| {"problem_id": "induction_nfactltnexpnm1ngt3", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- Show that for any integer $n \\geq 3$, we have $n! < n^{n-1}$.-/\ntheorem induction_nfactltnexpnm1ngt3 (n : β) (hβ : 3 β€ n) : n ! < n ^ (n - 1) := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem induction_nfactltnexpnm1ngt3 (n : β) (hβ : 3 β€ n) : n ! < n ^ (n - 1) := by \n have hβ : β (n : β), 3 β€ n β n ! < n ^ (n - 1) := by\n intro n hn\n have hβ : β (n : β), 3 β€ n β n ! < n ^ (n - 1) := by\n intro n hn\n induction' hn with n hn IH\n Β· \n norm_num [Nat.factorial]\n Β· \n cases n with\n | zero => contradiction \n | succ n =>\n cases n with\n | zero => contradiction \n | succ n =>\n cases n with\n | zero => contradiction \n | succ n =>\n simp_all [Nat.factorial, Nat.pow_succ, Nat.mul_assoc]\n <;>\n (try norm_num at *) <;>\n (try norm_num) <;>\n (try ring_nf at *) <;>\n (try ring_nf) <;>\n (try nlinarith) <;>\n (try\n {\n have hβ : n + 3 > 0 := by linarith\n have hβ : (n + 3 : β) ^ (n + 2) > 0 := by positivity\n have hβ
: (n + 3 : β) ^ (n + 2) > 0 := by positivity\n nlinarith [pow_pos (by linarith : (0 : β) < n + 3) (n + 2)]\n }) <;>\n (try\n {\n have hβ : (n + 3 : β) ^ (n + 2) > 0 := by positivity\n have hβ : (n + 3 : β) ^ (n + 2) > 0 := by positivity\n nlinarith [pow_pos (by linarith : (0 : β) < n + 3) (n + 2)]\n })\n exact hβ n hn\n have hβ : n ! < n ^ (n - 1) := by\n exact hβ n hβ\n exact hβ", "repl_code": "\n\n\n\ntheorem induction_nfactltnexpnm1ngt3 (n : β) (hβ : 3 β€ n) : n ! < n ^ (n - 1) := by \n have hβ : β (n : β), 3 β€ n β n ! < n ^ (n - 1) := by\n intro n hn\n have hβ : β (n : β), 3 β€ n β n ! < n ^ (n - 1) := by\n intro n hn\n induction' hn with n hn IH\n Β· \n norm_num [Nat.factorial]\n Β· \n cases n with\n | zero => contradiction \n | succ n =>\n cases n with\n | zero => contradiction \n | succ n =>\n cases n with\n | zero => contradiction \n | succ n =>\n simp_all [Nat.factorial, Nat.pow_succ, Nat.mul_assoc]\n <;>\n (try norm_num at *) <;>\n (try norm_num) <;>\n (try ring_nf at *) <;>\n (try ring_nf) <;>\n (try nlinarith) <;>\n (try\n {\n have hβ : n + 3 > 0 := by linarith\n have hβ : (n + 3 : β) ^ (n + 2) > 0 := by positivity\n have hβ
: (n + 3 : β) ^ (n + 2) > 0 := by positivity\n nlinarith [pow_pos (by linarith : (0 : β) < n + 3) (n + 2)]\n }) <;>\n (try\n {\n have hβ : (n + 3 : β) ^ (n + 2) > 0 := by positivity\n have hβ : (n + 3 : β) ^ (n + 2) > 0 := by positivity\n nlinarith [pow_pos (by linarith : (0 : β) < n + 3) (n + 2)]\n })\n exact hβ n hn\n have hβ : n ! < n ^ (n - 1) := by\n exact hβ n hβ\n exact hβ", "repl_errors": [{"pos": {"line": 22, "column": 21}, "endPos": {"line": 42, "column": 18}, "data": "unsolved goals\ncase step.succ.succ.succ\nnβΒ² : β\nhβ : 3 β€ nβΒ²\nnβΒΉ : β\nhn : 3 β€ nβΒΉ\nnβ n : β\nIH :\n n * n ! * 11 + n ^ 2 * n ! * 6 + n ^ 3 * n ! + n ! * 6 < n * (3 + n) ^ n * 6 + n ^ 2 * (3 + n) ^ n + (3 + n) ^ n * 9\nβ’ n * n ! * 50 + n ^ 2 * n ! * 35 + n ^ 3 * n ! * 10 + n ^ 4 * n ! + n ! * 24 <\n n * (4 + n) ^ n * 48 + n ^ 2 * (4 + n) ^ n * 12 + n ^ 3 * (4 + n) ^ n + (4 + n) ^ n * 64"}]} |
| {"problem_id": "numbertheory_notequiv2i2jasqbsqdiv8", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- Let $a$ and $b$ be two integer numbers. Show that the following statement is false: $a$ and $b$ are both even iff $8 \\mid a^2 + b^2$.-/\ntheorem numbertheory_notequiv2i2jasqbsqdiv8 :\n Β¬β a b : β€, (β i j, a = 2 * i β§ b = 2 * j) β β k, a ^ 2 + b ^ 2 = 8 * k := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem numbertheory_notequiv2i2jasqbsqdiv8 :\n Β¬β a b : β€, (β i j, a = 2 * i β§ b = 2 * j) β β k, a ^ 2 + b ^ 2 = 8 * k := by \n have h_main : β (a b : β€), Β¬((β i j, a = 2 * i β§ b = 2 * j) β β k, a ^ 2 + b ^ 2 = 8 * k) := by\n use 2, 0\n intro h\n have hβ : (β i j, (2 : β€) = 2 * i β§ (0 : β€) = 2 * j) := by\n refine' β¨1, 0, _β©\n <;> norm_num\n have hβ : β k, (2 : β€) ^ 2 + (0 : β€) ^ 2 = 8 * k := by\n rw [h] at *\n exact by simpa [hβ] using h\n rcases hβ with β¨k, hkβ©\n have hβ : (2 : β€) ^ 2 + (0 : β€) ^ 2 = 8 * k := by simpa using hk\n have hβ : (8 : β€) * k = 4 := by linarith\n have hβ
: (k : β€) = 1 / 2 := by\n omega\n have hβ : False := by\n have hβ : (k : β€) * 8 = 4 := by linarith\n have hβ : (k : β€) = 1 / 2 := by omega\n norm_num at hβ β’\n <;>\n (try omega) <;>\n (try\n {\n ring_nf at hβ β’\n <;> norm_cast at hβ β’\n <;> omega\n })\n exact hβ\n \n have h_final : Β¬β a b : β€, (β i j, a = 2 * i β§ b = 2 * j) β β k, a ^ 2 + b ^ 2 = 8 * k := by\n intro h\n rcases h_main with β¨a, b, hββ©\n have hβ := h a b\n exact hβ hβ\n \n exact h_final", "repl_code": "\n\n\n\ntheorem numbertheory_notequiv2i2jasqbsqdiv8 :\n Β¬β a b : β€, (β i j, a = 2 * i β§ b = 2 * j) β β k, a ^ 2 + b ^ 2 = 8 * k := by \n have h_main : β (a b : β€), Β¬((β i j, a = 2 * i β§ b = 2 * j) β β k, a ^ 2 + b ^ 2 = 8 * k) := by\n use 2, 0\n intro h\n have hβ : (β i j, (2 : β€) = 2 * i β§ (0 : β€) = 2 * j) := by\n refine' β¨1, 0, _β©\n <;> norm_num\n have hβ : β k, (2 : β€) ^ 2 + (0 : β€) ^ 2 = 8 * k := by\n rw [h] at *\n exact by simpa [hβ] using h\n rcases hβ with β¨k, hkβ©\n have hβ : (2 : β€) ^ 2 + (0 : β€) ^ 2 = 8 * k := by simpa using hk\n have hβ : (8 : β€) * k = 4 := by linarith\n have hβ
: (k : β€) = 1 / 2 := by\n omega\n have hβ : False := by\n have hβ : (k : β€) * 8 = 4 := by linarith\n have hβ : (k : β€) = 1 / 2 := by omega\n norm_num at hβ β’\n <;>\n (try omega) <;>\n (try\n {\n ring_nf at hβ β’\n <;> norm_cast at hβ β’\n <;> omega\n })\n exact hβ\n \n have h_final : Β¬β a b : β€, (β i j, a = 2 * i β§ b = 2 * j) β β k, a ^ 2 + b ^ 2 = 8 * k := by\n intro h\n rcases h_main with β¨a, b, hββ©\n have hβ := h a b\n exact hβ hβ\n \n exact h_final", "repl_errors": [{"pos": {"line": 15, "column": 15}, "endPos": {"line": 15, "column": 33}, "data": "type mismatch\n hβ\nhas type\n True : Prop\nbut is expected to have type\n β k, 4 = 8 * k : Prop"}]} |
| {"problem_id": "imosl_2007_algebra_p6", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- For a series $\\{a_n\\}$, we have $\\sum_{n=0}^{99} a_{n+1}^2 = 1$. Show that $\\sum_{n=0}^{98} (a_{n+1}^2 a_{n+2}) + a_{100}^2 * a_1 < \\frac{12}{25}$.-/\ntheorem imosl_2007_algebra_p6 (a : β β NNReal) (hβ : (β x in Finset.range 100, a (x + 1) ^ 2) = 1) :\n (β x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) + a 100 ^ 2 * a 1 < 12 / 25 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem imosl_2007_algebra_p6 (a : β β NNReal) (hβ : (β x in Finset.range 100, a (x + 1) ^ 2) = 1) :\n (β x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) + a 100 ^ 2 * a 1 < 12 / 25 := by \n have hβ : False := by\n \n have hβ : a 100 ^ 2 * a 1 β₯ 0 := by positivity\n have hβ : (β x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) β₯ 0 := by positivity\n \n \n \n \n \n \n \n \n \n \n \n have hβ : (β x in Finset.range 100, a (x + 1) ^ 2) = 1 := hβ\n have hβ
: a 100 ^ 2 β€ 1 := by\n have hβ
β : a 100 ^ 2 β€ β x in Finset.range 100, a (x + 1) ^ 2 := by\n have hβ
β : a 100 ^ 2 β€ β x in Finset.range 100, a (x + 1) ^ 2 := by\n calc\n a 100 ^ 2 β€ β x in Finset.range 100, a (x + 1) ^ 2 := by\n \n \n \n \n \n \n exact Finset.single_le_sum (fun i _ => sq_nonneg (a (i + 1))) (Finset.mem_range.mpr (by norm_num : (99 : β) < 100))\n _ = β x in Finset.range 100, a (x + 1) ^ 2 := by rfl\n exact hβ
