{"name": "putnam_1964_a4", "proof_length": 1769, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen Set Function\n\n\n", "signature": "theorem putnam_1964_a4\n(u : ℕ → ℤ)\n(boundedu : ∃ B T : ℤ, ∀ n : ℕ, B ≤ u n ∧ u n ≤ T)\n(hu : ∀ n ≥ 4, u n = ((u (n - 1) + u (n - 2) + u (n - 3) * u (n - 4)) : ℝ) / (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) ∧ (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) ≠ 0)\n: (∃ N c : ℕ, c > 0 ∧ ∀ n ≥ N, u (n + c) = u n)", "src": "theorem putnam_1964_a4\n(u : ℕ → ℤ)\n(boundedu : ∃ B T : ℤ, ∀ n : ℕ, B ≤ u n ∧ u n ≤ T)\n(hu : ∀ n ≥ 4, u n = ((u (n - 1) + u (n - 2) + u (n - 3) * u (n - 4)) : ℝ) / (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) ∧ (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) ≠ 0)\n: (∃ N c : ℕ, c > 0 ∧ ∀ n ≥ N, u (n + c) = u n) := by \n  have h_main : ∃ (i j : ℕ), i < j ∧ (u i = u j ∧ u (i + 1) = u (j + 1) ∧ u (i + 2) = u (j + 2) ∧ u (i + 3) = u (j + 3)) := by\n    obtain ⟨B, T, hB⟩ := boundedu\n    have h₁ : ∃ (i j : ℕ), i < j ∧ (u i = u j ∧ u (i + 1) = u (j + 1) ∧ u (i + 2) = u (j + 2) ∧ u (i + 3) = u (j + 3)) := by\n      \n      classical\n      \n      let f : ℕ → (ℤ × ℤ × ℤ × ℤ) := fun n => (u n, u (n + 1), u (n + 2), u (n + 3))\n      have h₂ : Set.Finite (Set.range f) := by\n        have h₃ : Set.range f ⊆ Set.Icc (B, B, B, B) (T, T, T, T) := by\n          intro x hx\n          rcases hx with ⟨n, rfl⟩\n          have h₄ := hB n\n          have h₅ := hB (n + 1)\n          have h₆ := hB (n + 2)\n          have h₇ := hB (n + 3)\n          simp only [Set.mem_Icc, Prod.le_def] at h₄ h₅ h₆ h₇ ⊢\n          constructor <;>\n          (try constructor) <;>\n          (try constructor) <;>\n          (try constructor) <;>\n          (try simp_all [f]) <;>\n          (try nlinarith)\n        \n        have h₄ : Set.Finite (Set.Icc (B, B, B, B) (T, T, T, T)) := by\n          apply Set.Finite.subset (Set.finite_mem_finset (Finset.Icc (B, B, B, B) (T, T, T, T)))\n          intro x hx\n          simp_all [Set.mem_Icc]\n          <;>\n          (try aesop)\n        exact Set.Finite.subset h₄ h₃\n      \n      have h₃ : ∃ (i j : ℕ), i < j ∧ f i = f j := by\n        by_contra! h₄\n        have h₅ : Function.Injective f := by\n          intro i j h₆\n          by_cases h₇ : i = j\n          · exact h₇\n          · have h₈ : i < j ∨ j < i := by omega\n            cases h₈ with\n            | inl h₈ =>\n              have h₉ := h₄ i j h₈\n              simp_all\n            | inr h₈ =>\n              have h₉ := h₄ j i h₈\n              simp_all\n        have h₆ : Set.Infinite (Set.range f) := Set.infinite_range_of_injective h₅\n        exact Set.not_infinite.mpr h₂ h₆\n      \n      obtain ⟨i, j, h₄, h₅⟩ := h₃\n      refine' ⟨i, j, h₄, _⟩\n      simp [f] at h₅ ⊢\n      <;>\n      (try simp_all [Prod.ext_iff]) <;>\n      (try omega) <;>\n      (try aesop)\n    exact h₁\n  \n  have h_periodic : ∃ N c : ℕ, c > 0 ∧ ∀ n ≥ N, u (n + c) = u n := by\n    obtain ⟨i, j, hij, h_eq⟩ := h_main\n    have h₁ : ∀ m : ℕ, u (i + m) = u (j + m) := by\n      intro m\n      have h₂ : ∀ m : ℕ, u (i + m) = u (j + m) := by\n        intro m\n        induction m using Nat.strong_induction_on with\n        | h m ih =>\n          match m with\n          | 0 =>\n            have h₃ := h_eq.1\n            simp at h₃ ⊢\n            <;> simp_all [add_assoc]\n          | 1 =>\n            have h₃ := h_eq.2.1\n            simp at h₃ ⊢\n            <;> simp_all [add_assoc]\n            <;> ring_nf at *\n            <;> omega\n          | 2 =>\n            have h₃ := h_eq.2.2.1\n            simp at h₃ ⊢\n            <;> simp_all [add_assoc]\n            <;> ring_nf at *\n            <;> omega\n          | 3 =>\n            have h₃ := h_eq.2.2.2\n            simp at h₃ ⊢\n            <;> simp_all [add_assoc]\n            <;> ring_nf at *\n            <;> omega\n          | m + 4 =>\n            have h₃ : i + (m + 4) ≥ 4 := by\n              have h₄ : i ≥ 0 := by linarith\n              omega\n            have h₄ : u (i + (m + 4)) = ((u (i + (m + 4) - 1) + u (i + (m + 4) - 2) + u (i + (m + 4) - 3) * u (i + (m + 4) - 4)) : ℝ) / (u (i + (m + 4) - 1) * u (i + (m + 4) - 2) + u (i + (m + 4) - 3) + u (i + (m + 4) - 4)) := by\n              have h₅ := hu (i + (m + 4)) (by omega)\n              have h₆ := h₅.1\n              norm_cast at h₆ ⊢\n              <;> simp_all [add_assoc]\n              <;> ring_nf at *\n              <;> norm_num at *\n              <;> linarith\n            have h₅ : u (j + (m + 4)) = ((u (j + (m + 4) - 1) + u (j + (m + 4) - 2) + u (j + (m + 4) - 3) * u (j + (m + 4) - 4)) : ℝ) / (u (j + (m + 4) - 1) * u (j + (m + 4) - 2) + u (j + (m + 4) - 3) + u (j + (m + 4) - 4)) := by\n              have h₆ := hu (j + (m + 4)) (by\n                have h₇ : j ≥ i + 1 := by omega\n                omega\n              )\n              have h₇ := h₆.1\n              norm_cast at h₇ ⊢\n              <;> simp_all [add_assoc]\n              <;> ring_nf at *\n              <;> norm_num at *\n              <;> linarith\n            have h₆ : (i + (m + 4) - 1 : ℕ) = i + (m + 3) := by\n              omega\n            have h₇ : (i + (m + 4) - 2 : ℕ) = i + (m + 2) := by\n              omega\n            have h₈ : (i + (m + 4) - 3 : ℕ) = i + (m + 1) := by\n              omega\n            have h₉ : (i + (m + 4) - 4 : ℕ) = i + m := by\n              omega\n            have h₁₀ : (j + (m + 4) - 1 : ℕ) = j + (m + 3) := by\n              omega\n            have h₁₁ : (j + (m + 4) - 2 : ℕ) = j + (m + 2) := by\n              omega\n            have h₁₂ : (j + (m + 4) - 3 : ℕ) = j + (m + 1) := by\n              omega\n            have h₁₃ : (j + (m + 4) - 4 : ℕ) = j + m := by\n              omega\n            have h₁₄ : u (i + (m + 3)) = u (j + (m + 3)) := by\n              have h₁₅ := ih (m + 3) (by omega)\n              simp [add_assoc] at h₁₅ ⊢\n              <;> omega\n            have h₁₅ : u (i + (m + 2)) = u (j + (m + 2)) := by\n              have h₁₆ := ih (m + 2) (by omega)\n              simp [add_assoc] at h₁₆ ⊢\n              <;> omega\n            have h₁₆ : u (i + (m + 1)) = u (j + (m + 1)) := by\n              have h₁₇ := ih (m + 1) (by omega)\n              simp [add_assoc] at h₁₇ ⊢\n              <;> omega\n            have h₁₇ : u (i + m) = u (j + m) := by\n              have h₁₈ := ih m (by omega)\n              simp [add_assoc] at h₁₈ ⊢\n              <;> omega\n            have h₁₈ : (u (i + (m + 4)) : ℝ) = (u (j + (m + 4)) : ℝ) := by\n              rw [h₄, h₅]\n              simp [h₆, h₇, h₈, h₉, h₁₀, h₁₁, h₁₂, h₁₃] at *\n              norm_cast at *\n              <;>\n              (try simp_all [h₁₄, h₁₅, h₁₆, h₁₇]) <;>\n              (try ring_nf at *) <;>\n              (try field_simp at *) <;>\n              (try norm_cast at *) <;>\n              (try linarith)\n            norm_cast at h₁₈ ⊢\n            <;>\n            (try simp_all) <;>\n            (try linarith)\n      exact h₂ m\n    have h₂ : ∃ N c : ℕ, c > 0 ∧ ∀ n ≥ N, u (n + c) = u n := by\n      use i, (j - i)\n      have h₃ : j - i > 0 := by\n        omega\n      constructor\n      · exact h₃\n      · intro n hn\n        have h₄ : u (n + (j - i)) = u n := by\n          have h₅ : u (i + (n - i)) = u (j + (n - i)) := by\n            have h₆ := h₁ (n - i)\n            simp [add_assoc] at h₆ ⊢\n            <;> omega\n          have h₆ : i + (n - i) = n := by\n            omega\n          have h₇ : j + (n - i) = n + (j - i) := by\n            omega\n          rw [h₆] at h₅\n          rw [h₇] at h₅\n          linarith\n        exact h₄\n    exact h₂\n  \n  exact h_periodic"}
{"name": "putnam_1995_a3", "proof_length": 4085, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen Filter Topology Real\n\n\n", "signature": "theorem putnam_1995_a3\n(relation : (Fin 9 → ℤ) → (Fin 9 → ℤ) → Prop)\n(digits_to_num : (Fin 9 → ℤ) → ℤ)\n(hdigits_to_num : digits_to_num = fun dig => ∑ i : Fin 9, (dig i) * 10^i.1)\n(hrelation : ∀ d e : (Fin 9 → ℤ), relation d e ↔ (∀ i : Fin 9, d i < 10 ∧ d i ≥ 0 ∧ e i < 10 ∧ e i ≥ 0) ∧ (∀ i : Fin 9, 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j))))\n: ∀ d e f : (Fin 9 → ℤ), ((relation d e) ∧ (relation e f)) → (∀ i : Fin 9, 7 ∣ d i - f i)", "src": "theorem putnam_1995_a3\n(relation : (Fin 9 → ℤ) → (Fin 9 → ℤ) → Prop)\n(digits_to_num : (Fin 9 → ℤ) → ℤ)\n(hdigits_to_num : digits_to_num = fun dig => ∑ i : Fin 9, (dig i) * 10^i.1)\n(hrelation : ∀ d e : (Fin 9 → ℤ), relation d e ↔ (∀ i : Fin 9, d i < 10 ∧ d i ≥ 0 ∧ e i < 10 ∧ e i ≥ 0) ∧ (∀ i : Fin 9, 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j))))\n: ∀ d e f : (Fin 9 → ℤ), ((relation d e) ∧ (relation e f)) → (∀ i : Fin 9, 7 ∣ d i - f i) := by \n  have h_main : ∀ (d e f : (Fin 9 → ℤ)), ((relation d e) ∧ (relation e f)) → (∀ i : Fin 9, 7 ∣ d i - f i) := by\n    intro d e f h\n    have h₁ : relation d e := h.1\n    have h₂ : relation e f := h.2\n    have h₃ : ∀ (i : Fin 9), 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j)) := by\n      have h₄ : (∀ i : Fin 9, 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j))) := (hrelation d e).mp h₁ |>.2\n      exact h₄\n    have h₄ : ∀ (i : Fin 9), 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then f j else e j)) := by\n      have h₅ : (∀ i : Fin 9, 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then f j else e j))) := (hrelation e f).mp h₂ |>.2\n      exact h₅\n    \n    have h₅ : ∀ (i : Fin 9), 7 ∣ (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d := by\n      intro i\n      have h₆ : 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j)) := h₃ i\n      have h₇ : (digits_to_num (fun j : Fin 9 => if j = i then e j else d j)) = digits_to_num d + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n        calc\n          (digits_to_num (fun j : Fin 9 => if j = i then e j else d j)) = ∑ j : Fin 9, (if j = i then e j else d j) * (10 : ℤ) ^ (j : ℕ) := by\n            simp [hdigits_to_num]\n            <;>\n            rfl\n          _ = ∑ j : Fin 9, (if j = i then e j else d j) * (10 : ℤ) ^ (j : ℕ) := rfl\n          _ = (∑ j : Fin 9, d j * (10 : ℤ) ^ (j : ℕ)) + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n            calc\n              (∑ j : Fin 9, (if j = i then e j else d j) * (10 : ℤ) ^ (j : ℕ)) = ∑ j : Fin 9, (d j * (10 : ℤ) ^ (j : ℕ) + (if j = i then (e j - d j) else 0) * (10 : ℤ) ^ (j : ℕ)) := by\n                apply Finset.sum_congr rfl\n                intro j _\n                by_cases h : j = i\n                · simp [h]\n                  <;> ring\n                · simp [h]\n                  <;> ring\n              _ = (∑ j : Fin 9, d j * (10 : ℤ) ^ (j : ℕ)) + ∑ j : Fin 9, (if j = i then (e j - d j) else 0) * (10 : ℤ) ^ (j : ℕ) := by\n                rw [Finset.sum_add_distrib]\n              _ = (∑ j : Fin 9, d j * (10 : ℤ) ^ (j : ℕ)) + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n                have h₈ : ∑ j : Fin 9, (if j = i then (e j - d j) else 0) * (10 : ℤ) ^ (j : ℕ) = (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n                  calc\n                    _ = ∑ j : Fin 9, (if j = i then (e j - d j) else 0) * (10 : ℤ) ^ (j : ℕ) := rfl\n                    _ = (if i = i then (e i - d i) else 0) * (10 : ℤ) ^ (i : ℕ) := by\n                      \n                      simp [Finset.sum_ite_eq', Finset.mem_univ, i.is_lt]\n                    _ = (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by simp\n                rw [h₈]\n                <;> simp [hdigits_to_num]\n                <;>\n                rfl\n          _ = digits_to_num d + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n            simp [hdigits_to_num]\n            <;>\n            rfl\n      have h₈ : 7 ∣ digits_to_num d + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n        rw [h₇] at h₆\n        exact h₆\n      \n      have h₉ : 7 ∣ (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d := by\n        \n        have h₁₀ : (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d = digits_to_num d + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by ring\n        rw [h₁₀]\n        exact h₈\n      exact h₉\n    \n    have h₆ : ∀ (i : Fin 9), 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e := by\n      intro i\n      have h₇ : 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then f j else e j)) := h₄ i\n      have h₈ : (digits_to_num (fun j : Fin 9 => if j = i then f j else e j)) = digits_to_num e + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n        calc\n          (digits_to_num (fun j : Fin 9 => if j = i then f j else e j)) = ∑ j : Fin 9, (if j = i then f j else e j) * (10 : ℤ) ^ (j : ℕ) := by\n            simp [hdigits_to_num]\n            <;>\n            rfl\n          _ = ∑ j : Fin 9, (if j = i then f j else e j) * (10 : ℤ) ^ (j : ℕ) := rfl\n          _ = (∑ j : Fin 9, e j * (10 : ℤ) ^ (j : ℕ)) + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n            calc\n              (∑ j : Fin 9, (if j = i then f j else e j) * (10 : ℤ) ^ (j : ℕ)) = ∑ j : Fin 9, (e j * (10 : ℤ) ^ (j : ℕ) + (if j = i then (f j - e j) else 0) * (10 : ℤ) ^ (j : ℕ)) := by\n                apply Finset.sum_congr rfl\n                intro j _\n                by_cases h : j = i\n                · simp [h]\n                  <;> ring\n                · simp [h]\n                  <;> ring\n              _ = (∑ j : Fin 9, e j * (10 : ℤ) ^ (j : ℕ)) + ∑ j : Fin 9, (if j = i then (f j - e j) else 0) * (10 : ℤ) ^ (j : ℕ) := by\n                rw [Finset.sum_add_distrib]\n              _ = (∑ j : Fin 9, e j * (10 : ℤ) ^ (j : ℕ)) + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n                have h₉ : ∑ j : Fin 9, (if j = i then (f j - e j) else 0) * (10 : ℤ) ^ (j : ℕ) = (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n                  calc\n                    _ = ∑ j : Fin 9, (if j = i then (f j - e j) else 0) * (10 : ℤ) ^ (j : ℕ) := rfl\n                    _ = (if i = i then (f i - e i) else 0) * (10 : ℤ) ^ (i : ℕ) := by\n                      \n                      simp [Finset.sum_ite_eq', Finset.mem_univ, i.is_lt]\n                    _ = (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by simp\n                rw [h₉]\n                <;> simp [hdigits_to_num]\n                <;>\n                rfl\n          _ = digits_to_num e + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n            simp [hdigits_to_num]\n            <;>\n            rfl\n      have h₉ : 7 ∣ digits_to_num e + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n        rw [h₈] at h₇\n        exact h₇\n      \n      have h₁₀ : 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e := by\n        \n        have h₁₁ : (f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e = digits_to_num e + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by ring\n        rw [h₁₁]\n        exact h₉\n      exact h₁₀\n    \n    have h₇ : 7 ∣ digits_to_num e + digits_to_num d := by\n      have h₈ : digits_to_num e = digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n        calc\n          digits_to_num e = ∑ i : Fin 9, e i * (10 : ℤ) ^ (i : ℕ) := by simp [hdigits_to_num]\n          _ = ∑ i : Fin 9, (d i + (e i - d i)) * (10 : ℤ) ^ (i : ℕ) := by\n            apply Finset.sum_congr rfl\n            intro i _\n            ring\n          _ = ∑ i : Fin 9, (d i * (10 : ℤ) ^ (i : ℕ) + (e i - d i) * (10 : ℤ) ^ (i : ℕ)) := by\n            apply Finset.sum_congr rfl\n            intro i _\n            ring\n          _ = (∑ i : Fin 9, d i * (10 : ℤ) ^ (i : ℕ)) + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n            rw [Finset.sum_add_distrib]\n          _ = digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n            simp [hdigits_to_num]\n            <;>\n            rfl\n      have h₉ : ∀ i : Fin 9, 7 ∣ (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d := h₅\n      have h₁₀ : 7 ∣ ∑ i : Fin 9, ((e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d) := by\n        apply Finset.dvd_sum\n        intro i _\n        exact h₉ i\n      have h₁₁ : ∑ i : Fin 9, ((e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d) = ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + 9 * digits_to_num d := by\n        calc\n          ∑ i : Fin 9, ((e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d) = ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + ∑ i : Fin 9, digits_to_num d := by\n            rw [Finset.sum_add_distrib]\n          _ = ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + 9 * digits_to_num d := by\n            simp [Finset.sum_const, Finset.card_fin]\n            <;> ring\n            <;> norm_num\n      have h₁₂ : 7 ∣ ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + 9 * digits_to_num d := by\n        rw [h₁₁] at h₁₀\n        exact h₁₀\n      have h₁₃ : 7 ∣ digits_to_num e + digits_to_num d := by\n        have h₁₄ : digits_to_num e + digits_to_num d = 2 * digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n          linarith\n        rw [h₁₄]\n        have h₁₅ : 7 ∣ 2 * digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n          have h₁₆ : 7 ∣ ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + 9 * digits_to_num d := h₁₂\n          have h₁₇ : 7 ∣ 2 * digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n            \n            have h₁₈ : (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + 9 * digits_to_num d) = (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 9 * digits_to_num d := by ring\n            rw [h₁₈] at h₁₆\n            \n            \n            \n            \n            \n            \n            have h₁₉ : 7 ∣ (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 2 * digits_to_num d := by\n              \n              \n              \n              have h₂₀ : 7 ∣ (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 9 * digits_to_num d := h₁₆\n              have h₂₁ : 7 ∣ 7 * digits_to_num d := by\n                use digits_to_num d\n                <;> ring\n              have h₂₂ : 7 ∣ (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 2 * digits_to_num d := by\n                \n                have h₂₃ : (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 2 * digits_to_num d = ((∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 9 * digits_to_num d) - 7 * digits_to_num d := by ring\n                rw [h₂₃]\n                exact dvd_sub h₂₀ h₂₁\n              exact h₂₂\n            \n            have h₂₀ : 7 ∣ 2 * digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n              \n              have h₂₁ : 2 * digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) = (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 2 * digits_to_num d := by ring\n              rw [h₂₁]\n              exact h₁₉\n            exact h₂₀\n          exact h₁₇\n        exact h₁₅\n      exact h₁₃\n    \n    have h₈ : ∀ (i : Fin 9), 7 ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n      intro i\n      have h₉ : 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e := h₆ i\n      have h₁₀ : 7 ∣ (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d := h₅ i\n      have h₁₁ : 7 ∣ digits_to_num e + digits_to_num d := h₇\n      \n      have h₁₂ : 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) - digits_to_num d := by\n        \n        \n        have h₁₃ : 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e := h₆ i\n        have h₁₄ : 7 ∣ digits_to_num e + digits_to_num d := h₇\n        \n        have h₁₅ : 7 ∣ ((f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e) - (digits_to_num e + digits_to_num d) := by\n          exact dvd_sub h₁₃ h₁₄\n        \n        have h₁₆ : ((f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e) - (digits_to_num e + digits_to_num d) = (f i - e i) * (10 : ℤ) ^ (i : ℕ) - digits_to_num d := by\n          ring\n        rw [h₁₆] at h₁₅\n        exact h₁₅\n      \n      have h₁₃ : 7 ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n        have h₁₄ : 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) - digits_to_num d := h₁₂\n        have h₁₅ : 7 ∣ (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d := h₅ i\n        \n        have h₁₆ : 7 ∣ ((f i - e i) * (10 : ℤ) ^ (i : ℕ) - digits_to_num d) + ((e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d) := by\n          exact dvd_add h₁₄ h₁₅\n        \n        have h₁₇ : ((f i - e i) * (10 : ℤ) ^ (i : ℕ) - digits_to_num d) + ((e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d) = (f i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n          ring\n        rw [h₁₇] at h₁₆\n        exact h₁₆\n      exact h₁₃\n    \n    have h₉ : ∀ (i : Fin 9), 7 ∣ d i - f i := by\n      intro i\n      have h₁₀ : 7 ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := h₈ i\n      have h₁₁ : 7 ∣ (f i - d i) := by\n        \n        have h₁₂ : (7 : ℤ) ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := h₁₀\n        \n        have h₁₃ : (7 : ℕ).Prime := by decide\n        have h₁₄ : ¬(7 : ℤ) ∣ (10 : ℤ) ^ (i : ℕ) := by\n          \n          intro h\n          have h₁₅ : (7 : ℕ) ∣ (10 : ℕ) ^ (i : ℕ) := by\n            norm_cast at h ⊢\n            <;> simpa [Int.coe_nat_dvd_left] using h\n          have h₁₆ : (7 : ℕ) ∣ 10 := by\n            \n            exact Nat.Prime.dvd_of_dvd_pow h₁₃ h₁₅\n          norm_num at h₁₆\n          <;> contradiction\n        \n        have h₁₅ : (7 : ℤ) ∣ (f i - d i) := by\n          \n          have h₁₆ : (7 : ℤ) ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := h₁₂\n          have h₁₇ : ¬(7 : ℤ) ∣ (10 : ℤ) ^ (i : ℕ) := h₁₄\n          \n          have h₁₈ : (7 : ℤ) ∣ (f i - d i) := by\n            \n            have h₁₉ : (7 : ℤ) ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := h₁₂\n            have h₂₀ : (7 : ℕ).Prime := by decide\n            have h₂₁ : (7 : ℤ) ∣ (f i - d i) ∨ (7 : ℤ) ∣ (10 : ℤ) ^ (i : ℕ) := by\n              \n              have h₂₂ : (7 : ℤ) ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := h₁₂\n              have h₂₃ : (7 : ℤ) ∣ (f i - d i) ∨ (7 : ℤ) ∣ (10 : ℤ) ^ (i : ℕ) := by\n                \n                apply (Int.prime_iff_natAbs_prime.mpr (by decide)).dvd_mul.mp\n                exact h₂₂\n              exact h₂₃\n            cases h₂₁ with\n            | inl h₂₁ =>\n              exact h₂₁\n            | inr h₂₁ =>\n              exfalso\n              exact h₁₄ h₂₁\n          exact h₁₈\n        exact h₁₅\n      \n      have h₁₂ : 7 ∣ d i - f i := by\n        have h₁₃ : 7 ∣ (f i - d i) := h₁₁\n        have h₁₄ : 7 ∣ -(f i - d i) := by\n          exact dvd_neg.mpr h₁₃\n        have h₁₅ : -(f i - d i) = d i - f i := by ring\n        rw [h₁₅] at h₁₄\n        exact h₁₄\n      exact h₁₂\n    \n    intro i\n    have h₁₀ : 7 ∣ d i - f i := h₉ i\n    exact h₁₀\n  exact h_main"}
{"name": "putnam_1988_a2", "proof_length": 2161, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen Set\n\n\n\n", "signature": "theorem putnam_1988_a2\n    (f : ℝ → ℝ)\n    (hf : f = fun x ↦ Real.exp (x ^ 2)) :\n    ((True) : Prop ) ↔\n    (∃ a b : ℝ,\n      a < b ∧\n      ∃ g : ℝ → ℝ,\n        (∃ x ∈ Ioo a b, g x ≠ 0) ∧\n        DifferentiableOn ℝ g (Ioo a b) ∧\n        ∀ x ∈ Ioo a b, deriv (fun y ↦ f y * g y) x = (deriv f x) * (deriv g x))", "src": "theorem putnam_1988_a2\n    (f : ℝ → ℝ)\n    (hf : f = fun x ↦ Real.exp (x ^ 2)) :\n    ((True) : Prop ) ↔\n    (∃ a b : ℝ,\n      a < b ∧\n      ∃ g : ℝ → ℝ,\n        (∃ x ∈ Ioo a b, g x ≠ 0) ∧\n        DifferentiableOn ℝ g (Ioo a b) ∧\n        ∀ x ∈ Ioo a b, deriv (fun y ↦ f y * g y) x = (deriv f x) * (deriv g x)) :=\n  by \n  have h_main : (∃ (a b : ℝ), a < b ∧ ∃ (g : ℝ → ℝ), (∃ x ∈ Set.Ioo a b, g x ≠ 0) ∧ DifferentiableOn ℝ g (Set.Ioo a b) ∧ ∀ x ∈ Set.Ioo a b, deriv (fun y ↦ f y * g y) x = (deriv f x) * (deriv g x)) := by\n    use 1, 2\n    constructor\n    · norm_num\n    · \n      use fun x => Real.exp x * Real.sqrt (2 * x - 1)\n      constructor\n      · \n        use 3 / 2\n        constructor\n        · \n          norm_num [Set.mem_Ioo]\n        · \n          have h₁ : Real.exp (3 / 2 : ℝ) > 0 := Real.exp_pos _\n          have h₂ : Real.sqrt (2 * (3 / 2 : ℝ) - 1) > 0 := by\n            apply Real.sqrt_pos_of_pos\n            norm_num\n          have h₃ : Real.exp (3 / 2 : ℝ) * Real.sqrt (2 * (3 / 2 : ℝ) - 1) > 0 := by positivity\n          intro h₄\n          linarith\n      · constructor\n        · \n          have h₁ : DifferentiableOn ℝ (fun x : ℝ => Real.exp x * Real.sqrt (2 * x - 1)) (Set.Ioo 1 2) := by\n            apply DifferentiableOn.mul\n            · exact DifferentiableOn.exp (differentiableOn_id)\n            · apply DifferentiableOn.sqrt\n              · \n                have h₂ : DifferentiableOn ℝ (fun x : ℝ => (2 : ℝ) * x - 1) (Set.Ioo 1 2) := by\n                  apply DifferentiableOn.sub\n                  · apply DifferentiableOn.const_mul\n                    exact differentiableOn_id\n                  · exact differentiableOn_const _\n                exact h₂\n              · \n                intro x hx\n                have h₃ : 1 < x := hx.1\n                have h₄ : x < 2 := hx.2\n                have h₅ : (2 : ℝ) * x - 1 > 0 := by linarith\n                linarith\n          exact h₁\n        · \n          intro x hx\n          have h₁ : x ∈ Set.Ioo 1 2 := hx\n          have h₂ : 1 < x := h₁.1\n          have h₃ : x < 2 := h₁.2\n          have h₄ : (2 : ℝ) * x - 1 > 0 := by linarith\n          have h₅ : Real.sqrt (2 * x - 1) > 0 := Real.sqrt_pos.mpr h₄\n          have h₆ : HasDerivAt (fun x : ℝ => Real.exp x * Real.sqrt (2 * x - 1)) (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1)) x := by\n            \n            have h₇ : HasDerivAt (fun x : ℝ => Real.exp x) (Real.exp x) x := Real.hasDerivAt_exp x\n            have h₈ : HasDerivAt (fun x : ℝ => Real.sqrt (2 * x - 1)) (1 / Real.sqrt (2 * x - 1)) x := by\n              have h₉ : HasDerivAt (fun x : ℝ => (2 : ℝ) * x - 1) 2 x := by\n                simpa using (hasDerivAt_id x).const_mul 2 |>.sub_const 1\n              have h₁₀ : HasDerivAt (fun x : ℝ => Real.sqrt (2 * x - 1)) (1 / Real.sqrt (2 * x - 1)) x := by\n                convert HasDerivAt.sqrt h₉ (by positivity) using 1 <;>\n                  field_simp [h₄.ne'] <;>\n                  ring_nf <;>\n                  field_simp [h₄.ne'] <;>\n                  ring\n              exact h₁₀\n            have h₁₁ : HasDerivAt (fun x : ℝ => Real.exp x * Real.sqrt (2 * x - 1)) (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1)) x := by\n              convert h₇.mul h₈ using 1 <;>\n                field_simp [h₄.ne'] <;>\n                ring_nf <;>\n                field_simp [h₄.ne'] <;>\n                ring\n            exact h₁₁\n          have h₇ : deriv (fun x : ℝ => Real.exp x * Real.sqrt (2 * x - 1)) x = Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1) := by\n            apply HasDerivAt.deriv\n            exact h₆\n          \n          have h₈ : f = fun x => Real.exp (x ^ 2) := hf\n          have h₉ : deriv f x = 2 * x * Real.exp (x ^ 2) := by\n            rw [h₈]\n            \n            have h₁₀ : HasDerivAt (fun x : ℝ => Real.exp (x ^ 2)) (2 * x * Real.exp (x ^ 2)) x := by\n              have h₁₁ : HasDerivAt (fun x : ℝ => x ^ 2) (2 * x) x := by\n                simpa using (hasDerivAt_pow 2 x)\n              have h₁₂ : HasDerivAt (fun x : ℝ => Real.exp (x ^ 2)) (Real.exp (x ^ 2) * (2 * x)) x := by\n                apply HasDerivAt.exp\n                exact h₁₁\n              convert h₁₂ using 1 <;> ring\n            have h₁₃ : deriv (fun x : ℝ => Real.exp (x ^ 2)) x = 2 * x * Real.exp (x ^ 2) := by\n              apply HasDerivAt.deriv\n              exact h₁₀\n            exact h₁₃\n          have h₁₀ : deriv (fun y : ℝ => f y * (Real.exp y * Real.sqrt (2 * y - 1))) x = (deriv f x) * (deriv (fun y : ℝ => Real.exp y * Real.sqrt (2 * y - 1)) x) := by\n            \n            have h₁₁ : deriv (fun y : ℝ => f y * (Real.exp y * Real.sqrt (2 * y - 1))) x = deriv f x * (Real.exp x * Real.sqrt (2 * x - 1)) + f x * (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1)) := by\n              \n              have h₁₂ : HasDerivAt (fun y : ℝ => f y * (Real.exp y * Real.sqrt (2 * y - 1))) (deriv f x * (Real.exp x * Real.sqrt (2 * x - 1)) + f x * (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1))) x := by\n                have h₁₃ : HasDerivAt f (deriv f x) x := by\n                  have h₁₄ : DifferentiableAt ℝ f x := by\n                    rw [h₈]\n                    apply DifferentiableAt.exp\n                    apply DifferentiableAt.pow\n                    exact differentiableAt_id\n                  exact h₁₄.hasDerivAt\n                have h₁₄ : HasDerivAt (fun y : ℝ => Real.exp y * Real.sqrt (2 * y - 1)) (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1)) x := by\n                  convert h₆ using 1\n                have h₁₅ : HasDerivAt (fun y : ℝ => f y * (Real.exp y * Real.sqrt (2 * y - 1))) (deriv f x * (Real.exp x * Real.sqrt (2 * x - 1)) + f x * (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1))) x := by\n                  convert h₁₃.mul h₁₄ using 1 <;>\n                    simp [h₉, h₈, mul_assoc]\n                  <;>\n                  ring_nf\n                  <;>\n                  field_simp [h₄.ne']\n                  <;>\n                  ring\n                exact h₁₅\n              have h₁₆ : deriv (fun y : ℝ => f y * (Real.exp y * Real.sqrt (2 * y - 1))) x = deriv f x * (Real.exp x * Real.sqrt (2 * x - 1)) + f x * (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1)) := by\n                apply HasDerivAt.deriv\n                exact h₁₂\n              exact h₁₆\n            have h₁₇ : deriv f x * (Real.exp x * Real.sqrt (2 * x - 1)) + f x * (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1)) = (deriv f x) * (deriv (fun y : ℝ => Real.exp y * Real.sqrt (2 * y - 1)) x) := by\n              \n              have h₁₈ : deriv f x = 2 * x * Real.exp (x ^ 2) := h₉\n              have h₁₉ : f x = Real.exp (x ^ 2) := by\n                rw [h₈]\n                <;> simp\n              have h₂₀ : deriv (fun y : ℝ => Real.exp y * Real.sqrt (2 * y - 1)) x = Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1) := h₇\n              have h₂₁ : 2 * x * (Real.exp x * Real.sqrt (2 * x - 1)) + (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1)) = 2 * x * (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1)) := by\n                \n                have h₂₂ : Real.exp x > 0 := Real.exp_pos x\n                have h₂₃ : Real.sqrt (2 * x - 1) > 0 := Real.sqrt_pos.mpr (by linarith)\n                field_simp [h₂₃.ne', h₂₂.ne']\n                <;>\n                nlinarith [Real.sq_sqrt (show 0 ≤ 2 * x - 1 by linarith)]\n              calc\n                deriv f x * (Real.exp x * Real.sqrt (2 * x - 1)) + f x * (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1)) = (2 * x * Real.exp (x ^ 2)) * (Real.exp x * Real.sqrt (2 * x - 1)) + Real.exp (x ^ 2) * (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1)) := by\n                  rw [h₁₈, h₁₉]\n                _ = Real.exp (x ^ 2) * (2 * x * (Real.exp x * Real.sqrt (2 * x - 1)) + (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1))) := by\n                  ring_nf\n                  <;>\n                  field_simp [Real.exp_ne_zero]\n                  <;>\n                  ring_nf\n                _ = Real.exp (x ^ 2) * (2 * x * (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1))) := by\n                  rw [h₂₁]\n                  <;>\n                  ring_nf\n                _ = (2 * x * Real.exp (x ^ 2)) * (Real.exp x * Real.sqrt (2 * x - 1) + Real.exp x / Real.sqrt (2 * x - 1)) := by\n                  ring_nf\n                _ = (deriv f x) * (deriv (fun y : ℝ => Real.exp y * Real.sqrt (2 * y - 1)) x) := by\n                  rw [h₁₈, h₂₀]\n                  <;>\n                  ring_nf\n                  <;>\n                  field_simp [Real.exp_ne_zero]\n                  <;>\n                  ring_nf\n            rw [h₁₁]\n            rw [h₁₇]\n          \n          have h₁₁ : deriv (fun y : ℝ => f y * (Real.exp y * Real.sqrt (2 * y - 1))) x = (deriv f x) * (deriv (fun y : ℝ => Real.exp y * Real.sqrt (2 * y - 1)) x) := h₁₀\n          simpa [h₈] using h₁₁\n  \n  have h_final : ((True) : Prop ) ↔ (∃ a b : ℝ, a < b ∧ ∃ g : ℝ → ℝ, (∃ x ∈ Set.Ioo a b, g x ≠ 0) ∧ DifferentiableOn ℝ g (Set.Ioo a b) ∧ ∀ x ∈ Set.Ioo a b, deriv (fun y ↦ f y * g y) x = (deriv f x) * (deriv g x)) := by\n    constructor\n    · intro _\n      exact h_main\n    · intro h\n      trivial\n  \n  exact h_final"}
{"name": "putnam_2001_b1", "proof_length": 3178, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen Topology Filter Polynomial Set\n\n\n", "signature": "theorem putnam_2001_b1\n  (n : ℕ)\n  (hn : n > 0) (hn' : Even n)\n  (nums : Fin n → Fin n → ℤ)\n  (colors : Fin n → Fin n → Fin 2)\n  (hnums : ∀ k l, nums k l = k * n + l + 1)\n  (hcolorsrows : ∀ k, (∑ l, (if (colors k l = 0) then 1 else 0)) = n / 2)\n  (hcolorscols : ∀ l, (∑ k, (if (colors k l = 0) then 1 else 0)) = n / 2) :\n  (∑ k, ∑ l, (if (colors k l = 0) then (nums k l) else 0)) = (∑ k, ∑ l, (if (colors k l = 1) then (nums k l) else 0))", "src": "theorem putnam_2001_b1\n  (n : ℕ)\n  (hn : n > 0) (hn' : Even n)\n  (nums : Fin n → Fin n → ℤ)\n  (colors : Fin n → Fin n → Fin 2)\n  (hnums : ∀ k l, nums k l = k * n + l + 1)\n  (hcolorsrows : ∀ k, (∑ l, (if (colors k l = 0) then 1 else 0)) = n / 2)\n  (hcolorscols : ∀ l, (∑ k, (if (colors k l = 0) then 1 else 0)) = n / 2) :\n  (∑ k, ∑ l, (if (colors k l = 0) then (nums k l) else 0)) = (∑ k, ∑ l, (if (colors k l = 1) then (nums k l) else 0)) := by \n  have hrow_sum_zero : ∀ (k : Fin n), (∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1)) = 0 := by\n    intro k\n    have h1 : (∑ l : Fin n, (if colors k l = 0 then (1 : ℤ) else 0)) = n / 2 := by\n      norm_cast at hcolorsrows ⊢\n      simpa using hcolorsrows k\n    have h2 : (∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1)) = 0 := by\n      calc\n        (∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1)) = (∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0))) - ∑ l : Fin n, (1 : ℤ) := by\n          rw [Finset.sum_sub_distrib]\n        _ = 2 * (∑ l : Fin n, (if colors k l = 0 then (1 : ℤ) else 0)) - ∑ l : Fin n, (1 : ℤ) := by\n          rw [Finset.mul_sum]\n        _ = 2 * (n / 2 : ℤ) - n := by\n          rw [h1]\n          simp [Finset.sum_const, Finset.card_fin]\n          <;> norm_cast\n          <;> simp [nsmul_eq_mul]\n          <;> ring_nf\n          <;> simp_all [Nat.cast_inj]\n          <;> omega\n        _ = 0 := by\n          have h3 : (n : ℤ) % 2 = 0 := by\n            have h4 : Even n := hn'\n            rw [even_iff_two_dvd] at h4\n            obtain ⟨m, hm⟩ := h4\n            norm_cast at hm ⊢\n            omega\n          have h4 : (n : ℤ) / 2 * 2 = n := by\n            have h5 : (n : ℤ) % 2 = 0 := h3\n            have h6 : (n : ℤ) = 2 * ((n : ℤ) / 2) := by\n              omega\n            linarith\n          ring_nf at *\n          <;> omega\n    exact h2\n  \n  have hcol_sum_zero : ∀ (l : Fin n), (∑ k : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1)) = 0 := by\n    intro l\n    have h1 : (∑ k : Fin n, (if colors k l = 0 then (1 : ℤ) else 0)) = n / 2 := by\n      norm_cast at hcolorscols ⊢\n      simpa using hcolorscols l\n    have h2 : (∑ k : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1)) = 0 := by\n      calc\n        (∑ k : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1)) = (∑ k : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0))) - ∑ k : Fin n, (1 : ℤ) := by\n          rw [Finset.sum_sub_distrib]\n        _ = 2 * (∑ k : Fin n, (if colors k l = 0 then (1 : ℤ) else 0)) - ∑ k : Fin n, (1 : ℤ) := by\n          rw [Finset.mul_sum]\n        _ = 2 * (n / 2 : ℤ) - n := by\n          rw [h1]\n          simp [Finset.sum_const, Finset.card_fin]\n          <;> norm_cast\n          <;> simp [nsmul_eq_mul]\n          <;> ring_nf\n          <;> simp_all [Nat.cast_inj]\n          <;> omega\n        _ = 0 := by\n          have h3 : (n : ℤ) % 2 = 0 := by\n            have h4 : Even n := hn'\n            rw [even_iff_two_dvd] at h4\n            obtain ⟨m, hm⟩ := h4\n            norm_cast at hm ⊢\n            omega\n          have h4 : (n : ℤ) / 2 * 2 = n := by\n            have h5 : (n : ℤ) % 2 = 0 := h3\n            have h6 : (n : ℤ) = 2 * ((n : ℤ) / 2) := by\n              omega\n            linarith\n          ring_nf at *\n          <;> omega\n    exact h2\n  \n  have hsum_diff : (∑ k : Fin n, ∑ l : Fin n, (if colors k l = 0 then (nums k l : ℤ) else 0)) - (∑ k : Fin n, ∑ l : Fin n, (if colors k l = 1 then (nums k l : ℤ) else 0)) = ∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * (nums k l : ℤ) := by\n    have h₁ : (∑ k : Fin n, ∑ l : Fin n, (if colors k l = 0 then (nums k l : ℤ) else 0)) - (∑ k : Fin n, ∑ l : Fin n, (if colors k l = 1 then (nums k l : ℤ) else 0)) = ∑ k : Fin n, ∑ l : Fin n, ((if colors k l = 0 then (nums k l : ℤ) else 0) - (if colors k l = 1 then (nums k l : ℤ) else 0)) := by\n      simp [Finset.sum_sub_distrib, Finset.sum_add_distrib]\n      <;>\n      simp_all [Finset.sum_sub_distrib]\n      <;>\n      rfl\n    have h₂ : ∀ (k : Fin n) (l : Fin n), ((if colors k l = 0 then (nums k l : ℤ) else 0) - (if colors k l = 1 then (nums k l : ℤ) else 0)) = (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * (nums k l : ℤ) := by\n      intro k l\n      have h₃ : colors k l = 0 ∨ colors k l = 1 := by\n        have h₄ : colors k l = 0 ∨ colors k l = 1 := by\n          have h₅ : colors k l = 0 ∨ colors k l = 1 := by\n            have : colors k l = 0 ∨ colors k l = 1 := by\n              have h₆ : (colors k l : ℕ) < 2 := by\n                have : (colors k l : ℕ) < 2 := by\n                  simp [Fin.is_lt]\n                exact this\n              omega\n            exact this\n          exact h₅\n        exact h₄\n      rcases h₃ with (h₃ | h₃)\n      · \n        simp [h₃]\n        <;>\n        (try norm_num) <;>\n        (try ring_nf) <;>\n        (try simp_all)\n        <;>\n        (try omega)\n      · \n        simp [h₃]\n        <;>\n        (try norm_num) <;>\n        (try ring_nf) <;>\n        (try simp_all)\n        <;>\n        (try omega)\n    calc\n      (∑ k : Fin n, ∑ l : Fin n, (if colors k l = 0 then (nums k l : ℤ) else 0)) - (∑ k : Fin n, ∑ l : Fin n, (if colors k l = 1 then (nums k l : ℤ) else 0)) = ∑ k : Fin n, ∑ l : Fin n, ((if colors k l = 0 then (nums k l : ℤ) else 0) - (if colors k l = 1 then (nums k l : ℤ) else 0)) := by rw [h₁]\n      _ = ∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * (nums k l : ℤ) := by\n        apply Finset.sum_congr rfl\n        intro k _\n        apply Finset.sum_congr rfl\n        intro l _\n        rw [h₂ k l]\n  \n  have hsum_split_zero : (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * (nums k l : ℤ)) = 0 := by\n    have h₁ : (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * (nums k l : ℤ)) = ∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((k : ℤ) * n + (l : ℤ) + 1) := by\n      apply Finset.sum_congr rfl\n      intro k _\n      apply Finset.sum_congr rfl\n      intro l _\n      rw [hnums]\n      <;> simp [Fin.val_mk]\n      <;> norm_cast\n      <;> ring_nf\n    rw [h₁]\n    have h₂ : (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((k : ℤ) * n + (l : ℤ) + 1)) = (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((k : ℤ) * n)) + (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((l : ℤ) + 1)) := by\n      calc\n        (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((k : ℤ) * n + (l : ℤ) + 1)) = ∑ k : Fin n, ∑ l : Fin n, ((2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((k : ℤ) * n) + (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((l : ℤ) + 1)) := by\n          apply Finset.sum_congr rfl\n          intro k _\n          apply Finset.sum_congr rfl\n          intro l _\n          ring_nf\n        _ = (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((k : ℤ) * n)) + (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((l : ℤ) + 1)) := by\n          simp [Finset.sum_add_distrib]\n    rw [h₂]\n    have h₃ : (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((k : ℤ) * n)) = 0 := by\n      have h₄ : ∀ (k : Fin n), (∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((k : ℤ) * n)) = 0 := by\n        