{"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 : ℕ) (h₀ : 3 ≤ 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 : ℕ) (h₀ : 3 ≤ n) : n ! < n ^ (n - 1) := by \n have h₁ : ∀ (n : ℕ), 3 ≤ n → n ! < n ^ (n - 1) := by\n intro n hn\n have h₂ : ∀ (n : ℕ), 3 ≤ n → n ! < n ^ (n - 1) := by\n intro n hn\n induction' hn with n hn IH\n · \n norm_num [Nat.factorial]\n · \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 h₃ : n + 3 > 0 := by linarith\n have h₄ : (n + 3 : ℕ) ^ (n + 2) > 0 := by positivity\n have h₅ : (n + 3 : ℕ) ^ (n + 2) > 0 := by positivity\n nlinarith [pow_pos (by linarith : (0 : ℕ) < n + 3) (n + 2)]\n }) <;>\n (try\n {\n have h₃ : (n + 3 : ℕ) ^ (n + 2) > 0 := by positivity\n have h₄ : (n + 3 : ℕ) ^ (n + 2) > 0 := by positivity\n nlinarith [pow_pos (by linarith : (0 : ℕ) < n + 3) (n + 2)]\n })\n exact h₂ n hn\n have h₂ : n ! < n ^ (n - 1) := by\n exact h₁ n h₀\n exact h₂", "repl_code": "\n\n\n\ntheorem induction_nfactltnexpnm1ngt3 (n : ℕ) (h₀ : 3 ≤ n) : n ! < n ^ (n - 1) := by \n have h₁ : ∀ (n : ℕ), 3 ≤ n → n ! < n ^ (n - 1) := by\n intro n hn\n have h₂ : ∀ (n : ℕ), 3 ≤ n → n ! < n ^ (n - 1) := by\n intro n hn\n induction' hn with n hn IH\n · \n norm_num [Nat.factorial]\n · \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 h₃ : n + 3 > 0 := by linarith\n have h₄ : (n + 3 : ℕ) ^ (n + 2) > 0 := by positivity\n have h₅ : (n + 3 : ℕ) ^ (n + 2) > 0 := by positivity\n nlinarith [pow_pos (by linarith : (0 : ℕ) < n + 3) (n + 2)]\n }) <;>\n (try\n {\n have h₃ : (n + 3 : ℕ) ^ (n + 2) > 0 := by positivity\n have h₄ : (n + 3 : ℕ) ^ (n + 2) > 0 := by positivity\n nlinarith [pow_pos (by linarith : (0 : ℕ) < n + 3) (n + 2)]\n })\n exact h₂ n hn\n have h₂ : n ! < n ^ (n - 1) := by\n exact h₁ n h₀\n exact h₂", "repl_errors": [{"pos": {"line": 22, "column": 21}, "endPos": {"line": 42, "column": 18}, "data": "unsolved goals\ncase step.succ.succ.succ\nn✝² : ℕ\nh₀ : 3 ≤ n✝²\nn✝¹ : ℕ\nhn : 3 ≤ n✝¹\nn✝ n : ℕ\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\n⊢ 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 ¬∀ a b : ℤ, (∃ i j, a = 2 * i ∧ b = 2 * j) ↔ ∃ 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 ¬∀ a b : ℤ, (∃ i j, a = 2 * i ∧ b = 2 * j) ↔ ∃ k, a ^ 2 + b ^ 2 = 8 * k := by \n have h_main : ∃ (a b : ℤ), ¬((∃ i j, a = 2 * i ∧ b = 2 * j) ↔ ∃ k, a ^ 2 + b ^ 2 = 8 * k) := by\n use 2, 0\n intro h\n have h₁ : (∃ i j, (2 : ℤ) = 2 * i ∧ (0 : ℤ) = 2 * j) := by\n refine' ⟨1, 0, _⟩\n <;> norm_num\n have h₂ : ∃ k, (2 : ℤ) ^ 2 + (0 : ℤ) ^ 2 = 8 * k := by\n rw [h] at *\n exact by simpa [h₁] using h\n rcases h₂ with ⟨k, hk⟩\n have h₃ : (2 : ℤ) ^ 2 + (0 : ℤ) ^ 2 = 8 * k := by simpa using hk\n have h₄ : (8 : ℤ) * k = 4 := by linarith\n have h₅ : (k : ℤ) = 1 / 2 := by\n omega\n have h₆ : False := by\n have h₇ : (k : ℤ) * 8 = 4 := by linarith\n have h₈ : (k : ℤ) = 1 / 2 := by omega\n norm_num at h₈ ⊢\n <;>\n (try omega) <;>\n (try\n {\n ring_nf at h₈ ⊢\n <;> norm_cast at h₈ ⊢\n <;> omega\n })\n exact h₆\n \n have h_final : ¬∀ a b : ℤ, (∃ i j, a = 2 * i ∧ b = 2 * j) ↔ ∃ k, a ^ 2 + b ^ 2 = 8 * k := by\n intro h\n rcases h_main with ⟨a, b, h₁⟩\n have h₂ := h a b\n exact h₁ h₂\n \n exact h_final", "repl_code": "\n\n\n\ntheorem numbertheory_notequiv2i2jasqbsqdiv8 :\n ¬∀ a b : ℤ, (∃ i j, a = 2 * i ∧ b = 2 * j) ↔ ∃ k, a ^ 2 + b ^ 2 = 8 * k := by \n have h_main : ∃ (a b : ℤ), ¬((∃ i j, a = 2 * i ∧ b = 2 * j) ↔ ∃ k, a ^ 2 + b ^ 2 = 8 * k) := by\n use 2, 0\n intro h\n have h₁ : (∃ i j, (2 : ℤ) = 2 * i ∧ (0 : ℤ) = 2 * j) := by\n refine' ⟨1, 0, _⟩\n <;> norm_num\n have h₂ : ∃ k, (2 : ℤ) ^ 2 + (0 : ℤ) ^ 2 = 8 * k := by\n rw [h] at *\n exact by simpa [h₁] using h\n rcases h₂ with ⟨k, hk⟩\n have h₃ : (2 : ℤ) ^ 2 + (0 : ℤ) ^ 2 = 8 * k := by simpa using hk\n have h₄ : (8 : ℤ) * k = 4 := by linarith\n have h₅ : (k : ℤ) = 1 / 2 := by\n omega\n have h₆ : False := by\n have h₇ : (k : ℤ) * 8 = 4 := by linarith\n have h₈ : (k : ℤ) = 1 / 2 := by omega\n norm_num at h₈ ⊢\n <;>\n (try omega) <;>\n (try\n {\n ring_nf at h₈ ⊢\n <;> norm_cast at h₈ ⊢\n <;> omega\n })\n exact h₆\n \n have h_final : ¬∀ a b : ℤ, (∃ i j, a = 2 * i ∧ b = 2 * j) ↔ ∃ k, a ^ 2 + b ^ 2 = 8 * k := by\n intro h\n rcases h_main with ⟨a, b, h₁⟩\n have h₂ := h a b\n exact h₁ h₂\n \n exact h_final", "repl_errors": [{"pos": {"line": 15, "column": 15}, "endPos": {"line": 15, "column": 33}, "data": "type mismatch\n h✝\nhas type\n True : Prop\nbut is expected to have type\n ∃ 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 : ℕ → NNReal) (h₀ : (∑ x in Finset.range 100, a (x + 1) ^ 2) = 1) :\n (∑ 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 : ℕ → NNReal) (h₀ : (∑ x in Finset.range 100, a (x + 1) ^ 2) = 1) :\n (∑ x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) + a 100 ^ 2 * a 1 < 12 / 25 := by \n have h₁ : False := by\n \n have h₂ : a 100 ^ 2 * a 1 ≥ 0 := by positivity\n have h₃ : (∑ x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) ≥ 0 := by positivity\n \n \n \n \n \n \n \n \n \n \n \n have h₄ : (∑ x in Finset.range 100, a (x + 1) ^ 2) = 1 := h₀\n have h₅ : a 100 ^ 2 ≤ 1 := by\n have h₅₁ : a 100 ^ 2 ≤ ∑ x in Finset.range 100, a (x + 1) ^ 2 := by\n have h₅₂ : a 100 ^ 2 ≤ ∑ x in Finset.range 