β\n calc\n a 100 ^ 2 β€ β x in Finset.range 100, a (x + 1) ^ 2 := hβ
β\n _ = 1 := hβ\n \n \n \n \n \n \n have hβ : False := by\n \n \n \n \n \n \n norm_num [Finset.sum_range_succ, Finset.sum_range_zero, Nat.cast_zero, Nat.cast_add, Nat.cast_one] at hβ hβ
β’\n <;>\n (try contradiction) <;>\n (try norm_num) <;>\n (try linarith) <;>\n (try ring_nf at hβ hβ
β’) <;>\n (try simp_all [Finset.sum_range_succ, Finset.sum_range_zero, Nat.cast_zero, Nat.cast_add, Nat.cast_one]) <;>\n (try norm_num at hβ hβ
β’) <;>\n (try linarith)\n exact hβ\n \n have hβ : (β x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) + a 100 ^ 2 * a 1 < 12 / 25 := by\n exfalso\n exact hβ\n \n exact hβ", "repl_code": "\n\n\n\ntheorem imosl_2007_algebra_p6 (a : β β NNReal) (hβ : (β x in Finset.range 100, a (x + 1) ^ 2) = 1) :\n (β x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) + a 100 ^ 2 * a 1 < 12 / 25 := by \n have hβ : False := by\n \n have hβ : a 100 ^ 2 * a 1 β₯ 0 := by positivity\n have hβ : (β x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) β₯ 0 := by positivity\n \n \n \n \n \n \n \n \n \n \n \n have hβ : (β x in Finset.range 100, a (x + 1) ^ 2) = 1 := hβ\n have hβ
: a 100 ^ 2 β€ 1 := by\n have hβ
β : a 100 ^ 2 β€ β x in Finset.range 100, a (x + 1) ^ 2 := by\n have hβ
β : a 100 ^ 2 β€ β x in Finset.range 100, a (x + 1) ^ 2 := by\n calc\n a 100 ^ 2 β€ β x in Finset.range 100, a (x + 1) ^ 2 := by\n \n \n \n \n \n \n exact Finset.single_le_sum (fun i _ => sq_nonneg (a (i + 1))) (Finset.mem_range.mpr (by norm_num : (99 : β) < 100))\n _ = β x in Finset.range 100, a (x + 1) ^ 2 := by rfl\n exact hβ
β\n calc\n a 100 ^ 2 β€ β x in Finset.range 100, a (x + 1) ^ 2 := hβ
β\n _ = 1 := hβ\n \n \n \n \n \n \n have hβ : False := by\n \n \n \n \n \n \n norm_num [Finset.sum_range_succ, Finset.sum_range_zero, Nat.cast_zero, Nat.cast_add, Nat.cast_one] at hβ hβ
β’\n <;>\n (try contradiction) <;>\n (try norm_num) <;>\n (try linarith) <;>\n (try ring_nf at hβ hβ
β’) <;>\n (try simp_all [Finset.sum_range_succ, Finset.sum_range_zero, Nat.cast_zero, Nat.cast_add, Nat.cast_one]) <;>\n (try norm_num at hβ hβ
β’) <;>\n (try linarith)\n exact hβ\n \n have hβ : (β x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) + a 100 ^ 2 * a 1 < 12 / 25 := by\n exfalso\n exact hβ\n \n exact hβ", "repl_errors": [{"pos": {"line": 46, "column": 23}, "endPos": {"line": 61, "column": 20}, "data": "unsolved goals\na : β β NNReal\nhβ
: a 100 ^ 2 β€ 1\nhβ :\n a 1 ^ 2 + a 2 ^ 2 + a 3 ^ 2 + a 4 ^ 2 + a 5 ^ 2 + a 6 ^ 2 + a 7 ^ 2 + a 8 ^ 2 + a 9 ^ 2 + a 10 ^ 2 + a 11 ^ 2 +\n a\n 12 ^\n 2 +\n a\n 13 ^\n 2 +\n a\n 14 ^\n 2 +\n a\n 15 ^\n 2 +\n a\n 16 ^\n 2 +\n a\n 17 ^\n 2 +\n a\n 18 ^\n 2 +\n a\n 19 ^\n 2 +\n a\n 20 ^\n 2 +\n a\n 21 ^\n 2 +\n a\n 22 ^\n 2 +\n a\n 23 ^\n 2 +\n a\n 24 ^\n 2 +\n a\n 25 ^\n 2 +\n a\n 26 ^\n 2 +\n a\n 27 ^\n 2 +\n a\n 28 ^\n 2 +\n a\n 29 ^\n 2 +\n a\n 30 ^\n 2 +\n a\n 31 ^\n 2 +\n a\n 32 ^\n 2 +\n a\n 33 ^\n 2 +\n a\n 34 ^\n 2 +\n a\n 35 ^\n 2 +\n a\n 36 ^\n 2 +\n a\n 37 ^\n 2 +\n a\n 38 ^\n 2 +\n a\n 39 ^\n 2 +\n a\n 40 ^\n 2 +\n a\n 41 ^\n 2 +\n a\n 42 ^\n 2 +\n a\n 43 ^\n 2 +\n a\n 44 ^\n 2 +\n a\n 45 ^\n 2 +\n a 46 ^\n 2 +\n a 47 ^\n 2 +\n a 48 ^ 2 +\n a 49 ^ 2 +\n a 50 ^ 2 +\n a 51 ^ 2 +\n a 52 ^ 2 +\n a 53 ^ 2 +\n a 54 ^ 2 +\n a 55 ^ 2 +\n a 56 ^ 2 +\n a 57 ^ 2 +\n a 58 ^ 2 +\n a 59 ^ 2 +\n a 60 ^ 2 +\n a 61 ^ 2 +\n a 62 ^ 2 +\n a 63 ^ 2 +\n a 64 ^ 2 +\n a 65 ^ 2 +\n a 66 ^ 2 +\n a 67 ^ 2 +\n a 68 ^ 2 +\n a 69 ^ 2 +\n a 70 ^ 2 +\n a 71 ^ 2 +\n a 72 ^ 2 +\n a 73 ^ 2 +\n a 74 ^ 2 +\n a 75 ^ 2 +\n a 76 ^ 2 +\n a 77 ^ 2 +\n a 78 ^ 2 +\n a 79 ^ 2 +\n a 80 ^ 2 +\n a 81 ^ 2 +\n a 82 ^ 2 +\n a 83 ^ 2 +\n a 84 ^ 2 +\n a 85 ^ 2 +\n a 86 ^ 2 +\n a 87 ^ 2 +\n a 88 ^ 2 +\n a 89 ^ 2 +\n a 90 ^ 2 +\n a 91 ^ 2 +\n a 92 ^ 2 +\n a 93 ^ 2 +\n a 94 ^ 2 +\n a 95 ^ 2 +\n a 96 ^ 2 +\n a 97 ^ 2 +\n a 98 ^ 2 +\n a 99 ^ 2 +\n a 100 ^ 2 =\n 1\nβ’ False"}]} |
| {"problem_id": "numbertheory_fxeq4powxp6powxp9powx_f2powmdvdf2pown", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- Let $f(x) = 4^x + 6^x + 9^x$. Show that if $m$ and $n$ are positive integers with $m \\leq n$, then $f(2^m)$ divides $f(2^n)$.-/\ntheorem numbertheory_fxeq4powxp6powxp9powx_f2powmdvdf2pown (m n : β) (f : β β β)\n (hβ : β x, f x = 4 ^ x + 6 ^ x + 9 ^ x) (hβ : 0 < m β§ 0 < n) (hβ : m β€ n) :\n f (2 ^ m) β£ f (2 ^ n) := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem numbertheory_fxeq4powxp6powxp9powx_f2powmdvdf2pown (m n : β) (f : β β β)\n (hβ : β x, f x = 4 ^ x + 6 ^ x + 9 ^ x) (hβ : 0 < m β§ 0 < n) (hβ : m β€ n) :\n f (2 ^ m) β£ f (2 ^ n) := by \n have h_main : False := by\n have hβ := hβ.1\n have hβ := hβ.2\n have hβ
:= hβ\n have hβ : m β€ n := hβ\n have hβ : 0 < m := hβ.1\n have hβ : 0 < n := hβ.2\n \n have hβ := hβ 2\n have hββ := hβ 3\n have hββ := hβ 4\n have hββ := hβ 8\n norm_num [pow_succ, pow_zero, pow_one, pow_two, pow_three] at hβ hββ hββ hββ\n <;>\n (try omega) <;>\n (try\n {\n have hββ := hβ 1\n have hββ := hβ 0\n norm_num at hββ hββ\n <;>\n omega\n }) <;>\n (try\n {\n have hββ
: m β€ 3 := by\n by_contra h\n have hββ : m β₯ 4 := by omega\n have hββ : n β₯ m := by omega\n have hββ : n β₯ 4 := by omega\n omega\n interval_cases m <;> interval_cases n <;> norm_num [hβ, pow_succ, pow_zero, pow_one, pow_two, pow_three] at hβ hββ hββ hββ β’ <;>\n (try omega) <;>\n (try {\n norm_num at hβ hββ hββ hββ β’\n <;>\n omega\n }) <;>\n (try {\n simp_all [Nat.dvd_iff_mod_eq_zero]\n <;>\n omega\n })\n }) <;>\n (try\n {\n have hββ
: m β€ 4 := by\n by_contra h\n have hββ : m β₯ 5 := by omega\n have hββ : n β₯ m := by omega\n have hββ : n β₯ 5 := by omega\n omega\n interval_cases m <;> interval_cases n <;> norm_num [hβ, pow_succ, pow_zero, pow_one, pow_two, pow_three] at hβ hββ hββ hββ β’ <;>\n (try omega) <;>\n (try {\n norm_num at hβ hββ hββ hββ β’\n <;>\n omega\n }) <;>\n (try {\n simp_all [Nat.dvd_iff_mod_eq_zero]\n <;>\n omega\n })\n }) <;>\n (try\n {\n omega\n })\n <;>\n (try\n {\n aesop\n })\n <;>\n (try\n {\n simp_all [hβ, pow_succ, pow_zero, pow_one, pow_two, pow_three]\n <;>\n ring_nf at *\n <;>\n norm_num at *\n <;>\n omega\n })\n <;>\n (try\n {\n omega\n })\n <;>\n (try\n {\n aesop\n })\n <;>\n (try\n {\n norm_num at *\n <;>\n omega\n })\n <;>\n (try\n {\n aesop\n })\n <;>\n (try\n {\n omega\n })\n <;>\n (try\n {\n aesop\n })\n <;>\n (try\n {\n norm_num at *\n <;>\n omega\n })\n <;>\n (try\n {\n aesop\n })\n \n have h_final : f (2 ^ m) β£ f (2 ^ n) := by\n exfalso\n exact h_main\n \n exact h_final", "repl_code": "\n\n\n\ntheorem numbertheory_fxeq4powxp6powxp9powx_f2powmdvdf2pown (m n : β) (f : β β β)\n (hβ : β x, f x = 4 ^ x + 6 ^ x + 9 ^ x) (hβ : 0 < m β§ 0 < n) (hβ : m β€ n) :\n f (2 ^ m) β£ f (2 ^ n) := by \n have h_main : False := by\n have hβ := hβ.1\n have hβ := hβ.2\n have hβ
:= hβ\n have hβ : m β€ n := hβ\n have hβ : 0 < m := hβ.1\n have hβ : 0 < n := hβ.2\n \n have hβ := hβ 2\n have hββ := hβ 3\n have hββ := hβ 4\n have hββ := hβ 8\n norm_num [pow_succ, pow_zero, pow_one, pow_two, pow_three] at hβ hββ hββ hββ\n <;>\n (try omega) <;>\n (try\n {\n have hββ := hβ 1\n have hββ := hβ 0\n norm_num at hββ hββ\n <;>\n omega\n }) <;>\n (try\n {\n have hββ
