{"mode": "clear", "steps_before": 211, "alphabet": 86, "compiles": false, "id": "amc12a_2021_p19", "proof_after": "theorem amc12a_2021_p19 (S : Finset \u211d)\n    (h\u2080 :\n      \u2200 x : \u211d,\n        x \u2208 S \u2194\n          0 \u2264 x \u2227\n            x \u2264 Real.pi \u2227\n              Real.sin (Real.pi / 2 * Real.cos x) = Real.cos (Real.pi / 2 * Real.sin x)) :\n    S.card = 2  :=  by\n  have h\u2081 : S = {0, Real.pi / 2} := by\n    apply Finset.ext\n    intro x\n    simp only [Finset.mem_insert, Finset.mem_singleton, h\u2080]\n    constructor\n    \u00b7 \n      intro h\n      have h\u2082 : 0 \u2264 x := h.1\n      have h\u2083 : x \u2264 Real.pi := h.2.1\n      have h\u2084 : Real.sin (Real.pi / 2 * Real.cos x) = Real.cos (Real.pi / 2 * Real.sin x) := h.2.2\n      have h\u2085 : x = 0 \u2228 x = Real.pi / 2 := by\n        have h\u2085\u2081 : Real.sin (Real.pi / 2 * Real.cos x) = Real.cos (Real.pi / 2 * (1 - Real.cos x)) := by\n          have h\u2085\u2082 : Real.sin (Real.pi / 2 * Real.cos x) = Real.cos (Real.pi / 2 - Real.pi / 2 * Real.cos x) := by\n            rw [\u2190 Real.cos_pi_div_two_sub]\n            <;> ring_nf\n          rw [h\u2085\u2082]\n          <;> ring_nf\n          <;> field_simp\n          <;> ring_nf\n          <;> linarith [Real.cos_le_one x, Real.cos_le_one (Real.pi / 2 - Real.pi / 2 * Real.cos x)]\n        rw [h\u2085\u2081] at h\u2084\n        have h\u2085\u2083 : Real.cos (Real.pi / 2 * (1 - Real.cos x)) = Real.cos (Real.pi / 2 * Real.sin x) := h\u2084\n        have h\u2085\u2084 : Real.pi / 2 * (1 - Real.cos x) = Real.pi / 2 * Real.sin x := by\n          have h\u2085\u2084\u2081 : Real.pi / 2 * (1 - Real.cos x) \u2265 0 := by\n            have h\u2085\u2084\u2082 : Real.cos x \u2264 1 := Real.cos_le_one x\n            have h\u2085\u2084\u2083 : Real.cos x \u2265 -1 := Real.neg_one_le_cos x\n            have h\u2085\u2084\u2084 : 1 - Real.cos x \u2265 0 := by linarith\n            have h\u2085\u2084\u2085 : Real.pi / 2 * (1 - Real.cos x) \u2265 0 := by\n              nlinarith [Real.pi_gt_three]\n            linarith\n          have h\u2085\u2084\u2082 : Real.pi / 2 * Real.sin x \u2265 0 := by\n            have h\u2085\u2084\u2083 : Real.sin x \u2265 0 := by\n              apply Real.sin_nonneg_of_mem_Icc\n              constructor <;> nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n            nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n          have h\u2085\u2084\u2083 : Real.pi / 2 * (1 - Real.cos x) \u2264 Real.pi := by\n            have h\u2085\u2084\u2084 : Real.cos x \u2265 -1 := Real.neg_one_le_cos x\n            have h\u2085\u2084\u2085 : 1 - Real.cos x \u2264 2 := by linarith\n            nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n          have h\u2085\u2084\u2084 : Real.pi / 2 * Real.sin x \u2264 Real.pi := by\n            have h\u2085\u2084\u2085 : Real.sin x \u2264 1 := Real.sin_le_one x\n            have h\u2085\u2084\u2086 : Real.pi / 2 * Real.sin x \u2264 Real.pi / 2 * 1 := by\n              nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n            nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n          have h\u2085\u2084\u2085 : Real.pi / 2 * (1 - Real.cos x) = Real.pi / 2 * Real.sin x := by\n            apply (injOn_cos.eq_iff \u27e8by nlinarith, by nlinarith\u27e9 \u27e8by nlinarith, by nlinarith\u27e9).1\n            exact h\u2085\u2083\n          exact h\u2085\u2084\u2085\n        have h\u2085\u2085 : 1 - Real.cos x = Real.sin x := by\n          apply mul_left_cancel\u2080 (show (Real.pi / 2 : \u211d) \u2260 0 by linarith [Real.pi_gt_three])\n          linarith\n        have h\u2085\u2086 : Real.sin x = 1 - Real.cos x := by linarith\n        have h\u2085\u2087 : Real.sin x ^ 2 + Real.cos x ^ 2 = 1 := Real.sin_sq_add_cos_sq x\n        have h\u2085\u2088 : Real.sin x \u2265 0 := by\n          apply Real.sin_nonneg_of_mem_Icc\n          constructor <;> nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n        have h\u2085\u2088 : (1 - Real.cos x) ^ 2 = 1 - Real.cos x ^ 2 := by\n          nlinarith [Real.sin_sq_add_cos_sq x]\n        have h\u2085\u2089 : Real.cos x = 0 \u2228 Real.cos x = 1 := by\n          have h\u2085\u2089\u2081 : Real.cos x = 0 \u2228 Real.cos x = 1 := by\n            have h\u2085\u2089\u2082 : Real.cos x * (Real.cos x - 1) = 0 := by\n              nlinarith [Real.sin_sq_add_cos_sq x]\n            have h\u2085\u2089\u2083 : Real.cos x = 0 \u2228 Real.cos x - 1 = 0 := by\n              apply eq_zero_or_eq_zero_of_mul_eq_zero h\u2085\u2089\u2082\n            cases h\u2085\u2089\u2083 with\n            | inl h\u2085\u2089\u2083 =>\n              exact Or.inl h\u2085\u2089\u2083\n            | inr h\u2085\u2089\u2083 =>\n              have h\u2085\u2089\u2084 : Real.cos x = 1 := by linarith\n              exact Or.inr h\u2085\u2089\u2084\n          exact h\u2085\u2089\u2081\n        cases h\u2085\u2089 with\n        | inl h\u2085\u2089 =>\n          have h\u2085\u2081\u2080 : Real.cos x = 0 := h\u2085\u2089\n          have h\u2085\u2081\u2081 : Real.sin x = 1 := by\n            nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n          have h\u2085\u2081\u2082 : x = Real.pi / 2 := by\n            have h\u2085\u2081\u2083 : Real.cos x = 0 := h\u2085\u2089\n            have h\u2085\u2081\u2084 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n            have h\u2085\u2081\u2085 : x = Real.pi / 2 := by\n              have h\u2085\u2081\u2085\u2081 : Real.cos x = 0 := h\u2085\u2089\n              have h\u2085\u2081\u2085\u2082 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n              have h\u2085\u2081\u2085\u2083 : x = Real.pi / 2 := by\n                have h\u2085\u2081\u2085\u2084 : Real.cos x = 0 := h\u2085\u2089\n                have h\u2085\u2081\u2085\u2085 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n                have h\u2085\u2081\u2085\u2086 : x = Real.pi / 2 := by\n                  have h\u2085\u2081\u2085\u2087 : Real.cos x = 0 := h\u2085\u2089\n                  have h\u2085\u2081\u2085\u2088 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n                  have h\u2085\u2081\u2085\u2089 : x = Real.pi / 2 := by\n                    have h\u2085\u2081\u2086\u2080 : Real.cos x = 0 := h\u2085\u2089\n                    have h\u2085\u2081\u2086\u2081 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n                    have h\u2085\u2081\u2086\u2082 : x = Real.pi / 2 := by\n                      have h\u2085\u2081\u2086\u2083 : Real.cos x = 0 := h\u2085\u2089\n                      have h\u2085\u2081\u2086\u2084 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n                      have h\u2085\u2081\u2086\u2085 : x = Real.pi / 2 := by\n                        have h\u2085\u2081\u2086\u2086 : Real.cos x = Real.cos (Real.pi / 2) := by\n                          norm_num [h\u2085\u2081\u2086\u2083]\n                        have h\u2085\u2081\u2086\u2087 : Real.sin x = Real.sin (Real.pi / 2) := by\n                          norm_num [h\u2085\u2081\u2086\u2084]\n                        have h\u2085\u2081\u2086\u2088 : x = Real.pi / 2 := by\n                          apply (injOn_cos.eq_iff \u27e8by linarith [h\u2082, h\u2083, Real.pi_pos], by linarith [h\u2082, h\u2083, Real.pi_pos]\u27e9 \u27e8by linarith [Real.pi_pos], by linarith [Real.pi_pos]\u27e9).1\n                          <;> simp_all [h\u2085\u2081\u2086\u2086, h\u2085\u2081\u2086\u2087]\n                          <;> linarith [Real.pi_gt_three]\n                        exact h\u2085\u2081\u2086\u2088\n                      exact h\u2085\u2081\u2086\u2085\n                    exact h\u2085\u2081\u2086\u2082\n                  exact h\u2085\u2081\u2085\u2089\n                exact h\u2085\u2081\u2085\u2086\n              exact h\u2085\u2081\u2085\u2083\n            exact h\u2085\u2081\u2085\n          exact Or.inr h\u2085\u2081\u2082\n        | inr h\u2085\u2089 =>\n          have h\u2085\u2081\u2080 : Real.cos x = 1 := h\u2085\u2089\n          have h\u2085\u2081\u2081 : Real.sin x = 0 := by\n            nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n          have h\u2085\u2081\u2082 : x = 0 := by\n            have h\u2085\u2081\u2083 : Real.cos x = 1 := h\u2085\u2089\n            have h\u2085\u2081\u2084 : Real.sin x = 0 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n            have h\u2085\u2081\u2085 : x = 0 := by\n              have h\u2085\u2081\u2085\u2081 : Real.cos x = 1 := h\u2085\u2089\n              have h\u2085\u2081\u2085\u2082 : Real.sin x = 0 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n              have h\u2085\u2081\u2085\u2083 : x = 0 := by\n                have h\u2085\u2081\u2085\u2084 : Real.cos x = 1 := h\u2085\u2089\n                have h\u2085\u2081\u2085\u2085 : Real.sin x = 0 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n                have h\u2085\u2081\u2085\u2086 : x = 0 := by\n                  have h\u2085\u2081\u2085\u2087 : Real.cos x = Real.cos 0 := by\n                    norm_num [h\u2085\u2081\u2085\u2084]\n                  have h\u2085\u2081\u2085\u2088 : Real.sin x = Real.sin 0 := by\n                    norm_num [h\u2085\u2081\u2085\u2085]\n                  have h\u2085\u2081\u2085\u2089 : x = 0 := by\n                    apply (injOn_cos.eq_iff \u27e8by linarith [h\u2082, h\u2083, Real.pi_pos], by linarith [h\u2082, h\u2083, Real.pi_pos]\u27e9 \u27e8by linarith [Real.pi_pos], by linarith [Real.pi_pos]\u27e9).1\n                    <;> simp_all [h\u2085\u2081\u2085\u2087, h\u2085\u2081\u2085\u2088]\n                    <;> linarith [Real.pi_gt_three]\n                  exact h\u2085\u2081\u2085\u2089\n                exact h\u2085\u2081\u2085\u2086\n              exact h\u2085\u2081\u2085\u2083\n            exact h\u2085\u2081\u2085\n          exact Or.inl h\u2085\u2081\u2082\n      cases h\u2085 with\n      | inl h\u2085 =>\n        simp_all\n      | inr h\u2085 =>\n        simp_all\n    \u00b7 \n      intro h\n      cases h with\n      | inl h =>\n        rw [h]\n        constructor\n        \u00b7 \n          linarith [Real.pi_pos]\n        \u00b7 constructor\n          \u00b7 \n            linarith [Real.pi_pos]\n          \u00b7 \n            have h\u2082 : Real.sin (Real.pi / 2 * Real.cos 0) = Real.cos (Real.pi / 2 * Real.sin 0) := by\n              norm_num [Real.cos_zero, Real.sin_zero]\n            exact h\u2082\n      | inr h =>\n        rw [h]\n        constructor\n        \u00b7 \n          linarith [Real.pi_pos]\n        \u00b7 constructor\n          \u00b7 \n            linarith [Real.pi_pos]\n          \u00b7 \n            have h\u2082 : Real.sin (Real.pi / 2 * Real.cos (Real.pi / 2)) = Real.cos (Real.pi / 2 * Real.sin (Real.pi / 2)) := by\n              have h\u2083 : Real.cos (Real.pi / 2) = 0 := by norm_num\n              have h\u2084 : Real.sin (Real.pi / 2) = 1 := by norm_num\n              rw [h\u2083, h\u2084]\n              <;> norm_num [Real.sin_zero, Real.cos_zero]\n            exact h\u2082\n  have h\u2082 : S.card = 2 := by\n    rw [h\u2081]\n    have h\u2083 : (0 : \u211d) \u2260 Real.pi / 2 := by\n      have h\u2084 : Real.pi > 0 := Real.pi_pos\n      have h\u2085 : Real.pi / 2 > 0 := by linarith\n      linarith [Real.pi_gt_three]\n    norm_num [Finset.card_insert_of_not_mem, Finset.mem_singleton, h\u2083]\n    <;>\n    (try norm_num) <;>\n    (try linarith [Real.pi_gt_three])\n    <;>\n    (try field_simp [Real.pi_ne_zero, Real.pi_pos.ne'])\n    <;>\n    (try ring_nf)\n    <;>\n    (try norm_num)\n    <;>\n    (try linarith [Real.pi_gt_three])\n  exact h\u2082\n", "proof_before": "theorem amc12a_2021_p19 (S : Finset \u211d)\n    (h\u2080 :\n      \u2200 x : \u211d,\n        x \u2208 S \u2194\n          0 \u2264 x \u2227\n            x \u2264 Real.pi \u2227\n              Real.sin (Real.pi / 2 * Real.cos x) = Real.cos (Real.pi / 2 * Real.sin x)) :\n    S.card = 2  :=  by\n  have h\u2081 : S = {0, Real.pi / 2} := by\n    apply Finset.ext\n    intro x\n    simp only [Finset.mem_insert, Finset.mem_singleton, h\u2080]\n    constructor\n    \u00b7 \n      intro h\n      have h\u2082 : 0 \u2264 x := h.1\n      have h\u2083 : x \u2264 Real.pi := h.2.1\n      have h\u2084 : Real.sin (Real.pi / 2 * Real.cos x) = Real.cos (Real.pi / 2 * Real.sin x) := h.2.2\n      have h\u2085 : x = 0 \u2228 x = Real.pi / 2 := by\n        have h\u2085\u2081 : Real.sin (Real.pi / 2 * Real.cos x) = Real.cos (Real.pi / 2 * (1 - Real.cos x)) := by\n          have h\u2085\u2082 : Real.sin (Real.pi / 2 * Real.cos x) = Real.cos (Real.pi / 2 - Real.pi / 2 * Real.cos x) := by\n            rw [\u2190 Real.cos_pi_div_two_sub]\n            <;> ring_nf\n          rw [h\u2085\u2082]\n          <;> ring_nf\n          <;> field_simp\n          <;> ring_nf\n          <;> linarith [Real.cos_le_one x, Real.cos_le_one (Real.pi / 2 - Real.pi / 2 * Real.cos x)]\n        rw [h\u2085\u2081] at h\u2084\n        have h\u2085\u2083 : Real.cos (Real.pi / 2 * (1 - Real.cos x)) = Real.cos (Real.pi / 2 * Real.sin x) := h\u2084\n        have h\u2085\u2084 : Real.pi / 2 * (1 - Real.cos x) = Real.pi / 2 * Real.sin x := by\n          have h\u2085\u2084\u2081 : Real.pi / 2 * (1 - Real.cos x) \u2265 0 := by\n            have h\u2085\u2084\u2082 : Real.cos x \u2264 1 := Real.cos_le_one x\n            have h\u2085\u2084\u2083 : Real.cos x \u2265 -1 := Real.neg_one_le_cos x\n            have h\u2085\u2084\u2084 : 1 - Real.cos x \u2265 0 := by linarith\n            have h\u2085\u2084\u2085 : Real.pi / 2 * (1 - Real.cos x) \u2265 0 := by\n              nlinarith [Real.pi_gt_three]\n            linarith\n          have h\u2085\u2084\u2082 : Real.pi / 2 * Real.sin x \u2265 0 := by\n            have h\u2085\u2084\u2083 : Real.sin x \u2265 0 := by\n              apply Real.sin_nonneg_of_mem_Icc\n              constructor <;> nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n            nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n          have h\u2085\u2084\u2083 : Real.pi / 2 * (1 - Real.cos x) \u2264 Real.pi := by\n            have h\u2085\u2084\u2084 : Real.cos x \u2265 -1 := Real.neg_one_le_cos x\n            have h\u2085\u2084\u2085 : 1 - Real.cos x \u2264 2 := by linarith\n            nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n          have h\u2085\u2084\u2084 : Real.pi / 2 * Real.sin x \u2264 Real.pi := by\n            have h\u2085\u2084\u2085 : Real.sin x \u2264 1 := Real.sin_le_one x\n            have h\u2085\u2084\u2086 : Real.pi / 2 * Real.sin x \u2264 Real.pi / 2 * 1 := by\n              nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n            nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n          have h\u2085\u2084\u2085 : Real.pi / 2 * (1 - Real.cos x) = Real.pi / 2 * Real.sin x := by\n            apply (injOn_cos.eq_iff \u27e8by nlinarith, by nlinarith\u27e9 \u27e8by nlinarith, by nlinarith\u27e9).1\n            exact h\u2085\u2083\n          exact h\u2085\u2084\u2085\n        have h\u2085\u2085 : 1 - Real.cos x = Real.sin x := by\n          apply mul_left_cancel\u2080 (show (Real.pi / 2 : \u211d) \u2260 0 by linarith [Real.pi_gt_three])\n          linarith\n        have h\u2085\u2086 : Real.sin x = 1 - Real.cos x := by linarith\n        have h\u2085\u2087 : Real.sin x ^ 2 + Real.cos x ^ 2 = 1 := Real.sin_sq_add_cos_sq x\n        have h\u2085\u2088 : Real.sin x \u2265 0 := by\n          apply Real.sin_nonneg_of_mem_Icc\n          constructor <;> nlinarith [Real.pi_gt_three, Real.pi_pos, Real.two_le_pi]\n        have h\u2085\u2088 : (1 - Real.cos x) ^ 2 = 1 - Real.cos x ^ 2 := by\n          nlinarith [Real.sin_sq_add_cos_sq x]\n        have h\u2085\u2089 : Real.cos x = 0 \u2228 Real.cos x = 1 := by\n          have h\u2085\u2089\u2081 : Real.cos x = 0 \u2228 Real.cos x = 1 := by\n            have h\u2085\u2089\u2082 : Real.cos x * (Real.cos x - 1) = 0 := by\n              nlinarith [Real.sin_sq_add_cos_sq x]\n            have h\u2085\u2089\u2083 : Real.cos x = 0 \u2228 Real.cos x - 1 = 0 := by\n              apply eq_zero_or_eq_zero_of_mul_eq_zero h\u2085\u2089\u2082\n            cases h\u2085\u2089\u2083 with\n            | inl h\u2085\u2089\u2083 =>\n              exact Or.inl h\u2085\u2089\u2083\n            | inr h\u2085\u2089\u2083 =>\n              have h\u2085\u2089\u2084 : Real.cos x = 1 := by linarith\n              exact Or.inr h\u2085\u2089\u2084\n          exact h\u2085\u2089\u2081\n        cases h\u2085\u2089 with\n        | inl h\u2085\u2089 =>\n          have h\u2085\u2081\u2080 : Real.cos x = 0 := h\u2085\u2089\n          have h\u2085\u2081\u2081 : Real.sin x = 1 := by\n            nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n          have h\u2085\u2081\u2082 : x = Real.pi / 2 := by\n            have h\u2085\u2081\u2083 : Real.cos x = 0 := h\u2085\u2089\n            have h\u2085\u2081\u2084 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n            have h\u2085\u2081\u2085 : x = Real.pi / 2 := by\n              have h\u2085\u2081\u2085\u2081 : Real.cos x = 0 := h\u2085\u2089\n              have h\u2085\u2081\u2085\u2082 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n              have h\u2085\u2081\u2085\u2083 : x = Real.pi / 2 := by\n                have h\u2085\u2081\u2085\u2084 : Real.cos x = 0 := h\u2085\u2089\n                have h\u2085\u2081\u2085\u2085 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n                have h\u2085\u2081\u2085\u2086 : x = Real.pi / 2 := by\n                  have h\u2085\u2081\u2085\u2087 : Real.cos x = 0 := h\u2085\u2089\n                  have h\u2085\u2081\u2085\u2088 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n                  have h\u2085\u2081\u2085\u2089 : x = Real.pi / 2 := by\n                    have h\u2085\u2081\u2086\u2080 : Real.cos x = 0 := h\u2085\u2089\n                    have h\u2085\u2081\u2086\u2081 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n                    have h\u2085\u2081\u2086\u2082 : x = Real.pi / 2 := by\n                      have h\u2085\u2081\u2086\u2083 : Real.cos x = 0 := h\u2085\u2089\n                      have h\u2085\u2081\u2086\u2084 : Real.sin x = 1 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n                      have h\u2085\u2081\u2086\u2085 : x = Real.pi / 2 := by\n                        have h\u2085\u2081\u2086\u2086 : Real.cos x = Real.cos (Real.pi / 2) := by\n                          norm_num [h\u2085\u2081\u2086\u2083]\n                        have h\u2085\u2081\u2086\u2087 : Real.sin x = Real.sin (Real.pi / 2) := by\n                          norm_num [h\u2085\u2081\u2086\u2084]\n                        have h\u2085\u2081\u2086\u2088 : x = Real.pi / 2 := by\n                          apply (injOn_cos.eq_iff \u27e8by linarith [h\u2082, h\u2083, Real.pi_pos], by linarith [h\u2082, h\u2083, Real.pi_pos]\u27e9 \u27e8by linarith [Real.pi_pos], by linarith [Real.pi_pos]\u27e9).1\n                          <;> simp_all [h\u2085\u2081\u2086\u2086, h\u2085\u2081\u2086\u2087]\n                          <;> linarith [Real.pi_gt_three]\n                        exact h\u2085\u2081\u2086\u2088\n                      exact h\u2085\u2081\u2086\u2085\n                    exact h\u2085\u2081\u2086\u2082\n                  exact h\u2085\u2081\u2085\u2089\n                exact h\u2085\u2081\u2085\u2086\n              exact h\u2085\u2081\u2085\u2083\n            exact h\u2085\u2081\u2085\n          exact Or.inr h\u2085\u2081\u2082\n        | inr h\u2085\u2089 =>\n          have h\u2085\u2081\u2080 : Real.cos x = 1 := h\u2085\u2089\n          have h\u2085\u2081\u2081 : Real.sin x = 0 := by\n            nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n          have h\u2085\u2081\u2082 : x = 0 := by\n            have h\u2085\u2081\u2083 : Real.cos x = 1 := h\u2085\u2089\n            have h\u2085\u2081\u2084 : Real.sin x = 0 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n            have h\u2085\u2081\u2085 : x = 0 := by\n              have h\u2085\u2081\u2085\u2081 : Real.cos x = 1 := h\u2085\u2089\n              have h\u2085\u2081\u2085\u2082 : Real.sin x = 0 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n              have h\u2085\u2081\u2085\u2083 : x = 0 := by\n                have h\u2085\u2081\u2085\u2084 : Real.cos x = 1 := h\u2085\u2089\n                have h\u2085\u2081\u2085\u2085 : Real.sin x = 0 := by nlinarith [Real.sin_sq_add_cos_sq x, Real.sin_le_one x, Real.cos_le_one x]\n                have h\u2085\u2081\u2085\u2086 : x = 0 := by\n                  have h\u2085\u2081\u2085\u2087 : Real.cos x = Real.cos 0 := by\n                    norm_num [h\u2085\u2081\u2085\u2084]\n                  have h\u2085\u2081\u2085\u2088 : Real.sin x = Real.sin 0 := by\n                    norm_num [h\u2085\u2081\u2085\u2085]\n                  have h\u2085\u2081\u2085\u2089 : x = 0 := by\n                    apply (injOn_cos.eq_iff \u27e8by linarith [h\u2082, h\u2083, Real.pi_pos], by linarith [h\u2082, h\u2083, Real.pi_pos]\u27e9 \u27e8by linarith [Real.pi_pos], by linarith [Real.pi_pos]\u27e9).1\n                    <;> simp_all [h\u2085\u2081\u2085\u2087, h\u2085\u2081\u2085\u2088]\n                    <;> linarith [Real.pi_gt_three]\n                  exact h\u2085\u2081\u2085\u2089\n                exact h\u2085\u2081\u2085\u2086\n              exact h\u2085\u2081\u2085\u2083\n            exact h\u2085\u2081\u2085\n          exact Or.inl h\u2085\u2081\u2082\n      cases h\u2085 with\n      | inl h\u2085 =>\n        simp_all\n      | inr h\u2085 =>\n        simp_all\n    \u00b7 \n      intro h\n      cases h with\n      | inl h =>\n        rw [h]\n        constructor\n        \u00b7 \n          linarith [Real.pi_pos]\n        \u00b7 constructor\n          \u00b7 \n            linarith [Real.pi_pos]\n          \u00b7 \n            have h\u2082 : Real.sin (Real.pi / 2 * Real.cos 0) = Real.cos (Real.pi / 2 * Real.sin 0) := by\n              norm_num [Real.cos_zero, Real.sin_zero]\n            exact h\u2082\n      | inr h =>\n        rw [h]\n        constructor\n        \u00b7 \n          linarith [Real.pi_pos]\n        \u00b7 constructor\n          \u00b7 \n            linarith [Real.pi_pos]\n          \u00b7 \n            have h\u2082 : Real.sin (Real.pi / 2 * Real.cos (Real.pi / 2)) = Real.cos (Real.pi / 2 * Real.sin (Real.pi / 2)) := by\n              have h\u2083 : Real.cos (Real.pi / 2) = 0 := by norm_num\n              have h\u2084 : Real.sin (Real.pi / 2) = 1 := by norm_num\n              rw [h\u2083, h\u2084]\n              <;> norm_num [Real.sin_zero, Real.cos_zero]\n            exact h\u2082\n  have h\u2082 : S.card = 2 := by\n    rw [h\u2081]\n    have h\u2083 : (0 : \u211d) \u2260 Real.pi / 2 := by\n      have h\u2084 : Real.pi > 0 := Real.pi_pos\n      have h\u2085 : Real.pi / 2 > 0 := by linarith\n      linarith [Real.pi_gt_three]\n    norm_num [Finset.card_insert_of_not_mem, Finset.mem_singleton, h\u2083]\n    <;>\n    (try norm_num) <;>\n    (try linarith [Real.pi_gt_three])\n    <;>\n    (try field_simp [Real.pi_ne_zero, Real.pi_pos.ne'])\n    <;>\n    (try ring_nf)\n    <;>\n    (try norm_num)\n    <;>\n    (try linarith [Real.pi_gt_three])\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 14, "alphabet": 17, "compiles": false, "id": "mathd_numbertheory_233", "proof_after": "theorem mathd_numbertheory_233 (b : ZMod (11 ^ 2)) (h\u2080 : b = 24\u207b\u00b9) : b = 116  :=  by\n  have h\u2081 : (24 : ZMod (11 ^ 2)) * 116 = 1 := by\n    norm_num [ZMod.val_mul, ZMod.val_nat_cast, ZMod.val_one, pow_succ]\n    <;> rfl\n  have h\u2082 : (24 : ZMod (11 ^ 2))\u207b\u00b9 = 116 := by\n    apply Eq.symm\n    apply Eq.symm\n    rw [\u2190 mul_one (116 : ZMod (11 ^ 2)), \u2190 h\u2081]\n    simp [mul_assoc, mul_comm, mul_left_comm, inv_mul_cancel_left]\n    <;> norm_num\n    <;> rfl\n  have h\u2083 : b = 116 := by\n    rw [h\u2080, h\u2082]\n    <;> rfl\n  exact h\u2083\n", "proof_before": "theorem mathd_numbertheory_233 (b : ZMod (11 ^ 2)) (h\u2080 : b = 24\u207b\u00b9) : b = 116  :=  by\n  have h\u2081 : (24 : ZMod (11 ^ 2)) * 116 = 1 := by\n    norm_num [ZMod.val_mul, ZMod.val_nat_cast, ZMod.val_one, pow_succ]\n    <;> rfl\n  have h\u2082 : (24 : ZMod (11 ^ 2))\u207b\u00b9 = 116 := by\n    apply Eq.symm\n    apply Eq.symm\n    rw [\u2190 mul_one (116 : ZMod (11 ^ 2)), \u2190 h\u2081]\n    simp [mul_assoc, mul_comm, mul_left_comm, inv_mul_cancel_left]\n    <;> norm_num\n    <;> rfl\n  have h\u2083 : b = 116 := by\n    rw [h\u2080, h\u2082]\n    <;> rfl\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 12, "alphabet": 14, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 246, "chars_after": 11, "seconds": 0.0, "id": "mathd_algebra_160", "proof_after": "theorem mathd_algebra_160 (n x : \u211d) (h\u2080 : n + x = 97) (h\u2081 : n + 5 * x = 265) : n + 2 * x = 139  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_160 (n x : \u211d) (h\u2080 : n + x = 97) (h\u2081 : n + 5 * x = 265) : n + 2 * x = 139  :=  by\n  have h\u2082 : x = 42 := by\n    have h\u2082\u2081 : 4 * x = 168 := by\n      linarith\n    linarith\n  have h\u2083 : n = 55 := by\n    have h\u2083\u2081 : n = 55 := by\n      linarith\n    exact h\u2083\u2081\n  have h\u2084 : n + 2 * x = 139 := by\n    rw [h\u2083, h\u2082]\n    <;> norm_num\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 114, "alphabet": 50, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 114, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 5234, "chars_after": 5234, "seconds": 244.9, "id": "numbertheory_fxeq4powxp6powxp9powx_f2powmdvdf2pown", "proof_after": "theorem numbertheory_fxeq4powxp6powxp9powx_f2powmdvdf2pown (m n : \u2115) (f : \u2115 \u2192 \u2115)\n    (h\u2080 : \u2200 x, f x = 4 ^ x + 6 ^ x + 9 ^ x) (h\u2081 : 0 < m \u2227 0 < n) (h\u2082 : m \u2264 n) :\n    f (2 ^ m) \u2223 f (2 ^ n)  :=  by\n  have h_main : \u2200 k : \u2115, f (2 * k) = f k * (f k - 2 * 6 ^ k) := by\n    intro k\n    have h\u2081 : f (2 * k) = 4 ^ (2 * k) + 6 ^ (2 * k) + 9 ^ (2 * k) := by\n      rw [h\u2080]\n    have h\u2082 : f k = 4 ^ k + 6 ^ k + 9 ^ k := by\n      rw [h\u2080]\n    rw [h\u2081, h\u2082]\n    have h\u2083 : (4 : \u2115) ^ (2 * k) = (4 : \u2115) ^ k * (4 : \u2115) ^ k := by\n      rw [show (2 * k : \u2115) = k + k by ring]\n      rw [pow_add]\n      <;> ring\n    have h\u2084 : (6 : \u2115) ^ (2 * k) = (6 : \u2115) ^ k * (6 : \u2115) ^ k := by\n      rw [show (2 * k : \u2115) = k + k by ring]\n      rw [pow_add]\n      <;> ring\n    have h\u2085 : (9 : \u2115) ^ (2 * k) = (9 : \u2115) ^ k * (9 : \u2115) ^ k := by\n      rw [show (2 * k : \u2115) = k + k by ring]\n      rw [pow_add]\n      <;> ring\n    rw [h\u2083, h\u2084, h\u2085]\n    have h\u2086 : (4 : \u2115) ^ k * (4 : \u2115) ^ k + (6 : \u2115) ^ k * (6 : \u2115) ^ k + (9 : \u2115) ^ k * (9 : \u2115) ^ k = ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) * ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) - 2 * ((4 : \u2115) ^ k * (6 : \u2115) ^ k + (4 : \u2115) ^ k * (9 : \u2115) ^ k + (6 : \u2115) ^ k * (9 : \u2115) ^ k) := by\n      have h\u2087 : ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) * ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) = (4 : \u2115) ^ k * (4 : \u2115) ^ k + (6 : \u2115) ^ k * (6 : \u2115) ^ k + (9 : \u2115) ^ k * (9 : \u2115) ^ k + 2 * ((4 : \u2115) ^ k * (6 : \u2115) ^ k + (4 : \u2115) ^ k * (9 : \u2115) ^ k + (6 : \u2115) ^ k * (9 : \u2115) ^ k) := by\n        ring\n      have h\u2088 : (4 : \u2115) ^ k * (6 : \u2115) ^ k + (4 : \u2115) ^ k * (9 : \u2115) ^ k + (6 : \u2115) ^ k * (9 : \u2115) ^ k = (6 : \u2115) ^ k * ((4 : \u2115) ^ k + (9 : \u2115) ^ k) + (4 : \u2115) ^ k * (9 : \u2115) ^ k := by\n        ring\n      omega\n    have h\u2087 : (4 : \u2115) ^ k * (6 : \u2115) ^ k + (4 : \u2115) ^ k * (9 : \u2115) ^ k + (6 : \u2115) ^ k * (9 : \u2115) ^ k = (6 : \u2115) ^ k * ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) := by\n      have h\u2088 : (4 : \u2115) ^ k * (6 : \u2115) ^ k + (4 : \u2115) ^ k * (9 : \u2115) ^ k + (6 : \u2115) ^ k * (9 : \u2115) ^ k = (6 : \u2115) ^ k * ((4 : \u2115) ^ k + (9 : \u2115) ^ k) + (4 : \u2115) ^ k * (9 : \u2115) ^ k := by\n        ring\n      have h\u2089 : (6 : \u2115) ^ k * ((4 : \u2115) ^ k + (9 : \u2115) ^ k) + (4 : \u2115) ^ k * (9 : \u2115) ^ k = (6 : \u2115) ^ k * ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) := by\n        have h\u2081\u2080 : (6 : \u2115) ^ k = (2 : \u2115) ^ k * (3 : \u2115) ^ k := by\n          rw [show (6 : \u2115) = 2 * 3 by norm_num]\n          rw [mul_pow]\n          <;> ring\n        rw [h\u2081\u2080]\n        have h\u2081\u2081 : (4 : \u2115) ^ k = (2 : \u2115) ^ (2 * k) := by\n          rw [show (4 : \u2115) = 2 ^ 2 by norm_num]\n          rw [\u2190 pow_mul]\n          <;> ring_nf\n        have h\u2081\u2082 : (9 : \u2115) ^ k = (3 : \u2115) ^ (2 * k) := by\n          rw [show (9 : \u2115) = 3 ^ 2 by norm_num]\n          rw [\u2190 pow_mul]\n          <;> ring_nf\n        rw [h\u2081\u2081, h\u2081\u2082]\n        have h\u2081\u2083 : (2 : \u2115) ^ k * (3 : \u2115) ^ k * ((2 : \u2115) ^ (2 * k) + (3 : \u2115) ^ (2 * k)) + (2 : \u2115) ^ (2 * k) * (3 : \u2115) ^ (2 * k) = (2 : \u2115) ^ k * (3 : \u2115) ^ k * ((2 : \u2115) ^ (2 * k) + (2 : \u2115) ^ k * (3 : \u2115) ^ k + (3 : \u2115) ^ (2 * k)) := by\n          have h\u2081\u2084 : (2 : \u2115) ^ (2 * k) = (2 : \u2115) ^ k * (2 : \u2115) ^ k := by\n            rw [show (2 * k : \u2115) = k + k by ring]\n            rw [pow_add]\n            <;> ring\n          have h\u2081\u2085 : (3 : \u2115) ^ (2 * k) = (3 : \u2115) ^ k * (3 : \u2115) ^ k := by\n            rw [show (2 * k : \u2115) = k + k by ring]\n            rw [pow_add]\n            <;> ring\n          rw [h\u2081\u2084, h\u2081\u2085]\n          ring_nf\n          <;> nlinarith [pow_pos (by norm_num : 0 < (2 : \u2115)) k, pow_pos (by norm_num : 0 < (3 : \u2115)) k]\n        rw [h\u2081\u2083]\n        <;> ring_nf\n        <;> nlinarith [pow_pos (by norm_num : 0 < (2 : \u2115)) k, pow_pos (by norm_num : 0 < (3 : \u2115)) k]\n      omega\n    rw [h\u2086, h\u2087]\n    <;> cases k with\n    | zero =>\n      norm_num [h\u2080, pow_zero, Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib]\n    | succ k =>\n      simp_all [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.add_assoc]\n      <;> ring_nf at *\n      <;> norm_num at *\n      <;> omega\n  have h_div : \u2200 k : \u2115, f k \u2223 f (2 * k) := by\n    intro k\n    have h\u2081 : f (2 * k) = f k * (f k - 2 * 6 ^ k) := h_main k\n    rw [h\u2081]\n    exact \u27e8f k - 2 * 6 ^ k, by\n      cases k with\n      | zero =>\n        simp [h\u2080, pow_zero, Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib]\n      | succ k =>\n        simp_all [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.add_assoc]\n        <;> ring_nf at *\n        <;> norm_num at *\n        <;> omega\n    \u27e9\n  have h_chain : \u2200 t : \u2115, f (2 ^ m) \u2223 f (2 ^ (m + t)) := by\n    intro t\n    have h\u2083 : \u2200 t : \u2115, f (2 ^ m) \u2223 f (2 ^ (m + t)) := by\n      intro t\n      induction' t with t ih\n      \u00b7 \n        simp [h\u2080]\n      \u00b7 \n        have h\u2084 : f (2 ^ (m + t)) \u2223 f (2 * 2 ^ (m + t)) := h_div _\n        have h\u2085 : f (2 * 2 ^ (m + t)) = f (2 ^ (m + t + 1)) := by\n          rw [show (2 * 2 ^ (m + t) : \u2115) = 2 ^ (m + t + 1) by\n            ring_nf\n            <;> simp [pow_add, pow_one, mul_add, mul_one, add_mul, one_mul]]\n        rw [h\u2085] at h\u2084\n        have h\u2086 : m + t + 1 = m + (t + 1) := by ring\n        rw [h\u2086] at h\u2084\n        exact dvd_trans ih h\u2084\n    exact h\u2083 t\n  have h_final : f (2 ^ m) \u2223 f (2 ^ n) := by\n    have h\u2083 : \u2203 t, n = m + t := by\n      use n - m\n      have h\u2084 : m \u2264 n := h\u2082\n      have h\u2085 : n - m + m = n := by\n        have h\u2086 : m \u2264 n := h\u2082\n        have h\u2087 : n - m + m = n := by\n          omega\n        exact h\u2087\n      omega\n    obtain \u27e8t, ht\u27e9 := h\u2083\n    have h\u2084 : f (2 ^ m) \u2223 f (2 ^ (m + t)) := h_chain t\n    have h\u2085 : f (2 ^ (m + t)) = f (2 ^ n) := by\n      rw [ht]\n      <;> simp [pow_add]\n      <;> ring_nf\n    rw [h\u2085] at h\u2084\n    exact h\u2084\n  exact h_final\n", "proof_before": "theorem numbertheory_fxeq4powxp6powxp9powx_f2powmdvdf2pown (m n : \u2115) (f : \u2115 \u2192 \u2115)\n    (h\u2080 : \u2200 x, f x = 4 ^ x + 6 ^ x + 9 ^ x) (h\u2081 : 0 < m \u2227 0 < n) (h\u2082 : m \u2264 n) :\n    f (2 ^ m) \u2223 f (2 ^ n)  :=  by\n  have h_main : \u2200 k : \u2115, f (2 * k) = f k * (f k - 2 * 6 ^ k) := by\n    intro k\n    have h\u2081 : f (2 * k) = 4 ^ (2 * k) + 6 ^ (2 * k) + 9 ^ (2 * k) := by\n      rw [h\u2080]\n    have h\u2082 : f k = 4 ^ k + 6 ^ k + 9 ^ k := by\n      rw [h\u2080]\n    rw [h\u2081, h\u2082]\n    have h\u2083 : (4 : \u2115) ^ (2 * k) = (4 : \u2115) ^ k * (4 : \u2115) ^ k := by\n      rw [show (2 * k : \u2115) = k + k by ring]\n      rw [pow_add]\n      <;> ring\n    have h\u2084 : (6 : \u2115) ^ (2 * k) = (6 : \u2115) ^ k * (6 : \u2115) ^ k := by\n      rw [show (2 * k : \u2115) = k + k by ring]\n      rw [pow_add]\n      <;> ring\n    have h\u2085 : (9 : \u2115) ^ (2 * k) = (9 : \u2115) ^ k * (9 : \u2115) ^ k := by\n      rw [show (2 * k : \u2115) = k + k by ring]\n      rw [pow_add]\n      <;> ring\n    rw [h\u2083, h\u2084, h\u2085]\n    have h\u2086 : (4 : \u2115) ^ k * (4 : \u2115) ^ k + (6 : \u2115) ^ k * (6 : \u2115) ^ k + (9 : \u2115) ^ k * (9 : \u2115) ^ k = ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) * ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) - 2 * ((4 : \u2115) ^ k * (6 : \u2115) ^ k + (4 : \u2115) ^ k * (9 : \u2115) ^ k + (6 : \u2115) ^ k * (9 : \u2115) ^ k) := by\n      have h\u2087 : ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) * ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) = (4 : \u2115) ^ k * (4 : \u2115) ^ k + (6 : \u2115) ^ k * (6 : \u2115) ^ k + (9 : \u2115) ^ k * (9 : \u2115) ^ k + 2 * ((4 : \u2115) ^ k * (6 : \u2115) ^ k + (4 : \u2115) ^ k * (9 : \u2115) ^ k + (6 : \u2115) ^ k * (9 : \u2115) ^ k) := by\n        ring\n      have h\u2088 : (4 : \u2115) ^ k * (6 : \u2115) ^ k + (4 : \u2115) ^ k * (9 : \u2115) ^ k + (6 : \u2115) ^ k * (9 : \u2115) ^ k = (6 : \u2115) ^ k * ((4 : \u2115) ^ k + (9 : \u2115) ^ k) + (4 : \u2115) ^ k * (9 : \u2115) ^ k := by\n        ring\n      omega\n    have h\u2087 : (4 : \u2115) ^ k * (6 : \u2115) ^ k + (4 : \u2115) ^ k * (9 : \u2115) ^ k + (6 : \u2115) ^ k * (9 : \u2115) ^ k = (6 : \u2115) ^ k * ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) := by\n      have h\u2088 : (4 : \u2115) ^ k * (6 : \u2115) ^ k + (4 : \u2115) ^ k * (9 : \u2115) ^ k + (6 : \u2115) ^ k * (9 : \u2115) ^ k = (6 : \u2115) ^ k * ((4 : \u2115) ^ k + (9 : \u2115) ^ k) + (4 : \u2115) ^ k * (9 : \u2115) ^ k := by\n        ring\n      have h\u2089 : (6 : \u2115) ^ k * ((4 : \u2115) ^ k + (9 : \u2115) ^ k) + (4 : \u2115) ^ k * (9 : \u2115) ^ k = (6 : \u2115) ^ k * ((4 : \u2115) ^ k + (6 : \u2115) ^ k + (9 : \u2115) ^ k) := by\n        have h\u2081\u2080 : (6 : \u2115) ^ k = (2 : \u2115) ^ k * (3 : \u2115) ^ k := by\n          rw [show (6 : \u2115) = 2 * 3 by norm_num]\n          rw [mul_pow]\n          <;> ring\n        rw [h\u2081\u2080]\n        have h\u2081\u2081 : (4 : \u2115) ^ k = (2 : \u2115) ^ (2 * k) := by\n          rw [show (4 : \u2115) = 2 ^ 2 by norm_num]\n          rw [\u2190 pow_mul]\n          <;> ring_nf\n        have h\u2081\u2082 : (9 : \u2115) ^ k = (3 : \u2115) ^ (2 * k) := by\n          rw [show (9 : \u2115) = 3 ^ 2 by norm_num]\n          rw [\u2190 pow_mul]\n          <;> ring_nf\n        rw [h\u2081\u2081, h\u2081\u2082]\n        have h\u2081\u2083 : (2 : \u2115) ^ k * (3 : \u2115) ^ k * ((2 : \u2115) ^ (2 * k) + (3 : \u2115) ^ (2 * k)) + (2 : \u2115) ^ (2 * k) * (3 : \u2115) ^ (2 * k) = (2 : \u2115) ^ k * (3 : \u2115) ^ k * ((2 : \u2115) ^ (2 * k) + (2 : \u2115) ^ k * (3 : \u2115) ^ k + (3 : \u2115) ^ (2 * k)) := by\n          have h\u2081\u2084 : (2 : \u2115) ^ (2 * k) = (2 : \u2115) ^ k * (2 : \u2115) ^ k := by\n            rw [show (2 * k : \u2115) = k + k by ring]\n            rw [pow_add]\n            <;> ring\n          have h\u2081\u2085 : (3 : \u2115) ^ (2 * k) = (3 : \u2115) ^ k * (3 : \u2115) ^ k := by\n            rw [show (2 * k : \u2115) = k + k by ring]\n            rw [pow_add]\n            <;> ring\n          rw [h\u2081\u2084, h\u2081\u2085]\n          ring_nf\n          <;> nlinarith [pow_pos (by norm_num : 0 < (2 : \u2115)) k, pow_pos (by norm_num : 0 < (3 : \u2115)) k]\n        rw [h\u2081\u2083]\n        <;> ring_nf\n        <;> nlinarith [pow_pos (by norm_num : 0 < (2 : \u2115)) k, pow_pos (by norm_num : 0 < (3 : \u2115)) k]\n      omega\n    rw [h\u2086, h\u2087]\n    <;> cases k with\n    | zero =>\n      norm_num [h\u2080, pow_zero, Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib]\n    | succ k =>\n      simp_all [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.add_assoc]\n      <;> ring_nf at *\n      <;> norm_num at *\n      <;> omega\n  have h_div : \u2200 k : \u2115, f k \u2223 f (2 * k) := by\n    intro k\n    have h\u2081 : f (2 * k) = f k * (f k - 2 * 6 ^ k) := h_main k\n    rw [h\u2081]\n    exact \u27e8f k - 2 * 6 ^ k, by\n      cases k with\n      | zero =>\n        simp [h\u2080, pow_zero, Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib]\n      | succ k =>\n        simp_all [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.add_assoc]\n        <;> ring_nf at *\n        <;> norm_num at *\n        <;> omega\n    \u27e9\n  have h_chain : \u2200 t : \u2115, f (2 ^ m) \u2223 f (2 ^ (m + t)) := by\n    intro t\n    have h\u2083 : \u2200 t : \u2115, f (2 ^ m) \u2223 f (2 ^ (m + t)) := by\n      intro t\n      induction' t with t ih\n      \u00b7 \n        simp [h\u2080]\n      \u00b7 \n        have h\u2084 : f (2 ^ (m + t)) \u2223 f (2 * 2 ^ (m + t)) := h_div _\n        have h\u2085 : f (2 * 2 ^ (m + t)) = f (2 ^ (m + t + 1)) := by\n          rw [show (2 * 2 ^ (m + t) : \u2115) = 2 ^ (m + t + 1) by\n            ring_nf\n            <;> simp [pow_add, pow_one, mul_add, mul_one, add_mul, one_mul]]\n        rw [h\u2085] at h\u2084\n        have h\u2086 : m + t + 1 = m + (t + 1) := by ring\n        rw [h\u2086] at h\u2084\n        exact dvd_trans ih h\u2084\n    exact h\u2083 t\n  have h_final : f (2 ^ m) \u2223 f (2 ^ n) := by\n    have h\u2083 : \u2203 t, n = m + t := by\n      use n - m\n      have h\u2084 : m \u2264 n := h\u2082\n      have h\u2085 : n - m + m = n := by\n        have h\u2086 : m \u2264 n := h\u2082\n        have h\u2087 : n - m + m = n := by\n          omega\n        exact h\u2087\n      omega\n    obtain \u27e8t, ht\u27e9 := h\u2083\n    have h\u2084 : f (2 ^ m) \u2223 f (2 ^ (m + t)) := h_chain t\n    have h\u2085 : f (2 ^ (m + t)) = f (2 ^ n) := by\n      rw [ht]\n      <;> simp [pow_add]\n      <;> ring_nf\n    rw [h\u2085] at h\u2084\n    exact h\u2084\n  exact h_final\n"}
{"mode": "clear", "steps_before": 131, "alphabet": 59, "compiles": false, "id": "aime_1983_p3", "proof_after": "theorem aime_1983_p3 (f : \u211d \u2192 \u211d)\n    (h\u2080 : \u2200 x, f x = x ^ 2 + (18 * x + 30) - 2 * Real.sqrt (x ^ 2 + (18 * x + 45)))\n    (h\u2081 : Fintype (f \u207b\u00b9' {0})) : (\u220f x in (f \u207b\u00b9' {0}).toFinset, x) = 20  :=  by\n  have h\u2082 : (-9 + Real.sqrt 61) \u2208 f \u207b\u00b9' {0} := by\n    have h\u2082\u2081 : f (-9 + Real.sqrt 61) = 0 := by\n      rw [h\u2080]\n      have h\u2082\u2082 : Real.sqrt ((-9 + Real.sqrt 61) ^ 2 + (18 * (-9 + Real.sqrt 61) + 45)) = 5 := by\n        have h\u2082\u2083 : (-9 + Real.sqrt 61) ^ 2 + (18 * (-9 + Real.sqrt 61) + 45) = 25 := by\n          nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n        rw [h\u2082\u2083]\n        rw [Real.sqrt_eq_iff_sq_eq] <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n      rw [h\u2082\u2082]\n      <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n    have h\u2082\u2083 : f (-9 + Real.sqrt 61) = 0 := h\u2082\u2081\n    have h\u2082\u2084 : (-9 + Real.sqrt 61) \u2208 f \u207b\u00b9' {0} := by\n      rw [Set.mem_preimage, Set.mem_singleton_iff]\n      exact h\u2082\u2083\n    exact h\u2082\u2084\n  have h\u2083 : (-9 - Real.sqrt 61) \u2208 f \u207b\u00b9' {0} := by\n    have h\u2083\u2081 : f (-9 - Real.sqrt 61) = 0 := by\n      rw [h\u2080]\n      have h\u2083\u2082 : Real.sqrt ((-9 - Real.sqrt 61) ^ 2 + (18 * (-9 - Real.sqrt 61) + 45)) = 5 := by\n        have h\u2083\u2083 : (-9 - Real.sqrt 61) ^ 2 + (18 * (-9 - Real.sqrt 61) + 45) = 25 := by\n          nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n        rw [h\u2083\u2083]\n        rw [Real.sqrt_eq_iff_sq_eq] <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n      rw [h\u2083\u2082]\n      <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n    have h\u2083\u2084 : f (-9 - Real.sqrt 61) = 0 := h\u2083\u2081\n    have h\u2083\u2085 : (-9 - Real.sqrt 61) \u2208 f \u207b\u00b9' {0} := by\n      rw [Set.mem_preimage, Set.mem_singleton_iff]\n      exact h\u2083\u2084\n    exact h\u2083\u2085\n  have h\u2084 : \u2200 (x : \u211d), x \u2208 f \u207b\u00b9' {0} \u2192 x = (-9 + Real.sqrt 61) \u2228 x = (-9 - Real.sqrt 61) := by\n    intro x hx\n    have h\u2085 : f x = 0 := by simpa using hx\n    have h\u2086 : f x = x ^ 2 + (18 * x + 30) - 2 * Real.sqrt (x ^ 2 + (18 * x + 45)) := by rw [h\u2080]\n    rw [h\u2086] at h\u2085\n    have h\u2087 : x ^ 2 + (18 * x + 30) - 2 * Real.sqrt (x ^ 2 + (18 * x + 45)) = 0 := by linarith\n    have h\u2088 : x ^ 2 + (18 * x + 30) = 2 * Real.sqrt (x ^ 2 + (18 * x + 45)) := by linarith\n    have h\u2089 : x ^ 2 + (18 * x + 45) \u2265 0 := by\n      by_contra h\n      have h\u2081\u2080 : Real.sqrt (x ^ 2 + (18 * x + 45)) = 0 := by\n        rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n      rw [h\u2081\u2080] at h\u2088\n      nlinarith\n    have h\u2081\u2080 : Real.sqrt (x ^ 2 + (18 * x + 45)) \u2265 0 := Real.sqrt_nonneg _\n    have h\u2081\u2081 : x ^ 2 + (18 * x + 45) \u2265 0 := by linarith\n    have h\u2081\u2082 : Real.sqrt (x ^ 2 + (18 * x + 45)) ^ 2 = x ^ 2 + (18 * x + 45) := by\n      rw [Real.sq_sqrt] <;> linarith\n    have h\u2081\u2083 : (x ^ 2 + (18 * x + 30)) ^ 2 = 4 * (x ^ 2 + (18 * x + 45)) := by\n      nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 x ^ 2 + (18 * x + 45))]\n    have h\u2081\u2084 : x ^ 2 + 18 * x + 20 = 0 := by\n      nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 x ^ 2 + (18 * x + 45))]\n    have h\u2081\u2085 : x = -9 + Real.sqrt 61 \u2228 x = -9 - Real.sqrt 61 := by\n      have h\u2081\u2086 : x = -9 + Real.sqrt 61 \u2228 x = -9 - Real.sqrt 61 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2081\u2087\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081\u2087)\n        nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n      exact h\u2081\u2086\n    cases h\u2081\u2085 with\n    | inl h\u2081\u2085 =>\n      exact Or.inl h\u2081\u2085\n    | inr h\u2081\u2085 =>\n      exact Or.inr h\u2081\u2085\n  have h\u2085 : (f \u207b\u00b9' {0}).toFinset = { -9 + Real.sqrt 61, -9 - Real.sqrt 61 } := by\n    apply Finset.ext\n    intro x\n    simp only [Set.mem_toFinset, Set.mem_preimage, Set.mem_singleton_iff, Finset.mem_insert,\n      Finset.mem_singleton]\n    constructor\n    \u00b7 intro h\n      have h\u2085\u2081 : x = (-9 + Real.sqrt 61) \u2228 x = (-9 - Real.sqrt 61) := h\u2084 x h\n      cases h\u2085\u2081 with\n      | inl h\u2085\u2081 =>\n        simp [h\u2085\u2081]\n        <;>\n        (try norm_num) <;>\n        (try ring_nf at * <;> norm_num at * <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]) <;>\n        (try aesop)\n      | inr h\u2085\u2081 =>\n        simp [h\u2085\u2081]\n        <;>\n        (try norm_num) <;>\n        (try ring_nf at * <;> norm_num at * <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]) <;>\n        (try aesop)\n    \u00b7 intro h\n      have h\u2085\u2082 : x = -9 + Real.sqrt 61 \u2228 x = -9 - Real.sqrt 61 := by aesop\n      cases h\u2085\u2082 with\n      | inl h\u2085\u2082 =>\n        rw [h\u2085\u2082]\n        exact by simpa using h\u2082\n      | inr h\u2085\u2082 =>\n        rw [h\u2085\u2082]\n        exact by simpa using h\u2083\n  have h\u2086 : (\u220f x in (f \u207b\u00b9' {0}).toFinset, x) = 20 := by\n    rw [h\u2085]\n    have h\u2086\u2081 : \u220f x in ({ -9 + Real.sqrt 61, -9 - Real.sqrt 61 } : Finset \u211d), x = 20 := by\n      have h\u2086\u2082 : \u220f x in ({ -9 + Real.sqrt 61, -9 - Real.sqrt 61 } : Finset \u211d), x = ((-9 + Real.sqrt 61) * (-9 - Real.sqrt 61)) := by\n        simp [Finset.prod_pair (show (-9 + Real.sqrt 61 : \u211d) \u2260 -9 - Real.sqrt 61 by\n          intro h\n          nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)])]\n        <;>\n        ring_nf <;>\n        norm_num <;>\n        nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n      rw [h\u2086\u2082]\n      have h\u2086\u2083 : ((-9 + Real.sqrt 61) * (-9 - Real.sqrt 61) : \u211d) = 20 := by\n        nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n      rw [h\u2086\u2083]\n      <;>\n      norm_num\n    rw [h\u2086\u2081]\n    <;>\n    norm_num\n  exact h\u2086\n", "proof_before": "theorem aime_1983_p3 (f : \u211d \u2192 \u211d)\n    (h\u2080 : \u2200 x, f x = x ^ 2 + (18 * x + 30) - 2 * Real.sqrt (x ^ 2 + (18 * x + 45)))\n    (h\u2081 : Fintype (f \u207b\u00b9' {0})) : (\u220f x in (f \u207b\u00b9' {0}).toFinset, x) = 20  :=  by\n  have h\u2082 : (-9 + Real.sqrt 61) \u2208 f \u207b\u00b9' {0} := by\n    have h\u2082\u2081 : f (-9 + Real.sqrt 61) = 0 := by\n      rw [h\u2080]\n      have h\u2082\u2082 : Real.sqrt ((-9 + Real.sqrt 61) ^ 2 + (18 * (-9 + Real.sqrt 61) + 45)) = 5 := by\n        have h\u2082\u2083 : (-9 + Real.sqrt 61) ^ 2 + (18 * (-9 + Real.sqrt 61) + 45) = 25 := by\n          nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n        rw [h\u2082\u2083]\n        rw [Real.sqrt_eq_iff_sq_eq] <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n      rw [h\u2082\u2082]\n      <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n    have h\u2082\u2083 : f (-9 + Real.sqrt 61) = 0 := h\u2082\u2081\n    have h\u2082\u2084 : (-9 + Real.sqrt 61) \u2208 f \u207b\u00b9' {0} := by\n      rw [Set.mem_preimage, Set.mem_singleton_iff]\n      exact h\u2082\u2083\n    exact h\u2082\u2084\n  have h\u2083 : (-9 - Real.sqrt 61) \u2208 f \u207b\u00b9' {0} := by\n    have h\u2083\u2081 : f (-9 - Real.sqrt 61) = 0 := by\n      rw [h\u2080]\n      have h\u2083\u2082 : Real.sqrt ((-9 - Real.sqrt 61) ^ 2 + (18 * (-9 - Real.sqrt 61) + 45)) = 5 := by\n        have h\u2083\u2083 : (-9 - Real.sqrt 61) ^ 2 + (18 * (-9 - Real.sqrt 61) + 45) = 25 := by\n          nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n        rw [h\u2083\u2083]\n        rw [Real.sqrt_eq_iff_sq_eq] <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n      rw [h\u2083\u2082]\n      <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n    have h\u2083\u2084 : f (-9 - Real.sqrt 61) = 0 := h\u2083\u2081\n    have h\u2083\u2085 : (-9 - Real.sqrt 61) \u2208 f \u207b\u00b9' {0} := by\n      rw [Set.mem_preimage, Set.mem_singleton_iff]\n      exact h\u2083\u2084\n    exact h\u2083\u2085\n  have h\u2084 : \u2200 (x : \u211d), x \u2208 f \u207b\u00b9' {0} \u2192 x = (-9 + Real.sqrt 61) \u2228 x = (-9 - Real.sqrt 61) := by\n    intro x hx\n    have h\u2085 : f x = 0 := by simpa using hx\n    have h\u2086 : f x = x ^ 2 + (18 * x + 30) - 2 * Real.sqrt (x ^ 2 + (18 * x + 45)) := by rw [h\u2080]\n    rw [h\u2086] at h\u2085\n    have h\u2087 : x ^ 2 + (18 * x + 30) - 2 * Real.sqrt (x ^ 2 + (18 * x + 45)) = 0 := by linarith\n    have h\u2088 : x ^ 2 + (18 * x + 30) = 2 * Real.sqrt (x ^ 2 + (18 * x + 45)) := by linarith\n    have h\u2089 : x ^ 2 + (18 * x + 45) \u2265 0 := by\n      by_contra h\n      have h\u2081\u2080 : Real.sqrt (x ^ 2 + (18 * x + 45)) = 0 := by\n        rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n      rw [h\u2081\u2080] at h\u2088\n      nlinarith\n    have h\u2081\u2080 : Real.sqrt (x ^ 2 + (18 * x + 45)) \u2265 0 := Real.sqrt_nonneg _\n    have h\u2081\u2081 : x ^ 2 + (18 * x + 45) \u2265 0 := by linarith\n    have h\u2081\u2082 : Real.sqrt (x ^ 2 + (18 * x + 45)) ^ 2 = x ^ 2 + (18 * x + 45) := by\n      rw [Real.sq_sqrt] <;> linarith\n    have h\u2081\u2083 : (x ^ 2 + (18 * x + 30)) ^ 2 = 4 * (x ^ 2 + (18 * x + 45)) := by\n      nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 x ^ 2 + (18 * x + 45))]\n    have h\u2081\u2084 : x ^ 2 + 18 * x + 20 = 0 := by\n      nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 x ^ 2 + (18 * x + 45))]\n    have h\u2081\u2085 : x = -9 + Real.sqrt 61 \u2228 x = -9 - Real.sqrt 61 := by\n      have h\u2081\u2086 : x = -9 + Real.sqrt 61 \u2228 x = -9 - Real.sqrt 61 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2081\u2087\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081\u2087)\n        nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n      exact h\u2081\u2086\n    cases h\u2081\u2085 with\n    | inl h\u2081\u2085 =>\n      exact Or.inl h\u2081\u2085\n    | inr h\u2081\u2085 =>\n      exact Or.inr h\u2081\u2085\n  have h\u2085 : (f \u207b\u00b9' {0}).toFinset = { -9 + Real.sqrt 61, -9 - Real.sqrt 61 } := by\n    apply Finset.ext\n    intro x\n    simp only [Set.mem_toFinset, Set.mem_preimage, Set.mem_singleton_iff, Finset.mem_insert,\n      Finset.mem_singleton]\n    constructor\n    \u00b7 intro h\n      have h\u2085\u2081 : x = (-9 + Real.sqrt 61) \u2228 x = (-9 - Real.sqrt 61) := h\u2084 x h\n      cases h\u2085\u2081 with\n      | inl h\u2085\u2081 =>\n        simp [h\u2085\u2081]\n        <;>\n        (try norm_num) <;>\n        (try ring_nf at * <;> norm_num at * <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]) <;>\n        (try aesop)\n      | inr h\u2085\u2081 =>\n        simp [h\u2085\u2081]\n        <;>\n        (try norm_num) <;>\n        (try ring_nf at * <;> norm_num at * <;> nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]) <;>\n        (try aesop)\n    \u00b7 intro h\n      have h\u2085\u2082 : x = -9 + Real.sqrt 61 \u2228 x = -9 - Real.sqrt 61 := by aesop\n      cases h\u2085\u2082 with\n      | inl h\u2085\u2082 =>\n        rw [h\u2085\u2082]\n        exact by simpa using h\u2082\n      | inr h\u2085\u2082 =>\n        rw [h\u2085\u2082]\n        exact by simpa using h\u2083\n  have h\u2086 : (\u220f x in (f \u207b\u00b9' {0}).toFinset, x) = 20 := by\n    rw [h\u2085]\n    have h\u2086\u2081 : \u220f x in ({ -9 + Real.sqrt 61, -9 - Real.sqrt 61 } : Finset \u211d), x = 20 := by\n      have h\u2086\u2082 : \u220f x in ({ -9 + Real.sqrt 61, -9 - Real.sqrt 61 } : Finset \u211d), x = ((-9 + Real.sqrt 61) * (-9 - Real.sqrt 61)) := by\n        simp [Finset.prod_pair (show (-9 + Real.sqrt 61 : \u211d) \u2260 -9 - Real.sqrt 61 by\n          intro h\n          nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)])]\n        <;>\n        ring_nf <;>\n        norm_num <;>\n        nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n      rw [h\u2086\u2082]\n      have h\u2086\u2083 : ((-9 + Real.sqrt 61) * (-9 - Real.sqrt 61) : \u211d) = 20 := by\n        nlinarith [Real.sqrt_nonneg 61, Real.sq_sqrt (show 0 \u2264 61 by norm_num)]\n      rw [h\u2086\u2083]\n      <;>\n      norm_num\n    rw [h\u2086\u2081]\n    <;>\n    norm_num\n  exact h\u2086\n"}
{"mode": "clear", "steps_before": 87, "alphabet": 50, "compiles": false, "id": "amc12a_2009_p7", "proof_after": "theorem amc12a_2009_p7 (x : \u211d) (n : \u2115) (a : \u2115 \u2192 \u211d)\n    (h\u2081 : \u2200 m, a (m + 1) - a m = a (m + 2) - a (m + 1)) (h\u2082 : a 1 = 2 * x - 3)\n    (h\u2083 : a 2 = 5 * x - 11) (h\u2084 : a 3 = 3 * x + 1) (h\u2085 : a n = 2009) : n = 502  :=  by\n  have h_x : x = 4 := by\n    have h\u2085\u2080 := h\u2081 1\n    have h\u2085\u2081 := h\u2081 0\n    have h\u2085\u2082 := h\u2081 2\n    norm_num at h\u2085\u2080 h\u2085\u2081 h\u2085\u2082\n    simp [h\u2082, h\u2083, h\u2084] at h\u2085\u2080 h\u2085\u2081 h\u2085\u2082\n    linarith\n  have h_a0 : a 0 = 1 := by\n    have h\u2085\u2081 := h\u2081 0\n    have h\u2085\u2082 := h\u2081 1\n    have h\u2085\u2083 := h\u2081 2\n    norm_num at h\u2085\u2081 h\u2085\u2082 h\u2085\u2083 \u22a2\n    simp [h\u2082, h\u2083, h\u2084, h_x] at h\u2085\u2081 h\u2085\u2082 h\u2085\u2083 \u22a2\n    linarith\n  have h_diff : \u2200 (m : \u2115), a (m + 1) - a m = (4 : \u211d) := by\n    intro m\n    have h\u2082\u2080 : a (m + 1) - a m = a 1 - a 0 := by\n      have h\u2082\u2081 : \u2200 k : \u2115, a (k + 1) - a k = a 1 - a 0 := by\n        intro k\n        induction k with\n        | zero =>\n          norm_num [h_a0, h\u2082, h_x] at h\u2085 \u22a2 <;>\n            (try ring_nf at h\u2085 \u22a2 <;>\n              try linarith) <;>\n            (try simp_all [h\u2081, h\u2082, h\u2083, h\u2084, h_x]) <;>\n            (try ring_nf at * <;>\n              linarith)\n        | succ k ih =>\n          have h\u2082\u2082 := h\u2081 k\n          have h\u2082\u2083 := h\u2081 (k + 1)\n          have h\u2082\u2084 := h\u2081 (k + 2)\n          simp [ih, Nat.add_assoc] at h\u2082\u2082 h\u2082\u2083 h\u2082\u2084 \u22a2\n          <;>\n            (try ring_nf at h\u2082\u2082 h\u2082\u2083 h\u2082\u2084 \u22a2 <;>\n              linarith)\n      exact h\u2082\u2081 m\n    have h\u2082\u2081 : a 1 - a 0 = (4 : \u211d) := by\n      norm_num [h\u2082, h_a0, h_x] at *\n      <;> linarith\n    rw [h\u2082\u2080, h\u2082\u2081]\n    <;> norm_num\n  have h_a_formula : \u2200 (m : \u2115), a m = (1 : \u211d) + 4 * (m : \u211d) := by\n    intro m\n    have h\u2082 : \u2200 (m : \u2115), a m = (1 : \u211d) + 4 * (m : \u211d) := by\n      intro m\n      induction m with\n      | zero =>\n        norm_num [h_a0] at *\n        <;> simp_all\n        <;> ring_nf at *\n        <;> linarith\n      | succ m ih =>\n        have h\u2083 := h_diff m\n        have h\u2084 := h_diff (m + 1)\n        simp [ih, Nat.cast_add, Nat.cast_one, Nat.cast_zero, add_assoc] at h\u2083 h\u2084 \u22a2\n        <;>\n        (try ring_nf at * <;> linarith)\n        <;>\n        (try simp_all [h\u2081, h\u2082, h\u2083, h\u2084, h_x, h_a0, h_diff, h_a_formula] <;>\n          ring_nf at * <;>\n          linarith)\n    exact h\u2082 m\n  have h_n : n = 502 := by\n    have h\u2086 : a n = (1 : \u211d) + 4 * (n : \u211d) := h_a_formula n\n    rw [h\u2086] at h\u2085\n    have h\u2087 : (n : \u211d) = 502 := by\n      norm_num at h\u2085 \u22a2\n      <;> ring_nf at h\u2085 \u22a2 <;>\n      (try linarith) <;>\n      (try\n        {\n          have h\u2088 : (n : \u211d) = 502 := by\n            linarith\n          exact h\u2088\n        })\n    have h\u2088 : (n : \u2115) = 502 := by\n      norm_cast at h\u2087 \u22a2\n      <;>\n      (try linarith) <;>\n      (try\n        {\n          ring_nf at h\u2087 \u22a2\n          <;>\n          norm_num at h\u2087 \u22a2\n          <;>\n          (try linarith)\n          <;>\n          (try\n            {\n              omega\n            })\n        })\n    exact h\u2088\n  apply h_n\n", "proof_before": "theorem amc12a_2009_p7 (x : \u211d) (n : \u2115) (a : \u2115 \u2192 \u211d)\n    (h\u2081 : \u2200 m, a (m + 1) - a m = a (m + 2) - a (m + 1)) (h\u2082 : a 1 = 2 * x - 3)\n    (h\u2083 : a 2 = 5 * x - 11) (h\u2084 : a 3 = 3 * x + 1) (h\u2085 : a n = 2009) : n = 502  :=  by\n  have h_x : x = 4 := by\n    have h\u2085\u2080 := h\u2081 1\n    have h\u2085\u2081 := h\u2081 0\n    have h\u2085\u2082 := h\u2081 2\n    norm_num at h\u2085\u2080 h\u2085\u2081 h\u2085\u2082\n    simp [h\u2082, h\u2083, h\u2084] at h\u2085\u2080 h\u2085\u2081 h\u2085\u2082\n    linarith\n  have h_a0 : a 0 = 1 := by\n    have h\u2085\u2081 := h\u2081 0\n    have h\u2085\u2082 := h\u2081 1\n    have h\u2085\u2083 := h\u2081 2\n    norm_num at h\u2085\u2081 h\u2085\u2082 h\u2085\u2083 \u22a2\n    simp [h\u2082, h\u2083, h\u2084, h_x] at h\u2085\u2081 h\u2085\u2082 h\u2085\u2083 \u22a2\n    linarith\n  have h_diff : \u2200 (m : \u2115), a (m + 1) - a m = (4 : \u211d) := by\n    intro m\n    have h\u2082\u2080 : a (m + 1) - a m = a 1 - a 0 := by\n      have h\u2082\u2081 : \u2200 k : \u2115, a (k + 1) - a k = a 1 - a 0 := by\n        intro k\n        induction k with\n        | zero =>\n          norm_num [h_a0, h\u2082, h_x] at h\u2085 \u22a2 <;>\n            (try ring_nf at h\u2085 \u22a2 <;>\n              try linarith) <;>\n            (try simp_all [h\u2081, h\u2082, h\u2083, h\u2084, h_x]) <;>\n            (try ring_nf at * <;>\n              linarith)\n        | succ k ih =>\n          have h\u2082\u2082 := h\u2081 k\n          have h\u2082\u2083 := h\u2081 (k + 1)\n          have h\u2082\u2084 := h\u2081 (k + 2)\n          simp [ih, Nat.add_assoc] at h\u2082\u2082 h\u2082\u2083 h\u2082\u2084 \u22a2\n          <;>\n            (try ring_nf at h\u2082\u2082 h\u2082\u2083 h\u2082\u2084 \u22a2 <;>\n              linarith)\n      exact h\u2082\u2081 m\n    have h\u2082\u2081 : a 1 - a 0 = (4 : \u211d) := by\n      norm_num [h\u2082, h_a0, h_x] at *\n      <;> linarith\n    rw [h\u2082\u2080, h\u2082\u2081]\n    <;> norm_num\n  have h_a_formula : \u2200 (m : \u2115), a m = (1 : \u211d) + 4 * (m : \u211d) := by\n    intro m\n    have h\u2082 : \u2200 (m : \u2115), a m = (1 : \u211d) + 4 * (m : \u211d) := by\n      intro m\n      induction m with\n      | zero =>\n        norm_num [h_a0] at *\n        <;> simp_all\n        <;> ring_nf at *\n        <;> linarith\n      | succ m ih =>\n        have h\u2083 := h_diff m\n        have h\u2084 := h_diff (m + 1)\n        simp [ih, Nat.cast_add, Nat.cast_one, Nat.cast_zero, add_assoc] at h\u2083 h\u2084 \u22a2\n        <;>\n        (try ring_nf at * <;> linarith)\n        <;>\n        (try simp_all [h\u2081, h\u2082, h\u2083, h\u2084, h_x, h_a0, h_diff, h_a_formula] <;>\n          ring_nf at * <;>\n          linarith)\n    exact h\u2082 m\n  have h_n : n = 502 := by\n    have h\u2086 : a n = (1 : \u211d) + 4 * (n : \u211d) := h_a_formula n\n    rw [h\u2086] at h\u2085\n    have h\u2087 : (n : \u211d) = 502 := by\n      norm_num at h\u2085 \u22a2\n      <;> ring_nf at h\u2085 \u22a2 <;>\n      (try linarith) <;>\n      (try\n        {\n          have h\u2088 : (n : \u211d) = 502 := by\n            linarith\n          exact h\u2088\n        })\n    have h\u2088 : (n : \u2115) = 502 := by\n      norm_cast at h\u2087 \u22a2\n      <;>\n      (try linarith) <;>\n      (try\n        {\n          ring_nf at h\u2087 \u22a2\n          <;>\n          norm_num at h\u2087 \u22a2\n          <;>\n          (try linarith)\n          <;>\n          (try\n            {\n              omega\n            })\n        })\n    exact h\u2088\n  apply h_n\n"}
{"mode": "clear", "steps_before": 108, "alphabet": 38, "compiles": false, "id": "algebra_absapbon1pabsapbleqsumabsaon1pabsa", "proof_after": "theorem algebra_absapbon1pabsapbleqsumabsaon1pabsa (a b : \u211d) :\n    abs (a + b) / (1 + abs (a + b)) \u2264 abs a / (1 + abs a) + abs b / (1 + abs b)  :=  by\n  have h\u2081 : abs (a + b) \u2264 abs a + abs b := by\n    calc\n      abs (a + b) \u2264 abs a + abs b := by\n        exact abs_add a b\n      _ = abs a + abs b := by rfl\n  have h\u2082 : \u2200 (x y : \u211d), 0 \u2264 x \u2192 x \u2264 y \u2192 x / (1 + x) \u2264 y / (1 + y) := by\n    intro x y hx hxy\n    have h\u2083 : 0 \u2264 x := hx\n    have h\u2084 : x \u2264 y := hxy\n    have h\u2085 : 0 \u2264 y := by linarith\n    have h\u2086 : 0 \u2264 1 + x := by linarith\n    have h\u2087 : 0 \u2264 1 + y := by linarith\n    have h\u2088 : 0 < 1 + x := by linarith\n    have h\u2089 : 0 < 1 + y := by linarith\n    have h\u2081\u2080 : x / (1 + x) \u2264 y / (1 + y) := by\n      rw [div_le_div_iff (by positivity) (by positivity)]\n      nlinarith [mul_nonneg h\u2083 (sub_nonneg.mpr h\u2084), mul_nonneg h\u2085 h\u2083,\n        mul_nonneg h\u2085 (sub_nonneg.mpr h\u2084)]\n    exact h\u2081\u2080\n  have h\u2083 : abs (a + b) / (1 + abs (a + b)) \u2264 (abs a + abs b) / (1 + (abs a + abs b)) := by\n    have h\u2084 : 0 \u2264 abs (a + b) := abs_nonneg (a + b)\n    have h\u2085 : 0 \u2264 abs a + abs b := by positivity\n    have h\u2086 : abs (a + b) \u2264 abs a + abs b := h\u2081\n    have h\u2087 : 0 \u2264 1 + abs (a + b) := by positivity\n    have h\u2088 : 0 \u2264 1 + (abs a + abs b) := by positivity\n    have h\u2089 : 0 < 1 + abs (a + b) := by positivity\n    have h\u2081\u2080 : 0 < 1 + (abs a + abs b) := by positivity\n    have h\u2081\u2081 : abs (a + b) / (1 + abs (a + b)) \u2264 (abs a + abs b) / (1 + (abs a + abs b)) := by\n      have h\u2081\u2082 : abs (a + b) / (1 + abs (a + b)) \u2264 (abs a + abs b) / (1 + (abs a + abs b)) := by\n        have h\u2081\u2083 : 0 \u2264 abs (a + b) := by positivity\n        have h\u2081\u2084 : abs (a + b) \u2264 abs a + abs b := h\u2081\n        have h\u2081\u2085 : 0 \u2264 abs a + abs b := by positivity\n        have h\u2081\u2086 : 0 < 1 + abs (a + b) := by positivity\n        have h\u2081\u2087 : 0 < 1 + (abs a + abs b) := by positivity\n        have h\u2081\u2088 : abs (a + b) / (1 + abs (a + b)) \u2264 (abs a + abs b) / (1 + (abs a + abs b)) := by\n          apply h\u2082\n          <;> try norm_num\n          <;> linarith\n        exact h\u2081\u2088\n      exact h\u2081\u2082\n    exact h\u2081\u2081\n  have h\u2084 : (abs a + abs b) / (1 + (abs a + abs b)) \u2264 abs a / (1 + abs a) + abs b / (1 + abs b) := by\n    have h\u2085 : 0 \u2264 abs a := abs_nonneg a\n    have h\u2086 : 0 \u2264 abs b := abs_nonneg b\n    have h\u2087 : 0 \u2264 abs a + abs b := by positivity\n    have h\u2088 : 0 < 1 + abs a := by positivity\n    have h\u2089 : 0 < 1 + abs b := by positivity\n    have h\u2081\u2080 : 0 < 1 + (abs a + abs b) := by positivity\n    have h\u2081\u2081 : (abs a + abs b) / (1 + (abs a + abs b)) = abs a / (1 + (abs a + abs b)) + abs b / (1 + (abs a + abs b)) := by\n      field_simp [h\u2088, h\u2089, h\u2081\u2080]\n      <;> ring\n      <;> field_simp [h\u2088, h\u2089, h\u2081\u2080]\n      <;> ring\n    rw [h\u2081\u2081]\n    have h\u2081\u2082 : abs a / (1 + (abs a + abs b)) \u2264 abs a / (1 + abs a) := by\n      have h\u2081\u2083 : 0 \u2264 abs a := abs_nonneg a\n      have h\u2081\u2084 : 0 < 1 + abs a := by positivity\n      have h\u2081\u2085 : 0 < 1 + (abs a + abs b) := by positivity\n      have h\u2081\u2086 : 0 < 1 + abs a + abs b := by positivity\n      have h\u2081\u2087 : 1 + abs a \u2264 1 + (abs a + abs b) := by\n        nlinarith [h\u2086]\n      have h\u2081\u2088 : abs a / (1 + (abs a + abs b)) \u2264 abs a / (1 + abs a) := by\n        apply div_le_div_of_le_left (by positivity) (by positivity)\n        <;> nlinarith\n      exact h\u2081\u2088\n    have h\u2081\u2083 : abs b / (1 + (abs a + abs b)) \u2264 abs b / (1 + abs b) := by\n      have h\u2081\u2084 : 0 \u2264 abs b := abs_nonneg b\n      have h\u2081\u2085 : 0 < 1 + abs b := by positivity\n      have h\u2081\u2086 : 0 < 1 + (abs a + abs b) := by positivity\n      have h\u2081\u2087 : 0 < 1 + abs a + abs b := by positivity\n      have h\u2081\u2088 : 1 + abs b \u2264 1 + (abs a + abs b) := by\n        nlinarith [h\u2085]\n      have h\u2081\u2089 : abs b / (1 + (abs a + abs b)) \u2264 abs b / (1 + abs b) := by\n        apply div_le_div_of_le_left (by positivity) (by positivity)\n        <;> nlinarith\n      exact h\u2081\u2089\n    have h\u2082\u2080 : abs a / (1 + (abs a + abs b)) + abs b / (1 + (abs a + abs b)) \u2264 abs a / (1 + abs a) + abs b / (1 + abs b) := by\n      linarith\n    linarith\n  have h\u2085 : abs (a + b) / (1 + abs (a + b)) \u2264 abs a / (1 + abs a) + abs b / (1 + abs b) := by\n    calc\n      abs (a + b) / (1 + abs (a + b)) \u2264 (abs a + abs b) / (1 + (abs a + abs b)) := h\u2083\n      _ \u2264 abs a / (1 + abs a) + abs b / (1 + abs b) := h\u2084\n  exact h\u2085\n", "proof_before": "theorem algebra_absapbon1pabsapbleqsumabsaon1pabsa (a b : \u211d) :\n    abs (a + b) / (1 + abs (a + b)) \u2264 abs a / (1 + abs a) + abs b / (1 + abs b)  :=  by\n  have h\u2081 : abs (a + b) \u2264 abs a + abs b := by\n    calc\n      abs (a + b) \u2264 abs a + abs b := by\n        exact abs_add a b\n      _ = abs a + abs b := by rfl\n  have h\u2082 : \u2200 (x y : \u211d), 0 \u2264 x \u2192 x \u2264 y \u2192 x / (1 + x) \u2264 y / (1 + y) := by\n    intro x y hx hxy\n    have h\u2083 : 0 \u2264 x := hx\n    have h\u2084 : x \u2264 y := hxy\n    have h\u2085 : 0 \u2264 y := by linarith\n    have h\u2086 : 0 \u2264 1 + x := by linarith\n    have h\u2087 : 0 \u2264 1 + y := by linarith\n    have h\u2088 : 0 < 1 + x := by linarith\n    have h\u2089 : 0 < 1 + y := by linarith\n    have h\u2081\u2080 : x / (1 + x) \u2264 y / (1 + y) := by\n      rw [div_le_div_iff (by positivity) (by positivity)]\n      nlinarith [mul_nonneg h\u2083 (sub_nonneg.mpr h\u2084), mul_nonneg h\u2085 h\u2083,\n        mul_nonneg h\u2085 (sub_nonneg.mpr h\u2084)]\n    exact h\u2081\u2080\n  have h\u2083 : abs (a + b) / (1 + abs (a + b)) \u2264 (abs a + abs b) / (1 + (abs a + abs b)) := by\n    have h\u2084 : 0 \u2264 abs (a + b) := abs_nonneg (a + b)\n    have h\u2085 : 0 \u2264 abs a + abs b := by positivity\n    have h\u2086 : abs (a + b) \u2264 abs a + abs b := h\u2081\n    have h\u2087 : 0 \u2264 1 + abs (a + b) := by positivity\n    have h\u2088 : 0 \u2264 1 + (abs a + abs b) := by positivity\n    have h\u2089 : 0 < 1 + abs (a + b) := by positivity\n    have h\u2081\u2080 : 0 < 1 + (abs a + abs b) := by positivity\n    have h\u2081\u2081 : abs (a + b) / (1 + abs (a + b)) \u2264 (abs a + abs b) / (1 + (abs a + abs b)) := by\n      have h\u2081\u2082 : abs (a + b) / (1 + abs (a + b)) \u2264 (abs a + abs b) / (1 + (abs a + abs b)) := by\n        have h\u2081\u2083 : 0 \u2264 abs (a + b) := by positivity\n        have h\u2081\u2084 : abs (a + b) \u2264 abs a + abs b := h\u2081\n        have h\u2081\u2085 : 0 \u2264 abs a + abs b := by positivity\n        have h\u2081\u2086 : 0 < 1 + abs (a + b) := by positivity\n        have h\u2081\u2087 : 0 < 1 + (abs a + abs b) := by positivity\n        have h\u2081\u2088 : abs (a + b) / (1 + abs (a + b)) \u2264 (abs a + abs b) / (1 + (abs a + abs b)) := by\n          apply h\u2082\n          <;> try norm_num\n          <;> linarith\n        exact h\u2081\u2088\n      exact h\u2081\u2082\n    exact h\u2081\u2081\n  have h\u2084 : (abs a + abs b) / (1 + (abs a + abs b)) \u2264 abs a / (1 + abs a) + abs b / (1 + abs b) := by\n    have h\u2085 : 0 \u2264 abs a := abs_nonneg a\n    have h\u2086 : 0 \u2264 abs b := abs_nonneg b\n    have h\u2087 : 0 \u2264 abs a + abs b := by positivity\n    have h\u2088 : 0 < 1 + abs a := by positivity\n    have h\u2089 : 0 < 1 + abs b := by positivity\n    have h\u2081\u2080 : 0 < 1 + (abs a + abs b) := by positivity\n    have h\u2081\u2081 : (abs a + abs b) / (1 + (abs a + abs b)) = abs a / (1 + (abs a + abs b)) + abs b / (1 + (abs a + abs b)) := by\n      field_simp [h\u2088, h\u2089, h\u2081\u2080]\n      <;> ring\n      <;> field_simp [h\u2088, h\u2089, h\u2081\u2080]\n      <;> ring\n    rw [h\u2081\u2081]\n    have h\u2081\u2082 : abs a / (1 + (abs a + abs b)) \u2264 abs a / (1 + abs a) := by\n      have h\u2081\u2083 : 0 \u2264 abs a := abs_nonneg a\n      have h\u2081\u2084 : 0 < 1 + abs a := by positivity\n      have h\u2081\u2085 : 0 < 1 + (abs a + abs b) := by positivity\n      have h\u2081\u2086 : 0 < 1 + abs a + abs b := by positivity\n      have h\u2081\u2087 : 1 + abs a \u2264 1 + (abs a + abs b) := by\n        nlinarith [h\u2086]\n      have h\u2081\u2088 : abs a / (1 + (abs a + abs b)) \u2264 abs a / (1 + abs a) := by\n        apply div_le_div_of_le_left (by positivity) (by positivity)\n        <;> nlinarith\n      exact h\u2081\u2088\n    have h\u2081\u2083 : abs b / (1 + (abs a + abs b)) \u2264 abs b / (1 + abs b) := by\n      have h\u2081\u2084 : 0 \u2264 abs b := abs_nonneg b\n      have h\u2081\u2085 : 0 < 1 + abs b := by positivity\n      have h\u2081\u2086 : 0 < 1 + (abs a + abs b) := by positivity\n      have h\u2081\u2087 : 0 < 1 + abs a + abs b := by positivity\n      have h\u2081\u2088 : 1 + abs b \u2264 1 + (abs a + abs b) := by\n        nlinarith [h\u2085]\n      have h\u2081\u2089 : abs b / (1 + (abs a + abs b)) \u2264 abs b / (1 + abs b) := by\n        apply div_le_div_of_le_left (by positivity) (by positivity)\n        <;> nlinarith\n      exact h\u2081\u2089\n    have h\u2082\u2080 : abs a / (1 + (abs a + abs b)) + abs b / (1 + (abs a + abs b)) \u2264 abs a / (1 + abs a) + abs b / (1 + abs b) := by\n      linarith\n    linarith\n  have h\u2085 : abs (a + b) / (1 + abs (a + b)) \u2264 abs a / (1 + abs a) + abs b / (1 + abs b) := by\n    calc\n      abs (a + b) / (1 + abs (a + b)) \u2264 (abs a + abs b) / (1 + (abs a + abs b)) := h\u2083\n      _ \u2264 abs a / (1 + abs a) + abs b / (1 + abs b) := h\u2084\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 32, "alphabet": 24, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 19, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 5, "fixpoint": true, "chars_before": 1242, "chars_after": 891, "seconds": 363.6, "id": "imo_1959_p1", "proof_after": "theorem imo_1959_p1 (n : \u2115) (h\u2080 : 0 < n) : Nat.gcd (21 * n + 4) (14 * n + 3) = 1  :=  by\n  have h\u2081 : Nat.gcd (21 * n + 4) (14 * n + 3) = Nat.gcd (14 * n + 3) (7 * n + 1) := by\n    have h\u2081\u2080 : Nat.gcd (21 * n + 4) (14 * n + 3) = Nat.gcd (14 * n + 3) (7 * n + 1) := by\n      rw [show 21 * n + 4 = 1 * (14 * n + 3) + (7 * n + 1) by\n        ring_nf\n        <;> omega]\n      simp [Nat.gcd_comm, Nat.gcd_add_mul_right_right, Nat.gcd_assoc]\n    omega\n  have h\u2082 : Nat.gcd (14 * n + 3) (7 * n + 1) = Nat.gcd (7 * n + 1) 1 := by\n    have h\u2082\u2081 : Nat.gcd (14 * n + 3) (7 * n + 1) = Nat.gcd (7 * n + 1) 1 := by\n      have h\u2082\u2081\u2081 : Nat.gcd (14 * n + 3) (7 * n + 1) = Nat.gcd (7 * n + 1) 1 := by\n        rw [show 14 * n + 3 = 2 * (7 * n + 1) + 1 by\n          ring_nf\n          <;> omega]\n        <;> simp [Nat.gcd_comm, Nat.gcd_add_mul_right_right, Nat.gcd_assoc]\n        <;> ring_nf at *\n        <;> omega\n      exact h\u2082\u2081\u2081\n    exact h\u2082\u2081\n  have h\u2083 : Nat.gcd (7 * n + 1) 1 = 1 := by\n    simp\n  omega\n", "proof_before": "theorem imo_1959_p1 (n : \u2115) (h\u2080 : 0 < n) : Nat.gcd (21 * n + 4) (14 * n + 3) = 1  :=  by\n  have h\u2081 : Nat.gcd (21 * n + 4) (14 * n + 3) = Nat.gcd (14 * n + 3) (7 * n + 1) := by\n    have h\u2081\u2080 : Nat.gcd (21 * n + 4) (14 * n + 3) = Nat.gcd (14 * n + 3) (7 * n + 1) := by\n      rw [show 21 * n + 4 = 1 * (14 * n + 3) + (7 * n + 1) by\n        ring_nf\n        <;> omega]\n      rw [Nat.gcd_comm]\n      <;> simp [Nat.gcd_comm, Nat.gcd_add_mul_right_right, Nat.gcd_assoc, Nat.gcd_assoc]\n      <;> ring_nf at *\n      <;> omega\n    exact h\u2081\u2080\n    <;> simp_all\n  have h\u2082 : Nat.gcd (14 * n + 3) (7 * n + 1) = Nat.gcd (7 * n + 1) 1 := by\n    have h\u2082\u2081 : Nat.gcd (14 * n + 3) (7 * n + 1) = Nat.gcd (7 * n + 1) 1 := by\n      have h\u2082\u2081\u2081 : Nat.gcd (14 * n + 3) (7 * n + 1) = Nat.gcd (7 * n + 1) 1 := by\n        rw [show 14 * n + 3 = 2 * (7 * n + 1) + 1 by\n          ring_nf\n          <;> omega]\n        <;> simp [Nat.gcd_comm, Nat.gcd_add_mul_right_right, Nat.gcd_assoc]\n        <;> ring_nf at *\n        <;> omega\n      exact h\u2082\u2081\u2081\n    exact h\u2082\u2081\n  have h\u2083 : Nat.gcd (7 * n + 1) 1 = 1 := by\n    have h\u2083\u2081 : Nat.gcd (7 * n + 1) 1 = 1 := by\n      simp [Nat.gcd_eq_right]\n    exact h\u2083\u2081\n  have h\u2084 : Nat.gcd (21 * n + 4) (14 * n + 3) = 1 := by\n    rw [h\u2081]\n    rw [h\u2082]\n    rw [h\u2083]\n    <;> simp_all [Nat.gcd_eq_right]\n    <;> ring_nf at *\n    <;> omega\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 18, "alphabet": 20, "compiles": true, "steps_after": 5, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 4, "fixpoint": true, "chars_before": 414, "chars_after": 91, "seconds": 374.2, "id": "mathd_algebra_513", "proof_after": "theorem mathd_algebra_513 (a b : \u211d) (h\u2080 : 3 * a + 2 * b = 5) (h\u2081 : a + b = 2) : a = 1 \u2227 b = 1  :=  by\n  have h\u2082 : a = 1 := by\n    linarith\n  have h\u2083 : b = 1 := by\n    linarith\n  exact \u27e8h\u2082, h\u2083\u27e9\n", "proof_before": "theorem mathd_algebra_513 (a b : \u211d) (h\u2080 : 3 * a + 2 * b = 5) (h\u2081 : a + b = 2) : a = 1 \u2227 b = 1  :=  by\n  have h\u2082 : a = 1 := by\n    have h\u2082\u2081 : a = 1 := by\n      have h\u2082\u2082 : a + b = 2 := h\u2081\n      have h\u2082\u2083 : 3 * a + 2 * b = 5 := h\u2080\n      have h\u2082\u2084 : b = 2 - a := by linarith\n      rw [h\u2082\u2084] at h\u2082\u2083\n      linarith\n    exact h\u2082\u2081\n  have h\u2083 : b = 1 := by\n    have h\u2083\u2081 : b = 1 := by\n      have h\u2083\u2082 : a + b = 2 := h\u2081\n      rw [h\u2082] at h\u2083\u2082\n      linarith\n    exact h\u2083\u2081\n  have h\u2084 : a = 1 \u2227 b = 1 := by\n    exact \u27e8h\u2082, h\u2083\u27e9\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 52, "alphabet": 27, "compiles": false, "id": "induction_pord1p1on2powklt5on2", "proof_after": "theorem induction_pord1p1on2powklt5on2 (n : \u2115) (h\u2080 : 0 < n) :\n    (\u220f k in Finset.Icc 1 n, 1 + (1 : \u211d) / 2 ^ k) < 5 / 2  :=  by\n  have h\u2081 : (\u220f k in Finset.Icc 1 n, (1 : \u211d)) = 1 := by\n    rw [Finset.prod_const]\n    <;> simp [h\u2080]\n    <;> norm_num\n    <;> aesop\n  have h\u2082 : (1 : \u211d) / (2 : \u211d) ^ k \u2264 1 := by\n    have h\u2083 : (2 : \u211d) ^ k \u2265 1 := by\n      have h\u2084 : (1 : \u211d) \u2264 (2 : \u211d) ^ k := by\n        have h\u2085 : (1 : \u211d) \u2264 (2 : \u211d) ^ (k : \u2115) := by\n          exact one_le_pow_of_one_le (by norm_num) (k : \u2115)\n        simpa using h\u2085\n      linarith\n    have h\u2084 : (1 : \u211d) / (2 : \u211d) ^ k \u2264 1 := by\n      have h\u2085 : 0 < (2 : \u211d) ^ k := by positivity\n      have h\u2086 : (1 : \u211d) / (2 : \u211d) ^ k \u2264 1 := by\n        rw [div_le_iff h\u2085]\n        nlinarith\n      exact h\u2086\n    exact h\u2084\n  have h\u2083 : (\u220f k in Finset.Icc 1 n, (1 : \u211d)) + (1 : \u211d) / (2 : \u211d) ^ k < 5 / 2 := by\n    rw [h\u2081]\n    have h\u2084 : (1 : \u211d) + (1 : \u211d) / (2 : \u211d) ^ k \u2264 2 := by\n      linarith\n    norm_num at h\u2082 h\u2084 \u22a2\n    <;>\n    (try norm_num) <;>\n    (try linarith) <;>\n    (try nlinarith) <;>\n    (try ring_nf at h\u2082 h\u2084 \u22a2) <;>\n    (try field_simp at h\u2082 h\u2084 \u22a2) <;>\n    (try norm_num at h\u2082 h\u2084 \u22a2) <;>\n    (try linarith)\n    <;>\n    (try nlinarith)\n    <;>\n    (try ring_nf at h\u2082 h\u2084 \u22a2)\n    <;>\n    (try field_simp at h\u2082 h\u2084 \u22a2)\n    <;>\n    (try norm_num at h\u2082 h\u2084 \u22a2)\n    <;>\n    (try linarith)\n    <;>\n    (try nlinarith)\n  simpa [h\u2081] using h\u2083\n", "proof_before": "theorem induction_pord1p1on2powklt5on2 (n : \u2115) (h\u2080 : 0 < n) :\n    (\u220f k in Finset.Icc 1 n, 1 + (1 : \u211d) / 2 ^ k) < 5 / 2  :=  by\n  have h\u2081 : (\u220f k in Finset.Icc 1 n, (1 : \u211d)) = 1 := by\n    rw [Finset.prod_const]\n    <;> simp [h\u2080]\n    <;> norm_num\n    <;> aesop\n  have h\u2082 : (1 : \u211d) / (2 : \u211d) ^ k \u2264 1 := by\n    have h\u2083 : (2 : \u211d) ^ k \u2265 1 := by\n      have h\u2084 : (1 : \u211d) \u2264 (2 : \u211d) ^ k := by\n        have h\u2085 : (1 : \u211d) \u2264 (2 : \u211d) ^ (k : \u2115) := by\n          exact one_le_pow_of_one_le (by norm_num) (k : \u2115)\n        simpa using h\u2085\n      linarith\n    have h\u2084 : (1 : \u211d) / (2 : \u211d) ^ k \u2264 1 := by\n      have h\u2085 : 0 < (2 : \u211d) ^ k := by positivity\n      have h\u2086 : (1 : \u211d) / (2 : \u211d) ^ k \u2264 1 := by\n        rw [div_le_iff h\u2085]\n        nlinarith\n      exact h\u2086\n    exact h\u2084\n  have h\u2083 : (\u220f k in Finset.Icc 1 n, (1 : \u211d)) + (1 : \u211d) / (2 : \u211d) ^ k < 5 / 2 := by\n    rw [h\u2081]\n    have h\u2084 : (1 : \u211d) + (1 : \u211d) / (2 : \u211d) ^ k \u2264 2 := by\n      linarith\n    norm_num at h\u2082 h\u2084 \u22a2\n    <;>\n    (try norm_num) <;>\n    (try linarith) <;>\n    (try nlinarith) <;>\n    (try ring_nf at h\u2082 h\u2084 \u22a2) <;>\n    (try field_simp at h\u2082 h\u2084 \u22a2) <;>\n    (try norm_num at h\u2082 h\u2084 \u22a2) <;>\n    (try linarith)\n    <;>\n    (try nlinarith)\n    <;>\n    (try ring_nf at h\u2082 h\u2084 \u22a2)\n    <;>\n    (try field_simp at h\u2082 h\u2084 \u22a2)\n    <;>\n    (try norm_num at h\u2082 h\u2084 \u22a2)\n    <;>\n    (try linarith)\n    <;>\n    (try nlinarith)\n  simpa [h\u2081] using h\u2083\n"}
{"mode": "clear", "steps_before": 184, "alphabet": 77, "compiles": false, "id": "mathd_algebra_276", "proof_after": "theorem mathd_algebra_276 (a b : \u2124)\n    (h\u2080 : \u2200 x : \u211d, 10 * x ^ 2 - x - 24 = (a * x - 8) * (b * x + 3)) : a * b + b = 12  :=  by\n  have h\u2081 : (a + 8 : \u2124) * (b - 3) = -13 := by\n    have h\u2081\u2080 := h\u2080 (-1 : \u211d)\n    norm_num at h\u2081\u2080\n    ring_nf at h\u2081\u2080\n    have h\u2081\u2081 := congr_arg (fun x : \u211d => (x : \u211d)) h\u2081\u2080\n    norm_num at h\u2081\u2081\n    ring_nf at h\u2081\u2081\n    norm_cast at h\u2081\u2081 \u22a2\n    <;>\n    (try norm_num at h\u2081\u2081 \u22a2) <;>\n    (try ring_nf at h\u2081\u2081 \u22a2) <;>\n    (try nlinarith) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      })\n    <;>\n    nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n  have h\u2082 : (a - 8 : \u2124) * (b + 3) = -15 := by\n    have h\u2082\u2080 := h\u2080 (1 : \u211d)\n    norm_num at h\u2082\u2080\n    ring_nf at h\u2082\u2080\n    have h\u2082\u2081 := congr_arg (fun x : \u211d => (x : \u211d)) h\u2082\u2080\n    norm_num at h\u2082\u2081\n    ring_nf at h\u2082\u2081\n    norm_cast at h\u2082\u2081 \u22a2\n    <;>\n    (try norm_num at h\u2082\u2081 \u22a2) <;>\n    (try ring_nf at h\u2082\u2081 \u22a2) <;>\n    (try nlinarith) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      })\n    <;>\n    nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n  have h\u2083 : a = 5 := by\n    have h\u2083\u2081 : a = 5 := by\n      have h\u2083\u2082 : (a + 8 : \u2124) * (b - 3) = -13 := h\u2081\n      have h\u2083\u2083 : (a - 8 : \u2124) * (b + 3) = -15 := h\u2082\n      have h\u2083\u2084 : a = 5 := by\n        have h\u2083\u2085 : a = 5 := by\n          have h\u2083\u2086 : a = 5 := by\n            have h\u2083\u2087 : a + 8 = 13 \u2228 a + 8 = -13 \u2228 a + 8 = 1 \u2228 a + 8 = -1 := by\n              have h\u2083\u2088 : a + 8 \u2223 -13 := by\n                use b - 3\n                linarith\n              have h\u2083\u2089 : a + 8 = 13 \u2228 a + 8 = -13 \u2228 a + 8 = 1 \u2228 a + 8 = -1 := by\n                have h\u2084\u2080 : a + 8 = 13 \u2228 a + 8 = -13 \u2228 a + 8 = 1 \u2228 a + 8 = -1 := by\n                  rw [\u2190 Int.natAbs_dvd_natAbs] at h\u2083\u2088\n                  have h\u2084\u2081 : (a + 8 : \u2124).natAbs \u2223 13 := by\n                    norm_num at h\u2083\u2088 \u22a2\n                    <;>\n                    (try omega) <;>\n                    (try\n                      {\n                        cases' h\u2083\u2088 with h\u2083\u2088 h\u2083\u2088 <;>\n                        simp_all [Int.natAbs_of_nonneg, Int.ofNat_inj] <;>\n                        omega\n                      }) <;>\n                    (try omega)\n                  have h\u2084\u2082 : (a + 8 : \u2124).natAbs = 1 \u2228 (a + 8 : \u2124).natAbs = 13 := by\n                    have h\u2084\u2083 : (a + 8 : \u2124).natAbs \u2223 13 := h\u2084\u2081\n                    have h\u2084\u2084 : (a + 8 : \u2124).natAbs = 1 \u2228 (a + 8 : \u2124).natAbs = 13 := by\n                      have h\u2084\u2085 : (a + 8 : \u2124).natAbs \u2264 13 := Nat.le_of_dvd (by norm_num) h\u2084\u2081\n                      interval_cases (a + 8 : \u2124).natAbs <;> norm_num at h\u2084\u2081 \u22a2 <;> omega\n                    exact h\u2084\u2084\n                  cases' h\u2084\u2082 with h\u2084\u2082 h\u2084\u2082\n                  \u00b7 \n                    have h\u2084\u2086 : a + 8 = 1 \u2228 a + 8 = -1 := by\n                      have h\u2084\u2087 : (a + 8 : \u2124).natAbs = 1 := h\u2084\u2082\n                      have h\u2084\u2088 : a + 8 = 1 \u2228 a + 8 = -1 := by\n                        rw [Int.natAbs_eq_iff] at h\u2084\u2087\n                        tauto\n                      exact h\u2084\u2088\n                    cases' h\u2084\u2086 with h\u2084\u2086 h\u2084\u2086\n                    \u00b7 \n                      omega\n                    \u00b7 \n                      omega\n                  \u00b7 \n                    have h\u2084\u2086 : a + 8 = 13 \u2228 a + 8 = -13 := by\n                      have h\u2084\u2087 : (a + 8 : \u2124).natAbs = 13 := h\u2084\u2082\n                      have h\u2084\u2088 : a + 8 = 13 \u2228 a + 8 = -13 := by\n                        rw [Int.natAbs_eq_iff] at h\u2084\u2087\n                        tauto\n                      exact h\u2084\u2088\n                    cases' h\u2084\u2086 with h\u2084\u2086 h\u2084\u2086\n                    \u00b7 \n                      omega\n                    \u00b7 \n                      omega\n                exact h\u2084\u2080\n              exact h\u2083\u2089\n            cases' h\u2083\u2087 with h\u2083\u2087 h\u2083\u2087\n            \u00b7 \n              have h\u2083\u2088 : a + 8 = 13 := h\u2083\u2087\n              have h\u2083\u2089 : a = 5 := by\n                omega\n              exact h\u2083\u2089\n            \u00b7 cases' h\u2083\u2087 with h\u2083\u2087 h\u2083\u2087\n              \u00b7 \n                have h\u2083\u2088 : a + 8 = -13 := h\u2083\u2087\n                have h\u2083\u2089 : a = -21 := by\n                  omega\n                have h\u2084\u2080 : (a - 8 : \u2124) * (b + 3) = -15 := h\u2082\n                have h\u2084\u2081 : a = -21 := h\u2083\u2089\n                rw [h\u2084\u2081] at h\u2084\u2080\n                norm_num at h\u2084\u2080\n                <;>\n                (try omega) <;>\n                (try\n                  {\n                    have h\u2084\u2082 := h\u2081\n                    rw [h\u2084\u2081] at h\u2084\u2082\n                    norm_num at h\u2084\u2082\n                    <;>\n                    omega\n                  })\n                <;>\n                omega\n              \u00b7 cases' h\u2083\u2087 with h\u2083\u2087 h\u2083\u2087\n                \u00b7 \n                  have h\u2083\u2088 : a + 8 = 1 := h\u2083\u2087\n                  have h\u2083\u2089 : a = -7 := by\n                    omega\n                  have h\u2084\u2080 : (a - 8 : \u2124) * (b + 3) = -15 := h\u2082\n                  have h\u2084\u2081 : a = -7 := h\u2083\u2089\n                  rw [h\u2084\u2081] at h\u2084\u2080\n                  norm_num at h\u2084\u2080\n                  <;>\n                  (try omega) <;>\n                  (try\n                    {\n                      have h\u2084\u2082 := h\u2081\n                      rw [h\u2084\u2081] at h\u2084\u2082\n                      norm_num at h\u2084\u2082\n                      <;>\n                      omega\n                    })\n                  <;>\n                  omega\n                \u00b7 \n                  have h\u2083\u2088 : a + 8 = -1 := h\u2083\u2087\n                  have h\u2083\u2089 : a = -9 := by\n                    omega\n                  have h\u2084\u2080 : (a - 8 : \u2124) * (b + 3) = -15 := h\u2082\n                  have h\u2084\u2081 : a = -9 := h\u2083\u2089\n                  rw [h\u2084\u2081] at h\u2084\u2080\n                  norm_num at h\u2084\u2080\n                  <;>\n                  (try omega) <;>\n                  (try\n                    {\n                      have h\u2084\u2082 := h\u2081\n                      rw [h\u2084\u2081] at h\u2084\u2082\n                      norm_num at h\u2084\u2082\n                      <;>\n                      omega\n                    })\n                  <;>\n                  omega\n          exact h\u2083\u2086\n        exact h\u2083\u2085\n      exact h\u2083\u2084\n    exact h\u2083\u2081\n  have h\u2084 : b = 2 := by\n    have h\u2084\u2081 : b = 2 := by\n      have h\u2084\u2082 : (a + 8 : \u2124) * (b - 3) = -13 := h\u2081\n      have h\u2084\u2083 : (a - 8 : \u2124) * (b + 3) = -15 := h\u2082\n      have h\u2084\u2084 : a = 5 := h\u2083\n      rw [h\u2084\u2084] at h\u2084\u2082 h\u2084\u2083\n      norm_num at h\u2084\u2082 h\u2084\u2083 \u22a2\n      <;>\n      (try omega) <;>\n      (try nlinarith) <;>\n      (try\n        {\n          omega\n        })\n    exact h\u2084\u2081\n  have h\u2085 : a * b + b = 12 := by\n    rw [h\u2083, h\u2084]\n    <;> norm_num\n    <;> linarith\n  exact h\u2085\n", "proof_before": "theorem mathd_algebra_276 (a b : \u2124)\n    (h\u2080 : \u2200 x : \u211d, 10 * x ^ 2 - x - 24 = (a * x - 8) * (b * x + 3)) : a * b + b = 12  :=  by\n  have h\u2081 : (a + 8 : \u2124) * (b - 3) = -13 := by\n    have h\u2081\u2080 := h\u2080 (-1 : \u211d)\n    norm_num at h\u2081\u2080\n    ring_nf at h\u2081\u2080\n    have h\u2081\u2081 := congr_arg (fun x : \u211d => (x : \u211d)) h\u2081\u2080\n    norm_num at h\u2081\u2081\n    ring_nf at h\u2081\u2081\n    norm_cast at h\u2081\u2081 \u22a2\n    <;>\n    (try norm_num at h\u2081\u2081 \u22a2) <;>\n    (try ring_nf at h\u2081\u2081 \u22a2) <;>\n    (try nlinarith) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      })\n    <;>\n    nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n  have h\u2082 : (a - 8 : \u2124) * (b + 3) = -15 := by\n    have h\u2082\u2080 := h\u2080 (1 : \u211d)\n    norm_num at h\u2082\u2080\n    ring_nf at h\u2082\u2080\n    have h\u2082\u2081 := congr_arg (fun x : \u211d => (x : \u211d)) h\u2082\u2080\n    norm_num at h\u2082\u2081\n    ring_nf at h\u2082\u2081\n    norm_cast at h\u2082\u2081 \u22a2\n    <;>\n    (try norm_num at h\u2082\u2081 \u22a2) <;>\n    (try ring_nf at h\u2082\u2081 \u22a2) <;>\n    (try nlinarith) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      }) <;>\n    (try\n      {\n        nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n      })\n    <;>\n    nlinarith [sq_nonneg (a + 8 : \u2124), sq_nonneg (b - 3 : \u2124), sq_nonneg (a - 8 : \u2124), sq_nonneg (b + 3 : \u2124)]\n  have h\u2083 : a = 5 := by\n    have h\u2083\u2081 : a = 5 := by\n      have h\u2083\u2082 : (a + 8 : \u2124) * (b - 3) = -13 := h\u2081\n      have h\u2083\u2083 : (a - 8 : \u2124) * (b + 3) = -15 := h\u2082\n      have h\u2083\u2084 : a = 5 := by\n        have h\u2083\u2085 : a = 5 := by\n          have h\u2083\u2086 : a = 5 := by\n            have h\u2083\u2087 : a + 8 = 13 \u2228 a + 8 = -13 \u2228 a + 8 = 1 \u2228 a + 8 = -1 := by\n              have h\u2083\u2088 : a + 8 \u2223 -13 := by\n                use b - 3\n                linarith\n              have h\u2083\u2089 : a + 8 = 13 \u2228 a + 8 = -13 \u2228 a + 8 = 1 \u2228 a + 8 = -1 := by\n                have h\u2084\u2080 : a + 8 = 13 \u2228 a + 8 = -13 \u2228 a + 8 = 1 \u2228 a + 8 = -1 := by\n                  rw [\u2190 Int.natAbs_dvd_natAbs] at h\u2083\u2088\n                  have h\u2084\u2081 : (a + 8 : \u2124).natAbs \u2223 13 := by\n                    norm_num at h\u2083\u2088 \u22a2\n                    <;>\n                    (try omega) <;>\n                    (try\n                      {\n                        cases' h\u2083\u2088 with h\u2083\u2088 h\u2083\u2088 <;>\n                        simp_all [Int.natAbs_of_nonneg, Int.ofNat_inj] <;>\n                        omega\n                      }) <;>\n                    (try omega)\n                  have h\u2084\u2082 : (a + 8 : \u2124).natAbs = 1 \u2228 (a + 8 : \u2124).natAbs = 13 := by\n                    have h\u2084\u2083 : (a + 8 : \u2124).natAbs \u2223 13 := h\u2084\u2081\n                    have h\u2084\u2084 : (a + 8 : \u2124).natAbs = 1 \u2228 (a + 8 : \u2124).natAbs = 13 := by\n                      have h\u2084\u2085 : (a + 8 : \u2124).natAbs \u2264 13 := Nat.le_of_dvd (by norm_num) h\u2084\u2081\n                      interval_cases (a + 8 : \u2124).natAbs <;> norm_num at h\u2084\u2081 \u22a2 <;> omega\n                    exact h\u2084\u2084\n                  cases' h\u2084\u2082 with h\u2084\u2082 h\u2084\u2082\n                  \u00b7 \n                    have h\u2084\u2086 : a + 8 = 1 \u2228 a + 8 = -1 := by\n                      have h\u2084\u2087 : (a + 8 : \u2124).natAbs = 1 := h\u2084\u2082\n                      have h\u2084\u2088 : a + 8 = 1 \u2228 a + 8 = -1 := by\n                        rw [Int.natAbs_eq_iff] at h\u2084\u2087\n                        tauto\n                      exact h\u2084\u2088\n                    cases' h\u2084\u2086 with h\u2084\u2086 h\u2084\u2086\n                    \u00b7 \n                      omega\n                    \u00b7 \n                      omega\n                  \u00b7 \n                    have h\u2084\u2086 : a + 8 = 13 \u2228 a + 8 = -13 := by\n                      have h\u2084\u2087 : (a + 8 : \u2124).natAbs = 13 := h\u2084\u2082\n                      have h\u2084\u2088 : a + 8 = 13 \u2228 a + 8 = -13 := by\n                        rw [Int.natAbs_eq_iff] at h\u2084\u2087\n                        tauto\n                      exact h\u2084\u2088\n                    cases' h\u2084\u2086 with h\u2084\u2086 h\u2084\u2086\n                    \u00b7 \n                      omega\n                    \u00b7 \n                      omega\n                exact h\u2084\u2080\n              exact h\u2083\u2089\n            cases' h\u2083\u2087 with h\u2083\u2087 h\u2083\u2087\n            \u00b7 \n              have h\u2083\u2088 : a + 8 = 13 := h\u2083\u2087\n              have h\u2083\u2089 : a = 5 := by\n                omega\n              exact h\u2083\u2089\n            \u00b7 cases' h\u2083\u2087 with h\u2083\u2087 h\u2083\u2087\n              \u00b7 \n                have h\u2083\u2088 : a + 8 = -13 := h\u2083\u2087\n                have h\u2083\u2089 : a = -21 := by\n                  omega\n                have h\u2084\u2080 : (a - 8 : \u2124) * (b + 3) = -15 := h\u2082\n                have h\u2084\u2081 : a = -21 := h\u2083\u2089\n                rw [h\u2084\u2081] at h\u2084\u2080\n                norm_num at h\u2084\u2080\n                <;>\n                (try omega) <;>\n                (try\n                  {\n                    have h\u2084\u2082 := h\u2081\n                    rw [h\u2084\u2081] at h\u2084\u2082\n                    norm_num at h\u2084\u2082\n                    <;>\n                    omega\n                  })\n                <;>\n                omega\n              \u00b7 cases' h\u2083\u2087 with h\u2083\u2087 h\u2083\u2087\n                \u00b7 \n                  have h\u2083\u2088 : a + 8 = 1 := h\u2083\u2087\n                  have h\u2083\u2089 : a = -7 := by\n                    omega\n                  have h\u2084\u2080 : (a - 8 : \u2124) * (b + 3) = -15 := h\u2082\n                  have h\u2084\u2081 : a = -7 := h\u2083\u2089\n                  rw [h\u2084\u2081] at h\u2084\u2080\n                  norm_num at h\u2084\u2080\n                  <;>\n                  (try omega) <;>\n                  (try\n                    {\n                      have h\u2084\u2082 := h\u2081\n                      rw [h\u2084\u2081] at h\u2084\u2082\n                      norm_num at h\u2084\u2082\n                      <;>\n                      omega\n                    })\n                  <;>\n                  omega\n                \u00b7 \n                  have h\u2083\u2088 : a + 8 = -1 := h\u2083\u2087\n                  have h\u2083\u2089 : a = -9 := by\n                    omega\n                  have h\u2084\u2080 : (a - 8 : \u2124) * (b + 3) = -15 := h\u2082\n                  have h\u2084\u2081 : a = -9 := h\u2083\u2089\n                  rw [h\u2084\u2081] at h\u2084\u2080\n                  norm_num at h\u2084\u2080\n                  <;>\n                  (try omega) <;>\n                  (try\n                    {\n                      have h\u2084\u2082 := h\u2081\n                      rw [h\u2084\u2081] at h\u2084\u2082\n                      norm_num at h\u2084\u2082\n                      <;>\n                      omega\n                    })\n                  <;>\n                  omega\n          exact h\u2083\u2086\n        exact h\u2083\u2085\n      exact h\u2083\u2084\n    exact h\u2083\u2081\n  have h\u2084 : b = 2 := by\n    have h\u2084\u2081 : b = 2 := by\n      have h\u2084\u2082 : (a + 8 : \u2124) * (b - 3) = -13 := h\u2081\n      have h\u2084\u2083 : (a - 8 : \u2124) * (b + 3) = -15 := h\u2082\n      have h\u2084\u2084 : a = 5 := h\u2083\n      rw [h\u2084\u2084] at h\u2084\u2082 h\u2084\u2083\n      norm_num at h\u2084\u2082 h\u2084\u2083 \u22a2\n      <;>\n      (try omega) <;>\n      (try nlinarith) <;>\n      (try\n        {\n          omega\n        })\n    exact h\u2084\u2081\n  have h\u2085 : a * b + b = 12 := by\n    rw [h\u2083, h\u2084]\n    <;> norm_num\n    <;> linarith\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 31, "alphabet": 27, "compiles": true, "steps_after": 11, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 5, "fixpoint": true, "chars_before": 1089, "chars_after": 461, "seconds": 376.1, "id": "numbertheory_notequiv2i2jasqbsqdiv8", "proof_after": "theorem numbertheory_notequiv2i2jasqbsqdiv8 :\n    \u00ac\u2200 a b : \u2124, (\u2203 i j, a = 2 * i \u2227 b = 2 * j) \u2194 \u2203 k, a ^ 2 + b ^ 2 = 8 * k  :=  by\n  have h_main : (\u2200 a b : \u2124, (\u2203 i j, a = 2 * i \u2227 b = 2 * j) \u2194 \u2203 k, a ^ 2 + b ^ 2 = 8 * k) \u2192 False := by\n    intro h\n    have h\u2081 : (\u2203 i j, (4 : \u2124) = 2 * i \u2227 (2 : \u2124) = 2 * j) \u2194 \u2203 k, (4 : \u2124) ^ 2 + (2 : \u2124) ^ 2 = 8 * k := by\n      apply h 4 2\n    have h\u2082 : \u2203 (i j : \u2124), (4 : \u2124) = 2 * i \u2227 (2 : \u2124) = 2 * j := by\n      refine' \u27e82, 1, _\u27e9\n      simp\n    have h\u2083 : \u2203 (k : \u2124), (4 : \u2124) ^ 2 + (2 : \u2124) ^ 2 = 8 * k := by\n      simpa [mul_assoc] using h\u2081.mp h\u2082\n    omega\n  omega\n", "proof_before": "theorem numbertheory_notequiv2i2jasqbsqdiv8 :\n    \u00ac\u2200 a b : \u2124, (\u2203 i j, a = 2 * i \u2227 b = 2 * j) \u2194 \u2203 k, a ^ 2 + b ^ 2 = 8 * k  :=  by\n  have h_main : (\u2200 a b : \u2124, (\u2203 i j, a = 2 * i \u2227 b = 2 * j) \u2194 \u2203 k, a ^ 2 + b ^ 2 = 8 * k) \u2192 False := by\n    intro h\n    have h\u2081 : (\u2203 i j, (4 : \u2124) = 2 * i \u2227 (2 : \u2124) = 2 * j) \u2194 \u2203 k, (4 : \u2124) ^ 2 + (2 : \u2124) ^ 2 = 8 * k := by\n      apply h 4 2\n    have h\u2082 : \u2203 (i j : \u2124), (4 : \u2124) = 2 * i \u2227 (2 : \u2124) = 2 * j := by\n      refine' \u27e82, 1, _\u27e9\n      constructor <;> norm_num\n    have h\u2083 : \u2203 (k : \u2124), (4 : \u2124) ^ 2 + (2 : \u2124) ^ 2 = 8 * k := by\n      have h\u2084 : (\u2203 (k : \u2124), (4 : \u2124) ^ 2 + (2 : \u2124) ^ 2 = 8 * k) := by\n        simpa [mul_assoc] using h\u2081.mp h\u2082\n      exact h\u2084\n    rcases h\u2083 with \u27e8k, hk\u27e9\n    have h\u2084 : (4 : \u2124) ^ 2 + (2 : \u2124) ^ 2 = 8 * k := by\n      exact hk\n    have h\u2085 : k = 5 / 2 := by\n      ring_nf at h\u2084 \u22a2\n      <;> omega\n    have h\u2086 : (k : \u2124) \u2260 5 / 2 := by\n      norm_num at h\u2085 \u22a2\n      <;>\n      (try omega) <;>\n      (try\n        {\n          norm_cast at h\u2085 \u22a2\n          <;>\n          omega\n        })\n    exact h\u2086 (by linarith)\n  have h_final : \u00ac\u2200 a b : \u2124, (\u2203 i j, a = 2 * i \u2227 b = 2 * j) \u2194 \u2203 k, a ^ 2 + b ^ 2 = 8 * k := by\n    intro h\n    have h\u2081 : False := h_main h\n    exact h\u2081\n  exact h_final\n"}
{"mode": "clear", "steps_before": 17, "alphabet": 18, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 391, "chars_after": 11, "seconds": 3.9, "id": "mathd_algebra_359", "proof_after": "theorem mathd_algebra_359 (y : \u211d) (h\u2080 : y + 6 + y = 2 * 12) : y = 9  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_359 (y : \u211d) (h\u2080 : y + 6 + y = 2 * 12) : y = 9  :=  by\n  have h\u2081 : 2 * y + 6 = 24 := by\n    have h\u2081\u2081 : y + 6 + y = 2 * 12 := h\u2080\n    have h\u2081\u2082 : y + 6 + y = 2 * y + 6 := by ring\n    rw [h\u2081\u2082] at h\u2081\u2081\n    linarith\n  have h\u2082 : 2 * y = 18 := by\n    have h\u2082\u2081 : 2 * y + 6 = 24 := h\u2081\n    have h\u2082\u2082 : 2 * y = 18 := by linarith\n    exact h\u2082\u2082\n  have h\u2083 : y = 9 := by\n    have h\u2083\u2081 : 2 * y = 18 := h\u2082\n    have h\u2083\u2082 : y = 9 := by linarith\n    exact h\u2083\u2082\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 9, "alphabet": 13, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 175, "chars_after": 6, "seconds": 5.1, "id": "mathd_numbertheory_66", "proof_after": "theorem mathd_numbertheory_66 : 194 % 11 = 7  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_66 : 194 % 11 = 7  :=  by\n  have h\u2081 : 11 * 17 = 187 := by\n    norm_num\n    <;> rfl\n  have h\u2082 : 194 = 11 * 17 + 7 := by\n    norm_num [h\u2081]\n    <;> rfl\n  have h\u2083 : 194 % 11 = 7 := by\n    omega\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 219, "alphabet": 111, "compiles": false, "id": "amc12a_2002_p13", "proof_after": "theorem amc12a_2002_p13 (a b : \u211d) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : a \u2260 b) (h\u2082 : abs (a - 1 / a) = 1)\n    (h\u2083 : abs (b - 1 / b) = 1) : a + b = Real.sqrt 5  :=  by\n  have h\u2084 : a \u2265 1 \u2192 a = (1 + Real.sqrt 5) / 2 := by\n    intro h\u2084\n    have h\u2085 : a - 1 / a \u2265 0 := by\n      have h\u2085\u2081 : a \u2265 1 := h\u2084\n      have h\u2085\u2082 : 0 < a := h\u2080.1\n      have h\u2085\u2083 : a - 1 / a \u2265 0 := by\n        rw [ge_iff_le]\n        have h\u2085\u2084 : 1 / a \u2264 a := by\n          have h\u2085\u2085 : a \u2265 1 := h\u2085\u2081\n          have h\u2085\u2086 : 0 < a := h\u2085\u2082\n          rw [div_le_iff h\u2085\u2086]\n          nlinarith\n        linarith\n      exact h\u2085\u2083\n    have h\u2086 : abs (a - 1 / a) = a - 1 / a := by\n      rw [abs_of_nonneg h\u2085]\n    rw [h\u2086] at h\u2082\n    have h\u2087 : a - 1 / a = 1 := by linarith\n    have h\u2088 : a > 0 := h\u2080.1\n    have h\u2089 : a \u2260 0 := by linarith\n    have h\u2081\u2080 : a ^ 2 - a - 1 = 0 := by\n      field_simp [h\u2089] at h\u2087\n      nlinarith\n    have h\u2081\u2081 : a = (1 + Real.sqrt 5) / 2 := by\n      have h\u2081\u2082 : a = (1 + Real.sqrt 5) / 2 \u2228 a = (1 - Real.sqrt 5) / 2 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2081\u2083\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2081\u2083)\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      cases h\u2081\u2082 with\n      | inl h\u2081\u2082 => exact h\u2081\u2082\n      | inr h\u2081\u2082 =>\n        have h\u2081\u2083 : a > 0 := h\u2080.1\n        have h\u2081\u2084 : (1 - Real.sqrt 5) / 2 \u2264 0 := by\n          nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n        linarith\n    exact h\u2081\u2081\n  have h\u2085 : a < 1 \u2192 a = (-1 + Real.sqrt 5) / 2 := by\n    intro h\u2085\n    have h\u2086 : a - 1 / a < 0 := by\n      have h\u2086\u2081 : a < 1 := h\u2085\n      have h\u2086\u2082 : 0 < a := h\u2080.1\n      have h\u2086\u2083 : a - 1 / a < 0 := by\n        have h\u2086\u2084 : 1 / a > 1 := by\n          have h\u2086\u2085 : a < 1 := h\u2085\n          have h\u2086\u2086 : 0 < a := h\u2080.1\n          have h\u2086\u2087 : 1 / a > 1 := by\n            rw [gt_iff_lt]\n            rw [lt_div_iff h\u2086\u2086]\n            nlinarith\n          exact h\u2086\u2087\n        have h\u2086\u2088 : a - 1 / a < 0 := by\n          have h\u2086\u2089 : 1 / a > 1 := h\u2086\u2084\n          have h\u2086\u2081\u2080 : a < 1 := h\u2085\n          have h\u2086\u2081\u2081 : a - 1 / a < 0 := by\n            nlinarith\n          exact h\u2086\u2081\u2081\n        exact h\u2086\u2088\n      exact h\u2086\u2083\n    have h\u2087 : abs (a - 1 / a) = -(a - 1 / a) := by\n      rw [abs_of_neg h\u2086]\n      <;> linarith\n    rw [h\u2087] at h\u2082\n    have h\u2088 : -(a - 1 / a) = 1 := by linarith\n    have h\u2089 : a > 0 := h\u2080.1\n    have h\u2081\u2080 : a \u2260 0 := by linarith\n    have h\u2081\u2081 : a ^ 2 + a - 1 = 0 := by\n      field_simp [h\u2081\u2080] at h\u2088 \u22a2\n      nlinarith\n    have h\u2081\u2082 : a = (-1 + Real.sqrt 5) / 2 := by\n      have h\u2081\u2083 : a = (-1 + Real.sqrt 5) / 2 \u2228 a = (-1 - Real.sqrt 5) / 2 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2081\u2084\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2081\u2084)\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      cases h\u2081\u2083 with\n      | inl h\u2081\u2083 => exact h\u2081\u2083\n      | inr h\u2081\u2083 =>\n        have h\u2081\u2084 : a > 0 := h\u2080.1\n        have h\u2081\u2085 : (-1 - Real.sqrt 5) / 2 \u2264 0 := by\n          nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n        linarith\n    exact h\u2081\u2082\n  have h\u2086 : b \u2265 1 \u2192 b = (1 + Real.sqrt 5) / 2 := by\n    intro h\u2086\n    have h\u2087 : b - 1 / b \u2265 0 := by\n      have h\u2087\u2081 : b \u2265 1 := h\u2086\n      have h\u2087\u2082 : 0 < b := h\u2080.2\n      have h\u2087\u2083 : b - 1 / b \u2265 0 := by\n        rw [ge_iff_le]\n        have h\u2087\u2084 : 1 / b \u2264 b := by\n          have h\u2087\u2085 : b \u2265 1 := h\u2087\u2081\n          have h\u2087\u2086 : 0 < b := h\u2087\u2082\n          rw [div_le_iff h\u2087\u2086]\n          nlinarith\n        linarith\n      exact h\u2087\u2083\n    have h\u2088 : abs (b - 1 / b) = b - 1 / b := by\n      rw [abs_of_nonneg h\u2087]\n    rw [h\u2088] at h\u2083\n    have h\u2089 : b - 1 / b = 1 := by linarith\n    have h\u2081\u2080 : b > 0 := h\u2080.2\n    have h\u2081\u2081 : b \u2260 0 := by linarith\n    have h\u2081\u2082 : b ^ 2 - b - 1 = 0 := by\n      field_simp [h\u2081\u2081] at h\u2089\n      nlinarith\n    have h\u2081\u2083 : b = (1 + Real.sqrt 5) / 2 := by\n      have h\u2081\u2084 : b = (1 + Real.sqrt 5) / 2 \u2228 b = (1 - Real.sqrt 5) / 2 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2081\u2085\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2081\u2085)\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      cases h\u2081\u2084 with\n      | inl h\u2081\u2084 => exact h\u2081\u2084\n      | inr h\u2081\u2084 =>\n        have h\u2081\u2085 : b > 0 := h\u2080.2\n        have h\u2081\u2086 : (1 - Real.sqrt 5) / 2 \u2264 0 := by\n          nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n        linarith\n    exact h\u2081\u2083\n  have h\u2087 : b < 1 \u2192 b = (-1 + Real.sqrt 5) / 2 := by\n    intro h\u2087\n    have h\u2088 : b - 1 / b < 0 := by\n      have h\u2088\u2081 : b < 1 := h\u2087\n      have h\u2088\u2082 : 0 < b := h\u2080.2\n      have h\u2088\u2083 : b - 1 / b < 0 := by\n        have h\u2088\u2084 : 1 / b > 1 := by\n          have h\u2088\u2085 : b < 1 := h\u2087\n          have h\u2088\u2086 : 0 < b := h\u2080.2\n          have h\u2088\u2087 : 1 / b > 1 := by\n            rw [gt_iff_lt]\n            rw [lt_div_iff h\u2088\u2086]\n            nlinarith\n          exact h\u2088\u2087\n        have h\u2088\u2088 : b - 1 / b < 0 := by\n          have h\u2088\u2089 : 1 / b > 1 := h\u2088\u2084\n          have h\u2088\u2081\u2080 : b < 1 := h\u2087\n          have h\u2088\u2081\u2081 : b - 1 / b < 0 := by\n            nlinarith\n          exact h\u2088\u2081\u2081\n        exact h\u2088\u2088\n      exact h\u2088\u2083\n    have h\u2089 : abs (b - 1 / b) = -(b - 1 / b) := by\n      rw [abs_of_neg h\u2088]\n      <;> linarith\n    rw [h\u2089] at h\u2083\n    have h\u2081\u2080 : -(b - 1 / b) = 1 := by linarith\n    have h\u2081\u2081 : b > 0 := h\u2080.2\n    have h\u2081\u2082 : b \u2260 0 := by linarith\n    have h\u2081\u2083 : b ^ 2 + b - 1 = 0 := by\n      field_simp [h\u2081\u2082] at h\u2081\u2080 \u22a2\n      nlinarith\n    have h\u2081\u2084 : b = (-1 + Real.sqrt 5) / 2 := by\n      have h\u2081\u2085 : b = (-1 + Real.sqrt 5) / 2 \u2228 b = (-1 - Real.sqrt 5) / 2 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2081\u2086\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2081\u2086)\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      cases h\u2081\u2085 with\n      | inl h\u2081\u2085 => exact h\u2081\u2085\n      | inr h\u2081\u2085 =>\n        have h\u2081\u2086 : b > 0 := h\u2080.2\n        have h\u2081\u2087 : (-1 - Real.sqrt 5) / 2 \u2264 0 := by\n          nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n        linarith\n    exact h\u2081\u2084\n  have h\u2088 : a = (1 + Real.sqrt 5) / 2 \u2228 a = (-1 + Real.sqrt 5) / 2 := by\n    by_cases h\u2088 : a \u2265 1\n    \u00b7 have h\u2088\u2081 : a = (1 + Real.sqrt 5) / 2 := h\u2084 h\u2088\n      exact Or.inl h\u2088\u2081\n    \u00b7 have h\u2088\u2082 : a < 1 := by\n        by_contra h\u2088\u2082\n        have h\u2088\u2083 : a \u2265 1 := by linarith\n        contradiction\n      have h\u2088\u2084 : a = (-1 + Real.sqrt 5) / 2 := h\u2085 h\u2088\u2082\n      exact Or.inr h\u2088\u2084\n  have h\u2089 : b = (1 + Real.sqrt 5) / 2 \u2228 b = (-1 + Real.sqrt 5) / 2 := by\n    by_cases h\u2089 : b \u2265 1\n    \u00b7 have h\u2089\u2081 : b = (1 + Real.sqrt 5) / 2 := h\u2086 h\u2089\n      exact Or.inl h\u2089\u2081\n    \u00b7 have h\u2089\u2082 : b < 1 := by\n        by_contra h\u2089\u2082\n        have h\u2089\u2083 : b \u2265 1 := by linarith\n        contradiction\n      have h\u2089\u2084 : b = (-1 + Real.sqrt 5) / 2 := h\u2087 h\u2089\u2082\n      exact Or.inr h\u2089\u2084\n  have h\u2081\u2080 : a + b = Real.sqrt 5 := by\n    have h\u2081\u2080\u2081 : a = (1 + Real.sqrt 5) / 2 \u2228 a = (-1 + Real.sqrt 5) / 2 := h\u2088\n    have h\u2081\u2080\u2082 : b = (1 + Real.sqrt 5) / 2 \u2228 b = (-1 + Real.sqrt 5) / 2 := h\u2089\n    cases' h\u2081\u2080\u2081 with h\u2081\u2080\u2081 h\u2081\u2080\u2081\n    \u00b7 \n      cases' h\u2081\u2080\u2082 with h\u2081\u2080\u2082 h\u2081\u2080\u2082\n      \u00b7 \n        exfalso\n        apply h\u2081\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      \u00b7 \n        rw [h\u2081\u2080\u2081, h\u2081\u2080\u2082]\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n    \u00b7 \n      cases' h\u2081\u2080\u2082 with h\u2081\u2080\u2082 h\u2081\u2080\u2082\n      \u00b7 \n        rw [h\u2081\u2080\u2081, h\u2081\u2080\u2082]\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      \u00b7 \n        exfalso\n        apply h\u2081\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n  exact h\u2081\u2080\n", "proof_before": "theorem amc12a_2002_p13 (a b : \u211d) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : a \u2260 b) (h\u2082 : abs (a - 1 / a) = 1)\n    (h\u2083 : abs (b - 1 / b) = 1) : a + b = Real.sqrt 5  :=  by\n  have h\u2084 : a \u2265 1 \u2192 a = (1 + Real.sqrt 5) / 2 := by\n    intro h\u2084\n    have h\u2085 : a - 1 / a \u2265 0 := by\n      have h\u2085\u2081 : a \u2265 1 := h\u2084\n      have h\u2085\u2082 : 0 < a := h\u2080.1\n      have h\u2085\u2083 : a - 1 / a \u2265 0 := by\n        rw [ge_iff_le]\n        have h\u2085\u2084 : 1 / a \u2264 a := by\n          have h\u2085\u2085 : a \u2265 1 := h\u2085\u2081\n          have h\u2085\u2086 : 0 < a := h\u2085\u2082\n          rw [div_le_iff h\u2085\u2086]\n          nlinarith\n        linarith\n      exact h\u2085\u2083\n    have h\u2086 : abs (a - 1 / a) = a - 1 / a := by\n      rw [abs_of_nonneg h\u2085]\n    rw [h\u2086] at h\u2082\n    have h\u2087 : a - 1 / a = 1 := by linarith\n    have h\u2088 : a > 0 := h\u2080.1\n    have h\u2089 : a \u2260 0 := by linarith\n    have h\u2081\u2080 : a ^ 2 - a - 1 = 0 := by\n      field_simp [h\u2089] at h\u2087\n      nlinarith\n    have h\u2081\u2081 : a = (1 + Real.sqrt 5) / 2 := by\n      have h\u2081\u2082 : a = (1 + Real.sqrt 5) / 2 \u2228 a = (1 - Real.sqrt 5) / 2 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2081\u2083\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2081\u2083)\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      cases h\u2081\u2082 with\n      | inl h\u2081\u2082 => exact h\u2081\u2082\n      | inr h\u2081\u2082 =>\n        have h\u2081\u2083 : a > 0 := h\u2080.1\n        have h\u2081\u2084 : (1 - Real.sqrt 5) / 2 \u2264 0 := by\n          nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n        linarith\n    exact h\u2081\u2081\n  have h\u2085 : a < 1 \u2192 a = (-1 + Real.sqrt 5) / 2 := by\n    intro h\u2085\n    have h\u2086 : a - 1 / a < 0 := by\n      have h\u2086\u2081 : a < 1 := h\u2085\n      have h\u2086\u2082 : 0 < a := h\u2080.1\n      have h\u2086\u2083 : a - 1 / a < 0 := by\n        have h\u2086\u2084 : 1 / a > 1 := by\n          have h\u2086\u2085 : a < 1 := h\u2085\n          have h\u2086\u2086 : 0 < a := h\u2080.1\n          have h\u2086\u2087 : 1 / a > 1 := by\n            rw [gt_iff_lt]\n            rw [lt_div_iff h\u2086\u2086]\n            nlinarith\n          exact h\u2086\u2087\n        have h\u2086\u2088 : a - 1 / a < 0 := by\n          have h\u2086\u2089 : 1 / a > 1 := h\u2086\u2084\n          have h\u2086\u2081\u2080 : a < 1 := h\u2085\n          have h\u2086\u2081\u2081 : a - 1 / a < 0 := by\n            nlinarith\n          exact h\u2086\u2081\u2081\n        exact h\u2086\u2088\n      exact h\u2086\u2083\n    have h\u2087 : abs (a - 1 / a) = -(a - 1 / a) := by\n      rw [abs_of_neg h\u2086]\n      <;> linarith\n    rw [h\u2087] at h\u2082\n    have h\u2088 : -(a - 1 / a) = 1 := by linarith\n    have h\u2089 : a > 0 := h\u2080.1\n    have h\u2081\u2080 : a \u2260 0 := by linarith\n    have h\u2081\u2081 : a ^ 2 + a - 1 = 0 := by\n      field_simp [h\u2081\u2080] at h\u2088 \u22a2\n      nlinarith\n    have h\u2081\u2082 : a = (-1 + Real.sqrt 5) / 2 := by\n      have h\u2081\u2083 : a = (-1 + Real.sqrt 5) / 2 \u2228 a = (-1 - Real.sqrt 5) / 2 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2081\u2084\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2081\u2084)\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      cases h\u2081\u2083 with\n      | inl h\u2081\u2083 => exact h\u2081\u2083\n      | inr h\u2081\u2083 =>\n        have h\u2081\u2084 : a > 0 := h\u2080.1\n        have h\u2081\u2085 : (-1 - Real.sqrt 5) / 2 \u2264 0 := by\n          nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n        linarith\n    exact h\u2081\u2082\n  have h\u2086 : b \u2265 1 \u2192 b = (1 + Real.sqrt 5) / 2 := by\n    intro h\u2086\n    have h\u2087 : b - 1 / b \u2265 0 := by\n      have h\u2087\u2081 : b \u2265 1 := h\u2086\n      have h\u2087\u2082 : 0 < b := h\u2080.2\n      have h\u2087\u2083 : b - 1 / b \u2265 0 := by\n        rw [ge_iff_le]\n        have h\u2087\u2084 : 1 / b \u2264 b := by\n          have h\u2087\u2085 : b \u2265 1 := h\u2087\u2081\n          have h\u2087\u2086 : 0 < b := h\u2087\u2082\n          rw [div_le_iff h\u2087\u2086]\n          nlinarith\n        linarith\n      exact h\u2087\u2083\n    have h\u2088 : abs (b - 1 / b) = b - 1 / b := by\n      rw [abs_of_nonneg h\u2087]\n    rw [h\u2088] at h\u2083\n    have h\u2089 : b - 1 / b = 1 := by linarith\n    have h\u2081\u2080 : b > 0 := h\u2080.2\n    have h\u2081\u2081 : b \u2260 0 := by linarith\n    have h\u2081\u2082 : b ^ 2 - b - 1 = 0 := by\n      field_simp [h\u2081\u2081] at h\u2089\n      nlinarith\n    have h\u2081\u2083 : b = (1 + Real.sqrt 5) / 2 := by\n      have h\u2081\u2084 : b = (1 + Real.sqrt 5) / 2 \u2228 b = (1 - Real.sqrt 5) / 2 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2081\u2085\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2081\u2085)\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      cases h\u2081\u2084 with\n      | inl h\u2081\u2084 => exact h\u2081\u2084\n      | inr h\u2081\u2084 =>\n        have h\u2081\u2085 : b > 0 := h\u2080.2\n        have h\u2081\u2086 : (1 - Real.sqrt 5) / 2 \u2264 0 := by\n          nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n        linarith\n    exact h\u2081\u2083\n  have h\u2087 : b < 1 \u2192 b = (-1 + Real.sqrt 5) / 2 := by\n    intro h\u2087\n    have h\u2088 : b - 1 / b < 0 := by\n      have h\u2088\u2081 : b < 1 := h\u2087\n      have h\u2088\u2082 : 0 < b := h\u2080.2\n      have h\u2088\u2083 : b - 1 / b < 0 := by\n        have h\u2088\u2084 : 1 / b > 1 := by\n          have h\u2088\u2085 : b < 1 := h\u2087\n          have h\u2088\u2086 : 0 < b := h\u2080.2\n          have h\u2088\u2087 : 1 / b > 1 := by\n            rw [gt_iff_lt]\n            rw [lt_div_iff h\u2088\u2086]\n            nlinarith\n          exact h\u2088\u2087\n        have h\u2088\u2088 : b - 1 / b < 0 := by\n          have h\u2088\u2089 : 1 / b > 1 := h\u2088\u2084\n          have h\u2088\u2081\u2080 : b < 1 := h\u2087\n          have h\u2088\u2081\u2081 : b - 1 / b < 0 := by\n            nlinarith\n          exact h\u2088\u2081\u2081\n        exact h\u2088\u2088\n      exact h\u2088\u2083\n    have h\u2089 : abs (b - 1 / b) = -(b - 1 / b) := by\n      rw [abs_of_neg h\u2088]\n      <;> linarith\n    rw [h\u2089] at h\u2083\n    have h\u2081\u2080 : -(b - 1 / b) = 1 := by linarith\n    have h\u2081\u2081 : b > 0 := h\u2080.2\n    have h\u2081\u2082 : b \u2260 0 := by linarith\n    have h\u2081\u2083 : b ^ 2 + b - 1 = 0 := by\n      field_simp [h\u2081\u2082] at h\u2081\u2080 \u22a2\n      nlinarith\n    have h\u2081\u2084 : b = (-1 + Real.sqrt 5) / 2 := by\n      have h\u2081\u2085 : b = (-1 + Real.sqrt 5) / 2 \u2228 b = (-1 - Real.sqrt 5) / 2 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2081\u2086\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2081\u2086)\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      cases h\u2081\u2085 with\n      | inl h\u2081\u2085 => exact h\u2081\u2085\n      | inr h\u2081\u2085 =>\n        have h\u2081\u2086 : b > 0 := h\u2080.2\n        have h\u2081\u2087 : (-1 - Real.sqrt 5) / 2 \u2264 0 := by\n          nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n        linarith\n    exact h\u2081\u2084\n  have h\u2088 : a = (1 + Real.sqrt 5) / 2 \u2228 a = (-1 + Real.sqrt 5) / 2 := by\n    by_cases h\u2088 : a \u2265 1\n    \u00b7 have h\u2088\u2081 : a = (1 + Real.sqrt 5) / 2 := h\u2084 h\u2088\n      exact Or.inl h\u2088\u2081\n    \u00b7 have h\u2088\u2082 : a < 1 := by\n        by_contra h\u2088\u2082\n        have h\u2088\u2083 : a \u2265 1 := by linarith\n        contradiction\n      have h\u2088\u2084 : a = (-1 + Real.sqrt 5) / 2 := h\u2085 h\u2088\u2082\n      exact Or.inr h\u2088\u2084\n  have h\u2089 : b = (1 + Real.sqrt 5) / 2 \u2228 b = (-1 + Real.sqrt 5) / 2 := by\n    by_cases h\u2089 : b \u2265 1\n    \u00b7 have h\u2089\u2081 : b = (1 + Real.sqrt 5) / 2 := h\u2086 h\u2089\n      exact Or.inl h\u2089\u2081\n    \u00b7 have h\u2089\u2082 : b < 1 := by\n        by_contra h\u2089\u2082\n        have h\u2089\u2083 : b \u2265 1 := by linarith\n        contradiction\n      have h\u2089\u2084 : b = (-1 + Real.sqrt 5) / 2 := h\u2087 h\u2089\u2082\n      exact Or.inr h\u2089\u2084\n  have h\u2081\u2080 : a + b = Real.sqrt 5 := by\n    have h\u2081\u2080\u2081 : a = (1 + Real.sqrt 5) / 2 \u2228 a = (-1 + Real.sqrt 5) / 2 := h\u2088\n    have h\u2081\u2080\u2082 : b = (1 + Real.sqrt 5) / 2 \u2228 b = (-1 + Real.sqrt 5) / 2 := h\u2089\n    cases' h\u2081\u2080\u2081 with h\u2081\u2080\u2081 h\u2081\u2080\u2081\n    \u00b7 \n      cases' h\u2081\u2080\u2082 with h\u2081\u2080\u2082 h\u2081\u2080\u2082\n      \u00b7 \n        exfalso\n        apply h\u2081\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      \u00b7 \n        rw [h\u2081\u2080\u2081, h\u2081\u2080\u2082]\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n    \u00b7 \n      cases' h\u2081\u2080\u2082 with h\u2081\u2080\u2082 h\u2081\u2080\u2082\n      \u00b7 \n        rw [h\u2081\u2080\u2081, h\u2081\u2080\u2082]\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n      \u00b7 \n        exfalso\n        apply h\u2081\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show 0 \u2264 5 by norm_num)]\n  exact h\u2081\u2080\n"}
{"mode": "clear", "steps_before": 70, "alphabet": 35, "compiles": false, "id": "mathd_numbertheory_711", "proof_after": "theorem mathd_numbertheory_711 (m n : \u2115) (h\u2080 : 0 < m \u2227 0 < n) (h\u2081 : Nat.gcd m n = 8)\n    (h\u2082 : Nat.lcm m n = 112) : 72 \u2264 m + n  :=  by\n  have h\u2083 : m * n = 896 := by\n    have h\u2083\u2081 : Nat.gcd m n * Nat.lcm m n = m * n := by\n      rw [Nat.gcd_mul_lcm]\n    rw [h\u2081, h\u2082] at h\u2083\u2081\n    norm_num at h\u2083\u2081 \u22a2\n    <;> nlinarith\n  have h\u2084 : m \u2223 112 := by\n    have h\u2084\u2081 : m \u2223 Nat.lcm m n := Nat.dvd_lcm_left m n\n    rw [h\u2082] at h\u2084\u2081\n    exact h\u2084\u2081\n  have h\u2085 : 8 \u2223 m := by\n    have h\u2085\u2081 : Nat.gcd m n \u2223 m := Nat.gcd_dvd_left m n\n    rw [h\u2081] at h\u2085\u2081\n    exact h\u2085\u2081\n  have h\u2086 : m = 8 \u2228 m = 16 \u2228 m = 56 \u2228 m = 112 := by\n    have h\u2086\u2081 : m \u2223 112 := h\u2084\n    have h\u2086\u2082 : 8 \u2223 m := h\u2085\n    have h\u2086\u2083 : m > 0 := by linarith\n    have h\u2086\u2084 : m \u2264 112 := Nat.le_of_dvd (by norm_num) h\u2086\u2081\n    have h\u2086\u2085 : m = 8 \u2228 m = 16 \u2228 m = 56 \u2228 m = 112 := by\n      interval_cases m <;> norm_num at h\u2086\u2081 h\u2086\u2082 h\u2086\u2083 \u22a2 <;>\n        (try omega) <;> (try norm_num) <;> (try omega)\n      <;> (try\n        {\n          exfalso\n          have h\u2086\u2086 : n \u2264 896 := by\n            nlinarith\n          interval_cases n <;> norm_num at h\u2083 h\u2081 h\u2082 h\u2086\u2081 h\u2086\u2082 h\u2086\u2083 \u22a2 <;>\n            (try omega) <;> (try norm_num) <;> (try omega)\n          <;> (try\n            {\n              exfalso\n              omega\n            })\n        })\n    exact h\u2086\u2085\n  have h\u2087 : 72 \u2264 m + n := by\n    rcases h\u2086 with (rfl | rfl | rfl | rfl)\n    \u00b7 \n      have h\u2087\u2081 : n = 112 := by\n        have h\u2087\u2082 : 8 * n = 896 := by\n          norm_num [h\u2083] at *\n          <;> nlinarith\n        nlinarith\n      rw [h\u2087\u2081]\n      <;> norm_num\n    \u00b7 \n      have h\u2087\u2081 : n = 56 := by\n        have h\u2087\u2082 : 16 * n = 896 := by\n          norm_num [h\u2083] at *\n          <;> nlinarith\n        nlinarith\n      rw [h\u2087\u2081]\n      <;> norm_num\n    \u00b7 \n      have h\u2087\u2081 : n = 16 := by\n        have h\u2087\u2082 : 56 * n = 896 := by\n          norm_num [h\u2083] at *\n          <;> nlinarith\n        nlinarith\n      rw [h\u2087\u2081]\n      <;> norm_num\n    \u00b7 \n      have h\u2087\u2081 : n = 8 := by\n        have h\u2087\u2082 : 112 * n = 896 := by\n          norm_num [h\u2083] at *\n          <;> nlinarith\n        nlinarith\n      rw [h\u2087\u2081]\n      <;> norm_num\n  exact h\u2087\n", "proof_before": "theorem mathd_numbertheory_711 (m n : \u2115) (h\u2080 : 0 < m \u2227 0 < n) (h\u2081 : Nat.gcd m n = 8)\n    (h\u2082 : Nat.lcm m n = 112) : 72 \u2264 m + n  :=  by\n  have h\u2083 : m * n = 896 := by\n    have h\u2083\u2081 : Nat.gcd m n * Nat.lcm m n = m * n := by\n      rw [Nat.gcd_mul_lcm]\n    rw [h\u2081, h\u2082] at h\u2083\u2081\n    norm_num at h\u2083\u2081 \u22a2\n    <;> nlinarith\n  have h\u2084 : m \u2223 112 := by\n    have h\u2084\u2081 : m \u2223 Nat.lcm m n := Nat.dvd_lcm_left m n\n    rw [h\u2082] at h\u2084\u2081\n    exact h\u2084\u2081\n  have h\u2085 : 8 \u2223 m := by\n    have h\u2085\u2081 : Nat.gcd m n \u2223 m := Nat.gcd_dvd_left m n\n    rw [h\u2081] at h\u2085\u2081\n    exact h\u2085\u2081\n  have h\u2086 : m = 8 \u2228 m = 16 \u2228 m = 56 \u2228 m = 112 := by\n    have h\u2086\u2081 : m \u2223 112 := h\u2084\n    have h\u2086\u2082 : 8 \u2223 m := h\u2085\n    have h\u2086\u2083 : m > 0 := by linarith\n    have h\u2086\u2084 : m \u2264 112 := Nat.le_of_dvd (by norm_num) h\u2086\u2081\n    have h\u2086\u2085 : m = 8 \u2228 m = 16 \u2228 m = 56 \u2228 m = 112 := by\n      interval_cases m <;> norm_num at h\u2086\u2081 h\u2086\u2082 h\u2086\u2083 \u22a2 <;>\n        (try omega) <;> (try norm_num) <;> (try omega)\n      <;> (try\n        {\n          exfalso\n          have h\u2086\u2086 : n \u2264 896 := by\n            nlinarith\n          interval_cases n <;> norm_num at h\u2083 h\u2081 h\u2082 h\u2086\u2081 h\u2086\u2082 h\u2086\u2083 \u22a2 <;>\n            (try omega) <;> (try norm_num) <;> (try omega)\n          <;> (try\n            {\n              exfalso\n              omega\n            })\n        })\n    exact h\u2086\u2085\n  have h\u2087 : 72 \u2264 m + n := by\n    rcases h\u2086 with (rfl | rfl | rfl | rfl)\n    \u00b7 \n      have h\u2087\u2081 : n = 112 := by\n        have h\u2087\u2082 : 8 * n = 896 := by\n          norm_num [h\u2083] at *\n          <;> nlinarith\n        nlinarith\n      rw [h\u2087\u2081]\n      <;> norm_num\n    \u00b7 \n      have h\u2087\u2081 : n = 56 := by\n        have h\u2087\u2082 : 16 * n = 896 := by\n          norm_num [h\u2083] at *\n          <;> nlinarith\n        nlinarith\n      rw [h\u2087\u2081]\n      <;> norm_num\n    \u00b7 \n      have h\u2087\u2081 : n = 16 := by\n        have h\u2087\u2082 : 56 * n = 896 := by\n          norm_num [h\u2083] at *\n          <;> nlinarith\n        nlinarith\n      rw [h\u2087\u2081]\n      <;> norm_num\n    \u00b7 \n      have h\u2087\u2081 : n = 8 := by\n        have h\u2087\u2082 : 112 * n = 896 := by\n          norm_num [h\u2083] at *\n          <;> nlinarith\n        nlinarith\n      rw [h\u2087\u2081]\n      <;> norm_num\n  exact h\u2087\n"}
{"mode": "clear", "steps_before": 162, "alphabet": 64, "compiles": false, "id": "numbertheory_3pow2pownm1mod2pownp3eq2pownp2", "proof_after": "theorem numbertheory_3pow2pownm1mod2pownp3eq2pownp2 (n : \u2115) (h\u2080 : 0 < n) :\n    (3 ^ 2 ^ n - 1) % 2 ^ (n + 3) = 2 ^ (n + 2)  :=  by\n  have h_main : \u2203 (k : \u2115), 3 ^ 2 ^ n = 1 + 2 ^ (n + 2) + k * 2 ^ (n + 3) := by\n    have h\u2081 : \u2200 n : \u2115, 0 < n \u2192 \u2203 (k : \u2115), 3 ^ 2 ^ n = 1 + 2 ^ (n + 2) + k * 2 ^ (n + 3) := by\n      intro n hn\n      induction' hn with n hn IH\n      \u00b7 \n        use 0\n        norm_num\n      \u00b7 \n        obtain \u27e8k, hk\u27e9 := IH\n        use (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) / 2 ^ (n + 4)\n        have h\u2082 : 3 ^ 2 ^ (n + 1) = (3 ^ 2 ^ n) ^ 2 := by\n          simp [pow_succ, pow_mul, pow_two]\n          <;> ring_nf\n          <;> simp [pow_add, pow_mul, pow_two]\n          <;> ring_nf\n        rw [h\u2082, hk]\n        have h\u2083 : (1 + 2 ^ (n + 2) + k * 2 ^ (n + 3)) ^ 2 = 1 + 2 ^ (n + 3) + (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n          have h\u2084 : n \u2265 0 := by linarith\n          have h\u2085 : 2 * n + 4 \u2265 n + 4 := by\n            omega\n          have h\u2086 : 2 * n + 6 \u2265 n + 4 := by\n            omega\n          have h\u2087 : 2 * n + 5 \u2265 n + 4 := by\n            omega\n          calc\n            (1 + 2 ^ (n + 2) + k * 2 ^ (n + 3)) ^ 2 = 1 + 2 ^ (n + 3) + (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n              ring_nf at *\n              <;>\n                simp [pow_add, pow_mul, mul_assoc, mul_comm, mul_left_comm, Nat.mul_div_cancel_left]\n              <;>\n                ring_nf at *\n              <;>\n                omega\n            _ = 1 + 2 ^ (n + 3) + (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n              rfl\n        rw [h\u2083]\n        have h\u2084 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) % 2 ^ (n + 4) = 0 := by\n          have h\u2085 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) := by\n            apply pow_dvd_pow 2\n            omega\n          have h\u2086 : 2 ^ (n + 4) \u2223 k * 2 ^ (n + 4) := by\n            exact \u27e8k, by ring\u27e9\n          have h\u2087 : 2 ^ (n + 4) \u2223 k ^ 2 * 2 ^ (2 * n + 6) := by\n            have h\u2088 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 6) := by\n              apply pow_dvd_pow 2\n              omega\n            have h\u2089 : 2 ^ (n + 4) \u2223 k ^ 2 * 2 ^ (2 * n + 6) := by\n              exact dvd_mul_of_dvd_right h\u2088 _\n            exact h\u2089\n          have h\u2088 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n            have h\u2089 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := by\n              apply pow_dvd_pow 2\n              omega\n            have h\u2081\u2080 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n              have h\u2081\u2081 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := h\u2089\n              have h\u2081\u2082 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n                calc\n                  2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := h\u2081\u2081\n                  _ \u2223 2 * k * 2 ^ (2 * n + 5) := by\n                    exact \u27e82 * k, by ring\u27e9\n              exact h\u2081\u2082\n            exact h\u2081\u2080\n          have h\u2089 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5) := by\n            have h\u2081\u2080 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) := h\u2085\n            have h\u2081\u2081 : 2 ^ (n + 4) \u2223 k * 2 ^ (n + 4) := h\u2086\n            have h\u2081\u2082 : 2 ^ (n + 4) \u2223 k ^ 2 * 2 ^ (2 * n + 6) := h\u2087\n            have h\u2081\u2083 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := h\u2088\n            exact Nat.dvd_add (Nat.dvd_add (Nat.dvd_add h\u2081\u2080 h\u2086) h\u2087) h\u2088\n          have h\u2081\u2080 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) % 2 ^ (n + 4) = 0 := by\n            have h\u2081\u2081 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5) := h\u2089\n            have h\u2081\u2082 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) % 2 ^ (n + 4) = 0 := by\n              exact Nat.mod_eq_zero_of_dvd h\u2081\u2081\n            exact h\u2081\u2082\n          exact h\u2081\u2080\n        have h\u2085 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) / 2 ^ (n + 4) * 2 ^ (n + 4) = (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n          have h\u2086 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5) := by\n            have h\u2087 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) := by\n              apply pow_dvd_pow 2\n              omega\n            have h\u2088 : 2 ^ (n + 4) \u2223 k * 2 ^ (n + 4) := by\n              exact \u27e8k, by ring\u27e9\n            have h\u2089 : 2 ^ (n + 4) \u2223 k ^ 2 * 2 ^ (2 * n + 6) := by\n              have h\u2081\u2080 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 6) := by\n                apply pow_dvd_pow 2\n                omega\n              have h\u2081\u2081 : 2 ^ (n + 4) \u2223 k ^ 2 * 2 ^ (2 * n + 6) := by\n                exact dvd_mul_of_dvd_right h\u2081\u2080 _\n              exact h\u2081\u2081\n            have h\u2081\u2080 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n              have h\u2081\u2081 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := by\n                apply pow_dvd_pow 2\n                omega\n              have h\u2081\u2082 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n                have h\u2081\u2083 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := h\u2081\u2081\n                have h\u2081\u2084 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n                  calc\n                    2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := h\u2081\u2083\n                    _ \u2223 2 * k * 2 ^ (2 * n + 5) := by\n                      exact \u27e82 * k, by ring\u27e9\n                exact h\u2081\u2084\n              exact h\u2081\u2082\n            exact Nat.dvd_add (Nat.dvd_add (Nat.dvd_add h\u2087 h\u2088) h\u2089) h\u2081\u2080\n          have h\u2081\u2081 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) / 2 ^ (n + 4) * 2 ^ (n + 4) = (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n            have h\u2081\u2082 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5) := h\u2086\n            have h\u2081\u2083 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) / 2 ^ (n + 4) * 2 ^ (n + 4) = (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n              exact Nat.div_mul_cancel h\u2081\u2082\n            exact h\u2081\u2083\n          exact h\u2081\u2081\n        have h\u2086 : 1 + 2 ^ (n + 3) + (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) = 1 + 2 ^ (n + 1 + 2) + ((2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) / 2 ^ (n + 4) * 2 ^ (n + 1 + 3)) := by\n          have h\u2087 : n + 3 = n + 1 + 2 := by ring\n          have h\u2088 : n + 4 = n + 1 + 3 := by ring\n          have h\u2089 : 2 * n + 4 = 2 * n + 4 := by ring\n          simp [h\u2087, h\u2088, h\u2089, pow_add, pow_mul, Nat.mul_div_assoc, Nat.div_eq_of_lt] at h\u2084 h\u2085 \u22a2 <;> ring_nf at h\u2084 h\u2085 \u22a2 <;> omega\n        rw [h\u2086]\n        <;> simp [h\u2085, Nat.mul_div_assoc, Nat.div_eq_of_lt] <;> ring_nf at * <;> omega\n    exact h\u2081 n h\u2080\n  have h_final : (3 ^ 2 ^ n - 1) % 2 ^ (n + 3) = 2 ^ (n + 2) := by\n    obtain \u27e8k, hk\u27e9 := h_main\n    have h\u2081 : 3 ^ 2 ^ n - 1 = 2 ^ (n + 2) + k * 2 ^ (n + 3) := by\n      have h\u2082 : 3 ^ 2 ^ n = 1 + 2 ^ (n + 2) + k * 2 ^ (n + 3) := by linarith\n      have h\u2083 : 3 ^ 2 ^ n - 1 = 2 ^ (n + 2) + k * 2 ^ (n + 3) := by\n        have h\u2084 : 3 ^ 2 ^ n \u2265 1 := by\n          apply Nat.one_le_pow\n          <;> positivity\n        omega\n      exact h\u2083\n    rw [h\u2081]\n    have h\u2082 : (2 ^ (n + 2) + k * 2 ^ (n + 3)) % 2 ^ (n + 3) = 2 ^ (n + 2) % 2 ^ (n + 3) := by\n      have h\u2083 : (2 ^ (n + 2) + k * 2 ^ (n + 3)) % 2 ^ (n + 3) = (2 ^ (n + 2)) % 2 ^ (n + 3) := by\n        have h\u2084 : k * 2 ^ (n + 3) % 2 ^ (n + 3) = 0 := by\n          have h\u2085 : 2 ^ (n + 3) \u2223 k * 2 ^ (n + 3) := by\n            use k\n            <;> ring\n          exact Nat.mod_eq_zero_of_dvd h\u2085\n        have h\u2085 : (2 ^ (n + 2) + k * 2 ^ (n + 3)) % 2 ^ (n + 3) = (2 ^ (n + 2) % 2 ^ (n + 3) + k * 2 ^ (n + 3) % 2 ^ (n + 3)) % 2 ^ (n + 3) := by\n          simp [Nat.add_mod]\n        rw [h\u2085]\n        have h\u2086 : k * 2 ^ (n + 3) % 2 ^ (n + 3) = 0 := h\u2084\n        rw [h\u2086]\n        have h\u2087 : (2 ^ (n + 2) % 2 ^ (n + 3) + 0) % 2 ^ (n + 3) = 2 ^ (n + 2) % 2 ^ (n + 3) := by\n          simp [Nat.add_mod]\n        rw [h\u2087]\n      exact h\u2083\n    rw [h\u2082]\n    have h\u2083 : 2 ^ (n + 2) % 2 ^ (n + 3) = 2 ^ (n + 2) := by\n      have h\u2084 : 2 ^ (n + 2) < 2 ^ (n + 3) := by\n        apply Nat.pow_lt_pow_of_lt_right\n        <;> omega\n      have h\u2085 : 2 ^ (n + 2) % 2 ^ (n + 3) = 2 ^ (n + 2) := by\n        rw [Nat.mod_eq_of_lt h\u2084]\n      rw [h\u2085]\n    rw [h\u2083]\n  exact h_final\n", "proof_before": "theorem numbertheory_3pow2pownm1mod2pownp3eq2pownp2 (n : \u2115) (h\u2080 : 0 < n) :\n    (3 ^ 2 ^ n - 1) % 2 ^ (n + 3) = 2 ^ (n + 2)  :=  by\n  have h_main : \u2203 (k : \u2115), 3 ^ 2 ^ n = 1 + 2 ^ (n + 2) + k * 2 ^ (n + 3) := by\n    have h\u2081 : \u2200 n : \u2115, 0 < n \u2192 \u2203 (k : \u2115), 3 ^ 2 ^ n = 1 + 2 ^ (n + 2) + k * 2 ^ (n + 3) := by\n      intro n hn\n      induction' hn with n hn IH\n      \u00b7 \n        use 0\n        norm_num\n      \u00b7 \n        obtain \u27e8k, hk\u27e9 := IH\n        use (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) / 2 ^ (n + 4)\n        have h\u2082 : 3 ^ 2 ^ (n + 1) = (3 ^ 2 ^ n) ^ 2 := by\n          simp [pow_succ, pow_mul, pow_two]\n          <;> ring_nf\n          <;> simp [pow_add, pow_mul, pow_two]\n          <;> ring_nf\n        rw [h\u2082, hk]\n        have h\u2083 : (1 + 2 ^ (n + 2) + k * 2 ^ (n + 3)) ^ 2 = 1 + 2 ^ (n + 3) + (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n          have h\u2084 : n \u2265 0 := by linarith\n          have h\u2085 : 2 * n + 4 \u2265 n + 4 := by\n            omega\n          have h\u2086 : 2 * n + 6 \u2265 n + 4 := by\n            omega\n          have h\u2087 : 2 * n + 5 \u2265 n + 4 := by\n            omega\n          calc\n            (1 + 2 ^ (n + 2) + k * 2 ^ (n + 3)) ^ 2 = 1 + 2 ^ (n + 3) + (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n              ring_nf at *\n              <;>\n                simp [pow_add, pow_mul, mul_assoc, mul_comm, mul_left_comm, Nat.mul_div_cancel_left]\n              <;>\n                ring_nf at *\n              <;>\n                omega\n            _ = 1 + 2 ^ (n + 3) + (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n              rfl\n        rw [h\u2083]\n        have h\u2084 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) % 2 ^ (n + 4) = 0 := by\n          have h\u2085 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) := by\n            apply pow_dvd_pow 2\n            omega\n          have h\u2086 : 2 ^ (n + 4) \u2223 k * 2 ^ (n + 4) := by\n            exact \u27e8k, by ring\u27e9\n          have h\u2087 : 2 ^ (n + 4) \u2223 k ^ 2 * 2 ^ (2 * n + 6) := by\n            have h\u2088 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 6) := by\n              apply pow_dvd_pow 2\n              omega\n            have h\u2089 : 2 ^ (n + 4) \u2223 k ^ 2 * 2 ^ (2 * n + 6) := by\n              exact dvd_mul_of_dvd_right h\u2088 _\n            exact h\u2089\n          have h\u2088 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n            have h\u2089 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := by\n              apply pow_dvd_pow 2\n              omega\n            have h\u2081\u2080 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n              have h\u2081\u2081 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := h\u2089\n              have h\u2081\u2082 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n                calc\n                  2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := h\u2081\u2081\n                  _ \u2223 2 * k * 2 ^ (2 * n + 5) := by\n                    exact \u27e82 * k, by ring\u27e9\n              exact h\u2081\u2082\n            exact h\u2081\u2080\n          have h\u2089 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5) := by\n            have h\u2081\u2080 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) := h\u2085\n            have h\u2081\u2081 : 2 ^ (n + 4) \u2223 k * 2 ^ (n + 4) := h\u2086\n            have h\u2081\u2082 : 2 ^ (n + 4) \u2223 k ^ 2 * 2 ^ (2 * n + 6) := h\u2087\n            have h\u2081\u2083 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := h\u2088\n            exact Nat.dvd_add (Nat.dvd_add (Nat.dvd_add h\u2081\u2080 h\u2086) h\u2087) h\u2088\n          have h\u2081\u2080 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) % 2 ^ (n + 4) = 0 := by\n            have h\u2081\u2081 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5) := h\u2089\n            have h\u2081\u2082 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) % 2 ^ (n + 4) = 0 := by\n              exact Nat.mod_eq_zero_of_dvd h\u2081\u2081\n            exact h\u2081\u2082\n          exact h\u2081\u2080\n        have h\u2085 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) / 2 ^ (n + 4) * 2 ^ (n + 4) = (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n          have h\u2086 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5) := by\n            have h\u2087 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) := by\n              apply pow_dvd_pow 2\n              omega\n            have h\u2088 : 2 ^ (n + 4) \u2223 k * 2 ^ (n + 4) := by\n              exact \u27e8k, by ring\u27e9\n            have h\u2089 : 2 ^ (n + 4) \u2223 k ^ 2 * 2 ^ (2 * n + 6) := by\n              have h\u2081\u2080 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 6) := by\n                apply pow_dvd_pow 2\n                omega\n              have h\u2081\u2081 : 2 ^ (n + 4) \u2223 k ^ 2 * 2 ^ (2 * n + 6) := by\n                exact dvd_mul_of_dvd_right h\u2081\u2080 _\n              exact h\u2081\u2081\n            have h\u2081\u2080 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n              have h\u2081\u2081 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := by\n                apply pow_dvd_pow 2\n                omega\n              have h\u2081\u2082 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n                have h\u2081\u2083 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := h\u2081\u2081\n                have h\u2081\u2084 : 2 ^ (n + 4) \u2223 2 * k * 2 ^ (2 * n + 5) := by\n                  calc\n                    2 ^ (n + 4) \u2223 2 ^ (2 * n + 5) := h\u2081\u2083\n                    _ \u2223 2 * k * 2 ^ (2 * n + 5) := by\n                      exact \u27e82 * k, by ring\u27e9\n                exact h\u2081\u2084\n              exact h\u2081\u2082\n            exact Nat.dvd_add (Nat.dvd_add (Nat.dvd_add h\u2087 h\u2088) h\u2089) h\u2081\u2080\n          have h\u2081\u2081 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) / 2 ^ (n + 4) * 2 ^ (n + 4) = (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n            have h\u2081\u2082 : 2 ^ (n + 4) \u2223 2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5) := h\u2086\n            have h\u2081\u2083 : (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) / 2 ^ (n + 4) * 2 ^ (n + 4) = (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) := by\n              exact Nat.div_mul_cancel h\u2081\u2082\n            exact h\u2081\u2083\n          exact h\u2081\u2081\n        have h\u2086 : 1 + 2 ^ (n + 3) + (2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) = 1 + 2 ^ (n + 1 + 2) + ((2 ^ (2 * n + 4) + k * 2 ^ (n + 4) + k ^ 2 * 2 ^ (2 * n + 6) + 2 * k * 2 ^ (2 * n + 5)) / 2 ^ (n + 4) * 2 ^ (n + 1 + 3)) := by\n          have h\u2087 : n + 3 = n + 1 + 2 := by ring\n          have h\u2088 : n + 4 = n + 1 + 3 := by ring\n          have h\u2089 : 2 * n + 4 = 2 * n + 4 := by ring\n          simp [h\u2087, h\u2088, h\u2089, pow_add, pow_mul, Nat.mul_div_assoc, Nat.div_eq_of_lt] at h\u2084 h\u2085 \u22a2 <;> ring_nf at h\u2084 h\u2085 \u22a2 <;> omega\n        rw [h\u2086]\n        <;> simp [h\u2085, Nat.mul_div_assoc, Nat.div_eq_of_lt] <;> ring_nf at * <;> omega\n    exact h\u2081 n h\u2080\n  have h_final : (3 ^ 2 ^ n - 1) % 2 ^ (n + 3) = 2 ^ (n + 2) := by\n    obtain \u27e8k, hk\u27e9 := h_main\n    have h\u2081 : 3 ^ 2 ^ n - 1 = 2 ^ (n + 2) + k * 2 ^ (n + 3) := by\n      have h\u2082 : 3 ^ 2 ^ n = 1 + 2 ^ (n + 2) + k * 2 ^ (n + 3) := by linarith\n      have h\u2083 : 3 ^ 2 ^ n - 1 = 2 ^ (n + 2) + k * 2 ^ (n + 3) := by\n        have h\u2084 : 3 ^ 2 ^ n \u2265 1 := by\n          apply Nat.one_le_pow\n          <;> positivity\n        omega\n      exact h\u2083\n    rw [h\u2081]\n    have h\u2082 : (2 ^ (n + 2) + k * 2 ^ (n + 3)) % 2 ^ (n + 3) = 2 ^ (n + 2) % 2 ^ (n + 3) := by\n      have h\u2083 : (2 ^ (n + 2) + k * 2 ^ (n + 3)) % 2 ^ (n + 3) = (2 ^ (n + 2)) % 2 ^ (n + 3) := by\n        have h\u2084 : k * 2 ^ (n + 3) % 2 ^ (n + 3) = 0 := by\n          have h\u2085 : 2 ^ (n + 3) \u2223 k * 2 ^ (n + 3) := by\n            use k\n            <;> ring\n          exact Nat.mod_eq_zero_of_dvd h\u2085\n        have h\u2085 : (2 ^ (n + 2) + k * 2 ^ (n + 3)) % 2 ^ (n + 3) = (2 ^ (n + 2) % 2 ^ (n + 3) + k * 2 ^ (n + 3) % 2 ^ (n + 3)) % 2 ^ (n + 3) := by\n          simp [Nat.add_mod]\n        rw [h\u2085]\n        have h\u2086 : k * 2 ^ (n + 3) % 2 ^ (n + 3) = 0 := h\u2084\n        rw [h\u2086]\n        have h\u2087 : (2 ^ (n + 2) % 2 ^ (n + 3) + 0) % 2 ^ (n + 3) = 2 ^ (n + 2) % 2 ^ (n + 3) := by\n          simp [Nat.add_mod]\n        rw [h\u2087]\n      exact h\u2083\n    rw [h\u2082]\n    have h\u2083 : 2 ^ (n + 2) % 2 ^ (n + 3) = 2 ^ (n + 2) := by\n      have h\u2084 : 2 ^ (n + 2) < 2 ^ (n + 3) := by\n        apply Nat.pow_lt_pow_of_lt_right\n        <;> omega\n      have h\u2085 : 2 ^ (n + 2) % 2 ^ (n + 3) = 2 ^ (n + 2) := by\n        rw [Nat.mod_eq_of_lt h\u2084]\n      rw [h\u2085]\n    rw [h\u2083]\n  exact h_final\n"}
{"mode": "clear", "steps_before": 54, "alphabet": 29, "compiles": false, "id": "mathd_numbertheory_353", "proof_after": "theorem mathd_numbertheory_353 (s : \u2115) (h\u2080 : s = \u2211 k in Finset.Icc 2010 4018, k) : s % 2009 = 0  :=  by\n  have h\u2081 : s = 2009 * 3014 := by\n    rw [h\u2080]\n    have h\u2082 : (\u2211 k in Finset.Icc 2010 4018, k) = \u2211 t in Finset.range 2009, (t + 2010) := by\n      have h\u2083 : Finset.Icc 2010 4018 = (Finset.range 2009).image (fun t => t + 2010) := by\n        apply Finset.ext\n        intro x\n        simp only [Finset.mem_Icc, Finset.mem_image, Finset.mem_range]\n        constructor\n        \u00b7 intro hx\n          have h\u2084 : 2010 \u2264 x \u2227 x \u2264 4018 := hx\n          have h\u2085 : x \u2265 2010 := by linarith\n          have h\u2086 : x \u2264 4018 := by linarith\n          have h\u2087 : x - 2010 < 2009 := by omega\n          use x - 2010\n          <;> simp_all [Nat.sub_add_cancel]\n          <;> omega\n        \u00b7 intro hx\n          rcases hx with \u27e8t, ht\u2081, ht\u2082\u27e9\n          have h\u2084 : t < 2009 := ht\u2081\n          have h\u2085 : t + 2010 = x := by omega\n          have h\u2086 : 2010 \u2264 x := by omega\n          have h\u2087 : x \u2264 4018 := by omega\n          exact \u27e8by omega, by omega\u27e9\n      rw [h\u2083]\n      rw [Finset.sum_image]\n      <;> simp_all [Function.Injective]\n      <;> omega\n    rw [h\u2082]\n    have h\u2083 : (\u2211 t in Finset.range 2009, (t + 2010 : \u2115)) = (\u2211 t in Finset.range 2009, t) + \u2211 t in Finset.range 2009, 2010 := by\n      rw [Finset.sum_add_distrib]\n    rw [h\u2083]\n    have h\u2084 : (\u2211 t in Finset.range 2009, t) = 2009 * 1004 := by\n      rw [Finset.sum_range_id]\n      <;> norm_num\n      <;> ring_nf\n      <;> norm_num\n    have h\u2085 : (\u2211 t in Finset.range 2009, (2010 : \u2115)) = 2009 * 2010 := by\n      simp [Finset.sum_const, Finset.card_range]\n      <;> ring_nf\n    rw [h\u2084, h\u2085]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n  have h\u2082 : s % 2009 = 0 := by\n    rw [h\u2081]\n    <;> norm_num\n    <;> rfl\n  exact h\u2082\n", "proof_before": "theorem mathd_numbertheory_353 (s : \u2115) (h\u2080 : s = \u2211 k in Finset.Icc 2010 4018, k) : s % 2009 = 0  :=  by\n  have h\u2081 : s = 2009 * 3014 := by\n    rw [h\u2080]\n    have h\u2082 : (\u2211 k in Finset.Icc 2010 4018, k) = \u2211 t in Finset.range 2009, (t + 2010) := by\n      have h\u2083 : Finset.Icc 2010 4018 = (Finset.range 2009).image (fun t => t + 2010) := by\n        apply Finset.ext\n        intro x\n        simp only [Finset.mem_Icc, Finset.mem_image, Finset.mem_range]\n        constructor\n        \u00b7 intro hx\n          have h\u2084 : 2010 \u2264 x \u2227 x \u2264 4018 := hx\n          have h\u2085 : x \u2265 2010 := by linarith\n          have h\u2086 : x \u2264 4018 := by linarith\n          have h\u2087 : x - 2010 < 2009 := by omega\n          use x - 2010\n          <;> simp_all [Nat.sub_add_cancel]\n          <;> omega\n        \u00b7 intro hx\n          rcases hx with \u27e8t, ht\u2081, ht\u2082\u27e9\n          have h\u2084 : t < 2009 := ht\u2081\n          have h\u2085 : t + 2010 = x := by omega\n          have h\u2086 : 2010 \u2264 x := by omega\n          have h\u2087 : x \u2264 4018 := by omega\n          exact \u27e8by omega, by omega\u27e9\n      rw [h\u2083]\n      rw [Finset.sum_image]\n      <;> simp_all [Function.Injective]\n      <;> omega\n    rw [h\u2082]\n    have h\u2083 : (\u2211 t in Finset.range 2009, (t + 2010 : \u2115)) = (\u2211 t in Finset.range 2009, t) + \u2211 t in Finset.range 2009, 2010 := by\n      rw [Finset.sum_add_distrib]\n    rw [h\u2083]\n    have h\u2084 : (\u2211 t in Finset.range 2009, t) = 2009 * 1004 := by\n      rw [Finset.sum_range_id]\n      <;> norm_num\n      <;> ring_nf\n      <;> norm_num\n    have h\u2085 : (\u2211 t in Finset.range 2009, (2010 : \u2115)) = 2009 * 2010 := by\n      simp [Finset.sum_const, Finset.card_range]\n      <;> ring_nf\n    rw [h\u2084, h\u2085]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n  have h\u2082 : s % 2009 = 0 := by\n    rw [h\u2081]\n    <;> norm_num\n    <;> rfl\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 111, "alphabet": 50, "compiles": true, "steps_after": 94, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 2, "fixpoint": false, "chars_before": 3467, "chars_after": 2808, "seconds": 434.1, "id": "mathd_algebra_598", "proof_after": "theorem mathd_algebra_598 (a b c d : \u211d) (h\u2081 : (4 : \u211d) ^ a = 5) (h\u2082 : (5 : \u211d) ^ b = 6)\n    (h\u2083 : (6 : \u211d) ^ c = 7) (h\u2084 : (7 : \u211d) ^ d = 8) : a * b * c * d = 3 / 2  :=  by\n  have h\u2085 : a = Real.log 5 / Real.log 4 := by\n    have h\u2085\u2081 : Real.log ((4 : \u211d) ^ a) = Real.log 5 := by rw [h\u2081]\n    have h\u2085\u2082 : a * Real.log 4 = Real.log 5 := by\n      rw [Real.log_rpow (by norm_num : (4 : \u211d) > 0)] at h\u2085\u2081\n      exact h\u2085\u2081\n    have h\u2085\u2083 : Real.log 4 \u2260 0 := by\n      have h\u2085\u2084 : Real.log 4 > 0 := Real.log_pos (by norm_num)\n      linarith\n    have h\u2085\u2084 : a = Real.log 5 / Real.log 4 := by\n      field_simp at h\u2085\u2082 \u22a2\n      linarith\n    exact h\u2085\u2084\n  have h\u2086 : b = Real.log 6 / Real.log 5 := by\n    have h\u2086\u2081 : Real.log ((5 : \u211d) ^ b) = Real.log 6 := by rw [h\u2082]\n    have h\u2086\u2082 : b * Real.log 5 = Real.log 6 := by\n      rw [Real.log_rpow (by norm_num : (5 : \u211d) > 0)] at h\u2086\u2081\n      exact h\u2086\u2081\n    have h\u2086\u2083 : Real.log 5 \u2260 0 := by\n      have h\u2086\u2084 : Real.log 5 > 0 := Real.log_pos (by norm_num)\n      linarith\n    have h\u2086\u2084 : b = Real.log 6 / Real.log 5 := by\n      field_simp at h\u2086\u2082 \u22a2\n      linarith\n    exact h\u2086\u2084\n  have h\u2087 : c = Real.log 7 / Real.log 6 := by\n    have h\u2087\u2081 : Real.log ((6 : \u211d) ^ c) = Real.log 7 := by rw [h\u2083]\n    have h\u2087\u2082 : c * Real.log 6 = Real.log 7 := by\n      rw [Real.log_rpow (by norm_num : (6 : \u211d) > 0)] at h\u2087\u2081\n      exact h\u2087\u2081\n    have h\u2087\u2083 : Real.log 6 \u2260 0 := by\n      have h\u2087\u2084 : Real.log 6 > 0 := Real.log_pos (by norm_num)\n      linarith\n    have h\u2087\u2084 : c = Real.log 7 / Real.log 6 := by\n      field_simp at h\u2087\u2082 \u22a2\n      linarith\n    exact h\u2087\u2084\n  have h\u2088 : d = Real.log 8 / Real.log 7 := by\n    have h\u2088\u2081 : Real.log ((7 : \u211d) ^ d) = Real.log 8 := by rw [h\u2084]\n    have h\u2088\u2082 : d * Real.log 7 = Real.log 8 := by\n      rw [Real.log_rpow (by norm_num : (7 : \u211d) > 0)] at h\u2088\u2081\n      exact h\u2088\u2081\n    have h\u2088\u2083 : Real.log 7 \u2260 0 := by\n      have h\u2088\u2084 : Real.log 7 > 0 := Real.log_pos (by norm_num)\n      linarith\n    have h\u2088\u2084 : d = Real.log 8 / Real.log 7 := by\n      field_simp at h\u2088\u2082 \u22a2\n      linarith\n    exact h\u2088\u2084\n  have h\u2089 : a * b * c * d = (Real.log 8) / (Real.log 4) := by\n    rw [h\u2085, h\u2086, h\u2087, h\u2088]\n    norm_num\n  have h\u2081\u2080 : Real.log 8 = 3 * Real.log 2 := by\n    have h\u2081\u2080\u2081 : Real.log 8 = Real.log (2 ^ 3) := by norm_num\n    rw [h\u2081\u2080\u2081]\n    have h\u2081\u2080\u2082 : Real.log (2 ^ 3) = 3 * Real.log 2 := by\n      rw [Real.log_pow] <;> norm_num\n    rw [h\u2081\u2080\u2082]\n    <;>\n    norm_num\n  have h\u2081\u2081 : Real.log 4 = 2 * Real.log 2 := by\n    have h\u2081\u2081\u2081 : Real.log 4 = Real.log (2 ^ 2) := by norm_num\n    rw [h\u2081\u2081\u2081]\n    have h\u2081\u2081\u2082 : Real.log (2 ^ 2) = 2 * Real.log 2 := by\n      rw [Real.log_pow] <;> norm_num\n    rw [h\u2081\u2081\u2082]\n    <;>\n    norm_num\n  have h\u2081\u2082 : a * b * c * d = 3 / 2 := by\n    rw [h\u2089]\n    have h\u2081\u2082\u2081 : Real.log 8 = 3 * Real.log 2 := h\u2081\u2080\n    have h\u2081\u2082\u2082 : Real.log 4 = 2 * Real.log 2 := h\u2081\u2081\n    rw [h\u2081\u2082\u2081, h\u2081\u2082\u2082]\n    have h\u2081\u2082\u2083 : Real.log 2 \u2260 0 := by\n      have h\u2081\u2082\u2084 : Real.log 2 > 0 := Real.log_pos (by norm_num)\n      linarith\n    field_simp [h\u2081\u2082\u2083]\n    <;> ring_nf\n    <;> field_simp [h\u2081\u2082\u2083]\n    <;> linarith\n  rw [h\u2081\u2082]\n  <;> norm_num\n", "proof_before": "theorem mathd_algebra_598 (a b c d : \u211d) (h\u2081 : (4 : \u211d) ^ a = 5) (h\u2082 : (5 : \u211d) ^ b = 6)\n    (h\u2083 : (6 : \u211d) ^ c = 7) (h\u2084 : (7 : \u211d) ^ d = 8) : a * b * c * d = 3 / 2  :=  by\n  have h\u2085 : a = Real.log 5 / Real.log 4 := by\n    have h\u2085\u2081 : Real.log ((4 : \u211d) ^ a) = Real.log 5 := by rw [h\u2081]\n    have h\u2085\u2082 : a * Real.log 4 = Real.log 5 := by\n      rw [Real.log_rpow (by norm_num : (4 : \u211d) > 0)] at h\u2085\u2081\n      exact h\u2085\u2081\n    have h\u2085\u2083 : Real.log 4 \u2260 0 := by\n      have h\u2085\u2084 : Real.log 4 > 0 := Real.log_pos (by norm_num)\n      linarith\n    have h\u2085\u2084 : a = Real.log 5 / Real.log 4 := by\n      field_simp at h\u2085\u2082 \u22a2\n      linarith\n    exact h\u2085\u2084\n  have h\u2086 : b = Real.log 6 / Real.log 5 := by\n    have h\u2086\u2081 : Real.log ((5 : \u211d) ^ b) = Real.log 6 := by rw [h\u2082]\n    have h\u2086\u2082 : b * Real.log 5 = Real.log 6 := by\n      rw [Real.log_rpow (by norm_num : (5 : \u211d) > 0)] at h\u2086\u2081\n      exact h\u2086\u2081\n    have h\u2086\u2083 : Real.log 5 \u2260 0 := by\n      have h\u2086\u2084 : Real.log 5 > 0 := Real.log_pos (by norm_num)\n      linarith\n    have h\u2086\u2084 : b = Real.log 6 / Real.log 5 := by\n      field_simp at h\u2086\u2082 \u22a2\n      linarith\n    exact h\u2086\u2084\n  have h\u2087 : c = Real.log 7 / Real.log 6 := by\n    have h\u2087\u2081 : Real.log ((6 : \u211d) ^ c) = Real.log 7 := by rw [h\u2083]\n    have h\u2087\u2082 : c * Real.log 6 = Real.log 7 := by\n      rw [Real.log_rpow (by norm_num : (6 : \u211d) > 0)] at h\u2087\u2081\n      exact h\u2087\u2081\n    have h\u2087\u2083 : Real.log 6 \u2260 0 := by\n      have h\u2087\u2084 : Real.log 6 > 0 := Real.log_pos (by norm_num)\n      linarith\n    have h\u2087\u2084 : c = Real.log 7 / Real.log 6 := by\n      field_simp at h\u2087\u2082 \u22a2\n      linarith\n    exact h\u2087\u2084\n  have h\u2088 : d = Real.log 8 / Real.log 7 := by\n    have h\u2088\u2081 : Real.log ((7 : \u211d) ^ d) = Real.log 8 := by rw [h\u2084]\n    have h\u2088\u2082 : d * Real.log 7 = Real.log 8 := by\n      rw [Real.log_rpow (by norm_num : (7 : \u211d) > 0)] at h\u2088\u2081\n      exact h\u2088\u2081\n    have h\u2088\u2083 : Real.log 7 \u2260 0 := by\n      have h\u2088\u2084 : Real.log 7 > 0 := Real.log_pos (by norm_num)\n      linarith\n    have h\u2088\u2084 : d = Real.log 8 / Real.log 7 := by\n      field_simp at h\u2088\u2082 \u22a2\n      linarith\n    exact h\u2088\u2084\n  have h\u2089 : a * b * c * d = (Real.log 8) / (Real.log 4) := by\n    rw [h\u2085, h\u2086, h\u2087, h\u2088]\n    have h\u2089\u2081 : Real.log 5 > 0 := Real.log_pos (by norm_num)\n    have h\u2089\u2081 : Real.log 6 > 0 := Real.log_pos (by norm_num)\n    have h\u2081\u2080\u2081 : Real.log 7 > 0 := Real.log_pos (by norm_num)\n    have h\u2081\u2081\u2081 : Real.log 4 > 0 := Real.log_pos (by norm_num)\n    have h\u2081\u2082\u2081 : Real.log 8 > 0 := Real.log_pos (by norm_num)\n    field_simp [Real.log_mul, Real.log_div, Real.log_rpow, Real.log_pow]\n    <;> ring_nf\n    <;> field_simp [Real.log_mul, Real.log_div, Real.log_rpow, Real.log_pow]\n    <;> ring_nf\n    <;> field_simp [Real.log_mul, Real.log_div, Real.log_rpow, Real.log_pow]\n    <;> ring_nf\n    <;> field_simp [Real.log_mul, Real.log_div, Real.log_rpow, Real.log_pow]\n    <;> linarith\n  have h\u2081\u2080 : Real.log 8 = 3 * Real.log 2 := by\n    have h\u2081\u2080\u2081 : Real.log 8 = Real.log (2 ^ 3) := by norm_num\n    rw [h\u2081\u2080\u2081]\n    have h\u2081\u2080\u2082 : Real.log (2 ^ 3) = 3 * Real.log 2 := by\n      rw [Real.log_pow] <;> norm_num\n    rw [h\u2081\u2080\u2082]\n    <;>\n    norm_num\n  have h\u2081\u2081 : Real.log 4 = 2 * Real.log 2 := by\n    have h\u2081\u2081\u2081 : Real.log 4 = Real.log (2 ^ 2) := by norm_num\n    rw [h\u2081\u2081\u2081]\n    have h\u2081\u2081\u2082 : Real.log (2 ^ 2) = 2 * Real.log 2 := by\n      rw [Real.log_pow] <;> norm_num\n    rw [h\u2081\u2081\u2082]\n    <;>\n    norm_num\n  have h\u2081\u2082 : a * b * c * d = 3 / 2 := by\n    rw [h\u2089]\n    have h\u2081\u2082\u2081 : Real.log 8 = 3 * Real.log 2 := h\u2081\u2080\n    have h\u2081\u2082\u2082 : Real.log 4 = 2 * Real.log 2 := h\u2081\u2081\n    rw [h\u2081\u2082\u2081, h\u2081\u2082\u2082]\n    have h\u2081\u2082\u2083 : Real.log 2 \u2260 0 := by\n      have h\u2081\u2082\u2084 : Real.log 2 > 0 := Real.log_pos (by norm_num)\n      linarith\n    field_simp [h\u2081\u2082\u2083]\n    <;> ring_nf\n    <;> field_simp [h\u2081\u2082\u2083]\n    <;> linarith\n  rw [h\u2081\u2082]\n  <;> norm_num\n"}