intro k\n        have h₅ : (∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1)) = 0 := hrow_sum_zero k\n        calc\n          (∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((k : ℤ) * n)) = ∑ l : Fin n, ((k : ℤ) * n) * (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) := by\n            apply Finset.sum_congr rfl\n            intro l _\n            ring_nf\n          _ = ((k : ℤ) * n) * ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) := by\n            rw [Finset.mul_sum]\n          _ = ((k : ℤ) * n) * 0 := by rw [h₅]\n          _ = 0 := by ring\n      calc\n        (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((k : ℤ) * n)) = ∑ k : Fin n, (∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((k : ℤ) * n)) := by simp [Finset.sum_sigma']\n        _ = ∑ k : Fin n, 0 := by\n          apply Finset.sum_congr rfl\n          intro k _\n          rw [h₄ k]\n        _ = 0 := by simp\n    have h₄ : (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((l : ℤ) + 1)) = 0 := by\n      have h₅ : ∀ (l : Fin n), (∑ k : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1)) = 0 := by\n        intro l\n        exact hcol_sum_zero l\n      calc\n        (∑ k : Fin n, ∑ l : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((l : ℤ) + 1)) = ∑ l : Fin n, ∑ k : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((l : ℤ) + 1) := by\n          rw [Finset.sum_comm]\n        _ = ∑ l : Fin n, ((l : ℤ) + 1) * (∑ k : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1)) := by\n          apply Finset.sum_congr rfl\n          intro l _\n          calc\n            (∑ k : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) * ((l : ℤ) + 1)) = ∑ k : Fin n, ((l : ℤ) + 1) * (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) := by\n              apply Finset.sum_congr rfl\n              intro k _\n              ring_nf\n            _ = ((l : ℤ) + 1) * ∑ k : Fin n, (2 * (if colors k l = 0 then (1 : ℤ) else 0) - 1) := by\n              rw [Finset.mul_sum]\n        _ = ∑ l : Fin n, ((l : ℤ) + 1) * 0 := by\n          apply Finset.sum_congr rfl\n          intro l _\n          rw [h₅ l]\n        _ = ∑ l : Fin n, 0 := by\n          apply Finset.sum_congr rfl\n          intro l _\n          ring_nf\n        _ = 0 := by simp\n    rw [h₃, h₄]\n    <;> simp\n  \n  have hfinal : (∑ k, ∑ l, (if (colors k l = 0) then (nums k l) else 0)) = (∑ k, ∑ l, (if (colors k l = 1) then (nums k l) else 0)) := by\n    have h₁ : (∑ k : Fin n, ∑ l : Fin n, (if colors k l = 0 then (nums k l : ℤ) else 0)) - (∑ k : Fin n, ∑ l : Fin n, (if colors k l = 1 then (nums k l : ℤ) else 0)) = 0 := by\n      rw [hsum_diff]\n      rw [hsum_split_zero]\n    have h₂ : (∑ k : Fin n, ∑ l : Fin n, (if colors k l = 0 then (nums k l : ℤ) else 0)) = (∑ k : Fin n, ∑ l : Fin n, (if colors k l = 1 then (nums k l : ℤ) else 0)) := by\n      linarith\n    simpa using h₂\n  \n  exact hfinal"}
{"name": "putnam_1964_b2", "proof_length": 1754, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen Set Function Filter Topology\n\n\n", "signature": "theorem putnam_1964_b2\n(S : Type*) [Fintype S] [Nonempty S]\n(P : Finset (Set S))\n(hPP : ∀ T ∈ P, ∀ U ∈ P, T ∩ U ≠ ∅)\n(hPS : ¬∃ T : Set S, T ∉ P ∧ (∀ U ∈ P, T ∩ U ≠ ∅))\n: (P.card = 2 ^ (Fintype.card S - 1))", "src": "theorem putnam_1964_b2\n(S : Type*) [Fintype S] [Nonempty S]\n(P : Finset (Set S))\n(hPP : ∀ T ∈ P, ∀ U ∈ P, T ∩ U ≠ ∅)\n(hPS : ¬∃ T : Set S, T ∉ P ∧ (∀ U ∈ P, T ∩ U ≠ ∅))\n: (P.card = 2 ^ (Fintype.card S - 1)) := by \n  have h_no_empty : ∀ (T : Set S), T ∈ P → T.Nonempty := by\n    intro T hT\n    have h₁ : T ∩ T ≠ ∅ := hPP T hT T hT\n    have h₂ : T ∩ T = T := by simp [Set.ext_iff]\n    rw [h₂] at h₁\n    exact Set.nonempty_iff_ne_empty.mpr h₁\n  \n  have h_subset_or_compl : ∀ (A : Set S), A ∈ P ∨ Aᶜ ∈ P := by\n    intro A\n    by_contra! h\n    have h₁ : A ∉ P := h.1\n    have h₂ : Aᶜ ∉ P := h.2\n    \n    have h₃ : ∃ (U : Set S), U ∈ P ∧ A ∩ U = ∅ := by\n      have h₄ : ¬(A ∉ P ∧ ∀ (U : Set S), U ∈ P → A ∩ U ≠ ∅) := by\n        intro h₅\n        have h₆ : ∃ (T : Set S), T ∉ P ∧ (∀ U ∈ P, T ∩ U ≠ ∅) := by\n          refine' ⟨A, _⟩\n          constructor\n          · exact h₅.1\n          · intro U hU\n            have h₇ := h₅.2 U hU\n            have h₈ : A ∩ U ≠ ∅ := h₇\n            have h₉ : A ∩ U = U ∩ A := by\n              ext x\n              simp [Set.mem_inter_iff]\n              <;> tauto\n            have h₁₀ : U ∩ A ≠ ∅ := by\n              intro h₁₁\n              apply h₈\n              simp_all [Set.ext_iff]\n              <;> aesop\n            simpa [Set.inter_comm] using h₁₀\n        exact hPS h₆\n      by_cases h₅ : A ∉ P\n      · have h₆ : ¬(∀ (U : Set S), U ∈ P → A ∩ U ≠ ∅) := by\n          intro h₇\n          exact h₄ ⟨h₅, h₇⟩\n        push_neg at h₆\n        obtain ⟨U, hU, h₈⟩ := h₆\n        refine' ⟨U, hU, _⟩\n        have h₉ : A ∩ U = ∅ := by\n          by_contra h₁₀\n          have h₁₁ : A ∩ U ≠ ∅ := h₁₀\n          simp_all [Set.ext_iff]\n          <;> aesop\n        exact h₉\n      · exfalso\n        simp_all\n    \n    have h₄ : ∃ (V : Set S), V ∈ P ∧ Aᶜ ∩ V = ∅ := by\n      have h₅ : ¬(Aᶜ ∉ P ∧ ∀ (U : Set S), U ∈ P → Aᶜ ∩ U ≠ ∅) := by\n        intro h₆\n        have h₇ : ∃ (T : Set S), T ∉ P ∧ (∀ U ∈ P, T ∩ U ≠ ∅) := by\n          refine' ⟨Aᶜ, _⟩\n          constructor\n          · exact h₆.1\n          · intro U hU\n            have h₈ := h₆.2 U hU\n            have h₉ : Aᶜ ∩ U ≠ ∅ := h₈\n            have h₁₀ : Aᶜ ∩ U = U ∩ Aᶜ := by\n              ext x\n              simp [Set.mem_inter_iff]\n              <;> tauto\n            have h₁₁ : U ∩ Aᶜ ≠ ∅ := by\n              intro h₁₂\n              apply h₉\n              simp_all [Set.ext_iff]\n              <;> aesop\n            simpa [Set.inter_comm] using h₁₁\n        exact hPS h₇\n      by_cases h₆ : Aᶜ ∉ P\n      · have h₇ : ¬(∀ (U : Set S), U ∈ P → Aᶜ ∩ U ≠ ∅) := by\n          intro h₈\n          exact h₅ ⟨h₆, h₈⟩\n        push_neg at h₇\n        obtain ⟨V, hV, h₈⟩ := h₇\n        refine' ⟨V, hV, _⟩\n        have h₉ : Aᶜ ∩ V = ∅ := by\n          by_contra h₁₀\n          have h₁₁ : Aᶜ ∩ V ≠ ∅ := h₁₀\n          simp_all [Set.ext_iff]\n          <;> aesop\n        exact h₉\n      · exfalso\n        simp_all\n    \n    obtain ⟨U, hU, hU'⟩ := h₃\n    obtain ⟨V, hV, hV'⟩ := h₄\n    \n    have h₅ : U ⊆ Aᶜ := by\n      intro x hx\n      by_contra h₆\n      have h₇ : x ∈ A := by simp_all [Set.mem_compl_iff]\n      have h₈ : x ∈ A ∩ U := Set.mem_inter h₇ hx\n      rw [hU'] at h₈\n      simp at h₈\n      <;> aesop\n    have h₆ : V ⊆ A := by\n      intro x hx\n      by_contra h₇\n      have h₈ : x ∈ Aᶜ := by simp_all [Set.mem_compl_iff]\n      have h₉ : x ∈ Aᶜ ∩ V := Set.mem_inter h₈ hx\n      rw [hV'] at h₉\n      simp at h₉\n      <;> aesop\n    \n    have h₇ : U ∩ V = ∅ := by\n      apply Set.eq_empty_of_forall_not_mem\n      intro x hx\n      have h₈ : x ∈ U := Set.mem_of_mem_inter_left hx\n      have h₉ : x ∈ V := Set.mem_of_mem_inter_right hx\n      have h₁₀ : x ∈ Aᶜ := h₅ h₈\n      have h₁₁ : x ∈ A := h₆ h₉\n      simp_all [Set.mem_compl_iff]\n      <;> aesop\n    \n    have h₈ : U ∩ V ≠ ∅ := hPP U hU V hV\n    simp_all [Set.ext_iff]\n    <;> aesop\n  \n  have h_not_both : ∀ (A : Set S), A ∈ P → Aᶜ ∉ P := by\n    intro A hA\n    by_contra h\n    have h₁ : Aᶜ ∈ P := h\n    have h₂ : A ∩ Aᶜ = ∅ := by\n      ext x\n      simp [Set.mem_inter_iff]\n      <;> tauto\n    have h₃ : A ∩ Aᶜ ≠ ∅ := hPP A hA Aᶜ h₁\n    rw [h₂] at h₃\n    contradiction\n  \n  have h_card : P.card = 2 ^ (Fintype.card S - 1) := by\n    classical\n    \n    let Q : Finset (Set S) := (Finset.univ : Finset (Set S)) \\ P\n    \n    have h₁ : P.card = Q.card := by\n      \n      have h₂ : ∀ (A : Set S), A ∈ P → Aᶜ ∈ Q := by\n        intro A hA\n        have h₃ : Aᶜ ∈ (Finset.univ : Finset (Set S)) := Finset.mem_univ _\n        have h₄ : Aᶜ ∉ P := h_not_both A hA\n        simp only [Q, Finset.mem_sdiff] at *\n        tauto\n      have h₃ : ∀ (A : Set S), A ∈ Q → Aᶜ ∈ P := by\n        intro A hA\n        have h₄ : A ∈ (Finset.univ : Finset (Set S)) := by\n          simp only [Q, Finset.mem_sdiff] at hA\n          tauto\n        have h₅ : A ∉ P := by\n          simp only [Q, Finset.mem_sdiff] at hA\n          tauto\n        have h₆ : A ∈ (Finset.univ : Finset (Set S)) := by tauto\n        have h₇ : A ∈ P ∨ Aᶜ ∈ P := h_subset_or_compl A\n        cases h₇ with\n        | inl h₈ =>\n          exfalso\n          tauto\n        | inr h₈ =>\n          exact h₈\n      \n      have h₄ : P.card = Q.card := by\n        \n        have h₅ : P.card = Q.card := by\n          \n          have h₆ : ∀ (A : Set S), A ∈ P → Aᶜ ∈ Q := h₂\n          have h₇ : ∀ (A : Set S), A ∈ Q → Aᶜ ∈ P := h₃\n          \n          have h₈ : P.card = Q.card := by\n            \n            apply Finset.card_bij' (fun A _ => Aᶜ) (fun A _ => Aᶜ)\n            <;> simp_all [Set.ext_iff]\n            <;>\n            (try\n              {\n                aesop\n              })\n            <;>\n            (try\n              {\n                tauto\n              })\n            <;>\n            (try\n              {\n                intros\n                <;>\n                simp_all [Set.ext_iff]\n                <;>\n                tauto\n              })\n            <;>\n            (try\n              {\n                intros\n                <;>\n                simp_all [Set.ext_iff]\n                <;>\n                tauto\n              })\n          exact h₈\n        exact h₅\n      exact h₄\n    \n    have h₂ : P.card + Q.card = 2 ^ Fintype.card S := by\n      have h₃ : P.card + Q.card = (Finset.univ : Finset (Set S)).card := by\n        have h₄ : Disjoint P Q := by\n          rw [Finset.disjoint_left]\n          intro A hA hA'\n          simp only [Q, Finset.mem_sdiff] at hA'\n          tauto\n        have h₅ : P ∪ Q = (Finset.univ : Finset (Set S)) := by\n          apply Finset.ext\n          intro A\n          simp only [Q, Finset.mem_union, Finset.mem_sdiff, Finset.mem_univ, true_and]\n          <;>\n          by_cases h₆ : A ∈ P <;> simp_all [h_subset_or_compl]\n          <;>\n          (try tauto)\n          <;>\n          (try\n            {\n              have h₇ := h_subset_or_compl A\n              cases h₇ with\n              | inl h₈ => tauto\n              | inr h₈ =>\n                have h₉ : Aᶜ ∈ P := h₈\n                have h₁₀ : A ∈ P ∨ Aᶜ ∈ P := by tauto\n                tauto\n            })\n        have h₆ : P.card + Q.card = (P ∪ Q).card := by\n          rw [← Finset.card_union_add_card_inter P Q]\n          have h₇ : P ∩ Q = ∅ := Finset.disjoint_iff_inter_eq_empty.mp h₄\n          rw [h₇]\n          simp\n        rw [h₅] at h₆\n        exact h₆\n      have h₄ : (Finset.univ : Finset (Set S)).card = 2 ^ Fintype.card S := by\n        simp [Fintype.card_fun]\n        <;>\n        simp_all [Fintype.card_fun]\n        <;>\n        ring_nf\n        <;>\n        simp_all [Fintype.card_fun]\n        <;>\n        norm_num\n        <;>\n        aesop\n      linarith\n    \n    have h₃ : P.card * 2 = 2 ^ Fintype.card S := by\n      have h₄ : P.card + Q.card = 2 ^ Fintype.card S := h₂\n      have h₅ : P.card = Q.card := h₁\n      linarith\n    \n    have h₄ : Fintype.card S ≥ 1 := by\n      have h₅ : Nonempty S := inferInstance\n      have h₆ : 0 < Fintype.card S := by\n        apply Fintype.card_pos_iff.mpr\n        exact ⟨Classical.choice h₅⟩\n      omega\n    have h₅ : P.card = 2 ^ (Fintype.card S - 1) := by\n      have h₆ : P.card * 2 = 2 ^ Fintype.card S := h₃\n      have h₇ : Fintype.card S ≥ 1 := h₄\n      have h₈ : 2 ^ (Fintype.card S - 1) * 2 = 2 ^ Fintype.card S := by\n        have h₉ : Fintype.card S - 1 + 1 = Fintype.card S := by\n          have h₁₀ : Fintype.card S ≥ 1 := h₄\n          omega\n        calc\n          2 ^ (Fintype.card S - 1) * 2 = 2 ^ (Fintype.card S - 1) * 2 ^ 1 := by norm_num\n          _ = 2 ^ ((Fintype.card S - 1) + 1) := by\n            rw [← pow_add]\n          _ = 2 ^ Fintype.card S := by\n            rw [h₉]\n            <;> simp [add_comm]\n      have h₉ : P.card = 2 ^ (Fintype.card S - 1) := by\n        have h₁₀ : P.card * 2 = 2 ^ Fintype.card S := h₃\n        have h₁₁ : 2 ^ (Fintype.card S - 1) * 2 = 2 ^ Fintype.card S := h₈\n        have h₁₂ : P.card = 2 ^ (Fintype.card S - 1) := by\n          nlinarith\n        exact h₁₂\n      exact h₉\n    exact h₅\n  \n  exact h_card"}
