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5697766 | 1 2 3 4 5 6 7 | {"text": "[GOAL]\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np✝ q✝ p q : R\nhp : p ≠ 0\n⊢ p / GCDMonoid.gcd p q ≠ 0\n[PROOFSTEP]\nobtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q\n[GOAL]\ncase intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np✝ q✝ p q : R\nhp : p ≠ 0\nr : R\nhr : p = GCDMonoid.gcd p q * r\n⊢ p / GCDMonoid.gcd p q ≠ 0\n[PROOFSTEP]\nobtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np✝ q✝ p q : R\nhp : p ≠ 0\nr : R\nhr : p = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ p / GCDMonoid.gcd p q ≠ 0\n[PROOFSTEP]\nnth_rw 1 [hr]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np✝ q✝ p q : R\nhp : p ≠ 0\nr : R\nhr : p = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0\n[PROOFSTEP]\nrw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np✝ q✝ p q : R\nhp : p ≠ 0\nr : R\nhr : p = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ r ≠ 0\n[PROOFSTEP]\nexact r0\n[GOAL]\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np✝ q✝ p q : R\nhq : q ≠ 0\n⊢ q / GCDMonoid.gcd p q ≠ 0\n[PROOFSTEP]\nobtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q\n[GOAL]\ncase intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np✝ q✝ p q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\n⊢ q / GCDMonoid.gcd p q ≠ 0\n[PROOFSTEP]\nobtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np✝ q✝ p q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ q / GCDMonoid.gcd p q ≠ 0\n[PROOFSTEP]\nnth_rw 1 [hr]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np✝ q✝ p q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0\n[PROOFSTEP]\nrw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np✝ q✝ p q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ r ≠ 0\n[PROOFSTEP]\nexact r0\n[GOAL]\nR : Type ?u.6736\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ Associated (gcd a b * lcm a b) (a * b)\n[PROOFSTEP]\nrw [EuclideanDomain.gcd_mul_lcm]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.EuclideanDomain", "llama_tokens": 1494, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099070133672955, "lm_q2_score": 0.8519528076067262, "lm_q1q2_score": 0.7751978346993184}}