100, a (x + 1) ^ 2 := by\n calc\n a 100 ^ 2 ≤ ∑ 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 : ℕ) < 100))\n _ = ∑ x in Finset.range 100, a (x + 1) ^ 2 := by rfl\n exact h₅₂\n calc\n a 100 ^ 2 ≤ ∑ x in Finset.range 100, a (x + 1) ^ 2 := h₅₁\n _ = 1 := h₄\n \n \n \n \n \n \n have h₆ : 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 h₄ h₅ ⊢\n <;>\n (try contradiction) <;>\n (try norm_num) <;>\n (try linarith) <;>\n (try ring_nf at h₄ h₅ ⊢) <;>\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 h₄ h₅ ⊢) <;>\n (try linarith)\n exact h₆\n \n have h₂ : (∑ x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) + a 100 ^ 2 * a 1 < 12 / 25 := by\n exfalso\n exact h₁\n \n exact h₂", "repl_code": "\n\n\n\ntheorem imosl_2007_algebra_p6 (a : ℕ → NNReal) (h₀ : (∑ x in Finset.range 100, a (x + 1) ^ 2) = 1) :\n (∑ x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) + a 100 ^ 2 * a 1 < 12 / 25 := by \n have h₁ : False := by\n \n have h₂ : a 100 ^ 2 * a 1 ≥ 0 := by positivity\n have h₃ : (∑ x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) ≥ 0 := by positivity\n \n \n \n \n \n \n \n \n \n \n \n have h₄ : (∑ x in Finset.range 100, a (x + 1) ^ 2) = 1 := h₀\n have h₅ : a 100 ^ 2 ≤ 1 := by\n have h₅₁ : a 100 ^ 2 ≤ ∑ x in Finset.range 100, a (x + 1) ^ 2 := by\n have h₅₂ : a 100 ^ 2 ≤ ∑ x in Finset.range 100, a (x + 1) ^ 2 := by\n calc\n a 100 ^ 2 ≤ ∑ 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 : ℕ) < 100))\n _ = ∑ x in Finset.range 100, a (x + 1) ^ 2 := by rfl\n exact h₅₂\n calc\n a 100 ^ 2 ≤ ∑ x in Finset.range 100, a (x + 1) ^ 2 := h₅₁\n _ = 1 := h₄\n \n \n \n \n \n \n have h₆ : 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 h₄ h₅ ⊢\n <;>\n (try contradiction) <;>\n (try norm_num) <;>\n (try linarith) <;>\n (try ring_nf at h₄ h₅ ⊢) <;>\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 h₄ h₅ ⊢) <;>\n (try linarith)\n exact h₆\n \n have h₂ : (∑ x in Finset.range 99, a (x + 1) ^ 2 * a (x + 2)) + a 100 ^ 2 * a 1 < 12 / 25 := by\n exfalso\n exact h₁\n \n exact h₂", "repl_errors": [{"pos": {"line": 46, "column": 23}, "endPos": {"line": 61, "column": 20}, "data": "unsolved goals\na : ℕ → NNReal\nh₅ : a 100 ^ 2 ≤ 1\nh₄ :\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\n⊢ 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 : ℕ) (f : ℕ → ℕ)\n (h₀ : ∀ x, f x = 4 ^ x + 6 ^ x + 9 ^ x) (h₁ : 0 < m ∧ 0 < n) (h₂ : m ≤ n) :\n f (2 ^ m) ∣ 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 : ℕ) (f : ℕ → ℕ)\n (h₀ : ∀ x, f x = 4 ^ x + 6 ^ x + 9 ^ x) (h₁ : 0 < m ∧ 0 < n) (h₂ : m ≤ n) :\n f (2 ^ m) ∣ f (2 ^ n) := by \n have h_main : False := by\n have h₃ := h₁.1\n have h₄ := h₁.2\n have h₅ := h₂\n have h₆ : m ≤ n := h₂\n have h₇ : 0 < m := h₁.1\n have h₈ : 0 < n := h₁.2\n \n have h₉ := h₀ 2\n have h₁₀ := h₀ 3\n have h₁₁ := h₀ 4\n have h₁₂ := h₀ 8\n norm_num [pow_succ, pow_zero, pow_one, pow_two, pow_three] at h₉ h₁₀ h₁₁ h₁₂\n <;>\n (try omega) <;>\n (try\n {\n have h₁₃ := h₀ 1\n have h₁₄ := h₀ 0\n norm_num at h₁₃ h₁₄\n <;>\n omega\n }) <;>\n (try\n {\n have h₁₅ : m ≤ 3 := by\n by_contra h\n have h₁₆ : m ≥ 4 := by omega\n have h₁₇ : n ≥ m := by omega\n have h₁₈ : n ≥ 4 := by omega\n omega\n interval_cases m <;> interval_cases n <;> norm_num [h₀, pow_succ, pow_zero, pow_one, pow_two, pow_three] at h₉ h₁₀ h₁₁ h₁₂ ⊢ <;>\n (try omega) <;>\n (try {\n norm_num at h₉ h₁₀ h₁₁ h₁₂ ⊢\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 h₁₅ : m ≤ 4 := by\n by_contra h\n have h₁₆ : m ≥ 5 := by omega\n have h₁₇ : n ≥ m := by omega\n have h₁₈ : n ≥ 5 := by omega\n omega\n interval_cases m <;> interval_cases n <;> norm_num [h₀, pow_succ, pow_zero, pow_one, pow_two, pow_three] at h₉ h₁₀ h₁₁ h₁₂ ⊢ <;>\n (try omega) <;>\n (try {\n norm_num at h₉ h₁₀ h₁₁ h₁₂ ⊢\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 [h₀, 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) ∣ 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 : ℕ) (f : ℕ → ℕ)\n (h₀ : ∀ x, f x = 4 ^ x + 6 ^ x + 9 ^ x) (h₁ : 0 < m ∧ 0 < n) (h₂ : m ≤ n) :\n f (2 ^ m) ∣ f (2 ^ n) := by \n have h_main : False := by\n have h₃ := h₁.1\n have h₄ := h₁.2\n have h₅ := h₂\n have h₆ : m ≤ n := h₂\n have h₇ : 0 < m := h₁.1\n have h₈ : 0 < n := h₁.2\n \n have h₉ := h₀ 2\n have h₁₀ := h₀ 3\n have h₁₁ := h₀ 4\n have h₁₂ := h₀ 8\n norm_num [pow_succ, pow_zero, pow_one, pow_two, pow_three] at h₉ h₁₀ h₁₁ h₁₂\n <;>\n (try omega) <;>\n (try\n {\n have h₁₃ := h₀ 1\n have h₁₄ := h₀ 0\n norm_num at h₁₃ h₁₄\n <;>\n omega\n }) <;>\n (try\n {\n have h₁₅ : m ≤ 3 := by\n by_contra h\n have h₁₆ : m ≥ 4 := by omega\n have h₁₇ : n ≥ m := by omega\n have h₁₈ : n ≥ 4 := by omega\n omega\n interval_cases m <;> interval_cases n <;> norm_num [h₀, pow_succ, pow_zero, pow_one, pow_two, pow_three] at h₉ h₁₀ h₁₁ h₁₂ ⊢ <;>\n (try omega) <;>\n (try {\n norm_num at h₉ h₁₀ h₁₁ h₁₂ ⊢\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 h₁₅ : m ≤ 4 := by\n by_contra h\n have h₁₆ : m ≥ 5 := by omega\n have h₁₇ : n ≥ m := by omega\n have h₁₈ : n ≥ 5 := by omega\n omega\n interval_cases m <;> interval_cases n <;> norm_num [h₀, pow_succ, pow_zero, pow_one, pow_two, pow_three] at h₉ h₁₀ h₁₁ h₁₂ ⊢ <;>\n (try omega) <;>\n (try {\n norm_num