: m β€ 3 := by\n by_contra h\n have hββ : m β₯ 4 := by omega\n have hββ : n β₯ m := by omega\n have hββ : n β₯ 4 := by omega\n omega\n interval_cases m <;> interval_cases n <;> norm_num [hβ, pow_succ, pow_zero, pow_one, pow_two, pow_three] at hβ hββ hββ hββ β’ <;>\n (try omega) <;>\n (try {\n norm_num at hβ hββ hββ hββ β’\n <;>\n omega\n }) <;>\n (try {\n simp_all [Nat.dvd_iff_mod_eq_zero]\n <;>\n omega\n })\n }) <;>\n (try\n {\n have hββ
: m β€ 4 := by\n by_contra h\n have hββ : m β₯ 5 := by omega\n have hββ : n β₯ m := by omega\n have hββ : n β₯ 5 := by omega\n omega\n interval_cases m <;> interval_cases n <;> norm_num [hβ, pow_succ, pow_zero, pow_one, pow_two, pow_three] at hβ hββ hββ hββ β’ <;>\n (try omega) <;>\n (try {\n norm_num at hβ hββ hββ hββ β’\n <;>\n omega\n }) <;>\n (try {\n simp_all [Nat.dvd_iff_mod_eq_zero]\n <;>\n omega\n })\n }) <;>\n (try\n {\n omega\n })\n <;>\n (try\n {\n aesop\n })\n <;>\n (try\n {\n simp_all [hβ, pow_succ, pow_zero, pow_one, pow_two, pow_three]\n <;>\n ring_nf at *\n <;>\n norm_num at *\n <;>\n omega\n })\n <;>\n (try\n {\n omega\n })\n <;>\n (try\n {\n aesop\n })\n <;>\n (try\n {\n norm_num at *\n <;>\n omega\n })\n <;>\n (try\n {\n aesop\n })\n <;>\n (try\n {\n omega\n })\n <;>\n (try\n {\n aesop\n })\n <;>\n (try\n {\n norm_num at *\n <;>\n omega\n })\n <;>\n (try\n {\n aesop\n })\n \n have h_final : f (2 ^ m) β£ f (2 ^ n) := by\n exfalso\n exact h_main\n \n exact h_final", "repl_errors": [{"pos": {"line": 8, "column": 25}, "endPos": {"line": 136, "column": 8}, "data": "unsolved goals\nm n : β\nf : β β β\nhβ : β (x : β), f x = 4 ^ x + 6 ^ x + 9 ^ x\nhβ : 0 < m β§ 0 < n\nhβ : m β€ n\nhβ : 0 < m\nhβ : 0 < n\nhβ
hβ : m β€ n\nhβ : 0 < m\nhβ : 0 < n\nhβ : f 2 = 133\nhββ : f 3 = 1009\nhββ : f 4 = 8113\nhββ : f 8 = 44791873\nβ’ False"}]} |
| {"problem_id": "aime_1987_p5", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- Find $3x^2 y^2$ if $x$ and $y$ are [[integer]]s such that $y^2 + 3x^2 y^2 = 30x^2 + 517$. Show that it is 588.-/\ntheorem aime_1987_p5 (x y : β€) (hβ : y ^ 2 + 3 * (x ^ 2 * y ^ 2) = 30 * x ^ 2 + 517) :\n 3 * (x ^ 2 * y ^ 2) = 588 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem aime_1987_p5 (x y : β€) (hβ : y ^ 2 + 3 * (x ^ 2 * y ^ 2) = 30 * x ^ 2 + 517) :\n 3 * (x ^ 2 * y ^ 2) = 588 := by \n have h_main : 3 * (x ^ 2 * y ^ 2) = 588 := by\n have hβ : y ^ 2 + 3 * (x ^ 2 * y ^ 2) = 30 * x ^ 2 + 517 := hβ\n have hβ : y ^ 2 = 30 * x ^ 2 - 71 := by\n \n have hβ := hβ\n ring_nf at hβ β’\n \n nlinarith [sq_nonneg (x ^ 2 - 4), sq_nonneg (x ^ 2 + 4), sq_nonneg (y - 7), sq_nonneg (y + 7),\n sq_nonneg (x - 2), sq_nonneg (x + 2), sq_nonneg (x ^ 2 - 2), sq_nonneg (x ^ 2 + 2)]\n have hβ : x β€ 14 := by\n nlinarith [sq_nonneg (x - 14), sq_nonneg (x + 14), sq_nonneg (y - 7), sq_nonneg (y + 7),\n sq_nonneg (x - 2), sq_nonneg (x + 2), sq_nonneg (x ^ 2 - 4), sq_nonneg (x ^ 2 + 4)]\n have hβ : x β₯ -14 := by\n nlinarith [sq_nonneg (x - 14), sq_nonneg (x + 14), sq_nonneg (y - 7), sq_nonneg (y + 7),\n sq_nonneg (x - 2), sq_nonneg (x + 2), sq_nonneg (x ^ 2 - 4), sq_nonneg (x ^ 2 + 4)]\n \n interval_cases x <;> (try omega) <;> (try\n {\n have hβ
: y β€ 14 := by\n nlinarith [sq_nonneg (y - 7), sq_nonneg (y + 7), sq_nonneg (y - 14), sq_nonneg (y + 14)]\n have hβ : y β₯ -14 := by\n nlinarith [sq_nonneg (y - 7), sq_nonneg (y + 7), sq_nonneg (y - 14), sq_nonneg (y + 14)]\n interval_cases y <;> norm_num at hβ β’ <;> (try omega) <;> (try nlinarith)\n }) <;> (try omega) <;> (try nlinarith) <;> (try\n {\n norm_num at hβ β’\n <;>\n (try omega) <;>\n (try nlinarith) <;>\n (try\n {\n have hβ
: y β€ 14 := by nlinarith\n have hβ : y β₯ -14 := by nlinarith\n interval_cases y <;> norm_num at hβ β’ <;> omega\n })\n })\n <;>\n (try omega) <;>\n (try nlinarith) <;>\n (try\n {\n have hβ
: y β€ 14 := by nlinarith\n have hβ : y β₯ -14 := by nlinarith\n interval_cases y <;> norm_num at hβ β’ <;> omega\n })\n <;>\n (try omega) <;>\n (try nlinarith)\n <;>\n (try\n {\n have hβ
: y β€ 14 := by nlinarith\n have hβ : y β₯ -14 := by nlinarith\n interval_cases y <;> norm_num at hβ β’ <;> omega\n })\n exact h_main", "repl_code": "\n\n\n\ntheorem aime_1987_p5 (x y : β€) (hβ : y ^ 2 + 3 * (x ^ 2 * y ^ 2) = 30 * x ^ 2 + 517) :\n 3 * (x ^ 2 * y ^ 2) = 588 := by \n have h_main : 3 * (x ^ 2 * y ^ 2) = 588 := by\n have hβ : y ^ 2 + 3 * (x ^ 2 * y ^ 2) = 30 * x ^ 2 + 517 := hβ\n have hβ : y ^ 2 = 30 * x ^ 2 - 71 := by\n \n have hβ := hβ\n ring_nf at hβ β’\n \n nlinarith [sq_nonneg (x ^ 2 - 4), sq_nonneg (x ^ 2 + 4), sq_nonneg (y - 7), sq_nonneg (y + 7),\n sq_nonneg (x - 2), sq_nonneg (x + 2), sq_nonneg (x ^ 2 - 2), sq_nonneg (x ^ 2 + 2)]\n have hβ : x β€ 14 := by\n nlinarith [sq_nonneg (x - 14), sq_nonneg (x + 14), sq_nonneg (y - 7), sq_nonneg (y + 7),\n sq_nonneg (x - 2), sq_nonneg (x + 2), sq_nonneg (x ^ 2 - 4), sq_nonneg (x ^ 2 + 4)]\n have hβ : x β₯ -14 := by\n nlinarith [sq_nonneg (x - 14), sq_nonneg (x + 14), sq_nonneg (y - 7), sq_nonneg (y + 7),\n sq_nonneg (x - 2), sq_nonneg (x + 2), sq_nonneg (x ^ 2 - 4), sq_nonneg (x ^ 2 + 4)]\n \n interval_cases x <;> (try omega) <;> (try\n {\n have hβ
: y β€ 14 := by\n nlinarith [sq_nonneg (y - 7), sq_nonneg (y + 7), sq_nonneg (y - 14), sq_nonneg (y + 14)]\n have hβ : y β₯ -14 := by\n nlinarith [sq_nonneg (y - 7), sq_nonneg (y + 7), sq_nonneg (y - 14), sq_nonneg (y + 14)]\n interval_cases y <;> norm_num at hβ β’ <;> (try omega) <;> (try nlinarith)\n }) <;> (try omega) <;> (try nlinarith) <;> (try\n {\n norm_num at hβ β’\n <;>\n (try omega) <;>\n (try nlinarith) <;>\n (try\n {\n have hβ
: y β€ 14 := by nlinarith\n have hβ : y β₯ -14 := by nlinarith\n interval_cases y <;> norm_num at hβ β’ <;> omega\n })\n })\n <;>\n (try omega) <;>\n (try nlinarith) <;>\n (try\n {\n have hβ
: y β€ 14 := by nlinarith\n have hβ : y β₯ -14 := by nlinarith\n interval_cases y <;> norm_num at hβ β’ <;> omega\n })\n <;>\n (try omega) <;>\n (try nlinarith)\n <;>\n (try\n {\n have hβ
: y β€ 14 := by nlinarith\n have hβ : y β₯ -14 := by nlinarith\n interval_cases y <;> norm_num at hβ β’ <;> omega\n })\n exact h_main", "repl_errors": [{"pos": {"line": 14, "column": 6}, "endPos": {"line": 15, "column": 91}, "data": "linarith failed to find a contradiction\ncase h1.h\nx y : β€\nhβ hβ : y ^ 2 + 3 * (x ^ 2 * y ^ 2) = 30 * x ^ 2 + 517\nhβ : y ^ 2 + y ^ 2 * x ^ 2 * 3 = 517 + x ^ 2 * 30\naβ : y ^ 2 < -71 + x ^ 2 * 30\nβ’ False\nfailed"}]} |
| {"problem_id": "mathd_algebra_184", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- We have two geometric sequences of positive real numbers: $$6,a,b\\text{ and }\\frac{1}{b},a,54$$Solve for $a$. Show that it is 3\\sqrt{2}.-/\ntheorem mathd_algebra_184 (a b : NNReal) (hβ : 0 < a β§ 0 < b) (hβ : a ^ 2 = 6 * b)\n (hβ : a ^ 2 = 54 / b) : a = 3 * NNReal.sqrt 2 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem mathd_algebra_184 (a b : NNReal) (hβ : 0 < a β§ 0 < b) (hβ : a ^ 2 = 6 * b)\n (hβ : a ^ 2 = 54 / b) : a = 3 * NNReal.sqrt 2 := by \n have h_b_pos : 0 < b := by\n exact hβ.2\n \n have h_a_sq : a ^ 2 = 18 := by\n have hβ : a ^ 2 = 6 * b := hβ\n have hβ : a ^ 2 = 54 / b := hβ\n have hβ
: b > 0 := by exact_mod_cast h_b_pos\n have hβ : (a ^ 2 : β) = 6 * b := by exact_mod_cast hβ\n have hβ : (a ^ 2 : β) = 54 / b := by exact_mod_cast hβ\n have hβ : (a ^ 2 : β) = 18 := by\n have hβ : (b : β) > 0 := by exact_mod_cast hβ
\n field_simp [hβ.ne'] at hβ hβ β’\n nlinarith [sq_nonneg ((a : β) ^ 2 - 18)]\n exact_mod_cast hβ\n \n have h_a : a = 3 * NNReal.sqrt 2 := by\n have hβ : a ^ 2 = 18 := h_a_sq\n have hβ : 0 < a := by exact_mod_cast hβ.1\n have hβ