{"text": "[GOAL]\nn : ℕ\n⊢ (n + 1)‼ = (n + 1) * (n - 1)‼\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n⊢ (zero + 1)‼ = (zero + 1) * (zero - 1)‼\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn✝ : ℕ\n⊢ (succ n✝ + 1)‼ = (succ n✝ + 1) * (succ n✝ - 1)‼\n[PROOFSTEP]\nrfl\n[GOAL]\nk : ℕ\n⊢ (k + 1 + 1)! = (k + 1 + 1)‼ * (k + 1)‼\n[PROOFSTEP]\nrw [doubleFactorial_add_two, factorial, factorial_eq_mul_doubleFactorial _, mul_comm _ k‼, mul_assoc]\n[GOAL]\nn : ℕ\n⊢ (2 * (n + 1))‼ = 2 ^ (n + 1) * (n + 1)!\n[PROOFSTEP]\nrw [mul_add, mul_one, doubleFactorial_add_two, factorial, pow_succ, doubleFactorial_two_mul _, succ_eq_add_one]\n[GOAL]\nn : ℕ\n⊢ (2 * n + 2) * (2 ^ n * n !) = 2 ^ n * 2 * ((n + 1) * n !)\n[PROOFSTEP]\nring\n[GOAL]\nn : ℕ\n⊢ (2 * (n + 1))‼ = ∏ i in Finset.range (n + 1), 2 * (i + 1)\n[PROOFSTEP]\nrw [Finset.prod_range_succ, ← doubleFactorial_eq_prod_even _, mul_comm (2 * n)‼, (by ring : 2 * (n + 1) = 2 * n + 2)]\n[GOAL]\nn : ℕ\n⊢ 2 * (n + 1) = 2 * n + 2\n[PROOFSTEP]\nring\n[GOAL]\nn : ℕ\n⊢ (2 * n + 2)‼ = (2 * n + 2) * (2 * n)‼\n[PROOFSTEP]\nrfl\n[GOAL]\nn : ℕ\n⊢ (2 * (n + 1) + 1)‼ = ∏ i in Finset.range (n + 1), (2 * (i + 1) + 1)\n[PROOFSTEP]\nrw [Finset.prod_range_succ, ← doubleFactorial_eq_prod_odd _, mul_comm (2 * n + 1)‼,\n (by ring : 2 * (n + 1) + 1 = 2 * n + 1 + 2)]\n[GOAL]\nn : ℕ\n⊢ 2 * (n + 1) + 1 = 2 * n + 1 + 2\n[PROOFSTEP]\nring\n[GOAL]\nn : ℕ\n⊢ (2 * n + 1 + 2)‼ = (2 * n + 1 + 2) * (2 * n + 1)‼\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Factorial.DoubleFactorial", "llama_tokens": 850, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.933430812881347, "lm_q2_score": 0.8221891327004132, "lm_q1q2_score": 0.7674566704787563}}
{"text": "[GOAL]\nx✝ y x : ℝ\n⊢ exp (arsinh x) = x + sqrt (1 + x ^ 2)\n[PROOFSTEP]\napply exp_log\n[GOAL]\ncase hx\nx✝ y x : ℝ\n⊢ 0 < x + sqrt (1 + x ^ 2)\n[PROOFSTEP]\nrw [← neg_lt_iff_pos_add']\n[GOAL]\ncase hx\nx✝ y x : ℝ\n⊢ -x < sqrt (1 + x ^ 2)\n[PROOFSTEP]\napply lt_sqrt_of_sq_lt\n[GOAL]\ncase hx.h\nx✝ y x : ℝ\n⊢ (-x) ^ 2 < 1 + x ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx y : ℝ\n⊢ arsinh 0 = 0\n[PROOFSTEP]\nsimp [arsinh]\n[GOAL]\nx✝ y x : ℝ\n⊢ arsinh (-x) = -arsinh x\n[PROOFSTEP]\nrw [← exp_eq_exp, exp_arsinh, exp_neg, exp_arsinh]\n[GOAL]\nx✝ y x : ℝ\n⊢ -x + sqrt (1 + (-x) ^ 2) = (x + sqrt (1 + x ^ 2))⁻¹\n[PROOFSTEP]\napply eq_inv_of_mul_eq_one_left\n[GOAL]\ncase h\nx✝ y x : ℝ\n⊢ (-x + sqrt (1 + (-x) ^ 2)) * (x + sqrt (1 + x ^ 2)) = 1\n[PROOFSTEP]\nrw [neg_sq, neg_add_eq_sub, add_comm x, mul_comm, ← sq_sub_sq, sq_sqrt, add_sub_cancel]\n[GOAL]\ncase h\nx✝ y x : ℝ\n⊢ 0 ≤ 1 + x ^ 2\n[PROOFSTEP]\nexact add_nonneg zero_le_one (sq_nonneg _)\n[GOAL]\nx✝ y x : ℝ\n⊢ sinh (arsinh x) = x\n[PROOFSTEP]\nrw [sinh_eq, ← arsinh_neg, exp_arsinh, exp_arsinh, neg_sq]\n[GOAL]\nx✝ y x : ℝ\n⊢ (x + sqrt (1 + x ^ 2) - (-x + sqrt (1 + x ^ 2))) / 2 = x\n[PROOFSTEP]\nfield_simp\n[GOAL]\nx✝ y x : ℝ\n⊢ cosh (arsinh x) = sqrt (1 + x ^ 2)\n[PROOFSTEP]\nrw [← sqrt_sq (cosh_pos _).le, cosh_sq', sinh_arsinh]\n[GOAL]\nx y : ℝ\n⊢ 0 ≤ arsinh x ↔ 0 ≤ x\n[PROOFSTEP]\nrw [← sinh_le_sinh, sinh_zero, sinh_arsinh]\n[GOAL]\nx y : ℝ\n⊢ arsinh x ≤ 0 ↔ x ≤ 0\n[PROOFSTEP]\nrw [← sinh_le_sinh, sinh_zero, sinh_arsinh]\n[GOAL]\nx✝ y x : ℝ\n⊢ HasStrictDerivAt arsinh (sqrt (1 + x ^ 2))⁻¹ x\n[PROOFSTEP]\nconvert\n sinhHomeomorph.toLocalHomeomorph.hasStrictDerivAt_symm (mem_univ x) (cosh_pos _).ne' (hasStrictDerivAt_sinh _) using 2\n[GOAL]\ncase h.e'_7.h.e'_3\nx✝ y x : ℝ\n⊢ sqrt (1 + x ^ 2) = cosh (↑(LocalHomeomorph.symm (Homeomorph.toLocalHomeomorph sinhHomeomorph)) x)\n[PROOFSTEP]\nexact (cosh_arsinh _).symm\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Arsinh", "llama_tokens": 1034, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026528034425, "lm_q2_score": 0.8354835452961427, "lm_q1q2_score": 0.7663912724737768}}
{"text": "[GOAL]\na : ℝ\n⊢ interior {z | z.re ≤ a} = {z | z.re < a}\n[PROOFSTEP]\nsimpa only [interior_Iic] using interior_preimage_re (Iic a)\n[GOAL]\na : ℝ\n⊢ interior {z | z.im ≤ a} = {z | z.im < a}\n[PROOFSTEP]\nsimpa only [interior_Iic] using interior_preimage_im (Iic a)\n[GOAL]\na : ℝ\n⊢ interior {z | a ≤ z.re} = {z | a < z.re}\n[PROOFSTEP]\nsimpa only [interior_Ici] using interior_preimage_re (Ici a)\n[GOAL]\na : ℝ\n⊢ interior {z | a ≤ z.im} = {z | a < z.im}\n[PROOFSTEP]\nsimpa only [interior_Ici] using interior_preimage_im (Ici a)\n[GOAL]\na : ℝ\n⊢ closure {z | z.re < a} = {z | z.re ≤ a}\n[PROOFSTEP]\nsimpa only [closure_Iio] using closure_preimage_re (Iio a)\n[GOAL]\na : ℝ\n⊢ closure {z | z.im < a} = {z | z.im ≤ a}\n[PROOFSTEP]\nsimpa only [closure_Iio] using closure_preimage_im (Iio a)\n[GOAL]\na : ℝ\n⊢ closure {z | a < z.re} = {z | a ≤ z.re}\n[PROOFSTEP]\nsimpa only [closure_Ioi] using closure_preimage_re (Ioi a)\n[GOAL]\na : ℝ\n⊢ closure {z | a < z.im} = {z | a ≤ z.im}\n[PROOFSTEP]\nsimpa only [closure_Ioi] using closure_preimage_im (Ioi a)\n[GOAL]\na : ℝ\n⊢ frontier {z | z.re ≤ a} = {z | z.re = a}\n[PROOFSTEP]\nsimpa only [frontier_Iic] using frontier_preimage_re (Iic a)\n[GOAL]\na : ℝ\n⊢ frontier {z | z.im ≤ a} = {z | z.im = a}\n[PROOFSTEP]\nsimpa only [frontier_Iic] using frontier_preimage_im (Iic a)\n[GOAL]\na : ℝ\n⊢ frontier {z | a ≤ z.re} = {z | z.re = a}\n[PROOFSTEP]\nsimpa only [frontier_Ici] using frontier_preimage_re (Ici a)\n[GOAL]\na : ℝ\n⊢ frontier {z | a ≤ z.im} = {z | z.im = a}\n[PROOFSTEP]\nsimpa only [frontier_Ici] using frontier_preimage_im (Ici a)\n[GOAL]\na : ℝ\n⊢ frontier {z | z.re < a} = {z | z.re = a}\n[PROOFSTEP]\nsimpa only [frontier_Iio] using