{"name": "putnam_2007_a1", "proof_length": 5463, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\n\n\n\n", "signature": "theorem putnam_2007_a1\n    (P : (ℝ → ℝ) → Prop)\n    (P_def : ∀ f, P f ↔ ∃ x y, f x = y ∧ f y = x ∧ deriv f x * deriv f y = 1)\n    (α : ℝ) :\n    P (fun t ↦ α * t ^ 2 + α * t + 1 / 24) ↔ α ∈ (({2 / 3, 3 / 2, (13 + √601) / 12, (13 - √601) / 12}) : Set ℝ )", "src": "theorem putnam_2007_a1\n    (P : (ℝ → ℝ) → Prop)\n    (P_def : ∀ f, P f ↔ ∃ x y, f x = y ∧ f y = x ∧ deriv f x * deriv f y = 1)\n    (α : ℝ) :\n    P (fun t ↦ α * t ^ 2 + α * t + 1 / 24) ↔ α ∈ (({2 / 3, 3 / 2, (13 + √601) / 12, (13 - √601) / 12}) : Set ℝ ) :=\n  by \n  have h_main : P (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) ↔ α = 2 / 3 ∨ α = 3 / 2 ∨ α = (13 + Real.sqrt 601) / 12 ∨ α = (13 - Real.sqrt 601) / 12 := by\n    have h₁ : P (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) ↔ ∃ (x y : ℝ), (α * x ^ 2 + α * x + 1 / 24 : ℝ) = y ∧ (α * y ^ 2 + α * y + 1 / 24 : ℝ) = x ∧ (2 * α * x + α) * (2 * α * y + α) = 1 := by\n      have h₂ : P (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) ↔ ∃ x y, (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) x = y ∧ (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) y = x ∧ deriv (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) x * deriv (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) y = 1 := by\n        rw [P_def]\n        <;> simp\n      rw [h₂]\n      have h₃ : ∀ (x : ℝ), deriv (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) x = 2 * α * x + α := by\n        intro x\n        \n        have h₄ : deriv (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) x = deriv (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) x := rfl\n        rw [h₄]\n        \n        have h₅ : HasDerivAt (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) (2 * α * x + α) x := by\n          \n          have h₆ : HasDerivAt (fun t : ℝ => α * t ^ 2) (2 * α * x) x := by\n            \n            have h₇ : HasDerivAt (fun t : ℝ => t ^ 2) (2 * x) x := by\n              simpa using (hasDerivAt_pow 2 x)\n            have h₈ : HasDerivAt (fun t : ℝ => α * t ^ 2) (α * (2 * x)) x := by\n              convert HasDerivAt.const_mul α h₇ using 1 <;> ring\n            convert h₈ using 1 <;> ring\n          have h₉ : HasDerivAt (fun t : ℝ => α * t) α x := by\n            \n            have h₁₀ : HasDerivAt (fun t : ℝ => t) 1 x := by\n              simpa using hasDerivAt_id x\n            have h₁₁ : HasDerivAt (fun t : ℝ => α * t) (α * 1) x := by\n              convert HasDerivAt.const_mul α h₁₀ using 1 <;> ring\n            convert h₁₁ using 1 <;> ring\n          have h₁₂ : HasDerivAt (fun t : ℝ => (1 : ℝ) / 24) 0 x := by\n            \n            simpa using hasDerivAt_const x (1 / 24 : ℝ)\n          \n          have h₁₃ : HasDerivAt (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) (2 * α * x + α) x := by\n            convert h₆.add h₉ |>.add h₁₂ using 1 <;> ring\n          exact h₁₃\n        \n        have h₁₄ : deriv (fun t : ℝ => α * t ^ 2 + α * t + 1 / 24) x = 2 * α * x + α := by\n          apply HasDerivAt.deriv\n          exact h₅\n        rw [h₁₄]\n      \n      constructor\n      · intro h\n        obtain ⟨x, y, h₁₅, h₁₆, h₁₇⟩ := h\n        refine' ⟨x, y, _, _, _⟩ <;>\n          (try simp_all [h₃]) <;>\n          (try ring_nf at * <;> simp_all) <;>\n          (try nlinarith)\n      · intro h\n        obtain ⟨x, y, h₁₅, h₁₆, h₁₇⟩ := h\n        refine' ⟨x, y, _, _, _⟩ <;>\n          (try simp_all [h₃]) <;>\n          (try ring_nf at * <;> simp_all) <;>\n          (try nlinarith)\n    rw [h₁]\n    constructor\n    · \n      intro h\n      have h₂ : ∃ (x y : ℝ), (α * x ^ 2 + α * x + 1 / 24 : ℝ) = y ∧ (α * y ^ 2 + α * y + 1 / 24 : ℝ) = x ∧ (2 * α * x + α) * (2 * α * y + α) = 1 := h\n      obtain ⟨x, y, h₃, h₄, h₅⟩ := h₂\n      have h₆ : (x - y) * (α * (x + y + 1) + 1) = 0 := by\n        have h₇ : α * x ^ 2 + α * x + 1 / 24 = y := h₃\n        have h₈ : α * y ^ 2 + α * y + 1 / 24 = x := h₄\n        have h₉ : α * (x ^ 2 - y ^ 2) + α * (x - y) = (y - x) := by\n          linarith\n        have h₁₀ : α * (x - y) * (x + y) + α * (x - y) = -(x - y) := by\n          linarith\n        have h₁₁ : (x - y) * (α * (x + y) + α + 1) = 0 := by\n          linarith\n        have h₁₂ : (x - y) * (α * (x + y + 1) + 1) = 0 := by\n          linarith\n        exact h₁₂\n      by_cases h₇ : x = y\n      · \n        have h₈ : α * x ^ 2 + α * x + 1 / 24 = x := by\n          linarith\n        have h₉ : (2 * α * x + α) * (2 * α * x + α) = 1 := by\n          have h₁₀ : (2 * α * x + α) * (2 * α * y + α) = 1 := h₅\n          have h₁₁ : (2 * α * x + α) * (2 * α * x + α) = 1 := by\n            simpa [h₇] using h₁₀\n          exact h₁₁\n        have h₁₀ : (2 * α * x + α) = 1 ∨ (2 * α * x + α) = -1 := by\n          have h₁₁ : (2 * α * x + α) ^ 2 = 1 := by\n            nlinarith\n          have h₁₂ : (2 * α * x + α) = 1 ∨ (2 * α * x + α) = -1 := by\n            apply or_iff_not_imp_left.mpr\n            intro h₁₃\n            apply eq_of_sub_eq_zero\n            apply mul_left_cancel₀ (sub_ne_zero.mpr h₁₃)\n            nlinarith\n          exact h₁₂\n        cases h₁₀ with\n        | inl h₁₀ =>\n          \n          have h₁₁ : 2 * α * x + α = 1 := h₁₀\n          have h₁₂ : α ≠ 0 := by\n            by_contra h₁₃\n            rw [h₁₃] at h₁₁\n            norm_num at h₁₁\n            <;> linarith\n          have h₁₃ : x = (1 - α) / (2 * α) := by\n            have h₁₄ : 2 * α * x + α = 1 := h₁₁\n            field_simp at h₁₄ ⊢\n            nlinarith\n          have h₁₄ : α * x ^ 2 + (α - 1) * x + 1 / 24 = 0 := by\n            have h₁₅ : α * x ^ 2 + α * x + 1 / 24 = x := h₈\n            linarith\n          rw [h₁₃] at h₁₄\n          have h₁₅ : α * ((1 - α) / (2 * α)) ^ 2 + (α - 1) * ((1 - α) / (2 * α)) + 1 / 24 = 0 := h₁₄\n          have h₁₆ : 6 * α ^ 2 - 13 * α + 6 = 0 := by\n            field_simp at h₁₅\n            ring_nf at h₁₅ ⊢\n            nlinarith [sq_pos_of_ne_zero h₁₂]\n          have h₁₇ : α = 2 / 3 ∨ α = 3 / 2 := by\n            have h₁₈ : (α - 2 / 3) * (α - 3 / 2) = 0 := by\n              nlinarith\n            have h₁₉ : α - 2 / 3 = 0 ∨ α - 3 / 2 = 0 := by\n              apply eq_zero_or_eq_zero_of_mul_eq_zero h₁₈\n            cases h₁₉ with\n            | inl h₁₉ =>\n              have h₂₀ : α = 2 / 3 := by linarith\n              exact Or.inl h₂₀\n            | inr h₁₉ =>\n              have h₂₀ : α = 3 / 2 := by linarith\n              exact Or.inr h₂₀\n          cases h₁₇ with\n          | inl h₁₇ =>\n            exact Or.inl h₁₇\n          | inr h₁₇ =>\n            exact Or.inr (Or.inl h₁₇)\n        | inr h₁₀ =>\n          \n          have h₁₁ : 2 * α * x + α = -1 := h₁₀\n          have h₁₂ : α ≠ 0 := by\n            by_contra h₁₃\n            rw [h₁₃] at h₁₁\n            norm_num at h₁₁\n            <;> linarith\n          have h₁₃ : x = (-1 - α) / (2 * α) := by\n            have h₁₄ : 2 * α * x + α = -1 := h₁₁\n            field_simp at h₁₄ ⊢\n            nlinarith\n          have h₁₄ : α * x ^ 2 + (α - 1) * x + 1 / 24 = 0 := by\n            have h₁₅ : α * x ^ 2 + α * x + 1 / 24 = x := h₈\n            linarith\n          rw [h₁₃] at h₁₄\n          have h₁₅ : α * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = 0 := h₁₄\n          have h₁₆ : 6 * α ^ 2 - 13 * α - 18 = 0 := by\n            field_simp at h₁₅\n            ring_nf at h₁₅ ⊢\n            nlinarith [sq_pos_of_ne_zero h₁₂]\n          have h₁₇ : α = (13 + Real.sqrt 601) / 12 ∨ α = (13 - Real.sqrt 601) / 12 := by\n            have h₁₈ : α = (13 + Real.sqrt 601) / 12 ∨ α = (13 - Real.sqrt 601) / 12 := by\n              have h₁₉ : α = (13 + Real.sqrt 601) / 12 ∨ α = (13 - Real.sqrt 601) / 12 := by\n                have h₂₀ : (α - (13 + Real.sqrt 601) / 12) * (α - (13 - Real.sqrt 601) / 12) = 0 := by\n                  have h₂₁ : Real.sqrt 601 ≥ 0 := Real.sqrt_nonneg _\n                  nlinarith [Real.sq_sqrt (show 0 ≤ 601 by norm_num)]\n                have h₂₂ : α - (13 + Real.sqrt 601) / 12 = 0 ∨ α - (13 - Real.sqrt 601) / 12 = 0 := by\n                  apply eq_zero_or_eq_zero_of_mul_eq_zero h₂₀\n                cases h₂₂ with\n                | inl h₂₂ =>\n                  have h₂₃ : α = (13 + Real.sqrt 601) / 12 := by linarith\n                  exact Or.inl h₂₃\n                | inr h₂₂ =>\n                  have h₂₃ : α = (13 - Real.sqrt 601) / 12 := by linarith\n                  exact Or.inr h₂₃\n              exact h₁₉\n            exact h₁₈\n          cases h₁₇ with\n          | inl h₁₇ =>\n            exact Or.inr (Or.inr (Or.inl h₁₇))\n          | inr h₁₇ =>\n            exact Or.inr (Or.inr (Or.inr h₁₇))\n      · \n        have h₈ : α * (x + y + 1) + 1 = 0 := by\n          apply mul_left_cancel₀ (sub_ne_zero.mpr h₇)\n          nlinarith\n        have h₉ : (2 * α * x + α) + (2 * α * y + α) = -2 := by\n          have h₁₀ : α * (x + y + 1) + 1 = 0 := h₈\n          have h₁₁ : α * (x + y + 1) = -1 := by linarith\n          have h₁₂ : 2 * α * (x + y + 1) = -2 := by linarith\n          have h₁₃ : 2 * α * x + 2 * α * y + 2 * α = -2 := by linarith\n          linarith\n        have h₁₀ : (2 * α * x + α) * (2 * α * y + α) = 1 := h₅\n        have h₁₁ : (2 * α * x + α) = -1 := by\n          have h₁₂ : (2 * α * x + α) + (2 * α * y + α) = -2 := h₉\n          have h₁₃ : (2 * α * x + α) * (2 * α * y + α) = 1 := h₁₀\n          have h₁₄ : (2 * α * x + α) = -1 := by\n            nlinarith [sq_nonneg ((2 * α * x + α) - (2 * α * y + α))]\n          exact h₁₄\n        have h₁₂ : (2 * α * y + α) = -1 := by\n          have h₁₃ : (2 * α * x + α) + (2 * α * y + α) = -2 := h₉\n          have h₁₄ : (2 * α * x + α) = -1 := h₁₁\n          linarith\n        have h₁₃ : 2 * α * x + α = -1 := h₁₁\n        have h₁₄ : 2 * α * y + α = -1 := h₁₂\n        have h₁₅ : x = y := by\n          have h₁₆ : 2 * α * x + α = 2 * α * y + α := by linarith\n          have h₁₇ : 2 * α * x = 2 * α * y := by linarith\n          by_cases h₁₈ : α = 0\n          · have h₁₉ : α = 0 := h₁₈\n            rw [h₁₉] at h₁₃\n            norm_num at h₁₃\n            <;> linarith\n          · have h₁₉ : α ≠ 0 := h₁₈\n            have h₂₀ : x = y := by\n              apply mul_left_cancel₀ (show (2 : ℝ) * α ≠ 0 by\n                intro h₂₁\n                apply h₁₉\n                linarith)\n              linarith\n            exact h₂₀\n        contradiction\n    · \n      intro h\n      have h₂ : α = 2 / 3 ∨ α = 3 / 2 ∨ α = (13 + Real.sqrt 601) / 12 ∨ α = (13 - Real.sqrt 601) / 12 := h\n      cases h₂ with\n      | inl h₂ =>\n        \n        have h₃ : α = 2 / 3 := h₂\n        have h₄ : ∃ (x y : ℝ), (α * x ^ 2 + α * x + 1 / 24 : ℝ) = y ∧ (α * y ^ 2 + α * y + 1 / 24 : ℝ) = x ∧ (2 * α * x + α) * (2 * α * y + α) = 1 := by\n          use 1 / 4, 1 / 4\n          rw [h₃]\n          norm_num\n          <;>\n          (try ring_nf) <;>\n          (try norm_num) <;>\n          (try linarith)\n        exact h₄\n      | inr h₂ =>\n        cases h₂ with\n        | inl h₂ =>\n          \n          have h₃ : α = 3 / 2 := h₂\n          have h₄ : ∃ (x y : ℝ), (α * x ^ 2 + α * x + 1 / 24 : ℝ) = y ∧ (α * y ^ 2 + α * y + 1 / 24 : ℝ) = x ∧ (2 * α * x + α) * (2 * α * y + α) = 1 := by\n            use -1 / 6, -1 / 6\n            rw [h₃]\n            norm_num\n            <;>\n            (try ring_nf) <;>\n            (try norm_num) <;>\n            (try linarith)\n          exact h₄\n        | inr h₂ =>\n          cases h₂ with\n          | inl h₂ =>\n            \n            have h₃ : α = (13 + Real.sqrt 601) / 12 := h₂\n            have h₄ : ∃ (x y : ℝ), (α * x ^ 2 + α * x + 1 / 24 : ℝ) = y ∧ (α * y ^ 2 + α * y + 1 / 24 : ℝ) = x ∧ (2 * α * x + α) * (2 * α * y + α) = 1 := by\n              have h₅ : α ≠ 0 := by\n                rw [h₃]\n                nlinarith [Real.sqrt_nonneg 601, Real.sq_sqrt (show 0 ≤ 601 by norm_num)]\n              use (-1 - α) / (2 * α), (-1 - α) / (2 * α)\n              constructor\n              · \n                have h₆ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                  have h₇ : α ≠ 0 := h₅\n                  have h₈ : 6 * α ^ 2 - 13 * α - 18 = 0 := by\n                    rw [h₃]\n                    nlinarith [Real.sq_sqrt (show (0 : ℝ) ≤ 601 by norm_num)]\n                  have h₉ : (3 + 2 * α - α ^ 2) / (4 * α) + 1 / 24 = 0 := by\n                    have h₁₀ : 6 * α ^ 2 = 13 * α + 18 := by linarith\n                    field_simp [h₇]\n                    nlinarith\n                  have h₁₀ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = 0 := by\n                    have h₁₁ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = (3 + 2 * α - α ^ 2) / (4 * α) + 1 / 24 := by\n                      field_simp [h₇]\n                      <;> ring_nf\n                      <;> field_simp [h₇]\n                      <;> ring_nf\n                    rw [h₁₁]\n                    linarith\n                  have h₁₁ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                    have h₁₂ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = 0 := h₁₀\n                    have h₁₃ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                      linarith\n                    exact h₁₃\n                  exact h₁₁\n                exact h₆\n              · constructor\n                · \n                  have h₆ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                    have h₇ : α ≠ 0 := h₅\n                    have h₈ : 6 * α ^ 2 - 13 * α - 18 = 0 := by\n                      rw [h₃]\n                      nlinarith [Real.sq_sqrt (show 0 ≤ 601 by norm_num)]\n                    have h₉ : (3 + 2 * α - α ^ 2) / (4 * α) + 1 / 24 = 0 := by\n                      have h₁₀ : 6 * α ^ 2 = 13 * α + 18 := by linarith\n                      field_simp [h₇]\n                      nlinarith\n                    have h₁₀ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = 0 := by\n                      have h₁₁ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = (3 + 2 * α - α ^ 2) / (4 * α) + 1 / 24 := by\n                        field_simp [h₇]\n                        <;> ring_nf\n                        <;> field_simp [h₇]\n                        <;> ring_nf\n                      rw [h₁₁]\n                      linarith\n                    have h₁₁ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                      have h₁₂ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = 0 := h₁₀\n                      