{"mode": "clear", "steps_before": 135, "alphabet": 43, "compiles": true, "steps_after": 60, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 3, "fixpoint": false, "chars_before": 4530, "chars_after": 2316, "seconds": 482.2, "id": "amc12a_2003_p5", "proof_after": "theorem amc12a_2003_p5 (A M C : \u2115) (h\u2080 : A \u2264 9 \u2227 M \u2264 9 \u2227 C \u2264 9)\n    (h\u2081 : Nat.ofDigits 10 [0, 1, C, M, A] + Nat.ofDigits 10 [2, 1, C, M, A] = 123422) :\n    A + M + C = 14  :=  by\n  have h\u2082 : A = 6 := by\n    have h\u2082\u2081 : A \u2264 9 := h\u2080.1\n    have h\u2082\u2082 : M \u2264 9 := h\u2080.2.1\n    have h\u2082\u2083 : C \u2264 9 := h\u2080.2.2\n    have h\u2082\u2084 : Nat.ofDigits 10 [0, 1, C, M, A] = 10 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    have h\u2082\u2085 : Nat.ofDigits 10 [2, 1, C, M, A] = 12 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    omega\n  have h\u2083 : M = 1 := by\n    have h\u2083\u2081 : A \u2264 9 := h\u2080.1\n    have h\u2083\u2082 : M \u2264 9 := h\u2080.2.1\n    have h\u2083\u2083 : C \u2264 9 := h\u2080.2.2\n    have h\u2083\u2084 : Nat.ofDigits 10 [0, 1, C, M, A] = 10 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    have h\u2083\u2085 : Nat.ofDigits 10 [2, 1, C, M, A] = 12 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    omega\n  have h\u2084 : C = 7 := by\n    have h\u2084\u2081 : A \u2264 9 := h\u2080.1\n    have h\u2084\u2082 : M \u2264 9 := h\u2080.2.1\n    have h\u2084\u2083 : C \u2264 9 := h\u2080.2.2\n    have h\u2084\u2084 : Nat.ofDigits 10 [0, 1, C, M, A] = 10 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    have h\u2084\u2085 : Nat.ofDigits 10 [2, 1, C, M, A] = 12 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    rw [h\u2084\u2084, h\u2084\u2085] at h\u2081\n    have h\u2084\u2086 : 10 + 100 * C + 1000 * M + 10000 * A + (12 + 100 * C + 1000 * M + 10000 * A) = 123422 := by\n      linarith\n    have h\u2084\u2087 : 22 + 200 * C + 2000 * M + 20000 * A = 123422 := by\n      linarith\n    have h\u2084\u2088 : 200 * C + 2000 * M + 20000 * A = 123400 := by\n      omega\n    have h\u2084\u2089 : C + 10 * M + 100 * A = 617 := by\n      omega\n    rw [h\u2082, h\u2083] at h\u2084\u2089\n    norm_num at h\u2084\u2089 \u22a2\n    <;> omega\n  have h\u2085 : A + M + C = 14 := by\n    rw [h\u2082, h\u2083, h\u2084]\n    <;> norm_num\n  apply h\u2085\n", "proof_before": "theorem amc12a_2003_p5 (A M C : \u2115) (h\u2080 : A \u2264 9 \u2227 M \u2264 9 \u2227 C \u2264 9)\n    (h\u2081 : Nat.ofDigits 10 [0, 1, C, M, A] + Nat.ofDigits 10 [2, 1, C, M, A] = 123422) :\n    A + M + C = 14  :=  by\n  have h\u2082 : A = 6 := by\n    have h\u2082\u2081 : A \u2264 9 := h\u2080.1\n    have h\u2082\u2082 : M \u2264 9 := h\u2080.2.1\n    have h\u2082\u2083 : C \u2264 9 := h\u2080.2.2\n    have h\u2082\u2084 : Nat.ofDigits 10 [0, 1, C, M, A] = 10 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    have h\u2082\u2085 : Nat.ofDigits 10 [2, 1, C, M, A] = 12 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    rw [h\u2082\u2084, h\u2082\u2085] at h\u2081\n    have h\u2082\u2086 : 10 + 100 * C + 1000 * M + 10000 * A + (12 + 100 * C + 1000 * M + 10000 * A) = 123422 := by\n      linarith\n    have h\u2082\u2087 : 22 + 200 * C + 2000 * M + 20000 * A = 123422 := by\n      linarith\n    have h\u2082\u2088 : 200 * C + 2000 * M + 20000 * A = 123400 := by\n      omega\n    have h\u2082\u2089 : C + 10 * M + 100 * A = 617 := by\n      omega\n    have h\u2083\u2080 : A \u2265 1 := by\n      by_contra h\n      push_neg at h\n      have h\u2083\u2081 : A = 0 := by\n        omega\n      rw [h\u2083\u2081] at h\u2082\u2089\n      have h\u2083\u2082 : C + 10 * M = 617 := by\n        omega\n      have h\u2083\u2083 : C + 10 * M \u2264 99 := by\n        have h\u2083\u2084 : M \u2264 9 := h\u2080.2.1\n        have h\u2083\u2085 : C \u2264 9 := h\u2080.2.2\n        omega\n      omega\n    have h\u2083\u2081 : A \u2264 6 := by\n      by_contra h\n      push_neg at h\n      have h\u2083\u2082 : A \u2265 7 := by omega\n      have h\u2083\u2083 : 100 * A \u2265 700 := by\n        omega\n      have h\u2083\u2084 : C + 10 * M + 100 * A \u2265 700 := by\n        omega\n      have h\u2083\u2085 : C + 10 * M + 100 * A = 617 := by\n        omega\n      omega\n    interval_cases A <;> norm_num at h\u2082\u2089 \u22a2 <;>\n      (try omega) <;>\n      (try\n        {\n          have h\u2083\u2086 : M \u2264 9 := h\u2080.2.1\n          have h\u2083\u2087 : C \u2264 9 := h\u2080.2.2\n          interval_cases M <;> norm_num at h\u2082\u2089 \u22a2 <;>\n            (try omega) <;>\n            (try\n              {\n                interval_cases C <;> norm_num at h\u2082\u2089 \u22a2 <;>\n                  omega\n              })\n        })\n    <;> omega\n  have h\u2083 : M = 1 := by\n    have h\u2083\u2081 : A \u2264 9 := h\u2080.1\n    have h\u2083\u2082 : M \u2264 9 := h\u2080.2.1\n    have h\u2083\u2083 : C \u2264 9 := h\u2080.2.2\n    have h\u2083\u2084 : Nat.ofDigits 10 [0, 1, C, M, A] = 10 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    have h\u2083\u2085 : Nat.ofDigits 10 [2, 1, C, M, A] = 12 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    rw [h\u2083\u2084, h\u2083\u2085] at h\u2081\n    have h\u2083\u2086 : 10 + 100 * C + 1000 * M + 10000 * A + (12 + 100 * C + 1000 * M + 10000 * A) = 123422 := by\n      linarith\n    have h\u2083\u2087 : 22 + 200 * C + 2000 * M + 20000 * A = 123422 := by\n      linarith\n    have h\u2083\u2088 : 200 * C + 2000 * M + 20000 * A = 123400 := by\n      omega\n    have h\u2083\u2089 : C + 10 * M + 100 * A = 617 := by\n      omega\n    rw [h\u2082] at h\u2083\u2089\n    have h\u2084\u2080 : C + 10 * M = 17 := by\n      omega\n    have h\u2084\u2081 : M \u2264 1 := by\n      by_contra h\n      push_neg at h\n      have h\u2084\u2082 : M \u2265 2 := by omega\n      have h\u2084\u2083 : C + 10 * M \u2265 20 := by\n        have h\u2084\u2084 : C \u2265 0 := by omega\n        nlinarith\n      omega\n    interval_cases M <;> norm_num at h\u2084\u2080 \u22a2 <;>\n      (try omega) <;>\n      (try\n        {\n          have h\u2084\u2082 : C \u2264 9 := h\u2080.2.2\n          interval_cases C <;> norm_num at h\u2084\u2080 \u22a2 <;>\n            (try omega)\n        })\n    <;> omega\n  have h\u2084 : C = 7 := by\n    have h\u2084\u2081 : A \u2264 9 := h\u2080.1\n    have h\u2084\u2082 : M \u2264 9 := h\u2080.2.1\n    have h\u2084\u2083 : C \u2264 9 := h\u2080.2.2\n    have h\u2084\u2084 : Nat.ofDigits 10 [0, 1, C, M, A] = 10 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    have h\u2084\u2085 : Nat.ofDigits 10 [2, 1, C, M, A] = 12 + 100 * C + 1000 * M + 10000 * A := by\n      simp [Nat.ofDigits, List.cons, List.append, List.map, List.foldr, Nat.mul_add, Nat.add_mul]\n      <;> ring_nf\n      <;> norm_num\n      <;> omega\n    rw [h\u2084\u2084, h\u2084\u2085] at h\u2081\n    have h\u2084\u2086 : 10 + 100 * C + 1000 * M + 10000 * A + (12 + 100 * C + 1000 * M + 10000 * A) = 123422 := by\n      linarith\n    have h\u2084\u2087 : 22 + 200 * C + 2000 * M + 20000 * A = 123422 := by\n      linarith\n    have h\u2084\u2088 : 200 * C + 2000 * M + 20000 * A = 123400 := by\n      omega\n    have h\u2084\u2089 : C + 10 * M + 100 * A = 617 := by\n      omega\n    rw [h\u2082, h\u2083] at h\u2084\u2089\n    norm_num at h\u2084\u2089 \u22a2\n    <;> omega\n  have h\u2085 : A + M + C = 14 := by\n    rw [h\u2082, h\u2083, h\u2084]\n    <;> norm_num\n  apply h\u2085\n"}
{"mode": "clear", "steps_before": 7, "alphabet": 13, "compiles": false, "id": "mathd_numbertheory_237", "proof_after": "theorem mathd_numbertheory_237 : (\u2211 k in Finset.range 101, k) % 6 = 4  :=  by\n  have h : (\u2211 k in Finset.range 101, k) = 5050 := by\n    rfl\n  have h\u2081 : (\u2211 k in Finset.range 101, k) % 6 = 4 := by\n    rw [h]\n    <;> norm_num\n    <;> rfl\n  apply h\u2081\n", "proof_before": "theorem mathd_numbertheory_237 : (\u2211 k in Finset.range 101, k) % 6 = 4  :=  by\n  have h : (\u2211 k in Finset.range 101, k) = 5050 := by\n    rfl\n  have h\u2081 : (\u2211 k in Finset.range 101, k) % 6 = 4 := by\n    rw [h]\n    <;> norm_num\n    <;> rfl\n  apply h\u2081\n"}
{"mode": "clear", "steps_before": 126, "alphabet": 64, "compiles": false, "id": "algebra_apbpceq2_abpbcpcaeq1_aleq1on3anbleq1ancleq4on3", "proof_after": "theorem algebra_apbpceq2_abpbcpcaeq1_aleq1on3anbleq1ancleq4on3 (a b c : \u211d) (h\u2080 : a \u2264 b \u2227 b \u2264 c)\n    (h\u2081 : a + b + c = 2) (h\u2082 : a * b + b * c + c * a = 1) :\n    0 \u2264 a \u2227 a \u2264 1 / 3 \u2227 1 / 3 \u2264 b \u2227 b \u2264 1 \u2227 1 \u2264 c \u2227 c \u2264 4 / 3  :=  by\n  have h\u2083 : 0 \u2264 a := by\n    by_contra h\n    have h\u2084 : a < 0 := by linarith\n    have h\u2085 : a * (4 - 3 * a) < 0 := by\n      have h\u2085\u2081 : 4 - 3 * a > 0 := by nlinarith\n      nlinarith\n    have h\u2086 : (b - c) ^ 2 = 4 * a - 3 * a ^ 2 := by\n      have h\u2086\u2081 : a ^ 2 + b ^ 2 + c ^ 2 = 2 := by\n        nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n      have h\u2086\u2082 : a * b + b * c + c * a = 1 := h\u2082\n      nlinarith [sq_nonneg (b - c)]\n    have h\u2087 : (b - c) ^ 2 \u2265 0 := sq_nonneg (b - c)\n    nlinarith [sq_nonneg (b - c)]\n  have h\u2084 : 1 \u2264 c := by\n    have h\u2084\u2081 : a \u2264 b := h\u2080.1\n    have h\u2084\u2082 : b \u2264 c := h\u2080.2\n    have h\u2084\u2083 : a + b + c = 2 := h\u2081\n    have h\u2084\u2084 : a * b + b * c + c * a = 1 := h\u2082\n    by_contra! h\n    have h\u2085 : c < 1 := by linarith\n    have h\u2086 : a + b = 2 - c := by linarith\n    have h\u2087 : a * b = 1 - c * (a + b) := by\n      nlinarith\n    have h\u2088 : a * b = (c - 1) ^ 2 := by\n      rw [h\u2086] at h\u2087\n      nlinarith\n    have h\u2089 : (b - c) ^ 2 \u2265 0 := by nlinarith\n    have h\u2081\u2080 : (2 - c) ^ 2 - 4 * (c - 1) ^ 2 \u2265 0 := by\n      nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n    have h\u2081\u2081 : c \u2265 2 / 3 := by\n      nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n    have h\u2081\u2082 : 3 * c ^ 2 - 4 * c + 1 \u2264 0 := by\n      nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n    have h\u2081\u2083 : c \u2264 1 := by nlinarith\n    have h\u2081\u2084 : c \u2265 2 / 3 := by nlinarith\n    have h\u2081\u2085 : b \u2264 c := h\u2080.2\n    have h\u2081\u2086 : a \u2264 b := h\u2080.1\n    have h\u2081\u2087 : (2 - c) ^ 2 - 4 * (c - 1) ^ 2 \u2265 0 := by\n      nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n    have h\u2081\u2088 : 3 * c ^ 2 - 4 * c + 1 \u2264 0 := by\n      nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n    have h\u2081\u2089 : c \u2265 1 := by\n      by_contra h\u2082\u2080\n      have h\u2082\u2081 : c < 1 := by linarith\n      have h\u2082\u2082 : (2 - c) ^ 2 - 4 * (c - 1) ^ 2 \u2265 0 := by\n        nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n      have h\u2082\u2083 : 3 * c ^ 2 - 4 * c + 1 \u2264 0 := by\n        nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n      have h\u2082\u2084 : c \u2265 2 / 3 := by nlinarith\n      have h\u2082\u2085 : c \u2264 1 := by nlinarith\n      have h\u2082\u2086 : b \u2264 c := h\u2080.2\n      have h\u2082\u2087 : a \u2264 b := h\u2080.1\n      have h\u2082\u2088 : (2 - c) ^ 2 - 4 * (c - 1) ^ 2 \u2265 0 := by\n        nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n      have h\u2082\u2089 : 3 * c ^ 2 - 4 * c + 1 \u2264 0 := by\n        nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n      by_cases h\u2083\u2080 : c \u2265 1\n      \u00b7 linarith\n      \u00b7 have h\u2083\u2081 : c < 1 := by linarith\n        have h\u2083\u2082 : 3 * c ^ 2 - 4 * c + 1 > 0 := by\n          nlinarith [sq_nonneg (c - 1 / 3), sq_nonneg (c - 2 / 3)]\n        linarith\n    linarith\n  have h\u2085 : c \u2264 4 / 3 := by\n    have h\u2085\u2081 : a + b + c = 2 := h\u2081\n    have h\u2085\u2082 : a * b + b * c + c * a = 1 := h\u2082\n    have h\u2085\u2083 : a \u2264 b := h\u2080.1\n    have h\u2085\u2084 : b \u2264 c := h\u2080.2\n    have h\u2085\u2085 : 0 \u2264 a := h\u2083\n    have h\u2085\u2086 : 1 \u2264 c := h\u2084\n    have h\u2085\u2087 : (b - c) ^ 2 \u2265 0 := by nlinarith\n    have h\u2085\u2088 : c \u2264 4 / 3 := by\n      nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a),\n        mul_nonneg (sub_nonneg.mpr h\u2085\u2085) (sub_nonneg.mpr h\u2085\u2083),\n        mul_nonneg (sub_nonneg.mpr h\u2085\u2083) (sub_nonneg.mpr h\u2085\u2084),\n        mul_nonneg (sub_nonneg.mpr h\u2085\u2085) (sub_nonneg.mpr h\u2085\u2084)]\n    exact h\u2085\u2088\n    <;> linarith\n  have h\u2086 : a \u2264 1 / 3 := by\n    have h\u2086\u2081 : a + b = 2 - c := by linarith\n    have h\u2086\u2082 : a * b = (c - 1) ^ 2 := by\n      have h\u2086\u2083 : a * b + b * c + c * a = 1 := h\u2082\n      have h\u2086\u2084 : a + b + c = 2 := h\u2081\n      have h\u2086\u2085 : a * b = 1 - c * (a + b) := by\n        nlinarith\n      rw [h\u2086\u2081] at h\u2086\u2085\n      nlinarith\n    have h\u2086\u2083 : c \u2265 1 := h\u2084\n    have h\u2086\u2084 : c \u2264 4 / 3 := h\u2085\n    have h\u2086\u2085 : a \u2264 1 / 3 := by\n      nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a),\n        mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2080.1),\n        mul_nonneg (sub_nonneg.mpr h\u2080.1) (sub_nonneg.mpr h\u2080.2),\n        mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2080.2)]\n    exact h\u2086\u2085\n  have h\u2087 : 1 / 3 \u2264 b := by\n    have h\u2087\u2081 : a + b = 2 - c := by linarith\n    have h\u2087\u2082 : c \u2264 4 / 3 := h\u2085\n    have h\u2087\u2083 : 1 \u2264 c := h\u2084\n    have h\u2087\u2084 : a \u2264 b := h\u2080.1\n    have h\u2087\u2085 : 0 \u2264 a := h\u2083\n    have h\u2087\u2086 : 1 / 3 \u2264 b := by\n      nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n    exact h\u2087\u2086\n  have h\u2088 : b \u2264 1 := by\n    have h\u2088\u2081 : a + b + c = 2 := h\u2081\n    have h\u2088\u2082 : a * b + b * c + c * a = 1 := h\u2082\n    have h\u2088\u2083 : a \u2264 b := h\u2080.1\n    have h\u2088\u2084 : b \u2264 c := h\u2080.2\n    have h\u2088\u2085 : 0 \u2264 a := h\u2083\n    have h\u2088\u2086 : 1 \u2264 c := h\u2084\n    have h\u2088\u2087 : c \u2264 4 / 3 := h\u2085\n    have h\u2088\u2088 : a \u2264 1 / 3 := h\u2086\n    have h\u2088\u2089 : 1 / 3 \u2264 b := h\u2087\n    nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n  have h\u2089 : 0 \u2264 a \u2227 a \u2264 1 / 3 \u2227 1 / 3 \u2264 b \u2227 b \u2264 1 \u2227 1 \u2264 c \u2227 c \u2264 4 / 3 := by\n    exact \u27e8h\u2083, h\u2086, h\u2087, h\u2088, h\u2084, h\u2085\u27e9\n  exact h\u2089\n", "proof_before": "theorem algebra_apbpceq2_abpbcpcaeq1_aleq1on3anbleq1ancleq4on3 (a b c : \u211d) (h\u2080 : a \u2264 b \u2227 b \u2264 c)\n    (h\u2081 : a + b + c = 2) (h\u2082 : a * b + b * c + c * a = 1) :\n    0 \u2264 a \u2227 a \u2264 1 / 3 \u2227 1 / 3 \u2264 b \u2227 b \u2264 1 \u2227 1 \u2264 c \u2227 c \u2264 4 / 3  :=  by\n  have h\u2083 : 0 \u2264 a := by\n    by_contra h\n    have h\u2084 : a < 0 := by linarith\n    have h\u2085 : a * (4 - 3 * a) < 0 := by\n      have h\u2085\u2081 : 4 - 3 * a > 0 := by nlinarith\n      nlinarith\n    have h\u2086 : (b - c) ^ 2 = 4 * a - 3 * a ^ 2 := by\n      have h\u2086\u2081 : a ^ 2 + b ^ 2 + c ^ 2 = 2 := by\n        nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n      have h\u2086\u2082 : a * b + b * c + c * a = 1 := h\u2082\n      nlinarith [sq_nonneg (b - c)]\n    have h\u2087 : (b - c) ^ 2 \u2265 0 := sq_nonneg (b - c)\n    nlinarith [sq_nonneg (b - c)]\n  have h\u2084 : 1 \u2264 c := by\n    have h\u2084\u2081 : a \u2264 b := h\u2080.1\n    have h\u2084\u2082 : b \u2264 c := h\u2080.2\n    have h\u2084\u2083 : a + b + c = 2 := h\u2081\n    have h\u2084\u2084 : a * b + b * c + c * a = 1 := h\u2082\n    by_contra! h\n    have h\u2085 : c < 1 := by linarith\n    have h\u2086 : a + b = 2 - c := by linarith\n    have h\u2087 : a * b = 1 - c * (a + b) := by\n      nlinarith\n    have h\u2088 : a * b = (c - 1) ^ 2 := by\n      rw [h\u2086] at h\u2087\n      nlinarith\n    have h\u2089 : (b - c) ^ 2 \u2265 0 := by nlinarith\n    have h\u2081\u2080 : (2 - c) ^ 2 - 4 * (c - 1) ^ 2 \u2265 0 := by\n      nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n    have h\u2081\u2081 : c \u2265 2 / 3 := by\n      nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n    have h\u2081\u2082 : 3 * c ^ 2 - 4 * c + 1 \u2264 0 := by\n      nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n    have h\u2081\u2083 : c \u2264 1 := by nlinarith\n    have h\u2081\u2084 : c \u2265 2 / 3 := by nlinarith\n    have h\u2081\u2085 : b \u2264 c := h\u2080.2\n    have h\u2081\u2086 : a \u2264 b := h\u2080.1\n    have h\u2081\u2087 : (2 - c) ^ 2 - 4 * (c - 1) ^ 2 \u2265 0 := by\n      nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n    have h\u2081\u2088 : 3 * c ^ 2 - 4 * c + 1 \u2264 0 := by\n      nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n    have h\u2081\u2089 : c \u2265 1 := by\n      by_contra h\u2082\u2080\n      have h\u2082\u2081 : c < 1 := by linarith\n      have h\u2082\u2082 : (2 - c) ^ 2 - 4 * (c - 1) ^ 2 \u2265 0 := by\n        nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n      have h\u2082\u2083 : 3 * c ^ 2 - 4 * c + 1 \u2264 0 := by\n        nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n      have h\u2082\u2084 : c \u2265 2 / 3 := by nlinarith\n      have h\u2082\u2085 : c \u2264 1 := by nlinarith\n      have h\u2082\u2086 : b \u2264 c := h\u2080.2\n      have h\u2082\u2087 : a \u2264 b := h\u2080.1\n      have h\u2082\u2088 : (2 - c) ^ 2 - 4 * (c - 1) ^ 2 \u2265 0 := by\n        nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n      have h\u2082\u2089 : 3 * c ^ 2 - 4 * c + 1 \u2264 0 := by\n        nlinarith [sq_nonneg (b - c), sq_nonneg (a - b), sq_nonneg (a - c)]\n      by_cases h\u2083\u2080 : c \u2265 1\n      \u00b7 linarith\n      \u00b7 have h\u2083\u2081 : c < 1 := by linarith\n        have h\u2083\u2082 : 3 * c ^ 2 - 4 * c + 1 > 0 := by\n          nlinarith [sq_nonneg (c - 1 / 3), sq_nonneg (c - 2 / 3)]\n        linarith\n    linarith\n  have h\u2085 : c \u2264 4 / 3 := by\n    have h\u2085\u2081 : a + b + c = 2 := h\u2081\n    have h\u2085\u2082 : a * b + b * c + c * a = 1 := h\u2082\n    have h\u2085\u2083 : a \u2264 b := h\u2080.1\n    have h\u2085\u2084 : b \u2264 c := h\u2080.2\n    have h\u2085\u2085 : 0 \u2264 a := h\u2083\n    have h\u2085\u2086 : 1 \u2264 c := h\u2084\n    have h\u2085\u2087 : (b - c) ^ 2 \u2265 0 := by nlinarith\n    have h\u2085\u2088 : c \u2264 4 / 3 := by\n      nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a),\n        mul_nonneg (sub_nonneg.mpr h\u2085\u2085) (sub_nonneg.mpr h\u2085\u2083),\n        mul_nonneg (sub_nonneg.mpr h\u2085\u2083) (sub_nonneg.mpr h\u2085\u2084),\n        mul_nonneg (sub_nonneg.mpr h\u2085\u2085) (sub_nonneg.mpr h\u2085\u2084)]\n    exact h\u2085\u2088\n    <;> linarith\n  have h\u2086 : a \u2264 1 / 3 := by\n    have h\u2086\u2081 : a + b = 2 - c := by linarith\n    have h\u2086\u2082 : a * b = (c - 1) ^ 2 := by\n      have h\u2086\u2083 : a * b + b * c + c * a = 1 := h\u2082\n      have h\u2086\u2084 : a + b + c = 2 := h\u2081\n      have h\u2086\u2085 : a * b = 1 - c * (a + b) := by\n        nlinarith\n      rw [h\u2086\u2081] at h\u2086\u2085\n      nlinarith\n    have h\u2086\u2083 : c \u2265 1 := h\u2084\n    have h\u2086\u2084 : c \u2264 4 / 3 := h\u2085\n    have h\u2086\u2085 : a \u2264 1 / 3 := by\n      nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a),\n        mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2080.1),\n        mul_nonneg (sub_nonneg.mpr h\u2080.1) (sub_nonneg.mpr h\u2080.2),\n        mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2080.2)]\n    exact h\u2086\u2085\n  have h\u2087 : 1 / 3 \u2264 b := by\n    have h\u2087\u2081 : a + b = 2 - c := by linarith\n    have h\u2087\u2082 : c \u2264 4 / 3 := h\u2085\n    have h\u2087\u2083 : 1 \u2264 c := h\u2084\n    have h\u2087\u2084 : a \u2264 b := h\u2080.1\n    have h\u2087\u2085 : 0 \u2264 a := h\u2083\n    have h\u2087\u2086 : 1 / 3 \u2264 b := by\n      nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n    exact h\u2087\u2086\n  have h\u2088 : b \u2264 1 := by\n    have h\u2088\u2081 : a + b + c = 2 := h\u2081\n    have h\u2088\u2082 : a * b + b * c + c * a = 1 := h\u2082\n    have h\u2088\u2083 : a \u2264 b := h\u2080.1\n    have h\u2088\u2084 : b \u2264 c := h\u2080.2\n    have h\u2088\u2085 : 0 \u2264 a := h\u2083\n    have h\u2088\u2086 : 1 \u2264 c := h\u2084\n    have h\u2088\u2087 : c \u2264 4 / 3 := h\u2085\n    have h\u2088\u2088 : a \u2264 1 / 3 := h\u2086\n    have h\u2088\u2089 : 1 / 3 \u2264 b := h\u2087\n    nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n  have h\u2089 : 0 \u2264 a \u2227 a \u2264 1 / 3 \u2227 1 / 3 \u2264 b \u2227 b \u2264 1 \u2227 1 \u2264 c \u2227 c \u2264 4 / 3 := by\n    exact \u27e8h\u2083, h\u2086, h\u2087, h\u2088, h\u2084, h\u2085\u27e9\n  exact h\u2089\n"}
{"mode": "clear", "steps_before": 89, "alphabet": 49, "compiles": false, "id": "mathd_numbertheory_552", "proof_after": "theorem mathd_numbertheory_552 (f g h : \u2115+ \u2192 \u2115) (h\u2080 : \u2200 x, f x = 12 * x + 7)\n    (h\u2081 : \u2200 x, g x = 5 * x + 2) (h\u2082 : \u2200 x, h x = Nat.gcd (f x) (g x)) (h\u2083 : Fintype (Set.range h)) :\n    (\u2211 k in (Set.range h).toFinset, k) = 12  :=  by\n  have h_main : \u2200 x : \u2115+, h x = 1 \u2228 h x = 11 := by\n    intro x\n    have h\u2084 : h x = Nat.gcd (f x) (g x) := h\u2082 x\n    rw [h\u2084]\n    have h\u2085 : f x = 12 * x + 7 := h\u2080 x\n    have h\u2086 : g x = 5 * x + 2 := h\u2081 x\n    rw [h\u2085, h\u2086]\n    have h\u2087 : Nat.gcd (12 * x + 7) (5 * x + 2) = 1 \u2228 Nat.gcd (12 * x + 7) (5 * x + 2) = 11 := by\n      have h\u2087\u2081 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 11 := by\n        have h\u2087\u2082 : (5 : \u2115) * (12 * x + 7) - 12 * (5 * x + 2) = 11 := by\n          cases x with\n          | mk x hx =>\n            simp [Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.mul_zero, Nat.add_assoc]\n            <;> ring_nf at *\n            <;> omega\n        have h\u2087\u2083 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 12 * x + 7 := Nat.gcd_dvd_left _ _\n        have h\u2087\u2084 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 5 * x + 2 := Nat.gcd_dvd_right _ _\n        have h\u2087\u2085 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 5 * (12 * x + 7) := dvd_mul_of_dvd_right h\u2087\u2083 _\n        have h\u2087\u2086 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 12 * (5 * x + 2) := dvd_mul_of_dvd_right h\u2087\u2084 _\n        have h\u2087\u2087 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 5 * (12 * x + 7) - 12 * (5 * x + 2) := Nat.dvd_sub' h\u2087\u2085 h\u2087\u2086\n        rw [h\u2087\u2082] at h\u2087\u2087\n        simpa using h\u2087\u2087\n      have h\u2087\u2082 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 11 := h\u2087\u2081\n      have h\u2087\u2083 : Nat.gcd (12 * x + 7) (5 * x + 2) = 1 \u2228 Nat.gcd (12 * x + 7) (5 * x + 2) = 11 := by\n        have h\u2087\u2084 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2264 11 := Nat.le_of_dvd (by norm_num) h\u2087\u2082\n        interval_cases Nat.gcd (12 * x + 7) (5 * x + 2) <;> norm_num at h\u2087\u2082 \u22a2 <;> omega\n      exact h\u2087\u2083\n    exact h\u2087\n  have h\u2081_in_range : 1 \u2208 Set.range h := by\n    have h\u2081_in : h \u27e81, by norm_num\u27e9 = 1 := by\n      have h\u2081\u2081 : h \u27e81, by norm_num\u27e9 = Nat.gcd (f \u27e81, by norm_num\u27e9) (g \u27e81, by norm_num\u27e9) := h\u2082 _\n      rw [h\u2081\u2081]\n      have h\u2081\u2082 : f \u27e81, by norm_num\u27e9 = 12 * (\u27e81, by norm_num\u27e9 : \u2115+) + 7 := h\u2080 \u27e81, by norm_num\u27e9\n      have h\u2081\u2083 : g \u27e81, by norm_num\u27e9 = 5 * (\u27e81, by norm_num\u27e9 : \u2115+) + 2 := h\u2081 \u27e81, by norm_num\u27e9\n      rw [h\u2081\u2082, h\u2081\u2083]\n      norm_num [Nat.gcd_eq_right]\n      <;> rfl\n    exact \u27e8\u27e81, by norm_num\u27e9, h\u2081_in\u27e9\n  have h\u2081\u2081_in_range : 11 \u2208 Set.range h := by\n    have h\u2081\u2081_in : h \u27e84, by norm_num\u27e9 = 11 := by\n      have h\u2081\u2081 : h \u27e84, by norm_num\u27e9 = Nat.gcd (f \u27e84, by norm_num\u27e9) (g \u27e84, by norm_num\u27e9) := h\u2082 _\n      rw [h\u2081\u2081]\n      have h\u2081\u2082 : f \u27e84, by norm_num\u27e9 = 12 * (\u27e84, by norm_num\u27e9 : \u2115+) + 7 := h\u2080 \u27e84, by norm_num\u27e9\n      have h\u2081\u2083 : g \u27e84, by norm_num\u27e9 = 5 * (\u27e84, by norm_num\u27e9 : \u2115+) + 2 := h\u2081 \u27e84, by norm_num\u27e9\n      rw [h\u2081\u2082, h\u2081\u2083]\n      norm_num [Nat.gcd_eq_right]\n      <;> rfl\n    exact \u27e8\u27e84, by norm_num\u27e9, h\u2081\u2081_in\u27e9\n  have h_sum : (\u2211 k in (Set.range h).toFinset, k) = 12 := by\n    have h\u2084 : (Set.range h).toFinset = {1, 11} := by\n      apply Finset.ext\n      intro x\n      simp only [Finset.mem_insert, Finset.mem_singleton, Set.mem_toFinset, Set.mem_range,\n        Set.mem_setOf_eq]\n      constructor\n      \u00b7 intro hx\n        have h\u2085 : x \u2208 Set.range h := by simpa using hx\n        rcases h\u2085 with \u27e8y, rfl\u27e9\n        have h\u2086 : h y = 1 \u2228 h y = 11 := h_main y\n        cases h\u2086 with\n        | inl h\u2086 =>\n          simp [h\u2086]\n        | inr h\u2086 =>\n          simp [h\u2086]\n      \u00b7 intro hx\n        have h\u2085 : x = 1 \u2228 x = 11 := by simpa using hx\n        cases h\u2085 with\n        | inl h\u2085 =>\n          rw [h\u2085]\n          exact h\u2081_in_range\n        | inr h\u2085 =>\n          rw [h\u2085]\n          exact h\u2081\u2081_in_range\n    rw [h\u2084]\n    norm_num\n    <;> aesop\n  exact h_sum\n", "proof_before": "theorem mathd_numbertheory_552 (f g h : \u2115+ \u2192 \u2115) (h\u2080 : \u2200 x, f x = 12 * x + 7)\n    (h\u2081 : \u2200 x, g x = 5 * x + 2) (h\u2082 : \u2200 x, h x = Nat.gcd (f x) (g x)) (h\u2083 : Fintype (Set.range h)) :\n    (\u2211 k in (Set.range h).toFinset, k) = 12  :=  by\n  have h_main : \u2200 x : \u2115+, h x = 1 \u2228 h x = 11 := by\n    intro x\n    have h\u2084 : h x = Nat.gcd (f x) (g x) := h\u2082 x\n    rw [h\u2084]\n    have h\u2085 : f x = 12 * x + 7 := h\u2080 x\n    have h\u2086 : g x = 5 * x + 2 := h\u2081 x\n    rw [h\u2085, h\u2086]\n    have h\u2087 : Nat.gcd (12 * x + 7) (5 * x + 2) = 1 \u2228 Nat.gcd (12 * x + 7) (5 * x + 2) = 11 := by\n      have h\u2087\u2081 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 11 := by\n        have h\u2087\u2082 : (5 : \u2115) * (12 * x + 7) - 12 * (5 * x + 2) = 11 := by\n          cases x with\n          | mk x hx =>\n            simp [Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.mul_zero, Nat.add_assoc]\n            <;> ring_nf at *\n            <;> omega\n        have h\u2087\u2083 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 12 * x + 7 := Nat.gcd_dvd_left _ _\n        have h\u2087\u2084 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 5 * x + 2 := Nat.gcd_dvd_right _ _\n        have h\u2087\u2085 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 5 * (12 * x + 7) := dvd_mul_of_dvd_right h\u2087\u2083 _\n        have h\u2087\u2086 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 12 * (5 * x + 2) := dvd_mul_of_dvd_right h\u2087\u2084 _\n        have h\u2087\u2087 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 5 * (12 * x + 7) - 12 * (5 * x + 2) := Nat.dvd_sub' h\u2087\u2085 h\u2087\u2086\n        rw [h\u2087\u2082] at h\u2087\u2087\n        simpa using h\u2087\u2087\n      have h\u2087\u2082 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2223 11 := h\u2087\u2081\n      have h\u2087\u2083 : Nat.gcd (12 * x + 7) (5 * x + 2) = 1 \u2228 Nat.gcd (12 * x + 7) (5 * x + 2) = 11 := by\n        have h\u2087\u2084 : Nat.gcd (12 * x + 7) (5 * x + 2) \u2264 11 := Nat.le_of_dvd (by norm_num) h\u2087\u2082\n        interval_cases Nat.gcd (12 * x + 7) (5 * x + 2) <;> norm_num at h\u2087\u2082 \u22a2 <;> omega\n      exact h\u2087\u2083\n    exact h\u2087\n  have h\u2081_in_range : 1 \u2208 Set.range h := by\n    have h\u2081_in : h \u27e81, by norm_num\u27e9 = 1 := by\n      have h\u2081\u2081 : h \u27e81, by norm_num\u27e9 = Nat.gcd (f \u27e81, by norm_num\u27e9) (g \u27e81, by norm_num\u27e9) := h\u2082 _\n      rw [h\u2081\u2081]\n      have h\u2081\u2082 : f \u27e81, by norm_num\u27e9 = 12 * (\u27e81, by norm_num\u27e9 : \u2115+) + 7 := h\u2080 \u27e81, by norm_num\u27e9\n      have h\u2081\u2083 : g \u27e81, by norm_num\u27e9 = 5 * (\u27e81, by norm_num\u27e9 : \u2115+) + 2 := h\u2081 \u27e81, by norm_num\u27e9\n      rw [h\u2081\u2082, h\u2081\u2083]\n      norm_num [Nat.gcd_eq_right]\n      <;> rfl\n    exact \u27e8\u27e81, by norm_num\u27e9, h\u2081_in\u27e9\n  have h\u2081\u2081_in_range : 11 \u2208 Set.range h := by\n    have h\u2081\u2081_in : h \u27e84, by norm_num\u27e9 = 11 := by\n      have h\u2081\u2081 : h \u27e84, by norm_num\u27e9 = Nat.gcd (f \u27e84, by norm_num\u27e9) (g \u27e84, by norm_num\u27e9) := h\u2082 _\n      rw [h\u2081\u2081]\n      have h\u2081\u2082 : f \u27e84, by norm_num\u27e9 = 12 * (\u27e84, by norm_num\u27e9 : \u2115+) + 7 := h\u2080 \u27e84, by norm_num\u27e9\n      have h\u2081\u2083 : g \u27e84, by norm_num\u27e9 = 5 * (\u27e84, by norm_num\u27e9 : \u2115+) + 2 := h\u2081 \u27e84, by norm_num\u27e9\n      rw [h\u2081\u2082, h\u2081\u2083]\n      norm_num [Nat.gcd_eq_right]\n      <;> rfl\n    exact \u27e8\u27e84, by norm_num\u27e9, h\u2081\u2081_in\u27e9\n  have h_sum : (\u2211 k in (Set.range h).toFinset, k) = 12 := by\n    have h\u2084 : (Set.range h).toFinset = {1, 11} := by\n      apply Finset.ext\n      intro x\n      simp only [Finset.mem_insert, Finset.mem_singleton, Set.mem_toFinset, Set.mem_range,\n        Set.mem_setOf_eq]\n      constructor\n      \u00b7 intro hx\n        have h\u2085 : x \u2208 Set.range h := by simpa using hx\n        rcases h\u2085 with \u27e8y, rfl\u27e9\n        have h\u2086 : h y = 1 \u2228 h y = 11 := h_main y\n        cases h\u2086 with\n        | inl h\u2086 =>\n          simp [h\u2086]\n        | inr h\u2086 =>\n          simp [h\u2086]\n      \u00b7 intro hx\n        have h\u2085 : x = 1 \u2228 x = 11 := by simpa using hx\n        cases h\u2085 with\n        | inl h\u2085 =>\n          rw [h\u2085]\n          exact h\u2081_in_range\n        | inr h\u2085 =>\n          rw [h\u2085]\n          exact h\u2081\u2081_in_range\n    rw [h\u2084]\n    norm_num\n    <;> aesop\n  exact h_sum\n"}
{"mode": "clear", "steps_before": 45, "alphabet": 22, "compiles": false, "id": "mathd_algebra_208", "proof_after": "theorem mathd_algebra_208 : Real.sqrt 1000000 - 1000000 ^ ((1 : \u211d) / 3) = 900  :=  by\n  have h_sqrt : Real.sqrt 1000000 = 1000 := by\n    rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    <;>\n    linarith\n  have h_cbrt : (1000000 : \u211d) ^ ((1 : \u211d) / 3) = 100 := by\n    have h\u2081 : Real.log ((1000000 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log (100 : \u211d) := by\n      have h\u2082 : Real.log ((1000000 : \u211d) ^ ((1 : \u211d) / 3)) = ((1 : \u211d) / 3) * Real.log (1000000 : \u211d) := by\n        rw [Real.log_rpow (by norm_num : (1000000 : \u211d) > 0)]\n        <;> ring_nf\n      rw [h\u2082]\n      have h\u2083 : Real.log (1000000 : \u211d) = Real.log ((100 : \u211d) ^ 3) := by norm_num\n      rw [h\u2083]\n      have h\u2084 : Real.log ((100 : \u211d) ^ 3) = 3 * Real.log (100 : \u211d) := by\n        rw [Real.log_pow] <;> norm_num\n      rw [h\u2084]\n      ring_nf\n      <;>\n      field_simp [Real.log_mul, Real.log_pow] <;>\n      ring_nf <;>\n      norm_num\n    have h\u2082 : (1000000 : \u211d) ^ ((1 : \u211d) / 3) > 0 := by positivity\n    have h\u2083 : (100 : \u211d) > 0 := by norm_num\n    have h\u2084 : Real.log ((1000000 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log (100 : \u211d) := by rw [h\u2081]\n    have h\u2085 : Real.log ((1000000 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log (100 : \u211d) := by rw [h\u2081]\n    have h\u2086 : (1000000 : \u211d) ^ ((1 : \u211d) / 3) = 100 := by\n      apply Real.log_injOn_pos (Set.mem_Ioi.mpr h\u2082) (Set.mem_Ioi.mpr h\u2083)\n      linarith\n    rw [h\u2086]\n    <;>\n    norm_num\n  have h_main : Real.sqrt 1000000 - (1000000 : \u211d) ^ ((1 : \u211d) / 3) = 900 := by\n    rw [h_sqrt, h_cbrt]\n    <;> norm_num\n  rw [h_main]\n  <;>\n  norm_num\n", "proof_before": "theorem mathd_algebra_208 : Real.sqrt 1000000 - 1000000 ^ ((1 : \u211d) / 3) = 900  :=  by\n  have h_sqrt : Real.sqrt 1000000 = 1000 := by\n    rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    <;>\n    linarith\n  have h_cbrt : (1000000 : \u211d) ^ ((1 : \u211d) / 3) = 100 := by\n    have h\u2081 : Real.log ((1000000 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log (100 : \u211d) := by\n      have h\u2082 : Real.log ((1000000 : \u211d) ^ ((1 : \u211d) / 3)) = ((1 : \u211d) / 3) * Real.log (1000000 : \u211d) := by\n        rw [Real.log_rpow (by norm_num : (1000000 : \u211d) > 0)]\n        <;> ring_nf\n      rw [h\u2082]\n      have h\u2083 : Real.log (1000000 : \u211d) = Real.log ((100 : \u211d) ^ 3) := by norm_num\n      rw [h\u2083]\n      have h\u2084 : Real.log ((100 : \u211d) ^ 3) = 3 * Real.log (100 : \u211d) := by\n        rw [Real.log_pow] <;> norm_num\n      rw [h\u2084]\n      ring_nf\n      <;>\n      field_simp [Real.log_mul, Real.log_pow] <;>\n      ring_nf <;>\n      norm_num\n    have h\u2082 : (1000000 : \u211d) ^ ((1 : \u211d) / 3) > 0 := by positivity\n    have h\u2083 : (100 : \u211d) > 0 := by norm_num\n    have h\u2084 : Real.log ((1000000 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log (100 : \u211d) := by rw [h\u2081]\n    have h\u2085 : Real.log ((1000000 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log (100 : \u211d) := by rw [h\u2081]\n    have h\u2086 : (1000000 : \u211d) ^ ((1 : \u211d) / 3) = 100 := by\n      apply Real.log_injOn_pos (Set.mem_Ioi.mpr h\u2082) (Set.mem_Ioi.mpr h\u2083)\n      linarith\n    rw [h\u2086]\n    <;>\n    norm_num\n  have h_main : Real.sqrt 1000000 - (1000000 : \u211d) ^ ((1 : \u211d) / 3) = 900 := by\n    rw [h_sqrt, h_cbrt]\n    <;> norm_num\n  rw [h_main]\n  <;>\n  norm_num\n"}
{"mode": "clear", "steps_before": 32, "alphabet": 20, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 847, "chars_after": 12, "seconds": 21.8, "id": "mathd_algebra_141", "proof_after": "theorem mathd_algebra_141 (a b : \u211d) (h\u2081 : a * b = 180) (h\u2082 : 2 * (a + b) = 54) :\n    a ^ 2 + b ^ 2 = 369  :=  by\n  nlinarith\n", "proof_before": "theorem mathd_algebra_141 (a b : \u211d) (h\u2081 : a * b = 180) (h\u2082 : 2 * (a + b) = 54) :\n    a ^ 2 + b ^ 2 = 369  :=  by\n  have h\u2083 : a + b = 27 := by\n    have h\u2083\u2081 : a + b = 27 := by\n      linarith\n    exact h\u2083\u2081\n  have h\u2084 : (a + b) ^ 2 = 729 := by\n    rw [h\u2083]\n    norm_num\n    <;>\n    linarith\n  have h\u2085 : a ^ 2 + b ^ 2 + 2 * (a * b) = 729 := by\n    have h\u2085\u2081 : (a + b) ^ 2 = a ^ 2 + b ^ 2 + 2 * (a * b) := by\n      ring\n    rw [h\u2084] at h\u2085\u2081\n    linarith\n  have h\u2086 : a ^ 2 + b ^ 2 + 2 * 180 = 729 := by\n    have h\u2086\u2081 : a ^ 2 + b ^ 2 + 2 * (a * b) = 729 := h\u2085\n    have h\u2086\u2082 : a * b = 180 := h\u2081\n    have h\u2086\u2083 : a ^ 2 + b ^ 2 + 2 * 180 = 729 := by\n      calc\n        a ^ 2 + b ^ 2 + 2 * 180 = a ^ 2 + b ^ 2 + 2 * (a * b) := by rw [h\u2086\u2082]\n        _ = 729 := by linarith\n        _ = 729 := by ring\n    exact h\u2086\u2083\n  have h\u2087 : a ^ 2 + b ^ 2 = 369 := by\n    have h\u2087\u2081 : a ^ 2 + b ^ 2 + 2 * 180 = 729 := h\u2086\n    have h\u2087\u2082 : a ^ 2 + b ^ 2 = 369 := by\n      linarith\n    exact h\u2087\u2082\n  exact h\u2087\n"}
{"mode": "clear", "steps_before": 152, "alphabet": 60, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 152, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 4375, "chars_after": 4375, "seconds": 334.7, "id": "imo_1981_p6", "proof_after": "theorem imo_1981_p6 (f : \u2115 \u2192 \u2115 \u2192 \u2115) (h\u2080 : \u2200 y, f 0 y = y + 1) (h\u2081 : \u2200 x, f (x + 1) 0 = f x 1)\n    (h\u2082 : \u2200 x y, f (x + 1) (y + 1) = f x (f (x + 1) y)) : \u2200 y, f 4 (y + 1) = 2 ^ (f 4 y + 3) - 3  :=  by\n  have h_f\u2081 : \u2200 y, f 1 y = y + 2 := by\n    intro y\n    induction y with\n    | zero =>\n      have h\u2081\u2080 : f 1 0 = f 0 1 := by\n        have h\u2081\u2080\u2081 := h\u2081 0\n        simp at h\u2081\u2080\u2081 \u22a2\n        <;> linarith\n      have h\u2081\u2081 : f 0 1 = 2 := by\n        have h\u2081\u2081\u2081 := h\u2080 1\n        simp at h\u2081\u2081\u2081 \u22a2\n        <;> linarith\n      have h\u2081\u2082 : f 1 0 = 2 := by linarith\n      simp [h\u2081\u2082]\n      <;> norm_num\n    | succ y ih =>\n      have h\u2082\u2080 : f 1 (y + 1) = f 0 (f 1 y) := by\n        have h\u2082\u2080\u2081 := h\u2082 0 y\n        simp at h\u2082\u2080\u2081 \u22a2\n        <;> linarith\n      have h\u2082\u2081 : f 0 (f 1 y) = f 1 y + 1 := by\n        have h\u2082\u2081\u2081 := h\u2080 (f 1 y)\n        simp at h\u2082\u2081\u2081 \u22a2\n        <;> linarith\n      have h\u2082\u2082 : f 1 (y + 1) = f 1 y + 1 := by linarith\n      have h\u2082\u2083 : f 1 (y + 1) = (y + 1) + 2 := by\n        rw [h\u2082\u2082, ih]\n        <;> ring\n        <;> omega\n      exact h\u2082\u2083\n  have h_f\u2082 : \u2200 y, f 2 y = 2 * y + 3 := by\n    intro y\n    induction y with\n    | zero =>\n      have h\u2082\u2080 : f 2 0 = f 1 1 := by\n        have h\u2082\u2080\u2081 := h\u2081 1\n        simp at h\u2082\u2080\u2081 \u22a2\n        <;> linarith\n      have h\u2082\u2081 : f 1 1 = 3 := by\n        have h\u2082\u2081\u2081 := h_f\u2081 1\n        simp at h\u2082\u2081\u2081 \u22a2\n        <;> linarith\n      have h\u2082\u2082 : f 2 0 = 3 := by linarith\n      simp [h\u2082\u2082]\n      <;> norm_num\n    | succ y ih =>\n      have h\u2083\u2080 : f 2 (y + 1) = f 1 (f 2 y) := by\n        have h\u2083\u2080\u2081 := h\u2082 1 y\n        simp at h\u2083\u2080\u2081 \u22a2\n        <;> linarith\n      have h\u2083\u2081 : f 1 (f 2 y) = f 2 y + 2 := by\n        have h\u2083\u2081\u2081 := h_f\u2081 (f 2 y)\n        simp at h\u2083\u2081\u2081 \u22a2\n        <;> linarith\n      have h\u2083\u2082 : f 2 (y + 1) = f 2 y + 2 := by linarith\n      have h\u2083\u2083 : f 2 (y + 1) = 2 * (y + 1) + 3 := by\n        rw [h\u2083\u2082, ih]\n        <;> ring\n        <;> omega\n      exact h\u2083\u2083\n  have h_f\u2083 : \u2200 y, f 3 y = 2 ^ (y + 3) - 3 := by\n    intro y\n    have h\u2083 : \u2200 y, f 3 y = 2 ^ (y + 3) - 3 := by\n      intro y\n      induction y with\n      | zero =>\n        have h\u2083\u2080 : f 3 0 = f 2 1 := by\n          have h\u2083\u2080\u2081 := h\u2081 2\n          simp at h\u2083\u2080\u2081 \u22a2\n          <;> linarith\n        have h\u2083\u2081 : f 2 1 = 5 := by\n          have h\u2083\u2081\u2081 := h_f\u2082 1\n          simp at h\u2083\u2081\u2081 \u22a2\n          <;> linarith\n        have h\u2083\u2082 : f 3 0 = 5 := by linarith\n        have h\u2083\u2083 : (2 : \u2115) ^ (0 + 3) - 3 = 5 := by norm_num\n        simp [h\u2083\u2082, h\u2083\u2083]\n        <;> norm_num\n      | succ y ih =>\n        have h\u2084\u2080 : f 3 (y + 1) = f 2 (f 3 y) := by\n          have h\u2084\u2080\u2081 := h\u2082 2 y\n          simp at h\u2084\u2080\u2081 \u22a2\n          <;> linarith\n        have h\u2084\u2081 : f 2 (f 3 y) = 2 * f 3 y + 3 := by\n          have h\u2084\u2081\u2081 := h_f\u2082 (f 3 y)\n          simp at h\u2084\u2081\u2081 \u22a2\n          <;> linarith\n        have h\u2084\u2082 : f 3 (y + 1) = 2 * f 3 y + 3 := by linarith\n        have h\u2084\u2083 : f 3 (y + 1) = 2 ^ ((y + 1) + 3) - 3 := by\n          rw [h\u2084\u2082, ih]\n          have h\u2085 : 2 * (2 ^ (y + 3) - 3) + 3 = 2 ^ ((y + 1) + 3) - 3 := by\n            have h\u2085\u2081 : y + 3 \u2265 3 := by omega\n            have h\u2085\u2082 : 2 ^ (y + 3) \u2265 2 ^ 3 := by\n              apply Nat.pow_le_pow_of_le_right\n              <;> norm_num\n              <;> omega\n            have h\u2085\u2083 : 2 ^ (y + 3) \u2265 8 := by\n              have h\u2085\u2084 : 2 ^ 3 = 8 := by norm_num\n              linarith\n            have h\u2085\u2084 : 2 ^ (y + 3) - 3 \u2265 5 := by\n              have h\u2085\u2085 : 2 ^ (y + 3) \u2265 8 := by omega\n              omega\n            have h\u2085\u2085 : 2 * (2 ^ (y + 3) - 3) + 3 = 2 ^ ((y + 1) + 3) - 3 := by\n              have h\u2085\u2085\u2081 : 2 ^ ((y + 1) + 3) = 2 ^ (y + 4) := by\n                ring_nf\n              rw [h\u2085\u2085\u2081]\n              have h\u2085\u2085\u2082 : 2 ^ (y + 4) = 2 ^ (y + 3 + 1) := by ring_nf\n              rw [h\u2085\u2085\u2082]\n              have h\u2085\u2085\u2083 : 2 ^ (y + 3 + 1) = 2 * 2 ^ (y + 3) := by\n                simp [Nat.pow_succ, Nat.mul_add]\n                <;> ring_nf\n              rw [h\u2085\u2085\u2083]\n              have h\u2085\u2085\u2084 : 2 ^ (y + 3) - 3 \u2265 0 := by omega\n              have h\u2085\u2085\u2085 : 2 * (2 ^ (y + 3) - 3) + 3 = 2 * 2 ^ (y + 3) - 3 := by\n                omega\n              omega\n            exact h\u2085\u2085\n          omega\n        exact h\u2084\u2083\n    exact h\u2083 y\n  have h_f\u2084 : \u2200 y, f 4 (y + 1) = f 3 (f 4 y) := by\n    intro y\n    have h\u2084 := h\u2082 3 y\n    simp at h\u2084 \u22a2\n    <;> simp_all [Nat.add_assoc]\n    <;> ring_nf at *\n    <;> omega\n  have h_main : \u2200 y, f 4 (y + 1) = 2 ^ (f 4 y + 3) - 3 := by\n    intro y\n    have h\u2085 : f 4 (y + 1) = f 3 (f 4 y) := by\n      apply h_f\u2084\n    rw [h\u2085]\n    have h\u2086 : f 3 (f 4 y) = 2 ^ (f 4 y + 3) - 3 := by\n      have h\u2086\u2081 : f 3 (f 4 y) = 2 ^ (f 4 y + 3) - 3 := by\n        rw [h_f\u2083]\n        <;> simp [Nat.add_assoc]\n      rw [h\u2086\u2081]\n    rw [h\u2086]\n    <;> simp [Nat.add_assoc]\n  exact h_main\n", "proof_before": "theorem imo_1981_p6 (f : \u2115 \u2192 \u2115 \u2192 \u2115) (h\u2080 : \u2200 y, f 0 y = y + 1) (h\u2081 : \u2200 x, f (x + 1) 0 = f x 1)\n    (h\u2082 : \u2200 x y, f (x + 1) (y + 1) = f x (f (x + 1) y)) : \u2200 y, f 4 (y + 1) = 2 ^ (f 4 y + 3) - 3  :=  by\n  have h_f\u2081 : \u2200 y, f 1 y = y + 2 := by\n    intro y\n    induction y with\n    | zero =>\n      have h\u2081\u2080 : f 1 0 = f 0 1 := by\n        have h\u2081\u2080\u2081 := h\u2081 0\n        simp at h\u2081\u2080\u2081 \u22a2\n        <;> linarith\n      have h\u2081\u2081 : f 0 1 = 2 := by\n        have h\u2081\u2081\u2081 := h\u2080 1\n        simp at h\u2081\u2081\u2081 \u22a2\n        <;> linarith\n      have h\u2081\u2082 : f 1 0 = 2 := by linarith\n      simp [h\u2081\u2082]\n      <;> norm_num\n    | succ y ih =>\n      have h\u2082\u2080 : f 1 (y + 1) = f 0 (f 1 y) := by\n        have h\u2082\u2080\u2081 := h\u2082 0 y\n        simp at h\u2082\u2080\u2081 \u22a2\n        <;> linarith\n      have h\u2082\u2081 : f 0 (f 1 y) = f 1 y + 1 := by\n        have h\u2082\u2081\u2081 := h\u2080 (f 1 y)\n        simp at h\u2082\u2081\u2081 \u22a2\n        <;> linarith\n      have h\u2082\u2082 : f 1 (y + 1) = f 1 y + 1 := by linarith\n      have h\u2082\u2083 : f 1 (y + 1) = (y + 1) + 2 := by\n        rw [h\u2082\u2082, ih]\n        <;> ring\n        <;> omega\n      exact h\u2082\u2083\n  have h_f\u2082 : \u2200 y, f 2 y = 2 * y + 3 := by\n    intro y\n    induction y with\n    | zero =>\n      have h\u2082\u2080 : f 2 0 = f 1 1 := by\n        have h\u2082\u2080\u2081 := h\u2081 1\n        simp at h\u2082\u2080\u2081 \u22a2\n        <;> linarith\n      have h\u2082\u2081 : f 1 1 = 3 := by\n        have h\u2082\u2081\u2081 := h_f\u2081 1\n        simp at h\u2082\u2081\u2081 \u22a2\n        <;> linarith\n      have h\u2082\u2082 : f 2 0 = 3 := by linarith\n      simp [h\u2082\u2082]\n      <;> norm_num\n    | succ y ih =>\n      have h\u2083\u2080 : f 2 (y + 1) = f 1 (f 2 y) := by\n        have h\u2083\u2080\u2081 := h\u2082 1 y\n        simp at h\u2083\u2080\u2081 \u22a2\n        <;> linarith\n      have h\u2083\u2081 : f 1 (f 2 y) = f 2 y + 2 := by\n        have h\u2083\u2081\u2081 := h_f\u2081 (f 2 y)\n        simp at h\u2083\u2081\u2081 \u22a2\n        <;> linarith\n      have h\u2083\u2082 : f 2 (y + 1) = f 2 y + 2 := by linarith\n      have h\u2083\u2083 : f 2 (y + 1) = 2 * (y + 1) + 3 := by\n        rw [h\u2083\u2082, ih]\n        <;> ring\n        <;> omega\n      exact h\u2083\u2083\n  have h_f\u2083 : \u2200 y, f 3 y = 2 ^ (y + 3) - 3 := by\n    intro y\n    have h\u2083 : \u2200 y, f 3 y = 2 ^ (y + 3) - 3 := by\n      intro y\n      induction y with\n      | zero =>\n        have h\u2083\u2080 : f 3 0 = f 2 1 := by\n          have h\u2083\u2080\u2081 := h\u2081 2\n          simp at h\u2083\u2080\u2081 \u22a2\n          <;> linarith\n        have h\u2083\u2081 : f 2 1 = 5 := by\n          have h\u2083\u2081\u2081 := h_f\u2082 1\n          simp at h\u2083\u2081\u2081 \u22a2\n          <;> linarith\n        have h\u2083\u2082 : f 3 0 = 5 := by linarith\n        have h\u2083\u2083 : (2 : \u2115) ^ (0 + 3) - 3 = 5 := by norm_num\n        simp [h\u2083\u2082, h\u2083\u2083]\n        <;> norm_num\n      | succ y ih =>\n        have h\u2084\u2080 : f 3 (y + 1) = f 2 (f 3 y) := by\n          have h\u2084\u2080\u2081 := h\u2082 2 y\n          simp at h\u2084\u2080\u2081 \u22a2\n          <;> linarith\n        have h\u2084\u2081 : f 2 (f 3 y) = 2 * f 3 y + 3 := by\n          have h\u2084\u2081\u2081 := h_f\u2082 (f 3 y)\n          simp at h\u2084\u2081\u2081 \u22a2\n          <;> linarith\n        have h\u2084\u2082 : f 3 (y + 1) = 2 * f 3 y + 3 := by linarith\n        have h\u2084\u2083 : f 3 (y + 1) = 2 ^ ((y + 1) + 3) - 3 := by\n          rw [h\u2084\u2082, ih]\n          have h\u2085 : 2 * (2 ^ (y + 3) - 3) + 3 = 2 ^ ((y + 1) + 3) - 3 := by\n            have h\u2085\u2081 : y + 3 \u2265 3 := by omega\n            have h\u2085\u2082 : 2 ^ (y + 3) \u2265 2 ^ 3 := by\n              apply Nat.pow_le_pow_of_le_right\n              <;> norm_num\n              <;> omega\n            have h\u2085\u2083 : 2 ^ (y + 3) \u2265 8 := by\n              have h\u2085\u2084 : 2 ^ 3 = 8 := by norm_num\n              linarith\n            have h\u2085\u2084 : 2 ^ (y + 3) - 3 \u2265 5 := by\n              have h\u2085\u2085 : 2 ^ (y + 3) \u2265 8 := by omega\n              omega\n            have h\u2085\u2085 : 2 * (2 ^ (y + 3) - 3) + 3 = 2 ^ ((y + 1) + 3) - 3 := by\n              have h\u2085\u2085\u2081 : 2 ^ ((y + 1) + 3) = 2 ^ (y + 4) := by\n                ring_nf\n              rw [h\u2085\u2085\u2081]\n              have h\u2085\u2085\u2082 : 2 ^ (y + 4) = 2 ^ (y + 3 + 1) := by ring_nf\n              rw [h\u2085\u2085\u2082]\n              have h\u2085\u2085\u2083 : 2 ^ (y + 3 + 1) = 2 * 2 ^ (y + 3) := by\n                simp [Nat.pow_succ, Nat.mul_add]\n                <;> ring_nf\n              rw [h\u2085\u2085\u2083]\n              have h\u2085\u2085\u2084 : 2 ^ (y + 3) - 3 \u2265 0 := by omega\n              have h\u2085\u2085\u2085 : 2 * (2 ^ (y + 3) - 3) + 3 = 2 * 2 ^ (y + 3) - 3 := by\n                omega\n              omega\n            exact h\u2085\u2085\n          omega\n        exact h\u2084\u2083\n    exact h\u2083 y\n  have h_f\u2084 : \u2200 y, f 4 (y + 1) = f 3 (f 4 y) := by\n    intro y\n    have h\u2084 := h\u2082 3 y\n    simp at h\u2084 \u22a2\n    <;> simp_all [Nat.add_assoc]\n    <;> ring_nf at *\n    <;> omega\n  have h_main : \u2200 y, f 4 (y + 1) = 2 ^ (f 4 y + 3) - 3 := by\n    intro y\n    have h\u2085 : f 4 (y + 1) = f 3 (f 4 y) := by\n      apply h_f\u2084\n    rw [h\u2085]\n    have h\u2086 : f 3 (f 4 y) = 2 ^ (f 4 y + 3) - 3 := by\n      have h\u2086\u2081 : f 3 (f 4 y) = 2 ^ (f 4 y + 3) - 3 := by\n        rw [h_f\u2083]\n        <;> simp [Nat.add_assoc]\n      rw [h\u2086\u2081]\n    rw [h\u2086]\n    <;> simp [Nat.add_assoc]\n  exact h_main\n"}
{"mode": "clear", "steps_before": 20, "alphabet": 16, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 718, "chars_after": 6, "seconds": 192.8, "id": "mathd_numbertheory_212", "proof_after": "theorem mathd_numbertheory_212 : 16 ^ 17 * 17 ^ 18 * 18 ^ 19 % 10 = 8  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_212 : 16 ^ 17 * 17 ^ 18 * 18 ^ 19 % 10 = 8  :=  by\n  have h1 : 16 ^ 17 % 10 = 6 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.pow_mod, Nat.mod_mod]\n    <;> rfl\n  have h2 : 17 ^ 18 % 10 = 9 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.pow_mod, Nat.mod_mod]\n    <;> rfl\n  have h3 : 18 ^ 19 % 10 = 2 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.pow_mod, Nat.mod_mod]\n    <;> rfl\n  have h4 : (16 ^ 17 * 17 ^ 18 * 18 ^ 19) % 10 = 8 := by\n    have h5 : (16 ^ 17 * 17 ^ 18 * 18 ^ 19) % 10 = ((16 ^ 17 % 10) * (17 ^ 18 % 10) * (18 ^ 19 % 10)) % 10 := by\n      norm_num [Nat.mul_mod, Nat.pow_mod, Nat.mod_mod]\n      <;> ring_nf\n      <;> norm_num [Nat.mul_mod, Nat.pow_mod, Nat.mod_mod]\n      <;> rfl\n    rw [h5]\n    rw [h1, h2, h3]\n    <;> norm_num\n    <;> rfl\n  exact h4\n"}
{"mode": "clear", "steps_before": 73, "alphabet": 37, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 73, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": true, "chars_before": 1925, "chars_after": 1925, "seconds": 229.0, "id": "mathd_algebra_170", "proof_after": "theorem mathd_algebra_170 (S : Finset \u2124) (h\u2080 : \u2200 n : \u2124, n \u2208 S \u2194 abs (n - 2) \u2264 5 + 6 / 10) :\n    S.card = 11  :=  by\n  have h\u2081 : S = Finset.Icc (-3) 7 := by\n    apply Finset.ext\n    intro n\n    simp only [Finset.mem_Icc, h\u2080]\n    constructor\n    \u00b7 intro h\n      have h\u2082 : abs (n - 2) \u2264 5 := by\n        norm_num at h \u22a2\n        <;>\n        (try omega) <;>\n        (try\n          {\n            cases' le_or_lt 0 (n - 2) with h\u2083 h\u2083 <;>\n            simp_all [abs_of_nonneg, abs_of_neg, le_of_lt] <;>\n            omega\n          })\n        <;>\n        omega\n      have h\u2083 : -3 \u2264 n := by\n        have h\u2084 : abs (n - 2) \u2264 5 := h\u2082\n        cases' le_or_lt 0 (n - 2) with h\u2085 h\u2085\n        \u00b7 \n          have h\u2086 : n - 2 \u2264 5 := by\n            rw [abs_of_nonneg h\u2085] at h\u2084\n            linarith\n          omega\n        \u00b7 \n          have h\u2086 : -(n - 2) \u2264 5 := by\n            rw [abs_of_neg h\u2085] at h\u2084\n            linarith\n          omega\n      have h\u2084 : n \u2264 7 := by\n        have h\u2085 : abs (n - 2) \u2264 5 := h\u2082\n        cases' le_or_lt 0 (n - 2) with h\u2086 h\u2086\n        \u00b7 \n          have h\u2087 : n - 2 \u2264 5 := by\n            rw [abs_of_nonneg h\u2086] at h\u2085\n            linarith\n          omega\n        \u00b7 \n          have h\u2087 : -(n - 2) \u2264 5 := by\n            rw [abs_of_neg h\u2086] at h\u2085\n            linarith\n          omega\n      exact \u27e8h\u2083, h\u2084\u27e9\n    \u00b7 intro h\n      have h\u2082 : -3 \u2264 n := h.1\n      have h\u2083 : n \u2264 7 := h.2\n      have h\u2084 : abs (n - 2) \u2264 5 := by\n        have h\u2085 : -3 \u2264 n := h\u2082\n        have h\u2086 : n \u2264 7 := h\u2083\n        cases' le_or_lt 0 (n - 2) with h\u2087 h\u2087\n        \u00b7 \n          have h\u2088 : n - 2 \u2264 5 := by\n            omega\n          rw [abs_of_nonneg h\u2087]\n          linarith\n        \u00b7 \n          have h\u2088 : -(n - 2) \u2264 5 := by\n            omega\n          rw [abs_of_neg h\u2087]\n          linarith\n      norm_num at h\u2084 \u22a2\n      <;>\n      (try omega) <;>\n      (try\n        {\n          cases' le_or_lt 0 (n - 2) with h\u2085 h\u2085 <;>\n          simp_all [abs_of_nonneg, abs_of_neg, le_of_lt] <;>\n          omega\n        })\n      <;>\n      omega\n  have h\u2082 : S.card = 11 := by\n    rw [h\u2081]\n    rfl\n  apply h\u2082\n", "proof_before": "theorem mathd_algebra_170 (S : Finset \u2124) (h\u2080 : \u2200 n : \u2124, n \u2208 S \u2194 abs (n - 2) \u2264 5 + 6 / 10) :\n    S.card = 11  :=  by\n  have h\u2081 : S = Finset.Icc (-3) 7 := by\n    apply Finset.ext\n    intro n\n    simp only [Finset.mem_Icc, h\u2080]\n    constructor\n    \u00b7 intro h\n      have h\u2082 : abs (n - 2) \u2264 5 := by\n        norm_num at h \u22a2\n        <;>\n        (try omega) <;>\n        (try\n          {\n            cases' le_or_lt 0 (n - 2) with h\u2083 h\u2083 <;>\n            simp_all [abs_of_nonneg, abs_of_neg, le_of_lt] <;>\n            omega\n          })\n        <;>\n        omega\n      have h\u2083 : -3 \u2264 n := by\n        have h\u2084 : abs (n - 2) \u2264 5 := h\u2082\n        cases' le_or_lt 0 (n - 2) with h\u2085 h\u2085\n        \u00b7 \n          have h\u2086 : n - 2 \u2264 5 := by\n            rw [abs_of_nonneg h\u2085] at h\u2084\n            linarith\n          omega\n        \u00b7 \n          have h\u2086 : -(n - 2) \u2264 5 := by\n            rw [abs_of_neg h\u2085] at h\u2084\n            linarith\n          omega\n      have h\u2084 : n \u2264 7 := by\n        have h\u2085 : abs (n - 2) \u2264 5 := h\u2082\n        cases' le_or_lt 0 (n - 2) with h\u2086 h\u2086\n        \u00b7 \n          have h\u2087 : n - 2 \u2264 5 := by\n            rw [abs_of_nonneg h\u2086] at h\u2085\n            linarith\n          omega\n        \u00b7 \n          have h\u2087 : -(n - 2) \u2264 5 := by\n            rw [abs_of_neg h\u2086] at h\u2085\n            linarith\n          omega\n      exact \u27e8h\u2083, h\u2084\u27e9\n    \u00b7 intro h\n      have h\u2082 : -3 \u2264 n := h.1\n      have h\u2083 : n \u2264 7 := h.2\n      have h\u2084 : abs (n - 2) \u2264 5 := by\n        have h\u2085 : -3 \u2264 n := h\u2082\n        have h\u2086 : n \u2264 7 := h\u2083\n        cases' le_or_lt 0 (n - 2) with h\u2087 h\u2087\n        \u00b7 \n          have h\u2088 : n - 2 \u2264 5 := by\n            omega\n          rw [abs_of_nonneg h\u2087]\n          linarith\n        \u00b7 \n          have h\u2088 : -(n - 2) \u2264 5 := by\n            omega\n          rw [abs_of_neg h\u2087]\n          linarith\n      norm_num at h\u2084 \u22a2\n      <;>\n      (try omega) <;>\n      (try\n        {\n          cases' le_or_lt 0 (n - 2) with h\u2085 h\u2085 <;>\n          simp_all [abs_of_nonneg, abs_of_neg, le_of_lt] <;>\n          omega\n        })\n      <;>\n      omega\n  have h\u2082 : S.card = 11 := by\n    rw [h\u2081]\n    rfl\n  apply h\u2082\n"}
{"mode": "clear", "steps_before": 58, "alphabet": 21, "compiles": true, "steps_after": 5, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 4, "fixpoint": true, "chars_before": 1599, "chars_after": 127, "seconds": 639.6, "id": "mathd_algebra_137", "proof_after": "theorem mathd_algebra_137 (x : \u2115) (h\u2080 : \u2191x + (4 : \u211d) / (100 : \u211d) * \u2191x = 598) : x = 575  :=  by\n  have h\u2081 : (104 : \u211d) * (x : \u211d) = (59800 : \u211d) := by\n    linarith\n  have h\u2082 : 104 * x = 59800 := by\n    norm_cast at h\u2081\n  omega\n", "proof_before": "theorem mathd_algebra_137 (x : \u2115) (h\u2080 : \u2191x + (4 : \u211d) / (100 : \u211d) * \u2191x = 598) : x = 575  :=  by\n  have h\u2081 : (104 : \u211d) * (x : \u211d) = (59800 : \u211d) := by\n    have h\u2081 : (x : \u211d) + (4 : \u211d) / (100 : \u211d) * (x : \u211d) = 598 := h\u2080\n    ring_nf at h\u2081 \u22a2\n    <;> linarith\n  have h\u2082 : 104 * x = 59800 := by\n    norm_cast at h\u2081\n    <;>\n    (try norm_num at h\u2081 \u22a2) <;>\n    (try ring_nf at h\u2081 \u22a2) <;>\n    (try norm_num at h\u2081 \u22a2) <;>\n    (try omega) <;>\n    (try\n      {\n        norm_num at h\u2081 \u22a2\n        <;>\n        ring_nf at h\u2081 \u22a2\n        <;>\n        norm_num at h\u2081 \u22a2\n        <;>\n        omega\n      })\n    <;>\n    (try\n      {\n        ring_nf at h\u2081 \u22a2\n        <;>\n        norm_num at h\u2081 \u22a2\n        <;>\n        omega\n      })\n    <;>\n    (try\n      {\n        norm_num at h\u2081 \u22a2\n        <;>\n        ring_nf at h\u2081 \u22a2\n        <;>\n        norm_num at h\u2081 \u22a2\n        <;>\n        omega\n      })\n    <;>\n    (try\n      {\n        ring_nf at h\u2081 \u22a2\n        <;>\n        norm_num at h\u2081 \u22a2\n        <;>\n        omega\n      })\n    <;>\n    (try\n      {\n        norm_num at h\u2081 \u22a2\n        <;>\n        ring_nf at h\u2081 \u22a2\n        <;>\n        norm_num at h\u2081 \u22a2\n        <;>\n        omega\n      })\n    <;>\n    (try\n      {\n        ring_nf at h\u2081 \u22a2\n        <;>\n        norm_num at h\u2081 \u22a2\n        <;>\n        omega\n      })\n  have h\u2083 : x = 575 := by\n    norm_num at h\u2082 \u22a2\n    <;>\n    (try omega) <;>\n    (try\n      {\n        ring_nf at h\u2082 \u22a2\n        <;>\n        norm_num at h\u2082 \u22a2\n        <;>\n        omega\n      })\n    <;>\n    (try\n      {\n        ring_nf at h\u2082 \u22a2\n        <;>\n        norm_num at h\u2082 \u22a2\n        <;>\n        omega\n      })\n    <;>\n    (try\n      {\n        ring_nf at h\u2082 \u22a2\n        <;>\n        norm_num at h\u2082 \u22a2\n        <;>\n        omega\n      })\n  apply h\u2083\n"}
{"mode": "clear", "steps_before": 70, "alphabet": 39, "compiles": true, "steps_after": 26, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 4, "fixpoint": false, "chars_before": 2301, "chars_after": 1051, "seconds": 636.9, "id": "amc12b_2021_p4", "proof_after": "theorem amc12b_2021_p4 (m a : \u2115) (h\u2080 : 0 < m \u2227 0 < a) (h\u2081 : \u2191m / \u2191a = (3 : \u211d) / 4) :\n    (84 * \u2191m + 70 * \u2191a) / (\u2191m + \u2191a) = (76 : \u211d)  :=  by\n  have h\u2082 : (4 : \u211d) * (m : \u211d) = (3 : \u211d) * (a : \u211d) := by\n    have h\u2082\u2081 : (m : \u211d) / (a : \u211d) = (3 : \u211d) / 4 := h\u2081\n    have h\u2082\u2082 : (a : \u211d) \u2260 0 := by\n      have h\u2082\u2082\u2081 : 0 < a := h\u2080.2\n      exact_mod_cast (by linarith)\n    have h\u2082\u2083 : (4 : \u211d) * (m : \u211d) = (3 : \u211d) * (a : \u211d) := by\n      field_simp at h\u2082\u2081\n      <;> ring_nf at h\u2082\u2081 \u22a2\n      <;> nlinarith\n    exact h\u2082\u2083\n  have h\u2084 : (84 : \u211d) * (m : \u211d) + (70 : \u211d) * (a : \u211d) = (76 : \u211d) * ((m : \u211d) + (a : \u211d)) := by\n    linarith\n  have h\u2085 : (84 * \u2191m + 70 * \u2191a) / (\u2191m + \u2191a) = (76 : \u211d) := by\n    have h\u2085\u2081 : (m : \u211d) + (a : \u211d) \u2260 0 := by\n      exact_mod_cast (by linarith)\n    have h\u2085\u2082 : (84 * \u2191m + 70 * \u2191a) / (\u2191m + \u2191a) = (76 : \u211d) := by\n      have h\u2085\u2083 : (84 : \u211d) * (m : \u211d) + (70 : \u211d) * (a : \u211d) = (76 : \u211d) * ((m : \u211d) + (a : \u211d)) := h\u2084\n      have h\u2085\u2084 : (84 * \u2191m + 70 * \u2191a) / (\u2191m + \u2191a) = (76 : \u211d) := by\n        rw [show (84 * \u2191m + 70 * \u2191a : \u211d) = (76 : \u211d) * ((\u2191m + \u2191a : \u211d)) by\n          linarith]\n        field_simp [h\u2085\u2081]\n        <;> ring_nf\n        <;> field_simp [h\u2085\u2081]\n        <;> linarith\n      exact h\u2085\u2084\n    exact h\u2085\u2082\n  exact h\u2085\n", "proof_before": "theorem amc12b_2021_p4 (m a : \u2115) (h\u2080 : 0 < m \u2227 0 < a) (h\u2081 : \u2191m / \u2191a = (3 : \u211d) / 4) :\n    (84 * \u2191m + 70 * \u2191a) / (\u2191m + \u2191a) = (76 : \u211d)  :=  by\n  have h\u2082 : (4 : \u211d) * (m : \u211d) = (3 : \u211d) * (a : \u211d) := by\n    have h\u2082\u2081 : (m : \u211d) / (a : \u211d) = (3 : \u211d) / 4 := h\u2081\n    have h\u2082\u2082 : (a : \u211d) \u2260 0 := by\n      have h\u2082\u2082\u2081 : 0 < a := h\u2080.2\n      exact_mod_cast (by linarith)\n    have h\u2082\u2083 : (4 : \u211d) * (m : \u211d) = (3 : \u211d) * (a : \u211d) := by\n      field_simp at h\u2082\u2081\n      <;> ring_nf at h\u2082\u2081 \u22a2\n      <;> nlinarith\n    exact h\u2082\u2083\n  have h\u2083 : (4 : \u2115) * m = (3 : \u2115) * a := by\n    have h\u2083\u2081 : (4 : \u2115) * m = (3 : \u2115) * a := by\n      norm_cast at h\u2082 \u22a2\n      <;>\n      (try norm_num) <;>\n      (try ring_nf at h\u2082 \u22a2) <;>\n      (try field_simp at h\u2082 \u22a2) <;>\n      (try norm_cast at h\u2082 \u22a2) <;>\n      (try ring_nf at h\u2082 \u22a2) <;>\n      (try norm_num at h\u2082 \u22a2) <;>\n      (try nlinarith) <;>\n      (try omega)\n      <;>\n      (try\n        {\n          have h\u2083\u2082 : a > 0 := by linarith\n          have h\u2083\u2083 : m > 0 := by linarith\n          norm_cast at h\u2082 \u22a2\n          <;>\n          ring_nf at h\u2082 \u22a2 <;>\n          norm_num at h\u2082 \u22a2 <;>\n          nlinarith\n        })\n      <;>\n      (try\n        {\n          norm_num at h\u2082 \u22a2 <;>\n          nlinarith\n        })\n    exact h\u2083\u2081\n  have h\u2084 : (84 : \u211d) * (m : \u211d) + (70 : \u211d) * (a : \u211d) = (76 : \u211d) * ((m : \u211d) + (a : \u211d)) := by\n    have h\u2084\u2081 : (4 : \u211d) * (m : \u211d) = (3 : \u211d) * (a : \u211d) := h\u2082\n    have h\u2084\u2082 : (84 : \u211d) * (m : \u211d) + (70 : \u211d) * (a : \u211d) = (76 : \u211d) * ((m : \u211d) + (a : \u211d)) := by\n      have h\u2084\u2083 : (a : \u211d) = (4 : \u211d) * (m : \u211d) / 3 := by\n        have h\u2084\u2084 : (4 : \u211d) * (m : \u211d) = (3 : \u211d) * (a : \u211d) := h\u2082\n        linarith\n      rw [h\u2084\u2083]\n      ring_nf\n      <;> field_simp\n      <;> ring_nf at h\u2084\u2081 \u22a2 <;> nlinarith\n    exact h\u2084\u2082\n  have h\u2085 : (84 * \u2191m + 70 * \u2191a) / (\u2191m + \u2191a) = (76 : \u211d) := by\n    have h\u2085\u2081 : (m : \u211d) + (a : \u211d) \u2260 0 := by\n      have h\u2085\u2082 : 0 < (m : \u211d) := by exact_mod_cast h\u2080.1\n      have h\u2085\u2083 : 0 < (a : \u211d) := by exact_mod_cast h\u2080.2\n      have h\u2085\u2084 : 0 < (m : \u211d) + (a : \u211d) := by linarith\n      exact ne_of_gt h\u2085\u2084\n    have h\u2085\u2082 : (84 * \u2191m + 70 * \u2191a) / (\u2191m + \u2191a) = (76 : \u211d) := by\n      have h\u2085\u2083 : (84 : \u211d) * (m : \u211d) + (70 : \u211d) * (a : \u211d) = (76 : \u211d) * ((m : \u211d) + (a : \u211d)) := h\u2084\n      have h\u2085\u2084 : (84 * \u2191m + 70 * \u2191a) / (\u2191m + \u2191a) = (76 : \u211d) := by\n        rw [show (84 * \u2191m + 70 * \u2191a : \u211d) = (76 : \u211d) * ((\u2191m + \u2191a : \u211d)) by\n          linarith]\n        field_simp [h\u2085\u2081]\n        <;> ring_nf\n        <;> field_simp [h\u2085\u2081]\n        <;> linarith\n      exact h\u2085\u2084\n    exact h\u2085\u2082\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 4, "alphabet": 13, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 286, "chars_after": 6, "seconds": 161.7, "id": "mathd_numbertheory_299", "proof_after": "theorem mathd_numbertheory_299 : 1 * 3 * 5 * 7 * 9 * 11 * 13 % 10 = 5  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_299 : 1 * 3 * 5 * 7 * 9 * 11 * 13 % 10 = 5  :=  by\n  have h\u2080 : (1 * 3 * 5 * 7 * 9 * 11 * 13 : \u2115) % 10 = 5 := by\n    norm_num [mul_assoc, Nat.mul_mod, Nat.add_mod, Nat.mod_mod, Nat.mod_eq_of_lt,\n      Nat.mod_eq_of_lt, Nat.mod_eq_of_lt, Nat.mod_eq_of_lt, Nat.mod_eq_of_lt,\n      Nat.mod_eq_of_lt, Nat.mod_eq_of_lt]\n    <;> rfl\n  exact h\u2080\n"}
{"mode": "clear", "steps_before": 58, "alphabet": 34, "compiles": false, "id": "mathd_numbertheory_314", "proof_after": "theorem mathd_numbertheory_314 (r n : \u2115) (h\u2080 : r = 1342 % 13) (h\u2081 : 0 < n) (h\u2082 : 1342 \u2223 n)\n    (h\u2083 : n % 13 < r) : 6710 \u2264 n  :=  by\n  have h\u2084 : r = 3 := by\n    norm_num [h\u2080, Nat.pow_succ, Nat.pow_zero, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;>\n    rfl\n  have h\u2085 : n % 13 < 3 := by\n    rw [h\u2084] at h\u2083\n    exact h\u2083\n  have h\u2086 : \u00ac(n < 6710) := by\n    intro h\u2086\n    have h\u2087 : n < 6710 := h\u2086\n    have h\u2088 : 1342 \u2223 n := h\u2082\n    have h\u2089 : n % 13 < 3 := h\u2085\n    have h\u2081\u2080 : n = 1342 \u2228 n = 2684 \u2228 n = 4026 \u2228 n = 5368 := by\n      have h\u2081\u2081 : n < 6710 := h\u2087\n      have h\u2081\u2082 : 1342 \u2223 n := h\u2082\n      have h\u2081\u2083 : n > 0 := h\u2081\n      have h\u2081\u2084 : n = 1342 \u2228 n = 2684 \u2228 n = 4026 \u2228 n = 5368 := by\n        have h\u2081\u2085 : n < 6710 := h\u2087\n        have h\u2081\u2086 : 1342 \u2223 n := h\u2082\n        obtain \u27e8k, hk\u27e9 := h\u2081\u2086\n        have h\u2081\u2087 : n = 1342 * k := by linarith\n        have h\u2081\u2088 : k > 0 := by\n          by_contra h\n          have h\u2081\u2089 : k = 0 := by omega\n          rw [h\u2081\u2089] at h\u2081\u2087\n          omega\n        have h\u2082\u2080 : k < 5 := by\n          by_contra h\n          have h\u2082\u2081 : k \u2265 5 := by omega\n          have h\u2082\u2082 : n \u2265 1342 * 5 := by\n            nlinarith\n          omega\n        interval_cases k <;> norm_num [h\u2081\u2087] at h\u2081\u2085 \u22a2 <;> omega\n      exact h\u2081\u2084\n    rcases h\u2081\u2080 with (rfl | rfl | rfl | rfl)\n    \u00b7 \n      norm_num at h\u2089 \u22a2 <;> omega\n    \u00b7 \n      norm_num at h\u2089 \u22a2 <;> omega\n    \u00b7 \n      norm_num at h\u2089 \u22a2 <;> omega\n    \u00b7 \n      norm_num at h\u2089 \u22a2 <;> omega\n  have h\u2087 : 6710 \u2264 n := by\n    by_contra h\u2088\n    have h\u2089 : n < 6710 := by omega\n    exact h\u2086 h\u2089\n  exact h\u2087\n", "proof_before": "theorem mathd_numbertheory_314 (r n : \u2115) (h\u2080 : r = 1342 % 13) (h\u2081 : 0 < n) (h\u2082 : 1342 \u2223 n)\n    (h\u2083 : n % 13 < r) : 6710 \u2264 n  :=  by\n  have h\u2084 : r = 3 := by\n    norm_num [h\u2080, Nat.pow_succ, Nat.pow_zero, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;>\n    rfl\n  have h\u2085 : n % 13 < 3 := by\n    rw [h\u2084] at h\u2083\n    exact h\u2083\n  have h\u2086 : \u00ac(n < 6710) := by\n    intro h\u2086\n    have h\u2087 : n < 6710 := h\u2086\n    have h\u2088 : 1342 \u2223 n := h\u2082\n    have h\u2089 : n % 13 < 3 := h\u2085\n    have h\u2081\u2080 : n = 1342 \u2228 n = 2684 \u2228 n = 4026 \u2228 n = 5368 := by\n      have h\u2081\u2081 : n < 6710 := h\u2087\n      have h\u2081\u2082 : 1342 \u2223 n := h\u2082\n      have h\u2081\u2083 : n > 0 := h\u2081\n      have h\u2081\u2084 : n = 1342 \u2228 n = 2684 \u2228 n = 4026 \u2228 n = 5368 := by\n        have h\u2081\u2085 : n < 6710 := h\u2087\n        have h\u2081\u2086 : 1342 \u2223 n := h\u2082\n        obtain \u27e8k, hk\u27e9 := h\u2081\u2086\n        have h\u2081\u2087 : n = 1342 * k := by linarith\n        have h\u2081\u2088 : k > 0 := by\n          by_contra h\n          have h\u2081\u2089 : k = 0 := by omega\n          rw [h\u2081\u2089] at h\u2081\u2087\n          omega\n        have h\u2082\u2080 : k < 5 := by\n          by_contra h\n          have h\u2082\u2081 : k \u2265 5 := by omega\n          have h\u2082\u2082 : n \u2265 1342 * 5 := by\n            nlinarith\n          omega\n        interval_cases k <;> norm_num [h\u2081\u2087] at h\u2081\u2085 \u22a2 <;> omega\n      exact h\u2081\u2084\n    rcases h\u2081\u2080 with (rfl | rfl | rfl | rfl)\n    \u00b7 \n      norm_num at h\u2089 \u22a2 <;> omega\n    \u00b7 \n      norm_num at h\u2089 \u22a2 <;> omega\n    \u00b7 \n      norm_num at h\u2089 \u22a2 <;> omega\n    \u00b7 \n      norm_num at h\u2089 \u22a2 <;> omega\n  have h\u2087 : 6710 \u2264 n := by\n    by_contra h\u2088\n    have h\u2089 : n < 6710 := by omega\n    exact h\u2086 h\u2089\n  exact h\u2087\n"}
{"mode": "clear", "steps_before": 41, "alphabet": 19, "compiles": true, "steps_after": 3, "lower_bound": 3, "exhausted_depth": 2, "certified": true, "rounds": 2, "fixpoint": true, "chars_before": 1631, "chars_after": 216, "seconds": 496.7, "id": "aime_1990_p15", "proof_after": "theorem aime_1990_p15 (a b x y : \u211d) (h\u2080 : a * x + b * y = 3) (h\u2081 : a * x ^ 2 + b * y ^ 2 = 7)\n    (h\u2082 : a * x ^ 3 + b * y ^ 3 = 16) (h\u2083 : a * x ^ 4 + b * y ^ 4 = 42) :\n    a * x ^ 5 + b * y ^ 5 = 20  :=  by\n  have h_sum_xy : x + y = -14 := by\n    nlinarith [sq_nonneg (x + y), sq_nonneg (x - y), sq_nonneg (a + b), sq_nonneg (a - b)]\n  nlinarith [sq_nonneg (x + y), sq_nonneg (x - y), sq_nonneg (a + b), sq_nonneg (a - b)]\n", "proof_before": "theorem aime_1990_p15 (a b x y : \u211d) (h\u2080 : a * x + b * y = 3) (h\u2081 : a * x ^ 2 + b * y ^ 2 = 7)\n    (h\u2082 : a * x ^ 3 + b * y ^ 3 = 16) (h\u2083 : a * x ^ 4 + b * y ^ 4 = 42) :\n    a * x ^ 5 + b * y ^ 5 = 20  :=  by\n  have h_sum_xy : x + y = -14 := by\n    have h\u2084 : 7 * (x + y) - 3 * (x * y) = 16 := by\n      have h\u2084\u2081 : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - x * y * (a * x + b * y) := by\n        ring\n      rw [h\u2082, h\u2081, h\u2080] at h\u2084\u2081\n      linarith\n    have h\u2085 : 16 * (x + y) - 7 * (x * y) = 42 := by\n      have h\u2085\u2081 : a * x ^ 4 + b * y ^ 4 = (x + y) * (a * x ^ 3 + b * y ^ 3) - x * y * (a * x ^ 2 + b * y ^ 2) := by\n        ring\n      rw [h\u2083, h\u2082, h\u2081] at h\u2085\u2081\n      linarith\n    have h\u2086 : x + y = -14 := by\n      linarith\n    exact h\u2086\n  have h_prod_xy : x * y = -38 := by\n    have h\u2084 : 7 * (x + y) - 3 * (x * y) = 16 := by\n      have h\u2084\u2081 : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - x * y * (a * x + b * y) := by\n        ring\n      rw [h\u2082, h\u2081, h\u2080] at h\u2084\u2081\n      linarith\n    have h\u2085 : 16 * (x + y) - 7 * (x * y) = 42 := by\n      have h\u2085\u2081 : a * x ^ 4 + b * y ^ 4 = (x + y) * (a * x ^ 3 + b * y ^ 3) - x * y * (a * x ^ 2 + b * y ^ 2) := by\n        ring\n      rw [h\u2083, h\u2082, h\u2081] at h\u2085\u2081\n      linarith\n    have h\u2086 : x * y = -38 := by\n      linarith\n    exact h\u2086\n  have h_sum_ab : a + b = 49 / 38 := by\n    have h\u2084 : (x + y) * (a * x + b * y) = (a * x ^ 2 + b * y ^ 2) + x * y * (a + b) := by\n      ring\n    rw [h\u2080, h\u2081, h_sum_xy, h_prod_xy] at h\u2084\n    <;> nlinarith [sq_nonneg (x + y), sq_nonneg (x - y), sq_nonneg (a + b), sq_nonneg (a - b)]\n  have h_recurrence : a * x ^ 5 + b * y ^ 5 = 20 := by\n    have h\u2085 : a * x ^ 5 + b * y ^ 5 = (x + y) * (a * x ^ 4 + b * y ^ 4) - x * y * (a * x ^ 3 + b * y ^ 3) := by\n      ring\n    rw [h\u2085]\n    rw [h\u2083, h\u2082, h_sum_xy, h_prod_xy]\n    <;> ring_nf\n    <;> linarith\n  exact h_recurrence\n"}
{"mode": "clear", "steps_before": 21, "alphabet": 14, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 554, "chars_after": 103, "seconds": 66.6, "id": "mathd_algebra_302", "proof_after": "theorem mathd_algebra_302 : (Complex.I / 2) ^ 2 = -(1 / 4)  :=  by\n  norm_num [Complex.ext_iff, pow_two, Complex.I_mul_I, Complex.div_re, Complex.div_im, Complex.normSq]\n", "proof_before": "theorem mathd_algebra_302 : (Complex.I / 2) ^ 2 = -(1 / 4)  :=  by\n  have h\u2080 : (Complex.I / 2 : \u2102) ^ 2 = -(1 / 4 : \u2102) := by\n    norm_num [Complex.ext_iff, pow_two, Complex.I_mul_I, Complex.div_re, Complex.div_im, Complex.normSq]\n    <;> ring_nf\n    <;> norm_num\n    <;>\n    simp [Complex.ext_iff, pow_two, Complex.I_mul_I, Complex.div_re, Complex.div_im, Complex.normSq]\n    <;>\n    norm_num\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    simp [Complex.ext_iff, pow_two, Complex.I_mul_I, Complex.div_re, Complex.div_im, Complex.normSq]\n    <;>\n    norm_num\n    <;>\n    ring_nf\n    <;>\n    norm_num\n  simpa using h\u2080\n"}
{"mode": "clear", "steps_before": 81, "alphabet": 30, "compiles": true, "steps_after": 81, "lower_bound": null, "exhausted_depth": 2, "certified": false, "rounds": 8, "fixpoint": true, "chars_before": 2527, "chars_after": 2527, "seconds": 953.6, "id": "numbertheory_4x3m7y3neq2003", "proof_after": "theorem numbertheory_4x3m7y3neq2003 (x y : \u2124) : 4 * x ^ 3 - 7 * y ^ 3 \u2260 2003  :=  by\n  have h_main : (4 * x ^ 3 - 7 * y ^ 3) % 7 \u2260 2003 % 7 := by\n    have h\u2081 : (4 * x ^ 3 - 7 * y ^ 3 : \u2124) % 7 = (4 * x ^ 3 : \u2124) % 7 := by\n      have h\u2082 : (4 * x ^ 3 - 7 * y ^ 3 : \u2124) % 7 = (4 * x ^ 3 : \u2124) % 7 - (7 * y ^ 3 : \u2124) % 7 := by\n        omega\n      have h\u2083 : (7 * y ^ 3 : \u2124) % 7 = 0 := by\n        have h\u2084 : (7 : \u2124) % 7 = 0 := by norm_num\n        have h\u2085 : (7 * y ^ 3 : \u2124) % 7 = 0 := by\n          simp [h\u2084, Int.mul_emod, Int.pow_succ, Int.pow_zero]\n        exact h\u2085\n      omega\n    have h\u2082 : (4 * x ^ 3 : \u2124) % 7 = (4 * (x % 7) ^ 3 : \u2124) % 7 := by\n      have h\u2083 : (x : \u2124) % 7 = (x % 7 : \u2124) := by simp [Int.emod_emod]\n      have h\u2084 : (4 * x ^ 3 : \u2124) % 7 = (4 * (x % 7) ^ 3 : \u2124) % 7 := by\n        have h\u2085 : (x : \u2124) % 7 = (x % 7 : \u2124) := by simp [Int.emod_emod]\n        have h\u2086 : (x : \u2124) ^ 3 % 7 = ((x % 7 : \u2124) ^ 3) % 7 := by\n          have h\u2087 : (x : \u2124) % 7 = (x % 7 : \u2124) := by simp [Int.emod_emod]\n          have h\u2088 : (x : \u2124) ^ 3 % 7 = ((x % 7 : \u2124) ^ 3) % 7 := by\n            norm_num [pow_three, Int.mul_emod, Int.add_emod, Int.emod_emod]\n            <;>\n            (try omega) <;>\n            (try ring_nf) <;>\n            (try omega) <;>\n            (try norm_num) <;>\n            (try omega)\n          exact h\u2088\n        have h\u2089 : (4 * x ^ 3 : \u2124) % 7 = (4 * (x % 7) ^ 3 : \u2124) % 7 := by\n          norm_num [h\u2086, Int.mul_emod, Int.add_emod, Int.emod_emod]\n          <;>\n          (try omega) <;>\n          (try ring_nf) <;>\n          (try omega) <;>\n          (try norm_num) <;>\n          (try omega)\n        exact h\u2089\n      exact h\u2084\n    have h\u2083 : (2003 : \u2124) % 7 = 1 := by norm_num\n    have h\u2084 : (4 * x ^ 3 - 7 * y ^ 3 : \u2124) % 7 = (4 * (x % 7) ^ 3 : \u2124) % 7 := by\n      rw [h\u2081, h\u2082]\n    have h\u2085 : (4 * (x % 7) ^ 3 : \u2124) % 7 \u2260 1 := by\n      have h\u2086 : x % 7 = 0 \u2228 x % 7 = 1 \u2228 x % 7 = 2 \u2228 x % 7 = 3 \u2228 x % 7 = 4 \u2228 x % 7 = 5 \u2228 x % 7 = 6 := by\n        omega\n      rcases h\u2086 with (h\u2086 | h\u2086 | h\u2086 | h\u2086 | h\u2086 | h\u2086 | h\u2086) <;>\n        (try {\n          simp [h\u2086, pow_three, Int.mul_emod, Int.add_emod, Int.emod_emod, Int.sub_emod]\n          <;> norm_num <;> omega\n        }) <;>\n        (try omega) <;>\n        (try ring_nf) <;>\n        (try omega) <;>\n        (try norm_num) <;>\n        (try omega)\n    have h\u2086 : (4 * x ^ 3 - 7 * y ^ 3 : \u2124) % 7 \u2260 1 := by\n      rw [h\u2084]\n      exact h\u2085\n    omega\n  have h_final : 4 * x ^ 3 - 7 * y ^ 3 \u2260 2003 := by\n    intro h\n    have h\u2081 : (4 * x ^ 3 - 7 * y ^ 3 : \u2124) % 7 = 2003 % 7 := by\n      rw [h]\n      <;> norm_num\n    have h\u2082 : (4 * x ^ 3 - 7 * y ^ 3) % 7 \u2260 2003 % 7 := h_main\n    contradiction\n  exact h_final\n", "proof_before": "theorem numbertheory_4x3m7y3neq2003 (x y : \u2124) : 4 * x ^ 3 - 7 * y ^ 3 \u2260 2003  :=  by\n  have h_main : (4 * x ^ 3 - 7 * y ^ 3) % 7 \u2260 2003 % 7 := by\n    have h\u2081 : (4 * x ^ 3 - 7 * y ^ 3 : \u2124) % 7 = (4 * x ^ 3 : \u2124) % 7 := by\n      have h\u2082 : (4 * x ^ 3 - 7 * y ^ 3 : \u2124) % 7 = (4 * x ^ 3 : \u2124) % 7 - (7 * y ^ 3 : \u2124) % 7 := by\n        omega\n      have h\u2083 : (7 * y ^ 3 : \u2124) % 7 = 0 := by\n        have h\u2084 : (7 : \u2124) % 7 = 0 := by norm_num\n        have h\u2085 : (7 * y ^ 3 : \u2124) % 7 = 0 := by\n          simp [h\u2084, Int.mul_emod, Int.pow_succ, Int.pow_zero]\n        exact h\u2085\n      omega\n    have h\u2082 : (4 * x ^ 3 : \u2124) % 7 = (4 * (x % 7) ^ 3 : \u2124) % 7 := by\n      have h\u2083 : (x : \u2124) % 7 = (x % 7 : \u2124) := by simp [Int.emod_emod]\n      have h\u2084 : (4 * x ^ 3 : \u2124) % 7 = (4 * (x % 7) ^ 3 : \u2124) % 7 := by\n        have h\u2085 : (x : \u2124) % 7 = (x % 7 : \u2124) := by simp [Int.emod_emod]\n        have h\u2086 : (x : \u2124) ^ 3 % 7 = ((x % 7 : \u2124) ^ 3) % 7 := by\n          have h\u2087 : (x : \u2124) % 7 = (x % 7 : \u2124) := by simp [Int.emod_emod]\n          have h\u2088 : (x : \u2124) ^ 3 % 7 = ((x % 7 : \u2124) ^ 3) % 7 := by\n            norm_num [pow_three, Int.mul_emod, Int.add_emod, Int.emod_emod]\n            <;>\n            (try omega) <;>\n            (try ring_nf) <;>\n            (try omega) <;>\n            (try norm_num) <;>\n            (try omega)\n          exact h\u2088\n        have h\u2089 : (4 * x ^ 3 : \u2124) % 7 = (4 * (x % 7) ^ 3 : \u2124) % 7 := by\n          norm_num [h\u2086, Int.mul_emod, Int.add_emod, Int.emod_emod]\n          <;>\n          (try omega) <;>\n          (try ring_nf) <;>\n          (try omega) <;>\n          (try norm_num) <;>\n          (try omega)\n        exact h\u2089\n      exact h\u2084\n    have h\u2083 : (2003 : \u2124) % 7 = 1 := by norm_num\n    have h\u2084 : (4 * x ^ 3 - 7 * y ^ 3 : \u2124) % 7 = (4 * (x % 7) ^ 3 : \u2124) % 7 := by\n      rw [h\u2081, h\u2082]\n    have h\u2085 : (4 * (x % 7) ^ 3 : \u2124) % 7 \u2260 1 := by\n      have h\u2086 : x % 7 = 0 \u2228 x % 7 = 1 \u2228 x % 7 = 2 \u2228 x % 7 = 3 \u2228 x % 7 = 4 \u2228 x % 7 = 5 \u2228 x % 7 = 6 := by\n        omega\n      rcases h\u2086 with (h\u2086 | h\u2086 | h\u2086 | h\u2086 | h\u2086 | h\u2086 | h\u2086) <;>\n        (try {\n          simp [h\u2086, pow_three, Int.mul_emod, Int.add_emod, Int.emod_emod, Int.sub_emod]\n          <;> norm_num <;> omega\n        }) <;>\n        (try omega) <;>\n        (try ring_nf) <;>\n        (try omega) <;>\n        (try norm_num) <;>\n        (try omega)\n    have h\u2086 : (4 * x ^ 3 - 7 * y ^ 3 : \u2124) % 7 \u2260 1 := by\n      rw [h\u2084]\n      exact h\u2085\n    omega\n  have h_final : 4 * x ^ 3 - 7 * y ^ 3 \u2260 2003 := by\n    intro h\n    have h\u2081 : (4 * x ^ 3 - 7 * y ^ 3 : \u2124) % 7 = 2003 % 7 := by\n      rw [h]\n      <;> norm_num\n    have h\u2082 : (4 * x ^ 3 - 7 * y ^ 3) % 7 \u2260 2003 % 7 := h_main\n    contradiction\n  exact h_final\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 90, "alphabet": 31, "compiles": true, "steps_after": 77, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 2, "fixpoint": false, "chars_before": 5275, "chars_after": 4392, "seconds": 637.0, "id": "imo_1963_p5", "proof_after": "theorem imo_1963_p5 : Real.cos (Real.pi / 7) - Real.cos (2 * Real.pi / 7) + Real.cos (3 * Real.pi / 7) = 1 / 2  :=  by\n  have h1 : 2 * Real.sin (Real.pi / 7) * (Real.cos (Real.pi / 7) - Real.cos (2 * Real.pi / 7) + Real.cos (3 * Real.pi / 7)) = Real.sin (Real.pi / 7) := by\n    have h1\u2081 : 2 * Real.sin (Real.pi / 7) * Real.cos (Real.pi / 7) = Real.sin (2 * Real.pi / 7) := by\n      have h1\u2081\u2081 : Real.sin (2 * Real.pi / 7) = 2 * Real.sin (Real.pi / 7) * Real.cos (Real.pi / 7) := by\n        have h1\u2081\u2082 : Real.sin (2 * Real.pi / 7) = Real.sin (2 * (Real.pi / 7)) := by ring\n        rw [h1\u2081\u2082]\n        have h1\u2081\u2083 : Real.sin (2 * (Real.pi / 7)) = 2 * Real.sin (Real.pi / 7) * Real.cos (Real.pi / 7) := by\n          rw [Real.sin_two_mul]\n          <;> ring\n        rw [h1\u2081\u2083]\n        <;> ring\n      linarith\n    have h1\u2082 : 2 * Real.sin (Real.pi / 7) * Real.cos (2 * Real.pi / 7) = Real.sin (3 * Real.pi / 7) - Real.sin (Real.pi / 7) := by\n      have h1\u2082\u2081 : 2 * Real.sin (Real.pi / 7) * Real.cos (2 * Real.pi / 7) = Real.sin (3 * Real.pi / 7) - Real.sin (Real.pi / 7) := by\n        have h1\u2082\u2082 : Real.sin (3 * Real.pi / 7) - Real.sin (Real.pi / 7) = 2 * Real.sin (Real.pi / 7) * Real.cos (2 * Real.pi / 7) := by\n          have h1\u2082\u2083 : Real.sin (3 * Real.pi / 7) - Real.sin (Real.pi / 7) = 2 * Real.sin ((3 * Real.pi / 7 - Real.pi / 7) / 2) * Real.cos ((3 * Real.pi / 7 + Real.pi / 7) / 2) := by\n            have h1\u2082\u2084 : Real.sin (3 * Real.pi / 7) - Real.sin (Real.pi / 7) = 2 * Real.sin ((3 * Real.pi / 7 - Real.pi / 7) / 2) * Real.cos ((3 * Real.pi / 7 + Real.pi / 7) / 2) := by\n              rw [\u2190 Real.sin_sub_sin]\n              <;> ring_nf\n              <;> field_simp\n              <;> ring_nf\n            rw [h1\u2082\u2084]\n          rw [h1\u2082\u2083]\n          have h1\u2082\u2085 : (3 * Real.pi / 7 - Real.pi / 7) / 2 = Real.pi / 7 := by ring\n          have h1\u2082\u2086 : (3 * Real.pi / 7 + Real.pi / 7) / 2 = 2 * Real.pi / 7 := by ring\n          rw [h1\u2082\u2085, h1\u2082\u2086]\n          <;> ring_nf\n          <;> field_simp\n          <;> ring_nf\n        linarith\n      linarith\n    have h1\u2083 : 2 * Real.sin (Real.pi / 7) * Real.cos (3 * Real.pi / 7) = Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7) := by\n      have h1\u2083\u2081 : 2 * Real.sin (Real.pi / 7) * Real.cos (3 * Real.pi / 7) = Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7) := by\n        have h1\u2083\u2082 : Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7) = 2 * Real.sin (Real.pi / 7) * Real.cos (3 * Real.pi / 7) := by\n          have h1\u2083\u2083 : Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7) = 2 * Real.sin ((4 * Real.pi / 7 - 2 * Real.pi / 7) / 2) * Real.cos ((4 * Real.pi / 7 + 2 * Real.pi / 7) / 2) := by\n            have h1\u2083\u2084 : Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7) = 2 * Real.sin ((4 * Real.pi / 7 - 2 * Real.pi / 7) / 2) * Real.cos ((4 * Real.pi / 7 + 2 * Real.pi / 7) / 2) := by\n              rw [\u2190 Real.sin_sub_sin]\n              <;> ring_nf\n              <;> field_simp\n              <;> ring_nf\n            rw [h1\u2083\u2084]\n          rw [h1\u2083\u2083]\n          have h1\u2083\u2085 : (4 * Real.pi / 7 - 2 * Real.pi / 7) / 2 = Real.pi / 7 := by ring\n          have h1\u2083\u2086 : (4 * Real.pi / 7 + 2 * Real.pi / 7) / 2 = 3 * Real.pi / 7 := by ring\n          rw [h1\u2083\u2085, h1\u2083\u2086]\n          <;> ring_nf\n          <;> field_simp\n          <;> ring_nf\n        linarith\n      linarith\n    have h1\u2084 : Real.sin (4 * Real.pi / 7) = Real.sin (3 * Real.pi / 7) := by\n      have h1\u2084\u2081 : Real.sin (4 * Real.pi / 7) = Real.sin (Real.pi - 3 * Real.pi / 7) := by\n        have h1\u2084\u2082 : 4 * Real.pi / 7 = Real.pi - 3 * Real.pi / 7 := by ring\n        rw [h1\u2084\u2082]\n        <;> simp [Real.sin_pi_sub]\n      rw [h1\u2084\u2081]\n      have h1\u2084\u2083 : Real.sin (Real.pi - 3 * Real.pi / 7) = Real.sin (3 * Real.pi / 7) := by\n        simp [Real.sin_pi_sub]\n      rw [h1\u2084\u2083]\n    linarith\n  have h2 : Real.sin (Real.pi / 7) > 0 := by\n    apply Real.sin_pos_of_pos_of_lt_pi\n    \u00b7 linarith [Real.pi_pos, Real.pi_gt_three]\n    \u00b7 linarith [Real.pi_pos, Real.pi_gt_three]\n  have h3 : Real.cos (Real.pi / 7) - Real.cos (2 * Real.pi / 7) + Real.cos (3 * Real.pi / 7) = 1 / 2 := by\n    have h3\u2081 : 2 * Real.sin (Real.pi / 7) > 0 := by linarith\n    have h3\u2082 : Real.cos (Real.pi / 7) - Real.cos (2 * Real.pi / 7) + Real.cos (3 * Real.pi / 7) = 1 / 2 := by\n      apply mul_left_cancel\u2080 (show (2 * Real.sin (Real.pi / 7) : \u211d) \u2260 0 by linarith)\n      nlinarith [Real.sin_le_one (Real.pi / 7), Real.sin_le_one (2 * Real.pi / 7), Real.sin_le_one (3 * Real.pi / 7),\n        Real.sin_le_one (4 * Real.pi / 7), Real.sin_le_one (Real.pi / 7)]\n    exact h3\u2082\n  apply h3\n", "proof_before": "theorem imo_1963_p5 : Real.cos (Real.pi / 7) - Real.cos (2 * Real.pi / 7) + Real.cos (3 * Real.pi / 7) = 1 / 2  :=  by\n  have h1 : 2 * Real.sin (Real.pi / 7) * (Real.cos (Real.pi / 7) - Real.cos (2 * Real.pi / 7) + Real.cos (3 * Real.pi / 7)) = Real.sin (Real.pi / 7) := by\n    have h1\u2081 : 2 * Real.sin (Real.pi / 7) * Real.cos (Real.pi / 7) = Real.sin (2 * Real.pi / 7) := by\n      have h1\u2081\u2081 : Real.sin (2 * Real.pi / 7) = 2 * Real.sin (Real.pi / 7) * Real.cos (Real.pi / 7) := by\n        have h1\u2081\u2082 : Real.sin (2 * Real.pi / 7) = Real.sin (2 * (Real.pi / 7)) := by ring\n        rw [h1\u2081\u2082]\n        have h1\u2081\u2083 : Real.sin (2 * (Real.pi / 7)) = 2 * Real.sin (Real.pi / 7) * Real.cos (Real.pi / 7) := by\n          rw [Real.sin_two_mul]\n          <;> ring\n        rw [h1\u2081\u2083]\n        <;> ring\n      linarith\n    have h1\u2082 : 2 * Real.sin (Real.pi / 7) * Real.cos (2 * Real.pi / 7) = Real.sin (3 * Real.pi / 7) - Real.sin (Real.pi / 7) := by\n      have h1\u2082\u2081 : 2 * Real.sin (Real.pi / 7) * Real.cos (2 * Real.pi / 7) = Real.sin (3 * Real.pi / 7) - Real.sin (Real.pi / 7) := by\n        have h1\u2082\u2082 : Real.sin (3 * Real.pi / 7) - Real.sin (Real.pi / 7) = 2 * Real.sin (Real.pi / 7) * Real.cos (2 * Real.pi / 7) := by\n          have h1\u2082\u2083 : Real.sin (3 * Real.pi / 7) - Real.sin (Real.pi / 7) = 2 * Real.sin ((3 * Real.pi / 7 - Real.pi / 7) / 2) * Real.cos ((3 * Real.pi / 7 + Real.pi / 7) / 2) := by\n            have h1\u2082\u2084 : Real.sin (3 * Real.pi / 7) - Real.sin (Real.pi / 7) = 2 * Real.sin ((3 * Real.pi / 7 - Real.pi / 7) / 2) * Real.cos ((3 * Real.pi / 7 + Real.pi / 7) / 2) := by\n              rw [\u2190 Real.sin_sub_sin]\n              <;> ring_nf\n              <;> field_simp\n              <;> ring_nf\n            rw [h1\u2082\u2084]\n          rw [h1\u2082\u2083]\n          have h1\u2082\u2085 : (3 * Real.pi / 7 - Real.pi / 7) / 2 = Real.pi / 7 := by ring\n          have h1\u2082\u2086 : (3 * Real.pi / 7 + Real.pi / 7) / 2 = 2 * Real.pi / 7 := by ring\n          rw [h1\u2082\u2085, h1\u2082\u2086]\n          <;> ring_nf\n          <;> field_simp\n          <;> ring_nf\n        linarith\n      linarith\n    have h1\u2083 : 2 * Real.sin (Real.pi / 7) * Real.cos (3 * Real.pi / 7) = Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7) := by\n      have h1\u2083\u2081 : 2 * Real.sin (Real.pi / 7) * Real.cos (3 * Real.pi / 7) = Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7) := by\n        have h1\u2083\u2082 : Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7) = 2 * Real.sin (Real.pi / 7) * Real.cos (3 * Real.pi / 7) := by\n          have h1\u2083\u2083 : Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7) = 2 * Real.sin ((4 * Real.pi / 7 - 2 * Real.pi / 7) / 2) * Real.cos ((4 * Real.pi / 7 + 2 * Real.pi / 7) / 2) := by\n            have h1\u2083\u2084 : Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7) = 2 * Real.sin ((4 * Real.pi / 7 - 2 * Real.pi / 7) / 2) * Real.cos ((4 * Real.pi / 7 + 2 * Real.pi / 7) / 2) := by\n              rw [\u2190 Real.sin_sub_sin]\n              <;> ring_nf\n              <;> field_simp\n              <;> ring_nf\n            rw [h1\u2083\u2084]\n          rw [h1\u2083\u2083]\n          have h1\u2083\u2085 : (4 * Real.pi / 7 - 2 * Real.pi / 7) / 2 = Real.pi / 7 := by ring\n          have h1\u2083\u2086 : (4 * Real.pi / 7 + 2 * Real.pi / 7) / 2 = 3 * Real.pi / 7 := by ring\n          rw [h1\u2083\u2085, h1\u2083\u2086]\n          <;> ring_nf\n          <;> field_simp\n          <;> ring_nf\n        linarith\n      linarith\n    have h1\u2084 : Real.sin (4 * Real.pi / 7) = Real.sin (3 * Real.pi / 7) := by\n      have h1\u2084\u2081 : Real.sin (4 * Real.pi / 7) = Real.sin (Real.pi - 3 * Real.pi / 7) := by\n        have h1\u2084\u2082 : 4 * Real.pi / 7 = Real.pi - 3 * Real.pi / 7 := by ring\n        rw [h1\u2084\u2082]\n        <;> simp [Real.sin_pi_sub]\n      rw [h1\u2084\u2081]\n      have h1\u2084\u2083 : Real.sin (Real.pi - 3 * Real.pi / 7) = Real.sin (3 * Real.pi / 7) := by\n        simp [Real.sin_pi_sub]\n      rw [h1\u2084\u2083]\n    calc\n      2 * Real.sin (Real.pi / 7) * (Real.cos (Real.pi / 7) - Real.cos (2 * Real.pi / 7) + Real.cos (3 * Real.pi / 7)) = 2 * Real.sin (Real.pi / 7) * Real.cos (Real.pi / 7) - 2 * Real.sin (Real.pi / 7) * Real.cos (2 * Real.pi / 7) + 2 * Real.sin (Real.pi / 7) * Real.cos (3 * Real.pi / 7) := by\n        ring\n      _ = Real.sin (2 * Real.pi / 7) - (Real.sin (3 * Real.pi / 7) - Real.sin (Real.pi / 7)) + (Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7)) := by\n        rw [h1\u2081, h1\u2082, h1\u2083]\n        <;> ring\n      _ = Real.sin (2 * Real.pi / 7) - Real.sin (3 * Real.pi / 7) + Real.sin (Real.pi / 7) + (Real.sin (4 * Real.pi / 7) - Real.sin (2 * Real.pi / 7)) := by\n        ring\n      _ = Real.sin (Real.pi / 7) := by\n        rw [h1\u2084]\n        <;> ring_nf\n        <;> field_simp\n        <;> ring_nf\n        <;> linarith [Real.sin_le_one (3 * Real.pi / 7), Real.sin_le_one (Real.pi / 7)]\n  have h2 : Real.sin (Real.pi / 7) > 0 := by\n    apply Real.sin_pos_of_pos_of_lt_pi\n    \u00b7 linarith [Real.pi_pos, Real.pi_gt_three]\n    \u00b7 linarith [Real.pi_pos, Real.pi_gt_three]\n  have h3 : Real.cos (Real.pi / 7) - Real.cos (2 * Real.pi / 7) + Real.cos (3 * Real.pi / 7) = 1 / 2 := by\n    have h3\u2081 : 2 * Real.sin (Real.pi / 7) > 0 := by linarith\n    have h3\u2082 : Real.cos (Real.pi / 7) - Real.cos (2 * Real.pi / 7) + Real.cos (3 * Real.pi / 7) = 1 / 2 := by\n      apply mul_left_cancel\u2080 (show (2 * Real.sin (Real.pi / 7) : \u211d) \u2260 0 by linarith)\n      nlinarith [Real.sin_le_one (Real.pi / 7), Real.sin_le_one (2 * Real.pi / 7), Real.sin_le_one (3 * Real.pi / 7),\n        Real.sin_le_one (4 * Real.pi / 7), Real.sin_le_one (Real.pi / 7)]\n    exact h3\u2082\n  apply h3\n"}
{"mode": "clear", "steps_before": 180, "alphabet": 61, "compiles": false, "id": "algebra_abpbcpcageq3_sumaonsqrtapbgeq3onsqrt2", "proof_after": "theorem algebra_abpbcpcageq3_sumaonsqrtapbgeq3onsqrt2 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c)\n    (h\u2081 : 3 \u2264 a * b + b * c + c * a) :\n    3 / Real.sqrt 2 \u2264 a / Real.sqrt (a + b) + b / Real.sqrt (b + c) + c / Real.sqrt (c + a)  :=  by\n  have h\u2082 : a + b + c \u2265 3 := by\n    have h\u2082\u2081 : (a + b + c) ^ 2 \u2265 3 * (a * b + b * c + c * a) := by\n      nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n    have h\u2082\u2082 : 3 * (a * b + b * c + c * a) \u2265 9 := by linarith\n    have h\u2082\u2083 : (a + b + c) ^ 2 \u2265 9 := by linarith\n    have h\u2082\u2084 : a + b + c \u2265 3 := by\n      nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n    exact h\u2082\u2084\n  have h\u2083 : \u2200 (x y : \u211d), 0 < x \u2192 0 < y \u2192 Real.sqrt (x + y) \u2264 (x + y + 2) / (2 * Real.sqrt 2) := by\n    intro x y hx hy\n    have h\u2084 : 0 < x + y := by linarith\n    have h\u2085 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n    have h\u2086 : 0 < Real.sqrt 2 * (x + y) := by positivity\n    have h\u2087 : 0 < Real.sqrt 2 * 2 := by positivity\n    have h\u2088 : Real.sqrt (x + y) \u2264 (x + y + 2) / (2 * Real.sqrt 2) := by\n      have h\u2088\u2081 : Real.sqrt (x + y) \u2264 (x + y + 2) / (2 * Real.sqrt 2) := by\n        rw [Real.sqrt_le_left (by positivity)]\n        field_simp [h\u2085.ne']\n        rw [le_div_iff (by positivity)]\n        nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num), sq_nonneg (x + y - 2),\n          sq_nonneg (Real.sqrt 2 * Real.sqrt (x + y) - 2),\n          Real.sq_sqrt (show 0 \u2264 x + y by positivity), sq_nonneg (x + y - 2)]\n      exact h\u2088\u2081\n    exact h\u2088\n  have h\u2084 : a / Real.sqrt (a + b) \u2265 (2 * Real.sqrt 2 * a) / (a + b + 2) := by\n    have h\u2084\u2081 : 0 < a := by linarith\n    have h\u2084\u2082 : 0 < b := by linarith\n    have h\u2084\u2083 : 0 < a + b := by linarith\n    have h\u2084\u2084 : 0 < Real.sqrt (a + b) := Real.sqrt_pos.mpr h\u2084\u2083\n    have h\u2084\u2085 : Real.sqrt (a + b) \u2264 (a + b + 2) / (2 * Real.sqrt 2) := h\u2083 a b h\u2084\u2081 h\u2084\u2082\n    have h\u2084\u2086 : a / Real.sqrt (a + b) \u2265 a / ((a + b + 2) / (2 * Real.sqrt 2)) := by\n      have h\u2084\u2087 : 0 < (a + b + 2) / (2 * Real.sqrt 2) := by positivity\n      have h\u2084\u2088 : 0 < Real.sqrt (a + b) := Real.sqrt_pos.mpr h\u2084\u2083\n      have h\u2084\u2089 : Real.sqrt (a + b) \u2264 (a + b + 2) / (2 * Real.sqrt 2) := h\u2084\u2085\n      have h\u2084\u2081\u2080 : a / Real.sqrt (a + b) \u2265 a / ((a + b + 2) / (2 * Real.sqrt 2)) := by\n        apply (div_le_div_iff (by positivity) (by positivity)).mpr\n        nlinarith [Real.sqrt_nonneg (a + b), Real.sq_sqrt (show 0 \u2264 a + b by positivity),\n          Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      exact h\u2084\u2081\u2080\n    have h\u2084\u2081\u2081 : a / ((a + b + 2) / (2 * Real.sqrt 2)) = (2 * Real.sqrt 2 * a) / (a + b + 2) := by\n      field_simp [h\u2084\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> field_simp [h\u2084\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    rw [h\u2084\u2081\u2081] at h\u2084\u2086\n    exact h\u2084\u2086\n  have h\u2085 : b / Real.sqrt (b + c) \u2265 (2 * Real.sqrt 2 * b) / (b + c + 2) := by\n    have h\u2085\u2081 : 0 < b := by linarith\n    have h\u2085\u2082 : 0 < c := by linarith\n    have h\u2085\u2083 : 0 < b + c := by linarith\n    have h\u2085\u2084 : 0 < Real.sqrt (b + c) := Real.sqrt_pos.mpr h\u2085\u2083\n    have h\u2085\u2085 : Real.sqrt (b + c) \u2264 (b + c + 2) / (2 * Real.sqrt 2) := h\u2083 b c h\u2085\u2081 h\u2085\u2082\n    have h\u2085\u2086 : b / Real.sqrt (b + c) \u2265 b / ((b + c + 2) / (2 * Real.sqrt 2)) := by\n      have h\u2085\u2087 : 0 < (b + c + 2) / (2 * Real.sqrt 2) := by positivity\n      have h\u2085\u2088 : 0 < Real.sqrt (b + c) := Real.sqrt_pos.mpr h\u2085\u2083\n      have h\u2085\u2089 : Real.sqrt (b + c) \u2264 (b + c + 2) / (2 * Real.sqrt 2) := h\u2085\u2085\n      have h\u2085\u2081\u2080 : b / Real.sqrt (b + c) \u2265 b / ((b + c + 2) / (2 * Real.sqrt 2)) := by\n        apply (div_le_div_iff (by positivity) (by positivity)).mpr\n        nlinarith [Real.sqrt_nonneg (b + c), Real.sq_sqrt (show 0 \u2264 b + c by positivity),\n          Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      exact h\u2085\u2081\u2080\n    have h\u2085\u2081\u2081 : b / ((b + c + 2) / (2 * Real.sqrt 2)) = (2 * Real.sqrt 2 * b) / (b + c + 2) := by\n      field_simp [h\u2085\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> field_simp [h\u2085\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    rw [h\u2085\u2081\u2081] at h\u2085\u2086\n    exact h\u2085\u2086\n  have h\u2086 : c / Real.sqrt (c + a) \u2265 (2 * Real.sqrt 2 * c) / (c + a + 2) := by\n    have h\u2086\u2081 : 0 < c := by linarith\n    have h\u2086\u2082 : 0 < a := by linarith\n    have h\u2086\u2083 : 0 < c + a := by linarith\n    have h\u2086\u2084 : 0 < Real.sqrt (c + a) := Real.sqrt_pos.mpr h\u2086\u2083\n    have h\u2086\u2085 : Real.sqrt (c + a) \u2264 (c + a + 2) / (2 * Real.sqrt 2) := h\u2083 c a h\u2086\u2081 h\u2086\u2082\n    have h\u2086\u2086 : c / Real.sqrt (c + a) \u2265 c / ((c + a + 2) / (2 * Real.sqrt 2)) := by\n      have h\u2086\u2087 : 0 < (c + a + 2) / (2 * Real.sqrt 2) := by positivity\n      have h\u2086\u2088 : 0 < Real.sqrt (c + a) := Real.sqrt_pos.mpr h\u2086\u2083\n      have h\u2086\u2089 : Real.sqrt (c + a) \u2264 (c + a + 2) / (2 * Real.sqrt 2) := h\u2086\u2085\n      have h\u2086\u2081\u2080 : c / Real.sqrt (c + a) \u2265 c / ((c + a + 2) / (2 * Real.sqrt 2)) := by\n        apply (div_le_div_iff (by positivity) (by positivity)).mpr\n        nlinarith [Real.sqrt_nonneg (c + a), Real.sq_sqrt (show 0 \u2264 c + a by positivity),\n          Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      exact h\u2086\u2081\u2080\n    have h\u2086\u2081\u2081 : c / ((c + a + 2) / (2 * Real.sqrt 2)) = (2 * Real.sqrt 2 * c) / (c + a + 2) := by\n      field_simp [h\u2086\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> field_simp [h\u2086\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    rw [h\u2086\u2081\u2081] at h\u2086\u2086\n    linarith\n  have h\u2087 : a / Real.sqrt (a + b) + b / Real.sqrt (b + c) + c / Real.sqrt (c + a) \u2265 (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) := by\n    have h\u2087\u2081 : a / Real.sqrt (a + b) \u2265 (2 * Real.sqrt 2 * a) / (a + b + 2) := h\u2084\n    have h\u2087\u2082 : b / Real.sqrt (b + c) \u2265 (2 * Real.sqrt 2 * b) / (b + c + 2) := h\u2085\n    have h\u2087\u2083 : c / Real.sqrt (c + a) \u2265 (2 * Real.sqrt 2 * c) / (c + a + 2) := h\u2086\n    have h\u2087\u2084 : a / Real.sqrt (a + b) + b / Real.sqrt (b + c) + c / Real.sqrt (c + a) \u2265 (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) := by\n      linarith\n    exact h\u2087\u2084\n  have h\u2088 : (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) \u2265 3 / Real.sqrt 2 := by\n    have h\u2088\u2081 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n    have h\u2088\u2082 : 0 < 2 * Real.sqrt 2 := by positivity\n    have h\u2088\u2083 : 0 < a := by linarith\n    have h\u2088\u2084 : 0 < b := by linarith\n    have h\u2088\u2085 : 0 < c := by linarith\n    have h\u2088\u2086 : 0 < a + b + 2 := by linarith\n    have h\u2088\u2087 : 0 < b + c + 2 := by linarith\n    have h\u2088\u2088 : 0 < c + a + 2 := by linarith\n    have h\u2088\u2089 : 0 < (a + b + 2) * (b + c + 2) * (c + a + 2) := by positivity\n    have h\u2089\u2080 : a / (a + b + 2) + b / (b + c + 2) + c / (c + a + 2) \u2265 3 / 4 := by\n      have h\u2089\u2081 : 0 < a * b := by positivity\n      have h\u2089\u2082 : 0 < b * c := by positivity\n      have h\u2089\u2083 : 0 < c * a := by positivity\n      field_simp\n      rw [div_le_div_iff (by positivity) (by positivity)]\n      nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a),\n        sq_nonneg (a - 1), sq_nonneg (b - 1), sq_nonneg (c - 1)]\n    have h\u2089\u2081 : (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) = 2 * Real.sqrt 2 * (a / (a + b + 2) + b / (b + c + 2) + c / (c + a + 2)) := by\n      field_simp [h\u2088\u2083.ne', h\u2088\u2084.ne', h\u2088\u2083.ne', h\u2088\u2084.ne', h\u2088\u2083.ne', h\u2088\u2084.ne']\n      <;> ring_nf\n      <;> field_simp [h\u2088\u2083.ne', h\u2088\u2084.ne', h\u2088\u2083.ne', h\u2088\u2084.ne', h\u2088\u2083.ne', h\u2088\u2084.ne']\n      <;> ring_nf\n      <;> nlinarith\n    rw [h\u2089\u2081]\n    have h\u2089\u2082 : 2 * Real.sqrt 2 * (a / (a + b + 2) + b / (b + c + 2) + c / (c + a + 2)) \u2265 2 * Real.sqrt 2 * (3 / 4) := by\n      gcongr\n      <;> linarith\n    have h\u2089\u2083 : 2 * Real.sqrt 2 * (3 / 4) = 3 / Real.sqrt 2 * (Real.sqrt 2 * Real.sqrt 2 / 2) := by\n      field_simp [Real.sqrt_eq_iff_sq_eq, mul_comm]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq, mul_comm]\n      <;> nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    have h\u2089\u2084 : 3 / Real.sqrt 2 * (Real.sqrt 2 * Real.sqrt 2 / 2) = 3 / Real.sqrt 2 * (1) := by\n      have h\u2089\u2085 : Real.sqrt 2 * Real.sqrt 2 = 2 := by\n        nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      rw [h\u2089\u2085]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq, mul_comm]\n      <;> nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    have h\u2089\u2085 : 3 / Real.sqrt 2 * (1 : \u211d) = 3 / Real.sqrt 2 := by\n      ring\n    have h\u2089\u2086 : 2 * Real.sqrt 2 * (3 / 4) \u2265 3 / Real.sqrt 2 := by\n      have h\u2089\u2087 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n      have h\u2089\u2088 : 0 < Real.sqrt 2 ^ 2 := by positivity\n      field_simp [h\u2089\u2087.ne']\n      rw [div_le_div_iff (by positivity) (by positivity)]\n      nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    linarith\n  have h\u2089 : 3 / Real.sqrt 2 \u2264 a / Real.sqrt (a + b) + b / Real.sqrt (b + c) + c / Real.sqrt (c + a) := by\n    have h\u2089\u2081 : a / Real.sqrt (a + b) + b / Real.sqrt (b + c) + c / Real.sqrt (c + a) \u2265 (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) := h\u2087\n    have h\u2089\u2082 : (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) \u2265 3 / Real.sqrt 2 := h\u2088\n    linarith\n  linarith\n", "proof_before": "theorem algebra_abpbcpcageq3_sumaonsqrtapbgeq3onsqrt2 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c)\n    (h\u2081 : 3 \u2264 a * b + b * c + c * a) :\n    3 / Real.sqrt 2 \u2264 a / Real.sqrt (a + b) + b / Real.sqrt (b + c) + c / Real.sqrt (c + a)  :=  by\n  have h\u2082 : a + b + c \u2265 3 := by\n    have h\u2082\u2081 : (a + b + c) ^ 2 \u2265 3 * (a * b + b * c + c * a) := by\n      nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n    have h\u2082\u2082 : 3 * (a * b + b * c + c * a) \u2265 9 := by linarith\n    have h\u2082\u2083 : (a + b + c) ^ 2 \u2265 9 := by linarith\n    have h\u2082\u2084 : a + b + c \u2265 3 := by\n      nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n    exact h\u2082\u2084\n  have h\u2083 : \u2200 (x y : \u211d), 0 < x \u2192 0 < y \u2192 Real.sqrt (x + y) \u2264 (x + y + 2) / (2 * Real.sqrt 2) := by\n    intro x y hx hy\n    have h\u2084 : 0 < x + y := by linarith\n    have h\u2085 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n    have h\u2086 : 0 < Real.sqrt 2 * (x + y) := by positivity\n    have h\u2087 : 0 < Real.sqrt 2 * 2 := by positivity\n    have h\u2088 : Real.sqrt (x + y) \u2264 (x + y + 2) / (2 * Real.sqrt 2) := by\n      have h\u2088\u2081 : Real.sqrt (x + y) \u2264 (x + y + 2) / (2 * Real.sqrt 2) := by\n        rw [Real.sqrt_le_left (by positivity)]\n        field_simp [h\u2085.ne']\n        rw [le_div_iff (by positivity)]\n        nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num), sq_nonneg (x + y - 2),\n          sq_nonneg (Real.sqrt 2 * Real.sqrt (x + y) - 2),\n          Real.sq_sqrt (show 0 \u2264 x + y by positivity), sq_nonneg (x + y - 2)]\n      exact h\u2088\u2081\n    exact h\u2088\n  have h\u2084 : a / Real.sqrt (a + b) \u2265 (2 * Real.sqrt 2 * a) / (a + b + 2) := by\n    have h\u2084\u2081 : 0 < a := by linarith\n    have h\u2084\u2082 : 0 < b := by linarith\n    have h\u2084\u2083 : 0 < a + b := by linarith\n    have h\u2084\u2084 : 0 < Real.sqrt (a + b) := Real.sqrt_pos.mpr h\u2084\u2083\n    have h\u2084\u2085 : Real.sqrt (a + b) \u2264 (a + b + 2) / (2 * Real.sqrt 2) := h\u2083 a b h\u2084\u2081 h\u2084\u2082\n    have h\u2084\u2086 : a / Real.sqrt (a + b) \u2265 a / ((a + b + 2) / (2 * Real.sqrt 2)) := by\n      have h\u2084\u2087 : 0 < (a + b + 2) / (2 * Real.sqrt 2) := by positivity\n      have h\u2084\u2088 : 0 < Real.sqrt (a + b) := Real.sqrt_pos.mpr h\u2084\u2083\n      have h\u2084\u2089 : Real.sqrt (a + b) \u2264 (a + b + 2) / (2 * Real.sqrt 2) := h\u2084\u2085\n      have h\u2084\u2081\u2080 : a / Real.sqrt (a + b) \u2265 a / ((a + b + 2) / (2 * Real.sqrt 2)) := by\n        apply (div_le_div_iff (by positivity) (by positivity)).mpr\n        nlinarith [Real.sqrt_nonneg (a + b), Real.sq_sqrt (show 0 \u2264 a + b by positivity),\n          Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      exact h\u2084\u2081\u2080\n    have h\u2084\u2081\u2081 : a / ((a + b + 2) / (2 * Real.sqrt 2)) = (2 * Real.sqrt 2 * a) / (a + b + 2) := by\n      field_simp [h\u2084\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> field_simp [h\u2084\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    rw [h\u2084\u2081\u2081] at h\u2084\u2086\n    exact h\u2084\u2086\n  have h\u2085 : b / Real.sqrt (b + c) \u2265 (2 * Real.sqrt 2 * b) / (b + c + 2) := by\n    have h\u2085\u2081 : 0 < b := by linarith\n    have h\u2085\u2082 : 0 < c := by linarith\n    have h\u2085\u2083 : 0 < b + c := by linarith\n    have h\u2085\u2084 : 0 < Real.sqrt (b + c) := Real.sqrt_pos.mpr h\u2085\u2083\n    have h\u2085\u2085 : Real.sqrt (b + c) \u2264 (b + c + 2) / (2 * Real.sqrt 2) := h\u2083 b c h\u2085\u2081 h\u2085\u2082\n    have h\u2085\u2086 : b / Real.sqrt (b + c) \u2265 b / ((b + c + 2) / (2 * Real.sqrt 2)) := by\n      have h\u2085\u2087 : 0 < (b + c + 2) / (2 * Real.sqrt 2) := by positivity\n      have h\u2085\u2088 : 0 < Real.sqrt (b + c) := Real.sqrt_pos.mpr h\u2085\u2083\n      have h\u2085\u2089 : Real.sqrt (b + c) \u2264 (b + c + 2) / (2 * Real.sqrt 2) := h\u2085\u2085\n      have h\u2085\u2081\u2080 : b / Real.sqrt (b + c) \u2265 b / ((b + c + 2) / (2 * Real.sqrt 2)) := by\n        apply (div_le_div_iff (by positivity) (by positivity)).mpr\n        nlinarith [Real.sqrt_nonneg (b + c), Real.sq_sqrt (show 0 \u2264 b + c by positivity),\n          Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      exact h\u2085\u2081\u2080\n    have h\u2085\u2081\u2081 : b / ((b + c + 2) / (2 * Real.sqrt 2)) = (2 * Real.sqrt 2 * b) / (b + c + 2) := by\n      field_simp [h\u2085\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> field_simp [h\u2085\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    rw [h\u2085\u2081\u2081] at h\u2085\u2086\n    exact h\u2085\u2086\n  have h\u2086 : c / Real.sqrt (c + a) \u2265 (2 * Real.sqrt 2 * c) / (c + a + 2) := by\n    have h\u2086\u2081 : 0 < c := by linarith\n    have h\u2086\u2082 : 0 < a := by linarith\n    have h\u2086\u2083 : 0 < c + a := by linarith\n    have h\u2086\u2084 : 0 < Real.sqrt (c + a) := Real.sqrt_pos.mpr h\u2086\u2083\n    have h\u2086\u2085 : Real.sqrt (c + a) \u2264 (c + a + 2) / (2 * Real.sqrt 2) := h\u2083 c a h\u2086\u2081 h\u2086\u2082\n    have h\u2086\u2086 : c / Real.sqrt (c + a) \u2265 c / ((c + a + 2) / (2 * Real.sqrt 2)) := by\n      have h\u2086\u2087 : 0 < (c + a + 2) / (2 * Real.sqrt 2) := by positivity\n      have h\u2086\u2088 : 0 < Real.sqrt (c + a) := Real.sqrt_pos.mpr h\u2086\u2083\n      have h\u2086\u2089 : Real.sqrt (c + a) \u2264 (c + a + 2) / (2 * Real.sqrt 2) := h\u2086\u2085\n      have h\u2086\u2081\u2080 : c / Real.sqrt (c + a) \u2265 c / ((c + a + 2) / (2 * Real.sqrt 2)) := by\n        apply (div_le_div_iff (by positivity) (by positivity)).mpr\n        nlinarith [Real.sqrt_nonneg (c + a), Real.sq_sqrt (show 0 \u2264 c + a by positivity),\n          Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      exact h\u2086\u2081\u2080\n    have h\u2086\u2081\u2081 : c / ((c + a + 2) / (2 * Real.sqrt 2)) = (2 * Real.sqrt 2 * c) / (c + a + 2) := by\n      field_simp [h\u2086\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> field_simp [h\u2086\u2083.ne', Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    rw [h\u2086\u2081\u2081] at h\u2086\u2086\n    linarith\n  have h\u2087 : a / Real.sqrt (a + b) + b / Real.sqrt (b + c) + c / Real.sqrt (c + a) \u2265 (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) := by\n    have h\u2087\u2081 : a / Real.sqrt (a + b) \u2265 (2 * Real.sqrt 2 * a) / (a + b + 2) := h\u2084\n    have h\u2087\u2082 : b / Real.sqrt (b + c) \u2265 (2 * Real.sqrt 2 * b) / (b + c + 2) := h\u2085\n    have h\u2087\u2083 : c / Real.sqrt (c + a) \u2265 (2 * Real.sqrt 2 * c) / (c + a + 2) := h\u2086\n    have h\u2087\u2084 : a / Real.sqrt (a + b) + b / Real.sqrt (b + c) + c / Real.sqrt (c + a) \u2265 (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) := by\n      linarith\n    exact h\u2087\u2084\n  have h\u2088 : (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) \u2265 3 / Real.sqrt 2 := by\n    have h\u2088\u2081 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n    have h\u2088\u2082 : 0 < 2 * Real.sqrt 2 := by positivity\n    have h\u2088\u2083 : 0 < a := by linarith\n    have h\u2088\u2084 : 0 < b := by linarith\n    have h\u2088\u2085 : 0 < c := by linarith\n    have h\u2088\u2086 : 0 < a + b + 2 := by linarith\n    have h\u2088\u2087 : 0 < b + c + 2 := by linarith\n    have h\u2088\u2088 : 0 < c + a + 2 := by linarith\n    have h\u2088\u2089 : 0 < (a + b + 2) * (b + c + 2) * (c + a + 2) := by positivity\n    have h\u2089\u2080 : a / (a + b + 2) + b / (b + c + 2) + c / (c + a + 2) \u2265 3 / 4 := by\n      have h\u2089\u2081 : 0 < a * b := by positivity\n      have h\u2089\u2082 : 0 < b * c := by positivity\n      have h\u2089\u2083 : 0 < c * a := by positivity\n      field_simp\n      rw [div_le_div_iff (by positivity) (by positivity)]\n      nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a),\n        sq_nonneg (a - 1), sq_nonneg (b - 1), sq_nonneg (c - 1)]\n    have h\u2089\u2081 : (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) = 2 * Real.sqrt 2 * (a / (a + b + 2) + b / (b + c + 2) + c / (c + a + 2)) := by\n      field_simp [h\u2088\u2083.ne', h\u2088\u2084.ne', h\u2088\u2083.ne', h\u2088\u2084.ne', h\u2088\u2083.ne', h\u2088\u2084.ne']\n      <;> ring_nf\n      <;> field_simp [h\u2088\u2083.ne', h\u2088\u2084.ne', h\u2088\u2083.ne', h\u2088\u2084.ne', h\u2088\u2083.ne', h\u2088\u2084.ne']\n      <;> ring_nf\n      <;> nlinarith\n    rw [h\u2089\u2081]\n    have h\u2089\u2082 : 2 * Real.sqrt 2 * (a / (a + b + 2) + b / (b + c + 2) + c / (c + a + 2)) \u2265 2 * Real.sqrt 2 * (3 / 4) := by\n      gcongr\n      <;> linarith\n    have h\u2089\u2083 : 2 * Real.sqrt 2 * (3 / 4) = 3 / Real.sqrt 2 * (Real.sqrt 2 * Real.sqrt 2 / 2) := by\n      field_simp [Real.sqrt_eq_iff_sq_eq, mul_comm]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq, mul_comm]\n      <;> nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    have h\u2089\u2084 : 3 / Real.sqrt 2 * (Real.sqrt 2 * Real.sqrt 2 / 2) = 3 / Real.sqrt 2 * (1) := by\n      have h\u2089\u2085 : Real.sqrt 2 * Real.sqrt 2 = 2 := by\n        nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      rw [h\u2089\u2085]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq, mul_comm]\n      <;> nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    have h\u2089\u2085 : 3 / Real.sqrt 2 * (1 : \u211d) = 3 / Real.sqrt 2 := by\n      ring\n    have h\u2089\u2086 : 2 * Real.sqrt 2 * (3 / 4) \u2265 3 / Real.sqrt 2 := by\n      have h\u2089\u2087 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n      have h\u2089\u2088 : 0 < Real.sqrt 2 ^ 2 := by positivity\n      field_simp [h\u2089\u2087.ne']\n      rw [div_le_div_iff (by positivity) (by positivity)]\n      nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n    linarith\n  have h\u2089 : 3 / Real.sqrt 2 \u2264 a / Real.sqrt (a + b) + b / Real.sqrt (b + c) + c / Real.sqrt (c + a) := by\n    have h\u2089\u2081 : a / Real.sqrt (a + b) + b / Real.sqrt (b + c) + c / Real.sqrt (c + a) \u2265 (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) := h\u2087\n    have h\u2089\u2082 : (2 * Real.sqrt 2 * a) / (a + b + 2) + (2 * Real.sqrt 2 * b) / (b + c + 2) + (2 * Real.sqrt 2 * c) / (c + a + 2) \u2265 3 / Real.sqrt 2 := h\u2088\n    linarith\n  linarith\n"}
{"mode": "clear", "steps_before": 46, "alphabet": 24, "compiles": true, "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1238, "chars_after": 24, "seconds": 132.3, "id": "mathd_algebra_33", "proof_after": "theorem mathd_algebra_33 (x y z : \u211d) (h\u2080 : x \u2260 0) (h\u2081 : 2 * x = 5 * y) (h\u2082 : 7 * y = 10 * z) :\n    z / x = 7 / 25  :=  by\n  field_simp\n  linarith\n", "proof_before": "theorem mathd_algebra_33 (x y z : \u211d) (h\u2080 : x \u2260 0) (h\u2081 : 2 * x = 5 * y) (h\u2082 : 7 * y = 10 * z) :\n    z / x = 7 / 25  :=  by\n  have h\u2083 : 14 * x = 35 * y := by\n    have h\u2083\u2081 : 14 * x = 35 * y := by\n      have h\u2083\u2082 : 2 * x = 5 * y := h\u2081\n      have h\u2083\u2083 : 7 * (2 * x) = 7 * (5 * y) := by rw [h\u2083\u2082]\n      have h\u2083\u2084 : 7 * (2 * x) = 14 * x := by ring\n      have h\u2083\u2085 : 7 * (5 * y) = 35 * y := by ring\n      linarith\n    exact h\u2083\u2081\n  have h\u2084 : 35 * y = 50 * z := by\n    have h\u2084\u2081 : 7 * y = 10 * z := h\u2082\n    have h\u2084\u2082 : 5 * (7 * y) = 5 * (10 * z) := by rw [h\u2084\u2081]\n    have h\u2084\u2083 : 5 * (7 * y) = 35 * y := by ring\n    have h\u2084\u2084 : 5 * (10 * z) = 50 * z := by ring\n    linarith\n  have h\u2085 : 14 * x = 50 * z := by\n    linarith\n  have h\u2086 : z = (7 : \u211d) / 25 * x := by\n    have h\u2086\u2081 : 14 * x = 50 * z := h\u2085\n    have h\u2086\u2082 : z = (7 : \u211d) / 25 * x := by\n      have h\u2086\u2083 : 50 * z = 14 * x := by linarith\n      have h\u2086\u2084 : z = (14 : \u211d) / 50 * x := by\n        apply mul_left_cancel\u2080 (show (50 : \u211d) \u2260 0 by norm_num)\n        linarith\n      have h\u2086\u2085 : z = (7 : \u211d) / 25 * x := by\n        rw [h\u2086\u2084]\n        <;> ring_nf at *\n        <;> linarith\n      exact h\u2086\u2085\n    exact h\u2086\u2082\n  have h\u2087 : z / x = 7 / 25 := by\n    rw [h\u2086]\n    have h\u2087\u2081 : (7 : \u211d) / 25 * x / x = 7 / 25 := by\n      have h\u2087\u2082 : x \u2260 0 := h\u2080\n      field_simp [h\u2087\u2082]\n      <;> ring_nf\n      <;> field_simp [h\u2087\u2082]\n      <;> ring_nf\n    rw [h\u2087\u2081]\n  apply h\u2087\n"}
{"mode": "clear", "steps_before": 161, "alphabet": 42, "compiles": false, "id": "mathd_algebra_293", "proof_after": "theorem mathd_algebra_293 (x : NNReal) :\n    Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = 36 * x * Real.sqrt (35 * x)  :=  by\n  have h\u2080 : Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) := by\n    have h\u2081 : Real.sqrt (60 * (x : \u211d)) * Real.sqrt (12 * (x : \u211d)) * Real.sqrt (63 * (x : \u211d)) = Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) := by\n      have h\u2082 : 0 \u2264 (60 : \u211d) * (x : \u211d) := by positivity\n      have h\u2083 : 0 \u2264 (12 : \u211d) * (x : \u211d) := by positivity\n      have h\u2084 : 0 \u2264 (63 : \u211d) * (x : \u211d) := by positivity\n      have h\u2085 : 0 \u2264 (60 : \u211d) * (x : \u211d) * (12 * (x : \u211d)) := by positivity\n      have h\u2086 : 0 \u2264 (60 : \u211d) * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d)) := by positivity\n      have h\u2087 : Real.sqrt (60 * (x : \u211d)) * Real.sqrt (12 * (x : \u211d)) = Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d))) := by\n        rw [\u2190 Real.sqrt_mul (by positivity)]\n      rw [h\u2087]\n      have h\u2088 : Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d))) * Real.sqrt (63 * (x : \u211d)) = Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) := by\n        rw [\u2190 Real.sqrt_mul (by positivity)]\n      rw [h\u2088]\n    have h\u2089 : Real.sqrt (60 * x) = Real.sqrt (60 * (x : \u211d)) := by simp [Real.sqrt_eq_iff_sq_eq]\n    have h\u2081\u2080 : Real.sqrt (12 * x) = Real.sqrt (12 * (x : \u211d)) := by simp [Real.sqrt_eq_iff_sq_eq]\n    have h\u2081\u2081 : Real.sqrt (63 * x) = Real.sqrt (63 * (x : \u211d)) := by simp [Real.sqrt_eq_iff_sq_eq]\n    rw [h\u2089, h\u2081\u2080, h\u2081\u2081]\n    exact h\u2081\n  have h\u2081 : Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) = Real.sqrt (45360 * (x : \u211d)^3) := by\n    have h\u2081\u2080 : (60 : \u211d) * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d)) = 45360 * (x : \u211d) ^ 3 := by\n      ring_nf\n      <;> field_simp\n      <;> ring_nf\n      <;> norm_cast\n      <;> simp [mul_assoc]\n      <;> ring_nf\n      <;> norm_num\n      <;> nlinarith\n    rw [h\u2081\u2080]\n    <;>\n    simp [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2082 : Real.sqrt (45360 * (x : \u211d)^3) = Real.sqrt ((36 : \u211d)^2 * 35 * (x : \u211d)^3) := by\n    have h\u2082\u2081 : (45360 : \u211d) * (x : \u211d) ^ 3 = (36 : \u211d) ^ 2 * 35 * (x : \u211d) ^ 3 := by\n      ring_nf\n      <;> norm_num\n      <;>\n      simp [mul_assoc]\n      <;>\n      ring_nf\n      <;>\n      norm_num\n      <;>\n      nlinarith\n    rw [h\u2082\u2081]\n    <;>\n    simp [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2083 : Real.sqrt ((36 : \u211d)^2 * 35 * (x : \u211d)^3) = 36 * Real.sqrt (35 * (x : \u211d)^3) := by\n    have h\u2083\u2081 : 0 \u2264 (36 : \u211d) := by positivity\n    have h\u2083\u2082 : 0 \u2264 (35 : \u211d) * (x : \u211d) ^ 3 := by positivity\n    have h\u2083\u2083 : 0 \u2264 (x : \u211d) := by positivity\n    have h\u2083\u2084 : 0 \u2264 (x : \u211d) ^ 3 := by positivity\n    have h\u2083\u2085 : 0 \u2264 (35 : \u211d) * (x : \u211d) ^ 3 := by positivity\n    have h\u2083\u2086 : Real.sqrt ((36 : \u211d) ^ 2 * 35 * (x : \u211d) ^ 3) = 36 * Real.sqrt (35 * (x : \u211d) ^ 3) := by\n      rw [Real.sqrt_eq_iff_sq_eq (by positivity) (by positivity)]\n      have h\u2083\u2087 : (36 : \u211d) ^ 2 = 1296 := by norm_num\n      have h\u2083\u2088 : (36 : \u211d) ^ 2 * 35 * (x : \u211d) ^ 3 = (36 : \u211d) ^ 2 * (35 * (x : \u211d) ^ 3) := by ring\n      rw [h\u2083\u2088]\n      have h\u2083\u2089 : Real.sqrt (35 * (x : \u211d) ^ 3) \u2265 0 := Real.sqrt_nonneg _\n      nlinarith [Real.sq_sqrt (by positivity : (0 : \u211d) \u2264 35 * (x : \u211d) ^ 3)]\n    rw [h\u2083\u2086]\n    <;>\n    simp [h\u2083\u2081, h\u2083\u2082, h\u2083\u2083, h\u2083\u2084, h\u2083\u2085]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2084 : Real.sqrt (35 * (x : \u211d)^3) = (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n    have h\u2084\u2081 : 0 \u2264 (x : \u211d) := by exact_mod_cast x.prop\n    have h\u2084\u2082 : Real.sqrt (35 * (x : \u211d) ^ 3) = (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n      have h\u2084\u2083 : Real.sqrt (35 * (x : \u211d) ^ 3) = Real.sqrt (35 * (x : \u211d) * (x : \u211d) ^ 2) := by\n        ring_nf\n        <;> field_simp <;> ring_nf\n      rw [h\u2084\u2083]\n      have h\u2084\u2084 : Real.sqrt (35 * (x : \u211d) * (x : \u211d) ^ 2) = (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n        have h\u2084\u2085 : 0 \u2264 (x : \u211d) ^ 2 := by positivity\n        have h\u2084\u2086 : 0 \u2264 35 * (x : \u211d) := by positivity\n        have h\u2084\u2087 : 0 \u2264 35 * (x : \u211d) * (x : \u211d) ^ 2 := by positivity\n        have h\u2084\u2088 : Real.sqrt (35 * (x : \u211d) * (x : \u211d) ^ 2) = (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n          rw [Real.sqrt_eq_iff_sq_eq (by positivity) (by positivity)]\n          nlinarith [Real.sq_sqrt (by positivity : 0 \u2264 (35 : \u211d) * (x : \u211d)), mul_self_nonneg (x : \u211d),\n            mul_nonneg h\u2084\u2081 (sq_nonneg (x : \u211d))]\n        rw [h\u2084\u2088]\n      rw [h\u2084\u2084]\n    rw [h\u2084\u2082]\n    <;>\n    simp_all [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2085 : 36 * Real.sqrt (35 * (x : \u211d)^3) = 36 * (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n    rw [h\u2084]\n    <;> ring_nf\n    <;> field_simp\n    <;> ring_nf\n  have h\u2086 : Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) = 36 * (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n    rw [h\u2081]\n    rw [h\u2082]\n    rw [h\u2083]\n    rw [h\u2085]\n    <;>\n    simp_all [mul_assoc]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2087 : Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = 36 * x * Real.sqrt (35 * x) := by\n    have h\u2087\u2081 : Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) := by\n      exact h\u2080\n    rw [h\u2087\u2081]\n    have h\u2087\u2082 : Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) = 36 * (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n      exact h\u2086\n    rw [h\u2087\u2082]\n    <;>\n    norm_cast\n    <;>\n    field_simp [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    nlinarith\n  exact h\u2087\n", "proof_before": "theorem mathd_algebra_293 (x : NNReal) :\n    Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = 36 * x * Real.sqrt (35 * x)  :=  by\n  have h\u2080 : Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) := by\n    have h\u2081 : Real.sqrt (60 * (x : \u211d)) * Real.sqrt (12 * (x : \u211d)) * Real.sqrt (63 * (x : \u211d)) = Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) := by\n      have h\u2082 : 0 \u2264 (60 : \u211d) * (x : \u211d) := by positivity\n      have h\u2083 : 0 \u2264 (12 : \u211d) * (x : \u211d) := by positivity\n      have h\u2084 : 0 \u2264 (63 : \u211d) * (x : \u211d) := by positivity\n      have h\u2085 : 0 \u2264 (60 : \u211d) * (x : \u211d) * (12 * (x : \u211d)) := by positivity\n      have h\u2086 : 0 \u2264 (60 : \u211d) * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d)) := by positivity\n      have h\u2087 : Real.sqrt (60 * (x : \u211d)) * Real.sqrt (12 * (x : \u211d)) = Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d))) := by\n        rw [\u2190 Real.sqrt_mul (by positivity)]\n      rw [h\u2087]\n      have h\u2088 : Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d))) * Real.sqrt (63 * (x : \u211d)) = Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) := by\n        rw [\u2190 Real.sqrt_mul (by positivity)]\n      rw [h\u2088]\n    have h\u2089 : Real.sqrt (60 * x) = Real.sqrt (60 * (x : \u211d)) := by simp [Real.sqrt_eq_iff_sq_eq]\n    have h\u2081\u2080 : Real.sqrt (12 * x) = Real.sqrt (12 * (x : \u211d)) := by simp [Real.sqrt_eq_iff_sq_eq]\n    have h\u2081\u2081 : Real.sqrt (63 * x) = Real.sqrt (63 * (x : \u211d)) := by simp [Real.sqrt_eq_iff_sq_eq]\n    rw [h\u2089, h\u2081\u2080, h\u2081\u2081]\n    exact h\u2081\n  have h\u2081 : Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) = Real.sqrt (45360 * (x : \u211d)^3) := by\n    have h\u2081\u2080 : (60 : \u211d) * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d)) = 45360 * (x : \u211d) ^ 3 := by\n      ring_nf\n      <;> field_simp\n      <;> ring_nf\n      <;> norm_cast\n      <;> simp [mul_assoc]\n      <;> ring_nf\n      <;> norm_num\n      <;> nlinarith\n    rw [h\u2081\u2080]\n    <;>\n    simp [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2082 : Real.sqrt (45360 * (x : \u211d)^3) = Real.sqrt ((36 : \u211d)^2 * 35 * (x : \u211d)^3) := by\n    have h\u2082\u2081 : (45360 : \u211d) * (x : \u211d) ^ 3 = (36 : \u211d) ^ 2 * 35 * (x : \u211d) ^ 3 := by\n      ring_nf\n      <;> norm_num\n      <;>\n      simp [mul_assoc]\n      <;>\n      ring_nf\n      <;>\n      norm_num\n      <;>\n      nlinarith\n    rw [h\u2082\u2081]\n    <;>\n    simp [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2083 : Real.sqrt ((36 : \u211d)^2 * 35 * (x : \u211d)^3) = 36 * Real.sqrt (35 * (x : \u211d)^3) := by\n    have h\u2083\u2081 : 0 \u2264 (36 : \u211d) := by positivity\n    have h\u2083\u2082 : 0 \u2264 (35 : \u211d) * (x : \u211d) ^ 3 := by positivity\n    have h\u2083\u2083 : 0 \u2264 (x : \u211d) := by positivity\n    have h\u2083\u2084 : 0 \u2264 (x : \u211d) ^ 3 := by positivity\n    have h\u2083\u2085 : 0 \u2264 (35 : \u211d) * (x : \u211d) ^ 3 := by positivity\n    have h\u2083\u2086 : Real.sqrt ((36 : \u211d) ^ 2 * 35 * (x : \u211d) ^ 3) = 36 * Real.sqrt (35 * (x : \u211d) ^ 3) := by\n      rw [Real.sqrt_eq_iff_sq_eq (by positivity) (by positivity)]\n      have h\u2083\u2087 : (36 : \u211d) ^ 2 = 1296 := by norm_num\n      have h\u2083\u2088 : (36 : \u211d) ^ 2 * 35 * (x : \u211d) ^ 3 = (36 : \u211d) ^ 2 * (35 * (x : \u211d) ^ 3) := by ring\n      rw [h\u2083\u2088]\n      have h\u2083\u2089 : Real.sqrt (35 * (x : \u211d) ^ 3) \u2265 0 := Real.sqrt_nonneg _\n      nlinarith [Real.sq_sqrt (by positivity : (0 : \u211d) \u2264 35 * (x : \u211d) ^ 3)]\n    rw [h\u2083\u2086]\n    <;>\n    simp [h\u2083\u2081, h\u2083\u2082, h\u2083\u2083, h\u2083\u2084, h\u2083\u2085]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2084 : Real.sqrt (35 * (x : \u211d)^3) = (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n    have h\u2084\u2081 : 0 \u2264 (x : \u211d) := by exact_mod_cast x.prop\n    have h\u2084\u2082 : Real.sqrt (35 * (x : \u211d) ^ 3) = (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n      have h\u2084\u2083 : Real.sqrt (35 * (x : \u211d) ^ 3) = Real.sqrt (35 * (x : \u211d) * (x : \u211d) ^ 2) := by\n        ring_nf\n        <;> field_simp <;> ring_nf\n      rw [h\u2084\u2083]\n      have h\u2084\u2084 : Real.sqrt (35 * (x : \u211d) * (x : \u211d) ^ 2) = (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n        have h\u2084\u2085 : 0 \u2264 (x : \u211d) ^ 2 := by positivity\n        have h\u2084\u2086 : 0 \u2264 35 * (x : \u211d) := by positivity\n        have h\u2084\u2087 : 0 \u2264 35 * (x : \u211d) * (x : \u211d) ^ 2 := by positivity\n        have h\u2084\u2088 : Real.sqrt (35 * (x : \u211d) * (x : \u211d) ^ 2) = (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n          rw [Real.sqrt_eq_iff_sq_eq (by positivity) (by positivity)]\n          nlinarith [Real.sq_sqrt (by positivity : 0 \u2264 (35 : \u211d) * (x : \u211d)), mul_self_nonneg (x : \u211d),\n            mul_nonneg h\u2084\u2081 (sq_nonneg (x : \u211d))]\n        rw [h\u2084\u2088]\n      rw [h\u2084\u2084]\n    rw [h\u2084\u2082]\n    <;>\n    simp_all [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2085 : 36 * Real.sqrt (35 * (x : \u211d)^3) = 36 * (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n    rw [h\u2084]\n    <;> ring_nf\n    <;> field_simp\n    <;> ring_nf\n  have h\u2086 : Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) = 36 * (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n    rw [h\u2081]\n    rw [h\u2082]\n    rw [h\u2083]\n    rw [h\u2085]\n    <;>\n    simp_all [mul_assoc]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2087 : Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = 36 * x * Real.sqrt (35 * x) := by\n    have h\u2087\u2081 : Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) := by\n      exact h\u2080\n    rw [h\u2087\u2081]\n    have h\u2087\u2082 : Real.sqrt (60 * (x : \u211d) * (12 * (x : \u211d)) * (63 * (x : \u211d))) = 36 * (x : \u211d) * Real.sqrt (35 * (x : \u211d)) := by\n      exact h\u2086\n    rw [h\u2087\u2082]\n    <;>\n    norm_cast\n    <;>\n    field_simp [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    nlinarith\n  exact h\u2087\n"}
{"mode": "clear", "steps_before": 242, "alphabet": 74, "compiles": false, "id": "amc12a_2008_p25", "proof_after": "theorem amc12a_2008_p25 (a b : \u2115 \u2192 \u211d) (h\u2080 : \u2200 n, a (n + 1) = Real.sqrt 3 * a n - b n)\n    (h\u2081 : \u2200 n, b (n + 1) = Real.sqrt 3 * b n + a n) (h\u2082 : a 100 = 2) (h\u2083 : b 100 = 4) :\n    a 1 + b 1 = 1 / 2 ^ 98  :=  by\n  have h\u2084 : \u2200 n, a (n + 3) = -8 * b n := by\n    intro n\n    have h\u2084\u2081 : a (n + 3) = -8 * b n := by\n      have h\u2084\u2082 : a (n + 3) = Real.sqrt 3 * a (n + 2) - b (n + 2) := by rw [h\u2080]\n      rw [h\u2084\u2082]\n      have h\u2084\u2083 : a (n + 2) = Real.sqrt 3 * a (n + 1) - b (n + 1) := by rw [h\u2080]\n      have h\u2084\u2084 : b (n + 2) = Real.sqrt 3 * b (n + 1) + a (n + 1) := by rw [h\u2081]\n      rw [h\u2084\u2083, h\u2084\u2084]\n      have h\u2084\u2085 : a (n + 1) = Real.sqrt 3 * a n - b n := by rw [h\u2080]\n      have h\u2084\u2086 : b (n + 1) = Real.sqrt 3 * b n + a n := by rw [h\u2081]\n      rw [h\u2084\u2085, h\u2084\u2086]\n      ring_nf\n      <;>\n      simp [Real.sqrt_eq_iff_sq_eq, pow_three, mul_assoc, mul_comm, mul_left_comm]\n      <;>\n      ring_nf at *\n      <;>\n      nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show 0 \u2264 3 by norm_num)]\n    exact h\u2084\u2081\n    <;>\n    simp_all\n    <;>\n    linarith\n  have h\u2085 : \u2200 n, b (n + 3) = 8 * a n := by\n    intro n\n    have h\u2085\u2081 : b (n + 3) = 8 * a n := by\n      have h\u2085\u2082 : b (n + 3) = Real.sqrt 3 * b (n + 2) + a (n + 2) := by rw [h\u2081]\n      rw [h\u2085\u2082]\n      have h\u2085\u2083 : b (n + 2) = Real.sqrt 3 * b (n + 1) + a (n + 1) := by rw [h\u2081]\n      have h\u2085\u2084 : a (n + 2) = Real.sqrt 3 * a (n + 1) - b (n + 1) := by rw [h\u2080]\n      rw [h\u2085\u2083, h\u2085\u2084]\n      have h\u2085\u2085 : b (n + 1) = Real.sqrt 3 * b n + a n := by rw [h\u2081]\n      have h\u2085\u2086 : a (n + 1) = Real.sqrt 3 * a n - b n := by rw [h\u2080]\n      rw [h\u2085\u2085, h\u2085\u2086]\n      ring_nf\n      <;>\n      simp [Real.sqrt_eq_iff_sq_eq, pow_three, mul_assoc, mul_comm, mul_left_comm]\n      <;>\n      ring_nf at *\n      <;>\n      nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show 0 \u2264 3 by norm_num)]\n    exact h\u2085\u2081\n    <;>\n    simp_all\n    <;>\n    linarith\n  have h\u2086 : \u2200 n, a (n + 6) = -64 * a n := by\n    intro n\n    have h\u2086\u2081 : a (n + 6) = -64 * a n := by\n      have h\u2086\u2082 : a (n + 6) = a ((n + 3) + 3) := by ring_nf\n      rw [h\u2086\u2082]\n      have h\u2086\u2083 : a ((n + 3) + 3) = -8 * b (n + 3) := by rw [h\u2084]\n      rw [h\u2086\u2083]\n      have h\u2086\u2084 : b (n + 3) = 8 * a n := by rw [h\u2085]\n      rw [h\u2086\u2084]\n      <;> ring_nf\n      <;> simp [h\u2084, h\u2085]\n      <;> ring_nf\n      <;> linarith\n    exact h\u2086\u2081\n  have h\u2087 : \u2200 n, b (n + 6) = -64 * b n := by\n    intro n\n    have h\u2087\u2081 : b (n + 6) = b ((n + 3) + 3) := by ring_nf\n    rw [h\u2087\u2081]\n    have h\u2087\u2082 : b ((n + 3) + 3) = 8 * a (n + 3) := by rw [h\u2085]\n    rw [h\u2087\u2082]\n    have h\u2087\u2083 : a (n + 3) = -8 * b n := by rw [h\u2084]\n    rw [h\u2087\u2083]\n    <;> ring_nf\n    <;> simp [h\u2084, h\u2085]\n    <;> ring_nf\n    <;> linarith\n  have h\u2088 : a 4 = 1 / (2 ^ 95 : \u211d) := by\n    have h\u2088\u2081 : a 100 = 2 := h\u2082\n    have h\u2088\u2082 : b 100 = 4 := h\u2083\n    have h\u2088\u2083 : a 100 = (2 : \u211d) ^ 96 * a 4 := by\n      have h\u2088\u2084 : a 100 = (2 : \u211d) ^ 96 * a 4 := by\n        have h\u2088\u2085 : a 100 = (2 : \u211d) ^ 96 * a 4 := by\n          have h\u2088\u2086 : a 100 = (-64 : \u211d) ^ 16 * a 4 := by\n            have h\u2088\u2087 : a 100 = (-64 : \u211d) ^ 16 * a 4 := by\n              norm_num [h\u2086, h\u2087, pow_succ, mul_assoc] at h\u2088\u2081 h\u2088\u2082 \u22a2\n              <;>\n                (try ring_nf at * <;> try norm_num at * <;> try linarith)\n              <;>\n                (try ring_nf at * <;> try norm_num at * <;> try linarith)\n              <;>\n                (try ring_nf at * <;> try norm_num at * <;> try linarith)\n              <;>\n                (try ring_nf at * <;> try norm_num at * <;> try linarith)\n            exact h\u2088\u2087\n          have h\u2088\u2088 : (-64 : \u211d) ^ 16 = (2 : \u211d) ^ 96 := by\n            norm_num [pow_succ, mul_assoc]\n            <;>\n            ring_nf\n            <;>\n            norm_num\n            <;>\n            linarith\n          rw [h\u2088\u2086, h\u2088\u2088]\n          <;>\n          ring_nf\n          <;>\n          norm_num\n          <;>\n          linarith\n        exact h\u2088\u2085\n      exact h\u2088\u2084\n    have h\u2088\u2089 : a 4 = 1 / (2 ^ 95 : \u211d) := by\n      have h\u2088\u2081\u2080 : (2 : \u211d) ^ 96 * a 4 = 2 := by\n        linarith\n      have h\u2088\u2081\u2081 : a 4 = 1 / (2 ^ 95 : \u211d) := by\n        have h\u2088\u2081\u2082 : a 4 = 1 / (2 ^ 95 : \u211d) := by\n          field_simp at h\u2088\u2081\u2080 \u22a2\n          ring_nf at h\u2088\u2081\u2080 \u22a2\n          <;> nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n        exact h\u2088\u2081\u2082\n      exact h\u2088\u2081\u2081\n    exact h\u2088\u2089\n  have h\u2089 : b 4 = 1 / (2 ^ 94 : \u211d) := by\n    have h\u2089\u2081 : b 100 = (2 : \u211d) ^ 96 * b 4 := by\n      have h\u2089\u2082 : b 100 = (-64 : \u211d) ^ 16 * b 4 := by\n        norm_num [h\u2086, h\u2087, pow_succ, mul_assoc] at h\u2082 h\u2083 \u22a2\n        <;>\n        (try ring_nf at * <;> try norm_num at * <;> try linarith)\n        <;>\n        (try ring_nf at * <;> try norm_num at * <;> try linarith)\n        <;>\n        (try ring_nf at * <;> try norm_num at * <;> try linarith)\n        <;>\n        (try ring_nf at * <;> try norm_num at * <;> try linarith)\n      have h\u2089\u2083 : (-64 : \u211d) ^ 16 = (2 : \u211d) ^ 96 := by\n        norm_num [pow_succ, mul_assoc]\n        <;>\n        ring_nf\n        <;>\n        norm_num\n        <;>\n        linarith\n      rw [h\u2089\u2082, h\u2089\u2083]\n      <;>\n      ring_nf\n      <;>\n      norm_num\n      <;>\n      linarith\n    have h\u2089\u2084 : b 4 = 1 / (2 ^ 94 : \u211d) := by\n      have h\u2089\u2085 : (2 : \u211d) ^ 96 * b 4 = 4 := by\n        linarith\n      have h\u2089\u2086 : b 4 = 1 / (2 ^ 94 : \u211d) := by\n        field_simp at h\u2089\u2085 \u22a2\n        ring_nf at h\u2089\u2085 \u22a2\n        <;> nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n      exact h\u2089\u2086\n    exact h\u2089\u2084\n  have h\u2081\u2080 : a 1 = 1 / (2 ^ 97 : \u211d) := by\n    have h\u2081\u2080\u2081 : a 4 = -8 * b 1 := by\n      have h\u2081\u2080\u2082 := h\u2084 1\n      have h\u2081\u2080\u2083 := h\u2085 1\n      norm_num at h\u2081\u2080\u2082 h\u2081\u2080\u2083 \u22a2\n      <;> linarith\n    have h\u2081\u2080\u2084 : a 4 = 1 / (2 ^ 95 : \u211d) := h\u2088\n    have h\u2081\u2080\u2085 : b 1 = -1 / (2 ^ 98 : \u211d) := by\n      have h\u2081\u2080\u2085\u2081 : a 4 = -8 * b 1 := h\u2081\u2080\u2081\n      rw [h\u2081\u2080\u2084] at h\u2081\u2080\u2085\u2081\n      ring_nf at h\u2081\u2080\u2085\u2081 \u22a2\n      <;> nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n    have h\u2081\u2080\u2086 : b 4 = 8 * a 1 := by\n      have h\u2081\u2080\u2086\u2081 := h\u2085 1\n      have h\u2081\u2080\u2086\u2082 := h\u2084 1\n      norm_num at h\u2081\u2080\u2086\u2081 h\u2081\u2080\u2086\u2082 \u22a2\n      <;> linarith\n    have h\u2081\u2080\u2087 : b 4 = 1 / (2 ^ 94 : \u211d) := h\u2089\n    have h\u2081\u2080\u2088 : a 1 = 1 / (2 ^ 97 : \u211d) := by\n      rw [h\u2081\u2080\u2086] at h\u2081\u2080\u2087\n      have h\u2081\u2080\u2088 : (8 : \u211d) * a 1 = 1 / (2 ^ 94 : \u211d) := by linarith\n      ring_nf at h\u2081\u2080\u2088 \u22a2\n      <;> nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n    exact h\u2081\u2080\u2088\n  have h\u2081\u2081 : b 1 = -1 / (2 ^ 98 : \u211d) := by\n    have h\u2081\u2081\u2081 : a 4 = -8 * b 1 := by\n      have h\u2081\u2081\u2082 := h\u2084 1\n      have h\u2081\u2081\u2083 := h\u2085 1\n      norm_num at h\u2081\u2081\u2082 h\u2081\u2081\u2083 \u22a2\n      <;> linarith\n    have h\u2081\u2081\u2084 : a 4 = 1 / (2 ^ 95 : \u211d) := h\u2088\n    have h\u2081\u2081\u2085 : b 1 = -1 / (2 ^ 98 : \u211d) := by\n      rw [h\u2081\u2081\u2084] at h\u2081\u2081\u2081\n      ring_nf at h\u2081\u2081\u2081 \u22a2\n      <;> nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n    exact h\u2081\u2081\u2085\n  have h\u2081\u2082 : a 1 + b 1 = 1 / (2 ^ 98 : \u211d) := by\n    rw [h\u2081\u2080, h\u2081\u2081]\n    <;> ring_nf <;> field_simp <;> ring_nf <;> norm_num\n    <;>\n    linarith\n  rw [h\u2081\u2082]\n  <;>\n  norm_num\n  <;>\n  simp_all [pow_add, pow_mul, mul_assoc]\n  <;>\n  ring_nf at *\n  <;>\n  norm_num at *\n  <;>\n  linarith\n", "proof_before": "theorem amc12a_2008_p25 (a b : \u2115 \u2192 \u211d) (h\u2080 : \u2200 n, a (n + 1) = Real.sqrt 3 * a n - b n)\n    (h\u2081 : \u2200 n, b (n + 1) = Real.sqrt 3 * b n + a n) (h\u2082 : a 100 = 2) (h\u2083 : b 100 = 4) :\n    a 1 + b 1 = 1 / 2 ^ 98  :=  by\n  have h\u2084 : \u2200 n, a (n + 3) = -8 * b n := by\n    intro n\n    have h\u2084\u2081 : a (n + 3) = -8 * b n := by\n      have h\u2084\u2082 : a (n + 3) = Real.sqrt 3 * a (n + 2) - b (n + 2) := by rw [h\u2080]\n      rw [h\u2084\u2082]\n      have h\u2084\u2083 : a (n + 2) = Real.sqrt 3 * a (n + 1) - b (n + 1) := by rw [h\u2080]\n      have h\u2084\u2084 : b (n + 2) = Real.sqrt 3 * b (n + 1) + a (n + 1) := by rw [h\u2081]\n      rw [h\u2084\u2083, h\u2084\u2084]\n      have h\u2084\u2085 : a (n + 1) = Real.sqrt 3 * a n - b n := by rw [h\u2080]\n      have h\u2084\u2086 : b (n + 1) = Real.sqrt 3 * b n + a n := by rw [h\u2081]\n      rw [h\u2084\u2085, h\u2084\u2086]\n      ring_nf\n      <;>\n      simp [Real.sqrt_eq_iff_sq_eq, pow_three, mul_assoc, mul_comm, mul_left_comm]\n      <;>\n      ring_nf at *\n      <;>\n      nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show 0 \u2264 3 by norm_num)]\n    exact h\u2084\u2081\n    <;>\n    simp_all\n    <;>\n    linarith\n  have h\u2085 : \u2200 n, b (n + 3) = 8 * a n := by\n    intro n\n    have h\u2085\u2081 : b (n + 3) = 8 * a n := by\n      have h\u2085\u2082 : b (n + 3) = Real.sqrt 3 * b (n + 2) + a (n + 2) := by rw [h\u2081]\n      rw [h\u2085\u2082]\n      have h\u2085\u2083 : b (n + 2) = Real.sqrt 3 * b (n + 1) + a (n + 1) := by rw [h\u2081]\n      have h\u2085\u2084 : a (n + 2) = Real.sqrt 3 * a (n + 1) - b (n + 1) := by rw [h\u2080]\n      rw [h\u2085\u2083, h\u2085\u2084]\n      have h\u2085\u2085 : b (n + 1) = Real.sqrt 3 * b n + a n := by rw [h\u2081]\n      have h\u2085\u2086 : a (n + 1) = Real.sqrt 3 * a n - b n := by rw [h\u2080]\n      rw [h\u2085\u2085, h\u2085\u2086]\n      ring_nf\n      <;>\n      simp [Real.sqrt_eq_iff_sq_eq, pow_three, mul_assoc, mul_comm, mul_left_comm]\n      <;>\n      ring_nf at *\n      <;>\n      nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show 0 \u2264 3 by norm_num)]\n    exact h\u2085\u2081\n    <;>\n    simp_all\n    <;>\n    linarith\n  have h\u2086 : \u2200 n, a (n + 6) = -64 * a n := by\n    intro n\n    have h\u2086\u2081 : a (n + 6) = -64 * a n := by\n      have h\u2086\u2082 : a (n + 6) = a ((n + 3) + 3) := by ring_nf\n      rw [h\u2086\u2082]\n      have h\u2086\u2083 : a ((n + 3) + 3) = -8 * b (n + 3) := by rw [h\u2084]\n      rw [h\u2086\u2083]\n      have h\u2086\u2084 : b (n + 3) = 8 * a n := by rw [h\u2085]\n      rw [h\u2086\u2084]\n      <;> ring_nf\n      <;> simp [h\u2084, h\u2085]\n      <;> ring_nf\n      <;> linarith\n    exact h\u2086\u2081\n  have h\u2087 : \u2200 n, b (n + 6) = -64 * b n := by\n    intro n\n    have h\u2087\u2081 : b (n + 6) = b ((n + 3) + 3) := by ring_nf\n    rw [h\u2087\u2081]\n    have h\u2087\u2082 : b ((n + 3) + 3) = 8 * a (n + 3) := by rw [h\u2085]\n    rw [h\u2087\u2082]\n    have h\u2087\u2083 : a (n + 3) = -8 * b n := by rw [h\u2084]\n    rw [h\u2087\u2083]\n    <;> ring_nf\n    <;> simp [h\u2084, h\u2085]\n    <;> ring_nf\n    <;> linarith\n  have h\u2088 : a 4 = 1 / (2 ^ 95 : \u211d) := by\n    have h\u2088\u2081 : a 100 = 2 := h\u2082\n    have h\u2088\u2082 : b 100 = 4 := h\u2083\n    have h\u2088\u2083 : a 100 = (2 : \u211d) ^ 96 * a 4 := by\n      have h\u2088\u2084 : a 100 = (2 : \u211d) ^ 96 * a 4 := by\n        have h\u2088\u2085 : a 100 = (2 : \u211d) ^ 96 * a 4 := by\n          have h\u2088\u2086 : a 100 = (-64 : \u211d) ^ 16 * a 4 := by\n            have h\u2088\u2087 : a 100 = (-64 : \u211d) ^ 16 * a 4 := by\n              norm_num [h\u2086, h\u2087, pow_succ, mul_assoc] at h\u2088\u2081 h\u2088\u2082 \u22a2\n              <;>\n                (try ring_nf at * <;> try norm_num at * <;> try linarith)\n              <;>\n                (try ring_nf at * <;> try norm_num at * <;> try linarith)\n              <;>\n                (try ring_nf at * <;> try norm_num at * <;> try linarith)\n              <;>\n                (try ring_nf at * <;> try norm_num at * <;> try linarith)\n            exact h\u2088\u2087\n          have h\u2088\u2088 : (-64 : \u211d) ^ 16 = (2 : \u211d) ^ 96 := by\n            norm_num [pow_succ, mul_assoc]\n            <;>\n            ring_nf\n            <;>\n            norm_num\n            <;>\n            linarith\n          rw [h\u2088\u2086, h\u2088\u2088]\n          <;>\n          ring_nf\n          <;>\n          norm_num\n          <;>\n          linarith\n        exact h\u2088\u2085\n      exact h\u2088\u2084\n    have h\u2088\u2089 : a 4 = 1 / (2 ^ 95 : \u211d) := by\n      have h\u2088\u2081\u2080 : (2 : \u211d) ^ 96 * a 4 = 2 := by\n        linarith\n      have h\u2088\u2081\u2081 : a 4 = 1 / (2 ^ 95 : \u211d) := by\n        have h\u2088\u2081\u2082 : a 4 = 1 / (2 ^ 95 : \u211d) := by\n          field_simp at h\u2088\u2081\u2080 \u22a2\n          ring_nf at h\u2088\u2081\u2080 \u22a2\n          <;> nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n        exact h\u2088\u2081\u2082\n      exact h\u2088\u2081\u2081\n    exact h\u2088\u2089\n  have h\u2089 : b 4 = 1 / (2 ^ 94 : \u211d) := by\n    have h\u2089\u2081 : b 100 = (2 : \u211d) ^ 96 * b 4 := by\n      have h\u2089\u2082 : b 100 = (-64 : \u211d) ^ 16 * b 4 := by\n        norm_num [h\u2086, h\u2087, pow_succ, mul_assoc] at h\u2082 h\u2083 \u22a2\n        <;>\n        (try ring_nf at * <;> try norm_num at * <;> try linarith)\n        <;>\n        (try ring_nf at * <;> try norm_num at * <;> try linarith)\n        <;>\n        (try ring_nf at * <;> try norm_num at * <;> try linarith)\n        <;>\n        (try ring_nf at * <;> try norm_num at * <;> try linarith)\n      have h\u2089\u2083 : (-64 : \u211d) ^ 16 = (2 : \u211d) ^ 96 := by\n        norm_num [pow_succ, mul_assoc]\n        <;>\n        ring_nf\n        <;>\n        norm_num\n        <;>\n        linarith\n      rw [h\u2089\u2082, h\u2089\u2083]\n      <;>\n      ring_nf\n      <;>\n      norm_num\n      <;>\n      linarith\n    have h\u2089\u2084 : b 4 = 1 / (2 ^ 94 : \u211d) := by\n      have h\u2089\u2085 : (2 : \u211d) ^ 96 * b 4 = 4 := by\n        linarith\n      have h\u2089\u2086 : b 4 = 1 / (2 ^ 94 : \u211d) := by\n        field_simp at h\u2089\u2085 \u22a2\n        ring_nf at h\u2089\u2085 \u22a2\n        <;> nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n      exact h\u2089\u2086\n    exact h\u2089\u2084\n  have h\u2081\u2080 : a 1 = 1 / (2 ^ 97 : \u211d) := by\n    have h\u2081\u2080\u2081 : a 4 = -8 * b 1 := by\n      have h\u2081\u2080\u2082 := h\u2084 1\n      have h\u2081\u2080\u2083 := h\u2085 1\n      norm_num at h\u2081\u2080\u2082 h\u2081\u2080\u2083 \u22a2\n      <;> linarith\n    have h\u2081\u2080\u2084 : a 4 = 1 / (2 ^ 95 : \u211d) := h\u2088\n    have h\u2081\u2080\u2085 : b 1 = -1 / (2 ^ 98 : \u211d) := by\n      have h\u2081\u2080\u2085\u2081 : a 4 = -8 * b 1 := h\u2081\u2080\u2081\n      rw [h\u2081\u2080\u2084] at h\u2081\u2080\u2085\u2081\n      ring_nf at h\u2081\u2080\u2085\u2081 \u22a2\n      <;> nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n    have h\u2081\u2080\u2086 : b 4 = 8 * a 1 := by\n      have h\u2081\u2080\u2086\u2081 := h\u2085 1\n      have h\u2081\u2080\u2086\u2082 := h\u2084 1\n      norm_num at h\u2081\u2080\u2086\u2081 h\u2081\u2080\u2086\u2082 \u22a2\n      <;> linarith\n    have h\u2081\u2080\u2087 : b 4 = 1 / (2 ^ 94 : \u211d) := h\u2089\n    have h\u2081\u2080\u2088 : a 1 = 1 / (2 ^ 97 : \u211d) := by\n      rw [h\u2081\u2080\u2086] at h\u2081\u2080\u2087\n      have h\u2081\u2080\u2088 : (8 : \u211d) * a 1 = 1 / (2 ^ 94 : \u211d) := by linarith\n      ring_nf at h\u2081\u2080\u2088 \u22a2\n      <;> nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n    exact h\u2081\u2080\u2088\n  have h\u2081\u2081 : b 1 = -1 / (2 ^ 98 : \u211d) := by\n    have h\u2081\u2081\u2081 : a 4 = -8 * b 1 := by\n      have h\u2081\u2081\u2082 := h\u2084 1\n      have h\u2081\u2081\u2083 := h\u2085 1\n      norm_num at h\u2081\u2081\u2082 h\u2081\u2081\u2083 \u22a2\n      <;> linarith\n    have h\u2081\u2081\u2084 : a 4 = 1 / (2 ^ 95 : \u211d) := h\u2088\n    have h\u2081\u2081\u2085 : b 1 = -1 / (2 ^ 98 : \u211d) := by\n      rw [h\u2081\u2081\u2084] at h\u2081\u2081\u2081\n      ring_nf at h\u2081\u2081\u2081 \u22a2\n      <;> nlinarith [Real.sqrt_nonneg 3, Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n    exact h\u2081\u2081\u2085\n  have h\u2081\u2082 : a 1 + b 1 = 1 / (2 ^ 98 : \u211d) := by\n    rw [h\u2081\u2080, h\u2081\u2081]\n    <;> ring_nf <;> field_simp <;> ring_nf <;> norm_num\n    <;>\n    linarith\n  rw [h\u2081\u2082]\n  <;>\n  norm_num\n  <;>\n  simp_all [pow_add, pow_mul, mul_assoc]\n  <;>\n  ring_nf at *\n  <;>\n  norm_num at *\n  <;>\n  linarith\n"}
{"mode": "clear", "steps_before": 35, "alphabet": 20, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1138, "chars_after": 8, "seconds": 25.3, "id": "mathd_numbertheory_582", "proof_after": "theorem mathd_numbertheory_582 (n : \u2115) (h\u2080 : 0 < n) (h\u2081 : 3 \u2223 n) :\n    (n + 4 + (n + 6) + (n + 8)) % 9 = 0  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_582 (n : \u2115) (h\u2080 : 0 < n) (h\u2081 : 3 \u2223 n) :\n    (n + 4 + (n + 6) + (n + 8)) % 9 = 0  :=  by\n  have h\u2082 : (n + 4 + (n + 6) + (n + 8)) = 3 * n + 18 := by\n    have h\u2082\u2081 : n + 4 + (n + 6) + (n + 8) = 3 * n + 18 := by\n      have h\u2082\u2082 : n + 4 + (n + 6) + (n + 8) = (n + 4 + (n + 6)) + (n + 8) := by\n        ring\n      rw [h\u2082\u2082]\n      have h\u2082\u2083 : n + 4 + (n + 6) = 2 * n + 10 := by\n        ring\n      rw [h\u2082\u2083]\n      have h\u2082\u2084 : (2 * n + 10) + (n + 8) = 3 * n + 18 := by\n        ring\n      rw [h\u2082\u2084]\n      <;> ring\n    rw [h\u2082\u2081]\n    <;> ring\n  have h\u2083 : (n + 4 + (n + 6) + (n + 8)) % 9 = 0 := by\n    rw [h\u2082]\n    have h\u2084 : 3 \u2223 n := h\u2081\n    have h\u2085 : n % 9 = 0 \u2228 n % 9 = 3 \u2228 n % 9 = 6 := by\n      have h\u2085\u2081 : n % 9 = 0 \u2228 n % 9 = 3 \u2228 n % 9 = 6 := by\n        omega\n      exact h\u2085\u2081\n    rcases h\u2085 with (h\u2085 | h\u2085 | h\u2085)\n    \u00b7 \n      simp [h\u2085, Nat.add_mod, Nat.mul_mod, Nat.mod_mod, Nat.mod_self, Nat.zero_add,\n        Nat.add_assoc]\n      <;> norm_num <;> omega\n    \u00b7 \n      simp [h\u2085, Nat.add_mod, Nat.mul_mod, Nat.mod_mod, Nat.mod_self, Nat.zero_add,\n        Nat.add_assoc]\n      <;> norm_num <;> omega\n    \u00b7 \n      simp [h\u2085, Nat.add_mod, Nat.mul_mod, Nat.mod_mod, Nat.mod_self, Nat.zero_add,\n        Nat.add_assoc]\n      <;> norm_num <;> omega\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 78, "alphabet": 37, "compiles": false, "id": "imo_1960_p2", "proof_after": "theorem imo_1960_p2 (x : \u211d) (h\u2080 : 0 \u2264 1 + 2 * x) (h\u2081 : (1 - Real.sqrt (1 + 2 * x)) ^ 2 \u2260 0)\n    (h\u2082 : 4 * x ^ 2 / (1 - Real.sqrt (1 + 2 * x)) ^ 2 < 2 * x + 9) : -(1 / 2) \u2264 x \u2227 x < 45 / 8  :=  by\n  have h\u2083 : -(1 / 2 : \u211d) \u2264 x := by\n    have h\u2083\u2081 : 0 \u2264 1 + 2 * x := h\u2080\n    have h\u2083\u2082 : -(1 / 2 : \u211d) \u2264 x := by linarith\n    exact h\u2083\u2082\n  have h\u2084 : x < 45 / 8 := by\n    by_cases hx : x > 0\n    \u00b7 \n      have h\u2085 : Real.sqrt (1 + 2 * x) > 1 := by\n        apply Real.lt_sqrt_of_sq_lt\n        nlinarith\n      have h\u2086 : Real.sqrt (1 + 2 * x) \u2260 1 := by\n        intro h\n        have h\u2087 : Real.sqrt (1 + 2 * x) = 1 := h\n        have h\u2088 : (Real.sqrt (1 + 2 * x)) ^ 2 = (1 : \u211d) ^ 2 := by rw [h\u2087]\n        have h\u2089 : (Real.sqrt (1 + 2 * x)) ^ 2 = 1 + 2 * x := by\n          rw [Real.sq_sqrt] <;> nlinarith\n        have h\u2081\u2080 : 1 + 2 * x = 1 := by nlinarith\n        have h\u2081\u2081 : x = 0 := by nlinarith\n        have h\u2081\u2082 : x > 0 := hx\n        linarith\n      have h\u2087 : Real.sqrt (1 + 2 * x) < 7 / 2 := by\n        by_contra h\u2087\n        have h\u2088 : Real.sqrt (1 + 2 * x) \u2265 7 / 2 := by linarith\n        have h\u2089 : (Real.sqrt (1 + 2 * x) - 1) ^ 2 \u2265 (7 / 2 - 1) ^ 2 := by\n          nlinarith [Real.sqrt_nonneg (1 + 2 * x), Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x)]\n        have h\u2081\u2080 : 4 * x ^ 2 / (1 - Real.sqrt (1 + 2 * x)) ^ 2 \u2265 2 * x + 9 := by\n          have h\u2081\u2081 : (1 - Real.sqrt (1 + 2 * x)) ^ 2 = (Real.sqrt (1 + 2 * x) - 1) ^ 2 := by\n            ring_nf\n            <;>\n            nlinarith [Real.sqrt_nonneg (1 + 2 * x), Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x)]\n          rw [h\u2081\u2081]\n          have h\u2081\u2082 : x = ((Real.sqrt (1 + 2 * x)) ^ 2 - 1) / 2 := by\n            nlinarith [Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x), Real.sqrt_nonneg (1 + 2 * x)]\n          rw [h\u2081\u2082]\n          have h\u2081\u2083 : 0 < Real.sqrt (1 + 2 * x) - 1 := by nlinarith [Real.sqrt_nonneg (1 + 2 * x), Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x)]\n          have h\u2081\u2084 : 0 < (Real.sqrt (1 + 2 * x) - 1) ^ 2 := by nlinarith\n          field_simp [h\u2081\u2084.ne']\n          rw [le_div_iff (by positivity)]\n          nlinarith [Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x), Real.sqrt_nonneg (1 + 2 * x),\n            sq_nonneg (Real.sqrt (1 + 2 * x) - 7 / 2)]\n        linarith\n      have h\u2088 : x < 45 / 8 := by\n        nlinarith [Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x), Real.sqrt_nonneg (1 + 2 * x)]\n      exact h\u2088\n    \u00b7 \n      have h\u2085 : x < 45 / 8 := by\n        by_cases h\u2085\u2081 : x < 0\n        \u00b7 \n          linarith\n        \u00b7 \n          have h\u2085\u2082 : x = 0 := by\n            by_contra h\u2085\u2082\n            have h\u2085\u2083 : x > 0 := by\n              by_contra h\u2085\u2084\n              have h\u2085\u2085 : x < 0 := by\n                cases' lt_or_gt_of_ne h\u2085\u2082 with h\u2085\u2085 h\u2085\u2085\n                \u00b7 linarith\n                \u00b7 exfalso\n                  linarith\n              linarith\n            linarith\n          exfalso\n          have h\u2085\u2083 : (1 - Real.sqrt (1 + 2 * x)) ^ 2 = 0 := by\n            rw [h\u2085\u2082]\n            have h\u2085\u2084 : Real.sqrt (1 + 2 * (0 : \u211d)) = 1 := by norm_num [Real.sqrt_eq_iff_sq_eq]\n            rw [h\u2085\u2084]\n            <;> norm_num\n          exact h\u2081 h\u2085\u2083\n      exact h\u2085\n  exact \u27e8h\u2083, h\u2084\u27e9\n", "proof_before": "theorem imo_1960_p2 (x : \u211d) (h\u2080 : 0 \u2264 1 + 2 * x) (h\u2081 : (1 - Real.sqrt (1 + 2 * x)) ^ 2 \u2260 0)\n    (h\u2082 : 4 * x ^ 2 / (1 - Real.sqrt (1 + 2 * x)) ^ 2 < 2 * x + 9) : -(1 / 2) \u2264 x \u2227 x < 45 / 8  :=  by\n  have h\u2083 : -(1 / 2 : \u211d) \u2264 x := by\n    have h\u2083\u2081 : 0 \u2264 1 + 2 * x := h\u2080\n    have h\u2083\u2082 : -(1 / 2 : \u211d) \u2264 x := by linarith\n    exact h\u2083\u2082\n  have h\u2084 : x < 45 / 8 := by\n    by_cases hx : x > 0\n    \u00b7 \n      have h\u2085 : Real.sqrt (1 + 2 * x) > 1 := by\n        apply Real.lt_sqrt_of_sq_lt\n        nlinarith\n      have h\u2086 : Real.sqrt (1 + 2 * x) \u2260 1 := by\n        intro h\n        have h\u2087 : Real.sqrt (1 + 2 * x) = 1 := h\n        have h\u2088 : (Real.sqrt (1 + 2 * x)) ^ 2 = (1 : \u211d) ^ 2 := by rw [h\u2087]\n        have h\u2089 : (Real.sqrt (1 + 2 * x)) ^ 2 = 1 + 2 * x := by\n          rw [Real.sq_sqrt] <;> nlinarith\n        have h\u2081\u2080 : 1 + 2 * x = 1 := by nlinarith\n        have h\u2081\u2081 : x = 0 := by nlinarith\n        have h\u2081\u2082 : x > 0 := hx\n        linarith\n      have h\u2087 : Real.sqrt (1 + 2 * x) < 7 / 2 := by\n        by_contra h\u2087\n        have h\u2088 : Real.sqrt (1 + 2 * x) \u2265 7 / 2 := by linarith\n        have h\u2089 : (Real.sqrt (1 + 2 * x) - 1) ^ 2 \u2265 (7 / 2 - 1) ^ 2 := by\n          nlinarith [Real.sqrt_nonneg (1 + 2 * x), Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x)]\n        have h\u2081\u2080 : 4 * x ^ 2 / (1 - Real.sqrt (1 + 2 * x)) ^ 2 \u2265 2 * x + 9 := by\n          have h\u2081\u2081 : (1 - Real.sqrt (1 + 2 * x)) ^ 2 = (Real.sqrt (1 + 2 * x) - 1) ^ 2 := by\n            ring_nf\n            <;>\n            nlinarith [Real.sqrt_nonneg (1 + 2 * x), Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x)]\n          rw [h\u2081\u2081]\n          have h\u2081\u2082 : x = ((Real.sqrt (1 + 2 * x)) ^ 2 - 1) / 2 := by\n            nlinarith [Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x), Real.sqrt_nonneg (1 + 2 * x)]\n          rw [h\u2081\u2082]\n          have h\u2081\u2083 : 0 < Real.sqrt (1 + 2 * x) - 1 := by nlinarith [Real.sqrt_nonneg (1 + 2 * x), Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x)]\n          have h\u2081\u2084 : 0 < (Real.sqrt (1 + 2 * x) - 1) ^ 2 := by nlinarith\n          field_simp [h\u2081\u2084.ne']\n          rw [le_div_iff (by positivity)]\n          nlinarith [Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x), Real.sqrt_nonneg (1 + 2 * x),\n            sq_nonneg (Real.sqrt (1 + 2 * x) - 7 / 2)]\n        linarith\n      have h\u2088 : x < 45 / 8 := by\n        nlinarith [Real.sq_sqrt (by nlinarith : 0 \u2264 1 + 2 * x), Real.sqrt_nonneg (1 + 2 * x)]\n      exact h\u2088\n    \u00b7 \n      have h\u2085 : x < 45 / 8 := by\n        by_cases h\u2085\u2081 : x < 0\n        \u00b7 \n          linarith\n        \u00b7 \n          have h\u2085\u2082 : x = 0 := by\n            by_contra h\u2085\u2082\n            have h\u2085\u2083 : x > 0 := by\n              by_contra h\u2085\u2084\n              have h\u2085\u2085 : x < 0 := by\n                cases' lt_or_gt_of_ne h\u2085\u2082 with h\u2085\u2085 h\u2085\u2085\n                \u00b7 linarith\n                \u00b7 exfalso\n                  linarith\n              linarith\n            linarith\n          exfalso\n          have h\u2085\u2083 : (1 - Real.sqrt (1 + 2 * x)) ^ 2 = 0 := by\n            rw [h\u2085\u2082]\n            have h\u2085\u2084 : Real.sqrt (1 + 2 * (0 : \u211d)) = 1 := by norm_num [Real.sqrt_eq_iff_sq_eq]\n            rw [h\u2085\u2084]\n            <;> norm_num\n          exact h\u2081 h\u2085\u2083\n      exact h\u2085\n  exact \u27e8h\u2083, h\u2084\u27e9\n"}
{"mode": "clear", "steps_before": 66, "alphabet": 21, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 18, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 6, "fixpoint": true, "chars_before": 1704, "chars_after": 327, "seconds": 819.7, "id": "mathd_numbertheory_135", "proof_after": "theorem mathd_numbertheory_135 (n A B C : \u2115) (h\u2080 : n = 3 ^ 17 + 3 ^ 10) (h\u2081 : 11 \u2223 n + 1)\n    (h\u2082 : [A, B, C].Pairwise (\u00b7 \u2260 \u00b7)) (h\u2083 : {A, B, C} \u2282 Finset.Icc 0 9) (h\u2084 : Odd A \u2227 Odd C)\n    (h\u2085 : \u00ac3 \u2223 B) (h\u2086 : Nat.digits 10 n = [B, A, B, C, C, A, C, B, A]) :\n    100 * A + 10 * B + C = 129  :=  by\n  have h\u2087 : n = 129199212 := by\n    omega\n  have h\u2088 : B = 2 := by\n    rw [h\u2087] at h\u2086\n    norm_num [Nat.digits_len] at h\u2086 <;>\n    omega\n  have h\u2089 : A = 1 := by\n    rw [h\u2087] at h\u2086\n    norm_num [Nat.digits_len] at h\u2086 <;>\n    omega\n  have h\u2081\u2080 : C = 9 := by\n    rw [h\u2087] at h\u2086\n    norm_num [Nat.digits_len] at h\u2086 <;>\n    omega\n  omega\n", "proof_before": "theorem mathd_numbertheory_135 (n A B C : \u2115) (h\u2080 : n = 3 ^ 17 + 3 ^ 10) (h\u2081 : 11 \u2223 n + 1)\n    (h\u2082 : [A, B, C].Pairwise (\u00b7 \u2260 \u00b7)) (h\u2083 : {A, B, C} \u2282 Finset.Icc 0 9) (h\u2084 : Odd A \u2227 Odd C)\n    (h\u2085 : \u00ac3 \u2223 B) (h\u2086 : Nat.digits 10 n = [B, A, B, C, C, A, C, B, A]) :\n    100 * A + 10 * B + C = 129  :=  by\n  have h\u2087 : n = 129199212 := by\n    rw [h\u2080]\n    norm_num\n    <;> rfl\n  have h\u2088 : B = 2 := by\n    rw [h\u2087] at h\u2086\n    norm_num [Nat.digits_len] at h\u2086 <;>\n    (try omega) <;>\n    (try\n      {\n        aesop\n      }) <;>\n    (try\n      {\n        simp_all [Finset.ext_iff, Finset.mem_Icc, List.Pairwise]\n        <;> omega\n      }) <;>\n    (try\n      {\n        norm_num at h\u2086 \u22a2\n        <;> aesop\n      })\n    <;>\n    (try\n      {\n        rcases h\u2084 with \u27e8h\u2084\u2081, h\u2084\u2082\u27e9\n        cases' h\u2084\u2081 with k hk <;> cases' h\u2084\u2082 with m hm <;> omega\n      })\n    <;>\n    (try\n      {\n        omega\n      })\n  have h\u2089 : A = 1 := by\n    rw [h\u2087] at h\u2086\n    norm_num [Nat.digits_len] at h\u2086 <;>\n    (try omega) <;>\n    (try\n      {\n        aesop\n      }) <;>\n    (try\n      {\n        simp_all [Finset.ext_iff, Finset.mem_Icc, List.Pairwise]\n        <;> omega\n      }) <;>\n    (try\n      {\n        norm_num at h\u2086 \u22a2\n        <;> aesop\n      })\n    <;>\n    (try\n      {\n        rcases h\u2084 with \u27e8h\u2084\u2081, h\u2084\u2082\u27e9\n        cases' h\u2084\u2081 with k hk <;> cases' h\u2084\u2082 with m hm <;> omega\n      })\n    <;>\n    (try\n      {\n        omega\n      })\n  have h\u2081\u2080 : C = 9 := by\n    rw [h\u2087] at h\u2086\n    norm_num [Nat.digits_len] at h\u2086 <;>\n    (try omega) <;>\n    (try\n      {\n        aesop\n      }) <;>\n    (try\n      {\n        simp_all [Finset.ext_iff, Finset.mem_Icc, List.Pairwise]\n        <;> omega\n      }) <;>\n    (try\n      {\n        norm_num at h\u2086 \u22a2\n        <;> aesop\n      })\n    <;>\n    (try\n      {\n        rcases h\u2084 with \u27e8h\u2084\u2081, h\u2084\u2082\u27e9\n        cases' h\u2084\u2081 with k hk <;> cases' h\u2084\u2082 with m hm <;> omega\n      })\n    <;>\n    (try\n      {\n        omega\n      })\n  have h\u2081\u2081 : 100 * A + 10 * B + C = 129 := by\n    rw [h\u2089, h\u2088, h\u2081\u2080]\n    <;> norm_num\n    <;> aesop\n  apply h\u2081\u2081\n"}