frontier_preimage_re (Iio a)\n[GOAL]\na : ℝ\n⊢ frontier {z | z.im < a} = {z | z.im = a}\n[PROOFSTEP]\nsimpa only [frontier_Iio] using frontier_preimage_im (Iio a)\n[GOAL]\na : ℝ\n⊢ frontier {z | a < z.re} = {z | z.re = a}\n[PROOFSTEP]\nsimpa only [frontier_Ioi] using frontier_preimage_re (Ioi a)\n[GOAL]\na : ℝ\n⊢ frontier {z | a < z.im} = {z | z.im = a}\n[PROOFSTEP]\nsimpa only [frontier_Ioi] using frontier_preimage_im (Ioi a)\n[GOAL]\ns t : Set ℝ\n⊢ closure (s ×ℂ t) = closure s ×ℂ closure t\n[PROOFSTEP]\nsimpa only [← preimage_eq_preimage equivRealProdClm.symm.toHomeomorph.surjective,\n equivRealProdClm.symm.toHomeomorph.preimage_closure] using @closure_prod_eq _ _ _ _ s t\n[GOAL]\ns t : Set ℝ\n⊢ interior (s ×ℂ t) = interior s ×ℂ interior t\n[PROOFSTEP]\nrw [Set.reProdIm, Set.reProdIm, interior_inter, interior_preimage_re, interior_preimage_im]\n[GOAL]\ns t : Set ℝ\n⊢ frontier (s ×ℂ t) = closure s ×ℂ frontier t ∪ frontier s ×ℂ closure t\n[PROOFSTEP]\nsimpa only [← preimage_eq_preimage equivRealProdClm.symm.toHomeomorph.surjective,\n equivRealProdClm.symm.toHomeomorph.preimage_frontier] using frontier_prod_eq s t\n[GOAL]\na b : ℝ\n⊢ frontier {z | a ≤ z.re ∧ b ≤ z.im} = {z | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ b ≤ z.im}\n[PROOFSTEP]\nsimpa only [closure_Ici, frontier_Ici] using frontier_reProdIm (Ici a) (Ici b)\n[GOAL]\na b : ℝ\n⊢ frontier {z | a ≤ z.re ∧ z.im ≤ b} = {z | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ z.im ≤ b}\n[PROOFSTEP]\nsimpa only [closure_Ici, closure_Iic, frontier_Ici, frontier_Iic] using frontier_reProdIm (Ici a) (Iic b)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Complex.ReImTopology", "llama_tokens": 1521, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9149009526726544, "lm_q2_score": 0.8376199694135332, "lm_q1q2_score": 0.7663393079940811}}
{"text": "[GOAL]\na : ℤ\nn : ℕ\n⊢ succ^[n + 1] a = a + ↑(n + 1)\n[PROOFSTEP]\nrw [Function.iterate_succ', Int.ofNat_succ, ← add_assoc]\n[GOAL]\na : ℤ\nn : ℕ\n⊢ (succ ∘ succ^[n]) a = a + ↑n + 1\n[PROOFSTEP]\nexact congr_arg _ (succ_iterate a n)\n[GOAL]\na : ℤ\nn : ℕ\n⊢ pred^[n + 1] a = a - ↑(n + 1)\n[PROOFSTEP]\nrw [Function.iterate_succ', Int.ofNat_succ, ← sub_sub]\n[GOAL]\na : ℤ\nn : ℕ\n⊢ (pred ∘ pred^[n]) a = a - ↑n - 1\n[PROOFSTEP]\nexact congr_arg _ (pred_iterate a n)\n[GOAL]\na b : ℤ\nh : a ≤ b\n⊢ Order.succ^[toNat (b - a)] a = b\n[PROOFSTEP]\nrw [succ_eq_succ, succ_iterate, toNat_sub_of_le h, ← add_sub_assoc, add_sub_cancel']\n[GOAL]\na b : ℤ\nh : a ≤ b\n⊢ Order.pred^[toNat (b - a)] b = a\n[PROOFSTEP]\nrw [pred_eq_pred, pred_iterate, toNat_sub_of_le