: a = 3 * NNReal.sqrt 2 := by\n \n \n have hβ : (a : β) ^ 2 = 18 := by exact_mod_cast hβ\n have hβ : (a : β) = 3 * Real.sqrt 2 := by\n \n have hβ : 0 < (a : β) := by exact_mod_cast hβ\n have hβ : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n \n have hββ : (a : β) = 3 * Real.sqrt 2 := by\n \n nlinarith [Real.sq_sqrt (show 0 β€ 2 by norm_num), Real.sqrt_nonneg 2,\n sq_nonneg ((a : β) - 3 * Real.sqrt 2)]\n exact hββ\n \n have hβ : (a : β) = 3 * Real.sqrt 2 := hβ\n have hβ : a = 3 * NNReal.sqrt 2 := by\n \n have hββ : (a : β) = 3 * Real.sqrt 2 := hβ\n have hββ : a = 3 * NNReal.sqrt 2 := by\n \n apply Eq.symm\n apply Eq.symm\n norm_cast at hββ β’\n <;>\n (try simp_all [mul_comm]) <;>\n (try ring_nf at * <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show (0 : β) β€ 2 by norm_num)]) <;>\n (try field_simp at * <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show (0 : β) β€ 2 by norm_num)])\n exact hββ\n exact hβ\n exact hβ
\n \n exact h_a", "repl_code": "\n\n\n\ntheorem mathd_algebra_184 (a b : NNReal) (hβ : 0 < a β§ 0 < b) (hβ : a ^ 2 = 6 * b)\n (hβ : a ^ 2 = 54 / b) : a = 3 * NNReal.sqrt 2 := by \n have h_b_pos : 0 < b := by\n exact hβ.2\n \n have h_a_sq : a ^ 2 = 18 := by\n have hβ : a ^ 2 = 6 * b := hβ\n have hβ : a ^ 2 = 54 / b := hβ\n have hβ
: b > 0 := by exact_mod_cast h_b_pos\n have hβ : (a ^ 2 : β) = 6 * b := by exact_mod_cast hβ\n have hβ : (a ^ 2 : β) = 54 / b := by exact_mod_cast hβ\n have hβ : (a ^ 2 : β) = 18 := by\n have hβ : (b : β) > 0 := by exact_mod_cast hβ
\n field_simp [hβ.ne'] at hβ hβ β’\n nlinarith [sq_nonneg ((a : β) ^ 2 - 18)]\n exact_mod_cast hβ\n \n have h_a : a = 3 * NNReal.sqrt 2 := by\n have hβ : a ^ 2 = 18 := h_a_sq\n have hβ : 0 < a := by exact_mod_cast hβ.1\n have hβ
: a = 3 * NNReal.sqrt 2 := by\n \n \n have hβ : (a : β) ^ 2 = 18 := by exact_mod_cast hβ\n have hβ : (a : β) = 3 * Real.sqrt 2 := by\n \n have hβ : 0 < (a : β) := by exact_mod_cast hβ\n have hβ : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n \n have hββ : (a : β) = 3 * Real.sqrt 2 := by\n \n nlinarith [Real.sq_sqrt (show 0 β€ 2 by norm_num), Real.sqrt_nonneg 2,\n sq_nonneg ((a : β) - 3 * Real.sqrt 2)]\n exact hββ\n \n have hβ : (a : β) = 3 * Real.sqrt 2 := hβ\n have hβ : a = 3 * NNReal.sqrt 2 := by\n \n have hββ : (a : β) = 3 * Real.sqrt 2 := hβ\n have hββ : a = 3 * NNReal.sqrt 2 := by\n \n apply Eq.symm\n apply Eq.symm\n norm_cast at hββ β’\n <;>\n (try simp_all [mul_comm]) <;>\n (try ring_nf at * <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show (0 : β) β€ 2 by norm_num)]) <;>\n (try field_simp at * <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show (0 : β) β€ 2 by norm_num)])\n exact hββ\n exact hβ\n exact hβ
\n \n exact h_a", "repl_errors": [{"pos": {"line": 44, "column": 44}, "endPos": {"line": 52, "column": 113}, "data": "unsolved goals\ncase h.h\na b : NNReal\nhβ : 54 / b = b * 6\nhβ : 18 = b * 6\nh_b_pos : 0 < b\nhβ : a ^ 2 = b * 6\nhβ : 0 < a\nhβ : (3 * β2) ^ 2 = 18\nhββ : βa = 3 * β2\nβ’ a = 3 * NNReal.sqrt 2"}]} |
| {"problem_id": "mathd_numbertheory_135", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- Let $n = 3^{17} + 3^{10}$. It is known that $11$ divides into $n+1$. If $n$ can be written in base $10$ as $ABCACCBAB$, where $A,B,C$ are distinct digits such that $A$ and $C$ are odd and $B$ is not divisible by $3$, find $100A + 10B + C$. Show that it is 129.-/\ntheorem mathd_numbertheory_135 (n A B C : β) (hβ : n = 3 ^ 17 + 3 ^ 10) (hβ : 11 β£ n + 1)\n (hβ : [A, B, C].Pairwise (Β· β Β·)) (hβ : {A, B, C} β Finset.Icc 0 9) (hβ : Odd A β§ Odd C)\n (hβ
: Β¬3 β£ B) (hβ : Nat.digits 10 n = [B, A, B, C, C, A, C, B, A]) :\n 100 * A + 10 * B + C = 129 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem mathd_numbertheory_135 (n A B C : β) (hβ : n = 3 ^ 17 + 3 ^ 10) (hβ : 11 β£ n + 1)\n (hβ : [A, B, C].Pairwise (Β· β Β·)) (hβ : {A, B, C} β Finset.Icc 0 9) (hβ : Odd A β§ Odd C)\n (hβ
: Β¬3 β£ B) (hβ : Nat.digits 10 n = [B, A, B, C, C, A, C, B, A]) :\n 100 * A + 10 * B + C = 129 := by \n have h_false : False := by\n have hβ : n = 3 ^ 17 + 3 ^ 10 := hβ\n rw [hβ] at hβ\n norm_num [Nat.dvd_iff_mod_eq_zero, Nat.pow_mod, Nat.add_mod, Nat.mod_eq_of_lt] at hβ\n <;>\n (try omega) <;>\n (try\n {\n simp_all [List.Pairwise, List.map_cons, List.map_nil, List.map_singleton, List.cons_eq_append,\n List.append_assoc]\n <;>\n (try omega) <;>\n (try\n {\n rcases A with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n rcases B with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n rcases C with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n simp_all (config := {decide := true}) [Nat.odd_iff_not_even, Nat.even_iff, Nat.pow_succ] <;>\n norm_num at * <;>\n omega\n })\n })\n <;>\n (try\n {\n omega\n })\n <;>\n (try\n {\n simp_all [Finset.subset_iff, Finset.mem_Icc, Nat.dvd_iff_mod_eq_zero]\n <;> omega\n })\n \n have h_main : 100 * A + 10 * B + C = 129 := by\n exfalso\n exact h_false\n \n exact h_main", "repl_code": "\n\n\n\ntheorem mathd_numbertheory_135 (n A B C : β) (hβ : n = 3 ^ 17 + 3 ^ 10) (hβ : 11 β£ n + 1)\n (hβ : [A, B, C].Pairwise (Β· β Β·)) (hβ : {A, B, C} β Finset.Icc 0 9) (hβ : Odd A β§ Odd C)\n (hβ
: Β¬3 β£ B) (hβ : Nat.digits 10 n = [B, A, B, C, C, A, C, B, A]) :\n 100 * A + 10 * B + C = 129 := by \n have h_false : False := by\n have hβ : n = 3 ^ 17 + 3 ^ 10 := hβ\n rw [hβ] at hβ\n norm_num [Nat.dvd_iff_mod_eq_zero, Nat.pow_mod, Nat.add_mod, Nat.mod_eq_of_lt] at hβ\n <;>\n (try omega) <;>\n (try\n {\n simp_all [List.Pairwise, List.map_cons, List.map_nil, List.map_singleton, List.cons_eq_append,\n List.append_assoc]\n <;>\n (try omega) <;>\n (try\n {\n rcases A with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n rcases B with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n rcases C with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n simp_all (config := {decide := true}) [Nat.odd_iff_not_even, Nat.even_iff, Nat.pow_succ] <;>\n norm_num at * <;>\n omega\n })\n })\n <;>\n (try\n {\n omega\n })\n <;>\n (try\n {\n simp_all [Finset.subset_iff, Finset.mem_Icc, Nat.dvd_iff_mod_eq_zero]\n <;> omega\n })\n \n have h_main : 100 * A + 10 * B + C = 129 := by\n exfalso\n exact h_false\n \n exact h_main", "repl_errors": [{"pos": {"line": 9, "column": 26}, "endPos": {"line": 41, "column": 8}, "data": "unsolved goals\nn A B C : β\nhβ : n = 3 ^ 17 + 3 ^ 10\nhβ : List.Pairwise (fun x x_1 => x β x_1) [A, B, C]\nhβ : {A, B, C} β Finset.Icc 0 9\nhβ : Odd A β§ Odd C\nhβ
: Β¬3 β£ B\nhβ : digits 10 n = [B, A, B, C, C, A, C, B, A]\nhβ : n = 3 ^ 17 + 3 ^ 10\nhβ : True\nβ’ False"}]} |
| {"problem_id": "amc12b_2020_p6", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- For all integers $n \\geq 9,$ the value of\n$\\frac{(n+2)!-(n+1)!}{n!}$is always which of the following?\n\n$\\textbf{(A) } \\text{a multiple of 4} \\qquad \\textbf{(B) } \\text{a multiple of 10} \\qquad \\textbf{(C) } \\text{a prime number} \\qquad \\textbf{(D) } \\text{a perfect square} \\qquad \\textbf{(E) } \\text{a perfect cube}$ Show that it is \\textbf{(D) } \\text{a perfect square}.-/\ntheorem amc12b_2020_p6 (n : β) (hβ : 9 β€ n) : β x : β, (x : β) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem amc12b_2020_p6 (n : β) (hβ : 9 β€ n) : β x : β, (x : β) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by \n have h_main : β x : β, (x : β) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by\n use n + 1\n have hβ : ((n + 2)! - (n + 1)!) / n ! = (n + 1 : β) ^ 2 := by\n have hβ : (n + 2)! = (n + 2) * (n + 1)! := by\n simp [Nat.factorial_succ, Nat.mul_comm]\n <;> ring\n rw [hβ]\n have hβ : (n + 2 : β) * (n + 1)! - (n + 1)! = (n + 1 : β) * (n + 1)! := by\n ring_nf\n <;> field_simp\n <;> ring_nf\n <;> nlinarith\n rw [hβ]\n have hβ : (n + 1 : β) * (n + 1)! / n ! = (n + 1 : β) ^ 2 := by\n have hβ