{"mode": "clear", "steps_before": 215, "alphabet": 79, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 215, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 5704, "chars_after": 5704, "seconds": 366.5, "id": "aime_1988_p8", "proof_after": "theorem aime_1988_p8 (f : \u2115 \u2192 \u2115 \u2192 \u211d) (h\u2080 : \u2200 x, 0 < x \u2192 f x x = x)\n    (h\u2081 : \u2200 x y, 0 < x \u2227 0 < y \u2192 f x y = f y x)\n    (h\u2082 : \u2200 x y, 0 < x \u2227 0 < y \u2192 (\u2191x + \u2191y) * f x y = y * f x (x + y)) : f 14 52 = 364  :=  by\n  have h\u2083 : f 2 2 = 2 := by\n    have h\u2083\u2081 : f 2 2 = 2 := by\n      have h\u2083\u2082 : f 2 2 = (2 : \u211d) := by\n        have h\u2083\u2083 : (0 : \u2115) < 2 := by decide\n        have h\u2083\u2084 : f 2 2 = (2 : \u211d) := by\n          simpa using h\u2080 2 (by decide)\n        exact h\u2083\u2084\n      exact_mod_cast h\u2083\u2082\n    exact h\u2083\u2081\n  have h\u2084 : f 2 4 = 4 := by\n    have h\u2084\u2081 : (\u21912 + \u21912 : \u211d) * f 2 2 = (2 : \u211d) * f 2 4 := by\n      have h\u2084\u2082 : 0 < (2 : \u2115) \u2227 0 < (2 : \u2115) := by norm_num\n      have h\u2084\u2083 := h\u2082 2 2 h\u2084\u2082\n      norm_num at h\u2084\u2083 \u22a2\n      <;> linarith\n    have h\u2084\u2082 : (\u21912 + \u21912 : \u211d) * f 2 2 = (2 : \u211d) * f 2 4 := by\n      exact h\u2084\u2081\n    have h\u2084\u2083 : f 2 2 = (2 : \u211d) := by exact_mod_cast h\u2083\n    have h\u2084\u2084 : (\u21912 + \u21912 : \u211d) * f 2 2 = (2 : \u211d) * f 2 4 := by exact h\u2084\u2082\n    have h\u2084\u2085 : f 2 4 = 4 := by\n      rw [h\u2084\u2083] at h\u2084\u2084\n      ring_nf at h\u2084\u2084 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2084\u2085\n  have h\u2085 : f 4 2 = 4 := by\n    have h\u2085\u2081 : f 4 2 = f 2 4 := by\n      apply h\u2081 4 2\n      <;> norm_num\n    rw [h\u2085\u2081]\n    exact_mod_cast h\u2084\n  have h\u2086 : f 4 6 = 12 := by\n    have h\u2086\u2081 : (\u21914 + \u21912 : \u211d) * f 4 2 = (\u21912 : \u211d) * f 4 6 := by\n      have h\u2086\u2082 : 0 < (4 : \u2115) \u2227 0 < (2 : \u2115) := by norm_num\n      have h\u2086\u2083 := h\u2082 4 2 h\u2086\u2082\n      norm_num at h\u2086\u2083 \u22a2 <;> ring_nf at h\u2086\u2083 \u22a2 <;>\n        (try norm_num at h\u2086\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2086\u2083 \u22a2 <;> norm_num at h\u2086\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2086\u2084 : (\u21914 + \u21912 : \u211d) * f 4 2 = (\u21912 : \u211d) * f 4 6 := by\n      exact h\u2086\u2081\n    have h\u2086\u2085 : f 4 2 = (4 : \u211d) := by exact_mod_cast h\u2085\n    have h\u2086\u2086 : (\u21914 + \u21912 : \u211d) * f 4 2 = (\u21912 : \u211d) * f 4 6 := by exact h\u2086\u2081\n    have h\u2086\u2087 : f 4 6 = 12 := by\n      rw [h\u2086\u2085] at h\u2086\u2086\n      ring_nf at h\u2086\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2086\u2087\n  have h\u2087 : f 4 10 = 20 := by\n    have h\u2087\u2081 : (\u21914 + \u21916 : \u211d) * f 4 6 = (\u21916 : \u211d) * f 4 10 := by\n      have h\u2087\u2082 : 0 < (4 : \u2115) \u2227 0 < (6 : \u2115) := by norm_num\n      have h\u2087\u2083 := h\u2082 4 6 h\u2087\u2082\n      norm_num at h\u2087\u2083 \u22a2 <;> ring_nf at h\u2087\u2083 \u22a2 <;>\n        (try norm_num at h\u2087\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2087\u2083 \u22a2 <;> norm_num at h\u2087\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2087\u2084 : (\u21914 + \u21916 : \u211d) * f 4 6 = (\u21916 : \u211d) * f 4 10 := by\n      exact h\u2087\u2081\n    have h\u2087\u2085 : f 4 6 = (12 : \u211d) := by exact_mod_cast h\u2086\n    have h\u2087\u2086 : (\u21914 + \u21916 : \u211d) * f 4 6 = (\u21916 : \u211d) * f 4 10 := by exact h\u2087\u2081\n    have h\u2087\u2087 : f 4 10 = 20 := by\n      rw [h\u2087\u2085] at h\u2087\u2086\n      ring_nf at h\u2087\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2087\u2087\n  have h\u2088 : f 10 4 = 20 := by\n    have h\u2088\u2081 : f 10 4 = f 4 10 := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2081]\n    exact_mod_cast h\u2087\n  have h\u2089 : f 10 14 = 70 := by\n    have h\u2089\u2081 : (\u219110 + \u21914 : \u211d) * f 10 4 = (\u21914 : \u211d) * f 10 14 := by\n      have h\u2089\u2082 : 0 < (10 : \u2115) \u2227 0 < (4 : \u2115) := by norm_num\n      have h\u2089\u2083 := h\u2082 10 4 h\u2089\u2082\n      norm_num at h\u2089\u2083 \u22a2 <;> ring_nf at h\u2089\u2083 \u22a2 <;>\n        (try norm_num at h\u2089\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2089\u2083 \u22a2 <;> norm_num at h\u2089\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2089\u2084 : (\u219110 + \u21914 : \u211d) * f 10 4 = (\u21914 : \u211d) * f 10 14 := by\n      exact h\u2089\u2081\n    have h\u2089\u2085 : f 10 4 = (20 : \u211d) := by exact_mod_cast h\u2088\n    have h\u2089\u2086 : (\u219110 + \u21914 : \u211d) * f 10 4 = (\u21914 : \u211d) * f 10 14 := by exact h\u2089\u2081\n    have h\u2089\u2087 : f 10 14 = 70 := by\n      rw [h\u2089\u2085] at h\u2089\u2086\n      ring_nf at h\u2089\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2089\u2087\n  have h\u2081\u2080 : f 14 10 = 70 := by\n    have h\u2081\u2080\u2081 : f 14 10 = f 10 14 := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2081]\n    exact_mod_cast h\u2089\n  have h\u2081\u2081 : f 14 24 = 168 := by\n    have h\u2081\u2081\u2081 : (\u219114 + \u219110 : \u211d) * f 14 10 = (\u219110 : \u211d) * f 14 24 := by\n      have h\u2081\u2081\u2082 : 0 < (14 : \u2115) \u2227 0 < (10 : \u2115) := by norm_num\n      have h\u2081\u2081\u2083 := h\u2082 14 10 h\u2081\u2081\u2082\n      norm_num at h\u2081\u2081\u2083 \u22a2 <;> ring_nf at h\u2081\u2081\u2083 \u22a2 <;>\n        (try norm_num at h\u2081\u2081\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2081\u2081\u2083 \u22a2 <;> norm_num at h\u2081\u2081\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2081\u2081\u2084 : (\u219114 + \u219110 : \u211d) * f 14 10 = (\u219110 : \u211d) * f 14 24 := by\n      exact h\u2081\u2081\u2081\n    have h\u2081\u2081\u2085 : f 14 10 = (70 : \u211d) := by exact_mod_cast h\u2081\u2080\n    have h\u2081\u2081\u2086 : (\u219114 + \u219110 : \u211d) * f 14 10 = (\u219110 : \u211d) * f 14 24 := by exact h\u2081\u2081\u2081\n    have h\u2081\u2081\u2087 : f 14 24 = 168 := by\n      rw [h\u2081\u2081\u2085] at h\u2081\u2081\u2086\n      ring_nf at h\u2081\u2081\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2081\u2081\u2087\n  have h\u2081\u2082 : f 14 38 = 266 := by\n    have h\u2081\u2082\u2081 : (\u219114 + \u219124 : \u211d) * f 14 24 = (\u219124 : \u211d) * f 14 38 := by\n      have h\u2081\u2082\u2082 : 0 < (14 : \u2115) \u2227 0 < (24 : \u2115) := by norm_num\n      have h\u2081\u2082\u2083 := h\u2082 14 24 h\u2081\u2082\u2082\n      norm_num at h\u2081\u2082\u2083 \u22a2 <;> ring_nf at h\u2081\u2082\u2083 \u22a2 <;>\n        (try norm_num at h\u2081\u2082\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2081\u2082\u2083 \u22a2 <;> norm_num at h\u2081\u2082\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2081\u2082\u2084 : (\u219114 + \u219124 : \u211d) * f 14 24 = (\u219124 : \u211d) * f 14 38 := by\n      exact h\u2081\u2082\u2081\n    have h\u2081\u2082\u2085 : f 14 24 = (168 : \u211d) := by exact_mod_cast h\u2081\u2081\n    have h\u2081\u2082\u2086 : (\u219114 + \u219124 : \u211d) * f 14 24 = (\u219124 : \u211d) * f 14 38 := by exact h\u2081\u2082\u2081\n    have h\u2081\u2082\u2087 : f 14 38 = 266 := by\n      rw [h\u2081\u2082\u2085] at h\u2081\u2082\u2086\n      ring_nf at h\u2081\u2082\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2081\u2082\u2087\n  have h\u2081\u2083 : f 14 52 = 364 := by\n    have h\u2081\u2083\u2081 : (\u219114 + \u219138 : \u211d) * f 14 38 = (\u219138 : \u211d) * f 14 52 := by\n      have h\u2081\u2083\u2082 : 0 < (14 : \u2115) \u2227 0 < (38 : \u2115) := by norm_num\n      have h\u2081\u2083\u2083 := h\u2082 14 38 h\u2081\u2083\u2082\n      norm_num at h\u2081\u2083\u2083 \u22a2 <;> ring_nf at h\u2081\u2083\u2083 \u22a2 <;>\n        (try norm_num at h\u2081\u2083\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2081\u2083\u2083 \u22a2 <;> norm_num at h\u2081\u2083\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2081\u2083\u2084 : (\u219114 + \u219138 : \u211d) * f 14 38 = (\u219138 : \u211d) * f 14 52 := by\n      exact h\u2081\u2083\u2081\n    have h\u2081\u2083\u2085 : f 14 38 = (266 : \u211d) := by exact_mod_cast h\u2081\u2082\n    have h\u2081\u2083\u2086 : (\u219114 + \u219138 : \u211d) * f 14 38 = (\u219138 : \u211d) * f 14 52 := by exact h\u2081\u2083\u2081\n    have h\u2081\u2083\u2087 : f 14 52 = 364 := by\n      rw [h\u2081\u2083\u2085] at h\u2081\u2083\u2086\n      ring_nf at h\u2081\u2083\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2081\u2083\u2087\n  exact h\u2081\u2083\n", "proof_before": "theorem aime_1988_p8 (f : \u2115 \u2192 \u2115 \u2192 \u211d) (h\u2080 : \u2200 x, 0 < x \u2192 f x x = x)\n    (h\u2081 : \u2200 x y, 0 < x \u2227 0 < y \u2192 f x y = f y x)\n    (h\u2082 : \u2200 x y, 0 < x \u2227 0 < y \u2192 (\u2191x + \u2191y) * f x y = y * f x (x + y)) : f 14 52 = 364  :=  by\n  have h\u2083 : f 2 2 = 2 := by\n    have h\u2083\u2081 : f 2 2 = 2 := by\n      have h\u2083\u2082 : f 2 2 = (2 : \u211d) := by\n        have h\u2083\u2083 : (0 : \u2115) < 2 := by decide\n        have h\u2083\u2084 : f 2 2 = (2 : \u211d) := by\n          simpa using h\u2080 2 (by decide)\n        exact h\u2083\u2084\n      exact_mod_cast h\u2083\u2082\n    exact h\u2083\u2081\n  have h\u2084 : f 2 4 = 4 := by\n    have h\u2084\u2081 : (\u21912 + \u21912 : \u211d) * f 2 2 = (2 : \u211d) * f 2 4 := by\n      have h\u2084\u2082 : 0 < (2 : \u2115) \u2227 0 < (2 : \u2115) := by norm_num\n      have h\u2084\u2083 := h\u2082 2 2 h\u2084\u2082\n      norm_num at h\u2084\u2083 \u22a2\n      <;> linarith\n    have h\u2084\u2082 : (\u21912 + \u21912 : \u211d) * f 2 2 = (2 : \u211d) * f 2 4 := by\n      exact h\u2084\u2081\n    have h\u2084\u2083 : f 2 2 = (2 : \u211d) := by exact_mod_cast h\u2083\n    have h\u2084\u2084 : (\u21912 + \u21912 : \u211d) * f 2 2 = (2 : \u211d) * f 2 4 := by exact h\u2084\u2082\n    have h\u2084\u2085 : f 2 4 = 4 := by\n      rw [h\u2084\u2083] at h\u2084\u2084\n      ring_nf at h\u2084\u2084 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2084\u2085\n  have h\u2085 : f 4 2 = 4 := by\n    have h\u2085\u2081 : f 4 2 = f 2 4 := by\n      apply h\u2081 4 2\n      <;> norm_num\n    rw [h\u2085\u2081]\n    exact_mod_cast h\u2084\n  have h\u2086 : f 4 6 = 12 := by\n    have h\u2086\u2081 : (\u21914 + \u21912 : \u211d) * f 4 2 = (\u21912 : \u211d) * f 4 6 := by\n      have h\u2086\u2082 : 0 < (4 : \u2115) \u2227 0 < (2 : \u2115) := by norm_num\n      have h\u2086\u2083 := h\u2082 4 2 h\u2086\u2082\n      norm_num at h\u2086\u2083 \u22a2 <;> ring_nf at h\u2086\u2083 \u22a2 <;>\n        (try norm_num at h\u2086\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2086\u2083 \u22a2 <;> norm_num at h\u2086\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2086\u2084 : (\u21914 + \u21912 : \u211d) * f 4 2 = (\u21912 : \u211d) * f 4 6 := by\n      exact h\u2086\u2081\n    have h\u2086\u2085 : f 4 2 = (4 : \u211d) := by exact_mod_cast h\u2085\n    have h\u2086\u2086 : (\u21914 + \u21912 : \u211d) * f 4 2 = (\u21912 : \u211d) * f 4 6 := by exact h\u2086\u2081\n    have h\u2086\u2087 : f 4 6 = 12 := by\n      rw [h\u2086\u2085] at h\u2086\u2086\n      ring_nf at h\u2086\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2086\u2087\n  have h\u2087 : f 4 10 = 20 := by\n    have h\u2087\u2081 : (\u21914 + \u21916 : \u211d) * f 4 6 = (\u21916 : \u211d) * f 4 10 := by\n      have h\u2087\u2082 : 0 < (4 : \u2115) \u2227 0 < (6 : \u2115) := by norm_num\n      have h\u2087\u2083 := h\u2082 4 6 h\u2087\u2082\n      norm_num at h\u2087\u2083 \u22a2 <;> ring_nf at h\u2087\u2083 \u22a2 <;>\n        (try norm_num at h\u2087\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2087\u2083 \u22a2 <;> norm_num at h\u2087\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2087\u2084 : (\u21914 + \u21916 : \u211d) * f 4 6 = (\u21916 : \u211d) * f 4 10 := by\n      exact h\u2087\u2081\n    have h\u2087\u2085 : f 4 6 = (12 : \u211d) := by exact_mod_cast h\u2086\n    have h\u2087\u2086 : (\u21914 + \u21916 : \u211d) * f 4 6 = (\u21916 : \u211d) * f 4 10 := by exact h\u2087\u2081\n    have h\u2087\u2087 : f 4 10 = 20 := by\n      rw [h\u2087\u2085] at h\u2087\u2086\n      ring_nf at h\u2087\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2087\u2087\n  have h\u2088 : f 10 4 = 20 := by\n    have h\u2088\u2081 : f 10 4 = f 4 10 := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2081]\n    exact_mod_cast h\u2087\n  have h\u2089 : f 10 14 = 70 := by\n    have h\u2089\u2081 : (\u219110 + \u21914 : \u211d) * f 10 4 = (\u21914 : \u211d) * f 10 14 := by\n      have h\u2089\u2082 : 0 < (10 : \u2115) \u2227 0 < (4 : \u2115) := by norm_num\n      have h\u2089\u2083 := h\u2082 10 4 h\u2089\u2082\n      norm_num at h\u2089\u2083 \u22a2 <;> ring_nf at h\u2089\u2083 \u22a2 <;>\n        (try norm_num at h\u2089\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2089\u2083 \u22a2 <;> norm_num at h\u2089\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2089\u2084 : (\u219110 + \u21914 : \u211d) * f 10 4 = (\u21914 : \u211d) * f 10 14 := by\n      exact h\u2089\u2081\n    have h\u2089\u2085 : f 10 4 = (20 : \u211d) := by exact_mod_cast h\u2088\n    have h\u2089\u2086 : (\u219110 + \u21914 : \u211d) * f 10 4 = (\u21914 : \u211d) * f 10 14 := by exact h\u2089\u2081\n    have h\u2089\u2087 : f 10 14 = 70 := by\n      rw [h\u2089\u2085] at h\u2089\u2086\n      ring_nf at h\u2089\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2089\u2087\n  have h\u2081\u2080 : f 14 10 = 70 := by\n    have h\u2081\u2080\u2081 : f 14 10 = f 10 14 := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2081]\n    exact_mod_cast h\u2089\n  have h\u2081\u2081 : f 14 24 = 168 := by\n    have h\u2081\u2081\u2081 : (\u219114 + \u219110 : \u211d) * f 14 10 = (\u219110 : \u211d) * f 14 24 := by\n      have h\u2081\u2081\u2082 : 0 < (14 : \u2115) \u2227 0 < (10 : \u2115) := by norm_num\n      have h\u2081\u2081\u2083 := h\u2082 14 10 h\u2081\u2081\u2082\n      norm_num at h\u2081\u2081\u2083 \u22a2 <;> ring_nf at h\u2081\u2081\u2083 \u22a2 <;>\n        (try norm_num at h\u2081\u2081\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2081\u2081\u2083 \u22a2 <;> norm_num at h\u2081\u2081\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2081\u2081\u2084 : (\u219114 + \u219110 : \u211d) * f 14 10 = (\u219110 : \u211d) * f 14 24 := by\n      exact h\u2081\u2081\u2081\n    have h\u2081\u2081\u2085 : f 14 10 = (70 : \u211d) := by exact_mod_cast h\u2081\u2080\n    have h\u2081\u2081\u2086 : (\u219114 + \u219110 : \u211d) * f 14 10 = (\u219110 : \u211d) * f 14 24 := by exact h\u2081\u2081\u2081\n    have h\u2081\u2081\u2087 : f 14 24 = 168 := by\n      rw [h\u2081\u2081\u2085] at h\u2081\u2081\u2086\n      ring_nf at h\u2081\u2081\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2081\u2081\u2087\n  have h\u2081\u2082 : f 14 38 = 266 := by\n    have h\u2081\u2082\u2081 : (\u219114 + \u219124 : \u211d) * f 14 24 = (\u219124 : \u211d) * f 14 38 := by\n      have h\u2081\u2082\u2082 : 0 < (14 : \u2115) \u2227 0 < (24 : \u2115) := by norm_num\n      have h\u2081\u2082\u2083 := h\u2082 14 24 h\u2081\u2082\u2082\n      norm_num at h\u2081\u2082\u2083 \u22a2 <;> ring_nf at h\u2081\u2082\u2083 \u22a2 <;>\n        (try norm_num at h\u2081\u2082\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2081\u2082\u2083 \u22a2 <;> norm_num at h\u2081\u2082\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2081\u2082\u2084 : (\u219114 + \u219124 : \u211d) * f 14 24 = (\u219124 : \u211d) * f 14 38 := by\n      exact h\u2081\u2082\u2081\n    have h\u2081\u2082\u2085 : f 14 24 = (168 : \u211d) := by exact_mod_cast h\u2081\u2081\n    have h\u2081\u2082\u2086 : (\u219114 + \u219124 : \u211d) * f 14 24 = (\u219124 : \u211d) * f 14 38 := by exact h\u2081\u2082\u2081\n    have h\u2081\u2082\u2087 : f 14 38 = 266 := by\n      rw [h\u2081\u2082\u2085] at h\u2081\u2082\u2086\n      ring_nf at h\u2081\u2082\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2081\u2082\u2087\n  have h\u2081\u2083 : f 14 52 = 364 := by\n    have h\u2081\u2083\u2081 : (\u219114 + \u219138 : \u211d) * f 14 38 = (\u219138 : \u211d) * f 14 52 := by\n      have h\u2081\u2083\u2082 : 0 < (14 : \u2115) \u2227 0 < (38 : \u2115) := by norm_num\n      have h\u2081\u2083\u2083 := h\u2082 14 38 h\u2081\u2083\u2082\n      norm_num at h\u2081\u2083\u2083 \u22a2 <;> ring_nf at h\u2081\u2083\u2083 \u22a2 <;>\n        (try norm_num at h\u2081\u2083\u2083 \u22a2) <;>\n        (try linarith) <;>\n        (try ring_nf at h\u2081\u2083\u2083 \u22a2 <;> norm_num at h\u2081\u2083\u2083 \u22a2 <;> linarith)\n      <;>\n      (try linarith)\n    have h\u2081\u2083\u2084 : (\u219114 + \u219138 : \u211d) * f 14 38 = (\u219138 : \u211d) * f 14 52 := by\n      exact h\u2081\u2083\u2081\n    have h\u2081\u2083\u2085 : f 14 38 = (266 : \u211d) := by exact_mod_cast h\u2081\u2082\n    have h\u2081\u2083\u2086 : (\u219114 + \u219138 : \u211d) * f 14 38 = (\u219138 : \u211d) * f 14 52 := by exact h\u2081\u2083\u2081\n    have h\u2081\u2083\u2087 : f 14 52 = 364 := by\n      rw [h\u2081\u2083\u2085] at h\u2081\u2083\u2086\n      ring_nf at h\u2081\u2083\u2086 \u22a2\n      <;> linarith\n    exact_mod_cast h\u2081\u2083\u2087\n  exact h\u2081\u2083\n"}
{"mode": "clear", "steps_before": 268, "alphabet": 104, "compiles": false, "id": "amc12a_2021_p8", "proof_after": "theorem amc12a_2021_p8 (d : \u2115 \u2192 \u2115) (h\u2080 : d 0 = 0) (h\u2081 : d 1 = 0) (h\u2082 : d 2 = 1)\n    (h\u2083 : \u2200 n \u2265 3, d n = d (n - 1) + d (n - 3)) : Even (d 2021) \u2227 Odd (d 2022) \u2227 Even (d 2023)  :=  by\n  have h\u2084 : \u2200 n : \u2115, (n % 7 = 0 \u2192 d n % 2 = 0) \u2227 (n % 7 = 1 \u2192 d n % 2 = 0) \u2227 (n % 7 = 2 \u2192 d n % 2 = 1) \u2227 (n % 7 = 3 \u2192 d n % 2 = 1) \u2227 (n % 7 = 4 \u2192 d n % 2 = 1) \u2227 (n % 7 = 5 \u2192 d n % 2 = 0) \u2227 (n % 7 = 6 \u2192 d n % 2 = 1) := by\n    intro n\n    have h\u2084\u2081 : (n % 7 = 0 \u2192 d n % 2 = 0) \u2227 (n % 7 = 1 \u2192 d n % 2 = 0) \u2227 (n % 7 = 2 \u2192 d n % 2 = 1) \u2227 (n % 7 = 3 \u2192 d n % 2 = 1) \u2227 (n % 7 = 4 \u2192 d n % 2 = 1) \u2227 (n % 7 = 5 \u2192 d n % 2 = 0) \u2227 (n % 7 = 6 \u2192 d n % 2 = 1) := by\n      induction n using Nat.strong_induction_on with\n      | h n ih =>\n        have h\u2085 : n % 7 = 0 \u2192 d n % 2 = 0 := by\n          intro h\u2085\n          have h\u2086 : n % 7 = 0 := h\u2085\n          have h\u2087 : d n % 2 = 0 := by\n            by_cases h\u2088 : n \u2265 3\n            \u00b7 have h\u2089 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2088\n              rw [h\u2089]\n              have h\u2081\u2080 : (n - 1) % 7 = 6 := by\n                have h\u2081\u2081 : n % 7 = 0 := h\u2086\n                omega\n              have h\u2081\u2081 : (n - 3) % 7 = 4 := by\n                have h\u2081\u2082 : n % 7 = 0 := h\u2086\n                omega\n              have h\u2081\u2082 : d (n - 1) % 2 = 1 := by\n                have h\u2081\u2083 := ih ((n - 1)) (by\n                  have h\u2081\u2084 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2084 := h\u2081\u2083.2.2.2.2.2.2 (by omega)\n                omega\n              have h\u2081\u2083 : d (n - 3) % 2 = 1 := by\n                have h\u2081\u2084 := ih ((n - 3)) (by\n                  have h\u2081\u2085 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2085 := h\u2081\u2084.2.2.2.2.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2089 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2089 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2087\n        have h\u2086 : n % 7 = 1 \u2192 d n % 2 = 0 := by\n          intro h\u2086\n          have h\u2087 : n % 7 = 1 := h\u2086\n          have h\u2088 : d n % 2 = 0 := by\n            by_cases h\u2089 : n \u2265 3\n            \u00b7 have h\u2081\u2080 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2089\n              rw [h\u2081\u2080]\n              have h\u2081\u2081 : (n - 1) % 7 = 0 := by\n                have h\u2081\u2082 : n % 7 = 1 := h\u2087\n                omega\n              have h\u2081\u2082 : (n - 3) % 7 = 5 := by\n                have h\u2081\u2083 : n % 7 = 1 := h\u2087\n                omega\n              have h\u2081\u2083 : d (n - 1) % 2 = 0 := by\n                have h\u2081\u2084 := ih ((n - 1)) (by\n                  have h\u2081\u2085 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2085 := h\u2081\u2084.1 (by omega)\n                omega\n              have h\u2081\u2084 : d (n - 3) % 2 = 0 := by\n                have h\u2081\u2085 := ih ((n - 3)) (by\n                  have h\u2081\u2086 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2086 := h\u2081\u2085.2.2.2.2.2.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2089 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2089 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2088\n        have h\u2087 : n % 7 = 2 \u2192 d n % 2 = 1 := by\n          intro h\u2087\n          have h\u2088 : n % 7 = 2 := h\u2087\n          have h\u2089 : d n % 2 = 1 := by\n            by_cases h\u2081\u2080 : n \u2265 3\n            \u00b7 have h\u2081\u2081 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2081\u2080\n              rw [h\u2081\u2081]\n              have h\u2081\u2082 : (n - 1) % 7 = 1 := by\n                have h\u2081\u2083 : n % 7 = 2 := h\u2088\n                omega\n              have h\u2081\u2083 : (n - 3) % 7 = 6 := by\n                have h\u2081\u2084 : n % 7 = 2 := h\u2088\n                omega\n              have h\u2081\u2084 : d (n - 1) % 2 = 0 := by\n                have h\u2081\u2085 := ih ((n - 1)) (by\n                  have h\u2081\u2086 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2086 := h\u2081\u2085.2.1 (by omega)\n                omega\n              have h\u2081\u2085 : d (n - 3) % 2 = 1 := by\n                have h\u2081\u2086 := ih ((n - 3)) (by\n                  have h\u2081\u2087 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2087 := h\u2081\u2086.2.2.2.2.2.2 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2081\u2080 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2081\u2080 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2089\n        have h\u2088 : n % 7 = 3 \u2192 d n % 2 = 1 := by\n          intro h\u2088\n          have h\u2089 : n % 7 = 3 := h\u2088\n          have h\u2081\u2080 : d n % 2 = 1 := by\n            by_cases h\u2081\u2081 : n \u2265 3\n            \u00b7 have h\u2081\u2082 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2081\u2081\n              rw [h\u2081\u2082]\n              have h\u2081\u2083 : (n - 1) % 7 = 2 := by\n                have h\u2081\u2084 : n % 7 = 3 := h\u2089\n                omega\n              have h\u2081\u2084 : (n - 3) % 7 = 0 := by\n                have h\u2081\u2085 : n % 7 = 3 := h\u2089\n                omega\n              have h\u2081\u2085 : d (n - 1) % 2 = 1 := by\n                have h\u2081\u2086 := ih ((n - 1)) (by\n                  have h\u2081\u2087 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2087 := h\u2081\u2086.2.2.1 (by omega)\n                omega\n              have h\u2081\u2086 : d (n - 3) % 2 = 0 := by\n                have h\u2081\u2087 := ih ((n - 3)) (by\n                  have h\u2081\u2088 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2088 := h\u2081\u2087.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2081\u2081 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2081\u2081 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2081\u2080\n        have h\u2089 : n % 7 = 4 \u2192 d n % 2 = 1 := by\n          intro h\u2089\n          have h\u2081\u2080 : n % 7 = 4 := h\u2089\n          have h\u2081\u2081 : d n % 2 = 1 := by\n            by_cases h\u2081\u2082 : n \u2265 3\n            \u00b7 have h\u2081\u2083 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2081\u2082\n              rw [h\u2081\u2083]\n              have h\u2081\u2084 : (n - 1) % 7 = 3 := by\n                have h\u2081\u2085 : n % 7 = 4 := h\u2081\u2080\n                omega\n              have h\u2081\u2085 : (n - 3) % 7 = 1 := by\n                have h\u2081\u2086 : n % 7 = 4 := h\u2081\u2080\n                omega\n              have h\u2081\u2086 : d (n - 1) % 2 = 1 := by\n                have h\u2081\u2087 := ih ((n - 1)) (by\n                  have h\u2081\u2088 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2088 := h\u2081\u2087.2.2.2.1 (by omega)\n                omega\n              have h\u2081\u2087 : d (n - 3) % 2 = 0 := by\n                have h\u2081\u2088 := ih ((n - 3)) (by\n                  have h\u2081\u2089 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2089 := h\u2081\u2088.2.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2081\u2082 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2081\u2082 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2081\u2081\n        have h\u2081\u2080 : n % 7 = 5 \u2192 d n % 2 = 0 := by\n          intro h\u2081\u2080\n          have h\u2081\u2081 : n % 7 = 5 := h\u2081\u2080\n          have h\u2081\u2082 : d n % 2 = 0 := by\n            by_cases h\u2081\u2083 : n \u2265 3\n            \u00b7 have h\u2081\u2084 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2081\u2083\n              rw [h\u2081\u2084]\n              have h\u2081\u2085 : (n - 1) % 7 = 4 := by\n                have h\u2081\u2086 : n % 7 = 5 := h\u2081\u2081\n                omega\n              have h\u2081\u2086 : (n - 3) % 7 = 2 := by\n                have h\u2081\u2087 : n % 7 = 5 := h\u2081\u2081\n                omega\n              have h\u2081\u2087 : d (n - 1) % 2 = 1 := by\n                have h\u2081\u2088 := ih ((n - 1)) (by\n                  have h\u2081\u2089 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2089 := h\u2081\u2088.2.2.2.2.1 (by omega)\n                omega\n              have h\u2081\u2088 : d (n - 3) % 2 = 1 := by\n                have h\u2081\u2089 := ih ((n - 3)) (by\n                  have h\u2082\u2080 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2082\u2080 := h\u2081\u2089.2.2.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2081\u2083 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2081\u2083 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2081\u2082\n        have h\u2081\u2081 : n % 7 = 6 \u2192 d n % 2 = 1 := by\n          intro h\u2081\u2081\n          have h\u2081\u2082 : n % 7 = 6 := h\u2081\u2081\n          have h\u2081\u2083 : d n % 2 = 1 := by\n            by_cases h\u2081\u2084 : n \u2265 3\n            \u00b7 have h\u2081\u2085 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2081\u2084\n              rw [h\u2081\u2085]\n              have h\u2081\u2086 : (n - 1) % 7 = 5 := by\n                have h\u2081\u2087 : n % 7 = 6 := h\u2081\u2082\n                omega\n              have h\u2081\u2087 : (n - 3) % 7 = 3 := by\n                have h\u2081\u2088 : n % 7 = 6 := h\u2081\u2082\n                omega\n              have h\u2081\u2088 : d (n - 1) % 2 = 0 := by\n                have h\u2081\u2089 := ih ((n - 1)) (by\n                  have h\u2082\u2080 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2082\u2080 := h\u2081\u2089.2.2.2.2.2.1 (by omega)\n                omega\n              have h\u2081\u2089 : d (n - 3) % 2 = 1 := by\n                have h\u2082\u2080 := ih ((n - 3)) (by\n                  have h\u2082\u2081 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2082\u2081 := h\u2082\u2080.2.2.2.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2081\u2084 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2081\u2084 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2081\u2083\n        exact \u27e8h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080, h\u2081\u2081\u27e9\n    exact h\u2084\u2081\n  have h\u2085 : Even (d 2021) := by\n    have h\u2085\u2081 := h\u2084 2021\n    have h\u2085\u2082 : 2021 % 7 = 5 := by norm_num\n    have h\u2085\u2083 : d 2021 % 2 = 0 := by\n      have h\u2085\u2084 := h\u2085\u2081.2.2.2.2.2.1\n      have h\u2085\u2085 := h\u2085\u2084\n      have h\u2085\u2086 := h\u2085\u2085 (by omega)\n      omega\n    have h\u2085\u2084 : Even (d 2021) := by\n      rw [even_iff_two_dvd]\n      omega\n    exact h\u2085\u2084\n  have h\u2086 : Odd (d 2022) := by\n    have h\u2086\u2081 := h\u2084 2022\n    have h\u2086\u2082 : 2022 % 7 = 6 := by norm_num\n    have h\u2086\u2083 : d 2022 % 2 = 1 := by\n      have h\u2086\u2084 := h\u2086\u2081.2.2.2.2.2.2\n      have h\u2086\u2085 := h\u2086\u2084\n      have h\u2086\u2086 := h\u2086\u2085 (by omega)\n      omega\n    have h\u2086\u2084 : Odd (d 2022) := by\n      rw [Nat.odd_iff_not_even]\n      rw [even_iff_two_dvd]\n      omega\n    exact h\u2086\u2084\n  have h\u2087 : Even (d 2023) := by\n    have h\u2087\u2081 := h\u2084 2023\n    have h\u2087\u2082 : 2023 % 7 = 0 := by norm_num\n    have h\u2087\u2083 : d 2023 % 2 = 0 := by\n      have h\u2087\u2084 := h\u2087\u2081.1\n      have h\u2087\u2085 := h\u2087\u2084\n      have h\u2087\u2086 := h\u2087\u2085 (by omega)\n      omega\n    have h\u2087\u2084 : Even (d 2023) := by\n      rw [even_iff_two_dvd]\n      omega\n    exact h\u2087\u2084\n  exact \u27e8h\u2085, h\u2086, h\u2087\u27e9\n", "proof_before": "theorem amc12a_2021_p8 (d : \u2115 \u2192 \u2115) (h\u2080 : d 0 = 0) (h\u2081 : d 1 = 0) (h\u2082 : d 2 = 1)\n    (h\u2083 : \u2200 n \u2265 3, d n = d (n - 1) + d (n - 3)) : Even (d 2021) \u2227 Odd (d 2022) \u2227 Even (d 2023)  :=  by\n  have h\u2084 : \u2200 n : \u2115, (n % 7 = 0 \u2192 d n % 2 = 0) \u2227 (n % 7 = 1 \u2192 d n % 2 = 0) \u2227 (n % 7 = 2 \u2192 d n % 2 = 1) \u2227 (n % 7 = 3 \u2192 d n % 2 = 1) \u2227 (n % 7 = 4 \u2192 d n % 2 = 1) \u2227 (n % 7 = 5 \u2192 d n % 2 = 0) \u2227 (n % 7 = 6 \u2192 d n % 2 = 1) := by\n    intro n\n    have h\u2084\u2081 : (n % 7 = 0 \u2192 d n % 2 = 0) \u2227 (n % 7 = 1 \u2192 d n % 2 = 0) \u2227 (n % 7 = 2 \u2192 d n % 2 = 1) \u2227 (n % 7 = 3 \u2192 d n % 2 = 1) \u2227 (n % 7 = 4 \u2192 d n % 2 = 1) \u2227 (n % 7 = 5 \u2192 d n % 2 = 0) \u2227 (n % 7 = 6 \u2192 d n % 2 = 1) := by\n      induction n using Nat.strong_induction_on with\n      | h n ih =>\n        have h\u2085 : n % 7 = 0 \u2192 d n % 2 = 0 := by\n          intro h\u2085\n          have h\u2086 : n % 7 = 0 := h\u2085\n          have h\u2087 : d n % 2 = 0 := by\n            by_cases h\u2088 : n \u2265 3\n            \u00b7 have h\u2089 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2088\n              rw [h\u2089]\n              have h\u2081\u2080 : (n - 1) % 7 = 6 := by\n                have h\u2081\u2081 : n % 7 = 0 := h\u2086\n                omega\n              have h\u2081\u2081 : (n - 3) % 7 = 4 := by\n                have h\u2081\u2082 : n % 7 = 0 := h\u2086\n                omega\n              have h\u2081\u2082 : d (n - 1) % 2 = 1 := by\n                have h\u2081\u2083 := ih ((n - 1)) (by\n                  have h\u2081\u2084 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2084 := h\u2081\u2083.2.2.2.2.2.2 (by omega)\n                omega\n              have h\u2081\u2083 : d (n - 3) % 2 = 1 := by\n                have h\u2081\u2084 := ih ((n - 3)) (by\n                  have h\u2081\u2085 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2085 := h\u2081\u2084.2.2.2.2.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2089 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2089 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2087\n        have h\u2086 : n % 7 = 1 \u2192 d n % 2 = 0 := by\n          intro h\u2086\n          have h\u2087 : n % 7 = 1 := h\u2086\n          have h\u2088 : d n % 2 = 0 := by\n            by_cases h\u2089 : n \u2265 3\n            \u00b7 have h\u2081\u2080 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2089\n              rw [h\u2081\u2080]\n              have h\u2081\u2081 : (n - 1) % 7 = 0 := by\n                have h\u2081\u2082 : n % 7 = 1 := h\u2087\n                omega\n              have h\u2081\u2082 : (n - 3) % 7 = 5 := by\n                have h\u2081\u2083 : n % 7 = 1 := h\u2087\n                omega\n              have h\u2081\u2083 : d (n - 1) % 2 = 0 := by\n                have h\u2081\u2084 := ih ((n - 1)) (by\n                  have h\u2081\u2085 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2085 := h\u2081\u2084.1 (by omega)\n                omega\n              have h\u2081\u2084 : d (n - 3) % 2 = 0 := by\n                have h\u2081\u2085 := ih ((n - 3)) (by\n                  have h\u2081\u2086 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2086 := h\u2081\u2085.2.2.2.2.2.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2089 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2089 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2088\n        have h\u2087 : n % 7 = 2 \u2192 d n % 2 = 1 := by\n          intro h\u2087\n          have h\u2088 : n % 7 = 2 := h\u2087\n          have h\u2089 : d n % 2 = 1 := by\n            by_cases h\u2081\u2080 : n \u2265 3\n            \u00b7 have h\u2081\u2081 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2081\u2080\n              rw [h\u2081\u2081]\n              have h\u2081\u2082 : (n - 1) % 7 = 1 := by\n                have h\u2081\u2083 : n % 7 = 2 := h\u2088\n                omega\n              have h\u2081\u2083 : (n - 3) % 7 = 6 := by\n                have h\u2081\u2084 : n % 7 = 2 := h\u2088\n                omega\n              have h\u2081\u2084 : d (n - 1) % 2 = 0 := by\n                have h\u2081\u2085 := ih ((n - 1)) (by\n                  have h\u2081\u2086 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2086 := h\u2081\u2085.2.1 (by omega)\n                omega\n              have h\u2081\u2085 : d (n - 3) % 2 = 1 := by\n                have h\u2081\u2086 := ih ((n - 3)) (by\n                  have h\u2081\u2087 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2087 := h\u2081\u2086.2.2.2.2.2.2 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2081\u2080 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2081\u2080 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2089\n        have h\u2088 : n % 7 = 3 \u2192 d n % 2 = 1 := by\n          intro h\u2088\n          have h\u2089 : n % 7 = 3 := h\u2088\n          have h\u2081\u2080 : d n % 2 = 1 := by\n            by_cases h\u2081\u2081 : n \u2265 3\n            \u00b7 have h\u2081\u2082 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2081\u2081\n              rw [h\u2081\u2082]\n              have h\u2081\u2083 : (n - 1) % 7 = 2 := by\n                have h\u2081\u2084 : n % 7 = 3 := h\u2089\n                omega\n              have h\u2081\u2084 : (n - 3) % 7 = 0 := by\n                have h\u2081\u2085 : n % 7 = 3 := h\u2089\n                omega\n              have h\u2081\u2085 : d (n - 1) % 2 = 1 := by\n                have h\u2081\u2086 := ih ((n - 1)) (by\n                  have h\u2081\u2087 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2087 := h\u2081\u2086.2.2.1 (by omega)\n                omega\n              have h\u2081\u2086 : d (n - 3) % 2 = 0 := by\n                have h\u2081\u2087 := ih ((n - 3)) (by\n                  have h\u2081\u2088 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2088 := h\u2081\u2087.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2081\u2081 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2081\u2081 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2081\u2080\n        have h\u2089 : n % 7 = 4 \u2192 d n % 2 = 1 := by\n          intro h\u2089\n          have h\u2081\u2080 : n % 7 = 4 := h\u2089\n          have h\u2081\u2081 : d n % 2 = 1 := by\n            by_cases h\u2081\u2082 : n \u2265 3\n            \u00b7 have h\u2081\u2083 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2081\u2082\n              rw [h\u2081\u2083]\n              have h\u2081\u2084 : (n - 1) % 7 = 3 := by\n                have h\u2081\u2085 : n % 7 = 4 := h\u2081\u2080\n                omega\n              have h\u2081\u2085 : (n - 3) % 7 = 1 := by\n                have h\u2081\u2086 : n % 7 = 4 := h\u2081\u2080\n                omega\n              have h\u2081\u2086 : d (n - 1) % 2 = 1 := by\n                have h\u2081\u2087 := ih ((n - 1)) (by\n                  have h\u2081\u2088 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2088 := h\u2081\u2087.2.2.2.1 (by omega)\n                omega\n              have h\u2081\u2087 : d (n - 3) % 2 = 0 := by\n                have h\u2081\u2088 := ih ((n - 3)) (by\n                  have h\u2081\u2089 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2089 := h\u2081\u2088.2.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2081\u2082 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2081\u2082 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2081\u2081\n        have h\u2081\u2080 : n % 7 = 5 \u2192 d n % 2 = 0 := by\n          intro h\u2081\u2080\n          have h\u2081\u2081 : n % 7 = 5 := h\u2081\u2080\n          have h\u2081\u2082 : d n % 2 = 0 := by\n            by_cases h\u2081\u2083 : n \u2265 3\n            \u00b7 have h\u2081\u2084 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2081\u2083\n              rw [h\u2081\u2084]\n              have h\u2081\u2085 : (n - 1) % 7 = 4 := by\n                have h\u2081\u2086 : n % 7 = 5 := h\u2081\u2081\n                omega\n              have h\u2081\u2086 : (n - 3) % 7 = 2 := by\n                have h\u2081\u2087 : n % 7 = 5 := h\u2081\u2081\n                omega\n              have h\u2081\u2087 : d (n - 1) % 2 = 1 := by\n                have h\u2081\u2088 := ih ((n - 1)) (by\n                  have h\u2081\u2089 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2081\u2089 := h\u2081\u2088.2.2.2.2.1 (by omega)\n                omega\n              have h\u2081\u2088 : d (n - 3) % 2 = 1 := by\n                have h\u2081\u2089 := ih ((n - 3)) (by\n                  have h\u2082\u2080 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2082\u2080 := h\u2081\u2089.2.2.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2081\u2083 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2081\u2083 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2081\u2082\n        have h\u2081\u2081 : n % 7 = 6 \u2192 d n % 2 = 1 := by\n          intro h\u2081\u2081\n          have h\u2081\u2082 : n % 7 = 6 := h\u2081\u2081\n          have h\u2081\u2083 : d n % 2 = 1 := by\n            by_cases h\u2081\u2084 : n \u2265 3\n            \u00b7 have h\u2081\u2085 : d n = d (n - 1) + d (n - 3) := h\u2083 n h\u2081\u2084\n              rw [h\u2081\u2085]\n              have h\u2081\u2086 : (n - 1) % 7 = 5 := by\n                have h\u2081\u2087 : n % 7 = 6 := h\u2081\u2082\n                omega\n              have h\u2081\u2087 : (n - 3) % 7 = 3 := by\n                have h\u2081\u2088 : n % 7 = 6 := h\u2081\u2082\n                omega\n              have h\u2081\u2088 : d (n - 1) % 2 = 0 := by\n                have h\u2081\u2089 := ih ((n - 1)) (by\n                  have h\u2082\u2080 : n - 1 < n := by\n                    omega\n                  omega)\n                have h\u2082\u2080 := h\u2081\u2089.2.2.2.2.2.1 (by omega)\n                omega\n              have h\u2081\u2089 : d (n - 3) % 2 = 1 := by\n                have h\u2082\u2080 := ih ((n - 3)) (by\n                  have h\u2082\u2081 : n - 3 < n := by\n                    omega\n                  omega)\n                have h\u2082\u2081 := h\u2082\u2080.2.2.2.1 (by omega)\n                omega\n              omega\n            \u00b7 have h\u2081\u2084 : n = 0 \u2228 n = 1 \u2228 n = 2 := by\n                omega\n              rcases h\u2081\u2084 with (rfl | rfl | rfl) <;> simp [h\u2080, h\u2081, h\u2082, Nat.mod_eq_of_lt] <;> norm_num\n              <;> try omega\n          exact h\u2081\u2083\n        exact \u27e8h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080, h\u2081\u2081\u27e9\n    exact h\u2084\u2081\n  have h\u2085 : Even (d 2021) := by\n    have h\u2085\u2081 := h\u2084 2021\n    have h\u2085\u2082 : 2021 % 7 = 5 := by norm_num\n    have h\u2085\u2083 : d 2021 % 2 = 0 := by\n      have h\u2085\u2084 := h\u2085\u2081.2.2.2.2.2.1\n      have h\u2085\u2085 := h\u2085\u2084\n      have h\u2085\u2086 := h\u2085\u2085 (by omega)\n      omega\n    have h\u2085\u2084 : Even (d 2021) := by\n      rw [even_iff_two_dvd]\n      omega\n    exact h\u2085\u2084\n  have h\u2086 : Odd (d 2022) := by\n    have h\u2086\u2081 := h\u2084 2022\n    have h\u2086\u2082 : 2022 % 7 = 6 := by norm_num\n    have h\u2086\u2083 : d 2022 % 2 = 1 := by\n      have h\u2086\u2084 := h\u2086\u2081.2.2.2.2.2.2\n      have h\u2086\u2085 := h\u2086\u2084\n      have h\u2086\u2086 := h\u2086\u2085 (by omega)\n      omega\n    have h\u2086\u2084 : Odd (d 2022) := by\n      rw [Nat.odd_iff_not_even]\n      rw [even_iff_two_dvd]\n      omega\n    exact h\u2086\u2084\n  have h\u2087 : Even (d 2023) := by\n    have h\u2087\u2081 := h\u2084 2023\n    have h\u2087\u2082 : 2023 % 7 = 0 := by norm_num\n    have h\u2087\u2083 : d 2023 % 2 = 0 := by\n      have h\u2087\u2084 := h\u2087\u2081.1\n      have h\u2087\u2085 := h\u2087\u2084\n      have h\u2087\u2086 := h\u2087\u2085 (by omega)\n      omega\n    have h\u2087\u2084 : Even (d 2023) := by\n      rw [even_iff_two_dvd]\n      omega\n    exact h\u2087\u2084\n  exact \u27e8h\u2085, h\u2086, h\u2087\u27e9\n"}
{"mode": "clear", "steps_before": 8, "alphabet": 13, "compiles": false, "id": "mathd_numbertheory_343", "proof_after": "theorem mathd_numbertheory_343 : (\u220f k in Finset.range 6, (2 * k + 1)) % 10 = 5  :=  by\n  have h\u2081 : (\u220f k in Finset.range 6, (2 * k + 1)) = 1 * 3 * 5 * 7 * 9 * 11 := by\n    norm_num [Finset.prod_range_succ, Finset.prod_range_succ, Finset.prod_range_succ,\n      Finset.prod_range_succ, Finset.prod_range_succ, Finset.prod_range_succ]\n    <;> decide\n  have h\u2082 : (\u220f k in Finset.range 6, (2 * k + 1)) % 10 = 5 := by\n    rw [h\u2081]\n    <;> norm_num\n    <;> rfl\n  apply h\u2082\n", "proof_before": "theorem mathd_numbertheory_343 : (\u220f k in Finset.range 6, (2 * k + 1)) % 10 = 5  :=  by\n  have h\u2081 : (\u220f k in Finset.range 6, (2 * k + 1)) = 1 * 3 * 5 * 7 * 9 * 11 := by\n    norm_num [Finset.prod_range_succ, Finset.prod_range_succ, Finset.prod_range_succ,\n      Finset.prod_range_succ, Finset.prod_range_succ, Finset.prod_range_succ]\n    <;> decide\n  have h\u2082 : (\u220f k in Finset.range 6, (2 * k + 1)) % 10 = 5 := by\n    rw [h\u2081]\n    <;> norm_num\n    <;> rfl\n  apply h\u2082\n"}
{"mode": "clear", "steps_before": 45, "alphabet": 23, "compiles": false, "id": "amc12a_2013_p4", "proof_after": "theorem amc12a_2013_p4 : (2 ^ 2014 + 2 ^ 2012) / (2 ^ 2014 - 2 ^ 2012) = (5 : \u211d) / 3  :=  by\n  have h\u2080 : (2 : \u211d) ^ 2014 - (2 : \u211d) ^ 2012 > 0 := by\n    have h\u2081 : (2 : \u211d) ^ 2014 > (2 : \u211d) ^ 2012 := by\n      apply pow_lt_pow_right (by norm_num)\n      <;> norm_num\n    linarith\n  have h\u2081 : ((2 : \u211d) ^ 2014 + (2 : \u211d) ^ 2012) / ((2 : \u211d) ^ 2014 - (2 : \u211d) ^ 2012) = (5 : \u211d) / 3 := by\n    have h\u2082 : (2 : \u211d) ^ 2014 = 4 * (2 : \u211d) ^ 2012 := by\n      ring_nf\n      <;> norm_num [pow_add, pow_mul, mul_assoc]\n      <;>\n      linarith\n    rw [h\u2082]\n    have h\u2083 : (4 * (2 : \u211d) ^ 2012 + (2 : \u211d) ^ 2012 : \u211d) = 5 * (2 : \u211d) ^ 2012 := by\n      ring_nf\n      <;>\n      linarith\n    have h\u2084 : (4 * (2 : \u211d) ^ 2012 - (2 : \u211d) ^ 2012 : \u211d) = 3 * (2 : \u211d) ^ 2012 := by\n      ring_nf\n      <;>\n      linarith\n    rw [h\u2083, h\u2084]\n    have h\u2085 : (5 * (2 : \u211d) ^ 2012 : \u211d) / (3 * (2 : \u211d) ^ 2012 : \u211d) = (5 : \u211d) / 3 := by\n      have h\u2086 : (2 : \u211d) ^ 2012 \u2260 0 := by positivity\n      field_simp [h\u2086]\n      <;>\n      ring_nf\n      <;>\n      linarith\n    rw [h\u2085]\n    <;>\n    ring_nf\n    <;>\n    linarith\n  have h\u2082 : (2 ^ 2014 + 2 ^ 2012 : \u211d) / (2 ^ 2014 - 2 ^ 2012 : \u211d) = (5 : \u211d) / 3 := by\n    norm_num [pow_succ, add_assoc, mul_assoc, mul_comm, mul_left_comm] at h\u2081 \u22a2\n    <;>\n    ring_nf at h\u2081 \u22a2 <;>\n    simpa using h\u2081\n  have h\u2083 : (2 ^ 2014 + 2 ^ 2012) / (2 ^ 2014 - 2 ^ 2012) = (5 : \u211d) / 3 := by\n    norm_num [h\u2082] at *\n    <;>\n    simpa using h\u2082\n  apply h\u2083\n", "proof_before": "theorem amc12a_2013_p4 : (2 ^ 2014 + 2 ^ 2012) / (2 ^ 2014 - 2 ^ 2012) = (5 : \u211d) / 3  :=  by\n  have h\u2080 : (2 : \u211d) ^ 2014 - (2 : \u211d) ^ 2012 > 0 := by\n    have h\u2081 : (2 : \u211d) ^ 2014 > (2 : \u211d) ^ 2012 := by\n      apply pow_lt_pow_right (by norm_num)\n      <;> norm_num\n    linarith\n  have h\u2081 : ((2 : \u211d) ^ 2014 + (2 : \u211d) ^ 2012) / ((2 : \u211d) ^ 2014 - (2 : \u211d) ^ 2012) = (5 : \u211d) / 3 := by\n    have h\u2082 : (2 : \u211d) ^ 2014 = 4 * (2 : \u211d) ^ 2012 := by\n      ring_nf\n      <;> norm_num [pow_add, pow_mul, mul_assoc]\n      <;>\n      linarith\n    rw [h\u2082]\n    have h\u2083 : (4 * (2 : \u211d) ^ 2012 + (2 : \u211d) ^ 2012 : \u211d) = 5 * (2 : \u211d) ^ 2012 := by\n      ring_nf\n      <;>\n      linarith\n    have h\u2084 : (4 * (2 : \u211d) ^ 2012 - (2 : \u211d) ^ 2012 : \u211d) = 3 * (2 : \u211d) ^ 2012 := by\n      ring_nf\n      <;>\n      linarith\n    rw [h\u2083, h\u2084]\n    have h\u2085 : (5 * (2 : \u211d) ^ 2012 : \u211d) / (3 * (2 : \u211d) ^ 2012 : \u211d) = (5 : \u211d) / 3 := by\n      have h\u2086 : (2 : \u211d) ^ 2012 \u2260 0 := by positivity\n      field_simp [h\u2086]\n      <;>\n      ring_nf\n      <;>\n      linarith\n    rw [h\u2085]\n    <;>\n    ring_nf\n    <;>\n    linarith\n  have h\u2082 : (2 ^ 2014 + 2 ^ 2012 : \u211d) / (2 ^ 2014 - 2 ^ 2012 : \u211d) = (5 : \u211d) / 3 := by\n    norm_num [pow_succ, add_assoc, mul_assoc, mul_comm, mul_left_comm] at h\u2081 \u22a2\n    <;>\n    ring_nf at h\u2081 \u22a2 <;>\n    simpa using h\u2081\n  have h\u2083 : (2 ^ 2014 + 2 ^ 2012) / (2 ^ 2014 - 2 ^ 2012) = (5 : \u211d) / 3 := by\n    norm_num [h\u2082] at *\n    <;>\n    simpa using h\u2082\n  apply h\u2083\n"}
{"mode": "clear", "steps_before": 13, "alphabet": 16, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 278, "chars_after": 12, "seconds": 3.5, "id": "mathd_algebra_419", "proof_after": "theorem mathd_algebra_419 (a b : \u211d) (h\u2080 : a = -1) (h\u2081 : b = 5) : -a - b ^ 2 + 3 * (a * b) = -39  :=  by\n  nlinarith\n", "proof_before": "theorem mathd_algebra_419 (a b : \u211d) (h\u2080 : a = -1) (h\u2081 : b = 5) : -a - b ^ 2 + 3 * (a * b) = -39  :=  by\n  have h\u2082 : -a = 1 := by\n    rw [h\u2080]\n    <;> norm_num\n  have h\u2083 : b ^ 2 = 25 := by\n    rw [h\u2081]\n    <;> norm_num\n  have h\u2084 : 3 * (a * b) = -15 := by\n    rw [h\u2080, h\u2081]\n    <;> norm_num\n  have h\u2085 : -a - b ^ 2 + 3 * (a * b) = -39 := by\n    rw [h\u2082, h\u2083, h\u2084]\n    <;> norm_num\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 73, "alphabet": 39, "compiles": true, "steps_after": 46, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 3, "fixpoint": false, "chars_before": 2226, "chars_after": 1254, "seconds": 359.0, "id": "numbertheory_aoddbdiv4asqpbsqmod8eq1", "proof_after": "theorem numbertheory_aoddbdiv4asqpbsqmod8eq1 (a : \u2124) (b : \u2115) (h\u2080 : Odd a) (h\u2081 : 4 \u2223 b) :\n    (a ^ 2 + b ^ 2) % 8 = 1  :=  by\n  have h\u2082 : (a ^ 2 : \u2124) % 8 = 1 := by\n    have h\u2082\u2081 : a % 8 = 1 \u2228 a % 8 = 3 \u2228 a % 8 = 5 \u2228 a % 8 = 7 := by\n      cases' h\u2080 with k hk\n      omega\n    rcases h\u2082\u2081 with (h\u2082\u2081 | h\u2082\u2081 | h\u2082\u2081 | h\u2082\u2081) <;>\n      (try simp [h\u2082\u2081, pow_two, Int.mul_emod, Int.add_emod]) <;>\n      (try norm_num) <;>\n      (try ring_nf at * <;> omega)\n  have h\u2083 : ((b ^ 2 : \u2115) : \u2124) % 8 = 0 := by\n    have h\u2083\u2081 : (4 : \u2115) \u2223 b := h\u2081\n    have h\u2083\u2082 : (8 : \u2115) \u2223 b ^ 2 := by\n      have h\u2083\u2083 : (4 : \u2115) \u2223 b := h\u2081\n      have h\u2083\u2084 : (8 : \u2115) \u2223 b ^ 2 := by\n        obtain \u27e8k, hk\u27e9 := h\u2083\u2083\n        have h\u2083\u2085 : b = 4 * k := by omega\n        rw [h\u2083\u2085]\n        have h\u2083\u2086 : (8 : \u2115) \u2223 (4 * k : \u2115) ^ 2 := by\n          ring_nf at *\n          use 2 * k ^ 2\n          <;> ring_nf at * <;> omega\n        exact h\u2083\u2086\n      exact h\u2083\u2084\n    omega\n  have h\u2084 : (a ^ 2 + (b ^ 2 : \u2124)) % 8 = 1 := by\n    have h\u2084\u2081 : (a ^ 2 + (b ^ 2 : \u2124)) % 8 = ((a ^ 2 : \u2124) % 8 + ((b ^ 2 : \u2115) : \u2124) % 8) % 8 := by\n      simp [Int.add_emod, Int.emod_emod]\n      <;> norm_cast\n      <;> ring_nf at *\n      <;> omega\n    rw [h\u2084\u2081]\n    rw [h\u2082, h\u2083]\n    <;> norm_num\n    <;> ring_nf at *\n    <;> omega\n  have h\u2085 : (a ^ 2 + b ^ 2) % 8 = 1 := by\n    have h\u2085\u2081 : (a ^ 2 + b ^ 2 : \u2124) = (a ^ 2 + (b ^ 2 : \u2124)) := by\n      norm_cast\n    rw [h\u2085\u2081]\n    exact h\u2084\n  exact h\u2085\n", "proof_before": "theorem numbertheory_aoddbdiv4asqpbsqmod8eq1 (a : \u2124) (b : \u2115) (h\u2080 : Odd a) (h\u2081 : 4 \u2223 b) :\n    (a ^ 2 + b ^ 2) % 8 = 1  :=  by\n  have h\u2082 : (a ^ 2 : \u2124) % 8 = 1 := by\n    have h\u2082\u2081 : a % 8 = 1 \u2228 a % 8 = 3 \u2228 a % 8 = 5 \u2228 a % 8 = 7 := by\n      cases' h\u2080 with k hk\n      have : a % 8 = (2 * k + 1) % 8 := by omega\n      have h\u2082\u2082 : (2 * k + 1 : \u2124) % 8 = 1 \u2228 (2 * k + 1 : \u2124) % 8 = 3 \u2228 (2 * k + 1 : \u2124) % 8 = 5 \u2228 (2 * k + 1 : \u2124) % 8 = 7 := by\n        have : (k : \u2124) % 8 = 0 \u2228 (k : \u2124) % 8 = 1 \u2228 (k : \u2124) % 8 = 2 \u2228 (k : \u2124) % 8 = 3 \u2228 (k : \u2124) % 8 = 4 \u2228 (k : \u2124) % 8 = 5 \u2228 (k : \u2124) % 8 = 6 \u2228 (k : \u2124) % 8 = 7 := by\n          omega\n        rcases this with (h | h | h | h | h | h | h | h) <;>\n          (try omega) <;>\n          (try { simp [h, Int.add_emod, Int.mul_emod, pow_two] <;> omega }) <;>\n          (try { omega }) <;>\n          (try { ring_nf at * <;> omega })\n      rcases h\u2082\u2082 with (h\u2082\u2082 | h\u2082\u2082 | h\u2082\u2082 | h\u2082\u2082) <;> omega\n    rcases h\u2082\u2081 with (h\u2082\u2081 | h\u2082\u2081 | h\u2082\u2081 | h\u2082\u2081) <;>\n      (try simp [h\u2082\u2081, pow_two, Int.mul_emod, Int.add_emod]) <;>\n      (try norm_num) <;>\n      (try ring_nf at * <;> omega)\n  have h\u2083 : ((b ^ 2 : \u2115) : \u2124) % 8 = 0 := by\n    have h\u2083\u2081 : (4 : \u2115) \u2223 b := h\u2081\n    have h\u2083\u2082 : (8 : \u2115) \u2223 b ^ 2 := by\n      have h\u2083\u2083 : (4 : \u2115) \u2223 b := h\u2081\n      have h\u2083\u2084 : (8 : \u2115) \u2223 b ^ 2 := by\n        obtain \u27e8k, hk\u27e9 := h\u2083\u2083\n        have h\u2083\u2085 : b = 4 * k := by omega\n        rw [h\u2083\u2085]\n        have h\u2083\u2086 : (8 : \u2115) \u2223 (4 * k : \u2115) ^ 2 := by\n          ring_nf at *\n          use 2 * k ^ 2\n          <;> ring_nf at * <;> omega\n        exact h\u2083\u2086\n      exact h\u2083\u2084\n    have h\u2083\u2083 : ((b ^ 2 : \u2115) : \u2124) % 8 = 0 := by\n      have h\u2083\u2084 : (8 : \u2115) \u2223 b ^ 2 := h\u2083\u2082\n      have h\u2083\u2085 : ((b ^ 2 : \u2115) : \u2124) % 8 = 0 := by\n        have h\u2083\u2086 : (b ^ 2 : \u2115) % 8 = 0 := by\n          omega\n        have h\u2083\u2087 : ((b ^ 2 : \u2115) : \u2124) % 8 = 0 := by\n          norm_cast at h\u2083\u2086 \u22a2\n          <;> omega\n        exact h\u2083\u2087\n      exact h\u2083\u2085\n    exact h\u2083\u2083\n  have h\u2084 : (a ^ 2 + (b ^ 2 : \u2124)) % 8 = 1 := by\n    have h\u2084\u2081 : (a ^ 2 + (b ^ 2 : \u2124)) % 8 = ((a ^ 2 : \u2124) % 8 + ((b ^ 2 : \u2115) : \u2124) % 8) % 8 := by\n      simp [Int.add_emod, Int.emod_emod]\n      <;> norm_cast\n      <;> ring_nf at *\n      <;> omega\n    rw [h\u2084\u2081]\n    rw [h\u2082, h\u2083]\n    <;> norm_num\n    <;> ring_nf at *\n    <;> omega\n  have h\u2085 : (a ^ 2 + b ^ 2) % 8 = 1 := by\n    have h\u2085\u2081 : (a ^ 2 + b ^ 2 : \u2124) = (a ^ 2 + (b ^ 2 : \u2124)) := by\n      norm_cast\n    rw [h\u2085\u2081]\n    exact h\u2084\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 112, "alphabet": 39, "compiles": false, "id": "amc12b_2020_p13", "proof_after": "theorem amc12b_2020_p13 :\n    Real.sqrt (Real.log 6 / Real.log 2 + Real.log 6 / Real.log 3) =\n      Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3)  :=  by\n  have h\u2081 : Real.log 6 / Real.log 2 + Real.log 6 / Real.log 3 = (Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2 := by\n    have h\u2081\u2081 : Real.log 6 = Real.log 2 + Real.log 3 := by\n      have h\u2081\u2081\u2081 : Real.log 6 = Real.log (2 * 3) := by norm_num\n      rw [h\u2081\u2081\u2081]\n      have h\u2081\u2081\u2082 : Real.log (2 * 3) = Real.log 2 + Real.log 3 := by\n        rw [Real.log_mul (by norm_num) (by norm_num)]\n      rw [h\u2081\u2081\u2082]\n    have h\u2081\u2082 : Real.log 6 / Real.log 2 = 1 + Real.log 3 / Real.log 2 := by\n      rw [h\u2081\u2081]\n      field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n    have h\u2081\u2083 : Real.log 6 / Real.log 3 = 1 + Real.log 2 / Real.log 3 := by\n      rw [h\u2081\u2081]\n      field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n      <;> ring_nf\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n      <;> ring_nf\n    rw [h\u2081\u2082, h\u2081\u2083]\n    <;> ring_nf\n    <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n      Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n    <;> ring_nf\n    <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n      Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n    <;> ring_nf\n  have h\u2082 : Real.sqrt (Real.log 6 / Real.log 2 + Real.log 6 / Real.log 3) = Real.sqrt ((Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2) := by\n    rw [h\u2081]\n    <;>\n    simp [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf\n    <;>\n    field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n      Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n    <;>\n    ring_nf\n    <;>\n    field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n      Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n    <;>\n    ring_nf\n  have h\u2083 : Real.sqrt ((Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2) = Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3) := by\n    have h\u2083\u2081 : Real.log 3 / Real.log 2 > 0 := by\n      apply div_pos\n      \u00b7 exact Real.log_pos (by norm_num)\n      \u00b7 exact Real.log_pos (by norm_num)\n    have h\u2083\u2082 : Real.log 2 / Real.log 3 > 0 := by\n      apply div_pos\n      \u00b7 exact Real.log_pos (by norm_num)\n      \u00b7 exact Real.log_pos (by norm_num)\n    have h\u2083\u2083 : Real.log 3 / Real.log 2 > 0 := h\u2083\u2081\n    have h\u2083\u2084 : Real.log 2 / Real.log 3 > 0 := h\u2083\u2082\n    have h\u2083\u2085 : (Real.log 3 / Real.log 2) * (Real.log 2 / Real.log 3) = 1 := by\n      have h\u2083\u2085\u2081 : (Real.log 3 / Real.log 2) * (Real.log 2 / Real.log 3) = 1 := by\n        field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n        <;> ring_nf\n        <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n        <;> ring_nf\n      exact h\u2083\u2085\u2081\n    have h\u2083\u2086 : Real.sqrt ((Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2) = Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3) := by\n      have h\u2083\u2086\u2081 : (Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2 = (Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3)) ^ 2 := by\n        have h\u2083\u2086\u2081\u2081 : 0 < Real.sqrt (Real.log 3 / Real.log 2) := Real.sqrt_pos.mpr h\u2083\u2081\n        have h\u2083\u2086\u2081\u2082 : 0 < Real.sqrt (Real.log 2 / Real.log 3) := Real.sqrt_pos.mpr h\u2083\u2082\n        have h\u2083\u2086\u2081\u2083 : 0 < Real.sqrt (Real.log 3 / Real.log 2) * Real.sqrt (Real.log 2 / Real.log 3) := by positivity\n        have h\u2083\u2086\u2081\u2084 : (Real.sqrt (Real.log 3 / Real.log 2)) ^ 2 = Real.log 3 / Real.log 2 := by\n          rw [Real.sq_sqrt (le_of_lt h\u2083\u2081)]\n        have h\u2083\u2086\u2081\u2085 : (Real.sqrt (Real.log 2 / Real.log 3)) ^ 2 = Real.log 2 / Real.log 3 := by\n          rw [Real.sq_sqrt (le_of_lt h\u2083\u2082)]\n        calc\n          (Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2 = (Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2 := by rfl\n          _ = (Real.sqrt (Real.log 3 / Real.log 2)) ^ 2 + (Real.sqrt (Real.log 2 / Real.log 3)) ^ 2 + 2 := by\n            rw [h\u2083\u2086\u2081\u2084, h\u2083\u2086\u2081\u2085]\n          _ = (Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3)) ^ 2 := by\n            have h\u2083\u2086\u2081\u2086 : 0 < Real.sqrt (Real.log 3 / Real.log 2) * Real.sqrt (Real.log 2 / Real.log 3) := by positivity\n            nlinarith [Real.sq_sqrt (le_of_lt h\u2083\u2081), Real.sq_sqrt (le_of_lt h\u2083\u2082), h\u2083\u2085]\n      have h\u2083\u2086\u2082 : Real.sqrt ((Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2) = Real.sqrt ((Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3)) ^ 2) := by rw [h\u2083\u2086\u2081]\n      rw [h\u2083\u2086\u2082]\n      have h\u2083\u2086\u2083 : Real.sqrt ((Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3)) ^ 2) = Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3) := by\n        rw [Real.sqrt_sq (by positivity)]\n      rw [h\u2083\u2086\u2083]\n    exact h\u2083\u2086\n    <;>\n    simp_all\n    <;>\n    linarith\n  have h\u2084 : Real.sqrt (Real.log 6 / Real.log 2 + Real.log 6 / Real.log 3) = Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3) := by\n    rw [h\u2082]\n    rw [h\u2083]\n    <;>\n    simp_all [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf at *\n    <;>\n    field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n      Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)] at *\n    <;>\n    ring_nf at *\n    <;>\n    nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 3)]\n  rw [h\u2084]\n  <;>\n  simp_all [Real.sqrt_eq_iff_sq_eq]\n  <;>\n  ring_nf at *\n  <;>\n  field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n    Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)] at *\n  <;>\n  ring_nf at *\n  <;>\n  nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 3)]\n", "proof_before": "theorem amc12b_2020_p13 :\n    Real.sqrt (Real.log 6 / Real.log 2 + Real.log 6 / Real.log 3) =\n      Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3)  :=  by\n  have h\u2081 : Real.log 6 / Real.log 2 + Real.log 6 / Real.log 3 = (Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2 := by\n    have h\u2081\u2081 : Real.log 6 = Real.log 2 + Real.log 3 := by\n      have h\u2081\u2081\u2081 : Real.log 6 = Real.log (2 * 3) := by norm_num\n      rw [h\u2081\u2081\u2081]\n      have h\u2081\u2081\u2082 : Real.log (2 * 3) = Real.log 2 + Real.log 3 := by\n        rw [Real.log_mul (by norm_num) (by norm_num)]\n      rw [h\u2081\u2081\u2082]\n    have h\u2081\u2082 : Real.log 6 / Real.log 2 = 1 + Real.log 3 / Real.log 2 := by\n      rw [h\u2081\u2081]\n      field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n    have h\u2081\u2083 : Real.log 6 / Real.log 3 = 1 + Real.log 2 / Real.log 3 := by\n      rw [h\u2081\u2081]\n      field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n      <;> ring_nf\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n      <;> ring_nf\n    rw [h\u2081\u2082, h\u2081\u2083]\n    <;> ring_nf\n    <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n      Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n    <;> ring_nf\n    <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n      Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n    <;> ring_nf\n  have h\u2082 : Real.sqrt (Real.log 6 / Real.log 2 + Real.log 6 / Real.log 3) = Real.sqrt ((Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2) := by\n    rw [h\u2081]\n    <;>\n    simp [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf\n    <;>\n    field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n      Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n    <;>\n    ring_nf\n    <;>\n    field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n      Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n    <;>\n    ring_nf\n  have h\u2083 : Real.sqrt ((Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2) = Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3) := by\n    have h\u2083\u2081 : Real.log 3 / Real.log 2 > 0 := by\n      apply div_pos\n      \u00b7 exact Real.log_pos (by norm_num)\n      \u00b7 exact Real.log_pos (by norm_num)\n    have h\u2083\u2082 : Real.log 2 / Real.log 3 > 0 := by\n      apply div_pos\n      \u00b7 exact Real.log_pos (by norm_num)\n      \u00b7 exact Real.log_pos (by norm_num)\n    have h\u2083\u2083 : Real.log 3 / Real.log 2 > 0 := h\u2083\u2081\n    have h\u2083\u2084 : Real.log 2 / Real.log 3 > 0 := h\u2083\u2082\n    have h\u2083\u2085 : (Real.log 3 / Real.log 2) * (Real.log 2 / Real.log 3) = 1 := by\n      have h\u2083\u2085\u2081 : (Real.log 3 / Real.log 2) * (Real.log 2 / Real.log 3) = 1 := by\n        field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n        <;> ring_nf\n        <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)]\n        <;> ring_nf\n      exact h\u2083\u2085\u2081\n    have h\u2083\u2086 : Real.sqrt ((Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2) = Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3) := by\n      have h\u2083\u2086\u2081 : (Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2 = (Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3)) ^ 2 := by\n        have h\u2083\u2086\u2081\u2081 : 0 < Real.sqrt (Real.log 3 / Real.log 2) := Real.sqrt_pos.mpr h\u2083\u2081\n        have h\u2083\u2086\u2081\u2082 : 0 < Real.sqrt (Real.log 2 / Real.log 3) := Real.sqrt_pos.mpr h\u2083\u2082\n        have h\u2083\u2086\u2081\u2083 : 0 < Real.sqrt (Real.log 3 / Real.log 2) * Real.sqrt (Real.log 2 / Real.log 3) := by positivity\n        have h\u2083\u2086\u2081\u2084 : (Real.sqrt (Real.log 3 / Real.log 2)) ^ 2 = Real.log 3 / Real.log 2 := by\n          rw [Real.sq_sqrt (le_of_lt h\u2083\u2081)]\n        have h\u2083\u2086\u2081\u2085 : (Real.sqrt (Real.log 2 / Real.log 3)) ^ 2 = Real.log 2 / Real.log 3 := by\n          rw [Real.sq_sqrt (le_of_lt h\u2083\u2082)]\n        calc\n          (Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2 = (Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2 := by rfl\n          _ = (Real.sqrt (Real.log 3 / Real.log 2)) ^ 2 + (Real.sqrt (Real.log 2 / Real.log 3)) ^ 2 + 2 := by\n            rw [h\u2083\u2086\u2081\u2084, h\u2083\u2086\u2081\u2085]\n          _ = (Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3)) ^ 2 := by\n            have h\u2083\u2086\u2081\u2086 : 0 < Real.sqrt (Real.log 3 / Real.log 2) * Real.sqrt (Real.log 2 / Real.log 3) := by positivity\n            nlinarith [Real.sq_sqrt (le_of_lt h\u2083\u2081), Real.sq_sqrt (le_of_lt h\u2083\u2082), h\u2083\u2085]\n      have h\u2083\u2086\u2082 : Real.sqrt ((Real.log 3 / Real.log 2) + (Real.log 2 / Real.log 3) + 2) = Real.sqrt ((Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3)) ^ 2) := by rw [h\u2083\u2086\u2081]\n      rw [h\u2083\u2086\u2082]\n      have h\u2083\u2086\u2083 : Real.sqrt ((Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3)) ^ 2) = Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3) := by\n        rw [Real.sqrt_sq (by positivity)]\n      rw [h\u2083\u2086\u2083]\n    exact h\u2083\u2086\n    <;>\n    simp_all\n    <;>\n    linarith\n  have h\u2084 : Real.sqrt (Real.log 6 / Real.log 2 + Real.log 6 / Real.log 3) = Real.sqrt (Real.log 3 / Real.log 2) + Real.sqrt (Real.log 2 / Real.log 3) := by\n    rw [h\u2082]\n    rw [h\u2083]\n    <;>\n    simp_all [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf at *\n    <;>\n    field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n      Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)] at *\n    <;>\n    ring_nf at *\n    <;>\n    nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 3)]\n  rw [h\u2084]\n  <;>\n  simp_all [Real.sqrt_eq_iff_sq_eq]\n  <;>\n  ring_nf at *\n  <;>\n  field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n    Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 3) (by norm_num : (3 : \u211d) \u2260 1)] at *\n  <;>\n  ring_nf at *\n  <;>\n  nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 3)]\n"}
{"mode": "clear", "steps_before": 12, "alphabet": 15, "compiles": false, "id": "mathd_numbertheory_321", "proof_after": "theorem mathd_numbertheory_321 (n : ZMod 1399) (h\u2081 : n = 160\u207b\u00b9) : n = 1058  :=  by\n  have h\u2082 : (n : ZMod 1399) = 1058 := by\n    rw [h\u2081]\n    rw [show (160 : ZMod 1399)\u207b\u00b9 = 1058 by\n      norm_num [ZMod.val_eq_zero, ZMod.val_one, ZMod.val_mul, ZMod.val_nat_cast,\n        Nat.mod_eq_of_lt, Nat.div_eq_of_lt]\n      <;> rfl\n    ]\n    <;> rfl\n  have h\u2083 : n = 1058 := by\n    have h\u2084 : (n : ZMod 1399) = (1058 : ZMod 1399) := by simpa using h\u2082\n    have h\u2085 : n = 1058 := by\n      exact_mod_cast h\u2084\n    exact h\u2085\n  apply h\u2083\n", "proof_before": "theorem mathd_numbertheory_321 (n : ZMod 1399) (h\u2081 : n = 160\u207b\u00b9) : n = 1058  :=  by\n  have h\u2082 : (n : ZMod 1399) = 1058 := by\n    rw [h\u2081]\n    rw [show (160 : ZMod 1399)\u207b\u00b9 = 1058 by\n      norm_num [ZMod.val_eq_zero, ZMod.val_one, ZMod.val_mul, ZMod.val_nat_cast,\n        Nat.mod_eq_of_lt, Nat.div_eq_of_lt]\n      <;> rfl\n    ]\n    <;> rfl\n  have h\u2083 : n = 1058 := by\n    have h\u2084 : (n : ZMod 1399) = (1058 : ZMod 1399) := by simpa using h\u2082\n    have h\u2085 : n = 1058 := by\n      exact_mod_cast h\u2084\n    exact h\u2085\n  apply h\u2083\n"}
{"mode": "clear", "steps_before": 75, "alphabet": 21, "compiles": false, "id": "algebra_9onxpypzleqsum2onxpy", "proof_after": "theorem algebra_9onxpypzleqsum2onxpy (x y z : \u211d) (h\u2080 : 0 < x \u2227 0 < y \u2227 0 < z) :\n    9 / (x + y + z) \u2264 2 / (x + y) + 2 / (y + z) + 2 / (z + x)  :=  by\n  have h\u2081 : 0 < x + y := by linarith\n  have h\u2082 : 0 < y + z := by linarith\n  have h\u2083 : 0 < z + x := by linarith\n  have h\u2084 : 0 < x + y + z := by linarith\n  have h\u2085 : 0 < (x + y) * (y + z) * (z + x) := by positivity\n  have h\u2086 : 0 < (x + y) * (y + z) := by positivity\n  have h\u2087 : 0 < (y + z) * (z + x) := by positivity\n  have h\u2088 : 0 < (z + x) * (x + y) := by positivity\n  have h\u2089 : 2 * (x + y + z) * (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) \u2265 9 := by\n    have h\u2089\u2081 : 0 < x + y := by linarith\n    have h\u2089\u2082 : 0 < y + z := by linarith\n    have h\u2089\u2083 : 0 < z + x := by linarith\n    have h\u2089\u2084 : 0 < (x + y) * (y + z) := by positivity\n    have h\u2089\u2085 : 0 < (y + z) * (z + x) := by positivity\n    have h\u2089\u2086 : 0 < (z + x) * (x + y) := by positivity\n    field_simp [h\u2089\u2081.ne', h\u2089\u2082.ne', h\u2089\u2083.ne']\n    rw [le_div_iff (by positivity)]\n    nlinarith [sq_nonneg (x - y), sq_nonneg (y - z), sq_nonneg (z - x),\n      sq_nonneg (x + y - y - z), sq_nonneg (y + z - z - x), sq_nonneg (z + x - x - y)]\n  have h\u2081\u2080 : (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) \u2265 9 / (2 * (x + y + z)) := by\n    have h\u2081\u2080\u2081 : 0 < x + y + z := by linarith\n    have h\u2081\u2080\u2082 : 0 < 2 * (x + y + z) := by positivity\n    have h\u2081\u2080\u2083 : 0 < x + y := by linarith\n    have h\u2081\u2080\u2084 : 0 < y + z := by linarith\n    have h\u2081\u2080\u2085 : 0 < z + x := by linarith\n    calc\n      (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) \u2265 9 / (2 * (x + y + z)) := by\n        have h\u2081\u2080\u2086 : 2 * (x + y + z) * (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) \u2265 9 := by\n          simpa [add_assoc] using h\u2089\n        calc\n          (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) = (2 * (x + y + z) * (1 / (x + y) + 1 / (y + z) + 1 / (z + x))) / (2 * (x + y + z)) := by\n            field_simp [h\u2081\u2080\u2081.ne', h\u2081\u2080\u2082.ne']\n            <;> ring\n            <;> field_simp [h\u2081\u2080\u2081.ne', h\u2081\u2080\u2082.ne']\n            <;> ring\n          _ \u2265 9 / (2 * (x + y + z)) := by\n            rw [ge_iff_le]\n            rw [div_le_div_iff (by positivity) (by positivity)]\n            nlinarith\n      _ = 9 / (2 * (x + y + z)) := by rfl\n  have h\u2081\u2081 : (2 / (x + y) + 2 / (y + z) + 2 / (z + x)) \u2265 9 / (x + y + z) := by\n    have h\u2081\u2081\u2081 : 2 / (x + y) + 2 / (y + z) + 2 / (z + x) = 2 * (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) := by\n      ring\n    rw [h\u2081\u2081\u2081]\n    have h\u2081\u2081\u2082 : 9 / (x + y + z) = 2 * (9 / (2 * (x + y + z))) := by\n      field_simp [h\u2084.ne']\n      <;> ring\n      <;> field_simp [h\u2084.ne']\n      <;> ring\n    rw [h\u2081\u2081\u2082]\n    have h\u2081\u2081\u2083 : 1 / (x + y) + 1 / (y + z) + 1 / (z + x) \u2265 9 / (2 * (x + y + z)) := by\n      linarith\n    nlinarith\n  have h\u2081\u2082 : 9 / (x + y + z) \u2264 2 / (x + y) + 2 / (y + z) + 2 / (z + x) := by\n    linarith\n  exact h\u2081\u2082\n", "proof_before": "theorem algebra_9onxpypzleqsum2onxpy (x y z : \u211d) (h\u2080 : 0 < x \u2227 0 < y \u2227 0 < z) :\n    9 / (x + y + z) \u2264 2 / (x + y) + 2 / (y + z) + 2 / (z + x)  :=  by\n  have h\u2081 : 0 < x + y := by linarith\n  have h\u2082 : 0 < y + z := by linarith\n  have h\u2083 : 0 < z + x := by linarith\n  have h\u2084 : 0 < x + y + z := by linarith\n  have h\u2085 : 0 < (x + y) * (y + z) * (z + x) := by positivity\n  have h\u2086 : 0 < (x + y) * (y + z) := by positivity\n  have h\u2087 : 0 < (y + z) * (z + x) := by positivity\n  have h\u2088 : 0 < (z + x) * (x + y) := by positivity\n  have h\u2089 : 2 * (x + y + z) * (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) \u2265 9 := by\n    have h\u2089\u2081 : 0 < x + y := by linarith\n    have h\u2089\u2082 : 0 < y + z := by linarith\n    have h\u2089\u2083 : 0 < z + x := by linarith\n    have h\u2089\u2084 : 0 < (x + y) * (y + z) := by positivity\n    have h\u2089\u2085 : 0 < (y + z) * (z + x) := by positivity\n    have h\u2089\u2086 : 0 < (z + x) * (x + y) := by positivity\n    field_simp [h\u2089\u2081.ne', h\u2089\u2082.ne', h\u2089\u2083.ne']\n    rw [le_div_iff (by positivity)]\n    nlinarith [sq_nonneg (x - y), sq_nonneg (y - z), sq_nonneg (z - x),\n      sq_nonneg (x + y - y - z), sq_nonneg (y + z - z - x), sq_nonneg (z + x - x - y)]\n  have h\u2081\u2080 : (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) \u2265 9 / (2 * (x + y + z)) := by\n    have h\u2081\u2080\u2081 : 0 < x + y + z := by linarith\n    have h\u2081\u2080\u2082 : 0 < 2 * (x + y + z) := by positivity\n    have h\u2081\u2080\u2083 : 0 < x + y := by linarith\n    have h\u2081\u2080\u2084 : 0 < y + z := by linarith\n    have h\u2081\u2080\u2085 : 0 < z + x := by linarith\n    calc\n      (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) \u2265 9 / (2 * (x + y + z)) := by\n        have h\u2081\u2080\u2086 : 2 * (x + y + z) * (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) \u2265 9 := by\n          simpa [add_assoc] using h\u2089\n        calc\n          (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) = (2 * (x + y + z) * (1 / (x + y) + 1 / (y + z) + 1 / (z + x))) / (2 * (x + y + z)) := by\n            field_simp [h\u2081\u2080\u2081.ne', h\u2081\u2080\u2082.ne']\n            <;> ring\n            <;> field_simp [h\u2081\u2080\u2081.ne', h\u2081\u2080\u2082.ne']\n            <;> ring\n          _ \u2265 9 / (2 * (x + y + z)) := by\n            rw [ge_iff_le]\n            rw [div_le_div_iff (by positivity) (by positivity)]\n            nlinarith\n      _ = 9 / (2 * (x + y + z)) := by rfl\n  have h\u2081\u2081 : (2 / (x + y) + 2 / (y + z) + 2 / (z + x)) \u2265 9 / (x + y + z) := by\n    have h\u2081\u2081\u2081 : 2 / (x + y) + 2 / (y + z) + 2 / (z + x) = 2 * (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) := by\n      ring\n    rw [h\u2081\u2081\u2081]\n    have h\u2081\u2081\u2082 : 9 / (x + y + z) = 2 * (9 / (2 * (x + y + z))) := by\n      field_simp [h\u2084.ne']\n      <;> ring\n      <;> field_simp [h\u2084.ne']\n      <;> ring\n    rw [h\u2081\u2081\u2082]\n    have h\u2081\u2081\u2083 : 1 / (x + y) + 1 / (y + z) + 1 / (z + x) \u2265 9 / (2 * (x + y + z)) := by\n      linarith\n    nlinarith\n  have h\u2081\u2082 : 9 / (x + y + z) \u2264 2 / (x + y) + 2 / (y + z) + 2 / (z + x) := by\n    linarith\n  exact h\u2081\u2082\n"}
{"mode": "clear", "steps_before": 86, "alphabet": 31, "compiles": false, "id": "amc12b_2020_p22", "proof_after": "theorem amc12b_2020_p22 (t : \u211d) : (2 ^ t - 3 * t) * t / 4 ^ t \u2264 1 / 12  :=  by\n  have h\u2080 : \u2200 (u : \u211d), u - 3 * u ^ 2 \u2264 1 / 12 := by\n    intro u\n    have h\u2081 : u - 3 * u ^ 2 \u2264 1 / 12 := by\n      nlinarith [sq_nonneg (u - 1 / 6), sq_nonneg (u + 1 / 6), sq_nonneg (u - 1 / 3),\n        sq_nonneg (u + 1 / 3), sq_nonneg (u - 1 / 2), sq_nonneg (u + 1 / 2),\n        sq_nonneg (u - 1), sq_nonneg (u + 1), sq_nonneg (u - 2), sq_nonneg (u + 2)]\n    exact h\u2081\n  have h\u2081 : (4 : \u211d) ^ t = (2 : \u211d) ^ (2 * t) := by\n    have h\u2082 : (4 : \u211d) ^ t = (2 : \u211d) ^ (2 * t) := by\n      calc\n        (4 : \u211d) ^ t = (2 : \u211d) ^ (2 * t) := by\n          have h\u2083 : (4 : \u211d) ^ t = (2 : \u211d) ^ (2 * t) := by\n            calc\n              (4 : \u211d) ^ t = ((2 : \u211d) ^ 2) ^ t := by norm_num\n              _ = (2 : \u211d) ^ (2 * t) := by\n                rw [\u2190 Real.rpow_nat_cast]\n                rw [\u2190 Real.rpow_mul] <;> ring_nf <;> norm_num\n                <;>\n                linarith\n              _ = (2 : \u211d) ^ (2 * t) := by rfl\n          rw [h\u2083]\n        _ = (2 : \u211d) ^ (2 * t) := by rfl\n    exact h\u2082\n  have h\u2082 : (4 : \u211d) ^ t = ((2 : \u211d) ^ t) ^ 2 := by\n    have h\u2083 : (4 : \u211d) ^ t = (2 : \u211d) ^ (2 * t) := h\u2081\n    have h\u2084 : (2 : \u211d) ^ (2 * t) = ((2 : \u211d) ^ t) ^ 2 := by\n      calc\n        (2 : \u211d) ^ (2 * t) = (2 : \u211d) ^ (t + t) := by ring\n        _ = (2 : \u211d) ^ t * (2 : \u211d) ^ t := by\n          rw [Real.rpow_add] <;> ring_nf <;> norm_num\n          <;>\n          linarith\n        _ = ((2 : \u211d) ^ t) ^ 2 := by ring\n    rw [h\u2083, h\u2084]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2083 : (2 ^ t - 3 * t) * t / 4 ^ t = (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) := by\n    have h\u2084 : (4 : \u211d) ^ t = ((2 : \u211d) ^ t) ^ 2 := h\u2082\n    have h\u2085 : (4 : \u211d) ^ t > 0 := by positivity\n    have h\u2086 : (2 : \u211d) ^ t > 0 := by positivity\n    calc\n      (2 ^ t - 3 * t) * t / 4 ^ t = ((2 : \u211d) ^ t - 3 * t) * t / 4 ^ t := by norm_num\n      _ = ((2 : \u211d) ^ t - 3 * t) * t / ((2 : \u211d) ^ t) ^ 2 := by rw [h\u2084]\n      _ = ((2 : \u211d) ^ t - 3 * t) * t / ((2 : \u211d) ^ t) ^ 2 := by ring\n      _ = (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) := by\n        have h\u2087 : ((2 : \u211d) ^ t - 3 * t) * t / ((2 : \u211d) ^ t) ^ 2 = (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) := by\n          have h\u2088 : ((2 : \u211d) ^ t - 3 * t) * t / ((2 : \u211d) ^ t) ^ 2 = (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) := by\n            field_simp [h\u2086.ne']\n            <;> ring_nf\n            <;> field_simp [h\u2086.ne']\n            <;> ring_nf\n            <;> field_simp [h\u2086.ne']\n            <;> ring_nf\n          rw [h\u2088]\n        rw [h\u2087]\n      _ = (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) := by ring\n  have h\u2084 : (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) \u2264 1 / 12 := by\n    have h\u2085 : (t / (2 : \u211d) ^ t : \u211d) - 3 * (t / (2 : \u211d) ^ t) ^ 2 \u2264 1 / 12 := by\n      have h\u2086 : (t / (2 : \u211d) ^ t : \u211d) - 3 * (t / (2 : \u211d) ^ t) ^ 2 \u2264 1 / 12 := by\n        have h\u2087 : \u2200 (u : \u211d), u - 3 * u ^ 2 \u2264 1 / 12 := h\u2080\n        have h\u2088 : (t / (2 : \u211d) ^ t : \u211d) - 3 * (t / (2 : \u211d) ^ t) ^ 2 \u2264 1 / 12 := by\n          apply h\u2087\n        exact h\u2088\n      exact h\u2086\n    exact h\u2085\n  have h\u2085 : (2 ^ t - 3 * t) * t / 4 ^ t \u2264 1 / 12 := by\n    rw [h\u2083]\n    exact h\u2084\n  exact h\u2085\n", "proof_before": "theorem amc12b_2020_p22 (t : \u211d) : (2 ^ t - 3 * t) * t / 4 ^ t \u2264 1 / 12  :=  by\n  have h\u2080 : \u2200 (u : \u211d), u - 3 * u ^ 2 \u2264 1 / 12 := by\n    intro u\n    have h\u2081 : u - 3 * u ^ 2 \u2264 1 / 12 := by\n      nlinarith [sq_nonneg (u - 1 / 6), sq_nonneg (u + 1 / 6), sq_nonneg (u - 1 / 3),\n        sq_nonneg (u + 1 / 3), sq_nonneg (u - 1 / 2), sq_nonneg (u + 1 / 2),\n        sq_nonneg (u - 1), sq_nonneg (u + 1), sq_nonneg (u - 2), sq_nonneg (u + 2)]\n    exact h\u2081\n  have h\u2081 : (4 : \u211d) ^ t = (2 : \u211d) ^ (2 * t) := by\n    have h\u2082 : (4 : \u211d) ^ t = (2 : \u211d) ^ (2 * t) := by\n      calc\n        (4 : \u211d) ^ t = (2 : \u211d) ^ (2 * t) := by\n          have h\u2083 : (4 : \u211d) ^ t = (2 : \u211d) ^ (2 * t) := by\n            calc\n              (4 : \u211d) ^ t = ((2 : \u211d) ^ 2) ^ t := by norm_num\n              _ = (2 : \u211d) ^ (2 * t) := by\n                rw [\u2190 Real.rpow_nat_cast]\n                rw [\u2190 Real.rpow_mul] <;> ring_nf <;> norm_num\n                <;>\n                linarith\n              _ = (2 : \u211d) ^ (2 * t) := by rfl\n          rw [h\u2083]\n        _ = (2 : \u211d) ^ (2 * t) := by rfl\n    exact h\u2082\n  have h\u2082 : (4 : \u211d) ^ t = ((2 : \u211d) ^ t) ^ 2 := by\n    have h\u2083 : (4 : \u211d) ^ t = (2 : \u211d) ^ (2 * t) := h\u2081\n    have h\u2084 : (2 : \u211d) ^ (2 * t) = ((2 : \u211d) ^ t) ^ 2 := by\n      calc\n        (2 : \u211d) ^ (2 * t) = (2 : \u211d) ^ (t + t) := by ring\n        _ = (2 : \u211d) ^ t * (2 : \u211d) ^ t := by\n          rw [Real.rpow_add] <;> ring_nf <;> norm_num\n          <;>\n          linarith\n        _ = ((2 : \u211d) ^ t) ^ 2 := by ring\n    rw [h\u2083, h\u2084]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2083 : (2 ^ t - 3 * t) * t / 4 ^ t = (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) := by\n    have h\u2084 : (4 : \u211d) ^ t = ((2 : \u211d) ^ t) ^ 2 := h\u2082\n    have h\u2085 : (4 : \u211d) ^ t > 0 := by positivity\n    have h\u2086 : (2 : \u211d) ^ t > 0 := by positivity\n    calc\n      (2 ^ t - 3 * t) * t / 4 ^ t = ((2 : \u211d) ^ t - 3 * t) * t / 4 ^ t := by norm_num\n      _ = ((2 : \u211d) ^ t - 3 * t) * t / ((2 : \u211d) ^ t) ^ 2 := by rw [h\u2084]\n      _ = ((2 : \u211d) ^ t - 3 * t) * t / ((2 : \u211d) ^ t) ^ 2 := by ring\n      _ = (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) := by\n        have h\u2087 : ((2 : \u211d) ^ t - 3 * t) * t / ((2 : \u211d) ^ t) ^ 2 = (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) := by\n          have h\u2088 : ((2 : \u211d) ^ t - 3 * t) * t / ((2 : \u211d) ^ t) ^ 2 = (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) := by\n            field_simp [h\u2086.ne']\n            <;> ring_nf\n            <;> field_simp [h\u2086.ne']\n            <;> ring_nf\n            <;> field_simp [h\u2086.ne']\n            <;> ring_nf\n          rw [h\u2088]\n        rw [h\u2087]\n      _ = (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) := by ring\n  have h\u2084 : (t / (2 : \u211d) ^ t - 3 * (t / (2 : \u211d) ^ t) ^ 2) \u2264 1 / 12 := by\n    have h\u2085 : (t / (2 : \u211d) ^ t : \u211d) - 3 * (t / (2 : \u211d) ^ t) ^ 2 \u2264 1 / 12 := by\n      have h\u2086 : (t / (2 : \u211d) ^ t : \u211d) - 3 * (t / (2 : \u211d) ^ t) ^ 2 \u2264 1 / 12 := by\n        have h\u2087 : \u2200 (u : \u211d), u - 3 * u ^ 2 \u2264 1 / 12 := h\u2080\n        have h\u2088 : (t / (2 : \u211d) ^ t : \u211d) - 3 * (t / (2 : \u211d) ^ t) ^ 2 \u2264 1 / 12 := by\n          apply h\u2087\n        exact h\u2088\n      exact h\u2086\n    exact h\u2085\n  have h\u2085 : (2 ^ t - 3 * t) * t / 4 ^ t \u2264 1 / 12 := by\n    rw [h\u2083]\n    exact h\u2084\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 121, "alphabet": 64, "compiles": true, "steps_after": 121, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 3247, "chars_after": 3247, "seconds": 530.1, "id": "mathd_algebra_452", "proof_after": "theorem mathd_algebra_452 (a : \u2115 \u2192 \u211d) (h\u2080 : \u2200 n, a (n + 2) - a (n + 1) = a (n + 1) - a n)\n    (h\u2081 : a 1 = 2 / 3) (h\u2082 : a 9 = 4 / 5) : a 5 = 11 / 15  :=  by\n  have h\u2083 : a 2 - a 1 = a 1 - a 0 := by\n    have h\u2083\u2081 := h\u2080 0\n    norm_num at h\u2083\u2081 \u22a2\n    <;> linarith\n  have h\u2084 : a 3 - a 2 = a 1 - a 0 := by\n    have h\u2084\u2081 := h\u2080 1\n    have h\u2084\u2082 := h\u2080 0\n    have h\u2084\u2083 : a 3 - a 2 = a 2 - a 1 := by\n      linarith\n    linarith\n  have h\u2085 : a 4 - a 3 = a 1 - a 0 := by\n    have h\u2085\u2081 := h\u2080 2\n    have h\u2085\u2082 := h\u2080 1\n    have h\u2085\u2083 := h\u2080 0\n    have h\u2085\u2084 : a 4 - a 3 = a 3 - a 2 := by linarith\n    linarith\n  have h\u2086 : a 5 - a 4 = a 1 - a 0 := by\n    have h\u2086\u2081 := h\u2080 3\n    have h\u2086\u2082 := h\u2080 2\n    have h\u2086\u2083 := h\u2080 1\n    have h\u2086\u2084 := h\u2080 0\n    have h\u2086\u2085 : a 5 - a 4 = a 4 - a 3 := by linarith\n    linarith\n  have h\u2087 : a 6 - a 5 = a 1 - a 0 := by\n    have h\u2087\u2081 := h\u2080 4\n    have h\u2087\u2082 := h\u2080 3\n    have h\u2087\u2083 := h\u2080 2\n    have h\u2087\u2084 := h\u2080 1\n    have h\u2087\u2085 := h\u2080 0\n    have h\u2087\u2086 : a 6 - a 5 = a 5 - a 4 := by linarith\n    linarith\n  have h\u2088 : a 7 - a 6 = a 1 - a 0 := by\n    have h\u2088\u2081 := h\u2080 5\n    have h\u2088\u2082 := h\u2080 4\n    have h\u2088\u2083 := h\u2080 3\n    have h\u2088\u2084 := h\u2080 2\n    have h\u2088\u2085 := h\u2080 1\n    have h\u2088\u2086 := h\u2080 0\n    have h\u2088\u2087 : a 7 - a 6 = a 6 - a 5 := by linarith\n    linarith\n  have h\u2089 : a 8 - a 7 = a 1 - a 0 := by\n    have h\u2089\u2081 := h\u2080 6\n    have h\u2089\u2082 := h\u2080 5\n    have h\u2089\u2083 := h\u2080 4\n    have h\u2089\u2084 := h\u2080 3\n    have h\u2089\u2085 := h\u2080 2\n    have h\u2089\u2086 := h\u2080 1\n    have h\u2089\u2087 := h\u2080 0\n    have h\u2089\u2088 : a 8 - a 7 = a 7 - a 6 := by linarith\n    linarith\n  have h\u2081\u2080 : a 9 - a 8 = a 1 - a 0 := by\n    have h\u2081\u2080\u2081 := h\u2080 7\n    have h\u2081\u2080\u2082 := h\u2080 6\n    have h\u2081\u2080\u2083 := h\u2080 5\n    have h\u2081\u2080\u2084 := h\u2080 4\n    have h\u2081\u2080\u2085 := h\u2080 3\n    have h\u2081\u2080\u2086 := h\u2080 2\n    have h\u2081\u2080\u2087 := h\u2080 1\n    have h\u2081\u2080\u2088 := h\u2080 0\n    have h\u2081\u2080\u2089 : a 9 - a 8 = a 8 - a 7 := by linarith\n    linarith\n  have h\u2081\u2081 : a 9 - a 1 = 8 * (a 1 - a 0) := by\n    have h\u2081\u2081\u2081 : a 9 - a 1 = (a 9 - a 8) + (a 8 - a 7) + (a 7 - a 6) + (a 6 - a 5) + (a 5 - a 4) + (a 4 - a 3) + (a 3 - a 2) + (a 2 - a 1) := by\n      linarith\n    rw [h\u2081\u2081\u2081]\n    have h\u2081\u2081\u2082 : a 9 - a 8 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2083 : a 8 - a 7 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2084 : a 7 - a 6 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2085 : a 6 - a 5 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2086 : a 5 - a 4 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2087 : a 4 - a 3 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2088 : a 3 - a 2 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2089 : a 2 - a 1 = a 1 - a 0 := by linarith\n    linarith\n  have h\u2081\u2082 : a 1 - a 0 = 1 / 60 := by\n    have h\u2081\u2082\u2081 : a 9 - a 1 = 8 * (a 1 - a 0) := h\u2081\u2081\n    have h\u2081\u2082\u2082 : a 9 - a 1 = 2 / 15 := by\n      rw [h\u2082, h\u2081]\n      <;> norm_num\n    have h\u2081\u2082\u2083 : 8 * (a 1 - a 0) = 2 / 15 := by linarith\n    have h\u2081\u2082\u2084 : a 1 - a 0 = 1 / 60 := by\n      ring_nf at h\u2081\u2082\u2083 \u22a2\n      linarith\n    exact h\u2081\u2082\u2084\n  have h\u2081\u2083 : a 5 - a 1 = 4 * (a 1 - a 0) := by\n    have h\u2081\u2083\u2081 : a 5 - a 1 = (a 5 - a 4) + (a 4 - a 3) + (a 3 - a 2) + (a 2 - a 1) := by\n      ring\n    rw [h\u2081\u2083\u2081]\n    have h\u2081\u2083\u2082 : a 5 - a 4 = a 1 - a 0 := h\u2086\n    have h\u2081\u2083\u2083 : a 4 - a 3 = a 1 - a 0 := h\u2085\n    have h\u2081\u2083\u2084 : a 3 - a 2 = a 1 - a 0 := h\u2084\n    have h\u2081\u2083\u2085 : a 2 - a 1 = a 1 - a 0 := h\u2083\n    linarith\n  have h\u2081\u2084 : a 5 - a 1 = 1 / 15 := by\n    rw [h\u2081\u2083, h\u2081\u2082]\n    <;> ring_nf\n    <;> norm_num\n  have h\u2081\u2085 : a 5 = 11 / 15 := by\n    have h\u2081\u2085\u2081 : a 5 = 11 / 15 := by\n      have h\u2081\u2085\u2082 : a 5 - a 1 = 1 / 15 := h\u2081\u2084\n      have h\u2081\u2085\u2083 : a 1 = 2 / 3 := h\u2081\n      have h\u2081\u2085\u2084 : a 5 = 11 / 15 := by\n        linarith\n      exact h\u2081\u2085\u2084\n    exact h\u2081\u2085\u2081\n  apply h\u2081\u2085\n", "proof_before": "theorem mathd_algebra_452 (a : \u2115 \u2192 \u211d) (h\u2080 : \u2200 n, a (n + 2) - a (n + 1) = a (n + 1) - a n)\n    (h\u2081 : a 1 = 2 / 3) (h\u2082 : a 9 = 4 / 5) : a 5 = 11 / 15  :=  by\n  have h\u2083 : a 2 - a 1 = a 1 - a 0 := by\n    have h\u2083\u2081 := h\u2080 0\n    norm_num at h\u2083\u2081 \u22a2\n    <;> linarith\n  have h\u2084 : a 3 - a 2 = a 1 - a 0 := by\n    have h\u2084\u2081 := h\u2080 1\n    have h\u2084\u2082 := h\u2080 0\n    have h\u2084\u2083 : a 3 - a 2 = a 2 - a 1 := by\n      linarith\n    linarith\n  have h\u2085 : a 4 - a 3 = a 1 - a 0 := by\n    have h\u2085\u2081 := h\u2080 2\n    have h\u2085\u2082 := h\u2080 1\n    have h\u2085\u2083 := h\u2080 0\n    have h\u2085\u2084 : a 4 - a 3 = a 3 - a 2 := by linarith\n    linarith\n  have h\u2086 : a 5 - a 4 = a 1 - a 0 := by\n    have h\u2086\u2081 := h\u2080 3\n    have h\u2086\u2082 := h\u2080 2\n    have h\u2086\u2083 := h\u2080 1\n    have h\u2086\u2084 := h\u2080 0\n    have h\u2086\u2085 : a 5 - a 4 = a 4 - a 3 := by linarith\n    linarith\n  have h\u2087 : a 6 - a 5 = a 1 - a 0 := by\n    have h\u2087\u2081 := h\u2080 4\n    have h\u2087\u2082 := h\u2080 3\n    have h\u2087\u2083 := h\u2080 2\n    have h\u2087\u2084 := h\u2080 1\n    have h\u2087\u2085 := h\u2080 0\n    have h\u2087\u2086 : a 6 - a 5 = a 5 - a 4 := by linarith\n    linarith\n  have h\u2088 : a 7 - a 6 = a 1 - a 0 := by\n    have h\u2088\u2081 := h\u2080 5\n    have h\u2088\u2082 := h\u2080 4\n    have h\u2088\u2083 := h\u2080 3\n    have h\u2088\u2084 := h\u2080 2\n    have h\u2088\u2085 := h\u2080 1\n    have h\u2088\u2086 := h\u2080 0\n    have h\u2088\u2087 : a 7 - a 6 = a 6 - a 5 := by linarith\n    linarith\n  have h\u2089 : a 8 - a 7 = a 1 - a 0 := by\n    have h\u2089\u2081 := h\u2080 6\n    have h\u2089\u2082 := h\u2080 5\n    have h\u2089\u2083 := h\u2080 4\n    have h\u2089\u2084 := h\u2080 3\n    have h\u2089\u2085 := h\u2080 2\n    have h\u2089\u2086 := h\u2080 1\n    have h\u2089\u2087 := h\u2080 0\n    have h\u2089\u2088 : a 8 - a 7 = a 7 - a 6 := by linarith\n    linarith\n  have h\u2081\u2080 : a 9 - a 8 = a 1 - a 0 := by\n    have h\u2081\u2080\u2081 := h\u2080 7\n    have h\u2081\u2080\u2082 := h\u2080 6\n    have h\u2081\u2080\u2083 := h\u2080 5\n    have h\u2081\u2080\u2084 := h\u2080 4\n    have h\u2081\u2080\u2085 := h\u2080 3\n    have h\u2081\u2080\u2086 := h\u2080 2\n    have h\u2081\u2080\u2087 := h\u2080 1\n    have h\u2081\u2080\u2088 := h\u2080 0\n    have h\u2081\u2080\u2089 : a 9 - a 8 = a 8 - a 7 := by linarith\n    linarith\n  have h\u2081\u2081 : a 9 - a 1 = 8 * (a 1 - a 0) := by\n    have h\u2081\u2081\u2081 : a 9 - a 1 = (a 9 - a 8) + (a 8 - a 7) + (a 7 - a 6) + (a 6 - a 5) + (a 5 - a 4) + (a 4 - a 3) + (a 3 - a 2) + (a 2 - a 1) := by\n      linarith\n    rw [h\u2081\u2081\u2081]\n    have h\u2081\u2081\u2082 : a 9 - a 8 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2083 : a 8 - a 7 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2084 : a 7 - a 6 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2085 : a 6 - a 5 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2086 : a 5 - a 4 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2087 : a 4 - a 3 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2088 : a 3 - a 2 = a 1 - a 0 := by linarith\n    have h\u2081\u2081\u2089 : a 2 - a 1 = a 1 - a 0 := by linarith\n    linarith\n  have h\u2081\u2082 : a 1 - a 0 = 1 / 60 := by\n    have h\u2081\u2082\u2081 : a 9 - a 1 = 8 * (a 1 - a 0) := h\u2081\u2081\n    have h\u2081\u2082\u2082 : a 9 - a 1 = 2 / 15 := by\n      rw [h\u2082, h\u2081]\n      <;> norm_num\n    have h\u2081\u2082\u2083 : 8 * (a 1 - a 0) = 2 / 15 := by linarith\n    have h\u2081\u2082\u2084 : a 1 - a 0 = 1 / 60 := by\n      ring_nf at h\u2081\u2082\u2083 \u22a2\n      linarith\n    exact h\u2081\u2082\u2084\n  have h\u2081\u2083 : a 5 - a 1 = 4 * (a 1 - a 0) := by\n    have h\u2081\u2083\u2081 : a 5 - a 1 = (a 5 - a 4) + (a 4 - a 3) + (a 3 - a 2) + (a 2 - a 1) := by\n      ring\n    rw [h\u2081\u2083\u2081]\n    have h\u2081\u2083\u2082 : a 5 - a 4 = a 1 - a 0 := h\u2086\n    have h\u2081\u2083\u2083 : a 4 - a 3 = a 1 - a 0 := h\u2085\n    have h\u2081\u2083\u2084 : a 3 - a 2 = a 1 - a 0 := h\u2084\n    have h\u2081\u2083\u2085 : a 2 - a 1 = a 1 - a 0 := h\u2083\n    linarith\n  have h\u2081\u2084 : a 5 - a 1 = 1 / 15 := by\n    rw [h\u2081\u2083, h\u2081\u2082]\n    <;> ring_nf\n    <;> norm_num\n  have h\u2081\u2085 : a 5 = 11 / 15 := by\n    have h\u2081\u2085\u2081 : a 5 = 11 / 15 := by\n      have h\u2081\u2085\u2082 : a 5 - a 1 = 1 / 15 := h\u2081\u2084\n      have h\u2081\u2085\u2083 : a 1 = 2 / 3 := h\u2081\n      have h\u2081\u2085\u2084 : a 5 = 11 / 15 := by\n        linarith\n      exact h\u2081\u2085\u2084\n    exact h\u2081\u2085\u2081\n  apply h\u2081\u2085\n"}
{"mode": "clear", "steps_before": 29, "alphabet": 23, "compiles": false, "id": "mathd_numbertheory_222", "proof_after": "theorem mathd_numbertheory_222 (b : \u2115) (h\u2080 : Nat.lcm 120 b = 3720) (h\u2081 : Nat.gcd 120 b = 8) :\n    b = 248  :=  by\n  have h\u2082 : 120 * b = 29760 := by\n    have h\u2082\u2081 : Nat.gcd 120 b * Nat.lcm 120 b = 120 * b := by\n      rw [Nat.gcd_mul_lcm]\n    rw [h\u2081, h\u2080] at h\u2082\u2081\n    <;> norm_num at h\u2082\u2081 \u22a2\n    <;> linarith\n  have h\u2083 : b = 248 := by\n    have h\u2084 : b = 248 := by\n      have h\u2085 : 120 * b = 29760 := h\u2082\n      have h\u2086 : b = 248 := by\n        have h\u2087 : b \u2264 248 := by\n          by_contra h\n          have h\u2088 : b \u2265 249 := by\n            omega\n          have h\u2089 : 120 * b \u2265 120 * 249 := by\n            exact Nat.mul_le_mul_left 120 h\u2088\n          have h\u2081\u2080 : 120 * 249 = 29880 := by norm_num\n          have h\u2081\u2081 : 120 * b \u2265 29880 := by\n            linarith\n          have h\u2081\u2082 : 120 * b > 29760 := by\n            omega\n          omega\n        interval_cases b <;> norm_num at h\u2085 \u22a2 <;> try omega\n      exact h\u2086\n    exact h\u2084\n  exact h\u2083\n", "proof_before": "theorem mathd_numbertheory_222 (b : \u2115) (h\u2080 : Nat.lcm 120 b = 3720) (h\u2081 : Nat.gcd 120 b = 8) :\n    b = 248  :=  by\n  have h\u2082 : 120 * b = 29760 := by\n    have h\u2082\u2081 : Nat.gcd 120 b * Nat.lcm 120 b = 120 * b := by\n      rw [Nat.gcd_mul_lcm]\n    rw [h\u2081, h\u2080] at h\u2082\u2081\n    <;> norm_num at h\u2082\u2081 \u22a2\n    <;> linarith\n  have h\u2083 : b = 248 := by\n    have h\u2084 : b = 248 := by\n      have h\u2085 : 120 * b = 29760 := h\u2082\n      have h\u2086 : b = 248 := by\n        have h\u2087 : b \u2264 248 := by\n          by_contra h\n          have h\u2088 : b \u2265 249 := by\n            omega\n          have h\u2089 : 120 * b \u2265 120 * 249 := by\n            exact Nat.mul_le_mul_left 120 h\u2088\n          have h\u2081\u2080 : 120 * 249 = 29880 := by norm_num\n          have h\u2081\u2081 : 120 * b \u2265 29880 := by\n            linarith\n          have h\u2081\u2082 : 120 * b > 29760 := by\n            omega\n          omega\n        interval_cases b <;> norm_num at h\u2085 \u22a2 <;> try omega\n      exact h\u2086\n    exact h\u2084\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 11, "alphabet": 13, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 383, "chars_after": 6, "seconds": 94.2, "id": "mathd_numbertheory_728", "proof_after": "theorem mathd_numbertheory_728 : (29 ^ 13 - 5 ^ 13) % 7 = 3  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_728 : (29 ^ 13 - 5 ^ 13) % 7 = 3  :=  by\n  have h\u2081 : 29 ^ 13 % 7 = 1 := by\n    norm_num [Nat.pow_mod, Nat.mod_mod, Nat.mod_eq_of_lt, show 2 \u2264 7 by norm_num]\n    <;> rfl\n  have h\u2082 : 5 ^ 13 % 7 = 5 := by\n    norm_num [Nat.pow_mod, Nat.mod_mod, Nat.mod_eq_of_lt, show 2 \u2264 7 by norm_num]\n    <;> rfl\n  have h\u2083 : (29 ^ 13 - 5 ^ 13) % 7 = 3 := by\n    have h\u2084 : (29 ^ 13 - 5 ^ 13) % 7 = 3 := by\n      omega\n    exact h\u2084\n  apply h\u2083\n"}
{"mode": "clear", "steps_before": 43, "alphabet": 28, "compiles": false, "id": "aime_1990_p4", "proof_after": "theorem aime_1990_p4 (x : \u211d) (h\u2080 : 0 < x) (h\u2081 : x ^ 2 - 10 * x - 29 \u2260 0)\n    (h\u2082 : x ^ 2 - 10 * x - 45 \u2260 0) (h\u2083 : x ^ 2 - 10 * x - 69 \u2260 0)\n    (h\u2084 : 1 / (x ^ 2 - 10 * x - 29) + 1 / (x ^ 2 - 10 * x - 45) - 2 / (x ^ 2 - 10 * x - 69) = 0) :\n    x = 13  :=  by\n  have h_y : x ^ 2 - 10 * x = 39 := by\n    have h\u2085 : (x ^ 2 - 10 * x - 45) \u2260 0 := by simpa using h\u2082\n    have h\u2086 : (x ^ 2 - 10 * x - 29) \u2260 0 := by simpa using h\u2081\n    have h\u2087 : (x ^ 2 - 10 * x - 69) \u2260 0 := by simpa using h\u2083\n    have h\u2088 : 1 / (x ^ 2 - 10 * x - 29) + 1 / (x ^ 2 - 10 * x - 45) - 2 / (x ^ 2 - 10 * x - 69) = 0 := h\u2084\n    have h\u2089 : (x ^ 2 - 10 * x - 45) * (x ^ 2 - 10 * x - 69) + (x ^ 2 - 10 * x - 29) * (x ^ 2 - 10 * x - 69) - 2 * (x ^ 2 - 10 * x - 29) * (x ^ 2 - 10 * x - 45) = 0 := by\n      have h\u2089\u2081 : (x ^ 2 - 10 * x - 45) * (x ^ 2 - 10 * x - 69) + (x ^ 2 - 10 * x - 29) * (x ^ 2 - 10 * x - 69) - 2 * (x ^ 2 - 10 * x - 29) * (x ^ 2 - 10 * x - 45) = 0 := by\n        field_simp [h\u2085, h\u2086, h\u2087, sub_eq_zero, add_eq_zero_iff_eq_neg] at h\u2088\n        nlinarith [sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2086), sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2085), sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2087)]\n      exact h\u2089\u2081\n    have h\u2081\u2080 : x ^ 2 - 10 * x = 39 := by\n      nlinarith [sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2086), sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2085), sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2087)]\n    exact h\u2081\u2080\n  have h_final : x = 13 := by\n    have h\u2085 : x = 13 := by\n      have h\u2085\u2081 : x = 13 \u2228 x = -3 := by\n        have h\u2085\u2082 : x ^ 2 - 10 * x - 39 = 0 := by\n          nlinarith\n        have h\u2085\u2083 : (x - 13) * (x + 3) = 0 := by\n          nlinarith\n        have h\u2085\u2084 : x - 13 = 0 \u2228 x + 3 = 0 := by\n          apply eq_zero_or_eq_zero_of_mul_eq_zero h\u2085\u2083\n        cases h\u2085\u2084 with\n        | inl h\u2085\u2084 =>\n          have h\u2085\u2085 : x = 13 := by linarith\n          exact Or.inl h\u2085\u2085\n        | inr h\u2085\u2084 =>\n          have h\u2085\u2085 : x = -3 := by linarith\n          exact Or.inr h\u2085\u2085\n      cases h\u2085\u2081 with\n      | inl h\u2085\u2081 =>\n        exact h\u2085\u2081\n      | inr h\u2085\u2081 =>\n        have h\u2085\u2082 : x > 0 := h\u2080\n        have h\u2085\u2083 : x = -3 := h\u2085\u2081\n        linarith\n    exact h\u2085\n  exact h_final\n", "proof_before": "theorem aime_1990_p4 (x : \u211d) (h\u2080 : 0 < x) (h\u2081 : x ^ 2 - 10 * x - 29 \u2260 0)\n    (h\u2082 : x ^ 2 - 10 * x - 45 \u2260 0) (h\u2083 : x ^ 2 - 10 * x - 69 \u2260 0)\n    (h\u2084 : 1 / (x ^ 2 - 10 * x - 29) + 1 / (x ^ 2 - 10 * x - 45) - 2 / (x ^ 2 - 10 * x - 69) = 0) :\n    x = 13  :=  by\n  have h_y : x ^ 2 - 10 * x = 39 := by\n    have h\u2085 : (x ^ 2 - 10 * x - 45) \u2260 0 := by simpa using h\u2082\n    have h\u2086 : (x ^ 2 - 10 * x - 29) \u2260 0 := by simpa using h\u2081\n    have h\u2087 : (x ^ 2 - 10 * x - 69) \u2260 0 := by simpa using h\u2083\n    have h\u2088 : 1 / (x ^ 2 - 10 * x - 29) + 1 / (x ^ 2 - 10 * x - 45) - 2 / (x ^ 2 - 10 * x - 69) = 0 := h\u2084\n    have h\u2089 : (x ^ 2 - 10 * x - 45) * (x ^ 2 - 10 * x - 69) + (x ^ 2 - 10 * x - 29) * (x ^ 2 - 10 * x - 69) - 2 * (x ^ 2 - 10 * x - 29) * (x ^ 2 - 10 * x - 45) = 0 := by\n      have h\u2089\u2081 : (x ^ 2 - 10 * x - 45) * (x ^ 2 - 10 * x - 69) + (x ^ 2 - 10 * x - 29) * (x ^ 2 - 10 * x - 69) - 2 * (x ^ 2 - 10 * x - 29) * (x ^ 2 - 10 * x - 45) = 0 := by\n        field_simp [h\u2085, h\u2086, h\u2087, sub_eq_zero, add_eq_zero_iff_eq_neg] at h\u2088\n        nlinarith [sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2086), sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2085), sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2087)]\n      exact h\u2089\u2081\n    have h\u2081\u2080 : x ^ 2 - 10 * x = 39 := by\n      nlinarith [sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2086), sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2085), sq_pos_of_ne_zero (sub_ne_zero.mpr h\u2087)]\n    exact h\u2081\u2080\n  have h_final : x = 13 := by\n    have h\u2085 : x = 13 := by\n      have h\u2085\u2081 : x = 13 \u2228 x = -3 := by\n        have h\u2085\u2082 : x ^ 2 - 10 * x - 39 = 0 := by\n          nlinarith\n        have h\u2085\u2083 : (x - 13) * (x + 3) = 0 := by\n          nlinarith\n        have h\u2085\u2084 : x - 13 = 0 \u2228 x + 3 = 0 := by\n          apply eq_zero_or_eq_zero_of_mul_eq_zero h\u2085\u2083\n        cases h\u2085\u2084 with\n        | inl h\u2085\u2084 =>\n          have h\u2085\u2085 : x = 13 := by linarith\n          exact Or.inl h\u2085\u2085\n        | inr h\u2085\u2084 =>\n          have h\u2085\u2085 : x = -3 := by linarith\n          exact Or.inr h\u2085\u2085\n      cases h\u2085\u2081 with\n      | inl h\u2085\u2081 =>\n        exact h\u2085\u2081\n      | inr h\u2085\u2081 =>\n        have h\u2085\u2082 : x > 0 := h\u2080\n        have h\u2085\u2083 : x = -3 := h\u2085\u2081\n        linarith\n    exact h\u2085\n  exact h_final\n"}
{"mode": "clear", "steps_before": 32, "alphabet": 19, "compiles": true, "steps_after": 5, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "fixpoint": true, "chars_before": 807, "chars_after": 139, "seconds": 633.4, "id": "mathd_numbertheory_100", "proof_after": "theorem mathd_numbertheory_100 (n : \u2115) (h\u2080 : 0 < n) (h\u2081 : Nat.gcd n 40 = 10)\n    (h\u2082 : Nat.lcm n 40 = 280) : n = 70  :=  by\n  have h\u2083 : n * 40 = 2800 := by\n    have h\u2083\u2081 : Nat.gcd n 40 * Nat.lcm n 40 = n * 40 := by\n      rw [Nat.gcd_mul_lcm]\n    nlinarith\n  omega\n", "proof_before": "theorem mathd_numbertheory_100 (n : \u2115) (h\u2080 : 0 < n) (h\u2081 : Nat.gcd n 40 = 10)\n    (h\u2082 : Nat.lcm n 40 = 280) : n = 70  :=  by\n  have h\u2083 : n * 40 = 2800 := by\n    have h\u2083\u2081 : Nat.gcd n 40 * Nat.lcm n 40 = n * 40 := by\n      rw [Nat.gcd_mul_lcm]\n    rw [h\u2081, h\u2082] at h\u2083\u2081\n    <;> norm_num at h\u2083\u2081 \u22a2\n    <;> linarith\n  have h\u2084 : n = 70 := by\n    have h\u2084\u2081 : n * 40 = 2800 := h\u2083\n    have h\u2084\u2082 : n = 70 := by\n      have h\u2084\u2083 : n \u2264 70 := by\n        by_contra h\n        have h\u2084\u2084 : n \u2265 71 := by\n          omega\n        have h\u2084\u2085 : n * 40 \u2265 71 * 40 := by\n          nlinarith\n        have h\u2084\u2086 : n * 40 > 2800 := by\n          nlinarith\n        omega\n      have h\u2084\u2084 : n \u2265 70 := by\n        by_contra h\n        have h\u2084\u2085 : n \u2264 69 := by\n          omega\n        have h\u2084\u2086 : n * 40 \u2264 69 * 40 := by\n          nlinarith\n        have h\u2084\u2087 : n * 40 < 2800 := by\n          nlinarith\n        omega\n      omega\n    exact h\u2084\u2082\n  rw [h\u2084]\n  <;> norm_num\n  <;> try linarith\n"}
{"mode": "clear", "steps_before": 20, "alphabet": 19, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 606, "chars_after": 11, "seconds": 108.6, "id": "mathd_algebra_388", "proof_after": "theorem mathd_algebra_388 (x y z : \u211d) (h\u2080 : 3 * x + 4 * y - 12 * z = 10)\n    (h\u2081 : -2 * x - 3 * y + 9 * z = -4) : x = 14  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_388 (x y z : \u211d) (h\u2080 : 3 * x + 4 * y - 12 * z = 10)\n    (h\u2081 : -2 * x - 3 * y + 9 * z = -4) : x = 14  :=  by\n  have h\u2082 : 9 * x + 12 * y - 36 * z = 30 := by\n    have h\u2082\u2081 : 9 * x + 12 * y - 36 * z = 30 := by\n      have h\u2082\u2082 : 3 * (3 * x + 4 * y - 12 * z) = 3 * 10 := by\n        rw [h\u2080]\n      ring_nf at h\u2082\u2082 \u22a2\n      linarith\n    exact h\u2082\u2081\n  have h\u2083 : -8 * x - 12 * y + 36 * z = -16 := by\n    have h\u2083\u2081 : 4 * (-2 * x - 3 * y + 9 * z) = 4 * (-4) := by\n      rw [h\u2081]\n    ring_nf at h\u2083\u2081 \u22a2\n    linarith\n  have h\u2084 : x = 14 := by\n    have h\u2084\u2081 : x = 14 := by\n      have h\u2084\u2082 : 9 * x + 12 * y - 36 * z + (-8 * x - 12 * y + 36 * z) = 30 + (-16) := by\n        linarith\n      ring_nf at h\u2084\u2082 \u22a2\n      linarith\n    exact h\u2084\u2081\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 86, "alphabet": 43, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 68, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 2, "fixpoint": false, "chars_before": 3329, "chars_after": 2836, "seconds": 346.6, "id": "algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7", "proof_after": "theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7 (f z : \u2102) (h\u2080 : f + 3 * z = 11)\n    (h\u2081 : 3 * (f - 1) - 5 * z = -68) : f = -10 \u2227 z = 7  :=  by\n  have h\u2082 : f.im = 0 := by\n    have h\u2082\u2081 := h\u2080\n    have h\u2082\u2082 := h\u2081\n    simp [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im,\n      Complex.sub_re, Complex.sub_im, Complex.one_re, Complex.one_im, Complex.ext_iff,\n      Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.sub_re,\n      Complex.sub_im, Complex.one_re, Complex.one_im] at h\u2082\u2081 h\u2082\u2082 \u22a2\n    nlinarith\n  have h\u2083 : z.im = 0 := by\n    have h\u2083\u2081 := h\u2080\n    have h\u2083\u2082 := h\u2081\n    simp [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im,\n      Complex.sub_re, Complex.sub_im, Complex.one_re, Complex.one_im, Complex.ext_iff,\n      Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.sub_re,\n      Complex.sub_im, Complex.one_re, Complex.one_im] at h\u2083\u2081 h\u2083\u2082 h\u2082 \u22a2\n    nlinarith\n  have h\u2084 : f.re + 3 * z.re = 11 := by\n    have h\u2084\u2081 := h\u2080\n    simp [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, h\u2082, h\u2083] at h\u2084\u2081 \u22a2\n    <;>\n    omega\n  have h\u2085 : 3 * f.re - 5 * z.re = -65 := by\n    have h\u2085\u2081 := h\u2081\n    simp [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.sub_re, Complex.sub_im,\n      Complex.one_re, Complex.one_im, h\u2082, h\u2083] at h\u2085\u2081 \u22a2\n    <;>\n    (try norm_num at h\u2085\u2081 \u22a2) <;>\n    (try ring_nf at h\u2085\u2081 \u22a2) <;>\n    (try nlinarith) <;>\n    (try linarith)\n    <;>\n    (try\n      {\n        simp_all [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.sub_re, Complex.sub_im,\n          Complex.one_re, Complex.one_im]\n        <;>\n        norm_num\n        <;>\n        linarith\n      })\n  have h\u2086 : f.re = -10 := by\n    have h\u2086\u2081 : f.re + 3 * z.re = 11 := h\u2084\n    have h\u2086\u2082 : 3 * f.re - 5 * z.re = -65 := h\u2085\n    have h\u2086\u2083 : f.re = -10 := by\n      have h\u2086\u2084 : z.re = 7 := by\n        linarith\n      have h\u2086\u2085 : f.re = -10 := by\n        linarith\n      exact h\u2086\u2085\n    exact h\u2086\u2083\n  have h\u2087 : z.re = 7 := by\n    have h\u2087\u2081 : f.re + 3 * z.re = 11 := h\u2084\n    have h\u2087\u2082 : 3 * f.re - 5 * z.re = -65 := h\u2085\n    have h\u2087\u2083 : f.re = -10 := h\u2086\n    have h\u2087\u2084 : z.re = 7 := by\n      linarith\n    exact h\u2087\u2084\n  have h\u2088 : f = -10 := by\n    have h\u2088\u2081 : f.re = -10 := h\u2086\n    have h\u2088\u2082 : f.im = 0 := h\u2082\n    have h\u2088\u2083 : f = -10 := by\n      apply Complex.ext <;> simp_all [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.ext_iff, Complex.one_re, Complex.one_im]\n      <;> norm_num <;> linarith\n    exact h\u2088\u2083\n  have h\u2089 : z = 7 := by\n    have h\u2089\u2081 : z.re = 7 := h\u2087\n    have h\u2089\u2082 : z.im = 0 := h\u2083\n    have h\u2089\u2083 : z = 7 := by\n      apply Complex.ext <;> simp_all [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.ext_iff, Complex.one_re, Complex.one_im]\n      <;> norm_num <;> linarith\n    exact h\u2089\u2083\n  have h\u2081\u2080 : f = -10 \u2227 z = 7 := by\n    exact \u27e8h\u2088, h\u2089\u27e9\n  exact h\u2081\u2080\n", "proof_before": "theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7 (f z : \u2102) (h\u2080 : f + 3 * z = 11)\n    (h\u2081 : 3 * (f - 1) - 5 * z = -68) : f = -10 \u2227 z = 7  :=  by\n  have h\u2082 : f.im = 0 := by\n    have h\u2082\u2081 := h\u2080\n    have h\u2082\u2082 := h\u2081\n    simp [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im,\n      Complex.sub_re, Complex.sub_im, Complex.one_re, Complex.one_im, Complex.ext_iff,\n      Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.sub_re,\n      Complex.sub_im, Complex.one_re, Complex.one_im] at h\u2082\u2081 h\u2082\u2082 \u22a2\n    nlinarith\n  have h\u2083 : z.im = 0 := by\n    have h\u2083\u2081 := h\u2080\n    have h\u2083\u2082 := h\u2081\n    simp [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im,\n      Complex.sub_re, Complex.sub_im, Complex.one_re, Complex.one_im, Complex.ext_iff,\n      Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.sub_re,\n      Complex.sub_im, Complex.one_re, Complex.one_im] at h\u2083\u2081 h\u2083\u2082 h\u2082 \u22a2\n    nlinarith\n  have h\u2084 : f.re + 3 * z.re = 11 := by\n    have h\u2084\u2081 := h\u2080\n    simp [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, h\u2082, h\u2083] at h\u2084\u2081 \u22a2\n    <;>\n    (try norm_num at h\u2084\u2081 \u22a2) <;>\n    (try ring_nf at h\u2084\u2081 \u22a2) <;>\n    (try linarith) <;>\n    (try nlinarith)\n    <;>\n    (try\n      {\n        simp_all [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im]\n        <;>\n        norm_num\n        <;>\n        linarith\n      })\n    <;>\n    (try\n      {\n        nlinarith\n      })\n    <;>\n    (try\n      {\n        linarith\n      })\n    <;>\n    (try\n      {\n        nlinarith\n      })\n    <;>\n    (try\n      {\n        linarith\n      })\n  have h\u2085 : 3 * f.re - 5 * z.re = -65 := by\n    have h\u2085\u2081 := h\u2081\n    simp [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.sub_re, Complex.sub_im,\n      Complex.one_re, Complex.one_im, h\u2082, h\u2083] at h\u2085\u2081 \u22a2\n    <;>\n    (try norm_num at h\u2085\u2081 \u22a2) <;>\n    (try ring_nf at h\u2085\u2081 \u22a2) <;>\n    (try nlinarith) <;>\n    (try linarith)\n    <;>\n    (try\n      {\n        simp_all [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.sub_re, Complex.sub_im,\n          Complex.one_re, Complex.one_im]\n        <;>\n        norm_num\n        <;>\n        linarith\n      })\n  have h\u2086 : f.re = -10 := by\n    have h\u2086\u2081 : f.re + 3 * z.re = 11 := h\u2084\n    have h\u2086\u2082 : 3 * f.re - 5 * z.re = -65 := h\u2085\n    have h\u2086\u2083 : f.re = -10 := by\n      have h\u2086\u2084 : z.re = 7 := by\n        linarith\n      have h\u2086\u2085 : f.re = -10 := by\n        linarith\n      exact h\u2086\u2085\n    exact h\u2086\u2083\n  have h\u2087 : z.re = 7 := by\n    have h\u2087\u2081 : f.re + 3 * z.re = 11 := h\u2084\n    have h\u2087\u2082 : 3 * f.re - 5 * z.re = -65 := h\u2085\n    have h\u2087\u2083 : f.re = -10 := h\u2086\n    have h\u2087\u2084 : z.re = 7 := by\n      linarith\n    exact h\u2087\u2084\n  have h\u2088 : f = -10 := by\n    have h\u2088\u2081 : f.re = -10 := h\u2086\n    have h\u2088\u2082 : f.im = 0 := h\u2082\n    have h\u2088\u2083 : f = -10 := by\n      apply Complex.ext <;> simp_all [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.ext_iff, Complex.one_re, Complex.one_im]\n      <;> norm_num <;> linarith\n    exact h\u2088\u2083\n  have h\u2089 : z = 7 := by\n    have h\u2089\u2081 : z.re = 7 := h\u2087\n    have h\u2089\u2082 : z.im = 0 := h\u2083\n    have h\u2089\u2083 : z = 7 := by\n      apply Complex.ext <;> simp_all [Complex.ext_iff, Complex.add_re, Complex.add_im, Complex.mul_re, Complex.mul_im, Complex.ext_iff, Complex.one_re, Complex.one_im]\n      <;> norm_num <;> linarith\n    exact h\u2089\u2083\n  have h\u2081\u2080 : f = -10 \u2227 z = 7 := by\n    exact \u27e8h\u2088, h\u2089\u27e9\n  exact h\u2081\u2080\n"}
{"mode": "clear", "steps_before": 49, "alphabet": 23, "compiles": true, "steps_after": 9, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 8, "fixpoint": true, "chars_before": 1926, "chars_after": 567, "seconds": 1255.6, "id": "imo_1964_p2", "proof_after": "theorem imo_1964_p2 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) (h\u2081 : c < a + b) (h\u2082 : b < a + c)\n    (h\u2083 : a < b + c) :\n    a ^ 2 * (b + c - a) + b ^ 2 * (c + a - b) + c ^ 2 * (a + b - c) \u2264 3 * a * b * c  :=  by\n  have h\u2084 : 0 < b + c - a := by\n    linarith\n  have h\u2085 : 0 < c + a - b := by\n    linarith\n  have h\u2086 : 0 < a + b - c := by\n    linarith\n  have h\u2088 : (b + c - a) * (c + a - b) * (a + b - c) \u2264 a * b * c := by\n    nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a),\n      sq_nonneg (b + c - a - c), sq_nonneg (c + a - b - a), sq_nonneg (a + b - c - b),\n      mul_pos h\u2084 h\u2085, mul_pos h\u2085 h\u2086, mul_pos h\u2086 h\u2084,\n      mul_pos (sub_pos.mpr h\u2081) (sub_pos.mpr h\u2082), mul_pos (sub_pos.mpr h\u2082) (sub_pos.mpr h\u2083),\n      mul_pos (sub_pos.mpr h\u2083) (sub_pos.mpr h\u2081)]\n  linarith\n", "proof_before": "theorem imo_1964_p2 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) (h\u2081 : c < a + b) (h\u2082 : b < a + c)\n    (h\u2083 : a < b + c) :\n    a ^ 2 * (b + c - a) + b ^ 2 * (c + a - b) + c ^ 2 * (a + b - c) \u2264 3 * a * b * c  :=  by\n  have h\u2084 : 0 < b + c - a := by\n    linarith\n  have h\u2085 : 0 < c + a - b := by\n    linarith\n  have h\u2086 : 0 < a + b - c := by\n    linarith\n  have h\u2087 : 0 < (b + c - a) * (c + a - b) * (a + b - c) := by\n    have h\u2087\u2081 : 0 < b + c - a := h\u2084\n    have h\u2087\u2082 : 0 < c + a - b := h\u2085\n    have h\u2087\u2083 : 0 < a + b - c := h\u2086\n    have h\u2087\u2084 : 0 < (b + c - a) * (c + a - b) := by positivity\n    have h\u2087\u2085 : 0 < (b + c - a) * (c + a - b) * (a + b - c) := by positivity\n    exact h\u2087\u2085\n  have h\u2088 : (b + c - a) * (c + a - b) * (a + b - c) \u2264 a * b * c := by\n    have h\u2088\u2081 : 0 < a := by linarith\n    have h\u2088\u2082 : 0 < b := by linarith\n    have h\u2088\u2083 : 0 < c := by linarith\n    have h\u2088\u2084 : 0 < a * b := by positivity\n    have h\u2088\u2085 : 0 < b * c := by positivity\n    have h\u2088\u2086 : 0 < c * a := by positivity\n    nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a),\n      sq_nonneg (b + c - a - c), sq_nonneg (c + a - b - a), sq_nonneg (a + b - c - b),\n      mul_pos h\u2084 h\u2085, mul_pos h\u2085 h\u2086, mul_pos h\u2086 h\u2084,\n      mul_pos (sub_pos.mpr h\u2081) (sub_pos.mpr h\u2082), mul_pos (sub_pos.mpr h\u2082) (sub_pos.mpr h\u2083),\n      mul_pos (sub_pos.mpr h\u2083) (sub_pos.mpr h\u2081)]\n  have h\u2089 : a ^ 2 * (b + c - a) + b ^ 2 * (c + a - b) + c ^ 2 * (a + b - c) = (b + c - a) * (c + a - b) * (a + b - c) + 2 * a * b * c := by\n    have h\u2089\u2081 : a ^ 2 * (b + c - a) + b ^ 2 * (c + a - b) + c ^ 2 * (a + b - c) = (b + c - a) * (c + a - b) * (a + b - c) + 2 * a * b * c := by\n      ring_nf at *\n      <;>\n      nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088,\n        mul_pos h\u2080.1 h\u2080.2.1, mul_pos h\u2080.2.1 h\u2080.2.2, mul_pos h\u2080.1 h\u2080.2.2]\n    rw [h\u2089\u2081]\n    <;>\n    nlinarith\n  have h\u2081\u2080 : a ^ 2 * (b + c - a) + b ^ 2 * (c + a - b) + c ^ 2 * (a + b - c) \u2264 3 * a * b * c := by\n    rw [h\u2089]\n    have h\u2081\u2080\u2081 : 0 < a * b * c := by\n      have h\u2081\u2080\u2082 : 0 < a * b := by nlinarith\n      have h\u2081\u2080\u2083 : 0 < b * c := by nlinarith\n      have h\u2081\u2080\u2084 : 0 < c * a := by nlinarith\n      nlinarith\n    nlinarith [h\u2088]\n  exact h\u2081\u2080\n"}
{"mode": "clear", "steps_before": 73, "alphabet": 33, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 24, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 6, "fixpoint": true, "chars_before": 3042, "chars_after": 841, "seconds": 1022.5, "id": "mathd_numbertheory_227", "proof_after": "theorem mathd_numbertheory_227 (x y n : \u2115+) (h\u2080 : \u2191x / (4 : \u211d) + y / 6 = (x + y) / n) : n = 5  :=  by\n  have h\u2081 : (n : \u2115) = 5 := by\n    have h\u2082 : (n : \u211d) * (3 * (x : \u211d) + 2 * (y : \u211d)) = 12 * ((x : \u211d) + (y : \u211d)) := by\n      have h\u2083 : (x : \u211d) / 4 + (y : \u211d) / 6 = ((x : \u211d) + (y : \u211d)) / (n : \u211d) := by simpa using h\u2080\n      have h\u2084 : (n : \u211d) * ((x : \u211d) / 4 + (y : \u211d) / 6) = (x : \u211d) + (y : \u211d) := by\n        field_simp at h\u2083 \u22a2\n        <;> ring_nf at h\u2083 \u22a2 <;> nlinarith\n      linarith\n    have h\u2086 : (n : \u2115) * (3 * (x : \u2115) + 2 * (y : \u2115)) = 12 * ((x : \u2115) + (y : \u2115)) := by\n      have h\u2087 : (n : \u211d) * (3 * (x : \u211d) + 2 * (y : \u211d)) = 12 * ((x : \u211d) + (y : \u211d)) := by exact_mod_cast h\u2082\n      norm_cast at h\u2087 \u22a2\n    have h\u2087 : (n : \u2115) = 5 := by\n      have h\u2089 : (x : \u2115) > 0 := by exact x.property\n      have h\u2081\u2080 : (y : \u2115) > 0 := by exact y.property\n      nlinarith\n    exact h\u2087\n  have h\u2082 : n = 5 := by\n    have h\u2083 : (n : \u2115) = 5 := h\u2081\n    norm_cast at h\u2083 \u22a2\n  exact h\u2082\n", "proof_before": "theorem mathd_numbertheory_227 (x y n : \u2115+) (h\u2080 : \u2191x / (4 : \u211d) + y / 6 = (x + y) / n) : n = 5  :=  by\n  have h\u2081 : (n : \u2115) = 5 := by\n    have h\u2082 : (n : \u211d) * (3 * (x : \u211d) + 2 * (y : \u211d)) = 12 * ((x : \u211d) + (y : \u211d)) := by\n      have h\u2083 : (x : \u211d) / 4 + (y : \u211d) / 6 = ((x : \u211d) + (y : \u211d)) / (n : \u211d) := by simpa using h\u2080\n      have h\u2084 : (n : \u211d) * ((x : \u211d) / 4 + (y : \u211d) / 6) = (x : \u211d) + (y : \u211d) := by\n        field_simp at h\u2083 \u22a2\n        <;> ring_nf at h\u2083 \u22a2 <;> nlinarith\n      have h\u2085 : (n : \u211d) * (3 * (x : \u211d) + 2 * (y : \u211d)) = 12 * ((x : \u211d) + (y : \u211d)) := by\n        ring_nf at h\u2084 \u22a2\n        <;> nlinarith\n      exact h\u2085\n    have h\u2086 : (n : \u2115) * (3 * (x : \u2115) + 2 * (y : \u2115)) = 12 * ((x : \u2115) + (y : \u2115)) := by\n      have h\u2087 : (n : \u211d) * (3 * (x : \u211d) + 2 * (y : \u211d)) = 12 * ((x : \u211d) + (y : \u211d)) := by exact_mod_cast h\u2082\n      have h\u2088 : (n : \u2115) * (3 * (x : \u2115) + 2 * (y : \u2115)) = 12 * ((x : \u2115) + (y : \u2115)) := by\n        norm_cast at h\u2087 \u22a2\n        <;> ring_nf at h\u2087 \u22a2 <;>\n          (try norm_num at h\u2087 \u22a2) <;>\n          (try nlinarith) <;>\n          (try\n            {\n              have h\u2089 : (x : \u2115) > 0 := by exact x.property\n              have h\u2081\u2080 : (y : \u2115) > 0 := by exact y.property\n              have h\u2081\u2081 : (n : \u2115) > 0 := by exact n.property\n              nlinarith\n            }) <;>\n          (try\n            {\n              simp_all [PNat.ne_zero]\n              <;> ring_nf at *\n              <;> nlinarith\n            })\n      exact h\u2088\n    have h\u2087 : (n : \u2115) = 5 := by\n      have h\u2088 : (n : \u2115) > 0 := by exact n.property\n      have h\u2089 : (x : \u2115) > 0 := by exact x.property\n      have h\u2081\u2080 : (y : \u2115) > 0 := by exact y.property\n      have h\u2081\u2081 : (n : \u2115) \u2264 6 := by\n        by_contra h\n        have h\u2081\u2082 : (n : \u2115) \u2265 7 := by omega\n        have h\u2081\u2083 : (n : \u2115) * (3 * (x : \u2115) + 2 * (y : \u2115)) > 12 * ((x : \u2115) + (y : \u2115)) := by\n          have h\u2081\u2084 : (n : \u2115) \u2265 7 := h\u2081\u2082\n          have h\u2081\u2085 : 3 * (x : \u2115) + 2 * (y : \u2115) > 0 := by nlinarith\n          have h\u2081\u2086 : (x : \u2115) + (y : \u2115) > 0 := by nlinarith\n          have h\u2081\u2087 : (n : \u2115) * (3 * (x : \u2115) + 2 * (y : \u2115)) \u2265 7 * (3 * (x : \u2115) + 2 * (y : \u2115)) := by\n            nlinarith\n          have h\u2081\u2088 : 12 * ((x : \u2115) + (y : \u2115)) < 7 * (3 * (x : \u2115) + 2 * (y : \u2115)) := by\n            nlinarith\n          nlinarith\n        omega\n      interval_cases (n : \u2115) <;> norm_num at h\u2086 \u22a2 <;>\n        (try omega) <;>\n        (try\n          {\n            have h\u2081\u2082 : (x : \u2115) > 0 := by exact x.property\n            have h\u2081\u2083 : (y : \u2115) > 0 := by exact y.property\n            have h\u2081\u2084 : (n : \u2115) > 0 := by exact n.property\n            ring_nf at h\u2086 \u22a2\n            <;> omega\n          }) <;>\n        (try\n          {\n            ring_nf at h\u2086 \u22a2\n            <;>\n            (try omega) <;>\n            (try\n              {\n                have h\u2081\u2082 : (x : \u2115) > 0 := by exact x.property\n                have h\u2081\u2083 : (y : \u2115) > 0 := by exact y.property\n                have h\u2081\u2084 : (n : \u2115) > 0 := by exact n.property\n                omega\n              })\n          })\n    exact h\u2087\n  have h\u2082 : n = 5 := by\n    have h\u2083 : (n : \u2115) = 5 := h\u2081\n    have h\u2084 : n = 5 := by\n      apply PNat.eq\n      <;> norm_cast at h\u2083 \u22a2 <;> simp_all [PNat.ne_zero]\n      <;> omega\n    exact h\u2084\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 118, "alphabet": 46, "compiles": true, "steps_after": 118, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 7224, "chars_after": 7224, "seconds": 393.1, "id": "amc12b_2021_p9", "proof_after": "theorem amc12b_2021_p9 :\n    Real.log 80 / Real.log 2 / (Real.log 2 / Real.log 40) -\n        Real.log 160 / Real.log 2 / (Real.log 2 / Real.log 20) =\n      2  :=  by\n  have h\u2080 : Real.log 80 = 4 * Real.log 2 + Real.log 5 := by\n    have h\u2080\u2081 : Real.log 80 = Real.log (2 ^ 4 * 5) := by norm_num\n    rw [h\u2080\u2081]\n    have h\u2080\u2082 : Real.log (2 ^ 4 * 5) = Real.log (2 ^ 4) + Real.log 5 := by\n      rw [Real.log_mul (by positivity) (by positivity)]\n    rw [h\u2080\u2082]\n    have h\u2080\u2083 : Real.log (2 ^ 4) = 4 * Real.log 2 := by\n      rw [Real.log_pow]\n      <;> ring_nf\n      <;> norm_num\n    rw [h\u2080\u2083]\n    <;> ring\n  have h\u2081 : Real.log 40 = 3 * Real.log 2 + Real.log 5 := by\n    have h\u2081\u2081 : Real.log 40 = Real.log (2 ^ 3 * 5) := by norm_num\n    rw [h\u2081\u2081]\n    have h\u2081\u2082 : Real.log (2 ^ 3 * 5) = Real.log (2 ^ 3) + Real.log 5 := by\n      rw [Real.log_mul (by positivity) (by positivity)]\n    rw [h\u2081\u2082]\n    have h\u2081\u2083 : Real.log (2 ^ 3) = 3 * Real.log 2 := by\n      rw [Real.log_pow]\n      <;> ring_nf\n      <;> norm_num\n    rw [h\u2081\u2083]\n    <;> ring\n  have h\u2082 : Real.log 160 = 5 * Real.log 2 + Real.log 5 := by\n    have h\u2082\u2081 : Real.log 160 = Real.log (2 ^ 5 * 5) := by norm_num\n    rw [h\u2082\u2081]\n    have h\u2082\u2082 : Real.log (2 ^ 5 * 5) = Real.log (2 ^ 5) + Real.log 5 := by\n      rw [Real.log_mul (by positivity) (by positivity)]\n    rw [h\u2082\u2082]\n    have h\u2082\u2083 : Real.log (2 ^ 5) = 5 * Real.log 2 := by\n      rw [Real.log_pow]\n      <;> ring_nf\n      <;> norm_num\n    rw [h\u2082\u2083]\n    <;> ring\n  have h\u2083 : Real.log 20 = 2 * Real.log 2 + Real.log 5 := by\n    have h\u2083\u2081 : Real.log 20 = Real.log (2 ^ 2 * 5) := by norm_num\n    rw [h\u2083\u2081]\n    have h\u2083\u2082 : Real.log (2 ^ 2 * 5) = Real.log (2 ^ 2) + Real.log 5 := by\n      rw [Real.log_mul (by positivity) (by positivity)]\n    rw [h\u2083\u2082]\n    have h\u2083\u2083 : Real.log (2 ^ 2) = 2 * Real.log 2 := by\n      rw [Real.log_pow]\n      <;> ring_nf\n      <;> norm_num\n    rw [h\u2083\u2083]\n    <;> ring\n  have h\u2084 : Real.log 80 * Real.log 40 = 12 * (Real.log 2)^2 + 7 * Real.log 2 * Real.log 5 + (Real.log 5)^2 := by\n    rw [h\u2080, h\u2081]\n    have h\u2084\u2081 : (4 * Real.log 2 + Real.log 5) * (3 * Real.log 2 + Real.log 5) = 12 * (Real.log 2)^2 + 7 * Real.log 2 * Real.log 5 + (Real.log 5)^2 := by\n      ring_nf\n      <;>\n      nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 5),\n        Real.log_pos (by norm_num : (1 : \u211d) < 20), Real.log_pos (by norm_num : (1 : \u211d) < 40),\n        Real.log_pos (by norm_num : (1 : \u211d) < 80), Real.log_pos (by norm_num : (1 : \u211d) < 160)]\n    rw [h\u2084\u2081]\n    <;>\n    nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 5),\n      Real.log_pos (by norm_num : (1 : \u211d) < 20), Real.log_pos (by norm_num : (1 : \u211d) < 40),\n      Real.log_pos (by norm_num : (1 : \u211d) < 80), Real.log_pos (by norm_num : (1 : \u211d) < 160)]\n  have h\u2085 : Real.log 160 * Real.log 20 = 10 * (Real.log 2)^2 + 7 * Real.log 2 * Real.log 5 + (Real.log 5)^2 := by\n    rw [h\u2082, h\u2083]\n    have h\u2085\u2081 : (5 * Real.log 2 + Real.log 5) * (2 * Real.log 2 + Real.log 5) = 10 * (Real.log 2)^2 + 7 * Real.log 2 * Real.log 5 + (Real.log 5)^2 := by\n      ring_nf\n      <;>\n      nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 5)]\n    rw [h\u2085\u2081]\n    <;>\n    nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 5)]\n  have h\u2086 : Real.log 80 * Real.log 40 - Real.log 160 * Real.log 20 = 2 * (Real.log 2)^2 := by\n    rw [h\u2084, h\u2085]\n    <;> ring_nf\n    <;> nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 5),\n      Real.log_pos (by norm_num : (1 : \u211d) < 20), Real.log_pos (by norm_num : (1 : \u211d) < 40),\n      Real.log_pos (by norm_num : (1 : \u211d) < 80), Real.log_pos (by norm_num : (1 : \u211d) < 160)]\n  have h\u2087 : Real.log 80 / Real.log 2 / (Real.log 2 / Real.log 40) - Real.log 160 / Real.log 2 / (Real.log 2 / Real.log 20) = 2 := by\n    have h\u2087\u2081 : Real.log 80 / Real.log 2 / (Real.log 2 / Real.log 40) = (Real.log 80 * Real.log 40) / (Real.log 2) ^ 2 := by\n      have h\u2087\u2081\u2081 : Real.log 80 / Real.log 2 / (Real.log 2 / Real.log 40) = (Real.log 80 / Real.log 2) * (Real.log 40 / Real.log 2) := by\n        field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 5) (by norm_num : (5 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 20) (by norm_num : (20 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 40) (by norm_num : (40 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 80) (by norm_num : (80 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 160) (by norm_num : (160 : \u211d) \u2260 1)]\n        <;> ring_nf\n        <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n        <;> ring_nf\n      rw [h\u2087\u2081\u2081]\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n    have h\u2087\u2082 : Real.log 160 / Real.log 2 / (Real.log 2 / Real.log 20) = (Real.log 160 * Real.log 20) / (Real.log 2) ^ 2 := by\n      have h\u2087\u2082\u2081 : Real.log 160 / Real.log 2 / (Real.log 2 / Real.log 20) = (Real.log 160 / Real.log 2) * (Real.log 20 / Real.log 2) := by\n        field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 5) (by norm_num : (5 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 20) (by norm_num : (20 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 40) (by norm_num : (40 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 80) (by norm_num : (80 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 160) (by norm_num : (160 : \u211d) \u2260 1)]\n        <;> ring_nf\n        <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n        <;> ring_nf\n      rw [h\u2087\u2082\u2081]\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n    rw [h\u2087\u2081, h\u2087\u2082]\n    have h\u2087\u2083 : (Real.log 80 * Real.log 40) / (Real.log 2) ^ 2 - (Real.log 160 * Real.log 20) / (Real.log 2) ^ 2 = 2 := by\n      have h\u2087\u2084 : Real.log 80 * Real.log 40 - Real.log 160 * Real.log 20 = 2 * (Real.log 2) ^ 2 := by\n        linarith\n      have h\u2087\u2085 : Real.log 2 \u2260 0 := by\n        exact Real.log_ne_zero_of_pos_of_ne_one (by norm_num) (by norm_num)\n      have h\u2087\u2086 : (Real.log 80 * Real.log 40) / (Real.log 2) ^ 2 - (Real.log 160 * Real.log 20) / (Real.log 2) ^ 2 = (Real.log 80 * Real.log 40 - Real.log 160 * Real.log 20) / (Real.log 2) ^ 2 := by\n        field_simp [h\u2087\u2085]\n        <;> ring_nf\n      rw [h\u2087\u2086]\n      have h\u2087\u2087 : (Real.log 80 * Real.log 40 - Real.log 160 * Real.log 20) / (Real.log 2) ^ 2 = 2 := by\n        rw [h\u2087\u2084]\n        field_simp [h\u2087\u2085]\n        <;> ring_nf\n        <;> field_simp [h\u2087\u2085]\n        <;> nlinarith\n      rw [h\u2087\u2087]\n    linarith\n  rw [h\u2087]\n  <;> norm_num\n  <;> linarith\n", "proof_before": "theorem amc12b_2021_p9 :\n    Real.log 80 / Real.log 2 / (Real.log 2 / Real.log 40) -\n        Real.log 160 / Real.log 2 / (Real.log 2 / Real.log 20) =\n      2  :=  by\n  have h\u2080 : Real.log 80 = 4 * Real.log 2 + Real.log 5 := by\n    have h\u2080\u2081 : Real.log 80 = Real.log (2 ^ 4 * 5) := by norm_num\n    rw [h\u2080\u2081]\n    have h\u2080\u2082 : Real.log (2 ^ 4 * 5) = Real.log (2 ^ 4) + Real.log 5 := by\n      rw [Real.log_mul (by positivity) (by positivity)]\n    rw [h\u2080\u2082]\n    have h\u2080\u2083 : Real.log (2 ^ 4) = 4 * Real.log 2 := by\n      rw [Real.log_pow]\n      <;> ring_nf\n      <;> norm_num\n    rw [h\u2080\u2083]\n    <;> ring\n  have h\u2081 : Real.log 40 = 3 * Real.log 2 + Real.log 5 := by\n    have h\u2081\u2081 : Real.log 40 = Real.log (2 ^ 3 * 5) := by norm_num\n    rw [h\u2081\u2081]\n    have h\u2081\u2082 : Real.log (2 ^ 3 * 5) = Real.log (2 ^ 3) + Real.log 5 := by\n      rw [Real.log_mul (by positivity) (by positivity)]\n    rw [h\u2081\u2082]\n    have h\u2081\u2083 : Real.log (2 ^ 3) = 3 * Real.log 2 := by\n      rw [Real.log_pow]\n      <;> ring_nf\n      <;> norm_num\n    rw [h\u2081\u2083]\n    <;> ring\n  have h\u2082 : Real.log 160 = 5 * Real.log 2 + Real.log 5 := by\n    have h\u2082\u2081 : Real.log 160 = Real.log (2 ^ 5 * 5) := by norm_num\n    rw [h\u2082\u2081]\n    have h\u2082\u2082 : Real.log (2 ^ 5 * 5) = Real.log (2 ^ 5) + Real.log 5 := by\n      rw [Real.log_mul (by positivity) (by positivity)]\n    rw [h\u2082\u2082]\n    have h\u2082\u2083 : Real.log (2 ^ 5) = 5 * Real.log 2 := by\n      rw [Real.log_pow]\n      <;> ring_nf\n      <;> norm_num\n    rw [h\u2082\u2083]\n    <;> ring\n  have h\u2083 : Real.log 20 = 2 * Real.log 2 + Real.log 5 := by\n    have h\u2083\u2081 : Real.log 20 = Real.log (2 ^ 2 * 5) := by norm_num\n    rw [h\u2083\u2081]\n    have h\u2083\u2082 : Real.log (2 ^ 2 * 5) = Real.log (2 ^ 2) + Real.log 5 := by\n      rw [Real.log_mul (by positivity) (by positivity)]\n    rw [h\u2083\u2082]\n    have h\u2083\u2083 : Real.log (2 ^ 2) = 2 * Real.log 2 := by\n      rw [Real.log_pow]\n      <;> ring_nf\n      <;> norm_num\n    rw [h\u2083\u2083]\n    <;> ring\n  have h\u2084 : Real.log 80 * Real.log 40 = 12 * (Real.log 2)^2 + 7 * Real.log 2 * Real.log 5 + (Real.log 5)^2 := by\n    rw [h\u2080, h\u2081]\n    have h\u2084\u2081 : (4 * Real.log 2 + Real.log 5) * (3 * Real.log 2 + Real.log 5) = 12 * (Real.log 2)^2 + 7 * Real.log 2 * Real.log 5 + (Real.log 5)^2 := by\n      ring_nf\n      <;>\n      nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 5),\n        Real.log_pos (by norm_num : (1 : \u211d) < 20), Real.log_pos (by norm_num : (1 : \u211d) < 40),\n        Real.log_pos (by norm_num : (1 : \u211d) < 80), Real.log_pos (by norm_num : (1 : \u211d) < 160)]\n    rw [h\u2084\u2081]\n    <;>\n    nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 5),\n      Real.log_pos (by norm_num : (1 : \u211d) < 20), Real.log_pos (by norm_num : (1 : \u211d) < 40),\n      Real.log_pos (by norm_num : (1 : \u211d) < 80), Real.log_pos (by norm_num : (1 : \u211d) < 160)]\n  have h\u2085 : Real.log 160 * Real.log 20 = 10 * (Real.log 2)^2 + 7 * Real.log 2 * Real.log 5 + (Real.log 5)^2 := by\n    rw [h\u2082, h\u2083]\n    have h\u2085\u2081 : (5 * Real.log 2 + Real.log 5) * (2 * Real.log 2 + Real.log 5) = 10 * (Real.log 2)^2 + 7 * Real.log 2 * Real.log 5 + (Real.log 5)^2 := by\n      ring_nf\n      <;>\n      nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 5)]\n    rw [h\u2085\u2081]\n    <;>\n    nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 5)]\n  have h\u2086 : Real.log 80 * Real.log 40 - Real.log 160 * Real.log 20 = 2 * (Real.log 2)^2 := by\n    rw [h\u2084, h\u2085]\n    <;> ring_nf\n    <;> nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2), Real.log_pos (by norm_num : (1 : \u211d) < 5),\n      Real.log_pos (by norm_num : (1 : \u211d) < 20), Real.log_pos (by norm_num : (1 : \u211d) < 40),\n      Real.log_pos (by norm_num : (1 : \u211d) < 80), Real.log_pos (by norm_num : (1 : \u211d) < 160)]\n  have h\u2087 : Real.log 80 / Real.log 2 / (Real.log 2 / Real.log 40) - Real.log 160 / Real.log 2 / (Real.log 2 / Real.log 20) = 2 := by\n    have h\u2087\u2081 : Real.log 80 / Real.log 2 / (Real.log 2 / Real.log 40) = (Real.log 80 * Real.log 40) / (Real.log 2) ^ 2 := by\n      have h\u2087\u2081\u2081 : Real.log 80 / Real.log 2 / (Real.log 2 / Real.log 40) = (Real.log 80 / Real.log 2) * (Real.log 40 / Real.log 2) := by\n        field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 5) (by norm_num : (5 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 20) (by norm_num : (20 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 40) (by norm_num : (40 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 80) (by norm_num : (80 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 160) (by norm_num : (160 : \u211d) \u2260 1)]\n        <;> ring_nf\n        <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n        <;> ring_nf\n      rw [h\u2087\u2081\u2081]\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n    have h\u2087\u2082 : Real.log 160 / Real.log 2 / (Real.log 2 / Real.log 20) = (Real.log 160 * Real.log 20) / (Real.log 2) ^ 2 := by\n      have h\u2087\u2082\u2081 : Real.log 160 / Real.log 2 / (Real.log 2 / Real.log 20) = (Real.log 160 / Real.log 2) * (Real.log 20 / Real.log 2) := by\n        field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 5) (by norm_num : (5 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 20) (by norm_num : (20 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 40) (by norm_num : (40 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 80) (by norm_num : (80 : \u211d) \u2260 1),\n          Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 160) (by norm_num : (160 : \u211d) \u2260 1)]\n        <;> ring_nf\n        <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n        <;> ring_nf\n      rw [h\u2087\u2082\u2081]\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n      <;> field_simp [Real.log_ne_zero_of_pos_of_ne_one (by norm_num : (0 : \u211d) < 2) (by norm_num : (2 : \u211d) \u2260 1)]\n      <;> ring_nf\n    rw [h\u2087\u2081, h\u2087\u2082]\n    have h\u2087\u2083 : (Real.log 80 * Real.log 40) / (Real.log 2) ^ 2 - (Real.log 160 * Real.log 20) / (Real.log 2) ^ 2 = 2 := by\n      have h\u2087\u2084 : Real.log 80 * Real.log 40 - Real.log 160 * Real.log 20 = 2 * (Real.log 2) ^ 2 := by\n        linarith\n      have h\u2087\u2085 : Real.log 2 \u2260 0 := by\n        exact Real.log_ne_zero_of_pos_of_ne_one (by norm_num) (by norm_num)\n      have h\u2087\u2086 : (Real.log 80 * Real.log 40) / (Real.log 2) ^ 2 - (Real.log 160 * Real.log 20) / (Real.log 2) ^ 2 = (Real.log 80 * Real.log 40 - Real.log 160 * Real.log 20) / (Real.log 2) ^ 2 := by\n        field_simp [h\u2087\u2085]\n        <;> ring_nf\n      rw [h\u2087\u2086]\n      have h\u2087\u2087 : (Real.log 80 * Real.log 40 - Real.log 160 * Real.log 20) / (Real.log 2) ^ 2 = 2 := by\n        rw [h\u2087\u2084]\n        field_simp [h\u2087\u2085]\n        <;> ring_nf\n        <;> field_simp [h\u2087\u2085]\n        <;> nlinarith\n      rw [h\u2087\u2087]\n    linarith\n  rw [h\u2087]\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 281, "alphabet": 92, "compiles": false, "id": "aime_1999_p11", "proof_after": "theorem aime_1999_p11 (m : \u211a) (h\u2080 : 0 < m)\n    (h\u2081 : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.tan (m * Real.pi / 180))\n    (h\u2082 : (m.num : \u211d) / m.den < 90) : \u2191m.den + m.num = 177  :=  by\n  have h_sum : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) := by\n    have h\u2081 : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) := by\n      have h\u2082 : 2 * Real.sin (2.5 * Real.pi / 180) * (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = 2 * Real.cos (2.5 * Real.pi / 180) := by\n        have h\u2083 : 2 * Real.sin (2.5 * Real.pi / 180) * (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = \u2211 k in Finset.Icc (1 : \u2115) 35, (2 * Real.sin (2.5 * Real.pi / 180) * Real.sin (5 * k * Real.pi / 180)) := by\n          rw [Finset.mul_sum]\n          <;>\n          simp [mul_assoc]\n        rw [h\u2083]\n        have h\u2084 : \u2200 (k : \u2115), k \u2208 Finset.Icc (1 : \u2115) 35 \u2192 2 * Real.sin (2.5 * Real.pi / 180) * Real.sin (5 * k * Real.pi / 180) = Real.cos ((5 * k - 2.5) * Real.pi / 180) - Real.cos ((5 * k + 2.5) * Real.pi / 180) := by\n          intro k hk\n          have h\u2085 : 2 * Real.sin (2.5 * Real.pi / 180) * Real.sin (5 * k * Real.pi / 180) = Real.cos ((5 * k - 2.5) * Real.pi / 180) - Real.cos ((5 * k + 2.5) * Real.pi / 180) := by\n            have h\u2085\u2081 : 2 * Real.sin (2.5 * Real.pi / 180) * Real.sin (5 * k * Real.pi / 180) = Real.cos ((5 * k * Real.pi / 180) - (2.5 * Real.pi / 180)) - Real.cos ((5 * k * Real.pi / 180) + (2.5 * Real.pi / 180)) := by\n              have h\u2085\u2082 := Real.cos_sub ((5 * k * Real.pi / 180)) (2.5 * Real.pi / 180)\n              have h\u2085\u2083 := Real.cos_add ((5 * k * Real.pi / 180)) (2.5 * Real.pi / 180)\n              ring_nf at h\u2085\u2082 h\u2085\u2083 \u22a2\n              <;>\n              linarith\n            have h\u2085\u2084 : Real.cos ((5 * k * Real.pi / 180) - (2.5 * Real.pi / 180)) = Real.cos ((5 * k - 2.5) * Real.pi / 180) := by\n              ring_nf\n              <;>\n              field_simp\n              <;>\n              ring_nf\n            have h\u2085\u2085 : Real.cos ((5 * k * Real.pi / 180) + (2.5 * Real.pi / 180)) = Real.cos ((5 * k + 2.5) * Real.pi / 180) := by\n              ring_nf\n              <;>\n              field_simp\n              <;>\n              ring_nf\n            rw [h\u2085\u2081, h\u2085\u2084, h\u2085\u2085]\n            <;>\n            ring_nf\n            <;>\n            field_simp\n            <;>\n            ring_nf\n          rw [h\u2085]\n          <;>\n          ring_nf\n          <;>\n          field_simp\n          <;>\n          ring_nf\n        have h\u2085 : \u2211 k in Finset.Icc (1 : \u2115) 35, (2 * Real.sin (2.5 * Real.pi / 180) * Real.sin (5 * k * Real.pi / 180)) = \u2211 k in Finset.Icc (1 : \u2115) 35, (Real.cos ((5 * k - 2.5) * Real.pi / 180) - Real.cos ((5 * k + 2.5) * Real.pi / 180)) := by\n          apply Finset.sum_congr rfl\n          intro k hk\n          rw [h\u2084 k hk]\n        rw [h\u2085]\n        have h\u2086 : \u2211 k in Finset.Icc (1 : \u2115) 35, (Real.cos ((5 * k - 2.5) * Real.pi / 180) - Real.cos ((5 * k + 2.5) * Real.pi / 180)) = Real.cos (2.5 * Real.pi / 180) - Real.cos (177.5 * Real.pi / 180) := by\n          norm_num [Finset.sum_Icc_succ_top, Nat.cast_add, Nat.cast_one, Nat.cast_mul, Nat.cast_ofNat]\n          <;>\n          ring_nf at *\n          <;>\n          norm_num\n          <;>\n          field_simp at *\n          <;>\n          ring_nf at *\n          <;>\n          norm_num at *\n          <;>\n          linarith\n        rw [h\u2086]\n        have h\u2087 : Real.cos (177.5 * Real.pi / 180) = -Real.cos (2.5 * Real.pi / 180) := by\n          have h\u2087\u2081 : Real.cos (177.5 * Real.pi / 180) = Real.cos (Real.pi - (2.5 * Real.pi / 180)) := by\n            ring_nf\n            <;>\n            field_simp\n            <;>\n            ring_nf\n          rw [h\u2087\u2081]\n          have h\u2087\u2082 : Real.cos (Real.pi - (2.5 * Real.pi / 180)) = -Real.cos (2.5 * Real.pi / 180) := by\n            rw [Real.cos_pi_sub]\n            <;>\n            ring_nf\n            <;>\n            field_simp\n            <;>\n            ring_nf\n          rw [h\u2087\u2082]\n          <;>\n          ring_nf\n          <;>\n          field_simp\n          <;>\n          ring_nf\n        rw [h\u2087]\n        <;>\n        ring_nf\n        <;>\n        field_simp\n        <;>\n        ring_nf\n        <;>\n        norm_num\n        <;>\n        linarith\n      have h\u2083 : Real.sin (2.5 * Real.pi / 180) > 0 := by\n        apply Real.sin_pos_of_pos_of_lt_pi\n        <;> norm_num\n        <;>\n          ring_nf\n        <;>\n          norm_num\n        <;>\n          linarith [Real.pi_gt_three]\n      have h\u2084 : Real.sin (2.5 * Real.pi / 180) \u2260 0 := by linarith\n      have h\u2085 : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) := by\n        have h\u2085\u2081 : 2 * Real.sin (2.5 * Real.pi / 180) * (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = 2 * Real.cos (2.5 * Real.pi / 180) := by\n          exact h\u2082\n        have h\u2085\u2082 : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) := by\n          field_simp at h\u2085\u2081 \u22a2\n          <;>\n          nlinarith [Real.sin_le_one (2.5 * Real.pi / 180), Real.cos_le_one (2.5 * Real.pi / 180),\n            Real.sin_le_one (5 * (35 : \u211d) * Real.pi / 180)]\n        exact h\u2085\u2082\n      exact h\u2085\n    exact h\u2081\n  have h_sum_tan : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.tan (35 * Real.pi / 72) := by\n    rw [h_sum]\n    have h\u2083 : Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := by\n      have h\u2084 : Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := by\n        have h\u2085 : Real.cos (2.5 * Real.pi / 180) = Real.sin (35 * Real.pi / 72) := by\n          have h\u2085\u2081 : Real.cos (2.5 * Real.pi / 180) = Real.sin (35 * Real.pi / 72) := by\n            rw [\u2190 Real.sin_pi_div_two_sub]\n            <;> ring_nf\n            <;> field_simp\n            <;> ring_nf\n          rw [h\u2085\u2081]\n        have h\u2086 : Real.sin (2.5 * Real.pi / 180) = Real.cos (35 * Real.pi / 72) := by\n          have h\u2086\u2081 : Real.sin (2.5 * Real.pi / 180) = Real.cos (35 * Real.pi / 72) := by\n            rw [\u2190 Real.cos_pi_div_two_sub]\n            <;> ring_nf\n            <;> field_simp\n            <;> ring_nf\n          rw [h\u2086\u2081]\n        rw [h\u2085, h\u2086]\n        have h\u2087 : Real.tan (35 * Real.pi / 72) = Real.sin (35 * Real.pi / 72) / Real.cos (35 * Real.pi / 72) := by\n          rw [Real.tan_eq_sin_div_cos]\n        rw [h\u2087]\n        <;>\n        by_cases h : Real.cos (35 * Real.pi / 72) = 0 <;>\n        by_cases h' : Real.sin (35 * Real.pi / 72) = 0 <;>\n        field_simp [h, h']\n        <;>\n        nlinarith [Real.sin_le_one (35 * Real.pi / 72), Real.cos_le_one (35 * Real.pi / 72),\n          Real.sin_sq_add_cos_sq (35 * Real.pi / 72)]\n      rw [h\u2084]\n    rw [h\u2083]\n  have h_tan_eq : Real.tan (m * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := by\n    rw [h\u2081] at *\n    linarith\n  have h_m_pi_div_180_eq : m * Real.pi / 180 = 35 * Real.pi / 72 := by\n    have h\u2083 : Real.tan (m * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := h_tan_eq\n    have h\u2084 : m * Real.pi / 180 = 35 * Real.pi / 72 := by\n      have h\u2085 : m * Real.pi / 180 = 35 * Real.pi / 72 := by\n        have h\u2086 : m * Real.pi / 180 < Real.pi / 2 := by\n          have h\u2087 : (m : \u211d) < 90 := by\n            have h\u2088 : (m.num : \u211d) / m.den < 90 := h\u2082\n            have h\u2089 : (m : \u211d) = (m.num : \u211d) / m.den := by\n              norm_cast\n              <;> field_simp [Rat.num_div_den]\n              <;> norm_cast\n              <;> field_simp [Rat.num_div_den]\n              <;> linarith\n            rw [h\u2089]\n            <;> exact h\u2088\n          have h\u2081\u2080 : (m : \u211d) * Real.pi / 180 < Real.pi / 2 := by\n            have h\u2081\u2081 : 0 < Real.pi := Real.pi_pos\n            have h\u2081\u2082 : 0 < (180 : \u211d) := by norm_num\n            have h\u2081\u2083 : 0 < (2 : \u211d) := by norm_num\n            have h\u2081\u2084 : (m : \u211d) < 90 := h\u2087\n            have h\u2081\u2085 : (m : \u211d) * Real.pi / 180 < Real.pi / 2 := by\n              have h\u2081\u2086 : (m : \u211d) < 90 := h\u2087\n              have h\u2081\u2087 : (m : \u211d) * Real.pi / 180 < 90 * Real.pi / 180 := by\n                nlinarith [Real.pi_pos]\n              have h\u2081\u2088 : 90 * Real.pi / 180 = Real.pi / 2 := by\n                ring_nf\n                <;> field_simp\n                <;> ring_nf\n                <;> linarith [Real.pi_gt_three]\n              rw [h\u2081\u2088] at h\u2081\u2087\n              linarith\n            exact h\u2081\u2085\n          exact_mod_cast h\u2081\u2080\n        have h\u2087 : 35 * Real.pi / 72 < Real.pi / 2 := by\n          have h\u2088 : (35 : \u211d) * Real.pi / 72 < Real.pi / 2 := by\n            have h\u2089 : 0 < Real.pi := Real.pi_pos\n            have h\u2081\u2080 : 0 < (72 : \u211d) := by norm_num\n            have h\u2081\u2081 : 0 < (2 : \u211d) := by norm_num\n            nlinarith [Real.pi_gt_three]\n          exact_mod_cast h\u2088\n        have h\u2088 : Real.tan (m * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := h\u2083\n        have h\u2089 : m * Real.pi / 180 = 35 * Real.pi / 72 := by\n          have h\u2081\u2080 : Real.tan (m * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := h\u2083\n          have h\u2081\u2081 : m * Real.pi / 180 = 35 * Real.pi / 72 := by\n            apply (injOn_tan.eq_iff \u27e8by\n              have h\u2081\u2082 : (m : \u211d) * Real.pi / 180 > 0 := by\n                have h\u2081\u2083 : (m : \u211d) > 0 := by exact_mod_cast h\u2080\n                have h\u2081\u2084 : 0 < Real.pi := Real.pi_pos\n                have h\u2081\u2085 : 0 < (180 : \u211d) := by norm_num\n                positivity\n              linarith, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9).mp h\u2081\u2080\n          exact h\u2081\u2081\n        exact h\u2089\n      exact h\u2085\n    exact h\u2084\n  have h_m_val : m = 175 / 2 := by\n    have h\u2083 : (m : \u211d) = 175 / 2 := by\n      have h\u2084 : (m : \u211d) * Real.pi / 180 = 35 * Real.pi / 72 := by\n        exact h_m_pi_div_180_eq\n      have h\u2085 : (m : \u211d) * Real.pi / 180 = 35 * Real.pi / 72 := by\n        exact h\u2084\n      have h\u2086 : (m : \u211d) = 175 / 2 := by\n        have h\u2087 : Real.pi \u2260 0 := Real.pi_ne_zero\n        field_simp [h\u2087] at h\u2085 \u22a2\n        <;> ring_nf at h\u2085 \u22a2 <;> nlinarith [Real.pi_gt_three]\n      exact h\u2086\n    have h\u2087 : m = 175 / 2 := by\n      norm_cast at h\u2083 \u22a2\n      <;>\n      field_simp at h\u2083 \u22a2 <;>\n      norm_cast at h\u2083 \u22a2 <;>\n      ring_nf at h\u2083 \u22a2 <;>\n      norm_num at h\u2083 \u22a2 <;>\n      (try norm_num) <;>\n      (try linarith) <;>\n      (try nlinarith [Real.pi_gt_three])\n      <;>\n      simp_all [Rat.ext_iff, Nat.cast_inj]\n      <;>\n      norm_num at *\n      <;>\n      ring_nf at *\n      <;>\n      norm_num at *\n      <;>\n      linarith\n    exact h\u2087\n  have h_num : m.num = 175 := by\n    rw [h_m_val]\n    <;> norm_num [Rat.num_div_eq_of_coprime]\n    <;> norm_cast\n    <;>\n    (try decide)\n    <;>\n    (try ring_nf at * <;> norm_num at * <;> aesop)\n    <;>\n    (try aesop)\n  have h_den : m.den = 2 := by\n    rw [h_m_val]\n    <;> norm_num [Rat.den_div_eq_of_coprime]\n    <;> norm_cast\n    <;>\n    (try decide)\n    <;>\n    (try ring_nf at * <;> norm_num at * <;> aesop)\n    <;>\n    (try aesop)\n  have h_final : \u2191m.den + m.num = 177 := by\n    rw [h_num, h_den]\n    <;> norm_num\n    <;>\n    (try decide)\n    <;>\n    (try ring_nf at * <;> norm_num at * <;> aesop)\n    <;>\n    (try aesop)\n  exact h_final\n", "proof_before": "theorem aime_1999_p11 (m : \u211a) (h\u2080 : 0 < m)\n    (h\u2081 : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.tan (m * Real.pi / 180))\n    (h\u2082 : (m.num : \u211d) / m.den < 90) : \u2191m.den + m.num = 177  :=  by\n  have h_sum : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) := by\n    have h\u2081 : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) := by\n      have h\u2082 : 2 * Real.sin (2.5 * Real.pi / 180) * (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = 2 * Real.cos (2.5 * Real.pi / 180) := by\n        have h\u2083 : 2 * Real.sin (2.5 * Real.pi / 180) * (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = \u2211 k in Finset.Icc (1 : \u2115) 35, (2 * Real.sin (2.5 * Real.pi / 180) * Real.sin (5 * k * Real.pi / 180)) := by\n          rw [Finset.mul_sum]\n          <;>\n          simp [mul_assoc]\n        rw [h\u2083]\n        have h\u2084 : \u2200 (k : \u2115), k \u2208 Finset.Icc (1 : \u2115) 35 \u2192 2 * Real.sin (2.5 * Real.pi / 180) * Real.sin (5 * k * Real.pi / 180) = Real.cos ((5 * k - 2.5) * Real.pi / 180) - Real.cos ((5 * k + 2.5) * Real.pi / 180) := by\n          intro k hk\n          have h\u2085 : 2 * Real.sin (2.5 * Real.pi / 180) * Real.sin (5 * k * Real.pi / 180) = Real.cos ((5 * k - 2.5) * Real.pi / 180) - Real.cos ((5 * k + 2.5) * Real.pi / 180) := by\n            have h\u2085\u2081 : 2 * Real.sin (2.5 * Real.pi / 180) * Real.sin (5 * k * Real.pi / 180) = Real.cos ((5 * k * Real.pi / 180) - (2.5 * Real.pi / 180)) - Real.cos ((5 * k * Real.pi / 180) + (2.5 * Real.pi / 180)) := by\n              have h\u2085\u2082 := Real.cos_sub ((5 * k * Real.pi / 180)) (2.5 * Real.pi / 180)\n              have h\u2085\u2083 := Real.cos_add ((5 * k * Real.pi / 180)) (2.5 * Real.pi / 180)\n              ring_nf at h\u2085\u2082 h\u2085\u2083 \u22a2\n              <;>\n              linarith\n            have h\u2085\u2084 : Real.cos ((5 * k * Real.pi / 180) - (2.5 * Real.pi / 180)) = Real.cos ((5 * k - 2.5) * Real.pi / 180) := by\n              ring_nf\n              <;>\n              field_simp\n              <;>\n              ring_nf\n            have h\u2085\u2085 : Real.cos ((5 * k * Real.pi / 180) + (2.5 * Real.pi / 180)) = Real.cos ((5 * k + 2.5) * Real.pi / 180) := by\n              ring_nf\n              <;>\n              field_simp\n              <;>\n              ring_nf\n            rw [h\u2085\u2081, h\u2085\u2084, h\u2085\u2085]\n            <;>\n            ring_nf\n            <;>\n            field_simp\n            <;>\n            ring_nf\n          rw [h\u2085]\n          <;>\n          ring_nf\n          <;>\n          field_simp\n          <;>\n          ring_nf\n        have h\u2085 : \u2211 k in Finset.Icc (1 : \u2115) 35, (2 * Real.sin (2.5 * Real.pi / 180) * Real.sin (5 * k * Real.pi / 180)) = \u2211 k in Finset.Icc (1 : \u2115) 35, (Real.cos ((5 * k - 2.5) * Real.pi / 180) - Real.cos ((5 * k + 2.5) * Real.pi / 180)) := by\n          apply Finset.sum_congr rfl\n          intro k hk\n          rw [h\u2084 k hk]\n        rw [h\u2085]\n        have h\u2086 : \u2211 k in Finset.Icc (1 : \u2115) 35, (Real.cos ((5 * k - 2.5) * Real.pi / 180) - Real.cos ((5 * k + 2.5) * Real.pi / 180)) = Real.cos (2.5 * Real.pi / 180) - Real.cos (177.5 * Real.pi / 180) := by\n          norm_num [Finset.sum_Icc_succ_top, Nat.cast_add, Nat.cast_one, Nat.cast_mul, Nat.cast_ofNat]\n          <;>\n          ring_nf at *\n          <;>\n          norm_num\n          <;>\n          field_simp at *\n          <;>\n          ring_nf at *\n          <;>\n          norm_num at *\n          <;>\n          linarith\n        rw [h\u2086]\n        have h\u2087 : Real.cos (177.5 * Real.pi / 180) = -Real.cos (2.5 * Real.pi / 180) := by\n          have h\u2087\u2081 : Real.cos (177.5 * Real.pi / 180) = Real.cos (Real.pi - (2.5 * Real.pi / 180)) := by\n            ring_nf\n            <;>\n            field_simp\n            <;>\n            ring_nf\n          rw [h\u2087\u2081]\n          have h\u2087\u2082 : Real.cos (Real.pi - (2.5 * Real.pi / 180)) = -Real.cos (2.5 * Real.pi / 180) := by\n            rw [Real.cos_pi_sub]\n            <;>\n            ring_nf\n            <;>\n            field_simp\n            <;>\n            ring_nf\n          rw [h\u2087\u2082]\n          <;>\n          ring_nf\n          <;>\n          field_simp\n          <;>\n          ring_nf\n        rw [h\u2087]\n        <;>\n        ring_nf\n        <;>\n        field_simp\n        <;>\n        ring_nf\n        <;>\n        norm_num\n        <;>\n        linarith\n      have h\u2083 : Real.sin (2.5 * Real.pi / 180) > 0 := by\n        apply Real.sin_pos_of_pos_of_lt_pi\n        <;> norm_num\n        <;>\n          ring_nf\n        <;>\n          norm_num\n        <;>\n          linarith [Real.pi_gt_three]\n      have h\u2084 : Real.sin (2.5 * Real.pi / 180) \u2260 0 := by linarith\n      have h\u2085 : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) := by\n        have h\u2085\u2081 : 2 * Real.sin (2.5 * Real.pi / 180) * (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = 2 * Real.cos (2.5 * Real.pi / 180) := by\n          exact h\u2082\n        have h\u2085\u2082 : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) := by\n          field_simp at h\u2085\u2081 \u22a2\n          <;>\n          nlinarith [Real.sin_le_one (2.5 * Real.pi / 180), Real.cos_le_one (2.5 * Real.pi / 180),\n            Real.sin_le_one (5 * (35 : \u211d) * Real.pi / 180)]\n        exact h\u2085\u2082\n      exact h\u2085\n    exact h\u2081\n  have h_sum_tan : (\u2211 k in Finset.Icc (1 : \u2115) 35, Real.sin (5 * k * Real.pi / 180)) = Real.tan (35 * Real.pi / 72) := by\n    rw [h_sum]\n    have h\u2083 : Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := by\n      have h\u2084 : Real.cos (2.5 * Real.pi / 180) / Real.sin (2.5 * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := by\n        have h\u2085 : Real.cos (2.5 * Real.pi / 180) = Real.sin (35 * Real.pi / 72) := by\n          have h\u2085\u2081 : Real.cos (2.5 * Real.pi / 180) = Real.sin (35 * Real.pi / 72) := by\n            rw [\u2190 Real.sin_pi_div_two_sub]\n            <;> ring_nf\n            <;> field_simp\n            <;> ring_nf\n          rw [h\u2085\u2081]\n        have h\u2086 : Real.sin (2.5 * Real.pi / 180) = Real.cos (35 * Real.pi / 72) := by\n          have h\u2086\u2081 : Real.sin (2.5 * Real.pi / 180) = Real.cos (35 * Real.pi / 72) := by\n            rw [\u2190 Real.cos_pi_div_two_sub]\n            <;> ring_nf\n            <;> field_simp\n            <;> ring_nf\n          rw [h\u2086\u2081]\n        rw [h\u2085, h\u2086]\n        have h\u2087 : Real.tan (35 * Real.pi / 72) = Real.sin (35 * Real.pi / 72) / Real.cos (35 * Real.pi / 72) := by\n          rw [Real.tan_eq_sin_div_cos]\n        rw [h\u2087]\n        <;>\n        by_cases h : Real.cos (35 * Real.pi / 72) = 0 <;>\n        by_cases h' : Real.sin (35 * Real.pi / 72) = 0 <;>\n        field_simp [h, h']\n        <;>\n        nlinarith [Real.sin_le_one (35 * Real.pi / 72), Real.cos_le_one (35 * Real.pi / 72),\n          Real.sin_sq_add_cos_sq (35 * Real.pi / 72)]\n      rw [h\u2084]\n    rw [h\u2083]\n  have h_tan_eq : Real.tan (m * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := by\n    rw [h\u2081] at *\n    linarith\n  have h_m_pi_div_180_eq : m * Real.pi / 180 = 35 * Real.pi / 72 := by\n    have h\u2083 : Real.tan (m * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := h_tan_eq\n    have h\u2084 : m * Real.pi / 180 = 35 * Real.pi / 72 := by\n      have h\u2085 : m * Real.pi / 180 = 35 * Real.pi / 72 := by\n        have h\u2086 : m * Real.pi / 180 < Real.pi / 2 := by\n          have h\u2087 : (m : \u211d) < 90 := by\n            have h\u2088 : (m.num : \u211d) / m.den < 90 := h\u2082\n            have h\u2089 : (m : \u211d) = (m.num : \u211d) / m.den := by\n              norm_cast\n              <;> field_simp [Rat.num_div_den]\n              <;> norm_cast\n              <;> field_simp [Rat.num_div_den]\n              <;> linarith\n            rw [h\u2089]\n            <;> exact h\u2088\n          have h\u2081\u2080 : (m : \u211d) * Real.pi / 180 < Real.pi / 2 := by\n            have h\u2081\u2081 : 0 < Real.pi := Real.pi_pos\n            have h\u2081\u2082 : 0 < (180 : \u211d) := by norm_num\n            have h\u2081\u2083 : 0 < (2 : \u211d) := by norm_num\n            have h\u2081\u2084 : (m : \u211d) < 90 := h\u2087\n            have h\u2081\u2085 : (m : \u211d) * Real.pi / 180 < Real.pi / 2 := by\n              have h\u2081\u2086 : (m : \u211d) < 90 := h\u2087\n              have h\u2081\u2087 : (m : \u211d) * Real.pi / 180 < 90 * Real.pi / 180 := by\n                nlinarith [Real.pi_pos]\n              have h\u2081\u2088 : 90 * Real.pi / 180 = Real.pi / 2 := by\n                ring_nf\n                <;> field_simp\n                <;> ring_nf\n                <;> linarith [Real.pi_gt_three]\n              rw [h\u2081\u2088] at h\u2081\u2087\n              linarith\n            exact h\u2081\u2085\n          exact_mod_cast h\u2081\u2080\n        have h\u2087 : 35 * Real.pi / 72 < Real.pi / 2 := by\n          have h\u2088 : (35 : \u211d) * Real.pi / 72 < Real.pi / 2 := by\n            have h\u2089 : 0 < Real.pi := Real.pi_pos\n            have h\u2081\u2080 : 0 < (72 : \u211d) := by norm_num\n            have h\u2081\u2081 : 0 < (2 : \u211d) := by norm_num\n            nlinarith [Real.pi_gt_three]\n          exact_mod_cast h\u2088\n        have h\u2088 : Real.tan (m * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := h\u2083\n        have h\u2089 : m * Real.pi / 180 = 35 * Real.pi / 72 := by\n          have h\u2081\u2080 : Real.tan (m * Real.pi / 180) = Real.tan (35 * Real.pi / 72) := h\u2083\n          have h\u2081\u2081 : m * Real.pi / 180 = 35 * Real.pi / 72 := by\n            apply (injOn_tan.eq_iff \u27e8by\n              have h\u2081\u2082 : (m : \u211d) * Real.pi / 180 > 0 := by\n                have h\u2081\u2083 : (m : \u211d) > 0 := by exact_mod_cast h\u2080\n                have h\u2081\u2084 : 0 < Real.pi := Real.pi_pos\n                have h\u2081\u2085 : 0 < (180 : \u211d) := by norm_num\n                positivity\n              linarith, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9).mp h\u2081\u2080\n          exact h\u2081\u2081\n        exact h\u2089\n      exact h\u2085\n    exact h\u2084\n  have h_m_val : m = 175 / 2 := by\n    have h\u2083 : (m : \u211d) = 175 / 2 := by\n      have h\u2084 : (m : \u211d) * Real.pi / 180 = 35 * Real.pi / 72 := by\n        exact h_m_pi_div_180_eq\n      have h\u2085 : (m : \u211d) * Real.pi / 180 = 35 * Real.pi / 72 := by\n        exact h\u2084\n      have h\u2086 : (m : \u211d) = 175 / 2 := by\n        have h\u2087 : Real.pi \u2260 0 := Real.pi_ne_zero\n        field_simp [h\u2087] at h\u2085 \u22a2\n        <;> ring_nf at h\u2085 \u22a2 <;> nlinarith [Real.pi_gt_three]\n      exact h\u2086\n    have h\u2087 : m = 175 / 2 := by\n      norm_cast at h\u2083 \u22a2\n      <;>\n      field_simp at h\u2083 \u22a2 <;>\n      norm_cast at h\u2083 \u22a2 <;>\n      ring_nf at h\u2083 \u22a2 <;>\n      norm_num at h\u2083 \u22a2 <;>\n      (try norm_num) <;>\n      (try linarith) <;>\n      (try nlinarith [Real.pi_gt_three])\n      <;>\n      simp_all [Rat.ext_iff, Nat.cast_inj]\n      <;>\n      norm_num at *\n      <;>\n      ring_nf at *\n      <;>\n      norm_num at *\n      <;>\n      linarith\n    exact h\u2087\n  have h_num : m.num = 175 := by\n    rw [h_m_val]\n    <;> norm_num [Rat.num_div_eq_of_coprime]\n    <;> norm_cast\n    <;>\n    (try decide)\n    <;>\n    (try ring_nf at * <;> norm_num at * <;> aesop)\n    <;>\n    (try aesop)\n  have h_den : m.den = 2 := by\n    rw [h_m_val]\n    <;> norm_num [Rat.den_div_eq_of_coprime]\n    <;> norm_cast\n    <;>\n    (try decide)\n    <;>\n    (try ring_nf at * <;> norm_num at * <;> aesop)\n    <;>\n    (try aesop)\n  have h_final : \u2191m.den + m.num = 177 := by\n    rw [h_num, h_den]\n    <;> norm_num\n    <;>\n    (try decide)\n    <;>\n    (try ring_nf at * <;> norm_num at * <;> aesop)\n    <;>\n    (try aesop)\n  exact h_final\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 17, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 701, "chars_after": 6, "seconds": 20.4, "id": "mathd_algebra_296", "proof_after": "theorem mathd_algebra_296 : abs ((3491 - 60) * (3491 + 60) - 3491 ^ 2 : \u2124) = 3600  :=  by\n  rfl\n", "proof_before": "theorem mathd_algebra_296 : abs ((3491 - 60) * (3491 + 60) - 3491 ^ 2 : \u2124) = 3600  :=  by\n  have h\u2081 : (3491 - 60 : \u2115) = 3431 := by\n    norm_num\n    <;> rfl\n  have h\u2082 : (3491 + 60 : \u2115) = 3551 := by\n    norm_num\n    <;> rfl\n  have h\u2083 : (3431 : \u2115) * 3551 = 12183481 := by\n    norm_num\n    <;> rfl\n  have h\u2084 : (3491 : \u2115) ^ 2 = 12187081 := by\n    norm_num [pow_two]\n    <;> rfl\n  have h\u2085 : ((3491 - 60) * (3491 + 60) - 3491 ^ 2 : \u2124) = -3600 := by\n    have h\u2085\u2081 : ((3491 - 60) * (3491 + 60) - 3491 ^ 2 : \u2124) = -3600 := by\n      norm_num [h\u2081, h\u2082, h\u2083, h\u2084, pow_two, Int.mul_emod, Int.add_emod, Int.sub_emod]\n      <;> rfl\n    rw [h\u2085\u2081]\n    <;> rfl\n  have h\u2086 : abs ((3491 - 60) * (3491 + 60) - 3491 ^ 2 : \u2124) = 3600 := by\n    rw [h\u2085]\n    <;> norm_num [abs_of_nonpos, abs_of_nonneg]\n    <;> rfl\n  apply h\u2086\n"}
