 0.7068993296586958}} {"text": "[STATEMENT]\nlemma t_heap_of_A_log_bound:\n \"t_heap_of_A xs \\ length xs * (nlog2 (length xs + 1) + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. t_heap_of_A xs \\ length xs * (nat \\log 2 (real (length xs + 1))\\ + 1)\n[PROOF STEP]\nusing t_heap_of_A_bound[of xs]\n acomplete_if_braun[OF braun_heap_of_A, of xs]\n[PROOF STATE]\nproof (prove)\nusing this:\nt_heap_of_A xs \\ length xs * (height (heap_of_A xs) + 1)\nacomplete (heap_of_A xs)\n\ngoal (1 subgoal):\n 1. t_heap_of_A xs \\ length xs * (nat \\log 2 (real (length xs + 1))\\ + 1)\n[PROOF STEP]\nby (simp add: height_acomplete size1_size size_heap_of_A)", "meta": {"llama_tokens": 309, "file": "Priority_Queue_Braun_Sorting_Braun", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314677809303, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7068993275328719}} {"text": "[STATEMENT]\nlemma between_nth_primes_imp_nonprime:\n assumes \"n > nth_prime k\" \"n < nth_prime (Suc k)\"\n shows \"\\prime n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ prime n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nnth_prime k < n\nn < nth_prime (Suc k)\n\ngoal (1 subgoal):\n 1. \\ prime n\n[PROOF STEP]\nby (metis Suc_leI not_le nth_prime_Suc smallest_prime_beyond_smallest)", "meta": {"llama_tokens": 184, "file": "Prime_Number_Theorem_Prime_Counting_Functions", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314677809303, "lm_q2_score": 0.7981867729389246, "lm_q1q2_score": 0.706899323281224}} {"text": "[STATEMENT]\nlemma sin_i_times:\n fixes z :: complex\n shows \"sin(\\ * z) = \\ * ((exp z - inverse (exp z)) / 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin (\\ * z) = \\ * ((exp z - inverse (exp z)) / 2)\n[PROOF STEP]\nusing sinh_complex\n[PROOF STATE]\nproof (prove)\nusing this:\n(exp ?z - inverse (exp ?z)) / 2 = - \\ * sin (\\ * ?z)\n\ngoal (1 subgoal):\n 1. sin (\\ * z) = \\ * ((exp z - inverse (exp z)) / 2)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 205, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314677809304, "lm_q2_score": 0.7981867729389246, "lm_q1q2_score": 0.706899323281224}} {"text": "[STATEMENT]\nlemma trace_minus_linear:\n fixes A B :: \"'a::comm_ring mat\"\n assumes A: \"A \\ carrier_mat n n\" and B: \"B \\ carrier_mat n n\"\n shows \"trace (A - B) = trace A - trace B\" (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (A - B) = trace A - trace B\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. trace (A - B) = trace A - trace B\n[PROOF STEP]\nhave \"?lhs = (\\i=0..i = 0..i = 0..i = 0.. carrier_mat n n\nB \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. (\\i = 0..i = 0..i = 0..i = 0.. = (\\i=0..i=0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i=0..i=0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i=0..i=0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0.. carrier_mat n n\nB \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. (\\i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0.. x\" \"x \\ 1\"\n shows \"arcsin x + arccos x = pi/2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arcsin x + arccos x = pi / 2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. arcsin x + arccos x = pi / 2\n[PROOF STEP]\nhave \"arcsin x = pi/2 - arccos x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arcsin x = pi / 2 - arccos x\n[PROOF STEP]\napply (rule sin_inj_pi)\n[PROOF STATE]\nproof (prove)\ngoal (5 subgoals):\n 1. - (pi / 2) \\ arcsin x\n 2. arcsin x \\ pi / 2\n 3. - (pi / 2) \\ pi / 2 - arccos x\n 4. pi / 2 - arccos x \\ pi / 2\n 5. sin (arcsin x) = sin (pi / 2 - arccos x)\n[PROOF STEP]\nusing assms arcsin [OF assms] arccos [OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n- 1 \\ x\nx \\ 1\n- (pi / 2) \\ arcsin x \\ arcsin x \\ pi / 2 \\ sin (arcsin x) = x\n0 \\ arccos x \\ arccos x \\ pi \\ cos (arccos x) = x\n\ngoal (5 subgoals):\n 1. - (pi / 2) \\ arcsin x\n 2. arcsin x \\ pi / 2\n 3. - (pi / 2) \\ pi / 2 - arccos x\n 4. pi / 2 - arccos x \\ pi / 2\n 5. sin (arcsin x) = sin (pi / 2 - arccos x)\n[PROOF STEP]\nby (auto simp: algebra_simps sin_diff)\n[PROOF STATE]\nproof (state)\nthis:\narcsin x = pi / 2 - arccos x\n\ngoal (1 subgoal):\n 1. arcsin x + arccos x = pi / 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\narcsin x = pi / 2 - arccos x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\narcsin x = pi / 2 - arccos x\n\ngoal (1 subgoal):\n 1. arcsin x + arccos x = pi / 2\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\narcsin x + arccos x = pi / 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 843, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199633332891, "lm_q2_score": 0.8438951104066293, "lm_q1q2_score": 0.7068633914359429}} {"text": "[STATEMENT]\nlemma (in Module) mHom_ker_eq:\"\\R module N; f \\ mHom R M N; a \\ carrier M; \n b\\ carrier M; f a = f b\\ \\ a \\ (-\\<^sub>a b) \\ ker\\<^bsub>M,N\\<^esub> f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\R module N; f \\ mHom R M N; a \\ carrier M; b \\ carrier M; f a = f b\\ \\ a \\ -\\<^sub>a b \\ ker\\<^bsub>M,N\\<^esub> f\n[PROOF STEP]\napply (simp add:ker_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\R module N; f \\ mHom R M N; a \\ carrier M; b \\ carrier M; f a = f b\\ \\ a \\ -\\<^sub>a b \\ carrier M \\ f (a \\ -\\<^sub>a b) = \\\\<^bsub>N\\<^esub>\n[PROOF STEP]\napply (frule ag_mOp_closed[of b])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\R module N; f \\ mHom R M N; a \\ carrier M; b \\ carrier M; f a = f b; -\\<^sub>a b \\ carrier M\\ \\ a \\ -\\<^sub>a b \\ carrier M \\ f (a \\ -\\<^sub>a b) = \\\\<^bsub>N\\<^esub>\n[PROOF STEP]\napply (simp add:ag_pOp_closed)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\R module N; f \\ mHom R M N; a \\ carrier M; b \\ carrier M; f a = f b; -\\<^sub>a b \\ carrier M\\ \\ f (a \\ -\\<^sub>a b) = \\\\<^bsub>N\\<^esub>\n[PROOF STEP]\napply (simp add:mHom_add mHom_inv)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\R module N; f \\ mHom R M N; a \\ carrier M; b \\ carrier M; f a = f b; -\\<^sub>a b \\ carrier M\\ \\ f b \\\\<^bsub>N\\<^esub> -\\<^sub>a\\<^bsub>N\\<^esub> f b = \\\\<^bsub>N\\<^esub>\n[PROOF STEP]\napply (frule mHom_mem [of N f b], assumption+)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\R module N; f \\ mHom R M N; a \\ carrier M; b \\ carrier M; f a = f b; -\\<^sub>a b \\ carrier M; f b \\ carrier N\\ \\ f b \\\\<^bsub>N\\<^esub> -\\<^sub>a\\<^bsub>N\\<^esub> f b = \\\\<^bsub>N\\<^esub>\n[PROOF STEP]\napply (frule_tac R = R and M = N in Module.module_is_ag,\n simp add:aGroup.ag_r_inv1)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1009, "file": "Group-Ring-Module_Algebra7", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8376199592797929, "lm_q2_score": 0.8438950947024555, "lm_q1q2_score": 0.7068633748610877}} {"text": "[STATEMENT]\nlemma cong_iff_lin_poly: \"([(a::'b::{field_gcd} poly) = b] (mod m)) = (\\k. b = a + m * k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [a = b] (mod m) = (\\k. b = a + m * k)\n[PROOF STEP]\nusing cong_diff_iff_cong_0 [of b a m]\n[PROOF STATE]\nproof (prove)\nusing this:\n[b - a = 0] (mod m) = [b = a] (mod m)\n\ngoal (1 subgoal):\n 1. [a = b] (mod m) = (\\k. b = a + m * k)\n[PROOF STEP]\nby (auto simp add: cong_0_iff dvd_def algebra_simps dest: cong_sym)", "meta": {"llama_tokens": 236, "file": "Berlekamp_Zassenhaus_Chinese_Remainder_Poly", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357666736773, "lm_q2_score": 0.8152324848629215, "lm_q1q2_score": 0.7068357225304103}} {"text": "[STATEMENT]\ntheorem Units_bijective:\n \"Units = {\\ \\ S \\\\<^sub>E S. bij_betw \\ S S}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Units = {\\ \\ S \\\\<^sub>E S. bij_betw \\ S S}\n[PROOF STEP]\nunfolding Units_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {u \\ S \\\\<^sub>E S. invertible u} = {\\ \\ S \\\\<^sub>E S. bij_betw \\ S S}\n[PROOF STEP]\nby (auto simp add: invertible_is_bijective)", "meta": {"llama_tokens": 202, "file": "Jacobson_Basic_Algebra_Group_Theory", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637648915617, "lm_q2_score": 0.8221891283434877, "lm_q1q2_score": 0.706806201524674}} {"text": "[STATEMENT]\nlemma onorm_id: \"onorm (\\x. x::'a::{real_normed_vector, perfect_space}) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. onorm (\\x. x) = 1\n[PROOF STEP]\nproof (rule antisym[OF onorm_id_le])\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 1 \\ onorm (\\x. x)\n[PROOF STEP]\nhave \"{0::'a} \\ UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {0::'a} \\ UNIV\n[PROOF STEP]\nby (metis not_open_singleton open_UNIV)\n[PROOF STATE]\nproof (state)\nthis:\n{0::'a} \\ UNIV\n\ngoal (1 subgoal):\n 1. 1 \\ onorm (\\x. x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{0::'a} \\ UNIV\n[PROOF STEP]\nobtain x :: 'a where \"x \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{0::'a} \\ UNIV\n\ngoal (1 subgoal):\n 1. (\\x. x \\ (0::'a) \\ thesis) \\ thesis\n[PROOF STEP]\nby fast\n[PROOF STATE]\nproof (state)\nthis:\nx \\ (0::'a)\n\ngoal (1 subgoal):\n 1. 1 \\ onorm (\\x. x)\n[PROOF STEP]\nhence \"1 \\ norm x / norm x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ (0::'a)\n\ngoal (1 subgoal):\n 1. 1 \\ norm x / norm x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ norm x / norm x\n\ngoal (1 subgoal):\n 1. 1 \\ onorm (\\x. x)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ norm x / norm x\n\ngoal (1 subgoal):\n 1. 1 \\ onorm (\\x. x)\n[PROOF STEP]\nhave \"\\ \\ onorm (\\x::'a. x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm x / norm x \\ onorm (\\x. x)\n[PROOF STEP]\nby (rule le_onorm) (rule bounded_linear_ident)\n[PROOF STATE]\nproof (state)\nthis:\nnorm x / norm x \\ onorm (\\x. x)\n\ngoal (1 subgoal):\n 1. 1 \\ onorm (\\x. x)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n1 \\ onorm (\\x. x)\n[PROOF STEP]\nshow \"1 \\ onorm (\\x::'a. x)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ onorm (\\x. x)\n\ngoal (1 subgoal):\n 1. 1 \\ onorm (\\x. x)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ onorm (\\x. x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1007, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637577007393, "lm_q2_score": 0.8221891283434876, "lm_q1q2_score": 0.7068061956124579}} {"text": "[STATEMENT]\nlemma span_not_univ_orthogonal:\n fixes S :: \"'a::euclidean_space set\"\n assumes sU: \"span S \\ UNIV\"\n shows \"\\a::'a. a \\ 0 \\ (\\x \\ span S. a \\ x = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n[PROOF STEP]\nfrom sU\n[PROOF STATE]\nproof (chain)\npicking this:\nspan S \\ UNIV\n[PROOF STEP]\nobtain a where a: \"a \\ span S\"\n[PROOF STATE]\nproof (prove)\nusing this:\nspan S \\ UNIV\n\ngoal (1 subgoal):\n 1. (\\a. a \\ span S \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\na \\ span S\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n[PROOF STEP]\nfrom orthogonal_basis_exists\n[PROOF STATE]\nproof (chain)\npicking this:\n\\B. independent B \\ B \\ span ?V \\ ?V \\ span B \\ card B = dim ?V \\ pairwise orthogonal B\n[PROOF STEP]\nobtain B where\n B: \"independent B\" \"B \\ span S\" \"S \\ span B\"\n \"card B = dim S\" \"pairwise orthogonal B\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\B. independent B \\ B \\ span ?V \\ ?V \\ span B \\ card B = dim ?V \\ pairwise orthogonal B\n\ngoal (1 subgoal):\n 1. (\\B. \\independent B; B \\ span S; S \\ span B; card B = dim S; pairwise orthogonal B\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nindependent B\nB \\ span S\nS \\ span B\ncard B = dim S\npairwise orthogonal B\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n[PROOF STEP]\nfrom B\n[PROOF STATE]\nproof (chain)\npicking this:\nindependent B\nB \\ span S\nS \\ span B\ncard B = dim S\npairwise orthogonal B\n[PROOF STEP]\nhave fB: \"finite B\" \"card B = dim S\"\n[PROOF STATE]\nproof (prove)\nusing this:\nindependent B\nB \\ span S\nS \\ span B\ncard B = dim S\npairwise orthogonal B\n\ngoal (1 subgoal):\n 1. finite B &&& card B = dim S\n[PROOF STEP]\nusing independent_bound\n[PROOF STATE]\nproof (prove)\nusing this:\nindependent B\nB \\ span S\nS \\ span B\ncard B = dim S\npairwise orthogonal B\nindependent ?S \\ finite ?S \\ card ?S \\ DIM(?'a)\n\ngoal (1 subgoal):\n 1. finite B &&& card B = dim S\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite B\ncard B = dim S\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n[PROOF STEP]\nfrom span_mono[OF B(2)] span_mono[OF B(3)]\n[PROOF STATE]\nproof (chain)\npicking this:\nspan B \\ span (span S)\nspan S \\ span (span B)\n[PROOF STEP]\nhave sSB: \"span S = span B\"\n[PROOF STATE]\nproof (prove)\nusing this:\nspan B \\ span (span S)\nspan S \\ span (span B)\n\ngoal (1 subgoal):\n 1. span S = span B\n[PROOF STEP]\nby (simp add: span_span)\n[PROOF STATE]\nproof (state)\nthis:\nspan S = span B\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n[PROOF STEP]\nlet ?a = \"a - sum (\\b. (a \\ b / (b \\ b)) *\\<^sub>R b) B\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n[PROOF STEP]\nhave \"sum (\\b. (a \\ b / (b \\ b)) *\\<^sub>R b) B \\ span S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b) \\ span S\n[PROOF STEP]\nunfolding sSB\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b) \\ span B\n[PROOF STEP]\napply (rule span_sum)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ (a \\ x / (x \\ x)) *\\<^sub>R x \\ span B\n[PROOF STEP]\napply (rule span_scale)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ x \\ span B\n[PROOF STEP]\napply (rule span_base)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ x \\ B\n[PROOF STEP]\napply assumption\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b) \\ span S\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n[PROOF STEP]\nwith a\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ span S\n(\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b) \\ span S\n[PROOF STEP]\nhave a0:\"?a \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ span S\n(\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b) \\ span S\n\ngoal (1 subgoal):\n 1. a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b) \\ (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b) \\ (0::'a)\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n[PROOF STEP]\nhave \"?a \\ x = 0\" if \"x\\span B\" for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nproof (rule span_induct [OF that])\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. subspace {b. (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ b = 0}\n 2. \\x. x \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nshow \"subspace {x. ?a \\ x = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. subspace {x. (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0}\n[PROOF STEP]\nby (auto simp add: subspace_def inner_add)\n[PROOF STATE]\nproof (state)\nthis:\nsubspace {x. (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0}\n\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nassume x: \"x \\ B\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ B\n\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nfrom x\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ B\n[PROOF STEP]\nhave B': \"B = insert x (B - {x})\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ B\n\ngoal (1 subgoal):\n 1. B = insert x (B - {x})\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nB = insert x (B - {x})\n\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nhave fth: \"finite (B - {x})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (B - {x})\n[PROOF STEP]\nusing fB\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite B\ncard B = dim S\n\ngoal (1 subgoal):\n 1. finite (B - {x})\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfinite (B - {x})\n\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nhave \"?a \\ x = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\napply (subst B')\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a - (\\b\\insert x (B - {x}). (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nusing fB fth\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite B\ncard B = dim S\nfinite (B - {x})\n\ngoal (1 subgoal):\n 1. (a - (\\b\\insert x (B - {x}). (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nunfolding sum_clauses(2)[OF fth]\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite B\ncard B = dim S\nfinite (B - {x})\n\ngoal (1 subgoal):\n 1. (a - (if x \\ B - {x} then \\b\\B - {x}. (a \\ b / (b \\ b)) *\\<^sub>R b else (a \\ x / (x \\ x)) *\\<^sub>R x + (\\b\\B - {x}. (a \\ b / (b \\ b)) *\\<^sub>R b))) \\ x = 0\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite B; card B = dim S\\ \\ (a - ((a \\ x / (x \\ x)) *\\<^sub>R x + (\\b\\B - {x}. (a \\ b / (b \\ b)) *\\<^sub>R b))) \\ x = 0\n[PROOF STEP]\nunfolding inner_simps\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite B; card B = dim S\\ \\ a \\ x - (a \\ x / (x \\ x) * (x \\ x) + (\\b\\B - {x}. (a \\ b / (b \\ b)) *\\<^sub>R b) \\ x) = 0\n[PROOF STEP]\napply (clarsimp simp add: inner_add inner_sum_left)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite B; card B = dim S; x \\ (0::'a)\\ \\ (\\b\\B - {x}. a \\ b * (b \\ x) / (b \\ b)) = 0\n[PROOF STEP]\napply (rule sum.neutral, rule ballI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\finite B; card B = dim S; x \\ (0::'a); x \\ B - {x}\\ \\ a \\ x * (x \\ x) / (x \\ x) = 0\n[PROOF STEP]\napply (simp only: inner_commute)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\finite B; card B = dim S; x \\ (0::'a); x \\ B - {x}\\ \\ x \\ a * (x \\ x) / (x \\ x) = 0\n[PROOF STEP]\napply (auto simp add: x field_simps\n intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n?xa2 \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ ?xa2 = 0\n\ngoal (1 subgoal):\n 1. \\x. x \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n?xa2 \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ ?xa2 = 0\n[PROOF STEP]\nshow \"?a \\ x = 0\" if \"x \\ B\" for x\n[PROOF STATE]\nproof (prove)\nusing this:\n?xa2 \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ ?xa2 = 0\n\ngoal (1 subgoal):\n 1. (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\n?xa2 \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ ?xa2 = 0\nx \\ B\n\ngoal (1 subgoal):\n 1. (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ x = 0\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n?x \\ B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ ?x = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n?x \\ span B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ ?x = 0\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n[PROOF STEP]\nwith a0\n[PROOF STATE]\nproof (chain)\npicking this:\na - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b) \\ (0::'a)\n?x \\ span B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ ?x = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b) \\ (0::'a)\n?x \\ span B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ ?x = 0\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n[PROOF STEP]\nunfolding sSB\n[PROOF STATE]\nproof (prove)\nusing this:\na - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b) \\ (0::'a)\n?x \\ span B \\ (a - (\\b\\B. (a \\ b / (b \\ b)) *\\<^sub>R b)) \\ ?x = 0\n\ngoal (1 subgoal):\n 1. \\a. a \\ (0::'a) \\ (\\x\\span B. a \\ x = 0)\n[PROOF STEP]\nby (auto intro: exI[where x=\"?a\"])\n[PROOF STATE]\nproof (state)\nthis:\n\\a. a \\ (0::'a) \\ (\\x\\span S. a \\ x = 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 6134, "file": null, "length": 61, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637433190939, "lm_q2_score": 0.8221891305219504, "lm_q1q2_score": 0.706806185660771}} {"text": "[STATEMENT]\nlemma cInf_asclose:\n fixes S :: \"'a::{linordered_idom,conditionally_complete_linorder} set\"\n assumes S: \"S \\ {}\"\n and b: \"\\x\\S. \\x - l\\ \\ e\"\n shows \"\\Inf S - l\\ \\ e\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nhave *: \"\\x - l\\ \\ e \\ l - e \\ x \\ x \\ l + e\" for x l e :: 'a\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x - l\\ \\ e) = (l - e \\ x \\ x \\ l + e)\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\n(\\?x - ?l\\ \\ ?e) = (?l - ?e \\ ?x \\ ?x \\ ?l + ?e)\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nhave \"bdd_below S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bdd_below S\n[PROOF STEP]\nusing b\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\S. \\x - l\\ \\ e\n\ngoal (1 subgoal):\n 1. bdd_below S\n[PROOF STEP]\nby (auto intro!: bdd_belowI[of _ \"l - e\"])\n[PROOF STATE]\nproof (state)\nthis:\nbdd_below S\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nwith S b\n[PROOF STATE]\nproof (chain)\npicking this:\nS \\ {}\n\\x\\S. \\x - l\\ \\ e\nbdd_below S\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ {}\n\\x\\S. \\x - l\\ \\ e\nbdd_below S\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nunfolding *\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ {}\n\\x\\S. l - e \\ x \\ x \\ l + e\nbdd_below S\n\ngoal (1 subgoal):\n 1. l - e \\ Inf S \\ Inf S \\ l + e\n[PROOF STEP]\nby (auto intro!: cInf_lower2 cInf_greatest)\n[PROOF STATE]\nproof (state)\nthis:\n\\Inf S - l\\ \\ e\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 916, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424528443251, "lm_q2_score": 0.8354835411997897, "lm_q1q2_score": 0.7067709961536129}} {"text": "[STATEMENT]\nlemma cInf_asclose:\n fixes S :: \"'a::{linordered_idom,conditionally_complete_linorder} set\"\n assumes S: \"S \\ {}\"\n and b: \"\\x\\S. \\x - l\\ \\ e\"\n shows \"\\Inf S - l\\ \\ e\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nhave *: \"\\x - l\\ \\ e \\ l - e \\ x \\ x \\ l + e\" for x l e :: 'a\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x - l\\ \\ e) = (l - e \\ x \\ x \\ l + e)\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\n(\\?x - ?l\\ \\ ?e) = (?l - ?e \\ ?x \\ ?x \\ ?l + ?e)\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nhave \"bdd_below S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bdd_below S\n[PROOF STEP]\nusing b\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\S. \\x - l\\ \\ e\n\ngoal (1 subgoal):\n 1. bdd_below S\n[PROOF STEP]\nby (auto intro!: bdd_belowI[of _ \"l - e\"])\n[PROOF STATE]\nproof (state)\nthis:\nbdd_below S\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nwith S b\n[PROOF STATE]\nproof (chain)\npicking this:\nS \\ {}\n\\x\\S. \\x - l\\ \\ e\nbdd_below S\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ {}\n\\x\\S. \\x - l\\ \\ e\nbdd_below S\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nunfolding *\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ {}\n\\x\\S. l - e \\ x \\ x \\ l + e\nbdd_below S\n\ngoal (1 subgoal):\n 1. l - e \\ Inf S \\ Inf S \\ l + e\n[PROOF STEP]\nby (auto intro!: cInf_lower2 cInf_greatest)\n[PROOF STATE]\nproof (state)\nthis:\n\\Inf S - l\\ \\ e\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 916, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424528443251, "lm_q2_score": 0.8354835371034368, "lm_q1q2_score": 0.7067709926883341}} {"text": "[STATEMENT]\nlemma cInf_asclose:\n fixes S :: \"'a::{linordered_idom,conditionally_complete_linorder} set\"\n assumes S: \"S \\ {}\"\n and b: \"\\x\\S. \\x - l\\ \\ e\"\n shows \"\\Inf S - l\\ \\ e\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nhave *: \"\\x - l\\ \\ e \\ l - e \\ x \\ x \\ l + e\" for x l e :: 'a\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x - l\\ \\ e) = (l - e \\ x \\ x \\ l + e)\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\n(\\?x - ?l\\ \\ ?e) = (?l - ?e \\ ?x \\ ?x \\ ?l + ?e)\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nhave \"bdd_below S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bdd_below S\n[PROOF STEP]\nusing b\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\S. \\x - l\\ \\ e\n\ngoal (1 subgoal):\n 1. bdd_below S\n[PROOF STEP]\nby (auto intro!: bdd_belowI[of _ \"l - e\"])\n[PROOF STATE]\nproof (state)\nthis:\nbdd_below S\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nwith S b\n[PROOF STATE]\nproof (chain)\npicking this:\nS \\ {}\n\\x\\S. \\x - l\\ \\ e\nbdd_below S\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ {}\n\\x\\S. \\x - l\\ \\ e\nbdd_below S\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nunfolding *\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ {}\n\\x\\S. l - e \\ x \\ x \\ l + e\nbdd_below S\n\ngoal (1 subgoal):\n 1. l - e \\ Inf S \\ Inf S \\ l + e\n[PROOF STEP]\nby (auto intro!: cInf_lower2 cInf_greatest)\n[PROOF STATE]\nproof (state)\nthis:\n\\Inf S - l\\ \\ e\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 916, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424528443251, "lm_q2_score": 0.8354835309589073, "lm_q1q2_score": 0.7067709874904157}} {"text": "[STATEMENT]\nlemma cInf_asclose:\n fixes S :: \"'a::{linordered_idom,conditionally_complete_linorder} set\"\n assumes S: \"S \\ {}\"\n and b: \"\\x\\S. \\x - l\\ \\ e\"\n shows \"\\Inf S - l\\ \\ e\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nhave *: \"\\x - l\\ \\ e \\ l - e \\ x \\ x \\ l + e\" for x l e :: 'a\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x - l\\ \\ e) = (l - e \\ x \\ x \\ l + e)\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\n(\\?x - ?l\\ \\ ?e) = (?l - ?e \\ ?x \\ ?x \\ ?l + ?e)\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nhave \"bdd_below S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bdd_below S\n[PROOF STEP]\nusing b\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\S. \\x - l\\ \\ e\n\ngoal (1 subgoal):\n 1. bdd_below S\n[PROOF STEP]\nby (auto intro!: bdd_belowI[of _ \"l - e\"])\n[PROOF STATE]\nproof (state)\nthis:\nbdd_below S\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nwith S b\n[PROOF STATE]\nproof (chain)\npicking this:\nS \\ {}\n\\x\\S. \\x - l\\ \\ e\nbdd_below S\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ {}\n\\x\\S. \\x - l\\ \\ e\nbdd_below S\n\ngoal (1 subgoal):\n 1. \\Inf S - l\\ \\ e\n[PROOF STEP]\nunfolding *\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ {}\n\\x\\S. l - e \\ x \\ x \\ l + e\nbdd_below S\n\ngoal (1 subgoal):\n 1. l - e \\ Inf S \\ Inf S \\ l + e\n[PROOF STEP]\nby (auto intro!: cInf_lower2 cInf_greatest)\n[PROOF STATE]\nproof (state)\nthis:\n\\Inf S - l\\ \\ e\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 916, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424528443251, "lm_q2_score": 0.8354835309589073, "lm_q1q2_score": 0.7067709874904157}} {"text": "[STATEMENT]\nlemma Func_singleton:\nfixes x :: 'b and A :: \"'a set\"\nshows \"|Func A {x}| =o |{x}|\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. |Func A {x}| =o |{x}|\n[PROOF STEP]\nproof (rule ordIso_symmetric)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. |{x}| =o |Func A {x}|\n[PROOF STEP]\ndefine f where [abs_def]: \"f y a = (if y = x \\ a \\ A then x else undefined)\" for y a\n[PROOF STATE]\nproof (state)\nthis:\nf \\ \\y a. if y = x \\ a \\ A then x else undefined\n\ngoal (1 subgoal):\n 1. |{x}| =o |Func A {x}|\n[PROOF STEP]\nhave \"Func A {x} \\ f ` {x}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Func A {x} \\ f ` {x}\n[PROOF STEP]\nunfolding f_def Func_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {f. (\\a\\A. f a \\ {x}) \\ (\\a. a \\ A \\ f a = undefined)} \\ (\\y a. if y = x \\ a \\ A then x else undefined) ` {x}\n[PROOF STEP]\nby (force simp: fun_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\nFunc A {x} \\ f ` {x}\n\ngoal (1 subgoal):\n 1. |{x}| =o |Func A {x}|\n[PROOF STEP]\nhence \"bij_betw f {x} (Func A {x})\"\n[PROOF STATE]\nproof (prove)\nusing this:\nFunc A {x} \\ f ` {x}\n\ngoal (1 subgoal):\n 1. bij_betw f {x} (Func A {x})\n[PROOF STEP]\nunfolding bij_betw_def inj_on_def f_def Func_def\n[PROOF STATE]\nproof (prove)\nusing this:\n{f. (\\a\\A. f a \\ {x}) \\ (\\a. a \\ A \\ f a = undefined)} \\ (\\y a. if y = x \\ a \\ A then x else undefined) ` {x}\n\ngoal (1 subgoal):\n 1. (\\xa\\{x}. \\y\\{x}. (\\a. if xa = x \\ a \\ A then x else undefined) = (\\a. if y = x \\ a \\ A then x else undefined) \\ xa = y) \\ (\\y a. if y = x \\ a \\ A then x else undefined) ` {x} = {f. (\\a\\A. f a \\ {x}) \\ (\\a. a \\ A \\ f a = undefined)}\n[PROOF STEP]\nby (auto split: if_split_asm)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw f {x} (Func A {x})\n\ngoal (1 subgoal):\n 1. |{x}| =o |Func A {x}|\n[PROOF STEP]\nthus \"|{x}| =o |Func A {x}|\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw f {x} (Func A {x})\n\ngoal (1 subgoal):\n 1. |{x}| =o |Func A {x}|\n[PROOF STEP]\nusing card_of_ordIso\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw f {x} (Func A {x})\n(\\f. bij_betw f ?A ?B) = (|?A| =o |?B|)\n\ngoal (1 subgoal):\n 1. |{x}| =o |Func A {x}|\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n|{x}| =o |Func A {x}|\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1169, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835371034369, "lm_q2_score": 0.8459424314825853, "lm_q1q2_score": 0.7067709748409522}} {"text": "[STATEMENT]\ntheorem (in group) group_inverse_times: \"inverse (x * y) = inverse y * inverse x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse (x * y) = inverse y * inverse x\n[PROOF STEP]\nproof (rule group_inverse_equality)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\nshow \"(inverse y * inverse x) * (x * y) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\nhave \"(inverse y * inverse x) * (x * y) =\n (inverse y * (inverse x * x)) * y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = inverse y * (inverse x * x) * y\n[PROOF STEP]\nby (simp only: group_assoc)\n[PROOF STATE]\nproof (state)\nthis:\ninverse y * inverse x * (x * y) = inverse y * (inverse x * x) * y\n\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ninverse y * inverse x * (x * y) = inverse y * (inverse x * x) * y\n\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\nhave \"\\ = (inverse y * 1) * y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse y * (inverse x * x) * y = inverse y * (1::'a) * y\n[PROOF STEP]\nby (simp only: group_left_inverse)\n[PROOF STATE]\nproof (state)\nthis:\ninverse y * (inverse x * x) * y = inverse y * (1::'a) * y\n\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ninverse y * (inverse x * x) * y = inverse y * (1::'a) * y\n\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\nhave \"\\ = inverse y * y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse y * (1::'a) * y = inverse y * y\n[PROOF STEP]\nby (simp only: group_right_one)\n[PROOF STATE]\nproof (state)\nthis:\ninverse y * (1::'a) * y = inverse y * y\n\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ninverse y * (1::'a) * y = inverse y * y\n\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\nhave \"\\ = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse y * y = (1::'a)\n[PROOF STEP]\nby (simp only: group_left_inverse)\n[PROOF STATE]\nproof (state)\nthis:\ninverse y * y = (1::'a)\n\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ninverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse y * inverse x * (x * y) = (1::'a)\n\ngoal (1 subgoal):\n 1. inverse y * inverse x * (x * y) = (1::'a)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ninverse y * inverse x * (x * y) = (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ninverse y * inverse x * (x * y) = (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1351, "file": null, "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424334245618, "lm_q2_score": 0.8354835309589074, "lm_q1q2_score": 0.7067709712655232}} {"text": "[STATEMENT]\nlemma pr_index_enumerator_mono2: \"m1 < m2 \\ pr_index_enumerator n m1 < pr_index_enumerator n m2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. m1 < m2 \\ pr_index_enumerator n m1 < pr_index_enumerator n m2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. m1 < m2 \\ pr_index_enumerator n m1 < pr_index_enumerator n m2\n[PROOF STEP]\nassume A: \"m1 < m2\"\n[PROOF STATE]\nproof (state)\nthis:\nm1 < m2\n\ngoal (1 subgoal):\n 1. m1 < m2 \\ pr_index_enumerator n m1 < pr_index_enumerator n m2\n[PROOF STEP]\ndefine f where \"f x = pr_index_enumerator n x\" for x\n[PROOF STATE]\nproof (state)\nthis:\nf ?x = pr_index_enumerator n ?x\n\ngoal (1 subgoal):\n 1. m1 < m2 \\ pr_index_enumerator n m1 < pr_index_enumerator n m2\n[PROOF STEP]\nhave f_inc: \"\\ x. f x < f (x+1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. f x < f (x + 1)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. f x < f (x + 1)\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. f x < f (x + 1)\n[PROOF STEP]\nshow \"f x < f (x+1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f x < f (x + 1)\n[PROOF STEP]\nby (unfold f_def, rule pr_index_enumerator_increase2)\n[PROOF STATE]\nproof (state)\nthis:\nf x < f (x + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\x. f x < f (x + 1)\n\ngoal (1 subgoal):\n 1. m1 < m2 \\ pr_index_enumerator n m1 < pr_index_enumerator n m2\n[PROOF STEP]\nfrom f_inc\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x. f x < f (x + 1)\n[PROOF STEP]\nhave \"\\ x y. (x < y \\ f x < f y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. f x < f (x + 1)\n\ngoal (1 subgoal):\n 1. \\x y. x < y \\ f x < f y\n[PROOF STEP]\nby (rule f_inc_mono)\n[PROOF STATE]\nproof (state)\nthis:\n\\x y. x < y \\ f x < f y\n\ngoal (1 subgoal):\n 1. m1 < m2 \\ pr_index_enumerator n m1 < pr_index_enumerator n m2\n[PROOF STEP]\nwith A f_def\n[PROOF STATE]\nproof (chain)\npicking this:\nm1 < m2\nf ?x = pr_index_enumerator n ?x\n\\x y. x < y \\ f x < f y\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nm1 < m2\nf ?x = pr_index_enumerator n ?x\n\\x y. x < y \\ f x < f y\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n m1 < pr_index_enumerator n m2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\npr_index_enumerator n m1 < pr_index_enumerator n m2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1184, "file": "Recursion-Theory-I_PRecFun2", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677660619633, "lm_q2_score": 0.8333245891029456, "lm_q1q2_score": 0.7066323902261283}} {"text": "[STATEMENT]\nlemma integral_subset_negligible:\n fixes f :: \"'a :: euclidean_space \\ 'b :: banach\"\n assumes \"S \\ T\" \"negligible (T - S)\"\n shows \"integral S f = integral T f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral S f = integral T f\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. integral S f = integral T f\n[PROOF STEP]\nhave \"integral T f = integral T (\\x. if x \\ S then f x else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral T f = integral T (\\x. if x \\ S then f x else (0::'b))\n[PROOF STEP]\nby (rule integral_spike[of \"T - S\"]) (use assms in auto)\n[PROOF STATE]\nproof (state)\nthis:\nintegral T f = integral T (\\x. if x \\ S then f x else (0::'b))\n\ngoal (1 subgoal):\n 1. integral S f = integral T f\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nintegral T f = integral T (\\x. if x \\ S then f x else (0::'b))\n\ngoal (1 subgoal):\n 1. integral S f = integral T f\n[PROOF STEP]\nhave \"\\ = integral (S \\ T) f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral T (\\x. if x \\ S then f x else (0::'b)) = integral (S \\ T) f\n[PROOF STEP]\nby (subst integral_restrict_Int) auto\n[PROOF STATE]\nproof (state)\nthis:\nintegral T (\\x. if x \\ S then f x else (0::'b)) = integral (S \\ T) f\n\ngoal (1 subgoal):\n 1. integral S f = integral T f\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nintegral T (\\x. if x \\ S then f x else (0::'b)) = integral (S \\ T) f\n\ngoal (1 subgoal):\n 1. integral S f = integral T f\n[PROOF STEP]\nhave \"S \\ T = S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. S \\ T = S\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ T\nnegligible (T - S)\n\ngoal (1 subgoal):\n 1. S \\ T = S\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nS \\ T = S\n\ngoal (1 subgoal):\n 1. integral S f = integral T f\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nintegral T f = integral S f\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nintegral T f = integral S f\n\ngoal (1 subgoal):\n 1. integral S f = integral T f\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nintegral S f = integral T f\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 957, "file": "Prime_Number_Theorem_Prime_Number_Theorem_Library", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333245953120233, "lm_q2_score": 0.8479677564567913, "lm_q1q2_score": 0.706632387487}} {"text": "[STATEMENT]\nlemma q63_upto_def: \"q63 = (\\k. k\\<^sup>2 mod 63) ` {..<63}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. q63 = (\\k. k\\<^sup>2 mod 63) ` {..<63}\n[PROOF STEP]\nby (simp add: q63_def lessThan_nat_numeral lessThan_Suc insert_commute)", "meta": {"llama_tokens": 130, "file": "Pell_Efficient_Discrete_Sqrt", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8479677583778258, "lm_q2_score": 0.8333245932423308, "lm_q1q2_score": 0.7066323873328126}} {"text": "[STATEMENT]\nlemma row_all1_matrix:\nassumes \"i < nr\"\nshows \"row (all1_matrix nr nc) i = all1_vec nc\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row (all1_matrix nr nc) i = all1_vec nc\n[PROOF STEP]\napply (rule eq_vecI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\ia. ia < dim_vec (all1_vec nc) \\ row (all1_matrix nr nc) i $ ia = all1_vec nc $ ia\n 2. dim_vec (row (all1_matrix nr nc) i) = dim_vec (all1_vec nc)\n[PROOF STEP]\napply (simp add: all1_matrix_def all1_vec_def assms)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_vec (row (all1_matrix nr nc) i) = dim_vec (all1_vec nc)\n[PROOF STEP]\nby (simp add: all1_matrix_def all1_vec_def)", "meta": {"llama_tokens": 314, "file": "Deep_Learning_DL_Concrete_Matrices", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333245787544824, "lm_q2_score": 0.8479677602988601, "lm_q1q2_score": 0.7066323766484295}} {"text": "[STATEMENT]\nlemma unitary_is_length_preserving [simp]:\n fixes U:: \"complex mat\" and v:: \"complex vec\"\n assumes \"unitary U\" and \"dim_vec v = dim_col U\"\n shows \"\\U * |v\\\\ = \\v\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\col_fst (U * |v\\)\\ = \\v\\\n[PROOF STEP]\nusing assms unitary_is_sq_length_preserving\n[PROOF STATE]\nproof (prove)\nusing this:\nunitary U\ndim_vec v = dim_col U\n\\unitary ?U; dim_vec ?v = dim_col ?U\\ \\ \\col_fst (?U * |?v\\)\\\\<^sup>2 = \\?v\\\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\col_fst (U * |v\\)\\ = \\v\\\n[PROOF STEP]\nby (metis cpx_vec_length_inner_prod inner_prod_csqrt of_real_hom.injectivity)", "meta": {"llama_tokens": 325, "file": "Isabelle_Marries_Dirac_Quantum", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.88242786954645, "lm_q2_score": 0.8006920116079209, "lm_q1q2_score": 0.7065529459660391}} {"text": "[STATEMENT]\nlemma norm_ge_square: \"norm x \\ a \\ a \\ 0 \\ inner x x \\ a\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a \\ norm x) = (a \\ 0 \\ a\\<^sup>2 \\ inner x x)\n[PROOF STEP]\napply (simp add: dot_square_norm abs_le_square_iff[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a \\ norm x) = (a \\ 0 \\ \\a\\ \\ norm x)\n[PROOF STEP]\nusing norm_ge_zero[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ norm x\n\ngoal (1 subgoal):\n 1. (a \\ norm x) = (a \\ 0 \\ \\a\\ \\ norm x)\n[PROOF STEP]\napply arith\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 317, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.882427872638409, "lm_q2_score": 0.8006920068519376, "lm_q1q2_score": 0.7065529442449338}} {"text": "[STATEMENT]\nlemma uedge_eq_iff: \"uedge (a,b) = uedge (c,d) \\ a=c \\ b=d \\ a=d \\ b=c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (uedge (a, b) = uedge (c, d)) = (a = c \\ b = d \\ a = d \\ b = c)\n[PROOF STEP]\nby (auto simp: uedge_def doubleton_eq_iff)", "meta": {"llama_tokens": 145, "file": "Prim_Dijkstra_Simple_Undirected_Graph", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278602705732, "lm_q2_score": 0.8006920116079209, "lm_q1q2_score": 0.7065529385389187}} {"text": "[STATEMENT]\nlemma Sigma_mset_plus_distrib2[simp]:\n \"Sigma_mset A (\\i. B i + C i) = Sigma_mset A B + Sigma_mset A C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (SIGMAMSET i\\#A. B i + C i) = Sigma_mset A B + Sigma_mset A C\n[PROOF STEP]\nunfolding Sigma_mset_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a\\#A. image_mset (Pair a) (B a + C a)) = (\\a\\#A. image_mset (Pair a) (B a)) + (\\a\\#A. image_mset (Pair a) (C a))\n[PROOF STEP]\nby (induction A) (auto simp: multiset_eq_iff)", "meta": {"llama_tokens": 256, "file": "Nested_Multisets_Ordinals_Multiset_More", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278664544912, "lm_q2_score": 0.8006919973399709, "lm_q1q2_score": 0.7065529308998957}} {"text": "[STATEMENT]\nlemma lm143: \n assumes \"card (Pow A) = 1\" \n shows \"A = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A = {}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (Pow A) = 1\n\ngoal (1 subgoal):\n 1. A = {}\n[PROOF STEP]\nby (metis Pow_bottom Pow_top cardinalityOneTheElemIdentity singletonD)", "meta": {"llama_tokens": 146, "file": "Vickrey_Clarke_Groves_MiscTools", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278540866548, "lm_q2_score": 0.8006920068519376, "lm_q1q2_score": 0.7065529293906925}} {"text": "[STATEMENT]\nlemma log_ceil_idem: \n assumes\"(x::real) \\ 1\" \n shows \"\\lb x \\ = \\lb \\x\\\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n[PROOF STEP]\nhave \"\\log 2 x \\ \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ \\log 2 x\\\n[PROOF STEP]\nby (smt (verit, ccfv_SIG) assms zero_le_ceiling zero_le_log_cancel_iff)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ \\log 2 x\\\n\ngoal (1 subgoal):\n 1. \\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n[PROOF STEP]\nhave \" \\log 2 x \\ -1 < log 2 x \\log 2 x \\ \\log 2 x \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_of_int (\\log 2 x\\ - 1) < log 2 x \\ log 2 x \\ real_of_int \\log 2 x\\\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int (\\log 2 x\\ - 1) < log 2 x \\ log 2 x \\ real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. \\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int (\\log 2 x\\ - 1) < log 2 x \\ log 2 x \\ real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. \\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n[PROOF STEP]\nhence \"2 powr (\\log 2 x \\ -1) < x \\ x \\ 2 powr ( \\log 2 x \\)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal_of_int (\\log 2 x\\ - 1) < log 2 x \\ log 2 x \\ real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. 2 powr real_of_int (\\log 2 x\\ - 1) < x \\ x \\ 2 powr real_of_int \\log 2 x\\\n[PROOF STEP]\nby (smt (verit, ccfv_SIG) assms less_log_iff real_nat_ceiling_ge)\n[PROOF STATE]\nproof (state)\nthis:\n2 powr real_of_int (\\log 2 x\\ - 1) < x \\ x \\ 2 powr real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. \\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n2 powr real_of_int (\\log 2 x\\ - 1) < x \\ x \\ 2 powr real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. \\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n[PROOF STEP]\nhence \"2 powr ((\\log 2 x \\ -1)) < \\x\\\" and \" \\x\\ \\ 2 powr (\\log 2 x \\)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 powr real_of_int (\\log 2 x\\ - 1) < x \\ x \\ 2 powr real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. 2 powr real_of_int (\\log 2 x\\ - 1) < real_of_int \\x\\ &&& real_of_int \\x\\ \\ 2 powr real_of_int \\log 2 x\\\n[PROOF STEP]\napply linarith\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_of_int \\x\\ \\ 2 powr real_of_int \\log 2 x\\\n[PROOF STEP]\nusing \\0 \\ \\log 2 x\\\\ calculation(2) ceiling_mono powr_int\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ \\log 2 x\\\n2 powr real_of_int (\\log 2 x\\ - 1) < x \\ x \\ 2 powr real_of_int \\log 2 x\\\n?y \\ ?x \\ \\?y\\ \\ \\?x\\\n0 < ?x \\ ?x powr real_of_int ?i = (if 0 \\ ?i then ?x ^ nat ?i else 1 / ?x ^ nat (- ?i))\n\ngoal (1 subgoal):\n 1. real_of_int \\x\\ \\ 2 powr real_of_int \\log 2 x\\\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n2 powr real_of_int (\\log 2 x\\ - 1) < real_of_int \\x\\\nreal_of_int \\x\\ \\ 2 powr real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. \\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n2 powr real_of_int (\\log 2 x\\ - 1) < real_of_int \\x\\\nreal_of_int \\x\\ \\ 2 powr real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. \\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n[PROOF STEP]\nhence \" \\log 2 x \\ -1 < log 2 \\ x \\ \\log 2 \\x\\ \\ \\log 2 x \\\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 powr real_of_int (\\log 2 x\\ - 1) < real_of_int \\x\\\nreal_of_int \\x\\ \\ 2 powr real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. real_of_int (\\log 2 x\\ - 1) < log 2 (real_of_int \\x\\) \\ log 2 (real_of_int \\x\\) \\ real_of_int \\log 2 x\\\n[PROOF STEP]\nby (smt (verit, best) assms ceiling_correct less_log_iff)\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int (\\log 2 x\\ - 1) < log 2 (real_of_int \\x\\) \\ log 2 (real_of_int \\x\\) \\ real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. \\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nreal_of_int (\\log 2 x\\ - 1) < log 2 x \\ log 2 x \\ real_of_int \\log 2 x\\\n2 powr real_of_int (\\log 2 x\\ - 1) < x \\ x \\ 2 powr real_of_int \\log 2 x\\\n2 powr real_of_int (\\log 2 x\\ - 1) < real_of_int \\x\\\nreal_of_int \\x\\ \\ 2 powr real_of_int \\log 2 x\\\nreal_of_int (\\log 2 x\\ - 1) < log 2 (real_of_int \\x\\) \\ log 2 (real_of_int \\x\\) \\ real_of_int \\log 2 x\\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal_of_int (\\log 2 x\\ - 1) < log 2 x \\ log 2 x \\ real_of_int \\log 2 x\\\n2 powr real_of_int (\\log 2 x\\ - 1) < x \\ x \\ 2 powr real_of_int \\log 2 x\\\n2 powr real_of_int (\\log 2 x\\ - 1) < real_of_int \\x\\\nreal_of_int \\x\\ \\ 2 powr real_of_int \\log 2 x\\\nreal_of_int (\\log 2 x\\ - 1) < log 2 (real_of_int \\x\\) \\ log 2 (real_of_int \\x\\) \\ real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. \\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\n\\log 2 x\\ = \\log 2 (real_of_int \\x\\)\\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3348, "file": "Van_Emde_Boas_Trees_VEBT_Height", "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711870587668, "lm_q2_score": 0.8311430436757312, "lm_q1q2_score": 0.7064476394486977}} {"text": "[STATEMENT]\nlemma rtrancl_on_converseI:\n assumes \"(y, x) \\ rtrancl_on F r\" shows \"(x, y) \\ rtrancl_on F (r\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x, y) \\ rtrancl_on F (r\\)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(y, x) \\ rtrancl_on F r\n\ngoal (1 subgoal):\n 1. (x, y) \\ rtrancl_on F (r\\)\n[PROOF STEP]\nproof induct\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. y \\ F \\ (y, y) \\ rtrancl_on F (r\\)\n 2. \\ya z. \\(y, ya) \\ rtrancl_on F r; (ya, z) \\ r; ya \\ F; z \\ F; (ya, y) \\ rtrancl_on F (r\\)\\ \\ (z, y) \\ rtrancl_on F (r\\)\n[PROOF STEP]\ncase (step a b)\n[PROOF STATE]\nproof (state)\nthis:\n(y, a) \\ rtrancl_on F r\n(a, b) \\ r\na \\ F\nb \\ F\n(a, y) \\ rtrancl_on F (r\\)\n\ngoal (2 subgoals):\n 1. y \\ F \\ (y, y) \\ rtrancl_on F (r\\)\n 2. \\ya z. \\(y, ya) \\ rtrancl_on F r; (ya, z) \\ r; ya \\ F; z \\ F; (ya, y) \\ rtrancl_on F (r\\)\\ \\ (z, y) \\ rtrancl_on F (r\\)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(y, a) \\ rtrancl_on F r\n(a, b) \\ r\na \\ F\nb \\ F\n(a, y) \\ rtrancl_on F (r\\)\n[PROOF STEP]\nhave \"(b,b) \\ rtrancl_on F (r\\)\" \"(b,a) \\ r\\\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(y, a) \\ rtrancl_on F r\n(a, b) \\ r\na \\ F\nb \\ F\n(a, y) \\ rtrancl_on F (r\\)\n\ngoal (1 subgoal):\n 1. (b, b) \\ rtrancl_on F (r\\) &&& (b, a) \\ r\\\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(b, b) \\ rtrancl_on F (r\\)\n(b, a) \\ r\\\n\ngoal (2 subgoals):\n 1. y \\ F \\ (y, y) \\ rtrancl_on F (r\\)\n 2. \\ya z. \\(y, ya) \\ rtrancl_on F r; (ya, z) \\ r; ya \\ F; z \\ F; (ya, y) \\ rtrancl_on F (r\\)\\ \\ (z, y) \\ rtrancl_on F (r\\)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(b, b) \\ rtrancl_on F (r\\)\n(b, a) \\ r\\\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(b, b) \\ rtrancl_on F (r\\)\n(b, a) \\ r\\\n\ngoal (1 subgoal):\n 1. (b, y) \\ rtrancl_on F (r\\)\n[PROOF STEP]\nusing step\n[PROOF STATE]\nproof (prove)\nusing this:\n(b, b) \\ rtrancl_on F (r\\)\n(b, a) \\ r\\\n(y, a) \\ rtrancl_on F r\n(a, b) \\ r\na \\ F\nb \\ F\n(a, y) \\ rtrancl_on F (r\\)\n\ngoal (1 subgoal):\n 1. (b, y) \\ rtrancl_on F (r\\)\n[PROOF STEP]\nby (metis rtrancl_on_trans rtrancl_on_into_rtrancl_on)\n[PROOF STATE]\nproof (state)\nthis:\n(b, y) \\ rtrancl_on F (r\\)\n\ngoal (1 subgoal):\n 1. y \\ F \\ (y, y) \\ rtrancl_on F (r\\)\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 1502, "file": "Graph_Theory_Rtrancl_On", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711680567799, "lm_q2_score": 0.831143054132195, "lm_q1q2_score": 0.7064476325430212}} {"text": "[STATEMENT]\nlemma fib_rec_odd:\n fixes \\ \\ :: real\n defines \"\\ \\ (1 + sqrt 5) / 2\"\n and \"\\ \\ (1 - sqrt 5) / 2\"\n shows \"fib (Suc (2 * n)) = fib n^2 + fib (Suc n)^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nhave \"of_nat (fib n^2 + fib (Suc n)^2) = ((\\ ^ n - \\ ^ n)\\<^sup>2 + (\\ * \\ ^ n - \\ * \\ ^ n)\\<^sup>2)/5\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real ((fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2) = ((\\ ^ n - \\ ^ n)\\<^sup>2 + (\\ * \\ ^ n - \\ * \\ ^ n)\\<^sup>2) / 5\n[PROOF STEP]\nby (simp add: fib_closed_form[folded \\_def \\_def] field_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\nreal ((fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2) = ((\\ ^ n - \\ ^ n)\\<^sup>2 + (\\ * \\ ^ n - \\ * \\ ^ n)\\<^sup>2) / 5\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal ((fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2) = ((\\ ^ n - \\ ^ n)\\<^sup>2 + (\\ * \\ ^ n - \\ * \\ ^ n)\\<^sup>2) / 5\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nlet ?A = \"\\^(2 * n) + \\^(2 * n) - 2*(\\ * \\)^n + \\^(2 * n + 2) + \\^(2 * n + 2) - 2*(\\ * \\)^(n + 1)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nhave \"(\\ ^ n - \\ ^ n)\\<^sup>2 + (\\ * \\ ^ n - \\ * \\ ^ n)\\<^sup>2 = ?A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ ^ n - \\ ^ n)\\<^sup>2 + (\\ * \\ ^ n - \\ * \\ ^ n)\\<^sup>2 = \\ ^ (2 * n) + \\ ^ (2 * n) - 2 * (\\ * \\) ^ n + \\ ^ (2 * n + 2) + \\ ^ (2 * n + 2) - 2 * (\\ * \\) ^ (n + 1)\n[PROOF STEP]\nby (simp add: power2_eq_square algebra_simps power_mult power_mult_distrib)\n[PROOF STATE]\nproof (state)\nthis:\n(\\ ^ n - \\ ^ n)\\<^sup>2 + (\\ * \\ ^ n - \\ * \\ ^ n)\\<^sup>2 = \\ ^ (2 * n) + \\ ^ (2 * n) - 2 * (\\ * \\) ^ n + \\ ^ (2 * n + 2) + \\ ^ (2 * n + 2) - 2 * (\\ * \\) ^ (n + 1)\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\ ^ n - \\ ^ n)\\<^sup>2 + (\\ * \\ ^ n - \\ * \\ ^ n)\\<^sup>2 = \\ ^ (2 * n) + \\ ^ (2 * n) - 2 * (\\ * \\) ^ n + \\ ^ (2 * n + 2) + \\ ^ (2 * n + 2) - 2 * (\\ * \\) ^ (n + 1)\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nhave \"\\ * \\ = -1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ * \\ = - 1\n[PROOF STEP]\nby (simp add: \\_def \\_def field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\ * \\ = - 1\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ * \\ = - 1\n[PROOF STEP]\nhave \"?A = \\^(2 * n + 1) * (\\ + inverse \\) + \\^(2 * n + 1) * (\\ + inverse \\)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ * \\ = - 1\n\ngoal (1 subgoal):\n 1. \\ ^ (2 * n) + \\ ^ (2 * n) - 2 * (\\ * \\) ^ n + \\ ^ (2 * n + 2) + \\ ^ (2 * n + 2) - 2 * (\\ * \\) ^ (n + 1) = \\ ^ (2 * n + 1) * (\\ + inverse \\) + \\ ^ (2 * n + 1) * (\\ + inverse \\)\n[PROOF STEP]\nby (auto simp: field_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n\\ ^ (2 * n) + \\ ^ (2 * n) - 2 * (\\ * \\) ^ n + \\ ^ (2 * n + 2) + \\ ^ (2 * n + 2) - 2 * (\\ * \\) ^ (n + 1) = \\ ^ (2 * n + 1) * (\\ + inverse \\) + \\ ^ (2 * n + 1) * (\\ + inverse \\)\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\ ^ (2 * n) + \\ ^ (2 * n) - 2 * (\\ * \\) ^ n + \\ ^ (2 * n + 2) + \\ ^ (2 * n + 2) - 2 * (\\ * \\) ^ (n + 1) = \\ ^ (2 * n + 1) * (\\ + inverse \\) + \\ ^ (2 * n + 1) * (\\ + inverse \\)\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nhave \"1 + sqrt 5 > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < 1 + sqrt 5\n[PROOF STEP]\nby (auto intro: add_pos_pos)\n[PROOF STATE]\nproof (state)\nthis:\n0 < 1 + sqrt 5\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < 1 + sqrt 5\n[PROOF STEP]\nhave \"\\ + inverse \\ = sqrt 5\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 + sqrt 5\n\ngoal (1 subgoal):\n 1. \\ + inverse \\ = sqrt 5\n[PROOF STEP]\nby (simp add: \\_def field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\ + inverse \\ = sqrt 5\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\ + inverse \\ = sqrt 5\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nhave \"\\ + inverse \\ = -sqrt 5\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ + inverse \\ = - sqrt 5\n[PROOF STEP]\nby (simp add: \\_def field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\ + inverse \\ = - sqrt 5\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\ + inverse \\ = - sqrt 5\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nhave \"(\\ ^ (2 * n + 1) * sqrt 5 + \\ ^ (2 * n + 1) * - sqrt 5) / 5 =\n (\\ ^ (2 * n + 1) - \\ ^ (2 * n + 1)) * (sqrt 5 / 5)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ ^ (2 * n + 1) * sqrt 5 + \\ ^ (2 * n + 1) * - sqrt 5) / 5 = (\\ ^ (2 * n + 1) - \\ ^ (2 * n + 1)) * (sqrt 5 / 5)\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\ ^ (2 * n + 1) * sqrt 5 + \\ ^ (2 * n + 1) * - sqrt 5) / 5 = (\\ ^ (2 * n + 1) - \\ ^ (2 * n + 1)) * (sqrt 5 / 5)\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\ ^ (2 * n + 1) * sqrt 5 + \\ ^ (2 * n + 1) * - sqrt 5) / 5 = (\\ ^ (2 * n + 1) - \\ ^ (2 * n + 1)) * (sqrt 5 / 5)\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nhave \"sqrt 5 / 5 = inverse (sqrt 5)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt 5 / 5 = inverse (sqrt 5)\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nsqrt 5 / 5 = inverse (sqrt 5)\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsqrt 5 / 5 = inverse (sqrt 5)\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nhave \"(\\ ^ (2 * n + 1) - \\ ^ (2 * n + 1)) * \\ = of_nat (fib (Suc (2 * n)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ ^ (2 * n + 1) - \\ ^ (2 * n + 1)) * inverse (sqrt 5) = real (fib (Suc (2 * n)))\n[PROOF STEP]\nby (simp add: fib_closed_form[folded \\_def \\_def] divide_inverse)\n[PROOF STATE]\nproof (state)\nthis:\n(\\ ^ (2 * n + 1) - \\ ^ (2 * n + 1)) * inverse (sqrt 5) = real (fib (Suc (2 * n)))\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal ((fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2) = real (fib (Suc (2 * n)))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal ((fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2) = real (fib (Suc (2 * n)))\n\ngoal (1 subgoal):\n 1. fib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n[PROOF STEP]\nby (simp only: of_nat_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\nfib (Suc (2 * n)) = (fib n)\\<^sup>2 + (fib (Suc n))\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4244, "file": null, "length": 35, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9252299653388754, "lm_q2_score": 0.7634837743174788, "lm_q1q2_score": 0.7063980660485547}} {"text": "[STATEMENT]\ntheorem is_Sup_binary: \"is_Sup {x, y} sup = is_sup x y sup\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_Sup {x, y} sup = is_sup x y sup\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. is_Sup {x, y} sup = is_sup x y sup\n[PROOF STEP]\nhave \"is_Sup {x, y} sup = is_Inf (dual ` {x, y}) (dual sup)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_Sup {x, y} sup = is_Inf (dual ` {x, y}) (dual sup)\n[PROOF STEP]\nby (simp only: dual_Inf)\n[PROOF STATE]\nproof (state)\nthis:\nis_Sup {x, y} sup = is_Inf (dual ` {x, y}) (dual sup)\n\ngoal (1 subgoal):\n 1. is_Sup {x, y} sup = is_sup x y sup\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nis_Sup {x, y} sup = is_Inf (dual ` {x, y}) (dual sup)\n\ngoal (1 subgoal):\n 1. is_Sup {x, y} sup = is_sup x y sup\n[PROOF STEP]\nhave \"dual ` {x, y} = {dual x, dual y}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dual ` {x, y} = {dual x, dual y}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndual ` {x, y} = {dual x, dual y}\n\ngoal (1 subgoal):\n 1. is_Sup {x, y} sup = is_sup x y sup\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndual ` {x, y} = {dual x, dual y}\n\ngoal (1 subgoal):\n 1. is_Sup {x, y} sup = is_sup x y sup\n[PROOF STEP]\nhave \"is_Inf \\ (dual sup) = is_inf (dual x) (dual y) (dual sup)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_Inf {dual x, dual y} (dual sup) = is_inf (dual x) (dual y) (dual sup)\n[PROOF STEP]\nby (rule is_Inf_binary)\n[PROOF STATE]\nproof (state)\nthis:\nis_Inf {dual x, dual y} (dual sup) = is_inf (dual x) (dual y) (dual sup)\n\ngoal (1 subgoal):\n 1. is_Sup {x, y} sup = is_sup x y sup\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nis_Inf {dual x, dual y} (dual sup) = is_inf (dual x) (dual y) (dual sup)\n\ngoal (1 subgoal):\n 1. is_Sup {x, y} sup = is_sup x y sup\n[PROOF STEP]\nhave \"\\ = is_sup x y sup\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_inf (dual x) (dual y) (dual sup) = is_sup x y sup\n[PROOF STEP]\nby (simp only: dual_inf)\n[PROOF STATE]\nproof (state)\nthis:\nis_inf (dual x) (dual y) (dual sup) = is_sup x y sup\n\ngoal (1 subgoal):\n 1. is_Sup {x, y} sup = is_sup x y sup\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nis_Sup {x, y} sup = is_sup x y sup\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nis_Sup {x, y} sup = is_sup x y sup\n\ngoal (1 subgoal):\n 1. is_Sup {x, y} sup = is_sup x y sup\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nis_Sup {x, y} sup = is_sup x y sup\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1234, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382165412808, "lm_q2_score": 0.819893340314393, "lm_q1q2_score": 0.7063694461685356}} {"text": "[STATEMENT]\nlemma geometric_sums_times_norm:\n fixes c::\"'a::{banach,real_normed_field}\"\n assumes \"norm c < 1\"\n shows \"(\\n. norm (c^n * of_nat n)) sums (norm c / (1 - norm c)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. norm (c ^ n * of_nat n)) sums (norm c / (1 - norm c)\\<^sup>2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\n. norm (c ^ n * of_nat n)) sums (norm c / (1 - norm c)\\<^sup>2)\n[PROOF STEP]\nhave \"norm (c^n * of_nat n) = (norm c) ^ n * of_nat n\" for n::nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (c ^ n * of_nat n) = norm c ^ n * real n\n[PROOF STEP]\nby (simp add: norm_power norm_mult)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (c ^ ?n * of_nat ?n) = norm c ^ ?n * real ?n\n\ngoal (1 subgoal):\n 1. (\\n. norm (c ^ n * of_nat n)) sums (norm c / (1 - norm c)\\<^sup>2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (c ^ ?n * of_nat ?n) = norm c ^ ?n * real ?n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (c ^ ?n * of_nat ?n) = norm c ^ ?n * real ?n\n\ngoal (1 subgoal):\n 1. (\\n. norm (c ^ n * of_nat n)) sums (norm c / (1 - norm c)\\<^sup>2)\n[PROOF STEP]\nusing geometric_sums_times_n[of \"norm c\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (c ^ ?n * of_nat ?n) = norm c ^ ?n * real ?n\nnorm (norm c) < 1 \\ (\\n. norm c ^ n * real n) sums (norm c / (1 - norm c)\\<^sup>2)\nnorm c < 1\n\ngoal (1 subgoal):\n 1. (\\n. norm (c ^ n * of_nat n)) sums (norm c / (1 - norm c)\\<^sup>2)\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. norm (c ^ n * of_nat n)) sums (norm c / (1 - norm c)\\<^sup>2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 785, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382165412809, "lm_q2_score": 0.8198933381139645, "lm_q1q2_score": 0.7063694442727824}} {"text": "[STATEMENT]\nlemma cartesian_product_complement_left:\n assumes \"B \\ carrier (R\\<^bsup>m\\<^esup>)\"\n assumes \"A \\ carrier (R\\<^bsup>n\\<^esup>)\"\n shows \"cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - (cartesian_product B A) = \n cartesian_product ((carrier (R\\<^bsup>m\\<^esup>)) - B) A \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A = cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A\n 2. cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n[PROOF STEP]\nshow \"cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A \\ cartesian_product ((carrier (R\\<^bsup>m\\<^esup>)) - B) A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A\n[PROOF STEP]\nunfolding cartesian_product_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {xs. \\as\\carrier (R\\<^bsup>m\\<^esup>). \\bs\\A. xs = as @ bs} - {xs. \\as\\B. \\bs\\A. xs = as @ bs} \\ {xs. \\as\\carrier (R\\<^bsup>m\\<^esup>) - B. \\bs\\A. xs = as @ bs}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ncartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A\n\ngoal (1 subgoal):\n 1. cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n[PROOF STEP]\nshow \"cartesian_product ((carrier (R\\<^bsup>m\\<^esup>)) - B) A \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A \\ x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A \\ x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n[PROOF STEP]\nassume A: \"x \\ cartesian_product ((carrier (R\\<^bsup>m\\<^esup>)) - B) A\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A\n\ngoal (1 subgoal):\n 1. \\x. x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A \\ x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n[PROOF STEP]\nhave 0: \"x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A\n[PROOF STEP]\nusing A\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A\n\ngoal (1 subgoal):\n 1. x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A\n[PROOF STEP]\nunfolding cartesian_product_def\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ {xs. \\as\\carrier (R\\<^bsup>m\\<^esup>) - B. \\bs\\A. xs = as @ bs}\n\ngoal (1 subgoal):\n 1. x \\ {xs. \\as\\carrier (R\\<^bsup>m\\<^esup>). \\bs\\A. xs = as @ bs}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nx \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A\n\ngoal (1 subgoal):\n 1. \\x. x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A \\ x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n[PROOF STEP]\nhave 1: \"take m x \\ (carrier (R\\<^bsup>m\\<^esup>)) - B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. take m x \\ carrier (R\\<^bsup>m\\<^esup>) - B\n[PROOF STEP]\nusing A cartesian_product_memE[of x \"((carrier (R\\<^bsup>m\\<^esup>)) - B)\" A R m]\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A\n\\x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A; carrier (R\\<^bsup>m\\<^esup>) - B \\ carrier (R\\<^bsup>m\\<^esup>)\\ \\ take m x \\ carrier (R\\<^bsup>m\\<^esup>) - B\n\\x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A; carrier (R\\<^bsup>m\\<^esup>) - B \\ carrier (R\\<^bsup>m\\<^esup>)\\ \\ drop m x \\ A\n\ngoal (1 subgoal):\n 1. take m x \\ carrier (R\\<^bsup>m\\<^esup>) - B\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ntake m x \\ carrier (R\\<^bsup>m\\<^esup>) - B\n\ngoal (1 subgoal):\n 1. \\x. x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A \\ x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n[PROOF STEP]\nhave 2: \"drop m x \\ A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. drop m x \\ A\n[PROOF STEP]\nusing cartesian_product_memE[of x \"((carrier (R\\<^bsup>m\\<^esup>)) - B)\" A R m]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A; carrier (R\\<^bsup>m\\<^esup>) - B \\ carrier (R\\<^bsup>m\\<^esup>)\\ \\ take m x \\ carrier (R\\<^bsup>m\\<^esup>) - B\n\\x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A; carrier (R\\<^bsup>m\\<^esup>) - B \\ carrier (R\\<^bsup>m\\<^esup>)\\ \\ drop m x \\ A\n\ngoal (1 subgoal):\n 1. drop m x \\ A\n[PROOF STEP]\nby (metis A Diff_subset)\n[PROOF STATE]\nproof (state)\nthis:\ndrop m x \\ A\n\ngoal (1 subgoal):\n 1. \\x. x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A \\ x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n[PROOF STEP]\nshow \"x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n[PROOF STEP]\napply(rule ccontr)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A \\ False\n[PROOF STEP]\nusing A 0 1 2 cartesian_product_memE[of x B A R m] assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A\nx \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A\ntake m x \\ carrier (R\\<^bsup>m\\<^esup>) - B\ndrop m x \\ A\n\\x \\ cartesian_product B A; B \\ carrier (R\\<^bsup>m\\<^esup>)\\ \\ take m x \\ B\n\\x \\ cartesian_product B A; B \\ carrier (R\\<^bsup>m\\<^esup>)\\ \\ drop m x \\ A\nB \\ carrier (R\\<^bsup>m\\<^esup>)\nA \\ carrier (R\\<^bsup>n\\<^esup>)\n\ngoal (1 subgoal):\n 1. x \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A \\ False\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nx \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncartesian_product (carrier (R\\<^bsup>m\\<^esup>) - B) A \\ cartesian_product (carrier (R\\<^bsup>m\\<^esup>)) A - cartesian_product B A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3306, "file": "Padic_Field_Ring_Powers", "length": 24, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382200964035, "lm_q2_score": 0.8198933337131076, "lm_q1q2_score": 0.7063694433960972}} {"text": "[STATEMENT]\nlemma integral_bound:\n fixes f::\"real \\ 'a::banach\"\n assumes \"a \\ b\"\n assumes \"continuous_on {a .. b} f\"\n assumes \"\\t. t \\ {a .. b} \\ norm (f t) \\ B\"\n shows \"norm (integral {a .. b} f) \\ B * (b - a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (integral {a..b} f) \\ B * (b - a)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm (integral {a..b} f) \\ B * (b - a)\n[PROOF STEP]\nnote continuous_on_imp_absolutely_integrable_on[OF assms(2)]\n[PROOF STATE]\nproof (state)\nthis:\nnorm (integral {a..b} f) \\ integral {a..b} (\\x. norm (f x))\n\ngoal (1 subgoal):\n 1. norm (integral {a..b} f) \\ B * (b - a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnorm (integral {a..b} f) \\ integral {a..b} (\\x. norm (f x))\n\ngoal (1 subgoal):\n 1. norm (integral {a..b} f) \\ B * (b - a)\n[PROOF STEP]\nhave \"integral {a..b} (\\x. norm (f x)) \\ integral {a..b} (\\_. B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral {a..b} (\\x. norm (f x)) \\ integral {a..b} (\\_. B)\n[PROOF STEP]\nby (rule integral_le)\n (auto intro!: integrable_continuous_real continuous_intros assms)\n[PROOF STATE]\nproof (state)\nthis:\nintegral {a..b} (\\x. norm (f x)) \\ integral {a..b} (\\_. B)\n\ngoal (1 subgoal):\n 1. norm (integral {a..b} f) \\ B * (b - a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nintegral {a..b} (\\x. norm (f x)) \\ integral {a..b} (\\_. B)\n\ngoal (1 subgoal):\n 1. norm (integral {a..b} f) \\ B * (b - a)\n[PROOF STEP]\nhave \"\\ = B * (b - a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral {a..b} (\\_. B) = B * (b - a)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ b\ncontinuous_on {a..b} f\n?t \\ {a..b} \\ norm (f ?t) \\ B\n\ngoal (1 subgoal):\n 1. integral {a..b} (\\_. B) = B * (b - a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nintegral {a..b} (\\_. B) = B * (b - a)\n\ngoal (1 subgoal):\n 1. norm (integral {a..b} f) \\ B * (b - a)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (integral {a..b} f) \\ B * (b - a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (integral {a..b} f) \\ B * (b - a)\n\ngoal (1 subgoal):\n 1. norm (integral {a..b} f) \\ B * (b - a)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nnorm (integral {a..b} f) \\ B * (b - a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1163, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382129861583, "lm_q2_score": 0.8198933293122506, "lm_q1q2_score": 0.7063694337749481}} {"text": "[STATEMENT]\nlemma normalize_monom [simp]: \"normalize (monom a n) = monom (normalize a) n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. normalize (monom a n) = monom (normalize a) n\n[PROOF STEP]\nby (cases \"a = 0\") (simp_all add: map_poly_monom normalize_poly_eq_map_poly degree_monom_eq)", "meta": {"llama_tokens": 121, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772482857833, "lm_q2_score": 0.8080672204860317, "lm_q1q2_score": 0.7063131725123719}} {"text": "[STATEMENT]\nlemma fds_abs_converges_ln: \n assumes \"fds_abs_converges (fds_deriv f / f) s\"\n shows \"fds_abs_converges (fds_ln l f) s\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fds_abs_converges (fds_ln l f) s\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfds_abs_converges (fds_deriv f / f) s\n\ngoal (1 subgoal):\n 1. fds_abs_converges (fds_ln l f) s\n[PROOF STEP]\nunfolding fds_ln_def\n[PROOF STATE]\nproof (prove)\nusing this:\nfds_abs_converges (fds_deriv f / f) s\n\ngoal (1 subgoal):\n 1. fds_abs_converges (fds_integral l (fds_deriv f / f)) s\n[PROOF STEP]\nby (intro fds_abs_converges_integral)", "meta": {"llama_tokens": 301, "file": "Dirichlet_Series_Dirichlet_Series", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772482857833, "lm_q2_score": 0.8080672158638528, "lm_q1q2_score": 0.7063131684722305}} {"text": "[STATEMENT]\nlemma openin_topology\\<^sub>N:\n \"openin (topology\\<^sub>N N) U \\ (\\x\\U. \\e>0. \\y. eNorm N (y-x) < e \\ y \\ U)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. openin (topology\\<^sub>N N) U = (\\x\\U. \\e>0. \\y. eNorm N (y - x) < e \\ y \\ U)\n[PROOF STEP]\nunfolding topology\\<^sub>N_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. openin (topology (\\U. \\x\\U. \\e>0. \\y. eNorm N (y - x) < e \\ y \\ U)) U = (\\x\\U. \\e>0. \\y. eNorm N (y - x) < e \\ y \\ U)\n[PROOF STEP]\nusing istopology_topology\\<^sub>N[of N]\n[PROOF STATE]\nproof (prove)\nusing this:\nistopology (\\U. \\x\\U. \\e>0. \\y. eNorm N (y - x) < e \\ y \\ U)\n\ngoal (1 subgoal):\n 1. openin (topology (\\U. \\x\\U. \\e>0. \\y. eNorm N (y - x) < e \\ y \\ U)) U = (\\x\\U. \\e>0. \\y. eNorm N (y - x) < e \\ y \\ U)\n[PROOF STEP]\nby (simp add: topology_inverse')", "meta": {"llama_tokens": 508, "file": "Lp_Functional_Spaces", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772253241803, "lm_q2_score": 0.8080672135527632, "lm_q1q2_score": 0.7063131478976411}} {"text": "[STATEMENT]\nlemma round_up_shift: \"round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. round_up p (x * 2 powr real_of_int k) = 2 powr real_of_int k * round_up (p + k) x\n[PROOF STEP]\nunfolding round_up_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_of_int \\x * 2 powr real_of_int k * 2 powr real_of_int p\\ * 2 powr - real_of_int p = 2 powr real_of_int k * (real_of_int \\x * 2 powr real_of_int (p + k)\\ * 2 powr - real_of_int (p + k))\n[PROOF STEP]\nby (simp add: powr_add powr_mult field_simps powr_diff)\n (simp flip: powr_add)", "meta": {"llama_tokens": 296, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267830311355, "lm_q2_score": 0.8128673201042493, "lm_q1q2_score": 0.7062408987573152}} {"text": "[STATEMENT]\nlemma reciprocal_add: \n fixes P Q :: \"'a::comm_semiring_0 poly\"\n assumes \"degree P \\ p\" and \"degree Q \\ p\" \n shows \"reciprocal_poly p (P + Q) = reciprocal_poly p P + reciprocal_poly p Q\" \n(is \"?L = ?R\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. reciprocal_poly p (P + Q) = reciprocal_poly p P + reciprocal_poly p Q\n[PROOF STEP]\nproof (rule poly_eqI, cases)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\n. ?P2 n \\ coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n 2. \\n. \\ ?P2 n \\ coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\n. ?P2 n \\ coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n 2. \\n. \\ ?P2 n \\ coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n[PROOF STEP]\nassume \"n \\ p\"\n[PROOF STATE]\nproof (state)\nthis:\nn \\ p\n\ngoal (2 subgoals):\n 1. \\n. ?P2 n \\ coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n 2. \\n. \\ ?P2 n \\ coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn \\ p\n[PROOF STEP]\nshow \"coeff ?L n = coeff ?R n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\ p\n\ngoal (1 subgoal):\n 1. coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\ p\ndegree P \\ p\ndegree Q \\ p\n\ngoal (1 subgoal):\n 1. coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n[PROOF STEP]\nby (auto simp: degree_add_le coeff_reciprocal)\n[PROOF STATE]\nproof (state)\nthis:\ncoeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n\ngoal (1 subgoal):\n 1. \\n. \\ n \\ p \\ coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\ n \\ p \\ coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\ n \\ p \\ coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n[PROOF STEP]\nassume \"\\n \\ p\"\n[PROOF STATE]\nproof (state)\nthis:\n\\ n \\ p\n\ngoal (1 subgoal):\n 1. \\n. \\ n \\ p \\ coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ n \\ p\n[PROOF STEP]\nshow \"coeff ?L n = coeff ?R n\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ n \\ p\n\ngoal (1 subgoal):\n 1. coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ n \\ p\ndegree P \\ p\ndegree Q \\ p\n\ngoal (1 subgoal):\n 1. coeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n[PROOF STEP]\nby (auto simp: degree_add_le coeff_reciprocal_less)\n[PROOF STATE]\nproof (state)\nthis:\ncoeff (reciprocal_poly p (P + Q)) n = coeff (reciprocal_poly p P + reciprocal_poly p Q) n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1569, "file": "Three_Circles_RRI_Misc", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267762381844, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.7062408873275011}} {"text": "[STATEMENT]\nlemma powr_of_int:\n fixes z::complex and n::int\n assumes \"z\\(0::complex)\"\n shows \"z powr of_int n = (if n\\0 then z^nat n else inverse (z^nat (-n)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z powr of_int n = (if 0 \\ n then z ^ nat n else inverse (z ^ nat (- n)))\n[PROOF STEP]\nby (metis assms not_le of_int_of_nat powr_complexpow powr_minus)", "meta": {"llama_tokens": 159, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267626522813, "lm_q2_score": 0.8128673110375457, "lm_q1q2_score": 0.7062408743146159}} {"text": "[STATEMENT]\ntheorem pr_index_enumerator_is_pr: \"pr_index_enumerator \\ PrimRec2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pr_index_enumerator \\ PrimRec2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. pr_index_enumerator \\ PrimRec2\n[PROOF STEP]\ndefine g where \"g x = x\" for x :: nat\n[PROOF STATE]\nproof (state)\nthis:\ng ?x = ?x\n\ngoal (1 subgoal):\n 1. pr_index_enumerator \\ PrimRec2\n[PROOF STEP]\nhave g_is_pr: \"g \\ PrimRec1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. g \\ PrimRec1\n[PROOF STEP]\nby (unfold g_def, rule pr_id1_1)\n[PROOF STATE]\nproof (state)\nthis:\ng \\ PrimRec1\n\ngoal (1 subgoal):\n 1. pr_index_enumerator \\ PrimRec2\n[PROOF STEP]\ndefine h where \"h a b c = comp_by_index index_of_id b\" for a b c :: nat\n[PROOF STATE]\nproof (state)\nthis:\nh ?a ?b ?c = comp_by_index index_of_id ?b\n\ngoal (1 subgoal):\n 1. pr_index_enumerator \\ PrimRec2\n[PROOF STEP]\nfrom comp_by_index_is_pr\n[PROOF STATE]\nproof (chain)\npicking this:\ncomp_by_index \\ PrimRec2\n[PROOF STEP]\nhave h_is_pr: \"h \\ PrimRec3\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncomp_by_index \\ PrimRec2\n\ngoal (1 subgoal):\n 1. h \\ PrimRec3\n[PROOF STEP]\nunfolding h_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncomp_by_index \\ PrimRec2\n\ngoal (1 subgoal):\n 1. (\\a b c. comp_by_index index_of_id b) \\ PrimRec3\n[PROOF STEP]\nby prec\n[PROOF STATE]\nproof (state)\nthis:\nh \\ PrimRec3\n\ngoal (1 subgoal):\n 1. pr_index_enumerator \\ PrimRec2\n[PROOF STEP]\nlet ?f = \"pr_index_enumerator\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. pr_index_enumerator \\ PrimRec2\n[PROOF STEP]\nfrom g_def\n[PROOF STATE]\nproof (chain)\npicking this:\ng ?x = ?x\n[PROOF STEP]\nhave f_at_0: \"\\ x. ?f x 0 = g x\"\n[PROOF STATE]\nproof (prove)\nusing this:\ng ?x = ?x\n\ngoal (1 subgoal):\n 1. \\x. pr_index_enumerator x 0 = g x\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\x. pr_index_enumerator x 0 = g x\n\ngoal (1 subgoal):\n 1. pr_index_enumerator \\ PrimRec2\n[PROOF STEP]\nfrom h_def\n[PROOF STATE]\nproof (chain)\npicking this:\nh ?a ?b ?c = comp_by_index index_of_id ?b\n[PROOF STEP]\nhave f_at_Suc: \"\\ x y. ?f x (Suc y) = h x (?f x y) y\"\n[PROOF STATE]\nproof (prove)\nusing this:\nh ?a ?b ?c = comp_by_index index_of_id ?b\n\ngoal (1 subgoal):\n 1. \\x y. pr_index_enumerator x (Suc y) = h x (pr_index_enumerator x y) y\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\x y. pr_index_enumerator x (Suc y) = h x (pr_index_enumerator x y) y\n\ngoal (1 subgoal):\n 1. pr_index_enumerator \\ PrimRec2\n[PROOF STEP]\nfrom g_is_pr h_is_pr f_at_0 f_at_Suc\n[PROOF STATE]\nproof (chain)\npicking this:\ng \\ PrimRec1\nh \\ PrimRec3\n\\x. pr_index_enumerator x 0 = g x\n\\x y. pr_index_enumerator x (Suc y) = h x (pr_index_enumerator x y) y\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ng \\ PrimRec1\nh \\ PrimRec3\n\\x. pr_index_enumerator x 0 = g x\n\\x y. pr_index_enumerator x (Suc y) = h x (pr_index_enumerator x y) y\n\ngoal (1 subgoal):\n 1. pr_index_enumerator \\ PrimRec2\n[PROOF STEP]\nby (rule pr_rec_last_scheme)\n[PROOF STATE]\nproof (state)\nthis:\npr_index_enumerator \\ PrimRec2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1497, "file": "Recursion-Theory-I_PRecFun2", "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.851952809486198, "lm_q2_score": 0.8289388125473628, "lm_q1q2_score": 0.7062167502418786}} {"text": "[STATEMENT]\nlemma ex_inverse:\n assumes coprime: \"coprime (e :: nat) ((P-1)*(Q-1))\" \n and \"prime P\" \n and \"prime Q\" \n and \"P \\ Q\" \n shows \"\\ d. [e*d = 1] (mod (P-1)) \\ d \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n[PROOF STEP]\nhave \"coprime e (P-1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coprime e (P - 1)\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime e ((P - 1) * (Q - 1))\n\ngoal (1 subgoal):\n 1. coprime e (P - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncoprime e (P - 1)\n\ngoal (1 subgoal):\n 1. \\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncoprime e (P - 1)\n[PROOF STEP]\nobtain d where d: \"[e*d = 1] (mod (P-1))\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime e (P - 1)\n\ngoal (1 subgoal):\n 1. (\\d. [e * d = 1] (mod P - 1) \\ thesis) \\ thesis\n[PROOF STEP]\nusing cong_solve_coprime_nat\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime e (P - 1)\ncoprime ?a ?n \\ \\x. [?a * x = Suc 0] (mod ?n)\n\ngoal (1 subgoal):\n 1. (\\d. [e * d = 1] (mod P - 1) \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n[e * d = 1] (mod P - 1)\n\ngoal (1 subgoal):\n 1. \\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[e * d = 1] (mod P - 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n[e * d = 1] (mod P - 1)\n\ngoal (1 subgoal):\n 1. \\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n[PROOF STEP]\nby (metis cong_0_1_nat cong_1 mult_0_right zero_neq_one)\n[PROOF STATE]\nproof (state)\nthis:\n\\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1005, "file": "Multi_Party_Computation_Number_Theory_Aux", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8519528019683106, "lm_q2_score": 0.8289388146603364, "lm_q1q2_score": 0.7062167458101637}} {"text": "[STATEMENT]\nlemma absolutely_integrable_sin_product:\n assumes \"f absolutely_integrable_on {-pi..pi}\"\n shows \"(\\x. sin(k * x) * f x) absolutely_integrable_on {-pi..pi}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. sin (k * x) * f x) absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nproof (rule absolutely_integrable_bounded_measurable_product_real)\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. (\\x. sin (k * x)) \\ borel_measurable (lebesgue_on {- pi..pi})\n 2. {- pi..pi} \\ sets lebesgue\n 3. bounded ((\\x. sin (k * x)) ` {- pi..pi})\n 4. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nshow \"(\\x. sin (k * x)) \\ borel_measurable (lebesgue_on {-pi..pi})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. sin (k * x)) \\ borel_measurable (lebesgue_on {- pi..pi})\n[PROOF STEP]\nby (metis borel_measurable_integrable integrable_sin_cx mult_commute_abs)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. sin (k * x)) \\ borel_measurable (lebesgue_on {- pi..pi})\n\ngoal (3 subgoals):\n 1. {- pi..pi} \\ sets lebesgue\n 2. bounded ((\\x. sin (k * x)) ` {- pi..pi})\n 3. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nshow \"bounded ((\\x. sin (k * x)) ` {-pi..pi})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bounded ((\\x. sin (k * x)) ` {- pi..pi})\n[PROOF STEP]\nby (metis (mono_tags, lifting) abs_sin_le_one bounded_iff imageE real_norm_def)\n[PROOF STATE]\nproof (state)\nthis:\nbounded ((\\x. sin (k * x)) ` {- pi..pi})\n\ngoal (2 subgoals):\n 1. {- pi..pi} \\ sets lebesgue\n 2. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nqed (auto simp: assms)", "meta": {"llama_tokens": 717, "file": "Fourier_Fourier", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527869325345, "lm_q2_score": 0.8289388146603365, "lm_q1q2_score": 0.7062167333464253}} {"text": "[STATEMENT]\nlemma lower_bound:\n fixes b::nat\n assumes \"b>0\"\n shows \"\\(c::int) \\ -(2^(b-1)). (-c + 2^(b-1) - 1 < 2^b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\c\\- (2 ^ (b - 1)). - c + 2 ^ (b - 1) - 1 < 2 ^ b\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\c\\- (2 ^ (b - 1)). - c + 2 ^ (b - 1) - 1 < 2 ^ b\n[PROOF STEP]\nhave a1: \"\\P. (\\b::nat. P b) \\ (\\b>0. P ((b-1)::nat))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\P. \\b. P b \\ \\b>0. P (b - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\b. ?P b \\ \\b>0. ?P (b - 1)\n\ngoal (1 subgoal):\n 1. \\c\\- (2 ^ (b - 1)). - c + 2 ^ (b - 1) - 1 < 2 ^ b\n[PROOF STEP]\nhave b2: \"\\b::nat. \\(c::int) \\ -(2^b). (-c + (2^b) - 1) < 2^(Suc b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\b c. - (2 ^ b) \\ c \\ - c + 2 ^ b - 1 < 2 ^ Suc b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\b c. - (2 ^ b) \\ c \\ - c + 2 ^ b - 1 < 2 ^ Suc b\n\ngoal (1 subgoal):\n 1. \\c\\- (2 ^ (b - 1)). - c + 2 ^ (b - 1) - 1 < 2 ^ b\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\c\\- (2 ^ (b - 1)). - c + 2 ^ (b - 1) - 1 < 2 ^ b\n[PROOF STEP]\nusing a1[OF b2] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\b>0. \\c\\- (2 ^ (b - 1)). - c + 2 ^ (b - 1) - 1 < 2 ^ Suc (b - 1)\n0 < b\n\ngoal (1 subgoal):\n 1. \\c\\- (2 ^ (b - 1)). - c + 2 ^ (b - 1) - 1 < 2 ^ b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\c\\- (2 ^ (b - 1)). - c + 2 ^ (b - 1) - 1 < 2 ^ b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 924, "file": "Solidity_Valuetypes", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527906914787, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.7062167310618988}} {"text": "[STATEMENT]\nlemma integral_sin_and_cos:\n fixes m n::int\n shows\n \"integral\\<^sup>L (lebesgue_on {-pi..pi}) (\\x. cos(m * x) * cos(n * x)) = (if \\m\\ = \\n\\ then if n = 0 then 2 * pi else pi else 0)\"\n \"integral\\<^sup>L (lebesgue_on {-pi..pi}) (\\x. cos(m * x) * sin(n * x)) = 0\"\n \"integral\\<^sup>L (lebesgue_on {-pi..pi}) (\\x. sin(m * x) * cos(n * x)) = 0\"\n \"\\m \\ 0; n \\ 0\\ \\ integral\\<^sup>L (lebesgue_on {-pi..pi}) (\\x. sin (m * x) * sin (n * x)) = (if m = n \\ n \\ 0 then pi else 0)\"\n \"\\integral\\<^sup>L (lebesgue_on {-pi..pi}) (\\x. sin (m * x) * sin (n * x))\\ = (if \\m\\ = \\n\\ \\ n \\ 0 then pi else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (LINT x|lebesgue_on {- pi..pi}. cos (real_of_int m * x) * cos (real_of_int n * x) = (if \\m\\ = \\n\\ then if n = 0 then 2 * pi else pi else 0) &&& LINT x|lebesgue_on {- pi..pi}. cos (real_of_int m * x) * sin (real_of_int n * x) = 0) &&& LINT x|lebesgue_on {- pi..pi}. sin (real_of_int m * x) * cos (real_of_int n * x) = 0 &&& (\\0 \\ m; 0 \\ n\\ \\ LINT x|lebesgue_on {- pi..pi}. sin (real_of_int m * x) * sin (real_of_int n * x) = (if m = n \\ n \\ 0 then pi else 0)) &&& \\LINT x|lebesgue_on {- pi..pi}. sin (real_of_int m * x) * sin (real_of_int n * x)\\ = (if \\m\\ = \\n\\ \\ n \\ 0 then pi else 0)\n[PROOF STEP]\nby (simp_all add: abs_if sin_times_sin cos_times_sin sin_times_cos cos_times_cos\n integrable_sin_cx integrable_cos_cx mult_ac\n flip: distrib_left distrib_right left_diff_distrib right_diff_distrib)", "meta": {"llama_tokens": 770, "file": "Fourier_Fourier", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094060543488, "lm_q2_score": 0.7905303186696747, "lm_q1q2_score": 0.7061881694387623}} {"text": "[STATEMENT]\nlemma dirichlet_prod_assoc: \n \"dirichlet_prod (dirichlet_prod f g) h = dirichlet_prod f (dirichlet_prod g h)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dirichlet_prod (dirichlet_prod f g) h = dirichlet_prod f (dirichlet_prod g h)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. dirichlet_prod (dirichlet_prod f g) h x = dirichlet_prod f (dirichlet_prod g h) x\n[PROOF STEP]\nfix n :: nat\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. dirichlet_prod (dirichlet_prod f g) h x = dirichlet_prod f (dirichlet_prod g h) x\n[PROOF STEP]\nshow \"dirichlet_prod (dirichlet_prod f g) h n = dirichlet_prod f (dirichlet_prod g h) n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dirichlet_prod (dirichlet_prod f g) h n = dirichlet_prod f (dirichlet_prod g h) n\n[PROOF STEP]\nby (cases \"n = 0\") (simp_all add: dirichlet_prod_assoc_aux1 dirichlet_prod_assoc_aux2)\n[PROOF STATE]\nproof (state)\nthis:\ndirichlet_prod (dirichlet_prod f g) h n = dirichlet_prod f (dirichlet_prod g h) n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 466, "file": "Dirichlet_Series_Dirichlet_Product", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933093975331751, "lm_q2_score": 0.7905303087996143, "lm_q1q2_score": 0.7061881538854983}} {"text": "[STATEMENT]\nlemma integral_cos: \"t \\ 0 \\ LBINT x=a..b. cos (t * x) = sin (t * b) / t - sin (t * a) / t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. t \\ 0 \\ LBINT x=ereal a..ereal b. cos (t * x) = sin (t * b) / t - sin (t * a) / t\n[PROOF STEP]\napply (intro interval_integral_FTC_finite continuous_intros)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\t \\ 0; min a b \\ x; x \\ max a b\\ \\ ((\\a. sin (t * a) / t) has_vector_derivative cos (t * x)) (at x within {min a b..max a b})\n[PROOF STEP]\nby (auto intro!: derivative_eq_intros simp: has_real_derivative_iff_has_vector_derivative[symmetric])", "meta": {"llama_tokens": 303, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513842182775, "lm_q2_score": 0.7879312006227324, "lm_q1q2_score": 0.7061844292268933}} {"text": "[STATEMENT]\nlemma limsup_root_powser:\n fixes f :: \"nat \\ 'a :: {banach, real_normed_div_algebra}\"\n shows \"limsup (\\n. ereal (root n (norm (f n * z ^ n)))) =\n limsup (\\n. ereal (root n (norm (f n)))) * ereal (norm z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. limsup (\\n. ereal (root n (norm (f n * z ^ n)))) = limsup (\\n. ereal (root n (norm (f n)))) * ereal (norm z)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. limsup (\\n. ereal (root n (norm (f n * z ^ n)))) = limsup (\\n. ereal (root n (norm (f n)))) * ereal (norm z)\n[PROOF STEP]\nhave A: \"(\\n. ereal (root n (norm (f n * z ^ n)))) =\n (\\n. ereal (root n (norm (f n))) * ereal (norm z))\" (is \"?g = ?h\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. ereal (root n (norm (f n * z ^ n)))) = (\\n. ereal (root n (norm (f n))) * ereal (norm z))\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. ereal (root n (norm (f n * z ^ n))) = ereal (root n (norm (f n))) * ereal (norm z)\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. ereal (root n (norm (f n * z ^ n))) = ereal (root n (norm (f n))) * ereal (norm z)\n[PROOF STEP]\nshow \"?g n = ?h n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ereal (root n (norm (f n * z ^ n))) = ereal (root n (norm (f n))) * ereal (norm z)\n[PROOF STEP]\nby (cases \"n = 0\") (simp_all add: norm_mult real_root_mult real_root_pos2 norm_power)\n[PROOF STATE]\nproof (state)\nthis:\nereal (root n (norm (f n * z ^ n))) = ereal (root n (norm (f n))) * ereal (norm z)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. ereal (root n (norm (f n * z ^ n)))) = (\\n. ereal (root n (norm (f n))) * ereal (norm z))\n\ngoal (1 subgoal):\n 1. limsup (\\n. ereal (root n (norm (f n * z ^ n)))) = limsup (\\n. ereal (root n (norm (f n)))) * ereal (norm z)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. limsup (\\n. ereal (root n (norm (f n * z ^ n)))) = limsup (\\n. ereal (root n (norm (f n)))) * ereal (norm z)\n[PROOF STEP]\nby (subst A, subst limsup_ereal_mult_right) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nlimsup (\\n. ereal (root n (norm (f n * z ^ n)))) = limsup (\\n. ereal (root n (norm (f n)))) * ereal (norm z)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1039, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513814471135, "lm_q2_score": 0.7879312006227324, "lm_q1q2_score": 0.7061844270434066}} {"text": "[STATEMENT]\nlemma Min_subset: \"\\ A \\ {}; A \\ B; finite B \\ \\ Min B \\ Min A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ {}; A \\ B; finite B\\ \\ Min B \\ Min A\n[PROOF STEP]\nby (rule linorder_class.Min_antimono)", "meta": {"llama_tokens": 126, "file": "List-Infinite_CommonSet_SetInterval2", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.896251378675949, "lm_q2_score": 0.7879311956428947, "lm_q1q2_score": 0.7061844203967332}} {"text": "[STATEMENT]\nlemma pdevs_val_add: \"pdevs_val (\\i. e i + f i) xs = pdevs_val e xs + pdevs_val f xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pdevs_val (\\i. e i + f i) xs = pdevs_val e xs + pdevs_val f xs\n[PROOF STEP]\nby (auto simp: pdevs_val_pdevs_domain algebra_simps sum.distrib)", "meta": {"llama_tokens": 140, "file": "Affine_Arithmetic_Affine_Form", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467738423873, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7061076539728831}} {"text": "[STATEMENT]\nlemma sinh_minus_cosh: \"sinh x - cosh x = -exp (-x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sinh x - cosh x = - exp (- x)\n[PROOF STEP]\nusing cosh_minus_sinh[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\ncosh x - sinh x = exp (- x)\n\ngoal (1 subgoal):\n 1. sinh x - cosh x = - exp (- x)\n[PROOF STEP]\nby (simp add: algebra_simps)", "meta": {"llama_tokens": 160, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.879146780175245, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7061076507753857}} {"text": "[STATEMENT]\nlemma finite_Bounded_vec_Max:\n assumes A: \"A \\ carrier_vec n\"\n and fin: \"finite A\"\n shows \"A \\ Bounded_vec (Max { abs (a $ i) | a i. a \\ A \\ i < n})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A \\ Bounded_vec (Max {\\a $ i\\ |a i. a \\ A \\ i < n})\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ A \\ x \\ Bounded_vec (Max {\\a $ i\\ |a i. a \\ A \\ i < n})\n[PROOF STEP]\nlet ?B = \"{ abs (a $ i) | a i. a \\ A \\ i < n}\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ A \\ x \\ Bounded_vec (Max {\\a $ i\\ |a i. a \\ A \\ i < n})\n[PROOF STEP]\nhave fin: \"finite ?B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {\\a $ i\\ |a i. a \\ A \\ i < n}\n[PROOF STEP]\nby (rule finite_subset[of _ \"(\\ (a,i). abs (a $ i)) ` (A \\ {0 ..< n})\"], insert fin, auto)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {\\a $ i\\ |a i. a \\ A \\ i < n}\n\ngoal (1 subgoal):\n 1. \\x. x \\ A \\ x \\ Bounded_vec (Max {\\a $ i\\ |a i. a \\ A \\ i < n})\n[PROOF STEP]\nfix a\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ A \\ x \\ Bounded_vec (Max {\\a $ i\\ |a i. a \\ A \\ i < n})\n[PROOF STEP]\nassume a: \"a \\ A\"\n[PROOF STATE]\nproof (state)\nthis:\na \\ A\n\ngoal (1 subgoal):\n 1. \\x. x \\ A \\ x \\ Bounded_vec (Max {\\a $ i\\ |a i. a \\ A \\ i < n})\n[PROOF STEP]\nshow \"a \\ Bounded_vec (Max ?B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ Bounded_vec (Max {\\a $ i\\ |a i. a \\ A \\ i < n})\n[PROOF STEP]\nunfolding Bounded_vec_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ {x. \\ix $ i\\ \\ Max {\\a $ i\\ |a i. a \\ A \\ i < n}}\n[PROOF STEP]\nby (standard, intro allI impI Max_ge[OF fin], insert a A, force)\n[PROOF STATE]\nproof (state)\nthis:\na \\ Bounded_vec (Max {\\a $ i\\ |a i. a \\ A \\ i < n})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1023, "file": "Linear_Inequalities_Integral_Bounded_Vectors", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467738423873, "lm_q2_score": 0.8031737940012417, "lm_q1q2_score": 0.7061076498309418}} {"text": "[STATEMENT]\nlemma finite_list: \"finite {xs::('a::finite) list. length xs = k}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {xs. length xs = k}\n[PROOF STEP]\napply (induct k)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. finite {xs. length xs = 0}\n 2. \\k. finite {xs. length xs = k} \\ finite {xs. length xs = Suc k}\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\k. finite {xs. length xs = k} \\ finite {xs. length xs = Suc k}\n[PROOF STEP]\napply (subgoal_tac \"{xs::('a::finite) list. length xs = Suc k} = (\\x. Cons x ` {xs. length xs = k})\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\k. \\finite {xs. length xs = k}; {xs. length xs = Suc k} = (\\x. (#) x ` {xs. length xs = k})\\ \\ finite {xs. length xs = Suc k}\n 2. \\k. finite {xs. length xs = k} \\ {xs. length xs = Suc k} = (\\x. (#) x ` {xs. length xs = k})\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\k x. \\finite {xs. length xs = k}; length x = Suc k\\ \\ \\xa. x \\ (#) xa ` {xs. length xs = k}\n[PROOF STEP]\napply (case_tac x)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\k x. \\finite {xs. length xs = k}; length x = Suc k; x = []\\ \\ \\xa. x \\ (#) xa ` {xs. length xs = k}\n 2. \\k x a list. \\finite {xs. length xs = k}; length x = Suc k; x = a # list\\ \\ \\xa. x \\ (#) xa ` {xs. length xs = k}\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 703, "file": "Presburger-Automata_Presburger_Automata", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467548438124, "lm_q2_score": 0.8031738010682209, "lm_q1q2_score": 0.7061076407846961}} {"text": "[STATEMENT]\ntheorem empty_if_no_b_accessible: \n \"ta_lang TA = {} \\ ta_initial TA \\ b_accessible (ta_rules TA) = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (ta_lang TA = {}) = (ta_initial TA \\ b_accessible (ta_rules TA) = {})\n[PROOF STEP]\nby (auto \n simp add: ta_lang_def \n intro: accs_is_b_accessible b_accessible_is_accs)", "meta": {"llama_tokens": 159, "file": "Tree-Automata_Ta", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942348544447, "lm_q2_score": 0.7931059536292271, "lm_q1q2_score": 0.7060976581448374}} {"text": "[STATEMENT]\nlemma collinear_ex_real:\n shows \"collinear z1 z2 z3 \\\n (\\ k::real. z1 = z2 \\ z3 - z1 = complex_of_real k * (z2 - z1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. collinear z1 z2 z3 = (\\k. z1 = z2 \\ z3 - z1 = cor k * (z2 - z1))\n[PROOF STEP]\nunfolding collinear_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (z1 = z2 \\ is_real ((z3 - z1) / (z2 - z1))) = (\\k. z1 = z2 \\ z3 - z1 = cor k * (z2 - z1))\n[PROOF STEP]\nby (metis Im_complex_of_real add_diff_cancel_right' complex_eq diff_zero legacy_Complex_simps(15) nonzero_mult_div_cancel_right right_minus_eq times_divide_eq_left zero_complex.code)", "meta": {"llama_tokens": 301, "file": "Complex_Geometry_Elementary_Complex_Geometry", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942290328345, "lm_q2_score": 0.7931059560743422, "lm_q1q2_score": 0.7060976557045556}} {"text": "[STATEMENT]\nlemma transpose_matrix_mult: \"transpose_matrix ((A::('a::comm_ring) matrix)*B) = (transpose_matrix B) * (transpose_matrix A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. transpose_matrix (A * B) = transpose_matrix B * transpose_matrix A\n[PROOF STEP]\napply (simp add: times_matrix_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. transpose_matrix (mult_matrix (*) (+) A B) = mult_matrix (*) (+) (transpose_matrix B) (transpose_matrix A)\n[PROOF STEP]\napply (subst transpose_mult_matrix)\n[PROOF STATE]\nproof (prove)\ngoal (5 subgoals):\n 1. \\a. (0::'a) * a = (0::'a)\n 2. \\a. a * (0::'a) = (0::'a)\n 3. (0::'a) + (0::'a) = (0::'a)\n 4. \\x y. y * x = x * y\n 5. mult_matrix (*) (+) (transpose_matrix B) (transpose_matrix A) = mult_matrix (*) (+) (transpose_matrix B) (transpose_matrix A)\n[PROOF STEP]\napply (simp_all add: mult.commute)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 403, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.7931059560743422, "lm_q1q2_score": 0.7060976487788245}} {"text": "[STATEMENT]\nlemma set_integral_mono_AE:\n fixes f g :: \"_ \\ real\"\n assumes \"set_integrable M A f\" \"set_integrable M A g\"\n \"AE x \\ A in M. f x \\ g x\"\n shows \"(LINT x:A|M. f x) \\ (LINT x:A|M. g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_lebesgue_integral M A f \\ set_lebesgue_integral M A g\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nset_integrable M A f\nset_integrable M A g\nAE x\\A in M. f x \\ g x\n\ngoal (1 subgoal):\n 1. set_lebesgue_integral M A f \\ set_lebesgue_integral M A g\n[PROOF STEP]\nunfolding set_integrable_def set_lebesgue_integral_def\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable M (\\x. indicat_real A x *\\<^sub>R f x)\nintegrable M (\\x. indicat_real A x *\\<^sub>R g x)\nAE x\\A in M. f x \\ g x\n\ngoal (1 subgoal):\n 1. LINT x|M. indicat_real A x *\\<^sub>R f x \\ LINT x|M. indicat_real A x *\\<^sub>R g x\n[PROOF STEP]\nby (auto intro: integral_mono_AE split: split_indicator)", "meta": {"llama_tokens": 442, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942319436397, "lm_q2_score": 0.7931059438487663, "lm_q1q2_score": 0.7060976471287728}} {"text": "[STATEMENT]\nlemma take_bit_Suc_bit1:\n \\take_bit (Suc n) (numeral (Num.Bit1 k)) = take_bit n (numeral k) * 2 + 1\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. take_bit (Suc n) (numeral (num.Bit1 k)) = take_bit n (numeral k) * (2::'a) + (1::'a)\n[PROOF STEP]\nby (simp add: take_bit_Suc numeral_Bit1_div_2 mod_2_eq_odd)", "meta": {"llama_tokens": 161, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.7931059438487663, "lm_q1q2_score": 0.706097642511619}} {"text": "[STATEMENT]\nlemma card_roots_unity_eq:\n assumes \"n > 0\"\n shows \"card {z::complex. z ^ n = 1} = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {z. z ^ n = 1} = n\n[PROOF STEP]\nusing bij_betw_same_card [OF bij_betw_roots_unity [OF assms]]\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {..GDERIV f x :> df; GDERIV g x :> dg\\\n \\ GDERIV (\\x. f x * g x) x :> scaleR (f x) dg + scaleR (g x) df\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\GDERIV f x :> df; GDERIV g x :> dg\\ \\ GDERIV (\\x. f x * g x) x :> f x *\\<^sub>R dg + g x *\\<^sub>R df\n[PROOF STEP]\nunfolding gderiv_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_derivative (\\h. inner h df)) (at x); (g has_derivative (\\h. inner h dg)) (at x)\\ \\ ((\\x. f x * g x) has_derivative (\\h. inner h (f x *\\<^sub>R dg + g x *\\<^sub>R df))) (at x)\n[PROOF STEP]\napply (rule has_derivative_subst)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\(f has_derivative (\\h. inner h df)) (at x); (g has_derivative (\\h. inner h dg)) (at x)\\ \\ ((\\x. f x * g x) has_derivative ?df) (at x)\n 2. \\(f has_derivative (\\h. inner h df)) (at x); (g has_derivative (\\h. inner h dg)) (at x)\\ \\ ?df = (\\h. inner h (f x *\\<^sub>R dg + g x *\\<^sub>R df))\n[PROOF STEP]\napply (erule (1) has_derivative_mult)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_derivative (\\h. inner h df)) (at x); (g has_derivative (\\h. inner h dg)) (at x)\\ \\ (\\h. f x * inner h dg + inner h df * g x) = (\\h. inner h (f x *\\<^sub>R dg + g x *\\<^sub>R df))\n[PROOF STEP]\napply (simp add: inner_add ac_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 727, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942144788076, "lm_q2_score": 0.7931059487389968, "lm_q1q2_score": 0.7060976376310546}} {"text": "[STATEMENT]\nlemma std_normal_distribution_even_moments:\n fixes k :: nat\n shows \"(LINT x|std_normal_distribution. x^(2 * k)) = fact (2 * k) / (2^k * fact k)\"\n and \"integrable std_normal_distribution (\\x. x^(2 * k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LINT x|std_normal_distribution. x ^ (2 * k) = fact (2 * k) / (2 ^ k * fact k) &&& integrable std_normal_distribution (\\x. x ^ (2 * k))\n[PROOF STEP]\nusing integral_std_normal_moment_even[of k]\n[PROOF STATE]\nproof (prove)\nusing this:\nLBINT x. std_normal_density x * x ^ (2 * k) = fact (2 * k) / (2 ^ k * fact k)\n\ngoal (1 subgoal):\n 1. LINT x|std_normal_distribution. x ^ (2 * k) = fact (2 * k) / (2 ^ k * fact k) &&& integrable std_normal_distribution (\\x. x ^ (2 * k))\n[PROOF STEP]\nby (subst integral_density)\n (auto simp: normal_density_nonneg integrable_density\n intro: integrable.intros std_normal_moment_even)", "meta": {"llama_tokens": 360, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213826762114, "lm_q2_score": 0.7853085758631159, "lm_q1q2_score": 0.7060877325575312}} {"text": "[STATEMENT]\nlemma echelon_form_condition2:\n assumes r: \"echelon_form A\"\n shows \"(\\i. i \\ (is_zero_row i A) \\ \\ (is_zero_row j A) \n \\ ((LEAST n. A $ i $ n \\ 0) < (LEAST n. A $ j $ n \\ 0)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i < j \\ \\ is_zero_row i A \\ \\ is_zero_row j A \\ (LEAST n. A $ i $ n \\ (0::'a)) < (LEAST n. A $ j $ n \\ (0::'a))\n[PROOF STEP]\nusing r\n[PROOF STATE]\nproof (prove)\nusing this:\nechelon_form A\n\ngoal (1 subgoal):\n 1. \\i. i < j \\ \\ is_zero_row i A \\ \\ is_zero_row j A \\ (LEAST n. A $ i $ n \\ (0::'a)) < (LEAST n. A $ j $ n \\ (0::'a))\n[PROOF STEP]\nunfolding echelon_form_def\n[PROOF STATE]\nproof (prove)\nusing this:\nechelon_form_upt_k A (ncols A)\n\ngoal (1 subgoal):\n 1. \\i. i < j \\ \\ is_zero_row i A \\ \\ is_zero_row j A \\ (LEAST n. A $ i $ n \\ (0::'a)) < (LEAST n. A $ j $ n \\ (0::'a))\n[PROOF STEP]\nby (metis echelon_form_upt_k_condition2 is_zero_row_def)", "meta": {"llama_tokens": 518, "file": "Echelon_Form_Echelon_Form", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213826762113, "lm_q2_score": 0.7853085758631159, "lm_q1q2_score": 0.7060877325575311}} {"text": "[STATEMENT]\nlemma Im_Ln_tendsto_at_top: \"((\\x. Im (Ln (Complex a x))) \\ pi/2 ) at_top \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\nproof (cases \"a=0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. a = 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n 2. a \\ 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\na \\ 0\n\ngoal (2 subgoals):\n 1. a = 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n 2. a \\ 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\ndefine f where \"f=(\\x. if a>0 then arctan (x/a) else arctan (x/a) + pi)\"\n[PROOF STATE]\nproof (state)\nthis:\nf = (\\x. if 0 < a then arctan (x / a) else arctan (x / a) + pi)\n\ngoal (2 subgoals):\n 1. a = 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n 2. a \\ 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\ndefine g where \"g=(\\x. Im (Ln (Complex a x)))\"\n[PROOF STATE]\nproof (state)\nthis:\ng = (\\x. Im (Ln (Complex a x)))\n\ngoal (2 subgoals):\n 1. a = 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n 2. a \\ 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\nhave \"(f \\ pi / 2) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f \\ pi / 2) at_top\n[PROOF STEP]\nproof (cases \"a>0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 0 < a \\ (f \\ pi / 2) at_top\n 2. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\n0 < a\n\ngoal (2 subgoals):\n 1. 0 < a \\ (f \\ pi / 2) at_top\n 2. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < a\n[PROOF STEP]\nhave \"(f \\ pi / 2) at_top \\ ((\\x. arctan (x * inverse a)) \\ pi / 2) at_top\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n\ngoal (1 subgoal):\n 1. (f \\ pi / 2) at_top = ((\\x. arctan (x * inverse a)) \\ pi / 2) at_top\n[PROOF STEP]\nunfolding f_def field_class.field_divide_inverse\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n\ngoal (1 subgoal):\n 1. ((\\x. if 0 < a then arctan (x * inverse a) else arctan (x * inverse a) + pi) \\ pi * inverse 2) at_top = ((\\x. arctan (x * inverse a)) \\ pi * inverse 2) at_top\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(f \\ pi / 2) at_top = ((\\x. arctan (x * inverse a)) \\ pi / 2) at_top\n\ngoal (2 subgoals):\n 1. 0 < a \\ (f \\ pi / 2) at_top\n 2. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(f \\ pi / 2) at_top = ((\\x. arctan (x * inverse a)) \\ pi / 2) at_top\n\ngoal (2 subgoals):\n 1. 0 < a \\ (f \\ pi / 2) at_top\n 2. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nhave \"... \\ (arctan \\ pi / 2) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. arctan (x * inverse a)) \\ pi / 2) at_top = (arctan \\ pi / 2) at_top\n[PROOF STEP]\napply (subst filterlim_at_top_linear_iff[of \"inverse a\" arctan 0 \"nhds (pi/2)\",simplified])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. a \\ 0\n 2. (if 0 < a then (arctan \\ pi / 2) at_top else (arctan \\ pi / 2) at_bot) = (arctan \\ pi / 2) at_top\n[PROOF STEP]\nusing True\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n\ngoal (2 subgoals):\n 1. a \\ 0\n 2. (if 0 < a then (arctan \\ pi / 2) at_top else (arctan \\ pi / 2) at_bot) = (arctan \\ pi / 2) at_top\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. arctan (x * inverse a)) \\ pi / 2) at_top = (arctan \\ pi / 2) at_top\n\ngoal (2 subgoals):\n 1. 0 < a \\ (f \\ pi / 2) at_top\n 2. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. arctan (x * inverse a)) \\ pi / 2) at_top = (arctan \\ pi / 2) at_top\n\ngoal (2 subgoals):\n 1. 0 < a \\ (f \\ pi / 2) at_top\n 2. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nhave \"...\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (arctan \\ pi / 2) at_top\n[PROOF STEP]\nusing tendsto_arctan_at_top\n[PROOF STATE]\nproof (prove)\nusing this:\n(arctan \\ pi / 2) at_top\n\ngoal (1 subgoal):\n 1. (arctan \\ pi / 2) at_top\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(arctan \\ pi / 2) at_top\n\ngoal (2 subgoals):\n 1. 0 < a \\ (f \\ pi / 2) at_top\n 2. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(f \\ pi / 2) at_top\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ pi / 2) at_top\n\ngoal (1 subgoal):\n 1. (f \\ pi / 2) at_top\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(f \\ pi / 2) at_top\n\ngoal (1 subgoal):\n 1. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ 0 < a\n\ngoal (1 subgoal):\n 1. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ 0 < a\n[PROOF STEP]\nhave \"(f \\ pi / 2) at_top \\ ((\\x. arctan (x * inverse a) + pi) \\ pi / 2) at_top\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 0 < a\n\ngoal (1 subgoal):\n 1. (f \\ pi / 2) at_top = ((\\x. arctan (x * inverse a) + pi) \\ pi / 2) at_top\n[PROOF STEP]\nunfolding f_def field_class.field_divide_inverse\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 0 < a\n\ngoal (1 subgoal):\n 1. ((\\x. if 0 < a then arctan (x * inverse a) else arctan (x * inverse a) + pi) \\ pi * inverse 2) at_top = ((\\x. arctan (x * inverse a) + pi) \\ pi * inverse 2) at_top\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(f \\ pi / 2) at_top = ((\\x. arctan (x * inverse a) + pi) \\ pi / 2) at_top\n\ngoal (1 subgoal):\n 1. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(f \\ pi / 2) at_top = ((\\x. arctan (x * inverse a) + pi) \\ pi / 2) at_top\n\ngoal (1 subgoal):\n 1. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nhave \"... \\ ((\\x. arctan (x * inverse a)) \\ - pi / 2) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. arctan (x * inverse a) + pi) \\ pi / 2) at_top = ((\\x. arctan (x * inverse a)) \\ - pi / 2) at_top\n[PROOF STEP]\napply (subst tendsto_add_const_iff[of \"-pi\",symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. - pi + (arctan (x * inverse a) + pi)) \\ - pi + pi / 2) at_top = ((\\x. arctan (x * inverse a)) \\ - pi / 2) at_top\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. arctan (x * inverse a) + pi) \\ pi / 2) at_top = ((\\x. arctan (x * inverse a)) \\ - pi / 2) at_top\n\ngoal (1 subgoal):\n 1. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. arctan (x * inverse a) + pi) \\ pi / 2) at_top = ((\\x. arctan (x * inverse a)) \\ - pi / 2) at_top\n\ngoal (1 subgoal):\n 1. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nhave \"... \\ (arctan \\ - pi / 2) at_bot\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. arctan (x * inverse a)) \\ - pi / 2) at_top = (arctan \\ - pi / 2) at_bot\n[PROOF STEP]\napply (subst filterlim_at_top_linear_iff[of \"inverse a\" arctan 0,simplified])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. a \\ 0\n 2. (if 0 < a then (arctan \\ - pi / 2) at_top else (arctan \\ - pi / 2) at_bot) = (arctan \\ - pi / 2) at_bot\n[PROOF STEP]\nusing False \\a\\0\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 0 < a\na \\ 0\n\ngoal (2 subgoals):\n 1. a \\ 0\n 2. (if 0 < a then (arctan \\ - pi / 2) at_top else (arctan \\ - pi / 2) at_bot) = (arctan \\ - pi / 2) at_bot\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. arctan (x * inverse a)) \\ - pi / 2) at_top = (arctan \\ - pi / 2) at_bot\n\ngoal (1 subgoal):\n 1. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. arctan (x * inverse a)) \\ - pi / 2) at_top = (arctan \\ - pi / 2) at_bot\n\ngoal (1 subgoal):\n 1. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nhave \"...\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (arctan \\ - pi / 2) at_bot\n[PROOF STEP]\nusing tendsto_arctan_at_bot\n[PROOF STATE]\nproof (prove)\nusing this:\n(arctan \\ - (pi / 2)) at_bot\n\ngoal (1 subgoal):\n 1. (arctan \\ - pi / 2) at_bot\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(arctan \\ - pi / 2) at_bot\n\ngoal (1 subgoal):\n 1. \\ 0 < a \\ (f \\ pi / 2) at_top\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(f \\ pi / 2) at_top\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ pi / 2) at_top\n\ngoal (1 subgoal):\n 1. (f \\ pi / 2) at_top\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(f \\ pi / 2) at_top\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(f \\ pi / 2) at_top\n\ngoal (2 subgoals):\n 1. a = 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n 2. a \\ 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(f \\ pi / 2) at_top\n\ngoal (2 subgoals):\n 1. a = 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n 2. a \\ 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\nhave \"\\\\<^sub>F x in at_top. f x = g x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. f x = g x\n[PROOF STEP]\nunfolding f_def g_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. (if 0 < a then arctan (x / a) else arctan (x / a) + pi) = Im (Ln (Complex a x))\n[PROOF STEP]\nusing \\a\\0\\\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ 0\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. (if 0 < a then arctan (x / a) else arctan (x / a) + pi) = Im (Ln (Complex a x))\n[PROOF STEP]\napply (subst Im_Ln_eq)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. a \\ 0 \\ Complex a x \\ 0\n 2. a \\ 0 \\ \\\\<^sub>F x in at_top. (if 0 < a then arctan (x / a) else arctan (x / a) + pi) = (if Re (Complex a x) \\ 0 then if 0 < Re (Complex a x) then arctan (Im (Complex a x) / Re (Complex a x)) else if 0 \\ Im (Complex a x) then arctan (Im (Complex a x) / Re (Complex a x)) + pi else arctan (Im (Complex a x) / Re (Complex a x)) - pi else if 0 < Im (Complex a x) then pi / 2 else - pi / 2)\n[PROOF STEP]\nsubgoal for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ 0 \\ Complex a x \\ 0\n[PROOF STEP]\nusing Complex_eq_0\n[PROOF STATE]\nproof (prove)\nusing this:\n(Complex ?a ?b = 0) = (?a = 0 \\ ?b = 0)\n\ngoal (1 subgoal):\n 1. a \\ 0 \\ Complex a x \\ 0\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ 0 \\ \\\\<^sub>F x in at_top. (if 0 < a then arctan (x / a) else arctan (x / a) + pi) = (if Re (Complex a x) \\ 0 then if 0 < Re (Complex a x) then arctan (Im (Complex a x) / Re (Complex a x)) else if 0 \\ Im (Complex a x) then arctan (Im (Complex a x) / Re (Complex a x)) + pi else arctan (Im (Complex a x) / Re (Complex a x)) - pi else if 0 < Im (Complex a x) then pi / 2 else - pi / 2)\n[PROOF STEP]\nsubgoal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ 0 \\ \\\\<^sub>F x in at_top. (if 0 < a then arctan (x / a) else arctan (x / a) + pi) = (if Re (Complex a x) \\ 0 then if 0 < Re (Complex a x) then arctan (Im (Complex a x) / Re (Complex a x)) else if 0 \\ Im (Complex a x) then arctan (Im (Complex a x) / Re (Complex a x)) + pi else arctan (Im (Complex a x) / Re (Complex a x)) - pi else if 0 < Im (Complex a x) then pi / 2 else - pi / 2)\n[PROOF STEP]\nunfolding eventually_at_top_linorder\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ 0 \\ \\N. \\n\\N. (if 0 < a then arctan (n / a) else arctan (n / a) + pi) = (if Re (Complex a n) \\ 0 then if 0 < Re (Complex a n) then arctan (Im (Complex a n) / Re (Complex a n)) else if 0 \\ Im (Complex a n) then arctan (Im (Complex a n) / Re (Complex a n)) + pi else arctan (Im (Complex a n) / Re (Complex a n)) - pi else if 0 < Im (Complex a n) then pi / 2 else - pi / 2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in at_top. f x = g x\n\ngoal (2 subgoals):\n 1. a = 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n 2. a \\ 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(f \\ pi / 2) at_top\n\\\\<^sub>F x in at_top. f x = g x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ pi / 2) at_top\n\\\\<^sub>F x in at_top. f x = g x\n\ngoal (1 subgoal):\n 1. ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\nusing tendsto_cong[of f g at_top]\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ pi / 2) at_top\n\\\\<^sub>F x in at_top. f x = g x\n\\\\<^sub>F x in at_top. f x = g x \\ (f \\ ?c) at_top = (g \\ ?c) at_top\n\ngoal (1 subgoal):\n 1. ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\nunfolding g_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ pi / 2) at_top\n\\\\<^sub>F x in at_top. f x = Im (Ln (Complex a x))\n\\\\<^sub>F x in at_top. f x = Im (Ln (Complex a x)) \\ (f \\ ?c) at_top = ((\\x. Im (Ln (Complex a x))) \\ ?c) at_top\n\ngoal (1 subgoal):\n 1. ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n\ngoal (1 subgoal):\n 1. a = 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. a = 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\na = 0\n\ngoal (1 subgoal):\n 1. a = 0 \\ ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n[PROOF STEP]\napply (rule tendsto_eventually)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. Im (Ln (Complex a x)) = pi / 2\n[PROOF STEP]\napply (rule eventually_at_top_linorderI[of 1])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. 1 \\ x \\ Im (Ln (Complex a x)) = pi / 2\n[PROOF STEP]\nusing True\n[PROOF STATE]\nproof (prove)\nusing this:\na = 0\n\ngoal (1 subgoal):\n 1. \\x. 1 \\ x \\ Im (Ln (Complex a x)) = pi / 2\n[PROOF STEP]\nby (subst Im_Ln_eq,auto simp add:Complex_eq_0)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. Im (Ln (Complex a x))) \\ pi / 2) at_top\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 7564, "file": "Count_Complex_Roots_Count_Half_Plane", "length": 71, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127417985636, "lm_q2_score": 0.8267118026095991, "lm_q1q2_score": 0.7059397420435957}} {"text": "[STATEMENT]\nlemma ceiling_one [simp]: \"\\1\\ = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\1::'a\\ = 1\n[PROOF STEP]\nusing ceiling_of_int [of 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\of_int 1\\ = 1\n\ngoal (1 subgoal):\n 1. \\1::'a\\ = 1\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 158, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.8267117962054049, "lm_q1q2_score": 0.7059397396484504}} {"text": "[STATEMENT]\nlemma ceiling_one [simp]: \"\\1\\ = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\1::'a\\ = 1\n[PROOF STEP]\nusing ceiling_of_int [of 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\of_int 1\\ = 1\n\ngoal (1 subgoal):\n 1. \\1::'a\\ = 1\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 158, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.8267117876664789, "lm_q1q2_score": 0.7059397323569526}} {"text": "[STATEMENT]\nlemma continuous_on_real_range:\n fixes f :: \"'a::real_normed_vector \\ real\"\n shows \"continuous_on s f \\\n (\\x \\ s. \\e>0. \\d>0. (\\x' \\ s. norm(x' - x) < d \\ \\f x' - f x\\ < e))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous_on s f = (\\x\\s. \\e>0. \\d>0. \\x'\\s. norm (x' - x) < d \\ \\f x' - f x\\ < e)\n[PROOF STEP]\nunfolding continuous_on_iff dist_norm\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\s. \\e>0. \\d>0. \\x'\\s. norm (x' - x) < d \\ norm (f x' - f x) < e) = (\\x\\s. \\e>0. \\d>0. \\x'\\s. norm (x' - x) < d \\ \\f x' - f x\\ < e)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 374, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127380808499, "lm_q2_score": 0.8267117940706734, "lm_q1q2_score": 0.7059397316786205}} {"text": "[STATEMENT]\nlemma CDERIV_quotient:\n \"DERIV f x :> d \\ DERIV g x :> e \\ g(x) \\ 0 \\\n DERIV (\\y. f y / g y) x :> (d * g x - (e * f x)) / (g x)\\<^sup>2\"\n for x :: complex\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative d) (at x); (g has_field_derivative e) (at x); g x \\ 0\\ \\ ((\\y. f y / g y) has_field_derivative (d * g x - e * f x) / (g x)\\<^sup>2) (at x)\n[PROOF STEP]\nunfolding numeral_2_eq_2\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative d) (at x); (g has_field_derivative e) (at x); g x \\ 0\\ \\ ((\\y. f y / g y) has_field_derivative (d * g x - e * f x) / g x ^ Suc (Suc 0)) (at x)\n[PROOF STEP]\nby (rule DERIV_quotient)", "meta": {"llama_tokens": 378, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872046056466901, "lm_q2_score": 0.7956580927949806, "lm_q1q2_score": 0.7059115244477683}} {"text": "[STATEMENT]\nlemma binomial_deriv1:\n \"(\\k\\n. (of_nat k * of_nat (n choose k)) * a^(k-1) * b^(n-k)) = real_of_nat n * (a+b)^(n-1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\n. real k * real (n choose k) * a ^ (k - 1) * b ^ (n - k)) = real n * (a + b) ^ (n - 1)\n[PROOF STEP]\napply (rule DERIV_unique [where f = \"\\a. (a+b)^n\" and x=a])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. ((\\a. (a + b) ^ n) has_real_derivative (\\k\\n. real k * real (n choose k) * a ^ (k - 1) * b ^ (n - k))) (at a)\n 2. ((\\a. (a + b) ^ n) has_real_derivative real n * (a + b) ^ (n - 1)) (at a)\n[PROOF STEP]\napply (subst binomial_ring)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. ((\\a. \\k\\n. real (n choose k) * a ^ k * b ^ (n - k)) has_real_derivative (\\k\\n. real k * real (n choose k) * a ^ (k - 1) * b ^ (n - k))) (at a)\n 2. ((\\a. (a + b) ^ n) has_real_derivative real n * (a + b) ^ (n - 1)) (at a)\n[PROOF STEP]\napply (rule derivative_eq_intros sum.cong | simp add: atMost_atLeast0)+\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 549, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.90192067652954, "lm_q2_score": 0.7826624840223698, "lm_q1q2_score": 0.7058994770837461}} {"text": "[STATEMENT]\nlemma (in strategic_space_2p) U\\<^sub>A\\<^sub>_is_gate:\n shows \"gate 1 U\\<^sub>A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. gate 1 (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]]) = 2 ^ 1\n 2. square_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n 3. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n[PROOF STEP]\nshow \"dim_row U\\<^sub>A = 2^1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]]) = 2 ^ 1\n[PROOF STEP]\nusing mat_of_cols_list_def\n[PROOF STATE]\nproof (prove)\nusing this:\nTensor.mat_of_cols_list ?nr ?cs = Matrix.mat ?nr (length ?cs) (\\(i, j). ?cs ! j ! i)\n\ngoal (1 subgoal):\n 1. dim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]]) = 2 ^ 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]]) = 2 ^ 1\n\ngoal (2 subgoals):\n 1. square_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n 2. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]]) = 2 ^ 1\n\ngoal (2 subgoals):\n 1. square_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n 2. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n[PROOF STEP]\nshow \"square_mat U\\<^sub>A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. square_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n[PROOF STEP]\nusing mat_of_cols_list_def\n[PROOF STATE]\nproof (prove)\nusing this:\nTensor.mat_of_cols_list ?nr ?cs = Matrix.mat ?nr (length ?cs) (\\(i, j). ?cs ! j ! i)\n\ngoal (1 subgoal):\n 1. square_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsquare_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n\ngoal (1 subgoal):\n 1. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ndim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]]) = 2 ^ 1\nsquare_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n[PROOF STEP]\nshow \"unitary U\\<^sub>A\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]]) = 2 ^ 1\nsquare_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n\ngoal (1 subgoal):\n 1. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n[PROOF STEP]\nusing mat_of_cols_list_def unitary_def U\\<^sub>A_cnj_times_U\\<^sub>A U\\<^sub>A_times_U\\<^sub>A_cnj\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_row (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]]) = 2 ^ 1\nsquare_mat (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\nTensor.mat_of_cols_list ?nr ?cs = Matrix.mat ?nr (length ?cs) (\\(i, j). ?cs ! j ! i)\nunitary ?M \\ ?M\\<^sup>\\ * ?M = 1\\<^sub>m (dim_col ?M) \\ ?M * ?M\\<^sup>\\ = 1\\<^sub>m (dim_row ?M)\nTensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]]\\<^sup>\\ * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]] = 1\\<^sub>m 2\nTensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]] * Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]]\\<^sup>\\ = 1\\<^sub>m 2\n\ngoal (1 subgoal):\n 1. unitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nunitary (Tensor.mat_of_cols_list 2 [exp (\\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2)) # map complex_of_real [- sin (\\\\<^sub>A / 2)], [complex_of_real (sin (\\\\<^sub>A / 2)), exp (- \\ * complex_of_real \\\\<^sub>A) * complex_of_real (cos (\\\\<^sub>A / 2))]])\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4557, "file": "Isabelle_Marries_Dirac_Quantum_Prisoners_Dilemma", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206765295399, "lm_q2_score": 0.7826624840223698, "lm_q1q2_score": 0.705899477083746}} {"text": "[STATEMENT]\nlemma measure_mono_eq_prod:\n assumes \"f \\ 0\" \"g \\ 0\"\n shows \"mahler_measure_monic (f * g) = mahler_measure_monic f * mahler_measure_monic g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mahler_measure_monic (f * g) = mahler_measure_monic f * mahler_measure_monic g\n[PROOF STEP]\nunfolding mahler_measure_monic_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a\\complex_roots_complex (f * g). max 1 (cmod a)) = (\\a\\complex_roots_complex f. max 1 (cmod a)) * (\\a\\complex_roots_complex g. max 1 (cmod a))\n[PROOF STEP]\nusing mset_mult_add_2[OF complex_roots_complex_prod[OF assms],of \"\\ a. max 1 (cmod a)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\a\\complex_roots_complex (f * g). max 1 (cmod a)) = (\\a\\complex_roots_complex f. max 1 (cmod a)) * (\\a\\complex_roots_complex g. max 1 (cmod a))\n\ngoal (1 subgoal):\n 1. (\\a\\complex_roots_complex (f * g). max 1 (cmod a)) = (\\a\\complex_roots_complex f. max 1 (cmod a)) * (\\a\\complex_roots_complex g. max 1 (cmod a))\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 507, "file": "Berlekamp_Zassenhaus_Mahler_Measure", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206870747658, "lm_q2_score": 0.7826624738835051, "lm_q1q2_score": 0.7058994761926468}} {"text": "[STATEMENT]\nlemma continuous_on_dist[continuous_intros]:\n fixes f g :: \"_ \\ 'a :: metric_space\"\n shows \"continuous_on s f \\ continuous_on s g \\ continuous_on s (\\x. dist (f x) (g x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\continuous_on s f; continuous_on s g\\ \\ continuous_on s (\\x. dist (f x) (g x))\n[PROOF STEP]\nunfolding continuous_on_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x\\s. (f \\ f x) (at x within s); \\x\\s. (g \\ g x) (at x within s)\\ \\ \\x\\s. ((\\x. dist (f x) (g x)) \\ dist (f x) (g x)) (at x within s)\n[PROOF STEP]\nby (auto intro: tendsto_dist)", "meta": {"llama_tokens": 306, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206659843132, "lm_q2_score": 0.7826624840223698, "lm_q1q2_score": 0.7058994688303927}} {"text": "[STATEMENT]\nlemma continuous_on_dist[continuous_intros]:\n fixes f g :: \"_ \\ 'a :: metric_space\"\n shows \"continuous_on s f \\ continuous_on s g \\ continuous_on s (\\x. dist (f x) (g x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\continuous_on s f; continuous_on s g\\ \\ continuous_on s (\\x. dist (f x) (g x))\n[PROOF STEP]\nunfolding continuous_on_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x\\s. (f \\ f x) (at x within s); \\x\\s. (g \\ g x) (at x within s)\\ \\ \\x\\s. ((\\x. dist (f x) (g x)) \\ dist (f x) (g x)) (at x within s)\n[PROOF STEP]\nby (auto intro: tendsto_dist)", "meta": {"llama_tokens": 306, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206659843132, "lm_q2_score": 0.7826624789529375, "lm_q1q2_score": 0.7058994642581669}} {"text": "[STATEMENT]\nlemma all_to_meta: \"Trueprop (\\a. P a) \\ (\\a. P a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a. P a \\ (\\a. P a)\n[PROOF STEP]\napply rule\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\a. \\a. P a \\ P a\n 2. (\\a. P a) \\ \\a. P a\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 162, "file": "Automatic_Refinement_Lib_Misc", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8633916240341031, "lm_q2_score": 0.8175744850834649, "lm_q1q2_score": 0.7058869624450583}} {"text": "[STATEMENT]\nlemma fds_left_inverse_unique:\n assumes \"f * g = (1 :: 'a :: field fds)\"\n shows \"f = inverse g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f = inverse g\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. f = inverse g\n[PROOF STEP]\nhave \"fds_nth (f * g) 1 = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fds_nth (f * g) 1 = (1::'a)\n[PROOF STEP]\nby (subst assms) simp\n[PROOF STATE]\nproof (state)\nthis:\nfds_nth (f * g) 1 = (1::'a)\n\ngoal (1 subgoal):\n 1. f = inverse g\n[PROOF STEP]\nhence \"fds_nth g 1 \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfds_nth (f * g) 1 = (1::'a)\n\ngoal (1 subgoal):\n 1. fds_nth g 1 \\ (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfds_nth g 1 \\ (0::'a)\n\ngoal (1 subgoal):\n 1. f = inverse g\n[PROOF STEP]\nhence \"(f - inverse g) * g = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfds_nth g 1 \\ (0::'a)\n\ngoal (1 subgoal):\n 1. (f - inverse g) * g = 0\n[PROOF STEP]\nunfolding ring_distribs\n[PROOF STATE]\nproof (prove)\nusing this:\nfds_nth g 1 \\ (0::'a)\n\ngoal (1 subgoal):\n 1. f * g - inverse g * g = 0\n[PROOF STEP]\nby (subst fds_left_inverse) (simp_all add: assms)\n[PROOF STATE]\nproof (state)\nthis:\n(f - inverse g) * g = 0\n\ngoal (1 subgoal):\n 1. f = inverse g\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(f - inverse g) * g = 0\n\ngoal (1 subgoal):\n 1. f = inverse g\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nf * g = 1\n[PROOF STEP]\nhave \"g \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nf * g = 1\n\ngoal (1 subgoal):\n 1. g \\ 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ng \\ 0\n\ngoal (1 subgoal):\n 1. f = inverse g\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(f - inverse g) * g = 0\ng \\ 0\n[PROOF STEP]\nshow \"f = inverse g\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(f - inverse g) * g = 0\ng \\ 0\n\ngoal (1 subgoal):\n 1. f = inverse g\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nf = inverse g\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 994, "file": "Dirichlet_Series_Dirichlet_Series", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744850834648, "lm_q2_score": 0.8633916082162403, "lm_q1q2_score": 0.7058869495127772}} {"text": "[STATEMENT]\nlemma Inf_pres_map_dual_var: \n \"Inf_pres f = Sup_pres (\\\\<^sub>F f)\"\n for f :: \"'a::complete_lattice_with_dual \\ 'b::complete_lattice_with_dual\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Inf_pres f = Sup_pres (\\\\<^sub>F f)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Inf_pres f = Sup_pres (\\\\<^sub>F f)\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Inf_pres f = Sup_pres (\\\\<^sub>F f)\n[PROOF STEP]\nfix x :: \"'a set\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Inf_pres f = Sup_pres (\\\\<^sub>F f)\n[PROOF STEP]\nassume \"\\ (f (\\ (\\ ` x))) = (\\y\\x. \\ (f (\\ y)))\" for x\n[PROOF STATE]\nproof (state)\nthis:\n\\ (f (\\ (\\ ` ?x))) = (\\y\\?x. \\ (f (\\ y)))\n\ngoal (1 subgoal):\n 1. Inf_pres f = Sup_pres (\\\\<^sub>F f)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ (f (\\ (\\ ` ?x))) = (\\y\\?x. \\ (f (\\ y)))\n[PROOF STEP]\nhave \"\\ (f ` \\ ` A) = f (\\ (\\ A))\" for A\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (f (\\ (\\ ` ?x))) = (\\y\\?x. \\ (f (\\ y)))\n\ngoal (1 subgoal):\n 1. \\ (f ` \\ ` A) = f (\\ (\\ A))\n[PROOF STEP]\nby (metis (no_types) Sup_dual_def_var image_image invol_dual_var subset_dual)\n[PROOF STATE]\nproof (state)\nthis:\n\\ (f ` \\ ` ?A) = f (\\ (\\ ?A))\n\ngoal (1 subgoal):\n 1. Inf_pres f = Sup_pres (\\\\<^sub>F f)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ (f ` \\ ` ?A) = f (\\ (\\ ?A))\n[PROOF STEP]\nhave \"\\ (f ` x) = f (\\ x)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (f ` \\ ` ?A) = f (\\ (\\ ?A))\n\ngoal (1 subgoal):\n 1. \\ (f ` x) = f (\\ x)\n[PROOF STEP]\nby (metis Sup_dual_def_var subset_dual)\n[PROOF STATE]\nproof (state)\nthis:\n\\ (f ` x) = f (\\ x)\n\ngoal (1 subgoal):\n 1. Inf_pres f = Sup_pres (\\\\<^sub>F f)\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. \\ (f (\\ (\\ ` x))) = (\\y\\x. \\ (f (\\ y)))) \\ \\ (f ` ?x2) = f (\\ ?x2)\n\ngoal (1 subgoal):\n 1. Inf_pres f = Sup_pres (\\\\<^sub>F f)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x. \\ (f (\\ (\\ ` x))) = (\\y\\x. \\ (f (\\ y)))) \\ \\ (f ` ?x2) = f (\\ ?x2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. \\ (f (\\ (\\ ` x))) = (\\y\\x. \\ (f (\\ y)))) \\ \\ (f ` ?x2) = f (\\ ?x2)\n\ngoal (1 subgoal):\n 1. Inf_pres f = Sup_pres (\\\\<^sub>F f)\n[PROOF STEP]\nby (auto simp add: map_dual_def fun_eq_iff Inf_dual_var Sup_dual_def_var image_comp)\n[PROOF STATE]\nproof (state)\nthis:\nInf_pres f = Sup_pres (\\\\<^sub>F f)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1425, "file": "Order_Lattice_Props_Order_Lattice_Props", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.863391617003942, "lm_q2_score": 0.8175744761936437, "lm_q1q2_score": 0.7058869490219808}} {"text": "[STATEMENT]\nlemma homeomorphism_add:\n \"homeomorphism UNIV UNIV (\\x. x + c) (\\x. x - c)\"\n for c::\"_::real_normed_vector\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. homeomorphism UNIV UNIV (\\x. x + c) (\\x. x - c)\n[PROOF STEP]\nunfolding homeomorphism_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\UNIV. x + c - c = x) \\ surj (\\x. x + c) \\ continuous_on UNIV (\\x. x + c) \\ (\\y\\UNIV. y - c + c = y) \\ surj (\\x. x - c) \\ continuous_on UNIV (\\x. x - c)\n[PROOF STEP]\nby (auto simp: algebra_simps continuous_intros intro!: image_eqI[where x=\"x - c\" for x])", "meta": {"llama_tokens": 282, "file": "Smooth_Manifolds_Analysis_More", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391624034103, "lm_q2_score": 0.8175744695262775, "lm_q1q2_score": 0.705886949013113}} {"text": "[STATEMENT]\nlemma bound_intersect_2d_nonempty:\n assumes \"bound_intersect_2d prec Xs x = Some (m, M)\"\n shows \"(\\(p1, p2) \\ set Xs. closed_segment p1 p2) \\ {p. fst p = x} \\ {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(p1, p2)\\set Xs. closed_segment p1 p2) \\ {p. fst p = x} \\ {}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\(p1, p2)\\set Xs. closed_segment p1 p2) \\ {p. fst p = x} \\ {}\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nbound_intersect_2d prec Xs x = Some (m, M)\n[PROOF STEP]\nhave \"bound_intersect_2d prec Xs x \\ None\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbound_intersect_2d prec Xs x = Some (m, M)\n\ngoal (1 subgoal):\n 1. bound_intersect_2d prec Xs x \\ None\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nbound_intersect_2d prec Xs x \\ None\n\ngoal (1 subgoal):\n 1. (\\(p1, p2)\\set Xs. closed_segment p1 p2) \\ {p. fst p = x} \\ {}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nbound_intersect_2d prec Xs x \\ None\n[PROOF STEP]\nobtain p1 p2 where \"(p1, p2) \\ set Xs\" \"intersect_segment_xline prec (p1, p2) x \\ None\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbound_intersect_2d prec Xs x \\ None\n\ngoal (1 subgoal):\n 1. (\\p1 p2. \\(p1, p2) \\ set Xs; intersect_segment_xline prec (p1, p2) x \\ None\\ \\ thesis) \\ thesis\n[PROOF STEP]\nunfolding bound_intersect_2d_eq_None_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (\\X\\set Xs. intersect_segment_xline prec X x = None)\n\ngoal (1 subgoal):\n 1. (\\p1 p2. \\(p1, p2) \\ set Xs; intersect_segment_xline prec (p1, p2) x \\ None\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(p1, p2) \\ set Xs\nintersect_segment_xline prec (p1, p2) x \\ None\n\ngoal (1 subgoal):\n 1. (\\(p1, p2)\\set Xs. closed_segment p1 p2) \\ {p. fst p = x} \\ {}\n[PROOF STEP]\nhence \"closed_segment p1 p2 \\ {p. fst p = x} \\ {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(p1, p2) \\ set Xs\nintersect_segment_xline prec (p1, p2) x \\ None\n\ngoal (1 subgoal):\n 1. closed_segment p1 p2 \\ {p. fst p = x} \\ {}\n[PROOF STEP]\nby (simp add: intersect_segment_xline_None_iff)\n[PROOF STATE]\nproof (state)\nthis:\nclosed_segment p1 p2 \\ {p. fst p = x} \\ {}\n\ngoal (1 subgoal):\n 1. (\\(p1, p2)\\set Xs. closed_segment p1 p2) \\ {p. fst p = x} \\ {}\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nclosed_segment p1 p2 \\ {p. fst p = x} \\ {}\n\ngoal (1 subgoal):\n 1. (\\(p1, p2)\\set Xs. closed_segment p1 p2) \\ {p. fst p = x} \\ {}\n[PROOF STEP]\nusing \\(p1, p2) \\ set Xs\\\n[PROOF STATE]\nproof (prove)\nusing this:\nclosed_segment p1 p2 \\ {p. fst p = x} \\ {}\n(p1, p2) \\ set Xs\n\ngoal (1 subgoal):\n 1. (\\(p1, p2)\\set Xs. closed_segment p1 p2) \\ {p. fst p = x} \\ {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\(p1, p2)\\set Xs. closed_segment p1 p2) \\ {p. fst p = x} \\ {}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1504, "file": "Affine_Arithmetic_Intersection", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.863391599428538, "lm_q2_score": 0.8175744695262775, "lm_q1q2_score": 0.7058869288962313}} {"text": "[STATEMENT]\nlemma prime_factors_primorial [simp]:\n \"prime_factors (primorial x) = {p. prime p \\ real p \\ x}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prime_factors (primorial x) = {p. prime p \\ real p \\ x}\n[PROOF STEP]\nunfolding prime_factorization_primorial\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_mset (mset_set {p. prime p \\ real p \\ x}) = {p. prime p \\ real p \\ x}\n[PROOF STEP]\nusing finite_primes_le[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {p. prime p \\ real p \\ x}\n\ngoal (1 subgoal):\n 1. set_mset (mset_set {p. prime p \\ real p \\ x}) = {p. prime p \\ real p \\ x}\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 293, "file": "Prime_Distribution_Elementary_Primorial", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9111797051879431, "lm_q2_score": 0.7745833841649233, "lm_q1q2_score": 0.7057846596268741}} {"text": "[STATEMENT]\ntheorem set_plus_rearrange4: \"C + (a +o D) = a +o (C + D)\"\n for a :: \"'a::comm_monoid_add\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. C + a +o D = a +o (C + D)\n[PROOF STEP]\nby (metis add.commute set_plus_rearrange3)", "meta": {"llama_tokens": 114, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111796979521252, "lm_q2_score": 0.7745833841649233, "lm_q1q2_score": 0.7057846540221298}} {"text": "[STATEMENT]\ntheorem set_plus_rearrange4: \"C + (a +o D) = a +o (C + D)\"\n for a :: \"'a::comm_monoid_add\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. C + a +o D = a +o (C + D)\n[PROOF STEP]\nby (metis add.commute set_plus_rearrange3)", "meta": {"llama_tokens": 114, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111796979521252, "lm_q2_score": 0.7745833789613196, "lm_q1q2_score": 0.7057846492807117}} {"text": "[STATEMENT]\nlemma log_mult_eq:\n \"log b (A * B) = (if 0 < A * B then log b \\A\\ + log b \\B\\ else log b 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. log b (A * B) = (if 0 < A * B then log b \\A\\ + log b \\B\\ else log b 0)\n[PROOF STEP]\nusing log_mult[of b \"\\A\\\" \"\\B\\\"] b_gt_1 log_neg_const[of \"A * B\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < b; b \\ 1; 0 < \\A\\; 0 < \\B\\\\ \\ log b (\\A\\ * \\B\\) = log b \\A\\ + log b \\B\\\n1 < b\nA * B \\ 0 \\ log b (A * B) = log b 0\n\ngoal (1 subgoal):\n 1. log b (A * B) = (if 0 < A * B then log b \\A\\ + log b \\B\\ else log b 0)\n[PROOF STEP]\nby (auto simp: zero_less_mult_iff mult_le_0_iff)", "meta": {"llama_tokens": 381, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392817460332, "lm_q2_score": 0.7981867729389246, "lm_q1q2_score": 0.705628461448111}} {"text": "[STATEMENT]\nlemma lm087: \n assumes \"\\ x. (P x \\ (f x = g x))\" \n shows \"Union {f x|x. P x} = Union {g x | x. P x}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ {f x |x. P x} = \\ {g x |x. P x}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. P x \\ f x = g x\n\ngoal (1 subgoal):\n 1. \\ {f x |x. P x} = \\ {g x |x. P x}\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 205, "file": "Vickrey_Clarke_Groves_Universes", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392756357327, "lm_q2_score": 0.7981867777396211, "lm_q1q2_score": 0.7056284608149542}} {"text": "[STATEMENT]\nlemma sum_union_disjoint_finite_set:\n fixes C::\"nat set\" and g::\"nat \\ int\"\n assumes \"finite C\"\n shows \"\\A B. A \\ B = {} \\ A \\ B = C \\ (\\k\\C. g k) = (\\k\\A. g k) + (\\k\\B. g k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A B. A \\ B = {} \\ A \\ B = C \\ sum g C = sum g A + sum g B\n[PROOF STEP]\nusing assms sum.union_disjoint\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite C\n\\finite ?A; finite ?B; ?A \\ ?B = {}\\ \\ sum ?g (?A \\ ?B) = sum ?g ?A + sum ?g ?B\n\ngoal (1 subgoal):\n 1. \\A B. A \\ B = {} \\ A \\ B = C \\ sum g C = sum g A + sum g B\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 324, "file": "Isabelle_Marries_Dirac_Deutsch_Jozsa", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392756357327, "lm_q2_score": 0.798186775339273, "lm_q1q2_score": 0.7056284586929522}} {"text": "[STATEMENT]\nlemma abs_zero_equality: assumes \"abs (x - y) = 0\" shows \"x = y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x = y\n[PROOF STEP]\nusing assms abs_zero eq_iff_diff_eq_0\n[PROOF STATE]\nproof (prove)\nusing this:\nPreliminaries.abs (x - y) = 0\nPreliminaries.abs ?x = 0 \\ ?x = 0\n(?a = ?b) = (?a - ?b = (0::?'a))\n\ngoal (1 subgoal):\n 1. x = y\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 181, "file": "Knot_Theory_Preliminaries", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392817460333, "lm_q2_score": 0.7981867681382279, "lm_q1q2_score": 0.7056284572041066}} {"text": "[STATEMENT]\nlemma Gromov_product_at_diff:\n \"\\Gromov_product_at x y z - Gromov_product_at a b c\\ \\ dist x a + dist y b + dist z c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Gromov_product_at x y z - Gromov_product_at a b c\\ \\ dist x a + dist y b + dist z c\n[PROOF STEP]\nunfolding Gromov_product_at_def abs_le_iff\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist x y + dist x z - dist y z) / 2 - (dist a b + dist a c - dist b c) / 2 \\ dist x a + dist y b + dist z c \\ - ((dist x y + dist x z - dist y z) / 2 - (dist a b + dist a c - dist b c) / 2) \\ dist x a + dist y b + dist z c\n[PROOF STEP]\napply (auto simp add: divide_simps)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. dist x y * 2 + dist x z * 2 - dist y z * 2 - (dist a b * 2 + dist a c * 2 - dist b c * 2) \\ dist x a * 4 + dist y b * 4 + dist z c * 4\n 2. dist a b * 2 + dist a c * 2 - dist b c * 2 - (dist x y * 2 + dist x z * 2 - dist y z * 2) \\ dist x a * 4 + dist y b * 4 + dist z c * 4\n[PROOF STEP]\nby (smt dist_commute dist_triangle4)+", "meta": {"llama_tokens": 477, "file": "Gromov_Hyperbolicity_Gromov_Hyperbolicity", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392695254318, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7056284559377929}} {"text": "[STATEMENT]\nlemma nonneg_power_le:\n assumes \"0 \\ a\" and \"0 \\ b\" shows \"a^n \\ b^n \\ n = 0 \\ a \\ b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) \\ a\n(0::'a) \\ b\n\ngoal (1 subgoal):\n 1. (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\nproof (cases rule: nonneg_linorder_cases)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. a < b \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n 2. a = b \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n 3. b < a \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\ncase less\n[PROOF STATE]\nproof (state)\nthis:\na < b\n\ngoal (3 subgoals):\n 1. a < b \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n 2. a = b \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n 3. b < a \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\nwith power_strict_mono[OF this, of n] assms\n[PROOF STATE]\nproof (chain)\npicking this:\n\\(0::'a) \\ a; 0 < n\\ \\ a ^ n < b ^ n\n(0::'a) \\ a\n(0::'a) \\ b\na < b\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\(0::'a) \\ a; 0 < n\\ \\ a ^ n < b ^ n\n(0::'a) \\ a\n(0::'a) \\ b\na < b\n\ngoal (1 subgoal):\n 1. (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\nby (cases n, auto)\n[PROOF STATE]\nproof (state)\nthis:\n(a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n\ngoal (2 subgoals):\n 1. a = b \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n 2. b < a \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. a = b \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n 2. b < a \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\ncase eq\n[PROOF STATE]\nproof (state)\nthis:\na = b\n\ngoal (2 subgoals):\n 1. a = b \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n 2. b < a \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na = b\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na = b\n\ngoal (1 subgoal):\n 1. (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n\ngoal (1 subgoal):\n 1. b < a \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. b < a \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\ncase greater\n[PROOF STATE]\nproof (state)\nthis:\nb < a\n\ngoal (1 subgoal):\n 1. b < a \\ (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\nwith power_strict_mono[OF this, of n] assms\n[PROOF STATE]\nproof (chain)\npicking this:\n\\(0::'a) \\ b; 0 < n\\ \\ b ^ n < a ^ n\n(0::'a) \\ a\n(0::'a) \\ b\nb < a\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\(0::'a) \\ b; 0 < n\\ \\ b ^ n < a ^ n\n(0::'a) \\ a\n(0::'a) \\ b\nb < a\n\ngoal (1 subgoal):\n 1. (a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n[PROOF STEP]\nby (cases n, auto)\n[PROOF STATE]\nproof (state)\nthis:\n(a ^ n \\ b ^ n) = (n = 0 \\ a \\ b)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1729, "file": "LLL_Basis_Reduction_Norms", "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8244619436290699, "lm_q2_score": 0.8558511524823263, "lm_q1q2_score": 0.7056167046327583}} {"text": "[STATEMENT]\nlemma ninv_inj:\"\\f \\ {i. i \\ n} \\ {i. i \\ n}; inj_on f {i. i \\ n}\\ \\\n inj_on (ninv n f) {i. i \\ n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ {i. i \\ n} \\ {i. i \\ n}; inj_on f {i. i \\ n}\\ \\ inj_on (ninv n f) {i. i \\ n}\n[PROOF STEP]\napply (subst inj_on_def, simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ {i. i \\ n} \\ {i. i \\ n}; inj_on f {i. i \\ n}\\ \\ \\x\\n. \\y\\n. ninv n f x = ninv n f y \\ x = y\n[PROOF STEP]\napply ((rule allI, rule impI)+, rule impI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x y. \\f \\ {i. i \\ n} \\ {i. i \\ n}; inj_on f {i. i \\ n}; x \\ n; y \\ n; ninv n f x = ninv n f y\\ \\ x = y\n[PROOF STEP]\napply (frule ninv_hom[of f n], assumption,\n frule_tac x = x in funcset_mem[of \"ninv n f\" \"{i. i \\ n}\" \"{i. i \\ n}\"], simp,\n frule_tac x = y in funcset_mem[of \"ninv n f\" \"{i. i \\ n}\" \"{i. i \\ n}\"],\n simp,\n frule_tac b = x in ninv_r_inv [of f n], assumption+)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x y. \\f \\ {i. i \\ n} \\ {i. i \\ n}; inj_on f {i. i \\ n}; x \\ n; y \\ n; ninv n f x = ninv n f y; ninv n f \\ {i. i \\ n} \\ {i. i \\ n}; ninv n f x \\ {i. i \\ n}; ninv n f y \\ {i. i \\ n}; f (ninv n f x) = x\\ \\ x = y\n[PROOF STEP]\napply (simp add:ninv_r_inv)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 809, "file": "Group-Ring-Module_Algebra1", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8558511579973932, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.7056166944141495}} {"text": "[STATEMENT]\nlemma smalloI_tendsto:\n assumes lim: \"((\\x. f x / g x) \\ 0) F\"\n assumes \"eventually (\\x. g x \\ 0) F\"\n shows \"f \\ o[F](g)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f \\ o[F](g)\n[PROOF STEP]\nproof (rule landau_o.smallI)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\c. 0 < c \\ \\\\<^sub>F x in F. norm (f x) \\ c * norm (g x)\n[PROOF STEP]\nfix c :: real\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\c. 0 < c \\ \\\\<^sub>F x in F. norm (f x) \\ c * norm (g x)\n[PROOF STEP]\nassume c_pos: \"c > 0\"\n[PROOF STATE]\nproof (state)\nthis:\n0 < c\n\ngoal (1 subgoal):\n 1. \\c. 0 < c \\ \\\\<^sub>F x in F. norm (f x) \\ c * norm (g x)\n[PROOF STEP]\nfrom c_pos and lim\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < c\n((\\x. f x / g x) \\ (0::'b)) F\n[PROOF STEP]\nhave ev: \"eventually (\\x. norm (f x / g x) < c) F\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < c\n((\\x. f x / g x) \\ (0::'b)) F\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. norm (f x / g x) < c\n[PROOF STEP]\nby (subst (asm) tendsto_iff) (simp add: dist_real_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in F. norm (f x / g x) < c\n\ngoal (1 subgoal):\n 1. \\c. 0 < c \\ \\\\<^sub>F x in F. norm (f x) \\ c * norm (g x)\n[PROOF STEP]\nwith assms(2)\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F x in F. g x \\ (0::'b)\n\\\\<^sub>F x in F. norm (f x / g x) < c\n[PROOF STEP]\nshow \"eventually (\\x. (norm (f x)) \\ c * (norm (g x))) F\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in F. g x \\ (0::'b)\n\\\\<^sub>F x in F. norm (f x / g x) < c\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. norm (f x) \\ c * norm (g x)\n[PROOF STEP]\nby eventually_elim (simp add: field_simps norm_divide)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in F. norm (f x) \\ c * norm (g x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 941, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8244619263765707, "lm_q2_score": 0.855851143290548, "lm_q1q2_score": 0.7056166822889156}} {"text": "[STATEMENT]\nlemma poly_pos:\n \"(\\x::real. poly p x > 0) \\ poly_inf p = 1 \\ (\\x. poly p x \\ 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. 0 < poly p x) = (poly_inf p = 1 \\ (\\x. poly p x \\ 0))\n[PROOF STEP]\nproof (intro iffI conjI)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\x. 0 < poly p x \\ poly_inf p = 1\n 2. \\x. 0 < poly p x \\ \\x. poly p x \\ 0\n 3. poly_inf p = 1 \\ (\\x. poly p x \\ 0) \\ \\x. 0 < poly p x\n[PROOF STEP]\nassume A: \"\\x::real. poly p x > 0\"\n[PROOF STATE]\nproof (state)\nthis:\n\\x. 0 < poly p x\n\ngoal (3 subgoals):\n 1. \\x. 0 < poly p x \\ poly_inf p = 1\n 2. \\x. 0 < poly p x \\ \\x. poly p x \\ 0\n 3. poly_inf p = 1 \\ (\\x. poly p x \\ 0) \\ \\x. 0 < poly p x\n[PROOF STEP]\nhave \"\\x. poly p (x::real) > 0 \\ poly p x \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. 0 < poly p x \\ poly p x \\ 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < poly p ?x \\ poly p ?x \\ 0\n\ngoal (3 subgoals):\n 1. \\x. 0 < poly p x \\ poly_inf p = 1\n 2. \\x. 0 < poly p x \\ \\x. poly p x \\ 0\n 3. poly_inf p = 1 \\ (\\x. poly p x \\ 0) \\ \\x. 0 < poly p x\n[PROOF STEP]\nwith A\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x. 0 < poly p x\n0 < poly p ?x \\ poly p ?x \\ 0\n[PROOF STEP]\nshow \"\\x::real. poly p x \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. 0 < poly p x\n0 < poly p ?x \\ poly p ?x \\ 0\n\ngoal (1 subgoal):\n 1. \\x. poly p x \\ 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\x. poly p x \\ 0\n\ngoal (2 subgoals):\n 1. \\x. 0 < poly p x \\ poly_inf p = 1\n 2. poly_inf p = 1 \\ (\\x. poly p x \\ 0) \\ \\x. 0 < poly p x\n[PROOF STEP]\nfrom poly_lim_inf\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F x in at_top. sgn (poly ?p x) = poly_inf ?p\n[PROOF STEP]\nobtain x where \"sgn (poly p x) = poly_inf p\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in at_top. sgn (poly ?p x) = poly_inf ?p\n\ngoal (1 subgoal):\n 1. (\\x. sgn (poly p x) = poly_inf p \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: eventually_at_top_linorder)\n[PROOF STATE]\nproof (state)\nthis:\nsgn (poly p x) = poly_inf p\n\ngoal (2 subgoals):\n 1. \\x. 0 < poly p x \\ poly_inf p = 1\n 2. poly_inf p = 1 \\ (\\x. poly p x \\ 0) \\ \\x. 0 < poly p x\n[PROOF STEP]\nwith A\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x. 0 < poly p x\nsgn (poly p x) = poly_inf p\n[PROOF STEP]\nshow \"poly_inf p = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. 0 < poly p x\nsgn (poly p x) = poly_inf p\n\ngoal (1 subgoal):\n 1. poly_inf p = 1\n[PROOF STEP]\nby (simp add: sgn_real_def split: if_split_asm)\n[PROOF STATE]\nproof (state)\nthis:\npoly_inf p = 1\n\ngoal (1 subgoal):\n 1. poly_inf p = 1 \\ (\\x. poly p x \\ 0) \\ \\x. 0 < poly p x\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. poly_inf p = 1 \\ (\\x. poly p x \\ 0) \\ \\x. 0 < poly p x\n[PROOF STEP]\nassume \"poly_inf p = 1 \\ (\\x. poly p x \\ 0)\"\n[PROOF STATE]\nproof (state)\nthis:\npoly_inf p = 1 \\ (\\x. poly p x \\ 0)\n\ngoal (1 subgoal):\n 1. poly_inf p = 1 \\ (\\x. poly p x \\ 0) \\ \\x. 0 < poly p x\n[PROOF STEP]\nhence A: \"poly_inf p = 1\" and B: \"(\\x. poly p x \\ 0)\"\n[PROOF STATE]\nproof (prove)\nusing this:\npoly_inf p = 1 \\ (\\x. poly p x \\ 0)\n\ngoal (1 subgoal):\n 1. poly_inf p = 1 &&& \\x. poly p x \\ 0\n[PROOF STEP]\nby simp_all\n[PROOF STATE]\nproof (state)\nthis:\npoly_inf p = 1\n\\x. poly p x \\ 0\n\ngoal (1 subgoal):\n 1. poly_inf p = 1 \\ (\\x. poly p x \\ 0) \\ \\x. 0 < poly p x\n[PROOF STEP]\nfrom poly_lim_inf\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F x in at_top. sgn (poly ?p x) = poly_inf ?p\n[PROOF STEP]\nobtain x where C: \"sgn (poly p x) = poly_inf p\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in at_top. sgn (poly ?p x) = poly_inf ?p\n\ngoal (1 subgoal):\n 1. (\\x. sgn (poly p x) = poly_inf p \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: eventually_at_top_linorder)\n[PROOF STATE]\nproof (state)\nthis:\nsgn (poly p x) = poly_inf p\n\ngoal (1 subgoal):\n 1. poly_inf p = 1 \\ (\\x. poly p x \\ 0) \\ \\x. 0 < poly p x\n[PROOF STEP]\nshow \"\\x. poly p x > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. 0 < poly p x\n[PROOF STEP]\nproof (rule ccontr)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ (\\x. 0 < poly p x) \\ False\n[PROOF STEP]\nassume \"\\(\\x. poly p x > 0)\"\n[PROOF STATE]\nproof (state)\nthis:\n\\ (\\x. 0 < poly p x)\n\ngoal (1 subgoal):\n 1. \\ (\\x. 0 < poly p x) \\ False\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ (\\x. 0 < poly p x)\n[PROOF STEP]\nobtain x' where \"poly p x' \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (\\x. 0 < poly p x)\n\ngoal (1 subgoal):\n 1. (\\x'. poly p x' \\ 0 \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: not_less)\n[PROOF STATE]\nproof (state)\nthis:\npoly p x' \\ 0\n\ngoal (1 subgoal):\n 1. \\ (\\x. 0 < poly p x) \\ False\n[PROOF STEP]\nwith A and C\n[PROOF STATE]\nproof (chain)\npicking this:\npoly_inf p = 1\nsgn (poly p x) = poly_inf p\npoly p x' \\ 0\n[PROOF STEP]\nhave \"sgn (poly p x') \\ sgn (poly p x)\"\n[PROOF STATE]\nproof (prove)\nusing this:\npoly_inf p = 1\nsgn (poly p x) = poly_inf p\npoly p x' \\ 0\n\ngoal (1 subgoal):\n 1. sgn (poly p x') \\ sgn (poly p x)\n[PROOF STEP]\nby (auto simp: sgn_real_def split: if_split_asm)\n[PROOF STATE]\nproof (state)\nthis:\nsgn (poly p x') \\ sgn (poly p x)\n\ngoal (1 subgoal):\n 1. \\ (\\x. 0 < poly p x) \\ False\n[PROOF STEP]\nfrom poly_different_sign_imp_root'[OF this] and B\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x. poly p x = 0\n\\x. poly p x \\ 0\n[PROOF STEP]\nshow False\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. poly p x = 0\n\\x. poly p x \\ 0\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\x. 0 < poly p x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3085, "file": "Sturm_Sequences_Lib_Misc_Polynomial", "length": 34, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.855851135937125, "lm_q2_score": 0.8244619285331332, "lm_q1q2_score": 0.7056166780719948}} {"text": "[STATEMENT]\ntheorem banach_fix:\\ \\TODO: rename to \\Banach_fix\\\\\n assumes s: \"complete s\" \"s \\ {}\"\n and c: \"0 \\ c\" \"c < 1\"\n and f: \"f ` s \\ s\"\n and lipschitz: \"\\x\\s. \\y\\s. dist (f x) (f y) \\ c * dist x y\"\n shows \"\\!x\\s. f x = x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nfrom c\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\nc < 1\n[PROOF STEP]\nhave \"1 - c > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\n\ngoal (1 subgoal):\n 1. 0 < 1 - c\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < 1 - c\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nfrom s(2)\n[PROOF STATE]\nproof (chain)\npicking this:\ns \\ {}\n[PROOF STEP]\nobtain z0 where z0: \"z0 \\ s\"\n[PROOF STATE]\nproof (prove)\nusing this:\ns \\ {}\n\ngoal (1 subgoal):\n 1. (\\z0. z0 \\ s \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nz0 \\ s\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\ndefine z where \"z n = (f ^^ n) z0\" for n\n[PROOF STATE]\nproof (state)\nthis:\nz ?n = (f ^^ ?n) z0\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nwith f z0\n[PROOF STATE]\nproof (chain)\npicking this:\nf ` s \\ s\nz0 \\ s\nz ?n = (f ^^ ?n) z0\n[PROOF STEP]\nhave z_in_s: \"z n \\ s\" for n :: nat\n[PROOF STATE]\nproof (prove)\nusing this:\nf ` s \\ s\nz0 \\ s\nz ?n = (f ^^ ?n) z0\n\ngoal (1 subgoal):\n 1. z n \\ s\n[PROOF STEP]\nby (induct n) auto\n[PROOF STATE]\nproof (state)\nthis:\nz ?n \\ s\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\ndefine d where \"d = dist (z 0) (z 1)\"\n[PROOF STATE]\nproof (state)\nthis:\nd = dist (z 0) (z 1)\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nhave fzn: \"f (z n) = z (Suc n)\" for n\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f (z n) = z (Suc n)\n[PROOF STEP]\nby (simp add: z_def)\n[PROOF STATE]\nproof (state)\nthis:\nf (z ?n) = z (Suc ?n)\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nhave cf_z: \"dist (z n) (z (Suc n)) \\ (c ^ n) * d\" for n :: nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist (z n) (z (Suc n)) \\ c ^ n * d\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. dist (z 0) (z (Suc 0)) \\ c ^ 0 * d\n 2. \\n. dist (z n) (z (Suc n)) \\ c ^ n * d \\ dist (z (Suc n)) (z (Suc (Suc n))) \\ c ^ Suc n * d\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. dist (z 0) (z (Suc 0)) \\ c ^ 0 * d\n 2. \\n. dist (z n) (z (Suc n)) \\ c ^ n * d \\ dist (z (Suc n)) (z (Suc (Suc n))) \\ c ^ Suc n * d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist (z 0) (z (Suc 0)) \\ c ^ 0 * d\n[PROOF STEP]\nby (simp add: d_def)\n[PROOF STATE]\nproof (state)\nthis:\ndist (z 0) (z (Suc 0)) \\ c ^ 0 * d\n\ngoal (1 subgoal):\n 1. \\n. dist (z n) (z (Suc n)) \\ c ^ n * d \\ dist (z (Suc n)) (z (Suc (Suc n))) \\ c ^ Suc n * d\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. dist (z n) (z (Suc n)) \\ c ^ n * d \\ dist (z (Suc n)) (z (Suc (Suc n))) \\ c ^ Suc n * d\n[PROOF STEP]\ncase (Suc m)\n[PROOF STATE]\nproof (state)\nthis:\ndist (z m) (z (Suc m)) \\ c ^ m * d\n\ngoal (1 subgoal):\n 1. \\n. dist (z n) (z (Suc n)) \\ c ^ n * d \\ dist (z (Suc n)) (z (Suc (Suc n))) \\ c ^ Suc n * d\n[PROOF STEP]\nwith \\0 \\ c\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\ndist (z m) (z (Suc m)) \\ c ^ m * d\n[PROOF STEP]\nhave \"c * dist (z m) (z (Suc m)) \\ c ^ Suc m * d\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\ndist (z m) (z (Suc m)) \\ c ^ m * d\n\ngoal (1 subgoal):\n 1. c * dist (z m) (z (Suc m)) \\ c ^ Suc m * d\n[PROOF STEP]\nusing mult_left_mono[of \"dist (z m) (z (Suc m))\" \"c ^ m * d\" c]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\ndist (z m) (z (Suc m)) \\ c ^ m * d\n\\dist (z m) (z (Suc m)) \\ c ^ m * d; 0 \\ c\\ \\ c * dist (z m) (z (Suc m)) \\ c * (c ^ m * d)\n\ngoal (1 subgoal):\n 1. c * dist (z m) (z (Suc m)) \\ c ^ Suc m * d\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nc * dist (z m) (z (Suc m)) \\ c ^ Suc m * d\n\ngoal (1 subgoal):\n 1. \\n. dist (z n) (z (Suc n)) \\ c ^ n * d \\ dist (z (Suc n)) (z (Suc (Suc n))) \\ c ^ Suc n * d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc * dist (z m) (z (Suc m)) \\ c ^ Suc m * d\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nc * dist (z m) (z (Suc m)) \\ c ^ Suc m * d\n\ngoal (1 subgoal):\n 1. dist (z (Suc m)) (z (Suc (Suc m))) \\ c ^ Suc m * d\n[PROOF STEP]\nusing lipschitz[THEN bspec[where x=\"z m\"], OF z_in_s, THEN bspec[where x=\"z (Suc m)\"], OF z_in_s]\n[PROOF STATE]\nproof (prove)\nusing this:\nc * dist (z m) (z (Suc m)) \\ c ^ Suc m * d\ndist (f (z m)) (f (z (Suc m))) \\ c * dist (z m) (z (Suc m))\n\ngoal (1 subgoal):\n 1. dist (z (Suc m)) (z (Suc (Suc m))) \\ c ^ Suc m * d\n[PROOF STEP]\nby (simp add: fzn mult_le_cancel_left)\n[PROOF STATE]\nproof (state)\nthis:\ndist (z (Suc m)) (z (Suc (Suc m))) \\ c ^ Suc m * d\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ndist (z ?n) (z (Suc ?n)) \\ c ^ ?n * d\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nhave cf_z2: \"(1 - c) * dist (z m) (z (m + n)) \\ (c ^ m) * d * (1 - c ^ n)\" for n m :: nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (1 - c) * dist (z m) (z (m + 0)) \\ c ^ m * d * (1 - c ^ 0)\n 2. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. (1 - c) * dist (z m) (z (m + 0)) \\ c ^ m * d * (1 - c ^ 0)\n 2. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1 - c) * dist (z m) (z (m + 0)) \\ c ^ m * d * (1 - c ^ 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(1 - c) * dist (z m) (z (m + 0)) \\ c ^ m * d * (1 - c ^ 0)\n\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\ncase (Suc k)\n[PROOF STATE]\nproof (state)\nthis:\n(1 - c) * dist (z m) (z (m + k)) \\ c ^ m * d * (1 - c ^ k)\n\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nfrom c\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\nc < 1\n[PROOF STEP]\nhave \"(1 - c) * dist (z m) (z (m + Suc k)) \\\n (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\n\ngoal (1 subgoal):\n 1. (1 - c) * dist (z m) (z (m + Suc k)) \\ (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))\n[PROOF STEP]\nby (simp add: dist_triangle)\n[PROOF STATE]\nproof (state)\nthis:\n(1 - c) * dist (z m) (z (m + Suc k)) \\ (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))\n\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(1 - c) * dist (z m) (z (m + Suc k)) \\ (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))\n\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nfrom c cf_z[of \"m + k\"]\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\nc < 1\ndist (z (m + k)) (z (Suc (m + k))) \\ c ^ (m + k) * d\n[PROOF STEP]\nhave \"\\ \\ (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\ndist (z (m + k)) (z (Suc (m + k))) \\ c ^ (m + k) * d\n\ngoal (1 subgoal):\n 1. (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k)))) \\ (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k)))) \\ (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)\n\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k)))) \\ (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)\n\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nfrom Suc\n[PROOF STATE]\nproof (chain)\npicking this:\n(1 - c) * dist (z m) (z (m + k)) \\ c ^ m * d * (1 - c ^ k)\n[PROOF STEP]\nhave \"\\ \\ c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(1 - c) * dist (z m) (z (m + k)) \\ c ^ m * d * (1 - c ^ k)\n\ngoal (1 subgoal):\n 1. (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d) \\ c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d) \\ c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d\n\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d) \\ c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d\n\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nhave \"\\ = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d = c ^ m * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)\n[PROOF STEP]\nby (simp add: power_add field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nc ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d = c ^ m * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)\n\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nc ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d = c ^ m * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)\n\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nfrom c\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\nc < 1\n[PROOF STEP]\nhave \"\\ \\ (c ^ m) * d * (1 - c ^ Suc k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\n\ngoal (1 subgoal):\n 1. c ^ m * (d * (1 - c ^ k) + (1 - c) * c ^ k * d) \\ c ^ m * d * (1 - c ^ Suc k)\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nc ^ m * (d * (1 - c ^ k) + (1 - c) * c ^ k * d) \\ c ^ m * d * (1 - c ^ Suc k)\n\ngoal (1 subgoal):\n 1. \\n. (1 - c) * dist (z m) (z (m + n)) \\ c ^ m * d * (1 - c ^ n) \\ (1 - c) * dist (z m) (z (m + Suc n)) \\ c ^ m * d * (1 - c ^ Suc n)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(1 - c) * dist (z m) (z (m + Suc k)) \\ c ^ m * d * (1 - c ^ Suc k)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(1 - c) * dist (z m) (z (m + Suc k)) \\ c ^ m * d * (1 - c ^ Suc k)\n\ngoal (1 subgoal):\n 1. (1 - c) * dist (z m) (z (m + Suc k)) \\ c ^ m * d * (1 - c ^ Suc k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(1 - c) * dist (z m) (z (m + Suc k)) \\ c ^ m * d * (1 - c ^ Suc k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(1 - c) * dist (z ?m) (z (?m + ?n)) \\ c ^ ?m * d * (1 - c ^ ?n)\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nhave \"\\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\" if \"e > 0\" for e\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nproof (cases \"d = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. d = 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n 2. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nd = 0\n\ngoal (2 subgoals):\n 1. d = 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n 2. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nfrom \\1 - c > 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < 1 - c\n[PROOF STEP]\nhave \"(1 - c) * x \\ 0 \\ x \\ 0\" for x\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 - c\n\ngoal (1 subgoal):\n 1. ((1 - c) * x \\ 0) = (x \\ 0)\n[PROOF STEP]\nby (simp add: mult_le_0_iff)\n[PROOF STATE]\nproof (state)\nthis:\n((1 - c) * ?x \\ 0) = (?x \\ 0)\n\ngoal (2 subgoals):\n 1. d = 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n 2. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nwith c cf_z2[of 0] True\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\nc < 1\n(1 - c) * dist (z 0) (z (0 + ?n)) \\ c ^ 0 * d * (1 - c ^ ?n)\nd = 0\n((1 - c) * ?x \\ 0) = (?x \\ 0)\n[PROOF STEP]\nhave \"z n = z0\" for n\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\n(1 - c) * dist (z 0) (z (0 + ?n)) \\ c ^ 0 * d * (1 - c ^ ?n)\nd = 0\n((1 - c) * ?x \\ 0) = (?x \\ 0)\n\ngoal (1 subgoal):\n 1. z n = z0\n[PROOF STEP]\nby (simp add: z_def)\n[PROOF STATE]\nproof (state)\nthis:\nz ?n = z0\n\ngoal (2 subgoals):\n 1. d = 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n 2. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nwith \\e > 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < e\nz ?n = z0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < e\nz ?n = z0\n\ngoal (1 subgoal):\n 1. \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n\ngoal (1 subgoal):\n 1. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nd \\ 0\n\ngoal (1 subgoal):\n 1. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nwith zero_le_dist[of \"z 0\" \"z 1\"]\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ dist (z 0) (z 1)\nd \\ 0\n[PROOF STEP]\nhave \"d > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ dist (z 0) (z 1)\nd \\ 0\n\ngoal (1 subgoal):\n 1. 0 < d\n[PROOF STEP]\nby (metis d_def less_le)\n[PROOF STATE]\nproof (state)\nthis:\n0 < d\n\ngoal (1 subgoal):\n 1. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nwith \\1 - c > 0\\ \\e > 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < 1 - c\n0 < e\n0 < d\n[PROOF STEP]\nhave \"0 < e * (1 - c) / d\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 - c\n0 < e\n0 < d\n\ngoal (1 subgoal):\n 1. 0 < e * (1 - c) / d\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < e * (1 - c) / d\n\ngoal (1 subgoal):\n 1. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nwith c\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\nc < 1\n0 < e * (1 - c) / d\n[PROOF STEP]\nobtain N where N: \"c ^ N < e * (1 - c) / d\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\n0 < e * (1 - c) / d\n\ngoal (1 subgoal):\n 1. (\\N. c ^ N < e * (1 - c) / d \\ thesis) \\ thesis\n[PROOF STEP]\nusing real_arch_pow_inv[of \"e * (1 - c) / d\" c]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\n0 < e * (1 - c) / d\n\\0 < e * (1 - c) / d; c < 1\\ \\ \\n. c ^ n < e * (1 - c) / d\n\ngoal (1 subgoal):\n 1. (\\N. c ^ N < e * (1 - c) / d \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nc ^ N < e * (1 - c) / d\n\ngoal (1 subgoal):\n 1. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nhave *: \"dist (z m) (z n) < e\" if \"m > n\" and as: \"m \\ N\" \"n \\ N\" for m n :: nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nfrom c \\n \\ N\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\nc < 1\nN \\ n\n[PROOF STEP]\nhave *: \"c ^ n \\ c ^ N\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\nN \\ n\n\ngoal (1 subgoal):\n 1. c ^ n \\ c ^ N\n[PROOF STEP]\nusing power_decreasing[OF \\n\\N\\, of c]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\nN \\ n\n\\0 \\ c; c \\ 1\\ \\ c ^ n \\ c ^ N\n\ngoal (1 subgoal):\n 1. c ^ n \\ c ^ N\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nc ^ n \\ c ^ N\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nfrom c \\m > n\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\nc < 1\nn < m\n[PROOF STEP]\nhave \"1 - c ^ (m - n) > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\nn < m\n\ngoal (1 subgoal):\n 1. 0 < 1 - c ^ (m - n)\n[PROOF STEP]\nusing power_strict_mono[of c 1 \"m - n\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\nn < m\n\\c < 1; 0 \\ c; 0 < m - n\\ \\ c ^ (m - n) < 1 ^ (m - n)\n\ngoal (1 subgoal):\n 1. 0 < 1 - c ^ (m - n)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < 1 - c ^ (m - n)\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nwith \\d > 0\\ \\0 < 1 - c\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < d\n0 < 1 - c\n0 < 1 - c ^ (m - n)\n[PROOF STEP]\nhave **: \"d * (1 - c ^ (m - n)) / (1 - c) > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n0 < 1 - c\n0 < 1 - c ^ (m - n)\n\ngoal (1 subgoal):\n 1. 0 < d * (1 - c ^ (m - n)) / (1 - c)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < d * (1 - c ^ (m - n)) / (1 - c)\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nfrom cf_z2[of n \"m - n\"] \\m > n\\\n[PROOF STATE]\nproof (chain)\npicking this:\n(1 - c) * dist (z n) (z (n + (m - n))) \\ c ^ n * d * (1 - c ^ (m - n))\nn < m\n[PROOF STEP]\nhave \"dist (z m) (z n) \\ c ^ n * d * (1 - c ^ (m - n)) / (1 - c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(1 - c) * dist (z n) (z (n + (m - n))) \\ c ^ n * d * (1 - c ^ (m - n))\nn < m\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) \\ c ^ n * d * (1 - c ^ (m - n)) / (1 - c)\n[PROOF STEP]\nby (simp add: pos_le_divide_eq[OF \\1 - c > 0\\] mult.commute dist_commute)\n[PROOF STATE]\nproof (state)\nthis:\ndist (z m) (z n) \\ c ^ n * d * (1 - c ^ (m - n)) / (1 - c)\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndist (z m) (z n) \\ c ^ n * d * (1 - c ^ (m - n)) / (1 - c)\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nhave \"\\ \\ c ^ N * d * (1 - c ^ (m - n)) / (1 - c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c ^ n * d * (1 - c ^ (m - n)) / (1 - c) \\ c ^ N * d * (1 - c ^ (m - n)) / (1 - c)\n[PROOF STEP]\nusing mult_right_mono[OF * order_less_imp_le[OF **]]\n[PROOF STATE]\nproof (prove)\nusing this:\nc ^ n * (d * (1 - c ^ (m - n)) / (1 - c)) \\ c ^ N * (d * (1 - c ^ (m - n)) / (1 - c))\n\ngoal (1 subgoal):\n 1. c ^ n * d * (1 - c ^ (m - n)) / (1 - c) \\ c ^ N * d * (1 - c ^ (m - n)) / (1 - c)\n[PROOF STEP]\nby (simp add: mult.assoc)\n[PROOF STATE]\nproof (state)\nthis:\nc ^ n * d * (1 - c ^ (m - n)) / (1 - c) \\ c ^ N * d * (1 - c ^ (m - n)) / (1 - c)\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nc ^ n * d * (1 - c ^ (m - n)) / (1 - c) \\ c ^ N * d * (1 - c ^ (m - n)) / (1 - c)\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nhave \"\\ < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c ^ N * d * (1 - c ^ (m - n)) / (1 - c) < e * (1 - c) / d * d * (1 - c ^ (m - n)) / (1 - c)\n[PROOF STEP]\nusing mult_strict_right_mono[OF N **]\n[PROOF STATE]\nproof (prove)\nusing this:\nc ^ N * (d * (1 - c ^ (m - n)) / (1 - c)) < e * (1 - c) / d * (d * (1 - c ^ (m - n)) / (1 - c))\n\ngoal (1 subgoal):\n 1. c ^ N * d * (1 - c ^ (m - n)) / (1 - c) < e * (1 - c) / d * d * (1 - c ^ (m - n)) / (1 - c)\n[PROOF STEP]\nby (auto simp: mult.assoc)\n[PROOF STATE]\nproof (state)\nthis:\nc ^ N * d * (1 - c ^ (m - n)) / (1 - c) < e * (1 - c) / d * d * (1 - c ^ (m - n)) / (1 - c)\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nc ^ N * d * (1 - c ^ (m - n)) / (1 - c) < e * (1 - c) / d * d * (1 - c ^ (m - n)) / (1 - c)\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nfrom c \\d > 0\\ \\1 - c > 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\nc < 1\n0 < d\n0 < 1 - c\n[PROOF STEP]\nhave \"\\ = e * (1 - c ^ (m - n))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\n0 < d\n0 < 1 - c\n\ngoal (1 subgoal):\n 1. e * (1 - c) / d * d * (1 - c ^ (m - n)) / (1 - c) = e * (1 - c ^ (m - n))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ne * (1 - c) / d * d * (1 - c ^ (m - n)) / (1 - c) = e * (1 - c ^ (m - n))\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ne * (1 - c) / d * d * (1 - c ^ (m - n)) / (1 - c) = e * (1 - c ^ (m - n))\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nfrom c \\1 - c ^ (m - n) > 0\\ \\e > 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\nc < 1\n0 < 1 - c ^ (m - n)\n0 < e\n[PROOF STEP]\nhave \"\\ \\ e\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\n0 < 1 - c ^ (m - n)\n0 < e\n\ngoal (1 subgoal):\n 1. e * (1 - c ^ (m - n)) \\ e\n[PROOF STEP]\nusing mult_right_le_one_le[of e \"1 - c ^ (m - n)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\n0 < 1 - c ^ (m - n)\n0 < e\n\\0 \\ e; 0 \\ 1 - c ^ (m - n); 1 - c ^ (m - n) \\ 1\\ \\ e * (1 - c ^ (m - n)) \\ e\n\ngoal (1 subgoal):\n 1. e * (1 - c ^ (m - n)) \\ e\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ne * (1 - c ^ (m - n)) \\ e\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndist (z m) (z n) < e\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndist (z m) (z n) < e\n\ngoal (1 subgoal):\n 1. dist (z m) (z n) < e\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndist (z m) (z n) < e\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\?n < ?m; N \\ ?m; N \\ ?n\\ \\ dist (z ?m) (z ?n) < e\n\ngoal (1 subgoal):\n 1. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nhave \"dist (z n) (z m) < e\" if \"N \\ m\" \"N \\ n\" for m n :: nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist (z n) (z m) < e\n[PROOF STEP]\nproof (cases \"n = m\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. n = m \\ dist (z n) (z m) < e\n 2. n \\ m \\ dist (z n) (z m) < e\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nn = m\n\ngoal (2 subgoals):\n 1. n = m \\ dist (z n) (z m) < e\n 2. n \\ m \\ dist (z n) (z m) < e\n[PROOF STEP]\nwith \\e > 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < e\nn = m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < e\nn = m\n\ngoal (1 subgoal):\n 1. dist (z n) (z m) < e\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndist (z n) (z m) < e\n\ngoal (1 subgoal):\n 1. n \\ m \\ dist (z n) (z m) < e\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. n \\ m \\ dist (z n) (z m) < e\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nn \\ m\n\ngoal (1 subgoal):\n 1. n \\ m \\ dist (z n) (z m) < e\n[PROOF STEP]\nwith *[of n m] *[of m n] and that\n[PROOF STATE]\nproof (chain)\npicking this:\n\\n < m; N \\ m; N \\ n\\ \\ dist (z m) (z n) < e\n\\m < n; N \\ n; N \\ m\\ \\ dist (z n) (z m) < e\nN \\ m\nN \\ n\nn \\ m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n < m; N \\ m; N \\ n\\ \\ dist (z m) (z n) < e\n\\m < n; N \\ n; N \\ m\\ \\ dist (z n) (z m) < e\nN \\ m\nN \\ n\nn \\ m\n\ngoal (1 subgoal):\n 1. dist (z n) (z m) < e\n[PROOF STEP]\nby (auto simp: dist_commute nat_neq_iff)\n[PROOF STATE]\nproof (state)\nthis:\ndist (z n) (z m) < e\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\N \\ ?m; N \\ ?n\\ \\ dist (z ?n) (z ?m) < e\n\ngoal (1 subgoal):\n 1. d \\ 0 \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\N \\ ?m; N \\ ?n\\ \\ dist (z ?n) (z ?m) < e\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\N \\ ?m; N \\ ?n\\ \\ dist (z ?n) (z ?m) < e\n\ngoal (1 subgoal):\n 1. \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < e\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n0 < ?e \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < ?e\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < ?e \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < ?e\n[PROOF STEP]\nhave \"Cauchy z\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < ?e \\ \\N. \\m n. N \\ m \\ N \\ n \\ dist (z m) (z n) < ?e\n\ngoal (1 subgoal):\n 1. Cauchy z\n[PROOF STEP]\nby (simp add: cauchy_def)\n[PROOF STATE]\nproof (state)\nthis:\nCauchy z\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nCauchy z\n[PROOF STEP]\nobtain x where \"x\\s\" and x:\"(z \\ x) sequentially\"\n[PROOF STATE]\nproof (prove)\nusing this:\nCauchy z\n\ngoal (1 subgoal):\n 1. (\\x. \\x \\ s; z \\ x\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s\n[PROOF STATE]\nproof (prove)\nusing this:\nCauchy z\n(\\n. z n \\ s) \\ Cauchy z \\ (\\l\\s. z \\ l)\nz ?n \\ s\n\ngoal (1 subgoal):\n 1. (\\x. \\x \\ s; z \\ x\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx \\ s\nz \\ x\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\ndefine e where \"e = dist (f x) x\"\n[PROOF STATE]\nproof (state)\nthis:\ne = dist (f x) x\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nhave \"e = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. e = 0\n[PROOF STEP]\nproof (rule ccontr)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. e \\ 0 \\ False\n[PROOF STEP]\nassume \"e \\ 0\"\n[PROOF STATE]\nproof (state)\nthis:\ne \\ 0\n\ngoal (1 subgoal):\n 1. e \\ 0 \\ False\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ne \\ 0\n[PROOF STEP]\nhave \"e > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\ne \\ 0\n\ngoal (1 subgoal):\n 1. 0 < e\n[PROOF STEP]\nunfolding e_def\n[PROOF STATE]\nproof (prove)\nusing this:\ndist (f x) x \\ 0\n\ngoal (1 subgoal):\n 1. 0 < dist (f x) x\n[PROOF STEP]\nusing zero_le_dist[of \"f x\" x]\n[PROOF STATE]\nproof (prove)\nusing this:\ndist (f x) x \\ 0\n0 \\ dist (f x) x\n\ngoal (1 subgoal):\n 1. 