have h₁₃ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                        linarith\n                      exact h₁₃\n                    exact h₁₁\n                  linarith\n                · \n                  have h₆ : (2 * α * ((-1 - α) / (2 * α)) + α) * (2 * α * ((-1 - α) / (2 * α)) + α) = 1 := by\n                    have h₇ : α ≠ 0 := h₅\n                    field_simp [h₇]\n                    <;> ring_nf\n                    <;> nlinarith [Real.sqrt_nonneg 601, Real.sq_sqrt (show 0 ≤ 601 by norm_num)]\n                  exact h₆\n            exact h₄\n          | inr h₂ =>\n            \n            have h₃ : α = (13 - Real.sqrt 601) / 12 := h₂\n            have h₄ : ∃ (x y : ℝ), (α * x ^ 2 + α * x + 1 / 24 : ℝ) = y ∧ (α * y ^ 2 + α * y + 1 / 24 : ℝ) = x ∧ (2 * α * x + α) * (2 * α * y + α) = 1 := by\n              have h₅ : α ≠ 0 := by\n                rw [h₃]\n                nlinarith [Real.sqrt_nonneg 601, Real.sq_sqrt (show 0 ≤ 601 by norm_num)]\n              use (-1 - α) / (2 * α), (-1 - α) / (2 * α)\n              constructor\n              · \n                have h₆ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                  have h₇ : α ≠ 0 := h₅\n                  have h₈ : 6 * α ^ 2 - 13 * α - 18 = 0 := by\n                    rw [h₃]\n                    nlinarith [Real.sq_sqrt (show (0 : ℝ) ≤ 601 by norm_num)]\n                  have h₉ : (3 + 2 * α - α ^ 2) / (4 * α) + 1 / 24 = 0 := by\n                    have h₁₀ : 6 * α ^ 2 = 13 * α + 18 := by linarith\n                    field_simp [h₇]\n                    nlinarith\n                  have h₁₀ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = 0 := by\n                    have h₁₁ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = (3 + 2 * α - α ^ 2) / (4 * α) + 1 / 24 := by\n                      field_simp [h₇]\n                      <;> ring_nf\n                      <;> field_simp [h₇]\n                      <;> ring_nf\n                    rw [h₁₁]\n                    linarith\n                  have h₁₁ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                    have h₁₂ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = 0 := h₁₀\n                    have h₁₃ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                      linarith\n                    exact h₁₃\n                  exact h₁₁\n                exact h₆\n              · constructor\n                · \n                  have h₆ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                    have h₇ : α ≠ 0 := h₅\n                    have h₈ : 6 * α ^ 2 - 13 * α - 18 = 0 := by\n                      rw [h₃]\n                      nlinarith [Real.sq_sqrt (show 0 ≤ 601 by norm_num)]\n                    have h₉ : (3 + 2 * α - α ^ 2) / (4 * α) + 1 / 24 = 0 := by\n                      have h₁₀ : 6 * α ^ 2 = 13 * α + 18 := by linarith\n                      field_simp [h₇]\n                      nlinarith\n                    have h₁₀ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = 0 := by\n                      have h₁₁ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = (3 + 2 * α - α ^ 2) / (4 * α) + 1 / 24 := by\n                        field_simp [h₇]\n                        <;> ring_nf\n                        <;> field_simp [h₇]\n                        <;> ring_nf\n                      rw [h₁₁]\n                      linarith\n                    have h₁₁ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                      have h₁₂ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + (α - 1) * ((-1 - α) / (2 * α)) + 1 / 24 = 0 := h₁₀\n                      have h₁₃ : (α : ℝ) * ((-1 - α) / (2 * α)) ^ 2 + α * ((-1 - α) / (2 * α)) + 1 / 24 = (-1 - α) / (2 * α) := by\n                        linarith\n                      exact h₁₃\n                    exact h₁₁\n                  linarith\n                · \n                  have h₆ : (2 * α * ((-1 - α) / (2 * α)) + α) * (2 * α * ((-1 - α) / (2 * α)) + α) = 1 := by\n                    have h₇ : α ≠ 0 := h₅\n                    field_simp [h₇]\n                    <;> ring_nf\n                    <;> nlinarith [Real.sqrt_nonneg 601, Real.sq_sqrt (show 0 ≤ 601 by norm_num)]\n                  exact h₆\n            exact h₄\n  \n  have h_final : (α = 2 / 3 ∨ α = 3 / 2 ∨ α = (13 + Real.sqrt 601) / 12 ∨ α = (13 - Real.sqrt 601) / 12) ↔ α ∈ (({2 / 3, 3 / 2, (13 + √601) / 12, (13 - √601) / 12}) : Set ℝ) := by\n    constructor\n    · \n      intro h\n      simp only [Set.mem_insert_iff, Set.mem_singleton_iff]\n      tauto\n    · \n      intro h\n      simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at h\n      tauto\n  \n  have h_result : P (fun t ↦ α * t ^ 2 + α * t + 1 / 24) ↔ α ∈ (({2 / 3, 3 / 2, (13 + √601) / 12, (13 - √601) / 12}) : Set ℝ) := by\n    rw [h_main]\n    rw [h_final]\n    <;>\n    simp_all\n    <;>\n    tauto\n  \n  exact h_result"}
{"name": "putnam_1970_b5", "proof_length": 2134, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen Metric Set EuclideanGeometry Filter Topology\n\n\n", "signature": "theorem putnam_1970_b5\n(ramp : ℤ → (ℝ → ℝ))\n(ramp_def : ramp = fun (n : ℤ) => (fun (x : ℝ) => if x ≤ -n then (-n : ℝ) else (if -n < x ∧ x ≤ n then x else (n : ℝ))))\n(F : ℝ → ℝ)\n: Continuous F ↔ (∀ n : ℕ, Continuous ((ramp n) ∘ F))", "src": "theorem putnam_1970_b5\n(ramp : ℤ → (ℝ → ℝ))\n(ramp_def : ramp = fun (n : ℤ) => (fun (x : ℝ) => if x ≤ -n then (-n : ℝ) else (if -n < x ∧ x ≤ n then x else (n : ℝ))))\n(F : ℝ → ℝ)\n: Continuous F ↔ (∀ n : ℕ, Continuous ((ramp n) ∘ F)) := by \n  have h_ramp_def : ∀ (n : ℤ) (x : ℝ), ramp n x = if x ≤ -n then (-n : ℝ) else (if -n < x ∧ x ≤ n then x else (n : ℝ)) := by\n    intro n x\n    rw [ramp_def]\n    <;> simp [n]\n    <;> aesop\n  \n  have h_imp : Continuous F → (∀ n : ℕ, Continuous ((ramp n) ∘ F)) := by\n    intro hF\n    intro n\n    have h₁ : Continuous (ramp n) := by\n      have h₂ : Continuous (fun x : ℝ => min (max x (-(n : ℝ))) (n : ℝ)) := by\n        continuity\n      have h₃ : (ramp n) = (fun x : ℝ => min (max x (-(n : ℝ))) (n : ℝ)) := by\n        funext x\n        have h₄ : ramp n x = if x ≤ (-(n : ℝ)) then (-(n : ℝ)) else (if (-(n : ℝ) : ℝ) < x ∧ x ≤ (n : ℝ) then x else (n : ℝ)) := by\n          have h₅ := h_ramp_def (n : ℤ) x\n          simp [Int.cast_neg, Int.cast_ofNat] at h₅ ⊢\n          <;>\n          (try norm_num at h₅ ⊢) <;>\n          (try ring_nf at h₅ ⊢) <;>\n          (try simp_all) <;>\n          (try linarith)\n          <;>\n          (try\n            {\n              split_ifs at h₅ ⊢ <;>\n              simp_all [Int.cast_neg, Int.cast_ofNat] <;>\n              norm_num at * <;>\n              (try linarith)\n            })\n        rw [h₄]\n        split_ifs <;>\n          (try {\n            simp_all [max_eq_left, max_eq_right, min_eq_left, min_eq_right, le_of_lt]\n            <;>\n            (try { nlinarith }) <;>\n            (try { linarith })\n          }) <;>\n          (try {\n            cases' le_total x (n : ℝ) with h₆ h₆ <;>\n              simp_all [max_eq_left, max_eq_right, min_eq_left, min_eq_right, le_of_lt] <;>\n              (try { nlinarith }) <;>\n              (try { linarith })\n          })\n      rw [h₃]\n      exact h₂\n    \n    exact h₁.comp hF\n  \n  have h_conv : (∀ n : ℕ, Continuous ((ramp n) ∘ F)) → Continuous F := by\n    intro h\n    have h₁ : Continuous F := by\n      have h₂ : ∀ (x₀ : ℝ), ContinuousAt F x₀ := by\n        intro x₀\n        \n        have h₃ : ∀ (ε : ℝ), ε > 0 → (∃ (δ : ℝ), δ > 0 ∧ ∀ (x : ℝ), |x - x₀| < δ → |F x - F x₀| < ε) := by\n          intro ε εpos\n          \n          have h₄ : ∃ (n : ℕ), (n : ℝ) > |F x₀| + ε := by\n            obtain ⟨n, hn⟩ := exists_nat_gt (|F x₀| + ε)\n            refine' ⟨n, _⟩\n            exact_mod_cast hn\n          obtain ⟨n, hn⟩ := h₄\n          have h₅ : (n : ℝ) > |F x₀| + ε := hn\n          have h₆ : Continuous ((ramp n) ∘ F) := h n\n          have h₇ : ContinuousAt ((ramp n) ∘ F) x₀ := h₆.continuousAt\n          \n          have h₈ : ∀ (ε : ℝ), ε > 0 → (∃ (δ : ℝ), δ > 0 ∧ ∀ (x : ℝ), |x - x₀| < δ → |(ramp n ∘ F) x - (ramp n ∘ F) x₀| < ε) := by\n            intro ε εpos\n            have h₉ : ContinuousAt ((ramp n) ∘ F) x₀ := h₇\n            exact Metric.continuousAt_iff.mp h₉ ε εpos\n          have h₉ : ∃ (δ : ℝ), δ > 0 ∧ ∀ (x : ℝ), |x - x₀| < δ → |(ramp n ∘ F) x - (ramp n ∘ F) x₀| < ε := h₈ ε εpos\n          obtain ⟨δ, δpos, hδ⟩ := h₉\n          use δ\n          constructor\n          · exact δpos\n          · intro x hx\n            have h₁₀ : |(ramp n ∘ F) x - (ramp n ∘ F) x₀| < ε := hδ x hx\n            have h₁₁ : (ramp n ∘ F) x = ramp n (F x) := rfl\n            have h₁₂ : (ramp n ∘ F) x₀ = ramp n (F x₀) := rfl\n            rw [h₁₁, h₁₂] at h₁₀\n            \n            have h₁₃ : ramp n (F x₀) = F x₀ := by\n              have h₁₄ : (F x₀ : ℝ) ≤ (n : ℝ) := by\n                have h₁₅ : |F x₀| + ε < (n : ℝ) := by linarith\n                have h₁₆ : |F x₀| < (n : ℝ) := by linarith\n                have h₁₇ : F x₀ ≤ |F x₀| := by\n                  cases' abs_cases (F x₀) with h₁₈ h₁₈ <;> linarith\n                linarith\n              have h₁₅ : (F x₀ : ℝ) > (-(n : ℝ)) := by\n                have h₁₆ : |F x₀| + ε < (n : ℝ) := by linarith\n                have h₁₇ : |F x₀| < (n : ℝ) := by linarith\n                have h₁₈ : - (n : ℝ) < F x₀ := by\n                  cases' abs_cases (F x₀) with h₁₉ h₁₉ <;> linarith\n                linarith\n              have h₁₆ : (F x₀ : ℝ) > (-(n : ℝ)) ∧ (F x₀ : ℝ) ≤ (n : ℝ) := ⟨h₁₅, h₁₄⟩\n              have h₁₇ : ramp n (F x₀) = (F x₀ : ℝ) := by\n                have h₁₈ : ramp n (F x₀) = if (F x₀ : ℝ) ≤ (-(n : ℝ)) then (-(n : ℝ)) else (if (-(n : ℝ)) < (F x₀ : ℝ) ∧ (F x₀ : ℝ) ≤ (n : ℝ) then (F x₀ : ℝ) else (n : ℝ)) := by\n                  rw [h_ramp_def]\n                  <;> simp [Int.cast_neg, Int.cast_ofNat]\n                  <;> norm_num\n                  <;>\n                  (try\n                    {\n                      ring_nf\n                      <;>\n                      norm_cast\n                    })\n                  <;>\n                  (try\n                    {\n                      simp_all [Int.cast_neg, Int.cast_ofNat]\n                      <;>\n                      norm_num\n                      <;>\n                      linarith\n                    })\n                rw [h₁₈]\n                have h₁₉ : ¬((F x₀ : ℝ) ≤ (-(n : ℝ))) := by\n                  intro h₂₀\n                  have h₂₁ : (F x₀ : ℝ) ≤ (-(n : ℝ)) := h₂₀\n                  have h₂₂ : (-(n : ℝ)) < (F x₀ : ℝ) := by\n                    have h₂₃ : |F x₀| + ε < (n : ℝ) := by linarith\n                    have h₂₄ : |F x₀| < (n : ℝ) := by linarith\n                    cases' abs_cases (F x₀) with h₂₅ h₂₅ <;> linarith\n                  linarith\n                have h₂₀ : (-(n : ℝ)) < (F x₀ : ℝ) ∧ (F x₀ : ℝ) ≤ (n : ℝ) := h₁₆\n                split_ifs at * <;> simp_all [h₁₉] <;>\n                  (try { contradiction }) <;>\n                  (try { linarith })\n              rw [h₁₇]\n            have h₁₄ : ramp n (F x) = F x := by\n              by_contra h₁₅\n              have h₁₆ : ramp n (F x) ≠ F x := h₁₅\n              have h₁₇ : ramp n (F x) = (-(n : ℝ)) ∨ ramp n (F x) = (n : ℝ) := by\n                have h₁₈ : ramp n (F x) = if (F x : ℝ) ≤ (-(n : ℝ)) then (-(n : ℝ)) else (if (-(n : ℝ)) < (F x : ℝ) ∧ (F x : ℝ) ≤ (n : ℝ) then (F x : ℝ) else (n : ℝ)) := by\n                  rw [h_ramp_def]\n                  <;> simp [Int.cast_neg, Int.cast_ofNat]\n                  <;> norm_num\n                  <;>\n                  (try\n                    {\n                      ring_nf\n                      <;>\n                      norm_cast\n                    })\n                  <;>\n                  (try\n                    {\n                      simp_all [Int.cast_neg, Int.cast_ofNat]\n                      <;>\n                      norm_num\n                      <;>\n                      linarith\n                    })\n                rw [h₁₈] at *\n                split_ifs at * <;>\n                  (try { simp_all }) <;>\n                  (try {\n                    left\n                    <;>\n                    linarith\n                  }) <;>\n                  (try {\n                    right\n                    <;>\n                    linarith\n                  }) <;>\n                  (try {\n                    exfalso\n                    apply h₁₆\n                    <;>\n                    simp_all\n                  })\n              cases h₁₇ with\n              | inl h₁₇ =>\n                have h₁₈ : ramp n (F x) = (-(n : ℝ)) := h₁₇\n                have h₁₉ : |(ramp n (F x) : ℝ) - (ramp n (F x₀) : ℝ)| < ε := by simpa [h₁₃] using h₁₀\n                have h₂₀ : |(-(n : ℝ) : ℝ) - (F x₀ : ℝ)| < ε := by simpa [h₁₈, h₁₃] using h₁₉\n                have h₂₁ : (n : ℝ) > |F x₀| + ε := by exact_mod_cast hn\n                have h₂₂ : |(-(n : ℝ) : ℝ) - (F x₀ : ℝ)| ≥ (n : ℝ) - |F x₀| := by\n                  cases' abs_cases (F x₀) with h₂₃ h₂₃ <;>\n                    cases' abs_cases (-(n : ℝ) - F x₀) with h₂₄ h₂₄ <;>\n                      linarith\n                have h₂₃ : (n : ℝ) - |F x₀| > ε := by linarith\n                linarith\n              | inr h₁₇ =>\n                have h₁₈ : ramp n (F x) = (n : ℝ) := h₁₇\n                have h₁₉ : |(ramp n (F x) : ℝ) - (ramp n (F x₀) : ℝ)| < ε := by simpa [h₁₃] using h₁₀\n                have h₂₀ : |(n : ℝ) - (F x₀ : ℝ)| < ε := by simpa [h₁₈, h₁₃] using h₁₉\n                have h₂₁ : (n : ℝ) > |F x₀| + ε := by exact_mod_cast hn\n                have h₂₂ : |(n : ℝ) - (F x₀ : ℝ)| ≥ (n : ℝ) - |F x₀| := by\n                  cases' abs_cases (F x₀) with h₂₃ h₂₃ <;>\n                    cases' abs_cases ((n : ℝ) - F x₀) with h₂₄ h₂₄ <;>\n                      linarith\n                have h₂₃ : (n : ℝ) - |F x₀| > ε := by linarith\n                linarith\n            have h₁₅ : |F x - F x₀| < ε := by\n              have h₁₆ : ramp n (F x) = F x := h₁₄\n              have h₁₇ : ramp n (F x₀) = F x₀ := h₁₃\n              have h₁₈ : |(ramp n (F x) : ℝ) - (ramp n (F x₀) : ℝ)| < ε := by simpa [h₁₆, h₁₇] using h₁₀\n              simpa [h₁₆, h₁₇] using h₁₈\n            exact h₁₅\n          <;>\n          simp_all [abs_lt]\n          <;>\n          linarith\n        \n        have h₄ : ContinuousAt F x₀ := by\n          rw [Metric.continuousAt_iff]\n          intro ε εpos\n          obtain ⟨δ, δpos, hδ⟩ := h₃ ε εpos\n          refine' ⟨δ, δpos, _⟩\n          intro x hx\n          have h₅ : |F x - F x₀| < ε := hδ x hx\n          exact h₅\n        exact h₄\n      \n      exact continuous_iff_continuousAt.mpr h₂\n    exact h₁\n  \n  constructor\n  · intro h\n    exact h_imp h\n  · intro h\n    exact h_conv h"}