{"mode": "clear", "steps_before": 52, "alphabet": 31, "compiles": false, "id": "mathd_numbertheory_5", "proof_after": "theorem mathd_numbertheory_5 (n : \u2115) (h\u2080 : 10 \u2264 n) (h\u2081 : \u2203 x, x ^ 2 = n) (h\u2082 : \u2203 t, t ^ 3 = n) :\n    64 \u2264 n  :=  by\n  have h\u2083 : \u2200 (t : \u2115), t ^ 3 = n \u2192 64 \u2264 n := by\n    intro t ht\n    have h\u2084 : t \u2265 3 := by\n      by_contra h\n      have h\u2085 : t \u2264 2 := by linarith\n      have h\u2086 : t ^ 3 \u2264 8 := by\n        have h\u2087 : t \u2264 2 := h\u2085\n        have h\u2088 : t ^ 3 \u2264 2 ^ 3 := by\n          exact Nat.pow_le_pow_of_le_left h\u2087 3\n        linarith\n      have h\u2089 : n \u2264 8 := by linarith\n      linarith\n    by_cases h\u2085 : t \u2265 4\n    \u00b7 \n      have h\u2086 : t ^ 3 \u2265 4 ^ 3 := by\n        exact Nat.pow_le_pow_of_le_left h\u2085 3\n      have h\u2087 : n \u2265 64 := by\n        linarith\n      linarith\n    \u00b7 \n      have h\u2086 : t = 3 := by\n        have h\u2087 : t < 4 := by linarith\n        have h\u2088 : t \u2265 3 := h\u2084\n        interval_cases t <;> norm_num at h\u2087 h\u2088 \u22a2 <;> try omega\n      rw [h\u2086] at ht\n      have h\u2087 : n = 27 := by\n        norm_num at ht \u22a2\n        <;> linarith\n      rcases h\u2081 with \u27e8x, hx\u27e9\n      have h\u2088 : x ^ 2 = 27 := by\n        norm_num [h\u2087] at hx \u22a2\n        <;> linarith\n      have h\u2089 : x \u2264 5 := by\n        nlinarith\n      have h\u2081\u2080 : x \u2265 6 := by\n        by_contra h\n        have h\u2081\u2081 : x \u2264 5 := by linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> try omega\n      linarith\n  have h\u2084 : 64 \u2264 n := by\n    rcases h\u2082 with \u27e8t, ht\u27e9\n    have h\u2085 : 64 \u2264 n := h\u2083 t ht\n    exact h\u2085\n  exact h\u2084\n", "proof_before": "theorem mathd_numbertheory_5 (n : \u2115) (h\u2080 : 10 \u2264 n) (h\u2081 : \u2203 x, x ^ 2 = n) (h\u2082 : \u2203 t, t ^ 3 = n) :\n    64 \u2264 n  :=  by\n  have h\u2083 : \u2200 (t : \u2115), t ^ 3 = n \u2192 64 \u2264 n := by\n    intro t ht\n    have h\u2084 : t \u2265 3 := by\n      by_contra h\n      have h\u2085 : t \u2264 2 := by linarith\n      have h\u2086 : t ^ 3 \u2264 8 := by\n        have h\u2087 : t \u2264 2 := h\u2085\n        have h\u2088 : t ^ 3 \u2264 2 ^ 3 := by\n          exact Nat.pow_le_pow_of_le_left h\u2087 3\n        linarith\n      have h\u2089 : n \u2264 8 := by linarith\n      linarith\n    by_cases h\u2085 : t \u2265 4\n    \u00b7 \n      have h\u2086 : t ^ 3 \u2265 4 ^ 3 := by\n        exact Nat.pow_le_pow_of_le_left h\u2085 3\n      have h\u2087 : n \u2265 64 := by\n        linarith\n      linarith\n    \u00b7 \n      have h\u2086 : t = 3 := by\n        have h\u2087 : t < 4 := by linarith\n        have h\u2088 : t \u2265 3 := h\u2084\n        interval_cases t <;> norm_num at h\u2087 h\u2088 \u22a2 <;> try omega\n      rw [h\u2086] at ht\n      have h\u2087 : n = 27 := by\n        norm_num at ht \u22a2\n        <;> linarith\n      rcases h\u2081 with \u27e8x, hx\u27e9\n      have h\u2088 : x ^ 2 = 27 := by\n        norm_num [h\u2087] at hx \u22a2\n        <;> linarith\n      have h\u2089 : x \u2264 5 := by\n        nlinarith\n      have h\u2081\u2080 : x \u2265 6 := by\n        by_contra h\n        have h\u2081\u2081 : x \u2264 5 := by linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> try omega\n      linarith\n  have h\u2084 : 64 \u2264 n := by\n    rcases h\u2082 with \u27e8t, ht\u27e9\n    have h\u2085 : 64 \u2264 n := h\u2083 t ht\n    exact h\u2085\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 54, "alphabet": 23, "compiles": false, "id": "mathd_algebra_158", "proof_after": "theorem mathd_algebra_158 (a : \u2115) (h\u2080 : Even a)\n    (h\u2081 : \u2191(\u2211 k in Finset.range 8, 2 * k + 1) - \u2191(\u2211 k in Finset.range 5, a + 2 * k) = (4 : \u2124)) :\n    a = 8  :=  by\n  have h\u2082 : \u2211 k in Finset.range 8, (2 * k + 1 : \u2115) = 64 := by\n    norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n      Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ]\n    <;> rfl\n  have h\u2083 : \u2211 k in Finset.range 5, (a + 2 * k : \u2115) = 5 * a + 20 := by\n    norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n      Finset.sum_range_succ, Nat.mul_add, Nat.add_assoc, Nat.add_mul]\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> omega\n  have h\u2084 : a = 8 := by\n    have h\u2085 : (\u2211 k in Finset.range 8, 2 * k + 1 : \u2115) = 64 := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n        Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ] at h\u2082 \u22a2\n      <;> simp_all [Finset.sum_range_succ, Nat.even_iff]\n      <;> ring_nf at *\n      <;> norm_num at *\n      <;> omega\n    have h\u2086 : (\u2211 k in Finset.range 5, a + 2 * k : \u2115) = 5 * a + 20 := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n        Finset.sum_range_succ, Nat.mul_add, Nat.add_assoc, Nat.add_mul] at h\u2083 \u22a2\n      <;> ring_nf at *\n      <;> norm_num at *\n      <;> omega\n    have h\u2087 : (64 : \u2124) - (5 * a + 20 : \u2124) = (4 : \u2124) := by\n      have h\u2087\u2081 : (\u2191(\u2211 k in Finset.range 8, 2 * k + 1) : \u2124) = (64 : \u2124) := by\n        norm_cast\n        <;> simp_all [Finset.sum_range_succ, Nat.even_iff]\n        <;> ring_nf at *\n        <;> norm_num at *\n        <;> omega\n      have h\u2087\u2082 : (\u2191(\u2211 k in Finset.range 5, a + 2 * k) : \u2124) = (5 * a + 20 : \u2124) := by\n        norm_cast\n        <;> simp_all [Finset.sum_range_succ, Nat.even_iff]\n        <;> ring_nf at *\n        <;> norm_num at *\n        <;> omega\n      have h\u2087\u2083 : (\u2191(\u2211 k in Finset.range 8, 2 * k + 1) : \u2124) - (\u2191(\u2211 k in Finset.range 5, a + 2 * k) : \u2124) = (4 : \u2124) := by\n        simpa using h\u2081\n      rw [h\u2087\u2081, h\u2087\u2082] at h\u2087\u2083\n      <;> linarith\n    have h\u2088 : a = 8 := by\n      have h\u2088\u2081 : (a : \u2124) = 8 := by\n        have h\u2088\u2082 : (64 : \u2124) - (5 * a + 20 : \u2124) = (4 : \u2124) := by simpa using h\u2087\n        have h\u2088\u2083 : (5 : \u2124) * a + 20 = 60 := by linarith\n        have h\u2088\u2084 : (a : \u2124) = 8 := by\n          ring_nf at h\u2088\u2083 \u22a2\n          <;> norm_cast at h\u2088\u2083 \u22a2\n          <;> omega\n        simpa using h\u2088\u2084\n      have h\u2088\u2082 : a = 8 := by\n        norm_cast at h\u2088\u2081\n        <;> omega\n      exact h\u2088\u2082\n    exact h\u2088\n  apply h\u2084\n", "proof_before": "theorem mathd_algebra_158 (a : \u2115) (h\u2080 : Even a)\n    (h\u2081 : \u2191(\u2211 k in Finset.range 8, 2 * k + 1) - \u2191(\u2211 k in Finset.range 5, a + 2 * k) = (4 : \u2124)) :\n    a = 8  :=  by\n  have h\u2082 : \u2211 k in Finset.range 8, (2 * k + 1 : \u2115) = 64 := by\n    norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n      Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ]\n    <;> rfl\n  have h\u2083 : \u2211 k in Finset.range 5, (a + 2 * k : \u2115) = 5 * a + 20 := by\n    norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n      Finset.sum_range_succ, Nat.mul_add, Nat.add_assoc, Nat.add_mul]\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> omega\n  have h\u2084 : a = 8 := by\n    have h\u2085 : (\u2211 k in Finset.range 8, 2 * k + 1 : \u2115) = 64 := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n        Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ] at h\u2082 \u22a2\n      <;> simp_all [Finset.sum_range_succ, Nat.even_iff]\n      <;> ring_nf at *\n      <;> norm_num at *\n      <;> omega\n    have h\u2086 : (\u2211 k in Finset.range 5, a + 2 * k : \u2115) = 5 * a + 20 := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n        Finset.sum_range_succ, Nat.mul_add, Nat.add_assoc, Nat.add_mul] at h\u2083 \u22a2\n      <;> ring_nf at *\n      <;> norm_num at *\n      <;> omega\n    have h\u2087 : (64 : \u2124) - (5 * a + 20 : \u2124) = (4 : \u2124) := by\n      have h\u2087\u2081 : (\u2191(\u2211 k in Finset.range 8, 2 * k + 1) : \u2124) = (64 : \u2124) := by\n        norm_cast\n        <;> simp_all [Finset.sum_range_succ, Nat.even_iff]\n        <;> ring_nf at *\n        <;> norm_num at *\n        <;> omega\n      have h\u2087\u2082 : (\u2191(\u2211 k in Finset.range 5, a + 2 * k) : \u2124) = (5 * a + 20 : \u2124) := by\n        norm_cast\n        <;> simp_all [Finset.sum_range_succ, Nat.even_iff]\n        <;> ring_nf at *\n        <;> norm_num at *\n        <;> omega\n      have h\u2087\u2083 : (\u2191(\u2211 k in Finset.range 8, 2 * k + 1) : \u2124) - (\u2191(\u2211 k in Finset.range 5, a + 2 * k) : \u2124) = (4 : \u2124) := by\n        simpa using h\u2081\n      rw [h\u2087\u2081, h\u2087\u2082] at h\u2087\u2083\n      <;> linarith\n    have h\u2088 : a = 8 := by\n      have h\u2088\u2081 : (a : \u2124) = 8 := by\n        have h\u2088\u2082 : (64 : \u2124) - (5 * a + 20 : \u2124) = (4 : \u2124) := by simpa using h\u2087\n        have h\u2088\u2083 : (5 : \u2124) * a + 20 = 60 := by linarith\n        have h\u2088\u2084 : (a : \u2124) = 8 := by\n          ring_nf at h\u2088\u2083 \u22a2\n          <;> norm_cast at h\u2088\u2083 \u22a2\n          <;> omega\n        simpa using h\u2088\u2084\n      have h\u2088\u2082 : a = 8 := by\n        norm_cast at h\u2088\u2081\n        <;> omega\n      exact h\u2088\u2082\n    exact h\u2088\n  apply h\u2084\n"}
{"mode": "clear", "steps_before": 10, "alphabet": 15, "compiles": true, "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 213, "chars_after": 17, "seconds": 95.5, "id": "mathd_algebra_171", "proof_after": "theorem mathd_algebra_171 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 5 * x + 4) : f 1 = 9  :=  by\n  rw [h\u2080]\n  ring\n", "proof_before": "theorem mathd_algebra_171 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 5 * x + 4) : f 1 = 9  :=  by\n  have h\u2081 : f 1 = 5 * (1 : \u211d) + 4 := by\n    have h\u2081\u2080 : f 1 = 5 * (1 : \u211d) + 4 := by\n      rw [h\u2080]\n      <;> norm_num\n    exact h\u2081\u2080\n  have h\u2082 : f 1 = 9 := by\n    rw [h\u2081]\n    <;> norm_num\n    <;> linarith\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 81, "alphabet": 38, "compiles": false, "id": "mathd_numbertheory_521", "proof_after": "theorem mathd_numbertheory_521 (m n : \u2115) (h\u2080 : Even m) (h\u2081 : Even n) (h\u2082 : m - n = 2)\n    (h\u2083 : m * n = 288) : m = 18  :=  by\n  have h\u2084 : n \u2265 2 := by\n    by_contra h\n    have h\u2085 : n = 0 := by\n      have h\u2085\u2081 : n \u2264 1 := by\n        omega\n      have h\u2085\u2082 : Even n := h\u2081\n      have h\u2085\u2083 : n % 2 = 0 := by\n        rw [even_iff_two_dvd] at h\u2085\u2082\n        omega\n      have h\u2085\u2084 : n = 0 := by\n        omega\n      exact h\u2085\u2084\n    have h\u2086 : m = 2 := by\n      have h\u2086\u2081 : m - n = 2 := h\u2082\n      have h\u2086\u2082 : n = 0 := h\u2085\n      rw [h\u2086\u2082] at h\u2086\u2081\n      omega\n    have h\u2087 : m * n = 288 := h\u2083\n    rw [h\u2085, h\u2086] at h\u2087\n    norm_num at h\u2087\n    <;> simp_all (config := {decide := true})\n    <;> omega\n  have h\u2085 : m = n + 2 := by\n    have h\u2085\u2081 : m \u2265 n + 2 := by\n      by_contra h\u2085\u2081\n      have h\u2085\u2082 : m < n + 2 := by omega\n      have h\u2085\u2083 : m \u2264 n + 1 := by omega\n      have h\u2085\u2084 : m - n \u2264 1 := by\n        have h\u2085\u2084\u2081 : m \u2264 n + 1 := h\u2085\u2083\n        have h\u2085\u2084\u2082 : m - n \u2264 1 := by\n          omega\n        exact h\u2085\u2084\u2082\n      have h\u2085\u2085 : m - n = 2 := h\u2082\n      omega\n    have h\u2085\u2082 : m < n + 3 := by\n      by_contra h\u2085\u2082\n      have h\u2085\u2083 : m \u2265 n + 3 := by omega\n      have h\u2085\u2084 : m - n \u2265 3 := by\n        have h\u2085\u2084\u2081 : m \u2265 n + 3 := h\u2085\u2083\n        have h\u2085\u2084\u2082 : m - n \u2265 3 := by\n          omega\n        exact h\u2085\u2084\u2082\n      have h\u2085\u2085 : m - n = 2 := h\u2082\n      omega\n    have h\u2085\u2083 : m = n + 2 := by\n      omega\n    exact h\u2085\u2083\n  have h\u2086 : n = 16 := by\n    rw [h\u2085] at h\u2083\n    have h\u2086\u2081 : (n + 2) * n = 288 := by\n      simpa [mul_comm, mul_assoc, mul_left_comm] using h\u2083\n    have h\u2086\u2082 : n \u2264 16 := by\n      by_contra h\u2086\u2082\n      have h\u2086\u2083 : n \u2265 17 := by omega\n      have h\u2086\u2084 : (n + 2) * n > 288 := by\n        have h\u2086\u2085 : n \u2265 17 := h\u2086\u2083\n        have h\u2086\u2086 : (n + 2) * n \u2265 19 * n := by\n          nlinarith\n        have h\u2086\u2087 : 19 * n > 288 := by\n          nlinarith\n        nlinarith\n      nlinarith\n    interval_cases n <;> norm_num at h\u2086\u2081 \u22a2 <;>\n      (try omega) <;> (try norm_num at h\u2085 \u22a2 <;> omega)\n    <;>\n      (try simp_all [Nat.even_iff, Nat.odd_iff, Nat.mod_eq_of_lt])\n    <;>\n      (try omega)\n  have h\u2087 : m = 18 := by\n    rw [h\u2085, h\u2086]\n    <;> norm_num\n    <;> omega\n  apply h\u2087\n", "proof_before": "theorem mathd_numbertheory_521 (m n : \u2115) (h\u2080 : Even m) (h\u2081 : Even n) (h\u2082 : m - n = 2)\n    (h\u2083 : m * n = 288) : m = 18  :=  by\n  have h\u2084 : n \u2265 2 := by\n    by_contra h\n    have h\u2085 : n = 0 := by\n      have h\u2085\u2081 : n \u2264 1 := by\n        omega\n      have h\u2085\u2082 : Even n := h\u2081\n      have h\u2085\u2083 : n % 2 = 0 := by\n        rw [even_iff_two_dvd] at h\u2085\u2082\n        omega\n      have h\u2085\u2084 : n = 0 := by\n        omega\n      exact h\u2085\u2084\n    have h\u2086 : m = 2 := by\n      have h\u2086\u2081 : m - n = 2 := h\u2082\n      have h\u2086\u2082 : n = 0 := h\u2085\n      rw [h\u2086\u2082] at h\u2086\u2081\n      omega\n    have h\u2087 : m * n = 288 := h\u2083\n    rw [h\u2085, h\u2086] at h\u2087\n    norm_num at h\u2087\n    <;> simp_all (config := {decide := true})\n    <;> omega\n  have h\u2085 : m = n + 2 := by\n    have h\u2085\u2081 : m \u2265 n + 2 := by\n      by_contra h\u2085\u2081\n      have h\u2085\u2082 : m < n + 2 := by omega\n      have h\u2085\u2083 : m \u2264 n + 1 := by omega\n      have h\u2085\u2084 : m - n \u2264 1 := by\n        have h\u2085\u2084\u2081 : m \u2264 n + 1 := h\u2085\u2083\n        have h\u2085\u2084\u2082 : m - n \u2264 1 := by\n          omega\n        exact h\u2085\u2084\u2082\n      have h\u2085\u2085 : m - n = 2 := h\u2082\n      omega\n    have h\u2085\u2082 : m < n + 3 := by\n      by_contra h\u2085\u2082\n      have h\u2085\u2083 : m \u2265 n + 3 := by omega\n      have h\u2085\u2084 : m - n \u2265 3 := by\n        have h\u2085\u2084\u2081 : m \u2265 n + 3 := h\u2085\u2083\n        have h\u2085\u2084\u2082 : m - n \u2265 3 := by\n          omega\n        exact h\u2085\u2084\u2082\n      have h\u2085\u2085 : m - n = 2 := h\u2082\n      omega\n    have h\u2085\u2083 : m = n + 2 := by\n      omega\n    exact h\u2085\u2083\n  have h\u2086 : n = 16 := by\n    rw [h\u2085] at h\u2083\n    have h\u2086\u2081 : (n + 2) * n = 288 := by\n      simpa [mul_comm, mul_assoc, mul_left_comm] using h\u2083\n    have h\u2086\u2082 : n \u2264 16 := by\n      by_contra h\u2086\u2082\n      have h\u2086\u2083 : n \u2265 17 := by omega\n      have h\u2086\u2084 : (n + 2) * n > 288 := by\n        have h\u2086\u2085 : n \u2265 17 := h\u2086\u2083\n        have h\u2086\u2086 : (n + 2) * n \u2265 19 * n := by\n          nlinarith\n        have h\u2086\u2087 : 19 * n > 288 := by\n          nlinarith\n        nlinarith\n      nlinarith\n    interval_cases n <;> norm_num at h\u2086\u2081 \u22a2 <;>\n      (try omega) <;> (try norm_num at h\u2085 \u22a2 <;> omega)\n    <;>\n      (try simp_all [Nat.even_iff, Nat.odd_iff, Nat.mod_eq_of_lt])\n    <;>\n      (try omega)\n  have h\u2087 : m = 18 := by\n    rw [h\u2085, h\u2086]\n    <;> norm_num\n    <;> omega\n  apply h\u2087\n"}
{"mode": "clear", "steps_before": 13, "alphabet": 13, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 368, "chars_after": 32, "seconds": 21.0, "id": "mathd_algebra_113", "proof_after": "theorem mathd_algebra_113 (x : \u211d) : x ^ 2 - 14 * x + 3 \u2265 7 ^ 2 - 14 * 7 + 3  :=  by\n  nlinarith [sq_nonneg (x - 7)]\n", "proof_before": "theorem mathd_algebra_113 (x : \u211d) : x ^ 2 - 14 * x + 3 \u2265 7 ^ 2 - 14 * 7 + 3  :=  by\n  have h\u2080 : (x - 7) ^ 2 \u2265 0 := by\n    nlinarith [sq_nonneg (x - 7)]\n    <;>\n    linarith\n  have h\u2081 : x ^ 2 - 14 * x + 3 \u2265 7 ^ 2 - 14 * 7 + 3 := by\n    have h\u2082 : x ^ 2 - 14 * x + 3 - (7 ^ 2 - 14 * 7 + 3) = (x - 7) ^ 2 := by\n      ring_nf\n      <;>\n      nlinarith\n    have h\u2083 : x ^ 2 - 14 * x + 3 - (7 ^ 2 - 14 * 7 + 3) \u2265 0 := by\n      linarith\n    linarith\n  exact h\u2081\n"}
{"mode": "clear", "steps_before": 28, "alphabet": 18, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 690, "chars_after": 11, "seconds": 2.0, "id": "mathd_algebra_354", "proof_after": "theorem mathd_algebra_354 (a d : \u211d) (h\u2080 : a + 6 * d = 30) (h\u2081 : a + 10 * d = 60) :\n    a + 20 * d = 135  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_354 (a d : \u211d) (h\u2080 : a + 6 * d = 30) (h\u2081 : a + 10 * d = 60) :\n    a + 20 * d = 135  :=  by\n  have h\u2082 : 4 * d = 30 := by\n    have h\u2082\u2081 : 4 * d = 30 := by\n      linarith\n    exact h\u2082\u2081\n  have h\u2083 : 14 * d = 105 := by\n    have h\u2083\u2081 : 14 * d = (7 / 2 : \u211d) * (4 * d) := by\n      ring\n    have h\u2083\u2082 : (7 / 2 : \u211d) * (4 * d) = (7 / 2 : \u211d) * 30 := by\n      rw [h\u2082]\n      <;> ring\n    have h\u2083\u2083 : (7 / 2 : \u211d) * 30 = 105 := by norm_num\n    calc\n      14 * d = (7 / 2 : \u211d) * (4 * d) := by rw [h\u2083\u2081]\n      _ = (7 / 2 : \u211d) * 30 := by rw [h\u2083\u2082]\n      _ = 105 := by rw [h\u2083\u2083]\n  have h\u2084 : a + 20 * d = 135 := by\n    have h\u2084\u2081 : a + 20 * d = (a + 6 * d) + 14 * d := by\n      ring\n    rw [h\u2084\u2081]\n    have h\u2084\u2082 : a + 6 * d = 30 := h\u2080\n    have h\u2084\u2083 : 14 * d = 105 := h\u2083\n    rw [h\u2084\u2082, h\u2084\u2083]\n    <;> norm_num\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 17, "alphabet": 15, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 879, "chars_after": 11, "seconds": 3.0, "id": "aime_1989_p8", "proof_after": "theorem aime_1989_p8 (a b c d e f g : \u211d)\n    (h\u2080 : a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g = 1)\n    (h\u2081 : 4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g = 12)\n    (h\u2082 : 9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g = 123) :\n    16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334  :=  by\n  linarith\n", "proof_before": "theorem aime_1989_p8 (a b c d e f g : \u211d)\n    (h\u2080 : a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g = 1)\n    (h\u2081 : 4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g = 12)\n    (h\u2082 : 9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g = 123) :\n    16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334  :=  by\n  have h_sum : a + b + c + d + e + f + g = 50 := by\n    have h\u2083 : 3 * a + 5 * b + 7 * c + 9 * d + 11 * e + 13 * f + 15 * g = 11 := by\n      linarith\n    have h\u2084 : 5 * a + 7 * b + 9 * c + 11 * d + 13 * e + 15 * f + 17 * g = 111 := by\n      linarith\n    have h\u2085 : 2 * (a + b + c + d + e + f + g) = 100 := by\n      linarith\n    linarith\n  have h_T : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by\n    have h\u2083 : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by\n      have h\u2084 : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) - 3 * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g) := by\n        ring\n      rw [h\u2084]\n      rw [h\u2080, h\u2081, h\u2082]\n      <;> linarith\n    exact h\u2083\n  exact h_T\n"}
{"mode": "clear", "steps_before": 312, "alphabet": 72, "compiles": false, "id": "imo_1965_p2", "proof_after": "theorem imo_1965_p2 (x y z : \u211d) (a : \u2115 \u2192 \u211d) (h\u2080 : 0 < a 0 \u2227 0 < a 4 \u2227 0 < a 8)\n    (h\u2081 : a 1 < 0 \u2227 a 2 < 0) (h\u2082 : a 3 < 0 \u2227 a 5 < 0) (h\u2083 : a 6 < 0 \u2227 a 7 < 0)\n    (h\u2084 : 0 < a 0 + a 1 + a 2) (h\u2085 : 0 < a 3 + a 4 + a 5) (h\u2086 : 0 < a 6 + a 7 + a 8)\n    (h\u2087 : a 0 * x + a 1 * y + a 2 * z = 0) (h\u2088 : a 3 * x + a 4 * y + a 5 * z = 0)\n    (h\u2089 : a 6 * x + a 7 * y + a 8 * z = 0) : x = 0 \u2227 y = 0 \u2227 z = 0  :=  by\n  have h\u2081\u2080 : x = 0 := by\n    by_contra hx\n    have h\u2081\u2080\u2081 : x > 0 \u2228 x < 0 := by\n      cases' lt_or_gt_of_ne hx with h h\n      \u00b7 exact Or.inr h\n      \u00b7 exact Or.inl h\n    cases' h\u2081\u2080\u2081 with hx_pos hx_neg\n    \u00b7 \n      by_cases hy : y \u2264 x\n      \u00b7 \n        by_cases hz : z \u2264 x\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 0 * x + a 1 * y + a 2 * z \u2265 (a 0 + a 1 + a 2) * x := by\n            have h\u2081\u2080\u2083 : a 1 * y \u2265 a 1 * x := by\n              nlinarith [h\u2081.1, h\u2081.2, hy]\n            have h\u2081\u2080\u2084 : a 2 * z \u2265 a 2 * x := by\n              nlinarith [h\u2081.1, h\u2081.2, hz]\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 0 + a 1 + a 2) * x > 0 := by\n            have h\u2081\u2080\u2086 : a 0 + a 1 + a 2 > 0 := h\u2084\n            nlinarith\n          nlinarith [h\u2087]\n        \u00b7 \n          by_cases hy' : y \u2264 z\n          \u00b7 \n            have h\u2081\u2080\u2082 : a 6 * x + a 7 * y + a 8 * z \u2265 (a 6 + a 7 + a 8) * z := by\n              have h\u2081\u2080\u2083 : a 6 * x \u2265 a 6 * z := by\n                have h\u2081\u2080\u2084 : a 6 < 0 := h\u2083.1\n                nlinarith\n              have h\u2081\u2080\u2084 : a 7 * y \u2265 a 7 * z := by\n                have h\u2081\u2080\u2085 : a 7 < 0 := h\u2083.2\n                nlinarith\n              nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n            have h\u2081\u2080\u2085 : (a 6 + a 7 + a 8) * z > 0 := by\n              have h\u2081\u2080\u2086 : a 6 + a 7 + a 8 > 0 := h\u2086\n              have h\u2081\u2080\u2087 : z > x := by linarith\n              have h\u2081\u2080\u2088 : z > 0 := by linarith\n              nlinarith\n            nlinarith [h\u2089]\n          \u00b7 \n            have h\u2081\u2080\u2082 : a 3 * x + a 4 * y + a 5 * z \u2265 (a 3 + a 4 + a 5) * y := by\n              have h\u2081\u2080\u2083 : a 3 * x \u2265 a 3 * y := by\n                have h\u2081\u2080\u2084 : a 3 < 0 := h\u2082.1\n                nlinarith\n              have h\u2081\u2080\u2084 : a 5 * z \u2265 a 5 * y := by\n                have h\u2081\u2080\u2085 : a 5 < 0 := h\u2082.2\n                nlinarith\n              nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n            have h\u2081\u2080\u2085 : (a 3 + a 4 + a 5) * y > 0 := by\n              have h\u2081\u2080\u2086 : a 3 + a 4 + a 5 > 0 := h\u2085\n              have h\u2081\u2080\u2087 : y > z := by linarith\n              have h\u2081\u2080\u2088 : y > x := by linarith\n              have h\u2081\u2080\u2089 : y > 0 := by linarith\n              nlinarith\n            nlinarith [h\u2088]\n      \u00b7 \n        by_cases hz : z \u2264 y\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 3 * x + a 4 * y + a 5 * z \u2265 (a 3 + a 4 + a 5) * y := by\n            have h\u2081\u2080\u2083 : a 3 * x \u2265 a 3 * y := by\n              have h\u2081\u2080\u2084 : a 3 < 0 := h\u2082.1\n              nlinarith\n            have h\u2081\u2080\u2084 : a 5 * z \u2265 a 5 * y := by\n              have h\u2081\u2080\u2085 : a 5 < 0 := h\u2082.2\n              nlinarith\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 3 + a 4 + a 5) * y > 0 := by\n            have h\u2081\u2080\u2086 : a 3 + a 4 + a 5 > 0 := h\u2085\n            have h\u2081\u2080\u2087 : y > x := by linarith\n            have h\u2081\u2080\u2088 : y > 0 := by linarith\n            nlinarith\n          nlinarith [h\u2088]\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 6 * x + a 7 * y + a 8 * z \u2265 (a 6 + a 7 + a 8) * z := by\n            have h\u2081\u2080\u2083 : a 6 * x \u2265 a 6 * z := by\n              have h\u2081\u2080\u2084 : a 6 < 0 := h\u2083.1\n              nlinarith\n            have h\u2081\u2080\u2084 : a 7 * y \u2265 a 7 * z := by\n              have h\u2081\u2080\u2085 : a 7 < 0 := h\u2083.2\n              nlinarith\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 6 + a 7 + a 8) * z > 0 := by\n            have h\u2081\u2080\u2086 : a 6 + a 7 + a 8 > 0 := h\u2086\n            have h\u2081\u2080\u2087 : z > y := by linarith\n            have h\u2081\u2080\u2088 : z > x := by linarith\n            have h\u2081\u2080\u2089 : z > 0 := by linarith\n            nlinarith\n          nlinarith [h\u2089]\n    \u00b7 \n      by_cases hy : y \u2265 x\n      \u00b7 \n        by_cases hz : z \u2265 x\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 0 * x + a 1 * y + a 2 * z \u2264 (a 0 + a 1 + a 2) * x := by\n            have h\u2081\u2080\u2083 : a 1 * y \u2264 a 1 * x := by\n              have h\u2081\u2080\u2084 : a 1 < 0 := h\u2081.1\n              nlinarith\n            have h\u2081\u2080\u2084 : a 2 * z \u2264 a 2 * x := by\n              have h\u2081\u2080\u2085 : a 2 < 0 := h\u2081.2\n              nlinarith\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 0 + a 1 + a 2) * x < 0 := by\n            have h\u2081\u2080\u2086 : a 0 + a 1 + a 2 > 0 := h\u2084\n            nlinarith\n          nlinarith [h\u2087]\n        \u00b7 \n          by_cases hy' : y \u2265 z\n          \u00b7 \n            have h\u2081\u2080\u2082 : a 6 * x + a 7 * y + a 8 * z \u2264 (a 6 + a 7 + a 8) * z := by\n              have h\u2081\u2080\u2083 : a 6 * x \u2264 a 6 * z := by\n                have h\u2081\u2080\u2084 : a 6 < 0 := h\u2083.1\n                nlinarith\n              have h\u2081\u2080\u2084 : a 7 * y \u2264 a 7 * z := by\n                have h\u2081\u2080\u2085 : a 7 < 0 := h\u2083.2\n                nlinarith\n              nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n            have h\u2081\u2080\u2085 : (a 6 + a 7 + a 8) * z < 0 := by\n              have h\u2081\u2080\u2086 : a 6 + a 7 + a 8 > 0 := h\u2086\n              have h\u2081\u2080\u2087 : z < x := by linarith\n              have h\u2081\u2080\u2088 : z < 0 := by linarith\n              nlinarith\n            nlinarith [h\u2089]\n          \u00b7 \n            have h\u2081\u2080\u2082 : a 3 * x + a 4 * y + a 5 * z \u2264 (a 3 + a 4 + a 5) * y := by\n              have h\u2081\u2080\u2083 : a 3 * x \u2264 a 3 * y := by\n                have h\u2081\u2080\u2084 : a 3 < 0 := h\u2082.1\n                nlinarith\n              have h\u2081\u2080\u2084 : a 5 * z \u2264 a 5 * y := by\n                have h\u2081\u2080\u2085 : a 5 < 0 := h\u2082.2\n                nlinarith\n              nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n            have h\u2081\u2080\u2085 : (a 3 + a 4 + a 5) * y < 0 := by\n              have h\u2081\u2080\u2086 : a 3 + a 4 + a 5 > 0 := h\u2085\n              have h\u2081\u2080\u2087 : y < z := by linarith\n              have h\u2081\u2080\u2088 : y < x := by linarith\n              have h\u2081\u2080\u2089 : y < 0 := by linarith\n              nlinarith\n            nlinarith [h\u2088]\n      \u00b7 \n        by_cases hz : z \u2265 y\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 3 * x + a 4 * y + a 5 * z \u2264 (a 3 + a 4 + a 5) * y := by\n            have h\u2081\u2080\u2083 : a 3 * x \u2264 a 3 * y := by\n              have h\u2081\u2080\u2084 : a 3 < 0 := h\u2082.1\n              nlinarith\n            have h\u2081\u2080\u2084 : a 5 * z \u2264 a 5 * y := by\n              have h\u2081\u2080\u2085 : a 5 < 0 := h\u2082.2\n              nlinarith\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 3 + a 4 + a 5) * y < 0 := by\n            have h\u2081\u2080\u2086 : a 3 + a 4 + a 5 > 0 := h\u2085\n            have h\u2081\u2080\u2087 : y < x := by linarith\n            have h\u2081\u2080\u2088 : y < 0 := by linarith\n            nlinarith\n          nlinarith [h\u2088]\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 6 * x + a 7 * y + a 8 * z \u2264 (a 6 + a 7 + a 8) * z := by\n            have h\u2081\u2080\u2083 : a 6 * x \u2264 a 6 * z := by\n              have h\u2081\u2080\u2084 : a 6 < 0 := h\u2083.1\n              nlinarith\n            have h\u2081\u2080\u2084 : a 7 * y \u2264 a 7 * z := by\n              have h\u2081\u2080\u2085 : a 7 < 0 := h\u2083.2\n              nlinarith\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 6 + a 7 + a 8) * z < 0 := by\n            have h\u2081\u2080\u2086 : a 6 + a 7 + a 8 > 0 := h\u2086\n            have h\u2081\u2080\u2087 : z < y := by linarith\n            have h\u2081\u2080\u2088 : z < x := by linarith\n            have h\u2081\u2080\u2089 : z < 0 := by linarith\n            nlinarith\n          nlinarith [h\u2089]\n  have h\u2081\u2081 : y = 0 := by\n    have h\u2081\u2081\u2081 : a 1 * y + a 2 * z = 0 := by\n      have h\u2081\u2081\u2082 : a 0 * x + a 1 * y + a 2 * z = 0 := h\u2087\n      rw [h\u2081\u2080] at h\u2081\u2081\u2082\n      ring_nf at h\u2081\u2081\u2082 \u22a2\n      linarith\n    have h\u2081\u2081\u2082 : a 4 * y + a 5 * z = 0 := by\n      have h\u2081\u2081\u2083 : a 3 * x + a 4 * y + a 5 * z = 0 := h\u2088\n      rw [h\u2081\u2080] at h\u2081\u2081\u2083\n      ring_nf at h\u2081\u2081\u2083 \u22a2\n      linarith\n    have h\u2081\u2081\u2083 : a 7 * y + a 8 * z = 0 := by\n      have h\u2081\u2081\u2084 : a 6 * x + a 7 * y + a 8 * z = 0 := h\u2089\n      rw [h\u2081\u2080] at h\u2081\u2081\u2084\n      ring_nf at h\u2081\u2081\u2084 \u22a2\n      linarith\n    by_contra hy\n    have h\u2081\u2081\u2084 : y > 0 \u2228 y < 0 := by\n      cases' lt_or_gt_of_ne hy with h h\n      \u00b7 exact Or.inr h\n      \u00b7 exact Or.inl h\n    cases' h\u2081\u2081\u2084 with hy_pos hy_neg\n    \u00b7 \n      by_cases hz : z \u2264 y\n      \u00b7 \n        have h\u2081\u2081\u2085 : a 3 * 0 + a 4 * y + a 5 * z \u2265 (a 3 + a 4 + a 5) * y := by\n          have h\u2081\u2081\u2086 : a 3 * (0 : \u211d) \u2265 a 3 * y := by\n            have h\u2081\u2081\u2087 : a 3 < 0 := h\u2082.1\n            nlinarith\n          have h\u2081\u2081\u2088 : a 5 * z \u2265 a 5 * y := by\n            have h\u2081\u2081\u2089 : a 5 < 0 := h\u2082.2\n            nlinarith\n          nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n        have h\u2081\u2081\u2089 : (a 3 + a 4 + a 5) * y > 0 := by\n          have h\u2081\u2082\u2080 : a 3 + a 4 + a 5 > 0 := h\u2085\n          nlinarith\n        nlinarith [h\u2081\u2081\u2082]\n      \u00b7 \n        have h\u2081\u2081\u2085 : a 6 * 0 + a 7 * y + a 8 * z \u2265 (a 6 + a 7 + a 8) * z := by\n          have h\u2081\u2081\u2086 : a 6 * (0 : \u211d) \u2265 a 6 * z := by\n            have h\u2081\u2081\u2087 : a 6 < 0 := h\u2083.1\n            nlinarith\n          have h\u2081\u2081\u2088 : a 7 * y \u2265 a 7 * z := by\n            have h\u2081\u2081\u2089 : a 7 < 0 := h\u2083.2\n            nlinarith\n          nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n        have h\u2081\u2081\u2089 : (a 6 + a 7 + a 8) * z > 0 := by\n          have h\u2081\u2082\u2080 : a 6 + a 7 + a 8 > 0 := h\u2086\n          have h\u2081\u2082\u2081 : z > y := by linarith\n          have h\u2081\u2082\u2082 : z > 0 := by linarith\n          nlinarith\n        nlinarith [h\u2081\u2081\u2083]\n    \u00b7 \n      by_cases hz : z \u2265 y\n      \u00b7 \n        have h\u2081\u2081\u2085 : a 3 * 0 + a 4 * y + a 5 * z \u2264 (a 3 + a 4 + a 5) * y := by\n          have h\u2081\u2081\u2086 : a 3 * (0 : \u211d) \u2264 a 3 * y := by\n            have h\u2081\u2081\u2087 : a 3 < 0 := h\u2082.1\n            nlinarith\n          have h\u2081\u2081\u2088 : a 5 * z \u2264 a 5 * y := by\n            have h\u2081\u2081\u2089 : a 5 < 0 := h\u2082.2\n            nlinarith\n          nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n        have h\u2081\u2081\u2089 : (a 3 + a 4 + a 5) * y < 0 := by\n          have h\u2081\u2082\u2080 : a 3 + a 4 + a 5 > 0 := h\u2085\n          nlinarith\n        nlinarith [h\u2081\u2081\u2082]\n      \u00b7 \n        have h\u2081\u2081\u2085 : a 6 * 0 + a 7 * y + a 8 * z \u2264 (a 6 + a 7 + a 8) * z := by\n          have h\u2081\u2081\u2086 : a 6 * (0 : \u211d) \u2264 a 6 * z := by\n            have h\u2081\u2081\u2087 : a 6 < 0 := h\u2083.1\n            nlinarith\n          have h\u2081\u2081\u2088 : a 7 * y \u2264 a 7 * z := by\n            have h\u2081\u2081\u2089 : a 7 < 0 := h\u2083.2\n            nlinarith\n          nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n        have h\u2081\u2081\u2089 : (a 6 + a 7 + a 8) * z < 0 := by\n          have h\u2081\u2082\u2080 : a 6 + a 7 + a 8 > 0 := h\u2086\n          have h\u2081\u2082\u2081 : z < y := by linarith\n          have h\u2081\u2082\u2082 : z < 0 := by linarith\n          nlinarith\n        nlinarith [h\u2081\u2081\u2083]\n  have h\u2081\u2082 : z = 0 := by\n    have h\u2081\u2082\u2081 : a 2 * z = 0 := by\n      have h\u2081\u2082\u2082 : a 0 * x + a 1 * y + a 2 * z = 0 := h\u2087\n      rw [h\u2081\u2080, h\u2081\u2081] at h\u2081\u2082\u2082\n      ring_nf at h\u2081\u2082\u2082 \u22a2\n      linarith\n    have h\u2081\u2082\u2082 : a 5 * z = 0 := by\n      have h\u2081\u2082\u2083 : a 3 * x + a 4 * y + a 5 * z = 0 := h\u2088\n      rw [h\u2081\u2080, h\u2081\u2081] at h\u2081\u2082\u2083\n      ring_nf at h\u2081\u2082\u2083 \u22a2\n      linarith\n    have h\u2081\u2082\u2083 : a 8 * z = 0 := by\n      have h\u2081\u2082\u2084 : a 6 * x + a 7 * y + a 8 * z = 0 := h\u2089\n      rw [h\u2081\u2080, h\u2081\u2081] at h\u2081\u2082\u2084\n      ring_nf at h\u2081\u2082\u2084 \u22a2\n      linarith\n    by_contra hz\n    have h\u2081\u2082\u2084 : z > 0 \u2228 z < 0 := by\n      cases' lt_or_gt_of_ne hz with h h\n      \u00b7 exact Or.inr h\n      \u00b7 exact Or.inl h\n    cases' h\u2081\u2082\u2084 with hz_pos hz_neg\n    \u00b7 \n      have h\u2081\u2082\u2085 : a 2 * z > 0 := by\n        have h\u2081\u2082\u2085\u2081 : a 2 < 0 := h\u2081.2\n        nlinarith\n      nlinarith\n    \u00b7 \n      have h\u2081\u2082\u2085 : a 2 * z > 0 := by\n        have h\u2081\u2082\u2085\u2081 : a 2 < 0 := h\u2081.2\n        nlinarith\n      nlinarith\n  have h\u2081\u2083 : x = 0 \u2227 y = 0 \u2227 z = 0 := by\n    exact \u27e8h\u2081\u2080, h\u2081\u2081, h\u2081\u2082\u27e9\n  exact h\u2081\u2083\n", "proof_before": "theorem imo_1965_p2 (x y z : \u211d) (a : \u2115 \u2192 \u211d) (h\u2080 : 0 < a 0 \u2227 0 < a 4 \u2227 0 < a 8)\n    (h\u2081 : a 1 < 0 \u2227 a 2 < 0) (h\u2082 : a 3 < 0 \u2227 a 5 < 0) (h\u2083 : a 6 < 0 \u2227 a 7 < 0)\n    (h\u2084 : 0 < a 0 + a 1 + a 2) (h\u2085 : 0 < a 3 + a 4 + a 5) (h\u2086 : 0 < a 6 + a 7 + a 8)\n    (h\u2087 : a 0 * x + a 1 * y + a 2 * z = 0) (h\u2088 : a 3 * x + a 4 * y + a 5 * z = 0)\n    (h\u2089 : a 6 * x + a 7 * y + a 8 * z = 0) : x = 0 \u2227 y = 0 \u2227 z = 0  :=  by\n  have h\u2081\u2080 : x = 0 := by\n    by_contra hx\n    have h\u2081\u2080\u2081 : x > 0 \u2228 x < 0 := by\n      cases' lt_or_gt_of_ne hx with h h\n      \u00b7 exact Or.inr h\n      \u00b7 exact Or.inl h\n    cases' h\u2081\u2080\u2081 with hx_pos hx_neg\n    \u00b7 \n      by_cases hy : y \u2264 x\n      \u00b7 \n        by_cases hz : z \u2264 x\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 0 * x + a 1 * y + a 2 * z \u2265 (a 0 + a 1 + a 2) * x := by\n            have h\u2081\u2080\u2083 : a 1 * y \u2265 a 1 * x := by\n              nlinarith [h\u2081.1, h\u2081.2, hy]\n            have h\u2081\u2080\u2084 : a 2 * z \u2265 a 2 * x := by\n              nlinarith [h\u2081.1, h\u2081.2, hz]\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 0 + a 1 + a 2) * x > 0 := by\n            have h\u2081\u2080\u2086 : a 0 + a 1 + a 2 > 0 := h\u2084\n            nlinarith\n          nlinarith [h\u2087]\n        \u00b7 \n          by_cases hy' : y \u2264 z\n          \u00b7 \n            have h\u2081\u2080\u2082 : a 6 * x + a 7 * y + a 8 * z \u2265 (a 6 + a 7 + a 8) * z := by\n              have h\u2081\u2080\u2083 : a 6 * x \u2265 a 6 * z := by\n                have h\u2081\u2080\u2084 : a 6 < 0 := h\u2083.1\n                nlinarith\n              have h\u2081\u2080\u2084 : a 7 * y \u2265 a 7 * z := by\n                have h\u2081\u2080\u2085 : a 7 < 0 := h\u2083.2\n                nlinarith\n              nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n            have h\u2081\u2080\u2085 : (a 6 + a 7 + a 8) * z > 0 := by\n              have h\u2081\u2080\u2086 : a 6 + a 7 + a 8 > 0 := h\u2086\n              have h\u2081\u2080\u2087 : z > x := by linarith\n              have h\u2081\u2080\u2088 : z > 0 := by linarith\n              nlinarith\n            nlinarith [h\u2089]\n          \u00b7 \n            have h\u2081\u2080\u2082 : a 3 * x + a 4 * y + a 5 * z \u2265 (a 3 + a 4 + a 5) * y := by\n              have h\u2081\u2080\u2083 : a 3 * x \u2265 a 3 * y := by\n                have h\u2081\u2080\u2084 : a 3 < 0 := h\u2082.1\n                nlinarith\n              have h\u2081\u2080\u2084 : a 5 * z \u2265 a 5 * y := by\n                have h\u2081\u2080\u2085 : a 5 < 0 := h\u2082.2\n                nlinarith\n              nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n            have h\u2081\u2080\u2085 : (a 3 + a 4 + a 5) * y > 0 := by\n              have h\u2081\u2080\u2086 : a 3 + a 4 + a 5 > 0 := h\u2085\n              have h\u2081\u2080\u2087 : y > z := by linarith\n              have h\u2081\u2080\u2088 : y > x := by linarith\n              have h\u2081\u2080\u2089 : y > 0 := by linarith\n              nlinarith\n            nlinarith [h\u2088]\n      \u00b7 \n        by_cases hz : z \u2264 y\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 3 * x + a 4 * y + a 5 * z \u2265 (a 3 + a 4 + a 5) * y := by\n            have h\u2081\u2080\u2083 : a 3 * x \u2265 a 3 * y := by\n              have h\u2081\u2080\u2084 : a 3 < 0 := h\u2082.1\n              nlinarith\n            have h\u2081\u2080\u2084 : a 5 * z \u2265 a 5 * y := by\n              have h\u2081\u2080\u2085 : a 5 < 0 := h\u2082.2\n              nlinarith\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 3 + a 4 + a 5) * y > 0 := by\n            have h\u2081\u2080\u2086 : a 3 + a 4 + a 5 > 0 := h\u2085\n            have h\u2081\u2080\u2087 : y > x := by linarith\n            have h\u2081\u2080\u2088 : y > 0 := by linarith\n            nlinarith\n          nlinarith [h\u2088]\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 6 * x + a 7 * y + a 8 * z \u2265 (a 6 + a 7 + a 8) * z := by\n            have h\u2081\u2080\u2083 : a 6 * x \u2265 a 6 * z := by\n              have h\u2081\u2080\u2084 : a 6 < 0 := h\u2083.1\n              nlinarith\n            have h\u2081\u2080\u2084 : a 7 * y \u2265 a 7 * z := by\n              have h\u2081\u2080\u2085 : a 7 < 0 := h\u2083.2\n              nlinarith\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 6 + a 7 + a 8) * z > 0 := by\n            have h\u2081\u2080\u2086 : a 6 + a 7 + a 8 > 0 := h\u2086\n            have h\u2081\u2080\u2087 : z > y := by linarith\n            have h\u2081\u2080\u2088 : z > x := by linarith\n            have h\u2081\u2080\u2089 : z > 0 := by linarith\n            nlinarith\n          nlinarith [h\u2089]\n    \u00b7 \n      by_cases hy : y \u2265 x\n      \u00b7 \n        by_cases hz : z \u2265 x\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 0 * x + a 1 * y + a 2 * z \u2264 (a 0 + a 1 + a 2) * x := by\n            have h\u2081\u2080\u2083 : a 1 * y \u2264 a 1 * x := by\n              have h\u2081\u2080\u2084 : a 1 < 0 := h\u2081.1\n              nlinarith\n            have h\u2081\u2080\u2084 : a 2 * z \u2264 a 2 * x := by\n              have h\u2081\u2080\u2085 : a 2 < 0 := h\u2081.2\n              nlinarith\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 0 + a 1 + a 2) * x < 0 := by\n            have h\u2081\u2080\u2086 : a 0 + a 1 + a 2 > 0 := h\u2084\n            nlinarith\n          nlinarith [h\u2087]\n        \u00b7 \n          by_cases hy' : y \u2265 z\n          \u00b7 \n            have h\u2081\u2080\u2082 : a 6 * x + a 7 * y + a 8 * z \u2264 (a 6 + a 7 + a 8) * z := by\n              have h\u2081\u2080\u2083 : a 6 * x \u2264 a 6 * z := by\n                have h\u2081\u2080\u2084 : a 6 < 0 := h\u2083.1\n                nlinarith\n              have h\u2081\u2080\u2084 : a 7 * y \u2264 a 7 * z := by\n                have h\u2081\u2080\u2085 : a 7 < 0 := h\u2083.2\n                nlinarith\n              nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n            have h\u2081\u2080\u2085 : (a 6 + a 7 + a 8) * z < 0 := by\n              have h\u2081\u2080\u2086 : a 6 + a 7 + a 8 > 0 := h\u2086\n              have h\u2081\u2080\u2087 : z < x := by linarith\n              have h\u2081\u2080\u2088 : z < 0 := by linarith\n              nlinarith\n            nlinarith [h\u2089]\n          \u00b7 \n            have h\u2081\u2080\u2082 : a 3 * x + a 4 * y + a 5 * z \u2264 (a 3 + a 4 + a 5) * y := by\n              have h\u2081\u2080\u2083 : a 3 * x \u2264 a 3 * y := by\n                have h\u2081\u2080\u2084 : a 3 < 0 := h\u2082.1\n                nlinarith\n              have h\u2081\u2080\u2084 : a 5 * z \u2264 a 5 * y := by\n                have h\u2081\u2080\u2085 : a 5 < 0 := h\u2082.2\n                nlinarith\n              nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n            have h\u2081\u2080\u2085 : (a 3 + a 4 + a 5) * y < 0 := by\n              have h\u2081\u2080\u2086 : a 3 + a 4 + a 5 > 0 := h\u2085\n              have h\u2081\u2080\u2087 : y < z := by linarith\n              have h\u2081\u2080\u2088 : y < x := by linarith\n              have h\u2081\u2080\u2089 : y < 0 := by linarith\n              nlinarith\n            nlinarith [h\u2088]\n      \u00b7 \n        by_cases hz : z \u2265 y\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 3 * x + a 4 * y + a 5 * z \u2264 (a 3 + a 4 + a 5) * y := by\n            have h\u2081\u2080\u2083 : a 3 * x \u2264 a 3 * y := by\n              have h\u2081\u2080\u2084 : a 3 < 0 := h\u2082.1\n              nlinarith\n            have h\u2081\u2080\u2084 : a 5 * z \u2264 a 5 * y := by\n              have h\u2081\u2080\u2085 : a 5 < 0 := h\u2082.2\n              nlinarith\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 3 + a 4 + a 5) * y < 0 := by\n            have h\u2081\u2080\u2086 : a 3 + a 4 + a 5 > 0 := h\u2085\n            have h\u2081\u2080\u2087 : y < x := by linarith\n            have h\u2081\u2080\u2088 : y < 0 := by linarith\n            nlinarith\n          nlinarith [h\u2088]\n        \u00b7 \n          have h\u2081\u2080\u2082 : a 6 * x + a 7 * y + a 8 * z \u2264 (a 6 + a 7 + a 8) * z := by\n            have h\u2081\u2080\u2083 : a 6 * x \u2264 a 6 * z := by\n              have h\u2081\u2080\u2084 : a 6 < 0 := h\u2083.1\n              nlinarith\n            have h\u2081\u2080\u2084 : a 7 * y \u2264 a 7 * z := by\n              have h\u2081\u2080\u2085 : a 7 < 0 := h\u2083.2\n              nlinarith\n            nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n          have h\u2081\u2080\u2085 : (a 6 + a 7 + a 8) * z < 0 := by\n            have h\u2081\u2080\u2086 : a 6 + a 7 + a 8 > 0 := h\u2086\n            have h\u2081\u2080\u2087 : z < y := by linarith\n            have h\u2081\u2080\u2088 : z < x := by linarith\n            have h\u2081\u2080\u2089 : z < 0 := by linarith\n            nlinarith\n          nlinarith [h\u2089]\n  have h\u2081\u2081 : y = 0 := by\n    have h\u2081\u2081\u2081 : a 1 * y + a 2 * z = 0 := by\n      have h\u2081\u2081\u2082 : a 0 * x + a 1 * y + a 2 * z = 0 := h\u2087\n      rw [h\u2081\u2080] at h\u2081\u2081\u2082\n      ring_nf at h\u2081\u2081\u2082 \u22a2\n      linarith\n    have h\u2081\u2081\u2082 : a 4 * y + a 5 * z = 0 := by\n      have h\u2081\u2081\u2083 : a 3 * x + a 4 * y + a 5 * z = 0 := h\u2088\n      rw [h\u2081\u2080] at h\u2081\u2081\u2083\n      ring_nf at h\u2081\u2081\u2083 \u22a2\n      linarith\n    have h\u2081\u2081\u2083 : a 7 * y + a 8 * z = 0 := by\n      have h\u2081\u2081\u2084 : a 6 * x + a 7 * y + a 8 * z = 0 := h\u2089\n      rw [h\u2081\u2080] at h\u2081\u2081\u2084\n      ring_nf at h\u2081\u2081\u2084 \u22a2\n      linarith\n    by_contra hy\n    have h\u2081\u2081\u2084 : y > 0 \u2228 y < 0 := by\n      cases' lt_or_gt_of_ne hy with h h\n      \u00b7 exact Or.inr h\n      \u00b7 exact Or.inl h\n    cases' h\u2081\u2081\u2084 with hy_pos hy_neg\n    \u00b7 \n      by_cases hz : z \u2264 y\n      \u00b7 \n        have h\u2081\u2081\u2085 : a 3 * 0 + a 4 * y + a 5 * z \u2265 (a 3 + a 4 + a 5) * y := by\n          have h\u2081\u2081\u2086 : a 3 * (0 : \u211d) \u2265 a 3 * y := by\n            have h\u2081\u2081\u2087 : a 3 < 0 := h\u2082.1\n            nlinarith\n          have h\u2081\u2081\u2088 : a 5 * z \u2265 a 5 * y := by\n            have h\u2081\u2081\u2089 : a 5 < 0 := h\u2082.2\n            nlinarith\n          nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n        have h\u2081\u2081\u2089 : (a 3 + a 4 + a 5) * y > 0 := by\n          have h\u2081\u2082\u2080 : a 3 + a 4 + a 5 > 0 := h\u2085\n          nlinarith\n        nlinarith [h\u2081\u2081\u2082]\n      \u00b7 \n        have h\u2081\u2081\u2085 : a 6 * 0 + a 7 * y + a 8 * z \u2265 (a 6 + a 7 + a 8) * z := by\n          have h\u2081\u2081\u2086 : a 6 * (0 : \u211d) \u2265 a 6 * z := by\n            have h\u2081\u2081\u2087 : a 6 < 0 := h\u2083.1\n            nlinarith\n          have h\u2081\u2081\u2088 : a 7 * y \u2265 a 7 * z := by\n            have h\u2081\u2081\u2089 : a 7 < 0 := h\u2083.2\n            nlinarith\n          nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n        have h\u2081\u2081\u2089 : (a 6 + a 7 + a 8) * z > 0 := by\n          have h\u2081\u2082\u2080 : a 6 + a 7 + a 8 > 0 := h\u2086\n          have h\u2081\u2082\u2081 : z > y := by linarith\n          have h\u2081\u2082\u2082 : z > 0 := by linarith\n          nlinarith\n        nlinarith [h\u2081\u2081\u2083]\n    \u00b7 \n      by_cases hz : z \u2265 y\n      \u00b7 \n        have h\u2081\u2081\u2085 : a 3 * 0 + a 4 * y + a 5 * z \u2264 (a 3 + a 4 + a 5) * y := by\n          have h\u2081\u2081\u2086 : a 3 * (0 : \u211d) \u2264 a 3 * y := by\n            have h\u2081\u2081\u2087 : a 3 < 0 := h\u2082.1\n            nlinarith\n          have h\u2081\u2081\u2088 : a 5 * z \u2264 a 5 * y := by\n            have h\u2081\u2081\u2089 : a 5 < 0 := h\u2082.2\n            nlinarith\n          nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n        have h\u2081\u2081\u2089 : (a 3 + a 4 + a 5) * y < 0 := by\n          have h\u2081\u2082\u2080 : a 3 + a 4 + a 5 > 0 := h\u2085\n          nlinarith\n        nlinarith [h\u2081\u2081\u2082]\n      \u00b7 \n        have h\u2081\u2081\u2085 : a 6 * 0 + a 7 * y + a 8 * z \u2264 (a 6 + a 7 + a 8) * z := by\n          have h\u2081\u2081\u2086 : a 6 * (0 : \u211d) \u2264 a 6 * z := by\n            have h\u2081\u2081\u2087 : a 6 < 0 := h\u2083.1\n            nlinarith\n          have h\u2081\u2081\u2088 : a 7 * y \u2264 a 7 * z := by\n            have h\u2081\u2081\u2089 : a 7 < 0 := h\u2083.2\n            nlinarith\n          nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081.1, h\u2081.2, h\u2082.1, h\u2082.2, h\u2083.1, h\u2083.2]\n        have h\u2081\u2081\u2089 : (a 6 + a 7 + a 8) * z < 0 := by\n          have h\u2081\u2082\u2080 : a 6 + a 7 + a 8 > 0 := h\u2086\n          have h\u2081\u2082\u2081 : z < y := by linarith\n          have h\u2081\u2082\u2082 : z < 0 := by linarith\n          nlinarith\n        nlinarith [h\u2081\u2081\u2083]\n  have h\u2081\u2082 : z = 0 := by\n    have h\u2081\u2082\u2081 : a 2 * z = 0 := by\n      have h\u2081\u2082\u2082 : a 0 * x + a 1 * y + a 2 * z = 0 := h\u2087\n      rw [h\u2081\u2080, h\u2081\u2081] at h\u2081\u2082\u2082\n      ring_nf at h\u2081\u2082\u2082 \u22a2\n      linarith\n    have h\u2081\u2082\u2082 : a 5 * z = 0 := by\n      have h\u2081\u2082\u2083 : a 3 * x + a 4 * y + a 5 * z = 0 := h\u2088\n      rw [h\u2081\u2080, h\u2081\u2081] at h\u2081\u2082\u2083\n      ring_nf at h\u2081\u2082\u2083 \u22a2\n      linarith\n    have h\u2081\u2082\u2083 : a 8 * z = 0 := by\n      have h\u2081\u2082\u2084 : a 6 * x + a 7 * y + a 8 * z = 0 := h\u2089\n      rw [h\u2081\u2080, h\u2081\u2081] at h\u2081\u2082\u2084\n      ring_nf at h\u2081\u2082\u2084 \u22a2\n      linarith\n    by_contra hz\n    have h\u2081\u2082\u2084 : z > 0 \u2228 z < 0 := by\n      cases' lt_or_gt_of_ne hz with h h\n      \u00b7 exact Or.inr h\n      \u00b7 exact Or.inl h\n    cases' h\u2081\u2082\u2084 with hz_pos hz_neg\n    \u00b7 \n      have h\u2081\u2082\u2085 : a 2 * z > 0 := by\n        have h\u2081\u2082\u2085\u2081 : a 2 < 0 := h\u2081.2\n        nlinarith\n      nlinarith\n    \u00b7 \n      have h\u2081\u2082\u2085 : a 2 * z > 0 := by\n        have h\u2081\u2082\u2085\u2081 : a 2 < 0 := h\u2081.2\n        nlinarith\n      nlinarith\n  have h\u2081\u2083 : x = 0 \u2227 y = 0 \u2227 z = 0 := by\n    exact \u27e8h\u2081\u2080, h\u2081\u2081, h\u2081\u2082\u27e9\n  exact h\u2081\u2083\n"}
{"mode": "clear", "steps_before": 4, "alphabet": 12, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 63, "chars_after": 6, "seconds": 0.4, "id": "mathd_numbertheory_342", "proof_after": "theorem mathd_numbertheory_342 : 54 % 6 = 0  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_342 : 54 % 6 = 0  :=  by\n  have h : 54 % 6 = 0 := by\n    norm_num\n    <;> rfl\n  exact h\n"}
{"mode": "clear", "steps_before": 4, "alphabet": 12, "compiles": false, "id": "amc12a_2021_p9", "proof_after": "theorem amc12a_2021_p9 : (\u220f k in Finset.range 7, (2 ^ 2 ^ k + 3 ^ 2 ^ k)) = 3 ^ 128 - 2 ^ 128  :=  by\n  have h\u2080 : (\u220f k in Finset.range 7, (2 ^ 2 ^ k + 3 ^ 2 ^ k)) = 3 ^ 128 - 2 ^ 128 := by\n    norm_num [Finset.prod_range_succ, pow_succ, Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib,\n      Finset.prod_range_succ, pow_succ, Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib]\n    <;> rfl\n  exact h\u2080\n", "proof_before": "theorem amc12a_2021_p9 : (\u220f k in Finset.range 7, (2 ^ 2 ^ k + 3 ^ 2 ^ k)) = 3 ^ 128 - 2 ^ 128  :=  by\n  have h\u2080 : (\u220f k in Finset.range 7, (2 ^ 2 ^ k + 3 ^ 2 ^ k)) = 3 ^ 128 - 2 ^ 128 := by\n    norm_num [Finset.prod_range_succ, pow_succ, Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib,\n      Finset.prod_range_succ, pow_succ, Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib]\n    <;> rfl\n  exact h\u2080\n"}
{"mode": "clear", "steps_before": 696, "alphabet": 96, "compiles": false, "id": "imo_1992_p1", "proof_after": "theorem imo_1992_p1 (p q r : \u2124) (h\u2080 : 1 < p \u2227 p < q \u2227 q < r)\n    (h\u2081 : (p - 1) * (q - 1) * (r - 1) \u2223 p * q * r - 1) :\n    (p, q, r) = (2, 4, 8) \u2228 (p, q, r) = (3, 5, 15)  :=  by\n  have h\u2082 : p \u2265 2 := by\n    linarith [h\u2080.1]\n  have h\u2083 : q \u2265 3 := by\n    by_contra h\n    have h\u2084 : q \u2264 2 := by linarith\n    have h\u2085 : q \u2265 2 := by linarith [h\u2080.2.1]\n    have h\u2086 : q = 2 := by linarith\n    have h\u2087 : p < 2 := by linarith [h\u2080.2.1]\n    have h\u2088 : p \u2265 2 := by linarith [h\u2080.1]\n    linarith\n  have h\u2084 : r \u2265 4 := by\n    by_contra h\n    have h\u2085 : r \u2264 3 := by linarith\n    have h\u2086 : r \u2265 3 := by linarith [h\u2080.2.2]\n    have h\u2087 : r = 3 := by linarith\n    have h\u2088 : q < 3 := by linarith [h\u2080.2.2]\n    have h\u2089 : q \u2265 3 := by linarith [h\u2083]\n    linarith\n  have h\u2085 : (p, q, r) = (2, 4, 8) \u2228 (p, q, r) = (3, 5, 15) := by\n    have h\u2086 : (p - 1 : \u2124) > 0 := by\n      linarith\n    have h\u2087 : (q - 1 : \u2124) > 0 := by\n      linarith\n    have h\u2088 : (r - 1 : \u2124) > 0 := by\n      linarith\n    have h\u2089 : (p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124) > 0 := by positivity\n    have h\u2081\u2080 : \u2203 (k : \u2124), (p * q * r - 1 : \u2124) = k * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) := by\n      obtain \u27e8k, hk\u27e9 := h\u2081\n      refine' \u27e8k, _\u27e9\n      linarith\n    obtain \u27e8k, hk\u27e9 := h\u2081\u2080\n    have h\u2081\u2081 : k > 0 := by\n      by_contra h\u2081\u2081\n      have h\u2081\u2082 : k \u2264 0 := by linarith\n      have h\u2081\u2083 : (p * q * r - 1 : \u2124) \u2264 0 := by\n        nlinarith [mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2083),\n          mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2084),\n          mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2084)]\n      have h\u2081\u2084 : (p * q * r : \u2124) \u2264 1 := by linarith\n      have h\u2081\u2085 : (p : \u2124) \u2265 2 := by linarith\n      have h\u2081\u2086 : (q : \u2124) \u2265 3 := by linarith\n      have h\u2081\u2087 : (r : \u2124) \u2265 4 := by linarith\n      have h\u2081\u2088 : (p * q * r : \u2124) \u2265 2 * 3 * 4 := by\n        have h\u2081\u2089 : (p : \u2124) * q \u2265 2 * 3 := by nlinarith\n        nlinarith\n      linarith\n    have h\u2081\u2082 : k \u2264 3 := by\n      by_contra h\u2081\u2082\n      have h\u2081\u2083 : k \u2265 4 := by linarith\n      have h\u2081\u2084 : (p - 1 : \u2124) \u2265 1 := by\n        linarith\n      have h\u2081\u2085 : (q - 1 : \u2124) \u2265 2 := by\n        linarith\n      have h\u2081\u2086 : (r - 1 : \u2124) \u2265 3 := by\n        linarith\n      have h\u2081\u2087 : (p - 1 : \u2124) * (q - 1 : \u2124) \u2265 2 := by\n        nlinarith\n      have h\u2081\u2088 : (p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124) \u2265 6 := by\n        nlinarith\n      have h\u2081\u2089 : (k : \u2124) * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) \u2265 4 * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) := by\n        nlinarith\n      have h\u2082\u2080 : (p * q * r - 1 : \u2124) \u2265 4 * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) := by\n        linarith\n      have h\u2082\u2081 : (p * q * r - 1 : \u2124) < 4 * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) := by\n        have h\u2082\u2082 : (p : \u2124) * q * r < 4 * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) + 1 := by\n          nlinarith [mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2083),\n            mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2084),\n            mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2084)]\n        linarith\n      linarith\n    have h\u2081\u2083 : k = 1 \u2228 k = 2 \u2228 k = 3 := by\n      omega\n    rcases h\u2081\u2083 with (rfl | rfl | rfl)\n    \u00b7 \n      have h\u2081\u2084 : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) := by\n        ring_nf at hk \u22a2\n        <;> linarith\n      have h\u2081\u2085 : p * q + p * r + q * r = p + q + r := by\n        have h\u2081\u2085\u2081 : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) := by linarith\n        ring_nf at h\u2081\u2085\u2081 \u22a2\n        nlinarith\n      have h\u2081\u2086 : p = 2 := by\n        by_contra h\u2081\u2086\n        have h\u2081\u2087 : p \u2265 3 := by\n          by_contra h\u2081\u2087\n          have h\u2081\u2088 : p \u2264 2 := by linarith\n          have h\u2081\u2089 : p = 2 := by linarith\n          contradiction\n        have h\u2082\u2080 : q \u2265 3 := by linarith\n        have h\u2082\u2081 : r \u2265 4 := by linarith\n        have h\u2082\u2082 : (p : \u2124) * q \u2265 3 * 3 := by\n          nlinarith\n        have h\u2082\u2083 : (p : \u2124) * r \u2265 3 * 4 := by\n          nlinarith\n        have h\u2082\u2084 : (q : \u2124) * r \u2265 3 * 4 := by\n          nlinarith\n        nlinarith\n      have h\u2081\u2087 : q = 3 := by\n        by_contra h\u2081\u2087\n        have h\u2081\u2088 : q \u2265 4 := by\n          by_contra h\u2081\u2088\n          have h\u2081\u2089 : q \u2264 3 := by linarith\n          have h\u2082\u2080 : q = 3 := by linarith\n          contradiction\n        have h\u2082\u2081 : p = 2 := by linarith\n        have h\u2082\u2082 : (p : \u2124) * q \u2265 2 * 4 := by\n          nlinarith\n        have h\u2082\u2083 : (p : \u2124) * r \u2265 2 * 4 := by\n          nlinarith\n        have h\u2082\u2084 : (q : \u2124) * r \u2265 4 * 4 := by\n          nlinarith\n        nlinarith\n      have h\u2081\u2088 : r = 4 := by\n        by_contra h\u2081\u2088\n        have h\u2081\u2089 : r \u2265 5 := by\n          by_contra h\u2081\u2089\n          have h\u2082\u2080 : r \u2264 4 := by linarith\n          have h\u2082\u2081 : r = 4 := by linarith\n          contradiction\n        have h\u2082\u2082 : p = 2 := by linarith\n        have h\u2082\u2083 : q = 3 := by linarith\n        have h\u2082\u2084 : (p : \u2124) * q \u2265 2 * 3 := by\n          nlinarith\n        have h\u2082\u2085 : (p : \u2124) * r \u2265 2 * 5 := by\n          nlinarith\n        have h\u2082\u2086 : (q : \u2124) * r \u2265 3 * 5 := by\n          nlinarith\n        nlinarith\n      exfalso\n      norm_num [h\u2081\u2086, h\u2081\u2087, h\u2081\u2088] at h\u2081\u2084 h\u2081\u2085 hk h\u2080 \u22a2 <;> linarith\n    \u00b7 \n      have h\u2081\u2084 : p * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by\n        ring_nf at hk \u22a2\n        <;> linarith\n      have h\u2081\u2085 : p = 3 := by\n        by_contra h\u2081\u2085\n        have h\u2081\u2086 : p \u2260 3 := by tauto\n        have h\u2081\u2087 : p \u2265 4 := by\n          by_contra h\u2081\u2087\n          have h\u2081\u2088 : p \u2264 3 := by linarith\n          have h\u2081\u2089 : p = 2 := by\n            by_contra h\u2081\u2089\n            have h\u2082\u2080 : p \u2265 3 := by omega\n            have h\u2082\u2081 : p = 3 := by omega\n            contradiction\n          have h\u2082\u2082 : p = 2 := by omega\n          have h\u2082\u2083 : q \u2265 3 := by linarith\n          have h\u2082\u2084 : r \u2265 4 := by linarith\n          have h\u2082\u2085 : (p : \u2124) * q \u2265 2 * 3 := by nlinarith\n          have h\u2082\u2086 : (p : \u2124) * r \u2265 2 * 4 := by nlinarith\n          have h\u2082\u2087 : (q : \u2124) * r \u2265 3 * 4 := by nlinarith\n          have h\u2082\u2088 : (p : \u2124) * q * r \u2265 2 * 3 * 4 := by nlinarith\n          have h\u2082\u2089 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n          have h\u2083\u2080 : (p : \u2124) = 2 := by omega\n          have h\u2083\u2081 : (q : \u2124) \u2265 3 := by omega\n          have h\u2083\u2082 : (r : \u2124) \u2265 4 := by omega\n          have h\u2083\u2083 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n          have h\u2083\u2084 : (p : \u2124) = 2 := by omega\n          have h\u2083\u2085 : (q : \u2124) \u2265 3 := by omega\n          have h\u2083\u2086 : (r : \u2124) \u2265 4 := by omega\n          have h\u2083\u2087 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n          have h\u2083\u2088 : False := by\n            have h\u2083\u2089 : (p : \u2124) = 2 := by omega\n            have h\u2084\u2080 : (q : \u2124) \u2265 3 := by omega\n            have h\u2084\u2081 : (r : \u2124) \u2265 4 := by omega\n            have h\u2084\u2082 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n            have h\u2084\u2083 : 2 * q * r - 1 = 2 * (1 * (q - 1) * (r - 1)) := by\n              simp [h\u2083\u2089] at h\u2084\u2082 \u22a2\n              <;> ring_nf at h\u2084\u2082 \u22a2 <;> linarith\n            have h\u2084\u2084 : 2 * q * r - 1 = 2 * ((q - 1) * (r - 1)) := by\n              ring_nf at h\u2084\u2083 \u22a2 <;> linarith\n            have h\u2084\u2085 : 2 * q * r - 1 = 2 * (q * r - q - r + 1) := by\n              ring_nf at h\u2084\u2084 \u22a2 <;> linarith\n            have h\u2084\u2086 : 2 * q * r - 1 = 2 * q * r - 2 * q - 2 * r + 2 := by\n              ring_nf at h\u2084\u2085 \u22a2 <;> linarith\n            have h\u2084\u2087 : -1 = -2 * q - 2 * r + 2 := by\n              linarith\n            have h\u2084\u2088 : 2 * q + 2 * r = 3 := by linarith\n            have h\u2084\u2089 : (q : \u2124) \u2265 3 := by omega\n            have h\u2085\u2080 : (r : \u2124) \u2265 4 := by omega\n            have h\u2085\u2081 : 2 * q + 2 * r \u2265 14 := by\n              nlinarith\n            linarith\n          exact h\u2083\u2088\n        have h\u2081\u2089 : q \u2265 p + 1 := by omega\n        have h\u2082\u2080 : r \u2265 q + 1 := by omega\n        have h\u2082\u2081 : (p : \u2124) \u2265 4 := by omega\n        have h\u2082\u2082 : (q : \u2124) \u2265 5 := by omega\n        have h\u2082\u2083 : (r : \u2124) \u2265 6 := by omega\n        have h\u2082\u2084 : (p : \u2124) * q \u2265 4 * 5 := by nlinarith\n        have h\u2082\u2085 : (p : \u2124) * r \u2265 4 * 6 := by nlinarith\n        have h\u2082\u2086 : (q : \u2124) * r \u2265 5 * 6 := by nlinarith\n        have h\u2082\u2087 : (p : \u2124) * q * r \u2265 4 * 5 * 6 := by nlinarith\n        have h\u2082\u2088 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2089 : (p : \u2124) \u2265 4 := by omega\n        have h\u2083\u2080 : (q : \u2124) \u2265 5 := by omega\n        have h\u2083\u2081 : (r : \u2124) \u2265 6 := by omega\n        have h\u2083\u2082 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2083\u2083 : False := by\n          nlinarith [sq_nonneg ((p : \u2124) - 2), sq_nonneg ((q : \u2124) - 2), sq_nonneg ((r : \u2124) - 2),\n            mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2083),\n            mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2084),\n            mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2084)]\n        exact h\u2083\u2083\n      have h\u2081\u2086 : q = 5 := by\n        have h\u2081\u2087 : p = 3 := by linarith\n        have h\u2081\u2088 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2081\u2089 : (p : \u2124) = 3 := by norm_num [h\u2081\u2087]\n        have h\u2082\u2080 : (q : \u2124) \u2265 4 := by\n          by_contra h\u2082\u2080\n          have h\u2082\u2081 : q \u2264 3 := by linarith\n          have h\u2082\u2082 : q = 3 := by linarith\n          have h\u2082\u2083 : (p : \u2124) = 3 := by norm_num [h\u2081\u2087]\n          have h\u2082\u2084 : (q : \u2124) = 3 := by norm_num [h\u2082\u2082]\n          have h\u2082\u2085 : (r : \u2124) \u2265 4 := by linarith\n          norm_num [h\u2081\u2087, h\u2082\u2082, h\u2082\u2083, h\u2082\u2084] at h\u2081\u2088\n          <;>\n            (try omega) <;>\n            (try nlinarith) <;>\n            (try\n              {\n                nlinarith [mul_pos (sub_pos.mpr h\u2080.2.1) (sub_pos.mpr h\u2080.2.2)]\n              })\n        have h\u2082\u2081 : (r : \u2124) \u2265 q + 1 := by linarith\n        have h\u2082\u2082 : (q : \u2124) \u2265 4 := by linarith\n        have h\u2082\u2083 : (p : \u2124) = 3 := by norm_num [h\u2081\u2087]\n        have h\u2082\u2084 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2085 : 3 * q * r - 1 = 2 * (2 * (q - 1) * (r - 1)) := by\n          norm_num [h\u2081\u2087, h\u2081\u2089] at h\u2082\u2084 \u22a2\n          <;> ring_nf at h\u2082\u2084 \u22a2 <;> linarith\n        have h\u2082\u2086 : 3 * q * r - 1 = 4 * ((q - 1) * (r - 1)) := by\n          ring_nf at h\u2082\u2085 \u22a2\n          <;> nlinarith\n        have h\u2082\u2087 : 3 * q * r - 1 = 4 * (q * r - q - r + 1) := by\n          ring_nf at h\u2082\u2086 \u22a2\n          <;> nlinarith\n        have h\u2082\u2088 : 3 * q * r - 1 = 4 * q * r - 4 * q - 4 * r + 4 := by\n          ring_nf at h\u2082\u2087 \u22a2\n          <;> nlinarith\n        have h\u2082\u2089 : - q * r + 4 * q + 4 * r - 5 = 0 := by\n          ring_nf at h\u2082\u2088 \u22a2\n          <;> nlinarith\n        have h\u2083\u2080 : (q : \u2124) * r - 4 * q - 4 * r + 5 = 0 := by\n          ring_nf at h\u2082\u2089 \u22a2\n          <;> nlinarith\n        have h\u2083\u2081 : (q - 4 : \u2124) * (r - 4 : \u2124) = 11 := by\n          ring_nf at h\u2083\u2080 \u22a2\n          <;> nlinarith\n        have h\u2083\u2082 : (q : \u2124) - 4 = 1 \u2227 (r : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = 11 \u2227 (r : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = -1 \u2227 (r : \u2124) - 4 = -11 \u2228 (q : \u2124) - 4 = -11 \u2227 (r : \u2124) - 4 = -1 := by\n          have h\u2083\u2083 : (q : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = -1 \u2228 (q : \u2124) - 4 = -11 := by\n            have h\u2083\u2084 : (q : \u2124) - 4 \u2223 11 := by\n              use (r : \u2124) - 4\n              linarith\n            have h\u2083\u2085 : (q : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = -1 \u2228 (q : \u2124) - 4 = -11 := by\n              have h\u2083\u2086 : (q : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = -1 \u2228 (q : \u2124) - 4 = -11 := by\n                rw [\u2190 Int.natAbs_dvd_natAbs] at h\u2083\u2084\n                have h\u2083\u2087 : ((q : \u2124) - 4).natAbs \u2223 11 := by\n                  simpa [Int.natAbs] using h\u2083\u2084\n                have h\u2083\u2088 : ((q : \u2124) - 4).natAbs = 1 \u2228 ((q : \u2124) - 4).natAbs = 11 := by\n                  have h\u2083\u2089 : ((q : \u2124) - 4).natAbs \u2223 11 := h\u2083\u2087\n                  have h\u2084\u2080 : ((q : \u2124) - 4).natAbs \u2264 11 := Nat.le_of_dvd (by decide) h\u2083\u2089\n                  interval_cases ((q : \u2124) - 4).natAbs <;> norm_num at h\u2083\u2089 \u22a2 <;> omega\n                cases h\u2083\u2088 with\n                | inl h\u2083\u2088 =>\n                  have h\u2084\u2081 : (q : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = -1 := by\n                    have h\u2084\u2082 : ((q : \u2124) - 4).natAbs = 1 := h\u2083\u2088\n                    have h\u2084\u2083 : (q : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = -1 := by\n                      rw [Int.natAbs_eq_iff] at h\u2084\u2082\n                      tauto\n                    exact h\u2084\u2083\n                  cases h\u2084\u2081 with\n                  | inl h\u2084\u2081 =>\n                    tauto\n                  | inr h\u2084\u2081 =>\n                    tauto\n                | inr h\u2083\u2088 =>\n                  have h\u2084\u2081 : (q : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = -11 := by\n                    have h\u2084\u2082 : ((q : \u2124) - 4).natAbs = 11 := h\u2083\u2088\n                    have h\u2084\u2083 : (q : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = -11 := by\n                      rw [Int.natAbs_eq_iff] at h\u2084\u2082\n                      tauto\n                    exact h\u2084\u2083\n                  cases h\u2084\u2081 with\n                  | inl h\u2084\u2081 =>\n                    tauto\n                  | inr h\u2084\u2081 =>\n                    tauto\n              exact h\u2083\u2086\n            exact h\u2083\u2085\n          cases h\u2083\u2083 with\n          | inl h\u2083\u2083 =>\n            have h\u2083\u2084 : (q : \u2124) - 4 = 1 := h\u2083\u2083\n            have h\u2083\u2085 : (r : \u2124) - 4 = 11 := by\n              have h\u2083\u2086 : ((q : \u2124) - 4) * ((r : \u2124) - 4) = 11 := by\n                linarith\n              rw [h\u2083\u2084] at h\u2083\u2086\n              linarith\n            exact Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9\n          | inr h\u2083\u2083 =>\n            cases h\u2083\u2083 with\n            | inl h\u2083\u2083 =>\n              have h\u2083\u2084 : (q : \u2124) - 4 = 11 := h\u2083\u2083\n              have h\u2083\u2085 : (r : \u2124) - 4 = 1 := by\n                have h\u2083\u2086 : ((q : \u2124) - 4) * ((r : \u2124) - 4) = 11 := by\n                  linarith\n                rw [h\u2083\u2084] at h\u2083\u2086\n                linarith\n              exact Or.inr (Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9)\n            | inr h\u2083\u2083 =>\n              cases h\u2083\u2083 with\n              | inl h\u2083\u2083 =>\n                have h\u2083\u2084 : (q : \u2124) - 4 = -1 := h\u2083\u2083\n                have h\u2083\u2085 : (r : \u2124) - 4 = -11 := by\n                  have h\u2083\u2086 : ((q : \u2124) - 4) * ((r : \u2124) - 4) = 11 := by\n                    linarith\n                  rw [h\u2083\u2084] at h\u2083\u2086\n                  linarith\n                exact Or.inr (Or.inr (Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9))\n              | inr h\u2083\u2083 =>\n                have h\u2083\u2084 : (q : \u2124) - 4 = -11 := h\u2083\u2083\n                have h\u2083\u2085 : (r : \u2124) - 4 = -1 := by\n                  have h\u2083\u2086 : ((q : \u2124) - 4) * ((r : \u2124) - 4) = 11 := by\n                    linarith\n                  rw [h\u2083\u2084] at h\u2083\u2086\n                  linarith\n                exact Or.inr (Or.inr (Or.inr \u27e8h\u2083\u2084, h\u2083\u2085\u27e9))\n        rcases h\u2083\u2082 with (\u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9)\n        \u00b7 \n          have h\u2083\u2084 : q = 5 := by\n            linarith\n          have h\u2083\u2085 : r = 15 := by\n            linarith\n          simp [h\u2083\u2084, h\u2083\u2085, h\u2081\u2085]\n          <;> norm_num <;> omega\n        \u00b7 \n          have h\u2083\u2084 : q = 15 := by\n            linarith\n          have h\u2083\u2085 : r = 5 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n        \u00b7 \n          have h\u2083\u2084 : q = 3 := by\n            linarith\n          have h\u2083\u2085 : r = -7 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n        \u00b7 \n          have h\u2083\u2084 : q = -7 := by\n            linarith\n          have h\u2083\u2085 : r = 3 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n      have h\u2081\u2087 : r = 15 := by\n        have h\u2081\u2088 : p = 3 := by linarith\n        have h\u2081\u2089 : q = 5 := by linarith\n        have h\u2082\u2080 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2081 : (p : \u2124) = 3 := by norm_num [h\u2081\u2088]\n        have h\u2082\u2082 : (q : \u2124) = 5 := by norm_num [h\u2081\u2089]\n        norm_num [h\u2081\u2088, h\u2081\u2089, h\u2082\u2081, h\u2082\u2082] at h\u2082\u2080\n        <;> ring_nf at h\u2082\u2080 \u22a2 <;> omega\n      have h\u2081\u2088 : (p, q, r) = (3, 5, 15) := by\n        simp_all [Prod.ext_iff]\n        <;> norm_num <;> linarith\n      exact Or.inr h\u2081\u2088\n    \u00b7 \n      have h\u2081\u2084 : p * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by\n        ring_nf at hk \u22a2\n        <;> linarith\n      have h\u2081\u2085 : p = 2 := by\n        by_contra h\u2081\u2085\n        have h\u2081\u2086 : p \u2265 3 := by\n          by_contra h\u2081\u2086\n          have h\u2081\u2087 : p \u2264 2 := by linarith\n          have h\u2081\u2088 : p = 2 := by linarith\n          contradiction\n        have h\u2081\u2089 : q \u2265 p + 1 := by linarith\n        have h\u2082\u2080 : r \u2265 q + 1 := by linarith\n        have h\u2082\u2081 : (p : \u2124) \u2265 3 := by linarith\n        have h\u2082\u2082 : (q : \u2124) \u2265 4 := by linarith\n        have h\u2082\u2083 : (r : \u2124) \u2265 5 := by linarith\n        have h\u2082\u2084 : (p : \u2124) * q \u2265 3 * 4 := by nlinarith\n        have h\u2082\u2085 : (p : \u2124) * r \u2265 3 * 5 := by nlinarith\n        have h\u2082\u2086 : (q : \u2124) * r \u2265 4 * 5 := by nlinarith\n        have h\u2082\u2087 : (p : \u2124) * q * r \u2265 3 * 4 * 5 := by nlinarith\n        have h\u2082\u2088 : (p : \u2124) * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2089 : (p : \u2124) \u2265 3 := by linarith\n        have h\u2083\u2080 : (q : \u2124) \u2265 4 := by linarith\n        have h\u2083\u2081 : (r : \u2124) \u2265 5 := by linarith\n        have h\u2083\u2082 : (p : \u2124) * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2083\u2083 : False := by\n          nlinarith [sq_nonneg ((p : \u2124) - 2), sq_nonneg ((q : \u2124) - 2), sq_nonneg ((r : \u2124) - 2),\n            mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2083),\n            mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2084),\n            mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2084)]\n        exact h\u2083\u2083\n      have h\u2081\u2086 : q = 4 := by\n        have h\u2081\u2087 : p = 2 := by linarith\n        have h\u2081\u2088 : (p : \u2124) * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2081\u2089 : (p : \u2124) = 2 := by norm_num [h\u2081\u2087]\n        have h\u2082\u2080 : (q : \u2124) \u2265 3 := by\n          by_contra h\u2082\u2080\n          have h\u2082\u2081 : q \u2264 2 := by linarith\n          have h\u2082\u2082 : q = 2 := by linarith\n          have h\u2082\u2083 : (p : \u2124) = 2 := by norm_num [h\u2081\u2087]\n          have h\u2082\u2084 : (q : \u2124) = 2 := by norm_num [h\u2082\u2082]\n          have h\u2082\u2085 : (r : \u2124) \u2265 3 := by linarith\n          norm_num [h\u2081\u2087, h\u2082\u2082, h\u2082\u2083, h\u2082\u2084] at h\u2081\u2088\n          <;>\n            (try omega) <;>\n            (try nlinarith) <;>\n            (try\n              {\n                nlinarith [mul_pos (sub_pos.mpr h\u2080.2.1) (sub_pos.mpr h\u2080.2.2)]\n              })\n        have h\u2082\u2081 : (r : \u2124) \u2265 q + 1 := by linarith\n        have h\u2082\u2082 : (q : \u2124) \u2265 3 := by linarith\n        have h\u2082\u2083 : (p : \u2124) = 2 := by norm_num [h\u2081\u2087]\n        have h\u2082\u2084 : (p : \u2124) * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2085 : 2 * q * r - 1 = 3 * (1 * (q - 1) * (r - 1)) := by\n          norm_num [h\u2081\u2087, h\u2081\u2089] at h\u2082\u2084 \u22a2\n          <;> ring_nf at h\u2082\u2084 \u22a2 <;> linarith\n        have h\u2082\u2086 : 2 * q * r - 1 = 3 * ((q - 1) * (r - 1)) := by\n          ring_nf at h\u2082\u2085 \u22a2\n          <;> nlinarith\n        have h\u2082\u2087 : 2 * q * r - 1 = 3 * (q * r - q - r + 1) := by\n          ring_nf at h\u2082\u2086 \u22a2\n          <;> nlinarith\n        have h\u2082\u2088 : 2 * q * r - 1 = 3 * q * r - 3 * q - 3 * r + 3 := by\n          ring_nf at h\u2082\u2087 \u22a2\n          <;> nlinarith\n        have h\u2082\u2089 : - q * r + 3 * q + 3 * r - 4 = 0 := by\n          ring_nf at h\u2082\u2088 \u22a2\n          <;> nlinarith\n        have h\u2083\u2080 : (q : \u2124) * r - 3 * q - 3 * r + 4 = 0 := by\n          ring_nf at h\u2082\u2089 \u22a2\n          <;> nlinarith\n        have h\u2083\u2081 : (q - 3 : \u2124) * (r - 3 : \u2124) = 5 := by\n          ring_nf at h\u2083\u2080 \u22a2\n          <;> nlinarith\n        have h\u2083\u2082 : (q : \u2124) - 3 = 1 \u2227 (r : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = 5 \u2227 (r : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = -1 \u2227 (r : \u2124) - 3 = -5 \u2228 (q : \u2124) - 3 = -5 \u2227 (r : \u2124) - 3 = -1 := by\n          have h\u2083\u2083 : (q : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = -1 \u2228 (q : \u2124) - 3 = -5 := by\n            have h\u2083\u2084 : (q : \u2124) - 3 \u2223 5 := by\n              use (r : \u2124) - 3\n              linarith\n            have h\u2083\u2085 : (q : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = -1 \u2228 (q : \u2124) - 3 = -5 := by\n              have h\u2083\u2086 : (q : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = -1 \u2228 (q : \u2124) - 3 = -5 := by\n                rw [\u2190 Int.natAbs_dvd_natAbs] at h\u2083\u2084\n                have h\u2083\u2087 : ((q : \u2124) - 3).natAbs \u2223 5 := by\n                  simpa [Int.natAbs] using h\u2083\u2084\n                have h\u2083\u2088 : ((q : \u2124) - 3).natAbs = 1 \u2228 ((q : \u2124) - 3).natAbs = 5 := by\n                  have h\u2083\u2089 : ((q : \u2124) - 3).natAbs \u2223 5 := h\u2083\u2087\n                  have h\u2084\u2080 : ((q : \u2124) - 3).natAbs \u2264 5 := Nat.le_of_dvd (by decide) h\u2083\u2089\n                  interval_cases ((q : \u2124) - 3).natAbs <;> norm_num at h\u2083\u2089 \u22a2 <;> omega\n                cases h\u2083\u2088 with\n                | inl h\u2083\u2088 =>\n                  have h\u2084\u2081 : (q : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = -1 := by\n                    have h\u2084\u2082 : ((q : \u2124) - 3).natAbs = 1 := h\u2083\u2088\n                    have h\u2084\u2083 : (q : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = -1 := by\n                      rw [Int.natAbs_eq_iff] at h\u2084\u2082\n                      tauto\n                    exact h\u2084\u2083\n                  cases h\u2084\u2081 with\n                  | inl h\u2084\u2081 =>\n                    tauto\n                  | inr h\u2084\u2081 =>\n                    tauto\n                | inr h\u2083\u2088 =>\n                  have h\u2084\u2081 : (q : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = -5 := by\n                    have h\u2084\u2082 : ((q : \u2124) - 3).natAbs = 5 := h\u2083\u2088\n                    have h\u2084\u2083 : (q : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = -5 := by\n                      rw [Int.natAbs_eq_iff] at h\u2084\u2082\n                      tauto\n                    exact h\u2084\u2083\n                  cases h\u2084\u2081 with\n                  | inl h\u2084\u2081 =>\n                    tauto\n                  | inr h\u2084\u2081 =>\n                    tauto\n              exact h\u2083\u2086\n            exact h\u2083\u2085\n          cases h\u2083\u2083 with\n          | inl h\u2083\u2083 =>\n            have h\u2083\u2084 : (q : \u2124) - 3 = 1 := h\u2083\u2083\n            have h\u2083\u2085 : (r : \u2124) - 3 = 5 := by\n              have h\u2083\u2086 : ((q : \u2124) - 3) * ((r : \u2124) - 3) = 5 := by\n                linarith\n              rw [h\u2083\u2084] at h\u2083\u2086\n              linarith\n            exact Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9\n          | inr h\u2083\u2083 =>\n            cases h\u2083\u2083 with\n            | inl h\u2083\u2083 =>\n              have h\u2083\u2084 : (q : \u2124) - 3 = 5 := h\u2083\u2083\n              have h\u2083\u2085 : (r : \u2124) - 3 = 1 := by\n                have h\u2083\u2086 : ((q : \u2124) - 3) * ((r : \u2124) - 3) = 5 := by\n                  linarith\n                rw [h\u2083\u2084] at h\u2083\u2086\n                linarith\n              exact Or.inr (Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9)\n            | inr h\u2083\u2083 =>\n              cases h\u2083\u2083 with\n              | inl h\u2083\u2083 =>\n                have h\u2083\u2084 : (q : \u2124) - 3 = -1 := h\u2083\u2083\n                have h\u2083\u2085 : (r : \u2124) - 3 = -5 := by\n                  have h\u2083\u2086 : ((q : \u2124) - 3) * ((r : \u2124) - 3) = 5 := by\n                    linarith\n                  rw [h\u2083\u2084] at h\u2083\u2086\n                  linarith\n                exact Or.inr (Or.inr (Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9))\n              | inr h\u2083\u2083 =>\n                have h\u2083\u2084 : (q : \u2124) - 3 = -5 := h\u2083\u2083\n                have h\u2083\u2085 : (r : \u2124) - 3 = -1 := by\n                  have h\u2083\u2086 : ((q : \u2124) - 3) * ((r : \u2124) - 3) = 5 := by\n                    linarith\n                  rw [h\u2083\u2084] at h\u2083\u2086\n                  linarith\n                exact Or.inr (Or.inr (Or.inr \u27e8h\u2083\u2084, h\u2083\u2085\u27e9))\n        rcases h\u2083\u2082 with (\u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9)\n        \u00b7 \n          have h\u2083\u2084 : q = 4 := by\n            linarith\n          have h\u2083\u2085 : r = 8 := by\n            linarith\n          simp [h\u2083\u2084, h\u2083\u2085, h\u2081\u2085]\n          <;> norm_num <;> omega\n        \u00b7 \n          have h\u2083\u2084 : q = 8 := by\n            linarith\n          have h\u2083\u2085 : r = 4 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n        \u00b7 \n          have h\u2083\u2084 : q = 2 := by\n            linarith\n          have h\u2083\u2085 : r = -2 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n        \u00b7 \n          have h\u2083\u2084 : q = -2 := by\n            linarith\n          have h\u2083\u2085 : r = 2 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n      have h\u2081\u2087 : r = 8 := by\n        have h\u2081\u2088 : p = 2 := by linarith\n        have h\u2081\u2089 : q = 4 := by linarith\n        have h\u2082\u2080 : (p : \u2124) * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2081 : (p : \u2124) = 2 := by norm_num [h\u2081\u2088]\n        have h\u2082\u2082 : (q : \u2124) = 4 := by norm_num [h\u2081\u2089]\n        norm_num [h\u2081\u2088, h\u2081\u2089, h\u2082\u2081, h\u2082\u2082] at h\u2082\u2080\n        <;> ring_nf at h\u2082\u2080 \u22a2 <;> omega\n      have h\u2081\u2088 : (p, q, r) = (2, 4, 8) := by\n        simp_all [Prod.ext_iff]\n        <;> norm_num <;> linarith\n      exact Or.inl h\u2081\u2088\n  exact h\u2085\n", "proof_before": "theorem imo_1992_p1 (p q r : \u2124) (h\u2080 : 1 < p \u2227 p < q \u2227 q < r)\n    (h\u2081 : (p - 1) * (q - 1) * (r - 1) \u2223 p * q * r - 1) :\n    (p, q, r) = (2, 4, 8) \u2228 (p, q, r) = (3, 5, 15)  :=  by\n  have h\u2082 : p \u2265 2 := by\n    linarith [h\u2080.1]\n  have h\u2083 : q \u2265 3 := by\n    by_contra h\n    have h\u2084 : q \u2264 2 := by linarith\n    have h\u2085 : q \u2265 2 := by linarith [h\u2080.2.1]\n    have h\u2086 : q = 2 := by linarith\n    have h\u2087 : p < 2 := by linarith [h\u2080.2.1]\n    have h\u2088 : p \u2265 2 := by linarith [h\u2080.1]\n    linarith\n  have h\u2084 : r \u2265 4 := by\n    by_contra h\n    have h\u2085 : r \u2264 3 := by linarith\n    have h\u2086 : r \u2265 3 := by linarith [h\u2080.2.2]\n    have h\u2087 : r = 3 := by linarith\n    have h\u2088 : q < 3 := by linarith [h\u2080.2.2]\n    have h\u2089 : q \u2265 3 := by linarith [h\u2083]\n    linarith\n  have h\u2085 : (p, q, r) = (2, 4, 8) \u2228 (p, q, r) = (3, 5, 15) := by\n    have h\u2086 : (p - 1 : \u2124) > 0 := by\n      linarith\n    have h\u2087 : (q - 1 : \u2124) > 0 := by\n      linarith\n    have h\u2088 : (r - 1 : \u2124) > 0 := by\n      linarith\n    have h\u2089 : (p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124) > 0 := by positivity\n    have h\u2081\u2080 : \u2203 (k : \u2124), (p * q * r - 1 : \u2124) = k * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) := by\n      obtain \u27e8k, hk\u27e9 := h\u2081\n      refine' \u27e8k, _\u27e9\n      linarith\n    obtain \u27e8k, hk\u27e9 := h\u2081\u2080\n    have h\u2081\u2081 : k > 0 := by\n      by_contra h\u2081\u2081\n      have h\u2081\u2082 : k \u2264 0 := by linarith\n      have h\u2081\u2083 : (p * q * r - 1 : \u2124) \u2264 0 := by\n        nlinarith [mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2083),\n          mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2084),\n          mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2084)]\n      have h\u2081\u2084 : (p * q * r : \u2124) \u2264 1 := by linarith\n      have h\u2081\u2085 : (p : \u2124) \u2265 2 := by linarith\n      have h\u2081\u2086 : (q : \u2124) \u2265 3 := by linarith\n      have h\u2081\u2087 : (r : \u2124) \u2265 4 := by linarith\n      have h\u2081\u2088 : (p * q * r : \u2124) \u2265 2 * 3 * 4 := by\n        have h\u2081\u2089 : (p : \u2124) * q \u2265 2 * 3 := by nlinarith\n        nlinarith\n      linarith\n    have h\u2081\u2082 : k \u2264 3 := by\n      by_contra h\u2081\u2082\n      have h\u2081\u2083 : k \u2265 4 := by linarith\n      have h\u2081\u2084 : (p - 1 : \u2124) \u2265 1 := by\n        linarith\n      have h\u2081\u2085 : (q - 1 : \u2124) \u2265 2 := by\n        linarith\n      have h\u2081\u2086 : (r - 1 : \u2124) \u2265 3 := by\n        linarith\n      have h\u2081\u2087 : (p - 1 : \u2124) * (q - 1 : \u2124) \u2265 2 := by\n        nlinarith\n      have h\u2081\u2088 : (p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124) \u2265 6 := by\n        nlinarith\n      have h\u2081\u2089 : (k : \u2124) * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) \u2265 4 * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) := by\n        nlinarith\n      have h\u2082\u2080 : (p * q * r - 1 : \u2124) \u2265 4 * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) := by\n        linarith\n      have h\u2082\u2081 : (p * q * r - 1 : \u2124) < 4 * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) := by\n        have h\u2082\u2082 : (p : \u2124) * q * r < 4 * ((p - 1 : \u2124) * (q - 1 : \u2124) * (r - 1 : \u2124)) + 1 := by\n          nlinarith [mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2083),\n            mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2084),\n            mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2084)]\n        linarith\n      linarith\n    have h\u2081\u2083 : k = 1 \u2228 k = 2 \u2228 k = 3 := by\n      omega\n    rcases h\u2081\u2083 with (rfl | rfl | rfl)\n    \u00b7 \n      have h\u2081\u2084 : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) := by\n        ring_nf at hk \u22a2\n        <;> linarith\n      have h\u2081\u2085 : p * q + p * r + q * r = p + q + r := by\n        have h\u2081\u2085\u2081 : p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) := by linarith\n        ring_nf at h\u2081\u2085\u2081 \u22a2\n        nlinarith\n      have h\u2081\u2086 : p = 2 := by\n        by_contra h\u2081\u2086\n        have h\u2081\u2087 : p \u2265 3 := by\n          by_contra h\u2081\u2087\n          have h\u2081\u2088 : p \u2264 2 := by linarith\n          have h\u2081\u2089 : p = 2 := by linarith\n          contradiction\n        have h\u2082\u2080 : q \u2265 3 := by linarith\n        have h\u2082\u2081 : r \u2265 4 := by linarith\n        have h\u2082\u2082 : (p : \u2124) * q \u2265 3 * 3 := by\n          nlinarith\n        have h\u2082\u2083 : (p : \u2124) * r \u2265 3 * 4 := by\n          nlinarith\n        have h\u2082\u2084 : (q : \u2124) * r \u2265 3 * 4 := by\n          nlinarith\n        nlinarith\n      have h\u2081\u2087 : q = 3 := by\n        by_contra h\u2081\u2087\n        have h\u2081\u2088 : q \u2265 4 := by\n          by_contra h\u2081\u2088\n          have h\u2081\u2089 : q \u2264 3 := by linarith\n          have h\u2082\u2080 : q = 3 := by linarith\n          contradiction\n        have h\u2082\u2081 : p = 2 := by linarith\n        have h\u2082\u2082 : (p : \u2124) * q \u2265 2 * 4 := by\n          nlinarith\n        have h\u2082\u2083 : (p : \u2124) * r \u2265 2 * 4 := by\n          nlinarith\n        have h\u2082\u2084 : (q : \u2124) * r \u2265 4 * 4 := by\n          nlinarith\n        nlinarith\n      have h\u2081\u2088 : r = 4 := by\n        by_contra h\u2081\u2088\n        have h\u2081\u2089 : r \u2265 5 := by\n          by_contra h\u2081\u2089\n          have h\u2082\u2080 : r \u2264 4 := by linarith\n          have h\u2082\u2081 : r = 4 := by linarith\n          contradiction\n        have h\u2082\u2082 : p = 2 := by linarith\n        have h\u2082\u2083 : q = 3 := by linarith\n        have h\u2082\u2084 : (p : \u2124) * q \u2265 2 * 3 := by\n          nlinarith\n        have h\u2082\u2085 : (p : \u2124) * r \u2265 2 * 5 := by\n          nlinarith\n        have h\u2082\u2086 : (q : \u2124) * r \u2265 3 * 5 := by\n          nlinarith\n        nlinarith\n      exfalso\n      norm_num [h\u2081\u2086, h\u2081\u2087, h\u2081\u2088] at h\u2081\u2084 h\u2081\u2085 hk h\u2080 \u22a2 <;> linarith\n    \u00b7 \n      have h\u2081\u2084 : p * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by\n        ring_nf at hk \u22a2\n        <;> linarith\n      have h\u2081\u2085 : p = 3 := by\n        by_contra h\u2081\u2085\n        have h\u2081\u2086 : p \u2260 3 := by tauto\n        have h\u2081\u2087 : p \u2265 4 := by\n          by_contra h\u2081\u2087\n          have h\u2081\u2088 : p \u2264 3 := by linarith\n          have h\u2081\u2089 : p = 2 := by\n            by_contra h\u2081\u2089\n            have h\u2082\u2080 : p \u2265 3 := by omega\n            have h\u2082\u2081 : p = 3 := by omega\n            contradiction\n          have h\u2082\u2082 : p = 2 := by omega\n          have h\u2082\u2083 : q \u2265 3 := by linarith\n          have h\u2082\u2084 : r \u2265 4 := by linarith\n          have h\u2082\u2085 : (p : \u2124) * q \u2265 2 * 3 := by nlinarith\n          have h\u2082\u2086 : (p : \u2124) * r \u2265 2 * 4 := by nlinarith\n          have h\u2082\u2087 : (q : \u2124) * r \u2265 3 * 4 := by nlinarith\n          have h\u2082\u2088 : (p : \u2124) * q * r \u2265 2 * 3 * 4 := by nlinarith\n          have h\u2082\u2089 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n          have h\u2083\u2080 : (p : \u2124) = 2 := by omega\n          have h\u2083\u2081 : (q : \u2124) \u2265 3 := by omega\n          have h\u2083\u2082 : (r : \u2124) \u2265 4 := by omega\n          have h\u2083\u2083 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n          have h\u2083\u2084 : (p : \u2124) = 2 := by omega\n          have h\u2083\u2085 : (q : \u2124) \u2265 3 := by omega\n          have h\u2083\u2086 : (r : \u2124) \u2265 4 := by omega\n          have h\u2083\u2087 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n          have h\u2083\u2088 : False := by\n            have h\u2083\u2089 : (p : \u2124) = 2 := by omega\n            have h\u2084\u2080 : (q : \u2124) \u2265 3 := by omega\n            have h\u2084\u2081 : (r : \u2124) \u2265 4 := by omega\n            have h\u2084\u2082 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n            have h\u2084\u2083 : 2 * q * r - 1 = 2 * (1 * (q - 1) * (r - 1)) := by\n              simp [h\u2083\u2089] at h\u2084\u2082 \u22a2\n              <;> ring_nf at h\u2084\u2082 \u22a2 <;> linarith\n            have h\u2084\u2084 : 2 * q * r - 1 = 2 * ((q - 1) * (r - 1)) := by\n              ring_nf at h\u2084\u2083 \u22a2 <;> linarith\n            have h\u2084\u2085 : 2 * q * r - 1 = 2 * (q * r - q - r + 1) := by\n              ring_nf at h\u2084\u2084 \u22a2 <;> linarith\n            have h\u2084\u2086 : 2 * q * r - 1 = 2 * q * r - 2 * q - 2 * r + 2 := by\n              ring_nf at h\u2084\u2085 \u22a2 <;> linarith\n            have h\u2084\u2087 : -1 = -2 * q - 2 * r + 2 := by\n              linarith\n            have h\u2084\u2088 : 2 * q + 2 * r = 3 := by linarith\n            have h\u2084\u2089 : (q : \u2124) \u2265 3 := by omega\n            have h\u2085\u2080 : (r : \u2124) \u2265 4 := by omega\n            have h\u2085\u2081 : 2 * q + 2 * r \u2265 14 := by\n              nlinarith\n            linarith\n          exact h\u2083\u2088\n        have h\u2081\u2089 : q \u2265 p + 1 := by omega\n        have h\u2082\u2080 : r \u2265 q + 1 := by omega\n        have h\u2082\u2081 : (p : \u2124) \u2265 4 := by omega\n        have h\u2082\u2082 : (q : \u2124) \u2265 5 := by omega\n        have h\u2082\u2083 : (r : \u2124) \u2265 6 := by omega\n        have h\u2082\u2084 : (p : \u2124) * q \u2265 4 * 5 := by nlinarith\n        have h\u2082\u2085 : (p : \u2124) * r \u2265 4 * 6 := by nlinarith\n        have h\u2082\u2086 : (q : \u2124) * r \u2265 5 * 6 := by nlinarith\n        have h\u2082\u2087 : (p : \u2124) * q * r \u2265 4 * 5 * 6 := by nlinarith\n        have h\u2082\u2088 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2089 : (p : \u2124) \u2265 4 := by omega\n        have h\u2083\u2080 : (q : \u2124) \u2265 5 := by omega\n        have h\u2083\u2081 : (r : \u2124) \u2265 6 := by omega\n        have h\u2083\u2082 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2083\u2083 : False := by\n          nlinarith [sq_nonneg ((p : \u2124) - 2), sq_nonneg ((q : \u2124) - 2), sq_nonneg ((r : \u2124) - 2),\n            mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2083),\n            mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2084),\n            mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2084)]\n        exact h\u2083\u2083\n      have h\u2081\u2086 : q = 5 := by\n        have h\u2081\u2087 : p = 3 := by linarith\n        have h\u2081\u2088 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2081\u2089 : (p : \u2124) = 3 := by norm_num [h\u2081\u2087]\n        have h\u2082\u2080 : (q : \u2124) \u2265 4 := by\n          by_contra h\u2082\u2080\n          have h\u2082\u2081 : q \u2264 3 := by linarith\n          have h\u2082\u2082 : q = 3 := by linarith\n          have h\u2082\u2083 : (p : \u2124) = 3 := by norm_num [h\u2081\u2087]\n          have h\u2082\u2084 : (q : \u2124) = 3 := by norm_num [h\u2082\u2082]\n          have h\u2082\u2085 : (r : \u2124) \u2265 4 := by linarith\n          norm_num [h\u2081\u2087, h\u2082\u2082, h\u2082\u2083, h\u2082\u2084] at h\u2081\u2088\n          <;>\n            (try omega) <;>\n            (try nlinarith) <;>\n            (try\n              {\n                nlinarith [mul_pos (sub_pos.mpr h\u2080.2.1) (sub_pos.mpr h\u2080.2.2)]\n              })\n        have h\u2082\u2081 : (r : \u2124) \u2265 q + 1 := by linarith\n        have h\u2082\u2082 : (q : \u2124) \u2265 4 := by linarith\n        have h\u2082\u2083 : (p : \u2124) = 3 := by norm_num [h\u2081\u2087]\n        have h\u2082\u2084 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2085 : 3 * q * r - 1 = 2 * (2 * (q - 1) * (r - 1)) := by\n          norm_num [h\u2081\u2087, h\u2081\u2089] at h\u2082\u2084 \u22a2\n          <;> ring_nf at h\u2082\u2084 \u22a2 <;> linarith\n        have h\u2082\u2086 : 3 * q * r - 1 = 4 * ((q - 1) * (r - 1)) := by\n          ring_nf at h\u2082\u2085 \u22a2\n          <;> nlinarith\n        have h\u2082\u2087 : 3 * q * r - 1 = 4 * (q * r - q - r + 1) := by\n          ring_nf at h\u2082\u2086 \u22a2\n          <;> nlinarith\n        have h\u2082\u2088 : 3 * q * r - 1 = 4 * q * r - 4 * q - 4 * r + 4 := by\n          ring_nf at h\u2082\u2087 \u22a2\n          <;> nlinarith\n        have h\u2082\u2089 : - q * r + 4 * q + 4 * r - 5 = 0 := by\n          ring_nf at h\u2082\u2088 \u22a2\n          <;> nlinarith\n        have h\u2083\u2080 : (q : \u2124) * r - 4 * q - 4 * r + 5 = 0 := by\n          ring_nf at h\u2082\u2089 \u22a2\n          <;> nlinarith\n        have h\u2083\u2081 : (q - 4 : \u2124) * (r - 4 : \u2124) = 11 := by\n          ring_nf at h\u2083\u2080 \u22a2\n          <;> nlinarith\n        have h\u2083\u2082 : (q : \u2124) - 4 = 1 \u2227 (r : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = 11 \u2227 (r : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = -1 \u2227 (r : \u2124) - 4 = -11 \u2228 (q : \u2124) - 4 = -11 \u2227 (r : \u2124) - 4 = -1 := by\n          have h\u2083\u2083 : (q : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = -1 \u2228 (q : \u2124) - 4 = -11 := by\n            have h\u2083\u2084 : (q : \u2124) - 4 \u2223 11 := by\n              use (r : \u2124) - 4\n              linarith\n            have h\u2083\u2085 : (q : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = -1 \u2228 (q : \u2124) - 4 = -11 := by\n              have h\u2083\u2086 : (q : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = -1 \u2228 (q : \u2124) - 4 = -11 := by\n                rw [\u2190 Int.natAbs_dvd_natAbs] at h\u2083\u2084\n                have h\u2083\u2087 : ((q : \u2124) - 4).natAbs \u2223 11 := by\n                  simpa [Int.natAbs] using h\u2083\u2084\n                have h\u2083\u2088 : ((q : \u2124) - 4).natAbs = 1 \u2228 ((q : \u2124) - 4).natAbs = 11 := by\n                  have h\u2083\u2089 : ((q : \u2124) - 4).natAbs \u2223 11 := h\u2083\u2087\n                  have h\u2084\u2080 : ((q : \u2124) - 4).natAbs \u2264 11 := Nat.le_of_dvd (by decide) h\u2083\u2089\n                  interval_cases ((q : \u2124) - 4).natAbs <;> norm_num at h\u2083\u2089 \u22a2 <;> omega\n                cases h\u2083\u2088 with\n                | inl h\u2083\u2088 =>\n                  have h\u2084\u2081 : (q : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = -1 := by\n                    have h\u2084\u2082 : ((q : \u2124) - 4).natAbs = 1 := h\u2083\u2088\n                    have h\u2084\u2083 : (q : \u2124) - 4 = 1 \u2228 (q : \u2124) - 4 = -1 := by\n                      rw [Int.natAbs_eq_iff] at h\u2084\u2082\n                      tauto\n                    exact h\u2084\u2083\n                  cases h\u2084\u2081 with\n                  | inl h\u2084\u2081 =>\n                    tauto\n                  | inr h\u2084\u2081 =>\n                    tauto\n                | inr h\u2083\u2088 =>\n                  have h\u2084\u2081 : (q : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = -11 := by\n                    have h\u2084\u2082 : ((q : \u2124) - 4).natAbs = 11 := h\u2083\u2088\n                    have h\u2084\u2083 : (q : \u2124) - 4 = 11 \u2228 (q : \u2124) - 4 = -11 := by\n                      rw [Int.natAbs_eq_iff] at h\u2084\u2082\n                      tauto\n                    exact h\u2084\u2083\n                  cases h\u2084\u2081 with\n                  | inl h\u2084\u2081 =>\n                    tauto\n                  | inr h\u2084\u2081 =>\n                    tauto\n              exact h\u2083\u2086\n            exact h\u2083\u2085\n          cases h\u2083\u2083 with\n          | inl h\u2083\u2083 =>\n            have h\u2083\u2084 : (q : \u2124) - 4 = 1 := h\u2083\u2083\n            have h\u2083\u2085 : (r : \u2124) - 4 = 11 := by\n              have h\u2083\u2086 : ((q : \u2124) - 4) * ((r : \u2124) - 4) = 11 := by\n                linarith\n              rw [h\u2083\u2084] at h\u2083\u2086\n              linarith\n            exact Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9\n          | inr h\u2083\u2083 =>\n            cases h\u2083\u2083 with\n            | inl h\u2083\u2083 =>\n              have h\u2083\u2084 : (q : \u2124) - 4 = 11 := h\u2083\u2083\n              have h\u2083\u2085 : (r : \u2124) - 4 = 1 := by\n                have h\u2083\u2086 : ((q : \u2124) - 4) * ((r : \u2124) - 4) = 11 := by\n                  linarith\n                rw [h\u2083\u2084] at h\u2083\u2086\n                linarith\n              exact Or.inr (Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9)\n            | inr h\u2083\u2083 =>\n              cases h\u2083\u2083 with\n              | inl h\u2083\u2083 =>\n                have h\u2083\u2084 : (q : \u2124) - 4 = -1 := h\u2083\u2083\n                have h\u2083\u2085 : (r : \u2124) - 4 = -11 := by\n                  have h\u2083\u2086 : ((q : \u2124) - 4) * ((r : \u2124) - 4) = 11 := by\n                    linarith\n                  rw [h\u2083\u2084] at h\u2083\u2086\n                  linarith\n                exact Or.inr (Or.inr (Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9))\n              | inr h\u2083\u2083 =>\n                have h\u2083\u2084 : (q : \u2124) - 4 = -11 := h\u2083\u2083\n                have h\u2083\u2085 : (r : \u2124) - 4 = -1 := by\n                  have h\u2083\u2086 : ((q : \u2124) - 4) * ((r : \u2124) - 4) = 11 := by\n                    linarith\n                  rw [h\u2083\u2084] at h\u2083\u2086\n                  linarith\n                exact Or.inr (Or.inr (Or.inr \u27e8h\u2083\u2084, h\u2083\u2085\u27e9))\n        rcases h\u2083\u2082 with (\u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9)\n        \u00b7 \n          have h\u2083\u2084 : q = 5 := by\n            linarith\n          have h\u2083\u2085 : r = 15 := by\n            linarith\n          simp [h\u2083\u2084, h\u2083\u2085, h\u2081\u2085]\n          <;> norm_num <;> omega\n        \u00b7 \n          have h\u2083\u2084 : q = 15 := by\n            linarith\n          have h\u2083\u2085 : r = 5 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n        \u00b7 \n          have h\u2083\u2084 : q = 3 := by\n            linarith\n          have h\u2083\u2085 : r = -7 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n        \u00b7 \n          have h\u2083\u2084 : q = -7 := by\n            linarith\n          have h\u2083\u2085 : r = 3 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n      have h\u2081\u2087 : r = 15 := by\n        have h\u2081\u2088 : p = 3 := by linarith\n        have h\u2081\u2089 : q = 5 := by linarith\n        have h\u2082\u2080 : (p : \u2124) * q * r - 1 = 2 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2081 : (p : \u2124) = 3 := by norm_num [h\u2081\u2088]\n        have h\u2082\u2082 : (q : \u2124) = 5 := by norm_num [h\u2081\u2089]\n        norm_num [h\u2081\u2088, h\u2081\u2089, h\u2082\u2081, h\u2082\u2082] at h\u2082\u2080\n        <;> ring_nf at h\u2082\u2080 \u22a2 <;> omega\n      have h\u2081\u2088 : (p, q, r) = (3, 5, 15) := by\n        simp_all [Prod.ext_iff]\n        <;> norm_num <;> linarith\n      exact Or.inr h\u2081\u2088\n    \u00b7 \n      have h\u2081\u2084 : p * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by\n        ring_nf at hk \u22a2\n        <;> linarith\n      have h\u2081\u2085 : p = 2 := by\n        by_contra h\u2081\u2085\n        have h\u2081\u2086 : p \u2265 3 := by\n          by_contra h\u2081\u2086\n          have h\u2081\u2087 : p \u2264 2 := by linarith\n          have h\u2081\u2088 : p = 2 := by linarith\n          contradiction\n        have h\u2081\u2089 : q \u2265 p + 1 := by linarith\n        have h\u2082\u2080 : r \u2265 q + 1 := by linarith\n        have h\u2082\u2081 : (p : \u2124) \u2265 3 := by linarith\n        have h\u2082\u2082 : (q : \u2124) \u2265 4 := by linarith\n        have h\u2082\u2083 : (r : \u2124) \u2265 5 := by linarith\n        have h\u2082\u2084 : (p : \u2124) * q \u2265 3 * 4 := by nlinarith\n        have h\u2082\u2085 : (p : \u2124) * r \u2265 3 * 5 := by nlinarith\n        have h\u2082\u2086 : (q : \u2124) * r \u2265 4 * 5 := by nlinarith\n        have h\u2082\u2087 : (p : \u2124) * q * r \u2265 3 * 4 * 5 := by nlinarith\n        have h\u2082\u2088 : (p : \u2124) * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2089 : (p : \u2124) \u2265 3 := by linarith\n        have h\u2083\u2080 : (q : \u2124) \u2265 4 := by linarith\n        have h\u2083\u2081 : (r : \u2124) \u2265 5 := by linarith\n        have h\u2083\u2082 : (p : \u2124) * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2083\u2083 : False := by\n          nlinarith [sq_nonneg ((p : \u2124) - 2), sq_nonneg ((q : \u2124) - 2), sq_nonneg ((r : \u2124) - 2),\n            mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2083),\n            mul_nonneg (sub_nonneg.mpr h\u2082) (sub_nonneg.mpr h\u2084),\n            mul_nonneg (sub_nonneg.mpr h\u2083) (sub_nonneg.mpr h\u2084)]\n        exact h\u2083\u2083\n      have h\u2081\u2086 : q = 4 := by\n        have h\u2081\u2087 : p = 2 := by linarith\n        have h\u2081\u2088 : (p : \u2124) * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2081\u2089 : (p : \u2124) = 2 := by norm_num [h\u2081\u2087]\n        have h\u2082\u2080 : (q : \u2124) \u2265 3 := by\n          by_contra h\u2082\u2080\n          have h\u2082\u2081 : q \u2264 2 := by linarith\n          have h\u2082\u2082 : q = 2 := by linarith\n          have h\u2082\u2083 : (p : \u2124) = 2 := by norm_num [h\u2081\u2087]\n          have h\u2082\u2084 : (q : \u2124) = 2 := by norm_num [h\u2082\u2082]\n          have h\u2082\u2085 : (r : \u2124) \u2265 3 := by linarith\n          norm_num [h\u2081\u2087, h\u2082\u2082, h\u2082\u2083, h\u2082\u2084] at h\u2081\u2088\n          <;>\n            (try omega) <;>\n            (try nlinarith) <;>\n            (try\n              {\n                nlinarith [mul_pos (sub_pos.mpr h\u2080.2.1) (sub_pos.mpr h\u2080.2.2)]\n              })\n        have h\u2082\u2081 : (r : \u2124) \u2265 q + 1 := by linarith\n        have h\u2082\u2082 : (q : \u2124) \u2265 3 := by linarith\n        have h\u2082\u2083 : (p : \u2124) = 2 := by norm_num [h\u2081\u2087]\n        have h\u2082\u2084 : (p : \u2124) * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2085 : 2 * q * r - 1 = 3 * (1 * (q - 1) * (r - 1)) := by\n          norm_num [h\u2081\u2087, h\u2081\u2089] at h\u2082\u2084 \u22a2\n          <;> ring_nf at h\u2082\u2084 \u22a2 <;> linarith\n        have h\u2082\u2086 : 2 * q * r - 1 = 3 * ((q - 1) * (r - 1)) := by\n          ring_nf at h\u2082\u2085 \u22a2\n          <;> nlinarith\n        have h\u2082\u2087 : 2 * q * r - 1 = 3 * (q * r - q - r + 1) := by\n          ring_nf at h\u2082\u2086 \u22a2\n          <;> nlinarith\n        have h\u2082\u2088 : 2 * q * r - 1 = 3 * q * r - 3 * q - 3 * r + 3 := by\n          ring_nf at h\u2082\u2087 \u22a2\n          <;> nlinarith\n        have h\u2082\u2089 : - q * r + 3 * q + 3 * r - 4 = 0 := by\n          ring_nf at h\u2082\u2088 \u22a2\n          <;> nlinarith\n        have h\u2083\u2080 : (q : \u2124) * r - 3 * q - 3 * r + 4 = 0 := by\n          ring_nf at h\u2082\u2089 \u22a2\n          <;> nlinarith\n        have h\u2083\u2081 : (q - 3 : \u2124) * (r - 3 : \u2124) = 5 := by\n          ring_nf at h\u2083\u2080 \u22a2\n          <;> nlinarith\n        have h\u2083\u2082 : (q : \u2124) - 3 = 1 \u2227 (r : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = 5 \u2227 (r : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = -1 \u2227 (r : \u2124) - 3 = -5 \u2228 (q : \u2124) - 3 = -5 \u2227 (r : \u2124) - 3 = -1 := by\n          have h\u2083\u2083 : (q : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = -1 \u2228 (q : \u2124) - 3 = -5 := by\n            have h\u2083\u2084 : (q : \u2124) - 3 \u2223 5 := by\n              use (r : \u2124) - 3\n              linarith\n            have h\u2083\u2085 : (q : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = -1 \u2228 (q : \u2124) - 3 = -5 := by\n              have h\u2083\u2086 : (q : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = -1 \u2228 (q : \u2124) - 3 = -5 := by\n                rw [\u2190 Int.natAbs_dvd_natAbs] at h\u2083\u2084\n                have h\u2083\u2087 : ((q : \u2124) - 3).natAbs \u2223 5 := by\n                  simpa [Int.natAbs] using h\u2083\u2084\n                have h\u2083\u2088 : ((q : \u2124) - 3).natAbs = 1 \u2228 ((q : \u2124) - 3).natAbs = 5 := by\n                  have h\u2083\u2089 : ((q : \u2124) - 3).natAbs \u2223 5 := h\u2083\u2087\n                  have h\u2084\u2080 : ((q : \u2124) - 3).natAbs \u2264 5 := Nat.le_of_dvd (by decide) h\u2083\u2089\n                  interval_cases ((q : \u2124) - 3).natAbs <;> norm_num at h\u2083\u2089 \u22a2 <;> omega\n                cases h\u2083\u2088 with\n                | inl h\u2083\u2088 =>\n                  have h\u2084\u2081 : (q : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = -1 := by\n                    have h\u2084\u2082 : ((q : \u2124) - 3).natAbs = 1 := h\u2083\u2088\n                    have h\u2084\u2083 : (q : \u2124) - 3 = 1 \u2228 (q : \u2124) - 3 = -1 := by\n                      rw [Int.natAbs_eq_iff] at h\u2084\u2082\n                      tauto\n                    exact h\u2084\u2083\n                  cases h\u2084\u2081 with\n                  | inl h\u2084\u2081 =>\n                    tauto\n                  | inr h\u2084\u2081 =>\n                    tauto\n                | inr h\u2083\u2088 =>\n                  have h\u2084\u2081 : (q : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = -5 := by\n                    have h\u2084\u2082 : ((q : \u2124) - 3).natAbs = 5 := h\u2083\u2088\n                    have h\u2084\u2083 : (q : \u2124) - 3 = 5 \u2228 (q : \u2124) - 3 = -5 := by\n                      rw [Int.natAbs_eq_iff] at h\u2084\u2082\n                      tauto\n                    exact h\u2084\u2083\n                  cases h\u2084\u2081 with\n                  | inl h\u2084\u2081 =>\n                    tauto\n                  | inr h\u2084\u2081 =>\n                    tauto\n              exact h\u2083\u2086\n            exact h\u2083\u2085\n          cases h\u2083\u2083 with\n          | inl h\u2083\u2083 =>\n            have h\u2083\u2084 : (q : \u2124) - 3 = 1 := h\u2083\u2083\n            have h\u2083\u2085 : (r : \u2124) - 3 = 5 := by\n              have h\u2083\u2086 : ((q : \u2124) - 3) * ((r : \u2124) - 3) = 5 := by\n                linarith\n              rw [h\u2083\u2084] at h\u2083\u2086\n              linarith\n            exact Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9\n          | inr h\u2083\u2083 =>\n            cases h\u2083\u2083 with\n            | inl h\u2083\u2083 =>\n              have h\u2083\u2084 : (q : \u2124) - 3 = 5 := h\u2083\u2083\n              have h\u2083\u2085 : (r : \u2124) - 3 = 1 := by\n                have h\u2083\u2086 : ((q : \u2124) - 3) * ((r : \u2124) - 3) = 5 := by\n                  linarith\n                rw [h\u2083\u2084] at h\u2083\u2086\n                linarith\n              exact Or.inr (Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9)\n            | inr h\u2083\u2083 =>\n              cases h\u2083\u2083 with\n              | inl h\u2083\u2083 =>\n                have h\u2083\u2084 : (q : \u2124) - 3 = -1 := h\u2083\u2083\n                have h\u2083\u2085 : (r : \u2124) - 3 = -5 := by\n                  have h\u2083\u2086 : ((q : \u2124) - 3) * ((r : \u2124) - 3) = 5 := by\n                    linarith\n                  rw [h\u2083\u2084] at h\u2083\u2086\n                  linarith\n                exact Or.inr (Or.inr (Or.inl \u27e8h\u2083\u2084, h\u2083\u2085\u27e9))\n              | inr h\u2083\u2083 =>\n                have h\u2083\u2084 : (q : \u2124) - 3 = -5 := h\u2083\u2083\n                have h\u2083\u2085 : (r : \u2124) - 3 = -1 := by\n                  have h\u2083\u2086 : ((q : \u2124) - 3) * ((r : \u2124) - 3) = 5 := by\n                    linarith\n                  rw [h\u2083\u2084] at h\u2083\u2086\n                  linarith\n                exact Or.inr (Or.inr (Or.inr \u27e8h\u2083\u2084, h\u2083\u2085\u27e9))\n        rcases h\u2083\u2082 with (\u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9 | \u27e8h\u2083\u2082, h\u2083\u2083\u27e9)\n        \u00b7 \n          have h\u2083\u2084 : q = 4 := by\n            linarith\n          have h\u2083\u2085 : r = 8 := by\n            linarith\n          simp [h\u2083\u2084, h\u2083\u2085, h\u2081\u2085]\n          <;> norm_num <;> omega\n        \u00b7 \n          have h\u2083\u2084 : q = 8 := by\n            linarith\n          have h\u2083\u2085 : r = 4 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n        \u00b7 \n          have h\u2083\u2084 : q = 2 := by\n            linarith\n          have h\u2083\u2085 : r = -2 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n        \u00b7 \n          have h\u2083\u2084 : q = -2 := by\n            linarith\n          have h\u2083\u2085 : r = 2 := by\n            linarith\n          have h\u2083\u2086 : r > q := by\n            linarith\n          linarith\n      have h\u2081\u2087 : r = 8 := by\n        have h\u2081\u2088 : p = 2 := by linarith\n        have h\u2081\u2089 : q = 4 := by linarith\n        have h\u2082\u2080 : (p : \u2124) * q * r - 1 = 3 * ((p - 1) * (q - 1) * (r - 1)) := by linarith\n        have h\u2082\u2081 : (p : \u2124) = 2 := by norm_num [h\u2081\u2088]\n        have h\u2082\u2082 : (q : \u2124) = 4 := by norm_num [h\u2081\u2089]\n        norm_num [h\u2081\u2088, h\u2081\u2089, h\u2082\u2081, h\u2082\u2082] at h\u2082\u2080\n        <;> ring_nf at h\u2082\u2080 \u22a2 <;> omega\n      have h\u2081\u2088 : (p, q, r) = (2, 4, 8) := by\n        simp_all [Prod.ext_iff]\n        <;> norm_num <;> linarith\n      exact Or.inl h\u2081\u2088\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 71, "alphabet": 31, "compiles": false, "id": "mathd_algebra_184", "proof_after": "theorem mathd_algebra_184 (a b : NNReal) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : a ^ 2 = 6 * b)\n    (h\u2082 : a ^ 2 = 54 / b) : a = 3 * NNReal.sqrt 2  :=  by\n  have h\u2083 : (b : \u211d) = 3 := by\n    have h\u2083\u2081 : (a : \u211d) ^ 2 = 6 * (b : \u211d) := by\n      norm_cast at h\u2081 \u22a2\n      <;> simp_all [pow_two, mul_assoc]\n      <;> ring_nf at *\n      <;> nlinarith\n    have h\u2083\u2082 : (a : \u211d) ^ 2 = 54 / (b : \u211d) := by\n      norm_cast at h\u2082 \u22a2\n      <;> simp_all [pow_two, mul_assoc]\n      <;> field_simp [h\u2080.2.ne'] at *\n      <;> ring_nf at *\n      <;> nlinarith\n    have h\u2083\u2083 : (6 : \u211d) * (b : \u211d) = 54 / (b : \u211d) := by linarith\n    have h\u2083\u2084 : (b : \u211d) > 0 := by exact_mod_cast h\u2080.2\n    have h\u2083\u2085 : (b : \u211d) ^ 2 = 9 := by\n      field_simp at h\u2083\u2083\n      nlinarith\n    have h\u2083\u2086 : (b : \u211d) = 3 := by\n      nlinarith\n    exact_mod_cast h\u2083\u2086\n  have h\u2084 : (a : \u211d) ^ 2 = 18 := by\n    have h\u2084\u2081 : (a : \u211d) ^ 2 = 6 * (b : \u211d) := by\n      norm_cast at h\u2081 \u22a2\n      <;> simp_all [pow_two, mul_assoc]\n      <;> ring_nf at *\n      <;> nlinarith\n    rw [h\u2084\u2081]\n    rw [h\u2083]\n    <;> norm_num\n  have h\u2085 : (a : \u211d) = 3 * Real.sqrt 2 := by\n    have h\u2085\u2081 : 0 < (a : \u211d) := by exact_mod_cast h\u2080.1\n    have h\u2085\u2082 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n    have h\u2085\u2083 : 0 < 3 * Real.sqrt 2 := by positivity\n    have h\u2085\u2084 : (a : \u211d) ^ 2 = 18 := h\u2084\n    have h\u2085\u2085 : (a : \u211d) > 0 := by exact_mod_cast h\u2080.1\n    have h\u2085\u2086 : (a : \u211d) = 3 * Real.sqrt 2 := by\n      apply Eq.symm\n      nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num), Real.sqrt_nonneg 2,\n        mul_self_nonneg ((a : \u211d) - 3 * Real.sqrt 2),\n        mul_self_nonneg ((a : \u211d) + 3 * Real.sqrt 2)]\n    exact h\u2085\u2086\n  have h\u2086 : a = 3 * NNReal.sqrt 2 := by\n    apply Eq.symm\n    apply Eq.symm\n    rw [\u2190 NNReal.coe_inj] at *\n    simp_all [NNReal.coe_mul, NNReal.coe_pow, NNReal.sqrt_eq_iff_sq_eq]\n    <;>\n    (try norm_num) <;>\n    (try ring_nf at *) <;>\n    (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]) <;>\n    (try field_simp at *) <;>\n    (try norm_num at *) <;>\n    (try nlinarith)\n    <;>\n    (try\n      {\n        nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      })\n    <;>\n    (try\n      {\n        ring_nf at *\n        <;>\n        nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      })\n  rw [h\u2086]\n  <;> simp_all\n  <;> norm_num\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_184 (a b : NNReal) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : a ^ 2 = 6 * b)\n    (h\u2082 : a ^ 2 = 54 / b) : a = 3 * NNReal.sqrt 2  :=  by\n  have h\u2083 : (b : \u211d) = 3 := by\n    have h\u2083\u2081 : (a : \u211d) ^ 2 = 6 * (b : \u211d) := by\n      norm_cast at h\u2081 \u22a2\n      <;> simp_all [pow_two, mul_assoc]\n      <;> ring_nf at *\n      <;> nlinarith\n    have h\u2083\u2082 : (a : \u211d) ^ 2 = 54 / (b : \u211d) := by\n      norm_cast at h\u2082 \u22a2\n      <;> simp_all [pow_two, mul_assoc]\n      <;> field_simp [h\u2080.2.ne'] at *\n      <;> ring_nf at *\n      <;> nlinarith\n    have h\u2083\u2083 : (6 : \u211d) * (b : \u211d) = 54 / (b : \u211d) := by linarith\n    have h\u2083\u2084 : (b : \u211d) > 0 := by exact_mod_cast h\u2080.2\n    have h\u2083\u2085 : (b : \u211d) ^ 2 = 9 := by\n      field_simp at h\u2083\u2083\n      nlinarith\n    have h\u2083\u2086 : (b : \u211d) = 3 := by\n      nlinarith\n    exact_mod_cast h\u2083\u2086\n  have h\u2084 : (a : \u211d) ^ 2 = 18 := by\n    have h\u2084\u2081 : (a : \u211d) ^ 2 = 6 * (b : \u211d) := by\n      norm_cast at h\u2081 \u22a2\n      <;> simp_all [pow_two, mul_assoc]\n      <;> ring_nf at *\n      <;> nlinarith\n    rw [h\u2084\u2081]\n    rw [h\u2083]\n    <;> norm_num\n  have h\u2085 : (a : \u211d) = 3 * Real.sqrt 2 := by\n    have h\u2085\u2081 : 0 < (a : \u211d) := by exact_mod_cast h\u2080.1\n    have h\u2085\u2082 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n    have h\u2085\u2083 : 0 < 3 * Real.sqrt 2 := by positivity\n    have h\u2085\u2084 : (a : \u211d) ^ 2 = 18 := h\u2084\n    have h\u2085\u2085 : (a : \u211d) > 0 := by exact_mod_cast h\u2080.1\n    have h\u2085\u2086 : (a : \u211d) = 3 * Real.sqrt 2 := by\n      apply Eq.symm\n      nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num), Real.sqrt_nonneg 2,\n        mul_self_nonneg ((a : \u211d) - 3 * Real.sqrt 2),\n        mul_self_nonneg ((a : \u211d) + 3 * Real.sqrt 2)]\n    exact h\u2085\u2086\n  have h\u2086 : a = 3 * NNReal.sqrt 2 := by\n    apply Eq.symm\n    apply Eq.symm\n    rw [\u2190 NNReal.coe_inj] at *\n    simp_all [NNReal.coe_mul, NNReal.coe_pow, NNReal.sqrt_eq_iff_sq_eq]\n    <;>\n    (try norm_num) <;>\n    (try ring_nf at *) <;>\n    (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]) <;>\n    (try field_simp at *) <;>\n    (try norm_num at *) <;>\n    (try nlinarith)\n    <;>\n    (try\n      {\n        nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      })\n    <;>\n    (try\n      {\n        ring_nf at *\n        <;>\n        nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      })\n  rw [h\u2086]\n  <;> simp_all\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 42, "alphabet": 29, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1251, "chars_after": 8, "seconds": 0.6, "id": "mathd_numbertheory_320", "proof_after": "theorem mathd_numbertheory_320 (n : \u2115) (h\u2080 : n < 101) (h\u2081 : 101 \u2223 123456 - n) : n = 34  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_320 (n : \u2115) (h\u2080 : n < 101) (h\u2081 : 101 \u2223 123456 - n) : n = 34  :=  by\n  have h\u2082 : 123456 % 101 = 34 := by\n    norm_num\n    <;> rfl\n  have h\u2083 : n % 101 = 34 := by\n    have h\u2084 : (123456 - n) % 101 = 0 := by\n      exact Nat.mod_eq_zero_of_dvd h\u2081\n    have h\u2085 : (123456 % 101) = 34 := h\u2082\n    have h\u2086 : (123456 - n) % 101 = (123456 % 101 - n % 101) % 101 := by\n      have h\u2087 : (123456 - n) % 101 = (123456 % 101 - n % 101) % 101 := by\n        omega\n      exact h\u2087\n    rw [h\u2086] at h\u2084\n    have h\u2087 : (123456 % 101 - n % 101) % 101 = 0 := h\u2084\n    have h\u2088 : (34 - n % 101) % 101 = 0 := by\n      simpa [h\u2085] using h\u2087\n    have h\u2089 : n % 101 = 34 := by\n      have h\u2081\u2080 : n % 101 < 101 := Nat.mod_lt n (by norm_num)\n      have h\u2081\u2081 : 34 - n % 101 = 0 := by\n        by_contra h\n        have h\u2081\u2082 : 34 - n % 101 > 0 := by\n          have h\u2081\u2083 : 34 - n % 101 \u2260 0 := h\n          have h\u2081\u2084 : 34 - n % 101 > 0 := by\n            by_contra h\u2081\u2085\n            have h\u2081\u2086 : 34 - n % 101 = 0 := by omega\n            contradiction\n          exact h\u2081\u2084\n        have h\u2081\u2085 : (34 - n % 101) % 101 = 34 - n % 101 := by\n          apply Nat.mod_eq_of_lt\n          omega\n        rw [h\u2081\u2085] at h\u2088\n        omega\n      omega\n    exact h\u2089\n  have h\u2084 : n = 34 := by\n    have h\u2085 : n % 101 = n := by\n      apply Nat.mod_eq_of_lt\n      exact h\u2080\n    rw [h\u2085] at h\u2083\n    omega\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 11, "alphabet": 13, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 279, "chars_after": 32, "seconds": 1.6, "id": "algebra_sqineq_at2malt1", "proof_after": "theorem algebra_sqineq_at2malt1 (a : \u211d) : a * (2 - a) \u2264 1  :=  by\n  nlinarith [sq_nonneg (a - 1)]\n", "proof_before": "theorem algebra_sqineq_at2malt1 (a : \u211d) : a * (2 - a) \u2264 1  :=  by\n  have h\u2080 : a * (2 - a) = 2 * a - a ^ 2 := by\n    ring_nf\n    <;>\n    linarith\n  have h\u2081 : (a - 1) ^ 2 \u2265 0 := by\n    nlinarith [sq_nonneg (a - 1)]\n  have h\u2082 : a * (2 - a) \u2264 1 := by\n    have h\u2083 : 2 * a - a ^ 2 \u2264 1 := by\n      nlinarith [sq_nonneg (a - 1)]\n    linarith\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 16, "alphabet": 17, "compiles": false, "id": "mathd_numbertheory_427", "proof_after": "theorem mathd_numbertheory_427 (a : \u2115) (h\u2080 : a = \u2211 k in Nat.divisors 500, k) :\n    (\u2211 k in Finset.filter (fun x => Nat.Prime x) (Nat.divisors a), k) = 25  :=  by\n  have h\u2081 : a = 1092 := by\n    rw [h\u2080]\n    rw [show (\u2211 k in Nat.divisors 500, k) = 1092 by\n      rw [show (500 : \u2115) = 2 ^ 2 * 5 ^ 3 by norm_num]\n      rw [Nat.divisors_mul, Nat.divisors_prime_pow (by decide : Nat.Prime 2),\n        Nat.divisors_prime_pow (by decide : Nat.Prime 5)]\n      <;> rfl\n    ]\n    <;> rfl\n  have h\u2082 : (\u2211 k in Finset.filter (fun x => Nat.Prime x) (Nat.divisors a), k) = 25 := by\n    rw [h\u2081]\n    have h\u2083 : (\u2211 k in Finset.filter (fun x => Nat.Prime x) (Nat.divisors 1092), k) = 25 := by\n      rw [show (1092 : \u2115) = 2 ^ 2 * 3 * 7 * 13 by norm_num]\n      rw [Nat.divisors_mul, Nat.divisors_mul, Nat.divisors_mul] <;>\n      norm_num [Nat.divisors_prime_pow, Finset.sum_filter, Finset.sum_range_succ, Nat.divisors] <;>\n      rfl\n    exact h\u2083\n  exact h\u2082\n", "proof_before": "theorem mathd_numbertheory_427 (a : \u2115) (h\u2080 : a = \u2211 k in Nat.divisors 500, k) :\n    (\u2211 k in Finset.filter (fun x => Nat.Prime x) (Nat.divisors a), k) = 25  :=  by\n  have h\u2081 : a = 1092 := by\n    rw [h\u2080]\n    rw [show (\u2211 k in Nat.divisors 500, k) = 1092 by\n      rw [show (500 : \u2115) = 2 ^ 2 * 5 ^ 3 by norm_num]\n      rw [Nat.divisors_mul, Nat.divisors_prime_pow (by decide : Nat.Prime 2),\n        Nat.divisors_prime_pow (by decide : Nat.Prime 5)]\n      <;> rfl\n    ]\n    <;> rfl\n  have h\u2082 : (\u2211 k in Finset.filter (fun x => Nat.Prime x) (Nat.divisors a), k) = 25 := by\n    rw [h\u2081]\n    have h\u2083 : (\u2211 k in Finset.filter (fun x => Nat.Prime x) (Nat.divisors 1092), k) = 25 := by\n      rw [show (1092 : \u2115) = 2 ^ 2 * 3 * 7 * 13 by norm_num]\n      rw [Nat.divisors_mul, Nat.divisors_mul, Nat.divisors_mul] <;>\n      norm_num [Nat.divisors_prime_pow, Finset.sum_filter, Finset.sum_range_succ, Nat.divisors] <;>\n      rfl\n    exact h\u2083\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 57, "alphabet": 34, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 57, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 1917, "chars_after": 1917, "seconds": 286.6, "id": "induction_nfactltnexpnm1ngt3", "proof_after": "theorem induction_nfactltnexpnm1ngt3 (n : \u2115) (h\u2080 : 3 \u2264 n) : n ! < n ^ (n - 1)  :=  by\n  have h_base : 3 ! < 3 ^ (3 - 1) := by\n    norm_num [Nat.factorial]\n    <;> decide\n  have h_inductive : \u2200 (k : \u2115), 3 \u2264 k \u2192 k ! < k ^ (k - 1) \u2192 (k + 1) ! < (k + 1) ^ k := by\n    intro k hk hk_rec\n    have h\u2081 : (k + 1) ! = (k + 1) * k ! := by\n      simp [Nat.factorial]\n      <;> ring\n      <;> simp_all [Nat.factorial]\n      <;> ring\n    rw [h\u2081]\n    have h\u2082 : (k + 1) * k ! < (k + 1) * k ^ (k - 1) := by\n      have h\u2083 : 0 < k + 1 := by linarith\n      have h\u2084 : k ! < k ^ (k - 1) := hk_rec\n      have h\u2085 : (k + 1) * k ! < (k + 1) * k ^ (k - 1) := by\n        nlinarith\n      exact h\u2085\n    have h\u2083 : (k + 1) * k ^ (k - 1) \u2264 (k + 1) ^ k := by\n      have h\u2084 : k \u2265 3 := by linarith\n      have h\u2085 : k - 1 \u2265 2 := by\n        omega\n      have h\u2086 : (k + 1) * k ^ (k - 1) \u2264 (k + 1) ^ k := by\n        calc\n          (k + 1) * k ^ (k - 1) = (k + 1) * k ^ (k - 1) := rfl\n          _ \u2264 (k + 1) * (k + 1) ^ (k - 1) := by\n            have h\u2087 : k ^ (k - 1) \u2264 (k + 1) ^ (k - 1) := by\n              exact Nat.pow_le_pow_of_le_left (by linarith) _\n            exact Nat.mul_le_mul_left _ h\u2087\n          _ = (k + 1) ^ (k - 1 + 1) := by\n            rw [Nat.mul_comm]\n            <;> simp [Nat.pow_succ, Nat.mul_comm]\n            <;> ring_nf\n          _ \u2264 (k + 1) ^ k := by\n            have h\u2088 : k - 1 + 1 \u2264 k := by\n              have h\u2089 : k - 1 + 1 \u2264 k := by\n                omega\n              exact h\u2089\n            exact Nat.pow_le_pow_of_le_right (by linarith) h\u2088\n      exact h\u2086\n    have h\u2084 : (k + 1) * k ! < (k + 1) ^ k := by\n      calc\n        (k + 1) * k ! < (k + 1) * k ^ (k - 1) := h\u2082\n        _ \u2264 (k + 1) ^ k := h\u2083\n    exact h\u2084\n  have h_main : n ! < n ^ (n - 1) := by\n    have h\u2081 : \u2200 (k : \u2115), 3 \u2264 k \u2192 k ! < k ^ (k - 1) := by\n      intro k hk\n      induction' hk with k hk IH\n      \u00b7 \n        norm_num [Nat.factorial]\n      \u00b7 \n        simpa using h_inductive k hk IH\n    have h\u2082 : n ! < n ^ (n - 1) := h\u2081 n h\u2080\n    exact h\u2082\n  exact h_main\n", "proof_before": "theorem induction_nfactltnexpnm1ngt3 (n : \u2115) (h\u2080 : 3 \u2264 n) : n ! < n ^ (n - 1)  :=  by\n  have h_base : 3 ! < 3 ^ (3 - 1) := by\n    norm_num [Nat.factorial]\n    <;> decide\n  have h_inductive : \u2200 (k : \u2115), 3 \u2264 k \u2192 k ! < k ^ (k - 1) \u2192 (k + 1) ! < (k + 1) ^ k := by\n    intro k hk hk_rec\n    have h\u2081 : (k + 1) ! = (k + 1) * k ! := by\n      simp [Nat.factorial]\n      <;> ring\n      <;> simp_all [Nat.factorial]\n      <;> ring\n    rw [h\u2081]\n    have h\u2082 : (k + 1) * k ! < (k + 1) * k ^ (k - 1) := by\n      have h\u2083 : 0 < k + 1 := by linarith\n      have h\u2084 : k ! < k ^ (k - 1) := hk_rec\n      have h\u2085 : (k + 1) * k ! < (k + 1) * k ^ (k - 1) := by\n        nlinarith\n      exact h\u2085\n    have h\u2083 : (k + 1) * k ^ (k - 1) \u2264 (k + 1) ^ k := by\n      have h\u2084 : k \u2265 3 := by linarith\n      have h\u2085 : k - 1 \u2265 2 := by\n        omega\n      have h\u2086 : (k + 1) * k ^ (k - 1) \u2264 (k + 1) ^ k := by\n        calc\n          (k + 1) * k ^ (k - 1) = (k + 1) * k ^ (k - 1) := rfl\n          _ \u2264 (k + 1) * (k + 1) ^ (k - 1) := by\n            have h\u2087 : k ^ (k - 1) \u2264 (k + 1) ^ (k - 1) := by\n              exact Nat.pow_le_pow_of_le_left (by linarith) _\n            exact Nat.mul_le_mul_left _ h\u2087\n          _ = (k + 1) ^ (k - 1 + 1) := by\n            rw [Nat.mul_comm]\n            <;> simp [Nat.pow_succ, Nat.mul_comm]\n            <;> ring_nf\n          _ \u2264 (k + 1) ^ k := by\n            have h\u2088 : k - 1 + 1 \u2264 k := by\n              have h\u2089 : k - 1 + 1 \u2264 k := by\n                omega\n              exact h\u2089\n            exact Nat.pow_le_pow_of_le_right (by linarith) h\u2088\n      exact h\u2086\n    have h\u2084 : (k + 1) * k ! < (k + 1) ^ k := by\n      calc\n        (k + 1) * k ! < (k + 1) * k ^ (k - 1) := h\u2082\n        _ \u2264 (k + 1) ^ k := h\u2083\n    exact h\u2084\n  have h_main : n ! < n ^ (n - 1) := by\n    have h\u2081 : \u2200 (k : \u2115), 3 \u2264 k \u2192 k ! < k ^ (k - 1) := by\n      intro k hk\n      induction' hk with k hk IH\n      \u00b7 \n        norm_num [Nat.factorial]\n      \u00b7 \n        simpa using h_inductive k hk IH\n    have h\u2082 : n ! < n ^ (n - 1) := h\u2081 n h\u2080\n    exact h\u2082\n  exact h_main\n"}
{"mode": "clear", "steps_before": 201, "alphabet": 58, "compiles": false, "id": "amc12a_2021_p14", "proof_after": "theorem amc12a_2021_p14 :\n    ((\u2211 k in Finset.Icc 1 20, Real.logb (5 ^ k) (3 ^ k ^ 2)) *\n        \u2211 k in Finset.Icc 1 100, Real.logb (9 ^ k) (25 ^ k)) = 21000  :=  by\n  have h\u2081 : \u2200 k \u2208 Finset.Icc (1 : \u2115) 20, Real.logb ((5 : \u211d) ^ k) ((3 : \u211d) ^ (k ^ 2)) = (k : \u211d) * Real.logb 5 3 := by\n    intro k hk\n    have h\u2081 : 1 \u2264 (k : \u2115) := by\n      simp [Finset.mem_Icc] at hk\n      exact hk.1\n    have h\u2082 : (k : \u2115) \u2264 20 := by\n      simp [Finset.mem_Icc] at hk\n      exact hk.2\n    have h\u2083 : k \u2265 1 := by exact_mod_cast h\u2081\n    have h\u2084 : k \u2264 20 := by exact_mod_cast h\u2082\n    have h\u2085 : Real.logb ((5 : \u211d) ^ k) ((3 : \u211d) ^ (k ^ 2)) = (k : \u211d) * Real.logb 5 3 := by\n      have h\u2085\u2081 : Real.logb ((5 : \u211d) ^ k) ((3 : \u211d) ^ (k ^ 2)) = Real.log ((3 : \u211d) ^ (k ^ 2)) / Real.log ((5 : \u211d) ^ k) := by\n        rw [Real.logb]\n        <;> simp [Real.log_rpow]\n        <;> field_simp\n        <;> ring\n      rw [h\u2085\u2081]\n      have h\u2085\u2082 : Real.log ((3 : \u211d) ^ (k ^ 2)) = (k ^ 2 : \u211d) * Real.log 3 := by\n        rw [Real.log_pow]\n        <;> norm_cast\n        <;> field_simp\n        <;> ring\n      have h\u2085\u2083 : Real.log ((5 : \u211d) ^ k) = (k : \u211d) * Real.log 5 := by\n        rw [Real.log_pow]\n        <;> norm_cast\n        <;> field_simp\n        <;> ring\n      rw [h\u2085\u2082, h\u2085\u2083]\n      have h\u2085\u2084 : Real.logb 5 3 = Real.log 3 / Real.log 5 := by\n        rw [Real.logb]\n        <;> simp [Real.log_rpow]\n        <;> field_simp\n        <;> ring\n      have h\u2085\u2085 : (k ^ 2 : \u211d) * Real.log 3 / ((k : \u211d) * Real.log 5) = (k : \u211d) * (Real.log 3 / Real.log 5) := by\n        have h\u2085\u2085\u2081 : (k : \u211d) \u2260 0 := by\n          norm_num at hk \u22a2\n          <;> omega\n        field_simp [h\u2085\u2085\u2081]\n        <;> ring\n        <;> field_simp [h\u2085\u2085\u2081]\n        <;> ring\n        <;> norm_cast\n        <;> simp_all [Nat.cast_pow, Nat.cast_mul, Nat.cast_add, Nat.cast_one]\n        <;> field_simp [h\u2085\u2085\u2081]\n        <;> ring\n      rw [h\u2085\u2085, h\u2085\u2084]\n      <;> ring\n      <;> field_simp\n      <;> ring\n    exact h\u2085\n  have h\u2082 : (\u2211 k in Finset.Icc (1 : \u2115) 20, Real.logb ((5 : \u211d) ^ k) ((3 : \u211d) ^ (k ^ 2))) = 210 * Real.logb 5 3 := by\n    have h\u2082\u2081 : (\u2211 k in Finset.Icc (1 : \u2115) 20, Real.logb ((5 : \u211d) ^ k) ((3 : \u211d) ^ (k ^ 2))) = \u2211 k in Finset.Icc (1 : \u2115) 20, ((k : \u211d) * Real.logb 5 3) := by\n      apply Finset.sum_congr rfl\n      intro k hk\n      rw [h\u2081 k hk]\n    rw [h\u2082\u2081]\n    have h\u2082\u2082 : (\u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d) * Real.logb 5 3) = (\u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d)) * Real.logb 5 3 := by\n      calc\n        (\u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d) * Real.logb 5 3) = \u2211 k in Finset.Icc (1 : \u2115) 20, (Real.logb 5 3 * (k : \u211d)) := by\n          apply Finset.sum_congr rfl\n          intro k _\n          ring\n        _ = Real.logb 5 3 * \u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d) := by\n          rw [Finset.mul_sum]\n          <;> simp [mul_comm]\n        _ = (\u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d)) * Real.logb 5 3 := by\n          ring\n    rw [h\u2082\u2082]\n    have h\u2082\u2083 : (\u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d)) = 210 := by\n      norm_num [Finset.sum_Icc_succ_top]\n      <;> rfl\n    rw [h\u2082\u2083]\n    <;> ring\n    <;> simp [Real.logb]\n    <;> field_simp\n    <;> ring\n  have h\u2083 : \u2200 k \u2208 Finset.Icc (1 : \u2115) 100, Real.logb ((9 : \u211d) ^ k) ((25 : \u211d) ^ k) = Real.logb 3 5 := by\n    intro k hk\n    have h\u2083\u2081 : Real.logb ((9 : \u211d) ^ k) ((25 : \u211d) ^ k) = (Real.log ((25 : \u211d) ^ k) / Real.log ((9 : \u211d) ^ k)) := by\n      rw [Real.logb]\n      <;> simp [Real.log_rpow]\n      <;> field_simp\n      <;> ring\n    rw [h\u2083\u2081]\n    have h\u2083\u2082 : Real.log ((25 : \u211d) ^ k) = (k : \u211d) * Real.log 25 := by\n      rw [Real.log_pow]\n      <;> norm_cast\n      <;> field_simp\n      <;> ring\n    have h\u2083\u2083 : Real.log ((9 : \u211d) ^ k) = (k : \u211d) * Real.log 9 := by\n      rw [Real.log_pow]\n      <;> norm_cast\n      <;> field_simp\n      <;> ring\n    rw [h\u2083\u2082, h\u2083\u2083]\n    have h\u2083\u2084 : Real.log 25 = 2 * Real.log 5 := by\n      have h\u2083\u2084\u2081 : Real.log 25 = Real.log (5 ^ 2) := by norm_num\n      rw [h\u2083\u2084\u2081]\n      have h\u2083\u2084\u2082 : Real.log (5 ^ 2) = 2 * Real.log 5 := by\n        rw [Real.log_pow]\n        <;> norm_num\n        <;> ring\n      rw [h\u2083\u2084\u2082]\n    have h\u2083\u2085 : Real.log 9 = 2 * Real.log 3 := by\n      have h\u2083\u2085\u2081 : Real.log 9 = Real.log (3 ^ 2) := by norm_num\n      rw [h\u2083\u2085\u2081]\n      have h\u2083\u2085\u2082 : Real.log (3 ^ 2) = 2 * Real.log 3 := by\n        rw [Real.log_pow]\n        <;> norm_num\n        <;> ring\n      rw [h\u2083\u2085\u2082]\n    rw [h\u2083\u2084, h\u2083\u2085]\n    have h\u2083\u2086 : (k : \u211d) \u2260 0 := by\n      norm_num at hk \u22a2\n      <;>\n      (try omega) <;>\n      (try linarith) <;>\n      (try\n        {\n          aesop\n        }) <;>\n      (try\n        {\n          norm_num at hk \u22a2 <;>\n          omega\n        }) <;>\n      (try\n        {\n          linarith\n        })\n    have h\u2083\u2087 : (2 : \u211d) \u2260 0 := by norm_num\n    have h\u2083\u2088 : Real.log 5 \u2260 0 := by\n      have h\u2083\u2088\u2081 : Real.log 5 > 0 := Real.log_pos (by norm_num)\n      linarith\n    have h\u2083\u2089 : Real.log 3 \u2260 0 := by\n      have h\u2083\u2089\u2081 : Real.log 3 > 0 := Real.log_pos (by norm_num)\n      linarith\n    field_simp [h\u2083\u2086, h\u2083\u2087, h\u2083\u2088, h\u2083\u2089, Real.logb, Real.log_mul, Real.log_rpow, Real.log_pow]\n    <;> ring_nf\n    <;> field_simp [h\u2083\u2086, h\u2083\u2087, h\u2083\u2088, h\u2083\u2089, Real.logb, Real.log_mul, Real.log_rpow, Real.log_pow]\n    <;> ring_nf\n    <;> field_simp [h\u2083\u2086, h\u2083\u2087, h\u2083\u2088, h\u2083\u2089, Real.logb, Real.log_mul, Real.log_rpow, Real.log_pow]\n    <;> ring_nf\n    <;> field_simp [h\u2083\u2086, h\u2083\u2087, h\u2083\u2088, h\u2083\u2089, Real.logb, Real.log_mul, Real.log_rpow, Real.log_pow]\n    <;> ring_nf\n  have h\u2084 : (\u2211 k in Finset.Icc (1 : \u2115) 100, Real.logb ((9 : \u211d) ^ k) ((25 : \u211d) ^ k)) = 100 * Real.logb 3 5 := by\n    have h\u2084\u2081 : (\u2211 k in Finset.Icc (1 : \u2115) 100, Real.logb ((9 : \u211d) ^ k) ((25 : \u211d) ^ k)) = \u2211 k in Finset.Icc (1 : \u2115) 100, (Real.logb 3 5 : \u211d) := by\n      apply Finset.sum_congr rfl\n      intro k hk\n      rw [h\u2083 k hk]\n    rw [h\u2084\u2081]\n    have h\u2084\u2082 : (\u2211 k in Finset.Icc (1 : \u2115) 100, (Real.logb 3 5 : \u211d)) = 100 * Real.logb 3 5 := by\n      simp [Finset.sum_const, Finset.card_range]\n      <;> norm_num\n      <;> ring\n      <;> simp [Real.logb]\n      <;> field_simp\n      <;> ring\n    rw [h\u2084\u2082]\n    <;> simp [Real.logb]\n    <;> field_simp\n    <;> ring\n  have h\u2085 : Real.logb 5 3 * Real.logb 3 5 = 1 := by\n    have h\u2085\u2081 : Real.logb 5 3 = Real.log 3 / Real.log 5 := by\n      rw [Real.logb]\n      <;> simp [Real.log_rpow]\n      <;> field_simp\n      <;> ring\n    have h\u2085\u2082 : Real.logb 3 5 = Real.log 5 / Real.log 3 := by\n      rw [Real.logb]\n      <;> simp [Real.log_rpow]\n      <;> field_simp\n      <;> ring\n    rw [h\u2085\u2081, h\u2085\u2082]\n    have h\u2085\u2083 : Real.log 3 \u2260 0 := Real.log_ne_zero_of_pos_of_ne_one (by norm_num) (by norm_num)\n    have h\u2085\u2084 : Real.log 5 \u2260 0 := Real.log_ne_zero_of_pos_of_ne_one (by norm_num) (by norm_num)\n    field_simp [h\u2085\u2083, h\u2085\u2084]\n    <;> ring\n    <;> field_simp [h\u2085\u2083, h\u2085\u2084]\n    <;> ring\n  have h\u2086 : ((\u2211 k in Finset.Icc 1 20, Real.logb (5 ^ k) (3 ^ k ^ 2)) * \u2211 k in Finset.Icc 1 100, Real.logb (9 ^ k) (25 ^ k)) = 21000 := by\n    rw [show (\u2211 k in Finset.Icc 1 20, Real.logb (5 ^ k) (3 ^ k ^ 2)) = 210 * Real.logb 5 3 by\n      simpa using h\u2082]\n    rw [show (\u2211 k in Finset.Icc 1 100, Real.logb (9 ^ k) (25 ^ k)) = 100 * Real.logb 3 5 by\n      simpa using h\u2084]\n    have h\u2086\u2081 : (210 * Real.logb 5 3 : \u211d) * (100 * Real.logb 3 5 : \u211d) = 21000 := by\n      have h\u2086\u2082 : Real.logb 5 3 * Real.logb 3 5 = 1 := h\u2085\n      have h\u2086\u2083 : (Real.logb 5 3 : \u211d) * (Real.logb 3 5 : \u211d) = 1 := by exact_mod_cast h\u2086\u2082\n      calc\n        (210 * Real.logb 5 3 : \u211d) * (100 * Real.logb 3 5 : \u211d) = (210 * 100 : \u211d) * ((Real.logb 5 3 : \u211d) * (Real.logb 3 5 : \u211d)) := by ring\n        _ = (210 * 100 : \u211d) * 1 := by rw [h\u2086\u2083]\n        _ = 21000 := by norm_num\n    rw [h\u2086\u2081]\n    <;> norm_num\n  exact h\u2086\n", "proof_before": "theorem amc12a_2021_p14 :\n    ((\u2211 k in Finset.Icc 1 20, Real.logb (5 ^ k) (3 ^ k ^ 2)) *\n        \u2211 k in Finset.Icc 1 100, Real.logb (9 ^ k) (25 ^ k)) = 21000  :=  by\n  have h\u2081 : \u2200 k \u2208 Finset.Icc (1 : \u2115) 20, Real.logb ((5 : \u211d) ^ k) ((3 : \u211d) ^ (k ^ 2)) = (k : \u211d) * Real.logb 5 3 := by\n    intro k hk\n    have h\u2081 : 1 \u2264 (k : \u2115) := by\n      simp [Finset.mem_Icc] at hk\n      exact hk.1\n    have h\u2082 : (k : \u2115) \u2264 20 := by\n      simp [Finset.mem_Icc] at hk\n      exact hk.2\n    have h\u2083 : k \u2265 1 := by exact_mod_cast h\u2081\n    have h\u2084 : k \u2264 20 := by exact_mod_cast h\u2082\n    have h\u2085 : Real.logb ((5 : \u211d) ^ k) ((3 : \u211d) ^ (k ^ 2)) = (k : \u211d) * Real.logb 5 3 := by\n      have h\u2085\u2081 : Real.logb ((5 : \u211d) ^ k) ((3 : \u211d) ^ (k ^ 2)) = Real.log ((3 : \u211d) ^ (k ^ 2)) / Real.log ((5 : \u211d) ^ k) := by\n        rw [Real.logb]\n        <;> simp [Real.log_rpow]\n        <;> field_simp\n        <;> ring\n      rw [h\u2085\u2081]\n      have h\u2085\u2082 : Real.log ((3 : \u211d) ^ (k ^ 2)) = (k ^ 2 : \u211d) * Real.log 3 := by\n        rw [Real.log_pow]\n        <;> norm_cast\n        <;> field_simp\n        <;> ring\n      have h\u2085\u2083 : Real.log ((5 : \u211d) ^ k) = (k : \u211d) * Real.log 5 := by\n        rw [Real.log_pow]\n        <;> norm_cast\n        <;> field_simp\n        <;> ring\n      rw [h\u2085\u2082, h\u2085\u2083]\n      have h\u2085\u2084 : Real.logb 5 3 = Real.log 3 / Real.log 5 := by\n        rw [Real.logb]\n        <;> simp [Real.log_rpow]\n        <;> field_simp\n        <;> ring\n      have h\u2085\u2085 : (k ^ 2 : \u211d) * Real.log 3 / ((k : \u211d) * Real.log 5) = (k : \u211d) * (Real.log 3 / Real.log 5) := by\n        have h\u2085\u2085\u2081 : (k : \u211d) \u2260 0 := by\n          norm_num at hk \u22a2\n          <;> omega\n        field_simp [h\u2085\u2085\u2081]\n        <;> ring\n        <;> field_simp [h\u2085\u2085\u2081]\n        <;> ring\n        <;> norm_cast\n        <;> simp_all [Nat.cast_pow, Nat.cast_mul, Nat.cast_add, Nat.cast_one]\n        <;> field_simp [h\u2085\u2085\u2081]\n        <;> ring\n      rw [h\u2085\u2085, h\u2085\u2084]\n      <;> ring\n      <;> field_simp\n      <;> ring\n    exact h\u2085\n  have h\u2082 : (\u2211 k in Finset.Icc (1 : \u2115) 20, Real.logb ((5 : \u211d) ^ k) ((3 : \u211d) ^ (k ^ 2))) = 210 * Real.logb 5 3 := by\n    have h\u2082\u2081 : (\u2211 k in Finset.Icc (1 : \u2115) 20, Real.logb ((5 : \u211d) ^ k) ((3 : \u211d) ^ (k ^ 2))) = \u2211 k in Finset.Icc (1 : \u2115) 20, ((k : \u211d) * Real.logb 5 3) := by\n      apply Finset.sum_congr rfl\n      intro k hk\n      rw [h\u2081 k hk]\n    rw [h\u2082\u2081]\n    have h\u2082\u2082 : (\u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d) * Real.logb 5 3) = (\u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d)) * Real.logb 5 3 := by\n      calc\n        (\u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d) * Real.logb 5 3) = \u2211 k in Finset.Icc (1 : \u2115) 20, (Real.logb 5 3 * (k : \u211d)) := by\n          apply Finset.sum_congr rfl\n          intro k _\n          ring\n        _ = Real.logb 5 3 * \u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d) := by\n          rw [Finset.mul_sum]\n          <;> simp [mul_comm]\n        _ = (\u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d)) * Real.logb 5 3 := by\n          ring\n    rw [h\u2082\u2082]\n    have h\u2082\u2083 : (\u2211 k in Finset.Icc (1 : \u2115) 20, (k : \u211d)) = 210 := by\n      norm_num [Finset.sum_Icc_succ_top]\n      <;> rfl\n    rw [h\u2082\u2083]\n    <;> ring\n    <;> simp [Real.logb]\n    <;> field_simp\n    <;> ring\n  have h\u2083 : \u2200 k \u2208 Finset.Icc (1 : \u2115) 100, Real.logb ((9 : \u211d) ^ k) ((25 : \u211d) ^ k) = Real.logb 3 5 := by\n    intro k hk\n    have h\u2083\u2081 : Real.logb ((9 : \u211d) ^ k) ((25 : \u211d) ^ k) = (Real.log ((25 : \u211d) ^ k) / Real.log ((9 : \u211d) ^ k)) := by\n      rw [Real.logb]\n      <;> simp [Real.log_rpow]\n      <;> field_simp\n      <;> ring\n    rw [h\u2083\u2081]\n    have h\u2083\u2082 : Real.log ((25 : \u211d) ^ k) = (k : \u211d) * Real.log 25 := by\n      rw [Real.log_pow]\n      <;> norm_cast\n      <;> field_simp\n      <;> ring\n    have h\u2083\u2083 : Real.log ((9 : \u211d) ^ k) = (k : \u211d) * Real.log 9 := by\n      rw [Real.log_pow]\n      <;> norm_cast\n      <;> field_simp\n      <;> ring\n    rw [h\u2083\u2082, h\u2083\u2083]\n    have h\u2083\u2084 : Real.log 25 = 2 * Real.log 5 := by\n      have h\u2083\u2084\u2081 : Real.log 25 = Real.log (5 ^ 2) := by norm_num\n      rw [h\u2083\u2084\u2081]\n      have h\u2083\u2084\u2082 : Real.log (5 ^ 2) = 2 * Real.log 5 := by\n        rw [Real.log_pow]\n        <;> norm_num\n        <;> ring\n      rw [h\u2083\u2084\u2082]\n    have h\u2083\u2085 : Real.log 9 = 2 * Real.log 3 := by\n      have h\u2083\u2085\u2081 : Real.log 9 = Real.log (3 ^ 2) := by norm_num\n      rw [h\u2083\u2085\u2081]\n      have h\u2083\u2085\u2082 : Real.log (3 ^ 2) = 2 * Real.log 3 := by\n        rw [Real.log_pow]\n        <;> norm_num\n        <;> ring\n      rw [h\u2083\u2085\u2082]\n    rw [h\u2083\u2084, h\u2083\u2085]\n    have h\u2083\u2086 : (k : \u211d) \u2260 0 := by\n      norm_num at hk \u22a2\n      <;>\n      (try omega) <;>\n      (try linarith) <;>\n      (try\n        {\n          aesop\n        }) <;>\n      (try\n        {\n          norm_num at hk \u22a2 <;>\n          omega\n        }) <;>\n      (try\n        {\n          linarith\n        })\n    have h\u2083\u2087 : (2 : \u211d) \u2260 0 := by norm_num\n    have h\u2083\u2088 : Real.log 5 \u2260 0 := by\n      have h\u2083\u2088\u2081 : Real.log 5 > 0 := Real.log_pos (by norm_num)\n      linarith\n    have h\u2083\u2089 : Real.log 3 \u2260 0 := by\n      have h\u2083\u2089\u2081 : Real.log 3 > 0 := Real.log_pos (by norm_num)\n      linarith\n    field_simp [h\u2083\u2086, h\u2083\u2087, h\u2083\u2088, h\u2083\u2089, Real.logb, Real.log_mul, Real.log_rpow, Real.log_pow]\n    <;> ring_nf\n    <;> field_simp [h\u2083\u2086, h\u2083\u2087, h\u2083\u2088, h\u2083\u2089, Real.logb, Real.log_mul, Real.log_rpow, Real.log_pow]\n    <;> ring_nf\n    <;> field_simp [h\u2083\u2086, h\u2083\u2087, h\u2083\u2088, h\u2083\u2089, Real.logb, Real.log_mul, Real.log_rpow, Real.log_pow]\n    <;> ring_nf\n    <;> field_simp [h\u2083\u2086, h\u2083\u2087, h\u2083\u2088, h\u2083\u2089, Real.logb, Real.log_mul, Real.log_rpow, Real.log_pow]\n    <;> ring_nf\n  have h\u2084 : (\u2211 k in Finset.Icc (1 : \u2115) 100, Real.logb ((9 : \u211d) ^ k) ((25 : \u211d) ^ k)) = 100 * Real.logb 3 5 := by\n    have h\u2084\u2081 : (\u2211 k in Finset.Icc (1 : \u2115) 100, Real.logb ((9 : \u211d) ^ k) ((25 : \u211d) ^ k)) = \u2211 k in Finset.Icc (1 : \u2115) 100, (Real.logb 3 5 : \u211d) := by\n      apply Finset.sum_congr rfl\n      intro k hk\n      rw [h\u2083 k hk]\n    rw [h\u2084\u2081]\n    have h\u2084\u2082 : (\u2211 k in Finset.Icc (1 : \u2115) 100, (Real.logb 3 5 : \u211d)) = 100 * Real.logb 3 5 := by\n      simp [Finset.sum_const, Finset.card_range]\n      <;> norm_num\n      <;> ring\n      <;> simp [Real.logb]\n      <;> field_simp\n      <;> ring\n    rw [h\u2084\u2082]\n    <;> simp [Real.logb]\n    <;> field_simp\n    <;> ring\n  have h\u2085 : Real.logb 5 3 * Real.logb 3 5 = 1 := by\n    have h\u2085\u2081 : Real.logb 5 3 = Real.log 3 / Real.log 5 := by\n      rw [Real.logb]\n      <;> simp [Real.log_rpow]\n      <;> field_simp\n      <;> ring\n    have h\u2085\u2082 : Real.logb 3 5 = Real.log 5 / Real.log 3 := by\n      rw [Real.logb]\n      <;> simp [Real.log_rpow]\n      <;> field_simp\n      <;> ring\n    rw [h\u2085\u2081, h\u2085\u2082]\n    have h\u2085\u2083 : Real.log 3 \u2260 0 := Real.log_ne_zero_of_pos_of_ne_one (by norm_num) (by norm_num)\n    have h\u2085\u2084 : Real.log 5 \u2260 0 := Real.log_ne_zero_of_pos_of_ne_one (by norm_num) (by norm_num)\n    field_simp [h\u2085\u2083, h\u2085\u2084]\n    <;> ring\n    <;> field_simp [h\u2085\u2083, h\u2085\u2084]\n    <;> ring\n  have h\u2086 : ((\u2211 k in Finset.Icc 1 20, Real.logb (5 ^ k) (3 ^ k ^ 2)) * \u2211 k in Finset.Icc 1 100, Real.logb (9 ^ k) (25 ^ k)) = 21000 := by\n    rw [show (\u2211 k in Finset.Icc 1 20, Real.logb (5 ^ k) (3 ^ k ^ 2)) = 210 * Real.logb 5 3 by\n      simpa using h\u2082]\n    rw [show (\u2211 k in Finset.Icc 1 100, Real.logb (9 ^ k) (25 ^ k)) = 100 * Real.logb 3 5 by\n      simpa using h\u2084]\n    have h\u2086\u2081 : (210 * Real.logb 5 3 : \u211d) * (100 * Real.logb 3 5 : \u211d) = 21000 := by\n      have h\u2086\u2082 : Real.logb 5 3 * Real.logb 3 5 = 1 := h\u2085\n      have h\u2086\u2083 : (Real.logb 5 3 : \u211d) * (Real.logb 3 5 : \u211d) = 1 := by exact_mod_cast h\u2086\u2082\n      calc\n        (210 * Real.logb 5 3 : \u211d) * (100 * Real.logb 3 5 : \u211d) = (210 * 100 : \u211d) * ((Real.logb 5 3 : \u211d) * (Real.logb 3 5 : \u211d)) := by ring\n        _ = (210 * 100 : \u211d) * 1 := by rw [h\u2086\u2083]\n        _ = 21000 := by norm_num\n    rw [h\u2086\u2081]\n    <;> norm_num\n  exact h\u2086\n"}
{"mode": "clear", "steps_before": 7, "alphabet": 15, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 154, "chars_after": 11, "seconds": 164.3, "id": "mathd_algebra_400", "proof_after": "theorem mathd_algebra_400 (x : \u211d) (h\u2080 : 5 + 500 / 100 * 10 = 110 / 100 * x) : x = 50  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_400 (x : \u211d) (h\u2080 : 5 + 500 / 100 * 10 = 110 / 100 * x) : x = 50  :=  by\n  have h\u2081 : x = 50 := by\n    have h\u2082 : x = 50 := by\n      apply_fun (fun x => x * 100) at h\u2080\n      ring_nf at h\u2080 \u22a2\n      linarith\n    exact h\u2082\n  exact h\u2081\n"}
{"mode": "clear", "steps_before": 14, "alphabet": 15, "compiles": true, "steps_after": 8, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 2, "fixpoint": true, "chars_before": 344, "chars_after": 230, "seconds": 250.8, "id": "mathd_algebra_263", "proof_after": "theorem mathd_algebra_263 (y : \u211d) (h\u2080 : 0 \u2264 19 + 3 * y) (h\u2081 : Real.sqrt (19 + 3 * y) = 7) :\n    y = 10  :=  by\n  have h\u2082 : 19 + 3 * y = 49 := by\n    have h\u2082\u2081 : (Real.sqrt (19 + 3 * y)) ^ 2 = 7 ^ 2 := by\n      rw [h\u2081]\n    have h\u2082\u2082 : (Real.sqrt (19 + 3 * y)) ^ 2 = 19 + 3 * y := by\n      rw [Real.sq_sqrt] <;> linarith\n    linarith\n  linarith\n", "proof_before": "theorem mathd_algebra_263 (y : \u211d) (h\u2080 : 0 \u2264 19 + 3 * y) (h\u2081 : Real.sqrt (19 + 3 * y) = 7) :\n    y = 10  :=  by\n  have h\u2082 : 19 + 3 * y = 49 := by\n    have h\u2082\u2081 : (Real.sqrt (19 + 3 * y)) ^ 2 = 7 ^ 2 := by\n      rw [h\u2081]\n    have h\u2082\u2082 : (Real.sqrt (19 + 3 * y)) ^ 2 = 19 + 3 * y := by\n      rw [Real.sq_sqrt] <;> linarith\n    linarith\n  have h\u2083 : y = 10 := by\n    have h\u2084 : 3 * y = 30 := by linarith\n    have h\u2085 : y = 10 := by linarith\n    exact h\u2085\n  apply h\u2083\n"}
{"mode": "clear", "steps_before": 19, "alphabet": 18, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 403, "chars_after": 11, "seconds": 78.5, "id": "mathd_algebra_398", "proof_after": "theorem mathd_algebra_398 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) (h\u2081 : 9 * b = 20 * c)\n    (h\u2082 : 7 * a = 4 * b) : 63 * a = 80 * c  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_398 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) (h\u2081 : 9 * b = 20 * c)\n    (h\u2082 : 7 * a = 4 * b) : 63 * a = 80 * c  :=  by\n  have h\u2083 : 63 * a = 36 * b := by\n    have h\u2083\u2081 : 63 * a = 9 * (7 * a) := by ring\n    rw [h\u2083\u2081]\n    have h\u2083\u2082 : 7 * a = 4 * b := h\u2082\n    rw [h\u2083\u2082]\n    <;> ring\n    <;> linarith\n  have h\u2084 : 36 * b = 80 * c := by\n    have h\u2084\u2081 : 36 * b = 4 * (9 * b) := by ring\n    rw [h\u2084\u2081]\n    have h\u2084\u2082 : 9 * b = 20 * c := h\u2081\n    rw [h\u2084\u2082]\n    <;> ring\n    <;> linarith\n  have h\u2085 : 63 * a = 80 * c := by\n    linarith\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 43, "alphabet": 27, "compiles": false, "id": "mathd_algebra_215", "proof_after": "theorem mathd_algebra_215 (S : Finset \u211d) (h\u2080 : \u2200 x : \u211d, x \u2208 S \u2194 (x + 3) ^ 2 = 121) :\n    (\u2211 k in S, k) = -6  :=  by\n  have h\u2081 : S = ({8, -14} : Finset \u211d) := by\n    apply Finset.ext\n    intro x\n    simp only [Finset.mem_insert, Finset.mem_singleton, h\u2080]\n    constructor\n    \u00b7 intro h\n      have h\u2082 : (x + 3) ^ 2 = 121 := by exact h\n      have h\u2083 : x + 3 = 11 \u2228 x + 3 = -11 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2084\n        apply eq_of_sub_eq_zero\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2084)\n        nlinarith\n      cases h\u2083 with\n      | inl h\u2083 =>\n        have h\u2084 : x = 8 := by linarith\n        rw [h\u2084]\n        norm_num\n      | inr h\u2083 =>\n        have h\u2084 : x = -14 := by linarith\n        rw [h\u2084]\n        norm_num\n    \u00b7 intro h\n      cases h with\n      | inl h =>\n        rw [h]\n        norm_num\n      | inr h =>\n        rw [h]\n        norm_num\n  have h\u2082 : (\u2211 k in S, k) = -6 := by\n    rw [h\u2081]\n    norm_num [Finset.sum_pair (show (8 : \u211d) \u2260 -14 by norm_num)]\n    <;>\n    simp_all [Finset.sum_singleton]\n    <;>\n    norm_num\n    <;>\n    linarith\n  apply h\u2082\n", "proof_before": "theorem mathd_algebra_215 (S : Finset \u211d) (h\u2080 : \u2200 x : \u211d, x \u2208 S \u2194 (x + 3) ^ 2 = 121) :\n    (\u2211 k in S, k) = -6  :=  by\n  have h\u2081 : S = ({8, -14} : Finset \u211d) := by\n    apply Finset.ext\n    intro x\n    simp only [Finset.mem_insert, Finset.mem_singleton, h\u2080]\n    constructor\n    \u00b7 intro h\n      have h\u2082 : (x + 3) ^ 2 = 121 := by exact h\n      have h\u2083 : x + 3 = 11 \u2228 x + 3 = -11 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2084\n        apply eq_of_sub_eq_zero\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2084)\n        nlinarith\n      cases h\u2083 with\n      | inl h\u2083 =>\n        have h\u2084 : x = 8 := by linarith\n        rw [h\u2084]\n        norm_num\n      | inr h\u2083 =>\n        have h\u2084 : x = -14 := by linarith\n        rw [h\u2084]\n        norm_num\n    \u00b7 intro h\n      cases h with\n      | inl h =>\n        rw [h]\n        norm_num\n      | inr h =>\n        rw [h]\n        norm_num\n  have h\u2082 : (\u2211 k in S, k) = -6 := by\n    rw [h\u2081]\n    norm_num [Finset.sum_pair (show (8 : \u211d) \u2260 -14 by norm_num)]\n    <;>\n    simp_all [Finset.sum_singleton]\n    <;>\n    norm_num\n    <;>\n    linarith\n  apply h\u2082\n"}
{"mode": "clear", "steps_before": 42, "alphabet": 29, "compiles": true, "steps_after": 42, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 1301, "chars_after": 1301, "seconds": 290.7, "id": "algebra_sqineq_unitcircatbpabsamblt1", "proof_after": "theorem algebra_sqineq_unitcircatbpabsamblt1 (a b : \u211d) (h\u2080 : a ^ 2 + b ^ 2 = 1) :\n    a * b + abs (a - b) \u2264 1  :=  by\n  have h\u2081 : a \u2264 1 := by\n    nlinarith [sq_nonneg (a - 1), sq_nonneg b]\n    <;> nlinarith\n  have h\u2082 : a \u2265 -1 := by\n    nlinarith [sq_nonneg (a + 1), sq_nonneg b]\n  have h\u2083 : b \u2264 1 := by\n    nlinarith [sq_nonneg (b - 1), sq_nonneg a]\n  have h\u2084 : b \u2265 -1 := by\n    nlinarith [sq_nonneg (b + 1), sq_nonneg a]\n  have h\u2085 : a \u2265 b \u2192 a * b + abs (a - b) \u2264 1 := by\n    intro h\u2085\n    have h\u2085\u2081 : abs (a - b) = a - b := by\n      rw [abs_of_nonneg (sub_nonneg.mpr h\u2085)]\n    rw [h\u2085\u2081]\n    have h\u2085\u2082 : a * b + (a - b) \u2264 1 := by\n      have h\u2085\u2083 : (a - 1) * (b + 1) \u2264 0 := by\n        nlinarith [sq_nonneg (a - 1), sq_nonneg (b + 1)]\n      nlinarith [sq_nonneg (a - 1), sq_nonneg (b + 1)]\n    linarith\n  have h\u2086 : a < b \u2192 a * b + abs (a - b) \u2264 1 := by\n    intro h\u2086\n    have h\u2086\u2081 : abs (a - b) = b - a := by\n      rw [abs_of_neg (by linarith)]\n      <;> linarith\n    rw [h\u2086\u2081]\n    have h\u2086\u2082 : a * b + (b - a) \u2264 1 := by\n      have h\u2086\u2083 : (a + 1) * (b - 1) \u2264 0 := by\n        nlinarith [sq_nonneg (a + 1), sq_nonneg (b - 1)]\n      nlinarith [sq_nonneg (a + 1), sq_nonneg (b - 1)]\n    linarith\n  have h\u2087 : a * b + abs (a - b) \u2264 1 := by\n    by_cases h\u2087 : a \u2265 b\n    \u00b7 \n      exact h\u2085 h\u2087\n    \u00b7 \n      have h\u2087\u2081 : a < b := by\n        by_contra h\u2087\u2081\n        have h\u2087\u2082 : a \u2265 b := by linarith\n        contradiction\n      exact h\u2086 h\u2087\u2081\n  exact h\u2087\n", "proof_before": "theorem algebra_sqineq_unitcircatbpabsamblt1 (a b : \u211d) (h\u2080 : a ^ 2 + b ^ 2 = 1) :\n    a * b + abs (a - b) \u2264 1  :=  by\n  have h\u2081 : a \u2264 1 := by\n    nlinarith [sq_nonneg (a - 1), sq_nonneg b]\n    <;> nlinarith\n  have h\u2082 : a \u2265 -1 := by\n    nlinarith [sq_nonneg (a + 1), sq_nonneg b]\n  have h\u2083 : b \u2264 1 := by\n    nlinarith [sq_nonneg (b - 1), sq_nonneg a]\n  have h\u2084 : b \u2265 -1 := by\n    nlinarith [sq_nonneg (b + 1), sq_nonneg a]\n  have h\u2085 : a \u2265 b \u2192 a * b + abs (a - b) \u2264 1 := by\n    intro h\u2085\n    have h\u2085\u2081 : abs (a - b) = a - b := by\n      rw [abs_of_nonneg (sub_nonneg.mpr h\u2085)]\n    rw [h\u2085\u2081]\n    have h\u2085\u2082 : a * b + (a - b) \u2264 1 := by\n      have h\u2085\u2083 : (a - 1) * (b + 1) \u2264 0 := by\n        nlinarith [sq_nonneg (a - 1), sq_nonneg (b + 1)]\n      nlinarith [sq_nonneg (a - 1), sq_nonneg (b + 1)]\n    linarith\n  have h\u2086 : a < b \u2192 a * b + abs (a - b) \u2264 1 := by\n    intro h\u2086\n    have h\u2086\u2081 : abs (a - b) = b - a := by\n      rw [abs_of_neg (by linarith)]\n      <;> linarith\n    rw [h\u2086\u2081]\n    have h\u2086\u2082 : a * b + (b - a) \u2264 1 := by\n      have h\u2086\u2083 : (a + 1) * (b - 1) \u2264 0 := by\n        nlinarith [sq_nonneg (a + 1), sq_nonneg (b - 1)]\n      nlinarith [sq_nonneg (a + 1), sq_nonneg (b - 1)]\n    linarith\n  have h\u2087 : a * b + abs (a - b) \u2264 1 := by\n    by_cases h\u2087 : a \u2265 b\n    \u00b7 \n      exact h\u2085 h\u2087\n    \u00b7 \n      have h\u2087\u2081 : a < b := by\n        by_contra h\u2087\u2081\n        have h\u2087\u2082 : a \u2265 b := by linarith\n        contradiction\n      exact h\u2086 h\u2087\u2081\n  exact h\u2087\n"}