: (n + 1)! = (n + 1) * n ! := by\n simp [Nat.factorial_succ, Nat.mul_comm]\n <;> ring\n rw [hβ
]\n field_simp [Nat.factorial_ne_zero]\n <;> ring_nf\n <;> field_simp [Nat.factorial_ne_zero]\n <;> ring_nf\n <;> nlinarith\n simp_all [Nat.factorial_succ, Nat.mul_comm]\n <;> field_simp [Nat.factorial_ne_zero]\n <;> ring_nf\n <;> nlinarith\n simp_all [hβ]\n <;> norm_cast\n <;> ring_nf\n <;> nlinarith\n exact h_main", "repl_code": "\n\n\n\ntheorem amc12b_2020_p6 (n : β) (hβ : 9 β€ n) : β x : β, (x : β) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by \n have h_main : β x : β, (x : β) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by\n use n + 1\n have hβ : ((n + 2)! - (n + 1)!) / n ! = (n + 1 : β) ^ 2 := by\n have hβ : (n + 2)! = (n + 2) * (n + 1)! := by\n simp [Nat.factorial_succ, Nat.mul_comm]\n <;> ring\n rw [hβ]\n have hβ : (n + 2 : β) * (n + 1)! - (n + 1)! = (n + 1 : β) * (n + 1)! := by\n ring_nf\n <;> field_simp\n <;> ring_nf\n <;> nlinarith\n rw [hβ]\n have hβ : (n + 1 : β) * (n + 1)! / n ! = (n + 1 : β) ^ 2 := by\n have hβ
: (n + 1)! = (n + 1) * n ! := by\n simp [Nat.factorial_succ, Nat.mul_comm]\n <;> ring\n rw [hβ
]\n field_simp [Nat.factorial_ne_zero]\n <;> ring_nf\n <;> field_simp [Nat.factorial_ne_zero]\n <;> ring_nf\n <;> nlinarith\n simp_all [Nat.factorial_succ, Nat.mul_comm]\n <;> field_simp [Nat.factorial_ne_zero]\n <;> ring_nf\n <;> nlinarith\n simp_all [hβ]\n <;> norm_cast\n <;> ring_nf\n <;> nlinarith\n exact h_main", "repl_errors": [{"pos": {"line": 18, "column": 10}, "endPos": {"line": 18, "column": 12}, "data": "tactic 'rewrite' failed, did not find instance of the pattern in the target expression\n (βn + 2) * β(n + 1)! - β(n + 1)!\nn : β\nhβ : 9 β€ n\nhβ : (n + 2)! = (n + 2) * (n + 1)!\nhβ : (βn + 2) * β(n + 1)! - β(n + 1)! = (βn + 1) * β(n + 1)!\nβ’ (β((n + 2) * (n + 1)!) - β(n + 1)!) / βn ! = (βn + 1) ^ 2"}]} |
| {"problem_id": "mathd_numbertheory_321", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- Notice that \\[35\\cdot40=1400.\\]Find some integer $n$ with $0\\leq n<1399$ such that $n$ is the multiplicative inverse to 160 modulo 1399. Show that it is 1058.-/\ntheorem mathd_numbertheory_321 (n : ZMod 1399) (hβ : n = 160β»ΒΉ) : n = 1058 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem mathd_numbertheory_321 (n : ZMod 1399) (hβ : n = 160β»ΒΉ) : n = 1058 := by \n have hβ : (160 : ZMod 1399) * 1058 = 1 := by\n norm_num [ZMod.nat_cast_self]\n <;> rfl\n \n have hβ : (160 : ZMod 1399)β»ΒΉ = 1058 := by\n have hβ : (160 : ZMod 1399) * 1058 = 1 := hβ\n have hβ
: (160 : ZMod 1399) β 0 := by\n norm_num [ZMod.nat_cast_self]\n <;> decide\n \n apply Eq.symm\n apply Eq.symm\n apply inv_eq_of_mul_eq_one_right\n <;> simp_all [hβ, hβ
]\n <;> norm_num [ZMod.nat_cast_self]\n <;> rfl\n \n have hβ : n = 1058 := by\n rw [hβ]\n rw [hβ]\n <;> rfl\n \n apply hβ", "repl_code": "\n\n\n\ntheorem mathd_numbertheory_321 (n : ZMod 1399) (hβ : n = 160β»ΒΉ) : n = 1058 := by \n have hβ : (160 : ZMod 1399) * 1058 = 1 := by\n norm_num [ZMod.nat_cast_self]\n <;> rfl\n \n have hβ : (160 : ZMod 1399)β»ΒΉ = 1058 := by\n have hβ : (160 : ZMod 1399) * 1058 = 1 := hβ\n have hβ
: (160 : ZMod 1399) β 0 := by\n norm_num [ZMod.nat_cast_self]\n <;> decide\n \n apply Eq.symm\n apply Eq.symm\n apply inv_eq_of_mul_eq_one_right\n <;> simp_all [hβ, hβ
]\n <;> norm_num [ZMod.nat_cast_self]\n <;> rfl\n \n have hβ : n = 1058 := by\n rw [hβ]\n rw [hβ]\n <;> rfl\n \n apply hβ", "repl_errors": [{"pos": {"line": 18, "column": 4}, "endPos": {"line": 18, "column": 36}, "data": "tactic 'apply' failed, failed to unify\n @Inv.inv ?G DivInvMonoid.toInv ?a = ?b\nwith\n @Inv.inv (ZMod 1399) (ZMod.instInv 1399) 160 = 1058\ncase h.h\nn : ZMod 1399\nhβ : n = 160β»ΒΉ\nhβ hβ : 160 * 1058 = 1\nhβ
: 160 β 0\nβ’ 160β»ΒΉ = 1058"}]} |
| {"problem_id": "mathd_algebra_329", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- In a rectangular coordinate system, the line $3y = x$ intersects the line $2x + 5y = 11$ at point $A$. What is the sum of the coordinates of point $A$? Show that it is 4.-/\ntheorem mathd_algebra_329 (x y : β) (hβ : 3 * y = x) (hβ : 2 * x + 5 * y = 11) : x + y = 4 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem mathd_algebra_329 (x y : β) (hβ : 3 * y = x) (hβ : 2 * x + 5 * y = 11) : x + y = 4 := by \n have h_y : y = 1 := by\n have hβ : 2 * x + 5 * y = 11 := hβ\n have hβ : 3 * y = x := hβ\n \n have hβ : 2 * (3 * y) + 5 * y = 11 := by\n rw [hβ] at hβ\n linarith\n \n have hβ
: 6 * y + 5 * y = 11 := by linarith\n have hβ : 11 * y = 11 := by linarith\n have hβ : y = 1 := by\n \n apply mul_left_cancelβ (show (11 : β) β 0 by norm_num)\n nlinarith\n exact hβ\n \n have h_x : x = 3 := by\n have hβ : x = 3 * y := by linarith\n rw [hβ, h_y]\n <;> ring\n <;> linarith\n \n have h_sum : x + y = 4 := by\n have hβ : x = 3 := h_x\n have hβ : y = 1 := h_y\n rw [hβ, hβ]\n <;> norm_num\n \n apply h_sum", "repl_code": "\n\n\n\ntheorem mathd_algebra_329 (x y : β) (hβ : 3 * y = x) (hβ : 2 * x + 5 * y = 11) : x + y = 4 := by \n have h_y : y = 1 := by\n have hβ : 2 * x + 5 * y = 11 := hβ\n have hβ : 3 * y = x := hβ\n \n have hβ : 2 * (3 * y) + 5 * y = 11 := by\n rw [hβ] at hβ\n linarith\n \n have hβ
: 6 * y + 5 * y = 11 := by linarith\n have hβ : 11 * y = 11 := by linarith\n have hβ : y = 1 := by\n \n apply mul_left_cancelβ (show (11 : β) β 0 by norm_num)\n nlinarith\n exact hβ\n \n have h_x : x = 3 := by\n have hβ : x = 3 * y := by linarith\n rw [hβ, h_y]\n <;> ring\n <;> linarith\n \n have h_sum : x + y = 4 := by\n have hβ : x = 3 := h_x\n have hβ : y = 1 := h_y\n rw [hβ, hβ]\n <;> norm_num\n \n apply h_sum", "repl_errors": [{"pos": {"line": 11, "column": 10}, "endPos": {"line": 11, "column": 12}, "data": "tactic 'rewrite' failed, did not find instance of the pattern in the target expression\n 3 * y\nx y : β\nhβ : 3 * y = x\nhβ hβ : 2 * x + 5 * y = 11\nhβ : 3 * y = x\nβ’ 2 * (3 * y) + 5 * y = 11"}]} |
| {"problem_id": "amc12b_2021_p3", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- Suppose$2+\\frac{1}{1+\\frac{1}{2+\\frac{2}{3+x}}}=\\frac{144}{53}.$What is the value of $x?$\n\n$\\textbf{(A) }\\frac34 \\qquad \\textbf{(B) }\\frac78 \\qquad \\textbf{(C) }\\frac{14}{15} \\qquad \\textbf{(D) }\\frac{37}{38} \\qquad \\textbf{(E) }\\frac{52}{53}$ Show that it is \\text{A}.-/\ntheorem amc12b_2021_p3 (x : β) (hβ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53) : x = 3 / 4 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem amc12b_2021_p3 (x : β) (hβ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53) : x = 3 / 4 := by \n have h_main : x = 3 / 4 := by\n have hβ : 3 + x β 0 := by\n by_contra h\n have hβ : x = -3 := by linarith\n rw [hβ] at hβ\n norm_num at hβ β’\n <;>\n (try contradiction) <;>\n (try linarith) <;>\n (try\n {\n field_simp at hβ β’ <;> ring_nf at hβ β’ <;> nlinarith\n })\n have hβ : 2 + 2 / (3 + x) β 0 := by\n by_contra h\n have hβ : 2 + 2 / (3 + x) = 0 := by linarith\n have hβ : 2 / (3 + x) = -2 := by linarith\n have hβ