h, sub_sub_cancel]\n[GOAL]\nz : ℤ\n⊢ z - 1 ⋖ z\n[PROOFSTEP]\nrw [Int.covby_iff_succ_eq, sub_add_cancel]\n[GOAL]\na b : ℕ\n⊢ ↑a ⋖ ↑b ↔ a ⋖ b\n[PROOFSTEP]\nrw [Nat.covby_iff_succ_eq, Int.covby_iff_succ_eq]\n[GOAL]\na b : ℕ\n⊢ ↑a + 1 = ↑b ↔ a + 1 = b\n[PROOFSTEP]\nexact Int.coe_nat_inj'\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.SuccPred", "llama_tokens": 585, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9207896715436483, "lm_q2_score": 0.8311430562234878, "lm_q1q2_score": 0.7653079417458094}}
{"text": "[GOAL]\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl : List α\na : α\np : α → Bool\n⊢ all l p = true ↔ ∀ (a : α), a ∈ l → p a = true\n[PROOFSTEP]\ninduction' l with a l ih\n[GOAL]\ncase nil\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl : List α\na : α\np : α → Bool\n⊢ all [] p = true ↔ ∀ (a : α), a ∈ [] → p a = true\n[PROOFSTEP]\nexact iff_of_true rfl (forall_mem_nil _)\n[GOAL]\ncase cons\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl✝ : List α\na✝ : α\np : α → Bool\na : α\nl : List α\nih : all l p = true ↔ ∀ (a : α), a ∈ l → p a = true\n⊢ all (a :: l) p = true ↔ ∀ (a_1 : α), a_1 ∈ a :: l → p a_1 = true\n[PROOFSTEP]\nsimp only [all_cons, Bool.and_coe_iff, ih, forall_mem_cons]\n[GOAL]\nα : Type u_1\np : α → Prop\ninst✝ : DecidablePred p\nl : List α\na : α\n⊢ (all l fun a => decide (p a)) = true ↔ ∀ (a : α), a ∈ l → p a\n[PROOFSTEP]\nsimp only [all_iff_forall, Bool.of_decide_iff]\n[GOAL]\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl : List α\na : α\np : α → Bool\n⊢ any l p = true ↔ ∃ a, a ∈ l ∧ p a = true\n[PROOFSTEP]\ninduction' l with a l ih\n[GOAL]\ncase nil\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl : List α\na : α\np : α → Bool\n⊢ any [] p = true ↔ ∃ a, a ∈ [] ∧ p a = true\n[PROOFSTEP]\nexact iff_of_false Bool.not_false' (not_exists_mem_nil _)\n[GOAL]\ncase cons\nα : Type u_1\np✝ : α → Prop\ninst✝ : DecidablePred p✝\nl✝ : List α\na✝ : α\np : α → Bool\na : α\nl : List α\nih : any l p = true ↔ ∃ a, a ∈ l ∧ p a = true\n⊢ any (a :: l) p = true ↔ ∃ a_1, a_1 ∈ a :: l ∧ p a_1 = true\n[PROOFSTEP]\nsimp only [any_cons, Bool.or_coe_iff, ih, exists_mem_cons_iff]\n[GOAL]\nα : Type u_1\np : α → Prop\ninst✝ : DecidablePred p\nl : List α\na : α\n⊢ (any l fun a => decide (p a)) = true ↔ ∃ a, a ∈ l ∧ p a\n[PROOFSTEP]\nsimp [any_iff_exists]\n", "meta": {"mathlib_filename": "Mathlib.Data.Bool.AllAny", "llama_tokens": 892, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.8438950986284991, "lm_q1q2_score": 0.7513149337616332}}
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