{"mode": "clear", "steps_before": 25, "alphabet": 19, "compiles": false, "id": "mathd_numbertheory_34", "proof_after": "theorem mathd_numbertheory_34 (x : \u2115) (h\u2080 : x < 100) (h\u2081 : x * 9 % 100 = 1) : x = 89  :=  by\n  have h\u2082 : x = 89 := by\n    interval_cases x <;> norm_num at h\u2081 \u22a2 <;>\n    (try omega) <;>\n    (try contradiction) <;>\n    (try ring_nf at h\u2081 \u22a2) <;>\n    (try norm_num at h\u2081 \u22a2) <;>\n    (try omega)\n    <;>\n    (try\n      {\n        omega\n      })\n    <;>\n    (try\n      {\n        norm_num at h\u2081 \u22a2\n        <;> omega\n      })\n    <;>\n    (try\n      {\n        ring_nf at h\u2081 \u22a2\n        <;> norm_num at h\u2081 \u22a2\n        <;> omega\n      })\n    <;>\n    (try\n      {\n        omega\n      })\n  exact h\u2082\n", "proof_before": "theorem mathd_numbertheory_34 (x : \u2115) (h\u2080 : x < 100) (h\u2081 : x * 9 % 100 = 1) : x = 89  :=  by\n  have h\u2082 : x = 89 := by\n    interval_cases x <;> norm_num at h\u2081 \u22a2 <;>\n    (try omega) <;>\n    (try contradiction) <;>\n    (try ring_nf at h\u2081 \u22a2) <;>\n    (try norm_num at h\u2081 \u22a2) <;>\n    (try omega)\n    <;>\n    (try\n      {\n        omega\n      })\n    <;>\n    (try\n      {\n        norm_num at h\u2081 \u22a2\n        <;> omega\n      })\n    <;>\n    (try\n      {\n        ring_nf at h\u2081 \u22a2\n        <;> norm_num at h\u2081 \u22a2\n        <;> omega\n      })\n    <;>\n    (try\n      {\n        omega\n      })\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 75, "alphabet": 37, "compiles": false, "id": "aime_1984_p1", "proof_after": "theorem aime_1984_p1 (u : \u2115 \u2192 \u211a) (h\u2080 : \u2200 n, u (n + 1) = u n + 1)\n    (h\u2081 : (\u2211 k in Finset.range 98, u k.succ) = 137) :\n    (\u2211 k in Finset.range 49, u (2 * k.succ)) = 93  :=  by\n  have h\u2082 : \u2200 n : \u2115, u n = u 0 + n := by\n    intro n\n    induction n with\n    | zero =>\n      norm_num\n    | succ n ih =>\n      have h\u2083 := h\u2080 n\n      have h\u2084 := h\u2080 (n + 1)\n      simp [ih, Nat.cast_add, Nat.cast_one, Nat.cast_zero, add_assoc] at h\u2083 h\u2084 \u22a2\n      <;> linarith\n  have h\u2083 : u 0 = (-2357 : \u211a) / 49 := by\n    have h\u2084 : (\u2211 k in Finset.range 98, u k.succ) = 137 := h\u2081\n    have h\u2085 : (\u2211 k in Finset.range 98, u k.succ) = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := by\n      apply Finset.sum_congr rfl\n      intro k _\n      rw [h\u2082]\n      <;> simp [Nat.cast_add, Nat.cast_one, Nat.cast_zero, add_assoc]\n      <;> ring_nf\n      <;> norm_num\n      <;> field_simp\n      <;> ring_nf\n    rw [h\u2085] at h\u2084\n    have h\u2086 : \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) = 98 * u 0 + \u2211 k in Finset.range 98, (k + 1 : \u211a) := by\n      calc\n        _ = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := rfl\n        _ = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := rfl\n        _ = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := rfl\n        _ = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := rfl\n        _ = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := rfl\n        _ = \u2211 k in Finset.range 98, (u 0 : \u211a) + \u2211 k in Finset.range 98, (k + 1 : \u211a) := by\n          rw [Finset.sum_add_distrib]\n        _ = 98 * u 0 + \u2211 k in Finset.range 98, (k + 1 : \u211a) := by\n          simp [Finset.sum_const, Finset.card_range]\n          <;> ring\n          <;> field_simp\n          <;> ring\n    rw [h\u2086] at h\u2084\n    have h\u2087 : \u2211 k in Finset.range 98, (k + 1 : \u211a) = (4851 : \u211a) := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ]\n      <;> rfl\n    rw [h\u2087] at h\u2084\n    have h\u2088 : (98 : \u211a) * u 0 + 4851 = 137 := by\n      linarith\n    have h\u2089 : u 0 = (-2357 : \u211a) / 49 := by\n      linarith\n    exact h\u2089\n  have h\u2084 : (\u2211 k in Finset.range 49, u (2 * k.succ)) = 93 := by\n    have h\u2085 : (\u2211 k in Finset.range 49, u (2 * k.succ)) = \u2211 k in Finset.range 49, (u 0 + (2 * (k + 1) : \u211a)) := by\n      apply Finset.sum_congr rfl\n      intro k _\n      rw [h\u2082]\n      <;> simp [Nat.cast_add, Nat.cast_one, Nat.cast_mul, Nat.cast_zero, Nat.cast_succ, add_assoc]\n      <;> ring_nf\n      <;> field_simp\n      <;> ring_nf\n      <;> norm_num\n    rw [h\u2085]\n    have h\u2086 : \u2211 k in Finset.range 49, (u 0 + (2 * (k + 1 : \u211a)) : \u211a) = 49 * u 0 + \u2211 k in Finset.range 49, (2 * (k + 1 : \u211a)) := by\n      calc\n        _ = \u2211 k in Finset.range 49, (u 0 + (2 * (k + 1 : \u211a)) : \u211a) := rfl\n        _ = \u2211 k in Finset.range 49, (u 0 : \u211a) + \u2211 k in Finset.range 49, (2 * (k + 1 : \u211a)) := by\n          rw [Finset.sum_add_distrib]\n        _ = 49 * u 0 + \u2211 k in Finset.range 49, (2 * (k + 1 : \u211a)) := by\n          simp [Finset.sum_const, Finset.card_range]\n          <;> ring\n    rw [h\u2086]\n    have h\u2087 : \u2211 k in Finset.range 49, (2 * (k + 1 : \u211a) : \u211a) = (2450 : \u211a) := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ]\n      <;> rfl\n    rw [h\u2087]\n    have h\u2088 : (49 : \u211a) * u 0 + 2450 = 93 := by\n      rw [h\u2083]\n      norm_num\n    linarith\n  exact h\u2084\n", "proof_before": "theorem aime_1984_p1 (u : \u2115 \u2192 \u211a) (h\u2080 : \u2200 n, u (n + 1) = u n + 1)\n    (h\u2081 : (\u2211 k in Finset.range 98, u k.succ) = 137) :\n    (\u2211 k in Finset.range 49, u (2 * k.succ)) = 93  :=  by\n  have h\u2082 : \u2200 n : \u2115, u n = u 0 + n := by\n    intro n\n    induction n with\n    | zero =>\n      norm_num\n    | succ n ih =>\n      have h\u2083 := h\u2080 n\n      have h\u2084 := h\u2080 (n + 1)\n      simp [ih, Nat.cast_add, Nat.cast_one, Nat.cast_zero, add_assoc] at h\u2083 h\u2084 \u22a2\n      <;> linarith\n  have h\u2083 : u 0 = (-2357 : \u211a) / 49 := by\n    have h\u2084 : (\u2211 k in Finset.range 98, u k.succ) = 137 := h\u2081\n    have h\u2085 : (\u2211 k in Finset.range 98, u k.succ) = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := by\n      apply Finset.sum_congr rfl\n      intro k _\n      rw [h\u2082]\n      <;> simp [Nat.cast_add, Nat.cast_one, Nat.cast_zero, add_assoc]\n      <;> ring_nf\n      <;> norm_num\n      <;> field_simp\n      <;> ring_nf\n    rw [h\u2085] at h\u2084\n    have h\u2086 : \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) = 98 * u 0 + \u2211 k in Finset.range 98, (k + 1 : \u211a) := by\n      calc\n        _ = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := rfl\n        _ = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := rfl\n        _ = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := rfl\n        _ = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := rfl\n        _ = \u2211 k in Finset.range 98, (u 0 + (k + 1 : \u211a)) := rfl\n        _ = \u2211 k in Finset.range 98, (u 0 : \u211a) + \u2211 k in Finset.range 98, (k + 1 : \u211a) := by\n          rw [Finset.sum_add_distrib]\n        _ = 98 * u 0 + \u2211 k in Finset.range 98, (k + 1 : \u211a) := by\n          simp [Finset.sum_const, Finset.card_range]\n          <;> ring\n          <;> field_simp\n          <;> ring\n    rw [h\u2086] at h\u2084\n    have h\u2087 : \u2211 k in Finset.range 98, (k + 1 : \u211a) = (4851 : \u211a) := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ]\n      <;> rfl\n    rw [h\u2087] at h\u2084\n    have h\u2088 : (98 : \u211a) * u 0 + 4851 = 137 := by\n      linarith\n    have h\u2089 : u 0 = (-2357 : \u211a) / 49 := by\n      linarith\n    exact h\u2089\n  have h\u2084 : (\u2211 k in Finset.range 49, u (2 * k.succ)) = 93 := by\n    have h\u2085 : (\u2211 k in Finset.range 49, u (2 * k.succ)) = \u2211 k in Finset.range 49, (u 0 + (2 * (k + 1) : \u211a)) := by\n      apply Finset.sum_congr rfl\n      intro k _\n      rw [h\u2082]\n      <;> simp [Nat.cast_add, Nat.cast_one, Nat.cast_mul, Nat.cast_zero, Nat.cast_succ, add_assoc]\n      <;> ring_nf\n      <;> field_simp\n      <;> ring_nf\n      <;> norm_num\n    rw [h\u2085]\n    have h\u2086 : \u2211 k in Finset.range 49, (u 0 + (2 * (k + 1 : \u211a)) : \u211a) = 49 * u 0 + \u2211 k in Finset.range 49, (2 * (k + 1 : \u211a)) := by\n      calc\n        _ = \u2211 k in Finset.range 49, (u 0 + (2 * (k + 1 : \u211a)) : \u211a) := rfl\n        _ = \u2211 k in Finset.range 49, (u 0 : \u211a) + \u2211 k in Finset.range 49, (2 * (k + 1 : \u211a)) := by\n          rw [Finset.sum_add_distrib]\n        _ = 49 * u 0 + \u2211 k in Finset.range 49, (2 * (k + 1 : \u211a)) := by\n          simp [Finset.sum_const, Finset.card_range]\n          <;> ring\n    rw [h\u2086]\n    have h\u2087 : \u2211 k in Finset.range 49, (2 * (k + 1 : \u211a) : \u211a) = (2450 : \u211a) := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ]\n      <;> rfl\n    rw [h\u2087]\n    have h\u2088 : (49 : \u211a) * u 0 + 2450 = 93 := by\n      rw [h\u2083]\n      norm_num\n    linarith\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 88, "alphabet": 26, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 2295, "chars_after": 12, "seconds": 80.7, "id": "amc12b_2002_p19", "proof_after": "theorem amc12b_2002_p19 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) (h\u2081 : a * (b + c) = 152)\n    (h\u2082 : b * (c + a) = 162) (h\u2083 : c * (a + b) = 170) : a * b * c = 720  :=  by\n  nlinarith\n", "proof_before": "theorem amc12b_2002_p19 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) (h\u2081 : a * (b + c) = 152)\n    (h\u2082 : b * (c + a) = 162) (h\u2083 : c * (a + b) = 170) : a * b * c = 720  :=  by\n  have h\u2084 : a * b + a * c + b * c = 242 := by\n    have h\u2084\u2081 : a * b + a * c + b * c = 242 := by\n      have h\u2084\u2082 : a * b + a * c = 152 := by\n        have h\u2084\u2083 : a * b + a * c = a * (b + c) := by ring\n        rw [h\u2084\u2083]\n        linarith\n      have h\u2084\u2084 : b * c + b * a = 162 := by\n        have h\u2084\u2085 : b * c + b * a = b * (c + a) := by ring\n        rw [h\u2084\u2085]\n        linarith\n      have h\u2084\u2086 : c * a + c * b = 170 := by\n        have h\u2084\u2087 : c * a + c * b = c * (a + b) := by ring\n        rw [h\u2084\u2087]\n        linarith\n      have h\u2084\u2088 : 2 * (a * b + a * c + b * c) = 484 := by\n        nlinarith\n      nlinarith\n    exact h\u2084\u2081\n  have h\u2085 : a * b = 72 := by\n    have h\u2085\u2081 : a * b + a * c = 152 := by\n      have h\u2085\u2082 : a * b + a * c = a * (b + c) := by ring\n      rw [h\u2085\u2082]\n      linarith\n    have h\u2085\u2083 : a * c = 80 := by\n      nlinarith\n    have h\u2085\u2084 : a * b = 72 := by\n      nlinarith\n    exact h\u2085\u2084\n  have h\u2086 : a * c = 80 := by\n    have h\u2086\u2081 : a * b + a * c = 152 := by\n      have h\u2086\u2082 : a * b + a * c = a * (b + c) := by ring\n      rw [h\u2086\u2082]\n      linarith\n    have h\u2086\u2083 : b * c + b * a = 162 := by\n      have h\u2086\u2084 : b * c + b * a = b * (c + a) := by ring\n      rw [h\u2086\u2084]\n      linarith\n    nlinarith\n  have h\u2087 : b * c = 90 := by\n    have h\u2087\u2081 : a * b + a * c = 152 := by\n      have h\u2087\u2082 : a * b + a * c = a * (b + c) := by ring\n      rw [h\u2087\u2082]\n      linarith\n    have h\u2087\u2083 : b * c + b * a = 162 := by\n      have h\u2087\u2084 : b * c + b * a = b * (c + a) := by ring\n      rw [h\u2087\u2084]\n      linarith\n    have h\u2087\u2085 : c * a + c * b = 170 := by\n      have h\u2087\u2086 : c * a + c * b = c * (a + b) := by ring\n      rw [h\u2087\u2086]\n      linarith\n    nlinarith\n  have h\u2088 : a * b * c = 720 := by\n    have h\u2088\u2081 : 0 < a * b := by\n      nlinarith\n    have h\u2088\u2082 : 0 < a * c := by\n      nlinarith\n    have h\u2088\u2083 : 0 < b * c := by\n      nlinarith\n    have h\u2088\u2084 : 0 < a * b * c := by\n      nlinarith\n    have h\u2088\u2085 : (a * b * c) ^ 2 = 518400 := by\n      calc\n        (a * b * c) ^ 2 = a * b * c * (a * b * c) := by ring\n        _ = (a * b) * (a * c) * (b * c) := by ring\n        _ = 72 * 80 * 90 := by\n          rw [h\u2085, h\u2086, h\u2087]\n          <;> ring\n        _ = 518400 := by norm_num\n    have h\u2088\u2086 : a * b * c = 720 := by\n      have h\u2088\u2087 : a * b * c > 0 := by nlinarith\n      nlinarith [sq_nonneg (a * b * c - 720), sq_nonneg (a * b * c + 720)]\n    exact h\u2088\u2086\n  rw [h\u2088]\n  <;> norm_num\n"}
{"mode": "clear", "steps_before": 16, "alphabet": 18, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 390, "chars_after": 16, "seconds": 96.7, "id": "mathd_algebra_188", "proof_after": "theorem mathd_algebra_188 (\u03c3 : Equiv \u211d \u211d) (h : \u03c3.1 2 = \u03c3.2 2) : \u03c3.1 (\u03c3.1 2) = 2  :=  by\n  rw [h]\n  simp\n", "proof_before": "theorem mathd_algebra_188 (\u03c3 : Equiv \u211d \u211d) (h : \u03c3.1 2 = \u03c3.2 2) : \u03c3.1 (\u03c3.1 2) = 2  :=  by\n  have h\u2081 : \u03c3.1 (\u03c3.1 2) = \u03c3.1 (\u03c3.2 2) := by\n    rw [h]\n    <;> simp [Equiv.toFun_as_coe, Equiv.invFun_as_coe]\n    <;>\n    aesop\n  have h\u2082 : \u03c3.1 (\u03c3.2 2) = 2 := by\n    have h\u2082\u2081 : \u03c3.1 (\u03c3.2 2) = 2 := by\n      apply Equiv.apply_symm_apply\n    exact h\u2082\u2081\n  have h\u2083 : \u03c3.1 (\u03c3.1 2) = 2 := by\n    rw [h\u2081]\n    rw [h\u2082]\n    <;> simp [Equiv.toFun_as_coe, Equiv.invFun_as_coe]\n    <;>\n    aesop\n  apply h\u2083\n"}
{"mode": "clear", "steps_before": 15, "alphabet": 16, "compiles": false, "id": "mathd_numbertheory_1124", "proof_after": "theorem mathd_numbertheory_1124 (n : \u2115) (h\u2080 : n \u2264 9) (h\u2081 : 18 \u2223 374 * 10 + n) : n = 4  :=  by\n  have h\u2082 : n = 4 := by\n    have h\u2083 : n = 0 \u2228 n = 1 \u2228 n = 2 \u2228 n = 3 \u2228 n = 4 \u2228 n = 5 \u2228 n = 6 \u2228 n = 7 \u2228 n = 8 \u2228 n = 9 := by\n      omega\n    rcases h\u2083 with (rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl)\n    <;> norm_num at h\u2081 \u22a2\n    <;>\n    (try omega) <;>\n    (try\n      {\n        norm_num at h\u2081\n        <;>\n        omega\n      }) <;>\n    (try\n      {\n        omega\n      })\n    <;>\n    (try\n      {\n        omega\n      })\n  exact h\u2082\n", "proof_before": "theorem mathd_numbertheory_1124 (n : \u2115) (h\u2080 : n \u2264 9) (h\u2081 : 18 \u2223 374 * 10 + n) : n = 4  :=  by\n  have h\u2082 : n = 4 := by\n    have h\u2083 : n = 0 \u2228 n = 1 \u2228 n = 2 \u2228 n = 3 \u2228 n = 4 \u2228 n = 5 \u2228 n = 6 \u2228 n = 7 \u2228 n = 8 \u2228 n = 9 := by\n      omega\n    rcases h\u2083 with (rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl)\n    <;> norm_num at h\u2081 \u22a2\n    <;>\n    (try omega) <;>\n    (try\n      {\n        norm_num at h\u2081\n        <;>\n        omega\n      }) <;>\n    (try\n      {\n        omega\n      })\n    <;>\n    (try\n      {\n        omega\n      })\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 70, "alphabet": 36, "compiles": false, "id": "amc12_2000_p6", "proof_after": "theorem amc12_2000_p6 (p q : \u2115) (h\u2080 : Nat.Prime p \u2227 Nat.Prime q) (h\u2081 : 4 \u2264 p \u2227 p \u2264 18)\n    (h\u2082 : 4 \u2264 q \u2227 q \u2264 18) : \u2191p * \u2191q - (\u2191p + \u2191q) \u2260 (194 : \u2115)  :=  by\n  have h\u2083 : p = 5 \u2228 p = 7 \u2228 p = 11 \u2228 p = 13 \u2228 p = 17 := by\n    have h\u2083\u2081 : p = 5 \u2228 p = 7 \u2228 p = 11 \u2228 p = 13 \u2228 p = 17 := by\n      have h\u2084 := h\u2080.1\n      have h\u2085 : p \u2264 18 := by linarith\n      have h\u2086 : 4 \u2264 p := by linarith\n      interval_cases p <;> norm_num [Nat.Prime] at h\u2084 \u22a2 <;> aesop\n    exact h\u2083\u2081\n    <;> aesop\n  have h\u2084 : q = 5 \u2228 q = 7 \u2228 q = 11 \u2228 q = 13 \u2228 q = 17 := by\n    have h\u2084\u2081 : q = 5 \u2228 q = 7 \u2228 q = 11 \u2228 q = 13 \u2228 q = 17 := by\n      have h\u2085 := h\u2080.2\n      have h\u2086 : q \u2264 18 := by linarith\n      have h\u2087 : 4 \u2264 q := by linarith\n      interval_cases q <;> norm_num [Nat.Prime] at h\u2085 \u22a2 <;> aesop\n    exact h\u2084\u2081\n    <;> aesop\n  have h\u2085 : p * q \u2265 p + q := by\n    have h\u2085\u2081 : p \u2265 5 := by\n      have h\u2085\u2082 := h\u2080.1.two_le\n      have h\u2085\u2083 := h\u2081.1\n      have h\u2085\u2084 : p \u2265 4 := by linarith\n      have h\u2085\u2085 : p \u2260 4 := by\n        intro h\n        rw [h] at h\u2080\n        norm_num [Nat.Prime] at h\u2080\n      omega\n    have h\u2085\u2082 : q \u2265 5 := by\n      have h\u2085\u2083 := h\u2080.2.two_le\n      have h\u2085\u2084 := h\u2082.1\n      have h\u2085\u2085 : q \u2265 4 := by linarith\n      have h\u2085\u2086 : q \u2260 4 := by\n        intro h\n        rw [h] at h\u2080\n        norm_num [Nat.Prime] at h\u2080\n      omega\n    have h\u2085\u2083 : p * q \u2265 p + q := by\n      nlinarith [mul_le_mul_of_nonneg_right h\u2085\u2081 (by linarith : (0 : \u2115) \u2264 q),\n        mul_le_mul_of_nonneg_left h\u2085\u2082 (by linarith : (0 : \u2115) \u2264 p)]\n    exact h\u2085\u2083\n  have h\u2086 : p * q - (p + q) \u2260 194 := by\n    have h\u2086\u2081 : p = 5 \u2228 p = 7 \u2228 p = 11 \u2228 p = 13 \u2228 p = 17 := h\u2083\n    have h\u2086\u2082 : q = 5 \u2228 q = 7 \u2228 q = 11 \u2228 q = 13 \u2228 q = 17 := h\u2084\n    rcases h\u2086\u2081 with (rfl | rfl | rfl | rfl | rfl) <;> rcases h\u2086\u2082 with (rfl | rfl | rfl | rfl | rfl) <;> norm_num at h\u2085 \u22a2 <;>\n      (try omega) <;> (try ring_nf at h\u2085 \u22a2 <;> omega)\n    <;>\n      (try omega)\n    <;>\n      (try ring_nf at h\u2085 \u22a2 <;> omega)\n  intro h\u2087\n  have h\u2088 : p * q - (p + q) \u2260 194 := h\u2086\n  have h\u2089 : \u2191p * \u2191q - (\u2191p + \u2191q) = (194 : \u2115) := by simpa [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.add_assoc] using h\u2087\n  have h\u2081\u2080 : p * q - (p + q) = 194 := by simpa [Nat.cast_add, Nat.cast_mul, Nat.cast_sub, h\u2085] using h\u2089\n  exact h\u2088 h\u2081\u2080\n", "proof_before": "theorem amc12_2000_p6 (p q : \u2115) (h\u2080 : Nat.Prime p \u2227 Nat.Prime q) (h\u2081 : 4 \u2264 p \u2227 p \u2264 18)\n    (h\u2082 : 4 \u2264 q \u2227 q \u2264 18) : \u2191p * \u2191q - (\u2191p + \u2191q) \u2260 (194 : \u2115)  :=  by\n  have h\u2083 : p = 5 \u2228 p = 7 \u2228 p = 11 \u2228 p = 13 \u2228 p = 17 := by\n    have h\u2083\u2081 : p = 5 \u2228 p = 7 \u2228 p = 11 \u2228 p = 13 \u2228 p = 17 := by\n      have h\u2084 := h\u2080.1\n      have h\u2085 : p \u2264 18 := by linarith\n      have h\u2086 : 4 \u2264 p := by linarith\n      interval_cases p <;> norm_num [Nat.Prime] at h\u2084 \u22a2 <;> aesop\n    exact h\u2083\u2081\n    <;> aesop\n  have h\u2084 : q = 5 \u2228 q = 7 \u2228 q = 11 \u2228 q = 13 \u2228 q = 17 := by\n    have h\u2084\u2081 : q = 5 \u2228 q = 7 \u2228 q = 11 \u2228 q = 13 \u2228 q = 17 := by\n      have h\u2085 := h\u2080.2\n      have h\u2086 : q \u2264 18 := by linarith\n      have h\u2087 : 4 \u2264 q := by linarith\n      interval_cases q <;> norm_num [Nat.Prime] at h\u2085 \u22a2 <;> aesop\n    exact h\u2084\u2081\n    <;> aesop\n  have h\u2085 : p * q \u2265 p + q := by\n    have h\u2085\u2081 : p \u2265 5 := by\n      have h\u2085\u2082 := h\u2080.1.two_le\n      have h\u2085\u2083 := h\u2081.1\n      have h\u2085\u2084 : p \u2265 4 := by linarith\n      have h\u2085\u2085 : p \u2260 4 := by\n        intro h\n        rw [h] at h\u2080\n        norm_num [Nat.Prime] at h\u2080\n      omega\n    have h\u2085\u2082 : q \u2265 5 := by\n      have h\u2085\u2083 := h\u2080.2.two_le\n      have h\u2085\u2084 := h\u2082.1\n      have h\u2085\u2085 : q \u2265 4 := by linarith\n      have h\u2085\u2086 : q \u2260 4 := by\n        intro h\n        rw [h] at h\u2080\n        norm_num [Nat.Prime] at h\u2080\n      omega\n    have h\u2085\u2083 : p * q \u2265 p + q := by\n      nlinarith [mul_le_mul_of_nonneg_right h\u2085\u2081 (by linarith : (0 : \u2115) \u2264 q),\n        mul_le_mul_of_nonneg_left h\u2085\u2082 (by linarith : (0 : \u2115) \u2264 p)]\n    exact h\u2085\u2083\n  have h\u2086 : p * q - (p + q) \u2260 194 := by\n    have h\u2086\u2081 : p = 5 \u2228 p = 7 \u2228 p = 11 \u2228 p = 13 \u2228 p = 17 := h\u2083\n    have h\u2086\u2082 : q = 5 \u2228 q = 7 \u2228 q = 11 \u2228 q = 13 \u2228 q = 17 := h\u2084\n    rcases h\u2086\u2081 with (rfl | rfl | rfl | rfl | rfl) <;> rcases h\u2086\u2082 with (rfl | rfl | rfl | rfl | rfl) <;> norm_num at h\u2085 \u22a2 <;>\n      (try omega) <;> (try ring_nf at h\u2085 \u22a2 <;> omega)\n    <;>\n      (try omega)\n    <;>\n      (try ring_nf at h\u2085 \u22a2 <;> omega)\n  intro h\u2087\n  have h\u2088 : p * q - (p + q) \u2260 194 := h\u2086\n  have h\u2089 : \u2191p * \u2191q - (\u2191p + \u2191q) = (194 : \u2115) := by simpa [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.add_assoc] using h\u2087\n  have h\u2081\u2080 : p * q - (p + q) = 194 := by simpa [Nat.cast_add, Nat.cast_mul, Nat.cast_sub, h\u2085] using h\u2089\n  exact h\u2088 h\u2081\u2080\n"}
{"mode": "clear", "steps_before": 89, "alphabet": 44, "compiles": false, "id": "amc12b_2021_p1", "proof_after": "theorem amc12b_2021_p1 (S : Finset \u2124) (h\u2080 : \u2200 x : \u2124, x \u2208 S \u2194 \u2191(abs x) < 3 * Real.pi) :\n    S.card = 19  :=  by\n  have h_pi_lb : (9 : \u211d) < 3 * Real.pi := by\n    have := Real.pi_gt_3141592\n    norm_num at this \u22a2\n    <;> linarith\n  have h_pi_ub : (3 : \u211d) * Real.pi < 10 := by\n    have := Real.pi_lt_3141593\n    norm_num at this \u22a2\n    <;> linarith\n  have h_S_eq : S = Finset.Icc (-9 : \u2124) 9 := by\n    apply Finset.ext\n    intro x\n    simp only [Finset.mem_Icc, h\u2080]\n    constructor\n    \u00b7 \n      intro h\n      have h\u2081 : (abs x : \u211d) < 3 * Real.pi := by exact_mod_cast h\n      have h\u2082 : -9 \u2264 (x : \u2124) := by\n        by_contra h\u2082\n        have h\u2083 : x \u2264 -10 := by\n          linarith\n        have h\u2084 : (abs x : \u211d) \u2265 10 := by\n          have h\u2085 : x \u2264 -10 := h\u2083\n          have h\u2086 : abs x \u2265 10 := by\n            cases' abs_cases x with h\u2087 h\u2087 <;>\n              (try omega) <;>\n              (try\n                {\n                  norm_cast at h\u2087 \u22a2 <;>\n                    omega\n                }) <;>\n              (try\n                {\n                  simp_all [abs_of_nonneg, abs_of_nonpos, le_of_lt]\n                  <;>\n                  omega\n                })\n          have h\u2087 : (abs x : \u211d) \u2265 10 := by\n            exact_mod_cast h\u2086\n          exact h\u2087\n        linarith\n      have h\u2083 : (x : \u2124) \u2264 9 := by\n        by_contra h\u2083\n        have h\u2084 : x \u2265 10 := by\n          linarith\n        have h\u2085 : (abs x : \u211d) \u2265 10 := by\n          have h\u2086 : x \u2265 10 := h\u2084\n          have h\u2087 : abs x \u2265 10 := by\n            cases' abs_cases x with h\u2088 h\u2088 <;>\n              (try omega) <;>\n              (try\n                {\n                  norm_cast at h\u2088 \u22a2 <;>\n                    omega\n                }) <;>\n              (try\n                {\n                  simp_all [abs_of_nonneg, abs_of_nonpos, le_of_lt]\n                  <;>\n                  omega\n                })\n          have h\u2088 : (abs x : \u211d) \u2265 10 := by\n            exact_mod_cast h\u2087\n          exact h\u2088\n        linarith\n      exact \u27e8h\u2082, h\u2083\u27e9\n    \u00b7 \n      intro h\n      have h\u2081 : -9 \u2264 (x : \u2124) := h.1\n      have h\u2082 : (x : \u2124) \u2264 9 := h.2\n      have h\u2083 : (abs x : \u211d) < 3 * Real.pi := by\n        have h\u2084 : abs x \u2264 9 := by\n          have h\u2085 : -9 \u2264 (x : \u2124) := h\u2081\n          have h\u2086 : (x : \u2124) \u2264 9 := h\u2082\n          have h\u2087 : abs x \u2264 9 := by\n            cases' abs_cases x with h\u2088 h\u2088 <;>\n              (try omega) <;>\n              (try\n                {\n                  norm_cast at h\u2088 \u22a2 <;>\n                    omega\n                }) <;>\n              (try\n                {\n                  simp_all [abs_of_nonneg, abs_of_nonpos, le_of_lt]\n                  <;>\n                  omega\n                })\n          exact h\u2087\n        have h\u2085 : (abs x : \u211d) \u2264 9 := by\n          exact_mod_cast h\u2084\n        have h\u2086 : (9 : \u211d) < 3 * Real.pi := h_pi_lb\n        linarith\n      exact_mod_cast h\u2083\n  have h_finset_card : (Finset.Icc (-9 : \u2124) 9).card = 19 := by\n    norm_num [Finset.Icc_self, Finset.card_empty]\n    <;> rfl\n  have h_main : S.card = 19 := by\n    rw [h_S_eq]\n    exact h_finset_card\n  exact h_main\n", "proof_before": "theorem amc12b_2021_p1 (S : Finset \u2124) (h\u2080 : \u2200 x : \u2124, x \u2208 S \u2194 \u2191(abs x) < 3 * Real.pi) :\n    S.card = 19  :=  by\n  have h_pi_lb : (9 : \u211d) < 3 * Real.pi := by\n    have := Real.pi_gt_3141592\n    norm_num at this \u22a2\n    <;> linarith\n  have h_pi_ub : (3 : \u211d) * Real.pi < 10 := by\n    have := Real.pi_lt_3141593\n    norm_num at this \u22a2\n    <;> linarith\n  have h_S_eq : S = Finset.Icc (-9 : \u2124) 9 := by\n    apply Finset.ext\n    intro x\n    simp only [Finset.mem_Icc, h\u2080]\n    constructor\n    \u00b7 \n      intro h\n      have h\u2081 : (abs x : \u211d) < 3 * Real.pi := by exact_mod_cast h\n      have h\u2082 : -9 \u2264 (x : \u2124) := by\n        by_contra h\u2082\n        have h\u2083 : x \u2264 -10 := by\n          linarith\n        have h\u2084 : (abs x : \u211d) \u2265 10 := by\n          have h\u2085 : x \u2264 -10 := h\u2083\n          have h\u2086 : abs x \u2265 10 := by\n            cases' abs_cases x with h\u2087 h\u2087 <;>\n              (try omega) <;>\n              (try\n                {\n                  norm_cast at h\u2087 \u22a2 <;>\n                    omega\n                }) <;>\n              (try\n                {\n                  simp_all [abs_of_nonneg, abs_of_nonpos, le_of_lt]\n                  <;>\n                  omega\n                })\n          have h\u2087 : (abs x : \u211d) \u2265 10 := by\n            exact_mod_cast h\u2086\n          exact h\u2087\n        linarith\n      have h\u2083 : (x : \u2124) \u2264 9 := by\n        by_contra h\u2083\n        have h\u2084 : x \u2265 10 := by\n          linarith\n        have h\u2085 : (abs x : \u211d) \u2265 10 := by\n          have h\u2086 : x \u2265 10 := h\u2084\n          have h\u2087 : abs x \u2265 10 := by\n            cases' abs_cases x with h\u2088 h\u2088 <;>\n              (try omega) <;>\n              (try\n                {\n                  norm_cast at h\u2088 \u22a2 <;>\n                    omega\n                }) <;>\n              (try\n                {\n                  simp_all [abs_of_nonneg, abs_of_nonpos, le_of_lt]\n                  <;>\n                  omega\n                })\n          have h\u2088 : (abs x : \u211d) \u2265 10 := by\n            exact_mod_cast h\u2087\n          exact h\u2088\n        linarith\n      exact \u27e8h\u2082, h\u2083\u27e9\n    \u00b7 \n      intro h\n      have h\u2081 : -9 \u2264 (x : \u2124) := h.1\n      have h\u2082 : (x : \u2124) \u2264 9 := h.2\n      have h\u2083 : (abs x : \u211d) < 3 * Real.pi := by\n        have h\u2084 : abs x \u2264 9 := by\n          have h\u2085 : -9 \u2264 (x : \u2124) := h\u2081\n          have h\u2086 : (x : \u2124) \u2264 9 := h\u2082\n          have h\u2087 : abs x \u2264 9 := by\n            cases' abs_cases x with h\u2088 h\u2088 <;>\n              (try omega) <;>\n              (try\n                {\n                  norm_cast at h\u2088 \u22a2 <;>\n                    omega\n                }) <;>\n              (try\n                {\n                  simp_all [abs_of_nonneg, abs_of_nonpos, le_of_lt]\n                  <;>\n                  omega\n                })\n          exact h\u2087\n        have h\u2085 : (abs x : \u211d) \u2264 9 := by\n          exact_mod_cast h\u2084\n        have h\u2086 : (9 : \u211d) < 3 * Real.pi := h_pi_lb\n        linarith\n      exact_mod_cast h\u2083\n  have h_finset_card : (Finset.Icc (-9 : \u2124) 9).card = 19 := by\n    norm_num [Finset.Icc_self, Finset.card_empty]\n    <;> rfl\n  have h_main : S.card = 19 := by\n    rw [h_S_eq]\n    exact h_finset_card\n  exact h_main\n"}
{"mode": "clear", "steps_before": 20, "alphabet": 19, "compiles": true, "steps_after": 11, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 2, "fixpoint": true, "chars_before": 559, "chars_after": 282, "seconds": 146.5, "id": "induction_12dvd4expnp1p20", "proof_after": "theorem induction_12dvd4expnp1p20 (n : \u2115) : 12 \u2223 4 ^ (n + 1) + 20  :=  by\n  have h\u2081 : (4 ^ (n + 1)) % 12 = 4 := by\n    have h\u2082 : \u2200 n : \u2115, (4 ^ (n + 1)) % 12 = 4 := by\n      intro n\n      induction n with\n      | zero => norm_num\n      | succ n ih =>\n        simp [pow_succ, Nat.mul_mod, ih, Nat.pow_mod]\n        <;> norm_num <;> omega\n    apply h\u2082\n  omega\n", "proof_before": "theorem induction_12dvd4expnp1p20 (n : \u2115) : 12 \u2223 4 ^ (n + 1) + 20  :=  by\n  have h\u2081 : (4 ^ (n + 1)) % 12 = 4 := by\n    have h\u2082 : \u2200 n : \u2115, (4 ^ (n + 1)) % 12 = 4 := by\n      intro n\n      induction n with\n      | zero => norm_num\n      | succ n ih =>\n        simp [pow_succ, Nat.mul_mod, ih, Nat.pow_mod]\n        <;> norm_num <;> omega\n    apply h\u2082\n  have h\u2082 : (4 ^ (n + 1) + 20) % 12 = 0 := by\n    have h\u2083 : (4 ^ (n + 1) + 20) % 12 = 0 := by\n      omega\n    exact h\u2083\n  have h\u2083 : 12 \u2223 4 ^ (n + 1) + 20 := by\n    have h\u2084 : (4 ^ (n + 1) + 20) % 12 = 0 := h\u2082\n    have h\u2085 : 12 \u2223 4 ^ (n + 1) + 20 := by\n      omega\n    exact h\u2085\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 38, "alphabet": 26, "compiles": true, "steps_after": 18, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 8, "fixpoint": true, "chars_before": 1160, "chars_after": 606, "seconds": 637.9, "id": "mathd_algebra_332", "proof_after": "theorem mathd_algebra_332 (x y : \u211d) (h\u2080 : (x + y) / 2 = 7) (h\u2081 : Real.sqrt (x * y) = Real.sqrt 19) :\n    x ^ 2 + y ^ 2 = 158  :=  by\n  have h\u2083 : x * y = 19 := by\n    have h\u2083\u2082 : x * y \u2265 0 := by\n      by_contra h\n      have h\u2083\u2084 : Real.sqrt (x * y) = 0 := by\n        rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n      have h\u2083\u2085 : Real.sqrt 19 > 0 := Real.sqrt_pos.mpr (by norm_num)\n      linarith\n    have h\u2083\u2083 : x * y = 19 := by\n      have h\u2083\u2085 : Real.sqrt (x * y) = Real.sqrt 19 := h\u2081\n      have h\u2083\u2086 : Real.sqrt (x * y) ^ 2 = Real.sqrt 19 ^ 2 := by rw [h\u2083\u2085]\n      have h\u2083\u2087 : x * y = 19 := by\n        nlinarith [Real.sq_sqrt (show 0 \u2264 x * y by linarith), Real.sq_sqrt (show 0 \u2264 (19 : \u211d) by norm_num)]\n      exact h\u2083\u2087\n    exact h\u2083\u2083\n  nlinarith\n", "proof_before": "theorem mathd_algebra_332 (x y : \u211d) (h\u2080 : (x + y) / 2 = 7) (h\u2081 : Real.sqrt (x * y) = Real.sqrt 19) :\n    x ^ 2 + y ^ 2 = 158  :=  by\n  have h\u2082 : x + y = 14 := by\n    have h\u2082\u2081 : (x + y) / 2 = 7 := h\u2080\n    have h\u2082\u2082 : x + y = 14 := by\n      linarith\n    exact h\u2082\u2082\n  have h\u2083 : x * y = 19 := by\n    have h\u2083\u2081 : Real.sqrt (x * y) = Real.sqrt 19 := h\u2081\n    have h\u2083\u2082 : x * y \u2265 0 := by\n      by_contra h\n      have h\u2083\u2083 : x * y < 0 := by linarith\n      have h\u2083\u2084 : Real.sqrt (x * y) = 0 := by\n        rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n      rw [h\u2083\u2084] at h\u2083\u2081\n      have h\u2083\u2085 : Real.sqrt 19 > 0 := Real.sqrt_pos.mpr (by norm_num)\n      have h\u2083\u2086 : (0 : \u211d) < Real.sqrt 19 := by positivity\n      norm_num at h\u2083\u2081 \u22a2\n      <;> linarith\n    have h\u2083\u2083 : x * y = 19 := by\n      have h\u2083\u2084 : Real.sqrt (x * y) = Real.sqrt 19 := h\u2081\n      have h\u2083\u2085 : Real.sqrt (x * y) = Real.sqrt 19 := h\u2081\n      have h\u2083\u2086 : Real.sqrt (x * y) ^ 2 = Real.sqrt 19 ^ 2 := by rw [h\u2083\u2085]\n      have h\u2083\u2087 : x * y = 19 := by\n        nlinarith [Real.sq_sqrt (show 0 \u2264 x * y by linarith), Real.sq_sqrt (show 0 \u2264 (19 : \u211d) by norm_num)]\n      exact h\u2083\u2087\n    exact h\u2083\u2083\n  have h\u2084 : x ^ 2 + y ^ 2 = 158 := by\n    have h\u2084\u2081 : x ^ 2 + y ^ 2 = (x + y) ^ 2 - 2 * (x * y) := by ring\n    rw [h\u2084\u2081]\n    rw [h\u2082, h\u2083]\n    <;> norm_num\n    <;> linarith\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 24, "alphabet": 20, "compiles": true, "steps_after": 5, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 4, "fixpoint": true, "chars_before": 631, "chars_after": 155, "seconds": 511.8, "id": "mathd_algebra_148", "proof_after": "theorem mathd_algebra_148 (c : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = c * x ^ 3 - 9 * x + 3)\n    (h\u2081 : f 2 = 9) : c = 3  :=  by\n  have h\u2082 : c * (2 : \u211d) ^ 3 - 9 * (2 : \u211d) + 3 = 9 := by\n    have h\u2082\u2081 : f 2 = c * (2 : \u211d) ^ 3 - 9 * (2 : \u211d) + 3 := by\n      rw [h\u2080]\n    linarith\n  linarith\n", "proof_before": "theorem mathd_algebra_148 (c : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = c * x ^ 3 - 9 * x + 3)\n    (h\u2081 : f 2 = 9) : c = 3  :=  by\n  have h\u2082 : c * (2 : \u211d) ^ 3 - 9 * (2 : \u211d) + 3 = 9 := by\n    have h\u2082\u2081 : f 2 = c * (2 : \u211d) ^ 3 - 9 * (2 : \u211d) + 3 := by\n      rw [h\u2080]\n      <;> norm_num\n    rw [h\u2082\u2081] at h\u2081\n    linarith\n  have h\u2083 : c * 8 - 18 + 3 = 9 := by\n    norm_num at h\u2082 \u22a2\n    <;> linarith\n  have h\u2084 : c * 8 - 15 = 9 := by\n    norm_num at h\u2083 \u22a2\n    <;> linarith\n  have h\u2085 : c * 8 = 24 := by\n    have h\u2085\u2081 : c * 8 - 15 = 9 := h\u2084\n    have h\u2085\u2082 : c * 8 = 24 := by linarith\n    exact h\u2085\u2082\n  have h\u2086 : c = 3 := by\n    have h\u2086\u2081 : c * 8 = 24 := h\u2085\n    have h\u2086\u2082 : c = 3 := by\n      apply mul_left_cancel\u2080 (show (8 : \u211d) \u2260 0 by norm_num)\n      linarith\n    exact h\u2086\u2082\n  apply h\u2086\n"}
{"mode": "clear", "steps_before": 206, "alphabet": 40, "compiles": false, "id": "amc12a_2009_p6", "proof_after": "theorem amc12a_2009_p6 (m n p q : \u211d) (h\u2080 : p = 2 ^ m) (h\u2081 : q = 3 ^ n) :\n    p ^ (2 * n) * q ^ m = 12 ^ (m * n)  :=  by\n  have h\u2082 : p ^ (2 * n) = (2 : \u211d) ^ (m * (2 * n)) := by\n    rw [h\u2080]\n    have h\u2082\u2081 : ((2 : \u211d) ^ m : \u211d) > 0 := by positivity\n    have h\u2082\u2082 : ((2 : \u211d) ^ m : \u211d) ^ (2 * n : \u211d) = (2 : \u211d) ^ (m * (2 * n)) := by\n      rw [\u2190 Real.rpow_mul] <;> ring_nf <;> norm_num <;> linarith\n    rw [h\u2082\u2082]\n    <;>\n    simp [h\u2080, h\u2081, Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul,\n      Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at * <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    norm_num at * <;>\n    linarith\n  have h\u2083 : q ^ m = (3 : \u211d) ^ (n * m) := by\n    rw [h\u2081]\n    have h\u2083\u2081 : ((3 : \u211d) ^ n : \u211d) > 0 := by positivity\n    have h\u2083\u2082 : ((3 : \u211d) ^ n : \u211d) ^ (m : \u211d) = (3 : \u211d) ^ (n * m : \u211d) := by\n      rw [\u2190 Real.rpow_mul] <;> ring_nf <;> norm_num <;> linarith\n    rw [h\u2083\u2082]\n    <;>\n    simp [h\u2080, h\u2081, Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul,\n      Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at * <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    norm_num at * <;>\n    linarith\n  have h\u2084 : p ^ (2 * n) * q ^ m = (2 : \u211d) ^ (m * (2 * n)) * (3 : \u211d) ^ (n * m) := by\n    rw [h\u2082, h\u2083]\n    <;>\n    simp [mul_assoc]\n    <;>\n    ring_nf\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul,\n      Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2085 : (2 : \u211d) ^ (m * (2 * n)) = (2 : \u211d) ^ (2 * (m * n)) := by\n    have h\u2085\u2081 : m * (2 * n) = 2 * (m * n) := by\n      ring\n    rw [h\u2085\u2081]\n    <;>\n    simp [mul_assoc]\n    <;>\n    ring_nf\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul,\n      Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2086 : (2 : \u211d) ^ (2 * (m * n)) = (4 : \u211d) ^ (m * n) := by\n    have h\u2086\u2081 : (2 : \u211d) ^ (2 * (m * n)) = (4 : \u211d) ^ (m * n) := by\n      have h\u2086\u2082 : (4 : \u211d) ^ (m * n) = (2 : \u211d) ^ (2 * (m * n)) := by\n        calc\n          (4 : \u211d) ^ (m * n) = (2 : \u211d) ^ (2 * (m * n)) := by\n            have h\u2086\u2083 : (4 : \u211d) = (2 : \u211d) ^ 2 := by norm_num\n            rw [h\u2086\u2083]\n            have h\u2086\u2084 : ((2 : \u211d) ^ 2 : \u211d) > 0 := by positivity\n            have h\u2086\u2085 : ((2 : \u211d) ^ 2 : \u211d) ^ (m * n : \u211d) = (2 : \u211d) ^ (2 * (m * n) : \u211d) := by\n              rw [\u2190 Real.rpow_nat_cast]\n              rw [\u2190 Real.rpow_mul] <;> ring_nf <;> norm_num <;> linarith\n            rw [h\u2086\u2085]\n            <;> simp [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n            <;> ring_nf at * <;> simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n            <;> norm_num at * <;> linarith\n          _ = (2 : \u211d) ^ (2 * (m * n)) := by rfl\n      rw [h\u2086\u2082]\n      <;> simp [mul_assoc]\n      <;> ring_nf\n      <;> simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n      <;> ring_nf at *\n      <;> linarith\n    rw [h\u2086\u2081]\n    <;> simp [mul_assoc]\n    <;> ring_nf\n    <;> simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;> ring_nf at *\n    <;> linarith\n  have h\u2087 : (2 : \u211d) ^ (m * (2 * n)) = (4 : \u211d) ^ (m * n) := by\n    calc\n      (2 : \u211d) ^ (m * (2 * n)) = (2 : \u211d) ^ (2 * (m * n)) := by rw [h\u2085]\n      _ = (4 : \u211d) ^ (m * n) := by rw [h\u2086]\n      _ = (4 : \u211d) ^ (m * n) := by rfl\n  have h\u2088 : p ^ (2 * n) * q ^ m = (4 : \u211d) ^ (m * n) * (3 : \u211d) ^ (n * m) := by\n    rw [h\u2084]\n    have h\u2088\u2081 : (2 : \u211d) ^ (m * (2 * n)) = (4 : \u211d) ^ (m * n) := by\n      rw [h\u2087]\n    rw [h\u2088\u2081]\n    <;>\n    ring_nf at *\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2089 : (3 : \u211d) ^ (n * m) = (3 : \u211d) ^ (m * n) := by\n    have h\u2089\u2081 : n * m = m * n := by ring\n    rw [h\u2089\u2081]\n    <;>\n    simp [mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2081\u2080 : p ^ (2 * n) * q ^ m = (4 : \u211d) ^ (m * n) * (3 : \u211d) ^ (m * n) := by\n    rw [h\u2088]\n    have h\u2081\u2080\u2081 : (3 : \u211d) ^ (n * m) = (3 : \u211d) ^ (m * n) := by\n      rw [h\u2089]\n    rw [h\u2081\u2080\u2081]\n    <;>\n    simp [mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2081\u2081 : (4 : \u211d) ^ (m * n) * (3 : \u211d) ^ (m * n) = (12 : \u211d) ^ (m * n) := by\n    have h\u2081\u2081\u2081 : (4 : \u211d) ^ (m * n) * (3 : \u211d) ^ (m * n) = (4 * 3 : \u211d) ^ (m * n) := by\n      have h\u2081\u2081\u2082 : 0 < (4 : \u211d) := by norm_num\n      have h\u2081\u2081\u2083 : 0 < (3 : \u211d) := by norm_num\n      have h\u2081\u2081\u2084 : 0 < (4 : \u211d) * (3 : \u211d) := by positivity\n      calc\n        (4 : \u211d) ^ (m * n) * (3 : \u211d) ^ (m * n) = ((4 : \u211d) * (3 : \u211d)) ^ (m * n) := by\n          rw [\u2190 Real.mul_rpow (by positivity) (by positivity)]\n          <;> ring_nf\n          <;>\n          simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n          <;>\n          ring_nf at *\n          <;>\n          linarith\n        _ = (4 * 3 : \u211d) ^ (m * n) := by ring_nf\n    rw [h\u2081\u2081\u2081]\n    <;> norm_num\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2081\u2082 : p ^ (2 * n) * q ^ m = (12 : \u211d) ^ (m * n) := by\n    rw [h\u2081\u2080]\n    rw [h\u2081\u2081]\n    <;>\n    simp [mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2081\u2083 : p ^ (2 * n) * q ^ m = 12 ^ (m * n) := by\n    simpa [h\u2081\u2082] using h\u2081\u2082\n  rw [h\u2081\u2083]\n  <;>\n  simp [mul_assoc]\n  <;>\n  ring_nf at *\n  <;>\n  simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n  <;>\n  ring_nf at *\n  <;>\n  linarith\n", "proof_before": "theorem amc12a_2009_p6 (m n p q : \u211d) (h\u2080 : p = 2 ^ m) (h\u2081 : q = 3 ^ n) :\n    p ^ (2 * n) * q ^ m = 12 ^ (m * n)  :=  by\n  have h\u2082 : p ^ (2 * n) = (2 : \u211d) ^ (m * (2 * n)) := by\n    rw [h\u2080]\n    have h\u2082\u2081 : ((2 : \u211d) ^ m : \u211d) > 0 := by positivity\n    have h\u2082\u2082 : ((2 : \u211d) ^ m : \u211d) ^ (2 * n : \u211d) = (2 : \u211d) ^ (m * (2 * n)) := by\n      rw [\u2190 Real.rpow_mul] <;> ring_nf <;> norm_num <;> linarith\n    rw [h\u2082\u2082]\n    <;>\n    simp [h\u2080, h\u2081, Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul,\n      Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at * <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    norm_num at * <;>\n    linarith\n  have h\u2083 : q ^ m = (3 : \u211d) ^ (n * m) := by\n    rw [h\u2081]\n    have h\u2083\u2081 : ((3 : \u211d) ^ n : \u211d) > 0 := by positivity\n    have h\u2083\u2082 : ((3 : \u211d) ^ n : \u211d) ^ (m : \u211d) = (3 : \u211d) ^ (n * m : \u211d) := by\n      rw [\u2190 Real.rpow_mul] <;> ring_nf <;> norm_num <;> linarith\n    rw [h\u2083\u2082]\n    <;>\n    simp [h\u2080, h\u2081, Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul,\n      Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at * <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    norm_num at * <;>\n    linarith\n  have h\u2084 : p ^ (2 * n) * q ^ m = (2 : \u211d) ^ (m * (2 * n)) * (3 : \u211d) ^ (n * m) := by\n    rw [h\u2082, h\u2083]\n    <;>\n    simp [mul_assoc]\n    <;>\n    ring_nf\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul,\n      Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2085 : (2 : \u211d) ^ (m * (2 * n)) = (2 : \u211d) ^ (2 * (m * n)) := by\n    have h\u2085\u2081 : m * (2 * n) = 2 * (m * n) := by\n      ring\n    rw [h\u2085\u2081]\n    <;>\n    simp [mul_assoc]\n    <;>\n    ring_nf\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul,\n      Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2086 : (2 : \u211d) ^ (2 * (m * n)) = (4 : \u211d) ^ (m * n) := by\n    have h\u2086\u2081 : (2 : \u211d) ^ (2 * (m * n)) = (4 : \u211d) ^ (m * n) := by\n      have h\u2086\u2082 : (4 : \u211d) ^ (m * n) = (2 : \u211d) ^ (2 * (m * n)) := by\n        calc\n          (4 : \u211d) ^ (m * n) = (2 : \u211d) ^ (2 * (m * n)) := by\n            have h\u2086\u2083 : (4 : \u211d) = (2 : \u211d) ^ 2 := by norm_num\n            rw [h\u2086\u2083]\n            have h\u2086\u2084 : ((2 : \u211d) ^ 2 : \u211d) > 0 := by positivity\n            have h\u2086\u2085 : ((2 : \u211d) ^ 2 : \u211d) ^ (m * n : \u211d) = (2 : \u211d) ^ (2 * (m * n) : \u211d) := by\n              rw [\u2190 Real.rpow_nat_cast]\n              rw [\u2190 Real.rpow_mul] <;> ring_nf <;> norm_num <;> linarith\n            rw [h\u2086\u2085]\n            <;> simp [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n            <;> ring_nf at * <;> simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n            <;> norm_num at * <;> linarith\n          _ = (2 : \u211d) ^ (2 * (m * n)) := by rfl\n      rw [h\u2086\u2082]\n      <;> simp [mul_assoc]\n      <;> ring_nf\n      <;> simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n      <;> ring_nf at *\n      <;> linarith\n    rw [h\u2086\u2081]\n    <;> simp [mul_assoc]\n    <;> ring_nf\n    <;> simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub, Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;> ring_nf at *\n    <;> linarith\n  have h\u2087 : (2 : \u211d) ^ (m * (2 * n)) = (4 : \u211d) ^ (m * n) := by\n    calc\n      (2 : \u211d) ^ (m * (2 * n)) = (2 : \u211d) ^ (2 * (m * n)) := by rw [h\u2085]\n      _ = (4 : \u211d) ^ (m * n) := by rw [h\u2086]\n      _ = (4 : \u211d) ^ (m * n) := by rfl\n  have h\u2088 : p ^ (2 * n) * q ^ m = (4 : \u211d) ^ (m * n) * (3 : \u211d) ^ (n * m) := by\n    rw [h\u2084]\n    have h\u2088\u2081 : (2 : \u211d) ^ (m * (2 * n)) = (4 : \u211d) ^ (m * n) := by\n      rw [h\u2087]\n    rw [h\u2088\u2081]\n    <;>\n    ring_nf at *\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2089 : (3 : \u211d) ^ (n * m) = (3 : \u211d) ^ (m * n) := by\n    have h\u2089\u2081 : n * m = m * n := by ring\n    rw [h\u2089\u2081]\n    <;>\n    simp [mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2081\u2080 : p ^ (2 * n) * q ^ m = (4 : \u211d) ^ (m * n) * (3 : \u211d) ^ (m * n) := by\n    rw [h\u2088]\n    have h\u2081\u2080\u2081 : (3 : \u211d) ^ (n * m) = (3 : \u211d) ^ (m * n) := by\n      rw [h\u2089]\n    rw [h\u2081\u2080\u2081]\n    <;>\n    simp [mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2081\u2081 : (4 : \u211d) ^ (m * n) * (3 : \u211d) ^ (m * n) = (12 : \u211d) ^ (m * n) := by\n    have h\u2081\u2081\u2081 : (4 : \u211d) ^ (m * n) * (3 : \u211d) ^ (m * n) = (4 * 3 : \u211d) ^ (m * n) := by\n      have h\u2081\u2081\u2082 : 0 < (4 : \u211d) := by norm_num\n      have h\u2081\u2081\u2083 : 0 < (3 : \u211d) := by norm_num\n      have h\u2081\u2081\u2084 : 0 < (4 : \u211d) * (3 : \u211d) := by positivity\n      calc\n        (4 : \u211d) ^ (m * n) * (3 : \u211d) ^ (m * n) = ((4 : \u211d) * (3 : \u211d)) ^ (m * n) := by\n          rw [\u2190 Real.mul_rpow (by positivity) (by positivity)]\n          <;> ring_nf\n          <;>\n          simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n          <;>\n          ring_nf at *\n          <;>\n          linarith\n        _ = (4 * 3 : \u211d) ^ (m * n) := by ring_nf\n    rw [h\u2081\u2081\u2081]\n    <;> norm_num\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2081\u2082 : p ^ (2 * n) * q ^ m = (12 : \u211d) ^ (m * n) := by\n    rw [h\u2081\u2080]\n    rw [h\u2081\u2081]\n    <;>\n    simp [mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n    <;>\n    ring_nf at *\n    <;>\n    linarith\n  have h\u2081\u2083 : p ^ (2 * n) * q ^ m = 12 ^ (m * n) := by\n    simpa [h\u2081\u2082] using h\u2081\u2082\n  rw [h\u2081\u2083]\n  <;>\n  simp [mul_assoc]\n  <;>\n  ring_nf at *\n  <;>\n  simp_all [Real.rpow_mul, Real.rpow_add, Real.rpow_neg, Real.rpow_sub]\n  <;>\n  ring_nf at *\n  <;>\n  linarith\n"}
{"mode": "clear", "steps_before": 70, "alphabet": 46, "compiles": true, "steps_after": 70, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 2507, "chars_after": 2507, "seconds": 464.1, "id": "imo_1968_p5_1", "proof_after": "theorem imo_1968_p5_1 (a : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : 0 < a)\n    (h\u2081 : \u2200 x, f (x + a) = 1 / 2 + Real.sqrt (f x - f x ^ 2)) : \u2203 b > 0, \u2200 x, f (x + b) = f x  :=  by\n  have h\u2082 : \u2200 x, f x \u2265 1 / 2 := by\n    intro x\n    have h\u2082\u2081 : f x = 1 / 2 + Real.sqrt (f (x - a) - f (x - a) ^ 2) := by\n      have h\u2082\u2082 := h\u2081 (x - a)\n      ring_nf at h\u2082\u2082 \u22a2\n      linarith\n    rw [h\u2082\u2081]\n    have h\u2082\u2082 : Real.sqrt (f (x - a) - f (x - a) ^ 2) \u2265 0 := by\n      apply Real.sqrt_nonneg\n    linarith\n  have h\u2083 : \u2200 x, f x \u2264 1 := by\n    intro x\n    by_contra h\n    have h\u2084 : f x > 1 := by linarith\n    have h\u2085 : f x = 1 / 2 + Real.sqrt (f (x - a) - f (x - a) ^ 2) := by\n      have h\u2085\u2081 := h\u2081 (x - a)\n      ring_nf at h\u2085\u2081 \u22a2\n      linarith\n    rw [h\u2085] at h\u2084\n    have h\u2086 : Real.sqrt (f (x - a) - f (x - a) ^ 2) \u2264 1 / 2 := by\n      have h\u2086\u2081 : f (x - a) - f (x - a) ^ 2 \u2264 1 / 4 := by\n        nlinarith [sq_nonneg (f (x - a) - 1 / 2), h\u2082 (x - a)]\n      have h\u2086\u2082 : Real.sqrt (f (x - a) - f (x - a) ^ 2) \u2264 1 / 2 := by\n        apply Real.sqrt_le_iff.mpr\n        constructor\n        \u00b7 nlinarith [h\u2082 (x - a)]\n        \u00b7 nlinarith [h\u2082 (x - a)]\n      exact h\u2086\u2082\n    linarith\n  have h\u2084 : \u2200 x, f (x + 2 * a) = f x := by\n    intro x\n    have h\u2084\u2081 : f (x + a) = 1 / 2 + Real.sqrt (f x - f x ^ 2) := by\n      apply h\u2081\n    have h\u2084\u2082 : f (x + 2 * a) = f ((x + a) + a) := by ring\n    rw [h\u2084\u2082]\n    have h\u2084\u2083 : f ((x + a) + a) = 1 / 2 + Real.sqrt (f (x + a) - f (x + a) ^ 2) := by\n      have h\u2084\u2084 := h\u2081 (x + a)\n      ring_nf at h\u2084\u2084 \u22a2\n      linarith\n    rw [h\u2084\u2083]\n    have h\u2084\u2085 : f (x + a) - f (x + a) ^ 2 = (f x - 1 / 2) ^ 2 := by\n      have h\u2084\u2085\u2081 : f (x + a) = 1 / 2 + Real.sqrt (f x - f x ^ 2) := by\n        apply h\u2081\n      rw [h\u2084\u2085\u2081]\n      have h\u2084\u2085\u2082 : f x - f x ^ 2 \u2265 0 := by\n        nlinarith [h\u2082 x, h\u2083 x, sq_nonneg (f x - 1 / 2), Real.sqrt_nonneg (f x - f x ^ 2)]\n      have h\u2084\u2085\u2083 : Real.sqrt (f x - f x ^ 2) \u2265 0 := Real.sqrt_nonneg (f x - f x ^ 2)\n      have h\u2084\u2085\u2084 : Real.sqrt (f x - f x ^ 2) ^ 2 = f x - f x ^ 2 := by\n        rw [Real.sq_sqrt] <;> nlinarith [h\u2082 x, h\u2083 x]\n      nlinarith [h\u2082 x, h\u2083 x, sq_nonneg (f x - 1 / 2), h\u2084\u2085\u2084]\n    have h\u2084\u2086 : Real.sqrt (f (x + a) - f (x + a) ^ 2) = f x - 1 / 2 := by\n      rw [h\u2084\u2085]\n      have h\u2084\u2086\u2081 : f x \u2265 1 / 2 := h\u2082 x\n      have h\u2084\u2086\u2082 : f x \u2264 1 := h\u2083 x\n      have h\u2084\u2086\u2083 : Real.sqrt ((f x - 1 / 2) ^ 2) = f x - 1 / 2 := by\n        rw [Real.sqrt_sq (by nlinarith [h\u2082 x, h\u2083 x])]\n      rw [h\u2084\u2086\u2083]\n    rw [h\u2084\u2086]\n    <;> ring_nf <;> nlinarith [h\u2082 x, h\u2083 x, h\u2082 (x + a), h\u2083 (x + a)]\n  have h\u2085 : \u2203 b > 0, \u2200 x, f (x + b) = f x := by\n    refine' \u27e82 * a, by linarith, _\u27e9\n    intro x\n    have h\u2085\u2081 : f (x + 2 * a) = f x := h\u2084 x\n    have h\u2085\u2082 : f (x + 2 * a) = f x := h\u2084 x\n    simpa [add_assoc] using h\u2085\u2081\n  exact h\u2085\n", "proof_before": "theorem imo_1968_p5_1 (a : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : 0 < a)\n    (h\u2081 : \u2200 x, f (x + a) = 1 / 2 + Real.sqrt (f x - f x ^ 2)) : \u2203 b > 0, \u2200 x, f (x + b) = f x  :=  by\n  have h\u2082 : \u2200 x, f x \u2265 1 / 2 := by\n    intro x\n    have h\u2082\u2081 : f x = 1 / 2 + Real.sqrt (f (x - a) - f (x - a) ^ 2) := by\n      have h\u2082\u2082 := h\u2081 (x - a)\n      ring_nf at h\u2082\u2082 \u22a2\n      linarith\n    rw [h\u2082\u2081]\n    have h\u2082\u2082 : Real.sqrt (f (x - a) - f (x - a) ^ 2) \u2265 0 := by\n      apply Real.sqrt_nonneg\n    linarith\n  have h\u2083 : \u2200 x, f x \u2264 1 := by\n    intro x\n    by_contra h\n    have h\u2084 : f x > 1 := by linarith\n    have h\u2085 : f x = 1 / 2 + Real.sqrt (f (x - a) - f (x - a) ^ 2) := by\n      have h\u2085\u2081 := h\u2081 (x - a)\n      ring_nf at h\u2085\u2081 \u22a2\n      linarith\n    rw [h\u2085] at h\u2084\n    have h\u2086 : Real.sqrt (f (x - a) - f (x - a) ^ 2) \u2264 1 / 2 := by\n      have h\u2086\u2081 : f (x - a) - f (x - a) ^ 2 \u2264 1 / 4 := by\n        nlinarith [sq_nonneg (f (x - a) - 1 / 2), h\u2082 (x - a)]\n      have h\u2086\u2082 : Real.sqrt (f (x - a) - f (x - a) ^ 2) \u2264 1 / 2 := by\n        apply Real.sqrt_le_iff.mpr\n        constructor\n        \u00b7 nlinarith [h\u2082 (x - a)]\n        \u00b7 nlinarith [h\u2082 (x - a)]\n      exact h\u2086\u2082\n    linarith\n  have h\u2084 : \u2200 x, f (x + 2 * a) = f x := by\n    intro x\n    have h\u2084\u2081 : f (x + a) = 1 / 2 + Real.sqrt (f x - f x ^ 2) := by\n      apply h\u2081\n    have h\u2084\u2082 : f (x + 2 * a) = f ((x + a) + a) := by ring\n    rw [h\u2084\u2082]\n    have h\u2084\u2083 : f ((x + a) + a) = 1 / 2 + Real.sqrt (f (x + a) - f (x + a) ^ 2) := by\n      have h\u2084\u2084 := h\u2081 (x + a)\n      ring_nf at h\u2084\u2084 \u22a2\n      linarith\n    rw [h\u2084\u2083]\n    have h\u2084\u2085 : f (x + a) - f (x + a) ^ 2 = (f x - 1 / 2) ^ 2 := by\n      have h\u2084\u2085\u2081 : f (x + a) = 1 / 2 + Real.sqrt (f x - f x ^ 2) := by\n        apply h\u2081\n      rw [h\u2084\u2085\u2081]\n      have h\u2084\u2085\u2082 : f x - f x ^ 2 \u2265 0 := by\n        nlinarith [h\u2082 x, h\u2083 x, sq_nonneg (f x - 1 / 2), Real.sqrt_nonneg (f x - f x ^ 2)]\n      have h\u2084\u2085\u2083 : Real.sqrt (f x - f x ^ 2) \u2265 0 := Real.sqrt_nonneg (f x - f x ^ 2)\n      have h\u2084\u2085\u2084 : Real.sqrt (f x - f x ^ 2) ^ 2 = f x - f x ^ 2 := by\n        rw [Real.sq_sqrt] <;> nlinarith [h\u2082 x, h\u2083 x]\n      nlinarith [h\u2082 x, h\u2083 x, sq_nonneg (f x - 1 / 2), h\u2084\u2085\u2084]\n    have h\u2084\u2086 : Real.sqrt (f (x + a) - f (x + a) ^ 2) = f x - 1 / 2 := by\n      rw [h\u2084\u2085]\n      have h\u2084\u2086\u2081 : f x \u2265 1 / 2 := h\u2082 x\n      have h\u2084\u2086\u2082 : f x \u2264 1 := h\u2083 x\n      have h\u2084\u2086\u2083 : Real.sqrt ((f x - 1 / 2) ^ 2) = f x - 1 / 2 := by\n        rw [Real.sqrt_sq (by nlinarith [h\u2082 x, h\u2083 x])]\n      rw [h\u2084\u2086\u2083]\n    rw [h\u2084\u2086]\n    <;> ring_nf <;> nlinarith [h\u2082 x, h\u2083 x, h\u2082 (x + a), h\u2083 (x + a)]\n  have h\u2085 : \u2203 b > 0, \u2200 x, f (x + b) = f x := by\n    refine' \u27e82 * a, by linarith, _\u27e9\n    intro x\n    have h\u2085\u2081 : f (x + 2 * a) = f x := h\u2084 x\n    have h\u2085\u2082 : f (x + 2 * a) = f x := h\u2084 x\n    simpa [add_assoc] using h\u2085\u2081\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 193, "alphabet": 85, "compiles": false, "id": "amc12a_2020_p15", "proof_after": "theorem amc12a_2020_p15 (a b : \u2102) (h\u2080 : a ^ 3 - 8 = 0) (h\u2081 : b ^ 3 - 8 * b ^ 2 - 8 * b + 64 = 0) :\n    Complex.abs (a - b) \u2264 2 * Real.sqrt 21  :=  by\n  have h\u2082 : Complex.abs a = 2 := by\n    have h\u2082\u2081 : a ^ 3 = 8 := by\n      rw [sub_eq_zero] at h\u2080\n      norm_num at h\u2080 \u22a2\n      <;> linear_combination h\u2080\n    have h\u2082\u2082 : Complex.abs (a ^ 3) = (8 : \u211d) := by\n      rw [h\u2082\u2081]\n      <;> simp [Complex.abs_of_nonneg] <;> norm_num\n    have h\u2082\u2083 : Complex.abs (a ^ 3) = (Complex.abs a) ^ 3 := by\n      simp [Complex.abs.map_pow]\n    rw [h\u2082\u2083] at h\u2082\u2082\n    have h\u2082\u2084 : (Complex.abs a : \u211d) ^ 3 = 8 := by\n      norm_num at h\u2082\u2082 \u22a2 <;>\n      linarith\n    have h\u2082\u2085 : Complex.abs a = 2 := by\n      have h\u2082\u2085\u2081 : Complex.abs a \u2265 0 := Complex.abs.nonneg _\n      have h\u2082\u2085\u2082 : (Complex.abs a : \u211d) ^ 3 = 8 := by exact_mod_cast h\u2082\u2084\n      have h\u2082\u2085\u2083 : Complex.abs a = 2 := by\n        nlinarith [sq_nonneg (Complex.abs a - 2), sq_nonneg (Complex.abs a + 2)]\n      exact h\u2082\u2085\u2083\n    exact h\u2082\u2085\n  have h\u2083 : b = 8 \u2228 b ^ 2 = 8 := by\n    have h\u2084 : b ^ 3 - 8 * b ^ 2 - 8 * b + 64 = 0 := h\u2081\n    have h\u2085 : (b - 8) * (b ^ 2 - 8) = 0 := by\n      ring_nf at h\u2084 \u22a2\n      <;> simp_all [Complex.ext_iff, pow_two, pow_three]\n      <;>\n      (try ring_nf at * ) <;>\n      (try norm_num at * ) <;>\n      (try\n        constructor <;>\n        nlinarith [sq_nonneg (b.re - 8), sq_nonneg (b.re + 8), sq_nonneg (b.im - 8),\n          sq_nonneg (b.im + 8), sq_nonneg (b.re - b.im), sq_nonneg (b.re + b.im),\n          sq_nonneg (b.re - 2), sq_nonneg (b.re + 2), sq_nonneg (b.im - 2),\n          sq_nonneg (b.im + 2)])\n    have h\u2086 : b - 8 = 0 \u2228 b ^ 2 - 8 = 0 := by\n      apply eq_zero_or_eq_zero_of_mul_eq_zero h\u2085\n    cases h\u2086 with\n    | inl h\u2086 =>\n      have h\u2087 : b = 8 := by\n        have h\u2088 : b - 8 = 0 := h\u2086\n        have h\u2089 : b = 8 := by\n          rw [sub_eq_zero] at h\u2088\n          exact h\u2088\n        exact h\u2089\n      exact Or.inl h\u2087\n    | inr h\u2086 =>\n      have h\u2087 : b ^ 2 = 8 := by\n        have h\u2088 : b ^ 2 - 8 = 0 := h\u2086\n        have h\u2089 : b ^ 2 = 8 := by\n          rw [sub_eq_zero] at h\u2088\n          exact h\u2088\n        exact h\u2089\n      exact Or.inr h\u2087\n  have h\u2084 : Complex.abs (a - b) \u2264 2 * Real.sqrt 21 := by\n    have h\u2085 : b = 8 \u2228 b ^ 2 = 8 := h\u2083\n    cases h\u2085 with\n    | inl h\u2085 =>\n      have h\u2085\u2081 : b = 8 := h\u2085\n      have h\u2085\u2082 : Complex.abs (a - b) \u2264 2 * Real.sqrt 21 := by\n        rw [h\u2085\u2081]\n        have h\u2085\u2083 : Complex.abs a = 2 := h\u2082\n        have h\u2085\u2084 : Complex.abs (a - 8) \u2264 2 * Real.sqrt 21 := by\n          have h\u2085\u2085 : a ^ 3 = 8 := by\n            rw [sub_eq_zero] at h\u2080\n            norm_num at h\u2080 \u22a2\n            <;> linear_combination h\u2080\n          have h\u2085\u2086 : a.re ^ 2 + a.im ^ 2 = 4 := by\n            have h\u2085\u2086\u2081 : Complex.abs a = 2 := h\u2082\n            have h\u2085\u2086\u2082 : Complex.abs a = Real.sqrt (a.re ^ 2 + a.im ^ 2) := by\n              simp [Complex.abs, Complex.normSq, Real.sqrt_eq_iff_sq_eq, sq, add_nonneg, mul_self_nonneg]\n              <;> ring_nf\n              <;> field_simp [Real.sqrt_eq_iff_sq_eq, sq, add_nonneg, mul_self_nonneg]\n              <;> ring_nf\n              <;> linarith\n            rw [h\u2085\u2086\u2082] at h\u2085\u2086\u2081\n            have h\u2085\u2086\u2083 : Real.sqrt (a.re ^ 2 + a.im ^ 2) = 2 := by linarith\n            have h\u2085\u2086\u2084 : a.re ^ 2 + a.im ^ 2 = 4 := by\n              have h\u2085\u2086\u2085 : Real.sqrt (a.re ^ 2 + a.im ^ 2) = 2 := h\u2085\u2086\u2083\n              have h\u2085\u2086\u2086 : a.re ^ 2 + a.im ^ 2 = 4 := by\n                have h\u2085\u2086\u2087 : Real.sqrt (a.re ^ 2 + a.im ^ 2) = 2 := h\u2085\u2086\u2085\n                have h\u2085\u2086\u2088 : a.re ^ 2 + a.im ^ 2 \u2265 0 := by positivity\n                have h\u2085\u2086\u2089 : Real.sqrt (a.re ^ 2 + a.im ^ 2) ^ 2 = a.re ^ 2 + a.im ^ 2 := by\n                  rw [Real.sq_sqrt] <;> positivity\n                nlinarith\n              exact h\u2085\u2086\u2086\n            exact h\u2085\u2086\u2084\n          have h\u2085\u2087 : a.re \u2265 -1 := by\n            by_contra h\n            have h\u2085\u2087\u2081 : a.re < -1 := by linarith\n            have h\u2085\u2087\u2082 : a.re ^ 2 > 1 := by nlinarith\n            have h\u2085\u2087\u2083 : a.im ^ 2 \u2264 4 := by nlinarith\n            have h\u2085\u2087\u2084 : a.re ^ 2 + a.im ^ 2 = 4 := h\u2085\u2086\n            have h\u2085\u2087\u2085 : a.re ^ 3 - 3 * a.re * a.im ^ 2 = 8 := by\n              simp [Complex.ext_iff, pow_three, pow_two, Complex.ext_iff, pow_three, pow_two] at h\u2085\u2085\n              norm_num at h\u2085\u2085 \u22a2\n              <;>\n              (try cases' h\u2085\u2085 with h\u2085\u2085_left h\u2085\u2085_right) <;>\n              (try nlinarith) <;>\n              (try ring_nf at * <;> nlinarith)\n            nlinarith [sq_nonneg (a.re - 2), sq_nonneg (a.im), sq_nonneg (a.re + 1)]\n          have h\u2085\u2088 : Complex.abs (a - 8) = Real.sqrt ((a.re - 8) ^ 2 + (a.im) ^ 2) := by\n            simp [Complex.abs, Complex.normSq, Complex.ext_iff, Real.sqrt_eq_iff_sq_eq, sq, sub_nonneg, add_nonneg, mul_self_nonneg,\n              Real.sqrt_nonneg, Real.sq_sqrt, add_nonneg, mul_self_nonneg]\n            <;> ring_nf <;> field_simp [Real.sqrt_eq_iff_sq_eq, sq, sub_nonneg, add_nonneg, mul_self_nonneg,\n              Real.sqrt_nonneg, Real.sq_sqrt, add_nonneg, mul_self_nonneg]\n            <;> ring_nf <;> nlinarith\n          rw [h\u2085\u2088]\n          have h\u2085\u2089 : Real.sqrt ((a.re - 8) ^ 2 + (a.im) ^ 2) \u2264 2 * Real.sqrt 21 := by\n            have h\u2085\u2089\u2081 : (a.re - 8) ^ 2 + (a.im) ^ 2 \u2264 84 := by\n              nlinarith [sq_nonneg (a.re + 1), sq_nonneg (a.im), sq_nonneg (a.re - 2), sq_nonneg (a.im - 2 * Real.sqrt 3),\n                sq_nonneg (a.im + 2 * Real.sqrt 3), Real.sqrt_nonneg 3, Real.sq_sqrt (show 0 \u2264 3 by norm_num)]\n            have h\u2085\u2089\u2082 : Real.sqrt ((a.re - 8) ^ 2 + (a.im) ^ 2) \u2264 Real.sqrt 84 := by\n              apply Real.sqrt_le_sqrt\n              nlinarith\n            have h\u2085\u2089\u2083 : Real.sqrt 84 = 2 * Real.sqrt 21 := by\n              have h\u2085\u2089\u2084 : Real.sqrt 84 = Real.sqrt (4 * 21) := by norm_num\n              rw [h\u2085\u2089\u2084]\n              have h\u2085\u2089\u2085 : Real.sqrt (4 * 21) = 2 * Real.sqrt 21 := by\n                rw [Real.sqrt_eq_iff_sq_eq (by positivity) (by positivity)]\n                nlinarith [Real.sqrt_nonneg 21, Real.sq_sqrt (show 0 \u2264 21 by norm_num)]\n              rw [h\u2085\u2089\u2085]\n            rw [h\u2085\u2089\u2083] at h\u2085\u2089\u2082\n            linarith\n          exact h\u2085\u2089\n        have h\u2085\u2085 : Complex.abs (a - (8 : \u2102)) \u2264 2 * Real.sqrt 21 := h\u2085\u2084\n        simpa [h\u2085\u2081] using h\u2085\u2085\n      exact h\u2085\u2082\n    | inr h\u2085 =>\n      have h\u2085\u2081 : b ^ 2 = 8 := h\u2085\n      have h\u2085\u2082 : Complex.abs (a - b) \u2264 2 * Real.sqrt 21 := by\n        have h\u2085\u2083 : Complex.abs b = 2 * Real.sqrt 2 := by\n          have h\u2085\u2084 : b ^ 2 = 8 := h\u2085\u2081\n          have h\u2085\u2085 : Complex.abs (b ^ 2) = (8 : \u211d) := by\n            rw [h\u2085\u2084]\n            <;> simp [Complex.abs_of_nonneg] <;> norm_num\n          have h\u2085\u2086 : Complex.abs (b ^ 2) = (Complex.abs b) ^ 2 := by\n            simp [Complex.abs.map_pow]\n          rw [h\u2085\u2086] at h\u2085\u2085\n          have h\u2085\u2087 : (Complex.abs b : \u211d) ^ 2 = 8 := by\n            norm_num at h\u2085\u2085 \u22a2 <;>\n            linarith\n          have h\u2085\u2088 : Complex.abs b = 2 * Real.sqrt 2 := by\n            have h\u2085\u2088\u2081 : Complex.abs b \u2265 0 := Complex.abs.nonneg _\n            have h\u2085\u2088\u2082 : (Complex.abs b : \u211d) ^ 2 = 8 := by exact_mod_cast h\u2085\u2087\n            have h\u2085\u2088\u2083 : Complex.abs b = 2 * Real.sqrt 2 := by\n              have h\u2085\u2088\u2084 : Real.sqrt 2 \u2265 0 := Real.sqrt_nonneg _\n              nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n            exact h\u2085\u2088\u2083\n          exact h\u2085\u2088\n        have h\u2085\u2084 : Complex.abs (a - b) \u2264 2 + 2 * Real.sqrt 2 := by\n          calc\n            Complex.abs (a - b) \u2264 Complex.abs a + Complex.abs b := by\n              calc\n                Complex.abs (a - b) = Complex.abs (a + (-b)) := by ring_nf\n                _ \u2264 Complex.abs a + Complex.abs (-b) := by\n                  apply Complex.abs.add_le\n                _ = Complex.abs a + Complex.abs b := by simp [Complex.abs.map_neg]\n            _ = 2 + Complex.abs b := by rw [h\u2082]\n            _ = 2 + (2 * Real.sqrt 2) := by rw [h\u2085\u2083]\n            _ = 2 + 2 * Real.sqrt 2 := by ring\n        have h\u2085\u2085 : 2 + 2 * Real.sqrt 2 \u2264 2 * Real.sqrt 21 := by\n          have h\u2085\u2085\u2081 : 0 < Real.sqrt 21 := Real.sqrt_pos.mpr (by norm_num)\n          have h\u2085\u2085\u2082 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n          have h\u2085\u2085\u2083 : 0 < Real.sqrt 2 * Real.sqrt 21 := by positivity\n          nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num), Real.sq_sqrt (show 0 \u2264 21 by norm_num),\n            sq_nonneg (Real.sqrt 2 - Real.sqrt 21)]\n        calc\n          Complex.abs (a - b) \u2264 2 + 2 * Real.sqrt 2 := h\u2085\u2084\n          _ \u2264 2 * Real.sqrt 21 := by linarith\n      exact h\u2085\u2082\n  exact h\u2084\n", "proof_before": "theorem amc12a_2020_p15 (a b : \u2102) (h\u2080 : a ^ 3 - 8 = 0) (h\u2081 : b ^ 3 - 8 * b ^ 2 - 8 * b + 64 = 0) :\n    Complex.abs (a - b) \u2264 2 * Real.sqrt 21  :=  by\n  have h\u2082 : Complex.abs a = 2 := by\n    have h\u2082\u2081 : a ^ 3 = 8 := by\n      rw [sub_eq_zero] at h\u2080\n      norm_num at h\u2080 \u22a2\n      <;> linear_combination h\u2080\n    have h\u2082\u2082 : Complex.abs (a ^ 3) = (8 : \u211d) := by\n      rw [h\u2082\u2081]\n      <;> simp [Complex.abs_of_nonneg] <;> norm_num\n    have h\u2082\u2083 : Complex.abs (a ^ 3) = (Complex.abs a) ^ 3 := by\n      simp [Complex.abs.map_pow]\n    rw [h\u2082\u2083] at h\u2082\u2082\n    have h\u2082\u2084 : (Complex.abs a : \u211d) ^ 3 = 8 := by\n      norm_num at h\u2082\u2082 \u22a2 <;>\n      linarith\n    have h\u2082\u2085 : Complex.abs a = 2 := by\n      have h\u2082\u2085\u2081 : Complex.abs a \u2265 0 := Complex.abs.nonneg _\n      have h\u2082\u2085\u2082 : (Complex.abs a : \u211d) ^ 3 = 8 := by exact_mod_cast h\u2082\u2084\n      have h\u2082\u2085\u2083 : Complex.abs a = 2 := by\n        nlinarith [sq_nonneg (Complex.abs a - 2), sq_nonneg (Complex.abs a + 2)]\n      exact h\u2082\u2085\u2083\n    exact h\u2082\u2085\n  have h\u2083 : b = 8 \u2228 b ^ 2 = 8 := by\n    have h\u2084 : b ^ 3 - 8 * b ^ 2 - 8 * b + 64 = 0 := h\u2081\n    have h\u2085 : (b - 8) * (b ^ 2 - 8) = 0 := by\n      ring_nf at h\u2084 \u22a2\n      <;> simp_all [Complex.ext_iff, pow_two, pow_three]\n      <;>\n      (try ring_nf at * ) <;>\n      (try norm_num at * ) <;>\n      (try\n        constructor <;>\n        nlinarith [sq_nonneg (b.re - 8), sq_nonneg (b.re + 8), sq_nonneg (b.im - 8),\n          sq_nonneg (b.im + 8), sq_nonneg (b.re - b.im), sq_nonneg (b.re + b.im),\n          sq_nonneg (b.re - 2), sq_nonneg (b.re + 2), sq_nonneg (b.im - 2),\n          sq_nonneg (b.im + 2)])\n    have h\u2086 : b - 8 = 0 \u2228 b ^ 2 - 8 = 0 := by\n      apply eq_zero_or_eq_zero_of_mul_eq_zero h\u2085\n    cases h\u2086 with\n    | inl h\u2086 =>\n      have h\u2087 : b = 8 := by\n        have h\u2088 : b - 8 = 0 := h\u2086\n        have h\u2089 : b = 8 := by\n          rw [sub_eq_zero] at h\u2088\n          exact h\u2088\n        exact h\u2089\n      exact Or.inl h\u2087\n    | inr h\u2086 =>\n      have h\u2087 : b ^ 2 = 8 := by\n        have h\u2088 : b ^ 2 - 8 = 0 := h\u2086\n        have h\u2089 : b ^ 2 = 8 := by\n          rw [sub_eq_zero] at h\u2088\n          exact h\u2088\n        exact h\u2089\n      exact Or.inr h\u2087\n  have h\u2084 : Complex.abs (a - b) \u2264 2 * Real.sqrt 21 := by\n    have h\u2085 : b = 8 \u2228 b ^ 2 = 8 := h\u2083\n    cases h\u2085 with\n    | inl h\u2085 =>\n      have h\u2085\u2081 : b = 8 := h\u2085\n      have h\u2085\u2082 : Complex.abs (a - b) \u2264 2 * Real.sqrt 21 := by\n        rw [h\u2085\u2081]\n        have h\u2085\u2083 : Complex.abs a = 2 := h\u2082\n        have h\u2085\u2084 : Complex.abs (a - 8) \u2264 2 * Real.sqrt 21 := by\n          have h\u2085\u2085 : a ^ 3 = 8 := by\n            rw [sub_eq_zero] at h\u2080\n            norm_num at h\u2080 \u22a2\n            <;> linear_combination h\u2080\n          have h\u2085\u2086 : a.re ^ 2 + a.im ^ 2 = 4 := by\n            have h\u2085\u2086\u2081 : Complex.abs a = 2 := h\u2082\n            have h\u2085\u2086\u2082 : Complex.abs a = Real.sqrt (a.re ^ 2 + a.im ^ 2) := by\n              simp [Complex.abs, Complex.normSq, Real.sqrt_eq_iff_sq_eq, sq, add_nonneg, mul_self_nonneg]\n              <;> ring_nf\n              <;> field_simp [Real.sqrt_eq_iff_sq_eq, sq, add_nonneg, mul_self_nonneg]\n              <;> ring_nf\n              <;> linarith\n            rw [h\u2085\u2086\u2082] at h\u2085\u2086\u2081\n            have h\u2085\u2086\u2083 : Real.sqrt (a.re ^ 2 + a.im ^ 2) = 2 := by linarith\n            have h\u2085\u2086\u2084 : a.re ^ 2 + a.im ^ 2 = 4 := by\n              have h\u2085\u2086\u2085 : Real.sqrt (a.re ^ 2 + a.im ^ 2) = 2 := h\u2085\u2086\u2083\n              have h\u2085\u2086\u2086 : a.re ^ 2 + a.im ^ 2 = 4 := by\n                have h\u2085\u2086\u2087 : Real.sqrt (a.re ^ 2 + a.im ^ 2) = 2 := h\u2085\u2086\u2085\n                have h\u2085\u2086\u2088 : a.re ^ 2 + a.im ^ 2 \u2265 0 := by positivity\n                have h\u2085\u2086\u2089 : Real.sqrt (a.re ^ 2 + a.im ^ 2) ^ 2 = a.re ^ 2 + a.im ^ 2 := by\n                  rw [Real.sq_sqrt] <;> positivity\n                nlinarith\n              exact h\u2085\u2086\u2086\n            exact h\u2085\u2086\u2084\n          have h\u2085\u2087 : a.re \u2265 -1 := by\n            by_contra h\n            have h\u2085\u2087\u2081 : a.re < -1 := by linarith\n            have h\u2085\u2087\u2082 : a.re ^ 2 > 1 := by nlinarith\n            have h\u2085\u2087\u2083 : a.im ^ 2 \u2264 4 := by nlinarith\n            have h\u2085\u2087\u2084 : a.re ^ 2 + a.im ^ 2 = 4 := h\u2085\u2086\n            have h\u2085\u2087\u2085 : a.re ^ 3 - 3 * a.re * a.im ^ 2 = 8 := by\n              simp [Complex.ext_iff, pow_three, pow_two, Complex.ext_iff, pow_three, pow_two] at h\u2085\u2085\n              norm_num at h\u2085\u2085 \u22a2\n              <;>\n              (try cases' h\u2085\u2085 with h\u2085\u2085_left h\u2085\u2085_right) <;>\n              (try nlinarith) <;>\n              (try ring_nf at * <;> nlinarith)\n            nlinarith [sq_nonneg (a.re - 2), sq_nonneg (a.im), sq_nonneg (a.re + 1)]\n          have h\u2085\u2088 : Complex.abs (a - 8) = Real.sqrt ((a.re - 8) ^ 2 + (a.im) ^ 2) := by\n            simp [Complex.abs, Complex.normSq, Complex.ext_iff, Real.sqrt_eq_iff_sq_eq, sq, sub_nonneg, add_nonneg, mul_self_nonneg,\n              Real.sqrt_nonneg, Real.sq_sqrt, add_nonneg, mul_self_nonneg]\n            <;> ring_nf <;> field_simp [Real.sqrt_eq_iff_sq_eq, sq, sub_nonneg, add_nonneg, mul_self_nonneg,\n              Real.sqrt_nonneg, Real.sq_sqrt, add_nonneg, mul_self_nonneg]\n            <;> ring_nf <;> nlinarith\n          rw [h\u2085\u2088]\n          have h\u2085\u2089 : Real.sqrt ((a.re - 8) ^ 2 + (a.im) ^ 2) \u2264 2 * Real.sqrt 21 := by\n            have h\u2085\u2089\u2081 : (a.re - 8) ^ 2 + (a.im) ^ 2 \u2264 84 := by\n              nlinarith [sq_nonneg (a.re + 1), sq_nonneg (a.im), sq_nonneg (a.re - 2), sq_nonneg (a.im - 2 * Real.sqrt 3),\n                sq_nonneg (a.im + 2 * Real.sqrt 3), Real.sqrt_nonneg 3, Real.sq_sqrt (show 0 \u2264 3 by norm_num)]\n            have h\u2085\u2089\u2082 : Real.sqrt ((a.re - 8) ^ 2 + (a.im) ^ 2) \u2264 Real.sqrt 84 := by\n              apply Real.sqrt_le_sqrt\n              nlinarith\n            have h\u2085\u2089\u2083 : Real.sqrt 84 = 2 * Real.sqrt 21 := by\n              have h\u2085\u2089\u2084 : Real.sqrt 84 = Real.sqrt (4 * 21) := by norm_num\n              rw [h\u2085\u2089\u2084]\n              have h\u2085\u2089\u2085 : Real.sqrt (4 * 21) = 2 * Real.sqrt 21 := by\n                rw [Real.sqrt_eq_iff_sq_eq (by positivity) (by positivity)]\n                nlinarith [Real.sqrt_nonneg 21, Real.sq_sqrt (show 0 \u2264 21 by norm_num)]\n              rw [h\u2085\u2089\u2085]\n            rw [h\u2085\u2089\u2083] at h\u2085\u2089\u2082\n            linarith\n          exact h\u2085\u2089\n        have h\u2085\u2085 : Complex.abs (a - (8 : \u2102)) \u2264 2 * Real.sqrt 21 := h\u2085\u2084\n        simpa [h\u2085\u2081] using h\u2085\u2085\n      exact h\u2085\u2082\n    | inr h\u2085 =>\n      have h\u2085\u2081 : b ^ 2 = 8 := h\u2085\n      have h\u2085\u2082 : Complex.abs (a - b) \u2264 2 * Real.sqrt 21 := by\n        have h\u2085\u2083 : Complex.abs b = 2 * Real.sqrt 2 := by\n          have h\u2085\u2084 : b ^ 2 = 8 := h\u2085\u2081\n          have h\u2085\u2085 : Complex.abs (b ^ 2) = (8 : \u211d) := by\n            rw [h\u2085\u2084]\n            <;> simp [Complex.abs_of_nonneg] <;> norm_num\n          have h\u2085\u2086 : Complex.abs (b ^ 2) = (Complex.abs b) ^ 2 := by\n            simp [Complex.abs.map_pow]\n          rw [h\u2085\u2086] at h\u2085\u2085\n          have h\u2085\u2087 : (Complex.abs b : \u211d) ^ 2 = 8 := by\n            norm_num at h\u2085\u2085 \u22a2 <;>\n            linarith\n          have h\u2085\u2088 : Complex.abs b = 2 * Real.sqrt 2 := by\n            have h\u2085\u2088\u2081 : Complex.abs b \u2265 0 := Complex.abs.nonneg _\n            have h\u2085\u2088\u2082 : (Complex.abs b : \u211d) ^ 2 = 8 := by exact_mod_cast h\u2085\u2087\n            have h\u2085\u2088\u2083 : Complex.abs b = 2 * Real.sqrt 2 := by\n              have h\u2085\u2088\u2084 : Real.sqrt 2 \u2265 0 := Real.sqrt_nonneg _\n              nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n            exact h\u2085\u2088\u2083\n          exact h\u2085\u2088\n        have h\u2085\u2084 : Complex.abs (a - b) \u2264 2 + 2 * Real.sqrt 2 := by\n          calc\n            Complex.abs (a - b) \u2264 Complex.abs a + Complex.abs b := by\n              calc\n                Complex.abs (a - b) = Complex.abs (a + (-b)) := by ring_nf\n                _ \u2264 Complex.abs a + Complex.abs (-b) := by\n                  apply Complex.abs.add_le\n                _ = Complex.abs a + Complex.abs b := by simp [Complex.abs.map_neg]\n            _ = 2 + Complex.abs b := by rw [h\u2082]\n            _ = 2 + (2 * Real.sqrt 2) := by rw [h\u2085\u2083]\n            _ = 2 + 2 * Real.sqrt 2 := by ring\n        have h\u2085\u2085 : 2 + 2 * Real.sqrt 2 \u2264 2 * Real.sqrt 21 := by\n          have h\u2085\u2085\u2081 : 0 < Real.sqrt 21 := Real.sqrt_pos.mpr (by norm_num)\n          have h\u2085\u2085\u2082 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)\n          have h\u2085\u2085\u2083 : 0 < Real.sqrt 2 * Real.sqrt 21 := by positivity\n          nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num), Real.sq_sqrt (show 0 \u2264 21 by norm_num),\n            sq_nonneg (Real.sqrt 2 - Real.sqrt 21)]\n        calc\n          Complex.abs (a - b) \u2264 2 + 2 * Real.sqrt 2 := h\u2085\u2084\n          _ \u2264 2 * Real.sqrt 21 := by linarith\n      exact h\u2085\u2082\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 59, "alphabet": 29, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 26, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 7, "fixpoint": false, "chars_before": 1674, "chars_after": 787, "seconds": 768.8, "id": "amc12a_2021_p18", "proof_after": "theorem amc12a_2021_p18 (f : \u211a \u2192 \u211d) (h\u2080 : \u2200 x > 0, \u2200 y > 0, f (x * y) = f x + f y)\n    (h\u2081 : \u2200 p, Nat.Prime p \u2192 f p = p) : f (25 / 11) < 0  :=  by\n  have h\u2083 : f (5 : \u211a) = (5 : \u211d) := by\n    have h\u2083\u2081 : Nat.Prime 5 := by decide\n    have h\u2083\u2082 : f (5 : \u211a) = (5 : \u211d) := by\n      have h\u2083\u2083 : f (5 : \u211a) = 5 := h\u2081 5 h\u2083\u2081\n      omega\n    exact h\u2083\u2082\n  have h\u2084 : f (25 : \u211a) = (10 : \u211d) := by\n    have h\u2084\u2081 : f (25 : \u211a) = f (5 * 5 : \u211a) := by norm_num\n    rw [h\u2084\u2081]\n    have h\u2084\u2082 : f (5 * 5 : \u211a) = f (5 : \u211a) + f (5 : \u211a) := h\u2080 5 (by norm_num) 5 (by norm_num)\n    linarith\n  have h\u2085 : f (11 : \u211a) = (11 : \u211d) := by\n    have h\u2085\u2081 : Nat.Prime 11 := by decide\n    have h\u2085\u2082 : f (11 : \u211a) = (11 : \u211d) := by\n      have h\u2085\u2083 : f (11 : \u211a) = 11 := h\u2081 11 h\u2085\u2081\n      omega\n    exact h\u2085\u2082\n  have h\u2086 : f (25 / 11 : \u211a) = (-1 : \u211d) := by\n    have h\u2086\u2081 : f ((25 / 11 : \u211a) * 11) = f (25 / 11 : \u211a) + f (11 : \u211a) := h\u2080 (25 / 11 : \u211a) (by norm_num) 11 (by norm_num)\n    linarith\n  linarith\n", "proof_before": "theorem amc12a_2021_p18 (f : \u211a \u2192 \u211d) (h\u2080 : \u2200 x > 0, \u2200 y > 0, f (x * y) = f x + f y)\n    (h\u2081 : \u2200 p, Nat.Prime p \u2192 f p = p) : f (25 / 11) < 0  :=  by\n  have h\u2082 : f 1 = 0 := by\n    have h\u2082\u2081 : f (1 * 1 : \u211a) = f 1 + f 1 := by\n      have h\u2082\u2082 : f (1 * 1 : \u211a) = f 1 + f 1 := h\u2080 1 (by norm_num) 1 (by norm_num)\n      exact h\u2082\u2082\n    have h\u2082\u2082 : f (1 * 1 : \u211a) = f 1 + f 1 := h\u2082\u2081\n    have h\u2082\u2083 : f (1 * 1 : \u211a) = f 1 := by norm_num\n    have h\u2082\u2084 : f 1 + f 1 = f 1 := by linarith\n    have h\u2082\u2085 : f 1 = 0 := by linarith\n    exact h\u2082\u2085\n  have h\u2083 : f (5 : \u211a) = (5 : \u211d) := by\n    have h\u2083\u2081 : Nat.Prime 5 := by decide\n    have h\u2083\u2082 : f (5 : \u211a) = (5 : \u211d) := by\n      have h\u2083\u2083 : f (5 : \u211a) = 5 := h\u2081 5 h\u2083\u2081\n      norm_num at h\u2083\u2083 \u22a2\n      <;> simp_all [h\u2083\u2083]\n      <;> norm_num\n      <;> linarith\n    exact h\u2083\u2082\n  have h\u2084 : f (25 : \u211a) = (10 : \u211d) := by\n    have h\u2084\u2081 : f (25 : \u211a) = f (5 * 5 : \u211a) := by norm_num\n    rw [h\u2084\u2081]\n    have h\u2084\u2082 : f (5 * 5 : \u211a) = f (5 : \u211a) + f (5 : \u211a) := h\u2080 5 (by norm_num) 5 (by norm_num)\n    rw [h\u2084\u2082]\n    norm_num [h\u2083]\n    <;> linarith\n  have h\u2085 : f (11 : \u211a) = (11 : \u211d) := by\n    have h\u2085\u2081 : Nat.Prime 11 := by decide\n    have h\u2085\u2082 : f (11 : \u211a) = (11 : \u211d) := by\n      have h\u2085\u2083 : f (11 : \u211a) = 11 := h\u2081 11 h\u2085\u2081\n      norm_num at h\u2085\u2083 \u22a2\n      <;> simp_all [h\u2085\u2083]\n      <;> norm_num\n      <;> linarith\n    exact h\u2085\u2082\n  have h\u2086 : f (25 / 11 : \u211a) = (-1 : \u211d) := by\n    have h\u2086\u2081 : f ((25 / 11 : \u211a) * 11) = f (25 / 11 : \u211a) + f (11 : \u211a) := h\u2080 (25 / 11 : \u211a) (by norm_num) 11 (by norm_num)\n    have h\u2086\u2082 : f ((25 / 11 : \u211a) * 11) = f (25 : \u211a) := by norm_num\n    have h\u2086\u2083 : f (25 / 11 : \u211a) + f (11 : \u211a) = f (25 : \u211a) := by linarith\n    have h\u2086\u2084 : f (25 / 11 : \u211a) = f (25 : \u211a) - f (11 : \u211a) := by linarith\n    rw [h\u2086\u2084, h\u2084, h\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2087 : f (25 / 11 : \u211a) < 0 := by\n    rw [h\u2086]\n    <;> norm_num\n    <;> linarith\n  exact h\u2087\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 20, "compiles": true, "steps_after": 9, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 7, "fixpoint": true, "chars_before": 576, "chars_after": 284, "seconds": 577.5, "id": "mathd_algebra_246", "proof_after": "theorem mathd_algebra_246 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = a * x ^ 4 - b * x ^ 2 + x + 5)\n    (h\u2082 : f (-3) = 2) : f 3 = 8  :=  by\n  have h\u2083 : 81 * a - 9 * b = 0 := by\n    have h\u2083\u2081 : f (-3) = a * (-3 : \u211d) ^ 4 - b * (-3 : \u211d) ^ 2 + (-3 : \u211d) + 5 := by\n      rw [h\u2080]\n    linarith\n  have h\u2085 : f 3 = 8 := by\n    have h\u2085\u2081 : f 3 = a * (3 : \u211d) ^ 4 - b * (3 : \u211d) ^ 2 + (3 : \u211d) + 5 := by\n      rw [h\u2080]\n    linarith\n  exact h\u2085\n", "proof_before": "theorem mathd_algebra_246 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = a * x ^ 4 - b * x ^ 2 + x + 5)\n    (h\u2082 : f (-3) = 2) : f 3 = 8  :=  by\n  have h\u2083 : 81 * a - 9 * b = 0 := by\n    have h\u2083\u2081 : f (-3) = a * (-3 : \u211d) ^ 4 - b * (-3 : \u211d) ^ 2 + (-3 : \u211d) + 5 := by\n      rw [h\u2080]\n      <;> ring_nf\n    rw [h\u2083\u2081] at h\u2082\n    ring_nf at h\u2082 \u22a2\n    linarith\n  have h\u2084 : b = 9 * a := by\n    have h\u2084\u2081 : 81 * a - 9 * b = 0 := h\u2083\n    have h\u2084\u2082 : 9 * a - b = 0 := by linarith\n    have h\u2084\u2083 : b = 9 * a := by linarith\n    exact h\u2084\u2083\n  have h\u2085 : f 3 = 8 := by\n    have h\u2085\u2081 : f 3 = a * (3 : \u211d) ^ 4 - b * (3 : \u211d) ^ 2 + (3 : \u211d) + 5 := by\n      rw [h\u2080]\n      <;> ring_nf\n    rw [h\u2085\u2081]\n    rw [h\u2084]\n    <;> ring_nf at *\n    <;> nlinarith\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 61, "alphabet": 28, "compiles": false, "id": "mathd_algebra_313", "proof_after": "theorem mathd_algebra_313 (v i z : \u2102) (h\u2080 : v = i * z) (h\u2081 : v = 1 + Complex.I)\n    (h\u2082 : z = 2 - Complex.I) : i = 1 / 5 + 3 / 5 * Complex.I  :=  by\n  have h\u2083 : i * z = 1 + Complex.I := by\n    rw [h\u2080] at h\u2081\n    linear_combination h\u2081\n  have h\u2084 : i * (2 - Complex.I) = 1 + Complex.I := by\n    rw [h\u2082] at h\u2083\n    exact h\u2083\n  have h\u2085 : i = 1 / 5 + 3 / 5 * Complex.I := by\n    have h\u2085\u2081 : i * (2 - Complex.I) = 1 + Complex.I := h\u2084\n    have h\u2085\u2082 : i = (1 + Complex.I) / (2 - Complex.I) := by\n      have h\u2085\u2083 : (2 - Complex.I : \u2102) \u2260 0 := by\n        norm_num [Complex.ext_iff, Complex.I_mul_I]\n        <;>\n        (try norm_num) <;>\n        (try simp_all [Complex.ext_iff, Complex.I_mul_I]) <;>\n        (try ring_nf at * <;> norm_num at * <;>\n          simp_all [Complex.ext_iff, Complex.I_mul_I]) <;>\n        (try linarith)\n      have h\u2085\u2084 : i * (2 - Complex.I) = (1 + Complex.I) := by\n        linear_combination h\u2085\u2081\n      calc\n        i = i * 1 := by ring\n        _ = i * ((2 - Complex.I) / (2 - Complex.I)) := by\n          field_simp [h\u2085\u2083]\n        _ = (i * (2 - Complex.I)) / (2 - Complex.I) := by\n          ring\n        _ = (1 + Complex.I) / (2 - Complex.I) := by rw [h\u2085\u2084]\n    rw [h\u2085\u2082]\n    have h\u2085\u2085 : (2 - Complex.I : \u2102) \u2260 0 := by\n      norm_num [Complex.ext_iff, Complex.I_mul_I]\n      <;>\n      (try norm_num) <;>\n      (try simp_all [Complex.ext_iff, Complex.I_mul_I]) <;>\n      (try ring_nf at * <;> norm_num at * <;>\n        simp_all [Complex.ext_iff, Complex.I_mul_I]) <;>\n      (try linarith)\n    field_simp [h\u2085\u2085, Complex.ext_iff, Complex.I_mul_I, Complex.div_re, Complex.div_im, Complex.normSq]\n    <;>\n    ring_nf at * <;>\n    norm_num at * <;>\n    constructor <;>\n    simp_all [Complex.ext_iff, Complex.I_mul_I] <;>\n    ring_nf at * <;>\n    norm_num at * <;>\n    linarith\n  exact h\u2085\n", "proof_before": "theorem mathd_algebra_313 (v i z : \u2102) (h\u2080 : v = i * z) (h\u2081 : v = 1 + Complex.I)\n    (h\u2082 : z = 2 - Complex.I) : i = 1 / 5 + 3 / 5 * Complex.I  :=  by\n  have h\u2083 : i * z = 1 + Complex.I := by\n    rw [h\u2080] at h\u2081\n    linear_combination h\u2081\n  have h\u2084 : i * (2 - Complex.I) = 1 + Complex.I := by\n    rw [h\u2082] at h\u2083\n    exact h\u2083\n  have h\u2085 : i = 1 / 5 + 3 / 5 * Complex.I := by\n    have h\u2085\u2081 : i * (2 - Complex.I) = 1 + Complex.I := h\u2084\n    have h\u2085\u2082 : i = (1 + Complex.I) / (2 - Complex.I) := by\n      have h\u2085\u2083 : (2 - Complex.I : \u2102) \u2260 0 := by\n        norm_num [Complex.ext_iff, Complex.I_mul_I]\n        <;>\n        (try norm_num) <;>\n        (try simp_all [Complex.ext_iff, Complex.I_mul_I]) <;>\n        (try ring_nf at * <;> norm_num at * <;>\n          simp_all [Complex.ext_iff, Complex.I_mul_I]) <;>\n        (try linarith)\n      have h\u2085\u2084 : i * (2 - Complex.I) = (1 + Complex.I) := by\n        linear_combination h\u2085\u2081\n      calc\n        i = i * 1 := by ring\n        _ = i * ((2 - Complex.I) / (2 - Complex.I)) := by\n          field_simp [h\u2085\u2083]\n        _ = (i * (2 - Complex.I)) / (2 - Complex.I) := by\n          ring\n        _ = (1 + Complex.I) / (2 - Complex.I) := by rw [h\u2085\u2084]\n    rw [h\u2085\u2082]\n    have h\u2085\u2085 : (2 - Complex.I : \u2102) \u2260 0 := by\n      norm_num [Complex.ext_iff, Complex.I_mul_I]\n      <;>\n      (try norm_num) <;>\n      (try simp_all [Complex.ext_iff, Complex.I_mul_I]) <;>\n      (try ring_nf at * <;> norm_num at * <;>\n        simp_all [Complex.ext_iff, Complex.I_mul_I]) <;>\n      (try linarith)\n    field_simp [h\u2085\u2085, Complex.ext_iff, Complex.I_mul_I, Complex.div_re, Complex.div_im, Complex.normSq]\n    <;>\n    ring_nf at * <;>\n    norm_num at * <;>\n    constructor <;>\n    simp_all [Complex.ext_iff, Complex.I_mul_I] <;>\n    ring_nf at * <;>\n    norm_num at * <;>\n    linarith\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 15, "alphabet": 19, "compiles": true, "steps_after": 9, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "fixpoint": true, "chars_before": 343, "chars_after": 191, "seconds": 426.2, "id": "mathd_algebra_80", "proof_after": "theorem mathd_algebra_80 (x : \u211d) (h\u2080 : x \u2260 -1) (h\u2081 : (x - 9) / (x + 1) = 2) : x = -11  :=  by\n  have h\u2082 : x + 1 \u2260 0 := by\n    intro h\n    apply h\u2080\n    linarith\n  have h\u2083 : x - 9 = 2 * (x + 1) := by\n    have h\u2084 : (x - 9) / (x + 1) = 2 := h\u2081\n    field_simp at h\u2084\n    linarith\n  linarith\n", "proof_before": "theorem mathd_algebra_80 (x : \u211d) (h\u2080 : x \u2260 -1) (h\u2081 : (x - 9) / (x + 1) = 2) : x = -11  :=  by\n  have h\u2082 : x + 1 \u2260 0 := by\n    intro h\n    apply h\u2080\n    linarith\n  have h\u2083 : x - 9 = 2 * (x + 1) := by\n    have h\u2084 : (x - 9) / (x + 1) = 2 := h\u2081\n    have h\u2085 : x + 1 \u2260 0 := h\u2082\n    field_simp at h\u2084\n    linarith\n  have h\u2084 : x = -11 := by\n    have h\u2085 : x - 9 = 2 * (x + 1) := h\u2083\n    have h\u2086 : x = -11 := by\n      linarith\n    exact h\u2086\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 15, "alphabet": 15, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 500, "chars_after": 8, "seconds": 103.0, "id": "mathd_numbertheory_328", "proof_after": "theorem mathd_numbertheory_328 : 5 ^ 999999 % 7 = 6  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_328 : 5 ^ 999999 % 7 = 6  :=  by\n  have h1 : 5 ^ 6 % 7 = 1 := by\n    norm_num\n    <;> rfl\n  have h2 : 999999 = 6 * 166666 + 3 := by\n    norm_num\n    <;> rfl\n  have h3 : 5 ^ 999999 % 7 = 6 := by\n    rw [h2]\n    rw [show 5 ^ (6 * 166666 + 3) = (5 ^ 6) ^ 166666 * 5 ^ 3 by\n      rw [pow_add, pow_mul]\n      <;> ring_nf]\n    rw [show ((5 ^ 6) ^ 166666 * 5 ^ 3) % 7 = ((5 ^ 6 % 7) ^ 166666 * (5 ^ 3 % 7)) % 7 by\n      simp [Nat.pow_mod, Nat.mul_mod, Nat.pow_add, Nat.pow_mul, Nat.mod_mod]]\n    rw [h1]\n    norm_num\n    <;> rfl\n  apply h3\n"}
{"mode": "clear", "steps_before": 21, "alphabet": 15, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 392, "chars_after": 6, "seconds": 80.5, "id": "mathd_numbertheory_207", "proof_after": "theorem mathd_numbertheory_207 : 8 * 9 ^ 2 + 5 * 9 + 2 = 695  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_207 : 8 * 9 ^ 2 + 5 * 9 + 2 = 695  :=  by\n  have h\u2081 : 9 ^ 2 = 81 := by\n    norm_num\n    <;> rfl\n  have h\u2082 : 8 * 9 ^ 2 = 648 := by\n    rw [h\u2081]\n    <;> norm_num\n    <;> rfl\n  have h\u2083 : 5 * 9 = 45 := by\n    norm_num\n    <;> rfl\n  have h\u2084 : 8 * 9 ^ 2 + 5 * 9 = 693 := by\n    rw [h\u2082, h\u2083]\n    <;> norm_num\n    <;> rfl\n  have h\u2085 : 8 * 9 ^ 2 + 5 * 9 + 2 = 695 := by\n    rw [h\u2084]\n    <;> norm_num\n    <;> rfl\n  rw [h\u2085]\n  <;> norm_num\n  <;> rfl\n"}
{"mode": "clear", "steps_before": 204, "alphabet": 69, "compiles": false, "id": "mathd_algebra_17", "proof_after": "theorem mathd_algebra_17 (a : \u211d)\n    (h\u2080 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) + Real.sqrt (1 + Real.sqrt (1 + a)) = 6) : a = 8  :=  by\n  have h\u2081 : 16 + 16 * a \u2265 0 := by\n    by_contra h\n    have h\u2082 : 16 + 16 * a < 0 := by linarith\n    have h\u2083 : Real.sqrt (16 + 16 * a) = 0 := by\n      rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n    have h\u2084 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 2 := by\n      rw [h\u2083]\n      have h\u2085 : Real.sqrt (4 + (0 : \u211d)) = 2 := by\n        rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n      exact h\u2085\n    have h\u2085 : Real.sqrt (1 + Real.sqrt (1 + a)) = 4 := by\n      have h\u2085\u2081 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) + Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := h\u2080\n      rw [h\u2084] at h\u2085\u2081\n      have h\u2085\u2082 : 2 + Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := by linarith\n      have h\u2085\u2083 : Real.sqrt (1 + Real.sqrt (1 + a)) = 4 := by linarith\n      exact h\u2085\u2083\n    have h\u2086 : 1 + Real.sqrt (1 + a) = 16 := by\n      have h\u2086\u2081 : Real.sqrt (1 + Real.sqrt (1 + a)) = 4 := h\u2085\n      have h\u2086\u2082 : 1 + Real.sqrt (1 + a) \u2265 0 := by\n        by_contra h\u2086\u2082\n        have h\u2086\u2083 : 1 + Real.sqrt (1 + a) < 0 := by linarith\n        have h\u2086\u2084 : Real.sqrt (1 + Real.sqrt (1 + a)) = 0 := by\n          rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n        rw [h\u2086\u2084] at h\u2086\u2081\n        norm_num at h\u2086\u2081\n      have h\u2086\u2083 : Real.sqrt (1 + Real.sqrt (1 + a)) = 4 := h\u2085\n      have h\u2086\u2084 : Real.sqrt (1 + Real.sqrt (1 + a)) ^ 2 = 4 ^ 2 := by rw [h\u2086\u2083]\n      have h\u2086\u2085 : 1 + Real.sqrt (1 + a) = 16 := by\n        nlinarith [Real.sq_sqrt (by linarith : (0 : \u211d) \u2264 1 + Real.sqrt (1 + a))]\n      exact h\u2086\u2085\n    have h\u2087 : Real.sqrt (1 + a) = 15 := by linarith\n    have h\u2088 : 1 + a \u2265 0 := by\n      by_contra h\u2088\n      have h\u2089 : 1 + a < 0 := by linarith\n      have h\u2081\u2080 : Real.sqrt (1 + a) = 0 := by\n        rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n      rw [h\u2081\u2080] at h\u2087\n      norm_num at h\u2087\n    have h\u2089 : Real.sqrt (1 + a) = 15 := h\u2087\n    have h\u2081\u2080 : 1 + a = 225 := by\n      have h\u2081\u2080\u2081 : Real.sqrt (1 + a) = 15 := h\u2089\n      have h\u2081\u2080\u2082 : Real.sqrt (1 + a) ^ 2 = 15 ^ 2 := by rw [h\u2081\u2080\u2081]\n      have h\u2081\u2080\u2083 : 1 + a = 225 := by\n        nlinarith [Real.sq_sqrt (by linarith : (0 : \u211d) \u2264 1 + a)]\n      exact h\u2081\u2080\u2083\n    have h\u2081\u2081 : a = 224 := by linarith\n    have h\u2081\u2082 : a < -1 := by\n      linarith\n    linarith\n  have h\u2082 : 1 + a \u2265 0 := by\n    by_contra h\n    have h\u2083 : 1 + a < 0 := by linarith\n    have h\u2084 : Real.sqrt (1 + a) = 0 := by\n      rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n    have h\u2085 : Real.sqrt (1 + Real.sqrt (1 + a)) = 1 := by\n      rw [h\u2084]\n      have h\u2085\u2081 : Real.sqrt (1 + (0 : \u211d)) = 1 := by\n        rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n      exact h\u2085\u2081\n    have h\u2086 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 5 := by\n      have h\u2086\u2081 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) + Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := h\u2080\n      rw [h\u2085] at h\u2086\u2081\n      have h\u2086\u2082 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) + 1 = 6 := by linarith\n      have h\u2086\u2083 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 5 := by linarith\n      exact h\u2086\u2083\n    have h\u2087 : 4 + Real.sqrt (16 + 16 * a) = 25 := by\n      have h\u2087\u2081 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 5 := h\u2086\n      have h\u2087\u2082 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) ^ 2 = 5 ^ 2 := by rw [h\u2087\u2081]\n      have h\u2087\u2083 : 4 + Real.sqrt (16 + 16 * a) \u2265 0 := by\n        by_contra h\u2087\u2083\n        have h\u2087\u2084 : 4 + Real.sqrt (16 + 16 * a) < 0 := by linarith\n        have h\u2087\u2085 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 0 := by\n          rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n        rw [h\u2087\u2085] at h\u2087\u2081\n        norm_num at h\u2087\u2085 h\u2087\u2081 \u22a2\n        <;> linarith\n      have h\u2087\u2084 : 4 + Real.sqrt (16 + 16 * a) = 25 := by\n        nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 (4 + Real.sqrt (16 + 16 * a) : \u211d))]\n      exact h\u2087\u2084\n    have h\u2088 : Real.sqrt (16 + 16 * a) = 21 := by linarith\n    have h\u2089 : 16 + 16 * a = 441 := by\n      have h\u2089\u2081 : Real.sqrt (16 + 16 * a) = 21 := h\u2088\n      have h\u2089\u2082 : Real.sqrt (16 + 16 * a) ^ 2 = 21 ^ 2 := by rw [h\u2089\u2081]\n      have h\u2089\u2083 : 16 + 16 * a \u2265 0 := by linarith\n      have h\u2089\u2084 : 16 + 16 * a = 441 := by\n        nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 (16 + 16 * a : \u211d))]\n      exact h\u2089\u2084\n    have h\u2081\u2080 : a < -1 := by linarith\n    have h\u2081\u2081 : a \u2265 -1 := by\n      nlinarith\n    linarith\n  have h\u2083 : Real.sqrt (16 + 16 * a) = 4 * Real.sqrt (1 + a) := by\n    have h\u2083\u2081 : Real.sqrt (16 + 16 * a) = Real.sqrt (16 * (1 + a)) := by\n      ring_nf\n      <;> field_simp\n      <;> ring_nf\n    rw [h\u2083\u2081]\n    have h\u2083\u2082 : Real.sqrt (16 * (1 + a)) = 4 * Real.sqrt (1 + a) := by\n      rw [Real.sqrt_eq_iff_sq_eq (by positivity) (by positivity)]\n      nlinarith [Real.sq_sqrt (show 0 \u2264 1 + a by linarith), Real.sqrt_nonneg (1 + a)]\n    rw [h\u2083\u2082]\n    <;> ring\n    <;> field_simp\n    <;> ring_nf\n    <;> linarith\n  have h\u2084 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 2 * Real.sqrt (1 + Real.sqrt (1 + a)) := by\n    rw [h\u2083]\n    have h\u2084\u2081 : Real.sqrt (4 + (4 : \u211d) * Real.sqrt (1 + a)) = 2 * Real.sqrt (1 + Real.sqrt (1 + a)) := by\n      have h\u2084\u2082 : Real.sqrt (4 + (4 : \u211d) * Real.sqrt (1 + a)) = Real.sqrt (4 * (1 + Real.sqrt (1 + a))) := by\n        have h\u2084\u2083 : (4 : \u211d) + (4 : \u211d) * Real.sqrt (1 + a) = 4 * (1 + Real.sqrt (1 + a)) := by ring\n        rw [h\u2084\u2083]\n        <;> ring_nf\n        <;> field_simp\n        <;> ring_nf\n      rw [h\u2084\u2082]\n      have h\u2084\u2084 : Real.sqrt (4 * (1 + Real.sqrt (1 + a))) = 2 * Real.sqrt (1 + Real.sqrt (1 + a)) := by\n        have h\u2084\u2085 : Real.sqrt (4 * (1 + Real.sqrt (1 + a))) = Real.sqrt 4 * Real.sqrt (1 + Real.sqrt (1 + a)) := by\n          rw [\u2190 Real.sqrt_mul] <;> nlinarith [Real.sqrt_nonneg (1 + a), Real.sqrt_nonneg (1 + Real.sqrt (1 + a)), Real.sq_sqrt (show 0 \u2264 1 + a by linarith), Real.sq_sqrt (show 0 \u2264 1 + Real.sqrt (1 + a) by positivity)]\n        rw [h\u2084\u2085]\n        have h\u2084\u2086 : Real.sqrt 4 = 2 := by norm_num [Real.sqrt_eq_iff_sq_eq]\n        rw [h\u2084\u2086]\n        <;> ring_nf\n        <;> field_simp\n        <;> ring_nf\n      rw [h\u2084\u2084]\n      <;> ring_nf\n      <;> field_simp\n      <;> ring_nf\n    rw [h\u2084\u2081]\n    <;> ring_nf\n    <;> field_simp\n    <;> ring_nf\n  have h\u2085 : Real.sqrt (1 + Real.sqrt (1 + a)) = 2 := by\n    have h\u2085\u2081 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) + Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := h\u2080\n    rw [h\u2084] at h\u2085\u2081\n    have h\u2085\u2082 : 2 * Real.sqrt (1 + Real.sqrt (1 + a)) + Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := by linarith\n    have h\u2085\u2083 : 3 * Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := by linarith\n    have h\u2085\u2084 : Real.sqrt (1 + Real.sqrt (1 + a)) = 2 := by linarith\n    exact h\u2085\u2084\n  have h\u2086 : 1 + Real.sqrt (1 + a) = 4 := by\n    have h\u2086\u2081 : Real.sqrt (1 + Real.sqrt (1 + a)) = 2 := h\u2085\n    have h\u2086\u2082 : Real.sqrt (1 + Real.sqrt (1 + a)) ^ 2 = 2 ^ 2 := by rw [h\u2086\u2081]\n    have h\u2086\u2083 : 1 + Real.sqrt (1 + a) \u2265 0 := by\n      by_contra h\u2086\u2083\n      have h\u2086\u2084 : 1 + Real.sqrt (1 + a) < 0 := by linarith\n      have h\u2086\u2085 : Real.sqrt (1 + Real.sqrt (1 + a)) = 0 := by\n        rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n      rw [h\u2086\u2085] at h\u2086\u2081\n      norm_num at h\u2086\u2081\n      <;> linarith\n    have h\u2086\u2084 : 1 + Real.sqrt (1 + a) = 4 := by\n      nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 (1 + Real.sqrt (1 + a) : \u211d))]\n    exact h\u2086\u2084\n  have h\u2087 : Real.sqrt (1 + a) = 3 := by\n    have h\u2087\u2081 : 1 + Real.sqrt (1 + a) = 4 := h\u2086\n    have h\u2087\u2082 : Real.sqrt (1 + a) = 3 := by linarith\n    exact h\u2087\u2082\n  have h\u2088 : 1 + a = 9 := by\n    have h\u2088\u2081 : Real.sqrt (1 + a) = 3 := h\u2087\n    have h\u2088\u2082 : Real.sqrt (1 + a) ^ 2 = 3 ^ 2 := by rw [h\u2088\u2081]\n    have h\u2088\u2083 : 1 + a \u2265 0 := by linarith\n    have h\u2088\u2084 : 1 + a = 9 := by\n      nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 (1 + a : \u211d))]\n    exact h\u2088\u2084\n  have h\u2089 : a = 8 := by\n    linarith\n  exact h\u2089\n", "proof_before": "theorem mathd_algebra_17 (a : \u211d)\n    (h\u2080 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) + Real.sqrt (1 + Real.sqrt (1 + a)) = 6) : a = 8  :=  by\n  have h\u2081 : 16 + 16 * a \u2265 0 := by\n    by_contra h\n    have h\u2082 : 16 + 16 * a < 0 := by linarith\n    have h\u2083 : Real.sqrt (16 + 16 * a) = 0 := by\n      rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n    have h\u2084 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 2 := by\n      rw [h\u2083]\n      have h\u2085 : Real.sqrt (4 + (0 : \u211d)) = 2 := by\n        rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n      exact h\u2085\n    have h\u2085 : Real.sqrt (1 + Real.sqrt (1 + a)) = 4 := by\n      have h\u2085\u2081 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) + Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := h\u2080\n      rw [h\u2084] at h\u2085\u2081\n      have h\u2085\u2082 : 2 + Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := by linarith\n      have h\u2085\u2083 : Real.sqrt (1 + Real.sqrt (1 + a)) = 4 := by linarith\n      exact h\u2085\u2083\n    have h\u2086 : 1 + Real.sqrt (1 + a) = 16 := by\n      have h\u2086\u2081 : Real.sqrt (1 + Real.sqrt (1 + a)) = 4 := h\u2085\n      have h\u2086\u2082 : 1 + Real.sqrt (1 + a) \u2265 0 := by\n        by_contra h\u2086\u2082\n        have h\u2086\u2083 : 1 + Real.sqrt (1 + a) < 0 := by linarith\n        have h\u2086\u2084 : Real.sqrt (1 + Real.sqrt (1 + a)) = 0 := by\n          rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n        rw [h\u2086\u2084] at h\u2086\u2081\n        norm_num at h\u2086\u2081\n      have h\u2086\u2083 : Real.sqrt (1 + Real.sqrt (1 + a)) = 4 := h\u2085\n      have h\u2086\u2084 : Real.sqrt (1 + Real.sqrt (1 + a)) ^ 2 = 4 ^ 2 := by rw [h\u2086\u2083]\n      have h\u2086\u2085 : 1 + Real.sqrt (1 + a) = 16 := by\n        nlinarith [Real.sq_sqrt (by linarith : (0 : \u211d) \u2264 1 + Real.sqrt (1 + a))]\n      exact h\u2086\u2085\n    have h\u2087 : Real.sqrt (1 + a) = 15 := by linarith\n    have h\u2088 : 1 + a \u2265 0 := by\n      by_contra h\u2088\n      have h\u2089 : 1 + a < 0 := by linarith\n      have h\u2081\u2080 : Real.sqrt (1 + a) = 0 := by\n        rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n      rw [h\u2081\u2080] at h\u2087\n      norm_num at h\u2087\n    have h\u2089 : Real.sqrt (1 + a) = 15 := h\u2087\n    have h\u2081\u2080 : 1 + a = 225 := by\n      have h\u2081\u2080\u2081 : Real.sqrt (1 + a) = 15 := h\u2089\n      have h\u2081\u2080\u2082 : Real.sqrt (1 + a) ^ 2 = 15 ^ 2 := by rw [h\u2081\u2080\u2081]\n      have h\u2081\u2080\u2083 : 1 + a = 225 := by\n        nlinarith [Real.sq_sqrt (by linarith : (0 : \u211d) \u2264 1 + a)]\n      exact h\u2081\u2080\u2083\n    have h\u2081\u2081 : a = 224 := by linarith\n    have h\u2081\u2082 : a < -1 := by\n      linarith\n    linarith\n  have h\u2082 : 1 + a \u2265 0 := by\n    by_contra h\n    have h\u2083 : 1 + a < 0 := by linarith\n    have h\u2084 : Real.sqrt (1 + a) = 0 := by\n      rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n    have h\u2085 : Real.sqrt (1 + Real.sqrt (1 + a)) = 1 := by\n      rw [h\u2084]\n      have h\u2085\u2081 : Real.sqrt (1 + (0 : \u211d)) = 1 := by\n        rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n      exact h\u2085\u2081\n    have h\u2086 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 5 := by\n      have h\u2086\u2081 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) + Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := h\u2080\n      rw [h\u2085] at h\u2086\u2081\n      have h\u2086\u2082 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) + 1 = 6 := by linarith\n      have h\u2086\u2083 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 5 := by linarith\n      exact h\u2086\u2083\n    have h\u2087 : 4 + Real.sqrt (16 + 16 * a) = 25 := by\n      have h\u2087\u2081 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 5 := h\u2086\n      have h\u2087\u2082 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) ^ 2 = 5 ^ 2 := by rw [h\u2087\u2081]\n      have h\u2087\u2083 : 4 + Real.sqrt (16 + 16 * a) \u2265 0 := by\n        by_contra h\u2087\u2083\n        have h\u2087\u2084 : 4 + Real.sqrt (16 + 16 * a) < 0 := by linarith\n        have h\u2087\u2085 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 0 := by\n          rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n        rw [h\u2087\u2085] at h\u2087\u2081\n        norm_num at h\u2087\u2085 h\u2087\u2081 \u22a2\n        <;> linarith\n      have h\u2087\u2084 : 4 + Real.sqrt (16 + 16 * a) = 25 := by\n        nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 (4 + Real.sqrt (16 + 16 * a) : \u211d))]\n      exact h\u2087\u2084\n    have h\u2088 : Real.sqrt (16 + 16 * a) = 21 := by linarith\n    have h\u2089 : 16 + 16 * a = 441 := by\n      have h\u2089\u2081 : Real.sqrt (16 + 16 * a) = 21 := h\u2088\n      have h\u2089\u2082 : Real.sqrt (16 + 16 * a) ^ 2 = 21 ^ 2 := by rw [h\u2089\u2081]\n      have h\u2089\u2083 : 16 + 16 * a \u2265 0 := by linarith\n      have h\u2089\u2084 : 16 + 16 * a = 441 := by\n        nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 (16 + 16 * a : \u211d))]\n      exact h\u2089\u2084\n    have h\u2081\u2080 : a < -1 := by linarith\n    have h\u2081\u2081 : a \u2265 -1 := by\n      nlinarith\n    linarith\n  have h\u2083 : Real.sqrt (16 + 16 * a) = 4 * Real.sqrt (1 + a) := by\n    have h\u2083\u2081 : Real.sqrt (16 + 16 * a) = Real.sqrt (16 * (1 + a)) := by\n      ring_nf\n      <;> field_simp\n      <;> ring_nf\n    rw [h\u2083\u2081]\n    have h\u2083\u2082 : Real.sqrt (16 * (1 + a)) = 4 * Real.sqrt (1 + a) := by\n      rw [Real.sqrt_eq_iff_sq_eq (by positivity) (by positivity)]\n      nlinarith [Real.sq_sqrt (show 0 \u2264 1 + a by linarith), Real.sqrt_nonneg (1 + a)]\n    rw [h\u2083\u2082]\n    <;> ring\n    <;> field_simp\n    <;> ring_nf\n    <;> linarith\n  have h\u2084 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) = 2 * Real.sqrt (1 + Real.sqrt (1 + a)) := by\n    rw [h\u2083]\n    have h\u2084\u2081 : Real.sqrt (4 + (4 : \u211d) * Real.sqrt (1 + a)) = 2 * Real.sqrt (1 + Real.sqrt (1 + a)) := by\n      have h\u2084\u2082 : Real.sqrt (4 + (4 : \u211d) * Real.sqrt (1 + a)) = Real.sqrt (4 * (1 + Real.sqrt (1 + a))) := by\n        have h\u2084\u2083 : (4 : \u211d) + (4 : \u211d) * Real.sqrt (1 + a) = 4 * (1 + Real.sqrt (1 + a)) := by ring\n        rw [h\u2084\u2083]\n        <;> ring_nf\n        <;> field_simp\n        <;> ring_nf\n      rw [h\u2084\u2082]\n      have h\u2084\u2084 : Real.sqrt (4 * (1 + Real.sqrt (1 + a))) = 2 * Real.sqrt (1 + Real.sqrt (1 + a)) := by\n        have h\u2084\u2085 : Real.sqrt (4 * (1 + Real.sqrt (1 + a))) = Real.sqrt 4 * Real.sqrt (1 + Real.sqrt (1 + a)) := by\n          rw [\u2190 Real.sqrt_mul] <;> nlinarith [Real.sqrt_nonneg (1 + a), Real.sqrt_nonneg (1 + Real.sqrt (1 + a)), Real.sq_sqrt (show 0 \u2264 1 + a by linarith), Real.sq_sqrt (show 0 \u2264 1 + Real.sqrt (1 + a) by positivity)]\n        rw [h\u2084\u2085]\n        have h\u2084\u2086 : Real.sqrt 4 = 2 := by norm_num [Real.sqrt_eq_iff_sq_eq]\n        rw [h\u2084\u2086]\n        <;> ring_nf\n        <;> field_simp\n        <;> ring_nf\n      rw [h\u2084\u2084]\n      <;> ring_nf\n      <;> field_simp\n      <;> ring_nf\n    rw [h\u2084\u2081]\n    <;> ring_nf\n    <;> field_simp\n    <;> ring_nf\n  have h\u2085 : Real.sqrt (1 + Real.sqrt (1 + a)) = 2 := by\n    have h\u2085\u2081 : Real.sqrt (4 + Real.sqrt (16 + 16 * a)) + Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := h\u2080\n    rw [h\u2084] at h\u2085\u2081\n    have h\u2085\u2082 : 2 * Real.sqrt (1 + Real.sqrt (1 + a)) + Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := by linarith\n    have h\u2085\u2083 : 3 * Real.sqrt (1 + Real.sqrt (1 + a)) = 6 := by linarith\n    have h\u2085\u2084 : Real.sqrt (1 + Real.sqrt (1 + a)) = 2 := by linarith\n    exact h\u2085\u2084\n  have h\u2086 : 1 + Real.sqrt (1 + a) = 4 := by\n    have h\u2086\u2081 : Real.sqrt (1 + Real.sqrt (1 + a)) = 2 := h\u2085\n    have h\u2086\u2082 : Real.sqrt (1 + Real.sqrt (1 + a)) ^ 2 = 2 ^ 2 := by rw [h\u2086\u2081]\n    have h\u2086\u2083 : 1 + Real.sqrt (1 + a) \u2265 0 := by\n      by_contra h\u2086\u2083\n      have h\u2086\u2084 : 1 + Real.sqrt (1 + a) < 0 := by linarith\n      have h\u2086\u2085 : Real.sqrt (1 + Real.sqrt (1 + a)) = 0 := by\n        rw [Real.sqrt_eq_zero_of_nonpos] <;> linarith\n      rw [h\u2086\u2085] at h\u2086\u2081\n      norm_num at h\u2086\u2081\n      <;> linarith\n    have h\u2086\u2084 : 1 + Real.sqrt (1 + a) = 4 := by\n      nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 (1 + Real.sqrt (1 + a) : \u211d))]\n    exact h\u2086\u2084\n  have h\u2087 : Real.sqrt (1 + a) = 3 := by\n    have h\u2087\u2081 : 1 + Real.sqrt (1 + a) = 4 := h\u2086\n    have h\u2087\u2082 : Real.sqrt (1 + a) = 3 := by linarith\n    exact h\u2087\u2082\n  have h\u2088 : 1 + a = 9 := by\n    have h\u2088\u2081 : Real.sqrt (1 + a) = 3 := h\u2087\n    have h\u2088\u2082 : Real.sqrt (1 + a) ^ 2 = 3 ^ 2 := by rw [h\u2088\u2081]\n    have h\u2088\u2083 : 1 + a \u2265 0 := by linarith\n    have h\u2088\u2084 : 1 + a = 9 := by\n      nlinarith [Real.sq_sqrt (by linarith : 0 \u2264 (1 + a : \u211d))]\n    exact h\u2088\u2084\n  have h\u2089 : a = 8 := by\n    linarith\n  exact h\u2089\n"}
{"mode": "clear", "steps_before": 4, "alphabet": 12, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 221, "chars_after": 6, "seconds": 1.2, "id": "mathd_numbertheory_254", "proof_after": "theorem mathd_numbertheory_254 : (239 + 174 + 83) % 10 = 6  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_254 : (239 + 174 + 83) % 10 = 6  :=  by\n  have h\u2081 : (239 + 174 + 83) % 10 = 6 := by\n    norm_num [Nat.add_mod, Nat.mod_mod, Nat.mod_eq_of_lt, Nat.mod_eq_of_lt,\n      Nat.mod_eq_of_lt, Nat.mod_eq_of_lt, Nat.mod_eq_of_lt, Nat.mod_eq_of_lt]\n    <;> rfl\n  apply h\u2081\n"}
{"mode": "clear", "steps_before": 130, "alphabet": 68, "compiles": false, "id": "mathd_numbertheory_495", "proof_after": "theorem mathd_numbertheory_495 (a b : \u2115) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : a % 10 = 2) (h\u2082 : b % 10 = 4)\n    (h\u2083 : Nat.gcd a b = 6) : 108 \u2264 Nat.lcm a b  :=  by\n  have h\u2084 : 12 \u2264 a := by\n    by_contra h\n    have h\u2085 : a \u2264 11 := by linarith\n    have h\u2086 : a % 10 = 2 := h\u2081\n    have h\u2087 : 6 \u2223 a := by\n      have h\u2087\u2081 : Nat.gcd a b = 6 := h\u2083\n      have h\u2087\u2082 : 6 \u2223 a := by\n        have h\u2087\u2083 : 6 \u2223 Nat.gcd a b := by simp [h\u2083]\n        exact Nat.dvd_trans h\u2087\u2083 (Nat.gcd_dvd_left a b)\n      exact h\u2087\u2082\n    interval_cases a <;> norm_num at h\u2086 h\u2087 \u22a2 <;> omega\n  have h\u2085 : 24 \u2264 b := by\n    by_contra h\n    have h\u2085\u2081 : b \u2264 23 := by linarith\n    have h\u2085\u2082 : b % 10 = 4 := h\u2082\n    have h\u2085\u2083 : 6 \u2223 b := by\n      have h\u2085\u2084 : 6 \u2223 Nat.gcd a b := by simp [h\u2083]\n      have h\u2085\u2085 : 6 \u2223 b := by\n        have h\u2085\u2085\u2081 : 6 \u2223 Nat.gcd a b := h\u2085\u2084\n        have h\u2085\u2085\u2082 : 6 \u2223 b := Nat.dvd_trans h\u2085\u2085\u2081 (Nat.gcd_dvd_right a b)\n        exact h\u2085\u2085\u2082\n      exact h\u2085\u2085\n    interval_cases b <;> norm_num at h\u2085\u2082 h\u2085\u2083 \u22a2 <;> omega\n  have h\u2086 : a \u2260 0 := by linarith\n  have h\u2087 : b \u2260 0 := by linarith\n  have h\u2088 : a * b = 6 * Nat.lcm a b := by\n    have h\u2088\u2081 : Nat.gcd a b = 6 := h\u2083\n    have h\u2088 : a * b = Nat.gcd a b * Nat.lcm a b := by\n      rw [Nat.gcd_mul_lcm]\n    rw [h\u2088\u2081] at h\u2088\n    <;> linarith\n  have h\u2089 : a = 12 \u2192 54 \u2264 b := by\n    intro h\u2089\u2081\n    by_contra h\u2089\u2082\n    have h\u2089\u2083 : b \u2264 53 := by linarith\n    have h\u2089\u2084 : 6 \u2223 b := by\n      have h\u2089\u2085 : 6 \u2223 Nat.gcd a b := by simp [h\u2083]\n      have h\u2089\u2086 : 6 \u2223 b := by\n        have h\u2089\u2086\u2081 : 6 \u2223 Nat.gcd a b := h\u2089\u2085\n        have h\u2089\u2086\u2082 : 6 \u2223 b := Nat.dvd_trans h\u2089\u2086\u2081 (Nat.gcd_dvd_right a b)\n        exact h\u2089\u2086\u2082\n      exact h\u2089\u2086\n    have h\u2089\u2085 : b % 10 = 4 := h\u2082\n    have h\u2089\u2086 : a = 12 := h\u2089\u2081\n    have h\u2089\u2087 : Nat.gcd a b = 6 := h\u2083\n    have h\u2089\u2088 : a = 12 := h\u2089\u2081\n    have h\u2089\u2089 : Nat.gcd 12 b = 6 := by simpa [h\u2089\u2088] using h\u2083\n    interval_cases b <;> norm_num [Nat.gcd_eq_right, Nat.gcd_eq_left] at h\u2089\u2084 h\u2089\u2085 h\u2089\u2089 \u22a2 <;>\n      (try omega) <;> (try contradiction) <;> (try nlinarith)\n  have h\u2081\u2080 : a = 12 \u2192 648 \u2264 a * b := by\n    intro h\u2081\u2080\u2081\n    have h\u2081\u2080\u2082 : 54 \u2264 b := h\u2089 h\u2081\u2080\u2081\n    have h\u2081\u2080\u2083 : a = 12 := h\u2081\u2080\u2081\n    have h\u2081\u2080\u2084 : a * b \u2265 12 * 54 := by\n      calc\n        a * b = 12 * b := by rw [h\u2081\u2080\u2083]\n        _ \u2265 12 * 54 := by\n          have h\u2081\u2080\u2085 : 54 \u2264 b := h\u2081\u2080\u2082\n          nlinarith\n        _ = 12 * 54 := by ring\n    nlinarith\n  have h\u2081\u2081 : a > 12 \u2192 42 \u2264 a := by\n    intro h\u2081\u2081\u2081\n    have h\u2081\u2081\u2082 : a \u2265 12 := by linarith\n    have h\u2081\u2081\u2083 : a % 10 = 2 := h\u2081\n    have h\u2081\u2081\u2084 : 6 \u2223 a := by\n      have h\u2081\u2081\u2085 : 6 \u2223 Nat.gcd a b := by simp [h\u2083]\n      have h\u2081\u2081\u2086 : 6 \u2223 a := by\n        have h\u2081\u2081\u2086\u2081 : 6 \u2223 Nat.gcd a b := h\u2081\u2081\u2085\n        have h\u2081\u2081\u2086\u2082 : 6 \u2223 a := Nat.dvd_trans h\u2081\u2081\u2086\u2081 (Nat.gcd_dvd_left a b)\n        exact h\u2081\u2081\u2086\u2082\n      exact h\u2081\u2081\u2086\n    by_contra h\u2081\u2081\u2085\n    have h\u2081\u2081\u2086 : a < 42 := by linarith\n    have h\u2081\u2081\u2087 : a > 12 := h\u2081\u2081\u2081\n    have h\u2081\u2081\u2088 : a \u2264 41 := by linarith\n    interval_cases a <;> norm_num at h\u2081\u2081\u2083 h\u2081\u2081\u2084 \u22a2 <;> omega\n  have h\u2081\u2082 : a > 12 \u2192 1008 \u2264 a * b := by\n    intro h\u2081\u2082\u2081\n    have h\u2081\u2082\u2082 : 42 \u2264 a := h\u2081\u2081 h\u2081\u2082\u2081\n    have h\u2081\u2082\u2083 : 24 \u2264 b := h\u2085\n    have h\u2081\u2082\u2084 : a * b \u2265 42 * 24 := by\n      nlinarith\n    nlinarith\n  have h\u2081\u2083 : 648 \u2264 a * b := by\n    by_cases h\u2081\u2083\u2081 : a = 12\n    \u00b7 have h\u2081\u2083\u2082 : 648 \u2264 a * b := h\u2081\u2080 h\u2081\u2083\u2081\n      exact h\u2081\u2083\u2082\n    \u00b7 have h\u2081\u2083\u2083 : a > 12 := by\n        by_contra h\u2081\u2083\u2084\n        have h\u2081\u2083\u2085 : a \u2264 12 := by linarith\n        have h\u2081\u2083\u2086 : a \u2265 12 := h\u2084\n        have h\u2081\u2083\u2087 : a = 12 := by omega\n        contradiction\n      have h\u2081\u2083\u2084 : 1008 \u2264 a * b := h\u2081\u2082 h\u2081\u2083\u2083\n      linarith\n  have h\u2081\u2084 : 108 \u2264 Nat.lcm a b := by\n    have h\u2081\u2084\u2081 : a * b = 6 * Nat.lcm a b := h\u2088\n    have h\u2081\u2084\u2082 : 648 \u2264 a * b := h\u2081\u2083\n    have h\u2081\u2084\u2083 : 108 \u2264 Nat.lcm a b := by\n      omega\n    exact h\u2081\u2084\u2083\n  exact h\u2081\u2084\n", "proof_before": "theorem mathd_numbertheory_495 (a b : \u2115) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : a % 10 = 2) (h\u2082 : b % 10 = 4)\n    (h\u2083 : Nat.gcd a b = 6) : 108 \u2264 Nat.lcm a b  :=  by\n  have h\u2084 : 12 \u2264 a := by\n    by_contra h\n    have h\u2085 : a \u2264 11 := by linarith\n    have h\u2086 : a % 10 = 2 := h\u2081\n    have h\u2087 : 6 \u2223 a := by\n      have h\u2087\u2081 : Nat.gcd a b = 6 := h\u2083\n      have h\u2087\u2082 : 6 \u2223 a := by\n        have h\u2087\u2083 : 6 \u2223 Nat.gcd a b := by simp [h\u2083]\n        exact Nat.dvd_trans h\u2087\u2083 (Nat.gcd_dvd_left a b)\n      exact h\u2087\u2082\n    interval_cases a <;> norm_num at h\u2086 h\u2087 \u22a2 <;> omega\n  have h\u2085 : 24 \u2264 b := by\n    by_contra h\n    have h\u2085\u2081 : b \u2264 23 := by linarith\n    have h\u2085\u2082 : b % 10 = 4 := h\u2082\n    have h\u2085\u2083 : 6 \u2223 b := by\n      have h\u2085\u2084 : 6 \u2223 Nat.gcd a b := by simp [h\u2083]\n      have h\u2085\u2085 : 6 \u2223 b := by\n        have h\u2085\u2085\u2081 : 6 \u2223 Nat.gcd a b := h\u2085\u2084\n        have h\u2085\u2085\u2082 : 6 \u2223 b := Nat.dvd_trans h\u2085\u2085\u2081 (Nat.gcd_dvd_right a b)\n        exact h\u2085\u2085\u2082\n      exact h\u2085\u2085\n    interval_cases b <;> norm_num at h\u2085\u2082 h\u2085\u2083 \u22a2 <;> omega\n  have h\u2086 : a \u2260 0 := by linarith\n  have h\u2087 : b \u2260 0 := by linarith\n  have h\u2088 : a * b = 6 * Nat.lcm a b := by\n    have h\u2088\u2081 : Nat.gcd a b = 6 := h\u2083\n    have h\u2088 : a * b = Nat.gcd a b * Nat.lcm a b := by\n      rw [Nat.gcd_mul_lcm]\n    rw [h\u2088\u2081] at h\u2088\n    <;> linarith\n  have h\u2089 : a = 12 \u2192 54 \u2264 b := by\n    intro h\u2089\u2081\n    by_contra h\u2089\u2082\n    have h\u2089\u2083 : b \u2264 53 := by linarith\n    have h\u2089\u2084 : 6 \u2223 b := by\n      have h\u2089\u2085 : 6 \u2223 Nat.gcd a b := by simp [h\u2083]\n      have h\u2089\u2086 : 6 \u2223 b := by\n        have h\u2089\u2086\u2081 : 6 \u2223 Nat.gcd a b := h\u2089\u2085\n        have h\u2089\u2086\u2082 : 6 \u2223 b := Nat.dvd_trans h\u2089\u2086\u2081 (Nat.gcd_dvd_right a b)\n        exact h\u2089\u2086\u2082\n      exact h\u2089\u2086\n    have h\u2089\u2085 : b % 10 = 4 := h\u2082\n    have h\u2089\u2086 : a = 12 := h\u2089\u2081\n    have h\u2089\u2087 : Nat.gcd a b = 6 := h\u2083\n    have h\u2089\u2088 : a = 12 := h\u2089\u2081\n    have h\u2089\u2089 : Nat.gcd 12 b = 6 := by simpa [h\u2089\u2088] using h\u2083\n    interval_cases b <;> norm_num [Nat.gcd_eq_right, Nat.gcd_eq_left] at h\u2089\u2084 h\u2089\u2085 h\u2089\u2089 \u22a2 <;>\n      (try omega) <;> (try contradiction) <;> (try nlinarith)\n  have h\u2081\u2080 : a = 12 \u2192 648 \u2264 a * b := by\n    intro h\u2081\u2080\u2081\n    have h\u2081\u2080\u2082 : 54 \u2264 b := h\u2089 h\u2081\u2080\u2081\n    have h\u2081\u2080\u2083 : a = 12 := h\u2081\u2080\u2081\n    have h\u2081\u2080\u2084 : a * b \u2265 12 * 54 := by\n      calc\n        a * b = 12 * b := by rw [h\u2081\u2080\u2083]\n        _ \u2265 12 * 54 := by\n          have h\u2081\u2080\u2085 : 54 \u2264 b := h\u2081\u2080\u2082\n          nlinarith\n        _ = 12 * 54 := by ring\n    nlinarith\n  have h\u2081\u2081 : a > 12 \u2192 42 \u2264 a := by\n    intro h\u2081\u2081\u2081\n    have h\u2081\u2081\u2082 : a \u2265 12 := by linarith\n    have h\u2081\u2081\u2083 : a % 10 = 2 := h\u2081\n    have h\u2081\u2081\u2084 : 6 \u2223 a := by\n      have h\u2081\u2081\u2085 : 6 \u2223 Nat.gcd a b := by simp [h\u2083]\n      have h\u2081\u2081\u2086 : 6 \u2223 a := by\n        have h\u2081\u2081\u2086\u2081 : 6 \u2223 Nat.gcd a b := h\u2081\u2081\u2085\n        have h\u2081\u2081\u2086\u2082 : 6 \u2223 a := Nat.dvd_trans h\u2081\u2081\u2086\u2081 (Nat.gcd_dvd_left a b)\n        exact h\u2081\u2081\u2086\u2082\n      exact h\u2081\u2081\u2086\n    by_contra h\u2081\u2081\u2085\n    have h\u2081\u2081\u2086 : a < 42 := by linarith\n    have h\u2081\u2081\u2087 : a > 12 := h\u2081\u2081\u2081\n    have h\u2081\u2081\u2088 : a \u2264 41 := by linarith\n    interval_cases a <;> norm_num at h\u2081\u2081\u2083 h\u2081\u2081\u2084 \u22a2 <;> omega\n  have h\u2081\u2082 : a > 12 \u2192 1008 \u2264 a * b := by\n    intro h\u2081\u2082\u2081\n    have h\u2081\u2082\u2082 : 42 \u2264 a := h\u2081\u2081 h\u2081\u2082\u2081\n    have h\u2081\u2082\u2083 : 24 \u2264 b := h\u2085\n    have h\u2081\u2082\u2084 : a * b \u2265 42 * 24 := by\n      nlinarith\n    nlinarith\n  have h\u2081\u2083 : 648 \u2264 a * b := by\n    by_cases h\u2081\u2083\u2081 : a = 12\n    \u00b7 have h\u2081\u2083\u2082 : 648 \u2264 a * b := h\u2081\u2080 h\u2081\u2083\u2081\n      exact h\u2081\u2083\u2082\n    \u00b7 have h\u2081\u2083\u2083 : a > 12 := by\n        by_contra h\u2081\u2083\u2084\n        have h\u2081\u2083\u2085 : a \u2264 12 := by linarith\n        have h\u2081\u2083\u2086 : a \u2265 12 := h\u2084\n        have h\u2081\u2083\u2087 : a = 12 := by omega\n        contradiction\n      have h\u2081\u2083\u2084 : 1008 \u2264 a * b := h\u2081\u2082 h\u2081\u2083\u2083\n      linarith\n  have h\u2081\u2084 : 108 \u2264 Nat.lcm a b := by\n    have h\u2081\u2084\u2081 : a * b = 6 * Nat.lcm a b := h\u2088\n    have h\u2081\u2084\u2082 : 648 \u2264 a * b := h\u2081\u2083\n    have h\u2081\u2084\u2083 : 108 \u2264 Nat.lcm a b := by\n      omega\n    exact h\u2081\u2084\u2083\n  exact h\u2081\u2084\n"}
{"mode": "clear", "steps_before": 150, "alphabet": 69, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 150, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 4502, "chars_after": 4502, "seconds": 370.6, "id": "aime_1991_p9", "proof_after": "theorem aime_1991_p9 (x : \u211d) (m : \u211a) (h\u2080 : 1 / Real.cos x + Real.tan x = 22 / 7)\n    (h\u2081 : 1 / Real.sin x + 1 / Real.tan x = m) : \u2191m.den + m.num = 44  :=  by\n  have h_cos_ne_zero : Real.cos x \u2260 0 := by\n    by_contra h\n    have h\u2082 : Real.cos x = 0 := by simpa using h\n    have h\u2083 : Real.tan x = 0 := by\n      rw [Real.tan_eq_sin_div_cos]\n      rw [h\u2082]\n      simp\n    have h\u2084 : 1 / Real.cos x + Real.tan x = 0 := by\n      rw [h\u2082, h\u2083]\n      <;> norm_num\n    rw [h\u2084] at h\u2080\n    norm_num at h\u2080\n    <;> linarith\n  have h_sin_ne_zero : Real.sin x \u2260 0 := by\n    by_contra h\n    have h\u2082 : Real.sin x = 0 := by simpa using h\n    have h\u2083 : Real.tan x = 0 := by\n      rw [Real.tan_eq_sin_div_cos]\n      rw [h\u2082]\n      simp\n    have h\u2084 : 1 / Real.cos x + Real.tan x = 22 / 7 := h\u2080\n    rw [h\u2083] at h\u2084\n    have h\u2085 : 1 / Real.cos x = 22 / 7 := by linarith\n    have h\u2086 : Real.cos x = 7 / 22 := by\n      have h\u2087 : Real.cos x \u2260 0 := h_cos_ne_zero\n      field_simp at h\u2085 \u22a2\n      <;> nlinarith [Real.sin_sq_add_cos_sq x]\n    have h\u2087 : Real.sin x ^ 2 + Real.cos x ^ 2 = 1 := by\n      rw [Real.sin_sq_add_cos_sq]\n    rw [h\u2082, h\u2086] at h\u2087\n    norm_num at h\u2087\n    <;> linarith\n  have h\u2082 : 1 + Real.sin x = (22 / 7 : \u211d) * Real.cos x := by\n    have h\u2083 : 1 / Real.cos x + Real.tan x = 22 / 7 := h\u2080\n    have h\u2084 : Real.tan x = Real.sin x / Real.cos x := by\n      rw [Real.tan_eq_sin_div_cos]\n    rw [h\u2084] at h\u2083\n    have h\u2085 : 1 / Real.cos x + Real.sin x / Real.cos x = 22 / 7 := h\u2083\n    have h\u2086 : (1 + Real.sin x) / Real.cos x = 22 / 7 := by\n      have h\u2087 : 1 / Real.cos x + Real.sin x / Real.cos x = (1 + Real.sin x) / Real.cos x := by\n        field_simp [h_cos_ne_zero]\n        <;> ring\n      rw [h\u2087] at h\u2085\n      exact h\u2085\n    have h\u2087 : Real.cos x \u2260 0 := h_cos_ne_zero\n    have h\u2088 : (1 + Real.sin x) = (22 / 7 : \u211d) * Real.cos x := by\n      field_simp [h\u2087] at h\u2086 \u22a2\n      <;> nlinarith [Real.sin_le_one (x), Real.cos_le_one (x), Real.sin_sq_add_cos_sq x]\n    exact h\u2088\n  have h\u2083 : Real.sin x = (22 / 7 : \u211d) * Real.cos x - 1 := by\n    linarith\n  have h\u2084 : Real.cos x = 308 / 533 := by\n    have h\u2085 : Real.sin x ^ 2 + Real.cos x ^ 2 = 1 := Real.sin_sq_add_cos_sq x\n    rw [h\u2083] at h\u2085\n    ring_nf at h\u2085 \u22a2\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h_cos_ne_zero)\n    nlinarith [Real.sin_le_one x, Real.cos_le_one x, Real.sin_le_one x, Real.cos_le_one x]\n  have h\u2085 : Real.sin x = 435 / 533 := by\n    rw [h\u2083]\n    rw [h\u2084]\n    <;> norm_num <;>\n    linarith [Real.sin_le_one x, Real.cos_le_one x]\n  have h\u2086 : (m : \u211d) = 841 / 435 := by\n    have h\u2087 : (m : \u211d) = 1 / Real.sin x + 1 / Real.tan x := by\n      rw [h\u2081]\n      <;> simp [h\u2085, h\u2084]\n      <;> ring_nf at *\n      <;> norm_num at *\n      <;> linarith\n    rw [h\u2087]\n    have h\u2088 : 1 / Real.tan x = Real.cos x / Real.sin x := by\n      rw [Real.tan_eq_sin_div_cos]\n      <;> field_simp [h_cos_ne_zero, h_sin_ne_zero]\n      <;> ring\n    rw [h\u2088]\n    have h\u2089 : (1 : \u211d) / Real.sin x + Real.cos x / Real.sin x = (1 + Real.cos x) / Real.sin x := by\n      field_simp [h_sin_ne_zero]\n      <;> ring\n    rw [h\u2089]\n    rw [h\u2085, h\u2084]\n    <;> norm_num\n    <;> field_simp\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have h\u2087 : m = 29 / 15 := by\n    norm_cast at h\u2086 \u22a2\n    <;>\n    norm_num [h\u2085, h\u2084, Real.sin_le_one, Real.cos_le_one] at h\u2086 \u22a2 <;>\n    field_simp [h_cos_ne_zero, h_sin_ne_zero, Real.tan_eq_sin_div_cos] at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    ring_nf at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    norm_num at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    norm_cast at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    field_simp at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    norm_cast at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    ring_nf at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    norm_num at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    norm_cast at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    (try norm_num) <;>\n    (try ring_nf at * ) <;>\n    (try norm_num at * ) <;>\n    (try linarith) <;>\n    (try\n      {\n        rcases m with \u27e8n, d, hn, hd\u27e9\n        norm_num at *\n        <;>\n        (try omega)\n      }) <;>\n    (try\n      {\n        rfl\n      })\n    <;>\n    (try\n      {\n        norm_num at *\n        <;>\n        linarith\n      })\n    <;>\n    (try\n      {\n        field_simp at *\n        <;>\n        ring_nf at *\n        <;>\n        norm_num at *\n        <;>\n        linarith\n      })\n    <;>\n    (try\n      {\n        norm_cast at *\n        <;>\n        ring_nf at *\n        <;>\n        norm_num at *\n        <;>\n        linarith\n      })\n    <;>\n    (try\n      {\n        ring_nf at *\n        <;>\n        norm_num at *\n        <;>\n        linarith\n      })\n  have h\u2088 : \u2191m.den + m.num = 44 := by\n    rw [h\u2087]\n    <;> norm_num [Rat.den_nz, Rat.num_div_den]\n    <;> rfl\n    <;> norm_cast\n    <;> simp_all [Rat.den_nz, Rat.num_div_den]\n    <;> norm_num\n    <;> rfl\n  exact h\u2088\n", "proof_before": "theorem aime_1991_p9 (x : \u211d) (m : \u211a) (h\u2080 : 1 / Real.cos x + Real.tan x = 22 / 7)\n    (h\u2081 : 1 / Real.sin x + 1 / Real.tan x = m) : \u2191m.den + m.num = 44  :=  by\n  have h_cos_ne_zero : Real.cos x \u2260 0 := by\n    by_contra h\n    have h\u2082 : Real.cos x = 0 := by simpa using h\n    have h\u2083 : Real.tan x = 0 := by\n      rw [Real.tan_eq_sin_div_cos]\n      rw [h\u2082]\n      simp\n    have h\u2084 : 1 / Real.cos x + Real.tan x = 0 := by\n      rw [h\u2082, h\u2083]\n      <;> norm_num\n    rw [h\u2084] at h\u2080\n    norm_num at h\u2080\n    <;> linarith\n  have h_sin_ne_zero : Real.sin x \u2260 0 := by\n    by_contra h\n    have h\u2082 : Real.sin x = 0 := by simpa using h\n    have h\u2083 : Real.tan x = 0 := by\n      rw [Real.tan_eq_sin_div_cos]\n      rw [h\u2082]\n      simp\n    have h\u2084 : 1 / Real.cos x + Real.tan x = 22 / 7 := h\u2080\n    rw [h\u2083] at h\u2084\n    have h\u2085 : 1 / Real.cos x = 22 / 7 := by linarith\n    have h\u2086 : Real.cos x = 7 / 22 := by\n      have h\u2087 : Real.cos x \u2260 0 := h_cos_ne_zero\n      field_simp at h\u2085 \u22a2\n      <;> nlinarith [Real.sin_sq_add_cos_sq x]\n    have h\u2087 : Real.sin x ^ 2 + Real.cos x ^ 2 = 1 := by\n      rw [Real.sin_sq_add_cos_sq]\n    rw [h\u2082, h\u2086] at h\u2087\n    norm_num at h\u2087\n    <;> linarith\n  have h\u2082 : 1 + Real.sin x = (22 / 7 : \u211d) * Real.cos x := by\n    have h\u2083 : 1 / Real.cos x + Real.tan x = 22 / 7 := h\u2080\n    have h\u2084 : Real.tan x = Real.sin x / Real.cos x := by\n      rw [Real.tan_eq_sin_div_cos]\n    rw [h\u2084] at h\u2083\n    have h\u2085 : 1 / Real.cos x + Real.sin x / Real.cos x = 22 / 7 := h\u2083\n    have h\u2086 : (1 + Real.sin x) / Real.cos x = 22 / 7 := by\n      have h\u2087 : 1 / Real.cos x + Real.sin x / Real.cos x = (1 + Real.sin x) / Real.cos x := by\n        field_simp [h_cos_ne_zero]\n        <;> ring\n      rw [h\u2087] at h\u2085\n      exact h\u2085\n    have h\u2087 : Real.cos x \u2260 0 := h_cos_ne_zero\n    have h\u2088 : (1 + Real.sin x) = (22 / 7 : \u211d) * Real.cos x := by\n      field_simp [h\u2087] at h\u2086 \u22a2\n      <;> nlinarith [Real.sin_le_one (x), Real.cos_le_one (x), Real.sin_sq_add_cos_sq x]\n    exact h\u2088\n  have h\u2083 : Real.sin x = (22 / 7 : \u211d) * Real.cos x - 1 := by\n    linarith\n  have h\u2084 : Real.cos x = 308 / 533 := by\n    have h\u2085 : Real.sin x ^ 2 + Real.cos x ^ 2 = 1 := Real.sin_sq_add_cos_sq x\n    rw [h\u2083] at h\u2085\n    ring_nf at h\u2085 \u22a2\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h_cos_ne_zero)\n    nlinarith [Real.sin_le_one x, Real.cos_le_one x, Real.sin_le_one x, Real.cos_le_one x]\n  have h\u2085 : Real.sin x = 435 / 533 := by\n    rw [h\u2083]\n    rw [h\u2084]\n    <;> norm_num <;>\n    linarith [Real.sin_le_one x, Real.cos_le_one x]\n  have h\u2086 : (m : \u211d) = 841 / 435 := by\n    have h\u2087 : (m : \u211d) = 1 / Real.sin x + 1 / Real.tan x := by\n      rw [h\u2081]\n      <;> simp [h\u2085, h\u2084]\n      <;> ring_nf at *\n      <;> norm_num at *\n      <;> linarith\n    rw [h\u2087]\n    have h\u2088 : 1 / Real.tan x = Real.cos x / Real.sin x := by\n      rw [Real.tan_eq_sin_div_cos]\n      <;> field_simp [h_cos_ne_zero, h_sin_ne_zero]\n      <;> ring\n    rw [h\u2088]\n    have h\u2089 : (1 : \u211d) / Real.sin x + Real.cos x / Real.sin x = (1 + Real.cos x) / Real.sin x := by\n      field_simp [h_sin_ne_zero]\n      <;> ring\n    rw [h\u2089]\n    rw [h\u2085, h\u2084]\n    <;> norm_num\n    <;> field_simp\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have h\u2087 : m = 29 / 15 := by\n    norm_cast at h\u2086 \u22a2\n    <;>\n    norm_num [h\u2085, h\u2084, Real.sin_le_one, Real.cos_le_one] at h\u2086 \u22a2 <;>\n    field_simp [h_cos_ne_zero, h_sin_ne_zero, Real.tan_eq_sin_div_cos] at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    ring_nf at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    norm_num at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    norm_cast at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    field_simp at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    norm_cast at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    ring_nf at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    norm_num at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    norm_cast at h\u2080 h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2 <;>\n    (try norm_num) <;>\n    (try ring_nf at * ) <;>\n    (try norm_num at * ) <;>\n    (try linarith) <;>\n    (try\n      {\n        rcases m with \u27e8n, d, hn, hd\u27e9\n        norm_num at *\n        <;>\n        (try omega)\n      }) <;>\n    (try\n      {\n        rfl\n      })\n    <;>\n    (try\n      {\n        norm_num at *\n        <;>\n        linarith\n      })\n    <;>\n    (try\n      {\n        field_simp at *\n        <;>\n        ring_nf at *\n        <;>\n        norm_num at *\n        <;>\n        linarith\n      })\n    <;>\n    (try\n      {\n        norm_cast at *\n        <;>\n        ring_nf at *\n        <;>\n        norm_num at *\n        <;>\n        linarith\n      })\n    <;>\n    (try\n      {\n        ring_nf at *\n        <;>\n        norm_num at *\n        <;>\n        linarith\n      })\n  have h\u2088 : \u2191m.den + m.num = 44 := by\n    rw [h\u2087]\n    <;> norm_num [Rat.den_nz, Rat.num_div_den]\n    <;> rfl\n    <;> norm_cast\n    <;> simp_all [Rat.den_nz, Rat.num_div_den]\n    <;> norm_num\n    <;> rfl\n  exact h\u2088\n"}
{"mode": "clear", "steps_before": 108, "alphabet": 44, "compiles": false, "id": "algebra_sum1onsqrt2to1onsqrt10000lt198", "proof_after": "theorem algebra_sum1onsqrt2to1onsqrt10000lt198 :\n    (\u2211 k in Finset.Icc (2 : \u2115) 10000, 1 / Real.sqrt k) < 198  :=  by\n  have h\u2081 : \u2200 (k : \u2115), k \u2208 Finset.Icc (2 : \u2115) 10000 \u2192 (1 : \u211d) / Real.sqrt k < 2 * (Real.sqrt k - Real.sqrt (k - 1)) := by\n    intro k hk\n    have h\u2082 : 2 \u2264 k \u2227 k \u2264 10000 := by simpa using Finset.mem_Icc.mp hk\n    have h\u2083 : (k : \u211d) \u2265 2 := by exact_mod_cast h\u2082.1\n    have h\u2084 : (k : \u211d) \u2264 10000 := by exact_mod_cast h\u2082.2\n    have h\u2085 : (k : \u211d) - 1 \u2265 1 := by\n      have h\u2085\u2081 : (k : \u211d) \u2265 2 := h\u2083\n      linarith\n    have h\u2086 : Real.sqrt (k : \u211d) \u2265 0 := Real.sqrt_nonneg _\n    have h\u2087 : Real.sqrt ((k : \u211d) - 1) \u2265 0 := Real.sqrt_nonneg _\n    have h\u2088 : Real.sqrt (k : \u211d) > 0 := Real.sqrt_pos.mpr (by linarith)\n    have h\u2089 : Real.sqrt ((k : \u211d) - 1) > 0 := Real.sqrt_pos.mpr (by linarith)\n    have h\u2081\u2080 : (1 : \u211d) / Real.sqrt k < 2 * (Real.sqrt k - Real.sqrt (k - 1)) := by\n      have h\u2081\u2080\u2081 : Real.sqrt k > Real.sqrt ((k : \u211d) - 1) := by\n        apply Real.sqrt_lt_sqrt\n        <;> nlinarith\n      have h\u2081\u2080\u2082 : Real.sqrt k - Real.sqrt ((k : \u211d) - 1) > 0 := by linarith\n      have h\u2081\u2080\u2083 : 2 * (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) > 0 := by positivity\n      have h\u2081\u2080\u2084 : (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) * (Real.sqrt k + Real.sqrt ((k : \u211d) - 1)) = 1 := by\n        nlinarith [Real.sq_sqrt (show 0 \u2264 (k : \u211d) by linarith),\n          Real.sq_sqrt (show 0 \u2264 (k : \u211d) - 1 by linarith),\n          mul_nonneg h\u2086 h\u2087, h\u2088.le, h\u2089.le]\n      have h\u2081\u2080\u2085 : (Real.sqrt k + Real.sqrt ((k : \u211d) - 1)) > 0 := by positivity\n      have h\u2081\u2080\u2086 : 2 * (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) > 1 / Real.sqrt k := by\n        have h\u2081\u2080\u2087 : (1 : \u211d) / Real.sqrt k < 2 * (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) := by\n          have h\u2081\u2080\u2088 : (1 : \u211d) / Real.sqrt k < 2 * (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) := by\n            have h\u2081\u2080\u2089 : (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) = 1 / (Real.sqrt k + Real.sqrt ((k : \u211d) - 1)) := by\n              field_simp [h\u2081\u2080\u2085.ne']\n              <;> nlinarith [Real.sq_sqrt (show 0 \u2264 (k : \u211d) by linarith),\n                Real.sq_sqrt (show 0 \u2264 (k : \u211d) - 1 by linarith),\n                mul_nonneg h\u2086 h\u2087, h\u2088.le, h\u2089.le]\n            rw [h\u2081\u2080\u2089]\n            have h\u2081\u2081\u2080 : (1 : \u211d) / Real.sqrt k < 2 * (1 / (Real.sqrt k + Real.sqrt ((k : \u211d) - 1))) := by\n              have h\u2081\u2081\u2081 : 0 < Real.sqrt k + Real.sqrt ((k : \u211d) - 1) := by positivity\n              have h\u2081\u2081\u2082 : 0 < Real.sqrt k * (Real.sqrt k + Real.sqrt ((k : \u211d) - 1)) := by positivity\n              field_simp [h\u2081\u2081\u2081.ne']\n              rw [div_lt_div_iff (by positivity) (by positivity)]\n              nlinarith [Real.sq_sqrt (show 0 \u2264 (k : \u211d) by linarith),\n                Real.sq_sqrt (show 0 \u2264 (k : \u211d) - 1 by linarith),\n                mul_nonneg h\u2086 h\u2087, h\u2088.le, h\u2089.le]\n            linarith\n          linarith\n        linarith\n      linarith\n    exact h\u2081\u2080\n  have h\u2082 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 198 := by\n    have h\u2083 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 2 * (Real.sqrt (10000 : \u211d) - Real.sqrt (1 : \u211d)) := by\n      have h\u2084 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 2 * (Real.sqrt (10000 : \u211d) - Real.sqrt (1 : \u211d)) := by\n        have h\u2085 : \u2200 (n : \u2115), n \u2265 2 \u2192 (\u2211 k in Finset.Icc (2 : \u2115) n, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 2 * (Real.sqrt (n : \u211d) - Real.sqrt (1 : \u211d)) := by\n          intro n hn\n          induction' hn with n hn IH\n          \u00b7 norm_num [Finset.sum_Icc_succ_top]\n          \u00b7 cases n with\n            | zero => contradiction\n            | succ n =>\n              cases n with\n              | zero => contradiction\n              | succ n =>\n                simp_all [Finset.sum_Icc_succ_top, Nat.cast_add, Nat.cast_one, Nat.cast_zero, Nat.cast_succ]\n                <;> ring_nf at *\n                <;> field_simp at *\n                <;> ring_nf at *\n                <;> norm_num at *\n                <;>\n                (try\n                  {\n                    nlinarith [Real.sqrt_nonneg (n + 1 + 1 : \u211d), Real.sqrt_nonneg (n + 1 : \u211d),\n                      Real.sqrt_nonneg (n + 1 + 1 - 1 : \u211d), Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 - 1 : \u211d) by\n                        {\n                          norm_num\n                          <;> nlinarith\n                        })]\n                  })\n                <;>\n                (try\n                  {\n                    nlinarith [Real.sqrt_nonneg (n + 1 + 1 : \u211d), Real.sqrt_nonneg (n + 1 : \u211d),\n                      Real.sqrt_nonneg (n + 1 + 1 - 1 : \u211d), Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 - 1 : \u211d) by\n                        {\n                          norm_num\n                          <;> nlinarith\n                        })]\n                  })\n                <;>\n                (try\n                  {\n                    nlinarith [Real.sqrt_nonneg (n + 1 + 1 : \u211d), Real.sqrt_nonneg (n + 1 : \u211d),\n                      Real.sqrt_nonneg (n + 1 + 1 - 1 : \u211d), Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 - 1 : \u211d) by\n                        {\n                          norm_num\n                          <;> nlinarith\n                        })]\n                  })\n        have h\u2086 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 2 * (Real.sqrt (10000 : \u211d) - Real.sqrt (1 : \u211d)) := by\n          apply h\u2085\n          <;> norm_num\n        rw [h\u2086]\n      rw [h\u2084]\n    rw [h\u2083]\n    have h\u2084 : Real.sqrt (10000 : \u211d) = 100 := by\n      rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    have h\u2085 : Real.sqrt (1 : \u211d) = 1 := by\n      rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    rw [h\u2084, h\u2085]\n    <;> norm_num\n  have h\u2083 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 1 / Real.sqrt k) < 198 := by\n    have h\u2084 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, (1 : \u211d) / Real.sqrt k) < 198 := by\n      have h\u2085 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, (1 : \u211d) / Real.sqrt k) < \u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1)) := by\n        have h\u2085\u2081 : \u2200 (k : \u2115), k \u2208 Finset.Icc (2 : \u2115) 10000 \u2192 (1 : \u211d) / Real.sqrt k < 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1)) := by\n          intro k hk\n          exact h\u2081 k hk\n        have h\u2085\u2082 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, (1 : \u211d) / Real.sqrt k) < \u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1)) := by\n          apply Finset.sum_lt_sum_of_nonempty (Finset.nonempty_of_ne_empty (by decide))\n          intro k hk\n          exact h\u2085\u2081 k hk\n        simpa [Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.mul_sum] using h\u2085\u2082\n      have h\u2085\u2083 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 198 := by\n        exact h\u2082\n      have h\u2085\u2084 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, (1 : \u211d) / Real.sqrt k) < 198 := by\n        linarith\n      exact h\u2085\u2084\n    exact h\u2084\n  exact h\u2083\n", "proof_before": "theorem algebra_sum1onsqrt2to1onsqrt10000lt198 :\n    (\u2211 k in Finset.Icc (2 : \u2115) 10000, 1 / Real.sqrt k) < 198  :=  by\n  have h\u2081 : \u2200 (k : \u2115), k \u2208 Finset.Icc (2 : \u2115) 10000 \u2192 (1 : \u211d) / Real.sqrt k < 2 * (Real.sqrt k - Real.sqrt (k - 1)) := by\n    intro k hk\n    have h\u2082 : 2 \u2264 k \u2227 k \u2264 10000 := by simpa using Finset.mem_Icc.mp hk\n    have h\u2083 : (k : \u211d) \u2265 2 := by exact_mod_cast h\u2082.1\n    have h\u2084 : (k : \u211d) \u2264 10000 := by exact_mod_cast h\u2082.2\n    have h\u2085 : (k : \u211d) - 1 \u2265 1 := by\n      have h\u2085\u2081 : (k : \u211d) \u2265 2 := h\u2083\n      linarith\n    have h\u2086 : Real.sqrt (k : \u211d) \u2265 0 := Real.sqrt_nonneg _\n    have h\u2087 : Real.sqrt ((k : \u211d) - 1) \u2265 0 := Real.sqrt_nonneg _\n    have h\u2088 : Real.sqrt (k : \u211d) > 0 := Real.sqrt_pos.mpr (by linarith)\n    have h\u2089 : Real.sqrt ((k : \u211d) - 1) > 0 := Real.sqrt_pos.mpr (by linarith)\n    have h\u2081\u2080 : (1 : \u211d) / Real.sqrt k < 2 * (Real.sqrt k - Real.sqrt (k - 1)) := by\n      have h\u2081\u2080\u2081 : Real.sqrt k > Real.sqrt ((k : \u211d) - 1) := by\n        apply Real.sqrt_lt_sqrt\n        <;> nlinarith\n      have h\u2081\u2080\u2082 : Real.sqrt k - Real.sqrt ((k : \u211d) - 1) > 0 := by linarith\n      have h\u2081\u2080\u2083 : 2 * (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) > 0 := by positivity\n      have h\u2081\u2080\u2084 : (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) * (Real.sqrt k + Real.sqrt ((k : \u211d) - 1)) = 1 := by\n        nlinarith [Real.sq_sqrt (show 0 \u2264 (k : \u211d) by linarith),\n          Real.sq_sqrt (show 0 \u2264 (k : \u211d) - 1 by linarith),\n          mul_nonneg h\u2086 h\u2087, h\u2088.le, h\u2089.le]\n      have h\u2081\u2080\u2085 : (Real.sqrt k + Real.sqrt ((k : \u211d) - 1)) > 0 := by positivity\n      have h\u2081\u2080\u2086 : 2 * (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) > 1 / Real.sqrt k := by\n        have h\u2081\u2080\u2087 : (1 : \u211d) / Real.sqrt k < 2 * (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) := by\n          have h\u2081\u2080\u2088 : (1 : \u211d) / Real.sqrt k < 2 * (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) := by\n            have h\u2081\u2080\u2089 : (Real.sqrt k - Real.sqrt ((k : \u211d) - 1)) = 1 / (Real.sqrt k + Real.sqrt ((k : \u211d) - 1)) := by\n              field_simp [h\u2081\u2080\u2085.ne']\n              <;> nlinarith [Real.sq_sqrt (show 0 \u2264 (k : \u211d) by linarith),\n                Real.sq_sqrt (show 0 \u2264 (k : \u211d) - 1 by linarith),\n                mul_nonneg h\u2086 h\u2087, h\u2088.le, h\u2089.le]\n            rw [h\u2081\u2080\u2089]\n            have h\u2081\u2081\u2080 : (1 : \u211d) / Real.sqrt k < 2 * (1 / (Real.sqrt k + Real.sqrt ((k : \u211d) - 1))) := by\n              have h\u2081\u2081\u2081 : 0 < Real.sqrt k + Real.sqrt ((k : \u211d) - 1) := by positivity\n              have h\u2081\u2081\u2082 : 0 < Real.sqrt k * (Real.sqrt k + Real.sqrt ((k : \u211d) - 1)) := by positivity\n              field_simp [h\u2081\u2081\u2081.ne']\n              rw [div_lt_div_iff (by positivity) (by positivity)]\n              nlinarith [Real.sq_sqrt (show 0 \u2264 (k : \u211d) by linarith),\n                Real.sq_sqrt (show 0 \u2264 (k : \u211d) - 1 by linarith),\n                mul_nonneg h\u2086 h\u2087, h\u2088.le, h\u2089.le]\n            linarith\n          linarith\n        linarith\n      linarith\n    exact h\u2081\u2080\n  have h\u2082 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 198 := by\n    have h\u2083 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 2 * (Real.sqrt (10000 : \u211d) - Real.sqrt (1 : \u211d)) := by\n      have h\u2084 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 2 * (Real.sqrt (10000 : \u211d) - Real.sqrt (1 : \u211d)) := by\n        have h\u2085 : \u2200 (n : \u2115), n \u2265 2 \u2192 (\u2211 k in Finset.Icc (2 : \u2115) n, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 2 * (Real.sqrt (n : \u211d) - Real.sqrt (1 : \u211d)) := by\n          intro n hn\n          induction' hn with n hn IH\n          \u00b7 norm_num [Finset.sum_Icc_succ_top]\n          \u00b7 cases n with\n            | zero => contradiction\n            | succ n =>\n              cases n with\n              | zero => contradiction\n              | succ n =>\n                simp_all [Finset.sum_Icc_succ_top, Nat.cast_add, Nat.cast_one, Nat.cast_zero, Nat.cast_succ]\n                <;> ring_nf at *\n                <;> field_simp at *\n                <;> ring_nf at *\n                <;> norm_num at *\n                <;>\n                (try\n                  {\n                    nlinarith [Real.sqrt_nonneg (n + 1 + 1 : \u211d), Real.sqrt_nonneg (n + 1 : \u211d),\n                      Real.sqrt_nonneg (n + 1 + 1 - 1 : \u211d), Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 - 1 : \u211d) by\n                        {\n                          norm_num\n                          <;> nlinarith\n                        })]\n                  })\n                <;>\n                (try\n                  {\n                    nlinarith [Real.sqrt_nonneg (n + 1 + 1 : \u211d), Real.sqrt_nonneg (n + 1 : \u211d),\n                      Real.sqrt_nonneg (n + 1 + 1 - 1 : \u211d), Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 - 1 : \u211d) by\n                        {\n                          norm_num\n                          <;> nlinarith\n                        })]\n                  })\n                <;>\n                (try\n                  {\n                    nlinarith [Real.sqrt_nonneg (n + 1 + 1 : \u211d), Real.sqrt_nonneg (n + 1 : \u211d),\n                      Real.sqrt_nonneg (n + 1 + 1 - 1 : \u211d), Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 : \u211d) by positivity),\n                      Real.sq_sqrt (show (0 : \u211d) \u2264 (n + 1 + 1 - 1 : \u211d) by\n                        {\n                          norm_num\n                          <;> nlinarith\n                        })]\n                  })\n        have h\u2086 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 2 * (Real.sqrt (10000 : \u211d) - Real.sqrt (1 : \u211d)) := by\n          apply h\u2085\n          <;> norm_num\n        rw [h\u2086]\n      rw [h\u2084]\n    rw [h\u2083]\n    have h\u2084 : Real.sqrt (10000 : \u211d) = 100 := by\n      rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    have h\u2085 : Real.sqrt (1 : \u211d) = 1 := by\n      rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    rw [h\u2084, h\u2085]\n    <;> norm_num\n  have h\u2083 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 1 / Real.sqrt k) < 198 := by\n    have h\u2084 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, (1 : \u211d) / Real.sqrt k) < 198 := by\n      have h\u2085 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, (1 : \u211d) / Real.sqrt k) < \u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1)) := by\n        have h\u2085\u2081 : \u2200 (k : \u2115), k \u2208 Finset.Icc (2 : \u2115) 10000 \u2192 (1 : \u211d) / Real.sqrt k < 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1)) := by\n          intro k hk\n          exact h\u2081 k hk\n        have h\u2085\u2082 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, (1 : \u211d) / Real.sqrt k) < \u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1)) := by\n          apply Finset.sum_lt_sum_of_nonempty (Finset.nonempty_of_ne_empty (by decide))\n          intro k hk\n          exact h\u2085\u2081 k hk\n        simpa [Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.mul_sum] using h\u2085\u2082\n      have h\u2085\u2083 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, 2 * (Real.sqrt (k : \u211d) - Real.sqrt ((k : \u211d) - 1))) = 198 := by\n        exact h\u2082\n      have h\u2085\u2084 : (\u2211 k in Finset.Icc (2 : \u2115) 10000, (1 : \u211d) / Real.sqrt k) < 198 := by\n        linarith\n      exact h\u2085\u2084\n    exact h\u2084\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 15, "alphabet": 18, "compiles": true, "steps_after": 5, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 4, "fixpoint": true, "chars_before": 314, "chars_after": 91, "seconds": 419.3, "id": "mathd_algebra_44", "proof_after": "theorem mathd_algebra_44 (s t : \u211d) (h\u2080 : s = 9 - 2 * t) (h\u2081 : t = 3 * s + 1) : s = 1 \u2227 t = 4  :=  by\n  have h\u2082 : s = 1 := by\n    linarith\n  have h\u2083 : t = 4 := by\n    linarith\n  exact \u27e8h\u2082, h\u2083\u27e9\n", "proof_before": "theorem mathd_algebra_44 (s t : \u211d) (h\u2080 : s = 9 - 2 * t) (h\u2081 : t = 3 * s + 1) : s = 1 \u2227 t = 4  :=  by\n  have h\u2082 : s = 1 := by\n    have h\u2082\u2081 : s = 7 - 6 * s := by\n      rw [h\u2081] at h\u2080\n      ring_nf at h\u2080 \u22a2\n      linarith\n    have h\u2082\u2082 : s = 1 := by\n      linarith\n    exact h\u2082\u2082\n  have h\u2083 : t = 4 := by\n    rw [h\u2082] at h\u2081\n    norm_num at h\u2081 \u22a2\n    <;> linarith\n  have h\u2084 : s = 1 \u2227 t = 4 := by\n    exact \u27e8h\u2082, h\u2083\u27e9\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 24, "alphabet": 21, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 608, "chars_after": 11, "seconds": 54.3, "id": "mathd_algebra_142", "proof_after": "theorem mathd_algebra_142 (m b : \u211d) (h\u2080 : m * 7 + b = -1) (h\u2081 : m * -1 + b = 7) : m + b = 5  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_142 (m b : \u211d) (h\u2080 : m * 7 + b = -1) (h\u2081 : m * -1 + b = 7) : m + b = 5  :=  by\n  have h\u2082 : m = -1 := by\n    have h\u2082\u2081 : 8 * m = -8 := by\n      have h\u2082\u2082 : m * 7 + b = -1 := h\u2080\n      have h\u2082\u2083 : m * -1 + b = 7 := h\u2081\n      have h\u2082\u2084 : (m * 7 + b) - (m * -1 + b) = -1 - 7 := by\n        linarith\n      have h\u2082\u2085 : (m * 7 + b) - (m * -1 + b) = 8 * m := by\n        ring_nf at h\u2082\u2084 \u22a2 <;> linarith\n      have h\u2082\u2086 : 8 * m = -8 := by linarith\n      exact h\u2082\u2086\n    have h\u2082\u2087 : m = -1 := by linarith\n    exact h\u2082\u2087\n  have h\u2083 : b = 6 := by\n    have h\u2083\u2081 : m = -1 := h\u2082\n    have h\u2083\u2082 : m * -1 + b = 7 := h\u2081\n    rw [h\u2083\u2081] at h\u2083\u2082\n    linarith\n  have h\u2084 : m + b = 5 := by\n    rw [h\u2082, h\u2083]\n    <;> norm_num\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 21, "alphabet": 20, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 546, "chars_after": 11, "seconds": 65.4, "id": "mathd_algebra_107", "proof_after": "theorem mathd_algebra_107 (x y : \u211d) (h\u2080 : x ^ 2 + 8 * x + y ^ 2 - 6 * y = 0) :\n    (x + 4) ^ 2 + (y - 3) ^ 2 = 5 ^ 2  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_107 (x y : \u211d) (h\u2080 : x ^ 2 + 8 * x + y ^ 2 - 6 * y = 0) :\n    (x + 4) ^ 2 + (y - 3) ^ 2 = 5 ^ 2  :=  by\n  have h\u2081 : (x + 4) ^ 2 + (y - 3) ^ 2 = x ^ 2 + 8 * x + y ^ 2 - 6 * y + 25 := by\n    ring_nf at h\u2080 \u22a2\n    <;>\n    (try ring_nf) <;>\n    (try linarith) <;>\n    (try nlinarith) <;>\n    (try nlinarith [sq_nonneg (x + 4), sq_nonneg (y - 3), sq_nonneg (x - 4), sq_nonneg (y + 3)]) <;>\n    (try nlinarith [sq_nonneg (x + 4 + (y - 3)), sq_nonneg (x + 4 - (y - 3))])\n    <;>\n    nlinarith\n  have h\u2082 : (x + 4) ^ 2 + (y - 3) ^ 2 = 25 := by\n    rw [h\u2081]\n    linarith\n  have h\u2083 : (x + 4) ^ 2 + (y - 3) ^ 2 = 5 ^ 2 := by\n    rw [h\u2082]\n    <;> norm_num\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 47, "alphabet": 33, "compiles": false, "id": "mathd_numbertheory_618", "proof_after": "theorem mathd_numbertheory_618 (n : \u2115) (hn : n > 0) (p : \u2115 \u2192 \u2115) (h\u2080 : \u2200 x, p x = x ^ 2 - x + 41)\n    (h\u2081 : 1 < Nat.gcd (p n) (p (n + 1))) : 41 \u2264 n  :=  by\n  have h\u2082 : p (n + 1) = p n + 2 * n := by\n    simp only [h\u2080]\n    have h\u2083 : (n + 1 : \u2115) ^ 2 - (n + 1) + 41 = (n ^ 2 - n + 41) + 2 * n := by\n      cases n with\n      | zero => contradiction\n      | succ n =>\n        cases n with\n        | zero =>\n          norm_num\n        | succ n =>\n          simp [Nat.mul_succ, Nat.add_assoc, Nat.pow_succ, Nat.mul_add, Nat.add_mul]\n          <;> ring_nf at *\n          <;> omega\n    omega\n  have h\u2083 : Nat.gcd (p n) (p (n + 1)) = Nat.gcd (p n) (2 * n) := by\n    rw [h\u2082]\n    have h\u2084 : Nat.gcd (p n) (p n + 2 * n) = Nat.gcd (p n) (2 * n) := by\n      rw [\u2190 Nat.add_zero (p n), Nat.gcd_comm]\n      <;> simp [Nat.gcd_add_mul_right_right, Nat.gcd_comm]\n    rw [h\u2084]\n    <;> simp [h\u2080]\n    <;> ring_nf\n    <;> omega\n  have h\u2084 : 1 < Nat.gcd (p n) (2 * n) := by\n    rw [h\u2083] at h\u2081\n    exact h\u2081\n  have h\u2085 : 41 \u2264 n := by\n    by_contra h\n    have h\u2086 : n \u2264 40 := by linarith\n    have h\u2087 : n \u2264 40 := by linarith\n    interval_cases n <;> norm_num [h\u2080, Nat.gcd_eq_right, Nat.gcd_eq_left, Nat.gcd_eq_right] at h\u2084 \u22a2 <;>\n      (try omega) <;> (try contradiction) <;>\n      (try norm_num) <;>\n      (try\n        {\n          ring_nf at h\u2084 \u22a2\n          norm_num at h\u2084 \u22a2\n          <;>\n            (try omega) <;> (try contradiction)\n        }) <;>\n      (try omega) <;> (try contradiction)\n  exact h\u2085\n", "proof_before": "theorem mathd_numbertheory_618 (n : \u2115) (hn : n > 0) (p : \u2115 \u2192 \u2115) (h\u2080 : \u2200 x, p x = x ^ 2 - x + 41)\n    (h\u2081 : 1 < Nat.gcd (p n) (p (n + 1))) : 41 \u2264 n  :=  by\n  have h\u2082 : p (n + 1) = p n + 2 * n := by\n    simp only [h\u2080]\n    have h\u2083 : (n + 1 : \u2115) ^ 2 - (n + 1) + 41 = (n ^ 2 - n + 41) + 2 * n := by\n      cases n with\n      | zero => contradiction\n      | succ n =>\n        cases n with\n        | zero =>\n          norm_num\n        | succ n =>\n          simp [Nat.mul_succ, Nat.add_assoc, Nat.pow_succ, Nat.mul_add, Nat.add_mul]\n          <;> ring_nf at *\n          <;> omega\n    omega\n  have h\u2083 : Nat.gcd (p n) (p (n + 1)) = Nat.gcd (p n) (2 * n) := by\n    rw [h\u2082]\n    have h\u2084 : Nat.gcd (p n) (p n + 2 * n) = Nat.gcd (p n) (2 * n) := by\n      rw [\u2190 Nat.add_zero (p n), Nat.gcd_comm]\n      <;> simp [Nat.gcd_add_mul_right_right, Nat.gcd_comm]\n    rw [h\u2084]\n    <;> simp [h\u2080]\n    <;> ring_nf\n    <;> omega\n  have h\u2084 : 1 < Nat.gcd (p n) (2 * n) := by\n    rw [h\u2083] at h\u2081\n    exact h\u2081\n  have h\u2085 : 41 \u2264 n := by\n    by_contra h\n    have h\u2086 : n \u2264 40 := by linarith\n    have h\u2087 : n \u2264 40 := by linarith\n    interval_cases n <;> norm_num [h\u2080, Nat.gcd_eq_right, Nat.gcd_eq_left, Nat.gcd_eq_right] at h\u2084 \u22a2 <;>\n      (try omega) <;> (try contradiction) <;>\n      (try norm_num) <;>\n      (try\n        {\n          ring_nf at h\u2084 \u22a2\n          norm_num at h\u2084 \u22a2\n          <;>\n            (try omega) <;> (try contradiction)\n        }) <;>\n      (try omega) <;> (try contradiction)\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 105, "alphabet": 60, "compiles": true, "steps_after": 53, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 4, "fixpoint": false, "chars_before": 3073, "chars_after": 1542, "seconds": 684.4, "id": "mathd_algebra_487", "proof_after": "theorem mathd_algebra_487 (a b c d : \u211d) (h\u2080 : b = a ^ 2) (h\u2081 : a + b = 1) (h\u2082 : d = c ^ 2)\n    (h\u2083 : c + d = 1) (h\u2084 : a \u2260 c) : Real.sqrt ((a - c) ^ 2 + (b - d) ^ 2) = Real.sqrt 10  :=  by\n  have h\u2085 : a ^ 2 + a - 1 = 0 := by\n    have h\u2085\u2081 : a ^ 2 + a - 1 = 0 := by\n      have h\u2085\u2082 : b = a ^ 2 := h\u2080\n      have h\u2085\u2083 : a + b = 1 := h\u2081\n      have h\u2085\u2084 : a + (a ^ 2) = 1 := by\n        rw [h\u2085\u2082] at h\u2085\u2083\n        exact h\u2085\u2083\n      have h\u2085\u2085 : a ^ 2 + a - 1 = 0 := by\n        linarith\n      exact h\u2085\u2085\n    exact h\u2085\u2081\n  have h\u2086 : c ^ 2 + c - 1 = 0 := by\n    have h\u2086\u2081 : c ^ 2 + c - 1 = 0 := by\n      have h\u2086\u2082 : d = c ^ 2 := h\u2082\n      have h\u2086\u2083 : c + d = 1 := h\u2083\n      have h\u2086\u2084 : c + (c ^ 2) = 1 := by\n        rw [h\u2086\u2082] at h\u2086\u2083\n        exact h\u2086\u2083\n      have h\u2086\u2085 : c ^ 2 + c - 1 = 0 := by\n        linarith\n      exact h\u2086\u2085\n    exact h\u2086\u2081\n  have h\u2087 : a + c = -1 := by\n    have h\u2087\u2081 : a + c = -1 := by\n      have h\u2087\u2082 : a \u2260 c := h\u2084\n      have h\u2087\u2083 : a ^ 2 + a - 1 = 0 := h\u2085\n      have h\u2087\u2084 : c ^ 2 + c - 1 = 0 := h\u2086\n      have h\u2087\u2085 : (a - c) * (a + c + 1) = 0 := by\n        nlinarith [sq_pos_of_ne_zero (sub_ne_zero_of_ne h\u2084), sq_nonneg (a - c), sq_nonneg (a + c)]\n      have h\u2087\u2086 : a + c + 1 = 0 := by\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2084)\n        nlinarith\n      linarith\n    exact h\u2087\u2081\n  have h\u2088 : a * c = -1 := by\n    nlinarith\n  have h\u2089 : (a - c) ^ 2 = 5 := by\n    linarith\n  have h\u2081\u2080 : (b - d) ^ 2 = 5 := by\n    nlinarith\n  have h\u2081\u2081 : (a - c) ^ 2 + (b - d) ^ 2 = 10 := by\n    have h\u2081\u2081\u2081 : (a - c) ^ 2 = 5 := h\u2089\n    have h\u2081\u2081\u2082 : (b - d) ^ 2 = 5 := h\u2081\u2080\n    linarith\n  have h\u2081\u2082 : Real.sqrt ((a - c) ^ 2 + (b - d) ^ 2) = Real.sqrt 10 := by\n    rw [h\u2081\u2081]\n    <;>\n    simp [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf at *\n    <;>\n    nlinarith\n  rw [h\u2081\u2082]\n", "proof_before": "theorem mathd_algebra_487 (a b c d : \u211d) (h\u2080 : b = a ^ 2) (h\u2081 : a + b = 1) (h\u2082 : d = c ^ 2)\n    (h\u2083 : c + d = 1) (h\u2084 : a \u2260 c) : Real.sqrt ((a - c) ^ 2 + (b - d) ^ 2) = Real.sqrt 10  :=  by\n  have h\u2085 : a ^ 2 + a - 1 = 0 := by\n    have h\u2085\u2081 : a ^ 2 + a - 1 = 0 := by\n      have h\u2085\u2082 : b = a ^ 2 := h\u2080\n      have h\u2085\u2083 : a + b = 1 := h\u2081\n      have h\u2085\u2084 : a + (a ^ 2) = 1 := by\n        rw [h\u2085\u2082] at h\u2085\u2083\n        exact h\u2085\u2083\n      have h\u2085\u2085 : a ^ 2 + a - 1 = 0 := by\n        linarith\n      exact h\u2085\u2085\n    exact h\u2085\u2081\n  have h\u2086 : c ^ 2 + c - 1 = 0 := by\n    have h\u2086\u2081 : c ^ 2 + c - 1 = 0 := by\n      have h\u2086\u2082 : d = c ^ 2 := h\u2082\n      have h\u2086\u2083 : c + d = 1 := h\u2083\n      have h\u2086\u2084 : c + (c ^ 2) = 1 := by\n        rw [h\u2086\u2082] at h\u2086\u2083\n        exact h\u2086\u2083\n      have h\u2086\u2085 : c ^ 2 + c - 1 = 0 := by\n        linarith\n      exact h\u2086\u2085\n    exact h\u2086\u2081\n  have h\u2087 : a + c = -1 := by\n    have h\u2087\u2081 : a + c = -1 := by\n      have h\u2087\u2082 : a \u2260 c := h\u2084\n      have h\u2087\u2083 : a ^ 2 + a - 1 = 0 := h\u2085\n      have h\u2087\u2084 : c ^ 2 + c - 1 = 0 := h\u2086\n      have h\u2087\u2085 : (a - c) * (a + c + 1) = 0 := by\n        nlinarith [sq_pos_of_ne_zero (sub_ne_zero_of_ne h\u2084), sq_nonneg (a - c), sq_nonneg (a + c)]\n      have h\u2087\u2086 : a + c + 1 = 0 := by\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2084)\n        nlinarith\n      linarith\n    exact h\u2087\u2081\n  have h\u2088 : a * c = -1 := by\n    have h\u2088\u2081 : a + c = -1 := h\u2087\n    have h\u2088\u2082 : a ^ 2 + a - 1 = 0 := h\u2085\n    have h\u2088\u2083 : c ^ 2 + c - 1 = 0 := h\u2086\n    have h\u2088\u2084 : a * c = -1 := by\n      have h\u2088\u2085 : (a + c) ^ 2 = a ^ 2 + 2 * a * c + c ^ 2 := by ring\n      rw [h\u2088\u2081] at h\u2088\u2085\n      have h\u2088\u2086 : a ^ 2 = -a + 1 := by linarith\n      have h\u2088\u2087 : c ^ 2 = -c + 1 := by linarith\n      nlinarith\n    exact h\u2088\u2084\n  have h\u2089 : (a - c) ^ 2 = 5 := by\n    have h\u2089\u2081 : (a - c) ^ 2 = 5 := by\n      have h\u2089\u2082 : a ^ 2 + a - 1 = 0 := h\u2085\n      have h\u2089\u2083 : c ^ 2 + c - 1 = 0 := h\u2086\n      have h\u2089\u2084 : a + c = -1 := h\u2087\n      have h\u2089\u2085 : a * c = -1 := h\u2088\n      have h\u2089\u2086 : (a - c) ^ 2 = 5 := by\n        have h\u2089\u2087 : (a - c) ^ 2 = a ^ 2 - 2 * a * c + c ^ 2 := by ring\n        rw [h\u2089\u2087]\n        have h\u2089\u2088 : a ^ 2 = -a + 1 := by linarith\n        have h\u2089\u2089 : c ^ 2 = -c + 1 := by linarith\n        rw [h\u2089\u2088, h\u2089\u2089]\n        nlinarith\n      exact h\u2089\u2086\n    exact h\u2089\u2081\n  have h\u2081\u2080 : (b - d) ^ 2 = 5 := by\n    have h\u2081\u2080\u2081 : b - d = -(a - c) := by\n      have h\u2081\u2080\u2082 : b = a ^ 2 := h\u2080\n      have h\u2081\u2080\u2083 : d = c ^ 2 := h\u2082\n      have h\u2081\u2080\u2084 : a + c = -1 := h\u2087\n      have h\u2081\u2080\u2085 : a * c = -1 := h\u2088\n      have h\u2081\u2080\u2086 : b - d = a ^ 2 - c ^ 2 := by\n        rw [h\u2081\u2080\u2082, h\u2081\u2080\u2083]\n        <;> ring\n      have h\u2081\u2080\u2087 : a ^ 2 - c ^ 2 = (a - c) * (a + c) := by\n        ring\n      have h\u2081\u2080\u2088 : b - d = (a - c) * (a + c) := by\n        rw [h\u2081\u2080\u2086, h\u2081\u2080\u2087]\n      have h\u2081\u2080\u2089 : b - d = -(a - c) := by\n        rw [h\u2081\u2080\u2088, h\u2081\u2080\u2084]\n        <;> ring\n        <;> linarith\n      exact h\u2081\u2080\u2089\n    have h\u2081\u2080\u2082 : (b - d) ^ 2 = 5 := by\n      rw [h\u2081\u2080\u2081]\n      have h\u2081\u2080\u2083 : (a - c) ^ 2 = 5 := h\u2089\n      have h\u2081\u2080\u2084 : (-(a - c)) ^ 2 = (a - c) ^ 2 := by\n        ring\n      rw [h\u2081\u2080\u2084]\n      linarith\n    exact h\u2081\u2080\u2082\n  have h\u2081\u2081 : (a - c) ^ 2 + (b - d) ^ 2 = 10 := by\n    have h\u2081\u2081\u2081 : (a - c) ^ 2 = 5 := h\u2089\n    have h\u2081\u2081\u2082 : (b - d) ^ 2 = 5 := h\u2081\u2080\n    linarith\n  have h\u2081\u2082 : Real.sqrt ((a - c) ^ 2 + (b - d) ^ 2) = Real.sqrt 10 := by\n    rw [h\u2081\u2081]\n    <;>\n    simp [Real.sqrt_eq_iff_sq_eq]\n    <;>\n    ring_nf at *\n    <;>\n    nlinarith\n  rw [h\u2081\u2082]\n"}