0 < dist (f x) x\n[PROOF STEP]\nby (metis dist_eq_0_iff dist_nz e_def)\n[PROOF STATE]\nproof (state)\nthis:\n0 < e\n\ngoal (1 subgoal):\n 1. e \\ 0 \\ False\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < e\n[PROOF STEP]\nobtain N where N:\"\\n\\N. dist (z n) x < e/2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < e\n\ngoal (1 subgoal):\n 1. (\\N. \\n\\N. dist (z n) x < e / 2 \\ thesis) \\ thesis\n[PROOF STEP]\nusing x[unfolded lim_sequentially, THEN spec[where x=\"e/2\"]]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < e\n0 < e / 2 \\ (\\no. \\n\\no. dist (z n) x < e / 2)\n\ngoal (1 subgoal):\n 1. (\\N. \\n\\N. dist (z n) x < e / 2 \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\n\\N. dist (z n) x < e / 2\n\ngoal (1 subgoal):\n 1. e \\ 0 \\ False\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\n\\N. dist (z n) x < e / 2\n[PROOF STEP]\nhave N':\"dist (z N) x < e/2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n\\N. dist (z n) x < e / 2\n\ngoal (1 subgoal):\n 1. dist (z N) x < e / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndist (z N) x < e / 2\n\ngoal (1 subgoal):\n 1. e \\ 0 \\ False\n[PROOF STEP]\nhave *: \"c * dist (z N) x \\ dist (z N) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c * dist (z N) x \\ dist (z N) x\n[PROOF STEP]\nunfolding mult_le_cancel_right2\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0 < dist (z N) x \\ c \\ 1) \\ (dist (z N) x < 0 \\ 1 \\ c)\n[PROOF STEP]\nusing zero_le_dist[of \"z N\" x] and c\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ dist (z N) x\n0 \\ c\nc < 1\n\ngoal (1 subgoal):\n 1. (0 < dist (z N) x \\ c \\ 1) \\ (dist (z N) x < 0 \\ 1 \\ c)\n[PROOF STEP]\nby (metis dist_eq_0_iff dist_nz order_less_asym less_le)\n[PROOF STATE]\nproof (state)\nthis:\nc * dist (z N) x \\ dist (z N) x\n\ngoal (1 subgoal):\n 1. e \\ 0 \\ False\n[PROOF STEP]\nhave \"dist (f (z N)) (f x) \\ c * dist (z N) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist (f (z N)) (f x) \\ c * dist (z N) x\n[PROOF STEP]\nusing lipschitz[THEN bspec[where x=\"z N\"], THEN bspec[where x=x]]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\z N \\ s; x \\ s\\ \\ dist (f (z N)) (f x) \\ c * dist (z N) x\n\ngoal (1 subgoal):\n 1. dist (f (z N)) (f x) \\ c * dist (z N) x\n[PROOF STEP]\nusing z_in_s[of N] \\x\\s\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\z N \\ s; x \\ s\\ \\ dist (f (z N)) (f x) \\ c * dist (z N) x\nz N \\ s\nx \\ s\n\ngoal (1 subgoal):\n 1. dist (f (z N)) (f x) \\ c * dist (z N) x\n[PROOF STEP]\nusing c\n[PROOF STATE]\nproof (prove)\nusing this:\n\\z N \\ s; x \\ s\\ \\ dist (f (z N)) (f x) \\ c * dist (z N) x\nz N \\ s\nx \\ s\n0 \\ c\nc < 1\n\ngoal (1 subgoal):\n 1. dist (f (z N)) (f x) \\ c * dist (z N) x\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndist (f (z N)) (f x) \\ c * dist (z N) x\n\ngoal (1 subgoal):\n 1. e \\ 0 \\ False\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndist (f (z N)) (f x) \\ c * dist (z N) x\n\ngoal (1 subgoal):\n 1. e \\ 0 \\ False\n[PROOF STEP]\nhave \"\\ < e/2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c * dist (z N) x < e / 2\n[PROOF STEP]\nusing N' and c\n[PROOF STATE]\nproof (prove)\nusing this:\ndist (z N) x < e / 2\n0 \\ c\nc < 1\n\ngoal (1 subgoal):\n 1. c * dist (z N) x < e / 2\n[PROOF STEP]\nusing *\n[PROOF STATE]\nproof (prove)\nusing this:\ndist (z N) x < e / 2\n0 \\ c\nc < 1\nc * dist (z N) x \\ dist (z N) x\n\ngoal (1 subgoal):\n 1. c * dist (z N) x < e / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nc * dist (z N) x < e / 2\n\ngoal (1 subgoal):\n 1. e \\ 0 \\ False\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndist (f (z N)) (f x) < e / 2\n[PROOF STEP]\nshow False\n[PROOF STATE]\nproof (prove)\nusing this:\ndist (f (z N)) (f x) < e / 2\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nunfolding fzn\n[PROOF STATE]\nproof (prove)\nusing this:\ndist (z (Suc N)) (f x) < e / 2\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nusing N[THEN spec[where x=\"Suc N\"]] and dist_triangle_half_r[of \"z (Suc N)\" \"f x\" e x]\n[PROOF STATE]\nproof (prove)\nusing this:\ndist (z (Suc N)) (f x) < e / 2\nN \\ Suc N \\ dist (z (Suc N)) x < e / 2\n\\dist (z (Suc N)) (f x) < e / 2; dist (z (Suc N)) x < e / 2\\ \\ dist (f x) x < e\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nunfolding e_def\n[PROOF STATE]\nproof (prove)\nusing this:\ndist (z (Suc N)) (f x) < dist (f x) x / 2\nN \\ Suc N \\ dist (z (Suc N)) x < dist (f x) x / 2\n\\dist (z (Suc N)) (f x) < dist (f x) x / 2; dist (z (Suc N)) x < dist (f x) x / 2\\ \\ dist (f x) x < dist (f x) x\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ne = 0\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ne = 0\n[PROOF STEP]\nhave \"f x = x\"\n[PROOF STATE]\nproof (prove)\nusing this:\ne = 0\n\ngoal (1 subgoal):\n 1. f x = x\n[PROOF STEP]\nby (auto simp: e_def)\n[PROOF STATE]\nproof (state)\nthis:\nf x = x\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nf x = x\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nhave \"y = x\" if \"f y = y\" \"y \\ s\" for y\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. y = x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. y = x\n[PROOF STEP]\nfrom \\x \\ s\\ \\f x = x\\ that\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ s\nf x = x\nf y = y\ny \\ s\n[PROOF STEP]\nhave \"dist x y \\ c * dist x y\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ s\nf x = x\nf y = y\ny \\ s\n\ngoal (1 subgoal):\n 1. dist x y \\ c * dist x y\n[PROOF STEP]\nusing lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ s\nf x = x\nf y = y\ny \\ s\n\\x \\ s; y \\ s\\ \\ dist (f x) (f y) \\ c * dist x y\n\ngoal (1 subgoal):\n 1. dist x y \\ c * dist x y\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndist x y \\ c * dist x y\n\ngoal (1 subgoal):\n 1. y = x\n[PROOF STEP]\nwith c and zero_le_dist[of x y]\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ c\nc < 1\n0 \\ dist x y\ndist x y \\ c * dist x y\n[PROOF STEP]\nhave \"dist x y = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ c\nc < 1\n0 \\ dist x y\ndist x y \\ c * dist x y\n\ngoal (1 subgoal):\n 1. dist x y = 0\n[PROOF STEP]\nby (simp add: mult_le_cancel_right1)\n[PROOF STATE]\nproof (state)\nthis:\ndist x y = 0\n\ngoal (1 subgoal):\n 1. y = x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndist x y = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndist x y = 0\n\ngoal (1 subgoal):\n 1. y = x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ny = x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\f ?y = ?y; ?y \\ s\\ \\ ?y = x\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nf x = x\n\\f ?y = ?y; ?y \\ s\\ \\ ?y = x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nf x = x\n\\f ?y = ?y; ?y \\ s\\ \\ ?y = x\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nusing \\x\\s\\\n[PROOF STATE]\nproof (prove)\nusing this:\nf x = x\n\\f ?y = ?y; ?y \\ s\\ \\ ?y = x\nx \\ s\n\ngoal (1 subgoal):\n 1. \\!x. x \\ s \\ f x = x\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\!x. x \\ s \\ f x = x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 19104, "file": null, "length": 203, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.855851135937125, "lm_q2_score": 0.8244619242200082, "lm_q1q2_score": 0.7056166743806019}} {"text": "[STATEMENT]\nlemma poincare_distance_sym:\n assumes \"u \\ unit_disc\" and \"v \\ unit_disc\"\n shows \"poincare_distance u v = poincare_distance v u\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poincare_distance u v = poincare_distance v u\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ unit_disc\nv \\ unit_disc\n\ngoal (1 subgoal):\n 1. poincare_distance u v = poincare_distance v u\n[PROOF STEP]\nusing poincare_distance_formula[OF assms(1) assms(2)]\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ unit_disc\nv \\ unit_disc\npoincare_distance u v = poincare_distance_formula (to_complex u) (to_complex v)\n\ngoal (1 subgoal):\n 1. poincare_distance u v = poincare_distance v u\n[PROOF STEP]\nusing poincare_distance_formula[OF assms(2) assms(1)]\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ unit_disc\nv \\ unit_disc\npoincare_distance u v = poincare_distance_formula (to_complex u) (to_complex v)\npoincare_distance v u = poincare_distance_formula (to_complex v) (to_complex u)\n\ngoal (1 subgoal):\n 1. poincare_distance u v = poincare_distance v u\n[PROOF STEP]\nby (simp add: mult.commute norm_minus_commute)", "meta": {"llama_tokens": 467, "file": "Poincare_Disc_Poincare_Distance", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972818382005, "lm_q2_score": 0.8104789086703225, "lm_q1q2_score": 0.7056007348755738}} {"text": "[STATEMENT]\nlemma lookup_monic: \"lookup (monic p) v = (lookup p v) / lc p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lookup (monic p) v = lookup p v / lc p\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. lookup (monic p) v = lookup p v / lc p\n[PROOF STEP]\nhave \"lookup (monic p) (0 \\ v) = (1 / lc p) * (lookup p v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lookup (monic p) ((0::'a) \\ v) = (1::'b) / lc p * lookup p v\n[PROOF STEP]\nunfolding monic_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lookup (monom_mult ((1::'b) / lc p) (0::'a) p) ((0::'a) \\ v) = (1::'b) / lc p * lookup p v\n[PROOF STEP]\nby (rule lookup_monom_mult_plus)\n[PROOF STATE]\nproof (state)\nthis:\nlookup (monic p) ((0::'a) \\ v) = (1::'b) / lc p * lookup p v\n\ngoal (1 subgoal):\n 1. lookup (monic p) v = lookup p v / lc p\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlookup (monic p) ((0::'a) \\ v) = (1::'b) / lc p * lookup p v\n\ngoal (1 subgoal):\n 1. lookup (monic p) v = lookup p v / lc p\n[PROOF STEP]\nby (simp add: term_simps)\n[PROOF STATE]\nproof (state)\nthis:\nlookup (monic p) v = lookup p v / lc p\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 560, "file": "Groebner_Bases_More_MPoly_Type_Class", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972616934408, "lm_q2_score": 0.8104789178257654, "lm_q1q2_score": 0.7056007265193746}} {"text": "[STATEMENT]\nlemma enumeration:\"\\f \\ {i. i \\ (n::nat)} \\ {i. i \\ m}; inj_on f {i. i \\ n}\\\n \\ n \\ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ {i. i \\ n} \\ {i. i \\ m}; inj_on f {i. i \\ n}\\ \\ n \\ m\n[PROOF STEP]\napply (frule image_sub[of f \"{i. i \\ n}\" \"{i. i \\ m}\" \"{i. i \\ n}\"])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\f \\ {i. i \\ n} \\ {i. i \\ m}; inj_on f {i. i \\ n}\\ \\ {i. i \\ n} \\ {i. i \\ n}\n 2. \\f \\ {i. i \\ n} \\ {i. i \\ m}; inj_on f {i. i \\ n}; f ` {i. i \\ n} \\ {i. i \\ m}\\ \\ n \\ m\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ {i. i \\ n} \\ {i. i \\ m}; inj_on f {i. i \\ n}; f ` {i. i \\ n} \\ {i. i \\ m}\\ \\ n \\ m\n[PROOF STEP]\napply (frule card_image[of f \"{i. i \\ n}\"])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ {i. i \\ n} \\ {i. i \\ m}; inj_on f {i. i \\ n}; f ` {i. i \\ n} \\ {i. i \\ m}; card (f ` {i. i \\ n}) = card {i. i \\ n}\\ \\ n \\ m\n[PROOF STEP]\napply(drule card_mono[OF finite_Collect_le_nat])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ {i. i \\ n} \\ {i. i \\ m}; inj_on f {i. i \\ n}; card (f ` {i. i \\ n}) = card {i. i \\ n}; card (f ` {i. i \\ n}) \\ card {i. i \\ m}\\ \\ n \\ m\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 850, "file": "Group-Ring-Module_Algebra1", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757870013740061, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7055622010417942}} {"text": "[STATEMENT]\nlemma mssnth_sin_series_stream_aux:\n \"mssnth (sin_series_stream_aux b (fact m) m) n = \n (if b then -1 else 1) * (-1) ^ n / (fact (2 * n + m))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mssnth (sin_series_stream_aux b (fact m) (real m)) n = (if b then - 1 else 1) * (- 1) ^ n / fact (2 * n + m)\n[PROOF STEP]\nproof (induction n arbitrary: b m)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\b m. mssnth (sin_series_stream_aux b (fact m) (real m)) 0 = (if b then - 1 else 1) * (- 1) ^ 0 / fact (2 * 0 + m)\n 2. \\n b m. (\\b m. mssnth (sin_series_stream_aux b (fact m) (real m)) n = (if b then - 1 else 1) * (- 1) ^ n / fact (2 * n + m)) \\ mssnth (sin_series_stream_aux b (fact m) (real m)) (Suc n) = (if b then - 1 else 1) * (- 1) ^ Suc n / fact (2 * Suc n + m)\n[PROOF STEP]\ncase (0 b m)\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\b m. mssnth (sin_series_stream_aux b (fact m) (real m)) 0 = (if b then - 1 else 1) * (- 1) ^ 0 / fact (2 * 0 + m)\n 2. \\n b m. (\\b m. mssnth (sin_series_stream_aux b (fact m) (real m)) n = (if b then - 1 else 1) * (- 1) ^ n / fact (2 * n + m)) \\ mssnth (sin_series_stream_aux b (fact m) (real m)) (Suc n) = (if b then - 1 else 1) * (- 1) ^ Suc n / fact (2 * Suc n + m)\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mssnth (sin_series_stream_aux b (fact m) (real m)) 0 = (if b then - 1 else 1) * (- 1) ^ 0 / fact (2 * 0 + m)\n[PROOF STEP]\nby (subst sin_series_stream_aux.code) (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nmssnth (sin_series_stream_aux b (fact m) (real m)) 0 = (if b then - 1 else 1) * (- 1) ^ 0 / fact (2 * 0 + m)\n\ngoal (1 subgoal):\n 1. \\n b m. (\\b m. mssnth (sin_series_stream_aux b (fact m) (real m)) n = (if b then - 1 else 1) * (- 1) ^ n / fact (2 * n + m)) \\ mssnth (sin_series_stream_aux b (fact m) (real m)) (Suc n) = (if b then - 1 else 1) * (- 1) ^ Suc n / fact (2 * Suc n + m)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n b m. (\\b m. mssnth (sin_series_stream_aux b (fact m) (real m)) n = (if b then - 1 else 1) * (- 1) ^ n / fact (2 * n + m)) \\ mssnth (sin_series_stream_aux b (fact m) (real m)) (Suc n) = (if b then - 1 else 1) * (- 1) ^ Suc n / fact (2 * Suc n + m)\n[PROOF STEP]\ncase (Suc n b m)\n[PROOF STATE]\nproof (state)\nthis:\nmssnth (sin_series_stream_aux ?b (fact ?m) (real ?m)) n = (if ?b then - 1 else 1) * (- 1) ^ n / fact (2 * n + ?m)\n\ngoal (1 subgoal):\n 1. \\n b m. (\\b m. mssnth (sin_series_stream_aux b (fact m) (real m)) n = (if b then - 1 else 1) * (- 1) ^ n / fact (2 * n + m)) \\ mssnth (sin_series_stream_aux b (fact m) (real m)) (Suc n) = (if b then - 1 else 1) * (- 1) ^ Suc n / fact (2 * Suc n + m)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mssnth (sin_series_stream_aux b (fact m) (real m)) (Suc n) = (if b then - 1 else 1) * (- 1) ^ Suc n / fact (2 * Suc n + m)\n[PROOF STEP]\nusing Suc.IH[of \"\\b\" \"m + 2\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nmssnth (sin_series_stream_aux (\\ b) (fact (m + 2)) (real (m + 2))) n = (if \\ b then - 1 else 1) * (- 1) ^ n / fact (2 * n + (m + 2))\n\ngoal (1 subgoal):\n 1. mssnth (sin_series_stream_aux b (fact m) (real m)) (Suc n) = (if b then - 1 else 1) * (- 1) ^ Suc n / fact (2 * Suc n + m)\n[PROOF STEP]\nby (subst sin_series_stream_aux.code) (auto simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nmssnth (sin_series_stream_aux b (fact m) (real m)) (Suc n) = (if b then - 1 else 1) * (- 1) ^ Suc n / fact (2 * Suc n + m)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1698, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869948899665, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7055621958180432}} {"text": "[STATEMENT]\nlemma aux_comp_with_cmod:\n fixes g:: \"nat \\ nat\"\n assumes \"(\\x<2^n. g x = 0) \\ (\\x<2^n. g x = 1)\"\n shows \"(cmod (\\k<2^n. (-1)^(g k)))\\<^sup>2 = 2^(2*n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nproof(rule disjE)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. ?P \\ ?Q\n 2. ?P \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n 3. ?Q \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nshow \"(\\x<2^n. g x = 0) \\ (\\x<2^n. g x = 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x<2 ^ n. g x = 0) \\ (\\x<2 ^ n. g x = 1)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x<2 ^ n. g x = 0) \\ (\\x<2 ^ n. g x = 1)\n\ngoal (1 subgoal):\n 1. (\\x<2 ^ n. g x = 0) \\ (\\x<2 ^ n. g x = 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\x<2 ^ n. g x = 0) \\ (\\x<2 ^ n. g x = 1)\n\ngoal (2 subgoals):\n 1. \\x<2 ^ n. g x = 0 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n 2. \\x<2 ^ n. g x = 1 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x<2 ^ n. g x = 0 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n 2. \\x<2 ^ n. g x = 1 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nassume \"\\x<2^n. g x = 0\"\n[PROOF STATE]\nproof (state)\nthis:\n\\x<2 ^ n. g x = 0\n\ngoal (2 subgoals):\n 1. \\x<2 ^ n. g x = 0 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n 2. \\x<2 ^ n. g x = 1 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x<2 ^ n. g x = 0\n[PROOF STEP]\nhave \"(cmod (\\k<2^n. (-1)^(g k)))\\<^sup>2 = (2^n)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x<2 ^ n. g x = 0\n\ngoal (1 subgoal):\n 1. (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = (2 ^ n)\\<^sup>2\n[PROOF STEP]\nby (simp add: norm_power)\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = (2 ^ n)\\<^sup>2\n\ngoal (2 subgoals):\n 1. \\x<2 ^ n. g x = 0 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n 2. \\x<2 ^ n. g x = 1 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = (2 ^ n)\\<^sup>2\n[PROOF STEP]\nshow \"?thesis\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = (2 ^ n)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nby (simp add: power_even_eq)\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n\ngoal (1 subgoal):\n 1. \\x<2 ^ n. g x = 1 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x<2 ^ n. g x = 1 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nassume \"\\x<2^n. g x = 1\"\n[PROOF STATE]\nproof (state)\nthis:\n\\x<2 ^ n. g x = 1\n\ngoal (1 subgoal):\n 1. \\x<2 ^ n. g x = 1 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x<2 ^ n. g x = 1\n[PROOF STEP]\nhave \"(cmod (\\k<2^n. (-1)^(g k)))\\<^sup>2 = (2^n)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x<2 ^ n. g x = 1\n\ngoal (1 subgoal):\n 1. (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = (2 ^ n)\\<^sup>2\n[PROOF STEP]\nby (simp add: norm_power)\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = (2 ^ n)\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\x<2 ^ n. g x = 1 \\ (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = (2 ^ n)\\<^sup>2\n[PROOF STEP]\nshow \"?thesis\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = (2 ^ n)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n[PROOF STEP]\nby (simp add: power_even_eq)\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (\\k<2 ^ n. (- 1) ^ g k))\\<^sup>2 = 2 ^ (2 * n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2415, "file": "Isabelle_Marries_Dirac_Deutsch_Jozsa", "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869948899665, "lm_q2_score": 0.8056321889812553, "lm_q1q2_score": 0.7055621937745191}} {"text": "[STATEMENT]\nlemma real_root_mult_exp_cancel:\n \"\\ 0 < x; 0 < m; 0 < n \\ \n \\ root (m * n) (x ^ (k * n)) = root m (x ^ k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < x; 0 < m; 0 < n\\ \\ root (m * n) (x ^ (k * n)) = root m (x ^ k)\n[PROOF STEP]\nby (simp add: power_mult real_root_pos_unique)", "meta": {"llama_tokens": 167, "file": "Real_Power_RatPower", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869916479466, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7055621932061676}} {"text": "[STATEMENT]\nlemma ortho_c_scalprod0:\n assumes \"z1 \\ 0\" and \"z2 \\ 0\"\n shows \"\\c z1 z2 = pi/2 \\ scalprod z1 z2 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\c z1 z2 = pi / 2) = (scalprod z1 z2 = 0)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\c z1 z2 = pi / 2) = (scalprod z1 z2 = 0)\n[PROOF STEP]\nhave \"\\ z1 z2 = pi / 2 \\ \\ z1 z2 = - pi / 2 \\ \\c z1 z2 = pi/2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ z1 z2 = pi / 2 \\ \\ z1 z2 = - pi / 2) = (\\c z1 z2 = pi / 2)\n[PROOF STEP]\nunfolding ang_vec_c_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ z1 z2 = pi / 2 \\ \\ z1 z2 = - pi / 2) = (\\\\ z1 z2\\ = pi / 2)\n[PROOF STEP]\nusing arctan\n[PROOF STATE]\nproof (prove)\nusing this:\n- (pi / 2) < arctan ?y \\ arctan ?y < pi / 2 \\ tan (arctan ?y) = ?y\n\ngoal (1 subgoal):\n 1. (\\ z1 z2 = pi / 2 \\ \\ z1 z2 = - pi / 2) = (\\\\ z1 z2\\ = pi / 2)\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n(\\ z1 z2 = pi / 2 \\ \\ z1 z2 = - pi / 2) = (\\c z1 z2 = pi / 2)\n\ngoal (1 subgoal):\n 1. (\\c z1 z2 = pi / 2) = (scalprod z1 z2 = 0)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\ z1 z2 = pi / 2 \\ \\ z1 z2 = - pi / 2) = (\\c z1 z2 = pi / 2)\n\ngoal (1 subgoal):\n 1. (\\c z1 z2 = pi / 2) = (scalprod z1 z2 = 0)\n[PROOF STEP]\nusing ortho_scalprod0[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\ z1 z2 = pi / 2 \\ \\ z1 z2 = - pi / 2) = (\\c z1 z2 = pi / 2)\n(\\ z1 z2 = pi / 2 \\ \\ z1 z2 = - pi / 2) = (scalprod z1 z2 = 0)\n\ngoal (1 subgoal):\n 1. (\\c z1 z2 = pi / 2) = (scalprod z1 z2 = 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\c z1 z2 = pi / 2) = (scalprod z1 z2 = 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 988, "file": "Complex_Geometry_Angles", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.875787001374006, "lm_q2_score": 0.805632181981183, "lm_q1q2_score": 0.7055621928676978}} {"text": "[STATEMENT]\nlemma C1_differentiable_on_mult [simp, derivative_intros]:\n fixes f g :: \"real \\ 'a :: real_normed_algebra\"\n shows \"f C1_differentiable_on S \\ g C1_differentiable_on S \\ (\\x. f x * g x) C1_differentiable_on S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f C1_differentiable_on S; g C1_differentiable_on S\\ \\ (\\x. f x * g x) C1_differentiable_on S\n[PROOF STEP]\nunfolding C1_differentiable_on_eq\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(\\x\\S. f differentiable at x) \\ continuous_on S (\\x. vector_derivative f (at x)); (\\x\\S. g differentiable at x) \\ continuous_on S (\\x. vector_derivative g (at x))\\ \\ (\\x\\S. (\\x. f x * g x) differentiable at x) \\ continuous_on S (\\x. vector_derivative (\\x. f x * g x) (at x))\n[PROOF STEP]\nby (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)", "meta": {"llama_tokens": 418, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869916479466, "lm_q2_score": 0.8056321843145404, "lm_q1q2_score": 0.7055621870755954}} {"text": "[STATEMENT]\nlemma \"((\\x::real. x powr 1.5 * (sqrt (x + 1) + sqrt (x - 1) - 2 * sqrt x)) \\ -1/4) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. x powr (15 / 10) * (sqrt (x + 1) + sqrt (x - 1) - 2 * sqrt x)) \\ - 1 / 4) at_top\n[PROOF STEP]\nby real_asymp", "meta": {"llama_tokens": 158, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026641072386, "lm_q2_score": 0.7690802370707281, "lm_q1q2_score": 0.7054793503772055}} {"text": "[STATEMENT]\nlemma infsum_Im: \n assumes \"f summable_on M\"\n shows \"infsum (\\x. Im (f x)) M = Im (infsum f M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sub>\\x\\M. Im (f x)) = Im (infsum f M)\n[PROOF STEP]\napply (rule infsum_comm_additive[where f=Im, unfolded o_def])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. additive Im\n 2. isCont Im (infsum f M)\n 3. f summable_on M\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nf summable_on M\n\ngoal (3 subgoals):\n 1. additive Im\n 2. isCont Im (infsum f M)\n 3. f summable_on M\n[PROOF STEP]\nby (auto intro!: additive.intro)", "meta": {"llama_tokens": 277, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240825770432, "lm_q2_score": 0.815232489352, "lm_q1q2_score": 0.7053587826865834}} {"text": "[STATEMENT]\nlemma ZFact_one: \"carrier (ZFact 1) = {UNIV}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. carrier (ZFact 1) = {UNIV}\n[PROOF STEP]\nunfolding ZFact_defs A_RCOSETS_defs r_coset_def ring_record_simps int.genideal_one\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a. {\\h. {h + a}}) = {UNIV}\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\a. {\\h. {h + a}}) \\ {UNIV}\n 2. {UNIV} \\ (\\a. {\\h. {h + a}})\n[PROOF STEP]\nhave \"\\a b::int. \\x. b = x + a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a b. \\x. b = x + a\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\n\\x. ?b = x + ?a\n\ngoal (2 subgoals):\n 1. (\\a. {\\h. {h + a}}) \\ {UNIV}\n 2. {UNIV} \\ (\\a. {\\h. {h + a}})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x. ?b = x + ?a\n[PROOF STEP]\nshow \"(\\a::int. {\\h. {h + a}}) \\ {UNIV}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. ?b = x + ?a\n\ngoal (1 subgoal):\n 1. (\\a. {\\h. {h + a}}) \\ {UNIV}\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n(\\a. {\\h. {h + a}}) \\ {UNIV}\n\ngoal (1 subgoal):\n 1. {UNIV} \\ (\\a. {\\h. {h + a}})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\a. {\\h. {h + a}}) \\ {UNIV}\n[PROOF STEP]\nshow \"{UNIV} \\ (\\a::int. {\\h. {h + a}})\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\a. {\\h. {h + a}}) \\ {UNIV}\n\ngoal (1 subgoal):\n 1. {UNIV} \\ (\\a. {\\h. {h + a}})\n[PROOF STEP]\nby (metis (no_types, lifting) UNIV_I UN_I singletonD singletonI subset_iff)\n[PROOF STATE]\nproof (state)\nthis:\n{UNIV} \\ (\\a. {\\h. {h + a}})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 897, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240791017536, "lm_q2_score": 0.8152324848629215, "lm_q1q2_score": 0.7053587759693556}} {"text": "[STATEMENT]\nlemma set_to_list_length:\n assumes \"finite S\"\n shows \"length (set_to_list S) = card S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (sorted_list_of_set S) = card S\n[PROOF STEP]\nby (metis assms length_remdups_card_conv length_sort set_sorted_list_of_set sorted_list_of_set_sort_remdups)", "meta": {"llama_tokens": 126, "file": "Padic_Field_Indices", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240825770432, "lm_q2_score": 0.8152324803738429, "lm_q1q2_score": 0.7053587749184655}} {"text": "[STATEMENT]\nlemma monomial_Sum_any:\n assumes \"finite {c. f c \\ 0}\"\n shows \"monomial (Sum_any f) a = (\\c. monomial (f c) a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. monomial (Sum_any f) a = (\\c. monomial (f c) a)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. monomial (Sum_any f) a = (\\c. monomial (f c) a)\n[PROOF STEP]\nhave \"{c. monomial (f c) a \\ 0} \\ {c. f c \\ 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {c. monomial (f c) a \\ 0} \\ {c. f c \\ (0::'b)}\n[PROOF STEP]\nby (rule, auto)\n[PROOF STATE]\nproof (state)\nthis:\n{c. monomial (f c) a \\ 0} \\ {c. f c \\ (0::'b)}\n\ngoal (1 subgoal):\n 1. monomial (Sum_any f) a = (\\c. monomial (f c) a)\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite {c. f c \\ (0::'b)}\n{c. monomial (f c) a \\ 0} \\ {c. f c \\ (0::'b)}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {c. f c \\ (0::'b)}\n{c. monomial (f c) a \\ 0} \\ {c. f c \\ (0::'b)}\n\ngoal (1 subgoal):\n 1. monomial (Sum_any f) a = (\\c. monomial (f c) a)\n[PROOF STEP]\nby (simp add: Groups_Big_Fun.comm_monoid_add_class.Sum_any.expand_superset monomial_sum)\n[PROOF STATE]\nproof (state)\nthis:\nmonomial (Sum_any f) a = (\\c. monomial (f c) a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 666, "file": "Polynomials_MPoly_Type_Class", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.865224072151174, "lm_q2_score": 0.8152324848629215, "lm_q1q2_score": 0.7053587703030173}} {"text": "[STATEMENT]\nlemma integrable_bij_count_space:\n fixes f :: \"'a \\ 'b::{banach, second_countable_topology}\"\n assumes g: \"bij_betw g A B\"\n shows \"integrable (count_space A) (\\x. f (g x)) \\ integrable (count_space B) f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integrable (count_space A) (\\x. f (g x)) = integrable (count_space B) f\n[PROOF STEP]\nunfolding integrable_iff_bounded\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. f (g x)) \\ borel_measurable (count_space A) \\ \\\\<^sup>+ x. ennreal (norm (f (g x))) \\count_space A < \\) = (f \\ borel_measurable (count_space B) \\ \\\\<^sup>+ x. ennreal (norm (f x)) \\count_space B < \\)\n[PROOF STEP]\nby (subst nn_integral_bij_count_space[OF g]) auto", "meta": {"llama_tokens": 329, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9073122138417878, "lm_q2_score": 0.7772998560157665, "lm_q1q2_score": 0.705253653180568}} {"text": "[STATEMENT]\nlemma exp_lower_taylor_even: \n fixes x::real\n shows \"even n \\ (\\m exp x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. even n \\ (\\m exp x\n[PROOF STEP]\nusing Maclaurin_exp_le [of x n]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\t. \\t\\ \\ \\x\\ \\ exp x = (\\m (\\m exp x\n[PROOF STEP]\nby (auto simp add: zero_le_even_power)", "meta": {"llama_tokens": 254, "file": "Special_Function_Bounds_Exp_Bounds", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681122619883, "lm_q2_score": 0.8221891327004132, "lm_q1q2_score": 0.7052476202787549}} {"text": "[STATEMENT]\nlemma ex_by_sq_aux:\n fixes x :: int and n :: nat\n assumes \"n mod 2 = 1\"\n shows \"x * (x * x) ^ (n div 2) = x ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * (x * x) ^ (n div 2) = x ^ n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x * (x * x) ^ (n div 2) = x ^ n\n[PROOF STEP]\nhave \"n > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nn mod 2 = 1\n\ngoal (1 subgoal):\n 1. 0 < n\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\n0 < n\n\ngoal (1 subgoal):\n 1. x * (x * x) ^ (n div 2) = x ^ n\n[PROOF STEP]\nhave \"2 * (n div 2) = n - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * (n div 2) = n - 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nn mod 2 = 1\n\ngoal (1 subgoal):\n 1. 2 * (n div 2) = n - 1\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\n2 * (n div 2) = n - 1\n\ngoal (1 subgoal):\n 1. x * (x * x) ^ (n div 2) = x ^ n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n2 * (n div 2) = n - 1\n[PROOF STEP]\nhave \"(x * x) ^ (n div 2) = x ^ (n - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * (n div 2) = n - 1\n\ngoal (1 subgoal):\n 1. (x * x) ^ (n div 2) = x ^ (n - 1)\n[PROOF STEP]\nby (simp add: semiring_normalization_rules)\n[PROOF STATE]\nproof (state)\nthis:\n(x * x) ^ (n div 2) = x ^ (n - 1)\n\ngoal (1 subgoal):\n 1. x * (x * x) ^ (n div 2) = x ^ n\n[PROOF STEP]\nwith \\0 < n\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < n\n(x * x) ^ (n div 2) = x ^ (n - 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\n(x * x) ^ (n div 2) = x ^ (n - 1)\n\ngoal (1 subgoal):\n 1. x * (x * x) ^ (n div 2) = x ^ n\n[PROOF STEP]\nby simp (metis Suc_pred power.simps(2))\n[PROOF STATE]\nproof (state)\nthis:\nx * (x * x) ^ (n div 2) = x ^ n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 960, "file": "IMP2_doc_Examples", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681122619885, "lm_q2_score": 0.8221891327004132, "lm_q1q2_score": 0.7052476202787549}} {"text": "[STATEMENT]\nlemma length_transpose:\n fixes xs :: \"'a list list\"\n shows \"length (transpose xs) = foldr (\\xs. max (length xs)) xs 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (transpose xs) = foldr (\\xs. max (length xs)) xs 0\n[PROOF STEP]\nby (induct rule: transpose.induct)\n (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max\n max_Suc_Suc[symmetric] simp del: max_Suc_Suc)", "meta": {"llama_tokens": 166, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577681049901037, "lm_q2_score": 0.8221891348788759, "lm_q1q2_score": 0.7052476161685062}} {"text": "[STATEMENT]\nlemma pr_index_enumerator_increase1: \"pr_index_enumerator n m < pr_index_enumerator (n+1) m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pr_index_enumerator n m < pr_index_enumerator (n + 1) m\n[PROOF STEP]\nproof (induct m)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. pr_index_enumerator n 0 < pr_index_enumerator (n + 1) 0\n 2. \\m. pr_index_enumerator n m < pr_index_enumerator (n + 1) m \\ pr_index_enumerator n (Suc m) < pr_index_enumerator (n + 1) (Suc m)\n[PROOF STEP]\nshow \"pr_index_enumerator n 0 < pr_index_enumerator (n + 1) 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pr_index_enumerator n 0 < pr_index_enumerator (n + 1) 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\npr_index_enumerator n 0 < pr_index_enumerator (n + 1) 0\n\ngoal (1 subgoal):\n 1. \\m. pr_index_enumerator n m < pr_index_enumerator (n + 1) m \\ pr_index_enumerator n (Suc m) < pr_index_enumerator (n + 1) (Suc m)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m. pr_index_enumerator n m < pr_index_enumerator (n + 1) m \\ pr_index_enumerator n (Suc m) < pr_index_enumerator (n + 1) (Suc m)\n[PROOF STEP]\nfix na\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m. pr_index_enumerator n m < pr_index_enumerator (n + 1) m \\ pr_index_enumerator n (Suc m) < pr_index_enumerator (n + 1) (Suc m)\n[PROOF STEP]\nassume A: \"pr_index_enumerator n na < pr_index_enumerator (n + 1) na\"\n[PROOF STATE]\nproof (state)\nthis:\npr_index_enumerator n na < pr_index_enumerator (n + 1) na\n\ngoal (1 subgoal):\n 1. \\m. pr_index_enumerator n m < pr_index_enumerator (n + 1) m \\ pr_index_enumerator n (Suc m) < pr_index_enumerator (n + 1) (Suc m)\n[PROOF STEP]\nshow \"pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n[PROOF STEP]\nlet ?a = \"pr_index_enumerator n na\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n[PROOF STEP]\nlet ?b = \"pr_index_enumerator (n+1) na\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n[PROOF STEP]\nhave S1: \"pr_index_enumerator n (Suc na) = comp_by_index index_of_id ?a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) = comp_by_index index_of_id (pr_index_enumerator n na)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\npr_index_enumerator n (Suc na) = comp_by_index index_of_id (pr_index_enumerator n na)\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n[PROOF STEP]\nhave L1: \"pr_index_enumerator (n+1) (Suc na) = comp_by_index index_of_id ?b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pr_index_enumerator (n + 1) (Suc na) = comp_by_index index_of_id (pr_index_enumerator (n + 1) na)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\npr_index_enumerator (n + 1) (Suc na) = comp_by_index index_of_id (pr_index_enumerator (n + 1) na)\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n[PROOF STEP]\nfrom A\n[PROOF STATE]\nproof (chain)\npicking this:\npr_index_enumerator n na < pr_index_enumerator (n + 1) na\n[PROOF STEP]\nhave \"c_pair index_of_id ?a < c_pair index_of_id ?b\"\n[PROOF STATE]\nproof (prove)\nusing this:\npr_index_enumerator n na < pr_index_enumerator (n + 1) na\n\ngoal (1 subgoal):\n 1. c_pair index_of_id (pr_index_enumerator n na) < c_pair index_of_id (pr_index_enumerator (n + 1) na)\n[PROOF STEP]\nby (rule c_pair_strict_mono2)\n[PROOF STATE]\nproof (state)\nthis:\nc_pair index_of_id (pr_index_enumerator n na) < c_pair index_of_id (pr_index_enumerator (n + 1) na)\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc_pair index_of_id (pr_index_enumerator n na) < c_pair index_of_id (pr_index_enumerator (n + 1) na)\n[PROOF STEP]\nhave \"c_pair 4 (c_pair index_of_id ?a) < c_pair 4 (c_pair index_of_id ?b)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc_pair index_of_id (pr_index_enumerator n na) < c_pair index_of_id (pr_index_enumerator (n + 1) na)\n\ngoal (1 subgoal):\n 1. c_pair 4 (c_pair index_of_id (pr_index_enumerator n na)) < c_pair 4 (c_pair index_of_id (pr_index_enumerator (n + 1) na))\n[PROOF STEP]\nby (rule c_pair_strict_mono2)\n[PROOF STATE]\nproof (state)\nthis:\nc_pair 4 (c_pair index_of_id (pr_index_enumerator n na)) < c_pair 4 (c_pair index_of_id (pr_index_enumerator (n + 1) na))\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc_pair 4 (c_pair index_of_id (pr_index_enumerator n na)) < c_pair 4 (c_pair index_of_id (pr_index_enumerator (n + 1) na))\n[PROOF STEP]\nhave \"comp_by_index index_of_id ?a < c_pair 4 (c_pair index_of_id ?b)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc_pair 4 (c_pair index_of_id (pr_index_enumerator n na)) < c_pair 4 (c_pair index_of_id (pr_index_enumerator (n + 1) na))\n\ngoal (1 subgoal):\n 1. comp_by_index index_of_id (pr_index_enumerator n na) < c_pair 4 (c_pair index_of_id (pr_index_enumerator (n + 1) na))\n[PROOF STEP]\nby (simp add: comp_by_index_def)\n[PROOF STATE]\nproof (state)\nthis:\ncomp_by_index index_of_id (pr_index_enumerator n na) < c_pair 4 (c_pair index_of_id (pr_index_enumerator (n + 1) na))\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncomp_by_index index_of_id (pr_index_enumerator n na) < c_pair 4 (c_pair index_of_id (pr_index_enumerator (n + 1) na))\n[PROOF STEP]\nhave \"comp_by_index index_of_id ?a < comp_by_index index_of_id ?b\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncomp_by_index index_of_id (pr_index_enumerator n na) < c_pair 4 (c_pair index_of_id (pr_index_enumerator (n + 1) na))\n\ngoal (1 subgoal):\n 1. comp_by_index index_of_id (pr_index_enumerator n na) < comp_by_index index_of_id (pr_index_enumerator (n + 1) na)\n[PROOF STEP]\nby (simp add: comp_by_index_def)\n[PROOF STATE]\nproof (state)\nthis:\ncomp_by_index index_of_id (pr_index_enumerator n na) < comp_by_index index_of_id (pr_index_enumerator (n + 1) na)\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n[PROOF STEP]\nwith S1 L1\n[PROOF STATE]\nproof (chain)\npicking this:\npr_index_enumerator n (Suc na) = comp_by_index index_of_id (pr_index_enumerator n na)\npr_index_enumerator (n + 1) (Suc na) = comp_by_index index_of_id (pr_index_enumerator (n + 1) na)\ncomp_by_index index_of_id (pr_index_enumerator n na) < comp_by_index index_of_id (pr_index_enumerator (n + 1) na)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\npr_index_enumerator n (Suc na) = comp_by_index index_of_id (pr_index_enumerator n na)\npr_index_enumerator (n + 1) (Suc na) = comp_by_index index_of_id (pr_index_enumerator (n + 1) na)\ncomp_by_index index_of_id (pr_index_enumerator n na) < comp_by_index index_of_id (pr_index_enumerator (n + 1) na)\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\npr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\npr_index_enumerator n (Suc na) < pr_index_enumerator (n + 1) (Suc na)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3542, "file": "Recursion-Theory-I_PRecFun2", "length": 31, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681158979306, "lm_q2_score": 0.8221891239865619, "lm_q1q2_score": 0.7052476157937232}} {"text": "[STATEMENT]\nlemma triangle_num_less_iff: \"triangle_num m < triangle_num n \\ m < n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (triangle_num m < triangle_num n) = (m < n)\n[PROOF STEP]\nusing strict_mono_less[OF strict_mono_triangle_num]\n[PROOF STATE]\nproof (prove)\nusing this:\n(triangle_num ?x < triangle_num ?y) = (?x < ?y)\n\ngoal (1 subgoal):\n 1. (triangle_num m < triangle_num n) = (m < n)\n[PROOF STEP]\n.", "meta": {"llama_tokens": 178, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577680977182186, "lm_q2_score": 0.822189121808099, "lm_q1q2_score": 0.7052475989779458}} {"text": "[STATEMENT]\nlemma set_of_vector_cart_basis': \n shows \"(set_of_vector cart_basis') = {axis i 1 :: 'a::{field}^'n | i. i \\ (UNIV :: 'n set)}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_of_vector cart_basis' = {axis i (1::'a) |i. i \\ UNIV}\n[PROOF STEP]\nunfolding set_of_vector_def cart_basis'_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {(\\i. axis i (1::'a)) $ i |i. i \\ UNIV} = {axis i (1::'a) |i. i \\ UNIV}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 224, "file": "Gauss_Jordan_Linear_Maps", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577680904463333, "lm_q2_score": 0.822189121808099, "lm_q1q2_score": 0.7052475929990808}} {"text": "[STATEMENT]\nlemma Suc_0_mod_card [simp]: \"Suc 0 mod CARD('a::nontriv) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc 0 mod CARD('a) = 1\n[PROOF STEP]\nusing one_mod_card\n[PROOF STATE]\nproof (prove)\nusing this:\n1 mod CARD(?'a) = 1\n\ngoal (1 subgoal):\n 1. Suc 0 mod CARD('a) = 1\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 153, "file": "Berlekamp_Zassenhaus_Finite_Field", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970842359877, "lm_q2_score": 0.8006919949619793, "lm_q1q2_score": 0.7052471745336075}} {"text": "[STATEMENT]\nlemma prime_elem_multiplicity_prod_distrib:\n assumes \"prime_elem p\" \"0 \\ f ` A\" \"finite A\"\n shows \"multiplicity p (prod f A) = (\\x\\A. multiplicity p (f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. multiplicity p (prod f A) = (\\x\\A. multiplicity p (f x))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. multiplicity p (prod f A) = (\\x\\A. multiplicity p (f x))\n[PROOF STEP]\nhave \"multiplicity p (prod f A) = (\\x\\#mset_set A. multiplicity p (f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. multiplicity p (prod f A) = (\\x\\#mset_set A. multiplicity p (f x))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nprime_elem p\n(0::'a) \\ f ` A\nfinite A\n\ngoal (1 subgoal):\n 1. multiplicity p (prod f A) = (\\x\\#mset_set A. multiplicity p (f x))\n[PROOF STEP]\nby (subst prod_unfold_prod_mset)\n (simp_all add: prime_elem_multiplicity_prod_mset_distrib sum_unfold_sum_mset\n multiset.map_comp o_def)\n[PROOF STATE]\nproof (state)\nthis:\nmultiplicity p (prod f A) = (\\x\\#mset_set A. multiplicity p (f x))\n\ngoal (1 subgoal):\n 1. multiplicity p (prod f A) = (\\x\\A. multiplicity p (f x))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmultiplicity p (prod f A) = (\\x\\#mset_set A. multiplicity p (f x))\n\ngoal (1 subgoal):\n 1. multiplicity p (prod f A) = (\\x\\A. multiplicity p (f x))\n[PROOF STEP]\nfrom \\finite A\\\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\n[PROOF STEP]\nhave \"\\ = (\\x\\A. multiplicity p (f x))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. (\\x\\#mset_set A. multiplicity p (f x)) = (\\x\\A. multiplicity p (f x))\n[PROOF STEP]\nby (induction A rule: finite_induct) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\#mset_set A. multiplicity p (f x)) = (\\x\\A. multiplicity p (f x))\n\ngoal (1 subgoal):\n 1. multiplicity p (prod f A) = (\\x\\A. multiplicity p (f x))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nmultiplicity p (prod f A) = (\\x\\A. multiplicity p (f x))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nmultiplicity p (prod f A) = (\\x\\A. multiplicity p (f x))\n\ngoal (1 subgoal):\n 1. multiplicity p (prod f A) = (\\x\\A. multiplicity p (f x))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nmultiplicity p (prod f A) = (\\x\\A. multiplicity p (f x))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1084, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970811069351, "lm_q2_score": 0.8006919949619792, "lm_q1q2_score": 0.7052471720282001}} {"text": "[STATEMENT]\nlemma degree_m_mult_le: \n assumes eq: \"f =m g * h\" \n shows \"degree_m f \\ degree_m g + degree_m h\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. degree_m f \\ degree_m g + degree_m h\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. degree_m f \\ degree_m g + degree_m h\n[PROOF STEP]\nhave \"degree_m f = degree_m (Mp g * Mp h)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. degree_m f = degree_m (Mp g * Mp h)\n[PROOF STEP]\nusing eq\n[PROOF STATE]\nproof (prove)\nusing this:\nf =m g * h\n\ngoal (1 subgoal):\n 1. degree_m f = degree_m (Mp g * Mp h)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndegree_m f = degree_m (Mp g * Mp h)\n\ngoal (1 subgoal):\n 1. degree_m f \\ degree_m g + degree_m h\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndegree_m f = degree_m (Mp g * Mp h)\n\ngoal (1 subgoal):\n 1. degree_m f \\ degree_m g + degree_m h\n[PROOF STEP]\nhave \"\\ \\ degree (Mp g * Mp h)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. degree_m (Mp g * Mp h) \\ degree (Mp g * Mp h)\n[PROOF STEP]\nby (rule degree_m_le)\n[PROOF STATE]\nproof (state)\nthis:\ndegree_m (Mp g * Mp h) \\ degree (Mp g * Mp h)\n\ngoal (1 subgoal):\n 1. degree_m f \\ degree_m g + degree_m h\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndegree_m (Mp g * Mp h) \\ degree (Mp g * Mp h)\n\ngoal (1 subgoal):\n 1. degree_m f \\ degree_m g + degree_m h\n[PROOF STEP]\nhave \"\\ \\ degree_m g + degree_m h\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. degree (Mp g * Mp h) \\ degree_m g + degree_m h\n[PROOF STEP]\nby (rule degree_mult_le)\n[PROOF STATE]\nproof (state)\nthis:\ndegree (Mp g * Mp h) \\ degree_m g + degree_m h\n\ngoal (1 subgoal):\n 1. degree_m f \\ degree_m g + degree_m h\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndegree_m f \\ degree_m g + degree_m h\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndegree_m f \\ degree_m g + degree_m h\n\ngoal (1 subgoal):\n 1. degree_m f \\ degree_m g + degree_m h\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndegree_m f \\ degree_m g + degree_m h\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1006, "file": "Berlekamp_Zassenhaus_Poly_Mod", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339516289534, "lm_q2_score": 0.839733963661418, "lm_q1q2_score": 0.7051531196224464}} {"text": "[STATEMENT]\ntheorem (in group) \"x * 1 = x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nhave \"x * 1 = x * (inverse x * x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * (1::'a) = x * (inverse x * x)\n[PROOF STEP]\nby (simp only: group_left_inverse)\n[PROOF STATE]\nproof (state)\nthis:\nx * (1::'a) = x * (inverse x * x)\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nnote calculation = this\n \\ \\first calculational step: init calculation register\\\n[PROOF STATE]\nproof (state)\nthis:\nx * (1::'a) = x * (inverse x * x)\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nhave \"\\ = x * inverse x * x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * (inverse x * x) = x * inverse x * x\n[PROOF STEP]\nby (simp only: group_assoc)\n[PROOF STATE]\nproof (state)\nthis:\nx * (inverse x * x) = x * inverse x * x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nnote calculation = trans [OF calculation this]\n \\ \\general calculational step: compose with transitivity rule\\\n[PROOF STATE]\nproof (state)\nthis:\nx * (1::'a) = x * inverse x * x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nhave \"\\ = 1 * x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * inverse x * x = (1::'a) * x\n[PROOF STEP]\nby (simp only: group_right_inverse)\n[PROOF STATE]\nproof (state)\nthis:\nx * inverse x * x = (1::'a) * x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nnote calculation = trans [OF calculation this]\n \\ \\general calculational step: compose with transitivity rule\\\n[PROOF STATE]\nproof (state)\nthis:\nx * (1::'a) = (1::'a) * x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nhave \"\\ = x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1::'a) * x = x\n[PROOF STEP]\nby (simp only: group_left_one)\n[PROOF STATE]\nproof (state)\nthis:\n(1::'a) * x = x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nnote calculation = trans [OF calculation this]\n \\ \\final calculational step: compose with transitivity rule \\dots\\\n[PROOF STATE]\nproof (state)\nthis:\nx * (1::'a) = x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nfrom calculation\n \\ \\\\dots\\ and pick up the final result\\\n[PROOF STATE]\nproof (chain)\npicking this:\nx * (1::'a) = x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx * (1::'a) = x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nx * (1::'a) = x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1163, "file": null, "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199795472731, "lm_q2_score": 0.8418256492357359, "lm_q1q2_score": 0.7051299830952069}} {"text": "[STATEMENT]\nlemma deg_pm_minus:\n assumes \"s adds (t::_ \\\\<^sub>0 _::comm_monoid_add)\"\n shows \"deg_pm (t - s) = deg_pm t - deg_pm s\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. deg_pm (t - s) = deg_pm t - deg_pm s\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. deg_pm (t - s) = deg_pm t - deg_pm s\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\ns adds t\n[PROOF STEP]\nhave \"(t - s) + s = t\"\n[PROOF STATE]\nproof (prove)\nusing this:\ns adds t\n\ngoal (1 subgoal):\n 1. t - s + s = t\n[PROOF STEP]\nby (rule adds_minus)\n[PROOF STATE]\nproof (state)\nthis:\nt - s + s = t\n\ngoal (1 subgoal):\n 1. deg_pm (t - s) = deg_pm t - deg_pm s\n[PROOF STEP]\nhence \"deg_pm t = deg_pm ((t - s) + s)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nt - s + s = t\n\ngoal (1 subgoal):\n 1. deg_pm t = deg_pm (t - s + s)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndeg_pm t = deg_pm (t - s + s)\n\ngoal (1 subgoal):\n 1. deg_pm (t - s) = deg_pm t - deg_pm s\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndeg_pm t = deg_pm (t - s + s)\n\ngoal (1 subgoal):\n 1. deg_pm (t - s) = deg_pm t - deg_pm s\n[PROOF STEP]\nhave \"\\ = deg_pm (t - s) + deg_pm s\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. deg_pm (t - s + s) = deg_pm (t - s) + deg_pm s\n[PROOF STEP]\nby (simp only: deg_pm_plus)\n[PROOF STATE]\nproof (state)\nthis:\ndeg_pm (t - s + s) = deg_pm (t - s) + deg_pm s\n\ngoal (1 subgoal):\n 1. deg_pm (t - s) = deg_pm t - deg_pm s\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndeg_pm t = deg_pm (t - s) + deg_pm s\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndeg_pm t = deg_pm (t - s) + deg_pm s\n\ngoal (1 subgoal):\n 1. deg_pm (t - s) = deg_pm t - deg_pm s\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndeg_pm (t - s) = deg_pm t - deg_pm s\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 897, "file": "Polynomials_MPoly_PM", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256472515683, "lm_q2_score": 0.8376199552262967, "lm_q1q2_score": 0.7051299609592069}} {"text": "[STATEMENT]\nlemma check_all_const_deg_prop: \n shows \"check_all_const_deg l = True \\ (\\p \\ set(l). degree p = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (check_all_const_deg l = True) = (\\p\\set l. degree p = 0)\n[PROOF STEP]\nproof (induct l)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (check_all_const_deg [] = True) = (\\p\\set []. degree p = 0)\n 2. \\a l. (check_all_const_deg l = True) = (\\p\\set l. degree p = 0) \\ (check_all_const_deg (a # l) = True) = (\\p\\set (a # l). degree p = 0)\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. (check_all_const_deg [] = True) = (\\p\\set []. degree p = 0)\n 2. \\a l. (check_all_const_deg l = True) = (\\p\\set l. degree p = 0) \\ (check_all_const_deg (a # l) = True) = (\\p\\set (a # l). degree p = 0)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (check_all_const_deg [] = True) = (\\p\\set []. degree p = 0)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(check_all_const_deg [] = True) = (\\p\\set []. degree p = 0)\n\ngoal (1 subgoal):\n 1. \\a l. (check_all_const_deg l = True) = (\\p\\set l. degree p = 0) \\ (check_all_const_deg (a # l) = True) = (\\p\\set (a # l). degree p = 0)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a l. (check_all_const_deg l = True) = (\\p\\set l. degree p = 0) \\ (check_all_const_deg (a # l) = True) = (\\p\\set (a # l). degree p = 0)\n[PROOF STEP]\ncase (Cons a l)\n[PROOF STATE]\nproof (state)\nthis:\n(check_all_const_deg l = True) = (\\p\\set l. degree p = 0)\n\ngoal (1 subgoal):\n 1. \\a l. (check_all_const_deg l = True) = (\\p\\set l. degree p = 0) \\ (check_all_const_deg (a # l) = True) = (\\p\\set (a # l). degree p = 0)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(check_all_const_deg l = True) = (\\p\\set l. degree p = 0)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(check_all_const_deg l = True) = (\\p\\set l. degree p = 0)\n\ngoal (1 subgoal):\n 1. (check_all_const_deg (a # l) = True) = (\\p\\set (a # l). degree p = 0)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(check_all_const_deg (a # l) = True) = (\\p\\set (a # l). degree p = 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1134, "file": "BenOr_Kozen_Reif_Renegar_Decision", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438951025545426, "lm_q2_score": 0.8354835371034368, "lm_q1q2_score": 0.7050604652265368}} {"text": "[STATEMENT]\nlemma pderiv_eq_0_iff: \"pderiv p = 0 \\ degree p = 0\"\n for p :: \"'a::{comm_semiring_1,semiring_no_zero_divisors,semiring_char_0} poly\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (pderiv p = 0) = (degree p = 0)\n[PROOF STEP]\nproof (cases \"degree p\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. degree p = 0 \\ (pderiv p = 0) = (degree p = 0)\n 2. \\nat. degree p = Suc nat \\ (pderiv p = 0) = (degree p = 0)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\ndegree p = 0\n\ngoal (2 subgoals):\n 1. degree p = 0 \\ (pderiv p = 0) = (degree p = 0)\n 2. \\nat. degree p = Suc nat \\ (pderiv p = 0) = (degree p = 0)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndegree p = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndegree p = 0\n\ngoal (1 subgoal):\n 1. (pderiv p = 0) = (degree p = 0)\n[PROOF STEP]\nby (metis degree_eq_zeroE pderiv.simps)\n[PROOF STATE]\nproof (state)\nthis:\n(pderiv p = 0) = (degree p = 0)\n\ngoal (1 subgoal):\n 1. \\nat. degree p = Suc nat \\ (pderiv p = 0) = (degree p = 0)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\nat. degree p = Suc nat \\ (pderiv p = 0) = (degree p = 0)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\ndegree p = Suc n\n\ngoal (1 subgoal):\n 1. \\nat. degree p = Suc nat \\ (pderiv p = 0) = (degree p = 0)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndegree p = Suc n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndegree p = Suc n\n\ngoal (1 subgoal):\n 1. (pderiv p = 0) = (degree p = 0)\n[PROOF STEP]\nusing coeff_0 coeff_pderiv degree_0 leading_coeff_0_iff mult_eq_0_iff nat.distinct(1) of_nat_eq_0_iff\n[PROOF STATE]\nproof (prove)\nusing this:\ndegree p = Suc n\ncoeff 0 ?n = (0::?'a)\ncoeff (pderiv ?p) ?n = of_nat (Suc ?n) * coeff ?p (Suc ?n)\ndegree 0 = 0\n(lead_coeff ?p = (0::?'a)) = (?p = 0)\n(?a * ?b = (0::?'a)) = (?a = (0::?'a) \\ ?b = (0::?'a))\n0 \\ Suc ?x2.0\n(of_nat ?m = (0::?'a)) = (?m = 0)\n\ngoal (1 subgoal):\n 1. (pderiv p = 0) = (degree p = 0)\n[PROOF STEP]\nby (metis coeff_0 coeff_pderiv degree_0 leading_coeff_0_iff mult_eq_0_iff nat.distinct(1) of_nat_eq_0_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(pderiv p = 0) = (degree p = 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1147, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835371034368, "lm_q2_score": 0.8438950986284991, "lm_q1q2_score": 0.705060461946392}} {"text": "[STATEMENT]\nlemma Polygamma_approx_1_complex: \n \"Re x > 0 \\ \n Polygamma_approx (Suc 0) m x = ln x - 1 / (2*x) + stirling_sum (Suc 0) m x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < Re x \\ Polygamma_approx (Suc 0) m x = Ln x - 1 / (2 * x) + stirling_sum (Suc 0) m x\n[PROOF STEP]\nunfolding Polygamma_approx_Suc Polygamma_approx_0\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < Re x \\ deriv (\\x. (x - 1 / 2) * Ln x - x + complex_of_real (ln (2 * pi)) / 2 + stirling_sum 0 m x) x = Ln x - 1 / (2 * x) + stirling_sum (Suc 0) m x\n[PROOF STEP]\nby (intro DERIV_imp_deriv) \n (auto intro!: derivative_eq_intros elim!: nonpos_Reals_cases simp: field_simps)", "meta": {"llama_tokens": 322, "file": "Stirling_Formula_Gamma_Asymptotics", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110511888303, "lm_q2_score": 0.7905303260722198, "lm_q1q2_score": 0.7050036810911152}} {"text": "[STATEMENT]\nlemma arith_modZero2:\n \"Suc (n + (t + n * t)) mod Suc n = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc (n + (t + n * t)) mod Suc n = 0\n[PROOF STEP]\nby (metis add_Suc_right add_Suc_shift mod_mult_self1_is_0 mult_Suc mult.commute)", "meta": {"llama_tokens": 123, "file": "FocusStreamsCaseStudies_arith_hints", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110569397306, "lm_q2_score": 0.7905303186696748, "lm_q1q2_score": 0.7050036790357047}} {"text": "[STATEMENT]\nlemma Z_ZmX_rho_psim[simp]:\n shows \"Complex_Matrix.trace (rho_psim * Z_ZmX) = -1/ (sqrt 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (rho_psim * Z_ZmX) = complex_of_real (- 1 / sqrt 2)\n[PROOF STEP]\napply (auto simp add: rho_psim_def ket_psim_def ket_10_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (rank_1_proj (1 / complex_of_real (sqrt 2) \\\\<^sub>v (ket_01 - (ket_1 \\ ket_0))) * Z_ZmX) = - (1 / complex_of_real (sqrt 2))\n[PROOF STEP]\napply (auto simp add: Z_ZmX_def Z_def ZmX_def X_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (rank_1_proj (1 / complex_of_real (sqrt 2) \\\\<^sub>v (ket_01 - (ket_1 \\ ket_0))) * (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) \\ 1 / complex_of_real (sqrt 2) \\\\<^sub>m (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) - Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1)))) = - (1 / complex_of_real (sqrt 2))\n[PROOF STEP]\napply (auto simp add: rank_1_proj_def outer_prod_def ket_01_def ket_1_def ket_0_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (Matrix.mat 4 (Suc 0) (\\(i, j). (1 / complex_of_real (sqrt 2) \\\\<^sub>v ((unit_vec 2 0 \\ unit_vec 2 (Suc 0)) - (unit_vec 2 (Suc 0) \\ unit_vec 2 0))) $ i) * Matrix.mat (Suc 0) 4 (\\(i, j). conjugate (1 / complex_of_real (sqrt 2) \\\\<^sub>v ((unit_vec 2 0 \\ unit_vec 2 (Suc 0)) - (unit_vec 2 (Suc 0) \\ unit_vec 2 0))) $ j) * (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) \\ 1 / complex_of_real (sqrt 2) \\\\<^sub>m (Matrix.mat 2 2 (\\(i, j). if i \\ j then 0 else if i = 0 then 1 else - 1) - Matrix.mat 2 2 (\\(i, j). if i = j then 0 else 1)))) = - (1 / complex_of_real (sqrt 2))\n[PROOF STEP]\napply (auto simp add: Complex_Matrix.trace_def sum_4_elems)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 / (complex_of_real (sqrt 2) * (complex_of_real (sqrt 2) * complex_of_real (sqrt 2))) = 1 / complex_of_real (sqrt 2)\n[PROOF STEP]\napply (simp add: csqrt_2_sq)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1039, "file": "Projective_Measurements_CHSH_Inequality", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.89181104831338, "lm_q2_score": 0.7905303236047049, "lm_q1q2_score": 0.7050036766174274}} {"text": "[STATEMENT]\nlemma list_of_set [simp] :\n assumes fin_s: \"finite s\"\n shows \"set (list_of_set s) = s\"\n \"distinct (list_of_set s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (list_of_set s) = s &&& distinct (list_of_set s)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. set (list_of_set s) = s\n 2. distinct (list_of_set s)\n[PROOF STEP]\nfrom fin_s list_of_set_set_is_empty[of s]\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite s\n(list_of_set_set s = {}) = infinite s\n[PROOF STEP]\nhave \"\\ (list_of_set_set s = {})\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite s\n(list_of_set_set s = {}) = infinite s\n\ngoal (1 subgoal):\n 1. list_of_set_set s \\ {}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlist_of_set_set s \\ {}\n\ngoal (2 subgoals):\n 1. set (list_of_set s) = s\n 2. distinct (list_of_set s)\n[PROOF STEP]\nhence \"list_of_set s \\ list_of_set_set s\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlist_of_set_set s \\ {}\n\ngoal (1 subgoal):\n 1. list_of_set s \\ list_of_set_set s\n[PROOF STEP]\nunfolding list_of_set_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlist_of_set_set s \\ {}\n\ngoal (1 subgoal):\n 1. set_choose (list_of_set_set s) \\ list_of_set_set s\n[PROOF STEP]\nby (rule set_choose_thm)\n[PROOF STATE]\nproof (state)\nthis:\nlist_of_set s \\ list_of_set_set s\n\ngoal (2 subgoals):\n 1. set (list_of_set s) = s\n 2. distinct (list_of_set s)\n[PROOF STEP]\nthus \"set (list_of_set s) = s\"\n \"distinct (list_of_set s)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlist_of_set s \\ list_of_set_set s\n\ngoal (1 subgoal):\n 1. set (list_of_set s) = s &&& distinct (list_of_set s)\n[PROOF STEP]\nunfolding list_of_set_set_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlist_of_set s \\ {l. set l = s \\ distinct l}\n\ngoal (1 subgoal):\n 1. set (list_of_set s) = s &&& distinct (list_of_set s)\n[PROOF STEP]\nby simp_all\n[PROOF STATE]\nproof (state)\nthis:\nset (list_of_set s) = s\ndistinct (list_of_set s)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 925, "file": "CakeML_generated_LemExtraDefs", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424373085146, "lm_q2_score": 0.8333245932423308, "lm_q1q2_score": 0.7049446374765438}} {"text": "[STATEMENT]\nlemma one_element_vec_dotP:\n assumes \"dim_vec x = n\"\n shows \"(one_element_vec n e) \\ x = (\\i\\{0 ..< dim_vec x}. e * x $ i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. one_element_vec n e \\ x = (\\i = 0..x. p(a) \\ (p(x) \\ p(b)) \\ p(c)) \\\n (\\x. (\\p(a) \\ p(x) \\ p(c)) \\ (\\p(a) \\ \\p(b) \\ p(c)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. p a \\ (p x \\ p b) \\ p c) = (\\x. (\\ p a \\ p x \\ p c) \\ (\\ p a \\ \\ p b \\ p c))\n[PROOF STEP]\nby auto2", "meta": {"llama_tokens": 209, "file": "Auto2_HOL_HOL_Pelletier", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.7931059560743422, "lm_q1q2_score": 0.7048798926323139}} {"text": "[STATEMENT]\nlemma (in group) FactGroup_order:\n assumes \"subgroup H G\" \"finite H\"\n shows \"order G = order (G Mod H) * card H\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. order G = order (G Mod H) * card H\n[PROOF STEP]\nusing lagrange assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsubgroup ?H G \\ card (rcosets ?H) * card ?H = order G\nsubgroup H G\nfinite H\n\ngoal (1 subgoal):\n 1. order G = order (G Mod H) * card H\n[PROOF STEP]\nunfolding FactGroup_def order_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsubgroup ?H G \\ card (rcosets ?H) * card ?H = card (carrier G)\nsubgroup H G\nfinite H\n\ngoal (1 subgoal):\n 1. card (carrier G) = card (carrier \\carrier = rcosets H, monoid.mult = (<#>), one = H\\) * card H\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 308, "file": "Probabilistic_Prime_Tests_Algebraic_Auxiliaries", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588023318196, "lm_q2_score": 0.7931059462938815, "lm_q1q2_score": 0.7048798909503946}} {"text": "[STATEMENT]\nlemma fold_random_permutation_foldl:\n assumes \"finite A\"\n shows \"fold_random_permutation f x A =\n map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nproof (induction f x A rule: fold_random_permutation.induct [case_names empty infinite remove])\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\f x. finite {} \\ fold_random_permutation f x {} = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set {}))\n 2. \\A f x. \\infinite A; finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n 3. \\A f x. \\finite A; A \\ {}; \\xa. \\xa \\ set_pmf (pmf_of_set A); finite (A - {xa})\\ \\ fold_random_permutation f (f xa x) (A - {xa}) = map_pmf (foldl (\\x y. f y x) (f xa x)) (pmf_of_set (permutations_of_set (A - {xa}))); finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\ncase (remove A f x)\n[PROOF STATE]\nproof (state)\nthis:\nfinite A\nA \\ {}\n\\?xa \\ set_pmf (pmf_of_set A); finite (A - {?xa})\\ \\ fold_random_permutation f (f ?xa x) (A - {?xa}) = map_pmf (foldl (\\x y. f y x) (f ?xa x)) (pmf_of_set (permutations_of_set (A - {?xa})))\nfinite A\n\ngoal (3 subgoals):\n 1. \\f x. finite {} \\ fold_random_permutation f x {} = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set {}))\n 2. \\A f x. \\infinite A; finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n 3. \\A f x. \\finite A; A \\ {}; \\xa. \\xa \\ set_pmf (pmf_of_set A); finite (A - {xa})\\ \\ fold_random_permutation f (f xa x) (A - {xa}) = map_pmf (foldl (\\x y. f y x) (f xa x)) (pmf_of_set (permutations_of_set (A - {xa}))); finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nfrom remove\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nA \\ {}\n\\?xa \\ set_pmf (pmf_of_set A); finite (A - {?xa})\\ \\ fold_random_permutation f (f ?xa x) (A - {?xa}) = map_pmf (foldl (\\x y. f y x) (f ?xa x)) (pmf_of_set (permutations_of_set (A - {?xa})))\nfinite A\n[PROOF STEP]\nhave \"fold_random_permutation f x A = \n pmf_of_set A \\ (\\a. fold_random_permutation f (f a x) (A - {a}))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nA \\ {}\n\\?xa \\ set_pmf (pmf_of_set A); finite (A - {?xa})\\ \\ fold_random_permutation f (f ?xa x) (A - {?xa}) = map_pmf (foldl (\\x y. f y x) (f ?xa x)) (pmf_of_set (permutations_of_set (A - {?xa})))\nfinite A\n\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = pmf_of_set A \\ (\\a. fold_random_permutation f (f a x) (A - {a}))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfold_random_permutation f x A = pmf_of_set A \\ (\\a. fold_random_permutation f (f a x) (A - {a}))\n\ngoal (3 subgoals):\n 1. \\f x. finite {} \\ fold_random_permutation f x {} = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set {}))\n 2. \\A f x. \\infinite A; finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n 3. \\A f x. \\finite A; A \\ {}; \\xa. \\xa \\ set_pmf (pmf_of_set A); finite (A - {xa})\\ \\ fold_random_permutation f (f xa x) (A - {xa}) = map_pmf (foldl (\\x y. f y x) (f xa x)) (pmf_of_set (permutations_of_set (A - {xa}))); finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfold_random_permutation f x A = pmf_of_set A \\ (\\a. fold_random_permutation f (f a x) (A - {a}))\n\ngoal (3 subgoals):\n 1. \\f x. finite {} \\ fold_random_permutation f x {} = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set {}))\n 2. \\A f x. \\infinite A; finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n 3. \\A f x. \\finite A; A \\ {}; \\xa. \\xa \\ set_pmf (pmf_of_set A); finite (A - {xa})\\ \\ fold_random_permutation f (f xa x) (A - {xa}) = map_pmf (foldl (\\x y. f y x) (f xa x)) (pmf_of_set (permutations_of_set (A - {xa}))); finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nfrom remove\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nA \\ {}\n\\?xa \\ set_pmf (pmf_of_set A); finite (A - {?xa})\\ \\ fold_random_permutation f (f ?xa x) (A - {?xa}) = map_pmf (foldl (\\x y. f y x) (f ?xa x)) (pmf_of_set (permutations_of_set (A - {?xa})))\nfinite A\n[PROOF STEP]\nhave \"\\ = pmf_of_set A \\ (\\a. map_pmf (foldl (\\x y. f y x) x)\n (map_pmf ((#) a) (pmf_of_set (permutations_of_set (A - {a})))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nA \\ {}\n\\?xa \\ set_pmf (pmf_of_set A); finite (A - {?xa})\\ \\ fold_random_permutation f (f ?xa x) (A - {?xa}) = map_pmf (foldl (\\x y. f y x) (f ?xa x)) (pmf_of_set (permutations_of_set (A - {?xa})))\nfinite A\n\ngoal (1 subgoal):\n 1. pmf_of_set A \\ (\\a. fold_random_permutation f (f a x) (A - {a})) = pmf_of_set A \\ (\\a. map_pmf (foldl (\\x y. f y x) x) (map_pmf ((#) a) (pmf_of_set (permutations_of_set (A - {a})))))\n[PROOF STEP]\nby (intro bind_pmf_cong) (simp_all add: pmf.map_comp o_def)\n[PROOF STATE]\nproof (state)\nthis:\npmf_of_set A \\ (\\a. fold_random_permutation f (f a x) (A - {a})) = pmf_of_set A \\ (\\a. map_pmf (foldl (\\x y. f y x) x) (map_pmf ((#) a) (pmf_of_set (permutations_of_set (A - {a})))))\n\ngoal (3 subgoals):\n 1. \\f x. finite {} \\ fold_random_permutation f x {} = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set {}))\n 2. \\A f x. \\infinite A; finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n 3. \\A f x. \\finite A; A \\ {}; \\xa. \\xa \\ set_pmf (pmf_of_set A); finite (A - {xa})\\ \\ fold_random_permutation f (f xa x) (A - {xa}) = map_pmf (foldl (\\x y. f y x) (f xa x)) (pmf_of_set (permutations_of_set (A - {xa}))); finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\npmf_of_set A \\ (\\a. fold_random_permutation f (f a x) (A - {a})) = pmf_of_set A \\ (\\a. map_pmf (foldl (\\x y. f y x) x) (map_pmf ((#) a) (pmf_of_set (permutations_of_set (A - {a})))))\n\ngoal (3 subgoals):\n 1. \\f x. finite {} \\ fold_random_permutation f x {} = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set {}))\n 2. \\A f x. \\infinite A; finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n 3. \\A f x. \\finite A; A \\ {}; \\xa. \\xa \\ set_pmf (pmf_of_set A); finite (A - {xa})\\ \\ fold_random_permutation f (f xa x) (A - {xa}) = map_pmf (foldl (\\x y. f y x) (f xa x)) (pmf_of_set (permutations_of_set (A - {xa}))); finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nfrom remove\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nA \\ {}\n\\?xa \\ set_pmf (pmf_of_set A); finite (A - {?xa})\\ \\ fold_random_permutation f (f ?xa x) (A - {?xa}) = map_pmf (foldl (\\x y. f y x) (f ?xa x)) (pmf_of_set (permutations_of_set (A - {?xa})))\nfinite A\n[PROOF STEP]\nhave \"\\ = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nA \\ {}\n\\?xa \\ set_pmf (pmf_of_set A); finite (A - {?xa})\\ \\ fold_random_permutation f (f ?xa x) (A - {?xa}) = map_pmf (foldl (\\x y. f y x) (f ?xa x)) (pmf_of_set (permutations_of_set (A - {?xa})))\nfinite A\n\ngoal (1 subgoal):\n 1. pmf_of_set A \\ (\\a. map_pmf (foldl (\\x y. f y x) x) (map_pmf ((#) a) (pmf_of_set (permutations_of_set (A - {a}))))) = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nby (simp_all add: random_permutation_of_set map_bind_pmf map_pmf_def [symmetric])\n[PROOF STATE]\nproof (state)\nthis:\npmf_of_set A \\ (\\a. map_pmf (foldl (\\x y. f y x) x) (map_pmf ((#) a) (pmf_of_set (permutations_of_set (A - {a}))))) = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n\ngoal (3 subgoals):\n 1. \\f x. finite {} \\ fold_random_permutation f x {} = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set {}))\n 2. \\A f x. \\infinite A; finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n 3. \\A f x. \\finite A; A \\ {}; \\xa. \\xa \\ set_pmf (pmf_of_set A); finite (A - {xa})\\ \\ fold_random_permutation f (f xa x) (A - {xa}) = map_pmf (foldl (\\x y. f y x) (f xa x)) (pmf_of_set (permutations_of_set (A - {xa}))); finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nfold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nfold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n\ngoal (1 subgoal):\n 1. fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nfold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n\ngoal (2 subgoals):\n 1. \\f x. finite {} \\ fold_random_permutation f x {} = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set {}))\n 2. \\A f x. \\infinite A; finite A\\ \\ fold_random_permutation f x A = map_pmf (foldl (\\x y. f y x) x) (pmf_of_set (permutations_of_set A))\n[PROOF STEP]\nqed (simp_all add: pmf_of_set_singleton)", "meta": {"llama_tokens": 5238, "file": null, "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587846530938, "lm_q2_score": 0.7931059609645724, "lm_q1q2_score": 0.7048798899679974}} {"text": "[STATEMENT]\nlemma set_of_subset_iff: \"set_of X \\ set_of Y \\ lower Y \\ lower X \\ upper X \\ upper Y\"\n for X Y::\"'a::linorder interval\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (set_of X \\ set_of Y) = (lower Y \\ lower X \\ upper X \\ upper Y)\n[PROOF STEP]\nby (auto simp: set_of_eq subset_iff)", "meta": {"llama_tokens": 143, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587993853655, "lm_q2_score": 0.7931059438487663, "lm_q1q2_score": 0.7048798864404267}} {"text": "[STATEMENT]\nlemma openin_prod_topology_alt:\n \"openin (prod_topology X Y) S \\\n (\\x y. (x,y) \\ S \\ (\\U V. openin X U \\ openin Y V \\ x \\ U \\ y \\ V \\ U \\ V \\ S))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. openin (prod_topology X Y) S = (\\x y. (x, y) \\ S \\ (\\U V. openin X U \\ openin Y V \\ x \\ U \\ y \\ V \\ U \\ V \\ S))\n[PROOF STEP]\napply (auto simp: openin_prod_topology arbitrary_union_of_alt, fastforce)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a b. \\\\x y. (x, y) \\ S \\ (\\U. openin X U \\ (\\V. openin Y V \\ x \\ U \\ y \\ V \\ U \\ V \\ S)); (a, b) \\ S\\ \\ \\U. (\\S T. U = S \\ T \\ openin X S \\ openin Y T) \\ (a, b) \\ U \\ U \\ S\n[PROOF STEP]\nby (metis mem_Sigma_iff)", "meta": {"llama_tokens": 425, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587964389113, "lm_q2_score": 0.7931059462938815, "lm_q1q2_score": 0.704879886276694}} {"text": "[STATEMENT]\nlemma cos_x_y_le_one: \"\\x / sqrt (x\\<^sup>2 + y\\<^sup>2)\\ \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x / sqrt (x\\<^sup>2 + y\\<^sup>2)\\ \\ 1\n[PROOF STEP]\nby (rule power2_le_imp_le [OF _ zero_le_one])\n (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)", "meta": {"llama_tokens": 152, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587817066391, "lm_q2_score": 0.7931059585194573, "lm_q1q2_score": 0.7048798854580292}} {"text": "[STATEMENT]\nlemma LIMSEQ_prod_nonneg: \n fixes f :: \"nat \\ 'a::{linordered_semidom,linorder_topology}\"\n assumes 0: \"\\n. 0 \\ f n\" and a: \"(\\n. prod f {..n}) \\ a\"\n shows \"a \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) \\ a\n[PROOF STEP]\nby (simp add: \"0\" prod_nonneg LIMSEQ_le_const [OF a])", "meta": {"llama_tokens": 164, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587875995482, "lm_q2_score": 0.7931059511841119, "lm_q1q2_score": 0.7048798836123777}} {"text": "[STATEMENT]\nlemma closure_eq_rel_interior_eq:\n fixes S1 S2 :: \"'n::euclidean_space set\"\n assumes \"convex S1\"\n and \"convex S2\"\n shows \"closure S1 = closure S2 \\ rel_interior S1 = rel_interior S2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (closure S1 = closure S2) = (rel_interior S1 = rel_interior S2)\n[PROOF STEP]\nby (metis convex_rel_interior_closure convex_closure_rel_interior assms)", "meta": {"llama_tokens": 163, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587875995482, "lm_q2_score": 0.7931059438487663, "lm_q1q2_score": 0.7048798770930249}} {"text": "[STATEMENT]\nlemma frontier_inside_subset:\n fixes S :: \"'a::real_normed_vector set\"\n assumes \"closed S\"\n shows \"frontier(inside S) \\ S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. frontier (inside S) \\ S\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. frontier (inside S) \\ S\n[PROOF STEP]\nhave \"closure (inside S) \\ - inside S = closure (inside S) - interior (inside S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. closure (inside S) \\ - inside S = closure (inside S) - interior (inside S)\n[PROOF STEP]\nby (metis (no_types) Diff_Compl assms closure_closed interior_closure open_closed open_inside)\n[PROOF STATE]\nproof (state)\nthis:\nclosure (inside S) \\ - inside S = closure (inside S) - interior (inside S)\n\ngoal (1 subgoal):\n 1. frontier (inside S) \\ S\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nclosure (inside S) \\ - inside S = closure (inside S) - interior (inside S)\n\ngoal (1 subgoal):\n 1. frontier (inside S) \\ S\n[PROOF STEP]\nhave \"- inside S \\ - outside S = S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - inside S \\ - outside S = S\n[PROOF STEP]\nby (metis (no_types) compl_sup double_compl inside_Un_outside)\n[PROOF STATE]\nproof (state)\nthis:\n- inside S \\ - outside S = S\n\ngoal (1 subgoal):\n 1. frontier (inside S) \\ S\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n- inside S \\ - outside S = S\n\ngoal (1 subgoal):\n 1. frontier (inside S) \\ S\n[PROOF STEP]\nhave \"closure (inside S) \\ - outside S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. closure (inside S) \\ - outside S\n[PROOF STEP]\nby (metis (no_types) assms closure_inside_subset union_with_inside)\n[PROOF STATE]\nproof (state)\nthis:\nclosure (inside S) \\ - outside S\n\ngoal (1 subgoal):\n 1. frontier (inside S) \\ S\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nclosure (inside S) \\ - inside S = closure (inside S) - interior (inside S)\n- inside S \\ - outside S = S\nclosure (inside S) \\ - outside S\n[PROOF STEP]\nhave \"closure (inside S) - interior (inside S) \\ S\"\n[PROOF STATE]\nproof (prove)\nusing this:\nclosure (inside S) \\ - inside S = closure (inside S) - interior (inside S)\n- inside S \\ - outside S = S\nclosure (inside S) \\ - outside S\n\ngoal (1 subgoal):\n 1. closure (inside S) - interior (inside S) \\ S\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nclosure (inside S) - interior (inside S) \\ S\n\ngoal (1 subgoal):\n 1. frontier (inside S) \\ S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nclosure (inside S) - interior (inside S) \\ S\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nclosure (inside S) - interior (inside S) \\ S\n\ngoal (1 subgoal):\n 1. frontier (inside S) \\ S\n[PROOF STEP]\nby (simp add: frontier_def open_inside interior_open)\n[PROOF STATE]\nproof (state)\nthis:\nfrontier (inside S) \\ S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1159, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637612961505, "lm_q2_score": 0.8198933359135361, "lm_q1q2_score": 0.7048325890130787}} {"text": "[STATEMENT]\nlemma le_floor_add: \"\\x\\ + \\y\\ \\ \\x + y\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\ + \\y\\ \\ \\x + y\\\n[PROOF STEP]\nby (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)", "meta": {"llama_tokens": 138, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297914570319, "lm_q2_score": 0.782662489091802, "lm_q1q2_score": 0.704810888083082}} {"text": "[STATEMENT]\nlemma setPaireq: \" {x, y} = {a, b} \\ x = a \\ y = b \\ x = b \\ y = a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x, y} = {a, b} \\ x = a \\ y = b \\ x = b \\ y = a\n[PROOF STEP]\nby (metis doubleton_eq_iff)", "meta": {"llama_tokens": 128, "file": "UPF_Firewall_FWNormalisation_NormalisationGenericProofs", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297914570319, "lm_q2_score": 0.7826624789529376, "lm_q1q2_score": 0.7048108789527325}} {"text": "[STATEMENT]\nlemma le_floor_add: \"\\x\\ + \\y\\ \\ \\x + y\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\ + \\y\\ \\ \\x + y\\\n[PROOF STEP]\nby (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)", "meta": {"llama_tokens": 138, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297914570319, "lm_q2_score": 0.7826624738835051, "lm_q1q2_score": 0.7048108743875575}} {"text": "[STATEMENT]\ntheorem has_absolute_integral_change_of_variables_compact:\n fixes f :: \"real^'m::{finite,wellorder} \\ real^'n\" and g :: \"real^'m::_ \\ real^'m::_\"\n assumes \"compact S\"\n and der_g: \"\\x. x \\ S \\ (g has_derivative g' x) (at x within S)\"\n and inj: \"inj_on g S\"\n shows \"((\\x. \\det (matrix (g' x))\\ *\\<^sub>R f(g x)) absolutely_integrable_on S \\\n integral S (\\x. \\det (matrix (g' x))\\ *\\<^sub>R f(g x)) = b\n \\ f absolutely_integrable_on (g ` S) \\ integral (g ` S) f = b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) absolutely_integrable_on S \\ integral S (\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \\ integral (g ` S) f = b)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) absolutely_integrable_on S \\ integral S (\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \\ integral (g ` S) f = b)\n[PROOF STEP]\nobtain h where hg: \"\\x. x \\ S \\ h(g x) = x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\h. (\\x. x \\ S \\ h (g x) = x) \\ thesis) \\ thesis\n[PROOF STEP]\nusing inj\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on g S\n\ngoal (1 subgoal):\n 1. (\\h. (\\x. x \\ S \\ h (g x) = x) \\ thesis) \\ thesis\n[PROOF STEP]\nby (metis the_inv_into_f_f)\n[PROOF STATE]\nproof (state)\nthis:\n?x \\ S \\ h (g ?x) = ?x\n\ngoal (1 subgoal):\n 1. ((\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) absolutely_integrable_on S \\ integral S (\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \\ integral (g ` S) f = b)\n[PROOF STEP]\nhave conth: \"continuous_on (g ` S) h\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous_on (g ` S) h\n[PROOF STEP]\nby (metis \\compact S\\ continuous_on_inv der_g has_derivative_continuous_on hg)\n[PROOF STATE]\nproof (state)\nthis:\ncontinuous_on (g ` S) h\n\ngoal (1 subgoal):\n 1. ((\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) absolutely_integrable_on S \\ integral S (\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \\ integral (g ` S) f = b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) absolutely_integrable_on S \\ integral S (\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \\ integral (g ` S) f = b)\n[PROOF STEP]\nby (rule has_absolute_integral_change_of_variables_invertible [OF der_g hg conth])\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) absolutely_integrable_on S \\ integral S (\\x. \\det (matrix (g' x))\\ *\\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \\ integral (g ` S) f = b)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1408, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.909907001151883, "lm_q2_score": 0.7745833893685269, "lm_q1q2_score": 0.7047988489623777}} {"text": "[STATEMENT]\nlemma proj2_no_3_Col_span:\n assumes \"proj2_no_3_Col S\" and \"p \\ S\"\n shows \"span (proj2_rep ` (S - {p})) = UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. span (proj2_rep ` (S - {p})) = UNIV\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. span (proj2_rep ` (S - {p})) = UNIV\n[PROOF STEP]\nfrom \\proj2_no_3_Col S\\\n[PROOF STATE]\nproof (chain)\npicking this:\nproj2_no_3_Col S\n[PROOF STEP]\nhave \"card S = 4\"\n[PROOF STATE]\nproof (prove)\nusing this:\nproj2_no_3_Col S\n\ngoal (1 subgoal):\n 1. card S = 4\n[PROOF STEP]\nunfolding proj2_no_3_Col_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncard S = 4 \\ (\\p\\S. \\ proj2_set_Col (S - {p}))\n\ngoal (1 subgoal):\n 1. card S = 4\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\ncard S = 4\n\ngoal (1 subgoal):\n 1. span (proj2_rep ` (S - {p})) = UNIV\n[PROOF STEP]\nwith \\p \\ S\\ and \\card S = 4\\ and card_gt_0_diff_singleton [of S p]\n[PROOF STATE]\nproof (chain)\npicking this:\np \\ S\ncard S = 4\n\\0 < card S; p \\ S\\ \\ card (S - {p}) = card S - 1\ncard S = 4\n[PROOF STEP]\nhave \"card (S - {p}) = 3\"\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ S\ncard S = 4\n\\0 < card S; p \\ S\\ \\ card (S - {p}) = card S - 1\ncard S = 4\n\ngoal (1 subgoal):\n 1. card (S - {p}) = 3\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (S - {p}) = 3\n\ngoal (1 subgoal):\n 1. span (proj2_rep ` (S - {p})) = UNIV\n[PROOF STEP]\nfrom \\proj2_no_3_Col S\\ and \\p \\ S\\\n[PROOF STATE]\nproof (chain)\npicking this:\nproj2_no_3_Col S\np \\ S\n[PROOF STEP]\nhave \"\\ proj2_set_Col (S - {p})\"\n[PROOF STATE]\nproof (prove)\nusing this:\nproj2_no_3_Col S\np \\ S\n\ngoal (1 subgoal):\n 1. \\ proj2_set_Col (S - {p})\n[PROOF STEP]\nunfolding proj2_no_3_Col_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncard S = 4 \\ (\\p\\S. \\ proj2_set_Col (S - {p}))\np \\ S\n\ngoal (1 subgoal):\n 1. \\ proj2_set_Col (S - {p})\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\ proj2_set_Col (S - {p})\n\ngoal (1 subgoal):\n 1. span (proj2_rep ` (S - {p})) = UNIV\n[PROOF STEP]\nwith \\card (S - {p}) = 3\\ and not_proj2_set_Col_iff_span\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (S - {p}) = 3\ncard ?S = 3 \\ (\\ proj2_set_Col ?S) = (span (proj2_rep ` ?S) = UNIV)\n\\ proj2_set_Col (S - {p})\n[PROOF STEP]\nshow \"span (proj2_rep ` (S - {p})) = UNIV\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (S - {p}) = 3\ncard ?S = 3 \\ (\\ proj2_set_Col ?S) = (span (proj2_rep ` ?S) = UNIV)\n\\ proj2_set_Col (S - {p})\n\ngoal (1 subgoal):\n 1. span (proj2_rep ` (S - {p})) = UNIV\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nspan (proj2_rep ` (S - {p})) = UNIV\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1380, "file": "Tarskis_Geometry_Projective", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357666736773, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7047850420558542}} {"text": "[STATEMENT]\nlemma integer_partitions_cardinality_aux:\n \"card (integer_partitions n) = (\\k\\n. Partition n k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (integer_partitions n) = sum (Partition n) {..n}\n[PROOF STEP]\nusing card_partitions_number_partition integer_partitions_number_partition_eq card_partitions\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {p. p partitions ?n} = card {N. number_partition ?n N}\ninteger_partitions ?n = {N. number_partition ?n N}\ncard {p. p partitions ?n} = sum (Partition ?n) {..?n}\n\ngoal (1 subgoal):\n 1. card (integer_partitions n) = sum (Partition n) {..n}\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 240, "file": "Combinatorial_Enumeration_Algorithms_Integer_Partitions", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357666736772, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7047850420558541}} {"text": "[STATEMENT]\nlemma H_without_scalar_prod:\n \"H = mat 2 2 (\\(i,j). if i\\j then 1/sqrt(2) else (if i=0 then 1/sqrt(2) else -(1/sqrt(2))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. H = mat 2 2 (\\x. complex_of_real (case x of (i, j) \\ if i \\ j then 1 / sqrt 2 else if i = 0 then 1 / sqrt 2 else - (1 / sqrt 2)))\n[PROOF STEP]\nusing cong_mat\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?nr = ?nr'; ?nc = ?nc'; \\i j. \\i < ?nr; j < ?nc\\ \\ ?f (i, j) = ?f' (i, j)\\ \\ mat ?nr ?nc ?f = mat ?nr' ?nc' ?f'\n\ngoal (1 subgoal):\n 1. H = mat 2 2 (\\x. complex_of_real (case x of (i, j) \\ if i \\ j then 1 / sqrt 2 else if i = 0 then 1 / sqrt 2 else - (1 / sqrt 2)))\n[PROOF STEP]\nby (auto simp: H_def)", "meta": {"llama_tokens": 379, "file": "Isabelle_Marries_Dirac_Quantum", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357666736772, "lm_q2_score": 0.8128673155708975, "lm_q1q2_score": 0.704785036159987}} {"text": "[STATEMENT]\nlemma has_integral_spike_set_eq:\n fixes f :: \"'n::euclidean_space \\ 'a::banach\"\n assumes \"negligible {x \\ S - T. f x \\ 0}\" \"negligible {x \\ T - S. f x \\ 0}\"\n shows \"(f has_integral y) S \\ (f has_integral y) T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f has_integral y) S = (f has_integral y) T\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (f has_integral y) S = (f has_integral y) T\n[PROOF STEP]\nhave \"((\\x. if x \\ S then f x else 0) has_integral y) UNIV =\n ((\\x. if x \\ T then f x else 0) has_integral y) UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. if x \\ S then f x else (0::'a)) has_integral y) UNIV = ((\\x. if x \\ T then f x else (0::'a)) has_integral y) UNIV\n[PROOF STEP]\nproof (rule has_integral_spike_eq)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. negligible ?S\n 2. \\x. x \\ UNIV - ?S \\ (if x \\ T then f x else (0::'a)) = (if x \\ S then f x else (0::'a))\n[PROOF STEP]\nshow \"negligible ({x \\ S - T. f x \\ 0} \\ {x \\ T - S. f x \\ 0})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. negligible ({x \\ S - T. f x \\ (0::'a)} \\ {x \\ T - S. f x \\ (0::'a)})\n[PROOF STEP]\nby (rule negligible_Un [OF assms])\n[PROOF STATE]\nproof (state)\nthis:\nnegligible ({x \\ S - T. f x \\ (0::'a)} \\ {x \\ T - S. f x \\ (0::'a)})\n\ngoal (1 subgoal):\n 1. \\x. x \\ UNIV - ({x \\ S - T. f x \\ (0::'a)} \\ {x \\ T - S. f x \\ (0::'a)}) \\ (if x \\ T then f x else (0::'a)) = (if x \\ S then f x else (0::'a))\n[PROOF STEP]\nqed auto\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. if x \\ S then f x else (0::'a)) has_integral y) UNIV = ((\\x. if x \\ T then f x else (0::'a)) has_integral y) UNIV\n\ngoal (1 subgoal):\n 1. (f has_integral y) S = (f has_integral y) T\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n((\\x. if x \\ S then f x else (0::'a)) has_integral y) UNIV = ((\\x. if x \\ T then f x else (0::'a)) has_integral y) UNIV\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. if x \\ S then f x else (0::'a)) has_integral y) UNIV = ((\\x. if x \\ T then f x else (0::'a)) has_integral y) UNIV\n\ngoal (1 subgoal):\n 1. (f has_integral y) S = (f has_integral y) T\n[PROOF STEP]\nby (simp add: has_integral_restrict_UNIV)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_integral y) S = (f has_integral y) T\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1210, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357494949105, "lm_q2_score": 0.8128673246376009, "lm_q1q2_score": 0.704785030057085}} {"text": "[STATEMENT]\nlemma (in comm_monoid) finprod_swap:\n assumes \"finite A\" \"finite B\" \"\\ a b. a \\ A \\ b \\ B \\ g a b \\ carrier G\"\n shows \"finprod G (\\ (b,a). g a b) (B \\ A) = finprod G (\\ (a,b). g a b) (A \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(b, a)\\B \\ A. g a b) = (\\(a, b)\\A \\ B. g a b)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\(b, a)\\B \\ A. g a b) = (\\(a, b)\\A \\ B. g a b)\n[PROOF STEP]\nhave [simp]: \"(\\(a, b). (b, a)) ` (A \\ B) = B \\ A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(a, b). (b, a)) ` (A \\ B) = B \\ A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\(a, b). (b, a)) ` (A \\ B) = B \\ A\n\ngoal (1 subgoal):\n 1. (\\(b, a)\\B \\ A. g a b) = (\\(a, b)\\A \\ B. g a b)\n[PROOF STEP]\nhave [simp]: \"(\\ x. case case x of (a, b) \\ (b, a) of (a, b) \\ g b a) = (\\ (a,b). g a b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. case case x of (a, b) \\ (b, a) of (a, b) \\ g b a) = (\\(a, b). g a b)\n[PROOF STEP]\nby (intro ext, auto)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. case case x of (a, b) \\ (b, a) of (a, b) \\ g b a) = (\\(a, b). g a b)\n\ngoal (1 subgoal):\n 1. (\\(b, a)\\B \\ A. g a b) = (\\(a, b)\\A \\ B. g a b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(b, a)\\B \\ A. g a b) = (\\(a, b)\\A \\ B. g a b)\n[PROOF STEP]\nby (rule trans[OF trans[OF _ finprod_reindex[of \"\\ (a,b). g b a\" \"\\ (a,b). (b,a)\"]]],\n insert assms, auto simp: inj_on_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\(b, a)\\B \\ A. g a b) = (\\(a, b)\\A \\ B. g a b)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 954, "file": "Jordan_Normal_Form_Missing_Ring", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8479677583778257, "lm_q2_score": 0.8311430394931456, "lm_q1q2_score": 0.7047825000903353}} {"text": "[STATEMENT]\nlemma in_set_permutations:\n assumes \"distinct xs\"\n shows \"ys \\ set (permutations xs) \\ distinct ys \\ set xs = set ys\" (is \"?L xs ys \\ ?R xs ys\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndistinct xs\n\ngoal (1 subgoal):\n 1. (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n[PROOF STEP]\nproof (induct \"length xs\" arbitrary: xs ys)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\xs ys. \\0 = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n 2. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys); Suc x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n0 = length xs\ndistinct xs\n\ngoal (2 subgoals):\n 1. \\xs ys. \\0 = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n 2. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys); Suc x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 = length xs\ndistinct xs\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n0 = length xs\ndistinct xs\n\ngoal (1 subgoal):\n 1. (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n\ngoal (1 subgoal):\n 1. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys); Suc x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys); Suc x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n\\n = length ?xs; distinct ?xs\\ \\ (?ys \\ set (permutations ?xs)) = (distinct ?ys \\ set ?xs = set ?ys)\nSuc n = length xs\ndistinct xs\n\ngoal (1 subgoal):\n 1. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys); Suc x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\n = length ?xs; distinct ?xs\\ \\ (?ys \\ set (permutations ?xs)) = (distinct ?ys \\ set ?xs = set ?ys)\nSuc n = length xs\ndistinct xs\n[PROOF STEP]\nhave \"xs \\ []\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n = length ?xs; distinct ?xs\\ \\ (?ys \\ set (permutations ?xs)) = (distinct ?ys \\ set ?xs = set ?ys)\nSuc n = length xs\ndistinct xs\n\ngoal (1 subgoal):\n 1. xs \\ []\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nxs \\ []\n\ngoal (1 subgoal):\n 1. \\x xs ys. \\\\xs ys. \\x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys); Suc x = length xs; distinct xs\\ \\ (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ys \\ set (permutations xs) \\ distinct ys \\ set xs = set ys\n 2. distinct ys \\ set xs = set ys \\ ys \\ set (permutations xs)\n[PROOF STEP]\nassume \"?L xs ys\"\n[PROOF STATE]\nproof (state)\nthis:\nys \\ set (permutations xs)\n\ngoal (2 subgoals):\n 1. ys \\ set (permutations xs) \\ distinct ys \\ set xs = set ys\n 2. distinct ys \\ set xs = set ys \\ ys \\ set (permutations xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nys \\ set (permutations xs)\n[PROOF STEP]\nobtain y ys' where \"ys = y # ys'\" \"y \\ set xs\" \"ys' \\ set (permutations (remove1 (hd ys) xs))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nys \\ set (permutations xs)\n\ngoal (1 subgoal):\n 1. (\\y ys'. \\ys = y # ys'; y \\ set xs; ys' \\ set (permutations (remove1 (hd ys) xs))\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing \\xs \\ []\\\n[PROOF STATE]\nproof (prove)\nusing this:\nys \\ set (permutations xs)\nxs \\ []\n\ngoal (1 subgoal):\n 1. (\\y ys'. \\ys = y # ys'; y \\ set xs; ys' \\ set (permutations (remove1 (hd ys) xs))\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: permutations_not_nil)\n[PROOF STATE]\nproof (state)\nthis:\nys = y # ys'\ny \\ set xs\nys' \\ set (permutations (remove1 (hd ys) xs))\n\ngoal (2 subgoals):\n 1. ys \\ set (permutations xs) \\ distinct ys \\ set xs = set ys\n 2. distinct ys \\ set xs = set ys \\ ys \\ set (permutations xs)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nys = y # ys'\ny \\ set xs\nys' \\ set (permutations (remove1 (hd ys) xs))\n\ngoal (2 subgoals):\n 1. ys \\ set (permutations xs) \\ distinct ys \\ set xs = set ys\n 2. distinct ys \\ set xs = set ys \\ ys \\ set (permutations xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nys = y # ys'\ny \\ set xs\nys' \\ set (permutations (remove1 (hd ys) xs))\n[PROOF STEP]\nhave \"?R (remove1 y xs) ys'\"\n[PROOF STATE]\nproof (prove)\nusing this:\nys = y # ys'\ny \\ set xs\nys' \\ set (permutations (remove1 (hd ys) xs))\n\ngoal (1 subgoal):\n 1. distinct ys' \\ set (remove1 y xs) = set ys'\n[PROOF STEP]\nusing Suc.prems Suc.hyps(2)\n[PROOF STATE]\nproof (prove)\nusing this:\nys = y # ys'\ny \\ set xs\nys' \\ set (permutations (remove1 (hd ys) xs))\ndistinct xs\nSuc n = length xs\n\ngoal (1 subgoal):\n 1. distinct ys' \\ set (remove1 y xs) = set ys'\n[PROOF STEP]\nby (intro Suc.hyps(1)[THEN iffD1]) (auto simp: length_remove1)\n[PROOF STATE]\nproof (state)\nthis:\ndistinct ys' \\ set (remove1 y xs) = set ys'\n\ngoal (2 subgoals):\n 1. ys \\ set (permutations xs) \\ distinct ys \\ set xs = set ys\n 2. distinct ys \\ set xs = set ys \\ ys \\ set (permutations xs)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nys = y # ys'\ny \\ set xs\nys' \\ set (permutations (remove1 (hd ys) xs))\ndistinct ys' \\ set (remove1 y xs) = set ys'\n[PROOF STEP]\nshow \"?R xs ys\"\n[PROOF STATE]\nproof (prove)\nusing this:\nys = y # ys'\ny \\ set xs\nys' \\ set (permutations (remove1 (hd ys) xs))\ndistinct ys' \\ set (remove1 y xs) = set ys'\n\ngoal (1 subgoal):\n 1. distinct ys \\ set xs = set ys\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\nys = y # ys'\ny \\ set xs\nys' \\ set (permutations (remove1 (hd ys) xs))\ndistinct ys' \\ set (remove1 y xs) = set ys'\n\\n = length ?xs; distinct ?xs\\ \\ (?ys \\ set (permutations ?xs)) = (distinct ?ys \\ set ?xs = set ?ys)\nSuc n = length xs\ndistinct xs\n\ngoal (1 subgoal):\n 1. distinct ys \\ set xs = set ys\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndistinct ys \\ set xs = set ys\n\ngoal (1 subgoal):\n 1. distinct ys \\ set xs = set ys \\ ys \\ set (permutations xs)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. distinct ys \\ set xs = set ys \\ ys \\ set (permutations xs)\n[PROOF STEP]\nassume \"?R xs ys\"\n[PROOF STATE]\nproof (state)\nthis:\ndistinct ys \\ set xs = set ys\n\ngoal (1 subgoal):\n 1. distinct ys \\ set xs = set ys \\ ys \\ set (permutations xs)\n[PROOF STEP]\nwith \\xs \\ []\\\n[PROOF STATE]\nproof (chain)\npicking this:\nxs \\ []\ndistinct ys \\ set xs = set ys\n[PROOF STEP]\nobtain y ys' where \"ys = y # ys'\" \"y \\ set xs\"\n[PROOF STATE]\nproof (prove)\nusing this:\nxs \\ []\ndistinct ys \\ set xs = set ys\n\ngoal (1 subgoal):\n 1. (\\y ys'. \\ys = y # ys'; y \\ set xs\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (cases ys) auto\n[PROOF STATE]\nproof (state)\nthis:\nys = y # ys'\ny \\ set xs\n\ngoal (1 subgoal):\n 1. distinct ys \\ set xs = set ys \\ ys \\ set (permutations xs)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nys = y # ys'\ny \\ set xs\n\ngoal (1 subgoal):\n 1. distinct ys \\ set xs = set ys \\ ys \\ set (permutations xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nys = y # ys'\ny \\ set xs\n[PROOF STEP]\nhave \"ys' \\ set (permutations (remove1 y xs))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nys = y # ys'\ny \\ set xs\n\ngoal (1 subgoal):\n 1. ys' \\ set (permutations (remove1 y xs))\n[PROOF STEP]\nusing Suc \\?R xs ys\\\n[PROOF STATE]\nproof (prove)\nusing this:\nys = y # ys'\ny \\ set xs\n\\n = length ?xs; distinct ?xs\\ \\ (?ys \\ set (permutations ?xs)) = (distinct ?ys \\ set ?xs = set ?ys)\nSuc n = length xs\ndistinct xs\ndistinct ys \\ set xs = set ys\n\ngoal (1 subgoal):\n 1. ys' \\ set (permutations (remove1 y xs))\n[PROOF STEP]\nby (intro Suc.hyps(1)[THEN iffD2]) (auto simp: length_remove1)\n[PROOF STATE]\nproof (state)\nthis:\nys' \\ set (permutations (remove1 y xs))\n\ngoal (1 subgoal):\n 1. distinct ys \\ set xs = set ys \\ ys \\ set (permutations xs)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nys = y # ys'\ny \\ set xs\nys' \\ set (permutations (remove1 y xs))\n[PROOF STEP]\nshow \"?L xs ys\"\n[PROOF STATE]\nproof (prove)\nusing this:\nys = y # ys'\ny \\ set xs\nys' \\ set (permutations (remove1 y xs))\n\ngoal (1 subgoal):\n 1. ys \\ set (permutations xs)\n[PROOF STEP]\nusing \\xs \\ []\\\n[PROOF STATE]\nproof (prove)\nusing this:\nys = y # ys'\ny \\ set xs\nys' \\ set (permutations (remove1 y xs))\nxs \\ []\n\ngoal (1 subgoal):\n 1. ys \\ set (permutations xs)\n[PROOF STEP]\nby (auto simp: permutations_not_nil)\n[PROOF STATE]\nproof (state)\nthis:\nys \\ set (permutations xs)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(ys \\ set (permutations xs)) = (distinct ys \\ set xs = set ys)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4699, "file": "Planarity_Certificates_Planarity_Executable_Permutations", "length": 43, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8479677468516187, "lm_q2_score": 0.8311430499496096, "lm_q1q2_score": 0.7047824993771529}} {"text": "[STATEMENT]\nlemma card_minus_subset_fset:\n assumes \"B |\\| A\"\n shows \"card_fset (A - B) = card_fset A - card_fset B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card_fset (A - B) = card_fset A - card_fset B\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nB |\\| A\n\ngoal (1 subgoal):\n 1. card_fset (A - B) = card_fset A - card_fset B\n[PROOF STEP]\nby (simp add: subset_fset card_fset card_Diff_subset)", "meta": {"llama_tokens": 196, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.877476793890012, "lm_q2_score": 0.8031738034238806, "lm_q1q2_score": 0.7047663739648334}} {"text": "[STATEMENT]\nlemma real_sqrt_less_iff [simp]: \"sqrt x < sqrt y \\ x < y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (sqrt x < sqrt y) = (x < y)\n[PROOF STEP]\nunfolding sqrt_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (root 2 x < root 2 y) = (x < y)\n[PROOF STEP]\nby (rule real_root_less_iff [OF pos2])", "meta": {"llama_tokens": 151, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8031738057795403, "lm_q1q2_score": 0.7047663657381196}} {"text": "[STATEMENT]\nlemma one_minus_const_fps_X_neg_power':\n fixes c :: \"'a :: field_char_0\"\n assumes \"n > 0\"\n shows \"inverse ((1 - fps_const c * fps_X) ^ n) = Abs_fps (\\k. of_nat ((n + k - 1) choose k) * c^k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse ((1 - fps_const c * fps_X) ^ n) = Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. inverse ((1 - fps_const c * fps_X) ^ n) = Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k)\n[PROOF STEP]\nhave \\

: \"\\j. Abs_fps (\\na. (- c) ^ na * fps_binomial (- of_nat n) $ na) $ j =\n Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k) $ j\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\j. Abs_fps (\\na. (- c) ^ na * fps_binomial (- of_nat n) $ na) $ j = Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k) $ j\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\n\ngoal (1 subgoal):\n 1. \\j. Abs_fps (\\na. (- c) ^ na * fps_binomial (- of_nat n) $ na) $ j = Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k) $ j\n[PROOF STEP]\nby (simp add: gbinomial_minus binomial_gbinomial of_nat_diff flip: power_mult_distrib mult.assoc)\n[PROOF STATE]\nproof (state)\nthis:\nAbs_fps (\\na. (- c) ^ na * fps_binomial (- of_nat n) $ na) $ ?j = Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k) $ ?j\n\ngoal (1 subgoal):\n 1. inverse ((1 - fps_const c * fps_X) ^ n) = Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse ((1 - fps_const c * fps_X) ^ n) = Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k)\n[PROOF STEP]\napply (rule fps_ext)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\na. inverse ((1 - fps_const c * fps_X) ^ n) $ na = Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k) $ na\n[PROOF STEP]\nusing \\
\n[PROOF STATE]\nproof (prove)\nusing this:\nAbs_fps (\\na. (- c) ^ na * fps_binomial (- of_nat n) $ na) $ ?j = Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k) $ ?j\n\ngoal (1 subgoal):\n 1. \\na. inverse ((1 - fps_const c * fps_X) ^ n) $ na = Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k) $ na\n[PROOF STEP]\nby (metis (no_types, lifting) one_minus_fps_X_const_neg_power fps_const_neg fps_compose_linear fps_nth_Abs_fps)\n[PROOF STATE]\nproof (state)\nthis:\ninverse ((1 - fps_const c * fps_X) ^ n) = Abs_fps (\\k. of_nat (n + k - 1 choose k) * c ^ k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1192, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767906859264, "lm_q2_score": 0.8031737940012418, "lm_q1q2_score": 0.704766363123249}} {"text": "[STATEMENT]\nlemma lead_coeff_map_poly_nz:\n assumes \"f (lead_coeff p) \\0\" \"f 0=0\"\n shows \"lead_coeff (map_poly f p) = f (lead_coeff p) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lead_coeff (map_poly f p) = f (lead_coeff p)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. lead_coeff (map_poly f p) = f (lead_coeff p)\n[PROOF STEP]\nhave \"lead_coeff (Poly (map f (coeffs p))) = f (lead_coeff p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lead_coeff (Poly (map f (coeffs p))) = f (lead_coeff p)\n[PROOF STEP]\nby (metis (mono_tags, lifting) antisym assms(1) assms(2) coeff_0_degree_minus_1 coeff_map_poly \n degree_Poly degree_eq_length_coeffs le_degree length_map map_poly_def)\n[PROOF STATE]\nproof (state)\nthis:\nlead_coeff (Poly (map f (coeffs p))) = f (lead_coeff p)\n\ngoal (1 subgoal):\n 1. lead_coeff (map_poly f p) = f (lead_coeff p)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlead_coeff (Poly (map f (coeffs p))) = f (lead_coeff p)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlead_coeff (Poly (map f (coeffs p))) = f (lead_coeff p)\n\ngoal (1 subgoal):\n 1. lead_coeff (map_poly f p) = f (lead_coeff p)\n[PROOF STEP]\nby (simp add: map_poly_def)\n[PROOF STATE]\nproof (state)\nthis:\nlead_coeff (map_poly f p) = f (lead_coeff p)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 640, "file": "Winding_Number_Eval_Missing_Algebraic", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736692, "lm_q2_score": 0.8031737916455819, "lm_q1q2_score": 0.7047663533358991}} {"text": "[STATEMENT]\nlemma inverse_mat_base_case_2: \n shows \"inverts_mat (mat_of_rows_list 2 [[1, 1], [1, -1]]::rat mat) (mat_of_rows_list 2 [[1/2, 1/2], [1/2, -(1/2)]]::rat mat) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverts_mat (mat_of_rows_list 2 [[1, 1], [1, - 1]]) (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]])\n[PROOF STEP]\nunfolding inverts_mat_def mat_of_rows_list_def mat_eq_iff\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_row (mat (length [[1, 1], [1, - 1]]) 2 (\\(i, y). [[1, 1], [1, - 1]] ! i ! y) * mat (length [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]]) 2 (\\(i, y). [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]] ! i ! y)) = dim_row (1\\<^sub>m (dim_row (mat (length [[1, 1], [1, - 1]]) 2 (\\(i, y). [[1, 1], [1, - 1]] ! i ! y)))) \\ dim_col (mat (length [[1, 1], [1, - 1]]) 2 (\\(i, y). [[1, 1], [1, - 1]] ! i ! y) * mat (length [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]]) 2 (\\(i, y). [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]] ! i ! y)) = dim_col (1\\<^sub>m (dim_row (mat (length [[1, 1], [1, - 1]]) 2 (\\(i, y). [[1, 1], [1, - 1]] ! i ! y)))) \\ (\\i j. i < dim_row (1\\<^sub>m (dim_row (mat (length [[1, 1], [1, - 1]]) 2 (\\(i, y). [[1, 1], [1, - 1]] ! i ! y)))) \\ j < dim_col (1\\<^sub>m (dim_row (mat (length [[1, 1], [1, - 1]]) 2 (\\(i, y). [[1, 1], [1, - 1]] ! i ! y)))) \\ (mat (length [[1, 1], [1, - 1]]) 2 (\\(i, y). [[1, 1], [1, - 1]] ! i ! y) * mat (length [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]]) 2 (\\(i, y). [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]] ! i ! y)) $$ (i, j) = 1\\<^sub>m (dim_row (mat (length [[1, 1], [1, - 1]]) 2 (\\(i, y). [[1, 1], [1, - 1]] ! i ! y))) $$ (i, j))\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\j. j < Suc (Suc 0) \\ vec 2 ((!) ([[1, 1], [1, - 1]] ! j)) \\ vec (Suc (Suc 0)) (\\i. [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]] ! i ! j) = 1\n 2. \\i j. \\i < Suc (Suc 0); i \\ j; j < Suc (Suc 0)\\ \\ vec 2 ((!) ([[1, 1], [1, - 1]] ! i)) \\ vec (Suc (Suc 0)) (\\i. [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]] ! i ! j) = 0\n[PROOF STEP]\nunfolding less_two\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\j. j = 0 \\ j = 1 \\ vec 2 ((!) ([[1, 1], [1, - 1]] ! j)) \\ vec (Suc (Suc 0)) (\\i. [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]] ! i ! j) = 1\n 2. \\i j. \\i = 0 \\ i = 1; i \\ j; j = 0 \\ j = 1\\ \\ vec 2 ((!) ([[1, 1], [1, - 1]] ! i)) \\ vec (Suc (Suc 0)) (\\i. [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]] ! i ! j) = 0\n[PROOF STEP]\nby (auto simp add: scalar_prod_def)", "meta": {"llama_tokens": 1433, "file": "BenOr_Kozen_Reif_BKR_Proofs", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314738181875, "lm_q2_score": 0.7956581097540519, "lm_q1q2_score": 0.7046598643968742}} {"text": "[STATEMENT]\nlemma dist_fs_finite:\n shows \"dist_fs (of_complex z1) (of_complex z2) = 2 * cmod(z1 - z2) / (sqrt (1+(cmod z1)\\<^sup>2) * sqrt (1+(cmod z2)\\<^sup>2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist_fs (of_complex z1) (of_complex z2) = 2 * cmod (z1 - z2) / (sqrt (1 + (cmod z1)\\<^sup>2) * sqrt (1 + (cmod z2)\\<^sup>2))\n[PROOF STEP]\napply transfer\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z1 z2. dist_fs_hcoords (of_complex_hcoords z1) (of_complex_hcoords z2) = 2 * cmod (z1 - z2) / (sqrt (1 + (cmod z1)\\<^sup>2) * sqrt (1 + (cmod z2)\\<^sup>2))\n[PROOF STEP]\napply transfer\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z1 z2. dist_fs_cvec (of_complex_cvec z1) (of_complex_cvec z2) = 2 * cmod (z1 - z2) / (sqrt (1 + (cmod z1)\\<^sup>2) * sqrt (1 + (cmod z2)\\<^sup>2))\n[PROOF STEP]\napply (subst cmod_square)+\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z1 z2. dist_fs_cvec (of_complex_cvec z1) (of_complex_cvec z2) = 2 * cmod (z1 - z2) / (sqrt (1 + Re (z1 * cnj z1)) * sqrt (1 + Re (z2 * cnj z2)))\n[PROOF STEP]\napply (simp add: real_sqrt_divide cmod_def power2_eq_square)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z1 z2. 2 * sqrt ((Re z1 - Re z2) * (Re z1 - Re z2) - (Im z1 - Im z2) * (Im z2 - Im z1)) / sqrt ((Re z1 * Re z1 + Im z1 * Im z1 + 1) * (Re z2 * Re z2 + Im z2 * Im z2 + 1)) = 2 * sqrt ((Re z1 - Re z2) * (Re z1 - Re z2) + (Im z1 - Im z2) * (Im z1 - Im z2)) / (sqrt (1 + (Re z1 * Re z1 + Im z1 * Im z1)) * sqrt (1 + (Re z2 * Re z2 + Im z2 * Im z2)))\n[PROOF STEP]\napply (subst real_sqrt_mult[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\z1 z2. 2 * sqrt ((Re z1 - Re z2) * (Re z1 - Re z2) - (Im z1 - Im z2) * (Im z2 - Im z1)) / sqrt ((Re z1 * Re z1 + Im z1 * Im z1 + 1) * (Re z2 * Re z2 + Im z2 * Im z2 + 1)) = 2 * sqrt ((Re z1 - Re z2) * (Re z1 - Re z2) + (Im z1 - Im z2) * (Im z1 - Im z2)) / sqrt ((1 + (Re z1 * Re z1 + Im z1 * Im z1)) * (1 + (Re z2 * Re z2 + Im z2 * Im z2)))\n[PROOF STEP]\napply (simp add: field_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1055, "file": "Complex_Geometry_Chordal_Metric", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314677809303, "lm_q2_score": 0.7956581049086031, "lm_q1q2_score": 0.7046598553019996}} {"text": "[STATEMENT]\nlemma invertible_matrix_mult_left_rank:\n fixes A::\"'a::{field}^'n::{mod_type}^'m::{mod_type}\" \n and P::\"'a::{field}^'m::{mod_type}^'m::{mod_type}\"\n assumes invertible_P: \"invertible P\"\n shows \"rank (P**A) = rank A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank (P ** A) = rank A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. rank (P ** A) = rank A\n[PROOF STEP]\ndefine TP where \"TP = (\\x. P *v x)\"\n[PROOF STATE]\nproof (state)\nthis:\nTP = (*v) P\n\ngoal (1 subgoal):\n 1. rank (P ** A) = rank A\n[PROOF STEP]\ndefine TA where \"TA = (\\x. A *v x)\"\n[PROOF STATE]\nproof (state)\nthis:\nTA = (*v) A\n\ngoal (1 subgoal):\n 1. rank (P ** A) = rank A\n[PROOF STEP]\ndefine TPA where \"TPA = (\\x. (P**A) *v x)\"\n[PROOF STATE]\nproof (state)\nthis:\nTPA = (*v) (P ** A)\n\ngoal (1 subgoal):\n 1. rank (P ** A) = rank A\n[PROOF STEP]\nhave sub: \"vec.subspace (range ((*v) A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.subspace (range ((*v) A))\n[PROOF STEP]\nby (metis vec.subspace_UNIV vec.subspace_image)\n[PROOF STATE]\nproof (state)\nthis:\nvec.subspace (range ((*v) A))\n\ngoal (1 subgoal):\n 1. rank (P ** A) = rank A\n[PROOF STEP]\nhave \"vec.dim (range TPA) = vec.dim (range (TP \\ TA))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (range TPA) = vec.dim (range (TP \\ TA))\n[PROOF STEP]\nunfolding TP_def TA_def TPA_def o_def matrix_vector_mul_assoc\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (range ((*v) (P ** A))) = vec.dim (range ((*v) (P ** A)))\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nvec.dim (range TPA) = vec.dim (range (TP \\ TA))\n\ngoal (1 subgoal):\n 1. rank (P ** A) = rank A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nvec.dim (range TPA) = vec.dim (range (TP \\ TA))\n\ngoal (1 subgoal):\n 1. rank (P ** A) = rank A\n[PROOF STEP]\nhave \"... = vec.dim (range TA)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (range (TP \\ TA)) = vec.dim (range TA)\n[PROOF STEP]\nusing dim_image_invertible_mat[OF invertible_P sub]\n[PROOF STATE]\nproof (prove)\nusing this:\nvec.dim ((*v) P ` range ((*v) A)) = vec.dim (range ((*v) A))\n\ngoal (1 subgoal):\n 1. vec.dim (range (TP \\ TA)) = vec.dim (range TA)\n[PROOF STEP]\nunfolding TP_def TA_def o_def fun.set_map[symmetric]\n[PROOF STATE]\nproof (prove)\nusing this:\nvec.dim (range (\\x. P *v (A *v x))) = vec.dim (range ((*v) A))\n\ngoal (1 subgoal):\n 1. vec.dim (range (\\x. P *v (A *v x))) = vec.dim (range ((*v) A))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nvec.dim (range (TP \\ TA)) = vec.dim (range TA)\n\ngoal (1 subgoal):\n 1. rank (P ** A) = rank A\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nvec.dim (range TPA) = vec.dim (range TA)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nvec.dim (range TPA) = vec.dim (range TA)\n\ngoal (1 subgoal):\n 1. rank (P ** A) = rank A\n[PROOF STEP]\nunfolding rank_eq_dim_image TPA_def TA_def\n[PROOF STATE]\nproof (prove)\nusing this:\nvec.dim (range ((*v) (P ** A))) = vec.dim (range ((*v) A))\n\ngoal (1 subgoal):\n 1. vec.dim (range ((*v) (P ** A))) = vec.dim (range ((*v) A))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nrank (P ** A) = rank A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1483, "file": "Gauss_Jordan_Linear_Maps", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.849971175657575, "lm_q2_score": 0.8289388146603365, "lm_q1q2_score": 0.7045740988450429}} {"text": "[STATEMENT]\nlemma sin_zero_pi_iff:\n fixes x::real\n assumes \"\\x\\ < pi\"\n shows \"sin x = 0 \\ x = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (sin x = 0) = (x = 0)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. sin x = 0 \\ x = 0\n 2. x = 0 \\ sin x = 0\n[PROOF STEP]\nshow \"x = 0\" if \"sin x = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x = 0\n[PROOF STEP]\nusing that assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsin x = 0\n\\x\\ < pi\n\ngoal (1 subgoal):\n 1. x = 0\n[PROOF STEP]\nby (auto simp: sin_zero_iff)\n[PROOF STATE]\nproof (state)\nthis:\nsin x = 0 \\ x = 0\n\ngoal (1 subgoal):\n 1. x = 0 \\ sin x = 0\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 350, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.849971175657575, "lm_q2_score": 0.8289388019824947, "lm_q1q2_score": 0.7045740880692427}} {"text": "[STATEMENT]\nlemma vangle_nonneg: \"vangle u v \\ 0\" and vangle_le_pi: \"vangle u v \\ pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ vangle u v &&& vangle u v \\ pi\n[PROOF STEP]\nusing Cauchy_Schwarz_ineq2[of u v]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\u \\ v\\ \\ norm u * norm v\n\ngoal (1 subgoal):\n 1. 0 \\ vangle u v &&& vangle u v \\ pi\n[PROOF STEP]\nby (auto simp: vangle_def field_simps intro!: arccos_lbound arccos_ubound)", "meta": {"llama_tokens": 217, "file": "Triangle_Angles", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8499711604559846, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7045740790599879}} {"text": "[STATEMENT]\nlemma kronecker_one:\n shows \"kronecker_product ((1\\<^sub>m x)::'a :: ring_1 mat) (1\\<^sub>m y) = 1\\<^sub>m (x*y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. kronecker_product (1\\<^sub>m x) (1\\<^sub>m y) = 1\\<^sub>m (x * y)\n[PROOF STEP]\nunfolding kronecker_product_def Let_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (dim_row (1\\<^sub>m x) * dim_row (1\\<^sub>m y)) (dim_col (1\\<^sub>m x) * dim_col (1\\<^sub>m y)) (\\(i, j). 1\\<^sub>m x $$ (i div dim_row (1\\<^sub>m y), j div dim_col (1\\<^sub>m y)) * 1\\<^sub>m y $$ (i mod dim_row (1\\<^sub>m y), j mod dim_col (1\\<^sub>m y))) = 1\\<^sub>m (x * y)\n[PROOF STEP]\napply (auto simp add:mat_eq_iff less_mult_imp_div_less less_mult_imp_mod_less)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. \\i < x * y; i div y = j div y; i mod y = j mod y; i \\ j; j < x * y\\ \\ False\n[PROOF STEP]\nby (metis div_mult_mod_eq)", "meta": {"llama_tokens": 439, "file": "BenOr_Kozen_Reif_More_Matrix", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942145139149, "lm_q2_score": 0.7799928951399098, "lm_q1q2_score": 0.7045630695418392}} {"text": "[STATEMENT]\nlemma \"((\\x::real. ((3 powr x + 4 powr x) / 4) powr (1/x)) \\ 4) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. ((3 powr x + 4 powr x) / 4) powr (1 / x)) \\ 4) at_top\n[PROOF STEP]\nby real_asymp", "meta": {"llama_tokens": 128, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9161096181702031, "lm_q2_score": 0.76908023177796, "lm_q1q2_score": 0.7045617974763583}} {"text": "[STATEMENT]\nlemma det_linear_row_sum:\n assumes fS: \"finite S\"\n shows \"det ((\\ i. if i = k then sum (a i) S else c i)::'a::comm_ring_1^'n^'n) =\n sum (\\j. det ((\\ i. if i = k then a i j else c i)::'a^'n^'n)) S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (\\i. if i = k then sum (a i) S else c i) = (\\j\\S. det (\\i. if i = k then a i j else c i))\n[PROOF STEP]\nusing fS\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\n\ngoal (1 subgoal):\n 1. det (\\i. if i = k then sum (a i) S else c i) = (\\j\\S. det (\\i. if i = k then a i j else c i))\n[PROOF STEP]\nby (induct rule: finite_induct; simp add: det_row_0 det_row_add cong: if_cong)", "meta": {"llama_tokens": 316, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9161096090086367, "lm_q2_score": 0.7690802370707281, "lm_q1q2_score": 0.7045617952791344}} {"text": "[STATEMENT]\nlemma eexp_strict_mono2 [mono_intros]:\n assumes \"x < y\"\n shows \"eexp x < eexp y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eexp x < eexp y\n[PROOF STEP]\nusing eexp_strict_mono assms\n[PROOF STATE]\nproof (prove)\nusing this:\nstrict_mono eexp\nx < y\n\ngoal (1 subgoal):\n 1. eexp x < eexp y\n[PROOF STEP]\nunfolding strict_mono_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x y. x < y \\ eexp x < eexp y\nx < y\n\ngoal (1 subgoal):\n 1. eexp x < eexp y\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 230, "file": "Gromov_Hyperbolicity_Eexp_Eln", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382165412808, "lm_q2_score": 0.8175744806385543, "lm_q1q2_score": 0.704371659939004}} {"text": "[STATEMENT]\nlemma eventually_nat_real:\n assumes \"eventually P (at_top :: real filter)\"\n shows \"eventually (\\x. P (real x)) (at_top :: nat filter)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in sequentially. P (real x)\n[PROOF STEP]\nusing assms filterlim_real_sequentially\n[PROOF STATE]\nproof (prove)\nusing this:\neventually P at_top\nfilterlim real at_top sequentially\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in sequentially. P (real x)\n[PROOF STEP]\nunfolding filterlim_def le_filter_def eventually_filtermap\n[PROOF STATE]\nproof (prove)\nusing this:\neventually P at_top\n\\P. eventually P at_top \\ (\\\\<^sub>F x in sequentially. P (real x))\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in sequentially. P (real x)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 290, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615382165412809, "lm_q2_score": 0.8175744739711883, "lm_q1q2_score": 0.7043716541948135}} {"text": "[STATEMENT]\nlemma length_distincts:\n assumes \"distincts xss\"\n shows \"length xss = card (set ` set xss)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length xss = card (set ` set xss)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndistincts xss\n\ngoal (1 subgoal):\n 1. length xss = card (set ` set xss)\n[PROOF STEP]\nproof (induct xss)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. distincts [] \\ length [] = card (set ` set [])\n 2. \\a xss. \\distincts xss \\ length xss = card (set ` set xss); distincts (a # xss)\\ \\ length (a # xss) = card (set ` set (a # xss))\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\ndistincts []\n\ngoal (2 subgoals):\n 1. distincts [] \\ length [] = card (set ` set [])\n 2. \\a xss. \\distincts xss \\ length xss = card (set ` set xss); distincts (a # xss)\\ \\ length (a # xss) = card (set ` set (a # xss))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndistincts []\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ndistincts []\n\ngoal (1 subgoal):\n 1. length [] = card (set ` set [])\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlength [] = card (set ` set [])\n\ngoal (1 subgoal):\n 1. \\a xss. \\distincts xss \\ length xss = card (set ` set xss); distincts (a # xss)\\ \\ length (a # xss) = card (set ` set (a # xss))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a xss. \\distincts xss \\ length xss = card (set ` set xss); distincts (a # xss)\\ \\ length (a # xss) = card (set ` set (a # xss))\n[PROOF STEP]\ncase (Cons xs xss)\n[PROOF STATE]\nproof (state)\nthis:\ndistincts xss \\ length xss = card (set ` set xss)\ndistincts (xs # xss)\n\ngoal (1 subgoal):\n 1. \\a xss. \\distincts xss \\ length xss = card (set ` set xss); distincts (a # xss)\\ \\ length (a # xss) = card (set ` set (a # xss))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndistincts xss \\ length xss = card (set ` set xss)\ndistincts (xs # xss)\n[PROOF STEP]\nhave \"set xs \\ set ` set xss\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndistincts xss \\ length xss = card (set ` set xss)\ndistincts (xs # xss)\n\ngoal (1 subgoal):\n 1. set xs \\ set ` set xss\n[PROOF STEP]\nusing equals0I[of \"set xs\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ndistincts xss \\ length xss = card (set ` set xss)\ndistincts (xs # xss)\n(\\y. y \\ set xs \\ False) \\ set xs = {}\n\ngoal (1 subgoal):\n 1. set xs \\ set ` set xss\n[PROOF STEP]\nby (auto simp: distincts_Cons disjoint_iff_not_equal )\n[PROOF STATE]\nproof (state)\nthis:\nset xs \\ set ` set xss\n\ngoal (1 subgoal):\n 1. \\a xss. \\distincts xss \\ length xss = card (set ` set xss); distincts (a # xss)\\ \\ length (a # xss) = card (set ` set (a # xss))\n[PROOF STEP]\nwith Cons\n[PROOF STATE]\nproof (chain)\npicking this:\ndistincts xss \\ length xss = card (set ` set xss)\ndistincts (xs # xss)\nset xs \\ set ` set xss\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ndistincts xss \\ length xss = card (set ` set xss)\ndistincts (xs # xss)\nset xs \\ set ` set xss\n\ngoal (1 subgoal):\n 1. length (xs # xss) = card (set ` set (xs # xss))\n[PROOF STEP]\nby (auto simp add: distincts_Cons)\n[PROOF STATE]\nproof (state)\nthis:\nlength (xs # xss) = card (set ` set (xs # xss))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1480, "file": "Planarity_Certificates_Planarity_Executable_Permutations", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744806385542, "lm_q2_score": 0.8615382040983515, "lm_q1q2_score": 0.7043716497659824}} {"text": "[STATEMENT]\nlemma inverse: assumes \"gcd x q = 1\" \n shows \"[x * inverse x q = 1] (mod q)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [int x * Number_Theory_Aux.inverse x q = 1] (mod int q)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. [int x * Number_Theory_Aux.inverse x q = 1] (mod int q)\n[PROOF STEP]\nhave 2: \"fst (bezw x q) * x + snd (bezw x q) * int q = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Number_Theory_Aux.inverse x q * int x + snd (bezw x q) * int q = 1\n[PROOF STEP]\nusing bezw_aux assms int_minus\n[PROOF STATE]\nproof (prove)\nusing this:\nint (gcd ?x ?y) = Number_Theory_Aux.inverse ?x ?y * int ?x + snd (bezw ?x ?y) * int ?y\ngcd x q = 1\nint (?n - ?m) = int (nat (int ?n - int ?m))\n\ngoal (1 subgoal):\n 1. Number_Theory_Aux.inverse x q * int x + snd (bezw x q) * int q = 1\n[PROOF STEP]\nby (metis Num.of_nat_simps(2))\n[PROOF STATE]\nproof (state)\nthis:\nNumber_Theory_Aux.inverse x q * int x + snd (bezw x q) * int q = 1\n\ngoal (1 subgoal):\n 1. [int x * Number_Theory_Aux.inverse x q = 1] (mod int q)\n[PROOF STEP]\nhence 3: \"(fst (bezw x q) * x + snd (bezw x q) * int q) mod q = 1 mod q\"\n[PROOF STATE]\nproof (prove)\nusing this:\nNumber_Theory_Aux.inverse x q * int x + snd (bezw x q) * int q = 1\n\ngoal (1 subgoal):\n 1. (Number_Theory_Aux.inverse x q * int x + snd (bezw x q) * int q) mod int q = int (1 mod q)\n[PROOF STEP]\nby (metis assms bezw_aux of_nat_mod)\n[PROOF STATE]\nproof (state)\nthis:\n(Number_Theory_Aux.inverse x q * int x + snd (bezw x q) * int q) mod int q = int (1 mod q)\n\ngoal (1 subgoal):\n 1. [int x * Number_Theory_Aux.inverse x q = 1] (mod int q)\n[PROOF STEP]\nhence 4: \"(fst (bezw x q) * x) mod q = 1 mod q\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(Number_Theory_Aux.inverse x q * int x + snd (bezw x q) * int q) mod int q = int (1 mod q)\n\ngoal (1 subgoal):\n 1. Number_Theory_Aux.inverse x q * int x mod int q = int (1 mod q)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nNumber_Theory_Aux.inverse x q * int x mod int q = int (1 mod q)\n\ngoal (1 subgoal):\n 1. [int x * Number_Theory_Aux.inverse x q = 1] (mod int q)\n[PROOF STEP]\nhence 5: \"[(fst (bezw x q)) * x = 1] (mod q)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nNumber_Theory_Aux.inverse x q * int x mod int q = int (1 mod q)\n\ngoal (1 subgoal):\n 1. [Number_Theory_Aux.inverse x q * int x = 1] (mod int q)\n[PROOF STEP]\nusing 2 3 cong_def\n[PROOF STATE]\nproof (prove)\nusing this:\nNumber_Theory_Aux.inverse x q * int x mod int q = int (1 mod q)\nNumber_Theory_Aux.inverse x q * int x + snd (bezw x q) * int q = 1\n(Number_Theory_Aux.inverse x q * int x + snd (bezw x q) * int q) mod int q = int (1 mod q)\n[?b = ?c] (mod ?a) = (?b mod ?a = ?c mod ?a)\n\ngoal (1 subgoal):\n 1. [Number_Theory_Aux.inverse x q * int x = 1] (mod int q)\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n[Number_Theory_Aux.inverse x q * int x = 1] (mod int q)\n\ngoal (1 subgoal):\n 1. [int x * Number_Theory_Aux.inverse x q = 1] (mod int q)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[Number_Theory_Aux.inverse x q * int x = 1] (mod int q)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n[Number_Theory_Aux.inverse x q * int x = 1] (mod int q)\n\ngoal (1 subgoal):\n 1. [int x * Number_Theory_Aux.inverse x q = 1] (mod int q)\n[PROOF STEP]\nby(simp add: mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\n[int x * Number_Theory_Aux.inverse x q = 1] (mod int q)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1601, "file": "Sigma_Commit_Crypto_Number_Theory_Aux", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382129861583, "lm_q2_score": 0.8175744695262775, "lm_q1q2_score": 0.7043716474587755}} {"text": "[STATEMENT]\nlemma eventually_nat_real:\n assumes \"eventually P (at_top :: real filter)\"\n shows \"eventually (\\x. P (real x)) (at_top :: nat filter)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in sequentially. P (real x)\n[PROOF STEP]\nusing assms filterlim_real_sequentially\n[PROOF STATE]\nproof (prove)\nusing this:\neventually P at_top\nfilterlim real at_top sequentially\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in sequentially. P (real x)\n[PROOF STEP]\nunfolding filterlim_def le_filter_def eventually_filtermap\n[PROOF STATE]\nproof (prove)\nusing this:\neventually P at_top\n\\P. eventually P at_top \\ (\\\\<^sub>F x in sequentially. P (real x))\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in sequentially. P (real x)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 290, "file": "Akra_Bazzi_Akra_Bazzi_Library", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.86153820232079, "lm_q2_score": 0.8175744761936437, "lm_q1q2_score": 0.7043716444832333}} {"text": "[STATEMENT]\nlemma bij_betw_eq:\n \"bij_betw f A B \\\n inj_on f A \\ (\\y\\B. \\x\\A. f(x)=y) \\ (\\x\\A. f x \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw f A B = (inj_on f A \\ (\\y\\B. \\x\\A. f x = y) \\ (\\x\\A. f x \\ B))\n[PROOF STEP]\nunfolding bij_betw_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (inj_on f A \\ f ` A = B) = (inj_on f A \\ (\\y\\B. \\x\\A. f x = y) \\ (\\x\\A. f x \\ B))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 287, "file": "Category_Yoneda", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278664544912, "lm_q2_score": 0.7981867753392728, "lm_q1q2_score": 0.7043422531948248}} {"text": "[STATEMENT]\nlemma ex_inverse:\n assumes coprime: \"coprime (e :: nat) ((P-1)*(Q-1))\" \n and \"prime P\" \n and \"prime Q\" \n and \"P \\ Q\" \n shows \"\\ d. [e*d = 1] (mod (P-1)) \\ d \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n[PROOF STEP]\nhave \"coprime e (P-1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coprime e (P - 1)\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime e ((P - 1) * (Q - 1))\n\ngoal (1 subgoal):\n 1. coprime e (P - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncoprime e (P - 1)\n\ngoal (1 subgoal):\n 1. \\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncoprime e (P - 1)\n[PROOF STEP]\nobtain d where d: \"[e*d = 1] (mod (P-1))\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime e (P - 1)\n\ngoal (1 subgoal):\n 1. (\\d. [e * d = 1] (mod P - 1) \\ thesis) \\ thesis\n[PROOF STEP]\nusing cong_solve_coprime_nat\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime e (P - 1)\ncoprime ?a ?n \\ \\x. [?a * x = Suc 0] (mod ?n)\n\ngoal (1 subgoal):\n 1. (\\d. [e * d = 1] (mod P - 1) \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n[e * d = 1] (mod P - 1)\n\ngoal (1 subgoal):\n 1. \\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[e * d = 1] (mod P - 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n[e * d = 1] (mod P - 1)\n\ngoal (1 subgoal):\n 1. \\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n[PROOF STEP]\nby (metis cong_0_1_nat cong_1 mult_0_right zero_neq_one)\n[PROOF STATE]\nproof (state)\nthis:\n\\d. [e * d = 1] (mod P - 1) \\ d \\ 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1005, "file": "Sigma_Commit_Crypto_Number_Theory_Aux", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8519528019683106, "lm_q2_score": 0.8267118004748677, "lm_q1q2_score": 0.7043194348348304}} {"text": "[STATEMENT]\nlemma euclidean_size_lcm_le1: \n assumes \"a \\ 0\" and \"b \\ 0\"\n shows \"euclidean_size a \\ euclidean_size (lcm a b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. euclidean_size a \\ euclidean_size (lcm a b)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. euclidean_size a \\ euclidean_size (lcm a b)\n[PROOF STEP]\nhave \"a dvd lcm a b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a dvd lcm a b\n[PROOF STEP]\nby (rule dvd_lcm1)\n[PROOF STATE]\nproof (state)\nthis:\na dvd lcm a b\n\ngoal (1 subgoal):\n 1. euclidean_size a \\ euclidean_size (lcm a b)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na dvd lcm a b\n[PROOF STEP]\nobtain c where A: \"lcm a b = a * c\"\n[PROOF STATE]\nproof (prove)\nusing this:\na dvd lcm a b\n\ngoal (1 subgoal):\n 1. (\\c. lcm a b = a * c \\ thesis) \\ thesis\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nlcm a b = a * c\n\ngoal (1 subgoal):\n 1. euclidean_size a \\ euclidean_size (lcm a b)\n[PROOF STEP]\nwith \\a \\ 0\\ and \\b \\ 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ (0::'a)\nb \\ (0::'a)\nlcm a b = a * c\n[PROOF STEP]\nhave \"c \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ (0::'a)\nb \\ (0::'a)\nlcm a b = a * c\n\ngoal (1 subgoal):\n 1. c \\ (0::'a)\n[PROOF STEP]\nby (auto simp: lcm_eq_0_iff)\n[PROOF STATE]\nproof (state)\nthis:\nc \\ (0::'a)\n\ngoal (1 subgoal):\n 1. euclidean_size a \\ euclidean_size (lcm a b)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc \\ (0::'a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'a)\n\ngoal (1 subgoal):\n 1. euclidean_size a \\ euclidean_size (lcm a b)\n[PROOF STEP]\nby (subst A, intro size_mult_mono)\n[PROOF STATE]\nproof (state)\nthis:\neuclidean_size a \\ euclidean_size (lcm a b)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 909, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117769928211, "lm_q2_score": 0.8519528076067262, "lm_q1q2_score": 0.7043194194905796}} {"text": "[STATEMENT]\nlemma vcard_vunion_vintersection: \n \"vcard (A \\\\<^sub>\\ B) \\ vcard (A \\\\<^sub>\\ B) = vcard A \\ vcard B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vcard (A \\\\<^sub>\\ B) \\ vcard (A \\\\<^sub>\\ B) = vcard A \\ vcard B\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. vcard (A \\\\<^sub>\\ B) \\ vcard (A \\\\<^sub>\\ B) = vcard A \\ vcard B\n[PROOF STEP]\nhave AB_ABB: \"A \\\\<^sub>\\ B = B \\\\<^sub>\\ (A -\\<^sub>\\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A \\\\<^sub>\\ B = B \\\\<^sub>\\ (A -\\<^sub>\\ B)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA \\\\<^sub>\\ B = B \\\\<^sub>\\ (A -\\<^sub>\\ B)\n\ngoal (1 subgoal):\n 1. vcard (A \\\\<^sub>\\ B) \\ vcard (A \\\\<^sub>\\ B) = vcard A \\ vcard B\n[PROOF STEP]\nhave ABB: \"vdisjnt B (A -\\<^sub>\\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vdisjnt B (A -\\<^sub>\\ B)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nvdisjnt B (A -\\<^sub>\\ B)\n\ngoal (1 subgoal):\n 1. vcard (A \\\\<^sub>\\ B) \\ vcard (A \\\\<^sub>\\ B) = vcard A \\ vcard B\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vcard (A \\\\<^sub>\\ B) \\ vcard (A \\\\<^sub>\\ B) = vcard A \\ vcard B\n[PROOF STEP]\nunfolding vcard_vunion[OF ABB, folded AB_ABB] cadd_assoc vcard_vdiff\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vcard B \\ vcard A = vcard A \\ vcard B\n[PROOF STEP]\nby (simp add: cadd_commute)\n[PROOF STATE]\nproof (state)\nthis:\nvcard (A \\\\<^sub>\\ B) \\ vcard (A \\\\<^sub>\\ B) = vcard A \\ vcard B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 811, "file": "CZH_Foundations_czh_sets_CZH_Sets_Cardinality", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772482857833, "lm_q2_score": 0.8056321889812553, "lm_q1q2_score": 0.7041847668751877}} {"text": "[STATEMENT]\ntheorem IF1map_cong:\n \"(\\a. a \\ IF1set x \\ f a = g a) \\ IF1map f x = IF1map g x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a. a \\ IF1set x \\ f a = g a) \\ IF1map f x = IF1map g x\n[PROOF STEP]\napply (rule mp)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (\\a. a \\ IF1set x \\ f a = g a) \\ ?P \\ IF1map f x = IF1map g x\n 2. (\\a. a \\ IF1set x \\ f a = g a) \\ ?P\n[PROOF STEP]\napply (rule conjunct1)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (\\a. a \\ IF1set x \\ f a = g a) \\ (?P \\ IF1map f x = IF1map g x) \\ ?Q3\n 2. (\\a. a \\ IF1set x \\ f a = g a) \\ ?P\n[PROOF STEP]\napply (rule IFmap_cong)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a. a \\ IF1set x \\ f a = g a) \\ \\a\\IF1set x. f a = g a\n[PROOF STEP]\napply (rule ballI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a. \\\\a. a \\ IF1set x \\ f a = g a; a \\ IF1set x\\ \\ f a = g a\n[PROOF STEP]\napply (tactic \\Goal.assume_rule_tac @{context} 1\\)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 584, "file": "BNF_Operations_LFP", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772417253255, "lm_q2_score": 0.8056321889812553, "lm_q1q2_score": 0.7041847615898719}} {"text": "[STATEMENT]\nlemma card_zfact_carr: \"card (carrier (ZFact (int n))) = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (carrier (ZFact (int n))) = n\n[PROOF STEP]\nusing bij_betw_same_card[OF zfact_iso_bij]\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {.. sqrt (real x) \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt_nat_ceiling x = \\sqrt (real x)\\\n[PROOF STEP]\nunfolding sqrt_nat_ceiling_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. root_nat_ceiling 2 x = \\sqrt (real x)\\\n[PROOF STEP]\nby (simp add: sqrt_def)", "meta": {"llama_tokens": 172, "file": "Sqrt_Babylonian_Sqrt_Babylonian", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.868826789824086, "lm_q2_score": 0.8104789086703224, "lm_q1q2_score": 0.7041657884401648}} {"text": "[STATEMENT]\nlemma smult_distrib_left_minus_mat:\n fixes A B :: \"'a::comm_ring_1 mat\"\n assumes \"A \\ carrier_mat n n\" \"B \\ carrier_mat n n\"\n shows \"c \\\\<^sub>m (B - A) = c \\\\<^sub>m B - c \\\\<^sub>m A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c \\\\<^sub>m (B - A) = c \\\\<^sub>m B - c \\\\<^sub>m A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n n\nB \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. c \\\\<^sub>m (B - A) = c \\\\<^sub>m B - c \\\\<^sub>m A\n[PROOF STEP]\nby (auto simp add: minus_add_uminus_mat add_smult_distrib_left_mat)", "meta": {"llama_tokens": 278, "file": "QHLProver_Complex_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267898240861, "lm_q2_score": 0.8104789018037399, "lm_q1q2_score": 0.7041657824742941}} {"text": "[STATEMENT]\nlemma linear_surj_adj_imp_inj:\n fixes f :: \"'m::euclidean_space \\ 'n::euclidean_space\"\n assumes \"linear f\" \"surj (adjoint f)\"\n shows \"inj f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj f\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. inj f\n[PROOF STEP]\nhave \"\\x. y = adjoint f x\" for y\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. y = adjoint f x\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlinear f\nsurj (adjoint f)\n\ngoal (1 subgoal):\n 1. \\x. y = adjoint f x\n[PROOF STEP]\nby (simp add: surjD)\n[PROOF STATE]\nproof (state)\nthis:\n\\x. ?y = adjoint f x\n\ngoal (1 subgoal):\n 1. inj f\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x. ?y = adjoint f x\n[PROOF STEP]\nshow \"inj f\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. ?y = adjoint f x\n\ngoal (1 subgoal):\n 1. inj f\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. ?y = adjoint f x\nlinear f\nsurj (adjoint f)\n\ngoal (1 subgoal):\n 1. inj f\n[PROOF STEP]\nunfolding inj_on_def image_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. ?y = adjoint f x\nlinear f\n{y. \\x\\UNIV. y = adjoint f x} = UNIV\n\ngoal (1 subgoal):\n 1. \\x\\UNIV. \\y\\UNIV. f x = f y \\ x = y\n[PROOF STEP]\nby (metis (no_types) adjoint_works euclidean_eqI)\n[PROOF STATE]\nproof (state)\nthis:\ninj f\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n\n\\ \\\\<^url>\\https://mathonline.wikidot.com/injectivity-and-surjectivity-of-the-adjoint-of-a-linear-map\\\\", "meta": {"llama_tokens": 713, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104789040926008, "lm_q2_score": 0.8688267830311354, "lm_q1q2_score": 0.7041657789573744}} {"text": "[STATEMENT]\ntheorem binop_dfa_accepts:\n assumes A: \"wf_dfa A n\"\n and B: \"wf_dfa B n\"\n and X: \"list_all (is_alph n) bss\"\n shows \"dfa_accepts (binop_dfa f A B) bss = f (dfa_accepts A bss) (dfa_accepts B bss)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dfa_accepts (binop_dfa f A B) bss = f (dfa_accepts A bss) (dfa_accepts B bss)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dfa_accepts (binop_dfa f A B) bss = f (dfa_accepts A bss) (dfa_accepts B bss)\n[PROOF STEP]\nfrom A B\n[PROOF STATE]\nproof (chain)\npicking this:\nwf_dfa A n\nwf_dfa B n\n[PROOF STEP]\ninterpret prod_DFS A B n\n[PROOF STATE]\nproof (prove)\nusing this:\nwf_dfa A n\nwf_dfa B n\n\ngoal (1 subgoal):\n 1. prod_DFS A B n\n[PROOF STEP]\nby unfold_locales\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dfa_accepts (binop_dfa f A B) bss = f (dfa_accepts A bss) (dfa_accepts B bss)\n[PROOF STEP]\nfrom X\n[PROOF STATE]\nproof (chain)\npicking this:\nlist_all (is_alph n) bss\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlist_all (is_alph n) bss\n\ngoal (1 subgoal):\n 1. dfa_accepts (binop_dfa f A B) bss = f (dfa_accepts A bss) (dfa_accepts B bss)\n[PROOF STEP]\nby (simp add: accepts_def dfa_accepting_def binop_dfa_steps)\n[PROOF STATE]\nproof (state)\nthis:\ndfa_accepts (binop_dfa f A B) bss = f (dfa_accepts A bss) (dfa_accepts B bss)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 671, "file": "Presburger-Automata_Presburger_Automata", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267830311354, "lm_q2_score": 0.8104789018037399, "lm_q1q2_score": 0.7041657769687508}} {"text": "[STATEMENT]\nlemma orthogonal_basis_exists:\n fixes V :: \"(real^'b) list\"\n assumes B: \"is_basis (set V)\"\n and d: \"distinct V\"\n shows \"vec.independent (set (Gram_Schmidt V)) \\ (set V) \\ vec.span (set (Gram_Schmidt V)) \n \\ (card (set (Gram_Schmidt V)) = vec.dim (set V)) \\ pairwise orthogonal (set (Gram_Schmidt V))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.independent (set (Gram_Schmidt V)) \\ set V \\ vec.span (set (Gram_Schmidt V)) \\ card (set (Gram_Schmidt V)) = vec.dim (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. vec.independent (set (Gram_Schmidt V)) \\ set V \\ vec.span (set (Gram_Schmidt V)) \\ card (set (Gram_Schmidt V)) = vec.dim (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nhave \"(set V) \\ vec.span (set (Gram_Schmidt V))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set V \\ vec.span (set (Gram_Schmidt V))\n[PROOF STEP]\nusing basis_orthogonal'[of V]\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (Gram_Schmidt V) = length V \\ span (set (Gram_Schmidt V)) = span (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. set V \\ vec.span (set (Gram_Schmidt V))\n[PROOF STEP]\nusing vec.span_superset[where ?'a=real, where ?'b='b]\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (Gram_Schmidt V) = length V \\ span (set (Gram_Schmidt V)) = span (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n?S \\ vec.span ?S\n\ngoal (1 subgoal):\n 1. set V \\ vec.span (set (Gram_Schmidt V))\n[PROOF STEP]\nby (auto simp: span_vec_eq)\n[PROOF STATE]\nproof (state)\nthis:\nset V \\ vec.span (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. vec.independent (set (Gram_Schmidt V)) \\ set V \\ vec.span (set (Gram_Schmidt V)) \\ card (set (Gram_Schmidt V)) = vec.dim (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nset V \\ vec.span (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. vec.independent (set (Gram_Schmidt V)) \\ set V \\ vec.span (set (Gram_Schmidt V)) \\ card (set (Gram_Schmidt V)) = vec.dim (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nhave \"pairwise orthogonal (set (Gram_Schmidt V))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nusing basis_orthogonal'[of V]\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (Gram_Schmidt V) = length V \\ span (set (Gram_Schmidt V)) = span (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\npairwise orthogonal (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. vec.independent (set (Gram_Schmidt V)) \\ set V \\ vec.span (set (Gram_Schmidt V)) \\ card (set (Gram_Schmidt V)) = vec.dim (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\npairwise orthogonal (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. vec.independent (set (Gram_Schmidt V)) \\ set V \\ vec.span (set (Gram_Schmidt V)) \\ card (set (Gram_Schmidt V)) = vec.dim (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nhave c: \"(card (set (Gram_Schmidt V)) = vec.dim (set V))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (set (Gram_Schmidt V)) = vec.dim (set V)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (set (Gram_Schmidt V)) = vec.dim (set V)\n[PROOF STEP]\nhave card_eq_dim: \"card (set V) = vec.dim (set V)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (set V) = vec.dim (set V)\n[PROOF STEP]\nby (metis B independent_is_basis vec.dim_span vec.indep_card_eq_dim_span)\n[PROOF STATE]\nproof (state)\nthis:\ncard (set V) = vec.dim (set V)\n\ngoal (1 subgoal):\n 1. card (set (Gram_Schmidt V)) = vec.dim (set V)\n[PROOF STEP]\nhave \"vec.dim (set V) \\ (card (set (Gram_Schmidt V)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (set V) \\ card (set (Gram_Schmidt V))\n[PROOF STEP]\nusing B\n[PROOF STATE]\nproof (prove)\nusing this:\nis_basis (set V)\n\ngoal (1 subgoal):\n 1. vec.dim (set V) \\ card (set (Gram_Schmidt V))\n[PROOF STEP]\nunfolding is_basis_def\n[PROOF STATE]\nproof (prove)\nusing this:\nvec.independent (set V) \\ vec.span (set V) = UNIV\n\ngoal (1 subgoal):\n 1. vec.dim (set V) \\ card (set (Gram_Schmidt V))\n[PROOF STEP]\nusing vec.independent_span_bound[of \"(set (Gram_Schmidt V))\" \"set V\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nvec.independent (set V) \\ vec.span (set V) = UNIV\n\\finite (set (Gram_Schmidt V)); vec.independent (set V); set V \\ vec.span (set (Gram_Schmidt V))\\ \\ finite (set V) \\ card (set V) \\ card (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. vec.dim (set V) \\ card (set (Gram_Schmidt V))\n[PROOF STEP]\nusing basis_orthogonal'[of V]\n[PROOF STATE]\nproof (prove)\nusing this:\nvec.independent (set V) \\ vec.span (set V) = UNIV\n\\finite (set (Gram_Schmidt V)); vec.independent (set V); set V \\ vec.span (set (Gram_Schmidt V))\\ \\ finite (set V) \\ card (set V) \\ card (set (Gram_Schmidt V))\nlength (Gram_Schmidt V) = length V \\ span (set (Gram_Schmidt V)) = span (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. vec.dim (set V) \\ card (set (Gram_Schmidt V))\n[PROOF STEP]\nby (simp add: calculation(1) local.card_eq_dim)\n[PROOF STATE]\nproof (state)\nthis:\nvec.dim (set V) \\ card (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. card (set (Gram_Schmidt V)) = vec.dim (set V)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nvec.dim (set V) \\ card (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. card (set (Gram_Schmidt V)) = vec.dim (set V)\n[PROOF STEP]\nhave \"(card (set (Gram_Schmidt V))) \\ vec.dim (set V)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (set (Gram_Schmidt V)) \\ vec.dim (set V)\n[PROOF STEP]\nusing card_Gram_Schmidt[OF d] card_eq_dim\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (set (Gram_Schmidt V)) \\ card (set V)\ncard (set V) = vec.dim (set V)\n\ngoal (1 subgoal):\n 1. card (set (Gram_Schmidt V)) \\ vec.dim (set V)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (set (Gram_Schmidt V)) \\ vec.dim (set V)\n\ngoal (1 subgoal):\n 1. card (set (Gram_Schmidt V)) = vec.dim (set V)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nvec.dim (set V) \\ card (set (Gram_Schmidt V))\ncard (set (Gram_Schmidt V)) \\ vec.dim (set V)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nvec.dim (set V) \\ card (set (Gram_Schmidt V))\ncard (set (Gram_Schmidt V)) \\ vec.dim (set V)\n\ngoal (1 subgoal):\n 1. card (set (Gram_Schmidt V)) = vec.dim (set V)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (set (Gram_Schmidt V)) = vec.dim (set V)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncard (set (Gram_Schmidt V)) = vec.dim (set V)\n\ngoal (1 subgoal):\n 1. vec.independent (set (Gram_Schmidt V)) \\ set V \\ vec.span (set (Gram_Schmidt V)) \\ card (set (Gram_Schmidt V)) = vec.dim (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncard (set (Gram_Schmidt V)) = vec.dim (set V)\n\ngoal (1 subgoal):\n 1. vec.independent (set (Gram_Schmidt V)) \\ set V \\ vec.span (set (Gram_Schmidt V)) \\ card (set (Gram_Schmidt V)) = vec.dim (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nhave \"vec.independent (set (Gram_Schmidt V))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.independent (set (Gram_Schmidt V))\n[PROOF STEP]\nproof (rule vec.card_le_dim_spanning[of _ \"UNIV::(real^'b) set\"])\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. set (Gram_Schmidt V) \\ UNIV\n 2. UNIV \\ vec.span (set (Gram_Schmidt V))\n 3. finite (set (Gram_Schmidt V))\n 4. card (set (Gram_Schmidt V)) \\ vec.dim UNIV\n[PROOF STEP]\nshow \"set (Gram_Schmidt V) \\ (UNIV::(real^'b) set)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (Gram_Schmidt V) \\ UNIV\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nset (Gram_Schmidt V) \\ UNIV\n\ngoal (3 subgoals):\n 1. UNIV \\ vec.span (set (Gram_Schmidt V))\n 2. finite (set (Gram_Schmidt V))\n 3. card (set (Gram_Schmidt V)) \\ vec.dim UNIV\n[PROOF STEP]\nshow \"UNIV \\ vec.span (set (Gram_Schmidt V))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. UNIV \\ vec.span (set (Gram_Schmidt V))\n[PROOF STEP]\nusing basis_orthogonal'[of V]\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (Gram_Schmidt V) = length V \\ span (set (Gram_Schmidt V)) = span (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. UNIV \\ vec.span (set (Gram_Schmidt V))\n[PROOF STEP]\nusing B\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (Gram_Schmidt V) = length V \\ span (set (Gram_Schmidt V)) = span (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\nis_basis (set V)\n\ngoal (1 subgoal):\n 1. UNIV \\ vec.span (set (Gram_Schmidt V))\n[PROOF STEP]\nunfolding is_basis_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (Gram_Schmidt V) = length V \\ span (set (Gram_Schmidt V)) = span (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\nvec.independent (set V) \\ vec.span (set V) = UNIV\n\ngoal (1 subgoal):\n 1. UNIV \\ vec.span (set (Gram_Schmidt V))\n[PROOF STEP]\nby (simp add: span_vec_eq)\n[PROOF STATE]\nproof (state)\nthis:\nUNIV \\ vec.span (set (Gram_Schmidt V))\n\ngoal (2 subgoals):\n 1. finite (set (Gram_Schmidt V))\n 2. card (set (Gram_Schmidt V)) \\ vec.dim UNIV\n[PROOF STEP]\nshow \"finite (set (Gram_Schmidt V))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (set (Gram_Schmidt V))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfinite (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. card (set (Gram_Schmidt V)) \\ vec.dim UNIV\n[PROOF STEP]\nshow \"card (set (Gram_Schmidt V)) \\ vec.dim (UNIV::(real^'b) set)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (set (Gram_Schmidt V)) \\ vec.dim UNIV\n[PROOF STEP]\nby (metis c top_greatest vec.dim_subset)\n[PROOF STATE]\nproof (state)\nthis:\ncard (set (Gram_Schmidt V)) \\ vec.dim UNIV\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nvec.independent (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. vec.independent (set (Gram_Schmidt V)) \\ set V \\ vec.span (set (Gram_Schmidt V)) \\ card (set (Gram_Schmidt V)) = vec.dim (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nset V \\ vec.span (set (Gram_Schmidt V))\npairwise orthogonal (set (Gram_Schmidt V))\ncard (set (Gram_Schmidt V)) = vec.dim (set V)\nvec.independent (set (Gram_Schmidt V))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nset V \\ vec.span (set (Gram_Schmidt V))\npairwise orthogonal (set (Gram_Schmidt V))\ncard (set (Gram_Schmidt V)) = vec.dim (set V)\nvec.independent (set (Gram_Schmidt V))\n\ngoal (1 subgoal):\n 1. vec.independent (set (Gram_Schmidt V)) \\ set V \\ vec.span (set (Gram_Schmidt V)) \\ card (set (Gram_Schmidt V)) = vec.dim (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nvec.independent (set (Gram_Schmidt V)) \\ set V \\ vec.span (set (Gram_Schmidt V)) \\ card (set (Gram_Schmidt V)) = vec.dim (set V) \\ pairwise orthogonal (set (Gram_Schmidt V))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4952, "file": "QR_Decomposition_Gram_Schmidt", "length": 47, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267762381844, "lm_q2_score": 0.8104789063814617, "lm_q1q2_score": 0.7041657754404546}} {"text": "[STATEMENT]\nlemma abs_eq_impl_unitfactor: \"\\a::int\\ = \\b\\ \\ \\ u. a = u*b \\ \\u\\=1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a\\ = \\b\\ \\ \\u. a = u * b \\ \\u\\ = 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a\\ = \\b\\ \\ \\u. a = u * b \\ \\u\\ = 1\n[PROOF STEP]\nassume \"\\a\\ = \\b\\\"\n[PROOF STATE]\nproof (state)\nthis:\n\\a\\ = \\b\\\n\ngoal (1 subgoal):\n 1. \\a\\ = \\b\\ \\ \\u. a = u * b \\ \\u\\ = 1\n[PROOF STEP]\nhence \"a = 1*b \\ a = (-1)*b\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a\\ = \\b\\\n\ngoal (1 subgoal):\n 1. a = 1 * b \\ a = - 1 * b\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\na = 1 * b \\ a = - 1 * b\n\ngoal (1 subgoal):\n 1. \\a\\ = \\b\\ \\ \\u. a = u * b \\ \\u\\ = 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na = 1 * b \\ a = - 1 * b\n[PROOF STEP]\nobtain u where \"a = u*b \\ (u=1 \\ u=-1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\na = 1 * b \\ a = - 1 * b\n\ngoal (1 subgoal):\n 1. (\\u. a = u * b \\ (u = 1 \\ u = - 1) \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\na = u * b \\ (u = 1 \\ u = - 1)\n\ngoal (1 subgoal):\n 1. \\a\\ = \\b\\ \\ \\u. a = u * b \\ \\u\\ = 1\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na = u * b \\ (u = 1 \\ u = - 1)\n\ngoal (1 subgoal):\n 1. \\u. a = u * b \\ \\u\\ = 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\u. a = u * b \\ \\u\\ = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 916, "file": "Fermat3_4_Quad_Form", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8244619436290699, "lm_q2_score": 0.853912747375134, "lm_q1q2_score": 0.7040185633905419}} {"text": "[STATEMENT]\nlemma eval_fds_add: \n assumes \"fds_converges f s\" \"fds_converges g s\"\n shows \"eval_fds (f + g) s = eval_fds f s + eval_fds g s\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eval_fds (f + g) s = eval_fds f s + eval_fds g s\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. eval_fds (f + g) s = eval_fds f s + eval_fds g s\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfds_converges f s\nfds_converges g s\n[PROOF STEP]\nhave \"(\\n. fds_nth f n / nat_power n s) sums eval_fds f s\"\n \"(\\n. fds_nth g n / nat_power n s) sums eval_fds g s\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfds_converges f s\nfds_converges g s\n\ngoal (1 subgoal):\n 1. (\\n. fds_nth f n / nat_power n s) sums eval_fds f s &&& (\\n. fds_nth g n / nat_power n s) sums eval_fds g s\n[PROOF STEP]\nby (simp_all add: fds_converges_def sums_iff eval_fds_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. fds_nth f n / nat_power n s) sums eval_fds f s\n(\\n. fds_nth g n / nat_power n s) sums eval_fds g s\n\ngoal (1 subgoal):\n 1. eval_fds (f + g) s = eval_fds f s + eval_fds g s\n[PROOF STEP]\nfrom sums_add[OF this]\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n. fds_nth f n / nat_power n s + fds_nth g n / nat_power n s) sums (eval_fds f s + eval_fds g s)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. fds_nth f n / nat_power n s + fds_nth g n / nat_power n s) sums (eval_fds f s + eval_fds g s)\n\ngoal (1 subgoal):\n 1. eval_fds (f + g) s = eval_fds f s + eval_fds g s\n[PROOF STEP]\nby (simp add: eval_fds_def sums_iff add_divide_distrib)\n[PROOF STATE]\nproof (state)\nthis:\neval_fds (f + g) s = eval_fds f s + eval_fds g s\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 816, "file": "Dirichlet_Series_Dirichlet_Series", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127641048444, "lm_q2_score": 0.8244619263765706, "lm_q1q2_score": 0.7040185624514221}} {"text": "[STATEMENT]\nlemma finite_nat_ex_max:\n assumes fin: \"finite (N::nat set)\"\n shows \"\\m. \\n\\N. n < m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m. \\n\\N. n < m\n[PROOF STEP]\nusing fin\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite N\n\ngoal (1 subgoal):\n 1. \\m. \\n\\N. n < m\n[PROOF STEP]\nproof (induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\m. \\n\\{}. n < m\n 2. \\x F. \\finite F; x \\ F; \\m. \\n\\F. n < m\\ \\ \\m. \\n\\insert x F. n < m\n[PROOF STEP]\ncase empty\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\m. \\n\\{}. n < m\n 2. \\x F. \\finite F; x \\ F; \\m. \\n\\F. n < m\\ \\ \\m. \\n\\insert x F. n < m\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m. \\n\\{}. n < m\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\m. \\n\\{}. n < m\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\m. \\n\\F. n < m\\ \\ \\m. \\n\\insert x F. n < m\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\m. \\n\\F. n < m\\ \\ \\m. \\n\\insert x F. n < m\n[PROOF STEP]\ncase (insert k N)\n[PROOF STATE]\nproof (state)\nthis:\nfinite N\nk \\ N\n\\m. \\n\\N. n < m\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\m. \\n\\F. n < m\\ \\ \\m. \\n\\insert x F. n < m\n[PROOF STEP]\nhave \"\\m. \\n\\N. n < m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m. \\n\\N. n < m\n[PROOF STEP]\nby fact\n[PROOF STATE]\nproof (state)\nthis:\n\\m. \\n\\N. n < m\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\m. \\n\\F. n < m\\ \\ \\m. \\n\\insert x F. n < m\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\m. \\n\\N. n < m\n[PROOF STEP]\nobtain m where m_max: \"\\n\\N. n < m\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\m. \\n\\N. n < m\n\ngoal (1 subgoal):\n 1. (\\m. \\n\\N. n < m \\ thesis) \\ thesis\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n\\n\\N. n < m\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; \\m. \\n\\F. n < m\\ \\ \\m. \\n\\insert x F. n < m\n[PROOF STEP]\nshow \"\\m. \\n\\insert k N. n < m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m. \\n\\insert k N. n < m\n[PROOF STEP]\nproof (rule exI [where x=\"Suc (max k m)\"])\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n\\insert k N. n < Suc (max k m)\n[PROOF STEP]\nqed (insert m_max, auto simp add: max_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\m. \\n\\insert k N. n < m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1472, "file": "Simpl_Simpl_Heap", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127566694178, "lm_q2_score": 0.8244619199068831, "lm_q1q2_score": 0.7040185507966473}} {"text": "[STATEMENT]\nlemma (in finite_borel_measure) emeasure_Ioc:\n assumes \"a \\ b\" shows \"emeasure M {a <.. b} = cdf M b - cdf M a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. emeasure M {a<..b} = ennreal (cdf M b - cdf M a)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. emeasure M {a<..b} = ennreal (cdf M b - cdf M a)\n[PROOF STEP]\nhave \"{a <.. b} = {..b} - {..a}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {a<..b} = {..b} - {..a}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{a<..b} = {..b} - {..a}\n\ngoal (1 subgoal):\n 1. emeasure M {a<..b} = ennreal (cdf M b - cdf M a)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n{a<..b} = {..b} - {..a}\n\ngoal (1 subgoal):\n 1. emeasure M {a<..b} = ennreal (cdf M b - cdf M a)\n[PROOF STEP]\nhave \"{..x} \\ sets M\" for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {..x} \\ sets M\n[PROOF STEP]\nusing atMost_borel[of x] M_is_borel\n[PROOF STATE]\nproof (prove)\nusing this:\n{..x} \\ sets borel\nsets M = sets borel\n\ngoal (1 subgoal):\n 1. {..x} \\ sets M\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{..?x} \\ sets M\n\ngoal (1 subgoal):\n 1. emeasure M {a<..b} = ennreal (cdf M b - cdf M a)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n{..?x} \\ sets M\n\ngoal (1 subgoal):\n 1. emeasure M {a<..b} = ennreal (cdf M b - cdf M a)\n[PROOF STEP]\nnote \\a \\ b\\\n[PROOF STATE]\nproof (state)\nthis:\na \\ b\n\ngoal (1 subgoal):\n 1. emeasure M {a<..b} = ennreal (cdf M b - cdf M a)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n{a<..b} = {..b} - {..a}\n{..?x} \\ sets M\na \\ b\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n{a<..b} = {..b} - {..a}\n{..?x} \\ sets M\na \\ b\n\ngoal (1 subgoal):\n 1. emeasure M {a<..b} = ennreal (cdf M b - cdf M a)\n[PROOF STEP]\nby (simp add: emeasure_eq_measure finite_measure_Diff cdf_def)\n[PROOF STATE]\nproof (state)\nthis:\nemeasure M {a<..b} = ennreal (cdf M b - cdf M a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1006, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.853912749233991, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.7040185501909697}} {"text": "[STATEMENT]\nlemma Union_atLeastAtMost_real_of_int:\n assumes \"a < b\" \n shows \"(\\n\\{a..n\\{a..x. x \\ (\\n\\{a.. x \\ {real_of_int a..real_of_int b}\n 2. \\x. x \\ {real_of_int a..real_of_int b} \\ x \\ (\\n\\{a..x. x \\ (\\n\\{a.. x \\ {real_of_int a..real_of_int b}\n 2. \\x. x \\ {real_of_int a..real_of_int b} \\ x \\ (\\n\\{a.. {real_of_int a..real_of_int b}\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ {real_of_int a..real_of_int b}\n\ngoal (2 subgoals):\n 1. \\x. x \\ (\\n\\{a.. x \\ {real_of_int a..real_of_int b}\n 2. \\x. x \\ {real_of_int a..real_of_int b} \\ x \\ (\\n\\{a.. (\\n\\{a.. {real_of_int a..real_of_int b}\n\ngoal (1 subgoal):\n 1. x \\ (\\n\\{a..x \\ {real_of_int a..real_of_int b}; x = real_of_int b\\ \\ x \\ (\\n\\{a..x \\ {real_of_int a..real_of_int b}; x \\ real_of_int b\\ \\ x \\ (\\n\\{a..x \\ {real_of_int a..real_of_int b}; x = real_of_int b\\ \\ x \\ (\\n\\{a..x \\ {real_of_int a..real_of_int b}; x \\ real_of_int b\\ \\ x \\ (\\n\\{a.. (\\n\\{a.. (\\n\\{a..x \\ {real_of_int a..real_of_int b}; x \\ real_of_int b\\ \\ x \\ (\\n\\{a..x \\ {real_of_int a..real_of_int b}; x \\ real_of_int b\\ \\ x \\ (\\n\\{a.. real_of_int b\n\ngoal (1 subgoal):\n 1. \\x \\ {real_of_int a..real_of_int b}; x \\ real_of_int b\\ \\ x \\ (\\n\\{a.. {real_of_int a..real_of_int b}\nx \\ real_of_int b\n[PROOF STEP]\nhave x: \"x \\ real_of_int a\" \"x < real_of_int b\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ {real_of_int a..real_of_int b}\nx \\ real_of_int b\n\ngoal (1 subgoal):\n 1. real_of_int a \\ x &&& x < real_of_int b\n[PROOF STEP]\nby simp_all\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int a \\ x\nx < real_of_int b\n\ngoal (1 subgoal):\n 1. \\x \\ {real_of_int a..real_of_int b}; x \\ real_of_int b\\ \\ x \\ (\\n\\{a.. of_int \\x\\\" \"x \\ of_int \\x\\ + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal_of_int a \\ x\nx < real_of_int b\n\ngoal (1 subgoal):\n 1. real_of_int \\x\\ \\ x &&& x \\ real_of_int \\x\\ + 1\n[PROOF STEP]\nby linarith+\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int \\x\\ \\ x\nx \\ real_of_int \\x\\ + 1\n\ngoal (1 subgoal):\n 1. \\x \\ {real_of_int a..real_of_int b}; x \\ real_of_int b\\ \\ x \\ (\\n\\{a..x\\ \\ x\nx \\ real_of_int \\x\\ + 1\n\ngoal (1 subgoal):\n 1. \\x \\ {real_of_int a..real_of_int b}; x \\ real_of_int b\\ \\ x \\ (\\n\\{a.. x\nx < real_of_int b\n[PROOF STEP]\nhave \"\\x\\ \\ a\" \"\\x\\ < b\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal_of_int a \\ x\nx < real_of_int b\n\ngoal (1 subgoal):\n 1. a \\ \\x\\ &&& \\x\\ < b\n[PROOF STEP]\nby linarith+\n[PROOF STATE]\nproof (state)\nthis:\na \\ \\x\\\n\\x\\ < b\n\ngoal (1 subgoal):\n 1. \\x \\ {real_of_int a..real_of_int b}; x \\ real_of_int b\\ \\ x \\ (\\n\\{a..x\\ \\ x\nx \\ real_of_int \\x\\ + 1\na \\ \\x\\\n\\x\\ < b\n[PROOF STEP]\nhave \"\\n\\{a.. {of_int n..of_int (n + 1)}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal_of_int \\x\\ \\ x\nx \\ real_of_int \\x\\ + 1\na \\ \\x\\\n\\x\\ < b\n\ngoal (1 subgoal):\n 1. \\n\\{a.. {real_of_int n..real_of_int (n + 1)}\n[PROOF STEP]\nby (intro bexI[of _ \"\\x\\\"]) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n\\n\\{a.. {real_of_int n..real_of_int (n + 1)}\n\ngoal (1 subgoal):\n 1. \\x \\ {real_of_int a..real_of_int b}; x \\ real_of_int b\\ \\ x \\ (\\n\\{a..n\\{a.. {real_of_int n..real_of_int (n + 1)}\n\ngoal (1 subgoal):\n 1. x \\ (\\n\\{a.. (\\n\\{a.. (\\n\\{a..x. x \\ (\\n\\{a.. x \\ {real_of_int a..real_of_int b}\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 3618, "file": "Euler_MacLaurin_Euler_MacLaurin", "length": 27, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127380808499, "lm_q2_score": 0.824461932846258, "lm_q1q2_score": 0.7040185465201779}} {"text": "[STATEMENT]\nlemma (in Module) mHom_tOp_commute:\"\\R module N; f \\ mHom R M N; \n g \\ mHom R M N\\ \\ tOp_mHom R M N f g = tOp_mHom R M N g f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\R module N; f \\ mHom R M N; g \\ mHom R M N\\ \\ tOp_mHom R M N f g = tOp_mHom R M N g f\n[PROOF STEP]\napply (frule_tac f = f and g = g in tOp_mHom_closed [of N], assumption+,\n frule_tac f = g and g = f in tOp_mHom_closed [of N], assumption+)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\R module N; f \\ mHom R M N; g \\ mHom R M N; tOp_mHom R M N f g \\ mHom R M N; tOp_mHom R M N g f \\ mHom R M N\\ \\ tOp_mHom R M N f g = tOp_mHom R M N g f\n[PROOF STEP]\napply (rule mHom_eq, assumption+)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\R module N; f \\ mHom R M N; g \\ mHom R M N; tOp_mHom R M N f g \\ mHom R M N; tOp_mHom R M N g f \\ mHom R M N\\ \\ \\m\\carrier M. tOp_mHom R M N f g m = tOp_mHom R M N g f m\n[PROOF STEP]\napply (rule ballI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m. \\R module N; f \\ mHom R M N; g \\ mHom R M N; tOp_mHom R M N f g \\ mHom R M N; tOp_mHom R M N g f \\ mHom R M N; m \\ carrier M\\ \\ tOp_mHom R M N f g m = tOp_mHom R M N g f m\n[PROOF STEP]\napply (thin_tac \"tOp_mHom R M N f g \\ mHom R M N\",\n thin_tac \"tOp_mHom R M N g f \\ mHom R M N\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m. \\R module N; f \\ mHom R M N; g \\ mHom R M N; m \\ carrier M\\ \\ tOp_mHom R M N f g m = tOp_mHom R M N g f m\n[PROOF STEP]\napply (simp add:tOp_mHom_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m. \\R module N; f \\ mHom R M N; g \\ mHom R M N; m \\ carrier M\\ \\ f m \\\\<^bsub>N\\<^esub> g m = g m \\\\<^bsub>N\\<^esub> f m\n[PROOF STEP]\napply (frule_tac f = f and m = m in mHom_mem [of N], assumption+,\n frule_tac f = g and m = m in mHom_mem [of N], assumption+,\n frule Module.module_is_ag [of N])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m. \\R module N; f \\ mHom R M N; g \\ mHom R M N; m \\ carrier M; f m \\ carrier N; g m \\ carrier N; aGroup N\\ \\ f m \\\\<^bsub>N\\<^esub> g m = g m \\\\<^bsub>N\\<^esub> f m\n[PROOF STEP]\napply (simp add:aGroup.ag_pOp_commute)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1187, "file": "Group-Ring-Module_Algebra7", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127417985636, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.7040185440607429}} {"text": "[STATEMENT]\nlemma convex_scaled:\n assumes \"convex S\"\n shows \"convex ((\\x. x *\\<^sub>R c) ` S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex ((\\x. x *\\<^sub>R c) ` S)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. convex ((\\x. x *\\<^sub>R c) ` S)\n[PROOF STEP]\nhave \"linear (\\x. x *\\<^sub>R c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. linear (\\x. x *\\<^sub>R c)\n[PROOF STEP]\nby (simp add: linearI scaleR_add_left)\n[PROOF STATE]\nproof (state)\nthis:\nlinear (\\x. x *\\<^sub>R c)\n\ngoal (1 subgoal):\n 1. convex ((\\x. x *\\<^sub>R c) ` S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlinear (\\x. x *\\<^sub>R c)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlinear (\\x. x *\\<^sub>R c)\n\ngoal (1 subgoal):\n 1. convex ((\\x. x *\\<^sub>R c) ` S)\n[PROOF STEP]\nusing \\convex S\\\n[PROOF STATE]\nproof (prove)\nusing this:\nlinear (\\x. x *\\<^sub>R c)\nconvex S\n\ngoal (1 subgoal):\n 1. convex ((\\x. x *\\<^sub>R c) ` S)\n[PROOF STEP]\nby (rule convex_linear_image)\n[PROOF STATE]\nproof (state)\nthis:\nconvex ((\\x. x *\\<^sub>R c) ` S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 564, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467738423873, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.7039257904840602}} {"text": "[STATEMENT]\nlemma row_space_eq:\n assumes A: \"A \\ carrier_mat nr n\"\n shows \"row_space A = {w\\carrier_vec (dim_col A). \\y\\carrier_vec (dim_row A). A\\<^sup>T *\\<^sub>v y = w}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row_space A = {w \\ carrier_vec (dim_col A). \\y\\carrier_vec (dim_row A). A\\<^sup>T *\\<^sub>v y = w}\n[PROOF STEP]\nusing A col_space_eq\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat nr n\n?A \\ carrier_mat n ?nc \\ col_space ?A = {y \\ carrier_vec (dim_row ?A). \\x\\carrier_vec (dim_col ?A). ?A *\\<^sub>v x = y}\n\ngoal (1 subgoal):\n 1. row_space A = {w \\ carrier_vec (dim_col A). \\y\\carrier_vec (dim_row A). A\\<^sup>T *\\<^sub>v y = w}\n[PROOF STEP]\nunfolding row_space_eq_col_space_transpose\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat nr n\n?A \\ carrier_mat n ?nc \\ col_space ?A = {y \\ carrier_vec (dim_row ?A). \\x\\carrier_vec (dim_col ?A). ?A *\\<^sub>v x = y}\n\ngoal (1 subgoal):\n 1. col_space A\\<^sup>T = {w \\ carrier_vec (dim_col A). \\y\\carrier_vec (dim_row A). A\\<^sup>T *\\<^sub>v y = w}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 530, "file": "Jordan_Normal_Form_VS_Connect", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467580102419, "lm_q2_score": 0.800692004473946, "lm_q1q2_score": 0.7039257798979917}} {"text": "[STATEMENT]\nlemma poly_split_eval: assumes \"poly_split m p = (c,q)\" \n shows \"eval_poly \\ p = (eval_monom \\ m * c) + eval_poly \\ q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eval_poly \\ p = eval_monom \\ m * c + eval_poly \\ q\n[PROOF STEP]\nusing poly_split[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\np =p (m, c) # q\n\ngoal (1 subgoal):\n 1. eval_poly \\ p = eval_monom \\ m * c + eval_poly \\ q\n[PROOF STEP]\nunfolding eq_poly_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\. eval_poly \\ p = eval_poly \\ ((m, c) # q)\n\ngoal (1 subgoal):\n 1. eval_poly \\ p = eval_monom \\ m * c + eval_poly \\ q\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 304, "file": "Polynomials_Polynomials", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467611766711, "lm_q2_score": 0.8006919997179627, "lm_q1q2_score": 0.703925778252119}} {"text": "[STATEMENT]\nlemma measurable_less_count_space [measurable (raw)]:\n assumes \"countable A\"\n assumes [measurable]: \"a \\ B \\\\<^sub>M count_space A\"\n assumes [measurable]: \"b \\ B \\\\<^sub>M count_space A\"\n shows \"Measurable.pred B (\\x. a x < b x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Measurable.pred B (\\x. a x < b x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Measurable.pred B (\\x. a x < b x)\n[PROOF STEP]\nhave \"Measurable.pred (count_space (A \\ A)) (\\x. fst x < snd x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Measurable.pred (count_space (A \\ A)) (\\x. fst x < snd x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nMeasurable.pred (count_space (A \\ A)) (\\x. fst x < snd x)\n\ngoal (1 subgoal):\n 1. Measurable.pred B (\\x. a x < b x)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nMeasurable.pred (count_space (A \\ A)) (\\x. fst x < snd x)\n\ngoal (1 subgoal):\n 1. Measurable.pred B (\\x. a x < b x)\n[PROOF STEP]\nhave \"count_space (A \\ A) = count_space A \\\\<^sub>M count_space A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. count_space (A \\ A) = count_space A \\\\<^sub>M count_space A\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\ncountable A\n\ngoal (1 subgoal):\n 1. count_space (A \\ A) = count_space A \\\\<^sub>M count_space A\n[PROOF STEP]\nby (simp add: pair_measure_countable)\n[PROOF STATE]\nproof (state)\nthis:\ncount_space (A \\ A) = count_space A \\\\<^sub>M count_space A\n\ngoal (1 subgoal):\n 1. Measurable.pred B (\\x. a x < b x)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nMeasurable.pred (count_space A \\\\<^sub>M count_space A) (\\x. fst x < snd x)\n[PROOF STEP]\nhave \"Measurable.pred B ((\\x. fst x < snd x) \\ (\\x. (a x, b x)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nMeasurable.pred (count_space A \\\\<^sub>M count_space A) (\\x. fst x < snd x)\n\ngoal (1 subgoal):\n 1. Measurable.pred B ((\\x. fst x < snd x) \\ (\\x. (a x, b x)))\n[PROOF STEP]\nby measurable\n[PROOF STATE]\nproof (state)\nthis:\nMeasurable.pred B ((\\x. fst x < snd x) \\ (\\x. (a x, b x)))\n\ngoal (1 subgoal):\n 1. Measurable.pred B (\\x. a x < b x)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nMeasurable.pred B ((\\x. fst x < snd x) \\ (\\x. (a x, b x)))\n\ngoal (1 subgoal):\n 1. Measurable.pred B (\\x. a x < b x)\n[PROOF STEP]\nby (simp add: o_def)\n[PROOF STATE]\nproof (state)\nthis:\nMeasurable.pred B (\\x. a x < b x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1184, "file": "Treaps_Random_Treap", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467611766711, "lm_q2_score": 0.800691997339971, "lm_q1q2_score": 0.7039257761615153}} {"text": "[STATEMENT]\nlemma vector_scalar_commute:\n fixes A :: \"'a::{field}^'m^'n\"\n shows \"A *v (c *s x) = c *s (A *v x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *v (c *s x) = c *s (A *v x)\n[PROOF STEP]\nby (simp add: vector_scalar_mult_def matrix_vector_mult_def mult_ac sum_distrib_left)", "meta": {"llama_tokens": 136, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467548438126, "lm_q2_score": 0.8006919997179627, "lm_q1q2_score": 0.7039257731814498}} {"text": "[STATEMENT]\nlemma ex2_6: \"\\integral {0 .. 1} (\\x::real. arctan (sqrt (x\\<^sup>2 + 2)) / (sqrt (x\\<^sup>2 + 2) * (x\\<^sup>2 + 1))) - 5*pi\\<^sup>2/96\\ \\ 1 / 10^6\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\integral {0..1} (\\x. arctan (sqrt (x\\<^sup>2 + 2)) / (sqrt (x\\<^sup>2 + 2) * (x\\<^sup>2 + 1))) - 5 * pi\\<^sup>2 / 96\\ \\ 1 / 10 ^ 6\n[PROOF STEP]\nby (rule abs_minus_leI) (tactic \\integral_bnds_tac 1 40 10 6 26 \"\" @{thms pow_gt_zero} @{context} 1\\)", "meta": {"llama_tokens": 267, "file": "Ordinary_Differential_Equations_Ex_Examples_Integral", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9324533126145178, "lm_q2_score": 0.7549149813536518, "lm_q1q2_score": 0.7039229751055395}} {"text": "[STATEMENT]\nlemma f_geom_series:\n assumes \"b = 2^(Suc c)\"\n shows \"(F q c b = f) \\ ( (b-1) * f = 2^c * (b^(Suc q) - 1) )\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (F q c b = f) = ((b - 1) * f = 2 ^ c * (b ^ Suc q - 1))\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (F q c b = f) = ((b - 1) * f = 2 ^ c * (b ^ Suc q - 1))\n[PROOF STEP]\nhave \"F q c b = 2^c * E q b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. F q c b = 2 ^ c * E q b\n[PROOF STEP]\nby (auto simp: E_def F_def sum_distrib_left sum_distrib_right)\n[PROOF STATE]\nproof (state)\nthis:\nF q c b = 2 ^ c * E q b\n\ngoal (1 subgoal):\n 1. (F q c b = f) = ((b - 1) * f = 2 ^ c * (b ^ Suc q - 1))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nF q c b = 2 ^ c * E q b\n\ngoal (1 subgoal):\n 1. (F q c b = f) = ((b - 1) * f = 2 ^ c * (b ^ Suc q - 1))\n[PROOF STEP]\nhave \"b \\ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 \\ b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nb = 2 ^ Suc c\n\ngoal (1 subgoal):\n 1. 2 \\ b\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ b\n\ngoal (1 subgoal):\n 1. (F q c b = f) = ((b - 1) * f = 2 ^ c * (b ^ Suc q - 1))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nF q c b = 2 ^ c * E q b\n2 \\ b\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nF q c b = 2 ^ c * E q b\n2 \\ b\n\ngoal (1 subgoal):\n 1. (F q c b = f) = ((b - 1) * f = 2 ^ c * (b ^ Suc q - 1))\n[PROOF STEP]\nby (smt e_geom_series mult.left_commute mult_cancel_left)\n[PROOF STATE]\nproof (state)\nthis:\n(F q c b = f) = ((b - 1) * f = 2 ^ c * (b ^ Suc q - 1))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 873, "file": "DPRM_Theorem_Register_Machine_MachineMasking", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916170039421, "lm_q2_score": 0.8152324915965392, "lm_q1q2_score": 0.7038648991536886}} {"text": "[STATEMENT]\nlemma hlog_mult:\n \"\\a x y. 0 < a \\ a \\ 1 \\ 0 < x \\ 0 < y \\ hlog a (x * y) = hlog a x + hlog a y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a x y. \\0 < a; a \\ 1; 0 < x; 0 < y\\ \\ hlog a (x * y) = hlog a x + hlog a y\n[PROOF STEP]\nby transfer (rule log_mult)", "meta": {"llama_tokens": 170, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391617003942, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.7038648952778558}} {"text": "[STATEMENT]\ntheorem ln_lower_13_eq: \"0\n ln_lower_13 x = (1/70)*(10 + 1921*x + 20149*x^2 + 50774*x^3 + 38524*x^4 + 8389*x^5 + 353*x^6)*(x - 1) /\n (x*(7 + 126*x + 525*x^2 + 700*x^3 + 315*x^4 + 42*x^5 + x^6))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ ln_lower_13 x = 1 / 70 * (10 + 1921 * x + 20149 * x\\<^sup>2 + 50774 * x ^ 3 + 38524 * x ^ 4 + 8389 * x ^ 5 + 353 * x ^ 6) * (x - 1) / (x * (7 + 126 * x + 525 * x\\<^sup>2 + 700 * x ^ 3 + 315 * x ^ 4 + 42 * x ^ 5 + x ^ 6))\n[PROOF STEP]\nunfolding ln_lower_13_def ln_upper_13_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ - ((353 + 8389 * inverse x + 20149 * inverse x ^ 4 + 50774 * inverse x ^ 3 + 38524 * (inverse x)\\<^sup>2 + 1921 * inverse x ^ 5 + 10 * inverse x ^ 6) * (inverse x - 1) / (70 * (1 + 42 * inverse x + 525 * inverse x ^ 4 + 700 * inverse x ^ 3 + 315 * (inverse x)\\<^sup>2 + 126 * inverse x ^ 5 + 7 * inverse x ^ 6))) = 1 / 70 * (10 + 1921 * x + 20149 * x\\<^sup>2 + 50774 * x ^ 3 + 38524 * x ^ 4 + 8389 * x ^ 5 + 353 * x ^ 6) * (x - 1) / (x * (7 + 126 * x + 525 * x\\<^sup>2 + 700 * x ^ 3 + 315 * x ^ 4 + 42 * x ^ 5 + x ^ 6))\n[PROOF STEP]\nby (simp add: zero_less_mult_iff add_pos_pos dual_order.strict_implies_not_eq divide_simps)\n algebra", "meta": {"llama_tokens": 763, "file": "Special_Function_Bounds_Log_CF_Bounds", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.863391617003942, "lm_q2_score": 0.8152324848629215, "lm_q1q2_score": 0.7038648933399395}} {"text": "[STATEMENT]\nlemma scaleC_le_0_iff:\n \"a *\\<^sub>C b \\ 0 \\ 0 < a \\ b \\ 0 \\ a < 0 \\ 0 \\ b \\ a = 0\"\n if \"a \\ \\\" (* Not present in Real_Vector_Spaces *)\n for b::\"'a::ordered_complex_vector\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a *\\<^sub>C b \\ (0::'a)) = (0 < a \\ b \\ (0::'a) \\ a < 0 \\ (0::'a) \\ b \\ a = 0)\n[PROOF STEP]\napply (insert zero_le_scaleC_iff [of \"-a\" b])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (- a \\ \\ \\ ((0::'a) \\ - a *\\<^sub>C b) = (0 < - a \\ (0::'a) \\ b \\ - a < 0 \\ b \\ (0::'a) \\ - a = 0)) \\ (a *\\<^sub>C b \\ (0::'a)) = (0 < a \\ b \\ (0::'a) \\ a < 0 \\ (0::'a) \\ b \\ a = 0)\n[PROOF STEP]\nusing less_complex_def that\n[PROOF STATE]\nproof (prove)\nusing this:\n(?x < ?y) = (Re ?x < Re ?y \\ Im ?x = Im ?y)\na \\ \\\n\ngoal (1 subgoal):\n 1. (- a \\ \\ \\ ((0::'a) \\ - a *\\<^sub>C b) = (0 < - a \\ (0::'a) \\ b \\ - a < 0 \\ b \\ (0::'a) \\ - a = 0)) \\ (a *\\<^sub>C b \\ (0::'a)) = (0 < a \\ b \\ (0::'a) \\ a < 0 \\ (0::'a) \\ b \\ a = 0)\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 626, "file": "Complex_Bounded_Operators_Complex_Vector_Spaces0", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391595913457, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7038648838979562}} {"text": "[STATEMENT]\nlemma Im_abs[simp]: \"Im (abs x) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Im \\x\\ = 0\n[PROOF STEP]\nusing abs_complex_real complex_is_Real_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x\\ \\ \\\n(?z \\ \\) = (Im ?z = 0)\n\ngoal (1 subgoal):\n 1. Im \\x\\ = 0\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 164, "file": "Complex_Bounded_Operators_extra_Extra_General", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8633916029436189, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.7038648838154237}} {"text": "[STATEMENT]\nlemma mat_mult_assoc_n:\n assumes wf1: \"mat n n m1\"\n and wf2: \"mat n n m2\"\n and wf3: \"mat n n m3\"\n shows \"mat_mult n (mat_mult n m1 m2) m3 = mat_mult n m1 (mat_mult n m2 m3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat_mult n (mat_mult n m1 m2) m3 = mat_mult n m1 (mat_mult n m2 m3)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nmat n n m1\nmat n n m2\nmat n n m3\n\ngoal (1 subgoal):\n 1. mat_mult n (mat_mult n m1 m2) m3 = mat_mult n m1 (mat_mult n m2 m3)\n[PROOF STEP]\nby (rule mat_mult_assoc)", "meta": {"llama_tokens": 260, "file": "Matrix_Matrix_Legacy", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513814471134, "lm_q2_score": 0.7853085859124002, "lm_q1q2_score": 0.7038339049862677}} {"text": "[STATEMENT]\ntheorem (in group) action_by_conjugation_on_subgroups_set:\n \"group_action G {H. subgroup H G} (\\g. \\H \\ {H. subgroup H G}. g <# H #> (inv g))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. group_action G {H. subgroup H G} (\\g. \\H\\{H. subgroup H G}. g <# H #> inv g)\n[PROOF STEP]\nunfolding group_action_def group_hom_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Group.group G \\ Group.group (BijGroup {H. subgroup H G}) \\ group_hom_axioms G (BijGroup {H. subgroup H G}) (\\g. \\H\\{H. subgroup H G}. g <# H #> inv g)\n[PROOF STEP]\nusing subgroup_conjugation_is_hom\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\g. \\H\\{H. subgroup H G}. g <# H #> inv g) \\ hom G (BijGroup {H. subgroup H G})\n\ngoal (1 subgoal):\n 1. Group.group G \\ Group.group (BijGroup {H. subgroup H G}) \\ group_hom_axioms G (BijGroup {H. subgroup H G}) (\\g. \\H\\{H. subgroup H G}. g <# H #> inv g)\n[PROOF STEP]\nby (simp add: group_BijGroup group_hom_axioms.intro is_group)", "meta": {"llama_tokens": 436, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513620489619, "lm_q2_score": 0.7853085909370422, "lm_q1q2_score": 0.7038338942560751}} {"text": "[STATEMENT]\nlemma affinity_inverses:\n assumes m0: \"m \\ (0::'a::field)\"\n shows \"(\\x. m *s x + c) \\ (\\x. inverse(m) *s x + (-(inverse(m) *s c))) = id\"\n \"(\\x. inverse(m) *s x + (-(inverse(m) *s c))) \\ (\\x. m *s x + c) = id\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. m *s x + c) \\ (\\x. inverse m *s x + - (inverse m *s c)) = id &&& (\\x. inverse m *s x + - (inverse m *s c)) \\ (\\x. m *s x + c) = id\n[PROOF STEP]\nusing m0\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ (0::'a)\n\ngoal (1 subgoal):\n 1. (\\x. m *s x + c) \\ (\\x. inverse m *s x + - (inverse m *s c)) = id &&& (\\x. inverse m *s x + - (inverse m *s c)) \\ (\\x. m *s x + c) = id\n[PROOF STEP]\nby (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)", "meta": {"llama_tokens": 402, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513703624558, "lm_q2_score": 0.785308580887758, "lm_q1q2_score": 0.7038338917780486}} {"text": "[STATEMENT]\ntheorem (in group) action_by_conjugation_on_power_set:\n \"group_action G {H. H \\ carrier G} (\\g. \\H \\ {H. H \\ carrier G}. g <# H #> (inv g))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. group_action G {H. H \\ carrier G} (\\g. \\H\\{H. H \\ carrier G}. g <# H #> inv g)\n[PROOF STEP]\nunfolding group_action_def group_hom_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Group.group G \\ Group.group (BijGroup {H. H \\ carrier G}) \\ group_hom_axioms G (BijGroup {H. H \\ carrier G}) (\\g. \\H\\{H. H \\ carrier G}. g <# H #> inv g)\n[PROOF STEP]\nusing subset_conjugation_is_hom\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\g. \\H\\{H. H \\ carrier G}. g <# H #> inv g) \\ hom G (BijGroup {H. H \\ carrier G})\n\ngoal (1 subgoal):\n 1. Group.group G \\ Group.group (BijGroup {H. H \\ carrier G}) \\ group_hom_axioms G (BijGroup {H. H \\ carrier G}) (\\g. \\H\\{H. H \\ carrier G}. g <# H #> inv g)\n[PROOF STEP]\nby (simp add: group_BijGroup group_hom_axioms.intro is_group)", "meta": {"llama_tokens": 483, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.7905303285397348, "lm_q1q2_score": 0.7038045870732768}} {"text": "[STATEMENT]\nlemma n_inter_num_choose: \"finite b1 \\ finite b2 \\ b1 |\\|\\<^sub>n b2 = (card (b1 \\ b2) choose n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite b1; finite b2\\ \\ (b1 |\\|\\<^sub>n b2) = card (b1 \\ b2) choose n\n[PROOF STEP]\nusing n_subsets n_intersect_num_subset_def\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite ?A \\ card {B. B \\ ?A \\ card B = ?k} = card ?A choose ?k\n(?b1.0 |\\|\\<^sub>?n ?b2.0) = card {x. x \\ ?b1.0 \\ ?b2.0 \\ card x = ?n}\n\ngoal (1 subgoal):\n 1. \\finite b1; finite b2\\ \\ (b1 |\\|\\<^sub>n b2) = card (b1 \\ b2) choose n\n[PROOF STEP]\nby (metis (full_types) finite_Int)", "meta": {"llama_tokens": 340, "file": "Design_Theory_Design_Basics", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.7905303211371898, "lm_q1q2_score": 0.7038045758806739}} {"text": "[STATEMENT]\nlemma card_bijections_range_permutation_eq_1:\n assumes \"finite A\" \"finite B\"\n assumes \"card A = card B\"\n shows \"card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // range_permutation A B) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // range_permutation A B) = 1\n[PROOF STEP]\nusing bij_betw_quotient_range_permutation_eq[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n{f \\ A \\\\<^sub>E B. bij_betw f A B} // range_permutation A B = {{f \\ A \\\\<^sub>E B. bij_betw f A B}}\n\ngoal (1 subgoal):\n 1. card ({f \\ A \\\\<^sub>E B. bij_betw f A B} // range_permutation A B) = 1\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 301, "file": "Twelvefold_Way_Card_Bijections_Direct", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7038045738923904}} {"text": "[STATEMENT]\nlemma not_empty_means_seq:\"\\i \\ {S . (\\ (m::nat) n . {m..n }=S) } . i \\ {}\n \\ ( \\m n . m \\ n \\ {m..n} = i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i\\{S. \\m n. {m..n} = S}. i \\ {} \\ (\\m n. m \\ n \\ {m..n} = i)\n[PROOF STEP]\nusing atLeastatMost_empty_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n({?a..?b} = {}) = (\\ ?a \\ ?b)\n\ngoal (1 subgoal):\n 1. \\i\\{S. \\m n. {m..n} = S}. i \\ {} \\ (\\m n. m \\ n \\ {m..n} = i)\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 296, "file": "Hybrid_Multi_Lane_Spatial_Logic_NatInt", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.7905303112671295, "lm_q1q2_score": 0.7038045670934162}} {"text": "[STATEMENT]\nlemma set_quicksort: \"set (quicksort xs) = set xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (quicksort xs) = set xs\n[PROOF STEP]\nby(rule mset_eq_setD[OF mset_quicksort])", "meta": {"llama_tokens": 82, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213718636754, "lm_q2_score": 0.782662489091802, "lm_q1q2_score": 0.7037085708984598}} {"text": "[STATEMENT]\nlemma list_rel_append2: \"(l,as @ bs) \\ \\R\\list_rel \n \\ (\\cs ds. l = cs@ds \\ (cs,as)\\\\R\\list_rel \\ (ds,bs)\\\\R\\list_rel)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((l, as @ bs) \\ \\R\\list_rel) = (\\cs ds. l = cs @ ds \\ (cs, as) \\ \\R\\list_rel \\ (ds, bs) \\ \\R\\list_rel)\n[PROOF STEP]\napply (simp add: list_rel_def list_all2_append2)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\us vs. l = us @ vs \\ length us = length as \\ length vs = length bs \\ list_all2 (\\x x'. (x, x') \\ R) us as \\ list_all2 (\\x x'. (x, x') \\ R) vs bs) = (\\cs ds. l = cs @ ds \\ list_all2 (\\x x'. (x, x') \\ R) cs as \\ list_all2 (\\x x'. (x, x') \\ R) ds bs)\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\cs ds. \\l = cs @ ds; list_all2 (\\x x'. (x, x') \\ R) cs as; list_all2 (\\x x'. (x, x') \\ R) ds bs\\ \\ \\us vs. cs @ ds = us @ vs \\ length us = length as \\ length vs = length bs \\ list_all2 (\\x x'. (x, x') \\ R) us as \\ list_all2 (\\x x'. (x, x') \\ R) vs bs\n[PROOF STEP]\napply (metis list_all2_lengthD)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 622, "file": "Automatic_Refinement_Parametricity_Param_HOL", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8558511469672594, "lm_q2_score": 0.8221891261650247, "lm_q1q2_score": 0.7036715066523451}} {"text": "[STATEMENT]\nlemma sum_list_two_pow_aux:\n \"(\\x\\[0..< n]. (2::nat) ^ (n - x)) + 2 ^ (0 - 1) + 2 ^ 0 = 2 ^ (Suc n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\[0..x\\[0..<0]. 2 ^ (0 - x)) + 2 ^ (0 - 1) + 2 ^ 0 = 2 ^ Suc 0\n 2. \\n. (\\x\\[0.. (\\x\\[0..x\\[0..<0]. 2 ^ (0 - x)) + 2 ^ (0 - 1) + 2 ^ 0 = 2 ^ Suc 0\n 2. \\n. (\\x\\[0.. (\\x\\[0..x\\[0..<0]. 2 ^ (0 - x)) + 2 ^ (0 - 1) + 2 ^ 0 = 2 ^ Suc 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\[0..<0]. 2 ^ (0 - x)) + 2 ^ (0 - 1) + 2 ^ 0 = 2 ^ Suc 0\n\ngoal (1 subgoal):\n 1. \\n. (\\x\\[0.. (\\x\\[0..n. (\\x\\[0.. (\\x\\[0..x\\[0..n. (\\x\\[0.. (\\x\\[0.. n \\ 2 ^ (Suc n - x) = 2 * 2^ (n - x)\" for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ n \\ (2::'a) ^ (Suc n - x) = (2::'a) * (2::'a) ^ (n - x)\n[PROOF STEP]\nby (simp add: Suc_diff_le)\n[PROOF STATE]\nproof (state)\nthis:\n?x1 \\ n \\ (2::?'a2) ^ (Suc n - ?x1) = (2::?'a2) * (2::?'a2) ^ (n - ?x1)\n\ngoal (1 subgoal):\n 1. \\n. (\\x\\[0.. (\\x\\[0.. n \\ (2::?'a2) ^ (Suc n - ?x1) = (2::?'a2) * (2::?'a2) ^ (n - ?x1)\n\ngoal (1 subgoal):\n 1. \\n. (\\x\\[0.. (\\x\\[0.. set [0.. x \\ n\" for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ set [0.. x \\ n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n?x1 \\ set [0.. ?x1 \\ n\n\ngoal (1 subgoal):\n 1. \\n. (\\x\\[0.. (\\x\\[0.. n \\ (2::?'a2) ^ (Suc n - ?x1) = (2::?'a2) * (2::?'a2) ^ (n - ?x1)\n?x1 \\ set [0.. ?x1 \\ n\n[PROOF STEP]\nhave \"(\\x\\[0..x\\[0.. n \\ (2::?'a2) ^ (Suc n - ?x1) = (2::?'a2) * (2::?'a2) ^ (n - ?x1)\n?x1 \\ set [0.. ?x1 \\ n\n\ngoal (1 subgoal):\n 1. (\\x\\[0..x\\[0..x\\[0..x\\[0..n. (\\x\\[0.. (\\x\\[0..x\\[0..x\\[0..n. (\\x\\[0.. (\\x\\[0..x\\[0..< n]. 2* 2 ^ (n - x)) + 2* 2 ^ (0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..n. (\\x\\[0.. (\\x\\[0..x\\[0..x\\[0..n. (\\x\\[0.. (\\x\\[0..x\\[0..< n]. (2::nat)* (2::nat) ^ (n - x)) = 2*(\\x\\[0..< n]. 2 ^ (n - x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\[0..x\\[0..x\\?xs. ?c * ?f x) = ?c * sum_list (map ?f ?xs)\n\ngoal (1 subgoal):\n 1. (\\x\\[0..x\\[0..x\\[0..x\\[0..n. (\\x\\[0.. (\\x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..< n]. 2 ^ (n - x)) + 2* 2 ^ (0)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..n. (\\x\\[0.. (\\x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0..x\\[0.. w) = (- v) \\ w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (v \\ w) = - v \\ w\n[PROOF STEP]\nunfolding scalar_prod_def uminus_vec_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (\\i = 0..i = 0..i. - v $ i) $ i * w $ i)\n[PROOF STEP]\napply (subst sum_negf[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x = 0..i = 0..i. - v $ i) $ i * w $ i)\n[PROOF STEP]\nproof (rule sum.cong[OF refl])\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ {0.. - (v $ x * w $ x) = vec (dim_vec v) (\\i. - v $ i) $ x * w $ x\n[PROOF STEP]\nfix i\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x \\ {0.. - (v $ x * w $ x) = vec (dim_vec v) (\\i. - v $ i) $ x * w $ x\n[PROOF STEP]\nassume i: \"i : {0 .. {0..x. x \\ {0.. - (v $ x * w $ x) = vec (dim_vec v) (\\i. - v $ i) $ x * w $ x\n[PROOF STEP]\nhave [simp]: \"dim_vec v = n\" \"dim_vec w = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_vec v = n &&& dim_vec w = n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec v = n\ndim_vec w = n\n\ngoal (1 subgoal):\n 1. \\x. x \\ {0.. - (v $ x * w $ x) = vec (dim_vec v) (\\i. - v $ i) $ x * w $ x\n[PROOF STEP]\nshow \"- (v $ i * w $ i) = vec (dim_vec v) (\\i. - v $ i) $ i * w $ i\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (v $ i * w $ i) = vec (dim_vec v) (\\i. - v $ i) $ i * w $ i\n[PROOF STEP]\nunfolding minus_mult_left\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - v $ i * w $ i = vec (dim_vec v) (\\i. - v $ i) $ i * w $ i\n[PROOF STEP]\nusing i\n[PROOF STATE]\nproof (prove)\nusing this:\ni \\ {0..i. - v $ i) $ i * w $ i\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n- (v $ i * w $ i) = vec (dim_vec v) (\\i. - v $ i) $ i * w $ i\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1146, "file": "Jordan_Normal_Form_Matrix", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511396138365, "lm_q2_score": 0.8221891239865619, "lm_q1q2_score": 0.7036714987420009}} {"text": "[STATEMENT]\nlemma affine_basis_exists:\n fixes V :: \"'n::real_vector set\"\n shows \"\\B. B \\ V \\ \\ affine_dependent B \\ affine hull V = affine hull B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n[PROOF STEP]\nproof (cases \"V = {}\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. V = {} \\ \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n 2. V \\ {} \\ \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nV = {}\n\ngoal (2 subgoals):\n 1. V = {} \\ \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n 2. V \\ {} \\ \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nV = {}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nV = {}\n\ngoal (1 subgoal):\n 1. \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n[PROOF STEP]\nusing affine_independent_0\n[PROOF STATE]\nproof (prove)\nusing this:\nV = {}\n\\ affine_dependent {}\n\ngoal (1 subgoal):\n 1. \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n\ngoal (1 subgoal):\n 1. V \\ {} \\ \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. V \\ {} \\ \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nV \\ {}\n\ngoal (1 subgoal):\n 1. V \\ {} \\ \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nV \\ {}\n[PROOF STEP]\nobtain x where \"x \\ V\"\n[PROOF STATE]\nproof (prove)\nusing this:\nV \\ {}\n\ngoal (1 subgoal):\n 1. (\\x. x \\ V \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx \\ V\n\ngoal (1 subgoal):\n 1. V \\ {} \\ \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ V\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ V\n\ngoal (1 subgoal):\n 1. \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n[PROOF STEP]\nusing affine_dependent_def[of \"{x}\"] extend_to_affine_basis_nonempty[of \"{x}\" V]\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ V\naffine_dependent {x} = (\\xa\\{x}. xa \\ affine hull ({x} - {xa}))\n\\\\ affine_dependent {x}; {x} \\ V; {x} \\ {}\\ \\ \\T. \\ affine_dependent T \\ {x} \\ T \\ T \\ V \\ affine hull T = affine hull V\n\ngoal (1 subgoal):\n 1. \\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\B\\V. \\ affine_dependent B \\ affine hull V = affine hull B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1443, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8558511322604133, "lm_q2_score": 0.8221891305219504, "lm_q1q2_score": 0.703671498289416}} {"text": "[STATEMENT]\nlemma inj_on_UnI: \"\\ inj_on f A; inj_on f B; f ` (A - B) \\ f ` (B - A) = {} \\ \\ inj_on f (A \\ B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\inj_on f A; inj_on f B; f ` (A - B) \\ f ` (B - A) = {}\\ \\ inj_on f (A \\ B)\n[PROOF STEP]\nby (auto iff: inj_on_Un)", "meta": {"llama_tokens": 170, "file": "SenSocialChoice_FSext", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045966995027, "lm_q2_score": 0.7931059585194573, "lm_q1q2_score": 0.7036472520682276}} {"text": "[STATEMENT]\nlemma hnorm_triangle_ineq2: \"\\a b::'a::real_normed_vector star. hnorm a - hnorm b \\ hnorm (a - b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a b. hnorm a - hnorm b \\ hnorm (a - b)\n[PROOF STEP]\nby transfer (rule norm_triangle_ineq2)", "meta": {"llama_tokens": 117, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045966995028, "lm_q2_score": 0.7931059462938815, "lm_q1q2_score": 0.7036472412216407}} {"text": "[STATEMENT]\nlemma \"((\\x::real. ((3 powr x + 4 powr x) / 4) powr (1/x)) \\ 4) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. ((3 powr x + 4 powr x) / 4) powr (1 / x)) \\ 4) at_top\n[PROOF STEP]\nby real_asymp", "meta": {"llama_tokens": 128, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9149009619539554, "lm_q2_score": 0.7690802476562641, "lm_q1q2_score": 0.7036322584005024}} {"text": "[STATEMENT]\nlemma \"((\\x::real. x powr (3/2) * (sqrt (x + 1) + sqrt (x - 1) - 2 * sqrt x)) \\ -1/4) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. x powr (3 / 2) * (sqrt (x + 1) + sqrt (x - 1) - 2 * sqrt x)) \\ - 1 / 4) at_top\n[PROOF STEP]\nby real_asymp", "meta": {"llama_tokens": 157, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9149009619539553, "lm_q2_score": 0.7690802370707283, "lm_q1q2_score": 0.7036322487157853}} {"text": "[STATEMENT]\nlemma finestpartSubset: \n assumes \"X \\ Y\" \n shows \"finestpart X \\ finestpart Y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finestpart X \\ finestpart Y\n[PROOF STEP]\nusing assms finestpart_def\n[PROOF STATE]\nproof (prove)\nusing this:\nX \\ Y\nfinestpart ?X = (\\x. {x}) ` ?X\n\ngoal (1 subgoal):\n 1. finestpart X \\ finestpart Y\n[PROOF STEP]\nby (metis image_mono)", "meta": {"llama_tokens": 174, "file": "Vickrey_Clarke_Groves_MiscTools", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972751232809, "lm_q2_score": 0.8080672158638527, "lm_q1q2_score": 0.7035011162475261}} {"text": "[STATEMENT]\nlemma ordered_nsets_2_eq:\n fixes A :: \"'a::linorder set\"\n shows \"nsets A 2 = {{x,y} | x y. x \\ A \\ y \\ A \\ x2\\<^esup> = {{x, y} |x y. x \\ A \\ y \\ A \\ x < y}\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. [A]\\<^bsup>2\\<^esup> \\ {{x, y} |x y. x \\ A \\ y \\ A \\ x < y}\n 2. {{x, y} |x y. x \\ A \\ y \\ A \\ x < y} \\ [A]\\<^bsup>2\\<^esup>\n[PROOF STEP]\nshow \"nsets A 2 \\ ?rhs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [A]\\<^bsup>2\\<^esup> \\ {{x, y} |x y. x \\ A \\ y \\ A \\ x < y}\n[PROOF STEP]\nunfolding numeral_nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [A]\\<^bsup>Suc (Suc 0)\\<^esup> \\ {{x, y} |x y. x \\ A \\ y \\ A \\ x < y}\n[PROOF STEP]\napply (clarsimp simp add: nsets_def card_Suc_eq Set.doubleton_eq_iff not_less)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\b ba. \\\\x y. y \\ A \\ x \\ A \\ (b = x \\ ba \\ y) \\ (b = y \\ ba \\ x) \\ y \\ x; b \\ A; ba \\ A\\ \\ b = ba\n[PROOF STEP]\nby (metis antisym)\n[PROOF STATE]\nproof (state)\nthis:\n[A]\\<^bsup>2\\<^esup> \\ {{x, y} |x y. x \\ A \\ y \\ A \\ x < y}\n\ngoal (1 subgoal):\n 1. {{x, y} |x y. x \\ A \\ y \\ A \\ x < y} \\ [A]\\<^bsup>2\\<^esup>\n[PROOF STEP]\nshow \"?rhs \\ nsets A 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {{x, y} |x y. x \\ A \\ y \\ A \\ x < y} \\ [A]\\<^bsup>2\\<^esup>\n[PROOF STEP]\nunfolding numeral_nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {{x, y} |x y. x \\ A \\ y \\ A \\ x < y} \\ [A]\\<^bsup>Suc (Suc 0)\\<^esup>\n[PROOF STEP]\nby (auto simp: nsets_def card_Suc_eq)\n[PROOF STATE]\nproof (state)\nthis:\n{{x, y} |x y. x \\ A \\ y \\ A \\ x < y} \\ [A]\\<^bsup>2\\<^esup>\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1030, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972684083609, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.7035011148454758}} {"text": "[STATEMENT]\nlemma interior_convex_hull_3_minimal:\n fixes a :: \"'a::euclidean_space\"\n assumes \"\\ collinear{a,b,c}\" and 2: \"DIM('a) = 2\"\n shows \"interior(convex hull {a,b,c}) =\n {v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v}\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. interior (convex hull {a, b, c}) = {v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v}\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. interior (convex hull {a, b, c}) \\ {v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v}\n 2. {v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v} \\ interior (convex hull {a, b, c})\n[PROOF STEP]\nhave abc: \"a \\ b\" \"a \\ c\" \"b \\ c\" \"\\ affine_dependent {a, b, c}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a \\ b &&& a \\ c) &&& b \\ c &&& \\ affine_dependent {a, b, c}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ collinear {a, b, c}\nDIM('a) = 2\n\ngoal (1 subgoal):\n 1. (a \\ b &&& a \\ c) &&& b \\ c &&& \\ affine_dependent {a, b, c}\n[PROOF STEP]\nby (auto simp: collinear_3_eq_affine_dependent)\n[PROOF STATE]\nproof (state)\nthis:\na \\ b\na \\ c\nb \\ c\n\\ affine_dependent {a, b, c}\n\ngoal (2 subgoals):\n 1. interior (convex hull {a, b, c}) \\ {v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v}\n 2. {v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v} \\ interior (convex hull {a, b, c})\n[PROOF STEP]\nwith 2\n[PROOF STATE]\nproof (chain)\npicking this:\nDIM('a) = 2\na \\ b\na \\ c\nb \\ c\n\\ affine_dependent {a, b, c}\n[PROOF STEP]\nshow \"?lhs \\ ?rhs\"\n[PROOF STATE]\nproof (prove)\nusing this:\nDIM('a) = 2\na \\ b\na \\ c\nb \\ c\n\\ affine_dependent {a, b, c}\n\ngoal (1 subgoal):\n 1. interior (convex hull {a, b, c}) \\ {v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v}\n[PROOF STEP]\nby (fastforce simp add: interior_convex_hull_explicit_minimal)\n[PROOF STATE]\nproof (state)\nthis:\ninterior (convex hull {a, b, c}) \\ {v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v}\n\ngoal (1 subgoal):\n 1. {v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v} \\ interior (convex hull {a, b, c})\n[PROOF STEP]\nshow \"?rhs \\ ?lhs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v} \\ interior (convex hull {a, b, c})\n[PROOF STEP]\nusing abc 2\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ b\na \\ c\nb \\ c\n\\ affine_dependent {a, b, c}\nDIM('a) = 2\n\ngoal (1 subgoal):\n 1. {v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v} \\ interior (convex hull {a, b, c})\n[PROOF STEP]\napply (clarsimp simp add: interior_convex_hull_explicit_minimal)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xa y z. \\a \\ b; a \\ c; b \\ c; \\ affine_dependent {a, b, c}; DIM('a) = 2; 0 < xa; 0 < y; 0 < z; xa + y + z = 1\\ \\ \\u. 0 < u a \\ 0 < u b \\ 0 < u c \\ u a + (u b + u c) = 1 \\ u a *\\<^sub>R a + (u b *\\<^sub>R b + u c *\\<^sub>R c) = xa *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c\n[PROOF STEP]\nsubgoal for x y z\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a \\ b; a \\ c; b \\ c; \\ affine_dependent {a, b, c}; DIM('a) = 2; 0 < x; 0 < y; 0 < z; x + y + z = 1\\ \\ \\u. 0 < u a \\ 0 < u b \\ 0 < u c \\ u a + (u b + u c) = 1 \\ u a *\\<^sub>R a + (u b *\\<^sub>R b + u c *\\<^sub>R c) = x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c\n[PROOF STEP]\nby (rule_tac x=\"\\r. (if r=a then x else if r=b then y else if r=c then z else 0)\" in exI) auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n{v. \\x y z. 0 < x \\ 0 < y \\ 0 < z \\ x + y + z = 1 \\ x *\\<^sub>R a + y *\\<^sub>R b + z *\\<^sub>R c = v} \\ interior (convex hull {a, b, c})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2396, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972784807408, "lm_q2_score": 0.8080672066194945, "lm_q1q2_score": 0.7035011109124664}} {"text": "[STATEMENT]\nlemma is_unit_diagonal: \n fixes U::\"'a::{comm_ring_1, algebraic_semidom}^'n::{finite, wellorder}^'n::{finite, wellorder}\"\n assumes U: \"upper_triangular U\"\n and det_U: \"is_unit (det U)\"\n shows \"\\i. is_unit (U $ i $ i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. is_unit (U $ i $ i)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i. is_unit (U $ i $ i)\n[PROOF STEP]\nhave \"is_unit (prod (\\i. U $ i $ i) UNIV)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_unit (\\i\\UNIV. U $ i $ i)\n[PROOF STEP]\nusing det_U det_upperdiagonal[of U] U\n[PROOF STATE]\nproof (prove)\nusing this:\nis_unit (det U)\n(\\i j. j < i \\ U $ i $ j = (0::'a)) \\ det U = (\\i\\UNIV. U $ i $ i)\nupper_triangular U\n\ngoal (1 subgoal):\n 1. is_unit (\\i\\UNIV. U $ i $ i)\n[PROOF STEP]\nunfolding upper_triangular_def\n[PROOF STATE]\nproof (prove)\nusing this:\nis_unit (det U)\n(\\i j. j < i \\ U $ i $ j = (0::'a)) \\ det U = (\\i\\UNIV. U $ i $ i)\n\\i j. j < i \\ U $ i $ j = (0::'a)\n\ngoal (1 subgoal):\n 1. is_unit (\\i\\UNIV. U $ i $ i)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nis_unit (\\i\\UNIV. U $ i $ i)\n\ngoal (1 subgoal):\n 1. \\i. is_unit (U $ i $ i)\n[PROOF STEP]\nhence \"\\i\\UNIV. is_unit (U $ i $ i)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nis_unit (\\i\\UNIV. U $ i $ i)\n\ngoal (1 subgoal):\n 1. \\i\\UNIV. is_unit (U $ i $ i)\n[PROOF STEP]\nusing unit_prod[of UNIV U]\n[PROOF STATE]\nproof (prove)\nusing this:\nis_unit (\\i\\UNIV. U $ i $ i)\nfinite UNIV \\ is_unit (\\i\\UNIV. U $ i $ i) = (\\i\\UNIV. is_unit (U $ i $ i))\n\ngoal (1 subgoal):\n 1. \\i\\UNIV. is_unit (U $ i $ i)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\i\\UNIV. is_unit (U $ i $ i)\n\ngoal (1 subgoal):\n 1. \\i. is_unit (U $ i $ i)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i\\UNIV. is_unit (U $ i $ i)\n\ngoal (1 subgoal):\n 1. \\i. is_unit (U $ i $ i)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\i. is_unit (U $ i $ i)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1076, "file": "Hermite_Hermite", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972684083609, "lm_q2_score": 0.8080672135527632, "lm_q1q2_score": 0.7035011088093913}} {"text": "[STATEMENT]\nlemma lead_coeff_div: \n fixes p q::\"('a::{idom_divide, inverse}) poly\" \n assumes \"q dvd p\"\n shows \"lead_coeff (p div q) = lead_coeff p / lead_coeff q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lead_coeff (p div q) = lead_coeff p / lead_coeff q\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nq dvd p\n\ngoal (1 subgoal):\n 1. lead_coeff (p div q) = lead_coeff p / lead_coeff q\n[PROOF STEP]\nby (smt (z3) div_by_0 dvd_div_mult_self lead_coeff_mult leading_coeff_0_iff\n nonzero_mult_div_cancel_right)", "meta": {"llama_tokens": 241, "file": "Three_Circles_RRI_Misc", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972616934408, "lm_q2_score": 0.8080672181749422, "lm_q1q2_score": 0.7035011074073408}} {"text": "[STATEMENT]\nlemma coplanar_4 :\n assumes \"rk {A, B, C} = 3\" and \"rk {B, C, \\} = 2\" \n shows \"rk {A, B, C, \\} = 3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rk {A, B, C, \\} = 3\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. rk {A, B, C, \\} = 3\n[PROOF STEP]\nhave f1:\"rk {A, B, C, \\} \\ 3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 3 \\ rk {A, B, C, \\}\n[PROOF STEP]\nusing matroid_ax_2\n[PROOF STATE]\nproof (prove)\nusing this:\n?X \\ ?Y \\ rk ?X \\ rk ?Y\n\ngoal (1 subgoal):\n 1. 