at h₉ h₁₀ h₁₁ h₁₂ ⊢\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 [h₀, 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) ∣ 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 : ℕ\nf : ℕ → ℕ\nh₀ : ∀ (x : ℕ), f x = 4 ^ x + 6 ^ x + 9 ^ x\nh₁ : 0 < m ∧ 0 < n\nh₂ : m ≤ n\nh₃ : 0 < m\nh₄ : 0 < n\nh₅ h₆ : m ≤ n\nh₇ : 0 < m\nh₈ : 0 < n\nh₉ : f 2 = 133\nh₁₀ : f 3 = 1009\nh₁₁ : f 4 = 8113\nh₁₂ : f 8 = 44791873\n⊢ 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 : ℤ) (h₀ : 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 : ℤ) (h₀ : 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 h₁ : y ^ 2 + 3 * (x ^ 2 * y ^ 2) = 30 * x ^ 2 + 517 := h₀\n have h₂ : y ^ 2 = 30 * x ^ 2 - 71 := by\n \n have h₃ := h₁\n ring_nf at h₃ ⊢\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 h₃ : x ≤ 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 h₄ : x ≥ -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 h₅ : y ≤ 14 := by\n nlinarith [sq_nonneg (y - 7), sq_nonneg (y + 7), sq_nonneg (y - 14), sq_nonneg (y + 14)]\n have h₆ : y ≥ -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 h₂ ⊢ <;> (try omega) <;> (try nlinarith)\n }) <;> (try omega) <;> (try nlinarith) <;> (try\n {\n norm_num at h₂ ⊢\n <;>\n (try omega) <;>\n (try nlinarith) <;>\n (try\n {\n have h₅ : y ≤ 14 := by nlinarith\n have h₆ : y ≥ -14 := by nlinarith\n interval_cases y <;> norm_num at h₂ ⊢ <;> omega\n })\n })\n <;>\n (try omega) <;>\n (try nlinarith) <;>\n (try\n {\n have h₅ : y ≤ 14 := by nlinarith\n have h₆ : y ≥ -14 := by nlinarith\n interval_cases y <;> norm_num at h₂ ⊢ <;> omega\n })\n <;>\n (try omega) <;>\n (try nlinarith)\n <;>\n (try\n {\n have h₅ : y ≤ 14 := by nlinarith\n have h₆ : y ≥ -14 := by nlinarith\n interval_cases y <;> norm_num at h₂ ⊢ <;> omega\n })\n exact h_main", "repl_code": "\n\n\n\ntheorem aime_1987_p5 (x y : ℤ) (h₀ : 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 h₁ : y ^ 2 + 3 * (x ^ 2 * y ^ 2) = 30 * x ^ 2 + 517 := h₀\n have h₂ : y ^ 2 = 30 * x ^ 2 - 71 := by\n \n have h₃ := h₁\n ring_nf at h₃ ⊢\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 h₃ : x ≤ 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 h₄ : x ≥ -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 h₅ : y ≤ 14 := by\n nlinarith [sq_nonneg (y - 7), sq_nonneg (y + 7), sq_nonneg (y - 14), sq_nonneg (y + 14)]\n have h₆ : y ≥ -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 h₂ ⊢ <;> (try omega) <;> (try nlinarith)\n }) <;> (try omega) <;> (try nlinarith) <;> (try\n {\n norm_num at h₂ ⊢\n <;>\n (try omega) <;>\n (try nlinarith) <;>\n (try\n {\n have h₅ : y ≤ 14 := by nlinarith\n have h₆ : y ≥ -14 := by nlinarith\n interval_cases y <;> norm_num at h₂ ⊢ <;> omega\n })\n })\n <;>\n (try omega) <;>\n (try nlinarith) <;>\n (try\n {\n have h₅ : y ≤ 14 := by nlinarith\n have h₆ : y ≥ -14 := by nlinarith\n interval_cases y <;> norm_num at h₂ ⊢ <;> omega\n })\n <;>\n (try omega) <;>\n (try nlinarith)\n <;>\n (try\n {\n have h₅ : y ≤ 14 := by nlinarith\n have h₆ : y ≥ -14 := by nlinarith\n interval_cases y <;> norm_num at h₂ ⊢ <;> 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 : ℤ\nh₀ h₁ : y ^ 2 + 3 * (x ^ 2 * y ^ 2) = 30 * x ^ 2 + 517\nh₃ : y ^ 2 + y ^ 2 * x ^ 2 * 3 = 517 + x ^ 2 * 30\na✝ : y ^ 2 < -71 + x ^ 2 * 30\n⊢ 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) (h₀ : 0 < a ∧ 0 < b) (h₁ : a ^ 2 = 6 * b)\n (h₂ : 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) (h₀ : 0 < a ∧ 0 < b) (h₁ : a ^ 2 = 6 * b)\n (h₂ : a ^ 2 = 54 / b) : a = 3 * NNReal.sqrt 2 := by \n have h_b_pos : 0 < b := by\n exact h₀.2\n \n have h_a_sq : a ^ 2 = 18 := by\n have h₃ : a ^ 2 = 6 * b := h₁\n have h₄ : a ^ 2 = 54 / b := h₂\n have h₅ : b > 0 := by exact_mod_cast h_b_pos\n have h₆ : (a ^ 2 : ℝ) = 6 * b := by exact_mod_cast h₃\n have h₇ : (a ^ 2 : ℝ) = 54 / b := by exact_mod_cast h₄\n have h₈ : (a ^ 2 : ℝ) = 18 := by\n have h₉ : (b : ℝ) > 0 := by exact_mod_cast h₅\n field_simp [h₉.ne'] at h₆ h₇ ⊢\n nlinarith [sq_nonneg ((a : ℝ) ^ 2 - 18)]\n exact_mod_cast h₈\n \n have h_a : a = 3 * NNReal.sqrt 2 := by\n have h₃ : a ^ 2 = 18 := h_a_sq\n have h₄ : 0 < a := by exact_mod_cast h₀.1\n have h₅ : a = 3 * NNReal.sqrt 2 := by\n \n \n have h₆ : (a : ℝ) ^ 2 = 18 := by exact_mod_cast h₃\n have h₇ : (a : ℝ) = 3 * Real.sqrt 2 := by\n \n have h₈ : 0 < (a : ℝ) := by exact_mod_cast h₄\n have h₉ : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n \n have h₁₀ : (a : ℝ) = 3 * Real.sqrt 2 := by\n \n nlinarith [Real.sq_sqrt (show 0 ≤ 2 by norm_num), Real.sqrt_nonneg 2,\n sq_nonneg ((a : ℝ) - 3 * Real.sqrt 2)]\n exact h₁₀\n \n have h₈ : (a : ℝ) = 3 * Real.sqrt 2 := h₇\n have h₉ : a = 3 * NNReal.sqrt 2 := by\n \n have h₁₀ : (a : ℝ) = 3 * Real.sqrt 2 := h₈\n have h₁₁ : a = 3 * NNReal.sqrt 2 := by\n \n apply Eq.symm\n apply Eq.symm\n norm_cast at h₁₀ ⊢\n <;>\n (try simp_all [mul_comm]) <;>\n (try ring_nf at * <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show (0 : ℝ) ≤ 2 by norm_num)]) <;>\n (try field_simp at * <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show (0 : ℝ) ≤ 2 by norm_num)])\n exact h₁₁\n exact h₉\n exact h₅\n \n exact h_a", "repl_code": "\n\n\n\ntheorem mathd_algebra_184 (a b : NNReal) (h₀ : 0 < a ∧ 0 < b) (h₁ : a ^ 2 = 6 * b)\n (h₂ : a ^ 2 = 54 / b) : a = 3 * NNReal.sqrt 2 := by \n have h_b_pos : 0 < b := by\n exact h₀.2\n \n have h_a_sq : a ^ 2 = 18 := by\n have h₃ : a ^ 2 = 6 * b := h₁\n have h₄ : a ^ 2 = 54 / b := h₂\n have h₅ : b > 0 := by exact_mod_cast h_b_pos\n have h₆ : (a ^ 2 : ℝ) = 6 * b := by exact_mod_cast h₃\n have h₇ : (a ^ 2 : ℝ) = 54 / b := by exact_mod_cast h₄\n have h₈ : (a ^ 2 : ℝ) = 18 := by\n have h₉ : (b : ℝ) > 0 := by exact_mod_cast h₅\n field_simp [h₉.ne'] at h₆ h₇ ⊢\n nlinarith [sq_nonneg ((a : ℝ) ^ 2 - 18)]\n exact_mod_cast h₈\n \n have h_a : a = 3 * NNReal.sqrt 2 := by\n have h₃ : a ^ 2 = 18 := h_a_sq\n have h₄ : 0 < a := by exact_mod_cast h₀.1\n have h₅ : a = 3 * NNReal.sqrt 2 := by\n \n \n have h₆ : (a : ℝ) ^ 2 = 18 := by exact_mod_cast h₃\n have h₇ : (a : ℝ) = 3 * Real.sqrt 2 := by\n \n have h₈ : 0 < (a : ℝ) := by exact_mod_cast h₄\n have h₉ : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n \n have h₁₀ : (a : ℝ) = 3 * Real.sqrt 2 := by\n \n nlinarith [Real.sq_sqrt (show 0 ≤ 2 by norm_num), Real.sqrt_nonneg 2,\n sq_nonneg ((a : ℝ) - 3 * Real.sqrt 2)]\n exact h₁₀\n \n have h₈ : (a : ℝ) = 3 * Real.sqrt 2 := h₇\n have h₉ : a = 3 * NNReal.sqrt 2 := by\n \n have h₁₀ : (a : ℝ) = 3 * Real.sqrt 2 := h₈\n have h₁₁ : a = 3 * NNReal.sqrt 2 := by\n \n apply Eq.symm\n apply Eq.symm\n norm_cast at h₁₀ ⊢\n <;>\n (try simp_all [mul_comm]) <;>\n (try ring_nf at * <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show (0 : ℝ) ≤ 2 by norm_num)]) <;>\n (try field_simp at * <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show (0 : ℝ) ≤ 2 by norm_num)])\n exact h₁₁\n exact h₉\n exact h₅\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\nh₁ : 54 / b = b * 6\nh₂ : 18 = b * 6\nh_b_pos : 0 < b\nh₃ : a ^ 2 = b * 6\nh₄ : 0 < a\nh₆ : (3 * √2) ^ 2 = 18\nh₁₀ : ↑a = 3 * √2\n⊢ 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 : ℕ) (h₀ : n = 3 ^ 17 + 3 ^ 10) (h₁ : 11 ∣ n + 1)\n (h₂ : [A, B, C].Pairwise (· ≠ ·)) (h₃ : {A, B, C} ⊂ Finset.Icc 0 9) (h₄ : Odd A ∧ Odd C)\n (h₅ : ¬3 ∣ B) (h₆ : 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 : ℕ) (h₀ : n = 3 ^ 17 + 3 ^ 10) (h₁ : 11 ∣ n + 1)\n (h₂ : [A, B, C].Pairwise (· ≠ ·)) (h₃ : {A, B, C} ⊂ Finset.Icc 0 9) (h₄ : Odd A ∧ Odd C)\n (h₅ : ¬3 ∣ B) (h₆ : 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 h₇ : n = 3 ^ 17 + 3 ^ 10 := h₀\n rw [h₇] at h₁\n norm_num [Nat.dvd_iff_mod_eq_zero, Nat.pow_mod, Nat.add_mod, Nat.mod_eq_of_lt] at h₁\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 : ℕ) (h₀ : n = 3 ^ 17 + 3 ^ 10) (h₁ : 11 ∣ n + 1)\n (h₂ : [A, B, C].Pairwise (· ≠ ·)) (h₃ : {A, B, C} ⊂ Finset.Icc 0 9) (h₄ : Odd A ∧ Odd C)\n (h₅ : ¬3 ∣ B) (h₆ : 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 h₇ : n = 3 ^ 17 + 3 ^ 10 := h₀\n rw [h₇] at h₁\n norm_num [Nat.dvd_iff_mod_eq_zero, Nat.pow_mod, Nat.add_mod, Nat.mod_eq_of_lt] at h₁\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 : ℕ\nh₀ : n = 3 ^ 17 + 3 ^ 10\nh₂ : List.Pairwise (fun x x_1 => x ≠ x_1) [A, B, C]\nh₃ : {A, B, C} ⊂ Finset.Icc 0 9\nh₄ : Odd A ∧ Odd C\nh₅ : ¬3 ∣ B\nh₆ : digits 10 n = [B, A, B, C, C, A, C, B, A]\nh₇ : n = 3 ^ 17 + 3 ^ 10\nh₁ : True\n⊢ 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 : ℕ) (h₀ : 9 ≤ n) : ∃ x : ℕ, (x : ℝ) ^ 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 : ℕ) (h₀ : 9 ≤ n) : ∃ x : ℕ, (x : ℝ) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by \n have h_main : ∃ x : ℕ, (x : ℝ) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by\n use n + 1\n have h₁ : ((n + 2)! - (n + 1)!) / n ! = (n + 1 : ℝ) ^ 2 := by\n have h₂ : (n + 2)! = (n + 2) * (n + 1)! := by\n simp [Nat.factorial_succ, Nat.mul_comm]\n <;> ring\n rw [h₂]\n have h₃ : (n + 2 : ℝ) * (n + 1)! - (n + 1)! = (n + 1 : ℝ) * (n + 1)! := by\n ring_nf\n <;> field_simp\n <;> ring_nf\n <;> nlinarith\n rw [h₃]\n have h₄ : (n + 1 : ℝ) * (n + 1)! / n ! = (n + 1 : ℝ) ^ 2 := by\n have h₅ : (n + 1)! = (n + 1) * n ! := by\n simp [Nat.factorial_succ, Nat.mul_comm]\n <;> ring\n rw [h₅]\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 [h₁]\n <;> norm_cast\n <;> ring_nf\n <;> nlinarith\n exact h_main", "repl_code": "\n\n\n\ntheorem amc12b_2020_p6 (n : ℕ) (h₀ : 9 ≤ n) : ∃ x : ℕ, (x : ℝ) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by \n have h_main : ∃ x : ℕ, (x : ℝ) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by\n use n + 1\n have h₁ : ((n + 2)! - (n + 1)!) / n ! = (n + 1 : ℝ) ^ 2 := by\n have h₂ : (n + 2)! = (n + 2) * (n + 1)! := by\n simp [Nat.factorial_succ, Nat.mul_comm]\n <;> ring\n rw [h₂]\n have h₃ : (n + 2 : ℝ) * (n + 1)! - (n + 1)! = (n + 1 : ℝ) * (n + 1)! := by\n ring_nf\n <;> field_simp\n <;> ring_nf\n <;> nlinarith\n rw [h₃]\n have h₄ : (n + 1 : ℝ) * (n + 1)! / n ! = (n + 1 : ℝ) ^ 2 := by\n have h₅ : (n + 1)! = (n + 1) * n ! := by\n simp [Nat.factorial_succ, Nat.mul_comm]\n <;> ring\n rw [h₅]\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 [h₁]\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 (↑n + 2) * ↑(n + 1)! - ↑(n + 1)!\nn : ℕ\nh₀ : 9 ≤ n\nh₂ : (n + 2)! = (n + 2) * (n + 1)!\nh₃ : (↑n + 2) * ↑(n + 1)! - ↑(n + 1)! = (↑n + 1) * ↑(n + 1)!