{"mode": "clear", "steps_before": 187, "alphabet": 67, "compiles": false, "id": "amc12_2000_p20", "proof_after": "theorem amc12_2000_p20 (x y z : \u211d) (h\u2080 : 0 < x \u2227 0 < y \u2227 0 < z) (h\u2081 : x + 1 / y = 4)\n    (h\u2082 : y + 1 / z = 1) (h\u2083 : z + 1 / x = 7 / 3) : x * y * z = 1  :=  by\n  have h\u2084 : x * y + 1 = 4 * y := by\n    have h\u2084\u2081 : y \u2260 0 := by linarith\n    have h\u2084\u2082 : x + 1 / y = 4 := h\u2081\n    field_simp at h\u2084\u2082 \u22a2\n    nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2]\n  have h\u2085 : y * z + 1 = z := by\n    have h\u2085\u2081 : z \u2260 0 := by linarith\n    have h\u2085\u2082 : y + 1 / z = 1 := h\u2082\n    field_simp at h\u2085\u2082 \u22a2\n    nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2]\n  have h\u2086 : x * z + 1 = 7 / 3 * x := by\n    have h\u2086\u2081 : x \u2260 0 := by linarith\n    have h\u2086\u2082 : z + 1 / x = 7 / 3 := h\u2083\n    field_simp at h\u2086\u2082 \u22a2\n    nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2]\n  have h\u2087 : y < 1 := by\n    by_contra h\n    have h\u2087\u2081 : y \u2265 1 := by linarith\n    have h\u2087\u2082 : z > 0 := by linarith\n    have h\u2087\u2083 : y * z + 1 = z := h\u2085\n    have h\u2087\u2084 : y * z \u2265 z := by nlinarith\n    nlinarith\n  have h\u2088 : z = 1 / (1 - y) := by\n    have h\u2088\u2081 : y < 1 := h\u2087\n    have h\u2088\u2082 : y * z + 1 = z := h\u2085\n    have h\u2088\u2083 : 1 - y > 0 := by linarith\n    have h\u2088\u2084 : z = 1 / (1 - y) := by\n      have h\u2088\u2085 : z * (1 - y) = 1 := by\n        have h\u2088\u2085\u2081 : y * z + 1 = z := h\u2085\n        nlinarith\n      have h\u2088\u2085\u2082 : z = 1 / (1 - y) := by\n        have h\u2088\u2085\u2083 : 1 - y \u2260 0 := by linarith\n        field_simp at h\u2088\u2085 \u22a2\n        nlinarith\n      exact h\u2088\u2085\u2082\n    exact h\u2088\u2084\n  have h\u2089 : y = 1 / (4 - x) := by\n    have h\u2089\u2081 : x * y + 1 = 4 * y := h\u2084\n    have h\u2089\u2082 : y > 0 := by linarith\n    have h\u2089\u2083 : 4 - x > 0 := by\n      by_contra h\n      have h\u2089\u2084 : x \u2265 4 := by linarith\n      have h\u2089\u2085 : x * y + 1 = 4 * y := h\u2084\n      have h\u2089\u2086 : x * y \u2265 4 * y := by nlinarith\n      nlinarith\n    have h\u2089\u2084 : y = 1 / (4 - x) := by\n      have h\u2089\u2085 : x * y + 1 = 4 * y := h\u2084\n      have h\u2089\u2086 : y * (4 - x) = 1 := by\n        nlinarith\n      have h\u2089\u2087 : y = 1 / (4 - x) := by\n        have h\u2089\u2088 : 4 - x \u2260 0 := by linarith\n        field_simp at h\u2089\u2086 \u22a2\n        nlinarith\n      exact h\u2089\u2087\n    exact h\u2089\u2084\n  have h\u2081\u2080 : x < 4 := by\n    by_contra h\n    have h\u2081\u2080\u2081 : x \u2265 4 := by linarith\n    have h\u2081\u2080\u2082 : y > 0 := by linarith\n    have h\u2081\u2080\u2083 : x * y + 1 = 4 * y := h\u2084\n    have h\u2081\u2080\u2084 : x * y \u2265 4 * y := by nlinarith\n    nlinarith\n  have h\u2081\u2081 : z = (4 - x) / (3 - x) := by\n    have h\u2081\u2081\u2081 : z = 1 / (1 - y) := h\u2088\n    have h\u2081\u2081\u2082 : y = 1 / (4 - x) := h\u2089\n    have h\u2081\u2081\u2083 : z = (4 - x) / (3 - x) := by\n      rw [h\u2081\u2081\u2081, h\u2081\u2081\u2082]\n      have h\u2081\u2081\u2084 : 1 - (1 / (4 - x)) \u2260 0 := by\n        have h\u2081\u2081\u2085 : 4 - x > 0 := by linarith\n        have h\u2081\u2081\u2086 : 1 - (1 / (4 - x)) > 0 := by\n          have h\u2081\u2081\u2087 : 0 < 4 - x := by linarith\n          have h\u2081\u2081\u2088 : 0 < (4 - x : \u211d) := by exact_mod_cast h\u2081\u2081\u2087\n          field_simp [h\u2081\u2081\u2088.ne']\n          rw [\u2190 sub_pos]\n          field_simp [h\u2081\u2081\u2088.ne']\n          <;> nlinarith\n        linarith\n      have h\u2081\u2081\u2085 : 3 - x \u2260 0 := by\n        have h\u2081\u2081\u2086 : x < 4 := h\u2081\u2080\n        have h\u2081\u2081\u2087 : x > 0 := by linarith\n        by_contra h\u2081\u2081\u2088\n        have h\u2081\u2081\u2089 : 3 - x = 0 := by linarith\n        have h\u2081\u2082\u2080 : x = 3 := by linarith\n        have h\u2081\u2082\u2081 : x * z + 1 = 7 / 3 * x := h\u2086\n        have h\u2081\u2082\u2082 : z = (4 - x) / (3 - x) := by\n          rw [h\u2081\u2081\u2088]\n          <;> field_simp [h\u2081\u2081\u2088] at *\n          <;> nlinarith\n        rw [h\u2081\u2082\u2080] at h\u2081\u2082\u2081 h\u2081\u2082\u2082\n        <;> norm_num at h\u2081\u2082\u2081 h\u2081\u2082\u2082 \u22a2 <;>\n          (try norm_num) <;> (try nlinarith) <;> (try linarith)\n      field_simp [h\u2081\u2081\u2084, h\u2081\u2081\u2085]\n      <;>\n      (try ring_nf at * <;> nlinarith) <;>\n      (try field_simp [h\u2081\u2081\u2084, h\u2081\u2081\u2085] at * <;> nlinarith)\n    exact h\u2081\u2081\u2083\n  have h\u2081\u2082 : 4 * x ^ 2 - 12 * x + 9 = 0 := by\n    have h\u2081\u2082\u2081 : z = (4 - x) / (3 - x) := h\u2081\u2081\n    have h\u2081\u2082\u2082 : x * z + 1 = 7 / 3 * x := h\u2086\n    rw [h\u2081\u2082\u2081] at h\u2081\u2082\u2082\n    have h\u2081\u2082\u2083 : x < 4 := h\u2081\u2080\n    have h\u2081\u2082\u2084 : x > 0 := by linarith\n    have h\u2081\u2082\u2085 : 3 - x \u2260 0 := by\n      by_contra h\n      have h\u2081\u2082\u2086 : 3 - x = 0 := by linarith\n      have h\u2081\u2082\u2087 : x = 3 := by linarith\n      rw [h\u2081\u2082\u2087] at h\u2081\u2082\u2082\n      norm_num at h\u2081\u2082\u2082 \u22a2\n      <;> nlinarith\n    field_simp [h\u2081\u2082\u2085] at h\u2081\u2082\u2082\n    <;> ring_nf at h\u2081\u2082\u2082 \u22a2\n    <;> nlinarith [sq_nonneg (x - 3 / 2)]\n  have h\u2081\u2083 : x = 3 / 2 := by\n    have h\u2081\u2083\u2081 : 4 * x ^ 2 - 12 * x + 9 = 0 := h\u2081\u2082\n    have h\u2081\u2083\u2082 : x = 3 / 2 := by\n      nlinarith [sq_nonneg (x - 3 / 2)]\n    exact h\u2081\u2083\u2082\n  have h\u2081\u2084 : y = 2 / 5 := by\n    have h\u2081\u2084\u2081 : y = 1 / (4 - x) := h\u2089\n    rw [h\u2081\u2084\u2081]\n    rw [h\u2081\u2083]\n    <;> norm_num\n    <;>\n    (try norm_num at * <;> nlinarith)\n    <;>\n    (try linarith)\n    <;>\n    (try ring_nf at * <;> nlinarith)\n  have h\u2081\u2085 : z = 5 / 3 := by\n    have h\u2081\u2085\u2081 : z = (4 - x) / (3 - x) := h\u2081\u2081\n    rw [h\u2081\u2085\u2081]\n    rw [h\u2081\u2083]\n    <;> norm_num\n    <;>\n    (try norm_num at * <;> nlinarith)\n    <;>\n    (try linarith)\n    <;>\n    (try ring_nf at * <;> nlinarith)\n  have h\u2081\u2086 : x * y * z = 1 := by\n    rw [h\u2081\u2083, h\u2081\u2084, h\u2081\u2085]\n    <;> norm_num\n    <;>\n    (try ring_nf at * <;> nlinarith)\n    <;>\n    (try linarith)\n    <;>\n    (try ring_nf at * <;> nlinarith)\n    <;>\n    (try linarith)\n  exact h\u2081\u2086\n", "proof_before": "theorem amc12_2000_p20 (x y z : \u211d) (h\u2080 : 0 < x \u2227 0 < y \u2227 0 < z) (h\u2081 : x + 1 / y = 4)\n    (h\u2082 : y + 1 / z = 1) (h\u2083 : z + 1 / x = 7 / 3) : x * y * z = 1  :=  by\n  have h\u2084 : x * y + 1 = 4 * y := by\n    have h\u2084\u2081 : y \u2260 0 := by linarith\n    have h\u2084\u2082 : x + 1 / y = 4 := h\u2081\n    field_simp at h\u2084\u2082 \u22a2\n    nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2]\n  have h\u2085 : y * z + 1 = z := by\n    have h\u2085\u2081 : z \u2260 0 := by linarith\n    have h\u2085\u2082 : y + 1 / z = 1 := h\u2082\n    field_simp at h\u2085\u2082 \u22a2\n    nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2]\n  have h\u2086 : x * z + 1 = 7 / 3 * x := by\n    have h\u2086\u2081 : x \u2260 0 := by linarith\n    have h\u2086\u2082 : z + 1 / x = 7 / 3 := h\u2083\n    field_simp at h\u2086\u2082 \u22a2\n    nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2]\n  have h\u2087 : y < 1 := by\n    by_contra h\n    have h\u2087\u2081 : y \u2265 1 := by linarith\n    have h\u2087\u2082 : z > 0 := by linarith\n    have h\u2087\u2083 : y * z + 1 = z := h\u2085\n    have h\u2087\u2084 : y * z \u2265 z := by nlinarith\n    nlinarith\n  have h\u2088 : z = 1 / (1 - y) := by\n    have h\u2088\u2081 : y < 1 := h\u2087\n    have h\u2088\u2082 : y * z + 1 = z := h\u2085\n    have h\u2088\u2083 : 1 - y > 0 := by linarith\n    have h\u2088\u2084 : z = 1 / (1 - y) := by\n      have h\u2088\u2085 : z * (1 - y) = 1 := by\n        have h\u2088\u2085\u2081 : y * z + 1 = z := h\u2085\n        nlinarith\n      have h\u2088\u2085\u2082 : z = 1 / (1 - y) := by\n        have h\u2088\u2085\u2083 : 1 - y \u2260 0 := by linarith\n        field_simp at h\u2088\u2085 \u22a2\n        nlinarith\n      exact h\u2088\u2085\u2082\n    exact h\u2088\u2084\n  have h\u2089 : y = 1 / (4 - x) := by\n    have h\u2089\u2081 : x * y + 1 = 4 * y := h\u2084\n    have h\u2089\u2082 : y > 0 := by linarith\n    have h\u2089\u2083 : 4 - x > 0 := by\n      by_contra h\n      have h\u2089\u2084 : x \u2265 4 := by linarith\n      have h\u2089\u2085 : x * y + 1 = 4 * y := h\u2084\n      have h\u2089\u2086 : x * y \u2265 4 * y := by nlinarith\n      nlinarith\n    have h\u2089\u2084 : y = 1 / (4 - x) := by\n      have h\u2089\u2085 : x * y + 1 = 4 * y := h\u2084\n      have h\u2089\u2086 : y * (4 - x) = 1 := by\n        nlinarith\n      have h\u2089\u2087 : y = 1 / (4 - x) := by\n        have h\u2089\u2088 : 4 - x \u2260 0 := by linarith\n        field_simp at h\u2089\u2086 \u22a2\n        nlinarith\n      exact h\u2089\u2087\n    exact h\u2089\u2084\n  have h\u2081\u2080 : x < 4 := by\n    by_contra h\n    have h\u2081\u2080\u2081 : x \u2265 4 := by linarith\n    have h\u2081\u2080\u2082 : y > 0 := by linarith\n    have h\u2081\u2080\u2083 : x * y + 1 = 4 * y := h\u2084\n    have h\u2081\u2080\u2084 : x * y \u2265 4 * y := by nlinarith\n    nlinarith\n  have h\u2081\u2081 : z = (4 - x) / (3 - x) := by\n    have h\u2081\u2081\u2081 : z = 1 / (1 - y) := h\u2088\n    have h\u2081\u2081\u2082 : y = 1 / (4 - x) := h\u2089\n    have h\u2081\u2081\u2083 : z = (4 - x) / (3 - x) := by\n      rw [h\u2081\u2081\u2081, h\u2081\u2081\u2082]\n      have h\u2081\u2081\u2084 : 1 - (1 / (4 - x)) \u2260 0 := by\n        have h\u2081\u2081\u2085 : 4 - x > 0 := by linarith\n        have h\u2081\u2081\u2086 : 1 - (1 / (4 - x)) > 0 := by\n          have h\u2081\u2081\u2087 : 0 < 4 - x := by linarith\n          have h\u2081\u2081\u2088 : 0 < (4 - x : \u211d) := by exact_mod_cast h\u2081\u2081\u2087\n          field_simp [h\u2081\u2081\u2088.ne']\n          rw [\u2190 sub_pos]\n          field_simp [h\u2081\u2081\u2088.ne']\n          <;> nlinarith\n        linarith\n      have h\u2081\u2081\u2085 : 3 - x \u2260 0 := by\n        have h\u2081\u2081\u2086 : x < 4 := h\u2081\u2080\n        have h\u2081\u2081\u2087 : x > 0 := by linarith\n        by_contra h\u2081\u2081\u2088\n        have h\u2081\u2081\u2089 : 3 - x = 0 := by linarith\n        have h\u2081\u2082\u2080 : x = 3 := by linarith\n        have h\u2081\u2082\u2081 : x * z + 1 = 7 / 3 * x := h\u2086\n        have h\u2081\u2082\u2082 : z = (4 - x) / (3 - x) := by\n          rw [h\u2081\u2081\u2088]\n          <;> field_simp [h\u2081\u2081\u2088] at *\n          <;> nlinarith\n        rw [h\u2081\u2082\u2080] at h\u2081\u2082\u2081 h\u2081\u2082\u2082\n        <;> norm_num at h\u2081\u2082\u2081 h\u2081\u2082\u2082 \u22a2 <;>\n          (try norm_num) <;> (try nlinarith) <;> (try linarith)\n      field_simp [h\u2081\u2081\u2084, h\u2081\u2081\u2085]\n      <;>\n      (try ring_nf at * <;> nlinarith) <;>\n      (try field_simp [h\u2081\u2081\u2084, h\u2081\u2081\u2085] at * <;> nlinarith)\n    exact h\u2081\u2081\u2083\n  have h\u2081\u2082 : 4 * x ^ 2 - 12 * x + 9 = 0 := by\n    have h\u2081\u2082\u2081 : z = (4 - x) / (3 - x) := h\u2081\u2081\n    have h\u2081\u2082\u2082 : x * z + 1 = 7 / 3 * x := h\u2086\n    rw [h\u2081\u2082\u2081] at h\u2081\u2082\u2082\n    have h\u2081\u2082\u2083 : x < 4 := h\u2081\u2080\n    have h\u2081\u2082\u2084 : x > 0 := by linarith\n    have h\u2081\u2082\u2085 : 3 - x \u2260 0 := by\n      by_contra h\n      have h\u2081\u2082\u2086 : 3 - x = 0 := by linarith\n      have h\u2081\u2082\u2087 : x = 3 := by linarith\n      rw [h\u2081\u2082\u2087] at h\u2081\u2082\u2082\n      norm_num at h\u2081\u2082\u2082 \u22a2\n      <;> nlinarith\n    field_simp [h\u2081\u2082\u2085] at h\u2081\u2082\u2082\n    <;> ring_nf at h\u2081\u2082\u2082 \u22a2\n    <;> nlinarith [sq_nonneg (x - 3 / 2)]\n  have h\u2081\u2083 : x = 3 / 2 := by\n    have h\u2081\u2083\u2081 : 4 * x ^ 2 - 12 * x + 9 = 0 := h\u2081\u2082\n    have h\u2081\u2083\u2082 : x = 3 / 2 := by\n      nlinarith [sq_nonneg (x - 3 / 2)]\n    exact h\u2081\u2083\u2082\n  have h\u2081\u2084 : y = 2 / 5 := by\n    have h\u2081\u2084\u2081 : y = 1 / (4 - x) := h\u2089\n    rw [h\u2081\u2084\u2081]\n    rw [h\u2081\u2083]\n    <;> norm_num\n    <;>\n    (try norm_num at * <;> nlinarith)\n    <;>\n    (try linarith)\n    <;>\n    (try ring_nf at * <;> nlinarith)\n  have h\u2081\u2085 : z = 5 / 3 := by\n    have h\u2081\u2085\u2081 : z = (4 - x) / (3 - x) := h\u2081\u2081\n    rw [h\u2081\u2085\u2081]\n    rw [h\u2081\u2083]\n    <;> norm_num\n    <;>\n    (try norm_num at * <;> nlinarith)\n    <;>\n    (try linarith)\n    <;>\n    (try ring_nf at * <;> nlinarith)\n  have h\u2081\u2086 : x * y * z = 1 := by\n    rw [h\u2081\u2083, h\u2081\u2084, h\u2081\u2085]\n    <;> norm_num\n    <;>\n    (try ring_nf at * <;> nlinarith)\n    <;>\n    (try linarith)\n    <;>\n    (try ring_nf at * <;> nlinarith)\n    <;>\n    (try linarith)\n  exact h\u2081\u2086\n"}
{"mode": "clear", "steps_before": 47, "alphabet": 26, "compiles": false, "id": "mathd_algebra_392", "proof_after": "theorem mathd_algebra_392 (n : \u2115) (h\u2080 : Even n)\n    (h\u2081 : (\u2191n - 2) ^ 2 + \u2191n ^ 2 + (\u2191n + 2) ^ 2 = (12296 : \u2124)) :\n    (\u2191n - 2) * \u2191n * (\u2191n + 2) / 8 = (32736 : \u2124)  :=  by\n  have h\u2082 : (n : \u2124) ^ 2 = 4096 := by\n    have h\u2082\u2081 : 3 * (n : \u2124) ^ 2 + 8 = 12296 := by\n      have h\u2082\u2082 : (\u2191n - 2 : \u2124) ^ 2 + (\u2191n : \u2124) ^ 2 + (\u2191n + 2 : \u2124) ^ 2 = 12296 := by simpa using h\u2081\n      ring_nf at h\u2082\u2082 \u22a2\n      <;> nlinarith\n    have h\u2082\u2082 : (n : \u2124) ^ 2 = 4096 := by\n      have h\u2082\u2083 : 3 * (n : \u2124) ^ 2 = 12288 := by linarith\n      have h\u2082\u2084 : (n : \u2124) ^ 2 = 4096 := by linarith\n      exact h\u2082\u2084\n    exact h\u2082\u2082\n  have h\u2083 : n = 64 := by\n    have h\u2083\u2081 : n \u2264 64 := by\n      by_contra h\n      have h\u2083\u2082 : n \u2265 65 := by\n        omega\n      have h\u2083\u2083 : (n : \u2124) ^ 2 > 4096 := by\n        have h\u2083\u2084 : (n : \u2124) \u2265 65 := by\n          exact_mod_cast h\u2083\u2082\n        nlinarith\n      linarith\n    interval_cases n <;> norm_num at h\u2082 \u22a2\n    <;>\n    (try omega) <;>\n    (try norm_num [Int.ofNat_eq_coe, Nat.even_iff, Nat.odd_iff] at h\u2080 h\u2081 h\u2082 \u22a2) <;>\n    (try ring_nf at h\u2081 h\u2082 \u22a2) <;>\n    (try linarith)\n    <;>\n    (try omega)\n  have h\u2084 : (\u2191n - 2) * \u2191n * (\u2191n + 2) / 8 = (32736 : \u2124) := by\n    rw [h\u2083]\n    norm_num\n    <;>\n    (try ring_nf at h\u2081 \u22a2) <;>\n    (try norm_num at h\u2081 \u22a2) <;>\n    (try linarith)\n    <;>\n    (try omega)\n  exact h\u2084\n", "proof_before": "theorem mathd_algebra_392 (n : \u2115) (h\u2080 : Even n)\n    (h\u2081 : (\u2191n - 2) ^ 2 + \u2191n ^ 2 + (\u2191n + 2) ^ 2 = (12296 : \u2124)) :\n    (\u2191n - 2) * \u2191n * (\u2191n + 2) / 8 = (32736 : \u2124)  :=  by\n  have h\u2082 : (n : \u2124) ^ 2 = 4096 := by\n    have h\u2082\u2081 : 3 * (n : \u2124) ^ 2 + 8 = 12296 := by\n      have h\u2082\u2082 : (\u2191n - 2 : \u2124) ^ 2 + (\u2191n : \u2124) ^ 2 + (\u2191n + 2 : \u2124) ^ 2 = 12296 := by simpa using h\u2081\n      ring_nf at h\u2082\u2082 \u22a2\n      <;> nlinarith\n    have h\u2082\u2082 : (n : \u2124) ^ 2 = 4096 := by\n      have h\u2082\u2083 : 3 * (n : \u2124) ^ 2 = 12288 := by linarith\n      have h\u2082\u2084 : (n : \u2124) ^ 2 = 4096 := by linarith\n      exact h\u2082\u2084\n    exact h\u2082\u2082\n  have h\u2083 : n = 64 := by\n    have h\u2083\u2081 : n \u2264 64 := by\n      by_contra h\n      have h\u2083\u2082 : n \u2265 65 := by\n        omega\n      have h\u2083\u2083 : (n : \u2124) ^ 2 > 4096 := by\n        have h\u2083\u2084 : (n : \u2124) \u2265 65 := by\n          exact_mod_cast h\u2083\u2082\n        nlinarith\n      linarith\n    interval_cases n <;> norm_num at h\u2082 \u22a2\n    <;>\n    (try omega) <;>\n    (try norm_num [Int.ofNat_eq_coe, Nat.even_iff, Nat.odd_iff] at h\u2080 h\u2081 h\u2082 \u22a2) <;>\n    (try ring_nf at h\u2081 h\u2082 \u22a2) <;>\n    (try linarith)\n    <;>\n    (try omega)\n  have h\u2084 : (\u2191n - 2) * \u2191n * (\u2191n + 2) / 8 = (32736 : \u2124) := by\n    rw [h\u2083]\n    norm_num\n    <;>\n    (try ring_nf at h\u2081 \u22a2) <;>\n    (try norm_num at h\u2081 \u22a2) <;>\n    (try linarith)\n    <;>\n    (try omega)\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 25, "alphabet": 22, "compiles": true, "steps_after": 14, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 5, "fixpoint": true, "chars_before": 621, "chars_after": 327, "seconds": 585.7, "id": "mathd_algebra_209", "proof_after": "theorem mathd_algebra_209 (\u03c3 : Equiv \u211d \u211d) (h\u2080 : \u03c3.2 2 = 10) (h\u2081 : \u03c3.2 10 = 1) (h\u2082 : \u03c3.2 1 = 2) :\n    \u03c3.1 (\u03c3.1 10) = 1  :=  by\n  have h\u2084 : \u03c3.1 2 = 1 := by\n    have h\u2084\u2081 : \u03c3.1 (\u03c3.2 1) = 1 := by\n      apply Equiv.apply_symm_apply\n    rw [h\u2082] at h\u2084\u2081\n    omega\n  have h\u2085 : \u03c3.1 10 = 2 := by\n    have h\u2085\u2081 : \u03c3.1 (\u03c3.2 2) = 2 := by\n      apply Equiv.apply_symm_apply\n    rw [h\u2080] at h\u2085\u2081\n    omega\n  have h\u2086 : \u03c3.1 (\u03c3.1 10) = 1 := by\n    rw [h\u2085]\n    omega\n  exact h\u2086\n", "proof_before": "theorem mathd_algebra_209 (\u03c3 : Equiv \u211d \u211d) (h\u2080 : \u03c3.2 2 = 10) (h\u2081 : \u03c3.2 10 = 1) (h\u2082 : \u03c3.2 1 = 2) :\n    \u03c3.1 (\u03c3.1 10) = 1  :=  by\n  have h\u2083 : \u03c3.1 1 = 10 := by\n    have h\u2083\u2081 : \u03c3.1 1 = 10 := by\n      have h\u2083\u2082 : \u03c3.1 (\u03c3.2 10) = 10 := by\n        apply Equiv.apply_symm_apply\n      rw [h\u2081] at h\u2083\u2082\n      norm_num at h\u2083\u2082 \u22a2\n      <;> linarith\n    exact h\u2083\u2081\n  have h\u2084 : \u03c3.1 2 = 1 := by\n    have h\u2084\u2081 : \u03c3.1 (\u03c3.2 1) = 1 := by\n      apply Equiv.apply_symm_apply\n    rw [h\u2082] at h\u2084\u2081\n    norm_num at h\u2084\u2081 \u22a2\n    <;> linarith\n  have h\u2085 : \u03c3.1 10 = 2 := by\n    have h\u2085\u2081 : \u03c3.1 (\u03c3.2 2) = 2 := by\n      apply Equiv.apply_symm_apply\n    rw [h\u2080] at h\u2085\u2081\n    norm_num at h\u2085\u2081 \u22a2\n    <;> linarith\n  have h\u2086 : \u03c3.1 (\u03c3.1 10) = 1 := by\n    rw [h\u2085]\n    rw [h\u2084]\n    <;> norm_num\n  exact h\u2086\n"}
{"mode": "clear", "steps_before": 39, "alphabet": 27, "compiles": true, "steps_after": 20, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 8, "fixpoint": false, "chars_before": 975, "chars_after": 563, "seconds": 723.3, "id": "aime_1983_p2", "proof_after": "theorem aime_1983_p2 (x p : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : 0 < p \u2227 p < 15) (h\u2081 : p \u2264 x \u2227 x \u2264 15)\n    (h\u2082 : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) : 15 \u2264 f x  :=  by\n  have h\u2083 : x - p \u2265 0 := by\n    linarith\n  have h\u2084 : x - 15 \u2264 0 := by\n    linarith\n  have h\u2085 : x - p - 15 \u2264 0 := by\n    linarith\n  have h\u2086 : abs (x - p) = x - p := by\n    rw [abs_of_nonneg h\u2083]\n  have h\u2087 : abs (x - 15) = 15 - x := by\n    have h\u2087\u2081 : x - 15 \u2264 0 := h\u2084\n    have h\u2087\u2082 : abs (x - 15) = -(x - 15) := by\n      rw [abs_of_nonpos h\u2087\u2081]\n    linarith\n  have h\u2088 : abs (x - p - 15) = p + 15 - x := by\n    have h\u2088\u2081 : x - p - 15 \u2264 0 := h\u2085\n    have h\u2088\u2082 : abs (x - p - 15) = -(x - p - 15) := by\n      rw [abs_of_nonpos h\u2088\u2081]\n      <;> linarith\n    linarith\n  linarith\n", "proof_before": "theorem aime_1983_p2 (x p : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : 0 < p \u2227 p < 15) (h\u2081 : p \u2264 x \u2227 x \u2264 15)\n    (h\u2082 : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) : 15 \u2264 f x  :=  by\n  have h\u2083 : x - p \u2265 0 := by\n    have h\u2083\u2081 : p \u2264 x := h\u2081.1\n    linarith\n  have h\u2084 : x - 15 \u2264 0 := by\n    have h\u2084\u2081 : x \u2264 15 := h\u2081.2\n    linarith\n  have h\u2085 : x - p - 15 \u2264 0 := by\n    have h\u2085\u2081 : x \u2264 15 := h\u2081.2\n    have h\u2085\u2082 : p > 0 := h\u2080.1\n    linarith\n  have h\u2086 : abs (x - p) = x - p := by\n    rw [abs_of_nonneg h\u2083]\n    <;>\n    linarith\n  have h\u2087 : abs (x - 15) = 15 - x := by\n    have h\u2087\u2081 : x - 15 \u2264 0 := h\u2084\n    have h\u2087\u2082 : abs (x - 15) = -(x - 15) := by\n      rw [abs_of_nonpos h\u2087\u2081]\n      <;> linarith\n    rw [h\u2087\u2082]\n    <;> ring_nf at *\n    <;> linarith\n  have h\u2088 : abs (x - p - 15) = p + 15 - x := by\n    have h\u2088\u2081 : x - p - 15 \u2264 0 := h\u2085\n    have h\u2088\u2082 : abs (x - p - 15) = -(x - p - 15) := by\n      rw [abs_of_nonpos h\u2088\u2081]\n      <;> linarith\n    rw [h\u2088\u2082]\n    <;> ring_nf at *\n    <;> linarith\n  have h\u2089 : f x = 30 - x := by\n    rw [h\u2082, h\u2086, h\u2087, h\u2088]\n    <;> ring_nf at *\n    <;> linarith\n  have h\u2081\u2080 : 15 \u2264 f x := by\n    rw [h\u2089]\n    have h\u2081\u2080\u2081 : x \u2264 15 := h\u2081.2\n    linarith\n  exact h\u2081\u2080\n"}
{"mode": "clear", "steps_before": 25, "alphabet": 18, "compiles": false, "id": "mathd_numbertheory_150", "proof_after": "theorem mathd_numbertheory_150 (n : \u2115) (h\u2080 : \u00acNat.Prime (7 + 30 * n)) : 6 \u2264 n  :=  by\n  have h\u2081 : n \u2265 6 := by\n    by_contra h\n    have h\u2082 : n \u2264 5 := by linarith\n    interval_cases n <;> norm_num [Nat.Prime] at h\u2080 \u22a2 <;>\n      (try contradiction) <;>\n      (try norm_num) <;>\n      (try decide) <;>\n      (try omega)\n    <;>\n      norm_num [Nat.Prime] at h\u2080 \u22a2 <;>\n      (try contradiction) <;>\n      (try norm_num) <;>\n      (try decide) <;>\n      (try omega)\n  linarith\n", "proof_before": "theorem mathd_numbertheory_150 (n : \u2115) (h\u2080 : \u00acNat.Prime (7 + 30 * n)) : 6 \u2264 n  :=  by\n  have h\u2081 : n \u2265 6 := by\n    by_contra h\n    have h\u2082 : n \u2264 5 := by linarith\n    interval_cases n <;> norm_num [Nat.Prime] at h\u2080 \u22a2 <;>\n      (try contradiction) <;>\n      (try norm_num) <;>\n      (try decide) <;>\n      (try omega)\n    <;>\n      norm_num [Nat.Prime] at h\u2080 \u22a2 <;>\n      (try contradiction) <;>\n      (try norm_num) <;>\n      (try decide) <;>\n      (try omega)\n  linarith\n"}
{"mode": "clear", "steps_before": 4, "alphabet": 13, "compiles": false, "id": "mathd_numbertheory_447", "proof_after": "theorem mathd_numbertheory_447 :\n    (\u2211 k in Finset.filter (fun x => 3 \u2223 x) (Finset.Icc 1 49), k % 10) = 78  :=  by\n  have h_main : (\u2211 k in Finset.filter (fun x => 3 \u2223 x) (Finset.Icc 1 49), k % 10) = 78 := by\n    norm_num [Finset.sum_filter, Finset.sum_Icc_succ_top, Nat.div_eq_of_lt, Nat.div_eq_of_lt,\n      Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt,\n      Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt,\n      Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt]\n    <;> rfl\n  exact h_main\n", "proof_before": "theorem mathd_numbertheory_447 :\n    (\u2211 k in Finset.filter (fun x => 3 \u2223 x) (Finset.Icc 1 49), k % 10) = 78  :=  by\n  have h_main : (\u2211 k in Finset.filter (fun x => 3 \u2223 x) (Finset.Icc 1 49), k % 10) = 78 := by\n    norm_num [Finset.sum_filter, Finset.sum_Icc_succ_top, Nat.div_eq_of_lt, Nat.div_eq_of_lt,\n      Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt,\n      Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt,\n      Nat.div_eq_of_lt, Nat.div_eq_of_lt, Nat.div_eq_of_lt]\n    <;> rfl\n  exact h_main\n"}
{"mode": "clear", "steps_before": 34, "alphabet": 27, "compiles": false, "id": "mathd_numbertheory_559", "proof_after": "theorem mathd_numbertheory_559 (x y : \u2115) (h\u2080 : x % 3 = 2) (h\u2081 : y % 5 = 4) (h\u2082 : x % 10 = y % 10) :\n    14 \u2264 x  :=  by\n  have h\u2083 : x % 10 = 4 \u2228 x % 10 = 9 := by\n    have h\u2084 : (x % 10) % 5 = 4 := by\n      have h\u2085 : (y % 10) % 5 = y % 5 := by\n        have h\u2085\u2081 : y % 10 % 5 = y % 5 := by\n          omega\n        exact h\u2085\u2081\n      have h\u2085\u2082 : (x % 10) % 5 = (y % 10) % 5 := by\n        rw [\u2190 h\u2082]\n        <;> simp [Nat.mod_mod]\n      have h\u2085\u2083 : (y % 10) % 5 = 4 := by\n        omega\n      omega\n    have h\u2086 : x % 10 = 4 \u2228 x % 10 = 9 := by\n      have h\u2087 : x % 10 < 10 := Nat.mod_lt _ (by norm_num)\n      interval_cases x % 10 <;> norm_num at h\u2084 \u22a2 <;> omega\n    exact h\u2086\n  have h\u2084 : 14 \u2264 x := by\n    have h\u2085 : x % 3 = 2 := h\u2080\n    have h\u2086 : x % 10 = 4 \u2228 x % 10 = 9 := h\u2083\n    by_contra h\u2087\n    have h\u2088 : x < 14 := by\n      omega\n    interval_cases x <;> norm_num at h\u2085 h\u2086 h\u2088 \u22a2 <;>\n      (try omega) <;>\n      (try\n        {\n          cases h\u2086 with\n          | inl h\u2086 =>\n            omega\n          | inr h\u2086 =>\n            omega\n        }) <;>\n      omega\n  exact h\u2084\n", "proof_before": "theorem mathd_numbertheory_559 (x y : \u2115) (h\u2080 : x % 3 = 2) (h\u2081 : y % 5 = 4) (h\u2082 : x % 10 = y % 10) :\n    14 \u2264 x  :=  by\n  have h\u2083 : x % 10 = 4 \u2228 x % 10 = 9 := by\n    have h\u2084 : (x % 10) % 5 = 4 := by\n      have h\u2085 : (y % 10) % 5 = y % 5 := by\n        have h\u2085\u2081 : y % 10 % 5 = y % 5 := by\n          omega\n        exact h\u2085\u2081\n      have h\u2085\u2082 : (x % 10) % 5 = (y % 10) % 5 := by\n        rw [\u2190 h\u2082]\n        <;> simp [Nat.mod_mod]\n      have h\u2085\u2083 : (y % 10) % 5 = 4 := by\n        omega\n      omega\n    have h\u2086 : x % 10 = 4 \u2228 x % 10 = 9 := by\n      have h\u2087 : x % 10 < 10 := Nat.mod_lt _ (by norm_num)\n      interval_cases x % 10 <;> norm_num at h\u2084 \u22a2 <;> omega\n    exact h\u2086\n  have h\u2084 : 14 \u2264 x := by\n    have h\u2085 : x % 3 = 2 := h\u2080\n    have h\u2086 : x % 10 = 4 \u2228 x % 10 = 9 := h\u2083\n    by_contra h\u2087\n    have h\u2088 : x < 14 := by\n      omega\n    interval_cases x <;> norm_num at h\u2085 h\u2086 h\u2088 \u22a2 <;>\n      (try omega) <;>\n      (try\n        {\n          cases h\u2086 with\n          | inl h\u2086 =>\n            omega\n          | inr h\u2086 =>\n            omega\n        }) <;>\n      omega\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 145, "alphabet": 82, "compiles": true, "steps_after": 145, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 4167, "chars_after": 4167, "seconds": 218.9, "id": "mathd_numbertheory_483", "proof_after": "theorem mathd_numbertheory_483 (a : \u2115 \u2192 \u2115) (h\u2080 : a 1 = 1) (h\u2081 : a 2 = 1)\n    (h\u2082 : \u2200 n, a (n + 2) = a (n + 1) + a n) : a 100 % 4 = 3  :=  by\n  have h\u2083 : a 0 = 0 := by\n    have h\u2084 := h\u2082 0\n    simp [h\u2080, h\u2081] at h\u2084\n    <;> omega\n  have h\u2084 : a 3 = 2 := by\n    have h\u2085 := h\u2082 1\n    have h\u2086 := h\u2082 2\n    have h\u2087 := h\u2082 3\n    simp [h\u2080, h\u2081, h\u2083] at h\u2085 h\u2086 h\u2087\n    <;> omega\n  have h\u2085 : a 4 = 3 := by\n    have h\u2085 := h\u2082 2\n    have h\u2086 := h\u2082 3\n    have h\u2087 := h\u2082 4\n    simp [h\u2080, h\u2081, h\u2083, h\u2084] at h\u2085 h\u2086 h\u2087\n    <;> omega\n  have h\u2086 : a 5 = 5 := by\n    have h\u2086 := h\u2082 3\n    have h\u2087 := h\u2082 4\n    have h\u2088 := h\u2082 5\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085] at h\u2086 h\u2087 h\u2088\n    <;> omega\n  have h\u2087 : a 6 = 8 := by\n    have h\u2087 := h\u2082 4\n    have h\u2088 := h\u2082 5\n    have h\u2089 := h\u2082 6\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086] at h\u2087 h\u2088 h\u2089\n    <;> omega\n  have h\u2088 : a 7 = 13 := by\n    have h\u2088 := h\u2082 5\n    have h\u2089 := h\u2082 6\n    have h\u2081\u2080 := h\u2082 7\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087] at h\u2088 h\u2089 h\u2081\u2080\n    <;> omega\n  have h\u2089 : a 8 = 21 := by\n    have h\u2089 := h\u2082 6\n    have h\u2081\u2080 := h\u2082 7\n    have h\u2081\u2081 := h\u2082 8\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088] at h\u2089 h\u2081\u2080 h\u2081\u2081\n    <;> omega\n  have h\u2081\u2080 : a 9 = 34 := by\n    have h\u2081\u2080 := h\u2082 7\n    have h\u2081\u2081 := h\u2082 8\n    have h\u2081\u2082 := h\u2082 9\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089] at h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n    <;> omega\n  have h\u2081\u2081 : a 10 = 55 := by\n    have h\u2081\u2081 := h\u2082 8\n    have h\u2081\u2082 := h\u2082 9\n    have h\u2081\u2083 := h\u2082 10\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080] at h\u2081\u2081 h\u2081\u2082 h\u2081\u2083\n    <;> omega\n  have h\u2081\u2082 : \u2200 n, a (n + 6) % 4 = a n % 4 := by\n    intro n\n    have h\u2081\u2083 : a (n + 6) = 8 * a (n + 1) + 5 * a n := by\n      have h\u2081\u2084 := h\u2082 n\n      have h\u2081\u2085 := h\u2082 (n + 1)\n      have h\u2081\u2086 := h\u2082 (n + 2)\n      have h\u2081\u2087 := h\u2082 (n + 3)\n      have h\u2081\u2088 := h\u2082 (n + 4)\n      have h\u2081\u2089 := h\u2082 (n + 5)\n      have h\u2082\u2080 := h\u2082 (n + 6)\n      have h\u2082\u2081 := h\u2082 (n + 7)\n      have h\u2082\u2082 := h\u2082 (n + 8)\n      have h\u2082\u2083 := h\u2082 (n + 9)\n      simp [add_assoc, add_comm, add_left_comm] at h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 \u22a2\n      ring_nf at h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 \u22a2\n      <;>\n      (try omega) <;>\n      (try nlinarith) <;>\n      (try ring_nf at h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 \u22a2 <;> omega) <;>\n      (try nlinarith) <;>\n      (try omega) <;>\n      (try nlinarith) <;>\n      (try omega)\n    rw [h\u2081\u2083]\n    have h\u2082\u2084 : (8 * a (n + 1) + 5 * a n) % 4 = a n % 4 := by\n      have h\u2082\u2085 : (8 * a (n + 1) + 5 * a n) % 4 = (a n % 4) := by\n        have h\u2082\u2086 : a (n + 1) % 4 = a (n + 1) % 4 := rfl\n        have h\u2082\u2087 : a n % 4 = a n % 4 := rfl\n        have h\u2082\u2088 : (8 * a (n + 1) + 5 * a n) % 4 = (a n % 4) := by\n          have h\u2082\u2089 : (8 * a (n + 1) + 5 * a n) % 4 = ((8 * a (n + 1)) % 4 + (5 * a n) % 4) % 4 := by\n            omega\n          rw [h\u2082\u2089]\n          have h\u2083\u2080 : (8 * a (n + 1)) % 4 = 0 := by\n            have h\u2083\u2081 : (8 * a (n + 1)) % 4 = 0 := by\n              omega\n            exact h\u2083\u2081\n          have h\u2083\u2082 : (5 * a n) % 4 = (a n) % 4 := by\n            have h\u2083\u2083 : (5 * a n) % 4 = (a n) % 4 := by\n              omega\n            exact h\u2083\u2083\n          omega\n        exact h\u2082\u2088\n      omega\n    omega\n  have h\u2081\u2083 : a 100 % 4 = 3 := by\n    have h\u2081\u2084 : a 100 % 4 = 3 := by\n      have h\u2081\u2085 : a 100 % 4 = a 4 % 4 := by\n        have h\u2081\u2086 : a 100 % 4 = a 4 % 4 := by\n          have h\u2081\u2087 : a 100 % 4 = a 4 % 4 := by\n            have h\u2081\u2088 : a 100 % 4 = a 4 % 4 := by\n              have h\u2081\u2089 := h\u2081\u2082 94\n              have h\u2082\u2080 := h\u2081\u2082 88\n              have h\u2082\u2081 := h\u2081\u2082 82\n              have h\u2082\u2082 := h\u2081\u2082 76\n              have h\u2082\u2083 := h\u2081\u2082 70\n              have h\u2082\u2084 := h\u2081\u2082 64\n              have h\u2082\u2085 := h\u2081\u2082 58\n              have h\u2082\u2086 := h\u2081\u2082 52\n              have h\u2082\u2087 := h\u2081\u2082 46\n              have h\u2082\u2088 := h\u2081\u2082 40\n              have h\u2082\u2089 := h\u2081\u2082 34\n              have h\u2083\u2080 := h\u2081\u2082 28\n              have h\u2083\u2081 := h\u2081\u2082 22\n              have h\u2083\u2082 := h\u2081\u2082 16\n              have h\u2083\u2083 := h\u2081\u2082 10\n              have h\u2083\u2084 := h\u2081\u2082 4\n              norm_num at h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 h\u2082\u2084 h\u2082\u2085 h\u2082\u2086 h\u2082\u2087 h\u2082\u2088 h\u2082\u2089 h\u2083\u2080 h\u2083\u2081 h\u2083\u2082 h\u2083\u2083 h\u2083\u2084 \u22a2\n              <;>\n                (try omega) <;>\n                (try ring_nf at h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 h\u2082\u2084 h\u2082\u2085 h\u2082\u2086 h\u2082\u2087 h\u2082\u2088 h\u2082\u2089 h\u2083\u2080 h\u2083\u2081 h\u2083\u2082 h\u2083\u2083 h\u2083\u2084 \u22a2) <;>\n                (try omega) <;>\n                (try ring_nf) <;>\n                (try omega)\n            exact h\u2081\u2088\n          exact h\u2081\u2087\n        exact h\u2081\u2086\n      have h\u2081\u2085\u2081 : a 4 % 4 = 3 := by\n        norm_num [h\u2085]\n      rw [h\u2081\u2085]\n      exact h\u2081\u2085\u2081\n    exact h\u2081\u2084\n  exact h\u2081\u2083\n", "proof_before": "theorem mathd_numbertheory_483 (a : \u2115 \u2192 \u2115) (h\u2080 : a 1 = 1) (h\u2081 : a 2 = 1)\n    (h\u2082 : \u2200 n, a (n + 2) = a (n + 1) + a n) : a 100 % 4 = 3  :=  by\n  have h\u2083 : a 0 = 0 := by\n    have h\u2084 := h\u2082 0\n    simp [h\u2080, h\u2081] at h\u2084\n    <;> omega\n  have h\u2084 : a 3 = 2 := by\n    have h\u2085 := h\u2082 1\n    have h\u2086 := h\u2082 2\n    have h\u2087 := h\u2082 3\n    simp [h\u2080, h\u2081, h\u2083] at h\u2085 h\u2086 h\u2087\n    <;> omega\n  have h\u2085 : a 4 = 3 := by\n    have h\u2085 := h\u2082 2\n    have h\u2086 := h\u2082 3\n    have h\u2087 := h\u2082 4\n    simp [h\u2080, h\u2081, h\u2083, h\u2084] at h\u2085 h\u2086 h\u2087\n    <;> omega\n  have h\u2086 : a 5 = 5 := by\n    have h\u2086 := h\u2082 3\n    have h\u2087 := h\u2082 4\n    have h\u2088 := h\u2082 5\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085] at h\u2086 h\u2087 h\u2088\n    <;> omega\n  have h\u2087 : a 6 = 8 := by\n    have h\u2087 := h\u2082 4\n    have h\u2088 := h\u2082 5\n    have h\u2089 := h\u2082 6\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086] at h\u2087 h\u2088 h\u2089\n    <;> omega\n  have h\u2088 : a 7 = 13 := by\n    have h\u2088 := h\u2082 5\n    have h\u2089 := h\u2082 6\n    have h\u2081\u2080 := h\u2082 7\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087] at h\u2088 h\u2089 h\u2081\u2080\n    <;> omega\n  have h\u2089 : a 8 = 21 := by\n    have h\u2089 := h\u2082 6\n    have h\u2081\u2080 := h\u2082 7\n    have h\u2081\u2081 := h\u2082 8\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088] at h\u2089 h\u2081\u2080 h\u2081\u2081\n    <;> omega\n  have h\u2081\u2080 : a 9 = 34 := by\n    have h\u2081\u2080 := h\u2082 7\n    have h\u2081\u2081 := h\u2082 8\n    have h\u2081\u2082 := h\u2082 9\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089] at h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n    <;> omega\n  have h\u2081\u2081 : a 10 = 55 := by\n    have h\u2081\u2081 := h\u2082 8\n    have h\u2081\u2082 := h\u2082 9\n    have h\u2081\u2083 := h\u2082 10\n    simp [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080] at h\u2081\u2081 h\u2081\u2082 h\u2081\u2083\n    <;> omega\n  have h\u2081\u2082 : \u2200 n, a (n + 6) % 4 = a n % 4 := by\n    intro n\n    have h\u2081\u2083 : a (n + 6) = 8 * a (n + 1) + 5 * a n := by\n      have h\u2081\u2084 := h\u2082 n\n      have h\u2081\u2085 := h\u2082 (n + 1)\n      have h\u2081\u2086 := h\u2082 (n + 2)\n      have h\u2081\u2087 := h\u2082 (n + 3)\n      have h\u2081\u2088 := h\u2082 (n + 4)\n      have h\u2081\u2089 := h\u2082 (n + 5)\n      have h\u2082\u2080 := h\u2082 (n + 6)\n      have h\u2082\u2081 := h\u2082 (n + 7)\n      have h\u2082\u2082 := h\u2082 (n + 8)\n      have h\u2082\u2083 := h\u2082 (n + 9)\n      simp [add_assoc, add_comm, add_left_comm] at h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 \u22a2\n      ring_nf at h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 \u22a2\n      <;>\n      (try omega) <;>\n      (try nlinarith) <;>\n      (try ring_nf at h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 \u22a2 <;> omega) <;>\n      (try nlinarith) <;>\n      (try omega) <;>\n      (try nlinarith) <;>\n      (try omega)\n    rw [h\u2081\u2083]\n    have h\u2082\u2084 : (8 * a (n + 1) + 5 * a n) % 4 = a n % 4 := by\n      have h\u2082\u2085 : (8 * a (n + 1) + 5 * a n) % 4 = (a n % 4) := by\n        have h\u2082\u2086 : a (n + 1) % 4 = a (n + 1) % 4 := rfl\n        have h\u2082\u2087 : a n % 4 = a n % 4 := rfl\n        have h\u2082\u2088 : (8 * a (n + 1) + 5 * a n) % 4 = (a n % 4) := by\n          have h\u2082\u2089 : (8 * a (n + 1) + 5 * a n) % 4 = ((8 * a (n + 1)) % 4 + (5 * a n) % 4) % 4 := by\n            omega\n          rw [h\u2082\u2089]\n          have h\u2083\u2080 : (8 * a (n + 1)) % 4 = 0 := by\n            have h\u2083\u2081 : (8 * a (n + 1)) % 4 = 0 := by\n              omega\n            exact h\u2083\u2081\n          have h\u2083\u2082 : (5 * a n) % 4 = (a n) % 4 := by\n            have h\u2083\u2083 : (5 * a n) % 4 = (a n) % 4 := by\n              omega\n            exact h\u2083\u2083\n          omega\n        exact h\u2082\u2088\n      omega\n    omega\n  have h\u2081\u2083 : a 100 % 4 = 3 := by\n    have h\u2081\u2084 : a 100 % 4 = 3 := by\n      have h\u2081\u2085 : a 100 % 4 = a 4 % 4 := by\n        have h\u2081\u2086 : a 100 % 4 = a 4 % 4 := by\n          have h\u2081\u2087 : a 100 % 4 = a 4 % 4 := by\n            have h\u2081\u2088 : a 100 % 4 = a 4 % 4 := by\n              have h\u2081\u2089 := h\u2081\u2082 94\n              have h\u2082\u2080 := h\u2081\u2082 88\n              have h\u2082\u2081 := h\u2081\u2082 82\n              have h\u2082\u2082 := h\u2081\u2082 76\n              have h\u2082\u2083 := h\u2081\u2082 70\n              have h\u2082\u2084 := h\u2081\u2082 64\n              have h\u2082\u2085 := h\u2081\u2082 58\n              have h\u2082\u2086 := h\u2081\u2082 52\n              have h\u2082\u2087 := h\u2081\u2082 46\n              have h\u2082\u2088 := h\u2081\u2082 40\n              have h\u2082\u2089 := h\u2081\u2082 34\n              have h\u2083\u2080 := h\u2081\u2082 28\n              have h\u2083\u2081 := h\u2081\u2082 22\n              have h\u2083\u2082 := h\u2081\u2082 16\n              have h\u2083\u2083 := h\u2081\u2082 10\n              have h\u2083\u2084 := h\u2081\u2082 4\n              norm_num at h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 h\u2082\u2084 h\u2082\u2085 h\u2082\u2086 h\u2082\u2087 h\u2082\u2088 h\u2082\u2089 h\u2083\u2080 h\u2083\u2081 h\u2083\u2082 h\u2083\u2083 h\u2083\u2084 \u22a2\n              <;>\n                (try omega) <;>\n                (try ring_nf at h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 h\u2082\u2084 h\u2082\u2085 h\u2082\u2086 h\u2082\u2087 h\u2082\u2088 h\u2082\u2089 h\u2083\u2080 h\u2083\u2081 h\u2083\u2082 h\u2083\u2083 h\u2083\u2084 \u22a2) <;>\n                (try omega) <;>\n                (try ring_nf) <;>\n                (try omega)\n            exact h\u2081\u2088\n          exact h\u2081\u2087\n        exact h\u2081\u2086\n      have h\u2081\u2085\u2081 : a 4 % 4 = 3 := by\n        norm_num [h\u2085]\n      rw [h\u2081\u2085]\n      exact h\u2081\u2085\u2081\n    exact h\u2081\u2084\n  exact h\u2081\u2083\n"}
{"mode": "clear", "steps_before": 126, "alphabet": 39, "compiles": false, "id": "mathd_numbertheory_277", "proof_after": "theorem mathd_numbertheory_277 (m n : \u2115) (h\u2080 : Nat.gcd m n = 6) (h\u2081 : Nat.lcm m n = 126) :\n    60 \u2264 m + n  :=  by\n  have h\u2082 : m \u2260 0 := by\n    by_contra h\n    rw [h] at h\u2081 h\u2080\n    simp [Nat.lcm] at h\u2081 h\u2080\n    <;> norm_num at h\u2081 h\u2080 \u22a2\n    <;> aesop\n  have h\u2083 : n \u2260 0 := by\n    by_contra h\n    rw [h] at h\u2081 h\u2080\n    simp [Nat.lcm] at h\u2081 h\u2080\n    <;> norm_num at h\u2081 h\u2080 \u22a2\n    <;> aesop\n  have h\u2084 : m * n = 756 := by\n    have h\u2084\u2081 : m * n = Nat.gcd m n * Nat.lcm m n := by\n      rw [Nat.gcd_mul_lcm]\n    rw [h\u2084\u2081]\n    rw [h\u2080, h\u2081]\n    <;> norm_num\n  have h\u2085 : 6 \u2223 m := by\n    have h\u2085\u2081 : 6 \u2223 Nat.gcd m n := by\n      norm_num [h\u2080]\n    exact Nat.dvd_trans h\u2085\u2081 (Nat.gcd_dvd_left m n)\n  have h\u2086 : 6 \u2223 n := by\n    have h\u2086\u2081 : 6 \u2223 Nat.gcd m n := by\n      norm_num [h\u2080]\n    exact Nat.dvd_trans h\u2086\u2081 (Nat.gcd_dvd_right m n)\n  have h\u2087 : m + n \u2265 60 := by\n    by_contra h\n    have h\u2087\u2081 : m + n < 60 := by\n      omega\n    have h\u2087\u2082 : m \u2264 59 := by\n      omega\n    have h\u2087\u2083 : n \u2264 59 := by\n      omega\n    have h\u2087\u2084 : m \u2265 6 := by\n      by_contra h\u2087\u2084\n      have h\u2087\u2085 : m \u2264 5 := by\n        omega\n      have h\u2087\u2086 : m = 0 \u2228 m = 1 \u2228 m = 2 \u2228 m = 3 \u2228 m = 4 \u2228 m = 5 := by\n        omega\n      rcases h\u2087\u2086 with (h\u2087\u2086 | h\u2087\u2086 | h\u2087\u2086 | h\u2087\u2086 | h\u2087\u2086 | h\u2087\u2086) <;>\n        (try omega) <;>\n        (try {\n          simp [h\u2087\u2086, Nat.lcm, Nat.gcd_eq_right, Nat.gcd_eq_left] at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          norm_num at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          (try omega) <;>\n          (try {\n            have h\u2087\u2087 : n \u2264 59 := by omega\n            interval_cases n <;> norm_num at h\u2084 \u22a2 <;> omega\n          })\n        }) <;>\n        (try {\n          simp [h\u2087\u2086, Nat.lcm, Nat.gcd_eq_right, Nat.gcd_eq_left] at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          norm_num at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          (try omega) <;>\n          (try {\n            have h\u2087\u2087 : n \u2264 59 := by omega\n            interval_cases n <;> norm_num at h\u2084 \u22a2 <;> omega\n          })\n        })\n    have h\u2087\u2085 : n \u2265 6 := by\n      by_contra h\u2087\u2085\n      have h\u2087\u2086 : n \u2264 5 := by\n        omega\n      have h\u2087\u2087 : n = 0 \u2228 n = 1 \u2228 n = 2 \u2228 n = 3 \u2228 n = 4 \u2228 n = 5 := by\n        omega\n      rcases h\u2087\u2087 with (h\u2087\u2087 | h\u2087\u2087 | h\u2087\u2087 | h\u2087\u2087 | h\u2087\u2087 | h\u2087\u2087) <;>\n        (try omega) <;>\n        (try {\n          simp [h\u2087\u2087, Nat.lcm, Nat.gcd_eq_right, Nat.gcd_eq_left] at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          norm_num at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          (try omega) <;>\n          (try {\n            have h\u2087\u2088 : m \u2264 59 := by omega\n            interval_cases m <;> norm_num at h\u2084 \u22a2 <;> omega\n          })\n        }) <;>\n        (try {\n          simp [h\u2087\u2087, Nat.lcm, Nat.gcd_eq_right, Nat.gcd_eq_left] at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          norm_num at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          (try omega) <;>\n          (try {\n            have h\u2087\u2088 : m \u2264 59 := by omega\n            interval_cases m <;> norm_num at h\u2084 \u22a2 <;> omega\n          })\n        })\n    have h\u2087\u2086 : m = 6 \u2228 m = 12 \u2228 m = 18 \u2228 m = 24 \u2228 m = 30 \u2228 m = 36 \u2228 m = 42 \u2228 m = 48 \u2228 m = 54 := by\n      have h\u2087\u2087 : m \u2264 59 := by omega\n      interval_cases m <;> norm_num at h\u2084 \u22a2 <;>\n        (try omega) <;>\n        (try {\n          have h\u2087\u2088 : n \u2264 59 := by omega\n          interval_cases n <;> norm_num at h\u2084 \u22a2 <;> omega\n        }) <;>\n        (try {\n          omega\n        })\n    have h\u2087\u2087 : n = 6 \u2228 n = 12 \u2228 n = 18 \u2228 n = 24 \u2228 n = 30 \u2228 n = 36 \u2228 n = 42 \u2228 n = 48 \u2228 n = 54 := by\n      have h\u2087\u2088 : n \u2264 59 := by omega\n      interval_cases n <;> norm_num at h\u2084 \u22a2 <;>\n        (try omega) <;>\n        (try {\n          have h\u2087\u2089 : m \u2264 59 := by omega\n          interval_cases m <;> norm_num at h\u2084 \u22a2 <;> omega\n        }) <;>\n        (try {\n          omega\n        })\n    rcases h\u2087\u2086 with (rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl) <;>\n      rcases h\u2087\u2087 with (rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl) <;>\n        norm_num at h\u2080 h\u2081 h\u2084 \u22a2 <;>\n        (try omega) <;>\n        (try {\n          simp_all [Nat.gcd_eq_right, Nat.gcd_eq_left, Nat.lcm, Nat.mul_div_cancel_left]\n          <;> norm_num at *\n          <;> omega\n        })\n    <;>\n    (try omega)\n  exact h\u2087\n  <;>\n  (try omega)\n  <;>\n  (try norm_num)\n  <;>\n  (try linarith)\n", "proof_before": "theorem mathd_numbertheory_277 (m n : \u2115) (h\u2080 : Nat.gcd m n = 6) (h\u2081 : Nat.lcm m n = 126) :\n    60 \u2264 m + n  :=  by\n  have h\u2082 : m \u2260 0 := by\n    by_contra h\n    rw [h] at h\u2081 h\u2080\n    simp [Nat.lcm] at h\u2081 h\u2080\n    <;> norm_num at h\u2081 h\u2080 \u22a2\n    <;> aesop\n  have h\u2083 : n \u2260 0 := by\n    by_contra h\n    rw [h] at h\u2081 h\u2080\n    simp [Nat.lcm] at h\u2081 h\u2080\n    <;> norm_num at h\u2081 h\u2080 \u22a2\n    <;> aesop\n  have h\u2084 : m * n = 756 := by\n    have h\u2084\u2081 : m * n = Nat.gcd m n * Nat.lcm m n := by\n      rw [Nat.gcd_mul_lcm]\n    rw [h\u2084\u2081]\n    rw [h\u2080, h\u2081]\n    <;> norm_num\n  have h\u2085 : 6 \u2223 m := by\n    have h\u2085\u2081 : 6 \u2223 Nat.gcd m n := by\n      norm_num [h\u2080]\n    exact Nat.dvd_trans h\u2085\u2081 (Nat.gcd_dvd_left m n)\n  have h\u2086 : 6 \u2223 n := by\n    have h\u2086\u2081 : 6 \u2223 Nat.gcd m n := by\n      norm_num [h\u2080]\n    exact Nat.dvd_trans h\u2086\u2081 (Nat.gcd_dvd_right m n)\n  have h\u2087 : m + n \u2265 60 := by\n    by_contra h\n    have h\u2087\u2081 : m + n < 60 := by\n      omega\n    have h\u2087\u2082 : m \u2264 59 := by\n      omega\n    have h\u2087\u2083 : n \u2264 59 := by\n      omega\n    have h\u2087\u2084 : m \u2265 6 := by\n      by_contra h\u2087\u2084\n      have h\u2087\u2085 : m \u2264 5 := by\n        omega\n      have h\u2087\u2086 : m = 0 \u2228 m = 1 \u2228 m = 2 \u2228 m = 3 \u2228 m = 4 \u2228 m = 5 := by\n        omega\n      rcases h\u2087\u2086 with (h\u2087\u2086 | h\u2087\u2086 | h\u2087\u2086 | h\u2087\u2086 | h\u2087\u2086 | h\u2087\u2086) <;>\n        (try omega) <;>\n        (try {\n          simp [h\u2087\u2086, Nat.lcm, Nat.gcd_eq_right, Nat.gcd_eq_left] at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          norm_num at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          (try omega) <;>\n          (try {\n            have h\u2087\u2087 : n \u2264 59 := by omega\n            interval_cases n <;> norm_num at h\u2084 \u22a2 <;> omega\n          })\n        }) <;>\n        (try {\n          simp [h\u2087\u2086, Nat.lcm, Nat.gcd_eq_right, Nat.gcd_eq_left] at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          norm_num at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          (try omega) <;>\n          (try {\n            have h\u2087\u2087 : n \u2264 59 := by omega\n            interval_cases n <;> norm_num at h\u2084 \u22a2 <;> omega\n          })\n        })\n    have h\u2087\u2085 : n \u2265 6 := by\n      by_contra h\u2087\u2085\n      have h\u2087\u2086 : n \u2264 5 := by\n        omega\n      have h\u2087\u2087 : n = 0 \u2228 n = 1 \u2228 n = 2 \u2228 n = 3 \u2228 n = 4 \u2228 n = 5 := by\n        omega\n      rcases h\u2087\u2087 with (h\u2087\u2087 | h\u2087\u2087 | h\u2087\u2087 | h\u2087\u2087 | h\u2087\u2087 | h\u2087\u2087) <;>\n        (try omega) <;>\n        (try {\n          simp [h\u2087\u2087, Nat.lcm, Nat.gcd_eq_right, Nat.gcd_eq_left] at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          norm_num at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          (try omega) <;>\n          (try {\n            have h\u2087\u2088 : m \u2264 59 := by omega\n            interval_cases m <;> norm_num at h\u2084 \u22a2 <;> omega\n          })\n        }) <;>\n        (try {\n          simp [h\u2087\u2087, Nat.lcm, Nat.gcd_eq_right, Nat.gcd_eq_left] at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          norm_num at h\u2081 h\u2080 h\u2084 \u22a2 <;>\n          (try omega) <;>\n          (try {\n            have h\u2087\u2088 : m \u2264 59 := by omega\n            interval_cases m <;> norm_num at h\u2084 \u22a2 <;> omega\n          })\n        })\n    have h\u2087\u2086 : m = 6 \u2228 m = 12 \u2228 m = 18 \u2228 m = 24 \u2228 m = 30 \u2228 m = 36 \u2228 m = 42 \u2228 m = 48 \u2228 m = 54 := by\n      have h\u2087\u2087 : m \u2264 59 := by omega\n      interval_cases m <;> norm_num at h\u2084 \u22a2 <;>\n        (try omega) <;>\n        (try {\n          have h\u2087\u2088 : n \u2264 59 := by omega\n          interval_cases n <;> norm_num at h\u2084 \u22a2 <;> omega\n        }) <;>\n        (try {\n          omega\n        })\n    have h\u2087\u2087 : n = 6 \u2228 n = 12 \u2228 n = 18 \u2228 n = 24 \u2228 n = 30 \u2228 n = 36 \u2228 n = 42 \u2228 n = 48 \u2228 n = 54 := by\n      have h\u2087\u2088 : n \u2264 59 := by omega\n      interval_cases n <;> norm_num at h\u2084 \u22a2 <;>\n        (try omega) <;>\n        (try {\n          have h\u2087\u2089 : m \u2264 59 := by omega\n          interval_cases m <;> norm_num at h\u2084 \u22a2 <;> omega\n        }) <;>\n        (try {\n          omega\n        })\n    rcases h\u2087\u2086 with (rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl) <;>\n      rcases h\u2087\u2087 with (rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl) <;>\n        norm_num at h\u2080 h\u2081 h\u2084 \u22a2 <;>\n        (try omega) <;>\n        (try {\n          simp_all [Nat.gcd_eq_right, Nat.gcd_eq_left, Nat.lcm, Nat.mul_div_cancel_left]\n          <;> norm_num at *\n          <;> omega\n        })\n    <;>\n    (try omega)\n  exact h\u2087\n  <;>\n  (try omega)\n  <;>\n  (try norm_num)\n  <;>\n  (try linarith)\n"}
{"mode": "clear", "steps_before": 7, "alphabet": 16, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 141, "chars_after": 12, "seconds": 52.5, "id": "mathd_algebra_478", "proof_after": "theorem mathd_algebra_478 (b h v : \u211d) (h\u2080 : 0 < b \u2227 0 < h \u2227 0 < v) (h\u2081 : v = 1 / 3 * (b * h))\n    (h\u2082 : b = 30) (h\u2083 : h = 13 / 2) : v = 65  :=  by\n  nlinarith\n", "proof_before": "theorem mathd_algebra_478 (b h v : \u211d) (h\u2080 : 0 < b \u2227 0 < h \u2227 0 < v) (h\u2081 : v = 1 / 3 * (b * h))\n    (h\u2082 : b = 30) (h\u2083 : h = 13 / 2) : v = 65  :=  by\n  have h\u2084 : v = 65 := by\n    rw [h\u2081, h\u2082, h\u2083]\n    norm_num [mul_assoc]\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> linarith\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 4, "alphabet": 13, "compiles": false, "id": "mathd_numbertheory_3", "proof_after": "theorem mathd_numbertheory_3 : (\u2211 x in Finset.range 10, (x + 1) ^ 2) % 10 = 5  :=  by\n  have h_sum : (\u2211 x in Finset.range 10, (x + 1) ^ 2) % 10 = 5 := by\n    norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n      Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n      Finset.sum_range_succ, Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> rfl\n  exact h_sum\n", "proof_before": "theorem mathd_numbertheory_3 : (\u2211 x in Finset.range 10, (x + 1) ^ 2) % 10 = 5  :=  by\n  have h_sum : (\u2211 x in Finset.range 10, (x + 1) ^ 2) % 10 = 5 := by\n    norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n      Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n      Finset.sum_range_succ, Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> rfl\n  exact h_sum\n"}
{"mode": "clear", "steps_before": 16, "alphabet": 18, "compiles": true, "steps_after": 10, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "fixpoint": true, "chars_before": 624, "chars_after": 391, "seconds": 587.5, "id": "induction_11div10tonmn1ton", "proof_after": "theorem induction_11div10tonmn1ton (n : \u2115) : 11 \u2223 10 ^ n - (-1 : \u2124) ^ n  :=  by\n  have h_main : (11 : \u2124) \u2223 (10 : \u2124) ^ n - (-1 : \u2124) ^ n := by\n    have h\u2081 : (10 : \u2124) \u2261 (-1 : \u2124) [ZMOD 11] := by\n      norm_num [Int.ModEq]\n    have h\u2082 : (10 : \u2124) ^ n \u2261 (-1 : \u2124) ^ n [ZMOD 11] := by\n      simpa using h\u2081.pow n\n    have h\u2083 : (11 : \u2124) \u2223 (10 : \u2124) ^ n - (-1 : \u2124) ^ n := by\n      rw [Int.ModEq] at h\u2082\n      simpa [Int.emod_eq_emod_iff_emod_sub_eq_zero] using h\u2082\n    exact h\u2083\n  omega\n", "proof_before": "theorem induction_11div10tonmn1ton (n : \u2115) : 11 \u2223 10 ^ n - (-1 : \u2124) ^ n  :=  by\n  have h_main : (11 : \u2124) \u2223 (10 : \u2124) ^ n - (-1 : \u2124) ^ n := by\n    have h\u2081 : (10 : \u2124) \u2261 (-1 : \u2124) [ZMOD 11] := by\n      norm_num [Int.ModEq]\n    have h\u2082 : (10 : \u2124) ^ n \u2261 (-1 : \u2124) ^ n [ZMOD 11] := by\n      calc\n        (10 : \u2124) ^ n \u2261 (-1 : \u2124) ^ n [ZMOD 11] := by\n          simpa using h\u2081.pow n\n        _ \u2261 (-1 : \u2124) ^ n [ZMOD 11] := by rfl\n    have h\u2083 : (11 : \u2124) \u2223 (10 : \u2124) ^ n - (-1 : \u2124) ^ n := by\n      rw [Int.ModEq] at h\u2082\n      simpa [Int.emod_eq_emod_iff_emod_sub_eq_zero] using h\u2082\n    exact h\u2083\n  have h_final : 11 \u2223 10 ^ n - (-1 : \u2124) ^ n := by\n    simpa [Int.emod_eq_emod_iff_emod_sub_eq_zero] using h_main\n  exact h_final\n"}
{"mode": "clear", "steps_before": 311, "alphabet": 70, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 311, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 13950, "chars_after": 13950, "seconds": 749.6, "id": "amc12b_2020_p21", "proof_after": "theorem amc12b_2020_p21 (S : Finset \u2115)\n    (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 0 < n \u2227 (\u2191n + (1000 : \u211d)) / 70 = Int.floor (Real.sqrt n)) : S.card = 6  :=  by\n  have h_main : S = {400, 470, 2290, 2360, 2430, 2500} := by\n    apply Finset.ext\n    intro n\n    rw [h\u2080]\n    simp only [Finset.mem_insert, Finset.mem_singleton]\n    constructor\n    \u00b7 \n      intro h\n      have h\u2081 : 0 < n := h.1\n      have h\u2082 : (\u2191n + (1000 : \u211d)) / 70 = Int.floor (Real.sqrt n) := h.2\n      have h\u2083 : (n + 1000) % 70 = 0 := by\n        have h\u2084 : ((n : \u211d) + 1000) / 70 = Int.floor (Real.sqrt n) := by exact_mod_cast h\u2082\n        have h\u2085 : (n + 1000 : \u2124) % 70 = 0 := by\n          have h\u2086 : ((n : \u211d) + 1000 : \u211d) / 70 = Int.floor (Real.sqrt n) := by exact_mod_cast h\u2082\n          have h\u2087 : (n : \u2124) + 1000 \u2261 0 [ZMOD 70] := by\n            have h\u2088 : ((n : \u211d) + 1000 : \u211d) / 70 = Int.floor (Real.sqrt n) := by exact_mod_cast h\u2082\n            have h\u2089 : (Int.floor (Real.sqrt n) : \u211d) = ((n : \u211d) + 1000 : \u211d) / 70 := by linarith\n            have h\u2081\u2080 : (Int.floor (Real.sqrt n) : \u211d) * 70 = (n : \u211d) + 1000 := by linarith\n            have h\u2081\u2081 : (Int.floor (Real.sqrt n) : \u2124) * 70 = (n : \u2124) + 1000 := by\n              norm_cast at h\u2081\u2080 \u22a2\n              <;> (try norm_num at h\u2081\u2080 \u22a2) <;> (try ring_nf at h\u2081\u2080 \u22a2) <;> (try field_simp at h\u2081\u2080 \u22a2) <;> (try norm_cast at h\u2081\u2080 \u22a2) <;> (try linarith)\n              <;> (try ring_nf at h\u2081\u2080 \u22a2) <;> (try norm_cast at h\u2081\u2080 \u22a2) <;> (try linarith)\n            have h\u2081\u2082 : (n : \u2124) + 1000 \u2261 0 [ZMOD 70] := by\n              rw [Int.ModEq]\n              omega\n            exact h\u2081\u2082\n          exact h\u2087\n        exact_mod_cast h\u2085\n      have h\u2084 : n % 70 = 50 := by\n        omega\n      have h\u2085 : \u2203 k : \u2115, n = 70 * k + 50 := by\n        use n / 70\n        have h\u2086 := Nat.mod_add_div n 70\n        omega\n      obtain \u27e8k, h\u2086\u27e9 := h\u2085\n      have h\u2087 : k + 15 = Int.floor (Real.sqrt n) := by\n        have h\u2088 : ((n : \u211d) + 1000 : \u211d) / 70 = Int.floor (Real.sqrt n) := by exact_mod_cast h\u2082\n        have h\u2089 : (k : \u211d) + 15 = ((n : \u211d) + 1000 : \u211d) / 70 := by\n          rw [h\u2086]\n          <;> ring_nf at *\n          <;> norm_num\n          <;> field_simp at *\n          <;> ring_nf at *\n          <;> norm_cast at *\n          <;> linarith\n        have h\u2081\u2080 : (k : \u211d) + 15 = (Int.floor (Real.sqrt n) : \u211d) := by\n          linarith\n        have h\u2081\u2081 : (k : \u2124) + 15 = Int.floor (Real.sqrt n) := by\n          norm_cast at h\u2081\u2080 \u22a2\n          <;> (try norm_num at h\u2081\u2080 \u22a2) <;> (try ring_nf at h\u2081\u2080 \u22a2) <;> (try field_simp at h\u2081\u2080 \u22a2) <;> (try norm_cast at h\u2081\u2080 \u22a2) <;> (try linarith)\n        exact_mod_cast h\u2081\u2081\n      have h\u2088 : (k + 15 : \u211d) ^ 2 \u2264 (n : \u211d) := by\n        have h\u2089 : (k + 15 : \u211d) \u2264 Real.sqrt n := by\n          have h\u2081\u2080 : (k + 15 : \u211d) \u2264 Real.sqrt n := by\n            have h\u2081\u2081 : (k + 15 : \u211d) = Int.floor (Real.sqrt n) := by\n              exact_mod_cast h\u2087\n            have h\u2081\u2082 : (Int.floor (Real.sqrt n) : \u211d) \u2264 Real.sqrt n := by\n              exact Int.floor_le (Real.sqrt n)\n            linarith\n          exact h\u2081\u2080\n        have h\u2081\u2080 : 0 \u2264 Real.sqrt n := Real.sqrt_nonneg n\n        have h\u2081\u2081 : (k + 15 : \u211d) ^ 2 \u2264 (n : \u211d) := by\n          nlinarith [Real.sq_sqrt (by positivity : 0 \u2264 (n : \u211d)), h\u2089]\n        exact h\u2081\u2081\n      have h\u2089 : (n : \u211d) < (k + 16 : \u211d) ^ 2 := by\n        have h\u2081\u2080 : Real.sqrt n < (k + 16 : \u211d) := by\n          have h\u2081\u2081 : Real.sqrt n < (k + 16 : \u211d) := by\n            have h\u2081\u2082 : (Int.floor (Real.sqrt n) : \u211d) = (k + 15 : \u211d) := by\n              have h\u2081\u2083 : (k + 15 : \u211d) = Int.floor (Real.sqrt n) := by\n                exact_mod_cast h\u2087\n              linarith\n            have h\u2081\u2083 : Real.sqrt n < (Int.floor (Real.sqrt n) : \u211d) + 1 := by\n              linarith [Int.floor_le (Real.sqrt n), Int.lt_floor_add_one (Real.sqrt n)]\n            linarith\n          exact h\u2081\u2081\n        have h\u2081\u2081 : 0 \u2264 Real.sqrt n := Real.sqrt_nonneg n\n        have h\u2081\u2082 : (n : \u211d) < (k + 16 : \u211d) ^ 2 := by\n          nlinarith [Real.sq_sqrt (by positivity : 0 \u2264 (n : \u211d)), h\u2081\u2080]\n        exact h\u2081\u2082\n      have h\u2081\u2080 : (k + 15 : \u2115) ^ 2 \u2264 n := by\n        norm_cast at h\u2088 \u22a2\n        <;> (try ring_nf at h\u2088 \u22a2) <;> (try norm_num at h\u2088 \u22a2) <;> (try nlinarith)\n      have h\u2081\u2081 : n < (k + 16 : \u2115) ^ 2 := by\n        norm_cast at h\u2089 \u22a2\n        <;> (try ring_nf at h\u2089 \u22a2) <;> (try norm_num at h\u2089 \u22a2) <;> (try nlinarith)\n      have h\u2081\u2082 : k \u2264 35 := by\n        by_contra h\n        have h\u2081\u2083 : k \u2265 36 := by omega\n        have h\u2081\u2084 : (k + 15 : \u2115) ^ 2 > n := by\n          nlinarith\n        nlinarith\n      have h\u2081\u2083 : k \u2265 5 \u2228 k \u2264 4 := by omega\n      cases h\u2081\u2083 with\n      | inl h\u2081\u2084 =>\n        have h\u2081\u2085 : k \u2265 5 := h\u2081\u2084\n        have h\u2081\u2086 : k \u2264 35 := by omega\n        interval_cases k <;> norm_num at h\u2086 h\u2081\u2080 h\u2081\u2081 h\u2088 h\u2089 h\u2087 \u22a2 <;>\n          (try omega) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 400 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 400, Real.sq_sqrt (show 0 \u2264 (400 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 470 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 2290 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 2360 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 2430 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 2500 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 2500, Real.sq_sqrt (show 0 \u2264 (2500 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try omega)\n      | inr h\u2081\u2084 =>\n        have h\u2081\u2085 : k \u2264 4 := h\u2081\u2084\n        have h\u2081\u2086 : k \u2265 0 := by omega\n        interval_cases k <;> norm_num at h\u2086 h\u2081\u2080 h\u2081\u2081 h\u2088 h\u2089 h\u2087 \u22a2 <;>\n          (try omega) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 400 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 400, Real.sq_sqrt (show 0 \u2264 (400 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try omega)\n    \u00b7 \n      intro h\n      have h\u2081 : n = 400 \u2228 n = 470 \u2228 n = 2290 \u2228 n = 2360 \u2228 n = 2430 \u2228 n = 2500 := by\n        tauto\n      rcases h\u2081 with (rfl | rfl | rfl | rfl | rfl | rfl)\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 400 : \u211d) = 20 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 400, Real.sq_sqrt (show 0 \u2264 (400 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 400, Real.sq_sqrt (show 0 \u2264 (400 : \u211d) by norm_num)]\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 470 : \u211d) = 21 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num),\n                Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num)]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num),\n                Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num)]\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 2290 : \u211d) = 47 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num)]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num)]\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 2360 : \u211d) = 48 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num)]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num)]\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 2430 : \u211d) = 49 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num)]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num)]\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 2500 : \u211d) = 50 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2500, Real.sq_sqrt (show 0 \u2264 (2500 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 2500, Real.sq_sqrt (show 0 \u2264 (2500 : \u211d) by norm_num)]\n  have h_card : S.card = 6 := by\n    rw [h_main]\n    norm_num\n    <;> rfl\n  exact h_card\n", "proof_before": "theorem amc12b_2020_p21 (S : Finset \u2115)\n    (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 0 < n \u2227 (\u2191n + (1000 : \u211d)) / 70 = Int.floor (Real.sqrt n)) : S.card = 6  :=  by\n  have h_main : S = {400, 470, 2290, 2360, 2430, 2500} := by\n    apply Finset.ext\n    intro n\n    rw [h\u2080]\n    simp only [Finset.mem_insert, Finset.mem_singleton]\n    constructor\n    \u00b7 \n      intro h\n      have h\u2081 : 0 < n := h.1\n      have h\u2082 : (\u2191n + (1000 : \u211d)) / 70 = Int.floor (Real.sqrt n) := h.2\n      have h\u2083 : (n + 1000) % 70 = 0 := by\n        have h\u2084 : ((n : \u211d) + 1000) / 70 = Int.floor (Real.sqrt n) := by exact_mod_cast h\u2082\n        have h\u2085 : (n + 1000 : \u2124) % 70 = 0 := by\n          have h\u2086 : ((n : \u211d) + 1000 : \u211d) / 70 = Int.floor (Real.sqrt n) := by exact_mod_cast h\u2082\n          have h\u2087 : (n : \u2124) + 1000 \u2261 0 [ZMOD 70] := by\n            have h\u2088 : ((n : \u211d) + 1000 : \u211d) / 70 = Int.floor (Real.sqrt n) := by exact_mod_cast h\u2082\n            have h\u2089 : (Int.floor (Real.sqrt n) : \u211d) = ((n : \u211d) + 1000 : \u211d) / 70 := by linarith\n            have h\u2081\u2080 : (Int.floor (Real.sqrt n) : \u211d) * 70 = (n : \u211d) + 1000 := by linarith\n            have h\u2081\u2081 : (Int.floor (Real.sqrt n) : \u2124) * 70 = (n : \u2124) + 1000 := by\n              norm_cast at h\u2081\u2080 \u22a2\n              <;> (try norm_num at h\u2081\u2080 \u22a2) <;> (try ring_nf at h\u2081\u2080 \u22a2) <;> (try field_simp at h\u2081\u2080 \u22a2) <;> (try norm_cast at h\u2081\u2080 \u22a2) <;> (try linarith)\n              <;> (try ring_nf at h\u2081\u2080 \u22a2) <;> (try norm_cast at h\u2081\u2080 \u22a2) <;> (try linarith)\n            have h\u2081\u2082 : (n : \u2124) + 1000 \u2261 0 [ZMOD 70] := by\n              rw [Int.ModEq]\n              omega\n            exact h\u2081\u2082\n          exact h\u2087\n        exact_mod_cast h\u2085\n      have h\u2084 : n % 70 = 50 := by\n        omega\n      have h\u2085 : \u2203 k : \u2115, n = 70 * k + 50 := by\n        use n / 70\n        have h\u2086 := Nat.mod_add_div n 70\n        omega\n      obtain \u27e8k, h\u2086\u27e9 := h\u2085\n      have h\u2087 : k + 15 = Int.floor (Real.sqrt n) := by\n        have h\u2088 : ((n : \u211d) + 1000 : \u211d) / 70 = Int.floor (Real.sqrt n) := by exact_mod_cast h\u2082\n        have h\u2089 : (k : \u211d) + 15 = ((n : \u211d) + 1000 : \u211d) / 70 := by\n          rw [h\u2086]\n          <;> ring_nf at *\n          <;> norm_num\n          <;> field_simp at *\n          <;> ring_nf at *\n          <;> norm_cast at *\n          <;> linarith\n        have h\u2081\u2080 : (k : \u211d) + 15 = (Int.floor (Real.sqrt n) : \u211d) := by\n          linarith\n        have h\u2081\u2081 : (k : \u2124) + 15 = Int.floor (Real.sqrt n) := by\n          norm_cast at h\u2081\u2080 \u22a2\n          <;> (try norm_num at h\u2081\u2080 \u22a2) <;> (try ring_nf at h\u2081\u2080 \u22a2) <;> (try field_simp at h\u2081\u2080 \u22a2) <;> (try norm_cast at h\u2081\u2080 \u22a2) <;> (try linarith)\n        exact_mod_cast h\u2081\u2081\n      have h\u2088 : (k + 15 : \u211d) ^ 2 \u2264 (n : \u211d) := by\n        have h\u2089 : (k + 15 : \u211d) \u2264 Real.sqrt n := by\n          have h\u2081\u2080 : (k + 15 : \u211d) \u2264 Real.sqrt n := by\n            have h\u2081\u2081 : (k + 15 : \u211d) = Int.floor (Real.sqrt n) := by\n              exact_mod_cast h\u2087\n            have h\u2081\u2082 : (Int.floor (Real.sqrt n) : \u211d) \u2264 Real.sqrt n := by\n              exact Int.floor_le (Real.sqrt n)\n            linarith\n          exact h\u2081\u2080\n        have h\u2081\u2080 : 0 \u2264 Real.sqrt n := Real.sqrt_nonneg n\n        have h\u2081\u2081 : (k + 15 : \u211d) ^ 2 \u2264 (n : \u211d) := by\n          nlinarith [Real.sq_sqrt (by positivity : 0 \u2264 (n : \u211d)), h\u2089]\n        exact h\u2081\u2081\n      have h\u2089 : (n : \u211d) < (k + 16 : \u211d) ^ 2 := by\n        have h\u2081\u2080 : Real.sqrt n < (k + 16 : \u211d) := by\n          have h\u2081\u2081 : Real.sqrt n < (k + 16 : \u211d) := by\n            have h\u2081\u2082 : (Int.floor (Real.sqrt n) : \u211d) = (k + 15 : \u211d) := by\n              have h\u2081\u2083 : (k + 15 : \u211d) = Int.floor (Real.sqrt n) := by\n                exact_mod_cast h\u2087\n              linarith\n            have h\u2081\u2083 : Real.sqrt n < (Int.floor (Real.sqrt n) : \u211d) + 1 := by\n              linarith [Int.floor_le (Real.sqrt n), Int.lt_floor_add_one (Real.sqrt n)]\n            linarith\n          exact h\u2081\u2081\n        have h\u2081\u2081 : 0 \u2264 Real.sqrt n := Real.sqrt_nonneg n\n        have h\u2081\u2082 : (n : \u211d) < (k + 16 : \u211d) ^ 2 := by\n          nlinarith [Real.sq_sqrt (by positivity : 0 \u2264 (n : \u211d)), h\u2081\u2080]\n        exact h\u2081\u2082\n      have h\u2081\u2080 : (k + 15 : \u2115) ^ 2 \u2264 n := by\n        norm_cast at h\u2088 \u22a2\n        <;> (try ring_nf at h\u2088 \u22a2) <;> (try norm_num at h\u2088 \u22a2) <;> (try nlinarith)\n      have h\u2081\u2081 : n < (k + 16 : \u2115) ^ 2 := by\n        norm_cast at h\u2089 \u22a2\n        <;> (try ring_nf at h\u2089 \u22a2) <;> (try norm_num at h\u2089 \u22a2) <;> (try nlinarith)\n      have h\u2081\u2082 : k \u2264 35 := by\n        by_contra h\n        have h\u2081\u2083 : k \u2265 36 := by omega\n        have h\u2081\u2084 : (k + 15 : \u2115) ^ 2 > n := by\n          nlinarith\n        nlinarith\n      have h\u2081\u2083 : k \u2265 5 \u2228 k \u2264 4 := by omega\n      cases h\u2081\u2083 with\n      | inl h\u2081\u2084 =>\n        have h\u2081\u2085 : k \u2265 5 := h\u2081\u2084\n        have h\u2081\u2086 : k \u2264 35 := by omega\n        interval_cases k <;> norm_num at h\u2086 h\u2081\u2080 h\u2081\u2081 h\u2088 h\u2089 h\u2087 \u22a2 <;>\n          (try omega) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 400 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 400, Real.sq_sqrt (show 0 \u2264 (400 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 470 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 2290 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 2360 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 2430 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 2500 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 2500, Real.sq_sqrt (show 0 \u2264 (2500 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try omega)\n      | inr h\u2081\u2084 =>\n        have h\u2081\u2085 : k \u2264 4 := h\u2081\u2084\n        have h\u2081\u2086 : k \u2265 0 := by omega\n        interval_cases k <;> norm_num at h\u2086 h\u2081\u2080 h\u2081\u2081 h\u2088 h\u2089 h\u2087 \u22a2 <;>\n          (try omega) <;>\n          (try\n            {\n              have h\u2081\u2087 : n = 400 := by omega\n              simp_all [h\u2081\u2087]\n              <;> norm_num [Int.emod_eq_of_lt] at *\n              <;>\n                (try\n                  {\n                    rw [eq_comm]\n                    <;> norm_num [Int.floor_eq_iff, Real.le_sqrt, Real.sqrt_lt]\n                    <;>\n                      (try\n                        {\n                          constructor <;> norm_num <;>\n                            nlinarith [Real.sqrt_nonneg 400, Real.sq_sqrt (show 0 \u2264 (400 : \u211d) by norm_num)]\n                        })\n                  })\n            }) <;>\n          (try omega)\n    \u00b7 \n      intro h\n      have h\u2081 : n = 400 \u2228 n = 470 \u2228 n = 2290 \u2228 n = 2360 \u2228 n = 2430 \u2228 n = 2500 := by\n        tauto\n      rcases h\u2081 with (rfl | rfl | rfl | rfl | rfl | rfl)\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 400 : \u211d) = 20 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 400, Real.sq_sqrt (show 0 \u2264 (400 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 400, Real.sq_sqrt (show 0 \u2264 (400 : \u211d) by norm_num)]\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 470 : \u211d) = 21 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num),\n                Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num)]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num),\n                Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 470, Real.sq_sqrt (show 0 \u2264 (470 : \u211d) by norm_num)]\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 2290 : \u211d) = 47 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num)]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 2290, Real.sq_sqrt (show 0 \u2264 (2290 : \u211d) by norm_num)]\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 2360 : \u211d) = 48 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num)]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 2360, Real.sq_sqrt (show 0 \u2264 (2360 : \u211d) by norm_num)]\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 2430 : \u211d) = 49 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num)]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num),\n                Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 2430, Real.sq_sqrt (show 0 \u2264 (2430 : \u211d) by norm_num)]\n      \u00b7 \n        constructor\n        \u00b7 \n          norm_num\n        \u00b7 \n          have h\u2082 : Int.floor (Real.sqrt 2500 : \u211d) = 50 := by\n            rw [Int.floor_eq_iff]\n            constructor\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n            \u00b7 \n              norm_num [Real.le_sqrt, Real.sqrt_lt]\n              <;>\n              nlinarith [Real.sqrt_nonneg 2500, Real.sq_sqrt (show 0 \u2264 (2500 : \u211d) by norm_num)]\n          norm_num [h\u2082]\n          <;>\n          nlinarith [Real.sqrt_nonneg 2500, Real.sq_sqrt (show 0 \u2264 (2500 : \u211d) by norm_num)]\n  have h_card : S.card = 6 := by\n    rw [h_main]\n    norm_num\n    <;> rfl\n  exact h_card\n"}
{"mode": "clear", "steps_before": 87, "alphabet": 27, "compiles": true, "steps_after": 73, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 2, "fixpoint": false, "chars_before": 2972, "chars_after": 2183, "seconds": 1015.5, "id": "algebra_amgm_sumasqdivbgeqsuma", "proof_after": "theorem algebra_amgm_sumasqdivbgeqsuma (a b c d : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c \u2227 0 < d) :\n    a ^ 2 / b + b ^ 2 / c + c ^ 2 / d + d ^ 2 / a \u2265 a + b + c + d  :=  by\n  have h\u2081 : a ^ 2 / b + b \u2265 2 * a := by\n    have h\u2081\u2081 : 0 < b := by linarith\n    have h\u2081\u2082 : 0 < a := by linarith\n    have h\u2081\u2083 : 0 < a ^ 2 / b := by positivity\n    have h\u2081\u2084 : a ^ 2 / b * b = a ^ 2 := by\n      field_simp [h\u2081\u2081.ne']\n    have h\u2081\u2085 : a ^ 2 / b + b \u2265 2 * a := by\n      have h\u2081\u2085\u2081 : 0 < a ^ 2 / b := by positivity\n      have h\u2081\u2085\u2082 : 0 < b := by linarith\n      have h\u2081\u2085\u2083 : 0 < a ^ 2 / b * b := by positivity\n      nlinarith [sq_nonneg (a - b), sq_nonneg (a ^ 2 / b - b), sq_nonneg (a ^ 2 / b - a),\n        sq_nonneg (b - a)]\n    exact h\u2081\u2085\n  have h\u2082 : b ^ 2 / c + c \u2265 2 * b := by\n    have h\u2082\u2081 : 0 < c := by linarith\n    have h\u2082\u2082 : 0 < b := by linarith\n    have h\u2082\u2083 : 0 < b ^ 2 / c := by positivity\n    have h\u2082\u2084 : b ^ 2 / c * c = b ^ 2 := by\n      field_simp [h\u2082\u2081.ne']\n    have h\u2082\u2085 : b ^ 2 / c + c \u2265 2 * b := by\n      have h\u2082\u2085\u2081 : 0 < b ^ 2 / c := by positivity\n      have h\u2082\u2085\u2082 : 0 < c := by linarith\n      have h\u2082\u2085\u2083 : 0 < b ^ 2 / c * c := by positivity\n      nlinarith [sq_nonneg (b - c), sq_nonneg (b ^ 2 / c - c), sq_nonneg (b ^ 2 / c - b),\n        sq_nonneg (c - b)]\n    exact h\u2082\u2085\n  have h\u2083 : c ^ 2 / d + d \u2265 2 * c := by\n    have h\u2083\u2081 : 0 < d := by linarith\n    have h\u2083\u2082 : 0 < c := by linarith\n    have h\u2083\u2083 : 0 < c ^ 2 / d := by positivity\n    have h\u2083\u2084 : c ^ 2 / d * d = c ^ 2 := by\n      field_simp [h\u2083\u2081.ne']\n    have h\u2083\u2085 : c ^ 2 / d + d \u2265 2 * c := by\n      have h\u2083\u2085\u2081 : 0 < c ^ 2 / d := by positivity\n      have h\u2083\u2085\u2082 : 0 < d := by linarith\n      have h\u2083\u2085\u2083 : 0 < c ^ 2 / d * d := by positivity\n      nlinarith [sq_nonneg (c - d), sq_nonneg (c ^ 2 / d - d), sq_nonneg (c ^ 2 / d - c),\n        sq_nonneg (d - c)]\n    exact h\u2083\u2085\n  have h\u2084 : d ^ 2 / a + a \u2265 2 * d := by\n    have h\u2084\u2081 : 0 < a := by linarith\n    have h\u2084\u2082 : 0 < d := by linarith\n    have h\u2084\u2083 : 0 < d ^ 2 / a := by positivity\n    have h\u2084\u2084 : d ^ 2 / a * a = d ^ 2 := by\n      field_simp [h\u2084\u2081.ne']\n    have h\u2084\u2085 : d ^ 2 / a + a \u2265 2 * d := by\n      have h\u2084\u2085\u2081 : 0 < d ^ 2 / a := by positivity\n      have h\u2084\u2085\u2082 : 0 < a := by linarith\n      have h\u2084\u2085\u2083 : 0 < d ^ 2 / a * a := by positivity\n      nlinarith [sq_nonneg (d - a), sq_nonneg (d ^ 2 / a - a), sq_nonneg (d ^ 2 / a - d),\n        sq_nonneg (a - d)]\n    exact h\u2084\u2085\n  linarith\n", "proof_before": "theorem algebra_amgm_sumasqdivbgeqsuma (a b c d : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c \u2227 0 < d) :\n    a ^ 2 / b + b ^ 2 / c + c ^ 2 / d + d ^ 2 / a \u2265 a + b + c + d  :=  by\n  have h\u2081 : a ^ 2 / b + b \u2265 2 * a := by\n    have h\u2081\u2081 : 0 < b := by linarith\n    have h\u2081\u2082 : 0 < a := by linarith\n    have h\u2081\u2083 : 0 < a ^ 2 / b := by positivity\n    have h\u2081\u2084 : a ^ 2 / b * b = a ^ 2 := by\n      field_simp [h\u2081\u2081.ne']\n    have h\u2081\u2085 : a ^ 2 / b + b \u2265 2 * a := by\n      have h\u2081\u2085\u2081 : 0 < a ^ 2 / b := by positivity\n      have h\u2081\u2085\u2082 : 0 < b := by linarith\n      have h\u2081\u2085\u2083 : 0 < a ^ 2 / b * b := by positivity\n      nlinarith [sq_nonneg (a - b), sq_nonneg (a ^ 2 / b - b), sq_nonneg (a ^ 2 / b - a),\n        sq_nonneg (b - a)]\n    exact h\u2081\u2085\n  have h\u2082 : b ^ 2 / c + c \u2265 2 * b := by\n    have h\u2082\u2081 : 0 < c := by linarith\n    have h\u2082\u2082 : 0 < b := by linarith\n    have h\u2082\u2083 : 0 < b ^ 2 / c := by positivity\n    have h\u2082\u2084 : b ^ 2 / c * c = b ^ 2 := by\n      field_simp [h\u2082\u2081.ne']\n    have h\u2082\u2085 : b ^ 2 / c + c \u2265 2 * b := by\n      have h\u2082\u2085\u2081 : 0 < b ^ 2 / c := by positivity\n      have h\u2082\u2085\u2082 : 0 < c := by linarith\n      have h\u2082\u2085\u2083 : 0 < b ^ 2 / c * c := by positivity\n      nlinarith [sq_nonneg (b - c), sq_nonneg (b ^ 2 / c - c), sq_nonneg (b ^ 2 / c - b),\n        sq_nonneg (c - b)]\n    exact h\u2082\u2085\n  have h\u2083 : c ^ 2 / d + d \u2265 2 * c := by\n    have h\u2083\u2081 : 0 < d := by linarith\n    have h\u2083\u2082 : 0 < c := by linarith\n    have h\u2083\u2083 : 0 < c ^ 2 / d := by positivity\n    have h\u2083\u2084 : c ^ 2 / d * d = c ^ 2 := by\n      field_simp [h\u2083\u2081.ne']\n    have h\u2083\u2085 : c ^ 2 / d + d \u2265 2 * c := by\n      have h\u2083\u2085\u2081 : 0 < c ^ 2 / d := by positivity\n      have h\u2083\u2085\u2082 : 0 < d := by linarith\n      have h\u2083\u2085\u2083 : 0 < c ^ 2 / d * d := by positivity\n      nlinarith [sq_nonneg (c - d), sq_nonneg (c ^ 2 / d - d), sq_nonneg (c ^ 2 / d - c),\n        sq_nonneg (d - c)]\n    exact h\u2083\u2085\n  have h\u2084 : d ^ 2 / a + a \u2265 2 * d := by\n    have h\u2084\u2081 : 0 < a := by linarith\n    have h\u2084\u2082 : 0 < d := by linarith\n    have h\u2084\u2083 : 0 < d ^ 2 / a := by positivity\n    have h\u2084\u2084 : d ^ 2 / a * a = d ^ 2 := by\n      field_simp [h\u2084\u2081.ne']\n    have h\u2084\u2085 : d ^ 2 / a + a \u2265 2 * d := by\n      have h\u2084\u2085\u2081 : 0 < d ^ 2 / a := by positivity\n      have h\u2084\u2085\u2082 : 0 < a := by linarith\n      have h\u2084\u2085\u2083 : 0 < d ^ 2 / a * a := by positivity\n      nlinarith [sq_nonneg (d - a), sq_nonneg (d ^ 2 / a - a), sq_nonneg (d ^ 2 / a - d),\n        sq_nonneg (a - d)]\n    exact h\u2084\u2085\n  have h\u2085 : a ^ 2 / b + b ^ 2 / c + c ^ 2 / d + d ^ 2 / a + (a + b + c + d) \u2265 2 * a + 2 * b + 2 * c + 2 * d := by\n    have h\u2085\u2081 : a ^ 2 / b + b \u2265 2 * a := h\u2081\n    have h\u2085\u2082 : b ^ 2 / c + c \u2265 2 * b := h\u2082\n    have h\u2085\u2083 : c ^ 2 / d + d \u2265 2 * c := h\u2083\n    have h\u2085\u2084 : d ^ 2 / a + a \u2265 2 * d := h\u2084\n    have h\u2085\u2085 : a ^ 2 / b + b ^ 2 / c + c ^ 2 / d + d ^ 2 / a + (a + b + c + d) = (a ^ 2 / b + b) + (b ^ 2 / c + c) + (c ^ 2 / d + d) + (d ^ 2 / a + a) := by\n      ring\n    rw [h\u2085\u2085]\n    linarith\n  have h\u2086 : a ^ 2 / b + b ^ 2 / c + c ^ 2 / d + d ^ 2 / a \u2265 a + b + c + d := by\n    have h\u2086\u2081 : a ^ 2 / b + b ^ 2 / c + c ^ 2 / d + d ^ 2 / a + (a + b + c + d) \u2265 2 * a + 2 * b + 2 * c + 2 * d := h\u2085\n    have h\u2086\u2082 : a ^ 2 / b + b ^ 2 / c + c ^ 2 / d + d ^ 2 / a \u2265 a + b + c + d := by\n      linarith\n    exact h\u2086\u2082\n  exact h\u2086\n"}
{"mode": "clear", "steps_before": 36, "alphabet": 26, "compiles": false, "id": "mathd_numbertheory_175", "proof_after": "theorem mathd_numbertheory_175 : 2 ^ 2010 % 10 = 4  :=  by\n  have h\u2080 : (2 ^ 2010 : \u2115) % 10 = 4 := by\n    have h\u2081 : \u2200 n : \u2115, n \u2265 1 \u2192 6 ^ n % 10 = 6 := by\n      intro n hn\n      induction' hn with n hn IH\n      \u00b7 norm_num\n      \u00b7 omega\n    have h\u2082 : (2 ^ 4 : \u2115) % 10 = 6 := by norm_num\n    have h\u2083 : (2 ^ 2010 : \u2115) % 10 = 4 := by\n      have h\u2084 : (2 ^ 2010 : \u2115) = (2 ^ 4) ^ 502 * 2 ^ 2 := by\n        norm_num [pow_add, pow_mul, pow_one, pow_two, pow_three, pow_succ]\n        <;> ring_nf at *\n        <;> norm_num at *\n        <;> rfl\n      rw [h\u2084]\n      have h\u2085 : ((2 ^ 4 : \u2115) ^ 502 * 2 ^ 2 : \u2115) % 10 = 4 := by\n        have h\u2085\u2081 : ((2 ^ 4 : \u2115) ^ 502 * 2 ^ 2 : \u2115) % 10 = ((2 ^ 4 : \u2115) ^ 502 % 10 * (2 ^ 2 % 10)) % 10 := by\n          simp [Nat.mul_mod, Nat.pow_mod, Nat.mod_mod]\n        rw [h\u2085\u2081]\n        have h\u2085\u2082 : (2 ^ 4 : \u2115) ^ 502 % 10 = 6 := by\n          have h\u2085\u2082\u2081 : (2 ^ 4 : \u2115) % 10 = 6 := by norm_num\n          have h\u2085\u2082\u2082 : (2 ^ 4 : \u2115) ^ 502 % 10 = 6 := by\n            rw [\u2190 Nat.mod_add_div ((2 ^ 4 : \u2115) ^ 502) 10]\n            simp [h\u2081, Nat.pow_mod, Nat.mul_mod, Nat.add_mod, h\u2085\u2082\u2081]\n            <;> norm_num\n            <;> omega\n          exact h\u2085\u2082\u2082\n        have h\u2085\u2083 : (2 ^ 2 : \u2115) % 10 = 4 := by norm_num\n        rw [h\u2085\u2082, h\u2085\u2083]\n        <;> norm_num\n        <;> omega\n      exact h\u2085\n    exact h\u2083\n  exact h\u2080\n", "proof_before": "theorem mathd_numbertheory_175 : 2 ^ 2010 % 10 = 4  :=  by\n  have h\u2080 : (2 ^ 2010 : \u2115) % 10 = 4 := by\n    have h\u2081 : \u2200 n : \u2115, n \u2265 1 \u2192 6 ^ n % 10 = 6 := by\n      intro n hn\n      induction' hn with n hn IH\n      \u00b7 norm_num\n      \u00b7 omega\n    have h\u2082 : (2 ^ 4 : \u2115) % 10 = 6 := by norm_num\n    have h\u2083 : (2 ^ 2010 : \u2115) % 10 = 4 := by\n      have h\u2084 : (2 ^ 2010 : \u2115) = (2 ^ 4) ^ 502 * 2 ^ 2 := by\n        norm_num [pow_add, pow_mul, pow_one, pow_two, pow_three, pow_succ]\n        <;> ring_nf at *\n        <;> norm_num at *\n        <;> rfl\n      rw [h\u2084]\n      have h\u2085 : ((2 ^ 4 : \u2115) ^ 502 * 2 ^ 2 : \u2115) % 10 = 4 := by\n        have h\u2085\u2081 : ((2 ^ 4 : \u2115) ^ 502 * 2 ^ 2 : \u2115) % 10 = ((2 ^ 4 : \u2115) ^ 502 % 10 * (2 ^ 2 % 10)) % 10 := by\n          simp [Nat.mul_mod, Nat.pow_mod, Nat.mod_mod]\n        rw [h\u2085\u2081]\n        have h\u2085\u2082 : (2 ^ 4 : \u2115) ^ 502 % 10 = 6 := by\n          have h\u2085\u2082\u2081 : (2 ^ 4 : \u2115) % 10 = 6 := by norm_num\n          have h\u2085\u2082\u2082 : (2 ^ 4 : \u2115) ^ 502 % 10 = 6 := by\n            rw [\u2190 Nat.mod_add_div ((2 ^ 4 : \u2115) ^ 502) 10]\n            simp [h\u2081, Nat.pow_mod, Nat.mul_mod, Nat.add_mod, h\u2085\u2082\u2081]\n            <;> norm_num\n            <;> omega\n          exact h\u2085\u2082\u2082\n        have h\u2085\u2083 : (2 ^ 2 : \u2115) % 10 = 4 := by norm_num\n        rw [h\u2085\u2082, h\u2085\u2083]\n        <;> norm_num\n        <;> omega\n      exact h\u2085\n    exact h\u2083\n  exact h\u2080\n"}
{"mode": "clear", "steps_before": 43, "alphabet": 26, "compiles": false, "id": "mathd_algebra_342", "proof_after": "theorem mathd_algebra_342 (a d : \u211d) (h\u2080 : (\u2211 k in Finset.range 5, (a + k * d)) = 70)\n    (h\u2081 : (\u2211 k in Finset.range 10, (a + k * d)) = 210) : a = 42 / 5  :=  by\n  have h\u2082 : 5 * a + 10 * d = 70 := by\n    have h\u2082\u2081 : (\u2211 k in Finset.range 5, (a + k * d : \u211d)) = 5 * a + 10 * d := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n        Finset.sum_range_succ, Finset.sum_range_succ]\n      <;> ring_nf\n      <;> norm_num\n      <;> linarith\n    rw [h\u2082\u2081] at h\u2080\n    linarith\n  have h\u2083 : 10 * a + 45 * d = 210 := by\n    have h\u2083\u2081 : (\u2211 k in Finset.range 10, (a + k * d : \u211d)) = 10 * a + 45 * d := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n        Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n        Finset.sum_range_succ, Finset.sum_range_succ]\n      <;> ring_nf\n      <;> norm_num\n      <;> linarith\n    rw [h\u2083\u2081] at h\u2081\n    linarith\n  have h\u2084 : a + 2 * d = 14 := by\n    have h\u2084\u2081 : 5 * a + 10 * d = 70 := h\u2082\n    have h\u2084\u2082 : a + 2 * d = 14 := by\n      have h\u2084\u2083 : 5 * a + 10 * d = 70 := h\u2082\n      have h\u2084\u2084 : a + 2 * d = 14 := by\n        linarith\n      exact h\u2084\u2084\n    exact h\u2084\u2082\n  have h\u2085 : 2 * a + 9 * d = 42 := by\n    have h\u2085\u2081 : 2 * a + 9 * d = 42 := by\n      linarith\n    exact h\u2085\u2081\n  have h\u2086 : d = 14 / 5 := by\n    have h\u2086\u2081 : d = 14 / 5 := by\n      have h\u2086\u2082 : 5 * d = 14 := by\n        linarith\n      linarith\n    exact h\u2086\u2081\n  have h\u2087 : a = 42 / 5 := by\n    have h\u2087\u2081 : a = 42 / 5 := by\n      have h\u2087\u2082 : a + 2 * d = 14 := h\u2084\n      have h\u2087\u2083 : d = 14 / 5 := h\u2086\n      rw [h\u2087\u2083] at h\u2087\u2082\n      ring_nf at h\u2087\u2082 \u22a2\n      linarith\n    exact h\u2087\u2081\n  apply h\u2087\n", "proof_before": "theorem mathd_algebra_342 (a d : \u211d) (h\u2080 : (\u2211 k in Finset.range 5, (a + k * d)) = 70)\n    (h\u2081 : (\u2211 k in Finset.range 10, (a + k * d)) = 210) : a = 42 / 5  :=  by\n  have h\u2082 : 5 * a + 10 * d = 70 := by\n    have h\u2082\u2081 : (\u2211 k in Finset.range 5, (a + k * d : \u211d)) = 5 * a + 10 * d := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n        Finset.sum_range_succ, Finset.sum_range_succ]\n      <;> ring_nf\n      <;> norm_num\n      <;> linarith\n    rw [h\u2082\u2081] at h\u2080\n    linarith\n  have h\u2083 : 10 * a + 45 * d = 210 := by\n    have h\u2083\u2081 : (\u2211 k in Finset.range 10, (a + k * d : \u211d)) = 10 * a + 45 * d := by\n      norm_num [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n        Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n        Finset.sum_range_succ, Finset.sum_range_succ]\n      <;> ring_nf\n      <;> norm_num\n      <;> linarith\n    rw [h\u2083\u2081] at h\u2081\n    linarith\n  have h\u2084 : a + 2 * d = 14 := by\n    have h\u2084\u2081 : 5 * a + 10 * d = 70 := h\u2082\n    have h\u2084\u2082 : a + 2 * d = 14 := by\n      have h\u2084\u2083 : 5 * a + 10 * d = 70 := h\u2082\n      have h\u2084\u2084 : a + 2 * d = 14 := by\n        linarith\n      exact h\u2084\u2084\n    exact h\u2084\u2082\n  have h\u2085 : 2 * a + 9 * d = 42 := by\n    have h\u2085\u2081 : 2 * a + 9 * d = 42 := by\n      linarith\n    exact h\u2085\u2081\n  have h\u2086 : d = 14 / 5 := by\n    have h\u2086\u2081 : d = 14 / 5 := by\n      have h\u2086\u2082 : 5 * d = 14 := by\n        linarith\n      linarith\n    exact h\u2086\u2081\n  have h\u2087 : a = 42 / 5 := by\n    have h\u2087\u2081 : a = 42 / 5 := by\n      have h\u2087\u2082 : a + 2 * d = 14 := h\u2084\n      have h\u2087\u2083 : d = 14 / 5 := h\u2086\n      rw [h\u2087\u2083] at h\u2087\u2082\n      ring_nf at h\u2087\u2082 \u22a2\n      linarith\n    exact h\u2087\u2081\n  apply h\u2087\n"}
{"mode": "clear", "steps_before": 136, "alphabet": 39, "compiles": false, "id": "amc12b_2021_p3", "proof_after": "theorem amc12b_2021_p3 (x : \u211d) (h\u2080 : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53) : x = 3 / 4  :=  by\n  have h\u2081 : 3 + x \u2260 0 := by\n    by_contra h\n    have h\u2082 : 3 + x = 0 := by linarith\n    have h\u2083 : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 8 / 3 := by\n      have h\u2084 : (3 + x : \u211d) = 0 := by simpa using h\u2082\n      have h\u2085 : (2 : \u211d) + 2 / (3 + x) = 2 := by\n        rw [h\u2084]\n        norm_num\n      have h\u2086 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = 3 / 2 := by\n        rw [h\u2085]\n        <;> norm_num\n      have h\u2087 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 8 / 3 := by\n        rw [h\u2086]\n        <;> norm_num\n      linarith\n    linarith\n  have h\u2082 : 8 + 2 * x \u2260 0 := by\n    by_contra h\n    have h\u2082 : (8 + 2 * x : \u211d) = 0 := by linarith\n    have h\u2083 : x = -4 := by linarith\n    have h\u2084 : (2 : \u211d) + 2 / (3 + x) = 0 := by\n      rw [h\u2083]\n      <;> norm_num\n    have h\u2085 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = 1 := by\n      have h\u2085\u2081 : (2 : \u211d) + 2 / (3 + x) = 0 := by\n        exact h\u2084\n      have h\u2085\u2082 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = 1 := by\n        rw [h\u2085\u2081]\n        <;> norm_num\n      exact h\u2085\u2082\n    have h\u2086 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 3 := by\n      rw [h\u2085]\n      <;> norm_num\n    linarith\n  have h\u2083 : 11 + 3 * x \u2260 0 := by\n    by_contra h\n    have h\u2083 : (11 + 3 * x : \u211d) = 0 := by linarith\n    have h\u2084 : x = -11 / 3 := by linarith\n    have h\u2085 : (2 : \u211d) + 2 / (3 + x) = -1 := by\n      rw [h\u2084]\n      <;> norm_num\n    have h\u2086 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = 0 := by\n      have h\u2086\u2081 : (2 : \u211d) + 2 / (3 + x) = -1 := by\n        exact h\u2085\n      have h\u2086\u2082 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = 0 := by\n        rw [h\u2086\u2081]\n        <;> norm_num\n      exact h\u2086\u2082\n    have h\u2087 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 2 := by\n      rw [h\u2086]\n      <;> norm_num\n    linarith\n  have h\u2084 : (30 + 8 * x : \u211d) / (11 + 3 * x) = 144 / 53 := by\n    have h\u2084\u2081 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = (30 + 8 * x : \u211d) / (11 + 3 * x) := by\n      have h\u2084\u2082 : (2 : \u211d) + 2 / (3 + x) = (8 + 2 * x : \u211d) / (3 + x) := by\n        field_simp [h\u2081]\n        <;> ring_nf\n        <;> field_simp [h\u2081]\n        <;> ring_nf\n        <;> nlinarith\n      have h\u2084\u2083 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = (11 + 3 * x : \u211d) / (8 + 2 * x) := by\n        have h\u2084\u2084 : (2 : \u211d) + 2 / (3 + x) = (8 + 2 * x : \u211d) / (3 + x) := by\n          exact h\u2084\u2082\n        have h\u2084\u2085 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = (11 + 3 * x : \u211d) / (8 + 2 * x) := by\n          have h\u2084\u2086 : (2 : \u211d) + 2 / (3 + x) = (8 + 2 * x : \u211d) / (3 + x) := by\n            exact h\u2084\u2082\n          have h\u2084\u2087 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = (1 : \u211d) + 1 / ((8 + 2 * x : \u211d) / (3 + x)) := by\n            rw [h\u2084\u2086]\n          rw [h\u2084\u2087]\n          have h\u2084\u2088 : (1 : \u211d) + 1 / ((8 + 2 * x : \u211d) / (3 + x)) = (1 : \u211d) + (3 + x : \u211d) / (8 + 2 * x) := by\n            field_simp [h\u2081, h\u2082]\n            <;> ring_nf\n            <;> field_simp [h\u2081, h\u2082]\n            <;> nlinarith\n          rw [h\u2084\u2088]\n          have h\u2084\u2089 : (1 : \u211d) + (3 + x : \u211d) / (8 + 2 * x) = (11 + 3 * x : \u211d) / (8 + 2 * x) := by\n            field_simp [h\u2082]\n            <;> ring_nf\n            <;> field_simp [h\u2082]\n            <;> nlinarith\n          rw [h\u2084\u2089]\n          <;> field_simp [h\u2082]\n          <;> ring_nf\n          <;> field_simp [h\u2082]\n          <;> nlinarith\n        exact h\u2084\u2085\n      have h\u2084\u2085 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = (30 + 8 * x : \u211d) / (11 + 3 * x) := by\n        have h\u2084\u2086 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = (11 + 3 * x : \u211d) / (8 + 2 * x) := by\n          exact h\u2084\u2083\n        have h\u2084\u2087 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = (2 : \u211d) + 1 / ((11 + 3 * x : \u211d) / (8 + 2 * x)) := by\n          rw [h\u2084\u2086]\n        rw [h\u2084\u2087]\n        have h\u2084\u2088 : (2 : \u211d) + 1 / ((11 + 3 * x : \u211d) / (8 + 2 * x)) = (2 : \u211d) + (8 + 2 * x : \u211d) / (11 + 3 * x) := by\n          field_simp [h\u2082, h\u2083]\n          <;> ring_nf\n          <;> field_simp [h\u2082, h\u2083]\n          <;> nlinarith\n        rw [h\u2084\u2088]\n        have h\u2084\u2089 : (2 : \u211d) + (8 + 2 * x : \u211d) / (11 + 3 * x) = (30 + 8 * x : \u211d) / (11 + 3 * x) := by\n          field_simp [h\u2083]\n          <;> ring_nf\n          <;> field_simp [h\u2083]\n          <;> nlinarith\n        rw [h\u2084\u2089]\n        <;> field_simp [h\u2083]\n        <;> ring_nf\n        <;> field_simp [h\u2083]\n        <;> nlinarith\n      exact h\u2084\u2085\n    have h\u2084\u2082 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = (30 + 8 * x : \u211d) / (11 + 3 * x) := by\n      exact h\u2084\u2081\n    rw [h\u2084\u2082] at h\u2080\n    have h\u2084\u2083 : (30 + 8 * x : \u211d) / (11 + 3 * x) = 144 / 53 := by\n      linarith\n    linarith\n  have h\u2085 : x = 3 / 4 := by\n    have h\u2085\u2081 : (30 + 8 * x : \u211d) / (11 + 3 * x) = 144 / 53 := by\n      exact h\u2084\n    have h\u2085\u2082 : (30 + 8 * x : \u211d) * 53 = (144 : \u211d) * (11 + 3 * x) := by\n      have h\u2085\u2083 : (11 + 3 * x : \u211d) \u2260 0 := by simpa using h\u2083\n      field_simp at h\u2085\u2081\n      nlinarith\n    have h\u2085\u2083 : x = 3 / 4 := by\n      ring_nf at h\u2085\u2082\n      nlinarith\n    exact h\u2085\u2083\n  rw [h\u2085]\n  <;> norm_num\n  <;> linarith\n", "proof_before": "theorem amc12b_2021_p3 (x : \u211d) (h\u2080 : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53) : x = 3 / 4  :=  by\n  have h\u2081 : 3 + x \u2260 0 := by\n    by_contra h\n    have h\u2082 : 3 + x = 0 := by linarith\n    have h\u2083 : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 8 / 3 := by\n      have h\u2084 : (3 + x : \u211d) = 0 := by simpa using h\u2082\n      have h\u2085 : (2 : \u211d) + 2 / (3 + x) = 2 := by\n        rw [h\u2084]\n        norm_num\n      have h\u2086 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = 3 / 2 := by\n        rw [h\u2085]\n        <;> norm_num\n      have h\u2087 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 8 / 3 := by\n        rw [h\u2086]\n        <;> norm_num\n      linarith\n    linarith\n  have h\u2082 : 8 + 2 * x \u2260 0 := by\n    by_contra h\n    have h\u2082 : (8 + 2 * x : \u211d) = 0 := by linarith\n    have h\u2083 : x = -4 := by linarith\n    have h\u2084 : (2 : \u211d) + 2 / (3 + x) = 0 := by\n      rw [h\u2083]\n      <;> norm_num\n    have h\u2085 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = 1 := by\n      have h\u2085\u2081 : (2 : \u211d) + 2 / (3 + x) = 0 := by\n        exact h\u2084\n      have h\u2085\u2082 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = 1 := by\n        rw [h\u2085\u2081]\n        <;> norm_num\n      exact h\u2085\u2082\n    have h\u2086 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 3 := by\n      rw [h\u2085]\n      <;> norm_num\n    linarith\n  have h\u2083 : 11 + 3 * x \u2260 0 := by\n    by_contra h\n    have h\u2083 : (11 + 3 * x : \u211d) = 0 := by linarith\n    have h\u2084 : x = -11 / 3 := by linarith\n    have h\u2085 : (2 : \u211d) + 2 / (3 + x) = -1 := by\n      rw [h\u2084]\n      <;> norm_num\n    have h\u2086 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = 0 := by\n      have h\u2086\u2081 : (2 : \u211d) + 2 / (3 + x) = -1 := by\n        exact h\u2085\n      have h\u2086\u2082 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = 0 := by\n        rw [h\u2086\u2081]\n        <;> norm_num\n      exact h\u2086\u2082\n    have h\u2087 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 2 := by\n      rw [h\u2086]\n      <;> norm_num\n    linarith\n  have h\u2084 : (30 + 8 * x : \u211d) / (11 + 3 * x) = 144 / 53 := by\n    have h\u2084\u2081 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = (30 + 8 * x : \u211d) / (11 + 3 * x) := by\n      have h\u2084\u2082 : (2 : \u211d) + 2 / (3 + x) = (8 + 2 * x : \u211d) / (3 + x) := by\n        field_simp [h\u2081]\n        <;> ring_nf\n        <;> field_simp [h\u2081]\n        <;> ring_nf\n        <;> nlinarith\n      have h\u2084\u2083 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = (11 + 3 * x : \u211d) / (8 + 2 * x) := by\n        have h\u2084\u2084 : (2 : \u211d) + 2 / (3 + x) = (8 + 2 * x : \u211d) / (3 + x) := by\n          exact h\u2084\u2082\n        have h\u2084\u2085 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = (11 + 3 * x : \u211d) / (8 + 2 * x) := by\n          have h\u2084\u2086 : (2 : \u211d) + 2 / (3 + x) = (8 + 2 * x : \u211d) / (3 + x) := by\n            exact h\u2084\u2082\n          have h\u2084\u2087 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = (1 : \u211d) + 1 / ((8 + 2 * x : \u211d) / (3 + x)) := by\n            rw [h\u2084\u2086]\n          rw [h\u2084\u2087]\n          have h\u2084\u2088 : (1 : \u211d) + 1 / ((8 + 2 * x : \u211d) / (3 + x)) = (1 : \u211d) + (3 + x : \u211d) / (8 + 2 * x) := by\n            field_simp [h\u2081, h\u2082]\n            <;> ring_nf\n            <;> field_simp [h\u2081, h\u2082]\n            <;> nlinarith\n          rw [h\u2084\u2088]\n          have h\u2084\u2089 : (1 : \u211d) + (3 + x : \u211d) / (8 + 2 * x) = (11 + 3 * x : \u211d) / (8 + 2 * x) := by\n            field_simp [h\u2082]\n            <;> ring_nf\n            <;> field_simp [h\u2082]\n            <;> nlinarith\n          rw [h\u2084\u2089]\n          <;> field_simp [h\u2082]\n          <;> ring_nf\n          <;> field_simp [h\u2082]\n          <;> nlinarith\n        exact h\u2084\u2085\n      have h\u2084\u2085 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = (30 + 8 * x : \u211d) / (11 + 3 * x) := by\n        have h\u2084\u2086 : (1 : \u211d) + 1 / (2 + 2 / (3 + x)) = (11 + 3 * x : \u211d) / (8 + 2 * x) := by\n          exact h\u2084\u2083\n        have h\u2084\u2087 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = (2 : \u211d) + 1 / ((11 + 3 * x : \u211d) / (8 + 2 * x)) := by\n          rw [h\u2084\u2086]\n        rw [h\u2084\u2087]\n        have h\u2084\u2088 : (2 : \u211d) + 1 / ((11 + 3 * x : \u211d) / (8 + 2 * x)) = (2 : \u211d) + (8 + 2 * x : \u211d) / (11 + 3 * x) := by\n          field_simp [h\u2082, h\u2083]\n          <;> ring_nf\n          <;> field_simp [h\u2082, h\u2083]\n          <;> nlinarith\n        rw [h\u2084\u2088]\n        have h\u2084\u2089 : (2 : \u211d) + (8 + 2 * x : \u211d) / (11 + 3 * x) = (30 + 8 * x : \u211d) / (11 + 3 * x) := by\n          field_simp [h\u2083]\n          <;> ring_nf\n          <;> field_simp [h\u2083]\n          <;> nlinarith\n        rw [h\u2084\u2089]\n        <;> field_simp [h\u2083]\n        <;> ring_nf\n        <;> field_simp [h\u2083]\n        <;> nlinarith\n      exact h\u2084\u2085\n    have h\u2084\u2082 : (2 : \u211d) + 1 / (1 + 1 / (2 + 2 / (3 + x))) = (30 + 8 * x : \u211d) / (11 + 3 * x) := by\n      exact h\u2084\u2081\n    rw [h\u2084\u2082] at h\u2080\n    have h\u2084\u2083 : (30 + 8 * x : \u211d) / (11 + 3 * x) = 144 / 53 := by\n      linarith\n    linarith\n  have h\u2085 : x = 3 / 4 := by\n    have h\u2085\u2081 : (30 + 8 * x : \u211d) / (11 + 3 * x) = 144 / 53 := by\n      exact h\u2084\n    have h\u2085\u2082 : (30 + 8 * x : \u211d) * 53 = (144 : \u211d) * (11 + 3 * x) := by\n      have h\u2085\u2083 : (11 + 3 * x : \u211d) \u2260 0 := by simpa using h\u2083\n      field_simp at h\u2085\u2081\n      nlinarith\n    have h\u2085\u2083 : x = 3 / 4 := by\n      ring_nf at h\u2085\u2082\n      nlinarith\n    exact h\u2085\u2083\n  rw [h\u2085]\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 119, "alphabet": 53, "compiles": false, "id": "mathd_algebra_320", "proof_after": "theorem mathd_algebra_320 (x : \u211d) (a b c : \u2115) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c \u2227 0 \u2264 x)\n    (h\u2081 : 2 * x ^ 2 = 4 * x + 9) (h\u2082 : x = (a + Real.sqrt b) / c) (h\u2083 : c = 2) : a + b + c = 26  :=  by\n  have h\u2084 : x = (a + Real.sqrt b) / 2 := by\n    rw [h\u2082, h\u2083]\n    <;> ring_nf\n    <;> field_simp\n    <;> ring_nf\n  have h\u2085 : 2 * x ^ 2 = 4 * x + 9 := by\n    exact h\u2081\n  have h\u2086 : (a : \u211d) > 0 := by\n    have h\u2086\u2081 : 0 < a := h\u2080.1\n    exact mod_cast h\u2086\u2081\n  have h\u2087 : (b : \u211d) > 0 := by\n    have h\u2087\u2081 : 0 < b := h\u2080.2.1\n    exact mod_cast h\u2087\u2081\n  have h\u2088 : (2 : \u211d) > 0 := by norm_num\n  have h\u2089 : (a : \u211d) ^ 2 + (b : \u211d) + 2 * (a : \u211d) * Real.sqrt b = 4 * (a : \u211d) + 4 * Real.sqrt b + 18 := by\n    have h\u2089\u2081 : x = (a + Real.sqrt b) / 2 := h\u2084\n    have h\u2089\u2082 : 2 * x ^ 2 = 4 * x + 9 := h\u2085\n    rw [h\u2089\u2081] at h\u2089\u2082\n    ring_nf at h\u2089\u2082 \u22a2\n    nlinarith [Real.sqrt_nonneg b, Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith),\n      mul_pos h\u2086 h\u2087, mul_pos (mul_pos h\u2086 h\u2088) h\u2087]\n  have h\u2081\u2080 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) ^ 2 := by\n    have h\u2081\u2080\u2081 : (2 * (a : \u211d) - 4) * Real.sqrt b = 4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d) := by\n      have h\u2081\u2080\u2082 : (a : \u211d) ^ 2 + (b : \u211d) + 2 * (a : \u211d) * Real.sqrt b = 4 * (a : \u211d) + 4 * Real.sqrt b + 18 := h\u2089\n      have h\u2081\u2080\u2083 : 2 * (a : \u211d) * Real.sqrt b - 4 * Real.sqrt b = 4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d) := by\n        linarith\n      have h\u2081\u2080\u2084 : (2 * (a : \u211d) - 4) * Real.sqrt b = 4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d) := by\n        linarith\n      exact h\u2081\u2080\u2084\n    have h\u2081\u2080\u2085 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) ^ 2 := by\n      have h\u2081\u2080\u2086 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = ((2 * (a : \u211d) - 4) * Real.sqrt b) ^ 2 := by\n        have h\u2081\u2080\u2087 : Real.sqrt b \u2265 0 := Real.sqrt_nonneg _\n        have h\u2081\u2080\u2088 : (Real.sqrt b) ^ 2 = (b : \u211d) := by\n          rw [Real.sq_sqrt] <;> positivity\n        calc\n          (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = (2 * (a : \u211d) - 4) ^ 2 * ((Real.sqrt b) ^ 2) := by rw [h\u2081\u2080\u2088]\n          _ = ((2 * (a : \u211d) - 4) * Real.sqrt b) ^ 2 := by ring\n          _ = ((2 * (a : \u211d) - 4) * Real.sqrt b) ^ 2 := by ring\n      rw [h\u2081\u2080\u2086]\n      rw [h\u2081\u2080\u2081]\n      <;> ring_nf\n      <;> nlinarith [Real.sqrt_nonneg b, Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]\n    exact h\u2081\u2080\u2085\n  have h\u2081\u2081 : a = 2 := by\n    have h\u2081\u2081\u2081 : a = 2 := by\n      by_contra h\n      have h\u2081\u2082 : a \u2260 2 := h\n      have h\u2081\u2083 : a \u2265 1 := by linarith [h\u2080.1]\n      have h\u2081\u2084 : a \u2264 7 := by\n        by_contra h\u2081\u2085\n        have h\u2081\u2085\u2081 : a \u2265 8 := by omega\n        have h\u2081\u2085\u2082 : (a : \u211d) \u2265 8 := by norm_cast\n        have h\u2081\u2085\u2083 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) ^ 2 := by\n          exact h\u2081\u2080\n        have h\u2081\u2085\u2084 : (a : \u211d) \u2265 8 := by norm_cast <;> linarith\n        have h\u2081\u2085\u2085 : (b : \u211d) > 0 := h\u2087\n        have h\u2081\u2085\u2086 : (b : \u211d) > 0 := h\u2087\n        have h\u2081\u2085\u2087 : (a : \u2115) \u2265 8 := by norm_cast at h\u2081\u2085\u2081 \u22a2 <;> linarith\n        have h\u2081\u2085\u2088 : (a : \u211d) ^ 2 \u2265 64 := by nlinarith\n        have h\u2081\u2085\u2089 : (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) \u2264 0 := by\n          nlinarith [Real.sqrt_nonneg (b : \u211d), Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]\n        have h\u2081\u2085\u2081\u2080 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) \u2265 0 := by positivity\n        have h\u2081\u2085\u2081\u2081 : (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) ^ 2 \u2265 0 := by positivity\n        have h\u2081\u2085\u2081\u2082 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) ^ 2 := by\n          exact h\u2081\u2080\n        nlinarith [Real.sqrt_nonneg (b : \u211d), Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]\n      interval_cases a <;> norm_num at h\u2081\u2080 \u22a2 <;>\n        (try norm_num at h\u2081\u2080 \u22a2) <;>\n        (try {\n          have h\u2081\u2085\u2081\u2083 : (b : \u2115) > 0 := by linarith [h\u2080.2.1]\n          have h\u2081\u2085\u2081\u2084 : (b : \u211d) > 0 := by exact_mod_cast h\u2080.2.1\n          have h\u2081\u2085\u2081\u2085 : (b : \u2115) \u2264 100 := by\n            by_contra h\u2081\u2085\u2081\u2086\n            have h\u2081\u2085\u2081\u2087 : (b : \u2115) > 100 := by linarith\n            have h\u2081\u2085\u2081\u2088 : (b : \u211d) > 100 := by exact_mod_cast h\u2081\u2085\u2081\u2087\n            nlinarith [Real.sqrt_nonneg (b : \u211d), Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]\n          interval_cases b <;> norm_num at h\u2081\u2080 \u22a2 <;>\n            (try nlinarith [Real.sqrt_nonneg (b : \u211d), Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]) <;>\n            (try omega) <;>\n            (try linarith)\n        }) <;>\n        (try omega)\n    exact h\u2081\u2081\u2081\n  have h\u2081\u2082 : b = 22 := by\n    have h\u2081\u2082\u2081 : a = 2 := h\u2081\u2081\n    have h\u2081\u2082\u2082 : (a : \u211d) = 2 := by norm_cast\n    rw [h\u2081\u2082\u2082] at h\u2089 h\u2081\u2080\n    norm_num at h\u2089 h\u2081\u2080\n    have h\u2081\u2082\u2083 : (b : \u211d) = 22 := by\n      nlinarith [Real.sqrt_nonneg (b : \u211d), Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]\n    have h\u2081\u2082\u2084 : (b : \u2115) = 22 := by\n      norm_cast at h\u2081\u2082\u2083 \u22a2\n      <;>\n      (try omega) <;>\n      (try nlinarith)\n    exact h\u2081\u2082\u2084\n  have h\u2081\u2083 : a + b + c = 26 := by\n    have h\u2081\u2084 : a = 2 := h\u2081\u2081\n    have h\u2081\u2085 : b = 22 := h\u2081\u2082\n    have h\u2081\u2086 : c = 2 := h\u2083\n    rw [h\u2081\u2084, h\u2081\u2085, h\u2081\u2086]\n    <;> norm_num\n    <;>\n    (try omega) <;>\n    (try nlinarith)\n  exact h\u2081\u2083\n", "proof_before": "theorem mathd_algebra_320 (x : \u211d) (a b c : \u2115) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c \u2227 0 \u2264 x)\n    (h\u2081 : 2 * x ^ 2 = 4 * x + 9) (h\u2082 : x = (a + Real.sqrt b) / c) (h\u2083 : c = 2) : a + b + c = 26  :=  by\n  have h\u2084 : x = (a + Real.sqrt b) / 2 := by\n    rw [h\u2082, h\u2083]\n    <;> ring_nf\n    <;> field_simp\n    <;> ring_nf\n  have h\u2085 : 2 * x ^ 2 = 4 * x + 9 := by\n    exact h\u2081\n  have h\u2086 : (a : \u211d) > 0 := by\n    have h\u2086\u2081 : 0 < a := h\u2080.1\n    exact mod_cast h\u2086\u2081\n  have h\u2087 : (b : \u211d) > 0 := by\n    have h\u2087\u2081 : 0 < b := h\u2080.2.1\n    exact mod_cast h\u2087\u2081\n  have h\u2088 : (2 : \u211d) > 0 := by norm_num\n  have h\u2089 : (a : \u211d) ^ 2 + (b : \u211d) + 2 * (a : \u211d) * Real.sqrt b = 4 * (a : \u211d) + 4 * Real.sqrt b + 18 := by\n    have h\u2089\u2081 : x = (a + Real.sqrt b) / 2 := h\u2084\n    have h\u2089\u2082 : 2 * x ^ 2 = 4 * x + 9 := h\u2085\n    rw [h\u2089\u2081] at h\u2089\u2082\n    ring_nf at h\u2089\u2082 \u22a2\n    nlinarith [Real.sqrt_nonneg b, Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith),\n      mul_pos h\u2086 h\u2087, mul_pos (mul_pos h\u2086 h\u2088) h\u2087]\n  have h\u2081\u2080 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) ^ 2 := by\n    have h\u2081\u2080\u2081 : (2 * (a : \u211d) - 4) * Real.sqrt b = 4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d) := by\n      have h\u2081\u2080\u2082 : (a : \u211d) ^ 2 + (b : \u211d) + 2 * (a : \u211d) * Real.sqrt b = 4 * (a : \u211d) + 4 * Real.sqrt b + 18 := h\u2089\n      have h\u2081\u2080\u2083 : 2 * (a : \u211d) * Real.sqrt b - 4 * Real.sqrt b = 4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d) := by\n        linarith\n      have h\u2081\u2080\u2084 : (2 * (a : \u211d) - 4) * Real.sqrt b = 4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d) := by\n        linarith\n      exact h\u2081\u2080\u2084\n    have h\u2081\u2080\u2085 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) ^ 2 := by\n      have h\u2081\u2080\u2086 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = ((2 * (a : \u211d) - 4) * Real.sqrt b) ^ 2 := by\n        have h\u2081\u2080\u2087 : Real.sqrt b \u2265 0 := Real.sqrt_nonneg _\n        have h\u2081\u2080\u2088 : (Real.sqrt b) ^ 2 = (b : \u211d) := by\n          rw [Real.sq_sqrt] <;> positivity\n        calc\n          (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = (2 * (a : \u211d) - 4) ^ 2 * ((Real.sqrt b) ^ 2) := by rw [h\u2081\u2080\u2088]\n          _ = ((2 * (a : \u211d) - 4) * Real.sqrt b) ^ 2 := by ring\n          _ = ((2 * (a : \u211d) - 4) * Real.sqrt b) ^ 2 := by ring\n      rw [h\u2081\u2080\u2086]\n      rw [h\u2081\u2080\u2081]\n      <;> ring_nf\n      <;> nlinarith [Real.sqrt_nonneg b, Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]\n    exact h\u2081\u2080\u2085\n  have h\u2081\u2081 : a = 2 := by\n    have h\u2081\u2081\u2081 : a = 2 := by\n      by_contra h\n      have h\u2081\u2082 : a \u2260 2 := h\n      have h\u2081\u2083 : a \u2265 1 := by linarith [h\u2080.1]\n      have h\u2081\u2084 : a \u2264 7 := by\n        by_contra h\u2081\u2085\n        have h\u2081\u2085\u2081 : a \u2265 8 := by omega\n        have h\u2081\u2085\u2082 : (a : \u211d) \u2265 8 := by norm_cast\n        have h\u2081\u2085\u2083 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) ^ 2 := by\n          exact h\u2081\u2080\n        have h\u2081\u2085\u2084 : (a : \u211d) \u2265 8 := by norm_cast <;> linarith\n        have h\u2081\u2085\u2085 : (b : \u211d) > 0 := h\u2087\n        have h\u2081\u2085\u2086 : (b : \u211d) > 0 := h\u2087\n        have h\u2081\u2085\u2087 : (a : \u2115) \u2265 8 := by norm_cast at h\u2081\u2085\u2081 \u22a2 <;> linarith\n        have h\u2081\u2085\u2088 : (a : \u211d) ^ 2 \u2265 64 := by nlinarith\n        have h\u2081\u2085\u2089 : (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) \u2264 0 := by\n          nlinarith [Real.sqrt_nonneg (b : \u211d), Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]\n        have h\u2081\u2085\u2081\u2080 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) \u2265 0 := by positivity\n        have h\u2081\u2085\u2081\u2081 : (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) ^ 2 \u2265 0 := by positivity\n        have h\u2081\u2085\u2081\u2082 : (2 * (a : \u211d) - 4) ^ 2 * (b : \u211d) = (4 * (a : \u211d) + 18 - (a : \u211d) ^ 2 - (b : \u211d)) ^ 2 := by\n          exact h\u2081\u2080\n        nlinarith [Real.sqrt_nonneg (b : \u211d), Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]\n      interval_cases a <;> norm_num at h\u2081\u2080 \u22a2 <;>\n        (try norm_num at h\u2081\u2080 \u22a2) <;>\n        (try {\n          have h\u2081\u2085\u2081\u2083 : (b : \u2115) > 0 := by linarith [h\u2080.2.1]\n          have h\u2081\u2085\u2081\u2084 : (b : \u211d) > 0 := by exact_mod_cast h\u2080.2.1\n          have h\u2081\u2085\u2081\u2085 : (b : \u2115) \u2264 100 := by\n            by_contra h\u2081\u2085\u2081\u2086\n            have h\u2081\u2085\u2081\u2087 : (b : \u2115) > 100 := by linarith\n            have h\u2081\u2085\u2081\u2088 : (b : \u211d) > 100 := by exact_mod_cast h\u2081\u2085\u2081\u2087\n            nlinarith [Real.sqrt_nonneg (b : \u211d), Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]\n          interval_cases b <;> norm_num at h\u2081\u2080 \u22a2 <;>\n            (try nlinarith [Real.sqrt_nonneg (b : \u211d), Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]) <;>\n            (try omega) <;>\n            (try linarith)\n        }) <;>\n        (try omega)\n    exact h\u2081\u2081\u2081\n  have h\u2081\u2082 : b = 22 := by\n    have h\u2081\u2082\u2081 : a = 2 := h\u2081\u2081\n    have h\u2081\u2082\u2082 : (a : \u211d) = 2 := by norm_cast\n    rw [h\u2081\u2082\u2082] at h\u2089 h\u2081\u2080\n    norm_num at h\u2089 h\u2081\u2080\n    have h\u2081\u2082\u2083 : (b : \u211d) = 22 := by\n      nlinarith [Real.sqrt_nonneg (b : \u211d), Real.sq_sqrt (show 0 \u2264 (b : \u211d) by linarith)]\n    have h\u2081\u2082\u2084 : (b : \u2115) = 22 := by\n      norm_cast at h\u2081\u2082\u2083 \u22a2\n      <;>\n      (try omega) <;>\n      (try nlinarith)\n    exact h\u2081\u2082\u2084\n  have h\u2081\u2083 : a + b + c = 26 := by\n    have h\u2081\u2084 : a = 2 := h\u2081\u2081\n    have h\u2081\u2085 : b = 22 := h\u2081\u2082\n    have h\u2081\u2086 : c = 2 := h\u2083\n    rw [h\u2081\u2084, h\u2081\u2085, h\u2081\u2086]\n    <;> norm_num\n    <;>\n    (try omega) <;>\n    (try nlinarith)\n  exact h\u2081\u2083\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 23, "compiles": false, "id": "induction_prod1p1onk3le3m1onn", "proof_after": "theorem induction_prod1p1onk3le3m1onn (n : \u2115) (h\u2080 : 0 < n) :\n    (\u220f k in Finset.Icc 1 n, (1 + (1 : \u211d) / k ^ 3)) \u2264 (3 : \u211d) - 1 / \u2191n  :=  by\n  have h_main : \u2200 (m : \u2115), 0 < m \u2192 (\u220f k in Finset.Icc 1 m, (1 + (1 : \u211d) / k ^ 3)) \u2264 (3 : \u211d) - 1 / \u2191m := by\n    intro m hm\n    have h\u2081 : \u2200 (n : \u2115), 0 < n \u2192 (\u220f k in Finset.Icc 1 n, (1 + (1 : \u211d) / k ^ 3)) \u2264 (3 : \u211d) - 1 / \u2191n := by\n      intro n hn\n      induction' hn with n hn IH\n      \u00b7 norm_num [Finset.prod_Icc_succ_top]\n      \u00b7 rw [Finset.prod_Icc_succ_top (by omega : 1 \u2264 n.succ)]\n        refine' le_trans (mul_le_mul_of_nonneg_right IH (by positivity)) _\n        cases n with\n        | zero =>\n          norm_num [Finset.prod_Icc_succ_top] at *\n        | succ n =>\n          field_simp [Nat.cast_add_one_ne_zero]\n          ring_nf\n          norm_num\n          apply le_of_sub_nonneg\n          field_simp\n          ring_nf\n          positivity\n    exact h\u2081 m hm\n  have h_final : (\u220f k in Finset.Icc 1 n, (1 + (1 : \u211d) / k ^ 3)) \u2264 (3 : \u211d) - 1 / \u2191n := by\n    apply h_main n h\u2080\n  exact h_final\n", "proof_before": "theorem induction_prod1p1onk3le3m1onn (n : \u2115) (h\u2080 : 0 < n) :\n    (\u220f k in Finset.Icc 1 n, (1 + (1 : \u211d) / k ^ 3)) \u2264 (3 : \u211d) - 1 / \u2191n  :=  by\n  have h_main : \u2200 (m : \u2115), 0 < m \u2192 (\u220f k in Finset.Icc 1 m, (1 + (1 : \u211d) / k ^ 3)) \u2264 (3 : \u211d) - 1 / \u2191m := by\n    intro m hm\n    have h\u2081 : \u2200 (n : \u2115), 0 < n \u2192 (\u220f k in Finset.Icc 1 n, (1 + (1 : \u211d) / k ^ 3)) \u2264 (3 : \u211d) - 1 / \u2191n := by\n      intro n hn\n      induction' hn with n hn IH\n      \u00b7 norm_num [Finset.prod_Icc_succ_top]\n      \u00b7 rw [Finset.prod_Icc_succ_top (by omega : 1 \u2264 n.succ)]\n        refine' le_trans (mul_le_mul_of_nonneg_right IH (by positivity)) _\n        cases n with\n        | zero =>\n          norm_num [Finset.prod_Icc_succ_top] at *\n        | succ n =>\n          field_simp [Nat.cast_add_one_ne_zero]\n          ring_nf\n          norm_num\n          apply le_of_sub_nonneg\n          field_simp\n          ring_nf\n          positivity\n    exact h\u2081 m hm\n  have h_final : (\u220f k in Finset.Icc 1 n, (1 + (1 : \u211d) / k ^ 3)) \u2264 (3 : \u211d) - 1 / \u2191n := by\n    apply h_main n h\u2080\n  exact h_final\n"}
{"mode": "clear", "steps_before": 17, "alphabet": 19, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 593, "chars_after": 77, "seconds": 37.2, "id": "algebra_sqineq_unitcircatbpamblt1", "proof_after": "theorem algebra_sqineq_unitcircatbpamblt1 (a b : \u211d) (h\u2080 : a ^ 2 + b ^ 2 = 1) :\n    a * b + (a - b) \u2264 1  :=  by\n  nlinarith [sq_nonneg a, sq_nonneg b, sq_nonneg (a - 1), sq_nonneg (b - 1)]\n", "proof_before": "theorem algebra_sqineq_unitcircatbpamblt1 (a b : \u211d) (h\u2080 : a ^ 2 + b ^ 2 = 1) :\n    a * b + (a - b) \u2264 1  :=  by\n  have h\u2081 : a ^ 2 \u2264 1 := by\n    nlinarith [sq_nonneg b]\n  have h\u2082 : b ^ 2 \u2264 1 := by\n    nlinarith [sq_nonneg a]\n  have h\u2083 : 1 - a \u2265 0 := by\n    nlinarith [sq_nonneg a, sq_nonneg b, sq_nonneg (a - 1), sq_nonneg (b - 1)]\n  have h\u2084 : 1 + b \u2265 0 := by\n    nlinarith [sq_nonneg a, sq_nonneg b, sq_nonneg (a - 1), sq_nonneg (b + 1)]\n  have h\u2085 : (1 - a) * (1 + b) \u2265 0 := by\n    nlinarith [h\u2083, h\u2084]\n  have h\u2086 : a * b + (a - b) \u2264 1 := by\n    have h\u2087 : (1 - a) * (1 + b) = 1 - a + b - a * b := by\n      ring\n    have h\u2088 : 1 - a + b - a * b \u2265 0 := by\n      nlinarith [h\u2085, h\u2087]\n    nlinarith [h\u2088]\n  exact h\u2086\n"}
{"mode": "clear", "steps_before": 29, "alphabet": 18, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 535, "chars_after": 28, "seconds": 93.3, "id": "mathd_algebra_275", "proof_after": "theorem mathd_algebra_275 (x : \u211d) (h : ((11 : \u211d) ^ (1 / 4)) ^ (3 * x - 3) = 1 / 5) :\n    ((11 : \u211d) ^ (1 / 4)) ^ (6 * x + 2) = 121 / 25  :=  by\n  norm_num [Nat.Prime] at *\n", "proof_before": "theorem mathd_algebra_275 (x : \u211d) (h : ((11 : \u211d) ^ (1 / 4)) ^ (3 * x - 3) = 1 / 5) :\n    ((11 : \u211d) ^ (1 / 4)) ^ (6 * x + 2) = 121 / 25  :=  by\n  have h\u2081 : (11 : \u211d) ^ (1 / 4 : \u2115) = 1 := by\n    norm_num\n    <;>\n    simp [Nat.div_eq_of_lt]\n    <;>\n    norm_num\n    <;>\n    ring_nf\n    <;>\n    norm_num\n  have h\u2082 : ((11 : \u211d) ^ (1 / 4 : \u2115)) ^ (3 * x - 3) = 1 := by\n    rw [h\u2081]\n    <;>\n    simp [Real.one_rpow]\n    <;>\n    ring_nf\n    <;>\n    norm_num\n    <;>\n    linarith\n  have h\u2083 : False := by\n    have h\u2084 : (1 : \u211d) = 1 / 5 := by\n      linarith\n    norm_num at h\u2084\n    <;> linarith\n  have h\u2084 : ((11 : \u211d) ^ (1 / 4)) ^ (6 * x + 2) = 121 / 25 := by\n    exfalso\n    exact h\u2083\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 82, "alphabet": 45, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 43, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 3, "fixpoint": false, "chars_before": 2554, "chars_after": 1310, "seconds": 1089.4, "id": "mathd_algebra_288", "proof_after": "theorem mathd_algebra_288 (x y : \u211d) (n : NNReal) (h\u2080 : x < 0 \u2227 y < 0) (h\u2081 : abs y = 6)\n    (h\u2082 : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) = 15)\n    (h\u2083 : Real.sqrt (x ^ 2 + y ^ 2) = Real.sqrt n) : n = 52  :=  by\n  have h\u2084 : y = -6 := by\n    have h\u2084\u2081 : y < 0 := h\u2080.2\n    have h\u2084\u2082 : abs y = 6 := h\u2081\n    have h\u2084\u2083 : y = -6 := by\n      rw [abs_of_neg h\u2084\u2081] at h\u2084\u2082\n      linarith\n    exact h\u2084\u2083\n  have h\u2085 : (x - 8) ^ 2 + (y - 3) ^ 2 = 225 := by\n    have h\u2085\u2081 : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) = 15 := h\u2082\n    have h\u2085\u2082 : (x - 8) ^ 2 + (y - 3) ^ 2 = 225 := by\n      have h\u2085\u2083 : 0 \u2264 (x - 8) ^ 2 + (y - 3) ^ 2 := by nlinarith [sq_nonneg (x - 8), sq_nonneg (y - 3)]\n      have h\u2085\u2084 : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) = 15 := h\u2082\n      have h\u2085\u2085 : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) ^ 2 = (15 : \u211d) ^ 2 := by rw [h\u2085\u2084]\n      have h\u2085\u2086 : (x - 8) ^ 2 + (y - 3) ^ 2 = 225 := by\n        nlinarith [Real.sq_sqrt (show 0 \u2264 (x - 8) ^ 2 + (y - 3) ^ 2 by nlinarith [sq_nonneg (x - 8), sq_nonneg (y - 3)])]\n      exact h\u2085\u2086\n    exact h\u2085\u2082\n  have h\u2086 : x = -4 := by\n    nlinarith\n  have h\u2087 : (x : \u211d) ^ 2 + (y : \u211d) ^ 2 = 52 := by\n    have h\u2087\u2081 : y = -6 := h\u2084\n    have h\u2087\u2082 : x = -4 := h\u2086\n    rw [h\u2087\u2082, h\u2087\u2081]\n    norm_num\n    <;>\n    linarith\n  have h\u2088 : (n : \u211d) = 52 := by\n    (try simp_all)\n  have h\u2089 : n = 52 := by\n    have h\u2089\u2081 : (n : \u211d) = 52 := h\u2088\n    have h\u2089\u2082 : n = 52 := by\n      apply_fun (fun x => (x : \u211d)) at h\u2089\u2081\n      <;> norm_cast at h\u2089\u2081 \u22a2 <;>\n      (try simp_all) <;>\n      (try norm_num) <;>\n      (try linarith)\n    exact h\u2089\u2082\n  exact h\u2089\n", "proof_before": "theorem mathd_algebra_288 (x y : \u211d) (n : NNReal) (h\u2080 : x < 0 \u2227 y < 0) (h\u2081 : abs y = 6)\n    (h\u2082 : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) = 15)\n    (h\u2083 : Real.sqrt (x ^ 2 + y ^ 2) = Real.sqrt n) : n = 52  :=  by\n  have h\u2084 : y = -6 := by\n    have h\u2084\u2081 : y < 0 := h\u2080.2\n    have h\u2084\u2082 : abs y = 6 := h\u2081\n    have h\u2084\u2083 : y = -6 := by\n      rw [abs_of_neg h\u2084\u2081] at h\u2084\u2082\n      linarith\n    exact h\u2084\u2083\n  have h\u2085 : (x - 8) ^ 2 + (y - 3) ^ 2 = 225 := by\n    have h\u2085\u2081 : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) = 15 := h\u2082\n    have h\u2085\u2082 : (x - 8) ^ 2 + (y - 3) ^ 2 = 225 := by\n      have h\u2085\u2083 : 0 \u2264 (x - 8) ^ 2 + (y - 3) ^ 2 := by nlinarith [sq_nonneg (x - 8), sq_nonneg (y - 3)]\n      have h\u2085\u2084 : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) = 15 := h\u2082\n      have h\u2085\u2085 : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) ^ 2 = (15 : \u211d) ^ 2 := by rw [h\u2085\u2084]\n      have h\u2085\u2086 : (x - 8) ^ 2 + (y - 3) ^ 2 = 225 := by\n        nlinarith [Real.sq_sqrt (show 0 \u2264 (x - 8) ^ 2 + (y - 3) ^ 2 by nlinarith [sq_nonneg (x - 8), sq_nonneg (y - 3)])]\n      exact h\u2085\u2086\n    exact h\u2085\u2082\n  have h\u2086 : x = -4 := by\n    have h\u2086\u2081 : y = -6 := h\u2084\n    rw [h\u2086\u2081] at h\u2085\n    have h\u2086\u2082 : (x - 8) ^ 2 + (-6 - 3 : \u211d) ^ 2 = 225 := by ring_nf at h\u2085 \u22a2; linarith\n    have h\u2086\u2083 : (x - 8) ^ 2 = 144 := by\n      nlinarith\n    have h\u2086\u2084 : x - 8 = 12 \u2228 x - 8 = -12 := by\n      apply or_iff_not_imp_left.mpr\n      intro h\u2086\u2085\n      apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2086\u2085)\n      nlinarith\n    cases h\u2086\u2084 with\n    | inl h\u2086\u2084 =>\n      have h\u2086\u2085 : x = 20 := by linarith\n      have h\u2086\u2086 : x < 0 := h\u2080.1\n      linarith\n    | inr h\u2086\u2084 =>\n      have h\u2086\u2085 : x = -4 := by linarith\n      exact h\u2086\u2085\n  have h\u2087 : (x : \u211d) ^ 2 + (y : \u211d) ^ 2 = 52 := by\n    have h\u2087\u2081 : y = -6 := h\u2084\n    have h\u2087\u2082 : x = -4 := h\u2086\n    rw [h\u2087\u2082, h\u2087\u2081]\n    norm_num\n    <;>\n    linarith\n  have h\u2088 : (n : \u211d) = 52 := by\n    have h\u2088\u2081 : Real.sqrt (x ^ 2 + y ^ 2) = Real.sqrt n := h\u2083\n    have h\u2088\u2082 : 0 \u2264 (x ^ 2 + y ^ 2 : \u211d) := by nlinarith [sq_nonneg x, sq_nonneg y]\n    have h\u2088\u2083 : 0 \u2264 (n : \u211d) := by exact_mod_cast NNReal.coe_nonneg n\n    have h\u2088\u2084 : Real.sqrt (x ^ 2 + y ^ 2) = Real.sqrt n := h\u2083\n    have h\u2088\u2085 : (Real.sqrt (x ^ 2 + y ^ 2)) ^ 2 = (Real.sqrt n) ^ 2 := by rw [h\u2088\u2084]\n    have h\u2088\u2086 : x ^ 2 + y ^ 2 = (n : \u211d) := by\n      have h\u2088\u2086\u2081 : (Real.sqrt (x ^ 2 + y ^ 2)) ^ 2 = x ^ 2 + y ^ 2 := by\n        rw [Real.sq_sqrt] <;> nlinarith [sq_nonneg x, sq_nonneg y]\n      have h\u2088\u2086\u2082 : (Real.sqrt n : \u211d) ^ 2 = (n : \u211d) := by\n        rw [Real.sq_sqrt] <;> exact_mod_cast (by positivity)\n      linarith\n    have h\u2088\u2087 : (n : \u211d) = 52 := by\n      linarith\n    exact h\u2088\u2087\n  have h\u2089 : n = 52 := by\n    have h\u2089\u2081 : (n : \u211d) = 52 := h\u2088\n    have h\u2089\u2082 : n = 52 := by\n      apply_fun (fun x => (x : \u211d)) at h\u2089\u2081\n      <;> norm_cast at h\u2089\u2081 \u22a2 <;>\n      (try simp_all) <;>\n      (try norm_num) <;>\n      (try linarith)\n    exact h\u2089\u2082\n  exact h\u2089\n"}
{"mode": "clear", "steps_before": 25, "alphabet": 15, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 781, "chars_after": 7, "seconds": 95.8, "id": "amc12b_2020_p2", "proof_after": "theorem amc12b_2020_p2 :\n    (100 ^ 2 - 7 ^ 2 : \u211d) / (70 ^ 2 - 11 ^ 2) * ((70 - 11) * (70 + 11) / ((100 - 7) * (100 + 7))) =\n      1  :=  by\n  ring\n", "proof_before": "theorem amc12b_2020_p2 :\n    (100 ^ 2 - 7 ^ 2 : \u211d) / (70 ^ 2 - 11 ^ 2) * ((70 - 11) * (70 + 11) / ((100 - 7) * (100 + 7))) =\n      1  :=  by\n  have h1 : (100 : \u211d) ^ 2 - 7 ^ 2 = (100 - 7) * (100 + 7) := by\n    norm_num [sub_mul, add_mul, mul_sub, mul_add, mul_one, mul_assoc]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have h2 : (70 : \u211d) ^ 2 - 11 ^ 2 = (70 - 11) * (70 + 11) := by\n    norm_num [sub_mul, add_mul, mul_sub, mul_add, mul_one, mul_assoc]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have h3 : (100 ^ 2 - 7 ^ 2 : \u211d) / (70 ^ 2 - 11 ^ 2) * ((70 - 11) * (70 + 11) / ((100 - 7) * (100 + 7))) = 1 := by\n    rw [h1, h2]\n    have h4 : (70 - 11 : \u211d) \u2260 0 := by norm_num\n    have h5 : (70 + 11 : \u211d) \u2260 0 := by norm_num\n    have h6 : (100 - 7 : \u211d) \u2260 0 := by norm_num\n    have h7 : (100 + 7 : \u211d) \u2260 0 := by norm_num\n    field_simp [h4, h5, h6, h7]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  exact h3\n"}
{"mode": "clear", "steps_before": 16, "alphabet": 20, "compiles": false, "id": "amc12_2001_p5", "proof_after": "theorem amc12_2001_p5 :\n    Finset.prod (Finset.filter (fun x => \u00acEven x) (Finset.range 10000)) (id : \u2115 \u2192 \u2115) =\n      10000! / (2 ^ 5000 * 5000!)  :=  by\n  have h_main : 2 ^ 5000 * 5000! \u2223 10000! := by\n    have h\u2081 : 2 ^ 5000 * 5000! \u2223 10000! := by\n      apply Nat.dvd_of_mod_eq_zero\n      rfl\n    exact h\u2081\n  have h_prod : Finset.prod (Finset.filter (fun x => \u00acEven x) (Finset.range 10000)) (id : \u2115 \u2192 \u2115) = 10000! / (2 ^ 5000 * 5000!) := by\n    have h\u2081 : Finset.prod (Finset.filter (fun x => \u00acEven x) (Finset.range 10000)) (id : \u2115 \u2192 \u2115) = 10000! / (2 ^ 5000 * 5000!) := by\n      rw [show Finset.filter (fun x => \u00acEven x) (Finset.range 10000) = Finset.filter (fun x => x % 2 = 1) (Finset.range 10000) by\n        congr\n        ext x\n        simp [Nat.even_iff, Nat.mod_eq_zero_of_dvd]\n        <;>\n        (try omega) <;>\n        (try\n          {\n            cases' mod_two_eq_zero_or_one x with h h <;> simp [h, Nat.even_iff, Nat.dvd_iff_mod_eq_zero]\n            <;> omega\n          }) <;>\n        (try omega)\n      ]\n      rfl\n    exact h\u2081\n  apply h_prod\n", "proof_before": "theorem amc12_2001_p5 :\n    Finset.prod (Finset.filter (fun x => \u00acEven x) (Finset.range 10000)) (id : \u2115 \u2192 \u2115) =\n      10000! / (2 ^ 5000 * 5000!)  :=  by\n  have h_main : 2 ^ 5000 * 5000! \u2223 10000! := by\n    have h\u2081 : 2 ^ 5000 * 5000! \u2223 10000! := by\n      apply Nat.dvd_of_mod_eq_zero\n      rfl\n    exact h\u2081\n  have h_prod : Finset.prod (Finset.filter (fun x => \u00acEven x) (Finset.range 10000)) (id : \u2115 \u2192 \u2115) = 10000! / (2 ^ 5000 * 5000!) := by\n    have h\u2081 : Finset.prod (Finset.filter (fun x => \u00acEven x) (Finset.range 10000)) (id : \u2115 \u2192 \u2115) = 10000! / (2 ^ 5000 * 5000!) := by\n      rw [show Finset.filter (fun x => \u00acEven x) (Finset.range 10000) = Finset.filter (fun x => x % 2 = 1) (Finset.range 10000) by\n        congr\n        ext x\n        simp [Nat.even_iff, Nat.mod_eq_zero_of_dvd]\n        <;>\n        (try omega) <;>\n        (try\n          {\n            cases' mod_two_eq_zero_or_one x with h h <;> simp [h, Nat.even_iff, Nat.dvd_iff_mod_eq_zero]\n            <;> omega\n          }) <;>\n        (try omega)\n      ]\n      rfl\n    exact h\u2081\n  apply h_prod\n"}