3 \\ rk {A, B, C, \\}\n[PROOF STEP]\nby (metis assms(1) empty_subsetI insert_mono)\n[PROOF STATE]\nproof (state)\nthis:\n3 \\ rk {A, B, C, \\}\n\ngoal (1 subgoal):\n 1. rk {A, B, C, \\} = 3\n[PROOF STEP]\nhave \"rk {A, B, C, \\} + rk {B, C} \\ rk {A, B, C} + rk {B, C, \\}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rk {A, B, C, \\} + rk {B, C} \\ rk {A, B, C} + rk {B, C, \\}\n[PROOF STEP]\nusing matroid_ax_3_alt\n[PROOF STATE]\nproof (prove)\nusing this:\n?I \\ ?X \\ ?Y \\ rk (?X \\ ?Y) + rk ?I \\ rk ?X + rk ?Y\n\ngoal (1 subgoal):\n 1. rk {A, B, C, \\} + rk {B, C} \\ rk {A, B, C} + rk {B, C, \\}\n[PROOF STEP]\nby (metis Un_insert_right add_One_commute add_mono assms(1) assms(2) matroid_ax_2_alt \n nat_1_add_1 numeral_plus_one rk_singleton semiring_norm(3) sup_bot.right_neutral)\n[PROOF STATE]\nproof (state)\nthis:\nrk {A, B, C, \\} + rk {B, C} \\ rk {A, B, C} + rk {B, C, \\}\n\ngoal (1 subgoal):\n 1. rk {A, B, C, \\} = 3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nrk {A, B, C, \\} + rk {B, C} \\ rk {A, B, C} + rk {B, C, \\}\n[PROOF STEP]\nhave f2:\"rk {A, B, C, \\} \\ 3\"\n[PROOF STATE]\nproof (prove)\nusing this:\nrk {A, B, C, \\} + rk {B, C} \\ rk {A, B, C} + rk {B, C, \\}\n\ngoal (1 subgoal):\n 1. rk {A, B, C, \\} \\ 3\n[PROOF STEP]\nby (metis Un_insert_right add_One_commute assms(2) matroid_ax_2_alt numeral_plus_one \n semiring_norm(3) sup_bot.right_neutral)\n[PROOF STATE]\nproof (state)\nthis:\nrk {A, B, C, \\} \\ 3\n\ngoal (1 subgoal):\n 1. rk {A, B, C, \\} = 3\n[PROOF STEP]\nfrom f1 and f2\n[PROOF STATE]\nproof (chain)\npicking this:\n3 \\ rk {A, B, C, \\}\nrk {A, B, C, \\} \\ 3\n[PROOF STEP]\nshow \"rk {A, B, C, \\} = 3\"\n[PROOF STATE]\nproof (prove)\nusing this:\n3 \\ rk {A, B, C, \\}\nrk {A, B, C, \\} \\ 3\n\ngoal (1 subgoal):\n 1. rk {A, B, C, \\} = 3\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrk {A, B, C, \\} = 3\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1402, "file": "Projective_Geometry_Desargues_3D", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972583359805, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.7035011067063154}} {"text": "[STATEMENT]\ntheorem join_idem: \"x \\ x = x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ x = x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x \\ x = x\n[PROOF STEP]\nhave \"dual x \\ dual x = dual x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dual x \\ dual x = dual x\n[PROOF STEP]\nby (rule meet_idem)\n[PROOF STATE]\nproof (state)\nthis:\ndual x \\ dual x = dual x\n\ngoal (1 subgoal):\n 1. x \\ x = x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndual x \\ dual x = dual x\n[PROOF STEP]\nhave \"dual (x \\ x) = dual x\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndual x \\ dual x = dual x\n\ngoal (1 subgoal):\n 1. dual (x \\ x) = dual x\n[PROOF STEP]\nby (simp only: dual_join)\n[PROOF STATE]\nproof (state)\nthis:\ndual (x \\ x) = dual x\n\ngoal (1 subgoal):\n 1. x \\ x = x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndual (x \\ x) = dual x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndual (x \\ x) = dual x\n\ngoal (1 subgoal):\n 1. x \\ x = x\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nx \\ x = x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 576, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972684083609, "lm_q2_score": 0.8080672089305841, "lm_q1q2_score": 0.7035011047853349}} {"text": "[STATEMENT]\nlemma closer_point_lemma:\n assumes \"inner (y - x) (z - x) > 0\"\n shows \"\\u>0. u \\ 1 \\ dist (x + u *\\<^sub>R (z - x)) y < dist x y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\u>0. u \\ 1 \\ dist (x + u *\\<^sub>R (z - x)) y < dist x y\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\u>0. u \\ 1 \\ dist (x + u *\\<^sub>R (z - x)) y < dist x y\n[PROOF STEP]\nobtain u where \"u > 0\"\n and u: \"\\v. \\0 u\\ \\ norm (v *\\<^sub>R (z - x) - (y - x)) < norm (y - x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\u. \\0 < u; \\v. \\0 < v; v \\ u\\ \\ norm (v *\\<^sub>R (z - x) - (y - x)) < norm (y - x)\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing closer_points_lemma[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\u>0. \\v>0. v \\ u \\ norm (v *\\<^sub>R (z - x) - (y - x)) < norm (y - x)\n\ngoal (1 subgoal):\n 1. (\\u. \\0 < u; \\v. \\0 < v; v \\ u\\ \\ norm (v *\\<^sub>R (z - x) - (y - x)) < norm (y - x)\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < u\n\\0 < ?v; ?v \\ u\\ \\ norm (?v *\\<^sub>R (z - x) - (y - x)) < norm (y - x)\n\ngoal (1 subgoal):\n 1. \\u>0. u \\ 1 \\ dist (x + u *\\<^sub>R (z - x)) y < dist x y\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\u>0. u \\ 1 \\ dist (x + u *\\<^sub>R (z - x)) y < dist x y\n[PROOF STEP]\nusing u[of \"min u 1\"] and \\u > 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < min u 1; min u 1 \\ u\\ \\ norm (min u 1 *\\<^sub>R (z - x) - (y - x)) < norm (y - x)\n0 < u\n\ngoal (1 subgoal):\n 1. \\u>0. u \\ 1 \\ dist (x + u *\\<^sub>R (z - x)) y < dist x y\n[PROOF STEP]\nby (metis diff_diff_add dist_commute dist_norm less_eq_real_def not_less u zero_less_one)\n[PROOF STATE]\nproof (state)\nthis:\n\\u>0. u \\ 1 \\ dist (x + u *\\<^sub>R (z - x)) y < dist x y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1039, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972549785201, "lm_q2_score": 0.8080672158638527, "lm_q1q2_score": 0.7035010999692054}} {"text": "[STATEMENT]\nlemma mono_bound:\n fixes f g h :: \"real \\ real\"\n assumes \"mono h\" \"eventually (\\x. f x \\ g x) at_top\"\n shows \"eventually (\\x. h (f x) \\ h (g x)) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. h (f x) \\ h (g x)\n[PROOF STEP]\nby (intro eventually_mono[OF assms(2)] monoD[OF assms(1)])", "meta": {"llama_tokens": 165, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972616934406, "lm_q2_score": 0.808067204308405, "lm_q1q2_score": 0.7035010953351714}} {"text": "[STATEMENT]\nlemma open_segment_translation_eq [simp]:\n \"d + x \\ open_segment (d + a) (d + b) \\ x \\ open_segment a b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (d + x \\ open_segment (d + a) (d + b)) = (x \\ open_segment a b)\n[PROOF STEP]\nby (simp add: open_segment_def)", "meta": {"llama_tokens": 130, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869981319863, "lm_q2_score": 0.803173791645582, "lm_q1q2_score": 0.7034091639635697}} {"text": "[STATEMENT]\nlemma sinh_double: \"sinh (2 * x) = 2 * sinh x * cosh x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sinh ((2::'a) * x) = (2::'a) * sinh x * cosh x\n[PROOF STEP]\nusing sinh_add[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\nsinh (x + x) = sinh x * cosh x + cosh x * sinh x\n\ngoal (1 subgoal):\n 1. sinh ((2::'a) * x) = (2::'a) * sinh x * cosh x\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 189, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869916479466, "lm_q2_score": 0.8031737963569014, "lm_q1q2_score": 0.7034091628818712}} {"text": "[STATEMENT]\nlemma leq_ccsubspace_code[code]:\n \\ \\Code equation for deciding inclusion of one space in another.\n Uses the constant \\<^term>\\is_subspace_of_vec_list\\ which implements the actual\n computation by checking for each generator of A whether it is in the\n span of B (by orthogonal projection onto an orthonormal basis of B\n which is computed using Gram-Schmidt).\\\n \"SPAN A \\ (SPAN B :: 'a::onb_enum ccsubspace)\n \\ (let d = length (canonical_basis :: 'a list) in\n is_subspace_of_vec_list d\n (filter (\\v. dim_vec v = d) A)\n (filter (\\v. dim_vec v = d) B))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (SPAN A \\ SPAN B) = (let d = length canonical_basis in is_subspace_of_vec_list d (filter (\\v. dim_vec v = d) A) (filter (\\v. dim_vec v = d) B))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (SPAN A \\ SPAN B) = (let d = length canonical_basis in is_subspace_of_vec_list d (filter (\\v. dim_vec v = d) A) (filter (\\v. dim_vec v = d) B))\n[PROOF STEP]\ndefine d A' B' where \"d = length (canonical_basis :: 'a list)\"\n and \"A' = filter (\\v. dim_vec v = d) A\"\n and \"B' = filter (\\v. dim_vec v = d) B\"\n[PROOF STATE]\nproof (state)\nthis:\nd = length canonical_basis\nA' = filter (\\v. dim_vec v = d) A\nB' = filter (\\v. dim_vec v = d) B\n\ngoal (1 subgoal):\n 1. (SPAN A \\ SPAN B) = (let d = length canonical_basis in is_subspace_of_vec_list d (filter (\\v. dim_vec v = d) A) (filter (\\v. dim_vec v = d) B))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (SPAN A \\ SPAN B) = (let d = length canonical_basis in is_subspace_of_vec_list d (filter (\\v. dim_vec v = d) A) (filter (\\v. dim_vec v = d) B))\n[PROOF STEP]\nunfolding SPAN_def d_def[symmetric] filter_set Let_def\n A'_def[symmetric] B'_def[symmetric] image_set\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (ccspan (set (map basis_enum_of_vec A')) \\ ccspan (set (map basis_enum_of_vec B'))) = is_subspace_of_vec_list d A' B'\n[PROOF STEP]\napply (subst ccspan_leq_using_vec)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_subspace_of_vec_list (length canonical_basis) (map vec_of_basis_enum (map basis_enum_of_vec A')) (map vec_of_basis_enum (map basis_enum_of_vec B')) = is_subspace_of_vec_list d A' B'\n[PROOF STEP]\nunfolding d_def[symmetric] map_map o_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_subspace_of_vec_list d (map (\\x. vec_of_basis_enum (basis_enum_of_vec x)) A') (map (\\x. vec_of_basis_enum (basis_enum_of_vec x)) B') = is_subspace_of_vec_list d A' B'\n[PROOF STEP]\napply (subst map_cong[where xs=A', OF refl])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. x \\ set A' \\ vec_of_basis_enum (basis_enum_of_vec x) = ?g x\n 2. is_subspace_of_vec_list d (map ?g A') (map (\\x. vec_of_basis_enum (basis_enum_of_vec x)) B') = is_subspace_of_vec_list d A' B'\n[PROOF STEP]\napply (rule basis_enum_of_vec_inverse)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. x \\ set A' \\ dim_vec x = length canonical_basis\n 2. is_subspace_of_vec_list d (map (\\x. x) A') (map (\\x. vec_of_basis_enum (basis_enum_of_vec x)) B') = is_subspace_of_vec_list d A' B'\n[PROOF STEP]\napply (simp add: A'_def d_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_subspace_of_vec_list d (map (\\x. x) A') (map (\\x. vec_of_basis_enum (basis_enum_of_vec x)) B') = is_subspace_of_vec_list d A' B'\n[PROOF STEP]\napply (subst map_cong[where xs=B', OF refl])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. x \\ set B' \\ vec_of_basis_enum (basis_enum_of_vec x) = ?g3 x\n 2. is_subspace_of_vec_list d (map (\\x. x) A') (map ?g3 B') = is_subspace_of_vec_list d A' B'\n[PROOF STEP]\napply (rule basis_enum_of_vec_inverse)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x. x \\ set B' \\ dim_vec x = length canonical_basis\n 2. is_subspace_of_vec_list d (map (\\x. x) A') (map (\\x. x) B') = is_subspace_of_vec_list d A' B'\n[PROOF STEP]\nby (simp_all add: B'_def d_def)\n[PROOF STATE]\nproof (state)\nthis:\n(SPAN A \\ SPAN B) = (let d = length canonical_basis in is_subspace_of_vec_list d (filter (\\v. dim_vec v = d) A) (filter (\\v. dim_vec v = d) B))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1884, "file": "Complex_Bounded_Operators_Cblinfun_Code", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392939666335, "lm_q2_score": 0.7956581049086031, "lm_q1q2_score": 0.7033930293022311}} {"text": "[STATEMENT]\nlemma set_image: \"\\ f \\ A \\ B; A1 \\ A; A2 \\ A \\ \\\n f`(A1 \\ A2) \\ (f` A1) \\ (f` A2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ A \\ B; A1 \\ A; A2 \\ A\\ \\ f ` (A1 \\ A2) \\ f ` A1 \\ f ` A2\n[PROOF STEP]\napply (simp add: image_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ A \\ B; A1 \\ A; A2 \\ A\\ \\ {y. \\x\\A1 \\ A2. y = f x} \\ {y. \\x\\A1. y = f x} \\ {y. \\x\\A1 \\ A2. y = f x} \\ {y. \\x\\A2. y = f x}\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 374, "file": "Group-Ring-Module_Algebra1", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392817460333, "lm_q2_score": 0.7956581073313276, "lm_q1q2_score": 0.7033930217205951}} {"text": "[STATEMENT]\nlemma continuous_levelset_openin_cases:\n fixes f :: \"_ \\ 'b::t1_space\"\n shows \"connected s \\ continuous_on s f \\\n openin (top_of_set s) {x \\ s. f x = a}\n \\ (\\x \\ s. f x \\ a) \\ (\\x \\ s. f x = a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\connected s; continuous_on s f; openin (top_of_set s) {x \\ s. f x = a}\\ \\ (\\x\\s. f x \\ a) \\ (\\x\\s. f x = a)\n[PROOF STEP]\nunfolding connected_clopen\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\T. openin (top_of_set s) T \\ closedin (top_of_set s) T \\ T = {} \\ T = s; continuous_on s f; openin (top_of_set s) {x \\ s. f x = a}\\ \\ (\\x\\s. f x \\ a) \\ (\\x\\s. f x = a)\n[PROOF STEP]\nusing continuous_closedin_preimage_constant\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous_on ?S ?f \\ closedin (top_of_set ?S) {x \\ ?S. ?f x = ?a}\n\ngoal (1 subgoal):\n 1. \\\\T. openin (top_of_set s) T \\ closedin (top_of_set s) T \\ T = {} \\ T = s; continuous_on s f; openin (top_of_set s) {x \\ s. f x = a}\\ \\ (\\x\\s. f x \\ a) \\ (\\x\\s. f x = a)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 586, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392817460332, "lm_q2_score": 0.7956581073313275, "lm_q1q2_score": 0.7033930217205949}} {"text": "[STATEMENT]\nlemma higher_pderiv_0 [simp]: \"(pderiv ^^ n) 0 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (pderiv ^^ n) 0 = 0\n[PROOF STEP]\nby (induction n) simp_all", "meta": {"llama_tokens": 85, "file": "E_Transcendental_E_Transcendental", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392817460332, "lm_q2_score": 0.7956581000631542, "lm_q1q2_score": 0.7033930152952442}} {"text": "[STATEMENT]\nlemma poly_eval_minus: \"poly_eval a (p - q) = poly_eval a p - poly_eval (a::_::comm_ring_1) q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly_eval a (p - q) = poly_eval a p - poly_eval a q\n[PROOF STEP]\nby (simp only: poly_eval_def poly_subst_minus lookup_minus)", "meta": {"llama_tokens": 118, "file": "Polynomials_MPoly_PM", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392756357327, "lm_q2_score": 0.7956581049086031, "lm_q1q2_score": 0.7033930147171014}} {"text": "[STATEMENT]\nlemma winding_number_ivt_abs:\n assumes \\: \"valid_path \\\" and z: \"z \\ path_image \\\" and \"0 \\ w\" \"w \\ \\Re(winding_number \\ z)\\\"\n shows \"\\t \\ {0..1}. \\Re (winding_number (subpath 0 t \\) z)\\ = w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\t\\{0..1}. \\Re (winding_number (subpath 0 t \\) z)\\ = w\n[PROOF STEP]\nusing assms winding_number_ivt_pos [of \\ z w] winding_number_ivt_neg [of \\ z \"-w\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nvalid_path \\\nz \\ path_image \\\n0 \\ w\nw \\ \\Re (winding_number \\ z)\\\n\\valid_path \\; z \\ path_image \\; 0 \\ w; w \\ Re (winding_number \\ z)\\ \\ \\t\\{0..1}. Re (winding_number (subpath 0 t \\) z) = w\n\\valid_path \\; z \\ path_image \\; Re (winding_number \\ z) \\ - w; - w \\ 0\\ \\ \\t\\{0..1}. Re (winding_number (subpath 0 t \\) z) = - w\n\ngoal (1 subgoal):\n 1. \\t\\{0..1}. \\Re (winding_number (subpath 0 t \\) z)\\ = w\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 510, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392756357327, "lm_q2_score": 0.7956580927949806, "lm_q1q2_score": 0.7033930040081832}} {"text": "[STATEMENT]\ntheorem condensation_test:\n assumes mono: \"\\m. 0 < m \\ f (Suc m) \\ f m\"\n assumes nonneg: \"\\n. f n \\ 0\"\n shows \"summable f \\ summable (\\n. 2^n * f (2^n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\ndefine f' where \"f' n = (if n = 0 then 0 else f n)\" for n\n[PROOF STATE]\nproof (state)\nthis:\nf' ?n = (if ?n = 0 then 0 else f ?n)\n\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nfrom mono\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < ?m \\ f (Suc ?m) \\ f ?m\n[PROOF STEP]\nhave mono': \"decseq (\\n. f (Suc n))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < ?m \\ f (Suc ?m) \\ f ?m\n\ngoal (1 subgoal):\n 1. decseq (\\n. f (Suc n))\n[PROOF STEP]\nby (intro decseq_SucI) simp\n[PROOF STATE]\nproof (state)\nthis:\ndecseq (\\n. f (Suc n))\n\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhence mono': \"f n \\ f m\" if \"m \\ n\" \"m > 0\" for m n\n[PROOF STATE]\nproof (prove)\nusing this:\ndecseq (\\n. f (Suc n))\n\ngoal (1 subgoal):\n 1. f n \\ f m\n[PROOF STEP]\nusing that decseqD[OF mono', of \"m - 1\" \"n - 1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ndecseq (\\n. f (Suc n))\nm \\ n\n0 < m\nm - 1 \\ n - 1 \\ f (Suc (n - 1)) \\ f (Suc (m - 1))\n\ngoal (1 subgoal):\n 1. f n \\ f m\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\?m \\ ?n; 0 < ?m\\ \\ f ?n \\ f ?m\n\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhave \"(\\n. f (Suc n)) = (\\n. f' (Suc n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. f (Suc n)) = (\\n. f' (Suc n))\n[PROOF STEP]\nby (intro ext) (simp add: f'_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. f (Suc n)) = (\\n. f' (Suc n))\n\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhence \"summable f \\ summable f'\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. f (Suc n)) = (\\n. f' (Suc n))\n\ngoal (1 subgoal):\n 1. summable f = summable f'\n[PROOF STEP]\nby (subst (1 2) summable_Suc_iff [symmetric]) (simp only:)\n[PROOF STATE]\nproof (state)\nthis:\nsummable f = summable f'\n\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsummable f = summable f'\n\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhave \"\\ \\ convergent (\\n. \\kn. sum f' {..n. sum f' {..n. sum f' {..n. sum f' {..n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsummable f' = convergent (\\n. sum f' {..n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhave \"monoseq (\\n. \\kn. sum f' {..n. \\kn. sum f' {..n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhence \"convergent (\\n. \\k Bseq (\\n. \\kn. sum f' {..n. sum f' {..n. sum f' {..n. sum f' {..n. sum f' {..n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nconvergent (\\n. sum f' {..n. sum f' {..n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhave \"\\ \\ Bseq (\\n. \\k=1..n. sum f' {..n. sum f' {1..n. sum f' {..n. sum f' {Suc 0..n. sum f' {..n. sum f' {1..n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nBseq (\\n. sum f' {..n. sum f' {1..n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhave \"\\ \\ Bseq (\\n. \\k=1..n. sum f' {1..n. sum f {1..n. \\k = 1..n. sum f {1..n. sum f' {1..n. sum f {1..n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nBseq (\\n. sum f' {1..n. sum f {1..n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhave \"\\ \\ Bseq (\\n. \\k=1..<2^n. f k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Bseq (\\n. sum f {1..n. sum f {1..<2 ^ n})\n[PROOF STEP]\nby (rule nonneg_incseq_Bseq_subseq_iff[symmetric])\n (auto intro!: sum_nonneg incseq_SucI nonneg simp: strict_mono_def)\n[PROOF STATE]\nproof (state)\nthis:\nBseq (\\n. sum f {1..n. sum f {1..<2 ^ n})\n\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nBseq (\\n. sum f {1..n. sum f {1..<2 ^ n})\n\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhave \"\\ \\ Bseq (\\n. \\kn. sum f {1..<2 ^ n}) = Bseq (\\n. \\kn. sum f {1..<2 ^ n}) \\ Bseq (\\n. \\kn. \\k Bseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nassume A: \"Bseq (\\n. \\k=1..<2^n. f k)\"\n[PROOF STATE]\nproof (state)\nthis:\nBseq (\\n. sum f {1..<2 ^ n})\n\ngoal (2 subgoals):\n 1. Bseq (\\n. sum f {1..<2 ^ n}) \\ Bseq (\\n. \\kn. \\k Bseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nhave \"eventually (\\n. norm (\\k norm (\\k=1..<2^n. f k)) sequentially\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. norm (\\k norm (sum f {1..<2 ^ n})\n[PROOF STEP]\nproof (intro always_eventually allI)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. norm (\\k norm (sum f {1..<2 ^ x})\n[PROOF STEP]\nfix n :: nat\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. norm (\\k norm (sum f {1..<2 ^ x})\n[PROOF STEP]\nhave \"norm (\\kkkk\\k = (\\kkkx. norm (\\k norm (sum f {1..<2 ^ x})\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnorm (\\kkx. norm (\\k norm (sum f {1..<2 ^ x})\n[PROOF STEP]\nhave \"\\ \\ (\\k=1..<2^n. f k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k sum f {1..<2 ^ n}\n[PROOF STEP]\nby (subst condensation_condense2 [symmetric]) (intro condensation_inequality mono')\n[PROOF STATE]\nproof (state)\nthis:\n(\\k sum f {1..<2 ^ n}\n\ngoal (1 subgoal):\n 1. \\x. norm (\\k norm (sum f {1..<2 ^ x})\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k sum f {1..<2 ^ n}\n\ngoal (1 subgoal):\n 1. \\x. norm (\\k norm (sum f {1..<2 ^ x})\n[PROOF STEP]\nhave \"\\ = norm \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f {1..<2 ^ n} = norm (sum f {1..<2 ^ n})\n[PROOF STEP]\nunfolding real_norm_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f {1..<2 ^ n} = \\sum f {1..<2 ^ n}\\\n[PROOF STEP]\nby (intro abs_of_nonneg[symmetric] sum_nonneg ballI mult_nonneg_nonneg nonneg)\n[PROOF STATE]\nproof (state)\nthis:\nsum f {1..<2 ^ n} = norm (sum f {1..<2 ^ n})\n\ngoal (1 subgoal):\n 1. \\x. norm (\\k norm (sum f {1..<2 ^ x})\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (\\k norm (sum f {1..<2 ^ n})\n[PROOF STEP]\nshow \"norm (\\k norm (\\k=1..<2^n. f k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (\\k norm (sum f {1..<2 ^ n})\n\ngoal (1 subgoal):\n 1. norm (\\k norm (sum f {1..<2 ^ n})\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nnorm (\\k norm (sum f {1..<2 ^ n})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F n in sequentially. norm (\\k norm (sum f {1..<2 ^ n})\n\ngoal (2 subgoals):\n 1. Bseq (\\n. sum f {1..<2 ^ n}) \\ Bseq (\\n. \\kn. \\k Bseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nfrom this and A\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F n in sequentially. norm (\\k norm (sum f {1..<2 ^ n})\nBseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nhave \"Bseq (\\n. \\k\\<^sub>F n in sequentially. norm (\\k norm (sum f {1..<2 ^ n})\nBseq (\\n. sum f {1..<2 ^ n})\n\ngoal (1 subgoal):\n 1. Bseq (\\n. \\kn. \\kn. sum f {1..<2 ^ n}) \\ Bseq (\\n. \\kn. \\k Bseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nfrom Bseq_mult[OF Bfun_const[of 2] this]\n[PROOF STATE]\nproof (chain)\npicking this:\nBseq (\\x. 2 * (\\kn. \\kx. 2 * (\\kn. \\kn. \\kn. sum f {1..<2 ^ n}) \\ Bseq (\\n. \\kn. \\k Bseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nhence \"Bseq (\\n. (\\k=Suc 0..n. \\kn. (\\k = Suc 0..n. (\\k = Suc 0..n. sum f {1..<2 ^ n}) \\ Bseq (\\n. \\kn. \\k Bseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nhence \"Bseq (\\n. (\\k=0..n. (\\k = Suc 0..n. \\k = 0..n. \\k = 0..n. sum f {1..<2 ^ n}) \\ Bseq (\\n. \\kn. \\k Bseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nthus \"Bseq (\\n. (\\kn. \\k = 0..n. \\kn. \\kn. \\k Bseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Bseq (\\n. \\k Bseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nassume A: \"Bseq (\\n. (\\kn. \\kn. \\k Bseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nhave \"eventually (\\n. norm (\\k=1..<2^n. f k) \\ norm (\\k\\<^sub>F n in sequentially. norm (sum f {1..<2 ^ n}) \\ norm (\\kx. norm (sum f {1..<2 ^ x}) \\ norm (\\kx. norm (sum f {1..<2 ^ x}) \\ norm (\\kk=1..<2^n. f k) = (\\k=1..<2^n. f k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (sum f {1..<2 ^ n}) = sum f {1..<2 ^ n}\n[PROOF STEP]\nunfolding real_norm_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\sum f {1..<2 ^ n}\\ = sum f {1..<2 ^ n}\n[PROOF STEP]\nby (intro abs_of_nonneg sum_nonneg ballI mult_nonneg_nonneg nonneg)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (sum f {1..<2 ^ n}) = sum f {1..<2 ^ n}\n\ngoal (1 subgoal):\n 1. \\x. norm (sum f {1..<2 ^ x}) \\ norm (\\kx. norm (sum f {1..<2 ^ x}) \\ norm (\\k \\ (\\k (\\k (\\kx. norm (sum f {1..<2 ^ x}) \\ norm (\\k (\\kx. norm (sum f {1..<2 ^ x}) \\ norm (\\k = norm \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\kkk\\k\n[PROOF STEP]\nby (intro abs_of_nonneg [symmetric] sum_nonneg ballI mult_nonneg_nonneg nonneg) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(\\kkx. norm (sum f {1..<2 ^ x}) \\ norm (\\k norm (\\kk=1..<2^n. f k) \\ norm (\\k norm (\\k norm (\\k norm (\\k\\<^sub>F n in sequentially. norm (sum f {1..<2 ^ n}) \\ norm (\\kn. \\k Bseq (\\n. sum f {1..<2 ^ n})\n[PROOF STEP]\nfrom this and A\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F n in sequentially. norm (sum f {1..<2 ^ n}) \\ norm (\\kn. \\kn. \\k=1..<2^n. f k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F n in sequentially. norm (sum f {1..<2 ^ n}) \\ norm (\\kn. \\kn. sum f {1..<2 ^ n})\n[PROOF STEP]\nby (rule Bseq_eventually_mono)\n[PROOF STATE]\nproof (state)\nthis:\nBseq (\\n. sum f {1..<2 ^ n})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nBseq (\\n. sum f {1..<2 ^ n}) = Bseq (\\n. \\kn. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nBseq (\\n. sum f {1..<2 ^ n}) = Bseq (\\n. \\kn. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhave \"monoseq (\\n. (\\kn. \\kn. \\kn. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhence \"Bseq (\\n. (\\k convergent (\\n. (\\kn. \\kn. \\kn. \\kn. \\kn. \\kn. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nBseq (\\n. \\kn. \\kn. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nhave \"\\ \\ summable (\\k. 2^k * f (2^k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convergent (\\n. \\kk. 2 ^ k * f (2 ^ k))\n[PROOF STEP]\nby (simp only: summable_iff_convergent)\n[PROOF STATE]\nproof (state)\nthis:\nconvergent (\\n. \\kk. 2 ^ k * f (2 ^ k))\n\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsummable f = summable (\\k. 2 ^ k * f (2 ^ k))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable f = summable (\\k. 2 ^ k * f (2 ^ k))\n\ngoal (1 subgoal):\n 1. summable f = summable (\\n. 2 ^ n * f (2 ^ n))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nsummable f = summable (\\n. 2 ^ n * f (2 ^ n))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 10778, "file": null, "length": 101, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339756938819, "lm_q2_score": 0.8376199613065411, "lm_q1q2_score": 0.7033779402284973}} {"text": "[STATEMENT]\nlemma arg_complex_of_real_positive [simp]:\n assumes \"k > 0\"\n shows \"Arg (cor k) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Arg (cor k) = 0\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Arg (cor k) = 0\n[PROOF STEP]\nhave \"cos (Arg (Complex k 0)) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < cos (Arg (Complex k 0))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < k\n\ngoal (1 subgoal):\n 1. 0 < cos (Arg (Complex k 0))\n[PROOF STEP]\nusing rcis_cmod_Arg[of \"Complex k 0\"] Re_rcis[of \"cmod (Complex k 0)\" \"Arg (Complex k 0)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < k\nrcis (cmod (Complex k 0)) (Arg (Complex k 0)) = Complex k 0\nRe (rcis (cmod (Complex k 0)) (Arg (Complex k 0))) = cmod (Complex k 0) * cos (Arg (Complex k 0))\n\ngoal (1 subgoal):\n 1. 0 < cos (Arg (Complex k 0))\n[PROOF STEP]\nusing cmod_eq_Re\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < k\nrcis (cmod (Complex k 0)) (Arg (Complex k 0)) = Complex k 0\nRe (rcis (cmod (Complex k 0)) (Arg (Complex k 0))) = cmod (Complex k 0) * cos (Arg (Complex k 0))\nis_real ?z \\ cmod ?z = \\Re ?z\\\n\ngoal (1 subgoal):\n 1. 0 < cos (Arg (Complex k 0))\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n0 < cos (Arg (Complex k 0))\n\ngoal (1 subgoal):\n 1. Arg (cor k) = 0\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < cos (Arg (Complex k 0))\n\ngoal (1 subgoal):\n 1. Arg (cor k) = 0\n[PROOF STEP]\nusing assms is_real_arg2[of \"cor k\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < cos (Arg (Complex k 0))\n0 < k\nis_real (cor k) \\ Arg (cor k) = 0 \\ Arg (cor k) = pi\n\ngoal (1 subgoal):\n 1. Arg (cor k) = 0\n[PROOF STEP]\nunfolding complex_of_real_def\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < cos (Arg (Complex k 0))\n0 < k\nis_real (Complex k 0) \\ Arg (Complex k 0) = 0 \\ Arg (Complex k 0) = pi\n\ngoal (1 subgoal):\n 1. Arg (Complex k 0) = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nArg (cor k) = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 961, "file": "Complex_Geometry_More_Complex", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339676722393, "lm_q2_score": 0.8376199613065411, "lm_q1q2_score": 0.7033779335094094}} {"text": "[STATEMENT]\nlemma eval_fds_altdef:\n assumes \"fds_abs_converges f s\"\n shows \"eval_fds f s = (\\\\<^sub>an. fds_nth f n / nat_power n s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eval_fds f s = (\\\\<^sub>an. fds_nth f n / nat_power n s)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. eval_fds f s = (\\\\<^sub>an. fds_nth f n / nat_power n s)\n[PROOF STEP]\nhave \"fds_abs_converges f s \\ (\\n. fds_nth f n / nat_power n s) abs_summable_on UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fds_abs_converges f s = ((\\n. fds_nth f n / nat_power n s) abs_summable_on UNIV)\n[PROOF STEP]\nunfolding fds_abs_converges_altdef\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\n. fds_nth f n / nat_power n s) abs_summable_on {1..}) = ((\\n. fds_nth f n / nat_power n s) abs_summable_on UNIV)\n[PROOF STEP]\nby (intro abs_summable_on_cong_neutral) (auto simp: Suc_le_eq)\n[PROOF STATE]\nproof (state)\nthis:\nfds_abs_converges f s = ((\\n. fds_nth f n / nat_power n s) abs_summable_on UNIV)\n\ngoal (1 subgoal):\n 1. eval_fds f s = (\\\\<^sub>an. fds_nth f n / nat_power n s)\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfds_abs_converges f s\nfds_abs_converges f s = ((\\n. fds_nth f n / nat_power n s) abs_summable_on UNIV)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfds_abs_converges f s\nfds_abs_converges f s = ((\\n. fds_nth f n / nat_power n s) abs_summable_on UNIV)\n\ngoal (1 subgoal):\n 1. eval_fds f s = (\\\\<^sub>an. fds_nth f n / nat_power n s)\n[PROOF STEP]\nunfolding eval_fds_def fds_abs_converges_altdef\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. fds_nth f n / nat_power n s) abs_summable_on {1..}\n((\\n. fds_nth f n / nat_power n s) abs_summable_on {1..}) = ((\\n. fds_nth f n / nat_power n s) abs_summable_on UNIV)\n\ngoal (1 subgoal):\n 1. (\\n. fds_nth f n / nat_power n s) = (\\\\<^sub>an. fds_nth f n / nat_power n s)\n[PROOF STEP]\nby (intro infsetsum_nat' [symmetric]) simp_all\n[PROOF STATE]\nproof (state)\nthis:\neval_fds f s = (\\\\<^sub>an. fds_nth f n / nat_power n s)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1016, "file": "Dirichlet_Series_Dirichlet_Series_Analysis", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.839733963661418, "lm_q2_score": 0.8376199633332891, "lm_q1q2_score": 0.7033779318517944}} {"text": "[STATEMENT]\nlemma prod_less_prodinf2:\n fixes f :: \"nat \\ real\"\n assumes f: \"convergent_prod f\" and 1: \"\\m. m\\n \\ 1 \\ f m\" and 0: \"\\m. 0 < f m\" and i: \"n \\ i\" \"1 < f i\"\n shows \"prod f {.. prod f {.. prod f {.. prod f {.. prod f {.. prod f {.. prod f {..1 < f i; prod f {.. prod f {.. 1; 0 < prod f {.. \\ 1 * prod f {.. ?m \\ 1 \\ f ?m\n0 < f ?m\nn \\ i\n1 < f i\n\ngoal (1 subgoal):\n 1. prod f {.. prodinf f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod f {.. prodinf f\n[PROOF STEP]\nusing prod_le_prodinf[of f _ \"Suc i\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\f has_prod ?a; \\i. 0 \\ f i; \\ia. Suc i \\ ia \\ 1 \\ f ia\\ \\ prod f {.. prodinf f\n\ngoal (1 subgoal):\n 1. prod f {.. prodinf f\n[PROOF STEP]\nby (meson \"0\" \"1\" Suc_leD convergent_prod_has_prod f \\n \\ i\\ le_trans less_eq_real_def)\n[PROOF STATE]\nproof (state)\nthis:\nprod f {.. prodinf f\n\ngoal (1 subgoal):\n 1. prod f {.. prodinf f\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nprod f {.. prodinf f\n\ngoal (1 subgoal):\n 1. prod f {..y\\Y. card (all_edges_between X {y}))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (all_edges_between X Y) = (\\y\\Y. card (all_edges_between X {y}))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (all_edges_between X Y) = (\\y\\Y. card (all_edges_between X {y}))\n[PROOF STEP]\nhave \"all_edges_between X Y = (\\y\\Y. all_edges_between X {y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. all_edges_between X Y = (\\y\\Y. all_edges_between X {y})\n[PROOF STEP]\nby (auto simp: all_edges_between_def)\n[PROOF STATE]\nproof (state)\nthis:\nall_edges_between X Y = (\\y\\Y. all_edges_between X {y})\n\ngoal (1 subgoal):\n 1. card (all_edges_between X Y) = (\\y\\Y. card (all_edges_between X {y}))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nall_edges_between X Y = (\\y\\Y. all_edges_between X {y})\n\ngoal (1 subgoal):\n 1. card (all_edges_between X Y) = (\\y\\Y. card (all_edges_between X {y}))\n[PROOF STEP]\nhave \"disjoint_family_on (\\y. all_edges_between X {y}) Y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. disjoint_family_on (\\y. all_edges_between X {y}) Y\n[PROOF STEP]\nunfolding disjoint_family_on_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m\\Y. \\n\\Y. m \\ n \\ all_edges_between X {m} \\ all_edges_between X {n} = {}\n[PROOF STEP]\nby (auto simp: disjoint_family_on_def all_edges_between_def)\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on (\\y. all_edges_between X {y}) Y\n\ngoal (1 subgoal):\n 1. card (all_edges_between X Y) = (\\y\\Y. card (all_edges_between X {y}))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nall_edges_between X Y = (\\y\\Y. all_edges_between X {y})\ndisjoint_family_on (\\y. all_edges_between X {y}) Y\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nall_edges_between X Y = (\\y\\Y. all_edges_between X {y})\ndisjoint_family_on (\\y. all_edges_between X {y}) Y\n\ngoal (1 subgoal):\n 1. card (all_edges_between X Y) = (\\y\\Y. card (all_edges_between X {y}))\n[PROOF STEP]\nby (simp add: card_UN_disjoint' assms finite_all_edges_between')\n[PROOF STATE]\nproof (state)\nthis:\ncard (all_edges_between X Y) = (\\y\\Y. card (all_edges_between X {y}))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1086, "file": "Undirected_Graph_Theory_Undirected_Graph_Basics", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256631249076, "lm_q2_score": 0.8354835432479661, "lm_q1q2_score": 0.7033314878246665}} {"text": "[STATEMENT]\nlemma lemNullLine:\n assumes \"direction lineA = vecZero\" \n and \"inLine x lineA\"\n shows \"x = basepoint lineA\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x = basepoint lineA\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x = basepoint lineA\n[PROOF STEP]\ndefine bp where \"bp = basepoint lineA\"\n[PROOF STATE]\nproof (state)\nthis:\nbp = basepoint lineA\n\ngoal (1 subgoal):\n 1. x = basepoint lineA\n[PROOF STEP]\nhave \"collinear x (basepoint lineA) (basepoint lineA \\ direction lineA)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. collinear x (basepoint lineA) (basepoint lineA \\ direction lineA)\n[PROOF STEP]\nby (metis inLine.simps assms(2))\n[PROOF STATE]\nproof (state)\nthis:\ncollinear x (basepoint lineA) (basepoint lineA \\ direction lineA)\n\ngoal (1 subgoal):\n 1. x = basepoint lineA\n[PROOF STEP]\nhence \"collinear x bp (bp \\ vecZero)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncollinear x (basepoint lineA) (basepoint lineA \\ direction lineA)\n\ngoal (1 subgoal):\n 1. collinear x bp (bp \\ 0)\n[PROOF STEP]\nby (metis bp_def assms(1))\n[PROOF STATE]\nproof (state)\nthis:\ncollinear x bp (bp \\ 0)\n\ngoal (1 subgoal):\n 1. x = basepoint lineA\n[PROOF STEP]\nhence \"collinear x bp bp\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncollinear x bp (bp \\ 0)\n\ngoal (1 subgoal):\n 1. collinear x bp bp\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncollinear x bp bp\n\ngoal (1 subgoal):\n 1. x = basepoint lineA\n[PROOF STEP]\nhence \"\\a b.( (a + b = 1) \\ \n (positionVector x = a**(positionVector bp) \\ b**(positionVector bp)) )\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncollinear x bp bp\n\ngoal (1 subgoal):\n 1. \\a b. a + b = (1::'a) \\ positionVector x = a ** positionVector bp \\ b ** positionVector bp\n[PROOF STEP]\nby (metis collinear.simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\a b. a + b = (1::'a) \\ positionVector x = a ** positionVector bp \\ b ** positionVector bp\n\ngoal (1 subgoal):\n 1. x = basepoint lineA\n[PROOF STEP]\nhence \"positionVector x = positionVector bp\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a b. a + b = (1::'a) \\ positionVector x = a ** positionVector bp \\ b ** positionVector bp\n\ngoal (1 subgoal):\n 1. positionVector x = positionVector bp\n[PROOF STEP]\nby (metis lemScale1 lemAddOverScale)\n[PROOF STATE]\nproof (state)\nthis:\npositionVector x = positionVector bp\n\ngoal (1 subgoal):\n 1. x = basepoint lineA\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\npositionVector x = positionVector bp\n\ngoal (1 subgoal):\n 1. x = basepoint lineA\n[PROOF STEP]\nby (simp add: bp_def)\n[PROOF STATE]\nproof (state)\nthis:\nx = basepoint lineA\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1131, "file": "No_FTL_observers_SpaceTime", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256393148982, "lm_q2_score": 0.8354835411997897, "lm_q1q2_score": 0.703331466207588}} {"text": "[STATEMENT]\nlemma primepow_multD:\n assumes \"primepow (a * b :: nat)\"\n shows \"a = 1 \\ primepow a\" \"b = 1 \\ primepow b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a = 1 \\ primepow a &&& b = 1 \\ primepow b\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. a = 1 \\ primepow a\n 2. b = 1 \\ primepow b\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nprimepow (a * b)\n[PROOF STEP]\nobtain p k where k: \"k > 0\" \"a * b = p ^ k\" \"prime p\"\n[PROOF STATE]\nproof (prove)\nusing this:\nprimepow (a * b)\n\ngoal (1 subgoal):\n 1. (\\k p. \\0 < k; a * b = p ^ k; prime p\\ \\ thesis) \\ thesis\n[PROOF STEP]\nunfolding primepow_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\p k. prime p \\ 0 < k \\ a * b = p ^ k\n\ngoal (1 subgoal):\n 1. (\\k p. \\0 < k; a * b = p ^ k; prime p\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < k\na * b = p ^ k\nprime p\n\ngoal (2 subgoals):\n 1. a = 1 \\ primepow a\n 2. b = 1 \\ primepow b\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < k\na * b = p ^ k\nprime p\n[PROOF STEP]\nobtain i j where \"a = p ^ i\" \"b = p ^ j\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < k\na * b = p ^ k\nprime p\n\ngoal (1 subgoal):\n 1. (\\i j. \\a = p ^ i; b = p ^ j\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing prime_power_mult_nat[of p a b]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < k\na * b = p ^ k\nprime p\n\\prime p; a * b = p ^ ?k\\ \\ \\i j. a = p ^ i \\ b = p ^ j\n\ngoal (1 subgoal):\n 1. (\\i j. \\a = p ^ i; b = p ^ j\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\na = p ^ i\nb = p ^ j\n\ngoal (2 subgoals):\n 1. a = 1 \\ primepow a\n 2. b = 1 \\ primepow b\n[PROOF STEP]\nwith \\prime p\\\n[PROOF STATE]\nproof (chain)\npicking this:\nprime p\na = p ^ i\nb = p ^ j\n[PROOF STEP]\nshow \"a = 1 \\ primepow a\" \"b = 1 \\ primepow b\"\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\na = p ^ i\nb = p ^ j\n\ngoal (1 subgoal):\n 1. a = 1 \\ primepow a &&& b = 1 \\ primepow b\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na = 1 \\ primepow a\nb = 1 \\ primepow b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1094, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8354835371034368, "lm_q2_score": 0.8418256432832333, "lm_q1q2_score": 0.7033314660746518}} {"text": "[STATEMENT]\nlemma limpt_of_limpts: \"x islimpt {y. y islimpt S} \\ x islimpt S\"\n for x :: \"'a::metric_space\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x islimpt {y. y islimpt S} \\ x islimpt S\n[PROOF STEP]\napply (clarsimp simp add: islimpt_approachable)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\e. \\\\e>0. \\x'. (\\e>0. \\x'a\\S. x'a \\ x' \\ dist x'a x' < e) \\ x' \\ x \\ dist x' x < e; 0 < e\\ \\ \\x'\\S. x' \\ x \\ dist x' x < e\n[PROOF STEP]\napply (drule_tac x=\"e/2\" in spec)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\e. \\0 < e; 0 < e / 2 \\ (\\x'. (\\e>0. \\x'a\\S. x'a \\ x' \\ dist x'a x' < e) \\ x' \\ x \\ dist x' x < e / 2)\\ \\ \\x'\\S. x' \\ x \\ dist x' x < e\n[PROOF STEP]\napply (auto simp: simp del: less_divide_eq_numeral1)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\e x'. \\0 < e; \\e>0. \\x'a\\S. x'a \\ x' \\ dist x'a x' < e; x' \\ x; dist x' x < e / 2\\ \\ \\x'\\S. x' \\ x \\ dist x' x < e\n[PROOF STEP]\napply (drule_tac x=\"dist x' x\" in spec)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\e x'. \\0 < e; x' \\ x; dist x' x < e / 2; 0 < dist x' x \\ (\\x'a\\S. x'a \\ x' \\ dist x'a x' < dist x' x)\\ \\ \\x'\\S. x' \\ x \\ dist x' x < e\n[PROOF STEP]\napply (auto simp del: less_divide_eq_numeral1)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\e x' x'a. \\0 < e; x' \\ x; dist x' x < e / 2; x'a \\ S; x'a \\ x'; dist x'a x' < dist x' x\\ \\ \\x'\\S. x' \\ x \\ dist x' x < e\n[PROOF STEP]\napply metric\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 944, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835289107307, "lm_q2_score": 0.8418256412990658, "lm_q1q2_score": 0.7033314575200825}} {"text": "[STATEMENT]\nlemma plus_vec0[simp]: assumes \"vec nr u\" shows \"vec_plus u (vec0 nr) = u\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec_plus u (vec0 nr) = u\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nvec nr u\n\ngoal (1 subgoal):\n 1. vec_plus u (vec0 nr) = u\n[PROOF STEP]\nunfolding vec_def vec_plusI_def vec0I_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength u = nr\n\ngoal (1 subgoal):\n 1. map (\\xy. fst xy + snd xy) (zip u (replicate nr (0::'a))) = u\n[PROOF STEP]\nproof (induct nr arbitrary: u)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\u. length u = 0 \\ map (\\xy. fst xy + snd xy) (zip u (replicate 0 (0::'a))) = u\n 2. \\nr u. \\\\u. length u = nr \\ map (\\xy. fst xy + snd xy) (zip u (replicate nr (0::'a))) = u; length u = Suc nr\\ \\ map (\\xy. fst xy + snd xy) (zip u (replicate (Suc nr) (0::'a))) = u\n[PROOF STEP]\ncase (Suc nn)\n[PROOF STATE]\nproof (state)\nthis:\nlength ?u = nn \\ map (\\xy. fst xy + snd xy) (zip ?u (replicate nn (0::'a))) = ?u\nlength u = Suc nn\n\ngoal (2 subgoals):\n 1. \\u. length u = 0 \\ map (\\xy. fst xy + snd xy) (zip u (replicate 0 (0::'a))) = u\n 2. \\nr u. \\\\u. length u = nr \\ map (\\xy. fst xy + snd xy) (zip u (replicate nr (0::'a))) = u; length u = Suc nr\\ \\ map (\\xy. fst xy + snd xy) (zip u (replicate (Suc nr) (0::'a))) = u\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nlength ?u = nn \\ map (\\xy. fst xy + snd xy) (zip ?u (replicate nn (0::'a))) = ?u\nlength u = Suc nn\n\ngoal (1 subgoal):\n 1. map (\\xy. fst xy + snd xy) (zip u (replicate (Suc nn) (0::'a))) = u\n[PROOF STEP]\nby (cases u, auto)\n[PROOF STATE]\nproof (state)\nthis:\nmap (\\xy. fst xy + snd xy) (zip u (replicate (Suc nn) (0::'a))) = u\n\ngoal (1 subgoal):\n 1. \\u. length u = 0 \\ map (\\xy. fst xy + snd xy) (zip u (replicate 0 (0::'a))) = u\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 924, "file": "Matrix_Matrix_Legacy", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835207180243, "lm_q2_score": 0.8418256472515683, "lm_q1q2_score": 0.7033314555964699}} {"text": "[STATEMENT]\nlemma closure_sum:\n fixes S T :: \"'a::real_normed_vector set\"\n shows \"closure S + closure T \\ closure (S + T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. closure S + closure T \\ closure (S + T)\n[PROOF STEP]\nunfolding set_plus_image closure_Times [symmetric] split_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\p. fst p + snd p) ` closure (S \\ T) \\ closure ((\\p. fst p + snd p) ` (S \\ T))\n[PROOF STEP]\nby (intro closure_bounded_linear_image_subset bounded_linear_add\n bounded_linear_fst bounded_linear_snd)", "meta": {"llama_tokens": 224, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240895276223, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.7033123810604097}} {"text": "[STATEMENT]\nlemma card_mono_strict_subset:\n \"finite A \\ finite B \\ finite C \\ A \\ B \\ {} \\ C = A - B \\ card C < card A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite B; finite C; A \\ B \\ {}; C = A - B\\ \\ card C < card A\n[PROOF STEP]\nby (metis Diff_disjoint Diff_subset inf_commute less_le psubset_card_mono)", "meta": {"llama_tokens": 164, "file": "Timed_Automata_Regions", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240825770432, "lm_q2_score": 0.8128673178375735, "lm_q1q2_score": 0.7033123793328763}} {"text": "[STATEMENT]\nlemma bounded_degree_polynomials_length:\n \"bounded_degree_polynomials F n = {x. x \\ carrier (poly_ring F) \\ length x \\ n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bounded_degree_polynomials F n = {x \\ carrier (poly_ring F). length x \\ n}\n[PROOF STEP]\nunfolding bounded_degree_polynomials_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x \\ carrier (poly_ring F). degree x < n \\ x = []} = {x \\ carrier (poly_ring F). length x \\ n}\n[PROOF STEP]\nusing leI order_less_le_trans\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ ?x < ?y \\ ?y \\ ?x\n\\?x < ?y; ?y \\ ?z\\ \\ ?x < ?z\n\ngoal (1 subgoal):\n 1. {x \\ carrier (poly_ring F). degree x < n \\ x = []} = {x \\ carrier (poly_ring F). length x \\ n}\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 358, "file": "Interpolation_Polynomials_HOL_Algebra_Bounded_Degree_Polynomials", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577680977182187, "lm_q2_score": 0.8198933381139646, "lm_q1q2_score": 0.7032783489658557}} {"text": "[STATEMENT]\nlemma pr_index_enumerator_mono1: \"n1 < n2 \\ pr_index_enumerator n1 m < pr_index_enumerator n2 m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n1 < n2 \\ pr_index_enumerator n1 m < pr_index_enumerator n2 m\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. n1 < n2 \\ pr_index_enumerator n1 m < pr_index_enumerator n2 m\n[PROOF STEP]\nassume A: \"n1 < n2\"\n[PROOF STATE]\nproof (state)\nthis:\nn1 < n2\n\ngoal (1 subgoal):\n 1. n1 < n2 \\ pr_index_enumerator n1 m < pr_index_enumerator n2 m\n[PROOF STEP]\ndefine f where \"f x = pr_index_enumerator x m\" for x\n[PROOF STATE]\nproof (state)\nthis:\nf ?x = pr_index_enumerator ?x m\n\ngoal (1 subgoal):\n 1. n1 < n2 \\ pr_index_enumerator n1 m < pr_index_enumerator n2 m\n[PROOF STEP]\nhave f_inc: \"\\ x. f x < f (x+1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. f x < f (x + 1)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. f x < f (x + 1)\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. f x < f (x + 1)\n[PROOF STEP]\nshow \"f x < f (x+1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f x < f (x + 1)\n[PROOF STEP]\nby (unfold f_def, rule pr_index_enumerator_increase1)\n[PROOF STATE]\nproof (state)\nthis:\nf x < f (x + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\x. f x < f (x + 1)\n\ngoal (1 subgoal):\n 1. n1 < n2 \\ pr_index_enumerator n1 m < pr_index_enumerator n2 m\n[PROOF STEP]\nfrom f_inc\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x. f x < f (x + 1)\n[PROOF STEP]\nhave \"\\ x y. (x < y \\ f x < f y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. f x < f (x + 1)\n\ngoal (1 subgoal):\n 1. \\x y. x < y \\ f x < f y\n[PROOF STEP]\nby (rule f_inc_mono)\n[PROOF STATE]\nproof (state)\nthis:\n\\x y. x < y \\ f x < f y\n\ngoal (1 subgoal):\n 1. n1 < n2 \\ pr_index_enumerator n1 m < pr_index_enumerator n2 m\n[PROOF STEP]\nwith A f_def\n[PROOF STATE]\nproof (chain)\npicking this:\nn1 < n2\nf ?x = pr_index_enumerator ?x m\n\\x y. x < y \\ f x < f y\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn1 < n2\nf ?x = pr_index_enumerator ?x m\n\\x y. x < y \\ f x < f y\n\ngoal (1 subgoal):\n 1. pr_index_enumerator n1 m < pr_index_enumerator n2 m\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\npr_index_enumerator n1 m < pr_index_enumerator n2 m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1184, "file": "Recursion-Theory-I_PRecFun2", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438951104066293, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.703238553112003}} {"text": "[STATEMENT]\nlemma times_mtx3: \"mtx \n ([a\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3] # \n [a\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3] # \n [a\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3] # []) * mtx \n ([b\\<^sub>1\\<^sub>1, b\\<^sub>1\\<^sub>2, b\\<^sub>1\\<^sub>3] # \n [b\\<^sub>2\\<^sub>1, b\\<^sub>2\\<^sub>2, b\\<^sub>2\\<^sub>3] # \n [b\\<^sub>3\\<^sub>1, b\\<^sub>3\\<^sub>2, b\\<^sub>3\\<^sub>3] # []) = (mtx \n ([a\\<^sub>1\\<^sub>1*b\\<^sub>1\\<^sub>1+a\\<^sub>1\\<^sub>2*b\\<^sub>2\\<^sub>1+a\\<^sub>1\\<^sub>3*b\\<^sub>3\\<^sub>1, a\\<^sub>1\\<^sub>1*b\\<^sub>1\\<^sub>2+a\\<^sub>1\\<^sub>2*b\\<^sub>2\\<^sub>2+a\\<^sub>1\\<^sub>3*b\\<^sub>3\\<^sub>2, a\\<^sub>1\\<^sub>1*b\\<^sub>1\\<^sub>3+a\\<^sub>1\\<^sub>2*b\\<^sub>2\\<^sub>3+a\\<^sub>1\\<^sub>3*b\\<^sub>3\\<^sub>3] # \n [a\\<^sub>2\\<^sub>1*b\\<^sub>1\\<^sub>1+a\\<^sub>2\\<^sub>2*b\\<^sub>2\\<^sub>1+a\\<^sub>2\\<^sub>3*b\\<^sub>3\\<^sub>1, a\\<^sub>2\\<^sub>1*b\\<^sub>1\\<^sub>2+a\\<^sub>2\\<^sub>2*b\\<^sub>2\\<^sub>2+a\\<^sub>2\\<^sub>3*b\\<^sub>3\\<^sub>2, a\\<^sub>2\\<^sub>1*b\\<^sub>1\\<^sub>3+a\\<^sub>2\\<^sub>2*b\\<^sub>2\\<^sub>3+a\\<^sub>2\\<^sub>3*b\\<^sub>3\\<^sub>3] # \n [a\\<^sub>3\\<^sub>1*b\\<^sub>1\\<^sub>1+a\\<^sub>3\\<^sub>2*b\\<^sub>2\\<^sub>1+a\\<^sub>3\\<^sub>3*b\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>1*b\\<^sub>1\\<^sub>2+a\\<^sub>3\\<^sub>2*b\\<^sub>2\\<^sub>2+a\\<^sub>3\\<^sub>3*b\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>1*b\\<^sub>1\\<^sub>3+a\\<^sub>3\\<^sub>2*b\\<^sub>2\\<^sub>3+a\\<^sub>3\\<^sub>3*b\\<^sub>3\\<^sub>3] # [])::3 sq_mtx)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mtx [[a\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3], [a\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3], [a\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3]] * mtx [[b\\<^sub>1\\<^sub>1, b\\<^sub>1\\<^sub>2, b\\<^sub>1\\<^sub>3], [b\\<^sub>2\\<^sub>1, b\\<^sub>2\\<^sub>2, b\\<^sub>2\\<^sub>3], [b\\<^sub>3\\<^sub>1, b\\<^sub>3\\<^sub>2, b\\<^sub>3\\<^sub>3]] = mtx [[a\\<^sub>1\\<^sub>1 * b\\<^sub>1\\<^sub>1 + a\\<^sub>1\\<^sub>2 * b\\<^sub>2\\<^sub>1 + a\\<^sub>1\\<^sub>3 * b\\<^sub>3\\<^sub>1, a\\<^sub>1\\<^sub>1 * b\\<^sub>1\\<^sub>2 + a\\<^sub>1\\<^sub>2 * b\\<^sub>2\\<^sub>2 + a\\<^sub>1\\<^sub>3 * b\\<^sub>3\\<^sub>2, a\\<^sub>1\\<^sub>1 * b\\<^sub>1\\<^sub>3 + a\\<^sub>1\\<^sub>2 * b\\<^sub>2\\<^sub>3 + a\\<^sub>1\\<^sub>3 * b\\<^sub>3\\<^sub>3], [a\\<^sub>2\\<^sub>1 * b\\<^sub>1\\<^sub>1 + a\\<^sub>2\\<^sub>2 * b\\<^sub>2\\<^sub>1 + a\\<^sub>2\\<^sub>3 * b\\<^sub>3\\<^sub>1, a\\<^sub>2\\<^sub>1 * b\\<^sub>1\\<^sub>2 + a\\<^sub>2\\<^sub>2 * b\\<^sub>2\\<^sub>2 + a\\<^sub>2\\<^sub>3 * b\\<^sub>3\\<^sub>2, a\\<^sub>2\\<^sub>1 * b\\<^sub>1\\<^sub>3 + a\\<^sub>2\\<^sub>2 * b\\<^sub>2\\<^sub>3 + a\\<^sub>2\\<^sub>3 * b\\<^sub>3\\<^sub>3], [a\\<^sub>3\\<^sub>1 * b\\<^sub>1\\<^sub>1 + a\\<^sub>3\\<^sub>2 * b\\<^sub>2\\<^sub>1 + a\\<^sub>3\\<^sub>3 * b\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>1 * b\\<^sub>1\\<^sub>2 + a\\<^sub>3\\<^sub>2 * b\\<^sub>2\\<^sub>2 + a\\<^sub>3\\<^sub>3 * b\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>1 * b\\<^sub>1\\<^sub>3 + a\\<^sub>3\\<^sub>2 * b\\<^sub>2\\<^sub>3 + a\\<^sub>3\\<^sub>3 * b\\<^sub>3\\<^sub>3]]\n[PROOF STEP]\nunfolding sq_mtx_times_eq\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. to_mtx (\\i j. \\k\\UNIV. mtx [[a\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3], [a\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3], [a\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3]] $$ i $ k * mtx [[b\\<^sub>1\\<^sub>1, b\\<^sub>1\\<^sub>2, b\\<^sub>1\\<^sub>3], [b\\<^sub>2\\<^sub>1, b\\<^sub>2\\<^sub>2, b\\<^sub>2\\<^sub>3], [b\\<^sub>3\\<^sub>1, b\\<^sub>3\\<^sub>2, b\\<^sub>3\\<^sub>3]] $$ k $ j) = mtx [[a\\<^sub>1\\<^sub>1 * b\\<^sub>1\\<^sub>1 + a\\<^sub>1\\<^sub>2 * b\\<^sub>2\\<^sub>1 + a\\<^sub>1\\<^sub>3 * b\\<^sub>3\\<^sub>1, a\\<^sub>1\\<^sub>1 * b\\<^sub>1\\<^sub>2 + a\\<^sub>1\\<^sub>2 * b\\<^sub>2\\<^sub>2 + a\\<^sub>1\\<^sub>3 * b\\<^sub>3\\<^sub>2, a\\<^sub>1\\<^sub>1 * b\\<^sub>1\\<^sub>3 + a\\<^sub>1\\<^sub>2 * b\\<^sub>2\\<^sub>3 + a\\<^sub>1\\<^sub>3 * b\\<^sub>3\\<^sub>3], [a\\<^sub>2\\<^sub>1 * b\\<^sub>1\\<^sub>1 + a\\<^sub>2\\<^sub>2 * b\\<^sub>2\\<^sub>1 + a\\<^sub>2\\<^sub>3 * b\\<^sub>3\\<^sub>1, a\\<^sub>2\\<^sub>1 * b\\<^sub>1\\<^sub>2 + a\\<^sub>2\\<^sub>2 * b\\<^sub>2\\<^sub>2 + a\\<^sub>2\\<^sub>3 * b\\<^sub>3\\<^sub>2, a\\<^sub>2\\<^sub>1 * b\\<^sub>1\\<^sub>3 + a\\<^sub>2\\<^sub>2 * b\\<^sub>2\\<^sub>3 + a\\<^sub>2\\<^sub>3 * b\\<^sub>3\\<^sub>3], [a\\<^sub>3\\<^sub>1 * b\\<^sub>1\\<^sub>1 + a\\<^sub>3\\<^sub>2 * b\\<^sub>2\\<^sub>1 + a\\<^sub>3\\<^sub>3 * b\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>1 * b\\<^sub>1\\<^sub>2 + a\\<^sub>3\\<^sub>2 * b\\<^sub>2\\<^sub>2 + a\\<^sub>3\\<^sub>3 * b\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>1 * b\\<^sub>1\\<^sub>3 + a\\<^sub>3\\<^sub>2 * b\\<^sub>2\\<^sub>3 + a\\<^sub>3\\<^sub>3 * b\\<^sub>3\\<^sub>3]]\n[PROOF STEP]\nunfolding UNIV_3\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. to_mtx (\\i j. \\k\\{1, 2, 3}. mtx [[a\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3], [a\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3], [a\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3]] $$ i $ k * mtx [[b\\<^sub>1\\<^sub>1, b\\<^sub>1\\<^sub>2, b\\<^sub>1\\<^sub>3], [b\\<^sub>2\\<^sub>1, b\\<^sub>2\\<^sub>2, b\\<^sub>2\\<^sub>3], [b\\<^sub>3\\<^sub>1, b\\<^sub>3\\<^sub>2, b\\<^sub>3\\<^sub>3]] $$ k $ j) = mtx [[a\\<^sub>1\\<^sub>1 * b\\<^sub>1\\<^sub>1 + a\\<^sub>1\\<^sub>2 * b\\<^sub>2\\<^sub>1 + a\\<^sub>1\\<^sub>3 * b\\<^sub>3\\<^sub>1, a\\<^sub>1\\<^sub>1 * b\\<^sub>1\\<^sub>2 + a\\<^sub>1\\<^sub>2 * b\\<^sub>2\\<^sub>2 + a\\<^sub>1\\<^sub>3 * b\\<^sub>3\\<^sub>2, a\\<^sub>1\\<^sub>1 * b\\<^sub>1\\<^sub>3 + a\\<^sub>1\\<^sub>2 * b\\<^sub>2\\<^sub>3 + a\\<^sub>1\\<^sub>3 * b\\<^sub>3\\<^sub>3], [a\\<^sub>2\\<^sub>1 * b\\<^sub>1\\<^sub>1 + a\\<^sub>2\\<^sub>2 * b\\<^sub>2\\<^sub>1 + a\\<^sub>2\\<^sub>3 * b\\<^sub>3\\<^sub>1, a\\<^sub>2\\<^sub>1 * b\\<^sub>1\\<^sub>2 + a\\<^sub>2\\<^sub>2 * b\\<^sub>2\\<^sub>2 + a\\<^sub>2\\<^sub>3 * b\\<^sub>3\\<^sub>2, a\\<^sub>2\\<^sub>1 * b\\<^sub>1\\<^sub>3 + a\\<^sub>2\\<^sub>2 * b\\<^sub>2\\<^sub>3 + a\\<^sub>2\\<^sub>3 * b\\<^sub>3\\<^sub>3], [a\\<^sub>3\\<^sub>1 * b\\<^sub>1\\<^sub>1 + a\\<^sub>3\\<^sub>2 * b\\<^sub>2\\<^sub>1 + a\\<^sub>3\\<^sub>3 * b\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>1 * b\\<^sub>1\\<^sub>2 + a\\<^sub>3\\<^sub>2 * b\\<^sub>2\\<^sub>2 + a\\<^sub>3\\<^sub>3 * b\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>1 * b\\<^sub>1\\<^sub>3 + a\\<^sub>3\\<^sub>2 * b\\<^sub>2\\<^sub>3 + a\\<^sub>3\\<^sub>3 * b\\<^sub>3\\<^sub>3]]\n[PROOF STEP]\nby (simp add: sq_mtx_eq_iff)", "meta": {"llama_tokens": 3205, "file": "Matrices_for_ODEs_SQ_MTX", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505376715775, "lm_q2_score": 0.7772998560157665, "lm_q1q2_score": 0.7031847326767029}} {"text": "[STATEMENT]\nlemma sigma_sets_singletons_and_empty:\n assumes \"countable M\"\n shows \"sigma_sets M (insert {} ((\\k. {k}) ` M)) = Pow M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sigma_sets M (insert {} ((\\k. {k}) ` M)) = Pow M\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sigma_sets M (insert {} ((\\k. {k}) ` M)) = Pow M\n[PROOF STEP]\nhave \"sigma_sets M ((\\k. {k}) ` M) = Pow M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sigma_sets M ((\\k. {k}) ` M) = Pow M\n[PROOF STEP]\nusing assms sigma_sets_singletons\n[PROOF STATE]\nproof (prove)\nusing this:\ncountable M\ncountable ?S \\ sigma_sets ?S ((\\s. {s}) ` ?S) = Pow ?S\n\ngoal (1 subgoal):\n 1. sigma_sets M ((\\k. {k}) ` M) = Pow M\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsigma_sets M ((\\k. {k}) ` M) = Pow M\n\ngoal (1 subgoal):\n 1. sigma_sets M (insert {} ((\\k. {k}) ` M)) = Pow M\n[PROOF STEP]\nhence \"Pow M \\ sigma_sets M (insert {} ((\\k. {k}) ` M))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsigma_sets M ((\\k. {k}) ` M) = Pow M\n\ngoal (1 subgoal):\n 1. Pow M \\ sigma_sets M (insert {} ((\\k. {k}) ` M))\n[PROOF STEP]\nby (metis sigma_sets_subseteq subset_insertI)\n[PROOF STATE]\nproof (state)\nthis:\nPow M \\ sigma_sets M (insert {} ((\\k. {k}) ` M))\n\ngoal (1 subgoal):\n 1. sigma_sets M (insert {} ((\\k. {k}) ` M)) = Pow M\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nPow M \\ sigma_sets M (insert {} ((\\k. {k}) ` M))\n\ngoal (1 subgoal):\n 1. sigma_sets M (insert {} ((\\k. {k}) ` M)) = Pow M\n[PROOF STEP]\nhave \"(insert {} ((\\k. {k}) ` M)) \\ Pow M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. insert {} ((\\k. {k}) ` M) \\ Pow M\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ninsert {} ((\\k. {k}) ` M) \\ Pow M\n\ngoal (1 subgoal):\n 1. sigma_sets M (insert {} ((\\k. {k}) ` M)) = Pow M\n[PROOF STEP]\nhence \"sigma_sets M (insert {} ((\\k. {k}) ` M)) \\ Pow M\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninsert {} ((\\k. {k}) ` M) \\ Pow M\n\ngoal (1 subgoal):\n 1. sigma_sets M (insert {} ((\\k. {k}) ` M)) \\ Pow M\n[PROOF STEP]\nby (meson sigma_algebra.sigma_sets_subset sigma_algebra_Pow)\n[PROOF STATE]\nproof (state)\nthis:\nsigma_sets M (insert {} ((\\k. {k}) ` M)) \\ Pow M\n\ngoal (1 subgoal):\n 1. sigma_sets M (insert {} ((\\k. {k}) ` M)) = Pow M\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nPow M \\ sigma_sets M (insert {} ((\\k. {k}) ` M))\nsigma_sets M (insert {} ((\\k. {k}) ` M)) \\ Pow M\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nPow M \\ sigma_sets M (insert {} ((\\k. {k}) ` M))\nsigma_sets M (insert {} ((\\k. {k}) ` M)) \\ Pow M\n\ngoal (1 subgoal):\n 1. sigma_sets M (insert {} ((\\k. {k}) ` M)) = Pow M\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\nsigma_sets M (insert {} ((\\k. {k}) ` M)) = Pow M\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1390, "file": "Universal_Hash_Families_Preliminary_Results", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424334245618, "lm_q2_score": 0.8311430499496095, "lm_q1q2_score": 0.7030991741982848}} {"text": "[STATEMENT]\nlemma upper_bound2:\n fixes b::nat\n and c::int\n assumes \"b > 0\"\n and \"c < 2^b\"\n and \"c \\ 0\"\n shows \"c - (2^(b-1)) < 2^(b-1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c - 2 ^ (b - 1) < 2 ^ (b - 1)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. c - 2 ^ (b - 1) < 2 ^ (b - 1)\n[PROOF STEP]\nhave a1: \"\\P. (\\b::nat. P b) \\ (\\b>0. P ((b-1)::nat))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\P. \\b. P b \\ \\b>0. P (b - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\b. ?P b \\ \\b>0. ?P (b - 1)\n\ngoal (1 subgoal):\n 1. c - 2 ^ (b - 1) < 2 ^ (b - 1)\n[PROOF STEP]\nhave b2: \"\\b::nat. (\\(c::int)<2^(Suc b). c\\0 \\ (c - 2^b) < 2^b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\b c. c < 2 ^ Suc b \\ 0 \\ c \\ c - 2 ^ b < 2 ^ b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\b c. c < 2 ^ Suc b \\ 0 \\ c \\ c - 2 ^ b < 2 ^ b\n\ngoal (1 subgoal):\n 1. c - 2 ^ (b - 1) < 2 ^ (b - 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c - 2 ^ (b - 1) < 2 ^ (b - 1)\n[PROOF STEP]\nusing a1[OF b2] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\b>0. \\c<2 ^ Suc (b - 1). 0 \\ c \\ c - 2 ^ (b - 1) < 2 ^ (b - 1)\n0 < b\nc < 2 ^ b\n0 \\ c\n\ngoal (1 subgoal):\n 1. c - 2 ^ (b - 1) < 2 ^ (b - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nc - 2 ^ (b - 1) < 2 ^ (b - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 841, "file": "Solidity_Valuetypes", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424217727027, "lm_q2_score": 0.8311430415844384, "lm_q1q2_score": 0.70309915743747}} {"text": "[STATEMENT]\nlemma (in group) weak_group_morphism_ker:\n assumes \"weak_group_morphism f H G\" shows \"kernel G (image_group f G) f = H\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. kernel G (image_group f G) f = H\n[PROOF STEP]\nusing vimage_eq_rcoset(1)[OF assms one_closed] weak_group_morphism.axioms(1)[OF assms(1)]\n[PROOF STATE]\nproof (prove)\nusing this:\n{b \\ carrier G. f b = f \\} = H #> \\\nH \\ G\n\ngoal (1 subgoal):\n 1. kernel G (image_group f G) f = H\n[PROOF STEP]\nby (simp add: image_group_def kernel_def normal_def subgroup.subset)", "meta": {"llama_tokens": 231, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970904940925, "lm_q2_score": 0.7981867825403176, "lm_q1q2_score": 0.7030405957323527}} {"text": "[STATEMENT]\nlemma roots_of_complex_rf_polys: \n \"set (roots_of_complex_rf_polys ps) = {x. \\ p \\ set (polys_rf ps). poly (poly_rf p) x = 0 }\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (roots_of_complex_rf_polys ps) = {x. \\p\\set (polys_rf ps). poly (poly_rf p) x = 0}\n[PROOF STEP]\nunfolding roots_of_complex_rf_polys_def set_concat set_map image_comp o_def\n roots_of_complex_rf_poly\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\set (polys_rf ps). {xa. poly (poly_rf x) xa = 0}) = {x. \\p\\set (polys_rf ps). poly (poly_rf p) x = 0}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 282, "file": "Factor_Algebraic_Polynomial_Roots_of_Real_Complex_Poly", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970873650401, "lm_q2_score": 0.7981867849406659, "lm_q1q2_score": 0.7030405953490042}} {"text": "[STATEMENT]\nlemma Holder_inequality_sum:\n assumes \"p > (0::real)\" \"q > 0\" \"1/p + 1/q = 1\"\n assumes \"finite A\"\n shows \"\\\\x\\A. f x * g x\\ \\ (\\x\\A. \\f x\\ powr p) powr (1/p) * (\\x\\A. \\g x\\ powr q) powr (1/q)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x\\A. f x * g x\\ \\ (\\x\\A. \\f x\\ powr p) powr (1 / p) * (\\x\\A. \\g x\\ powr q) powr (1 / q)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\x\\A. f x * g x\\ \\ (\\x\\A. \\f x\\ powr p) powr (1 / p) * (\\x\\A. \\g x\\ powr q) powr (1 / q)\n[PROOF STEP]\nhave \"\\LINT x|count_space A. f x * g x\\ \\ \n (LINT x|count_space A. \\f x\\ powr p) powr (1 / p) * \n (LINT x|count_space A. \\g x\\ powr q) powr (1 / q)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\LINT x|count_space A. f x * g x\\ \\ (LINT x|count_space A. \\f x\\ powr p) powr (1 / p) * (LINT x|count_space A. \\g x\\ powr q) powr (1 / q)\n[PROOF STEP]\nusing assms integrable_count_space\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < p\n0 < q\n1 / p + 1 / q = 1\nfinite A\nfinite ?X \\ integrable (count_space ?X) ?f\n\ngoal (1 subgoal):\n 1. \\LINT x|count_space A. f x * g x\\ \\ (LINT x|count_space A. \\f x\\ powr p) powr (1 / p) * (LINT x|count_space A. \\g x\\ powr q) powr (1 / q)\n[PROOF STEP]\nby (intro Lp.Holder_inequality, auto)\n[PROOF STATE]\nproof (state)\nthis:\n\\LINT x|count_space A. f x * g x\\ \\ (LINT x|count_space A. \\f x\\ powr p) powr (1 / p) * (LINT x|count_space A. \\g x\\ powr q) powr (1 / q)\n\ngoal (1 subgoal):\n 1. \\\\x\\A. f x * g x\\ \\ (\\x\\A. \\f x\\ powr p) powr (1 / p) * (\\x\\A. \\g x\\ powr q) powr (1 / q)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\LINT x|count_space A. f x * g x\\ \\ (LINT x|count_space A. \\f x\\ powr p) powr (1 / p) * (LINT x|count_space A. \\g x\\ powr q) powr (1 / q)\n\ngoal (1 subgoal):\n 1. \\\\x\\A. f x * g x\\ \\ (\\x\\A. \\f x\\ powr p) powr (1 / p) * (\\x\\A. \\g x\\ powr q) powr (1 / q)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\LINT x|count_space A. f x * g x\\ \\ (LINT x|count_space A. \\f x\\ powr p) powr (1 / p) * (LINT x|count_space A. \\g x\\ powr q) powr (1 / q)\n0 < p\n0 < q\n1 / p + 1 / q = 1\nfinite A\n\ngoal (1 subgoal):\n 1. \\\\x\\A. f x * g x\\ \\ (\\x\\A. \\f x\\ powr p) powr (1 / p) * (\\x\\A. \\g x\\ powr q) powr (1 / q)\n[PROOF STEP]\nby (simp add: lebesgue_integral_count_space_finite[symmetric])\n[PROOF STATE]\nproof (state)\nthis:\n\\\\x\\A. f x * g x\\ \\ (\\x\\A. \\f x\\ powr p) powr (1 / p) * (\\x\\A. \\g x\\ powr q) powr (1 / q)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1510, "file": "Frequency_Moments_Frequency_Moment_k", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970842359877, "lm_q2_score": 0.7981867825403177, "lm_q1q2_score": 0.7030405907372163}} {"text": "[STATEMENT]\nlemma norm_cblinfun_ceuclidean_le:\n fixes a::\"'a::ceuclidean_space \\\\<^sub>C\\<^sub>L 'b::complex_normed_vector\"\n shows \"norm a \\ sum (\\x. norm (a x)) CBasis\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm a \\ (\\x\\CBasis. norm (cblinfun_apply a x))\n[PROOF STEP]\napply (rule norm_cblinfun_bound)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. 0 \\ (\\x\\CBasis. norm (cblinfun_apply a x))\n 2. \\x. norm (cblinfun_apply a x) \\ (\\x\\CBasis. norm (cblinfun_apply a x)) * norm x\n[PROOF STEP]\napply (simp add: sum_nonneg)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. norm (cblinfun_apply a x) \\ (\\x\\CBasis. norm (cblinfun_apply a x)) * norm x\n[PROOF STEP]\napply (subst ceuclidean_representation[symmetric, where 'a='a])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. norm (cblinfun_apply a (\\b\\CBasis. (b \\\\<^sub>C x) *\\<^sub>C b)) \\ (\\x\\CBasis. norm (cblinfun_apply a x)) * norm x\n[PROOF STEP]\napply (simp only: cblinfun.cbilinear_simps sum_distrib_right)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. norm (\\i\\CBasis. (i \\\\<^sub>C x) *\\<^sub>C cblinfun_apply a i) \\ (\\n\\CBasis. norm (cblinfun_apply a n) * norm x)\n[PROOF STEP]\napply (rule order.trans[OF norm_sum sum_mono])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x n. n \\ CBasis \\ norm ((n \\\\<^sub>C x) *\\<^sub>C cblinfun_apply a n) \\ norm (cblinfun_apply a n) * norm x\n[PROOF STEP]\napply (simp add: abs_mult mult_right_mono ac_simps CBasis_le_norm)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x n. n \\ CBasis \\ norm (cblinfun_apply a n) * cmod (n \\\\<^sub>C x) \\ norm x * norm (cblinfun_apply a n)\n[PROOF STEP]\nby (metis complex_inner_class.Cauchy_Schwarz_ineq2 mult.commute mult.left_neutral mult_right_mono norm_CBasis norm_ge_zero)", "meta": {"llama_tokens": 846, "file": "Complex_Bounded_Operators_Complex_Bounded_Linear_Function0", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970873650401, "lm_q2_score": 0.798186775339273, "lm_q1q2_score": 0.7030405868921252}} {"text": "[STATEMENT]\nlemma hnorm_triangle_ineq4: \"\\a b::'a::real_normed_vector star. hnorm (a - b) \\ hnorm a + hnorm b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a b. hnorm (a - b) \\ hnorm a + hnorm b\n[PROOF STEP]\nby transfer (rule norm_triangle_ineq4)", "meta": {"llama_tokens": 118, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970779778825, "lm_q2_score": 0.7981867825403176, "lm_q1q2_score": 0.7030405857420793}} {"text": "[STATEMENT]\nlemma Cauchy_iff: \"Cauchy X \\ (\\e>0. \\M. \\m\\M. \\n\\M. norm (X m - X n) < e)\"\n for X :: \"nat \\ 'a::real_normed_vector\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Cauchy X = (\\e>0. \\M. \\m\\M. \\n\\M. norm (X m - X n) < e)\n[PROOF STEP]\nunfolding Cauchy_def dist_norm\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\e>0. \\M. \\m\\M. \\n\\M. norm (X m - X n) < e) = (\\e>0. \\M. \\m\\M. \\n\\M. norm (X m - X n) < e)\n[PROOF STEP]\n..", "meta": {"llama_tokens": 290, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970779778825, "lm_q2_score": 0.7981867801399694, "lm_q1q2_score": 0.7030405836278596}} {"text": "[STATEMENT]\nlemma connected_nest:\n fixes S :: \"'a::linorder \\ 'b::euclidean_space set\"\n assumes S: \"\\n. compact(S n)\" \"\\n. connected(S n)\"\n and nest: \"\\m n. m \\ n \\ S n \\ S m\"\n shows \"connected(\\ (range S))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. connected (\\ (range S))\n[PROOF STEP]\nproof (rule connected_chain)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\Sa. Sa \\ range S \\ compact Sa \\ connected Sa\n 2. \\Sa T. Sa \\ range S \\ T \\ range S \\ Sa \\ T \\ T \\ Sa\n[PROOF STEP]\nshow \"\\A T. A \\ range S \\ T \\ range S \\ A \\ T \\ T \\ A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A T. A \\ range S \\ T \\ range S \\ A \\ T \\ T \\ A\n[PROOF STEP]\nby (metis image_iff le_cases nest)\n[PROOF STATE]\nproof (state)\nthis:\n?A \\ range S \\ ?T \\ range S \\ ?A \\ ?T \\ ?T \\ ?A\n\ngoal (1 subgoal):\n 1. \\Sa. Sa \\ range S \\ compact Sa \\ connected Sa\n[PROOF STEP]\nqed (use S in blast)", "meta": {"llama_tokens": 466, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.880797071719777, "lm_q2_score": 0.7981867849406659, "lm_q1q2_score": 0.7030405828611619}} {"text": "[STATEMENT]\nlemma poly_add_assoc: \"poly_add p1 (poly_add p2 p3) =p poly_add (poly_add p1 p2) p3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly_add p1 (poly_add p2 p3) =p poly_add (poly_add p1 p2) p3\n[PROOF STEP]\nunfolding eq_poly_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\. eval_poly \\ (poly_add p1 (poly_add p2 p3)) = eval_poly \\ (poly_add (poly_add p1 p2) p3)\n[PROOF STEP]\nby (auto simp: field_simps)", "meta": {"llama_tokens": 207, "file": "Polynomials_Polynomials", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970842359877, "lm_q2_score": 0.7981867705385762, "lm_q1q2_score": 0.7030405801661173}} {"text": "[STATEMENT]\nlemma Cauchy_iff: \"Cauchy X \\ (\\e>0. \\M. \\m\\M. \\n\\M. norm (X m - X n) < e)\"\n for X :: \"nat \\ 'a::real_normed_vector\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Cauchy X = (\\e>0. \\M. \\m\\M. \\n\\M. norm (X m - X n) < e)\n[PROOF STEP]\nunfolding Cauchy_def dist_norm\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\e>0. \\M. \\m\\M. \\n\\M. norm (X m - X n) < e) = (\\e>0. \\M. \\m\\M. \\n\\M. norm (X m - X n) < e)\n[PROOF STEP]\n..", "meta": {"llama_tokens": 290, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970779778825, "lm_q2_score": 0.7981867729389246, "lm_q1q2_score": 0.7030405772852004}} {"text": "[STATEMENT]\nlemma card_disjoint_shuffles:\n assumes \"set xs \\ set ys = {}\"\n shows \"card (shuffles xs ys) = (length xs + length ys) choose length xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles xs ys) = length xs + length ys choose length xs\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nset xs \\ set ys = {}\n\ngoal (1 subgoal):\n 1. card (shuffles xs ys) = length xs + length ys choose length xs\n[PROOF STEP]\nproof (induction xs ys rule: shuffles.induct)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\ncase (3 x xs y ys)\n[PROOF STATE]\nproof (state)\nthis:\nset xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\nset (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n[PROOF STEP]\nby (rule shuffles.simps)\n[PROOF STATE]\nproof (state)\nthis:\nshuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nshuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card \\ = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n[PROOF STEP]\nby (rule card_Un_disjoint) (insert \"3.prems\", auto)\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n[PROOF STEP]\nby (rule card_image) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"\\ = (length xs + length (y # ys)) choose length xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n[PROOF STEP]\nusing \"3.prems\"\n[PROOF STATE]\nproof (prove)\nusing this:\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (1 subgoal):\n 1. card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n[PROOF STEP]\nby (intro \"3.IH\") auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n[PROOF STEP]\nby (rule card_image) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"\\ = (length (x # xs) + length ys) choose length (x # xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n[PROOF STEP]\nusing \"3.prems\"\n[PROOF STATE]\nproof (prove)\nusing this:\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n[PROOF STEP]\nby (intro \"3.IH\") auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"length xs + length (y # ys) choose length xs + \\ =\n (length (x # xs) + length (y # ys)) choose length (x # xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length xs + length (y # ys) choose length xs + (length (x # xs) + length ys choose length (x # xs)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlength xs + length (y # ys) choose length xs + (length (x # xs) + length ys choose length (x # xs)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (2 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 6519, "file": null, "length": 29, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970654616711, "lm_q2_score": 0.7981867825403177, "lm_q1q2_score": 0.7030405757518049}} {"text": "[STATEMENT]\nlemma mod_minus: assumes \"a - b > 0\" and \"c - d > 0\" \n shows \"(a - b + (c - d mod q)) mod q = (a - b + (c - d)) mod q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a - b + (c - d mod q)) mod q = (a - b + (c - d)) mod q\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a - b\n0 < c - d\n\ngoal (1 subgoal):\n 1. (a - b + (c - d mod q)) mod q = (a - b + (c - d)) mod q\n[PROOF STEP]\nby (metis cong_def minus_mod mod_add_right_eq zero_less_diff)", "meta": {"llama_tokens": 224, "file": "Multi_Party_Computation_Secure_Multiplication", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970779778825, "lm_q2_score": 0.798186768138228, "lm_q1q2_score": 0.7030405730567608}} {"text": "[STATEMENT]\nlemma MinPredicate:\nfixes \n P::\"nat \\ bool\"\nassumes\n \"\\ n . P n\"\nshows \n \"(\\ n0 . (P n0) \\ (\\ n' . (P n') \\ (n' \\ n0)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n0. P n0 \\ (\\n'. P n' \\ n0 \\ n')\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n. P n\n\ngoal (1 subgoal):\n 1. \\n0. P n0 \\ (\\n'. P n' \\ n0 \\ n')\n[PROOF STEP]\nby (metis LeastI2_wellorder Suc_n_not_le_n)", "meta": {"llama_tokens": 247, "file": "FLP_ListUtilities", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.880797071719777, "lm_q2_score": 0.7981867729389246, "lm_q1q2_score": 0.7030405722900633}} {"text": "[STATEMENT]\nlemma card_disjoint_shuffles:\n assumes \"set xs \\ set ys = {}\"\n shows \"card (shuffles xs ys) = (length xs + length ys) choose length xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles xs ys) = length xs + length ys choose length xs\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nset xs \\ set ys = {}\n\ngoal (1 subgoal):\n 1. card (shuffles xs ys) = length xs + length ys choose length xs\n[PROOF STEP]\nproof (induction xs ys rule: shuffles.induct)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\ncase (3 x xs y ys)\n[PROOF STATE]\nproof (state)\nthis:\nset xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\nset (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n[PROOF STEP]\nby (rule shuffles.simps)\n[PROOF STATE]\nproof (state)\nthis:\nshuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nshuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card \\ = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n[PROOF STEP]\nby (rule card_Un_disjoint) (insert \"3.prems\", auto)\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n[PROOF STEP]\nby (rule card_image) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"\\ = (length xs + length (y # ys)) choose length xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n[PROOF STEP]\nusing \"3.prems\"\n[PROOF STATE]\nproof (prove)\nusing this:\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (1 subgoal):\n 1. card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n[PROOF STEP]\nby (intro \"3.IH\") auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n[PROOF STEP]\nby (rule card_image) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"\\ = (length (x # xs) + length ys) choose length (x # xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n[PROOF STEP]\nusing \"3.prems\"\n[PROOF STATE]\nproof (prove)\nusing this:\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n[PROOF STEP]\nby (intro \"3.IH\") auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"length xs + length (y # ys) choose length xs + \\ =\n (length (x # xs) + length (y # ys)) choose length (x # xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length xs + length (y # ys) choose length xs + (length (x # xs) + length ys choose length (x # xs)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlength xs + length (y # ys) choose length xs + (length (x # xs) + length ys choose length (x # xs)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (2 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 6519, "file": null, "length": 29, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970654616711, "lm_q2_score": 0.7981867777396211, "lm_q1q2_score": 0.7030405715233654}} {"text": "[STATEMENT]\nlemma sum_div_reduce:\n fixes d :: nat and f :: \"nat \\ complex\"\n assumes \"d dvd k\" \"d > 0\" \n shows \"(\\n | n \\ {1..k} \\ d dvd n. f n) = (\\c \\ {1..k div d}. f (c*d))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n | n \\ {1..k} \\ d dvd n. f n) = (\\c = 1..k div d. f (c * d))\n[PROOF STEP]\nby (rule sum.reindex_bij_witness[of _ \"\\k. k * d\" \"\\k. k div d\"])\n (use assms in \\fastforce simp: div_le_mono\\)+", "meta": {"llama_tokens": 231, "file": "Gauss_Sums_Gauss_Sums_Auxiliary", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970654616711, "lm_q2_score": 0.7981867753392728, "lm_q1q2_score": 0.7030405694091457}} {"text": "[STATEMENT]\nlemma card_disjoint_shuffles:\n assumes \"set xs \\ set ys = {}\"\n shows \"card (shuffles xs ys) = (length xs + length ys) choose length xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles xs ys) = length xs + length ys choose length xs\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nset xs \\ set ys = {}\n\ngoal (1 subgoal):\n 1. card (shuffles xs ys) = length xs + length ys choose length xs\n[PROOF STEP]\nproof (induction xs ys rule: shuffles.induct)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\ncase (3 x xs y ys)\n[PROOF STATE]\nproof (state)\nthis:\nset xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\nset (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n[PROOF STEP]\nby (rule shuffles.simps)\n[PROOF STATE]\nproof (state)\nthis:\nshuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nshuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card \\ = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n[PROOF STEP]\nby (rule card_Un_disjoint) (insert \"3.prems\", auto)\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n[PROOF STEP]\nby (rule card_image) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"\\ = (length xs + length (y # ys)) choose length xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n[PROOF STEP]\nusing \"3.prems\"\n[PROOF STATE]\nproof (prove)\nusing this:\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (1 subgoal):\n 1. card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n[PROOF STEP]\nby (intro \"3.IH\") auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n[PROOF STEP]\nby (rule card_image) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"\\ = (length (x # xs) + length ys) choose length (x # xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n[PROOF STEP]\nusing \"3.prems\"\n[PROOF STATE]\nproof (prove)\nusing this:\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n[PROOF STEP]\nby (intro \"3.IH\") auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"length xs + length (y # ys) choose length xs + \\ =\n (length (x # xs) + length (y # ys)) choose length (x # xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length xs + length (y # ys) choose length xs + (length (x # xs) + length ys choose length (x # xs)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlength xs + length (y # ys) choose length xs + (length (x # xs) + length ys choose length (x # xs)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (2 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 6519, "file": null, "length": 29, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970654616711, "lm_q2_score": 0.7981867705385762, "lm_q1q2_score": 0.7030405651807062}} {"text": "[STATEMENT]\nlemma card_disjoint_shuffles:\n assumes \"set xs \\ set ys = {}\"\n shows \"card (shuffles xs ys) = (length xs + length ys) choose length xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles xs ys) = length xs + length ys choose length xs\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nset xs \\ set ys = {}\n\ngoal (1 subgoal):\n 1. card (shuffles xs ys) = length xs + length ys choose length xs\n[PROOF STEP]\nproof (induction xs ys rule: shuffles.induct)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\ncase (3 x xs y ys)\n[PROOF STATE]\nproof (state)\nthis:\nset xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\nset (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n[PROOF STEP]\nby (rule shuffles.simps)\n[PROOF STATE]\nproof (state)\nthis:\nshuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nshuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card \\ = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n[PROOF STEP]\nby (rule card_Un_disjoint) (insert \"3.prems\", auto)\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys) \\ (#) y ` shuffles (x # xs) ys) = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n[PROOF STEP]\nby (rule card_image) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"\\ = (length xs + length (y # ys)) choose length xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n[PROOF STEP]\nusing \"3.prems\"\n[PROOF STATE]\nproof (prove)\nusing this:\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (1 subgoal):\n 1. card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n[PROOF STEP]\nby (intro \"3.IH\") auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"card ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n[PROOF STEP]\nby (rule card_image) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"\\ = (length (x # xs) + length ys) choose length (x # xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n[PROOF STEP]\nusing \"3.prems\"\n[PROOF STATE]\nproof (prove)\nusing this:\nset (x # xs) \\ set (y # ys) = {}\n\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n[PROOF STEP]\nby (intro \"3.IH\") auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nhave \"length xs + length (y # ys) choose length xs + \\ =\n (length (x # xs) + length (y # ys)) choose length (x # xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length xs + length (y # ys) choose length xs + (length (x # xs) + length ys choose length (x # xs)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlength xs + length (y # ys) choose length xs + (length (x # xs) + length ys choose length (x # xs)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (3 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n 3. \\x xs y ys. \\set xs \\ set (y # ys) = {} \\ card (shuffles xs (y # ys)) = length xs + length (y # ys) choose length xs; set (x # xs) \\ set ys = {} \\ card (shuffles (x # xs) ys) = length (x # xs) + length ys choose length (x # xs); set (x # xs) \\ set (y # ys) = {}\\ \\ card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (1 subgoal):\n 1. card (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard (shuffles (x # xs) (y # ys)) = length (x # xs) + length (y # ys) choose length (x # xs)\n\ngoal (2 subgoals):\n 1. \\ys. set [] \\ set ys = {} \\ card (shuffles [] ys) = length [] + length ys choose length []\n 2. \\xs. set xs \\ set [] = {} \\ card (shuffles xs []) = length xs + length [] choose length xs\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 6519, "file": null, "length": 29, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970654616711, "lm_q2_score": 0.7981867681382279, "lm_q1q2_score": 0.7030405630664864}} {"text": "[STATEMENT]\nlemma det_dependent_columns:\n assumes d: \"vec.dependent (columns (A::real^'n^'n))\"\n shows \"det A = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A = 0\n[PROOF STEP]\nby (metis d det_dependent_rows rows_transpose det_transpose)", "meta": {"llama_tokens": 103, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9207896715436482, "lm_q2_score": 0.7634837581726991, "lm_q1q2_score": 0.7030079589167497}} {"text": "[STATEMENT]\nlemma all1_vec_scalar_prod:\nshows \"all1_vec (length xs) \\ (vec_of_list xs) = sum_list xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. all1_vec (length xs) \\ vec_of_list xs = sum_list xs\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. all1_vec (length xs) \\ vec_of_list xs = sum_list xs\n[PROOF STEP]\nhave \"all1_vec (length xs) \\ (vec_of_list xs) = (\\i = 0.. vec_of_list xs = sum (($) (vec_of_list xs)) {0..i = 0.. vec_of_list xs = sum (($) (vec_of_list xs)) {0.. vec_of_list xs = sum_list xs\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nall1_vec (length xs) \\ vec_of_list xs = sum (($) (vec_of_list xs)) {0.. vec_of_list xs = sum_list xs\n[PROOF STEP]\nhave \"... = (\\i = 0.. vec_of_list xs = sum_list xs\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum (($) (vec_of_list xs)) {0.. vec_of_list xs = sum_list xs\n[PROOF STEP]\nhave \"... = sum_list xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((!) xs) {0.. vec_of_list xs = sum_list xs\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nall1_vec (length xs) \\ vec_of_list xs = sum_list xs\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nall1_vec (length xs) \\ vec_of_list xs = sum_list xs\n\ngoal (1 subgoal):\n 1. all1_vec (length xs) \\ vec_of_list xs = sum_list xs\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nall1_vec (length xs) \\ vec_of_list xs = sum_list xs\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1467, "file": "Deep_Learning_DL_Concrete_Matrices", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289388209992571, "lm_q2_score": 0.8479677526147223, "lm_q1q2_score": 0.7029133890978376}} {"text": "[STATEMENT]\nlemma catalan_eq_I: \"real (catalan n) = I n / (2 * pi)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (catalan n) = I n / (2 * pi)\n[PROOF STEP]\nproof (induction n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. real (catalan 0) = I 0 / (2 * pi)\n 2. \\n. real (catalan n) = I n / (2 * pi) \\ real (catalan (Suc n)) = I (Suc n) / (2 * pi)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. real (catalan 0) = I 0 / (2 * pi)\n 2. \\n. real (catalan n) = I n / (2 * pi) \\ real (catalan (Suc n)) = I (Suc n) / (2 * pi)\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (catalan 0) = I 0 / (2 * pi)\n[PROOF STEP]\nusing has_integral_I0\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. x powr - (1 / 2) * sqrt (4 - x)) has_integral 2 * pi) {0..4}\n\ngoal (1 subgoal):\n 1. real (catalan 0) = I 0 / (2 * pi)\n[PROOF STEP]\nby (simp add: I_def integral_unique)\n[PROOF STATE]\nproof (state)\nthis:\nreal (catalan 0) = I 0 / (2 * pi)\n\ngoal (1 subgoal):\n 1. \\n. real (catalan n) = I n / (2 * pi) \\ real (catalan (Suc n)) = I (Suc n) / (2 * pi)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. real (catalan n) = I n / (2 * pi) \\ real (catalan (Suc n)) = I (Suc n) / (2 * pi)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nreal (catalan n) = I n / (2 * pi)\n\ngoal (1 subgoal):\n 1. \\n. real (catalan n) = I n / (2 * pi) \\ real (catalan (Suc n)) = I (Suc n) / (2 * pi)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (catalan (Suc n)) = I (Suc n) / (2 * pi)\n[PROOF STEP]\nby (simp add: of_nat_catalan_Suc' Suc.IH I_Suc)\n[PROOF STATE]\nproof (state)\nthis:\nreal (catalan (Suc n)) = I (Suc n) / (2 * pi)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 880, "file": "Catalan_Numbers_Catalan_Numbers", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677660619633, "lm_q2_score": 0.8289388040954683, "lm_q1q2_score": 0.7029133859109097}} {"text": "[STATEMENT]\nlemma poincare_distance_zero_opposite [simp]:\n assumes \"of_complex z \\ unit_disc\"\n shows \"poincare_distance 0\\<^sub>h (of_complex (- z)) = poincare_distance 0\\<^sub>h (of_complex z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poincare_distance 0\\<^sub>h (of_complex (- z)) = poincare_distance 0\\<^sub>h (of_complex z)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. poincare_distance 0\\<^sub>h (of_complex (- z)) = poincare_distance 0\\<^sub>h (of_complex z)\n[PROOF STEP]\nhave *: \"of_complex (-z) \\ unit_disc\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_complex (- z) \\ unit_disc\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nof_complex z \\ unit_disc\n\ngoal (1 subgoal):\n 1. of_complex (- z) \\ unit_disc\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nof_complex (- z) \\ unit_disc\n\ngoal (1 subgoal):\n 1. poincare_distance 0\\<^sub>h (of_complex (- z)) = poincare_distance 0\\<^sub>h (of_complex z)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poincare_distance 0\\<^sub>h (of_complex (- z)) = poincare_distance 0\\<^sub>h (of_complex z)\n[PROOF STEP]\nusing poincare_distance_zero[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\npoincare_distance 0\\<^sub>h (of_complex z) = (let x' = to_complex (of_complex z) in \\ln (Re (cor ((1 - cmod x') / (1 + cmod x'))))\\)\n\ngoal (1 subgoal):\n 1. poincare_distance 0\\<^sub>h (of_complex (- z)) = poincare_distance 0\\<^sub>h (of_complex z)\n[PROOF STEP]\nusing poincare_distance_zero[OF *]\n[PROOF STATE]\nproof (prove)\nusing this:\npoincare_distance 0\\<^sub>h (of_complex z) = (let x' = to_complex (of_complex z) in \\ln (Re (cor ((1 - cmod x') / (1 + cmod x'))))\\)\npoincare_distance 0\\<^sub>h (of_complex (- z)) = (let x' = to_complex (of_complex (- z)) in \\ln (Re (cor ((1 - cmod x') / (1 + cmod x'))))\\)\n\ngoal (1 subgoal):\n 1. poincare_distance 0\\<^sub>h (of_complex (- z)) = poincare_distance 0\\<^sub>h (of_complex z)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\npoincare_distance 0\\<^sub>h (of_complex (- z)) = poincare_distance 0\\<^sub>h (of_complex z)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 938, "file": "Poincare_Disc_Poincare_Distance", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677545357569, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.702913379939857}} {"text": "[STATEMENT]\nlemma sum_choose_diagonal:\n assumes \"m \\ n\"\n shows \"(\\k\\m. (n - k) choose (m - k)) = Suc n choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nhave \"(\\k\\m. (n-k) choose (m - k)) = (\\k\\m. (n - m + k) choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n[PROOF STEP]\nusing sum.atLeastAtMost_rev [of \"\\k. (n - k) choose (m - k)\" 0 m] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k = 0..m. n - k choose (m - k)) = (\\i = 0..m. n - (m + 0 - i) choose (m - (m + 0 - i)))\nm \\ n\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n[PROOF STEP]\nby (simp add: atMost_atLeast0)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nhave \"\\ = Suc (n - m + m) choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\m. n - m + k choose k) = Suc (n - m + m) choose m\n[PROOF STEP]\nby (rule sum_choose_lower)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - m + k choose k) = Suc (n - m + m) choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - m + k choose k) = Suc (n - m + m) choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nhave \"\\ = Suc n choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc (n - m + m) choose m = Suc n choose m\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ n\n\ngoal (1 subgoal):\n 1. Suc (n - m + m) choose m = Suc n choose m\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nSuc (n - m + m) choose m = Suc n choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k\\m. n - k choose (m - k)) = Suc n choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - k choose (m - k)) = Suc n choose m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1347, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677468516188, "lm_q2_score": 0.8289388083214155, "lm_q1q2_score": 0.7029133735701767}} {"text": "[STATEMENT]\nlemma sum_choose_diagonal:\n assumes \"m \\ n\"\n shows \"(\\k\\m. (n - k) choose (m - k)) = Suc n choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nhave \"(\\k\\m. (n-k) choose (m - k)) = (\\k\\m. (n - m + k) choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n[PROOF STEP]\nusing sum.atLeastAtMost_rev [of \"\\k. (n - k) choose (m - k)\" 0 m] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k = 0..m. n - k choose (m - k)) = (\\i = 0..m. n - (m + 0 - i) choose (m - (m + 0 - i)))\nm \\ n\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n[PROOF STEP]\nby (simp add: atMost_atLeast0)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nhave \"\\ = Suc (n - m + m) choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\m. n - m + k choose k) = Suc (n - m + m) choose m\n[PROOF STEP]\nby (rule sum_choose_lower)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - m + k choose k) = Suc (n - m + m) choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - m + k choose k) = Suc (n - m + m) choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nhave \"\\ = Suc n choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc (n - m + m) choose m = Suc n choose m\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ n\n\ngoal (1 subgoal):\n 1. Suc (n - m + m) choose m = Suc n choose m\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nSuc (n - m + m) choose m = Suc n choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k\\m. n - k choose (m - k)) = Suc n choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - k choose (m - k)) = Suc n choose m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1347, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677468516188, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7029133717784432}} {"text": "[STATEMENT]\nlemma mult_ceiling_le:\n assumes \"0 \\ a\" and \"0 \\ b\"\n shows \"\\a * b\\ \\ \\a\\ * \\b\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a * b\\ \\ \\a\\ * \\b\\\n[PROOF STEP]\nby (metis assms ceiling_le_iff ceiling_mono le_of_int_ceiling mult_mono of_int_mult)", "meta": {"llama_tokens": 168, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637361282707, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7028391344994344}} {"text": "[STATEMENT]\nlemma mult_ceiling_le:\n assumes \"0 \\ a\" and \"0 \\ b\"\n shows \"\\a * b\\ \\ \\a\\ * \\b\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a * b\\ \\ \\a\\ * \\b\\\n[PROOF STEP]\nby (metis assms ceiling_le_iff ceiling_mono le_of_int_ceiling mult_mono of_int_mult)", "meta": {"llama_tokens": 168, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637361282707, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7028391344994344}} {"text": "[STATEMENT]\nlemma qtable_union: \"qtable n A P Q1 X \\ qtable n A P Q2 Y \\\n (\\x. wf_tuple n A x \\ P x \\ Q x \\ Q1 x \\ Q2 x) \\ qtable n A P Q (X \\ Y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\qtable n A P Q1 X; qtable n A P Q2 Y; \\x. \\wf_tuple n A x; P x\\ \\ Q x = (Q1 x \\ Q2 x)\\ \\ qtable n A P Q (X \\ Y)\n[PROOF STEP]\nunfolding qtable_def table_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Ball X (wf_tuple n A) \\ (\\x. (x \\ X \\ P x \\ Q1 x) \\ (wf_tuple n A x \\ P x \\ Q1 x \\ x \\ X)); Ball Y (wf_tuple n A) \\ (\\x. (x \\ Y \\ P x \\ Q2 x) \\ (wf_tuple n A x \\ P x \\ Q2 x \\ x \\ Y)); \\x. \\wf_tuple n A x; P x\\ \\ Q x = (Q1 x \\ Q2 x)\\ \\ Ball (X \\ Y) (wf_tuple n A) \\ (\\x. (x \\ X \\ Y \\ P x \\ Q x) \\ (wf_tuple n A x \\ P x \\ Q x \\ x \\ X \\ Y))\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 523, "file": "MFOTL_Monitor_Table", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637397236824, "lm_q2_score": 0.817574478416099, "lm_q1q2_score": 0.7028391336178227}} {"text": "[STATEMENT]\nlemma mult_ceiling_le:\n assumes \"0 \\ a\" and \"0 \\ b\"\n shows \"\\a * b\\ \\ \\a\\ * \\b\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a * b\\ \\ \\a\\ * \\b\\\n[PROOF STEP]\nby (metis assms ceiling_le_iff ceiling_mono le_of_int_ceiling mult_mono of_int_mult)", "meta": {"llama_tokens": 168, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637361282707, "lm_q2_score": 0.8175744695262775, "lm_q1q2_score": 0.7028391230360488}} {"text": "[STATEMENT]\nlemma lower_asymptotic_density_infinite_interval [simp]:\n \"lower_asymptotic_density {N..} = 1\"\n \"lower_asymptotic_density {N<..} = 1\"\n \"lower_asymptotic_density UNIV = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lower_asymptotic_density {N..} = 1 &&& lower_asymptotic_density {N<..} = 1 &&& lower_asymptotic_density UNIV = 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. lower_asymptotic_density {N..} = 1\n 2. lower_asymptotic_density {N<..} = 1\n 3. lower_asymptotic_density UNIV = 1\n[PROOF STEP]\nhave \"UNIV - {N..} = {.. sorted (xs) \\ distinct (xs) \\ set(xs) = A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_sorted_list_of_set A xs = (sorted xs \\ distinct xs \\ set xs = A)\n[PROOF STEP]\napply (auto intro: sorted_is_sorted_list_of_set)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x. \\is_sorted_list_of_set A xs; x \\ set xs\\ \\ x \\ A\n 2. \\x. \\is_sorted_list_of_set A xs; x \\ A\\ \\ x \\ set xs\n 3. \\sorted xs; distinct xs; A = set xs\\ \\ is_sorted_list_of_set (set xs) xs\n[PROOF STEP]\napply (auto simp add: is_sorted_list_of_set_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. \\sorted xs; distinct xs; A = set xs; i < length xs - Suc 0\\ \\ xs ! i < xs ! Suc i\n[PROOF STEP]\napply (metis Nat.add_0_right One_nat_def add_Suc_right sorted_distinct)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 456, "file": "UTP_toolkit_List_Extra", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473813156295, "lm_q2_score": 0.8056321866478979, "lm_q1q2_score": 0.7027911283258782}} {"text": "[STATEMENT]\nlemma vectorize_minus: \"vectorize_poly (p - q) = vectorize_poly p - vectorize_poly q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vectorize_poly (p - q) = vectorize_poly p - vectorize_poly q\n[PROOF STEP]\nby (rule poly_mapping_eqI, simp add: lookup_vectorize_poly lookup_minus proj_minus)", "meta": {"llama_tokens": 115, "file": "Polynomials_MPoly_Type_Class", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473813156294, "lm_q2_score": 0.8056321866478979, "lm_q1q2_score": 0.7027911283258781}} {"text": "[STATEMENT]\nlemma scalar_prod_Cauchy:\n fixes u v::\"'a :: {trivial_conjugatable_linordered_field} Matrix.vec\"\n assumes \"u \\ carrier_vec n\" \"v \\ carrier_vec n\"\n shows \"(u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2 \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nassume v_0: \"v \\ 0\\<^sub>v n\"\n[PROOF STATE]\nproof (state)\nthis:\nv \\ 0\\<^sub>v n\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nhave \"0 \\ (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v)\" for r\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) \\ (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v)\n[PROOF STEP]\nby (simp add: scalar_prod_ge_0)\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) \\ (u - ?r \\\\<^sub>v v) \\ (u - ?r \\\\<^sub>v v)\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) \\ (u - ?r \\\\<^sub>v v) \\ (u - ?r \\\\<^sub>v v)\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nhave \"(u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\" for r::'a\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nhave \"(u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = (u - r \\\\<^sub>v v) \\ u - (u - r \\\\<^sub>v v) \\ (r \\\\<^sub>v v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = (u - r \\\\<^sub>v v) \\ u - (u - r \\\\<^sub>v v) \\ (r \\\\<^sub>v v)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ carrier_vec n\nv \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = (u - r \\\\<^sub>v v) \\ u - (u - r \\\\<^sub>v v) \\ (r \\\\<^sub>v v)\n[PROOF STEP]\nby (subst scalar_prod_minus_distrib) auto\n[PROOF STATE]\nproof (state)\nthis:\n(u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = (u - r \\\\<^sub>v v) \\ u - (u - r \\\\<^sub>v v) \\ (r \\\\<^sub>v v)\n\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = (u - r \\\\<^sub>v v) \\ u - (u - r \\\\<^sub>v v) \\ (r \\\\<^sub>v v)\n\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nhave \"\\ = u \\ u - (r \\\\<^sub>v v) \\ u - r * ((u - r \\\\<^sub>v v) \\ v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ u - (u - r \\\\<^sub>v v) \\ (r \\\\<^sub>v v) = u \\ u - (r \\\\<^sub>v v) \\ u - r * ((u - r \\\\<^sub>v v) \\ v)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ carrier_vec n\nv \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ u - (u - r \\\\<^sub>v v) \\ (r \\\\<^sub>v v) = u \\ u - (r \\\\<^sub>v v) \\ u - r * ((u - r \\\\<^sub>v v) \\ v)\n[PROOF STEP]\nby (subst minus_scalar_prod_distrib) auto\n[PROOF STATE]\nproof (state)\nthis:\n(u - r \\\\<^sub>v v) \\ u - (u - r \\\\<^sub>v v) \\ (r \\\\<^sub>v v) = u \\ u - (r \\\\<^sub>v v) \\ u - r * ((u - r \\\\<^sub>v v) \\ v)\n\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(u - r \\\\<^sub>v v) \\ u - (u - r \\\\<^sub>v v) \\ (r \\\\<^sub>v v) = u \\ u - (r \\\\<^sub>v v) \\ u - r * ((u - r \\\\<^sub>v v) \\ v)\n\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nhave \"\\ = u \\ u - r * (v \\ u) - r * (u \\ v - r * (v \\ v))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. u \\ u - (r \\\\<^sub>v v) \\ u - r * ((u - r \\\\<^sub>v v) \\ v) = u \\ u - r * (v \\ u) - r * (u \\ v - r * (v \\ v))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ carrier_vec n\nv \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. u \\ u - (r \\\\<^sub>v v) \\ u - r * ((u - r \\\\<^sub>v v) \\ v) = u \\ u - r * (v \\ u) - r * (u \\ v - r * (v \\ v))\n[PROOF STEP]\nby (subst minus_scalar_prod_distrib) auto\n[PROOF STATE]\nproof (state)\nthis:\nu \\ u - (r \\\\<^sub>v v) \\ u - r * ((u - r \\\\<^sub>v v) \\ v) = u \\ u - r * (v \\ u) - r * (u \\ v - r * (v \\ v))\n\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nu \\ u - (r \\\\<^sub>v v) \\ u - r * ((u - r \\\\<^sub>v v) \\ v) = u \\ u - r * (v \\ u) - r * (u \\ v - r * (v \\ v))\n\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nhave \"\\ = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. u \\ u - r * (v \\ u) - r * (u \\ v - r * (v \\ v)) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nusing assms comm_scalar_prod\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ carrier_vec n\nv \\ carrier_vec n\n\\?v\\<^sub>1 \\ carrier_vec ?n; ?v\\<^sub>2 \\ carrier_vec ?n\\ \\ ?v\\<^sub>1 \\ ?v\\<^sub>2 = ?v\\<^sub>2 \\ ?v\\<^sub>1\n\ngoal (1 subgoal):\n 1. u \\ u - r * (v \\ u) - r * (u \\ v - r * (v \\ v)) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nby (auto simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nu \\ u - r * (v \\ u) - r * (u \\ v - r * (v \\ v)) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n\ngoal (1 subgoal):\n 1. (u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(u - r \\\\<^sub>v v) \\ (u - r \\\\<^sub>v v) = u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(u - ?r \\\\<^sub>v v) \\ (u - ?r \\\\<^sub>v v) = u \\ u - ?r * (u \\ v) - ?r * (u \\ v) + ?r * ?r * (v \\ v)\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(u - ?r \\\\<^sub>v v) \\ (u - ?r \\\\<^sub>v v) = u \\ u - ?r * (u \\ v) - ?r * (u \\ v) + ?r * ?r * (v \\ v)\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nhave \"u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v) = sq_norm u - (u \\ v)\\<^sup>2 / sq_norm v\"\n if \"r = (u \\ v) / (v \\ v)\" for r\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. u \\ u - r * (u \\ v) - r * (u \\ v) + r * r * (v \\ v) = \\u\\\\<^sup>2 - (u \\ v)\\<^sup>2 / \\v\\\\<^sup>2\n[PROOF STEP]\nunfolding that\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. u \\ u - u \\ v / (v \\ v) * (u \\ v) - u \\ v / (v \\ v) * (u \\ v) + u \\ v / (v \\ v) * (u \\ v / (v \\ v)) * (v \\ v) = \\u\\\\<^sup>2 - (u \\ v)\\<^sup>2 / \\v\\\\<^sup>2\n[PROOF STEP]\nby (auto simp add: sq_norm_vec_as_cscalar_prod power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n?r = u \\ v / (v \\ v) \\ u \\ u - ?r * (u \\ v) - ?r * (u \\ v) + ?r * ?r * (v \\ v) = \\u\\\\<^sup>2 - (u \\ v)\\<^sup>2 / \\v\\\\<^sup>2\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n?r1 = u \\ v / (v \\ v) \\ (0::'a) \\ \\u\\\\<^sup>2 - (u \\ v)\\<^sup>2 / \\v\\\\<^sup>2\n[PROOF STEP]\nhave \"0 \\ \\u\\\\<^sup>2 - (u \\ v)\\<^sup>2 / \\v\\\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?r1 = u \\ v / (v \\ v) \\ (0::'a) \\ \\u\\\\<^sup>2 - (u \\ v)\\<^sup>2 / \\v\\\\<^sup>2\n\ngoal (1 subgoal):\n 1. (0::'a) \\ \\u\\\\<^sup>2 - (u \\ v)\\<^sup>2 / \\v\\\\<^sup>2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) \\ \\u\\\\<^sup>2 - (u \\ v)\\<^sup>2 / \\v\\\\<^sup>2\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(0::'a) \\ \\u\\\\<^sup>2 - (u \\ v)\\<^sup>2 / \\v\\\\<^sup>2\n[PROOF STEP]\nhave \"(u \\ v)\\<^sup>2 / \\v\\\\<^sup>2 \\ \\u\\\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) \\ \\u\\\\<^sup>2 - (u \\ v)\\<^sup>2 / \\v\\\\<^sup>2\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 / \\v\\\\<^sup>2 \\ \\u\\\\<^sup>2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(u \\ v)\\<^sup>2 / \\v\\\\<^sup>2 \\ \\u\\\\<^sup>2\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(u \\ v)\\<^sup>2 / \\v\\\\<^sup>2 \\ \\u\\\\<^sup>2\n[PROOF STEP]\nhave \"(u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(u \\ v)\\<^sup>2 / \\v\\\\<^sup>2 \\ \\u\\\\<^sup>2\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nusing pos_divide_le_eq[of \"\\v\\\\<^sup>2\"] v_0 assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(u \\ v)\\<^sup>2 / \\v\\\\<^sup>2 \\ \\u\\\\<^sup>2\n(0::'a) < \\v\\\\<^sup>2 \\ (?b / \\v\\\\<^sup>2 \\ ?a) = (?b \\ ?a * \\v\\\\<^sup>2)\nv \\ 0\\<^sub>v n\nu \\ carrier_vec n\nv \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nby (auto)\n[PROOF STATE]\nproof (state)\nthis:\n(u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\nv \\ 0\\<^sub>v n \\ (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nv \\ 0\\<^sub>v n \\ (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ 0\\<^sub>v n \\ (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n\ngoal (1 subgoal):\n 1. (u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n[PROOF STEP]\nby (fastforce simp add: assms)\n[PROOF STATE]\nproof (state)\nthis:\n(u \\ v)\\<^sup>2 \\ \\u\\\\<^sup>2 * \\v\\\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 6490, "file": "LLL_Basis_Reduction_Norms", "length": 46, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473680407889, "lm_q2_score": 0.8056321866478979, "lm_q1q2_score": 0.7027911176312394}} {"text": "[STATEMENT]\nlemma finite_permutations:\n assumes \"finite S\"\n shows \"finite {p. p permutes S}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {p. p permutes S}\n[PROOF STEP]\nusing card_permutations[OF refl assms]\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {p. p permutes S} = fact (card S)\n\ngoal (1 subgoal):\n 1. finite {p. p permutes S}\n[PROOF STEP]\nby (auto intro: card_ge_0_finite)", "meta": {"llama_tokens": 163, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473614033683, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7027911163549161}} {"text": "[STATEMENT]\nlemma fekete:\n fixes u::\"nat \\ real\"\n assumes \"\\n m. u (m+n) \\ u m + u n\"\n \"bdd_below {u n/n | n. n>0}\"\n shows \"(\\n. u n/n) \\ Inf {u n/n | n. n>0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. u n / real n) \\ Inf {u n / real n |n. 0 < n}\n[PROOF STEP]\napply (rule subadditive_converges_bounded)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. subadditive u\n 2. bdd_below {u n / real n |n. 0 < n}\n[PROOF STEP]\nunfolding subadditive_def\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\m n. u (m + n) \\ u m + u n\n 2. bdd_below {u n / real n |n. 0 < n}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu (?m + ?n) \\ u ?m + u ?n\nbdd_below {u n / real n |n. 0 < n}\n\ngoal (2 subgoals):\n 1. \\m n. u (m + n) \\ u m + u n\n 2. bdd_below {u n / real n |n. 0 < n}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 442, "file": "Ergodic_Theory_Kohlberg_Neyman_Karlsson", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.907312213841788, "lm_q2_score": 0.7745833945721304, "lm_q1q2_score": 0.7027889745343269}} {"text": "[STATEMENT]\nlemma ang_vec_sym_pi:\n assumes \"\\ z1 z2 = pi\"\n shows \"\\ z1 z2 = \\ z2 z1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ z1 z2 = \\ z2 z1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ z1 z2 = pi\n\ngoal (1 subgoal):\n 1. \\ z1 z2 = \\ z2 z1\n[PROOF STEP]\nunfolding ang_vec_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\Arg z2 - Arg z1\\ = pi\n\ngoal (1 subgoal):\n 1. \\Arg z2 - Arg z1\\ = \\Arg z1 - Arg z2\\\n[PROOF STEP]\nusing canon_ang_uminus_pi[of \"Arg z2 - Arg z1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\Arg z2 - Arg z1\\ = pi\n\\Arg z2 - Arg z1\\ = pi \\ \\- (Arg z2 - Arg z1)\\ = \\Arg z2 - Arg z1\\\n\ngoal (1 subgoal):\n 1. \\Arg z2 - Arg z1\\ = \\Arg z1 - Arg z2\\\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 473, "file": "Complex_Geometry_Angles", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357735451835, "lm_q2_score": 0.8104789155369048, "lm_q1q2_score": 0.7027142134746016}} {"text": "[STATEMENT]\nlemma cardofpairs: \"S \\ [] \\ sorted S \\ distinct S \\ card {(x,y). x \\ set S \\ y\\set S \\ xS \\ []; sorted S; distinct S\\ \\ real (card {(x, y). x \\ set S \\ y \\ set S \\ x < y}) = real (length S * (length S - 1)) / 2\n[PROOF STEP]\nproof (induct S rule: list_nonempty_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\ncase (cons s ss)\n[PROOF STATE]\nproof (state)\nthis:\nss \\ []\n\\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\nsorted (s # ss)\ndistinct (s # ss)\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nss \\ []\n\\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\nsorted (s # ss)\ndistinct (s # ss)\n[PROOF STEP]\nhave \"sorted ss\" \"distinct ss\"\n[PROOF STATE]\nproof (prove)\nusing this:\nss \\ []\n\\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\nsorted (s # ss)\ndistinct (s # ss)\n\ngoal (1 subgoal):\n 1. sorted ss &&& distinct ss\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsorted ss\ndistinct ss\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nfrom cons(2)[OF this(1) this(2)]\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\n[PROOF STEP]\nhave iH: \"card {(x, y) . x \\ set ss \\ y \\ set ss \\ x < y}\n = (length ss * (length ss-1)) / 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\n\ngoal (1 subgoal):\n 1. real (card {(x, y). x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nreal (card {(x, y). x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nfrom cons\n[PROOF STATE]\nproof (chain)\npicking this:\nss \\ []\n\\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\nsorted (s # ss)\ndistinct (s # ss)\n[PROOF STEP]\nhave sss: \"s \\ set ss\"\n[PROOF STATE]\nproof (prove)\nusing this:\nss \\ []\n\\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\nsorted (s # ss)\ndistinct (s # ss)\n\ngoal (1 subgoal):\n 1. s \\ set ss\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ns \\ set ss\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nfrom cons\n[PROOF STATE]\nproof (chain)\npicking this:\nss \\ []\n\\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\nsorted (s # ss)\ndistinct (s # ss)\n[PROOF STEP]\nhave tt: \"(\\y\\set (s#ss). s \\ y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nss \\ []\n\\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\nsorted (s # ss)\ndistinct (s # ss)\n\ngoal (1 subgoal):\n 1. \\y\\set (s # ss). s \\ y\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\y\\set (s # ss). s \\ y\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nwith cons\n[PROOF STATE]\nproof (chain)\npicking this:\nss \\ []\n\\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\nsorted (s # ss)\ndistinct (s # ss)\n\\y\\set (s # ss). s \\ y\n[PROOF STEP]\nhave tt': \"(\\y\\set ss. s < y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nss \\ []\n\\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\nsorted (s # ss)\ndistinct (s # ss)\n\\y\\set (s # ss). s \\ y\n\ngoal (1 subgoal):\n 1. \\y\\set ss. s < y\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ss \\ []; \\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2; sorted (s # ss); distinct (s # ss); \\y\\set (s # ss). s \\ y\\ \\ \\y\\set ss. s < y\n[PROOF STEP]\nfrom sss\n[PROOF STATE]\nproof (chain)\npicking this:\ns \\ set ss\n[PROOF STEP]\nhave \"(\\y\\set ss. s \\ y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ns \\ set ss\n\ngoal (1 subgoal):\n 1. \\y\\set ss. s \\ y\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\y\\set ss. s \\ y\n\ngoal (1 subgoal):\n 1. \\ss \\ []; \\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2; sorted (s # ss); distinct (s # ss); \\y\\set (s # ss). s \\ y\\ \\ \\y\\set ss. s < y\n[PROOF STEP]\nwith tt\n[PROOF STATE]\nproof (chain)\npicking this:\n\\y\\set (s # ss). s \\ y\n\\y\\set ss. s \\ y\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\y\\set (s # ss). s \\ y\n\\y\\set ss. s \\ y\n\ngoal (1 subgoal):\n 1. \\y\\set ss. s < y\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n\\y\\set ss. s < y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\y\\set ss. s < y\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\y\\set ss. s < y\n[PROOF STEP]\nhave \"{(x, y) . x = s \\ y \\ set ss \\ x < y}\n = {(x, y) . x = s \\ y \\ set ss}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\y\\set ss. s < y\n\ngoal (1 subgoal):\n 1. {(x, y). x = s \\ y \\ set ss \\ x < y} = {(x, y). x = s \\ y \\ set ss}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{(x, y). x = s \\ y \\ set ss \\ x < y} = {(x, y). x = s \\ y \\ set ss}\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n{(x, y). x = s \\ y \\ set ss \\ x < y} = {(x, y). x = s \\ y \\ set ss}\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave \"\\ = {s}\\(set ss)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {(x, y). x = s \\ y \\ set ss} = {s} \\ set ss\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{(x, y). x = s \\ y \\ set ss} = {s} \\ set ss\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n{(x, y). x = s \\ y \\ set ss \\ x < y} = {s} \\ set ss\n[PROOF STEP]\nhave \"{(x, y) . x = s \\ y \\ set ss \\ x < y} = {s}\\(set ss)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{(x, y). x = s \\ y \\ set ss \\ x < y} = {s} \\ set ss\n\ngoal (1 subgoal):\n 1. {(x, y). x = s \\ y \\ set ss \\ x < y} = {s} \\ set ss\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n{(x, y). x = s \\ y \\ set ss \\ x < y} = {s} \\ set ss\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{(x, y). x = s \\ y \\ set ss \\ x < y} = {s} \\ set ss\n[PROOF STEP]\nhave \"card {(x, y) . x = s \\ y \\ set ss \\ x < y}\n = card (set ss)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{(x, y). x = s \\ y \\ set ss \\ x < y} = {s} \\ set ss\n\ngoal (1 subgoal):\n 1. card {(x, y). x = s \\ y \\ set ss \\ x < y} = card (set ss)\n[PROOF STEP]\nby(auto)\n[PROOF STATE]\nproof (state)\nthis:\ncard {(x, y). x = s \\ y \\ set ss \\ x < y} = card (set ss)\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {(x, y). x = s \\ y \\ set ss \\ x < y} = card (set ss)\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nfrom cons distinct_card\n[PROOF STATE]\nproof (chain)\npicking this:\nss \\ []\n\\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\nsorted (s # ss)\ndistinct (s # ss)\ndistinct ?xs \\ card (set ?xs) = length ?xs\n[PROOF STEP]\nhave \"\\ = length ss\"\n[PROOF STATE]\nproof (prove)\nusing this:\nss \\ []\n\\sorted ss; distinct ss\\ \\ real (card {a. case a of (x, y) \\ x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\nsorted (s # ss)\ndistinct (s # ss)\ndistinct ?xs \\ card (set ?xs) = length ?xs\n\ngoal (1 subgoal):\n 1. card (set ss) = length ss\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (set ss) = length ss\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {(x, y). x = s \\ y \\ set ss \\ x < y} = length ss\n[PROOF STEP]\nhave step: \"card {(x, y) . x = s \\ y \\ set ss \\ x < y} =\n length ss\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {(x, y). x = s \\ y \\ set ss \\ x < y} = length ss\n\ngoal (1 subgoal):\n 1. card {(x, y). x = s \\ y \\ set ss \\ x < y} = length ss\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {(x, y). x = s \\ y \\ set ss \\ x < y} = length ss\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave uni: \"{(x, y) . x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y}\n = {(x, y) . x \\ set ss \\ y \\ set ss \\ x < y}\n \\ {(x, y) . x = s \\ y \\ set ss \\ x < y}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {(x, y). x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y} = {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y}\n[PROOF STEP]\nusing tt\n[PROOF STATE]\nproof (prove)\nusing this:\n\\y\\set (s # ss). s \\ y\n\ngoal (1 subgoal):\n 1. {(x, y). x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y} = {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{(x, y). x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y} = {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y}\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave disj: \"{(x, y) . x \\ set ss \\ y \\ set ss \\ x < y}\n \\ {(x, y) . x = s \\ y \\ set ss \\ x < y} = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n[PROOF STEP]\nusing sss\n[PROOF STATE]\nproof (prove)\nusing this:\ns \\ set ss\n\ngoal (1 subgoal):\n 1. {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n[PROOF STEP]\nby(auto)\n[PROOF STATE]\nproof (state)\nthis:\n{(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave \"card {(x, y) . x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y}\n = card ({(x, y) . x \\ set ss \\ y \\ set ss \\ x < y}\n \\ {(x, y) . x = s \\ y \\ set ss \\ x < y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {(x, y). x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y} = card ({(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y})\n[PROOF STEP]\nusing uni\n[PROOF STATE]\nproof (prove)\nusing this:\n{(x, y). x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y} = {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y}\n\ngoal (1 subgoal):\n 1. card {(x, y). x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y} = card ({(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y})\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard {(x, y). x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y} = card ({(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y})\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {(x, y). x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y} = card ({(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y})\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave \"\\ = card {(x, y) . x \\ set ss \\ y \\ set ss \\ x < y}\n + card {(x, y) . x = s \\ y \\ set ss \\ x < y}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y}) = card {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} + card {(x, y). x = s \\ y \\ set ss \\ x < y}\n[PROOF STEP]\napply(rule card_Un_disjoint)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. finite {(x, y). x \\ set ss \\ y \\ set ss \\ x < y}\n 2. finite {(x, y). x = s \\ y \\ set ss \\ x < y}\n 3. {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n[PROOF STEP]\napply(rule finite_subset[where B=\"(set ss) \\ (set ss)\"])\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ set ss \\ set ss\n 2. finite (set ss \\ set ss)\n 3. finite {(x, y). x = s \\ y \\ set ss \\ x < y}\n 4. {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n[PROOF STEP]\napply(force)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. finite (set ss \\ set ss)\n 2. finite {(x, y). x = s \\ y \\ set ss \\ x < y}\n 3. {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n[PROOF STEP]\napply(simp)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. finite {(x, y). x = s \\ y \\ set ss \\ x < y}\n 2. {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n[PROOF STEP]\napply(rule finite_subset[where B=\"{s} \\ (set ss)\"])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. {(x, y). x = s \\ y \\ set ss \\ x < y} \\ {s} \\ set ss\n 2. finite ({s} \\ set ss)\n 3. {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n[PROOF STEP]\napply(force)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. finite ({s} \\ set ss)\n 2. {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n[PROOF STEP]\napply(simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n[PROOF STEP]\nusing disj\n[PROOF STATE]\nproof (prove)\nusing this:\n{(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n\ngoal (1 subgoal):\n 1. {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y} = {}\n[PROOF STEP]\napply(simp)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\ncard ({(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y}) = card {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} + card {(x, y). x = s \\ y \\ set ss \\ x < y}\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ({(x, y). x \\ set ss \\ y \\ set ss \\ x < y} \\ {(x, y). x = s \\ y \\ set ss \\ x < y}) = card {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} + card {(x, y). x = s \\ y \\ set ss \\ x < y}\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave \"\\ = (length ss * (length ss-1)) / 2\n + length ss\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (card {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} + card {(x, y). x = s \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2 + real (length ss)\n[PROOF STEP]\nusing iH step\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (card {(x, y). x \\ set ss \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2\ncard {(x, y). x = s \\ y \\ set ss \\ x < y} = length ss\n\ngoal (1 subgoal):\n 1. real (card {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} + card {(x, y). x = s \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2 + real (length ss)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nreal (card {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} + card {(x, y). x = s \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2 + real (length ss)\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (card {(x, y). x \\ set ss \\ y \\ set ss \\ x < y} + card {(x, y). x = s \\ y \\ set ss \\ x < y}) = real (length ss * (length ss - 1)) / 2 + real (length ss)\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave \"\\ = (length ss * (length ss-1) + 2*length ss) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (length ss * (length ss - 1)) / 2 + real (length ss) = real (length ss * (length ss - 1) + 2 * length ss) / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nreal (length ss * (length ss - 1)) / 2 + real (length ss) = real (length ss * (length ss - 1) + 2 * length ss) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (length ss * (length ss - 1)) / 2 + real (length ss) = real (length ss * (length ss - 1) + 2 * length ss) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave \"\\ = (length ss * (length ss-1) + length ss * 2) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (length ss * (length ss - 1) + 2 * length ss) / 2 = real (length ss * (length ss - 1) + length ss * 2) / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nreal (length ss * (length ss - 1) + 2 * length ss) / 2 = real (length ss * (length ss - 1) + length ss * 2) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (length ss * (length ss - 1) + 2 * length ss) / 2 = real (length ss * (length ss - 1) + length ss * 2) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave \"\\ = (length ss * (length ss-1+2)) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (length ss * (length ss - 1) + length ss * 2) / 2 = real (length ss * (length ss - 1 + 2)) / 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal (length ss * (length ss - 1) + length ss * 2) / 2 = real (length ss * (length ss - 1 + 2)) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (length ss * (length ss - 1) + length ss * 2) / 2 = real (length ss * (length ss - 1 + 2)) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave \"\\ = (length ss * (length ss+1)) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (length ss * (length ss - 1 + 2)) / 2 = real (length ss * (length ss + 1)) / 2\n[PROOF STEP]\nusing cons(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nss \\ []\n\ngoal (1 subgoal):\n 1. real (length ss * (length ss - 1 + 2)) / 2 = real (length ss * (length ss + 1)) / 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal (length ss * (length ss - 1 + 2)) / 2 = real (length ss * (length ss + 1)) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (length ss * (length ss - 1 + 2)) / 2 = real (length ss * (length ss + 1)) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave \"\\ = ((length ss+1) * length ss) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (length ss * (length ss + 1)) / 2 = real ((length ss + 1) * length ss) / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nreal (length ss * (length ss + 1)) / 2 = real ((length ss + 1) * length ss) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (length ss * (length ss + 1)) / 2 = real ((length ss + 1) * length ss) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nhave \"\\ = (length (s#ss) * (length (s#ss)-1)) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real ((length ss + 1) * length ss) / 2 = real (length (s # ss) * (length (s # ss) - 1)) / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nreal ((length ss + 1) * length ss) / 2 = real (length (s # ss) * (length (s # ss) - 1)) / 2\n\ngoal (2 subgoals):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n 2. \\x xs. \\xs \\ []; \\sorted xs; distinct xs\\ \\ real (card {(x, y). x \\ set xs \\ y \\ set xs \\ x < y}) = real (length xs * (length xs - 1)) / 2; sorted (x # xs); distinct (x # xs)\\ \\ real (card {(xa, y). xa \\ set (x # xs) \\ y \\ set (x # xs) \\ xa < y}) = real (length (x # xs) * (length (x # xs) - 1)) / 2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (card {(x, y). x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y}) = real (length (s # ss) * (length (s # ss) - 1)) / 2\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (card {(x, y). x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y}) = real (length (s # ss) * (length (s # ss) - 1)) / 2\n\ngoal (1 subgoal):\n 1. real (card {a. case a of (x, y) \\ x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y}) = real (length (s # ss) * (length (s # ss) - 1)) / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nreal (card {a. case a of (x, y) \\ x \\ set (s # ss) \\ y \\ set (s # ss) \\ x < y}) = real (length (s # ss) * (length (s # ss) - 1)) / 2\n\ngoal (1 subgoal):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n[PROOF STEP]\ncase single\n[PROOF STATE]\nproof (state)\nthis:\nsorted [x_]\ndistinct [x_]\n\ngoal (1 subgoal):\n 1. \\x. \\sorted [x]; distinct [x]\\ \\ real (card {(xa, y). xa \\ set [x] \\ y \\ set [x] \\ xa < y}) = real (length [x] * (length [x] - 1)) / 2\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nsorted [x_]\ndistinct [x_]\n\ngoal (1 subgoal):\n 1. real (card {a. case a of (xa, y) \\ xa \\ set [x_] \\ y \\ set [x_] \\ xa < y}) = real (length [x_] * (length [x_] - 1)) / 2\n[PROOF STEP]\nby(simp cong: conj_cong)\n[PROOF STATE]\nproof (state)\nthis:\nreal (card {a. case a of (xa, y) \\ xa \\ set [x_] \\ y \\ set [x_] \\ xa < y}) = real (length [x_] * (length [x_] - 1)) / 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 19586, "file": "List_Update_List_Factoring", "length": 95, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357701094303, "lm_q2_score": 0.81047890180374, "lm_q1q2_score": 0.7027141987828511}} {"text": "[STATEMENT]\nlemma list_add_assoc: fixes xs :: \"'a::monoid_add list\"\nshows \"(xs+ys)+zs = xs+(ys+zs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. xs + ys + zs = xs + (ys + zs)\n[PROOF STEP]\napply(induct xs arbitrary: ys zs)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\ys zs. [] + ys + zs = [] + (ys + zs)\n 2. \\a xs ys zs. (\\ys zs. xs + ys + zs = xs + (ys + zs)) \\ a # xs + ys + zs = a # xs + (ys + zs)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a xs ys zs. (\\ys zs. xs + ys + zs = xs + (ys + zs)) \\ a # xs + ys + zs = a # xs + (ys + zs)\n[PROOF STEP]\napply(case_tac ys)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\a xs ys zs. \\\\ys zs. xs + ys + zs = xs + (ys + zs); ys = []\\ \\ a # xs + ys + zs = a # xs + (ys + zs)\n 2. \\a xs ys zs aa list. \\\\ys zs. xs + ys + zs = xs + (ys + zs); ys = aa # list\\ \\ a # xs + ys + zs = a # xs + (ys + zs)\n[PROOF STEP]\napply(simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a xs ys zs aa list. \\\\ys zs. xs + ys + zs = xs + (ys + zs); ys = aa # list\\ \\ a # xs + ys + zs = a # xs + (ys + zs)\n[PROOF STEP]\napply(simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a xs ys zs aa list. \\\\ys zs. xs + ys + zs = xs + (ys + zs); ys = aa # list\\ \\ (a + aa) # xs + list + zs = a # xs + (aa # list + zs)\n[PROOF STEP]\napply(case_tac zs)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\a xs ys zs aa list. \\\\ys zs. xs + ys + zs = xs + (ys + zs); ys = aa # list; zs = []\\ \\ (a + aa) # xs + list + zs = a # xs + (aa # list + zs)\n 2. \\a xs ys zs aa list ab lista. \\\\ys zs. xs + ys + zs = xs + (ys + zs); ys = aa # list; zs = ab # lista\\ \\ (a + aa) # xs + list + zs = a # xs + (aa # list + zs)\n[PROOF STEP]\napply(simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a xs ys zs aa list ab lista. \\\\ys zs. xs + ys + zs = xs + (ys + zs); ys = aa # list; zs = ab # lista\\ \\ (a + aa) # xs + list + zs = a # xs + (aa # list + zs)\n[PROOF STEP]\napply(simp add: add.assoc)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1104, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357632379241, "lm_q2_score": 0.8104789063814616, "lm_q1q2_score": 0.7027141971826886}} {"text": "[STATEMENT]\nlemma homotopic_nearby_loops:\n fixes g h :: \"real \\ 'a::euclidean_space\"\n assumes \"path g\" \"open S\" \"path_image g \\ S\" \"pathfinish g = pathstart g\"\n shows \"\\e. 0 < e \\\n (\\h. path h \\ pathfinish h = pathstart h \\\n (\\t \\ {0..1}. norm(h t - g t) < e) \\ homotopic_loops S g h)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\e>0. \\h. path h \\ pathfinish h = pathstart h \\ (\\t\\{0..1}. norm (h t - g t) < e) \\ homotopic_loops S g h\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\e>0. \\h. path h \\ pathfinish h = pathstart h \\ (\\t\\{0..1}. norm (h t - g t) < e) \\ homotopic_loops S g h\n[PROOF STEP]\nobtain e where \"e > 0\" and e: \"\\x y. x \\ path_image g \\ y \\ - S \\ e \\ dist x y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\e. \\0 < e; \\x y. \\x \\ path_image g; y \\ - S\\ \\ e \\ dist x y\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing separate_compact_closed [of \"path_image g\" \"-S\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\compact (path_image g); closed (- S); path_image g \\ - S = {}\\ \\ \\d>0. \\x\\path_image g. \\y\\- S. d \\ dist x y\npath g\nopen S\npath_image g \\ S\npathfinish g = pathstart g\n\ngoal (1 subgoal):\n 1. (\\e. \\0 < e; \\x y. \\x \\ path_image g; y \\ - S\\ \\ e \\ dist x y\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n0 < e\n\\?x \\ path_image g; ?y \\ - S\\ \\ e \\ dist ?x ?y\n\ngoal (1 subgoal):\n 1. \\e>0. \\h. path h \\ pathfinish h = pathstart h \\ (\\t\\{0..1}. norm (h t - g t) < e) \\ homotopic_loops S g h\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\e>0. \\h. path h \\ pathfinish h = pathstart h \\ (\\t\\{0..1}. norm (h t - g t) < e) \\ homotopic_loops S g h\n[PROOF STEP]\nusing e [unfolded dist_norm] \\e > 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x \\ path_image g; ?y \\ - S\\ \\ e \\ norm (?x - ?y)\n0 < e\n\ngoal (1 subgoal):\n 1. \\e>0. \\h. path h \\ pathfinish h = pathstart h \\ (\\t\\{0..1}. norm (h t - g t) < e) \\ homotopic_loops S g h\n[PROOF STEP]\nby (fastforce simp: path_image_def intro!: homotopic_loops_nearby_explicit assms exI)\n[PROOF STATE]\nproof (state)\nthis:\n\\e>0. \\h. path h \\ pathfinish h = pathstart h \\ (\\t\\{0..1}. norm (h t - g t) < e) \\ homotopic_loops S g h\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1259, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357529306639, "lm_q2_score": 0.810478913248044, "lm_q1q2_score": 0.7027141947824441}} {"text": "[STATEMENT]\nlemma norm_blinfun_of_list:\n \"norm (blinfun_of_list xs::'a::executable_euclidean_space\\\\<^sub>L'a) \\ (\\x\\xs. abs x)\"\n if \"length xs = DIM('a)*DIM('a)\"\\ \\should work for all lists\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (blinfun_of_list xs) \\ sum_list (map abs xs)\n[PROOF STEP]\nunfolding blinfun_of_list_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (blinfun_of_matrix (\\i j. xs ! (index Basis_list i * DIM('a) + index Basis_list j))) \\ sum_list (map abs xs)\n[PROOF STEP]\napply (rule norm_blinfun_of_matrix[le])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\Basis. \\j\\Basis. \\xs ! (index Basis_list i * DIM('a) + index Basis_list j)\\) \\ sum_list (map abs xs)\n[PROOF STEP]\napply (auto simp: sum_Basis_sum_nth_Basis_list)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ijxs ! (i * DIM('a) + j)\\) \\ sum_list (map abs xs)\n[PROOF STEP]\napply (subst add.commute)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ijxs ! (j + i * DIM('a))\\) \\ sum_list (map abs xs)\n[PROOF STEP]\napply (subst sum_mult_product[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\axs ! a\\) \\ sum_list (map abs xs)\n[PROOF STEP]\nby (auto simp: sum_list_sum_nth that atLeast0LessThan)", "meta": {"llama_tokens": 648, "file": "Ordinary_Differential_Equations_Ex_Examples_Poincare_Map", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357563664174, "lm_q2_score": 0.8104788995148791, "lm_q1q2_score": 0.7027141856599047}} {"text": "[STATEMENT]\nlemma deriv_inverse [simp]:\n \"\\f field_differentiable at z; f z \\ 0\\\n \\ deriv (\\w. inverse (f w)) z = - deriv f z / f z ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f field_differentiable at z; f z \\ (0::'a)\\ \\ deriv (\\w. inverse (f w)) z = - deriv f z / (f z)\\<^sup>2\n[PROOF STEP]\nunfolding DERIV_deriv_iff_field_differentiable[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative deriv f z) (at z); f z \\ (0::'a)\\ \\ deriv (\\w. inverse (f w)) z = - deriv f z / (f z)\\<^sup>2\n[PROOF STEP]\nby (safe intro!: DERIV_imp_deriv derivative_eq_intros) (auto simp: field_split_simps power2_eq_square)", "meta": {"llama_tokens": 328, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894717137996, "lm_q2_score": 0.7853085859124002, "lm_q1q2_score": 0.7026858547208675}} {"text": "[STATEMENT]\nlemma unity_root_add: \"unity_root k (m + n) = unity_root k m * unity_root k n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. unity_root k (m + n) = unity_root k m * unity_root k n\n[PROOF STEP]\nby (simp add: unity_root_conv_exp add_divide_distrib algebra_simps exp_add)", "meta": {"llama_tokens": 116, "file": "Gauss_Sums_Complex_Roots_Of_Unity", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894745194281, "lm_q2_score": 0.7853085808877581, "lm_q1q2_score": 0.702685852428155}} {"text": "[STATEMENT]\nlemma makespan_correct:\n \"\\x \\ {1..m}. T x \\ makespan T\"\n \"\\x \\ {1..m}. T x = makespan T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\{1..m}. T x \\ makespan T &&& \\x\\{1..m}. T x = makespan T\n[PROOF STEP]\nusing f_Max\\<^sub>0_correct m_gt_0\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\{1..?m}. ?T x \\ f_Max\\<^sub>0 ?T ?m\n0 < ?m \\ \\x\\{1..?m}. ?T x = f_Max\\<^sub>0 ?T ?m\n0 < m\n\ngoal (1 subgoal):\n 1. \\x\\{1..m}. T x \\ makespan T &&& \\x\\{1..m}. T x = makespan T\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 288, "file": "Approximation_Algorithms_Approx_LB_Hoare", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894604912848, "lm_q2_score": 0.7853085859124002, "lm_q1q2_score": 0.7026858459077303}} {"text": "[STATEMENT]\nlemma iso_onto_image:\n assumes \"group G\" \"group H\"\n shows\n \"f \\ iso G (subgroup_generated H (f ` carrier G)) \\ f \\ hom G H \\ inj_on f (carrier G)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f \\ Group.iso G (subgroup_generated H (f ` carrier G))) = (f \\ hom G H \\ inj_on f (carrier G))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nGroup.group G\nGroup.group H\n\ngoal (1 subgoal):\n 1. (f \\ Group.iso G (subgroup_generated H (f ` carrier G))) = (f \\ hom G H \\ inj_on f (carrier G))\n[PROOF STEP]\napply (auto simp: iso_def bij_betw_def hom_into_subgroup_eq_gen carrier_subgroup_generated hom_carrier generate.incl Int_absorb1 Int_absorb2)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\Group.group G; Group.group H; f \\ hom G H; inj_on f (carrier G); x \\ generate H (f ` carrier G)\\ \\ x \\ f ` carrier G\n[PROOF STEP]\nby (metis group.generateI group.subgroupE(1) group.subgroup_self group_hom.generate_img group_hom.intro group_hom_axioms.intro)", "meta": {"llama_tokens": 438, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789454880027, "lm_q2_score": 0.7853085808877581, "lm_q1q2_score": 0.7026858370051647}} {"text": "[STATEMENT]\nlemma two_pow_r_le_1: \"0 < 1 - 2 powr - real r\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < 1 - 2 powr - real r\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 < 1 - 2 powr - real r\n[PROOF STEP]\nhave a: \"2 powr (0::real) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 powr 0 = 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 powr 0 = 1\n\ngoal (1 subgoal):\n 1. 0 < 1 - 2 powr - real r\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < 1 - 2 powr - real r\n[PROOF STEP]\nusing r_ge_23\n[PROOF STATE]\nproof (prove)\nusing this:\n23 \\ r\n\ngoal (1 subgoal):\n 1. 0 < 1 - 2 powr - real r\n[PROOF STEP]\nby (simp, subst a[symmetric], intro powr_less_mono, auto)\n[PROOF STATE]\nproof (state)\nthis:\n0 < 1 - 2 powr - real r\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 425, "file": "Frequency_Moments_Frequency_Moment_0", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110454379297, "lm_q2_score": 0.78793120560257, "lm_q1q2_score": 0.7026857522015962}} {"text": "[STATEMENT]\nlemma homeomorphic_open_imp_same_dimension:\n fixes S :: \"'a::euclidean_space set\" and T :: \"'b::euclidean_space set\"\n assumes \"S homeomorphic T\" \"open S\" \"S \\ {}\" \"open T\" \"T \\ {}\"\n shows \"DIM('a) = DIM('b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. DIM('a) = DIM('b)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nS homeomorphic T\nopen S\nS \\ {}\nopen T\nT \\ {}\n\ngoal (1 subgoal):\n 1. DIM('a) = DIM('b)\n[PROOF STEP]\napply (simp add: homeomorphic_minimal)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\f g. (\\x\\S. f x \\ T \\ g (f x) = x) \\ (\\y\\T. g y \\ S \\ f (g y) = y) \\ continuous_on S f \\ continuous_on T g; open S; S \\ {}; open T; T \\ {}\\ \\ DIM('a) = DIM('b)\n[PROOF STEP]\napply (rule order_antisym; metis inj_onI invariance_of_dimension)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 433, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110454379296, "lm_q2_score": 0.7879312006227324, "lm_q1q2_score": 0.702685747760522}} {"text": "[STATEMENT]\nlemma vangle_eqI:\n assumes \"u \\ 0\" \"v \\ 0\" \"w \\ 0\" \"x \\ 0\"\n assumes \"(u \\ v) * norm w * norm x = (w \\ x) * norm u * norm v\"\n shows \"vangle u v = vangle w x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vangle u v = vangle w x\n[PROOF STEP]\nusing assms Cauchy_Schwarz_ineq2[of u v] Cauchy_Schwarz_ineq2[of w x]\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ (0::'a)\nv \\ (0::'a)\nw \\ (0::'b)\nx \\ (0::'b)\nu \\ v * norm w * norm x = w \\ x * norm u * norm v\n\\u \\ v\\ \\ norm u * norm v\n\\w \\ x\\ \\ norm w * norm x\n\ngoal (1 subgoal):\n 1. vangle u v = vangle w x\n[PROOF STEP]\nunfolding vangle_def\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ (0::'a)\nv \\ (0::'a)\nw \\ (0::'b)\nx \\ (0::'b)\nu \\ v * norm w * norm x = w \\ x * norm u * norm v\n\\u \\ v\\ \\ norm u * norm v\n\\w \\ x\\ \\ norm w * norm x\n\ngoal (1 subgoal):\n 1. (if u = (0::'a) \\ v = (0::'a) then pi / 2 else arccos (u \\ v / (norm u * norm v))) = (if w = (0::'b) \\ x = (0::'b) then pi / 2 else arccos (w \\ x / (norm w * norm x)))\n[PROOF STEP]\nby (auto simp: arccos_eq_iff field_simps)", "meta": {"llama_tokens": 596, "file": "Triangle_Angles", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110569397306, "lm_q2_score": 0.787931185683219, "lm_q1q2_score": 0.7026857434999266}} {"text": "[STATEMENT]\nlemma homeomorphic_open_imp_same_dimension:\n fixes S :: \"'a::euclidean_space set\" and T :: \"'b::euclidean_space set\"\n assumes \"S homeomorphic T\" \"open S\" \"S \\ {}\" \"open T\" \"T \\ {}\"\n shows \"DIM('a) = DIM('b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. DIM('a) = DIM('b)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nS homeomorphic T\nopen S\nS \\ {}\nopen T\nT \\ {}\n\ngoal (1 subgoal):\n 1. DIM('a) = DIM('b)\n[PROOF STEP]\napply (simp add: homeomorphic_minimal)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\f g. (\\x\\S. f x \\ T \\ g (f x) = x) \\ (\\y\\T. g y \\ S \\ f (g y) = y) \\ continuous_on S f \\ continuous_on T g; open S; S \\ {}; open T; T \\ {}\\ \\ DIM('a) = DIM('b)\n[PROOF STEP]\napply (rule order_antisym; metis inj_onI invariance_of_dimension)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 433, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110454379296, "lm_q2_score": 0.7879311956428947, "lm_q1q2_score": 0.7026857433194478}} {"text": "[STATEMENT]\nlemma linear_bij_rank_eq_ncols:\n fixes f::\"('a::{field}^'n::{mod_type})=>('a::{field}^'n::{mod_type})\"\n assumes lf: \"linear ((*s)) ((*s)) f\"\n shows \"bij f \\ rank (matrix f) = ncols (matrix f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij f = (rank (matrix f) = ncols (matrix f))\n[PROOF STEP]\nunfolding bij_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (inj f \\ surj f) = (rank (matrix f) = ncols (matrix f))\n[PROOF STEP]\nusing lf linear_injective_rank_eq_ncols vec.linear_inj_imp_surj\n[PROOF STATE]\nproof (prove)\nusing this:\nlinear (*s) (*s) f\nlinear (*s) (*s) ?f \\ inj ?f = (rank (matrix ?f) = ncols (matrix ?f))\n\\linear (*s) (*s) ?f; inj ?f\\ \\ surj ?f\n\ngoal (1 subgoal):\n 1. (inj f \\ surj f) = (rank (matrix f) = ncols (matrix f))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 369, "file": "Gauss_Jordan_Linear_Maps", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110339361276, "lm_q2_score": 0.787931190663057, "lm_q1q2_score": 0.702685729815745}} {"text": "[STATEMENT]\nlemma bij_betw_nth_root_unity:\n fixes c :: complex and n :: nat\n assumes c: \"c \\ 0\" and n: \"n > 0\"\n defines \"c' \\ root n (norm c) * cis (Arg c / n)\"\n shows \"bij_betw (\\z. c' * z) {z. z ^ n = 1} {z. z ^ n = c}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw ((*) c') {z. z ^ n = 1} {z. z ^ n = c}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. bij_betw ((*) c') {z. z ^ n = 1} {z. z ^ n = c}\n[PROOF STEP]\nhave \"c' ^ n = of_real (root n (norm c) ^ n) * cis (Arg c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c' ^ n = complex_of_real (root n (cmod c) ^ n) * cis (Arg c)\n[PROOF STEP]\nunfolding of_real_power\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c' ^ n = complex_of_real (root n (cmod c)) ^ n * cis (Arg c)\n[PROOF STEP]\nusing n\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\n\ngoal (1 subgoal):\n 1. c' ^ n = complex_of_real (root n (cmod c)) ^ n * cis (Arg c)\n[PROOF STEP]\nby (simp add: c'_def power_mult_distrib DeMoivre)\n[PROOF STATE]\nproof (state)\nthis:\nc' ^ n = complex_of_real (root n (cmod c) ^ n) * cis (Arg c)\n\ngoal (1 subgoal):\n 1. bij_betw ((*) c') {z. z ^ n = 1} {z. z ^ n = c}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nc' ^ n = complex_of_real (root n (cmod c) ^ n) * cis (Arg c)\n\ngoal (1 subgoal):\n 1. bij_betw ((*) c') {z. z ^ n = 1} {z. z ^ n = c}\n[PROOF STEP]\nfrom n\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < n\n[PROOF STEP]\nhave \"root n (norm c) ^ n = norm c\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\n\ngoal (1 subgoal):\n 1. root n (cmod c) ^ n = cmod c\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nroot n (cmod c) ^ n = cmod c\n\ngoal (1 subgoal):\n 1. bij_betw ((*) c') {z. z ^ n = 1} {z. z ^ n = c}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nroot n (cmod c) ^ n = cmod c\n\ngoal (1 subgoal):\n 1. bij_betw ((*) c') {z. z ^ n = 1} {z. z ^ n = c}\n[PROOF STEP]\nfrom c\n[PROOF STATE]\nproof (chain)\npicking this:\nc \\ 0\n[PROOF STEP]\nhave \"of_real \\ * cis (Arg c) = c\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ 0\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod c) * cis (Arg c) = c\n[PROOF STEP]\nby (simp add: cis_Arg Complex.sgn_eq)\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (cmod c) * cis (Arg c) = c\n\ngoal (1 subgoal):\n 1. bij_betw ((*) c') {z. z ^ n = 1} {z. z ^ n = c}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nc' ^ n = c\n[PROOF STEP]\nhave [simp]: \"c' ^ n = c\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc' ^ n = c\n\ngoal (1 subgoal):\n 1. c' ^ n = c\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nc' ^ n = c\n\ngoal (1 subgoal):\n 1. bij_betw ((*) c') {z. z ^ n = 1} {z. z ^ n = c}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw ((*) c') {z. z ^ n = 1} {z. z ^ n = c}\n[PROOF STEP]\nunfolding bij_betw_def inj_on_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\{z. z ^ n = 1}. \\y\\{z. z ^ n = 1}. c' * x = c' * y \\ x = y) \\ (*) c' ` {z. z ^ n = 1} = {z. z ^ n = c}\n[PROOF STEP]\nproof safe\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x xa. xa ^ n = 1 \\ (c' * xa) ^ n = c\n 3. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nfix z :: complex\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x xa. xa ^ n = 1 \\ (c' * xa) ^ n = c\n 3. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nassume \"z ^ n = 1\"\n[PROOF STATE]\nproof (state)\nthis:\nz ^ n = 1\n\ngoal (3 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x xa. xa ^ n = 1 \\ (c' * xa) ^ n = c\n 3. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nhence \"(c' * z) ^ n = c' ^ n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz ^ n = 1\n\ngoal (1 subgoal):\n 1. (c' * z) ^ n = c' ^ n\n[PROOF STEP]\nby (simp add: power_mult_distrib)\n[PROOF STATE]\nproof (state)\nthis:\n(c' * z) ^ n = c' ^ n\n\ngoal (3 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x xa. xa ^ n = 1 \\ (c' * xa) ^ n = c\n 3. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(c' * z) ^ n = c' ^ n\n\ngoal (3 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x xa. xa ^ n = 1 \\ (c' * xa) ^ n = c\n 3. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nhave \"c' ^ n = of_real (root n (norm c) ^ n) * cis (Arg c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c' ^ n = complex_of_real (root n (cmod c) ^ n) * cis (Arg c)\n[PROOF STEP]\nunfolding of_real_power\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c' ^ n = complex_of_real (root n (cmod c)) ^ n * cis (Arg c)\n[PROOF STEP]\nusing n\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\n\ngoal (1 subgoal):\n 1. c' ^ n = complex_of_real (root n (cmod c)) ^ n * cis (Arg c)\n[PROOF STEP]\nby (simp add: c'_def power_mult_distrib DeMoivre)\n[PROOF STATE]\nproof (state)\nthis:\nc' ^ n = complex_of_real (root n (cmod c) ^ n) * cis (Arg c)\n\ngoal (3 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x xa. xa ^ n = 1 \\ (c' * xa) ^ n = c\n 3. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nc' ^ n = complex_of_real (root n (cmod c) ^ n) * cis (Arg c)\n\ngoal (3 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x xa. xa ^ n = 1 \\ (c' * xa) ^ n = c\n 3. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nfrom n\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < n\n[PROOF STEP]\nhave \"root n (norm c) ^ n = norm c\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\n\ngoal (1 subgoal):\n 1. root n (cmod c) ^ n = cmod c\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nroot n (cmod c) ^ n = cmod c\n\ngoal (3 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x xa. xa ^ n = 1 \\ (c' * xa) ^ n = c\n 3. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nroot n (cmod c) ^ n = cmod c\n\ngoal (3 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x xa. xa ^ n = 1 \\ (c' * xa) ^ n = c\n 3. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nfrom c\n[PROOF STATE]\nproof (chain)\npicking this:\nc \\ 0\n[PROOF STEP]\nhave \"\\ * cis (Arg c) = c\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ 0\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod c) * cis (Arg c) = c\n[PROOF STEP]\nby (simp add: cis_Arg Complex.sgn_eq)\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (cmod c) * cis (Arg c) = c\n\ngoal (3 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x xa. xa ^ n = 1 \\ (c' * xa) ^ n = c\n 3. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(c' * z) ^ n = c\n[PROOF STEP]\nshow \"(c' * z) ^ n = c\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(c' * z) ^ n = c\n\ngoal (1 subgoal):\n 1. (c' * z) ^ n = c\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(c' * z) ^ n = c\n\ngoal (2 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nfix z\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nassume z: \"c = z ^ n\"\n[PROOF STATE]\nproof (state)\nthis:\nc = z ^ n\n\ngoal (2 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\ndefine z' where \"z' = z / c'\"\n[PROOF STATE]\nproof (state)\nthis:\nz' = z / c'\n\ngoal (2 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nfrom c and n\n[PROOF STATE]\nproof (chain)\npicking this:\nc \\ 0\n0 < n\n[PROOF STEP]\nhave \"c' \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ 0\n0 < n\n\ngoal (1 subgoal):\n 1. c' \\ 0\n[PROOF STEP]\nby (auto simp: c'_def)\n[PROOF STATE]\nproof (state)\nthis:\nc' \\ 0\n\ngoal (2 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nwith n c\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < n\nc \\ 0\nc' \\ 0\n[PROOF STEP]\nhave \"z = c' * z'\" and \"z' ^ n = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\nc \\ 0\nc' \\ 0\n\ngoal (1 subgoal):\n 1. z = c' * z' &&& z' ^ n = 1\n[PROOF STEP]\nby (auto simp: z'_def power_divide z)\n[PROOF STATE]\nproof (state)\nthis:\nz = c' * z'\nz' ^ n = 1\n\ngoal (2 subgoals):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n 2. \\x. c = x ^ n \\ x \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nthus \"z \\ (\\z. c' * z) ` {z. z ^ n = 1}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz = c' * z'\nz' ^ n = 1\n\ngoal (1 subgoal):\n 1. z \\ (*) c' ` {z. z ^ n = 1}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nz \\ (*) c' ` {z. z ^ n = 1}\n\ngoal (1 subgoal):\n 1. \\x y. \\x ^ n = 1; y ^ n = 1; c' * x = c' * y\\ \\ x = y\n[PROOF STEP]\nqed (insert c n, auto simp: c'_def)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw ((*) c') {z. z ^ n = 1} {z. z ^ n = c}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5278, "file": null, "length": 53, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.84997116805678, "lm_q2_score": 0.8267117962054049, "lm_q1q2_score": 0.7026811910670266}} {"text": "[STATEMENT]\nlemma is_det_bound_ge_zero: assumes \"is_det_bound f\"\n and \"x \\ 0\" \n shows \"f n x \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) \\ f n x\n[PROOF STEP]\nusing assms(1)[unfolded is_det_bound_def, rule_format, of \"0\\<^sub>m n n\" n x]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0\\<^sub>m n n \\ carrier_mat n n; 0\\<^sub>m n n \\ Bounded_mat x\\ \\ \\det (0\\<^sub>m n n)\\ \\ f n x\n\ngoal (1 subgoal):\n 1. (0::'a) \\ f n x\n[PROOF STEP]\nusing assms(2)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0\\<^sub>m n n \\ carrier_mat n n; 0\\<^sub>m n n \\ Bounded_mat x\\ \\ \\det (0\\<^sub>m n n)\\ \\ f n x\n(0::'a) \\ x\n\ngoal (1 subgoal):\n 1. (0::'a) \\ f n x\n[PROOF STEP]\nunfolding Bounded_mat_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0\\<^sub>m n n \\ carrier_mat n n; 0\\<^sub>m n n \\ {A. \\ijA $$ (i, j)\\ \\ x}\\ \\ \\det (0\\<^sub>m n n)\\ \\ f n x\n(0::'a) \\ x\n\ngoal (1 subgoal):\n 1. (0::'a) \\ f n x\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 561, "file": "Linear_Inequalities_Integral_Bounded_Vectors", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8499711642563823, "lm_q2_score": 0.8267117962054049, "lm_q1q2_score": 0.7026811879251931}} {"text": "[STATEMENT]\nlemma pell_power_add: \"pell_power z (m + n) = pell_mul (pell_power z m) (pell_power z n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pell_power z (m + n) = pell_mul (pell_power z m) (pell_power z n)\n[PROOF STEP]\nby (induction m arbitrary: z )\n (simp_all add: funpow_add o_def pell_power_Suc pell_mul_assoc)", "meta": {"llama_tokens": 147, "file": "Pell_Pell", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976953030553434, "lm_q2_score": 0.7826624789529375, "lm_q1q2_score": 0.7025924312337035}} {"text": "[STATEMENT]\nlemma strict_SD_iff:\n assumes \"p \\ lotteries_on carrier\" \"q \\ lotteries_on carrier\"\n shows \"(p \\[SD(le)] q) \\\n (\\u. is_vnm_utility u \\ measure_pmf.expectation p u < measure_pmf.expectation q u)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p \\[SD le] q = (\\u. is_vnm_utility u \\ measure_pmf.expectation p u < measure_pmf.expectation q u)\n[PROOF STEP]\nusing not_strict_SD_iff[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\ p \\[SD le] q) = (\\u. is_vnm_utility u \\ measure_pmf.expectation q u \\ measure_pmf.expectation p u)\n\ngoal (1 subgoal):\n 1. p \\[SD le] q = (\\u. is_vnm_utility u \\ measure_pmf.expectation p u < measure_pmf.expectation q u)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 329, "file": "Randomised_Social_Choice_Stochastic_Dominance", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976953030553433, "lm_q2_score": 0.7826624738835052, "lm_q1q2_score": 0.7025924266828979}} {"text": "[STATEMENT]\nlemma ffact_monomial:\n \"ffact n a = (\\k = 0..n. (- 1) ^ (n - k) * of_nat (stirling n k) * a ^ k)\"\n for a :: \"'a :: comm_ring_1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ffact 0 a = (\\k = 0..0. (- (1::'a)) ^ (0 - k) * of_nat (stirling 0 k) * a ^ k)\n 2. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. ffact 0 a = (\\k = 0..0. (- (1::'a)) ^ (0 - k) * of_nat (stirling 0 k) * a ^ k)\n 2. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ffact 0 a = (\\k = 0..0. (- (1::'a)) ^ (0 - k) * of_nat (stirling 0 k) * a ^ k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nffact 0 a = (\\k = 0..0. (- (1::'a)) ^ (0 - k) * of_nat (stirling 0 k) * a ^ k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k)\n[PROOF STEP]\nhave \"ffact (Suc n) a = (a - of_nat n) * (\\k = 0..n. (- 1) ^ (n - k) * of_nat (stirling n k) * a ^ k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k)\n\ngoal (1 subgoal):\n 1. ffact (Suc n) a = (a - of_nat n) * (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k)\n[PROOF STEP]\nby (simp add: ffact_Suc_rev)\n[PROOF STATE]\nproof (state)\nthis:\nffact (Suc n) a = (a - of_nat n) * (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nffact (Suc n) a = (a - of_nat n) * (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nhave \"\\ = (\\k = 0..n. (- 1) ^ (n - k) * of_nat (stirling n k) * a ^ (Suc k)) +\n (\\k = 0..n. (- 1) * (- 1) ^ (n - k) * of_nat (n * (stirling n k)) * a ^ k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a - of_nat n) * (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k)\n[PROOF STEP]\nby (simp only: diff_conv_add_uminus distrib_right) (simp add: sum_distrib_left field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(a - of_nat n) * (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(a - of_nat n) * (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nhave \"\\ = (\\k = 0..n. (- 1) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) +\n (\\k = 0..n. (- 1) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n[PROOF STEP]\nhave \"(\\k = 0..n. (- 1) * (- 1) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) =\n (\\k = 0..n. (- 1) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k)\n[PROOF STEP]\nby (simp add: Suc_diff_le)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k)\n\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k)\n\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n[PROOF STEP]\nhave \"\\ = (\\k = Suc 0..Suc n. (- 1) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = Suc 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k)\n[PROOF STEP]\nby (simp add: sum.atLeast_Suc_atMost) (cases n; simp)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. (- (1::'a)) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = Suc 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k)\n\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. (- (1::'a)) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = Suc 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k)\n\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n[PROOF STEP]\nhave \"\\ = (\\k = 0..n. (- 1) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = Suc 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n[PROOF STEP]\nby (simp only: sum.shift_bounds_cl_Suc_ivl)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = Suc 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. - (1::'a) * (- (1::'a)) ^ (n - k) * of_nat (n * stirling n k) * a ^ k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nhave \"\\ = (\\k = 0..n. (- 1) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k) + stirling n k) * a ^ Suc k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k) + stirling n k) * a ^ Suc k)\n[PROOF STEP]\nby (simp add: sum.distrib algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k) + stirling n k) * a ^ Suc k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling n k) * a ^ Suc k) + (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k)) * a ^ Suc k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k) + stirling n k) * a ^ Suc k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nhave \"\\ = (\\k = 0..n. (- 1) ^ (Suc n - Suc k) * of_nat (stirling (Suc n) (Suc k)) * a ^ Suc k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k) + stirling n k) * a ^ Suc k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling (Suc n) (Suc k)) * a ^ Suc k)\n[PROOF STEP]\nby (simp only: stirling.simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k) + stirling n k) * a ^ Suc k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling (Suc n) (Suc k)) * a ^ Suc k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (n * stirling n (Suc k) + stirling n k) * a ^ Suc k) = (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling (Suc n) (Suc k)) * a ^ Suc k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nhave \"\\ = (\\k = Suc 0..Suc n. (- 1) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling (Suc n) (Suc k)) * a ^ Suc k) = (\\k = Suc 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nby (simp only: sum.shift_bounds_cl_Suc_ivl)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling (Suc n) (Suc k)) * a ^ Suc k) = (\\k = Suc 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..n. (- (1::'a)) ^ (Suc n - Suc k) * of_nat (stirling (Suc n) (Suc k)) * a ^ Suc k) = (\\k = Suc 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nhave \"\\ = (\\k = 0..Suc n. (- 1) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = Suc 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k) = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nby (simp add: sum.atLeast_Suc_atMost)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = Suc 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k) = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n\ngoal (1 subgoal):\n 1. \\n. ffact n a = (\\k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k) \\ ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n\ngoal (1 subgoal):\n 1. ffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nffact (Suc n) a = (\\k = 0..Suc n. (- (1::'a)) ^ (Suc n - k) * of_nat (stirling (Suc n) k) * a ^ k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 9819, "file": "Discrete_Summation_Factorials", "length": 43, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976952811593496, "lm_q2_score": 0.7826624789529375, "lm_q1q2_score": 0.7025924140965307}} {"text": "[STATEMENT]\nlemma cos_of_quarter_pi:\n fixes x:: real\n assumes \"x = pi/2\"\n shows \"cos (x/2) = (sqrt 2)/2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos (x / 2) = sqrt 2 / 2\n[PROOF STEP]\nby (auto simp add: assms cos_45)", "meta": {"llama_tokens": 110, "file": "Isabelle_Marries_Dirac_Basics", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.888758793492457, "lm_q2_score": 0.7905303162021596, "lm_q1q2_score": 0.702590770047042}} {"text": "[STATEMENT]\nlemma sin_cos_csqrt:\n assumes \"0 < sin(Re z) \\ sin(Re z) = 0 \\ 0 \\ Im z * cos(Re z)\"\n shows \"sin z = csqrt(1 - (cos z)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin z = csqrt (1 - (cos z)\\<^sup>2)\n[PROOF STEP]\nproof (rule csqrt_unique [THEN sym])\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (sin z)\\<^sup>2 = 1 - (cos z)\\<^sup>2\n 2. 0 < Re (sin z) \\ Re (sin z) = 0 \\ 0 \\ Im (sin z)\n[PROOF STEP]\nshow \"(sin z)\\<^sup>2 = 1 - (cos z)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (sin z)\\<^sup>2 = 1 - (cos z)\\<^sup>2\n[PROOF STEP]\nby (simp add: sin_squared_eq)\n[PROOF STATE]\nproof (state)\nthis:\n(sin z)\\<^sup>2 = 1 - (cos z)\\<^sup>2\n\ngoal (1 subgoal):\n 1. 0 < Re (sin z) \\ Re (sin z) = 0 \\ 0 \\ Im (sin z)\n[PROOF STEP]\nqed (use assms in \\auto simp: Re_sin Im_sin add_pos_pos mult_le_0_iff zero_le_mult_iff\\)", "meta": {"llama_tokens": 453, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587993853654, "lm_q2_score": 0.7905303087996143, "lm_q1q2_score": 0.7025907681264874}} {"text": "[STATEMENT]\nlemma interchange_rows_transpose:\n shows \"interchange_rows (transpose A) a b = transpose (interchange_columns A a b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. interchange_rows (Finite_Cartesian_Product.transpose A) a b = Finite_Cartesian_Product.transpose (interchange_columns A a b)\n[PROOF STEP]\nunfolding interchange_rows_def interchange_columns_def transpose_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i j. if i = a then (\\i j. A $ j $ i) $ b $ j else if i = b then (\\i j. A $ j $ i) $ a $ j else (\\i j. A $ j $ i) $ i $ j) = (\\i j. (\\i j. if j = a then A $ i $ b else if j = b then A $ i $ a else A $ i $ j) $ j $ i)\n[PROOF STEP]\nby vector", "meta": {"llama_tokens": 280, "file": "Gauss_Jordan_Elementary_Operations", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774768002981829, "lm_q2_score": 0.8006920092299293, "lm_q1q2_score": 0.7025886622834014}} {"text": "[STATEMENT]\nlemma length_filter_replace2:\n \"\\ x \\ set xs; P x \\ \\\n length(filter P (replace x ys xs)) =\n length(filter P xs) + length(filter P ys) - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x \\ set xs; P x\\ \\ |filter P (replace x ys xs)| = |filter P xs| + |filter P ys| - 1\n[PROOF STEP]\napply(induct xs)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\x \\ set []; P x\\ \\ |filter P (replace x ys [])| = |filter P []| + |filter P ys| - 1\n 2. \\a xs. \\\\x \\ set xs; P x\\ \\ |filter P (replace x ys xs)| = |filter P xs| + |filter P ys| - 1; x \\ set (a # xs); P x\\ \\ |filter P (replace x ys (a # xs))| = |filter P (a # xs)| + |filter P ys| - 1\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a xs. \\\\x \\ set xs; P x\\ \\ |filter P (replace x ys xs)| = |filter P xs| + |filter P ys| - 1; x \\ set (a # xs); P x\\ \\ |filter P (replace x ys (a # xs))| = |filter P (a # xs)| + |filter P ys| - 1\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a xs. \\|filter P (replace x ys xs)| = |filter P xs| + |filter P ys| - Suc 0; P x; x \\ set xs; P a; a \\ x\\ \\ Suc (|filter P xs| + |filter P ys| - Suc 0) = |filter P xs| + |filter P ys|\n[PROOF STEP]\napply(drule split_list)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a xs. \\|filter P (replace x ys xs)| = |filter P xs| + |filter P ys| - Suc 0; P x; P a; a \\ x; \\ys zs. xs = ys @ x # zs\\ \\ Suc (|filter P xs| + |filter P ys| - Suc 0) = |filter P xs| + |filter P ys|\n[PROOF STEP]\napply clarsimp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 826, "file": "Flyspeck-Tame_ListAux", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.877476793890012, "lm_q2_score": 0.8006920092299292, "lm_q1q2_score": 0.7025886571524301}} {"text": "[STATEMENT]\nlemma finite_mids: \"finite (mids xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (mids xs)\n[PROOF STEP]\nby (induction xs) (simp add: mids_def, simp add: mids_cons finite_fronts)", "meta": {"llama_tokens": 84, "file": "Maximum_Segment_Sum_Maximum_Segment_Sum", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767906859265, "lm_q2_score": 0.800692004473946, "lm_q1q2_score": 0.7025886504136797}} {"text": "[STATEMENT]\nlemma inverse_eq1:\n assumes \"eventually (\\x. g x \\ 0) F\"\n shows \"f \\ L F (\\x. inverse (g x)) \\ (\\x. f x * g x) \\ L F (\\_. 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f \\ L F (\\x. inverse (g x))) = ((\\x. f x * g x) \\ L F (\\_. 1::'b))\n[PROOF STEP]\nusing divide_eq1[of g F f \"\\_. 1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in F. g x \\ (0::'b) \\ (f \\ L F (\\x. (1::'b) / g x)) = ((\\x. f x * g x) \\ L F (\\_. 1::'b))\n\ngoal (1 subgoal):\n 1. (f \\ L F (\\x. inverse (g x))) = ((\\x. f x * g x) \\ L F (\\_. 1::'b))\n[PROOF STEP]\nby (simp add: divide_inverse assms)", "meta": {"llama_tokens": 351, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.7025886457615613}} {"text": "[STATEMENT]\nlemma finite_card_eq_imp_bij_betw:\n assumes \"finite A\"\n and \"card (f ` A) = card A\"\n shows \"bij_betw f A (f ` A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw f A (f ` A)\n[PROOF STEP]\nusing \\card (f ` A) = card A\\\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (f ` A) = card A\n\ngoal (1 subgoal):\n 1. bij_betw f A (f ` A)\n[PROOF STEP]\nunfolding inj_on_iff_eq_card [OF \\finite A\\, symmetric]\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on f A\n\ngoal (1 subgoal):\n 1. bij_betw f A (f ` A)\n[PROOF STEP]\nby (rule inj_on_imp_bij_betw)", "meta": {"llama_tokens": 268, "file": "First_Order_Terms_Fun_More", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767810736692, "lm_q2_score": 0.8006920044739461, "lm_q1q2_score": 0.7025886427172221}} {"text": "[STATEMENT]\nlemma poly_of_vec_add:\n assumes \"dim_vec a = dim_vec b\"\n shows \"poly_of_vec (a + b) = poly_of_vec a + poly_of_vec b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly_of_vec (a + b) = poly_of_vec a + poly_of_vec b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec a = dim_vec b\n\ngoal (1 subgoal):\n 1. poly_of_vec (a + b) = poly_of_vec a + poly_of_vec b\n[PROOF STEP]\nby (auto simp add: poly_eq_iff coeff_poly_of_vec)", "meta": {"llama_tokens": 209, "file": "LLL_Basis_Reduction_Missing_Lemmas", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767938900121, "lm_q2_score": 0.8006919925839875, "lm_q1q2_score": 0.7025886425460027}} {"text": "[STATEMENT]\nlemma has_vderiv_on_sq_mtx_affine:\n fixes t\\<^sub>0::real and A :: \"('a::finite) sq_mtx\"\n defines \"lSol c t \\ exp ((c * (t - t\\<^sub>0)) *\\<^sub>R A)\"\n shows \"D (\\t. lSol 1 t *\\<^sub>V s + lSol 1 t *\\<^sub>V (\\\\<^sub>t\\<^sub>0\\<^sup>t (lSol (-1) \\ *\\<^sub>V B) \\\\)) = \n (\\t. A *\\<^sub>V (lSol 1 t *\\<^sub>V s + lSol 1 t *\\<^sub>V (\\\\<^sub>t\\<^sub>0\\<^sup>t (lSol (-1) \\ *\\<^sub>V B) \\\\)) + B) on {t\\<^sub>0--t}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. D (\\t. lSol 1 t *\\<^sub>V s + lSol 1 t *\\<^sub>V (\\\\<^sub>t\\<^sub>0\\<^sup>t lSol (- 1) \\ *\\<^sub>V B\\\\)) = (\\t. A *\\<^sub>V (lSol 1 t *\\<^sub>V s + lSol 1 t *\\<^sub>V (\\\\<^sub>t\\<^sub>0\\<^sup>t lSol (- 1) \\ *\\<^sub>V B\\\\)) + B) on {t\\<^sub>0--t}\n[PROOF STEP]\nunfolding assms\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. D (\\t. exp ((1 * (t - t\\<^sub>0)) *\\<^sub>R A) *\\<^sub>V s + exp ((1 * (t - t\\<^sub>0)) *\\<^sub>R A) *\\<^sub>V (\\\\<^sub>t\\<^sub>0\\<^sup>t exp ((- 1 * (\\ - t\\<^sub>0)) *\\<^sub>R A) *\\<^sub>V B\\\\)) = (\\t. A *\\<^sub>V (exp ((1 * (t - t\\<^sub>0)) *\\<^sub>R A) *\\<^sub>V s + exp ((1 * (t - t\\<^sub>0)) *\\<^sub>R A) *\\<^sub>V (\\\\<^sub>t\\<^sub>0\\<^sup>t exp ((- 1 * (\\ - t\\<^sub>0)) *\\<^sub>R A) *\\<^sub>V B\\\\)) + B) on {t\\<^sub>0--t}\n[PROOF STEP]\napply(simp only: mult.left_neutral mult_minus1)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. D (\\t. exp ((t - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s + exp ((t - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V (\\\\<^sub>t\\<^sub>0\\<^sup>t exp (- (\\ - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V B\\\\)) = (\\t. A *\\<^sub>V (exp ((t - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s + exp ((t - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V (\\\\<^sub>t\\<^sub>0\\<^sup>t exp (- (\\ - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V B\\\\)) + B) on {t\\<^sub>0--t}\n[PROOF STEP]\napply(rule poly_derivatives, (force)?, (force)?, (force)?, (force)?)+\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ta. ta \\ {t\\<^sub>0--t} \\ (exp ((ta - t\\<^sub>0) *\\<^sub>R A) * A) *\\<^sub>V s + (exp ((ta - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V (exp ((t\\<^sub>0 - ta) *\\<^sub>R A) *\\<^sub>V B) + (exp ((ta - t\\<^sub>0) *\\<^sub>R A) * A) *\\<^sub>V (\\\\<^sub>t\\<^sub>0\\<^sup>ta exp ((t\\<^sub>0 - \\) *\\<^sub>R A) *\\<^sub>V B\\\\)) = A *\\<^sub>V (exp ((ta - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s + exp ((ta - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V (\\\\<^sub>t\\<^sub>0\\<^sup>ta exp (- (\\ - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V B\\\\)) + B\n[PROOF STEP]\nby (simp add: mtx_vec_mult_add_rdistl sq_mtx_times_vec_assoc[symmetric] \n exp_minus_inverse exp_times_scaleR_commute mult_exp_exp scale_left_distrib[symmetric])", "meta": {"llama_tokens": 1351, "file": "Matrices_for_ODEs_MTX_Flows", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297941266013, "lm_q2_score": 0.7799928900257126, "lm_q1q2_score": 0.7024068366750678}} {"text": "[STATEMENT]\nlemma compress_mod: \n assumes \"x\\{\\q-(q/(2*2^d))\\..q-1}\" \n \"of_nat d < \\(log 2 q)::real\\\" \n shows \"compress d x = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. compress d x = 0\n[PROOF STEP]\nunfolding compress_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. round (real_of_int (2 ^ d * x) / real_of_int q) mod 2 ^ d = 0\n[PROOF STEP]\nusing compress_2d[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nround (real_of_int (2 ^ d * x) / real_of_int q) = 2 ^ d\n\ngoal (1 subgoal):\n 1. round (real_of_int (2 ^ d * x) / real_of_int q) mod 2 ^ d = 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 296, "file": "CRYSTALS-Kyber_Compress", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297861178929, "lm_q2_score": 0.7799928951399098, "lm_q1q2_score": 0.7024068350338191}} {"text": "[STATEMENT]\nlemma trigonometric_set_mul_absolutely_integrable:\n assumes \"f absolutely_integrable_on {-pi..pi}\"\n shows \"(\\x. trigonometric_set n x * f x) absolutely_integrable_on {-pi..pi}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. trigonometric_set n x * f x) absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nproof (rule absolutely_integrable_bounded_measurable_product_real)\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. trigonometric_set n \\ borel_measurable (lebesgue_on {- pi..pi})\n 2. {- pi..pi} \\ sets lebesgue\n 3. bounded (trigonometric_set n ` {- pi..pi})\n 4. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nshow \"trigonometric_set n \\ borel_measurable (lebesgue_on {-pi..pi})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trigonometric_set n \\ borel_measurable (lebesgue_on {- pi..pi})\n[PROOF STEP]\nusing square_integrable_def square_integrable_trigonometric_set\n[PROOF STATE]\nproof (prove)\nusing this:\n?f square_integrable ?S \\ ?S \\ sets lebesgue \\ ?f \\ borel_measurable (lebesgue_on ?S) \\ integrable (lebesgue_on ?S) (\\x. (?f x)\\<^sup>2)\ntrigonometric_set ?i square_integrable {- pi..pi}\n\ngoal (1 subgoal):\n 1. trigonometric_set n \\ borel_measurable (lebesgue_on {- pi..pi})\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ntrigonometric_set n \\ borel_measurable (lebesgue_on {- pi..pi})\n\ngoal (3 subgoals):\n 1. {- pi..pi} \\ sets lebesgue\n 2. bounded (trigonometric_set n ` {- pi..pi})\n 3. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nshow \"bounded (trigonometric_set n ` {-pi..pi})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bounded (trigonometric_set n ` {- pi..pi})\n[PROOF STEP]\nunfolding bounded_iff\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a. \\x\\trigonometric_set n ` {- pi..pi}. norm x \\ a\n[PROOF STEP]\nusing pi_gt3 sqrt_pi_ge1\n[PROOF STATE]\nproof (prove)\nusing this:\n3 < pi\n1 \\ sqrt pi\n\ngoal (1 subgoal):\n 1. \\a. \\x\\trigonometric_set n ` {- pi..pi}. norm x \\ a\n[PROOF STEP]\nby (rule_tac x=1 in exI)\n (auto simp: trigonometric_set_def dist_real_def\n intro: order_trans [OF abs_sin_le_one] order_trans [OF abs_cos_le_one])\n[PROOF STATE]\nproof (state)\nthis:\nbounded (trigonometric_set n ` {- pi..pi})\n\ngoal (2 subgoals):\n 1. {- pi..pi} \\ sets lebesgue\n 2. f absolutely_integrable_on {- pi..pi}\n[PROOF STEP]\nqed (auto simp: assms)", "meta": {"llama_tokens": 1025, "file": "Fourier_Fourier", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8519527982093666, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.7024026451936042}} {"text": "[STATEMENT]\nlemma unit_vec_of_right_length_is_state [simp]:\n assumes \"i < 2^n\"\n shows \"unit_vec (2^n) i \\ {v| n v::complex vec. dim_vec v = 2^n \\ \\v\\ = 1}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. unit_vec (2 ^ n) i \\ {uu_. \\n v. uu_ = v \\ dim_vec v = 2 ^ n \\ \\v\\ = 1}\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. unit_vec (2 ^ n) i \\ {uu_. \\n v. uu_ = v \\ dim_vec v = 2 ^ n \\ \\v\\ = 1}\n[PROOF STEP]\nhave \"dim_vec (unit_vec (2^n) i) = 2^n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_vec (unit_vec (2 ^ n) i) = 2 ^ n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec (unit_vec (2 ^ n) i) = 2 ^ n\n\ngoal (1 subgoal):\n 1. unit_vec (2 ^ n) i \\ {uu_. \\n v. uu_ = v \\ dim_vec v = 2 ^ n \\ \\v\\ = 1}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec (unit_vec (2 ^ n) i) = 2 ^ n\n\ngoal (1 subgoal):\n 1. unit_vec (2 ^ n) i \\ {uu_. \\n v. uu_ = v \\ dim_vec v = 2 ^ n \\ \\v\\ = 1}\n[PROOF STEP]\nhave \"\\unit_vec (2^n) i\\ = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\unit_vec (2 ^ n) i\\ = 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ n\n\ngoal (1 subgoal):\n 1. \\unit_vec (2 ^ n) i\\ = 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\unit_vec (2 ^ n) i\\ = 1\n\ngoal (1 subgoal):\n 1. unit_vec (2 ^ n) i \\ {uu_. \\n v. uu_ = v \\ dim_vec v = 2 ^ n \\ \\v\\ = 1}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ndim_vec (unit_vec (2 ^ n) i) = 2 ^ n\n\\unit_vec (2 ^ n) i\\ = 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec (unit_vec (2 ^ n) i) = 2 ^ n\n\\unit_vec (2 ^ n) i\\ = 1\n\ngoal (1 subgoal):\n 1. unit_vec (2 ^ n) i \\ {uu_. \\n v. uu_ = v \\ dim_vec v = 2 ^ n \\ \\v\\ = 1}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nunit_vec (2 ^ n) i \\ {uu_. \\n v. uu_ = v \\ dim_vec v = 2 ^ n \\ \\v\\ = 1}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1089, "file": "Isabelle_Marries_Dirac_Quantum", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8519527869325346, "lm_q2_score": 0.8244619177503205, "lm_q1q2_score": 0.7024026285471276}} {"text": "[STATEMENT]\nlemma column_Gram_Schmidt_column_k:\n fixes A::\"'a::{real_inner}^'n::{mod_type}^'m::{mod_type}\"\n shows \"column k (Gram_Schmidt_column_k A (to_nat k)) = \n (column k A) - (\\x\\{column i A|i. i < k}. (x \\ (column k A) / (x \\ x)) *\\<^sub>R x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. column k (Gram_Schmidt_column_k A (mod_type_class.to_nat k)) = column k A - (\\x\\{column i A |i. i < k}. (x \\ column k A / (x \\ x)) *\\<^sub>R x)\n[PROOF STEP]\nunfolding Gram_Schmidt_column_k_def column_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. (\\a b. (if b = mod_type_class.from_nat (mod_type_class.to_nat k) then (\\i. A $ i $ b) - proj_onto (\\i. A $ i $ b) {\\ia. A $ ia $ i |i. i < b} else \\i. A $ i $ b) $ a) $ i $ k) = (\\i. A $ i $ k) - (\\x\\{\\ia. A $ ia $ i |i. i < k}. (x \\ (\\i. A $ i $ k) / (x \\ x)) *\\<^sub>R x)\n[PROOF STEP]\nunfolding from_nat_to_nat_id\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. (\\a b. (if b = k then (\\i. A $ i $ b) - proj_onto (\\i. A $ i $ b) {\\ia. A $ ia $ i |i. i < b} else \\i. A $ i $ b) $ a) $ i $ k) = (\\i. A $ i $ k) - (\\x\\{\\ia. A $ ia $ i |i. i < k}. (x \\ (\\i. A $ i $ k) / (x \\ x)) *\\<^sub>R x)\n[PROOF STEP]\nunfolding proj_onto_def proj_def[abs_def]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. (\\a b. (if b = k then (\\i. A $ i $ b) - (\\u\\{\\ia. A $ ia $ i |i. i < b}. ((\\i. A $ i $ b) \\ u / (u \\ u)) *\\<^sub>R u) else \\i. A $ i $ b) $ a) $ i $ k) = (\\i. A $ i $ k) - (\\x\\{\\ia. A $ ia $ i |i. i < k}. (x \\ (\\i. A $ i $ k) / (x \\ x)) *\\<^sub>R x)\n[PROOF STEP]\nunfolding proj_onto_sum_rw\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. (\\a b. (if b = k then (\\i. A $ i $ b) - (\\u\\{\\ia. A $ ia $ i |i. i < b}. ((\\i. A $ i $ b) \\ u / (u \\ u)) *\\<^sub>R u) else \\i. A $ i $ b) $ a) $ i $ k) = (\\i. A $ i $ k) - (\\x\\{\\ia. A $ ia $ i |i. i < k}. ((\\i. A $ i $ k) \\ x / (x \\ x)) *\\<^sub>R x)\n[PROOF STEP]\nby vector", "meta": {"llama_tokens": 1098, "file": "QR_Decomposition_Gram_Schmidt", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314738181875, "lm_q2_score": 0.7931059585194573, "lm_q1q2_score": 0.7023995989375733}} {"text": "[STATEMENT]\nlemma binary_expression_gteq0:\n assumes \"\\n. a n \\ {0,1 :: nat}\"\n shows \"0 \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\n[PROOF STEP]\nhave \"(\\n. 0) \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. 0) \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\n[PROOF STEP]\nusing binary_expression_summable[of a] summable_iff_shift[of \"\\n. real (a n) * (1 / 2) ^ Suc n\" k] suminf_le[of \"\\n. 0\" \"\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k)\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. a n \\ {0, 1}) \\ summable (\\n. real (a n) * (1 / 2) ^ Suc n)\nsummable (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k)) = summable (\\n. real (a n) * (1 / 2) ^ Suc n)\n\\\\n. 0 \\ real (a (n + k)) * (1 / 2) ^ Suc (n + k); summable (\\n. 0); summable (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\\ \\ (\\n. 0) \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\na ?n \\ {0, 1}\n\ngoal (1 subgoal):\n 1. (\\n. 0) \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. 0) \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\n\ngoal (1 subgoal):\n 1. 0 \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. 0) \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\n\ngoal (1 subgoal):\n 1. 0 \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ (\\n. real (a (n + k)) * (1 / 2) ^ Suc (n + k))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1025, "file": "Quasi_Borel_Spaces_StandardBorel", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314738181875, "lm_q2_score": 0.7931059511841119, "lm_q1q2_score": 0.7023995924411605}} {"text": "[STATEMENT]\nlemma coeff_Bernstein_sum: \n fixes b::\"nat \\ real\" and p::nat and c d::real\n defines \"P \\ (\\j = 0..p. (smult (b j) (Bernstein_Poly j p c d)))\"\n assumes \"i \\ p\" and \"c \\ d\"\n shows \"coeff ((reciprocal_poly p (P \\\\<^sub>p [:c, 1:] \n \\\\<^sub>p [:0, d-c:])) \\\\<^sub>p [:1, 1:]) (p - i) = (p choose i) * (b i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coeff (reciprocal_poly p (P \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:]) \\\\<^sub>p [:1, 1:]) (p - i) = real (p choose i) * b i\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. coeff (reciprocal_poly p (P \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:]) \\\\<^sub>p [:1, 1:]) (p - i) = real (p choose i) * b i\n[PROOF STEP]\nhave h: \"P \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d-c:] \n = (\\j = 0..p. (smult (b j) (Bernstein_Poly_01 j p)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. P \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:] = (\\j = 0..p. smult (b j) (Bernstein_Poly_01 j p))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nP \\ \\j = 0..p. smult (b j) (Bernstein_Poly j p c d)\ni \\ p\nc \\ d\n\ngoal (1 subgoal):\n 1. P \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:] = (\\j = 0..p. smult (b j) (Bernstein_Poly_01 j p))\n[PROOF STEP]\nby (auto simp: P_def pcompose_sum pcompose_smult \n pcompose_add Bernstein_Poly_rescale_01)\n[PROOF STATE]\nproof (state)\nthis:\nP \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:] = (\\j = 0..p. smult (b j) (Bernstein_Poly_01 j p))\n\ngoal (1 subgoal):\n 1. coeff (reciprocal_poly p (P \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:]) \\\\<^sub>p [:1, 1:]) (p - i) = real (p choose i) * b i\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nP \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:] = (\\j = 0..p. smult (b j) (Bernstein_Poly_01 j p))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nP \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:] = (\\j = 0..p. smult (b j) (Bernstein_Poly_01 j p))\n\ngoal (1 subgoal):\n 1. coeff (reciprocal_poly p (P \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:]) \\\\<^sub>p [:1, 1:]) (p - i) = real (p choose i) * b i\n[PROOF STEP]\nusing coeff_Bernstein_sum_01 assms\n[PROOF STATE]\nproof (prove)\nusing this:\nP \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:] = (\\j = 0..p. smult (b j) (Bernstein_Poly_01 j p))\n?i \\ ?p \\ coeff (reciprocal_poly ?p (\\x = 0..?p. smult (?b x) (Bernstein_Poly_01 x ?p)) \\\\<^sub>p [:1, 1:]) (?p - ?i) = real (?p choose ?i) * ?b ?i\nP \\ \\j = 0..p. smult (b j) (Bernstein_Poly j p c d)\ni \\ p\nc \\ d\n\ngoal (1 subgoal):\n 1. coeff (reciprocal_poly p (P \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:]) \\\\<^sub>p [:1, 1:]) (p - i) = real (p choose i) * b i\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncoeff (reciprocal_poly p (P \\\\<^sub>p [:c, 1:] \\\\<^sub>p [:0, d - c:]) \\\\<^sub>p [:1, 1:]) (p - i) = real (p choose i) * b i\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1511, "file": "Three_Circles_Bernstein", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314738181875, "lm_q2_score": 0.7931059511841119, "lm_q1q2_score": 0.7023995924411605}} {"text": "[STATEMENT]\nlemma homeomorphic_spheres_gen:\n fixes a :: \"'a::euclidean_space\" and b :: \"'b::euclidean_space\"\n assumes \"0 < r\" \"0 < s\" \"DIM('a::euclidean_space) = DIM('b::euclidean_space)\"\n shows \"(sphere a r homeomorphic sphere b s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sphere a r homeomorphic sphere b s\n[PROOF STEP]\nusing assms homeomorphic_trans [OF homeomorphic_spheres homeomorphic_spheres']\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < r\n0 < s\nDIM('a) = DIM('b)\n\\0 < ?d2; 0 < ?