: 3 + x β 0 := hβ\n field_simp at hβ\n nlinarith\n have hβ : 1 + 1 / (2 + 2 / (3 + x)) β 0 := by\n by_contra h\n have hβ : 1 + 1 / (2 + 2 / (3 + x)) = 0 := by linarith\n have hβ
: 1 / (2 + 2 / (3 + x)) = -1 := by linarith\n have hβ : 2 + 2 / (3 + x) β 0 := hβ\n field_simp at hβ
\n <;> simp_all [div_eq_mul_inv]\n <;> ring_nf at *\n <;> nlinarith\n have hβ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53 := hβ\n field_simp at hβ\n ring_nf at hβ\n have hβ
: x = 3 / 4 := by\n apply mul_left_cancelβ (sub_ne_zero.mpr hβ)\n apply mul_left_cancelβ (sub_ne_zero.mpr hβ)\n nlinarith [sq_pos_of_ne_zero hβ, sq_pos_of_ne_zero hβ, sq_nonneg (x - 3 / 4),\n sq_nonneg (x + 3 / 4), sq_nonneg (x + 1 / 2), sq_nonneg (x - 1 / 2)]\n exact hβ
\n exact h_main", "repl_code": "\n\n\n\ntheorem amc12b_2021_p3 (x : β) (hβ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53) : x = 3 / 4 := by \n have h_main : x = 3 / 4 := by\n have hβ : 3 + x β 0 := by\n by_contra h\n have hβ : x = -3 := by linarith\n rw [hβ] at hβ\n norm_num at hβ β’\n <;>\n (try contradiction) <;>\n (try linarith) <;>\n (try\n {\n field_simp at hβ β’ <;> ring_nf at hβ β’ <;> nlinarith\n })\n have hβ : 2 + 2 / (3 + x) β 0 := by\n by_contra h\n have hβ : 2 + 2 / (3 + x) = 0 := by linarith\n have hβ : 2 / (3 + x) = -2 := by linarith\n have hβ
: 3 + x β 0 := hβ\n field_simp at hβ\n nlinarith\n have hβ : 1 + 1 / (2 + 2 / (3 + x)) β 0 := by\n by_contra h\n have hβ : 1 + 1 / (2 + 2 / (3 + x)) = 0 := by linarith\n have hβ
: 1 / (2 + 2 / (3 + x)) = -1 := by linarith\n have hβ : 2 + 2 / (3 + x) β 0 := hβ\n field_simp at hβ
\n <;> simp_all [div_eq_mul_inv]\n <;> ring_nf at *\n <;> nlinarith\n have hβ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53 := hβ\n field_simp at hβ\n ring_nf at hβ\n have hβ
: x = 3 / 4 := by\n apply mul_left_cancelβ (sub_ne_zero.mpr hβ)\n apply mul_left_cancelβ (sub_ne_zero.mpr hβ)\n nlinarith [sq_pos_of_ne_zero hβ, sq_pos_of_ne_zero hβ, sq_nonneg (x - 3 / 4),\n sq_nonneg (x + 3 / 4), sq_nonneg (x + 1 / 2), sq_nonneg (x - 1 / 2)]\n exact hβ
\n exact h_main", "repl_errors": [{"pos": {"line": 25, "column": 6}, "endPos": {"line": 25, "column": 15}, "data": "linarith failed to find a contradiction\nx : β\nhβ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53\nhβ : 3 + x β 0\nh hβ : 2 + 2 / (3 + x) = 0\nhβ
: 3 + x β 0\nhβ : 2 = -(2 * (3 + x))\nβ’ False\nfailed"}, {"pos": {"line": 41, "column": 6}, "endPos": {"line": 42, "column": 76}, "data": "linarith failed to find a contradiction\ncase h1.h\nx : β\nhβ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53\nhβ : 3 + x β 0\nhβ : 2 + 2 / (3 + x) β 0\nhβ : 1 + 1 / (2 + 2 / (3 + x)) β 0\nhβ : 106 + (1 + x * (8 + x * 2)β»ΒΉ + (8 + x * 2)β»ΒΉ * 3)β»ΒΉ * 53 = 144\naβ : (2 + 2 / (3 + x) - 0) * ((3 + x - 0) * x) < (2 + 2 / (3 + x) - 0) * ((3 + x - 0) * (3 / 4))\nβ’ False\nfailed"}]} |
| {"problem_id": "imo_1977_p6", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- Let $f(n)$ be a function $f: \\mathbb{N}^{+}\\to\\mathbb{N}^{+}$. Prove that if $ f(n+1) > f(f(n)) $ for each positive integer $n$, then $f(n)=n$.-/\ntheorem imo_1977_p6 (f : β β β) (hβ : β n, 0 < f n) (hβ : β n, 0 < n β f (f n) < f (n + 1)) :\n β n, 0 < n β f n = n := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem imo_1977_p6 (f : β β β) (hβ : β n, 0 < f n) (hβ : β n, 0 < n β f (f n) < f (n + 1)) :\n β n, 0 < n β f n = n := by \n have h_main : β n, 0 < n β f n = n := by\n intro n hn\n by_contra h\n \n have hβ : f n > n := by\n \n have hβ : f n β n := h\n have hβ : f n > n := by\n by_contra hβ
\n \n have hβ : f n < n := by\n omega\n \n have hβ : 0 < f n := hβ n\n have hβ : f (f n) < f (n + 1) := hβ n hn\n have hβ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n \n have hββ : f n β€ n := by omega\n have hββ : f n < n := hβ\n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ
: f (f n) < f (n + 1) := hβ n hn\n \n have hββ : f n > 0 := hβ n\n have hββ : f (f n) > 0 := hβ (f n)\n have hββ : f (n + 1) > 0 := hβ (n + 1)\n \n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n \n omega\n exact hβ\n \n have hβ : 0 < f n := hβ n\n have hβ : f (f n) < f (n + 1) := hβ n hn\n have hβ
: f n > n := hβ\n have hβ : f (f n) < f (n + 1) := hβ n hn\n have hβ : f n > 0 := hβ n\n have hβ : f (f n) > 0 := hβ (f n)\n have hβ : f (n + 1) > 0 := hβ (n + 1)\n \n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n \n have hββ : f n > n := hβ\n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ
: f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n \n omega\n exact h_main", "repl_code": "\n\n\n\ntheorem imo_1977_p6 (f : β β β) (hβ : β n, 0 < f n) (hβ : β n, 0 < n β f (f n) < f (n + 1)) :\n β n, 0 < n β f n = n := by \n have h_main : β n, 0 < n β f n = n := by\n intro n hn\n by_contra h\n \n have hβ : f n > n := by\n \n have hβ : f n β n := h\n have hβ : f n > n := by\n by_contra hβ
\n \n have hβ : f n < n := by\n omega\n \n have hβ : 0 < f n := hβ n\n have hβ : f (f n) < f (n + 1) := hβ n hn\n have hβ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n \n have hββ : f n β€ n := by omega\n have hββ : f n < n := hβ\n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ
: f (f n) < f (n + 1) := hβ n hn\n \n have hββ : f n > 0 := hβ n\n have hββ : f (f n) > 0 := hβ (f n)\n have hββ : f (n + 1) > 0 := hβ (n + 1)\n \n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n \n omega\n exact hβ\n \n have hβ : 0 < f n := hβ n\n have hβ : f (f n) < f (n + 1) := hβ n hn\n have hβ
: f n > n := hβ\n have hβ : f (f n) < f (n + 1) := hβ n hn\n have hβ : f n > 0 := hβ n\n have hβ : f (f n) > 0 := hβ (f n)\n have hβ : f (n + 1) > 0 := hβ (n + 1)\n \n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n \n have hββ : f n > n := hβ\n have hββ : f (f n) < f (n + 1) := hβ n hn\n have hββ
: f (f n) < f (n + 1) := hβ n hn\n have hββ : f (f n) < f (n + 1) := hβ n hn\n \n omega\n exact h_main", "repl_errors": [{"pos": {"line": 39, "column": 8}, "endPos": {"line": 39, "column": 13}, "data": "omega could not prove the goal:\na possible counterexample may satisfy the constraints\n c β₯ 1\n a - b β₯ 1\n d β₯ 1\n b β₯ 1\n c - d β₯ 1\n a β₯ 1\nwhere\n a := βn\n b := β(f n)\n c := β(f (n + 1))\n d := β(f (f n))"}, {"pos": {"line": 59, "column": 4}, "endPos": {"line": 59, "column": 9}, "data": "omega could not prove the goal:\na possible counterexample may satisfy the constraints\n c β₯ 1\n a - b β€ -1\n d β₯ 1\n b β₯ 1\n c - d β₯ 1\n a β₯ 1\nwhere\n a := βn\n b := β(f n)\n c := β(f (n + 1))\n d := β(f (f n))"}]} |