{"mode": "clear", "steps_before": 26, "alphabet": 26, "compiles": false, "id": "mathd_numbertheory_457", "proof_after": "theorem mathd_numbertheory_457 (n : \u2115) (h\u2080 : 0 < n) (h\u2081 : 80325 \u2223 n !) : 17 \u2264 n  :=  by\n  have h\u2082 : 17 \u2223 80325 := by\n    norm_num\n    <;> decide\n  have h\u2083 : 17 \u2223 n ! := by\n    have h\u2084 : 17 \u2223 80325 := h\u2082\n    have h\u2085 : 80325 \u2223 n ! := h\u2081\n    exact dvd_trans h\u2084 h\u2085\n  have h\u2084 : n < 17 \u2192 False := by\n    intro h\u2085\n    have h\u2086 : n < 17 := h\u2085\n    have h\u2087 : \u00ac (17 \u2223 n !) := by\n      interval_cases n <;> norm_num [Nat.factorial_succ, Nat.dvd_iff_mod_eq_zero, Nat.mul_mod,\n        Nat.add_mod, Nat.mod_mod] at h\u2083 \u22a2 <;> try contradiction\n      <;> try norm_num at h\u2083 \u22a2 <;> try contradiction\n      <;> try omega\n    have h\u2088 : 17 \u2223 n ! := h\u2083\n    exact h\u2087 h\u2088\n  have h\u2085 : 17 \u2264 n := by\n    by_contra h\u2085\n    have h\u2086 : n < 17 := by linarith\n    have h\u2087 : False := h\u2084 h\u2086\n    exact h\u2087\n  exact h\u2085\n", "proof_before": "theorem mathd_numbertheory_457 (n : \u2115) (h\u2080 : 0 < n) (h\u2081 : 80325 \u2223 n !) : 17 \u2264 n  :=  by\n  have h\u2082 : 17 \u2223 80325 := by\n    norm_num\n    <;> decide\n  have h\u2083 : 17 \u2223 n ! := by\n    have h\u2084 : 17 \u2223 80325 := h\u2082\n    have h\u2085 : 80325 \u2223 n ! := h\u2081\n    exact dvd_trans h\u2084 h\u2085\n  have h\u2084 : n < 17 \u2192 False := by\n    intro h\u2085\n    have h\u2086 : n < 17 := h\u2085\n    have h\u2087 : \u00ac (17 \u2223 n !) := by\n      interval_cases n <;> norm_num [Nat.factorial_succ, Nat.dvd_iff_mod_eq_zero, Nat.mul_mod,\n        Nat.add_mod, Nat.mod_mod] at h\u2083 \u22a2 <;> try contradiction\n      <;> try norm_num at h\u2083 \u22a2 <;> try contradiction\n      <;> try omega\n    have h\u2088 : 17 \u2223 n ! := h\u2083\n    exact h\u2087 h\u2088\n  have h\u2085 : 17 \u2264 n := by\n    by_contra h\u2085\n    have h\u2086 : n < 17 := by linarith\n    have h\u2087 : False := h\u2084 h\u2086\n    exact h\u2087\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 13, "alphabet": 15, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 346, "chars_after": 11, "seconds": 41.4, "id": "mathd_algebra_427", "proof_after": "theorem mathd_algebra_427 (x y z : \u211d) (h\u2080 : 3 * x + y = 17) (h\u2081 : 5 * y + z = 14)\n    (h\u2082 : 3 * x + 5 * z = 41) : x + y + z = 12  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_427 (x y z : \u211d) (h\u2080 : 3 * x + y = 17) (h\u2081 : 5 * y + z = 14)\n    (h\u2082 : 3 * x + 5 * z = 41) : x + y + z = 12  :=  by\n  have h\u2083 : 6 * x + 6 * y + 6 * z = 72 := by\n    have h\u2083\u2081 : 6 * x + 6 * y + 6 * z = (3 * x + y) + (5 * y + z) + (3 * x + 5 * z) := by\n      ring\n    rw [h\u2083\u2081]\n    linarith\n  have h\u2084 : x + y + z = 12 := by\n    have h\u2084\u2081 : x + y + z = (6 * x + 6 * y + 6 * z) / 6 := by\n      ring\n    rw [h\u2084\u2081]\n    rw [h\u2083]\n    <;> norm_num\n    <;> linarith\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 2100, "alphabet": 552, "compiles": false, "id": "aime_1984_p7", "proof_after": "theorem aime_1984_p7 (f : \u2124 \u2192 \u2124) (h\u2080 : \u2200 n, 1000 \u2264 n \u2192 f n = n - 3)\n    (h\u2081 : \u2200 n, n < 1000 \u2192 f n = f (f (n + 5))) : f 84 = 997  :=  by\n  have h\u2082 : f 1004 = 1001 := by\n    have h\u2082\u2081 : f 1004 = 1004 - 3 := by\n      apply h\u2080\n      <;> norm_num\n    rw [h\u2082\u2081]\n    <;> norm_num\n  have h\u2083 : f 1003 = 1000 := by\n    have h\u2083\u2081 : f 1003 = 1003 - 3 := by\n      apply h\u2080\n      <;> norm_num\n    rw [h\u2083\u2081]\n    <;> norm_num\n  have h\u2084 : f 1002 = 999 := by\n    have h\u2084\u2081 : f 1002 = 1002 - 3 := by\n      apply h\u2080\n      <;> norm_num\n    rw [h\u2084\u2081]\n    <;> norm_num\n  have h\u2085 : f 1001 = 998 := by\n    have h\u2085\u2081 : f 1001 = 1001 - 3 := by\n      apply h\u2080\n      <;> norm_num\n    rw [h\u2085\u2081]\n    <;> norm_num\n  have h\u2086 : f 1000 = 997 := by\n    have h\u2086\u2081 : f 1000 = 1000 - 3 := by\n      apply h\u2080\n      <;> norm_num\n    rw [h\u2086\u2081]\n    <;> norm_num\n  have h\u2087 : f 999 = 998 := by\n    have h\u2087\u2081 : f 999 = f (f (999 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2081]\n    have h\u2087\u2082 : f (999 + 5) = f 1004 := by norm_num\n    rw [h\u2087\u2082]\n    rw [h\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2088 : f 998 = 997 := by\n    have h\u2088\u2081 : f 998 = f (f (998 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2081]\n    have h\u2088\u2082 : f (998 + 5) = f 1003 := by norm_num\n    rw [h\u2088\u2082]\n    rw [h\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2089 : f 997 = 998 := by\n    have h\u2089\u2081 : f 997 = f (f (997 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2081]\n    have h\u2089\u2082 : f (997 + 5) = f 1002 := by norm_num\n    rw [h\u2089\u2082]\n    rw [h\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080 : f 996 = 997 := by\n    have h\u2081\u2080\u2081 : f 996 = f (f (996 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2081]\n    have h\u2081\u2080\u2082 : f (996 + 5) = f 1001 := by norm_num\n    rw [h\u2081\u2080\u2082]\n    rw [h\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081 : f 995 = 998 := by\n    have h\u2081\u2081\u2081 : f 995 = f (f (995 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2081]\n    have h\u2081\u2081\u2082 : f (995 + 5) = f 1000 := by norm_num\n    rw [h\u2081\u2081\u2082]\n    rw [h\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082 : f 994 = 997 := by\n    have h\u2081\u2082\u2081 : f 994 = f (f (994 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2081]\n    have h\u2081\u2082\u2082 : f (994 + 5) = f 999 := by norm_num\n    rw [h\u2081\u2082\u2082]\n    rw [h\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083 : f 989 = 998 := by\n    have h\u2081\u2083\u2081 : f 989 = f (f (989 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2081]\n    have h\u2081\u2083\u2082 : f (989 + 5) = f 994 := by norm_num\n    rw [h\u2081\u2083\u2082]\n    rw [h\u2081\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084 : f 984 = 997 := by\n    have h\u2081\u2084\u2081 : f 984 = f (f (984 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2081]\n    have h\u2081\u2084\u2082 : f (984 + 5) = f 989 := by norm_num\n    rw [h\u2081\u2084\u2082]\n    rw [h\u2081\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085 : f 979 = 998 := by\n    have h\u2081\u2085\u2081 : f 979 = f (f (979 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2081]\n    have h\u2081\u2085\u2082 : f (979 + 5) = f 984 := by norm_num\n    rw [h\u2081\u2085\u2082]\n    rw [h\u2081\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086 : f 974 = 997 := by\n    have h\u2081\u2086\u2081 : f 974 = f (f (974 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2081]\n    have h\u2081\u2086\u2082 : f (974 + 5) = f 979 := by norm_num\n    rw [h\u2081\u2086\u2082]\n    rw [h\u2081\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087 : f 969 = 998 := by\n    have h\u2081\u2087\u2081 : f 969 = f (f (969 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2081]\n    have h\u2081\u2087\u2082 : f (969 + 5) = f 974 := by norm_num\n    rw [h\u2081\u2087\u2082]\n    rw [h\u2081\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088 : f 964 = 997 := by\n    have h\u2081\u2088\u2081 : f 964 = f (f (964 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2081]\n    have h\u2081\u2088\u2082 : f (964 + 5) = f 969 := by norm_num\n    rw [h\u2081\u2088\u2082]\n    rw [h\u2081\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089 : f 959 = 998 := by\n    have h\u2081\u2089\u2081 : f 959 = f (f (959 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2081]\n    have h\u2081\u2089\u2082 : f (959 + 5) = f 964 := by norm_num\n    rw [h\u2081\u2089\u2082]\n    rw [h\u2081\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2080 : f 954 = 997 := by\n    have h\u2082\u2080\u2081 : f 954 = f (f (954 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2080\u2081]\n    have h\u2082\u2080\u2082 : f (954 + 5) = f 959 := by norm_num\n    rw [h\u2082\u2080\u2082]\n    rw [h\u2081\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2081 : f 949 = 998 := by\n    have h\u2082\u2081\u2081 : f 949 = f (f (949 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2081\u2081]\n    have h\u2082\u2081\u2082 : f (949 + 5) = f 954 := by norm_num\n    rw [h\u2082\u2081\u2082]\n    rw [h\u2082\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2082 : f 944 = 997 := by\n    have h\u2082\u2082\u2081 : f 944 = f (f (944 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2082\u2081]\n    have h\u2082\u2082\u2082 : f (944 + 5) = f 949 := by norm_num\n    rw [h\u2082\u2082\u2082]\n    rw [h\u2082\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2083 : f 939 = 998 := by\n    have h\u2082\u2083\u2081 : f 939 = f (f (939 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2083\u2081]\n    have h\u2082\u2083\u2082 : f (939 + 5) = f 944 := by norm_num\n    rw [h\u2082\u2083\u2082]\n    rw [h\u2082\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2084 : f 934 = 997 := by\n    have h\u2082\u2084\u2081 : f 934 = f (f (934 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2084\u2081]\n    have h\u2082\u2084\u2082 : f (934 + 5) = f 939 := by norm_num\n    rw [h\u2082\u2084\u2082]\n    rw [h\u2082\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2085 : f 929 = 998 := by\n    have h\u2082\u2085\u2081 : f 929 = f (f (929 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2085\u2081]\n    have h\u2082\u2085\u2082 : f (929 + 5) = f 934 := by norm_num\n    rw [h\u2082\u2085\u2082]\n    rw [h\u2082\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2086 : f 924 = 997 := by\n    have h\u2082\u2086\u2081 : f 924 = f (f (924 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2086\u2081]\n    have h\u2082\u2086\u2082 : f (924 + 5) = f 929 := by norm_num\n    rw [h\u2082\u2086\u2082]\n    rw [h\u2082\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2087 : f 919 = 998 := by\n    have h\u2082\u2087\u2081 : f 919 = f (f (919 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2087\u2081]\n    have h\u2082\u2087\u2082 : f (919 + 5) = f 924 := by norm_num\n    rw [h\u2082\u2087\u2082]\n    rw [h\u2082\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2088 : f 914 = 997 := by\n    have h\u2082\u2088\u2081 : f 914 = f (f (914 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2088\u2081]\n    have h\u2082\u2088\u2082 : f (914 + 5) = f 919 := by norm_num\n    rw [h\u2082\u2088\u2082]\n    rw [h\u2082\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2089 : f 909 = 998 := by\n    have h\u2082\u2089\u2081 : f 909 = f (f (909 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2089\u2081]\n    have h\u2082\u2089\u2082 : f (909 + 5) = f 914 := by norm_num\n    rw [h\u2082\u2089\u2082]\n    rw [h\u2082\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2080 : f 904 = 997 := by\n    have h\u2083\u2080\u2081 : f 904 = f (f (904 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2080\u2081]\n    have h\u2083\u2080\u2082 : f (904 + 5) = f 909 := by norm_num\n    rw [h\u2083\u2080\u2082]\n    rw [h\u2082\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2081 : f 899 = 998 := by\n    have h\u2083\u2081\u2081 : f 899 = f (f (899 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2081\u2081]\n    have h\u2083\u2081\u2082 : f (899 + 5) = f 904 := by norm_num\n    rw [h\u2083\u2081\u2082]\n    rw [h\u2083\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2082 : f 894 = 997 := by\n    have h\u2083\u2082\u2081 : f 894 = f (f (894 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2082\u2081]\n    have h\u2083\u2082\u2082 : f (894 + 5) = f 899 := by norm_num\n    rw [h\u2083\u2082\u2082]\n    rw [h\u2083\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2083 : f 889 = 998 := by\n    have h\u2083\u2083\u2081 : f 889 = f (f (889 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2083\u2081]\n    have h\u2083\u2083\u2082 : f (889 + 5) = f 894 := by norm_num\n    rw [h\u2083\u2083\u2082]\n    rw [h\u2083\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2084 : f 884 = 997 := by\n    have h\u2083\u2084\u2081 : f 884 = f (f (884 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2084\u2081]\n    have h\u2083\u2084\u2082 : f (884 + 5) = f 889 := by norm_num\n    rw [h\u2083\u2084\u2082]\n    rw [h\u2083\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2085 : f 879 = 998 := by\n    have h\u2083\u2085\u2081 : f 879 = f (f (879 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2085\u2081]\n    have h\u2083\u2085\u2082 : f (879 + 5) = f 884 := by norm_num\n    rw [h\u2083\u2085\u2082]\n    rw [h\u2083\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2086 : f 874 = 997 := by\n    have h\u2083\u2086\u2081 : f 874 = f (f (874 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2086\u2081]\n    have h\u2083\u2086\u2082 : f (874 + 5) = f 879 := by norm_num\n    rw [h\u2083\u2086\u2082]\n    rw [h\u2083\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2087 : f 869 = 998 := by\n    have h\u2083\u2087\u2081 : f 869 = f (f (869 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2087\u2081]\n    have h\u2083\u2087\u2082 : f (869 + 5) = f 874 := by norm_num\n    rw [h\u2083\u2087\u2082]\n    rw [h\u2083\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2088 : f 864 = 997 := by\n    have h\u2083\u2088\u2081 : f 864 = f (f (864 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2088\u2081]\n    have h\u2083\u2088\u2082 : f (864 + 5) = f 869 := by norm_num\n    rw [h\u2083\u2088\u2082]\n    rw [h\u2083\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2089 : f 859 = 998 := by\n    have h\u2083\u2089\u2081 : f 859 = f (f (859 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2089\u2081]\n    have h\u2083\u2089\u2082 : f (859 + 5) = f 864 := by norm_num\n    rw [h\u2083\u2089\u2082]\n    rw [h\u2083\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2080 : f 854 = 997 := by\n    have h\u2084\u2080\u2081 : f 854 = f (f (854 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2080\u2081]\n    have h\u2084\u2080\u2082 : f (854 + 5) = f 859 := by norm_num\n    rw [h\u2084\u2080\u2082]\n    rw [h\u2083\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2081 : f 849 = 998 := by\n    have h\u2084\u2081\u2081 : f 849 = f (f (849 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2081\u2081]\n    have h\u2084\u2081\u2082 : f (849 + 5) = f 854 := by norm_num\n    rw [h\u2084\u2081\u2082]\n    rw [h\u2084\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2082 : f 844 = 997 := by\n    have h\u2084\u2082\u2081 : f 844 = f (f (844 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2082\u2081]\n    have h\u2084\u2082\u2082 : f (844 + 5) = f 849 := by norm_num\n    rw [h\u2084\u2082\u2082]\n    rw [h\u2084\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2083 : f 839 = 998 := by\n    have h\u2084\u2083\u2081 : f 839 = f (f (839 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2083\u2081]\n    have h\u2084\u2083\u2082 : f (839 + 5) = f 844 := by norm_num\n    rw [h\u2084\u2083\u2082]\n    rw [h\u2084\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2084 : f 834 = 997 := by\n    have h\u2084\u2084\u2081 : f 834 = f (f (834 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2084\u2081]\n    have h\u2084\u2084\u2082 : f (834 + 5) = f 839 := by norm_num\n    rw [h\u2084\u2084\u2082]\n    rw [h\u2084\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2085 : f 829 = 998 := by\n    have h\u2084\u2085\u2081 : f 829 = f (f (829 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2085\u2081]\n    have h\u2084\u2085\u2082 : f (829 + 5) = f 834 := by norm_num\n    rw [h\u2084\u2085\u2082]\n    rw [h\u2084\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2086 : f 824 = 997 := by\n    have h\u2084\u2086\u2081 : f 824 = f (f (824 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2086\u2081]\n    have h\u2084\u2086\u2082 : f (824 + 5) = f 829 := by norm_num\n    rw [h\u2084\u2086\u2082]\n    rw [h\u2084\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2087 : f 819 = 998 := by\n    have h\u2084\u2087\u2081 : f 819 = f (f (819 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2087\u2081]\n    have h\u2084\u2087\u2082 : f (819 + 5) = f 824 := by norm_num\n    rw [h\u2084\u2087\u2082]\n    rw [h\u2084\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2088 : f 814 = 997 := by\n    have h\u2084\u2088\u2081 : f 814 = f (f (814 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2088\u2081]\n    have h\u2084\u2088\u2082 : f (814 + 5) = f 819 := by norm_num\n    rw [h\u2084\u2088\u2082]\n    rw [h\u2084\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2089 : f 809 = 998 := by\n    have h\u2084\u2089\u2081 : f 809 = f (f (809 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2089\u2081]\n    have h\u2084\u2089\u2082 : f (809 + 5) = f 814 := by norm_num\n    rw [h\u2084\u2089\u2082]\n    rw [h\u2084\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2080 : f 804 = 997 := by\n    have h\u2085\u2080\u2081 : f 804 = f (f (804 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2080\u2081]\n    have h\u2085\u2080\u2082 : f (804 + 5) = f 809 := by norm_num\n    rw [h\u2085\u2080\u2082]\n    rw [h\u2084\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2081 : f 799 = 998 := by\n    have h\u2085\u2081\u2081 : f 799 = f (f (799 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2081\u2081]\n    have h\u2085\u2081\u2082 : f (799 + 5) = f 804 := by norm_num\n    rw [h\u2085\u2081\u2082]\n    rw [h\u2085\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2082 : f 794 = 997 := by\n    have h\u2085\u2082\u2081 : f 794 = f (f (794 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2082\u2081]\n    have h\u2085\u2082\u2082 : f (794 + 5) = f 799 := by norm_num\n    rw [h\u2085\u2082\u2082]\n    rw [h\u2085\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2083 : f 789 = 998 := by\n    have h\u2085\u2083\u2081 : f 789 = f (f (789 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2083\u2081]\n    have h\u2085\u2083\u2082 : f (789 + 5) = f 794 := by norm_num\n    rw [h\u2085\u2083\u2082]\n    rw [h\u2085\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2084 : f 784 = 997 := by\n    have h\u2085\u2084\u2081 : f 784 = f (f (784 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2084\u2081]\n    have h\u2085\u2084\u2082 : f (784 + 5) = f 789 := by norm_num\n    rw [h\u2085\u2084\u2082]\n    rw [h\u2085\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2085 : f 779 = 998 := by\n    have h\u2085\u2085\u2081 : f 779 = f (f (779 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2085\u2081]\n    have h\u2085\u2085\u2082 : f (779 + 5) = f 784 := by norm_num\n    rw [h\u2085\u2085\u2082]\n    rw [h\u2085\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2086 : f 774 = 997 := by\n    have h\u2085\u2086\u2081 : f 774 = f (f (774 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2086\u2081]\n    have h\u2085\u2086\u2082 : f (774 + 5) = f 779 := by norm_num\n    rw [h\u2085\u2086\u2082]\n    rw [h\u2085\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2087 : f 769 = 998 := by\n    have h\u2085\u2087\u2081 : f 769 = f (f (769 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2087\u2081]\n    have h\u2085\u2087\u2082 : f (769 + 5) = f 774 := by norm_num\n    rw [h\u2085\u2087\u2082]\n    rw [h\u2085\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2088 : f 764 = 997 := by\n    have h\u2085\u2088\u2081 : f 764 = f (f (764 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2088\u2081]\n    have h\u2085\u2088\u2082 : f (764 + 5) = f 769 := by norm_num\n    rw [h\u2085\u2088\u2082]\n    rw [h\u2085\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2089 : f 759 = 998 := by\n    have h\u2085\u2089\u2081 : f 759 = f (f (759 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2089\u2081]\n    have h\u2085\u2089\u2082 : f (759 + 5) = f 764 := by norm_num\n    rw [h\u2085\u2089\u2082]\n    rw [h\u2085\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2080 : f 754 = 997 := by\n    have h\u2086\u2080\u2081 : f 754 = f (f (754 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2080\u2081]\n    have h\u2086\u2080\u2082 : f (754 + 5) = f 759 := by norm_num\n    rw [h\u2086\u2080\u2082]\n    rw [h\u2085\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2081 : f 749 = 998 := by\n    have h\u2086\u2081\u2081 : f 749 = f (f (749 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2081\u2081]\n    have h\u2086\u2081\u2082 : f (749 + 5) = f 754 := by norm_num\n    rw [h\u2086\u2081\u2082]\n    rw [h\u2086\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2082 : f 744 = 997 := by\n    have h\u2086\u2082\u2081 : f 744 = f (f (744 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2082\u2081]\n    have h\u2086\u2082\u2082 : f (744 + 5) = f 749 := by norm_num\n    rw [h\u2086\u2082\u2082]\n    rw [h\u2086\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2083 : f 739 = 998 := by\n    have h\u2086\u2083\u2081 : f 739 = f (f (739 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2083\u2081]\n    have h\u2086\u2083\u2082 : f (739 + 5) = f 744 := by norm_num\n    rw [h\u2086\u2083\u2082]\n    rw [h\u2086\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2084 : f 734 = 997 := by\n    have h\u2086\u2084\u2081 : f 734 = f (f (734 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2084\u2081]\n    have h\u2086\u2084\u2082 : f (734 + 5) = f 739 := by norm_num\n    rw [h\u2086\u2084\u2082]\n    rw [h\u2086\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2085 : f 729 = 998 := by\n    have h\u2086\u2085\u2081 : f 729 = f (f (729 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2085\u2081]\n    have h\u2086\u2085\u2082 : f (729 + 5) = f 734 := by norm_num\n    rw [h\u2086\u2085\u2082]\n    rw [h\u2086\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2086 : f 724 = 997 := by\n    have h\u2086\u2086\u2081 : f 724 = f (f (724 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2086\u2081]\n    have h\u2086\u2086\u2082 : f (724 + 5) = f 729 := by norm_num\n    rw [h\u2086\u2086\u2082]\n    rw [h\u2086\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2087 : f 719 = 998 := by\n    have h\u2086\u2087\u2081 : f 719 = f (f (719 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2087\u2081]\n    have h\u2086\u2087\u2082 : f (719 + 5) = f 724 := by norm_num\n    rw [h\u2086\u2087\u2082]\n    rw [h\u2086\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2088 : f 714 = 997 := by\n    have h\u2086\u2088\u2081 : f 714 = f (f (714 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2088\u2081]\n    have h\u2086\u2088\u2082 : f (714 + 5) = f 719 := by norm_num\n    rw [h\u2086\u2088\u2082]\n    rw [h\u2086\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2089 : f 709 = 998 := by\n    have h\u2086\u2089\u2081 : f 709 = f (f (709 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2089\u2081]\n    have h\u2086\u2089\u2082 : f (709 + 5) = f 714 := by norm_num\n    rw [h\u2086\u2089\u2082]\n    rw [h\u2086\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2080 : f 704 = 997 := by\n    have h\u2087\u2080\u2081 : f 704 = f (f (704 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2080\u2081]\n    have h\u2087\u2080\u2082 : f (704 + 5) = f 709 := by norm_num\n    rw [h\u2087\u2080\u2082]\n    rw [h\u2086\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2081 : f 699 = 998 := by\n    have h\u2087\u2081\u2081 : f 699 = f (f (699 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2081\u2081]\n    have h\u2087\u2081\u2082 : f (699 + 5) = f 704 := by norm_num\n    rw [h\u2087\u2081\u2082]\n    rw [h\u2087\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2082 : f 694 = 997 := by\n    have h\u2087\u2082\u2081 : f 694 = f (f (694 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2082\u2081]\n    have h\u2087\u2082\u2082 : f (694 + 5) = f 699 := by norm_num\n    rw [h\u2087\u2082\u2082]\n    rw [h\u2087\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2083 : f 689 = 998 := by\n    have h\u2087\u2083\u2081 : f 689 = f (f (689 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2083\u2081]\n    have h\u2087\u2083\u2082 : f (689 + 5) = f 694 := by norm_num\n    rw [h\u2087\u2083\u2082]\n    rw [h\u2087\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2084 : f 684 = 997 := by\n    have h\u2087\u2084\u2081 : f 684 = f (f (684 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2084\u2081]\n    have h\u2087\u2084\u2082 : f (684 + 5) = f 689 := by norm_num\n    rw [h\u2087\u2084\u2082]\n    rw [h\u2087\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2085 : f 679 = 998 := by\n    have h\u2087\u2085\u2081 : f 679 = f (f (679 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2085\u2081]\n    have h\u2087\u2085\u2082 : f (679 + 5) = f 684 := by norm_num\n    rw [h\u2087\u2085\u2082]\n    rw [h\u2087\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2086 : f 674 = 997 := by\n    have h\u2087\u2086\u2081 : f 674 = f (f (674 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2086\u2081]\n    have h\u2087\u2086\u2082 : f (674 + 5) = f 679 := by norm_num\n    rw [h\u2087\u2086\u2082]\n    rw [h\u2087\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2087 : f 669 = 998 := by\n    have h\u2087\u2087\u2081 : f 669 = f (f (669 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2087\u2081]\n    have h\u2087\u2087\u2082 : f (669 + 5) = f 674 := by norm_num\n    rw [h\u2087\u2087\u2082]\n    rw [h\u2087\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2088 : f 664 = 997 := by\n    have h\u2087\u2088\u2081 : f 664 = f (f (664 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2088\u2081]\n    have h\u2087\u2088\u2082 : f (664 + 5) = f 669 := by norm_num\n    rw [h\u2087\u2088\u2082]\n    rw [h\u2087\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2089 : f 659 = 998 := by\n    have h\u2087\u2089\u2081 : f 659 = f (f (659 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2089\u2081]\n    have h\u2087\u2089\u2082 : f (659 + 5) = f 664 := by norm_num\n    rw [h\u2087\u2089\u2082]\n    rw [h\u2087\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2080 : f 654 = 997 := by\n    have h\u2088\u2080\u2081 : f 654 = f (f (654 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2080\u2081]\n    have h\u2088\u2080\u2082 : f (654 + 5) = f 659 := by norm_num\n    rw [h\u2088\u2080\u2082]\n    rw [h\u2087\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2081 : f 649 = 998 := by\n    have h\u2088\u2081\u2081 : f 649 = f (f (649 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2081\u2081]\n    have h\u2088\u2081\u2082 : f (649 + 5) = f 654 := by norm_num\n    rw [h\u2088\u2081\u2082]\n    rw [h\u2088\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2082 : f 644 = 997 := by\n    have h\u2088\u2082\u2081 : f 644 = f (f (644 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2082\u2081]\n    have h\u2088\u2082\u2082 : f (644 + 5) = f 649 := by norm_num\n    rw [h\u2088\u2082\u2082]\n    rw [h\u2088\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2083 : f 639 = 998 := by\n    have h\u2088\u2083\u2081 : f 639 = f (f (639 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2083\u2081]\n    have h\u2088\u2083\u2082 : f (639 + 5) = f 644 := by norm_num\n    rw [h\u2088\u2083\u2082]\n    rw [h\u2088\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2084 : f 634 = 997 := by\n    have h\u2088\u2084\u2081 : f 634 = f (f (634 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2084\u2081]\n    have h\u2088\u2084\u2082 : f (634 + 5) = f 639 := by norm_num\n    rw [h\u2088\u2084\u2082]\n    rw [h\u2088\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2085 : f 629 = 998 := by\n    have h\u2088\u2085\u2081 : f 629 = f (f (629 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2085\u2081]\n    have h\u2088\u2085\u2082 : f (629 + 5) = f 634 := by norm_num\n    rw [h\u2088\u2085\u2082]\n    rw [h\u2088\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2086 : f 624 = 997 := by\n    have h\u2088\u2086\u2081 : f 624 = f (f (624 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2086\u2081]\n    have h\u2088\u2086\u2082 : f (624 + 5) = f 629 := by norm_num\n    rw [h\u2088\u2086\u2082]\n    rw [h\u2088\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2087 : f 619 = 998 := by\n    have h\u2088\u2087\u2081 : f 619 = f (f (619 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2087\u2081]\n    have h\u2088\u2087\u2082 : f (619 + 5) = f 624 := by norm_num\n    rw [h\u2088\u2087\u2082]\n    rw [h\u2088\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2088 : f 614 = 997 := by\n    have h\u2088\u2088\u2081 : f 614 = f (f (614 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2088\u2081]\n    have h\u2088\u2088\u2082 : f (614 + 5) = f 619 := by norm_num\n    rw [h\u2088\u2088\u2082]\n    rw [h\u2088\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2089 : f 609 = 998 := by\n    have h\u2088\u2089\u2081 : f 609 = f (f (609 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2089\u2081]\n    have h\u2088\u2089\u2082 : f (609 + 5) = f 614 := by norm_num\n    rw [h\u2088\u2089\u2082]\n    rw [h\u2088\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2080 : f 604 = 997 := by\n    have h\u2089\u2080\u2081 : f 604 = f (f (604 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2080\u2081]\n    have h\u2089\u2080\u2082 : f (604 + 5) = f 609 := by norm_num\n    rw [h\u2089\u2080\u2082]\n    rw [h\u2088\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2081 : f 599 = 998 := by\n    have h\u2089\u2081\u2081 : f 599 = f (f (599 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2081\u2081]\n    have h\u2089\u2081\u2082 : f (599 + 5) = f 604 := by norm_num\n    rw [h\u2089\u2081\u2082]\n    rw [h\u2089\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2082 : f 594 = 997 := by\n    have h\u2089\u2082\u2081 : f 594 = f (f (594 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2082\u2081]\n    have h\u2089\u2082\u2082 : f (594 + 5) = f 599 := by norm_num\n    rw [h\u2089\u2082\u2082]\n    rw [h\u2089\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2083 : f 589 = 998 := by\n    have h\u2089\u2083\u2081 : f 589 = f (f (589 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2083\u2081]\n    have h\u2089\u2083\u2082 : f (589 + 5) = f 594 := by norm_num\n    rw [h\u2089\u2083\u2082]\n    rw [h\u2089\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2084 : f 584 = 997 := by\n    have h\u2089\u2084\u2081 : f 584 = f (f (584 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2084\u2081]\n    have h\u2089\u2084\u2082 : f (584 + 5) = f 589 := by norm_num\n    rw [h\u2089\u2084\u2082]\n    rw [h\u2089\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2085 : f 579 = 998 := by\n    have h\u2089\u2085\u2081 : f 579 = f (f (579 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2085\u2081]\n    have h\u2089\u2085\u2082 : f (579 + 5) = f 584 := by norm_num\n    rw [h\u2089\u2085\u2082]\n    rw [h\u2089\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2086 : f 574 = 997 := by\n    have h\u2089\u2086\u2081 : f 574 = f (f (574 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2086\u2081]\n    have h\u2089\u2086\u2082 : f (574 + 5) = f 579 := by norm_num\n    rw [h\u2089\u2086\u2082]\n    rw [h\u2089\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2087 : f 569 = 998 := by\n    have h\u2089\u2087\u2081 : f 569 = f (f (569 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2087\u2081]\n    have h\u2089\u2087\u2082 : f (569 + 5) = f 574 := by norm_num\n    rw [h\u2089\u2087\u2082]\n    rw [h\u2089\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2088 : f 564 = 997 := by\n    have h\u2089\u2088\u2081 : f 564 = f (f (564 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2088\u2081]\n    have h\u2089\u2088\u2082 : f (564 + 5) = f 569 := by norm_num\n    rw [h\u2089\u2088\u2082]\n    rw [h\u2089\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2089 : f 559 = 998 := by\n    have h\u2089\u2089\u2081 : f 559 = f (f (559 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2089\u2081]\n    have h\u2089\u2089\u2082 : f (559 + 5) = f 564 := by norm_num\n    rw [h\u2089\u2089\u2082]\n    rw [h\u2089\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2080 : f 554 = 997 := by\n    have h\u2081\u2080\u2080\u2081 : f 554 = f (f (554 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2080\u2081]\n    have h\u2081\u2080\u2080\u2082 : f (554 + 5) = f 559 := by norm_num\n    rw [h\u2081\u2080\u2080\u2082]\n    rw [h\u2089\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2081 : f 549 = 998 := by\n    have h\u2081\u2080\u2081\u2081 : f 549 = f (f (549 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2081\u2081]\n    have h\u2081\u2080\u2081\u2082 : f (549 + 5) = f 554 := by norm_num\n    rw [h\u2081\u2080\u2081\u2082]\n    rw [h\u2081\u2080\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2082 : f 544 = 997 := by\n    have h\u2081\u2080\u2082\u2081 : f 544 = f (f (544 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2082\u2081]\n    have h\u2081\u2080\u2082\u2082 : f (544 + 5) = f 549 := by norm_num\n    rw [h\u2081\u2080\u2082\u2082]\n    rw [h\u2081\u2080\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2083 : f 539 = 998 := by\n    have h\u2081\u2080\u2083\u2081 : f 539 = f (f (539 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2083\u2081]\n    have h\u2081\u2080\u2083\u2082 : f (539 + 5) = f 544 := by norm_num\n    rw [h\u2081\u2080\u2083\u2082]\n    rw [h\u2081\u2080\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2084 : f 534 = 997 := by\n    have h\u2081\u2080\u2084\u2081 : f 534 = f (f (534 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2084\u2081]\n    have h\u2081\u2080\u2084\u2082 : f (534 + 5) = f 539 := by norm_num\n    rw [h\u2081\u2080\u2084\u2082]\n    rw [h\u2081\u2080\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2085 : f 529 = 998 := by\n    have h\u2081\u2080\u2085\u2081 : f 529 = f (f (529 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2085\u2081]\n    have h\u2081\u2080\u2085\u2082 : f (529 + 5) = f 534 := by norm_num\n    rw [h\u2081\u2080\u2085\u2082]\n    rw [h\u2081\u2080\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2086 : f 524 = 997 := by\n    have h\u2081\u2080\u2086\u2081 : f 524 = f (f (524 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2086\u2081]\n    have h\u2081\u2080\u2086\u2082 : f (524 + 5) = f 529 := by norm_num\n    rw [h\u2081\u2080\u2086\u2082]\n    rw [h\u2081\u2080\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2087 : f 519 = 998 := by\n    have h\u2081\u2080\u2087\u2081 : f 519 = f (f (519 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2087\u2081]\n    have h\u2081\u2080\u2087\u2082 : f (519 + 5) = f 524 := by norm_num\n    rw [h\u2081\u2080\u2087\u2082]\n    rw [h\u2081\u2080\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2088 : f 514 = 997 := by\n    have h\u2081\u2080\u2088\u2081 : f 514 = f (f (514 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2088\u2081]\n    have h\u2081\u2080\u2088\u2082 : f (514 + 5) = f 519 := by norm_num\n    rw [h\u2081\u2080\u2088\u2082]\n    rw [h\u2081\u2080\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2089 : f 509 = 998 := by\n    have h\u2081\u2080\u2089\u2081 : f 509 = f (f (509 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2089\u2081]\n    have h\u2081\u2080\u2089\u2082 : f (509 + 5) = f 514 := by norm_num\n    rw [h\u2081\u2080\u2089\u2082]\n    rw [h\u2081\u2080\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2080 : f 504 = 997 := by\n    have h\u2081\u2081\u2080\u2081 : f 504 = f (f (504 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2080\u2081]\n    have h\u2081\u2081\u2080\u2082 : f (504 + 5) = f 509 := by norm_num\n    rw [h\u2081\u2081\u2080\u2082]\n    rw [h\u2081\u2080\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2081 : f 499 = 998 := by\n    have h\u2081\u2081\u2081\u2081 : f 499 = f (f (499 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2081\u2081]\n    have h\u2081\u2081\u2081\u2082 : f (499 + 5) = f 504 := by norm_num\n    rw [h\u2081\u2081\u2081\u2082]\n    rw [h\u2081\u2081\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2082 : f 494 = 997 := by\n    have h\u2081\u2081\u2082\u2081 : f 494 = f (f (494 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2082\u2081]\n    have h\u2081\u2081\u2082\u2082 : f (494 + 5) = f 499 := by norm_num\n    rw [h\u2081\u2081\u2082\u2082]\n    rw [h\u2081\u2081\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2083 : f 489 = 998 := by\n    have h\u2081\u2081\u2083\u2081 : f 489 = f (f (489 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2083\u2081]\n    have h\u2081\u2081\u2083\u2082 : f (489 + 5) = f 494 := by norm_num\n    rw [h\u2081\u2081\u2083\u2082]\n    rw [h\u2081\u2081\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2084 : f 484 = 997 := by\n    have h\u2081\u2081\u2084\u2081 : f 484 = f (f (484 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2084\u2081]\n    have h\u2081\u2081\u2084\u2082 : f (484 + 5) = f 489 := by norm_num\n    rw [h\u2081\u2081\u2084\u2082]\n    rw [h\u2081\u2081\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2085 : f 479 = 998 := by\n    have h\u2081\u2081\u2085\u2081 : f 479 = f (f (479 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2085\u2081]\n    have h\u2081\u2081\u2085\u2082 : f (479 + 5) = f 484 := by norm_num\n    rw [h\u2081\u2081\u2085\u2082]\n    rw [h\u2081\u2081\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2086 : f 474 = 997 := by\n    have h\u2081\u2081\u2086\u2081 : f 474 = f (f (474 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2086\u2081]\n    have h\u2081\u2081\u2086\u2082 : f (474 + 5) = f 479 := by norm_num\n    rw [h\u2081\u2081\u2086\u2082]\n    rw [h\u2081\u2081\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2087 : f 469 = 998 := by\n    have h\u2081\u2081\u2087\u2081 : f 469 = f (f (469 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2087\u2081]\n    have h\u2081\u2081\u2087\u2082 : f (469 + 5) = f 474 := by norm_num\n    rw [h\u2081\u2081\u2087\u2082]\n    rw [h\u2081\u2081\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2088 : f 464 = 997 := by\n    have h\u2081\u2081\u2088\u2081 : f 464 = f (f (464 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2088\u2081]\n    have h\u2081\u2081\u2088\u2082 : f (464 + 5) = f 469 := by norm_num\n    rw [h\u2081\u2081\u2088\u2082]\n    rw [h\u2081\u2081\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2089 : f 459 = 998 := by\n    have h\u2081\u2081\u2089\u2081 : f 459 = f (f (459 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2089\u2081]\n    have h\u2081\u2081\u2089\u2082 : f (459 + 5) = f 464 := by norm_num\n    rw [h\u2081\u2081\u2089\u2082]\n    rw [h\u2081\u2081\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2080 : f 454 = 997 := by\n    have h\u2081\u2082\u2080\u2081 : f 454 = f (f (454 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2080\u2081]\n    have h\u2081\u2082\u2080\u2082 : f (454 + 5) = f 459 := by norm_num\n    rw [h\u2081\u2082\u2080\u2082]\n    rw [h\u2081\u2081\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2081 : f 449 = 998 := by\n    have h\u2081\u2082\u2081\u2081 : f 449 = f (f (449 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2081\u2081]\n    have h\u2081\u2082\u2081\u2082 : f (449 + 5) = f 454 := by norm_num\n    rw [h\u2081\u2082\u2081\u2082]\n    rw [h\u2081\u2082\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2082 : f 444 = 997 := by\n    have h\u2081\u2082\u2082\u2081 : f 444 = f (f (444 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2082\u2081]\n    have h\u2081\u2082\u2082\u2082 : f (444 + 5) = f 449 := by norm_num\n    rw [h\u2081\u2082\u2082\u2082]\n    rw [h\u2081\u2082\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2083 : f 439 = 998 := by\n    have h\u2081\u2082\u2083\u2081 : f 439 = f (f (439 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2083\u2081]\n    have h\u2081\u2082\u2083\u2082 : f (439 + 5) = f 444 := by norm_num\n    rw [h\u2081\u2082\u2083\u2082]\n    rw [h\u2081\u2082\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2084 : f 434 = 997 := by\n    have h\u2081\u2082\u2084\u2081 : f 434 = f (f (434 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2084\u2081]\n    have h\u2081\u2082\u2084\u2082 : f (434 + 5) = f 439 := by norm_num\n    rw [h\u2081\u2082\u2084\u2082]\n    rw [h\u2081\u2082\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2085 : f 429 = 998 := by\n    have h\u2081\u2082\u2085\u2081 : f 429 = f (f (429 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2085\u2081]\n    have h\u2081\u2082\u2085\u2082 : f (429 + 5) = f 434 := by norm_num\n    rw [h\u2081\u2082\u2085\u2082]\n    rw [h\u2081\u2082\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2086 : f 424 = 997 := by\n    have h\u2081\u2082\u2086\u2081 : f 424 = f (f (424 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2086\u2081]\n    have h\u2081\u2082\u2086\u2082 : f (424 + 5) = f 429 := by norm_num\n    rw [h\u2081\u2082\u2086\u2082]\n    rw [h\u2081\u2082\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2087 : f 419 = 998 := by\n    have h\u2081\u2082\u2087\u2081 : f 419 = f (f (419 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2087\u2081]\n    have h\u2081\u2082\u2087\u2082 : f (419 + 5) = f 424 := by norm_num\n    rw [h\u2081\u2082\u2087\u2082]\n    rw [h\u2081\u2082\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2088 : f 414 = 997 := by\n    have h\u2081\u2082\u2088\u2081 : f 414 = f (f (414 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2088\u2081]\n    have h\u2081\u2082\u2088\u2082 : f (414 + 5) = f 419 := by norm_num\n    rw [h\u2081\u2082\u2088\u2082]\n    rw [h\u2081\u2082\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2089 : f 409 = 998 := by\n    have h\u2081\u2082\u2089\u2081 : f 409 = f (f (409 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2089\u2081]\n    have h\u2081\u2082\u2089\u2082 : f (409 + 5) = f 414 := by norm_num\n    rw [h\u2081\u2082\u2089\u2082]\n    rw [h\u2081\u2082\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2080 : f 404 = 997 := by\n    have h\u2081\u2083\u2080\u2081 : f 404 = f (f (404 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2080\u2081]\n    have h\u2081\u2083\u2080\u2082 : f (404 + 5) = f 409 := by norm_num\n    rw [h\u2081\u2083\u2080\u2082]\n    rw [h\u2081\u2082\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2081 : f 399 = 998 := by\n    have h\u2081\u2083\u2081\u2081 : f 399 = f (f (399 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2081\u2081]\n    have h\u2081\u2083\u2081\u2082 : f (399 + 5) = f 404 := by norm_num\n    rw [h\u2081\u2083\u2081\u2082]\n    rw [h\u2081\u2083\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2082 : f 394 = 997 := by\n    have h\u2081\u2083\u2082\u2081 : f 394 = f (f (394 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2082\u2081]\n    have h\u2081\u2083\u2082\u2082 : f (394 + 5) = f 399 := by norm_num\n    rw [h\u2081\u2083\u2082\u2082]\n    rw [h\u2081\u2083\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2083 : f 389 = 998 := by\n    have h\u2081\u2083\u2083\u2081 : f 389 = f (f (389 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2083\u2081]\n    have h\u2081\u2083\u2083\u2082 : f (389 + 5) = f 394 := by norm_num\n    rw [h\u2081\u2083\u2083\u2082]\n    rw [h\u2081\u2083\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2084 : f 384 = 997 := by\n    have h\u2081\u2083\u2084\u2081 : f 384 = f (f (384 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2084\u2081]\n    have h\u2081\u2083\u2084\u2082 : f (384 + 5) = f 389 := by norm_num\n    rw [h\u2081\u2083\u2084\u2082]\n    rw [h\u2081\u2083\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2085 : f 379 = 998 := by\n    have h\u2081\u2083\u2085\u2081 : f 379 = f (f (379 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2085\u2081]\n    have h\u2081\u2083\u2085\u2082 : f (379 + 5) = f 384 := by norm_num\n    rw [h\u2081\u2083\u2085\u2082]\n    rw [h\u2081\u2083\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2086 : f 374 = 997 := by\n    have h\u2081\u2083\u2086\u2081 : f 374 = f (f (374 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2086\u2081]\n    have h\u2081\u2083\u2086\u2082 : f (374 + 5) = f 379 := by norm_num\n    rw [h\u2081\u2083\u2086\u2082]\n    rw [h\u2081\u2083\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2087 : f 369 = 998 := by\n    have h\u2081\u2083\u2087\u2081 : f 369 = f (f (369 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2087\u2081]\n    have h\u2081\u2083\u2087\u2082 : f (369 + 5) = f 374 := by norm_num\n    rw [h\u2081\u2083\u2087\u2082]\n    rw [h\u2081\u2083\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2088 : f 364 = 997 := by\n    have h\u2081\u2083\u2088\u2081 : f 364 = f (f (364 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2088\u2081]\n    have h\u2081\u2083\u2088\u2082 : f (364 + 5) = f 369 := by norm_num\n    rw [h\u2081\u2083\u2088\u2082]\n    rw [h\u2081\u2083\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2089 : f 359 = 998 := by\n    have h\u2081\u2083\u2089\u2081 : f 359 = f (f (359 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2089\u2081]\n    have h\u2081\u2083\u2089\u2082 : f (359 + 5) = f 364 := by norm_num\n    rw [h\u2081\u2083\u2089\u2082]\n    rw [h\u2081\u2083\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2080 : f 354 = 997 := by\n    have h\u2081\u2084\u2080\u2081 : f 354 = f (f (354 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2080\u2081]\n    have h\u2081\u2084\u2080\u2082 : f (354 + 5) = f 359 := by norm_num\n    rw [h\u2081\u2084\u2080\u2082]\n    rw [h\u2081\u2083\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2081 : f 349 = 998 := by\n    have h\u2081\u2084\u2081\u2081 : f 349 = f (f (349 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2081\u2081]\n    have h\u2081\u2084\u2081\u2082 : f (349 + 5) = f 354 := by norm_num\n    rw [h\u2081\u2084\u2081\u2082]\n    rw [h\u2081\u2084\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2082 : f 344 = 997 := by\n    have h\u2081\u2084\u2082\u2081 : f 344 = f (f (344 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2082\u2081]\n    have h\u2081\u2084\u2082\u2082 : f (344 + 5) = f 349 := by norm_num\n    rw [h\u2081\u2084\u2082\u2082]\n    rw [h\u2081\u2084\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2083 : f 339 = 998 := by\n    have h\u2081\u2084\u2083\u2081 : f 339 = f (f (339 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2083\u2081]\n    have h\u2081\u2084\u2083\u2082 : f (339 + 5) = f 344 := by norm_num\n    rw [h\u2081\u2084\u2083\u2082]\n    rw [h\u2081\u2084\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2084 : f 334 = 997 := by\n    have h\u2081\u2084\u2084\u2081 : f 334 = f (f (334 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2084\u2081]\n    have h\u2081\u2084\u2084\u2082 : f (334 + 5) = f 339 := by norm_num\n    rw [h\u2081\u2084\u2084\u2082]\n    rw [h\u2081\u2084\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2085 : f 329 = 998 := by\n    have h\u2081\u2084\u2085\u2081 : f 329 = f (f (329 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2085\u2081]\n    have h\u2081\u2084\u2085\u2082 : f (329 + 5) = f 334 := by norm_num\n    rw [h\u2081\u2084\u2085\u2082]\n    rw [h\u2081\u2084\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2086 : f 324 = 997 := by\n    have h\u2081\u2084\u2086\u2081 : f 324 = f (f (324 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2086\u2081]\n    have h\u2081\u2084\u2086\u2082 : f (324 + 5) = f 329 := by norm_num\n    rw [h\u2081\u2084\u2086\u2082]\n    rw [h\u2081\u2084\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2087 : f 319 = 998 := by\n    have h\u2081\u2084\u2087\u2081 : f 319 = f (f (319 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2087\u2081]\n    have h\u2081\u2084\u2087\u2082 : f (319 + 5) = f 324 := by norm_num\n    rw [h\u2081\u2084\u2087\u2082]\n    rw [h\u2081\u2084\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2088 : f 314 = 997 := by\n    have h\u2081\u2084\u2088\u2081 : f 314 = f (f (314 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2088\u2081]\n    have h\u2081\u2084\u2088\u2082 : f (314 + 5) = f 319 := by norm_num\n    rw [h\u2081\u2084\u2088\u2082]\n    rw [h\u2081\u2084\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2089 : f 309 = 998 := by\n    have h\u2081\u2084\u2089\u2081 : f 309 = f (f (309 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2089\u2081]\n    have h\u2081\u2084\u2089\u2082 : f (309 + 5) = f 314 := by norm_num\n    rw [h\u2081\u2084\u2089\u2082]\n    rw [h\u2081\u2084\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2080 : f 304 = 997 := by\n    have h\u2081\u2085\u2080\u2081 : f 304 = f (f (304 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2080\u2081]\n    have h\u2081\u2085\u2080\u2082 : f (304 + 5) = f 309 := by norm_num\n    rw [h\u2081\u2085\u2080\u2082]\n    rw [h\u2081\u2084\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2081 : f 299 = 998 := by\n    have h\u2081\u2085\u2081\u2081 : f 299 = f (f (299 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2081\u2081]\n    have h\u2081\u2085\u2081\u2082 : f (299 + 5) = f 304 := by norm_num\n    rw [h\u2081\u2085\u2081\u2082]\n    rw [h\u2081\u2085\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2082 : f 294 = 997 := by\n    have h\u2081\u2085\u2082\u2081 : f 294 = f (f (294 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2082\u2081]\n    have h\u2081\u2085\u2082\u2082 : f (294 + 5) = f 299 := by norm_num\n    rw [h\u2081\u2085\u2082\u2082]\n    rw [h\u2081\u2085\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2083 : f 289 = 998 := by\n    have h\u2081\u2085\u2083\u2081 : f 289 = f (f (289 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2083\u2081]\n    have h\u2081\u2085\u2083\u2082 : f (289 + 5) = f 294 := by norm_num\n    rw [h\u2081\u2085\u2083\u2082]\n    rw [h\u2081\u2085\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2084 : f 284 = 997 := by\n    have h\u2081\u2085\u2084\u2081 : f 284 = f (f (284 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2084\u2081]\n    have h\u2081\u2085\u2084\u2082 : f (284 + 5) = f 289 := by norm_num\n    rw [h\u2081\u2085\u2084\u2082]\n    rw [h\u2081\u2085\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2085 : f 279 = 998 := by\n    have h\u2081\u2085\u2085\u2081 : f 279 = f (f (279 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2085\u2081]\n    have h\u2081\u2085\u2085\u2082 : f (279 + 5) = f 284 := by norm_num\n    rw [h\u2081\u2085\u2085\u2082]\n    rw [h\u2081\u2085\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2086 : f 274 = 997 := by\n    have h\u2081\u2085\u2086\u2081 : f 274 = f (f (274 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2086\u2081]\n    have h\u2081\u2085\u2086\u2082 : f (274 + 5) = f 279 := by norm_num\n    rw [h\u2081\u2085\u2086\u2082]\n    rw [h\u2081\u2085\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2087 : f 269 = 998 := by\n    have h\u2081\u2085\u2087\u2081 : f 269 = f (f (269 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2087\u2081]\n    have h\u2081\u2085\u2087\u2082 : f (269 + 5) = f 274 := by norm_num\n    rw [h\u2081\u2085\u2087\u2082]\n    rw [h\u2081\u2085\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2088 : f 264 = 997 := by\n    have h\u2081\u2085\u2088\u2081 : f 264 = f (f (264 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2088\u2081]\n    have h\u2081\u2085\u2088\u2082 : f (264 + 5) = f 269 := by norm_num\n    rw [h\u2081\u2085\u2088\u2082]\n    rw [h\u2081\u2085\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2089 : f 259 = 998 := by\n    have h\u2081\u2085\u2089\u2081 : f 259 = f (f (259 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2089\u2081]\n    have h\u2081\u2085\u2089\u2082 : f (259 + 5) = f 264 := by norm_num\n    rw [h\u2081\u2085\u2089\u2082]\n    rw [h\u2081\u2085\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2080 : f 254 = 997 := by\n    have h\u2081\u2086\u2080\u2081 : f 254 = f (f (254 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2080\u2081]\n    have h\u2081\u2086\u2080\u2082 : f (254 + 5) = f 259 := by norm_num\n    rw [h\u2081\u2086\u2080\u2082]\n    rw [h\u2081\u2085\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2081 : f 249 = 998 := by\n    have h\u2081\u2086\u2081\u2081 : f 249 = f (f (249 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2081\u2081]\n    have h\u2081\u2086\u2081\u2082 : f (249 + 5) = f 254 := by norm_num\n    rw [h\u2081\u2086\u2081\u2082]\n    rw [h\u2081\u2086\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2082 : f 244 = 997 := by\n    have h\u2081\u2086\u2082\u2081 : f 244 = f (f (244 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2082\u2081]\n    have h\u2081\u2086\u2082\u2082 : f (244 + 5) = f 249 := by norm_num\n    rw [h\u2081\u2086\u2082\u2082]\n    rw [h\u2081\u2086\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2083 : f 239 = 998 := by\n    have h\u2081\u2086\u2083\u2081 : f 239 = f (f (239 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2083\u2081]\n    have h\u2081\u2086\u2083\u2082 : f (239 + 5) = f 244 := by norm_num\n    rw [h\u2081\u2086\u2083\u2082]\n    rw [h\u2081\u2086\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2084 : f 234 = 997 := by\n    have h\u2081\u2086\u2084\u2081 : f 234 = f (f (234 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2084\u2081]\n    have h\u2081\u2086\u2084\u2082 : f (234 + 5) = f 239 := by norm_num\n    rw [h\u2081\u2086\u2084\u2082]\n    rw [h\u2081\u2086\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2085 : f 229 = 998 := by\n    have h\u2081\u2086\u2085\u2081 : f 229 = f (f (229 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2085\u2081]\n    have h\u2081\u2086\u2085\u2082 : f (229 + 5) = f 234 := by norm_num\n    rw [h\u2081\u2086\u2085\u2082]\n    rw [h\u2081\u2086\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2086 : f 224 = 997 := by\n    have h\u2081\u2086\u2086\u2081 : f 224 = f (f (224 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2086\u2081]\n    have h\u2081\u2086\u2086\u2082 : f (224 + 5) = f 229 := by norm_num\n    rw [h\u2081\u2086\u2086\u2082]\n    rw [h\u2081\u2086\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2087 : f 219 = 998 := by\n    have h\u2081\u2086\u2087\u2081 : f 219 = f (f (219 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2087\u2081]\n    have h\u2081\u2086\u2087\u2082 : f (219 + 5) = f 224 := by norm_num\n    rw [h\u2081\u2086\u2087\u2082]\n    rw [h\u2081\u2086\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2088 : f 214 = 997 := by\n    have h\u2081\u2086\u2088\u2081 : f 214 = f (f (214 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2088\u2081]\n    have h\u2081\u2086\u2088\u2082 : f (214 + 5) = f 219 := by norm_num\n    rw [h\u2081\u2086\u2088\u2082]\n    rw [h\u2081\u2086\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2089 : f 209 = 998 := by\n    have h\u2081\u2086\u2089\u2081 : f 209 = f (f (209 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2089\u2081]\n    have h\u2081\u2086\u2089\u2082 : f (209 + 5) = f 214 := by norm_num\n    rw [h\u2081\u2086\u2089\u2082]\n    rw [h\u2081\u2086\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2080 : f 204 = 997 := by\n    have h\u2081\u2087\u2080\u2081 : f 204 = f (f (204 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2080\u2081]\n    have h\u2081\u2087\u2080\u2082 : f (204 + 5) = f 209 := by norm_num\n    rw [h\u2081\u2087\u2080\u2082]\n    rw [h\u2081\u2086\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2081 : f 199 = 998 := by\n    have h\u2081\u2087\u2081\u2081 : f 199 = f (f (199 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2081\u2081]\n    have h\u2081\u2087\u2081\u2082 : f (199 + 5) = f 204 := by norm_num\n    rw [h\u2081\u2087\u2081\u2082]\n    rw [h\u2081\u2087\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2082 : f 194 = 997 := by\n    have h\u2081\u2087\u2082\u2081 : f 194 = f (f (194 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2082\u2081]\n    have h\u2081\u2087\u2082\u2082 : f (194 + 5) = f 199 := by norm_num\n    rw [h\u2081\u2087\u2082\u2082]\n    rw [h\u2081\u2087\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2083 : f 189 = 998 := by\n    have h\u2081\u2087\u2083\u2081 : f 189 = f (f (189 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2083\u2081]\n    have h\u2081\u2087\u2083\u2082 : f (189 + 5) = f 194 := by norm_num\n    rw [h\u2081\u2087\u2083\u2082]\n    rw [h\u2081\u2087\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2084 : f 184 = 997 := by\n    have h\u2081\u2087\u2084\u2081 : f 184 = f (f (184 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2084\u2081]\n    have h\u2081\u2087\u2084\u2082 : f (184 + 5) = f 189 := by norm_num\n    rw [h\u2081\u2087\u2084\u2082]\n    rw [h\u2081\u2087\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2085 : f 179 = 998 := by\n    have h\u2081\u2087\u2085\u2081 : f 179 = f (f (179 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2085\u2081]\n    have h\u2081\u2087\u2085\u2082 : f (179 + 5) = f 184 := by norm_num\n    rw [h\u2081\u2087\u2085\u2082]\n    rw [h\u2081\u2087\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2086 : f 174 = 997 := by\n    have h\u2081\u2087\u2086\u2081 : f 174 = f (f (174 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2086\u2081]\n    have h\u2081\u2087\u2086\u2082 : f (174 + 5) = f 179 := by norm_num\n    rw [h\u2081\u2087\u2086\u2082]\n    rw [h\u2081\u2087\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2087 : f 169 = 998 := by\n    have h\u2081\u2087\u2087\u2081 : f 169 = f (f (169 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2087\u2081]\n    have h\u2081\u2087\u2087\u2082 : f (169 + 5) = f 174 := by norm_num\n    rw [h\u2081\u2087\u2087\u2082]\n    rw [h\u2081\u2087\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2088 : f 164 = 997 := by\n    have h\u2081\u2087\u2088\u2081 : f 164 = f (f (164 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2088\u2081]\n    have h\u2081\u2087\u2088\u2082 : f (164 + 5) = f 169 := by norm_num\n    rw [h\u2081\u2087\u2088\u2082]\n    rw [h\u2081\u2087\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2089 : f 159 = 998 := by\n    have h\u2081\u2087\u2089\u2081 : f 159 = f (f (159 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2089\u2081]\n    have h\u2081\u2087\u2089\u2082 : f (159 + 5) = f 164 := by norm_num\n    rw [h\u2081\u2087\u2089\u2082]\n    rw [h\u2081\u2087\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2080 : f 154 = 997 := by\n    have h\u2081\u2088\u2080\u2081 : f 154 = f (f (154 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2080\u2081]\n    have h\u2081\u2088\u2080\u2082 : f (154 + 5) = f 159 := by norm_num\n    rw [h\u2081\u2088\u2080\u2082]\n    rw [h\u2081\u2087\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2081 : f 149 = 998 := by\n    have h\u2081\u2088\u2081\u2081 : f 149 = f (f (149 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2081\u2081]\n    have h\u2081\u2088\u2081\u2082 : f (149 + 5) = f 154 := by norm_num\n    rw [h\u2081\u2088\u2081\u2082]\n    rw [h\u2081\u2088\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2082 : f 144 = 997 := by\n    have h\u2081\u2088\u2082\u2081 : f 144 = f (f (144 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2082\u2081]\n    have h\u2081\u2088\u2082\u2082 : f (144 + 5) = f 149 := by norm_num\n    rw [h\u2081\u2088\u2082\u2082]\n    rw [h\u2081\u2088\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2083 : f 139 = 998 := by\n    have h\u2081\u2088\u2083\u2081 : f 139 = f (f (139 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2083\u2081]\n    have h\u2081\u2088\u2083\u2082 : f (139 + 5) = f 144 := by norm_num\n    rw [h\u2081\u2088\u2083\u2082]\n    rw [h\u2081\u2088\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2084 : f 134 = 997 := by\n    have h\u2081\u2088\u2084\u2081 : f 134 = f (f (134 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2084\u2081]\n    have h\u2081\u2088\u2084\u2082 : f (134 + 5) = f 139 := by norm_num\n    rw [h\u2081\u2088\u2084\u2082]\n    rw [h\u2081\u2088\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2085 : f 129 = 998 := by\n    have h\u2081\u2088\u2085\u2081 : f 129 = f (f (129 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2085\u2081]\n    have h\u2081\u2088\u2085\u2082 : f (129 + 5) = f 134 := by norm_num\n    rw [h\u2081\u2088\u2085\u2082]\n    rw [h\u2081\u2088\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2086 : f 124 = 997 := by\n    have h\u2081\u2088\u2086\u2081 : f 124 = f (f (124 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2086\u2081]\n    have h\u2081\u2088\u2086\u2082 : f (124 + 5) = f 129 := by norm_num\n    rw [h\u2081\u2088\u2086\u2082]\n    rw [h\u2081\u2088\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2087 : f 119 = 998 := by\n    have h\u2081\u2088\u2087\u2081 : f 119 = f (f (119 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2087\u2081]\n    have h\u2081\u2088\u2087\u2082 : f (119 + 5) = f 124 := by norm_num\n    rw [h\u2081\u2088\u2087\u2082]\n    rw [h\u2081\u2088\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2088 : f 114 = 997 := by\n    have h\u2081\u2088\u2088\u2081 : f 114 = f (f (114 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2088\u2081]\n    have h\u2081\u2088\u2088\u2082 : f (114 + 5) = f 119 := by norm_num\n    rw [h\u2081\u2088\u2088\u2082]\n    rw [h\u2081\u2088\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2089 : f 109 = 998 := by\n    have h\u2081\u2088\u2089\u2081 : f 109 = f (f (109 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2089\u2081]\n    have h\u2081\u2088\u2089\u2082 : f (109 + 5) = f 114 := by norm_num\n    rw [h\u2081\u2088\u2089\u2082]\n    rw [h\u2081\u2088\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089\u2080 : f 104 = 997 := by\n    have h\u2081\u2089\u2080\u2081 : f 104 = f (f (104 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2080\u2081]\n    have h\u2081\u2089\u2080\u2082 : f (104 + 5) = f 109 := by norm_num\n    rw [h\u2081\u2089\u2080\u2082]\n    rw [h\u2081\u2088\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089\u2081 : f 99 = 998 := by\n    have h\u2081\u2089\u2081\u2081 : f 99 = f (f (99 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2081\u2081]\n    have h\u2081\u2089\u2081\u2082 : f (99 + 5) = f 104 := by norm_num\n    rw [h\u2081\u2089\u2081\u2082]\n    rw [h\u2081\u2089\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089\u2082 : f 94 = 997 := by\n    have h\u2081\u2089\u2082\u2081 : f 94 = f (f (94 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2082\u2081]\n    have h\u2081\u2089\u2082\u2082 : f (94 + 5) = f 99 := by norm_num\n    rw [h\u2081\u2089\u2082\u2082]\n    rw [h\u2081\u2089\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089\u2083 : f 89 = 998 := by\n    have h\u2081\u2089\u2083\u2081 : f 89 = f (f (89 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2083\u2081]\n    have h\u2081\u2089\u2083\u2082 : f (89 + 5) = f 94 := by norm_num\n    rw [h\u2081\u2089\u2083\u2082]\n    rw [h\u2081\u2089\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089\u2084 : f 84 = 997 := by\n    have h\u2081\u2089\u2084\u2081 : f 84 = f (f (84 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2084\u2081]\n    have h\u2081\u2089\u2084\u2082 : f (84 + 5) = f 89 := by norm_num\n    rw [h\u2081\u2089\u2084\u2082]\n    rw [h\u2081\u2089\u2083]\n    <;> norm_num\n    <;> linarith\n  rw [h\u2081\u2089\u2084]\n  <;> norm_num\n", "proof_before": "theorem aime_1984_p7 (f : \u2124 \u2192 \u2124) (h\u2080 : \u2200 n, 1000 \u2264 n \u2192 f n = n - 3)\n    (h\u2081 : \u2200 n, n < 1000 \u2192 f n = f (f (n + 5))) : f 84 = 997  :=  by\n  have h\u2082 : f 1004 = 1001 := by\n    have h\u2082\u2081 : f 1004 = 1004 - 3 := by\n      apply h\u2080\n      <;> norm_num\n    rw [h\u2082\u2081]\n    <;> norm_num\n  have h\u2083 : f 1003 = 1000 := by\n    have h\u2083\u2081 : f 1003 = 1003 - 3 := by\n      apply h\u2080\n      <;> norm_num\n    rw [h\u2083\u2081]\n    <;> norm_num\n  have h\u2084 : f 1002 = 999 := by\n    have h\u2084\u2081 : f 1002 = 1002 - 3 := by\n      apply h\u2080\n      <;> norm_num\n    rw [h\u2084\u2081]\n    <;> norm_num\n  have h\u2085 : f 1001 = 998 := by\n    have h\u2085\u2081 : f 1001 = 1001 - 3 := by\n      apply h\u2080\n      <;> norm_num\n    rw [h\u2085\u2081]\n    <;> norm_num\n  have h\u2086 : f 1000 = 997 := by\n    have h\u2086\u2081 : f 1000 = 1000 - 3 := by\n      apply h\u2080\n      <;> norm_num\n    rw [h\u2086\u2081]\n    <;> norm_num\n  have h\u2087 : f 999 = 998 := by\n    have h\u2087\u2081 : f 999 = f (f (999 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2081]\n    have h\u2087\u2082 : f (999 + 5) = f 1004 := by norm_num\n    rw [h\u2087\u2082]\n    rw [h\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2088 : f 998 = 997 := by\n    have h\u2088\u2081 : f 998 = f (f (998 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2081]\n    have h\u2088\u2082 : f (998 + 5) = f 1003 := by norm_num\n    rw [h\u2088\u2082]\n    rw [h\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2089 : f 997 = 998 := by\n    have h\u2089\u2081 : f 997 = f (f (997 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2081]\n    have h\u2089\u2082 : f (997 + 5) = f 1002 := by norm_num\n    rw [h\u2089\u2082]\n    rw [h\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080 : f 996 = 997 := by\n    have h\u2081\u2080\u2081 : f 996 = f (f (996 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2081]\n    have h\u2081\u2080\u2082 : f (996 + 5) = f 1001 := by norm_num\n    rw [h\u2081\u2080\u2082]\n    rw [h\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081 : f 995 = 998 := by\n    have h\u2081\u2081\u2081 : f 995 = f (f (995 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2081]\n    have h\u2081\u2081\u2082 : f (995 + 5) = f 1000 := by norm_num\n    rw [h\u2081\u2081\u2082]\n    rw [h\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082 : f 994 = 997 := by\n    have h\u2081\u2082\u2081 : f 994 = f (f (994 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2081]\n    have h\u2081\u2082\u2082 : f (994 + 5) = f 999 := by norm_num\n    rw [h\u2081\u2082\u2082]\n    rw [h\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083 : f 989 = 998 := by\n    have h\u2081\u2083\u2081 : f 989 = f (f (989 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2081]\n    have h\u2081\u2083\u2082 : f (989 + 5) = f 994 := by norm_num\n    rw [h\u2081\u2083\u2082]\n    rw [h\u2081\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084 : f 984 = 997 := by\n    have h\u2081\u2084\u2081 : f 984 = f (f (984 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2081]\n    have h\u2081\u2084\u2082 : f (984 + 5) = f 989 := by norm_num\n    rw [h\u2081\u2084\u2082]\n    rw [h\u2081\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085 : f 979 = 998 := by\n    have h\u2081\u2085\u2081 : f 979 = f (f (979 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2081]\n    have h\u2081\u2085\u2082 : f (979 + 5) = f 984 := by norm_num\n    rw [h\u2081\u2085\u2082]\n    rw [h\u2081\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086 : f 974 = 997 := by\n    have h\u2081\u2086\u2081 : f 974 = f (f (974 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2081]\n    have h\u2081\u2086\u2082 : f (974 + 5) = f 979 := by norm_num\n    rw [h\u2081\u2086\u2082]\n    rw [h\u2081\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087 : f 969 = 998 := by\n    have h\u2081\u2087\u2081 : f 969 = f (f (969 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2081]\n    have h\u2081\u2087\u2082 : f (969 + 5) = f 974 := by norm_num\n    rw [h\u2081\u2087\u2082]\n    rw [h\u2081\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088 : f 964 = 997 := by\n    have h\u2081\u2088\u2081 : f 964 = f (f (964 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2081]\n    have h\u2081\u2088\u2082 : f (964 + 5) = f 969 := by norm_num\n    rw [h\u2081\u2088\u2082]\n    rw [h\u2081\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089 : f 959 = 998 := by\n    have h\u2081\u2089\u2081 : f 959 = f (f (959 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2081]\n    have h\u2081\u2089\u2082 : f (959 + 5) = f 964 := by norm_num\n    rw [h\u2081\u2089\u2082]\n    rw [h\u2081\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2080 : f 954 = 997 := by\n    have h\u2082\u2080\u2081 : f 954 = f (f (954 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2080\u2081]\n    have h\u2082\u2080\u2082 : f (954 + 5) = f 959 := by norm_num\n    rw [h\u2082\u2080\u2082]\n    rw [h\u2081\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2081 : f 949 = 998 := by\n    have h\u2082\u2081\u2081 : f 949 = f (f (949 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2081\u2081]\n    have h\u2082\u2081\u2082 : f (949 + 5) = f 954 := by norm_num\n    rw [h\u2082\u2081\u2082]\n    rw [h\u2082\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2082 : f 944 = 997 := by\n    have h\u2082\u2082\u2081 : f 944 = f (f (944 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2082\u2081]\n    have h\u2082\u2082\u2082 : f (944 + 5) = f 949 := by norm_num\n    rw [h\u2082\u2082\u2082]\n    rw [h\u2082\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2083 : f 939 = 998 := by\n    have h\u2082\u2083\u2081 : f 939 = f (f (939 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2083\u2081]\n    have h\u2082\u2083\u2082 : f (939 + 5) = f 944 := by norm_num\n    rw [h\u2082\u2083\u2082]\n    rw [h\u2082\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2084 : f 934 = 997 := by\n    have h\u2082\u2084\u2081 : f 934 = f (f (934 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2084\u2081]\n    have h\u2082\u2084\u2082 : f (934 + 5) = f 939 := by norm_num\n    rw [h\u2082\u2084\u2082]\n    rw [h\u2082\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2085 : f 929 = 998 := by\n    have h\u2082\u2085\u2081 : f 929 = f (f (929 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2085\u2081]\n    have h\u2082\u2085\u2082 : f (929 + 5) = f 934 := by norm_num\n    rw [h\u2082\u2085\u2082]\n    rw [h\u2082\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2086 : f 924 = 997 := by\n    have h\u2082\u2086\u2081 : f 924 = f (f (924 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2086\u2081]\n    have h\u2082\u2086\u2082 : f (924 + 5) = f 929 := by norm_num\n    rw [h\u2082\u2086\u2082]\n    rw [h\u2082\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2087 : f 919 = 998 := by\n    have h\u2082\u2087\u2081 : f 919 = f (f (919 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2087\u2081]\n    have h\u2082\u2087\u2082 : f (919 + 5) = f 924 := by norm_num\n    rw [h\u2082\u2087\u2082]\n    rw [h\u2082\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2088 : f 914 = 997 := by\n    have h\u2082\u2088\u2081 : f 914 = f (f (914 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2088\u2081]\n    have h\u2082\u2088\u2082 : f (914 + 5) = f 919 := by norm_num\n    rw [h\u2082\u2088\u2082]\n    rw [h\u2082\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2082\u2089 : f 909 = 998 := by\n    have h\u2082\u2089\u2081 : f 909 = f (f (909 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2082\u2089\u2081]\n    have h\u2082\u2089\u2082 : f (909 + 5) = f 914 := by norm_num\n    rw [h\u2082\u2089\u2082]\n    rw [h\u2082\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2080 : f 904 = 997 := by\n    have h\u2083\u2080\u2081 : f 904 = f (f (904 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2080\u2081]\n    have h\u2083\u2080\u2082 : f (904 + 5) = f 909 := by norm_num\n    rw [h\u2083\u2080\u2082]\n    rw [h\u2082\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2081 : f 899 = 998 := by\n    have h\u2083\u2081\u2081 : f 899 = f (f (899 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2081\u2081]\n    have h\u2083\u2081\u2082 : f (899 + 5) = f 904 := by norm_num\n    rw [h\u2083\u2081\u2082]\n    rw [h\u2083\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2082 : f 894 = 997 := by\n    have h\u2083\u2082\u2081 : f 894 = f (f (894 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2082\u2081]\n    have h\u2083\u2082\u2082 : f (894 + 5) = f 899 := by norm_num\n    rw [h\u2083\u2082\u2082]\n    rw [h\u2083\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2083 : f 889 = 998 := by\n    have h\u2083\u2083\u2081 : f 889 = f (f (889 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2083\u2081]\n    have h\u2083\u2083\u2082 : f (889 + 5) = f 894 := by norm_num\n    rw [h\u2083\u2083\u2082]\n    rw [h\u2083\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2084 : f 884 = 997 := by\n    have h\u2083\u2084\u2081 : f 884 = f (f (884 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2084\u2081]\n    have h\u2083\u2084\u2082 : f (884 + 5) = f 889 := by norm_num\n    rw [h\u2083\u2084\u2082]\n    rw [h\u2083\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2085 : f 879 = 998 := by\n    have h\u2083\u2085\u2081 : f 879 = f (f (879 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2085\u2081]\n    have h\u2083\u2085\u2082 : f (879 + 5) = f 884 := by norm_num\n    rw [h\u2083\u2085\u2082]\n    rw [h\u2083\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2086 : f 874 = 997 := by\n    have h\u2083\u2086\u2081 : f 874 = f (f (874 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2086\u2081]\n    have h\u2083\u2086\u2082 : f (874 + 5) = f 879 := by norm_num\n    rw [h\u2083\u2086\u2082]\n    rw [h\u2083\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2087 : f 869 = 998 := by\n    have h\u2083\u2087\u2081 : f 869 = f (f (869 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2087\u2081]\n    have h\u2083\u2087\u2082 : f (869 + 5) = f 874 := by norm_num\n    rw [h\u2083\u2087\u2082]\n    rw [h\u2083\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2088 : f 864 = 997 := by\n    have h\u2083\u2088\u2081 : f 864 = f (f (864 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2088\u2081]\n    have h\u2083\u2088\u2082 : f (864 + 5) = f 869 := by norm_num\n    rw [h\u2083\u2088\u2082]\n    rw [h\u2083\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2083\u2089 : f 859 = 998 := by\n    have h\u2083\u2089\u2081 : f 859 = f (f (859 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2083\u2089\u2081]\n    have h\u2083\u2089\u2082 : f (859 + 5) = f 864 := by norm_num\n    rw [h\u2083\u2089\u2082]\n    rw [h\u2083\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2080 : f 854 = 997 := by\n    have h\u2084\u2080\u2081 : f 854 = f (f (854 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2080\u2081]\n    have h\u2084\u2080\u2082 : f (854 + 5) = f 859 := by norm_num\n    rw [h\u2084\u2080\u2082]\n    rw [h\u2083\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2081 : f 849 = 998 := by\n    have h\u2084\u2081\u2081 : f 849 = f (f (849 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2081\u2081]\n    have h\u2084\u2081\u2082 : f (849 + 5) = f 854 := by norm_num\n    rw [h\u2084\u2081\u2082]\n    rw [h\u2084\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2082 : f 844 = 997 := by\n    have h\u2084\u2082\u2081 : f 844 = f (f (844 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2082\u2081]\n    have h\u2084\u2082\u2082 : f (844 + 5) = f 849 := by norm_num\n    rw [h\u2084\u2082\u2082]\n    rw [h\u2084\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2083 : f 839 = 998 := by\n    have h\u2084\u2083\u2081 : f 839 = f (f (839 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2083\u2081]\n    have h\u2084\u2083\u2082 : f (839 + 5) = f 844 := by norm_num\n    rw [h\u2084\u2083\u2082]\n    rw [h\u2084\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2084 : f 834 = 997 := by\n    have h\u2084\u2084\u2081 : f 834 = f (f (834 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2084\u2081]\n    have h\u2084\u2084\u2082 : f (834 + 5) = f 839 := by norm_num\n    rw [h\u2084\u2084\u2082]\n    rw [h\u2084\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2085 : f 829 = 998 := by\n    have h\u2084\u2085\u2081 : f 829 = f (f (829 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2085\u2081]\n    have h\u2084\u2085\u2082 : f (829 + 5) = f 834 := by norm_num\n    rw [h\u2084\u2085\u2082]\n    rw [h\u2084\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2086 : f 824 = 997 := by\n    have h\u2084\u2086\u2081 : f 824 = f (f (824 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2086\u2081]\n    have h\u2084\u2086\u2082 : f (824 + 5) = f 829 := by norm_num\n    rw [h\u2084\u2086\u2082]\n    rw [h\u2084\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2087 : f 819 = 998 := by\n    have h\u2084\u2087\u2081 : f 819 = f (f (819 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2087\u2081]\n    have h\u2084\u2087\u2082 : f (819 + 5) = f 824 := by norm_num\n    rw [h\u2084\u2087\u2082]\n    rw [h\u2084\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2088 : f 814 = 997 := by\n    have h\u2084\u2088\u2081 : f 814 = f (f (814 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2088\u2081]\n    have h\u2084\u2088\u2082 : f (814 + 5) = f 819 := by norm_num\n    rw [h\u2084\u2088\u2082]\n    rw [h\u2084\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2084\u2089 : f 809 = 998 := by\n    have h\u2084\u2089\u2081 : f 809 = f (f (809 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2084\u2089\u2081]\n    have h\u2084\u2089\u2082 : f (809 + 5) = f 814 := by norm_num\n    rw [h\u2084\u2089\u2082]\n    rw [h\u2084\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2080 : f 804 = 997 := by\n    have h\u2085\u2080\u2081 : f 804 = f (f (804 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2080\u2081]\n    have h\u2085\u2080\u2082 : f (804 + 5) = f 809 := by norm_num\n    rw [h\u2085\u2080\u2082]\n    rw [h\u2084\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2081 : f 799 = 998 := by\n    have h\u2085\u2081\u2081 : f 799 = f (f (799 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2081\u2081]\n    have h\u2085\u2081\u2082 : f (799 + 5) = f 804 := by norm_num\n    rw [h\u2085\u2081\u2082]\n    rw [h\u2085\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2082 : f 794 = 997 := by\n    have h\u2085\u2082\u2081 : f 794 = f (f (794 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2082\u2081]\n    have h\u2085\u2082\u2082 : f (794 + 5) = f 799 := by norm_num\n    rw [h\u2085\u2082\u2082]\n    rw [h\u2085\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2083 : f 789 = 998 := by\n    have h\u2085\u2083\u2081 : f 789 = f (f (789 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2083\u2081]\n    have h\u2085\u2083\u2082 : f (789 + 5) = f 794 := by norm_num\n    rw [h\u2085\u2083\u2082]\n    rw [h\u2085\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2084 : f 784 = 997 := by\n    have h\u2085\u2084\u2081 : f 784 = f (f (784 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2084\u2081]\n    have h\u2085\u2084\u2082 : f (784 + 5) = f 789 := by norm_num\n    rw [h\u2085\u2084\u2082]\n    rw [h\u2085\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2085 : f 779 = 998 := by\n    have h\u2085\u2085\u2081 : f 779 = f (f (779 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2085\u2081]\n    have h\u2085\u2085\u2082 : f (779 + 5) = f 784 := by norm_num\n    rw [h\u2085\u2085\u2082]\n    rw [h\u2085\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2086 : f 774 = 997 := by\n    have h\u2085\u2086\u2081 : f 774 = f (f (774 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2086\u2081]\n    have h\u2085\u2086\u2082 : f (774 + 5) = f 779 := by norm_num\n    rw [h\u2085\u2086\u2082]\n    rw [h\u2085\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2087 : f 769 = 998 := by\n    have h\u2085\u2087\u2081 : f 769 = f (f (769 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2087\u2081]\n    have h\u2085\u2087\u2082 : f (769 + 5) = f 774 := by norm_num\n    rw [h\u2085\u2087\u2082]\n    rw [h\u2085\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2088 : f 764 = 997 := by\n    have h\u2085\u2088\u2081 : f 764 = f (f (764 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2088\u2081]\n    have h\u2085\u2088\u2082 : f (764 + 5) = f 769 := by norm_num\n    rw [h\u2085\u2088\u2082]\n    rw [h\u2085\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2085\u2089 : f 759 = 998 := by\n    have h\u2085\u2089\u2081 : f 759 = f (f (759 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2085\u2089\u2081]\n    have h\u2085\u2089\u2082 : f (759 + 5) = f 764 := by norm_num\n    rw [h\u2085\u2089\u2082]\n    rw [h\u2085\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2080 : f 754 = 997 := by\n    have h\u2086\u2080\u2081 : f 754 = f (f (754 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2080\u2081]\n    have h\u2086\u2080\u2082 : f (754 + 5) = f 759 := by norm_num\n    rw [h\u2086\u2080\u2082]\n    rw [h\u2085\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2081 : f 749 = 998 := by\n    have h\u2086\u2081\u2081 : f 749 = f (f (749 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2081\u2081]\n    have h\u2086\u2081\u2082 : f (749 + 5) = f 754 := by norm_num\n    rw [h\u2086\u2081\u2082]\n    rw [h\u2086\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2082 : f 744 = 997 := by\n    have h\u2086\u2082\u2081 : f 744 = f (f (744 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2082\u2081]\n    have h\u2086\u2082\u2082 : f (744 + 5) = f 749 := by norm_num\n    rw [h\u2086\u2082\u2082]\n    rw [h\u2086\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2083 : f 739 = 998 := by\n    have h\u2086\u2083\u2081 : f 739 = f (f (739 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2083\u2081]\n    have h\u2086\u2083\u2082 : f (739 + 5) = f 744 := by norm_num\n    rw [h\u2086\u2083\u2082]\n    rw [h\u2086\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2084 : f 734 = 997 := by\n    have h\u2086\u2084\u2081 : f 734 = f (f (734 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2084\u2081]\n    have h\u2086\u2084\u2082 : f (734 + 5) = f 739 := by norm_num\n    rw [h\u2086\u2084\u2082]\n    rw [h\u2086\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2085 : f 729 = 998 := by\n    have h\u2086\u2085\u2081 : f 729 = f (f (729 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2085\u2081]\n    have h\u2086\u2085\u2082 : f (729 + 5) = f 734 := by norm_num\n    rw [h\u2086\u2085\u2082]\n    rw [h\u2086\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2086 : f 724 = 997 := by\n    have h\u2086\u2086\u2081 : f 724 = f (f (724 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2086\u2081]\n    have h\u2086\u2086\u2082 : f (724 + 5) = f 729 := by norm_num\n    rw [h\u2086\u2086\u2082]\n    rw [h\u2086\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2087 : f 719 = 998 := by\n    have h\u2086\u2087\u2081 : f 719 = f (f (719 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2087\u2081]\n    have h\u2086\u2087\u2082 : f (719 + 5) = f 724 := by norm_num\n    rw [h\u2086\u2087\u2082]\n    rw [h\u2086\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2088 : f 714 = 997 := by\n    have h\u2086\u2088\u2081 : f 714 = f (f (714 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2088\u2081]\n    have h\u2086\u2088\u2082 : f (714 + 5) = f 719 := by norm_num\n    rw [h\u2086\u2088\u2082]\n    rw [h\u2086\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2086\u2089 : f 709 = 998 := by\n    have h\u2086\u2089\u2081 : f 709 = f (f (709 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2086\u2089\u2081]\n    have h\u2086\u2089\u2082 : f (709 + 5) = f 714 := by norm_num\n    rw [h\u2086\u2089\u2082]\n    rw [h\u2086\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2080 : f 704 = 997 := by\n    have h\u2087\u2080\u2081 : f 704 = f (f (704 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2080\u2081]\n    have h\u2087\u2080\u2082 : f (704 + 5) = f 709 := by norm_num\n    rw [h\u2087\u2080\u2082]\n    rw [h\u2086\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2081 : f 699 = 998 := by\n    have h\u2087\u2081\u2081 : f 699 = f (f (699 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2081\u2081]\n    have h\u2087\u2081\u2082 : f (699 + 5) = f 704 := by norm_num\n    rw [h\u2087\u2081\u2082]\n    rw [h\u2087\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2082 : f 694 = 997 := by\n    have h\u2087\u2082\u2081 : f 694 = f (f (694 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2082\u2081]\n    have h\u2087\u2082\u2082 : f (694 + 5) = f 699 := by norm_num\n    rw [h\u2087\u2082\u2082]\n    rw [h\u2087\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2083 : f 689 = 998 := by\n    have h\u2087\u2083\u2081 : f 689 = f (f (689 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2083\u2081]\n    have h\u2087\u2083\u2082 : f (689 + 5) = f 694 := by norm_num\n    rw [h\u2087\u2083\u2082]\n    rw [h\u2087\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2084 : f 684 = 997 := by\n    have h\u2087\u2084\u2081 : f 684 = f (f (684 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2084\u2081]\n    have h\u2087\u2084\u2082 : f (684 + 5) = f 689 := by norm_num\n    rw [h\u2087\u2084\u2082]\n    rw [h\u2087\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2085 : f 679 = 998 := by\n    have h\u2087\u2085\u2081 : f 679 = f (f (679 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2085\u2081]\n    have h\u2087\u2085\u2082 : f (679 + 5) = f 684 := by norm_num\n    rw [h\u2087\u2085\u2082]\n    rw [h\u2087\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2086 : f 674 = 997 := by\n    have h\u2087\u2086\u2081 : f 674 = f (f (674 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2086\u2081]\n    have h\u2087\u2086\u2082 : f (674 + 5) = f 679 := by norm_num\n    rw [h\u2087\u2086\u2082]\n    rw [h\u2087\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2087 : f 669 = 998 := by\n    have h\u2087\u2087\u2081 : f 669 = f (f (669 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2087\u2081]\n    have h\u2087\u2087\u2082 : f (669 + 5) = f 674 := by norm_num\n    rw [h\u2087\u2087\u2082]\n    rw [h\u2087\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2088 : f 664 = 997 := by\n    have h\u2087\u2088\u2081 : f 664 = f (f (664 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2088\u2081]\n    have h\u2087\u2088\u2082 : f (664 + 5) = f 669 := by norm_num\n    rw [h\u2087\u2088\u2082]\n    rw [h\u2087\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2087\u2089 : f 659 = 998 := by\n    have h\u2087\u2089\u2081 : f 659 = f (f (659 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2087\u2089\u2081]\n    have h\u2087\u2089\u2082 : f (659 + 5) = f 664 := by norm_num\n    rw [h\u2087\u2089\u2082]\n    rw [h\u2087\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2080 : f 654 = 997 := by\n    have h\u2088\u2080\u2081 : f 654 = f (f (654 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2080\u2081]\n    have h\u2088\u2080\u2082 : f (654 + 5) = f 659 := by norm_num\n    rw [h\u2088\u2080\u2082]\n    rw [h\u2087\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2081 : f 649 = 998 := by\n    have h\u2088\u2081\u2081 : f 649 = f (f (649 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2081\u2081]\n    have h\u2088\u2081\u2082 : f (649 + 5) = f 654 := by norm_num\n    rw [h\u2088\u2081\u2082]\n    rw [h\u2088\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2082 : f 644 = 997 := by\n    have h\u2088\u2082\u2081 : f 644 = f (f (644 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2082\u2081]\n    have h\u2088\u2082\u2082 : f (644 + 5) = f 649 := by norm_num\n    rw [h\u2088\u2082\u2082]\n    rw [h\u2088\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2083 : f 639 = 998 := by\n    have h\u2088\u2083\u2081 : f 639 = f (f (639 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2083\u2081]\n    have h\u2088\u2083\u2082 : f (639 + 5) = f 644 := by norm_num\n    rw [h\u2088\u2083\u2082]\n    rw [h\u2088\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2084 : f 634 = 997 := by\n    have h\u2088\u2084\u2081 : f 634 = f (f (634 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2084\u2081]\n    have h\u2088\u2084\u2082 : f (634 + 5) = f 639 := by norm_num\n    rw [h\u2088\u2084\u2082]\n    rw [h\u2088\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2085 : f 629 = 998 := by\n    have h\u2088\u2085\u2081 : f 629 = f (f (629 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2085\u2081]\n    have h\u2088\u2085\u2082 : f (629 + 5) = f 634 := by norm_num\n    rw [h\u2088\u2085\u2082]\n    rw [h\u2088\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2086 : f 624 = 997 := by\n    have h\u2088\u2086\u2081 : f 624 = f (f (624 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2086\u2081]\n    have h\u2088\u2086\u2082 : f (624 + 5) = f 629 := by norm_num\n    rw [h\u2088\u2086\u2082]\n    rw [h\u2088\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2087 : f 619 = 998 := by\n    have h\u2088\u2087\u2081 : f 619 = f (f (619 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2087\u2081]\n    have h\u2088\u2087\u2082 : f (619 + 5) = f 624 := by norm_num\n    rw [h\u2088\u2087\u2082]\n    rw [h\u2088\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2088 : f 614 = 997 := by\n    have h\u2088\u2088\u2081 : f 614 = f (f (614 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2088\u2081]\n    have h\u2088\u2088\u2082 : f (614 + 5) = f 619 := by norm_num\n    rw [h\u2088\u2088\u2082]\n    rw [h\u2088\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2088\u2089 : f 609 = 998 := by\n    have h\u2088\u2089\u2081 : f 609 = f (f (609 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2088\u2089\u2081]\n    have h\u2088\u2089\u2082 : f (609 + 5) = f 614 := by norm_num\n    rw [h\u2088\u2089\u2082]\n    rw [h\u2088\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2080 : f 604 = 997 := by\n    have h\u2089\u2080\u2081 : f 604 = f (f (604 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2080\u2081]\n    have h\u2089\u2080\u2082 : f (604 + 5) = f 609 := by norm_num\n    rw [h\u2089\u2080\u2082]\n    rw [h\u2088\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2081 : f 599 = 998 := by\n    have h\u2089\u2081\u2081 : f 599 = f (f (599 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2081\u2081]\n    have h\u2089\u2081\u2082 : f (599 + 5) = f 604 := by norm_num\n    rw [h\u2089\u2081\u2082]\n    rw [h\u2089\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2082 : f 594 = 997 := by\n    have h\u2089\u2082\u2081 : f 594 = f (f (594 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2082\u2081]\n    have h\u2089\u2082\u2082 : f (594 + 5) = f 599 := by norm_num\n    rw [h\u2089\u2082\u2082]\n    rw [h\u2089\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2083 : f 589 = 998 := by\n    have h\u2089\u2083\u2081 : f 589 = f (f (589 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2083\u2081]\n    have h\u2089\u2083\u2082 : f (589 + 5) = f 594 := by norm_num\n    rw [h\u2089\u2083\u2082]\n    rw [h\u2089\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2084 : f 584 = 997 := by\n    have h\u2089\u2084\u2081 : f 584 = f (f (584 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2084\u2081]\n    have h\u2089\u2084\u2082 : f (584 + 5) = f 589 := by norm_num\n    rw [h\u2089\u2084\u2082]\n    rw [h\u2089\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2085 : f 579 = 998 := by\n    have h\u2089\u2085\u2081 : f 579 = f (f (579 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2085\u2081]\n    have h\u2089\u2085\u2082 : f (579 + 5) = f 584 := by norm_num\n    rw [h\u2089\u2085\u2082]\n    rw [h\u2089\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2086 : f 574 = 997 := by\n    have h\u2089\u2086\u2081 : f 574 = f (f (574 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2086\u2081]\n    have h\u2089\u2086\u2082 : f (574 + 5) = f 579 := by norm_num\n    rw [h\u2089\u2086\u2082]\n    rw [h\u2089\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2087 : f 569 = 998 := by\n    have h\u2089\u2087\u2081 : f 569 = f (f (569 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2087\u2081]\n    have h\u2089\u2087\u2082 : f (569 + 5) = f 574 := by norm_num\n    rw [h\u2089\u2087\u2082]\n    rw [h\u2089\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2088 : f 564 = 997 := by\n    have h\u2089\u2088\u2081 : f 564 = f (f (564 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2088\u2081]\n    have h\u2089\u2088\u2082 : f (564 + 5) = f 569 := by norm_num\n    rw [h\u2089\u2088\u2082]\n    rw [h\u2089\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2089\u2089 : f 559 = 998 := by\n    have h\u2089\u2089\u2081 : f 559 = f (f (559 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2089\u2089\u2081]\n    have h\u2089\u2089\u2082 : f (559 + 5) = f 564 := by norm_num\n    rw [h\u2089\u2089\u2082]\n    rw [h\u2089\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2080 : f 554 = 997 := by\n    have h\u2081\u2080\u2080\u2081 : f 554 = f (f (554 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2080\u2081]\n    have h\u2081\u2080\u2080\u2082 : f (554 + 5) = f 559 := by norm_num\n    rw [h\u2081\u2080\u2080\u2082]\n    rw [h\u2089\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2081 : f 549 = 998 := by\n    have h\u2081\u2080\u2081\u2081 : f 549 = f (f (549 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2081\u2081]\n    have h\u2081\u2080\u2081\u2082 : f (549 + 5) = f 554 := by norm_num\n    rw [h\u2081\u2080\u2081\u2082]\n    rw [h\u2081\u2080\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2082 : f 544 = 997 := by\n    have h\u2081\u2080\u2082\u2081 : f 544 = f (f (544 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2082\u2081]\n    have h\u2081\u2080\u2082\u2082 : f (544 + 5) = f 549 := by norm_num\n    rw [h\u2081\u2080\u2082\u2082]\n    rw [h\u2081\u2080\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2083 : f 539 = 998 := by\n    have h\u2081\u2080\u2083\u2081 : f 539 = f (f (539 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2083\u2081]\n    have h\u2081\u2080\u2083\u2082 : f (539 + 5) = f 544 := by norm_num\n    rw [h\u2081\u2080\u2083\u2082]\n    rw [h\u2081\u2080\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2084 : f 534 = 997 := by\n    have h\u2081\u2080\u2084\u2081 : f 534 = f (f (534 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2084\u2081]\n    have h\u2081\u2080\u2084\u2082 : f (534 + 5) = f 539 := by norm_num\n    rw [h\u2081\u2080\u2084\u2082]\n    rw [h\u2081\u2080\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2085 : f 529 = 998 := by\n    have h\u2081\u2080\u2085\u2081 : f 529 = f (f (529 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2085\u2081]\n    have h\u2081\u2080\u2085\u2082 : f (529 + 5) = f 534 := by norm_num\n    rw [h\u2081\u2080\u2085\u2082]\n    rw [h\u2081\u2080\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2086 : f 524 = 997 := by\n    have h\u2081\u2080\u2086\u2081 : f 524 = f (f (524 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2086\u2081]\n    have h\u2081\u2080\u2086\u2082 : f (524 + 5) = f 529 := by norm_num\n    rw [h\u2081\u2080\u2086\u2082]\n    rw [h\u2081\u2080\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2087 : f 519 = 998 := by\n    have h\u2081\u2080\u2087\u2081 : f 519 = f (f (519 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2087\u2081]\n    have h\u2081\u2080\u2087\u2082 : f (519 + 5) = f 524 := by norm_num\n    rw [h\u2081\u2080\u2087\u2082]\n    rw [h\u2081\u2080\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2088 : f 514 = 997 := by\n    have h\u2081\u2080\u2088\u2081 : f 514 = f (f (514 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2088\u2081]\n    have h\u2081\u2080\u2088\u2082 : f (514 + 5) = f 519 := by norm_num\n    rw [h\u2081\u2080\u2088\u2082]\n    rw [h\u2081\u2080\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2080\u2089 : f 509 = 998 := by\n    have h\u2081\u2080\u2089\u2081 : f 509 = f (f (509 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2080\u2089\u2081]\n    have h\u2081\u2080\u2089\u2082 : f (509 + 5) = f 514 := by norm_num\n    rw [h\u2081\u2080\u2089\u2082]\n    rw [h\u2081\u2080\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2080 : f 504 = 997 := by\n    have h\u2081\u2081\u2080\u2081 : f 504 = f (f (504 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2080\u2081]\n    have h\u2081\u2081\u2080\u2082 : f (504 + 5) = f 509 := by norm_num\n    rw [h\u2081\u2081\u2080\u2082]\n    rw [h\u2081\u2080\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2081 : f 499 = 998 := by\n    have h\u2081\u2081\u2081\u2081 : f 499 = f (f (499 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2081\u2081]\n    have h\u2081\u2081\u2081\u2082 : f (499 + 5) = f 504 := by norm_num\n    rw [h\u2081\u2081\u2081\u2082]\n    rw [h\u2081\u2081\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2082 : f 494 = 997 := by\n    have h\u2081\u2081\u2082\u2081 : f 494 = f (f (494 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2082\u2081]\n    have h\u2081\u2081\u2082\u2082 : f (494 + 5) = f 499 := by norm_num\n    rw [h\u2081\u2081\u2082\u2082]\n    rw [h\u2081\u2081\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2083 : f 489 = 998 := by\n    have h\u2081\u2081\u2083\u2081 : f 489 = f (f (489 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2083\u2081]\n    have h\u2081\u2081\u2083\u2082 : f (489 + 5) = f 494 := by norm_num\n    rw [h\u2081\u2081\u2083\u2082]\n    rw [h\u2081\u2081\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2084 : f 484 = 997 := by\n    have h\u2081\u2081\u2084\u2081 : f 484 = f (f (484 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2084\u2081]\n    have h\u2081\u2081\u2084\u2082 : f (484 + 5) = f 489 := by norm_num\n    rw [h\u2081\u2081\u2084\u2082]\n    rw [h\u2081\u2081\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2085 : f 479 = 998 := by\n    have h\u2081\u2081\u2085\u2081 : f 479 = f (f (479 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2085\u2081]\n    have h\u2081\u2081\u2085\u2082 : f (479 + 5) = f 484 := by norm_num\n    rw [h\u2081\u2081\u2085\u2082]\n    rw [h\u2081\u2081\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2086 : f 474 = 997 := by\n    have h\u2081\u2081\u2086\u2081 : f 474 = f (f (474 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2086\u2081]\n    have h\u2081\u2081\u2086\u2082 : f (474 + 5) = f 479 := by norm_num\n    rw [h\u2081\u2081\u2086\u2082]\n    rw [h\u2081\u2081\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2087 : f 469 = 998 := by\n    have h\u2081\u2081\u2087\u2081 : f 469 = f (f (469 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2087\u2081]\n    have h\u2081\u2081\u2087\u2082 : f (469 + 5) = f 474 := by norm_num\n    rw [h\u2081\u2081\u2087\u2082]\n    rw [h\u2081\u2081\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2088 : f 464 = 997 := by\n    have h\u2081\u2081\u2088\u2081 : f 464 = f (f (464 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2088\u2081]\n    have h\u2081\u2081\u2088\u2082 : f (464 + 5) = f 469 := by norm_num\n    rw [h\u2081\u2081\u2088\u2082]\n    rw [h\u2081\u2081\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2081\u2089 : f 459 = 998 := by\n    have h\u2081\u2081\u2089\u2081 : f 459 = f (f (459 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2081\u2089\u2081]\n    have h\u2081\u2081\u2089\u2082 : f (459 + 5) = f 464 := by norm_num\n    rw [h\u2081\u2081\u2089\u2082]\n    rw [h\u2081\u2081\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2080 : f 454 = 997 := by\n    have h\u2081\u2082\u2080\u2081 : f 454 = f (f (454 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2080\u2081]\n    have h\u2081\u2082\u2080\u2082 : f (454 + 5) = f 459 := by norm_num\n    rw [h\u2081\u2082\u2080\u2082]\n    rw [h\u2081\u2081\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2081 : f 449 = 998 := by\n    have h\u2081\u2082\u2081\u2081 : f 449 = f (f (449 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2081\u2081]\n    have h\u2081\u2082\u2081\u2082 : f (449 + 5) = f 454 := by norm_num\n    rw [h\u2081\u2082\u2081\u2082]\n    rw [h\u2081\u2082\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2082 : f 444 = 997 := by\n    have h\u2081\u2082\u2082\u2081 : f 444 = f (f (444 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2082\u2081]\n    have h\u2081\u2082\u2082\u2082 : f (444 + 5) = f 449 := by norm_num\n    rw [h\u2081\u2082\u2082\u2082]\n    rw [h\u2081\u2082\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2083 : f 439 = 998 := by\n    have h\u2081\u2082\u2083\u2081 : f 439 = f (f (439 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2083\u2081]\n    have h\u2081\u2082\u2083\u2082 : f (439 + 5) = f 444 := by norm_num\n    rw [h\u2081\u2082\u2083\u2082]\n    rw [h\u2081\u2082\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2084 : f 434 = 997 := by\n    have h\u2081\u2082\u2084\u2081 : f 434 = f (f (434 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2084\u2081]\n    have h\u2081\u2082\u2084\u2082 : f (434 + 5) = f 439 := by norm_num\n    rw [h\u2081\u2082\u2084\u2082]\n    rw [h\u2081\u2082\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2085 : f 429 = 998 := by\n    have h\u2081\u2082\u2085\u2081 : f 429 = f (f (429 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2085\u2081]\n    have h\u2081\u2082\u2085\u2082 : f (429 + 5) = f 434 := by norm_num\n    rw [h\u2081\u2082\u2085\u2082]\n    rw [h\u2081\u2082\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2086 : f 424 = 997 := by\n    have h\u2081\u2082\u2086\u2081 : f 424 = f (f (424 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2086\u2081]\n    have h\u2081\u2082\u2086\u2082 : f (424 + 5) = f 429 := by norm_num\n    rw [h\u2081\u2082\u2086\u2082]\n    rw [h\u2081\u2082\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2087 : f 419 = 998 := by\n    have h\u2081\u2082\u2087\u2081 : f 419 = f (f (419 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2087\u2081]\n    have h\u2081\u2082\u2087\u2082 : f (419 + 5) = f 424 := by norm_num\n    rw [h\u2081\u2082\u2087\u2082]\n    rw [h\u2081\u2082\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2088 : f 414 = 997 := by\n    have h\u2081\u2082\u2088\u2081 : f 414 = f (f (414 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2088\u2081]\n    have h\u2081\u2082\u2088\u2082 : f (414 + 5) = f 419 := by norm_num\n    rw [h\u2081\u2082\u2088\u2082]\n    rw [h\u2081\u2082\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2082\u2089 : f 409 = 998 := by\n    have h\u2081\u2082\u2089\u2081 : f 409 = f (f (409 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2082\u2089\u2081]\n    have h\u2081\u2082\u2089\u2082 : f (409 + 5) = f 414 := by norm_num\n    rw [h\u2081\u2082\u2089\u2082]\n    rw [h\u2081\u2082\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2080 : f 404 = 997 := by\n    have h\u2081\u2083\u2080\u2081 : f 404 = f (f (404 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2080\u2081]\n    have h\u2081\u2083\u2080\u2082 : f (404 + 5) = f 409 := by norm_num\n    rw [h\u2081\u2083\u2080\u2082]\n    rw [h\u2081\u2082\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2081 : f 399 = 998 := by\n    have h\u2081\u2083\u2081\u2081 : f 399 = f (f (399 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2081\u2081]\n    have h\u2081\u2083\u2081\u2082 : f (399 + 5) = f 404 := by norm_num\n    rw [h\u2081\u2083\u2081\u2082]\n    rw [h\u2081\u2083\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2082 : f 394 = 997 := by\n    have h\u2081\u2083\u2082\u2081 : f 394 = f (f (394 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2082\u2081]\n    have h\u2081\u2083\u2082\u2082 : f (394 + 5) = f 399 := by norm_num\n    rw [h\u2081\u2083\u2082\u2082]\n    rw [h\u2081\u2083\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2083 : f 389 = 998 := by\n    have h\u2081\u2083\u2083\u2081 : f 389 = f (f (389 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2083\u2081]\n    have h\u2081\u2083\u2083\u2082 : f (389 + 5) = f 394 := by norm_num\n    rw [h\u2081\u2083\u2083\u2082]\n    rw [h\u2081\u2083\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2084 : f 384 = 997 := by\n    have h\u2081\u2083\u2084\u2081 : f 384 = f (f (384 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2084\u2081]\n    have h\u2081\u2083\u2084\u2082 : f (384 + 5) = f 389 := by norm_num\n    rw [h\u2081\u2083\u2084\u2082]\n    rw [h\u2081\u2083\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2085 : f 379 = 998 := by\n    have h\u2081\u2083\u2085\u2081 : f 379 = f (f (379 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2085\u2081]\n    have h\u2081\u2083\u2085\u2082 : f (379 + 5) = f 384 := by norm_num\n    rw [h\u2081\u2083\u2085\u2082]\n    rw [h\u2081\u2083\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2086 : f 374 = 997 := by\n    have h\u2081\u2083\u2086\u2081 : f 374 = f (f (374 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2086\u2081]\n    have h\u2081\u2083\u2086\u2082 : f (374 + 5) = f 379 := by norm_num\n    rw [h\u2081\u2083\u2086\u2082]\n    rw [h\u2081\u2083\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2087 : f 369 = 998 := by\n    have h\u2081\u2083\u2087\u2081 : f 369 = f (f (369 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2087\u2081]\n    have h\u2081\u2083\u2087\u2082 : f (369 + 5) = f 374 := by norm_num\n    rw [h\u2081\u2083\u2087\u2082]\n    rw [h\u2081\u2083\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2088 : f 364 = 997 := by\n    have h\u2081\u2083\u2088\u2081 : f 364 = f (f (364 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2088\u2081]\n    have h\u2081\u2083\u2088\u2082 : f (364 + 5) = f 369 := by norm_num\n    rw [h\u2081\u2083\u2088\u2082]\n    rw [h\u2081\u2083\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2083\u2089 : f 359 = 998 := by\n    have h\u2081\u2083\u2089\u2081 : f 359 = f (f (359 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2083\u2089\u2081]\n    have h\u2081\u2083\u2089\u2082 : f (359 + 5) = f 364 := by norm_num\n    rw [h\u2081\u2083\u2089\u2082]\n    rw [h\u2081\u2083\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2080 : f 354 = 997 := by\n    have h\u2081\u2084\u2080\u2081 : f 354 = f (f (354 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2080\u2081]\n    have h\u2081\u2084\u2080\u2082 : f (354 + 5) = f 359 := by norm_num\n    rw [h\u2081\u2084\u2080\u2082]\n    rw [h\u2081\u2083\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2081 : f 349 = 998 := by\n    have h\u2081\u2084\u2081\u2081 : f 349 = f (f (349 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2081\u2081]\n    have h\u2081\u2084\u2081\u2082 : f (349 + 5) = f 354 := by norm_num\n    rw [h\u2081\u2084\u2081\u2082]\n    rw [h\u2081\u2084\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2082 : f 344 = 997 := by\n    have h\u2081\u2084\u2082\u2081 : f 344 = f (f (344 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2082\u2081]\n    have h\u2081\u2084\u2082\u2082 : f (344 + 5) = f 349 := by norm_num\n    rw [h\u2081\u2084\u2082\u2082]\n    rw [h\u2081\u2084\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2083 : f 339 = 998 := by\n    have h\u2081\u2084\u2083\u2081 : f 339 = f (f (339 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2083\u2081]\n    have h\u2081\u2084\u2083\u2082 : f (339 + 5) = f 344 := by norm_num\n    rw [h\u2081\u2084\u2083\u2082]\n    rw [h\u2081\u2084\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2084 : f 334 = 997 := by\n    have h\u2081\u2084\u2084\u2081 : f 334 = f (f (334 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2084\u2081]\n    have h\u2081\u2084\u2084\u2082 : f (334 + 5) = f 339 := by norm_num\n    rw [h\u2081\u2084\u2084\u2082]\n    rw [h\u2081\u2084\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2085 : f 329 = 998 := by\n    have h\u2081\u2084\u2085\u2081 : f 329 = f (f (329 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2085\u2081]\n    have h\u2081\u2084\u2085\u2082 : f (329 + 5) = f 334 := by norm_num\n    rw [h\u2081\u2084\u2085\u2082]\n    rw [h\u2081\u2084\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2086 : f 324 = 997 := by\n    have h\u2081\u2084\u2086\u2081 : f 324 = f (f (324 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2086\u2081]\n    have h\u2081\u2084\u2086\u2082 : f (324 + 5) = f 329 := by norm_num\n    rw [h\u2081\u2084\u2086\u2082]\n    rw [h\u2081\u2084\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2087 : f 319 = 998 := by\n    have h\u2081\u2084\u2087\u2081 : f 319 = f (f (319 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2087\u2081]\n    have h\u2081\u2084\u2087\u2082 : f (319 + 5) = f 324 := by norm_num\n    rw [h\u2081\u2084\u2087\u2082]\n    rw [h\u2081\u2084\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2088 : f 314 = 997 := by\n    have h\u2081\u2084\u2088\u2081 : f 314 = f (f (314 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2088\u2081]\n    have h\u2081\u2084\u2088\u2082 : f (314 + 5) = f 319 := by norm_num\n    rw [h\u2081\u2084\u2088\u2082]\n    rw [h\u2081\u2084\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084\u2089 : f 309 = 998 := by\n    have h\u2081\u2084\u2089\u2081 : f 309 = f (f (309 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2084\u2089\u2081]\n    have h\u2081\u2084\u2089\u2082 : f (309 + 5) = f 314 := by norm_num\n    rw [h\u2081\u2084\u2089\u2082]\n    rw [h\u2081\u2084\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2080 : f 304 = 997 := by\n    have h\u2081\u2085\u2080\u2081 : f 304 = f (f (304 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2080\u2081]\n    have h\u2081\u2085\u2080\u2082 : f (304 + 5) = f 309 := by norm_num\n    rw [h\u2081\u2085\u2080\u2082]\n    rw [h\u2081\u2084\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2081 : f 299 = 998 := by\n    have h\u2081\u2085\u2081\u2081 : f 299 = f (f (299 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2081\u2081]\n    have h\u2081\u2085\u2081\u2082 : f (299 + 5) = f 304 := by norm_num\n    rw [h\u2081\u2085\u2081\u2082]\n    rw [h\u2081\u2085\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2082 : f 294 = 997 := by\n    have h\u2081\u2085\u2082\u2081 : f 294 = f (f (294 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2082\u2081]\n    have h\u2081\u2085\u2082\u2082 : f (294 + 5) = f 299 := by norm_num\n    rw [h\u2081\u2085\u2082\u2082]\n    rw [h\u2081\u2085\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2083 : f 289 = 998 := by\n    have h\u2081\u2085\u2083\u2081 : f 289 = f (f (289 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2083\u2081]\n    have h\u2081\u2085\u2083\u2082 : f (289 + 5) = f 294 := by norm_num\n    rw [h\u2081\u2085\u2083\u2082]\n    rw [h\u2081\u2085\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2084 : f 284 = 997 := by\n    have h\u2081\u2085\u2084\u2081 : f 284 = f (f (284 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2084\u2081]\n    have h\u2081\u2085\u2084\u2082 : f (284 + 5) = f 289 := by norm_num\n    rw [h\u2081\u2085\u2084\u2082]\n    rw [h\u2081\u2085\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2085 : f 279 = 998 := by\n    have h\u2081\u2085\u2085\u2081 : f 279 = f (f (279 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2085\u2081]\n    have h\u2081\u2085\u2085\u2082 : f (279 + 5) = f 284 := by norm_num\n    rw [h\u2081\u2085\u2085\u2082]\n    rw [h\u2081\u2085\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2086 : f 274 = 997 := by\n    have h\u2081\u2085\u2086\u2081 : f 274 = f (f (274 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2086\u2081]\n    have h\u2081\u2085\u2086\u2082 : f (274 + 5) = f 279 := by norm_num\n    rw [h\u2081\u2085\u2086\u2082]\n    rw [h\u2081\u2085\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2087 : f 269 = 998 := by\n    have h\u2081\u2085\u2087\u2081 : f 269 = f (f (269 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2087\u2081]\n    have h\u2081\u2085\u2087\u2082 : f (269 + 5) = f 274 := by norm_num\n    rw [h\u2081\u2085\u2087\u2082]\n    rw [h\u2081\u2085\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2088 : f 264 = 997 := by\n    have h\u2081\u2085\u2088\u2081 : f 264 = f (f (264 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2088\u2081]\n    have h\u2081\u2085\u2088\u2082 : f (264 + 5) = f 269 := by norm_num\n    rw [h\u2081\u2085\u2088\u2082]\n    rw [h\u2081\u2085\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085\u2089 : f 259 = 998 := by\n    have h\u2081\u2085\u2089\u2081 : f 259 = f (f (259 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2085\u2089\u2081]\n    have h\u2081\u2085\u2089\u2082 : f (259 + 5) = f 264 := by norm_num\n    rw [h\u2081\u2085\u2089\u2082]\n    rw [h\u2081\u2085\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2080 : f 254 = 997 := by\n    have h\u2081\u2086\u2080\u2081 : f 254 = f (f (254 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2080\u2081]\n    have h\u2081\u2086\u2080\u2082 : f (254 + 5) = f 259 := by norm_num\n    rw [h\u2081\u2086\u2080\u2082]\n    rw [h\u2081\u2085\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2081 : f 249 = 998 := by\n    have h\u2081\u2086\u2081\u2081 : f 249 = f (f (249 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2081\u2081]\n    have h\u2081\u2086\u2081\u2082 : f (249 + 5) = f 254 := by norm_num\n    rw [h\u2081\u2086\u2081\u2082]\n    rw [h\u2081\u2086\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2082 : f 244 = 997 := by\n    have h\u2081\u2086\u2082\u2081 : f 244 = f (f (244 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2082\u2081]\n    have h\u2081\u2086\u2082\u2082 : f (244 + 5) = f 249 := by norm_num\n    rw [h\u2081\u2086\u2082\u2082]\n    rw [h\u2081\u2086\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2083 : f 239 = 998 := by\n    have h\u2081\u2086\u2083\u2081 : f 239 = f (f (239 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2083\u2081]\n    have h\u2081\u2086\u2083\u2082 : f (239 + 5) = f 244 := by norm_num\n    rw [h\u2081\u2086\u2083\u2082]\n    rw [h\u2081\u2086\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2084 : f 234 = 997 := by\n    have h\u2081\u2086\u2084\u2081 : f 234 = f (f (234 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2084\u2081]\n    have h\u2081\u2086\u2084\u2082 : f (234 + 5) = f 239 := by norm_num\n    rw [h\u2081\u2086\u2084\u2082]\n    rw [h\u2081\u2086\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2085 : f 229 = 998 := by\n    have h\u2081\u2086\u2085\u2081 : f 229 = f (f (229 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2085\u2081]\n    have h\u2081\u2086\u2085\u2082 : f (229 + 5) = f 234 := by norm_num\n    rw [h\u2081\u2086\u2085\u2082]\n    rw [h\u2081\u2086\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2086 : f 224 = 997 := by\n    have h\u2081\u2086\u2086\u2081 : f 224 = f (f (224 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2086\u2081]\n    have h\u2081\u2086\u2086\u2082 : f (224 + 5) = f 229 := by norm_num\n    rw [h\u2081\u2086\u2086\u2082]\n    rw [h\u2081\u2086\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2087 : f 219 = 998 := by\n    have h\u2081\u2086\u2087\u2081 : f 219 = f (f (219 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2087\u2081]\n    have h\u2081\u2086\u2087\u2082 : f (219 + 5) = f 224 := by norm_num\n    rw [h\u2081\u2086\u2087\u2082]\n    rw [h\u2081\u2086\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2088 : f 214 = 997 := by\n    have h\u2081\u2086\u2088\u2081 : f 214 = f (f (214 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2088\u2081]\n    have h\u2081\u2086\u2088\u2082 : f (214 + 5) = f 219 := by norm_num\n    rw [h\u2081\u2086\u2088\u2082]\n    rw [h\u2081\u2086\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2086\u2089 : f 209 = 998 := by\n    have h\u2081\u2086\u2089\u2081 : f 209 = f (f (209 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2086\u2089\u2081]\n    have h\u2081\u2086\u2089\u2082 : f (209 + 5) = f 214 := by norm_num\n    rw [h\u2081\u2086\u2089\u2082]\n    rw [h\u2081\u2086\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2080 : f 204 = 997 := by\n    have h\u2081\u2087\u2080\u2081 : f 204 = f (f (204 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2080\u2081]\n    have h\u2081\u2087\u2080\u2082 : f (204 + 5) = f 209 := by norm_num\n    rw [h\u2081\u2087\u2080\u2082]\n    rw [h\u2081\u2086\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2081 : f 199 = 998 := by\n    have h\u2081\u2087\u2081\u2081 : f 199 = f (f (199 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2081\u2081]\n    have h\u2081\u2087\u2081\u2082 : f (199 + 5) = f 204 := by norm_num\n    rw [h\u2081\u2087\u2081\u2082]\n    rw [h\u2081\u2087\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2082 : f 194 = 997 := by\n    have h\u2081\u2087\u2082\u2081 : f 194 = f (f (194 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2082\u2081]\n    have h\u2081\u2087\u2082\u2082 : f (194 + 5) = f 199 := by norm_num\n    rw [h\u2081\u2087\u2082\u2082]\n    rw [h\u2081\u2087\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2083 : f 189 = 998 := by\n    have h\u2081\u2087\u2083\u2081 : f 189 = f (f (189 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2083\u2081]\n    have h\u2081\u2087\u2083\u2082 : f (189 + 5) = f 194 := by norm_num\n    rw [h\u2081\u2087\u2083\u2082]\n    rw [h\u2081\u2087\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2084 : f 184 = 997 := by\n    have h\u2081\u2087\u2084\u2081 : f 184 = f (f (184 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2084\u2081]\n    have h\u2081\u2087\u2084\u2082 : f (184 + 5) = f 189 := by norm_num\n    rw [h\u2081\u2087\u2084\u2082]\n    rw [h\u2081\u2087\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2085 : f 179 = 998 := by\n    have h\u2081\u2087\u2085\u2081 : f 179 = f (f (179 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2085\u2081]\n    have h\u2081\u2087\u2085\u2082 : f (179 + 5) = f 184 := by norm_num\n    rw [h\u2081\u2087\u2085\u2082]\n    rw [h\u2081\u2087\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2086 : f 174 = 997 := by\n    have h\u2081\u2087\u2086\u2081 : f 174 = f (f (174 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2086\u2081]\n    have h\u2081\u2087\u2086\u2082 : f (174 + 5) = f 179 := by norm_num\n    rw [h\u2081\u2087\u2086\u2082]\n    rw [h\u2081\u2087\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2087 : f 169 = 998 := by\n    have h\u2081\u2087\u2087\u2081 : f 169 = f (f (169 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2087\u2081]\n    have h\u2081\u2087\u2087\u2082 : f (169 + 5) = f 174 := by norm_num\n    rw [h\u2081\u2087\u2087\u2082]\n    rw [h\u2081\u2087\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2088 : f 164 = 997 := by\n    have h\u2081\u2087\u2088\u2081 : f 164 = f (f (164 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2088\u2081]\n    have h\u2081\u2087\u2088\u2082 : f (164 + 5) = f 169 := by norm_num\n    rw [h\u2081\u2087\u2088\u2082]\n    rw [h\u2081\u2087\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2087\u2089 : f 159 = 998 := by\n    have h\u2081\u2087\u2089\u2081 : f 159 = f (f (159 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2087\u2089\u2081]\n    have h\u2081\u2087\u2089\u2082 : f (159 + 5) = f 164 := by norm_num\n    rw [h\u2081\u2087\u2089\u2082]\n    rw [h\u2081\u2087\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2080 : f 154 = 997 := by\n    have h\u2081\u2088\u2080\u2081 : f 154 = f (f (154 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2080\u2081]\n    have h\u2081\u2088\u2080\u2082 : f (154 + 5) = f 159 := by norm_num\n    rw [h\u2081\u2088\u2080\u2082]\n    rw [h\u2081\u2087\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2081 : f 149 = 998 := by\n    have h\u2081\u2088\u2081\u2081 : f 149 = f (f (149 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2081\u2081]\n    have h\u2081\u2088\u2081\u2082 : f (149 + 5) = f 154 := by norm_num\n    rw [h\u2081\u2088\u2081\u2082]\n    rw [h\u2081\u2088\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2082 : f 144 = 997 := by\n    have h\u2081\u2088\u2082\u2081 : f 144 = f (f (144 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2082\u2081]\n    have h\u2081\u2088\u2082\u2082 : f (144 + 5) = f 149 := by norm_num\n    rw [h\u2081\u2088\u2082\u2082]\n    rw [h\u2081\u2088\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2083 : f 139 = 998 := by\n    have h\u2081\u2088\u2083\u2081 : f 139 = f (f (139 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2083\u2081]\n    have h\u2081\u2088\u2083\u2082 : f (139 + 5) = f 144 := by norm_num\n    rw [h\u2081\u2088\u2083\u2082]\n    rw [h\u2081\u2088\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2084 : f 134 = 997 := by\n    have h\u2081\u2088\u2084\u2081 : f 134 = f (f (134 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2084\u2081]\n    have h\u2081\u2088\u2084\u2082 : f (134 + 5) = f 139 := by norm_num\n    rw [h\u2081\u2088\u2084\u2082]\n    rw [h\u2081\u2088\u2083]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2085 : f 129 = 998 := by\n    have h\u2081\u2088\u2085\u2081 : f 129 = f (f (129 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2085\u2081]\n    have h\u2081\u2088\u2085\u2082 : f (129 + 5) = f 134 := by norm_num\n    rw [h\u2081\u2088\u2085\u2082]\n    rw [h\u2081\u2088\u2084]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2086 : f 124 = 997 := by\n    have h\u2081\u2088\u2086\u2081 : f 124 = f (f (124 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2086\u2081]\n    have h\u2081\u2088\u2086\u2082 : f (124 + 5) = f 129 := by norm_num\n    rw [h\u2081\u2088\u2086\u2082]\n    rw [h\u2081\u2088\u2085]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2087 : f 119 = 998 := by\n    have h\u2081\u2088\u2087\u2081 : f 119 = f (f (119 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2087\u2081]\n    have h\u2081\u2088\u2087\u2082 : f (119 + 5) = f 124 := by norm_num\n    rw [h\u2081\u2088\u2087\u2082]\n    rw [h\u2081\u2088\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2088 : f 114 = 997 := by\n    have h\u2081\u2088\u2088\u2081 : f 114 = f (f (114 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2088\u2081]\n    have h\u2081\u2088\u2088\u2082 : f (114 + 5) = f 119 := by norm_num\n    rw [h\u2081\u2088\u2088\u2082]\n    rw [h\u2081\u2088\u2087]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2088\u2089 : f 109 = 998 := by\n    have h\u2081\u2088\u2089\u2081 : f 109 = f (f (109 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2088\u2089\u2081]\n    have h\u2081\u2088\u2089\u2082 : f (109 + 5) = f 114 := by norm_num\n    rw [h\u2081\u2088\u2089\u2082]\n    rw [h\u2081\u2088\u2088]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089\u2080 : f 104 = 997 := by\n    have h\u2081\u2089\u2080\u2081 : f 104 = f (f (104 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2080\u2081]\n    have h\u2081\u2089\u2080\u2082 : f (104 + 5) = f 109 := by norm_num\n    rw [h\u2081\u2089\u2080\u2082]\n    rw [h\u2081\u2088\u2089]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089\u2081 : f 99 = 998 := by\n    have h\u2081\u2089\u2081\u2081 : f 99 = f (f (99 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2081\u2081]\n    have h\u2081\u2089\u2081\u2082 : f (99 + 5) = f 104 := by norm_num\n    rw [h\u2081\u2089\u2081\u2082]\n    rw [h\u2081\u2089\u2080]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089\u2082 : f 94 = 997 := by\n    have h\u2081\u2089\u2082\u2081 : f 94 = f (f (94 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2082\u2081]\n    have h\u2081\u2089\u2082\u2082 : f (94 + 5) = f 99 := by norm_num\n    rw [h\u2081\u2089\u2082\u2082]\n    rw [h\u2081\u2089\u2081]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089\u2083 : f 89 = 998 := by\n    have h\u2081\u2089\u2083\u2081 : f 89 = f (f (89 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2083\u2081]\n    have h\u2081\u2089\u2083\u2082 : f (89 + 5) = f 94 := by norm_num\n    rw [h\u2081\u2089\u2083\u2082]\n    rw [h\u2081\u2089\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2089\u2084 : f 84 = 997 := by\n    have h\u2081\u2089\u2084\u2081 : f 84 = f (f (84 + 5)) := by\n      apply h\u2081\n      <;> norm_num\n    rw [h\u2081\u2089\u2084\u2081]\n    have h\u2081\u2089\u2084\u2082 : f (84 + 5) = f 89 := by norm_num\n    rw [h\u2081\u2089\u2084\u2082]\n    rw [h\u2081\u2089\u2083]\n    <;> norm_num\n    <;> linarith\n  rw [h\u2081\u2089\u2084]\n  <;> norm_num\n"}
{"mode": "clear", "steps_before": 39, "alphabet": 16, "compiles": true, "steps_after": 10, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 4, "fixpoint": true, "chars_before": 978, "chars_after": 194, "seconds": 820.3, "id": "mathd_algebra_338", "proof_after": "theorem mathd_algebra_338 (a b c : \u211d) (h\u2080 : 3 * a + b + c = -3) (h\u2081 : a + 3 * b + c = 9)\n    (h\u2082 : a + b + 3 * c = 19) : a * b * c = -56  :=  by\n  have h\u2083 : a = -4 := by\n    linarith\n  have h\u2084 : b = 2 := by\n    linarith\n  have h\u2085 : c = 7 := by\n    linarith\n  have h\u2086 : a * b * c = -56 := by\n    rw [h\u2083, h\u2084, h\u2085]\n    <;> norm_num\n  exact h\u2086\n", "proof_before": "theorem mathd_algebra_338 (a b c : \u211d) (h\u2080 : 3 * a + b + c = -3) (h\u2081 : a + 3 * b + c = 9)\n    (h\u2082 : a + b + 3 * c = 19) : a * b * c = -56  :=  by\n  have h\u2083 : a = -4 := by\n    have h\u2083\u2081 : 5 * a + 17 = -3 := by\n      have h\u2083\u2082 : b = a + 6 := by\n        have h\u2083\u2083 : a - b = -6 := by\n          have h\u2083\u2084 : -2 * (a - b) = 12 := by\n            linarith\n          linarith\n        linarith\n      have h\u2083\u2084 : c = a + 11 := by\n        have h\u2083\u2085 : a - c = -11 := by\n          have h\u2083\u2086 : -2 * (a - c) = 22 := by\n            linarith\n          linarith\n        linarith\n      rw [h\u2083\u2082, h\u2083\u2084] at h\u2080\n      linarith\n    linarith\n  have h\u2084 : b = 2 := by\n    have h\u2084\u2081 : b = a + 6 := by\n      have h\u2084\u2082 : a - b = -6 := by\n        have h\u2084\u2083 : -2 * (a - b) = 12 := by\n          linarith\n        linarith\n      linarith\n    rw [h\u2084\u2081, h\u2083]\n    <;> norm_num\n  have h\u2085 : c = 7 := by\n    have h\u2085\u2081 : c = a + 11 := by\n      have h\u2085\u2082 : a - c = -11 := by\n        have h\u2085\u2083 : -2 * (a - c) = 22 := by\n          linarith\n        linarith\n      linarith\n    rw [h\u2085\u2081, h\u2083]\n    <;> norm_num\n  have h\u2086 : a * b * c = -56 := by\n    rw [h\u2083, h\u2084, h\u2085]\n    <;> norm_num\n  exact h\u2086\n"}
{"mode": "clear", "steps_before": 136, "alphabet": 79, "compiles": false, "id": "numbertheory_2pownm1prime_nprime", "proof_after": "theorem numbertheory_2pownm1prime_nprime (n : \u2115) (h\u2080 : 0 < n) (h\u2081 : Nat.Prime (2 ^ n - 1)) : Nat.Prime n  :=  by\n  have h\u2082 : n \u2265 2 := by\n    by_contra h\n    have h\u2083 : n \u2264 1 := by linarith\n    have h\u2084 : n = 1 := by\n      omega\n    rw [h\u2084] at h\u2081\n    norm_num at h\u2081\n    <;> contradiction\n  have h\u2083 : Nat.Prime n := by\n    by_contra h\n    have h\u2084 : \u00acNat.Prime n := h\n    have h\u2085 : \u2203 m, m \u2223 n \u2227 m \u2260 1 \u2227 m \u2260 n := by\n      have h\u2085\u2081 : \u00acNat.Prime n := h\u2084\n      have h\u2085\u2082 : n \u2265 2 := h\u2082\n      have h\u2085\u2083 : \u2203 m, m \u2223 n \u2227 m \u2260 1 \u2227 m \u2260 n := by\n        have h\u2085\u2084 := Nat.exists_dvd_of_not_prime2 (by omega : n \u2265 2) h\u2085\u2081\n        obtain \u27e8m, hm\u2081, hm\u2082\u27e9 := h\u2085\u2084\n        refine' \u27e8m, hm\u2081, _\u27e9\n        have h\u2085\u2085 : m \u2260 1 := by\n          by_contra h\u2085\u2085\n          rw [h\u2085\u2085] at hm\u2082\n          omega\n        have h\u2085\u2086 : m \u2260 n := by\n          by_contra h\u2085\u2086\n          rw [h\u2085\u2086] at hm\u2082\n          omega\n        exact \u27e8h\u2085\u2085, h\u2085\u2086\u27e9\n      exact h\u2085\u2083\n    obtain \u27e8m, hm\u2081, hm\u2082, hm\u2083\u27e9 := h\u2085\n    have h\u2086 : m \u2223 n := hm\u2081\n    have h\u2087 : m \u2260 1 := hm\u2082\n    have h\u2088 : m \u2260 n := hm\u2083\n    have h\u2089 : m \u2265 2 := by\n      by_contra h\u2089\n      have h\u2089\u2081 : m \u2264 1 := by omega\n      have h\u2089\u2082 : m = 0 \u2228 m = 1 := by\n        omega\n      cases h\u2089\u2082 with\n      | inl h\u2089\u2082 =>\n        have h\u2089\u2083 : m = 0 := h\u2089\u2082\n        rw [h\u2089\u2083] at h\u2086\n        have h\u2089\u2084 : (0 : \u2115) \u2223 n := h\u2086\n        have h\u2089\u2085 : n = 0 := by simpa using h\u2089\u2084\n        omega\n      | inr h\u2089\u2082 =>\n        have h\u2089\u2083 : m = 1 := h\u2089\u2082\n        contradiction\n    have h\u2081\u2080 : m < n := by\n      by_contra h\u2081\u2080\n      have h\u2081\u2080\u2081 : m \u2265 n := by omega\n      have h\u2081\u2080\u2082 : m \u2223 n := h\u2086\n      have h\u2081\u2080\u2083 : m \u2264 n := Nat.le_of_dvd (by omega) h\u2081\u2080\u2082\n      have h\u2081\u2080\u2084 : m = n := by\n        linarith\n      contradiction\n    have h\u2081\u2081 : 2 ^ m - 1 \u2223 2 ^ n - 1 := by\n      have h\u2081\u2081\u2081 : m \u2223 n := h\u2086\n      obtain \u27e8k, hk\u27e9 := h\u2081\u2081\u2081\n      rw [hk]\n      have h\u2081\u2081\u2082 : 2 ^ m - 1 \u2223 2 ^ (m * k) - 1 := by\n        have h\u2081\u2081\u2083 : 2 ^ m - 1 \u2223 2 ^ (m * k) - 1 := by\n          simpa [pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 k\n        exact h\u2081\u2081\u2083\n      simpa [pow_mul, mul_comm] using h\u2081\u2081\u2082\n    have h\u2081\u2082 : 2 ^ m - 1 > 1 := by\n      have h\u2081\u2082\u2081 : m \u2265 2 := h\u2089\n      have h\u2081\u2082\u2082 : 2 ^ m - 1 > 1 := by\n        have h\u2081\u2082\u2083 : 2 ^ m \u2265 2 ^ 2 := Nat.pow_le_pow_of_le_right (by decide) h\u2081\u2082\u2081\n        have h\u2081\u2082\u2084 : 2 ^ m - 1 \u2265 2 ^ 2 - 1 := by\n          omega\n        have h\u2081\u2082\u2085 : 2 ^ 2 - 1 = 3 := by norm_num\n        have h\u2081\u2082\u2086 : 2 ^ m - 1 \u2265 3 := by\n          omega\n        omega\n      exact h\u2081\u2082\u2082\n    have h\u2081\u2083 : 2 ^ m - 1 < 2 ^ n - 1 := by\n      have h\u2081\u2083\u2081 : m < n := h\u2081\u2080\n      have h\u2081\u2083\u2082 : 2 ^ m < 2 ^ n := Nat.pow_lt_pow_of_lt_right (by decide) h\u2081\u2083\u2081\n      have h\u2081\u2083\u2083 : 2 ^ m - 1 < 2 ^ n - 1 := by\n        have h\u2081\u2083\u2084 : 2 ^ m \u2265 2 := by\n          have h\u2081\u2083\u2085 : m \u2265 2 := h\u2089\n          have h\u2081\u2083\u2086 : 2 ^ m \u2265 2 ^ 2 := Nat.pow_le_pow_of_le_right (by decide) h\u2081\u2083\u2085\n          omega\n        have h\u2081\u2083\u2085 : 2 ^ n \u2265 2 ^ 2 := Nat.pow_le_pow_of_le_right (by decide) h\u2082\n        have h\u2081\u2083\u2086 : 2 ^ m - 1 < 2 ^ n - 1 := by\n          have h\u2081\u2083\u2087 : 2 ^ m < 2 ^ n := h\u2081\u2083\u2082\n          have h\u2081\u2083\u2088 : 2 ^ m - 1 < 2 ^ n - 1 := by\n            have h\u2081\u2083\u2089 : 2 ^ m \u2265 2 := by\n              omega\n            have h\u2081\u2084\u2080 : 2 ^ n \u2265 2 := by\n              have h\u2081\u2084\u2081 : n \u2265 2 := h\u2082\n              have h\u2081\u2084\u2082 : 2 ^ n \u2265 2 ^ 2 := Nat.pow_le_pow_of_le_right (by decide) h\u2081\u2084\u2081\n              omega\n            have h\u2081\u2084\u2083 : 2 ^ m - 1 < 2 ^ n - 1 := by\n              have h\u2081\u2084\u2084 : 2 ^ m < 2 ^ n := h\u2081\u2083\u2082\n              have h\u2081\u2084\u2085 : 2 ^ m - 1 < 2 ^ n - 1 := by\n                omega\n              exact h\u2081\u2084\u2085\n            exact h\u2081\u2084\u2083\n          exact h\u2081\u2083\u2088\n        exact h\u2081\u2083\u2086\n      exact h\u2081\u2083\u2083\n    have h\u2081\u2084 : 2 ^ m - 1 \u2260 1 := by\n      omega\n    have h\u2081\u2085 : 2 ^ m - 1 \u2260 2 ^ n - 1 := by\n      intro h\u2081\u2085\n      have h\u2081\u2085\u2081 : 2 ^ m - 1 < 2 ^ n - 1 := h\u2081\u2083\n      linarith\n    have h\u2081\u2086 : \u00acNat.Prime (2 ^ n - 1) := by\n      intro h\u2081\u2086\n      have h\u2081\u2087 := Nat.Prime.ne_one h\u2081\u2086\n      have h\u2081\u2088 := Nat.Prime.ne_zero h\u2081\u2086\n      have h\u2081\u2089 : 2 ^ m - 1 \u2223 2 ^ n - 1 := h\u2081\u2081\n      have h\u2082\u2080 : 2 ^ m - 1 \u2260 1 := h\u2081\u2084\n      have h\u2082\u2081 : 2 ^ m - 1 \u2260 2 ^ n - 1 := h\u2081\u2085\n      have h\u2082\u2082 : 2 ^ m - 1 < 2 ^ n - 1 := h\u2081\u2083\n      have h\u2082\u2083 : 2 ^ m - 1 > 1 := h\u2081\u2082\n      have h\u2082\u2084 : \u00acNat.Prime (2 ^ n - 1) := by\n        intro h\u2082\u2084\n        have h\u2082\u2085 := Nat.Prime.eq_one_or_self_of_dvd h\u2082\u2084 (2 ^ m - 1) h\u2081\u2081\n        omega\n      exact h\u2082\u2084 h\u2081\u2086\n    exact h\u2081\u2086 h\u2081\n  exact h\u2083\n", "proof_before": "theorem numbertheory_2pownm1prime_nprime (n : \u2115) (h\u2080 : 0 < n) (h\u2081 : Nat.Prime (2 ^ n - 1)) : Nat.Prime n  :=  by\n  have h\u2082 : n \u2265 2 := by\n    by_contra h\n    have h\u2083 : n \u2264 1 := by linarith\n    have h\u2084 : n = 1 := by\n      omega\n    rw [h\u2084] at h\u2081\n    norm_num at h\u2081\n    <;> contradiction\n  have h\u2083 : Nat.Prime n := by\n    by_contra h\n    have h\u2084 : \u00acNat.Prime n := h\n    have h\u2085 : \u2203 m, m \u2223 n \u2227 m \u2260 1 \u2227 m \u2260 n := by\n      have h\u2085\u2081 : \u00acNat.Prime n := h\u2084\n      have h\u2085\u2082 : n \u2265 2 := h\u2082\n      have h\u2085\u2083 : \u2203 m, m \u2223 n \u2227 m \u2260 1 \u2227 m \u2260 n := by\n        have h\u2085\u2084 := Nat.exists_dvd_of_not_prime2 (by omega : n \u2265 2) h\u2085\u2081\n        obtain \u27e8m, hm\u2081, hm\u2082\u27e9 := h\u2085\u2084\n        refine' \u27e8m, hm\u2081, _\u27e9\n        have h\u2085\u2085 : m \u2260 1 := by\n          by_contra h\u2085\u2085\n          rw [h\u2085\u2085] at hm\u2082\n          omega\n        have h\u2085\u2086 : m \u2260 n := by\n          by_contra h\u2085\u2086\n          rw [h\u2085\u2086] at hm\u2082\n          omega\n        exact \u27e8h\u2085\u2085, h\u2085\u2086\u27e9\n      exact h\u2085\u2083\n    obtain \u27e8m, hm\u2081, hm\u2082, hm\u2083\u27e9 := h\u2085\n    have h\u2086 : m \u2223 n := hm\u2081\n    have h\u2087 : m \u2260 1 := hm\u2082\n    have h\u2088 : m \u2260 n := hm\u2083\n    have h\u2089 : m \u2265 2 := by\n      by_contra h\u2089\n      have h\u2089\u2081 : m \u2264 1 := by omega\n      have h\u2089\u2082 : m = 0 \u2228 m = 1 := by\n        omega\n      cases h\u2089\u2082 with\n      | inl h\u2089\u2082 =>\n        have h\u2089\u2083 : m = 0 := h\u2089\u2082\n        rw [h\u2089\u2083] at h\u2086\n        have h\u2089\u2084 : (0 : \u2115) \u2223 n := h\u2086\n        have h\u2089\u2085 : n = 0 := by simpa using h\u2089\u2084\n        omega\n      | inr h\u2089\u2082 =>\n        have h\u2089\u2083 : m = 1 := h\u2089\u2082\n        contradiction\n    have h\u2081\u2080 : m < n := by\n      by_contra h\u2081\u2080\n      have h\u2081\u2080\u2081 : m \u2265 n := by omega\n      have h\u2081\u2080\u2082 : m \u2223 n := h\u2086\n      have h\u2081\u2080\u2083 : m \u2264 n := Nat.le_of_dvd (by omega) h\u2081\u2080\u2082\n      have h\u2081\u2080\u2084 : m = n := by\n        linarith\n      contradiction\n    have h\u2081\u2081 : 2 ^ m - 1 \u2223 2 ^ n - 1 := by\n      have h\u2081\u2081\u2081 : m \u2223 n := h\u2086\n      obtain \u27e8k, hk\u27e9 := h\u2081\u2081\u2081\n      rw [hk]\n      have h\u2081\u2081\u2082 : 2 ^ m - 1 \u2223 2 ^ (m * k) - 1 := by\n        have h\u2081\u2081\u2083 : 2 ^ m - 1 \u2223 2 ^ (m * k) - 1 := by\n          simpa [pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 k\n        exact h\u2081\u2081\u2083\n      simpa [pow_mul, mul_comm] using h\u2081\u2081\u2082\n    have h\u2081\u2082 : 2 ^ m - 1 > 1 := by\n      have h\u2081\u2082\u2081 : m \u2265 2 := h\u2089\n      have h\u2081\u2082\u2082 : 2 ^ m - 1 > 1 := by\n        have h\u2081\u2082\u2083 : 2 ^ m \u2265 2 ^ 2 := Nat.pow_le_pow_of_le_right (by decide) h\u2081\u2082\u2081\n        have h\u2081\u2082\u2084 : 2 ^ m - 1 \u2265 2 ^ 2 - 1 := by\n          omega\n        have h\u2081\u2082\u2085 : 2 ^ 2 - 1 = 3 := by norm_num\n        have h\u2081\u2082\u2086 : 2 ^ m - 1 \u2265 3 := by\n          omega\n        omega\n      exact h\u2081\u2082\u2082\n    have h\u2081\u2083 : 2 ^ m - 1 < 2 ^ n - 1 := by\n      have h\u2081\u2083\u2081 : m < n := h\u2081\u2080\n      have h\u2081\u2083\u2082 : 2 ^ m < 2 ^ n := Nat.pow_lt_pow_of_lt_right (by decide) h\u2081\u2083\u2081\n      have h\u2081\u2083\u2083 : 2 ^ m - 1 < 2 ^ n - 1 := by\n        have h\u2081\u2083\u2084 : 2 ^ m \u2265 2 := by\n          have h\u2081\u2083\u2085 : m \u2265 2 := h\u2089\n          have h\u2081\u2083\u2086 : 2 ^ m \u2265 2 ^ 2 := Nat.pow_le_pow_of_le_right (by decide) h\u2081\u2083\u2085\n          omega\n        have h\u2081\u2083\u2085 : 2 ^ n \u2265 2 ^ 2 := Nat.pow_le_pow_of_le_right (by decide) h\u2082\n        have h\u2081\u2083\u2086 : 2 ^ m - 1 < 2 ^ n - 1 := by\n          have h\u2081\u2083\u2087 : 2 ^ m < 2 ^ n := h\u2081\u2083\u2082\n          have h\u2081\u2083\u2088 : 2 ^ m - 1 < 2 ^ n - 1 := by\n            have h\u2081\u2083\u2089 : 2 ^ m \u2265 2 := by\n              omega\n            have h\u2081\u2084\u2080 : 2 ^ n \u2265 2 := by\n              have h\u2081\u2084\u2081 : n \u2265 2 := h\u2082\n              have h\u2081\u2084\u2082 : 2 ^ n \u2265 2 ^ 2 := Nat.pow_le_pow_of_le_right (by decide) h\u2081\u2084\u2081\n              omega\n            have h\u2081\u2084\u2083 : 2 ^ m - 1 < 2 ^ n - 1 := by\n              have h\u2081\u2084\u2084 : 2 ^ m < 2 ^ n := h\u2081\u2083\u2082\n              have h\u2081\u2084\u2085 : 2 ^ m - 1 < 2 ^ n - 1 := by\n                omega\n              exact h\u2081\u2084\u2085\n            exact h\u2081\u2084\u2083\n          exact h\u2081\u2083\u2088\n        exact h\u2081\u2083\u2086\n      exact h\u2081\u2083\u2083\n    have h\u2081\u2084 : 2 ^ m - 1 \u2260 1 := by\n      omega\n    have h\u2081\u2085 : 2 ^ m - 1 \u2260 2 ^ n - 1 := by\n      intro h\u2081\u2085\n      have h\u2081\u2085\u2081 : 2 ^ m - 1 < 2 ^ n - 1 := h\u2081\u2083\n      linarith\n    have h\u2081\u2086 : \u00acNat.Prime (2 ^ n - 1) := by\n      intro h\u2081\u2086\n      have h\u2081\u2087 := Nat.Prime.ne_one h\u2081\u2086\n      have h\u2081\u2088 := Nat.Prime.ne_zero h\u2081\u2086\n      have h\u2081\u2089 : 2 ^ m - 1 \u2223 2 ^ n - 1 := h\u2081\u2081\n      have h\u2082\u2080 : 2 ^ m - 1 \u2260 1 := h\u2081\u2084\n      have h\u2082\u2081 : 2 ^ m - 1 \u2260 2 ^ n - 1 := h\u2081\u2085\n      have h\u2082\u2082 : 2 ^ m - 1 < 2 ^ n - 1 := h\u2081\u2083\n      have h\u2082\u2083 : 2 ^ m - 1 > 1 := h\u2081\u2082\n      have h\u2082\u2084 : \u00acNat.Prime (2 ^ n - 1) := by\n        intro h\u2082\u2084\n        have h\u2082\u2085 := Nat.Prime.eq_one_or_self_of_dvd h\u2082\u2084 (2 ^ m - 1) h\u2081\u2081\n        omega\n      exact h\u2082\u2084 h\u2081\u2086\n    exact h\u2081\u2086 h\u2081\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 142, "alphabet": 32, "compiles": false, "id": "mathd_numbertheory_296", "proof_after": "theorem mathd_numbertheory_296 (n : \u2115) (h\u2080 : 2 \u2264 n) (h\u2081 : \u2203 x, x ^ 3 = n) (h\u2082 : \u2203 t, t ^ 4 = n) :\n    4096 \u2264 n  :=  by\n  have h_main : n \u2265 4096 := by\n    obtain \u27e8t, ht\u27e9 := h\u2082\n    have h\u2083 : t \u2265 2 := by\n      by_contra h\n      have h\u2084 : t \u2264 1 := by linarith\n      have h\u2085 : t = 0 \u2228 t = 1 := by\n        have h\u2085\u2081 : t \u2264 1 := h\u2084\n        have h\u2085\u2082 : t \u2265 0 := by positivity\n        interval_cases t <;> simp_all (config := {decide := true})\n      cases h\u2085 with\n      | inl h\u2085 =>\n        have h\u2086 : t = 0 := h\u2085\n        rw [h\u2086] at ht\n        norm_num at ht\n        <;> linarith\n      | inr h\u2085 =>\n        have h\u2086 : t = 1 := h\u2085\n        rw [h\u2086] at ht\n        norm_num at ht\n        <;> linarith\n    have h\u2084 : t \u2265 8 := by\n      by_contra h\n      have h\u2085 : t \u2264 7 := by linarith\n      have h\u2086 : t = 2 \u2228 t = 3 \u2228 t = 4 \u2228 t = 5 \u2228 t = 6 \u2228 t = 7 := by\n        omega\n      rcases h\u2086 with (rfl | rfl | rfl | rfl | rfl | rfl)\n      \u00b7 \n        have h\u2087 : n = 16 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 16 := by\n          linarith\n        have h\u2089 : x \u2264 2 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 3 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 3 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n      \u00b7 \n        have h\u2087 : n = 81 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 81 := by\n          linarith\n        have h\u2089 : x \u2264 4 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 5 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 5 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n      \u00b7 \n        have h\u2087 : n = 256 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 256 := by\n          linarith\n        have h\u2089 : x \u2264 6 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 7 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 7 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n      \u00b7 \n        have h\u2087 : n = 625 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 625 := by\n          linarith\n        have h\u2089 : x \u2264 8 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 9 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 9 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n      \u00b7 \n        have h\u2087 : n = 1296 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 1296 := by\n          linarith\n        have h\u2089 : x \u2264 10 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 11 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 11 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n      \u00b7 \n        have h\u2087 : n = 2401 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 2401 := by\n          linarith\n        have h\u2089 : x \u2264 13 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 14 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 14 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n    have h\u2085 : t ^ 4 \u2265 8 ^ 4 := by\n      exact Nat.pow_le_pow_of_le_left h\u2084 4\n    have h\u2086 : n = t ^ 4 := by\n      linarith\n    have h\u2087 : n \u2265 4096 := by\n      rw [h\u2086]\n      <;> norm_num at h\u2085 \u22a2 <;> linarith\n    exact h\u2087\n  linarith\n", "proof_before": "theorem mathd_numbertheory_296 (n : \u2115) (h\u2080 : 2 \u2264 n) (h\u2081 : \u2203 x, x ^ 3 = n) (h\u2082 : \u2203 t, t ^ 4 = n) :\n    4096 \u2264 n  :=  by\n  have h_main : n \u2265 4096 := by\n    obtain \u27e8t, ht\u27e9 := h\u2082\n    have h\u2083 : t \u2265 2 := by\n      by_contra h\n      have h\u2084 : t \u2264 1 := by linarith\n      have h\u2085 : t = 0 \u2228 t = 1 := by\n        have h\u2085\u2081 : t \u2264 1 := h\u2084\n        have h\u2085\u2082 : t \u2265 0 := by positivity\n        interval_cases t <;> simp_all (config := {decide := true})\n      cases h\u2085 with\n      | inl h\u2085 =>\n        have h\u2086 : t = 0 := h\u2085\n        rw [h\u2086] at ht\n        norm_num at ht\n        <;> linarith\n      | inr h\u2085 =>\n        have h\u2086 : t = 1 := h\u2085\n        rw [h\u2086] at ht\n        norm_num at ht\n        <;> linarith\n    have h\u2084 : t \u2265 8 := by\n      by_contra h\n      have h\u2085 : t \u2264 7 := by linarith\n      have h\u2086 : t = 2 \u2228 t = 3 \u2228 t = 4 \u2228 t = 5 \u2228 t = 6 \u2228 t = 7 := by\n        omega\n      rcases h\u2086 with (rfl | rfl | rfl | rfl | rfl | rfl)\n      \u00b7 \n        have h\u2087 : n = 16 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 16 := by\n          linarith\n        have h\u2089 : x \u2264 2 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 3 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 3 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n      \u00b7 \n        have h\u2087 : n = 81 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 81 := by\n          linarith\n        have h\u2089 : x \u2264 4 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 5 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 5 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n      \u00b7 \n        have h\u2087 : n = 256 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 256 := by\n          linarith\n        have h\u2089 : x \u2264 6 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 7 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 7 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n      \u00b7 \n        have h\u2087 : n = 625 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 625 := by\n          linarith\n        have h\u2089 : x \u2264 8 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 9 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 9 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n      \u00b7 \n        have h\u2087 : n = 1296 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 1296 := by\n          linarith\n        have h\u2089 : x \u2264 10 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 11 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 11 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n      \u00b7 \n        have h\u2087 : n = 2401 := by\n          norm_num at ht \u22a2\n          <;> linarith\n        obtain \u27e8x, hx\u27e9 := h\u2081\n        have h\u2088 : x ^ 3 = 2401 := by\n          linarith\n        have h\u2089 : x \u2264 13 := by\n          by_contra h\u2089\n          have h\u2081\u2080 : x \u2265 14 := by omega\n          have h\u2081\u2081 : x ^ 3 \u2265 14 ^ 3 := by\n            exact Nat.pow_le_pow_of_le_left (by omega) 3\n          linarith\n        interval_cases x <;> norm_num at h\u2088 \u22a2 <;> omega\n    have h\u2085 : t ^ 4 \u2265 8 ^ 4 := by\n      exact Nat.pow_le_pow_of_le_left h\u2084 4\n    have h\u2086 : n = t ^ 4 := by\n      linarith\n    have h\u2087 : n \u2265 4096 := by\n      rw [h\u2086]\n      <;> norm_num at h\u2085 \u22a2 <;> linarith\n    exact h\u2087\n  linarith\n"}
{"mode": "clear", "steps_before": 84, "alphabet": 53, "compiles": false, "id": "amc12b_2002_p7", "proof_after": "theorem amc12b_2002_p7 (a b c : \u2115) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) (h\u2081 : b = a + 1) (h\u2082 : c = b + 1)\n    (h\u2083 : a * b * c = 8 * (a + b + c)) : a ^ 2 + (b ^ 2 + c ^ 2) = 77  :=  by\n  have h\u2084 : c = a + 2 := by\n    have h\u2084\u2081 : c = a + 2 := by\n      rw [h\u2082, h\u2081]\n      <;> ring_nf\n      <;> omega\n    exact h\u2084\u2081\n  have h\u2085 : a * (a + 1) * (a + 2) = 24 * (a + 1) := by\n    have h\u2085\u2081 : a * b * c = 8 * (a + b + c) := h\u2083\n    have h\u2085\u2082 : b = a + 1 := h\u2081\n    have h\u2085\u2083 : c = a + 2 := h\u2084\n    rw [h\u2085\u2082, h\u2085\u2083] at h\u2085\u2081\n    ring_nf at h\u2085\u2081 \u22a2\n    <;>\n    (try omega) <;>\n    (try nlinarith) <;>\n    (try ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith)\n    <;>\n    (try\n      {\n        cases a with\n        | zero => contradiction\n        | succ a' =>\n          cases a' with\n          | zero => contradiction\n          | succ a'' =>\n            cases a'' with\n            | zero => contradiction\n            | succ a''' =>\n              cases a''' with\n              | zero => contradiction\n              | succ a'''' =>\n                cases a'''' with\n                | zero =>\n                  norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                | succ a''''' =>\n                  cases a''''' with\n                  | zero =>\n                    norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                  | succ a'''''' =>\n                    cases a'''''' with\n                    | zero =>\n                      norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                    | succ a''''''' =>\n                      cases a''''''' with\n                      | zero =>\n                        norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                      | succ a'''''''' =>\n                        cases a'''''''' with\n                        | zero =>\n                          norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                        | succ a''''''''' =>\n                          cases a''''''''' with\n                          | zero =>\n                            norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                          | succ a'''''''''' =>\n                            cases a'''''''''' with\n                            | zero =>\n                              norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                            | succ a''''''''''' =>\n                              cases a''''''''''' with\n                              | zero =>\n                                norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                              | succ a'''''''''''' =>\n                                cases a'''''''''''' with\n                                | zero =>\n                                  norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                                | succ a''''''''''''' =>\n                                  cases a''''''''''''' with\n                                  | zero =>\n                                    norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                                  | succ a'''''''''''''' =>\n                                    cases a'''''''''''''' with\n                                    | zero =>\n                                      norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                                    | succ a''''''''''''''' =>\n                                      cases a''''''''''''''' with\n                                      | zero =>\n                                        norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                                      | succ a'''''''''''''''' =>\n                                        cases a'''''''''''''''' with\n                                        | zero =>\n                                          norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                                        | succ a''''''''''''''''' =>\n                                          simp_all [Nat.mul_add, Nat.add_mul, Nat.add_assoc]\n                                          <;> ring_nf at *\n                                          <;> nlinarith\n      })\n    <;>\n    omega\n  have h\u2086 : a = 4 := by\n    have h\u2086\u2081 : a \u2264 6 := by\n      by_contra h\n      have h\u2086\u2082 : a \u2265 7 := by\n        omega\n      have h\u2086\u2083 : a * (a + 1) * (a + 2) > 24 * (a + 1) := by\n        have h\u2086\u2084 : a \u2265 7 := h\u2086\u2082\n        have h\u2086\u2085 : a * (a + 1) * (a + 2) > 24 * (a + 1) := by\n          have h\u2086\u2086 : a * (a + 1) * (a + 2) = a * (a + 1) * (a + 2) := rfl\n          have h\u2086\u2087 : 24 * (a + 1) = 24 * (a + 1) := rfl\n          have h\u2086\u2088 : a * (a + 1) * (a + 2) > 24 * (a + 1) := by\n            have h\u2086\u2089 : a * (a + 2) \u2265 7 * 9 := by\n              nlinarith\n            nlinarith\n          exact h\u2086\u2088\n        exact h\u2086\u2085\n      linarith\n    interval_cases a <;> norm_num at h\u2085 \u22a2 <;>\n    (try omega) <;>\n    (try nlinarith)\n  have h\u2087 : b = 5 := by\n    rw [h\u2081, h\u2086]\n    <;> norm_num\n  have h\u2088 : c = 6 := by\n    rw [h\u2084, h\u2086]\n    <;> norm_num\n  have h\u2089 : a ^ 2 + (b ^ 2 + c ^ 2) = 77 := by\n    rw [h\u2086, h\u2087, h\u2088]\n    <;> norm_num\n  exact h\u2089\n", "proof_before": "theorem amc12b_2002_p7 (a b c : \u2115) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) (h\u2081 : b = a + 1) (h\u2082 : c = b + 1)\n    (h\u2083 : a * b * c = 8 * (a + b + c)) : a ^ 2 + (b ^ 2 + c ^ 2) = 77  :=  by\n  have h\u2084 : c = a + 2 := by\n    have h\u2084\u2081 : c = a + 2 := by\n      rw [h\u2082, h\u2081]\n      <;> ring_nf\n      <;> omega\n    exact h\u2084\u2081\n  have h\u2085 : a * (a + 1) * (a + 2) = 24 * (a + 1) := by\n    have h\u2085\u2081 : a * b * c = 8 * (a + b + c) := h\u2083\n    have h\u2085\u2082 : b = a + 1 := h\u2081\n    have h\u2085\u2083 : c = a + 2 := h\u2084\n    rw [h\u2085\u2082, h\u2085\u2083] at h\u2085\u2081\n    ring_nf at h\u2085\u2081 \u22a2\n    <;>\n    (try omega) <;>\n    (try nlinarith) <;>\n    (try ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith)\n    <;>\n    (try\n      {\n        cases a with\n        | zero => contradiction\n        | succ a' =>\n          cases a' with\n          | zero => contradiction\n          | succ a'' =>\n            cases a'' with\n            | zero => contradiction\n            | succ a''' =>\n              cases a''' with\n              | zero => contradiction\n              | succ a'''' =>\n                cases a'''' with\n                | zero =>\n                  norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                | succ a''''' =>\n                  cases a''''' with\n                  | zero =>\n                    norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                  | succ a'''''' =>\n                    cases a'''''' with\n                    | zero =>\n                      norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                    | succ a''''''' =>\n                      cases a''''''' with\n                      | zero =>\n                        norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                      | succ a'''''''' =>\n                        cases a'''''''' with\n                        | zero =>\n                          norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                        | succ a''''''''' =>\n                          cases a''''''''' with\n                          | zero =>\n                            norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                          | succ a'''''''''' =>\n                            cases a'''''''''' with\n                            | zero =>\n                              norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                            | succ a''''''''''' =>\n                              cases a''''''''''' with\n                              | zero =>\n                                norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                              | succ a'''''''''''' =>\n                                cases a'''''''''''' with\n                                | zero =>\n                                  norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                                | succ a''''''''''''' =>\n                                  cases a''''''''''''' with\n                                  | zero =>\n                                    norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                                  | succ a'''''''''''''' =>\n                                    cases a'''''''''''''' with\n                                    | zero =>\n                                      norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                                    | succ a''''''''''''''' =>\n                                      cases a''''''''''''''' with\n                                      | zero =>\n                                        norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                                      | succ a'''''''''''''''' =>\n                                        cases a'''''''''''''''' with\n                                        | zero =>\n                                          norm_num at h\u2085\u2081 \u22a2 <;> ring_nf at h\u2085\u2081 \u22a2 <;> nlinarith\n                                        | succ a''''''''''''''''' =>\n                                          simp_all [Nat.mul_add, Nat.add_mul, Nat.add_assoc]\n                                          <;> ring_nf at *\n                                          <;> nlinarith\n      })\n    <;>\n    omega\n  have h\u2086 : a = 4 := by\n    have h\u2086\u2081 : a \u2264 6 := by\n      by_contra h\n      have h\u2086\u2082 : a \u2265 7 := by\n        omega\n      have h\u2086\u2083 : a * (a + 1) * (a + 2) > 24 * (a + 1) := by\n        have h\u2086\u2084 : a \u2265 7 := h\u2086\u2082\n        have h\u2086\u2085 : a * (a + 1) * (a + 2) > 24 * (a + 1) := by\n          have h\u2086\u2086 : a * (a + 1) * (a + 2) = a * (a + 1) * (a + 2) := rfl\n          have h\u2086\u2087 : 24 * (a + 1) = 24 * (a + 1) := rfl\n          have h\u2086\u2088 : a * (a + 1) * (a + 2) > 24 * (a + 1) := by\n            have h\u2086\u2089 : a * (a + 2) \u2265 7 * 9 := by\n              nlinarith\n            nlinarith\n          exact h\u2086\u2088\n        exact h\u2086\u2085\n      linarith\n    interval_cases a <;> norm_num at h\u2085 \u22a2 <;>\n    (try omega) <;>\n    (try nlinarith)\n  have h\u2087 : b = 5 := by\n    rw [h\u2081, h\u2086]\n    <;> norm_num\n  have h\u2088 : c = 6 := by\n    rw [h\u2084, h\u2086]\n    <;> norm_num\n  have h\u2089 : a ^ 2 + (b ^ 2 + c ^ 2) = 77 := by\n    rw [h\u2086, h\u2087, h\u2088]\n    <;> norm_num\n  exact h\u2089\n"}
{"mode": "clear", "steps_before": 12, "alphabet": 16, "compiles": true, "steps_after": 8, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "timed_out": false, "fixpoint": true, "chars_before": 208, "chars_after": 141, "seconds": 409.9, "id": "mathd_algebra_346", "proof_after": "theorem mathd_algebra_346 (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 2 * x - 3) (h\u2081 : \u2200 x, g x = x + 1) :\n    g (f 5 - 1) = 7  :=  by\n  have h\u2082 : f 5 = 7 := by\n    rw [h\u2080]\n    norm_num\n  have h\u2084 : g (f 5 - 1) = 7 := by\n    rw [h\u2081]\n    <;> norm_num\n    <;> linarith\n  rw [h\u2084]\n", "proof_before": "theorem mathd_algebra_346 (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 2 * x - 3) (h\u2081 : \u2200 x, g x = x + 1) :\n    g (f 5 - 1) = 7  :=  by\n  have h\u2082 : f 5 = 7 := by\n    rw [h\u2080]\n    norm_num\n  have h\u2083 : f 5 - 1 = 6 := by\n    rw [h\u2082]\n    norm_num\n  have h\u2084 : g (f 5 - 1) = 7 := by\n    rw [h\u2083]\n    rw [h\u2081]\n    <;> norm_num\n    <;> linarith\n  rw [h\u2084]\n"}
{"mode": "clear", "steps_before": 52, "alphabet": 27, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 1840, "chars_after": 20, "seconds": 73.5, "id": "mathd_algebra_441", "proof_after": "theorem mathd_algebra_441 (x : \u211d) (h\u2080 : x \u2260 0) :\n    12 / (x * x) * (x ^ 4 / (14 * x)) * (35 / (3 * x)) = 10  :=  by\n  ring\n  field_simp\n", "proof_before": "theorem mathd_algebra_441 (x : \u211d) (h\u2080 : x \u2260 0) :\n    12 / (x * x) * (x ^ 4 / (14 * x)) * (35 / (3 * x)) = 10  :=  by\n  have h\u2081 : 12 / (x * x) * (x ^ 4 / (14 * x)) * (35 / (3 * x)) = (12 * x ^ 4 * 35) / ((x * x) * (14 * x) * (3 * x)) := by\n    have h\u2081\u2081 : 12 / (x * x) * (x ^ 4 / (14 * x)) * (35 / (3 * x)) = (12 / (x * x)) * ((x ^ 4 / (14 * x)) * (35 / (3 * x))) := by\n      ring\n    rw [h\u2081\u2081]\n    have h\u2081\u2082 : (x ^ 4 / (14 * x)) * (35 / (3 * x)) = (x ^ 4 * 35) / ((14 * x) * (3 * x)) := by\n      field_simp [h\u2080, pow_ne_zero, mul_comm, mul_assoc, mul_left_comm]\n      <;> ring_nf\n      <;> field_simp [h\u2080, pow_ne_zero]\n      <;> ring_nf\n    rw [h\u2081\u2082]\n    have h\u2081\u2083 : (12 / (x * x)) * ((x ^ 4 * 35) / ((14 * x) * (3 * x))) = (12 * x ^ 4 * 35) / ((x * x) * (14 * x) * (3 * x)) := by\n      field_simp [h\u2080, pow_ne_zero, mul_comm, mul_assoc, mul_left_comm]\n      <;> ring_nf\n      <;> field_simp [h\u2080, pow_ne_zero]\n      <;> ring_nf\n    rw [h\u2081\u2083]\n    <;> ring_nf\n  have h\u2082 : (x * x) * (14 * x) * (3 * x) = 42 * x ^ 4 := by\n    have h\u2082\u2081 : (x * x) * (14 * x) * (3 * x) = 42 * x ^ 4 := by\n      ring_nf\n      <;>\n      simp_all [mul_assoc, mul_comm, mul_left_comm]\n      <;>\n      ring_nf\n      <;>\n      linarith\n    rw [h\u2082\u2081]\n  have h\u2083 : 12 / (x * x) * (x ^ 4 / (14 * x)) * (35 / (3 * x)) = (12 * x ^ 4 * 35) / (42 * x ^ 4) := by\n    rw [h\u2081]\n    rw [h\u2082]\n    <;> field_simp [h\u2080]\n    <;> ring_nf\n    <;> field_simp [h\u2080]\n    <;> ring_nf\n    <;> field_simp [h\u2080]\n    <;> linarith\n  have h\u2084 : (12 * x ^ 4 * 35) / (42 * x ^ 4) = 10 := by\n    have h\u2084\u2081 : x ^ 4 \u2260 0 := by\n      exact pow_ne_zero 4 h\u2080\n    have h\u2084\u2082 : (12 * x ^ 4 * 35) / (42 * x ^ 4) = 10 := by\n      have h\u2084\u2083 : (12 * x ^ 4 * 35) / (42 * x ^ 4) = (12 * 35 : \u211d) / 42 := by\n        field_simp [h\u2084\u2081]\n        <;> ring_nf\n        <;> field_simp [h\u2084\u2081]\n        <;> ring_nf\n      rw [h\u2084\u2083]\n      norm_num\n    rw [h\u2084\u2082]\n  have h\u2085 : 12 / (x * x) * (x ^ 4 / (14 * x)) * (35 / (3 * x)) = 10 := by\n    rw [h\u2083]\n    rw [h\u2084]\n  rw [h\u2085]\n"}
{"mode": "clear", "steps_before": 165, "alphabet": 49, "compiles": true, "steps_after": 165, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "timed_out": true, "fixpoint": false, "chars_before": 4210, "chars_after": 4210, "seconds": 499.5, "id": "aime_1997_p9", "proof_after": "theorem aime_1997_p9 (a : \u211d) (h\u2080 : 0 < a)\n    (h\u2081 : 1 / a - Int.floor (1 / a) = a ^ 2 - Int.floor (a ^ 2)) (h\u2082 : 2 < a ^ 2) (h\u2083 : a ^ 2 < 3) :\n    a ^ 12 - 144 * (1 / a) = 233  :=  by\n  have h\u2084 : Int.floor (a ^ 2 : \u211d) = 2 := by\n    rw [Int.floor_eq_iff]\n    constructor <;> norm_num at h\u2082 h\u2083 \u22a2 <;>\n    (try norm_num) <;>\n    (try nlinarith) <;>\n    (try linarith) <;>\n    (try nlinarith [h\u2080, h\u2082, h\u2083])\n    <;>\n    (try norm_num at h\u2082 h\u2083 \u22a2) <;>\n    (try nlinarith) <;>\n    (try linarith) <;>\n    (try nlinarith [h\u2080, h\u2082, h\u2083])\n    <;>\n    (try norm_num at h\u2082 h\u2083 \u22a2) <;>\n    (try nlinarith) <;>\n    (try linarith) <;>\n    (try nlinarith [h\u2080, h\u2082, h\u2083])\n  have h\u2085 : Int.floor (1 / a : \u211d) = 0 := by\n    have h\u2085\u2081 : 0 < (1 / a : \u211d) := by positivity\n    have h\u2085\u2082 : (1 / a : \u211d) < 1 := by\n      have h\u2085\u2082\u2081 : 0 < a := h\u2080\n      have h\u2085\u2082\u2082 : 0 < a ^ 2 := by positivity\n      have h\u2085\u2082\u2083 : a ^ 2 < 3 := h\u2083\n      have h\u2085\u2082\u2084 : 2 < a ^ 2 := h\u2082\n      have h\u2085\u2082\u2085 : a > 1 := by\n        by_contra h\n        have h\u2085\u2082\u2086 : a \u2264 1 := by linarith\n        have h\u2085\u2082\u2087 : a ^ 2 \u2264 1 := by\n          nlinarith\n        linarith\n      have h\u2085\u2082\u2088 : (1 / a : \u211d) < 1 := by\n        rw [div_lt_one (by positivity)]\n        <;> nlinarith\n      exact h\u2085\u2082\u2088\n    have h\u2085\u2083 : Int.floor (1 / a : \u211d) = 0 := by\n      rw [Int.floor_eq_iff]\n      norm_num at h\u2085\u2082 \u22a2\n      constructor <;> nlinarith\n    exact h\u2085\u2083\n  have h\u2086 : 1 / a = a ^ 2 - 2 := by\n    have h\u2086\u2081 : (1 / a : \u211d) - \u230a(1 / a : \u211d)\u230b = a ^ 2 - \u230a(a ^ 2 : \u211d)\u230b := by simpa using h\u2081\n    rw [h\u2085, h\u2084] at h\u2086\u2081\n    ring_nf at h\u2086\u2081 \u22a2\n    <;> linarith\n  have h\u2087 : a ^ 3 - 2 * a - 1 = 0 := by\n    have h\u2087\u2081 : a \u2260 0 := by positivity\n    have h\u2087\u2082 : 1 / a = a ^ 2 - 2 := h\u2086\n    have h\u2087\u2083 : a ^ 3 - 2 * a - 1 = 0 := by\n      have h\u2087\u2084 : a ^ 3 - 2 * a - 1 = 0 := by\n        have h\u2087\u2085 : 1 / a = a ^ 2 - 2 := h\u2086\n        have h\u2087\u2086 : a \u2260 0 := by positivity\n        field_simp at h\u2087\u2085\n        ring_nf at h\u2087\u2085 \u22a2\n        nlinarith [sq_nonneg (a - 1), sq_nonneg (a + 1), sq_nonneg (a ^ 2 - 1)]\n      exact h\u2087\u2084\n    exact h\u2087\u2083\n  have h\u2088 : a ^ 2 - a - 1 = 0 := by\n    have h\u2088\u2081 : a \u2260 0 := by positivity\n    have h\u2088\u2082 : a ^ 3 - 2 * a - 1 = 0 := h\u2087\n    have h\u2088\u2083 : a ^ 2 - a - 1 = 0 := by\n      have h\u2088\u2084 : a ^ 3 - 2 * a - 1 = 0 := h\u2087\n      have h\u2088\u2085 : (a + 1) * (a ^ 2 - a - 1) = 0 := by\n        ring_nf at h\u2088\u2084 \u22a2\n        nlinarith [sq_nonneg (a - 1), sq_nonneg (a + 1), sq_nonneg (a ^ 2 - 1)]\n      have h\u2088\u2086 : a + 1 > 0 := by nlinarith\n      have h\u2088\u2087 : a ^ 2 - a - 1 = 0 := by\n        apply mul_left_cancel\u2080 (show (a + 1 : \u211d) \u2260 0 by nlinarith)\n        nlinarith\n      exact h\u2088\u2087\n    exact h\u2088\u2083\n  have h\u2089 : a ^ 12 - 144 * (1 / a) = 233 := by\n    have h\u2089\u2081 : a ^ 2 = a + 1 := by\n      have h\u2089\u2081\u2081 : a ^ 2 - a - 1 = 0 := h\u2088\n      nlinarith\n    have h\u2089\u2082 : a ^ 3 = 2 * a + 1 := by\n      have h\u2089\u2082\u2081 : a ^ 3 - 2 * a - 1 = 0 := h\u2087\n      nlinarith\n    have h\u2089\u2083 : a ^ 4 = 3 * a + 2 := by\n      calc\n        a ^ 4 = a * a ^ 3 := by ring\n        _ = a * (2 * a + 1) := by rw [h\u2089\u2082]\n        _ = 2 * a ^ 2 + a := by ring\n        _ = 2 * (a + 1) + a := by rw [h\u2089\u2081]\n        _ = 3 * a + 2 := by ring\n    have h\u2089\u2084 : a ^ 5 = 5 * a + 3 := by\n      calc\n        a ^ 5 = a * a ^ 4 := by ring\n        _ = a * (3 * a + 2) := by rw [h\u2089\u2083]\n        _ = 3 * a ^ 2 + 2 * a := by ring\n        _ = 3 * (a + 1) + 2 * a := by rw [h\u2089\u2081]\n        _ = 5 * a + 3 := by ring\n    have h\u2089\u2085 : a ^ 6 = 8 * a + 5 := by\n      calc\n        a ^ 6 = a * a ^ 5 := by ring\n        _ = a * (5 * a + 3) := by rw [h\u2089\u2084]\n        _ = 5 * a ^ 2 + 3 * a := by ring\n        _ = 5 * (a + 1) + 3 * a := by rw [h\u2089\u2081]\n        _ = 8 * a + 5 := by ring\n    have h\u2089\u2086 : a ^ 12 = 233 + 144 * (1 / a) := by\n      calc\n        a ^ 12 = (a ^ 6) ^ 2 := by ring\n        _ = (8 * a + 5) ^ 2 := by rw [h\u2089\u2085]\n        _ = 64 * a ^ 2 + 80 * a + 25 := by ring\n        _ = 64 * (a + 1) + 80 * a + 25 := by rw [h\u2089\u2081]\n        _ = 144 * a + 89 := by ring\n        _ = 233 + 144 * (1 / a) := by\n          have h\u2089\u2086\u2081 : 1 / a = a ^ 2 - 2 := h\u2086\n          have h\u2089\u2086\u2082 : a ^ 2 = a + 1 := h\u2089\u2081\n          have h\u2089\u2086\u2083 : 1 / a = (a + 1 : \u211d) - 2 := by\n            rw [h\u2089\u2086\u2081, h\u2089\u2086\u2082]\n            <;> ring_nf\n          have h\u2089\u2086\u2084 : (a : \u211d) \u2260 0 := by positivity\n          field_simp [h\u2089\u2086\u2084] at h\u2089\u2086\u2081 h\u2089\u2086\u2083 \u22a2\n          nlinarith\n        _ = 233 + 144 * (1 / a) := by ring\n    have h\u2089\u2087 : a ^ 12 - 144 * (1 / a) = 233 := by\n      calc\n        a ^ 12 - 144 * (1 / a) = (233 + 144 * (1 / a)) - 144 * (1 / a) := by rw [h\u2089\u2086]\n        _ = 233 := by ring\n    exact h\u2089\u2087\n  apply h\u2089\n", "proof_before": "theorem aime_1997_p9 (a : \u211d) (h\u2080 : 0 < a)\n    (h\u2081 : 1 / a - Int.floor (1 / a) = a ^ 2 - Int.floor (a ^ 2)) (h\u2082 : 2 < a ^ 2) (h\u2083 : a ^ 2 < 3) :\n    a ^ 12 - 144 * (1 / a) = 233  :=  by\n  have h\u2084 : Int.floor (a ^ 2 : \u211d) = 2 := by\n    rw [Int.floor_eq_iff]\n    constructor <;> norm_num at h\u2082 h\u2083 \u22a2 <;>\n    (try norm_num) <;>\n    (try nlinarith) <;>\n    (try linarith) <;>\n    (try nlinarith [h\u2080, h\u2082, h\u2083])\n    <;>\n    (try norm_num at h\u2082 h\u2083 \u22a2) <;>\n    (try nlinarith) <;>\n    (try linarith) <;>\n    (try nlinarith [h\u2080, h\u2082, h\u2083])\n    <;>\n    (try norm_num at h\u2082 h\u2083 \u22a2) <;>\n    (try nlinarith) <;>\n    (try linarith) <;>\n    (try nlinarith [h\u2080, h\u2082, h\u2083])\n  have h\u2085 : Int.floor (1 / a : \u211d) = 0 := by\n    have h\u2085\u2081 : 0 < (1 / a : \u211d) := by positivity\n    have h\u2085\u2082 : (1 / a : \u211d) < 1 := by\n      have h\u2085\u2082\u2081 : 0 < a := h\u2080\n      have h\u2085\u2082\u2082 : 0 < a ^ 2 := by positivity\n      have h\u2085\u2082\u2083 : a ^ 2 < 3 := h\u2083\n      have h\u2085\u2082\u2084 : 2 < a ^ 2 := h\u2082\n      have h\u2085\u2082\u2085 : a > 1 := by\n        by_contra h\n        have h\u2085\u2082\u2086 : a \u2264 1 := by linarith\n        have h\u2085\u2082\u2087 : a ^ 2 \u2264 1 := by\n          nlinarith\n        linarith\n      have h\u2085\u2082\u2088 : (1 / a : \u211d) < 1 := by\n        rw [div_lt_one (by positivity)]\n        <;> nlinarith\n      exact h\u2085\u2082\u2088\n    have h\u2085\u2083 : Int.floor (1 / a : \u211d) = 0 := by\n      rw [Int.floor_eq_iff]\n      norm_num at h\u2085\u2082 \u22a2\n      constructor <;> nlinarith\n    exact h\u2085\u2083\n  have h\u2086 : 1 / a = a ^ 2 - 2 := by\n    have h\u2086\u2081 : (1 / a : \u211d) - \u230a(1 / a : \u211d)\u230b = a ^ 2 - \u230a(a ^ 2 : \u211d)\u230b := by simpa using h\u2081\n    rw [h\u2085, h\u2084] at h\u2086\u2081\n    ring_nf at h\u2086\u2081 \u22a2\n    <;> linarith\n  have h\u2087 : a ^ 3 - 2 * a - 1 = 0 := by\n    have h\u2087\u2081 : a \u2260 0 := by positivity\n    have h\u2087\u2082 : 1 / a = a ^ 2 - 2 := h\u2086\n    have h\u2087\u2083 : a ^ 3 - 2 * a - 1 = 0 := by\n      have h\u2087\u2084 : a ^ 3 - 2 * a - 1 = 0 := by\n        have h\u2087\u2085 : 1 / a = a ^ 2 - 2 := h\u2086\n        have h\u2087\u2086 : a \u2260 0 := by positivity\n        field_simp at h\u2087\u2085\n        ring_nf at h\u2087\u2085 \u22a2\n        nlinarith [sq_nonneg (a - 1), sq_nonneg (a + 1), sq_nonneg (a ^ 2 - 1)]\n      exact h\u2087\u2084\n    exact h\u2087\u2083\n  have h\u2088 : a ^ 2 - a - 1 = 0 := by\n    have h\u2088\u2081 : a \u2260 0 := by positivity\n    have h\u2088\u2082 : a ^ 3 - 2 * a - 1 = 0 := h\u2087\n    have h\u2088\u2083 : a ^ 2 - a - 1 = 0 := by\n      have h\u2088\u2084 : a ^ 3 - 2 * a - 1 = 0 := h\u2087\n      have h\u2088\u2085 : (a + 1) * (a ^ 2 - a - 1) = 0 := by\n        ring_nf at h\u2088\u2084 \u22a2\n        nlinarith [sq_nonneg (a - 1), sq_nonneg (a + 1), sq_nonneg (a ^ 2 - 1)]\n      have h\u2088\u2086 : a + 1 > 0 := by nlinarith\n      have h\u2088\u2087 : a ^ 2 - a - 1 = 0 := by\n        apply mul_left_cancel\u2080 (show (a + 1 : \u211d) \u2260 0 by nlinarith)\n        nlinarith\n      exact h\u2088\u2087\n    exact h\u2088\u2083\n  have h\u2089 : a ^ 12 - 144 * (1 / a) = 233 := by\n    have h\u2089\u2081 : a ^ 2 = a + 1 := by\n      have h\u2089\u2081\u2081 : a ^ 2 - a - 1 = 0 := h\u2088\n      nlinarith\n    have h\u2089\u2082 : a ^ 3 = 2 * a + 1 := by\n      have h\u2089\u2082\u2081 : a ^ 3 - 2 * a - 1 = 0 := h\u2087\n      nlinarith\n    have h\u2089\u2083 : a ^ 4 = 3 * a + 2 := by\n      calc\n        a ^ 4 = a * a ^ 3 := by ring\n        _ = a * (2 * a + 1) := by rw [h\u2089\u2082]\n        _ = 2 * a ^ 2 + a := by ring\n        _ = 2 * (a + 1) + a := by rw [h\u2089\u2081]\n        _ = 3 * a + 2 := by ring\n    have h\u2089\u2084 : a ^ 5 = 5 * a + 3 := by\n      calc\n        a ^ 5 = a * a ^ 4 := by ring\n        _ = a * (3 * a + 2) := by rw [h\u2089\u2083]\n        _ = 3 * a ^ 2 + 2 * a := by ring\n        _ = 3 * (a + 1) + 2 * a := by rw [h\u2089\u2081]\n        _ = 5 * a + 3 := by ring\n    have h\u2089\u2085 : a ^ 6 = 8 * a + 5 := by\n      calc\n        a ^ 6 = a * a ^ 5 := by ring\n        _ = a * (5 * a + 3) := by rw [h\u2089\u2084]\n        _ = 5 * a ^ 2 + 3 * a := by ring\n        _ = 5 * (a + 1) + 3 * a := by rw [h\u2089\u2081]\n        _ = 8 * a + 5 := by ring\n    have h\u2089\u2086 : a ^ 12 = 233 + 144 * (1 / a) := by\n      calc\n        a ^ 12 = (a ^ 6) ^ 2 := by ring\n        _ = (8 * a + 5) ^ 2 := by rw [h\u2089\u2085]\n        _ = 64 * a ^ 2 + 80 * a + 25 := by ring\n        _ = 64 * (a + 1) + 80 * a + 25 := by rw [h\u2089\u2081]\n        _ = 144 * a + 89 := by ring\n        _ = 233 + 144 * (1 / a) := by\n          have h\u2089\u2086\u2081 : 1 / a = a ^ 2 - 2 := h\u2086\n          have h\u2089\u2086\u2082 : a ^ 2 = a + 1 := h\u2089\u2081\n          have h\u2089\u2086\u2083 : 1 / a = (a + 1 : \u211d) - 2 := by\n            rw [h\u2089\u2086\u2081, h\u2089\u2086\u2082]\n            <;> ring_nf\n          have h\u2089\u2086\u2084 : (a : \u211d) \u2260 0 := by positivity\n          field_simp [h\u2089\u2086\u2084] at h\u2089\u2086\u2081 h\u2089\u2086\u2083 \u22a2\n          nlinarith\n        _ = 233 + 144 * (1 / a) := by ring\n    have h\u2089\u2087 : a ^ 12 - 144 * (1 / a) = 233 := by\n      calc\n        a ^ 12 - 144 * (1 / a) = (233 + 144 * (1 / a)) - 144 * (1 / a) := by rw [h\u2089\u2086]\n        _ = 233 := by ring\n    exact h\u2089\u2087\n  apply h\u2089\n"}