\\1; 0 < ?\\1; DIM(?'a1) = DIM(?'b1)\\ \\ sphere ?a2 ?d2 homeomorphic sphere ?b1 ?\\1\n\ngoal (1 subgoal):\n 1. sphere a r homeomorphic sphere b s\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 290, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314677809302, "lm_q2_score": 0.7931059560743422, "lm_q1q2_score": 0.7023995919839177}} {"text": "[STATEMENT]\nlemma icard_inj_on_le: \"\\ inj_on f A; f ` A \\ B \\ \\ icard A \\ icard B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\inj_on f A; f ` A \\ B\\ \\ icard A \\ icard B\n[PROOF STEP]\napply (case_tac \"finite B\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\inj_on f A; f ` A \\ B; finite B\\ \\ icard A \\ icard B\n 2. \\inj_on f A; f ` A \\ B; infinite B\\ \\ icard A \\ icard B\n[PROOF STEP]\napply (metis icard_image icard_mono)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\inj_on f A; f ` A \\ B; infinite B\\ \\ icard A \\ icard B\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 366, "file": "List-Infinite_CommonSet_InfiniteSet2", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314647623016, "lm_q2_score": 0.793105951184112, "lm_q1q2_score": 0.7023995852588836}} {"text": "[STATEMENT]\nlemma Suc_unat_mask_div:\n \"Suc (unat (mask sz div word_size :: machine_word)) = 2 ^ (min sz word_bits - word_size_bits)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc (unat (mask sz div word_size)) = 2 ^ (min sz word_bits - word_size_bits)\n[PROOF STEP]\nby (simp add: word_size_word_size_bits unat_drop_bit_eq unat_mask_eq drop_bit_mask_eq Suc_mask_eq_exp\n flip: drop_bit_eq_div word_bits_conv)", "meta": {"llama_tokens": 177, "file": "Word_Lib_Machine_Word_32", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314647623015, "lm_q2_score": 0.793105951184112, "lm_q1q2_score": 0.7023995852588835}} {"text": "[STATEMENT]\nlemma Suc_unat_mask_div:\n \"Suc (unat (mask sz div word_size :: machine_word)) = 2 ^ (min sz word_bits - word_size_bits)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc (unat (mask sz div word_size)) = 2 ^ (min sz word_bits - word_size_bits)\n[PROOF STEP]\nby (simp add: word_size_word_size_bits unat_drop_bit_eq unat_mask_eq drop_bit_mask_eq Suc_mask_eq_exp\n flip: drop_bit_eq_div word_bits_conv)", "meta": {"llama_tokens": 177, "file": "Word_Lib_Machine_Word_64", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314647623015, "lm_q2_score": 0.7931059511841119, "lm_q1q2_score": 0.7023995852588834}} {"text": "[STATEMENT]\nlemma winding_number_homotopic_paths_null_eq:\n assumes \"path p\" and \\: \"\\ \\ path_image p\"\n shows \"winding_number p \\ = 0 \\ (\\a. homotopic_paths (-{\\}) p (\\t. a))\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (winding_number p \\ = 0) = (\\a. homotopic_paths (- {\\}) p (\\t. a))\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. winding_number p \\ = 0 \\ \\a. homotopic_paths (- {\\}) p (\\t. a)\n 2. \\a. homotopic_paths (- {\\}) p (\\t. a) \\ winding_number p \\ = 0\n[PROOF STEP]\nassume ?lhs\n[PROOF STATE]\nproof (state)\nthis:\nwinding_number p \\ = 0\n\ngoal (2 subgoals):\n 1. winding_number p \\ = 0 \\ \\a. homotopic_paths (- {\\}) p (\\t. a)\n 2. \\a. homotopic_paths (- {\\}) p (\\t. a) \\ winding_number p \\ = 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nwinding_number p \\ = 0\n[PROOF STEP]\nshow ?rhs\n[PROOF STATE]\nproof (prove)\nusing this:\nwinding_number p \\ = 0\n\ngoal (1 subgoal):\n 1. \\a. homotopic_paths (- {\\}) p (\\t. a)\n[PROOF STEP]\nby (auto simp: winding_number_homotopic_paths_null_explicit_eq [OF assms] linepath_refl)\n[PROOF STATE]\nproof (state)\nthis:\n\\a. homotopic_paths (- {\\}) p (\\t. a)\n\ngoal (1 subgoal):\n 1. \\a. homotopic_paths (- {\\}) p (\\t. a) \\ winding_number p \\ = 0\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\a. homotopic_paths (- {\\}) p (\\t. a) \\ winding_number p \\ = 0\n[PROOF STEP]\nassume ?rhs\n[PROOF STATE]\nproof (state)\nthis:\n\\a. homotopic_paths (- {\\}) p (\\t. a)\n\ngoal (1 subgoal):\n 1. \\a. homotopic_paths (- {\\}) p (\\t. a) \\ winding_number p \\ = 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\a. homotopic_paths (- {\\}) p (\\t. a)\n[PROOF STEP]\nshow ?lhs\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a. homotopic_paths (- {\\}) p (\\t. a)\n\ngoal (1 subgoal):\n 1. winding_number p \\ = 0\n[PROOF STEP]\nby (metis \\ homotopic_paths_imp_pathfinish pathfinish_def pathfinish_in_path_image winding_number_homotopic_paths winding_number_zero_const)\n[PROOF STATE]\nproof (state)\nthis:\nwinding_number p \\ = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1091, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382236515259, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7023539546068462}} {"text": "[STATEMENT]\nlemma eventually_nat_real:\n assumes \"eventually P (at_top :: real filter)\"\n shows \"eventually (\\x. P (real x)) (at_top :: nat filter)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in sequentially. P (real x)\n[PROOF STEP]\nusing assms filterlim_real_sequentially\n[PROOF STATE]\nproof (prove)\nusing this:\neventually P at_top\nfilterlim real at_top sequentially\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in sequentially. P (real x)\n[PROOF STEP]\nunfolding filterlim_def le_filter_def eventually_filtermap\n[PROOF STATE]\nproof (prove)\nusing this:\neventually P at_top\n\\P. eventually P at_top \\ (\\\\<^sub>F x in sequentially. P (real x))\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in sequentially. P (real x)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 290, "file": "Landau_Symbols_Landau_Library", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.86153820232079, "lm_q2_score": 0.8152324983301568, "lm_q1q2_score": 0.7023539410848497}} {"text": "[STATEMENT]\nlemma log_est: \"log 2 (real n + 1) \\ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. log 2 (real n + 1) \\ real n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. log 2 (real n + 1) \\ real n\n[PROOF STEP]\nhave \"1 + real n = real (n + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + real n = real (n + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 + real n = real (n + 1)\n\ngoal (1 subgoal):\n 1. log 2 (real n + 1) \\ real n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 + real n = real (n + 1)\n\ngoal (1 subgoal):\n 1. log 2 (real n + 1) \\ real n\n[PROOF STEP]\nhave \"... \\ real (2 ^ n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (n + 1) \\ real (2 ^ n)\n[PROOF STEP]\nby (intro of_nat_mono suc_n_le_2_pow_n)\n[PROOF STATE]\nproof (state)\nthis:\nreal (n + 1) \\ real (2 ^ n)\n\ngoal (1 subgoal):\n 1. log 2 (real n + 1) \\ real n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (n + 1) \\ real (2 ^ n)\n\ngoal (1 subgoal):\n 1. log 2 (real n + 1) \\ real n\n[PROOF STEP]\nhave \"... = 2 powr (real n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (2 ^ n) = 2 powr real n\n[PROOF STEP]\nby (simp add:powr_realpow)\n[PROOF STATE]\nproof (state)\nthis:\nreal (2 ^ n) = 2 powr real n\n\ngoal (1 subgoal):\n 1. log 2 (real n + 1) \\ real n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n1 + real n \\ 2 powr real n\n[PROOF STEP]\nhave \"1 + real n \\ 2 powr (real n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 + real n \\ 2 powr real n\n\ngoal (1 subgoal):\n 1. 1 + real n \\ 2 powr real n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 + real n \\ 2 powr real n\n\ngoal (1 subgoal):\n 1. log 2 (real n + 1) \\ real n\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n1 + real n \\ 2 powr real n\n\ngoal (1 subgoal):\n 1. log 2 (real n + 1) \\ real n\n[PROOF STEP]\nby (simp add: Transcendental.log_le_iff)\n[PROOF STATE]\nproof (state)\nthis:\nlog 2 (real n + 1) \\ real n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 996, "file": "Frequency_Moments_Frequency_Moments_Preliminary_Results", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.86153820232079, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7023539372173373}} {"text": "[STATEMENT]\nlemma compact_fip_Heine_Borel:\n fixes \\ :: \"'a::heine_borel set set\"\n assumes clof: \"\\T. T \\ \\ \\ compact T\"\n and none: \"\\\\'. \\finite \\'; \\' \\ \\\\ \\ \\\\' \\ {}\"\n shows \"\\\\ \\ {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ \\ \\ {}\n[PROOF STEP]\nby (metis InterI all_not_in_conv clof closed_fip_Heine_Borel compact_eq_bounded_closed none)", "meta": {"llama_tokens": 200, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.86153820232079, "lm_q2_score": 0.8152324915965392, "lm_q1q2_score": 0.7023539352835809}} {"text": "[STATEMENT]\ntheorem residue_primroot_bij_betw_primroots:\n assumes \"m > 1\" and \"residue_primroot m g\"\n shows \"bij_betw (\\i. g ^ i mod m) (totatives (totient m))\n {g\\totatives m. residue_primroot m g}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nproof (cases \"m = 2\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. m = 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n 2. m \\ 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\ncase [simp]: True\n[PROOF STATE]\nproof (state)\nthis:\nm = 2\n\ngoal (2 subgoals):\n 1. m = 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n 2. m \\ 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nhave [simp]: \"totatives 2 = {1}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. totatives 2 = {1}\n[PROOF STEP]\nby (auto simp: totatives_def elim!: oddE)\n[PROOF STATE]\nproof (state)\nthis:\ntotatives 2 = {1}\n\ngoal (2 subgoals):\n 1. m = 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n 2. m \\ 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < m\nresidue_primroot m g\n[PROOF STEP]\nhave \"odd g\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\nresidue_primroot m g\n\ngoal (1 subgoal):\n 1. odd g\n[PROOF STEP]\nby (auto simp: residue_primroot_def)\n[PROOF STATE]\nproof (state)\nthis:\nodd g\n\ngoal (2 subgoals):\n 1. m = 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n 2. m \\ 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nhence pow_eq: \"(\\i. g ^ i mod m) = (\\_. 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nodd g\n\ngoal (1 subgoal):\n 1. (\\i. g ^ i mod m) = (\\_. 1)\n[PROOF STEP]\nby (auto simp: fun_eq_iff mod_2_eq_odd)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i. g ^ i mod m) = (\\_. 1)\n\ngoal (2 subgoals):\n 1. m = 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n 2. m \\ 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nhave \"{g \\ totatives m. residue_primroot m g} = {1}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {g \\ totatives m. residue_primroot m g} = {1}\n[PROOF STEP]\nby (auto simp: residue_primroot_def)\n[PROOF STATE]\nproof (state)\nthis:\n{g \\ totatives m. residue_primroot m g} = {1}\n\ngoal (2 subgoals):\n 1. m = 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n 2. m \\ 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n{g \\ totatives m. residue_primroot m g} = {1}\n\ngoal (1 subgoal):\n 1. bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nusing pow_eq\n[PROOF STATE]\nproof (prove)\nusing this:\n{g \\ totatives m. residue_primroot m g} = {1}\n(\\i. g ^ i mod m) = (\\_. 1)\n\ngoal (1 subgoal):\n 1. bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nby (auto simp: bij_betw_def)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n\ngoal (1 subgoal):\n 1. m \\ 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. m \\ 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nm \\ 2\n\ngoal (1 subgoal):\n 1. m \\ 2 \\ bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ 2\n\ngoal (1 subgoal):\n 1. bij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nunfolding bij_betw_def\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ 2\n\ngoal (1 subgoal):\n 1. inj_on (\\i. g ^ i mod m) (totatives (totient m)) \\ (\\i. g ^ i mod m) ` totatives (totient m) = {g \\ totatives m. residue_primroot m g}\n[PROOF STEP]\nproof safe\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. m \\ 2 \\ inj_on (\\i. g ^ i mod m) (totatives (totient m))\n 2. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ g ^ i mod m \\ totatives m\n 3. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ residue_primroot m (g ^ i mod m)\n 4. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nfrom assms False\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < m\nresidue_primroot m g\nm \\ 2\n[PROOF STEP]\nhave \"m > 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\nresidue_primroot m g\nm \\ 2\n\ngoal (1 subgoal):\n 1. 2 < m\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n2 < m\n\ngoal (4 subgoals):\n 1. m \\ 2 \\ inj_on (\\i. g ^ i mod m) (totatives (totient m))\n 2. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ g ^ i mod m \\ totatives m\n 3. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ residue_primroot m (g ^ i mod m)\n 4. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nfrom assms \\m > 2\\\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < m\nresidue_primroot m g\n2 < m\n[PROOF STEP]\nhave \"totient m > 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\nresidue_primroot m g\n2 < m\n\ngoal (1 subgoal):\n 1. 1 < totient m\n[PROOF STEP]\nby (intro totient_gt_1) auto\n[PROOF STATE]\nproof (state)\nthis:\n1 < totient m\n\ngoal (4 subgoals):\n 1. m \\ 2 \\ inj_on (\\i. g ^ i mod m) (totatives (totient m))\n 2. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ g ^ i mod m \\ totatives m\n 3. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ residue_primroot m (g ^ i mod m)\n 4. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < m\nresidue_primroot m g\n[PROOF STEP]\nhave [simp]: \"ord m g = totient m\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\nresidue_primroot m g\n\ngoal (1 subgoal):\n 1. ord m g = totient m\n[PROOF STEP]\nby (simp add: residue_primroot_def)\n[PROOF STATE]\nproof (state)\nthis:\nord m g = totient m\n\ngoal (4 subgoals):\n 1. m \\ 2 \\ inj_on (\\i. g ^ i mod m) (totatives (totient m))\n 2. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ g ^ i mod m \\ totatives m\n 3. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ residue_primroot m (g ^ i mod m)\n 4. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < m\nresidue_primroot m g\n[PROOF STEP]\nhave \"coprime m g\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\nresidue_primroot m g\n\ngoal (1 subgoal):\n 1. coprime m g\n[PROOF STEP]\nby (simp add: residue_primroot_def)\n[PROOF STATE]\nproof (state)\nthis:\ncoprime m g\n\ngoal (4 subgoals):\n 1. m \\ 2 \\ inj_on (\\i. g ^ i mod m) (totatives (totient m))\n 2. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ g ^ i mod m \\ totatives m\n 3. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ residue_primroot m (g ^ i mod m)\n 4. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nhence \"inj_on (\\i. g ^ i mod m) {..i. g ^ i mod m) {..i. g ^ i mod m) {.. 2 \\ inj_on (\\i. g ^ i mod m) (totatives (totient m))\n 2. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ g ^ i mod m \\ totatives m\n 3. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ residue_primroot m (g ^ i mod m)\n 4. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nthus \"inj_on (\\i. g ^ i mod m) (totatives (totient m))\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on (\\i. g ^ i mod m) {..i. g ^ i mod m) (totatives (totient m))\n[PROOF STEP]\nby (rule inj_on_subset)\n (use assms \\totient m > 1\\ in \\auto simp: totatives_less residue_primroot_def\\)\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\i. g ^ i mod m) (totatives (totient m))\n\ngoal (3 subgoals):\n 1. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ g ^ i mod m \\ totatives m\n 2. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ residue_primroot m (g ^ i mod m)\n 3. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\i. g ^ i mod m) (totatives (totient m))\n\ngoal (3 subgoals):\n 1. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ g ^ i mod m \\ totatives m\n 2. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ residue_primroot m (g ^ i mod m)\n 3. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nfix i\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ g ^ i mod m \\ totatives m\n 2. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ residue_primroot m (g ^ i mod m)\n 3. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nassume i: \"i \\ totatives (totient m)\"\n[PROOF STATE]\nproof (state)\nthis:\ni \\ totatives (totient m)\n\ngoal (3 subgoals):\n 1. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ g ^ i mod m \\ totatives m\n 2. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ residue_primroot m (g ^ i mod m)\n 3. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nfrom \\coprime m g\\ and \\m > 2\\\n[PROOF STATE]\nproof (chain)\npicking this:\ncoprime m g\n2 < m\n[PROOF STEP]\nshow \"g ^ i mod m \\ totatives m\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime m g\n2 < m\n\ngoal (1 subgoal):\n 1. g ^ i mod m \\ totatives m\n[PROOF STEP]\nby (intro power_in_totatives) auto\n[PROOF STATE]\nproof (state)\nthis:\ng ^ i mod m \\ totatives m\n\ngoal (2 subgoals):\n 1. \\x i. \\m \\ 2; i \\ totatives (totient m)\\ \\ residue_primroot m (g ^ i mod m)\n 2. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nshow \"residue_primroot m (g ^ i mod m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. residue_primroot m (g ^ i mod m)\n[PROOF STEP]\nusing i \\m > 2\\ \\coprime m g\\\n[PROOF STATE]\nproof (prove)\nusing this:\ni \\ totatives (totient m)\n2 < m\ncoprime m g\n\ngoal (1 subgoal):\n 1. residue_primroot m (g ^ i mod m)\n[PROOF STEP]\nby (auto simp: residue_primroot_def coprime_commute ord_power totatives_def)\n[PROOF STATE]\nproof (state)\nthis:\nresidue_primroot m (g ^ i mod m)\n\ngoal (1 subgoal):\n 1. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n?i2 \\ totatives (totient m) \\ residue_primroot m (g ^ ?i2 mod m)\n\ngoal (1 subgoal):\n 1. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\n?i2 \\ totatives (totient m) \\ residue_primroot m (g ^ ?i2 mod m)\n\ngoal (1 subgoal):\n 1. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nassume x: \"x \\ totatives m\" \"residue_primroot m x\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ totatives m\nresidue_primroot m x\n\ngoal (1 subgoal):\n 1. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ totatives m\nresidue_primroot m x\n[PROOF STEP]\nobtain i where i: \"i < totient m\" \"x = (g ^ i mod m)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ totatives m\nresidue_primroot m x\n\ngoal (1 subgoal):\n 1. (\\i. \\i < totient m; x = g ^ i mod m\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing assms residue_primroot_is_generator[of m g]\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ totatives m\nresidue_primroot m x\n1 < m\nresidue_primroot m g\n\\1 < m; residue_primroot m g\\ \\ bij_betw (\\i. g ^ i mod m) {..i. \\i < totient m; x = g ^ i mod m\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (auto simp: bij_betw_def)\n[PROOF STATE]\nproof (state)\nthis:\ni < totient m\nx = g ^ i mod m\n\ngoal (1 subgoal):\n 1. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nfrom i x \\m > 2\\\n[PROOF STATE]\nproof (chain)\npicking this:\ni < totient m\nx = g ^ i mod m\nx \\ totatives m\nresidue_primroot m x\n2 < m\n[PROOF STEP]\nhave \"i > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < totient m\nx = g ^ i mod m\nx \\ totatives m\nresidue_primroot m x\n2 < m\n\ngoal (1 subgoal):\n 1. 0 < i\n[PROOF STEP]\nby (intro Nat.gr0I) (auto simp: residue_primroot_1_iff)\n[PROOF STATE]\nproof (state)\nthis:\n0 < i\n\ngoal (1 subgoal):\n 1. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nhave \"totient m div gcd i (totient m) = totient m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. totient m div gcd i (totient m) = totient m\n[PROOF STEP]\nusing x i \\coprime m g\\\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ totatives m\nresidue_primroot m x\ni < totient m\nx = g ^ i mod m\ncoprime m g\n\ngoal (1 subgoal):\n 1. totient m div gcd i (totient m) = totient m\n[PROOF STEP]\nby (auto simp add: residue_primroot_def ord_power)\n[PROOF STATE]\nproof (state)\nthis:\ntotient m div gcd i (totient m) = totient m\n\ngoal (1 subgoal):\n 1. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nhence \"coprime i (totient m)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ntotient m div gcd i (totient m) = totient m\n\ngoal (1 subgoal):\n 1. coprime i (totient m)\n[PROOF STEP]\nunfolding coprime_iff_gcd_eq_1\n[PROOF STATE]\nproof (prove)\nusing this:\ntotient m div gcd i (totient m) = totient m\n\ngoal (1 subgoal):\n 1. gcd i (totient m) = 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ntotient m div gcd i (totient m) = totient m\n1 < m\nresidue_primroot m g\n\ngoal (1 subgoal):\n 1. gcd i (totient m) = 1\n[PROOF STEP]\nby (subst (asm) dvd_div_eq_mult) auto\n[PROOF STATE]\nproof (state)\nthis:\ncoprime i (totient m)\n\ngoal (1 subgoal):\n 1. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nwith i \\i > 0\\\n[PROOF STATE]\nproof (chain)\npicking this:\ni < totient m\nx = g ^ i mod m\n0 < i\ncoprime i (totient m)\n[PROOF STEP]\nhave \"i \\ totatives (totient m)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < totient m\nx = g ^ i mod m\n0 < i\ncoprime i (totient m)\n\ngoal (1 subgoal):\n 1. i \\ totatives (totient m)\n[PROOF STEP]\nby (auto simp: totatives_def)\n[PROOF STATE]\nproof (state)\nthis:\ni \\ totatives (totient m)\n\ngoal (1 subgoal):\n 1. \\x. \\m \\ 2; x \\ totatives m; residue_primroot m x\\ \\ x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nthus \"x \\ (\\i. g ^ i mod m) ` totatives (totient m)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni \\ totatives (totient m)\n\ngoal (1 subgoal):\n 1. x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nusing i\n[PROOF STATE]\nproof (prove)\nusing this:\ni \\ totatives (totient m)\ni < totient m\nx = g ^ i mod m\n\ngoal (1 subgoal):\n 1. x \\ (\\i. g ^ i mod m) ` totatives (totient m)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx \\ (\\i. g ^ i mod m) ` totatives (totient m)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n\\?x2 \\ totatives m; residue_primroot m ?x2\\ \\ ?x2 \\ (\\i. g ^ i mod m) ` totatives (totient m)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw (\\i. g ^ i mod m) (totatives (totient m)) {g \\ totatives m. residue_primroot m g}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 8611, "file": null, "length": 71, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.86153820232079, "lm_q2_score": 0.8152324826183822, "lm_q1q2_score": 0.7023539275485557}} {"text": "[STATEMENT]\ntheorem sublist_imp_len_filter_le: \n fixes P l l'\n assumes \"subseq l' l\" \n shows \"length (filter P l') \\ length (filter P l)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (filter P l') \\ length (filter P l)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsubseq l' l\n\ngoal (1 subgoal):\n 1. length (filter P l') \\ length (filter P l)\n[PROOF STEP]\nby (simp add: sublist_length)\n\n\n\\ \\TODO showcase (non-trivial proof translation/ obscurity).\\", "meta": {"llama_tokens": 202, "file": "Factored_Transition_System_Bounding_FSSublist", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615381952105442, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.702353925619565}} {"text": "[STATEMENT]\nlemma sum_moebius_mu_divisors':\n \"(\\d | d dvd n. moebius_mu d) = (if n = 1 then 1 else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\d | d dvd n. moebius_mu d) = (if n = 1 then 1::'a else (0::'a))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\d | d dvd n. moebius_mu d) = (if n = 1 then 1::'a else (0::'a))\n[PROOF STEP]\nhave \"n = 0 \\ n = 1 \\ n > 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n = 0 \\ n = 1 \\ 1 < n\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\nn = 0 \\ n = 1 \\ 1 < n\n\ngoal (1 subgoal):\n 1. (\\d | d dvd n. moebius_mu d) = (if n = 1 then 1::'a else (0::'a))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 0 \\ n = 1 \\ 1 < n\n\ngoal (1 subgoal):\n 1. (\\d | d dvd n. moebius_mu d) = (if n = 1 then 1::'a else (0::'a))\n[PROOF STEP]\nusing sum_moebius_mu_divisors[of n]\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 0 \\ n = 1 \\ 1 < n\n1 < n \\ (\\d | d dvd n. moebius_mu d) = (0::?'a)\n\ngoal (1 subgoal):\n 1. (\\d | d dvd n. moebius_mu d) = (if n = 1 then 1::'a else (0::'a))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\d | d dvd n. moebius_mu d) = (if n = 1 then 1::'a else (0::'a))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 676, "file": "Dirichlet_Series_Moebius_Mu", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615381987656672, "lm_q2_score": 0.8152324826183822, "lm_q1q2_score": 0.7023539246503041}} {"text": "[STATEMENT]\ntheorem deleteMin_correct:\n assumes I: \"invar q\"\n and NE: \"q \\ Nil\"\n shows \"invar (deleteMin q)\"\n and \"queue_to_multiset (deleteMin q) = queue_to_multiset q - {#findMin q#}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. invar (deleteMin q) &&& queue_to_multiset (deleteMin q) = queue_to_multiset q - {#findMin q#}\n[PROOF STEP]\napply (rule deleteMin_invar[OF I NE])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. queue_to_multiset (deleteMin q) = queue_to_multiset q - {#findMin q#}\n[PROOF STEP]\nusing deleteMin_mset[of q] I NE\n[PROOF STATE]\nproof (prove)\nusing this:\n\\queue_invar q; queue_to_multiset q \\ {#}\\ \\ queue_to_multiset (deleteMin q) = queue_to_multiset q - {#findMin q#}\ninvar q\nq \\ []\n\ngoal (1 subgoal):\n 1. queue_to_multiset (deleteMin q) = queue_to_multiset q - {#findMin q#}\n[PROOF STEP]\nunfolding invar_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\queue_invar q; queue_to_multiset q \\ {#}\\ \\ queue_to_multiset (deleteMin q) = queue_to_multiset q - {#findMin q#}\nqueue_invar q \\ rank_skew_invar q\nq \\ []\n\ngoal (1 subgoal):\n 1. queue_to_multiset (deleteMin q) = queue_to_multiset q - {#findMin q#}\n[PROOF STEP]\napply (auto simp add: empty_correct)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 582, "file": "Binomial-Heaps_SkewBinomialHeap", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.903294209307224, "lm_q2_score": 0.7772998611746912, "lm_q1q2_score": 0.7021304634944077}} {"text": "[STATEMENT]\nlemma Proj_sup: \\orthogonal_spaces S T \\ Proj (sup S T) = Proj S + Proj T\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal_spaces S T \\ Proj (S \\ T) = Proj S + Proj T\n[PROOF STEP]\nunfolding orthogonal_spaces_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\space_as_set S. \\y\\space_as_set T. is_orthogonal x y \\ Proj (S \\ T) = Proj S + Proj T\n[PROOF STEP]\napply transfer\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\S T. \\closed_csubspace S; closed_csubspace T; \\x\\S. \\y\\T. is_orthogonal x y\\ \\ projection (S +\\<^sub>M T) = (\\x. projection S x + projection T x)\n[PROOF STEP]\nby (simp add: projection_plus)", "meta": {"llama_tokens": 323, "file": "Complex_Bounded_Operators_Complex_Bounded_Linear_Function", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942145139149, "lm_q2_score": 0.7772998560157665, "lm_q1q2_score": 0.7021304628815409}} {"text": "[STATEMENT]\nlemma Suc_Suc_div: \"\\0 < s; s mod 2 = Suc 0; Suc (Suc s) \\ 2 * n\\\n \\ (Suc (Suc (s div 2))) \\ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < s; s mod 2 = Suc 0; Suc (Suc s) \\ 2 * n\\ \\ Suc (Suc (s div 2)) \\ n\n[PROOF STEP]\nby(arith)", "meta": {"llama_tokens": 168, "file": "Universal_Turing_Machine_Abacus_Mopup", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278788223265, "lm_q2_score": 0.7956581097540519, "lm_q1q2_score": 0.7021108980580499}} {"text": "[STATEMENT]\nlemma ln_fact_conv_sum_upto: \"ln (fact n) = sum_upto ln n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ln (fact n) = sum_upto (\\x. ln (real x)) (real n)\n[PROOF STEP]\nby (induction n) (auto simp: sum_upto_plus1 add.commute[of 1] ln_mult)", "meta": {"llama_tokens": 123, "file": "Prime_Number_Theorem_Prime_Number_Theorem_Library", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278602705731, "lm_q2_score": 0.7956581097540519, "lm_q1q2_score": 0.7021108832971968}} {"text": "[STATEMENT]\nlemma even_of_intvl_intvl:\n fixes S :: \"nat set\"\n assumes \"S = {m..n} \\ {i. even i}\"\n shows \"\\m' n'. S = (\\i. i * 2) ` {m'..n'}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m' n'. S = (\\i. i * 2) ` {m'..n'}\n[PROOF STEP]\napply (rule exI[where x=\"Suc m div 2\"], rule exI[where x=\"n div 2\"])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. S = (\\i. i * 2) ` {Suc m div 2..n div 2}\n[PROOF STEP]\napply (fastforce simp add: assms mult.commute)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 267, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127603871312, "lm_q2_score": 0.8221891239865619, "lm_q1q2_score": 0.7020777844236422}} {"text": "[STATEMENT]\nlemma square_powr_half [simp]:\n fixes x::real shows \"x\\<^sup>2 powr (1/2) = \\x\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x\\<^sup>2 powr (1 / 2) = \\x\\\n[PROOF STEP]\nby (simp add: powr_half_sqrt)", "meta": {"llama_tokens": 109, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267830311354, "lm_q2_score": 0.8080672227971211, "lm_q1q2_score": 0.7020704456557264}} {"text": "[STATEMENT]\nlemma norm2_le_norm1_int: \"\\f :: int poly\\\\<^sup>2 \\ (norm1 f)^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f\\\\<^sup>2 \\ (norm1 f)\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\f\\\\<^sup>2 \\ (norm1 f)\\<^sup>2\n[PROOF STEP]\ndefine F where \"F = (!) (coeffs f)\"\n[PROOF STATE]\nproof (state)\nthis:\nF = (!) (coeffs f)\n\ngoal (1 subgoal):\n 1. \\f\\\\<^sup>2 \\ (norm1 f)\\<^sup>2\n[PROOF STEP]\ndefine n where \"n = length (coeffs f)\"\n[PROOF STATE]\nproof (state)\nthis:\nn = length (coeffs f)\n\ngoal (1 subgoal):\n 1. \\f\\\\<^sup>2 \\ (norm1 f)\\<^sup>2\n[PROOF STEP]\nhave 1: \"\\f\\\\<^sup>2 = (\\i = 0..f\\\\<^sup>2 = (\\i = 0..i = 0..f\\\\<^sup>2 = (\\i = 0..f\\\\<^sup>2 \\ (norm1 f)\\<^sup>2\n[PROOF STEP]\nhave 2: \"norm1 f = (\\i = 0..F i\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm1 f = (\\i = 0..F i\\)\n[PROOF STEP]\nunfolding norm1_def sq_norm_poly_def sum_list_sum_nth F_def n_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((!) (map abs (coeffs f))) {0..i = 0..coeffs f ! i\\)\n[PROOF STEP]\nby (subst sum.cong, auto)\n[PROOF STATE]\nproof (state)\nthis:\nnorm1 f = (\\i = 0..F i\\)\n\ngoal (1 subgoal):\n 1. \\f\\\\<^sup>2 \\ (norm1 f)\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f\\\\<^sup>2 \\ (norm1 f)\\<^sup>2\n[PROOF STEP]\nunfolding 1 2\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0.. (\\i = 0..F i\\)\\<^sup>2\n[PROOF STEP]\nby (rule norm2_norm1_main_inequality)\n[PROOF STATE]\nproof (state)\nthis:\n\\f\\\\<^sup>2 \\ (norm1 f)\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1149, "file": "LLL_Basis_Reduction_Norms", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267830311355, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.7020704436477901}} {"text": "[STATEMENT]\nlemma replicate_spec: \"\\\\i < n. xs ! i = x; n=length xs\\\n \\ replicate (length xs) x = xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\i \\ replicate (length xs) x = xs\n[PROOF STEP]\napply hypsubst_thin\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i replicate (length xs) x = xs\n[PROOF STEP]\napply (induct xs)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\i replicate (length []) x = []\n 2. \\a xs. \\\\i replicate (length xs) x = xs; \\i \\ replicate (length (a # xs)) x = a # xs\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a xs. \\\\i replicate (length xs) x = xs; \\i \\ replicate (length (a # xs)) x = a # xs\n[PROOF STEP]\napply force\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 484, "file": "BDD_LevellistProof", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267830311354, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.70207044364779}} {"text": "[STATEMENT]\nlemma assoc_as_fmset_eq:\n assumes \"wfactors G as a\"\n and \"wfactors G bs b\"\n and \"a \\ carrier G\"\n and \"b \\ carrier G\"\n and \"set as \\ carrier G\"\n and \"set bs \\ carrier G\"\n shows \"a \\ b \\ (fmset G as = fmset G bs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ b = (fmset G as = fmset G bs)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. a \\ b = (fmset G as = fmset G bs)\n[PROOF STEP]\nhave \"a \\ b \\ (a divides b \\ b divides a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ b = (a divides b \\ b divides a)\n[PROOF STEP]\nby (simp add:associated_def)\n[PROOF STATE]\nproof (state)\nthis:\na \\ b = (a divides b \\ b divides a)\n\ngoal (1 subgoal):\n 1. a \\ b = (fmset G as = fmset G bs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\na \\ b = (a divides b \\ b divides a)\n\ngoal (1 subgoal):\n 1. a \\ b = (fmset G as = fmset G bs)\n[PROOF STEP]\nhave \"... \\ \n (fmset G as \\# fmset G bs \\ fmset G bs \\# fmset G as)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a divides b \\ b divides a) = (fmset G as \\# fmset G bs \\ fmset G bs \\# fmset G as)\n[PROOF STEP]\nusing divides_as_fmsubset assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\wfactors G ?as ?a; wfactors G ?bs ?b; ?a \\ carrier G; ?b \\ carrier G; set ?as \\ carrier G; set ?bs \\ carrier G\\ \\ ?a divides ?b = (fmset G ?as \\# fmset G ?bs)\nwfactors G as a\nwfactors G bs b\na \\ carrier G\nb \\ carrier G\nset as \\ carrier G\nset bs \\ carrier G\n\ngoal (1 subgoal):\n 1. (a divides b \\ b divides a) = (fmset G as \\# fmset G bs \\ fmset G bs \\# fmset G as)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(a divides b \\ b divides a) = (fmset G as \\# fmset G bs \\ fmset G bs \\# fmset G as)\n\ngoal (1 subgoal):\n 1. a \\ b = (fmset G as = fmset G bs)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(a divides b \\ b divides a) = (fmset G as \\# fmset G bs \\ fmset G bs \\# fmset G as)\n\ngoal (1 subgoal):\n 1. a \\ b = (fmset G as = fmset G bs)\n[PROOF STEP]\nhave \"... \\ (fmset G as = fmset G bs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (fmset G as \\# fmset G bs \\ fmset G bs \\# fmset G as) = (fmset G as = fmset G bs)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(fmset G as \\# fmset G bs \\ fmset G bs \\# fmset G as) = (fmset G as = fmset G bs)\n\ngoal (1 subgoal):\n 1. a \\ b = (fmset G as = fmset G bs)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ b = (fmset G as = fmset G bs)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ b = (fmset G as = fmset G bs)\n\ngoal (1 subgoal):\n 1. a \\ b = (fmset G as = fmset G bs)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na \\ b = (fmset G as = fmset G bs)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1396, "file": "Finite_Fields_Finite_Fields_Factorization_Ext", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8688267694452331, "lm_q2_score": 0.8080672158638528, "lm_q1q2_score": 0.702070428653595}} {"text": "[STATEMENT]\nlemma Chebyshev_sum_upper_nat:\n fixes a b :: \"nat \\ nat\"\n shows \"(\\i j. \\ i\\j; j \\ a i \\ a j) \\\n (\\i j. \\ i\\j; j \\ b i \\ b j) \\\n n * (\\i=0.. (\\i=0..i=0..\\i j. \\i \\ j; j < n\\ \\ a i \\ a j; \\i j. \\i \\ j; j < n\\ \\ b j \\ b i\\ \\ n * (\\i = 0.. sum a {0..\\i j. \\i \\ j; j < n\\ \\ real (a i) \\ real (a j); \\i j. \\i \\ j; j < n\\ \\ real (b j) \\ real (b i)\\ \\ real n * (\\k = 0.. (\\k = 0..k = 0..\\i j. \\i \\ j; j < n\\ \\ a i \\ a j; \\i j. \\i \\ j; j < n\\ \\ b j \\ b i\\ \\ n * (\\i = 0.. sum a {0.. 0) = (x \\ (0::'a))\n[PROOF STEP]\napply (auto)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. singleton_matrix j i x \\ 0 \\ x \\ (0::'a)\n 2. x \\ (0::'a) \\ singleton_matrix j i x \\ 0\n[PROOF STEP]\napply (simp add: le_matrix_def)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\ja ia. (if j = ja \\ i = ia then x else (0::'a)) \\ (0::'a) \\ x \\ (0::'a)\n 2. x \\ (0::'a) \\ singleton_matrix j i x \\ 0\n[PROOF STEP]\napply (drule_tac j=j and i=i in spec2)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (if j = j \\ i = i then x else (0::'a)) \\ (0::'a) \\ x \\ (0::'a)\n 2. x \\ (0::'a) \\ singleton_matrix j i x \\ 0\n[PROOF STEP]\napply (simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ (0::'a) \\ singleton_matrix j i x \\ 0\n[PROOF STEP]\napply (simp add: le_matrix_def)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 539, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267694452331, "lm_q2_score": 0.8080672066194945, "lm_q1q2_score": 0.7020704206218491}} {"text": "[STATEMENT]\nlemma append_rows_mult_right:\n assumes A: \"(A::'a::comm_semiring_1 mat) \\ carrier_mat a n\" and B: \"B \\ carrier_mat b n\"\n and Q: \"Q\\ carrier_mat n n\"\n shows \"(A @\\<^sub>r B) * Q = (A * Q) @\\<^sub>r (B*Q)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (A @\\<^sub>r B) * Q = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (A @\\<^sub>r B) * Q = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nhave \"transpose_mat ((A @\\<^sub>r B) * Q) = Q\\<^sup>T * (A @\\<^sub>r B)\\<^sup>T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((A @\\<^sub>r B) * Q)\\<^sup>T = Q\\<^sup>T * (A @\\<^sub>r B)\\<^sup>T\n[PROOF STEP]\nby (rule transpose_mult, insert A B Q, auto)\n[PROOF STATE]\nproof (state)\nthis:\n((A @\\<^sub>r B) * Q)\\<^sup>T = Q\\<^sup>T * (A @\\<^sub>r B)\\<^sup>T\n\ngoal (1 subgoal):\n 1. (A @\\<^sub>r B) * Q = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n((A @\\<^sub>r B) * Q)\\<^sup>T = Q\\<^sup>T * (A @\\<^sub>r B)\\<^sup>T\n\ngoal (1 subgoal):\n 1. (A @\\<^sub>r B) * Q = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nhave \"... = Q\\<^sup>T * (A\\<^sup>T @\\<^sub>c B\\<^sup>T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Q\\<^sup>T * (A @\\<^sub>r B)\\<^sup>T = Q\\<^sup>T * (A\\<^sup>T @\\<^sub>c B\\<^sup>T)\n[PROOF STEP]\nusing transpose_mat_append_rows assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?A \\ carrier_mat ?a ?n; ?B \\ carrier_mat ?b ?n\\ \\ (?A @\\<^sub>r ?B)\\<^sup>T = ?A\\<^sup>T @\\<^sub>c ?B\\<^sup>T\nA \\ carrier_mat a n\nB \\ carrier_mat b n\nQ \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. Q\\<^sup>T * (A @\\<^sub>r B)\\<^sup>T = Q\\<^sup>T * (A\\<^sup>T @\\<^sub>c B\\<^sup>T)\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\nQ\\<^sup>T * (A @\\<^sub>r B)\\<^sup>T = Q\\<^sup>T * (A\\<^sup>T @\\<^sub>c B\\<^sup>T)\n\ngoal (1 subgoal):\n 1. (A @\\<^sub>r B) * Q = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nQ\\<^sup>T * (A @\\<^sub>r B)\\<^sup>T = Q\\<^sup>T * (A\\<^sup>T @\\<^sub>c B\\<^sup>T)\n\ngoal (1 subgoal):\n 1. (A @\\<^sub>r B) * Q = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nhave \"... = Q\\<^sup>T * A\\<^sup>T @\\<^sub>c Q\\<^sup>T * B\\<^sup>T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Q\\<^sup>T * (A\\<^sup>T @\\<^sub>c B\\<^sup>T) = Q\\<^sup>T * A\\<^sup>T @\\<^sub>c Q\\<^sup>T * B\\<^sup>T\n[PROOF STEP]\nusing append_cols_mult_left assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?A \\ carrier_mat ?n ?a; ?B \\ carrier_mat ?n ?b; ?P \\ carrier_mat ?n ?n\\ \\ ?P * (?A @\\<^sub>c ?B) = ?P * ?A @\\<^sub>c ?P * ?B\nA \\ carrier_mat a n\nB \\ carrier_mat b n\nQ \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. Q\\<^sup>T * (A\\<^sup>T @\\<^sub>c B\\<^sup>T) = Q\\<^sup>T * A\\<^sup>T @\\<^sub>c Q\\<^sup>T * B\\<^sup>T\n[PROOF STEP]\nby (metis transpose_carrier_mat)\n[PROOF STATE]\nproof (state)\nthis:\nQ\\<^sup>T * (A\\<^sup>T @\\<^sub>c B\\<^sup>T) = Q\\<^sup>T * A\\<^sup>T @\\<^sub>c Q\\<^sup>T * B\\<^sup>T\n\ngoal (1 subgoal):\n 1. (A @\\<^sub>r B) * Q = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nQ\\<^sup>T * (A\\<^sup>T @\\<^sub>c B\\<^sup>T) = Q\\<^sup>T * A\\<^sup>T @\\<^sub>c Q\\<^sup>T * B\\<^sup>T\n\ngoal (1 subgoal):\n 1. (A @\\<^sub>r B) * Q = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nhave \"transpose_mat ... = (A * Q) @\\<^sub>r (B*Q)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Q\\<^sup>T * A\\<^sup>T @\\<^sub>c Q\\<^sup>T * B\\<^sup>T)\\<^sup>T = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nby (smt A B Matrix.transpose_mult Matrix.transpose_transpose append_cols_def append_rows_def Q\n carrier_mat_triv index_mult_mat(2) index_transpose_mat(2) transpose_four_block_mat\n zero_carrier_mat zero_transpose_mat)\n[PROOF STATE]\nproof (state)\nthis:\n(Q\\<^sup>T * A\\<^sup>T @\\<^sub>c Q\\<^sup>T * B\\<^sup>T)\\<^sup>T = A * Q @\\<^sub>r B * Q\n\ngoal (1 subgoal):\n 1. (A @\\<^sub>r B) * Q = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n((A @\\<^sub>r B) * Q)\\<^sup>T\\<^sup>T = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n((A @\\<^sub>r B) * Q)\\<^sup>T\\<^sup>T = A * Q @\\<^sub>r B * Q\n\ngoal (1 subgoal):\n 1. (A @\\<^sub>r B) * Q = A * Q @\\<^sub>r B * Q\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(A @\\<^sub>r B) * Q = A * Q @\\<^sub>r B * Q\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2131, "file": "Smith_Normal_Form_SNF_Missing_Lemmas", "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267626522813, "lm_q2_score": 0.8080672066194946, "lm_q1q2_score": 0.7020704151326876}} {"text": "[STATEMENT]\nlemma poly_monom: \"poly (monom a n) x = a * x ^ n\"\n for a x :: \"'a::comm_semiring_1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly (monom a n) x = a * x ^ n\n[PROOF STEP]\nby (cases \"a = 0\", simp_all) (induct n, simp_all add: mult.left_commute poly_eq_fold_coeffs)", "meta": {"llama_tokens": 129, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772482857833, "lm_q2_score": 0.8031737940012418, "lm_q1q2_score": 0.702035939755858}} {"text": "[STATEMENT]\nlemma orthogonal_commute: \"orthogonal x y \\ orthogonal y x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal x y = orthogonal y x\n[PROOF STEP]\nby (simp add: orthogonal_def inner_commute)", "meta": {"llama_tokens": 78, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772450055545, "lm_q2_score": 0.8031737940012418, "lm_q1q2_score": 0.7020359371212642}} {"text": "[STATEMENT]\nlemma row_rank_eq_col_rank_rref:\n fixes A::\"'a::{field}^'m::{mod_type}^'n::{mod_type}\"\nassumes r: \"reduced_row_echelon_form A\"\nshows \"row_rank A = col_rank A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row_rank A = col_rank A\n[PROOF STEP]\nunfolding rref_row_rank[OF r] rref_col_rank[OF r]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {column (LEAST n. A $ i $ n \\ (0::'a)) A |i. row i A \\ 0} = card {column (LEAST n. A $ i $ n \\ (0::'a)) A |i. row i A \\ 0}\n[PROOF STEP]\n..", "meta": {"llama_tokens": 253, "file": "Gauss_Jordan_Gauss_Jordan", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.874077222043951, "lm_q2_score": 0.8031738057795403, "lm_q1q2_score": 0.7020359289742485}} {"text": "[STATEMENT]\nlemma echelon_form_upt_row_condition2_explicit:\n assumes \"echelon_form_upt_row A k\"\n and \"ia < j\" and \"to_nat j < k\" and \"\\ is_zero_row ia A\" and \"\\ is_zero_row j A\"\n shows \"(LEAST n. A $ ia $ n \\ 0) < (LEAST n. A $ j $ n \\ 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (LEAST n. A $ ia $ n \\ (0::'a)) < (LEAST n. A $ j $ n \\ (0::'a))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nechelon_form_upt_row A k\nia < j\nmod_type_class.to_nat j < k\n\\ is_zero_row ia A\n\\ is_zero_row j A\n\ngoal (1 subgoal):\n 1. (LEAST n. A $ ia $ n \\ (0::'a)) < (LEAST n. A $ j $ n \\ (0::'a))\n[PROOF STEP]\nunfolding echelon_form_upt_row_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i. mod_type_class.to_nat i < k \\ is_zero_row i A \\ \\ (\\j>i. mod_type_class.to_nat j < k \\ \\ is_zero_row j A)) \\ (\\i j. i < j \\ mod_type_class.to_nat j < k \\ \\ is_zero_row i A \\ \\ is_zero_row j A \\ (LEAST n. A $ i $ n \\ (0::'a)) < (LEAST n. A $ j $ n \\ (0::'a)))\nia < j\nmod_type_class.to_nat j < k\n\\ is_zero_row ia A\n\\ is_zero_row j A\n\ngoal (1 subgoal):\n 1. (LEAST n. A $ ia $ n \\ (0::'a)) < (LEAST n. A $ j $ n \\ (0::'a))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 632, "file": "Hermite_Hermite", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772220439509, "lm_q2_score": 0.8031737892899221, "lm_q1q2_score": 0.7020359145610486}} {"text": "[STATEMENT]\nlemma \"((x::int) - y) ^ 2 = x ^ 2 + y ^ 2 - 2 * x * y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x - y)\\<^sup>2 = x\\<^sup>2 + y\\<^sup>2 - 2 * x * y\n[PROOF STEP]\nby ring", "meta": {"llama_tokens": 100, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9263037363973295, "lm_q2_score": 0.7577943767446202, "lm_q1q2_score": 0.7019477625994274}} {"text": "[STATEMENT]\nlemma Icc_plus_Icc:\n \"{a .. b} + {c .. d} = {a + c .. b + d}\"\n if \"a \\ b\" \"c \\ d\"\n for a b c d::\"'a::linordered_ab_group_add\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {a..b} + {c..d} = {a + c..b + d}\n[PROOF STEP]\nusing ex_sum_in_interval[OF that]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a + c \\ ?x; ?x \\ b + d\\ \\ \\xa\\a. xa \\ b \\ (\\xb\\c. xb \\ d \\ ?x = xa + xb)\n\ngoal (1 subgoal):\n 1. {a..b} + {c..d} = {a + c..b + d}\n[PROOF STEP]\nby (auto intro: add_mono simp: atLeastAtMost_iff Bex_def set_plus_def)", "meta": {"llama_tokens": 306, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391595913457, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7018228147277316}} {"text": "[STATEMENT]\ntheorem meet_commute: \"x \\ y = y \\ x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y = y \\ x\n[PROOF STEP]\nproof (rule meetI)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. y \\ x \\ x\n 2. y \\ x \\ y\n 3. \\z. \\z \\ x; z \\ y\\ \\ z \\ y \\ x\n[PROOF STEP]\nshow \"y \\ x \\ x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. y \\ x \\ x\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\ny \\ x \\ x\n\ngoal (2 subgoals):\n 1. y \\ x \\ y\n 2. \\z. \\z \\ x; z \\ y\\ \\ z \\ y \\ x\n[PROOF STEP]\nshow \"y \\ x \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. y \\ x \\ y\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\ny \\ x \\ y\n\ngoal (1 subgoal):\n 1. \\z. \\z \\ x; z \\ y\\ \\ z \\ y \\ x\n[PROOF STEP]\nfix z\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\z. \\z \\ x; z \\ y\\ \\ z \\ y \\ x\n[PROOF STEP]\nassume \"z \\ y\" and \"z \\ x\"\n[PROOF STATE]\nproof (state)\nthis:\nz \\ y\nz \\ x\n\ngoal (1 subgoal):\n 1. \\z. \\z \\ x; z \\ y\\ \\ z \\ y \\ x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nz \\ y\nz \\ x\n[PROOF STEP]\nshow \"z \\ y \\ x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ y\nz \\ x\n\ngoal (1 subgoal):\n 1. z \\ y \\ x\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nz \\ y \\ x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 851, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673087708698, "lm_q2_score": 0.8633916064587, "lm_q1q2_score": 0.7018228115574413}} {"text": "[STATEMENT]\nlemma (in poly_mod) inverse_mod_coprime_exp:\n assumes m: \"m = p^n\" and p: \"prime p\" \n and n: \"n \\ 0\" and cop: \"coprime x p\"\n shows \"M (inverse_mod x m * x) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M (inverse_mod x m * x) = 1\n[PROOF STEP]\nunfolding M_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse_mod x m * x mod m = 1\n[PROOF STEP]\nunfolding m\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse_mod x (p ^ n) * x mod p ^ n = 1\n[PROOF STEP]\nusing inverse_mod_pow[OF cop _ n] p\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < p \\ inverse_mod x (p ^ n) * x mod p ^ n = 1\nprime p\n\ngoal (1 subgoal):\n 1. inverse_mod x (p ^ n) * x mod p ^ n = 1\n[PROOF STEP]\nby (auto simp: prime_int_iff)", "meta": {"llama_tokens": 346, "file": "Berlekamp_Zassenhaus_Poly_Mod", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.905989810230102, "lm_q2_score": 0.774583389368527, "lm_q1q2_score": 0.701764657941381}} {"text": "[STATEMENT]\nlemma sin_kpi [simp]:\n fixes k::int\n shows \"sin (k * pi) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin (real_of_int k * pi) = 0\n[PROOF STEP]\nby (simp add: sin_zero_iff_int2)", "meta": {"llama_tokens": 94, "file": "Complex_Geometry_More_Transcendental", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9124361652391386, "lm_q2_score": 0.7690802423634961, "lm_q1q2_score": 0.7017366271033357}} {"text": "[STATEMENT]\nlemma homeomorphic_path_connectedness:\n \"S homeomorphic T \\ path_connected S \\ path_connected T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. S homeomorphic T \\ path_connected S = path_connected T\n[PROOF STEP]\nunfolding homeomorphic_def homeomorphism_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f g. (\\x\\S. g (f x) = x) \\ f ` S = T \\ continuous_on S f \\ (\\y\\T. f (g y) = y) \\ g ` T = S \\ continuous_on T g \\ path_connected S = path_connected T\n[PROOF STEP]\nby (metis path_connected_continuous_image)", "meta": {"llama_tokens": 231, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467738423874, "lm_q2_score": 0.798186775339273, "lm_q1q2_score": 0.7017233284631803}} {"text": "[STATEMENT]\nlemma ex_kleene_qfp:\n assumes comp: \"omega_complete A (\\)\"\n shows \"\\p. extreme_bound A (\\) Fn p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Ex (extreme_bound A (\\) {f^n \\ |. n})\n[PROOF STEP]\nusing fn_monotone\n[PROOF STATE]\nproof (prove)\nusing this:\nmonotone (\\) (\\) (\\n. f^n \\)\n\ngoal (1 subgoal):\n 1. Ex (extreme_bound A (\\) {f^n \\ |. n})\n[PROOF STEP]\napply (intro comp[unfolded omega_complete_def, THEN completeD, OF FnA])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. monotone (\\) (\\) (\\n. f^n \\) \\ {f^n \\ |. n} \\ {range f |f. monotone (\\) (\\) f}\n[PROOF STEP]\nby fast", "meta": {"llama_tokens": 306, "file": "Complete_Non_Orders_Kleene_Fixed_Point", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467738423873, "lm_q2_score": 0.798186775339273, "lm_q1q2_score": 0.7017233284631802}} {"text": "[STATEMENT]\nlemma locally_path_connected_space_im_kleinen:\n \"locally_path_connected_space X \\\n (\\V x. openin X V \\ x \\ V\n \\ (\\U. openin X U \\\n x \\ U \\ U \\ V \\\n (\\y \\ U. \\c. path_connectedin X c \\\n c \\ V \\ x \\ c \\ y \\ c)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. locally_path_connected_space X = (\\V x. openin X V \\ x \\ V \\ (\\U. openin X U \\ x \\ U \\ U \\ V \\ (\\y\\U. \\c. path_connectedin X c \\ c \\ V \\ x \\ c \\ y \\ c)))\n[PROOF STEP]\napply (simp add: locally_path_connected_space_def neighbourhood_base_of_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\topspace X. neighbourhood_base_at x (path_connectedin X) X) = (\\V x. openin X V \\ x \\ V \\ (\\U. openin X U \\ x \\ U \\ U \\ V \\ (\\y\\U. \\c. path_connectedin X c \\ c \\ V \\ x \\ c \\ y \\ c)))\n[PROOF STEP]\napply (simp add: weakly_locally_path_connected_at flip: weakly_locally_path_connected_at_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\topspace X. \\V. openin X V \\ x \\ V \\ (\\U. openin X U \\ x \\ U \\ U \\ V \\ (\\y\\U. \\C. path_connectedin X C \\ C \\ V \\ x \\ C \\ y \\ C))) = (\\V x. openin X V \\ x \\ V \\ (\\U. openin X U \\ x \\ U \\ U \\ V \\ (\\y\\U. \\c. path_connectedin X c \\ c \\ V \\ x \\ c \\ y \\ c)))\n[PROOF STEP]\nusing openin_subset\n[PROOF STATE]\nproof (prove)\nusing this:\nopenin ?U ?S \\ ?S \\ topspace ?U\n\ngoal (1 subgoal):\n 1. (\\x\\topspace X. \\V. openin X V \\ x \\ V \\ (\\U. openin X U \\ x \\ U \\ U \\ V \\ (\\y\\U. \\C. path_connectedin X C \\ C \\ V \\ x \\ C \\ y \\ C))) = (\\V x. openin X V \\ x \\ V \\ (\\U. openin X U \\ x \\ U \\ U \\ V \\ (\\y\\U. \\c. path_connectedin X c \\ c \\ V \\ x \\ c \\ y \\ c)))\n[PROOF STEP]\napply force\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1018, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467611766711, "lm_q2_score": 0.7981867825403176, "lm_q1q2_score": 0.7017233246843482}} {"text": "[STATEMENT]\nlemma max_Sup_absorb_right:\n fixes f g::\\'a \\ real\\\n assumes \\X \\ {}\\ and \\bdd_above (f ` X)\\ and \\bdd_above (g ` X)\\ and \\Sup (f ` X) \\ Sup (g ` X)\\\n shows \\Sup ((pointwise_max f g) ` X) = Sup (g ` X)\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sup (pointwise_max f g ` X) = Sup (g ` X)\n[PROOF STEP]\ntext \\\n Explanation: For real-valued functions \\<^term>\\f\\ and \\<^term>\\g\\ and a nonempty set \\<^term>\\X\\, such that \n the \\<^term>\\f\\ and \\<^term>\\g\\ are bounded above on \\<^term>\\X\\, if the supremum of \\<^term>\\f\\ on \\<^term>\\X\\ is \n lower-equal the supremum of \\<^term>\\g\\ on \\<^term>\\X\\, then the supremum of \\<^term>\\pointwise_max f g\\ on \\<^term>\\X\\\n equals the supremum of \\<^term>\\g\\. This is the right analog of @{text max_Sup_absorb_left}.\n\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sup (pointwise_max f g ` X) = Sup (g ` X)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Sup (pointwise_max f g ` X) = Sup (g ` X)\n[PROOF STEP]\nhave \\Sup ((pointwise_max g f) ` X) = Sup (g ` X)\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sup (pointwise_max g f ` X) = Sup (g ` X)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nX \\ {}\nbdd_above (f ` X)\nbdd_above (g ` X)\nSup (f ` X) \\ Sup (g ` X)\n\ngoal (1 subgoal):\n 1. Sup (pointwise_max g f ` X) = Sup (g ` X)\n[PROOF STEP]\nby (simp add: max_Sup_absorb_left)\n[PROOF STATE]\nproof (state)\nthis:\nSup (pointwise_max g f ` X) = Sup (g ` X)\n\ngoal (1 subgoal):\n 1. Sup (pointwise_max f g ` X) = Sup (g ` X)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nSup (pointwise_max g f ` X) = Sup (g ` X)\n\ngoal (1 subgoal):\n 1. Sup (pointwise_max f g ` X) = Sup (g ` X)\n[PROOF STEP]\nhave \\pointwise_max g f = pointwise_max f g\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pointwise_max g f = pointwise_max f g\n[PROOF STEP]\nunfolding pointwise_max_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. max (g x) (f x)) = (\\x. max (f x) (g x))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\npointwise_max g f = pointwise_max f g\n\ngoal (1 subgoal):\n 1. Sup (pointwise_max f g ` X) = Sup (g ` X)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nSup (pointwise_max g f ` X) = Sup (g ` X)\npointwise_max g f = pointwise_max f g\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nSup (pointwise_max g f ` X) = Sup (g ` X)\npointwise_max g f = pointwise_max f g\n\ngoal (1 subgoal):\n 1. Sup (pointwise_max f g ` X) = Sup (g ` X)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nSup (pointwise_max f g ` X) = Sup (g ` X)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1280, "file": "Banach_Steinhaus_Banach_Steinhaus_Missing", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8376199754937772, "lm_q2_score": 0.8376199552262967, "lm_q1q2_score": 0.7016072063697494}} {"text": "[STATEMENT]\nlemma affine_diff_divide:\n assumes \"affine S\" \"k \\ 0\" \"k \\ 1\" and xy: \"x \\ S\" \"y /\\<^sub>R (1 - k) \\ S\"\n shows \"(x - y) /\\<^sub>R k \\ S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x - y) /\\<^sub>R k \\ S\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (x - y) /\\<^sub>R k \\ S\n[PROOF STEP]\nhave \"inverse(k) *\\<^sub>R (x - y) = (1 - inverse k) *\\<^sub>R inverse(1 - k) *\\<^sub>R y + inverse(k) *\\<^sub>R x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x - y) /\\<^sub>R k = (1 - inverse k) *\\<^sub>R (y /\\<^sub>R (1 - k)) + x /\\<^sub>R k\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\naffine S\nk \\ 0\nk \\ 1\nx \\ S\ny /\\<^sub>R (1 - k) \\ S\n\ngoal (1 subgoal):\n 1. (x - y) /\\<^sub>R k = (1 - inverse k) *\\<^sub>R (y /\\<^sub>R (1 - k)) + x /\\<^sub>R k\n[PROOF STEP]\nby (simp add: algebra_simps) (simp add: scaleR_left_distrib [symmetric] field_split_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(x - y) /\\<^sub>R k = (1 - inverse k) *\\<^sub>R (y /\\<^sub>R (1 - k)) + x /\\<^sub>R k\n\ngoal (1 subgoal):\n 1. (x - y) /\\<^sub>R k \\ S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(x - y) /\\<^sub>R k = (1 - inverse k) *\\<^sub>R (y /\\<^sub>R (1 - k)) + x /\\<^sub>R k\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(x - y) /\\<^sub>R k = (1 - inverse k) *\\<^sub>R (y /\\<^sub>R (1 - k)) + x /\\<^sub>R k\n\ngoal (1 subgoal):\n 1. (x - y) /\\<^sub>R k \\ S\n[PROOF STEP]\nusing \\affine S\\ xy\n[PROOF STATE]\nproof (prove)\nusing this:\n(x - y) /\\<^sub>R k = (1 - inverse k) *\\<^sub>R (y /\\<^sub>R (1 - k)) + x /\\<^sub>R k\naffine S\nx \\ S\ny /\\<^sub>R (1 - k) \\ S\n\ngoal (1 subgoal):\n 1. (x - y) /\\<^sub>R k \\ S\n[PROOF STEP]\nby (auto simp: affine_alt)\n[PROOF STATE]\nproof (state)\nthis:\n(x - y) /\\<^sub>R k \\ S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 934, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835452961425, "lm_q2_score": 0.8397339736884711, "lm_q1q2_score": 0.7015839174428615}} {"text": "[STATEMENT]\nlemma q65_upto_def: \"q65 = (\\k. k\\<^sup>2 mod 65) ` {..<65}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. q65 = (\\k. k\\<^sup>2 mod 65) ` {..<65}\n[PROOF STEP]\nby (simp add: q65_def lessThan_nat_numeral lessThan_Suc insert_commute)", "meta": {"llama_tokens": 130, "file": "Pell_Efficient_Discrete_Sqrt", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8397339756938818, "lm_q2_score": 0.8354835391516132, "lm_q1q2_score": 0.7015839139585791}} {"text": "[STATEMENT]\nlemma double_add_less_zero_iff_single_less_zero [simp]: \"a + a < 0 \\ a < 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a + a < (0::'a)) = (a < (0::'a))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (a + a < (0::'a)) = (a < (0::'a))\n[PROOF STEP]\nhave \"a + a < 0 \\ 0 < - (a + a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a + a < (0::'a)) = ((0::'a) < - (a + a))\n[PROOF STEP]\nby (subst less_minus_iff) simp\n[PROOF STATE]\nproof (state)\nthis:\n(a + a < (0::'a)) = ((0::'a) < - (a + a))\n\ngoal (1 subgoal):\n 1. (a + a < (0::'a)) = (a < (0::'a))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(a + a < (0::'a)) = ((0::'a) < - (a + a))\n\ngoal (1 subgoal):\n 1. (a + a < (0::'a)) = (a < (0::'a))\n[PROOF STEP]\nhave \"\\ \\ a < 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((0::'a) < - (a + a)) = (a < (0::'a))\n[PROOF STEP]\nby (simp only: minus_add_distrib zero_less_double_add_iff_zero_less_single_add) simp\n[PROOF STATE]\nproof (state)\nthis:\n((0::'a) < - (a + a)) = (a < (0::'a))\n\ngoal (1 subgoal):\n 1. (a + a < (0::'a)) = (a < (0::'a))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(a + a < (0::'a)) = ((0::'a) < - (a + a))\n((0::'a) < - (a + a)) = (a < (0::'a))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(a + a < (0::'a)) = ((0::'a) < - (a + a))\n((0::'a) < - (a + a)) = (a < (0::'a))\n\ngoal (1 subgoal):\n 1. (a + a < (0::'a)) = (a < (0::'a))\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n(a + a < (0::'a)) = (a < (0::'a))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 840, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835452961425, "lm_q2_score": 0.839733963661418, "lm_q1q2_score": 0.7015839090654236}} {"text": "[STATEMENT]\nlemma natpermute_finite: \"finite (natpermute n k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (natpermute n k)\n[PROOF STEP]\nproof (induct k arbitrary: n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\n. finite (natpermute n 0)\n 2. \\k n. (\\n. finite (natpermute n k)) \\ finite (natpermute n (Suc k))\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\n. finite (natpermute n 0)\n 2. \\k n. (\\n. finite (natpermute n k)) \\ finite (natpermute n (Suc k))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (natpermute n 0)\n[PROOF STEP]\nby (simp add: natpermute_0)\n[PROOF STATE]\nproof (state)\nthis:\nfinite (natpermute n 0)\n\ngoal (1 subgoal):\n 1. \\k n. (\\n. finite (natpermute n k)) \\ finite (natpermute n (Suc k))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\k n. (\\n. finite (natpermute n k)) \\ finite (natpermute n (Suc k))\n[PROOF STEP]\ncase (Suc k)\n[PROOF STATE]\nproof (state)\nthis:\nfinite (natpermute ?n k)\n\ngoal (1 subgoal):\n 1. \\k n. (\\n. finite (natpermute n k)) \\ finite (natpermute n (Suc k))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (natpermute ?n k)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (natpermute ?n k)\n\ngoal (1 subgoal):\n 1. finite (natpermute n (Suc k))\n[PROOF STEP]\nusing natpermute_split [of k \"Suc k\"] finite_UN_I\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (natpermute ?n k)\nk \\ Suc k \\ natpermute ?n (Suc k) = (\\m\\{0..?n}. {l1 @ l2 |l1 l2. l1 \\ natpermute m k \\ l2 \\ natpermute (?n - m) (Suc k - k)})\n\\finite ?A; \\a. a \\ ?A \\ finite (?B a)\\ \\ finite (\\ (?B ` ?A))\n\ngoal (1 subgoal):\n 1. finite (natpermute n (Suc k))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfinite (natpermute n (Suc k))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 913, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339716830606, "lm_q2_score": 0.8354835350552604, "lm_q1q2_score": 0.7015839071677574}} {"text": "[STATEMENT]\nlemma Sum_any_diff:\n assumes \"finite {x. f x \\ 0}\"\n assumes \"finite {x. g x \\ 0}\"\n shows \"Sum_any (\\x. f x - g x :: 'a :: ab_group_add) = Sum_any f - Sum_any g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. f x - g x) = Sum_any f - Sum_any g\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x. f x - g x) = Sum_any f - Sum_any g\n[PROOF STEP]\nhave \"{x. f x - g x \\ 0} \\ {x. f x \\ 0} \\ {x. g x \\ 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x. f x - g x \\ (0::'a)} \\ {x. f x \\ (0::'a)} \\ {x. g x \\ (0::'a)}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{x. f x - g x \\ (0::'a)} \\ {x. f x \\ (0::'a)} \\ {x. g x \\ (0::'a)}\n\ngoal (1 subgoal):\n 1. (\\x. f x - g x) = Sum_any f - Sum_any g\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n{x. f x - g x \\ (0::'a)} \\ {x. f x \\ (0::'a)} \\ {x. g x \\ (0::'a)}\n\ngoal (1 subgoal):\n 1. (\\x. f x - g x) = Sum_any f - Sum_any g\n[PROOF STEP]\nhave \"finite ({x. f x \\ 0} \\ {x. g x \\ 0})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite ({x. f x \\ (0::'a)} \\ {x. g x \\ (0::'a)})\n[PROOF STEP]\nby (subst finite_Un) (insert assms, auto)\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({x. f x \\ (0::'a)} \\ {x. g x \\ (0::'a)})\n\ngoal (1 subgoal):\n 1. (\\x. f x - g x) = Sum_any f - Sum_any g\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n{x. f x - g x \\ (0::'a)} \\ {x. f x \\ (0::'a)} \\ {x. g x \\ (0::'a)}\nfinite ({x. f x \\ (0::'a)} \\ {x. g x \\ (0::'a)})\n[PROOF STEP]\nhave \"finite {x. f x - g x \\ 0}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{x. f x - g x \\ (0::'a)} \\ {x. f x \\ (0::'a)} \\ {x. g x \\ (0::'a)}\nfinite ({x. f x \\ (0::'a)} \\ {x. g x \\ (0::'a)})\n\ngoal (1 subgoal):\n 1. finite {x. f x - g x \\ (0::'a)}\n[PROOF STEP]\nby (rule finite_subset)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {x. f x - g x \\ (0::'a)}\n\ngoal (1 subgoal):\n 1. (\\x. f x - g x) = Sum_any f - Sum_any g\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite {x. f x \\ (0::'a)}\nfinite {x. g x \\ (0::'a)}\nfinite {x. f x - g x \\ (0::'a)}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {x. f x \\ (0::'a)}\nfinite {x. g x \\ (0::'a)}\nfinite {x. f x - g x \\ (0::'a)}\n\ngoal (1 subgoal):\n 1. (\\x. f x - g x) = Sum_any f - Sum_any g\n[PROOF STEP]\nby (simp add: algebra_simps Sum_any.distrib [symmetric])\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. f x - g x) = Sum_any f - Sum_any g\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1403, "file": "Symmetric_Polynomials_Symmetric_Polynomials", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835371034369, "lm_q2_score": 0.8397339616560072, "lm_q1q2_score": 0.7015839005102428}} {"text": "[STATEMENT]\ntheorem ln_lower_1_eq: \"0 ln_lower_1 x = (x - 1)/x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ ln_lower_1 x = (x - 1) / x\n[PROOF STEP]\nby (auto simp: ln_lower_1_def divide_simps)", "meta": {"llama_tokens": 112, "file": "Special_Function_Bounds_Log_CF_Bounds", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094174159129, "lm_q2_score": 0.7853085758631159, "lm_q1q2_score": 0.7015235463960002}} {"text": "[STATEMENT]\nlemma arccos_arcsin_sqrt_pos: \"0 \\ x \\ x \\ 1 \\ arccos x = arcsin(sqrt(1 - x\\<^sup>2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 \\ x; x \\ 1\\ \\ arccos x = arcsin (sqrt (1 - x\\<^sup>2))\n[PROOF STEP]\napply (simp add: abs_square_le_1 arcsin_eq_Re_Arcsin arccos_eq_Re_Arccos)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 \\ x; x \\ 1\\ \\ Re (Arccos (complex_of_real x)) = Re (Arcsin (complex_of_real (sqrt (1 - x\\<^sup>2))))\n[PROOF STEP]\napply (subst Arccos_Arcsin_csqrt_pos)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\0 \\ x; x \\ 1\\ \\ 0 < Re (complex_of_real x) \\ Re (complex_of_real x) = 0 \\ 0 \\ Im (complex_of_real x)\n 2. \\0 \\ x; x \\ 1\\ \\ Re (Arcsin (csqrt (1 - (complex_of_real x)\\<^sup>2))) = Re (Arcsin (complex_of_real (sqrt (1 - x\\<^sup>2))))\n[PROOF STEP]\napply (auto simp: power_le_one csqrt_1_diff_eq)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 511, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094032139577, "lm_q2_score": 0.7853085859124003, "lm_q1q2_score": 0.7015235442202034}} {"text": "[STATEMENT]\nlemma summable_rabs_comparison_test: \"\\N. \\n\\N. \\f n\\ \\ g n \\ summable g \\ summable (\\n. \\f n\\)\"\n for f :: \"nat \\ real\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\N. \\n\\N. \\f n\\ \\ g n; summable g\\ \\ summable (\\n. \\f n\\)\n[PROOF STEP]\nby (rule summable_comparison_test) auto", "meta": {"llama_tokens": 184, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8418256631249077, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.7015140317891596}} {"text": "[STATEMENT]\nlemma summable_rabs_comparison_test: \"\\N. \\n\\N. \\f n\\ \\ g n \\ summable g \\ summable (\\n. \\f n\\)\"\n for f :: \"nat \\ real\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\N. \\n\\N. \\f n\\ \\ g n; summable g\\ \\ summable (\\n. \\f n\\)\n[PROOF STEP]\nby (rule summable_comparison_test) auto", "meta": {"llama_tokens": 184, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8418256631249077, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.7015140317891596}} {"text": "[STATEMENT]\nlemma summable_rabs_comparison_test: \"\\N. \\n\\N. \\f n\\ \\ g n \\ summable g \\ summable (\\n. \\f n\\)\"\n for f :: \"nat \\ real\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\N. \\n\\N. \\f n\\ \\ g n; summable g\\ \\ summable (\\n. \\f n\\)\n[PROOF STEP]\nby (rule summable_comparison_test) auto", "meta": {"llama_tokens": 184, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8418256631249077, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.7015140317891596}} {"text": "[STATEMENT]\nlemma summable_rabs_comparison_test: \"\\N. \\n\\N. \\f n\\ \\ g n \\ summable g \\ summable (\\n. \\f n\\)\"\n for f :: \"nat \\ real\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\N. \\n\\N. \\f n\\ \\ g n; summable g\\ \\ summable (\\n. \\f n\\)\n[PROOF STEP]\nby (rule summable_comparison_test) auto", "meta": {"llama_tokens": 184, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8418256631249077, "lm_q2_score": 0.8333245932423308, "lm_q1q2_score": 0.701514028304519}} {"text": "[STATEMENT]\nlemma express_poly :\n assumes h : \"MPoly_Type.degree (p::real mpoly) var = 1 \\ MPoly_Type.degree p var = 2\"\n shows \"p =\n (isolate_variable_sparse p var 2)*(Var var)^2\n +(isolate_variable_sparse p var 1)*(Var var)\n +(isolate_variable_sparse p var 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. p = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nhave h1a: \"MPoly_Type.degree p var = 1 \\ p =\n isolate_variable_sparse p var 0 + \n isolate_variable_sparse p var 1 * Var var\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. MPoly_Type.degree p var = 1 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var\n[PROOF STEP]\nusing sum_over_zero[where mp=\"p\",where x=\"var\"]\n[PROOF STATE]\nproof (prove)\nusing this:\np = (\\i\\MPoly_Type.degree p var. isolate_variable_sparse p var i * Var var ^ i)\n\ngoal (1 subgoal):\n 1. MPoly_Type.degree p var = 1 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nMPoly_Type.degree p var = 1 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var\n\ngoal (1 subgoal):\n 1. p = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nhave h1b: \"MPoly_Type.degree p var = 1 \\ isolate_variable_sparse p var 2 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. MPoly_Type.degree p var = 1 \\ isolate_variable_sparse p var 2 = 0\n[PROOF STEP]\nusing isovar_greater_degree\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i>MPoly_Type.degree ?p ?var. isolate_variable_sparse ?p ?var i = 0\n\ngoal (1 subgoal):\n 1. MPoly_Type.degree p var = 1 \\ isolate_variable_sparse p var 2 = 0\n[PROOF STEP]\nby (simp add: isovar_greater_degree)\n[PROOF STATE]\nproof (state)\nthis:\nMPoly_Type.degree p var = 1 \\ isolate_variable_sparse p var 2 = 0\n\ngoal (1 subgoal):\n 1. p = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nhave h1: \"MPoly_Type.degree p var = 1 \\ p =\n isolate_variable_sparse p var 0 + \n isolate_variable_sparse p var 1 * Var var\n + isolate_variable_sparse p var 2 * (Var var)^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. MPoly_Type.degree p var = 1 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\n[PROOF STEP]\nusing h1a h1b\n[PROOF STATE]\nproof (prove)\nusing this:\nMPoly_Type.degree p var = 1 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var\nMPoly_Type.degree p var = 1 \\ isolate_variable_sparse p var 2 = 0\n\ngoal (1 subgoal):\n 1. MPoly_Type.degree p var = 1 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nMPoly_Type.degree p var = 1 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\n\ngoal (1 subgoal):\n 1. p = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nhave h2a: \"MPoly_Type.degree p var = 2 \\ p = (\\i::nat \\ 2. isolate_variable_sparse p var i * Var var^i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. MPoly_Type.degree p var = 2 \\ p = (\\i\\2. isolate_variable_sparse p var i * Var var ^ i)\n[PROOF STEP]\nusing sum_over_zero[where mp=\"p\", where x=\"var\"]\n[PROOF STATE]\nproof (prove)\nusing this:\np = (\\i\\MPoly_Type.degree p var. isolate_variable_sparse p var i * Var var ^ i)\n\ngoal (1 subgoal):\n 1. MPoly_Type.degree p var = 2 \\ p = (\\i\\2. isolate_variable_sparse p var i * Var var ^ i)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nMPoly_Type.degree p var = 2 \\ p = (\\i\\2. isolate_variable_sparse p var i * Var var ^ i)\n\ngoal (1 subgoal):\n 1. p = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nhave h2b: \"(\\i::nat \\ 2. isolate_variable_sparse p var i * Var var^i) =\n (\\i::nat \\ 1. isolate_variable_sparse p var i * Var var^i) +\n isolate_variable_sparse p var 2 * (Var var)^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\2. isolate_variable_sparse p var i * Var var ^ i) = (\\i\\1. isolate_variable_sparse p var i * Var var ^ i) + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\2. isolate_variable_sparse p var i * Var var ^ i) = isolate_variable_sparse p var 0 + isolate_variable_sparse p var (Suc 0) * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\n[PROOF STEP]\nby (simp add: numerals(2))\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\2. isolate_variable_sparse p var i * Var var ^ i) = (\\i\\1. isolate_variable_sparse p var i * Var var ^ i) + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\n\ngoal (1 subgoal):\n 1. p = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nhave h2: \"MPoly_Type.degree p var = 2 \\ p =\n isolate_variable_sparse p var 0 + \n isolate_variable_sparse p var 1 * Var var + \n isolate_variable_sparse p var 2 * (Var var)^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. MPoly_Type.degree p var = 2 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\n[PROOF STEP]\nusing h2a h2b\n[PROOF STATE]\nproof (prove)\nusing this:\nMPoly_Type.degree p var = 2 \\ p = (\\i\\2. isolate_variable_sparse p var i * Var var ^ i)\n(\\i\\2. isolate_variable_sparse p var i * Var var ^ i) = (\\i\\1. isolate_variable_sparse p var i * Var var ^ i) + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\n\ngoal (1 subgoal):\n 1. MPoly_Type.degree p var = 2 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nMPoly_Type.degree p var = 2 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\n\ngoal (1 subgoal):\n 1. p = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nhave h3: \"isolate_variable_sparse p var 0 + \n isolate_variable_sparse p var 1 * Var var + \n isolate_variable_sparse p var 2 * (Var var)^2 = \n isolate_variable_sparse p var 2 * (Var var)^2 +\n isolate_variable_sparse p var 1 * Var var + \n isolate_variable_sparse p var 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nby (simp add: add.commute)\n[PROOF STATE]\nproof (state)\nthis:\nisolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n\ngoal (1 subgoal):\n 1. p = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nusing h h1 h2 h3\n[PROOF STATE]\nproof (prove)\nusing this:\nMPoly_Type.degree p var = 1 \\ MPoly_Type.degree p var = 2\nMPoly_Type.degree p var = 1 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\nMPoly_Type.degree p var = 2 \\ p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2\nisolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n\ngoal (1 subgoal):\n 1. p = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\np = isolate_variable_sparse p var 2 * (Var var)\\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3923, "file": "Virtual_Substitution_ExecutiblePolyProps", "length": 25, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8333246077301781, "lm_q2_score": 0.8418256412990657, "lm_q1q2_score": 0.7015140223127495}} {"text": "[STATEMENT]\ntheorem has_derivative_inverse_on:\n fixes f :: \"'n::euclidean_space \\ 'n\"\n assumes \"open S\"\n and derf: \"\\x. x \\ S \\ (f has_derivative f'(x)) (at x)\"\n and \"\\x. x \\ S \\ g (f x) = x\"\n and \"f' x \\ g' x = id\"\n and \"x \\ S\"\n shows \"(g has_derivative g'(x)) (at (f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (g has_derivative g' x) (at (f x))\n[PROOF STEP]\nproof (rule has_derivative_inverse_strong[where g'=\"g' x\" and f=f])\n[PROOF STATE]\nproof (state)\ngoal (6 subgoals):\n 1. open ?S\n 2. x \\ ?S\n 3. continuous_on ?S f\n 4. \\x. x \\ ?S \\ g (f x) = x\n 5. (f has_derivative ?f') (at x)\n 6. ?f' \\ g' x = id\n[PROOF STEP]\nshow \"continuous_on S f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous_on S f\n[PROOF STEP]\nunfolding continuous_on_eq_continuous_at[OF \\open S\\]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\S. isCont f x\n[PROOF STEP]\nusing derf has_derivative_continuous\n[PROOF STATE]\nproof (prove)\nusing this:\n?x \\ S \\ (f has_derivative f' ?x) (at ?x)\n(?f has_derivative ?f') (at ?x within ?s) \\ continuous (at ?x within ?s) ?f\n\ngoal (1 subgoal):\n 1. \\x\\S. isCont f x\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ncontinuous_on S f\n\ngoal (5 subgoals):\n 1. open S\n 2. x \\ S\n 3. \\x. x \\ S \\ g (f x) = x\n 4. (f has_derivative ?f') (at x)\n 5. ?f' \\ g' x = id\n[PROOF STEP]\nqed (use assms in auto)", "meta": {"llama_tokens": 701, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256472515683, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.7015140185615155}} {"text": "[STATEMENT]\nlemma DERIV_fun_powr2:\n fixes a::real\n assumes a_pos: \"a > 0\"\n and f: \"DERIV f x :> r\"\n shows \"DERIV (\\x. a.^(f x)) x :> a.^(f x) * r * ln a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. a .^ f x) has_real_derivative a .^ f x * r * ln a) (at x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\x. a .^ f x) has_real_derivative a .^ f x * r * ln a) (at x)\n[PROOF STEP]\nlet ?g = \"(\\x. a)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\x. a .^ f x) has_real_derivative a .^ f x * r * ln a) (at x)\n[PROOF STEP]\nhave g: \"DERIV ?g x :> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. a) has_real_derivative 0) (at x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. a) has_real_derivative 0) (at x)\n\ngoal (1 subgoal):\n 1. ((\\x. a .^ f x) has_real_derivative a .^ f x * r * ln a) (at x)\n[PROOF STEP]\nhave pos: \"?g x > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < a\n[PROOF STEP]\nby (simp add: a_pos)\n[PROOF STATE]\nproof (state)\nthis:\n0 < a\n\ngoal (1 subgoal):\n 1. ((\\x. a .^ f x) has_real_derivative a .^ f x * r * ln a) (at x)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. a .^ f x) has_real_derivative a .^ f x * r * ln a) (at x)\n[PROOF STEP]\nusing DERIV_powr[OF g pos f] a_pos\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\x. a .^ f x) has_real_derivative a .^ f x * (r * ln a + 0 * f x / a)) (at x)\n0 < a\n\ngoal (1 subgoal):\n 1. ((\\x. a .^ f x) has_real_derivative a .^ f x * r * ln a) (at x)\n[PROOF STEP]\nby (auto simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. a .^ f x) has_real_derivative a .^ f x * r * ln a) (at x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 899, "file": "Actuarial_Mathematics_Preliminaries", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256472515684, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.7015140185615155}} {"text": "[STATEMENT]\nlemma cts_step_uniformly_continuous:\n assumes [arith]: \"a < b\"\n shows \"uniformly_continuous_on UNIV (cts_step a b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. isUCont (cts_step a b)\n[PROOF STEP]\nunfolding uniformly_continuous_on_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\e>0. \\d>0. \\x\\UNIV. \\x'\\UNIV. dist x' x < d \\ dist (cts_step a b x') (cts_step a b x) < e\n[PROOF STEP]\nproof clarsimp\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\e. 0 < e \\ \\d>0. \\x x'. dist x' x < d \\ dist (cts_step a b x') (cts_step a b x) < e\n[PROOF STEP]\nfix e :: real\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\e. 0 < e \\ \\d>0. \\x x'. dist x' x < d \\ dist (cts_step a b x') (cts_step a b x) < e\n[PROOF STEP]\nassume [arith]: \"0 < e\"\n[PROOF STATE]\nproof (state)\nthis:\n0 < e\n\ngoal (1 subgoal):\n 1. \\e. 0 < e \\ \\d>0. \\x x'. dist x' x < d \\ dist (cts_step a b x') (cts_step a b x) < e\n[PROOF STEP]\nlet ?d = \"min (e * (b - a)) (b - a)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\e. 0 < e \\ \\d>0. \\x x'. dist x' x < d \\ dist (cts_step a b x') (cts_step a b x) < e\n[PROOF STEP]\nhave \"?d > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < min (e * (b - a)) (b - a)\n[PROOF STEP]\nby (auto simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n0 < min (e * (b - a)) (b - a)\n\ngoal (1 subgoal):\n 1. \\e. 0 < e \\ \\d>0. \\x x'. dist x' x < d \\ dist (cts_step a b x') (cts_step a b x) < e\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n0 < min (e * (b - a)) (b - a)\n\ngoal (1 subgoal):\n 1. \\e. 0 < e \\ \\d>0. \\x x'. dist x' x < d \\ dist (cts_step a b x') (cts_step a b x) < e\n[PROOF STEP]\nhave \"dist x' x < ?d \\ dist (cts_step a b x') (cts_step a b x) < e\" for x x'\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist x' x < min (e * (b - a)) (b - a) \\ dist (cts_step a b x') (cts_step a b x) < e\n[PROOF STEP]\nby (auto simp: dist_real_def divide_simps cts_step_def)\n[PROOF STATE]\nproof (state)\nthis:\ndist ?x' ?x < min (e * (b - a)) (b - a) \\ dist (cts_step a b ?x') (cts_step a b ?x) < e\n\ngoal (1 subgoal):\n 1. \\e. 0 < e \\ \\d>0. \\x x'. dist x' x < d \\ dist (cts_step a b x') (cts_step a b x) < e\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < min (e * (b - a)) (b - a)\ndist ?x' ?x < min (e * (b - a)) (b - a) \\ dist (cts_step a b ?x') (cts_step a b ?x) < e\n[PROOF STEP]\nshow \"\\d > 0. \\x x'. dist x' x < d \\ dist (cts_step a b x') (cts_step a b x) < e\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < min (e * (b - a)) (b - a)\ndist ?x' ?x < min (e * (b - a)) (b - a) \\ dist (cts_step a b ?x') (cts_step a b ?x) < e\n\ngoal (1 subgoal):\n 1. \\d>0. \\x x'. dist x' x < d \\ dist (cts_step a b x') (cts_step a b x) < e\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\d>0. \\x x'. dist x' x < d \\ dist (cts_step a b x') (cts_step a b x) < e\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1524, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256393148982, "lm_q2_score": 0.8333245911726382, "lm_q1q2_score": 0.7015140067207324}} {"text": "[STATEMENT]\nlemma cscalar_prod_conjugate_transpose:\n fixes x y ::\"'a :: {conjugatable_ring, comm_ring} vec\"\n assumes \"A \\ carrier_mat nr nc\"\n assumes \"x \\ carrier_vec nr\"\n assumes \"y \\ carrier_vec nc\"\n shows \"x \\c (A *\\<^sub>v y) = (A\\<^sup>H *\\<^sub>v x) \\c y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\c (A *\\<^sub>v y) = (A\\<^sup>H *\\<^sub>v x) \\c y\n[PROOF STEP]\nunfolding mult_mat_vec_def scalar_prod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..i. \\ia = 0..i. \\ia = 0..i = 0..H) (\\i. \\ia = 0..H i $ ia * x $ ia) $ i * conjugate y $ i)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat nr nc\nx \\ carrier_vec nr\ny \\ carrier_vec nc\n\ngoal (1 subgoal):\n 1. (\\i = 0..i. \\ia = 0..i. \\ia = 0..i = 0..H) (\\i. \\ia = 0..H i $ ia * x $ ia) $ i * conjugate y $ i)\n[PROOF STEP]\napply (auto simp add: sum_distrib_left sum_distrib_right sum_conjugate conjugate_dist_mul)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ carrier_mat nr nc; x \\ carrier_vec nr; y \\ carrier_vec nc\\ \\ (\\i = 0..n = 0..i = 0..n = 0..A \\ carrier_mat nr nc; x \\ carrier_vec nr; y \\ carrier_vec nc\\ \\ (\\j = 0..i = 0..i = 0..n = 0.. finite C; \\c. c \\ C \\ card c = k; \\c1 c2. \\c1 \\ C; c2 \\ C; c1 \\ c2\\ \\ c1 \\ c2 = {} \\ \\\n card (\\C) = k * card C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite C; \\c. c \\ C \\ card c = k; \\c1 c2. \\c1 \\ C; c2 \\ C; c1 \\ c2\\ \\ c1 \\ c2 = {}\\ \\ card (\\ C) = k * card C\n[PROOF STEP]\nby (metis card.infinite card_partition finite_Union mult_eq_if)", "meta": {"llama_tokens": 264, "file": "List-Infinite_CommonSet_SetInterval2", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942173896132, "lm_q2_score": 0.787931190663057, "lm_q1q2_score": 0.7014905827482324}} {"text": "[STATEMENT]\nlemma tan_zero [simp]: \"tan 0 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tan (0::'a) = (0::'a)\n[PROOF STEP]\nby (simp add: tan_def)", "meta": {"llama_tokens": 76, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942144788076, "lm_q2_score": 0.7879311856832191, "lm_q1q2_score": 0.701490576021197}} {"text": "[STATEMENT]\nlemma deriv_power_fun [simp]:\n assumes \"f field_differentiable at x\"\n shows \"deriv (\\x. f x ^ n) x = of_nat n * deriv f x * f x ^ (n - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. deriv (\\x. f x ^ n) x = of_nat n * deriv f x * f x ^ (n - 1)\n[PROOF STEP]\nusing DERIV_power[of f \"deriv f x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_field_derivative deriv f x) (at ?x within ?s) \\ ((\\x. f x ^ ?n) has_field_derivative of_nat ?n * (deriv f x * f ?x ^ (?n - Suc 0))) (at ?x within ?s)\n\ngoal (1 subgoal):\n 1. deriv (\\x. f x ^ n) x = of_nat n * deriv f x * f x ^ (n - 1)\n[PROOF STEP]\nby (simp add: DERIV_imp_deriv assms field_differentiable_derivI mult.assoc [symmetric])", "meta": {"llama_tokens": 322, "file": "Hyperdual_TwiceFieldDifferentiable", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513731336202, "lm_q2_score": 0.7826624840223698, "lm_q1q2_score": 0.7014623260052191}} {"text": "[STATEMENT]\nlemma divisor_count_primorial: \"divisor_count (primorial x) = 2 powr primes_pi x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (divisor_count (primorial x)) = 2 powr primes_pi x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. real (divisor_count (primorial x)) = 2 powr primes_pi x\n[PROOF STEP]\nhave \"divisor_count (primorial x) = (\\p | prime p \\ real p \\ x. divisor_count p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. divisor_count (primorial x) = prod divisor_count {p. prime p \\ real p \\ x}\n[PROOF STEP]\nunfolding primorial_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. divisor_count (\\{p. prime p \\ real p \\ x}) = prod divisor_count {p. prime p \\ real p \\ x}\n[PROOF STEP]\nby (subst divisor_count.prod_coprime) (auto simp: primes_coprime)\n[PROOF STATE]\nproof (state)\nthis:\ndivisor_count (primorial x) = prod divisor_count {p. prime p \\ real p \\ x}\n\ngoal (1 subgoal):\n 1. real (divisor_count (primorial x)) = 2 powr primes_pi x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndivisor_count (primorial x) = prod divisor_count {p. prime p \\ real p \\ x}\n\ngoal (1 subgoal):\n 1. real (divisor_count (primorial x)) = 2 powr primes_pi x\n[PROOF STEP]\nhave \"\\ = (\\p | prime p \\ real p \\ x. 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod divisor_count {p. prime p \\ real p \\ x} = (\\p\\{p. prime p \\ real p \\ x}. 2)\n[PROOF STEP]\nby (intro prod.cong divisor_count.prime) auto\n[PROOF STATE]\nproof (state)\nthis:\nprod divisor_count {p. prime p \\ real p \\ x} = (\\p\\{p. prime p \\ real p \\ x}. 2)\n\ngoal (1 subgoal):\n 1. real (divisor_count (primorial x)) = 2 powr primes_pi x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nprod divisor_count {p. prime p \\ real p \\ x} = (\\p\\{p. prime p \\ real p \\ x}. 2)\n\ngoal (1 subgoal):\n 1. real (divisor_count (primorial x)) = 2 powr primes_pi x\n[PROOF STEP]\nhave \"\\ = 2 powr primes_pi x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (\\p\\{p. prime p \\ real p \\ x}. 2) = 2 powr primes_pi x\n[PROOF STEP]\nby (simp add: primes_pi_def prime_sum_upto_def powr_realpow)\n[PROOF STATE]\nproof (state)\nthis:\nreal (\\p\\{p. prime p \\ real p \\ x}. 2) = 2 powr primes_pi x\n\ngoal (1 subgoal):\n 1. real (divisor_count (primorial x)) = 2 powr primes_pi x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (divisor_count (primorial x)) = 2 powr primes_pi x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (divisor_count (primorial x)) = 2 powr primes_pi x\n\ngoal (1 subgoal):\n 1. real (divisor_count (primorial x)) = 2 powr primes_pi x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nreal (divisor_count (primorial x)) = 2 powr primes_pi x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1267, "file": "Prime_Distribution_Elementary_Primorial", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513648201267, "lm_q2_score": 0.7826624840223698, "lm_q1q2_score": 0.7014623194985595}} {"text": "[STATEMENT]\nlemma val_ultrametric':\n assumes \"x \\ carrier Q\\<^sub>p\"\n assumes \"y \\ carrier Q\\<^sub>p\"\n shows \" min (val x) (val y) \\ val (x \\ y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. min (val x) (val y) \\ val (x \\ y)\n[PROOF STEP]\nusing val_ultrametric[of x \"\\y\"]\n val_minus[of y]\n assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x \\ carrier Q\\<^sub>p; \\ y \\ carrier Q\\<^sub>p\\ \\ min (val x) (val (\\ y)) \\ val (x \\ \\ y)\ny \\ carrier Q\\<^sub>p \\ val y = val (\\ y)\nx \\ carrier Q\\<^sub>p\ny \\ carrier Q\\<^sub>p\n\ngoal (1 subgoal):\n 1. min (val x) (val y) \\ val (x \\ y)\n[PROOF STEP]\nby (metis Qp.domain_axioms a_minus_def cring.cring_simprules(3) domain.axioms(1))", "meta": {"llama_tokens": 365, "file": "Padic_Field_Padic_Fields", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513675912912, "lm_q2_score": 0.7826624789529375, "lm_q1q2_score": 0.7014623171239605}} {"text": "[STATEMENT]\nlemma iso_hom_induced_relativization_contractible:\n assumes \"contractible_space(subtopology X S)\" \"contractible_space(subtopology X T)\" \"T \\ S\" \"topspace X \\ T \\ {}\"\n shows \"(hom_induced p X T X S id) \\ iso (relative_homology_group p X T) (relative_homology_group p X S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. hom_induced p X T X S id \\ Group.iso (relative_homology_group p X T) (relative_homology_group p X S)\n[PROOF STEP]\nproof (rule very_short_exact_sequence)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. exact_seq ([?D, relative_homology_group p X S, relative_homology_group p X T, ?A], [?h, hom_induced p X T X S id, ?f])\n 2. trivial_group ?A\n 3. trivial_group ?D\n[PROOF STEP]\nshow \"exact_seq\n ([relative_homology_group(p - 1) (subtopology X S) T, relative_homology_group p X S, relative_homology_group p X T, relative_homology_group p (subtopology X S) T],\n [hom_relboundary p X S T, hom_induced p X T X S id, hom_induced p (subtopology X S) T X T id])\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exact_seq ([relative_homology_group (p - 1) (subtopology X S) T, relative_homology_group p X S, relative_homology_group p X T, relative_homology_group p (subtopology X S) T], [hom_relboundary p X S T, hom_induced p X T X S id, hom_induced p (subtopology X S) T X T id])\n[PROOF STEP]\nusing homology_exactness_triple_1 [OF \\T \\ S\\] homology_exactness_triple_3 [OF \\T \\ S\\]\n[PROOF STATE]\nproof (prove)\nusing this:\nexact_seq ([relative_homology_group (?p - 1) (subtopology ?X S) T, relative_homology_group ?p ?X S, relative_homology_group ?p ?X T], [hom_relboundary ?p ?X S T, hom_induced ?p ?X T ?X S id])\nexact_seq ([relative_homology_group ?p ?X S, relative_homology_group ?p ?X T, relative_homology_group ?p (subtopology ?X S) T], [hom_induced ?p ?X T ?X S id, hom_induced ?p (subtopology ?X S) T ?X T id])\n\ngoal (1 subgoal):\n 1. exact_seq ([relative_homology_group (p - 1) (subtopology X S) T, relative_homology_group p X S, relative_homology_group p X T, relative_homology_group p (subtopology X S) T], [hom_relboundary p X S T, hom_induced p X T X S id, hom_induced p (subtopology X S) T X T id])\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nexact_seq ([relative_homology_group (p - 1) (subtopology X S) T, relative_homology_group p X S, relative_homology_group p X T, relative_homology_group p (subtopology X S) T], [hom_relboundary p X S T, hom_induced p X T X S id, hom_induced p (subtopology X S) T X T id])\n\ngoal (2 subgoals):\n 1. trivial_group (relative_homology_group p (subtopology X S) T)\n 2. trivial_group (relative_homology_group (p - 1) (subtopology X S) T)\n[PROOF STEP]\nshow \"trivial_group (relative_homology_group p (subtopology X S) T)\" \"trivial_group (relative_homology_group(p - 1) (subtopology X S) T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trivial_group (relative_homology_group p (subtopology X S) T) &&& trivial_group (relative_homology_group (p - 1) (subtopology X S) T)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncontractible_space (subtopology X S)\ncontractible_space (subtopology X T)\nT \\ S\ntopspace X \\ T \\ {}\n\ngoal (1 subgoal):\n 1. trivial_group (relative_homology_group p (subtopology X S) T) &&& trivial_group (relative_homology_group (p - 1) (subtopology X S) T)\n[PROOF STEP]\nby (force simp: inf.absorb_iff2 subtopology_subtopology topspace_subtopology intro!: trivial_relative_homology_group_contractible_spaces)+\n[PROOF STATE]\nproof (state)\nthis:\ntrivial_group (relative_homology_group p (subtopology X S) T)\ntrivial_group (relative_homology_group (p - 1) (subtopology X S) T)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1471, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513703624557, "lm_q2_score": 0.7826624688140726, "lm_q1q2_score": 0.7014623102058754}} {"text": "[STATEMENT]\nlemma rat_floor_lemma: \"of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nhave \"Fract a b = of_int (a div b) + Fract (a mod b) b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Fract a b = of_int (a div b) + Fract (a mod b) b\n[PROOF STEP]\nby (cases \"b = 0\") (simp, simp add: of_int_rat)\n[PROOF STATE]\nproof (state)\nthis:\nFract a b = of_int (a div b) + Fract (a mod b) b\n\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nFract a b = of_int (a div b) + Fract (a mod b) b\n\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nhave \"0 \\ Fract (a mod b) b \\ Fract (a mod b) b < 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ Fract (a mod b) b \\ Fract (a mod b) b < 1\n[PROOF STEP]\nunfolding Fract_of_int_quotient\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ of_int (a mod b) / of_int b \\ of_int (a mod b) / of_int b < 1\n[PROOF STEP]\nby (rule linorder_cases [of b 0]) (simp_all add: divide_nonpos_neg)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ Fract (a mod b) b \\ Fract (a mod b) b < 1\n\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nFract a b = of_int (a div b) + Fract (a mod b) b\n0 \\ Fract (a mod b) b \\ Fract (a mod b) b < 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nFract a b = of_int (a div b) + Fract (a mod b) b\n0 \\ Fract (a mod b) b \\ Fract (a mod b) b < 1\n\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nof_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1041, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438951025545426, "lm_q2_score": 0.8311430562234877, "lm_q1q2_score": 0.7013975546692162}} {"text": "[STATEMENT]\nlemma card_Pow_filter_one: \n assumes \"finite A\" \n shows \"card {x \\ Pow A . card x = 1} = card (A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {x \\ Pow A. card x = 1} = card A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. card {x \\ Pow A. card x = 1} = card A\n[PROOF STEP]\nproof (induct rule: finite_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. card {x \\ Pow {}. card x = 1} = card {}\n 2. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\ncase empty\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. card {x \\ Pow {}. card x = 1} = card {}\n 2. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {x \\ Pow {}. card x = 1} = card {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard {x \\ Pow {}. card x = 1} = card {}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\ncase (insert x F)\n[PROOF STATE]\nproof (state)\nthis:\nfinite F\nx \\ F\ncard {x \\ Pow F. card x = 1} = card F\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nhave \"Pow (insert x F) = Pow F \\ insert x ` Pow F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Pow (insert x F) = Pow F \\ insert x ` Pow F\n[PROOF STEP]\nby (simp add: Pow_insert)\n[PROOF STATE]\nproof (state)\nthis:\nPow (insert x F) = Pow F \\ insert x ` Pow F\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nPow (insert x F) = Pow F \\ insert x ` Pow F\n[PROOF STEP]\nhave split: \"{y \\ Pow (insert x F) . card y = 1} = \n {y \\ (Pow F) . card y = 1} \\ {y \\ (insert x ` Pow F) . card y = 1}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nPow (insert x F) = Pow F \\ insert x ` Pow F\n\ngoal (1 subgoal):\n 1. {y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {y \\ insert x ` Pow F. card y = 1}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {y \\ insert x ` Pow F. card y = 1}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nhave \"\\ y . y \\ (insert x ` Pow F) \\ finite y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\y. y \\ insert x ` Pow F \\ finite y\n[PROOF STEP]\nusing finite_subset insert.hyps(1)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?A \\ ?B; finite ?B\\ \\ finite ?A\nfinite F\n\ngoal (1 subgoal):\n 1. \\y. y \\ insert x ` Pow F \\ finite y\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n?y \\ insert x ` Pow F \\ finite ?y\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n?y \\ insert x ` Pow F \\ finite ?y\n[PROOF STEP]\nhave single: \"\\ y . y \\ (insert x ` Pow F) \\ card y = 1 \\ y = {x}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?y \\ insert x ` Pow F \\ finite ?y\n\ngoal (1 subgoal):\n 1. \\y. \\y \\ insert x ` Pow F; card y = 1\\ \\ y = {x}\n[PROOF STEP]\nby (metis card_1_singletonE empty_iff image_iff insertCI insertE)\n[PROOF STATE]\nproof (state)\nthis:\n\\?y \\ insert x ` Pow F; card ?y = 1\\ \\ ?y = {x}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\?y \\ insert x ` Pow F; card ?y = 1\\ \\ ?y = {x}\n[PROOF STEP]\nhave \"card {y \\ (insert x ` Pow F) . card y = 1} = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?y \\ insert x ` Pow F; card ?y = 1\\ \\ ?y = {x}\n\ngoal (1 subgoal):\n 1. card {y \\ insert x ` Pow F. card y = 1} = 1\n[PROOF STEP]\nusing empty_iff imageI is_singletonI is_singletonI' is_singleton_altdef\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?y \\ insert x ` Pow F; card ?y = 1\\ \\ ?y = {x}\n(?c \\ {}) = False\n?x \\ ?A \\ ?f ?x \\ ?f ` ?A\nis_singleton {?x}\n\\?A \\ {}; \\x y. \\x \\ ?A; y \\ ?A\\ \\ x = y\\ \\ is_singleton ?A\nis_singleton ?A = (card ?A = 1)\n\ngoal (1 subgoal):\n 1. card {y \\ insert x ` Pow F. card y = 1} = 1\n[PROOF STEP]\n(* LONG *)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?y \\ insert x ` Pow F; card ?y = 1\\ \\ ?y = {x}\n(?c \\ {}) = False\n?x \\ ?A \\ ?f ?x \\ ?f ` ?A\nis_singleton {?x}\n\\?A \\ {}; \\x y. \\x \\ ?A; y \\ ?A\\ \\ x = y\\ \\ is_singleton ?A\nis_singleton ?A = (card ?A = 1)\n\ngoal (1 subgoal):\n 1. card {y \\ insert x ` Pow F. card y = 1} = 1\n[PROOF STEP]\nby (metis (full_types, lifting) Collect_empty_eq_bot Pow_bottom bot_empty_eq mem_Collect_eq)\n[PROOF STATE]\nproof (state)\nthis:\ncard {y \\ insert x ` Pow F. card y = 1} = 1\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {y \\ insert x ` Pow F. card y = 1} = 1\n[PROOF STEP]\nhave \" {y \\ (insert x ` Pow F) . card y = 1} = {{x}}\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {y \\ insert x ` Pow F. card y = 1} = 1\n\ngoal (1 subgoal):\n 1. {y \\ insert x ` Pow F. card y = 1} = {{x}}\n[PROOF STEP]\nusing single card_1_singletonE card_eq_0_iff\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {y \\ insert x ` Pow F. card y = 1} = 1\n\\?y \\ insert x ` Pow F; card ?y = 1\\ \\ ?y = {x}\n\\card ?A = 1; \\x. ?A = {x} \\ ?thesis\\ \\ ?thesis\n(card ?A = 0) = (?A = {} \\ infinite ?A)\n\ngoal (1 subgoal):\n 1. {y \\ insert x ` Pow F. card y = 1} = {{x}}\n[PROOF STEP]\nby (smt empty_Collect_eq mem_Collect_eq singletonD zero_neq_one)\n[PROOF STATE]\nproof (state)\nthis:\n{y \\ insert x ` Pow F. card y = 1} = {{x}}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{y \\ insert x ` Pow F. card y = 1} = {{x}}\n[PROOF STEP]\nhave split2:\"{y \\ Pow (insert x F) . card y = 1} = {y \\ (Pow F) . card y = 1} \\ {{x}}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{y \\ insert x ` Pow F. card y = 1} = {{x}}\n\ngoal (1 subgoal):\n 1. {y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}\n[PROOF STEP]\nusing split\n[PROOF STATE]\nproof (prove)\nusing this:\n{y \\ insert x ` Pow F. card y = 1} = {{x}}\n{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {y \\ insert x ` Pow F. card y = 1}\n\ngoal (1 subgoal):\n 1. {y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {x \\ Pow F. card x = 1} = card F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}\n\ngoal (1 subgoal):\n 1. card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nproof (cases \"x \\ F\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}; x \\ F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n 2. \\{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}; x \\ F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nx \\ F\n\ngoal (2 subgoals):\n 1. \\{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}; x \\ F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n 2. \\{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}; x \\ F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ F\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ F\n\ngoal (1 subgoal):\n 1. card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nusing insert.hyps(2)\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ F\nx \\ F\n\ngoal (1 subgoal):\n 1. card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n\ngoal (1 subgoal):\n 1. \\{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}; x \\ F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}; x \\ F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nx \\ F\n\ngoal (1 subgoal):\n 1. \\{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}; x \\ F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ F\n[PROOF STEP]\nhave \"{y \\ (Pow F) . card y = 1} \\ {{x}} = {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ F\n\ngoal (1 subgoal):\n 1. {y \\ Pow F. card y = 1} \\ {{x}} = {}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n{y \\ Pow F. card y = 1} \\ {{x}} = {}\n\ngoal (1 subgoal):\n 1. \\{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}; x \\ F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{y \\ Pow F. card y = 1} \\ {{x}} = {}\n[PROOF STEP]\nhave fact:\"card {y \\ Pow (insert x F) . card y = 1} = \n card {y \\ (Pow F) . card y = 1} + card {{x}}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{y \\ Pow F. card y = 1} \\ {{x}} = {}\n\ngoal (1 subgoal):\n 1. card {y \\ Pow (insert x F). card y = 1} = card {y \\ Pow F. card y = 1} + card {{x}}\n[PROOF STEP]\nusing split2 card_Un_disjoint insert.hyps(1)\n[PROOF STATE]\nproof (prove)\nusing this:\n{y \\ Pow F. card y = 1} \\ {{x}} = {}\n{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}\n\\finite ?A; finite ?B; ?A \\ ?B = {}\\ \\ card (?A \\ ?B) = card ?A + card ?B\nfinite F\n\ngoal (1 subgoal):\n 1. card {y \\ Pow (insert x F). card y = 1} = card {y \\ Pow F. card y = 1} + card {{x}}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard {y \\ Pow (insert x F). card y = 1} = card {y \\ Pow F. card y = 1} + card {{x}}\n\ngoal (1 subgoal):\n 1. \\{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}; x \\ F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nhave \"card (insert x F) = card F + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (insert x F) = card F + 1\n[PROOF STEP]\nusing False card_insert_disjoint\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ F\n\\finite ?A; ?x \\ ?A\\ \\ card (insert ?x ?A) = Suc (card ?A)\n\ngoal (1 subgoal):\n 1. card (insert x F) = card F + 1\n[PROOF STEP]\nby (metis Suc_eq_plus1 insert.hyps(1))\n[PROOF STATE]\nproof (state)\nthis:\ncard (insert x F) = card F + 1\n\ngoal (1 subgoal):\n 1. \\{y \\ Pow (insert x F). card y = 1} = {y \\ Pow F. card y = 1} \\ {{x}}; x \\ F\\ \\ card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (insert x F) = card F + 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (insert x F) = card F + 1\n\ngoal (1 subgoal):\n 1. card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nusing fact insert.hyps(3)\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (insert x F) = card F + 1\ncard {y \\ Pow (insert x F). card y = 1} = card {y \\ Pow F. card y = 1} + card {{x}}\ncard {x \\ Pow F. card x = 1} = card F\n\ngoal (1 subgoal):\n 1. card {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncard {xa \\ Pow (insert x F). card xa = 1} = card (insert x F)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 6648, "file": "Design_Theory_Multisets_Extras", "length": 58, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438951025545427, "lm_q2_score": 0.8311430499496096, "lm_q1q2_score": 0.7013975493747212}} {"text": "[STATEMENT]\nlemma rat_floor_lemma: \"of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nhave \"Fract a b = of_int (a div b) + Fract (a mod b) b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Fract a b = of_int (a div b) + Fract (a mod b) b\n[PROOF STEP]\nby (cases \"b = 0\") (simp, simp add: of_int_rat)\n[PROOF STATE]\nproof (state)\nthis:\nFract a b = of_int (a div b) + Fract (a mod b) b\n\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nFract a b = of_int (a div b) + Fract (a mod b) b\n\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nhave \"0 \\ Fract (a mod b) b \\ Fract (a mod b) b < 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ Fract (a mod b) b \\ Fract (a mod b) b < 1\n[PROOF STEP]\nunfolding Fract_of_int_quotient\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ of_int (a mod b) / of_int b \\ of_int (a mod b) / of_int b < 1\n[PROOF STEP]\nby (rule linorder_cases [of b 0]) (simp_all add: divide_nonpos_neg)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ Fract (a mod b) b \\ Fract (a mod b) b < 1\n\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nFract a b = of_int (a div b) + Fract (a mod b) b\n0 \\ Fract (a mod b) b \\ Fract (a mod b) b < 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nFract a b = of_int (a div b) + Fract (a mod b) b\n0 \\ Fract (a mod b) b \\ Fract (a mod b) b < 1\n\ngoal (1 subgoal):\n 1. of_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nof_int (a div b) \\ Fract a b \\ Fract a b < of_int (a div b + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1041, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438951025545426, "lm_q2_score": 0.8311430499496096, "lm_q1q2_score": 0.7013975493747211}} {"text": "[STATEMENT]\nlemma ignore_one:\n fixes P :: \"_ \\ nat \\ bool\"\n assumes \"P 1 n\" \"1 < n\"\n shows \"card {a \\ {1.. a \\ a < n \\ P a n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {a \\ {1.. a \\ a < n \\ P a n}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {a \\ {1.. a \\ a < n \\ P a n}\n[PROOF STEP]\nhave \"insert 1 {a. 2 \\ a \\ a < n \\ P a n} = {a. 1 \\ a \\ a < n \\ P a n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. insert 1 {a. 2 \\ a \\ a < n \\ P a n} = {a. 1 \\ a \\ a < n \\ P a n}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nP 1 n\n1 < n\n\ngoal (1 subgoal):\n 1. insert 1 {a. 2 \\ a \\ a < n \\ P a n} = {a. 1 \\ a \\ a < n \\ P a n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninsert 1 {a. 2 \\ a \\ a < n \\ P a n} = {a. 1 \\ a \\ a < n \\ P a n}\n\ngoal (1 subgoal):\n 1. card {a \\ {1.. a \\ a < n \\ P a n}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ninsert 1 {a. 2 \\ a \\ a < n \\ P a n} = {a. 1 \\ a \\ a < n \\ P a n}\n\ngoal (1 subgoal):\n 1. card {a \\ {1.. a \\ a < n \\ P a n}\n[PROOF STEP]\nhave \"card (insert 1 {a. 2 \\ a \\ a < n \\ P a n}) = Suc (card {a. 2 \\ a \\ a < n \\ P a n})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (insert 1 {a. 2 \\ a \\ a < n \\ P a n}) = Suc (card {a. 2 \\ a \\ a < n \\ P a n})\n[PROOF STEP]\nusing card_insert_disjoint\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite ?A; ?x \\ ?A\\ \\ card (insert ?x ?A) = Suc (card ?A)\n\ngoal (1 subgoal):\n 1. card (insert 1 {a. 2 \\ a \\ a < n \\ P a n}) = Suc (card {a. 2 \\ a \\ a < n \\ P a n})\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (insert 1 {a. 2 \\ a \\ a < n \\ P a n}) = Suc (card {a. 2 \\ a \\ a < n \\ P a n})\n\ngoal (1 subgoal):\n 1. card {a \\ {1.. a \\ a < n \\ P a n}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ninsert 1 {a. 2 \\ a \\ a < n \\ P a n} = {a. 1 \\ a \\ a < n \\ P a n}\ncard (insert 1 {a. 2 \\ a \\ a < n \\ P a n}) = Suc (card {a. 2 \\ a \\ a < n \\ P a n})\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ninsert 1 {a. 2 \\ a \\ a < n \\ P a n} = {a. 1 \\ a \\ a < n \\ P a n}\ncard (insert 1 {a. 2 \\ a \\ a < n \\ P a n}) = Suc (card {a. 2 \\ a \\ a < n \\ P a n})\n\ngoal (1 subgoal):\n 1. card {a \\ {1.. a \\ a < n \\ P a n}\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\ncard {a \\ {1.. a \\ a < n \\ P a n}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1550, "file": "Probabilistic_Prime_Tests_Fermat_Witness", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430478583169, "lm_q2_score": 0.8438950966654774, "lm_q1q2_score": 0.7013975427152338}} {"text": "[STATEMENT]\nlemma to_fun_smult:\n assumes \"f \\ carrier P\"\n assumes \"b \\ carrier R\"\n assumes \"c \\ carrier R\"\n shows \"to_fun (c \\\\<^bsub>P\\<^esub> f) b = c \\(to_fun f b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. to_fun (c \\\\<^bsub>P\\<^esub> f) b = c \\ to_fun f b\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. to_fun (c \\\\<^bsub>P\\<^esub> f) b = c \\ to_fun f b\n[PROOF STEP]\nhave \"(c \\\\<^bsub>P\\<^esub> f) = (to_poly c) \\\\<^bsub>P\\<^esub> f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c \\\\<^bsub>P\\<^esub> f = to_polynomial R c \\\\<^bsub>P\\<^esub> f\n[PROOF STEP]\nby (metis P_def assms(1) assms(3) monom_mult_is_smult to_polynomial_def)\n[PROOF STATE]\nproof (state)\nthis:\nc \\\\<^bsub>P\\<^esub> f = to_polynomial R c \\\\<^bsub>P\\<^esub> f\n\ngoal (1 subgoal):\n 1. to_fun (c \\\\<^bsub>P\\<^esub> f) b = c \\ to_fun f b\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc \\\\<^bsub>P\\<^esub> f = to_polynomial R c \\\\<^bsub>P\\<^esub> f\n[PROOF STEP]\nhave \"to_fun (c \\\\<^bsub>P\\<^esub> f) b = to_fun (to_poly c) b \\ to_fun f b\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\\\<^bsub>P\\<^esub> f = to_polynomial R c \\\\<^bsub>P\\<^esub> f\n\ngoal (1 subgoal):\n 1. to_fun (c \\\\<^bsub>P\\<^esub> f) b = to_fun (to_polynomial R c) b \\ to_fun f b\n[PROOF STEP]\nby (simp add: assms(1) assms(2) assms(3) to_fun_mult to_poly_closed)\n[PROOF STATE]\nproof (state)\nthis:\nto_fun (c \\\\<^bsub>P\\<^esub> f) b = to_fun (to_polynomial R c) b \\ to_fun f b\n\ngoal (1 subgoal):\n 1. to_fun (c \\\\<^bsub>P\\<^esub> f) b = c \\ to_fun f b\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nto_fun (c \\\\<^bsub>P\\<^esub> f) b = to_fun (to_polynomial R c) b \\ to_fun f b\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nto_fun (c \\\\<^bsub>P\\<^esub> f) b = to_fun (to_polynomial R c) b \\ to_fun f b\n\ngoal (1 subgoal):\n 1. to_fun (c \\\\<^bsub>P\\<^esub> f) b = c \\ to_fun f b\n[PROOF STEP]\nby (simp add: assms(2) assms(3) to_fun_to_poly)\n[PROOF STATE]\nproof (state)\nthis:\nto_fun (c \\\\<^bsub>P\\<^esub> f) b = c \\ to_fun f b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1058, "file": "Padic_Ints_Cring_Poly", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972784807408, "lm_q2_score": 0.8056321936479701, "lm_q1q2_score": 0.7013811952463919}} {"text": "[STATEMENT]\nlemma d_OUT_monotone_convergence_SUP:\n assumes \"incseq (\\n y. f n (x, y))\"\n shows \"d_OUT (\\e. SUP n. f n e) x = (SUP n. d_OUT (f n) x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. d_OUT (\\e. \\n. f n e) x = (\\n. d_OUT (f n) x)\n[PROOF STEP]\nunfolding d_OUT_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sup>+ y. \\n. f n (x, y)) = (\\n. \\\\<^sup>+ y. f n (x, y))\n[PROOF STEP]\nby(rule nn_integral_monotone_convergence_SUP[OF assms]) simp", "meta": {"llama_tokens": 262, "file": "MFMC_Countable_MFMC_Network", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972751232809, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7013811905100995}} {"text": "[STATEMENT]\nlemma dist_fs_cvec_zero:\n assumes \"z \\ vec_zero\" and \"w \\ vec_zero\"\n shows \"dist_fs_cvec z w = 0 \\ (cmod \\z,w\\)\\<^sup>2 = (\\z\\\\<^sup>2 * \\w\\\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist_fs_cvec z w = 0) = ((cmod \\z,w\\)\\<^sup>2 = \\z\\\\<^sup>2 * \\w\\\\<^sup>2)\n[PROOF STEP]\nusing assms norm_cvec_gt_0[of z] norm_cvec_gt_0[of w]\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ vec_zero\nw \\ vec_zero\nz \\ vec_zero \\ 0 < \\z\\\nw \\ vec_zero \\ 0 < \\w\\\n\ngoal (1 subgoal):\n 1. (dist_fs_cvec z w = 0) = ((cmod \\z,w\\)\\<^sup>2 = \\z\\\\<^sup>2 * \\w\\\\<^sup>2)\n[PROOF STEP]\nby (subst dist_fs_cvec_iff) auto", "meta": {"llama_tokens": 385, "file": "Complex_Geometry_Chordal_Metric", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972684083609, "lm_q2_score": 0.8056321959813274, "lm_q1q2_score": 0.7013811891631729}} {"text": "[STATEMENT]\nlemma card_gcd_eq_totient:\n \"n > 0 \\ d dvd n \\ card {k\\{0<..n}. gcd k n = d} = totient (n div d)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < n; d dvd n\\ \\ card {k \\ {0<..n}. gcd k n = d} = totient (n div d)\n[PROOF STEP]\nunfolding totient_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < n; d dvd n\\ \\ card {k \\ {0<..n}. gcd k n = d} = card (totatives (n div d))\n[PROOF STEP]\nby (rule sym, rule bij_betw_same_card[OF bij_betw_totatives_gcd_eq])", "meta": {"llama_tokens": 259, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972684083609, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7013811851003439}} {"text": "[STATEMENT]\ntheorem derangements_formula':\n assumes \"n \\ 0\" \"finite S\" \"card S = n\"\n shows \"card (derangements S) = nat (round (fact n / exp 1 :: real))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (derangements S) = nat (round (fact n / exp 1))\n[PROOF STEP]\nusing derangements_formula[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nint (card (derangements S)) = round (fact n / exp 1)\n\ngoal (1 subgoal):\n 1. card (derangements S) = nat (round (fact n / exp 1))\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 204, "file": "Derangements_Derangements", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972549785201, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7013811742808318}} {"text": "[STATEMENT]\nlemma mat_adjoint_def': \"mat_adjoint M = transpose_mat (map_mat conjugate M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat_adjoint M = (map_mat conjugate M)\\<^sup>T\n[PROOF STEP]\napply (rule mat_eq_iff[THEN iffD2])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_row (mat_adjoint M) = dim_row (map_mat conjugate M)\\<^sup>T \\ dim_col (mat_adjoint M) = dim_col (map_mat conjugate M)\\<^sup>T \\ (\\i j. i < dim_row (map_mat conjugate M)\\<^sup>T \\ j < dim_col (map_mat conjugate M)\\<^sup>T \\ mat_adjoint M $$ (i, j) = (map_mat conjugate M)\\<^sup>T $$ (i, j))\n[PROOF STEP]\napply (auto simp: mat_adjoint_def transpose_mat_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_col M; j < dim_row M\\ \\ mat_of_rows (dim_row M) (map conjugate (cols M)) $$ (i, j) = conjugate (M $$ (j, i))\n[PROOF STEP]\napply (subst mat_of_rows_index)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_col M; j < dim_row M\\ \\ i < length (map conjugate (cols M))\n 2. \\i j. \\i < dim_col M; j < dim_row M\\ \\ j < dim_row M\n 3. \\i j. \\i < dim_col M; j < dim_row M\\ \\ map conjugate (cols M) ! i $ j = conjugate (M $$ (j, i))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 581, "file": "Complex_Bounded_Operators_extra_Extra_Jordan_Normal_Form", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972549785201, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7013811742808318}} {"text": "[STATEMENT]\nlemma d_OUT_current_of_bipartite:\n assumes f: \"current (bipartite_web_of \\) f\"\n shows \"d_OUT (current_of_bipartite f) x = d_OUT f (Inl x) - f (Inl x, Inr x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. d_OUT (current_of_bipartite f) x = d_OUT f (Inl x) - f (Inl x, Inr x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. d_OUT (current_of_bipartite f) x = d_OUT f (Inl x) - f (Inl x, Inr x)\n[PROOF STEP]\nhave \"d_OUT (current_of_bipartite f) x = \\\\<^sup>+ y. f (Inl x, y) * indicator \\<^bold>E (x, projr y) \\count_space (range Inr)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. d_OUT (current_of_bipartite f) x = (\\\\<^sup>+ y\\range Inr. f (Inl x, y) * indicator \\<^bold>E (x, projr y))\n[PROOF STEP]\nby(simp add: d_OUT_def nn_integral_count_space_reindex)\n[PROOF STATE]\nproof (state)\nthis:\nd_OUT (current_of_bipartite f) x = (\\\\<^sup>+ y\\range Inr. f (Inl x, y) * indicator \\<^bold>E (x, projr y))\n\ngoal (1 subgoal):\n 1. d_OUT (current_of_bipartite f) x = d_OUT f (Inl x) - f (Inl x, Inr x)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nd_OUT (current_of_bipartite f) x = (\\\\<^sup>+ y\\range Inr. f (Inl x, y) * indicator \\<^bold>E (x, projr y))\n\ngoal (1 subgoal):\n 1. d_OUT (current_of_bipartite f) x = d_OUT f (Inl x) - f (Inl x, Inr x)\n[PROOF STEP]\nhave \"\\ = d_OUT f (Inl x) - \\\\<^sup>+ y. f (Inl x, y) * indicator {Inr x} y \\count_space UNIV\" (is \"_ = _ - ?rest\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sup>+ y\\range Inr. f (Inl x, y) * indicator \\<^bold>E (x, projr y)) = d_OUT f (Inl x) - (\\\\<^sup>+y\\{Inr x}. f (Inl x, y)\\count_space UNIV)\n[PROOF STEP]\nunfolding d_OUT_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sup>+ y\\range Inr. f (Inl x, y) * indicator \\<^bold>E (x, projr y)) = (\\\\<^sup>+ y. f (Inl x, y)) - (\\\\<^sup>+y\\{Inr x}. f (Inl x, y)\\count_space UNIV)\n[PROOF STEP]\nby(subst nn_integral_diff[symmetric])(auto 4 4 simp add: current_bipartite_web_finite[OF f] AE_count_space nn_integral_count_space_indicator no_loop split: split_indicator intro!: nn_integral_cong intro: currentD_outside[OF f] elim: edge_bipartite_webE)\n[PROOF STATE]\nproof (state)\nthis:\n(\\\\<^sup>+ y\\range Inr. f (Inl x, y) * indicator \\<^bold>E (x, projr y)) = d_OUT f (Inl x) - (\\\\<^sup>+y\\{Inr x}. f (Inl x, y)\\count_space UNIV)\n\ngoal (1 subgoal):\n 1. d_OUT (current_of_bipartite f) x = d_OUT f (Inl x) - f (Inl x, Inr x)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nd_OUT (current_of_bipartite f) x = d_OUT f (Inl x) - (\\\\<^sup>+y\\{Inr x}. f (Inl x, y)\\count_space UNIV)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nd_OUT (current_of_bipartite f) x = d_OUT f (Inl x) - (\\\\<^sup>+y\\{Inr x}. f (Inl x, y)\\count_space UNIV)\n\ngoal (1 subgoal):\n 1. d_OUT (current_of_bipartite f) x = d_OUT f (Inl x) - f (Inl x, Inr x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nd_OUT (current_of_bipartite f) x = d_OUT f (Inl x) - f (Inl x, Inr x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1472, "file": "MFMC_Countable_MFMC_Reduction", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972583359805, "lm_q2_score": 0.805632181981183, "lm_q1q2_score": 0.7013811688600516}} {"text": "[STATEMENT]\nlemma distinct_concat:\n \"\\ distinct xs;\n \\ ys. ys \\ set xs \\ distinct ys;\n \\ ys zs. \\ ys \\ set xs ; zs \\ set xs ; ys \\ zs \\ \\ set ys \\ set zs = {}\n \\ \\ distinct (concat xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\distinct xs; \\ys. ys \\ set xs \\ distinct ys; \\ys zs. \\ys \\ set xs; zs \\ set xs; ys \\ zs\\ \\ set ys \\ set zs = {}\\ \\ distinct (concat xs)\n[PROOF STEP]\nby (induct xs) auto", "meta": {"llama_tokens": 251, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045996818986, "lm_q2_score": 0.7905303285397348, "lm_q1q2_score": 0.7013621436684951}} {"text": "[STATEMENT]\nlemma \"nnf ((Atom (k::nat)) \\<^bold>\\ (Not ((Atom l) \\<^bold>\\ (Not (Atom m))))) = \\<^bold>\\ (Atom k) \\<^bold>\\ (\\<^bold>\\ (Atom l) \\<^bold>\\ Atom m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. nnf (Atom k \\<^bold>\\ \\<^bold>\\ (Atom l \\<^bold>\\ \\<^bold>\\ (Atom m))) = \\<^bold>\\ (Atom k) \\<^bold>\\ (\\<^bold>\\ (Atom l) \\<^bold>\\ Atom m)\n[PROOF STEP]\nby code_simp", "meta": {"llama_tokens": 199, "file": "Propositional_Proof_Systems_CNF_Formulas", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045996818986, "lm_q2_score": 0.7905303236047049, "lm_q1q2_score": 0.7013621392901139}} {"text": "[STATEMENT]\nlemma eval_fps_diff:\n fixes f g :: \"'a :: {banach, real_normed_div_algebra} fps\"\n assumes \"norm z < fps_conv_radius f\" \"norm z < fps_conv_radius g\"\n shows \"eval_fps (f - g) z = eval_fps f z - eval_fps g z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eval_fps (f - g) z = eval_fps f z - eval_fps g z\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nereal (norm z) < fps_conv_radius f\nereal (norm z) < fps_conv_radius g\n\ngoal (1 subgoal):\n 1. eval_fps (f - g) z = eval_fps f z - eval_fps g z\n[PROOF STEP]\nunfolding eval_fps_def\n[PROOF STATE]\nproof (prove)\nusing this:\nereal (norm z) < fps_conv_radius f\nereal (norm z) < fps_conv_radius g\n\ngoal (1 subgoal):\n 1. (\\n. fps_nth (f - g) n * z ^ n) = (\\n. fps_nth f n * z ^ n) - (\\n. fps_nth g n * z ^ n)\n[PROOF STEP]\nby (subst suminf_diff) (auto simp: ring_distribs intro!: summable_fps)", "meta": {"llama_tokens": 404, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045937171068, "lm_q2_score": 0.7905303112671295, "lm_q1q2_score": 0.7013621236288116}} {"text": "[STATEMENT]\nlemma infsetsum_cdiv:\n fixes f :: \"'a \\ 'b :: {banach, real_normed_field, second_countable_topology}\"\n assumes \"c \\ 0 \\ f abs_summable_on A\"\n shows \"infsetsum (\\x. f x / c) A = infsetsum f A / c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. f x / c) = infsetsum f A / c\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'b) \\ f abs_summable_on A\n\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. f x / c) = infsetsum f A / c\n[PROOF STEP]\nunfolding infsetsum_def abs_summable_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'b) \\ integrable (count_space A) f\n\ngoal (1 subgoal):\n 1. LINT x|count_space A. f x / c = integral\\<^sup>L (count_space A) f / c\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 352, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045877523147, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7013621211026533}} {"text": "[STATEMENT]\nlemma rec_exec_pr_Suc_simps: \"rec_exec (Pr n f g) (xs @ [Suc y])\n = rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rec_exec (Pr n f g) (xs @ [Suc y]) = rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])\n[PROOF STEP]\napply(induct y)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. rec_exec (Pr n f g) (xs @ [Suc 0]) = rec_exec g (xs @ [0, rec_exec (Pr n f g) (xs @ [0])])\n 2. \\y. rec_exec (Pr n f g) (xs @ [Suc y]) = rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) \\ rec_exec (Pr n f g) (xs @ [Suc (Suc y)]) = rec_exec g (xs @ [Suc y, rec_exec (Pr n f g) (xs @ [Suc y])])\n[PROOF STEP]\napply(simp add: rec_exec.simps)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\y. rec_exec (Pr n f g) (xs @ [Suc y]) = rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) \\ rec_exec (Pr n f g) (xs @ [Suc (Suc y)]) = rec_exec g (xs @ [Suc y, rec_exec (Pr n f g) (xs @ [Suc y])])\n[PROOF STEP]\napply(simp add: rec_exec.simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 529, "file": "Universal_Turing_Machine_Recursive", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045817875224, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7013621163873042}} {"text": "[STATEMENT]\nlemma sin_plus_cos_eq_45:\n \"sin x + cos x = sqrt 2 * sin (x + pi/4)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin x + cos x = sqrt 2 * sin (x + pi / 4)\n[PROOF STEP]\napply (simp add: sin_add sin_45 cos_45 )\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin x + cos x = sqrt 2 * (sin x * sqrt 2 / 2 + cos x * sqrt 2 / 2)\n[PROOF STEP]\nby (simp add: field_simps)", "meta": {"llama_tokens": 186, "file": "Hyperdual_AnalyticTestFunction", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9086178969328287, "lm_q2_score": 0.7718435083355187, "lm_q1q2_score": 0.7013108253050753}} {"text": "[STATEMENT]\nlemma infsetsum_Sigma':\n fixes A :: \"'a set\" and B :: \"'a \\ 'b set\"\n assumes [simp]: \"countable A\" and \"\\i. countable (B i)\"\n assumes summable: \"(\\(x,y). f x y) abs_summable_on (Sigma A B)\"\n shows \"infsetsum (\\x. infsetsum (\\y. f x y) (B x)) A = infsetsum (\\(x,y). f x y) (Sigma A B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. infsetsum (f x) (B x)) = (\\\\<^sub>a(x, y)\\Sigma A B. f x y)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncountable A\ncountable (B ?i)\n(\\(x, y). f x y) abs_summable_on Sigma A B\n\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. infsetsum (f x) (B x)) = (\\\\<^sub>a(x, y)\\Sigma A B. f x y)\n[PROOF STEP]\nby (subst infsetsum_Sigma) auto", "meta": {"llama_tokens": 347, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9086178969328287, "lm_q2_score": 0.7718434873426302, "lm_q1q2_score": 0.7013108062305611}} {"text": "[STATEMENT]\ntheorem prim_pyth_triple_iff:\n \"prim_pyth_triple x y z \\\n (\\u v. 0 < v \\ v < u \\ coprime u v \\ \\(odd u \\ odd v) \\\n (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2)\"\n (is \"_ \\ ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prim_pyth_triple x y z = (\\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. prim_pyth_triple x y z \\ \\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2\n 2. \\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2 \\ prim_pyth_triple x y z\n[PROOF STEP]\nassume \"prim_pyth_triple x y z\"\n[PROOF STATE]\nproof (state)\nthis:\nprim_pyth_triple x y z\n\ngoal (2 subgoals):\n 1. prim_pyth_triple x y z \\ \\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2\n 2. \\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2 \\ prim_pyth_triple x y z\n[PROOF STEP]\nfrom prim_pyth_tripleE[OF this]\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\u v. \\0 < v; v < u; coprime u v; \\ (odd u \\ odd v); z = u\\<^sup>2 + v\\<^sup>2; x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v\\ \\ ?thesis) \\ ?thesis\n[PROOF STEP]\nshow ?rhs\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\u v. \\0 < v; v < u; coprime u v; \\ (odd u \\ odd v); z = u\\<^sup>2 + v\\<^sup>2; x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v\\ \\ ?thesis) \\ ?thesis\n\ngoal (1 subgoal):\n 1. \\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\n\\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2 \\ prim_pyth_triple x y z\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2 \\ prim_pyth_triple x y z\n[PROOF STEP]\nassume ?rhs\n[PROOF STATE]\nproof (state)\nthis:\n\\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2 \\ prim_pyth_triple x y z\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2\n[PROOF STEP]\nobtain u v where uv: \"0 < v\" \"v < u\" \"coprime u v\" \"\\(odd u \\ odd v)\" \"z = u\\<^sup>2 + v\\<^sup>2\" and\n eq: \"x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2\n\ngoal (1 subgoal):\n 1. (\\v u. \\0 < v; v < u; coprime u v; \\ (odd u \\ odd v); z = u\\<^sup>2 + v\\<^sup>2; x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\n0 < v\nv < u\ncoprime u v\n\\ (odd u \\ odd v)\nz = u\\<^sup>2 + v\\<^sup>2\nx = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v\n\ngoal (1 subgoal):\n 1. \\u v. 0 < v \\ v < u \\ coprime u v \\ \\ (odd u \\ odd v) \\ (x = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v) \\ z = u\\<^sup>2 + v\\<^sup>2 \\ prim_pyth_triple x y z\n[PROOF STEP]\nthus \"prim_pyth_triple x y z\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < v\nv < u\ncoprime u v\n\\ (odd u \\ odd v)\nz = u\\<^sup>2 + v\\<^sup>2\nx = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v\n\ngoal (1 subgoal):\n 1. prim_pyth_triple x y z\n[PROOF STEP]\nusing uv prim_pyth_tripleI1[OF uv(1-4)] prim_pyth_tripleI2[OF uv(1-4)] uv(5) eq\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < v\nv < u\ncoprime u v\n\\ (odd u \\ odd v)\nz = u\\<^sup>2 + v\\<^sup>2\nx = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v\n0 < v\nv < u\ncoprime u v\n\\ (odd u \\ odd v)\nz = u\\<^sup>2 + v\\<^sup>2\nprim_pyth_triple (2 * u * v) (u\\<^sup>2 - v\\<^sup>2) (u\\<^sup>2 + v\\<^sup>2)\nprim_pyth_triple (u\\<^sup>2 - v\\<^sup>2) (2 * u * v) (u\\<^sup>2 + v\\<^sup>2)\nz = u\\<^sup>2 + v\\<^sup>2\nx = 2 * u * v \\ y = u\\<^sup>2 - v\\<^sup>2 \\ x = u\\<^sup>2 - v\\<^sup>2 \\ y = 2 * u * v\n\ngoal (1 subgoal):\n 1. prim_pyth_triple x y z\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nprim_pyth_triple x y z\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3416, "file": "Gaussian_Integers_Gaussian_Integers_Pythagorean_Triples", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213853793452, "lm_q2_score": 0.7799928900257126, "lm_q1q2_score": 0.7013082878659579}} {"text": "[STATEMENT]\nlemma eventually_at_left_to_right:\n \"eventually P (at_left a) \\ eventually (\\x. P (- x)) (at_right (-a))\"\n for a :: real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eventually P (at_left a) = (\\\\<^sub>F x in at_right (- a). P (- x))\n[PROOF STEP]\nunfolding at_left_minus[of a]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eventually P (filtermap uminus (at_right (- a))) = (\\\\<^sub>F x in at_right (- a). P (- x))\n[PROOF STEP]\nby (simp add: eventually_filtermap)", "meta": {"llama_tokens": 205, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577681195338728, "lm_q2_score": 0.817574478416099, "lm_q1q2_score": 0.7012893229298641}} {"text": "[STATEMENT]\nlemma senior_states_card_le:\n \"card (senior_states p n) < card (senior_states q n) \\ senior_states p n \\ senior_states q n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (card (senior_states p n) < card (senior_states q n)) = (senior_states p n \\ senior_states q n)\n[PROOF STEP]\nby (metis card_mono not_less senior_states_cases_subseteq senior_states_finite senior_states_psubset_card_mono subset_not_subset_eq)", "meta": {"llama_tokens": 168, "file": "LTL_to_DRA_Semi_Mojmir", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577681122619883, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7012893207972595}} {"text": "[STATEMENT]\nlemma mult_ceiling_le:\n assumes \"0 \\ a\" and \"0 \\ b\"\n shows \"\\a * b\\ \\ \\a\\ * \\b\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a * b\\ \\ \\a\\ * \\b\\\n[PROOF STEP]\nby (metis assms ceiling_le_iff ceiling_mono le_of_int_ceiling mult_mono of_int_mult)", "meta": {"llama_tokens": 168, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577681122619883, "lm_q2_score": 0.817574478416099, "lm_q1q2_score": 0.701289316984557}} {"text": "[STATEMENT]\nlemma eventually_at_left_to_right:\n \"eventually P (at_left a) \\ eventually (\\x. P (- x)) (at_right (-a))\"\n for a :: real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eventually P (at_left a) = (\\\\<^sub>F x in at_right (- a). P (- x))\n[PROOF STEP]\nunfolding at_left_minus[of a]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. eventually P (filtermap uminus (at_right (- a))) = (\\\\<^sub>F x in at_right (- a). P (- x))\n[PROOF STEP]\nby (simp add: eventually_filtermap)", "meta": {"llama_tokens": 205, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577681195338728, "lm_q2_score": 0.8175744673038222, "lm_q1q2_score": 0.7012893133981073}} {"text": "[STATEMENT]\nlemma finite_From_empty:\n assumes \"\\s. finite (LV r s)\"\n shows \"finite (LV (From r n) s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (LV (From r n) s)\n[PROOF STEP]\napply(rule finite_subset)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. LV (From r n) s \\ ?B\n 2. finite ?B\n[PROOF STEP]\napply(rule LV_From_5)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (Stars_Append (LV (Star r) s) (\\i\\n. LV (From r i) []))\n[PROOF STEP]\napply(rule finite_Stars_Append)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. finite (LV (Star r) s)\n 2. finite (\\i\\n. LV (From r i) [])\n[PROOF STEP]\napply(rule LV_STAR_finite)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\s. finite (LV r s)\n 2. finite (\\i\\n. LV (From r i) [])\n[PROOF STEP]\napply(rule assms)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (\\i\\n. LV (From r i) [])\n[PROOF STEP]\napply(rule finite_UN_I)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. finite {..n}\n 2. \\i. i \\ {..n} \\ finite (LV (From r i) [])\n[PROOF STEP]\napply(auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i \\ n \\ finite (LV (From r i) [])\n[PROOF STEP]\nby (simp add: assms finite_Stars_Pow LV_From_empty)", "meta": {"llama_tokens": 599, "file": "Posix-Lexing_Extensions_LexicalVals3", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577681013541613, "lm_q2_score": 0.8175744806385543, "lm_q1q2_score": 0.7012893099729473}} {"text": "[STATEMENT]\nlemma rel_interior_interior:\n fixes S :: \"'n::euclidean_space set\"\n assumes \"affine hull S = UNIV\"\n shows \"rel_interior S = interior S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rel_interior S = interior S\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\naffine hull S = UNIV\n\ngoal (1 subgoal):\n 1. rel_interior S = interior S\n[PROOF STEP]\nunfolding rel_interior interior_def\n[PROOF STATE]\nproof (prove)\nusing this:\naffine hull S = UNIV\n\ngoal (1 subgoal):\n 1. {x \\ S. \\T. open T \\ x \\ T \\ T \\ affine hull S \\ S} = \\ {T. open T \\ T \\ S}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 269, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577681049901037, "lm_q2_score": 0.8175744739711883, "lm_q1q2_score": 0.701289307226547}} {"text": "[STATEMENT]\nlemma map_uminus_psums: \n fixes xs :: \"'a :: ab_group_add list\"\n shows \"map uminus (psums xs) = psums (map uminus xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. map uminus (psums xs) = psums (map uminus xs)\n[PROOF STEP]\nby (induction xs rule: psums.induct) (simp_all)", "meta": {"llama_tokens": 122, "file": "Descartes_Sign_Rule_Descartes_Sign_Rule", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240964782012, "lm_q2_score": 0.8104789063814617, "lm_q1q2_score": 0.7012458794885408}} {"text": "[STATEMENT]\nlemma ket_psim_dim:\n shows \"dim_vec ket_psim = 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_vec ket_psim = 4\n[PROOF STEP]\nusing ket_01_dim ket_10_dim\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec ket_01 = 4\ndim_vec ket_10 = 4\n\ngoal (1 subgoal):\n 1. dim_vec ket_psim = 4\n[PROOF STEP]\nunfolding ket_psim_def\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec ket_01 = 4\ndim_vec ket_10 = 4\n\ngoal (1 subgoal):\n 1. dim_vec (complex_of_real (1 / sqrt 2) \\\\<^sub>v (ket_01 - ket_10)) = 4\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 275, "file": "Projective_Measurements_CHSH_Inequality", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240895276223, "lm_q2_score": 0.8104789040926008, "lm_q1q2_score": 0.7012458718748656}} {"text": "[STATEMENT]\nlemma card_cart_basis: \"card (cart_basis::('a::zero_neq_one^'i) set) = CARD('i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card cart_basis = CARD('i)\n[PROOF STEP]\nunfolding cart_basis_def Setcompr_eq_image\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (range (\\i. axis i (1::'a))) = CARD('i)\n[PROOF STEP]\nby (rule card_image) (auto simp: inj_on_def axis_eq_axis)", "meta": {"llama_tokens": 178, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240756264639, "lm_q2_score": 0.8104789155369047, "lm_q1q2_score": 0.7012458705101572}} {"text": "[STATEMENT]\nlemma aHom_inv_inv:\"\\aGroup F; aGroup G; f \\ aHom F G; a \\ carrier F\\ \\\n f (-\\<^sub>a\\<^bsub>F\\<^esub> a) = -\\<^sub>a\\<^bsub>G\\<^esub> (f a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\aGroup F; aGroup G; f \\ aHom F G; a \\ carrier F\\ \\ f (-\\<^sub>a\\<^bsub>F\\<^esub> a) = -\\<^sub>a\\<^bsub>G\\<^esub> f a\n[PROOF STEP]\napply (frule aGroup.ag_l_inv1 [of \"F\" \"a\"], assumption+,\n frule sym, thin_tac \"-\\<^sub>a\\<^bsub>F\\<^esub> a \\\\<^bsub>F\\<^esub> a = \\\\<^bsub>F\\<^esub>\",\n frule aHom_0_0[of \"F\" \"G\" \"f\"], assumption+,\n frule aGroup.ag_mOp_closed[of \"F\" \"a\"], assumption+)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\aGroup F; aGroup G; f \\ aHom F G; a \\ carrier F; \\\\<^bsub>F\\<^esub> = -\\<^sub>a\\<^bsub>F\\<^esub> a \\\\<^bsub>F\\<^esub> a; f \\\\<^bsub>F\\<^esub> = \\\\<^bsub>G\\<^esub>; -\\<^sub>a\\<^bsub>F\\<^esub> a \\ carrier F\\ \\ f (-\\<^sub>a\\<^bsub>F\\<^esub> a) = -\\<^sub>a\\<^bsub>G\\<^esub> f a\n[PROOF STEP]\napply (simp add:aHom_add, thin_tac \"\\\\<^bsub>F\\<^esub> = -\\<^sub>a\\<^bsub>F\\<^esub> a \\\\<^bsub>F\\<^esub> a\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\aGroup F; aGroup G; f \\ aHom F G; a \\ carrier F; f (-\\<^sub>a\\<^bsub>F\\<^esub> a) \\\\<^bsub>G\\<^esub> f a = \\\\<^bsub>G\\<^esub>; -\\<^sub>a\\<^bsub>F\\<^esub> a \\ carrier F\\ \\ f (-\\<^sub>a\\<^bsub>F\\<^esub> a) = -\\<^sub>a\\<^bsub>G\\<^esub> f a\n[PROOF STEP]\napply (frule aHom_mem[of \"F\" \"G\" \"f\" \"-\\<^sub>a\\<^bsub>F\\<^esub> a\"], assumption+,\n frule aHom_mem[of \"F\" \"G\" \"f\" \"a\"], assumption+,\n simp only:aGroup.ag_pOp_commute[of \"G\" \"f (-\\<^sub>a\\<^bsub>F\\<^esub> a)\" \"f a\"])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\aGroup F; aGroup G; f \\ aHom F G; a \\ carrier F; f a \\\\<^bsub>G\\<^esub> f (-\\<^sub>a\\<^bsub>F\\<^esub> a) = \\\\<^bsub>G\\<^esub>; -\\<^sub>a\\<^bsub>F\\<^esub> a \\ carrier F; f (-\\<^sub>a\\<^bsub>F\\<^esub> a) \\ carrier G; f a \\ carrier G\\ \\ f (-\\<^sub>a\\<^bsub>F\\<^esub> a) = -\\<^sub>a\\<^bsub>G\\<^esub> f a\n[PROOF STEP]\napply (rule aGroup.ag_inv_unique[of \"G\"], assumption+)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1106, "file": "Group-Ring-Module_Algebra4", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240756264638, "lm_q2_score": 0.8104789063814616, "lm_q1q2_score": 0.7012458625886474}} {"text": "[STATEMENT]\nlemma split_float_interval_bounds:\n shows\n lower_split_float_interval1: \"lower (fst (split_float_interval X)) = lower X\"\n and lower_split_float_interval2: \"lower (snd (split_float_interval X)) = mid X\"\n and upper_split_float_interval1: \"upper (fst (split_float_interval X)) = mid X\"\n and upper_split_float_interval2: \"upper (snd (split_float_interval X)) = upper X\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (lower (fst (split_float_interval X)) = lower X &&& lower (snd (split_float_interval X)) = mid X) &&& upper (fst (split_float_interval X)) = mid X &&& upper (snd (split_float_interval X)) = upper X\n[PROOF STEP]\nusing mid_le[of X]\n[PROOF STATE]\nproof (prove)\nusing this:\nlower X \\ mid X\nmid X \\ upper X\n\ngoal (1 subgoal):\n 1. (lower (fst (split_float_interval X)) = lower X &&& lower (snd (split_float_interval X)) = mid X) &&& upper (fst (split_float_interval X)) = mid X &&& upper (snd (split_float_interval X)) = upper X\n[PROOF STEP]\nby (auto simp: split_float_interval_def mid_def[symmetric] min_def max_def real_of_float_eq\n lower_split_interval1 lower_split_interval2\n upper_split_interval1 upper_split_interval2)", "meta": {"llama_tokens": 424, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240791017536, "lm_q2_score": 0.8104789018037399, "lm_q1q2_score": 0.7012458614445415}} {"text": "[STATEMENT]\nlemma complex_derivative_transform_within_open:\n \"\\f holomorphic_on s; g holomorphic_on s; open s; z \\ s; \\w. w \\ s \\ f w = g w\\\n \\ deriv f z = deriv g z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f holomorphic_on s; g holomorphic_on s; open s; z \\ s; \\w. w \\ s \\ f w = g w\\ \\ deriv f z = deriv g z\n[PROOF STEP]\nunfolding holomorphic_on_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x\\s. f field_differentiable at x within s; \\x\\s. g field_differentiable at x within s; open s; z \\ s; \\w. w \\ s \\ f w = g w\\ \\ deriv f z = deriv g z\n[PROOF STEP]\nby (rule DERIV_imp_deriv)\n (metis DERIV_deriv_iff_field_differentiable has_field_derivative_transform_within_open at_within_open)", "meta": {"llama_tokens": 362, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.875787001374006, "lm_q2_score": 0.8006920068519376, "lm_q1q2_score": 0.7012356517049936}} {"text": "[STATEMENT]\nlemma isShortestPath_min_dist_def: \n \"isShortestPath u p v \\ isPath u p v \\ length p = min_dist u v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. isShortestPath u p v = (isPath u p v \\ length p = min_dist u v)\n[PROOF STEP]\nunfolding isShortestPath_def min_dist_def dist_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (isPath u p v \\ (\\p'. isPath u p' v \\ length p \\ length p')) = (isPath u p v \\ length p = (LEAST d. \\p. isPath u p v \\ length p = d))\n[PROOF STEP]\napply (rule iffI; clarsimp)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\isPath u p v; \\p'. isPath u p' v \\ length p \\ length p'\\ \\ length p = (LEAST d. \\p. isPath u p v \\ length p = d)\n 2. \\p'. \\isPath u p v; length p = (LEAST d. \\p. isPath u p v \\ length p = d); isPath u p' v\\ \\ (LEAST d. \\p. isPath u p v \\ length p = d) \\ length p'\n[PROOF STEP]\napply (rule Least_equality[symmetric]; auto; fail)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\p'. \\isPath u p v; length p = (LEAST d. \\p. isPath u p v \\ length p = d); isPath u p' v\\ \\ (LEAST d. \\p. isPath u p v \\ length p = d) \\ length p'\n[PROOF STEP]\napply (rule Least_le; auto; fail)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 604, "file": "Flow_Networks_Graph", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.875787001374006, "lm_q2_score": 0.8006920068519378, "lm_q1q2_score": 0.7012356517049936}} {"text": "[STATEMENT]\nlemma dvd_multiplicity_eq:\n \"x \\ 0 \\ y \\ 0 \\ x dvd y \\ (\\p. multiplicity p x \\ multiplicity p y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x \\ (0::'a); y \\ (0::'a)\\ \\ (x dvd y) = (\\p. multiplicity p x \\ multiplicity p y)\n[PROOF STEP]\nby (auto intro: dvd_imp_multiplicity_le multiplicity_le_imp_dvd)", "meta": {"llama_tokens": 179, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869981319863, "lm_q2_score": 0.8006919997179627, "lm_q1q2_score": 0.7012356428612918}} {"text": "[STATEMENT]\nlemma octo_inverse_mult: \"inverse (x * y) = inverse y * inverse x\" for x y::octo\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse (x * y) = inverse y * inverse x\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. inverse (x * y) = inverse y * inverse x\n[PROOF STEP]\nhave \"inverse (x * y) = (cnj y * cnj x) /\\<^sub>R (norm (x * y) ^ 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse (x * y) = Octonions.cnj y * Octonions.cnj x /\\<^sub>R (norm (x * y))\\<^sup>2\n[PROOF STEP]\nby (simp add: octo_inverse_cnj)\n[PROOF STATE]\nproof (state)\nthis:\ninverse (x * y) = Octonions.cnj y * Octonions.cnj x /\\<^sub>R (norm (x * y))\\<^sup>2\n\ngoal (1 subgoal):\n 1. inverse (x * y) = inverse y * inverse x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ninverse (x * y) = Octonions.cnj y * Octonions.cnj x /\\<^sub>R (norm (x * y))\\<^sup>2\n\ngoal (1 subgoal):\n 1. inverse (x * y) = inverse y * inverse x\n[PROOF STEP]\nhave \"\\ = (cnj y /\\<^sub>R norm y ^ 2) * (cnj x /\\<^sub>R norm x ^ 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Octonions.cnj y * Octonions.cnj x /\\<^sub>R (norm (x * y))\\<^sup>2 = Octonions.cnj y /\\<^sub>R (norm y)\\<^sup>2 * (Octonions.cnj x /\\<^sub>R (norm x)\\<^sup>2)\n[PROOF STEP]\nby (simp add: octo_mult_scaleR_left octo_mult_scaleR_right multiplicative_norm_octo\n power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\nOctonions.cnj y * Octonions.cnj x /\\<^sub>R (norm (x * y))\\<^sup>2 = Octonions.cnj y /\\<^sub>R (norm y)\\<^sup>2 * (Octonions.cnj x /\\<^sub>R (norm x)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. inverse (x * y) = inverse y * inverse x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nOctonions.cnj y * Octonions.cnj x /\\<^sub>R (norm (x * y))\\<^sup>2 = Octonions.cnj y /\\<^sub>R (norm y)\\<^sup>2 * (Octonions.cnj x /\\<^sub>R (norm x)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. inverse (x * y) = inverse y * inverse x\n[PROOF STEP]\nhave \"\\ = inverse y * inverse x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Octonions.cnj y /\\<^sub>R (norm y)\\<^sup>2 * (Octonions.cnj x /\\<^sub>R (norm x)\\<^sup>2) = inverse y * inverse x\n[PROOF STEP]\nby (simp add: octo_inverse_cnj)\n[PROOF STATE]\nproof (state)\nthis:\nOctonions.cnj y /\\<^sub>R (norm y)\\<^sup>2 * (Octonions.cnj x /\\<^sub>R (norm x)\\<^sup>2) = inverse y * inverse x\n\ngoal (1 subgoal):\n 1. inverse (x * y) = inverse y * inverse x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ninverse (x * y) = inverse y * inverse x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ninverse (x * y) = inverse y * inverse x\n\ngoal (1 subgoal):\n 1. inverse (x * y) = inverse y * inverse x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ninverse (x * y) = inverse y * inverse x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1233, "file": "Octonions_Octonions", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.875787001374006, "lm_q2_score": 0.8006919949619793, "lm_q1q2_score": 0.7012356412919226}} {"text": "[STATEMENT]\nlemma vderiv_on_mtx_vec_multI[poly_derivatives]:\n assumes \"D u = u' on T\" and \"D A = A' on T\"\n and \"g = (\\t. A t *\\<^sub>V u' t + A' t *\\<^sub>V u t )\"\n shows \"D (\\t. A t *\\<^sub>V u t) = g on T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. D (\\t. A t *\\<^sub>V u t) = g on T\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n D u = u' on T\n D A = A' on T\ng = (\\t. A t *\\<^sub>V u' t + A' t *\\<^sub>V u t)\n\ngoal (1 subgoal):\n 1. D (\\t. A t *\\<^sub>V u t) = g on T\n[PROOF STEP]\nunfolding has_vderiv_on_def has_vector_derivative_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\T. D u \\ (\\xa. xa *\\<^sub>R u' x) at x within T\n\\x\\T. D A \\ (\\xa. xa *\\<^sub>R A' x) at x within T\ng = (\\t. A t *\\<^sub>V u' t + A' t *\\<^sub>V u t)\n\ngoal (1 subgoal):\n 1. \\x\\T. D (\\t. A t *\\<^sub>V u t) \\ (\\xa. xa *\\<^sub>R g x) at x within T\n[PROOF STEP]\napply clarify\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\\\x\\T. D u \\ (\\xa. xa *\\<^sub>R u' x) at x within T; \\x\\T. D A \\ (\\xa. xa *\\<^sub>R A' x) at x within T; g = (\\t. A t *\\<^sub>V u' t + A' t *\\<^sub>V u t); x \\ T\\ \\ D (\\t. A t *\\<^sub>V u t) \\ (\\xa. xa *\\<^sub>R (A x *\\<^sub>V u' x + A' x *\\<^sub>V u x)) at x within T\n[PROOF STEP]\napply(erule_tac x=x in ballE, simp_all)+\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\g = (\\t. A t *\\<^sub>V u' t + A' t *\\<^sub>V u t); x \\ T; D u \\ (\\xa. xa *\\<^sub>R u' x) at x within T; D A \\ (\\xa. xa *\\<^sub>R A' x) at x within T\\ \\ D (\\t. A t *\\<^sub>V u t) \\ (\\xa. xa *\\<^sub>R (A x *\\<^sub>V u' x + A' x *\\<^sub>V u x)) at x within T\n[PROOF STEP]\napply(rule derivative_eq_intros)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x. \\g = (\\t. A t *\\<^sub>V u' t + A' t *\\<^sub>V u t); x \\ T; D u \\ (\\xa. xa *\\<^sub>R u' x) at x within T; D A \\ (\\xa. xa *\\<^sub>R A' x) at x within T\\ \\ D A \\ ?f'9 x at x within T\n 2. \\x. \\g = (\\t. A t *\\<^sub>V u' t + A' t *\\<^sub>V u t); x \\ T; D u \\ (\\xa. xa *\\<^sub>R u' x) at x within T; D A \\ (\\xa. xa *\\<^sub>R A' x) at x within T\\ \\ D u \\ ?g'9 x at x within T\n 3. \\x. \\g = (\\t. A t *\\<^sub>V u' t + A' t *\\<^sub>V u t); x \\ T; D u \\ (\\xa. xa *\\<^sub>R u' x) at x within T; D A \\ (\\xa. xa *\\<^sub>R A' x) at x within T\\ \\ (\\h. A x *\\<^sub>V ?g'9 x h + ?f'9 x h *\\<^sub>V u x) = (\\xa. xa *\\<^sub>R (A x *\\<^sub>V u' x + A' x *\\<^sub>V u x))\n[PROOF STEP]\nby (auto simp: fun_eq_iff mtx_vec_scaleR_commute pth_6 scaleR_mtx_vec_assoc)", "meta": {"llama_tokens": 1396, "file": "Matrices_for_ODEs_MTX_Flows", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.875787001374006, "lm_q2_score": 0.8006919949619792, "lm_q1q2_score": 0.7012356412919225}} {"text": "[STATEMENT]\nlemma not_prime_eq_prod_nat:\n assumes \"m > 1\" \"\\ prime (m::nat)\"\n shows \"\\n k. n = m * k \\ 1 < m \\ m < n \\ 1 < k \\ k < n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n k. n = m * k \\ 1 < m \\ m < n \\ 1 < k \\ k < n\n[PROOF STEP]\nusing assms irreducible_altdef[of m]\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < m\n\\ prime m\nirreducible m = (m \\ 0 \\ \\ is_unit m \\ (\\b. b dvd m \\ m dvd b \\ is_unit b))\n\ngoal (1 subgoal):\n 1. \\n k. n = m * k \\ 1 < m \\ m < n \\ 1 < k \\ k < n\n[PROOF STEP]\nby (auto simp: prime_elem_iff_irreducible irreducible_altdef)", "meta": {"llama_tokens": 312, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869981319862, "lm_q2_score": 0.800691997339971, "lm_q1q2_score": 0.7012356407786775}} {"text": "[STATEMENT]\nlemma matrix_construction_is_kronecker_product_R: \n fixes qs1 :: \"real poly list\"\n fixes subs1 subs2 :: \"(nat list*nat list) list\"\n fixes signs1 signs2 :: \"rat list list\"\n (* n1 is the number of polynomials in the \"1\" sets *)\n assumes \"\\l i. l \\ set subs1 \\ (i \\ set (fst l) \\ i \\ set (snd l)) \\ i < n1\"\n assumes \"\\j. j \\ set signs1 \\ length j = n1\"\n shows \"(matrix_A_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2)) =\n kronecker_product (matrix_A_R signs1 subs1) (matrix_A_R signs2 subs2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2)\n[PROOF STEP]\nunfolding mat_eq_iff dim_row_matrix_A_R dim_col_matrix_A_R\n length_subsets_smash_R length_signs_smash dim_kronecker\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length subs1 * length subs2 = length subs1 * length subs2 \\ length signs1 * length signs2 = length signs1 * length signs2 \\ (\\i j. i < length subs1 * length subs2 \\ j < length signs1 * length signs2 \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j))\n[PROOF STEP]\nproof safe\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nfix i j\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nassume i: \"i < length subs1 * length subs2\"\n[PROOF STATE]\nproof (state)\nthis:\ni < length subs1 * length subs2\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nassume j: \"j < length signs1 * length signs2\"\n[PROOF STATE]\nproof (state)\nthis:\nj < length signs1 * length signs2\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave ld: \"i div length subs2 < length subs1\"\n \"j div length signs2 < length signs1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i div length subs2 < length subs1 &&& j div length signs2 < length signs1\n[PROOF STEP]\nusing i j less_mult_imp_div_less\n[PROOF STATE]\nproof (prove)\nusing this:\ni < length subs1 * length subs2\nj < length signs1 * length signs2\n?m < ?i * ?n \\ ?m div ?n < ?i\n\ngoal (1 subgoal):\n 1. i div length subs2 < length subs1 &&& j div length signs2 < length signs1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ni div length subs2 < length subs1\nj div length signs2 < length signs1\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave lm: \"i mod length subs2 < length subs2\"\n \"j mod length signs2 < length signs2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i mod length subs2 < length subs2 &&& j mod length signs2 < length signs2\n[PROOF STEP]\nusing i j less_mult_imp_mod_less\n[PROOF STATE]\nproof (prove)\nusing this:\ni < length subs1 * length subs2\nj < length signs1 * length signs2\n?m < ?n * ?i \\ ?m mod ?i < ?i\n\ngoal (1 subgoal):\n 1. i mod length subs2 < length subs2 &&& j mod length signs2 < length signs2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ni mod length subs2 < length subs2\nj mod length signs2 < length signs2\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave n1: \"n1 = length (signs1 ! (j div length signs2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n1 = length (signs1 ! (j div length signs2))\n[PROOF STEP]\nusing assms(2) ld(2) nth_mem\n[PROOF STATE]\nproof (prove)\nusing this:\n?j \\ set signs1 \\ length ?j = n1\nj div length signs2 < length signs1\n?n < length ?xs \\ ?xs ! ?n \\ set ?xs\n\ngoal (1 subgoal):\n 1. n1 = length (signs1 ! (j div length signs2))\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nn1 = length (signs1 ! (j div length signs2))\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave 1: \"matrix_A_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) =\n z_R (subsets_smash_R n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = z_R (subsets_smash_R n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\n[PROOF STEP]\nunfolding mat_of_rows_list_def matrix_A_R_def mtx_row_R_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (length (map (\\index_list. map (z_R index_list) (signs_smash signs1 signs2)) (subsets_smash_R n1 subs1 subs2))) (length (signs_smash signs1 signs2)) (\\(i, y). map (\\index_list. map (z_R index_list) (signs_smash signs1 signs2)) (subsets_smash_R n1 subs1 subs2) ! i ! y) $$ (i, j) = z_R (subsets_smash_R n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\n[PROOF STEP]\nusing i j\n[PROOF STATE]\nproof (prove)\nusing this:\ni < length subs1 * length subs2\nj < length signs1 * length signs2\n\ngoal (1 subgoal):\n 1. mat (length (map (\\index_list. map (z_R index_list) (signs_smash signs1 signs2)) (subsets_smash_R n1 subs1 subs2))) (length (signs_smash signs1 signs2)) (\\(i, y). map (\\index_list. map (z_R index_list) (signs_smash signs1 signs2)) (subsets_smash_R n1 subs1 subs2) ! i ! y) $$ (i, j) = z_R (subsets_smash_R n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\n[PROOF STEP]\nby (auto simp add: length_signs_smash length_subsets_smash_R)\n[PROOF STATE]\nproof (state)\nthis:\nM_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = z_R (subsets_smash_R n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave 2: \" ... = z_R ((fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst(subs2 ! (i mod length subs2)))),\n (snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))))\n (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z_R (subsets_smash_R n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j) = z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\nunfolding signs_smash_def subsets_smash_R_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z_R (concat (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1) ! i) (concat (map (\\l1. map ((@) l1) signs2) signs1) ! j) = z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (subst length_eq_concat)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x. x \\ set (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1) \\ length x = ?n\n 2. i < ?n * length (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1)\n 3. z_R (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1 ! (i div ?n) ! (i mod ?n)) (concat (map (\\l1. map ((@) l1) signs2) signs1) ! j) = z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\nusing i\n[PROOF STATE]\nproof (prove)\nusing this:\ni < length subs1 * length subs2\n\ngoal (3 subgoals):\n 1. \\x. x \\ set (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1) \\ length x = ?n\n 2. i < ?n * length (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1)\n 3. z_R (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1 ! (i div ?n) ! (i mod ?n)) (concat (map (\\l1. map ((@) l1) signs2) signs1) ! j) = z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (auto simp add: mult.commute)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i < length subs1 * length subs2 \\ z_R (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1 ! (i div length subs2) ! (i mod length subs2)) (concat (map (\\l1. map ((@) l1) signs2) signs1) ! j) = z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (subst length_eq_concat)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\x. \\i < length subs1 * length subs2; x \\ set (map (\\l1. map ((@) l1) signs2) signs1)\\ \\ length x = ?n11\n 2. i < length subs1 * length subs2 \\ j < ?n11 * length (map (\\l1. map ((@) l1) signs2) signs1)\n 3. i < length subs1 * length subs2 \\ z_R (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1 ! (i div length subs2) ! (i mod length subs2)) (map (\\l1. map ((@) l1) signs2) signs1 ! (j div ?n11) ! (j mod ?n11)) = z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\nusing j\n[PROOF STATE]\nproof (prove)\nusing this:\nj < length signs1 * length signs2\n\ngoal (3 subgoals):\n 1. \\x. \\i < length subs1 * length subs2; x \\ set (map (\\l1. map ((@) l1) signs2) signs1)\\ \\ length x = ?n11\n 2. i < length subs1 * length subs2 \\ j < ?n11 * length (map (\\l1. map ((@) l1) signs2) signs1)\n 3. i < length subs1 * length subs2 \\ z_R (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1 ! (i div length subs2) ! (i mod length subs2)) (map (\\l1. map ((@) l1) signs2) signs1 ! (j div ?n11) ! (j mod ?n11)) = z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (auto simp add: mult.commute)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ z_R (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1 ! (i div length subs2) ! (i mod length subs2)) (map (\\l1. map ((@) l1) signs2) signs1 ! (j div length signs2) ! (j mod length signs2)) = z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\nusing ld lm\n[PROOF STATE]\nproof (prove)\nusing this:\ni div length subs2 < length subs1\nj div length signs2 < length signs1\ni mod length subs2 < length subs2\nj mod length signs2 < length signs2\n\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ z_R (map (\\l1. map (\\l2. (fst l1 @ map ((+) n1) (fst l2), snd l1 @ map ((+) n1) (snd l2))) subs2) subs1 ! (i div length subs2) ! (i mod length subs2)) (map (\\l1. map ((@) l1) signs2) signs1 ! (j div length signs2) ! (j mod length signs2)) = z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nz_R (subsets_smash_R n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j) = z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave 3: \"... =\n z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) *\n z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2)) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\nunfolding n1\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. z_R (fst (subs1 ! (i div length subs2)) @ map ((+) (length (signs1 ! (j div length signs2)))) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) (length (signs1 ! (j div length signs2)))) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2)) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (subst z_append_R)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\ia. ia \\ set (fst (subs1 ! (i div length subs2))) \\ ia < length (signs1 ! (j div length signs2))\n 2. \\ia. ia \\ set (snd (subs1 ! (i div length subs2))) \\ ia < length (signs1 ! (j div length signs2))\n 3. z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2)) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (auto simp add: n1[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\ia. ia \\ set (fst (subs1 ! (i div length subs2))) \\ ia < n1\n 2. \\ia. ia \\ set (snd (subs1 ! (i div length subs2))) \\ ia < n1\n[PROOF STEP]\nusing assms(1) ld(1) nth_mem\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?l \\ set subs1; ?i \\ set (fst ?l) \\ ?i \\ set (snd ?l)\\ \\ ?i < n1\ni div length subs2 < length subs1\n?n < length ?xs \\ ?xs ! ?n \\ set ?xs\n\ngoal (2 subgoals):\n 1. \\ia. ia \\ set (fst (subs1 ! (i div length subs2))) \\ ia < n1\n 2. \\ia. ia \\ set (snd (subs1 ! (i div length subs2))) \\ ia < n1\n[PROOF STEP]\napply blast\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ia. ia \\ set (snd (subs1 ! (i div length subs2))) \\ ia < n1\n[PROOF STEP]\nusing assms(1) ld(1) nth_mem\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?l \\ set subs1; ?i \\ set (fst ?l) \\ ?i \\ set (snd ?l)\\ \\ ?i < n1\ni div length subs2 < length subs1\n?n < length ?xs \\ ?xs ! ?n \\ set ?xs\n\ngoal (1 subgoal):\n 1. \\ia. ia \\ set (snd (subs1 ! (i div length subs2))) \\ ia < n1\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nz_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2)) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nhave 4: \"kronecker_product (matrix_A_R signs1 subs1) (matrix_A_R signs2 subs2) $$ (i,j) =\n z_R (subs1 ! (i div length subs2))\n (signs1 ! (j div length signs2)) *\n z_R (subs2 ! (i mod length subs2))\n (signs2 ! (j mod length signs2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\nunfolding kronecker_product_def matrix_A_R_def mat_of_rows_list_def mtx_row_R_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (let ra = dim_row (mat (length (map (\\index_list. map (z_R index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z_R index_list) signs1) subs1 ! i ! y)); ca = dim_col (mat (length (map (\\index_list. map (z_R index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z_R index_list) signs1) subs1 ! i ! y)); rb = dim_row (mat (length (map (\\index_list. map (z_R index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z_R index_list) signs2) subs2 ! i ! y)); cb = dim_col (mat (length (map (\\index_list. map (z_R index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z_R index_list) signs2) subs2 ! i ! y)) in mat (ra * rb) (ca * cb) (\\(i, j). mat (length (map (\\index_list. map (z_R index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z_R index_list) signs1) subs1 ! i ! y) $$ (i div rb, j div cb) * mat (length (map (\\index_list. map (z_R index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z_R index_list) signs2) subs2 ! i ! y) $$ (i mod rb, j mod cb))) $$ (i, j) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\nusing i j\n[PROOF STATE]\nproof (prove)\nusing this:\ni < length subs1 * length subs2\nj < length signs1 * length signs2\n\ngoal (1 subgoal):\n 1. (let ra = dim_row (mat (length (map (\\index_list. map (z_R index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z_R index_list) signs1) subs1 ! i ! y)); ca = dim_col (mat (length (map (\\index_list. map (z_R index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z_R index_list) signs1) subs1 ! i ! y)); rb = dim_row (mat (length (map (\\index_list. map (z_R index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z_R index_list) signs2) subs2 ! i ! y)); cb = dim_col (mat (length (map (\\index_list. map (z_R index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z_R index_list) signs2) subs2 ! i ! y)) in mat (ra * rb) (ca * cb) (\\(i, j). mat (length (map (\\index_list. map (z_R index_list) signs1) subs1)) (length signs1) (\\(i, y). map (\\index_list. map (z_R index_list) signs1) subs1 ! i ! y) $$ (i div rb, j div cb) * mat (length (map (\\index_list. map (z_R index_list) signs2) subs2)) (length signs2) (\\(i, y). map (\\index_list. map (z_R index_list) signs2) subs2 ! i ! y) $$ (i mod rb, j mod cb))) $$ (i, j) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (auto simp add: Let_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ mat (length subs1) (length signs1) (\\(i, y). map (\\index_list. map (z_R index_list) signs1) subs1 ! i ! y) $$ (i div length subs2, j div length signs2) * mat (length subs2) (length signs2) (\\(i, y). map (\\index_list. map (z_R index_list) signs2) subs2 ! i ! y) $$ (i mod length subs2, j mod length signs2) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (subst index_mat(1)[OF ld])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ (case (i div length subs2, j div length signs2) of (i, x) \\ map (\\index_list. map (z_R index_list) signs1) subs1 ! i ! x) * mat (length subs2) (length signs2) (\\(i, y). map (\\index_list. map (z_R index_list) signs2) subs2 ! i ! y) $$ (i mod length subs2, j mod length signs2) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\napply (subst index_mat(1)[OF lm])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ (case (i div length subs2, j div length signs2) of (i, x) \\ map (\\index_list. map (z_R index_list) signs1) subs1 ! i ! x) * (case (i mod length subs2, j mod length signs2) of (i, x) \\ map (\\index_list. map (z_R index_list) signs2) subs2 ! i ! x) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\nusing ld lm\n[PROOF STATE]\nproof (prove)\nusing this:\ni div length subs2 < length subs1\nj div length signs2 < length signs1\ni mod length subs2 < length subs2\nj mod length signs2 < length signs2\n\ngoal (1 subgoal):\n 1. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ (case (i div length subs2, j div length signs2) of (i, x) \\ map (\\index_list. map (z_R index_list) signs1) subs1 ! i ! x) * (case (i mod length subs2, j mod length signs2) of (i, x) \\ map (\\index_list. map (z_R index_list) signs2) subs2 ! i ! x) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nkronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n\ngoal (1 subgoal):\n 1. \\i j. \\i < length subs1 * length subs2; j < length signs1 * length signs2\\ \\ M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nshow \"matrix_A_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) =\n kronecker_product (matrix_A_R signs1 subs1) (matrix_A_R signs2 subs2) $$ (i, j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nusing 1 2 3 4\n[PROOF STATE]\nproof (prove)\nusing this:\nM_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = z_R (subsets_smash_R n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)\nz_R (subsets_smash_R n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j) = z_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))\nz_R (fst (subs1 ! (i div length subs2)) @ map ((+) n1) (fst (subs2 ! (i mod length subs2))), snd (subs1 ! (i div length subs2)) @ map ((+) n1) (snd (subs2 ! (i mod length subs2)))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2)) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\nkronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))\n\ngoal (1 subgoal):\n 1. M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nM_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 11414, "file": "BenOr_Kozen_Reif_Renegar_Proofs", "length": 48, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869851639066, "lm_q2_score": 0.8006920068519376, "lm_q1q2_score": 0.7012356387256965}} {"text": "[STATEMENT]\nlemma card_Compl:\n fixes S :: \"('a :: finite) set\"\n shows \"card (-S) = card (UNIV :: 'a set) - card S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (- S) = card UNIV - card S\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (- S) = card UNIV - card S\n[PROOF STEP]\nhave \"card S + card (-S) = card (UNIV :: 'a set)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card S + card (- S) = card UNIV\n[PROOF STEP]\nby(rule card_Un_disjoint[of S \"-S\", simplified Compl_partition, symmetric])\n (auto)\n[PROOF STATE]\nproof (state)\nthis:\ncard S + card (- S) = card UNIV\n\ngoal (1 subgoal):\n 1. card (- S) = card UNIV - card S\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard S + card (- S) = card UNIV\n\ngoal (1 subgoal):\n 1. card (- S) = card UNIV - card S\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (- S) = card UNIV - card S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 400, "file": "Consensus_Refined_Quorums", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869786798663, "lm_q2_score": 0.8006920116079209, "lm_q1q2_score": 0.7012356376992055}} {"text": "[STATEMENT]\nlemma Pythagoras:\n fixes A B C :: \"'a :: real_inner\"\n assumes \"orthogonal (A - C) (B - C)\"\n shows \"(dist B C) ^ 2 + (dist C A) ^ 2 = (dist A B) ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist B C)\\<^sup>2 + (dist C A)\\<^sup>2 = (dist A B)\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (dist B C)\\<^sup>2 + (dist C A)\\<^sup>2 = (dist A B)\\<^sup>2\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\northogonal (A - C) (B - C)\n[PROOF STEP]\nhave \"cos (angle A C B) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\northogonal (A - C) (B - C)\n\ngoal (1 subgoal):\n 1. cos (angle A C B) = 0\n[PROOF STEP]\nby (metis orthogonal_iff_angle cos_pi_half)\n[PROOF STATE]\nproof (state)\nthis:\ncos (angle A C B) = 0\n\ngoal (1 subgoal):\n 1. (dist B C)\\<^sup>2 + (dist C A)\\<^sup>2 = (dist A B)\\<^sup>2\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\ncos (angle A C B) = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncos (angle A C B) = 0\n\ngoal (1 subgoal):\n 1. (dist B C)\\<^sup>2 + (dist C A)\\<^sup>2 = (dist A B)\\<^sup>2\n[PROOF STEP]\nby (simp add: cosine_law_triangle[of A B C] dist_commute)\n[PROOF STATE]\nproof (state)\nthis:\n(dist B C)\\<^sup>2 + (dist C A)\\<^sup>2 = (dist A B)\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 627, "file": "Triangle_Triangle", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869916479466, "lm_q2_score": 0.8006919997179627, "lm_q1q2_score": 0.7012356376695731}} {"text": "[STATEMENT]\nlemma var_sum_pairwise_indep:\n fixes f :: \"'b \\ 'a \\ real\"\n assumes \"finite I\"\n assumes \"\\i. i \\ I \\ f i \\ borel_measurable M\"\n assumes \"\\i. i \\ I \\ integrable M (\\\\. f i \\^2)\"\n assumes \"\\i j. i \\ I \\ j \\ I \\ i \\ j \\ indep_var borel (f i) borel (f j)\"\n shows \"variance (\\\\. (\\i \\ I. f i \\)) = (\\i \\ I. variance (f i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. expectation (\\x. (f i x - expectation (f i))\\<^sup>2))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. expectation (\\x. (f i x - expectation (f i))\\<^sup>2))\n[PROOF STEP]\nhave \"\\i j. i \\ I \\ j \\ I - {i} \\ covariance (f i) (f j) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. \\i \\ I; j \\ I - {i}\\ \\ covariance (f i) (f j) = 0\n[PROOF STEP]\nusing covar_indep_eq_zero assms(4) square_integrable_imp_integrable[OF assms(2,3)]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\integrable M ?f; integrable M ?g; indep_var borel ?f borel ?g\\ \\ covariance ?f ?g = 0\n\\?i12 \\ I; ?j12 \\ I; ?i12 \\ ?j12\\ \\ indep_var borel (f ?i12) borel (f ?j12)\n\\?i13 \\ I; ?i13 \\ I\\ \\ integrable M (f ?i13)\n\ngoal (1 subgoal):\n 1. \\i j. \\i \\ I; j \\ I - {i}\\ \\ covariance (f i) (f j) = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\?i12 \\ I; ?j12 \\ I - {?i12}\\ \\ covariance (f ?i12) (f ?j12) = 0\n\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. expectation (\\x. (f i x - expectation (f i))\\<^sup>2))\n[PROOF STEP]\nhence a:\"(\\i \\ I. \\j \\ I - {i}. covariance (f i) (f j)) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?i12 \\ I; ?j12 \\ I - {?i12}\\ \\ covariance (f ?i12) (f ?j12) = 0\n\ngoal (1 subgoal):\n 1. (\\i\\I. \\j\\I - {i}. covariance (f i) (f j)) = 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\I. \\j\\I - {i}. covariance (f i) (f j)) = 0\n\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. expectation (\\x. (f i x - expectation (f i))\\<^sup>2))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i\\I. \\j\\I - {i}. covariance (f i) (f j)) = 0\n\ngoal (1 subgoal):\n 1. expectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. expectation (\\x. (f i x - expectation (f i))\\<^sup>2))\n[PROOF STEP]\nby (simp add: var_sum_2[OF assms(1,2,3)])\n[PROOF STATE]\nproof (state)\nthis:\nexpectation (\\x. ((\\i\\I. f i x) - expectation (\\\\. \\i\\I. f i \\))\\<^sup>2) = (\\i\\I. expectation (\\x. (f i x - expectation (f i))\\<^sup>2))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1613, "file": "Frequency_Moments_Probability_Ext", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869948899665, "lm_q2_score": 0.8006919949619792, "lm_q1q2_score": 0.7012356361002039}} {"text": "[STATEMENT]\nlemma card_Compl:\n fixes S :: \"('a :: finite) set\"\n shows \"card (-S) = card (UNIV :: 'a set) - card S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (- S) = card UNIV - card S\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (- S) = card UNIV - card S\n[PROOF STEP]\nhave \"card S + card (-S) = card (UNIV :: 'a set)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card S + card (- S) = card UNIV\n[PROOF STEP]\nby(rule card_Un_disjoint[of S \"-S\", simplified Compl_partition, symmetric])\n (auto)\n[PROOF STATE]\nproof (state)\nthis:\ncard S + card (- S) = card UNIV\n\ngoal (1 subgoal):\n 1. card (- S) = card UNIV - card S\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard S + card (- S) = card UNIV\n\ngoal (1 subgoal):\n 1. card (- S) = card UNIV - card S\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (- S) = card UNIV - card S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 400, "file": "Consensus_Refined_Voting_OneThirdRule_Proofs", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869786798663, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.7012356293687491}} {"text": "[STATEMENT]\nlemma arbitrary_intersection_of_Int:\n \"\\(arbitrary intersection_of P) S; (arbitrary intersection_of P) T\\\n \\ (arbitrary intersection_of P) (S \\ T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(arbitrary intersection_of P) S; (arbitrary intersection_of P) T\\ \\ (arbitrary intersection_of P) (S \\ T)\n[PROOF STEP]\nusing arbitrary_intersection_of_Inter [of \"{S,T}\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\Sa. Sa \\ {S, T} \\ (arbitrary intersection_of ?P) Sa) \\ (arbitrary intersection_of ?P) (\\ {S, T})\n\ngoal (1 subgoal):\n 1. \\(arbitrary intersection_of P) S; (arbitrary intersection_of P) T\\ \\ (arbitrary intersection_of P) (S \\ T)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 308, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869786798663, "lm_q2_score": 0.8006919925839875, "lm_q1q2_score": 0.7012356210382923}} {"text": "[STATEMENT]\nlemma normalized_vec_norm :\n fixes v :: \"complex vec\"\n assumes dim_v: \"v \\ carrier_vec n\" \n and neq0: \"v \\ 0\\<^sub>v n\"\n shows \"vec_normalize v \\c vec_normalize v = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inner_prod (vec_normalize v) (vec_normalize v) = 1\n[PROOF STEP]\nunfolding vec_normalize_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inner_prod (if v = 0\\<^sub>v (dim_vec v) then v else 1 / vec_norm v \\\\<^sub>v v) (if v = 0\\<^sub>v (dim_vec v) then v else 1 / vec_norm v \\\\<^sub>v v) = 1\n[PROOF STEP]\nproof (simp, rule conjI)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. v = 0\\<^sub>v (dim_vec v) \\ inner_prod v v = 1\n 2. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nshow \"v = 0\\<^sub>v (dim_vec v) \\ v \\c v = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v = 0\\<^sub>v (dim_vec v) \\ inner_prod v v = 1\n[PROOF STEP]\nusing neq0 dim_v\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ 0\\<^sub>v n\nv \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. v = 0\\<^sub>v (dim_vec v) \\ inner_prod v v = 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv = 0\\<^sub>v (dim_vec v) \\ inner_prod v v = 1\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nhave dim_a: \"(vec_normalize v) \\ carrier_vec n\" \"conjugate (vec_normalize v) \\ carrier_vec n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec_normalize v \\ carrier_vec n &&& conjugate (vec_normalize v) \\ carrier_vec n\n[PROOF STEP]\nusing dim_v vec_normalize_def\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ carrier_vec n\nvec_normalize ?v = (if ?v = 0\\<^sub>v (dim_vec ?v) then ?v else 1 / vec_norm ?v \\\\<^sub>v ?v)\n\ngoal (1 subgoal):\n 1. vec_normalize v \\ carrier_vec n &&& conjugate (vec_normalize v) \\ carrier_vec n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nvec_normalize v \\ carrier_vec n\nconjugate (vec_normalize v) \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nnote dim = dim_v dim_a\n[PROOF STATE]\nproof (state)\nthis:\nv \\ carrier_vec n\nvec_normalize v \\ carrier_vec n\nconjugate (vec_normalize v) \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nhave nvge0: \"vec_norm v > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < vec_norm v\n[PROOF STEP]\nusing vec_norm_ge_0 neq0 dim_v\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?v \\ carrier_vec ?n; ?v \\ 0\\<^sub>v ?n\\ \\ 0 < vec_norm ?v\nv \\ 0\\<^sub>v n\nv \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. 0 < vec_norm v\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < vec_norm v\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < vec_norm v\n[PROOF STEP]\nhave vvvv: \"v \\c v = (vec_norm v) * (vec_norm v)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < vec_norm v\n\ngoal (1 subgoal):\n 1. inner_prod v v = vec_norm v * vec_norm v\n[PROOF STEP]\nunfolding vec_norm_def\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < csqrt (inner_prod v v)\n\ngoal (1 subgoal):\n 1. inner_prod v v = csqrt (inner_prod v v) * csqrt (inner_prod v v)\n[PROOF STEP]\nby (metis power2_csqrt power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\ninner_prod v v = vec_norm v * vec_norm v\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nfrom nvge0\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < vec_norm v\n[PROOF STEP]\nhave \"conjugate (vec_norm v) = vec_norm v\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < vec_norm v\n\ngoal (1 subgoal):\n 1. conjugate (vec_norm v) = vec_norm v\n[PROOF STEP]\nby (simp add: complex_eq_iff complex_is_Real_iff less_complex_def)\n[PROOF STATE]\nproof (state)\nthis:\nconjugate (vec_norm v) = vec_norm v\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nconjugate (vec_norm v) = vec_norm v\n[PROOF STEP]\nhave \"v \\c (1 / vec_norm v \\\\<^sub>v v) = 1 / vec_norm v * (v \\c v)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nconjugate (vec_norm v) = vec_norm v\n\ngoal (1 subgoal):\n 1. inner_prod (1 / vec_norm v \\\\<^sub>v v) v = 1 / vec_norm v * inner_prod v v\n[PROOF STEP]\nby (subst conjugate_smult_vec, auto)\n[PROOF STATE]\nproof (state)\nthis:\ninner_prod (1 / vec_norm v \\\\<^sub>v v) v = 1 / vec_norm v * inner_prod v v\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ninner_prod (1 / vec_norm v \\\\<^sub>v v) v = 1 / vec_norm v * inner_prod v v\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nhave \"\\ = 1 / vec_norm v * vec_norm v * vec_norm v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 / vec_norm v * inner_prod v v = 1 / vec_norm v * vec_norm v * vec_norm v\n[PROOF STEP]\nusing vvvv\n[PROOF STATE]\nproof (prove)\nusing this:\ninner_prod v v = vec_norm v * vec_norm v\n\ngoal (1 subgoal):\n 1. 1 / vec_norm v * inner_prod v v = 1 / vec_norm v * vec_norm v * vec_norm v\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n1 / vec_norm v * inner_prod v v = 1 / vec_norm v * vec_norm v * vec_norm v\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 / vec_norm v * inner_prod v v = 1 / vec_norm v * vec_norm v * vec_norm v\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nhave \"\\ = vec_norm v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 / vec_norm v * vec_norm v * vec_norm v = vec_norm v\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n1 / vec_norm v * vec_norm v * vec_norm v = vec_norm v\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ninner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nhave \"v \\c (1 / vec_norm v \\\\<^sub>v v) = vec_norm v\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n\ngoal (1 subgoal):\n 1. inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ninner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ninner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nshow \"v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ v \\c (1 / vec_norm v \\\\<^sub>v v) = vec_norm v\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nusing neq0 nvge0\n[PROOF STATE]\nproof (prove)\nusing this:\ninner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\nv \\ 0\\<^sub>v n\n0 < vec_norm v\n\ngoal (1 subgoal):\n 1. v \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv \\ 0\\<^sub>v (dim_vec v) \\ vec_norm v \\ 0 \\ inner_prod (1 / vec_norm v \\\\<^sub>v v) v = vec_norm v\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3874, "file": "QHLProver_Complex_Matrix", "length": 37, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424489603726, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7012345237622571}} {"text": "[STATEMENT]\nlemma liminf_minus_ennreal:\n fixes u v::\"nat \\ ennreal\"\n shows \"(\\n. v n \\ u n) \\ liminf (\\n. u n - v n) \\ limsup u - limsup v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. v n \\ u n) \\ liminf (\\n. u n - v n) \\ limsup u - limsup v\n[PROOF STEP]\nunfolding liminf_SUP_INF limsup_INF_SUP\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. v n \\ u n) \\ (SUP n. INF n\\{n..}. u n - v n) \\ (INF n. Sup (u ` {n..})) - (INF n. Sup (v ` {n..}))\n[PROOF STEP]\nincluding ennreal.lifting\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. v n \\ u n) \\ (SUP n. INF n\\{n..}. u n - v n) \\ (INF n. Sup (u ` {n..})) - (INF n. Sup (v ` {n..}))\n[PROOF STEP]\nproof (transfer, clarsimp)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\v u. \\\\x. 0 \\ v x; \\x. 0 \\ u x; \\n. v n \\ u n\\ \\ (SUP n. INF n\\{n..}. max 0 (u n - v n)) \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\n[PROOF STEP]\nfix v u :: \"nat \\ ereal\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\v u. \\\\x. 0 \\ v x; \\x. 0 \\ u x; \\n. v n \\ u n\\ \\ (SUP n. INF n\\{n..}. max 0 (u n - v n)) \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\n[PROOF STEP]\nassume *: \"\\x. 0 \\ v x\" \"\\x. 0 \\ u x\" \"\\n. v n \\ u n\"\n[PROOF STATE]\nproof (state)\nthis:\n\\x. 0 \\ v x\n\\x. 0 \\ u x\nv ?n \\ u ?n\n\ngoal (1 subgoal):\n 1. \\v u. \\\\x. 0 \\ v x; \\x. 0 \\ u x; \\n. v n \\ u n\\ \\ (SUP n. INF n\\{n..}. max 0 (u n - v n)) \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\x. 0 \\ v x\n\\x. 0 \\ u x\nv ?n \\ u ?n\n\ngoal (1 subgoal):\n 1. \\v u. \\\\x. 0 \\ v x; \\x. 0 \\ u x; \\n. v n \\ u n\\ \\ (SUP n. INF n\\{n..}. max 0 (u n - v n)) \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\n[PROOF STEP]\nhave \"0 \\ limsup u - limsup v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ limsup u - limsup v\n[PROOF STEP]\nusing *\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. 0 \\ v x\n\\x. 0 \\ u x\nv ?n \\ u ?n\n\ngoal (1 subgoal):\n 1. 0 \\ limsup u - limsup v\n[PROOF STEP]\nby (intro ereal_diff_positive Limsup_mono always_eventually) simp\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ limsup u - limsup v\n\ngoal (1 subgoal):\n 1. \\v u. \\\\x. 0 \\ v x; \\x. 0 \\ u x; \\n. v n \\ u n\\ \\ (SUP n. INF n\\{n..}. max 0 (u n - v n)) \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ limsup u - limsup v\n\ngoal (1 subgoal):\n 1. \\v u. \\\\x. 0 \\ v x; \\x. 0 \\ u x; \\n. v n \\ u n\\ \\ (SUP n. INF n\\{n..}. max 0 (u n - v n)) \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\n[PROOF STEP]\nhave \"0 \\ Sup (u ` {x..})\" for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ Sup (u ` {x..})\n[PROOF STEP]\nusing *\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. 0 \\ v x\n\\x. 0 \\ u x\nv ?n \\ u ?n\n\ngoal (1 subgoal):\n 1. 0 \\ Sup (u ` {x..})\n[PROOF STEP]\nby (intro SUP_upper2[of x]) auto\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ Sup (u ` {?x..})\n\ngoal (1 subgoal):\n 1. \\v u. \\\\x. 0 \\ v x; \\x. 0 \\ u x; \\n. v n \\ u n\\ \\ (SUP n. INF n\\{n..}. max 0 (u n - v n)) \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ Sup (u ` {?x..})\n\ngoal (1 subgoal):\n 1. \\v u. \\\\x. 0 \\ v x; \\x. 0 \\ u x; \\n. v n \\ u n\\ \\ (SUP n. INF n\\{n..}. max 0 (u n - v n)) \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\n[PROOF STEP]\nhave \"0 \\ Sup (v ` {x..})\" for x\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ Sup (v ` {x..})\n[PROOF STEP]\nusing *\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. 0 \\ v x\n\\x. 0 \\ u x\nv ?n \\ u ?n\n\ngoal (1 subgoal):\n 1. 0 \\ Sup (v ` {x..})\n[PROOF STEP]\nby (intro SUP_upper2[of x]) auto\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ Sup (v ` {?x..})\n\ngoal (1 subgoal):\n 1. \\v u. \\\\x. 0 \\ v x; \\x. 0 \\ u x; \\n. v n \\ u n\\ \\ (SUP n. INF n\\{n..}. max 0 (u n - v n)) \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x. 0 \\ v x\n\\x. 0 \\ u x\nv ?n \\ u ?n\n0 \\ limsup u - limsup v\n0 \\ Sup (u ` {?x..})\n0 \\ Sup (v ` {?x..})\n[PROOF STEP]\nshow \"(SUP n. INF n\\{n..}. max 0 (u n - v n))\n \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. 0 \\ v x\n\\x. 0 \\ u x\nv ?n \\ u ?n\n0 \\ limsup u - limsup v\n0 \\ Sup (u ` {?x..})\n0 \\ Sup (v ` {?x..})\n\ngoal (1 subgoal):\n 1. (SUP n. INF n\\{n..}. max 0 (u n - v n)) \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\n[PROOF STEP]\nby (auto simp: * ereal_diff_positive max.absorb2 liminf_SUP_INF[symmetric] limsup_INF_SUP[symmetric] ereal_liminf_limsup_minus)\n[PROOF STATE]\nproof (state)\nthis:\n(SUP n. INF n\\{n..