\n⊢ (↑((n + 2) * (n + 1)!) - ↑(n + 1)!) / ↑n ! = (↑n + 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) (h₁ : n = 160⁻¹) : 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) (h₁ : n = 160⁻¹) : n = 1058 := by \n have h₂ : (160 : ZMod 1399) * 1058 = 1 := by\n norm_num [ZMod.nat_cast_self]\n <;> rfl\n \n have h₃ : (160 : ZMod 1399)⁻¹ = 1058 := by\n have h₄ : (160 : ZMod 1399) * 1058 = 1 := h₂\n have h₅ : (160 : ZMod 1399) ≠ 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 [h₄, h₅]\n <;> norm_num [ZMod.nat_cast_self]\n <;> rfl\n \n have h₄ : n = 1058 := by\n rw [h₁]\n rw [h₃]\n <;> rfl\n \n apply h₄", "repl_code": "\n\n\n\ntheorem mathd_numbertheory_321 (n : ZMod 1399) (h₁ : n = 160⁻¹) : n = 1058 := by \n have h₂ : (160 : ZMod 1399) * 1058 = 1 := by\n norm_num [ZMod.nat_cast_self]\n <;> rfl\n \n have h₃ : (160 : ZMod 1399)⁻¹ = 1058 := by\n have h₄ : (160 : ZMod 1399) * 1058 = 1 := h₂\n have h₅ : (160 : ZMod 1399) ≠ 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 [h₄, h₅]\n <;> norm_num [ZMod.nat_cast_self]\n <;> rfl\n \n have h₄ : n = 1058 := by\n rw [h₁]\n rw [h₃]\n <;> rfl\n \n apply h₄", "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\nh₁ : n = 160⁻¹\nh₂ h₄ : 160 * 1058 = 1\nh₅ : 160 ≠ 0\n⊢ 160⁻¹ = 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 : ℝ) (h₀ : 3 * y = x) (h₁ : 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 : ℝ) (h₀ : 3 * y = x) (h₁ : 2 * x + 5 * y = 11) : x + y = 4 := by \n have h_y : y = 1 := by\n have h₂ : 2 * x + 5 * y = 11 := h₁\n have h₃ : 3 * y = x := h₀\n \n have h₄ : 2 * (3 * y) + 5 * y = 11 := by\n rw [h₃] at h₂\n linarith\n \n have h₅ : 6 * y + 5 * y = 11 := by linarith\n have h₆ : 11 * y = 11 := by linarith\n have h₇ : y = 1 := by\n \n apply mul_left_cancel₀ (show (11 : ℝ) ≠ 0 by norm_num)\n nlinarith\n exact h₇\n \n have h_x : x = 3 := by\n have h₂ : x = 3 * y := by linarith\n rw [h₂, h_y]\n <;> ring\n <;> linarith\n \n have h_sum : x + y = 4 := by\n have h₃ : x = 3 := h_x\n have h₄ : y = 1 := h_y\n rw [h₃, h₄]\n <;> norm_num\n \n apply h_sum", "repl_code": "\n\n\n\ntheorem mathd_algebra_329 (x y : ℝ) (h₀ : 3 * y = x) (h₁ : 2 * x + 5 * y = 11) : x + y = 4 := by \n have h_y : y = 1 := by\n have h₂ : 2 * x + 5 * y = 11 := h₁\n have h₃ : 3 * y = x := h₀\n \n have h₄ : 2 * (3 * y) + 5 * y = 11 := by\n rw [h₃] at h₂\n linarith\n \n have h₅ : 6 * y + 5 * y = 11 := by linarith\n have h₆ : 11 * y = 11 := by linarith\n have h₇ : y = 1 := by\n \n apply mul_left_cancel₀ (show (11 : ℝ) ≠ 0 by norm_num)\n nlinarith\n exact h₇\n \n have h_x : x = 3 := by\n have h₂ : x = 3 * y := by linarith\n rw [h₂, h_y]\n <;> ring\n <;> linarith\n \n have h_sum : x + y = 4 := by\n have h₃ : x = 3 := h_x\n have h₄ : y = 1 := h_y\n rw [h₃, h₄]\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 : ℝ\nh₀ : 3 * y = x\nh₁ h₂ : 2 * x + 5 * y = 11\nh₃ : 3 * y = x\n⊢ 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 : ℝ) (h₀ : 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 : ℝ) (h₀ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53) : x = 3 / 4 := by \n have h_main : x = 3 / 4 := by\n have h₁ : 3 + x ≠ 0 := by\n by_contra h\n have h₂ : x = -3 := by linarith\n rw [h₂] at h₀\n norm_num at h₀ ⊢\n <;>\n (try contradiction) <;>\n (try linarith) <;>\n (try\n {\n field_simp at h₀ ⊢ <;> ring_nf at h₀ ⊢ <;> nlinarith\n })\n have h₂ : 2 + 2 / (3 + x) ≠ 0 := by\n by_contra h\n have h₃ : 2 + 2 / (3 + x) = 0 := by linarith\n have h₄ : 2 / (3 + x) = -2 := by linarith\n have h₅ : 3 + x ≠ 0 := h₁\n field_simp at h₄\n nlinarith\n have h₃ : 1 + 1 / (2 + 2 / (3 + x)) ≠ 0 := by\n by_contra h\n have h₄ : 1 + 1 / (2 + 2 / (3 + x)) = 0 := by linarith\n have h₅ : 1 / (2 + 2 / (3 + x)) = -1 := by linarith\n have h₆ : 2 + 2 / (3 + x) ≠ 0 := h₂\n field_simp at h₅\n <;> simp_all [div_eq_mul_inv]\n <;> ring_nf at *\n <;> nlinarith\n have h₄ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53 := h₀\n field_simp at h₄\n ring_nf at h₄\n have h₅ : x = 3 / 4 := by\n apply mul_left_cancel₀ (sub_ne_zero.mpr h₁)\n apply mul_left_cancel₀ (sub_ne_zero.mpr h₂)\n nlinarith [sq_pos_of_ne_zero h₁, sq_pos_of_ne_zero h₂, sq_nonneg (x - 3 / 4),\n sq_nonneg (x + 3 / 4), sq_nonneg (x + 1 / 2), sq_nonneg (x - 1 / 2)]\n exact h₅\n exact h_main", "repl_code": "\n\n\n\ntheorem amc12b_2021_p3 (x : ℝ) (h₀ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53) : x = 3 / 4 := by \n have h_main : x = 3 / 4 := by\n have h₁ : 3 + x ≠ 0 := by\n by_contra h\n have h₂ : x = -3 := by linarith\n rw [h₂] at h₀\n norm_num at h₀ ⊢\n <;>\n (try contradiction) <;>\n (try linarith) <;>\n (try\n {\n field_simp at h₀ ⊢ <;> ring_nf at h₀ ⊢ <;> nlinarith\n })\n have h₂ : 2 + 2 / (3 + x) ≠ 0 := by\n by_contra h\n have h₃ : 2 + 2 / (3 + x) = 0 := by linarith\n have h₄ : 2 / (3 + x) = -2 := by linarith\n have h₅ : 3 + x ≠ 0 := h₁\n field_simp at h₄\n nlinarith\n have h₃ : 1 + 1 / (2 + 2 / (3 + x)) ≠ 0 := by\n by_contra h\n have h₄ : 1 + 1 / (2 + 2 / (3 + x)) = 0 := by linarith\n have h₅ : 1 / (2 + 2 / (3 + x)) = -1 := by linarith\n have h₆ : 2 + 2 / (3 + x) ≠ 0 := h₂\n field_simp at h₅\n <;> simp_all [div_eq_mul_inv]\n <;> ring_nf at *\n <;> nlinarith\n have h₄ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53 := h₀\n field_simp at h₄\n ring_nf at h₄\n have h₅ : x = 3 / 4 := by\n apply mul_left_cancel₀ (sub_ne_zero.mpr h₁)\n apply mul_left_cancel₀ (sub_ne_zero.mpr h₂)\n nlinarith [sq_pos_of_ne_zero h₁, sq_pos_of_ne_zero h₂, sq_nonneg (x - 3 / 4),\n sq_nonneg (x + 3 / 4), sq_nonneg (x + 1 / 2), sq_nonneg (x - 1 / 2)]\n exact h₅\n exact h_main", "repl_errors": [{"pos": {"line": 25, "column": 6}, "endPos": {"line": 25, "column": 15}, "data": "linarith failed to find a contradiction\nx : ℝ\nh₀ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53\nh₁ : 3 + x ≠ 0\nh h₃ : 2 + 2 / (3 + x) = 0\nh₅ : 3 + x ≠ 0\nh₄ : 2 = -(2 * (3 + x))\n⊢ False\nfailed"}, {"pos": {"line": 41, "column": 6}, "endPos": {"line": 42, "column": 76}, "data": "linarith failed to find a contradiction\ncase h1.h\nx : ℝ\nh₀ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53\nh₁ : 3 + x ≠ 0\nh₂ : 2 + 2 / (3 + x) ≠ 0\nh₃ : 1 + 1 / (2 + 2 / (3 + x)) ≠ 0\nh₄ : 106 + (1 + x * (8 + x * 2)⁻¹ + (8 + x * 2)⁻¹ * 3)⁻¹ * 53 = 144\na✝ : (2 + 2 / (3 + x) - 0) * ((3 + x - 0) * x) < (2 + 2 / (3 + x) - 0) * ((3 + x - 0) * (3 / 4))\n⊢ 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 : ℕ → ℕ) (h₀ : ∀ n, 0 < f n) (h₁ : ∀ n, 0 < n → f (f n) < f (n + 1)) :\n ∀ n, 0 < n → 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 : ℕ → ℕ) (h₀ : ∀ n, 0 < f n) (h₁ : ∀ n, 0 < n → f (f n) < f (n + 1)) :\n ∀ n, 0 < n → f n = n := by \n have h_main : ∀ n, 0 < n → f n = n := by\n intro n hn\n by_contra h\n \n have h₂ : f n > n := by\n \n have h₃ : f n ≠ n := h\n have h₄ : f n > n := by\n by_contra h₅\n \n have h₆ : f n < n := by\n omega\n \n have h₇ : 0 < f n := h₀ n\n have h₈ : f (f n) < f (n + 1) := h₁ n hn\n have h₉ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₀ : f (f n) < f (n + 1) := h₁ n hn\n \n have h₁₁ : f n ≤ n := by omega\n have h₁₂ : f n < n := h₆\n have h₁₃ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₄ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₅ : f (f n) < f (n + 1) := h₁ n hn\n \n have h₁₆ : f n > 0 := h₀ n\n have h₁₇ : f (f n) > 0 := h₀ (f n)\n have h₁₈ : f (n + 1) > 0 := h₀ (n + 1)\n \n have h₁₉ : f (f n) < f (n + 1) := h₁ n hn\n have h₂₀ : f (f n) < f (n + 1) := h₁ n hn\n have h₂₁ : f (f n) < f (n + 1) := h₁ n hn\n \n omega\n exact h₄\n \n have h₃ : 0 < f n := h₀ n\n have h₄ : f (f n) < f (n + 1) := h₁ n hn\n have h₅ : f n > n := h₂\n have h₆ : f (f n) < f (n + 1) := h₁ n hn\n have h₇ : f n > 0 := h₀ n\n have h₈ : f (f n) > 0 := h₀ (f n)\n have h₉ : f (n + 1) > 0 := h₀ (n + 1)\n \n have h₁₀ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₁ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₂ : f (f n) < f (n + 1) := h₁ n hn\n \n have h₁₃ : f n > n := h₂\n have h₁₄ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₅ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₆ : f (f n) < f (n + 1) := h₁ n hn\n \n omega\n exact h_main", "repl_code": "\n\n\n\ntheorem imo_1977_p6 (f : ℕ → ℕ) (h₀ : ∀ n, 0 < f n) (h₁ : ∀ n, 0 < n → f (f n) < f (n + 1)) :\n ∀ n, 0 < n → f n = n := by \n have h_main : ∀ n, 0 < n → f n = n := by\n intro n hn\n by_contra h\n \n have h₂ : f n > n := by\n \n have h₃ : f n ≠ n := h\n have h₄ : f n > n := by\n by_contra h₅\n \n have h₆ : f n < n := by\n omega\n \n have h₇ : 0 < f n := h₀ n\n have h₈ : f (f n) < f (n + 1) := h₁ n hn\n have h₉ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₀ : f (f n) < f (n + 1) := h₁ n hn\n \n have h₁₁ : f n ≤ n := by omega\n have h₁₂ : f n < n := h₆\n have h₁₃ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₄ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₅ : f (f n) < f (n + 1) := h₁ n hn\n \n have h₁₆ : f n > 0 := h₀ n\n have h₁₇ : f (f n) > 0 := h₀ (f n)\n have h₁₈ : f (n + 1) > 0 := h₀ (n + 1)\n \n have h₁₉ : f (f n) < f (n + 1) := h₁ n hn\n have h₂₀ : f (f n) < f (n + 1) := h₁ n hn\n have h₂₁ : f (f n) < f (n + 1) := h₁ n hn\n \n omega\n exact h₄\n \n have h₃ : 0 < f n := h₀ n\n have h₄ : f (f n) < f (n + 1) := h₁ n hn\n have h₅ : f n > n := h₂\n have h₆ : f (f n) < f (n + 1) := h₁ n hn\n have h₇ : f n > 0 := h₀ n\n have h₈ : f (f n) > 0 := h₀ (f n)\n have h₉ : f (n + 1) > 0 := h₀ (n + 1)\n \n have h₁₀ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₁ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₂ : f (f n) < f (n + 1) := h₁ n hn\n \n have h₁₃ : f n > n := h₂\n have h₁₄ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₅ : f (f n) < f (n + 1) := h₁ n hn\n have h₁₆ : f (f n) < f (n + 1) := h₁ 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 ≥ 1\n a - b ≥ 1\n d ≥ 1\n b ≥ 1\n c - d ≥ 1\n a ≥ 1\nwhere\n a := ↑n\n b := ↑(f n)\n c := ↑(f (n + 1))\n d := ↑(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 ≥ 1\n a - b ≤ -1\n d ≥ 1\n b ≥ 1\n c - d ≥ 1\n a ≥ 1\nwhere\n a := ↑n\n b := ↑(f n)\n c := ↑(f (n + 1))\n d := ↑(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 : ℕ) :\n (∑ k in Finset.range n, k ^ 3) = (∑ 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 : ℕ) :\n (∑ k in Finset.range n, k ^ 3) = (∑ k in Finset.range n, k) ^ 2 := by \n have h_main : ∀ (n : ℕ), (∑ k in Finset.range n, k ^ 3) = (∑ k in Finset.range n, k) ^ 2 := by\n intro n\n have h : ∀ n : ℕ, (∑ k in Finset.range