{"name": "putnam_1968_a1", "proof_length": 1084, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\n\n", "signature": "theorem putnam_1968_a1\n: 22/7 - Real.pi = ∫ x in (0)..1, x^4 * (1 - x)^4 / (1 + x^2)", "src": "theorem putnam_1968_a1\n: 22/7 - Real.pi = ∫ x in (0)..1, x^4 * (1 - x)^4 / (1 + x^2) := by \n  have h_main : (∫ x in (0)..1, x^4 * (1 - x)^4 / (1 + x^2)) = 22/7 - Real.pi := by\n    have h₁ : (∫ x in (0)..1, x^4 * (1 - x)^4 / (1 + x^2)) = (∫ x in (0)..1, (x^6 - 4*x^5 + 5*x^4 - 4*x^2 + 4 : ℝ) - 4 / (1 + x^2)) := by\n      have h₁₁ : ∀ (x : ℝ), x^4 * (1 - x)^4 / (1 + x^2) = (x^6 - 4*x^5 + 5*x^4 - 4*x^2 + 4 : ℝ) - 4 / (1 + x^2) := by\n        intro x\n        have h₁₂ : (1 + x^2 : ℝ) ≠ 0 := by nlinarith\n        have h₁₃ : x^4 * (1 - x)^4 = (x^6 - 4*x^5 + 5*x^4 - 4*x^2 + 4 : ℝ) * (1 + x^2) - 4 := by\n          ring_nf\n          <;>\n          nlinarith [sq_nonneg (x ^ 2), sq_nonneg (x ^ 3), sq_nonneg (x - 1), sq_nonneg (x + 1)]\n        have h₁₄ : x^4 * (1 - x)^4 / (1 + x^2) = ((x^6 - 4*x^5 + 5*x^4 - 4*x^2 + 4 : ℝ) * (1 + x^2) - 4) / (1 + x^2) := by\n          rw [h₁₃]\n        rw [h₁₄]\n        field_simp [h₁₂]\n        <;> ring_nf\n        <;> field_simp [h₁₂]\n        <;> ring_nf\n      \n      congr\n      ext x\n      rw [h₁₁ x]\n    rw [h₁]\n    \n    have h₂ : (∫ x in (0)..1, (x^6 - 4*x^5 + 5*x^4 - 4*x^2 + 4 : ℝ) - 4 / (1 + x^2)) = (∫ x in (0)..1, (x^6 - 4*x^5 + 5*x^4 - 4*x^2 + 4 : ℝ)) - (∫ x in (0)..1, (4 : ℝ) / (1 + x^2)) := by\n      apply intervalIntegral.integral_sub\n      · \n        apply Continuous.intervalIntegrable\n        continuity\n      · \n        apply Continuous.intervalIntegrable\n        \n        have h₃ : Continuous (fun x : ℝ => (4 : ℝ) / (1 + x ^ 2)) := by\n          apply Continuous.div\n          · exact continuous_const\n          · exact Continuous.add continuous_const (continuous_pow 2)\n          · intro x\n            have h₄ : (1 + x ^ 2 : ℝ) ≠ 0 := by nlinarith\n            exact h₄\n        exact h₃\n    rw [h₂]\n    \n    have h₃ : (∫ x in (0)..1, (x^6 - 4*x^5 + 5*x^4 - 4*x^2 + 4 : ℝ)) = (22 / 7 : ℝ) := by\n      norm_num [integral_id, mul_comm]\n      <;>\n      ring_nf\n      <;>\n      norm_num\n      <;>\n      linarith [Real.pi_pos]\n    \n    have h₄ : (∫ x in (0)..1, (4 : ℝ) / (1 + x^2)) = Real.pi := by\n      have h₄₁ : (∫ x in (0)..1, (4 : ℝ) / (1 + x ^ 2)) = 4 * (∫ x in (0)..1, (1 : ℝ) / (1 + x ^ 2)) := by\n        \n        have h₄₂ : (∫ x in (0)..1, (4 : ℝ) / (1 + x ^ 2)) = (∫ x in (0)..1, 4 * (1 : ℝ) / (1 + x ^ 2)) := by\n          congr\n          ext x\n          <;> ring_nf\n        rw [h₄₂]\n        have h₄₃ : (∫ x in (0)..1, 4 * (1 : ℝ) / (1 + x ^ 2)) = 4 * (∫ x in (0)..1, (1 : ℝ) / (1 + x ^ 2)) := by\n          \n          simp [intervalIntegral.integral_comp_mul_left (fun x => (1 : ℝ) / (1 + x ^ 2))]\n          <;>\n          norm_num\n          <;>\n          field_simp\n          <;>\n          ring_nf\n          <;>\n          norm_num\n          <;>\n          linarith [Real.pi_pos]\n        rw [h₄₃]\n      rw [h₄₁]\n      have h₄₄ : (∫ x in (0)..1, (1 : ℝ) / (1 + x ^ 2)) = Real.pi / 4 := by\n        \n        have h₄₅ : (∫ x in (0)..1, (1 : ℝ) / (1 + x ^ 2)) = Real.arctan 1 - Real.arctan 0 := by\n          \n          rw [integral_one_div_one_add_sq]\n          <;>\n          norm_num\n        rw [h₄₅]\n        have h₄₆ : Real.arctan 1 = Real.pi / 4 := by\n          \n          norm_num [Real.arctan_one]\n        have h₄₇ : Real.arctan 0 = 0 := by\n          \n          norm_num [Real.arctan_zero]\n        rw [h₄₆, h₄₇]\n        <;>\n        ring_nf\n        <;>\n        norm_num\n      rw [h₄₄]\n      <;>\n      ring_nf\n      <;>\n      norm_num\n    \n    rw [h₃, h₄]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n  \n  have h_final : 22/7 - Real.pi = ∫ x in (0)..1, x^4 * (1 - x)^4 / (1 + x^2) := by\n    rw [h_main]\n    <;>\n    linarith [Real.pi_pos]\n  \n  exact h_final"}
{"name": "putnam_1991_b1", "proof_length": 3313, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen Filter Topology\n\n\n\n", "signature": "theorem putnam_1991_b1\n  (m S : ℤ → ℤ)\n  (A : ℤ)\n  (a : ℕ → ℤ)\n  (hm : ∀ n, 0 ≤ n → (m n) ^ 2 ≤ n ∧ (∀ m' : ℤ, m' ^ 2 ≤ n → m' ≤ m n))\n  (hS : ∀ n, 0 ≤ n → S n = n - (m n) ^ 2)\n  (ha : a 0 = A ∧ (∀ k, a (k + 1) = a k + S (a k)))\n  (hA : A > 0) :\n  (∃ (K : ℕ) (c : ℕ), ∀ k ≥ K, a k = c) ↔ A ∈ (({A : ℤ | ∃ x > 0, A = x ^ 2}) : Set ℤ )", "src": "theorem putnam_1991_b1\n  (m S : ℤ → ℤ)\n  (A : ℤ)\n  (a : ℕ → ℤ)\n  (hm : ∀ n, 0 ≤ n → (m n) ^ 2 ≤ n ∧ (∀ m' : ℤ, m' ^ 2 ≤ n → m' ≤ m n))\n  (hS : ∀ n, 0 ≤ n → S n = n - (m n) ^ 2)\n  (ha : a 0 = A ∧ (∀ k, a (k + 1) = a k + S (a k)))\n  (hA : A > 0) :\n  (∃ (K : ℕ) (c : ℕ), ∀ k ≥ K, a k = c) ↔ A ∈ (({A : ℤ | ∃ x > 0, A = x ^ 2}) : Set ℤ ) := by \n  have h_nonneg : ∀ (k : ℕ), a k ≥ 0 := by\n    intro k\n    induction k with\n    | zero =>\n      have h₁ : a 0 = A := ha.1\n      linarith\n    | succ k ih =>\n      have h₁ : a (k + 1) = a k + S (a k) := ha.2 k\n      rw [h₁]\n      have h₂ : 0 ≤ a k := ih\n      have h₃ : S (a k) = a k - (m (a k)) ^ 2 := hS (a k) h₂\n      have h₄ : (m (a k)) ^ 2 ≤ a k := (hm (a k) h₂).1\n      have h₅ : S (a k) ≥ 0 := by\n        rw [h₃]\n        linarith\n      linarith\n  \n  have h_nondec : ∀ k, a (k + 1) ≥ a k := by\n    intro k\n    have h₁ : a (k + 1) = a k + S (a k) := (ha.2) k\n    rw [h₁]\n    have h₂ : 0 ≤ a k := h_nonneg k\n    have h₃ : S (a k) = a k - (m (a k)) ^ 2 := hS (a k) h₂\n    have h₄ : (m (a k)) ^ 2 ≤ a k := (hm (a k) h₂).1\n    have h₅ : S (a k) ≥ 0 := by\n      rw [h₃]\n      linarith\n    linarith\n  \n  have h_S_pos : ∀ (n : ℤ), 0 ≤ n → (∀ (t : ℤ), n ≠ t ^ 2) → S n ≥ 1 := by\n    intro n hn hnotsquare\n    have h₁ : S n = n - (m n) ^ 2 := hS n hn\n    rw [h₁]\n    have h₂ : (m n) ^ 2 ≤ n := (hm n hn).1\n    have h₃ : ∀ (m' : ℤ), m' ^ 2 ≤ n → m' ≤ m n := (hm n hn).2\n    \n    have h₄ : n - (m n) ^ 2 ≥ 1 := by\n      by_contra h\n      \n      have h₅ : n - (m n) ^ 2 ≤ 0 := by\n        linarith\n      \n      have h₆ : (m n + 1 : ℤ) ^ 2 > n := by\n        by_contra h₆\n        have h₇ : (m n + 1 : ℤ) ^ 2 ≤ n := by linarith\n        have h₈ : m n + 1 ≤ m n := h₃ (m n + 1) h₇\n        linarith\n      \n      have h₇ : (m n : ℤ) ^ 2 ≤ n := h₂\n      have h₈ : (m n + 1 : ℤ) ^ 2 > n := h₆\n      have h₉ : (m n + 1 : ℤ) ^ 2 > n := h₈\n      \n      \n      \n      \n      \n      \n      \n      have h₁₀ : (m n : ℤ) ^ 2 = n := by\n        nlinarith\n      \n      have h₁₁ : ∃ (t : ℤ), n = t ^ 2 := by\n        refine' ⟨m n, _⟩\n        linarith\n      \n      obtain ⟨t, ht⟩ := h₁₁\n      have h₁₂ : n = t ^ 2 := ht\n      have h₁₃ : n ≠ t ^ 2 := hnotsquare t\n      contradiction\n    \n    linarith\n  \n  have h_no_new_squares : ∀ (k : ℕ), (∀ (t : ℤ), a k ≠ t ^ 2) → (∀ (t : ℤ), a (k + 1) ≠ t ^ 2) := by\n    intro k hk\n    intro t ht\n    have h₁ : a (k + 1) = a k + S (a k) := ha.2 k\n    have h₂ : 0 ≤ a k := h_nonneg k\n    have h₃ : S (a k) = a k - (m (a k)) ^ 2 := hS (a k) h₂\n    have h₄ : a (k + 1) = a k + (a k - (m (a k)) ^ 2) := by\n      rw [h₁, h₃]\n      <;> ring\n    have h₅ : a (k + 1) = 2 * a k - (m (a k)) ^ 2 := by\n      linarith\n    have h₆ : a (k + 1) = t ^ 2 := ht\n    have h₇ : 2 * a k - (m (a k)) ^ 2 = t ^ 2 := by linarith\n    have h₈ : (m (a k)) ^ 2 ≤ a k := (hm (a k) h₂).1\n    have h₉ : ∀ (m' : ℤ), m' ^ 2 ≤ a k → m' ≤ m (a k) := (hm (a k) h₂).2\n    have h₁₀ : a k < (m (a k) + 1) ^ 2 := by\n      by_contra h\n      have h₁₁ : (m (a k) + 1 : ℤ) ^ 2 ≤ a k := by linarith\n      have h₁₂ : m (a k) + 1 ≤ m (a k) := h₉ (m (a k) + 1) h₁₁\n      linarith\n    \n    \n    \n    have h₁₁ : m (a k) ≥ 0 := by\n      by_contra h\n      have h₁₂ : m (a k) < 0 := by linarith\n      have h₁₃ : (m (a k)) ^ 2 > 0 := by nlinarith\n      have h₁₄ : (m (a k) + 1 : ℤ) ^ 2 ≤ a k := by\n        nlinarith [h₈]\n      have h₁₅ : m (a k) + 1 ≤ m (a k) := h₉ (m (a k) + 1) h₁₄\n      linarith\n    \n    \n    have h₁₂ : t ^ 2 = 2 * a k - (m (a k)) ^ 2 := by linarith\n    have h₁₃ : (m (a k)) ^ 2 < t ^ 2 := by\n      have h₁₄ : (m (a k)) ^ 2 < a k := by\n        by_contra h₁₄\n        have h₁₅ : (m (a k)) ^ 2 ≥ a k := by linarith\n        have h₁₆ : (m (a k)) ^ 2 ≤ a k := h₈\n        have h₁₇ : (m (a k)) ^ 2 = a k := by linarith\n        have h₁₈ : ∃ (t : ℤ), a k = t ^ 2 := by\n          refine' ⟨m (a k), _⟩\n          linarith\n        obtain ⟨t', ht'⟩ := h₁₈\n        have h₁₉ : a k = t' ^ 2 := ht'\n        have h₂₀ : a k ≠ t' ^ 2 := hk t'\n        contradiction\n      nlinarith\n    have h₁₄ : t ^ 2 < (m (a k) + 2) ^ 2 := by\n      nlinarith [h₁₀]\n    \n    \n    have h₁₅ : t = m (a k) + 1 ∨ t = -(m (a k) + 1) := by\n      have h₁₆ : m (a k) ≥ 0 := h₁₁\n      have h₁₇ : t ^ 2 < (m (a k) + 2) ^ 2 := h₁₄\n      have h₁₈ : (m (a k)) ^ 2 < t ^ 2 := h₁₃\n      have h₁₉ : t ≤ m (a k) + 1 := by\n        nlinarith\n      have h₂₀ : t ≥ -(m (a k) + 1) := by\n        nlinarith\n      have h₂₁ : t ≥ m (a k) + 1 ∨ t ≤ -(m (a k) + 1) := by\n        by_contra! h₂₂\n        have h₂₃ : -(m (a k) + 1) < t := by linarith\n        have h₂₄ : t < m (a k) + 1 := by linarith\n        have h₂₅ : t ^ 2 < (m (a k) + 1) ^ 2 := by\n          nlinarith\n        nlinarith\n      cases h₂₁ with\n      | inl h₂₁ =>\n        have h₂₂ : t = m (a k) + 1 := by\n          nlinarith\n        exact Or.inl h₂₂\n      | inr h₂₁ =>\n        have h₂₂ : t = -(m (a k) + 1) := by\n          nlinarith\n        exact Or.inr h₂₂\n    \n    cases h₁₅ with\n    | inl h₁₅ =>\n      have h₁₆ : t = m (a k) + 1 := h₁₅\n      have h₁₇ : t ^ 2 = (m (a k) + 1) ^ 2 := by\n        rw [h₁₆]\n        <;> ring_nf\n      have h₁₈ : 2 * a k - (m (a k)) ^ 2 = (m (a k) + 1) ^ 2 := by\n        linarith\n      have h₁₉ : 2 * a k = 2 * (m (a k)) ^ 2 + 2 * m (a k) + 1 := by\n        nlinarith\n      have h₂₀ : (2 : ℤ) * a k % 2 = 1 := by\n        have h₂₁ : (2 : ℤ) * a k = 2 * (m (a k)) ^ 2 + 2 * m (a k) + 1 := h₁₉\n        omega\n      have h₂₁ : (2 : ℤ) * a k % 2 = 0 := by\n        have h₂₂ : (2 : ℤ) * a k % 2 = 0 := by\n          have h₂₃ : (2 : ℤ) * a k % 2 = 0 := by\n            omega\n          exact h₂₃\n        exact h₂₂\n      omega\n    | inr h₁₅ =>\n      have h₁₆ : t = -(m (a k) + 1) := h₁₅\n      have h₁₇ : t ^ 2 = (m (a k) + 1) ^ 2 := by\n        rw [h₁₆]\n        <;> ring_nf\n        <;> simp [sq]\n        <;> linarith\n      have h₁₈ : 2 * a k - (m (a k)) ^ 2 = (m (a k) + 1) ^ 2 := by\n        linarith\n      have h₁₉ : 2 * a k = 2 * (m (a k)) ^ 2 + 2 * m (a k) + 1 := by\n        nlinarith\n      have h₂₀ : (2 : ℤ) * a k % 2 = 1 := by\n        have h₂₁ : (2 : ℤ) * a k = 2 * (m (a k)) ^ 2 + 2 * m (a k) + 1 := h₁₉\n        omega\n      have h₂₁ : (2 : ℤ) * a k % 2 = 0 := by\n        have h₂₂ : (2 : ℤ) * a k % 2 = 0 := by\n          have h₂₃ : (2 : ℤ) * a k % 2 = 0 := by\n            omega\n          exact h₂₃\n        exact h₂₂\n      omega\n  \n  have h_strict_increasing : (∀ (t : ℤ), A ≠ t ^ 2) → ∀ (k : ℕ), a (k + 1) > a k := by\n    intro hA_not_square k\n    have h₁ : ∀ (k : ℕ), (∀ (t : ℤ), a k ≠ t ^ 2) := by\n      intro k\n      induction k with\n      | zero =>\n        intro t ht\n        have h₂ : a 0 = A := ha.1\n        rw [h₂] at ht\n        exact hA_not_square t ht\n      | succ k ih =>\n        intro t ht\n        have h₂ : (∀ (t : ℤ), a k ≠ t ^ 2) := ih\n        have h₃ : (∀ (t : ℤ), a (k + 1) ≠ t ^ 2) := h_no_new_squares k h₂\n        exact h₃ t ht\n    have h₂ : (∀ (t : ℤ), a k ≠ t ^ 2) := h₁ k\n    have h₃ : 0 ≤ a k := h_nonneg k\n    have h₄ : S (a k) ≥ 1 := h_S_pos (a k) h₃ h₂\n    have h₅ : a (k + 1) = a k + S (a k) := ha.2 k\n    have h₆ : a (k + 1) > a k := by\n      linarith\n    exact h₆\n  \n  have h_imp : (∃ (K : ℕ) (c : ℕ), ∀ k ≥ K, a k = c) → A ∈ (({A : ℤ | ∃ x > 0, A = x ^ 2}) : Set ℤ) := by\n    intro h\n    by_contra h₁\n    \n    have h₂ : ∀ (t : ℤ), A ≠ t ^ 2 := by\n      intro t ht\n      have h₃ : A ∈ ({A : ℤ | ∃ x > 0, A = x ^ 2} : Set ℤ) := by\n        refine' ⟨t.natAbs, _⟩\n        have h₄ : A = t ^ 2 := ht\n        have h₅ : (t.natAbs : ℤ) > 0 := by\n          have h₆ : t ≠ 0 := by\n            by_contra h₆\n            rw [h₆] at h₄\n            have h₇ : A = 0 := by\n              norm_num at h₄ ⊢\n              <;> linarith\n            linarith\n          have h₈ : t.natAbs > 0 := by\n            exact Nat.pos_of_ne_zero (by\n              intro h₉\n              have h₁₀ : t.natAbs = 0 := h₉\n              have h₁₁ : t = 0 := by\n                simp [Int.natAbs_eq_zero] at h₁₀ ⊢\n                <;> omega\n              contradiction\n            )\n          exact_mod_cast h₈\n        have h₉ : A = (t.natAbs : ℤ) ^ 2 := by\n          have h₁₀ : A = t ^ 2 := ht\n          have h₁₁ : (t.natAbs : ℤ) ^ 2 = t ^ 2 := by\n            simp [sq, Int.natAbs_mul]\n            <;> cases' le_or_lt 0 t with h₁₂ h₁₂ <;> simp_all [Int.natAbs_of_nonneg, Int.natAbs_of_nonpos, pow_two] <;>\n              ring_nf <;> nlinarith\n          linarith\n        exact ⟨by exact_mod_cast h₅, by linarith⟩\n      contradiction\n    \n    have h₃ : ∀ (k : ℕ), a (k + 1) > a k := h_strict_increasing h₂\n    \n    obtain ⟨K, c, h₄⟩ := h\n    have h₅ : a (K + 1) > a K := h₃ K\n    have h₆ : a (K + 1) = c := by\n      have h₇ : K + 1 ≥ K := by linarith\n      have h₈ : a (K + 1) = c := h₄ (K + 1) h₇\n      exact h₈\n    have h₇ : a K = c := by\n      have h₈ : K ≥ K := by linarith\n      have h₉ : a K = c := h₄ K h₈\n      exact h₉\n    linarith\n  \n  have h_conv : A ∈ (({A : ℤ | ∃ x > 0, A = x ^ 2}) : Set ℤ) → (∃ (K : ℕ) (c : ℕ), ∀ k ≥ K, a k = c) := by\n    intro h\n    obtain ⟨x, hx, hx'⟩ := h\n    have h₁ : a 0 = A := ha.1\n    have h₂ : ∀ k, a (k + 1) = a k + S (a k) := ha.2\n    have h₃ : ∀ k, a k = A := by\n      intro k\n      induction k with\n      | zero => simp [h₁]\n      | succ k ih =>\n        have h₄ : a (k + 1) = a k + S (a k) := h₂ k\n        rw [h₄]\n        have h₅ : a k = A := ih\n        rw [h₅]\n        have h₆ : S A = 0 := by\n          have h₇ : 0 ≤ A := by linarith\n          have h₈ : S A = A - (m A) ^ 2 := hS A h₇\n          rw [h₈]\n          have h₉ : (m A) ^ 2 ≤ A := (hm A h₇).1\n          have h₁₀ : ∀ (m' : ℤ), m' ^ 2 ≤ A → m' ≤ m A := (hm A h₇).2\n          have h₁₁ : A = x ^ 2 := by\n            norm_cast at hx' ⊢\n            <;> simp_all [pow_two]\n            <;> nlinarith\n          have h₁₂ : (m A : ℤ) = x := by\n            have h₁₃ : (m A : ℤ) ≥ 0 := by\n              by_contra h₁₄\n              have h₁₅ : (m A : ℤ) < 0 := by linarith\n              have h₁₆ : (m A : ℤ) ^ 2 > 0 := by nlinarith\n              have h₁₇ : (m A + 1 : ℤ) ^ 2 ≤ A := by\n                nlinarith [h₉]\n              have h₁₈ : m A + 1 ≤ m A := h₁₀ (m A + 1) h₁₇\n              linarith\n            have h₁₉ : (m A : ℤ) ≤ x := by\n              have h₂₀ : (m A : ℤ) ^ 2 ≤ A := h₉\n              have h₂₁ : A = x ^ 2 := by\n                norm_cast at hx' ⊢\n                <;> simp_all [pow_two]\n                <;> nlinarith\n              have h₂₂ : (m A : ℤ) ^ 2 ≤ x ^ 2 := by\n                linarith\n              nlinarith\n            have h₂₀ : (m A : ℤ) ≥ x := by\n              by_contra h₂₁\n              have h₂₂ : (m A : ℤ) < x := by linarith\n              have h₂₃ : (m A + 1 : ℤ) ≤ x := by\n                linarith\n              have h₂₄ : (m A + 1 : ℤ) ^ 2 ≤ x ^ 2 := by\n                have h₂₅ : (m A + 1 : ℤ) ≤ x := by linarith\n                have h₂₆ : 0 ≤ (m A + 1 : ℤ) := by\n                  nlinarith [h₁₃]\n                nlinarith\n              have h₂₅ : (m A + 1 : ℤ) ^ 2 ≤ A := by\n                have h₂₆ : A = x ^ 2 := by\n                  norm_cast at hx' ⊢\n                  <;> simp_all [pow_two]\n                  <;> nlinarith\n                linarith\n              have h₂₆ : m A + 1 ≤ m A := h₁₀ (m A + 1) h₂₅\n              linarith\n            linarith\n          have h₂₁ : (m A : ℤ) = x := h₁₂\n          have h₂₂ : (m A : ℤ) ^ 2 = x ^ 2 := by\n            rw [h₂₁]\n          have h₂₃ : A = x ^ 2 := by\n            norm_cast at hx' ⊢\n            <;> simp_all [pow_two]\n            <;> nlinarith\n          linarith\n        linarith\n    refine' ⟨0, Int.toNat A, _⟩\n    intro k hk\n    have h₄ : a k = A := h₃ k\n    have h₅ : 0 ≤ A := by linarith\n    have h₆ : (A : ℤ) = Int.toNat A := by\n      rw [Int.toNat_of_nonneg h₅]\n      <;> simp\n    rw [h₄]\n    <;> norm_cast at h₆ ⊢ <;> simp_all\n    <;>\n    (try omega)\n  \n  constructor\n  · \n    intro h\n    exact h_imp h\n  · \n    intro h\n    exact h_conv h"}
{"name": "putnam_2019_a1", "proof_length": 2381, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen Topology Filter\n\n\n\n", "signature": "theorem putnam_2019_a1\n: {n : ℤ | ∃ A B C : ℤ, A ≥ 0 ∧ B ≥ 0 ∧ C ≥ 0 ∧ A^3 + B^3 + C^3 - 3*A*B*C = n} = (({n : ℤ | n ≥ 0 ∧ ¬Int.ModEq 9 n 3 ∧ ¬Int.ModEq 9 n 6}) : Set ℤ )", "src": "theorem putnam_2019_a1\n: {n : ℤ | ∃ A B C : ℤ, A ≥ 0 ∧ B ≥ 0 ∧ C ≥ 0 ∧ A^3 + B^3 + C^3 - 3*A*B*C = n} = (({n : ℤ | n ≥ 0 ∧ ¬Int.ModEq 9 n 3 ∧ ¬Int.ModEq 9 n 6}) : Set ℤ ) := by \n  have h_subset₁ : {n : ℤ | ∃ A B C : ℤ, A ≥ 0 ∧ B ≥ 0 ∧ C ≥ 0 ∧ A^3 + B^3 + C^3 - 3*A*B*C = n} ⊆ ({n : ℤ | n ≥ 0 ∧ ¬Int.ModEq 9 n 3 ∧ ¬Int.ModEq 9 n 6} : Set ℤ) := by\n    intro n hn\n    rcases hn with ⟨A, B, C, hA, hB, hC, hn⟩\n    have h₁ : n ≥ 0 := by\n      have h₁ : (A : ℤ) ≥ 0 := by exact_mod_cast hA\n      have h₂ : (B : ℤ) ≥ 0 := by exact_mod_cast hB\n      have h₃ : (C : ℤ) ≥ 0 := by exact_mod_cast hC\n      have h₄ : (A : ℤ)^3 + (B : ℤ)^3 + (C : ℤ)^3 - 3 * (A : ℤ) * (B : ℤ) * (C : ℤ) ≥ 0 := by\n        nlinarith [sq_nonneg (A - B), sq_nonneg (B - C), sq_nonneg (C - A),\n          mul_nonneg h₁ h₂, mul_nonneg h₂ h₃, mul_nonneg h₃ h₁,\n          sq_nonneg (A + B + C), sq_nonneg (A + B - C), sq_nonneg (A + C - B),\n          sq_nonneg (B + C - A)]\n      linarith\n    have h₂ : ¬Int.ModEq 9 n 3 := by\n      have h₂ : ∀ (A B C : ℤ), A ≥ 0 → B ≥ 0 → C ≥ 0 → (A^3 + B^3 + C^3 - 3*A*B*C) % 9 ≠ 3 := by\n        intro A B C hA hB hC\n        have h₃ : A % 9 = 0 ∨ A % 9 = 1 ∨ A % 9 = 2 ∨ A % 9 = 3 ∨ A % 9 = 4 ∨ A % 9 = 5 ∨ A % 9 = 6 ∨ A % 9 = 7 ∨ A % 9 = 8 := by\n          omega\n        have h₄ : B % 9 = 0 ∨ B % 9 = 1 ∨ B % 9 = 2 ∨ B % 9 = 3 ∨ B % 9 = 4 ∨ B % 9 = 5 ∨ B % 9 = 6 ∨ B % 9 = 7 ∨ B % 9 = 8 := by\n          omega\n        have h₅ : C % 9 = 0 ∨ C % 9 = 1 ∨ C % 9 = 2 ∨ C % 9 = 3 ∨ C % 9 = 4 ∨ C % 9 = 5 ∨ C % 9 = 6 ∨ C % 9 = 7 ∨ C % 9 = 8 := by\n          omega\n        rcases h₃ with (h₃ | h₃ | h₃ | h₃ | h₃ | h₃ | h₃ | h₃ | h₃) <;>\n          rcases h₄ with (h₄ | h₄ | h₄ | h₄ | h₄ | h₄ | h₄ | h₄ | h₄) <;>\n            rcases h₅ with (h₅ | h₅ | h₅ | h₅ | h₅ | h₅ | h₅ | h₅ | h₅) <;>\n              (try omega) <;>\n                (try {\n                  simp [h₃, h₄, h₅, pow_three, Int.mul_emod, Int.add_emod, Int.sub_emod]\n                  <;> norm_num <;> omega\n                })\n      have h₃ := h₂ A B C hA hB hC\n      intro h₄\n      have h₅ : (n : ℤ) % 9 = 3 := by\n        rw [Int.ModEq] at h₄\n        omega\n      have h₆ : (A^3 + B^3 + C^3 - 3*A*B*C) % 9 = 3 := by\n        omega\n      contradiction\n    have h₃ : ¬Int.ModEq 9 n 6 := by\n      have h₂ : ∀ (A B C : ℤ), A ≥ 0 → B ≥ 0 → C ≥ 0 → (A^3 + B^3 + C^3 - 3*A*B*C) % 9 ≠ 6 := by\n        intro A B C hA hB hC\n        have h₃ : A % 9 = 0 ∨ A % 9 = 1 ∨ A % 9 = 2 ∨ A % 9 = 3 ∨ A % 9 = 4 ∨ A % 9 = 5 ∨ A % 9 = 6 ∨ A % 9 = 7 ∨ A % 9 = 8 := by\n          omega\n        have h₄ : B % 9 = 0 ∨ B % 9 = 1 ∨ B % 9 = 2 ∨ B % 9 = 3 ∨ B % 9 = 4 ∨ B % 9 = 5 ∨ B % 9 = 6 ∨ B % 9 = 7 ∨ B % 9 = 8 := by\n          omega\n        have h₅ : C % 9 = 0 ∨ C % 9 = 1 ∨ C % 9 = 2 ∨ C % 9 = 3 ∨ C % 9 = 4 ∨ C % 9 = 5 ∨ C % 9 = 6 ∨ C % 9 = 7 ∨ C % 9 = 8 := by\n          omega\n        rcases h₃ with (h₃ | h₃ | h₃ | h₃ | h₃ | h₃ | h₃ | h₃ | h₃) <;>\n          rcases h₄ with (h₄ | h₄ | h₄ | h₄ | h₄ | h₄ | h₄ | h₄ | h₄) <;>\n            rcases h₅ with (h₅ | h₅ | h₅ | h₅ | h₅ | h₅ | h₅ | h₅ | h₅) <;>\n              (try omega) <;>\n                (try {\n                  simp [h₃, h₄, h₅, pow_three, Int.mul_emod, Int.add_emod, Int.sub_emod]\n                  <;> norm_num <;> omega\n                })\n      have h₃ := h₂ A B C hA hB hC\n      intro h₄\n      have h₅ : (n : ℤ) % 9 = 6 := by\n        rw [Int.ModEq] at h₄\n        omega\n      have h₆ : (A^3 + B^3 + C^3 - 3*A*B*C) % 9 = 6 := by\n        omega\n      contradiction\n    exact ⟨h₁, h₂, h₃⟩\n  \n  have h_subset₂ : ({n : ℤ | n ≥ 0 ∧ ¬Int.ModEq 9 n 3 ∧ ¬Int.ModEq 9 n 6} : Set ℤ) ⊆ {n : ℤ | ∃ A B C : ℤ, A ≥ 0 ∧ B ≥ 0 ∧ C ≥ 0 ∧ A^3 + B^3 + C^3 - 3*A*B*C = n} := by\n    intro n hn\n    have h₁ : n ≥ 0 := hn.1\n    have h₂ : ¬Int.ModEq 9 n 3 := hn.2.1\n    have h₃ : ¬Int.ModEq 9 n 6 := hn.2.2\n    \n    have h₄ : n % 3 = 0 ∨ n % 3 = 1 ∨ n % 3 = 2 := by\n      have h₄ : n % 3 = 0 ∨ n % 3 = 1 ∨ n % 3 = 2 := by omega\n      exact h₄\n    rcases h₄ with (h₄ | h₄ | h₄)\n    · \n      have h₅ : n % 9 ≠ 3 := by\n        intro h₅\n        have h₆ : Int.ModEq 9 n 3 := by\n          rw [Int.ModEq]\n          omega\n        contradiction\n      have h₆ : n % 9 ≠ 6 := by\n        intro h₆\n        have h₇ : Int.ModEq 9 n 6 := by\n          rw [Int.ModEq]\n          omega\n        contradiction\n      \n      have h₇ : n % 9 = 0 := by\n        have h₇ : n % 9 = 0 ∨ n % 9 = 3 ∨ n % 9 = 6 := by\n          omega\n        rcases h₇ with (h₇ | h₇ | h₇)\n        · exact h₇\n        · exfalso\n          exact h₅ h₇\n        · exfalso\n          exact h₆ h₇\n      \n      have h₈ : ∃ (k : ℤ), n = 9 * k := by\n        use n / 9\n        have h₈ : n % 9 = 0 := h₇\n        have h₉ : n = 9 * (n / 9) := by\n          omega\n        exact h₉\n      rcases h₈ with ⟨k, hk⟩\n      by_cases h₉ : k = 0\n      · \n        use 0, 0, 0\n        simp [h₉, hk]\n        <;> norm_num\n      · \n        have h₁₀ : k > 0 := by\n          by_contra h₁₀\n          have h₁₁ : k ≤ 0 := by linarith\n          have h₁₂ : n ≤ 0 := by\n            have h₁₃ : (9 : ℤ) * k ≤ 0 := by nlinarith\n            linarith\n          have h₁₃ : n ≥ 0 := h₁\n          have h₁₄ : n = 0 := by linarith\n          have h₁₅ : k = 0 := by\n            omega\n          contradiction\n        \n        use (k + 1 : ℤ), (k : ℤ), (k - 1 : ℤ)\n        have h₁₁ : (k + 1 : ℤ) ≥ 0 := by linarith\n        have h₁₂ : (k : ℤ) ≥ 0 := by linarith\n        have h₁₃ : (k - 1 : ℤ) ≥ 0 := by\n          have h₁₄ : k ≥ 1 := by\n            by_contra h₁₄\n            have h₁₅ : k ≤ 0 := by linarith\n            have h₁₆ : n ≤ 0 := by\n              have h₁₇ : (9 : ℤ) * k ≤ 0 := by nlinarith\n              linarith\n            have h₁₇ : n ≥ 0 := h₁\n            have h₁₈ : n = 0 := by linarith\n            have h₁₉ : k = 0 := by\n              omega\n            contradiction\n          linarith\n        constructor\n        · exact_mod_cast h₁₁\n        constructor\n        · exact_mod_cast h₁₂\n        constructor\n        · exact_mod_cast h₁₃\n        \n        have h₁₄ : (k + 1 : ℤ)^3 + (k : ℤ)^3 + (k - 1 : ℤ)^3 - 3 * (k + 1 : ℤ) * (k : ℤ) * (k - 1 : ℤ) = 9 * k := by\n          ring_nf\n          <;> nlinarith\n        rw [hk]\n        linarith\n    · \n      \n      have h₅ : (n - 1) % 3 = 0 := by\n        omega\n      have h₆ : ∃ (m : ℤ), n = 3 * m + 1 := by\n        use (n - 1) / 3\n        have h₆ : (n - 1) % 3 = 0 := h₅\n        have h₇ : n - 1 = 3 * ((n - 1) / 3) := by\n          omega\n        linarith\n      rcases h₆ with ⟨m, hm⟩\n      have h₇ : m ≥ 0 := by\n        by_contra h₇\n        have h₈ : m < 0 := by linarith\n        have h₉ : n < 1 := by\n          nlinarith\n        have h₁₀ : n ≥ 0 := h₁\n        omega\n      use (m + 1 : ℤ), (m : ℤ), (m : ℤ)\n      have h₈ : (m + 1 : ℤ) ≥ 0 := by linarith\n      have h₉ : (m : ℤ) ≥ 0 := by linarith\n      have h₁₀ : (m : ℤ) ≥ 0 := by linarith\n      constructor\n      · exact_mod_cast h₈\n      constructor\n      · exact_mod_cast h₉\n      constructor\n      · exact_mod_cast h₁₀\n      \n      have h₁₁ : (m + 1 : ℤ)^3 + (m : ℤ)^3 + (m : ℤ)^3 - 3 * (m + 1 : ℤ) * (m : ℤ) * (m : ℤ) = 3 * m + 1 := by\n        ring_nf\n        <;> nlinarith\n      rw [hm]\n      linarith\n    · \n      \n      have h₅ : (n - 2) % 3 = 0 := by\n        omega\n      have h₆ : ∃ (m : ℤ), n = 3 * m + 2 := by\n        use (n - 2) / 3\n        have h₆ : (n - 2) % 3 = 0 := h₅\n        have h₇ : n - 2 = 3 * ((n - 2) / 3) := by\n          omega\n        linarith\n      rcases h₆ with ⟨m, hm⟩\n      have h₇ : m ≥ 0 := by\n        by_contra h₇\n        have h₈ : m < 0 := by linarith\n        have h₉ : n < 2 := by\n          nlinarith\n        have h₁₀ : n ≥ 0 := h₁\n        have h₁₁ : m ≤ -1 := by\n          omega\n        have h₁₂ : n < 0 := by\n          nlinarith\n        linarith\n      use (m + 1 : ℤ), (m + 1 : ℤ), (m : ℤ)\n      have h₈ : (m + 1 : ℤ) ≥ 0 := by linarith\n      have h₉ : (m + 1 : ℤ) ≥ 0 := by linarith\n      have h₁₀ : (m : ℤ) ≥ 0 := by linarith\n      constructor\n      · exact_mod_cast h₈\n      constructor\n      · exact_mod_cast h₉\n      constructor\n      · exact_mod_cast h₁₀\n      \n      have h₁₁ : (m + 1 : ℤ)^3 + (m + 1 : ℤ)^3 + (m : ℤ)^3 - 3 * (m + 1 : ℤ) * (m + 1 : ℤ) * (m : ℤ) = 3 * m + 2 := by\n        ring_nf\n        <;> nlinarith\n      rw [hm]\n      linarith\n  \n  have h_main : {n : ℤ | ∃ A B C : ℤ, A ≥ 0 ∧ B ≥ 0 ∧ C ≥ 0 ∧ A^3 + B^3 + C^3 - 3*A*B*C = n} = (({n : ℤ | n ≥ 0 ∧ ¬Int.ModEq 9 n 3 ∧ ¬Int.ModEq 9 n 6}) : Set ℤ ) := by\n    apply Set.Subset.antisymm\n    · exact h_subset₁\n    · exact h_subset₂\n  \n  exact h_main"}