}. max 0 (u n - v n)) \\ max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3004, "file": null, "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424295406087, "lm_q2_score": 0.8289388167733099, "lm_q1q2_score": 0.7012345166017313}} {"text": "[STATEMENT]\nlemma hd_sorted_list_of_fset:\n \"s \\ {||} \\ hd (sorted_list_of_fset s) = (fMin s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. s \\ {||} \\ hd (sorted_list_of_fset s) = fMin s\n[PROOF STEP]\napply (insert exists_fset_of_list[of s])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\s \\ {||}; \\l. s = fset_of_list l\\ \\ hd (sorted_list_of_fset s) = fMin s\n[PROOF STEP]\napply (erule exE)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\l. \\s \\ {||}; s = fset_of_list l\\ \\ hd (sorted_list_of_fset s) = fMin s\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\l. \\fset_of_list l \\ {||}; s = fset_of_list l\\ \\ hd (sorted_list_of_fset (fset_of_list l)) = fMin (fset_of_list l)\n[PROOF STEP]\napply (simp add: sorted_list_of_fset_sort fMin_Min hd_sort_remdups)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\l. \\fset_of_list l \\ {||}; s = fset_of_list l\\ \\ hd (sort l) = Min (set l)\n[PROOF STEP]\nby (metis fset_of_list_simps(1) hd_sort_Min)", "meta": {"llama_tokens": 549, "file": "Extended_Finite_State_Machines_FSet_Utils", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8459424373085146, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7012345141035798}} {"text": "[STATEMENT]\nlemma closed_segment_subset_interval: \"is_interval T \\ a \\ T \\ b \\ T \\ closed_segment a b \\ T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\is_interval T; a \\ T; b \\ T\\ \\ {a--b} \\ T\n[PROOF STEP]\nby (rule closed_segment_subset) (auto intro!: closed_segment_subset is_interval_convex)", "meta": {"llama_tokens": 145, "file": "Ordinary_Differential_Equations_IVP_Initial_Value_Problem", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392878563335, "lm_q2_score": 0.7931059462938815, "lm_q1q2_score": 0.7011368159562665}} {"text": "[STATEMENT]\nlemma homeomorphic_maps_prod:\n \"homeomorphic_maps (prod_topology X Y) (prod_topology X' Y') (\\(x,y). (f x, g y)) (\\(x,y). (f' x, g' y)) \\\n topspace(prod_topology X Y) = {} \\\n topspace(prod_topology X' Y') = {} \\\n homeomorphic_maps X X' f f' \\\n homeomorphic_maps Y Y' g g'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. homeomorphic_maps (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) (\\(x, y). (f' x, g' y)) = (topspace (prod_topology X Y) = {} \\ topspace (prod_topology X' Y') = {} \\ homeomorphic_maps X X' f f' \\ homeomorphic_maps Y Y' g g')\n[PROOF STEP]\nunfolding homeomorphic_maps_def continuous_map_prod_top\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g) \\ (topspace (prod_topology X' Y') = {} \\ continuous_map X' X f' \\ continuous_map Y' Y g') \\ (\\x\\topspace (prod_topology X Y). (case case x of (x, y) \\ (f x, g y) of (x, y) \\ (f' x, g' y)) = x) \\ (\\y\\topspace (prod_topology X' Y'). (case case y of (x, y) \\ (f' x, g' y) of (x, y) \\ (f x, g y)) = y)) = (topspace (prod_topology X Y) = {} \\ topspace (prod_topology X' Y') = {} \\ (continuous_map X X' f \\ continuous_map X' X f' \\ (\\x\\topspace X. f' (f x) = x) \\ (\\y\\topspace X'. f (f' y) = y)) \\ continuous_map Y Y' g \\ continuous_map Y' Y g' \\ (\\x\\topspace Y. g' (g x) = x) \\ (\\y\\topspace Y'. g (g' y) = y))\n[PROOF STEP]\nby (auto simp: continuous_map_def homeomorphic_maps_def continuous_map_prod_top)", "meta": {"llama_tokens": 713, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392756357327, "lm_q2_score": 0.7931059560743422, "lm_q1q2_score": 0.7011368149103467}} {"text": "[STATEMENT]\nlemma polyhedron_imp_convex:\n fixes S :: \"'a :: euclidean_space set\"\n shows \"polyhedron S \\ convex S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. polyhedron S \\ convex S\n[PROOF STEP]\nby (metis convex_Inter convex_halfspace_le polyhedron_def)", "meta": {"llama_tokens": 104, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206765295399, "lm_q2_score": 0.7772998611746912, "lm_q1q2_score": 0.7010628166569949}} {"text": "[STATEMENT]\nlemma cis_divide: \"cis a / cis b = cis (a - b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cis a / cis b = cis (a - b)\n[PROOF STEP]\nby (simp add: divide_complex_def cis_mult)", "meta": {"llama_tokens": 90, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9019206686206199, "lm_q2_score": 0.7772998560157665, "lm_q1q2_score": 0.7010628058564516}} {"text": "[STATEMENT]\nlemma orthogonal_basis_exists:\n fixes V :: \"('a::euclidean_space) set\"\n shows \"\\B. independent B \\ B \\ span V \\ V \\ span B \\\n (card B = dim V) \\ pairwise orthogonal B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nfrom basis_exists[of V]\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\B. \\B \\ V; independent B; V \\ span B; card B = dim V\\ \\ ?thesis) \\ ?thesis\n[PROOF STEP]\nobtain B where\n B: \"B \\ V\" \"independent B\" \"V \\ span B\" \"card B = dim V\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\B. \\B \\ V; independent B; V \\ span B; card B = dim V\\ \\ ?thesis) \\ ?thesis\n\ngoal (1 subgoal):\n 1. (\\B. \\B \\ V; independent B; V \\ span B; card B = dim V\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\nB \\ V\nindependent B\nV \\ span B\ncard B = dim V\n\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nfrom B\n[PROOF STATE]\nproof (chain)\npicking this:\nB \\ V\nindependent B\nV \\ span B\ncard B = dim V\n[PROOF STEP]\nhave fB: \"finite B\" \"card B = dim V\"\n[PROOF STATE]\nproof (prove)\nusing this:\nB \\ V\nindependent B\nV \\ span B\ncard B = dim V\n\ngoal (1 subgoal):\n 1. finite B &&& card B = dim V\n[PROOF STEP]\nusing independent_bound\n[PROOF STATE]\nproof (prove)\nusing this:\nB \\ V\nindependent B\nV \\ span B\ncard B = dim V\nindependent ?S \\ finite ?S \\ card ?S \\ DIM(?'a)\n\ngoal (1 subgoal):\n 1. finite B &&& card B = dim V\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite B\ncard B = dim V\n\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nfrom basis_orthogonal[OF fB(1)]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\C. finite C \\ card C \\ card B \\ span C = span B \\ pairwise orthogonal C\n[PROOF STEP]\nobtain C where\n C: \"finite C\" \"card C \\ card B\" \"span C = span B\" \"pairwise orthogonal C\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\C. finite C \\ card C \\ card B \\ span C = span B \\ pairwise orthogonal C\n\ngoal (1 subgoal):\n 1. (\\C. \\finite C; card C \\ card B; span C = span B; pairwise orthogonal C\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfinite C\ncard C \\ card B\nspan C = span B\npairwise orthogonal C\n\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nfrom C B\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite C\ncard C \\ card B\nspan C = span B\npairwise orthogonal C\nB \\ V\nindependent B\nV \\ span B\ncard B = dim V\n[PROOF STEP]\nhave CSV: \"C \\ span V\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite C\ncard C \\ card B\nspan C = span B\npairwise orthogonal C\nB \\ V\nindependent B\nV \\ span B\ncard B = dim V\n\ngoal (1 subgoal):\n 1. C \\ span V\n[PROOF STEP]\nby (metis span_superset span_mono subset_trans)\n[PROOF STATE]\nproof (state)\nthis:\nC \\ span V\n\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nfrom span_mono[OF B(3)] C\n[PROOF STATE]\nproof (chain)\npicking this:\nspan V \\ span (span B)\nfinite C\ncard C \\ card B\nspan C = span B\npairwise orthogonal C\n[PROOF STEP]\nhave SVC: \"span V \\ span C\"\n[PROOF STATE]\nproof (prove)\nusing this:\nspan V \\ span (span B)\nfinite C\ncard C \\ card B\nspan C = span B\npairwise orthogonal C\n\ngoal (1 subgoal):\n 1. span V \\ span C\n[PROOF STEP]\nby (simp add: span_span)\n[PROOF STATE]\nproof (state)\nthis:\nspan V \\ span C\n\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nfrom card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB\n[PROOF STATE]\nproof (chain)\npicking this:\ncard C \\ dim (span V) \\ independent C\ncard C \\ card B\nspan C = span B\nfinite B\ncard B = dim V\n[PROOF STEP]\nhave iC: \"independent C\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard C \\ dim (span V) \\ independent C\ncard C \\ card B\nspan C = span B\nfinite B\ncard B = dim V\n\ngoal (1 subgoal):\n 1. independent C\n[PROOF STEP]\nby (simp)\n[PROOF STATE]\nproof (state)\nthis:\nindependent C\n\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nfrom C fB\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite C\ncard C \\ card B\nspan C = span B\npairwise orthogonal C\nfinite B\ncard B = dim V\n[PROOF STEP]\nhave \"card C \\ dim V\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite C\ncard C \\ card B\nspan C = span B\npairwise orthogonal C\nfinite B\ncard B = dim V\n\ngoal (1 subgoal):\n 1. card C \\ dim V\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard C \\ dim V\n\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncard C \\ dim V\n\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nhave \"dim V \\ card C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim V \\ card C\n[PROOF STEP]\nusing span_card_ge_dim[OF CSV SVC C(1)]\n[PROOF STATE]\nproof (prove)\nusing this:\ndim (span V) \\ card C\n\ngoal (1 subgoal):\n 1. dim V \\ card C\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim V \\ card C\n\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncard C \\ dim V\ndim V \\ card C\n[PROOF STEP]\nhave CdV: \"card C = dim V\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard C \\ dim V\ndim V \\ card C\n\ngoal (1 subgoal):\n 1. card C = dim V\n[PROOF STEP]\nusing C(1)\n[PROOF STATE]\nproof (prove)\nusing this:\ncard C \\ dim V\ndim V \\ card C\nfinite C\n\ngoal (1 subgoal):\n 1. card C = dim V\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard C = dim V\n\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nfrom C B CSV CdV iC\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite C\ncard C \\ card B\nspan C = span B\npairwise orthogonal C\nB \\ V\nindependent B\nV \\ span B\ncard B = dim V\nC \\ span V\ncard C = dim V\nindependent C\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite C\ncard C \\ card B\nspan C = span B\npairwise orthogonal C\nB \\ V\nindependent B\nV \\ span B\ncard B = dim V\nC \\ span V\ncard C = dim V\nindependent C\n\ngoal (1 subgoal):\n 1. \\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\B. independent B \\ B \\ span V \\ V \\ span B \\ card B = dim V \\ pairwise orthogonal B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3128, "file": null, "length": 35, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8479677583778257, "lm_q2_score": 0.8267118004748677, "lm_q1q2_score": 0.7010249522731699}} {"text": "[STATEMENT]\nlemma mult_ffact:\n \"a * ffact n a = ffact (Suc n) a + of_nat n * ffact n a\"\n for a :: \"'a :: comm_ring_1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a * ffact n a = ffact (Suc n) a + of_nat n * ffact n a\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. a * ffact n a = ffact (Suc n) a + of_nat n * ffact n a\n[PROOF STEP]\nhave \"ffact (Suc n) a + of_nat n * (ffact n a) = (a - of_nat n) * (ffact n a) + of_nat n * (ffact n a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ffact (Suc n) a + of_nat n * ffact n a = (a - of_nat n) * ffact n a + of_nat n * ffact n a\n[PROOF STEP]\nusing ffact_Suc_rev [of n]\n[PROOF STATE]\nproof (prove)\nusing this:\nffact (Suc n) ?m = (?m - of_nat n) * ffact n ?m\n\ngoal (1 subgoal):\n 1. ffact (Suc n) a + of_nat n * ffact n a = (a - of_nat n) * ffact n a + of_nat n * ffact n a\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nffact (Suc n) a + of_nat n * ffact n a = (a - of_nat n) * ffact n a + of_nat n * ffact n a\n\ngoal (1 subgoal):\n 1. a * ffact n a = ffact (Suc n) a + of_nat n * ffact n a\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nffact (Suc n) a + of_nat n * ffact n a = (a - of_nat n) * ffact n a + of_nat n * ffact n a\n\ngoal (1 subgoal):\n 1. a * ffact n a = ffact (Suc n) a + of_nat n * ffact n a\n[PROOF STEP]\nhave \"\\ = a * ffact n a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a - of_nat n) * ffact n a + of_nat n * ffact n a = a * ffact n a\n[PROOF STEP]\nusing ffact_add_diff_assoc\n[PROOF STATE]\nproof (prove)\nusing this:\n(?a - of_nat ?n) * ffact ?n ?a + of_nat ?n * ffact ?n ?a = ?a * ffact ?n ?a\n\ngoal (1 subgoal):\n 1. (a - of_nat n) * ffact n a + of_nat n * ffact n a = a * ffact n a\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(a - of_nat n) * ffact n a + of_nat n * ffact n a = a * ffact n a\n\ngoal (1 subgoal):\n 1. a * ffact n a = ffact (Suc n) a + of_nat n * ffact n a\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nffact (Suc n) a + of_nat n * ffact n a = a * ffact n a\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nffact (Suc n) a + of_nat n * ffact n a = a * ffact n a\n\ngoal (1 subgoal):\n 1. a * ffact n a = ffact (Suc n) a + of_nat n * ffact n a\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na * ffact n a = ffact (Suc n) a + of_nat n * ffact n a\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1128, "file": "Discrete_Summation_Factorials", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677506936878, "lm_q2_score": 0.8267118004748677, "lm_q1q2_score": 0.7010249459206024}} {"text": "[STATEMENT]\nlemma sum_choose_diagonal:\n assumes \"m \\ n\"\n shows \"(\\k\\m. (n - k) choose (m - k)) = Suc n choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nhave \"(\\k\\m. (n-k) choose (m - k)) = (\\k\\m. (n - m + k) choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n[PROOF STEP]\nusing sum.atLeastAtMost_rev [of \"\\k. (n - k) choose (m - k)\" 0 m] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k = 0..m. n - k choose (m - k)) = (\\i = 0..m. n - (m + 0 - i) choose (m - (m + 0 - i)))\nm \\ n\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n[PROOF STEP]\nby (simp add: atMost_atLeast0)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - k choose (m - k)) = (\\k\\m. n - m + k choose k)\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nhave \"\\ = Suc (n - m + m) choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k\\m. n - m + k choose k) = Suc (n - m + m) choose m\n[PROOF STEP]\nby (rule sum_choose_lower)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - m + k choose k) = Suc (n - m + m) choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - m + k choose k) = Suc (n - m + m) choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nhave \"\\ = Suc n choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc (n - m + m) choose m = Suc n choose m\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ n\n\ngoal (1 subgoal):\n 1. Suc (n - m + m) choose m = Suc n choose m\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nSuc (n - m + m) choose m = Suc n choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k\\m. n - k choose (m - k)) = Suc n choose m\n\ngoal (1 subgoal):\n 1. (\\k\\m. n - k choose (m - k)) = Suc n choose m\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\k\\m. n - k choose (m - k)) = Suc n choose m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1347, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677468516188, "lm_q2_score": 0.8267117876664789, "lm_q1q2_score": 0.701024931883218}} {"text": "[STATEMENT]\nlemma (in module) lincomb_diff:\n assumes A_fin: \"finite A\" and AinC: \"A\\carrier M\" and a_fun: \"a\\A\\carrier R\" and \n b_fun: \"b\\A\\carrier R\" \n shows \"lincomb (\\v. a v \\\\<^bsub>R\\<^esub> b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lincomb (\\v. a v \\ b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. lincomb (\\v. a v \\ b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n[PROOF STEP]\nfrom A_fin AinC\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nA \\ carrier M\n[PROOF STEP]\ninterpret mh: mod_hom R \"func_space A\" M \"(\\a. lincomb a A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nA \\ carrier M\n\ngoal (1 subgoal):\n 1. mod_hom R (func_space A) M (\\a. lincomb a A)\n[PROOF STEP]\nby (rule \n lincomb_is_mod_hom)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. lincomb (\\v. a v \\ b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n[PROOF STEP]\nlet ?a=\"restrict a A\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. lincomb (\\v. a v \\ b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n[PROOF STEP]\nlet ?b=\"restrict b A\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. lincomb (\\v. a v \\ b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n[PROOF STEP]\nfrom a_fun b_fun\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ A \\ carrier R\nb \\ A \\ carrier R\n[PROOF STEP]\nhave ainC: \"?a\\carrier (LinearCombinations.ring.func_space R A)\" \n and binC: \"?b\\carrier (LinearCombinations.ring.func_space R A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ A \\ carrier R\nb \\ A \\ carrier R\n\ngoal (1 subgoal):\n 1. restrict a A \\ carrier (func_space A) &&& restrict b A \\ carrier (func_space A)\n[PROOF STEP]\nby (unfold func_space_def, auto)\n[PROOF STATE]\nproof (state)\nthis:\nrestrict a A \\ carrier (func_space A)\nrestrict b A \\ carrier (func_space A)\n\ngoal (1 subgoal):\n 1. lincomb (\\v. a v \\ b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n[PROOF STEP]\nfrom a_fun b_fun ainC binC A_fin AinC\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ A \\ carrier R\nb \\ A \\ carrier R\nrestrict a A \\ carrier (func_space A)\nrestrict b A \\ carrier (func_space A)\nfinite A\nA \\ carrier M\n[PROOF STEP]\nhave 1: \"LinearCombinations.module.lincomb M (?a\\\\<^bsub>(func_space A)\\<^esub> ?b) A\n = LinearCombinations.module.lincomb M (\\x. a x \\\\<^bsub>R\\<^esub> b x) A\"\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ A \\ carrier R\nb \\ A \\ carrier R\nrestrict a A \\ carrier (func_space A)\nrestrict b A \\ carrier (func_space A)\nfinite A\nA \\ carrier M\n\ngoal (1 subgoal):\n 1. lincomb (restrict a A \\\\<^bsub>func_space A\\<^esub> restrict b A) A = lincomb (\\x. a x \\ b x) A\n[PROOF STEP]\napply (subst mh.M.M.minus_eq)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a \\ A \\ carrier R; b \\ A \\ carrier R; restrict a A \\ carrier (func_space A); restrict b A \\ carrier (func_space A); finite A; A \\ carrier M\\ \\ lincomb (restrict a A \\\\<^bsub>func_space A\\<^esub> \\\\<^bsub>func_space A\\<^esub> restrict b A) A = lincomb (\\x. a x \\ b x) A\n[PROOF STEP]\napply (intro lincomb_cong, auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\v. \\a \\ A \\ carrier R; b \\ A \\ carrier R; restrict a A \\ carrier (func_space A); restrict b A \\ carrier (func_space A); finite A; A \\ carrier M; v \\ A\\ \\ (restrict a A \\\\<^bsub>func_space A\\<^esub> \\\\<^bsub>func_space A\\<^esub> restrict b A) v = a v \\ b v\n[PROOF STEP]\napply (subst func_space_neg, auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\v. \\a \\ A \\ carrier R; b \\ A \\ carrier R; restrict a A \\ carrier (func_space A); restrict b A \\ carrier (func_space A); finite A; A \\ carrier M; v \\ A\\ \\ (restrict a A \\\\<^bsub>func_space A\\<^esub> (\\v. if v \\ A then \\ restrict b A v else undefined)) v = a v \\ b v\n[PROOF STEP]\napply (simp add: restrict_def func_space_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\v. \\a \\ A \\ carrier R; b \\ A \\ carrier R; (\\x. if x \\ A then a x else undefined) \\ A \\\\<^sub>E carrier R; (\\x. if x \\ A then b x else undefined) \\ A \\\\<^sub>E carrier R; finite A; A \\ carrier M; v \\ A\\ \\ a v \\ \\ b v = a v \\ b v\n[PROOF STEP]\nby (subst R.minus_eq, auto)\n[PROOF STATE]\nproof (state)\nthis:\nlincomb (restrict a A \\\\<^bsub>func_space A\\<^esub> restrict b A) A = lincomb (\\x. a x \\ b x) A\n\ngoal (1 subgoal):\n 1. lincomb (\\v. a v \\ b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n[PROOF STEP]\nfrom a_fun b_fun A_fin AinC\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ A \\ carrier R\nb \\ A \\ carrier R\nfinite A\nA \\ carrier M\n[PROOF STEP]\nhave 2: \"LinearCombinations.module.lincomb M ?a A \\\\<^bsub>M\\<^esub> \n LinearCombinations.module.lincomb M ?b A = LinearCombinations.module.lincomb M a A \\\\<^bsub>M\\<^esub> \n LinearCombinations.module.lincomb M b A\"\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ A \\ carrier R\nb \\ A \\ carrier R\nfinite A\nA \\ carrier M\n\ngoal (1 subgoal):\n 1. lincomb (restrict a A) A \\\\<^bsub>M\\<^esub> lincomb (restrict b A) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n[PROOF STEP]\nby (simp cong: lincomb_cong)\n[PROOF STATE]\nproof (state)\nthis:\nlincomb (restrict a A) A \\\\<^bsub>M\\<^esub> lincomb (restrict b A) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n\ngoal (1 subgoal):\n 1. lincomb (\\v. a v \\ b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n[PROOF STEP]\nfrom ainC binC\n[PROOF STATE]\nproof (chain)\npicking this:\nrestrict a A \\ carrier (func_space A)\nrestrict b A \\ carrier (func_space A)\n[PROOF STEP]\nhave \"LinearCombinations.module.lincomb M (?a\\\\<^bsub>(LinearCombinations.ring.func_space R A)\\<^esub> ?b) A\n = LinearCombinations.module.lincomb M ?a A \\\\<^bsub>M\\<^esub> \n LinearCombinations.module.lincomb M ?b A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nrestrict a A \\ carrier (func_space A)\nrestrict b A \\ carrier (func_space A)\n\ngoal (1 subgoal):\n 1. lincomb (restrict a A \\\\<^bsub>func_space A\\<^esub> restrict b A) A = lincomb (restrict a A) A \\\\<^bsub>M\\<^esub> lincomb (restrict b A) A\n[PROOF STEP]\nby (simp cong: lincomb_cong)\n[PROOF STATE]\nproof (state)\nthis:\nlincomb (restrict a A \\\\<^bsub>func_space A\\<^esub> restrict b A) A = lincomb (restrict a A) A \\\\<^bsub>M\\<^esub> lincomb (restrict b A) A\n\ngoal (1 subgoal):\n 1. lincomb (\\v. a v \\ b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n[PROOF STEP]\nwith 1 2\n[PROOF STATE]\nproof (chain)\npicking this:\nlincomb (restrict a A \\\\<^bsub>func_space A\\<^esub> restrict b A) A = lincomb (\\x. a x \\ b x) A\nlincomb (restrict a A) A \\\\<^bsub>M\\<^esub> lincomb (restrict b A) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\nlincomb (restrict a A \\\\<^bsub>func_space A\\<^esub> restrict b A) A = lincomb (restrict a A) A \\\\<^bsub>M\\<^esub> lincomb (restrict b A) A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlincomb (restrict a A \\\\<^bsub>func_space A\\<^esub> restrict b A) A = lincomb (\\x. a x \\ b x) A\nlincomb (restrict a A) A \\\\<^bsub>M\\<^esub> lincomb (restrict b A) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\nlincomb (restrict a A \\\\<^bsub>func_space A\\<^esub> restrict b A) A = lincomb (restrict a A) A \\\\<^bsub>M\\<^esub> lincomb (restrict b A) A\n\ngoal (1 subgoal):\n 1. lincomb (\\v. a v \\ b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlincomb (\\v. a v \\ b v) A = lincomb a A \\\\<^bsub>M\\<^esub> lincomb b A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3518, "file": "VectorSpace_LinearCombinations", "length": 26, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970842359876, "lm_q2_score": 0.7956581097540519, "lm_q1q2_score": 0.7008133431200864}} {"text": "[STATEMENT]\nlemma content_cbox_cart:\n \"cbox a b \\ {} \\ content(cbox a b) = prod (\\i. b$i - a$i) UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cbox a b \\ {} \\ content (cbox a b) = (\\i\\UNIV. b $ i - a $ i)\n[PROOF STEP]\nby (simp add: content_cbox_if Basis_vec_def cart_eq_inner_axis axis_eq_axis prod.UNION_disjoint)", "meta": {"llama_tokens": 162, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970779778824, "lm_q2_score": 0.7956581073313275, "lm_q1q2_score": 0.7008133360068456}} {"text": "[STATEMENT]\nlemma r01_binary_expression'_sum_range:\n \"\\k::nat. (snd (snd (r01_binary_expansion'' r n))) = real k/2^(Suc n) \\\n k < 2^(Suc n) \\\n ((r01_binary_expansion' r n) = 0 \\ even k) \\\n ((r01_binary_expansion' r n) = 1 \\ odd k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\k. snd (snd (r01_binary_expansion'' r n)) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\k. snd (snd (r01_binary_expansion'' r n)) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k)\n[PROOF STEP]\nhave [simp]:\"(snd (snd (r01_binary_expansion'' r n))) = (\\i=0..n. real (r01_binary_expansion' r i) * ((1/2)^(Suc i)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. snd (snd (r01_binary_expansion'' r n)) = (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i)\n[PROOF STEP]\nusing r01_binary_expression_eq_lr[of r n]\n[PROOF STATE]\nproof (prove)\nusing this:\nsnd (snd (r01_binary_expansion'' r n)) = r01_binary_expression r n\n\ngoal (1 subgoal):\n 1. snd (snd (r01_binary_expansion'' r n)) = (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i)\n[PROOF STEP]\nby(simp add: r01_binary_expression_def r01_binary_sum_def)\n[PROOF STATE]\nproof (state)\nthis:\nsnd (snd (r01_binary_expansion'' r n)) = (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i)\n\ngoal (1 subgoal):\n 1. \\k. snd (snd (r01_binary_expansion'' r n)) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k)\n[PROOF STEP]\nhave \"\\k::nat. (\\i=0..n. real (r01_binary_expansion' r i) * ((1/2)^(Suc i))) = real k/2^(Suc n) \\\n k < 2^(Suc n) \\\n ((r01_binary_expansion' r n) = 0 \\ even k) \\\n ((r01_binary_expansion' r n) = 1 \\ odd k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k)\n[PROOF STEP]\nproof(induction n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\k. (\\i = 0..0. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc 0 \\ k < 2 ^ Suc 0 \\ (r01_binary_expansion' r 0 = 0 \\ even k) \\ (r01_binary_expansion' r 0 = 1 \\ odd k)\n 2. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\k. (\\i = 0..0. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc 0 \\ k < 2 ^ Suc 0 \\ (r01_binary_expansion' r 0 = 0 \\ even k) \\ (r01_binary_expansion' r 0 = 1 \\ odd k)\n 2. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nconsider \"r01_binary_expansion' r 0 = 0\" | \"r01_binary_expansion' r 0 = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\r01_binary_expansion' r 0 = 0 \\ thesis; r01_binary_expansion' r 0 = 1 \\ thesis\\ \\ thesis\n[PROOF STEP]\nusing real01_binary_expansion'_0or1[of r 0]\n[PROOF STATE]\nproof (prove)\nusing this:\nr01_binary_expansion' r 0 \\ {0, 1}\n\ngoal (1 subgoal):\n 1. \\r01_binary_expansion' r 0 = 0 \\ thesis; r01_binary_expansion' r 0 = 1 \\ thesis\\ \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\r01_binary_expansion' r 0 = 0 \\ ?thesis; r01_binary_expansion' r 0 = 1 \\ ?thesis\\ \\ ?thesis\n\ngoal (2 subgoals):\n 1. \\k. (\\i = 0..0. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc 0 \\ k < 2 ^ Suc 0 \\ (r01_binary_expansion' r 0 = 0 \\ even k) \\ (r01_binary_expansion' r 0 = 1 \\ odd k)\n 2. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\r01_binary_expansion' r 0 = 0 \\ ?thesis; r01_binary_expansion' r 0 = 1 \\ ?thesis\\ \\ ?thesis\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n\\r01_binary_expansion' r 0 = 0 \\ ?thesis; r01_binary_expansion' r 0 = 1 \\ ?thesis\\ \\ ?thesis\n\ngoal (1 subgoal):\n 1. \\k. (\\i = 0..0. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc 0 \\ k < 2 ^ Suc 0 \\ (r01_binary_expansion' r 0 = 0 \\ even k) \\ (r01_binary_expansion' r 0 = 1 \\ odd k)\n[PROOF STEP]\nby cases auto\n[PROOF STATE]\nproof (state)\nthis:\n\\k. (\\i = 0..0. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc 0 \\ k < 2 ^ Suc 0 \\ (r01_binary_expansion' r 0 = 0 \\ even k) \\ (r01_binary_expansion' r 0 = 1 \\ odd k)\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\ncase (Suc n')\n[PROOF STATE]\nproof (state)\nthis:\n\\k. (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n' \\ k < 2 ^ Suc n' \\ (r01_binary_expansion' r n' = 0 \\ even k) \\ (r01_binary_expansion' r n' = 1 \\ odd k)\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\k. (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n' \\ k < 2 ^ Suc n' \\ (r01_binary_expansion' r n' = 0 \\ even k) \\ (r01_binary_expansion' r n' = 1 \\ odd k)\n[PROOF STEP]\nobtain k :: nat where ih:\n \"(\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2^(Suc n') \\ k < 2^(Suc n')\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\k. (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n' \\ k < 2 ^ Suc n' \\ (r01_binary_expansion' r n' = 0 \\ even k) \\ (r01_binary_expansion' r n' = 1 \\ odd k)\n\ngoal (1 subgoal):\n 1. (\\k. (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n' \\ k < 2 ^ Suc n' \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n' \\ k < 2 ^ Suc n'\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nhave \"(\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n')\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n')\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nhave \"... = real k / 2^(Suc n') + (real (r01_binary_expansion' r (Suc n')))/ 2^ Suc (Suc n')\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n') = real k / 2 ^ Suc n' + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n') = real k / 2 ^ Suc n' + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n[PROOF STEP]\nhave \"\\r ra n. (r::real) * (1 / ra) ^ n = r / ra ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\r ra n. r * (1 / ra) ^ n = r / ra ^ n\n[PROOF STEP]\nby (simp add: power_one_over)\n[PROOF STATE]\nproof (state)\nthis:\n?r * (1 / ?ra) ^ ?n = ?r / ?ra ^ ?n\n\ngoal (1 subgoal):\n 1. (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n') = real k / 2 ^ Suc n' + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n?r * (1 / ?ra) ^ ?n = ?r / ?ra ^ ?n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n?r * (1 / ?ra) ^ ?n = ?r / ?ra ^ ?n\n\ngoal (1 subgoal):\n 1. (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n') = real k / 2 ^ Suc n' + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n[PROOF STEP]\nusing ih\n[PROOF STATE]\nproof (prove)\nusing this:\n?r * (1 / ?ra) ^ ?n = ?r / ?ra ^ ?n\n(\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n' \\ k < 2 ^ Suc n'\n\ngoal (1 subgoal):\n 1. (\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n') = real k / 2 ^ Suc n' + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n') = real k / 2 ^ Suc n' + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n') = real k / 2 ^ Suc n' + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) + real (r01_binary_expansion' r (Suc n')) * (1 / 2) ^ Suc (Suc n') = real k / 2 ^ Suc n' + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nhave \"... = (2*real k) / 2^(Suc (Suc n')) + (real (r01_binary_expansion' r (Suc n')))/ 2^ Suc (Suc n')\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real k / 2 ^ Suc n' + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n') = 2 * real k / 2 ^ Suc (Suc n') + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal k / 2 ^ Suc n' + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n') = 2 * real k / 2 ^ Suc (Suc n') + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal k / 2 ^ Suc n' + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n') = 2 * real k / 2 ^ Suc (Suc n') + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nhave \"... = (2*(real k) + real (r01_binary_expansion' r (Suc n')))/2 ^ Suc (Suc n')\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * real k / 2 ^ Suc (Suc n') + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n') = (2 * real k + real (r01_binary_expansion' r (Suc n'))) / 2 ^ Suc (Suc n')\n[PROOF STEP]\nby (simp add: add_divide_distrib)\n[PROOF STATE]\nproof (state)\nthis:\n2 * real k / 2 ^ Suc (Suc n') + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n') = (2 * real k + real (r01_binary_expansion' r (Suc n'))) / 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 * real k / 2 ^ Suc (Suc n') + real (r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n') = (2 * real k + real (r01_binary_expansion' r (Suc n'))) / 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nhave \"... = (real (2*k + r01_binary_expansion' r (Suc n')))/2 ^ Suc (Suc n')\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (2 * real k + real (r01_binary_expansion' r (Suc n'))) / 2 ^ Suc (Suc n') = real (2 * k + r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(2 * real k + real (r01_binary_expansion' r (Suc n'))) / 2 ^ Suc (Suc n') = real (2 * k + r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real (2 * k + r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n[PROOF STEP]\nhave \"(\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real (2 * k + r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real (2 * k + r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. (\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real (2 * k + r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real (2 * k + r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real (2 * k + r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nhave \"2 * k + r01_binary_expansion' r (Suc n') < 2^Suc (Suc n')\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * k + r01_binary_expansion' r (Suc n') < 2 ^ Suc (Suc n')\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 * k + r01_binary_expansion' r (Suc n') < 2 ^ Suc (Suc n')\n[PROOF STEP]\nhave \"k + 1 \\ 2^Suc n'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k + 1 \\ 2 ^ Suc n'\n[PROOF STEP]\nusing ih\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i = 0..n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n' \\ k < 2 ^ Suc n'\n\ngoal (1 subgoal):\n 1. k + 1 \\ 2 ^ Suc n'\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nk + 1 \\ 2 ^ Suc n'\n\ngoal (1 subgoal):\n 1. 2 * k + r01_binary_expansion' r (Suc n') < 2 ^ Suc (Suc n')\n[PROOF STEP]\nhence \"2*k + 2 \\ 2^Suc (Suc n')\"\n[PROOF STATE]\nproof (prove)\nusing this:\nk + 1 \\ 2 ^ Suc n'\n\ngoal (1 subgoal):\n 1. 2 * k + 2 \\ 2 ^ Suc (Suc n')\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * k + 2 \\ 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. 2 * k + r01_binary_expansion' r (Suc n') < 2 ^ Suc (Suc n')\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * k + 2 \\ 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. 2 * k + r01_binary_expansion' r (Suc n') < 2 ^ Suc (Suc n')\n[PROOF STEP]\nusing real01_binary_expansion'_0or1[of r \"Suc n'\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * k + 2 \\ 2 ^ Suc (Suc n')\nr01_binary_expansion' r (Suc n') \\ {0, 1}\n\ngoal (1 subgoal):\n 1. 2 * k + r01_binary_expansion' r (Suc n') < 2 ^ Suc (Suc n')\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n2 * k + r01_binary_expansion' r (Suc n') < 2 ^ Suc (Suc n')\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n2 * k + r01_binary_expansion' r (Suc n') < 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n2 * k + r01_binary_expansion' r (Suc n') < 2 ^ Suc (Suc n')\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nhave \"r01_binary_expansion' r (Suc n') = 0 \\ even (2 * k + r01_binary_expansion' r (Suc n'))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r01_binary_expansion' r (Suc n') = 0 \\ even (2 * k + r01_binary_expansion' r (Suc n'))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nr01_binary_expansion' r (Suc n') = 0 \\ even (2 * k + r01_binary_expansion' r (Suc n'))\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nr01_binary_expansion' r (Suc n') = 0 \\ even (2 * k + r01_binary_expansion' r (Suc n'))\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nhave \"r01_binary_expansion' r (Suc n') = 1 \\ odd (2 * k + r01_binary_expansion' r (Suc n'))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r01_binary_expansion' r (Suc n') = 1 \\ odd (2 * k + r01_binary_expansion' r (Suc n'))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nr01_binary_expansion' r (Suc n') = 1 \\ odd (2 * k + r01_binary_expansion' r (Suc n'))\n\ngoal (1 subgoal):\n 1. \\n. \\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k) \\ \\k. (\\i = 0..Suc n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n) \\ k < 2 ^ Suc (Suc n) \\ (r01_binary_expansion' r (Suc n) = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n) = 1 \\ odd k)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real (2 * k + r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n2 * k + r01_binary_expansion' r (Suc n') < 2 ^ Suc (Suc n')\nr01_binary_expansion' r (Suc n') = 0 \\ even (2 * k + r01_binary_expansion' r (Suc n'))\nr01_binary_expansion' r (Suc n') = 1 \\ odd (2 * k + r01_binary_expansion' r (Suc n'))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real (2 * k + r01_binary_expansion' r (Suc n')) / 2 ^ Suc (Suc n')\n2 * k + r01_binary_expansion' r (Suc n') < 2 ^ Suc (Suc n')\nr01_binary_expansion' r (Suc n') = 0 \\ even (2 * k + r01_binary_expansion' r (Suc n'))\nr01_binary_expansion' r (Suc n') = 1 \\ odd (2 * k + r01_binary_expansion' r (Suc n'))\n\ngoal (1 subgoal):\n 1. \\k. (\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n') \\ k < 2 ^ Suc (Suc n') \\ (r01_binary_expansion' r (Suc n') = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n') = 1 \\ odd k)\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n\\k. (\\i = 0..Suc n'. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc (Suc n') \\ k < 2 ^ Suc (Suc n') \\ (r01_binary_expansion' r (Suc n') = 0 \\ even k) \\ (r01_binary_expansion' r (Suc n') = 1 \\ odd k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k)\n\ngoal (1 subgoal):\n 1. \\k. snd (snd (r01_binary_expansion'' r n)) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\k. (\\i = 0..n. real (r01_binary_expansion' r i) * (1 / 2) ^ Suc i) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k)\n\ngoal (1 subgoal):\n 1. \\k. snd (snd (r01_binary_expansion'' r n)) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\k. snd (snd (r01_binary_expansion'' r n)) = real k / 2 ^ Suc n \\ k < 2 ^ Suc n \\ (r01_binary_expansion' r n = 0 \\ even k) \\ (r01_binary_expansion' r n = 1 \\ odd k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 15328, "file": "Quasi_Borel_Spaces_StandardBorel", "length": 67, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970748488297, "lm_q2_score": 0.7956581073313276, "lm_q1q2_score": 0.7008133335171896}} {"text": "[STATEMENT]\ntheorem heron:\n fixes A B C :: \"real ^ 2\"\n defines \"a \\ dist B C\" and \"b \\ dist A C\" and \"c \\ dist A B\"\n defines \"s \\ (a + b + c) / 2\"\n shows \"content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nhave [simp]: \"(UNIV :: 2 set) = {1, 2}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. UNIV = {1, 2}\n[PROOF STEP]\nusing exhaust_2\n[PROOF STATE]\nproof (prove)\nusing this:\n?x = 1 \\ ?x = 2\n\ngoal (1 subgoal):\n 1. UNIV = {1, 2}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nUNIV = {1, 2}\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nhave dist_eq: \"dist (A :: real ^ 2) B ^ 2 = (A $ 1 - B $ 1) ^ 2 + (A $ 2 - B $ 2) ^ 2\"\n for A B\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist A B)\\<^sup>2 = (A $ 1 - B $ 1)\\<^sup>2 + (A $ 2 - B $ 2)\\<^sup>2\n[PROOF STEP]\nby (simp add: dist_vec_def dist_real_def)\n[PROOF STATE]\nproof (state)\nthis:\n(dist ?A ?B)\\<^sup>2 = (?A $ 1 - ?B $ 1)\\<^sup>2 + (?A $ 2 - ?B $ 2)\\<^sup>2\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nhave nonneg: \"s * (s - a) * (s - b) * (s - c) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ s * (s - a) * (s - b) * (s - c)\n[PROOF STEP]\nusing dist_triangle[of A B C] dist_triangle[of A C B] dist_triangle[of B C A]\n[PROOF STATE]\nproof (prove)\nusing this:\ndist A B \\ dist A C + dist C B\ndist A C \\ dist A B + dist B C\ndist B C \\ dist B A + dist A C\n\ngoal (1 subgoal):\n 1. 0 \\ s * (s - a) * (s - b) * (s - c)\n[PROOF STEP]\nby (intro mult_nonneg_nonneg) (auto simp: s_def a_def b_def c_def dist_commute)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ s * (s - a) * (s - b) * (s - c)\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nhave \"16 * content (convex hull {A, B, C}) ^ 2 =\n 4 * ((C $ 1 - A $ 1) * (B $ 2 - A $ 2) - (B $ 1 - A $ 1) * (C $ 2 - A $ 2)) ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 16 * (content (convex hull {A, B, C}))\\<^sup>2 = 4 * ((C $ 1 - A $ 1) * (B $ 2 - A $ 2) - (B $ 1 - A $ 1) * (C $ 2 - A $ 2))\\<^sup>2\n[PROOF STEP]\nby (subst content_triangle) (simp add: power_divide)\n[PROOF STATE]\nproof (state)\nthis:\n16 * (content (convex hull {A, B, C}))\\<^sup>2 = 4 * ((C $ 1 - A $ 1) * (B $ 2 - A $ 2) - (B $ 1 - A $ 1) * (C $ 2 - A $ 2))\\<^sup>2\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n16 * (content (convex hull {A, B, C}))\\<^sup>2 = 4 * ((C $ 1 - A $ 1) * (B $ 2 - A $ 2) - (B $ 1 - A $ 1) * (C $ 2 - A $ 2))\\<^sup>2\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nhave \"\\ = (2 * (dist A B ^ 2 * dist A C ^ 2 + dist A B ^ 2 * dist B C ^ 2 + \n dist A C ^ 2 * dist B C ^ 2) - (dist A B ^ 2) ^ 2 - (dist A C ^ 2) ^ 2 - (dist B C ^ 2) ^ 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 4 * ((C $ 1 - A $ 1) * (B $ 2 - A $ 2) - (B $ 1 - A $ 1) * (C $ 2 - A $ 2))\\<^sup>2 = 2 * ((dist A B)\\<^sup>2 * (dist A C)\\<^sup>2 + (dist A B)\\<^sup>2 * (dist B C)\\<^sup>2 + (dist A C)\\<^sup>2 * (dist B C)\\<^sup>2) - ((dist A B)\\<^sup>2)\\<^sup>2 - ((dist A C)\\<^sup>2)\\<^sup>2 - ((dist B C)\\<^sup>2)\\<^sup>2\n[PROOF STEP]\nunfolding dist_eq\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 4 * ((C $ 1 - A $ 1) * (B $ 2 - A $ 2) - (B $ 1 - A $ 1) * (C $ 2 - A $ 2))\\<^sup>2 = 2 * (((A $ 1 - B $ 1)\\<^sup>2 + (A $ 2 - B $ 2)\\<^sup>2) * ((A $ 1 - C $ 1)\\<^sup>2 + (A $ 2 - C $ 2)\\<^sup>2) + ((A $ 1 - B $ 1)\\<^sup>2 + (A $ 2 - B $ 2)\\<^sup>2) * ((B $ 1 - C $ 1)\\<^sup>2 + (B $ 2 - C $ 2)\\<^sup>2) + ((A $ 1 - C $ 1)\\<^sup>2 + (A $ 2 - C $ 2)\\<^sup>2) * ((B $ 1 - C $ 1)\\<^sup>2 + (B $ 2 - C $ 2)\\<^sup>2)) - ((A $ 1 - B $ 1)\\<^sup>2 + (A $ 2 - B $ 2)\\<^sup>2)\\<^sup>2 - ((A $ 1 - C $ 1)\\<^sup>2 + (A $ 2 - C $ 2)\\<^sup>2)\\<^sup>2 - ((B $ 1 - C $ 1)\\<^sup>2 + (B $ 2 - C $ 2)\\<^sup>2)\\<^sup>2\n[PROOF STEP]\nunfolding power2_eq_square\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 4 * (((C $ 1 - A $ 1) * (B $ 2 - A $ 2) - (B $ 1 - A $ 1) * (C $ 2 - A $ 2)) * ((C $ 1 - A $ 1) * (B $ 2 - A $ 2) - (B $ 1 - A $ 1) * (C $ 2 - A $ 2))) = 2 * (((A $ 1 - B $ 1) * (A $ 1 - B $ 1) + (A $ 2 - B $ 2) * (A $ 2 - B $ 2)) * ((A $ 1 - C $ 1) * (A $ 1 - C $ 1) + (A $ 2 - C $ 2) * (A $ 2 - C $ 2)) + ((A $ 1 - B $ 1) * (A $ 1 - B $ 1) + (A $ 2 - B $ 2) * (A $ 2 - B $ 2)) * ((B $ 1 - C $ 1) * (B $ 1 - C $ 1) + (B $ 2 - C $ 2) * (B $ 2 - C $ 2)) + ((A $ 1 - C $ 1) * (A $ 1 - C $ 1) + (A $ 2 - C $ 2) * (A $ 2 - C $ 2)) * ((B $ 1 - C $ 1) * (B $ 1 - C $ 1) + (B $ 2 - C $ 2) * (B $ 2 - C $ 2))) - ((A $ 1 - B $ 1) * (A $ 1 - B $ 1) + (A $ 2 - B $ 2) * (A $ 2 - B $ 2)) * ((A $ 1 - B $ 1) * (A $ 1 - B $ 1) + (A $ 2 - B $ 2) * (A $ 2 - B $ 2)) - ((A $ 1 - C $ 1) * (A $ 1 - C $ 1) + (A $ 2 - C $ 2) * (A $ 2 - C $ 2)) * ((A $ 1 - C $ 1) * (A $ 1 - C $ 1) + (A $ 2 - C $ 2) * (A $ 2 - C $ 2)) - ((B $ 1 - C $ 1) * (B $ 1 - C $ 1) + (B $ 2 - C $ 2) * (B $ 2 - C $ 2)) * ((B $ 1 - C $ 1) * (B $ 1 - C $ 1) + (B $ 2 - C $ 2) * (B $ 2 - C $ 2))\n[PROOF STEP]\nby algebra\n[PROOF STATE]\nproof (state)\nthis:\n4 * ((C $ 1 - A $ 1) * (B $ 2 - A $ 2) - (B $ 1 - A $ 1) * (C $ 2 - A $ 2))\\<^sup>2 = 2 * ((dist A B)\\<^sup>2 * (dist A C)\\<^sup>2 + (dist A B)\\<^sup>2 * (dist B C)\\<^sup>2 + (dist A C)\\<^sup>2 * (dist B C)\\<^sup>2) - ((dist A B)\\<^sup>2)\\<^sup>2 - ((dist A C)\\<^sup>2)\\<^sup>2 - ((dist B C)\\<^sup>2)\\<^sup>2\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n4 * ((C $ 1 - A $ 1) * (B $ 2 - A $ 2) - (B $ 1 - A $ 1) * (C $ 2 - A $ 2))\\<^sup>2 = 2 * ((dist A B)\\<^sup>2 * (dist A C)\\<^sup>2 + (dist A B)\\<^sup>2 * (dist B C)\\<^sup>2 + (dist A C)\\<^sup>2 * (dist B C)\\<^sup>2) - ((dist A B)\\<^sup>2)\\<^sup>2 - ((dist A C)\\<^sup>2)\\<^sup>2 - ((dist B C)\\<^sup>2)\\<^sup>2\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nhave \"\\ = (a + b + c) * ((a + b + c) - 2 * a) * ((a + b + c) - 2 * b) *\n ((a + b + c) - 2 * c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * ((dist A B)\\<^sup>2 * (dist A C)\\<^sup>2 + (dist A B)\\<^sup>2 * (dist B C)\\<^sup>2 + (dist A C)\\<^sup>2 * (dist B C)\\<^sup>2) - ((dist A B)\\<^sup>2)\\<^sup>2 - ((dist A C)\\<^sup>2)\\<^sup>2 - ((dist B C)\\<^sup>2)\\<^sup>2 = (a + b + c) * (a + b + c - 2 * a) * (a + b + c - 2 * b) * (a + b + c - 2 * c)\n[PROOF STEP]\nunfolding power2_eq_square\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * (dist A B * dist A B * (dist A C * dist A C) + dist A B * dist A B * (dist B C * dist B C) + dist A C * dist A C * (dist B C * dist B C)) - dist A B * dist A B * (dist A B * dist A B) - dist A C * dist A C * (dist A C * dist A C) - dist B C * dist B C * (dist B C * dist B C) = (a + b + c) * (a + b + c - 2 * a) * (a + b + c - 2 * b) * (a + b + c - 2 * c)\n[PROOF STEP]\nby (simp add: s_def a_def b_def c_def algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n2 * ((dist A B)\\<^sup>2 * (dist A C)\\<^sup>2 + (dist A B)\\<^sup>2 * (dist B C)\\<^sup>2 + (dist A C)\\<^sup>2 * (dist B C)\\<^sup>2) - ((dist A B)\\<^sup>2)\\<^sup>2 - ((dist A C)\\<^sup>2)\\<^sup>2 - ((dist B C)\\<^sup>2)\\<^sup>2 = (a + b + c) * (a + b + c - 2 * a) * (a + b + c - 2 * b) * (a + b + c - 2 * c)\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 * ((dist A B)\\<^sup>2 * (dist A C)\\<^sup>2 + (dist A B)\\<^sup>2 * (dist B C)\\<^sup>2 + (dist A C)\\<^sup>2 * (dist B C)\\<^sup>2) - ((dist A B)\\<^sup>2)\\<^sup>2 - ((dist A C)\\<^sup>2)\\<^sup>2 - ((dist B C)\\<^sup>2)\\<^sup>2 = (a + b + c) * (a + b + c - 2 * a) * (a + b + c - 2 * b) * (a + b + c - 2 * c)\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nhave \"\\ = 16 * s * (s - a) * (s - b) * (s - c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a + b + c) * (a + b + c - 2 * a) * (a + b + c - 2 * b) * (a + b + c - 2 * c) = 16 * s * (s - a) * (s - b) * (s - c)\n[PROOF STEP]\nby (simp add: s_def field_split_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(a + b + c) * (a + b + c - 2 * a) * (a + b + c - 2 * b) * (a + b + c - 2 * c) = 16 * s * (s - a) * (s - b) * (s - c)\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n16 * (content (convex hull {A, B, C}))\\<^sup>2 = 16 * s * (s - a) * (s - b) * (s - c)\n[PROOF STEP]\nhave \"content (convex hull {A, B, C}) ^ 2 = s * (s - a) * (s - b) * (s - c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n16 * (content (convex hull {A, B, C}))\\<^sup>2 = 16 * s * (s - a) * (s - b) * (s - c)\n\ngoal (1 subgoal):\n 1. (content (convex hull {A, B, C}))\\<^sup>2 = s * (s - a) * (s - b) * (s - c)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(content (convex hull {A, B, C}))\\<^sup>2 = s * (s - a) * (s - b) * (s - c)\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(content (convex hull {A, B, C}))\\<^sup>2 = s * (s - a) * (s - b) * (s - c)\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nhave \"\\ = sqrt (s * (s - a) * (s - b) * (s - c)) ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. s * (s - a) * (s - b) * (s - c) = (sqrt (s * (s - a) * (s - b) * (s - c)))\\<^sup>2\n[PROOF STEP]\nby (intro real_sqrt_pow2 [symmetric] nonneg)\n[PROOF STATE]\nproof (state)\nthis:\ns * (s - a) * (s - b) * (s - c) = (sqrt (s * (s - a) * (s - b) * (s - c)))\\<^sup>2\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(content (convex hull {A, B, C}))\\<^sup>2 = (sqrt (s * (s - a) * (s - b) * (s - c)))\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(content (convex hull {A, B, C}))\\<^sup>2 = (sqrt (s * (s - a) * (s - b) * (s - c)))\\<^sup>2\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nusing nonneg\n[PROOF STATE]\nproof (prove)\nusing this:\n(content (convex hull {A, B, C}))\\<^sup>2 = (sqrt (s * (s - a) * (s - b) * (s - c)))\\<^sup>2\n0 \\ s * (s - a) * (s - b) * (s - c)\n\ngoal (1 subgoal):\n 1. content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n[PROOF STEP]\nby (subst (asm) power2_eq_iff_nonneg) auto\n[PROOF STATE]\nproof (state)\nthis:\ncontent (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5924, "file": null, "length": 34, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.880797071719777, "lm_q2_score": 0.7956581024858786, "lm_q1q2_score": 0.7008133267596761}} {"text": "[STATEMENT]\nlemma im_set_un:\"\\ f\\A \\ B; A1 \\ A; A2 \\ A \\ \\\n f`(A1 \\ A2) = (f`A1) \\ (f`A2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ A \\ B; A1 \\ A; A2 \\ A\\ \\ f ` (A1 \\ A2) = f ` A1 \\ f ` A2\n[PROOF STEP]\napply (simp add:image_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ A \\ B; A1 \\ A; A2 \\ A\\ \\ {y. \\x\\A1 \\ A2. y = f x} = {y. \\x\\A1. y = f x} \\ {y. \\x\\A2. y = f x}\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 344, "file": "Group-Ring-Module_Algebra1", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970654616711, "lm_q2_score": 0.795658095217705, "lm_q1q2_score": 0.7008133153785775}} {"text": "[STATEMENT]\nlemma linear_iff: \"linear s1 s2 f \\\n (vector_space s1 \\ vector_space s2 \\ (\\x y. f (x + y) = f x + f y) \\ (\\c x. f (s1 c x) = s2 c (f x)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. linear s1 s2 f = (vector_space s1 \\ vector_space s2 \\ (\\x y. f (x + y) = f x + f y) \\ (\\c x. f (s1 c x) = s2 c (f x)))\n[PROOF STEP]\nunfolding linear_def module_hom_iff vector_space_def module_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((((\\a x y. s1 a (x + y) = s1 a x + s1 a y) \\ (\\a b x. s1 (a + b) x = s1 a x + s1 b x)) \\ (\\a b x. s1 a (s1 b x) = s1 (a * b) x) \\ (\\x. s1 (1::'a) x = x)) \\ (((\\a x y. s2 a (x + y) = s2 a x + s2 a y) \\ (\\a b x. s2 (a + b) x = s2 a x + s2 b x)) \\ (\\a b x. s2 a (s2 b x) = s2 (a * b) x) \\ (\\x. s2 (1::'a) x = x)) \\ (((\\a x y. s1 a (x + y) = s1 a x + s1 a y) \\ (\\a b x. s1 (a + b) x = s1 a x + s1 b x)) \\ (\\a b x. s1 a (s1 b x) = s1 (a * b) x) \\ (\\x. s1 (1::'a) x = x)) \\ (((\\a x y. s2 a (x + y) = s2 a x + s2 a y) \\ (\\a b x. s2 (a + b) x = s2 a x + s2 b x)) \\ (\\a b x. s2 a (s2 b x) = s2 (a * b) x) \\ (\\x. s2 (1::'a) x = x)) \\ (\\x y. f (x + y) = f x + f y) \\ (\\c x. f (s1 c x) = s2 c (f x))) = ((((\\a x y. s1 a (x + y) = s1 a x + s1 a y) \\ (\\a b x. s1 (a + b) x = s1 a x + s1 b x)) \\ (\\a b x. s1 a (s1 b x) = s1 (a * b) x) \\ (\\x. s1 (1::'a) x = x)) \\ (((\\a x y. s2 a (x + y) = s2 a x + s2 a y) \\ (\\a b x. s2 (a + b) x = s2 a x + s2 b x)) \\ (\\a b x. s2 a (s2 b x) = s2 (a * b) x) \\ (\\x. s2 (1::'a) x = x)) \\ (\\x y. f (x + y) = f x + f y) \\ (\\c x. f (s1 c x) = s2 c (f x)))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 982, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9111797075998823, "lm_q2_score": 0.7690802370707281, "lm_q1q2_score": 0.7007703055349542}} {"text": "[STATEMENT]\nlemma integral_reflect_real[simp]:\n fixes f :: \"real \\ 'a::euclidean_space\"\n shows \"integral\\<^sup>L (lebesgue_on {-b .. -a}) (\\x. f(-x)) = integral\\<^sup>L (lebesgue_on {a..b::real}) f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LINT x|lebesgue_on {- b..- a}. f (- x) = integral\\<^sup>L (lebesgue_on {a..b}) f\n[PROOF STEP]\nusing has_bochner_integral_reflect_real [of b a f]\n[PROOF STATE]\nproof (prove)\nusing this:\nhas_bochner_integral (lebesgue_on {- b..- a}) (\\x. f (- x)) ?i = has_bochner_integral (lebesgue_on {a..b}) f ?i\n\ngoal (1 subgoal):\n 1. LINT x|lebesgue_on {- b..- a}. f (- x) = integral\\<^sup>L (lebesgue_on {a..b}) f\n[PROOF STEP]\nby (metis has_bochner_integral_iff not_integrable_integral_eq)", "meta": {"llama_tokens": 328, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9111797003640646, "lm_q2_score": 0.7690802370707281, "lm_q1q2_score": 0.7007702999700298}} {"text": "[STATEMENT]\nlemma scalar_prod_left_zero[simp]: \"v \\ carrier_vec n \\ 0\\<^sub>v n \\ v = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v \\ carrier_vec n \\ 0\\<^sub>v n \\ v = (0::'a)\n[PROOF STEP]\nunfolding scalar_prod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v \\ carrier_vec n \\ (\\i = 0..v n $ i * v $ i) = (0::'a)\n[PROOF STEP]\nby (rule sum.neutral, auto)", "meta": {"llama_tokens": 205, "file": "Jordan_Normal_Form_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8499711832583696, "lm_q2_score": 0.8244619306896955, "lm_q1q2_score": 0.7007688827798004}} {"text": "[STATEMENT]\nlemma DERIV_const_ratio_const2:\n fixes f :: \"real \\ real\"\n assumes \"a \\ b\" and df: \"\\x. DERIV f x :> k\"\n shows \"(f b - f a) / (b - a) = k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f b - f a) / (b - a) = k\n[PROOF STEP]\nusing DERIV_const_ratio_const [OF assms] \\a \\ b\\\n[PROOF STATE]\nproof (prove)\nusing this:\nf b - f a = (b - a) * k\na \\ b\n\ngoal (1 subgoal):\n 1. (f b - f a) / (b - a) = k\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 229, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8499711756575749, "lm_q2_score": 0.8244619350028204, "lm_q1q2_score": 0.7007688801792663}} {"text": "[STATEMENT]\nlemma DERIV_const_ratio_const2:\n fixes f :: \"real \\ real\"\n assumes \"a \\ b\" and df: \"\\x. DERIV f x :> k\"\n shows \"(f b - f a) / (b - a) = k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f b - f a) / (b - a) = k\n[PROOF STEP]\nusing DERIV_const_ratio_const [OF assms] \\a \\ b\\\n[PROOF STATE]\nproof (prove)\nusing this:\nf b - f a = (b - a) * k\na \\ b\n\ngoal (1 subgoal):\n 1. (f b - f a) / (b - a) = k\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 229, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8499711756575749, "lm_q2_score": 0.8244619350028204, "lm_q1q2_score": 0.7007688801792663}} {"text": "[STATEMENT]\nlemma DERIV_const_ratio_const2:\n fixes f :: \"real \\ real\"\n assumes \"a \\ b\" and df: \"\\x. DERIV f x :> k\"\n shows \"(f b - f a) / (b - a) = k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f b - f a) / (b - a) = k\n[PROOF STEP]\nusing DERIV_const_ratio_const [OF assms] \\a \\ b\\\n[PROOF STATE]\nproof (prove)\nusing this:\nf b - f a = (b - a) * k\na \\ b\n\ngoal (1 subgoal):\n 1. (f b - f a) / (b - a) = k\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 229, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8499711756575749, "lm_q2_score": 0.8244619199068831, "lm_q1q2_score": 0.7007688673481548}} {"text": "[STATEMENT]\nlemma countable_iff_vcard_less1: \"countable (elts x) \\ vcard x < \\1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. countable (elts x) = (vcard x < \\1)\n[PROOF STEP]\nby (simp add: countable_iff_le_Aleph0 lt_csucc_iff one_V_def)", "meta": {"llama_tokens": 122, "file": "ZFC_in_HOL_ZFC_Cardinals", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505402422645, "lm_q2_score": 0.7745833841649233, "lm_q1q2_score": 0.7007272769474795}} {"text": "[STATEMENT]\nlemma wens_single_FF_eq_UNIV:\n \"wens_single (range (\\x::int. (x, x + 1))) (atLeast k) = UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. wens_single (range (\\x. (x, x + 1))) {k..} = UNIV\n[PROOF STEP]\napply (auto simp add: wens_single_eq_Union)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\xa. x \\ wens_single_finite (range (\\x. (x, x + 1))) {k..} xa\n[PROOF STEP]\napply (rule_tac x=\"nat (k-x)\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ wens_single_finite (range (\\x. (x, x + 1))) {k..} (nat (k - x))\n[PROOF STEP]\napply (simp add: wens_single_finite_FF)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 337, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.904650527388829, "lm_q2_score": 0.7745833893685269, "lm_q1q2_score": 0.7007272716988645}} {"text": "[STATEMENT]\ntheorem (in abelian_group) a_lagrange:\n \"\\finite(carrier G); additive_subgroup H G\\\n \\ card(a_rcosets H) * card(H) = order(G)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite (carrier G); additive_subgroup H G\\ \\ card (a_rcosets H) * card H = order G\n[PROOF STEP]\nby (rule group.lagrange [OF a_group,\n folded A_RCOSETS_def, simplified monoid_record_simps order_def, folded order_def])\n (fast intro!: additive_subgroup.a_subgroup)+", "meta": {"llama_tokens": 203, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505248181418, "lm_q2_score": 0.7745833737577158, "lm_q1q2_score": 0.7007272555853244}} {"text": "[STATEMENT]\nlemma DERIV_inverse'[derivative_intros]:\n assumes \"(f has_field_derivative D) (at x within s)\"\n and \"f x \\ 0\"\n shows \"((\\x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x)))\n (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)\n[PROOF STEP]\nhave \"(f has_derivative (\\x. x * D)) = (f has_derivative (*) D)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f has_derivative (\\x. x * D)) = (f has_derivative (*) D)\n[PROOF STEP]\nby (rule arg_cong [of \"\\x. x * D\"]) (simp add: fun_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_derivative (\\x. x * D)) = (f has_derivative (*) D)\n\ngoal (1 subgoal):\n 1. ((\\x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\n(f has_field_derivative D) (at x within s)\nf x \\ (0::'a)\n(f has_derivative (\\x. x * D)) = (f has_derivative (*) D)\n[PROOF STEP]\nhave \"(f has_derivative (\\x. x * D)) (at x within s)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_field_derivative D) (at x within s)\nf x \\ (0::'a)\n(f has_derivative (\\x. x * D)) = (f has_derivative (*) D)\n\ngoal (1 subgoal):\n 1. (f has_derivative (\\x. x * D)) (at x within s)\n[PROOF STEP]\nby (auto dest!: has_field_derivative_imp_has_derivative)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_derivative (\\x. x * D)) (at x within s)\n\ngoal (1 subgoal):\n 1. ((\\x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(f has_derivative (\\x. x * D)) (at x within s)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_derivative (\\x. x * D)) (at x within s)\n\ngoal (1 subgoal):\n 1. ((\\x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)\n[PROOF STEP]\nusing \\f x \\ 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_derivative (\\x. x * D)) (at x within s)\nf x \\ (0::'a)\n\ngoal (1 subgoal):\n 1. ((\\x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)\n[PROOF STEP]\nby (auto intro: has_derivative_imp_has_field_derivative has_derivative_inverse)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1218, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473879530492, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7006465653792309}} {"text": "[STATEMENT]\nlemma fpower_Inf_comm: \n fixes f :: \"'a::complete_lattice \\ 'a\"\n shows \"Inf_pres f \\ f (\\i. fpower f i x) = (\\i. fpower f i (f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Inf_pres f \\ f (\\i. fpower f i x) = (\\i. fpower f i (f x))\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Inf_pres f \\ f (\\i. fpower f i x) = (\\i. fpower f i (f x))\n[PROOF STEP]\nassume \"Inf_pres f\"\n[PROOF STATE]\nproof (state)\nthis:\nInf_pres f\n\ngoal (1 subgoal):\n 1. Inf_pres f \\ f (\\i. fpower f i x) = (\\i. fpower f i (f x))\n[PROOF STEP]\nhence \"f (\\i. fpower f i x) = (\\i. fpower f (Suc i) x)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nInf_pres f\n\ngoal (1 subgoal):\n 1. f (\\i. fpower f i x) = (\\i. fpower f (Suc i) x)\n[PROOF STEP]\nby (simp add: fun_eq_iff image_comp)\n[PROOF STATE]\nproof (state)\nthis:\nf (\\i. fpower f i x) = (\\i. fpower f (Suc i) x)\n\ngoal (1 subgoal):\n 1. Inf_pres f \\ f (\\i. fpower f i x) = (\\i. fpower f i (f x))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nf (\\i. fpower f i x) = (\\i. fpower f (Suc i) x)\n\ngoal (1 subgoal):\n 1. Inf_pres f \\ f (\\i. fpower f i x) = (\\i. fpower f i (f x))\n[PROOF STEP]\nhave \"... = (\\i. fpower f i (f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. fpower f (Suc i) x) = (\\i. fpower f i (f x))\n[PROOF STEP]\nby (metis comp_eq_dest_lhs fun_mon.power_Suc2)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i. fpower f (Suc i) x) = (\\i. fpower f i (f x))\n\ngoal (1 subgoal):\n 1. Inf_pres f \\ f (\\i. fpower f i x) = (\\i. fpower f i (f x))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nf (\\i. fpower f i x) = (\\i. fpower f i (f x))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nf (\\i. fpower f i x) = (\\i. fpower f i (f x))\n\ngoal (1 subgoal):\n 1. f (\\i. fpower f i x) = (\\i. fpower f i (f x))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nf (\\i. fpower f i x) = (\\i. fpower f i (f x))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1083, "file": "Transformer_Semantics_Sup_Inf_Preserving_Transformers", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473813156293, "lm_q2_score": 0.8031738010682209, "lm_q1q2_score": 0.7006465621031827}} {"text": "[STATEMENT]\nlemma LIMSEQ_n_over_Suc_n: \"(\\n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. of_nat n / of_nat (Suc n)) \\ (1::'a)\n[PROOF STEP]\nproof (rule Lim_transform_eventually)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ?f \\ (1::'a)\n 2. \\\\<^sub>F x in sequentially. ?f x = of_nat x / of_nat (Suc x)\n[PROOF STEP]\nshow \"eventually (\\n. inverse (of_nat (Suc n) / of_nat n :: 'a) =\n of_nat n / of_nat (Suc n)) sequentially\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. inverse (of_nat (Suc n) / of_nat n) = of_nat n / of_nat (Suc n)\n[PROOF STEP]\nusing eventually_gt_at_top[of \"0::nat\"]\n[PROOF STATE]\nproof (prove)\nusing this:\neventually ((<) 0) sequentially\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. inverse (of_nat (Suc n) / of_nat n) = of_nat n / of_nat (Suc n)\n[PROOF STEP]\nby eventually_elim (simp add: field_simps del: of_nat_Suc)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F n in sequentially. inverse (of_nat (Suc n) / of_nat n) = of_nat n / of_nat (Suc n)\n\ngoal (1 subgoal):\n 1. (\\n. inverse (of_nat (Suc n) / of_nat n)) \\ (1::'a)\n[PROOF STEP]\nhave \"(\\n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \\ inverse 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. inverse (of_nat (Suc n) / of_nat n)) \\ inverse (1::'a)\n[PROOF STEP]\nby (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. inverse (of_nat (Suc n) / of_nat n)) \\ inverse (1::'a)\n\ngoal (1 subgoal):\n 1. (\\n. inverse (of_nat (Suc n) / of_nat n)) \\ (1::'a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n. inverse (of_nat (Suc n) / of_nat n)) \\ inverse (1::'a)\n[PROOF STEP]\nshow \"(\\n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \\ 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. inverse (of_nat (Suc n) / of_nat n)) \\ inverse (1::'a)\n\ngoal (1 subgoal):\n 1. (\\n. inverse (of_nat (Suc n) / of_nat n)) \\ (1::'a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. inverse (of_nat (Suc n) / of_nat n)) \\ (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1069, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473614033683, "lm_q2_score": 0.8031738034238806, "lm_q1q2_score": 0.7006465481651298}} {"text": "[STATEMENT]\nlemma pos_distrib_left:\n assumes \"c > 0\"\n shows \"(\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * (u z))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * u z)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * u z)\n[PROOF STEP]\nhave \"(\\z\\outcomes. pmf q z * (c * u z)) = (\\z\\outcomes. pmf q z * c * u z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\z\\outcomes. pmf q z * (c * u z)) = (\\z\\outcomes. pmf q z * c * u z)\n[PROOF STEP]\nby (simp add: ab_semigroup_mult_class.mult_ac(1))\n[PROOF STATE]\nproof (state)\nthis:\n(\\z\\outcomes. pmf q z * (c * u z)) = (\\z\\outcomes. pmf q z * c * u z)\n\ngoal (1 subgoal):\n 1. (\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * u z)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\z\\outcomes. pmf q z * (c * u z)) = (\\z\\outcomes. pmf q z * c * u z)\n\ngoal (1 subgoal):\n 1. (\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * u z)\n[PROOF STEP]\nhave \"... = (\\z\\outcomes. c * pmf q z * u z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\z\\outcomes. pmf q z * c * u z) = (\\z\\outcomes. c * pmf q z * u z)\n[PROOF STEP]\nby (simp add: mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\n(\\z\\outcomes. pmf q z * c * u z) = (\\z\\outcomes. c * pmf q z * u z)\n\ngoal (1 subgoal):\n 1. (\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * u z)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\z\\outcomes. pmf q z * c * u z) = (\\z\\outcomes. c * pmf q z * u z)\n\ngoal (1 subgoal):\n 1. (\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * u z)\n[PROOF STEP]\nhave \"... = c * (\\z\\outcomes. pmf q z * u z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\z\\outcomes. c * pmf q z * u z) = c * (\\z\\outcomes. pmf q z * u z)\n[PROOF STEP]\nby (simp add: ab_semigroup_mult_class.mult_ac(1) sum_distrib_left)\n[PROOF STATE]\nproof (state)\nthis:\n(\\z\\outcomes. c * pmf q z * u z) = c * (\\z\\outcomes. pmf q z * u z)\n\ngoal (1 subgoal):\n 1. (\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * u z)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * u z)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * u z)\n\ngoal (1 subgoal):\n 1. (\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * u z)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\z\\outcomes. pmf q z * (c * u z)) = c * (\\z\\outcomes. pmf q z * u z)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1486, "file": "Neumann_Morgenstern_Utility_Expected_Utility", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473713594992, "lm_q2_score": 0.8031737940012418, "lm_q1q2_score": 0.7006465479418192}} {"text": "[STATEMENT]\nlemma hyperpow_not_zero: \"\\r n. r \\ (0::'a::{field} star) \\ r pow n \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\r n. r \\ 0 \\ r pow n \\ 0\n[PROOF STEP]\nby transfer (rule power_not_zero)", "meta": {"llama_tokens": 116, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473614033683, "lm_q2_score": 0.8031738010682209, "lm_q1q2_score": 0.7006465461101764}} {"text": "[STATEMENT]\nlemma hausdorff_distance_sym:\n \"hausdorff_distance A B = hausdorff_distance B A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. hausdorff_distance A B = hausdorff_distance B A\n[PROOF STEP]\nunfolding hausdorff_distance_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if A = {} \\ B = {} \\ \\ bounded A \\ \\ bounded B then 0 else max (SUP x\\A. infdist x B) (SUP x\\B. infdist x A)) = (if B = {} \\ A = {} \\ \\ bounded B \\ \\ bounded A then 0 else max (SUP x\\B. infdist x A) (SUP x\\A. infdist x B))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 252, "file": "Gromov_Hyperbolicity_Hausdorff_Distance", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.872347368040789, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7006465411664109}} {"text": "[STATEMENT]\nlemma convex_imp_path_connected:\n fixes S :: \"'a::real_normed_vector set\"\n assumes \"convex S\"\n shows \"path_connected S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. path_connected S\n[PROOF STEP]\nunfolding path_connected_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\S. \\y\\S. \\g. path g \\ path_image g \\ S \\ pathstart g = x \\ pathfinish g = y\n[PROOF STEP]\nusing assms convex_contains_segment\n[PROOF STATE]\nproof (prove)\nusing this:\nconvex S\nconvex ?S = (\\a\\?S. \\b\\?S. closed_segment a b \\ ?S)\n\ngoal (1 subgoal):\n 1. \\x\\S. \\y\\S. \\g. path g \\ path_image g \\ S \\ pathstart g = x \\ pathfinish g = y\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 321, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473680407889, "lm_q2_score": 0.8031737869342624, "lm_q1q2_score": 0.7006465391114572}} {"text": "[STATEMENT]\nlemma LIMSEQ_n_over_Suc_n: \"(\\n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. of_nat n / of_nat (Suc n)) \\ (1::'a)\n[PROOF STEP]\nproof (rule Lim_transform_eventually)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ?f \\ (1::'a)\n 2. \\\\<^sub>F x in sequentially. ?f x = of_nat x / of_nat (Suc x)\n[PROOF STEP]\nshow \"eventually (\\n. inverse (of_nat (Suc n) / of_nat n :: 'a) =\n of_nat n / of_nat (Suc n)) sequentially\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. inverse (of_nat (Suc n) / of_nat n) = of_nat n / of_nat (Suc n)\n[PROOF STEP]\nusing eventually_gt_at_top[of \"0::nat\"]\n[PROOF STATE]\nproof (prove)\nusing this:\neventually ((<) 0) sequentially\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. inverse (of_nat (Suc n) / of_nat n) = of_nat n / of_nat (Suc n)\n[PROOF STEP]\nby eventually_elim (simp add: field_simps del: of_nat_Suc)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F n in sequentially. inverse (of_nat (Suc n) / of_nat n) = of_nat n / of_nat (Suc n)\n\ngoal (1 subgoal):\n 1. (\\n. inverse (of_nat (Suc n) / of_nat n)) \\ (1::'a)\n[PROOF STEP]\nhave \"(\\n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \\ inverse 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. inverse (of_nat (Suc n) / of_nat n)) \\ inverse (1::'a)\n[PROOF STEP]\nby (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. inverse (of_nat (Suc n) / of_nat n)) \\ inverse (1::'a)\n\ngoal (1 subgoal):\n 1. (\\n. inverse (of_nat (Suc n) / of_nat n)) \\ (1::'a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n. inverse (of_nat (Suc n) / of_nat n)) \\ inverse (1::'a)\n[PROOF STEP]\nshow \"(\\n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \\ 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. inverse (of_nat (Suc n) / of_nat n)) \\ inverse (1::'a)\n\ngoal (1 subgoal):\n 1. (\\n. inverse (of_nat (Suc n) / of_nat n)) \\ (1::'a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. inverse (of_nat (Suc n) / of_nat n)) \\ (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1069, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473614033683, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7006465358354086}} {"text": "[STATEMENT]\nlemma integral_norm_bound_integral_component:\n fixes f :: \"'n::euclidean_space \\ 'a::banach\"\n fixes g :: \"'n \\ 'b::euclidean_space\"\n assumes f: \"f integrable_on S\" and g: \"g integrable_on S\"\n and fg: \"\\x. x \\ S \\ norm(f x) \\ (g x)\\k\"\n shows \"norm (integral S f) \\ (integral S g)\\k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (integral S f) \\ integral S g \\ k\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm (integral S f) \\ integral S g \\ k\n[PROOF STEP]\nhave \"norm (integral S f) \\ integral S ((\\x. x \\ k) \\ g)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (integral S f) \\ integral S ((\\x. x \\ k) \\ g)\n[PROOF STEP]\nusing integral_norm_bound_integral[OF f integrable_linear[OF g]]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\bounded_linear ?h1; \\x. x \\ S \\ norm (f x) \\ (?h1 \\ g) x\\ \\ norm (integral S f) \\ integral S (?h1 \\ g)\n\ngoal (1 subgoal):\n 1. norm (integral S f) \\ integral S ((\\x. x \\ k) \\ g)\n[PROOF STEP]\nby (simp add: bounded_linear_inner_left fg)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (integral S f) \\ integral S ((\\x. x \\ k) \\ g)\n\ngoal (1 subgoal):\n 1. norm (integral S f) \\ integral S g \\ k\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (integral S f) \\ integral S ((\\x. x \\ k) \\ g)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (integral S f) \\ integral S ((\\x. x \\ k) \\ g)\n\ngoal (1 subgoal):\n 1. norm (integral S f) \\ integral S g \\ k\n[PROOF STEP]\nunfolding o_def integral_component_eq[OF g]\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (integral S f) \\ integral S g \\ k\n\ngoal (1 subgoal):\n 1. norm (integral S f) \\ integral S g \\ k\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nnorm (integral S f) \\ integral S g \\ k\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 885, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357701094303, "lm_q2_score": 0.8080672227971211, "lm_q1q2_score": 0.7006231868180904}} {"text": "[STATEMENT]\nlemma card_bijections_eq_zero:\n assumes \"finite A\" \"finite B\"\n assumes \"card A \\ card B\"\n shows \"card {f \\ A \\\\<^sub>E B. bij_betw f A B} = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {f \\ A \\\\<^sub>E B. bij_betw f A B} = 0\n[PROOF STEP]\nusing bij_betw_set_is_empty[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n{f \\ A \\\\<^sub>E B. bij_betw f A B} = {}\n\ngoal (1 subgoal):\n 1. card {f \\ A \\\\<^sub>E B. bij_betw f A B} = 0\n[PROOF STEP]\nby (simp only: card.empty)", "meta": {"llama_tokens": 244, "file": "Twelvefold_Way_Card_Bijections", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357666736772, "lm_q2_score": 0.8080672158638528, "lm_q1q2_score": 0.7006231780303794}} {"text": "[STATEMENT]\nlemma inf_3_elms: assumes \"infinite X\" shows \"(\\x\\X. \\y\\X. \\z\\X. x \\ y \\ y \\ z \\ x \\ z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\X. \\y\\X. \\z\\X. x \\ y \\ y \\ z \\ x \\ z\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x\\X. \\y\\X. \\z\\X. x \\ y \\ y \\ z \\ x \\ z\n[PROOF STEP]\nobtain x y where 1: \"x\\X\" \"y\\X\" \"y\\x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x y. \\x \\ X; y \\ X; y \\ x\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (metis assms finite.emptyI finite.insertI rev_finite_subset singleton_iff subsetI)\n[PROOF STATE]\nproof (state)\nthis:\nx \\ X\ny \\ X\ny \\ x\n\ngoal (1 subgoal):\n 1. \\x\\X. \\y\\X. \\z\\X. x \\ y \\ y \\ z \\ x \\ z\n[PROOF STEP]\nhave \"infinite (X-{x,y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. infinite (X - {x, y})\n[PROOF STEP]\nusing infinite_remove\n[PROOF STATE]\nproof (prove)\nusing this:\ninfinite ?S \\ infinite (?S - {?a})\n\ngoal (1 subgoal):\n 1. infinite (X - {x, y})\n[PROOF STEP]\nby (simp add: assms)\n[PROOF STATE]\nproof (state)\nthis:\ninfinite (X - {x, y})\n\ngoal (1 subgoal):\n 1. \\x\\X. \\y\\X. \\z\\X. x \\ y \\ y \\ z \\ x \\ z\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ninfinite (X - {x, y})\n[PROOF STEP]\nobtain z where 2: \"z\\X\" \"x\\z\" \"z\\y\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninfinite (X - {x, y})\n\ngoal (1 subgoal):\n 1. (\\z. \\z \\ X; x \\ z; z \\ y\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing infinite_imp_nonempty\n[PROOF STATE]\nproof (prove)\nusing this:\ninfinite (X - {x, y})\ninfinite ?S \\ ?S \\ {}\n\ngoal (1 subgoal):\n 1. (\\z. \\z \\ X; x \\ z; z \\ y\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby (metis Diff_eq_empty_iff insertCI subset_eq)\n[PROOF STATE]\nproof (state)\nthis:\nz \\ X\nx \\ z\nz \\ y\n\ngoal (1 subgoal):\n 1. \\x\\X. \\y\\X. \\z\\X. x \\ y \\ y \\ z \\ x \\ z\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\X. \\y\\X. \\z\\X. x \\ y \\ y \\ z \\ x \\ z\n[PROOF STEP]\nusing 1 2\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ X\ny \\ X\ny \\ x\nz \\ X\nx \\ z\nz \\ y\n\ngoal (1 subgoal):\n 1. \\x\\X. \\y\\X. \\z\\X. x \\ y \\ y \\ z \\ x \\ z\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\X. \\y\\X. \\z\\X. x \\ y \\ y \\ z \\ x \\ z\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1424, "file": "Schutz_Spacetime_Util", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357666736772, "lm_q2_score": 0.8080672066194945, "lm_q1q2_score": 0.7006231700151901}} {"text": "[STATEMENT]\nlemma discriminant_nonneg:\n fixes a b c x :: real\n assumes \"a \\ 0\"\n and \"discrim a b c \\ 0\"\n shows \"a * x\\<^sup>2 + b * x + c = 0 \\\n x = (-b + sqrt (discrim a b c)) / (2 * a) \\\n x = (-b - sqrt (discrim a b c)) / (2 * a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a * x\\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (a * x\\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n[PROOF STEP]\nfrom complete_square and plus_or_minus_sqrt and assms\n[PROOF STATE]\nproof (chain)\npicking this:\n?a \\ 0 \\ (?a * ?x\\<^sup>2 + ?b * ?x + ?c = 0) = ((2 * ?a * ?x + ?b)\\<^sup>2 = discrim ?a ?b ?c)\n0 \\ ?y \\ (?x\\<^sup>2 = ?y) = (?x = sqrt ?y \\ ?x = - sqrt ?y)\na \\ 0\n0 \\ discrim a b c\n[PROOF STEP]\nhave \"a * x\\<^sup>2 + b * x + c = 0 \\\n (2 * a) * x + b = sqrt (discrim a b c) \\\n (2 * a) * x + b = - sqrt (discrim a b c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?a \\ 0 \\ (?a * ?x\\<^sup>2 + ?b * ?x + ?c = 0) = ((2 * ?a * ?x + ?b)\\<^sup>2 = discrim ?a ?b ?c)\n0 \\ ?y \\ (?x\\<^sup>2 = ?y) = (?x = sqrt ?y \\ ?x = - sqrt ?y)\na \\ 0\n0 \\ discrim a b c\n\ngoal (1 subgoal):\n 1. (a * x\\<^sup>2 + b * x + c = 0) = (2 * a * x + b = sqrt (discrim a b c) \\ 2 * a * x + b = - sqrt (discrim a b c))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(a * x\\<^sup>2 + b * x + c = 0) = (2 * a * x + b = sqrt (discrim a b c) \\ 2 * a * x + b = - sqrt (discrim a b c))\n\ngoal (1 subgoal):\n 1. (a * x\\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(a * x\\<^sup>2 + b * x + c = 0) = (2 * a * x + b = sqrt (discrim a b c) \\ 2 * a * x + b = - sqrt (discrim a b c))\n\ngoal (1 subgoal):\n 1. (a * x\\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n[PROOF STEP]\nhave \"\\ \\ (2 * a) * x = (-b + sqrt (discrim a b c)) \\\n (2 * a) * x = (-b - sqrt (discrim a b c))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (2 * a * x + b = sqrt (discrim a b c) \\ 2 * a * x + b = - sqrt (discrim a b c)) = (2 * a * x = - b + sqrt (discrim a b c) \\ 2 * a * x = - b - sqrt (discrim a b c))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(2 * a * x + b = sqrt (discrim a b c) \\ 2 * a * x + b = - sqrt (discrim a b c)) = (2 * a * x = - b + sqrt (discrim a b c) \\ 2 * a * x = - b - sqrt (discrim a b c))\n\ngoal (1 subgoal):\n 1. (a * x\\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(2 * a * x + b = sqrt (discrim a b c) \\ 2 * a * x + b = - sqrt (discrim a b c)) = (2 * a * x = - b + sqrt (discrim a b c) \\ 2 * a * x = - b - sqrt (discrim a b c))\n\ngoal (1 subgoal):\n 1. (a * x\\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n[PROOF STEP]\nfrom \\a \\ 0\\ and divide_non_zero [of \"2 * a\" x]\n[PROOF STATE]\nproof (chain)\npicking this:\na \\ 0\n2 * a \\ 0 \\ (2 * a * x = ?z) = (x = ?z / (2 * a))\n[PROOF STEP]\nhave \"\\ \\ x = (-b + sqrt (discrim a b c)) / (2 * a) \\\n x = (-b - sqrt (discrim a b c)) / (2 * a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ 0\n2 * a \\ 0 \\ (2 * a * x = ?z) = (x = ?z / (2 * a))\n\ngoal (1 subgoal):\n 1. (2 * a * x = - b + sqrt (discrim a b c) \\ 2 * a * x = - b - sqrt (discrim a b c)) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(2 * a * x = - b + sqrt (discrim a b c) \\ 2 * a * x = - b - sqrt (discrim a b c)) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n\ngoal (1 subgoal):\n 1. (a * x\\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(a * x\\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n[PROOF STEP]\nshow \"a * x\\<^sup>2 + b * x + c = 0 \\\n x = (-b + sqrt (discrim a b c)) / (2 * a) \\\n x = (-b - sqrt (discrim a b c)) / (2 * a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(a * x\\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n\ngoal (1 subgoal):\n 1. (a * x\\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(a * x\\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \\ x = (- b - sqrt (discrim a b c)) / (2 * a))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2614, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.851952809486198, "lm_q2_score": 0.8221891283434876, "lm_q1q2_score": 0.7004663378212425}} {"text": "[STATEMENT]\nlemma powr_half_sqrt: \"0 \\ x \\ x powr (1/2) = sqrt x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ x \\ x powr (1 / 2) = sqrt x\n[PROOF STEP]\nby (simp add: powr_def root_powr_inverse sqrt_def)", "meta": {"llama_tokens": 111, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767874818409, "lm_q2_score": 0.7981867849406659, "lm_q1q2_score": 0.7003903758601945}} {"text": "[STATEMENT]\nlemma reduce_not0':\n assumes A: \"A \\ carrier_mat m n\" and a: \"a 0\"\n shows \"reduce a b 0 A $$ (a, 0) \\ 0\" (is \"?reduce_ab $$ (a,0) \\ _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. reduce a b 0 A $$ (a, 0) \\ 0\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. reduce a b 0 A $$ (a, 0) \\ 0\n[PROOF STEP]\nhave \"?reduce_ab $$ (a,0) = (let r = gcd (A $$ (a, 0)) (A $$ (b, 0)) in if 0 dvd r then 0 else r)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. reduce a b 0 A $$ (a, 0) = (let r = gcd (A $$ (a, 0)) (A $$ (b, 0)) in if 0 dvd r then 0 else r)\n[PROOF STEP]\nby (rule reduce_gcd[OF A _ j Aaj], insert a, simp)\n[PROOF STATE]\nproof (state)\nthis:\nreduce a b 0 A $$ (a, 0) = (let r = gcd (A $$ (a, 0)) (A $$ (b, 0)) in if 0 dvd r then 0 else r)\n\ngoal (1 subgoal):\n 1. reduce a b 0 A $$ (a, 0) \\ 0\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreduce a b 0 A $$ (a, 0) = (let r = gcd (A $$ (a, 0)) (A $$ (b, 0)) in if 0 dvd r then 0 else r)\n\ngoal (1 subgoal):\n 1. reduce a b 0 A $$ (a, 0) \\ 0\n[PROOF STEP]\nhave \"... \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (let r = gcd (A $$ (a, 0)) (A $$ (b, 0)) in if 0 dvd r then 0 else r) \\ 0\n[PROOF STEP]\nunfolding Let_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if 0 dvd gcd (A $$ (a, 0)) (A $$ (b, 0)) then 0 else gcd (A $$ (a, 0)) (A $$ (b, 0))) \\ 0\n[PROOF STEP]\nby (simp add: assms(6))\n[PROOF STATE]\nproof (state)\nthis:\n(let r = gcd (A $$ (a, 0)) (A $$ (b, 0)) in if 0 dvd r then 0 else r) \\ 0\n\ngoal (1 subgoal):\n 1. reduce a b 0 A $$ (a, 0) \\ 0\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreduce a b 0 A $$ (a, 0) \\ 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreduce a b 0 A $$ (a, 0) \\ 0\n\ngoal (1 subgoal):\n 1. reduce a b 0 A $$ (a, 0) \\ 0\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nreduce a b 0 A $$ (a, 0) \\ 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1076, "file": "Modular_arithmetic_LLL_and_HNF_algorithms_HNF_Mod_Det_Soundness", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.877476793890012, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7003903746563624}} {"text": "[STATEMENT]\nlemma cos_one_sin_zero:\n fixes x :: \"'a::{real_normed_field,banach}\"\n assumes \"cos x = 1\"\n shows \"sin x = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin x = (0::'a)\n[PROOF STEP]\nusing sin_cos_squared_add [of x, unfolded assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n(sin x)\\<^sup>2 + (1::'a)\\<^sup>2 = (1::'a)\n\ngoal (1 subgoal):\n 1. sin x = (0::'a)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 191, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.7981867801399694, "lm_q1q2_score": 0.7003903716476947}} {"text": "[STATEMENT]\nlemma roots1: fixes p :: \"'a :: field poly\"\n assumes p1: \"degree p = 1\" \n shows \"{x. poly p x = 0} = {roots1 p}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x. poly p x = (0::'a)} = {roots1 p}\n[PROOF STEP]\nusing degree1_coeffs[OF p1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a b. p = [:b, a:] \\ a \\ (0::'a)\n\ngoal (1 subgoal):\n 1. {x. poly p x = (0::'a)} = {roots1 p}\n[PROOF STEP]\nunfolding roots1_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a b. p = [:b, a:] \\ a \\ (0::'a)\n\ngoal (1 subgoal):\n 1. {x. poly p x = (0::'a)} = {- coeff p 0 / coeff p 1}\n[PROOF STEP]\nby (auto simp: add_eq_0_iff nonzero_neg_divide_eq_eq2)", "meta": {"llama_tokens": 329, "file": "Polynomial_Factorization_Explicit_Roots", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.7981867753392728, "lm_q1q2_score": 0.7003903674351949}} {"text": "[STATEMENT]\ntheorem IF2set_simps:\n \"IF2set (ctor2 x) = F2set1 x \\ ((\\a \\ F2set2 x. IF1set a) \\ (\\a \\ F2set3 x. IF2set a))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. IF2set (ctor2 x) = F2set1 x \\ (\\ (IF1set ` F2set2 x) \\ \\ (IF2set ` F2set3 x))\n[PROOF STEP]\napply (rule trans[OF o_eq_dest[OF IF2set]])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. IF2col (F2map id IF1set IF2set x) = F2set1 x \\ (\\ (IF1set ` F2set2 x) \\ \\ (IF2set ` F2set3 x))\n[PROOF STEP]\napply (rule arg_cong2[of _ _ _ _ \"(\\)\"])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. F2set1 (F2map id IF1set IF2set x) = F2set1 x\n 2. \\ (F2set2 (F2map id IF1set IF2set x)) \\ \\ (F2set3 (F2map id IF1set IF2set x)) = \\ (IF1set ` F2set2 x) \\ \\ (IF2set ` F2set3 x)\n[PROOF STEP]\napply (rule trans[OF F2.set_map(1) trans[OF fun_cong[OF image_id] id_apply]])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (F2set2 (F2map id IF1set IF2set x)) \\ \\ (F2set3 (F2map id IF1set IF2set x)) = \\ (IF1set ` F2set2 x) \\ \\ (IF2set ` F2set3 x)\n[PROOF STEP]\napply (rule arg_cong2[of _ _ _ _ \"(\\)\"])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\ (F2set2 (F2map id IF1set IF2set x)) = \\ (IF1set ` F2set2 x)\n 2. \\ (F2set3 (F2map id IF1set IF2set x)) = \\ (IF2set ` F2set3 x)\n[PROOF STEP]\napply (rule arg_cong[OF F2.set_map(2)])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (F2set3 (F2map id IF1set IF2set x)) = \\ (IF2set ` F2set3 x)\n[PROOF STEP]\napply (rule arg_cong[OF F2.set_map(3)])\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 828, "file": "BNF_Operations_LFP", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767906859264, "lm_q2_score": 0.7981867705385762, "lm_q1q2_score": 0.7003903657801539}} {"text": "[STATEMENT]\nlemma ceiling_add_le: \"\\x + y\\ \\ \\x\\ + \\y\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x + y\\ \\ \\x\\ + \\y\\\n[PROOF STEP]\nby (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)", "meta": {"llama_tokens": 141, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767746654976, "lm_q2_score": 0.798186784940666, "lm_q1q2_score": 0.7003903656303587}} {"text": "[STATEMENT]\nlemma tendsto_null_power: \"\\(f \\ 0) F; 0 < n\\ \\ ((\\x. f x ^ n) \\ 0) F\"\n for f :: \"'a \\ 'b::{power,real_normed_algebra_1}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f \\ (0::'b)) F; 0 < n\\ \\ ((\\x. f x ^ n) \\ (0::'b)) F\n[PROOF STEP]\nusing tendsto_power [of f 0 F n]\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ (0::'b)) F \\ ((\\x. f x ^ n) \\ (0::'b) ^ n) F\n\ngoal (1 subgoal):\n 1. \\(f \\ (0::'b)) F; 0 < n\\ \\ ((\\x. f x ^ n) \\ (0::'b)) F\n[PROOF STEP]\nby (simp add: power_0_left)", "meta": {"llama_tokens": 330, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.7981867729389245, "lm_q1q2_score": 0.7003903653289449}} {"text": "[STATEMENT]\nlemma ceiling_add_le: \"\\x + y\\ \\ \\x\\ + \\y\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x + y\\ \\ \\x\\ + \\y\\\n[PROOF STEP]\nby (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)", "meta": {"llama_tokens": 141, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767874818409, "lm_q2_score": 0.7981867705385763, "lm_q1q2_score": 0.7003903632226952}} {"text": "[STATEMENT]\nlemma tendsto_null_power: \"\\(f \\ 0) F; 0 < n\\ \\ ((\\x. f x ^ n) \\ 0) F\"\n for f :: \"'a \\ 'b::{power,real_normed_algebra_1}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f \\ (0::'b)) F; 0 < n\\ \\ ((\\x. f x ^ n) \\ (0::'b)) F\n[PROOF STEP]\nusing tendsto_power [of f 0 F n]\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ (0::'b)) F \\ ((\\x. f x ^ n) \\ (0::'b) ^ n) F\n\ngoal (1 subgoal):\n 1. \\(f \\ (0::'b)) F; 0 < n\\ \\ ((\\x. f x ^ n) \\ (0::'b)) F\n[PROOF STEP]\nby (simp add: power_0_left)", "meta": {"llama_tokens": 330, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.7981867705385762, "lm_q1q2_score": 0.7003903632226951}} {"text": "[STATEMENT]\nlemma col_mat_mult_index :\n assumes wf1: \"mat nr n m1\"\n and wf2: \"mat n nc m2\"\n and j: \"j < nc\"\n shows \"col (mat_multI ze pl ti nr m1 m2) j = map (\\ i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0 ..< nr]\" (is \"col ?l j = ?r\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. col (mat_multI ze pl ti nr m1 m2) j = map (\\i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0.. Ball (set m1) (vec nr)\nlength m2 = nc \\ Ball (set m2) (vec n)\nlength (mat_multI ze pl ti nr m1 m2) = nc \\ Ball (set (mat_multI ze pl ti nr m1 m2)) (vec nr)\nj < nc\n\ngoal (1 subgoal):\n 1. length (mat_multI ze pl ti nr m1 m2 ! j) = length (map (\\i. scalar_prodI ze pl ti (row m1 i) (m2 ! j)) [0..i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..i. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..x + y\\ \\ \\x\\ + \\y\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x + y\\ \\ \\x\\ + \\y\\\n[PROOF STEP]\nby (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)", "meta": {"llama_tokens": 141, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767746654976, "lm_q2_score": 0.7981867801399695, "lm_q1q2_score": 0.700390361417859}} {"text": "[STATEMENT]\nlemma ceiling_add_le: \"\\x + y\\ \\ \\x\\ + \\y\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x + y\\ \\ \\x\\ + \\y\\\n[PROOF STEP]\nby (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)", "meta": {"llama_tokens": 141, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767746654976, "lm_q2_score": 0.7981867801399695, "lm_q1q2_score": 0.700390361417859}} {"text": "[STATEMENT]\nlemma isUCont_prod_metric:\n fixes f :: \\('a::metric_space \\ 'b::metric_space) \\ 'c::metric_space\\\n shows \\isUCont f \\ (\\e>0. \\d>0. \\x. \\y. \\x'. \\y'. dist x y < d \\ dist x' y' < d \\ dist (f (x, x')) (f (y, y')) < e)\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. isUCont f = (\\e>0. \\d>0. \\x y x' y'. dist x y < d \\ dist x' y' < d \\ dist (f (x, x')) (f (y, y')) < e)\n[PROOF STEP]\nusing uniformly_continuous_on_prod_metric[of UNIV UNIV]\n[PROOF STATE]\nproof (prove)\nusing this:\nuniformly_continuous_on (UNIV \\ UNIV) ?f = (\\e>0. \\d>0. \\x\\UNIV. \\y\\UNIV. \\x'\\UNIV. \\y'\\UNIV. dist x y < d \\ dist x' y' < d \\ dist (?f (x, x')) (?f (y, y')) < e)\n\ngoal (1 subgoal):\n 1. isUCont f = (\\e>0. \\d>0. \\x y x' y'. dist x y < d \\ dist x' y' < d \\ dist (f (x, x')) (f (y, y')) < e)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 469, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767778695834, "lm_q2_score": 0.7981867753392728, "lm_q1q2_score": 0.7003903597628183}} {"text": "[STATEMENT]\nlemma times_iff_sum_squares: \"f*g = (f+g)\\<^sup>2/4 - (f-g)\\<^sup>2/(4::real)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f * g = (f + g)\\<^sup>2 / 4 - (f - g)\\<^sup>2 / 4\n[PROOF STEP]\nby (simp add: power2_eq_square field_simps)", "meta": {"llama_tokens": 128, "file": "Integration_RealRandVar", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110569397306, "lm_q2_score": 0.7853085859124002, "lm_q1q2_score": 0.7003468800263828}} {"text": "[STATEMENT]\nlemma coplanar_linear_image:\n fixes f :: \"'a::euclidean_space \\ 'b::real_normed_vector\"\n assumes \"coplanar S\" \"linear f\" shows \"coplanar(f ` S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coplanar (f ` S)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. coplanar (f ` S)\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. coplanar (f ` S)\n[PROOF STEP]\nfix u v w\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. coplanar (f ` S)\n[PROOF STEP]\nassume \"S \\ affine hull {u, v, w}\"\n[PROOF STATE]\nproof (state)\nthis:\nS \\ affine hull {u, v, w}\n\ngoal (1 subgoal):\n 1. coplanar (f ` S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nS \\ affine hull {u, v, w}\n[PROOF STEP]\nhave \"f ` S \\ f ` (affine hull {u, v, w})\"\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ affine hull {u, v, w}\n\ngoal (1 subgoal):\n 1. f ` S \\ f ` (affine hull {u, v, w})\n[PROOF STEP]\nby (simp add: image_mono)\n[PROOF STATE]\nproof (state)\nthis:\nf ` S \\ f ` (affine hull {u, v, w})\n\ngoal (1 subgoal):\n 1. coplanar (f ` S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nf ` S \\ f ` (affine hull {u, v, w})\n[PROOF STEP]\nhave \"f ` S \\ affine hull (f ` {u, v, w})\"\n[PROOF STATE]\nproof (prove)\nusing this:\nf ` S \\ f ` (affine hull {u, v, w})\n\ngoal (1 subgoal):\n 1. f ` S \\ affine hull f ` {u, v, w}\n[PROOF STEP]\nby (metis assms(2) linear_conv_bounded_linear affine_hull_linear_image)\n[PROOF STATE]\nproof (state)\nthis:\nf ` S \\ affine hull f ` {u, v, w}\n\ngoal (1 subgoal):\n 1. coplanar (f ` S)\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\nS \\ affine hull {?u2, ?v2, ?w2} \\ f ` S \\ affine hull f ` {?u2, ?v2, ?w2}\n\ngoal (1 subgoal):\n 1. coplanar (f ` S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nS \\ affine hull {?u2, ?v2, ?w2} \\ f ` S \\ affine hull f ` {?u2, ?v2, ?w2}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nS \\ affine hull {?u2, ?v2, ?w2} \\ f ` S \\ affine hull f ` {?u2, ?v2, ?w2}\n\ngoal (1 subgoal):\n 1. coplanar (f ` S)\n[PROOF STEP]\nby auto (meson assms(1) coplanar_def)\n[PROOF STATE]\nproof (state)\nthis:\ncoplanar (f ` S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1094, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110540642806, "lm_q2_score": 0.7853085758631159, "lm_q1q2_score": 0.7003468688062044}} {"text": "[STATEMENT]\nlemma hnorm_triangle_ineq: \"\\x y::'a::real_normed_vector star. hnorm (x + y) \\ hnorm x + hnorm y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x y. hnorm (x + y) \\ hnorm x + hnorm y\n[PROOF STEP]\nby transfer (rule norm_triangle_ineq)", "meta": {"llama_tokens": 116, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.89181104831338, "lm_q2_score": 0.785308580887758, "lm_q1q2_score": 0.7003468687710043}} {"text": "[STATEMENT]\nlemma ccSup_inter_less_eq: \"countable A \\ countable B \\ Sup (A \\ B) \\ inf (Sup A) (Sup B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\countable A; countable B\\ \\ Sup (A \\ B) \\ inf (Sup A) (Sup B)\n[PROOF STEP]\nby (auto intro: ccSup_least ccSup_upper)", "meta": {"llama_tokens": 143, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110454379297, "lm_q2_score": 0.7853085708384736, "lm_q1q2_score": 0.7003468575508256}} {"text": "[STATEMENT]\nlemma powr_lower_bound: \"\\(l::real) > 0; l \\ x; x \\ u\\ \\ min (l powr z) (u powr z) \\ x powr z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < l; l \\ x; x \\ u\\ \\ min (l powr z) (u powr z) \\ x powr z\n[PROOF STEP]\napply (cases \"z \\ 0\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\0 < l; l \\ x; x \\ u; 0 \\ z\\ \\ min (l powr z) (u powr z) \\ x powr z\n 2. \\0 < l; l \\ x; x \\ u; \\ 0 \\ z\\ \\ min (l powr z) (u powr z) \\ x powr z\n[PROOF STEP]\napply (rule order.trans[OF min.cobounded1 powr_mono2], simp_all) []\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < l; l \\ x; x \\ u; \\ 0 \\ z\\ \\ min (l powr z) (u powr z) \\ x powr z\n[PROOF STEP]\napply (rule order.trans[OF min.cobounded2 powr_mono2'], simp_all) []\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 484, "file": "Landau_Symbols_Landau_Library", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.782662489091802, "lm_q1q2_score": 0.7003181441653592}} {"text": "[STATEMENT]\nlemma complex_neq_0: \"z\\0 \\ (Re z)\\<^sup>2 + (Im z)\\<^sup>2 > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (z \\ 0) = (0 < (Re z)\\<^sup>2 + (Im z)\\<^sup>2)\n[PROOF STEP]\nby (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)", "meta": {"llama_tokens": 142, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894632969136, "lm_q2_score": 0.7826624840223698, "lm_q1q2_score": 0.7003181440210055}} {"text": "[STATEMENT]\nlemma L2_set_nonneg [simp]: \"0 \\ L2_set f A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ L2_set f A\n[PROOF STEP]\nunfolding L2_set_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ sqrt (\\i\\A. (f i)\\<^sup>2)\n[PROOF STEP]\nby (simp add: sum_nonneg)", "meta": {"llama_tokens": 152, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894689081711, "lm_q2_score": 0.7826624789529376, "lm_q1q2_score": 0.7003181438766517}} {"text": "[STATEMENT]\nlemma all_countings: \"all_countings a b = (a + b) choose a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. all_countings a b = a + b choose a\n[PROOF STEP]\nunfolding all_countings_set\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {V \\ Pow {0.. carrier (DirProds G I)\"\n shows \"(\\i \\ I. x i = \\\\<^bsub>G i\\<^esub>) \\ x = \\\\<^bsub>DirProds G I\\<^esub>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\I. x i = \\\\<^bsub>G i\\<^esub>) = (x = \\\\<^bsub>DirProds G I\\<^esub>)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ carrier (DirProds G I)\n\ngoal (1 subgoal):\n 1. (\\i\\I. x i = \\\\<^bsub>G i\\<^esub>) = (x = \\\\<^bsub>DirProds G I\\<^esub>)\n[PROOF STEP]\nunfolding DirProds_def\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ carrier \\carrier = Pi\\<^sub>E I (carrier \\ G), monoid.mult = \\x y. \\i\\I. x i \\\\<^bsub>G i\\<^esub> y i, one = \\i\\I. \\\\<^bsub>G i\\<^esub>\\\n\ngoal (1 subgoal):\n 1. (\\i\\I. x i = \\\\<^bsub>G i\\<^esub>) = (x = \\\\<^bsub>\\carrier = Pi\\<^sub>E I (carrier \\ G), monoid.mult = \\x y. \\i\\I. x i \\\\<^bsub>G i\\<^esub> y i, one = \\i\\I. \\\\<^bsub>G i\\<^esub>\\\\<^esub>)\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 518, "file": "Finitely_Generated_Abelian_Groups_DirProds", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789454880027, "lm_q2_score": 0.7826624840223698, "lm_q1q2_score": 0.7003181374334242}} {"text": "[STATEMENT]\nlemma fps_mult_commute_lemma:\n fixes n :: nat\n and f :: \"nat \\ nat \\ 'a::comm_monoid_add\"\n shows \"(\\i=0..n. f i (n - i)) = (\\i=0..n. f (n - i) i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..n. f i (n - i)) = (\\i = 0..n. f (n - i) i)\n[PROOF STEP]\nby (rule sum.reindex_bij_witness[where i=\"(-) n\" and j=\"(-) n\"]) auto", "meta": {"llama_tokens": 184, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894604912849, "lm_q2_score": 0.7826624789529376, "lm_q1q2_score": 0.7003181372890707}} {"text": "[STATEMENT]\nlemma measure_UNION_le:\n \"finite I \\ (\\i. i \\ I \\ F i \\ sets M) \\ measure M (\\i\\I. F i) \\ (\\i\\I. measure M (F i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite I; \\i. i \\ I \\ F i \\ sets M\\ \\ Sigma_Algebra.measure M (\\ (F ` I)) \\ (\\i\\I. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nproof (induction I rule: finite_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\i. i \\ {} \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` {})) \\ (\\i\\{}. Sigma_Algebra.measure M (F i))\n 2. \\x Fa. \\finite Fa; x \\ Fa; (\\i. i \\ Fa \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` Fa)) \\ (\\i\\Fa. Sigma_Algebra.measure M (F i)); \\i. i \\ insert x Fa \\ F i \\ sets M\\ \\ Sigma_Algebra.measure M (\\ (F ` insert x Fa)) \\ (\\i\\insert x Fa. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\ncase (insert i I)\n[PROOF STATE]\nproof (state)\nthis:\nfinite I\ni \\ I\n(\\i. i \\ I \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` I)) \\ (\\i\\I. Sigma_Algebra.measure M (F i))\n?i \\ insert i I \\ F ?i \\ sets M\n\ngoal (2 subgoals):\n 1. (\\i. i \\ {} \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` {})) \\ (\\i\\{}. Sigma_Algebra.measure M (F i))\n 2. \\x Fa. \\finite Fa; x \\ Fa; (\\i. i \\ Fa \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` Fa)) \\ (\\i\\Fa. Sigma_Algebra.measure M (F i)); \\i. i \\ insert x Fa \\ F i \\ sets M\\ \\ Sigma_Algebra.measure M (\\ (F ` insert x Fa)) \\ (\\i\\insert x Fa. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite I\ni \\ I\n(\\i. i \\ I \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` I)) \\ (\\i\\I. Sigma_Algebra.measure M (F i))\n?i \\ insert i I \\ F ?i \\ sets M\n[PROOF STEP]\nhave \"measure M (\\i\\insert i I. F i) = measure M (F i \\ \\ (F ` I))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite I\ni \\ I\n(\\i. i \\ I \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` I)) \\ (\\i\\I. Sigma_Algebra.measure M (F i))\n?i \\ insert i I \\ F ?i \\ sets M\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (\\ (F ` insert i I)) = Sigma_Algebra.measure M (F i \\ \\ (F ` I))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M (\\ (F ` insert i I)) = Sigma_Algebra.measure M (F i \\ \\ (F ` I))\n\ngoal (2 subgoals):\n 1. (\\i. i \\ {} \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` {})) \\ (\\i\\{}. Sigma_Algebra.measure M (F i))\n 2. \\x Fa. \\finite Fa; x \\ Fa; (\\i. i \\ Fa \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` Fa)) \\ (\\i\\Fa. Sigma_Algebra.measure M (F i)); \\i. i \\ insert x Fa \\ F i \\ sets M\\ \\ Sigma_Algebra.measure M (\\ (F ` insert x Fa)) \\ (\\i\\insert x Fa. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M (\\ (F ` insert i I)) = Sigma_Algebra.measure M (F i \\ \\ (F ` I))\n\ngoal (2 subgoals):\n 1. (\\i. i \\ {} \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` {})) \\ (\\i\\{}. Sigma_Algebra.measure M (F i))\n 2. \\x Fa. \\finite Fa; x \\ Fa; (\\i. i \\ Fa \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` Fa)) \\ (\\i\\Fa. Sigma_Algebra.measure M (F i)); \\i. i \\ insert x Fa \\ F i \\ sets M\\ \\ Sigma_Algebra.measure M (\\ (F ` insert x Fa)) \\ (\\i\\insert x Fa. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nfrom insert\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite I\ni \\ I\n(\\i. i \\ I \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` I)) \\ (\\i\\I. Sigma_Algebra.measure M (F i))\n?i \\ insert i I \\ F ?i \\ sets M\n[PROOF STEP]\nhave \"measure M (F i \\ \\ (F ` I)) \\ measure M (F i) + measure M (\\ (F ` I))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite I\ni \\ I\n(\\i. i \\ I \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` I)) \\ (\\i\\I. Sigma_Algebra.measure M (F i))\n?i \\ insert i I \\ F ?i \\ sets M\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (F i \\ \\ (F ` I)) \\ Sigma_Algebra.measure M (F i) + Sigma_Algebra.measure M (\\ (F ` I))\n[PROOF STEP]\nby (intro measure_Un_le sets.finite_Union) auto\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M (F i \\ \\ (F ` I)) \\ Sigma_Algebra.measure M (F i) + Sigma_Algebra.measure M (\\ (F ` I))\n\ngoal (2 subgoals):\n 1. (\\i. i \\ {} \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` {})) \\ (\\i\\{}. Sigma_Algebra.measure M (F i))\n 2. \\x Fa. \\finite Fa; x \\ Fa; (\\i. i \\ Fa \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` Fa)) \\ (\\i\\Fa. Sigma_Algebra.measure M (F i)); \\i. i \\ insert x Fa \\ F i \\ sets M\\ \\ Sigma_Algebra.measure M (\\ (F ` insert x Fa)) \\ (\\i\\insert x Fa. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M (F i \\ \\ (F ` I)) \\ Sigma_Algebra.measure M (F i) + Sigma_Algebra.measure M (\\ (F ` I))\n\ngoal (2 subgoals):\n 1. (\\i. i \\ {} \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` {})) \\ (\\i\\{}. Sigma_Algebra.measure M (F i))\n 2. \\x Fa. \\finite Fa; x \\ Fa; (\\i. i \\ Fa \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` Fa)) \\ (\\i\\Fa. Sigma_Algebra.measure M (F i)); \\i. i \\ insert x Fa \\ F i \\ sets M\\ \\ Sigma_Algebra.measure M (\\ (F ` insert x Fa)) \\ (\\i\\insert x Fa. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nhave \"measure M (\\i\\I. F i) \\ (\\i\\I. measure M (F i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (\\ (F ` I)) \\ (\\i\\I. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nusing insert\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite I\ni \\ I\n(\\i. i \\ I \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` I)) \\ (\\i\\I. Sigma_Algebra.measure M (F i))\n?i \\ insert i I \\ F ?i \\ sets M\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (\\ (F ` I)) \\ (\\i\\I. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M (\\ (F ` I)) \\ (\\i\\I. Sigma_Algebra.measure M (F i))\n\ngoal (2 subgoals):\n 1. (\\i. i \\ {} \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` {})) \\ (\\i\\{}. Sigma_Algebra.measure M (F i))\n 2. \\x Fa. \\finite Fa; x \\ Fa; (\\i. i \\ Fa \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` Fa)) \\ (\\i\\Fa. Sigma_Algebra.measure M (F i)); \\i. i \\ insert x Fa \\ F i \\ sets M\\ \\ Sigma_Algebra.measure M (\\ (F ` insert x Fa)) \\ (\\i\\insert x Fa. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x y. x \\ y \\ Sigma_Algebra.measure M (F i) + x \\ Sigma_Algebra.measure M (F i) + y) \\ Sigma_Algebra.measure M (\\ (F ` insert i I)) \\ Sigma_Algebra.measure M (F i) + (\\i\\I. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x y. x \\ y \\ Sigma_Algebra.measure M (F i) + x \\ Sigma_Algebra.measure M (F i) + y) \\ Sigma_Algebra.measure M (\\ (F ` insert i I)) \\ Sigma_Algebra.measure M (F i) + (\\i\\I. Sigma_Algebra.measure M (F i))\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (\\ (F ` insert i I)) \\ (\\i\\insert i I. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nusing insert\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x y. x \\ y \\ Sigma_Algebra.measure M (F i) + x \\ Sigma_Algebra.measure M (F i) + y) \\ Sigma_Algebra.measure M (\\ (F ` insert i I)) \\ Sigma_Algebra.measure M (F i) + (\\i\\I. Sigma_Algebra.measure M (F i))\nfinite I\ni \\ I\n(\\i. i \\ I \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` I)) \\ (\\i\\I. Sigma_Algebra.measure M (F i))\n?i \\ insert i I \\ F ?i \\ sets M\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (\\ (F ` insert i I)) \\ (\\i\\insert i I. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M (\\ (F ` insert i I)) \\ (\\i\\insert i I. Sigma_Algebra.measure M (F i))\n\ngoal (1 subgoal):\n 1. (\\i. i \\ {} \\ F i \\ sets M) \\ Sigma_Algebra.measure M (\\ (F ` {})) \\ (\\i\\{}. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 4103, "file": null, "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894604912848, "lm_q2_score": 0.7826624738835052, "lm_q1q2_score": 0.7003181327529959}} {"text": "[STATEMENT]\nlemma setVsList: \n assumes \"\\l \\ set (g1 G). set (g2 l N) = f2 (set l) N\" \n shows \"set [set (g2 l N). l <- (g1 G)] = {f2 P N| P. P \\ set (map set (g1 G))}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (map (\\l. set (g2 l N)) (g1 G)) = {f2 P N |P. P \\ set (map set (g1 G))}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\l\\set (g1 G). set (g2 l N) = f2 (set l) N\n\ngoal (1 subgoal):\n 1. set (map (\\l. set (g2 l N)) (g1 G)) = {f2 P N |P. P \\ set (map set (g1 G))}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 281, "file": "Vickrey_Clarke_Groves_MiscTools", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894520743981, "lm_q2_score": 0.7826624789529376, "lm_q1q2_score": 0.7003181307014892}} {"text": "[STATEMENT]\nlemma image_mset_eq_imp_set_eq: \n assumes \"image_mset f s = image_mset g s\" \n shows \"f`(set_mset s) = g`set_mset s\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f ` set_mset s = g ` set_mset s\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nimage_mset f s = image_mset g s\n\ngoal (1 subgoal):\n 1. f ` set_mset s = g ` set_mset s\n[PROOF STEP]\nby (metis set_image_mset)", "meta": {"llama_tokens": 190, "file": "IMP2_lib_IMP2_Aux_Lemmas", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382165412808, "lm_q2_score": 0.8128673155708976, "lm_q1q2_score": 0.7003162573416497}} {"text": "[STATEMENT]\nlemma Im_Arctan_of_real [simp]: \"Im (Arctan (of_real x)) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Im (Arctan (complex_of_real x)) = 0\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Im (Arctan (complex_of_real x)) = 0\n[PROOF STEP]\nhave ne: \"1 + x\\<^sup>2 \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + x\\<^sup>2 \\ 0\n[PROOF STEP]\nby (metis power_one sum_power2_eq_zero_iff zero_neq_one)\n[PROOF STATE]\nproof (state)\nthis:\n1 + x\\<^sup>2 \\ 0\n\ngoal (1 subgoal):\n 1. Im (Arctan (complex_of_real x)) = 0\n[PROOF STEP]\nhave ne1: \"1 + \\ * complex_of_real x \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + \\ * complex_of_real x \\ 0\n[PROOF STEP]\nusing Complex_eq complex_eq_cancel_iff2\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex ?a ?b = complex_of_real ?a + \\ * complex_of_real ?b\n(Complex ?x ?y = complex_of_real ?xa) = (?x = ?xa \\ ?y = 0)\n\ngoal (1 subgoal):\n 1. 1 + \\ * complex_of_real x \\ 0\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n1 + \\ * complex_of_real x \\ 0\n\ngoal (1 subgoal):\n 1. Im (Arctan (complex_of_real x)) = 0\n[PROOF STEP]\nhave \"Re (Ln ((1 - \\ * x) * inverse (1 + \\ * x))) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Re (Ln ((1 - \\ * complex_of_real x) * inverse (1 + \\ * complex_of_real x))) = 0\n[PROOF STEP]\napply (rule norm_exp_imaginary)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (exp (Ln ((1 - \\ * complex_of_real x) * inverse (1 + \\ * complex_of_real x)))) = 1\n[PROOF STEP]\nusing ne\n[PROOF STATE]\nproof (prove)\nusing this:\n1 + x\\<^sup>2 \\ 0\n\ngoal (1 subgoal):\n 1. cmod (exp (Ln ((1 - \\ * complex_of_real x) * inverse (1 + \\ * complex_of_real x)))) = 1\n[PROOF STEP]\napply (simp add: ne1 cmod_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + x\\<^sup>2 \\ 0 \\ (1 / (1 + x\\<^sup>2) - x * x / (1 + x\\<^sup>2))\\<^sup>2 + (2 * x / (1 + x\\<^sup>2))\\<^sup>2 = 1\n[PROOF STEP]\napply (auto simp: field_split_simps)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + x\\<^sup>2 \\ 0 \\ (1 - x * x)\\<^sup>2 + 4 * x\\<^sup>2 = (1 + x\\<^sup>2)\\<^sup>2\n[PROOF STEP]\napply algebra\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\nRe (Ln ((1 - \\ * complex_of_real x) * inverse (1 + \\ * complex_of_real x))) = 0\n\ngoal (1 subgoal):\n 1. Im (Arctan (complex_of_real x)) = 0\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nRe (Ln ((1 - \\ * complex_of_real x) * inverse (1 + \\ * complex_of_real x))) = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nRe (Ln ((1 - \\ * complex_of_real x) * inverse (1 + \\ * complex_of_real x))) = 0\n\ngoal (1 subgoal):\n 1. Im (Arctan (complex_of_real x)) = 0\n[PROOF STEP]\nunfolding Arctan_def divide_complex_def\n[PROOF STATE]\nproof (prove)\nusing this:\nRe (Ln ((1 - \\ * complex_of_real x) * inverse (1 + \\ * complex_of_real x))) = 0\n\ngoal (1 subgoal):\n 1. Im (\\ * inverse 2 * Ln ((1 - \\ * complex_of_real x) * inverse (1 + \\ * complex_of_real x))) = 0\n[PROOF STEP]\nby (simp add: complex_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\nIm (Arctan (complex_of_real x)) = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1536, "file": null, "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.86153820232079, "lm_q2_score": 0.8128673201042493, "lm_q1q2_score": 0.7003162496879332}} {"text": "[STATEMENT]\nlemma isDiagonal_eq_upt_k_min:\n\"isDiagonal A = isDiagonal_upt_k A (min (nrows A) (ncols A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. isDiagonal A = isDiagonal_upt_k A (min (nrows A) (ncols A))\n[PROOF STEP]\nunfolding isDiagonal_def isDiagonal_upt_k_def nrows_def ncols_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a b. to_nat a \\ to_nat b \\ A $ a $ b = (0::'a)) = (\\a b. to_nat a \\ to_nat b \\ (to_nat a < min CARD('c) CARD('b) \\ to_nat b < min CARD('c) CARD('b)) \\ A $ a $ b = (0::'a))\n[PROOF STEP]\nby (auto, meson less_trans not_less_iff_gr_or_eq to_nat_less_card)", "meta": {"llama_tokens": 303, "file": "Smith_Normal_Form_Diagonal_To_Smith", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615381987656672, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7003162487509177}} {"text": "[STATEMENT]\nlemma dist_round_complex_le: \"dist z (gauss2complex (round_complex z)) \\ sqrt 2 / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist z (gauss2complex (round_complex z)) \\ sqrt 2 / 2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dist z (gauss2complex (round_complex z)) \\ sqrt 2 / 2\n[PROOF STEP]\nhave \"dist z (gauss2complex (round_complex z)) ^ 2 =\n norm (z - gauss2complex (round_complex z)) ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist z (gauss2complex (round_complex z)))\\<^sup>2 = (cmod (z - gauss2complex (round_complex z)))\\<^sup>2\n[PROOF STEP]\nby (simp add: dist_norm)\n[PROOF STATE]\nproof (state)\nthis:\n(dist z (gauss2complex (round_complex z)))\\<^sup>2 = (cmod (z - gauss2complex (round_complex z)))\\<^sup>2\n\ngoal (1 subgoal):\n 1. dist z (gauss2complex (round_complex z)) \\ sqrt 2 / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(dist z (gauss2complex (round_complex z)))\\<^sup>2 = (cmod (z - gauss2complex (round_complex z)))\\<^sup>2\n\ngoal (1 subgoal):\n 1. dist z (gauss2complex (round_complex z)) \\ sqrt 2 / 2\n[PROOF STEP]\nhave \"\\ \\ 1 / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cmod (z - gauss2complex (round_complex z)))\\<^sup>2 \\ 1 / 2\n[PROOF STEP]\nby (rule norm_round_complex_le)\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (z - gauss2complex (round_complex z)))\\<^sup>2 \\ 1 / 2\n\ngoal (1 subgoal):\n 1. dist z (gauss2complex (round_complex z)) \\ sqrt 2 / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (z - gauss2complex (round_complex z)))\\<^sup>2 \\ 1 / 2\n\ngoal (1 subgoal):\n 1. dist z (gauss2complex (round_complex z)) \\ sqrt 2 / 2\n[PROOF STEP]\nhave \"\\ = (sqrt 2 / 2) ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 / 2 = (sqrt 2 / 2)\\<^sup>2\n[PROOF STEP]\nby (simp add: power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n1 / 2 = (sqrt 2 / 2)\\<^sup>2\n\ngoal (1 subgoal):\n 1. dist z (gauss2complex (round_complex z)) \\ sqrt 2 / 2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(dist z (gauss2complex (round_complex z)))\\<^sup>2 \\ (sqrt 2 / 2)\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(dist z (gauss2complex (round_complex z)))\\<^sup>2 \\ (sqrt 2 / 2)\\<^sup>2\n\ngoal (1 subgoal):\n 1. dist z (gauss2complex (round_complex z)) \\ sqrt 2 / 2\n[PROOF STEP]\nby (rule power2_le_imp_le) auto\n[PROOF STATE]\nproof (state)\nthis:\ndist z (gauss2complex (round_complex z)) \\ sqrt 2 / 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1149, "file": "Gaussian_Integers_Gaussian_Integers", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382058759128, "lm_q2_score": 0.8128673155708976, "lm_q1q2_score": 0.7003162486721206}} {"text": "[STATEMENT]\nlemma harm_mono: \"m \\ n \\ harm m \\ (harm n :: real)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. m \\ n \\ harm m \\ harm n\n[PROOF STEP]\nunfolding harm_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. m \\ n \\ (\\k = 1..m. inverse (real k)) \\ (\\k = 1..n. inverse (real k))\n[PROOF STEP]\nby (intro sum_mono2) auto", "meta": {"llama_tokens": 169, "file": "Quick_Sort_Cost_Randomised_Quick_Sort", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615382094310355, "lm_q2_score": 0.8128673110375457, "lm_q1q2_score": 0.7003162476563077}} {"text": "[STATEMENT]\nlemma aux_comp_with_sqrt2:\n shows \"(sqrt 2)^n * (sqrt 2)^n = 2^n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt 2 ^ n * sqrt 2 ^ n = 2 ^ n\n[PROOF STEP]\nby (smt power_mult_distrib real_sqrt_mult_self)", "meta": {"llama_tokens": 106, "file": "Isabelle_Marries_Dirac_Deutsch_Jozsa", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.907312213841788, "lm_q2_score": 0.7718435030872968, "lm_q1q2_score": 0.7003030375255361}} {"text": "[STATEMENT]\nlemma gcd_left_idem: \"gcd a (gcd a b) = gcd a b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. gcd a (gcd a b) = gcd a b\n[PROOF STEP]\nby (fact gcd.left_idem)", "meta": {"llama_tokens": 88, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.7879312031126512, "lm_q1q2_score": 0.7002807854334598}} {"text": "[STATEMENT]\nlemma padic_mult_comm0:\nassumes \"prime p\"\nshows \"(padic_mult p x y)= (padic_mult p y x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. padic_mult p x y = padic_mult p y x\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. padic_mult p x y = padic_mult p y x\n[PROOF STEP]\nunfolding padic_mult_def\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. (\\n. x n \\\\<^bsub>residue_ring (p ^ n)\\<^esub> y n) = (\\n. y n \\\\<^bsub>residue_ring (p ^ n)\\<^esub> x n)\n[PROOF STEP]\nusing padic_integers.residue_mult_comm[of p]\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\npadic_integers p \\ ?x \\\\<^bsub>residue_ring (p ^ ?k)\\<^esub> ?y = ?y \\\\<^bsub>residue_ring (p ^ ?k)\\<^esub> ?x\n\ngoal (1 subgoal):\n 1. (\\n. x n \\\\<^bsub>residue_ring (p ^ n)\\<^esub> y n) = (\\n. y n \\\\<^bsub>residue_ring (p ^ n)\\<^esub> x n)\n[PROOF STEP]\nby (simp add: padic_integers_def)", "meta": {"llama_tokens": 461, "file": "Padic_Ints_Padic_Integers", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.7879312006227323, "lm_q1q2_score": 0.7002807832205226}} {"text": "[STATEMENT]\ntheorem set_vebt_pred: \"invar_vebt t n \\ vebt_pred t x = Some px \\ is_pred_in_set (set_vebt t) x px\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. invar_vebt t n \\ (vebt_pred t x = Some px) = is_pred_in_set (set_vebt t) x px\n[PROOF STEP]\nby (simp add: pred_corr set_vebt_set_vebt'_valid)", "meta": {"llama_tokens": 151, "file": "Van_Emde_Boas_Trees_VEBT_Intf_Functional", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952975813453, "lm_q2_score": 0.7799929053683038, "lm_q1q2_score": 0.7001959632959376}} {"text": "[STATEMENT]\nlemma obtain_partition:\n assumes \"finite A\"\n assumes \"number_partition (card A) N\"\n shows \"\\P. partition_on A P \\ image_mset card (mset_set P) = N\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\P. partition_on A P \\ image_mset card (mset_set P) = N\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nnumber_partition (card A) N\n\ngoal (1 subgoal):\n 1. \\P. partition_on A P \\ image_mset card (mset_set P) = N\n[PROOF STEP]\nproof (induct N arbitrary: A)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\A. \\finite A; number_partition (card A) {#}\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = {#}\n 2. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\ncase empty\n[PROOF STATE]\nproof (state)\nthis:\nfinite A\nnumber_partition (card A) {#}\n\ngoal (2 subgoals):\n 1. \\A. \\finite A; number_partition (card A) {#}\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = {#}\n 2. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nnumber_partition (card A) {#}\n[PROOF STEP]\nhave \"A = {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nnumber_partition (card A) {#}\n\ngoal (1 subgoal):\n 1. A = {}\n[PROOF STEP]\nunfolding number_partition_def\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\\\\<^sub># {#} = card A \\ 0 \\# {#}\n\ngoal (1 subgoal):\n 1. A = {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA = {}\n\ngoal (2 subgoals):\n 1. \\A. \\finite A; number_partition (card A) {#}\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = {#}\n 2. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nA = {}\n[PROOF STEP]\nhave \"partition_on A {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA = {}\n\ngoal (1 subgoal):\n 1. partition_on A {}\n[PROOF STEP]\nby (simp add: partition_on_empty)\n[PROOF STATE]\nproof (state)\nthis:\npartition_on A {}\n\ngoal (2 subgoals):\n 1. \\A. \\finite A; number_partition (card A) {#}\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = {#}\n 2. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\npartition_on A {}\n\ngoal (2 subgoals):\n 1. \\A. \\finite A; number_partition (card A) {#}\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = {#}\n 2. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nhave \"image_mset card (mset_set {}) = {#}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. image_mset card (mset_set {}) = {#}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nimage_mset card (mset_set {}) = {#}\n\ngoal (2 subgoals):\n 1. \\A. \\finite A; number_partition (card A) {#}\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = {#}\n 2. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\npartition_on A {}\nimage_mset card (mset_set {}) = {#}\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\npartition_on A {}\nimage_mset card (mset_set {}) = {#}\n\ngoal (1 subgoal):\n 1. \\P. partition_on A P \\ image_mset card (mset_set P) = {#}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\P. partition_on A P \\ image_mset card (mset_set P) = {#}\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\ncase (add x N)\n[PROOF STATE]\nproof (state)\nthis:\n\\finite ?A; number_partition (card ?A) N\\ \\ \\P. partition_on ?A P \\ image_mset card (mset_set P) = N\nfinite A\nnumber_partition (card A) (add_mset x N)\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nfrom add.prems(2)\n[PROOF STATE]\nproof (chain)\npicking this:\nnumber_partition (card A) (add_mset x N)\n[PROOF STEP]\nhave \"0 \\# add_mset x N\" and \"sum_mset (add_mset x N) = card A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nnumber_partition (card A) (add_mset x N)\n\ngoal (1 subgoal):\n 1. 0 \\# add_mset x N &&& \\\\<^sub># (add_mset x N) = card A\n[PROOF STEP]\nunfolding number_partition_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub># (add_mset x N) = card A \\ 0 \\# add_mset x N\n\ngoal (1 subgoal):\n 1. 0 \\# add_mset x N &&& \\\\<^sub># (add_mset x N) = card A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 \\# add_mset x N\n\\\\<^sub># (add_mset x N) = card A\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\# add_mset x N\n\\\\<^sub># (add_mset x N) = card A\n[PROOF STEP]\nhave \"x \\ card A\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\# add_mset x N\n\\\\<^sub># (add_mset x N) = card A\n\ngoal (1 subgoal):\n 1. x \\ card A\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx \\ card A\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ card A\n[PROOF STEP]\nobtain X where \"X \\ A\" and \"card X = x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ card A\n\ngoal (1 subgoal):\n 1. (\\X. \\X \\ A; card X = x\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing subset_with_given_card_exists\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ card A\n?n \\ card ?A \\ \\B\\?A. card B = ?n\n\ngoal (1 subgoal):\n 1. (\\X. \\X \\ A; card X = x\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nX \\ A\ncard X = x\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nX \\ A\ncard X = x\n[PROOF STEP]\nhave \"X \\ {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nX \\ A\ncard X = x\n\ngoal (1 subgoal):\n 1. X \\ {}\n[PROOF STEP]\nusing \\0 \\# add_mset x N\\ \\finite A\\\n[PROOF STATE]\nproof (prove)\nusing this:\nX \\ A\ncard X = x\n0 \\# add_mset x N\nfinite A\n\ngoal (1 subgoal):\n 1. X \\ {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nX \\ {}\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nhave \"sum_mset N = card (A - X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub># N = card (A - X)\n[PROOF STEP]\nusing \\sum_mset (add_mset x N) = card A\\ \\card X = x\\ \\X \\ A\\\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub># (add_mset x N) = card A\ncard X = x\nX \\ A\n\ngoal (1 subgoal):\n 1. \\\\<^sub># N = card (A - X)\n[PROOF STEP]\nby (metis add.commute add.prems(1) add_diff_cancel_right' card_Diff_subset infinite_super sum_mset.add_mset)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub># N = card (A - X)\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nfrom this \\0 \\# add_mset x N\\\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub># N = card (A - X)\n0 \\# add_mset x N\n[PROOF STEP]\nhave \"number_partition (card (A - X)) N\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub># N = card (A - X)\n0 \\# add_mset x N\n\ngoal (1 subgoal):\n 1. number_partition (card (A - X)) N\n[PROOF STEP]\nunfolding number_partition_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub># N = card (A - X)\n0 \\# add_mset x N\n\ngoal (1 subgoal):\n 1. \\\\<^sub># N = card (A - X) \\ 0 \\# N\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nnumber_partition (card (A - X)) N\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nnumber_partition (card (A - X)) N\n[PROOF STEP]\nobtain P where \"partition_on (A - X) P\" and eq_N: \"image_mset card (mset_set P) = N\"\n[PROOF STATE]\nproof (prove)\nusing this:\nnumber_partition (card (A - X)) N\n\ngoal (1 subgoal):\n 1. (\\P. \\partition_on (A - X) P; image_mset card (mset_set P) = N\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing add.hyps \\finite A\\\n[PROOF STATE]\nproof (prove)\nusing this:\nnumber_partition (card (A - X)) N\n\\finite ?A; number_partition (card ?A) N\\ \\ \\P. partition_on ?A P \\ image_mset card (mset_set P) = N\nfinite A\n\ngoal (1 subgoal):\n 1. (\\P. \\partition_on (A - X) P; image_mset card (mset_set P) = N\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\npartition_on (A - X) P\nimage_mset card (mset_set P) = N\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nfrom \\partition_on (A - X) P\\\n[PROOF STATE]\nproof (chain)\npicking this:\npartition_on (A - X) P\n[PROOF STEP]\nhave \"finite P\"\n[PROOF STATE]\nproof (prove)\nusing this:\npartition_on (A - X) P\n\ngoal (1 subgoal):\n 1. finite P\n[PROOF STEP]\nusing \\finite A\\ finite_elements\n[PROOF STATE]\nproof (prove)\nusing this:\npartition_on (A - X) P\nfinite A\n\\finite ?A; partition_on ?A ?P\\ \\ finite ?P\n\ngoal (1 subgoal):\n 1. finite P\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfinite P\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nfrom \\partition_on (A - X) P\\\n[PROOF STATE]\nproof (chain)\npicking this:\npartition_on (A - X) P\n[PROOF STEP]\nhave \"X \\ P\"\n[PROOF STATE]\nproof (prove)\nusing this:\npartition_on (A - X) P\n\ngoal (1 subgoal):\n 1. X \\ P\n[PROOF STEP]\nusing \\X \\ {}\\ partition_onD1\n[PROOF STATE]\nproof (prove)\nusing this:\npartition_on (A - X) P\nX \\ {}\npartition_on ?A ?P \\ ?A = \\ ?P\n\ngoal (1 subgoal):\n 1. X \\ P\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nX \\ P\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nhave \"partition_on A (insert X P)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. partition_on A (insert X P)\n[PROOF STEP]\nusing \\partition_on (A - X) P\\ \\X \\ A\\ \\X \\ {}\\\n[PROOF STATE]\nproof (prove)\nusing this:\npartition_on (A - X) P\nX \\ A\nX \\ {}\n\ngoal (1 subgoal):\n 1. partition_on A (insert X P)\n[PROOF STEP]\nby (rule partition_on_insert')\n[PROOF STATE]\nproof (state)\nthis:\npartition_on A (insert X P)\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\npartition_on A (insert X P)\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nhave \"image_mset card (mset_set (insert X P)) = add_mset x N\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. image_mset card (mset_set (insert X P)) = add_mset x N\n[PROOF STEP]\nusing eq_N \\card X = x\\ \\finite P\\ \\X \\ P\\\n[PROOF STATE]\nproof (prove)\nusing this:\nimage_mset card (mset_set P) = N\ncard X = x\nfinite P\nX \\ P\n\ngoal (1 subgoal):\n 1. image_mset card (mset_set (insert X P)) = add_mset x N\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nimage_mset card (mset_set (insert X P)) = add_mset x N\n\ngoal (1 subgoal):\n 1. \\x N A. \\\\A. \\finite A; number_partition (card A) N\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = N; finite A; number_partition (card A) (add_mset x N)\\ \\ \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\npartition_on A (insert X P)\nimage_mset card (mset_set (insert X P)) = add_mset x N\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\npartition_on A (insert X P)\nimage_mset card (mset_set (insert X P)) = add_mset x N\n\ngoal (1 subgoal):\n 1. \\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\P. partition_on A P \\ image_mset card (mset_set P) = add_mset x N\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 7559, "file": "Twelvefold_Way_Twelvefold_Way_Core", "length": 63, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952975813454, "lm_q2_score": 0.7799928900257126, "lm_q1q2_score": 0.7001959495229656}} {"text": "[STATEMENT]\nlemma arcs_eq_2verts: \"card (arcs G) = 2 * (card (verts G) - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (arcs G) = 2 * (card (verts G) - 1)\n[PROOF STEP]\nusing psp_tree.arcs_undir_G_eq_2vertsG[OF psp_dir_tree_r undirected_tree_axioms] card_gt0\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?source \\ verts G; Suc (card (verts G) - 1) = card (verts G)\\ \\ card (arcs G) = 2 * (card (verts G) - 1)\n0 < card (verts G)\n\ngoal (1 subgoal):\n 1. card (arcs G) = 2 * (card (verts G) - 1)\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 269, "file": "Query_Optimization_Directed_Tree_Additions", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976952866333484, "lm_q2_score": 0.7799928900257127, "lm_q1q2_score": 0.7001959409836059}} {"text": "[STATEMENT]\nlemma ivl_integral_minus_sets':\n fixes f::\"real \\ 'a::banach\"\n shows \"f integrable_on (closed_segment a c) \\ f integrable_on (closed_segment b c) \\ f integrable_on (closed_segment a b) \\\n ivl_integral a c f - ivl_integral b c f = ivl_integral a b f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f integrable_on closed_segment a c; f integrable_on closed_segment b c; f integrable_on closed_segment a b\\ \\ ivl_integral a c f - ivl_integral b c f = ivl_integral a b f\n[PROOF STEP]\nusing ivl_integral_combine[of f a b c]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\f integrable_on closed_segment a b; f integrable_on closed_segment b c; f integrable_on closed_segment a c\\ \\ ivl_integral a b f + ivl_integral b c f = ivl_integral a c f\n\ngoal (1 subgoal):\n 1. \\f integrable_on closed_segment a c; f integrable_on closed_segment b c; f integrable_on closed_segment a b\\ \\ ivl_integral a c f - ivl_integral b c f = ivl_integral a b f\n[PROOF STEP]\nby (auto simp: algebra_simps closed_segment_commute)", "meta": {"llama_tokens": 427, "file": "Ordinary_Differential_Equations_Library_Interval_Integral_HK", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314798554445, "lm_q2_score": 0.79053032607222, "lm_q1q2_score": 0.7001185425499473}} {"text": "[STATEMENT]\nlemma index_list_set:\n\"set (index_list n) = {..n. set (index_list n) = {.. set (index_list (Suc n)) = {..n. set (index_list n) = {.. set (index_list (Suc n)) = {..q'. step_eps bs q q' \\ q' \\ X) \\\n step_eps_closure_set {q} bs = {q} \\ step_eps_closure_set X bs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\q'. step_eps bs q q' = (q' \\ X)) \\ step_eps_closure_set {q} bs = {q} \\ step_eps_closure_set X bs\n[PROOF STEP]\nunfolding step_eps_closure_set_def step_eps_closure_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\q'. step_eps bs q q' = (q' \\ X)) \\ (\\q\\{q}. Collect ((step_eps bs)\\<^sup>*\\<^sup>* q)) = {q} \\ (\\q\\X. Collect ((step_eps bs)\\<^sup>*\\<^sup>* q))\n[PROOF STEP]\nusing rtranclp_step[of \"step_eps bs\" q]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\(step_eps bs)\\<^sup>*\\<^sup>* q ?q''; \\q'. step_eps bs q q' = (q' \\ ?X)\\ \\ q = ?q'' \\ (\\q'\\?X. step_eps bs q q' \\ (step_eps bs)\\<^sup>*\\<^sup>* q' ?q'')\n\ngoal (1 subgoal):\n 1. (\\q'. step_eps bs q q' = (q' \\ X)) \\ (\\q\\{q}. Collect ((step_eps bs)\\<^sup>*\\<^sup>* q)) = {q} \\ (\\q\\X. Collect ((step_eps bs)\\<^sup>*\\<^sup>* q))\n[PROOF STEP]\nby (auto simp add: converse_rtranclp_into_rtranclp)", "meta": {"llama_tokens": 566, "file": "VYDRA_MDL_NFA", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127678225575, "lm_q2_score": 0.8198933403143929, "lm_q1q2_score": 0.7001173915471454}} {"text": "[STATEMENT]\nlemma bitwise_leq: \"(\\k. a \\ k \\ b \\ k) \\ a \\ b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k. a \\ k \\ b \\ k) \\ a \\ b\n[PROOF STEP]\nusing digitwise_leq[of 2]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\1 < 2; \\t. nth_digit ?x t 2 \\ nth_digit ?y t 2\\ \\ ?x \\ ?y\n\ngoal (1 subgoal):\n 1. (\\k. a \\ k \\ b \\ k) \\ a \\ b\n[PROOF STEP]\nby (simp add: nth_digit_base2_equiv)", "meta": {"llama_tokens": 256, "file": "Digit_Expansions_Binary_Operations", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.8198933359135361, "lm_q1q2_score": 0.700117369500427}} {"text": "[STATEMENT]\nlemma calc_x_axis_intersection_fun_mono:\n fixes x1 x2 :: real\n assumes \"x1 > 1\" and \"x2 > x1\"\n shows \"x1 - sqrt(x1\\<^sup>2 - 1) > x2 - sqrt(x2\\<^sup>2 - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < x1\nx1 < x2\n\ngoal (1 subgoal):\n 1. x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\1 < x1; x1 < x2\\ \\ x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nhave *: \"sqrt(x1\\<^sup>2 - 1) + sqrt(x2\\<^sup>2 - 1) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < x1\nx1 < x2\n\ngoal (1 subgoal):\n 1. 0 < sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)\n[PROOF STEP]\nby (smt one_less_power pos2 real_sqrt_gt_zero)\n[PROOF STATE]\nproof (state)\nthis:\n0 < sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)\n\ngoal (1 subgoal):\n 1. \\1 < x1; x1 < x2\\ \\ x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nhave \"sqrt(x1\\<^sup>2 - 1) < x1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (x1\\<^sup>2 - 1) < x1\n[PROOF STEP]\nusing real_sqrt_less_iff[of \"x1\\<^sup>2 - 1\" \"x1\\<^sup>2\"] \\x1 > 1\\\n[PROOF STATE]\nproof (prove)\nusing this:\n(sqrt (x1\\<^sup>2 - 1) < sqrt (x1\\<^sup>2)) = (x1\\<^sup>2 - 1 < x1\\<^sup>2)\n1 < x1\n\ngoal (1 subgoal):\n 1. sqrt (x1\\<^sup>2 - 1) < x1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (x1\\<^sup>2 - 1) < x1\n\ngoal (1 subgoal):\n 1. \\1 < x1; x1 < x2\\ \\ x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (x1\\<^sup>2 - 1) < x1\n\ngoal (1 subgoal):\n 1. \\1 < x1; x1 < x2\\ \\ x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nhave \"sqrt(x2\\<^sup>2 - 1) < x2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (x2\\<^sup>2 - 1) < x2\n[PROOF STEP]\nusing real_sqrt_less_iff[of \"x2\\<^sup>2 - 1\" \"x2\\<^sup>2\"] \\x1 > 1\\ \\x2 > x1\\\n[PROOF STATE]\nproof (prove)\nusing this:\n(sqrt (x2\\<^sup>2 - 1) < sqrt (x2\\<^sup>2)) = (x2\\<^sup>2 - 1 < x2\\<^sup>2)\n1 < x1\nx1 < x2\n\ngoal (1 subgoal):\n 1. sqrt (x2\\<^sup>2 - 1) < x2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (x2\\<^sup>2 - 1) < x2\n\ngoal (1 subgoal):\n 1. \\1 < x1; x1 < x2\\ \\ x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nsqrt (x1\\<^sup>2 - 1) < x1\nsqrt (x2\\<^sup>2 - 1) < x2\n[PROOF STEP]\nhave \"sqrt(x1\\<^sup>2 - 1) + sqrt(x2\\<^sup>2 - 1) < x1 + x2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (x1\\<^sup>2 - 1) < x1\nsqrt (x2\\<^sup>2 - 1) < x2\n\ngoal (1 subgoal):\n 1. sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1) < x1 + x2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1) < x1 + x2\n\ngoal (1 subgoal):\n 1. \\1 < x1; x1 < x2\\ \\ x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nhence \"(x1 + x2) / (sqrt(x1\\<^sup>2 - 1) + sqrt(x2\\<^sup>2 - 1)) > 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1) < x1 + x2\n\ngoal (1 subgoal):\n 1. 1 < (x1 + x2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\n[PROOF STEP]\nusing *\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1) < x1 + x2\n0 < sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)\n\ngoal (1 subgoal):\n 1. 1 < (x1 + x2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\n[PROOF STEP]\nusing less_divide_eq_1_pos[of \"sqrt(x1\\<^sup>2 - 1) + sqrt(x2\\<^sup>2 - 1)\" \"x1 + x2\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1) < x1 + x2\n0 < sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)\n0 < sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1) \\ (1 < (x1 + x2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))) = (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1) < x1 + x2)\n\ngoal (1 subgoal):\n 1. 1 < (x1 + x2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 < (x1 + x2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\n\ngoal (1 subgoal):\n 1. \\1 < x1; x1 < x2\\ \\ x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nhence \"(x2\\<^sup>2 - x1\\<^sup>2) / (sqrt(x1\\<^sup>2 - 1) + sqrt(x2\\<^sup>2 - 1)) > x2 - x1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < (x1 + x2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\n\ngoal (1 subgoal):\n 1. x2 - x1 < (x2\\<^sup>2 - x1\\<^sup>2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\n[PROOF STEP]\nusing \\x2 > x1\\\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < (x1 + x2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\nx1 < x2\n\ngoal (1 subgoal):\n 1. x2 - x1 < (x2\\<^sup>2 - x1\\<^sup>2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\n[PROOF STEP]\nusing mult_less_cancel_left_pos[of \"x2 - x1\" 1 \"(x2 + x1) / (sqrt(x1\\<^sup>2 - 1) + sqrt(x2\\<^sup>2 - 1))\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < (x1 + x2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\nx1 < x2\n0 < x2 - x1 \\ ((x2 - x1) * 1 < (x2 - x1) * ((x2 + x1) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)))) = (1 < (x2 + x1) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)))\n\ngoal (1 subgoal):\n 1. x2 - x1 < (x2\\<^sup>2 - x1\\<^sup>2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\n[PROOF STEP]\nby (simp add: power2_eq_square field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nx2 - x1 < (x2\\<^sup>2 - x1\\<^sup>2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\n\ngoal (1 subgoal):\n 1. \\1 < x1; x1 < x2\\ \\ x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nx2 - x1 < (x2\\<^sup>2 - x1\\<^sup>2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\n\ngoal (1 subgoal):\n 1. \\1 < x1; x1 < x2\\ \\ x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nhave \"(x2\\<^sup>2 - x1\\<^sup>2) = (sqrt(x1\\<^sup>2 - 1) + sqrt(x2\\<^sup>2 - 1)) * ((sqrt(x2\\<^sup>2 - 1) - sqrt(x1\\<^sup>2 - 1)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x2\\<^sup>2 - x1\\<^sup>2 = (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)) * (sqrt (x2\\<^sup>2 - 1) - sqrt (x1\\<^sup>2 - 1))\n[PROOF STEP]\nusing \\x1 > 1\\ \\x2 > x1\\\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < x1\nx1 < x2\n\ngoal (1 subgoal):\n 1. x2\\<^sup>2 - x1\\<^sup>2 = (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)) * (sqrt (x2\\<^sup>2 - 1) - sqrt (x1\\<^sup>2 - 1))\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nx2\\<^sup>2 - x1\\<^sup>2 = (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)) * (sqrt (x2\\<^sup>2 - 1) - sqrt (x1\\<^sup>2 - 1))\n\ngoal (1 subgoal):\n 1. \\1 < x1; x1 < x2\\ \\ x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nx2 - x1 < (x2\\<^sup>2 - x1\\<^sup>2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\nx2\\<^sup>2 - x1\\<^sup>2 = (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)) * (sqrt (x2\\<^sup>2 - 1) - sqrt (x1\\<^sup>2 - 1))\n[PROOF STEP]\nhave \"sqrt(x2\\<^sup>2 - 1) - sqrt(x1\\<^sup>2 - 1) > x2 - x1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx2 - x1 < (x2\\<^sup>2 - x1\\<^sup>2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\nx2\\<^sup>2 - x1\\<^sup>2 = (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)) * (sqrt (x2\\<^sup>2 - 1) - sqrt (x1\\<^sup>2 - 1))\n\ngoal (1 subgoal):\n 1. x2 - x1 < sqrt (x2\\<^sup>2 - 1) - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nusing *\n[PROOF STATE]\nproof (prove)\nusing this:\nx2 - x1 < (x2\\<^sup>2 - x1\\<^sup>2) / (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1))\nx2\\<^sup>2 - x1\\<^sup>2 = (sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)) * (sqrt (x2\\<^sup>2 - 1) - sqrt (x1\\<^sup>2 - 1))\n0 < sqrt (x1\\<^sup>2 - 1) + sqrt (x2\\<^sup>2 - 1)\n\ngoal (1 subgoal):\n 1. x2 - x1 < sqrt (x2\\<^sup>2 - 1) - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx2 - x1 < sqrt (x2\\<^sup>2 - 1) - sqrt (x1\\<^sup>2 - 1)\n\ngoal (1 subgoal):\n 1. \\1 < x1; x1 < x2\\ \\ x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx2 - x1 < sqrt (x2\\<^sup>2 - 1) - sqrt (x1\\<^sup>2 - 1)\n\ngoal (1 subgoal):\n 1. x2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx2 - sqrt (x2\\<^sup>2 - 1) < x1 - sqrt (x1\\<^sup>2 - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4760, "file": "Poincare_Disc_Poincare_Lines_Axis_Intersections", "length": 34, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127380808499, "lm_q2_score": 0.8198933359135361, "lm_q1q2_score": 0.7001173634041696}}

outcomes\\ \\ (?x \\ ?y \\ (?y, ?x) \\ relation) = (measure_pmf.expectation ?y u < measure_pmf.expectation ?x u)\n\ngoal (1 subgoal):\n 1. (p \\ q \\ (q, p) \\ relation) = ((\\z\\outcomes. pmf q z * u z) < (\\z\\outcomes. pmf p z * u z))\n[PROOF STEP]\nby presburger", "meta": {"llama_tokens": 534, "file": "Neumann_Morgenstern_Utility_Expected_Utility", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942119105695, "lm_q2_score": 0.805632181981183, "lm_q1q2_score": 0.7277228869124852}} {"text": "[STATEMENT]\nlemma prob_long_aux: \n shows \"2 / pi * ((l / d) - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)) \\ 0\"\n and \"Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = \n ennreal (2 / pi * ((l / d) - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ 2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)) &&& emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nusing emeasure_buffon_set'_long(1)\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d\n\ngoal (1 subgoal):\n 1. 0 \\ 2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)) &&& emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ 0 \\ 2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l))\n 2. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nhave *: \"l * sqrt ((l\\<^sup>2 - d\\<^sup>2) / l\\<^sup>2) + 0 \\ l + d * arccos (d / l)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. l * sqrt ((l\\<^sup>2 - d\\<^sup>2) / l\\<^sup>2) + 0 \\ l + d * arccos (d / l)\n[PROOF STEP]\nusing d l_ge_d\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\nd \\ l\n\ngoal (1 subgoal):\n 1. l * sqrt ((l\\<^sup>2 - d\\<^sup>2) / l\\<^sup>2) + 0 \\ l + d * arccos (d / l)\n[PROOF STEP]\nby (intro add_mono mult_nonneg_nonneg arccos_lbound) (auto simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nl * sqrt ((l\\<^sup>2 - d\\<^sup>2) / l\\<^sup>2) + 0 \\ l + d * arccos (d / l)\n\ngoal (2 subgoals):\n 1. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ 0 \\ 2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l))\n 2. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nhave \"l / d \\ sqrt ((l / d)\\<^sup>2 - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt ((l / d)\\<^sup>2 - 1) \\ l / d\n[PROOF STEP]\nusing l d l_ge_d\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < l\n0 < d\nd \\ l\n\ngoal (1 subgoal):\n 1. sqrt ((l / d)\\<^sup>2 - 1) \\ l / d\n[PROOF STEP]\nby (intro real_le_lsqrt) (auto simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nsqrt ((l / d)\\<^sup>2 - 1) \\ l / d\n\ngoal (2 subgoals):\n 1. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ 0 \\ 2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l))\n 2. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nthus \"2 / pi * ((l / d) - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)) \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt ((l / d)\\<^sup>2 - 1) \\ l / d\n\ngoal (1 subgoal):\n 1. 0 \\ 2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l))\n[PROOF STEP]\nusing d l l_ge_d\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt ((l / d)\\<^sup>2 - 1) \\ l / d\n0 < d\n0 < l\nd \\ l\n\ngoal (1 subgoal):\n 1. 0 \\ 2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l))\n[PROOF STEP]\nby (intro mult_nonneg_nonneg add_nonneg_nonneg arccos_lbound) (auto simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ 2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l))\n\ngoal (1 subgoal):\n 1. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nhave \"emeasure Buffon {(x,\\). needle l x \\ \\ {-d/2, d/2} \\ {}} = \n ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / ennreal (2 * d * pi)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / ennreal (2 * d * pi)\n[PROOF STEP]\nusing d l l_ge_d *\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n0 < l\nd \\ l\nl * sqrt ((l\\<^sup>2 - d\\<^sup>2) / l\\<^sup>2) + 0 \\ l + d * arccos (d / l)\n\ngoal (1 subgoal):\n 1. emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / ennreal (2 * d * pi)\n[PROOF STEP]\nunfolding buffon_prob_aux emeasure_set_long ennreal_numeral [symmetric]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n0 < l\nd \\ l\nl * sqrt ((l\\<^sup>2 - d\\<^sup>2) / l\\<^sup>2) + 0 \\ l + d * arccos (d / l)\n\ngoal (1 subgoal):\n 1. ennreal 4 * ennreal (l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d) / ennreal (2 * d * pi) = ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / ennreal (2 * d * pi)\n[PROOF STEP]\nby (subst ennreal_mult [symmetric])\n (auto intro!: add_nonneg_nonneg mult_nonneg_nonneg simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nemeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / ennreal (2 * d * pi)\n\ngoal (1 subgoal):\n 1. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nemeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / ennreal (2 * d * pi)\n\ngoal (1 subgoal):\n 1. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nhave \"\\ = ennreal ((4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / (2 * d * pi))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / ennreal (2 * d * pi) = ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d) / (2 * d * pi))\n[PROOF STEP]\nusing d l *\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n0 < l\nl * sqrt ((l\\<^sup>2 - d\\<^sup>2) / l\\<^sup>2) + 0 \\ l + d * arccos (d / l)\n\ngoal (1 subgoal):\n 1. ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / ennreal (2 * d * pi) = ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d) / (2 * d * pi))\n[PROOF STEP]\nby (subst divide_ennreal) (auto simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / ennreal (2 * d * pi) = ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d) / (2 * d * pi))\n\ngoal (1 subgoal):\n 1. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / ennreal (2 * d * pi) = ennreal (4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d) / (2 * d * pi))\n\ngoal (1 subgoal):\n 1. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nhave \"(4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d)) / (2 * d * pi) =\n 2 / pi * (l / d - l / d * sqrt ((d / l)^2 * ((l / d)^2 - 1)) + arccos (d / l))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d) / (2 * d * pi) = 2 / pi * (l / d - l / d * sqrt ((d / l)\\<^sup>2 * ((l / d)\\<^sup>2 - 1)) + arccos (d / l))\n[PROOF STEP]\nusing d l\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n0 < l\n\ngoal (1 subgoal):\n 1. 4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d) / (2 * d * pi) = 2 / pi * (l / d - l / d * sqrt ((d / l)\\<^sup>2 * ((l / d)\\<^sup>2 - 1)) + arccos (d / l))\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d) / (2 * d * pi) = 2 / pi * (l / d - l / d * sqrt ((d / l)\\<^sup>2 * ((l / d)\\<^sup>2 - 1)) + arccos (d / l))\n\ngoal (1 subgoal):\n 1. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n4 * (l - l * sqrt (1 - (d / l)\\<^sup>2) + arccos (d / l) * d) / (2 * d * pi) = 2 / pi * (l / d - l / d * sqrt ((d / l)\\<^sup>2 * ((l / d)\\<^sup>2 - 1)) + arccos (d / l))\n\ngoal (1 subgoal):\n 1. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nhave \"l / d * sqrt ((d / l)^2 * ((l / d)^2 - 1)) = sqrt ((l / d) ^ 2 - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. l / d * sqrt ((d / l)\\<^sup>2 * ((l / d)\\<^sup>2 - 1)) = sqrt ((l / d)\\<^sup>2 - 1)\n[PROOF STEP]\nusing d l l_ge_d\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n0 < l\nd \\ l\n\ngoal (1 subgoal):\n 1. l / d * sqrt ((d / l)\\<^sup>2 * ((l / d)\\<^sup>2 - 1)) = sqrt ((l / d)\\<^sup>2 - 1)\n[PROOF STEP]\nunfolding real_sqrt_mult real_sqrt_abs\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n0 < l\nd \\ l\n\ngoal (1 subgoal):\n 1. l / d * (\\d / l\\ * sqrt ((l / d)\\<^sup>2 - 1)) = sqrt ((l / d)\\<^sup>2 - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nl / d * sqrt ((d / l)\\<^sup>2 * ((l / d)\\<^sup>2 - 1)) = sqrt ((l / d)\\<^sup>2 - 1)\n\ngoal (1 subgoal):\n 1. 0 \\ l * (1 - sqrt (1 - (d / l)\\<^sup>2)) + arccos (d / l) * d \\ emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nemeasure local.Buffon {a. case a of (x, \\) \\ needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\nshow \"emeasure Buffon {(x,\\). needle l x \\ \\ {-d/2, d/2} \\ {}} = \n ennreal (2 / pi * ((l / d) - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nemeasure local.Buffon {a. case a of (x, \\) \\ needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n\ngoal (1 subgoal):\n 1. emeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nemeasure local.Buffon {(x, \\). needle l x \\ \\ {- d / 2, d / 2} \\ {}} = ennreal (2 / pi * (l / d - sqrt ((l / d)\\<^sup>2 - 1) + arccos (d / l)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 6294, "file": "Buffons_Needle_Buffons_Needle", "length": 32, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032941962904956, "lm_q2_score": 0.8056321796478255, "lm_q1q2_score": 0.7277228722207427}} {"text": "[STATEMENT]\nlemma padic_add_comm0:\nassumes \"prime p\"\nshows \"(padic_add p x y)= (padic_add p y x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. padic_add p x y = padic_add p y x\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. padic_add p x y = padic_add p y x\n[PROOF STEP]\nunfolding padic_add_def\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. (\\n. x n \\\\<^bsub>residue_ring (p ^ n)\\<^esub> y n) = (\\n. y n \\\\<^bsub>residue_ring (p ^ n)\\<^esub> x n)\n[PROOF STEP]\nusing padic_integers.residue_add_comm[of p]\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\npadic_integers p \\ ?x \\\\<^bsub>residue_ring (p ^ ?k)\\<^esub> ?y = ?y \\\\<^bsub>residue_ring (p ^ ?k)\\<^esub> ?x\n\ngoal (1 subgoal):\n 1. (\\n. x n \\\\<^bsub>residue_ring (p ^ n)\\<^esub> y n) = (\\n. y n \\\\<^bsub>residue_ring (p ^ n)\\<^esub> x n)\n[PROOF STEP]\nby (simp add: padic_integers_def)", "meta": {"llama_tokens": 461, "file": "Padic_Ints_Padic_Integers", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9059898254600903, "lm_q2_score": 0.8031737963569014, "lm_q1q2_score": 0.7276672875755072}} {"text": "[STATEMENT]\nlemma constraints_of_mat_vec_solution: \n assumes A: \"A \\ carrier_mat nr nc\" \n and b: \"b \\ carrier_vec nr\" \n shows \"(\\ x \\ carrier_vec nc. A *\\<^sub>v x \\ b) = \n (\\ v. \\ c \\ constraints_of_mat_vec A b. v \\\\<^sub>l\\<^sub>e c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\carrier_vec nc. A *\\<^sub>v x \\ b) = (\\v. \\c\\constraints_of_mat_vec A b. v \\\\<^sub>l\\<^sub>e c)\n[PROOF STEP]\nusing constraints_of_mat_vec_solution_1[OF assms] constraints_of_mat_vec_solution_2[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\carrier_vec nc. A *\\<^sub>v x \\ b \\ \\v. \\c\\constraints_of_mat_vec A b. v \\\\<^sub>l\\<^sub>e c\n\\v. \\c\\constraints_of_mat_vec A b. v \\\\<^sub>l\\<^sub>e c \\ \\x\\carrier_vec nc. A *\\<^sub>v x \\ b\n\ngoal (1 subgoal):\n 1. (\\x\\carrier_vec nc. A *\\<^sub>v x \\ b) = (\\v. \\c\\constraints_of_mat_vec A b. v \\\\<^sub>l\\<^sub>e c)\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 497, "file": "Farkas_Matrix_Farkas", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.905989815306765, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7276672730180113}} {"text": "[STATEMENT]\nlemma integral_le:\n fixes f :: \"'n::euclidean_space \\ real\"\n assumes \"f integrable_on S\"\n and \"g integrable_on S\"\n and \"\\x. x \\ S \\ f x \\ g x\"\n shows \"integral S f \\ integral S g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral S f \\ integral S g\n[PROOF STEP]\nby (rule has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)])", "meta": {"llama_tokens": 165, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976953003183443, "lm_q2_score": 0.810478913248044, "lm_q1q2_score": 0.7275631114298882}} {"text": "[STATEMENT]\nlemma pochhammer_times_pochhammer_half:\n fixes z :: \"'a::field_char_0\"\n shows \"pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n) = (\\k=0..2*n+1. z + of_nat k / 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. pochhammer z (Suc 0) * pochhammer (z + (1::'a) / (2::'a)) (Suc 0) = (\\k = 0..2 * 0 + 1. z + of_nat k / (2::'a))\n 2. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. pochhammer z (Suc 0) * pochhammer (z + (1::'a) / (2::'a)) (Suc 0) = (\\k = 0..2 * 0 + 1. z + of_nat k / (2::'a))\n 2. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pochhammer z (Suc 0) * pochhammer (z + (1::'a) / (2::'a)) (Suc 0) = (\\k = 0..2 * 0 + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nby (simp add: atLeast0_atMost_Suc)\n[PROOF STATE]\nproof (state)\nthis:\npochhammer z (Suc 0) * pochhammer (z + (1::'a) / (2::'a)) (Suc 0) = (\\k = 0..2 * 0 + 1. z + of_nat k / (2::'a))\n\ngoal (1 subgoal):\n 1. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\npochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a))\n\ngoal (1 subgoal):\n 1. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\ndefine n' where \"n' = Suc n\"\n[PROOF STATE]\nproof (state)\nthis:\nn' = Suc n\n\ngoal (1 subgoal):\n 1. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nhave \"pochhammer z (Suc n') * pochhammer (z + 1 / 2) (Suc n') =\n (pochhammer z n' * pochhammer (z + 1 / 2) n') * ((z + of_nat n') * (z + 1/2 + of_nat n'))\"\n (is \"_ = _ * ?A\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pochhammer z (Suc n') * pochhammer (z + (1::'a) / (2::'a)) (Suc n') = pochhammer z n' * pochhammer (z + (1::'a) / (2::'a)) n' * ((z + of_nat n') * (z + (1::'a) / (2::'a) + of_nat n'))\n[PROOF STEP]\nby (simp_all add: pochhammer_rec' mult_ac)\n[PROOF STATE]\nproof (state)\nthis:\npochhammer z (Suc n') * pochhammer (z + (1::'a) / (2::'a)) (Suc n') = pochhammer z n' * pochhammer (z + (1::'a) / (2::'a)) n' * ((z + of_nat n') * (z + (1::'a) / (2::'a) + of_nat n'))\n\ngoal (1 subgoal):\n 1. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\npochhammer z (Suc n') * pochhammer (z + (1::'a) / (2::'a)) (Suc n') = pochhammer z n' * pochhammer (z + (1::'a) / (2::'a)) n' * ((z + of_nat n') * (z + (1::'a) / (2::'a) + of_nat n'))\n\ngoal (1 subgoal):\n 1. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nhave \"?A = (z + of_nat (Suc (2 * n + 1)) / 2) * (z + of_nat (Suc (Suc (2 * n + 1))) / 2)\"\n (is \"_ = ?B\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (z + of_nat n') * (z + (1::'a) / (2::'a) + of_nat n') = (z + of_nat (Suc (2 * n + 1)) / (2::'a)) * (z + of_nat (Suc (Suc (2 * n + 1))) / (2::'a))\n[PROOF STEP]\nby (simp add: field_simps n'_def)\n[PROOF STATE]\nproof (state)\nthis:\n(z + of_nat n') * (z + (1::'a) / (2::'a) + of_nat n') = (z + of_nat (Suc (2 * n + 1)) / (2::'a)) * (z + of_nat (Suc (Suc (2 * n + 1))) / (2::'a))\n\ngoal (1 subgoal):\n 1. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(z + of_nat n') * (z + (1::'a) / (2::'a) + of_nat n') = (z + of_nat (Suc (2 * n + 1)) / (2::'a)) * (z + of_nat (Suc (Suc (2 * n + 1))) / (2::'a))\n\ngoal (1 subgoal):\n 1. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nnote Suc[folded n'_def]\n[PROOF STATE]\nproof (state)\nthis:\npochhammer z n' * pochhammer (z + (1::'a) / (2::'a)) n' = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a))\n\ngoal (1 subgoal):\n 1. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\npochhammer z n' * pochhammer (z + (1::'a) / (2::'a)) n' = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a))\n\ngoal (1 subgoal):\n 1. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nhave \"(\\k=0..2 * n + 1. z + of_nat k / 2) * ?B = (\\k=0..2 * Suc n + 1. z + of_nat k / 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) * ((z + of_nat (Suc (2 * n + 1)) / (2::'a)) * (z + of_nat (Suc (Suc (2 * n + 1))) / (2::'a))) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nby (simp add: atLeast0_atMost_Suc)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) * ((z + of_nat (Suc (2 * n + 1)) / (2::'a)) * (z + of_nat (Suc (Suc (2 * n + 1))) / (2::'a))) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n\ngoal (1 subgoal):\n 1. \\n. pochhammer z (Suc n) * pochhammer (z + (1::'a) / (2::'a)) (Suc n) = (\\k = 0..2 * n + 1. z + of_nat k / (2::'a)) \\ pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\npochhammer z (Suc n') * pochhammer (z + (1::'a) / (2::'a)) (Suc n') = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\npochhammer z (Suc n') * pochhammer (z + (1::'a) / (2::'a)) (Suc n') = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n\ngoal (1 subgoal):\n 1. pochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n[PROOF STEP]\nby (simp add: n'_def)\n[PROOF STATE]\nproof (state)\nthis:\npochhammer z (Suc (Suc n)) * pochhammer (z + (1::'a) / (2::'a)) (Suc (Suc n)) = (\\k = 0..2 * Suc n + 1. z + of_nat k / (2::'a))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4722, "file": null, "length": 22, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976952921073469, "lm_q2_score": 0.8104789155369047, "lm_q1q2_score": 0.7275631068297473}} {"text": "[STATEMENT]\nlemma cos_arcsin_nonzero: \"- 1 < x \\ x < 1 \\ cos (arcsin x) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\- 1 < x; x < 1\\ \\ cos (arcsin x) \\ 0\n[PROOF STEP]\nusing arcsin_lt_bounded cos_gt_zero_pi\n[PROOF STATE]\nproof (prove)\nusing this:\n\\- 1 < ?y; ?y < 1\\ \\ - (pi / 2) < arcsin ?y \\ arcsin ?y < pi / 2\n\\- (pi / 2) < ?x; ?x < pi / 2\\ \\ 0 < cos ?x\n\ngoal (1 subgoal):\n 1. \\- 1 < x; x < 1\\ \\ cos (arcsin x) \\ 0\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 294, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976953030553434, "lm_q2_score": 0.8104789040926008, "lm_q1q2_score": 0.7275631054293699}} {"text": "[STATEMENT]\nlemma card2 :\n shows \"card{a, b} \\ Suc 0 \\ card{a, b} \\ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc 0 \\ card {a, b} \\ card {a, b} \\ 2\n[PROOF STEP]\nby (simp add: card_insert_if)", "meta": {"llama_tokens": 112, "file": "Involutions2Squares_Involutions2Squares", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952811593495, "lm_q2_score": 0.8104789040926008, "lm_q1q2_score": 0.7275630876831287}} {"text": "[STATEMENT]\nlemma gbinomial_trinomial_revision:\n assumes \"k \\ m\"\n shows \"(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nhave \"(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ m\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n[PROOF STEP]\nby (simp add: binomial_gbinomial [symmetric] binomial_fact)\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nhave \"\\ = (a gchoose k) * (a - of_nat k gchoose (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ m\n\ngoal (1 subgoal):\n 1. (a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nby (simp add: gbinomial_pochhammer power_diff pochhammer_product)\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1242, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278633625322, "lm_q2_score": 0.824461932846258, "lm_q1q2_score": 0.727528181825267}} {"text": "[STATEMENT]\nlemma gbinomial_trinomial_revision:\n assumes \"k \\ m\"\n shows \"(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nhave \"(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ m\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n[PROOF STEP]\nby (simp add: binomial_gbinomial [symmetric] binomial_fact)\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nhave \"\\ = (a gchoose k) * (a - of_nat k gchoose (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ m\n\ngoal (1 subgoal):\n 1. (a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nby (simp add: gbinomial_pochhammer power_diff pochhammer_product)\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1242, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278633625322, "lm_q2_score": 0.824461932846258, "lm_q1q2_score": 0.727528181825267}} {"text": "[STATEMENT]\nlemma orthogonal_matrix_orthonormal_rows:\n fixes A :: \"real^'n^'n\"\n shows \"orthogonal_matrix A \\\n (\\i. norm(row i A) = 1) \\\n (\\i j. i \\ j \\ orthogonal (row i A) (row j A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal_matrix A = ((\\i. norm (row i A) = 1) \\ (\\i j. i \\ j \\ orthogonal (row i A) (row j A)))\n[PROOF STEP]\nusing orthogonal_matrix_orthonormal_columns [of \"transpose A\"]\n[PROOF STATE]\nproof (prove)\nusing this:\northogonal_matrix (Finite_Cartesian_Product.transpose A) = ((\\i. norm (column i (Finite_Cartesian_Product.transpose A)) = 1) \\ (\\i j. i \\ j \\ orthogonal (column i (Finite_Cartesian_Product.transpose A)) (column j (Finite_Cartesian_Product.transpose A))))\n\ngoal (1 subgoal):\n 1. orthogonal_matrix A = ((\\i. norm (row i A) = 1) \\ (\\i j. i \\ j \\ orthogonal (row i A) (row j A)))\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 384, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278540866547, "lm_q2_score": 0.8244619199068831, "lm_q1q2_score": 0.7275281627595942}} {"text": "[STATEMENT]\nlemma sum_lessThan_conv_atMost_nat:\n fixes f :: \"nat \\ 'b :: ab_group_add\"\n shows \"sum f {.. 0) + 1) in (rank_A \\ (if (\\a. P_mult_b $ a \\ 0) \n then (to_nat (GREATEST a. P_mult_b $ a \\ 0) + 1) else 0)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. consistent A b = (let GJ_P = Gauss_Jordan_PA A; P_mult_b = fst GJ_P *v b in Let (if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (snd GJ_P) \\ 0) + 1) ((\\) (if \\a. P_mult_b $ a \\ (0::'a) then mod_type_class.to_nat (GREATEST a. P_mult_b $ a \\ (0::'a)) + 1 else 0)))\n[PROOF STEP]\nunfolding consistent_eq_rank_ge Let_def rank_Gauss_Jordan_code\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((if \\a. (P_Gauss_Jordan A *v b) $ a \\ (0::'a) then mod_type_class.to_nat (GREATEST a. (P_Gauss_Jordan A *v b) $ a \\ (0::'a)) + 1 else 0) \\ (if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (Gauss_Jordan A) \\ 0) + 1)) = ((if \\a. (fst (Gauss_Jordan_PA A) *v b) $ a \\ (0::'a) then mod_type_class.to_nat (GREATEST a. (fst (Gauss_Jordan_PA A) *v b) $ a \\ (0::'a)) + 1 else 0) \\ (if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (snd (Gauss_Jordan_PA A)) \\ 0) + 1))\n[PROOF STEP]\nunfolding Gauss_Jordan_PA_eq P_Gauss_Jordan_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((if \\a. (fst (Gauss_Jordan_PA A) *v b) $ a \\ (0::'a) then mod_type_class.to_nat (GREATEST a. (fst (Gauss_Jordan_PA A) *v b) $ a \\ (0::'a)) + 1 else 0) \\ (if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (Gauss_Jordan A) \\ 0) + 1)) = ((if \\a. (fst (Gauss_Jordan_PA A) *v b) $ a \\ (0::'a) then mod_type_class.to_nat (GREATEST a. (fst (Gauss_Jordan_PA A) *v b) $ a \\ (0::'a)) + 1 else 0) \\ (if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (Gauss_Jordan A) \\ 0) + 1))\n[PROOF STEP]\n..", "meta": {"llama_tokens": 1068, "file": "Gauss_Jordan_System_Of_Equations", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9173026595857203, "lm_q2_score": 0.7931059462938815, "lm_q1q2_score": 0.727518193868627}} {"text": "[STATEMENT]\nlemma cosh_real_strict_mono:\n assumes \"0 \\ x\" and \"x < (y::real)\"\n shows \"cosh x < cosh y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cosh x < cosh y\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. cosh x < cosh y\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ x\nx < y\n[PROOF STEP]\nhave \"\\z>x. z < y \\ cosh y - cosh x = (y - x) * sinh z\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ x\nx < y\n\ngoal (1 subgoal):\n 1. \\z>x. z < y \\ cosh y - cosh x = (y - x) * sinh z\n[PROOF STEP]\nby (intro MVT2) (auto dest: connectedD_interval intro!: derivative_eq_intros)\n[PROOF STATE]\nproof (state)\nthis:\n\\z>x. z < y \\ cosh y - cosh x = (y - x) * sinh z\n\ngoal (1 subgoal):\n 1. cosh x < cosh y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\z>x. z < y \\ cosh y - cosh x = (y - x) * sinh z\n[PROOF STEP]\nobtain z where z: \"z > x\" \"z < y\" \"cosh y - cosh x = (y - x) * sinh z\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\z>x. z < y \\ cosh y - cosh x = (y - x) * sinh z\n\ngoal (1 subgoal):\n 1. (\\z. \\x < z; z < y; cosh y - cosh x = (y - x) * sinh z\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nx < z\nz < y\ncosh y - cosh x = (y - x) * sinh z\n\ngoal (1 subgoal):\n 1. cosh x < cosh y\n[PROOF STEP]\nnote \\cosh y - cosh x = (y - x) * sinh z\\\n[PROOF STATE]\nproof (state)\nthis:\ncosh y - cosh x = (y - x) * sinh z\n\ngoal (1 subgoal):\n 1. cosh x < cosh y\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncosh y - cosh x = (y - x) * sinh z\n\ngoal (1 subgoal):\n 1. cosh x < cosh y\n[PROOF STEP]\nfrom \\z > x\\ and assms\n[PROOF STATE]\nproof (chain)\npicking this:\nx < z\n0 \\ x\nx < y\n[PROOF STEP]\nhave \"(y - x) * sinh z > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx < z\n0 \\ x\nx < y\n\ngoal (1 subgoal):\n 1. 0 < (y - x) * sinh z\n[PROOF STEP]\nby (intro mult_pos_pos) auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < (y - x) * sinh z\n\ngoal (1 subgoal):\n 1. cosh x < cosh y\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < cosh y - cosh x\n[PROOF STEP]\nshow \"cosh x < cosh y\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < cosh y - cosh x\n\ngoal (1 subgoal):\n 1. cosh x < cosh y\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncosh x < cosh y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1156, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127455162773, "lm_q2_score": 0.8519528094861981, "lm_q1q2_score": 0.7274933625986654}} {"text": "[STATEMENT]\nlemma phi'_nonzero:\n assumes \"m > 0\"\n shows \"phi' m > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < phi' m\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 < phi' m\n[PROOF STEP]\nhave \"1 \\ {x. 1 \\ x \\ x \\ m \\ coprime x m}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 \\ {x. 1 \\ x \\ x \\ m \\ coprime x m}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < m\n\ngoal (1 subgoal):\n 1. 1 \\ {x. 1 \\ x \\ x \\ m \\ coprime x m}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ {x. 1 \\ x \\ x \\ m \\ coprime x m}\n\ngoal (1 subgoal):\n 1. 0 < phi' m\n[PROOF STEP]\nhence \"card {x. 1 \\ x \\ x \\ m \\ coprime x m} > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ {x. 1 \\ x \\ x \\ m \\ coprime x m}\n\ngoal (1 subgoal):\n 1. 0 < card {x. 1 \\ x \\ x \\ m \\ coprime x m}\n[PROOF STEP]\nby (auto simp: card_gt_0_iff)\n[PROOF STATE]\nproof (state)\nthis:\n0 < card {x. 1 \\ x \\ x \\ m \\ coprime x m}\n\ngoal (1 subgoal):\n 1. 0 < phi' m\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < card {x. 1 \\ x \\ x \\ m \\ coprime x m}\n\ngoal (1 subgoal):\n 1. 0 < phi' m\n[PROOF STEP]\nunfolding phi'_def\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < card {x. 1 \\ x \\ x \\ m \\ coprime x m}\n\ngoal (1 subgoal):\n 1. 0 < card {x. 1 \\ x \\ x \\ m \\ coprime x m}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < phi' m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 809, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711756575749, "lm_q2_score": 0.855851143290548, "lm_q1q2_score": 0.7274488024505467}} {"text": "[STATEMENT]\nlemma distinct_set_subseqs:\n assumes \"distinct xs\"\n shows \"distinct (map set (subseqs xs))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. distinct (map set (subseqs xs))\n[PROOF STEP]\nproof (rule card_distinct)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (set (map set (subseqs xs))) = length (map set (subseqs xs))\n[PROOF STEP]\nhave \"finite (set xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (set xs)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nfinite (set xs)\n\ngoal (1 subgoal):\n 1. card (set (map set (subseqs xs))) = length (map set (subseqs xs))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (set xs)\n[PROOF STEP]\nhave \"card (Pow (set xs)) = 2 ^ card (set xs)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (set xs)\n\ngoal (1 subgoal):\n 1. card (Pow (set xs)) = 2 ^ card (set xs)\n[PROOF STEP]\nby (rule card_Pow)\n[PROOF STATE]\nproof (state)\nthis:\ncard (Pow (set xs)) = 2 ^ card (set xs)\n\ngoal (1 subgoal):\n 1. card (set (map set (subseqs xs))) = length (map set (subseqs xs))\n[PROOF STEP]\nwith assms distinct_card [of xs]\n[PROOF STATE]\nproof (chain)\npicking this:\ndistinct xs\ndistinct xs \\ card (set xs) = length xs\ncard (Pow (set xs)) = 2 ^ card (set xs)\n[PROOF STEP]\nhave \"card (Pow (set xs)) = 2 ^ length xs\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndistinct xs\ndistinct xs \\ card (set xs) = length xs\ncard (Pow (set xs)) = 2 ^ card (set xs)\n\ngoal (1 subgoal):\n 1. card (Pow (set xs)) = 2 ^ length xs\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (Pow (set xs)) = 2 ^ length xs\n\ngoal (1 subgoal):\n 1. card (set (map set (subseqs xs))) = length (map set (subseqs xs))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (Pow (set xs)) = 2 ^ length xs\n[PROOF STEP]\nshow \"card (set (map set (subseqs xs))) = length (map set (subseqs xs))\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (Pow (set xs)) = 2 ^ length xs\n\ngoal (1 subgoal):\n 1. card (set (map set (subseqs xs))) = length (map set (subseqs xs))\n[PROOF STEP]\nby (simp add: subseqs_powset length_subseqs)\n[PROOF STATE]\nproof (state)\nthis:\ncard (set (map set (subseqs xs))) = length (map set (subseqs xs))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 929, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711642563823, "lm_q2_score": 0.8558511488056151, "lm_q1q2_score": 0.727448797380471}} {"text": "[STATEMENT]\nlemma representation:\n fixes k m :: nat\n assumes \"k > 0\" \"n = m mod k\" \"l = (m-n)div k\"\n shows \"m = n+k*l \\ 0\\n \\ n\\k-1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. m = n + k * l \\ 0 \\ n \\ n \\ k - 1\n[PROOF STEP]\nby (metis Suc_pred' assms le_add2 le_add_same_cancel2 \n less_Suc_eq_le minus_mod_eq_div_mult minus_mod_eq_mult_div mod_div_mult_eq\n mod_less_divisor neq0_conv nonzero_mult_div_cancel_left)", "meta": {"llama_tokens": 217, "file": "DPRM_Theorem_Diophantine_Exponentiation", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045907347108, "lm_q2_score": 0.8198933425148214, "lm_q1q2_score": 0.7274131373919762}} {"text": "[STATEMENT]\nlemma continuous_map_prod_top:\n \"continuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x,y). (f x, g y)) \\\n topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n[PROOF STEP]\nproof (cases \"topspace (prod_topology X Y) = {}\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. topspace (prod_topology X Y) = {} \\ continuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n 2. topspace (prod_topology X Y) \\ {} \\ continuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\ntopspace (prod_topology X Y) = {}\n\ngoal (2 subgoals):\n 1. topspace (prod_topology X Y) = {} \\ continuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n 2. topspace (prod_topology X Y) \\ {} \\ continuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ntopspace (prod_topology X Y) = {}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntopspace (prod_topology X Y) = {}\n\ngoal (1 subgoal):\n 1. continuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n[PROOF STEP]\nby (simp add: continuous_map_on_empty)\n[PROOF STATE]\nproof (state)\nthis:\ncontinuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n\ngoal (1 subgoal):\n 1. topspace (prod_topology X Y) \\ {} \\ continuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. topspace (prod_topology X Y) \\ {} \\ continuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\ntopspace (prod_topology X Y) \\ {}\n\ngoal (1 subgoal):\n 1. topspace (prod_topology X Y) \\ {} \\ continuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ntopspace (prod_topology X Y) \\ {}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntopspace (prod_topology X Y) \\ {}\n\ngoal (1 subgoal):\n 1. continuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n[PROOF STEP]\nby (simp add: continuous_map_paired case_prod_unfold continuous_map_of_fst [unfolded o_def] continuous_map_of_snd [unfolded o_def])\n[PROOF STATE]\nproof (state)\nthis:\ncontinuous_map (prod_topology X Y) (prod_topology X' Y') (\\(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \\ continuous_map X X' f \\ continuous_map Y Y' g)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1610, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045937171068, "lm_q2_score": 0.8198933337131077, "lm_q1q2_score": 0.7274131320283019}} {"text": "[STATEMENT]\nlemma r01_to_r01_r01_fst_def':\n \"r01_to_r01_r01_fst r = (\\n. real (r01_binary_expansion' r (2*n)) * (1/2)^(n+1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r01_to_r01_r01_fst r = (\\n. real (r01_binary_expansion' r (2 * n)) * (1 / 2) ^ (n + 1))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. r01_to_r01_r01_fst r = (\\n. real (r01_binary_expansion' r (2 * n)) * (1 / 2) ^ (n + 1))\n[PROOF STEP]\nhave \"r01_to_r01_r01_fst_sum r = (\\n. \\i=0..n. real (r01_binary_expansion' r (2*i)) * (1/2)^(i+1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. r01_to_r01_r01_fst_sum r = (\\n. \\i = 0..n. real (r01_binary_expansion' r (2 * i)) * (1 / 2) ^ (i + 1))\n[PROOF STEP]\nby(auto simp add: r01_to_r01_r01_fst_sum_def r01_binary_sum_def r01_to_r01_r01_fst'_def)\n[PROOF STATE]\nproof (state)\nthis:\nr01_to_r01_r01_fst_sum r = (\\n. \\i = 0..n. real (r01_binary_expansion' r (2 * i)) * (1 / 2) ^ (i + 1))\n\ngoal (1 subgoal):\n 1. r01_to_r01_r01_fst r = (\\n. real (r01_binary_expansion' r (2 * n)) * (1 / 2) ^ (n + 1))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nr01_to_r01_r01_fst_sum r = (\\n. \\i = 0..n. real (r01_binary_expansion' r (2 * i)) * (1 / 2) ^ (i + 1))\n\ngoal (1 subgoal):\n 1. r01_to_r01_r01_fst r = (\\n. real (r01_binary_expansion' r (2 * n)) * (1 / 2) ^ (n + 1))\n[PROOF STEP]\nusing lim_sum_ai real01_binary_expansion'_0or1\n[PROOF STATE]\nproof (prove)\nusing this:\nr01_to_r01_r01_fst_sum r = (\\n. \\i = 0..n. real (r01_binary_expansion' r (2 * i)) * (1 / 2) ^ (i + 1))\n(\\n. ?a n \\ {0, 1}) \\ lim (\\n. \\i = 0..n. real (?a i) * (1 / 2) ^ Suc i) = (\\n. real (?a n) * (1 / 2) ^ Suc n)\nr01_binary_expansion' ?r ?n \\ {0, 1}\n\ngoal (1 subgoal):\n 1. r01_to_r01_r01_fst r = (\\n. real (r01_binary_expansion' r (2 * n)) * (1 / 2) ^ (n + 1))\n[PROOF STEP]\nby(simp add: r01_to_r01_r01_fst_def)\n[PROOF STATE]\nproof (state)\nthis:\nr01_to_r01_r01_fst r = (\\n. real (r01_binary_expansion' r (2 * n)) * (1 / 2) ^ (n + 1))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1186, "file": "Quasi_Borel_Spaces_StandardBorel", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767906859264, "lm_q2_score": 0.8289388125473628, "lm_q1q2_score": 0.7273745689090627}} {"text": "[STATEMENT]\nlemma log2_of_power_le: \"\\ m \\ 2 ^ n; m > 0 \\ \\ log 2 m \\ n\" for m n :: nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\m \\ 2 ^ n; 0 < m\\ \\ log 2 (real m) \\ real n\n[PROOF STEP]\nusing log_of_power_le[of _ 2]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\real ?m \\ 2 ^ ?n; 1 < 2; 0 < ?m\\ \\ log 2 (real ?m) \\ real ?n\n\ngoal (1 subgoal):\n 1. \\m \\ 2 ^ n; 0 < m\\ \\ log 2 (real m) \\ real n\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 269, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767778695834, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7273745527228427}} {"text": "[STATEMENT]\nlemma det_QR_decomposition:\n fixes A::\"real^'n::{mod_type}^'n::{mod_type}\"\n assumes r: \"rank A = ncols A\"\n shows \"\\det A\\ = \\(prod (\\i. snd(QR_decomposition A)$i$i) (UNIV::'n set))\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nlet ?Q=\"fst(QR_decomposition A)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nlet ?R=\"snd(QR_decomposition A)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nhave det_R: \"det ?R = (prod (\\i. snd(QR_decomposition A)$i$i) (UNIV::'n set))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (snd (QR_decomposition A)) = (\\i\\UNIV. snd (QR_decomposition A) $ i $ i)\n[PROOF STEP]\napply (rule det_upperdiagonal)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. j < i \\ snd (QR_decomposition A) $ i $ j = 0\n[PROOF STEP]\nusing upper_triangular_snd_QR_decomposition[OF r]\n[PROOF STATE]\nproof (prove)\nusing this:\nupper_triangular (snd (QR_decomposition A))\n\ngoal (1 subgoal):\n 1. \\i j. j < i \\ snd (QR_decomposition A) $ i $ j = 0\n[PROOF STEP]\nunfolding upper_triangular_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i j. j < i \\ snd (QR_decomposition A) $ i $ j = 0\n\ngoal (1 subgoal):\n 1. \\i j. j < i \\ snd (QR_decomposition A) $ i $ j = 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndet (snd (QR_decomposition A)) = (\\i\\UNIV. snd (QR_decomposition A) $ i $ i)\n\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nhave \"\\det A\\ = \\det ?Q * det ?R\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\det A\\ = \\det (fst (QR_decomposition A)) * det (snd (QR_decomposition A))\\\n[PROOF STEP]\nby (metis QR_decomposition_mult det_mul r)\n[PROOF STATE]\nproof (state)\nthis:\n\\det A\\ = \\det (fst (QR_decomposition A)) * det (snd (QR_decomposition A))\\\n\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\det A\\ = \\det (fst (QR_decomposition A)) * det (snd (QR_decomposition A))\\\n\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nhave \"... = \\det ?Q\\ * \\det ?R\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\det (fst (QR_decomposition A)) * det (snd (QR_decomposition A))\\ = \\det (fst (QR_decomposition A))\\ * \\det (snd (QR_decomposition A))\\\n[PROOF STEP]\nunfolding abs_mult\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\det (fst (QR_decomposition A))\\ * \\det (snd (QR_decomposition A))\\ = \\det (fst (QR_decomposition A))\\ * \\det (snd (QR_decomposition A))\\\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n\\det (fst (QR_decomposition A)) * det (snd (QR_decomposition A))\\ = \\det (fst (QR_decomposition A))\\ * \\det (snd (QR_decomposition A))\\\n\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\det (fst (QR_decomposition A)) * det (snd (QR_decomposition A))\\ = \\det (fst (QR_decomposition A))\\ * \\det (snd (QR_decomposition A))\\\n\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nhave \"... = 1 * \\det ?R\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\det (fst (QR_decomposition A))\\ * \\det (snd (QR_decomposition A))\\ = 1 * \\det (snd (QR_decomposition A))\\\n[PROOF STEP]\nusing det_orthogonal_matrix[OF orthogonal_matrix_fst_QR_decomposition'[OF r]]\n[PROOF STATE]\nproof (prove)\nusing this:\ndet (fst (QR_decomposition A)) = 1 \\ det (fst (QR_decomposition A)) = - 1\n\ngoal (1 subgoal):\n 1. \\det (fst (QR_decomposition A))\\ * \\det (snd (QR_decomposition A))\\ = 1 * \\det (snd (QR_decomposition A))\\\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\det (fst (QR_decomposition A))\\ * \\det (snd (QR_decomposition A))\\ = 1 * \\det (snd (QR_decomposition A))\\\n\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\det (fst (QR_decomposition A))\\ * \\det (snd (QR_decomposition A))\\ = 1 * \\det (snd (QR_decomposition A))\\\n\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nhave \"... = \\det ?R\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 * \\det (snd (QR_decomposition A))\\ = \\det (snd (QR_decomposition A))\\\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 * \\det (snd (QR_decomposition A))\\ = \\det (snd (QR_decomposition A))\\\n\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 * \\det (snd (QR_decomposition A))\\ = \\det (snd (QR_decomposition A))\\\n\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nhave \"... = \\(prod (\\i. snd(QR_decomposition A)$i$i) (UNIV::'n set))\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\det (snd (QR_decomposition A))\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nunfolding det_R\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n\\det (snd (QR_decomposition A))\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n\ngoal (1 subgoal):\n 1. \\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\det A\\ = \\\\i\\UNIV. snd (QR_decomposition A) $ i $ i\\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3346, "file": "QR_Decomposition_QR_Decomposition", "length": 29, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767778695836, "lm_q2_score": 0.8289388019824946, "lm_q1q2_score": 0.7273745490146721}} {"text": "[STATEMENT]\nlemma rel_frontier_cball [simp]:\n fixes a :: \"'n::euclidean_space\"\n shows \"rel_frontier(cball a r) = (if r = 0 then {} else sphere a r)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\nproof (cases rule: linorder_cases [of r 0])\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. r < 0 \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n 2. r = 0 \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n 3. 0 < r \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\ncase less\n[PROOF STATE]\nproof (state)\nthis:\nr < 0\n\ngoal (3 subgoals):\n 1. r < 0 \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n 2. r = 0 \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n 3. 0 < r \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nr < 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nr < 0\n\ngoal (1 subgoal):\n 1. rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\nby (force simp: sphere_def)\n[PROOF STATE]\nproof (state)\nthis:\nrel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n\ngoal (2 subgoals):\n 1. r = 0 \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n 2. 0 < r \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. r = 0 \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n 2. 0 < r \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\ncase equal\n[PROOF STATE]\nproof (state)\nthis:\nr = 0\n\ngoal (2 subgoals):\n 1. r = 0 \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n 2. 0 < r \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nr = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nr = 0\n\ngoal (1 subgoal):\n 1. rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nrel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n\ngoal (1 subgoal):\n 1. 0 < r \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 < r \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\ncase greater\n[PROOF STATE]\nproof (state)\nthis:\n0 < r\n\ngoal (1 subgoal):\n 1. 0 < r \\ rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < r\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < r\n\ngoal (1 subgoal):\n 1. rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n[PROOF STEP]\nby simp (metis centre_in_ball empty_iff frontier_cball frontier_def interior_cball interior_rel_interior_gen rel_frontier_def)\n[PROOF STATE]\nproof (state)\nthis:\nrel_frontier (cball a r) = (if r = 0 then {} else sphere a r)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1392, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972650509008, "lm_q2_score": 0.8354835432479661, "lm_q1q2_score": 0.7273696877467154}} {"text": "[STATEMENT]\nlemma infsetsum_scaleR_left:\n assumes \"c \\ 0 \\ f abs_summable_on A\"\n shows \"infsetsum (\\x. f x *\\<^sub>R c) A = infsetsum f A *\\<^sub>R c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. f x *\\<^sub>R c) = infsetsum f A *\\<^sub>R c\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'a) \\ f abs_summable_on A\n\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. f x *\\<^sub>R c) = infsetsum f A *\\<^sub>R c\n[PROOF STEP]\nunfolding infsetsum_def abs_summable_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'a) \\ integrable (count_space A) f\n\ngoal (1 subgoal):\n 1. LINT x|count_space A. f x *\\<^sub>R c = integral\\<^sup>L (count_space A) f *\\<^sub>R c\n[PROOF STEP]\nby (rule Bochner_Integration.integral_scaleR_left)", "meta": {"llama_tokens": 370, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894632969137, "lm_q2_score": 0.8128673201042492, "lm_q1q2_score": 0.7273451130876817}} {"text": "[STATEMENT]\nlemma has_complex_derivative_funpow_1:\n \"\\(f has_field_derivative 1) (at z); f z = z\\ \\ (f^^n has_field_derivative 1) (at z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ n has_field_derivative (1::'a)) (at z)\n[PROOF STEP]\nproof (induction n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\(f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ 0 has_field_derivative (1::'a)) (at z)\n 2. \\n. \\\\(f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ n has_field_derivative (1::'a)) (at z); (f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ Suc n has_field_derivative (1::'a)) (at z)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n(f has_field_derivative (1::'a)) (at z)\nf z = z\n\ngoal (2 subgoals):\n 1. \\(f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ 0 has_field_derivative (1::'a)) (at z)\n 2. \\n. \\\\(f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ n has_field_derivative (1::'a)) (at z); (f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ Suc n has_field_derivative (1::'a)) (at z)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(f has_field_derivative (1::'a)) (at z)\nf z = z\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_field_derivative (1::'a)) (at z)\nf z = z\n\ngoal (1 subgoal):\n 1. (f ^^ 0 has_field_derivative (1::'a)) (at z)\n[PROOF STEP]\nby (simp add: id_def)\n[PROOF STATE]\nproof (state)\nthis:\n(f ^^ 0 has_field_derivative (1::'a)) (at z)\n\ngoal (1 subgoal):\n 1. \\n. \\\\(f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ n has_field_derivative (1::'a)) (at z); (f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ Suc n has_field_derivative (1::'a)) (at z)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\\\(f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ n has_field_derivative (1::'a)) (at z); (f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ Suc n has_field_derivative (1::'a)) (at z)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n\\(f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ n has_field_derivative (1::'a)) (at z)\n(f has_field_derivative (1::'a)) (at z)\nf z = z\n\ngoal (1 subgoal):\n 1. \\n. \\\\(f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ n has_field_derivative (1::'a)) (at z); (f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ Suc n has_field_derivative (1::'a)) (at z)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\(f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ n has_field_derivative (1::'a)) (at z)\n(f has_field_derivative (1::'a)) (at z)\nf z = z\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n\\(f has_field_derivative (1::'a)) (at z); f z = z\\ \\ (f ^^ n has_field_derivative (1::'a)) (at z)\n(f has_field_derivative (1::'a)) (at z)\nf z = z\n\ngoal (1 subgoal):\n 1. (f ^^ Suc n has_field_derivative (1::'a)) (at z)\n[PROOF STEP]\nby (metis DERIV_chain funpow_Suc_right mult.right_neutral)\n[PROOF STATE]\nproof (state)\nthis:\n(f ^^ Suc n has_field_derivative (1::'a)) (at z)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1704, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894604912848, "lm_q2_score": 0.8128673178375734, "lm_q1q2_score": 0.72734510877888}} {"text": "[STATEMENT]\nlemma prod_list_map_filter: \"prod_list (map g (filter f xs)) * prod_list (map g (filter (\\ x. \\ f x) xs)) \n = prod_list (map g xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod_list (map g (filter f xs)) * prod_list (map g (filter (\\x. \\ f x) xs)) = prod_list (map g xs)\n[PROOF STEP]\nby (induct xs, auto simp: ac_simps)", "meta": {"llama_tokens": 149, "file": "Berlekamp_Zassenhaus_Berlekamp_Type_Based", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894632969136, "lm_q2_score": 0.8128673110375458, "lm_q1q2_score": 0.7273451049748909}} {"text": "[STATEMENT]\nlemma cos_double_sin: \"cos (2 * w) = 1 - 2 * sin w ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos ((2::'a) * w) = (1::'a) - (2::'a) * (sin w)\\<^sup>2\n[PROOF STEP]\nby (simp add: cos_double sin_squared_eq)", "meta": {"llama_tokens": 114, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111796979521253, "lm_q2_score": 0.7981867825403176, "lm_q1q2_score": 0.7272915914244653}} {"text": "[STATEMENT]\nlemma fib_closed_form':\n fixes \\ \\ :: real\n defines \"\\ \\ (1 + sqrt 5) / 2\"\n and \"\\ \\ (1 - sqrt 5) / 2\"\n assumes \"n > 0\"\n shows \"fib n = round (\\ ^ n / sqrt 5)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. int (fib n) = round (\\ ^ n / sqrt 5)\n[PROOF STEP]\nproof (rule sym, rule round_unique')\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n[PROOF STEP]\nhave \"\\\\ ^ n / sqrt 5 - of_int (int (fib n))\\ = \\\\\\ ^ n / sqrt 5\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ = \\\\\\ ^ n / sqrt 5\n[PROOF STEP]\nby (simp add: fib_closed_form[folded \\_def \\_def] field_simps power_abs)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ = \\\\\\ ^ n / sqrt 5\n\ngoal (1 subgoal):\n 1. \\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ = \\\\\\ ^ n / sqrt 5\n\ngoal (1 subgoal):\n 1. \\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\n\\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ = \\\\\\ ^ n / sqrt 5\n\ngoal (1 subgoal):\n 1. \\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ \\ (1 + sqrt 5) / 2\n\\ \\ (1 - sqrt 5) / 2\n0 < n\n[PROOF STEP]\nhave \"\\\\\\^n \\ \\\\\\^1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ \\ (1 + sqrt 5) / 2\n\\ \\ (1 - sqrt 5) / 2\n0 < n\n\ngoal (1 subgoal):\n 1. \\\\\\ ^ n \\ \\\\\\ ^ 1\n[PROOF STEP]\nby (intro power_decreasing) (simp_all add: algebra_simps real_le_lsqrt)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\\\ ^ n \\ \\\\\\ ^ 1\n\ngoal (1 subgoal):\n 1. \\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\\\\\ ^ n \\ \\\\\\ ^ 1\n\ngoal (1 subgoal):\n 1. \\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n[PROOF STEP]\nhave \"\\ < sqrt 5 / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\\\ ^ 1 < sqrt 5 / 2\n[PROOF STEP]\nby (simp add: \\_def field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\\\ ^ 1 < sqrt 5 / 2\n\ngoal (1 subgoal):\n 1. \\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\\\ ^ n < sqrt 5 / 2\n[PROOF STEP]\nhave \"\\\\\\^n / sqrt 5 < 1/2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\\\ ^ n < sqrt 5 / 2\n\ngoal (1 subgoal):\n 1. \\\\\\ ^ n / sqrt 5 < 1 / 2\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\\\ ^ n / sqrt 5 < 1 / 2\n\ngoal (1 subgoal):\n 1. \\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\n\\\\\\ ^ n / sqrt 5 < 1 / 2\n\ngoal (1 subgoal):\n 1. \\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n[PROOF STEP]\nshow \"\\\\ ^ n / sqrt 5 - of_int (int (fib n))\\ < 1/2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n\ngoal (1 subgoal):\n 1. \\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\\\ ^ n / sqrt 5 - real_of_int (int (fib n))\\ < 1 / 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1901, "file": null, "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424411924673, "lm_q2_score": 0.8596637577007394, "lm_q1q2_score": 0.7272260577940532}} {"text": "[STATEMENT]\nlemma tanh_ln_real:\n \"tanh (ln x :: real) = (x ^ 2 - 1) / (x ^ 2 + 1)\" if \"x > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tanh (ln x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. tanh (ln x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n[PROOF STEP]\nfrom that\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < x\n[PROOF STEP]\nhave \"(x * 2 - inverse x * 2) * (x\\<^sup>2 + 1) =\n (x\\<^sup>2 - 1) * (2 * x + 2 * inverse x)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n\ngoal (1 subgoal):\n 1. (x * 2 - inverse x * 2) * (x\\<^sup>2 + 1) = (x\\<^sup>2 - 1) * (2 * x + 2 * inverse x)\n[PROOF STEP]\nby (simp add: field_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n(x * 2 - inverse x * 2) * (x\\<^sup>2 + 1) = (x\\<^sup>2 - 1) * (2 * x + 2 * inverse x)\n\ngoal (1 subgoal):\n 1. tanh (ln x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(x * 2 - inverse x * 2) * (x\\<^sup>2 + 1) = (x\\<^sup>2 - 1) * (2 * x + 2 * inverse x)\n\ngoal (1 subgoal):\n 1. tanh (ln x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n[PROOF STEP]\nhave \"x\\<^sup>2 + 1 > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x\\<^sup>2 + 1\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n\ngoal (1 subgoal):\n 1. 0 < x\\<^sup>2 + 1\n[PROOF STEP]\nby (simp add: ac_simps add_pos_nonneg)\n[PROOF STATE]\nproof (state)\nthis:\n0 < x\\<^sup>2 + 1\n\ngoal (1 subgoal):\n 1. tanh (ln x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n0 < x\\<^sup>2 + 1\n\ngoal (1 subgoal):\n 1. tanh (ln x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n[PROOF STEP]\nhave \"2 * x + 2 * inverse x > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < 2 * x + 2 * inverse x\n[PROOF STEP]\nusing that\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n\ngoal (1 subgoal):\n 1. 