n, k ^ 3) = (∑ 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 : ℕ) :\n (∑ k in Finset.range n, k ^ 3) = (∑ k in Finset.range n, k) ^ 2 := by \n have h_main : ∀ (n : ℕ), (∑ k in Finset.range n, k ^ 3) = (∑ k in Finset.range n, k) ^ 2 := by\n intro n\n have h : ∀ n : ℕ, (∑ k in Finset.range n, k ^ 3) = (∑ 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 ≤ 0\n e ≥ 0\n d ≥ 0\n a - 2*b - d ≥ 1\n c ≥ 0\n f ≥ 0\n a ≥ 0\n a - 2*b - 2*e - f ≤ -1\nwhere\n a := ↑(n ^ 3)\n b := ↑(∑ k ∈ Finset.range n, k) * ↑n\n c := ↑((∑ k ∈ Finset.range n, k) ^ 2)\n d := ↑(n ^ 2)\n e := ↑(∑ k ∈ Finset.range n, k)\n f := ↑n"}, {"pos": {"line": 18, "column": 21}, "endPos": {"line": 21, "column": 19}, "data": "unsolved goals\ncase succ.succ\nn✝¹ n✝ n : ℕ\nih : ∑ x ∈ Finset.range n, x ^ 3 + n ^ 3 = (∑ k ∈ Finset.range n, k) * n * 2 + (∑ k ∈ Finset.range n, k) ^ 2 + n ^ 2\nih' :\n ∑ x ∈ Finset.range n, x ^ 3 = (∑ k ∈ Finset.range n, k) ^ 2 →\n (∑ k ∈ Finset.range n, k) ^ 2 + n ^ 3 = (∑ k ∈ Finset.range n, k) * n * 2 + (∑ k ∈ Finset.range n, k) ^ 2 + n ^ 2\n⊢ 1 + (∑ k ∈ Finset.range n, k) * n * 2 + (∑ k ∈ Finset.range n, k) ^ 2 + n * 3 + n ^ 2 * 4 + n ^ 3 =\n 1 + (∑ k ∈ Finset.range n, k) * 2 + (∑ k ∈ Finset.range n, k) * n * 4 + (∑ k ∈ 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 : ℕ+) (h₀ : ↑x / (4 : ℝ) + 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 : ℕ+) (h₀ : ↑x / (4 : ℝ) + y / 6 = (x + y) / n) : n = 5 := by \n have h₁ : (n : ℝ) > 0 := by\n exact mod_cast n.prop\n \n have h₂ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y = 12 * (x : ℝ) + 12 * (y : ℝ) := by\n have h₂₁ : (x : ℝ) / 4 + (y : ℝ) / 6 = ((x : ℝ) + (y : ℝ)) / n := by simpa using h₀\n have h₂₂ : (n : ℝ) ≠ 0 := by linarith\n field_simp at h₂₁ ⊢\n ring_nf at h₂₁ ⊢\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) h₁, mul_pos (Nat.cast_pos.mpr y.prop) h₁]\n \n have h₃ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y = 12 * (x : ℝ) + 12 * (y : ℝ) := by\n exact h₂\n \n have h₄ : (n : ℕ) ≤ 4 := by\n have h₄₁ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y = 12 * (x : ℝ) + 12 * (y : ℝ) := h₂\n have h₄₂ : (n : ℕ) ≤ 4 := by\n by_contra h\n have h₄₃ : (n : ℕ) ≥ 5 := by\n omega\n have h₄₄ : (n : ℝ) ≥ 5 := by\n exact_mod_cast h₄₃\n have h₄₅ : (x : ℝ) > 0 := by exact_mod_cast x.prop\n have h₄₆ : (y : ℝ) > 0 := by exact_mod_cast y.prop\n have h₄₇ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y > 12 * (x : ℝ) + 12 * (y : ℝ) := by\n nlinarith [mul_pos h₄₅ h₄₆, mul_pos (by positivity : (0 : ℝ) < 3) h₄₅,\n mul_pos (by positivity : (0 : ℝ) < 2) h₄₆]\n linarith\n exact h₄₂\n \n have h₅ : (n : ℕ) ≥ 5 := by\n have h₅₁ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y = 12 * (x : ℝ) + 12 * (y : ℝ) := h₂\n have h₅₂ : (n : ℕ) ≥ 5 := by\n by_contra h\n have h₅₃ : (n : ℕ) ≤ 4 := by omega\n have h₅₄ : (n : ℕ) ≤ 4 := h₅₃\n have h₅₅ : (n : ℝ) ≤ 4 := by exact_mod_cast h₅₄\n have h₅₆ : (x : ℝ) > 0 := by exact_mod_cast x.prop\n have h₅₇ : (y : ℝ) > 0 := by exact_mod_cast y.prop\n have h₅₈ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y < 12 * (x : ℝ) + 12 * (y : ℝ) := by\n have h₅₉ : (n : ℝ) ≤ 4 := h₅₅\n have h₅₁₀ : (n : ℝ) > 0 := h₁\n have h₅₁₁ : (x : ℝ) > 0 := h₅₆\n have h₅₁₂ : (y : ℝ) > 0 := h₅₇\n nlinarith [mul_pos h₅₁₁ h₅₁₂, mul_pos h₅₁₁ h₅₁₀, mul_pos h₅₁₂ h₅₁₀]\n linarith\n exact h₅₂\n \n have h₆ : n = 5 := by\n have h₆₁ : (n : ℕ) ≤ 4 := h₄\n have h₆₂ : (n : ℕ) ≥ 5 := h₅\n have h₆₃ : (n : ℕ) = 5 := by\n omega\n have h₆₄ : n = 5 := by\n exact_mod_cast h₆₃\n exact h₆₄\n \n apply h₆", "repl_code": "\n\n\n\ntheorem mathd_numbertheory_227 (x y n : ℕ+) (h₀ : ↑x / (4 : ℝ) + y / 6 = (x + y) / n) : n = 5 := by \n have h₁ : (n : ℝ) > 0 := by\n exact mod_cast n.prop\n \n have h₂ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y = 12 * (x : ℝ) + 12 * (y : ℝ) := by\n have h₂₁ : (x : ℝ) / 4 + (y : ℝ) / 6 = ((x : ℝ) + (y : ℝ)) / n := by simpa using h₀\n have h₂₂ : (n : ℝ) ≠ 0 := by linarith\n field_simp at h₂₁ ⊢\n ring_nf at h₂₁ ⊢\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) h₁, mul_pos (Nat.cast_pos.mpr y.prop) h₁]\n \n have h₃ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y = 12 * (x : ℝ) + 12 * (y : ℝ) := by\n exact h₂\n \n have h₄ : (n : ℕ) ≤ 4 := by\n have h₄₁ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y = 12 * (x : ℝ) + 12 * (y : ℝ) := h₂\n have h₄₂ : (n : ℕ) ≤ 4 := by\n by_contra h\n have h₄₃ : (n : ℕ) ≥ 5 := by\n omega\n have h₄₄ : (n : ℝ) ≥ 5 := by\n exact_mod_cast h₄₃\n have h₄₅ : (x : ℝ) > 0 := by exact_mod_cast x.prop\n have h₄₆ : (y : ℝ) > 0 := by exact_mod_cast y.prop\n have h₄₇ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y > 12 * (x : ℝ) + 12 * (y : ℝ) := by\n nlinarith [mul_pos h₄₅ h₄₆, mul_pos (by positivity : (0 : ℝ) < 3) h₄₅,\n mul_pos (by positivity : (0 : ℝ) < 2) h₄₆]\n linarith\n exact h₄₂\n \n have h₅ : (n : ℕ) ≥ 5 := by\n have h₅₁ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y = 12 * (x : ℝ) + 12 * (y : ℝ) := h₂\n have h₅₂ : (n : ℕ) ≥ 5 := by\n by_contra h\n have h₅₃ : (n : ℕ) ≤ 4 := by omega\n have h₅₄ : (n : ℕ) ≤ 4 := h₅₃\n have h₅₅ : (n : ℝ) ≤ 4 := by exact_mod_cast h₅₄\n have h₅₆ : (x : ℝ) > 0 := by exact_mod_cast x.prop\n have