| {"problem_id": "induction_sumkexp3eqsumksq", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- Show that for all natural numbers $n$, we have $\\sum_{k=0}^{n-1}k^3 = \\left(\\sum_{k=0}^{n-1}k\\right)^2$.-/\ntheorem induction_sumkexp3eqsumksq (n : β) :\n (β k in Finset.range n, k ^ 3) = (β k in Finset.range n, k) ^ 2 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem induction_sumkexp3eqsumksq (n : β) :\n (β k in Finset.range n, k ^ 3) = (β k in Finset.range n, k) ^ 2 := by \n have h_main : β (n : β), (β k in Finset.range n, k ^ 3) = (β k in Finset.range n, k) ^ 2 := by\n intro n\n have h : β n : β, (β k in Finset.range n, k ^ 3) = (β k in Finset.range n, k) ^ 2 := by\n intro n\n induction n with\n | zero => simp\n | succ n ih =>\n rw [Finset.sum_range_succ, Finset.sum_range_succ, ih]\n simp [pow_two, Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.mul_zero, Nat.add_zero]\n <;> induction n with\n | zero => simp\n | succ n ih' =>\n simp_all [Finset.sum_range_succ, pow_two, Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.mul_zero, Nat.add_zero]\n <;> ring_nf at *\n <;> omega\n exact h n\n exact h_main n", "repl_code": "\n\n\n\ntheorem induction_sumkexp3eqsumksq (n : β) :\n (β k in Finset.range n, k ^ 3) = (β k in Finset.range n, k) ^ 2 := by \n have h_main : β (n : β), (β k in Finset.range n, k ^ 3) = (β k in Finset.range n, k) ^ 2 := by\n intro n\n have h : β n : β, (β k in Finset.range n, k ^ 3) = (β k in Finset.range n, k) ^ 2 := by\n intro n\n induction n with\n | zero => simp\n | succ n ih =>\n rw [Finset.sum_range_succ, Finset.sum_range_succ, ih]\n simp [pow_two, Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.mul_zero, Nat.add_zero]\n <;> induction n with\n | zero => simp\n | succ n ih' =>\n simp_all [Finset.sum_range_succ, pow_two, Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.mul_zero, Nat.add_zero]\n <;> ring_nf at *\n <;> omega\n exact h n\n exact h_main n", "repl_errors": [{"pos": {"line": 21, "column": 14}, "endPos": {"line": 21, "column": 19}, "data": "omega could not prove the goal:\na possible counterexample may satisfy the constraints\n a - 2*b - c - d β€ 0\n e β₯ 0\n d β₯ 0\n a - 2*b - d β₯ 1\n c β₯ 0\n f β₯ 0\n a β₯ 0\n a - 2*b - 2*e - f β€ -1\nwhere\n a := β(n ^ 3)\n b := β(β k β Finset.range n, k) * βn\n c := β((β k β Finset.range n, k) ^ 2)\n d := β(n ^ 2)\n e := β(β k β Finset.range n, k)\n f := βn"}, {"pos": {"line": 18, "column": 21}, "endPos": {"line": 21, "column": 19}, "data": "unsolved goals\ncase succ.succ\nnβΒΉ nβ n : β\nih : β x β Finset.range n, x ^ 3 + n ^ 3 = (β k β Finset.range n, k) * n * 2 + (β k β Finset.range n, k) ^ 2 + n ^ 2\nih' :\n β x β Finset.range n, x ^ 3 = (β k β Finset.range n, k) ^ 2 β\n (β k β Finset.range n, k) ^ 2 + n ^ 3 = (β k β Finset.range n, k) * n * 2 + (β k β Finset.range n, k) ^ 2 + n ^ 2\nβ’ 1 + (β k β Finset.range n, k) * n * 2 + (β k β Finset.range n, k) ^ 2 + n * 3 + n ^ 2 * 4 + n ^ 3 =\n 1 + (β k β Finset.range n, k) * 2 + (β k β Finset.range n, k) * n * 4 + (β k β Finset.range n, k) ^ 2 + n * 4 +\n n ^ 2 * 4"}]} |
| {"problem_id": "mathd_numbertheory_227", "kind": "failed", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family? Show that it is 5.-/\ntheorem mathd_numbertheory_227 (x y n : β+) (hβ : βx / (4 : β) + y / 6 = (x + y) / n) : n = 5 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem mathd_numbertheory_227 (x y n : β+) (hβ : βx / (4 : β) + y / 6 = (x + y) / n) : n = 5 := by \n have hβ : (n : β) > 0 := by\n exact mod_cast n.prop\n \n have hβ : (3 : β) * (n : β) * x + 2 * (n : β) * y = 12 * (x : β) + 12 * (y : β) := by\n have hββ : (x : β) / 4 + (y : β) / 6 = ((x : β) + (y : β)) / n := by simpa using hβ\n have hββ : (n : β) β 0 := by linarith\n field_simp at hββ β’\n ring_nf at hββ β’\n nlinarith [mul_pos (Nat.cast_pos.mpr x.prop) (Nat.cast_pos.mpr y.prop),\n mul_pos (Nat.cast_pos.mpr x.prop) hβ, mul_pos (Nat.cast_pos.mpr y.prop) hβ]\n \n have hβ : (3 : β) * (n : β) * x + 2 * (n : β) * y = 12 * (x : β) + 12 * (y : β) := by\n exact hβ\n \n have hβ : (n : β) β€ 4 := by\n have hββ : (3 : β) * (n : β) * x + 2 * (n : β) * y = 12 * (x : β) + 12 * (y : β) := hβ\n have hββ : (n : β) β€ 4 := by\n by_contra h\n have hββ : (n : β) β₯ 5 := by\n omega\n have hββ : (n : β) β₯ 5 := by\n exact_mod_cast hββ\n have hββ
: (x : β) > 0 := by exact_mod_cast x.prop\n have hββ : (y : β) > 0 := by exact_mod_cast y.prop\n have hββ : (3 : β) * (n : β) * x + 2 * (n : β) * y > 12 * (x : β) + 12 * (y : β) := by\n nlinarith [mul_pos hββ
hββ, mul_pos (by positivity : (0 : β) < 3) hββ
,\n mul_pos (by positivity : (0 : β) < 2) hββ]\n linarith\n exact hββ\n \n have hβ
: (n : β) β₯ 5 := by\n have hβ
β : (3 : β) * (n : β) * x + 2 * (n : β) * y = 12 * (x : β) + 12 * (y : β) := hβ\n have hβ
β : (n : β) β₯ 5 := by\n by_contra h\n have hβ
β : (n : β) β€ 4 := by omega\n have hβ
β : (n : β) β€ 4 := hβ
β\n have hβ
β
: (n : β) β€ 4 := by exact_mod_cast hβ
β\n have hβ
β : (x : β) > 0 := by exact_mod_cast x.prop\n have hβ
β : (y : β) > 0 := by exact_mod_cast y.prop\n have hβ
β : (3 : β) * (n : β) * x + 2 * (n : β) * y < 12 * (x : β) + 12 * (y : β) := by\n have hβ
β : (n : β) β€ 4 := hβ
β
\n have hβ
ββ : (n : β) > 0 := hβ\n have hβ
ββ : (x : β) > 0 := hβ
β\n have hβ
ββ : (y : β) > 0 := hβ
β\n nlinarith [mul_pos hβ
ββ hβ
ββ, mul_pos hβ
ββ hβ
ββ, mul_pos hβ
ββ hβ
ββ]\n linarith\n exact hβ
β\n \n have hβ : n = 5 := by\n have hββ : (n : β) β€ 4 := hβ\n have hββ : (n : β) β₯ 5 := hβ
\n have hββ : (n : β) = 5 := by\n omega\n have hββ : n = 5 := by\n exact_mod_cast hββ\n exact hββ\n \n apply hβ", "repl_code": "\n\n\n\ntheorem mathd_numbertheory_227 (x y n : β+) (hβ : βx / (4 : β) + y / 6 = (x + y) / n) : n = 5 := by \n have hβ : (n : β) > 0 := by\n exact mod_cast n.prop\n \n have hβ : (3 : β) * (n : β) * x + 2 * (n : β) * y = 12 * (x : β) + 12 * (y : β) := by\n have hββ : (x : β) / 4 + (y : β) / 6 = ((x : β) + (y : β)) / n := by simpa using hβ\n have hββ : (n : β) β 0 := by linarith\n field_simp at hββ β’\n ring_nf at hββ β’\n nlinarith [mul_pos (Nat.cast_pos.mpr x.prop) (Nat.cast_pos.mpr y.prop),\n mul_pos (Nat.cast_pos.mpr x.prop) hβ, mul_pos (Nat.cast_pos.mpr y.prop) hβ]\n \n have hβ : (3 : β) * (n : β) * x + 2 * (n : β) * y = 12 * (x : β) + 12 * (y : β) := by\n exact hβ\n \n have hβ : (n : β) β€ 4 := by\n have hββ : (3 : β) * (n : β) * x + 2 * (n : β) * y = 12 * (x : β) + 12 * (y : β) := hβ\n have hββ : (n : β) β€ 4 := by\n by_contra h\n have hββ : (n : β) β₯ 5 := by\n omega\n have hββ : (n : β) β₯ 5 := by\n exact_mod_cast hββ\n have hββ
: (x : β) > 0 := by exact_mod_cast x.prop\n have hββ : (y : β) > 0 := by exact_mod_cast y.prop\n have hββ : (3 : β) * (n : β) * x + 2 * (n : β) * y > 12 * (x : β) + 12 * (y : β) := by\n nlinarith [mul_pos hββ
hββ, mul_pos (by positivity : (0 : β) < 3) hββ
,\n mul_pos (by positivity : (0 : β) < 2) hββ]\n linarith\n exact hββ\n \n have hβ
: (n : β) β₯ 5 := by\n have hβ
β : (3 : β) * (n : β) * x + 2 * (n : β) * y = 12 * (x : β) + 12 * (y : β) := hβ\n have hβ
β : (n : β) β₯ 5 := by\n by_contra h\n have hβ
β : (n : β) β€ 4 := by omega\n have hβ
β : (n : β) β€ 4 := hβ
β\n have hβ
β
: (n : β) β€ 4 := by exact_mod_cast hβ
β\n have hβ
β : (x : β) > 0 := by exact_mod_cast x.prop\n have hβ
β : (y : β) > 0 := by exact_mod_cast y.prop\n have hβ
β : (3 : β) * (n : β) * x + 2 * (n : β) * y < 12 * (x : β) + 12 * (y : β) := by\n have hβ
β : (n : β) β€ 4 := hβ
β
\n have hβ
ββ : (n : β) > 0 := hβ\n have hβ
ββ : (x : β) > 0 := hβ
β\n have hβ
ββ : (y : β) > 0 := hβ
β\n nlinarith [mul_pos hβ
ββ hβ
ββ, mul_pos hβ
ββ hβ
ββ, mul_pos hβ
ββ hβ
ββ]\n linarith\n exact hβ
β\n \n have hβ : n = 5 := by\n have hββ : (n : β) β€ 4 := hβ\n have hββ : (n : β) β₯ 5 := hβ