0 < 2 * x + 2 * inverse x\n[PROOF STEP]\nby (simp add: add_pos_pos)\n[PROOF STATE]\nproof (state)\nthis:\n0 < 2 * x + 2 * inverse x\n\ngoal (1 subgoal):\n 1. tanh (ln x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(x * 2 - inverse x * 2) * (x\\<^sup>2 + 1) = (x\\<^sup>2 - 1) * (2 * x + 2 * inverse x)\n0 < x\\<^sup>2 + 1\n0 < 2 * x + 2 * inverse x\n[PROOF STEP]\nhave \"(x * 2 - inverse x * 2) /\n (2 * x + 2 * inverse x) =\n (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(x * 2 - inverse x * 2) * (x\\<^sup>2 + 1) = (x\\<^sup>2 - 1) * (2 * x + 2 * inverse x)\n0 < x\\<^sup>2 + 1\n0 < 2 * x + 2 * inverse x\n\ngoal (1 subgoal):\n 1. (x * 2 - inverse x * 2) / (2 * x + 2 * inverse x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n[PROOF STEP]\nby (simp add: frac_eq_eq)\n[PROOF STATE]\nproof (state)\nthis:\n(x * 2 - inverse x * 2) / (2 * x + 2 * inverse x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n\ngoal (1 subgoal):\n 1. tanh (ln x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n[PROOF STEP]\nwith that\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < x\n(x * 2 - inverse x * 2) / (2 * x + 2 * inverse x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n(x * 2 - inverse x * 2) / (2 * x + 2 * inverse x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n\ngoal (1 subgoal):\n 1. tanh (ln x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n[PROOF STEP]\nby (simp add: tanh_def sinh_ln_real cosh_ln_real)\n[PROOF STATE]\nproof (state)\nthis:\ntanh (ln x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1785, "file": null, "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615381952105442, "lm_q2_score": 0.8438951045175642, "lm_q1q2_score": 0.7270478652930759}} {"text": "[STATEMENT]\nlemma cosh_double: \"cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cosh ((2::'a) * x) = (cosh x)\\<^sup>2 + (sinh x)\\<^sup>2\n[PROOF STEP]\nusing cosh_add[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\ncosh (x + x) = cosh x * cosh x + sinh x * sinh x\n\ngoal (1 subgoal):\n 1. cosh ((2::'a) * x) = (cosh x)\\<^sup>2 + (sinh x)\\<^sup>2\n[PROOF STEP]\nby (simp add: power2_eq_square)", "meta": {"llama_tokens": 216, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615381952105442, "lm_q2_score": 0.8438951025545426, "lm_q1q2_score": 0.7270478636018577}} {"text": "[STATEMENT]\nlemma choose_rising_sum:\n \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))\"\n \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1) &&& (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n 2. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nshow \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n[PROOF STEP]\nby (induct m) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nhave \"\\ = (n + m + 1) choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n + m + 1 choose (n + 1) = n + m + 1 choose m\n[PROOF STEP]\nby (subst binomial_symmetric) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nn + m + 1 choose (n + 1) = n + m + 1 choose m\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nshow \"(\\j\\m. ((n + j) choose n)) = (n + m + 1) choose m\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 958, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.89181104831338, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.7270333389464286}} {"text": "[STATEMENT]\nlemma l2wi_negation_type_intersect_alt: \"wordinterval_to_set (l2wi_negation_type_intersect l) = \n wordinterval_to_set (wordinterval_setminus (l2wi_intersect (getPos l)) (l2wi (getNeg l)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. wordinterval_to_set (l2wi_negation_type_intersect l) = wordinterval_to_set (wordinterval_setminus (l2wi_intersect (getPos l)) (l2wi (getNeg l)))\n[PROOF STEP]\napply(simp add: l2wi_intersect l2wi)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. wordinterval_to_set (l2wi_negation_type_intersect l) = (\\x\\set (getPos l). case x of (i, j) \\ {i..j}) - (\\x\\set (getNeg l). case x of (i, j) \\ {i..j})\n[PROOF STEP]\napply(induction l rule :l2wi_negation_type_intersect.induct)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. wordinterval_to_set (l2wi_negation_type_intersect []) = (\\x\\set (getPos []). case x of (i, j) \\ {i..j}) - (\\x\\set (getNeg []). case x of (i, j) \\ {i..j})\n 2. \\s e ls. wordinterval_to_set (l2wi_negation_type_intersect ls) = (\\x\\set (getPos ls). case x of (i, j) \\ {i..j}) - (\\x\\set (getNeg ls). case x of (i, j) \\ {i..j}) \\ wordinterval_to_set (l2wi_negation_type_intersect (Pos (s, e) # ls)) = (\\x\\set (getPos (Pos (s, e) # ls)). case x of (i, j) \\ {i..j}) - (\\x\\set (getNeg (Pos (s, e) # ls)). case x of (i, j) \\ {i..j})\n 3. \\s e ls. wordinterval_to_set (l2wi_negation_type_intersect ls) = (\\x\\set (getPos ls). case x of (i, j) \\ {i..j}) - (\\x\\set (getNeg ls). case x of (i, j) \\ {i..j}) \\ wordinterval_to_set (l2wi_negation_type_intersect (Neg (s, e) # ls)) = (\\x\\set (getPos (Neg (s, e) # ls)). case x of (i, j) \\ {i..j}) - (\\x\\set (getNeg (Neg (s, e) # ls)). case x of (i, j) \\ {i..j})\n[PROOF STEP]\napply(simp_all)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\s e ls. wordinterval_to_set (l2wi_negation_type_intersect ls) = (\\x\\set (getPos ls). case x of (i, j) \\ {i..j}) - (\\x\\set (getNeg ls). case x of (i, j) \\ {i..j}) \\ {s..e} \\ ((\\x\\set (getPos ls). case x of (i, j) \\ {i..j}) - (\\x\\set (getNeg ls). case x of (i, j) \\ {i..j})) = {s..e} \\ (\\(x, y)\\set (getPos ls). {x..y}) - (\\x\\set (getNeg ls). case x of (i, j) \\ {i..j})\n 2. \\s e ls. wordinterval_to_set (l2wi_negation_type_intersect ls) = (\\x\\set (getPos ls). case x of (i, j) \\ {i..j}) - (\\x\\set (getNeg ls). case x of (i, j) \\ {i..j}) \\ (UNIV - {s..e}) \\ ((\\x\\set (getPos ls). case x of (i, j) \\ {i..j}) - (\\x\\set (getNeg ls). case x of (i, j) \\ {i..j})) = (\\x\\set (getPos ls). case x of (i, j) \\ {i..j}) - ({s..e} \\ (\\(x, y)\\set (getNeg ls). {x..y}))\n[PROOF STEP]\napply(fast)+\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1363, "file": "Iptables_Semantics_Common_WordInterval_Lists", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110368115781, "lm_q2_score": 0.8152324960856175, "lm_q1q2_score": 0.7270333375766054}} {"text": "[STATEMENT]\nlemma choose_rising_sum:\n \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))\"\n \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1) &&& (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n 2. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nshow \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n[PROOF STEP]\nby (induct m) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nhave \"\\ = (n + m + 1) choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n + m + 1 choose (n + 1) = n + m + 1 choose m\n[PROOF STEP]\nby (subst binomial_symmetric) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nn + m + 1 choose (n + 1) = n + m + 1 choose m\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nshow \"(\\j\\m. ((n + j) choose n)) = (n + m + 1) choose m\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 958, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.89181104831338, "lm_q2_score": 0.8152324826183822, "lm_q1q2_score": 0.7270333349430188}} {"text": "[STATEMENT]\nlemma exists_solve_consistent_rref:\nfixes A::\"'a::{field}^'cols::{mod_type}^'rows::{mod_type}\"\nassumes rref: \"reduced_row_echelon_form A\"\nshows \"(\\i. A $ i $ j = 1 \\ j = (LEAST n. A $ i $ n \\ 0)) \n = (IArray.exists (\\i. (matrix_to_iarray A) !! i !! (to_nat j) = 1\n \\ (to_nat j)=least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray[0..i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))) = IArray.exists (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0..i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a)) \\ IArray.exists (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0..i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0.. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nassume \"\\i. A $ i $ j = 1 \\ j = (LEAST n. A $ i $ n \\ 0)\"\n[PROOF STATE]\nproof (state)\nthis:\n\\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n\ngoal (2 subgoals):\n 1. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a)) \\ IArray.exists (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0..i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0.. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\n\\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nobtain i where Aij: \"A $ i $ j = 1\" and j_eq: \"j = (LEAST n. A $ i $ n \\ 0)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n\ngoal (1 subgoal):\n 1. (\\i. \\A $ i $ j = (1::'a); j = (LEAST n. A $ i $ n \\ (0::'a))\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nA $ i $ j = (1::'a)\nj = (LEAST n. A $ i $ n \\ (0::'a))\n\ngoal (2 subgoals):\n 1. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a)) \\ IArray.exists (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0..i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0.. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nshow \"IArray.exists (\\i. matrix_to_iarray A !! i !! to_nat j = 1 \\ to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A)))\n (IArray [0..i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0..i\\set (IArray.list_of (IArray [0.. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))\n[PROOF STEP]\napply (rule bexI[of _ \"to_nat i\"])+\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n 2. mod_type_class.to_nat i \\ set (IArray.list_of (IArray [0.. 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mod_type_class.to_nat j = mod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nunfolding j_eq\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a)) = mod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmod_type_class.to_nat j = mod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a))\n\ngoal (2 subgoals):\n 1. matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = (1::'a)\n 2. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmod_type_class.to_nat j = mod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a))\n\ngoal (2 subgoals):\n 1. matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = (1::'a)\n 2. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n[PROOF STEP]\nhave \"... = to_nat (LEAST n. A $ i $ n \\ 0 \\ 0\\n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a)) = mod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a) \\ (0::'cols) \\ n)\n[PROOF STEP]\nby (metis least_mod_type)\n[PROOF STATE]\nproof (state)\nthis:\nmod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a)) = mod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a) \\ (0::'cols) \\ n)\n\ngoal (2 subgoals):\n 1. matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = (1::'a)\n 2. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a)) = mod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a) \\ (0::'cols) \\ n)\n\ngoal (2 subgoals):\n 1. matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = (1::'a)\n 2. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n[PROOF STEP]\nhave \"...= least_non_zero_position_of_vector_from_index (vec_to_iarray (row i A)) (to_nat (0::'cols))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a) \\ (0::'cols) \\ n) = least_non_zero_position_of_vector_from_index (vec_to_iarray (row i A)) (mod_type_class.to_nat (0::'cols))\n[PROOF STEP]\nproof (rule vec_to_iarray_least_non_zero_position_of_vector_from_index''[symmetric, of \"0::'cols\" i A])\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ vector_all_zero_from_index (mod_type_class.to_nat (0::'cols), vec_to_iarray (row i A))\n[PROOF STEP]\nshow \"\\ vector_all_zero_from_index (to_nat (0::'cols), vec_to_iarray (row i A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ vector_all_zero_from_index (mod_type_class.to_nat (0::'cols), vec_to_iarray (row i A))\n[PROOF STEP]\nunfolding vector_all_zero_from_index_eq[symmetric, of \"0::'cols\" \"row i A\"]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (\\m\\0::'cols. row i A $ m = (0::'a))\n[PROOF STEP]\nunfolding row_def vec_nth_inverse\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (\\m\\0::'cols. A $ i $ m = (0::'a))\n[PROOF STEP]\nusing Aij least_mod_type[of j]\n[PROOF STATE]\nproof (prove)\nusing this:\nA $ i $ j = (1::'a)\n(0::'cols) \\ j\n\ngoal (1 subgoal):\n 1. \\ (\\m\\0::'cols. A $ i $ m = (0::'a))\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n\\ vector_all_zero_from_index (mod_type_class.to_nat (0::'cols), vec_to_iarray (row i A))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nmod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a) \\ (0::'cols) \\ n) = least_non_zero_position_of_vector_from_index (vec_to_iarray (row i A)) (mod_type_class.to_nat (0::'cols))\n\ngoal (2 subgoals):\n 1. matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = (1::'a)\n 2. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmod_type_class.to_nat (LEAST n. A $ i $ n \\ (0::'a) \\ (0::'cols) \\ n) = least_non_zero_position_of_vector_from_index (vec_to_iarray (row i A)) (mod_type_class.to_nat (0::'cols))\n\ngoal (2 subgoals):\n 1. matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = (1::'a)\n 2. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n[PROOF STEP]\nhave \"... = least_non_zero_position_of_vector (row_iarray (to_nat i) (matrix_to_iarray A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. least_non_zero_position_of_vector_from_index (vec_to_iarray (row i A)) (mod_type_class.to_nat (0::'cols)) = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n[PROOF STEP]\nunfolding vec_to_iarray_row least_non_zero_position_of_vector_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. least_non_zero_position_of_vector_from_index (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A)) (mod_type_class.to_nat (0::'cols)) = least_non_zero_position_of_vector_from_index (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A)) 0\n[PROOF STEP]\nunfolding to_nat_0\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. least_non_zero_position_of_vector_from_index (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A)) 0 = least_non_zero_position_of_vector_from_index (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A)) 0\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nleast_non_zero_position_of_vector_from_index (vec_to_iarray (row i A)) (mod_type_class.to_nat (0::'cols)) = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n\ngoal (2 subgoals):\n 1. matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = (1::'a)\n 2. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nmod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n[PROOF STEP]\nshow \"to_nat j = least_non_zero_position_of_vector (row_iarray (to_nat i) (matrix_to_iarray A))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n\ngoal (1 subgoal):\n 1. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nmod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A))\n\ngoal (1 subgoal):\n 1. matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = (1::'a)\n[PROOF STEP]\nshow \"matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = (1::'a)\n[PROOF STEP]\nunfolding matrix_to_iarray_nth\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A $ i $ j = (1::'a)\n[PROOF STEP]\nusing Aij\n[PROOF STATE]\nproof (prove)\nusing this:\nA $ i $ j = (1::'a)\n\ngoal (1 subgoal):\n 1. A $ i $ j = (1::'a)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nIArray.exists (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0..i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0.. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. IArray.exists (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0.. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nassume ex_eq: \"IArray.exists (\\i. matrix_to_iarray A !! i !! to_nat j = 1 \\ to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A)))\n (IArray [0..i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0..i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0.. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nhave \"\\y. List.find (\\i. matrix_to_iarray A !! i !! to_nat j = 1 \\ to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A)))\n [0..y. find (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) [0..x. x \\ set [0.. matrix_to_iarray A !! x !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray x (matrix_to_iarray A)) \\ False\n[PROOF STEP]\nassume\" \\ (\\x. x \\ set [0..\n matrix_to_iarray A !! x !! mod_type_class.to_nat j = 1 \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray x (matrix_to_iarray A)))\"\n[PROOF STATE]\nproof (state)\nthis:\n\\x. x \\ set [0.. matrix_to_iarray A !! x !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray x (matrix_to_iarray A))\n\ngoal (1 subgoal):\n 1. \\x. x \\ set [0.. matrix_to_iarray A !! x !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray x (matrix_to_iarray A)) \\ False\n[PROOF STEP]\nthus False\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. x \\ set [0.. matrix_to_iarray A !! x !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray x (matrix_to_iarray A))\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nusing ex_eq\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. x \\ set [0.. matrix_to_iarray A !! x !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray x (matrix_to_iarray A))\nIArray.exists (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0..x. x \\ set [0.. matrix_to_iarray A !! x !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray x (matrix_to_iarray A))\n\\i\\set (IArray.list_of (IArray [0.. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\y. find (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) [0..i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0.. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\n\\y. find (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) [0..i. matrix_to_iarray A !! i !! to_nat j = 1 \\ to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A)))\n [0..y. find (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) [0..y. find (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) [0.. thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfind (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) [0..i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0.. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nfind (\\i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) [0..ja (matrix_to_iarray A !! ([0..\n to_nat j = least_non_zero_position_of_vector (row_iarray ([0..i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) [0..i. \\i < length [0..ja (matrix_to_iarray A !! ([0.. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray ([0.. \\ thesis) \\ thesis\n[PROOF STEP]\nunfolding find_Some_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray ([0.. y = [0.. (\\ja (matrix_to_iarray A !! ([0.. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray ([0..i. \\i < length [0..ja (matrix_to_iarray A !! ([0.. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray ([0.. \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ni < length [0..ja (matrix_to_iarray A !! ([0.. mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray ([0..i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \\ mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0.. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nshow \"\\i. A $ i $ j = 1 \\ j = (LEAST n. A $ i $ n \\ 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n[PROOF STEP]\nproof (rule exI[of _ \"from_nat i\"], rule conjI)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. A $ mod_type_class.from_nat i $ j = (1::'a)\n 2. j = (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n[PROOF STEP]\nhave i_rw: \"[0.. [0.. [0.. (0::'a))\n[PROOF STEP]\nhave i_less_card: \"i < CARD ('rows)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i < CARD('rows)\n[PROOF STEP]\nusing i_less_length\n[PROOF STATE]\nproof (prove)\nusing this:\ni < length [0.. ($) A \\ mod_type_class.from_nat) [0.. (0::'a))\n[PROOF STEP]\nshow A_ij: \"A $ from_nat i $ j = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A $ mod_type_class.from_nat i $ j = (1::'a)\n[PROOF STEP]\nusing Aij_1\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix_to_iarray A !! ([0.. (0::'a))\n[PROOF STEP]\nhave \"to_nat j = least_non_zero_position_of_vector (row_iarray ([0.. (0::'a))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray ([0.. (0::'a))\n[PROOF STEP]\nhave \"... = least_non_zero_position_of_vector_from_index (row_iarray i (matrix_to_iarray A)) 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. least_non_zero_position_of_vector (row_iarray ([0.. (0::'a))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nleast_non_zero_position_of_vector (row_iarray ([0.. (0::'a))\n[PROOF STEP]\nhave \"... = least_non_zero_position_of_vector_from_index (vec_to_iarray (row (from_nat i) A)) (to_nat (0::'cols))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. least_non_zero_position_of_vector_from_index (row_iarray i (matrix_to_iarray A)) 0 = least_non_zero_position_of_vector_from_index (vec_to_iarray (row (mod_type_class.from_nat i) A)) (mod_type_class.to_nat (0::'cols))\n[PROOF STEP]\nunfolding vec_to_iarray_row\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. least_non_zero_position_of_vector_from_index (row_iarray i (matrix_to_iarray A)) 0 = least_non_zero_position_of_vector_from_index (row_iarray (mod_type_class.to_nat (mod_type_class.from_nat i)) (matrix_to_iarray A)) (mod_type_class.to_nat (0::'cols))\n[PROOF STEP]\nunfolding to_nat_from_nat_id[OF i_less_card]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. least_non_zero_position_of_vector_from_index (row_iarray i (matrix_to_iarray A)) 0 = least_non_zero_position_of_vector_from_index (row_iarray i (matrix_to_iarray A)) (mod_type_class.to_nat (0::'cols))\n[PROOF STEP]\nunfolding to_nat_0\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. least_non_zero_position_of_vector_from_index (row_iarray i (matrix_to_iarray A)) 0 = least_non_zero_position_of_vector_from_index (row_iarray i (matrix_to_iarray A)) 0\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nleast_non_zero_position_of_vector_from_index (row_iarray i (matrix_to_iarray A)) 0 = least_non_zero_position_of_vector_from_index (vec_to_iarray (row (mod_type_class.from_nat i) A)) (mod_type_class.to_nat (0::'cols))\n\ngoal (1 subgoal):\n 1. j = (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nleast_non_zero_position_of_vector_from_index (row_iarray i (matrix_to_iarray A)) 0 = least_non_zero_position_of_vector_from_index (vec_to_iarray (row (mod_type_class.from_nat i) A)) (mod_type_class.to_nat (0::'cols))\n\ngoal (1 subgoal):\n 1. j = (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n[PROOF STEP]\nhave \"... = to_nat (LEAST n. A $ (from_nat i) $ n \\ 0 \\ 0 \\ n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. least_non_zero_position_of_vector_from_index (vec_to_iarray (row (mod_type_class.from_nat i) A)) (mod_type_class.to_nat (0::'cols)) = mod_type_class.to_nat (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a) \\ (0::'cols) \\ n)\n[PROOF STEP]\nproof (rule vec_to_iarray_least_non_zero_position_of_vector_from_index'')\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ vector_all_zero_from_index (mod_type_class.to_nat (0::'cols), vec_to_iarray (row (mod_type_class.from_nat i) A))\n[PROOF STEP]\nshow \"\\ vector_all_zero_from_index (to_nat (0::'cols), vec_to_iarray (row (from_nat i) A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ vector_all_zero_from_index (mod_type_class.to_nat (0::'cols), vec_to_iarray (row (mod_type_class.from_nat i) A))\n[PROOF STEP]\nunfolding vector_all_zero_from_index_eq[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ (\\m\\0::'cols. row (mod_type_class.from_nat i) A $ m = (0::'a))\n[PROOF STEP]\nusing A_ij\n[PROOF STATE]\nproof (prove)\nusing this:\nA $ mod_type_class.from_nat i $ j = (1::'a)\n\ngoal (1 subgoal):\n 1. \\ (\\m\\0::'cols. row (mod_type_class.from_nat i) A $ m = (0::'a))\n[PROOF STEP]\nby (metis iarray_to_vec_vec_to_iarray least_mod_type vec_matrix vec_to_iarray_row' zero_neq_one)\n[PROOF STATE]\nproof (state)\nthis:\n\\ vector_all_zero_from_index (mod_type_class.to_nat (0::'cols), vec_to_iarray (row (mod_type_class.from_nat i) A))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nleast_non_zero_position_of_vector_from_index (vec_to_iarray (row (mod_type_class.from_nat i) A)) (mod_type_class.to_nat (0::'cols)) = mod_type_class.to_nat (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a) \\ (0::'cols) \\ n)\n\ngoal (1 subgoal):\n 1. j = (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nleast_non_zero_position_of_vector_from_index (vec_to_iarray (row (mod_type_class.from_nat i) A)) (mod_type_class.to_nat (0::'cols)) = mod_type_class.to_nat (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a) \\ (0::'cols) \\ n)\n\ngoal (1 subgoal):\n 1. j = (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n[PROOF STEP]\nhave \"... = to_nat (LEAST n. A $ (from_nat i) $ n \\ 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mod_type_class.to_nat (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a) \\ (0::'cols) \\ n) = mod_type_class.to_nat (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n[PROOF STEP]\nusing least_mod_type\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::?'a) \\ ?n\n\ngoal (1 subgoal):\n 1. mod_type_class.to_nat (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a) \\ (0::'cols) \\ n) = mod_type_class.to_nat (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\nmod_type_class.to_nat (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a) \\ (0::'cols) \\ n) = mod_type_class.to_nat (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n\ngoal (1 subgoal):\n 1. j = (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nmod_type_class.to_nat j = mod_type_class.to_nat (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n[PROOF STEP]\nshow \"j = (LEAST n. A $ from_nat i $ n \\ 0)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmod_type_class.to_nat j = mod_type_class.to_nat (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n\ngoal (1 subgoal):\n 1. j = (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n[PROOF STEP]\nunfolding to_nat_eq\n[PROOF STATE]\nproof (prove)\nusing this:\nj = (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n\ngoal (1 subgoal):\n 1. j = (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nj = (LEAST n. A $ mod_type_class.from_nat i $ n \\ (0::'a))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\i. A $ i $ j = (1::'a) \\ j = (LEAST n. A $ i $ n \\ (0::'a))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 17235, "file": "Gauss_Jordan_System_Of_Equations_IArrays", "length": 105, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.919642526773001, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7270052952135817}} {"text": "[STATEMENT]\nlemma real_triangle_num:\n \"real (triangle_num n) = real n * (real n + 1) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (triangle_num n) = real n * (real n + 1) / 2\n[PROOF STEP]\nby (simp add: triangle_num_def field_char_0_class.of_nat_div algebra_simps)", "meta": {"llama_tokens": 119, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9136765328159726, "lm_q2_score": 0.7956581024858786, "lm_q1q2_score": 0.7269741363862334}} {"text": "[STATEMENT]\nlemma unit_ball_vol_numeral:\n \"unit_ball_vol (numeral (Num.Bit0 n)) = pi ^ numeral n / fact (numeral n)\" (is ?th1)\n \"unit_ball_vol (numeral (Num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) /\n fact (2 * Suc (numeral n)) * pi ^ numeral n\" (is ?th2)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. unit_ball_vol (numeral (num.Bit0 n)) = pi ^ numeral n / fact (numeral n) &&& unit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. unit_ball_vol (numeral (num.Bit0 n)) = pi ^ numeral n / fact (numeral n)\n 2. unit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nhave \"numeral (Num.Bit0 n) = (2 * numeral n :: nat)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. numeral (num.Bit0 n) = 2 * numeral n\n[PROOF STEP]\nby (simp only: numeral_Bit0 mult_2 ring_distribs)\n[PROOF STATE]\nproof (state)\nthis:\nnumeral (num.Bit0 n) = 2 * numeral n\n\ngoal (2 subgoals):\n 1. unit_ball_vol (numeral (num.Bit0 n)) = pi ^ numeral n / fact (numeral n)\n 2. unit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnumeral (num.Bit0 n) = 2 * numeral n\n\ngoal (2 subgoals):\n 1. unit_ball_vol (numeral (num.Bit0 n)) = pi ^ numeral n / fact (numeral n)\n 2. unit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nhave \"unit_ball_vol \\ = pi ^ numeral n / fact (numeral n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. unit_ball_vol (real (2 * numeral n)) = pi ^ numeral n / fact (numeral n)\n[PROOF STEP]\nby (rule unit_ball_vol_even)\n[PROOF STATE]\nproof (state)\nthis:\nunit_ball_vol (real (2 * numeral n)) = pi ^ numeral n / fact (numeral n)\n\ngoal (2 subgoals):\n 1. unit_ball_vol (numeral (num.Bit0 n)) = pi ^ numeral n / fact (numeral n)\n 2. unit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nunit_ball_vol (real (numeral (num.Bit0 n))) = pi ^ numeral n / fact (numeral n)\n[PROOF STEP]\nshow ?th1\n[PROOF STATE]\nproof (prove)\nusing this:\nunit_ball_vol (real (numeral (num.Bit0 n))) = pi ^ numeral n / fact (numeral n)\n\ngoal (1 subgoal):\n 1. unit_ball_vol (numeral (num.Bit0 n)) = pi ^ numeral n / fact (numeral n)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nunit_ball_vol (numeral (num.Bit0 n)) = pi ^ numeral n / fact (numeral n)\n\ngoal (1 subgoal):\n 1. unit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. unit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nhave \"numeral (Num.Bit1 n) = (2 * numeral n + 1 :: nat)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. numeral (num.Bit1 n) = 2 * numeral n + 1\n[PROOF STEP]\nby (simp only: numeral_Bit1 mult_2)\n[PROOF STATE]\nproof (state)\nthis:\nnumeral (num.Bit1 n) = 2 * numeral n + 1\n\ngoal (1 subgoal):\n 1. unit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnumeral (num.Bit1 n) = 2 * numeral n + 1\n\ngoal (1 subgoal):\n 1. unit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nhave \"unit_ball_vol \\ = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) /\n fact (2 * Suc (numeral n)) * pi ^ numeral n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. unit_ball_vol (real (2 * numeral n + 1)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nby (rule unit_ball_vol_odd)\n[PROOF STATE]\nproof (state)\nthis:\nunit_ball_vol (real (2 * numeral n + 1)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n\ngoal (1 subgoal):\n 1. unit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nunit_ball_vol (real (numeral (num.Bit1 n))) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nshow ?th2\n[PROOF STATE]\nproof (prove)\nusing this:\nunit_ball_vol (real (numeral (num.Bit1 n))) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n\ngoal (1 subgoal):\n 1. unit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nunit_ball_vol (numeral (num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) / fact (2 * Suc (numeral n)) * pi ^ numeral n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2336, "file": null, "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9136765257642906, "lm_q2_score": 0.7956581073313275, "lm_q1q2_score": 0.7269741352026783}} {"text": "[STATEMENT]\nlemma cosine_rule:\n fixes a b c :: \"real ^ ('n::finite)\"\n shows \"(norm_dist a c)\\<^sup>2 =\n (norm_dist a b)\\<^sup>2 + (norm_dist b c)\\<^sup>2 + 2 * ((a - b) \\ (b - c))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (norm_dist a c)\\<^sup>2 = (norm_dist a b)\\<^sup>2 + (norm_dist b c)\\<^sup>2 + 2 * ((a - b) \\ (b - c))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (norm_dist a c)\\<^sup>2 = (norm_dist a b)\\<^sup>2 + (norm_dist b c)\\<^sup>2 + 2 * ((a - b) \\ (b - c))\n[PROOF STEP]\nhave \"(a - b) + (b - c) = a - c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a - b + (b - c) = a - c\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na - b + (b - c) = a - c\n\ngoal (1 subgoal):\n 1. (norm_dist a c)\\<^sup>2 = (norm_dist a b)\\<^sup>2 + (norm_dist b c)\\<^sup>2 + 2 * ((a - b) \\ (b - c))\n[PROOF STEP]\nwith dot_norm [of \"a - b\" \"b - c\"]\n[PROOF STATE]\nproof (chain)\npicking this:\n(a - b) \\ (b - c) = ((norm (a - b + (b - c)))\\<^sup>2 - (norm (a - b))\\<^sup>2 - (norm (b - c))\\<^sup>2) / 2\na - b + (b - c) = a - c\n[PROOF STEP]\nhave \"(a - b) \\ (b - c) =\n ((norm (a - c))\\<^sup>2 - (norm (a - b))\\<^sup>2 - (norm (b - c))\\<^sup>2) / 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(a - b) \\ (b - c) = ((norm (a - b + (b - c)))\\<^sup>2 - (norm (a - b))\\<^sup>2 - (norm (b - c))\\<^sup>2) / 2\na - b + (b - c) = a - c\n\ngoal (1 subgoal):\n 1. (a - b) \\ (b - c) = ((norm (a - c))\\<^sup>2 - (norm (a - b))\\<^sup>2 - (norm (b - c))\\<^sup>2) / 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(a - b) \\ (b - c) = ((norm (a - c))\\<^sup>2 - (norm (a - b))\\<^sup>2 - (norm (b - c))\\<^sup>2) / 2\n\ngoal (1 subgoal):\n 1. (norm_dist a c)\\<^sup>2 = (norm_dist a b)\\<^sup>2 + (norm_dist b c)\\<^sup>2 + 2 * ((a - b) \\ (b - c))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(a - b) \\ (b - c) = ((norm (a - c))\\<^sup>2 - (norm (a - b))\\<^sup>2 - (norm (b - c))\\<^sup>2) / 2\n\ngoal (1 subgoal):\n 1. (norm_dist a c)\\<^sup>2 = (norm_dist a b)\\<^sup>2 + (norm_dist b c)\\<^sup>2 + 2 * ((a - b) \\ (b - c))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(norm_dist a c)\\<^sup>2 = (norm_dist a b)\\<^sup>2 + (norm_dist b c)\\<^sup>2 + 2 * ((a - b) \\ (b - c))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1166, "file": "Tarskis_Geometry_Miscellany", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473846343393, "lm_q2_score": 0.8333245891029456, "lm_q1q2_score": 0.72694852585544}} {"text": "[STATEMENT]\nlemma arcsin_0 [simp]: \"arcsin 0 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arcsin 0 = 0\n[PROOF STEP]\nusing arcsin_sin [of 0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\- (pi / 2) \\ 0; 0 \\ pi / 2\\ \\ arcsin (sin 0) = 0\n\ngoal (1 subgoal):\n 1. arcsin 0 = 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 171, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473614033683, "lm_q2_score": 0.8333245973817158, "lm_q1q2_score": 0.726948513718464}} {"text": "[STATEMENT]\nlemma two_powrs_div:\n assumes \"j < (i::nat) \"\n shows \"((2^i) div ((2::nat)^(Suc j)))*2 = ((2^i) div (2^j))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ i div 2 ^ Suc j * 2 = 2 ^ i div 2 ^ j\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 ^ i div 2 ^ Suc j * 2 = 2 ^ i div 2 ^ j\n[PROOF STEP]\nhave \"((2::nat)^i) div (2^(Suc j)) = 2^(i -1) div(2^ j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ i div 2 ^ Suc j = 2 ^ (i - 1) div 2 ^ j\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nj < i\n\ngoal (1 subgoal):\n 1. 2 ^ i div 2 ^ Suc j = 2 ^ (i - 1) div 2 ^ j\n[PROOF STEP]\nby (smt (z3) One_nat_def add_le_cancel_left diff_Suc_Suc div_by_Suc_0 div_if less_nat_zero_code plus_1_eq_Suc power_diff_power_eq zero_neq_numeral)\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ i div 2 ^ Suc j = 2 ^ (i - 1) div 2 ^ j\n\ngoal (1 subgoal):\n 1. 2 ^ i div 2 ^ Suc j * 2 = 2 ^ i div 2 ^ j\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 ^ i div 2 ^ Suc j = 2 ^ (i - 1) div 2 ^ j\n\ngoal (1 subgoal):\n 1. 2 ^ i div 2 ^ Suc j * 2 = 2 ^ i div 2 ^ j\n[PROOF STEP]\nby (metis Suc_diff_Suc Suc_leI assms less_imp_le_nat mult.commute power_Suc power_diff_power_eq zero_neq_numeral)\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ i div 2 ^ Suc j * 2 = 2 ^ i div 2 ^ j\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 689, "file": "Number_Theoretic_Transform_Preliminary_Lemmas", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916134888614, "lm_q2_score": 0.8418256472515683, "lm_q1q2_score": 0.7268252038568366}} {"text": "[STATEMENT]\nlemma last_take_conv_nth: assumes \"n < length xs\" shows \"last (take (Suc n) xs) = xs!n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. last (take (Suc n) xs) = xs ! n\n[PROOF STEP]\nunfolding take_Suc_conv_app_nth[OF assms]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. last (take n xs \\ [xs ! n]) = xs ! n\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 155, "file": "Combinatorics_Words_CoWBasic", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391595913457, "lm_q2_score": 0.8418256532040707, "lm_q1q2_score": 0.726825194200751}} {"text": "[STATEMENT]\nlemma pos_bounded: \"\\K>0. \\a b. norm (a ** b) \\ norm a * norm b * K\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\K>0. \\a b. norm (a ** b) \\ norm a * norm b * K\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\K>0. \\a b. norm (a ** b) \\ norm a * norm b * K\n[PROOF STEP]\nobtain K where \"\\a b. norm (a ** b) \\ norm a * norm b * K\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\K. (\\a b. norm (a ** b) \\ norm a * norm b * K) \\ thesis) \\ thesis\n[PROOF STEP]\nusing bounded\n[PROOF STATE]\nproof (prove)\nusing this:\n\\K. \\a b. norm (a ** b) \\ norm a * norm b * K\n\ngoal (1 subgoal):\n 1. (\\K. (\\a b. norm (a ** b) \\ norm a * norm b * K) \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nnorm (?a ** ?b) \\ norm ?a * norm ?b * K\n\ngoal (1 subgoal):\n 1. \\K>0. \\a b. norm (a ** b) \\ norm a * norm b * K\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (?a ** ?b) \\ norm ?a * norm ?b * K\n[PROOF STEP]\nhave \"norm (a ** b) \\ norm a * norm b * (max 1 K)\" for a b\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (?a ** ?b) \\ norm ?a * norm ?b * K\n\ngoal (1 subgoal):\n 1. norm (a ** b) \\ norm a * norm b * max 1 K\n[PROOF STEP]\nby (rule order.trans) (simp add: mult_left_mono)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (?a ** ?b) \\ norm ?a * norm ?b * max 1 K\n\ngoal (1 subgoal):\n 1. \\K>0. \\a b. norm (a ** b) \\ norm a * norm b * K\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (?a ** ?b) \\ norm ?a * norm ?b * max 1 K\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (?a ** ?b) \\ norm ?a * norm ?b * max 1 K\n\ngoal (1 subgoal):\n 1. \\K>0. \\a b. norm (a ** b) \\ norm a * norm b * K\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n\\K>0. \\a b. norm (a ** b) \\ norm a * norm b * K\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 926, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467738423874, "lm_q2_score": 0.8267117962054049, "lm_q1q2_score": 0.7268010085314269}} {"text": "[STATEMENT]\nlemma reciprocal_mult: \n fixes P Q::\"'a::{ring_char_0,field} poly\" \n assumes \"degree (P * Q) \\ p\"\n and \"degree P \\ p\" and \"degree Q \\ p\"\n shows \"monom 1 p * reciprocal_poly p (P * Q) = \n reciprocal_poly p P * reciprocal_poly p Q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n[PROOF STEP]\nproof (cases \"P=0 \\ Q=0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. P = 0 \\ Q = 0 \\ monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n 2. \\ (P = 0 \\ Q = 0) \\ monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nP = 0 \\ Q = 0\n\ngoal (2 subgoals):\n 1. P = 0 \\ Q = 0 \\ monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n 2. \\ (P = 0 \\ Q = 0) \\ monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nP = 0 \\ Q = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nP = 0 \\ Q = 0\n\ngoal (1 subgoal):\n 1. monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nP = 0 \\ Q = 0\ndegree (P * Q) \\ p\n\ngoal (1 subgoal):\n 1. monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n[PROOF STEP]\nby (auto simp: reciprocal_fcompose fcompose_mult)\n[PROOF STATE]\nproof (state)\nthis:\nmonom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n\ngoal (1 subgoal):\n 1. \\ (P = 0 \\ Q = 0) \\ monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ (P = 0 \\ Q = 0) \\ monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ (P = 0 \\ Q = 0)\n\ngoal (1 subgoal):\n 1. \\ (P = 0 \\ Q = 0) \\ monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ (P = 0 \\ Q = 0)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (P = 0 \\ Q = 0)\n\ngoal (1 subgoal):\n 1. monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (P = 0 \\ Q = 0)\ndegree (P * Q) \\ p\ndegree P \\ p\ndegree Q \\ p\n\ngoal (1 subgoal):\n 1. monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n[PROOF STEP]\nby (auto simp: degree_mult_eq mult_monom reciprocal_fcompose fcompose_mult)\n[PROOF STATE]\nproof (state)\nthis:\nmonom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1445, "file": "Three_Circles_RRI_Misc", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467706759584, "lm_q2_score": 0.8267117940706734, "lm_q1q2_score": 0.7268010040369605}} {"text": "[STATEMENT]\nlemma sin_squared_le_one:\n fixes x:: real\n shows \"(sin x)\\<^sup>2 \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (sin x)\\<^sup>2 \\ 1\n[PROOF STEP]\nusing abs_sin_le_one abs_square_le_1\n[PROOF STATE]\nproof (prove)\nusing this:\n\\sin ?x\\ \\ 1\n(?x\\<^sup>2 \\ (1::?'a)) = (\\?x\\ \\ (1::?'a))\n\ngoal (1 subgoal):\n 1. (sin x)\\<^sup>2 \\ 1\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 205, "file": "Isabelle_Marries_Dirac_Basics", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467675095292, "lm_q2_score": 0.8267117962054048, "lm_q1q2_score": 0.7268010032959783}} {"text": "[STATEMENT]\nlemma power_minus_is_div:\n \"b \\ a \\ (2 :: nat) ^ (a - b) = 2 ^ a div 2 ^ b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b\n[PROOF STEP]\napply (induct a arbitrary: b)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\b. b \\ 0 \\ 2 ^ (0 - b) = 2 ^ 0 div 2 ^ b\n 2. \\a b. \\\\b. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b; b \\ Suc a\\ \\ 2 ^ (Suc a - b) = 2 ^ Suc a div 2 ^ b\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a b. \\\\b. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b; b \\ Suc a\\ \\ 2 ^ (Suc a - b) = 2 ^ Suc a div 2 ^ b\n[PROOF STEP]\napply (erule le_SucE)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\a b. \\\\b. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b; b \\ a\\ \\ 2 ^ (Suc a - b) = 2 ^ Suc a div 2 ^ b\n 2. \\a b. \\\\b. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b; b = Suc a\\ \\ 2 ^ (Suc a - b) = 2 ^ Suc a div 2 ^ b\n[PROOF STEP]\napply (clarsimp simp:Suc_diff_le le_iff_add power_add)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a b. \\\\b. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b; b = Suc a\\ \\ 2 ^ (Suc a - b) = 2 ^ Suc a div 2 ^ b\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 747, "file": "Van_Emde_Boas_Trees_VEBT_MinMax", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467580102418, "lm_q2_score": 0.8267118004748677, "lm_q1q2_score": 0.7268009991962898}} {"text": "[STATEMENT]\nlemma power_minus_is_div:\n \"b \\ a \\ (2 :: nat) ^ (a - b) = 2 ^ a div 2 ^ b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b\n[PROOF STEP]\napply (induct a arbitrary: b)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\b. b \\ 0 \\ 2 ^ (0 - b) = 2 ^ 0 div 2 ^ b\n 2. \\a b. \\\\b. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b; b \\ Suc a\\ \\ 2 ^ (Suc a - b) = 2 ^ Suc a div 2 ^ b\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a b. \\\\b. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b; b \\ Suc a\\ \\ 2 ^ (Suc a - b) = 2 ^ Suc a div 2 ^ b\n[PROOF STEP]\napply (erule le_SucE)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\a b. \\\\b. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b; b \\ a\\ \\ 2 ^ (Suc a - b) = 2 ^ Suc a div 2 ^ b\n 2. \\a b. \\\\b. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b; b = Suc a\\ \\ 2 ^ (Suc a - b) = 2 ^ Suc a div 2 ^ b\n[PROOF STEP]\napply (clarsimp simp:Suc_diff_le le_iff_add power_add)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a b. \\\\b. b \\ a \\ 2 ^ (a - b) = 2 ^ a div 2 ^ b; b = Suc a\\ \\ 2 ^ (Suc a - b) = 2 ^ Suc a div 2 ^ b\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 747, "file": "Word_Lib_More_Divides", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467580102418, "lm_q2_score": 0.8267117855317474, "lm_q1q2_score": 0.726800986059094}} {"text": "[STATEMENT]\nlemma mat_add_limit:\n fixes X :: \"nat \\ complex mat\" and A :: \"complex mat\" and m :: nat and B :: \"complex mat\"\n assumes dimB: \"B \\ carrier_mat m m\" and limX: \"limit_mat X A m\"\n shows \"limit_mat (mat_add_seq B X) (B + A) m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. limit_mat (mat_add_seq B X) (B + A) m\n[PROOF STEP]\nunfolding mat_add_seq_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. limit_mat (\\n. B + X n) (B + A) m\n[PROOF STEP]\nusing limit_mat_add limit_mat_const[OF dimB] limX\n[PROOF STATE]\nproof (prove)\nusing this:\n\\limit_mat ?X ?A ?m; limit_mat ?Y ?B ?m\\ \\ limit_mat (\\k. ?X k + ?Y k) (?A + ?B) ?m\nlimit_mat (\\k. B) B m\nlimit_mat X A m\n\ngoal (1 subgoal):\n 1. limit_mat (\\n. B + X n) (B + A) m\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 363, "file": "QHLProver_Matrix_Limit", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587817066391, "lm_q2_score": 0.8175744806385543, "lm_q1q2_score": 0.7266264993667597}} {"text": "[STATEMENT]\nlemma infsetsum_conv_nn_integral:\n assumes \"nn_integral (count_space A) f \\ \\\" \"\\x. x \\ A \\ f x \\ 0\"\n shows \"infsetsum f A = enn2real (nn_integral (count_space A) f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. infsetsum f A = enn2real (\\\\<^sup>+ x. ennreal (f x) \\count_space A)\n[PROOF STEP]\nunfolding infsetsum_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral\\<^sup>L (count_space A) f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\count_space A)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sup>+ x. ennreal (f x) \\count_space A \\ \\\n?x \\ A \\ 0 \\ f ?x\n\ngoal (1 subgoal):\n 1. integral\\<^sup>L (count_space A) f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\count_space A)\n[PROOF STEP]\nby (subst integral_eq_nn_integral) auto", "meta": {"llama_tokens": 385, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206818021529, "lm_q2_score": 0.8056321866478979, "lm_q1q2_score": 0.7266163310632314}} {"text": "[STATEMENT]\nlemma hnorm_triangle_ineq3: \"\\x y::'a::real_normed_vector star. \\hnorm x - hnorm y\\ \\ hnorm (x - y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x y. \\hnorm x - hnorm y\\ \\ hnorm (x - y)\n[PROOF STEP]\nby transfer (rule norm_triangle_ineq3)", "meta": {"llama_tokens": 129, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505402422645, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7265916109137504}} {"text": "[STATEMENT]\nlemma de_morgan_orthogonal_complement_inter:\n fixes A B::\"'a::chilbert_space set\"\n assumes a1: \\closed_csubspace A\\ and a2: \\closed_csubspace B\\\n shows \\orthogonal_complement (A \\ B) = orthogonal_complement A +\\<^sub>M orthogonal_complement B\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal_complement (A \\ B) = orthogonal_complement A +\\<^sub>M orthogonal_complement B\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. orthogonal_complement (A \\ B) = orthogonal_complement A +\\<^sub>M orthogonal_complement B\n[PROOF STEP]\nhave \\orthogonal_complement A +\\<^sub>M orthogonal_complement B\n = orthogonal_complement (orthogonal_complement (orthogonal_complement A +\\<^sub>M orthogonal_complement B))\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal_complement A +\\<^sub>M orthogonal_complement B = orthogonal_complement (orthogonal_complement (orthogonal_complement A +\\<^sub>M orthogonal_complement B))\n[PROOF STEP]\nby (simp add: closed_subspace_closed_sum)\n[PROOF STATE]\nproof (state)\nthis:\northogonal_complement A +\\<^sub>M orthogonal_complement B = orthogonal_complement (orthogonal_complement (orthogonal_complement A +\\<^sub>M orthogonal_complement B))\n\ngoal (1 subgoal):\n 1. orthogonal_complement (A \\ B) = orthogonal_complement A +\\<^sub>M orthogonal_complement B\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\northogonal_complement A +\\<^sub>M orthogonal_complement B = orthogonal_complement (orthogonal_complement (orthogonal_complement A +\\<^sub>M orthogonal_complement B))\n\ngoal (1 subgoal):\n 1. orthogonal_complement (A \\ B) = orthogonal_complement A +\\<^sub>M orthogonal_complement B\n[PROOF STEP]\nhave \\\\ = orthogonal_complement (orthogonal_complement (orthogonal_complement A) \\ orthogonal_complement (orthogonal_complement B))\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal_complement (orthogonal_complement (orthogonal_complement A +\\<^sub>M orthogonal_complement B)) = orthogonal_complement (orthogonal_complement (orthogonal_complement A) \\ orthogonal_complement (orthogonal_complement B))\n[PROOF STEP]\nby (simp add: de_morgan_orthogonal_complement_plus orthogonal_complementI)\n[PROOF STATE]\nproof (state)\nthis:\northogonal_complement (orthogonal_complement (orthogonal_complement A +\\<^sub>M orthogonal_complement B)) = orthogonal_complement (orthogonal_complement (orthogonal_complement A) \\ orthogonal_complement (orthogonal_complement B))\n\ngoal (1 subgoal):\n 1. orthogonal_complement (A \\ B) = orthogonal_complement A +\\<^sub>M orthogonal_complement B\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\northogonal_complement (orthogonal_complement (orthogonal_complement A +\\<^sub>M orthogonal_complement B)) = orthogonal_complement (orthogonal_complement (orthogonal_complement A) \\ orthogonal_complement (orthogonal_complement B))\n\ngoal (1 subgoal):\n 1. orthogonal_complement (A \\ B) = orthogonal_complement A +\\<^sub>M orthogonal_complement B\n[PROOF STEP]\nhave \\\\ = orthogonal_complement (A \\ B)\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal_complement (orthogonal_complement (orthogonal_complement A) \\ orthogonal_complement (orthogonal_complement B)) = orthogonal_complement (A \\ B)\n[PROOF STEP]\nby (simp add: a1 a2)\n[PROOF STATE]\nproof (state)\nthis:\northogonal_complement (orthogonal_complement (orthogonal_complement A) \\ orthogonal_complement (orthogonal_complement B)) = orthogonal_complement (A \\ B)\n\ngoal (1 subgoal):\n 1. orthogonal_complement (A \\ B) = orthogonal_complement A +\\<^sub>M orthogonal_complement B\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\northogonal_complement A +\\<^sub>M orthogonal_complement B = orthogonal_complement (A \\ B)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\northogonal_complement A +\\<^sub>M orthogonal_complement B = orthogonal_complement (A \\ B)\n\ngoal (1 subgoal):\n 1. orthogonal_complement (A \\ B) = orthogonal_complement A +\\<^sub>M orthogonal_complement B\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\northogonal_complement (A \\ B) = orthogonal_complement A +\\<^sub>M orthogonal_complement B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1449, "file": "Complex_Bounded_Operators_Complex_Inner_Product", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505299595162, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7265916026549163}} {"text": "[STATEMENT]\nlemma arcs_graph_G_ge_2vertsG:\n \"\\graph G; connected G\\ \\ card (arcs G) \\ 2 * (card (verts G) - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\graph G; Digraph_Component.connected G\\ \\ 2 * (card (verts G) - 1) \\ card (arcs G)\n[PROOF STEP]\nusing arcs_graph_G_ge_2vertsT connected_verts_G_eq_T\n[PROOF STATE]\nproof (prove)\nusing this:\ngraph G \\ 2 * (card (verts T) - 1) \\ card (arcs G)\n\\graph G; Digraph_Component.connected G\\ \\ verts T = verts G\n\ngoal (1 subgoal):\n 1. \\graph G; Digraph_Component.connected G\\ \\ 2 * (card (verts G) - 1) \\ card (arcs G)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 312, "file": "Query_Optimization_Directed_Tree_Additions", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9161096112990283, "lm_q2_score": 0.7931059609645724, "lm_q1q2_score": 0.7265719936181968}} {"text": "[STATEMENT]\nlemma summable_norm_exp: \"summable (\\n. norm (x^n /\\<^sub>R fact n))\"\n for x :: \"'a::{real_normed_algebra_1,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. summable (\\n. norm (x ^ n /\\<^sub>R fact n))\n[PROOF STEP]\nproof (rule summable_norm_comparison_test [OF exI, rule_format])\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\n. ?N \\ n \\ norm (x ^ n /\\<^sub>R fact n) \\ ?g n\n 2. summable ?g\n[PROOF STEP]\nshow \"summable (\\n. norm x^n /\\<^sub>R fact n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. summable (\\n. norm x ^ n /\\<^sub>R fact n)\n[PROOF STEP]\nby (rule summable_exp_generic)\n[PROOF STATE]\nproof (state)\nthis:\nsummable (\\n. norm x ^ n /\\<^sub>R fact n)\n\ngoal (1 subgoal):\n 1. \\n. ?N \\ n \\ norm (x ^ n /\\<^sub>R fact n) \\ norm x ^ n /\\<^sub>R fact n\n[PROOF STEP]\nshow \"norm (x^n /\\<^sub>R fact n) \\ norm x^n /\\<^sub>R fact n\" for n\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (x ^ n /\\<^sub>R fact n) \\ norm x ^ n /\\<^sub>R fact n\n[PROOF STEP]\nby (simp add: norm_power_ineq)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (x ^ ?n /\\<^sub>R fact ?n) \\ norm x ^ ?n /\\<^sub>R fact ?n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 559, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240964782011, "lm_q2_score": 0.839733963661418, "lm_q1q2_score": 0.7265580599910089}} {"text": "[STATEMENT]\nlemma vderiv_divI[poly_derivatives]:\n assumes \"\\t\\T. g t \\ (0::real)\" and \"D f = f' on T\"and \"D g = g' on T\" \n and \"h = (\\t. (f' t * g t - f t * (g' t)) / (g t)^2)\"\n shows \"D (\\t. (f t)/(g t)) = h on T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. D (\\t. f t / g t) = h on T\n[PROOF STEP]\napply(subgoal_tac \"(\\t. (f t)/(g t)) = (\\t. (f t) * (1/(g t)))\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (\\t. f t / g t) = (\\t. f t * (1 / g t)) \\ D (\\t. f t / g t) = h on T\n 2. (\\t. f t / g t) = (\\t. f t * (1 / g t))\n[PROOF STEP]\napply(erule ssubst, rule poly_derivatives(5)[OF assms(2)])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. D (\\x. 1 / g x) = ?g'4 on T\n 2. \\t. t \\ T \\ f t * ?g'4 t + f' t * (1 / g t) = h t\n 3. (\\t. f t / g t) = (\\t. f t * (1 / g t))\n[PROOF STEP]\napply(rule vderiv_on_composeI[where g=g and f=\"\\t. 1/t\" and f'=\"\\t. - 1/t^2\", OF _ assms(3)])\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. D (/) 1 = (\\x. - 1 / x\\<^sup>2) on g ` T\n 2. ?g'4 = (\\x. g' x *\\<^sub>R (- 1 / (g x)\\<^sup>2))\n 3. \\t. t \\ T \\ f t * ?g'4 t + f' t * (1 / g t) = h t\n 4. (\\t. f t / g t) = (\\t. f t * (1 / g t))\n[PROOF STEP]\napply(subst has_vderiv_on_def, subst has_vector_derivative_def, clarsimp)\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. \\x. x \\ T \\ D (/) 1 \\ (\\xa. - (xa / (g x)\\<^sup>2)) at (g x) within g ` T\n 2. ?g'4 = (\\x. g' x *\\<^sub>R (- 1 / (g x)\\<^sup>2))\n 3. \\t. t \\ T \\ f t * ?g'4 t + f' t * (1 / g t) = h t\n 4. (\\t. f t / g t) = (\\t. f t * (1 / g t))\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\t\\T. g t \\ 0\n\ngoal (4 subgoals):\n 1. \\x. x \\ T \\ D (/) 1 \\ (\\xa. - (xa / (g x)\\<^sup>2)) at (g x) within g ` T\n 2. ?g'4 = (\\x. g' x *\\<^sub>R (- 1 / (g x)\\<^sup>2))\n 3. \\t. t \\ T \\ f t * ?g'4 t + f' t * (1 / g t) = h t\n 4. (\\t. f t / g t) = (\\t. f t * (1 / g t))\n[PROOF STEP]\napply(force intro!: derivative_eq_intros simp: fun_eq_iff power2_eq_square)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. ?g'4 = (\\x. g' x *\\<^sub>R (- 1 / (g x)\\<^sup>2))\n 2. \\t. t \\ T \\ f t * ?g'4 t + f' t * (1 / g t) = h t\n 3. (\\t. f t / g t) = (\\t. f t * (1 / g t))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\t\\T. g t \\ 0\n D f = f' on T\n D g = g' on T\nh = (\\t. (f' t * g t - f t * g' t) / (g t)\\<^sup>2)\n\ngoal (3 subgoals):\n 1. ?g'4 = (\\x. g' x *\\<^sub>R (- 1 / (g x)\\<^sup>2))\n 2. \\t. t \\ T \\ f t * ?g'4 t + f' t * (1 / g t) = h t\n 3. (\\t. f t / g t) = (\\t. f t * (1 / g t))\n[PROOF STEP]\nby (auto simp: field_simps power2_eq_square)", "meta": {"llama_tokens": 1505, "file": "Hybrid_Systems_VCs_HS_Preliminaries", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8652240895276223, "lm_q2_score": 0.8397339616560072, "lm_q1q2_score": 0.7265580524192422}} {"text": "[STATEMENT]\nlemma det_four_block_mat_upper_right_zero: fixes A1 :: \"'a :: idom mat\" \n assumes A1: \"A1 \\ carrier_mat n n\"\n and A20: \"A2 = (0\\<^sub>m n m)\" and A3: \"A3 \\ carrier_mat m n\"\n and A4: \"A4 \\ carrier_mat m m\" \nshows \"det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nusing assms(2-)\n[PROOF STATE]\nproof (prove)\nusing this:\nA2 = 0\\<^sub>m n m\nA3 \\ carrier_mat m n\nA4 \\ carrier_mat m m\n\ngoal (1 subgoal):\n 1. det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nproof (induct m arbitrary: A2 A3 A4)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\A2 A3 A4. \\A2 = 0\\<^sub>m n 0; A3 \\ carrier_mat 0 n; A4 \\ carrier_mat 0 0\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n 2. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\ncase (0 A2 A3 A4)\n[PROOF STATE]\nproof (state)\nthis:\nA2 = 0\\<^sub>m n 0\nA3 \\ carrier_mat 0 n\nA4 \\ carrier_mat 0 0\n\ngoal (2 subgoals):\n 1. \\A2 A3 A4. \\A2 = 0\\<^sub>m n 0; A3 \\ carrier_mat 0 n; A4 \\ carrier_mat 0 0\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n 2. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nhence *: \"four_block_mat A1 A2 A3 A4 = A1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA2 = 0\\<^sub>m n 0\nA3 \\ carrier_mat 0 n\nA4 \\ carrier_mat 0 0\n\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 A3 A4 = A1\n[PROOF STEP]\nusing A1\n[PROOF STATE]\nproof (prove)\nusing this:\nA2 = 0\\<^sub>m n 0\nA3 \\ carrier_mat 0 n\nA4 \\ carrier_mat 0 0\nA1 \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 A3 A4 = A1\n[PROOF STEP]\nby (intro eq_matI, auto)\n[PROOF STATE]\nproof (state)\nthis:\nfour_block_mat A1 A2 A3 A4 = A1\n\ngoal (2 subgoals):\n 1. \\A2 A3 A4. \\A2 = 0\\<^sub>m n 0; A3 \\ carrier_mat 0 n; A4 \\ carrier_mat 0 0\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n 2. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nfrom 0\n[PROOF STATE]\nproof (chain)\npicking this:\nA2 = 0\\<^sub>m n 0\nA3 \\ carrier_mat 0 n\nA4 \\ carrier_mat 0 0\n[PROOF STEP]\nhave 4: \"A4 = 1\\<^sub>m 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA2 = 0\\<^sub>m n 0\nA3 \\ carrier_mat 0 n\nA4 \\ carrier_mat 0 0\n\ngoal (1 subgoal):\n 1. A4 = 1\\<^sub>m 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA4 = 1\\<^sub>m 0\n\ngoal (2 subgoals):\n 1. \\A2 A3 A4. \\A2 = 0\\<^sub>m n 0; A3 \\ carrier_mat 0 n; A4 \\ carrier_mat 0 0\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n 2. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nunfolding *\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A1 = det A1 * det A4\n[PROOF STEP]\nunfolding 4\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A1 = det A1 * det (1\\<^sub>m 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndet (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\ncase (Suc m A2 A3 A4)\n[PROOF STATE]\nproof (state)\nthis:\n\\?A2.0 = 0\\<^sub>m n m; ?A3.0 \\ carrier_mat m n; ?A4.0 \\ carrier_mat m m\\ \\ det (four_block_mat A1 ?A2.0 ?A3.0 ?A4.0) = det A1 * det ?A4.0\nA2 = 0\\<^sub>m n (Suc m)\nA3 \\ carrier_mat (Suc m) n\nA4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nlet ?m = \"Suc m\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nfrom Suc\n[PROOF STATE]\nproof (chain)\npicking this:\n\\?A2.0 = 0\\<^sub>m n m; ?A3.0 \\ carrier_mat m n; ?A4.0 \\ carrier_mat m m\\ \\ det (four_block_mat A1 ?A2.0 ?A3.0 ?A4.0) = det A1 * det ?A4.0\nA2 = 0\\<^sub>m n (Suc m)\nA3 \\ carrier_mat (Suc m) n\nA4 \\ carrier_mat (Suc m) (Suc m)\n[PROOF STEP]\nhave A2: \"A2 \\ carrier_mat n ?m\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?A2.0 = 0\\<^sub>m n m; ?A3.0 \\ carrier_mat m n; ?A4.0 \\ carrier_mat m m\\ \\ det (four_block_mat A1 ?A2.0 ?A3.0 ?A4.0) = det A1 * det ?A4.0\nA2 = 0\\<^sub>m n (Suc m)\nA3 \\ carrier_mat (Suc m) n\nA4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. A2 \\ carrier_mat n (Suc m)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA2 \\ carrier_mat n (Suc m)\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nnote A20 = Suc(2)\n[PROOF STATE]\nproof (state)\nthis:\nA2 = 0\\<^sub>m n (Suc m)\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nnote A34 = Suc(3-4)\n[PROOF STATE]\nproof (state)\nthis:\nA3 \\ carrier_mat (Suc m) n\nA4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nlet ?A = \"four_block_mat A1 A2 A3 A4\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nlet ?P = \"\\ B3 B4 v k. v \\ 0 \\ v * det ?A = det (four_block_mat A1 A2 B3 B4)\n \\ v * det A4 = det B4 \\ B3 \\ carrier_mat ?m n \\ B4 \\ carrier_mat ?m ?m \\ (\\ i < k. B4 $$ (i,m) = 0)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nhave \"k \\ m \\ \\ B3 B4 v. ?P B3 B4 v k\" for k\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k \\ m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i<0. B4 $$ (i, m) = (0::'a))\n 2. \\k. \\k \\ m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\\ \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\n\ngoal (2 subgoals):\n 1. 0 \\ m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i<0. B4 $$ (i, m) = (0::'a))\n 2. \\k. \\k \\ m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\\ \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ (1::'a) * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 A3 A4) \\ (1::'a) * det A4 = det A4 \\ A3 \\ carrier_mat (Suc m) n \\ A4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i<0. A4 $$ (i, m) = (0::'a))\n[PROOF STEP]\nusing A34\n[PROOF STATE]\nproof (prove)\nusing this:\nA3 \\ carrier_mat (Suc m) n\nA4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. (1::'a) \\ (0::'a) \\ (1::'a) * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 A3 A4) \\ (1::'a) * det A4 = det A4 \\ A3 \\ carrier_mat (Suc m) n \\ A4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i<0. A4 $$ (i, m) = (0::'a))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(1::'a) \\ (0::'a) \\ (1::'a) * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 A3 A4) \\ (1::'a) * det A4 = det A4 \\ A3 \\ carrier_mat (Suc m) n \\ A4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i<0. A4 $$ (i, m) = (0::'a))\n\ngoal (2 subgoals):\n 1. 0 \\ m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i<0. B4 $$ (i, m) = (0::'a))\n 2. \\k. \\k \\ m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\\ \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ (1::'a) * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 A3 A4) \\ (1::'a) * det A4 = det A4 \\ A3 \\ carrier_mat (Suc m) n \\ A4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i<0. A4 $$ (i, m) = (0::'a))\n\ngoal (1 subgoal):\n 1. \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i<0. B4 $$ (i, m) = (0::'a))\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i<0. B4 $$ (i, m) = (0::'a))\n\ngoal (1 subgoal):\n 1. \\k. \\k \\ m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\\ \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\ik. \\k \\ m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\\ \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\n\ngoal (1 subgoal):\n 1. \\k. \\k \\ m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\\ \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\n[PROOF STEP]\nobtain B3 B4 v where v: \"v \\ 0\" and det: \"v * det ?A = \n det (four_block_mat A1 A2 B3 B4)\" \"v * det A4 = det B4\" \n and B3: \"B3 \\ carrier_mat ?m n\" and B4: \"B4 \\ carrier_mat ?m ?m\" and 0: \"\\ i < k. B4 $$ (i,m) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\n\ngoal (1 subgoal):\n 1. (\\v B3 B4. \\v \\ (0::'a); v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4); v * det A4 = det B4; B3 \\ carrier_mat (Suc m) n; B4 \\ carrier_mat (Suc m) (Suc m); \\i \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv \\ (0::'a)\nv * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4)\nv * det A4 = det B4\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\n\\ik. \\k \\ m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\\ \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\iB3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\ii i < Suc k. B4 $$ (i,m) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\iii ?m = ?n)\n\ngoal (1 subgoal):\n 1. \\ii \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a)\nv * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4)\nv * det A4 = det B4\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\n\\i (0::'a)\nv * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4)\nv * det A4 = det B4\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\n\\i (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\iB3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\iB3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a)\n\ngoal (1 subgoal):\n 1. B4 $$ (k, m) \\ (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m\n[PROOF STEP]\nhave k: \"k < ?m\" \"Suc k < ?m\" \"k \\ Suc k\"\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc k \\ m\n\ngoal (1 subgoal):\n 1. k < Suc m &&& Suc k < Suc m &&& k \\ Suc k\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nk < Suc m\nSuc k < Suc m\nk \\ Suc k\n\ngoal (1 subgoal):\n 1. B4 $$ (k, m) \\ (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\iB3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a)\n[PROOF STEP]\nhave v: \"?v \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ (0::'a)\n\ngoal (1 subgoal):\n 1. - v \\ (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n- v \\ (0::'a)\n\ngoal (2 subgoals):\n 1. B4 $$ (Suc k, m) = (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i carrier_mat (Suc m) n\n[PROOF STEP]\nhave B3': \"?B3 \\ carrier_mat ?m n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nB3 \\ carrier_mat (Suc m) n\n\ngoal (1 subgoal):\n 1. swaprows k (Suc k) B3 \\ carrier_mat (Suc m) n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nswaprows k (Suc k) B3 \\ carrier_mat (Suc m) n\n\ngoal (2 subgoals):\n 1. B4 $$ (Suc k, m) = (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i carrier_mat (Suc m) (Suc m)\n[PROOF STEP]\nhave B4': \"?B4 \\ carrier_mat ?m ?m\"\n[PROOF STATE]\nproof (prove)\nusing this:\nB4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. swaprows k (Suc k) B4 \\ carrier_mat (Suc m) (Suc m)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nswaprows k (Suc k) B4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (2 subgoals):\n 1. B4 $$ (Suc k, m) = (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i = det (swaprows (n + k) (n + ?k) (four_block_mat A1 A2 B3 B4))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - det (four_block_mat A1 A2 B3 B4) = det (swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4))\n[PROOF STEP]\nby (rule sym, rule det_swaprows[of _ \"n + ?m\"], insert A1 A2 B3 B4 k, auto)\n[PROOF STATE]\nproof (state)\nthis:\n- det (four_block_mat A1 A2 B3 B4) = det (swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4))\n\ngoal (2 subgoals):\n 1. B4 $$ (Suc k, m) = (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\ii j. \\i < dim_row A1 + dim_row B4; j < dim_col A1 + dim_col B4\\ \\ swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n 2. dim_row A1 + dim_row B4 = dim_row A1 + dim_row B4\n 3. dim_col A1 + dim_col B4 = dim_col A1 + dim_col B4\n[PROOF STEP]\ncase (1 i j)\n[PROOF STATE]\nproof (state)\nthis:\ni < dim_row A1 + dim_row B4\nj < dim_col A1 + dim_col B4\n\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row A1 + dim_row B4; j < dim_col A1 + dim_col B4\\ \\ swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n 2. dim_row A1 + dim_row B4 = dim_row A1 + dim_row B4\n 3. dim_col A1 + dim_col B4 = dim_col A1 + dim_col B4\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\nproof (cases \"i < n\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. i < n \\ swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n 2. \\ i < n \\ swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\ni < n\n\ngoal (2 subgoals):\n 1. i < n \\ swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n 2. \\ i < n \\ swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n\n\ngoal (1 subgoal):\n 1. swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\nusing 1(2) A1 A2 B3 B4\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n\nj < dim_col A1 + dim_col B4\nA1 \\ carrier_mat n n\nA2 \\ carrier_mat n (Suc m)\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nswaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n\ngoal (1 subgoal):\n 1. \\ i < n \\ swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ i < n \\ swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ i < n\n\ngoal (1 subgoal):\n 1. \\ i < n \\ swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\nhence \"i = n + (i - n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ i < n\n\ngoal (1 subgoal):\n 1. i = n + (i - n)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ni = n + (i - n)\n\ngoal (1 subgoal):\n 1. \\ i < n \\ swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni = n + (i - n)\n[PROOF STEP]\nobtain d where \"i = n + d\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni = n + (i - n)\n\ngoal (1 subgoal):\n 1. (\\d. i = n + d \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ni = n + d\n\ngoal (1 subgoal):\n 1. \\ i < n \\ swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ni = n + d\n\ngoal (1 subgoal):\n 1. swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\nusing 1 A1 A2 B3 B4 k(2)\n[PROOF STATE]\nproof (prove)\nusing this:\ni = n + d\ni < dim_row A1 + dim_row B4\nj < dim_col A1 + dim_col B4\nA1 \\ carrier_mat n n\nA2 \\ carrier_mat n (Suc m)\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\nSuc k < Suc m\n\ngoal (1 subgoal):\n 1. swaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nswaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nswaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) $$ (i, j) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4) $$ (i, j)\n\ngoal (2 subgoals):\n 1. dim_row A1 + dim_row B4 = dim_row A1 + dim_row B4\n 2. dim_col A1 + dim_col B4 = dim_col A1 + dim_col B4\n[PROOF STEP]\nqed auto\n[PROOF STATE]\nproof (state)\nthis:\nswaprows (n + k) (n + Suc k) (four_block_mat A1 A2 B3 B4) = four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4)\n\ngoal (2 subgoals):\n 1. B4 $$ (Suc k, m) = (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\ii carrier_mat (Suc m) (Suc m)\nSuc k < Suc m\n[PROOF STEP]\nhave \"\\ i < Suc k. ?B4 $$ (i,m) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nB4 $$ (Suc k, m) = (0::'a)\n\\i carrier_mat (Suc m) (Suc m)\nSuc k < Suc m\n\ngoal (1 subgoal):\n 1. \\ii carrier_mat (Suc m) (Suc m)\nSuc k < m \\ Suc k = m\n\ngoal (1 subgoal):\n 1. \\i. i < k \\ i = k \\ swaprows k (Suc k) B4 $$ (i, m) = (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i carrier_mat (Suc m) n\nswaprows k (Suc k) B4 \\ carrier_mat (Suc m) (Suc m)\n- v \\ (0::'a)\n\\i carrier_mat (Suc m) n\nswaprows k (Suc k) B4 \\ carrier_mat (Suc m) (Suc m)\n- v \\ (0::'a)\n\\i (0::'a) \\ - v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4)) \\ - v * det A4 = det (swaprows k (Suc k) B4) \\ swaprows k (Suc k) B3 \\ carrier_mat (Suc m) n \\ swaprows k (Suc k) B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ - v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4)) \\ - v * det A4 = det (swaprows k (Suc k) B4) \\ swaprows k (Suc k) B3 \\ carrier_mat (Suc m) n \\ swaprows k (Suc k) B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ - v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 (swaprows k (Suc k) B3) (swaprows k (Suc k) B4)) \\ - v * det A4 = det (swaprows k (Suc k) B4) \\ swaprows k (Suc k) B3 \\ carrier_mat (Suc m) n \\ swaprows k (Suc k) B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\iB3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\iB3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a)\n\ngoal (1 subgoal):\n 1. B4 $$ (Suc k, m) \\ (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\ii carrier_mat (Suc m) (Suc m)\n[PROOF STEP]\nhave 0: \"\\ i < Suc k. ?B4 $$ (i,m) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. \\ii Suc k = m\nB4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. \\i. i < k \\ i = k \\ addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4) $$ (i, m) = (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a)\nv \\ (0::'a)\n[PROOF STEP]\nhave v: \"?v \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nB4 $$ (Suc k, m) \\ (0::'a)\nv \\ (0::'a)\n\ngoal (1 subgoal):\n 1. v * B4 $$ (Suc k, m) \\ (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv * B4 $$ (Suc k, m) \\ (0::'a)\n\ngoal (1 subgoal):\n 1. B4 $$ (Suc k, m) \\ (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i carrier_mat (Suc m) n\n[PROOF STEP]\nhave B3': \"?B3 \\ carrier_mat ?m n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nB3 \\ carrier_mat (Suc m) n\n\ngoal (1 subgoal):\n 1. addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3) \\ carrier_mat (Suc m) n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\naddrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3) \\ carrier_mat (Suc m) n\n\ngoal (1 subgoal):\n 1. B4 $$ (Suc k, m) \\ (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i carrier_mat (Suc m) (Suc m)\n[PROOF STEP]\nhave B4': \"?B4 \\ carrier_mat ?m ?m\"\n[PROOF STATE]\nproof (prove)\nusing this:\nB4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4) \\ carrier_mat (Suc m) (Suc m)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\naddrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4) \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. B4 $$ (Suc k, m) \\ (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i carrier_mat (n + ?m) (n + ?m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4) \\ carrier_mat (n + Suc m) (n + Suc m)\n[PROOF STEP]\nusing A1 A2 B3 B4 k\n[PROOF STATE]\nproof (prove)\nusing this:\nA1 \\ carrier_mat n n\nA2 \\ carrier_mat n (Suc m)\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\nk < Suc m\nSuc k < Suc m\nk \\ Suc k\n\ngoal (1 subgoal):\n 1. multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4) \\ carrier_mat (n + Suc m) (n + Suc m)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmultrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4) \\ carrier_mat (n + Suc m) (n + Suc m)\n\ngoal (1 subgoal):\n 1. B4 $$ (Suc k, m) \\ (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i = det (addrow (- ?b) (n + k) (n + ?k) ?B')\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. B4 $$ (Suc k, m) * det (four_block_mat A1 A2 B3 B4) = det (addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)))\n[PROOF STEP]\nby (subst det_addrow[OF _ _ B'], insert k(2), force, force, rule sym, rule det_multrow[of _ \"n + ?m\"],\n insert A1 A2 B3 B4 k, auto)\n[PROOF STATE]\nproof (state)\nthis:\nB4 $$ (Suc k, m) * det (four_block_mat A1 A2 B3 B4) = det (addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)))\n\ngoal (1 subgoal):\n 1. B4 $$ (Suc k, m) \\ (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\ii j. \\i < dim_row A1 + dim_row B4; j < dim_col A1 + dim_col B4\\ \\ addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n 2. dim_row A1 + dim_row B4 = dim_row A1 + dim_row B4\n 3. dim_col A1 + dim_col B4 = dim_col A1 + dim_col B4\n[PROOF STEP]\ncase (1 i j)\n[PROOF STATE]\nproof (state)\nthis:\ni < dim_row A1 + dim_row B4\nj < dim_col A1 + dim_col B4\n\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row A1 + dim_row B4; j < dim_col A1 + dim_col B4\\ \\ addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n 2. dim_row A1 + dim_row B4 = dim_row A1 + dim_row B4\n 3. dim_col A1 + dim_col B4 = dim_col A1 + dim_col B4\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\nproof (cases \"i < n\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. i < n \\ addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n 2. \\ i < n \\ addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\ni < n\n\ngoal (2 subgoals):\n 1. i < n \\ addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n 2. \\ i < n \\ addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n\n\ngoal (1 subgoal):\n 1. addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\nusing 1(2) A1 A2 B3 B4\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n\nj < dim_col A1 + dim_col B4\nA1 \\ carrier_mat n n\nA2 \\ carrier_mat n (Suc m)\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\naddrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n\ngoal (1 subgoal):\n 1. \\ i < n \\ addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ i < n \\ addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ i < n\n\ngoal (1 subgoal):\n 1. \\ i < n \\ addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\nhence \"i = n + (i - n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ i < n\n\ngoal (1 subgoal):\n 1. i = n + (i - n)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ni = n + (i - n)\n\ngoal (1 subgoal):\n 1. \\ i < n \\ addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni = n + (i - n)\n[PROOF STEP]\nobtain d where \"i = n + d\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni = n + (i - n)\n\ngoal (1 subgoal):\n 1. (\\d. i = n + d \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ni = n + d\n\ngoal (1 subgoal):\n 1. \\ i < n \\ addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ni = n + d\n\ngoal (1 subgoal):\n 1. addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\nusing 1 A1 A2 B3 B4 k(2)\n[PROOF STATE]\nproof (prove)\nusing this:\ni = n + d\ni < dim_row A1 + dim_row B4\nj < dim_col A1 + dim_col B4\nA1 \\ carrier_mat n n\nA2 \\ carrier_mat n (Suc m)\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\nSuc k < Suc m\n\ngoal (1 subgoal):\n 1. addrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\naddrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\naddrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) $$ (i, j) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) $$ (i, j)\n\ngoal (2 subgoals):\n 1. dim_row A1 + dim_row B4 = dim_row A1 + dim_row B4\n 2. dim_col A1 + dim_col B4 = dim_col A1 + dim_col B4\n[PROOF STEP]\nqed auto\n[PROOF STATE]\nproof (state)\nthis:\naddrow (- B4 $$ (k, m)) (n + k) (n + Suc k) (multrow (n + k) (B4 $$ (Suc k, m)) (four_block_mat A1 A2 B3 B4)) = four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4))\n\ngoal (1 subgoal):\n 1. B4 $$ (Suc k, m) \\ (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i carrier_mat (Suc m) n\naddrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4) \\ carrier_mat (Suc m) (Suc m)\nv * B4 $$ (Suc k, m) \\ (0::'a)\n\\i carrier_mat (Suc m) n\naddrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4) \\ carrier_mat (Suc m) (Suc m)\nv * B4 $$ (Suc k, m) \\ (0::'a)\n\\i (0::'a) \\ v * B4 $$ (Suc k, m) * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4))) \\ v * B4 $$ (Suc k, m) * det A4 = det (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) \\ addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3) \\ carrier_mat (Suc m) n \\ addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4) \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ v * B4 $$ (Suc k, m) * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4))) \\ v * B4 $$ (Suc k, m) * det A4 = det (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) \\ addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3) \\ carrier_mat (Suc m) n \\ addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4) \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) \\ v * B4 $$ (Suc k, m) * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3)) (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4))) \\ v * B4 $$ (Suc k, m) * det A4 = det (addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4)) \\ addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B3) \\ carrier_mat (Suc m) n \\ addrow (- B4 $$ (k, m)) k (Suc k) (multrow k (B4 $$ (Suc k, m)) B4) \\ carrier_mat (Suc m) (Suc m) \\ (\\iB3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\iB3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\iB3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\iB3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i m \\ \\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\im A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nfrom this[OF le_refl]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\B3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\iB3 B4 v. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\iv B3 B4. v \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nv \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\im A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nlet ?B = \"four_block_mat A1 A2 B3 B4\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nfrom P\n[PROOF STATE]\nproof (chain)\npicking this:\nv \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i 0\" and det: \"v * det ?A = det ?B\" \"v * det A4 = det B4\" \n and B3: \"B3 \\ carrier_mat ?m n\" and B4: \"B4 \\ carrier_mat ?m ?m\" and 0: \"\\ i. i < m \\ B4 $$ (i, m) = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ (0::'a) \\ v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) \\ v * det A4 = det B4 \\ B3 \\ carrier_mat (Suc m) n \\ B4 \\ carrier_mat (Suc m) (Suc m) \\ (\\i (0::'a) &&& v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4) &&& v * det A4 = det B4) &&& B3 \\ carrier_mat (Suc m) n &&& B4 \\ carrier_mat (Suc m) (Suc m) &&& (\\i. i < m \\ B4 $$ (i, m) = (0::'a))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv \\ (0::'a)\nv * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4)\nv * det A4 = det B4\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\n?i < m \\ B4 $$ (?i, m) = (0::'a)\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nlet ?A2 = \"0\\<^sub>m n m\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nlet ?A3 = \"mat m n (\\ ij. B3 $$ ij)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nlet ?A4 = \"mat m m (\\ ij. B4 $$ ij)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nlet ?B1 = \"four_block_mat A1 ?A2 ?A3 ?A4\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nlet ?B2 = \"0\\<^sub>m (n + m) 1\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nlet ?B3 = \"mat 1 (n + m) (\\ (i,j). if j < n then B3 $$ (m,j) else B4 $$ (m,j - n))\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nlet ?B4 = \"mat 1 1 (\\ _. B4 $$ (m,m))\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nhave B44: \"B4 = four_block_mat ?A4 (0\\<^sub>m m 1) (mat 1 m (\\ (i,j). B4 $$ (m,j))) ?B4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. B4 = four_block_mat (mat m m (($$) B4)) (0\\<^sub>m m 1) (mat 1 m (\\(i, j). B4 $$ (m, j))) (mat 1 1 (\\_. B4 $$ (m, m)))\n[PROOF STEP]\nproof (rule eq_matI, unfold index_mat_four_block dim_col_mat dim_row_mat, goal_cases)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\i j. \\i < m + 1; j < m + 1\\ \\ B4 $$ (i, j) = four_block_mat (mat m m (($$) B4)) (0\\<^sub>m m 1) (mat 1 m (\\(i, j). B4 $$ (m, j))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. dim_row B4 = m + 1\n 3. dim_col B4 = m + 1\n[PROOF STEP]\ncase (1 i j)\n[PROOF STATE]\nproof (state)\nthis:\ni < m + 1\nj < m + 1\n\ngoal (3 subgoals):\n 1. \\i j. \\i < m + 1; j < m + 1\\ \\ B4 $$ (i, j) = four_block_mat (mat m m (($$) B4)) (0\\<^sub>m m 1) (mat 1 m (\\(i, j). B4 $$ (m, j))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. dim_row B4 = m + 1\n 3. dim_col B4 = m + 1\n[PROOF STEP]\nhence [simp]: \"\\ i < m \\ i = m\" \"\\ j < m \\ j = m\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < m + 1\nj < m + 1\n\ngoal (1 subgoal):\n 1. (\\ i < m \\ i = m) &&& (\\ j < m \\ j = m)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\ i < m \\ i = m\n\\ j < m \\ j = m\n\ngoal (3 subgoals):\n 1. \\i j. \\i < m + 1; j < m + 1\\ \\ B4 $$ (i, j) = four_block_mat (mat m m (($$) B4)) (0\\<^sub>m m 1) (mat 1 m (\\(i, j). B4 $$ (m, j))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. dim_row B4 = m + 1\n 3. dim_col B4 = m + 1\n[PROOF STEP]\nfrom 1\n[PROOF STATE]\nproof (chain)\npicking this:\ni < m + 1\nj < m + 1\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ni < m + 1\nj < m + 1\n\ngoal (1 subgoal):\n 1. B4 $$ (i, j) = four_block_mat (mat m m (($$) B4)) (0\\<^sub>m m 1) (mat 1 m (\\(i, j). B4 $$ (m, j))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nusing B4 0\n[PROOF STATE]\nproof (prove)\nusing this:\ni < m + 1\nj < m + 1\nB4 \\ carrier_mat (Suc m) (Suc m)\n?i < m \\ B4 $$ (?i, m) = (0::'a)\n\ngoal (1 subgoal):\n 1. B4 $$ (i, j) = four_block_mat (mat m m (($$) B4)) (0\\<^sub>m m 1) (mat 1 m (\\(i, j). B4 $$ (m, j))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nB4 $$ (i, j) = four_block_mat (mat m m (($$) B4)) (0\\<^sub>m m 1) (mat 1 m (\\(i, j). B4 $$ (m, j))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n\ngoal (2 subgoals):\n 1. dim_row B4 = m + 1\n 2. dim_col B4 = m + 1\n[PROOF STEP]\nqed (insert B4, auto)\n[PROOF STATE]\nproof (state)\nthis:\nB4 = four_block_mat (mat m m (($$) B4)) (0\\<^sub>m m 1) (mat 1 m (\\(i, j). B4 $$ (m, j))) (mat 1 1 (\\_. B4 $$ (m, m)))\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nhave \"?B = four_block_mat ?B1 ?B2 ?B3 ?B4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 B3 B4 = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m)))\n[PROOF STEP]\nproof (rule eq_matI, unfold index_mat_four_block dim_col_mat dim_row_mat, goal_cases)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row A1 + m + 1; j < dim_col A1 + m + 1\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. dim_row A1 + dim_row B4 = dim_row A1 + m + 1\n 3. dim_col A1 + dim_col B4 = dim_col A1 + m + 1\n[PROOF STEP]\ncase (1 i j)\n[PROOF STATE]\nproof (state)\nthis:\ni < dim_row A1 + m + 1\nj < dim_col A1 + m + 1\n\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row A1 + m + 1; j < dim_col A1 + m + 1\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. dim_row A1 + dim_row B4 = dim_row A1 + m + 1\n 3. dim_col A1 + dim_col B4 = dim_col A1 + m + 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni < dim_row A1 + m + 1\nj < dim_col A1 + m + 1\n[PROOF STEP]\nconsider (UL) \"i < n + m\" \"j < n + m\" | (UR) \"i < n + m\" \"j = n + m\" \n | (LL) \"i = n + m\" \"j < n + m\" | (LR) \"i = n + m\" \"j = n + m\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_row A1 + m + 1\nj < dim_col A1 + m + 1\n\ngoal (1 subgoal):\n 1. \\\\i < n + m; j < n + m\\ \\ thesis; \\i < n + m; j = n + m\\ \\ thesis; \\i = n + m; j < n + m\\ \\ thesis; \\i = n + m; j = n + m\\ \\ thesis\\ \\ thesis\n[PROOF STEP]\nusing A1\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_row A1 + m + 1\nj < dim_col A1 + m + 1\nA1 \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. \\\\i < n + m; j < n + m\\ \\ thesis; \\i < n + m; j = n + m\\ \\ thesis; \\i = n + m; j < n + m\\ \\ thesis; \\i = n + m; j = n + m\\ \\ thesis\\ \\ thesis\n[PROOF STEP]\nby auto linarith\n[PROOF STATE]\nproof (state)\nthis:\n\\\\i < n + m; j < n + m\\ \\ ?thesis2; \\i < n + m; j = n + m\\ \\ ?thesis2; \\i = n + m; j < n + m\\ \\ ?thesis2; \\i = n + m; j = n + m\\ \\ ?thesis2\\ \\ ?thesis2\n\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row A1 + m + 1; j < dim_col A1 + m + 1\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. dim_row A1 + dim_row B4 = dim_row A1 + m + 1\n 3. dim_col A1 + dim_col B4 = dim_col A1 + m + 1\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\i < n + m; j < n + m\\ \\ ?thesis2; \\i < n + m; j = n + m\\ \\ ?thesis2; \\i = n + m; j < n + m\\ \\ ?thesis2; \\i = n + m; j = n + m\\ \\ ?thesis2\\ \\ ?thesis2\n\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nproof cases\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. \\i < n + m; j < n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 3. \\i = n + m; j < n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 4. \\i = n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\ncase UL\n[PROOF STATE]\nproof (state)\nthis:\ni < n + m\nj < n + m\n\ngoal (4 subgoals):\n 1. \\i < n + m; j < n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 3. \\i = n + m; j < n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 4. \\i = n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nhence [simp]: \"\\ i < n \\ i - n < m\" \n \"\\ j < n \\ j - n < m\" \"\\ j < n \\ j - n < Suc m\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n + m\nj < n + m\n\ngoal (1 subgoal):\n 1. (\\ i < n \\ i - n < m) &&& (\\ j < n \\ j - n < m) &&& (\\ j < n \\ j - n < Suc m)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\ i < n \\ i - n < m\n\\ j < n \\ j - n < m\n\\ j < n \\ j - n < Suc m\n\ngoal (4 subgoals):\n 1. \\i < n + m; j < n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 3. \\i = n + m; j < n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 4. \\i = n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nfrom UL\n[PROOF STATE]\nproof (chain)\npicking this:\ni < n + m\nj < n + m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n + m\nj < n + m\n\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nusing A1 A20 B3 B4\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n + m\nj < n + m\nA1 \\ carrier_mat n n\nA2 = 0\\<^sub>m n (Suc m)\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfour_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n\ngoal (3 subgoals):\n 1. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. \\i = n + m; j < n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 3. \\i = n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. \\i = n + m; j < n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 3. \\i = n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\ncase LL\n[PROOF STATE]\nproof (state)\nthis:\ni = n + m\nj < n + m\n\ngoal (3 subgoals):\n 1. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. \\i = n + m; j < n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 3. \\i = n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nhence [simp]: \"\\ j < n \\ j - n < m\" \"\\ j < n \\ j - n < Suc m\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni = n + m\nj < n + m\n\ngoal (1 subgoal):\n 1. (\\ j < n \\ j - n < m) &&& (\\ j < n \\ j - n < Suc m)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\ j < n \\ j - n < m\n\\ j < n \\ j - n < Suc m\n\ngoal (3 subgoals):\n 1. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. \\i = n + m; j < n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 3. \\i = n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nfrom LL\n[PROOF STATE]\nproof (chain)\npicking this:\ni = n + m\nj < n + m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ni = n + m\nj < n + m\n\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nusing A1 A2 B3 B4\n[PROOF STATE]\nproof (prove)\nusing this:\ni = n + m\nj < n + m\nA1 \\ carrier_mat n n\nA2 \\ carrier_mat n (Suc m)\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfour_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n\ngoal (2 subgoals):\n 1. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. \\i = n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. \\i = n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\ncase LR\n[PROOF STATE]\nproof (state)\nthis:\ni = n + m\nj = n + m\n\ngoal (2 subgoals):\n 1. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n 2. \\i = n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ni = n + m\nj = n + m\n\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nusing A1 A2 B3 B4\n[PROOF STATE]\nproof (prove)\nusing this:\ni = n + m\nj = n + m\nA1 \\ carrier_mat n n\nA2 \\ carrier_mat n (Suc m)\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfour_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n\ngoal (1 subgoal):\n 1. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\ncase UR\n[PROOF STATE]\nproof (state)\nthis:\ni < n + m\nj = n + m\n\ngoal (1 subgoal):\n 1. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nhence [simp]: \"\\ i < n \\ i - n < m\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n + m\nj = n + m\n\ngoal (1 subgoal):\n 1. \\ i < n \\ i - n < m\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\ i < n \\ i - n < m\n\ngoal (1 subgoal):\n 1. \\i < n + m; j = n + m\\ \\ four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nfrom UR\n[PROOF STATE]\nproof (chain)\npicking this:\ni < n + m\nj = n + m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n + m\nj = n + m\n\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nusing A1 A20 0 B3 B4\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n + m\nj = n + m\nA1 \\ carrier_mat n n\nA2 = 0\\<^sub>m n (Suc m)\n?i < m \\ B4 $$ (?i, m) = (0::'a)\nB3 \\ carrier_mat (Suc m) n\nB4 \\ carrier_mat (Suc m) (Suc m)\n\ngoal (1 subgoal):\n 1. four_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfour_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nfour_block_mat A1 A2 B3 B4 $$ (i, j) = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))) $$ (i, j)\n\ngoal (2 subgoals):\n 1. dim_row A1 + dim_row B4 = dim_row A1 + m + 1\n 2. dim_col A1 + dim_col B4 = dim_col A1 + m + 1\n[PROOF STEP]\nqed (insert B4, auto)\n[PROOF STATE]\nproof (state)\nthis:\nfour_block_mat A1 A2 B3 B4 = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m)))\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nhence \"det ?B = det (four_block_mat ?B1 ?B2 ?B3 ?B4)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfour_block_mat A1 A2 B3 B4 = four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m)))\n\ngoal (1 subgoal):\n 1. det (four_block_mat A1 A2 B3 B4) = det (four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndet (four_block_mat A1 A2 B3 B4) = det (four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))))\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndet (four_block_mat A1 A2 B3 B4) = det (four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m))))\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nhave \"\\ = det ?B1 * det ?B4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m)))) = det (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) * det (mat 1 1 (\\_. B4 $$ (m, m)))\n[PROOF STEP]\nby (rule det_four_block_mat_upper_right_zero_col[of _ \"n + m\"], insert A1 A2 B3 B4, auto)\n[PROOF STATE]\nproof (state)\nthis:\ndet (four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m)))) = det (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) * det (mat 1 1 (\\_. B4 $$ (m, m)))\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndet (four_block_mat (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) (0\\<^sub>m (n + m) 1) (mat 1 (n + m) (\\(i, j). if j < n then B3 $$ (m, j) else B4 $$ (m, j - n))) (mat 1 1 (\\_. B4 $$ (m, m)))) = det (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) * det (mat 1 1 (\\_. B4 $$ (m, m)))\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nhave \"det ?B1 = det A1 * det (mat m m (($$) B4))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) = det A1 * det (mat m m (($$) B4))\n[PROOF STEP]\nby (rule Suc(1), insert B3 B4, auto)\n[PROOF STATE]\nproof (state)\nthis:\ndet (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) = det A1 * det (mat m m (($$) B4))\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndet (four_block_mat A1 (0\\<^sub>m n m) (mat m n (($$) B3)) (mat m m (($$) B4))) = det A1 * det (mat m m (($$) B4))\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nhave \"\\ * det ?B4 = det A1 * (det (mat m m (($$) B4)) * det ?B4)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A1 * det (mat m m (($$) B4)) * det (mat 1 1 (\\_. B4 $$ (m, m))) = det A1 * (det (mat m m (($$) B4)) * det (mat 1 1 (\\_. B4 $$ (m, m))))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndet A1 * det (mat m m (($$) B4)) * det (mat 1 1 (\\_. B4 $$ (m, m))) = det A1 * (det (mat m m (($$) B4)) * det (mat 1 1 (\\_. B4 $$ (m, m))))\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndet A1 * det (mat m m (($$) B4)) * det (mat 1 1 (\\_. B4 $$ (m, m))) = det A1 * (det (mat m m (($$) B4)) * det (mat 1 1 (\\_. B4 $$ (m, m))))\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nhave \"det (mat m m (($$) B4)) * det ?B4 = det B4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (mat m m (($$) B4)) * det (mat 1 1 (\\_. B4 $$ (m, m))) = det B4\n[PROOF STEP]\nunfolding arg_cong[OF B44, of det]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (mat m m (($$) B4)) * det (mat 1 1 (\\_. B4 $$ (m, m))) = det (four_block_mat (mat m m (($$) B4)) (0\\<^sub>m m 1) (mat 1 m (\\(i, j). B4 $$ (m, j))) (mat 1 1 (\\_. B4 $$ (m, m))))\n[PROOF STEP]\nby (subst det_four_block_mat_upper_right_zero_col[OF _ refl], auto)\n[PROOF STATE]\nproof (state)\nthis:\ndet (mat m m (($$) B4)) * det (mat 1 1 (\\_. B4 $$ (m, m))) = det B4\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndet (four_block_mat A1 A2 B3 B4) = det A1 * det B4\n[PROOF STEP]\nhave id: \"det ?B = det A1 * det B4\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndet (four_block_mat A1 A2 B3 B4) = det A1 * det B4\n\ngoal (1 subgoal):\n 1. det (four_block_mat A1 A2 B3 B4) = det A1 * det B4\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ndet (four_block_mat A1 A2 B3 B4) = det A1 * det B4\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nfrom this[folded det]\n[PROOF STATE]\nproof (chain)\npicking this:\nv * det (four_block_mat A1 A2 A3 A4) = det A1 * (v * det A4)\n[PROOF STEP]\nhave \"v * det ?A = v * (det A1 * det A4)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv * det (four_block_mat A1 A2 A3 A4) = det A1 * (v * det A4)\n\ngoal (1 subgoal):\n 1. v * det (four_block_mat A1 A2 A3 A4) = v * (det A1 * det A4)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nv * det (four_block_mat A1 A2 A3 A4) = v * (det A1 * det A4)\n\ngoal (1 subgoal):\n 1. \\m A2 A3 A4. \\\\A2 A3 A4. \\A2 = 0\\<^sub>m n m; A3 \\ carrier_mat m n; A4 \\ carrier_mat m m\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4; A2 = 0\\<^sub>m n (Suc m); A3 \\ carrier_mat (Suc m) n; A4 \\ carrier_mat (Suc m) (Suc m)\\ \\ det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nwith v\n[PROOF STATE]\nproof (chain)\npicking this:\nv \\ (0::'a)\nv * det (four_block_mat A1 A2 A3 A4) = v * (det A1 * det A4)\n[PROOF STEP]\nshow \"det ?A = det A1 * det A4\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ (0::'a)\nv * det (four_block_mat A1 A2 A3 A4) = v * (det A1 * det A4)\n\ngoal (1 subgoal):\n 1. det (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndet (four_block_mat A1 A2 A3 A4) = det A1 * det A4\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 65153, "file": "Jordan_Normal_Form_Determinant", "length": 270, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.865224072151174, "lm_q2_score": 0.8397339676722393, "lm_q1q2_score": 0.7265580430330372}} {"text": "[STATEMENT]\nlemma sum_mat_add: assumes A: \"(A :: 'a :: comm_monoid_add mat) \\ carrier_mat nr nc\" and B: \"B \\ carrier_mat nr nc\"\n shows \"sum_mat (A + B) = sum_mat A + sum_mat B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_mat (A + B) = sum_mat A + sum_mat B\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum_mat (A + B) = sum_mat A + sum_mat B\n[PROOF STEP]\nfrom A B\n[PROOF STATE]\nproof (chain)\npicking this:\nA \\ carrier_mat nr nc\nB \\ carrier_mat nr nc\n[PROOF STEP]\nhave id: \"dim_row A = nr\" \"dim_row B = nr\" \"dim_col A = nc\" \"dim_col B = nc\"\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat nr nc\nB \\ carrier_mat nr nc\n\ngoal (1 subgoal):\n 1. (dim_row A = nr &&& dim_row B = nr) &&& dim_col A = nc &&& dim_col B = nc\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndim_row A = nr\ndim_row B = nr\ndim_col A = nc\ndim_col B = nc\n\ngoal (1 subgoal):\n 1. sum_mat (A + B) = sum_mat A + sum_mat B\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_mat (A + B) = sum_mat A + sum_mat B\n[PROOF STEP]\nunfolding sum_mat_def id\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum (($$) (A + B)) ({0.. {0.. {0.. {0.. B = {}\" \"p\\0\"\n shows \"proots_count p (A \\ B) = proots_count p A + proots_count p B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. proots_count p (A \\ B) = proots_count p A + proots_count p B\n[PROOF STEP]\nunfolding proots_count_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\r\\proots_within p (A \\ B). order r p) = (\\r\\proots_within p A. order r p) + (\\r\\proots_within p B. order r p)\n[PROOF STEP]\napply (subst proots_within_union[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\r\\proots_within p A \\ proots_within p B. order r p) = (\\r\\proots_within p A. order r p) + (\\r\\proots_within p B. order r p)\n[PROOF STEP]\napply (subst sum.union_disjoint)\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. finite (proots_within p A)\n 2. finite (proots_within p B)\n 3. proots_within p A \\ proots_within p B = {}\n 4. (\\r\\proots_within p A. order r p) + (\\r\\proots_within p B. order r p) = (\\r\\proots_within p A. order r p) + (\\r\\proots_within p B. order r p)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ B = {}\np \\ 0\n\ngoal (4 subgoals):\n 1. finite (proots_within p A)\n 2. finite (proots_within p B)\n 3. proots_within p A \\ proots_within p B = {}\n 4. (\\r\\proots_within p A. order r p) + (\\r\\proots_within p B. order r p) = (\\r\\proots_within p A. order r p) + (\\r\\proots_within p B. order r p)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 712, "file": "Budan_Fourier_BF_Misc", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.874077222043951, "lm_q2_score": 0.8311430520409023, "lm_q1q2_score": 0.7264832100490429}} {"text": "[STATEMENT]\nlemma iT_Mult_icard: \"0 < k \\ icard (I \\ k) = icard I\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < k \\ icard (I \\ k) = icard I\n[PROOF STEP]\napply (unfold iT_Mult_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < k \\ icard ((\\n. n * k) ` I) = icard I\n[PROOF STEP]\napply (rule icard_image)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < k \\ inj_on (\\n. n * k) I\n[PROOF STEP]\napply (rule inj_imp_inj_on)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < k \\ inj (\\n. n * k)\n[PROOF STEP]\napply (simp add: mult_right_inj)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 326, "file": "Nat-Interval-Logic_IL_IntervalOperators", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.907312221360624, "lm_q2_score": 0.8006920044739461, "lm_q1q2_score": 0.7264776412049466}} {"text": "[STATEMENT]\nlemma two_cong_0_iff: \"[2 = 0] (mod p) \\ p = 1 \\ p = 2\" for p :: nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [2 = 0] (mod p) = (p = 1 \\ p = 2)\n[PROOF STEP]\nunfolding cong_altdef_nat[of 0 2 p, simplified]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (p dvd 2) = (p = 1 \\ p = 2)\n[PROOF STEP]\nusing dvd_refl prime_nat_iff two_is_prime_nat\n[PROOF STATE]\nproof (prove)\nusing this:\n?a dvd ?a\nprime ?n = (1 < ?n \\ (\\m. m dvd ?n \\ m = 1 \\ m = ?n))\nprime 2\n\ngoal (1 subgoal):\n 1. (p dvd 2) = (p = 1 \\ p = 2)\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 305, "file": "Probabilistic_Prime_Tests_Legendre_Symbol", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9219218284193597, "lm_q2_score": 0.7879312006227324, "lm_q1q2_score": 0.7264109731467707}} {"text": "[STATEMENT]\nlemma left_null_space_orthogonal_complement_col_space:\n fixes A::\"'a^'cols::{finite, wellorder}^'rows::{finite, wellorder}\"\n shows \"left_null_space A = ROWS.v.orthogonal_complement (col_space (\\ i j. cnj (A $ i $ j)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. left_null_space A = ROWS.v.orthogonal_complement (col_space (\\i j. cnj_class.cnj (A $ i $ j)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. left_null_space A = ROWS.v.orthogonal_complement (col_space (\\i j. cnj_class.cnj (A $ i $ j)))\n[PROOF STEP]\ninterpret m: matrix inner_rows inner_cols\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Generalizations2.matrix inner_rows inner_cols\n[PROOF STEP]\nby unfold_locales\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. left_null_space A = ROWS.v.orthogonal_complement (col_space (\\i j. cnj_class.cnj (A $ i $ j)))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. left_null_space A = ROWS.v.orthogonal_complement (col_space (\\i j. cnj_class.cnj (A $ i $ j)))\n[PROOF STEP]\nusing m.null_space_orthogonal_complement_row_space[of \"transpose A\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nnull_space (Finite_Cartesian_Product.transpose A) = ROWS.v.orthogonal_complement (row_space (\\i j. cnj_class.cnj (Finite_Cartesian_Product.transpose A $ i $ j)))\n\ngoal (1 subgoal):\n 1. left_null_space A = ROWS.v.orthogonal_complement (col_space (\\i j. cnj_class.cnj (A $ i $ j)))\n[PROOF STEP]\nunfolding left_null_space_eq_null_space_transpose\n[PROOF STATE]\nproof (prove)\nusing this:\nnull_space (Finite_Cartesian_Product.transpose A) = ROWS.v.orthogonal_complement (row_space (\\i j. cnj_class.cnj (Finite_Cartesian_Product.transpose A $ i $ j)))\n\ngoal (1 subgoal):\n 1. null_space (Finite_Cartesian_Product.transpose A) = ROWS.v.orthogonal_complement (col_space (\\i j. cnj_class.cnj (A $ i $ j)))\n[PROOF STEP]\nunfolding col_space_eq_row_space_transpose\n[PROOF STATE]\nproof (prove)\nusing this:\nnull_space (Finite_Cartesian_Product.transpose A) = ROWS.v.orthogonal_complement (row_space (\\i j. cnj_class.cnj (Finite_Cartesian_Product.transpose A $ i $ j)))\n\ngoal (1 subgoal):\n 1. null_space (Finite_Cartesian_Product.transpose A) = ROWS.v.orthogonal_complement (row_space (Finite_Cartesian_Product.transpose (\\i j. cnj_class.cnj (A $ i $ j))))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nleft_null_space A = ROWS.v.orthogonal_complement (col_space (\\i j. cnj_class.cnj (A $ i $ j)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1050, "file": "QR_Decomposition_Generalizations2", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513731336202, "lm_q2_score": 0.8104789040926008, "lm_q1q2_score": 0.7263928306888252}} {"text": "[STATEMENT]\nlemma dist_riemann_sphere_ge_0 [simp]: \n shows \"dist_riemann_sphere' M1 M2 \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ dist_riemann_sphere' M1 M2\n[PROOF STEP]\napply transfer\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\M1 M2. \\M1 \\ unit_sphere; M2 \\ unit_sphere\\ \\ 0 \\ dist_riemann_sphere_r3 M1 M2\n[PROOF STEP]\nusing norm_ge_zero\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ norm ?x\n\ngoal (1 subgoal):\n 1. \\M1 M2. \\M1 \\ unit_sphere; M2 \\ unit_sphere\\ \\ 0 \\ dist_riemann_sphere_r3 M1 M2\n[PROOF STEP]\nby (simp add: split_def Let_def)", "meta": {"llama_tokens": 310, "file": "Complex_Geometry_Chordal_Metric", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513675912913, "lm_q2_score": 0.8104789063814616, "lm_q1q2_score": 0.7263928282482791}} {"text": "[STATEMENT]\ntheorem injections_equiv: \n assumes \"finite Y\" and \"distinct X\" \n shows \"set (injections_alg X Y) = injections (set X) Y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (injections_alg X Y) = injections (set X) Y\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. set (injections_alg X Y) = injections (set X) Y\n[PROOF STEP]\nlet ?P=\"\\ l. distinct l \\ (set (injections_alg l Y)=injections (set l) Y)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. set (injections_alg X Y) = injections (set X) Y\n[PROOF STEP]\nhave \"?P []\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. distinct [] \\ set (injections_alg [] Y) = injections (set []) Y\n[PROOF STEP]\nusing injectionsFromEmptyAreEmpty list.set(1) lm099\n[PROOF STATE]\nproof (prove)\nusing this:\nset (injections_alg [] ?Z) = {{}}\nset [] = {}\ninjections {} ?Y = {{}}\n\ngoal (1 subgoal):\n 1. distinct [] \\ set (injections_alg [] Y) = injections (set []) Y\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\ndistinct [] \\ set (injections_alg [] Y) = injections (set []) Y\n\ngoal (1 subgoal):\n 1. set (injections_alg X Y) = injections (set X) Y\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndistinct [] \\ set (injections_alg [] Y) = injections (set []) Y\n\ngoal (1 subgoal):\n 1. set (injections_alg X Y) = injections (set X) Y\n[PROOF STEP]\nhave \"\\x xs. ?P xs \\ ?P (x#xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x xs. (distinct xs \\ set (injections_alg xs Y) = injections (set xs) Y) \\ distinct (x # xs) \\ set (injections_alg (x # xs) Y) = injections (set (x # xs)) Y\n[PROOF STEP]\nusing assms(1) lm101\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite Y\n\\?x \\ set ?xs; set (injections_alg ?xs ?Y) = injections (set ?xs) ?Y; finite ?Y\\ \\ set (injections_alg (?x # ?xs) ?Y) = injections ({?x} \\ set ?xs) ?Y\n\ngoal (1 subgoal):\n 1. \\x xs. (distinct xs \\ set (injections_alg xs Y) = injections (set xs) Y) \\ distinct (x # xs) \\ set (injections_alg (x # xs) Y) = injections (set (x # xs)) Y\n[PROOF STEP]\nby (metis distinct.simps(2) insert_is_Un list.simps(15))\n[PROOF STATE]\nproof (state)\nthis:\n\\x xs. (distinct xs \\ set (injections_alg xs Y) = injections (set xs) Y) \\ distinct (x # xs) \\ set (injections_alg (x # xs) Y) = injections (set (x # xs)) Y\n\ngoal (1 subgoal):\n 1. set (injections_alg X Y) = injections (set X) Y\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ndistinct [] \\ set (injections_alg [] Y) = injections (set []) Y\n\\x xs. (distinct xs \\ set (injections_alg xs Y) = injections (set xs) Y) \\ distinct (x # xs) \\ set (injections_alg (x # xs) Y) = injections (set (x # xs)) Y\n[PROOF STEP]\nhave \"?P X\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndistinct [] \\ set (injections_alg [] Y) = injections (set []) Y\n\\x xs. (distinct xs \\ set (injections_alg xs Y) = injections (set xs) Y) \\ distinct (x # xs) \\ set (injections_alg (x # xs) Y) = injections (set (x # xs)) Y\n\ngoal (1 subgoal):\n 1. distinct X \\ set (injections_alg X Y) = injections (set X) Y\n[PROOF STEP]\nby (rule structInduct)\n[PROOF STATE]\nproof (state)\nthis:\ndistinct X \\ set (injections_alg X Y) = injections (set X) Y\n\ngoal (1 subgoal):\n 1. set (injections_alg X Y) = injections (set X) Y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndistinct X \\ set (injections_alg X Y) = injections (set X) Y\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndistinct X \\ set (injections_alg X Y) = injections (set X) Y\n\ngoal (1 subgoal):\n 1. set (injections_alg X Y) = injections (set X) Y\n[PROOF STEP]\nusing assms(2)\n[PROOF STATE]\nproof (prove)\nusing this:\ndistinct X \\ set (injections_alg X Y) = injections (set X) Y\ndistinct X\n\ngoal (1 subgoal):\n 1. set (injections_alg X Y) = injections (set X) Y\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nset (injections_alg X Y) = injections (set X) Y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1661, "file": "Vickrey_Clarke_Groves_Universes", "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.880797071719777, "lm_q2_score": 0.8244619285331332, "lm_q1q2_score": 0.7261836523964237}} {"text": "[STATEMENT]\nlemma cos_squared_le_one:\n fixes x:: real\n shows \"(cos x)\\<^sup>2 \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cos x)\\<^sup>2 \\ 1\n[PROOF STEP]\nusing abs_cos_le_one abs_square_le_1\n[PROOF STATE]\nproof (prove)\nusing this:\n\\cos ?x\\ \\ 1\n(?x\\<^sup>2 \\ (1::?'a)) = (\\?x\\ \\ (1::?'a))\n\ngoal (1 subgoal):\n 1. (cos x)\\<^sup>2 \\ 1\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 205, "file": "Isabelle_Marries_Dirac_Basics", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.880797071719777, "lm_q2_score": 0.824461919906883, "lm_q1q2_score": 0.7261836447984478}} {"text": "[STATEMENT]\nlemma subgroup_Inter:\n assumes \"\\I\\S. subgroup I\"\n shows \"subgroup (\\S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. subgroup (\\ S)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\I\\S. subgroup I\n\ngoal (1 subgoal):\n 1. subgroup (\\ S)\n[PROOF STEP]\nunfolding subgroup_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\I\\S. (0::'a) \\ I \\ (\\a\\I. \\b\\I. a + b \\ I) \\ (\\a\\I. - a \\ I)\n\ngoal (1 subgoal):\n 1. (0::'a) \\ \\ S \\ (\\a\\\\ S. \\b\\\\ S. a + b \\ \\ S) \\ (\\a\\\\ S. - a \\ \\ S)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 321, "file": "Echelon_Form_Rings2", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970654616711, "lm_q2_score": 0.8244619177503206, "lm_q1q2_score": 0.7261836377393841}} {"text": "[STATEMENT]\nlemma weighted_nesting_sum:\n fixes g :: \"nat \\ 'a::comm_ring_1\"\n shows \"(\\kik Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a \\ \\\n \\A. Gale_Shapley1 P\\<^sub>a P\\<^sub>b = Some(A) \\ Pref.matching P\\<^sub>a A {\n Pref.stable P\\<^sub>a P\\<^sub>b A { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a\\ \\ \\A. Gale_Shapley1 P\\<^sub>a P\\<^sub>b = Some A \\ Pref.matching P\\<^sub>a A { Pref.stable P\\<^sub>a P\\<^sub>b A { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b A\n[PROOF STEP]\nunfolding Gale_Shapley1_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a\\ \\ \\A. (if Pref P\\<^sub>a P\\<^sub>b then Some (fst (gs1 (length P\\<^sub>a) P\\<^sub>a (map ranking P\\<^sub>b))) else None) = Some A \\ Pref.matching P\\<^sub>a A { Pref.stable P\\<^sub>a P\\<^sub>b A { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b A\n[PROOF STEP]\nusing Pref.gs1\n[PROOF STATE]\nproof (prove)\nusing this:\n\\Pref ?P\\<^sub>a ?P\\<^sub>b; ?R\\<^sub>b = map ranking ?P\\<^sub>b\\ \\ gs1 (length ?P\\<^sub>a) ?P\\<^sub>a ?R\\<^sub>b = (?A, ?BMaia) \\ Pref.matching ?P\\<^sub>a ?A {a} \\ Pref.stable ?P\\<^sub>a ?P\\<^sub>b ?A {a} \\ Pref.opti\\<^sub>a ?P\\<^sub>a ?P\\<^sub>b ?A\n\ngoal (1 subgoal):\n 1. \\Pref P\\<^sub>a P\\<^sub>b; n = length P\\<^sub>a\\ \\ \\A. (if Pref P\\<^sub>a P\\<^sub>b then Some (fst (gs1 (length P\\<^sub>a) P\\<^sub>a (map ranking P\\<^sub>b))) else None) = Some A \\ Pref.matching P\\<^sub>a A { Pref.stable P\\<^sub>a P\\<^sub>b A { Pref.opti\\<^sub>a P\\<^sub>a P\\<^sub>b A\n[PROOF STEP]\nby (metis fst_conv surj_pair)", "meta": {"llama_tokens": 840, "file": "Gale_Shapley_Gale_Shapley1", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094145755219, "lm_q2_score": 0.8128673155708975, "lm_q1q2_score": 0.7261420258002145}} {"text": "[STATEMENT]\nlemma hermitian_decomp_decomp':\n fixes A::\"complex Matrix.mat\"\n assumes \"hermitian_decomp A B U\"\n shows \"real_diag_decomp A B U\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_diag_decomp A B U\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nhermitian_decomp A B U\n\ngoal (1 subgoal):\n 1. real_diag_decomp A B U\n[PROOF STEP]\nunfolding hermitian_decomp_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsimilar_mat_wit A B U (Complex_Matrix.adjoint U) \\ diagonal_mat B \\ diag_mat B = eigvals A \\ Complex_Matrix.unitary U \\ (\\i \\)\n\ngoal (1 subgoal):\n 1. real_diag_decomp A B U\n[PROOF STEP]\nby (metis real_diag_decomp_def unitarily_equiv_def unitary_diag_def)", "meta": {"llama_tokens": 324, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.893309411735131, "lm_q2_score": 0.8128673178375734, "lm_q1q2_score": 0.7261420255161964}} {"text": "[STATEMENT]\nlemma cos_zero_iff_int: \"cos x = 0 \\ (\\i. odd i \\ x = of_int i * (pi/2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cos x = 0) = (\\i. odd i \\ x = real_of_int i * (pi / 2))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (cos x = 0) = (\\i. odd i \\ x = real_of_int i * (pi / 2))\n[PROOF STEP]\nhave 1: \"\\n. odd n \\ \\i. odd i \\ real n = real_of_int i\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. odd n \\ \\i. odd i \\ real n = real_of_int i\n[PROOF STEP]\nby (metis even_of_nat_iff of_int_of_nat_eq)\n[PROOF STATE]\nproof (state)\nthis:\nodd ?n \\ \\i. odd i \\ real ?n = real_of_int i\n\ngoal (1 subgoal):\n 1. (cos x = 0) = (\\i. odd i \\ x = real_of_int i * (pi / 2))\n[PROOF STEP]\nhave 2: \"\\n. odd n \\ \\i. odd i \\ - (real n * pi) = real_of_int i * pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. odd n \\ \\i. odd i \\ - (real n * pi) = real_of_int i * pi\n[PROOF STEP]\nby (metis even_minus even_of_nat_iff mult.commute mult_minus_right of_int_minus of_int_of_nat_eq)\n[PROOF STATE]\nproof (state)\nthis:\nodd ?n \\ \\i. odd i \\ - (real ?n * pi) = real_of_int i * pi\n\ngoal (1 subgoal):\n 1. (cos x = 0) = (\\i. odd i \\ x = real_of_int i * (pi / 2))\n[PROOF STEP]\nhave 3: \"\\odd i; \\n. even n \\ real_of_int i \\ - (real n)\\\n \\ \\n. odd n \\ real_of_int i = real n\" for i\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\odd i; \\n. even n \\ real_of_int i \\ - real n\\ \\ \\n. odd n \\ real_of_int i = real n\n[PROOF STEP]\nby (cases i rule: int_cases2) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\odd ?i; \\n. even n \\ real_of_int ?i \\ - real n\\ \\ \\n. odd n \\ real_of_int ?i = real n\n\ngoal (1 subgoal):\n 1. (cos x = 0) = (\\i. odd i \\ x = real_of_int i * (pi / 2))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cos x = 0) = (\\i. odd i \\ x = real_of_int i * (pi / 2))\n[PROOF STEP]\nby (force simp: cos_zero_iff intro!: 1 2 3)\n[PROOF STATE]\nproof (state)\nthis:\n(cos x = 0) = (\\i. odd i \\ x = real_of_int i * (pi / 2))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1108, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094003735663, "lm_q2_score": 0.8128673246376008, "lm_q1q2_score": 0.7261420223552802}} {"text": "[STATEMENT]\nlemma effective_elements_matrix_distribution1:\n \"matrix_compose_cond A1 A2 B1 B2 i j \\\n ((matrix_mult A1 A2)\\(matrix_mult B1 B2))!j!i\n = f (scalar_product (row A1 (i div (row_length B1))) (col A2 (j div (length B2))))\n (scalar_product (row B1 (i mod (row_length B1))) (col B2 (j mod (length B2))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_compose_cond A1 A2 B1 B2 i j \\ (A1 \\ A2 \\ B1 \\ B2) ! j ! i = scalar_product (row A1 (i div row_length B1)) (col A2 (j div length B2)) * scalar_product (row B1 (i mod row_length B1)) (col B2 (j mod length B2))\n[PROOF STEP]\nusing elements_matrix_distribution_1 matrix_compose_cond_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\mat (row_length ?A1.0) (length ?A1.0) ?A1.0; mat (row_length ?A2.0) (length ?A2.0) ?A2.0; mat (row_length ?B1.0) (length ?B1.0) ?B1.0; mat (row_length ?B2.0) (length ?B2.0) ?B2.0; length ?A1.0 = row_length ?A2.0; length ?B1.0 = row_length ?B2.0; ?A1.0 \\ [] \\ ?A2.0 \\ [] \\ ?B1.0 \\ [] \\ ?B2.0 \\ []; ?i < row_length ?A1.0 * row_length ?B1.0; ?j < length ?A2.0 * length ?B2.0\\ \\ (?A1.0 \\ ?A2.0 \\ ?B1.0 \\ ?B2.0) ! ?j ! ?i = scalar_product (row ?A1.0 (?i div row_length ?B1.0)) (col ?A2.0 (?j div length ?B2.0)) * scalar_product (row ?B1.0 (?i mod row_length ?B1.0)) (col ?B2.0 (?j mod length ?B2.0))\nmatrix_compose_cond ?A1.0 ?A2.0 ?B1.0 ?B2.0 ?i ?j \\ mat (row_length ?A1.0) (length ?A1.0) ?A1.0 \\ mat (row_length ?A2.0) (length ?A2.0) ?A2.0 \\ mat (row_length ?B1.0) (length ?B1.0) ?B1.0 \\ mat (row_length ?B2.0) (length ?B2.0) ?B2.0 \\ length ?A1.0 = row_length ?A2.0 \\ length ?B1.0 = row_length ?B2.0 \\ ?A1.0 \\ [] \\ ?A2.0 \\ [] \\ ?B1.0 \\ [] \\ ?B2.0 \\ [] \\ ?i < row_length ?A1.0 * row_length ?B1.0 \\ ?j < length ?A2.0 * length ?B2.0\n\ngoal (1 subgoal):\n 1. matrix_compose_cond A1 A2 B1 B2 i j \\ (A1 \\ A2 \\ B1 \\ B2) ! j ! i = scalar_product (row A1 (i div row_length B1)) (col A2 (j div length B2)) * scalar_product (row B1 (i mod row_length B1)) (col B2 (j mod length B2))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 1032, "file": "Matrix_Tensor_Matrix_Tensor", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314647623015, "lm_q2_score": 0.8198933315126791, "lm_q1q2_score": 0.7261233321364172}} {"text": "[STATEMENT]\nlemma sum_equal: \"0 < (N::nat) \\ sum (\\ n. 1/N) {..< N} = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < N \\ (\\n real (card {..b. b \\ Basis \\ l \\ b \\ u \\ b\"\n shows \"emeasure lborel (box l u) = (\\b\\Basis. (u - l) \\ b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. emeasure lborel (box l u) = ennreal (prod ((\\) (u - l)) Basis)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. emeasure lborel (box l u) = ennreal (prod ((\\) (u - l)) Basis)\n[PROOF STEP]\nhave \"(\\x. \\b\\Basis. indicator {l\\b <..< u\\b} (x \\ b) :: ennreal) = indicator (box l u)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. \\b\\Basis. indicator {l \\ b<.. b} (x \\ b)) = indicator (box l u)\n[PROOF STEP]\nby (auto simp: fun_eq_iff box_def split: split_indicator)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. \\b\\Basis. indicator {l \\ b<.. b} (x \\ b)) = indicator (box l u)\n\ngoal (1 subgoal):\n 1. emeasure lborel (box l u) = ennreal (prod ((\\) (u - l)) Basis)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x. \\b\\Basis. indicator {l \\ b<.. b} (x \\ b)) = indicator (box l u)\n[PROOF STEP]\nhave \"emeasure lborel (box l u) = (\\\\<^sup>+x. (\\b\\Basis. indicator {l\\b <..< u\\b} (x \\ b)) \\lborel)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. \\b\\Basis. indicator {l \\ b<.. b} (x \\ b)) = indicator (box l u)\n\ngoal (1 subgoal):\n 1. emeasure lborel (box l u) = \\\\<^sup>+ x. (\\b\\Basis. indicator {l \\ b<.. b} (x \\ b)) \\lborel\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nemeasure lborel (box l u) = \\\\<^sup>+ x. (\\b\\Basis. indicator {l \\ b<.. b} (x \\ b)) \\lborel\n\ngoal (1 subgoal):\n 1. emeasure lborel (box l u) = ennreal (prod ((\\) (u - l)) Basis)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nemeasure lborel (box l u) = \\\\<^sup>+ x. (\\b\\Basis. indicator {l \\ b<.. b} (x \\ b)) \\lborel\n\ngoal (1 subgoal):\n 1. emeasure lborel (box l u) = ennreal (prod ((\\) (u - l)) Basis)\n[PROOF STEP]\nhave \"\\ = (\\b\\Basis. (u - l) \\ b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. (\\b\\Basis. indicator {l \\ b<.. b} (x \\ b)) \\lborel = ennreal (prod ((\\) (u - l)) Basis)\n[PROOF STEP]\nby (subst nn_integral_lborel_prod) (simp_all add: prod_ennreal inner_diff_left)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>+ x. (\\b\\Basis. indicator {l \\ b<.. b} (x \\ b)) \\lborel = ennreal (prod ((\\) (u - l)) Basis)\n\ngoal (1 subgoal):\n 1. emeasure lborel (box l u) = ennreal (prod ((\\) (u - l)) Basis)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nemeasure lborel (box l u) = ennreal (prod ((\\) (u - l)) Basis)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nemeasure lborel (box l u) = ennreal (prod ((\\) (u - l)) Basis)\n\ngoal (1 subgoal):\n 1. emeasure lborel (box l u) = ennreal (prod ((\\) (u - l)) Basis)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nemeasure lborel (box l u) = ennreal (prod ((\\) (u - l)) Basis)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1543, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9184802440252812, "lm_q2_score": 0.7905303137346446, "lm_q1q2_score": 0.7260864754683785}} {"text": "[STATEMENT]\nlemma odd_square_cong_4_int:\n fixes x :: int\n assumes \"odd x\"\n shows \"[x ^ 2 = 1] (mod 4)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 1] (mod 4)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 1] (mod 4)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nodd x\n[PROOF STEP]\nhave \"odd \\x\\\"\n[PROOF STATE]\nproof (prove)\nusing this:\nodd x\n\ngoal (1 subgoal):\n 1. odd \\x\\\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nodd \\x\\\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 1] (mod 4)\n[PROOF STEP]\nhence [simp]: \"\\x\\ mod 2 = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nodd \\x\\\n\ngoal (1 subgoal):\n 1. \\x\\ mod 2 = 1\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ mod 2 = 1\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 1] (mod 4)\n[PROOF STEP]\nhave \"(\\x\\ ^ 2) mod 4 = ((\\x\\ mod 4) ^ 2) mod 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\\\<^sup>2 mod 4 = (\\x\\ mod 4)\\<^sup>2 mod 4\n[PROOF STEP]\nby (simp add: power_mod)\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\\\<^sup>2 mod 4 = (\\x\\ mod 4)\\<^sup>2 mod 4\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 1] (mod 4)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\\\<^sup>2 mod 4 = (\\x\\ mod 4)\\<^sup>2 mod 4\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 1] (mod 4)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nodd x\n[PROOF STEP]\nhave \"\\x\\ mod 4 = 1 \\ \\x\\ mod 4 = 3\"\n[PROOF STATE]\nproof (prove)\nusing this:\nodd x\n\ngoal (1 subgoal):\n 1. \\x\\ mod 4 = 1 \\ \\x\\ mod 4 = 3\n[PROOF STEP]\nusing mod_double_modulus[of 2 \"\\x\\\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nodd x\n\\0 < 2; 0 \\ \\x\\\\ \\ \\x\\ mod (2 * 2) = \\x\\ mod 2 \\ \\x\\ mod (2 * 2) = \\x\\ mod 2 + 2\n\ngoal (1 subgoal):\n 1. \\x\\ mod 4 = 1 \\ \\x\\ mod 4 = 3\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\x\\ mod 4 = 1 \\ \\x\\ mod 4 = 3\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 1] (mod 4)\n[PROOF STEP]\nhence \"((\\x\\ mod 4) ^ 2) mod 4 = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\ mod 4 = 1 \\ \\x\\ mod 4 = 3\n\ngoal (1 subgoal):\n 1. (\\x\\ mod 4)\\<^sup>2 mod 4 = 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\x\\ mod 4)\\<^sup>2 mod 4 = 1\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 1] (mod 4)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\x\\\\<^sup>2 mod 4 = 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\\\<^sup>2 mod 4 = 1\n\ngoal (1 subgoal):\n 1. [x\\<^sup>2 = 1] (mod 4)\n[PROOF STEP]\nby (simp add: cong_def)\n[PROOF STATE]\nproof (state)\nthis:\n[x\\<^sup>2 = 1] (mod 4)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1493, "file": "Gaussian_Integers_Gaussian_Integers", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757870046160257, "lm_q2_score": 0.8289388146603364, "lm_q1q2_score": 0.7259738415013348}} {"text": "[STATEMENT]\nlemma (in group) finite_subgroup_card_neq_0:\n assumes \"subgroup H G\" \"finite H\"\n shows \"card H \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card H \\ 0\n[PROOF STEP]\nusing subgroup_nonempty assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ subgroup {} ?G\nsubgroup H G\nfinite H\n\ngoal (1 subgoal):\n 1. card H \\ 0\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 159, "file": "Finitely_Generated_Abelian_Groups_Miscellaneous_Groups", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869786798664, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7259738125997867}} {"text": "[STATEMENT]\nlemma inner_mult_diag_real:\n fixes B::\"complex Matrix.mat\"\n assumes \"diagonal_mat B\"\n and \"B \\ carrier_mat n n\"\n and \"\\i < n. B$$(i, i) \\ Reals\"\n and \"v \\ carrier_vec n\"\nshows \"inner_prod (B *\\<^sub>v v) v \\ Reals\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (B *\\<^sub>v v) v \\ \\\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (B *\\<^sub>v v) v \\ \\\n[PROOF STEP]\nhave \"inner_prod (B *\\<^sub>v v) v = \n (\\ i \\ {0 ..< n}. (conjugate (B $$ (i,i))) * (vec_index v i * \n (conjugate (vec_index v i))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (B *\\<^sub>v v) v = (\\i = 0..diagonal_mat ?B; ?B \\ carrier_mat ?n ?n; ?v \\ carrier_vec ?n\\ \\ Complex_Matrix.inner_prod (?B *\\<^sub>v ?v) ?v = (\\i = 0.. carrier_mat n n\n\\i \\\nv \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (B *\\<^sub>v v) v = (\\i = 0..v v) v = (\\i = 0..v v) v \\ \\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (B *\\<^sub>v v) v = (\\i = 0..v v) v \\ \\\n[PROOF STEP]\nhave \"... \\ Reals\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0.. \\\n[PROOF STEP]\nproof (rule real_sum_real)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i. i < n \\ conjugate (B $$ (i, i)) * (v $ i * conjugate (v $ i)) \\ \\\n[PROOF STEP]\nshow \"\\i. i < n \\ \n conjugate (B $$ (i, i)) * \n ((vec_index v i) * conjugate (vec_index v i)) \\ \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. i < n \\ conjugate (B $$ (i, i)) * (v $ i * conjugate (v $ i)) \\ \\\n[PROOF STEP]\nusing assms mult_conj_real\n[PROOF STATE]\nproof (prove)\nusing this:\ndiagonal_mat B\nB \\ carrier_mat n n\n\\i \\\nv \\ carrier_vec n\n?v * conjugate ?v \\ \\\n\ngoal (1 subgoal):\n 1. \\i. i < n \\ conjugate (B $$ (i, i)) * (v $ i * conjugate (v $ i)) \\ \\\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n?i < n \\ conjugate (B $$ (?i, ?i)) * (v $ ?i * conjugate (v $ ?i)) \\ \\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\i = 0.. \\\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (B *\\<^sub>v v) v \\ \\\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nComplex_Matrix.inner_prod (B *\\<^sub>v v) v \\ \\\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex_Matrix.inner_prod (B *\\<^sub>v v) v \\ \\\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (B *\\<^sub>v v) v \\ \\\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (B *\\<^sub>v v) v \\ \\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1680, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527944504226, "lm_q2_score": 0.8519527982093666, "lm_q1q2_score": 0.725823567174327}} {"text": "[STATEMENT]\nlemma convex_hull_empty[simp]:\n \"convex_hull {} = {}\"\n \"A \\ carrier_vec n \\ convex_hull A = {} \\ A = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex_hull {} = {} &&& (A \\ carrier_vec n \\ (convex_hull A = {}) = (A = {}))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. convex_hull {} = {}\n 2. A \\ carrier_vec n \\ (convex_hull A = {}) = (A = {})\n[PROOF STEP]\nshow \"convex_hull {} = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex_hull {} = {}\n[PROOF STEP]\nunfolding convex_hull_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x. \\Ws c. finite Ws \\ Ws \\ {} \\ convex_lincomb c Ws x} = {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nconvex_hull {} = {}\n\ngoal (1 subgoal):\n 1. A \\ carrier_vec n \\ (convex_hull A = {}) = (A = {})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nconvex_hull {} = {}\n[PROOF STEP]\nshow \"A \\ carrier_vec n \\ convex_hull A = {} \\ A = {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nconvex_hull {} = {}\n\ngoal (1 subgoal):\n 1. A \\ carrier_vec n \\ (convex_hull A = {}) = (A = {})\n[PROOF STEP]\nusing set_in_convex_hull[of A]\n[PROOF STATE]\nproof (prove)\nusing this:\nconvex_hull {} = {}\nA \\ carrier_vec n \\ A \\ convex_hull A\n\ngoal (1 subgoal):\n 1. A \\ carrier_vec n \\ (convex_hull A = {}) = (A = {})\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nA \\ carrier_vec n \\ (convex_hull A = {}) = (A = {})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 726, "file": "Linear_Inequalities_Convex_Hull", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.853912760387131, "lm_q2_score": 0.8499711794579723, "lm_q1q2_score": 0.7258012361004627}} {"text": "[STATEMENT]\nlemma integral_normal_moment_abs_odd:\n \"integral\\<^sup>L lborel (\\x. normal_density \\ \\ x * \\x - \\\\^(2 * k + 1)) = 2 ^ k * \\ ^ (2 * k + 1) * fact k * sqrt (2 / pi)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LBINT x. normal_density \\ \\ x * \\x - \\\\ ^ (2 * k + 1) = 2 ^ k * \\ ^ (2 * k + 1) * fact k * sqrt (2 / pi)\n[PROOF STEP]\nusing normal_moment_abs_odd\n[PROOF STATE]\nproof (prove)\nusing this:\nhas_bochner_integral lborel (\\x. normal_density \\ \\ x * \\x - \\\\ ^ (2 * k + 1)) (2 ^ k * \\ ^ (2 * k + 1) * fact k * sqrt (2 / pi))\n\ngoal (1 subgoal):\n 1. LBINT x. normal_density \\ \\ x * \\x - \\\\ ^ (2 * k + 1) = 2 ^ k * \\ ^ (2 * k + 1) * fact k * sqrt (2 / pi)\n[PROOF STEP]\nby (rule has_bochner_integral_integral_eq)", "meta": {"llama_tokens": 396, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9241418158002492, "lm_q2_score": 0.785308580887758, "lm_q1q2_score": 0.7257364979051296}} {"text": "[STATEMENT]\nlemma ccSup_union_distrib: \"countable A \\ countable B \\ Sup (A \\ B) = sup (Sup A) (Sup B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\countable A; countable B\\ \\ Sup (A \\ B) = sup (Sup A) (Sup B)\n[PROOF STEP]\nby (rule order.antisym) (auto intro: ccSup_least ccSup_upper le_supI1 le_supI2)", "meta": {"llama_tokens": 156, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.882427872638409, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7255226091975805}} {"text": "[STATEMENT]\nlemma closedin_compact_projection:\n fixes S :: \"'a::euclidean_space set\" and T :: \"'b::euclidean_space set\"\n assumes \"compact S\" and clo: \"closedin (top_of_set (S \\ T)) U\"\n shows \"closedin (top_of_set T) {y. \\x. x \\ S \\ (x, y) \\ U}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. closedin (top_of_set T) {y. \\x. x \\ S \\ (x, y) \\ U}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. closedin (top_of_set T) {y. \\x. x \\ S \\ (x, y) \\ U}\n[PROOF STEP]\nhave \"U \\ S \\ T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. U \\ S \\ T\n[PROOF STEP]\nby (metis clo closedin_imp_subset)\n[PROOF STATE]\nproof (state)\nthis:\nU \\ S \\ T\n\ngoal (1 subgoal):\n 1. closedin (top_of_set T) {y. \\x. x \\ S \\ (x, y) \\ U}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nU \\ S \\ T\n[PROOF STEP]\nhave \"{y. \\x. x \\ S \\ (x, y) \\ U} = snd ` U\"\n[PROOF STATE]\nproof (prove)\nusing this:\nU \\ S \\ T\n\ngoal (1 subgoal):\n 1. {y. \\x. x \\ S \\ (x, y) \\ U} = snd ` U\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n{y. \\x. x \\ S \\ (x, y) \\ U} = snd ` U\n\ngoal (1 subgoal):\n 1. closedin (top_of_set T) {y. \\x. x \\ S \\ (x, y) \\ U}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n{y. \\x. x \\ S \\ (x, y) \\ U} = snd ` U\n\ngoal (1 subgoal):\n 1. closedin (top_of_set T) {y. \\x. x \\ S \\ (x, y) \\ U}\n[PROOF STEP]\nhave \"closedin (top_of_set T) (snd ` U)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. closedin (top_of_set T) (snd ` U)\n[PROOF STEP]\nby (rule closed_map_snd [OF assms])\n[PROOF STATE]\nproof (state)\nthis:\nclosedin (top_of_set T) (snd ` U)\n\ngoal (1 subgoal):\n 1. closedin (top_of_set T) {y. \\x. x \\ S \\ (x, y) \\ U}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n{y. \\x. x \\ S \\ (x, y) \\ U} = snd ` U\nclosedin (top_of_set T) (snd ` U)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n{y. \\x. x \\ S \\ (x, y) \\ U} = snd ` U\nclosedin (top_of_set T) (snd ` U)\n\ngoal (1 subgoal):\n 1. closedin (top_of_set T) {y. \\x. x \\ S \\ (x, y) \\ U}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nclosedin (top_of_set T) {y. \\x. x \\ S \\ (x, y) \\ U}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1206, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278757303677, "lm_q2_score": 0.8221891305219504, "lm_q1q2_score": 0.7255226078950827}} {"text": "[STATEMENT]\nlemma gbinomial_trinomial_revision:\n assumes \"k \\ m\"\n shows \"(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nhave \"(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ m\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n[PROOF STEP]\nby (simp add: binomial_gbinomial [symmetric] binomial_fact)\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nhave \"\\ = (a gchoose k) * (a - of_nat k gchoose (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ m\n\ngoal (1 subgoal):\n 1. (a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nby (simp add: gbinomial_pochhammer power_diff pochhammer_product)\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1242, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278633625322, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7255226034933916}} {"text": "[STATEMENT]\nlemma gbinomial_trinomial_revision:\n assumes \"k \\ m\"\n shows \"(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nhave \"(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ m\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n[PROOF STEP]\nby (simp add: binomial_gbinomial [symmetric] binomial_fact)\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nhave \"\\ = (a gchoose k) * (a - of_nat k gchoose (m - k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ m\n\ngoal (1 subgoal):\n 1. (a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nby (simp add: gbinomial_pochhammer power_diff pochhammer_product)\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * fact m / (fact k * fact (m - k)) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal (1 subgoal):\n 1. (a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1242, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278633625322, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7255226034933916}} {"text": "[STATEMENT]\nlemma gbinomial_sum_up_index:\n \"(\\j = 0..n. (of_nat j gchoose k) :: 'a::field_char_0) = (of_nat n + 1) gchoose (k + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n 2. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n 2. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nusing gbinomial_Suc_Suc[of 0 k]\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::?'b1) + (1::?'b1) gchoose Suc k = (0::?'b1) gchoose k + ((0::?'b1) gchoose Suc k)\n\ngoal (1 subgoal):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nby (cases k) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nusing gbinomial_Suc_Suc[of \"of_nat (Suc n) :: 'a\" k]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\nof_nat (Suc n) + (1::'a) gchoose Suc k = of_nat (Suc n) gchoose k + (of_nat (Suc n) gchoose Suc k)\n\ngoal (1 subgoal):\n 1. (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nby (simp add: add_ac)\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1647, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.88242786954645, "lm_q2_score": 0.8221891283434876, "lm_q1q2_score": 0.7255226008883965}} {"text": "[STATEMENT]\nlemma gbinomial_sum_up_index:\n \"(\\j = 0..n. (of_nat j gchoose k) :: 'a::field_char_0) = (of_nat n + 1) gchoose (k + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n 2. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n 2. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nusing gbinomial_Suc_Suc[of 0 k]\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::?'b1) + (1::?'b1) gchoose Suc k = (0::?'b1) gchoose k + ((0::?'b1) gchoose Suc k)\n\ngoal (1 subgoal):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nby (cases k) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nusing gbinomial_Suc_Suc[of \"of_nat (Suc n) :: 'a\" k]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\nof_nat (Suc n) + (1::'a) gchoose Suc k = of_nat (Suc n) gchoose k + (of_nat (Suc n) gchoose Suc k)\n\ngoal (1 subgoal):\n 1. (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nby (simp add: add_ac)\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1647, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.88242786954645, "lm_q2_score": 0.8221891261650247, "lm_q1q2_score": 0.7255225989660602}} {"text": "[STATEMENT]\nlemma gbinomial_sum_up_index:\n \"(\\j = 0..n. (of_nat j gchoose k) :: 'a::field_char_0) = (of_nat n + 1) gchoose (k + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n 2. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n 2. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nusing gbinomial_Suc_Suc[of 0 k]\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::?'b1) + (1::?'b1) gchoose Suc k = (0::?'b1) gchoose k + ((0::?'b1) gchoose Suc k)\n\ngoal (1 subgoal):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nby (cases k) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nusing gbinomial_Suc_Suc[of \"of_nat (Suc n) :: 'a\" k]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\nof_nat (Suc n) + (1::'a) gchoose Suc k = of_nat (Suc n) gchoose k + (of_nat (Suc n) gchoose Suc k)\n\ngoal (1 subgoal):\n 1. (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nby (simp add: add_ac)\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1647, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.88242786954645, "lm_q2_score": 0.822189121808099, "lm_q1q2_score": 0.7255225951213875}} {"text": "[STATEMENT]\nlemma \"count_true (vec (5::nat) (\\i. True)) = 5\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. count_true (vec 5 (\\i. True)) = 5\n[PROOF STEP]\nunfolding count_true_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..i. True)). if vec 5 (\\i. True) $ i then 1 else 0) = 5\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 167, "file": "Simplicial_complexes_and_boolean_functions_Boolean_functions", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9207896824119663, "lm_q2_score": 0.7879311956428947, "lm_q1q2_score": 0.725518915398502}} {"text": "[STATEMENT]\nlemma scaleR_mtx3: \"k *\\<^sub>R mtx \n ([a\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3] # \n [a\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3] # \n [a\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3] # []) = mtx \n ([k*a\\<^sub>1\\<^sub>1, k*a\\<^sub>1\\<^sub>2, k*a\\<^sub>1\\<^sub>3] # \n [k*a\\<^sub>2\\<^sub>1, k*a\\<^sub>2\\<^sub>2, k*a\\<^sub>2\\<^sub>3] # \n [k*a\\<^sub>3\\<^sub>1, k*a\\<^sub>3\\<^sub>2, k*a\\<^sub>3\\<^sub>3] # [])\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k *\\<^sub>R mtx [[a\\<^sub>1\\<^sub>1, a\\<^sub>1\\<^sub>2, a\\<^sub>1\\<^sub>3], [a\\<^sub>2\\<^sub>1, a\\<^sub>2\\<^sub>2, a\\<^sub>2\\<^sub>3], [a\\<^sub>3\\<^sub>1, a\\<^sub>3\\<^sub>2, a\\<^sub>3\\<^sub>3]] = mtx [[k * a\\<^sub>1\\<^sub>1, k * a\\<^sub>1\\<^sub>2, k * a\\<^sub>1\\<^sub>3], [k * a\\<^sub>2\\<^sub>1, k * a\\<^sub>2\\<^sub>2, k * a\\<^sub>2\\<^sub>3], [k * a\\<^sub>3\\<^sub>1, k * a\\<^sub>3\\<^sub>2, k * a\\<^sub>3\\<^sub>3]]\n[PROOF STEP]\nby (simp add: sq_mtx_eq_iff)", "meta": {"llama_tokens": 520, "file": "Matrices_for_ODEs_SQ_MTX", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942119105696, "lm_q2_score": 0.8031737987125612, "lm_q1q2_score": 0.7255022435352815}} {"text": "[STATEMENT]\nlemma isometry_onD:\n assumes \"isometry_on X f\"\n \"x \\ X\" \"y \\ X\"\n shows \"dist (f x) (f y) = dist x y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist (f x) (f y) = dist x y\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nisometry_on X f\nx \\ X\ny \\ X\n\ngoal (1 subgoal):\n 1. dist (f x) (f y) = dist x y\n[PROOF STEP]\nunfolding isometry_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\X. \\y\\X. dist (f x) (f y) = dist x y\nx \\ X\ny \\ X\n\ngoal (1 subgoal):\n 1. dist (f x) (f y) = dist x y\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 282, "file": "Gromov_Hyperbolicity_Isometries", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972717658209, "lm_q2_score": 0.8333245994514084, "lm_q1q2_score": 0.7254901227777416}} {"text": "[STATEMENT]\nlemma inner_prod_distrib_right:\n fixes u v w :: \"('a::conjugatable_field) vec\"\n assumes dimu: \"u \\ carrier_vec n\" and dimv:\"v \\ carrier_vec n\" and dimw: \"w \\ carrier_vec n\" \n shows \"inner_prod u (v + w) = inner_prod u v + inner_prod u w\" (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inner_prod u (v + w) = inner_prod u v + inner_prod u w\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. inner_prod u (v + w) = inner_prod u v + inner_prod u w\n[PROOF STEP]\nhave dimvw: \"v + w \\ carrier_vec n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v + w \\ carrier_vec n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ carrier_vec n\nv \\ carrier_vec n\nw \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. v + w \\ carrier_vec n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv + w \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. inner_prod u (v + w) = inner_prod u v + inner_prod u w\n[PROOF STEP]\nhave dimcu: \"conjugate u \\ carrier_vec n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. conjugate u \\ carrier_vec n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\ carrier_vec n\nv \\ carrier_vec n\nw \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. conjugate u \\ carrier_vec n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nconjugate u \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. inner_prod u (v + w) = inner_prod u v + inner_prod u w\n[PROOF STEP]\nhave \"(v + w) \\ (conjugate u) = v \\ conjugate u + w \\ conjugate u\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inner_prod u (v + w) = inner_prod u v + inner_prod u w\n[PROOF STEP]\napply (simp add: comm_scalar_prod[OF dimvw dimcu])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. conjugate u \\ (v + w) = inner_prod u v + inner_prod u w\n[PROOF STEP]\napply (simp add: scalar_prod_add_distrib[OF dimcu dimv dimw])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. conjugate u \\ v + conjugate u \\ w = inner_prod u v + inner_prod u w\n[PROOF STEP]\napply (insert dimv dimw dimcu, simp add: comm_scalar_prod[of _ n])\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\ninner_prod u (v + w) = inner_prod u v + inner_prod u w\n\ngoal (1 subgoal):\n 1. inner_prod u (v + w) = inner_prod u v + inner_prod u w\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ninner_prod u (v + w) = inner_prod u v + inner_prod u w\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ninner_prod u (v + w) = inner_prod u v + inner_prod u w\n\ngoal (1 subgoal):\n 1. inner_prod u (v + w) = inner_prod u v + inner_prod u w\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninner_prod u (v + w) = inner_prod u v + inner_prod u w\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1233, "file": "QHLProver_Complex_Matrix", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637577007394, "lm_q2_score": 0.8438951064805861, "lm_q1q2_score": 0.7254660383423662}} {"text": "[STATEMENT]\nlemma abs_summable_on_diff [intro]:\n assumes \"f abs_summable_on A\" and \"g abs_summable_on A\"\n shows \"(\\x. f x - g x) abs_summable_on A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. f x - g x) abs_summable_on A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nf abs_summable_on A\ng abs_summable_on A\n\ngoal (1 subgoal):\n 1. (\\x. f x - g x) abs_summable_on A\n[PROOF STEP]\nunfolding abs_summable_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable (count_space A) f\nintegrable (count_space A) g\n\ngoal (1 subgoal):\n 1. integrable (count_space A) (\\x. f x - g x)\n[PROOF STEP]\nby (rule Bochner_Integration.integrable_diff)", "meta": {"llama_tokens": 309, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637505099167, "lm_q2_score": 0.8438951005915208, "lm_q1q2_score": 0.7254660272114503}} {"text": "[STATEMENT]\nlemma continuous_openin_preimage_gen:\n assumes \"continuous_on S f\" \"open T\"\n shows \"openin (top_of_set S) (S \\ f -` T)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. openin (top_of_set S) (S \\ f -` T)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. openin (top_of_set S) (S \\ f -` T)\n[PROOF STEP]\nhave *: \"(S \\ f -` T) = (S \\ f -` (T \\ f ` S))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. S \\ f -` T = S \\ f -` (T \\ f ` S)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nS \\ f -` T = S \\ f -` (T \\ f ` S)\n\ngoal (1 subgoal):\n 1. openin (top_of_set S) (S \\ f -` T)\n[PROOF STEP]\nhave \"openin (top_of_set (f ` S)) (T \\ f ` S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. openin (top_of_set (f ` S)) (T \\ f ` S)\n[PROOF STEP]\nusing openin_open_Int[of T \"f ` S\", OF assms(2)]\n[PROOF STATE]\nproof (prove)\nusing this:\nopenin (top_of_set (f ` S)) (f ` S \\ T)\n\ngoal (1 subgoal):\n 1. openin (top_of_set (f ` S)) (T \\ f ` S)\n[PROOF STEP]\nunfolding openin_open\n[PROOF STATE]\nproof (prove)\nusing this:\n\\Ta. open Ta \\ f ` S \\ T = f ` S \\ Ta\n\ngoal (1 subgoal):\n 1. \\Ta. open Ta \\ T \\ f ` S = f ` S \\ Ta\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nopenin (top_of_set (f ` S)) (T \\ f ` S)\n\ngoal (1 subgoal):\n 1. openin (top_of_set S) (S \\ f -` T)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nopenin (top_of_set (f ` S)) (T \\ f ` S)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nopenin (top_of_set (f ` S)) (T \\ f ` S)\n\ngoal (1 subgoal):\n 1. openin (top_of_set S) (S \\ f -` T)\n[PROOF STEP]\nusing assms(1)[unfolded continuous_on_open, THEN spec[where x=\"T \\ f ` S\"]]\n[PROOF STATE]\nproof (prove)\nusing this:\nopenin (top_of_set (f ` S)) (T \\ f ` S)\nopenin (top_of_set (f ` S)) (T \\ f ` S) \\ openin (top_of_set S) (S \\ f -` (T \\ f ` S))\n\ngoal (1 subgoal):\n 1. openin (top_of_set S) (S \\ f -` T)\n[PROOF STEP]\nusing *\n[PROOF STATE]\nproof (prove)\nusing this:\nopenin (top_of_set (f ` S)) (T \\ f ` S)\nopenin (top_of_set (f ` S)) (T \\ f ` S) \\ openin (top_of_set S) (S \\ f -` (T \\ f ` S))\nS \\ f -` T = S \\ f -` (T \\ f ` S)\n\ngoal (1 subgoal):\n 1. openin (top_of_set S) (S \\ f -` T)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nopenin (top_of_set S) (S \\ f -` T)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1218, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637469145053, "lm_q2_score": 0.8438950966654774, "lm_q1q2_score": 0.7254660208022229}} {"text": "[STATEMENT]\nlemma int_less_real_le: \"n < m \\ real_of_int n + 1 \\ real_of_int m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (n < m) = (real_of_int n + 1 \\ real_of_int m)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (n < m) = (real_of_int n + 1 \\ real_of_int m)\n[PROOF STEP]\nhave \"(0::real) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ 1\n[PROOF STEP]\nby (metis less_eq_real_def zero_less_one)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ 1\n\ngoal (1 subgoal):\n 1. (n < m) = (real_of_int n + 1 \\ real_of_int m)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ 1\n\ngoal (1 subgoal):\n 1. (n < m) = (real_of_int n + 1 \\ real_of_int m)\n[PROOF STEP]\nby (metis floor_of_int less_floor_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(n < m) = (real_of_int n + 1 \\ real_of_int m)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 485, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767938900121, "lm_q2_score": 0.8267117876664789, "lm_q1q2_score": 0.7254204089126624}} {"text": "[STATEMENT]\nlemma Discrete_sqrt_square_is_square:\n assumes \"is_square n\"\n shows \"Discrete.sqrt n ^ 2 = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Discrete.sqrt n)\\<^sup>2 = n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nis_square n\n\ngoal (1 subgoal):\n 1. (Discrete.sqrt n)\\<^sup>2 = n\n[PROOF STEP]\nunfolding is_nth_power_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\y. n = y\\<^sup>2\n\ngoal (1 subgoal):\n 1. (Discrete.sqrt n)\\<^sup>2 = n\n[PROOF STEP]\nby force", "meta": {"llama_tokens": 226, "file": "Pell_Pell_Algorithm", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.8267117876664789, "lm_q1q2_score": 0.7254204036149515}} {"text": "[STATEMENT]\nlemma half_sum:\n \"(\\n. ((1::real) / 2) ^ (Suc (n + k))) = (1/2)^k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. (1 / 2) ^ Suc (n + k)) = (1 / 2) ^ k\n[PROOF STEP]\nusing suminf_split_initial_segment[of \"\\n. ((1::real) / 2) ^ (Suc n)\" k] half_1_minus_sum[of k] power_half_series sums_unique[of \"\\n. (1 / 2) ^ Suc n\" 1] power_half_summable\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\n. (1 / 2) ^ Suc n) \\ (\\n. (1 / 2) ^ Suc n) = (\\n. (1 / 2) ^ Suc (n + k)) + (\\iin. (1 / 2) ^ Suc n) sums 1\n(\\n. ((1::?'a1) / (2::?'a1)) ^ Suc n) sums (1::?'a1) \\ (1::?'a1) = (\\n. ((1::?'a1) / (2::?'a1)) ^ Suc n)\nsummable (\\n. (1 / 2) ^ Suc n)\n\ngoal (1 subgoal):\n 1. (\\n. (1 / 2) ^ Suc (n + k)) = (1 / 2) ^ k\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 475, "file": "Quasi_Borel_Spaces_StandardBorel", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.905989822921759, "lm_q2_score": 0.8006920116079209, "lm_q1q2_score": 0.7254188138115273}} {"text": "[STATEMENT]\nlemma sumset_assoc:\n shows \"sumset (sumset A B) C = sumset A (sumset B C)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sumset (sumset A B) C = sumset A (sumset B C)\n[PROOF STEP]\nby (fastforce simp add: sumset_eq associative Bex_def)", "meta": {"llama_tokens": 104, "file": "Pluennecke_Ruzsa_Inequality_Pluennecke_Ruzsa_Inequality", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976952866333484, "lm_q2_score": 0.8080672181749422, "lm_q1q2_score": 0.7253981330385673}} {"text": "[STATEMENT]\nlemma det_col_add: \"det (upd_col M i (a + b)) = det (upd_col M i a) + det (upd_col M i b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Square_Matrix.det (upd_col M i (a + b)) = Square_Matrix.det (upd_col M i a) + Square_Matrix.det (upd_col M i b)\n[PROOF STEP]\nusing det_row_add[of \"transpose M\" i a b]\n[PROOF STATE]\nproof (prove)\nusing this:\nSquare_Matrix.det (upd_row (Square_Matrix.transpose M) i (a + b)) = Square_Matrix.det (upd_row (Square_Matrix.transpose M) i a) + Square_Matrix.det (upd_row (Square_Matrix.transpose M) i b)\n\ngoal (1 subgoal):\n 1. Square_Matrix.det (upd_col M i (a + b)) = Square_Matrix.det (upd_col M i a) + Square_Matrix.det (upd_col M i b)\n[PROOF STEP]\nby (simp add: upd_row_transpose det_transpose)", "meta": {"llama_tokens": 322, "file": "Cayley_Hamilton_Square_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.88720460564669, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7253558466534985}} {"text": "[STATEMENT]\ntheorem card_partition_on_at_most_size:\n assumes \"finite A\"\n shows \"card {P. partition_on A P \\ card P \\ k} = (\\j\\k. Stirling (card A) j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\ card P \\ k} = sum (Stirling (card A)) {..k}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\ card P \\ k} = sum (Stirling (card A)) {..k}\n[PROOF STEP]\nhave \"card {P. partition_on A P \\ card P \\ k} = card (\\j\\k. {P. partition_on A P \\ card P = j})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\ card P \\ k} = card (\\j\\k. {P. partition_on A P \\ card P = j})\n[PROOF STEP]\nby (rule arg_cong[where f=\"card\"]) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard {P. partition_on A P \\ card P \\ k} = card (\\j\\k. {P. partition_on A P \\ card P = j})\n\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\ card P \\ k} = sum (Stirling (card A)) {..k}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {P. partition_on A P \\ card P \\ k} = card (\\j\\k. {P. partition_on A P \\ card P = j})\n\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\ card P \\ k} = sum (Stirling (card A)) {..k}\n[PROOF STEP]\nhave \"\\ = (\\j\\k. card {P. partition_on A P \\ card P = j})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (\\j\\k. {P. partition_on A P \\ card P = j}) = (\\j\\k. card {P. partition_on A P \\ card P = j})\n[PROOF STEP]\nby (subst card_UN_disjoint) (auto simp add: \\finite A\\ finitely_many_partition_on)\n[PROOF STATE]\nproof (state)\nthis:\ncard (\\j\\k. {P. partition_on A P \\ card P = j}) = (\\j\\k. card {P. partition_on A P \\ card P = j})\n\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\ card P \\ k} = sum (Stirling (card A)) {..k}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (\\j\\k. {P. partition_on A P \\ card P = j}) = (\\j\\k. card {P. partition_on A P \\ card P = j})\n\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\ card P \\ k} = sum (Stirling (card A)) {..k}\n[PROOF STEP]\nhave \"(\\j\\k. card {P. partition_on A P \\ card P = j}) = (\\j\\k. Stirling (card A) j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j\\k. card {P. partition_on A P \\ card P = j}) = sum (Stirling (card A)) {..k}\n[PROOF STEP]\nusing \\finite A\\\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. (\\j\\k. card {P. partition_on A P \\ card P = j}) = sum (Stirling (card A)) {..k}\n[PROOF STEP]\nby (simp add: card_partition_on)\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\k. card {P. partition_on A P \\ card P = j}) = sum (Stirling (card A)) {..k}\n\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\ card P \\ k} = sum (Stirling (card A)) {..k}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {P. partition_on A P \\ card P \\ k} = sum (Stirling (card A)) {..k}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {P. partition_on A P \\ card P \\ k} = sum (Stirling (card A)) {..k}\n\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\ card P \\ k} = sum (Stirling (card A)) {..k}\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {P. partition_on A P \\ card P \\ k} = sum (Stirling (card A)) {..k}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1545, "file": "Card_Partitions_Card_Partitions", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872046056466901, "lm_q2_score": 0.817574478416099, "lm_q1q2_score": 0.7253558427099535}} {"text": "[STATEMENT]\nlemma sum_indicator_eq_card:\n assumes \"finite A\"\n shows \"(\\x \\ A. indicator B x) = card (A Int B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum (indicator B) A = card (A \\ B)\n[PROOF STEP]\nusing sum_mult_indicator [OF assms, of \"\\x. 1::nat\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x\\A. 1 * indicator ?B x) = (\\x\\A \\ ?B. 1)\n\ngoal (1 subgoal):\n 1. sum (indicator B) A = card (A \\ B)\n[PROOF STEP]\nunfolding card_eq_sum\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x\\A. 1 * indicator ?B x) = (\\x\\A \\ ?B. 1)\n\ngoal (1 subgoal):\n 1. sum (indicator B) A = (\\x\\A \\ B. 1)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 319, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045847699186, "lm_q2_score": 0.8175744850834648, "lm_q1q2_score": 0.7253558315569554}} {"text": "[STATEMENT]\nlemma choose_linear_sum: \"(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nproof (cases n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. n = 0 \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n 2. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nn = 0\n\ngoal (2 subgoals):\n 1. n = 0 \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n 2. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 0\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\ncase (Suc m)\n[PROOF STATE]\nproof (state)\nthis:\nn = Suc m\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nhave \"(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n[PROOF STEP]\nby (simp add: Suc)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nhave \"\\ = Suc m * 2 ^ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\Suc m. i * (Suc m choose i)) = Suc m * 2 ^ m\n[PROOF STEP]\nunfolding sum.atMost_Suc_shift Suc_times_binomial sum_distrib_left[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 * (Suc m choose 0) + Suc m * sum ((choose) m) {..m} = Suc m * 2 ^ m\n[PROOF STEP]\nby (simp add: choose_row_sum)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\Suc m. i * (Suc m choose i)) = Suc m * 2 ^ m\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\nn = Suc m\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1690, "file": null, "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045907347108, "lm_q2_score": 0.8175744739711883, "lm_q1q2_score": 0.7253558265747546}} {"text": "[STATEMENT]\nlemma choose_linear_sum: \"(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nproof (cases n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. n = 0 \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n 2. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nn = 0\n\ngoal (2 subgoals):\n 1. n = 0 \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n 2. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 0\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\ncase (Suc m)\n[PROOF STATE]\nproof (state)\nthis:\nn = Suc m\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nhave \"(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n[PROOF STEP]\nby (simp add: Suc)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nhave \"\\ = Suc m * 2 ^ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\Suc m. i * (Suc m choose i)) = Suc m * 2 ^ m\n[PROOF STEP]\nunfolding sum.atMost_Suc_shift Suc_times_binomial sum_distrib_left[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 * (Suc m choose 0) + Suc m * sum ((choose) m) {..m} = Suc m * 2 ^ m\n[PROOF STEP]\nby (simp add: choose_row_sum)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\Suc m. i * (Suc m choose i)) = Suc m * 2 ^ m\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\nn = Suc m\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1690, "file": null, "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045907347108, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.725355824602982}} {"text": "[STATEMENT]\nlemma choose_linear_sum: \"(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nproof (cases n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. n = 0 \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n 2. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nn = 0\n\ngoal (2 subgoals):\n 1. n = 0 \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n 2. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 0\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\ncase (Suc m)\n[PROOF STATE]\nproof (state)\nthis:\nn = Suc m\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nhave \"(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n[PROOF STEP]\nby (simp add: Suc)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nhave \"\\ = Suc m * 2 ^ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\Suc m. i * (Suc m choose i)) = Suc m * 2 ^ m\n[PROOF STEP]\nunfolding sum.atMost_Suc_shift Suc_times_binomial sum_distrib_left[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 * (Suc m choose 0) + Suc m * sum ((choose) m) {..m} = Suc m * 2 ^ m\n[PROOF STEP]\nby (simp add: choose_row_sum)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\Suc m. i * (Suc m choose i)) = Suc m * 2 ^ m\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\nn = Suc m\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1690, "file": null, "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045907347108, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.725355824602982}} {"text": "[STATEMENT]\nlemma choose_linear_sum: \"(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nproof (cases n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. n = 0 \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n 2. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nn = 0\n\ngoal (2 subgoals):\n 1. n = 0 \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n 2. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 0\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\ncase (Suc m)\n[PROOF STATE]\nproof (state)\nthis:\nn = Suc m\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nhave \"(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n[PROOF STEP]\nby (simp add: Suc)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = (\\i\\Suc m. i * (Suc m choose i))\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nhave \"\\ = Suc m * 2 ^ m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\Suc m. i * (Suc m choose i)) = Suc m * 2 ^ m\n[PROOF STEP]\nunfolding sum.atMost_Suc_shift Suc_times_binomial sum_distrib_left[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 * (Suc m choose 0) + Suc m * sum ((choose) m) {..m} = Suc m * 2 ^ m\n[PROOF STEP]\nby (simp add: choose_row_sum)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\Suc m. i * (Suc m choose i)) = Suc m * 2 ^ m\n\ngoal (1 subgoal):\n 1. \\nat. n = Suc nat \\ (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\i\\n. i * (n choose i)) = Suc m * 2 ^ m\nn = Suc m\n\ngoal (1 subgoal):\n 1. (\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\n. i * (n choose i)) = n * 2 ^ (n - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1690, "file": null, "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045907347108, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.725355824602982}} {"text": "[STATEMENT]\nlemma gchoose_row_sum_weighted:\n \"(\\k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))\"\n for r :: \"'a::field_char_0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..m. (r gchoose k) * (r / (2::'a) - of_nat k)) = of_nat (Suc m) / (2::'a) * (r gchoose Suc m)\n[PROOF STEP]\nby (induct m) (simp_all add: field_simps distrib gbinomial_mult_1)", "meta": {"llama_tokens": 199, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045877523148, "lm_q2_score": 0.8175744673038222, "lm_q1q2_score": 0.725355818221106}} {"text": "[STATEMENT]\nlemma vector_smult_distrib: \"(A *v ((a :: 'a :: comm_ring_1) *s x)) = a *s ((A *v x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *v (a *s x) = a *s (A *v x)\n[PROOF STEP]\nunfolding matrix_vector_mult_def vector_scalar_mult_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. \\j\\UNIV. A $h i $h j * (\\i. a * x $h i) $h j) = (\\i. a * (\\i. \\j\\UNIV. A $h i $h j * x $h j) $h i)\n[PROOF STEP]\nby (simp add: ac_simps sum_distrib_left)", "meta": {"llama_tokens": 243, "file": "Perron_Frobenius_Perron_Frobenius_Aux", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045877523147, "lm_q2_score": 0.8175744673038221, "lm_q1q2_score": 0.7253558182211057}} {"text": "[STATEMENT]\nlemma totient_prime_power_Suc:\n assumes \"prime p\"\n shows \"totient (p ^ Suc n) = p ^ n * (p - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. totient (p ^ Suc n) = p ^ n * (p - 1)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. totient (p ^ Suc n) = p ^ n * (p - 1)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nprime p\n[PROOF STEP]\nhave \"totient (p ^ Suc n) = card ({0<..p ^ Suc n} - (*) p ` {0<..p ^ n})\"\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. totient (p ^ Suc n) = card ({0<..p ^ Suc n} - (*) p ` {0<..p ^ n})\n[PROOF STEP]\nunfolding totient_def\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. card (totatives (p ^ Suc n)) = card ({0<..p ^ Suc n} - (*) p ` {0<..p ^ n})\n[PROOF STEP]\nby (subst totatives_prime_power_Suc) simp_all\n[PROOF STATE]\nproof (state)\nthis:\ntotient (p ^ Suc n) = card ({0<..p ^ Suc n} - (*) p ` {0<..p ^ n})\n\ngoal (1 subgoal):\n 1. totient (p ^ Suc n) = p ^ n * (p - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ntotient (p ^ Suc n) = card ({0<..p ^ Suc n} - (*) p ` {0<..p ^ n})\n\ngoal (1 subgoal):\n 1. totient (p ^ Suc n) = p ^ n * (p - 1)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nprime p\n[PROOF STEP]\nhave \"\\ = p ^ Suc n - card ((*) p ` {0<..p^n})\"\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. card ({0<..p ^ Suc n} - (*) p ` {0<..p ^ n}) = p ^ Suc n - card ((*) p ` {0<..p ^ n})\n[PROOF STEP]\nby (subst card_Diff_subset) (auto intro: prime_gt_0_nat)\n[PROOF STATE]\nproof (state)\nthis:\ncard ({0<..p ^ Suc n} - (*) p ` {0<..p ^ n}) = p ^ Suc n - card ((*) p ` {0<..p ^ n})\n\ngoal (1 subgoal):\n 1. totient (p ^ Suc n) = p ^ n * (p - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ({0<..p ^ Suc n} - (*) p ` {0<..p ^ n}) = p ^ Suc n - card ((*) p ` {0<..p ^ n})\n\ngoal (1 subgoal):\n 1. totient (p ^ Suc n) = p ^ n * (p - 1)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nprime p\n[PROOF STEP]\nhave \"card ((*) p ` {0<..p^n}) = p ^ n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. card ((*) p ` {0<..p ^ n}) = p ^ n\n[PROOF STEP]\nby (subst card_image) (auto simp: inj_on_def)\n[PROOF STATE]\nproof (state)\nthis:\ncard ((*) p ` {0<..p ^ n}) = p ^ n\n\ngoal (1 subgoal):\n 1. totient (p ^ Suc n) = p ^ n * (p - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((*) p ` {0<..p ^ n}) = p ^ n\n\ngoal (1 subgoal):\n 1. totient (p ^ Suc n) = p ^ n * (p - 1)\n[PROOF STEP]\nhave \"p ^ Suc n - p ^ n = p ^ n * (p - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p ^ Suc n - p ^ n = p ^ n * (p - 1)\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\np ^ Suc n - p ^ n = p ^ n * (p - 1)\n\ngoal (1 subgoal):\n 1. totient (p ^ Suc n) = p ^ n * (p - 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ntotient (p ^ Suc n) = p ^ n * (p - 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntotient (p ^ Suc n) = p ^ n * (p - 1)\n\ngoal (1 subgoal):\n 1. totient (p ^ Suc n) = p ^ n * (p - 1)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ntotient (p ^ Suc n) = p ^ n * (p - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1575, "file": null, "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382094310357, "lm_q2_score": 0.8418256532040707, "lm_q1q2_score": 0.725264965914547}} {"text": "[STATEMENT]\nlemma append_rows_split:\n assumes k: \"k\\dim_row A\"\n shows \"A = (mat_of_rows (dim_col A) [Matrix.row A i. i \\ [0..r\n (mat_of_rows (dim_col A) [Matrix.row A i. i \\ [k..r ?A2\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A = mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..i j. \\i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. \\ A $$ (i, j) = (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r ?A2) \\ carrier_mat (k + (dim_row A-k)) (dim_col A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. carrier_mat (k + (dim_row A - k)) (dim_col A)\n[PROOF STEP]\nby (rule carrier_append_rows, insert k, auto)\n[PROOF STATE]\nproof (state)\nthis:\nmat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. carrier_mat (k + (dim_row A - k)) (dim_col A)\n\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. \\ A $$ (i, j) = (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r ?A2) \\ carrier_mat (dim_row A) (dim_col A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. carrier_mat (k + (dim_row A - k)) (dim_col A)\n\ngoal (1 subgoal):\n 1. mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. carrier_mat (dim_row A) (dim_col A)\n[PROOF STEP]\nusing k\n[PROOF STATE]\nproof (prove)\nusing this:\nmat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. carrier_mat (k + (dim_row A - k)) (dim_col A)\nk \\ dim_row A\n\ngoal (1 subgoal):\n 1. mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. carrier_mat (dim_row A) (dim_col A)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. carrier_mat (dim_row A) (dim_col A)\n\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. \\ A $$ (i, j) = (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r ?A2)\" and \"dim_col A = dim_col (?A1 @\\<^sub>r ?A2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. carrier_mat (dim_row A) (dim_col A)\n\ngoal (1 subgoal):\n 1. dim_row A = dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..i j. \\i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. \\ A $$ (i, j) = (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..i j. \\i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. \\ A $$ (i, j) = (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r ?A2)\" and j: \"j < dim_col (?A1 @\\<^sub>r ?A2)\"\n[PROOF STATE]\nproof (state)\nthis:\ni < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..i j. \\i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. \\ A $$ (i, j) = (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r ?A2) $$ (i, j) = (if i < dim_row ?A1 then ?A1 $$(i,j) else ?A2$$(i-(dim_row ?A1),j))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..i j. \\i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. \\ A $$ (i, j) = (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..i j. \\i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. \\ A $$ (i, j) = (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. carrier_mat (dim_row A) (dim_col A)\n\\ i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0.. (if i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. carrier_mat (dim_row A) (dim_col A)\n\\ i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..i j. \\i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. \\ A $$ (i, j) = (mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r ?A2) $$ (i,j)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(mat_of_rows (dim_col A) (map (Matrix.row A) [0..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k..r mat_of_rows (dim_col A) (map (Matrix.row A) [k.. x > 0; b > 1 \\\n \\ \\log b x\\ = int k + 1 \\ b powr k < x \\ x \\ b powr (k + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < x; 1 < b\\ \\ (\\log b x\\ = int k + 1) = (b powr real k < x \\ x \\ b powr real (k + 1))\n[PROOF STEP]\nby (auto simp: ceiling_eq_iff powr_less_iff le_powr_iff)", "meta": {"llama_tokens": 208, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.8104789063814617, "lm_q1q2_score": 0.7252079811067317}} {"text": "[STATEMENT]\ntheorem card_number_partitions_with_atmost_k_parts:\n shows \"card {N. number_partition n N \\ size N \\ x} = Partition (n + x) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\ size N \\ x} = Partition (n + x) x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\ size N \\ x} = Partition (n + x) x\n[PROOF STEP]\nhave \"bij_betw count {N. number_partition n N \\ size N \\ x} {p. p partitions n \\ sum p {..n} \\ x}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw count {N. number_partition n N \\ size N \\ x} {p. p partitions n \\ sum p {..n} \\ x}\n[PROOF STEP]\nby (rule bij_betw_multiset_number_partition_with_atmost_size)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw count {N. number_partition n N \\ size N \\ x} {p. p partitions n \\ sum p {..n} \\ x}\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\ size N \\ x} = Partition (n + x) x\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nbij_betw count {N. number_partition n N \\ size N \\ x} {p. p partitions n \\ sum p {..n} \\ x}\n[PROOF STEP]\nhave \"card {N. number_partition n N \\ size N \\ x} = card {p. p partitions n \\ sum p {..n} \\ x}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw count {N. number_partition n N \\ size N \\ x} {p. p partitions n \\ sum p {..n} \\ x}\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\ size N \\ x} = card {p. p partitions n \\ sum p {..n} \\ x}\n[PROOF STEP]\nby (rule bij_betw_same_card)\n[PROOF STATE]\nproof (state)\nthis:\ncard {N. number_partition n N \\ size N \\ x} = card {p. p partitions n \\ sum p {..n} \\ x}\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\ size N \\ x} = Partition (n + x) x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {N. number_partition n N \\ size N \\ x} = card {p. p partitions n \\ sum p {..n} \\ x}\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\ size N \\ x} = Partition (n + x) x\n[PROOF STEP]\nhave \"card {p. p partitions n \\ sum p {..n} \\ x} = Partition (n + x) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {p. p partitions n \\ sum p {..n} \\ x} = Partition (n + x) x\n[PROOF STEP]\nby (rule card_partitions_atmost_k_parts)\n[PROOF STATE]\nproof (state)\nthis:\ncard {p. p partitions n \\ sum p {..n} \\ x} = Partition (n + x) x\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\ size N \\ x} = Partition (n + x) x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {N. number_partition n N \\ size N \\ x} = Partition (n + x) x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {N. number_partition n N \\ size N \\ x} = Partition (n + x) x\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\ size N \\ x} = Partition (n + x) x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {N. number_partition n N \\ size N \\ x} = Partition (n + x) x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1320, "file": "Card_Number_Partitions_Card_Number_Partitions", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894548800271, "lm_q2_score": 0.8104789086703225, "lm_q1q2_score": 0.7252079808808771}} {"text": "[STATEMENT]\nlemma sums_zeta_of_nat_offset:\n fixes r :: nat\n assumes n: \"n > 1\"\n shows \"(\\k. 1 / (r + k + 1) ^ n) sums (zeta (of_nat n) - (\\k=1..r. 1 / k ^ n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\k = 1..r. 1 / real (k ^ n)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\k = 1..r. 1 / real (k ^ n)))\n[PROOF STEP]\nhave \"(\\k. 1 / (k + 1) ^ n) sums zeta (of_nat n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. complex_of_real (1 / real ((x + 1) ^ n))) sums zeta (of_nat n)\n[PROOF STEP]\nusing sums_zeta[of \"of_nat n\"] n\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < Re (of_nat n) \\ (\\na. of_nat (Suc na) powr - of_nat n) sums zeta (of_nat n)\n1 < n\n\ngoal (1 subgoal):\n 1. (\\x. complex_of_real (1 / real ((x + 1) ^ n))) sums zeta (of_nat n)\n[PROOF STEP]\nby (simp add: powr_minus field_simps flip: of_nat_Suc)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. complex_of_real (1 / real ((x + 1) ^ n))) sums zeta (of_nat n)\n\ngoal (1 subgoal):\n 1. (\\x. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\k = 1..r. 1 / real (k ^ n)))\n[PROOF STEP]\nfrom sums_split_initial_segment[OF this, of r]\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\i. complex_of_real (1 / real ((i + r + 1) ^ n))) sums (zeta (of_nat n) - (\\ik. 1 / (r + k + 1) ^ n) sums (zeta (of_nat n) - (\\ki. complex_of_real (1 / real ((i + r + 1) ^ n))) sums (zeta (of_nat n) - (\\ix. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\kx. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\kx. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\k = 1..r. 1 / real (k ^ n)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\kx. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\k = 1..r. 1 / real (k ^ n)))\n[PROOF STEP]\nhave \"(\\kk=1..r. 1 / k ^ n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\kk = 1..r. 1 / real (k ^ n))\n[PROOF STEP]\nby (intro sum.reindex_bij_witness[of _ \"\\k. k - 1\" Suc]) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\kk = 1..r. 1 / real (k ^ n))\n\ngoal (1 subgoal):\n 1. (\\x. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\k = 1..r. 1 / real (k ^ n)))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\x. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\k = 1..r. 1 / real (k ^ n)))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\k = 1..r. 1 / real (k ^ n)))\n\ngoal (1 subgoal):\n 1. (\\x. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\k = 1..r. 1 / real (k ^ n)))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\\k = 1..r. 1 / real (k ^ n)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1896, "file": "Zeta_3_Irrational_Zeta_3_Irrational", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473879530492, "lm_q2_score": 0.8311430562234877, "lm_q1q2_score": 0.7250454741118738}} {"text": "[STATEMENT]\nlemma norm_sum_half:\n assumes \"norm(a + b) \\ e\"\n shows \"norm a \\ e/2 \\ norm b \\ e/2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. e / 2 \\ norm a \\ e / 2 \\ norm b\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. e / 2 \\ norm a \\ e / 2 \\ norm b\n[PROOF STEP]\nhave \"e \\ norm (- a - b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. e \\ norm (- a - b)\n[PROOF STEP]\nby (simp add: add.commute assms norm_minus_commute)\n[PROOF STATE]\nproof (state)\nthis:\ne \\ norm (- a - b)\n\ngoal (1 subgoal):\n 1. e / 2 \\ norm a \\ e / 2 \\ norm b\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ne \\ norm (- a - b)\n\ngoal (1 subgoal):\n 1. e / 2 \\ norm a \\ e / 2 \\ norm b\n[PROOF STEP]\nusing norm_triangle_ineq4 order_trans\n[PROOF STATE]\nproof (prove)\nusing this:\ne \\ norm (- a - b)\nnorm (?a - ?b) \\ norm ?a + norm ?b\n\\?x \\ ?y; ?y \\ ?z\\ \\ ?x \\ ?z\n\ngoal (1 subgoal):\n 1. e / 2 \\ norm a \\ e / 2 \\ norm b\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\ne / 2 \\ norm a \\ e / 2 \\ norm b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 571, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473746782093, "lm_q2_score": 0.8311430478583169, "lm_q1q2_score": 0.7250454557812481}} {"text": "[STATEMENT]\nlemma card_convert_triangle_rep_bound: \n fixes G :: \"ugraph\" and t :: real\n assumes \"X \\ uverts G\" and \"Y \\ uverts G\" and \"Z \\ uverts G\" and fin: \"finite (uverts G)\" \nand wf: \"uwellformed G\"\n assumes \"card {(x,y,z) \\ X \\ Y \\ Z . (triangle_in_graph x y z G)} \\ t\"\n shows \"card (triangle_set G) \\ 1/6 *t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 / 6 * t \\ real (card (triangle_set G))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 1 / 6 * t \\ real (card (triangle_set G))\n[PROOF STEP]\ndefine t' where \"t' \\ card {(x,y,z) \\ X \\ Y \\ Z . (triangle_in_graph x y z G)}\"\n[PROOF STATE]\nproof (state)\nthis:\nt' \\ card {(x, y, z). (x, y, z) \\ X \\ Y \\ Z \\ triangle_in_graph x y z G}\n\ngoal (1 subgoal):\n 1. 1 / 6 * t \\ real (card (triangle_set G))\n[PROOF STEP]\nhave \"t' \\ t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. t \\ real t'\n[PROOF STEP]\nusing assms t'_def\n[PROOF STATE]\nproof (prove)\nusing this:\nX \\ uverts G\nY \\ uverts G\nZ \\ uverts G\nfinite (uverts G)\nuwellformed G\nt \\ real (card {(x, y, z). (x, y, z) \\ X \\ Y \\ Z \\ triangle_in_graph x y z G})\nt' \\ card {(x, y, z). (x, y, z) \\ X \\ Y \\ Z \\ triangle_in_graph x y z G}\n\ngoal (1 subgoal):\n 1. t \\ real t'\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nt \\ real t'\n\ngoal (1 subgoal):\n 1. 1 / 6 * t \\ real (card (triangle_set G))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nt \\ real t'\n[PROOF STEP]\nhave tgt: \"1/6 * t' \\ 1/6 * t\"\n[PROOF STATE]\nproof (prove)\nusing this:\nt \\ real t'\n\ngoal (1 subgoal):\n 1. 1 / 6 * t \\ 1 / 6 * real t'\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 / 6 * t \\ 1 / 6 * real t'\n\ngoal (1 subgoal):\n 1. 1 / 6 * t \\ real (card (triangle_set G))\n[PROOF STEP]\nhave \"card (triangle_set G) \\ 1/6 *t'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 / 6 * real t' \\ real (card (triangle_set G))\n[PROOF STEP]\nusing t'_def card_convert_triangle_rep assms\n[PROOF STATE]\nproof (prove)\nusing this:\nt' \\ card {(x, y, z). (x, y, z) \\ X \\ Y \\ Z \\ triangle_in_graph x y z G}\n\\?X \\ uverts ?G; ?Y \\ uverts ?G; ?Z \\ uverts ?G; finite (uverts ?G); uwellformed ?G\\ \\ 1 / 6 * real (card {(x, y, z). (x, y, z) \\ ?X \\ ?Y \\ ?Z \\ triangle_in_graph x y z ?G}) \\ real (card (triangle_set ?G))\nX \\ uverts G\nY \\ uverts G\nZ \\ uverts G\nfinite (uverts G)\nuwellformed G\nt \\ real (card {(x, y, z). (x, y, z) \\ X \\ Y \\ Z \\ triangle_in_graph x y z G})\n\ngoal (1 subgoal):\n 1. 1 / 6 * real t' \\ real (card (triangle_set G))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n1 / 6 * real t' \\ real (card (triangle_set G))\n\ngoal (1 subgoal):\n 1. 1 / 6 * t \\ real (card (triangle_set G))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n1 / 6 * real t' \\ real (card (triangle_set G))\n\ngoal (1 subgoal):\n 1. 1 / 6 * t \\ real (card (triangle_set G))\n[PROOF STEP]\nusing tgt\n[PROOF STATE]\nproof (prove)\nusing this:\n1 / 6 * real t' \\ real (card (triangle_set G))\n1 / 6 * t \\ 1 / 6 * real t'\n\ngoal (1 subgoal):\n 1. 1 / 6 * t \\ real (card (triangle_set G))\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\n1 / 6 * t \\ real (card (triangle_set G))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1623, "file": "Roth_Arithmetic_Progressions_Roth_Arithmetic_Progressions", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473647220787, "lm_q2_score": 0.8311430436757313, "lm_q1q2_score": 0.7250454438576118}} {"text": "[STATEMENT]\nlemma t1_space_prod_topology:\n \"t1_space(prod_topology X Y) \\ topspace(prod_topology X Y) = {} \\ t1_space X \\ t1_space Y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n[PROOF STEP]\nproof (cases \"topspace (prod_topology X Y) = {}\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. topspace (prod_topology X Y) = {} \\ t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n 2. topspace (prod_topology X Y) \\ {} \\ t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\ntopspace (prod_topology X Y) = {}\n\ngoal (2 subgoals):\n 1. topspace (prod_topology X Y) = {} \\ t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n 2. topspace (prod_topology X Y) \\ {} \\ t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ntopspace (prod_topology X Y) = {}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntopspace (prod_topology X Y) = {}\n\ngoal (1 subgoal):\n 1. t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n[PROOF STEP]\nby (auto simp: t1_space_empty)\n[PROOF STATE]\nproof (state)\nthis:\nt1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n\ngoal (1 subgoal):\n 1. topspace (prod_topology X Y) \\ {} \\ t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. topspace (prod_topology X Y) \\ {} \\ t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\ntopspace (prod_topology X Y) \\ {}\n\ngoal (1 subgoal):\n 1. topspace (prod_topology X Y) \\ {} \\ t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n[PROOF STEP]\nhave eq: \"{(x,y)} = {x} \\ {y}\" for x y\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {(x, y)} = {x} \\ {y}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n{(?x1, ?y1)} = {?x1} \\ {?y1}\n\ngoal (1 subgoal):\n 1. topspace (prod_topology X Y) \\ {} \\ t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n[PROOF STEP]\nhave \"t1_space (prod_topology X Y) \\ (t1_space X \\ t1_space Y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. t1_space (prod_topology X Y) = (t1_space X \\ t1_space Y)\n[PROOF STEP]\nusing False\n[PROOF STATE]\nproof (prove)\nusing this:\ntopspace (prod_topology X Y) \\ {}\n\ngoal (1 subgoal):\n 1. t1_space (prod_topology X Y) = (t1_space X \\ t1_space Y)\n[PROOF STEP]\nby (force simp: t1_space_closedin_singleton closedin_prod_Times_iff eq simp del: insert_Times_insert)\n[PROOF STATE]\nproof (state)\nthis:\nt1_space (prod_topology X Y) = (t1_space X \\ t1_space Y)\n\ngoal (1 subgoal):\n 1. topspace (prod_topology X Y) \\ {} \\ t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n[PROOF STEP]\nwith False\n[PROOF STATE]\nproof (chain)\npicking this:\ntopspace (prod_topology X Y) \\ {}\nt1_space (prod_topology X Y) = (t1_space X \\ t1_space Y)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntopspace (prod_topology X Y) \\ {}\nt1_space (prod_topology X Y) = (t1_space X \\ t1_space Y)\n\ngoal (1 subgoal):\n 1. t1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nt1_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \\ t1_space X \\ t1_space Y)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1814, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916099737806, "lm_q2_score": 0.8397339716830605, "lm_q1q2_score": 0.7250192657611148}} {"text": "[STATEMENT]\nlemma has_real_derivative_powr:\n assumes \"z > 0\"\n shows \"((\\z. z powr r) has_real_derivative r * z powr (r - 1)) (at z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\z. z powr r) has_real_derivative r * z powr (r - 1)) (at z)\n[PROOF STEP]\nproof (subst DERIV_cong_ev[OF refl _ refl])\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\\\<^sub>F x in nhds z. x powr r = ?g x\n 2. (?g has_real_derivative r * z powr (r - 1)) (at z)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < z\n[PROOF STEP]\nhave \"eventually (\\z. z \\ 0) (nhds z)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < z\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F z in nhds z. z \\ 0\n[PROOF STEP]\nby (intro t1_space_nhds) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F z in nhds z. z \\ 0\n\ngoal (2 subgoals):\n 1. \\\\<^sub>F x in nhds z. x powr r = ?g x\n 2. (?g has_real_derivative r * z powr (r - 1)) (at z)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F z in nhds z. z \\ 0\n[PROOF STEP]\nshow \"eventually (\\z. z powr r = exp (r * ln z)) (nhds z)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F z in nhds z. z \\ 0\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F z in nhds z. z powr r = exp (r * ln z)\n[PROOF STEP]\nunfolding powr_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F z in nhds z. z \\ 0\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F z in nhds z. (if z = 0 then 0 else exp (r * ln z)) = exp (r * ln z)\n[PROOF STEP]\nby eventually_elim simp\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F z in nhds z. z powr r = exp (r * ln z)\n\ngoal (1 subgoal):\n 1. ((\\z. exp (r * ln z)) has_real_derivative r * z powr (r - 1)) (at z)\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < z\n[PROOF STEP]\nshow \"((\\z. exp (r * ln z)) has_real_derivative r * z powr (r - 1)) (at z)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < z\n\ngoal (1 subgoal):\n 1. ((\\z. exp (r * ln z)) has_real_derivative r * z powr (r - 1)) (at z)\n[PROOF STEP]\nby (auto intro!: derivative_eq_intros simp: powr_def field_simps exp_diff)\n[PROOF STATE]\nproof (state)\nthis:\n((\\z. exp (r * ln z)) has_real_derivative r * z powr (r - 1)) (at z)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1091, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110454379297, "lm_q2_score": 0.8128673201042492, "lm_q1q2_score": 0.7249240545444987}} {"text": "[STATEMENT]\nlemma choose_rising_sum:\n \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))\"\n \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1) &&& (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n 2. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nshow \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n[PROOF STEP]\nby (induct m) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nhave \"\\ = (n + m + 1) choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n + m + 1 choose (n + 1) = n + m + 1 choose m\n[PROOF STEP]\nby (subst binomial_symmetric) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nn + m + 1 choose (n + 1) = n + m + 1 choose m\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nshow \"(\\j\\m. ((n + j) choose n)) = (n + m + 1) choose m\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 958, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.89181104831338, "lm_q2_score": 0.8128673155708976, "lm_q1q2_score": 0.7249240528389653}} {"text": "[STATEMENT]\nlemma choose_rising_sum:\n \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))\"\n \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1) &&& (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n 2. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nshow \"(\\j\\m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n[PROOF STEP]\nby (induct m) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose (n + 1)\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nhave \"\\ = (n + m + 1) choose m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n + m + 1 choose (n + 1) = n + m + 1 choose m\n[PROOF STEP]\nby (subst binomial_symmetric) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nn + m + 1 choose (n + 1) = n + m + 1 choose m\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\nshow \"(\\j\\m. ((n + j) choose n)) = (n + m + 1) choose m\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n\ngoal (1 subgoal):\n 1. (\\j\\m. n + j choose n) = n + m + 1 choose m\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\j\\m. n + j choose n) = n + m + 1 choose m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 958, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.89181104831338, "lm_q2_score": 0.8128673155708976, "lm_q1q2_score": 0.7249240528389653}} {"text": "[STATEMENT]\nlemma fbox_fdia_demorgan: \"( |R] (f::'b \\ 'c::complete_boolean_algebra) x) = - |R\\ (\\y. -f y) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. |R] f x = - |R\\ (\\y. - f y) x\n[PROOF STEP]\napply (simp add: fbox_def fdia_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\{f y |y. (x, y) \\ R} = - \\{- f y |y. (x, y) \\ R}\n[PROOF STEP]\napply (rule antisym)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\{f y |y. (x, y) \\ R} \\ - \\{- f y |y. (x, y) \\ R}\n 2. - \\{- f y |y. (x, y) \\ R} \\ \\{f y |y. (x, y) \\ R}\n[PROOF STEP]\napply (simp add: uminus_Sup)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\{f y |y. (x, y) \\ R} \\ \\(uminus ` {- f y |y. (x, y) \\ R})\n 2. - \\{- f y |y. (x, y) \\ R} \\ \\{f y |y. (x, y) \\ R}\n[PROOF STEP]\napply (rule INF_greatest; rule Inf_lower)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\xa. xa \\ {- f y |y. (x, y) \\ R} \\ - xa \\ {f y |y. (x, y) \\ R}\n 2. - \\{- f y |y. (x, y) \\ R} \\ \\{f y |y. (x, y) \\ R}\n[PROOF STEP]\napply auto[1]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - \\{- f y |y. (x, y) \\ R} \\ \\{f y |y. (x, y) \\ R}\n[PROOF STEP]\napply (rule Inf_greatest)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xa. xa \\ {f y |y. (x, y) \\ R} \\ - \\{- f y |y. (x, y) \\ R} \\ xa\n[PROOF STEP]\nby (simp add: Sup_upper compl_le_swap2)", "meta": {"llama_tokens": 830, "file": "PSemigroupsConvolution_Unary_Modalities", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.89181104831338, "lm_q2_score": 0.8128673110375457, "lm_q1q2_score": 0.724924048796072}} {"text": "[STATEMENT]\nlemma enumerate_1:\"\\\\j \\ (n::nat). f j \\ A; \\j \\ (m::nat). g j \\ A;\n inj_on f {i. i \\ n}; inj_on g {j. j \\ m}; f `{j. j \\ n} = A;\n g ` {j. j \\ m} = A \\ \\ n = m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\j\\n. f j \\ A; \\j\\m. g j \\ A; inj_on f {i. i \\ n}; inj_on g {j. j \\ m}; f ` {j. j \\ n} = A; g ` {j. j \\ m} = A\\ \\ n = m\n[PROOF STEP]\napply (frule card_image[of f \"{i. i \\ n}\"],\n frule card_image[of g \"{i. i \\ m}\"])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\j\\n. f j \\ A; \\j\\m. g j \\ A; inj_on f {i. i \\ n}; inj_on g {j. j \\ m}; f ` {j. j \\ n} = A; g ` {j. j \\ m} = A; card (f ` {i. i \\ n}) = card {i. i \\ n}; card (g ` {i. i \\ m}) = card {i. i \\ m}\\ \\ n = m\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 509, "file": "Group-Ring-Module_Algebra1", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467580102418, "lm_q2_score": 0.8244619220634456, "lm_q1q2_score": 0.7248230258849708}} {"text": "[STATEMENT]\nlemma lim_mono:\n fixes X Y :: \"nat \\ 'a::linorder_topology\"\n assumes \"\\n. N \\ n \\ X n \\ Y n\"\n and \"X \\ x\"\n and \"Y \\ y\"\n shows \"x \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nN \\ ?n \\ X ?n \\ Y ?n\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nby (intro LIMSEQ_le[OF assms(2,3)]) auto", "meta": {"llama_tokens": 214, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392817460333, "lm_q2_score": 0.8198933381139645, "lm_q1q2_score": 0.7248179177346268}} {"text": "[STATEMENT]\nlemma lim_mono:\n fixes X Y :: \"nat \\ 'a::linorder_topology\"\n assumes \"\\n. N \\ n \\ X n \\ Y n\"\n and \"X \\ x\"\n and \"Y \\ y\"\n shows \"x \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nN \\ ?n \\ X ?n \\ Y ?n\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nby (intro LIMSEQ_le[OF assms(2,3)]) auto", "meta": {"llama_tokens": 214, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392817460333, "lm_q2_score": 0.819893335913536, "lm_q1q2_score": 0.7248179157893615}} {"text": "[STATEMENT]\nlemma nat_interval_union:\n assumes \\m \\ n\\\n shows \\{i::nat. i \\ n \\ P i}\n = {i::nat. i \\ m \\ P i} \\ {i::nat. m < i \\ i \\ n \\ P i}\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {i. i \\ n \\ P i} = {i. i \\ m \\ P i} \\ {i. m < i \\ i \\ n \\ P i}\n[PROOF STEP]\nusing assms le_cases nat_less_le\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ n\n\\?x \\ ?y \\ ?P; ?y \\ ?x \\ ?P\\ \\ ?P\n(?m < ?n) = (?m \\ ?n \\ ?m \\ ?n)\n\ngoal (1 subgoal):\n 1. {i. i \\ n \\ P i} = {i. i \\ m \\ P i} \\ {i. m < i \\ i \\ n \\ P i}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 356, "file": "TESL_Language_StutteringLemmas", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392817460333, "lm_q2_score": 0.8198933315126792, "lm_q1q2_score": 0.7248179118988313}} {"text": "[STATEMENT]\nlemma lim_mono:\n fixes X Y :: \"nat \\ 'a::linorder_topology\"\n assumes \"\\n. N \\ n \\ X n \\ Y n\"\n and \"X \\ x\"\n and \"Y \\ y\"\n shows \"x \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nN \\ ?n \\ X ?n \\ Y ?n\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nby (intro LIMSEQ_le[OF assms(2,3)]) auto", "meta": {"llama_tokens": 214, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392817460333, "lm_q2_score": 0.8198933315126791, "lm_q1q2_score": 0.7248179118988312}} {"text": "[STATEMENT]\ntheorem time_mergesort_tm_bigo:\n \"(\\xs. time (mergesort_tm f xs)) \\ O[length going_to at_top]((\\n. n * ln n) o length)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. real (time (mergesort_tm f x))) \\ O[length going_to sequentially]((\\n. real n * ln (real n)) \\ length)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\x. real (time (mergesort_tm f x))) \\ O[length going_to sequentially]((\\n. real n * ln (real n)) \\ length)\n[PROOF STEP]\nhave 0: \"\\xs. time (mergesort_tm f xs) \\ (mergesort_recurrence o length) xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xs. real (time (mergesort_tm f xs)) \\ (mergesort_recurrence \\ length) xs\n[PROOF STEP]\nunfolding comp_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\xs. real (time (mergesort_tm f xs)) \\ mergesort_recurrence (length xs)\n[PROOF STEP]\nusing time_mergesort_conv_mergesort_recurrence\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (time (mergesort_tm ?f ?xs)) \\ mergesort_recurrence (length ?xs)\n\ngoal (1 subgoal):\n 1. \\xs. real (time (mergesort_tm f xs)) \\ mergesort_recurrence (length xs)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nreal (time (mergesort_tm f ?xs)) \\ (mergesort_recurrence \\ length) ?xs\n\ngoal (1 subgoal):\n 1. (\\x. real (time (mergesort_tm f x))) \\ O[length going_to sequentially]((\\n. real n * ln (real n)) \\ length)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. real (time (mergesort_tm f x))) \\ O[length going_to sequentially]((\\n. real n * ln (real n)) \\ length)\n[PROOF STEP]\nusing bigo_measure_trans[OF 0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\mergesort_recurrence \\ O(?f); \\x. x \\ ?A \\ 0 \\ real (time (mergesort_tm f x))\\ \\ (\\x. real (time (mergesort_tm f x))) \\ O[length going_to sequentially within ?A](?f \\ length)\n\ngoal (1 subgoal):\n 1. (\\x. real (time (mergesort_tm f x))) \\ O[length going_to sequentially]((\\n. real n * ln (real n)) \\ length)\n[PROOF STEP]\nby (simp add: bigthetaD1 mergesort_recurrence)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. real (time (mergesort_tm f x))) \\ O[length going_to sequentially]((\\n. real n * ln (real n)) \\ length)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1013, "file": "Closest_Pair_Points_Common", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392695254319, "lm_q2_score": 0.8198933425148214, "lm_q1q2_score": 0.7248179116055674}} {"text": "[STATEMENT]\nlemma lim_mono:\n fixes X Y :: \"nat \\ 'a::linorder_topology\"\n assumes \"\\n. N \\ n \\ X n \\ Y n\"\n and \"X \\ x\"\n and \"Y \\ y\"\n shows \"x \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nN \\ ?n \\ X ?n \\ Y ?n\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nby (intro LIMSEQ_le[OF assms(2,3)]) auto", "meta": {"llama_tokens": 214, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392817460333, "lm_q2_score": 0.8198933293122507, "lm_q1q2_score": 0.724817909953566}} {"text": "[STATEMENT]\nlemma disjoint_orbits:\n assumes \"A \\ orbits f\" and \"B \\ orbits f\" and \"A \\ B\"\n shows \"A \\ B = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A \\ B = {}\n[PROOF STEP]\nusing \\A \\ orbits f\\\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ orbits f\n\ngoal (1 subgoal):\n 1. A \\ B = {}\n[PROOF STEP]\napply cases\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a. \\A = orbit f a; a \\ affected f\\ \\ A \\ B = {}\n[PROOF STEP]\nusing \\B \\ orbits f\\\n[PROOF STATE]\nproof (prove)\nusing this:\nB \\ orbits f\n\ngoal (1 subgoal):\n 1. \\a. \\A = orbit f a; a \\ affected f\\ \\ A \\ B = {}\n[PROOF STEP]\napply cases\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a aa. \\A = orbit f a; a \\ affected f; B = orbit f aa; aa \\ affected f\\ \\ A \\ B = {}\n[PROOF STEP]\nusing \\A \\ B\\\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ B\n\ngoal (1 subgoal):\n 1. \\a aa. \\A = orbit f a; a \\ affected f; B = orbit f aa; aa \\ affected f\\ \\ A \\ B = {}\n[PROOF STEP]\napply (simp_all add: orbit_disjoint)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 570, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240895276223, "lm_q2_score": 0.8376199572530448, "lm_q1q2_score": 0.7247289648844316}} {"text": "[STATEMENT]\ntheorem dual_Sup [iff?]:\n \"is_Sup (dual ` A) (dual inf) = is_Inf A inf\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_Sup (dual ` A) (dual inf) = is_Inf A inf\n[PROOF STEP]\nby (simp add: is_Inf_def is_Sup_def dual_all [symmetric] dual_leq)", "meta": {"llama_tokens": 120, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9136765328159725, "lm_q2_score": 0.7931059560743422, "lm_q1q2_score": 0.7246423001017019}} {"text": "[STATEMENT]\nlemma llength_lmirror_aux: \"llength (lmirror_aux acc xs) = 2 * llength xs + llength acc\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. llength (lmirror_aux acc xs) = 2 * llength xs + llength acc\n[PROOF STEP]\napply(coinduction arbitrary: acc xs rule: enat_coinduct)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\acc xs. (llength (lmirror_aux acc xs) = 0) = (2 * llength xs + llength acc = 0) \\ (llength (lmirror_aux acc xs) \\ 0 \\ 2 * llength xs + llength acc \\ 0 \\ (\\acca xsa. epred (llength (lmirror_aux acc xs)) = llength (lmirror_aux acca xsa) \\ epred (2 * llength xs + llength acc) = 2 * llength xsa + llength acca) \\ epred (llength (lmirror_aux acc xs)) = epred (2 * llength xs + llength acc))\n[PROOF STEP]\napply(auto simp add: iadd_is_0 epred_iadd1 mult_2 epred_llength ltl_lmirror_aux iadd_Suc_right)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\acc xs. \\\\ lnull xs; llength (lmirror_aux (LCons (lhd xs) acc) (ltl xs)) \\ llength (ltl xs) + llength xs + llength acc\\ \\ \\acca xsa. llength (lmirror_aux (LCons (lhd xs) acc) (ltl xs)) = llength (lmirror_aux acca xsa) \\ llength (ltl xs) + llength xs + llength acc = llength xsa + llength xsa + llength acca\n[PROOF STEP]\napply(rule exI conjI refl)+\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\acc xs. \\\\ lnull xs; llength (lmirror_aux (LCons (lhd xs) acc) (ltl xs)) \\ llength (ltl xs) + llength xs + llength acc\\ \\ llength (ltl xs) + llength xs + llength acc = llength (ltl xs) + llength (ltl xs) + llength (LCons (lhd xs) acc)\n[PROOF STEP]\napply(simp add: iadd_Suc_right llength_ltl)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\acc xs. \\\\ lnull xs; llength (lmirror_aux (LCons (lhd xs) acc) (ltl xs)) \\ epred (llength xs) + llength xs + llength acc\\ \\ epred (llength xs) + llength xs + llength acc = eSuc (epred (llength xs) + epred (llength xs) + llength acc)\n[PROOF STEP]\nby (metis (opaque_lifting, no_types) add.commute epred_llength iadd_Suc_right lhd_LCons_ltl llength_LCons)", "meta": {"llama_tokens": 899, "file": "Coinductive_Examples_LMirror", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772417253255, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.7245565471366613}} {"text": "[STATEMENT]\nlemma is_zero_row_JNF_multrow[simp]: \n fixes A::\"'a::comm_ring_1 mat\"\n assumes \"ijaj:(list_eq xs m) = (m * (real (length (list_eq xs m))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list (list_eq xs m) = m * real (length (list_eq xs m))\n[PROOF STEP]\napply (induct_tac xs)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. sum_list (list_eq [] m) = m * real (length (list_eq [] m))\n 2. \\a list. sum_list (list_eq list m) = m * real (length (list_eq list m)) \\ sum_list (list_eq (a # list) m) = m * real (length (list_eq (a # list) m))\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a list. sum_list (list_eq list m) = m * real (length (list_eq list m)) \\ sum_list (list_eq (a # list) m) = m * real (length (list_eq (a # list) m))\n[PROOF STEP]\napply (simp add:field_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 380, "file": "Cauchy_CauchysMeanTheorem", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887588052782737, "lm_q2_score": 0.8152324960856175, "lm_q1q2_score": 0.7245450592450784}} {"text": "[STATEMENT]\nlemma zeta_conv_hurwitz_zeta_multiplication:\n fixes k a :: nat and s :: complex\n assumes \"k > 0\" \"s \\ 1\"\n shows \"k powr s * zeta s = (\\n=1..k. hurwitz_zeta (n / k) s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat k powr s * zeta s = (\\n = 1..k. hurwitz_zeta (real n / real k) s)\n[PROOF STEP]\nusing perzeta_conv_hurwitz_zeta_multiplication[of k s 0]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < k; s \\ 1\\ \\ of_nat k powr s * perzeta (real_of_int 0 / real k) s = (\\n = 1..k. exp (complex_of_real (2 * pi * real n * real_of_int 0 / real k) * \\) * hurwitz_zeta (real n / real k) s)\n\ngoal (1 subgoal):\n 1. of_nat k powr s * zeta s = (\\n = 1..k. hurwitz_zeta (real n / real k) s)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\0 < k; s \\ 1\\ \\ of_nat k powr s * perzeta (real_of_int 0 / real k) s = (\\n = 1..k. exp (complex_of_real (2 * pi * real n * real_of_int 0 / real k) * \\) * hurwitz_zeta (real n / real k) s)\n0 < k\ns \\ 1\n\ngoal (1 subgoal):\n 1. of_nat k powr s * zeta s = (\\n = 1..k. hurwitz_zeta (real n / real k) s)\n[PROOF STEP]\nby (simp add: perzeta_int)", "meta": {"llama_tokens": 549, "file": "Zeta_Function_Zeta_Function", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588023318196, "lm_q2_score": 0.8152324960856175, "lm_q1q2_score": 0.7245450568430333}} {"text": "[STATEMENT]\nlemma ordIso_Pow_mono[simp]:\nassumes \"r \\o r'\"\nshows \"|Pow(Field r)| \\o |Pow(Field r')|\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. |Pow (Field r)| \\o |Pow (Field r')|\n[PROOF STEP]\nusing assms card_of_mono2 card_of_Pow_mono\n[PROOF STATE]\nproof (prove)\nusing this:\nr \\o r'\n?r \\o ?r' \\ |Field ?r| \\o |Field ?r'|\n|?A| \\o |?B| \\ |Pow ?A| \\o |Pow ?B|\n\ngoal (1 subgoal):\n 1. |Pow (Field r)| \\o |Pow (Field r')|\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 252, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.888758793492457, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7245450476420437}} {"text": "[STATEMENT]\nlemma nn_integral_neg_binomial_pmf_real:\n assumes p: \"p \\ {0<..1}\"\n shows \"nn_integral (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (n * (1 - p) / p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p)\n[PROOF STEP]\nproof (induction n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. integral\\<^sup>N (measure_pmf (neg_binomial_pmf 0 p)) of_nat = ennreal (real 0 * (1 - p) / p)\n 2. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. integral\\<^sup>N (measure_pmf (neg_binomial_pmf 0 p)) of_nat = ennreal (real 0 * (1 - p) / p)\n 2. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral\\<^sup>N (measure_pmf (neg_binomial_pmf 0 p)) of_nat = ennreal (real 0 * (1 - p) / p)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>N (measure_pmf (neg_binomial_pmf 0 p)) of_nat = ennreal (real 0 * (1 - p) / p)\n\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p)\n\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nhave \"nn_integral (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat =\n nn_integral (measure_pmf (geometric_pmf p)) of_nat +\n nn_integral (measure_pmf (neg_binomial_pmf n p)) of_nat\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = integral\\<^sup>N (measure_pmf (geometric_pmf p)) of_nat + integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat\n[PROOF STEP]\nby (simp add: neg_binomial_pmf_Suc case_prod_unfold nn_integral_add nn_integral_pair_pmf')\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = integral\\<^sup>N (measure_pmf (geometric_pmf p)) of_nat + integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat\n\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = integral\\<^sup>N (measure_pmf (geometric_pmf p)) of_nat + integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat\n\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nhave \"nn_integral (measure_pmf (geometric_pmf p)) of_nat = ennreal ((1-p) / p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral\\<^sup>N (measure_pmf (geometric_pmf p)) of_nat = ennreal ((1 - p) / p)\n[PROOF STEP]\nunfolding ennreal_of_nat_eq_real_of_nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ i. ennreal (real i) \\measure_pmf (geometric_pmf p) = ennreal ((1 - p) / p)\n[PROOF STEP]\nusing expectation_geometric_pmf[OF p] integrable_real_geometric_pmf[OF p]\n[PROOF STATE]\nproof (prove)\nusing this:\nmeasure_pmf.expectation (geometric_pmf p) real = (1 - p) / p\nintegrable (measure_pmf (geometric_pmf p)) real\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ i. ennreal (real i) \\measure_pmf (geometric_pmf p) = ennreal ((1 - p) / p)\n[PROOF STEP]\nby (subst nn_integral_eq_integral) auto\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>N (measure_pmf (geometric_pmf p)) of_nat = ennreal ((1 - p) / p)\n\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>N (measure_pmf (geometric_pmf p)) of_nat = ennreal ((1 - p) / p)\n\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nhave \"nn_integral (measure_pmf (neg_binomial_pmf n p)) of_nat = n * (1 - p) / p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p)\n[PROOF STEP]\nusing p\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ {0<..1}\n\ngoal (1 subgoal):\n 1. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p)\n[PROOF STEP]\nby (subst Suc.IH)\n (auto simp: ennreal_of_nat_eq_real_of_nat ennreal_mult simp flip: divide_ennreal ennreal_minus)\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p)\n\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p)\n\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nhave \"ennreal ((1 - p) / p) + ennreal (real n * (1 - p) / p) =\n ennreal ((1-p) / p + real n * (1 - p) / p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ennreal ((1 - p) / p) + ennreal (real n * (1 - p) / p) = ennreal ((1 - p) / p + real n * (1 - p) / p)\n[PROOF STEP]\nby (intro ennreal_plus [symmetric] divide_nonneg_pos mult_nonneg_nonneg) (use p in auto)\n[PROOF STATE]\nproof (state)\nthis:\nennreal ((1 - p) / p) + ennreal (real n * (1 - p) / p) = ennreal ((1 - p) / p + real n * (1 - p) / p)\n\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nennreal ((1 - p) / p) + ennreal (real n * (1 - p) / p) = ennreal ((1 - p) / p + real n * (1 - p) / p)\n\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nhave \"(1-p) / p + real n * (1 - p) / p = real (Suc n) * (1 - p) / p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1 - p) / p + real n * (1 - p) / p = real (Suc n) * (1 - p) / p\n[PROOF STEP]\nusing p\n[PROOF STATE]\nproof (prove)\nusing this:\np \\ {0<..1}\n\ngoal (1 subgoal):\n 1. (1 - p) / p + real n * (1 - p) / p = real (Suc n) * (1 - p) / p\n[PROOF STEP]\nby (auto simp: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(1 - p) / p + real n * (1 - p) / p = real (Suc n) * (1 - p) / p\n\ngoal (1 subgoal):\n 1. \\n. integral\\<^sup>N (measure_pmf (neg_binomial_pmf n p)) of_nat = ennreal (real n * (1 - p) / p) \\ integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nintegral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nintegral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n\ngoal (1 subgoal):\n 1. integral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n[PROOF STEP]\nby (simp add: ennreal_of_nat_eq_real_of_nat)\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>N (measure_pmf (neg_binomial_pmf (Suc n) p)) of_nat = ennreal (real (Suc n) * (1 - p) / p)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4243, "file": null, "length": 28, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9019206844384594, "lm_q2_score": 0.803173801068221, "lm_q1q2_score": 0.7243990643824889}} {"text": "[STATEMENT]\nlemma op_norm_le_transpose: \"\\A\\\\<^sub>o\\<^sub>p \\ \\transpose A\\\\<^sub>o\\<^sub>p\" for A :: \"real^'n^'n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ \\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ \\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nhave obs:\"\\x. \\x\\ = 1 \\ (\\A *v x\\) \\ sqrt ((\\transpose A ** A\\\\<^sub>o\\<^sub>p)) * (\\x\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\x\\ = 1 \\ \\A *v x\\ \\ sqrt (\\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p) * \\x\\\n[PROOF STEP]\nusing norm_matrix_vec_mult_le_transpose\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x\\ = 1 \\ \\?A *v ?x\\ \\ sqrt (\\Finite_Cartesian_Product.transpose ?A ** ?A\\\\<^sub>o\\<^sub>p) * \\?x\\\n\ngoal (1 subgoal):\n 1. \\x. \\x\\ = 1 \\ \\A *v x\\ \\ sqrt (\\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p) * \\x\\\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\x. \\x\\ = 1 \\ \\A *v x\\ \\ sqrt (\\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p) * \\x\\\n\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ \\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nhave \"(\\A\\\\<^sub>o\\<^sub>p) \\ sqrt ((\\transpose A ** A\\\\<^sub>o\\<^sub>p))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ sqrt (\\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p)\n[PROOF STEP]\nusing obs\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x. \\x\\ = 1 \\ \\A *v x\\ \\ sqrt (\\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p) * \\x\\\n\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ sqrt (\\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p)\n[PROOF STEP]\napply(unfold op_norm_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\x\\ = 1 \\ \\A *v x\\ \\ sqrt (Sup {\\Finite_Cartesian_Product.transpose A ** A *v x\\ |x. \\x\\ = 1}) * \\x\\ \\ Sup {\\A *v x\\ |x. \\x\\ = 1} \\ sqrt (Sup {\\Finite_Cartesian_Product.transpose A ** A *v x\\ |x. \\x\\ = 1})\n[PROOF STEP]\nby (rule cSup_least[OF op_norm_set_proptys(3)]) clarsimp\n[PROOF STATE]\nproof (state)\nthis:\n\\A\\\\<^sub>o\\<^sub>p \\ sqrt (\\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p)\n\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ \\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nhence \"((\\A\\\\<^sub>o\\<^sub>p))\\<^sup>2 \\ (\\transpose A ** A\\\\<^sub>o\\<^sub>p)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A\\\\<^sub>o\\<^sub>p \\ sqrt (\\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p)\n\ngoal (1 subgoal):\n 1. (\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ \\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nusing power_mono[of \"(\\A\\\\<^sub>o\\<^sub>p)\" _ 2] op_norm_ge_0\n[PROOF STATE]\nproof (prove)\nusing this:\n\\A\\\\<^sub>o\\<^sub>p \\ sqrt (\\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p)\n\\\\A\\\\<^sub>o\\<^sub>p \\ ?b; 0 \\ \\A\\\\<^sub>o\\<^sub>p\\ \\ (\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ ?b\\<^sup>2\n0 \\ \\?A\\\\<^sub>o\\<^sub>p\n\ngoal (1 subgoal):\n 1. (\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ \\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nby (metis not_le real_less_lsqrt)\n[PROOF STATE]\nproof (state)\nthis:\n(\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ \\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p\n\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ \\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ \\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p\n\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ \\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nhave \"... \\ (\\transpose A\\\\<^sub>o\\<^sub>p) * (\\A\\\\<^sub>o\\<^sub>p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p \\ (\\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p) * (\\A\\\\<^sub>o\\<^sub>p)\n[PROOF STEP]\nusing op_norm_matrix_matrix_mult_le\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?A ** ?B\\\\<^sub>o\\<^sub>p \\ (\\?A\\\\<^sub>o\\<^sub>p) * (\\?B\\\\<^sub>o\\<^sub>p)\n\ngoal (1 subgoal):\n 1. \\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p \\ (\\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p) * (\\A\\\\<^sub>o\\<^sub>p)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\Finite_Cartesian_Product.transpose A ** A\\\\<^sub>o\\<^sub>p \\ (\\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p) * (\\A\\\\<^sub>o\\<^sub>p)\n\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ \\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ (\\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p) * (\\A\\\\<^sub>o\\<^sub>p)\n[PROOF STEP]\nhave \"((\\A\\\\<^sub>o\\<^sub>p))\\<^sup>2 \\ (\\transpose A\\\\<^sub>o\\<^sub>p) * (\\A\\\\<^sub>o\\<^sub>p)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ (\\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p) * (\\A\\\\<^sub>o\\<^sub>p)\n\ngoal (1 subgoal):\n 1. (\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ (\\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p) * (\\A\\\\<^sub>o\\<^sub>p)\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\n(\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ (\\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p) * (\\A\\\\<^sub>o\\<^sub>p)\n\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ \\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nthus \"(\\A\\\\<^sub>o\\<^sub>p) \\ (\\transpose A\\\\<^sub>o\\<^sub>p)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ (\\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p) * (\\A\\\\<^sub>o\\<^sub>p)\n\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ \\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nusing sq_le_cancel[of \"(\\A\\\\<^sub>o\\<^sub>p)\"] op_norm_ge_0\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ (\\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p) * (\\A\\\\<^sub>o\\<^sub>p)\n\\0 \\ \\A\\\\<^sub>o\\<^sub>p; 0 \\ ?b; (\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ ?b * (\\A\\\\<^sub>o\\<^sub>p)\\ \\ \\A\\\\<^sub>o\\<^sub>p \\ ?b\n\\0 \\ \\A\\\\<^sub>o\\<^sub>p; 0 \\ ?b; (\\A\\\\<^sub>o\\<^sub>p)\\<^sup>2 \\ (\\A\\\\<^sub>o\\<^sub>p) * ?b\\ \\ \\A\\\\<^sub>o\\<^sub>p \\ ?b\n0 \\ \\?A\\\\<^sub>o\\<^sub>p\n\ngoal (1 subgoal):\n 1. \\A\\\\<^sub>o\\<^sub>p \\ \\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\n\\A\\\\<^sub>o\\<^sub>p \\ \\Finite_Cartesian_Product.transpose A\\\\<^sub>o\\<^sub>p\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3661, "file": "Matrices_for_ODEs_MTX_Norms", "length": 22, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357735451834, "lm_q2_score": 0.8354835350552604, "lm_q1q2_score": 0.7243941131009021}} {"text": "[STATEMENT]\nlemma ceiling_correct: \"of_int \\x\\ - 1 < x \\ x \\ of_int \\x\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int \\x\\ - (1::'a) < x \\ x \\ of_int \\x\\\n[PROOF STEP]\nunfolding ceiling_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int (- \\- x\\) - (1::'a) < x \\ x \\ of_int (- \\- x\\)\n[PROOF STEP]\nusing floor_correct [of \"- x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\- x\\ \\ - x \\ - x < of_int (\\- x\\ + 1)\n\ngoal (1 subgoal):\n 1. of_int (- \\- x\\) - (1::'a) < x \\ x \\ of_int (- \\- x\\)\n[PROOF STEP]\nby (simp add: le_minus_iff)", "meta": {"llama_tokens": 349, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213826762113, "lm_q2_score": 0.8056321936479701, "lm_q1q2_score": 0.7243611318812322}} {"text": "[STATEMENT]\nlemma ceiling_correct: \"of_int \\x\\ - 1 < x \\ x \\ of_int \\x\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int \\x\\ - (1::'a) < x \\ x \\ of_int \\x\\\n[PROOF STEP]\nunfolding ceiling_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int (- \\- x\\) - (1::'a) < x \\ x \\ of_int (- \\- x\\)\n[PROOF STEP]\nusing floor_correct [of \"- x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\- x\\ \\ - x \\ - x < of_int (\\- x\\ + 1)\n\ngoal (1 subgoal):\n 1. of_int (- \\- x\\) - (1::'a) < x \\ x \\ of_int (- \\- x\\)\n[PROOF STEP]\nby (simp add: le_minus_iff)", "meta": {"llama_tokens": 349, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213826762113, "lm_q2_score": 0.8056321889812553, "lm_q1q2_score": 0.724361127685289}} {"text": "[STATEMENT]\nlemma ceiling_correct: \"of_int \\x\\ - 1 < x \\ x \\ of_int \\x\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int \\x\\ - (1::'a) < x \\ x \\ of_int \\x\\\n[PROOF STEP]\nunfolding ceiling_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int (- \\- x\\) - (1::'a) < x \\ x \\ of_int (- \\- x\\)\n[PROOF STEP]\nusing floor_correct [of \"- x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\- x\\ \\ - x \\ - x < of_int (\\- x\\ + 1)\n\ngoal (1 subgoal):\n 1. of_int (- \\- x\\) - (1::'a) < x \\ x \\ of_int (- \\- x\\)\n[PROOF STEP]\nby (simp add: le_minus_iff)", "meta": {"llama_tokens": 349, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213826762113, "lm_q2_score": 0.8056321866478979, "lm_q1q2_score": 0.7243611255873176}} {"text": "[STATEMENT]\nlemma ceiling_correct: \"of_int \\x\\ - 1 < x \\ x \\ of_int \\x\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int \\x\\ - (1::'a) < x \\ x \\ of_int \\x\\\n[PROOF STEP]\nunfolding ceiling_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int (- \\- x\\) - (1::'a) < x \\ x \\ of_int (- \\- x\\)\n[PROOF STEP]\nusing floor_correct [of \"- x\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\- x\\ \\ - x \\ - x < of_int (\\- x\\ + 1)\n\ngoal (1 subgoal):\n 1. of_int (- \\- x\\) - (1::'a) < x \\ x \\ of_int (- \\- x\\)\n[PROOF STEP]\nby (simp add: le_minus_iff)", "meta": {"llama_tokens": 349, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213826762113, "lm_q2_score": 0.8056321866478979, "lm_q1q2_score": 0.7243611255873176}} {"text": "[STATEMENT]\nlemma log_of_power_less: \"\\ m < b ^ n; b > 1; m > 0 \\ \\ log b (real m) < n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\real m < b ^ n; 1 < b; 0 < m\\ \\ log b (real m) < real n\n[PROOF STEP]\nby (simp add: log_less_iff powr_realpow)", "meta": {"llama_tokens": 138, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213664574068, "lm_q2_score": 0.805632181981183, "lm_q1q2_score": 0.7243611083249836}} {"text": "[STATEMENT]\nlemma (in deutsch) deutsch_algo_result [simp]: \n shows \"deutsch_algo = \\\\<^sub>3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. deutsch_algo = Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) + (real (f 1) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2)), real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2) + real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2)]])\n[PROOF STEP]\nusing deutsch_algo_def H_on_ket_zero H_on_ket_one \\\\<^sub>0_to_\\\\<^sub>1 \\\\<^sub>1_to_\\\\<^sub>2 \\\\<^sub>2_to_\\\\<^sub>3\n[PROOF STATE]\nproof (prove)\nusing this:\ndeutsch_algo \\ (H \\ Quantum.Id 1) * (U\\<^sub>f * (H * |Deutsch.zero\\ \\ H * |Deutsch.one\\))\nH * |Deutsch.zero\\ = Tensor.mat_of_cols_list 2 (map (map complex_of_real) [[1 / sqrt 2, 1 / sqrt 2]])\nH * |Deutsch.one\\ = Tensor.mat_of_cols_list 2 (map (map complex_of_real) [[1 / sqrt 2, - 1 / sqrt 2]])\nTensor.mat_of_cols_list 2 (map (map complex_of_real) [[1 / sqrt 2, 1 / sqrt 2]]) \\ Tensor.mat_of_cols_list 2 (map (map complex_of_real) [[1 / sqrt 2, - 1 / sqrt 2]]) = \\\\<^sub>1\nU\\<^sub>f * \\\\<^sub>1 = Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[real (1 - f 0) / 2 - real (f 0) / 2, real (f 0) / 2 - real (1 - f 0) / 2, real (1 - f 1) / 2 - real (f 1) / 2, real (f 1) / 2 - real (1 - f 1) / 2]])\n(H \\ Quantum.Id 1) * Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[real (1 - f 0) / 2 - real (f 0) / 2, real (f 0) / 2 - real (1 - f 0) / 2, real (1 - f 1) / 2 - real (f 1) / 2, real (f 1) / 2 - real (1 - f 1) / 2]]) = Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) + (real (f 1) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2)), real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2) + real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2)]])\n\ngoal (1 subgoal):\n 1. deutsch_algo = Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) + (real (f 1) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2)), real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2) + real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2)]])\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 1561, "file": "Isabelle_Marries_Dirac_Deutsch", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505325302033, "lm_q2_score": 0.8006920092299292, "lm_q1q2_score": 0.7243464525425339}} {"text": "[STATEMENT]\nlemma eventually_floor_eq:\n fixes f::\"'a \\ 'b::{order_topology,floor_ceiling}\"\n assumes f: \"(f \\ l) F\"\n and l: \"l \\ \\\"\n shows \"\\\\<^sub>F x in F. floor (f x) = floor l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. \\f x\\ = \\l\\\n[PROOF STEP]\nusing eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in F. of_int \\l\\ < f x\n\\\\<^sub>F x in F. f x < of_int \\l\\\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. \\f x\\ = \\l\\\n[PROOF STEP]\nby eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)", "meta": {"llama_tokens": 323, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505299595162, "lm_q2_score": 0.8006920068519378, "lm_q1q2_score": 0.7243464483329541}} {"text": "[STATEMENT]\ntheorem dual_sup [iff?]:\n \"is_sup (dual x) (dual y) (dual inf) = is_inf x y inf\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. is_sup (dual x) (dual y) (dual inf) = is_inf x y inf\n[PROOF STEP]\nby (simp add: is_inf_def is_sup_def dual_all [symmetric] dual_leq)", "meta": {"llama_tokens": 126, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505428129514, "lm_q2_score": 0.8006919949619792, "lm_q1q2_score": 0.7243464478683395}} {"text": "[STATEMENT]\nlemma eventually_floor_eq:\n fixes f::\"'a \\ 'b::{order_topology,floor_ceiling}\"\n assumes f: \"(f \\ l) F\"\n and l: \"l \\ \\\"\n shows \"\\\\<^sub>F x in F. floor (f x) = floor l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. \\f x\\ = \\l\\\n[PROOF STEP]\nusing eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in F. of_int \\l\\ < f x\n\\\\<^sub>F x in F. f x < of_int \\l\\\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. \\f x\\ = \\l\\\n[PROOF STEP]\nby eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)", "meta": {"llama_tokens": 323, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505402422645, "lm_q2_score": 0.8006919949619793, "lm_q1q2_score": 0.7243464458100112}} {"text": "[STATEMENT]\nlemma unitary_col_norm:\n assumes \"unitary U\"\nand \"U\\ carrier_mat n n\"\n and \"i < n\"\nshows \"\\Matrix.col U i\\ = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\Matrix.col U i\\ = 1\n[PROOF STEP]\nusing assms unitary_col_norm_square cpx_vec_length_inner_prod \n inner_prod_csqrt\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex_Matrix.unitary U\nU \\ carrier_mat n n\ni < n\n\\Complex_Matrix.unitary ?U; ?U \\ carrier_mat ?n ?n; ?i < ?n\\ \\ \\Matrix.col ?U ?i\\\\<^sup>2 = 1\ncomplex_of_real (\\?v\\\\<^sup>2) = \\?v|?v\\\ncsqrt \\?v|?v\\ = complex_of_real \\?v\\\n\ngoal (1 subgoal):\n 1. \\Matrix.col U i\\ = 1\n[PROOF STEP]\nby (metis csqrt_1 of_real_eq_1_iff)", "meta": {"llama_tokens": 354, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505376715774, "lm_q2_score": 0.8006919925839876, "lm_q1q2_score": 0.724346441600431}} {"text": "[STATEMENT]\nlemma integrable_on_powr2_from_0:\n fixes a c :: real\n assumes a_pos: \"a > 0\" and a_neq_1: \"a \\ 1\" and c_nneg: \"c \\ 0\"\n shows \"(\\x. a.^x) integrable_on {0..c}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (.^) a integrable_on {0..c}\n[PROOF STEP]\nusing has_integral_powr2_from_0[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n((.^) a has_integral (a .^ c - 1) / ln a) {0..c}\n\ngoal (1 subgoal):\n 1. (.^) a integrable_on {0..c}\n[PROOF STEP]\nunfolding integrable_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\n((.^) a has_integral (a .^ c - 1) / ln a) {0..c}\n\ngoal (1 subgoal):\n 1. \\y. ((.^) a has_integral y) {0..c}\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 337, "file": "Actuarial_Mathematics_Preliminaries", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505248181418, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.7243464399137938}} {"text": "[STATEMENT]\nlemma four_block_carrier_mat[simp]:\n \"A \\ carrier_mat nr1 nc1 \\ D \\ carrier_mat nr2 nc2 \\\n four_block_mat A B C D \\ carrier_mat (nr1 + nr2) (nc1 + nc2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ carrier_mat nr1 nc1; D \\ carrier_mat nr2 nc2\\ \\ four_block_mat A B C D \\ carrier_mat (nr1 + nr2) (nc1 + nc2)\n[PROOF STEP]\nunfolding carrier_mat_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ {m. dim_row m = nr1 \\ dim_col m = nc1}; D \\ {m. dim_row m = nr2 \\ dim_col m = nc2}\\ \\ four_block_mat A B C D \\ {m. dim_row m = nr1 + nr2 \\ dim_col m = nc1 + nc2}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 339, "file": "Jordan_Normal_Form_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513731336204, "lm_q2_score": 0.8080672158638527, "lm_q1q2_score": 0.7242313518022395}} {"text": "[STATEMENT]\nlemma arccos_le_mono: \"\\x\\ \\ 1 \\ \\y\\ \\ 1 \\ arccos x \\ arccos y \\ y \\ x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x\\ \\ 1; \\y\\ \\ 1\\ \\ (arccos x \\ arccos y) = (y \\ x)\n[PROOF STEP]\nusing arccos_less_mono [of y x]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\y\\ \\ 1; \\x\\ \\ 1\\ \\ (arccos y < arccos x) = (x < y)\n\ngoal (1 subgoal):\n 1. \\\\x\\ \\ 1; \\y\\ \\ 1\\ \\ (arccos x \\ arccos y) = (y \\ x)\n[PROOF STEP]\nby (simp add: not_le [symmetric])", "meta": {"llama_tokens": 335, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513731336204, "lm_q2_score": 0.8080672112416737, "lm_q1q2_score": 0.7242313476596053}} {"text": "[STATEMENT]\nlemma boundedUn: \"bounded (A Un B) = (bounded A & bounded B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bounded (A \\ B) = (bounded A \\ bounded B)\n[PROOF STEP]\napply(auto simp add: bounded_def boundedByUn)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\N Na. \\boundedBy N A; boundedBy Na B\\ \\ \\N. boundedBy N A \\ boundedBy N B\n[PROOF STEP]\napply(rule_tac x=\"N+Na\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\N Na. \\boundedBy N A; boundedBy Na B\\ \\ boundedBy (N + Na) A \\ boundedBy (N + Na) B\n[PROOF STEP]\napply(blast intro: boundedByAdd1 boundedByAdd2)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 306, "file": "Completeness_Tree", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513620489619, "lm_q2_score": 0.8080672089305841, "lm_q1q2_score": 0.7242313366311391}} {"text": "[STATEMENT]\ntheorem (in linear_map) rank_nullity: \n assumes fd: \"V.fin_dim\"\n shows \"(vectorspace.dim K (W.vs imT)) + (vectorspace.dim K (V.vs kerT)) = V.dim\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vectorspace.dim K (W.vs imT) + vectorspace.dim K (V.vs kerT) = V.dim\n[PROOF STEP]\nby (rule rank_nullity_main[OF fd])", "meta": {"llama_tokens": 144, "file": "VectorSpace_VectorSpace", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9161096112990285, "lm_q2_score": 0.7905303236047049, "lm_q1q2_score": 0.7242124274776014}} {"text": "[STATEMENT]\nlemma quantum_payoff_simp:\n fixes x y:: real\n shows \"3 * (cmod (complex_of_real (sin x) * complex_of_real (cos y)))\\<^sup>2 +\n (cmod (complex_of_real (cos x) * complex_of_real (cos y)))\\<^sup>2 = \n 2 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + (cos y)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 3 * (cmod (complex_of_real (sin x) * complex_of_real (cos y)))\\<^sup>2 + (cmod (complex_of_real (cos x) * complex_of_real (cos y)))\\<^sup>2 = 2 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + (cos y)\\<^sup>2\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 3 * (cmod (complex_of_real (sin x) * complex_of_real (cos y)))\\<^sup>2 + (cmod (complex_of_real (cos x) * complex_of_real (cos y)))\\<^sup>2 = 2 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + (cos y)\\<^sup>2\n[PROOF STEP]\nhave \"3 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + (cos x)\\<^sup>2 * (cos y)\\<^sup>2 = \n (2 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2) + ((sin x)\\<^sup>2 + (cos x)\\<^sup>2) * (cos y)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 3 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + (cos x)\\<^sup>2 * (cos y)\\<^sup>2 = 2 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + ((sin x)\\<^sup>2 + (cos x)\\<^sup>2) * (cos y)\\<^sup>2\n[PROOF STEP]\nby (auto simp add: algebra_simps simp del: sin_cos_squared_add2)\n[PROOF STATE]\nproof (state)\nthis:\n3 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + (cos x)\\<^sup>2 * (cos y)\\<^sup>2 = 2 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + ((sin x)\\<^sup>2 + (cos x)\\<^sup>2) * (cos y)\\<^sup>2\n\ngoal (1 subgoal):\n 1. 3 * (cmod (complex_of_real (sin x) * complex_of_real (cos y)))\\<^sup>2 + (cmod (complex_of_real (cos x) * complex_of_real (cos y)))\\<^sup>2 = 2 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + (cos y)\\<^sup>2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n3 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + (cos x)\\<^sup>2 * (cos y)\\<^sup>2 = 2 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + ((sin x)\\<^sup>2 + (cos x)\\<^sup>2) * (cos y)\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n3 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + (cos x)\\<^sup>2 * (cos y)\\<^sup>2 = 2 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + ((sin x)\\<^sup>2 + (cos x)\\<^sup>2) * (cos y)\\<^sup>2\n\ngoal (1 subgoal):\n 1. 3 * (cmod (complex_of_real (sin x) * complex_of_real (cos y)))\\<^sup>2 + (cmod (complex_of_real (cos x) * complex_of_real (cos y)))\\<^sup>2 = 2 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + (cos y)\\<^sup>2\n[PROOF STEP]\nby (simp add: cmod_real_prod_squared power_mult_distrib)\n[PROOF STATE]\nproof (state)\nthis:\n3 * (cmod (complex_of_real (sin x) * complex_of_real (cos y)))\\<^sup>2 + (cmod (complex_of_real (cos x) * complex_of_real (cos y)))\\<^sup>2 = 2 * (sin x)\\<^sup>2 * (cos y)\\<^sup>2 + (cos y)\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1368, "file": "Isabelle_Marries_Dirac_Quantum_Prisoners_Dilemma", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.907312221360624, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7242046183716143}} {"text": "[STATEMENT]\nlemma top_ccsubspace_code[code]: \n \\ \\Code equation for \\<^term>\\top\\, the subspace containing everything.\n Top is represented as the span of the standard basis vectors.\\\n \"(top::'a ccsubspace) =\n (let n = length (canonical_basis :: 'a::onb_enum list) in SPAN (unit_vecs n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ = (let n = length canonical_basis in SPAN (unit_vecs n))\n[PROOF STEP]\nunfolding SPAN_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ = (let n = length canonical_basis; na = length canonical_basis in ccspan (basis_enum_of_vec ` Set.filter (\\v. dim_vec v = na) (set (unit_vecs n))))\n[PROOF STEP]\napply (simp only: index_unit_vec Let_def map_filter_map_filter filter_set image_set map_map_filter \n map_filter_map o_def unit_vecs_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\ = ccspan (set (List.map_filter (\\x. if length canonical_basis = length canonical_basis then Some (basis_enum_of_vec (unit_vec (length canonical_basis) x)) else None) [0.. = ccspan ((!) canonical_basis ` {0.. length canonical_basis\n 2. \\ = ccspan (set (take (length canonical_basis) canonical_basis))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 568, "file": "Complex_Bounded_Operators_Cblinfun_Code", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9073122188543453, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7242046163711359}} {"text": "[STATEMENT]\nlemma sin_pi_half [simp]: \"sin(pi/2) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin (pi / 2) = 1\n[PROOF STEP]\nusing sin_cos_squared_add2 [where x = \"pi/2\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(cos (pi / 2))\\<^sup>2 + (sin (pi / 2))\\<^sup>2 = 1\n\ngoal (1 subgoal):\n 1. sin (pi / 2) = 1\n[PROOF STEP]\nusing sin_gt_zero_02 [OF pi_half_gt_zero pi_half_less_two]\n[PROOF STATE]\nproof (prove)\nusing this:\n(cos (pi / 2))\\<^sup>2 + (sin (pi / 2))\\<^sup>2 = 1\n0 < sin (pi / 2)\n\ngoal (1 subgoal):\n 1. sin (pi / 2) = 1\n[PROOF STEP]\nby (simp add: power2_eq_1_iff)", "meta": {"llama_tokens": 291, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970842359877, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7241817946106068}} {"text": "[STATEMENT]\nlemma job_dist_lower_bound_makespan:\n assumes \"lb T A j\"\n shows \"(\\x \\ {1..j}. t x) / m \\ makespan T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (sum t {1..j}) / real m \\ real (makespan T)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. real (sum t {1..j}) / real m \\ real (makespan T)\n[PROOF STEP]\nhave \"(\\x \\ {1..j}. t x) \\ m * makespan T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum t {1..j} \\ m * makespan T\n[PROOF STEP]\nusing assms lb_impl_job_sum[symmetric]\n and sum_le_card_Max[of \"{1..m}\"] m_gt_0\n[PROOF STATE]\nproof (prove)\nusing this:\nlb T A j\nlb ?T ?A ?j \\ sum t {1..?j} = sum ?T {1..m}\n\\finite {1..m}; {1..m} \\ {}\\ \\ sum ?f {1..m} \\ card {1..m} * Max (?f ` {1..m})\n0 < m\n\ngoal (1 subgoal):\n 1. sum t {1..j} \\ m * makespan T\n[PROOF STEP]\nby (simp add: makespan_def')\n[PROOF STATE]\nproof (state)\nthis:\nsum t {1..j} \\ m * makespan T\n\ngoal (1 subgoal):\n 1. real (sum t {1..j}) / real m \\ real (makespan T)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nsum t {1..j} \\ m * makespan T\n[PROOF STEP]\nhave \"real (\\x \\ {1..j}. t x) \\ real m * real (makespan T)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsum t {1..j} \\ m * makespan T\n\ngoal (1 subgoal):\n 1. real (sum t {1..j}) \\ real m * real (makespan T)\n[PROOF STEP]\nusing of_nat_mono\n[PROOF STATE]\nproof (prove)\nusing this:\nsum t {1..j} \\ m * makespan T\n?i \\ ?j \\ of_nat ?i \\ of_nat ?j\n\ngoal (1 subgoal):\n 1. real (sum t {1..j}) \\ real m * real (makespan T)\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nreal (sum t {1..j}) \\ real m * real (makespan T)\n\ngoal (1 subgoal):\n 1. real (sum t {1..j}) / real m \\ real (makespan T)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (sum t {1..j}) \\ real m * real (makespan T)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (sum t {1..j}) \\ real m * real (makespan T)\n\ngoal (1 subgoal):\n 1. real (sum t {1..j}) / real m \\ real (makespan T)\n[PROOF STEP]\nby (simp add: field_simps m_gt_0)\n[PROOF STATE]\nproof (state)\nthis:\nreal (sum t {1..j}) / real m \\ real (makespan T)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1062, "file": "Approximation_Algorithms_Approx_LB_Hoare", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.880797081106935, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7241817920379336}} {"text": "[STATEMENT]\nlemma sum_less_0_iff:\n fixes f :: \"_ \\ 'a :: {comm_monoid_add,ordered_ab_group_add}\"\n shows \"finite A \\ (\\i. i \\ A \\ 0 \\ f i) \\ 0 < (\\i\\A. f i) \\ (\\i\\A. 0 < f i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; \\i. i \\ A \\ (0::'a) \\ f i\\ \\ ((0::'a) < sum f A) = (\\i\\A. (0::'a) < f i)\n[PROOF STEP]\nusing sum_nonneg[of A f] sum_eq_0_iff[of A f]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. x \\ A \\ (0::'a) \\ f x) \\ (0::'a) \\ sum f A\n\\finite A; \\i. i \\ A \\ (0::'a) \\ f i\\ \\ (sum f A = (0::'a)) = (\\i\\A. f i = (0::'a))\n\ngoal (1 subgoal):\n 1. \\finite A; \\i. i \\ A \\ (0::'a) \\ f i\\ \\ ((0::'a) < sum f A) = (\\i\\A. (0::'a) < f i)\n[PROOF STEP]\nby (simp add: less_le)", "meta": {"llama_tokens": 469, "file": "Probabilistic_Noninterference_Trace_Based", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8807970811069351, "lm_q2_score": 0.8221891239865619, "lm_q1q2_score": 0.7241817805252316}} {"text": "[STATEMENT]\nlemma product_partition:\n assumes \"partition_on A P\" and \"\\p. p \\ P \\ finite p\" \n shows \"card A = (\\p\\P. card p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card A = sum card P\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\npartition_on A P\n?p \\ P \\ finite ?p\n\ngoal (1 subgoal):\n 1. card A = sum card P\n[PROOF STEP]\nunfolding partition_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ P = A \\ disjoint P \\ {} \\ P\n?p \\ P \\ finite ?p\n\ngoal (1 subgoal):\n 1. card A = sum card P\n[PROOF STEP]\nby (meson card_Union_disjoint)", "meta": {"llama_tokens": 259, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970654616711, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7241817791745677}} {"text": "[STATEMENT]\nlemma card_singletons: \n assumes \"finite S\" shows \"card {{v} | v. v \\ S} = card S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {{v} |v. v \\ S} = card S\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\n\ngoal (1 subgoal):\n 1. card {{v} |v. v \\ S} = card S\n[PROOF STEP]\nproof (induct S rule: finite_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. card {{v} |v. v \\ {}} = card {}\n 2. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\ncase empty\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. card {{v} |v. v \\ {}} = card {}\n 2. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {{v} |v. v \\ {}} = card {}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard {{v} |v. v \\ {}} = card {}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\ncase (insert x F)\n[PROOF STATE]\nproof (state)\nthis:\nfinite F\nx \\ F\ncard {{v} |v. v \\ F} = card F\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite F\nx \\ F\ncard {{v} |v. v \\ F} = card F\n[PROOF STEP]\nhave disj: \"{{x}} \\ {{v} |v. v \\ F} = {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite F\nx \\ F\ncard {{v} |v. v \\ F} = card F\n\ngoal (1 subgoal):\n 1. {{x}} \\ {{v} |v. v \\ F} = {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{{x}} \\ {{v} |v. v \\ F} = {}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nhave \"{{v} |v. v \\ insert x F} = ({{x}} \\ {{v} |v. v \\ F})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {{v} |v. v \\ insert x F} = {{x}} \\ {{v} |v. v \\ F}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{{v} |v. v \\ insert x F} = {{x}} \\ {{v} |v. v \\ F}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{{v} |v. v \\ insert x F} = {{x}} \\ {{v} |v. v \\ F}\n[PROOF STEP]\nhave \"card {{v} |v. v \\ insert x F} = card ({{x}} \\ {{v} |v. v \\ F})\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{{v} |v. v \\ insert x F} = {{x}} \\ {{v} |v. v \\ F}\n\ngoal (1 subgoal):\n 1. card {{v} |v. v \\ insert x F} = card ({{x}} \\ {{v} |v. v \\ F})\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard {{v} |v. v \\ insert x F} = card ({{x}} \\ {{v} |v. v \\ F})\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {{v} |v. v \\ insert x F} = card ({{x}} \\ {{v} |v. v \\ F})\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nhave \"... = card {{x}} + card {{v} |v. v \\ F}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({{x}} \\ {{v} |v. v \\ F}) = card {{x}} + card {{v} |v. v \\ F}\n[PROOF STEP]\nusing card_Un_disjoint disj assms finite_subset\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite ?A; finite ?B; ?A \\ ?B = {}\\ \\ card (?A \\ ?B) = card ?A + card ?B\n{{x}} \\ {{v} |v. v \\ F} = {}\nfinite S\n\\?A \\ ?B; finite ?B\\ \\ finite ?A\n\ngoal (1 subgoal):\n 1. card ({{x}} \\ {{v} |v. v \\ F}) = card {{x}} + card {{v} |v. v \\ F}\n[PROOF STEP]\nusing insert.hyps(1)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite ?A; finite ?B; ?A \\ ?B = {}\\ \\ card (?A \\ ?B) = card ?A + card ?B\n{{x}} \\ {{v} |v. v \\ F} = {}\nfinite S\n\\?A \\ ?B; finite ?B\\ \\ finite ?A\nfinite F\n\ngoal (1 subgoal):\n 1. card ({{x}} \\ {{v} |v. v \\ F}) = card {{x}} + card {{v} |v. v \\ F}\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\ncard ({{x}} \\ {{v} |v. v \\ F}) = card {{x}} + card {{v} |v. v \\ F}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ({{x}} \\ {{v} |v. v \\ F}) = card {{x}} + card {{v} |v. v \\ F}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nhave \"... = 1 + card {{v} |v. v \\ F}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {{x}} + card {{v} |v. v \\ F} = 1 + card {{v} |v. v \\ F}\n[PROOF STEP]\nusing is_singleton_altdef\n[PROOF STATE]\nproof (prove)\nusing this:\nis_singleton ?A = (card ?A = 1)\n\ngoal (1 subgoal):\n 1. card {{x}} + card {{v} |v. v \\ F} = 1 + card {{v} |v. v \\ F}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard {{x}} + card {{v} |v. v \\ F} = 1 + card {{v} |v. v \\ F}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {{x}} + card {{v} |v. v \\ F} = 1 + card {{v} |v. v \\ F}\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nhave \"... = 1 + card F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + card {{v} |v. v \\ F} = 1 + card F\n[PROOF STEP]\nusing insert.hyps\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite F\nx \\ F\ncard {{v} |v. v \\ F} = card F\n\ngoal (1 subgoal):\n 1. 1 + card {{v} |v. v \\ F} = 1 + card F\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n1 + card {{v} |v. v \\ F} = 1 + card F\n\ngoal (1 subgoal):\n 1. \\x F. \\finite F; x \\ F; card {{v} |v. v \\ F} = card F\\ \\ card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {{v} |v. v \\ insert x F} = 1 + card F\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {{v} |v. v \\ insert x F} = 1 + card F\n\ngoal (1 subgoal):\n 1. card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nusing insert.hyps(1) insert.hyps(2)\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {{v} |v. v \\ insert x F} = 1 + card F\nfinite F\nx \\ F\n\ngoal (1 subgoal):\n 1. card {{v} |v. v \\ insert x F} = card (insert x F)\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\ncard {{v} |v. v \\ insert x F} = card (insert x F)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3663, "file": "Undirected_Graph_Theory_Undirected_Graph_Basics", "length": 34, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677622198946, "lm_q2_score": 0.8539127585282744, "lm_q1q2_score": 0.724090490980238}} {"text": "[STATEMENT]\nlemma binomial_symmetric:\n assumes kn: \"k \\ n\"\n shows \"n choose k = n choose (n - k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nhave kn': \"n - k \\ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n - k \\ n\n[PROOF STEP]\nusing kn\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ n\n\ngoal (1 subgoal):\n 1. n - k \\ n\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\nn - k \\ n\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nfrom binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']\n[PROOF STATE]\nproof (chain)\npicking this:\nfact k * fact (n - k) * (n choose k) = fact n\nfact (n - k) * fact (n - (n - k)) * (n choose (n - k)) = fact n\n[PROOF STEP]\nhave \"fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact n\nfact (n - k) * fact (n - (n - k)) * (n choose (n - k)) = fact n\n\ngoal (1 subgoal):\n 1. fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nusing kn\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\nk \\ n\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn choose k = n choose (n - k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 960, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127417985637, "lm_q2_score": 0.8479677660619633, "lm_q1q2_score": 0.7240904800747742}} {"text": "[STATEMENT]\nlemma binomial_symmetric:\n assumes kn: \"k \\ n\"\n shows \"n choose k = n choose (n - k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nhave kn': \"n - k \\ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n - k \\ n\n[PROOF STEP]\nusing kn\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ n\n\ngoal (1 subgoal):\n 1. n - k \\ n\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\nn - k \\ n\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nfrom binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']\n[PROOF STATE]\nproof (chain)\npicking this:\nfact k * fact (n - k) * (n choose k) = fact n\nfact (n - k) * fact (n - (n - k)) * (n choose (n - k)) = fact n\n[PROOF STEP]\nhave \"fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact n\nfact (n - k) * fact (n - (n - k)) * (n choose (n - k)) = fact n\n\ngoal (1 subgoal):\n 1. fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nusing kn\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\nk \\ n\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn choose k = n choose (n - k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 960, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127417985637, "lm_q2_score": 0.8479677622198947, "lm_q1q2_score": 0.7240904767939828}} {"text": "[STATEMENT]\nlemma binomial_symmetric:\n assumes kn: \"k \\ n\"\n shows \"n choose k = n choose (n - k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nhave kn': \"n - k \\ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n - k \\ n\n[PROOF STEP]\nusing kn\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ n\n\ngoal (1 subgoal):\n 1. n - k \\ n\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\nn - k \\ n\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nfrom binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']\n[PROOF STATE]\nproof (chain)\npicking this:\nfact k * fact (n - k) * (n choose k) = fact n\nfact (n - k) * fact (n - (n - k)) * (n choose (n - k)) = fact n\n[PROOF STEP]\nhave \"fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact n\nfact (n - k) * fact (n - (n - k)) * (n choose (n - k)) = fact n\n\ngoal (1 subgoal):\n 1. fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nusing kn\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\nk \\ n\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn choose k = n choose (n - k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 960, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127417985637, "lm_q2_score": 0.8479677622198947, "lm_q1q2_score": 0.7240904767939828}} {"text": "[STATEMENT]\nlemma binomial_symmetric:\n assumes kn: \"k \\ n\"\n shows \"n choose k = n choose (n - k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nhave kn': \"n - k \\ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n - k \\ n\n[PROOF STEP]\nusing kn\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\ n\n\ngoal (1 subgoal):\n 1. n - k \\ n\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\nn - k \\ n\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nfrom binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']\n[PROOF STATE]\nproof (chain)\npicking this:\nfact k * fact (n - k) * (n choose k) = fact n\nfact (n - k) * fact (n - (n - k)) * (n choose (n - k)) = fact n\n[PROOF STEP]\nhave \"fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact n\nfact (n - k) * fact (n - (n - k)) * (n choose (n - k)) = fact n\n\ngoal (1 subgoal):\n 1. fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nusing kn\n[PROOF STATE]\nproof (prove)\nusing this:\nfact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))\nk \\ n\n\ngoal (1 subgoal):\n 1. n choose k = n choose (n - k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn choose k = n choose (n - k)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 960, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127417985637, "lm_q2_score": 0.8479677564567912, "lm_q1q2_score": 0.7240904718727953}} {"text": "[STATEMENT]\nlemma borel_sigma_sets_Ioc: \"borel = sigma UNIV (range (\\(a, b). {a <.. b::real}))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. borel = sigma UNIV (range (\\(a, b). {a<..b}))\n[PROOF STEP]\nproof (rule borel_eq_sigmaI5[OF borel_eq_atMost])\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\a. {..a} \\ sets (sigma UNIV (range (\\(a, b). {a<..b})))\n 2. \\a b. {a<..b} \\ sets borel\n[PROOF STEP]\nfix i :: real\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\a. {..a} \\ sets (sigma UNIV (range (\\(a, b). {a<..b})))\n 2. \\a b. {a<..b} \\ sets borel\n[PROOF STEP]\nhave \"{..i} = (\\j::nat. {-j <.. i})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {..i} = (\\x. {- real x<..i})\n[PROOF STEP]\nby (auto simp: minus_less_iff reals_Archimedean2)\n[PROOF STATE]\nproof (state)\nthis:\n{..i} = (\\x. {- real x<..i})\n\ngoal (2 subgoals):\n 1. \\a. {..a} \\ sets (sigma UNIV (range (\\(a, b). {a<..b})))\n 2. \\a b. {a<..b} \\ sets borel\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n{..i} = (\\x. {- real x<..i})\n\ngoal (2 subgoals):\n 1. \\a. {..a} \\ sets (sigma UNIV (range (\\(a, b). {a<..b})))\n 2. \\a b. {a<..b} \\ sets borel\n[PROOF STEP]\nhave \"\\ \\ sets (sigma UNIV (range (\\(i, j). {i<..j})))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. {- real x<..i}) \\ sets (sigma UNIV (range (\\(i, j). {i<..j})))\n[PROOF STEP]\nby (intro sets.countable_nat_UN) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. {- real x<..i}) \\ sets (sigma UNIV (range (\\(i, j). {i<..j})))\n\ngoal (2 subgoals):\n 1. \\a. {..a} \\ sets (sigma UNIV (range (\\(a, b). {a<..b})))\n 2. \\a b. {a<..b} \\ sets borel\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n{..i} \\ sets (sigma UNIV (range (\\(i, j). {i<..j})))\n[PROOF STEP]\nshow \"{..i} \\ sets (sigma UNIV (range (\\(i, j). {i<..j})))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{..i} \\ sets (sigma UNIV (range (\\(i, j). {i<..j})))\n\ngoal (1 subgoal):\n 1. {..i} \\ sets (sigma UNIV (range (\\(i, j). {i<..j})))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n{..i} \\ sets (sigma UNIV (range (\\(i, j). {i<..j})))\n\ngoal (1 subgoal):\n 1. \\a b. {a<..b} \\ sets borel\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 1119, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314707995588, "lm_q2_score": 0.8175744761936437, "lm_q1q2_score": 0.7240696858395556}} {"text": "[STATEMENT]\nlemma dim_kernel_zero_jordan_block_pow: \n \"kernel.dim n ((jordan_block n (0 :: 'a)) ^\\<^sub>m k) = min k n\" (is \"kernel.dim _ ?A = ?c\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nhave A: \"?A \\ carrier_mat n n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. jordan_block n (0::'a) ^\\<^sub>m k \\ carrier_mat n n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\njordan_block n (0::'a) ^\\<^sub>m k \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nhence dim: \"dim_row ?A = n\"\n[PROOF STATE]\nproof (prove)\nusing this:\njordan_block n (0::'a) ^\\<^sub>m k \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. dim_row (jordan_block n (0::'a) ^\\<^sub>m k) = n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_row (jordan_block n (0::'a) ^\\<^sub>m k) = n\n\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nlet ?f = \"\\ i. min (k + i) n\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nhave piv: \"pivot_fun ?A ?f n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pivot_fun (jordan_block n (0::'a) ^\\<^sub>m k) (\\i. min (k + i) n) n\n[PROOF STEP]\nunfolding jordan_block_zero_pow\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pivot_fun (mat n n (\\(i, j). if i \\ j \\ j - i = k then 1::'a else (0::'a))) (\\i. min (k + i) n) n\n[PROOF STEP]\nby (intro pivot_funI, auto)\n[PROOF STATE]\nproof (state)\nthis:\npivot_fun (jordan_block n (0::'a) ^\\<^sub>m k) (\\i. min (k + i) n) n\n\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nhence row: \"row_echelon_form ?A\"\n[PROOF STATE]\nproof (prove)\nusing this:\npivot_fun (jordan_block n (0::'a) ^\\<^sub>m k) (\\i. min (k + i) n) n\n\ngoal (1 subgoal):\n 1. row_echelon_form (jordan_block n (0::'a) ^\\<^sub>m k)\n[PROOF STEP]\nunfolding row_echelon_form_def\n[PROOF STATE]\nproof (prove)\nusing this:\npivot_fun (jordan_block n (0::'a) ^\\<^sub>m k) (\\i. min (k + i) n) n\n\ngoal (1 subgoal):\n 1. \\f. pivot_fun (jordan_block n (0::'a) ^\\<^sub>m k) f (dim_col (jordan_block n (0::'a) ^\\<^sub>m k))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrow_echelon_form (jordan_block n (0::'a) ^\\<^sub>m k)\n\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nfrom find_base_vectors(5-6)[OF row A]\n[PROOF STATE]\nproof (chain)\npicking this:\nlength (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k)) = card {i. i < n \\ row (jordan_block n (0::'a) ^\\<^sub>m k) i \\ 0\\<^sub>v n}\nvectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = n - card {i. i < n \\ row (jordan_block n (0::'a) ^\\<^sub>m k) i \\ 0\\<^sub>v n}\n[PROOF STEP]\nhave \"kernel.dim n ?A = n - length (map fst (pivot_positions ?A))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k)) = card {i. i < n \\ row (jordan_block n (0::'a) ^\\<^sub>m k) i \\ 0\\<^sub>v n}\nvectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = n - card {i. i < n \\ row (jordan_block n (0::'a) ^\\<^sub>m k) i \\ 0\\<^sub>v n}\n\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = n - length (map fst (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nvectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = n - length (map fst (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k)))\n\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nvectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = n - length (map fst (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k)))\n\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nhave \"length (map fst (pivot_positions ?A)) = card (fst ` set (pivot_positions ?A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (map fst (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k))) = card (fst ` set (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k)))\n[PROOF STEP]\nby (subst distinct_card[OF pivot_positions(2)[OF A piv], symmetric], simp)\n[PROOF STATE]\nproof (state)\nthis:\nlength (map fst (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k))) = card (fst ` set (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k)))\n\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlength (map fst (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k))) = card (fst ` set (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k)))\n\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nhave \"fst ` set (pivot_positions ?A) = { 0 ..< (n - ?c)}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fst ` set (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k)) = {0.. min (k + i) n \\ n} = {0..m k)) = {0..carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfst ` set (pivot_positions (jordan_block n (0::'a) ^\\<^sub>m k)) = {0..carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nhave \"card \\ = n - ?c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {0..carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nvectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = n - (n - min k n)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nvectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = n - (n - min k n)\n\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nvectorspace.dim class_ring (module_vec TYPE('a) n\\carrier := mat_kernel (jordan_block n (0::'a) ^\\<^sub>m k)\\) = min k n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3978, "file": "Jordan_Normal_Form_Jordan_Normal_Form_Uniqueness", "length": 29, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314647623016, "lm_q2_score": 0.8175744739711884, "lm_q1q2_score": 0.7240696789353718}} {"text": "[STATEMENT]\nlemma Gcd_minors_dvd_diag_PAQ:\n fixes P A Q::\"'a::{semiring_Gcd,comm_ring_1} mat\"\n assumes A: \"A \\ carrier_mat m n\"\n and P: \"P \\ carrier_mat m m\"\n and Q: \"Q \\ carrier_mat n n\"\n and SNF: \"Smith_normal_form_mat (P*A*Q)\"\n and k: \"k\\min m n\"\n shows \"Gcd (minors A k) dvd (\\i=0..i = 0..i = 0..i = 0..i = 0..i=0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0..i = 0.. (x - proj x b) = 0\n[PROOF STEP]\nusing vector_sub_project_orthogonal\n[PROOF STATE]\nproof (prove)\nusing this:\n?b \\ (?x - (?b \\ ?x / (?b \\ ?b)) *\\<^sub>R ?b) = 0\n\ngoal (1 subgoal):\n 1. b \\ (x - proj x b) = 0\n[PROOF STEP]\nunfolding proj_def inner_commute[of x b]\n[PROOF STATE]\nproof (prove)\nusing this:\n?b \\ (?x - (?b \\ ?x / (?b \\ ?b)) *\\<^sub>R ?b) = 0\n\ngoal (1 subgoal):\n 1. b \\ (x - (b \\ x / (b \\ b)) *\\<^sub>R b) = 0\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 318, "file": "QR_Decomposition_Projections", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.893309411735131, "lm_q2_score": 0.8104789132480439, "lm_q1q2_score": 0.7240084412173383}} {"text": "[STATEMENT]\nlemma dim_left_null_space_space_iarray[code_unfold]:\n fixes A::\"'a::{field}^'columns::{mod_type}^'rows::{mod_type}\"\n shows \"vec.dim (left_null_space A) = nrows_iarray (matrix_to_iarray A) - rank_iarray (matrix_to_iarray A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (left_null_space A) = nrows_iarray (matrix_to_iarray A) - rank_iarray (matrix_to_iarray A)\n[PROOF STEP]\nunfolding dim_left_null_space nrows_eq_card_rows matrix_to_iarray_rank dimension_vector\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. CARD('rows) - rank_iarray (matrix_to_iarray A) = CARD('rows) - rank_iarray (matrix_to_iarray A)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 272, "file": "Gauss_Jordan_Gauss_Jordan_IArrays", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094003735663, "lm_q2_score": 0.8104789086703224, "lm_q1q2_score": 0.7240084279197081}} {"text": "[STATEMENT]\nlemma closure_aff_dim [simp]:\n fixes S :: \"'n::euclidean_space set\"\n shows \"aff_dim (closure S) = aff_dim S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. aff_dim (closure S) = aff_dim S\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. aff_dim (closure S) = aff_dim S\n[PROOF STEP]\nhave \"aff_dim S \\ aff_dim (closure S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. aff_dim S \\ aff_dim (closure S)\n[PROOF STEP]\nusing aff_dim_subset closure_subset\n[PROOF STATE]\nproof (prove)\nusing this:\n?S \\ ?T \\ aff_dim ?S \\ aff_dim ?T\n?S \\ closure ?S\n\ngoal (1 subgoal):\n 1. aff_dim S \\ aff_dim (closure S)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\naff_dim S \\ aff_dim (closure S)\n\ngoal (1 subgoal):\n 1. aff_dim (closure S) = aff_dim S\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\naff_dim S \\ aff_dim (closure S)\n\ngoal (1 subgoal):\n 1. aff_dim (closure S) = aff_dim S\n[PROOF STEP]\nhave \"aff_dim (closure S) \\ aff_dim (affine hull S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. aff_dim (closure S) \\ aff_dim (affine hull S)\n[PROOF STEP]\nusing aff_dim_subset closure_affine_hull\n[PROOF STATE]\nproof (prove)\nusing this:\n?S \\ ?T \\ aff_dim ?S \\ aff_dim ?T\nclosure ?S \\ affine hull ?S\n\ngoal (1 subgoal):\n 1. aff_dim (closure S) \\ aff_dim (affine hull S)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\naff_dim (closure S) \\ aff_dim (affine hull S)\n\ngoal (1 subgoal):\n 1. aff_dim (closure S) = aff_dim S\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\naff_dim (closure S) \\ aff_dim (affine hull S)\n\ngoal (1 subgoal):\n 1. aff_dim (closure S) = aff_dim S\n[PROOF STEP]\nhave \"aff_dim (affine hull S) = aff_dim S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. aff_dim (affine hull S) = aff_dim S\n[PROOF STEP]\nusing aff_dim_affine_hull\n[PROOF STATE]\nproof (prove)\nusing this:\naff_dim (affine hull ?S) = aff_dim ?S\n\ngoal (1 subgoal):\n 1. aff_dim (affine hull S) = aff_dim S\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\naff_dim (affine hull S) = aff_dim S\n\ngoal (1 subgoal):\n 1. aff_dim (closure S) = aff_dim S\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\naff_dim S \\ aff_dim (closure S)\naff_dim (closure S) \\ aff_dim (affine hull S)\naff_dim (affine hull S) = aff_dim S\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\naff_dim S \\ aff_dim (closure S)\naff_dim (closure S) \\ aff_dim (affine hull S)\naff_dim (affine hull S) = aff_dim S\n\ngoal (1 subgoal):\n 1. aff_dim (closure S) = aff_dim S\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\naff_dim (closure S) = aff_dim S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1198, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511396138365, "lm_q2_score": 0.8459424431344437, "lm_q1q2_score": 0.7240008040043268}} {"text": "[STATEMENT]\nlemma Cauchy_Schwarz_eq2_iff:\n \"\\inner x y\\ = norm x * norm y \\ ((\\k. x = k *\\<^sub>R y) \\ y = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x \\ y\\ = norm x * norm y) = ((\\k. x = k *\\<^sub>R y) \\ y = (0::'a))\n[PROOF STEP]\nusing Cauchy_Schwarz_eq_iff[of x y]\n[PROOF STATE]\nproof (prove)\nusing this:\n((x \\ y)\\<^sup>2 = x \\ x * (y \\ y)) = ((\\k. x = k *\\<^sub>R y) \\ y = (0::'a))\n\ngoal (1 subgoal):\n 1. (\\x \\ y\\ = norm x * norm y) = ((\\k. x = k *\\<^sub>R y) \\ y = (0::'a))\n[PROOF STEP]\nby (subst power_eq_iff_eq_base[symmetric, where n = 2])\n (simp_all add: dot_square_norm power_mult_distrib)", "meta": {"llama_tokens": 346, "file": "Ordinary_Differential_Equations_IVP_Cones", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099070035949657, "lm_q2_score": 0.7956581024858786, "lm_q1q2_score": 0.7239748799189819}} {"text": "[STATEMENT]\nlemma transpose_matrix_minus: \"transpose_matrix (-(A::('a::group_add) matrix)) = - transpose_matrix (A::'a matrix)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. transpose_matrix (- A) = - transpose_matrix A\n[PROOF STEP]\nby (simp add: minus_matrix_def transpose_apply_matrix)", "meta": {"llama_tokens": 108, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9184802462567087, "lm_q2_score": 0.7879311931529758, "lm_q1q2_score": 0.7236992363204875}} {"text": "[STATEMENT]\nlemma card_union': \"(finite s) \\ (finite t) \\ (disjnt s t) \\ (card (s \\ t) = card s + card t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite s \\ finite t \\ disjnt s t \\ card (s \\ t) = card s + card t\n[PROOF STEP]\nby (simp add: card_Un_disjoint disjnt_def)", "meta": {"llama_tokens": 132, "file": "Factored_Transition_System_Bounding_SetUtils", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.8128673178375734, "lm_q1q2_score": 0.723691079674092}} {"text": "[STATEMENT]\nlemma dim_null_space[code_unfold]:\n fixes A::\"'a::{field}^'cols::{mod_type}^'rows::{mod_type}\"\n shows \"vec.dim (null_space A) = (vec.dimension TYPE('a) TYPE('cols)) - rank (A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (null_space A) = finite_dimensional_vector_space.dimension cart_basis - rank A\n[PROOF STEP]\napply (rule add_implies_diff)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (null_space A) + rank A = finite_dimensional_vector_space.dimension cart_basis\n[PROOF STEP]\nusing rank_nullity_theorem_matrices\n[PROOF STATE]\nproof (prove)\nusing this:\nncols ?A = vec.dim (null_space ?A) + vec.dim (col_space ?A)\n\ngoal (1 subgoal):\n 1. vec.dim (null_space A) + rank A = finite_dimensional_vector_space.dimension cart_basis\n[PROOF STEP]\nunfolding rank_eq_dim_col_space[of A]\n[PROOF STATE]\nproof (prove)\nusing this:\nncols ?A = vec.dim (null_space ?A) + vec.dim (col_space ?A)\n\ngoal (1 subgoal):\n 1. vec.dim (null_space A) + vec.dim (col_space A) = finite_dimensional_vector_space.dimension cart_basis\n[PROOF STEP]\nunfolding dimension_vector ncols_def\n[PROOF STATE]\nproof (prove)\nusing this:\nCARD(?'cols) = vec.dim (null_space ?A) + vec.dim (col_space ?A)\n\ngoal (1 subgoal):\n 1. vec.dim (null_space A) + vec.dim (col_space A) = CARD('cols)\n[PROOF STEP]\n..", "meta": {"llama_tokens": 528, "file": "Gauss_Jordan_Gauss_Jordan", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.7236910756380751}} {"text": "[STATEMENT]\nlemma matrix_vector_right_distrib_minus:\n fixes A::\"'a::{ring_1}^'n^'m\"\n shows \"A *v (b - c) = (A *v b) - (A *v c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *v (b - c) = A *v b - A *v c\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. A *v (b - c) = A *v b - A *v c\n[PROOF STEP]\nhave \"A *v (b - c) = A *v (b + - c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *v (b - c) = A *v (b + - c)\n[PROOF STEP]\nby (metis diff_minus_eq_add minus_minus)\n[PROOF STATE]\nproof (state)\nthis:\nA *v (b - c) = A *v (b + - c)\n\ngoal (1 subgoal):\n 1. A *v (b - c) = A *v b - A *v c\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nA *v (b - c) = A *v (b + - c)\n\ngoal (1 subgoal):\n 1. A *v (b - c) = A *v b - A *v c\n[PROOF STEP]\nhave \"... = (A *v b) + (A *v (- c))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *v (b + - c) = A *v b + A *v - c\n[PROOF STEP]\nunfolding matrix_vector_right_distrib\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *v b + A *v - c = A *v b + A *v - c\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nA *v (b + - c) = A *v b + A *v - c\n\ngoal (1 subgoal):\n 1. A *v (b - c) = A *v b - A *v c\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nA *v (b + - c) = A *v b + A *v - c\n\ngoal (1 subgoal):\n 1. A *v (b - c) = A *v b - A *v c\n[PROOF STEP]\nhave \"... = (A *v b) - (A *v c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A *v b + A *v - c = A *v b - A *v c\n[PROOF STEP]\nby (metis (no_types, opaque_lifting) add.commute add_minus_cancel\n matrix_vector_right_distrib uminus_add_conv_diff)\n[PROOF STATE]\nproof (state)\nthis:\nA *v b + A *v - c = A *v b - A *v c\n\ngoal (1 subgoal):\n 1. A *v (b - c) = A *v b - A *v c\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nA *v (b - c) = A *v b - A *v c\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nA *v (b - c) = A *v b - A *v c\n\ngoal (1 subgoal):\n 1. A *v (b - c) = A *v b - A *v c\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nA *v (b - c) = A *v b - A *v c\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1055, "file": "QR_Decomposition_Miscellaneous_QR", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637397236823, "lm_q2_score": 0.8418256512199033, "lm_q1q2_score": 0.7236869875230263}} {"text": "[STATEMENT]\nlemma zipWithS_fix_ind_lemma:\n fixes P Q :: \"nat \\ nat \\ bool\"\n assumes P_0: \"\\j. P 0 j\" and P_Suc: \"\\i j. P i j \\ Q i j \\ P (Suc i) j\"\n assumes Q_0: \"\\i. Q i 0\" and Q_Suc: \"\\i j. P i j \\ Q i j \\ Q i (Suc j)\"\n shows \"P i j \\ Q i j\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. P i j \\ Q i j\n[PROOF STEP]\napply (induct n \\ \"i + j\" arbitrary: i j)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\i j. 0 = i + j \\ P i j \\ Q i j\n 2. \\n i j. \\\\i j. n = i + j \\ P i j \\ Q i j; Suc n = i + j\\ \\ P i j \\ Q i j\n[PROOF STEP]\napply (simp add: P_0 Q_0)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n i j. \\\\i j. n = i + j \\ P i j \\ Q i j; Suc n = i + j\\ \\ P i j \\ Q i j\n[PROOF STEP]\napply (rule conjI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\n i j. \\\\i j. n = i + j \\ P i j \\ Q i j; Suc n = i + j\\ \\ P i j\n 2. \\n i j. \\\\i j. n = i + j \\ P i j \\ Q i j; Suc n = i + j\\ \\ Q i j\n[PROOF STEP]\napply (case_tac i, simp add: P_0, simp add: P_Suc)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n i j. \\\\i j. n = i + j \\ P i j \\ Q i j; Suc n = i + j\\ \\ Q i j\n[PROOF STEP]\napply (case_tac j, simp add: Q_0, simp add: Q_Suc)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 741, "file": "Stream-Fusion_StreamFusion", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637397236824, "lm_q2_score": 0.8418256412990658, "lm_q1q2_score": 0.7236869789944421}} {"text": "[STATEMENT]\nlemma complex_quadratic_equation_monic_roots:\n fixes \\ :: complex\n assumes \"\\ = (-b + ccsqrt(b\\<^sup>2 - 4*c)) / 2 \\\n \\ = (-b - ccsqrt(b\\<^sup>2 - 4*c)) / 2\"\n shows \"\\\\<^sup>2 + b * \\ + c = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>2 + b * \\ + c = 0\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n\ngoal (1 subgoal):\n 1. \\\\<^sup>2 + b * \\ + c = 0\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\\\<^sup>2 + b * \\ + c = 0\n 2. \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\\\<^sup>2 + b * \\ + c = 0\n[PROOF STEP]\nassume *: \"\\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2\"\n[PROOF STATE]\nproof (state)\nthis:\n\\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n\ngoal (2 subgoals):\n 1. \\ = (- b + ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\\\<^sup>2 + b * \\ + c = 0\n 2. \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\\\<^sup>2 + b * \\ + c = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>2 + b * \\ + c = 0\n[PROOF STEP]\nby ((subst *)+) (subst power_divide, subst power2_sum, simp add: field_simps, simp add: power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>2 + b * \\ + c = 0\n\ngoal (1 subgoal):\n 1. \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\\\<^sup>2 + b * \\ + c = 0\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\\\<^sup>2 + b * \\ + c = 0\n[PROOF STEP]\nassume *: \"\\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\"\n[PROOF STATE]\nproof (state)\nthis:\n\\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2\n\ngoal (1 subgoal):\n 1. \\ = (- b - ccsqrt (b\\<^sup>2 - 4 * c)) / 2 \\ \\\\<^sup>2 + b * \\ + c = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>2 + b * \\ + c = 0\n[PROOF STEP]\nby ((subst *)+, subst power_divide, subst power2_diff, simp add: field_simps, simp add: power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>2 + b * \\ + c = 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1147, "file": "Complex_Geometry_Quadratic", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972616934408, "lm_q2_score": 0.8311430520409024, "lm_q1q2_score": 0.7235908651823386}} {"text": "[STATEMENT]\nlemma LIMSEQ_Suc_n_over_n: \"(\\n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. of_nat (Suc n) / of_nat n) \\ (1::'a)\n[PROOF STEP]\nproof (rule Lim_transform_eventually)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ?f \\ (1::'a)\n 2. \\\\<^sub>F x in sequentially. ?f x = of_nat (Suc x) / of_nat x\n[PROOF STEP]\nshow \"eventually (\\n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. (1::'a) + inverse (of_nat n) = of_nat (Suc n) / of_nat n\n[PROOF STEP]\nusing eventually_gt_at_top[of \"0::nat\"]\n[PROOF STATE]\nproof (prove)\nusing this:\neventually ((<) 0) sequentially\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. (1::'a) + inverse (of_nat n) = of_nat (Suc n) / of_nat n\n[PROOF STEP]\nby eventually_elim (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F n in sequentially. (1::'a) + inverse (of_nat n) = of_nat (Suc n) / of_nat n\n\ngoal (1 subgoal):\n 1. (\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a)\n[PROOF STEP]\nhave \"(\\n. 1 + inverse (of_nat n) :: 'a) \\ 1 + 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a) + (0::'a)\n[PROOF STEP]\nby (intro tendsto_add tendsto_const lim_inverse_n)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a) + (0::'a)\n\ngoal (1 subgoal):\n 1. (\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a) + (0::'a)\n[PROOF STEP]\nshow \"(\\n. 1 + inverse (of_nat n) :: 'a) \\ 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a) + (0::'a)\n\ngoal (1 subgoal):\n 1. (\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1043, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972616934406, "lm_q2_score": 0.8311430499496096, "lm_q1q2_score": 0.7235908633616647}} {"text": "[STATEMENT]\nlemma ceiling_diff_one [simp]: \"\\x - 1\\ = \\x\\ - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x - (1::'a)\\ = \\x\\ - 1\n[PROOF STEP]\nusing ceiling_diff_of_int [of x 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x - of_int 1\\ = \\x\\ - 1\n\ngoal (1 subgoal):\n 1. \\x - (1::'a)\\ = \\x\\ - 1\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 216, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972616934408, "lm_q2_score": 0.8311430478583168, "lm_q1q2_score": 0.7235908615409911}} {"text": "[STATEMENT]\nlemma ceiling_diff_one [simp]: \"\\x - 1\\ = \\x\\ - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x - (1::'a)\\ = \\x\\ - 1\n[PROOF STEP]\nusing ceiling_diff_of_int [of x 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x - of_int 1\\ = \\x\\ - 1\n\ngoal (1 subgoal):\n 1. \\x - (1::'a)\\ = \\x\\ - 1\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 216, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972616934408, "lm_q2_score": 0.8311430478583168, "lm_q1q2_score": 0.7235908615409911}} {"text": "[STATEMENT]\nlemma ceiling_diff_one [simp]: \"\\x - 1\\ = \\x\\ - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x - (1::'a)\\ = \\x\\ - 1\n[PROOF STEP]\nusing ceiling_diff_of_int [of x 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x - of_int 1\\ = \\x\\ - 1\n\ngoal (1 subgoal):\n 1. \\x - (1::'a)\\ = \\x\\ - 1\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 216, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972616934408, "lm_q2_score": 0.8311430478583168, "lm_q1q2_score": 0.7235908615409911}} {"text": "[STATEMENT]\nlemma ceiling_diff_one [simp]: \"\\x - 1\\ = \\x\\ - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x - (1::'a)\\ = \\x\\ - 1\n[PROOF STEP]\nusing ceiling_diff_of_int [of x 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x - of_int 1\\ = \\x\\ - 1\n\ngoal (1 subgoal):\n 1. \\x - (1::'a)\\ = \\x\\ - 1\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 216, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972616934408, "lm_q2_score": 0.8311430415844385, "lm_q1q2_score": 0.7235908560789698}} {"text": "[STATEMENT]\nlemma LIMSEQ_Suc_n_over_n: \"(\\n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. of_nat (Suc n) / of_nat n) \\ (1::'a)\n[PROOF STEP]\nproof (rule Lim_transform_eventually)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ?f \\ (1::'a)\n 2. \\\\<^sub>F x in sequentially. ?f x = of_nat (Suc x) / of_nat x\n[PROOF STEP]\nshow \"eventually (\\n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. (1::'a) + inverse (of_nat n) = of_nat (Suc n) / of_nat n\n[PROOF STEP]\nusing eventually_gt_at_top[of \"0::nat\"]\n[PROOF STATE]\nproof (prove)\nusing this:\neventually ((<) 0) sequentially\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. (1::'a) + inverse (of_nat n) = of_nat (Suc n) / of_nat n\n[PROOF STEP]\nby eventually_elim (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F n in sequentially. (1::'a) + inverse (of_nat n) = of_nat (Suc n) / of_nat n\n\ngoal (1 subgoal):\n 1. (\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a)\n[PROOF STEP]\nhave \"(\\n. 1 + inverse (of_nat n) :: 'a) \\ 1 + 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a) + (0::'a)\n[PROOF STEP]\nby (intro tendsto_add tendsto_const lim_inverse_n)\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a) + (0::'a)\n\ngoal (1 subgoal):\n 1. (\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a) + (0::'a)\n[PROOF STEP]\nshow \"(\\n. 1 + inverse (of_nat n) :: 'a) \\ 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a) + (0::'a)\n\ngoal (1 subgoal):\n 1. (\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\n. (1::'a) + inverse (of_nat n)) \\ (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1043, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972616934406, "lm_q2_score": 0.8311430415844385, "lm_q1q2_score": 0.7235908560789697}} {"text": "[STATEMENT]\nlemma fib_add: \"fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\"\n (is \"?P n\")\n \\ \\see @{cite \\page 280\\ \"Concrete-Math\"}\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\n[PROOF STEP]\nproof (induct n rule: fib_induct)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. fib (0 + k + 1) = fib (k + 1) * fib (0 + 1) + fib k * fib 0\n 2. fib (1 + k + 1) = fib (k + 1) * fib (1 + 1) + fib k * fib 1\n 3. \\n. \\fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1); fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\\ \\ fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nshow \"?P 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fib (0 + k + 1) = fib (k + 1) * fib (0 + 1) + fib k * fib 0\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfib (0 + k + 1) = fib (k + 1) * fib (0 + 1) + fib k * fib 0\n\ngoal (2 subgoals):\n 1. fib (1 + k + 1) = fib (k + 1) * fib (1 + 1) + fib k * fib 1\n 2. \\n. \\fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1); fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\\ \\ fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nshow \"?P 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fib (1 + k + 1) = fib (k + 1) * fib (1 + 1) + fib k * fib 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfib (1 + k + 1) = fib (k + 1) * fib (1 + 1) + fib k * fib 1\n\ngoal (1 subgoal):\n 1. \\n. \\fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1); fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\\ \\ fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1); fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\\ \\ fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nhave \"fib (n + 2 + k + 1)\n = fib (n + k + 1) + fib (n + 1 + k + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fib (n + 2 + k + 1) = fib (n + k + 1) + fib (n + 1 + k + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfib (n + 2 + k + 1) = fib (n + k + 1) + fib (n + 1 + k + 1)\n\ngoal (1 subgoal):\n 1. \\n. \\fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1); fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\\ \\ fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfib (n + 2 + k + 1) = fib (n + k + 1) + fib (n + 1 + k + 1)\n\ngoal (1 subgoal):\n 1. \\n. \\fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1); fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\\ \\ fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nassume \"fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\" (is \" _ = ?R1\")\n[PROOF STATE]\nproof (state)\nthis:\nfib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\n\ngoal (1 subgoal):\n 1. \\n. \\fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1); fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\\ \\ fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\n\ngoal (1 subgoal):\n 1. \\n. \\fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1); fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\\ \\ fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nassume \"fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1)\"\n (is \" _ = ?R2\")\n[PROOF STATE]\nproof (state)\nthis:\nfib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1)\n\ngoal (1 subgoal):\n 1. \\n. \\fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1); fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\\ \\ fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nfib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1)\n\ngoal (1 subgoal):\n 1. \\n. \\fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1); fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\\ \\ fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nhave \"?R1 + ?R2 = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fib (k + 1) * fib (n + 1) + fib k * fib n + (fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1)) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nby (simp add: add_mult_distrib2)\n[PROOF STATE]\nproof (state)\nthis:\nfib (k + 1) * fib (n + 1) + fib k * fib n + (fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1)) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n\ngoal (1 subgoal):\n 1. \\n. \\fib (n + 1 + k + 1) = fib (k + 1) * fib (n + 1 + 1) + fib k * fib (n + 1); fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n\\ \\ fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nfib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\nshow \"?P (n + 2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n\ngoal (1 subgoal):\n 1. fib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nfib (n + 2 + k + 1) = fib (k + 1) * fib (n + 2 + 1) + fib k * fib (n + 2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3335, "file": null, "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972583359805, "lm_q2_score": 0.8311430415844384, "lm_q1q2_score": 0.7235908532884399}} {"text": "[STATEMENT]\nlemma lim_mono:\n fixes X Y :: \"nat \\ 'a::linorder_topology\"\n assumes \"\\n. N \\ n \\ X n \\ Y n\"\n and \"X \\ x\"\n and \"Y \\ y\"\n shows \"x \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nN \\ ?n \\ X ?n \\ Y ?n\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nby (intro LIMSEQ_le[OF assms(2,3)]) auto", "meta": {"llama_tokens": 214, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.882427872638409, "lm_q2_score": 0.8198933337131076, "lm_q1q2_score": 0.7234967302588707}} {"text": "[STATEMENT]\nlemma gbinomial_sum_up_index:\n \"(\\j = 0..n. (of_nat j gchoose k) :: 'a::field_char_0) = (of_nat n + 1) gchoose (k + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n 2. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n 2. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nusing gbinomial_Suc_Suc[of 0 k]\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::?'b1) + (1::?'b1) gchoose Suc k = (0::?'b1) gchoose k + ((0::?'b1) gchoose Suc k)\n\ngoal (1 subgoal):\n 1. (\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nby (cases k) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..0. of_nat j gchoose k) = of_nat 0 + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. \\n. (\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1) \\ (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\n\ngoal (1 subgoal):\n 1. (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nusing gbinomial_Suc_Suc[of \"of_nat (Suc n) :: 'a\" k]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\j = 0..n. of_nat j gchoose k) = of_nat n + (1::'a) gchoose (k + 1)\nof_nat (Suc n) + (1::'a) gchoose Suc k = of_nat (Suc n) gchoose k + (of_nat (Suc n) gchoose Suc k)\n\ngoal (1 subgoal):\n 1. (\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n[PROOF STEP]\nby (simp add: add_ac)\n[PROOF STATE]\nproof (state)\nthis:\n(\\j = 0..Suc n. of_nat j gchoose k) = of_nat (Suc n) + (1::'a) gchoose (k + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1647, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.88242786954645, "lm_q2_score": 0.8198933337131076, "lm_q1q2_score": 0.7234967277237941}} {"text": "[STATEMENT]\nlemma less_power_add_imp_div_less [simp]:\n fixes i m n:: nat\n assumes \"i < 2^(m+n)\"\n shows \"i div 2^n < 2^m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i div 2 ^ n < 2 ^ m\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ni < 2 ^ (m + n)\n\ngoal (1 subgoal):\n 1. i div 2 ^ n < 2 ^ m\n[PROOF STEP]\nby (simp add: less_mult_imp_div_less power_add)", "meta": {"llama_tokens": 182, "file": "Isabelle_Marries_Dirac_Basics", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767938900121, "lm_q2_score": 0.8244619306896955, "lm_q1q2_score": 0.7234462116259635}} {"text": "[STATEMENT]\nlemma int_less_real_le: \"n < m \\ real_of_int n + 1 \\ real_of_int m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (n < m) = (real_of_int n + 1 \\ real_of_int m)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (n < m) = (real_of_int n + 1 \\ real_of_int m)\n[PROOF STEP]\nhave \"(0::real) \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ 1\n[PROOF STEP]\nby (metis less_eq_real_def zero_less_one)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ 1\n\ngoal (1 subgoal):\n 1. (n < m) = (real_of_int n + 1 \\ real_of_int m)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\ 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ 1\n\ngoal (1 subgoal):\n 1. (n < m) = (real_of_int n + 1 \\ real_of_int m)\n[PROOF STEP]\nby (metis floor_of_int less_floor_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(n < m) = (real_of_int n + 1 \\ real_of_int m)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 485, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767938900121, "lm_q2_score": 0.8244619306896955, "lm_q1q2_score": 0.7234462116259635}} {"text": "[STATEMENT]\nlemma (in finite_measure) measure_eq_bigunion_image:\n assumes \"range f \\ sets M\" \"range g \\ sets M\"\n assumes \"disjoint_family f\" \"disjoint_family g\"\n assumes \"\\ n :: nat. measure M (f n) = measure M (g n)\"\n shows \"measure M (\\i. f i) = measure M (\\i. g i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (\\ (range f)) = Sigma_Algebra.measure M (\\ (range g))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nrange f \\ sets M\nrange g \\ sets M\ndisjoint_family f\ndisjoint_family g\nSigma_Algebra.measure M (f ?n) = Sigma_Algebra.measure M (g ?n)\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (\\ (range f)) = Sigma_Algebra.measure M (\\ (range g))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\range f \\ sets M; range g \\ sets M; disjoint_family f; disjoint_family g; \\n. Sigma_Algebra.measure M (f n) = Sigma_Algebra.measure M (g n)\\ \\ Sigma_Algebra.measure M (\\ (range f)) = Sigma_Algebra.measure M (\\ (range g))\n[PROOF STEP]\nhave a: \"(\\ i. measure M (f i)) sums (measure M (\\i. f i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. Sigma_Algebra.measure M (f i)) sums Sigma_Algebra.measure M (\\ (range f))\n[PROOF STEP]\nby (rule finite_measure_UNION[OF assms(1,3)])\n[PROOF STATE]\nproof (state)\nthis:\n(\\i. Sigma_Algebra.measure M (f i)) sums Sigma_Algebra.measure M (\\ (range f))\n\ngoal (1 subgoal):\n 1. \\range f \\ sets M; range g \\ sets M; disjoint_family f; disjoint_family g; \\n. Sigma_Algebra.measure M (f n) = Sigma_Algebra.measure M (g n)\\ \\ Sigma_Algebra.measure M (\\ (range f)) = Sigma_Algebra.measure M (\\ (range g))\n[PROOF STEP]\nhave b: \"(\\ i. measure M (g i)) sums (measure M (\\i. g i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. Sigma_Algebra.measure M (g i)) sums Sigma_Algebra.measure M (\\ (range g))\n[PROOF STEP]\nby (rule finite_measure_UNION[OF assms(2,4)])\n[PROOF STATE]\nproof (state)\nthis:\n(\\i. Sigma_Algebra.measure M (g i)) sums Sigma_Algebra.measure M (\\ (range g))\n\ngoal (1 subgoal):\n 1. \\range f \\ sets M; range g \\ sets M; disjoint_family f; disjoint_family g; \\n. Sigma_Algebra.measure M (f n) = Sigma_Algebra.measure M (g n)\\ \\ Sigma_Algebra.measure M (\\ (range f)) = Sigma_Algebra.measure M (\\ (range g))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (\\ (range f)) = Sigma_Algebra.measure M (\\ (range g))\n[PROOF STEP]\nusing sums_unique[OF b] sums_unique[OF a] assms\n[PROOF STATE]\nproof (prove)\nusing this:\nSigma_Algebra.measure M (\\ (range g)) = (\\i. Sigma_Algebra.measure M (g i))\nSigma_Algebra.measure M (\\ (range f)) = (\\i. Sigma_Algebra.measure M (f i))\nrange f \\ sets M\nrange g \\ sets M\ndisjoint_family f\ndisjoint_family g\nSigma_Algebra.measure M (f ?n) = Sigma_Algebra.measure M (g ?n)\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (\\ (range f)) = Sigma_Algebra.measure M (\\ (range g))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M (\\ (range f)) = Sigma_Algebra.measure M (\\ (range g))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1351, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767906859264, "lm_q2_score": 0.8244619263765706, "lm_q1q2_score": 0.7234462051996498}} {"text": "[STATEMENT]\nlemma one_add_floor: \"\\x\\ + 1 = \\x + 1\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\ + 1 = \\x + (1::'a)\\\n[PROOF STEP]\nusing floor_add_int [of x 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\ + 1 = \\x + of_int 1\\\n\ngoal (1 subgoal):\n 1. \\x\\ + 1 = \\x + (1::'a)\\\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 209, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8244619350028204, "lm_q1q2_score": 0.7234462048440436}} {"text": "[STATEMENT]\nlemma bounded_sums:\n fixes S :: \"'a::real_normed_vector set\"\n assumes \"bounded S\" and \"bounded T\"\n shows \"bounded (\\x\\ S. \\y \\ T. {x + y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bounded (\\x\\S. \\y\\T. {x + y})\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nbounded S\nbounded T\n\ngoal (1 subgoal):\n 1. bounded (\\x\\S. \\y\\T. {x + y})\n[PROOF STEP]\nby (simp add: bounded_iff) (meson norm_triangle_mono)", "meta": {"llama_tokens": 215, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.877476793890012, "lm_q2_score": 0.8244619220634457, "lm_q1q2_score": 0.7234462040566293}} {"text": "[STATEMENT]\nlemma one_add_floor: \"\\x\\ + 1 = \\x + 1\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\ + 1 = \\x + (1::'a)\\\n[PROOF STEP]\nusing floor_add_int [of x 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\ + 1 = \\x + of_int 1\\\n\ngoal (1 subgoal):\n 1. \\x\\ + 1 = \\x + (1::'a)\\\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 209, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.7234461972747098}} {"text": "[STATEMENT]\nlemma diameter_ball [simp]:\n fixes a :: \"'a::euclidean_space\"\n shows \"diameter(ball a r) = (if r < 0 then 0 else 2*r)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. diameter (ball a r) = (if r < 0 then 0 else 2 * r)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. diameter (ball a r) = (if r < 0 then 0 else 2 * r)\n[PROOF STEP]\nhave \"diameter(ball a r) = 2*r\" if \"r > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. diameter (ball a r) = 2 * r\n[PROOF STEP]\nby (metis bounded_ball diameter_closure closure_ball diameter_cball less_eq_real_def linorder_not_less that)\n[PROOF STATE]\nproof (state)\nthis:\n0 < r \\ diameter (ball a r) = 2 * r\n\ngoal (1 subgoal):\n 1. diameter (ball a r) = (if r < 0 then 0 else 2 * r)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < r \\ diameter (ball a r) = 2 * r\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < r \\ diameter (ball a r) = 2 * r\n\ngoal (1 subgoal):\n 1. diameter (ball a r) = (if r < 0 then 0 else 2 * r)\n[PROOF STEP]\nby (simp add: diameter_def)\n[PROOF STATE]\nproof (state)\nthis:\ndiameter (ball a r) = (if r < 0 then 0 else 2 * r)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 535, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767746654974, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.723446191991416}} {"text": "[STATEMENT]\nlemma triangle_points_closer:\n fixes a::complex\n shows \"\\x \\ convex hull {a,b,c}; y \\ convex hull {a,b,c}\\\n \\ norm(x - y) \\ norm(a - b) \\\n norm(x - y) \\ norm(b - c) \\\n norm(x - y) \\ norm(c - a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x \\ convex hull {a, b, c}; y \\ convex hull {a, b, c}\\ \\ cmod (x - y) \\ cmod (a - b) \\ cmod (x - y) \\ cmod (b - c) \\ cmod (x - y) \\ cmod (c - a)\n[PROOF STEP]\nusing simplex_extremal_le [of \"{a,b,c}\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite {a, b, c}; {a, b, c} \\ {}\\ \\ \\u\\{a, b, c}. \\v\\{a, b, c}. \\x\\convex hull {a, b, c}. \\y\\convex hull {a, b, c}. cmod (x - y) \\ cmod (u - v)\n\ngoal (1 subgoal):\n 1. \\x \\ convex hull {a, b, c}; y \\ convex hull {a, b, c}\\ \\ cmod (x - y) \\ cmod (a - b) \\ cmod (x - y) \\ cmod (b - c) \\ cmod (x - y) \\ cmod (c - a)\n[PROOF STEP]\nby (auto simp: norm_minus_commute)", "meta": {"llama_tokens": 533, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297861178929, "lm_q2_score": 0.8031737963569016, "lm_q1q2_score": 0.7232819270487767}} {"text": "[STATEMENT]\nlemma has_contour_integral_bound_linepath:\n assumes \"(f has_contour_integral i) (linepath a b)\"\n \"0 \\ B\" and B: \"\\x. x \\ closed_segment a b \\ norm(f x) \\ B\"\n shows \"norm i \\ B * norm(b - a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod i \\ B * cmod (b - a)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. cmod i \\ B * cmod (b - a)\n[PROOF STEP]\nhave \"norm i \\ (B * norm (b - a)) * content (cbox 0 (1::real))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod i \\ B * cmod (b - a) * content (cbox 0 1)\n[PROOF STEP]\nproof (rule has_integral_bound\n [of _ \"\\x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})\"])\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. 0 \\ B * cmod (b - a)\n 2. ((\\x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})) has_integral i) (cbox 0 1)\n 3. \\x. x \\ cbox 0 1 \\ cmod (f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})) \\ B * cmod (b - a)\n[PROOF STEP]\nshow \"cmod (f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1}))\n \\ B * cmod (b - a)\"\n if \"x \\ cbox 0 1\" for x::real\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})) \\ B * cmod (b - a)\n[PROOF STEP]\nusing that box_real(2) norm_mult\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ cbox 0 1\ncbox ?a ?b = {?a..?b}\nnorm (?x * ?y) = norm ?x * norm ?y\n\ngoal (1 subgoal):\n 1. cmod (f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})) \\ B * cmod (b - a)\n[PROOF STEP]\nby (metis B linepath_in_path mult_right_mono norm_ge_zero vector_derivative_linepath_within)\n[PROOF STATE]\nproof (state)\nthis:\n?x \\ cbox 0 1 \\ cmod (f (linepath a b ?x) * vector_derivative (linepath a b) (at ?x within {0..1})) \\ B * cmod (b - a)\n\ngoal (2 subgoals):\n 1. 0 \\ B * cmod (b - a)\n 2. ((\\x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})) has_integral i) (cbox 0 1)\n[PROOF STEP]\nqed (use assms has_contour_integral_def in auto)\n[PROOF STATE]\nproof (state)\nthis:\ncmod i \\ B * cmod (b - a) * content (cbox 0 1)\n\ngoal (1 subgoal):\n 1. cmod i \\ B * cmod (b - a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncmod i \\ B * cmod (b - a) * content (cbox 0 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncmod i \\ B * cmod (b - a) * content (cbox 0 1)\n\ngoal (1 subgoal):\n 1. cmod i \\ B * cmod (b - a)\n[PROOF STEP]\nby (auto simp: content_real)\n[PROOF STATE]\nproof (state)\nthis:\ncmod i \\ B * cmod (b - a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1266, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045996818986, "lm_q2_score": 0.8152324960856175, "lm_q1q2_score": 0.7232780203373153}} {"text": "[STATEMENT]\nlemma gchoose_row_sum_weighted:\n \"(\\k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))\"\n for r :: \"'a::field_char_0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..m. (r gchoose k) * (r / (2::'a) - of_nat k)) = of_nat (Suc m) / (2::'a) * (r gchoose Suc m)\n[PROOF STEP]\nby (induct m) (simp_all add: field_simps distrib gbinomial_mult_1)", "meta": {"llama_tokens": 199, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045877523148, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7232780086205655}} {"text": "[STATEMENT]\nlemma gchoose_row_sum_weighted:\n \"(\\k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))\"\n for r :: \"'a::field_char_0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..m. (r gchoose k) * (r / (2::'a) - of_nat k)) = of_nat (Suc m) / (2::'a) * (r gchoose Suc m)\n[PROOF STEP]\nby (induct m) (simp_all add: field_simps distrib gbinomial_mult_1)", "meta": {"llama_tokens": 199, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045877523148, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.723278002646469}} {"text": "[STATEMENT]\nlemma gchoose_row_sum_weighted:\n \"(\\k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))\"\n for r :: \"'a::field_char_0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 0..m. (r gchoose k) * (r / (2::'a) - of_nat k)) = of_nat (Suc m) / (2::'a) * (r gchoose Suc m)\n[PROOF STEP]\nby (induct m) (simp_all add: field_simps distrib gbinomial_mult_1)", "meta": {"llama_tokens": 199, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045877523148, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.723278002646469}} {"text": "[STATEMENT]\ntheorem rqs_cost_exp_eq [code]: \"rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\ndefine F where \"F = (\\n. rqs_cost_exp n / (real n + 1))\"\n[PROOF STATE]\nproof (state)\nthis:\nF = (\\n. rqs_cost_exp n / (real n + 1))\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nhave [simp]: \"F 0 = 0\" \"F (Suc 0) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. F 0 = 0 &&& F (Suc 0) = 0\n[PROOF STEP]\nby (simp_all add: F_def)\n[PROOF STATE]\nproof (state)\nthis:\nF 0 = 0\nF (Suc 0) = 0\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nhave F_Suc: \"F (Suc m) = F m + real (2*m) / (real ((m+1)*(m+2)))\" if \"m > 0\" for m\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\nproof (cases m)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. m = 0 \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n 2. \\nat. m = Suc nat \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nm = Suc n\n\ngoal (2 subgoals):\n 1. m = 0 \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n 2. \\nat. m = Suc nat \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\nhave A: \"rqs_cost_exp (Suc (Suc n)) * real (Suc (Suc n)) = \n real ((n+1)*(n+2)) + 2 * (\\i\\n. rqs_cost_exp i) + 2 * rqs_cost_exp (Suc n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rqs_cost_exp (Suc (Suc n)) * real (Suc (Suc n)) = real ((n + 1) * (n + 2)) + 2 * sum rqs_cost_exp {..n} + 2 * rqs_cost_exp (Suc n)\n[PROOF STEP]\nby (subst rqs_cost_exp_Suc') (simp_all add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nrqs_cost_exp (Suc (Suc n)) * real (Suc (Suc n)) = real ((n + 1) * (n + 2)) + 2 * sum rqs_cost_exp {..n} + 2 * rqs_cost_exp (Suc n)\n\ngoal (2 subgoals):\n 1. m = 0 \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n 2. \\nat. m = Suc nat \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\nhave B: \"rqs_cost_exp (Suc n) * real (Suc n) = real (n*(n+1)) + 2 * (\\i\\n. rqs_cost_exp i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rqs_cost_exp (Suc n) * real (Suc n) = real (n * (n + 1)) + 2 * sum rqs_cost_exp {..n}\n[PROOF STEP]\nby (subst rqs_cost_exp_Suc') (simp_all add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nrqs_cost_exp (Suc n) * real (Suc n) = real (n * (n + 1)) + 2 * sum rqs_cost_exp {..n}\n\ngoal (2 subgoals):\n 1. m = 0 \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n 2. \\nat. m = Suc nat \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\nhave \"rqs_cost_exp (Suc (Suc n)) * real (Suc (Suc n)) - rqs_cost_exp (Suc n) * real (Suc n) =\n real ((n+1)*(n+2)) - real (n*(n+1)) + 2 * rqs_cost_exp (Suc n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rqs_cost_exp (Suc (Suc n)) * real (Suc (Suc n)) - rqs_cost_exp (Suc n) * real (Suc n) = real ((n + 1) * (n + 2)) - real (n * (n + 1)) + 2 * rqs_cost_exp (Suc n)\n[PROOF STEP]\nby (subst A, subst B) simp_all\n[PROOF STATE]\nproof (state)\nthis:\nrqs_cost_exp (Suc (Suc n)) * real (Suc (Suc n)) - rqs_cost_exp (Suc n) * real (Suc n) = real ((n + 1) * (n + 2)) - real (n * (n + 1)) + 2 * rqs_cost_exp (Suc n)\n\ngoal (2 subgoals):\n 1. m = 0 \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n 2. \\nat. m = Suc nat \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nrqs_cost_exp (Suc (Suc n)) * real (Suc (Suc n)) - rqs_cost_exp (Suc n) * real (Suc n) = real ((n + 1) * (n + 2)) - real (n * (n + 1)) + 2 * rqs_cost_exp (Suc n)\n\ngoal (2 subgoals):\n 1. m = 0 \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n 2. \\nat. m = Suc nat \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\nhave \"real ((n+1)*(n+2)) - real (n*(n+1)) = real (2*(n+1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real ((n + 1) * (n + 2)) - real (n * (n + 1)) = real (2 * (n + 1))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal ((n + 1) * (n + 2)) - real (n * (n + 1)) = real (2 * (n + 1))\n\ngoal (2 subgoals):\n 1. m = 0 \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n 2. \\nat. m = Suc nat \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nrqs_cost_exp (Suc (Suc n)) * real (Suc (Suc n)) - rqs_cost_exp (Suc n) * real (Suc n) = real (2 * (n + 1)) + 2 * rqs_cost_exp (Suc n)\n[PROOF STEP]\nhave \"rqs_cost_exp (Suc (Suc n)) * real (n+2) = rqs_cost_exp (Suc n) * real (n+3) + real (2*(n+1))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nrqs_cost_exp (Suc (Suc n)) * real (Suc (Suc n)) - rqs_cost_exp (Suc n) * real (Suc n) = real (2 * (n + 1)) + 2 * rqs_cost_exp (Suc n)\n\ngoal (1 subgoal):\n 1. rqs_cost_exp (Suc (Suc n)) * real (n + 2) = rqs_cost_exp (Suc n) * real (n + 3) + real (2 * (n + 1))\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nrqs_cost_exp (Suc (Suc n)) * real (n + 2) = rqs_cost_exp (Suc n) * real (n + 3) + real (2 * (n + 1))\n\ngoal (2 subgoals):\n 1. m = 0 \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n 2. \\nat. m = Suc nat \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\nhence \"rqs_cost_exp (Suc (Suc n)) / real (n+3) = \n rqs_cost_exp (Suc n) / real (n+2) + real (2*(n+1)) / (real (n+2)*real (n+3))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nrqs_cost_exp (Suc (Suc n)) * real (n + 2) = rqs_cost_exp (Suc n) * real (n + 3) + real (2 * (n + 1))\n\ngoal (1 subgoal):\n 1. rqs_cost_exp (Suc (Suc n)) / real (n + 3) = rqs_cost_exp (Suc n) / real (n + 2) + real (2 * (n + 1)) / (real (n + 2) * real (n + 3))\n[PROOF STEP]\nby (simp add: divide_simps del: of_nat_Suc of_nat_add)\n[PROOF STATE]\nproof (state)\nthis:\nrqs_cost_exp (Suc (Suc n)) / real (n + 3) = rqs_cost_exp (Suc n) / real (n + 2) + real (2 * (n + 1)) / (real (n + 2) * real (n + 3))\n\ngoal (2 subgoals):\n 1. m = 0 \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n 2. \\nat. m = Suc nat \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nrqs_cost_exp (Suc (Suc n)) / real (n + 3) = rqs_cost_exp (Suc n) / real (n + 2) + real (2 * (n + 1)) / (real (n + 2) * real (n + 3))\n\ngoal (1 subgoal):\n 1. F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\nby (simp add: F_def algebra_simps Suc)\n[PROOF STATE]\nproof (state)\nthis:\nF (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n\ngoal (1 subgoal):\n 1. m = 0 \\ F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))\n[PROOF STEP]\nqed simp_all\n[PROOF STATE]\nproof (state)\nthis:\n0 < ?m \\ F (Suc ?m) = F ?m + real (2 * ?m) / real ((?m + 1) * (?m + 2))\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nhave F_eq: \"F n = 2 * (\\k=1..n. real (k - 1) / real (k * (k + 1)))\" for n\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. F n = 2 * (\\k = 1..n. real (k - 1) / real (k * (k + 1)))\n[PROOF STEP]\nproof (cases \"n \\ 1\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 1 \\ n \\ F n = 2 * (\\k = 1..n. real (k - 1) / real (k * (k + 1)))\n 2. \\ 1 \\ n \\ F n = 2 * (\\k = 1..n. real (k - 1) / real (k * (k + 1)))\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\n1 \\ n\n\ngoal (2 subgoals):\n 1. 1 \\ n \\ F n = 2 * (\\k = 1..n. real (k - 1) / real (k * (k + 1)))\n 2. \\ 1 \\ n \\ F n = 2 * (\\k = 1..n. real (k - 1) / real (k * (k + 1)))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\ n\n\ngoal (1 subgoal):\n 1. F n = 2 * (\\k = 1..n. real (k - 1) / real (k * (k + 1)))\n[PROOF STEP]\nby (induction n rule: dec_induct) (simp_all add: F_Suc algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nF n = 2 * (\\k = 1..n. real (k - 1) / real (k * (k + 1)))\n\ngoal (1 subgoal):\n 1. \\ 1 \\ n \\ F n = 2 * (\\k = 1..n. real (k - 1) / real (k * (k + 1)))\n[PROOF STEP]\nqed (simp_all add: not_le)\n[PROOF STATE]\nproof (state)\nthis:\nF ?n = 2 * (\\k = 1..?n. real (k - 1) / real (k * (k + 1)))\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nhave \"F n = 2 * (\\k=1..n. real (k - 1) / real (k * (k + 1)))\" (is \"_ = 2 * ?S\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. F n = 2 * (\\k = 1..n. real (k - 1) / real (k * (k + 1)))\n[PROOF STEP]\nby (fact F_eq)\n[PROOF STATE]\nproof (state)\nthis:\nF n = 2 * (\\k = 1..n. real (k - 1) / real (k * (k + 1)))\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nF n = 2 * (\\k = 1..n. real (k - 1) / real (k * (k + 1)))\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nhave \"?S = (\\k=1..n. 2 / real (Suc k) - 1 / real k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 1..n. real (k - 1) / real (k * (k + 1))) = (\\k = 1..n. 2 / real (Suc k) - 1 / real k)\n[PROOF STEP]\nby (intro sum.cong) (simp_all add: field_simps of_nat_diff)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 1..n. real (k - 1) / real (k * (k + 1))) = (\\k = 1..n. 2 / real (Suc k) - 1 / real k)\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 1..n. real (k - 1) / real (k * (k + 1))) = (\\k = 1..n. 2 / real (Suc k) - 1 / real k)\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nhave \"\\ = 2 * (\\k=1..n. inverse (real (Suc k))) - harm n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 1..n. 2 / real (Suc k) - 1 / real k) = 2 * (\\k = 1..n. inverse (real (Suc k))) - harm n\n[PROOF STEP]\nby (subst sum_subtractf) (simp add: harm_def sum.distrib sum_distrib_left divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 1..n. 2 / real (Suc k) - 1 / real k) = 2 * (\\k = 1..n. inverse (real (Suc k))) - harm n\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 1..n. 2 / real (Suc k) - 1 / real k) = 2 * (\\k = 1..n. inverse (real (Suc k))) - harm n\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nhave \"(\\k=1..n. inverse (real (Suc k))) = (\\k=Suc 1..Suc n. inverse (real k))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = 1..n. inverse (real (Suc k))) = (\\k = Suc 1..Suc n. inverse (real k))\n[PROOF STEP]\nby (intro sum.reindex_bij_witness[of _ \"\\x. x - 1\" Suc]) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 1..n. inverse (real (Suc k))) = (\\k = Suc 1..Suc n. inverse (real k))\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = 1..n. inverse (real (Suc k))) = (\\k = Suc 1..Suc n. inverse (real k))\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nhave \"\\ = harm (Suc n) - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = Suc 1..Suc n. inverse (real k)) = harm (Suc n) - 1\n[PROOF STEP]\nunfolding harm_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\k = Suc 1..Suc n. inverse (real k)) = (\\k = 1..Suc n. inverse (real k)) - 1\n[PROOF STEP]\nby (subst (2) sum.atLeast_Suc_atMost) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n(\\k = Suc 1..Suc n. inverse (real k)) = harm (Suc n) - 1\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nF n = 2 * (2 * (harm (Suc n) - 1) - harm n)\n[PROOF STEP]\nhave \"F n = 2 * harm n + 4 * (1 / (n + 1) - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nF n = 2 * (2 * (harm (Suc n) - 1) - harm n)\n\ngoal (1 subgoal):\n 1. F n = 2 * harm n + 4 * (1 / real (n + 1) - 1)\n[PROOF STEP]\nby (simp add: harm_Suc field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nF n = 2 * harm n + 4 * (1 / real (n + 1) - 1)\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nF n = 2 * harm n + 4 * (1 / real (n + 1) - 1)\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nhave \"\\ * real (n + 1) = 2 * real (n + 1) * harm n - 4 * real n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (2 * harm n + 4 * (1 / real (n + 1) - 1)) * real (n + 1) = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(2 * harm n + 4 * (1 / real (n + 1) - 1)) * real (n + 1) = 2 * real (n + 1) * harm n - 4 * real n\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(2 * harm n + 4 * (1 / real (n + 1) - 1)) * real (n + 1) = 2 * real (n + 1) * harm n - 4 * real n\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nhave \"F n * real (n + 1) = rqs_cost_exp n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. F n * real (n + 1) = rqs_cost_exp n\n[PROOF STEP]\nby (simp add: F_def add_ac)\n[PROOF STATE]\nproof (state)\nthis:\nF n * real (n + 1) = rqs_cost_exp n\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nrqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nrqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n\ngoal (1 subgoal):\n 1. rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nrqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 7356, "file": "Quick_Sort_Cost_Randomised_Quick_Sort", "length": 58, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045877523147, "lm_q2_score": 0.8152324871074608, "lm_q1q2_score": 0.7232780026464689}} {"text": "[STATEMENT]\nlemma card_sameprod: \"finite X \\ card (X^\\) = card X choose 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite X \\ card X^\\ = card X choose 2\n[PROOF STEP]\nunfolding sameprod_altdef\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite X \\ card {Y. Y \\ X \\ card Y = 2} = card X choose 2\n[PROOF STEP]\nby (subst n_subsets, auto)", "meta": {"llama_tokens": 161, "file": "Clique_and_Monotone_Circuits_Clique_Large_Monotone_Circuits", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942171172603, "lm_q2_score": 0.8006920068519376, "lm_q1q2_score": 0.7232604594813691}} {"text": "[STATEMENT]\nlemma dim_left_null_space[code_unfold]:\n fixes A::\"'a::{field}^'cols::{mod_type}^'rows::{mod_type}\"\n shows \"vec.dim (left_null_space A) = (vec.dimension TYPE('a) TYPE('rows)) - rank (A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (left_null_space A) = finite_dimensional_vector_space.dimension cart_basis - rank A\n[PROOF STEP]\nunfolding left_null_space_eq_null_space_transpose\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (null_space (Finite_Cartesian_Product.transpose A)) = finite_dimensional_vector_space.dimension cart_basis - rank A\n[PROOF STEP]\nunfolding dim_null_space\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite_dimensional_vector_space.dimension cart_basis - rank (Finite_Cartesian_Product.transpose A) = finite_dimensional_vector_space.dimension cart_basis - rank A\n[PROOF STEP]\nunfolding rank_transpose\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite_dimensional_vector_space.dimension cart_basis - rank A = finite_dimensional_vector_space.dimension cart_basis - rank A\n[PROOF STEP]\n..", "meta": {"llama_tokens": 397, "file": "Gauss_Jordan_Gauss_Jordan", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032941988938413, "lm_q2_score": 0.8006920020959544, "lm_q1q2_score": 0.723260440593971}} {"text": "[STATEMENT]\nlemma permutation_mat_both: assumes A: \"A \\ carrier_mat n n\" and p: \"p permutes {.. (i,j). A $$ (p i, p j)) * permutation_mat n (inv p) = A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. permutation_mat n p * Matrix.mat n n (\\(i, j). A $$ (p i, p j)) * permutation_mat n (inv p) = A\n[PROOF STEP]\nunfolding permutation_mat_left[OF mat_carrier p]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix.mat n n (\\(i, j). Matrix.mat n n (\\(i, j). A $$ (p i, p j)) $$ (inv p i, j)) * permutation_mat n (inv p) = A\n[PROOF STEP]\nby (subst permutation_mat_right[OF _ permutes_inv[OF p], of _ n], force, insert A p, \n auto intro!: eq_matI simp: permutes_inverses permutes_lt[OF permutes_inv[OF p]])", "meta": {"llama_tokens": 327, "file": "Perron_Frobenius_Perron_Frobenius_General", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.914900957313305, "lm_q2_score": 0.7905303112671294, "lm_q1q2_score": 0.7232569385634816}} {"text": "[STATEMENT]\nlemma card_funcsetE: \"finite A \\ card (A \\\\<^sub>E B) = card B ^ card A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite A \\ card (A \\\\<^sub>E B) = card B ^ card A\n[PROOF STEP]\nby (subst card_PiE, auto)", "meta": {"llama_tokens": 101, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952975813453, "lm_q2_score": 0.8056321796478255, "lm_q1q2_score": 0.7232122192500625}} {"text": "[STATEMENT]\nlemma periodic_arithmetic_sum_periodic_arithmetic:\n assumes \"periodic_arithmetic f k\"\n shows \"(\\l \\ {m..n}. f l) = (\\l \\ {m+k..n+k}. f l)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f {m..n} = sum f {m + k..n + k}\n[PROOF STEP]\nusing periodic_arithmetic_def assms\n[PROOF STATE]\nproof (prove)\nusing this:\nperiodic_arithmetic ?f ?k = (\\n. ?f (n + ?k) = ?f n)\nperiodic_arithmetic f k\n\ngoal (1 subgoal):\n 1. sum f {m..n} = sum f {m + k..n + k}\n[PROOF STEP]\nby (intro sum.reindex_bij_witness\n [of \"{m..n}\" \"\\l. l-k\" \"\\l. l+k\" \"{m+k..n+k}\" f f])\n auto", "meta": {"llama_tokens": 281, "file": "Gauss_Sums_Periodic_Arithmetic", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.897695292107347, "lm_q2_score": 0.8056321843145404, "lm_q1q2_score": 0.7232122190293214}} {"text": "[STATEMENT]\nlemma integral_std_normal_moment_abs_odd:\n \"integral\\<^sup>L lborel (\\x. std_normal_density x * \\x\\^(2 * k + 1)) = sqrt (2 / pi) * 2^k * fact k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LBINT x. std_normal_density x * \\x\\ ^ (2 * k + 1) = sqrt (2 / pi) * 2 ^ k * fact k\n[PROOF STEP]\nusing std_normal_moment_abs_odd\n[PROOF STATE]\nproof (prove)\nusing this:\nhas_bochner_integral lborel (\\x. std_normal_density x * \\x\\ ^ (2 * k + 1)) (sqrt (2 / pi) * 2 ^ k * fact k)\n\ngoal (1 subgoal):\n 1. LBINT x. std_normal_density x * \\x\\ ^ (2 * k + 1) = sqrt (2 / pi) * 2 ^ k * fact k\n[PROOF STEP]\nby (rule has_bochner_integral_integral_eq)", "meta": {"llama_tokens": 322, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.897695283896349, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7232122186982091}} {"text": "[STATEMENT]\nlemma ennreal_prod_eq_top:\n fixes f :: \"'a \\ ennreal\"\n shows \"(\\i\\I. f i) = top \\ (finite I \\ ((\\i\\I. f i \\ 0) \\ (\\i\\I. f i = top)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (prod f I = top) = (finite I \\ (\\i\\I. f i \\ 0) \\ (\\i\\I. f i = top))\n[PROOF STEP]\nby (induction I rule: infinite_finite_induct) (auto simp: ennreal_mult_eq_top_iff)", "meta": {"llama_tokens": 211, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952866333484, "lm_q2_score": 0.8056321796478255, "lm_q1q2_score": 0.7232122104300039}} {"text": "[STATEMENT]\nlemma Arg_minus:\n assumes \"z \\ 0\"\n shows \"Arg (-z) = (if Arg z \\ 0 then Arg z + pi else Arg z - pi)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Arg (- z) = (if Arg z \\ 0 then Arg z + pi else Arg z - pi)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Arg (- z) = (if Arg z \\ 0 then Arg z + pi else Arg z - pi)\n[PROOF STEP]\nhave [simp]: \"cmod z * cos (Arg z) = Re z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod z * cos (Arg z) = Re z\n[PROOF STEP]\nusing assms Arg_eq [of z]\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ 0\nz \\ 0 \\ z = complex_of_real (cmod z) * exp (\\ * complex_of_real (Arg z))\n\ngoal (1 subgoal):\n 1. cmod z * cos (Arg z) = Re z\n[PROOF STEP]\nby (metis Re_exp exp_Ln norm_exp_eq_Re Arg_def)\n[PROOF STATE]\nproof (state)\nthis:\ncmod z * cos (Arg z) = Re z\n\ngoal (1 subgoal):\n 1. Arg (- z) = (if Arg z \\ 0 then Arg z + pi else Arg z - pi)\n[PROOF STEP]\nhave [simp]: \"cmod z * sin (Arg z) = Im z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod z * sin (Arg z) = Im z\n[PROOF STEP]\nusing assms Arg_eq [of z]\n[PROOF STATE]\nproof (prove)\nusing this:\nz \\ 0\nz \\ 0 \\ z = complex_of_real (cmod z) * exp (\\ * complex_of_real (Arg z))\n\ngoal (1 subgoal):\n 1. cmod z * sin (Arg z) = Im z\n[PROOF STEP]\nby (metis Im_exp exp_Ln norm_exp_eq_Re Arg_def)\n[PROOF STATE]\nproof (state)\nthis:\ncmod z * sin (Arg z) = Im z\n\ngoal (1 subgoal):\n 1. Arg (- z) = (if Arg z \\ 0 then Arg z + pi else Arg z - pi)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Arg (- z) = (if Arg z \\ 0 then Arg z + pi else Arg z - pi)\n[PROOF STEP]\napply (rule Arg_unique [of \"norm z\", OF complex_eqI])\n[PROOF STATE]\nproof (prove)\ngoal (5 subgoals):\n 1. Re (complex_of_real (cmod z) * exp (\\ * complex_of_real (if Arg z \\ 0 then Arg z + pi else Arg z - pi))) = Re (- z)\n 2. Im (complex_of_real (cmod z) * exp (\\ * complex_of_real (if Arg z \\ 0 then Arg z + pi else Arg z - pi))) = Im (- z)\n 3. 0 < cmod z\n 4. - pi < (if Arg z \\ 0 then Arg z + pi else Arg z - pi)\n 5. (if Arg z \\ 0 then Arg z + pi else Arg z - pi) \\ pi\n[PROOF STEP]\nusing mpi_less_Arg [of z] Arg_le_pi [of z] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n- pi < Arg z\nArg z \\ pi\nz \\ 0\n\ngoal (5 subgoals):\n 1. Re (complex_of_real (cmod z) * exp (\\ * complex_of_real (if Arg z \\ 0 then Arg z + pi else Arg z - pi))) = Re (- z)\n 2. Im (complex_of_real (cmod z) * exp (\\ * complex_of_real (if Arg z \\ 0 then Arg z + pi else Arg z - pi))) = Im (- z)\n 3. 0 < cmod z\n 4. - pi < (if Arg z \\ 0 then Arg z + pi else Arg z - pi)\n 5. (if Arg z \\ 0 then Arg z + pi else Arg z - pi) \\ pi\n[PROOF STEP]\nby (auto simp: Re_exp Im_exp)\n[PROOF STATE]\nproof (state)\nthis:\nArg (- z) = (if Arg z \\ 0 then Arg z + pi else Arg z - pi)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1297, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473779969193, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7231225941158308}} {"text": "[STATEMENT]\nlemma sum_single: \n assumes \"finite A\"\n assumes \"\\j. j \\ i \\ j\\A \\ f j = 0\"\n shows \"sum f A = (if i\\A then f i else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f A = (if i \\ A then f i else (0::'b))\n[PROOF STEP]\napply (subst sum.mono_neutral_cong_right[where S=\\A \\ {i}\\ and h=f])\n[PROOF STATE]\nproof (prove)\ngoal (5 subgoals):\n 1. finite A\n 2. A \\ {i} \\ A\n 3. \\i\\A - A \\ {i}. f i = (0::'b)\n 4. \\x. x \\ A \\ {i} \\ f x = f x\n 5. sum f (A \\ {i}) = (if i \\ A then f i else (0::'b))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\\?j \\ i; ?j \\ A\\ \\ f ?j = (0::'b)\n\ngoal (5 subgoals):\n 1. finite A\n 2. A \\ {i} \\ A\n 3. \\i\\A - A \\ {i}. f i = (0::'b)\n 4. \\x. x \\ A \\ {i} \\ f x = f x\n 5. sum f (A \\ {i}) = (if i \\ A then f i else (0::'b))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 492, "file": "Complex_Bounded_Operators_extra_Extra_General", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473647220787, "lm_q2_score": 0.8289388146603365, "lm_q1q2_score": 0.7231225904847882}} {"text": "[STATEMENT]\nlemma relative_polar_form_eq:\n \"Q + dist P Q *\\<^sub>R of_radiant (radiant_of (P - Q)) = P\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Q + dist P Q *\\<^sub>R of_radiant (radiant_of (P - Q)) = P\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Q + dist P Q *\\<^sub>R of_radiant (radiant_of (P - Q)) = P\n[PROOF STEP]\nhave \"norm (P - Q) *\\<^sub>R of_radiant (radiant_of (P - Q)) = P - Q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (P - Q) *\\<^sub>R of_radiant (radiant_of (P - Q)) = P - Q\n[PROOF STEP]\nunfolding polar_form_eq\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. P - Q = P - Q\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nnorm (P - Q) *\\<^sub>R of_radiant (radiant_of (P - Q)) = P - Q\n\ngoal (1 subgoal):\n 1. Q + dist P Q *\\<^sub>R of_radiant (radiant_of (P - Q)) = P\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nnorm (P - Q) *\\<^sub>R of_radiant (radiant_of (P - Q)) = P - Q\n\ngoal (1 subgoal):\n 1. Q + dist P Q *\\<^sub>R of_radiant (radiant_of (P - Q)) = P\n[PROOF STEP]\nhave \"dist P Q = norm (P - Q)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist P Q = norm (P - Q)\n[PROOF STEP]\nby (simp add: dist_norm)\n[PROOF STATE]\nproof (state)\nthis:\ndist P Q = norm (P - Q)\n\ngoal (1 subgoal):\n 1. Q + dist P Q *\\<^sub>R of_radiant (radiant_of (P - Q)) = P\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (P - Q) *\\<^sub>R of_radiant (radiant_of (P - Q)) = P - Q\ndist P Q = norm (P - Q)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (P - Q) *\\<^sub>R of_radiant (radiant_of (P - Q)) = P - Q\ndist P Q = norm (P - Q)\n\ngoal (1 subgoal):\n 1. Q + dist P Q *\\<^sub>R of_radiant (radiant_of (P - Q)) = P\n[PROOF STEP]\nby (metis add.commute diff_add_cancel)\n[PROOF STATE]\nproof (state)\nthis:\nQ + dist P Q *\\<^sub>R of_radiant (radiant_of (P - Q)) = P\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 912, "file": "Ptolemys_Theorem_Ptolemys_Theorem", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.872347368040789, "lm_q2_score": 0.8289388040954683, "lm_q1q2_score": 0.723122584019561}} {"text": "[STATEMENT]\nlemma tensor_smult0: fixes A::\"'a::ring tensor\"\nshows \"0 \\ A = tensor0 (dims A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) \\ A = tensor0 (dims A)\n[PROOF STEP]\nunfolding smult_def tensor0_def vec_smult_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tensor_from_vec (dims A) (map ((*) (0::'a)) (vec A)) = tensor_from_vec (dims A) (vec0 (prod_list (dims A)))\n[PROOF STEP]\nusing vec_smult0 length_vec\n[PROOF STATE]\nproof (prove)\nusing this:\nvec_smult (0::?'a) ?as = vec0 (length ?as)\nlength (vec ?A) = prod_list (dims ?A)\n\ngoal (1 subgoal):\n 1. tensor_from_vec (dims A) (map ((*) (0::'a)) (vec A)) = tensor_from_vec (dims A) (vec0 (prod_list (dims A)))\n[PROOF STEP]\nby (metis (no_types) vec_smult_def)", "meta": {"llama_tokens": 335, "file": "Deep_Learning_Tensor_Scalar_Mult", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789468908171, "lm_q2_score": 0.8080672112416737, "lm_q1q2_score": 0.7230500307890441}} {"text": "[STATEMENT]\nlemma norm_vec_le_norm_bfun: \n \"norm (vec_lambda (apply_bfun (x :: 'd::finite \\\\<^sub>b real))) \\ norm x * card (UNIV :: 'd set)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (vec_lambda (apply_bfun x)) \\ norm x * real CARD('d)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm (vec_lambda (apply_bfun x)) \\ norm x * real CARD('d)\n[PROOF STEP]\nhave \"norm (vec_lambda (apply_bfun x)) \\ (\\ i \\ UNIV . \\(apply_bfun x i)\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (vec_lambda (apply_bfun x)) \\ (\\i\\UNIV. \\apply_bfun x i\\)\n[PROOF STEP]\nusing L2_set_le_sum_abs\n[PROOF STATE]\nproof (prove)\nusing this:\nL2_set ?f ?A \\ (\\i\\?A. \\?f i\\)\n\ngoal (1 subgoal):\n 1. norm (vec_lambda (apply_bfun x)) \\ (\\i\\UNIV. \\apply_bfun x i\\)\n[PROOF STEP]\nunfolding norm_vec_def L2_set_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (\\i\\?A. (?f i)\\<^sup>2) \\ (\\i\\?A. \\?f i\\)\n\ngoal (1 subgoal):\n 1. sqrt (\\i\\UNIV. (norm (vec_lambda (apply_bfun x) $ i))\\<^sup>2) \\ (\\i\\UNIV. \\apply_bfun x i\\)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nnorm (vec_lambda (apply_bfun x)) \\ (\\i\\UNIV. \\apply_bfun x i\\)\n\ngoal (1 subgoal):\n 1. norm (vec_lambda (apply_bfun x)) \\ norm x * real CARD('d)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnorm (vec_lambda (apply_bfun x)) \\ (\\i\\UNIV. \\apply_bfun x i\\)\n\ngoal (1 subgoal):\n 1. norm (vec_lambda (apply_bfun x)) \\ norm x * real CARD('d)\n[PROOF STEP]\nhave \"\\ \\ (card (UNIV :: 'd set) * (\\xa. \\apply_bfun x xa\\))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\UNIV. \\apply_bfun x i\\) \\ real CARD('d) * (\\xa. \\apply_bfun x xa\\)\n[PROOF STEP]\nby (auto intro!: sum_bounded_above cSup_upper)\n[PROOF STATE]\nproof (state)\nthis:\n(\\i\\UNIV. \\apply_bfun x i\\) \\ real CARD('d) * (\\xa. \\apply_bfun x xa\\)\n\ngoal (1 subgoal):\n 1. norm (vec_lambda (apply_bfun x)) \\ norm x * real CARD('d)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nnorm (vec_lambda (apply_bfun x)) \\ real CARD('d) * (\\xa. \\apply_bfun x xa\\)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm (vec_lambda (apply_bfun x)) \\ real CARD('d) * (\\xa. \\apply_bfun x xa\\)\n\ngoal (1 subgoal):\n 1. norm (vec_lambda (apply_bfun x)) \\ norm x * real CARD('d)\n[PROOF STEP]\nby (simp add: norm_bfun_def dist_bfun_def mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (vec_lambda (apply_bfun x)) \\ norm x * real CARD('d)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1280, "file": "MDP-Rewards_Bounded_Functions", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.7230500299922517}} {"text": "[STATEMENT]\nlemma card_complementary:\n \"card (ceros_of_boolean_input v)\n + card {x. x < (dim_vec v) \\ (vec_index v x = True)} = (dim_vec v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (ceros_of_boolean_input v) + card {x. x < dim_vec v \\ v $ x = True} = dim_vec v\n[PROOF STEP]\nunfolding ceros_of_boolean_input_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {x. x < dim_vec v \\ v $ x = False} + card {x. x < dim_vec v \\ v $ x = True} = dim_vec v\n[PROOF STEP]\nusing card_UNIV_union [of v]\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {x. x < dim_vec v \\ v $ x = True} + card {x. x < dim_vec v \\ v $ x = False} = card {x. x < dim_vec v}\n\ngoal (1 subgoal):\n 1. card {x. x < dim_vec v \\ v $ x = False} + card {x. x < dim_vec v \\ v $ x = True} = dim_vec v\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 368, "file": "Simplicial_complexes_and_boolean_functions_Simplicial_complex", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240756264639, "lm_q2_score": 0.8354835350552604, "lm_q1q2_score": 0.722880469319318}} {"text": "[STATEMENT]\nlemma row_space_subset_span_basis_row_space:\nfixes A::\"'a::{field}^'cols::{mod_type}^'rows::{mod_type}\"\nshows \"row_space A \\ vec.span (basis_row_space A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row_space A \\ vec.span (basis_row_space A)\n[PROOF STEP]\nproof (rule vec.card_ge_dim_independent)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. basis_row_space A \\ row_space A\n 2. vec.independent (basis_row_space A)\n 3. vec.dim (row_space A) \\ card (basis_row_space A)\n[PROOF STEP]\nshow \"basis_row_space A \\ row_space A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. basis_row_space A \\ row_space A\n[PROOF STEP]\nby (rule basis_row_space_subset_row_space)\n[PROOF STATE]\nproof (state)\nthis:\nbasis_row_space A \\ row_space A\n\ngoal (2 subgoals):\n 1. vec.independent (basis_row_space A)\n 2. vec.dim (row_space A) \\ card (basis_row_space A)\n[PROOF STEP]\nshow \"vec.independent (basis_row_space A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.independent (basis_row_space A)\n[PROOF STEP]\nunfolding basis_row_space_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.independent {row i (Gauss_Jordan A) |i. row i (Gauss_Jordan A) \\ 0}\n[PROOF STEP]\nby (rule independent_not_zero_rows_rref[OF rref_Gauss_Jordan])\n[PROOF STATE]\nproof (state)\nthis:\nvec.independent (basis_row_space A)\n\ngoal (1 subgoal):\n 1. vec.dim (row_space A) \\ card (basis_row_space A)\n[PROOF STEP]\nshow \"vec.dim (row_space A) \\ card (basis_row_space A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (row_space A) \\ card (basis_row_space A)\n[PROOF STEP]\nunfolding basis_row_space_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.dim (row_space A) \\ card {row i (Gauss_Jordan A) |i. row i (Gauss_Jordan A) \\ 0}\n[PROOF STEP]\nusing rref_rank[OF rref_Gauss_Jordan, of A]\n[PROOF STATE]\nproof (prove)\nusing this:\nrank (Gauss_Jordan A) = card {row i (Gauss_Jordan A) |i. row i (Gauss_Jordan A) \\ 0}\n\ngoal (1 subgoal):\n 1. vec.dim (row_space A) \\ card {row i (Gauss_Jordan A) |i. row i (Gauss_Jordan A) \\ 0}\n[PROOF STEP]\nunfolding row_rank_def[symmetric] rank_def[symmetric] rank_Gauss_Jordan[symmetric]\n[PROOF STATE]\nproof (prove)\nusing this:\nrank A = card {row i (Gauss_Jordan A) |i. row i (Gauss_Jordan A) \\ 0}\n\ngoal (1 subgoal):\n 1. rank A \\ card {row i (Gauss_Jordan A) |i. row i (Gauss_Jordan A) \\ 0}\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nvec.dim (row_space A) \\ card (basis_row_space A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1160, "file": "Gauss_Jordan_Bases_Of_Fundamental_Subspaces", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.879146780175245, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.722824930623834}} {"text": "[STATEMENT]\nlemma count_vec_alt: \"count_vec v x = card { i. v $ i = x \\ i< dim_vec v}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. count_vec v x = card {i. v $ i = x \\ i < dim_vec v}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. count_vec v x = card {i. v $ i = x \\ i < dim_vec v}\n[PROOF STEP]\nhave \"count_vec v x = count (image_mset (($) v) (mset_set {.. i < dim_vec v}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncount_vec v x = count (image_mset (($) v) (mset_set {.. i < dim_vec v}\n[PROOF STEP]\nhave \"... = size {#a \\# (image_mset (($) v) (mset_set {.. i < dim_vec v}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncount (image_mset (($) v) (mset_set {.. i < dim_vec v}\n[PROOF STEP]\nhave \"... = size {#a \\# (mset_set {..# mset_set {..# mset_set {.. i < dim_vec v}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncount_vec v x = size {#a \\# mset_set {.. {..# mset_set {.. {.. {.. i < dim_vec v}\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncount_vec v x = card {a \\ {.. i < dim_vec v}\n[PROOF STEP]\nby (smt (verit) Collect_cong lessThan_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncount_vec v x = card {i. v $ i = x \\ i < dim_vec v}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1569, "file": "Fishers_Inequality_Matrix_Vector_Extras", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467643431002, "lm_q2_score": 0.822189121808099, "lm_q1q2_score": 0.7228249061156854}} {"text": "[STATEMENT]\nlemma adjoint_minus:\n fixes A B :: \"'a::conjugatable_field mat\"\n assumes \"A \\ carrier_mat n m\" \"B \\ carrier_mat n m\"\n shows \"adjoint (A - B) = adjoint A - adjoint B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. adjoint (A - B) = adjoint A - adjoint B\n[PROOF STEP]\napply (rule eq_matI)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (adjoint A - adjoint B); j < dim_col (adjoint A - adjoint B)\\ \\ adjoint (A - B) $$ (i, j) = (adjoint A - adjoint B) $$ (i, j)\n 2. dim_row (adjoint (A - B)) = dim_row (adjoint A - adjoint B)\n 3. dim_col (adjoint (A - B)) = dim_col (adjoint A - adjoint B)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\ carrier_mat n m\nB \\ carrier_mat n m\n\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row (adjoint A - adjoint B); j < dim_col (adjoint A - adjoint B)\\ \\ adjoint (A - B) $$ (i, j) = (adjoint A - adjoint B) $$ (i, j)\n 2. dim_row (adjoint (A - B)) = dim_row (adjoint A - adjoint B)\n 3. dim_col (adjoint (A - B)) = dim_col (adjoint A - adjoint B)\n[PROOF STEP]\napply(auto simp add: adjoint_eval)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i j. \\i < m; j < n; A \\ carrier_mat n m; B \\ carrier_mat n m\\ \\ conjugate (A $$ (j, i) - B $$ (j, i)) = conjugate (A $$ (j, i)) - conjugate (B $$ (j, i))\n[PROOF STEP]\nby (metis add_uminus_conv_diff conjugate_dist_add conjugate_neg)", "meta": {"llama_tokens": 648, "file": "QHLProver_Complex_Matrix", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467548438126, "lm_q2_score": 0.822189123986562, "lm_q1q2_score": 0.722824900220663}} {"text": "[STATEMENT]\nlemma cstrong_operator_topology_basis:\n fixes f::\"('a::complex_normed_vector \\\\<^sub>C\\<^sub>L'b::complex_normed_vector)\" and U::\"'i \\ 'b set\" and x::\"'i \\ 'a\"\n assumes \"finite I\" \"\\i. i \\ I \\ open (U i)\"\n shows \"openin cstrong_operator_topology {f. \\i\\I. cblinfun_apply f (x i) \\ U i}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. openin cstrong_operator_topology {f. \\i\\I. cblinfun_apply f (x i) \\ U i}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. openin cstrong_operator_topology {f. \\i\\I. cblinfun_apply f (x i) \\ U i}\n[PROOF STEP]\nhave \"open {g::('a\\'b). \\i\\I. g (x i) \\ U i}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. open {g. \\i\\I. g (x i) \\ U i}\n[PROOF STEP]\nby (rule product_topology_basis'[OF assms])\n[PROOF STATE]\nproof (state)\nthis:\nopen {g. \\i\\I. g (x i) \\ U i}\n\ngoal (1 subgoal):\n 1. openin cstrong_operator_topology {f. \\i\\I. cblinfun_apply f (x i) \\ U i}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nopen {g. \\i\\I. g (x i) \\ U i}\n\ngoal (1 subgoal):\n 1. openin cstrong_operator_topology {f. \\i\\I. cblinfun_apply f (x i) \\ U i}\n[PROOF STEP]\nhave \"{f. \\i\\I. cblinfun_apply f (x i) \\ U i}\n = cblinfun_apply-`{g::('a\\'b). \\i\\I. g (x i) \\ U i} \\ UNIV\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {f. \\i\\I. cblinfun_apply f (x i) \\ U i} = cblinfun_apply -` {g. \\i\\I. g (x i) \\ U i} \\ UNIV\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{f. \\i\\I. cblinfun_apply f (x i) \\ U i} = cblinfun_apply -` {g. \\i\\I. g (x i) \\ U i} \\ UNIV\n\ngoal (1 subgoal):\n 1. openin cstrong_operator_topology {f. \\i\\I. cblinfun_apply f (x i) \\ U i}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nopen {g. \\i\\I. g (x i) \\ U i}\n{f. \\i\\I. cblinfun_apply f (x i) \\ U i} = cblinfun_apply -` {g. \\i\\I. g (x i) \\ U i} \\ UNIV\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nopen {g. \\i\\I. g (x i) \\ U i}\n{f. \\i\\I. cblinfun_apply f (x i) \\ U i} = cblinfun_apply -` {g. \\i\\I. g (x i) \\ U i} \\ UNIV\n\ngoal (1 subgoal):\n 1. openin cstrong_operator_topology {f. \\i\\I. cblinfun_apply f (x i) \\ U i}\n[PROOF STEP]\nunfolding cstrong_operator_topology_def\n[PROOF STATE]\nproof (prove)\nusing this:\nopen {g. \\i\\I. g (x i) \\ U i}\n{f. \\i\\I. cblinfun_apply f (x i) \\ U i} = cblinfun_apply -` {g. \\i\\I. g (x i) \\ U i} \\ UNIV\n\ngoal (1 subgoal):\n 1. openin (pullback_topology UNIV cblinfun_apply euclidean) {f. \\i\\I. cblinfun_apply f (x i) \\ U i}\n[PROOF STEP]\nby (subst openin_pullback_topology) auto\n[PROOF STATE]\nproof (state)\nthis:\nopenin cstrong_operator_topology {f. \\i\\I. cblinfun_apply f (x i) \\ U i}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1459, "file": "Complex_Bounded_Operators_Complex_Bounded_Linear_Function0", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110540642804, "lm_q2_score": 0.8104789063814617, "lm_q1q2_score": 0.7227940477969166}} {"text": "[STATEMENT]\nlemma min_height_acomplete: assumes \"acomplete t\"\nshows \"min_height t = nat(floor(log 2 (size1 t)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. min_height t = nat \\log 2 (real (size1 t))\\\n[PROOF STEP]\nproof cases\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ?P \\ min_height t = nat \\log 2 (real (size1 t))\\\n 2. \\ ?P \\ min_height t = nat \\log 2 (real (size1 t))\\\n[PROOF STEP]\nassume *: \"complete t\"\n[PROOF STATE]\nproof (state)\nthis:\nTree.complete t\n\ngoal (2 subgoals):\n 1. ?P \\ min_height t = nat \\log 2 (real (size1 t))\\\n 2. \\ ?P \\ min_height t = nat \\log 2 (real (size1 t))\\\n[PROOF STEP]\nhence \"size1 t = 2 ^ min_height t\"\n[PROOF STATE]\nproof (prove)\nusing this:\nTree.complete t\n\ngoal (1 subgoal):\n 1. size1 t = 2 ^ min_height t\n[PROOF STEP]\nby (simp add: complete_iff_height size1_if_complete)\n[PROOF STATE]\nproof (state)\nthis:\nsize1 t = 2 ^ min_height t\n\ngoal (2 subgoals):\n 1. ?P \\ min_height t = nat \\log 2 (real (size1 t))\\\n 2. \\ ?P \\ min_height t = nat \\log 2 (real (size1 t))\\\n[PROOF STEP]\nfrom log2_of_power_eq[OF this]\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (min_height t) = log 2 (real (size1 t))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (min_height t) = log 2 (real (size1 t))\n\ngoal (1 subgoal):\n 1. min_height t = nat \\log 2 (real (size1 t))\\\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\nmin_height t = nat \\log 2 (real (size1 t))\\\n\ngoal (1 subgoal):\n 1. \\ Tree.complete t \\ min_height t = nat \\log 2 (real (size1 t))\\\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ Tree.complete t \\ min_height t = nat \\log 2 (real (size1 t))\\\n[PROOF STEP]\nassume *: \"\\ complete t\"\n[PROOF STATE]\nproof (state)\nthis:\n\\ Tree.complete t\n\ngoal (1 subgoal):\n 1. \\ Tree.complete t \\ min_height t = nat \\log 2 (real (size1 t))\\\n[PROOF STEP]\nhence \"height t = min_height t + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ Tree.complete t\n\ngoal (1 subgoal):\n 1. height t = min_height t + 1\n[PROOF STEP]\nusing assms min_height_le_height[of t]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ Tree.complete t\nacomplete t\nmin_height t \\ height t\n\ngoal (1 subgoal):\n 1. height t = min_height t + 1\n[PROOF STEP]\nby(auto simp: acomplete_def complete_iff_height)\n[PROOF STATE]\nproof (state)\nthis:\nheight t = min_height t + 1\n\ngoal (1 subgoal):\n 1. \\ Tree.complete t \\ min_height t = nat \\log 2 (real (size1 t))\\\n[PROOF STEP]\nhence \"size1 t < 2 ^ (min_height t + 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nheight t = min_height t + 1\n\ngoal (1 subgoal):\n 1. size1 t < 2 ^ (min_height t + 1)\n[PROOF STEP]\nby (metis * size1_height_if_incomplete)\n[PROOF STATE]\nproof (state)\nthis:\nsize1 t < 2 ^ (min_height t + 1)\n\ngoal (1 subgoal):\n 1. \\ Tree.complete t \\ min_height t = nat \\log 2 (real (size1 t))\\\n[PROOF STEP]\nfrom floor_log_nat_eq_if[OF min_height_size1 this]\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ 2 \\ \\log (real 2) (real (size1 t))\\ = int (min_height t)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ 2 \\ \\log (real 2) (real (size1 t))\\ = int (min_height t)\n\ngoal (1 subgoal):\n 1. min_height t = nat \\log 2 (real (size1 t))\\\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmin_height t = nat \\log 2 (real (size1 t))\\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1596, "file": null, "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110511888303, "lm_q2_score": 0.8104788995148791, "lm_q1q2_score": 0.7227940393427307}} {"text": "[STATEMENT]\nlemma norm_matrix_le_op_norm: \"\\x\\ = 1 \\ \\A *v x\\ \\ \\A\\\\<^sub>o\\<^sub>p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\ = 1 \\ \\A *v x\\ \\ \\A\\\\<^sub>o\\<^sub>p\n[PROOF STEP]\napply(unfold onorm_def, rule cSup_upper[OF _ onorm_set_proptys(2)])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\ = 1 \\ \\A *v x\\ \\ range (\\x. \\A *v x\\ / \\x\\)\n[PROOF STEP]\nunfolding image_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\ = 1 \\ \\A *v x\\ \\ {y. \\x\\UNIV. y = \\A *v x\\ / \\x\\}\n[PROOF STEP]\nby (clarsimp, rule_tac x=x in exI) simp", "meta": {"llama_tokens": 359, "file": "Matrices_for_ODEs_MTX_Norms", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392817460332, "lm_q2_score": 0.8175744673038222, "lm_q1q2_score": 0.7227679448491667}} {"text": "[STATEMENT]\nlemma minus_image_eq_vimage:\n fixes A :: \"'a::ab_group_add set\"\n shows \"(\\x. - x) ` A = (\\x. - x) -` A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. uminus ` A = uminus -` A\n[PROOF STEP]\nby (auto intro!: image_eqI [where f=\"\\x. - x\"])", "meta": {"llama_tokens": 123, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392756357327, "lm_q2_score": 0.817574471748733, "lm_q1q2_score": 0.7227679437830167}} {"text": "[STATEMENT]\nlemma sum_norm_allsubsets_bound_cart:\n fixes f:: \"'a \\ real ^'n\"\n assumes fP: \"finite P\" and fPs: \"\\Q. Q \\ P \\ norm (sum f Q) \\ e\"\n shows \"sum (\\x. norm (f x)) P \\ 2 * real CARD('n) * e\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x\\P. norm (f x)) \\ 2 * real CARD('n) * e\n[PROOF STEP]\nusing sum_norm_allsubsets_bound[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\Q. Q \\ P \\ Q \\ P) \\ (\\x\\P. norm (f x)) \\ 2 * real DIM((real, 'n) vec) * e\n\ngoal (1 subgoal):\n 1. (\\x\\P. norm (f x)) \\ 2 * real CARD('n) * e\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 305, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111797051879431, "lm_q2_score": 0.7931059585194573, "lm_q1q2_score": 0.7226620534665602}} {"text": "[STATEMENT]\nlemma matrix_sum_add_distrib:\n shows \"(\\k. k < n \\ f k \\ carrier_mat d d) \\ (\\k. k < n \\ g k \\ carrier_mat d d)\n \\ matrix_sum d (\\k. (f k) + (g k)) n = matrix_sum d f n + matrix_sum d g n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\\\k. k < 0 \\ f k \\ carrier_mat d d; \\k. k < 0 \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) 0 = matrix_sum d f 0 + matrix_sum d g 0\n 2. \\n. \\\\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n; \\k. k < Suc n \\ f k \\ carrier_mat d d; \\k. k < Suc n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) (Suc n) = matrix_sum d f (Suc n) + matrix_sum d g (Suc n)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n?k < 0 \\ f ?k \\ carrier_mat d d\n?k < 0 \\ g ?k \\ carrier_mat d d\n\ngoal (2 subgoals):\n 1. \\\\k. k < 0 \\ f k \\ carrier_mat d d; \\k. k < 0 \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) 0 = matrix_sum d f 0 + matrix_sum d g 0\n 2. \\n. \\\\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n; \\k. k < Suc n \\ f k \\ carrier_mat d d; \\k. k < Suc n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) (Suc n) = matrix_sum d f (Suc n) + matrix_sum d g (Suc n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n?k < 0 \\ f ?k \\ carrier_mat d d\n?k < 0 \\ g ?k \\ carrier_mat d d\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n?k < 0 \\ f ?k \\ carrier_mat d d\n?k < 0 \\ g ?k \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. matrix_sum d (\\k. f k + g k) 0 = matrix_sum d f 0 + matrix_sum d g 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_sum d (\\k. f k + g k) 0 = matrix_sum d f 0 + matrix_sum d g 0\n\ngoal (1 subgoal):\n 1. \\n. \\\\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n; \\k. k < Suc n \\ f k \\ carrier_mat d d; \\k. k < Suc n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) (Suc n) = matrix_sum d f (Suc n) + matrix_sum d g (Suc n)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. \\\\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n; \\k. k < Suc n \\ f k \\ carrier_mat d d; \\k. k < Suc n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) (Suc n) = matrix_sum d f (Suc n) + matrix_sum d g (Suc n)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n\n?k < Suc n \\ f ?k \\ carrier_mat d d\n?k < Suc n \\ g ?k \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. \\n. \\\\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n; \\k. k < Suc n \\ f k \\ carrier_mat d d; \\k. k < Suc n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) (Suc n) = matrix_sum d f (Suc n) + matrix_sum d g (Suc n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n\n?k < Suc n \\ f ?k \\ carrier_mat d d\n?k < Suc n \\ g ?k \\ carrier_mat d d\n[PROOF STEP]\nhave dfn: \"f n \\ carrier_mat d d\" and dgn: \"g n \\ carrier_mat d d\"\n and dfk: \"k < n \\ f k \\ carrier_mat d d\" and dgk: \"k < n \\ g k \\ carrier_mat d d\"\n and eq: \"matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n\" for k\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n\n?k < Suc n \\ f ?k \\ carrier_mat d d\n?k < Suc n \\ g ?k \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. (f n \\ carrier_mat d d &&& g n \\ carrier_mat d d) &&& (k < n \\ f k \\ carrier_mat d d) &&& (k < n \\ g k \\ carrier_mat d d) &&& matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nf n \\ carrier_mat d d\ng n \\ carrier_mat d d\n?k < n \\ f ?k \\ carrier_mat d d\n?k < n \\ g ?k \\ carrier_mat d d\nmatrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n\n\ngoal (1 subgoal):\n 1. \\n. \\\\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n; \\k. k < Suc n \\ f k \\ carrier_mat d d; \\k. k < Suc n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) (Suc n) = matrix_sum d f (Suc n) + matrix_sum d g (Suc n)\n[PROOF STEP]\nhave dsf: \"matrix_sum d f n \\ carrier_mat d d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_sum d f n \\ carrier_mat d d\n[PROOF STEP]\nusing matrix_sum_dim dfk\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. k < ?n \\ ?f k \\ carrier_mat ?d ?d) \\ matrix_sum ?d ?f ?n \\ carrier_mat ?d ?d\n?k < n \\ f ?k \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. matrix_sum d f n \\ carrier_mat d d\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_sum d f n \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. \\n. \\\\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n; \\k. k < Suc n \\ f k \\ carrier_mat d d; \\k. k < Suc n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) (Suc n) = matrix_sum d f (Suc n) + matrix_sum d g (Suc n)\n[PROOF STEP]\nhave dsg: \"matrix_sum d g n \\ carrier_mat d d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_sum d g n \\ carrier_mat d d\n[PROOF STEP]\nusing matrix_sum_dim dgk\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\k. k < ?n \\ ?f k \\ carrier_mat ?d ?d) \\ matrix_sum ?d ?f ?n \\ carrier_mat ?d ?d\n?k < n \\ g ?k \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. matrix_sum d g n \\ carrier_mat d d\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_sum d g n \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. \\n. \\\\\\k. k < n \\ f k \\ carrier_mat d d; \\k. k < n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) n = matrix_sum d f n + matrix_sum d g n; \\k. k < Suc n \\ f k \\ carrier_mat d d; \\k. k < Suc n \\ g k \\ carrier_mat d d\\ \\ matrix_sum d (\\k. f k + g k) (Suc n) = matrix_sum d f (Suc n) + matrix_sum d g (Suc n)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_sum d (\\k. f k + g k) (Suc n) = matrix_sum d f (Suc n) + matrix_sum d g (Suc n)\n[PROOF STEP]\nunfolding matrix_sum.simps eq\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f n + g n + (matrix_sum d f n + matrix_sum d g n) = f n + matrix_sum d f n + (g n + matrix_sum d g n)\n[PROOF STEP]\nusing dfn dgn dsf dsg\n[PROOF STATE]\nproof (prove)\nusing this:\nf n \\ carrier_mat d d\ng n \\ carrier_mat d d\nmatrix_sum d f n \\ carrier_mat d d\nmatrix_sum d g n \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. f n + g n + (matrix_sum d f n + matrix_sum d g n) = f n + matrix_sum d f n + (g n + matrix_sum d g n)\n[PROOF STEP]\nby (mat_assoc d)\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_sum d (\\k. f k + g k) (Suc n) = matrix_sum d f (Suc n) + matrix_sum d g (Suc n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4166, "file": "QHLProver_Matrix_Limit", "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772482857833, "lm_q2_score": 0.8267117940706735, "lm_q1q2_score": 0.7226099700866975}} {"text": "[STATEMENT]\nlemma l7: \"fact n = \\ {0 <.. int n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fact n = \\{0<..int n}\n[PROOF STEP]\nproof (induction n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. fact 0 = \\{0<..int 0}\n 2. \\n. fact n = \\{0<..int n} \\ fact (Suc n) = \\{0<..int (Suc n)}\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nfact n = \\{0<..int n}\n\ngoal (2 subgoals):\n 1. fact 0 = \\{0<..int 0}\n 2. \\n. fact n = \\{0<..int n} \\ fact (Suc n) = \\{0<..int (Suc n)}\n[PROOF STEP]\nhave \"int (Suc n) = int n + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. int (Suc n) = int n + 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nint (Suc n) = int n + 1\n\ngoal (2 subgoals):\n 1. fact 0 = \\{0<..int 0}\n 2. \\n. fact n = \\{0<..int n} \\ fact (Suc n) = \\{0<..int (Suc n)}\n[PROOF STEP]\nhence \"insert (int (Suc n)) {0<..int n} = {0<..int (Suc n)}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nint (Suc n) = int n + 1\n\ngoal (1 subgoal):\n 1. insert (int (Suc n)) {0<..int n} = {0<..int (Suc n)}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninsert (int (Suc n)) {0<..int n} = {0<..int (Suc n)}\n\ngoal (2 subgoals):\n 1. fact 0 = \\{0<..int 0}\n 2. \\n. fact n = \\{0<..int n} \\ fact (Suc n) = \\{0<..int (Suc n)}\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ninsert (int (Suc n)) {0<..int n} = {0<..int (Suc n)}\n\ngoal (1 subgoal):\n 1. fact (Suc n) = \\{0<..int (Suc n)}\n[PROOF STEP]\nusing prod.insert[of \"{0<..int n}\" \"int (Suc n)\" \"\\x. x\"] Suc fact_Suc\n[PROOF STATE]\nproof (prove)\nusing this:\ninsert (int (Suc n)) {0<..int n} = {0<..int (Suc n)}\n\\finite {0<..int n}; int (Suc n) \\ {0<..int n}\\ \\ \\(insert (int (Suc n)) {0<..int n}) = int (Suc n) * \\{0<..int n}\nfact n = \\{0<..int n}\nfact (Suc ?n) = of_nat (Suc ?n) * fact ?n\n\ngoal (1 subgoal):\n 1. fact (Suc n) = \\{0<..int (Suc n)}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfact (Suc n) = \\{0<..int (Suc n)}\n\ngoal (1 subgoal):\n 1. fact 0 = \\{0<..int 0}\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 1086, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772351648677, "lm_q2_score": 0.8267118004748677, "lm_q1q2_score": 0.7226099648372422}} {"text": "[STATEMENT]\nlemma tendsto_ln_powr_over_ln_powr:\n assumes \"(a::real) > 0\" \"c > 0\"\n shows \"((\\x. ln (a*x) powr d / ln (c*x) powr d) \\ 1) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. ln (a * x) powr d / ln (c * x) powr d) \\ 1) at_top\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\x. ln (a * x) powr d / ln (c * x) powr d) \\ 1) at_top\n[PROOF STEP]\nhave \"eventually (\\x. ln (a*x) powr d / ln (c*x) powr d = (ln (a*x) / ln (c*x)) powr d) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. ln (a * x) powr d / ln (c * x) powr d = (ln (a * x) / ln (c * x)) powr d\n[PROOF STEP]\nusing assms eventually_gt_at_top[of \"max (inverse a) (inverse c)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n0 < c\neventually ((<) (max (inverse a) (inverse c))) at_top\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. ln (a * x) powr d / ln (c * x) powr d = (ln (a * x) / ln (c * x)) powr d\n[PROOF STEP]\nby (auto elim!: eventually_mono simp: powr_divide field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in at_top. ln (a * x) powr d / ln (c * x) powr d = (ln (a * x) / ln (c * x)) powr d\n\ngoal (1 subgoal):\n 1. ((\\x. ln (a * x) powr d / ln (c * x) powr d) \\ 1) at_top\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in at_top. ln (a * x) powr d / ln (c * x) powr d = (ln (a * x) / ln (c * x)) powr d\n\ngoal (1 subgoal):\n 1. ((\\x. ln (a * x) powr d / ln (c * x) powr d) \\ 1) at_top\n[PROOF STEP]\nhave \"((\\x. (ln (a*x) / ln (c*x)) powr d) \\ 1) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. (ln (a * x) / ln (c * x)) powr d) \\ 1) at_top\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n0 < c\n\ngoal (1 subgoal):\n 1. ((\\x. (ln (a * x) / ln (c * x)) powr d) \\ 1) at_top\n[PROOF STEP]\nby (intro tendsto_eq_rhs[OF tendsto_powr[OF tendsto_ln_over_ln tendsto_const]]) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. (ln (a * x) / ln (c * x)) powr d) \\ 1) at_top\n\ngoal (1 subgoal):\n 1. ((\\x. ln (a * x) powr d / ln (c * x) powr d) \\ 1) at_top\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\\\<^sub>F x in at_top. ln (a * x) powr d / ln (c * x) powr d = (ln (a * x) / ln (c * x)) powr d\n((\\x. (ln (a * x) / ln (c * x)) powr d) \\ 1) at_top\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in at_top. ln (a * x) powr d / ln (c * x) powr d = (ln (a * x) / ln (c * x)) powr d\n((\\x. (ln (a * x) / ln (c * x)) powr d) \\ 1) at_top\n\ngoal (1 subgoal):\n 1. ((\\x. ln (a * x) powr d / ln (c * x) powr d) \\ 1) at_top\n[PROOF STEP]\nby (subst tendsto_cong)\n[PROOF STATE]\nproof (state)\nthis:\n((\\x. ln (a * x) powr d / ln (c * x) powr d) \\ 1) at_top\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1502, "file": "Landau_Symbols_Landau_Real_Products", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772351648677, "lm_q2_score": 0.8267117940706734, "lm_q1q2_score": 0.7226099592394817}} {"text": "[STATEMENT]\nlemma scalar_prod_double_sum_fn_vec:\n fixes c :: \"nat \\ ('a :: {comm_semiring_0})\"\n fixes f :: \"nat \\ 'a vec\"\n assumes \"\\ j . j < k \\ dim_vec (f j) = n\"\n shows \"(vec n (\\i. \\j = 0.. (vec n (\\i. \\j = 0.. j1 \\ {0.. (f j1))) + \n (\\ j1 \\ {0.. j2 \\ ({0..< k} - {j1}) . c j1 * c j2 * ((f j1) \\ (f j2))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec n (\\i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. vec n (\\i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nhave sum_simp: \"\\ j1 j2. (\\l \\ {0..l \\ {0..j1 j2. (\\l = 0..l = 0..j1 j2. (\\l = 0..l = 0..j1 j2. (\\l = 0..l = 0..l \\ {0..l \\ {0..l = 0..l = 0..?A = ?B; \\x. x \\ ?B \\ ?g x = ?h x\\ \\ sum ?g ?A = sum ?h ?B\n\ngoal (1 subgoal):\n 1. (\\l = 0..l = 0..l = 0..l = 0..j1 j2. (\\l = 0..l = 0..l = 0..l = 0..j1 j2. (\\l = 0..l = 0..l = 0..l = 0..l \\ {0..l \\ {0..l = 0..l = 0..l = 0..l = 0.. l. (f j1) $ l * (f j2) $ l\" \"{0..l = 0..l = 0..l = 0..n = 0..l = 0..l = 0..l = 0..l = 0..l = 0..l = 0..i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nhave \"(vec n (\\i. \\j = 0.. (vec n (\\i. \\j = 0.. l = 0..j1 = 0..j2 = 0..i. \\j = 0.. vec n (\\i. \\j = 0..l = 0..j1 = 0..j2 = 0..i = 0..i. \\j = 0..i. \\j = 0..i. \\j = 0..l = 0..j1 = 0..j2 = 0..i. \\j = 0.. vec n (\\i. \\j = 0..l = 0..j1 = 0..j2 = 0..i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nvec n (\\i. \\j = 0.. vec n (\\i. \\j = 0..l = 0..j1 = 0..j2 = 0..i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nhave \"... = (\\ l \\ {0.. j1 \\ {0.. j2 \\ {0..< k}. c j1 * (f j1) $ l * (c j2 * (f j2) $ l))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\l = 0..j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0..i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\l = 0..j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0..i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nhave \"... = (\\ j1 \\ {0.. j2 \\ {0..l \\ {0..l = 0..j1 = 0..j2 = 0..j1 = 0..j2 = 0..l = 0.. l j1 j2 .(c j1 * (f j1) $ l * (c j2 * (f j2) $ l))\" \"{0..l = 0..i = 0..j = 0..i = 0..j = 0..l = 0..l = 0..j1 = 0..j2 = 0..j1 = 0..j2 = 0..l = 0..l = 0..j1 = 0..j2 = 0..j1 = 0..j2 = 0..l = 0..i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\l = 0..j1 = 0..j2 = 0..j1 = 0..j2 = 0..l = 0..i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nhave \"... = (\\ j1 \\ {0.. j2 \\ {0..l \\ {0..j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0..l = 0..l = 0..l = 0..j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0..l = 0..i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0..l = 0..i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nhave \"... = (\\ j1 \\ {0.. j2 \\ {0.. (f j2))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0.. f j2))\n[PROOF STEP]\nunfolding scalar_prod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0..i = 0.. dim_vec (f ?j) = n\n\ngoal (1 subgoal):\n 1. (\\j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0..i = 0..j1 = 0..j2 = 0..l = 0..j1 = 0..j2 = 0.. f j2))\n\ngoal (1 subgoal):\n 1. vec n (\\i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nvec n (\\i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0..j2 = 0.. f j2))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nvec n (\\i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0..j2 = 0.. f j2))\n\ngoal (1 subgoal):\n 1. vec n (\\i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nusing double_sum_split_case\n[PROOF STATE]\nproof (prove)\nusing this:\nvec n (\\i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0..j2 = 0.. f j2))\nfinite ?A \\ (\\i\\?A. sum (?f i) ?A) = (\\i\\?A. ?f i i) + (\\i\\?A. sum (?f i) (?A - {i}))\n\ngoal (1 subgoal):\n 1. vec n (\\i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nvec n (\\i. \\j = 0.. vec n (\\i. \\j = 0..j1 = 0.. f j1)) + (\\j1 = 0..j2\\{0.. f j2))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 8493, "file": "Fishers_Inequality_Matrix_Vector_Extras", "length": 37, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772384450967, "lm_q2_score": 0.8267117898012105, "lm_q1q2_score": 0.7226099582194453}} {"text": "[STATEMENT]\nlemma dagger_of_dagger_is_id:\n fixes M :: \"complex Matrix.mat\"\n shows \"(M\\<^sup>\\)\\<^sup>\\ = M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M\\<^sup>\\\\<^sup>\\ = M\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\i j. \\i < dim_row M; j < dim_col M\\ \\ M\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)\n 2. dim_row M\\<^sup>\\\\<^sup>\\ = dim_row M\n 3. dim_col M\\<^sup>\\\\<^sup>\\ = dim_col M\n[PROOF STEP]\nshow \"dim_row ((M\\<^sup>\\)\\<^sup>\\) = dim_row M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_row M\\<^sup>\\\\<^sup>\\ = dim_row M\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_row M\\<^sup>\\\\<^sup>\\ = dim_row M\n\ngoal (2 subgoals):\n 1. \\i j. \\i < dim_row M; j < dim_col M\\ \\ M\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)\n 2. dim_col M\\<^sup>\\\\<^sup>\\ = dim_col M\n[PROOF STEP]\nshow \"dim_col ((M\\<^sup>\\)\\<^sup>\\) = dim_col M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_col M\\<^sup>\\\\<^sup>\\ = dim_col M\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_col M\\<^sup>\\\\<^sup>\\ = dim_col M\n\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_row M; j < dim_col M\\ \\ M\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)\n[PROOF STEP]\nfix i j\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_row M; j < dim_col M\\ \\ M\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)\n[PROOF STEP]\nassume a0:\"i < dim_row M\" and a1:\"j < dim_col M\"\n[PROOF STATE]\nproof (state)\nthis:\ni < dim_row M\nj < dim_col M\n\ngoal (1 subgoal):\n 1. \\i j. \\i < dim_row M; j < dim_col M\\ \\ M\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni < dim_row M\nj < dim_col M\n[PROOF STEP]\nshow \"(M\\<^sup>\\)\\<^sup>\\ $$ (i,j) = M $$ (i,j)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_row M\nj < dim_col M\n\ngoal (1 subgoal):\n 1. M\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\i < dim_row M; j < dim_col M\\ \\ M\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)\n[PROOF STEP]\nusing dagger_def a0 a1\n[PROOF STATE]\nproof (prove)\nusing this:\n?M\\<^sup>\\ \\ mat (dim_col ?M) (dim_row ?M) (\\(i, j). cnj (?M $$ (j, i)))\ni < dim_row M\nj < dim_col M\n\ngoal (1 subgoal):\n 1. M\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nM\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nM\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1391, "file": "Isabelle_Marries_Dirac_Quantum", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8670357563664174, "lm_q2_score": 0.8333245891029456, "lm_q1q2_score": 0.7225222154116064}} {"text": "[STATEMENT]\nlemma mset_subset_eq_mono_add_left_cancel: \"C + (A::'a multiset) \\# C + B \\ A \\# B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (C + A \\# C + B) = (A \\# B)\n[PROOF STEP]\nby (fact subset_mset.add_le_cancel_left)", "meta": {"llama_tokens": 123, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.8128673201042493, "lm_q1q2_score": 0.7224429786852994}} {"text": "[STATEMENT]\nlemma assoc_mult_vec_mat:\n assumes \"v \\ carrier_vec n1\" and \"A \\ carrier_mat n1 n2\" and \"B \\ carrier_mat n2 n3\"\n shows \"v \\<^sub>v* (A * B) = (v \\<^sub>v* A) \\<^sub>v* B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v \\<^sub>v* (A * B) = v \\<^sub>v* A \\<^sub>v* B\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\ carrier_vec n1\nA \\ carrier_mat n1 n2\nB \\ carrier_mat n2 n3\n\ngoal (1 subgoal):\n 1. v \\<^sub>v* (A * B) = v \\<^sub>v* A \\<^sub>v* B\n[PROOF STEP]\nby (intro eq_vecI, auto simp add: mult_vec_mat_def mult_mat_vec_def assoc_scalar_prod)", "meta": {"llama_tokens": 296, "file": "Groebner_Bases_Macaulay_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.888758793492457, "lm_q2_score": 0.8128673178375734, "lm_q1q2_score": 0.7224429766707713}} {"text": "[STATEMENT]\nlemma linepath_le_1:\n fixes a::\"'a::linordered_idom\" shows \"\\a \\ 1; b \\ 1; 0 \\ u; u \\ 1\\ \\ (1 - u) * a + u * b \\ 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\a \\ (1::'a); b \\ (1::'a); (0::'a) \\ u; u \\ (1::'a)\\ \\ ((1::'a) - u) * a + u * b \\ (1::'a)\n[PROOF STEP]\nusing mult_left_le [of a \"1-u\"] mult_left_le [of b u]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\a \\ (1::'a); (0::'a) \\ (1::'a) - u\\ \\ ((1::'a) - u) * a \\ (1::'a) - u\n\\b \\ (1::'a); (0::'a) \\ u\\ \\ u * b \\ u\n\ngoal (1 subgoal):\n 1. \\a \\ (1::'a); b \\ (1::'a); (0::'a) \\ u; u \\ (1::'a)\\ \\ ((1::'a) - u) * a + u * b \\ (1::'a)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 443, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.8128673110375458, "lm_q1q2_score": 0.722442970627187}} {"text": "[STATEMENT]\nlemma ivl_integral_norm_bound_integral:\n fixes f :: \"real \\ 'a::banach\"\n assumes \"f integrable_on (closed_segment a b)\"\n and \"g integrable_on (closed_segment a b)\"\n and \"\\x. x \\ closed_segment a b \\ norm (f x) \\ g x\"\n shows \"norm (ivl_integral a b f) \\ integral (closed_segment a b) g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (ivl_integral a b f) \\ integral (closed_segment a b) g\n[PROOF STEP]\nusing integral_norm_bound_integral[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\x. x \\ closed_segment a b \\ x \\ closed_segment a b) \\ norm (integral (closed_segment a b) f) \\ integral (closed_segment a b) g\n\ngoal (1 subgoal):\n 1. norm (ivl_integral a b f) \\ integral (closed_segment a b) g\n[PROOF STEP]\nby (auto simp: ivl_integral_def closed_segment_eq_real_ivl split: if_split_asm)", "meta": {"llama_tokens": 344, "file": "Ordinary_Differential_Equations_Library_Interval_Integral_HK", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9230391727723468, "lm_q2_score": 0.7826624738835052, "lm_q1q2_score": 0.7224281224533892}} {"text": "[STATEMENT]\nlemma dagger_of_prod:\n fixes M N::\"complex Matrix.mat\"\n assumes \"dim_col M = dim_row N\"\n shows \"(M * N)\\<^sup>\\ = N\\<^sup>\\ * (M\\<^sup>\\)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ = N\\<^sup>\\ * M\\<^sup>\\\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ = N\\<^sup>\\ * M\\<^sup>\\\n[PROOF STEP]\nhave \"(M * N)\\<^sup>\\ = ((M * N)\\<^sup>\\)\\<^sup>t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ = M * N\\<^sup>\\\\<^sup>t\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nM * N\\<^sup>\\ = M * N\\<^sup>\\\\<^sup>t\n\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ = N\\<^sup>\\ * M\\<^sup>\\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nM * N\\<^sup>\\ = M * N\\<^sup>\\\\<^sup>t\n\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ = N\\<^sup>\\ * M\\<^sup>\\\n[PROOF STEP]\nhave \"... = ((M\\<^sup>\\) * (N\\<^sup>\\))\\<^sup>t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\\\<^sup>t = M\\<^sup>\\ * N\\<^sup>\\\\<^sup>t\n[PROOF STEP]\nusing assms cpx_mat_cnj_prod\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_col M = dim_row N\ndim_col ?M = dim_row ?N \\ ?M * ?N\\<^sup>\\ = ?M\\<^sup>\\ * ?N\\<^sup>\\\n\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\\\<^sup>t = M\\<^sup>\\ * N\\<^sup>\\\\<^sup>t\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nM * N\\<^sup>\\\\<^sup>t = M\\<^sup>\\ * N\\<^sup>\\\\<^sup>t\n\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ = N\\<^sup>\\ * M\\<^sup>\\\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nM * N\\<^sup>\\\\<^sup>t = M\\<^sup>\\ * N\\<^sup>\\\\<^sup>t\n\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ = N\\<^sup>\\ * M\\<^sup>\\\n[PROOF STEP]\nhave \"... = (N\\<^sup>\\)\\<^sup>t * ((M\\<^sup>\\)\\<^sup>t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M\\<^sup>\\ * N\\<^sup>\\\\<^sup>t = N\\<^sup>\\\\<^sup>t * M\\<^sup>\\\\<^sup>t\n[PROOF STEP]\nusing assms transpose_of_prod\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_col M = dim_row N\ndim_col ?M = dim_row ?N \\ ?M * ?N\\<^sup>t = ?N\\<^sup>t * ?M\\<^sup>t\n\ngoal (1 subgoal):\n 1. M\\<^sup>\\ * N\\<^sup>\\\\<^sup>t = N\\<^sup>\\\\<^sup>t * M\\<^sup>\\\\<^sup>t\n[PROOF STEP]\nby (metis cnj_transpose_is_dagger dim_col_of_dagger dim_row_of_dagger index_transpose_mat(2) index_transpose_mat(3))\n[PROOF STATE]\nproof (state)\nthis:\nM\\<^sup>\\ * N\\<^sup>\\\\<^sup>t = N\\<^sup>\\\\<^sup>t * M\\<^sup>\\\\<^sup>t\n\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ = N\\<^sup>\\ * M\\<^sup>\\\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nM * N\\<^sup>\\ = N\\<^sup>\\\\<^sup>t * M\\<^sup>\\\\<^sup>t\n[PROOF STEP]\nshow \"(M * N)\\<^sup>\\ = N\\<^sup>\\ * (M\\<^sup>\\)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nM * N\\<^sup>\\ = N\\<^sup>\\\\<^sup>t * M\\<^sup>\\\\<^sup>t\n\ngoal (1 subgoal):\n 1. M * N\\<^sup>\\ = N\\<^sup>\\ * M\\<^sup>\\\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nM * N\\<^sup>\\ = N\\<^sup>\\ * M\\<^sup>\\\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1423, "file": "Isabelle_Marries_Dirac_Quantum", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127529517043, "lm_q2_score": 0.8459424353665382, "lm_q1q2_score": 0.7223610338225098}} {"text": "[STATEMENT]\ntheorem (in group) group_one_equality:\n assumes eq: \"e * x = x\"\n shows \"1 = e\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\nhave \"1 = x * inverse x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1::'a) = x * inverse x\n[PROOF STEP]\nby (simp only: group_right_inverse)\n[PROOF STATE]\nproof (state)\nthis:\n(1::'a) = x * inverse x\n\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(1::'a) = x * inverse x\n\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\nhave \"\\ = (e * x) * inverse x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * inverse x = e * x * inverse x\n[PROOF STEP]\nby (simp only: eq)\n[PROOF STATE]\nproof (state)\nthis:\nx * inverse x = e * x * inverse x\n\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx * inverse x = e * x * inverse x\n\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\nhave \"\\ = e * (x * inverse x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. e * x * inverse x = e * (x * inverse x)\n[PROOF STEP]\nby (simp only: group_assoc)\n[PROOF STATE]\nproof (state)\nthis:\ne * x * inverse x = e * (x * inverse x)\n\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ne * x * inverse x = e * (x * inverse x)\n\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\nhave \"\\ = e * 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. e * (x * inverse x) = e * (1::'a)\n[PROOF STEP]\nby (simp only: group_right_inverse)\n[PROOF STATE]\nproof (state)\nthis:\ne * (x * inverse x) = e * (1::'a)\n\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ne * (x * inverse x) = e * (1::'a)\n\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\nhave \"\\ = e\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. e * (1::'a) = e\n[PROOF STEP]\nby (simp only: group_right_one)\n[PROOF STATE]\nproof (state)\nthis:\ne * (1::'a) = e\n\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(1::'a) = e\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(1::'a) = e\n\ngoal (1 subgoal):\n 1. (1::'a) = e\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(1::'a) = e\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1120, "file": null, "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511469672594, "lm_q2_score": 0.8438951104066293, "lm_q1q2_score": 0.7222485981615757}} {"text": "[STATEMENT]\nlemma NSDERIV_mult:\n assumes \"NSDERIV g x :> Db\" \"NSDERIV f x :> Da\"\n shows \"NSDERIV (\\x. f x * g x) x :> (Da * g x) + (Db * f x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. NSDERIV (\\x. f x * g x) x :> Da * g x + Db * f x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. NSDERIV (\\x. f x * g x) x :> Da * g x + Db * f x\n[PROOF STEP]\nhave \"(f has_field_derivative Da) (at x)\" \"(g has_field_derivative Db) (at x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f has_field_derivative Da) (at x) &&& (g has_field_derivative Db) (at x)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nNSDERIV g x :> Db\nNSDERIV f x :> Da\n\ngoal (1 subgoal):\n 1. (f has_field_derivative Da) (at x) &&& (g has_field_derivative Db) (at x)\n[PROOF STEP]\nby (simp_all add: DERIV_NS_iff NSDERIV_NSLIM_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(f has_field_derivative Da) (at x)\n(g has_field_derivative Db) (at x)\n\ngoal (1 subgoal):\n 1. NSDERIV (\\x. f x * g x) x :> Da * g x + Db * f x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(f has_field_derivative Da) (at x)\n(g has_field_derivative Db) (at x)\n[PROOF STEP]\nhave \"((\\a. f a * g a) has_field_derivative Da * g x + Db * f x) (at x)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_field_derivative Da) (at x)\n(g has_field_derivative Db) (at x)\n\ngoal (1 subgoal):\n 1. ((\\a. f a * g a) has_field_derivative Da * g x + Db * f x) (at x)\n[PROOF STEP]\nusing DERIV_mult\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_field_derivative Da) (at x)\n(g has_field_derivative Db) (at x)\n\\(?f has_field_derivative ?Da) (at ?x within ?s); (?g has_field_derivative ?Db) (at ?x within ?s)\\ \\ ((\\x. ?f x * ?g x) has_field_derivative ?Da * ?g ?x + ?Db * ?f ?x) (at ?x within ?s)\n\ngoal (1 subgoal):\n 1. ((\\a. f a * g a) has_field_derivative Da * g x + Db * f x) (at x)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n((\\a. f a * g a) has_field_derivative Da * g x + Db * f x) (at x)\n\ngoal (1 subgoal):\n 1. NSDERIV (\\x. f x * g x) x :> Da * g x + Db * f x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n((\\a. f a * g a) has_field_derivative Da * g x + Db * f x) (at x)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\a. f a * g a) has_field_derivative Da * g x + Db * f x) (at x)\n\ngoal (1 subgoal):\n 1. NSDERIV (\\x. f x * g x) x :> Da * g x + Db * f x\n[PROOF STEP]\nby (simp add: DERIV_NS_iff NSDERIV_NSLIM_iff)\n[PROOF STATE]\nproof (state)\nthis:\nNSDERIV (\\x. f x * g x) x :> Da * g x + Db * f x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1250, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438951064805861, "lm_q2_score": 0.8558511451289037, "lm_q1q2_score": 0.7222485932500877}} {"text": "[STATEMENT]\nlemma rbt_height_le: assumes \"rbt t\" shows \"height t \\ 2 * log 2 (size1 t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (height t) \\ 2 * log 2 (real (size1 t))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. real (height t) \\ 2 * log 2 (real (size1 t))\n[PROOF STEP]\nhave \"2 powr (height t / 2) \\ 2 powr bheight t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 powr (real (height t) / 2) \\ 2 powr real (bheight t)\n[PROOF STEP]\nusing rbt_height_bheight[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (height t) / 2 \\ real (bheight t)\n\ngoal (1 subgoal):\n 1. 2 powr (real (height t) / 2) \\ 2 powr real (bheight t)\n[PROOF STEP]\nby (simp)\n[PROOF STATE]\nproof (state)\nthis:\n2 powr (real (height t) / 2) \\ 2 powr real (bheight t)\n\ngoal (1 subgoal):\n 1. real (height t) \\ 2 * log 2 (real (size1 t))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 powr (real (height t) / 2) \\ 2 powr real (bheight t)\n\ngoal (1 subgoal):\n 1. real (height t) \\ 2 * log 2 (real (size1 t))\n[PROOF STEP]\nhave \"\\ \\ size1 t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 powr real (bheight t) \\ real (size1 t)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nrbt t\n\ngoal (1 subgoal):\n 1. 2 powr real (bheight t) \\ real (size1 t)\n[PROOF STEP]\nby (simp add: powr_realpow bheight_size_bound rbt_def)\n[PROOF STATE]\nproof (state)\nthis:\n2 powr real (bheight t) \\ real (size1 t)\n\ngoal (1 subgoal):\n 1. real (height t) \\ 2 * log 2 (real (size1 t))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n2 powr (real (height t) / 2) \\ real (size1 t)\n[PROOF STEP]\nhave \"2 powr (height t / 2) \\ size1 t\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 powr (real (height t) / 2) \\ real (size1 t)\n\ngoal (1 subgoal):\n 1. 