h₅₇ : (y : ℝ) > 0 := by exact_mod_cast y.prop\n have h₅₈ : (3 : ℝ) * (n : ℝ) * x + 2 * (n : ℝ) * y < 12 * (x : ℝ) + 12 * (y : ℝ) := by\n have h₅₉ : (n : ℝ) ≤ 4 := h₅₅\n have h₅₁₀ : (n : ℝ) > 0 := h₁\n have h₅₁₁ : (x : ℝ) > 0 := h₅₆\n have h₅₁₂ : (y : ℝ) > 0 := h₅₇\n nlinarith [mul_pos h₅₁₁ h₅₁₂, mul_pos h₅₁₁ h₅₁₀, mul_pos h₅₁₂ h₅₁₀]\n linarith\n exact h₅₂\n \n have h₆ : n = 5 := by\n have h₆₁ : (n : ℕ) ≤ 4 := h₄\n have h₆₂ : (n : ℕ) ≥ 5 := h₅\n have h₆₃ : (n : ℕ) = 5 := by\n omega\n have h₆₄ : n = 5 := by\n exact_mod_cast h₆₃\n exact h₆₄\n \n apply h₆", "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 : ℕ+\nh₀ : ↑↑x / 4 + ↑↑y / 6 = (↑↑x + ↑↑y) / ↑↑n\nh₁ : ↑↑n > 0\nh₂ h₃ h₄₁ : 3 * ↑↑n * ↑↑x + 2 * ↑↑n * ↑↑y = 12 * ↑↑x + 12 * ↑↑y\nh : ¬↑n ≤ 4\nh₄₃ : ↑n ≥ 5\nh₄₄ : ↑↑n ≥ 5\nh₄₅ : ↑↑x > 0\nh₄₆ : ↑↑y > 0\na✝ : 12 * ↑↑x + 12 * ↑↑y ≥ 3 * ↑↑n * ↑↑x + 2 * ↑↑n * ↑↑y\n⊢ 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 : ℝ) (h₀ : x ≠ 0) (h₁ : 2 * x = 5 * y) (h₂ : 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 : ℝ) (h₀ : x ≠ 0) (h₁ : 2 * x = 5 * y) (h₂ : 7 * y = 10 * z) :\n z / x = 7 / 25 := by \n have h_y : y = (2 * x) / 5 := by\n have h₃ : y = (2 * x) / 5 := by\n \n apply Eq.symm\n field_simp\n linarith\n exact h₃\n \n have h_z : z = (7 * x) / 25 := by\n have h₃ : z = (7 * x) / 25 := by\n \n have h₄ : 7 * y = 10 * z := h₂\n rw [h_y] at h₄\n ring_nf at h₄ ⊢\n nlinarith\n exact h₃\n \n have h_main : z / x = 7 / 25 := by\n rw [h_z]\n have h₃ : x ≠ 0 := h₀\n field_simp [h₃]\n <;> ring_nf\n <;> field_simp [h₃] 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 : ℝ) (h₁ : a * b = 180) (h₂ : 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 : ℝ) (h₁ : a * b = 180) (h₂ : 2 * (a + b) = 54) :\n a ^ 2 + b ^ 2 = 369 := by \n have h_sum : a + b = 27 := by\n have h₃ : a + b = 27 := by\n \n have h₄ : 2 * (a + b) = 54 := h₂\n \n linarith\n exact h₃\n \n have h_main : a ^ 2 + b ^ 2 = 369 := by\n have h₃ : a ^ 2 + b ^ 2 = 369 := by\n have h₄ : (a + b) ^ 2 = 729 := by\n \n have h₅ : a + b = 27 := h_sum\n rw [h₅]\n norm_num\n \n have h₅ : a ^ 2 + b ^ 2 + 2 * (a * b) = 729 := by\n nlinarith\n \n nlinarith [sq_nonneg (a - b)]\n exact h₃\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 : ℕ) (h₀ : n ≤ 9) (h₁ : 18 ∣ 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 : ℕ) (h₀ : n ≤ 9) (h₁ : 18 ∣ 374 * 10 + n) : n = 4 := by \n have h₂ : 14 + n = 18 := by\n have h₂₁ : 18 ∣ 374 * 10 + n := h₁\n have h₂₂ : 374 * 10 + n = 3740 + n := by norm_num\n rw [h₂₂] at h₂₁\n have h₂₃ : 18 ∣ 3740 + n := h₂₁\n have h₂₄ : 3740 % 18 = 14 := by norm_num\n have h₂₅ : (3740 + n) % 18 = 0 := by\n exact Nat.mod_eq_zero_of_dvd h₂₃\n have h₂₆ : (14 + n) % 18 = 0 := by\n omega\n have h₂₇ : 14 + n = 18 := by\n have h₂₈ : n ≤ 9 := h₀\n have h₂₉ : n ≤ 23 := by omega\n interval_cases n <;> norm_num at h₂₆ ⊢ <;> omega\n exact h₂₇\n \n have h₃ : n = 4 := by\n have h₃₁ : 14 + n = 18 := h₂\n have h₃₂ : n = 4 := by omega\n exact h₃₂\n \n exact h₃", "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 : ℝ) / (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 : ℝ) / (70 ^ 2 - 11 ^ 2) * ((70 - 11) * (70 + 11) / ((100 - 7) * (100 + 7))) =\n 1 := by \n have h_main : (100 ^ 2 - 7 ^ 2 : ℝ) / (70 ^ 2 - 11 ^ 2) * ((70 - 11) * (70 + 11) / ((100 - 7) * (100 + 7))) = 1 := by\n have h₁ : (100 ^ 2 - 7 ^ 2 : ℝ) = 9951 := by\n norm_num\n have h₂ : (70 ^ 2 - 11 ^ 2 : ℝ) = 4779 := by\n norm_num\n have h₃ : ((70 - 11 : ℝ) * (70 + 11)) = 4779 := by\n norm_num\n have h₄ : ((100 - 7 : ℝ) * (100 + 7)) = 9951 := by\n norm_num\n rw [h₁, h₂, h₃, h₄]\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 (σ : Equiv ℝ ℝ) (h₀ : σ.2 2 = 10) (h₁ : σ.2 10 = 1) (h₂ : σ.2 1 = 2) :\n σ.1 (σ.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 (σ : Equiv ℝ ℝ) (h₀ : σ.2 2 = 10) (h₁ : σ.2 10 = 1) (h₂ : σ.2 1 = 2) :\n σ.1 (σ.1 10) = 1 := by \n have h₃ : σ.1 10 = 2 := by\n have h₃₁ : σ.1 (σ.2 2) = 2 := by\n apply Equiv.apply_symm_apply\n rw [h₀] at h₃₁\n exact h₃₁\n \n have h₄ : σ.1 (σ.1 10) = σ.1 2 := by\n rw [h₃]\n <;> simp [h₃]\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 h₅ : σ.1 2 = 1 := by\n have h₅₁ : σ.1 (σ.2 1) = 1 := by\n apply Equiv.apply_symm_apply\n rw [h₂] at h₅₁\n exact h₅₁\n \n have h₆ : σ.1 (σ.1 10) = 1 := by\n rw [h₄]\n rw [h₅]\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 h₆", "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 : ℝ) (h₀ : a = -1) (h₁ : 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 : ℝ) (h₀ : a = -1) (h₁ : b = 5) : -a - b ^ 2 + 3 * (a * b) = -39 := by \n have h₂ : -a = 1 := by\n rw [h₀]\n norm_num\n <;> simp_all\n <;> ring_nf\n <;> linarith\n \n have h₃ : -b ^ 2 = -25 := by\n rw [h₁]\n norm_num\n <;> simp_all\n <;> ring_nf\n <;> linarith\n \n have h₄ : 3 * (a * b) = -15 := by\n rw [h₀, h₁]\n norm_num\n <;> ring_nf\n <;> linarith\n \n have h₅ : -a - b ^ 2 + 3 * (a * b) = -39 := by\n have h₅₁ : -a - b ^ 2 + 3 * (a * b) = -a - b ^ 2 + 3 * (a * b) := by rfl\n rw [h₅₁]\n have h₅₂ : -a = 1 := h₂\n have h₅₃ : -b ^ 2 = -25 := h₃\n have h₅₄ : 3 * (a * b) = -15 := h₄\n \n nlinarith [sq_nonneg (a + 1), sq_nonneg (b - 5)]\n \n exact h₅", "repl_code": null, "repl_errors": null}