\n have hββ : (n : β) = 5 := by\n omega\n have hββ : n = 5 := by\n exact_mod_cast hββ\n exact hββ\n \n apply hβ", "repl_errors": [{"pos": {"line": 14, "column": 15}, "endPos": {"line": 14, "column": 74}, "data": "typeclass instance problem is stuck, it is often due to metavariables\n PosMulStrictMono ?m.21755"}, {"pos": {"line": 31, "column": 8}, "endPos": {"line": 32, "column": 52}, "data": "linarith failed to find a contradiction\ncase h\nx y n : β+\nhβ : ββx / 4 + ββy / 6 = (ββx + ββy) / ββn\nhβ : ββn > 0\nhβ hβ hββ : 3 * ββn * ββx + 2 * ββn * ββy = 12 * ββx + 12 * ββy\nh : Β¬βn β€ 4\nhββ : βn β₯ 5\nhββ : ββn β₯ 5\nhββ
: ββx > 0\nhββ : ββy > 0\naβ : 12 * ββx + 12 * ββy β₯ 3 * ββn * ββx + 2 * ββn * ββy\nβ’ False\nfailed"}]} |
| {"problem_id": "mathd_algebra_33", "kind": "accepted", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- If $2x = 5y$ and $7y = 10z$, what is the value of $\\frac{z}{x}$ expressed as a common fraction? Show that it is \\frac{7}{25}.-/\ntheorem mathd_algebra_33 (x y z : β) (hβ : x β 0) (hβ : 2 * x = 5 * y) (hβ : 7 * y = 10 * z) :\n z / x = 7 / 25 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem mathd_algebra_33 (x y z : β) (hβ : x β 0) (hβ : 2 * x = 5 * y) (hβ : 7 * y = 10 * z) :\n z / x = 7 / 25 := by \n have h_y : y = (2 * x) / 5 := by\n have hβ : y = (2 * x) / 5 := by\n \n apply Eq.symm\n field_simp\n linarith\n exact hβ\n \n have h_z : z = (7 * x) / 25 := by\n have hβ : z = (7 * x) / 25 := by\n \n have hβ : 7 * y = 10 * z := hβ\n rw [h_y] at hβ\n ring_nf at hβ β’\n nlinarith\n exact hβ\n \n have h_main : z / x = 7 / 25 := by\n rw [h_z]\n have hβ : x β 0 := hβ\n field_simp [hβ]\n <;> ring_nf\n <;> field_simp [hβ] at *\n <;> nlinarith\n \n apply h_main", "repl_code": null, "repl_errors": null} |
| {"problem_id": "mathd_algebra_141", "kind": "accepted", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- A rectangular patio has an area of $180$ square feet and a perimeter of $54$ feet. What is the length of the diagonal (in feet) squared? Show that it is 369.-/\ntheorem mathd_algebra_141 (a b : β) (hβ : a * b = 180) (hβ : 2 * (a + b) = 54) :\n a ^ 2 + b ^ 2 = 369 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem mathd_algebra_141 (a b : β) (hβ : a * b = 180) (hβ : 2 * (a + b) = 54) :\n a ^ 2 + b ^ 2 = 369 := by \n have h_sum : a + b = 27 := by\n have hβ : a + b = 27 := by\n \n have hβ : 2 * (a + b) = 54 := hβ\n \n linarith\n exact hβ\n \n have h_main : a ^ 2 + b ^ 2 = 369 := by\n have hβ : a ^ 2 + b ^ 2 = 369 := by\n have hβ : (a + b) ^ 2 = 729 := by\n \n have hβ
: a + b = 27 := h_sum\n rw [hβ
]\n norm_num\n \n have hβ
: a ^ 2 + b ^ 2 + 2 * (a * b) = 729 := by\n nlinarith\n \n nlinarith [sq_nonneg (a - b)]\n exact hβ\n exact h_main", "repl_code": null, "repl_errors": null} |
| {"problem_id": "mathd_numbertheory_1124", "kind": "accepted", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- The four-digit number $\\underline{374n}$ is divisible by 18. Find the units digit $n$. Show that it is 4.-/\ntheorem mathd_numbertheory_1124 (n : β) (hβ : n β€ 9) (hβ : 18 β£ 374 * 10 + n) : n = 4 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem mathd_numbertheory_1124 (n : β) (hβ : n β€ 9) (hβ : 18 β£ 374 * 10 + n) : n = 4 := by \n have hβ : 14 + n = 18 := by\n have hββ : 18 β£ 374 * 10 + n := hβ\n have hββ : 374 * 10 + n = 3740 + n := by norm_num\n rw [hββ] at hββ\n have hββ : 18 β£ 3740 + n := hββ\n have hββ : 3740 % 18 = 14 := by norm_num\n have hββ
: (3740 + n) % 18 = 0 := by\n exact Nat.mod_eq_zero_of_dvd hββ\n have hββ : (14 + n) % 18 = 0 := by\n omega\n have hββ : 14 + n = 18 := by\n have hββ : n β€ 9 := hβ\n have hββ : n β€ 23 := by omega\n interval_cases n <;> norm_num at hββ β’ <;> omega\n exact hββ\n \n have hβ : n = 4 := by\n have hββ : 14 + n = 18 := hβ\n have hββ : n = 4 := by omega\n exact hββ\n \n exact hβ", "repl_code": null, "repl_errors": null} |
| {"problem_id": "amc12b_2020_p2", "kind": "accepted", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- What is the value of the following expression?\n\n$\\frac{100^2-7^2}{70^2-11^2} \\cdot \\frac{(70-11)(70+11)}{(100-7)(100+7)}$\n\n$\\textbf{(A) } 1 \\qquad \\textbf{(B) } \\frac{9951}{9950} \\qquad \\textbf{(C) } \\frac{4780}{4779} \\qquad \\textbf{(D) } \\frac{108}{107} \\qquad \\textbf{(E) } \\frac{81}{80} $ Show that it is \\textbf{(A) } 1.-/\ntheorem amc12b_2020_p2 :\n (100 ^ 2 - 7 ^ 2 : β) / (70 ^ 2 - 11 ^ 2) * ((70 - 11) * (70 + 11) / ((100 - 7) * (100 + 7))) =\n 1 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem amc12b_2020_p2 :\n (100 ^ 2 - 7 ^ 2 : β) / (70 ^ 2 - 11 ^ 2) * ((70 - 11) * (70 + 11) / ((100 - 7) * (100 + 7))) =\n 1 := by \n have h_main : (100 ^ 2 - 7 ^ 2 : β) / (70 ^ 2 - 11 ^ 2) * ((70 - 11) * (70 + 11) / ((100 - 7) * (100 + 7))) = 1 := by\n have hβ : (100 ^ 2 - 7 ^ 2 : β) = 9951 := by\n norm_num\n have hβ : (70 ^ 2 - 11 ^ 2 : β) = 4779 := by\n norm_num\n have hβ : ((70 - 11 : β) * (70 + 11)) = 4779 := by\n norm_num\n have hβ : ((100 - 7 : β) * (100 + 7)) = 9951 := by\n norm_num\n rw [hβ, hβ, hβ, hβ]\n norm_num\n <;> field_simp\n <;> ring\n <;> norm_num\n <;> linarith\n exact h_main", "repl_code": null, "repl_errors": null} |
| {"problem_id": "mathd_algebra_209", "kind": "accepted", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- Suppose that $h(x)=f^{-1}(x)$. If $h(2)=10$, $h(10)=1$ and $h(1)=2$, what is $f(f(10))$? Show that it is 1.-/\ntheorem mathd_algebra_209 (Ο : Equiv β β) (hβ : Ο.2 2 = 10) (hβ : Ο.2 10 = 1) (hβ : Ο.2 1 = 2) :\n Ο.1 (Ο.1 10) = 1 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem mathd_algebra_209 (Ο : Equiv β β) (hβ : Ο.2 2 = 10) (hβ : Ο.2 10 = 1) (hβ : Ο.2 1 = 2) :\n Ο.1 (Ο.1 10) = 1 := by \n have hβ : Ο.1 10 = 2 := by\n have hββ : Ο.1 (Ο.2 2) = 2 := by\n apply Equiv.apply_symm_apply\n rw [hβ] at hββ\n exact hββ\n \n have hβ : Ο.1 (Ο.1 10) = Ο.1 2 := by\n rw [hβ]\n <;> simp [hβ]\n <;>\n (try simp_all [Equiv.symm_apply_apply, Equiv.apply_symm_apply])\n <;>\n (try norm_num)\n <;>\n (try linarith)\n <;>\n (try ring_nf at *)\n <;>\n (try nlinarith)\n \n have hβ
: Ο.1 2 = 1 := by\n have hβ
β : Ο.1 (Ο.2 1) = 1 := by\n apply Equiv.apply_symm_apply\n rw [hβ] at hβ
β\n exact hβ
β\n \n have hβ : Ο.1 (Ο.1 10) = 1 := by\n rw [hβ]\n rw [hβ
]\n <;>\n (try simp_all [Equiv.symm_apply_apply, Equiv.apply_symm_apply])\n <;>\n (try norm_num)\n <;>\n (try linarith)\n <;>\n (try ring_nf at *)\n <;>\n (try nlinarith)\n \n apply hβ", "repl_code": null, "repl_errors": null} |
| {"problem_id": "mathd_algebra_419", "kind": "accepted", "formal_statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n/-- What is the value of $-a-b^2+3ab$ if $a=-1$ and $b=5$? Show that it is -39.-/\ntheorem mathd_algebra_419 (a b : β) (hβ : a = -1) (hβ : b = 5) : -a - b ^ 2 + 3 * (a * b) = -39 := by sorry", "lean_code": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\ntheorem mathd_algebra_419 (a b : β) (hβ : a = -1) (hβ : b = 5) : -a - b ^ 2 + 3 * (a * b) = -39 := by \n have hβ : -a = 1 := by\n rw [hβ]\n norm_num\n <;> simp_all\n <;> ring_nf\n <;> linarith\n \n have hβ : -b ^ 2 = -25 := by\n rw [hβ]\n norm_num\n <;> simp_all\n <;> ring_nf\n <;> linarith\n \n have hβ : 3 * (a * b) = -15 := by\n rw [hβ, hβ]\n norm_num\n <;> ring_nf\n <;> linarith\n \n have hβ
: -a - b ^ 2 + 3 * (a * b) = -39 := by\n have hβ
β : -a - b ^ 2 + 3 * (a * b) = -a - b ^ 2 + 3 * (a * b) := by rfl\n rw [hβ
β]\n have hβ
β : -a = 1 := hβ\n have hβ
β : -b ^ 2 = -25 := hβ\n have hβ
β : 3 * (a * b) = -15 := hβ\n \n nlinarith [sq_nonneg (a + 1), sq_nonneg (b - 5)]\n \n exact hβ
", "repl_code": null, "repl_errors": null} |
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