2 powr (real (height t) / 2) \\ real (size1 t)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n2 powr (real (height t) / 2) \\ real (size1 t)\n\ngoal (1 subgoal):\n 1. real (height t) \\ 2 * log 2 (real (size1 t))\n[PROOF STEP]\nhence \"height t / 2 \\ log 2 (size1 t)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 powr (real (height t) / 2) \\ real (size1 t)\n\ngoal (1 subgoal):\n 1. real (height t) / 2 \\ log 2 (real (size1 t))\n[PROOF STEP]\nby (simp add: le_log_iff size1_size del: divide_le_eq_numeral1(1))\n[PROOF STATE]\nproof (state)\nthis:\nreal (height t) / 2 \\ log 2 (real (size1 t))\n\ngoal (1 subgoal):\n 1. real (height t) \\ 2 * log 2 (real (size1 t))\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (height t) / 2 \\ log 2 (real (size1 t))\n\ngoal (1 subgoal):\n 1. real (height t) \\ 2 * log 2 (real (size1 t))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal (height t) \\ 2 * log 2 (real (size1 t))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1310, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.855851143290548, "lm_q2_score": 0.8438951005915208, "lm_q1q2_score": 0.7222485866585451}} {"text": "[STATEMENT]\nlemma sqrt_sum_squares_le_sum_abs: \"sqrt (x\\<^sup>2 + y\\<^sup>2) \\ \\x\\ + \\y\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (x\\<^sup>2 + y\\<^sup>2) \\ \\x\\ + \\y\\\n[PROOF STEP]\nby (rule power2_le_imp_le) (simp_all add: power2_sum)", "meta": {"llama_tokens": 142, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206659843132, "lm_q2_score": 0.8006919997179627, "lm_q1q2_score": 0.7221606616339364}} {"text": "[STATEMENT]\ntheorem num_sylow_normalizer:\n assumes Psize:\"P \\ subgroups_of_size (p ^ a)\"\n shows \"card (rcosets\\<^bsub>G\\carrier := group_action.stabilizer G (conjugation_action (p ^ a)) P\\\\<^esub> P) * p ^ a = card (group_action.stabilizer G (conjugation_action (p ^ a)) P)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := group_action.stabilizer G (conjugation_action (p ^ a)) P\\\\<^esub> P) * p ^ a = card (group_action.stabilizer G (conjugation_action (p ^ a)) P)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := group_action.stabilizer G (conjugation_action (p ^ a)) P\\\\<^esub> P) * p ^ a = card (group_action.stabilizer G (conjugation_action (p ^ a)) P)\n[PROOF STEP]\nfrom finite_G\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (carrier G)\n[PROOF STEP]\ninterpret conj: group_action G \"(conjugation_action (p ^ a))\" \"(subgroups_of_size (p ^ a))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (carrier G)\n\ngoal (1 subgoal):\n 1. group_action G (conjugation_action (p ^ a)) (subgroups_of_size (p ^ a))\n[PROOF STEP]\nby (rule acts_on_subsets)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n[PROOF STEP]\nfrom Psize\n[PROOF STATE]\nproof (chain)\npicking this:\nP \\ subgroups_of_size (p ^ a)\n[PROOF STEP]\nhave PG:\"subgroup P G\" and cardP:\"card P = p ^ a\"\n[PROOF STATE]\nproof (prove)\nusing this:\nP \\ subgroups_of_size (p ^ a)\n\ngoal (1 subgoal):\n 1. subgroup P G &&& card P = p ^ a\n[PROOF STEP]\nunfolding subgroups_of_size_def\n[PROOF STATE]\nproof (prove)\nusing this:\nP \\ {H. subgroup H G \\ card H = p ^ a}\n\ngoal (1 subgoal):\n 1. subgroup P G &&& card P = p ^ a\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsubgroup P G\ncard P = p ^ a\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n[PROOF STEP]\nwith finite_G\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (carrier G)\nsubgroup P G\ncard P = p ^ a\n[PROOF STEP]\nhave \"order G = card (conj.orbit P) * card (conj.stabilizer P)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (carrier G)\nsubgroup P G\ncard P = p ^ a\n\ngoal (1 subgoal):\n 1. order G = card (conj.orbit P) * card (conj.stabilizer P)\n[PROOF STEP]\nby (metis Psize acts_on_subsets group_action.orbit_size)\n[PROOF STATE]\nproof (state)\nthis:\norder G = card (conj.orbit P) * card (conj.stabilizer P)\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n[PROOF STEP]\nwith order_G Psize\n[PROOF STATE]\nproof (chain)\npicking this:\norder G = p ^ a * m\nP \\ subgroups_of_size (p ^ a)\norder G = card (conj.orbit P) * card (conj.stabilizer P)\n[PROOF STEP]\nhave \"p ^ a * m = card (subgroups_of_size (p ^ a)) * card (conj.stabilizer P)\"\n[PROOF STATE]\nproof (prove)\nusing this:\norder G = p ^ a * m\nP \\ subgroups_of_size (p ^ a)\norder G = card (conj.orbit P) * card (conj.stabilizer P)\n\ngoal (1 subgoal):\n 1. p ^ a * m = card (subgroups_of_size (p ^ a)) * card (conj.stabilizer P)\n[PROOF STEP]\nby (metis num_eq_card_orbit)\n[PROOF STATE]\nproof (state)\nthis:\np ^ a * m = card (subgroups_of_size (p ^ a)) * card (conj.stabilizer P)\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\np ^ a * m = card (subgroups_of_size (p ^ a)) * card (conj.stabilizer P)\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n[PROOF STEP]\nfrom Psize\n[PROOF STATE]\nproof (chain)\npicking this:\nP \\ subgroups_of_size (p ^ a)\n[PROOF STEP]\ninterpret stabGroup: group \"G\\carrier := conj.stabilizer P\\\"\n[PROOF STATE]\nproof (prove)\nusing this:\nP \\ subgroups_of_size (p ^ a)\n\ngoal (1 subgoal):\n 1. Group.group (G\\carrier := conj.stabilizer P\\)\n[PROOF STEP]\nby (metis conj.stabilizer_is_subgroup subgroup_imp_group)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n[PROOF STEP]\nfrom finite_G Psize\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (carrier G)\nP \\ subgroups_of_size (p ^ a)\n[PROOF STEP]\nhave PStab:\"subgroup P (G\\carrier := conj.stabilizer P\\)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (carrier G)\nP \\ subgroups_of_size (p ^ a)\n\ngoal (1 subgoal):\n 1. subgroup P (G\\carrier := conj.stabilizer P\\)\n[PROOF STEP]\nby (rule stabilizer_supergrp_P)\n[PROOF STATE]\nproof (state)\nthis:\nsubgroup P (G\\carrier := conj.stabilizer P\\)\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n[PROOF STEP]\nfrom finite_G Psize\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (carrier G)\nP \\ subgroups_of_size (p ^ a)\n[PROOF STEP]\nhave \"finite (conj.stabilizer P)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (carrier G)\nP \\ subgroups_of_size (p ^ a)\n\ngoal (1 subgoal):\n 1. finite (conj.stabilizer P)\n[PROOF STEP]\nby (metis card.infinite conj.stabilizer_is_subgroup less_nat_zero_code subgroup.finite_imp_card_positive)\n[PROOF STATE]\nproof (state)\nthis:\nfinite (conj.stabilizer P)\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n[PROOF STEP]\nwith finite_G PStab stabGroup.lagrange\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (carrier G)\nsubgroup P (G\\carrier := conj.stabilizer P\\)\nsubgroup ?H (G\\carrier := conj.stabilizer P\\) \\ card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> ?H) * card ?H = order (G\\carrier := conj.stabilizer P\\)\nfinite (conj.stabilizer P)\n[PROOF STEP]\nhave \"card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * card P = order (G\\carrier := conj.stabilizer P\\)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (carrier G)\nsubgroup P (G\\carrier := conj.stabilizer P\\)\nsubgroup ?H (G\\carrier := conj.stabilizer P\\) \\ card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> ?H) * card ?H = order (G\\carrier := conj.stabilizer P\\)\nfinite (conj.stabilizer P)\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * card P = order (G\\carrier := conj.stabilizer P\\)\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\ncard (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * card P = order (G\\carrier := conj.stabilizer P\\)\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n[PROOF STEP]\nwith cardP\n[PROOF STATE]\nproof (chain)\npicking this:\ncard P = p ^ a\ncard (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * card P = order (G\\carrier := conj.stabilizer P\\)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard P = p ^ a\ncard (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * card P = order (G\\carrier := conj.stabilizer P\\)\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n[PROOF STEP]\nunfolding order_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncard P = p ^ a\ncard (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * card P = card (carrier (G\\carrier := conj.stabilizer P\\))\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (rcosets\\<^bsub>G\\carrier := conj.stabilizer P\\\\<^esub> P) * p ^ a = card (conj.stabilizer P)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3584, "file": "Secondary_Sylow_SndSylow", "length": 32, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.880797071719777, "lm_q2_score": 0.8198933337131076, "lm_q1q2_score": 0.7221596474570711}} {"text": "[STATEMENT]\nlemma le_err_antisym [rule_format]:\n \"order r \\ e1 <=_(le r) e2 \\ e2 <=_(le r) e1 \\ e1=e2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. order r \\ e1 \\\\<^bsub>le r\\<^esub> e2 \\ e2 \\\\<^bsub>le r\\<^esub> e1 \\ e1 = e2\n[PROOF STEP]\napply (unfold unfold_lesub_err le_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. order r \\ (case e2 of Err \\ True | OK y \\ case e1 of Err \\ False | OK x \\ x \\\\<^bsub>r\\<^esub> y) \\ (case e1 of Err \\ True | OK y \\ case e2 of Err \\ False | OK x \\ x \\\\<^bsub>r\\<^esub> y) \\ e1 = e2\n[PROOF STEP]\napply (simp split: err.split)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. order r \\ \\x2. e1 = OK x2 \\ (\\x2a. e2 = OK x2a \\ x2 \\\\<^bsub>r\\<^esub> x2a \\ x2a \\\\<^bsub>r\\<^esub> x2 \\ x2 = x2a)\n[PROOF STEP]\napply (blast intro: order_antisym)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 463, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.880797068590724, "lm_q2_score": 0.8198933337131076, "lm_q1q2_score": 0.7221596448915814}} {"text": "[STATEMENT]\nlemma (in finite_measure) finite_measure_Union:\n assumes sets: \"A \\ sets M\" \"B \\ sets M\" and \"A \\ B = {}\"\n shows \"measure M (A \\ B) = measure M A + measure M B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (A \\ B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M B\n[PROOF STEP]\nusing measure_Union[OF _ _ assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\emeasure M A \\ \\; emeasure M B \\ \\\\ \\ Sigma_Algebra.measure M (A \\ B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M B\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (A \\ B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M B\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 300, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213772699436, "lm_q2_score": 0.8031737892899221, "lm_q1q2_score": 0.7221507236134742}} {"text": "[STATEMENT]\nlemma Log_mult: \n \"\\ 0 < a; a \\ 1; 0 < x; 0 < y \\\n \\ Log a (x * y) = Log a x + Log a y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < a; a \\ 1; 0 < x; 0 < y\\ \\ Log a (x * y) = Log a x + Log a y\n[PROOF STEP]\nby (metis Log_powreal_cancel powreal_Log_cancel powreal_add)", "meta": {"llama_tokens": 169, "file": "Real_Power_Log", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213718636754, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7221507192713014}} {"text": "[STATEMENT]\nlemma K2_isometry_cosh_dist:\n assumes \"a \\ hyp2\" and \"b \\ hyp2\" and \"is_K2_isometry J\"\n shows \"cosh_dist (apply_cltn2 a J) (apply_cltn2 b J) = cosh_dist a b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cosh_dist (apply_cltn2 a J) (apply_cltn2 b J) = cosh_dist a b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ hyp2\nb \\ hyp2\nis_K2_isometry J\n\ngoal (1 subgoal):\n 1. cosh_dist (apply_cltn2 a J) (apply_cltn2 b J) = cosh_dist a b\n[PROOF STEP]\nby (unfold cosh_dist_def) (simp add: K2_isometry_exp_2dist)", "meta": {"llama_tokens": 263, "file": "Tarskis_Geometry_Hyperbolic_Tarski", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213691605412, "lm_q2_score": 0.8031737892899222, "lm_q1q2_score": 0.7221507171002148}} {"text": "[STATEMENT]\nlemma phytagorean_theorem_norm:\n assumes o: \"orthogonal x y\"\n shows \"norm (x+y)^2=norm x^2 + norm y^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nhave \"norm (x+y)^2 = (x+y) \\ (x+y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (x + y) \\ (x + y)\n[PROOF STEP]\nunfolding power2_norm_eq_inner\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x + y) \\ (x + y) = (x + y) \\ (x + y)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n(norm (x + y))\\<^sup>2 = (x + y) \\ (x + y)\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(norm (x + y))\\<^sup>2 = (x + y) \\ (x + y)\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nhave \"... = ((x+y) \\ x) + ((x+y) \\ y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x + y) \\ (x + y) = (x + y) \\ x + (x + y) \\ y\n[PROOF STEP]\nunfolding inner_right_distrib\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x + y) \\ x + (x + y) \\ y = (x + y) \\ x + (x + y) \\ y\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n(x + y) \\ (x + y) = (x + y) \\ x + (x + y) \\ y\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(x + y) \\ (x + y) = (x + y) \\ x + (x + y) \\ y\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nhave \"... = (x \\ x) + (x \\ y) + (y \\ x) + (y \\ y) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x + y) \\ x + (x + y) \\ y = x \\ x + x \\ y + y \\ x + y \\ y\n[PROOF STEP]\nunfolding real_inner_class.inner_add_left\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ x + y \\ x + (x \\ y + y \\ y) = x \\ x + x \\ y + y \\ x + y \\ y\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(x + y) \\ x + (x + y) \\ y = x \\ x + x \\ y + y \\ x + y \\ y\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(x + y) \\ x + (x + y) \\ y = x \\ x + x \\ y + y \\ x + y \\ y\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nhave \"... = (x \\ x) + (y \\ y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ x + x \\ y + y \\ x + y \\ y = x \\ x + y \\ y\n[PROOF STEP]\nusing o\n[PROOF STATE]\nproof (prove)\nusing this:\northogonal x y\n\ngoal (1 subgoal):\n 1. x \\ x + x \\ y + y \\ x + y \\ y = x \\ x + y \\ y\n[PROOF STEP]\nunfolding orthogonal_def\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ y = 0\n\ngoal (1 subgoal):\n 1. x \\ x + x \\ y + y \\ x + y \\ y = x \\ x + y \\ y\n[PROOF STEP]\nby (metis monoid_add_class.add.right_neutral inner_commute)\n[PROOF STATE]\nproof (state)\nthis:\nx \\ x + x \\ y + y \\ x + y \\ y = x \\ x + y \\ y\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx \\ x + x \\ y + y \\ x + y \\ y = x \\ x + y \\ y\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nhave \"... = norm x^2 + norm y^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ x + y \\ y = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nunfolding power2_norm_eq_inner\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ x + y \\ y = x \\ x + y \\ y\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nx \\ x + y \\ y = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(norm (x + y))\\<^sup>2 = (norm x)\\<^sup>2 + (norm y)\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2285, "file": "QR_Decomposition_Projections", "length": 25, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267830311354, "lm_q2_score": 0.8311430436757313, "lm_q1q2_score": 0.7221193368754921}} {"text": "[STATEMENT]\nlemma birkhoff_sum_cmult:\n fixes f::\"_ \\ real\"\n shows \"birkhoff_sum (\\x. c * f x) n x = c * birkhoff_sum f n x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. birkhoff_sum (\\x. c * f x) n x = c * birkhoff_sum f n x\n[PROOF STEP]\nunfolding birkhoff_sum_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\ii 0\" and a: \"is_real a\" and b: \"is_real b\"\n shows \"c * max a b = max (c * a) (c * b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c * max a b = max (c * a) (c * b)\n[PROOF STEP]\nby (rule hom_max, simp add: real_mult_le_cancel_left_pos[OF a b c])", "meta": {"llama_tokens": 158, "file": "LLL_Basis_Reduction_Norms", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757870046160257, "lm_q2_score": 0.824461932846258, "lm_q1q2_score": 0.7220530465873632}} {"text": "[STATEMENT]\ntheorem real_lebesgue_integral_def:\n assumes f[measurable]: \"integrable M f\"\n shows \"integral\\<^sup>L M f = enn2real (\\\\<^sup>+x. f x \\M) - enn2real (\\\\<^sup>+x. ennreal (- f x) \\M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. integral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\nhave \"integral\\<^sup>L M f = integral\\<^sup>L M (\\x. max 0 (f x) - max 0 (- f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integral\\<^sup>L M f = LINT x|M. max 0 (f x) - max 0 (- f x)\n[PROOF STEP]\nby (auto intro!: arg_cong[where f=\"integral\\<^sup>L M\"])\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>L M f = LINT x|M. max 0 (f x) - max 0 (- f x)\n\ngoal (1 subgoal):\n 1. integral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>L M f = LINT x|M. max 0 (f x) - max 0 (- f x)\n\ngoal (1 subgoal):\n 1. integral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\nhave \"\\ = integral\\<^sup>L M (\\x. max 0 (f x)) - integral\\<^sup>L M (\\x. max 0 (- f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LINT x|M. max 0 (f x) - max 0 (- f x) = (LINT x|M. max 0 (f x)) - (LINT x|M. max 0 (- f x))\n[PROOF STEP]\nby (intro integral_diff integrable_max integrable_minus integrable_zero f)\n[PROOF STATE]\nproof (state)\nthis:\nLINT x|M. max 0 (f x) - max 0 (- f x) = (LINT x|M. max 0 (f x)) - (LINT x|M. max 0 (- f x))\n\ngoal (1 subgoal):\n 1. integral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nLINT x|M. max 0 (f x) - max 0 (- f x) = (LINT x|M. max 0 (f x)) - (LINT x|M. max 0 (- f x))\n\ngoal (1 subgoal):\n 1. integral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\nhave \"integral\\<^sup>L M (\\x. max 0 (f x)) = enn2real (\\\\<^sup>+x. ennreal (f x) \\M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LINT x|M. max 0 (f x) = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M)\n[PROOF STEP]\nby (subst integral_eq_nn_integral) (auto intro!: arg_cong[where f=enn2real] nn_integral_cong simp: max_def ennreal_neg)\n[PROOF STATE]\nproof (state)\nthis:\nLINT x|M. max 0 (f x) = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M)\n\ngoal (1 subgoal):\n 1. integral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nLINT x|M. max 0 (f x) = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M)\n\ngoal (1 subgoal):\n 1. integral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\nhave \"integral\\<^sup>L M (\\x. max 0 (- f x)) = enn2real (\\\\<^sup>+x. ennreal (- f x) \\M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. LINT x|M. max 0 (- f x) = enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\nby (subst integral_eq_nn_integral) (auto intro!: arg_cong[where f=enn2real] nn_integral_cong simp: max_def ennreal_neg)\n[PROOF STATE]\nproof (state)\nthis:\nLINT x|M. max 0 (- f x) = enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n\ngoal (1 subgoal):\n 1. integral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nintegral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nintegral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n\ngoal (1 subgoal):\n 1. integral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nintegral\\<^sup>L M f = enn2real (\\\\<^sup>+ x. ennreal (f x) \\M) - enn2real (\\\\<^sup>+ x. ennreal (- f x) \\M)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2177, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869948899665, "lm_q2_score": 0.8244619263765706, "lm_q1q2_score": 0.7220530329025295}} {"text": "[STATEMENT]\nlemma null_space_is_preserved:\n fixes A::\"'a::{field}^'cols^'rows\"\n assumes P: \"invertible P\"\n shows \"null_space (P**A) = null_space A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. null_space (P ** A) = null_space A\n[PROOF STEP]\nunfolding null_space_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x. P ** A *v x = 0} = {x. A *v x = 0}\n[PROOF STEP]\nusing P matrix_inv_left matrix_left_invertible_ker matrix_vector_mul_assoc matrix_vector_mult_0_right\n[PROOF STATE]\nproof (prove)\nusing this:\ninvertible P\ninvertible ?M \\ matrix_inv ?M ** ?M = mat (1::?'a)\n(\\B. B ** ?A = mat (1::?'a)) = (\\x. ?A *v x = 0 \\ x = 0)\n?A *v (?B *v ?x) = ?A ** ?B *v ?x\n?A *v 0 = 0\n\ngoal (1 subgoal):\n 1. {x. P ** A *v x = 0} = {x. A *v x = 0}\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 375, "file": "Rank_Nullity_Theorem_Fundamental_Subspaces", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869851639066, "lm_q2_score": 0.8244619306896955, "lm_q1q2_score": 0.7220530286611422}} {"text": "[STATEMENT]\nlemma uniform_limit_null_comparison:\n assumes \"\\\\<^sub>F x in F. \\a\\S. norm (f x a) \\ g x a\"\n assumes \"uniform_limit S g (\\_. 0) F\"\n shows \"uniform_limit S f (\\_. 0) F\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. uniform_limit S f (\\_. 0::'c) F\n[PROOF STEP]\nusing assms(2)\n[PROOF STATE]\nproof (prove)\nusing this:\nuniform_limit S g (\\_. 0) F\n\ngoal (1 subgoal):\n 1. uniform_limit S f (\\_. 0::'c) F\n[PROOF STEP]\nproof (rule metric_uniform_limit_imp_uniform_limit)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. \\y\\S. dist (f x y) (0::'c) \\ dist (g x y) 0\n[PROOF STEP]\nshow \"\\\\<^sub>F x in F. \\y\\S. dist (f x y) 0 \\ dist (g x y) 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. \\y\\S. dist (f x y) (0::'c) \\ dist (g x y) 0\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\<^sub>F x in F. \\a\\S. norm (f x a) \\ g x a\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. \\y\\S. dist (f x y) (0::'c) \\ dist (g x y) 0\n[PROOF STEP]\nby (rule eventually_mono) (force simp add: dist_norm)\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sub>F x in F. \\y\\S. dist (f x y) (0::'c) \\ dist (g x y) 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 627, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8757869916479466, "lm_q2_score": 0.8244619199068831, "lm_q1q2_score": 0.7220530245635395}} {"text": "[STATEMENT]\nlemma integrable_density:\n fixes f :: \"'a \\ 'b::{banach, second_countable_topology}\" and g :: \"'a \\ real\"\n assumes [measurable]: \"f \\ borel_measurable M\" \"g \\ borel_measurable M\"\n and nn: \"AE x in M. 0 \\ g x\"\n shows \"integrable (density M g) f \\ integrable M (\\x. g x *\\<^sub>R f x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integrable (density M (\\x. ennreal (g x))) f = integrable M (\\x. g x *\\<^sub>R f x)\n[PROOF STEP]\nunfolding integrable_iff_bounded\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f \\ borel_measurable (density M (\\x. ennreal (g x))) \\ \\\\<^sup>+ x. ennreal (norm (f x)) \\density M (\\x. ennreal (g x)) < \\) = ((\\x. g x *\\<^sub>R f x) \\ borel_measurable M \\ \\\\<^sup>+ x. ennreal (norm (g x *\\<^sub>R f x)) \\M < \\)\n[PROOF STEP]\nusing nn\n[PROOF STATE]\nproof (prove)\nusing this:\nAE x in M. 0 \\ g x\n\ngoal (1 subgoal):\n 1. (f \\ borel_measurable (density M (\\x. ennreal (g x))) \\ \\\\<^sup>+ x. ennreal (norm (f x)) \\density M (\\x. ennreal (g x)) < \\) = ((\\x. g x *\\<^sub>R f x) \\ borel_measurable M \\ \\\\<^sup>+ x. ennreal (norm (g x *\\<^sub>R f x)) \\M < \\)\n[PROOF STEP]\napply (simp add: nn_integral_density less_top[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. AE x in M. 0 \\ g x \\ (\\\\<^sup>+ x. ennreal (g x) * ennreal (norm (f x)) \\M = top) = (\\\\<^sup>+ x. ennreal (\\g x\\ * norm (f x)) \\M = top)\n[PROOF STEP]\napply (intro arg_cong2[where f=\"(=)\"] refl nn_integral_cong_AE)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. AE x in M. 0 \\ g x \\ AE x in M. ennreal (g x) * ennreal (norm (f x)) = ennreal (\\g x\\ * norm (f x))\n[PROOF STEP]\napply (auto simp: ennreal_mult)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 877, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869786798663, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.7220530195379192}} {"text": "[STATEMENT]\nlemma card_UNIV_tau_i_below_zero: \n assumes \"finite (UNIV :: 'a set)\" and \"mono \\\"\n and \"(\\i :: nat. ((\\ :: ('a set \\ 'a set)) ^^ (i + (1 :: nat)))(UNIV :: 'a set) \\ (\\ ^^ i) UNIV)\"\n shows \"card((\\ ^^ (card (UNIV ::'a set))) (UNIV ::'a set)) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ((\\ ^^ card UNIV) UNIV) \\ 0\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ((\\ ^^ card UNIV) UNIV) \\ 0\n[PROOF STEP]\nhave \"(\\ i :: nat. card((\\ ^^ i) (UNIV ::'a set)) \\ (card (UNIV :: 'a set)) - i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\i. card ((\\ ^^ i) UNIV) \\ card UNIV - i\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite UNIV\nmono \\\n\\i. (\\ ^^ (i + 1)) UNIV \\ (\\ ^^ i) UNIV\n\ngoal (1 subgoal):\n 1. \\i. card ((\\ ^^ i) UNIV) \\ card UNIV - i\n[PROOF STEP]\nby (rule card_univ_subtract)\n[PROOF STATE]\nproof (state)\nthis:\n\\i. card ((\\ ^^ i) UNIV) \\ card UNIV - i\n\ngoal (1 subgoal):\n 1. card ((\\ ^^ card UNIV) UNIV) \\ 0\n[PROOF STEP]\nthus \"card((\\ ^^ (card (UNIV ::'a set))) (UNIV ::'a set)) \\ 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\i. card ((\\ ^^ i) UNIV) \\ card UNIV - i\n\ngoal (1 subgoal):\n 1. card ((\\ ^^ card UNIV) UNIV) \\ 0\n[PROOF STEP]\nby (drule_tac x = \"card (UNIV ::'a set)\" in spec, simp)\n[PROOF STATE]\nproof (state)\nthis:\ncard ((\\ ^^ card UNIV) UNIV) \\ 0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 748, "file": "Attack_Trees_MC", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314858927012, "lm_q2_score": 0.8152324983301567, "lm_q1q2_score": 0.7219955688441557}} {"text": "[STATEMENT]\nlemma sqrt_Max_power2_eq_max_abs:\n \"finite A \\ A \\ {} \\ sqrt (Max {(f i)\\<^sup>2|i. i \\ A}) = Max {\\f i\\ |i. i \\ A}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; A \\ {}\\ \\ sqrt (Max {(f i)\\<^sup>2 |i. i \\ A}) = Max {\\f i\\ |i. i \\ A}\n[PROOF STEP]\napply(subst cSup_eq_Max[symmetric], simp_all)+\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\finite A; A \\ {}\\ \\ sqrt (Sup {(f i)\\<^sup>2 |i. i \\ A}) = Sup {\\f i\\ |i. i \\ A}\n[PROOF STEP]\nusing sqrt_Sup_power2_eq_Sup_abs\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite ?A; ?A \\ {}\\ \\ sqrt (Sup {(?f i)\\<^sup>2 |i. i \\ ?A}) = Sup {\\?f i\\ |i. i \\ ?A}\n\ngoal (1 subgoal):\n 1. \\finite A; A \\ {}\\ \\ sqrt (Sup {(f i)\\<^sup>2 |i. i \\ A}) = Sup {\\f i\\ |i. i \\ A}\n[PROOF STEP]\n.", "meta": {"llama_tokens": 492, "file": "Matrices_for_ODEs_MTX_Norms", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.907312221360624, "lm_q2_score": 0.7956581000631542, "lm_q1q2_score": 0.721910318211874}} {"text": "[STATEMENT]\nlemma rank_argument_det: \n fixes M :: \"('c :: {conjugatable_ordered_field}) mat\"\n assumes \"M \\ carrier_mat x y\"\n assumes \"det (M* M\\<^sup>T) \\ 0\"\n shows \"x \\ y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nlet ?B = \"(M * M\\<^sup>T)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nhave Mt_car: \"M\\<^sup>T \\ carrier_mat y x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M\\<^sup>T \\ carrier_mat y x\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nM \\ carrier_mat x y\ndet (M * M\\<^sup>T) \\ (0::'c)\n\ngoal (1 subgoal):\n 1. M\\<^sup>T \\ carrier_mat y x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nM\\<^sup>T \\ carrier_mat y x\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nhave b_car: \"?B \\ carrier_mat x x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M * M\\<^sup>T \\ carrier_mat x x\n[PROOF STEP]\nusing transpose_carrier_mat assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(?A\\<^sup>T \\ carrier_mat ?nc ?nr) = (?A \\ carrier_mat ?nr ?nc)\nM \\ carrier_mat x y\ndet (M * M\\<^sup>T) \\ (0::'c)\n\ngoal (1 subgoal):\n 1. M * M\\<^sup>T \\ carrier_mat x x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nM * M\\<^sup>T \\ carrier_mat x x\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nM * M\\<^sup>T \\ carrier_mat x x\n[PROOF STEP]\nhave b_rank: \"vec_space.rank x ?B = x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nM * M\\<^sup>T \\ carrier_mat x x\n\ngoal (1 subgoal):\n 1. rank x (M * M\\<^sup>T) = x\n[PROOF STEP]\nusing vec_space.low_rank_det_zero assms(2)\n[PROOF STATE]\nproof (prove)\nusing this:\nM * M\\<^sup>T \\ carrier_mat x x\n\\?A \\ carrier_mat ?n ?n; det ?A \\ (0::?'a)\\ \\ rank ?n ?A = ?n\ndet (M * M\\<^sup>T) \\ (0::'c)\n\ngoal (1 subgoal):\n 1. rank x (M * M\\<^sup>T) = x\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nrank x (M * M\\<^sup>T) = x\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nrank x (M * M\\<^sup>T) = x\n[PROOF STEP]\nhave \"rank x ?B \\ min (rank x M) (rank y M\\<^sup>T)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nrank x (M * M\\<^sup>T) = x\n\ngoal (1 subgoal):\n 1. rank x (M * M\\<^sup>T) \\ min (rank x M) (rank y M\\<^sup>T)\n[PROOF STEP]\nusing rank_mat_mult_lt_min_rank_factor Mt_car b_car assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nrank x (M * M\\<^sup>T) = x\n\\?A \\ carrier_mat ?n ?m; ?B \\ carrier_mat ?m ?nc\\ \\ rank ?n (?A * ?B) \\ min (rank ?n ?A) (rank ?m ?B)\nM\\<^sup>T \\ carrier_mat y x\nM * M\\<^sup>T \\ carrier_mat x x\nM \\ carrier_mat x y\n\ngoal (1 subgoal):\n 1. rank x (M * M\\<^sup>T) \\ min (rank x M) (rank y M\\<^sup>T)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nrank x (M * M\\<^sup>T) \\ min (rank x M) (rank y M\\<^sup>T)\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nrank x (M * M\\<^sup>T) \\ min (rank x M) (rank y M\\<^sup>T)\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nusing le_trans vec_space.rank_le_nc\n[PROOF STATE]\nproof (prove)\nusing this:\nrank x (M * M\\<^sup>T) \\ min (rank x M) (rank y M\\<^sup>T)\n\\?i \\ ?j; ?j \\ ?k\\ \\ ?i \\ ?k\n?A \\ carrier_mat ?n ?nc \\ rank ?n ?A \\ ?nc\n\ngoal (1 subgoal):\n 1. x \\ y\n[PROOF STEP]\nby (metis assms(1) b_rank min.bounded_iff)\n[PROOF STATE]\nproof (state)\nthis:\nx \\ y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1744, "file": "Fishers_Inequality_Rank_Argument_General", "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637612961505, "lm_q2_score": 0.8397339656668287, "lm_q1q2_score": 0.7218888594132785}} {"text": "[STATEMENT]\nlemma infsetsum_cmult_left:\n fixes f :: \"'a \\ 'b :: {banach, real_normed_algebra, second_countable_topology}\"\n assumes \"c \\ 0 \\ f abs_summable_on A\"\n shows \"infsetsum (\\x. f x * c) A = infsetsum f A * c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. f x * c) = infsetsum f A * c\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'b) \\ f abs_summable_on A\n\ngoal (1 subgoal):\n 1. (\\\\<^sub>ax\\A. f x * c) = infsetsum f A * c\n[PROOF STEP]\nunfolding infsetsum_def abs_summable_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\ (0::'b) \\ integrable (count_space A) f\n\ngoal (1 subgoal):\n 1. LINT x|count_space A. f x * c = integral\\<^sup>L (count_space A) f * c\n[PROOF STEP]\nby (rule Bochner_Integration.integral_mult_left)", "meta": {"llama_tokens": 369, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094174159129, "lm_q2_score": 0.8080672066194945, "lm_q1q2_score": 0.7218540455781647}} {"text": "[STATEMENT]\nlemma odd_nodes_no_edge[simp]: \"finite (nodes g) \\ num_of_odd_nodes (g \\edges:={} \\) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (nodes g) \\ num_of_odd_nodes (g\\edges := {}\\) = 0\n[PROOF STEP]\nunfolding num_of_odd_nodes_def odd_nodes_set_def degree_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (nodes g) \\ card {v \\ nodes (g\\edges := {}\\). odd (card {e \\ edges (g\\edges := {}\\). fst e = v})} = 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 249, "file": "Koenigsberg_Friendship_MoreGraph", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.893309411735131, "lm_q2_score": 0.8080672066194945, "lm_q1q2_score": 0.7218540409877112}} {"text": "[STATEMENT]\nlemma im_set_mono: \"\\f \\A \\ B; A1 \\ A2; A2 \\ A \\ \\ (f ` A1) \\ (f ` A2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ A \\ B; A1 \\ A2; A2 \\ A\\ \\ f ` A1 \\ f ` A2\n[PROOF STEP]\napply (simp add:image_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ A \\ B; A1 \\ A2; A2 \\ A\\ \\ {y. \\x\\A1. y = f x} \\ {y. \\x\\A2. y = f x}\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 298, "file": "Group-Ring-Module_Algebra1", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094060543487, "lm_q2_score": 0.8080672112416737, "lm_q1q2_score": 0.7218540405262934}} {"text": "[STATEMENT]\nlemma DFT_lower:\n \"DFT (2 * m) a i =\n DFT m (%i. a (2 * i)) i +\n (root (2 * m)) ^ i * DFT m (%i. a (2 * i + 1)) i\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. DFT (2 * m) a i = DFT m (\\i. a (2 * i)) i + FFT.root (2 * m) ^ i * DFT m (\\i. a (2 * i + 1)) i\n[PROOF STEP]\nproof (unfold DFT_def)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\j = 0..<2 * m. FFT.root (2 * m) ^ (i * j) * a j) = (\\j = 0..j = 0..j = 0..<2 * m. root (2 * m) ^ (i * j) * a j) =\n (\\j = 0..j = 0..j = 0..<2 * m. FFT.root (2 * m) ^ (i * j) * a j) = (\\j = 0..j = 0..j = 0..<2 * m. FFT.root (2 * m) ^ (i * j) * a j) = (\\j = 0..j = 0..j = 0..<2 * m. FFT.root (2 * m) ^ (i * j) * a j) = (\\j = 0..j = 0..j = 0..<2 * m. FFT.root (2 * m) ^ (i * j) * a j) = (\\j = 0..j = 0..j = 0..<2 * m. FFT.root (2 * m) ^ (i * j) * a j) = (\\j = 0..j = 0..j = 0..j = 0..j = 0..j = 0..j = 0..j = 0..First pair of sums\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..j = 0..j = 0..j = 0..j = 0..j = 0..Second pair of sums\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j = 0..j = 0..j = 0..n = 0..j = 0..n = 0..j = 0..n = 0..j = 0..j = 0..j = 0..j = 0..j = 0..<2 * m. FFT.root (2 * m) ^ (i * j) * a j) = (\\j = 0..j = 0..j = 0..<2 * m. FFT.root (2 * m) ^ (i * j) * a j) = (\\j = 0..j = 0..j = 0..<2 * m. FFT.root (2 * m) ^ (i * j) * a j) = (\\j = 0..j = 0..j = 0..<2 * m. FFT.root (2 * m) ^ (i * j) * a j) = (\\j = 0..j = 0..j = 0..<2 * m. FFT.root (2 * m) ^ (i * j) * a j) = (\\j = 0..j = 0.. dfa_accepts N bs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dfa_accepts (and_dfa M N) bs = (dfa_accepts M bs \\ dfa_accepts N bs)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nwf_dfa M n\nwf_dfa N n\nlist_all (is_alph n) bs\n\ngoal (1 subgoal):\n 1. dfa_accepts (and_dfa M N) bs = (dfa_accepts M bs \\ dfa_accepts N bs)\n[PROOF STEP]\nby (simp add: binop_dfa_accepts and_dfa_def)", "meta": {"llama_tokens": 280, "file": "Presburger-Automata_Presburger_Automata", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933093946927838, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.7218540396034572}} {"text": "[STATEMENT]\nlemma complex_add_divide_simps[vector_add_divide_simps]: (* In Real_Vector_Spaces, these lemmas are unnamed *)\n \"v + (b / z) *\\<^sub>C w = (if z = 0 then v else (z *\\<^sub>C v + b *\\<^sub>C w) /\\<^sub>C z)\"\n \"a *\\<^sub>C v + (b / z) *\\<^sub>C w = (if z = 0 then a *\\<^sub>C v else ((a * z) *\\<^sub>C v + b *\\<^sub>C w) /\\<^sub>C z)\"\n \"(a / z) *\\<^sub>C v + w = (if z = 0 then w else (a *\\<^sub>C v + z *\\<^sub>C w) /\\<^sub>C z)\"\n \"(a / z) *\\<^sub>C v + b *\\<^sub>C w = (if z = 0 then b *\\<^sub>C w else (a *\\<^sub>C v + (b * z) *\\<^sub>C w) /\\<^sub>C z)\"\n \"v - (b / z) *\\<^sub>C w = (if z = 0 then v else (z *\\<^sub>C v - b *\\<^sub>C w) /\\<^sub>C z)\"\n \"a *\\<^sub>C v - (b / z) *\\<^sub>C w = (if z = 0 then a *\\<^sub>C v else ((a * z) *\\<^sub>C v - b *\\<^sub>C w) /\\<^sub>C z)\"\n \"(a / z) *\\<^sub>C v - w = (if z = 0 then -w else (a *\\<^sub>C v - z *\\<^sub>C w) /\\<^sub>C z)\"\n \"(a / z) *\\<^sub>C v - b *\\<^sub>C w = (if z = 0 then -b *\\<^sub>C w else (a *\\<^sub>C v - (b * z) *\\<^sub>C w) /\\<^sub>C z)\"\n for v :: \"'a :: complex_vector\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((v + (b / z) *\\<^sub>C w = (if z = 0 then v else (z *\\<^sub>C v + b *\\<^sub>C w) /\\<^sub>C z) &&& a *\\<^sub>C v + (b / z) *\\<^sub>C w = (if z = 0 then a *\\<^sub>C v else ((a * z) *\\<^sub>C v + b *\\<^sub>C w) /\\<^sub>C z)) &&& (a / z) *\\<^sub>C v + w = (if z = 0 then w else (a *\\<^sub>C v + z *\\<^sub>C w) /\\<^sub>C z) &&& (a / z) *\\<^sub>C v + b *\\<^sub>C w = (if z = 0 then b *\\<^sub>C w else (a *\\<^sub>C v + (b * z) *\\<^sub>C w) /\\<^sub>C z)) &&& (v - (b / z) *\\<^sub>C w = (if z = 0 then v else (z *\\<^sub>C v - b *\\<^sub>C w) /\\<^sub>C z) &&& a *\\<^sub>C v - (b / z) *\\<^sub>C w = (if z = 0 then a *\\<^sub>C v else ((a * z) *\\<^sub>C v - b *\\<^sub>C w) /\\<^sub>C z)) &&& (a / z) *\\<^sub>C v - w = (if z = 0 then - w else (a *\\<^sub>C v - z *\\<^sub>C w) /\\<^sub>C z) &&& (a / z) *\\<^sub>C v - b *\\<^sub>C w = (if z = 0 then - b *\\<^sub>C w else (a *\\<^sub>C v - (b * z) *\\<^sub>C w) /\\<^sub>C z)\n[PROOF STEP]\nby (simp_all add: divide_inverse_commute scaleC_add_right scaleC_diff_right)", "meta": {"llama_tokens": 1032, "file": "Complex_Bounded_Operators_Complex_Vector_Spaces0", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933093946927838, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.7218540396034572}} {"text": "[STATEMENT]\nlemma wordinterval_setminus'_set_eq:\n \"wordinterval_to_set (wordinterval_setminus' r1 r2) =\n wordinterval_to_set r1 - wordinterval_to_set r2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. wordinterval_to_set (wordinterval_setminus' r1 r2) = wordinterval_to_set r1 - wordinterval_to_set r2\n[PROOF STEP]\napply(induction rule: wordinterval_setminus'.induct)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\s e ms me. wordinterval_to_set (wordinterval_setminus' (WordInterval s e) (WordInterval ms me)) = wordinterval_to_set (WordInterval s e) - wordinterval_to_set (WordInterval ms me)\n 2. \\r1 r2 t. \\wordinterval_to_set (wordinterval_setminus' r1 t) = wordinterval_to_set r1 - wordinterval_to_set t; wordinterval_to_set (wordinterval_setminus' r2 t) = wordinterval_to_set r2 - wordinterval_to_set t\\ \\ wordinterval_to_set (wordinterval_setminus' (RangeUnion r1 r2) t) = wordinterval_to_set (RangeUnion r1 r2) - wordinterval_to_set t\n 3. \\v va r1 r2. \\wordinterval_to_set (wordinterval_setminus' (WordInterval v va) r1) = wordinterval_to_set (WordInterval v va) - wordinterval_to_set r1; wordinterval_to_set (wordinterval_setminus' (wordinterval_setminus' (WordInterval v va) r1) r2) = wordinterval_to_set (wordinterval_setminus' (WordInterval v va) r1) - wordinterval_to_set r2\\ \\ wordinterval_to_set (wordinterval_setminus' (WordInterval v va) (RangeUnion r1 r2)) = wordinterval_to_set (WordInterval v va) - wordinterval_to_set (RangeUnion r1 r2)\n[PROOF STEP]\nusing wordinterval_setminus'_rr_set_eq\n[PROOF STATE]\nproof (prove)\nusing this:\nwordinterval_to_set (wordinterval_setminus' (WordInterval ?s ?e) (WordInterval ?ms ?me)) = wordinterval_to_set (WordInterval ?s ?e) - wordinterval_to_set (WordInterval ?ms ?me)\n\ngoal (3 subgoals):\n 1. \\s e ms me. wordinterval_to_set (wordinterval_setminus' (WordInterval s e) (WordInterval ms me)) = wordinterval_to_set (WordInterval s e) - wordinterval_to_set (WordInterval ms me)\n 2. \\r1 r2 t. \\wordinterval_to_set (wordinterval_setminus' r1 t) = wordinterval_to_set r1 - wordinterval_to_set t; wordinterval_to_set (wordinterval_setminus' r2 t) = wordinterval_to_set r2 - wordinterval_to_set t\\ \\ wordinterval_to_set (wordinterval_setminus' (RangeUnion r1 r2) t) = wordinterval_to_set (RangeUnion r1 r2) - wordinterval_to_set t\n 3. \\v va r1 r2. \\wordinterval_to_set (wordinterval_setminus' (WordInterval v va) r1) = wordinterval_to_set (WordInterval v va) - wordinterval_to_set r1; wordinterval_to_set (wordinterval_setminus' (wordinterval_setminus' (WordInterval v va) r1) r2) = wordinterval_to_set (wordinterval_setminus' (WordInterval v va) r1) - wordinterval_to_set r2\\ \\ wordinterval_to_set (wordinterval_setminus' (WordInterval v va) (RangeUnion r1 r2)) = wordinterval_to_set (WordInterval v va) - wordinterval_to_set (RangeUnion r1 r2)\n[PROOF STEP]\napply blast\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\r1 r2 t. \\wordinterval_to_set (wordinterval_setminus' r1 t) = wordinterval_to_set r1 - wordinterval_to_set t; wordinterval_to_set (wordinterval_setminus' r2 t) = wordinterval_to_set r2 - wordinterval_to_set t\\ \\ wordinterval_to_set (wordinterval_setminus' (RangeUnion r1 r2) t) = wordinterval_to_set (RangeUnion r1 r2) - wordinterval_to_set t\n 2. \\v va r1 r2. \\wordinterval_to_set (wordinterval_setminus' (WordInterval v va) r1) = wordinterval_to_set (WordInterval v va) - wordinterval_to_set r1; wordinterval_to_set (wordinterval_setminus' (wordinterval_setminus' (WordInterval v va) r1) r2) = wordinterval_to_set (wordinterval_setminus' (WordInterval v va) r1) - wordinterval_to_set r2\\ \\ wordinterval_to_set (wordinterval_setminus' (WordInterval v va) (RangeUnion r1 r2)) = wordinterval_to_set (WordInterval v va) - wordinterval_to_set (RangeUnion r1 r2)\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1436, "file": "IP_Addresses_WordInterval", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933093946927837, "lm_q2_score": 0.8080672158638527, "lm_q1q2_score": 0.7218540354744212}} {"text": "[STATEMENT]\ntheorem ln_lower_11_eq: \"0\n ln_lower_11 x = (1/30)*(142*x^5 + 2272*x^4 + 6397*x^3 + 4397*x^2 + 647*x + 5)*(x - 1) /\n (x*(x^5 + 30*x^4 + 150*x^3 + 200*x^2 + 75*x + 6))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ ln_lower_11 x = 1 / 30 * (142 * x ^ 5 + 2272 * x ^ 4 + 6397 * x ^ 3 + 4397 * x\\<^sup>2 + 647 * x + 5) * (x - 1) / (x * (x ^ 5 + 30 * x ^ 4 + 150 * x ^ 3 + 200 * x\\<^sup>2 + 75 * x + 6))\n[PROOF STEP]\nunfolding ln_lower_11_def ln_upper_11_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < x \\ - ((5 * inverse x ^ 5 + 647 * inverse x ^ 4 + 4397 * inverse x ^ 3 + 6397 * (inverse x)\\<^sup>2 + 2272 * inverse x + 142) * (inverse x - 1) / (30 * (6 * inverse x ^ 5 + 75 * inverse x ^ 4 + 200 * inverse x ^ 3 + 150 * (inverse x)\\<^sup>2 + 30 * inverse x + 1))) = 1 / 30 * (142 * x ^ 5 + 2272 * x ^ 4 + 6397 * x ^ 3 + 4397 * x\\<^sup>2 + 647 * x + 5) * (x - 1) / (x * (x ^ 5 + 30 * x ^ 4 + 150 * x ^ 3 + 200 * x\\<^sup>2 + 75 * x + 6))\n[PROOF STEP]\nby (simp add: zero_less_mult_iff add_pos_pos dual_order.strict_implies_not_eq divide_simps)\n algebra", "meta": {"llama_tokens": 658, "file": "Special_Function_Bounds_Log_CF_Bounds", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094032139577, "lm_q2_score": 0.8080672066194946, "lm_q1q2_score": 0.7218540341020305}} {"text": "[STATEMENT]\nlemma coprime_crossproduct_int:\n fixes a b c d :: int\n assumes \"coprime a d\" and \"coprime b c\"\n shows \"\\a\\ * \\c\\ = \\b\\ * \\d\\ \\ \\a\\ = \\b\\ \\ \\c\\ = \\d\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\a\\ * \\c\\ = \\b\\ * \\d\\) = (\\a\\ = \\b\\ \\ \\c\\ = \\d\\)\n[PROOF STEP]\nusing assms coprime_crossproduct [of a d b c]\n[PROOF STATE]\nproof (prove)\nusing this:\ncoprime a d\ncoprime b c\n\\coprime a d; coprime b c\\ \\ (normalize a * normalize c = normalize b * normalize d) = (normalize a = normalize b \\ normalize c = normalize d)\n\ngoal (1 subgoal):\n 1. (\\a\\ * \\c\\ = \\b\\ * \\d\\) = (\\a\\ = \\b\\ \\ \\c\\ = \\d\\)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 407, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9161096181702032, "lm_q2_score": 0.7879311956428946, "lm_q1q2_score": 0.7218313467848039}} {"text": "[STATEMENT]\nlemma (in aGroup) ring_nsum_zeroTr:\"(\\j \\ (n::nat). f j \\ carrier A) \\ \n (\\j \\ n. f j = \\) \\ nsum A f n = \\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\j\\n. f j \\ carrier A) \\ (\\j\\n. f j = \\) \\ \\\\<^sub>e A f n = \\\n[PROOF STEP]\napply (induct_tac n)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (\\j\\0. f j \\ carrier A) \\ (\\j\\0. f j = \\) \\ \\\\<^sub>e A f 0 = \\\n 2. \\n. (\\j\\n. f j \\ carrier A) \\ (\\j\\n. f j = \\) \\ \\\\<^sub>e A f n = \\ \\ (\\j\\Suc n. f j \\ carrier A) \\ (\\j\\Suc n. f j = \\) \\ \\\\<^sub>e A f Suc n = \\\n[PROOF STEP]\napply (rule impI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (\\j\\0. f j \\ carrier A) \\ (\\j\\0. f j = \\) \\ \\\\<^sub>e A f 0 = \\\n 2. \\n. (\\j\\n. f j \\ carrier A) \\ (\\j\\n. f j = \\) \\ \\\\<^sub>e A f n = \\ \\ (\\j\\Suc n. f j \\ carrier A) \\ (\\j\\Suc n. f j = \\) \\ \\\\<^sub>e A f Suc n = \\\n[PROOF STEP]\napply (erule conjE)+\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\\\j\\0. f j \\ carrier A; \\j\\0. f j = \\\\ \\ \\\\<^sub>e A f 0 = \\\n 2. \\n. (\\j\\n. f j \\ carrier A) \\ (\\j\\n. f j = \\) \\ \\\\<^sub>e A f n = \\ \\ (\\j\\Suc n. f j \\ carrier A) \\ (\\j\\Suc n. f j = \\) \\ \\\\<^sub>e A f Suc n = \\\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. (\\j\\n. f j \\ carrier A) \\ (\\j\\n. f j = \\) \\ \\\\<^sub>e A f n = \\ \\ (\\j\\Suc n. f j \\ carrier A) \\ (\\j\\Suc n. f j = \\) \\ \\\\<^sub>e A f Suc n = \\\n[PROOF STEP]\napply (rule impI, (erule conjE)+)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. \\(\\j\\n. f j \\ carrier A) \\ (\\j\\n. f j = \\) \\ \\\\<^sub>e A f n = \\; \\j\\Suc n. f j \\ carrier A; \\j\\Suc n. f j = \\\\ \\ \\\\<^sub>e A f Suc n = \\\n[PROOF STEP]\napply (cut_tac n = n in Nsetn_sub_mem1, simp)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. \\((\\j. j \\ n) \\ \\ \\ carrier A) \\ \\\\<^sub>e A f n = \\; (\\j. j \\ Suc n) \\ \\ \\ carrier A; \\j\\Suc n. f j = \\\\ \\ \\\\<^sub>e A f n \\ \\ = \\\n[PROOF STEP]\napply (simp add:ag_inc_zero)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. \\\\\\<^sub>e A f n = \\; \\j\\Suc n. f j = \\\\ \\ \\ \\ \\ = \\\n[PROOF STEP]\napply (cut_tac ag_inc_zero,\n simp add:ag_r_zero)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1466, "file": "Group-Ring-Module_Algebra5", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972751232809, "lm_q2_score": 0.8289388040954683, "lm_q1q2_score": 0.7216718640894658}} {"text": "[STATEMENT]\nlemma iffExI:\n \"\\ \\x. P x \\ Q x; \\x. Q x \\ P x \\ \\ (\\x. P x) \\ (\\x. Q x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x. P x \\ Q x; \\x. Q x \\ P x\\ \\ (\\x. P x) = (\\x. Q x)\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 162, "file": "Automatic_Refinement_Lib_Misc", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972616934406, "lm_q2_score": 0.8289388040954683, "lm_q1q2_score": 0.7216718529569501}} {"text": "[STATEMENT]\nlemma polar_identity:\n includes notation_norm\n shows \\\\x + y\\^2 = \\x\\^2 + \\y\\^2 + 2 * Re (x \\\\<^sub>C y)\\\n \\ \\Shown in the proof of Corollary 1.5 in @{cite conway2013course}\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x + y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x + y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nhave \\(x \\\\<^sub>C y) + (y \\\\<^sub>C x) = (x \\\\<^sub>C y) + cnj (x \\\\<^sub>C y)\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\\\<^sub>C y + y \\\\<^sub>C x = x \\\\<^sub>C y + cnj (x \\\\<^sub>C y)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx \\\\<^sub>C y + y \\\\<^sub>C x = x \\\\<^sub>C y + cnj (x \\\\<^sub>C y)\n\ngoal (1 subgoal):\n 1. \\x + y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nhence \\(x \\\\<^sub>C y) + (y \\\\<^sub>C x) = 2 * Re (x \\\\<^sub>C y) \\\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\\\<^sub>C y + y \\\\<^sub>C x = x \\\\<^sub>C y + cnj (x \\\\<^sub>C y)\n\ngoal (1 subgoal):\n 1. x \\\\<^sub>C y + y \\\\<^sub>C x = complex_of_real (2 * Re (x \\\\<^sub>C y))\n[PROOF STEP]\nusing complex_add_cnj\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\\\<^sub>C y + y \\\\<^sub>C x = x \\\\<^sub>C y + cnj (x \\\\<^sub>C y)\n?z + cnj ?z = complex_of_real (2 * Re ?z)\n\ngoal (1 subgoal):\n 1. x \\\\<^sub>C y + y \\\\<^sub>C x = complex_of_real (2 * Re (x \\\\<^sub>C y))\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\nx \\\\<^sub>C y + y \\\\<^sub>C x = complex_of_real (2 * Re (x \\\\<^sub>C y))\n\ngoal (1 subgoal):\n 1. \\x + y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nhave \\\\x + y\\^2 = (x+y) \\\\<^sub>C (x+y)\\\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (\\x + y\\\\<^sup>2) = (x + y) \\\\<^sub>C (x + y)\n[PROOF STEP]\nby (simp add: cdot_square_norm)\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (\\x + y\\\\<^sup>2) = (x + y) \\\\<^sub>C (x + y)\n\ngoal (1 subgoal):\n 1. \\x + y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nhence \\\\x + y\\^2 = (x \\\\<^sub>C x) + (x \\\\<^sub>C y) + (y \\\\<^sub>C x) + (y \\\\<^sub>C y)\\\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (\\x + y\\\\<^sup>2) = (x + y) \\\\<^sub>C (x + y)\n\ngoal (1 subgoal):\n 1. complex_of_real (\\x + y\\\\<^sup>2) = x \\\\<^sub>C x + x \\\\<^sub>C y + y \\\\<^sub>C x + y \\\\<^sub>C y\n[PROOF STEP]\nby (simp add: cinner_add_left cinner_add_right)\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (\\x + y\\\\<^sup>2) = x \\\\<^sub>C x + x \\\\<^sub>C y + y \\\\<^sub>C x + y \\\\<^sub>C y\n\ngoal (1 subgoal):\n 1. \\x + y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (\\x + y\\\\<^sup>2) = x \\\\<^sub>C x + x \\\\<^sub>C y + y \\\\<^sub>C x + y \\\\<^sub>C y\n\ngoal (1 subgoal):\n 1. \\x + y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nusing \\(x \\\\<^sub>C y) + (y \\\\<^sub>C x) = 2 * Re (x \\\\<^sub>C y)\\\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (\\x + y\\\\<^sup>2) = x \\\\<^sub>C x + x \\\\<^sub>C y + y \\\\<^sub>C x + y \\\\<^sub>C y\nx \\\\<^sub>C y + y \\\\<^sub>C x = complex_of_real (2 * Re (x \\\\<^sub>C y))\n\ngoal (1 subgoal):\n 1. \\x + y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C y)\n[PROOF STEP]\nby (smt (verit, ccfv_SIG) Re_complex_of_real plus_complex.simps(1) power2_norm_eq_cinner')\n[PROOF STATE]\nproof (state)\nthis:\n\\x + y\\\\<^sup>2 = \\x\\\\<^sup>2 + \\y\\\\<^sup>2 + 2 * Re (x \\\\<^sub>C y)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2032, "file": "Complex_Bounded_Operators_Complex_Inner_Product", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615382165412809, "lm_q2_score": 0.8376199653600372, "lm_q1q2_score": 0.721641611095656}} {"text": "[STATEMENT]\nlemma stoch_non_neg_vec_norm1: assumes \"stoch_vec (v :: real ^ 'n)\" \"non_neg_vec v\" \n shows \"norm1 v = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm1 v = 1\n[PROOF STEP]\nunfolding assms(1)[unfolded stoch_vec_def, symmetric] norm1_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i\\UNIV. norm (v $h i)) = sum (($h) v) UNIV\n[PROOF STEP]\nby (rule sum.cong, insert assms(2)[unfolded non_neg_vec_def], auto)", "meta": {"llama_tokens": 201, "file": "Stochastic_Matrices_Stochastic_Matrix_Perron_Frobenius", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615382094310355, "lm_q2_score": 0.8376199694135332, "lm_q1q2_score": 0.7216416086322142}} {"text": "[STATEMENT]\nlemma row_transpose_scalar_prod_as_sum:\nassumes j: \"j < dim_col A\" and dim_v: \"dim_vec v = dim_row A\"\nshows \"row (transpose_mat A) j \\ v = (\\i = 0..T j \\ v = (\\i = 0..T j \\ v = (\\i = 0.. v = col A j \\ v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row A\\<^sup>T j \\ v = col A j \\ v\n[PROOF STEP]\nusing j row_transpose\n[PROOF STATE]\nproof (prove)\nusing this:\nj < dim_col A\n?j < dim_col ?A \\ row ?A\\<^sup>T ?j = col ?A ?j\n\ngoal (1 subgoal):\n 1. row A\\<^sup>T j \\ v = col A j \\ v\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrow A\\<^sup>T j \\ v = col A j \\ v\n\ngoal (1 subgoal):\n 1. row A\\<^sup>T j \\ v = (\\i = 0..T j \\ v = col A j \\ v\n\ngoal (1 subgoal):\n 1. row A\\<^sup>T j \\ v = (\\i = 0..i = 0.. v = (\\i = 0.. v = (\\i = 0..T j \\ v = (\\i = 0..T j \\ v = (\\i = 0..T j \\ v = (\\i = 0..T j \\ v = (\\i = 0..T j \\ v = (\\i = 0.. ?c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ cos (pi / 4)\n[PROOF STEP]\nby (simp add: cos_ge_zero)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\ cos (pi / 4)\n\ngoal (1 subgoal):\n 1. cos (pi / 4) = sqrt 2 / 2\n[PROOF STEP]\nhave \"0 = cos (pi / 4 + pi / 4)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 = cos (pi / 4 + pi / 4)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 = cos (pi / 4 + pi / 4)\n\ngoal (1 subgoal):\n 1. cos (pi / 4) = sqrt 2 / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n0 = cos (pi / 4 + pi / 4)\n\ngoal (1 subgoal):\n 1. cos (pi / 4) = sqrt 2 / 2\n[PROOF STEP]\nhave \"cos (pi / 4 + pi / 4) = ?c\\<^sup>2 - ?s\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos (pi / 4 + pi / 4) = (cos (pi / 4))\\<^sup>2 - (sin (pi / 4))\\<^sup>2\n[PROOF STEP]\nby (simp only: cos_add power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\ncos (pi / 4 + pi / 4) = (cos (pi / 4))\\<^sup>2 - (sin (pi / 4))\\<^sup>2\n\ngoal (1 subgoal):\n 1. cos (pi / 4) = sqrt 2 / 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncos (pi / 4 + pi / 4) = (cos (pi / 4))\\<^sup>2 - (sin (pi / 4))\\<^sup>2\n\ngoal (1 subgoal):\n 1. cos (pi / 4) = sqrt 2 / 2\n[PROOF STEP]\nhave \"\\ = 2 * ?c\\<^sup>2 - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cos (pi / 4))\\<^sup>2 - (sin (pi / 4))\\<^sup>2 = 2 * (cos (pi / 4))\\<^sup>2 - 1\n[PROOF STEP]\nby (simp add: sin_squared_eq)\n[PROOF STATE]\nproof (state)\nthis:\n(cos (pi / 4))\\<^sup>2 - (sin (pi / 4))\\<^sup>2 = 2 * (cos (pi / 4))\\<^sup>2 - 1\n\ngoal (1 subgoal):\n 1. cos (pi / 4) = sqrt 2 / 2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n0 = 2 * (cos (pi / 4))\\<^sup>2 - 1\n[PROOF STEP]\nhave \"?c\\<^sup>2 = (sqrt 2 / 2)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 = 2 * (cos (pi / 4))\\<^sup>2 - 1\n\ngoal (1 subgoal):\n 1. (cos (pi / 4))\\<^sup>2 = (sqrt 2 / 2)\\<^sup>2\n[PROOF STEP]\nby (simp add: power_divide)\n[PROOF STATE]\nproof (state)\nthis:\n(cos (pi / 4))\\<^sup>2 = (sqrt 2 / 2)\\<^sup>2\n\ngoal (1 subgoal):\n 1. cos (pi / 4) = sqrt 2 / 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(cos (pi / 4))\\<^sup>2 = (sqrt 2 / 2)\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(cos (pi / 4))\\<^sup>2 = (sqrt 2 / 2)\\<^sup>2\n\ngoal (1 subgoal):\n 1. cos (pi / 4) = sqrt 2 / 2\n[PROOF STEP]\nusing nonneg\n[PROOF STATE]\nproof (prove)\nusing this:\n(cos (pi / 4))\\<^sup>2 = (sqrt 2 / 2)\\<^sup>2\n0 \\ cos (pi / 4)\n\ngoal (1 subgoal):\n 1. cos (pi / 4) = sqrt 2 / 2\n[PROOF STEP]\nby (rule power2_eq_imp_eq) simp\n[PROOF STATE]\nproof (state)\nthis:\ncos (pi / 4) = sqrt 2 / 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1614, "file": null, "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199592797929, "lm_q2_score": 0.8615382165412809, "lm_q1q2_score": 0.7216416058572932}} {"text": "[STATEMENT]\nlemma lcm_div_gt_0:\n fixes a :: nat\n assumes \"a > 0\" and \"b > 0\"\n shows \"lcm a b div a > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < lcm a b div a\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 < lcm a b div a\n[PROOF STEP]\nhave \"lcm a b = (a * b) div (gcd a b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lcm a b = a * b div gcd a b\n[PROOF STEP]\nusing lcm_nat_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlcm ?x ?y = ?x * ?y div gcd ?x ?y\n\ngoal (1 subgoal):\n 1. lcm a b = a * b div gcd a b\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nlcm a b = a * b div gcd a b\n\ngoal (1 subgoal):\n 1. 0 < lcm a b div a\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nlcm a b = a * b div gcd a b\n\ngoal (1 subgoal):\n 1. 0 < lcm a b div a\n[PROOF STEP]\nhave \"\\ > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < a * b div gcd a b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\n0 < b\n\ngoal (1 subgoal):\n 1. 0 < a * b div gcd a b\n[PROOF STEP]\nby (metis assms calculation lcm_pos_nat)\n[PROOF STATE]\nproof (state)\nthis:\n0 < a * b div gcd a b\n\ngoal (1 subgoal):\n 1. 0 < lcm a b div a\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nlcm a b = a * b div gcd a b\n0 < a * b div gcd a b\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlcm a b = a * b div gcd a b\n0 < a * b div gcd a b\n\ngoal (1 subgoal):\n 1. 0 < lcm a b div a\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlcm a b = a * b div gcd a b\n0 < a * b div gcd a b\n0 < a\n0 < b\n\ngoal (1 subgoal):\n 1. 0 < lcm a b div a\n[PROOF STEP]\nby simp (metis div_greater_zero_iff div_le_mono2 div_mult_self_is_m gcd_le2_nat not_gr0)\n[PROOF STATE]\nproof (state)\nthis:\n0 < lcm a b div a\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 882, "file": "Diophantine_Eqns_Lin_Hom_Linear_Diophantine_Equations", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199471193039, "lm_q2_score": 0.8615382165412808, "lm_q1q2_score": 0.721641595380567}} {"text": "[STATEMENT]\nlemma inner_prod_swap:\n fixes x y :: \"complex vec\"\n assumes \"y \\ carrier_vec n\" and \"x \\ carrier_vec n\" \n shows \"inner_prod y x = conjugate (inner_prod x y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inner_prod y x = conjugate (inner_prod x y)\n[PROOF STEP]\napply (simp add: scalar_prod_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..xa = 0..xa. xa \\ {0.. x $ xa * cnj (y $ xa) = cnj (y $ xa) * x $ xa\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ny \\ carrier_vec n\nx \\ carrier_vec n\n\ngoal (2 subgoals):\n 1. {0..xa. xa \\ {0.. x $ xa * cnj (y $ xa) = cnj (y $ xa) * x $ xa\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 469, "file": "QHLProver_Complex_Matrix", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942290328344, "lm_q2_score": 0.8104789155369048, "lm_q1q2_score": 0.7215647012552964}} {"text": "[STATEMENT]\nlemma finite_nonempty_carrier_has_minimum:\n assumes \"carrier \\ {}\"\n shows \"\\e \\ carrier. \\m \\ carrier. m \\[relation] e\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\e\\carrier. \\m\\carrier. m \\ e\n[PROOF STEP]\nusing finite_nempty_preorder_has_min[of carrier relation] assms\n \\finite carrier\\ reflexivity total transitivity\n[PROOF STATE]\nproof (prove)\nusing this:\n\\finite carrier; carrier \\ {}; refl_on carrier relation; trans relation; total_on carrier relation\\ \\ \\x\\carrier. \\y\\carrier. y \\ x\ncarrier \\ {}\nfinite carrier\nrefl_on carrier relation\ntotal_on carrier relation\ntrans relation\n\ngoal (1 subgoal):\n 1. \\e\\carrier. \\m\\carrier. m \\ e\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 332, "file": "First_Welfare_Theorem_Preferences", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942144788076, "lm_q2_score": 0.8104789018037399, "lm_q1q2_score": 0.7215646772330073}} {"text": "[STATEMENT]\nlemma permutation_mat_id_2: assumes p: \"p permutes {..m n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. permutation_mat n (inv p) * permutation_mat n p = 1\\<^sub>m n\n[PROOF STEP]\nby (subst permutation_mat_right[OF _ p, of _ n], force, unfold permutation_mat_def, rule eq_matI, \n insert p, auto simp: permutes_lt[OF p] permutes_inverses)", "meta": {"llama_tokens": 173, "file": "Perron_Frobenius_Perron_Frobenius_General", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8221891392358014, "lm_q1q2_score": 0.7214518793303619}} {"text": "[STATEMENT]\nlemma one_add_floor: \"\\x\\ + 1 = \\x + 1\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\ + 1 = \\x + (1::'a)\\\n[PROOF STEP]\nusing floor_add_int [of x 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\ + 1 = \\x + of_int 1\\\n\ngoal (1 subgoal):\n 1. \\x\\ + 1 = \\x + (1::'a)\\\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 209, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8221891327004132, "lm_q1q2_score": 0.7214518735957105}} {"text": "[STATEMENT]\nlemma one_add_floor: \"\\x\\ + 1 = \\x + 1\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x\\ + 1 = \\x + (1::'a)\\\n[PROOF STEP]\nusing floor_add_int [of x 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\x\\ + 1 = \\x + of_int 1\\\n\ngoal (1 subgoal):\n 1. \\x\\ + 1 = \\x + (1::'a)\\\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 209, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8221891327004132, "lm_q1q2_score": 0.7214518735957105}} {"text": "[STATEMENT]\nlemma cindex_poly_smult_1: \n fixes p q::\"real poly\" and c::real\n shows \"cindex_poly a b (smult c q) p = (sign c) * cindex_poly a b q p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cindex_poly a b (smult c q) p = sign c * cindex_poly a b q p\n[PROOF STEP]\nunfolding cindex_poly_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x | poly p x = 0 \\ a < x \\ x < b. jump_poly (smult c q) p x) = sign c * (\\x | poly p x = 0 \\ a < x \\ x < b. jump_poly q p x)\n[PROOF STEP]\nusing sum_distrib_left[THEN sym, of \"sign c\" \"\\x. jump_poly q p x\"\n \"{x. poly p x = (0::real) \\ a < x \\ x < b}\"] jump_poly_smult_1\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\n\\{x. poly p x = 0 \\ a < x \\ x < b}. sign c * jump_poly q p n) = sign c * (\\x | poly p x = 0 \\ a < x \\ x < b. jump_poly q p x)\njump_poly (smult ?c ?q) ?p ?x = sign ?c * jump_poly ?q ?p ?x\n\ngoal (1 subgoal):\n 1. (\\x | poly p x = 0 \\ a < x \\ x < b. jump_poly (smult c q) p x) = sign c * (\\x | poly p x = 0 \\ a < x \\ x < b. jump_poly q p x)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 517, "file": "Sturm_Tarski_Sturm_Tarski", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767778695834, "lm_q2_score": 0.8221891305219504, "lm_q1q2_score": 0.7214518690497954}} {"text": "[STATEMENT]\nlemma dist_stereographic_infinite:\n assumes \"stereographic M1 = \\\\<^sub>h\" and \"stereographic M2 = of_complex m2\"\n shows \"dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nhave *: \"M1 = inv_stereographic \\\\<^sub>h\" \"M2 = inv_stereographic (of_complex m2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. M1 = inv_stereographic \\\\<^sub>h &&& M2 = inv_stereographic (of_complex m2)\n[PROOF STEP]\nusing inv_stereographic_is_inv assms\n[PROOF STATE]\nproof (prove)\nusing this:\ninv_stereographic = inv stereographic\nstereographic M1 = \\\\<^sub>h\nstereographic M2 = of_complex m2\n\ngoal (1 subgoal):\n 1. M1 = inv_stereographic \\\\<^sub>h &&& M2 = inv_stereographic (of_complex m2)\n[PROOF STEP]\nby (metis inv_stereographic_stereographic)+\n[PROOF STATE]\nproof (state)\nthis:\nM1 = inv_stereographic \\\\<^sub>h\nM2 = inv_stereographic (of_complex m2)\n\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nhave \"(1 + (cmod m2)\\<^sup>2) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + (cmod m2)\\<^sup>2 \\ 0\n[PROOF STEP]\nby (smt power2_less_0)\n[PROOF STATE]\nproof (state)\nthis:\n1 + (cmod m2)\\<^sup>2 \\ 0\n\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nhave \"(1 + (cmod m2)\\<^sup>2) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < 1 + (cmod m2)\\<^sup>2\n[PROOF STEP]\nby (smt realpow_square_minus_le)+\n[PROOF STATE]\nproof (state)\nthis:\n0 < 1 + (cmod m2)\\<^sup>2\n\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nhence \"sqrt (1 + cmod m2 * cmod m2) > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 + (cmod m2)\\<^sup>2\n\ngoal (1 subgoal):\n 1. 0 < sqrt (1 + cmod m2 * cmod m2)\n[PROOF STEP]\nusing real_sqrt_gt_0_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 + (cmod m2)\\<^sup>2\n(0 < sqrt ?y) = (0 < ?y)\n\ngoal (1 subgoal):\n 1. 0 < sqrt (1 + cmod m2 * cmod m2)\n[PROOF STEP]\nby (simp add: power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n0 < sqrt (1 + cmod m2 * cmod m2)\n\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nhence **: \"2 / sqrt (1 + cmod m2 * cmod m2) > 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < sqrt (1 + cmod m2 * cmod m2)\n\ngoal (1 subgoal):\n 1. 0 < 2 / sqrt (1 + cmod m2 * cmod m2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < 2 / sqrt (1 + cmod m2 * cmod m2)\n\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nhave \"(dist_riemann_sphere' M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist_riemann_sphere' M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\nusing *\n[PROOF STATE]\nproof (prove)\nusing this:\nM1 = inv_stereographic \\\\<^sub>h\nM2 = inv_stereographic (of_complex m2)\n\ngoal (1 subgoal):\n 1. (dist_riemann_sphere' M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\napply transfer\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\M1 M2 m2. \\M1 \\ unit_sphere; M1 = inv_stereographic_hcoords_r3 \\\\<^sub>h\\<^sub>c; M2 \\ unit_sphere; M2 = inv_stereographic_hcoords_r3 (of_complex_hcoords m2)\\ \\ (dist_riemann_sphere_r3 M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\napply transfer\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\M1 M2 m2. \\M1 \\ unit_sphere; M1 = inv_stereographic_cvec_r3 \\\\<^sub>v; M2 \\ unit_sphere; M2 = inv_stereographic_cvec_r3 (of_complex_cvec m2)\\ \\ (dist_riemann_sphere_r3 M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\M1 M2 m2. \\M1 \\ unit_sphere; M1 = inv_stereographic_cvec_r3 \\\\<^sub>v; M2 \\ unit_sphere; M2 = inv_stereographic_cvec_r3 (of_complex_cvec m2)\\ \\ (dist_riemann_sphere_r3 M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\nfix M1 M2 m2\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\M1 M2 m2. \\M1 \\ unit_sphere; M1 = inv_stereographic_cvec_r3 \\\\<^sub>v; M2 \\ unit_sphere; M2 = inv_stereographic_cvec_r3 (of_complex_cvec m2)\\ \\ (dist_riemann_sphere_r3 M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\nassume us: \"M1 \\ unit_sphere\" \"M2 \\ unit_sphere\" and\n *: \"M1 = inv_stereographic_cvec_r3 \\\\<^sub>v\" \"M2 = inv_stereographic_cvec_r3 (of_complex_cvec m2)\"\n[PROOF STATE]\nproof (state)\nthis:\nM1 \\ unit_sphere\nM2 \\ unit_sphere\nM1 = inv_stereographic_cvec_r3 \\\\<^sub>v\nM2 = inv_stereographic_cvec_r3 (of_complex_cvec m2)\n\ngoal (1 subgoal):\n 1. \\M1 M2 m2. \\M1 \\ unit_sphere; M1 = inv_stereographic_cvec_r3 \\\\<^sub>v; M2 \\ unit_sphere; M2 = inv_stereographic_cvec_r3 (of_complex_cvec m2)\\ \\ (dist_riemann_sphere_r3 M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\nhave \"(1 + (cmod m2)\\<^sup>2) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + (cmod m2)\\<^sup>2 \\ 0\n[PROOF STEP]\nby (smt power2_less_0)\n[PROOF STATE]\nproof (state)\nthis:\n1 + (cmod m2)\\<^sup>2 \\ 0\n\ngoal (1 subgoal):\n 1. \\M1 M2 m2. \\M1 \\ unit_sphere; M1 = inv_stereographic_cvec_r3 \\\\<^sub>v; M2 \\ unit_sphere; M2 = inv_stereographic_cvec_r3 (of_complex_cvec m2)\\ \\ (dist_riemann_sphere_r3 M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\nthus \"(dist_riemann_sphere_r3 M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 + (cmod m2)\\<^sup>2 \\ 0\n\ngoal (1 subgoal):\n 1. (dist_riemann_sphere_r3 M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\napply (subst dist_riemann_sphere_r3_inner[OF us])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + (cmod m2)\\<^sup>2 \\ 0 \\ (2 - 2 * inner M1 M2) * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\napply (subst *)+\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + (cmod m2)\\<^sup>2 \\ 0 \\ (2 - 2 * inner (inv_stereographic_cvec_r3 \\\\<^sub>v) (inv_stereographic_cvec_r3 (of_complex_cvec m2))) * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\napply (simp add: complex_mult_cnj_cmod)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 + (cmod m2)\\<^sup>2 \\ 0 \\ (2 - (2 * (cmod m2)\\<^sup>2 - 2) / (1 + (cmod m2)\\<^sup>2)) * (1 + (cmod m2)\\<^sup>2) = 4\n[PROOF STEP]\napply (subst left_diff_distrib[of 2], simp)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(dist_riemann_sphere_r3 M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(dist_riemann_sphere' M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nhence \"(dist_riemann_sphere' M1 M2)\\<^sup>2 = 4 / (1 + (cmod m2)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(dist_riemann_sphere' M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n\ngoal (1 subgoal):\n 1. (dist_riemann_sphere' M1 M2)\\<^sup>2 = 4 / (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nusing \\(1 + (cmod m2)\\<^sup>2) \\ 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\n(dist_riemann_sphere' M1 M2)\\<^sup>2 * (1 + (cmod m2)\\<^sup>2) = 4\n1 + (cmod m2)\\<^sup>2 \\ 0\n\ngoal (1 subgoal):\n 1. (dist_riemann_sphere' M1 M2)\\<^sup>2 = 4 / (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(dist_riemann_sphere' M1 M2)\\<^sup>2 = 4 / (1 + (cmod m2)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nthus \"dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(dist_riemann_sphere' M1 M2)\\<^sup>2 = 4 / (1 + (cmod m2)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nusing power2_eq_iff[of \"dist_riemann_sphere' M1 M2\" \"2 / sqrt (1 + (cmod m2)\\<^sup>2)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(dist_riemann_sphere' M1 M2)\\<^sup>2 = 4 / (1 + (cmod m2)\\<^sup>2)\n((dist_riemann_sphere' M1 M2)\\<^sup>2 = (2 / sqrt (1 + (cmod m2)\\<^sup>2))\\<^sup>2) = (dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2) \\ dist_riemann_sphere' M1 M2 = - (2 / sqrt (1 + (cmod m2)\\<^sup>2)))\n\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\nusing \\(1 + (cmod m2)\\<^sup>2) > 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\n(dist_riemann_sphere' M1 M2)\\<^sup>2 = 4 / (1 + (cmod m2)\\<^sup>2)\n((dist_riemann_sphere' M1 M2)\\<^sup>2 = (2 / sqrt (1 + (cmod m2)\\<^sup>2))\\<^sup>2) = (dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2) \\ dist_riemann_sphere' M1 M2 = - (2 / sqrt (1 + (cmod m2)\\<^sup>2)))\n0 < 1 + (cmod m2)\\<^sup>2\n\ngoal (1 subgoal):\n 1. dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n[PROOF STEP]\napply (auto simp add: power2_eq_square real_sqrt_mult[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\0 < 1 + cmod m2 * cmod m2; dist_riemann_sphere' M1 M2 = - (2 / sqrt (1 + cmod m2 * cmod m2))\\ \\ False\n[PROOF STEP]\nusing dist_riemann_sphere_ge_0[of M1 M2] **\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\ dist_riemann_sphere' M1 M2\n0 < 2 / sqrt (1 + cmod m2 * cmod m2)\n\ngoal (1 subgoal):\n 1. \\0 < 1 + cmod m2 * cmod m2; dist_riemann_sphere' M1 M2 = - (2 / sqrt (1 + cmod m2 * cmod m2))\\ \\ False\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\\<^sup>2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 5073, "file": "Complex_Geometry_Chordal_Metric", "length": 39, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278757303678, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7214505141623947}} {"text": "[STATEMENT]\nlemma linear_continuous_compose:\n fixes f :: \"'a::euclidean_space \\ 'b::euclidean_space\" and g :: \"'b \\ 'c::real_normed_vector\"\n assumes \"continuous F f\" \"linear g\"\n shows \"continuous F (\\x. g(f x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. continuous F (\\x. g (f x))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous F f\nlinear g\n\ngoal (1 subgoal):\n 1. continuous F (\\x. g (f x))\n[PROOF STEP]\nunfolding continuous_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(f \\ f (Lim F (\\x. x))) F\nlinear g\n\ngoal (1 subgoal):\n 1. ((\\x. g (f x)) \\ g (f (Lim F (\\x. x)))) F\n[PROOF STEP]\nby (rule Lim_linear)", "meta": {"llama_tokens": 296, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278664544911, "lm_q2_score": 0.8175744739711884, "lm_q1q2_score": 0.7214504987340487}} {"text": "[STATEMENT]\nlemma independent_card_le_dim: assumes \"B \\ V\" and \"independent B\" shows \"card B \\ dim V\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card B \\ dim V\n[PROOF STEP]\nby (subst dim_eq_card[symmetric, OF refl \\independent B\\]) (rule dim_subset[OF \\B \\ V\\])", "meta": {"llama_tokens": 122, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278602705732, "lm_q2_score": 0.817574478416099, "lm_q1q2_score": 0.7214504976005481}} {"text": "[STATEMENT]\nlemma independent_card_le_dim: assumes \"B \\ V\" and \"independent B\" shows \"card B \\ dim V\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card B \\ dim V\n[PROOF STEP]\nby (subst dim_eq_card[symmetric, OF refl \\independent B\\]) (rule dim_subset[OF \\B \\ V\\])", "meta": {"llama_tokens": 122, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278602705732, "lm_q2_score": 0.8175744761936437, "lm_q1q2_score": 0.7214504956393917}} {"text": "[STATEMENT]\nlemma independent_card_le_dim: assumes \"B \\ V\" and \"independent B\" shows \"card B \\ dim V\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card B \\ dim V\n[PROOF STEP]\nby (subst dim_eq_card[symmetric, OF refl \\independent B\\]) (rule dim_subset[OF \\B \\ V\\])", "meta": {"llama_tokens": 122, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278602705732, "lm_q2_score": 0.8175744739711883, "lm_q1q2_score": 0.7214504936782351}} {"text": "[STATEMENT]\nlemma square_integrable_imp_integrable_product:\n assumes \"f square_integrable S\" \"g square_integrable S\"\n shows \"integrable (lebesgue_on S) (\\x. f x * g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. integrable (lebesgue_on S) (\\x. f x * g x)\n[PROOF STEP]\nusing absolutely_integrable_measurable assms integrable_abs_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n?S \\ sets lebesgue \\ (?f absolutely_integrable_on ?S) = (?f \\ borel_measurable (lebesgue_on ?S) \\ integrable (lebesgue_on ?S) (norm \\ ?f))\nf square_integrable S\ng square_integrable S\n?f \\ borel_measurable ?M \\ integrable ?M (\\x. \\?f x\\) = integrable ?M ?f\n\ngoal (1 subgoal):\n 1. integrable (lebesgue_on S) (\\x. f x * g x)\n[PROOF STEP]\nby (metis (full_types) absolutely_integrable_measurable_real square_integrable_def square_integrable_imp_absolutely_integrable_product)", "meta": {"llama_tokens": 370, "file": "Fourier_Square_Integrable", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278602705731, "lm_q2_score": 0.8175744739711883, "lm_q1q2_score": 0.721450493678235}} {"text": "[STATEMENT]\nlemma subgroup_Int:\n assumes \"subgroup I G\" \"subgroup J G\"\n shows \"subgroup (I \\ J) G\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. subgroup (I \\ J) G\n[PROOF STEP]\nusing subgroup_Inter[ where ?A = \"{I,J}\"] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\\\H. H \\ {I, J} \\ subgroup H ?G; {I, J} \\ {}\\ \\ subgroup (\\ {I, J}) ?G\nsubgroup I G\nsubgroup J G\n\ngoal (1 subgoal):\n 1. subgroup (I \\ J) G\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 215, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.863391602943619, "lm_q2_score": 0.8354835391516132, "lm_q1q2_score": 0.7213494721011192}} {"text": "[STATEMENT]\ntheorem mset_quicksort [simp]: \"mset (quicksort R xs) = mset xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mset (quicksort R xs) = mset xs\n[PROOF STEP]\nby (induction R xs rule: quicksort.induct) (simp_all)", "meta": {"llama_tokens": 92, "file": "Quick_Sort_Cost_Quick_Sort_Average_Case", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9184802350995703, "lm_q2_score": 0.7853085859124002, "lm_q1q2_score": 0.7212904146145325}} {"text": "[STATEMENT]\nlemma mtx_times_scaleR_commute: \"A * (c *\\<^sub>R B) = c *\\<^sub>R (A * B)\" for A::\"('n::finite) sq_mtx\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A * c *\\<^sub>R B = c *\\<^sub>R (A * B)\n[PROOF STEP]\nunfolding sq_mtx_scaleR_eq sq_mtx_times_eq\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. to_mtx (\\i j. \\k\\UNIV. A $$ i $ k * to_mtx (\\i j. c *\\<^sub>R B $$ i $ j) $$ k $ j) = to_mtx (\\i j. c *\\<^sub>R to_mtx (\\i j. \\k\\UNIV. A $$ i $ k * B $$ k $ j) $$ i $ j)\n[PROOF STEP]\napply(simp add: to_mtx_inject)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\i. \\j. \\k\\UNIV. A $$ i $ k * (c * B $$ k $ j)) = (\\i. \\j. c * (\\k\\UNIV. A $$ i $ k * B $$ k $ j))\n[PROOF STEP]\napply(simp add: vec_eq_iff fun_eq_iff)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x i. (\\k\\UNIV. A $$ x $ k * (c * B $$ k $ i)) = c * (\\k\\UNIV. A $$ x $ k * B $$ k $ i)\n[PROOF STEP]\nby (simp add: semiring_normalization_rules(19) vector_space_over_itself.scale_sum_right)", "meta": {"llama_tokens": 520, "file": "Matrices_for_ODEs_SQ_MTX", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8723473713594991, "lm_q2_score": 0.8267117962054049, "lm_q1q2_score": 0.721179862291675}} {"text": "[STATEMENT]\nlemma comp_assoc:\"\\f \\ A \\ B; g \\ B \\ C; h \\ C \\ D \\ \\\n compose A h (compose A g f) = compose A (compose B h g) f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ A \\ B; g \\ B \\ C; h \\ C \\ D\\ \\ compose A h (compose A g f) = compose A (compose B h g) f\n[PROOF STEP]\napply (rule funcset_eq[of _ \"A\"])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\f \\ A \\ B; g \\ B \\ C; h \\ C \\ D\\ \\ compose A h (compose A g f) \\ extensional A\n 2. \\f \\ A \\ B; g \\ B \\ C; h \\ C \\ D\\ \\ compose A (compose B h g) f \\ extensional A\n 3. \\f \\ A \\ B; g \\ B \\ C; h \\ C \\ D\\ \\ \\x\\A. compose A h (compose A g f) x = compose A (compose B h g) f x\n[PROOF STEP]\napply (simp add:compose_def)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\f \\ A \\ B; g \\ B \\ C; h \\ C \\ D\\ \\ compose A (compose B h g) f \\ extensional A\n 2. \\f \\ A \\ B; g \\ B \\ C; h \\ C \\ D\\ \\ \\x\\A. compose A h (compose A g f) x = compose A (compose B h g) f x\n[PROOF STEP]\napply (simp add:compose_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\f \\ A \\ B; g \\ B \\ C; h \\ C \\ D\\ \\ \\x\\A. compose A h (compose A g f) x = compose A (compose B h g) f x\n[PROOF STEP]\napply (rule ballI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\f \\ A \\ B; g \\ B \\ C; h \\ C \\ D; x \\ A\\ \\ compose A h (compose A g f) x = compose A (compose B h g) f x\n[PROOF STEP]\napply (simp add:funcset_mem[of \"f\" \"A\" \"B\"] compose_def)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 868, "file": "Group-Ring-Module_Algebra1", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045907347107, "lm_q2_score": 0.8128673246376009, "lm_q1q2_score": 0.7211796220767219}} {"text": "[STATEMENT]\nlemma sum_list_mono:\n fixes xs ys::\"'a::ordered_ab_group_add list\"\n shows\n \"length xs = length ys \\ (\\x y. (x, y) \\ set (zip xs ys) \\ x \\ y) \\\n sum_list xs \\ sum_list ys\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\length xs = length ys; \\x y. (x, y) \\ set (zip xs ys) \\ x \\ y\\ \\ sum_list xs \\ sum_list ys\n[PROOF STEP]\nby (induct xs ys rule: list_induct2) (auto simp: algebra_simps intro: add_mono)", "meta": {"llama_tokens": 220, "file": "Affine_Arithmetic_Affine_Form", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045937171068, "lm_q2_score": 0.8128673110375457, "lm_q1q2_score": 0.7211796124349829}} {"text": "[STATEMENT]\nlemma gbinomial_r_part_sum: \"(\\k\\m. (2 * (of_nat m) + 1 gchoose k)) = 2 ^ (2 * m)\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nhave \"?lhs = of_nat (\\k\\m. (2 * m + 1) choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n[PROOF STEP]\nby (simp add: binomial_gbinomial add_ac)\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nhave \"\\ = of_nat (2 ^ (2 * m))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (sum ((choose) (2 * m + 1)) {..m}) = of_nat (2 ^ (2 * m))\n[PROOF STEP]\nby (subst binomial_r_part_sum) (rule refl)\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (sum ((choose) (2 * m + 1)) {..m}) = of_nat (2 ^ (2 * m))\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (2 ^ (2 * m))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (2 ^ (2 * m))\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1064, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045877523147, "lm_q2_score": 0.8128673155708976, "lm_q1q2_score": 0.7211796116084089}} {"text": "[STATEMENT]\nlemma gbinomial_r_part_sum: \"(\\k\\m. (2 * (of_nat m) + 1 gchoose k)) = 2 ^ (2 * m)\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nhave \"?lhs = of_nat (\\k\\m. (2 * m + 1) choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n[PROOF STEP]\nby (simp add: binomial_gbinomial add_ac)\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nhave \"\\ = of_nat (2 ^ (2 * m))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (sum ((choose) (2 * m + 1)) {..m}) = of_nat (2 ^ (2 * m))\n[PROOF STEP]\nby (subst binomial_r_part_sum) (rule refl)\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (sum ((choose) (2 * m + 1)) {..m}) = of_nat (2 ^ (2 * m))\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (2 ^ (2 * m))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (2 ^ (2 * m))\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1064, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045877523147, "lm_q2_score": 0.8128673155708976, "lm_q1q2_score": 0.7211796116084089}} {"text": "[STATEMENT]\nlemma gbinomial_r_part_sum: \"(\\k\\m. (2 * (of_nat m) + 1 gchoose k)) = 2 ^ (2 * m)\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nhave \"?lhs = of_nat (\\k\\m. (2 * m + 1) choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n[PROOF STEP]\nby (simp add: binomial_gbinomial add_ac)\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nhave \"\\ = of_nat (2 ^ (2 * m))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (sum ((choose) (2 * m + 1)) {..m}) = of_nat (2 ^ (2 * m))\n[PROOF STEP]\nby (subst binomial_r_part_sum) (rule refl)\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (sum ((choose) (2 * m + 1)) {..m}) = of_nat (2 ^ (2 * m))\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (2 ^ (2 * m))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (2 ^ (2 * m))\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1064, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045877523147, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.7211796095974037}} {"text": "[STATEMENT]\nlemma gbinomial_r_part_sum: \"(\\k\\m. (2 * (of_nat m) + 1 gchoose k)) = 2 ^ (2 * m)\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nhave \"?lhs = of_nat (\\k\\m. (2 * m + 1) choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n[PROOF STEP]\nby (simp add: binomial_gbinomial add_ac)\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (sum ((choose) (2 * m + 1)) {..m})\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nhave \"\\ = of_nat (2 ^ (2 * m))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (sum ((choose) (2 * m + 1)) {..m}) = of_nat (2 ^ (2 * m))\n[PROOF STEP]\nby (subst binomial_r_part_sum) (rule refl)\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (sum ((choose) (2 * m + 1)) {..m}) = of_nat (2 ^ (2 * m))\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (2 ^ (2 * m))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = of_nat (2 ^ (2 * m))\n\ngoal (1 subgoal):\n 1. sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1064, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045877523147, "lm_q2_score": 0.8128673133042217, "lm_q1q2_score": 0.7211796095974037}} {"text": "[STATEMENT]\nlemma arsinh_real_has_field_derivative [derivative_intros]:\n fixes x :: real\n shows \"(arsinh has_field_derivative (1 / (sqrt (x ^ 2 + 1)))) (at x within A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (arsinh has_real_derivative 1 / sqrt (x\\<^sup>2 + 1)) (at x within A)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (arsinh has_real_derivative 1 / sqrt (x\\<^sup>2 + 1)) (at x within A)\n[PROOF STEP]\nhave pos: \"1 + x ^ 2 > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < 1 + x\\<^sup>2\n[PROOF STEP]\nby (intro add_pos_nonneg) auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < 1 + x\\<^sup>2\n\ngoal (1 subgoal):\n 1. (arsinh has_real_derivative 1 / sqrt (x\\<^sup>2 + 1)) (at x within A)\n[PROOF STEP]\nfrom pos arsinh_real_aux[of x]\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < 1 + x\\<^sup>2\n0 < x + sqrt (x\\<^sup>2 + 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 + x\\<^sup>2\n0 < x + sqrt (x\\<^sup>2 + 1)\n\ngoal (1 subgoal):\n 1. (arsinh has_real_derivative 1 / sqrt (x\\<^sup>2 + 1)) (at x within A)\n[PROOF STEP]\nunfolding arsinh_def [abs_def]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < 1 + x\\<^sup>2\n0 < x + sqrt (x\\<^sup>2 + 1)\n\ngoal (1 subgoal):\n 1. ((\\x. ln (x + (x\\<^sup>2 + 1) powr of_real (1 / 2))) has_real_derivative 1 / sqrt (x\\<^sup>2 + 1)) (at x within A)\n[PROOF STEP]\nby (auto intro!: derivative_eq_intros simp: powr_minus powr_half_sqrt field_split_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(arsinh has_real_derivative 1 / sqrt (x\\<^sup>2 + 1)) (at x within A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 755, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240895276223, "lm_q2_score": 0.8333246015211009, "lm_q1q2_score": 0.7210125196320631}} {"text": "[STATEMENT]\nlemma chinese_remainder_nat:\n fixes m1 m2 :: nat\n assumes a: \"coprime m1 m2\"\n shows \"\\x. x mod m1 = u1 mod m1 \\ x mod m2 = u2 mod m2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. x mod m1 = u1 mod m1 \\ x mod m2 = u2 mod m2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x mod m1 = u1 mod m1 \\ x mod m2 = u2 mod m2\n[PROOF STEP]\nfrom chinese_remainder_aux_nat [OF a]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\b1 b2. b1 mod m1 = 1 mod m1 \\ b1 mod m2 = 0 mod m2 \\ b2 mod m1 = 0 mod m1 \\ b2 mod m2 = 1 mod m2\n[PROOF STEP]\nobtain b1 b2 where \"b1 mod m1 = 1 mod m1\" and \"b1 mod m2 = 0 mod m2\" and\n\"b2 mod m1 = 0 mod m1\" and \"b2 mod m2 = 1 mod m2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\b1 b2. b1 mod m1 = 1 mod m1 \\ b1 mod m2 = 0 mod m2 \\ b2 mod m1 = 0 mod m1 \\ b2 mod m2 = 1 mod m2\n\ngoal (1 subgoal):\n 1. (\\b1 b2. \\b1 mod m1 = 1 mod m1; b1 mod m2 = 0 mod m2; b2 mod m1 = 0 mod m1; b2 mod m2 = 1 mod m2\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\nb1 mod m1 = 1 mod m1\nb1 mod m2 = 0 mod m2\nb2 mod m1 = 0 mod m1\nb2 mod m2 = 1 mod m2\n\ngoal (1 subgoal):\n 1. \\x. x mod m1 = u1 mod m1 \\ x mod m2 = u2 mod m2\n[PROOF STEP]\nlet ?x = \"u1*b1+u2*b2\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\x. x mod m1 = u1 mod m1 \\ x mod m2 = u2 mod m2\n[PROOF STEP]\nhave \"?x mod m1 = (u1*1+u2*0) mod m1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (u1 * b1 + u2 * b2) mod m1 = (u1 * 1 + u2 * 0) mod m1\n[PROOF STEP]\napply (rule mod_add_cong)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. u1 * b1 mod m1 = u1 * 1 mod m1\n 2. u2 * b2 mod m1 = u2 * 0 mod m1\n[PROOF STEP]\napply(rule cong_scalar2_nat)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. b1 mod m1 = 1 mod m1\n 2. u2 * b2 mod m1 = u2 * 0 mod m1\n[PROOF STEP]\napply (rule \\b1 mod m1 = 1 mod m1\\)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. u2 * b2 mod m1 = u2 * 0 mod m1\n[PROOF STEP]\napply(rule cong_scalar2_nat)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. b2 mod m1 = 0 mod m1\n[PROOF STEP]\napply (rule \\b2 mod m1 = 0 mod m1\\)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(u1 * b1 + u2 * b2) mod m1 = (u1 * 1 + u2 * 0) mod m1\n\ngoal (1 subgoal):\n 1. \\x. x mod m1 = u1 mod m1 \\ x mod m2 = u2 mod m2\n[PROOF STEP]\nhence 1:\"?x mod m1 = u1 mod m1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(u1 * b1 + u2 * b2) mod m1 = (u1 * 1 + u2 * 0) mod m1\n\ngoal (1 subgoal):\n 1. (u1 * b1 + u2 * b2) mod m1 = u1 mod m1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(u1 * b1 + u2 * b2) mod m1 = u1 mod m1\n\ngoal (1 subgoal):\n 1. \\x. x mod m1 = u1 mod m1 \\ x mod m2 = u2 mod m2\n[PROOF STEP]\nhave \"?x mod m2 = (u1*0+u2*1) mod m2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (u1 * b1 + u2 * b2) mod m2 = (u1 * 0 + u2 * 1) mod m2\n[PROOF STEP]\napply (rule mod_add_cong)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. u1 * b1 mod m2 = u1 * 0 mod m2\n 2. u2 * b2 mod m2 = u2 * 1 mod m2\n[PROOF STEP]\napply(rule cong_scalar2_nat)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. b1 mod m2 = 0 mod m2\n 2. u2 * b2 mod m2 = u2 * 1 mod m2\n[PROOF STEP]\napply (rule \\b1 mod m2 = 0 mod m2\\)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. u2 * b2 mod m2 = u2 * 1 mod m2\n[PROOF STEP]\napply(rule cong_scalar2_nat)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. b2 mod m2 = 1 mod m2\n[PROOF STEP]\napply (rule \\b2 mod m2 = 1 mod m2\\)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(u1 * b1 + u2 * b2) mod m2 = (u1 * 0 + u2 * 1) mod m2\n\ngoal (1 subgoal):\n 1. \\x. x mod m1 = u1 mod m1 \\ x mod m2 = u2 mod m2\n[PROOF STEP]\nhence \"?x mod m2 = u2 mod m2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(u1 * b1 + u2 * b2) mod m2 = (u1 * 0 + u2 * 1) mod m2\n\ngoal (1 subgoal):\n 1. (u1 * b1 + u2 * b2) mod m2 = u2 mod m2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(u1 * b1 + u2 * b2) mod m2 = u2 mod m2\n\ngoal (1 subgoal):\n 1. \\x. x mod m1 = u1 mod m1 \\ x mod m2 = u2 mod m2\n[PROOF STEP]\nwith 1\n[PROOF STATE]\nproof (chain)\npicking this:\n(u1 * b1 + u2 * b2) mod m1 = u1 mod m1\n(u1 * b1 + u2 * b2) mod m2 = u2 mod m2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(u1 * b1 + u2 * b2) mod m1 = u1 mod m1\n(u1 * b1 + u2 * b2) mod m2 = u2 mod m2\n\ngoal (1 subgoal):\n 1. \\x. x mod m1 = u1 mod m1 \\ x mod m2 = u2 mod m2\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\x. x mod m1 = u1 mod m1 \\ x mod m2 = u2 mod m2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2431, "file": "DPRM_Theorem_Diophantine_Exponentiation", "length": 27, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240860523327, "lm_q2_score": 0.8333245994514084, "lm_q1q2_score": 0.721012514945271}} {"text": "[STATEMENT]\nlemma GDERIV_mult:\n \"\\cGDERIV f x :> df; cGDERIV g x :> dg\\\n \\ cGDERIV (\\x. f x * g x) x :> cnj (f x) *\\<^sub>C dg + cnj (g x) *\\<^sub>C df\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\cGDERIV f x :> df; cGDERIV g x :> dg\\ \\ cGDERIV (\\x. f x * g x) x :> cnj (f x) *\\<^sub>C dg + cnj (g x) *\\<^sub>C df\n[PROOF STEP]\nunfolding cgderiv_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_derivative cinner df) (at x); (g has_derivative cinner dg) (at x)\\ \\ ((\\x. f x * g x) has_derivative cinner (cnj (f x) *\\<^sub>C dg + cnj (g x) *\\<^sub>C df)) (at x)\n[PROOF STEP]\napply (rule has_derivative_subst)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\(f has_derivative cinner df) (at x); (g has_derivative cinner dg) (at x)\\ \\ ((\\x. f x * g x) has_derivative ?df) (at x)\n 2. \\(f has_derivative cinner df) (at x); (g has_derivative cinner dg) (at x)\\ \\ ?df = cinner (cnj (f x) *\\<^sub>C dg + cnj (g x) *\\<^sub>C df)\n[PROOF STEP]\napply (erule (1) has_derivative_mult)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_derivative cinner df) (at x); (g has_derivative cinner dg) (at x)\\ \\ (\\h. f x * cinner dg h + cinner df h * g x) = cinner (cnj (f x) *\\<^sub>C dg + cnj (g x) *\\<^sub>C df)\n[PROOF STEP]\napply (rule ext)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\h. \\(f has_derivative cinner df) (at x); (g has_derivative cinner dg) (at x)\\ \\ f x * cinner dg h + cinner df h * g x = cinner (cnj (f x) *\\<^sub>C dg + cnj (g x) *\\<^sub>C df) h\n[PROOF STEP]\nby (simp add: cinner_add ac_simps)", "meta": {"llama_tokens": 834, "file": "Complex_Bounded_Operators_Complex_Inner_Product0", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.897695283896349, "lm_q2_score": 0.8031738034238807, "lm_q1q2_score": 0.721005335482711}} {"text": "[STATEMENT]\nlemma Max_Int_le2: \"\\ (A \\ B) \\ {}; finite B \\ \\ Max (A \\ B) \\ Max B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\A \\ B \\ {}; finite B\\ \\ Max (A \\ B) \\ Max B\n[PROOF STEP]\nby (rule Max_subset[OF _ Int_lower2])", "meta": {"llama_tokens": 139, "file": "List-Infinite_CommonSet_SetInterval2", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942067038784, "lm_q2_score": 0.7981867753392728, "lm_q1q2_score": 0.7209974900316153}} {"text": "[STATEMENT]\nlemma aff_dim_convex_hull:\n fixes S :: \"'n::euclidean_space set\"\n shows \"aff_dim (convex hull S) = aff_dim S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. aff_dim (convex hull S) = aff_dim S\n[PROOF STEP]\nusing aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]\n hull_subset[of S \"convex\"] aff_dim_subset[of S \"convex hull S\"]\n aff_dim_subset[of \"convex hull S\" \"affine hull S\"]\n[PROOF STATE]\nproof (prove)\nusing this:\naff_dim (affine hull S) = aff_dim S\nconvex hull S \\ affine hull S\nS \\ convex hull S\nS \\ convex hull S \\ aff_dim S \\ aff_dim (convex hull S)\nconvex hull S \\ affine hull S \\ aff_dim (convex hull S) \\ aff_dim (affine hull S)\n\ngoal (1 subgoal):\n 1. aff_dim (convex hull S) = aff_dim S\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 342, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.914900950352329, "lm_q2_score": 0.7879311856832191, "lm_q1q2_score": 0.7208789905938146}} {"text": "[STATEMENT]\nlemma complex_mod_mult_cnj: \"cmod (z * cnj z) = (cmod z)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (z * cnj z) = (cmod z)\\<^sup>2\n[PROOF STEP]\nby (simp add: norm_mult power2_eq_square)", "meta": {"llama_tokens": 107, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894520743981, "lm_q2_score": 0.8056321936479701, "lm_q1q2_score": 0.7208711891277626}} {"text": "[STATEMENT]\nlemma inversion_id_iff: \n shows \"a = inversion a \\ a \\ \\\\<^sub>h \\ (to_complex a) * cnj (to_complex a) = 1\" (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a = inversion a) = (a \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. a = inversion a \\ a \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1\n 2. a \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1 \\ a = inversion a\n[PROOF STEP]\nassume \"a = inversion a\"\n[PROOF STATE]\nproof (state)\nthis:\na = inversion a\n\ngoal (2 subgoals):\n 1. a = inversion a \\ a \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1\n 2. a \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1 \\ a = inversion a\n[PROOF STEP]\nthus ?rhs\n[PROOF STATE]\nproof (prove)\nusing this:\na = inversion a\n\ngoal (1 subgoal):\n 1. a \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1\n[PROOF STEP]\nunfolding inversion_def\n[PROOF STATE]\nproof (prove)\nusing this:\na = (conjugate \\ reciprocal) a\n\ngoal (1 subgoal):\n 1. a \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1\n[PROOF STEP]\nusing inf_or_of_complex[of a]\n[PROOF STATE]\nproof (prove)\nusing this:\na = (conjugate \\ reciprocal) a\na = \\\\<^sub>h \\ (\\x. a = of_complex x)\n\ngoal (1 subgoal):\n 1. a \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1\n[PROOF STEP]\nby (metis (full_types) comp_apply complex_cnj_cancel_iff complex_cnj_zero inversion_def inversion_infty inversion_of_complex inversion_sym nonzero_eq_divide_eq of_complex_zero reciprocal_zero to_complex_of_complex zero_one_infty_not_equal(5))\n[PROOF STATE]\nproof (state)\nthis:\na \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1\n\ngoal (1 subgoal):\n 1. a \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1 \\ a = inversion a\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. a \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1 \\ a = inversion a\n[PROOF STEP]\nassume ?rhs\n[PROOF STATE]\nproof (state)\nthis:\na \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1\n\ngoal (1 subgoal):\n 1. a \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1 \\ a = inversion a\n[PROOF STEP]\nthus ?lhs\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1\n\ngoal (1 subgoal):\n 1. a = inversion a\n[PROOF STEP]\nusing inf_or_of_complex[of a]\n[PROOF STATE]\nproof (prove)\nusing this:\na \\ \\\\<^sub>h \\ to_complex a * cnj (to_complex a) = 1\na = \\\\<^sub>h \\ (\\x. a = of_complex x)\n\ngoal (1 subgoal):\n 1. a = inversion a\n[PROOF STEP]\nby (metis inversion_of_complex mult_not_zero nonzero_mult_div_cancel_right one_neq_zero to_complex_of_complex)\n[PROOF STATE]\nproof (state)\nthis:\na = inversion a\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1370, "file": "Complex_Geometry_Homogeneous_Coordinates", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467706759583, "lm_q2_score": 0.8198933315126792, "lm_q1q2_score": 0.7208065746981248}} {"text": "[STATEMENT]\ntheorem join_commute: \"x \\ y = y \\ x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y = y \\ x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x \\ y = y \\ x\n[PROOF STEP]\nhave \"dual (x \\ y) = dual x \\ dual y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dual (x \\ y) = dual x \\ dual y\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\ndual (x \\ y) = dual x \\ dual y\n\ngoal (1 subgoal):\n 1. x \\ y = y \\ x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndual (x \\ y) = dual x \\ dual y\n\ngoal (1 subgoal):\n 1. x \\ y = y \\ x\n[PROOF STEP]\nhave \"\\ = dual y \\ dual x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dual x \\ dual y = dual y \\ dual x\n[PROOF STEP]\nby (rule meet_commute)\n[PROOF STATE]\nproof (state)\nthis:\ndual x \\ dual y = dual y \\ dual x\n\ngoal (1 subgoal):\n 1. x \\ y = y \\ x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndual x \\ dual y = dual y \\ dual x\n\ngoal (1 subgoal):\n 1. x \\ y = y \\ x\n[PROOF STEP]\nhave \"\\ = dual (y \\ x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dual y \\ dual x = dual (y \\ x)\n[PROOF STEP]\nby (simp only: dual_join)\n[PROOF STATE]\nproof (state)\nthis:\ndual y \\ dual x = dual (y \\ x)\n\ngoal (1 subgoal):\n 1. x \\ y = y \\ x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndual (x \\ y) = dual (y \\ x)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndual (x \\ y) = dual (y \\ x)\n\ngoal (1 subgoal):\n 1. x \\ y = y \\ x\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nx \\ y = y \\ x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 890, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711794579723, "lm_q2_score": 0.8479677602988602, "lm_q1q2_score": 0.7207481573635574}} {"text": "[STATEMENT]\nlemma not_on_path_cball:\n fixes g :: \"real \\ 'a::heine_borel\"\n assumes \"path g\"\n and \"z \\ path_image g\"\n shows \"\\e>0. cball z e \\ (path_image g) = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\e>0. cball z e \\ path_image g = {}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\e>0. cball z e \\ path_image g = {}\n[PROOF STEP]\nobtain e where \"ball z e \\ path_image g = {}\" \"e > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\e. \\ball z e \\ path_image g = {}; 0 < e\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing not_on_path_ball[OF assms]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\e>0. ball z e \\ path_image g = {}\n\ngoal (1 subgoal):\n 1. (\\e. \\ball z e \\ path_image g = {}; 0 < e\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nball z e \\ path_image g = {}\n0 < e\n\ngoal (1 subgoal):\n 1. \\e>0. cball z e \\ path_image g = {}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nball z e \\ path_image g = {}\n0 < e\n\ngoal (1 subgoal):\n 1. \\e>0. cball z e \\ path_image g = {}\n[PROOF STEP]\nhave \"cball z (e/2) \\ ball z e\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cball z (e / 2) \\ ball z e\n[PROOF STEP]\nusing \\e > 0\\\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < e\n\ngoal (1 subgoal):\n 1. cball z (e / 2) \\ ball z e\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncball z (e / 2) \\ ball z e\n\ngoal (1 subgoal):\n 1. \\e>0. cball z e \\ path_image g = {}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nball z e \\ path_image g = {}\n0 < e\ncball z (e / 2) \\ ball z e\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nball z e \\ path_image g = {}\n0 < e\ncball z (e / 2) \\ ball z e\n\ngoal (1 subgoal):\n 1. \\e>0. cball z e \\ path_image g = {}\n[PROOF STEP]\nby (rule_tac x=\"e/2\" in exI) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\e>0. cball z e \\ path_image g = {}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 984, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8479677506936879, "lm_q2_score": 0.849971181358171, "lm_q1q2_score": 0.720748150810745}} {"text": "[STATEMENT]\nlemma choose_two: \"n choose 2 = n * (n - 1) div 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nproof (cases \"n \\ 2\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ 2 \\ n\n\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ 2 \\ n\n[PROOF STEP]\nhave \"n = 0 \\ n = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 2 \\ n\n\ngoal (1 subgoal):\n 1. n = 0 \\ n = 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nn = 0 \\ n = 1\n\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = 0 \\ n = 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 0 \\ n = 1\n\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nn choose 2 = n * (n - 1) div 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ n\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ndefine m where \"m = n - 2\"\n[PROOF STATE]\nproof (state)\nthis:\nm = n - 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nwith True\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ n\nm = n - 2\n[PROOF STEP]\nhave \"n = m + 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\nm = n - 2\n\ngoal (1 subgoal):\n 1. n = m + 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn = m + 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = m + 2\n[PROOF STEP]\nhave \"fact n = n * (n - 1) * fact (n - 2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn = m + 2\n\ngoal (1 subgoal):\n 1. fact n = n * (n - 1) * fact (n - 2)\n[PROOF STEP]\nby (simp add: fact_prod_Suc atLeast0_lessThan_Suc algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nfact n = n * (n - 1) * fact (n - 2)\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nwith True\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ n\nfact n = n * (n - 1) * fact (n - 2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\nfact n = n * (n - 1) * fact (n - 2)\n\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nby (simp add: binomial_fact')\n[PROOF STATE]\nproof (state)\nthis:\nn choose 2 = n * (n - 1) div 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1500, "file": null, "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527944504226, "lm_q2_score": 0.8459424411924673, "lm_q1q2_score": 0.7207030267181348}} {"text": "[STATEMENT]\nlemma choose_two: \"n choose 2 = n * (n - 1) div 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nproof (cases \"n \\ 2\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ 2 \\ n\n\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ 2 \\ n\n[PROOF STEP]\nhave \"n = 0 \\ n = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 2 \\ n\n\ngoal (1 subgoal):\n 1. n = 0 \\ n = 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nn = 0 \\ n = 1\n\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = 0 \\ n = 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 0 \\ n = 1\n\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nn choose 2 = n * (n - 1) div 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ n\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ndefine m where \"m = n - 2\"\n[PROOF STATE]\nproof (state)\nthis:\nm = n - 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nwith True\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ n\nm = n - 2\n[PROOF STEP]\nhave \"n = m + 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\nm = n - 2\n\ngoal (1 subgoal):\n 1. n = m + 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn = m + 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = m + 2\n[PROOF STEP]\nhave \"fact n = n * (n - 1) * fact (n - 2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn = m + 2\n\ngoal (1 subgoal):\n 1. fact n = n * (n - 1) * fact (n - 2)\n[PROOF STEP]\nby (simp add: fact_prod_Suc atLeast0_lessThan_Suc algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nfact n = n * (n - 1) * fact (n - 2)\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nwith True\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ n\nfact n = n * (n - 1) * fact (n - 2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\nfact n = n * (n - 1) * fact (n - 2)\n\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nby (simp add: binomial_fact')\n[PROOF STATE]\nproof (state)\nthis:\nn choose 2 = n * (n - 1) div 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1500, "file": null, "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527944504226, "lm_q2_score": 0.8459424353665382, "lm_q1q2_score": 0.7207030217547182}} {"text": "[STATEMENT]\nlemma choose_two: \"n choose 2 = n * (n - 1) div 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nproof (cases \"n \\ 2\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ 2 \\ n\n\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ 2 \\ n\n[PROOF STEP]\nhave \"n = 0 \\ n = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 2 \\ n\n\ngoal (1 subgoal):\n 1. n = 0 \\ n = 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nn = 0 \\ n = 1\n\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = 0 \\ n = 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 0 \\ n = 1\n\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nn choose 2 = n * (n - 1) div 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ n\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ndefine m where \"m = n - 2\"\n[PROOF STATE]\nproof (state)\nthis:\nm = n - 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nwith True\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ n\nm = n - 2\n[PROOF STEP]\nhave \"n = m + 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\nm = n - 2\n\ngoal (1 subgoal):\n 1. n = m + 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn = m + 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = m + 2\n[PROOF STEP]\nhave \"fact n = n * (n - 1) * fact (n - 2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn = m + 2\n\ngoal (1 subgoal):\n 1. fact n = n * (n - 1) * fact (n - 2)\n[PROOF STEP]\nby (simp add: fact_prod_Suc atLeast0_lessThan_Suc algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nfact n = n * (n - 1) * fact (n - 2)\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nwith True\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ n\nfact n = n * (n - 1) * fact (n - 2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\nfact n = n * (n - 1) * fact (n - 2)\n\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nby (simp add: binomial_fact')\n[PROOF STATE]\nproof (state)\nthis:\nn choose 2 = n * (n - 1) div 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1500, "file": null, "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527944504226, "lm_q2_score": 0.8459424353665381, "lm_q1q2_score": 0.7207030217547181}} {"text": "[STATEMENT]\nlemma powreal_IVT_lower_lemma:\n assumes \"a > (1::real)\" \n and \"x > 0\" \n shows \"\\n::nat. x < a pow\\<^sub>\\ (real n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\n. x < a pow\\<^sub>\\ real n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n. x < a pow\\<^sub>\\ real n\n[PROOF STEP]\nhave invx0: \"0 < inverse x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < inverse x\n[PROOF STEP]\nby (simp add: assms(2))\n[PROOF STATE]\nproof (state)\nthis:\n0 < inverse x\n\ngoal (1 subgoal):\n 1. \\n. x < a pow\\<^sub>\\ real n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < inverse x\n[PROOF STEP]\nhave \"\\n. a pow\\<^sub>\\ - real n < inverse x\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < inverse x\n\ngoal (1 subgoal):\n 1. \\n. a pow\\<^sub>\\ - real n < inverse x\n[PROOF STEP]\nusing assms(1) powreal_IVT_upper_lemma\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < inverse x\n1 < a\n\\1 < ?a; 0 < ?x\\ \\ \\n. ?a pow\\<^sub>\\ - real n < ?x\n\ngoal (1 subgoal):\n 1. \\n. a pow\\<^sub>\\ - real n < inverse x\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\n. a pow\\<^sub>\\ - real n < inverse x\n\ngoal (1 subgoal):\n 1. \\n. x < a pow\\<^sub>\\ real n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\n. a pow\\<^sub>\\ - real n < inverse x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n. a pow\\<^sub>\\ - real n < inverse x\n\ngoal (1 subgoal):\n 1. \\n. x < a pow\\<^sub>\\ real n\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\n. a pow\\<^sub>\\ - real n < inverse x\n1 < a\n\ngoal (1 subgoal):\n 1. \\n. x < a pow\\<^sub>\\ real n\n[PROOF STEP]\nby (auto dest: inverse_less_imp_less \n simp add: powreal_minus powreal_gt_zero )\n[PROOF STATE]\nproof (state)\nthis:\n\\n. x < a pow\\<^sub>\\ real n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 873, "file": "Real_Power_Log", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424256566558, "lm_q2_score": 0.8519528000888387, "lm_q1q2_score": 0.7207030182521321}} {"text": "[STATEMENT]\nlemma choose_two: \"n choose 2 = n * (n - 1) div 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nproof (cases \"n \\ 2\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ 2 \\ n\n\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\ 2 \\ n\n[PROOF STEP]\nhave \"n = 0 \\ n = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ 2 \\ n\n\ngoal (1 subgoal):\n 1. n = 0 \\ n = 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nn = 0 \\ n = 1\n\ngoal (2 subgoals):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n 2. \\ 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = 0 \\ n = 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 0 \\ n = 1\n\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nn choose 2 = n * (n - 1) div 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ n\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\ndefine m where \"m = n - 2\"\n[PROOF STATE]\nproof (state)\nthis:\nm = n - 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nwith True\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ n\nm = n - 2\n[PROOF STEP]\nhave \"n = m + 2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\nm = n - 2\n\ngoal (1 subgoal):\n 1. n = m + 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn = m + 2\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn = m + 2\n[PROOF STEP]\nhave \"fact n = n * (n - 1) * fact (n - 2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn = m + 2\n\ngoal (1 subgoal):\n 1. fact n = n * (n - 1) * fact (n - 2)\n[PROOF STEP]\nby (simp add: fact_prod_Suc atLeast0_lessThan_Suc algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nfact n = n * (n - 1) * fact (n - 2)\n\ngoal (1 subgoal):\n 1. 2 \\ n \\ n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nwith True\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ n\nfact n = n * (n - 1) * fact (n - 2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ n\nfact n = n * (n - 1) * fact (n - 2)\n\ngoal (1 subgoal):\n 1. n choose 2 = n * (n - 1) div 2\n[PROOF STEP]\nby (simp add: binomial_fact')\n[PROOF STATE]\nproof (state)\nthis:\nn choose 2 = n * (n - 1) div 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1500, "file": null, "length": 21, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527944504226, "lm_q2_score": 0.8459424295406088, "lm_q1q2_score": 0.7207030167913014}} {"text": "[STATEMENT]\nlemma cpx_vec_length_square:\n shows \"\\v\\\\<^sup>2 = (\\i = 0..2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\v\\\\<^sup>2 = (\\i = 0..2)\n[PROOF STEP]\nunfolding cpx_vec_length_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (sqrt (\\i2))\\<^sup>2 = (\\i = 0..2)\n[PROOF STEP]\nby (simp add: lessThan_atLeast0 sum_nonneg)", "meta": {"llama_tokens": 251, "file": "Projective_Measurements_Linear_Algebra_Complements", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392939666335, "lm_q2_score": 0.8152324871074607, "lm_q1q2_score": 0.7206975523211422}} {"text": "[STATEMENT]\nlemma higher_pderiv_monom:\n \"m \\ n + 1 \\ (pderiv ^^ m) (monom c n) = monom (pochhammer (int n - int m + 1) m * c) (n - m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. m \\ n + 1 \\ (pderiv ^^ m) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int m + 1) m * c) (n - m)\n[PROOF STEP]\nproof (induction m arbitrary: c n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\c n. 0 \\ n + 1 \\ (pderiv ^^ 0) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int 0 + 1) 0 * c) (n - 0)\n 2. \\m c n. \\\\c n. m \\ n + 1 \\ (pderiv ^^ m) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int m + 1) m * c) (n - m); Suc m \\ n + 1\\ \\ (pderiv ^^ Suc m) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int (Suc m) + 1) (Suc m) * c) (n - Suc m)\n[PROOF STEP]\ncase (Suc m)\n[PROOF STATE]\nproof (state)\nthis:\nm \\ ?n + 1 \\ (pderiv ^^ m) (Polynomial.monom ?c ?n) = Polynomial.monom (pochhammer (int ?n - int m + 1) m * ?c) (?n - m)\nSuc m \\ n + 1\n\ngoal (2 subgoals):\n 1. \\c n. 0 \\ n + 1 \\ (pderiv ^^ 0) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int 0 + 1) 0 * c) (n - 0)\n 2. \\m c n. \\\\c n. m \\ n + 1 \\ (pderiv ^^ m) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int m + 1) m * c) (n - m); Suc m \\ n + 1\\ \\ (pderiv ^^ Suc m) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int (Suc m) + 1) (Suc m) * c) (n - Suc m)\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\ ?n + 1 \\ (pderiv ^^ m) (Polynomial.monom ?c ?n) = Polynomial.monom (pochhammer (int ?n - int m + 1) m * ?c) (?n - m)\nSuc m \\ n + 1\n\ngoal (1 subgoal):\n 1. (pderiv ^^ Suc m) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int (Suc m) + 1) (Suc m) * c) (n - Suc m)\n[PROOF STEP]\nby (cases n)\n (simp_all del: funpow.simps add: funpow_Suc_right pderiv_monom pochhammer_rec' Suc.IH)\n[PROOF STATE]\nproof (state)\nthis:\n(pderiv ^^ Suc m) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int (Suc m) + 1) (Suc m) * c) (n - Suc m)\n\ngoal (1 subgoal):\n 1. \\c n. 0 \\ n + 1 \\ (pderiv ^^ 0) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int 0 + 1) 0 * c) (n - 0)\n[PROOF STEP]\nqed simp_all", "meta": {"llama_tokens": 1164, "file": "E_Transcendental_E_Transcendental", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392786908831, "lm_q2_score": 0.8152324938410783, "lm_q1q2_score": 0.7206975458206366}} {"text": "[STATEMENT]\ntheorem (in group) action_by_conjugation:\n \"group_action G (carrier G) (\\g. (\\h \\ carrier G. g \\ h \\ (inv g)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. group_action G (carrier G) (\\g. \\h\\carrier G. g \\ h \\ inv g)\n[PROOF STEP]\nunfolding group_action_def group_hom_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Group.group G \\ Group.group (BijGroup (carrier G)) \\ group_hom_axioms G (BijGroup (carrier G)) (\\g. \\h\\carrier G. g \\ h \\ inv g)\n[PROOF STEP]\nusing conjugation_is_hom\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\g. \\h\\carrier G. g \\ h \\ inv g) \\ hom G (BijGroup (carrier G))\n\ngoal (1 subgoal):\n 1. Group.group G \\ Group.group (BijGroup (carrier G)) \\ group_hom_axioms G (BijGroup (carrier G)) (\\g. \\h\\carrier G. g \\ h \\ inv g)\n[PROOF STEP]\nby (simp add: group_BijGroup group_hom_axioms.intro is_group)", "meta": {"llama_tokens": 415, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9207896671963206, "lm_q2_score": 0.7826624738835052, "lm_q1q2_score": 0.7206675188542417}} {"text": "[STATEMENT]\nlemma le_add_right_mono: \n assumes \n \"a <= b + (c::'a::ordered_ab_group_add)\"\n \"c <= d\" \n shows \"a <= b + d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\ b + d\n[PROOF STEP]\napply (rule_tac order_trans[where y = \"b+c\"])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. a \\ b + c\n 2. b + c \\ b + d\n[PROOF STEP]\napply (simp_all add: assms)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 212, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772384450967, "lm_q2_score": 0.8244619242200082, "lm_q1q2_score": 0.7206434019253554}} {"text": "[STATEMENT]\nlemma cos_cmod_scalprod:\n shows \"cmod z1 * cmod z2 * (cos (\\ z1 z2)) = Re (scalprod z1 z2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n[PROOF STEP]\nproof (cases \"z1 = 0 \\ z2 = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. z1 = 0 \\ z2 = 0 \\ cmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n 2. \\ (z1 = 0 \\ z2 = 0) \\ cmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nz1 = 0 \\ z2 = 0\n\ngoal (2 subgoals):\n 1. z1 = 0 \\ z2 = 0 \\ cmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n 2. \\ (z1 = 0 \\ z2 = 0) \\ cmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nz1 = 0 \\ z2 = 0\n\ngoal (1 subgoal):\n 1. cmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n\ngoal (1 subgoal):\n 1. \\ (z1 = 0 \\ z2 = 0) \\ cmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\ (z1 = 0 \\ z2 = 0) \\ cmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\ (z1 = 0 \\ z2 = 0)\n\ngoal (1 subgoal):\n 1. \\ (z1 = 0 \\ z2 = 0) \\ cmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\ (z1 = 0 \\ z2 = 0)\n\ngoal (1 subgoal):\n 1. cmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n[PROOF STEP]\nby (simp add: cos_diff cos_arg sin_arg field_simps)\n[PROOF STATE]\nproof (state)\nthis:\ncmod z1 * cmod z2 * cos (\\ z1 z2) = Re (scalprod z1 z2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 980, "file": "Complex_Geometry_Angles", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772253241802, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.7206433929926614}} {"text": "[STATEMENT]\nlemma continuous_image_subset_interior:\n fixes f :: \"'a::euclidean_space \\ 'b::euclidean_space\"\n assumes \"continuous_on S f\" \"inj_on f S\" \"DIM('b) \\ DIM('a)\"\n shows \"f ` (interior S) \\ interior(f ` S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f ` interior S \\ interior (f ` S)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. f ` interior S \\ interior (f ` S)\n[PROOF STEP]\nhave \"open (f ` interior S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. open (f ` interior S)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous_on S f\ninj_on f S\nDIM('b) \\ DIM('a)\n\ngoal (1 subgoal):\n 1. open (f ` interior S)\n[PROOF STEP]\nby (intro invariance_of_domain_gen) (auto simp: subset_inj_on interior_subset continuous_on_subset)\n[PROOF STATE]\nproof (state)\nthis:\nopen (f ` interior S)\n\ngoal (1 subgoal):\n 1. f ` interior S \\ interior (f ` S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nopen (f ` interior S)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nopen (f ` interior S)\n\ngoal (1 subgoal):\n 1. f ` interior S \\ interior (f ` S)\n[PROOF STEP]\nby (simp add: image_mono interior_maximal interior_subset)\n[PROOF STATE]\nproof (state)\nthis:\nf ` interior S \\ interior (f ` S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 562, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110540642805, "lm_q2_score": 0.808067204308405, "lm_q1q2_score": 0.720643265229055}} {"text": "[STATEMENT]\ntheorem n_sequences_card:\n assumes \"finite A\"\n shows \"card (n_sequences A n) = card A ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (n_sequences A n) = card A ^ n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (n_sequences A n) = card A ^ n\n[PROOF STEP]\nobtain xs where set: \"set xs = A\" and dis: \"distinct xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\xs. \\set xs = A; distinct xs\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing assms finite_distinct_list\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite ?A \\ \\xs. set xs = ?A \\ distinct xs\n\ngoal (1 subgoal):\n 1. (\\xs. \\set xs = A; distinct xs\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nset xs = A\ndistinct xs\n\ngoal (1 subgoal):\n 1. card (n_sequences A n) = card A ^ n\n[PROOF STEP]\nhave \"length (n_sequence_enum xs n) = (length xs) ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (n_sequence_enum xs n) = length xs ^ n\n[PROOF STEP]\nusing n_sequence_enum_distinct n_sequence_enum_length\n[PROOF STATE]\nproof (prove)\nusing this:\ndistinct ?xs \\ distinct (n_sequence_enum ?xs ?n)\nlength (n_sequence_enum ?xs ?n) = length ?xs ^ ?n\n\ngoal (1 subgoal):\n 1. length (n_sequence_enum xs n) = length xs ^ n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength (n_sequence_enum xs n) = length xs ^ n\n\ngoal (1 subgoal):\n 1. card (n_sequences A n) = card A ^ n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength (n_sequence_enum xs n) = length xs ^ n\n[PROOF STEP]\nhave \"card (set (n_sequence_enum xs n)) = card (set xs) ^ n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (n_sequence_enum xs n) = length xs ^ n\n\ngoal (1 subgoal):\n 1. card (set (n_sequence_enum xs n)) = card (set xs) ^ n\n[PROOF STEP]\nby (simp add: dis distinct_card n_sequence_enum_distinct)\n[PROOF STATE]\nproof (state)\nthis:\ncard (set (n_sequence_enum xs n)) = card (set xs) ^ n\n\ngoal (1 subgoal):\n 1. card (n_sequences A n) = card A ^ n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (set (n_sequence_enum xs n)) = card (set xs) ^ n\n[PROOF STEP]\nhave \"card (n_sequences (set xs) n) = card (set xs) ^ n\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (set (n_sequence_enum xs n)) = card (set xs) ^ n\n\ngoal (1 subgoal):\n 1. card (n_sequences (set xs) n) = card (set xs) ^ n\n[PROOF STEP]\nby (simp add: n_sequence_enum_correct)\n[PROOF STATE]\nproof (state)\nthis:\ncard (n_sequences (set xs) n) = card (set xs) ^ n\n\ngoal (1 subgoal):\n 1. card (n_sequences A n) = card A ^ n\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (n_sequences (set xs) n) = card (set xs) ^ n\n[PROOF STEP]\nshow \"card (n_sequences A n) = card A ^ n\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (n_sequences (set xs) n) = card (set xs) ^ n\n\ngoal (1 subgoal):\n 1. card (n_sequences A n) = card A ^ n\n[PROOF STEP]\nusing set\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (n_sequences (set xs) n) = card (set xs) ^ n\nset xs = A\n\ngoal (1 subgoal):\n 1. card (n_sequences A n) = card A ^ n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (n_sequences A n) = card A ^ n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1365, "file": "Combinatorial_Enumeration_Algorithms_n_Sequences", "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357701094303, "lm_q2_score": 0.8311430478583168, "lm_q1q2_score": 0.7206307525709349}} {"text": "[STATEMENT]\nlemma fps_integral0_by_parts:\n fixes a b :: \"'a::{division_ring,ring_char_0} fps\"\n shows\n \"fps_integral0 (a * b) =\n a * fps_integral0 b - fps_integral0 (fps_deriv a * fps_integral0 b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fps_integral0 (a * b) = a * fps_integral0 b - fps_integral0 (fps_deriv a * fps_integral0 b)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. fps_integral0 (a * b) = a * fps_integral0 b - fps_integral0 (fps_deriv a * fps_integral0 b)\n[PROOF STEP]\nhave \"fps_integral0 (fps_deriv (a * fps_integral0 b)) = a * fps_integral0 b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fps_integral0 (fps_deriv (a * fps_integral0 b)) = a * fps_integral0 b\n[PROOF STEP]\nusing fps_integral0_deriv[of \"(a * fps_integral0 b)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nfps_integral0 (fps_deriv (a * fps_integral0 b)) = a * fps_integral0 b - fps_const ((a * fps_integral0 b) $ 0)\n\ngoal (1 subgoal):\n 1. fps_integral0 (fps_deriv (a * fps_integral0 b)) = a * fps_integral0 b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfps_integral0 (fps_deriv (a * fps_integral0 b)) = a * fps_integral0 b\n\ngoal (1 subgoal):\n 1. fps_integral0 (a * b) = a * fps_integral0 b - fps_integral0 (fps_deriv a * fps_integral0 b)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfps_integral0 (fps_deriv (a * fps_integral0 b)) = a * fps_integral0 b\n\ngoal (1 subgoal):\n 1. fps_integral0 (a * b) = a * fps_integral0 b - fps_integral0 (fps_deriv a * fps_integral0 b)\n[PROOF STEP]\nhave\n \"fps_integral0 (a * b) =\n fps_integral0 (fps_deriv (a * fps_integral0 b)) -\n fps_integral0 (fps_deriv a * fps_integral0 b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fps_integral0 (a * b) = fps_integral0 (fps_deriv (a * fps_integral0 b)) - fps_integral0 (fps_deriv a * fps_integral0 b)\n[PROOF STEP]\nby (auto simp: fps_deriv_fps_integral fps_integral0_sub[symmetric])\n[PROOF STATE]\nproof (state)\nthis:\nfps_integral0 (a * b) = fps_integral0 (fps_deriv (a * fps_integral0 b)) - fps_integral0 (fps_deriv a * fps_integral0 b)\n\ngoal (1 subgoal):\n 1. fps_integral0 (a * b) = a * fps_integral0 b - fps_integral0 (fps_deriv a * fps_integral0 b)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nfps_integral0 (fps_deriv (a * fps_integral0 b)) = a * fps_integral0 b\nfps_integral0 (a * b) = fps_integral0 (fps_deriv (a * fps_integral0 b)) - fps_integral0 (fps_deriv a * fps_integral0 b)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfps_integral0 (fps_deriv (a * fps_integral0 b)) = a * fps_integral0 b\nfps_integral0 (a * b) = fps_integral0 (fps_deriv (a * fps_integral0 b)) - fps_integral0 (fps_deriv a * fps_integral0 b)\n\ngoal (1 subgoal):\n 1. fps_integral0 (a * b) = a * fps_integral0 b - fps_integral0 (fps_deriv a * fps_integral0 b)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfps_integral0 (a * b) = a * fps_integral0 b - fps_integral0 (fps_deriv a * fps_integral0 b)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1437, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357701094304, "lm_q2_score": 0.8311430457670241, "lm_q1q2_score": 0.7206307507577093}} {"text": "[STATEMENT]\nlemma Union_atLeastAtMost_real_of_nat:\n assumes \"a < b\"\n shows \"(\\n\\{a..n\\{a..x. x \\ (\\n\\{a.. x \\ {real a..real b}\n 2. \\x. x \\ {real a..real b} \\ x \\ (\\n\\{a..x. x \\ (\\n\\{a.. x \\ {real a..real b}\n 2. \\x. x \\ {real a..real b} \\ x \\ (\\n\\{a.. {real a..real b}\"\n[PROOF STATE]\nproof (state)\nthis:\nx \\ {real a..real b}\n\ngoal (2 subgoals):\n 1. \\x. x \\ (\\n\\{a.. x \\ {real a..real b}\n 2. \\x. x \\ {real a..real b} \\ x \\ (\\n\\{a.. (\\n\\{a.. {real a..real b}\n\ngoal (1 subgoal):\n 1. x \\ (\\n\\{a..x \\ {real a..real b}; x = real b\\ \\ x \\ (\\n\\{a..x \\ {real a..real b}; x \\ real b\\ \\ x \\ (\\n\\{a..x \\ {real a..real b}; x = real b\\ \\ x \\ (\\n\\{a..x \\ {real a..real b}; x \\ real b\\ \\ x \\ (\\n\\{a.. (\\n\\{a.. (\\n\\{a..x \\ {real a..real b}; x \\ real b\\ \\ x \\ (\\n\\{a..x \\ {real a..real b}; x \\ real b\\ \\ x \\ (\\n\\{a.. real b\n\ngoal (1 subgoal):\n 1. \\x \\ {real a..real b}; x \\ real b\\ \\ x \\ (\\n\\{a.. {real a..real b}\nx \\ real b\n[PROOF STEP]\nhave x: \"x \\ real a\" \"x < real b\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ {real a..real b}\nx \\ real b\n\ngoal (1 subgoal):\n 1. real a \\ x &&& x < real b\n[PROOF STEP]\nby simp_all\n[PROOF STATE]\nproof (state)\nthis:\nreal a \\ x\nx < real b\n\ngoal (1 subgoal):\n 1. \\x \\ {real a..real b}; x \\ real b\\ \\ x \\ (\\n\\{a.. real (nat \\x\\)\" \"x \\ real (Suc (nat \\x\\))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal a \\ x\nx < real b\n\ngoal (1 subgoal):\n 1. real (nat \\x\\) \\ x &&& x \\ real (Suc (nat \\x\\))\n[PROOF STEP]\nby linarith+\n[PROOF STATE]\nproof (state)\nthis:\nreal (nat \\x\\) \\ x\nx \\ real (Suc (nat \\x\\))\n\ngoal (1 subgoal):\n 1. \\x \\ {real a..real b}; x \\ real b\\ \\ x \\ (\\n\\{a..x\\) \\ x\nx \\ real (Suc (nat \\x\\))\n\ngoal (1 subgoal):\n 1. \\x \\ {real a..real b}; x \\ real b\\ \\ x \\ (\\n\\{a.. x\nx < real b\n[PROOF STEP]\nhave \"nat \\x\\ \\ a\" \"nat \\x\\ < b\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal a \\ x\nx < real b\n\ngoal (1 subgoal):\n 1. a \\ nat \\x\\ &&& nat \\x\\ < b\n[PROOF STEP]\nby linarith+\n[PROOF STATE]\nproof (state)\nthis:\na \\ nat \\x\\\nnat \\x\\ < b\n\ngoal (1 subgoal):\n 1. \\x \\ {real a..real b}; x \\ real b\\ \\ x \\ (\\n\\{a..x\\) \\ x\nx \\ real (Suc (nat \\x\\))\na \\ nat \\x\\\nnat \\x\\ < b\n[PROOF STEP]\nhave \"\\n\\{a.. {real n..real (n + 1)}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (nat \\x\\) \\ x\nx \\ real (Suc (nat \\x\\))\na \\ nat \\x\\\nnat \\x\\ < b\n\ngoal (1 subgoal):\n 1. \\n\\{a.. {real n..real (n + 1)}\n[PROOF STEP]\nby (intro bexI[of _ \"nat \\x\\\"]) simp_all\n[PROOF STATE]\nproof (state)\nthis:\n\\n\\{a.. {real n..real (n + 1)}\n\ngoal (1 subgoal):\n 1. \\x \\ {real a..real b}; x \\ real b\\ \\ x \\ (\\n\\{a..n\\{a.. {real n..real (n + 1)}\n\ngoal (1 subgoal):\n 1. x \\ (\\n\\{a.. (\\n\\{a.. (\\n\\{a..x. x \\ (\\n\\{a.. x \\ {real a..real b}\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 3034, "file": "Prime_Number_Theorem_Prime_Number_Theorem_Library", "length": 27, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430353105599, "lm_q2_score": 0.8670357666736772, "lm_q1q2_score": 0.7206307388359785}} {"text": "[STATEMENT]\nlemma BMax_rec_eq2:\n \"BMax_rec R x = Max ({z. z \\ x \\ R z} \\ {0})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. BMax_rec R x = Max ({z. z \\ x \\ R z} \\ {0})\n[PROOF STEP]\napply(induct x)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. BMax_rec R 0 = Max ({z. z \\ 0 \\ R z} \\ {0})\n 2. \\x. BMax_rec R x = Max ({z. z \\ x \\ R z} \\ {0}) \\ BMax_rec R (Suc x) = Max ({z. z \\ Suc x \\ R z} \\ {0})\n[PROOF STEP]\napply(auto intro: Max_eqI Max_eqI[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\BMax_rec R x = Max (insert 0 {z. z \\ x \\ R z}); \\ R (Suc x)\\ \\ Max (insert 0 {z. z \\ x \\ R z}) = Max (insert 0 {z. z \\ Suc x \\ R z})\n[PROOF STEP]\napply(simp add: le_Suc_eq)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\BMax_rec R x = Max (insert 0 {z. z \\ x \\ R z}); \\ R (Suc x)\\ \\ Max (insert 0 {z. z \\ x \\ R z}) = Max (insert 0 {z. (z \\ x \\ z = Suc x) \\ R z})\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 545, "file": "Universal_Turing_Machine_Recs_alt_Def", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357460591569, "lm_q2_score": 0.831143045767024, "lm_q1q2_score": 0.7206307307684917}} {"text": "[STATEMENT]\nlemma BMax_rec_eq2:\n \"BMax_rec R x = Max ({z. z \\ x \\ R z} \\ {0})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. BMax_rec R x = Max ({z. z \\ x \\ R z} \\ {0})\n[PROOF STEP]\napply(induct x)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. BMax_rec R 0 = Max ({z. z \\ 0 \\ R z} \\ {0})\n 2. \\x. BMax_rec R x = Max ({z. z \\ x \\ R z} \\ {0}) \\ BMax_rec R (Suc x) = Max ({z. z \\ Suc x \\ R z} \\ {0})\n[PROOF STEP]\napply(auto intro: Max_eqI Max_eqI[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\BMax_rec R x = Max (insert 0 {z. z \\ x \\ R z}); \\ R (Suc x)\\ \\ Max (insert 0 {z. z \\ x \\ R z}) = Max (insert 0 {z. z \\ Suc x \\ R z})\n[PROOF STEP]\napply(simp add: le_Suc_eq)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x. \\BMax_rec R x = Max (insert 0 {z. z \\ x \\ R z}); \\ R (Suc x)\\ \\ Max (insert 0 {z. z \\ x \\ R z}) = Max (insert 0 {z. (z \\ x \\ z = Suc x) \\ R z})\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 545, "file": "Universal_Turing_Machine_Recs", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357460591569, "lm_q2_score": 0.8311430436757312, "lm_q1q2_score": 0.720630728955266}} {"text": "[STATEMENT]\nlemma sum_less_split:\nassumes \"l1<(l2::nat)\"\nshows \"sum f {..h \\ circline_set (poincare_line (of_complex (-1)) (of_complex 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0\\<^sub>h \\ circline_set (poincare_line (of_complex (- 1)) (of_complex 1))\n[PROOF STEP]\nunfolding circline_set_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0\\<^sub>h \\ Collect (on_circline (poincare_line (of_complex (- 1)) (of_complex 1)))\n[PROOF STEP]\nby simp (transfer, transfer, simp add: vec_cnj_def)\n[PROOF STATE]\nproof (state)\nthis:\n0\\<^sub>h \\ circline_set (poincare_line (of_complex (- 1)) (of_complex 1))\n\ngoal (1 subgoal):\n 1. poincare_line (of_complex (- 1)) (of_complex 1) = x_axis\n[PROOF STEP]\nhence \"poincare_line 0\\<^sub>h (of_complex 1) = poincare_line (of_complex (-1)) (of_complex 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0\\<^sub>h \\ circline_set (poincare_line (of_complex (- 1)) (of_complex 1))\n\ngoal (1 subgoal):\n 1. poincare_line 0\\<^sub>h (of_complex 1) = poincare_line (of_complex (- 1)) (of_complex 1)\n[PROOF STEP]\nby (metis is_poincare_line_poincare_line is_poincare_line_trough_zero_trough_infty not_zero_on_unit_circle of_complex_inj of_complex_one one_neq_neg_one one_on_unit_circle poincare_line_0_real_is_x_axis poincare_line_circline_set(2) reciprocal_involution reciprocal_one reciprocal_zero unique_circline_01inf')\n[PROOF STATE]\nproof (state)\nthis:\npoincare_line 0\\<^sub>h (of_complex 1) = poincare_line (of_complex (- 1)) (of_complex 1)\n\ngoal (1 subgoal):\n 1. poincare_line (of_complex (- 1)) (of_complex 1) = x_axis\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\npoincare_line 0\\<^sub>h (of_complex 1) = poincare_line (of_complex (- 1)) (of_complex 1)\n\ngoal (1 subgoal):\n 1. poincare_line (of_complex (- 1)) (of_complex 1) = x_axis\n[PROOF STEP]\nusing poincare_line_0_real_is_x_axis[of \"of_complex 1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\npoincare_line 0\\<^sub>h (of_complex 1) = poincare_line (of_complex (- 1)) (of_complex 1)\n\\of_complex 1 \\ circline_set x_axis; of_complex 1 \\ 0\\<^sub>h; of_complex 1 \\ \\\\<^sub>h\\ \\ poincare_line 0\\<^sub>h (of_complex 1) = x_axis\n\ngoal (1 subgoal):\n 1. poincare_line (of_complex (- 1)) (of_complex 1) = x_axis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\npoincare_line (of_complex (- 1)) (of_complex 1) = x_axis\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1220, "file": "Poincare_Disc_Poincare_Lines", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.853912760387131, "lm_q2_score": 0.8438951005915208, "lm_q1q2_score": 0.7206127948232812}} {"text": "[STATEMENT]\nlemma abs_summable_on_add [intro]:\n assumes \"f abs_summable_on A\" and \"g abs_summable_on A\"\n shows \"(\\x. f x + g x) abs_summable_on A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. f x + g x) abs_summable_on A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nf abs_summable_on A\ng abs_summable_on A\n\ngoal (1 subgoal):\n 1. (\\x. f x + g x) abs_summable_on A\n[PROOF STEP]\nunfolding abs_summable_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable (count_space A) f\nintegrable (count_space A) g\n\ngoal (1 subgoal):\n 1. integrable (count_space A) (\\x. f x + g x)\n[PROOF STEP]\nby (rule Bochner_Integration.integrable_add)", "meta": {"llama_tokens": 309, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127529517042, "lm_q2_score": 0.8438951045175642, "lm_q1q2_score": 0.7206127919010594}} {"text": "[STATEMENT]\nlemma LIMSEQ_inv_powr:\n assumes \"0 < c\" \"0 < d\"\n shows \"(\\n :: nat. (c / n) powr d) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\n. (c / real n) powr d) \\ 0\n[PROOF STEP]\nproof (rule tendsto_zero_powrI)\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. (\\x. c / real x) \\ 0\n 2. (\\x. d) \\ ?b\n 3. \\\\<^sup>\\n. 0 \\ c / real n\n 4. 0 < ?b\n[PROOF STEP]\nfrom \\0 < c\\\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < c\n[PROOF STEP]\nhave \"\\x. 0 < x \\ 0 < c / x\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < c\n\ngoal (1 subgoal):\n 1. \\x. 0 < x \\ 0 < c / x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < ?x \\ 0 < c / ?x\n\ngoal (4 subgoals):\n 1. (\\x. c / real x) \\ 0\n 2. (\\x. d) \\ ?b\n 3. \\\\<^sup>\\n. 0 \\ c / real n\n 4. 0 < ?b\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 < ?x \\ 0 < c / ?x\n[PROOF STEP]\nshow \"\\\\<^sup>\\n. 0 \\ c / real n\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < ?x \\ 0 < c / ?x\n\ngoal (1 subgoal):\n 1. \\\\<^sup>\\n. 0 \\ c / real n\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < ?x \\ 0 < c / ?x\n0 < c\n\ngoal (1 subgoal):\n 1. \\\\<^sup>\\n. 0 \\ c / real n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\\\<^sup>\\n. 0 \\ c / real n\n\ngoal (3 subgoals):\n 1. (\\x. c / real x) \\ 0\n 2. (\\x. d) \\ ?b\n 3. 0 < ?b\n[PROOF STEP]\nshow \"(\\x. c / real x) \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. c / real x) \\ 0\n[PROOF STEP]\nby (intro tendsto_divide_0[OF tendsto_const] filterlim_at_top_imp_at_infinity\n filterlim_real_sequentially tendsto_divide_0)\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. c / real x) \\ 0\n\ngoal (2 subgoals):\n 1. (\\x. d) \\ ?b\n 2. 0 < ?b\n[PROOF STEP]\nshow \"0 < d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < d\n[PROOF STEP]\nby (rule assms)\n[PROOF STATE]\nproof (state)\nthis:\n0 < d\n\ngoal (1 subgoal):\n 1. (\\x. d) \\ d\n[PROOF STEP]\nshow \"(\\x. d) \\ d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\x. d) \\ d\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\x. d) \\ d\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1207, "file": "Girth_Chromatic_Girth_Chromatic_Misc", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.853912760387131, "lm_q2_score": 0.8438950947024555, "lm_q1q2_score": 0.7206127897945331}} {"text": "[STATEMENT]\nlemma length_of_bin_rep_aux:\n fixes n m:: nat\n assumes \"m < 2^n\"\n shows \"length (bin_rep_aux n m) = n+1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (bin_rep_aux n m) = n + 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nm < 2 ^ n\n\ngoal (1 subgoal):\n 1. length (bin_rep_aux n m) = n + 1\n[PROOF STEP]\nproof(induction n arbitrary: m)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\m. m < 2 ^ 0 \\ length (bin_rep_aux 0 m) = 0 + 1\n 2. \\n m. \\\\m. m < 2 ^ n \\ length (bin_rep_aux n m) = n + 1; m < 2 ^ Suc n\\ \\ length (bin_rep_aux (Suc n) m) = Suc n + 1\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nm < 2 ^ 0\n\ngoal (2 subgoals):\n 1. \\m. m < 2 ^ 0 \\ length (bin_rep_aux 0 m) = 0 + 1\n 2. \\n m. \\\\m. m < 2 ^ n \\ length (bin_rep_aux n m) = n + 1; m < 2 ^ Suc n\\ \\ length (bin_rep_aux (Suc n) m) = Suc n + 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nm < 2 ^ 0\n[PROOF STEP]\nshow \"length (bin_rep_aux 0 m) = 0 + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nm < 2 ^ 0\n\ngoal (1 subgoal):\n 1. length (bin_rep_aux 0 m) = 0 + 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlength (bin_rep_aux 0 m) = 0 + 1\n\ngoal (1 subgoal):\n 1. \\n m. \\\\m. m < 2 ^ n \\ length (bin_rep_aux n m) = n + 1; m < 2 ^ Suc n\\ \\ length (bin_rep_aux (Suc n) m) = Suc n + 1\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\n m. \\\\m. m < 2 ^ n \\ length (bin_rep_aux n m) = n + 1; m < 2 ^ Suc n\\ \\ length (bin_rep_aux (Suc n) m) = Suc n + 1\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n?m < 2 ^ n \\ length (bin_rep_aux n ?m) = n + 1\nm < 2 ^ Suc n\n\ngoal (1 subgoal):\n 1. \\n m. \\\\m. m < 2 ^ n \\ length (bin_rep_aux n m) = n + 1; m < 2 ^ Suc n\\ \\ length (bin_rep_aux (Suc n) m) = Suc n + 1\n[PROOF STEP]\nassume a0:\"\\m. m < 2^n \\ length (bin_rep_aux n m) = n + 1\" and \"m < 2^(Suc n)\"\n[PROOF STATE]\nproof (state)\nthis:\n?m < 2 ^ n \\ length (bin_rep_aux n ?m) = n + 1\nm < 2 ^ Suc n\n\ngoal (1 subgoal):\n 1. \\n m. \\\\m. m < 2 ^ n \\ length (bin_rep_aux n m) = n + 1; m < 2 ^ Suc n\\ \\ length (bin_rep_aux (Suc n) m) = Suc n + 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n?m < 2 ^ n \\ length (bin_rep_aux n ?m) = n + 1\nm < 2 ^ Suc n\n[PROOF STEP]\nshow \"length (bin_rep_aux (Suc n) m) = Suc n + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?m < 2 ^ n \\ length (bin_rep_aux n ?m) = n + 1\nm < 2 ^ Suc n\n\ngoal (1 subgoal):\n 1. length (bin_rep_aux (Suc n) m) = Suc n + 1\n[PROOF STEP]\nusing a0\n[PROOF STATE]\nproof (prove)\nusing this:\n?m < 2 ^ n \\ length (bin_rep_aux n ?m) = n + 1\nm < 2 ^ Suc n\n?m < 2 ^ n \\ length (bin_rep_aux n ?m) = n + 1\n\ngoal (1 subgoal):\n 1. length (bin_rep_aux (Suc n) m) = Suc n + 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlength (bin_rep_aux (Suc n) m) = Suc n + 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1558, "file": "Isabelle_Marries_Dirac_Binary_Nat", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.843895086850368, "lm_q2_score": 0.8539127529517043, "lm_q1q2_score": 0.7206127768148153}} {"text": "[STATEMENT]\ntheorem mset_quicksort [simp]: \"mset (quicksort R xs) = mset xs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mset (quicksort R xs) = mset xs\n[PROOF STEP]\nby (induction R xs rule: quicksort.induct) simp_all", "meta": {"llama_tokens": 90, "file": "Functional_Ordered_Resolution_Prover_Executable_Subsumption", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026641072386, "lm_q2_score": 0.7853085859124002, "lm_q1q2_score": 0.720365658003733}} {"text": "[STATEMENT]\nlemma real_sqrt_sum_squares_less: \"\\x\\ < u / sqrt 2 \\ \\y\\ < u / sqrt 2 \\ sqrt (x\\<^sup>2 + y\\<^sup>2) < u\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x\\ < u / sqrt 2; \\y\\ < u / sqrt 2\\ \\ sqrt (x\\<^sup>2 + y\\<^sup>2) < u\n[PROOF STEP]\napply (rule power2_less_imp_less)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\\\x\\ < u / sqrt 2; \\y\\ < u / sqrt 2\\ \\ (sqrt (x\\<^sup>2 + y\\<^sup>2))\\<^sup>2 < u\\<^sup>2\n 2. \\\\x\\ < u / sqrt 2; \\y\\ < u / sqrt 2\\ \\ 0 \\ u\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\\\x\\ < u / sqrt 2; \\y\\ < u / sqrt 2\\ \\ x\\<^sup>2 + y\\<^sup>2 < u\\<^sup>2\n 2. \\\\x\\ < u / sqrt 2; \\y\\ < u / sqrt 2\\ \\ 0 \\ u\n[PROOF STEP]\napply (drule power_strict_mono [OF _ abs_ge_zero pos2])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\\\y\\ < u / sqrt 2; \\x\\\\<^sup>2 < (u / sqrt 2)\\<^sup>2\\ \\ x\\<^sup>2 + y\\<^sup>2 < u\\<^sup>2\n 2. \\\\x\\ < u / sqrt 2; \\y\\ < u / sqrt 2\\ \\ 0 \\ u\n[PROOF STEP]\napply (drule power_strict_mono [OF _ abs_ge_zero pos2])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\\\x\\\\<^sup>2 < (u / sqrt 2)\\<^sup>2; \\y\\\\<^sup>2 < (u / sqrt 2)\\<^sup>2\\ \\ x\\<^sup>2 + y\\<^sup>2 < u\\<^sup>2\n 2. \\\\x\\ < u / sqrt 2; \\y\\ < u / sqrt 2\\ \\ 0 \\ u\n[PROOF STEP]\napply (simp add: power_divide)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\x\\ < u / sqrt 2; \\y\\ < u / sqrt 2\\ \\ 0 \\ u\n[PROOF STEP]\napply (drule order_le_less_trans [OF abs_ge_zero])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\\\y\\ < u / sqrt 2; 0 < u / sqrt 2\\ \\ 0 \\ u\n[PROOF STEP]\napply (simp add: zero_less_divide_iff)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1058, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587993853654, "lm_q2_score": 0.8104789132480439, "lm_q1q2_score": 0.7203202658654873}} {"text": "[STATEMENT]\nlemma lists_of_length_list_set : \n \"set (lists_of_length xs k) = {xs' . length xs' = k \\ set xs' \\ set xs}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (lists_of_length xs k) = {xs'. length xs' = k \\ set xs' \\ set xs}\n[PROOF STEP]\nusing lists_of_length_containment[of _ xs k] \n lists_of_length_length[of _ xs k] \n lists_of_length_elems[of _ xs k]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\set ?xs \\ set xs; length ?xs = k\\ \\ ?xs \\ set (lists_of_length xs k)\n?xs \\ set (lists_of_length xs k) \\ length ?xs = k\n?xs \\ set (lists_of_length xs k) \\ set ?xs \\ set xs\n\ngoal (1 subgoal):\n 1. set (lists_of_length xs k) = {xs'. length xs' = k \\ set xs' \\ set xs}\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 331, "file": "FSM_Tests_Util", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587875995483, "lm_q2_score": 0.8104789155369048, "lm_q1q2_score": 0.7203202583475762}} {"text": "[STATEMENT]\nlemma frontier_straddle:\n fixes a :: \"'a::metric_space\"\n shows \"a \\ frontier S \\ (\\e>0. (\\x\\S. dist a x < e) \\ (\\x. x \\ S \\ dist a x < e))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a \\ frontier S) = (\\e>0. (\\x\\S. dist a x < e) \\ (\\x. x \\ S \\ dist a x < e))\n[PROOF STEP]\nunfolding frontier_def closure_interior\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a \\ - interior (- S) - interior S) = (\\e>0. (\\x\\S. dist a x < e) \\ (\\x. x \\ S \\ dist a x < e))\n[PROOF STEP]\nby (auto simp: mem_interior subset_eq ball_def)", "meta": {"llama_tokens": 291, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587875995482, "lm_q2_score": 0.8104789086703225, "lm_q1q2_score": 0.7203202522448408}} {"text": "[STATEMENT]\nlemma set_le_two [simp]: \"card {a, b} \\ 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {a, b} \\ 2\n[PROOF STEP]\nby (simp add: card_insert_if)", "meta": {"llama_tokens": 82, "file": "Schutz_Spacetime_Util", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111797100118214, "lm_q2_score": 0.7905303260722198, "lm_q1q2_score": 0.7203151932660359}} {"text": "[STATEMENT]\nlemma upper_bound:\n fixes b::nat\n and c::int\n assumes \"b > 0\"\n and \"c < 2^(b-1)\"\n shows \"c + 2^(b-1) < 2^b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c + 2 ^ (b - 1) < 2 ^ b\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. c + 2 ^ (b - 1) < 2 ^ b\n[PROOF STEP]\nhave a1: \"\\P. (\\b::nat. P b) \\ (\\b>0. P ((b-1)::nat))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\P. \\b. P b \\ \\b>0. P (b - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\b. ?P b \\ \\b>0. ?P (b - 1)\n\ngoal (1 subgoal):\n 1. c + 2 ^ (b - 1) < 2 ^ b\n[PROOF STEP]\nhave b2: \"\\b::nat. (\\(c::int)<2^b. (c + 2^b) < 2^(Suc b))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\b c. c < 2 ^ b \\ c + 2 ^ b < 2 ^ Suc b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\b c. c < 2 ^ b \\ c + 2 ^ b < 2 ^ Suc b\n\ngoal (1 subgoal):\n 1. c + 2 ^ (b - 1) < 2 ^ b\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c + 2 ^ (b - 1) < 2 ^ b\n[PROOF STEP]\nusing a1[OF b2] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\b>0. \\c<2 ^ (b - 1). c + 2 ^ (b - 1) < 2 ^ Suc (b - 1)\n0 < b\nc < 2 ^ (b - 1)\n\ngoal (1 subgoal):\n 1. c + 2 ^ (b - 1) < 2 ^ b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nc + 2 ^ (b - 1) < 2 ^ b\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 759, "file": "Solidity_Valuetypes", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637612961506, "lm_q2_score": 0.8376199694135332, "lm_q1q2_score": 0.7200715334428046}} {"text": "[STATEMENT]\nlemma ex_countable_subset_image_inj:\n \"(\\T. countable T \\ T \\ f ` S \\ P T) \\\n (\\T. countable T \\ T \\ S \\ inj_on f T \\ P (f ` T))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\T. countable T \\ T \\ f ` S \\ P T) = (\\T. countable T \\ T \\ S \\ inj_on f T \\ P (f ` T))\n[PROOF STEP]\nby (metis countable_image_inj_eq subset_image_inj)", "meta": {"llama_tokens": 198, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8596637505099167, "lm_q2_score": 0.8376199572530448, "lm_q1q2_score": 0.7200715139541086}} {"text": "[STATEMENT]\nlemma \"((\\x::real. exp (exp x) / exp (exp x - exp (-exp (exp x)))) \\ 1) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\x. exp (exp x) / exp (exp x - exp (- exp (exp x)))) \\ 1) at_top\n[PROOF STEP]\nby real_asymp", "meta": {"llama_tokens": 118, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9294403999037782, "lm_q2_score": 0.7745833893685269, "lm_q1q2_score": 0.7199290951735076}} {"text": "[STATEMENT]\nlemma rank_Gauss_Jordan_code[code]:\n fixes A::\"'a::{field}^'n::{mod_type}^'m::{mod_type}\"\n shows \"rank A = (if A = 0 then 0 else (let A'=(Gauss_Jordan A) in to_nat (GREATEST a. row a A' \\ 0) + 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n[PROOF STEP]\nproof (cases \"A = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. A = 0 \\ rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n 2. A \\ 0 \\ rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nA = 0\n\ngoal (2 subgoals):\n 1. A = 0 \\ rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n 2. A \\ 0 \\ rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n[PROOF STEP]\nunfolding if_P[OF True]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank A = 0\n[PROOF STEP]\nunfolding True rank_0\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 = 0\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nrank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n\ngoal (1 subgoal):\n 1. A \\ 0 \\ rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. A \\ 0 \\ rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nA \\ 0\n\ngoal (1 subgoal):\n 1. A \\ 0 \\ rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n[PROOF STEP]\nunfolding if_not_P[OF False]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank A = (let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n[PROOF STEP]\nunfolding rank_eq_suc_to_nat_greatest[OF False] Let_def is_zero_row_eq_row_zero\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mod_type_class.to_nat (GREATEST a. row a (Gauss_Jordan A) \\ 0) + 1 = mod_type_class.to_nat (GREATEST a. row a (Gauss_Jordan A) \\ 0) + 1\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nrank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\ 0) + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1507, "file": "Gauss_Jordan_Gauss_Jordan", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213745668094, "lm_q2_score": 0.800691997339971, "lm_q1q2_score": 0.7199192892529588}} {"text": "[STATEMENT]\nlemma (in lower_semilattice) meet_assoc:\n assumes L: \"x \\ carrier L\" \"y \\ carrier L\" \"z \\ carrier L\"\n shows \"(x \\ y) \\ z = x \\ (y \\ z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\ y \\ z = x \\ (y \\ z)\n[PROOF STEP]\nusing weak_meet_assoc L\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x \\ carrier L; ?y \\ carrier L; ?z \\ carrier L\\ \\ ?x \\ ?y \\ ?z .= ?x \\ (?y \\ ?z)\nx \\ carrier L\ny \\ carrier L\nz \\ carrier L\n\ngoal (1 subgoal):\n 1. x \\ y \\ z = x \\ (y \\ z)\n[PROOF STEP]\nunfolding eq_is_equal\n[PROOF STATE]\nproof (prove)\nusing this:\n\\?x \\ carrier L; ?y \\ carrier L; ?z \\ carrier L\\ \\ ?x \\ ?y \\ ?z = ?x \\ (?y \\ ?z)\nx \\ carrier L\ny \\ carrier L\nz \\ carrier L\n\ngoal (1 subgoal):\n 1. x \\ y \\ z = x \\ (y \\ z)\n[PROOF STEP]\n.", "meta": {"llama_tokens": 471, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206738932334, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7199011564715877}} {"text": "[STATEMENT]\nlemma exp_sin_cos_squared_add:\n fixes x y :: real\n shows \"exp (- (\\ * x)) * cos (y) * (exp (\\ * x) * cos (y)) + sin(y) * sin(y) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp (- (\\ * complex_of_real x)) * complex_of_real (cos y) * (exp (\\ * complex_of_real x) * complex_of_real (cos y)) + complex_of_real (sin y * sin y) = 1\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. exp (- (\\ * complex_of_real x)) * complex_of_real (cos y) * (exp (\\ * complex_of_real x) * complex_of_real (cos y)) + complex_of_real (sin y * sin y) = 1\n[PROOF STEP]\nhave \"exp (- (\\ * x)) * cos (y) * (exp (\\ * x) * cos (y)) = cos(y) * cos(y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp (- (\\ * complex_of_real x)) * complex_of_real (cos y) * (exp (\\ * complex_of_real x) * complex_of_real (cos y)) = complex_of_real (cos y * cos y)\n[PROOF STEP]\nusing exp_minus_inverse\n[PROOF STATE]\nproof (prove)\nusing this:\nexp ?x * exp (- ?x) = (1::?'a)\n\ngoal (1 subgoal):\n 1. exp (- (\\ * complex_of_real x)) * complex_of_real (cos y) * (exp (\\ * complex_of_real x) * complex_of_real (cos y)) = complex_of_real (cos y * cos y)\n[PROOF STEP]\nby (auto simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nexp (- (\\ * complex_of_real x)) * complex_of_real (cos y) * (exp (\\ * complex_of_real x) * complex_of_real (cos y)) = complex_of_real (cos y * cos y)\n\ngoal (1 subgoal):\n 1. exp (- (\\ * complex_of_real x)) * complex_of_real (cos y) * (exp (\\ * complex_of_real x) * complex_of_real (cos y)) + complex_of_real (sin y * sin y) = 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nexp (- (\\ * complex_of_real x)) * complex_of_real (cos y) * (exp (\\ * complex_of_real x) * complex_of_real (cos y)) = complex_of_real (cos y * cos y)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nexp (- (\\ * complex_of_real x)) * complex_of_real (cos y) * (exp (\\ * complex_of_real x) * complex_of_real (cos y)) = complex_of_real (cos y * cos y)\n\ngoal (1 subgoal):\n 1. exp (- (\\ * complex_of_real x)) * complex_of_real (cos y) * (exp (\\ * complex_of_real x) * complex_of_real (cos y)) + complex_of_real (sin y * sin y) = 1\n[PROOF STEP]\nby (metis of_real_add of_real_hom.hom_one sin_cos_squared_add3)\n[PROOF STATE]\nproof (state)\nthis:\nexp (- (\\ * complex_of_real x)) * complex_of_real (cos y) * (exp (\\ * complex_of_real x) * complex_of_real (cos y)) + complex_of_real (sin y * sin y) = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1068, "file": "Isabelle_Marries_Dirac_Quantum_Prisoners_Dilemma", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9019206738932334, "lm_q2_score": 0.7981867777396211, "lm_q1q2_score": 0.7199011564715876}} {"text": "[STATEMENT]\nlemma mat_mult_limit:\n fixes X :: \"nat \\ complex mat\" and A B :: \"complex mat\" and m :: nat\n assumes dimB: \"B \\ carrier_mat m m\" and limX: \"limit_mat X A m\"\n shows \"limit_mat (mat_mult_seq B X) (B * A) m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. limit_mat (mat_mult_seq B X) (B * A) m\n[PROOF STEP]\nunfolding mat_mult_seq_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. limit_mat (\\n. B * X n) (B * A) m\n[PROOF STEP]\nusing limit_mat_mult limit_mat_const[OF dimB] limX\n[PROOF STATE]\nproof (prove)\nusing this:\n\\limit_mat ?X ?A ?m; limit_mat ?Y ?B ?m\\ \\ limit_mat (\\k. ?X k * ?Y k) (?A * ?B) ?m\nlimit_mat (\\k. B) B m\nlimit_mat X A m\n\ngoal (1 subgoal):\n 1. limit_mat (\\n. B * X n) (B * A) m\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 357, "file": "QHLProver_Matrix_Limit", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.90192067652954, "lm_q2_score": 0.7981867753392728, "lm_q1q2_score": 0.7199011564109289}} {"text": "[STATEMENT]\nlemma UNIV_subtract:\n shows \"UNIV - EVEN = ODD\"\n and \"UNIV - ODD = EVEN\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. UNIV - EVEN = ODD &&& UNIV - ODD = EVEN\n[PROOF STEP]\nusing EVEN_union_ODD EVEN_intersect_ODD\n[PROOF STATE]\nproof (prove)\nusing this:\nEVEN \\ ODD = UNIV\nEVEN \\ ODD = {}\n\ngoal (1 subgoal):\n 1. UNIV - EVEN = ODD &&& UNIV - ODD = EVEN\n[PROOF STEP]\nby (blast)+", "meta": {"llama_tokens": 197, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9019206712569268, "lm_q2_score": 0.7981867777396212, "lm_q1q2_score": 0.7199011543673226}} {"text": "[STATEMENT]\nlemma collinear_inv_rotation:\n assumes \"collinear (Ax, Ay) (Bx, By) (Cx, Cy)\" and \"c\\<^sup>2 + s\\<^sup>2 = 1\"\n shows \"collinear (Ax * c - Ay * s, Ay * c + Ax * s)\n (Bx * c - By * s, By * c + Bx * s) (Cx * c - Cy * s, Cy * c + Cx * s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. collinear (Ax * c - Ay * s, Ay * c + Ax * s) (Bx * c - By * s, By * c + Bx * s) (Cx * c - Cy * s, Cy * c + Cx * s)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncollinear (Ax, Ay) (Bx, By) (Cx, Cy)\nc\\<^sup>2 + s\\<^sup>2 = 1\n\ngoal (1 subgoal):\n 1. collinear (Ax * c - Ay * s, Ay * c + Ax * s) (Bx * c - By * s, By * c + Bx * s) (Cx * c - Cy * s, Cy * c + Cx * s)\n[PROOF STEP]\nby (algebra add: collinear_def split_def fst_conv snd_conv)", "meta": {"llama_tokens": 366, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9019206765295399, "lm_q2_score": 0.798186768138228, "lm_q1q2_score": 0.7199011499161576}} {"text": "[STATEMENT]\ntheorem IFNTT_inv_FNTT: \n assumes \"length numbers = n\"\n shows \"IFNTT (FNTT numbers) = map ((*) (of_int_mod_ring (int n))) numbers\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. IFNTT (FNTT numbers) = map ((*) (of_int_mod_ring (int n))) numbers\n[PROOF STEP]\nby (simp add: FNTT_correct IFNTT_correct assms length_NTT ntt_correct)", "meta": {"llama_tokens": 142, "file": "Number_Theoretic_Transform_Butterfly", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206659843132, "lm_q2_score": 0.7981867753392728, "lm_q1q2_score": 0.7199011479938684}} {"text": "[STATEMENT]\nlemma unitary_adj_eq_inv:\n shows \"unitary M \\ mat_det M \\ 0 \\ mat_adj M = mat_inv M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. unitary M = (mat_det M \\ 0 \\ mat_adj M = mat_inv M)\n[PROOF STEP]\nusing unitary_regular[of M] mult_mm_inv_r[of M \"mat_adj M\" eye] mat_eye_l[of \"mat_inv M\"] mat_inv_l[of M]\n[PROOF STATE]\nproof (prove)\nusing this:\nunitary M \\ mat_det M \\ 0\n\\mat_det M \\ 0; mat_adj M *\\<^sub>m\\<^sub>m M = eye\\ \\ mat_adj M = eye *\\<^sub>m\\<^sub>m mat_inv M\neye *\\<^sub>m\\<^sub>m mat_inv M = mat_inv M\nmat_det M \\ 0 \\ mat_inv M *\\<^sub>m\\<^sub>m M = eye\n\ngoal (1 subgoal):\n 1. unitary M = (mat_det M \\ 0 \\ mat_adj M = mat_inv M)\n[PROOF STEP]\nunfolding unitary_def\n[PROOF STATE]\nproof (prove)\nusing this:\nmat_adj M *\\<^sub>m\\<^sub>m M = eye \\ mat_det M \\ 0\n\\mat_det M \\ 0; mat_adj M *\\<^sub>m\\<^sub>m M = eye\\ \\ mat_adj M = eye *\\<^sub>m\\<^sub>m mat_inv M\neye *\\<^sub>m\\<^sub>m mat_inv M = mat_inv M\nmat_det M \\ 0 \\ mat_inv M *\\<^sub>m\\<^sub>m M = eye\n\ngoal (1 subgoal):\n 1. (mat_adj M *\\<^sub>m\\<^sub>m M = eye) = (mat_det M \\ 0 \\ mat_adj M = mat_inv M)\n[PROOF STEP]\nby - (rule, simp_all)", "meta": {"llama_tokens": 595, "file": "Complex_Geometry_Unitary_Matrices", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.885631476836816, "lm_q2_score": 0.8128673201042493, "lm_q1q2_score": 0.7199008851763111}} {"text": "[STATEMENT]\nlemma facet_of_convex_hull_affine_independent_alt:\n fixes S :: \"'a::euclidean_space set\"\n assumes \"\\ affine_dependent S\"\n shows \"(T facet_of (convex hull S) \\ 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u})))\"\n (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (T facet_of convex hull S) = (2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u})))\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nassume L: ?lhs\n[PROOF STATE]\nproof (state)\nthis:\nT facet_of convex hull S\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nT facet_of convex hull S\n[PROOF STEP]\nobtain x where\n \"x \\ S\" and x: \"T = convex hull (S - {x})\" and \"finite S\"\n[PROOF STATE]\nproof (prove)\nusing this:\nT facet_of convex hull S\n\ngoal (1 subgoal):\n 1. (\\x. \\x \\ S; T = convex hull (S - {x}); finite S\\ \\ thesis) \\ thesis\n[PROOF STEP]\nusing assms facet_of_convex_hull_affine_independent aff_independent_finite\n[PROOF STATE]\nproof (prove)\nusing this:\nT facet_of convex hull S\n\\ affine_dependent S\n\\ affine_dependent ?S \\ (?T facet_of convex hull ?S) = (?T \\ {} \\ (\\u. u \\ ?S \\ ?T = convex hull (?S - {u})))\n\\ affine_dependent ?B \\ finite ?B\n\ngoal (1 subgoal):\n 1. (\\x. \\x \\ S; T = convex hull (S - {x}); finite S\\ \\ thesis) \\ thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nx \\ S\nT = convex hull (S - {x})\nfinite S\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nx \\ S\nT = convex hull (S - {x})\nfinite S\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nhave \"Suc (Suc 0) \\ card S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Suc (Suc 0) \\ card S\n[PROOF STEP]\nusing L x \\x \\ S\\ \\finite S\\\n[PROOF STATE]\nproof (prove)\nusing this:\nT facet_of convex hull S\nT = convex hull (S - {x})\nx \\ S\nfinite S\n\ngoal (1 subgoal):\n 1. Suc (Suc 0) \\ card S\n[PROOF STEP]\nby (metis Suc_leI assms card.remove convex_hull_eq_empty card_gt_0_iff facet_of_convex_hull_affine_independent finite_Diff not_less_eq_eq)\n[PROOF STATE]\nproof (state)\nthis:\nSuc (Suc 0) \\ card S\n\ngoal (2 subgoals):\n 1. T facet_of convex hull S \\ 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n 2. 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\ S\nT = convex hull (S - {x})\nfinite S\nSuc (Suc 0) \\ card S\n[PROOF STEP]\nshow ?rhs\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\ S\nT = convex hull (S - {x})\nfinite S\nSuc (Suc 0) \\ card S\n\ngoal (1 subgoal):\n 1. 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n\ngoal (1 subgoal):\n 1. 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nassume ?rhs\n[PROOF STATE]\nproof (state)\nthis:\n2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n\ngoal (1 subgoal):\n 1. 2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u})) \\ T facet_of convex hull S\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n[PROOF STEP]\nshow ?lhs\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n\ngoal (1 subgoal):\n 1. T facet_of convex hull S\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n2 \\ card S \\ (\\u. u \\ S \\ T = convex hull (S - {u}))\n\\ affine_dependent S\n\ngoal (1 subgoal):\n 1. T facet_of convex hull S\n[PROOF STEP]\nby (auto simp: facet_of_convex_hull_affine_independent Set.subset_singleton_iff)\n[PROOF STATE]\nproof (state)\nthis:\nT facet_of convex hull S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2341, "file": null, "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314677809303, "lm_q2_score": 0.8128673201042492, "lm_q1q2_score": 0.7199008778150775}} {"text": "[STATEMENT]\nlemma (in group) DirProd_iso_set_trans:\n assumes \"g \\ iso G G2\"\n and \"h \\ iso H I\"\n shows \"(\\(x,y). (g x, h y)) \\ iso (G \\\\ H) (G2 \\\\ I)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ Group.iso (G \\\\ H) (G2 \\\\ I)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ Group.iso (G \\\\ H) (G2 \\\\ I)\n[PROOF STEP]\nhave \"(\\(x,y). (g x, h y)) \\ hom (G \\\\ H) (G2 \\\\ I)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ hom (G \\\\ H) (G2 \\\\ I)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ng \\ Group.iso G G2\nh \\ Group.iso H I\n\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ hom (G \\\\ H) (G2 \\\\ I)\n[PROOF STEP]\nunfolding iso_def hom_def\n[PROOF STATE]\nproof (prove)\nusing this:\ng \\ {h \\ {h \\ carrier G \\ carrier G2. \\x\\carrier G. \\y\\carrier G. h (x \\ y) = h x \\\\<^bsub>G2\\<^esub> h y}. bij_betw h (carrier G) (carrier G2)}\nh \\ {h \\ {h \\ carrier H \\ carrier I. \\x\\carrier H. \\y\\carrier H. h (x \\\\<^bsub>H\\<^esub> y) = h x \\\\<^bsub>I\\<^esub> h y}. bij_betw h (carrier H) (carrier I)}\n\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ {h \\ carrier (G \\\\ H) \\ carrier (G2 \\\\ I). \\x\\carrier (G \\\\ H). \\y\\carrier (G \\\\ H). h (x \\\\<^bsub>G \\\\ H\\<^esub> y) = h x \\\\<^bsub>G2 \\\\ I\\<^esub> h y}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\(x, y). (g x, h y)) \\ hom (G \\\\ H) (G2 \\\\ I)\n\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ Group.iso (G \\\\ H) (G2 \\\\ I)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(\\(x, y). (g x, h y)) \\ hom (G \\\\ H) (G2 \\\\ I)\n\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ Group.iso (G \\\\ H) (G2 \\\\ I)\n[PROOF STEP]\nhave \" inj_on (\\(x,y). (g x, h y)) (carrier (G \\\\ H))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on (\\(x, y). (g x, h y)) (carrier (G \\\\ H))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ng \\ Group.iso G G2\nh \\ Group.iso H I\n\ngoal (1 subgoal):\n 1. inj_on (\\(x, y). (g x, h y)) (carrier (G \\\\ H))\n[PROOF STEP]\nunfolding iso_def DirProd_def bij_betw_def inj_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\ng \\ {h \\ hom G G2. (\\x\\carrier G. \\y\\carrier G. h x = h y \\ x = y) \\ h ` carrier G = carrier G2}\nh \\ {h \\ hom H I. (\\x\\carrier H. \\y\\carrier H. h x = h y \\ x = y) \\ h ` carrier H = carrier I}\n\ngoal (1 subgoal):\n 1. \\x\\carrier \\carrier = carrier G \\ carrier H, mult = \\(g, h) (g', h'). (g \\ g', h \\\\<^bsub>H\\<^esub> h'), one = (\\, \\\\<^bsub>H\\<^esub>)\\. \\y\\carrier \\carrier = carrier G \\ carrier H, mult = \\(g, h) (g', h'). (g \\ g', h \\\\<^bsub>H\\<^esub> h'), one = (\\, \\\\<^bsub>H\\<^esub>)\\. (case x of (x, y) \\ (g x, h y)) = (case y of (x, y) \\ (g x, h y)) \\ x = y\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\(x, y). (g x, h y)) (carrier (G \\\\ H))\n\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ Group.iso (G \\\\ H) (G2 \\\\ I)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\(x, y). (g x, h y)) (carrier (G \\\\ H))\n\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ Group.iso (G \\\\ H) (G2 \\\\ I)\n[PROOF STEP]\nhave \"(\\(x, y). (g x, h y)) ` carrier (G \\\\ H) = carrier (G2 \\\\ I)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) ` carrier (G \\\\ H) = carrier (G2 \\\\ I)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ng \\ Group.iso G G2\nh \\ Group.iso H I\n\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) ` carrier (G \\\\ H) = carrier (G2 \\\\ I)\n[PROOF STEP]\nunfolding iso_def bij_betw_def image_def DirProd_def\n[PROOF STATE]\nproof (prove)\nusing this:\ng \\ {h \\ hom G G2. inj_on h (carrier G) \\ {y. \\x\\carrier G. y = h x} = carrier G2}\nh \\ {h \\ hom H I. inj_on h (carrier H) \\ {y. \\x\\carrier H. y = h x} = carrier I}\n\ngoal (1 subgoal):\n 1. {y. \\x\\carrier \\carrier = carrier G \\ carrier H, mult = \\(g, h) (g', h'). (g \\ g', h \\\\<^bsub>H\\<^esub> h'), one = (\\, \\\\<^bsub>H\\<^esub>)\\. y = (case x of (x, y) \\ (g x, h y))} = carrier \\carrier = carrier G2 \\ carrier I, mult = \\(g, h) (g', h'). (g \\\\<^bsub>G2\\<^esub> g', h \\\\<^bsub>I\\<^esub> h'), one = (\\\\<^bsub>G2\\<^esub>, \\\\<^bsub>I\\<^esub>)\\\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n(\\(x, y). (g x, h y)) ` carrier (G \\\\ H) = carrier (G2 \\\\ I)\n\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ Group.iso (G \\\\ H) (G2 \\\\ I)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\(x, y). (g x, h y)) \\ hom (G \\\\ H) (G2 \\\\ I)\ninj_on (\\(x, y). (g x, h y)) (carrier (G \\\\ H))\n(\\(x, y). (g x, h y)) ` carrier (G \\\\ H) = carrier (G2 \\\\ I)\n[PROOF STEP]\nshow \"(\\(x,y). (g x, h y)) \\ iso (G \\\\ H) (G2 \\\\ I)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\(x, y). (g x, h y)) \\ hom (G \\\\ H) (G2 \\\\ I)\ninj_on (\\(x, y). (g x, h y)) (carrier (G \\\\ H))\n(\\(x, y). (g x, h y)) ` carrier (G \\\\ H) = carrier (G2 \\\\ I)\n\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ Group.iso (G \\\\ H) (G2 \\\\ I)\n[PROOF STEP]\nunfolding iso_def bij_betw_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\(x, y). (g x, h y)) \\ hom (G \\\\ H) (G2 \\\\ I)\ninj_on (\\(x, y). (g x, h y)) (carrier (G \\\\ H))\n(\\(x, y). (g x, h y)) ` carrier (G \\\\ H) = carrier (G2 \\\\ I)\n\ngoal (1 subgoal):\n 1. (\\(x, y). (g x, h y)) \\ {h \\ hom (G \\\\ H) (G2 \\\\ I). inj_on h (carrier (G \\\\ H)) \\ h ` carrier (G \\\\ H) = carrier (G2 \\\\ I)}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\(x, y). (g x, h y)) \\ Group.iso (G \\\\ H) (G2 \\\\ I)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3235, "file": null, "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513786759491, "lm_q2_score": 0.803173791645582, "lm_q1q2_score": 0.7198456180787424}} {"text": "[STATEMENT]\nlemma mod_plus_minus_range: \n assumes \"b>0\"\n shows \"y mod+- b \\ {-\\b/2\\..\\b/2\\}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. y mod+- b \\ {- \\real_of_int b / 2\\..\\real_of_int b / 2\\}\n[PROOF STEP]\nunfolding mod_plus_minus_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (y + \\real_of_int b / 2\\) mod b - \\real_of_int b / 2\\ \\ {- \\real_of_int b / 2\\..\\real_of_int b / 2\\}\n[PROOF STEP]\nusing mod_range[OF assms, of \"(y + \\real_of_int b / 2\\)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(y + \\real_of_int b / 2\\) mod b \\ {0..b - 1}\n\ngoal (1 subgoal):\n 1. (y + \\real_of_int b / 2\\) mod b - \\real_of_int b / 2\\ \\ {- \\real_of_int b / 2\\..\\real_of_int b / 2\\}\n[PROOF STEP]\nby (auto)(linarith)", "meta": {"llama_tokens": 452, "file": "CRYSTALS-Kyber_Mod_Plus_Minus", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513675912913, "lm_q2_score": 0.8031738010682209, "lm_q1q2_score": 0.7198456176208887}} {"text": "[STATEMENT]\nlemma tan_double: \"cos x \\ 0 \\ cos (2 * x) \\ 0 \\ tan (2 * x) = (2 * tan x) / (1 - (tan x)\\<^sup>2)\"\n for x :: \"'a::{real_normed_field,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\cos x \\ (0::'a); cos ((2::'a) * x) \\ (0::'a)\\ \\ tan ((2::'a) * x) = (2::'a) * tan x / ((1::'a) - (tan x)\\<^sup>2)\n[PROOF STEP]\nusing tan_add [of x x]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\cos x \\ (0::'a); cos x \\ (0::'a); cos (x + x) \\ (0::'a)\\ \\ tan (x + x) = (tan x + tan x) / ((1::'a) - tan x * tan x)\n\ngoal (1 subgoal):\n 1. \\cos x \\ (0::'a); cos ((2::'a) * x) \\ (0::'a)\\ \\ tan ((2::'a) * x) = (2::'a) * tan x / ((1::'a) - (tan x)\\<^sup>2)\n[PROOF STEP]\nby (simp add: power2_eq_square)", "meta": {"llama_tokens": 423, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513620489619, "lm_q2_score": 0.8031737869342624, "lm_q1q2_score": 0.7198456005018554}} {"text": "[STATEMENT]\nlemma prod_id_cancel_nat:\n \\ \\Contributed by Manuel Eberl\\\n fixes A::\"nat set\"\n assumes \"B \\ A\" and \"finite A\" and \"0 \\ B\"\n shows \"\\A / \\B = \\(A-B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (\\A) / real (\\B) = real (\\(A - B))\n[PROOF STEP]\nusing assms(1-2)\n[PROOF STATE]\nproof (prove)\nusing this:\nB \\ A\nfinite A\n\ngoal (1 subgoal):\n 1. real (\\A) / real (\\B) = real (\\(A - B))\n[PROOF STEP]\nby (rule prod_cancel_nat) (metis assms(3))", "meta": {"llama_tokens": 242, "file": "Random_Graph_Subgraph_Threshold_Ugraph_Misc", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505402422645, "lm_q2_score": 0.7956581073313275, "lm_q1q2_score": 0.7197925366454231}} {"text": "[STATEMENT]\nlemma dist_riemann_sphere_r3_inner:\n assumes \"M1 \\ unit_sphere\" and \"M2 \\ unit_sphere\"\n shows \"(dist_riemann_sphere_r3 M1 M2)\\<^sup>2 = 2 - 2 * inner M1 M2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist_riemann_sphere_r3 M1 M2)\\<^sup>2 = 2 - 2 * inner M1 M2\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nM1 \\ unit_sphere\nM2 \\ unit_sphere\n\ngoal (1 subgoal):\n 1. (dist_riemann_sphere_r3 M1 M2)\\<^sup>2 = 2 - 2 * inner M1 M2\n[PROOF STEP]\napply (cases M1, cases M2)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x a b c. \\M2 \\ unit_sphere; M1 = Rep_riemann_sphere x; M1 \\ unit_sphere; M2 \\ unit_sphere; M2 = (a, b, c)\\ \\ (dist_riemann_sphere_r3 M1 M2)\\<^sup>2 = 2 - 2 * inner M1 M2\n[PROOF STEP]\napply (auto simp add: norm_prod_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\x a b c xa aa ba. \\a * a + b * b + c * c = 1; M1 = (xa, aa, ba); M2 = (a, b, c); Rep_riemann_sphere x = (xa, aa, ba); xa * xa + aa * aa + ba * ba = 1\\ \\ (xa - a)\\<^sup>2 + ((aa - b)\\<^sup>2 + (ba - c)\\<^sup>2) = 2 - (2 * (xa * a) + (2 * (aa * b) + 2 * (ba * c)))\n[PROOF STEP]\napply (simp add: power2_eq_square field_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 640, "file": "Complex_Geometry_Chordal_Metric", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505325302034, "lm_q2_score": 0.7956581024858786, "lm_q1q2_score": 0.7197925261258213}} {"text": "[STATEMENT]\nlemma DDH_bound2: \n shows \"\\spmf (ddh.DDH0 \\ \\) True - spmf (ddh.DDH1 \\ \\) True\\ \n \\ ddh.advantage \\ \\ + \\spmf (ddh.ddh_1 \\ \\) True - spmf (ddh.DDH1 \\ \\) True\\\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\spmf (DDH0 \\) True - spmf (DDH1 \\) True\\ \\ advantage \\ + \\spmf (ddh_1 \\) True - spmf (DDH1 \\) True\\\n[PROOF STEP]\nusing advantage_def DDH_bound1\n[PROOF STATE]\nproof (prove)\nusing this:\nadvantage ?\\ = \\spmf (ddh_0 ?\\) True - spmf (ddh_1 ?\\) True\\\n\\spmf (DDH0 ?\\) True - spmf (DDH1 ?\\) True\\ \\ \\spmf (ddh_0 ?\\) True - spmf (ddh_1 ?\\) True\\ + \\spmf (ddh_1 ?\\) True - spmf (DDH1 ?\\) True\\\n\ngoal (1 subgoal):\n 1. \\spmf (DDH0 \\) True - spmf (DDH1 \\) True\\ \\ advantage \\ + \\spmf (ddh_1 \\) True - spmf (DDH1 \\) True\\\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 477, "file": "Multi_Party_Computation_DH_Ext", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505325302033, "lm_q2_score": 0.7956580903722561, "lm_q1q2_score": 0.7197925151672261}} {"text": "[STATEMENT]\nlemma sum_list_map_power2_poly:\n shows \"poly (sum_list (map power2 (ls::real poly list))) x \\ (0::real)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ poly (sum_list (map power2 ls)) x\n[PROOF STEP]\napply (induct ls)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. 0 \\ poly (sum_list (map power2 [])) x\n 2. \\a ls. 0 \\ poly (sum_list (map power2 ls)) x \\ 0 \\ poly (sum_list (map power2 (a # ls))) x\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 205, "file": "BenOr_Kozen_Reif_Renegar_Proofs", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972616934406, "lm_q2_score": 0.8267118026095992, "lm_q1q2_score": 0.7197330315615653}} {"text": "[STATEMENT]\nlemma mult_adjugate_det: \"A * adjugate A = diag (det A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A * adjugate A = diag (Square_Matrix.det A)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. A * adjugate A = diag (Square_Matrix.det A)\n[PROOF STEP]\nhave \"transpose (transpose (A * adjugate A)) = transpose (diag (det A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Square_Matrix.transpose (Square_Matrix.transpose (A * adjugate A)) = Square_Matrix.transpose (diag (Square_Matrix.det A))\n[PROOF STEP]\nunfolding transpose_mult adjugate_transpose[symmetric] adjugate_mult_det det_transpose\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Square_Matrix.transpose (diag (Square_Matrix.det A)) = Square_Matrix.transpose (diag (Square_Matrix.det A))\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nSquare_Matrix.transpose (Square_Matrix.transpose (A * adjugate A)) = Square_Matrix.transpose (diag (Square_Matrix.det A))\n\ngoal (1 subgoal):\n 1. A * adjugate A = diag (Square_Matrix.det A)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nSquare_Matrix.transpose (Square_Matrix.transpose (A * adjugate A)) = Square_Matrix.transpose (diag (Square_Matrix.det A))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nSquare_Matrix.transpose (Square_Matrix.transpose (A * adjugate A)) = Square_Matrix.transpose (diag (Square_Matrix.det A))\n\ngoal (1 subgoal):\n 1. A * adjugate A = diag (Square_Matrix.det A)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nA * adjugate A = diag (Square_Matrix.det A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 650, "file": "Cayley_Hamilton_Square_Matrix", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972549785201, "lm_q2_score": 0.8267118004748677, "lm_q1q2_score": 0.7197330241517699}} {"text": "[STATEMENT]\nlemma closest_point_exists:\n assumes \"closed S\"\n and \"S \\ {}\"\n shows closest_point_in_set: \"closest_point S a \\ S\"\n and \"\\y\\S. dist a (closest_point S a) \\ dist a y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. closest_point S a \\ S &&& \\y\\S. dist a (closest_point S a) \\ dist a y\n[PROOF STEP]\nunfolding closest_point_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (SOME x. x \\ S \\ (\\y\\S. dist a x \\ dist a y)) \\ S &&& \\y\\S. dist a (SOME x. x \\ S \\ (\\y\\S. dist a x \\ dist a y)) \\ dist a y\n[PROOF STEP]\nby (rule_tac someI2_ex, auto intro: distance_attains_inf[OF assms(1,2), of a])+", "meta": {"llama_tokens": 312, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972583359805, "lm_q2_score": 0.826711791935942, "lm_q1q2_score": 0.7197330194934566}} {"text": "[STATEMENT]\nlemma \"(a::int) ^ 10 - b ^ 10 =\n (a - b) * (a ^ 9 + a ^ 8 * b + a ^ 7 * b ^ 2 + a ^ 6 * b ^ 3 + a ^ 5 * b ^ 4 +\n a ^ 4 * b ^ 5 + a ^ 3 * b ^ 6 + a ^ 2 * b ^ 7 + a * b ^ 8 + b ^ 9)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a ^ 10 - b ^ 10 = (a - b) * (a ^ 9 + a ^ 8 * b + a ^ 7 * b\\<^sup>2 + a ^ 6 * b ^ 3 + a ^ 5 * b ^ 4 + a ^ 4 * b ^ 5 + a ^ 3 * b ^ 6 + a\\<^sup>2 * b ^ 7 + a * b ^ 8 + b ^ 9)\n[PROOF STEP]\nby ring", "meta": {"llama_tokens": 257, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9324533126145178, "lm_q2_score": 0.7718435030872968, "lm_q1q2_score": 0.7197080312737437}} {"text": "[STATEMENT]\nlemma DERIV_quotient:\n \"(f has_field_derivative d) (at x within s) \\\n (g has_field_derivative e) (at x within s)\\ g x \\ 0 \\\n ((\\y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative d) (at x within s); (g has_field_derivative e) (at x within s); g x \\ (0::'a)\\ \\ ((\\y. f y / g y) has_field_derivative (d * g x - e * f x) / g x ^ Suc (Suc 0)) (at x within s)\n[PROOF STEP]\nby (drule (2) DERIV_divide) (simp add: mult.commute)", "meta": {"llama_tokens": 284, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094117351309, "lm_q2_score": 0.8056321936479701, "lm_q1q2_score": 0.7196788209825512}} {"text": "[STATEMENT]\nlemma DERIV_quotient:\n \"(f has_field_derivative d) (at x within s) \\\n (g has_field_derivative e) (at x within s)\\ g x \\ 0 \\\n ((\\y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative d) (at x within s); (g has_field_derivative e) (at x within s); g x \\ (0::'a)\\ \\ ((\\y. f y / g y) has_field_derivative (d * g x - e * f x) / g x ^ Suc (Suc 0)) (at x within s)\n[PROOF STEP]\nby (drule (2) DERIV_divide) (simp add: mult.commute)", "meta": {"llama_tokens": 284, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094117351309, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7196788188981411}} {"text": "[STATEMENT]\nlemma DERIV_quotient:\n \"(f has_field_derivative d) (at x within s) \\\n (g has_field_derivative e) (at x within s)\\ g x \\ 0 \\\n ((\\y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\(f has_field_derivative d) (at x within s); (g has_field_derivative e) (at x within s); g x \\ (0::'a)\\ \\ ((\\y. f y / g y) has_field_derivative (d * g x - e * f x) / g x ^ Suc (Suc 0)) (at x within s)\n[PROOF STEP]\nby (drule (2) DERIV_divide) (simp add: mult.commute)", "meta": {"llama_tokens": 284, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094117351309, "lm_q2_score": 0.8056321913146127, "lm_q1q2_score": 0.7196788188981411}} {"text": "[STATEMENT]\nlemma nth_weave:\nassumes \"i < length (weave A xs ys)\"\nshows \"weave A xs ys ! i = (if i\\A then xs!(card {a\\A. a-A. a A then xs ! card {a \\ A. a < i} else ys ! card {a \\ - A. a < i})\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. weave A xs ys ! i = (if i \\ A then xs ! card {a \\ A. a < i} else ys ! card {a \\ - A. a < i})\n[PROOF STEP]\nhave \"i < length xs + length ys\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. i < length xs + length ys\n[PROOF STEP]\nusing length_weave\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (weave ?A ?xs ?ys) = length ?xs + length ?ys\n\ngoal (1 subgoal):\n 1. i < length xs + length ys\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength (weave ?A ?xs ?ys) = length ?xs + length ?ys\ni < length (weave A xs ys)\n\ngoal (1 subgoal):\n 1. i < length xs + length ys\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\ni < length xs + length ys\n\ngoal (1 subgoal):\n 1. weave A xs ys ! i = (if i \\ A then xs ! card {a \\ A. a < i} else ys ! card {a \\ - A. a < i})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni < length xs + length ys\n[PROOF STEP]\nhave \"i < length [0.. A then xs ! card {a \\ A. a < i} else ys ! card {a \\ - A. a < i})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ni < length [0..i < length xs + length ys\\ add.left_neutral nth_upt)\n[PROOF STATE]\nproof (state)\nthis:\n[0.. A then xs ! card {a \\ A. a < i} else ys ! card {a \\ - A. a < i})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[0.. A then xs ! card {a \\ A. a < i} else ys ! card {a \\ - A. a < i})\n[PROOF STEP]\nunfolding weave_def nth_map[OF \\i < length [0..]\n[PROOF STATE]\nproof (prove)\nusing this:\n[0.. A then xs ! card {a \\ A. a < [0.. - A. a < [0.. A then xs ! card {a \\ A. a < i} else ys ! card {a \\ - A. a < i})\n[PROOF STEP]\nby presburger\n[PROOF STATE]\nproof (state)\nthis:\nweave A xs ys ! i = (if i \\ A then xs ! card {a \\ A. a < i} else ys ! card {a \\ - A. a < i})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1452, "file": "Jordan_Normal_Form_DL_Missing_Sublist", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094117351309, "lm_q2_score": 0.8056321866478978, "lm_q1q2_score": 0.7196788147293208}} {"text": "[STATEMENT]\nlemma in_closed_iff_infdist_zero:\n assumes \"closed A\" \"A \\ {}\"\n shows \"x \\ A \\ infdist x A = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x \\ A) = (infdist x A = 0)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (x \\ A) = (infdist x A = 0)\n[PROOF STEP]\nhave \"x \\ closure A \\ infdist x A = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x \\ closure A) = (infdist x A = 0)\n[PROOF STEP]\nby (rule in_closure_iff_infdist_zero) fact\n[PROOF STATE]\nproof (state)\nthis:\n(x \\ closure A) = (infdist x A = 0)\n\ngoal (1 subgoal):\n 1. (x \\ A) = (infdist x A = 0)\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\nclosed A\nA \\ {}\n(x \\ closure A) = (infdist x A = 0)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nclosed A\nA \\ {}\n(x \\ closure A) = (infdist x A = 0)\n\ngoal (1 subgoal):\n 1. (x \\ A) = (infdist x A = 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(x \\ A) = (infdist x A = 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 515, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916029436189, "lm_q2_score": 0.8333246035907933, "lm_q1q2_score": 0.7194854652666108}} {"text": "[STATEMENT]\ntheorem rtl_inter_closed: \n \"\\L1\\regular_languages A; L2\\regular_languages A\\ \\ \n L1\\L2 \\ regular_languages A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\L1 \\ regular_languages A; L2 \\ regular_languages A\\ \\ L1 \\ L2 \\ regular_languages A\n[PROOF STEP]\nproof (elim rtlE, goal_cases)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\TA TAa. \\L1 = ta_lang TA; ranked_tree_automaton TA A; L2 = ta_lang TAa; ranked_tree_automaton TAa A\\ \\ L1 \\ L2 \\ regular_languages A\n[PROOF STEP]\ncase (1 TA1 TA2)\n[PROOF STATE]\nproof (state)\nthis:\nL1 = ta_lang TA1\nranked_tree_automaton TA1 A\nL2 = ta_lang TA2\nranked_tree_automaton TA2 A\n\ngoal (1 subgoal):\n 1. \\TA TAa. \\L1 = ta_lang TA; ranked_tree_automaton TA A; L2 = ta_lang TAa; ranked_tree_automaton TAa A\\ \\ L1 \\ L2 \\ regular_languages A\n[PROOF STEP]\nwith ta_prod_correct[of TA1 TA2] ta_prod_rta[of TA1 A TA2]\n[PROOF STATE]\nproof (chain)\npicking this:\n\\tree_automaton TA1; tree_automaton TA2\\ \\ ta_lang (ta_prod TA1 TA2) = ta_lang TA1 \\ ta_lang TA2\n\\tree_automaton TA1; tree_automaton TA2\\ \\ tree_automaton (ta_prod TA1 TA2)\n\\ranked_tree_automaton TA1 A; ranked_tree_automaton TA2 A\\ \\ ranked_tree_automaton (ta_prod TA1 TA2) A\nL1 = ta_lang TA1\nranked_tree_automaton TA1 A\nL2 = ta_lang TA2\nranked_tree_automaton TA2 A\n[PROOF STEP]\nhave \n L: \"ta_lang (ta_prod TA1 TA2) = L1\\L2\" and\n A: \"ranked_tree_automaton (ta_prod TA1 TA2) A\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\tree_automaton TA1; tree_automaton TA2\\ \\ ta_lang (ta_prod TA1 TA2) = ta_lang TA1 \\ ta_lang TA2\n\\tree_automaton TA1; tree_automaton TA2\\ \\ tree_automaton (ta_prod TA1 TA2)\n\\ranked_tree_automaton TA1 A; ranked_tree_automaton TA2 A\\ \\ ranked_tree_automaton (ta_prod TA1 TA2) A\nL1 = ta_lang TA1\nranked_tree_automaton TA1 A\nL2 = ta_lang TA2\nranked_tree_automaton TA2 A\n\ngoal (1 subgoal):\n 1. ta_lang (ta_prod TA1 TA2) = L1 \\ L2 &&& ranked_tree_automaton (ta_prod TA1 TA2) A\n[PROOF STEP]\nby (simp_all add: ranked_tree_automaton.axioms)\n[PROOF STATE]\nproof (state)\nthis:\nta_lang (ta_prod TA1 TA2) = L1 \\ L2\nranked_tree_automaton (ta_prod TA1 TA2) A\n\ngoal (1 subgoal):\n 1. \\TA TAa. \\L1 = ta_lang TA; ranked_tree_automaton TA A; L2 = ta_lang TAa; ranked_tree_automaton TAa A\\ \\ L1 \\ L2 \\ regular_languages A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. L1 \\ L2 \\ regular_languages A\n[PROOF STEP]\nusing ranked_tree_automaton.rtlI[OF A]\n[PROOF STATE]\nproof (prove)\nusing this:\nta_lang (ta_prod TA1 TA2) \\ regular_languages A\n\ngoal (1 subgoal):\n 1. L1 \\ L2 \\ regular_languages A\n[PROOF STEP]\nby (simp add: L)\n[PROOF STATE]\nproof (state)\nthis:\nL1 \\ L2 \\ regular_languages A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1403, "file": "Tree-Automata_Ta", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.8198933381139645, "lm_q1q2_score": 0.7194373724060042}} {"text": "[STATEMENT]\nlemma mod_add_eq_imp_mod_0: \"\n ((n + k) mod (m::nat) = n mod m) = (k mod m = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((n + k) mod m = n mod m) = (k mod m = 0)\n[PROOF STEP]\nby (metis add_eq_if mod_add mod_add_self1 mod_self add.commute)", "meta": {"llama_tokens": 126, "file": "List-Infinite_CommonArith_Util_Div", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9161096158798117, "lm_q2_score": 0.7853085758631159, "lm_q1q2_score": 0.719428737781081}} {"text": "[STATEMENT]\nlemma consistentD:\n assumes \"consistent x t T B p incr\"\n assumes \"h \\ 0\" \"t + h \\ T\"\n shows \"dist (x (t + h)) (discrete_evolution incr (t + h) t (x t)) \\ B * h ^ (p + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist (x (t + h)) (discrete_evolution incr (t + h) t (x t)) \\ B * h ^ (p + 1)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nconsistent x t T B p incr\n0 \\ h\nt + h \\ T\n\ngoal (1 subgoal):\n 1. dist (x (t + h)) (discrete_evolution incr (t + h) t (x t)) \\ B * h ^ (p + 1)\n[PROOF STEP]\nunfolding consistent_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\h\\0. t + h \\ T \\ dist (x (t + h)) (discrete_evolution incr (t + h) t (x t)) \\ B * h ^ (p + 1)\n0 \\ h\nt + h \\ T\n\ngoal (1 subgoal):\n 1. dist (x (t + h)) (discrete_evolution incr (t + h) t (x t)) \\ B * h ^ (p + 1)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 420, "file": "Ordinary_Differential_Equations_Numerics_One_Step_Method", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9161096090086367, "lm_q2_score": 0.7853085708384736, "lm_q1q2_score": 0.7194287277819653}} {"text": "[STATEMENT]\ntheorem hoare_fixpoint_complete_mbt:\n \"F x = x\n \\ (!! w f . hoare (Sup_less p w) f q \\ hoare (p w) (F f) q) \n \\ hoare (Sup (range p)) x q\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\F x = x; \\w f. hoare (Sup_less p w) f q \\ hoare (p w) (F f) q\\ \\ hoare (\\ range p) x q\n[PROOF STEP]\napply (simp add: hoare_Sup Sup_less_def, safe)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\pa. \\F x = x; \\w f. \\pa. (\\v hoare pa f q \\ hoare (p w) (F f) q\\ \\ hoare (p pa) x q\n[PROOF STEP]\napply (rule_tac F = F in hoare_fixpoint_mbt)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\pa. \\F x = x; \\w f. \\pa. (\\v hoare pa f q \\ hoare (p w) (F f) q\\ \\ F x = x\n 2. \\pa w f. \\F x = x; \\w f. \\pa. (\\v hoare pa f q \\ hoare (p w) (F f) q; \\v. v < w \\ hoare (p v) f q\\ \\ hoare (p w) (F f) q\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 533, "file": "MonoBoolTranAlgebra_Statements", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942377652497, "lm_q2_score": 0.8080672112416737, "lm_q1q2_score": 0.7194175818954969}} {"text": "[STATEMENT]\ntheorem NonnegFirstPrices:\n assumes \"\\ X. b (n, X) \\ 0\" \n shows \"firstPriceP N \\ b r n \\ 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\ firstPriceP N \\ b r n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\X. 0 \\ b (n, X)\n\ngoal (1 subgoal):\n 1. 0 \\ firstPriceP N \\ b r n\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 178, "file": "Vickrey_Clarke_Groves_FirstPrice", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942261220294, "lm_q2_score": 0.8080672181749421, "lm_q1q2_score": 0.7194175786596412}} {"text": "[STATEMENT]\nlemma filtermap_times_pos_at_right:\n fixes c::\"'a::{linordered_field, linorder_topology}\"\n assumes \"c > 0\"\n shows \"filtermap (times c) (at_right p) = at_right (c * p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. filtermap ((*) c) (at_right p) = at_right (c * p)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) < c\n\ngoal (1 subgoal):\n 1. filtermap ((*) c) (at_right p) = at_right (c * p)\n[PROOF STEP]\nby (intro filtermap_fun_inverse[where g=\"\\x. inverse c * x\"])\n (auto intro!: filterlim_ident filterlim_times_pos)", "meta": {"llama_tokens": 239, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.882427872638409, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7193838752458878}} {"text": "[STATEMENT]\nlemma hom_component_minus: \"hom_component (p - q) n = hom_component p n - hom_component q n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. hom_component (p - q) n = hom_component p n - hom_component q n\n[PROOF STEP]\nby (rule poly_mapping_eqI) (simp add: hom_component_def lookup_except lookup_minus)", "meta": {"llama_tokens": 115, "file": "Polynomials_MPoly_PM", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278664544911, "lm_q2_score": 0.8152324915965392, "lm_q1q2_score": 0.719383868223913}} {"text": "[STATEMENT]\nlemma divideC_field_simps[field_simps]: (* In Real_Vector_Spaces, these lemmas are unnamed *)\n \"c \\ 0 \\ a = b /\\<^sub>C c \\ c *\\<^sub>C a = b\"\n \"c \\ 0 \\ b /\\<^sub>C c = a \\ b = c *\\<^sub>C a\"\n \"c \\ 0 \\ a + b /\\<^sub>C c = (c *\\<^sub>C a + b) /\\<^sub>C c\"\n \"c \\ 0 \\ a /\\<^sub>C c + b = (a + c *\\<^sub>C b) /\\<^sub>C c\"\n \"c \\ 0 \\ a - b /\\<^sub>C c = (c *\\<^sub>C a - b) /\\<^sub>C c\"\n \"c \\ 0 \\ a /\\<^sub>C c - b = (a - c *\\<^sub>C b) /\\<^sub>C c\"\n \"c \\ 0 \\ - (a /\\<^sub>C c) + b = (- a + c *\\<^sub>C b) /\\<^sub>C c\"\n \"c \\ 0 \\ - (a /\\<^sub>C c) - b = (- a - c *\\<^sub>C b) /\\<^sub>C c\"\n for a b :: \"'a :: complex_vector\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (((c \\ 0 \\ (a = b /\\<^sub>C c) = (c *\\<^sub>C a = b)) &&& (c \\ 0 \\ (b /\\<^sub>C c = a) = (b = c *\\<^sub>C a))) &&& (c \\ 0 \\ a + b /\\<^sub>C c = (c *\\<^sub>C a + b) /\\<^sub>C c) &&& (c \\ 0 \\ a /\\<^sub>C c + b = (a + c *\\<^sub>C b) /\\<^sub>C c)) &&& ((c \\ 0 \\ a - b /\\<^sub>C c = (c *\\<^sub>C a - b) /\\<^sub>C c) &&& (c \\ 0 \\ a /\\<^sub>C c - b = (a - c *\\<^sub>C b) /\\<^sub>C c)) &&& (c \\ 0 \\ - (a /\\<^sub>C c) + b = (- a + c *\\<^sub>C b) /\\<^sub>C c) &&& (c \\ 0 \\ - (a /\\<^sub>C c) - b = (- a - c *\\<^sub>C b) /\\<^sub>C c)\n[PROOF STEP]\nby (auto simp add: scaleC_add_right scaleC_add_left scaleC_diff_right scaleC_diff_left)", "meta": {"llama_tokens": 762, "file": "Complex_Bounded_Operators_Complex_Vector_Spaces0", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278602705732, "lm_q2_score": 0.815232480373843, "lm_q1q2_score": 0.7193838532793623}} {"text": "[STATEMENT]\nlemma dist_triangle_eq:\n fixes a b c :: \"'a :: real_inner\"\n shows \"(dist a c = dist a b + dist b c) \\ dist a b *\\<^sub>R (c - b) + dist b c *\\<^sub>R (a - b) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist a c = dist a b + dist b c) = (dist a b *\\<^sub>R (c - b) + dist b c *\\<^sub>R (a - b) = (0::'a))\n[PROOF STEP]\nusing norm_triangle_eq[of \"b - a\" \"c - b\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(norm (b - a + (c - b)) = norm (b - a) + norm (c - b)) = (norm (b - a) *\\<^sub>R (c - b) = norm (c - b) *\\<^sub>R (b - a))\n\ngoal (1 subgoal):\n 1. (dist a c = dist a b + dist b c) = (dist a b *\\<^sub>R (c - b) + dist b c *\\<^sub>R (a - b) = (0::'a))\n[PROOF STEP]\nby (simp add: dist_norm norm_minus_commute algebra_simps)", "meta": {"llama_tokens": 346, "file": "Triangle_Angles", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099069962657176, "lm_q2_score": 0.79053032607222, "lm_q1q2_score": 0.7193090744533319}} {"text": "[STATEMENT]\nlemma ocU_is_expected_utility_bernoulli:\n shows \"\\x \\ \\