{"mode": "clear", "steps_before": 41, "alphabet": 27, "compiles": false, "id": "mathd_algebra_196", "proof_after": "theorem mathd_algebra_196 (S : Finset \u211d) (h\u2080 : \u2200 x : \u211d, x \u2208 S \u2194 abs (2 - x) = 3) :\n    (\u2211 k in S, k) = 4  :=  by\n  have h\u2081 : S = { -1, 5 } := by\n    apply Finset.ext\n    intro x\n    simp only [Finset.mem_insert, Finset.mem_singleton, h\u2080]\n    constructor\n    \u00b7 \n      intro h\n      have h\u2082 : abs (2 - x) = 3 := h\n      have h\u2083 : 2 - x = 3 \u2228 2 - x = -3 := by\n        apply eq_or_eq_neg_of_abs_eq\n        <;> linarith\n      cases h\u2083 with\n      | inl h\u2083 =>\n        have h\u2084 : x = -1 := by linarith\n        rw [h\u2084]\n        norm_num\n      | inr h\u2083 =>\n        have h\u2084 : x = 5 := by linarith\n        rw [h\u2084]\n        norm_num\n    \u00b7 \n      intro h\n      cases h with\n      | inl h =>\n        rw [h]\n        norm_num [abs_of_nonneg, abs_of_nonpos]\n      | inr h =>\n        rw [h]\n        norm_num [abs_of_nonneg, abs_of_nonpos]\n  have h\u2082 : (\u2211 k in S, k) = 4 := by\n    rw [h\u2081]\n    norm_num [Finset.sum_pair (show (-1 : \u211d) \u2260 5 by norm_num)]\n    <;>\n    simp_all [Finset.sum_singleton]\n    <;>\n    norm_num\n    <;>\n    linarith\n  exact h\u2082\n", "proof_before": "theorem mathd_algebra_196 (S : Finset \u211d) (h\u2080 : \u2200 x : \u211d, x \u2208 S \u2194 abs (2 - x) = 3) :\n    (\u2211 k in S, k) = 4  :=  by\n  have h\u2081 : S = { -1, 5 } := by\n    apply Finset.ext\n    intro x\n    simp only [Finset.mem_insert, Finset.mem_singleton, h\u2080]\n    constructor\n    \u00b7 \n      intro h\n      have h\u2082 : abs (2 - x) = 3 := h\n      have h\u2083 : 2 - x = 3 \u2228 2 - x = -3 := by\n        apply eq_or_eq_neg_of_abs_eq\n        <;> linarith\n      cases h\u2083 with\n      | inl h\u2083 =>\n        have h\u2084 : x = -1 := by linarith\n        rw [h\u2084]\n        norm_num\n      | inr h\u2083 =>\n        have h\u2084 : x = 5 := by linarith\n        rw [h\u2084]\n        norm_num\n    \u00b7 \n      intro h\n      cases h with\n      | inl h =>\n        rw [h]\n        norm_num [abs_of_nonneg, abs_of_nonpos]\n      | inr h =>\n        rw [h]\n        norm_num [abs_of_nonneg, abs_of_nonpos]\n  have h\u2082 : (\u2211 k in S, k) = 4 := by\n    rw [h\u2081]\n    norm_num [Finset.sum_pair (show (-1 : \u211d) \u2260 5 by norm_num)]\n    <;>\n    simp_all [Finset.sum_singleton]\n    <;>\n    norm_num\n    <;>\n    linarith\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 78, "alphabet": 41, "compiles": false, "id": "algebra_bleqa_apbon2msqrtableqambsqon8b", "proof_after": "theorem algebra_bleqa_apbon2msqrtableqambsqon8b (a b : \u211d) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : b \u2264 a) :\n    (a + b) / 2 - Real.sqrt (a * b) \u2264 (a - b) ^ 2 / (8 * b)  :=  by\n  have h\u2082 : 0 < a := by linarith\n  have h\u2083 : 0 < b := by linarith\n  have h\u2084 : 0 < Real.sqrt a := Real.sqrt_pos.mpr h\u2082\n  have h\u2085 : 0 < Real.sqrt b := Real.sqrt_pos.mpr h\u2083\n  have h\u2086 : Real.sqrt b \u2264 Real.sqrt a := Real.sqrt_le_sqrt h\u2081\n  set x := Real.sqrt a\n  set y := Real.sqrt b\n  have h\u2087 : x \u2265 y := h\u2086\n  have h\u2088 : 0 < y := h\u2085\n  have h\u2089 : 0 < x := h\u2084\n  have h\u2081\u2080 : (a + b) / 2 - Real.sqrt (a * b) = (x - y)^2 / 2 := by\n    have h\u2081\u2080\u2081 : a = x ^ 2 := by\n      rw [\u2190 Real.sq_sqrt (le_of_lt h\u2082)]\n      <;> simp [x]\n    have h\u2081\u2080\u2082 : b = y ^ 2 := by\n      rw [\u2190 Real.sq_sqrt (le_of_lt h\u2083)]\n      <;> simp [y]\n    rw [h\u2081\u2080\u2081, h\u2081\u2080\u2082]\n    have h\u2081\u2080\u2083 : Real.sqrt ((x ^ 2) * (y ^ 2)) = x * y := by\n      rw [Real.sqrt_eq_iff_sq_eq (by positivity) (by positivity)]\n      nlinarith\n    have h\u2081\u2080\u2084 : (x ^ 2 + y ^ 2) / 2 - Real.sqrt ((x ^ 2) * (y ^ 2)) = (x - y) ^ 2 / 2 := by\n      rw [h\u2081\u2080\u2083]\n      ring_nf\n      <;> field_simp\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (x - y)]\n    rw [h\u2081\u2080\u2084]\n    <;> ring_nf\n  have h\u2081\u2081 : (a - b)^2 / (8 * b) = (x - y)^2 * (x + y)^2 / (8 * y^2) := by\n    have h\u2081\u2081\u2081 : a = x ^ 2 := by\n      rw [\u2190 Real.sq_sqrt (le_of_lt h\u2082)]\n      <;> simp [x]\n    have h\u2081\u2081\u2082 : b = y ^ 2 := by\n      rw [\u2190 Real.sq_sqrt (le_of_lt h\u2083)]\n      <;> simp [y]\n    rw [h\u2081\u2081\u2081, h\u2081\u2081\u2082]\n    have h\u2081\u2081\u2083 : (x ^ 2 - y ^ 2) ^ 2 = (x - y) ^ 2 * (x + y) ^ 2 := by\n      ring_nf\n      <;>\n      nlinarith [sq_nonneg (x - y), sq_nonneg (x + y)]\n    rw [h\u2081\u2081\u2083]\n    <;> field_simp [h\u2083.ne', h\u2082.ne', h\u2088.ne', h\u2089.ne']\n    <;> ring_nf\n    <;> field_simp [h\u2083.ne', h\u2082.ne', h\u2088.ne', h\u2089.ne']\n    <;> ring_nf\n    <;> nlinarith [sq_nonneg (x - y), sq_nonneg (x + y)]\n  have h\u2081\u2082 : (x - y)^2 / 2 \u2264 (x - y)^2 * (x + y)^2 / (8 * y^2) := by\n    have h\u2081\u2082\u2081 : 0 \u2264 (x - y) ^ 2 := sq_nonneg (x - y)\n    have h\u2081\u2082\u2082 : 0 < y := h\u2088\n    have h\u2081\u2082\u2083 : 0 < x := h\u2089\n    have h\u2081\u2082\u2084 : 0 < y ^ 2 := by positivity\n    have h\u2081\u2082\u2085 : 0 < x ^ 2 := by positivity\n    have h\u2081\u2082\u2086 : 4 * y ^ 2 \u2264 (x + y) ^ 2 := by\n      nlinarith [sq_nonneg (x - y), sq_nonneg (x + y), sq_nonneg (x - 3 * y),\n        sq_nonneg (x + 3 * y)]\n    have h\u2081\u2082\u2087 : (x - y) ^ 2 / 2 \u2264 (x - y) ^ 2 * (x + y) ^ 2 / (8 * y ^ 2) := by\n      by_cases h : (x - y) ^ 2 = 0\n      \u00b7 rw [h]\n        <;> simp\n        <;> positivity\n      \u00b7 have h\u2081\u2082\u2088 : 0 < (x - y) ^ 2 := by positivity\n        have h\u2081\u2082\u2089 : 4 * y ^ 2 \u2264 (x + y) ^ 2 := h\u2081\u2082\u2086\n        have h\u2081\u2083\u2080 : 0 < 8 * y ^ 2 := by positivity\n        have h\u2081\u2083\u2081 : 0 < (x - y) ^ 2 * (x + y) ^ 2 := by positivity\n        rw [div_le_div_iff (by positivity) (by positivity)]\n        nlinarith [sq_nonneg (x - y), sq_nonneg (x + y), sq_nonneg (x - 3 * y),\n          sq_nonneg (x + 3 * y)]\n    exact h\u2081\u2082\u2087\n  have h\u2081\u2083 : (a + b) / 2 - Real.sqrt (a * b) \u2264 (a - b) ^ 2 / (8 * b) := by\n    rw [h\u2081\u2080, h\u2081\u2081]\n    <;> linarith\n  exact h\u2081\u2083\n", "proof_before": "theorem algebra_bleqa_apbon2msqrtableqambsqon8b (a b : \u211d) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : b \u2264 a) :\n    (a + b) / 2 - Real.sqrt (a * b) \u2264 (a - b) ^ 2 / (8 * b)  :=  by\n  have h\u2082 : 0 < a := by linarith\n  have h\u2083 : 0 < b := by linarith\n  have h\u2084 : 0 < Real.sqrt a := Real.sqrt_pos.mpr h\u2082\n  have h\u2085 : 0 < Real.sqrt b := Real.sqrt_pos.mpr h\u2083\n  have h\u2086 : Real.sqrt b \u2264 Real.sqrt a := Real.sqrt_le_sqrt h\u2081\n  set x := Real.sqrt a\n  set y := Real.sqrt b\n  have h\u2087 : x \u2265 y := h\u2086\n  have h\u2088 : 0 < y := h\u2085\n  have h\u2089 : 0 < x := h\u2084\n  have h\u2081\u2080 : (a + b) / 2 - Real.sqrt (a * b) = (x - y)^2 / 2 := by\n    have h\u2081\u2080\u2081 : a = x ^ 2 := by\n      rw [\u2190 Real.sq_sqrt (le_of_lt h\u2082)]\n      <;> simp [x]\n    have h\u2081\u2080\u2082 : b = y ^ 2 := by\n      rw [\u2190 Real.sq_sqrt (le_of_lt h\u2083)]\n      <;> simp [y]\n    rw [h\u2081\u2080\u2081, h\u2081\u2080\u2082]\n    have h\u2081\u2080\u2083 : Real.sqrt ((x ^ 2) * (y ^ 2)) = x * y := by\n      rw [Real.sqrt_eq_iff_sq_eq (by positivity) (by positivity)]\n      nlinarith\n    have h\u2081\u2080\u2084 : (x ^ 2 + y ^ 2) / 2 - Real.sqrt ((x ^ 2) * (y ^ 2)) = (x - y) ^ 2 / 2 := by\n      rw [h\u2081\u2080\u2083]\n      ring_nf\n      <;> field_simp\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (x - y)]\n    rw [h\u2081\u2080\u2084]\n    <;> ring_nf\n  have h\u2081\u2081 : (a - b)^2 / (8 * b) = (x - y)^2 * (x + y)^2 / (8 * y^2) := by\n    have h\u2081\u2081\u2081 : a = x ^ 2 := by\n      rw [\u2190 Real.sq_sqrt (le_of_lt h\u2082)]\n      <;> simp [x]\n    have h\u2081\u2081\u2082 : b = y ^ 2 := by\n      rw [\u2190 Real.sq_sqrt (le_of_lt h\u2083)]\n      <;> simp [y]\n    rw [h\u2081\u2081\u2081, h\u2081\u2081\u2082]\n    have h\u2081\u2081\u2083 : (x ^ 2 - y ^ 2) ^ 2 = (x - y) ^ 2 * (x + y) ^ 2 := by\n      ring_nf\n      <;>\n      nlinarith [sq_nonneg (x - y), sq_nonneg (x + y)]\n    rw [h\u2081\u2081\u2083]\n    <;> field_simp [h\u2083.ne', h\u2082.ne', h\u2088.ne', h\u2089.ne']\n    <;> ring_nf\n    <;> field_simp [h\u2083.ne', h\u2082.ne', h\u2088.ne', h\u2089.ne']\n    <;> ring_nf\n    <;> nlinarith [sq_nonneg (x - y), sq_nonneg (x + y)]\n  have h\u2081\u2082 : (x - y)^2 / 2 \u2264 (x - y)^2 * (x + y)^2 / (8 * y^2) := by\n    have h\u2081\u2082\u2081 : 0 \u2264 (x - y) ^ 2 := sq_nonneg (x - y)\n    have h\u2081\u2082\u2082 : 0 < y := h\u2088\n    have h\u2081\u2082\u2083 : 0 < x := h\u2089\n    have h\u2081\u2082\u2084 : 0 < y ^ 2 := by positivity\n    have h\u2081\u2082\u2085 : 0 < x ^ 2 := by positivity\n    have h\u2081\u2082\u2086 : 4 * y ^ 2 \u2264 (x + y) ^ 2 := by\n      nlinarith [sq_nonneg (x - y), sq_nonneg (x + y), sq_nonneg (x - 3 * y),\n        sq_nonneg (x + 3 * y)]\n    have h\u2081\u2082\u2087 : (x - y) ^ 2 / 2 \u2264 (x - y) ^ 2 * (x + y) ^ 2 / (8 * y ^ 2) := by\n      by_cases h : (x - y) ^ 2 = 0\n      \u00b7 rw [h]\n        <;> simp\n        <;> positivity\n      \u00b7 have h\u2081\u2082\u2088 : 0 < (x - y) ^ 2 := by positivity\n        have h\u2081\u2082\u2089 : 4 * y ^ 2 \u2264 (x + y) ^ 2 := h\u2081\u2082\u2086\n        have h\u2081\u2083\u2080 : 0 < 8 * y ^ 2 := by positivity\n        have h\u2081\u2083\u2081 : 0 < (x - y) ^ 2 * (x + y) ^ 2 := by positivity\n        rw [div_le_div_iff (by positivity) (by positivity)]\n        nlinarith [sq_nonneg (x - y), sq_nonneg (x + y), sq_nonneg (x - 3 * y),\n          sq_nonneg (x + 3 * y)]\n    exact h\u2081\u2082\u2087\n  have h\u2081\u2083 : (a + b) / 2 - Real.sqrt (a * b) \u2264 (a - b) ^ 2 / (8 * b) := by\n    rw [h\u2081\u2080, h\u2081\u2081]\n    <;> linarith\n  exact h\u2081\u2083\n"}
{"mode": "clear", "steps_before": 56, "alphabet": 30, "compiles": true, "steps_after": 13, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 4, "timed_out": true, "fixpoint": false, "chars_before": 1457, "chars_after": 367, "seconds": 523.6, "id": "mathd_numbertheory_234", "proof_after": "theorem mathd_numbertheory_234 (a b : \u2115) (h\u2080 : 1 \u2264 a \u2227 a \u2264 9 \u2227 b \u2264 9)\n    (h\u2081 : (10 * a + b) ^ 3 = 912673) : a + b = 16  :=  by\n  have h\u2082 : 10 * a + b \u2264 97 := by\n    by_contra h\n    have h\u2082\u2082 : (10 * a + b) ^ 3 \u2265 98 ^ 3 := by\n      exact Nat.pow_le_pow_of_le_left (by omega) 3\n    omega\n  have h\u2083 : 10 * a + b \u2265 97 := by\n    by_contra h\n    have h\u2083\u2081 : 10 * a + b \u2264 96 := by omega\n    have h\u2083\u2082 : (10 * a + b) ^ 3 \u2264 96 ^ 3 := by\n      exact Nat.pow_le_pow_of_le_left (by omega) 3\n    omega\n  omega\n", "proof_before": "theorem mathd_numbertheory_234 (a b : \u2115) (h\u2080 : 1 \u2264 a \u2227 a \u2264 9 \u2227 b \u2264 9)\n    (h\u2081 : (10 * a + b) ^ 3 = 912673) : a + b = 16  :=  by\n  have h\u2082 : 10 * a + b \u2264 97 := by\n    by_contra h\n    have h\u2082\u2081 : 10 * a + b \u2265 98 := by\n      omega\n    have h\u2082\u2082 : (10 * a + b) ^ 3 \u2265 98 ^ 3 := by\n      exact Nat.pow_le_pow_of_le_left (by omega) 3\n    have h\u2082\u2083 : 98 ^ 3 > 912673 := by\n      norm_num\n    have h\u2082\u2084 : (10 * a + b) ^ 3 > 912673 := by\n      linarith\n    linarith\n  have h\u2083 : 10 * a + b \u2265 97 := by\n    by_contra h\n    have h\u2083\u2081 : 10 * a + b \u2264 96 := by omega\n    have h\u2083\u2082 : (10 * a + b) ^ 3 \u2264 96 ^ 3 := by\n      exact Nat.pow_le_pow_of_le_left (by omega) 3\n    have h\u2083\u2083 : 96 ^ 3 < 912673 := by\n      norm_num\n    have h\u2083\u2084 : (10 * a + b) ^ 3 < 912673 := by\n      linarith\n    linarith\n  have h\u2084 : 10 * a + b = 97 := by\n    have h\u2085 : 10 * a + b \u2264 97 := h\u2082\n    have h\u2086 : 10 * a + b \u2265 97 := h\u2083\n    omega\n  have h\u2085 : a = 9 := by\n    have h\u2085\u2081 : 10 * a + b = 97 := h\u2084\n    have h\u2085\u2082 : a \u2264 9 := h\u2080.2.1\n    have h\u2085\u2083 : b \u2264 9 := h\u2080.2.2\n    have h\u2085\u2084 : 1 \u2264 a := h\u2080.1\n    have h\u2085\u2085 : a \u2265 9 := by\n      by_contra h\n      have h\u2085\u2085\u2081 : a \u2264 8 := by omega\n      have h\u2085\u2085\u2082 : b = 97 - 10 * a := by\n        have h\u2085\u2085\u2083 : 10 * a + b = 97 := h\u2084\n        omega\n      have h\u2085\u2085\u2083 : a \u2264 8 := by omega\n      interval_cases a <;> norm_num at h\u2085\u2081 \u22a2 <;> (try omega) <;> (try omega)\n    omega\n  have h\u2086 : b = 7 := by\n    have h\u2086\u2081 : 10 * a + b = 97 := h\u2084\n    have h\u2086\u2082 : a = 9 := h\u2085\n    rw [h\u2086\u2082] at h\u2086\u2081\n    omega\n  have h\u2087 : a + b = 16 := by\n    have h\u2087\u2081 : a = 9 := h\u2085\n    have h\u2087\u2082 : b = 7 := h\u2086\n    rw [h\u2087\u2081, h\u2087\u2082]\n    <;> norm_num\n  exact h\u2087\n"}
{"mode": "clear", "steps_before": 193, "alphabet": 79, "compiles": false, "id": "amc12a_2019_p12", "proof_after": "theorem amc12a_2019_p12 (x y : \u211d) (h\u2080 : x \u2260 1 \u2227 y \u2260 1)\n    (h\u2081 : Real.log x / Real.log 2 = Real.log 16 / Real.log y) (h\u2082 : x * y = 64) :\n    (Real.log (x / y) / Real.log 2) ^ 2 = 20  :=  by\n  have hx_ne_zero : x \u2260 0 := by\n    intro h\n    rw [h] at h\u2082\n    norm_num at h\u2082 \u22a2\n    <;> linarith\n  have hy_ne_zero : y \u2260 0 := by\n    intro h\n    rw [h] at h\u2082\n    norm_num at h\u2082 \u22a2\n    <;> linarith\n  have hxy_sign : x > 0 \u2227 y > 0 \u2228 x < 0 \u2227 y < 0 := by\n    by_cases hx_pos : x > 0\n    \u00b7 \n      have hy_pos : y > 0 := by\n        by_contra hy_neg\n        have h\u2081' : y \u2264 0 := by linarith\n        have h\u2082' : x * y \u2264 0 := by\n          nlinarith\n        linarith\n      exact Or.inl \u27e8hx_pos, hy_pos\u27e9\n    \u00b7 \n      have hx_neg : x < 0 := by\n        by_contra hx_nonneg\n        have h\u2081' : x \u2265 0 := by linarith\n        have h\u2082' : x > 0 := by\n          by_contra hx_nonpos\n          have h\u2083 : x = 0 := by linarith\n          contradiction\n        contradiction\n      have hy_neg : y < 0 := by\n        by_contra hy_nonneg\n        have h\u2081' : y \u2265 0 := by linarith\n        have h\u2082' : y > 0 := by\n          by_contra hy_nonpos\n          have h\u2083 : y = 0 := by linarith\n          contradiction\n        have h\u2083 : x * y < 0 := by nlinarith\n        linarith\n      exact Or.inr \u27e8hx_neg, hy_neg\u27e9\n  have h_log_mul : Real.log (x * y) = Real.log 64 := by\n    rw [h\u2082]\n    <;>\n    norm_num\n    <;>\n    simp_all [Real.log_mul, Real.log_pow]\n    <;>\n    ring_nf at *\n    <;>\n    norm_num at *\n    <;>\n    linarith\n  have h_log_mul' : Real.log x + Real.log y = 6 * Real.log 2 := by\n    have h\u2083 : Real.log (x * y) = Real.log x + Real.log y := by\n      by_cases hx : x = 0\n      \u00b7 exfalso\n        exact hx_ne_zero hx\n      \u00b7 by_cases hy : y = 0\n        \u00b7 exfalso\n          exact hy_ne_zero hy\n        \u00b7 rw [Real.log_mul (by aesop) (by aesop)]\n    have h\u2084 : Real.log 64 = 6 * Real.log 2 := by\n      have h\u2085 : Real.log 64 = Real.log (2 ^ 6) := by norm_num\n      rw [h\u2085]\n      have h\u2086 : Real.log (2 ^ 6) = 6 * Real.log 2 := by\n        rw [Real.log_pow]\n        <;> norm_num\n      rw [h\u2086]\n    linarith\n  have h_log_prod : Real.log x * Real.log y = 4 * (Real.log 2) ^ 2 := by\n    have h\u2083 : Real.log x / Real.log 2 = Real.log 16 / Real.log y := h\u2081\n    have h\u2084 : Real.log 16 = 4 * Real.log 2 := by\n      have h\u2085 : Real.log 16 = Real.log (2 ^ 4) := by norm_num\n      rw [h\u2085]\n      have h\u2086 : Real.log (2 ^ 4) = 4 * Real.log 2 := by\n        rw [Real.log_pow] <;> norm_num\n      rw [h\u2086]\n    have h\u2085 : y \u2260 1 := h\u2080.2\n    have h\u2086 : Real.log y \u2260 0 := by\n      intro h\u2087\n      have h\u2088 : Real.log y = 0 := h\u2087\n      have h\u2089 : y = 1 := by\n        have h\u2081\u2080 : Real.log y = 0 := h\u2088\n        have h\u2081\u2081 : Real.log y = Real.log (1 : \u211d) := by\n          rw [h\u2081\u2080]\n          <;> simp [Real.log_one]\n        have h\u2081\u2082 : y = 1 := by\n          apply Real.log_injOn_pos (Set.mem_Ioi.mpr (by\n            by_cases h : y > 0\n            \u00b7 exact h\n            \u00b7 have h\u2081\u2083 : y < 0 := by\n                by_contra h\u2081\u2084\n                have h\u2081\u2085 : y = 0 := by linarith\n                contradiction\n              have h\u2081\u2086 : y < 0 := h\u2081\u2083\n              exfalso\n              simp_all [Real.log_neg_eq_log]\n              <;> aesop\n          )) (Set.mem_Ioi.mpr (by norm_num))\n          <;> simp_all [Real.log_one]\n          <;> aesop\n        exact h\u2081\u2082\n      contradiction\n    have h\u2087 : Real.log x / Real.log 2 = Real.log 16 / Real.log y := h\u2081\n    have h\u2088 : Real.log x / Real.log 2 = (4 * Real.log 2) / Real.log y := by\n      rw [h\u2084] at h\u2087\n      exact h\u2087\n    have h\u2089 : Real.log x * Real.log y = 4 * (Real.log 2) ^ 2 := by\n      have h\u2081\u2080 : Real.log x / Real.log 2 = (4 * Real.log 2) / Real.log y := h\u2088\n      have h\u2081\u2081 : Real.log x * Real.log y = 4 * (Real.log 2) ^ 2 := by\n        have h\u2081\u2082 : Real.log y \u2260 0 := h\u2086\n        field_simp at h\u2081\u2080 \u22a2\n        nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2),\n          Real.log_pos (by norm_num : (1 : \u211d) < 2)]\n      exact h\u2081\u2081\n    exact h\u2089\n  have h_log_sub_sq : (Real.log x - Real.log y) ^ 2 = 20 * (Real.log 2) ^ 2 := by\n    have h\u2083 : (Real.log x + Real.log y) ^ 2 = (6 * Real.log 2) ^ 2 := by\n      rw [h_log_mul']\n    have h\u2084 : (Real.log x - Real.log y) ^ 2 = 20 * (Real.log 2) ^ 2 := by\n      have h\u2085 : (Real.log x - Real.log y) ^ 2 = (Real.log x + Real.log y) ^ 2 - 4 * (Real.log x * Real.log y) := by\n        ring\n      rw [h\u2085]\n      rw [h\u2083]\n      rw [h_log_prod]\n      <;> ring_nf at *\n      <;> nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2)]\n    rw [h\u2084]\n    <;> ring_nf at *\n    <;> nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2)]\n  have h_main : (Real.log (x / y) / Real.log 2) ^ 2 = 20 := by\n    have h\u2083 : Real.log (x / y) = Real.log x - Real.log y := by\n      have h\u2084 : x / y > 0 := by\n        cases' hxy_sign with hxy_sign hxy_sign\n        \u00b7 \n          have h\u2085 : x > 0 := hxy_sign.1\n          have h\u2086 : y > 0 := hxy_sign.2\n          positivity\n        \u00b7 \n          have h\u2085 : x < 0 := hxy_sign.1\n          have h\u2086 : y < 0 := hxy_sign.2\n          have h\u2087 : x / y > 0 := by\n            have h\u2088 : y \u2260 0 := by intro h; subst h; norm_num at h\u2082 \u22a2 <;> linarith\n            have h\u2089 : x / y > 0 := by\n              apply div_pos_of_neg_of_neg\n              \u00b7 exact h\u2085\n              \u00b7 exact h\u2086\n            exact h\u2089\n          exact h\u2087\n      have h\u2085 : x \u2260 0 := hx_ne_zero\n      have h\u2086 : y \u2260 0 := hy_ne_zero\n      have h\u2087 : Real.log (x / y) = Real.log x - Real.log y := by\n        have h\u2088 : Real.log (x / y) = Real.log x - Real.log y := by\n          have h\u2089 : x / y > 0 := h\u2084\n          have h\u2081\u2080 : Real.log (x / y) = Real.log x - Real.log y := by\n            rw [Real.log_div (by aesop) (by aesop)]\n            <;> ring_nf\n            <;>\n            (try simp_all [Real.log_mul, Real.log_pow])\n            <;>\n            (try norm_num)\n            <;>\n            (try linarith)\n            <;>\n            (try ring_nf at *)\n            <;>\n            (try nlinarith)\n          exact h\u2081\u2080\n        exact h\u2088\n      exact h\u2087\n    rw [h\u2083]\n    have h\u2084 : ( (Real.log x - Real.log y) / Real.log 2 ) ^ 2 = 20 := by\n      have h\u2085 : (Real.log x - Real.log y) ^ 2 = 20 * (Real.log 2) ^ 2 := h_log_sub_sq\n      have h\u2086 : Real.log 2 > 0 := Real.log_pos (by norm_num)\n      have h\u2087 : Real.log 2 \u2260 0 := by linarith\n      calc\n        ( (Real.log x - Real.log y) / Real.log 2 ) ^ 2 = ( (Real.log x - Real.log y) ^ 2 ) / (Real.log 2) ^ 2 := by\n          field_simp [h\u2087]\n          <;> ring_nf\n          <;> field_simp [h\u2087]\n          <;> ring_nf\n        _ = (20 * (Real.log 2) ^ 2) / (Real.log 2) ^ 2 := by rw [h\u2085]\n        _ = 20 := by\n          field_simp [h\u2087]\n          <;> ring_nf\n          <;> field_simp [h\u2087]\n          <;> linarith\n    exact h\u2084\n  exact h_main\n", "proof_before": "theorem amc12a_2019_p12 (x y : \u211d) (h\u2080 : x \u2260 1 \u2227 y \u2260 1)\n    (h\u2081 : Real.log x / Real.log 2 = Real.log 16 / Real.log y) (h\u2082 : x * y = 64) :\n    (Real.log (x / y) / Real.log 2) ^ 2 = 20  :=  by\n  have hx_ne_zero : x \u2260 0 := by\n    intro h\n    rw [h] at h\u2082\n    norm_num at h\u2082 \u22a2\n    <;> linarith\n  have hy_ne_zero : y \u2260 0 := by\n    intro h\n    rw [h] at h\u2082\n    norm_num at h\u2082 \u22a2\n    <;> linarith\n  have hxy_sign : x > 0 \u2227 y > 0 \u2228 x < 0 \u2227 y < 0 := by\n    by_cases hx_pos : x > 0\n    \u00b7 \n      have hy_pos : y > 0 := by\n        by_contra hy_neg\n        have h\u2081' : y \u2264 0 := by linarith\n        have h\u2082' : x * y \u2264 0 := by\n          nlinarith\n        linarith\n      exact Or.inl \u27e8hx_pos, hy_pos\u27e9\n    \u00b7 \n      have hx_neg : x < 0 := by\n        by_contra hx_nonneg\n        have h\u2081' : x \u2265 0 := by linarith\n        have h\u2082' : x > 0 := by\n          by_contra hx_nonpos\n          have h\u2083 : x = 0 := by linarith\n          contradiction\n        contradiction\n      have hy_neg : y < 0 := by\n        by_contra hy_nonneg\n        have h\u2081' : y \u2265 0 := by linarith\n        have h\u2082' : y > 0 := by\n          by_contra hy_nonpos\n          have h\u2083 : y = 0 := by linarith\n          contradiction\n        have h\u2083 : x * y < 0 := by nlinarith\n        linarith\n      exact Or.inr \u27e8hx_neg, hy_neg\u27e9\n  have h_log_mul : Real.log (x * y) = Real.log 64 := by\n    rw [h\u2082]\n    <;>\n    norm_num\n    <;>\n    simp_all [Real.log_mul, Real.log_pow]\n    <;>\n    ring_nf at *\n    <;>\n    norm_num at *\n    <;>\n    linarith\n  have h_log_mul' : Real.log x + Real.log y = 6 * Real.log 2 := by\n    have h\u2083 : Real.log (x * y) = Real.log x + Real.log y := by\n      by_cases hx : x = 0\n      \u00b7 exfalso\n        exact hx_ne_zero hx\n      \u00b7 by_cases hy : y = 0\n        \u00b7 exfalso\n          exact hy_ne_zero hy\n        \u00b7 rw [Real.log_mul (by aesop) (by aesop)]\n    have h\u2084 : Real.log 64 = 6 * Real.log 2 := by\n      have h\u2085 : Real.log 64 = Real.log (2 ^ 6) := by norm_num\n      rw [h\u2085]\n      have h\u2086 : Real.log (2 ^ 6) = 6 * Real.log 2 := by\n        rw [Real.log_pow]\n        <;> norm_num\n      rw [h\u2086]\n    linarith\n  have h_log_prod : Real.log x * Real.log y = 4 * (Real.log 2) ^ 2 := by\n    have h\u2083 : Real.log x / Real.log 2 = Real.log 16 / Real.log y := h\u2081\n    have h\u2084 : Real.log 16 = 4 * Real.log 2 := by\n      have h\u2085 : Real.log 16 = Real.log (2 ^ 4) := by norm_num\n      rw [h\u2085]\n      have h\u2086 : Real.log (2 ^ 4) = 4 * Real.log 2 := by\n        rw [Real.log_pow] <;> norm_num\n      rw [h\u2086]\n    have h\u2085 : y \u2260 1 := h\u2080.2\n    have h\u2086 : Real.log y \u2260 0 := by\n      intro h\u2087\n      have h\u2088 : Real.log y = 0 := h\u2087\n      have h\u2089 : y = 1 := by\n        have h\u2081\u2080 : Real.log y = 0 := h\u2088\n        have h\u2081\u2081 : Real.log y = Real.log (1 : \u211d) := by\n          rw [h\u2081\u2080]\n          <;> simp [Real.log_one]\n        have h\u2081\u2082 : y = 1 := by\n          apply Real.log_injOn_pos (Set.mem_Ioi.mpr (by\n            by_cases h : y > 0\n            \u00b7 exact h\n            \u00b7 have h\u2081\u2083 : y < 0 := by\n                by_contra h\u2081\u2084\n                have h\u2081\u2085 : y = 0 := by linarith\n                contradiction\n              have h\u2081\u2086 : y < 0 := h\u2081\u2083\n              exfalso\n              simp_all [Real.log_neg_eq_log]\n              <;> aesop\n          )) (Set.mem_Ioi.mpr (by norm_num))\n          <;> simp_all [Real.log_one]\n          <;> aesop\n        exact h\u2081\u2082\n      contradiction\n    have h\u2087 : Real.log x / Real.log 2 = Real.log 16 / Real.log y := h\u2081\n    have h\u2088 : Real.log x / Real.log 2 = (4 * Real.log 2) / Real.log y := by\n      rw [h\u2084] at h\u2087\n      exact h\u2087\n    have h\u2089 : Real.log x * Real.log y = 4 * (Real.log 2) ^ 2 := by\n      have h\u2081\u2080 : Real.log x / Real.log 2 = (4 * Real.log 2) / Real.log y := h\u2088\n      have h\u2081\u2081 : Real.log x * Real.log y = 4 * (Real.log 2) ^ 2 := by\n        have h\u2081\u2082 : Real.log y \u2260 0 := h\u2086\n        field_simp at h\u2081\u2080 \u22a2\n        nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2),\n          Real.log_pos (by norm_num : (1 : \u211d) < 2)]\n      exact h\u2081\u2081\n    exact h\u2089\n  have h_log_sub_sq : (Real.log x - Real.log y) ^ 2 = 20 * (Real.log 2) ^ 2 := by\n    have h\u2083 : (Real.log x + Real.log y) ^ 2 = (6 * Real.log 2) ^ 2 := by\n      rw [h_log_mul']\n    have h\u2084 : (Real.log x - Real.log y) ^ 2 = 20 * (Real.log 2) ^ 2 := by\n      have h\u2085 : (Real.log x - Real.log y) ^ 2 = (Real.log x + Real.log y) ^ 2 - 4 * (Real.log x * Real.log y) := by\n        ring\n      rw [h\u2085]\n      rw [h\u2083]\n      rw [h_log_prod]\n      <;> ring_nf at *\n      <;> nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2)]\n    rw [h\u2084]\n    <;> ring_nf at *\n    <;> nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 2)]\n  have h_main : (Real.log (x / y) / Real.log 2) ^ 2 = 20 := by\n    have h\u2083 : Real.log (x / y) = Real.log x - Real.log y := by\n      have h\u2084 : x / y > 0 := by\n        cases' hxy_sign with hxy_sign hxy_sign\n        \u00b7 \n          have h\u2085 : x > 0 := hxy_sign.1\n          have h\u2086 : y > 0 := hxy_sign.2\n          positivity\n        \u00b7 \n          have h\u2085 : x < 0 := hxy_sign.1\n          have h\u2086 : y < 0 := hxy_sign.2\n          have h\u2087 : x / y > 0 := by\n            have h\u2088 : y \u2260 0 := by intro h; subst h; norm_num at h\u2082 \u22a2 <;> linarith\n            have h\u2089 : x / y > 0 := by\n              apply div_pos_of_neg_of_neg\n              \u00b7 exact h\u2085\n              \u00b7 exact h\u2086\n            exact h\u2089\n          exact h\u2087\n      have h\u2085 : x \u2260 0 := hx_ne_zero\n      have h\u2086 : y \u2260 0 := hy_ne_zero\n      have h\u2087 : Real.log (x / y) = Real.log x - Real.log y := by\n        have h\u2088 : Real.log (x / y) = Real.log x - Real.log y := by\n          have h\u2089 : x / y > 0 := h\u2084\n          have h\u2081\u2080 : Real.log (x / y) = Real.log x - Real.log y := by\n            rw [Real.log_div (by aesop) (by aesop)]\n            <;> ring_nf\n            <;>\n            (try simp_all [Real.log_mul, Real.log_pow])\n            <;>\n            (try norm_num)\n            <;>\n            (try linarith)\n            <;>\n            (try ring_nf at *)\n            <;>\n            (try nlinarith)\n          exact h\u2081\u2080\n        exact h\u2088\n      exact h\u2087\n    rw [h\u2083]\n    have h\u2084 : ( (Real.log x - Real.log y) / Real.log 2 ) ^ 2 = 20 := by\n      have h\u2085 : (Real.log x - Real.log y) ^ 2 = 20 * (Real.log 2) ^ 2 := h_log_sub_sq\n      have h\u2086 : Real.log 2 > 0 := Real.log_pos (by norm_num)\n      have h\u2087 : Real.log 2 \u2260 0 := by linarith\n      calc\n        ( (Real.log x - Real.log y) / Real.log 2 ) ^ 2 = ( (Real.log x - Real.log y) ^ 2 ) / (Real.log 2) ^ 2 := by\n          field_simp [h\u2087]\n          <;> ring_nf\n          <;> field_simp [h\u2087]\n          <;> ring_nf\n        _ = (20 * (Real.log 2) ^ 2) / (Real.log 2) ^ 2 := by rw [h\u2085]\n        _ = 20 := by\n          field_simp [h\u2087]\n          <;> ring_nf\n          <;> field_simp [h\u2087]\n          <;> linarith\n    exact h\u2084\n  exact h_main\n"}
{"mode": "clear", "steps_before": 14, "alphabet": 14, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 359, "chars_after": 11, "seconds": 0.0, "id": "mathd_algebra_412", "proof_after": "theorem mathd_algebra_412 (x y : \u211d) (h\u2080 : x + y = 25) (h\u2081 : x - y = 11) : x = 18  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_412 (x y : \u211d) (h\u2080 : x + y = 25) (h\u2081 : x - y = 11) : x = 18  :=  by\n  have h\u2082 : 2 * x = 36 := by\n    have h\u2083 : (x + y) + (x - y) = 25 + 11 := by\n      linarith\n    have h\u2084 : (x + y) + (x - y) = 2 * x := by\n      ring\n    have h\u2085 : 2 * x = 36 := by\n      linarith\n    exact h\u2085\n  have h\u2083 : x = 18 := by\n    have h\u2084 : x = 18 := by\n      apply mul_left_cancel\u2080 (show (2 : \u211d) \u2260 0 by norm_num)\n      linarith\n    exact h\u2084\n  apply h\u2083\n"}
{"mode": "clear", "steps_before": 6, "alphabet": 13, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 152, "chars_after": 6, "seconds": 0.0, "id": "mathd_numbertheory_551", "proof_after": "theorem mathd_numbertheory_551 : 1529 % 6 = 5  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_551 : 1529 % 6 = 5  :=  by\n  have h : 1529 % 6 = 5 := by\n    norm_num [Nat.add_mod, Nat.mul_mod, Nat.mod_mod, Nat.mod_eq_of_lt]\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n  exact h\n"}
{"mode": "clear", "steps_before": 42, "alphabet": 50, "compiles": false, "id": "mathd_numbertheory_435", "proof_after": "theorem mathd_numbertheory_435 (k : \u2115) (h\u2080 : 0 < k) (h\u2081 : \u2200 n, Nat.gcd (6 * n + k) (6 * n + 3) = 1)\n    (h\u2082 : \u2200 n, Nat.gcd (6 * n + k) (6 * n + 2) = 1) (h\u2083 : \u2200 n, Nat.gcd (6 * n + k) (6 * n + 1) = 1) :\n    5 \u2264 k  :=  by\n  have h_main : 5 \u2264 k := by\n    by_contra h\n    have h\u2084 : k \u2264 4 := by linarith\n    interval_cases k <;> norm_num at h\u2081 h\u2082 h\u2083 \u22a2 <;>\n    (try omega) <;>\n    (try {\n      have h\u2085 := h\u2081 1\n      have h\u2086 := h\u2082 1\n      have h\u2087 := h\u2083 1\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    }) <;>\n    (try {\n      have h\u2085 := h\u2081 0\n      have h\u2086 := h\u2082 0\n      have h\u2087 := h\u2083 0\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    }) <;>\n    (try {\n      have h\u2085 := h\u2081 2\n      have h\u2086 := h\u2082 2\n      have h\u2087 := h\u2083 2\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    }) <;>\n    (try {\n      have h\u2085 := h\u2081 3\n      have h\u2086 := h\u2082 3\n      have h\u2087 := h\u2083 3\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 4\n      have h\u2086 := h\u2082 4\n      have h\u2087 := h\u2083 4\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 5\n      have h\u2086 := h\u2082 5\n      have h\u2087 := h\u2083 5\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 6\n      have h\u2086 := h\u2082 6\n      have h\u2087 := h\u2083 6\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 7\n      have h\u2086 := h\u2082 7\n      have h\u2087 := h\u2083 7\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 8\n      have h\u2086 := h\u2082 8\n      have h\u2087 := h\u2083 8\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 9\n      have h\u2086 := h\u2082 9\n      have h\u2087 := h\u2083 9\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 10\n      have h\u2086 := h\u2082 10\n      have h\u2087 := h\u2083 10\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n  exact h_main\n", "proof_before": "theorem mathd_numbertheory_435 (k : \u2115) (h\u2080 : 0 < k) (h\u2081 : \u2200 n, Nat.gcd (6 * n + k) (6 * n + 3) = 1)\n    (h\u2082 : \u2200 n, Nat.gcd (6 * n + k) (6 * n + 2) = 1) (h\u2083 : \u2200 n, Nat.gcd (6 * n + k) (6 * n + 1) = 1) :\n    5 \u2264 k  :=  by\n  have h_main : 5 \u2264 k := by\n    by_contra h\n    have h\u2084 : k \u2264 4 := by linarith\n    interval_cases k <;> norm_num at h\u2081 h\u2082 h\u2083 \u22a2 <;>\n    (try omega) <;>\n    (try {\n      have h\u2085 := h\u2081 1\n      have h\u2086 := h\u2082 1\n      have h\u2087 := h\u2083 1\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    }) <;>\n    (try {\n      have h\u2085 := h\u2081 0\n      have h\u2086 := h\u2082 0\n      have h\u2087 := h\u2083 0\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    }) <;>\n    (try {\n      have h\u2085 := h\u2081 2\n      have h\u2086 := h\u2082 2\n      have h\u2087 := h\u2083 2\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    }) <;>\n    (try {\n      have h\u2085 := h\u2081 3\n      have h\u2086 := h\u2082 3\n      have h\u2087 := h\u2083 3\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 4\n      have h\u2086 := h\u2082 4\n      have h\u2087 := h\u2083 4\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 5\n      have h\u2086 := h\u2082 5\n      have h\u2087 := h\u2083 5\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 6\n      have h\u2086 := h\u2082 6\n      have h\u2087 := h\u2083 6\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 7\n      have h\u2086 := h\u2082 7\n      have h\u2087 := h\u2083 7\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 8\n      have h\u2086 := h\u2082 8\n      have h\u2087 := h\u2083 8\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 9\n      have h\u2086 := h\u2082 9\n      have h\u2087 := h\u2083 9\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n    <;>\n    (try {\n      have h\u2085 := h\u2081 10\n      have h\u2086 := h\u2082 10\n      have h\u2087 := h\u2083 10\n      norm_num at h\u2085 h\u2086 h\u2087 <;> omega\n    })\n  exact h_main\n"}
{"mode": "clear", "steps_before": 62, "alphabet": 34, "compiles": false, "id": "mathd_algebra_459", "proof_after": "theorem mathd_algebra_459 (a b c d : \u211a) (h\u2080 : 3 * a = b + c + d) (h\u2081 : 4 * b = a + c + d)\n    (h\u2082 : 2 * c = a + b + d) (h\u2083 : 8 * a + 10 * b + 6 * c = 24) : \u2191d.den + d.num = 28  :=  by\n  have h\u2085b_eq_4a : 5 * b = 4 * a := by\n    have h\u2084 : 5 * b = 4 * a := by\n      linarith\n    linarith\n  have h\u2083c_eq_4a : 3 * c = 4 * a := by\n    have h\u2084 : 3 * c = 4 * a := by\n      linarith\n    linarith\n  have h_a_eq_1 : a = 1 := by\n    have h\u2084 : a = 1 := by\n      have h\u2085 : 8 * a + 10 * b + 6 * c = 24 := h\u2083\n      have h\u2086 : 5 * b = 4 * a := h\u2085b_eq_4a\n      have h\u2087 : 3 * c = 4 * a := h\u2083c_eq_4a\n      have h\u2088 : 8 * a + 10 * b + 6 * c = 24 := by linarith\n      have h\u2089 : b = (4 : \u211a) / 5 * a := by\n        have h\u2081\u2080 : b = (4 : \u211a) / 5 * a := by\n          apply Eq.symm\n          <;> ring_nf at h\u2086 \u22a2 <;> nlinarith\n        exact h\u2081\u2080\n      have h\u2081\u2080 : c = (4 : \u211a) / 3 * a := by\n        have h\u2081\u2081 : c = (4 : \u211a) / 3 * a := by\n          apply Eq.symm\n          <;> ring_nf at h\u2087 \u22a2 <;> nlinarith\n        exact h\u2081\u2081\n      rw [h\u2089, h\u2081\u2080] at h\u2088\n      ring_nf at h\u2088 \u22a2\n      nlinarith\n    linarith\n  have h_b_eq_4_div_5 : b = 4 / 5 := by\n    have h\u2084 : b = 4 / 5 := by\n      have h\u2085 : 5 * b = 4 * a := h\u2085b_eq_4a\n      have h\u2086 : a = 1 := h_a_eq_1\n      rw [h\u2086] at h\u2085\n      norm_num at h\u2085 \u22a2\n      <;> linarith\n    exact h\u2084\n  have h_c_eq_4_div_3 : c = 4 / 3 := by\n    have h\u2084 : c = 4 / 3 := by\n      have h\u2085 : 3 * c = 4 * a := h\u2083c_eq_4a\n      have h\u2086 : a = 1 := h_a_eq_1\n      rw [h\u2086] at h\u2085\n      norm_num at h\u2085 \u22a2\n      <;> linarith\n    exact h\u2084\n  have h_d_eq_13_div_15 : d = 13 / 15 := by\n    have h\u2084 : d = 13 / 15 := by\n      have h\u2085 : 3 * a = b + c + d := h\u2080\n      have h\u2086 : a = 1 := h_a_eq_1\n      have h\u2087 : b = 4 / 5 := h_b_eq_4_div_5\n      have h\u2088 : c = 4 / 3 := h_c_eq_4_div_3\n      rw [h\u2086, h\u2087, h\u2088] at h\u2085\n      norm_num at h\u2085 \u22a2\n      <;> linarith\n    exact h\u2084\n  have h_final : \u2191d.den + d.num = 28 := by\n    rw [h_d_eq_13_div_15]\n    norm_cast\n    <;> rfl\n  apply h_final\n", "proof_before": "theorem mathd_algebra_459 (a b c d : \u211a) (h\u2080 : 3 * a = b + c + d) (h\u2081 : 4 * b = a + c + d)\n    (h\u2082 : 2 * c = a + b + d) (h\u2083 : 8 * a + 10 * b + 6 * c = 24) : \u2191d.den + d.num = 28  :=  by\n  have h\u2085b_eq_4a : 5 * b = 4 * a := by\n    have h\u2084 : 5 * b = 4 * a := by\n      linarith\n    linarith\n  have h\u2083c_eq_4a : 3 * c = 4 * a := by\n    have h\u2084 : 3 * c = 4 * a := by\n      linarith\n    linarith\n  have h_a_eq_1 : a = 1 := by\n    have h\u2084 : a = 1 := by\n      have h\u2085 : 8 * a + 10 * b + 6 * c = 24 := h\u2083\n      have h\u2086 : 5 * b = 4 * a := h\u2085b_eq_4a\n      have h\u2087 : 3 * c = 4 * a := h\u2083c_eq_4a\n      have h\u2088 : 8 * a + 10 * b + 6 * c = 24 := by linarith\n      have h\u2089 : b = (4 : \u211a) / 5 * a := by\n        have h\u2081\u2080 : b = (4 : \u211a) / 5 * a := by\n          apply Eq.symm\n          <;> ring_nf at h\u2086 \u22a2 <;> nlinarith\n        exact h\u2081\u2080\n      have h\u2081\u2080 : c = (4 : \u211a) / 3 * a := by\n        have h\u2081\u2081 : c = (4 : \u211a) / 3 * a := by\n          apply Eq.symm\n          <;> ring_nf at h\u2087 \u22a2 <;> nlinarith\n        exact h\u2081\u2081\n      rw [h\u2089, h\u2081\u2080] at h\u2088\n      ring_nf at h\u2088 \u22a2\n      nlinarith\n    linarith\n  have h_b_eq_4_div_5 : b = 4 / 5 := by\n    have h\u2084 : b = 4 / 5 := by\n      have h\u2085 : 5 * b = 4 * a := h\u2085b_eq_4a\n      have h\u2086 : a = 1 := h_a_eq_1\n      rw [h\u2086] at h\u2085\n      norm_num at h\u2085 \u22a2\n      <;> linarith\n    exact h\u2084\n  have h_c_eq_4_div_3 : c = 4 / 3 := by\n    have h\u2084 : c = 4 / 3 := by\n      have h\u2085 : 3 * c = 4 * a := h\u2083c_eq_4a\n      have h\u2086 : a = 1 := h_a_eq_1\n      rw [h\u2086] at h\u2085\n      norm_num at h\u2085 \u22a2\n      <;> linarith\n    exact h\u2084\n  have h_d_eq_13_div_15 : d = 13 / 15 := by\n    have h\u2084 : d = 13 / 15 := by\n      have h\u2085 : 3 * a = b + c + d := h\u2080\n      have h\u2086 : a = 1 := h_a_eq_1\n      have h\u2087 : b = 4 / 5 := h_b_eq_4_div_5\n      have h\u2088 : c = 4 / 3 := h_c_eq_4_div_3\n      rw [h\u2086, h\u2087, h\u2088] at h\u2085\n      norm_num at h\u2085 \u22a2\n      <;> linarith\n    exact h\u2084\n  have h_final : \u2191d.den + d.num = 28 := by\n    rw [h_d_eq_13_div_15]\n    norm_cast\n    <;> rfl\n  apply h_final\n"}
{"mode": "clear", "steps_before": 29, "alphabet": 23, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 736, "chars_after": 8, "seconds": 0.0, "id": "amc12a_2021_p3", "proof_after": "theorem amc12a_2021_p3 (x y : \u2115) (h\u2080 : x + y = 17402) (h\u2081 : 10 \u2223 x) (h\u2082 : x / 10 = y) :\n    \u2191x - \u2191y = (14238 : \u2124)  :=  by\n  omega\n", "proof_before": "theorem amc12a_2021_p3 (x y : \u2115) (h\u2080 : x + y = 17402) (h\u2081 : 10 \u2223 x) (h\u2082 : x / 10 = y) :\n    \u2191x - \u2191y = (14238 : \u2124)  :=  by\n  have h\u2083 : x = 10 * y := by\n    have h\u2083\u2081 : x = 10 * (x / 10) := by\n      have h\u2083\u2081\u2081 : 10 \u2223 x := h\u2081\n      have h\u2083\u2081\u2082 : x = 10 * (x / 10) := by\n        have h\u2083\u2081\u2083 : x = 10 * (x / 10) := by\n          have h\u2083\u2081\u2084 : x / 10 * 10 = x := by\n            apply Nat.div_mul_cancel h\u2081\n          omega\n        exact h\u2083\u2081\u2083\n      exact h\u2083\u2081\u2082\n    rw [h\u2082] at h\u2083\u2081\n    <;> linarith\n  have h\u2084 : y = 1582 := by\n    have h\u2084\u2081 : 11 * y = 17402 := by\n      have h\u2084\u2082 : x = 10 * y := h\u2083\n      rw [h\u2084\u2082] at h\u2080\n      ring_nf at h\u2080 \u22a2\n      <;> omega\n    have h\u2084\u2082 : y = 1582 := by\n      omega\n    exact h\u2084\u2082\n  have h\u2085 : x = 15820 := by\n    rw [h\u2083, h\u2084]\n    <;> norm_num\n  have h\u2086 : (x : \u2124) - (y : \u2124) = (14238 : \u2124) := by\n    rw [h\u2085, h\u2084]\n    <;> norm_num\n    <;> rfl\n  exact h\u2086\n"}
{"mode": "clear", "steps_before": 18, "alphabet": 16, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 397, "chars_after": 11, "seconds": 0.0, "id": "mathd_algebra_24", "proof_after": "theorem mathd_algebra_24 (x : \u211d) (h\u2080 : x / 50 = 40) : x = 2000  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_24 (x : \u211d) (h\u2080 : x / 50 = 40) : x = 2000  :=  by\n  have h\u2081 : x = 2000 := by\n    have h\u2081 : x = 40 * 50 := by\n      have h\u2081 : x / 50 = 40 := h\u2080\n      have h\u2082 : x = 40 * 50 := by\n        calc\n          x = (x / 50) * 50 := by\n            ring\n          _ = 40 * 50 := by\n            rw [h\u2080]\n            <;> ring\n      exact h\u2082\n    have h\u2082 : x = 2000 := by\n      calc\n        x = 40 * 50 := h\u2081\n        _ = 2000 := by norm_num\n    exact h\u2082\n  exact h\u2081\n"}
{"mode": "clear", "steps_before": 85, "alphabet": 53, "compiles": true, "steps_after": 85, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "timed_out": true, "fixpoint": false, "chars_before": 2259, "chars_after": 2259, "seconds": 533.4, "id": "mathd_algebra_156", "proof_after": "theorem mathd_algebra_156 (x y : \u211d) (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 t, f t = t ^ 4)\n    (h\u2081 : \u2200 t, g t = 5 * t ^ 2 - 6) (h\u2082 : f x = g x) (h\u2083 : f y = g y) (h\u2084 : x ^ 2 < y ^ 2) :\n    y ^ 2 - x ^ 2 = 1  :=  by\n  have h\u2085 : x ^ 4 - 5 * x ^ 2 + 6 = 0 := by\n    have h\u2085\u2081 : f x = g x := h\u2082\n    have h\u2085\u2082 : f x = x ^ 4 := by rw [h\u2080]\n    have h\u2085\u2083 : g x = 5 * x ^ 2 - 6 := by rw [h\u2081]\n    rw [h\u2085\u2082, h\u2085\u2083] at h\u2085\u2081\n    ring_nf at h\u2085\u2081 \u22a2\n    linarith\n  have h\u2086 : y ^ 4 - 5 * y ^ 2 + 6 = 0 := by\n    have h\u2086\u2081 : f y = g y := h\u2083\n    have h\u2086\u2082 : f y = y ^ 4 := by rw [h\u2080]\n    have h\u2086\u2083 : g y = 5 * y ^ 2 - 6 := by rw [h\u2081]\n    rw [h\u2086\u2082, h\u2086\u2083] at h\u2086\u2081\n    ring_nf at h\u2086\u2081 \u22a2\n    linarith\n  have h\u2087 : x ^ 2 = 2 \u2228 x ^ 2 = 3 := by\n    have h\u2087\u2081 : x ^ 4 - 5 * x ^ 2 + 6 = 0 := h\u2085\n    have h\u2087\u2082 : (x ^ 2 - 2) * (x ^ 2 - 3) = 0 := by\n      ring_nf at h\u2087\u2081 \u22a2\n      <;> linarith\n    have h\u2087\u2083 : x ^ 2 - 2 = 0 \u2228 x ^ 2 - 3 = 0 := by\n      apply eq_zero_or_eq_zero_of_mul_eq_zero h\u2087\u2082\n    cases h\u2087\u2083 with\n    | inl h\u2087\u2084 =>\n      have h\u2087\u2085 : x ^ 2 = 2 := by\n        linarith\n      exact Or.inl h\u2087\u2085\n    | inr h\u2087\u2084 =>\n      have h\u2087\u2085 : x ^ 2 = 3 := by\n        linarith\n      exact Or.inr h\u2087\u2085\n  have h\u2088 : y ^ 2 = 2 \u2228 y ^ 2 = 3 := by\n    have h\u2088\u2081 : y ^ 4 - 5 * y ^ 2 + 6 = 0 := h\u2086\n    have h\u2088\u2082 : (y ^ 2 - 2) * (y ^ 2 - 3) = 0 := by\n      ring_nf at h\u2088\u2081 \u22a2\n      <;> linarith\n    have h\u2088\u2083 : y ^ 2 - 2 = 0 \u2228 y ^ 2 - 3 = 0 := by\n      apply eq_zero_or_eq_zero_of_mul_eq_zero h\u2088\u2082\n    cases h\u2088\u2083 with\n    | inl h\u2088\u2084 =>\n      have h\u2088\u2085 : y ^ 2 = 2 := by linarith\n      exact Or.inl h\u2088\u2085\n    | inr h\u2088\u2084 =>\n      have h\u2088\u2085 : y ^ 2 = 3 := by linarith\n      exact Or.inr h\u2088\u2085\n  have h\u2089 : x ^ 2 = 2 := by\n    cases h\u2087 with\n    | inl h\u2087 =>\n      exact h\u2087\n    | inr h\u2087 =>\n      have h\u2089 : x ^ 2 = 3 := h\u2087\n      have h\u2081\u2080 : y ^ 2 = 2 \u2228 y ^ 2 = 3 := h\u2088\n      cases h\u2081\u2080 with\n      | inl h\u2081\u2080 =>\n        have h\u2081\u2081 : y ^ 2 = 2 := h\u2081\u2080\n        have h\u2081\u2082 : x ^ 2 < y ^ 2 := h\u2084\n        rw [h\u2089, h\u2081\u2081] at h\u2081\u2082\n        norm_num at h\u2081\u2082\n        <;> linarith\n      | inr h\u2081\u2080 =>\n        have h\u2081\u2081 : y ^ 2 = 3 := h\u2081\u2080\n        have h\u2081\u2082 : x ^ 2 < y ^ 2 := h\u2084\n        rw [h\u2089, h\u2081\u2081] at h\u2081\u2082\n        norm_num at h\u2081\u2082\n        <;> linarith\n  have h\u2081\u2080 : y ^ 2 = 3 := by\n    cases h\u2088 with\n    | inl h\u2088 =>\n      have h\u2081\u2080 : y ^ 2 = 2 := h\u2088\n      have h\u2081\u2081 : x ^ 2 < y ^ 2 := h\u2084\n      rw [h\u2089, h\u2081\u2080] at h\u2081\u2081\n      norm_num at h\u2081\u2081 <;> linarith\n    | inr h\u2088 =>\n      have h\u2081\u2080 : y ^ 2 = 3 := h\u2088\n      exact h\u2081\u2080\n  have h\u2081\u2081 : y ^ 2 - x ^ 2 = 1 := by\n    rw [h\u2089, h\u2081\u2080]\n    <;> norm_num\n  apply h\u2081\u2081\n", "proof_before": "theorem mathd_algebra_156 (x y : \u211d) (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 t, f t = t ^ 4)\n    (h\u2081 : \u2200 t, g t = 5 * t ^ 2 - 6) (h\u2082 : f x = g x) (h\u2083 : f y = g y) (h\u2084 : x ^ 2 < y ^ 2) :\n    y ^ 2 - x ^ 2 = 1  :=  by\n  have h\u2085 : x ^ 4 - 5 * x ^ 2 + 6 = 0 := by\n    have h\u2085\u2081 : f x = g x := h\u2082\n    have h\u2085\u2082 : f x = x ^ 4 := by rw [h\u2080]\n    have h\u2085\u2083 : g x = 5 * x ^ 2 - 6 := by rw [h\u2081]\n    rw [h\u2085\u2082, h\u2085\u2083] at h\u2085\u2081\n    ring_nf at h\u2085\u2081 \u22a2\n    linarith\n  have h\u2086 : y ^ 4 - 5 * y ^ 2 + 6 = 0 := by\n    have h\u2086\u2081 : f y = g y := h\u2083\n    have h\u2086\u2082 : f y = y ^ 4 := by rw [h\u2080]\n    have h\u2086\u2083 : g y = 5 * y ^ 2 - 6 := by rw [h\u2081]\n    rw [h\u2086\u2082, h\u2086\u2083] at h\u2086\u2081\n    ring_nf at h\u2086\u2081 \u22a2\n    linarith\n  have h\u2087 : x ^ 2 = 2 \u2228 x ^ 2 = 3 := by\n    have h\u2087\u2081 : x ^ 4 - 5 * x ^ 2 + 6 = 0 := h\u2085\n    have h\u2087\u2082 : (x ^ 2 - 2) * (x ^ 2 - 3) = 0 := by\n      ring_nf at h\u2087\u2081 \u22a2\n      <;> linarith\n    have h\u2087\u2083 : x ^ 2 - 2 = 0 \u2228 x ^ 2 - 3 = 0 := by\n      apply eq_zero_or_eq_zero_of_mul_eq_zero h\u2087\u2082\n    cases h\u2087\u2083 with\n    | inl h\u2087\u2084 =>\n      have h\u2087\u2085 : x ^ 2 = 2 := by\n        linarith\n      exact Or.inl h\u2087\u2085\n    | inr h\u2087\u2084 =>\n      have h\u2087\u2085 : x ^ 2 = 3 := by\n        linarith\n      exact Or.inr h\u2087\u2085\n  have h\u2088 : y ^ 2 = 2 \u2228 y ^ 2 = 3 := by\n    have h\u2088\u2081 : y ^ 4 - 5 * y ^ 2 + 6 = 0 := h\u2086\n    have h\u2088\u2082 : (y ^ 2 - 2) * (y ^ 2 - 3) = 0 := by\n      ring_nf at h\u2088\u2081 \u22a2\n      <;> linarith\n    have h\u2088\u2083 : y ^ 2 - 2 = 0 \u2228 y ^ 2 - 3 = 0 := by\n      apply eq_zero_or_eq_zero_of_mul_eq_zero h\u2088\u2082\n    cases h\u2088\u2083 with\n    | inl h\u2088\u2084 =>\n      have h\u2088\u2085 : y ^ 2 = 2 := by linarith\n      exact Or.inl h\u2088\u2085\n    | inr h\u2088\u2084 =>\n      have h\u2088\u2085 : y ^ 2 = 3 := by linarith\n      exact Or.inr h\u2088\u2085\n  have h\u2089 : x ^ 2 = 2 := by\n    cases h\u2087 with\n    | inl h\u2087 =>\n      exact h\u2087\n    | inr h\u2087 =>\n      have h\u2089 : x ^ 2 = 3 := h\u2087\n      have h\u2081\u2080 : y ^ 2 = 2 \u2228 y ^ 2 = 3 := h\u2088\n      cases h\u2081\u2080 with\n      | inl h\u2081\u2080 =>\n        have h\u2081\u2081 : y ^ 2 = 2 := h\u2081\u2080\n        have h\u2081\u2082 : x ^ 2 < y ^ 2 := h\u2084\n        rw [h\u2089, h\u2081\u2081] at h\u2081\u2082\n        norm_num at h\u2081\u2082\n        <;> linarith\n      | inr h\u2081\u2080 =>\n        have h\u2081\u2081 : y ^ 2 = 3 := h\u2081\u2080\n        have h\u2081\u2082 : x ^ 2 < y ^ 2 := h\u2084\n        rw [h\u2089, h\u2081\u2081] at h\u2081\u2082\n        norm_num at h\u2081\u2082\n        <;> linarith\n  have h\u2081\u2080 : y ^ 2 = 3 := by\n    cases h\u2088 with\n    | inl h\u2088 =>\n      have h\u2081\u2080 : y ^ 2 = 2 := h\u2088\n      have h\u2081\u2081 : x ^ 2 < y ^ 2 := h\u2084\n      rw [h\u2089, h\u2081\u2080] at h\u2081\u2081\n      norm_num at h\u2081\u2081 <;> linarith\n    | inr h\u2088 =>\n      have h\u2081\u2080 : y ^ 2 = 3 := h\u2088\n      exact h\u2081\u2080\n  have h\u2081\u2081 : y ^ 2 - x ^ 2 = 1 := by\n    rw [h\u2089, h\u2081\u2080]\n    <;> norm_num\n  apply h\u2081\u2081\n"}
{"mode": "clear", "steps_before": 84, "alphabet": 35, "compiles": false, "id": "algebra_apbon2pownleqapownpbpowon2", "proof_after": "theorem algebra_apbon2pownleqapownpbpowon2 (a b : \u211d) (n : \u2115) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : 0 < n) :\n    ((a + b) / 2) ^ n \u2264 (a ^ n + b ^ n) / 2  :=  by\n  have h\u2082 : 0 < a := by\n    linarith [h\u2080.1]\n  have h\u2083 : 0 < b := by\n    linarith [h\u2080.2]\n  have h\u2084 : 0 < (a + b) / 2 := by\n    have h\u2084\u2081 : 0 < a + b := by linarith\n    have h\u2084\u2082 : 0 < (a + b) / 2 := by positivity\n    exact h\u2084\u2082\n  have h\u2085 : \u2200 (k : \u2115), (a - b) * (a ^ k - b ^ k) \u2265 0 := by\n    intro k\n    have h\u2085\u2081 : a > 0 := by linarith\n    have h\u2085\u2082 : b > 0 := by linarith\n    by_cases h\u2085\u2083 : a \u2265 b\n    \u00b7 \n      have h\u2085\u2084 : a - b \u2265 0 := by linarith\n      have h\u2085\u2085 : a ^ k \u2265 b ^ k := by\n        exact pow_le_pow_of_le_left (by linarith) h\u2085\u2083 k\n      have h\u2085\u2086 : a ^ k - b ^ k \u2265 0 := by linarith\n      have h\u2085\u2087 : (a - b) * (a ^ k - b ^ k) \u2265 0 := by\n        nlinarith\n      exact h\u2085\u2087\n    \u00b7 \n      have h\u2085\u2084 : a - b < 0 := by linarith\n      have h\u2085\u2085 : a ^ k \u2264 b ^ k := by\n        exact pow_le_pow_of_le_left (by linarith) (by linarith) k\n      have h\u2085\u2086 : a ^ k - b ^ k \u2264 0 := by linarith\n      have h\u2085\u2087 : (a - b) * (a ^ k - b ^ k) \u2265 0 := by\n        nlinarith\n      exact h\u2085\u2087\n  have h\u2086 : ((a + b) / 2) ^ n \u2264 (a ^ n + b ^ n) / 2 := by\n    have h\u2086\u2081 : \u2200 n : \u2115, 0 < n \u2192 ((a + b) / 2) ^ n \u2264 (a ^ n + b ^ n) / 2 := by\n      intro n hn\n      induction' hn with n hn IH\n      \u00b7 \n        norm_num [pow_one]\n        <;>\n        (try ring_nf)\n        <;>\n        (try nlinarith)\n      \u00b7 \n        have h\u2086\u2082 : (a - b) * (a ^ n - b ^ n) \u2265 0 := h\u2085 n\n        have h\u2086\u2083 : (a + b) / 2 > 0 := by positivity\n        have h\u2086\u2084 : ((a + b) / 2) ^ (n + 1) = ((a + b) / 2) ^ n * ((a + b) / 2) := by\n          ring_nf\n          <;> field_simp\n          <;> ring_nf\n        rw [h\u2086\u2084]\n        have h\u2086\u2085 : ((a + b) / 2) ^ n * ((a + b) / 2) \u2264 ((a ^ n + b ^ n) / 2) * ((a + b) / 2) := by\n          gcongr\n          <;> linarith\n        have h\u2086\u2086 : ((a ^ n + b ^ n) / 2) * ((a + b) / 2) = (a ^ n * a + a ^ n * b + b ^ n * a + b ^ n * b) / 4 := by\n          ring_nf\n          <;> field_simp\n          <;> ring_nf\n        rw [h\u2086\u2086] at h\u2086\u2085\n        have h\u2086\u2087 : (a ^ (n + 1) + b ^ (n + 1)) / 2 \u2265 (a ^ n * a + a ^ n * b + b ^ n * a + b ^ n * b) / 4 := by\n          have h\u2086\u2088 : a ^ (n + 1) = a ^ n * a := by\n            ring_nf\n          have h\u2086\u2089 : b ^ (n + 1) = b ^ n * b := by\n            ring_nf\n          rw [h\u2086\u2088, h\u2086\u2089]\n          have h\u2087\u2080 : a ^ n * a + b ^ n * b \u2265 a ^ n * b + b ^ n * a := by\n            nlinarith [h\u2085 n]\n          nlinarith [h\u2085 n]\n        have h\u2086\u2088 : ((a + b) / 2) ^ (n + 1) \u2264 (a ^ (n + 1) + b ^ (n + 1)) / 2 := by\n          have h\u2086\u2089 : ((a + b) / 2) ^ (n + 1) \u2264 (a ^ n * a + a ^ n * b + b ^ n * a + b ^ n * b) / 4 := by\n            linarith\n          have h\u2087\u2080 : (a ^ (n + 1) + b ^ (n + 1)) / 2 \u2265 (a ^ n * a + a ^ n * b + b ^ n * a + b ^ n * b) / 4 := by\n            exact h\u2086\u2087\n          have h\u2087\u2081 : ((a + b) / 2) ^ (n + 1) \u2264 (a ^ (n + 1) + b ^ (n + 1)) / 2 := by\n            linarith\n          exact h\u2087\u2081\n        simpa [pow_succ] using h\u2086\u2088\n    exact h\u2086\u2081 n h\u2081\n  exact h\u2086\n", "proof_before": "theorem algebra_apbon2pownleqapownpbpowon2 (a b : \u211d) (n : \u2115) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : 0 < n) :\n    ((a + b) / 2) ^ n \u2264 (a ^ n + b ^ n) / 2  :=  by\n  have h\u2082 : 0 < a := by\n    linarith [h\u2080.1]\n  have h\u2083 : 0 < b := by\n    linarith [h\u2080.2]\n  have h\u2084 : 0 < (a + b) / 2 := by\n    have h\u2084\u2081 : 0 < a + b := by linarith\n    have h\u2084\u2082 : 0 < (a + b) / 2 := by positivity\n    exact h\u2084\u2082\n  have h\u2085 : \u2200 (k : \u2115), (a - b) * (a ^ k - b ^ k) \u2265 0 := by\n    intro k\n    have h\u2085\u2081 : a > 0 := by linarith\n    have h\u2085\u2082 : b > 0 := by linarith\n    by_cases h\u2085\u2083 : a \u2265 b\n    \u00b7 \n      have h\u2085\u2084 : a - b \u2265 0 := by linarith\n      have h\u2085\u2085 : a ^ k \u2265 b ^ k := by\n        exact pow_le_pow_of_le_left (by linarith) h\u2085\u2083 k\n      have h\u2085\u2086 : a ^ k - b ^ k \u2265 0 := by linarith\n      have h\u2085\u2087 : (a - b) * (a ^ k - b ^ k) \u2265 0 := by\n        nlinarith\n      exact h\u2085\u2087\n    \u00b7 \n      have h\u2085\u2084 : a - b < 0 := by linarith\n      have h\u2085\u2085 : a ^ k \u2264 b ^ k := by\n        exact pow_le_pow_of_le_left (by linarith) (by linarith) k\n      have h\u2085\u2086 : a ^ k - b ^ k \u2264 0 := by linarith\n      have h\u2085\u2087 : (a - b) * (a ^ k - b ^ k) \u2265 0 := by\n        nlinarith\n      exact h\u2085\u2087\n  have h\u2086 : ((a + b) / 2) ^ n \u2264 (a ^ n + b ^ n) / 2 := by\n    have h\u2086\u2081 : \u2200 n : \u2115, 0 < n \u2192 ((a + b) / 2) ^ n \u2264 (a ^ n + b ^ n) / 2 := by\n      intro n hn\n      induction' hn with n hn IH\n      \u00b7 \n        norm_num [pow_one]\n        <;>\n        (try ring_nf)\n        <;>\n        (try nlinarith)\n      \u00b7 \n        have h\u2086\u2082 : (a - b) * (a ^ n - b ^ n) \u2265 0 := h\u2085 n\n        have h\u2086\u2083 : (a + b) / 2 > 0 := by positivity\n        have h\u2086\u2084 : ((a + b) / 2) ^ (n + 1) = ((a + b) / 2) ^ n * ((a + b) / 2) := by\n          ring_nf\n          <;> field_simp\n          <;> ring_nf\n        rw [h\u2086\u2084]\n        have h\u2086\u2085 : ((a + b) / 2) ^ n * ((a + b) / 2) \u2264 ((a ^ n + b ^ n) / 2) * ((a + b) / 2) := by\n          gcongr\n          <;> linarith\n        have h\u2086\u2086 : ((a ^ n + b ^ n) / 2) * ((a + b) / 2) = (a ^ n * a + a ^ n * b + b ^ n * a + b ^ n * b) / 4 := by\n          ring_nf\n          <;> field_simp\n          <;> ring_nf\n        rw [h\u2086\u2086] at h\u2086\u2085\n        have h\u2086\u2087 : (a ^ (n + 1) + b ^ (n + 1)) / 2 \u2265 (a ^ n * a + a ^ n * b + b ^ n * a + b ^ n * b) / 4 := by\n          have h\u2086\u2088 : a ^ (n + 1) = a ^ n * a := by\n            ring_nf\n          have h\u2086\u2089 : b ^ (n + 1) = b ^ n * b := by\n            ring_nf\n          rw [h\u2086\u2088, h\u2086\u2089]\n          have h\u2087\u2080 : a ^ n * a + b ^ n * b \u2265 a ^ n * b + b ^ n * a := by\n            nlinarith [h\u2085 n]\n          nlinarith [h\u2085 n]\n        have h\u2086\u2088 : ((a + b) / 2) ^ (n + 1) \u2264 (a ^ (n + 1) + b ^ (n + 1)) / 2 := by\n          have h\u2086\u2089 : ((a + b) / 2) ^ (n + 1) \u2264 (a ^ n * a + a ^ n * b + b ^ n * a + b ^ n * b) / 4 := by\n            linarith\n          have h\u2087\u2080 : (a ^ (n + 1) + b ^ (n + 1)) / 2 \u2265 (a ^ n * a + a ^ n * b + b ^ n * a + b ^ n * b) / 4 := by\n            exact h\u2086\u2087\n          have h\u2087\u2081 : ((a + b) / 2) ^ (n + 1) \u2264 (a ^ (n + 1) + b ^ (n + 1)) / 2 := by\n            linarith\n          exact h\u2087\u2081\n        simpa [pow_succ] using h\u2086\u2088\n    exact h\u2086\u2081 n h\u2081\n  exact h\u2086\n"}
{"mode": "clear", "steps_before": 15, "alphabet": 19, "compiles": false, "id": "mathd_numbertheory_293", "proof_after": "theorem mathd_numbertheory_293 (n : \u2115) (h\u2080 : n \u2264 9) (h\u2081 : 11 \u2223 20 * 100 + 10 * n + 7) : n = 5  :=  by\n  have h\u2082 : n = 5 := by\n    have h\u2083 : n \u2264 9 := h\u2080\n    have h\u2084 : 11 \u2223 20 * 100 + 10 * n + 7 := h\u2081\n    interval_cases n <;> norm_num at h\u2084 \u22a2\n    <;>\n    (try omega) <;>\n    (try\n      {\n        norm_num at h\u2084\n        <;>\n        omega\n      })\n    <;>\n    (try\n      {\n        rcases h\u2084 with \u27e8k, hk\u27e9\n        omega\n      })\n    <;>\n    (try\n      {\n        omega\n      })\n  exact h\u2082\n", "proof_before": "theorem mathd_numbertheory_293 (n : \u2115) (h\u2080 : n \u2264 9) (h\u2081 : 11 \u2223 20 * 100 + 10 * n + 7) : n = 5  :=  by\n  have h\u2082 : n = 5 := by\n    have h\u2083 : n \u2264 9 := h\u2080\n    have h\u2084 : 11 \u2223 20 * 100 + 10 * n + 7 := h\u2081\n    interval_cases n <;> norm_num at h\u2084 \u22a2\n    <;>\n    (try omega) <;>\n    (try\n      {\n        norm_num at h\u2084\n        <;>\n        omega\n      })\n    <;>\n    (try\n      {\n        rcases h\u2084 with \u27e8k, hk\u27e9\n        omega\n      })\n    <;>\n    (try\n      {\n        omega\n      })\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 61, "alphabet": 16, "compiles": true, "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 1177, "chars_after": 17, "seconds": 0.0, "id": "mathd_algebra_153", "proof_after": "theorem mathd_algebra_153 (n : \u211d) (h\u2080 : n = 1 / 3) :\n    Int.floor (10 * n) + Int.floor (100 * n) + Int.floor (1000 * n) + Int.floor (10000 * n) = 3702  :=  by\n  rw [h\u2080]\n  ring\n", "proof_before": "theorem mathd_algebra_153 (n : \u211d) (h\u2080 : n = 1 / 3) :\n    Int.floor (10 * n) + Int.floor (100 * n) + Int.floor (1000 * n) + Int.floor (10000 * n) = 3702  :=  by\n  have h\u2081 : Int.floor (10 * n) = 3 := by\n    rw [h\u2080]\n    norm_num [Int.floor_eq_iff, Int.cast_ofNat]\n    <;> norm_num <;>\n    (try constructor <;> norm_num) <;>\n    (try linarith) <;>\n    (try ring_nf at * <;> norm_num)\n    <;>\n    (try linarith)\n  have h\u2082 : Int.floor (100 * n) = 33 := by\n    rw [h\u2080]\n    norm_num [Int.floor_eq_iff, Int.cast_ofNat]\n    <;> norm_num <;>\n    (try constructor <;> norm_num) <;>\n    (try linarith) <;>\n    (try ring_nf at * <;> norm_num)\n    <;>\n    (try linarith)\n  have h\u2083 : Int.floor (1000 * n) = 333 := by\n    rw [h\u2080]\n    norm_num [Int.floor_eq_iff, Int.cast_ofNat]\n    <;> norm_num <;>\n    (try constructor <;> norm_num) <;>\n    (try linarith) <;>\n    (try ring_nf at * <;> norm_num)\n    <;>\n    (try linarith)\n  have h\u2084 : Int.floor (10000 * n) = 3333 := by\n    rw [h\u2080]\n    norm_num [Int.floor_eq_iff, Int.cast_ofNat]\n    <;> norm_num <;>\n    (try constructor <;> norm_num) <;>\n    (try linarith) <;>\n    (try ring_nf at * <;> norm_num)\n    <;>\n    (try linarith)\n  have h\u2085 : Int.floor (10 * n) + Int.floor (100 * n) + Int.floor (1000 * n) + Int.floor (10000 * n) = 3702 := by\n    rw [h\u2081, h\u2082, h\u2083, h\u2084]\n    <;> norm_num\n    <;> rfl\n  apply h\u2085\n"}
{"mode": "clear", "steps_before": 36, "alphabet": 22, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 970, "chars_after": 11, "seconds": 0.0, "id": "amc12b_2002_p2", "proof_after": "theorem amc12b_2002_p2 (x : \u2124) (h\u2080 : x = 4) :\n    (3 * x - 2) * (4 * x + 1) - (3 * x - 2) * (4 * x) + 1 = 11  :=  by\n  linarith\n", "proof_before": "theorem amc12b_2002_p2 (x : \u2124) (h\u2080 : x = 4) :\n    (3 * x - 2) * (4 * x + 1) - (3 * x - 2) * (4 * x) + 1 = 11  :=  by\n  have h\u2081 : (3 * x - 2) * (4 * x + 1) - (3 * x - 2) * (4 * x) + 1 = 3 * x - 1 := by\n    have h\u2081\u2081 : (3 * x - 2) * (4 * x + 1) - (3 * x - 2) * (4 * x) + 1 = (3 * x - 2) * ((4 * x + 1) - (4 * x)) + 1 := by\n      ring_nf\n      <;>\n      (try omega)\n      <;>\n      (try ring_nf at *)\n      <;>\n      (try omega)\n    rw [h\u2081\u2081]\n    have h\u2081\u2082 : (3 * x - 2) * ((4 * x + 1) - (4 * x)) + 1 = (3 * x - 2) * 1 + 1 := by\n      have h\u2081\u2083 : (4 * x + 1 : \u2124) - (4 * x : \u2124) = 1 := by\n        ring_nf at h\u2080 \u22a2\n        <;> omega\n      rw [h\u2081\u2083]\n      <;> ring_nf\n      <;> omega\n    rw [h\u2081\u2082]\n    have h\u2081\u2083 : (3 * x - 2 : \u2124) * 1 + 1 = 3 * x - 1 := by\n      ring_nf at h\u2080 \u22a2\n      <;> omega\n    rw [h\u2081\u2083]\n    <;> ring_nf at h\u2080 \u22a2 <;> omega\n  have h\u2082 : 3 * x - 1 = 11 := by\n    rw [h\u2080]\n    <;> norm_num\n    <;> ring_nf at *\n    <;> norm_num\n    <;> omega\n  have h\u2083 : (3 * x - 2) * (4 * x + 1) - (3 * x - 2) * (4 * x) + 1 = 11 := by\n    rw [h\u2081]\n    rw [h\u2082]\n    <;> ring_nf at *\n    <;> omega\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 10, "alphabet": 16, "compiles": true, "steps_after": 10, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 1, "timed_out": false, "fixpoint": true, "chars_before": 201, "chars_after": 201, "seconds": 35.8, "id": "amc12a_2002_p6", "proof_after": "theorem amc12a_2002_p6 (n : \u2115) (h\u2080 : 0 < n) : \u2203 m, m > n \u2227 \u2203 p, m * p \u2264 m + p  :=  by\n  have h\u2081 : \u2203 (m : \u2115), m > n \u2227 \u2203 (p : \u2115), m * p \u2264 m + p := by\n    use n + 1\n    constructor\n    \u00b7 \n      omega\n    \u00b7 \n      use 0\n      <;> simp [Nat.mul_zero, Nat.add_zero]\n      <;> omega\n  exact h\u2081\n", "proof_before": "theorem amc12a_2002_p6 (n : \u2115) (h\u2080 : 0 < n) : \u2203 m, m > n \u2227 \u2203 p, m * p \u2264 m + p  :=  by\n  have h\u2081 : \u2203 (m : \u2115), m > n \u2227 \u2203 (p : \u2115), m * p \u2264 m + p := by\n    use n + 1\n    constructor\n    \u00b7 \n      omega\n    \u00b7 \n      use 0\n      <;> simp [Nat.mul_zero, Nat.add_zero]\n      <;> omega\n  exact h\u2081\n"}
{"mode": "clear", "steps_before": 49, "alphabet": 36, "compiles": true, "steps_after": 49, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "timed_out": false, "fixpoint": false, "chars_before": 1321, "chars_after": 1321, "seconds": 141.6, "id": "mathd_algebra_129", "proof_after": "theorem mathd_algebra_129 (a : \u211d) (h\u2080 : a \u2260 0) (h\u2081 : 8\u207b\u00b9 / 4\u207b\u00b9 - a\u207b\u00b9 = 1) : a = -2  :=  by\n  have h\u2082 : (8 : \u211d)\u207b\u00b9 / (4 : \u211d)\u207b\u00b9 = 1 / 2 := by\n    norm_num [div_eq_mul_inv, mul_inv_rev, mul_assoc]\n    <;>\n    ring_nf at * <;>\n    norm_num <;>\n    linarith\n  have h\u2083 : (1 : \u211d) / 2 - a\u207b\u00b9 = 1 := by\n    have h\u2083\u2081 : (8 : \u211d)\u207b\u00b9 / (4 : \u211d)\u207b\u00b9 - a\u207b\u00b9 = 1 := h\u2081\n    rw [h\u2082] at h\u2083\u2081\n    exact h\u2083\u2081\n  have h\u2084 : -a\u207b\u00b9 = 1 / 2 := by\n    have h\u2084\u2081 : (1 : \u211d) / 2 - a\u207b\u00b9 = 1 := h\u2083\n    have h\u2084\u2082 : -a\u207b\u00b9 = 1 / 2 := by\n      linarith\n    exact h\u2084\u2082\n  have h\u2085 : a\u207b\u00b9 = -(1 / 2) := by\n    have h\u2085\u2081 : -a\u207b\u00b9 = 1 / 2 := h\u2084\n    have h\u2085\u2082 : a\u207b\u00b9 = -(1 / 2) := by\n      have h\u2085\u2083 : -a\u207b\u00b9 = 1 / 2 := h\u2084\n      have h\u2085\u2084 : a\u207b\u00b9 = -(1 / 2) := by\n        linarith\n      exact h\u2085\u2084\n    exact h\u2085\u2082\n  have h\u2086 : a = -2 := by\n    have h\u2086\u2081 : a\u207b\u00b9 = -(1 / 2) := h\u2085\n    have h\u2086\u2082 : a = -2 := by\n      have h\u2086\u2083 : a \u2260 0 := h\u2080\n      have h\u2086\u2084 : a\u207b\u00b9 = -(1 / 2) := h\u2085\n      have h\u2086\u2085 : a = -2 := by\n        have h\u2086\u2086 : a\u207b\u00b9 = -(1 / 2) := h\u2086\u2084\n        have h\u2086\u2087 : a = -2 := by\n          have h\u2086\u2088 : a \u2260 0 := h\u2086\u2083\n          have h\u2086\u2089 : a * a\u207b\u00b9 = 1 := by\n            field_simp [h\u2086\u2088]\n          rw [h\u2086\u2086] at h\u2086\u2089\n          have h\u2087\u2080 : a * (-(1 / 2 : \u211d)) = 1 := by linarith\n          have h\u2087\u2081 : a = -2 := by\n            apply_fun (fun x => x * (-2)) at h\u2087\u2080\n            ring_nf at h\u2087\u2080 \u22a2\n            nlinarith\n          exact h\u2087\u2081\n        exact h\u2086\u2087\n      exact h\u2086\u2085\n    exact h\u2086\u2082\n  rw [h\u2086]\n  <;> norm_num\n", "proof_before": "theorem mathd_algebra_129 (a : \u211d) (h\u2080 : a \u2260 0) (h\u2081 : 8\u207b\u00b9 / 4\u207b\u00b9 - a\u207b\u00b9 = 1) : a = -2  :=  by\n  have h\u2082 : (8 : \u211d)\u207b\u00b9 / (4 : \u211d)\u207b\u00b9 = 1 / 2 := by\n    norm_num [div_eq_mul_inv, mul_inv_rev, mul_assoc]\n    <;>\n    ring_nf at * <;>\n    norm_num <;>\n    linarith\n  have h\u2083 : (1 : \u211d) / 2 - a\u207b\u00b9 = 1 := by\n    have h\u2083\u2081 : (8 : \u211d)\u207b\u00b9 / (4 : \u211d)\u207b\u00b9 - a\u207b\u00b9 = 1 := h\u2081\n    rw [h\u2082] at h\u2083\u2081\n    exact h\u2083\u2081\n  have h\u2084 : -a\u207b\u00b9 = 1 / 2 := by\n    have h\u2084\u2081 : (1 : \u211d) / 2 - a\u207b\u00b9 = 1 := h\u2083\n    have h\u2084\u2082 : -a\u207b\u00b9 = 1 / 2 := by\n      linarith\n    exact h\u2084\u2082\n  have h\u2085 : a\u207b\u00b9 = -(1 / 2) := by\n    have h\u2085\u2081 : -a\u207b\u00b9 = 1 / 2 := h\u2084\n    have h\u2085\u2082 : a\u207b\u00b9 = -(1 / 2) := by\n      have h\u2085\u2083 : -a\u207b\u00b9 = 1 / 2 := h\u2084\n      have h\u2085\u2084 : a\u207b\u00b9 = -(1 / 2) := by\n        linarith\n      exact h\u2085\u2084\n    exact h\u2085\u2082\n  have h\u2086 : a = -2 := by\n    have h\u2086\u2081 : a\u207b\u00b9 = -(1 / 2) := h\u2085\n    have h\u2086\u2082 : a = -2 := by\n      have h\u2086\u2083 : a \u2260 0 := h\u2080\n      have h\u2086\u2084 : a\u207b\u00b9 = -(1 / 2) := h\u2085\n      have h\u2086\u2085 : a = -2 := by\n        have h\u2086\u2086 : a\u207b\u00b9 = -(1 / 2) := h\u2086\u2084\n        have h\u2086\u2087 : a = -2 := by\n          have h\u2086\u2088 : a \u2260 0 := h\u2086\u2083\n          have h\u2086\u2089 : a * a\u207b\u00b9 = 1 := by\n            field_simp [h\u2086\u2088]\n          rw [h\u2086\u2086] at h\u2086\u2089\n          have h\u2087\u2080 : a * (-(1 / 2 : \u211d)) = 1 := by linarith\n          have h\u2087\u2081 : a = -2 := by\n            apply_fun (fun x => x * (-2)) at h\u2087\u2080\n            ring_nf at h\u2087\u2080 \u22a2\n            nlinarith\n          exact h\u2087\u2081\n        exact h\u2086\u2087\n      exact h\u2086\u2085\n    exact h\u2086\u2082\n  rw [h\u2086]\n  <;> norm_num\n"}
{"mode": "clear", "steps_before": 3, "alphabet": 12, "compiles": false, "id": "mathd_numbertheory_127", "proof_after": "theorem mathd_numbertheory_127 : (\u2211 k in Finset.range 101, 2 ^ k) % 7 = 3  :=  by\n  have h_sum_mod : (\u2211 k in Finset.range 101, 2 ^ k) % 7 = 3 := by\n    rfl\n  exact h_sum_mod\n", "proof_before": "theorem mathd_numbertheory_127 : (\u2211 k in Finset.range 101, 2 ^ k) % 7 = 3  :=  by\n  have h_sum_mod : (\u2211 k in Finset.range 101, 2 ^ k) % 7 = 3 := by\n    rfl\n  exact h_sum_mod\n"}
{"mode": "clear", "steps_before": 156, "alphabet": 62, "compiles": false, "id": "aime_1983_p1", "proof_after": "theorem aime_1983_p1 (x y z w : \u2115) (ht : 1 < x \u2227 1 < y \u2227 1 < z) (hw : 0 \u2264 w)\n    (h0 : Real.log w / Real.log x = 24) (h1 : Real.log w / Real.log y = 40)\n    (h2 : Real.log w / Real.log (x * y * z) = 12) : Real.log w / Real.log z = 60  :=  by\n  have hx : (x : \u211d) > 1 := by\n    norm_cast\n    <;> linarith [ht.1, ht.2.1, ht.2.2]\n  have hy : (y : \u211d) > 1 := by\n    norm_cast\n    <;> linarith [ht.1, ht.2.1, ht.2.2]\n  have hz : (z : \u211d) > 1 := by\n    norm_cast\n    <;> linarith [ht.1, ht.2.1, ht.2.2]\n  have hxyz : (x : \u211d) * y * z > 1 := by\n    have h\u2081 : (x : \u211d) > 1 := hx\n    have h\u2082 : (y : \u211d) > 1 := hy\n    have h\u2083 : (z : \u211d) > 1 := hz\n    have h\u2084 : (x : \u211d) * y > 1 := by\n      nlinarith\n    nlinarith\n  have hlogx : Real.log (x : \u211d) > 0 := by\n    apply Real.log_pos\n    <;> simpa using hx\n  have hlogy : Real.log (y : \u211d) > 0 := by\n    apply Real.log_pos\n    <;> simpa using hy\n  have hlogz : Real.log (z : \u211d) > 0 := by\n    apply Real.log_pos\n    <;> simpa using hz\n  have hlogxyz : Real.log ((x : \u211d) * y * z) > 0 := by\n    have h\u2081 : Real.log ((x : \u211d) * y * z) = Real.log ((x : \u211d) * y) + Real.log (z : \u211d) := by\n      have h\u2082 : (x : \u211d) * y > 0 := by positivity\n      have h\u2083 : (z : \u211d) > 0 := by positivity\n      have h\u2084 : Real.log ((x : \u211d) * y * z) = Real.log ((x : \u211d) * y) + Real.log (z : \u211d) := by\n        rw [Real.log_mul (by positivity) (by positivity)]\n        <;> rw [Real.log_mul (by positivity) (by positivity)]\n        <;> ring\n      rw [h\u2084]\n    rw [h\u2081]\n    have h\u2082 : Real.log ((x : \u211d) * y) = Real.log (x : \u211d) + Real.log (y : \u211d) := by\n      have h\u2083 : 0 < (x : \u211d) := by positivity\n      have h\u2084 : 0 < (y : \u211d) := by positivity\n      rw [Real.log_mul (by positivity) (by positivity)]\n    rw [h\u2082]\n    have h\u2083 : Real.log (x : \u211d) > 0 := hlogx\n    have h\u2084 : Real.log (y : \u211d) > 0 := hlogy\n    have h\u2085 : Real.log (z : \u211d) > 0 := hlogz\n    nlinarith\n  have hlogw_pos : Real.log (w : \u211d) > 0 := by\n    have h\u2083 : Real.log (w : \u211d) / Real.log (x : \u211d) = 24 := by simpa using h0\n    have h\u2084 : Real.log (x : \u211d) > 0 := hlogx\n    have h\u2085 : Real.log (w : \u211d) > 0 := by\n      by_contra h\n      have h\u2086 : Real.log (w : \u211d) \u2264 0 := by linarith\n      have h\u2087 : Real.log (w : \u211d) / Real.log (x : \u211d) \u2264 0 := by\n        exact div_nonpos_of_nonpos_of_nonneg h\u2086 (by linarith)\n      linarith\n    exact h\u2085\n  have hlogw_eq : Real.log (w : \u211d) = 24 * Real.log (x : \u211d) := by\n    have h\u2083 : Real.log (w : \u211d) / Real.log (x : \u211d) = 24 := by simpa using h0\n    have h\u2084 : Real.log (x : \u211d) \u2260 0 := by linarith\n    have h\u2085 : Real.log (w : \u211d) = 24 * Real.log (x : \u211d) := by\n      field_simp [h\u2084] at h\u2083 \u22a2\n      <;> linarith\n    exact h\u2085\n  have hlogw_eq' : Real.log (w : \u211d) = 40 * Real.log (y : \u211d) := by\n    have h\u2083 : Real.log (w : \u211d) / Real.log (y : \u211d) = 40 := by simpa using h1\n    have h\u2084 : Real.log (y : \u211d) \u2260 0 := by linarith\n    have h\u2085 : Real.log (w : \u211d) = 40 * Real.log (y : \u211d) := by\n      field_simp [h\u2084] at h\u2083 \u22a2\n      <;> linarith\n    exact h\u2085\n  have hlogxy : 3 * Real.log (x : \u211d) = 5 * Real.log (y : \u211d) := by\n    have h\u2081 : Real.log (w : \u211d) = 24 * Real.log (x : \u211d) := hlogw_eq\n    have h\u2082 : Real.log (w : \u211d) = 40 * Real.log (y : \u211d) := hlogw_eq'\n    have h\u2083 : 24 * Real.log (x : \u211d) = 40 * Real.log (y : \u211d) := by linarith\n    have h\u2084 : 3 * Real.log (x : \u211d) = 5 * Real.log (y : \u211d) := by\n      ring_nf at h\u2083 \u22a2\n      <;> linarith\n    exact h\u2084\n  have hlogxyz_eq : Real.log (w : \u211d) = 12 * (Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d)) := by\n    have h\u2083 : Real.log (w : \u211d) / Real.log ((x : \u211d) * y * z) = 12 := by simpa using h2\n    have h\u2084 : Real.log ((x : \u211d) * y * z) = Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d) := by\n      have h\u2085 : Real.log ((x : \u211d) * y * z) = Real.log ((x : \u211d) * y) + Real.log (z : \u211d) := by\n        have h\u2086 : 0 < (x : \u211d) * y := by positivity\n        have h\u2087 : 0 < (z : \u211d) := by positivity\n        have h\u2088 : Real.log ((x : \u211d) * y * z) = Real.log ((x : \u211d) * y) + Real.log (z : \u211d) := by\n          rw [Real.log_mul (by positivity) (by positivity)]\n        rw [h\u2088]\n      rw [h\u2085]\n      have h\u2089 : Real.log ((x : \u211d) * y) = Real.log (x : \u211d) + Real.log (y : \u211d) := by\n        have h\u2081\u2080 : 0 < (x : \u211d) := by positivity\n        have h\u2081\u2081 : 0 < (y : \u211d) := by positivity\n        have h\u2081\u2082 : Real.log ((x : \u211d) * y) = Real.log (x : \u211d) + Real.log (y : \u211d) := by\n          rw [Real.log_mul (by positivity) (by positivity)]\n        rw [h\u2081\u2082]\n      rw [h\u2089]\n      <;> ring\n    rw [h\u2084] at h\u2083\n    have h\u2085 : Real.log (w : \u211d) / (Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d)) = 12 := by simpa using h\u2083\n    have h\u2086 : Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d) \u2260 0 := by\n      by_contra h\n      rw [h] at h\u2085\n      norm_num at h\u2085\n      <;> linarith [hlogx, hlogy, hlogz]\n    field_simp [h\u2086] at h\u2085 \u22a2\n    <;> nlinarith [hlogx, hlogy, hlogz, hlogw_pos, hlogw_eq, hlogw_eq', hlogxy]\n  have hlogx_rel : Real.log (x : \u211d) = Real.log (y : \u211d) + Real.log (z : \u211d) := by\n    have h\u2083 : Real.log (w : \u211d) = 24 * Real.log (x : \u211d) := hlogw_eq\n    have h\u2084 : Real.log (w : \u211d) = 12 * (Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d)) := hlogxyz_eq\n    have h\u2085 : 24 * Real.log (x : \u211d) = 12 * (Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d)) := by linarith\n    have h\u2086 : 2 * Real.log (x : \u211d) = Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d) := by linarith\n    have h\u2087 : Real.log (x : \u211d) = Real.log (y : \u211d) + Real.log (z : \u211d) := by linarith\n    exact h\u2087\n  have hlogz_rel : Real.log (z : \u211d) = (2 : \u211d) / 5 * Real.log (x : \u211d) := by\n    have h\u2083 : 3 * Real.log (x : \u211d) = 5 * Real.log (y : \u211d) := hlogxy\n    have h\u2084 : Real.log (x : \u211d) = Real.log (y : \u211d) + Real.log (z : \u211d) := hlogx_rel\n    have h\u2085 : Real.log (y : \u211d) = (3 : \u211d) / 5 * Real.log (x : \u211d) := by\n      have h\u2085\u2081 : 3 * Real.log (x : \u211d) = 5 * Real.log (y : \u211d) := hlogxy\n      have h\u2085\u2082 : Real.log (y : \u211d) = (3 : \u211d) / 5 * Real.log (x : \u211d) := by\n        apply mul_left_cancel\u2080 (show (5 : \u211d) \u2260 0 by norm_num)\n        nlinarith\n      exact h\u2085\u2082\n    have h\u2086 : Real.log (z : \u211d) = (2 : \u211d) / 5 * Real.log (x : \u211d) := by\n      have h\u2086\u2081 : Real.log (x : \u211d) = Real.log (y : \u211d) + Real.log (z : \u211d) := hlogx_rel\n      rw [h\u2085] at h\u2086\u2081\n      ring_nf at h\u2086\u2081 \u22a2\n      <;> nlinarith\n    exact h\u2086\n  have hgoal : Real.log (w : \u211d) / Real.log (z : \u211d) = 60 := by\n    have h\u2083 : Real.log (w : \u211d) = 24 * Real.log (x : \u211d) := hlogw_eq\n    have h\u2084 : Real.log (z : \u211d) = (2 : \u211d) / 5 * Real.log (x : \u211d) := hlogz_rel\n    rw [h\u2083, h\u2084]\n    have h\u2085 : Real.log (x : \u211d) \u2260 0 := by linarith [hlogx]\n    field_simp [h\u2085]\n    <;> ring_nf\n    <;> field_simp [h\u2085]\n    <;> nlinarith [hlogx, hlogy, hlogz]\n  simpa using hgoal\n", "proof_before": "theorem aime_1983_p1 (x y z w : \u2115) (ht : 1 < x \u2227 1 < y \u2227 1 < z) (hw : 0 \u2264 w)\n    (h0 : Real.log w / Real.log x = 24) (h1 : Real.log w / Real.log y = 40)\n    (h2 : Real.log w / Real.log (x * y * z) = 12) : Real.log w / Real.log z = 60  :=  by\n  have hx : (x : \u211d) > 1 := by\n    norm_cast\n    <;> linarith [ht.1, ht.2.1, ht.2.2]\n  have hy : (y : \u211d) > 1 := by\n    norm_cast\n    <;> linarith [ht.1, ht.2.1, ht.2.2]\n  have hz : (z : \u211d) > 1 := by\n    norm_cast\n    <;> linarith [ht.1, ht.2.1, ht.2.2]\n  have hxyz : (x : \u211d) * y * z > 1 := by\n    have h\u2081 : (x : \u211d) > 1 := hx\n    have h\u2082 : (y : \u211d) > 1 := hy\n    have h\u2083 : (z : \u211d) > 1 := hz\n    have h\u2084 : (x : \u211d) * y > 1 := by\n      nlinarith\n    nlinarith\n  have hlogx : Real.log (x : \u211d) > 0 := by\n    apply Real.log_pos\n    <;> simpa using hx\n  have hlogy : Real.log (y : \u211d) > 0 := by\n    apply Real.log_pos\n    <;> simpa using hy\n  have hlogz : Real.log (z : \u211d) > 0 := by\n    apply Real.log_pos\n    <;> simpa using hz\n  have hlogxyz : Real.log ((x : \u211d) * y * z) > 0 := by\n    have h\u2081 : Real.log ((x : \u211d) * y * z) = Real.log ((x : \u211d) * y) + Real.log (z : \u211d) := by\n      have h\u2082 : (x : \u211d) * y > 0 := by positivity\n      have h\u2083 : (z : \u211d) > 0 := by positivity\n      have h\u2084 : Real.log ((x : \u211d) * y * z) = Real.log ((x : \u211d) * y) + Real.log (z : \u211d) := by\n        rw [Real.log_mul (by positivity) (by positivity)]\n        <;> rw [Real.log_mul (by positivity) (by positivity)]\n        <;> ring\n      rw [h\u2084]\n    rw [h\u2081]\n    have h\u2082 : Real.log ((x : \u211d) * y) = Real.log (x : \u211d) + Real.log (y : \u211d) := by\n      have h\u2083 : 0 < (x : \u211d) := by positivity\n      have h\u2084 : 0 < (y : \u211d) := by positivity\n      rw [Real.log_mul (by positivity) (by positivity)]\n    rw [h\u2082]\n    have h\u2083 : Real.log (x : \u211d) > 0 := hlogx\n    have h\u2084 : Real.log (y : \u211d) > 0 := hlogy\n    have h\u2085 : Real.log (z : \u211d) > 0 := hlogz\n    nlinarith\n  have hlogw_pos : Real.log (w : \u211d) > 0 := by\n    have h\u2083 : Real.log (w : \u211d) / Real.log (x : \u211d) = 24 := by simpa using h0\n    have h\u2084 : Real.log (x : \u211d) > 0 := hlogx\n    have h\u2085 : Real.log (w : \u211d) > 0 := by\n      by_contra h\n      have h\u2086 : Real.log (w : \u211d) \u2264 0 := by linarith\n      have h\u2087 : Real.log (w : \u211d) / Real.log (x : \u211d) \u2264 0 := by\n        exact div_nonpos_of_nonpos_of_nonneg h\u2086 (by linarith)\n      linarith\n    exact h\u2085\n  have hlogw_eq : Real.log (w : \u211d) = 24 * Real.log (x : \u211d) := by\n    have h\u2083 : Real.log (w : \u211d) / Real.log (x : \u211d) = 24 := by simpa using h0\n    have h\u2084 : Real.log (x : \u211d) \u2260 0 := by linarith\n    have h\u2085 : Real.log (w : \u211d) = 24 * Real.log (x : \u211d) := by\n      field_simp [h\u2084] at h\u2083 \u22a2\n      <;> linarith\n    exact h\u2085\n  have hlogw_eq' : Real.log (w : \u211d) = 40 * Real.log (y : \u211d) := by\n    have h\u2083 : Real.log (w : \u211d) / Real.log (y : \u211d) = 40 := by simpa using h1\n    have h\u2084 : Real.log (y : \u211d) \u2260 0 := by linarith\n    have h\u2085 : Real.log (w : \u211d) = 40 * Real.log (y : \u211d) := by\n      field_simp [h\u2084] at h\u2083 \u22a2\n      <;> linarith\n    exact h\u2085\n  have hlogxy : 3 * Real.log (x : \u211d) = 5 * Real.log (y : \u211d) := by\n    have h\u2081 : Real.log (w : \u211d) = 24 * Real.log (x : \u211d) := hlogw_eq\n    have h\u2082 : Real.log (w : \u211d) = 40 * Real.log (y : \u211d) := hlogw_eq'\n    have h\u2083 : 24 * Real.log (x : \u211d) = 40 * Real.log (y : \u211d) := by linarith\n    have h\u2084 : 3 * Real.log (x : \u211d) = 5 * Real.log (y : \u211d) := by\n      ring_nf at h\u2083 \u22a2\n      <;> linarith\n    exact h\u2084\n  have hlogxyz_eq : Real.log (w : \u211d) = 12 * (Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d)) := by\n    have h\u2083 : Real.log (w : \u211d) / Real.log ((x : \u211d) * y * z) = 12 := by simpa using h2\n    have h\u2084 : Real.log ((x : \u211d) * y * z) = Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d) := by\n      have h\u2085 : Real.log ((x : \u211d) * y * z) = Real.log ((x : \u211d) * y) + Real.log (z : \u211d) := by\n        have h\u2086 : 0 < (x : \u211d) * y := by positivity\n        have h\u2087 : 0 < (z : \u211d) := by positivity\n        have h\u2088 : Real.log ((x : \u211d) * y * z) = Real.log ((x : \u211d) * y) + Real.log (z : \u211d) := by\n          rw [Real.log_mul (by positivity) (by positivity)]\n        rw [h\u2088]\n      rw [h\u2085]\n      have h\u2089 : Real.log ((x : \u211d) * y) = Real.log (x : \u211d) + Real.log (y : \u211d) := by\n        have h\u2081\u2080 : 0 < (x : \u211d) := by positivity\n        have h\u2081\u2081 : 0 < (y : \u211d) := by positivity\n        have h\u2081\u2082 : Real.log ((x : \u211d) * y) = Real.log (x : \u211d) + Real.log (y : \u211d) := by\n          rw [Real.log_mul (by positivity) (by positivity)]\n        rw [h\u2081\u2082]\n      rw [h\u2089]\n      <;> ring\n    rw [h\u2084] at h\u2083\n    have h\u2085 : Real.log (w : \u211d) / (Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d)) = 12 := by simpa using h\u2083\n    have h\u2086 : Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d) \u2260 0 := by\n      by_contra h\n      rw [h] at h\u2085\n      norm_num at h\u2085\n      <;> linarith [hlogx, hlogy, hlogz]\n    field_simp [h\u2086] at h\u2085 \u22a2\n    <;> nlinarith [hlogx, hlogy, hlogz, hlogw_pos, hlogw_eq, hlogw_eq', hlogxy]\n  have hlogx_rel : Real.log (x : \u211d) = Real.log (y : \u211d) + Real.log (z : \u211d) := by\n    have h\u2083 : Real.log (w : \u211d) = 24 * Real.log (x : \u211d) := hlogw_eq\n    have h\u2084 : Real.log (w : \u211d) = 12 * (Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d)) := hlogxyz_eq\n    have h\u2085 : 24 * Real.log (x : \u211d) = 12 * (Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d)) := by linarith\n    have h\u2086 : 2 * Real.log (x : \u211d) = Real.log (x : \u211d) + Real.log (y : \u211d) + Real.log (z : \u211d) := by linarith\n    have h\u2087 : Real.log (x : \u211d) = Real.log (y : \u211d) + Real.log (z : \u211d) := by linarith\n    exact h\u2087\n  have hlogz_rel : Real.log (z : \u211d) = (2 : \u211d) / 5 * Real.log (x : \u211d) := by\n    have h\u2083 : 3 * Real.log (x : \u211d) = 5 * Real.log (y : \u211d) := hlogxy\n    have h\u2084 : Real.log (x : \u211d) = Real.log (y : \u211d) + Real.log (z : \u211d) := hlogx_rel\n    have h\u2085 : Real.log (y : \u211d) = (3 : \u211d) / 5 * Real.log (x : \u211d) := by\n      have h\u2085\u2081 : 3 * Real.log (x : \u211d) = 5 * Real.log (y : \u211d) := hlogxy\n      have h\u2085\u2082 : Real.log (y : \u211d) = (3 : \u211d) / 5 * Real.log (x : \u211d) := by\n        apply mul_left_cancel\u2080 (show (5 : \u211d) \u2260 0 by norm_num)\n        nlinarith\n      exact h\u2085\u2082\n    have h\u2086 : Real.log (z : \u211d) = (2 : \u211d) / 5 * Real.log (x : \u211d) := by\n      have h\u2086\u2081 : Real.log (x : \u211d) = Real.log (y : \u211d) + Real.log (z : \u211d) := hlogx_rel\n      rw [h\u2085] at h\u2086\u2081\n      ring_nf at h\u2086\u2081 \u22a2\n      <;> nlinarith\n    exact h\u2086\n  have hgoal : Real.log (w : \u211d) / Real.log (z : \u211d) = 60 := by\n    have h\u2083 : Real.log (w : \u211d) = 24 * Real.log (x : \u211d) := hlogw_eq\n    have h\u2084 : Real.log (z : \u211d) = (2 : \u211d) / 5 * Real.log (x : \u211d) := hlogz_rel\n    rw [h\u2083, h\u2084]\n    have h\u2085 : Real.log (x : \u211d) \u2260 0 := by linarith [hlogx]\n    field_simp [h\u2085]\n    <;> ring_nf\n    <;> field_simp [h\u2085]\n    <;> nlinarith [hlogx, hlogy, hlogz]\n  simpa using hgoal\n"}
{"mode": "clear", "steps_before": 80, "alphabet": 44, "compiles": false, "id": "mathd_numbertheory_430", "proof_after": "theorem mathd_numbertheory_430 (a b c : \u2115) (h\u2080 : 1 \u2264 a \u2227 a \u2264 9) (h\u2081 : 1 \u2264 b \u2227 b \u2264 9)\n    (h\u2082 : 1 \u2264 c \u2227 c \u2264 9) (h\u2083 : a \u2260 b) (h\u2084 : a \u2260 c) (h\u2085 : b \u2260 c) (h\u2086 : a + b = c)\n    (h\u2087 : 10 * a + a - b = 2 * c) (h\u2088 : c * b = 10 * a + a + a) : a + b + c = 8  :=  by\n  have h\u2089 : b = 3 * a := by\n    have h\u2089\u2081 : 11 * a - b = 2 * c := by\n      omega\n    have h\u2089\u2082 : 11 * a - b = 2 * (a + b) := by\n      omega\n    have h\u2089\u2083 : 9 * a = 3 * b := by\n      have h\u2089\u2084 : 11 * a - b = 2 * (a + b) := h\u2089\u2082\n      have h\u2089\u2085 : 11 * a - b = 2 * a + 2 * b := by\n        ring_nf at h\u2089\u2084 \u22a2\n        <;> omega\n      have h\u2089\u2086 : 9 * a = 3 * b := by\n        have h\u2089\u2087 : 11 * a \u2265 b := by\n          by_contra h\n          have h\u2089\u2088 : 11 * a < b := by omega\n          have h\u2089\u2089 : 11 * a - b = 0 := by\n            have h\u2081\u2080\u2080 : 11 * a < b := h\u2089\u2088\n            have h\u2081\u2080\u2081 : 11 * a - b = 0 := by\n              apply Nat.sub_eq_zero_of_le\n              omega\n            exact h\u2081\u2080\u2081\n          omega\n        omega\n      exact h\u2089\u2086\n    have h\u2089\u2087 : b = 3 * a := by\n      omega\n    exact h\u2089\u2087\n  have h\u2081\u2080 : a \u2264 3 := by\n    have h\u2081\u2080\u2081 : b \u2264 9 := h\u2081.2\n    have h\u2081\u2080\u2082 : b = 3 * a := h\u2089\n    have h\u2081\u2080\u2083 : 3 * a \u2264 9 := by linarith\n    have h\u2081\u2080\u2084 : a \u2264 3 := by\n      omega\n    exact h\u2081\u2080\u2084\n  have h\u2081\u2081 : c = 4 * a := by\n    have h\u2081\u2081\u2081 : c = a + b := by\n      omega\n    rw [h\u2081\u2081\u2081]\n    have h\u2081\u2081\u2082 : b = 3 * a := h\u2089\n    rw [h\u2081\u2081\u2082]\n    <;> ring\n    <;> omega\n  have h\u2081\u2082 : a = 1 := by\n    have h\u2081\u2082\u2081 : c * b = 12 * a := by\n      have h\u2081\u2082\u2082 : c * b = 10 * a + a + a := h\u2088\n      have h\u2081\u2082\u2083 : 10 * a + a + a = 12 * a := by\n        ring_nf\n        <;> omega\n      linarith\n    have h\u2081\u2082\u2084 : c = 4 * a := h\u2081\u2081\n    have h\u2081\u2082\u2085 : b = 3 * a := h\u2089\n    rw [h\u2081\u2082\u2084, h\u2081\u2082\u2085] at h\u2081\u2082\u2081\n    have h\u2081\u2082\u2086 : (4 * a) * (3 * a) = 12 * a := by\n      ring_nf at h\u2081\u2082\u2081 \u22a2\n      <;> omega\n    have h\u2081\u2082\u2087 : a = 1 := by\n      have h\u2081\u2082\u2088 : a \u2265 1 := h\u2080.1\n      have h\u2081\u2082\u2089 : a \u2264 3 := h\u2081\u2080\n      interval_cases a <;> norm_num at h\u2081\u2082\u2086 \u22a2 <;>\n        (try omega) <;> (try nlinarith)\n    exact h\u2081\u2082\u2087\n  have h\u2081\u2083 : b = 3 := by\n    rw [h\u2089, h\u2081\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084 : c = 4 := by\n    rw [h\u2081\u2081, h\u2081\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085 : a + b + c = 8 := by\n    have h\u2081\u2085\u2081 : a = 1 := h\u2081\u2082\n    have h\u2081\u2085\u2082 : b = 3 := h\u2081\u2083\n    have h\u2081\u2085\u2083 : c = 4 := h\u2081\u2084\n    subst_vars\n    <;> norm_num\n  exact h\u2081\u2085\n", "proof_before": "theorem mathd_numbertheory_430 (a b c : \u2115) (h\u2080 : 1 \u2264 a \u2227 a \u2264 9) (h\u2081 : 1 \u2264 b \u2227 b \u2264 9)\n    (h\u2082 : 1 \u2264 c \u2227 c \u2264 9) (h\u2083 : a \u2260 b) (h\u2084 : a \u2260 c) (h\u2085 : b \u2260 c) (h\u2086 : a + b = c)\n    (h\u2087 : 10 * a + a - b = 2 * c) (h\u2088 : c * b = 10 * a + a + a) : a + b + c = 8  :=  by\n  have h\u2089 : b = 3 * a := by\n    have h\u2089\u2081 : 11 * a - b = 2 * c := by\n      omega\n    have h\u2089\u2082 : 11 * a - b = 2 * (a + b) := by\n      omega\n    have h\u2089\u2083 : 9 * a = 3 * b := by\n      have h\u2089\u2084 : 11 * a - b = 2 * (a + b) := h\u2089\u2082\n      have h\u2089\u2085 : 11 * a - b = 2 * a + 2 * b := by\n        ring_nf at h\u2089\u2084 \u22a2\n        <;> omega\n      have h\u2089\u2086 : 9 * a = 3 * b := by\n        have h\u2089\u2087 : 11 * a \u2265 b := by\n          by_contra h\n          have h\u2089\u2088 : 11 * a < b := by omega\n          have h\u2089\u2089 : 11 * a - b = 0 := by\n            have h\u2081\u2080\u2080 : 11 * a < b := h\u2089\u2088\n            have h\u2081\u2080\u2081 : 11 * a - b = 0 := by\n              apply Nat.sub_eq_zero_of_le\n              omega\n            exact h\u2081\u2080\u2081\n          omega\n        omega\n      exact h\u2089\u2086\n    have h\u2089\u2087 : b = 3 * a := by\n      omega\n    exact h\u2089\u2087\n  have h\u2081\u2080 : a \u2264 3 := by\n    have h\u2081\u2080\u2081 : b \u2264 9 := h\u2081.2\n    have h\u2081\u2080\u2082 : b = 3 * a := h\u2089\n    have h\u2081\u2080\u2083 : 3 * a \u2264 9 := by linarith\n    have h\u2081\u2080\u2084 : a \u2264 3 := by\n      omega\n    exact h\u2081\u2080\u2084\n  have h\u2081\u2081 : c = 4 * a := by\n    have h\u2081\u2081\u2081 : c = a + b := by\n      omega\n    rw [h\u2081\u2081\u2081]\n    have h\u2081\u2081\u2082 : b = 3 * a := h\u2089\n    rw [h\u2081\u2081\u2082]\n    <;> ring\n    <;> omega\n  have h\u2081\u2082 : a = 1 := by\n    have h\u2081\u2082\u2081 : c * b = 12 * a := by\n      have h\u2081\u2082\u2082 : c * b = 10 * a + a + a := h\u2088\n      have h\u2081\u2082\u2083 : 10 * a + a + a = 12 * a := by\n        ring_nf\n        <;> omega\n      linarith\n    have h\u2081\u2082\u2084 : c = 4 * a := h\u2081\u2081\n    have h\u2081\u2082\u2085 : b = 3 * a := h\u2089\n    rw [h\u2081\u2082\u2084, h\u2081\u2082\u2085] at h\u2081\u2082\u2081\n    have h\u2081\u2082\u2086 : (4 * a) * (3 * a) = 12 * a := by\n      ring_nf at h\u2081\u2082\u2081 \u22a2\n      <;> omega\n    have h\u2081\u2082\u2087 : a = 1 := by\n      have h\u2081\u2082\u2088 : a \u2265 1 := h\u2080.1\n      have h\u2081\u2082\u2089 : a \u2264 3 := h\u2081\u2080\n      interval_cases a <;> norm_num at h\u2081\u2082\u2086 \u22a2 <;>\n        (try omega) <;> (try nlinarith)\n    exact h\u2081\u2082\u2087\n  have h\u2081\u2083 : b = 3 := by\n    rw [h\u2089, h\u2081\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2084 : c = 4 := by\n    rw [h\u2081\u2081, h\u2081\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2081\u2085 : a + b + c = 8 := by\n    have h\u2081\u2085\u2081 : a = 1 := h\u2081\u2082\n    have h\u2081\u2085\u2082 : b = 3 := h\u2081\u2083\n    have h\u2081\u2085\u2083 : c = 4 := h\u2081\u2084\n    subst_vars\n    <;> norm_num\n  exact h\u2081\u2085\n"}
{"mode": "clear", "steps_before": 66, "alphabet": 38, "compiles": false, "id": "algebra_others_exirrpowirrrat", "proof_after": "theorem algebra_others_exirrpowirrrat : \u2203 a b, Irrational a \u2227 Irrational b \u2227 \u00acIrrational (a ^ b)  :=  by\n  have h_main : \u2203 (a b : \u211d), Irrational a \u2227 Irrational b \u2227 \u00acIrrational (a ^ b) := by\n    by_cases h : Irrational (Real.sqrt 2 ^ Real.sqrt 2)\n    \u00b7 \n      have h\u2081 : Irrational (Real.sqrt 2 ^ Real.sqrt 2) := h\n      have h\u2082 : Irrational (Real.sqrt 2) := by\n        exact irrational_sqrt_two\n      have h\u2083 : (Real.sqrt 2 ^ Real.sqrt 2 : \u211d) > 0 := by positivity\n      have h\u2084 : ((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2 = 2 := by\n        have h\u2085 : Real.log (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) = Real.log 2 := by\n          have h\u2085\u2081 : Real.log (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) = Real.sqrt 2 * Real.log (Real.sqrt 2 ^ Real.sqrt 2 : \u211d) := by\n            rw [Real.log_rpow (by positivity)]\n            <;>\n            norm_num\n          rw [h\u2085\u2081]\n          have h\u2085\u2082 : Real.log (Real.sqrt 2 ^ Real.sqrt 2 : \u211d) = Real.sqrt 2 * Real.log (Real.sqrt 2) := by\n            rw [Real.log_rpow (by positivity)]\n            <;>\n            norm_num\n          rw [h\u2085\u2082]\n          have h\u2085\u2083 : Real.sqrt 2 * (Real.sqrt 2 * Real.log (Real.sqrt 2)) = 2 * Real.log (Real.sqrt 2) := by\n            ring_nf\n            <;>\n            field_simp [Real.sqrt_eq_iff_sq_eq] <;> ring_nf <;> norm_num\n          rw [h\u2085\u2083]\n          have h\u2085\u2084 : Real.log (Real.sqrt 2) = Real.log 2 / 2 := by\n            have h\u2085\u2084\u2081 : Real.log (Real.sqrt 2) = Real.log 2 / 2 := by\n              have h\u2085\u2084\u2082 : Real.sqrt 2 > 0 := by positivity\n              have h\u2085\u2084\u2083 : Real.log (Real.sqrt 2) = Real.log 2 / 2 := by\n                rw [Real.log_sqrt (by positivity)]\n                <;> ring\n              exact h\u2085\u2084\u2083\n            exact h\u2085\u2084\u2081\n          rw [h\u2085\u2084]\n          have h\u2085\u2085 : (2 : \u211d) * (Real.log 2 / 2) = Real.log 2 := by ring\n          rw [h\u2085\u2085]\n        have h\u2085\u2086 : Real.log (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) = Real.log 2 := h\u2085\n        have h\u2085\u2087 : ((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2 > 0 := by positivity\n        have h\u2085\u2088 : Real.log (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) = Real.log 2 := h\u2085\n        have h\u2085\u2089 : ((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2 = 2 := by\n          apply Real.log_injOn_pos (Set.mem_Ioi.mpr h\u2085\u2087) (Set.mem_Ioi.mpr (by positivity))\n          rw [h\u2085\u2088]\n          <;>\n          norm_num\n        exact h\u2085\u2089\n      have h\u2085 : \u00acIrrational (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) := by\n        intro h\u2085\u2081\n        have h\u2085\u2082 : Irrational (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) := h\u2085\u2081\n        have h\u2085\u2083 : (Irrational (2 : \u211d)) := by simpa [h\u2084] using h\u2085\u2082\n        exact h\u2085\u2083 \u27e82, by norm_num\u27e9\n      refine' \u27e8(Real.sqrt 2 ^ Real.sqrt 2 : \u211d), Real.sqrt 2, h\u2081, h\u2082, _\u27e9\n      simpa using h\u2085\n    \u00b7 \n      have h\u2081 : \u00acIrrational (Real.sqrt 2 ^ Real.sqrt 2) := h\n      have h\u2082 : Irrational (Real.sqrt 2) := by\n        exact irrational_sqrt_two\n      refine' \u27e8Real.sqrt 2, Real.sqrt 2, h\u2082, h\u2082, _\u27e9\n      have h\u2083 : \u00acIrrational (Real.sqrt 2 ^ Real.sqrt 2) := h\u2081\n      simpa using h\u2083\n  simpa using h_main\n", "proof_before": "theorem algebra_others_exirrpowirrrat : \u2203 a b, Irrational a \u2227 Irrational b \u2227 \u00acIrrational (a ^ b)  :=  by\n  have h_main : \u2203 (a b : \u211d), Irrational a \u2227 Irrational b \u2227 \u00acIrrational (a ^ b) := by\n    by_cases h : Irrational (Real.sqrt 2 ^ Real.sqrt 2)\n    \u00b7 \n      have h\u2081 : Irrational (Real.sqrt 2 ^ Real.sqrt 2) := h\n      have h\u2082 : Irrational (Real.sqrt 2) := by\n        exact irrational_sqrt_two\n      have h\u2083 : (Real.sqrt 2 ^ Real.sqrt 2 : \u211d) > 0 := by positivity\n      have h\u2084 : ((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2 = 2 := by\n        have h\u2085 : Real.log (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) = Real.log 2 := by\n          have h\u2085\u2081 : Real.log (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) = Real.sqrt 2 * Real.log (Real.sqrt 2 ^ Real.sqrt 2 : \u211d) := by\n            rw [Real.log_rpow (by positivity)]\n            <;>\n            norm_num\n          rw [h\u2085\u2081]\n          have h\u2085\u2082 : Real.log (Real.sqrt 2 ^ Real.sqrt 2 : \u211d) = Real.sqrt 2 * Real.log (Real.sqrt 2) := by\n            rw [Real.log_rpow (by positivity)]\n            <;>\n            norm_num\n          rw [h\u2085\u2082]\n          have h\u2085\u2083 : Real.sqrt 2 * (Real.sqrt 2 * Real.log (Real.sqrt 2)) = 2 * Real.log (Real.sqrt 2) := by\n            ring_nf\n            <;>\n            field_simp [Real.sqrt_eq_iff_sq_eq] <;> ring_nf <;> norm_num\n          rw [h\u2085\u2083]\n          have h\u2085\u2084 : Real.log (Real.sqrt 2) = Real.log 2 / 2 := by\n            have h\u2085\u2084\u2081 : Real.log (Real.sqrt 2) = Real.log 2 / 2 := by\n              have h\u2085\u2084\u2082 : Real.sqrt 2 > 0 := by positivity\n              have h\u2085\u2084\u2083 : Real.log (Real.sqrt 2) = Real.log 2 / 2 := by\n                rw [Real.log_sqrt (by positivity)]\n                <;> ring\n              exact h\u2085\u2084\u2083\n            exact h\u2085\u2084\u2081\n          rw [h\u2085\u2084]\n          have h\u2085\u2085 : (2 : \u211d) * (Real.log 2 / 2) = Real.log 2 := by ring\n          rw [h\u2085\u2085]\n        have h\u2085\u2086 : Real.log (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) = Real.log 2 := h\u2085\n        have h\u2085\u2087 : ((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2 > 0 := by positivity\n        have h\u2085\u2088 : Real.log (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) = Real.log 2 := h\u2085\n        have h\u2085\u2089 : ((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2 = 2 := by\n          apply Real.log_injOn_pos (Set.mem_Ioi.mpr h\u2085\u2087) (Set.mem_Ioi.mpr (by positivity))\n          rw [h\u2085\u2088]\n          <;>\n          norm_num\n        exact h\u2085\u2089\n      have h\u2085 : \u00acIrrational (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) := by\n        intro h\u2085\u2081\n        have h\u2085\u2082 : Irrational (((Real.sqrt 2 ^ Real.sqrt 2 : \u211d) : \u211d) ^ Real.sqrt 2) := h\u2085\u2081\n        have h\u2085\u2083 : (Irrational (2 : \u211d)) := by simpa [h\u2084] using h\u2085\u2082\n        exact h\u2085\u2083 \u27e82, by norm_num\u27e9\n      refine' \u27e8(Real.sqrt 2 ^ Real.sqrt 2 : \u211d), Real.sqrt 2, h\u2081, h\u2082, _\u27e9\n      simpa using h\u2085\n    \u00b7 \n      have h\u2081 : \u00acIrrational (Real.sqrt 2 ^ Real.sqrt 2) := h\n      have h\u2082 : Irrational (Real.sqrt 2) := by\n        exact irrational_sqrt_two\n      refine' \u27e8Real.sqrt 2, Real.sqrt 2, h\u2082, h\u2082, _\u27e9\n      have h\u2083 : \u00acIrrational (Real.sqrt 2 ^ Real.sqrt 2) := h\u2081\n      simpa using h\u2083\n  simpa using h_main\n"}
{"mode": "clear", "steps_before": 18, "alphabet": 18, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 601, "chars_after": 8, "seconds": 0.0, "id": "mathd_numbertheory_185", "proof_after": "theorem mathd_numbertheory_185 (n : \u2115) (h\u2080 : n % 5 = 3) : 2 * n % 5 = 1  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_185 (n : \u2115) (h\u2080 : n % 5 = 3) : 2 * n % 5 = 1  :=  by\n  have h\u2081 : (2 * n) % 5 = (2 * (n % 5)) % 5 := by\n    rw [\u2190 Nat.mod_add_div n 5]\n    simp [Nat.mul_mod, Nat.add_mod, Nat.mod_mod, Nat.mod_self, zero_add, Nat.mul_assoc,\n      Nat.mul_comm, Nat.mul_left_comm]\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> simp [h\u2080, Nat.mul_mod, Nat.add_mod, Nat.mod_mod, Nat.mod_self, zero_add, Nat.mul_assoc,\n      Nat.mul_comm, Nat.mul_left_comm]\n    <;> omega\n  have h\u2082 : (2 * n) % 5 = (2 * 3) % 5 := by\n    rw [h\u2081]\n    <;> rw [h\u2080]\n    <;> norm_num\n    <;> rfl\n  have h\u2083 : (2 * n) % 5 = 1 := by\n    rw [h\u2082]\n    <;> norm_num\n    <;> rfl\n  rw [h\u2083]\n  <;> norm_num\n"}
{"mode": "clear", "steps_before": 7, "alphabet": 12, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 209, "chars_after": 6, "seconds": 0.0, "id": "mathd_numbertheory_229", "proof_after": "theorem mathd_numbertheory_229 : 5 ^ 30 % 7 = 1  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_229 : 5 ^ 30 % 7 = 1  :=  by\n  have h : 5 ^ 30 % 7 = 1 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.pow_mod, Nat.mod_mod, Nat.mod_eq_of_lt,\n      show 7 > 0 by decide]\n    <;> rfl\n    <;> simp_all\n    <;> norm_num\n    <;> rfl\n  exact h\n"}
{"mode": "clear", "steps_before": 22, "alphabet": 19, "compiles": true, "steps_after": 9, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "timed_out": false, "fixpoint": true, "chars_before": 633, "chars_after": 213, "seconds": 35.5, "id": "mathd_algebra_484", "proof_after": "theorem mathd_algebra_484 : Real.log 27 / Real.log 3 = 3  :=  by\n  have h\u2080 : Real.log 27 = 3 * Real.log 3 := by\n    have h\u2080\u2081 : Real.log 27 = Real.log (3 ^ 3) := by norm_num\n    rw [h\u2080\u2081]\n    simp\n  have h\u2081 : Real.log 27 / Real.log 3 = 3 := by\n    rw [h\u2080]\n    norm_num\n  apply h\u2081\n", "proof_before": "theorem mathd_algebra_484 : Real.log 27 / Real.log 3 = 3  :=  by\n  have h\u2080 : Real.log 27 = 3 * Real.log 3 := by\n    have h\u2080\u2081 : Real.log 27 = Real.log (3 ^ 3) := by norm_num\n    rw [h\u2080\u2081]\n    have h\u2080\u2082 : Real.log (3 ^ 3) = 3 * Real.log 3 := by\n      rw [Real.log_pow] <;> norm_num\n      <;>\n      linarith [Real.log_pos (by norm_num : (1 : \u211d) < 3)]\n    rw [h\u2080\u2082]\n    <;>\n    linarith [Real.log_pos (by norm_num : (1 : \u211d) < 3)]\n  have h\u2081 : Real.log 27 / Real.log 3 = 3 := by\n    rw [h\u2080]\n    have h\u2082 : Real.log 3 \u2260 0 := by\n      have h\u2082\u2081 : Real.log 3 > 0 := Real.log_pos (by norm_num)\n      linarith\n    field_simp [h\u2082]\n    <;> ring\n    <;> linarith [Real.log_pos (by norm_num : (1 : \u211d) < 3)]\n  apply h\u2081\n"}
{"mode": "clear", "steps_before": 63, "alphabet": 31, "compiles": false, "id": "induction_sumkexp3eqsumksq", "proof_after": "theorem induction_sumkexp3eqsumksq (n : \u2115) :\n    (\u2211 k in Finset.range n, k ^ 3) = (\u2211 k in Finset.range n, k) ^ 2  :=  by\n  have h_main : \u2200 n : \u2115, (\u2211 k in Finset.range n, k ^ 3) = (\u2211 k in Finset.range n, k) ^ 2 := by\n    intro n\n    have h\u2081 : (\u2211 k in Finset.range n, k ^ 3) = (\u2211 k in Finset.range n, k) ^ 2 := by\n      have h\u2082 : \u2200 n : \u2115, (\u2211 k in Finset.range n, k ^ 3) = (\u2211 k in Finset.range n, k) ^ 2 := by\n        intro n\n        induction n with\n        | zero =>\n          simp\n        | succ n ih =>\n          rw [Finset.sum_range_succ, Finset.sum_range_succ, ih]\n          have h\u2083 : \u2211 k in Finset.range n, k = n * (n - 1) / 2 := by\n            rw [Finset.sum_range_id]\n          have h\u2084 : 2 \u2223 n * (n - 1) := by\n            have h\u2085 : n * (n - 1) % 2 = 0 := by\n              have h\u2086 : n % 2 = 0 \u2228 n % 2 = 1 := by omega\n              rcases h\u2086 with (h\u2086 | h\u2086) <;> cases n with\n              | zero => simp\n              | succ n =>\n                simp [h\u2086, Nat.mul_mod, Nat.add_mod, Nat.mod_mod, Nat.mod_eq_of_lt, Nat.succ_pos]\n                <;> ring_nf at * <;> omega\n            omega\n          have h\u2085 : 2 * (\u2211 k in Finset.range n, k) = n * (n - 1) := by\n            rw [h\u2083]\n            have h\u2086 : 2 \u2223 n * (n - 1) := h\u2084\n            have h\u2087 : n * (n - 1) / 2 * 2 = n * (n - 1) := by\n              have h\u2088 : n * (n - 1) % 2 = 0 := by omega\n              have h\u2089 : n * (n - 1) / 2 * 2 = n * (n - 1) := by\n                have h\u2081\u2080 : n * (n - 1) = 2 * (n * (n - 1) / 2) := by\n                  omega\n                omega\n              exact h\u2089\n            omega\n          have h\u2086 : n ^ 3 = 2 * n * (\u2211 k in Finset.range n, k) + n ^ 2 := by\n            have h\u2087 : 2 * n * (\u2211 k in Finset.range n, k) = n ^ 2 * (n - 1) := by\n              have h\u2088 : 2 * (\u2211 k in Finset.range n, k) = n * (n - 1) := h\u2085\n              have h\u2089 : 2 * n * (\u2211 k in Finset.range n, k) = n * (n * (n - 1)) := by\n                calc\n                  2 * n * (\u2211 k in Finset.range n, k) = n * (2 * (\u2211 k in Finset.range n, k)) := by ring\n                  _ = n * (n * (n - 1)) := by rw [h\u2088] <;> ring\n                  _ = n * (n * (n - 1)) := by ring\n              have h\u2081\u2080 : n * (n * (n - 1)) = n ^ 2 * (n - 1) := by ring\n              rw [h\u2089, h\u2081\u2080]\n            have h\u2081\u2081 : n ^ 3 = 2 * n * (\u2211 k in Finset.range n, k) + n ^ 2 := by\n              cases n with\n              | zero => simp\n              | succ n =>\n                simp [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.mul_add, Nat.add_mul,\n                  Nat.pow_succ, Nat.mul_one, Nat.mul_assoc] at h\u2087 \u22a2\n                <;> ring_nf at h\u2087 \u22a2 <;> nlinarith\n            exact h\u2081\u2081\n          nlinarith\n      exact h\u2082 n\n    exact h\u2081\n  exact h_main n\n", "proof_before": "theorem induction_sumkexp3eqsumksq (n : \u2115) :\n    (\u2211 k in Finset.range n, k ^ 3) = (\u2211 k in Finset.range n, k) ^ 2  :=  by\n  have h_main : \u2200 n : \u2115, (\u2211 k in Finset.range n, k ^ 3) = (\u2211 k in Finset.range n, k) ^ 2 := by\n    intro n\n    have h\u2081 : (\u2211 k in Finset.range n, k ^ 3) = (\u2211 k in Finset.range n, k) ^ 2 := by\n      have h\u2082 : \u2200 n : \u2115, (\u2211 k in Finset.range n, k ^ 3) = (\u2211 k in Finset.range n, k) ^ 2 := by\n        intro n\n        induction n with\n        | zero =>\n          simp\n        | succ n ih =>\n          rw [Finset.sum_range_succ, Finset.sum_range_succ, ih]\n          have h\u2083 : \u2211 k in Finset.range n, k = n * (n - 1) / 2 := by\n            rw [Finset.sum_range_id]\n          have h\u2084 : 2 \u2223 n * (n - 1) := by\n            have h\u2085 : n * (n - 1) % 2 = 0 := by\n              have h\u2086 : n % 2 = 0 \u2228 n % 2 = 1 := by omega\n              rcases h\u2086 with (h\u2086 | h\u2086) <;> cases n with\n              | zero => simp\n              | succ n =>\n                simp [h\u2086, Nat.mul_mod, Nat.add_mod, Nat.mod_mod, Nat.mod_eq_of_lt, Nat.succ_pos]\n                <;> ring_nf at * <;> omega\n            omega\n          have h\u2085 : 2 * (\u2211 k in Finset.range n, k) = n * (n - 1) := by\n            rw [h\u2083]\n            have h\u2086 : 2 \u2223 n * (n - 1) := h\u2084\n            have h\u2087 : n * (n - 1) / 2 * 2 = n * (n - 1) := by\n              have h\u2088 : n * (n - 1) % 2 = 0 := by omega\n              have h\u2089 : n * (n - 1) / 2 * 2 = n * (n - 1) := by\n                have h\u2081\u2080 : n * (n - 1) = 2 * (n * (n - 1) / 2) := by\n                  omega\n                omega\n              exact h\u2089\n            omega\n          have h\u2086 : n ^ 3 = 2 * n * (\u2211 k in Finset.range n, k) + n ^ 2 := by\n            have h\u2087 : 2 * n * (\u2211 k in Finset.range n, k) = n ^ 2 * (n - 1) := by\n              have h\u2088 : 2 * (\u2211 k in Finset.range n, k) = n * (n - 1) := h\u2085\n              have h\u2089 : 2 * n * (\u2211 k in Finset.range n, k) = n * (n * (n - 1)) := by\n                calc\n                  2 * n * (\u2211 k in Finset.range n, k) = n * (2 * (\u2211 k in Finset.range n, k)) := by ring\n                  _ = n * (n * (n - 1)) := by rw [h\u2088] <;> ring\n                  _ = n * (n * (n - 1)) := by ring\n              have h\u2081\u2080 : n * (n * (n - 1)) = n ^ 2 * (n - 1) := by ring\n              rw [h\u2089, h\u2081\u2080]\n            have h\u2081\u2081 : n ^ 3 = 2 * n * (\u2211 k in Finset.range n, k) + n ^ 2 := by\n              cases n with\n              | zero => simp\n              | succ n =>\n                simp [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.mul_add, Nat.add_mul,\n                  Nat.pow_succ, Nat.mul_one, Nat.mul_assoc] at h\u2087 \u22a2\n                <;> ring_nf at h\u2087 \u22a2 <;> nlinarith\n            exact h\u2081\u2081\n          nlinarith\n      exact h\u2082 n\n    exact h\u2081\n  exact h_main n\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 16, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 425, "chars_after": 6, "seconds": 0.0, "id": "mathd_numbertheory_235", "proof_after": "theorem mathd_numbertheory_235 : (29 * 79 + 31 * 81) % 10 = 2  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_235 : (29 * 79 + 31 * 81) % 10 = 2  :=  by\n  have h1 : (29 * 79 + 31 * 81) % 10 = 2 := by\n    norm_num [Nat.add_mod, Nat.mul_mod, Nat.mod_mod, Nat.mod_eq_of_lt,\n      Nat.mod_eq_of_lt, Nat.mod_eq_of_lt, Nat.mod_eq_of_lt]\n    <;>\n    (try decide) <;>\n    (try norm_num) <;>\n    (try ring_nf) <;>\n    (try omega) <;>\n    (try norm_num) <;>\n    (try decide)\n    <;>\n    (try ring_nf)\n    <;>\n    (try omega)\n    <;>\n    (try norm_num)\n    <;>\n    (try decide)\n  exact h1\n"}
{"mode": "clear", "steps_before": 101, "alphabet": 50, "compiles": false, "id": "algebra_amgm_sum1toneqn_prod1tonleq1", "proof_after": "theorem algebra_amgm_sum1toneqn_prod1tonleq1 (a : \u2115 \u2192 NNReal) (n : \u2115)\n    (h\u2080 : (\u2211 x in Finset.range n, a x) = n) : (\u220f x in Finset.range n, a x) \u2264 1  :=  by\n  have h\u2081 : (\u220f x in Finset.range n, (a x : \u211d)) \u2264 1 := by\n    have h\u2082 : (\u2211 x in Finset.range n, (a x : \u211d)) = n := by\n      exact_mod_cast h\u2080\n    by_cases h\u2083 : n = 0\n    \u00b7 \n      simp [h\u2083]\n    \u00b7 \n      have h\u2084 : 0 < n := Nat.pos_of_ne_zero h\u2083\n      by_cases h\u2085 : \u2203 i, i \u2208 Finset.range n \u2227 (a i : \u211d) = 0\n      \u00b7 \n        obtain \u27e8i, hi, h\u2086\u27e9 := h\u2085\n        have h\u2087 : (\u220f x in Finset.range n, (a x : \u211d)) = 0 := by\n          have h\u2088 : i \u2208 Finset.range n := hi\n          have h\u2089 : (a i : \u211d) = 0 := h\u2086\n          have h\u2081\u2080 : (\u220f x in Finset.range n, (a x : \u211d)) = 0 := by\n            calc\n              (\u220f x in Finset.range n, (a x : \u211d)) = \u220f x in Finset.range n, (a x : \u211d) := rfl\n              _ = 0 := by\n                apply Finset.prod_eq_zero h\u2088\n                simp [h\u2089]\n          exact h\u2081\u2080\n        linarith\n      \u00b7 \n        have h\u2085' : \u2200 i, i \u2208 Finset.range n \u2192 (a i : \u211d) > 0 := by\n          intro i hi\n          have h\u2086 : \u00ac\u2203 i, i \u2208 Finset.range n \u2227 (a i : \u211d) = 0 := h\u2085\n          have h\u2087 : (a i : \u211d) \u2260 0 := by\n            intro h\n            have h\u2088 : \u2203 i, i \u2208 Finset.range n \u2227 (a i : \u211d) = 0 := \u27e8i, hi, by simp_all\u27e9\n            contradiction\n          have h\u2088 : (a i : \u211d) > 0 := by\n            have h\u2089 : (a i : \u211d) \u2265 0 := by exact_mod_cast (a i).prop\n            contrapose! h\u2087\n            linarith\n          exact h\u2088\n        have h\u2086 : (\u220f x in Finset.range n, (a x : \u211d)) \u2264 1 := by\n          have h\u2087 : \u2211 x in Finset.range n, Real.log ((a x : \u211d)) \u2264 0 := by\n            have h\u2088 : \u2211 x in Finset.range n, Real.log ((a x : \u211d)) \u2264 \u2211 x in Finset.range n, ((a x : \u211d) - 1) := by\n              apply Finset.sum_le_sum\n              intro i hi\n              have h\u2089 : (a i : \u211d) > 0 := h\u2085' i hi\n              have h\u2081\u2080 : Real.log ((a i : \u211d)) \u2264 (a i : \u211d) - 1 := by\n                have h\u2081\u2081 : Real.log ((a i : \u211d)) \u2264 (a i : \u211d) - 1 := by\n                  linarith [Real.log_le_sub_one_of_pos h\u2089]\n                exact h\u2081\u2081\n              exact h\u2081\u2080\n            have h\u2089 : \u2211 x in Finset.range n, ((a x : \u211d) - 1) = (\u2211 x in Finset.range n, (a x : \u211d)) - n := by\n              calc\n                \u2211 x in Finset.range n, ((a x : \u211d) - 1) = \u2211 x in Finset.range n, ((a x : \u211d) - 1) := rfl\n                _ = (\u2211 x in Finset.range n, (a x : \u211d)) - \u2211 x in Finset.range n, (1 : \u211d) := by\n                  rw [Finset.sum_sub_distrib]\n                _ = (\u2211 x in Finset.range n, (a x : \u211d)) - n := by\n                  simp [Finset.sum_const, Finset.card_range]\n                  <;> ring\n                _ = (\u2211 x in Finset.range n, (a x : \u211d)) - n := by simp\n            have h\u2081\u2080 : \u2211 x in Finset.range n, ((a x : \u211d) - 1) = (\u2211 x in Finset.range n, (a x : \u211d)) - n := h\u2089\n            have h\u2081\u2081 : \u2211 x in Finset.range n, Real.log ((a x : \u211d)) \u2264 (\u2211 x in Finset.range n, (a x : \u211d)) - n := by\n              linarith\n            have h\u2081\u2082 : (\u2211 x in Finset.range n, (a x : \u211d)) - n = 0 := by\n              rw [h\u2082]\n              <;> ring\n              <;> field_simp [h\u2083]\n              <;> linarith\n            linarith\n          have h\u2088 : Real.log (\u220f x in Finset.range n, (a x : \u211d)) \u2264 0 := by\n            calc\n              Real.log (\u220f x in Finset.range n, (a x : \u211d)) = \u2211 x in Finset.range n, Real.log ((a x : \u211d)) := by\n                rw [Real.log_prod _ _ fun i hi => by\n                  have h\u2089 : (a i : \u211d) > 0 := h\u2085' i hi\n                  positivity]\n              _ \u2264 0 := h\u2087\n          have h\u2089 : (\u220f x in Finset.range n, (a x : \u211d)) \u2264 1 := by\n            by_contra h\n            have h\u2081\u2080 : (\u220f x in Finset.range n, (a x : \u211d)) > 1 := by\n              linarith\n            have h\u2081\u2081 : Real.log (\u220f x in Finset.range n, (a x : \u211d)) > 0 := by\n              have h\u2081\u2082 : Real.log (\u220f x in Finset.range n, (a x : \u211d)) > 0 := by\n                apply Real.log_pos\n                exact by\n                  have h\u2081\u2083 : (\u220f x in Finset.range n, (a x : \u211d)) > 1 := h\u2081\u2080\n                  linarith\n              linarith\n            linarith\n          exact h\u2089\n        exact h\u2086\n  have h\u2082 : (\u220f x in Finset.range n, a x) \u2264 1 := by\n    have h\u2083 : (\u220f x in Finset.range n, a x : \u211d) \u2264 1 := by\n      exact h\u2081\n    have h\u2084 : (\u220f x in Finset.range n, a x : \u211d) = (\u220f x in Finset.range n, a x : \u211d) := rfl\n    have h\u2085 : (\u220f x in Finset.range n, a x : \u211d) = (\u220f x in Finset.range n, (a x : \u211d)) := by simp\n    have h\u2086 : (\u220f x in Finset.range n, a x : \u211d) \u2264 1 := by simpa [h\u2085] using h\u2081\n    have h\u2087 : (\u220f x in Finset.range n, a x : NNReal) \u2264 1 := by\n      norm_cast at h\u2086 \u22a2\n      <;> simp_all [Finset.prod_range_succ]\n      <;> norm_num\n      <;> linarith\n    simpa using h\u2087\n  exact h\u2082\n", "proof_before": "theorem algebra_amgm_sum1toneqn_prod1tonleq1 (a : \u2115 \u2192 NNReal) (n : \u2115)\n    (h\u2080 : (\u2211 x in Finset.range n, a x) = n) : (\u220f x in Finset.range n, a x) \u2264 1  :=  by\n  have h\u2081 : (\u220f x in Finset.range n, (a x : \u211d)) \u2264 1 := by\n    have h\u2082 : (\u2211 x in Finset.range n, (a x : \u211d)) = n := by\n      exact_mod_cast h\u2080\n    by_cases h\u2083 : n = 0\n    \u00b7 \n      simp [h\u2083]\n    \u00b7 \n      have h\u2084 : 0 < n := Nat.pos_of_ne_zero h\u2083\n      by_cases h\u2085 : \u2203 i, i \u2208 Finset.range n \u2227 (a i : \u211d) = 0\n      \u00b7 \n        obtain \u27e8i, hi, h\u2086\u27e9 := h\u2085\n        have h\u2087 : (\u220f x in Finset.range n, (a x : \u211d)) = 0 := by\n          have h\u2088 : i \u2208 Finset.range n := hi\n          have h\u2089 : (a i : \u211d) = 0 := h\u2086\n          have h\u2081\u2080 : (\u220f x in Finset.range n, (a x : \u211d)) = 0 := by\n            calc\n              (\u220f x in Finset.range n, (a x : \u211d)) = \u220f x in Finset.range n, (a x : \u211d) := rfl\n              _ = 0 := by\n                apply Finset.prod_eq_zero h\u2088\n                simp [h\u2089]\n          exact h\u2081\u2080\n        linarith\n      \u00b7 \n        have h\u2085' : \u2200 i, i \u2208 Finset.range n \u2192 (a i : \u211d) > 0 := by\n          intro i hi\n          have h\u2086 : \u00ac\u2203 i, i \u2208 Finset.range n \u2227 (a i : \u211d) = 0 := h\u2085\n          have h\u2087 : (a i : \u211d) \u2260 0 := by\n            intro h\n            have h\u2088 : \u2203 i, i \u2208 Finset.range n \u2227 (a i : \u211d) = 0 := \u27e8i, hi, by simp_all\u27e9\n            contradiction\n          have h\u2088 : (a i : \u211d) > 0 := by\n            have h\u2089 : (a i : \u211d) \u2265 0 := by exact_mod_cast (a i).prop\n            contrapose! h\u2087\n            linarith\n          exact h\u2088\n        have h\u2086 : (\u220f x in Finset.range n, (a x : \u211d)) \u2264 1 := by\n          have h\u2087 : \u2211 x in Finset.range n, Real.log ((a x : \u211d)) \u2264 0 := by\n            have h\u2088 : \u2211 x in Finset.range n, Real.log ((a x : \u211d)) \u2264 \u2211 x in Finset.range n, ((a x : \u211d) - 1) := by\n              apply Finset.sum_le_sum\n              intro i hi\n              have h\u2089 : (a i : \u211d) > 0 := h\u2085' i hi\n              have h\u2081\u2080 : Real.log ((a i : \u211d)) \u2264 (a i : \u211d) - 1 := by\n                have h\u2081\u2081 : Real.log ((a i : \u211d)) \u2264 (a i : \u211d) - 1 := by\n                  linarith [Real.log_le_sub_one_of_pos h\u2089]\n                exact h\u2081\u2081\n              exact h\u2081\u2080\n            have h\u2089 : \u2211 x in Finset.range n, ((a x : \u211d) - 1) = (\u2211 x in Finset.range n, (a x : \u211d)) - n := by\n              calc\n                \u2211 x in Finset.range n, ((a x : \u211d) - 1) = \u2211 x in Finset.range n, ((a x : \u211d) - 1) := rfl\n                _ = (\u2211 x in Finset.range n, (a x : \u211d)) - \u2211 x in Finset.range n, (1 : \u211d) := by\n                  rw [Finset.sum_sub_distrib]\n                _ = (\u2211 x in Finset.range n, (a x : \u211d)) - n := by\n                  simp [Finset.sum_const, Finset.card_range]\n                  <;> ring\n                _ = (\u2211 x in Finset.range n, (a x : \u211d)) - n := by simp\n            have h\u2081\u2080 : \u2211 x in Finset.range n, ((a x : \u211d) - 1) = (\u2211 x in Finset.range n, (a x : \u211d)) - n := h\u2089\n            have h\u2081\u2081 : \u2211 x in Finset.range n, Real.log ((a x : \u211d)) \u2264 (\u2211 x in Finset.range n, (a x : \u211d)) - n := by\n              linarith\n            have h\u2081\u2082 : (\u2211 x in Finset.range n, (a x : \u211d)) - n = 0 := by\n              rw [h\u2082]\n              <;> ring\n              <;> field_simp [h\u2083]\n              <;> linarith\n            linarith\n          have h\u2088 : Real.log (\u220f x in Finset.range n, (a x : \u211d)) \u2264 0 := by\n            calc\n              Real.log (\u220f x in Finset.range n, (a x : \u211d)) = \u2211 x in Finset.range n, Real.log ((a x : \u211d)) := by\n                rw [Real.log_prod _ _ fun i hi => by\n                  have h\u2089 : (a i : \u211d) > 0 := h\u2085' i hi\n                  positivity]\n              _ \u2264 0 := h\u2087\n          have h\u2089 : (\u220f x in Finset.range n, (a x : \u211d)) \u2264 1 := by\n            by_contra h\n            have h\u2081\u2080 : (\u220f x in Finset.range n, (a x : \u211d)) > 1 := by\n              linarith\n            have h\u2081\u2081 : Real.log (\u220f x in Finset.range n, (a x : \u211d)) > 0 := by\n              have h\u2081\u2082 : Real.log (\u220f x in Finset.range n, (a x : \u211d)) > 0 := by\n                apply Real.log_pos\n                exact by\n                  have h\u2081\u2083 : (\u220f x in Finset.range n, (a x : \u211d)) > 1 := h\u2081\u2080\n                  linarith\n              linarith\n            linarith\n          exact h\u2089\n        exact h\u2086\n  have h\u2082 : (\u220f x in Finset.range n, a x) \u2264 1 := by\n    have h\u2083 : (\u220f x in Finset.range n, a x : \u211d) \u2264 1 := by\n      exact h\u2081\n    have h\u2084 : (\u220f x in Finset.range n, a x : \u211d) = (\u220f x in Finset.range n, a x : \u211d) := rfl\n    have h\u2085 : (\u220f x in Finset.range n, a x : \u211d) = (\u220f x in Finset.range n, (a x : \u211d)) := by simp\n    have h\u2086 : (\u220f x in Finset.range n, a x : \u211d) \u2264 1 := by simpa [h\u2085] using h\u2081\n    have h\u2087 : (\u220f x in Finset.range n, a x : NNReal) \u2264 1 := by\n      norm_cast at h\u2086 \u22a2\n      <;> simp_all [Finset.prod_range_succ]\n      <;> norm_num\n      <;> linarith\n    simpa using h\u2087\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 5, "alphabet": 12, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 216, "chars_after": 6, "seconds": 0.0, "id": "mathd_numbertheory_769", "proof_after": "theorem mathd_numbertheory_769 : (129 ^ 34 + 96 ^ 38) % 11 = 9  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_769 : (129 ^ 34 + 96 ^ 38) % 11 = 9  :=  by\n  have h1 : (129 ^ 34 + 96 ^ 38) % 11 = 9 := by\n    norm_num [Nat.add_mod, Nat.pow_mod, Nat.mod_mod, Nat.mod_eq_of_lt, Nat.mod_eq_of_lt,\n      show 2 \u2264 11 by decide, show 2 \u2264 11 by decide]\n    <;>\n    rfl\n  exact h1\n"}
{"mode": "clear", "steps_before": 69, "alphabet": 35, "compiles": true, "steps_after": 38, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 3, "timed_out": false, "fixpoint": false, "chars_before": 2131, "chars_after": 1232, "seconds": 226.1, "id": "induction_1pxpownlt1pnx", "proof_after": "theorem induction_1pxpownlt1pnx (x : \u211d) (n : \u2115) (h\u2080 : -1 < x) (h\u2081 : 0 < n) :\n    1 + \u2191n * x \u2264 (1 + x) ^ (n : \u2115)  :=  by\n  have h_inductive : \u2200 (k : \u2115), 0 < k \u2192 1 + (k : \u211d) * x \u2264 (1 + x) ^ k \u2192 1 + ((k + 1 : \u2115) : \u211d) * x \u2264 (1 + x) ^ (k + 1) := by\n    intro k hk_pos hk_ineq\n    have h\u2081_pos : 0 < (1 + x : \u211d) := by linarith\n    have h\u2082 : (1 + x : \u211d) ^ (k + 1 : \u2115) = (1 + x : \u211d) ^ k * (1 + x : \u211d) := by\n      ring_nf\n      <;> field_simp [h\u2081_pos.ne']\n      <;> ring_nf\n      <;> simp [pow_succ]\n      <;> ring_nf\n    rw [h\u2082]\n    have h\u2083 : (1 + (k : \u211d) * x : \u211d) * (1 + x : \u211d) = 1 + ((k + 1 : \u2115) : \u211d) * x + (k : \u211d) * x ^ 2 := by\n      ring_nf\n      <;> field_simp [h\u2081_pos.ne']\n      <;> ring_nf\n      <;> simp [add_assoc, add_comm, add_left_comm]\n      <;> ring_nf at *\n      <;> norm_cast\n      <;> ring_nf at *\n      <;> field_simp [h\u2081_pos.ne'] at *\n      <;> ring_nf at *\n      <;> norm_cast at *\n      <;> linarith\n    nlinarith\n  have h_main : 1 + (n : \u211d) * x \u2264 (1 + x) ^ n := by\n    have h\u2082 : \u2200 (k : \u2115), 0 < k \u2192 1 + (k : \u211d) * x \u2264 (1 + x) ^ k := by\n      intro k hk_pos\n      induction' hk_pos with k hk_pos IH\n      \u00b7 \n        simpa using h_base\n      \u00b7 \n        simpa using h_inductive k hk_pos IH\n    have h\u2083 : 0 < n := h\u2081\n    have h\u2084 : 1 + (n : \u211d) * x \u2264 (1 + x) ^ n := h\u2082 n h\u2083\n    exact h\u2084\n    <;> norm_num at *\n    <;> linarith\n  simpa using h_main\n", "proof_before": "theorem induction_1pxpownlt1pnx (x : \u211d) (n : \u2115) (h\u2080 : -1 < x) (h\u2081 : 0 < n) :\n    1 + \u2191n * x \u2264 (1 + x) ^ (n : \u2115)  :=  by\n  have h_base : 1 + (1 : \u211d) * x \u2264 (1 + x) ^ 1 := by\n    norm_num\n    <;>\n    (try ring_nf at * <;> linarith) <;>\n    (try nlinarith)\n    <;>\n    (try linarith)\n    <;>\n    (try nlinarith)\n    <;>\n    (try linarith)\n    <;>\n    (try nlinarith)\n  have h_inductive : \u2200 (k : \u2115), 0 < k \u2192 1 + (k : \u211d) * x \u2264 (1 + x) ^ k \u2192 1 + ((k + 1 : \u2115) : \u211d) * x \u2264 (1 + x) ^ (k + 1) := by\n    intro k hk_pos hk_ineq\n    have h\u2081_pos : 0 < (1 + x : \u211d) := by linarith\n    have h\u2082 : (1 + x : \u211d) ^ (k + 1 : \u2115) = (1 + x : \u211d) ^ k * (1 + x : \u211d) := by\n      ring_nf\n      <;> field_simp [h\u2081_pos.ne']\n      <;> ring_nf\n      <;> simp [pow_succ]\n      <;> ring_nf\n    rw [h\u2082]\n    have h\u2083 : (1 + (k : \u211d) * x : \u211d) * (1 + x : \u211d) = 1 + ((k + 1 : \u2115) : \u211d) * x + (k : \u211d) * x ^ 2 := by\n      ring_nf\n      <;> field_simp [h\u2081_pos.ne']\n      <;> ring_nf\n      <;> simp [add_assoc, add_comm, add_left_comm]\n      <;> ring_nf at *\n      <;> norm_cast\n      <;> ring_nf at *\n      <;> field_simp [h\u2081_pos.ne'] at *\n      <;> ring_nf at *\n      <;> norm_cast at *\n      <;> linarith\n    have h\u2084 : (1 + (k : \u211d) * x : \u211d) \u2264 (1 + x : \u211d) ^ k := by\n      exact hk_ineq\n    have h\u2085 : (1 + (k : \u211d) * x : \u211d) * (1 + x : \u211d) \u2264 (1 + x : \u211d) ^ k * (1 + x : \u211d) := by\n      exact mul_le_mul_of_nonneg_right h\u2084 (by linarith)\n    have h\u2086 : 1 + ((k + 1 : \u2115) : \u211d) * x + (k : \u211d) * x ^ 2 \u2264 (1 + x : \u211d) ^ k * (1 + x : \u211d) := by\n      linarith [sq_nonneg (x : \u211d)]\n    have h\u2087 : 1 + ((k + 1 : \u2115) : \u211d) * x \u2264 (1 + x : \u211d) ^ k * (1 + x : \u211d) := by\n      have h\u2088 : (k : \u211d) * x ^ 2 \u2265 0 := by\n        have h\u2089 : (k : \u211d) \u2265 0 := by exact_mod_cast Nat.zero_le k\n        have h\u2081\u2080 : x ^ 2 \u2265 0 := by nlinarith\n        nlinarith\n      linarith\n    exact h\u2087\n    <;> norm_num at *\n    <;> linarith\n  have h_main : 1 + (n : \u211d) * x \u2264 (1 + x) ^ n := by\n    have h\u2082 : \u2200 (k : \u2115), 0 < k \u2192 1 + (k : \u211d) * x \u2264 (1 + x) ^ k := by\n      intro k hk_pos\n      induction' hk_pos with k hk_pos IH\n      \u00b7 \n        simpa using h_base\n      \u00b7 \n        simpa using h_inductive k hk_pos IH\n    have h\u2083 : 0 < n := h\u2081\n    have h\u2084 : 1 + (n : \u211d) * x \u2264 (1 + x) ^ n := h\u2082 n h\u2083\n    exact h\u2084\n    <;> norm_num at *\n    <;> linarith\n  simpa using h_main\n"}
{"mode": "clear", "steps_before": 21, "alphabet": 18, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 329, "chars_after": 11, "seconds": 0.0, "id": "mathd_algebra_440", "proof_after": "theorem mathd_algebra_440 (x : \u211d) (h\u2080 : 3 / 2 / 3 = x / 10) : x = 5  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_440 (x : \u211d) (h\u2080 : 3 / 2 / 3 = x / 10) : x = 5  :=  by\n  have h\u2081 : x = 5 := by\n    have h\u2082 : x = 5 := by\n      norm_num at h\u2080 \u22a2\n      <;>\n      (try ring_nf at h\u2080 \u22a2) <;>\n      (try linarith) <;>\n      (try nlinarith) <;>\n      (try field_simp at h\u2080 \u22a2) <;>\n      (try ring_nf at h\u2080 \u22a2) <;>\n      (try linarith) <;>\n      (try nlinarith)\n      <;>\n      linarith\n    exact h\u2082\n  exact h\u2081\n"}
{"mode": "clear", "steps_before": 19, "alphabet": 20, "compiles": true, "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 441, "chars_after": 21, "seconds": 26.5, "id": "mathd_algebra_76", "proof_after": "theorem mathd_algebra_76 (f : \u2124 \u2192 \u2124) (h\u2080 : \u2200 n, Odd n \u2192 f n = n ^ 2)\n    (h\u2081 : \u2200 n, Even n \u2192 f n = n ^ 2 - 4 * n - 1) : f 4 = -1  :=  by\n  apply h\u2081\n  trivial\n", "proof_before": "theorem mathd_algebra_76 (f : \u2124 \u2192 \u2124) (h\u2080 : \u2200 n, Odd n \u2192 f n = n ^ 2)\n    (h\u2081 : \u2200 n, Even n \u2192 f n = n ^ 2 - 4 * n - 1) : f 4 = -1  :=  by\n  have h\u2082 : Even (4 : \u2124) := by\n    rw [Int.even_iff]\n    <;> norm_num\n    <;>\n    trivial\n  have h\u2083 : f 4 = (4 : \u2124) ^ 2 - 4 * (4 : \u2124) - 1 := by\n    have h\u2084 : f 4 = (4 : \u2124) ^ 2 - 4 * (4 : \u2124) - 1 := by\n      have h\u2085 : Even (4 : \u2124) := h\u2082\n      have h\u2086 : f 4 = (4 : \u2124) ^ 2 - 4 * (4 : \u2124) - 1 := by\n        apply h\u2081\n        exact h\u2085\n      rw [h\u2086]\n    exact h\u2084\n  have h\u2084 : f 4 = -1 := by\n    rw [h\u2083]\n    <;> norm_num\n    <;>\n    trivial\n  apply h\u2084\n"}
{"mode": "clear", "steps_before": 60, "alphabet": 27, "compiles": true, "steps_after": 38, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 5, "timed_out": false, "fixpoint": true, "chars_before": 2090, "chars_after": 1454, "seconds": 358.0, "id": "amc12b_2020_p6", "proof_after": "theorem amc12b_2020_p6 (n : \u2115) (h\u2080 : 9 \u2264 n) : \u2203 x : \u2115, (x : \u211d) ^ 2 = ((n + 2)! - (n + 1)!) / n !  :=  by\n  have h\u2081 : ((n + 1 : \u2115) : \u211d) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by\n    have h\u2082 : (n + 2 : \u2115) ! = (n + 2 : \u2115) * (n + 1 : \u2115) * n ! := by\n      have h\u2083 : (n + 2 : \u2115) ! = (n + 2 : \u2115) * (n + 1 : \u2115) * n ! := by\n        calc\n          (n + 2 : \u2115)! = (n + 2 : \u2115) * (n + 1 : \u2115)! := by\n            simp [Nat.factorial_succ, Nat.mul_succ]\n            <;> ring_nf\n          _ = (n + 2 : \u2115) * ((n + 1 : \u2115) * n !) := by\n            simp [Nat.factorial_succ, Nat.mul_succ]\n            <;> ring_nf\n          _ = (n + 2 : \u2115) * (n + 1 : \u2115) * n ! := by\n            ring_nf\n            <;> simp [Nat.mul_assoc]\n            <;> ring_nf\n      exact h\u2083\n    have h\u2084 : ((n + 2 : \u2115) ! : \u211d) = ((n + 2 : \u2115) * (n + 1 : \u2115) * n ! : \u211d) := by\n      norm_cast\n    have h\u2085 : ((n + 1 : \u2115) ! : \u211d) = ((n + 1 : \u2115) * n ! : \u211d) := by\n      norm_cast\n    have h\u2086 : (((n + 2 : \u2115) ! : \u211d) - ((n + 1 : \u2115) ! : \u211d)) = ((n + 1 : \u2115) ^ 2 * n ! : \u211d) := by\n      rw [h\u2084, h\u2085]\n      <;> norm_num\n      <;> ring_nf\n      <;> norm_cast\n      <;> simp [Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]\n      <;> ring_nf\n      <;> norm_num\n      <;> simp [Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]\n      <;> ring_nf\n      <;> norm_num\n    have h\u2087 : ((n + 1 : \u2115) : \u211d) ^ 2 = (((n + 2 : \u2115) ! : \u211d) - ((n + 1 : \u2115) ! : \u211d)) / (n ! : \u211d) := by\n      rw [h\u2086]\n      field_simp\n    simpa using h\u2087\n  have h\u2082 : \u2203 x : \u2115, (x : \u211d) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by\n    refine' \u27e8n + 1, _\u27e9\n    simpa using h\u2081\n  apply h\u2082\n", "proof_before": "theorem amc12b_2020_p6 (n : \u2115) (h\u2080 : 9 \u2264 n) : \u2203 x : \u2115, (x : \u211d) ^ 2 = ((n + 2)! - (n + 1)!) / n !  :=  by\n  have h\u2081 : ((n + 1 : \u2115) : \u211d) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by\n    have h\u2082 : (n + 2 : \u2115) ! = (n + 2 : \u2115) * (n + 1 : \u2115) * n ! := by\n      have h\u2083 : (n + 2 : \u2115) ! = (n + 2 : \u2115) * (n + 1 : \u2115) * n ! := by\n        calc\n          (n + 2 : \u2115)! = (n + 2 : \u2115) * (n + 1 : \u2115)! := by\n            simp [Nat.factorial_succ, Nat.mul_succ]\n            <;> ring_nf\n          _ = (n + 2 : \u2115) * ((n + 1 : \u2115) * n !) := by\n            simp [Nat.factorial_succ, Nat.mul_succ]\n            <;> ring_nf\n          _ = (n + 2 : \u2115) * (n + 1 : \u2115) * n ! := by\n            ring_nf\n            <;> simp [Nat.mul_assoc]\n            <;> ring_nf\n      exact h\u2083\n    have h\u2083 : (n + 1 : \u2115) ! = (n + 1 : \u2115) * n ! := by\n      have h\u2084 : (n + 1 : \u2115) ! = (n + 1 : \u2115) * n ! := by\n        calc\n          (n + 1 : \u2115)! = (n + 1 : \u2115) * n ! := by\n            simp [Nat.factorial_succ, Nat.mul_succ]\n            <;> ring_nf\n          _ = (n + 1 : \u2115) * n ! := by rfl\n      exact h\u2084\n    have h\u2084 : ((n + 2 : \u2115) ! : \u211d) = ((n + 2 : \u2115) * (n + 1 : \u2115) * n ! : \u211d) := by\n      norm_cast\n      <;> simp [h\u2082]\n      <;> ring_nf\n    have h\u2085 : ((n + 1 : \u2115) ! : \u211d) = ((n + 1 : \u2115) * n ! : \u211d) := by\n      norm_cast\n      <;> simp [h\u2083]\n      <;> ring_nf\n    have h\u2086 : (((n + 2 : \u2115) ! : \u211d) - ((n + 1 : \u2115) ! : \u211d)) = ((n + 1 : \u2115) ^ 2 * n ! : \u211d) := by\n      rw [h\u2084, h\u2085]\n      <;> norm_num\n      <;> ring_nf\n      <;> norm_cast\n      <;> simp [Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]\n      <;> ring_nf\n      <;> norm_num\n      <;> simp [Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]\n      <;> ring_nf\n      <;> norm_num\n    have h\u2087 : ((n + 1 : \u2115) : \u211d) ^ 2 = (((n + 2 : \u2115) ! : \u211d) - ((n + 1 : \u2115) ! : \u211d)) / (n ! : \u211d) := by\n      rw [h\u2086]\n      have h\u2088 : (n ! : \u211d) \u2260 0 := by\n        norm_cast\n        exact Nat.factorial_ne_zero n\n      field_simp [h\u2088]\n      <;> ring_nf\n      <;> norm_cast\n      <;> simp [Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]\n      <;> ring_nf\n      <;> norm_num\n      <;> linarith\n    simpa using h\u2087\n  have h\u2082 : \u2203 x : \u2115, (x : \u211d) ^ 2 = ((n + 2)! - (n + 1)!) / n ! := by\n    refine' \u27e8n + 1, _\u27e9\n    simpa using h\u2081\n  apply h\u2082\n"}
{"mode": "clear", "steps_before": 35, "alphabet": 18, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 817, "chars_after": 7, "seconds": 0.0, "id": "mathd_algebra_314", "proof_after": "theorem mathd_algebra_314 (n : \u2115) (h\u2080 : n = 11) : (1 / 4) ^ (n + 1) * 2 ^ (2 * n) = 1 / 4  :=  by\n  ring\n", "proof_before": "theorem mathd_algebra_314 (n : \u2115) (h\u2080 : n = 11) : (1 / 4) ^ (n + 1) * 2 ^ (2 * n) = 1 / 4  :=  by\n  have h\u2081 : 1 / 4 = 0 := by\n    norm_num\n    <;> simp_all [Nat.div_eq_of_lt]\n    <;> norm_num\n  have h\u2082 : (1 / 4) ^ (n + 1) = 0 := by\n    rw [h\u2081]\n    have h\u2082\u2081 : n + 1 > 0 := by\n      have h\u2082\u2081\u2081 : n \u2265 0 := by linarith\n      linarith\n    have h\u2082\u2082 : 0 ^ (n + 1) = 0 := by\n      have h\u2082\u2082\u2081 : n + 1 \u2260 0 := by linarith\n      simp [h\u2082\u2082\u2081, Nat.zero_pow]\n    simp_all [Nat.zero_pow]\n    <;> simp_all\n  have h\u2083 : (1 / 4) ^ (n + 1) * 2 ^ (2 * n) = 0 := by\n    rw [h\u2082]\n    <;> simp [Nat.mul_zero]\n    <;> norm_num\n    <;> simp_all\n    <;> ring_nf\n    <;> norm_num\n    <;> simp_all\n  have h\u2084 : 1 / 4 = 0 := by\n    norm_num\n    <;> simp_all [Nat.div_eq_of_lt]\n    <;> norm_num\n  have h\u2085 : (1 / 4) ^ (n + 1) * 2 ^ (2 * n) = 1 / 4 := by\n    rw [h\u2083, h\u2084]\n    <;> simp_all\n    <;> norm_num\n    <;> ring_nf at *\n    <;> simp_all\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 80, "alphabet": 38, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 80, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "timed_out": false, "fixpoint": false, "chars_before": 2481, "chars_after": 2481, "seconds": 92.0, "id": "mathd_algebra_756", "proof_after": "theorem mathd_algebra_756 (a b : \u211d) (h\u2080 : (2 : \u211d) ^ a = 32) (h\u2081 : a ^ b = 125) : b ^ a = 243  :=  by\n  have h\u2082 : a > 0 := by\n    by_contra h\n    have h\u2082\u2081 : a \u2264 0 := by linarith\n    have h\u2082\u2082 : (2 : \u211d) ^ a \u2264 1 := by\n      have h\u2082\u2082\u2081 : a \u2264 0 := h\u2082\u2081\n      have h\u2082\u2082\u2082 : (2 : \u211d) ^ a \u2264 (2 : \u211d) ^ (0 : \u211d) := by\n        apply Real.rpow_le_rpow_of_exponent_le\n        \u00b7 norm_num\n        \u00b7 linarith\n      norm_num at h\u2082\u2082\u2082 \u22a2\n      <;> linarith\n    have h\u2082\u2083 : (2 : \u211d) ^ a = 32 := h\u2080\n    linarith\n  have h\u2083 : a = 5 := by\n    have h\u2084 : Real.log ((2 : \u211d) ^ a) = Real.log (32 : \u211d) := by rw [h\u2080]\n    have h\u2085 : a * Real.log 2 = Real.log 32 := by\n      rw [Real.log_rpow (by norm_num : (2 : \u211d) > 0)] at h\u2084\n      <;> simp_all [Real.log_pow]\n      <;> ring_nf at *\n      <;> linarith\n    have h\u2086 : Real.log 32 = 5 * Real.log 2 := by\n      have h\u2086\u2081 : Real.log 32 = Real.log (2 ^ 5) := by norm_num\n      rw [h\u2086\u2081]\n      have h\u2086\u2082 : Real.log (2 ^ 5 : \u211d) = 5 * Real.log 2 := by\n        rw [Real.log_pow]\n        <;> norm_num\n        <;> linarith\n      rw [h\u2086\u2082]\n    rw [h\u2086] at h\u2085\n    have h\u2087 : a * Real.log 2 = 5 * Real.log 2 := by linarith\n    have h\u2088 : a = 5 := by\n      apply mul_left_cancel\u2080 (show (Real.log 2 : \u211d) \u2260 0 by\n        have h\u2089 : Real.log 2 > 0 := Real.log_pos (by norm_num)\n        linarith)\n      linarith\n    exact h\u2088\n  have h\u2084 : b = 3 := by\n    have h\u2085 : a > 0 := h\u2082\n    have h\u2086 : a = 5 := h\u2083\n    have h\u2087 : (a : \u211d) ^ b = 125 := h\u2081\n    rw [h\u2086] at h\u2087\n    have h\u2088 : (5 : \u211d) ^ b = 125 := by exact_mod_cast h\u2087\n    have h\u2089 : Real.log ((5 : \u211d) ^ b) = Real.log 125 := by rw [h\u2088]\n    have h\u2081\u2080 : b * Real.log 5 = Real.log 125 := by\n      have h\u2081\u2080\u2081 : Real.log ((5 : \u211d) ^ b) = b * Real.log 5 := by\n        rw [Real.log_rpow (by norm_num : (5 : \u211d) > 0)]\n      rw [h\u2081\u2080\u2081] at h\u2089\n      linarith\n    have h\u2081\u2081 : Real.log 125 = Real.log (5 ^ 3) := by norm_num\n    have h\u2081\u2082 : Real.log (5 ^ 3 : \u211d) = 3 * Real.log 5 := by\n      rw [Real.log_pow]\n      <;> norm_num\n      <;> linarith\n    have h\u2081\u2083 : Real.log 125 = 3 * Real.log 5 := by\n      rw [h\u2081\u2081, h\u2081\u2082]\n    rw [h\u2081\u2083] at h\u2081\u2080\n    have h\u2081\u2084 : b * Real.log 5 = 3 * Real.log 5 := by linarith\n    have h\u2081\u2085 : b = 3 := by\n      apply mul_left_cancel\u2080 (show (Real.log 5 : \u211d) \u2260 0 by\n        have h\u2081\u2085\u2081 : Real.log 5 > 0 := Real.log_pos (by norm_num)\n        linarith)\n      linarith\n    exact h\u2081\u2085\n  have h\u2085 : b ^ a = 243 := by\n    rw [h\u2084, h\u2083]\n    have h\u2085\u2081 : (3 : \u211d) ^ (5 : \u211d) = 243 := by\n      norm_num [Real.rpow_def_of_pos, Real.rpow_def_of_nonneg, Real.log_pow]\n      <;>\n      ring_nf\n      <;>\n      norm_num\n      <;>\n      linarith\n    rw [h\u2085\u2081]\n    <;> norm_num\n  exact h\u2085\n", "proof_before": "theorem mathd_algebra_756 (a b : \u211d) (h\u2080 : (2 : \u211d) ^ a = 32) (h\u2081 : a ^ b = 125) : b ^ a = 243  :=  by\n  have h\u2082 : a > 0 := by\n    by_contra h\n    have h\u2082\u2081 : a \u2264 0 := by linarith\n    have h\u2082\u2082 : (2 : \u211d) ^ a \u2264 1 := by\n      have h\u2082\u2082\u2081 : a \u2264 0 := h\u2082\u2081\n      have h\u2082\u2082\u2082 : (2 : \u211d) ^ a \u2264 (2 : \u211d) ^ (0 : \u211d) := by\n        apply Real.rpow_le_rpow_of_exponent_le\n        \u00b7 norm_num\n        \u00b7 linarith\n      norm_num at h\u2082\u2082\u2082 \u22a2\n      <;> linarith\n    have h\u2082\u2083 : (2 : \u211d) ^ a = 32 := h\u2080\n    linarith\n  have h\u2083 : a = 5 := by\n    have h\u2084 : Real.log ((2 : \u211d) ^ a) = Real.log (32 : \u211d) := by rw [h\u2080]\n    have h\u2085 : a * Real.log 2 = Real.log 32 := by\n      rw [Real.log_rpow (by norm_num : (2 : \u211d) > 0)] at h\u2084\n      <;> simp_all [Real.log_pow]\n      <;> ring_nf at *\n      <;> linarith\n    have h\u2086 : Real.log 32 = 5 * Real.log 2 := by\n      have h\u2086\u2081 : Real.log 32 = Real.log (2 ^ 5) := by norm_num\n      rw [h\u2086\u2081]\n      have h\u2086\u2082 : Real.log (2 ^ 5 : \u211d) = 5 * Real.log 2 := by\n        rw [Real.log_pow]\n        <;> norm_num\n        <;> linarith\n      rw [h\u2086\u2082]\n    rw [h\u2086] at h\u2085\n    have h\u2087 : a * Real.log 2 = 5 * Real.log 2 := by linarith\n    have h\u2088 : a = 5 := by\n      apply mul_left_cancel\u2080 (show (Real.log 2 : \u211d) \u2260 0 by\n        have h\u2089 : Real.log 2 > 0 := Real.log_pos (by norm_num)\n        linarith)\n      linarith\n    exact h\u2088\n  have h\u2084 : b = 3 := by\n    have h\u2085 : a > 0 := h\u2082\n    have h\u2086 : a = 5 := h\u2083\n    have h\u2087 : (a : \u211d) ^ b = 125 := h\u2081\n    rw [h\u2086] at h\u2087\n    have h\u2088 : (5 : \u211d) ^ b = 125 := by exact_mod_cast h\u2087\n    have h\u2089 : Real.log ((5 : \u211d) ^ b) = Real.log 125 := by rw [h\u2088]\n    have h\u2081\u2080 : b * Real.log 5 = Real.log 125 := by\n      have h\u2081\u2080\u2081 : Real.log ((5 : \u211d) ^ b) = b * Real.log 5 := by\n        rw [Real.log_rpow (by norm_num : (5 : \u211d) > 0)]\n      rw [h\u2081\u2080\u2081] at h\u2089\n      linarith\n    have h\u2081\u2081 : Real.log 125 = Real.log (5 ^ 3) := by norm_num\n    have h\u2081\u2082 : Real.log (5 ^ 3 : \u211d) = 3 * Real.log 5 := by\n      rw [Real.log_pow]\n      <;> norm_num\n      <;> linarith\n    have h\u2081\u2083 : Real.log 125 = 3 * Real.log 5 := by\n      rw [h\u2081\u2081, h\u2081\u2082]\n    rw [h\u2081\u2083] at h\u2081\u2080\n    have h\u2081\u2084 : b * Real.log 5 = 3 * Real.log 5 := by linarith\n    have h\u2081\u2085 : b = 3 := by\n      apply mul_left_cancel\u2080 (show (Real.log 5 : \u211d) \u2260 0 by\n        have h\u2081\u2085\u2081 : Real.log 5 > 0 := Real.log_pos (by norm_num)\n        linarith)\n      linarith\n    exact h\u2081\u2085\n  have h\u2085 : b ^ a = 243 := by\n    rw [h\u2084, h\u2083]\n    have h\u2085\u2081 : (3 : \u211d) ^ (5 : \u211d) = 243 := by\n      norm_num [Real.rpow_def_of_pos, Real.rpow_def_of_nonneg, Real.log_pow]\n      <;>\n      ring_nf\n      <;>\n      norm_num\n      <;>\n      linarith\n    rw [h\u2085\u2081]\n    <;> norm_num\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 25, "alphabet": 18, "compiles": true, "steps_after": 13, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 5, "timed_out": false, "fixpoint": true, "chars_before": 587, "chars_after": 371, "seconds": 172.9, "id": "mathd_algebra_270", "proof_after": "theorem mathd_algebra_270 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 (x) (_ : x \u2260 -2), f x = 1 / (x + 2)) :\n    f (f 1) = 3 / 7  :=  by\n  have h\u2081 : f 1 = 1 / 3 := by\n    have h\u2081\u2081 : (1 : \u211d) \u2260 -2 := by norm_num\n    have h\u2081\u2082 : f 1 = 1 / (1 + 2 : \u211d) := h\u2080 1 h\u2081\u2081\n    linarith\n  have h\u2082 : f (f 1) = f (1 / 3) := by\n    rw [h\u2081]\n  have h\u2083 : f (1 / 3) = 3 / 7 := by\n    have h\u2083\u2081 : (1 / 3 : \u211d) \u2260 -2 := by norm_num\n    have h\u2083\u2082 : f (1 / 3 : \u211d) = 1 / ((1 / 3 : \u211d) + 2 : \u211d) := h\u2080 (1 / 3 : \u211d) h\u2083\u2081\n    linarith\n  linarith\n", "proof_before": "theorem mathd_algebra_270 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 (x) (_ : x \u2260 -2), f x = 1 / (x + 2)) :\n    f (f 1) = 3 / 7  :=  by\n  have h\u2081 : f 1 = 1 / 3 := by\n    have h\u2081\u2081 : (1 : \u211d) \u2260 -2 := by norm_num\n    have h\u2081\u2082 : f 1 = 1 / (1 + 2 : \u211d) := h\u2080 1 h\u2081\u2081\n    rw [h\u2081\u2082]\n    <;> norm_num\n    <;> field_simp\n    <;> ring\n    <;> norm_num\n  have h\u2082 : f (f 1) = f (1 / 3) := by\n    rw [h\u2081]\n    <;> norm_num\n  have h\u2083 : f (1 / 3) = 3 / 7 := by\n    have h\u2083\u2081 : (1 / 3 : \u211d) \u2260 -2 := by norm_num\n    have h\u2083\u2082 : f (1 / 3 : \u211d) = 1 / ((1 / 3 : \u211d) + 2 : \u211d) := h\u2080 (1 / 3 : \u211d) h\u2083\u2081\n    rw [h\u2083\u2082]\n    <;> norm_num\n    <;> field_simp\n    <;> ring\n    <;> norm_num\n  have h\u2084 : f (f 1) = 3 / 7 := by\n    rw [h\u2082, h\u2083]\n    <;> norm_num\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 24, "alphabet": 21, "compiles": true, "steps_after": 10, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 8, "timed_out": false, "fixpoint": false, "chars_before": 1128, "chars_after": 850, "seconds": 242.1, "id": "imo_1983_p6", "proof_after": "theorem imo_1983_p6 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) (h\u2081 : c < a + b) (h\u2082 : b < a + c)\n    (h\u2083 : a < b + c) : 0 \u2264 a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a)  :=  by\n  have h\u2084 : 0 < b + c - a := by\n    linarith\n  have h\u2085 : 0 < c + a - b := by\n    linarith [h\u2082]\n    <;> linarith\n  have h\u2086 : 0 < a + b - c := by\n    linarith\n  have h\u2087 : 0 \u2264 a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by\n    nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a),\n      mul_pos h\u2080.1 h\u2080.2.1, mul_pos h\u2080.2.1 h\u2080.2.2, mul_pos h\u2080.2.2 h\u2080.1,\n      mul_pos (sub_pos.mpr h\u2081) (sub_pos.mpr h\u2082), mul_pos (sub_pos.mpr h\u2082) (sub_pos.mpr h\u2083),\n      mul_pos (sub_pos.mpr h\u2083) (sub_pos.mpr h\u2081), mul_pos h\u2084 h\u2085, mul_pos h\u2085 h\u2086,\n      mul_pos h\u2086 h\u2084, sq_nonneg (a - b + c), sq_nonneg (b - c + a),\n      sq_nonneg (c - a + b), mul_nonneg (sub_nonneg.mpr h\u2081.le) (sub_nonneg.mpr h\u2082.le),\n      mul_nonneg (sub_nonneg.mpr h\u2082.le) (sub_nonneg.mpr h\u2083.le),\n      mul_nonneg (sub_nonneg.mpr h\u2083.le) (sub_nonneg.mpr h\u2081.le)]\n  exact h\u2087\n", "proof_before": "theorem imo_1983_p6 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) (h\u2081 : c < a + b) (h\u2082 : b < a + c)\n    (h\u2083 : a < b + c) : 0 \u2264 a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a)  :=  by\n  have h\u2084 : 0 < b + c - a := by\n    linarith [h\u2083]\n    <;> linarith\n  have h\u2085 : 0 < c + a - b := by\n    linarith [h\u2082]\n    <;> linarith\n  have h\u2086 : 0 < a + b - c := by\n    linarith [h\u2081]\n    <;> linarith\n  have h\u2087 : 0 \u2264 a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by\n    have h\u2087\u2081 : 0 < a := by linarith\n    have h\u2087\u2082 : 0 < b := by linarith\n    have h\u2087\u2083 : 0 < c := by linarith\n    have h\u2087\u2084 : 0 < a * b := by positivity\n    have h\u2087\u2085 : 0 < b * c := by positivity\n    have h\u2087\u2086 : 0 < c * a := by positivity\n    nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a),\n      mul_pos h\u2080.1 h\u2080.2.1, mul_pos h\u2080.2.1 h\u2080.2.2, mul_pos h\u2080.2.2 h\u2080.1,\n      mul_pos (sub_pos.mpr h\u2081) (sub_pos.mpr h\u2082), mul_pos (sub_pos.mpr h\u2082) (sub_pos.mpr h\u2083),\n      mul_pos (sub_pos.mpr h\u2083) (sub_pos.mpr h\u2081), mul_pos h\u2084 h\u2085, mul_pos h\u2085 h\u2086,\n      mul_pos h\u2086 h\u2084, sq_nonneg (a - b + c), sq_nonneg (b - c + a),\n      sq_nonneg (c - a + b), mul_nonneg (sub_nonneg.mpr h\u2081.le) (sub_nonneg.mpr h\u2082.le),\n      mul_nonneg (sub_nonneg.mpr h\u2082.le) (sub_nonneg.mpr h\u2083.le),\n      mul_nonneg (sub_nonneg.mpr h\u2083.le) (sub_nonneg.mpr h\u2081.le)]\n  exact h\u2087\n"}
{"mode": "clear", "steps_before": 14, "alphabet": 13, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 317, "chars_after": 6, "seconds": 0.0, "id": "mathd_numbertheory_517", "proof_after": "theorem mathd_numbertheory_517 : 121 * 122 * 123 % 4 = 2  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_517 : 121 * 122 * 123 % 4 = 2  :=  by\n  have h\u2081 : 121 % 4 = 1 := by\n    norm_num\n    <;> rfl\n  have h\u2082 : 122 % 4 = 2 := by\n    norm_num\n    <;> rfl\n  have h\u2083 : 123 % 4 = 3 := by\n    norm_num\n    <;> rfl\n  have h\u2084 : 121 * 122 * 123 % 4 = 2 := by\n    norm_num [h\u2081, h\u2082, h\u2083, Nat.mul_mod, Nat.add_mod, Nat.mod_mod, Nat.mod_eq_of_lt]\n    <;>\n    rfl\n  apply h\u2084\n"}
{"mode": "clear", "steps_before": 117, "alphabet": 45, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 117, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "timed_out": true, "fixpoint": false, "chars_before": 5852, "chars_after": 5852, "seconds": 1027.4, "id": "amc12b_2021_p18", "proof_after": "theorem amc12b_2021_p18 (z : \u2102)\n    (h\u2080 : 12 * Complex.normSq z = 2 * Complex.normSq (z + 2) + Complex.normSq (z ^ 2 + 1) + 31) :\n    z + 6 / z = -2  :=  by\n  have h\u2081 : z.re = -1 := by\n    have h\u2081\u2081 : Complex.normSq z = z.re * z.re + z.im * z.im := by\n      simp [Complex.normSq_apply]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    have h\u2081\u2082 : Complex.normSq (z + 2) = (z.re + 2) * (z.re + 2) + z.im * z.im := by\n      simp [Complex.normSq_apply, Complex.ext_iff, pow_two]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    have h\u2081\u2083 : Complex.normSq (z ^ 2 + 1) = (z.re * z.re - z.im * z.im + 1) * (z.re * z.re - z.im * z.im + 1) + (2 * z.re * z.im) * (2 * z.re * z.im) := by\n      simp [Complex.normSq_apply, Complex.ext_iff, pow_two]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    rw [h\u2081\u2081, h\u2081\u2082, h\u2081\u2083] at h\u2080\n    have h\u2081\u2084 : z.re * z.re + z.im * z.im = (z.re * z.re + z.im * z.im) := by ring\n    have h\u2081\u2085 : 12 * (z.re * z.re + z.im * z.im) = 2 * ((z.re + 2) * (z.re + 2) + z.im * z.im) + ((z.re * z.re - z.im * z.im + 1) * (z.re * z.re - z.im * z.im + 1) + (2 * z.re * z.im) * (2 * z.re * z.im)) + 31 := by\n      ring_nf at h\u2080 \u22a2\n      <;> nlinarith\n    have h\u2081\u2086 : (z.re + 1) = 0 := by\n      nlinarith [sq_nonneg (z.re + 1), sq_nonneg (z.im), sq_nonneg (z.re - 1),\n        sq_nonneg (z.re * z.re + z.im * z.im - 6),\n        sq_nonneg (z.re * z.re + z.im * z.im - 2 * z.re),\n        sq_nonneg (z.re * z.re + z.im * z.im + 2 * z.re)]\n    nlinarith\n  have h\u2082 : z.re * z.re + z.im * z.im = 6 := by\n    have h\u2082\u2081 : Complex.normSq z = z.re * z.re + z.im * z.im := by\n      simp [Complex.normSq_apply]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    have h\u2082\u2082 : Complex.normSq (z + 2) = (z.re + 2) * (z.re + 2) + z.im * z.im := by\n      simp [Complex.normSq_apply, Complex.ext_iff, pow_two]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    have h\u2082\u2083 : Complex.normSq (z ^ 2 + 1) = (z.re * z.re - z.im * z.im + 1) * (z.re * z.re - z.im * z.im + 1) + (2 * z.re * z.im) * (2 * z.re * z.im) := by\n      simp [Complex.normSq_apply, Complex.ext_iff, pow_two]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    rw [h\u2082\u2081, h\u2082\u2082, h\u2082\u2083] at h\u2080\n    have h\u2082\u2084 : z.re * z.re + z.im * z.im = (z.re * z.re + z.im * z.im) := by ring\n    have h\u2082\u2085 : 12 * (z.re * z.re + z.im * z.im) = 2 * ((z.re + 2) * (z.re + 2) + z.im * z.im) + ((z.re * z.re - z.im * z.im + 1) * (z.re * z.re - z.im * z.im + 1) + (2 * z.re * z.im) * (2 * z.re * z.im)) + 31 := by\n      ring_nf at h\u2080 \u22a2\n      <;> nlinarith\n    nlinarith [sq_nonneg (z.re + 1), sq_nonneg (z.im), sq_nonneg (z.re - 1),\n      sq_nonneg (z.re * z.re + z.im * z.im - 6),\n      sq_nonneg (z.re * z.re + z.im * z.im - 2 * z.re),\n      sq_nonneg (z.re * z.re + z.im * z.im + 2 * z.re)]\n  have h\u2083 : z \u2260 0 := by\n    by_contra h\n    rw [h] at h\u2080\n    norm_num [Complex.normSq, Complex.ext_iff, pow_two] at h\u2080\n    <;>\n    (try norm_num at h\u2080) <;>\n    (try nlinarith)\n    <;>\n    (try simp_all [Complex.ext_iff, pow_two])\n    <;>\n    (try nlinarith)\n  have h\u2084 : z + 6 / z = -2 := by\n    have h\u2084\u2081 : z.im = Real.sqrt 5 \u2228 z.im = -Real.sqrt 5 := by\n      have h\u2084\u2082 : z.im * z.im = 5 := by\n        nlinarith\n      have h\u2084\u2083 : z.im = Real.sqrt 5 \u2228 z.im = -Real.sqrt 5 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2084\u2084\n        apply eq_of_sub_eq_zero\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2084\u2084)\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n      exact h\u2084\u2083\n    have h\u2084\u2085 : z + 6 / z = -2 := by\n      cases h\u2084\u2081 with\n      | inl h\u2084\u2081 =>\n        have h\u2084\u2086 : z.im = Real.sqrt 5 := h\u2084\u2081\n        have h\u2084\u2087 : z.re = -1 := h\u2081\n        have h\u2084\u2088 : z.re * z.re + z.im * z.im = 6 := h\u2082\n        have h\u2084\u2089 : z \u2260 0 := h\u2083\n        have h\u2085\u2080 : z + 6 / z = -2 := by\n          field_simp [Complex.ext_iff, Complex.normSq, pow_two, h\u2084\u2089, Complex.ext_iff, Complex.normSq, pow_two]\n          <;> simp_all [Complex.ext_iff, Complex.normSq, pow_two]\n          <;> ring_nf at *\n          <;> norm_num at *\n          <;>\n          (try\n            {\n              nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n          <;>\n          (try\n            {\n              constructor <;> nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n          <;>\n          (try\n            {\n              nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n        exact h\u2085\u2080\n      | inr h\u2084\u2081 =>\n        have h\u2084\u2086 : z.im = -Real.sqrt 5 := h\u2084\u2081\n        have h\u2084\u2087 : z.re = -1 := h\u2081\n        have h\u2084\u2088 : z.re * z.re + z.im * z.im = 6 := h\u2082\n        have h\u2084\u2089 : z \u2260 0 := h\u2083\n        have h\u2085\u2080 : z + 6 / z = -2 := by\n          field_simp [Complex.ext_iff, Complex.normSq, pow_two, h\u2084\u2089, Complex.ext_iff, Complex.normSq, pow_two]\n          <;> simp_all [Complex.ext_iff, Complex.normSq, pow_two]\n          <;> ring_nf at *\n          <;> norm_num at *\n          <;>\n          (try\n            {\n              nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n          <;>\n          (try\n            {\n              constructor <;> nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n          <;>\n          (try\n            {\n              nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n        exact h\u2085\u2080\n    exact h\u2084\u2085\n  exact h\u2084\n", "proof_before": "theorem amc12b_2021_p18 (z : \u2102)\n    (h\u2080 : 12 * Complex.normSq z = 2 * Complex.normSq (z + 2) + Complex.normSq (z ^ 2 + 1) + 31) :\n    z + 6 / z = -2  :=  by\n  have h\u2081 : z.re = -1 := by\n    have h\u2081\u2081 : Complex.normSq z = z.re * z.re + z.im * z.im := by\n      simp [Complex.normSq_apply]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    have h\u2081\u2082 : Complex.normSq (z + 2) = (z.re + 2) * (z.re + 2) + z.im * z.im := by\n      simp [Complex.normSq_apply, Complex.ext_iff, pow_two]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    have h\u2081\u2083 : Complex.normSq (z ^ 2 + 1) = (z.re * z.re - z.im * z.im + 1) * (z.re * z.re - z.im * z.im + 1) + (2 * z.re * z.im) * (2 * z.re * z.im) := by\n      simp [Complex.normSq_apply, Complex.ext_iff, pow_two]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    rw [h\u2081\u2081, h\u2081\u2082, h\u2081\u2083] at h\u2080\n    have h\u2081\u2084 : z.re * z.re + z.im * z.im = (z.re * z.re + z.im * z.im) := by ring\n    have h\u2081\u2085 : 12 * (z.re * z.re + z.im * z.im) = 2 * ((z.re + 2) * (z.re + 2) + z.im * z.im) + ((z.re * z.re - z.im * z.im + 1) * (z.re * z.re - z.im * z.im + 1) + (2 * z.re * z.im) * (2 * z.re * z.im)) + 31 := by\n      ring_nf at h\u2080 \u22a2\n      <;> nlinarith\n    have h\u2081\u2086 : (z.re + 1) = 0 := by\n      nlinarith [sq_nonneg (z.re + 1), sq_nonneg (z.im), sq_nonneg (z.re - 1),\n        sq_nonneg (z.re * z.re + z.im * z.im - 6),\n        sq_nonneg (z.re * z.re + z.im * z.im - 2 * z.re),\n        sq_nonneg (z.re * z.re + z.im * z.im + 2 * z.re)]\n    nlinarith\n  have h\u2082 : z.re * z.re + z.im * z.im = 6 := by\n    have h\u2082\u2081 : Complex.normSq z = z.re * z.re + z.im * z.im := by\n      simp [Complex.normSq_apply]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    have h\u2082\u2082 : Complex.normSq (z + 2) = (z.re + 2) * (z.re + 2) + z.im * z.im := by\n      simp [Complex.normSq_apply, Complex.ext_iff, pow_two]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    have h\u2082\u2083 : Complex.normSq (z ^ 2 + 1) = (z.re * z.re - z.im * z.im + 1) * (z.re * z.re - z.im * z.im + 1) + (2 * z.re * z.im) * (2 * z.re * z.im) := by\n      simp [Complex.normSq_apply, Complex.ext_iff, pow_two]\n      <;> ring_nf\n      <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n      <;> ring_nf\n      <;> nlinarith [sq_nonneg (z.re + z.im), sq_nonneg (z.re - z.im)]\n    rw [h\u2082\u2081, h\u2082\u2082, h\u2082\u2083] at h\u2080\n    have h\u2082\u2084 : z.re * z.re + z.im * z.im = (z.re * z.re + z.im * z.im) := by ring\n    have h\u2082\u2085 : 12 * (z.re * z.re + z.im * z.im) = 2 * ((z.re + 2) * (z.re + 2) + z.im * z.im) + ((z.re * z.re - z.im * z.im + 1) * (z.re * z.re - z.im * z.im + 1) + (2 * z.re * z.im) * (2 * z.re * z.im)) + 31 := by\n      ring_nf at h\u2080 \u22a2\n      <;> nlinarith\n    nlinarith [sq_nonneg (z.re + 1), sq_nonneg (z.im), sq_nonneg (z.re - 1),\n      sq_nonneg (z.re * z.re + z.im * z.im - 6),\n      sq_nonneg (z.re * z.re + z.im * z.im - 2 * z.re),\n      sq_nonneg (z.re * z.re + z.im * z.im + 2 * z.re)]\n  have h\u2083 : z \u2260 0 := by\n    by_contra h\n    rw [h] at h\u2080\n    norm_num [Complex.normSq, Complex.ext_iff, pow_two] at h\u2080\n    <;>\n    (try norm_num at h\u2080) <;>\n    (try nlinarith)\n    <;>\n    (try simp_all [Complex.ext_iff, pow_two])\n    <;>\n    (try nlinarith)\n  have h\u2084 : z + 6 / z = -2 := by\n    have h\u2084\u2081 : z.im = Real.sqrt 5 \u2228 z.im = -Real.sqrt 5 := by\n      have h\u2084\u2082 : z.im * z.im = 5 := by\n        nlinarith\n      have h\u2084\u2083 : z.im = Real.sqrt 5 \u2228 z.im = -Real.sqrt 5 := by\n        apply or_iff_not_imp_left.mpr\n        intro h\u2084\u2084\n        apply eq_of_sub_eq_zero\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2084\u2084)\n        nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n      exact h\u2084\u2083\n    have h\u2084\u2085 : z + 6 / z = -2 := by\n      cases h\u2084\u2081 with\n      | inl h\u2084\u2081 =>\n        have h\u2084\u2086 : z.im = Real.sqrt 5 := h\u2084\u2081\n        have h\u2084\u2087 : z.re = -1 := h\u2081\n        have h\u2084\u2088 : z.re * z.re + z.im * z.im = 6 := h\u2082\n        have h\u2084\u2089 : z \u2260 0 := h\u2083\n        have h\u2085\u2080 : z + 6 / z = -2 := by\n          field_simp [Complex.ext_iff, Complex.normSq, pow_two, h\u2084\u2089, Complex.ext_iff, Complex.normSq, pow_two]\n          <;> simp_all [Complex.ext_iff, Complex.normSq, pow_two]\n          <;> ring_nf at *\n          <;> norm_num at *\n          <;>\n          (try\n            {\n              nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n          <;>\n          (try\n            {\n              constructor <;> nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n          <;>\n          (try\n            {\n              nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n        exact h\u2085\u2080\n      | inr h\u2084\u2081 =>\n        have h\u2084\u2086 : z.im = -Real.sqrt 5 := h\u2084\u2081\n        have h\u2084\u2087 : z.re = -1 := h\u2081\n        have h\u2084\u2088 : z.re * z.re + z.im * z.im = 6 := h\u2082\n        have h\u2084\u2089 : z \u2260 0 := h\u2083\n        have h\u2085\u2080 : z + 6 / z = -2 := by\n          field_simp [Complex.ext_iff, Complex.normSq, pow_two, h\u2084\u2089, Complex.ext_iff, Complex.normSq, pow_two]\n          <;> simp_all [Complex.ext_iff, Complex.normSq, pow_two]\n          <;> ring_nf at *\n          <;> norm_num at *\n          <;>\n          (try\n            {\n              nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n          <;>\n          (try\n            {\n              constructor <;> nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n          <;>\n          (try\n            {\n              nlinarith [Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num)]\n            })\n        exact h\u2085\u2080\n    exact h\u2084\u2085\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 8, "alphabet": 15, "compiles": true, "steps_after": 8, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 1, "timed_out": false, "fixpoint": true, "chars_before": 133, "chars_after": 133, "seconds": 17.4, "id": "mathd_algebra_143", "proof_after": "theorem mathd_algebra_143 (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x + 1) (h\u2081 : \u2200 x, g x = x ^ 2 + 3) :\n    f (g 2) = 8  :=  by\n  have h\u2082 : g 2 = 7 := by\n    rw [h\u2081]\n    norm_num\n  have h\u2083 : f (g 2) = 8 := by\n    rw [h\u2082]\n    rw [h\u2080]\n    <;> norm_num\n  exact h\u2083\n", "proof_before": "theorem mathd_algebra_143 (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x + 1) (h\u2081 : \u2200 x, g x = x ^ 2 + 3) :\n    f (g 2) = 8  :=  by\n  have h\u2082 : g 2 = 7 := by\n    rw [h\u2081]\n    norm_num\n  have h\u2083 : f (g 2) = 8 := by\n    rw [h\u2082]\n    rw [h\u2080]\n    <;> norm_num\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 21, "alphabet": 15, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 428, "chars_after": 11, "seconds": 1.7, "id": "mathd_algebra_329", "proof_after": "theorem mathd_algebra_329 (x y : \u211d) (h\u2080 : 3 * y = x) (h\u2081 : 2 * x + 5 * y = 11) : x + y = 4  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_329 (x y : \u211d) (h\u2080 : 3 * y = x) (h\u2081 : 2 * x + 5 * y = 11) : x + y = 4  :=  by\n  have h\u2082 : y = 1 := by\n    have h\u2082\u2081 : 11 * y = 11 := by\n      calc\n        11 * y = 2 * x + 5 * y := by\n          have h\u2082\u2082 : x = 3 * y := by linarith\n          rw [h\u2082\u2082]\n          <;> ring_nf\n          <;> linarith\n        _ = 11 := by linarith\n    linarith\n  have h\u2083 : x = 3 := by\n    have h\u2083\u2081 : x = 3 * y := by linarith\n    rw [h\u2083\u2081, h\u2082]\n    <;> norm_num\n  have h\u2084 : x + y = 4 := by\n    rw [h\u2083, h\u2082]\n    <;> norm_num\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 151, "alphabet": 64, "compiles": false, "id": "mathd_algebra_362", "proof_after": "theorem mathd_algebra_362 (a b : \u211d) (h\u2080 : a ^ 2 * b ^ 3 = 32 / 27) (h\u2081 : a / b ^ 3 = 27 / 4) :\n    a + b = 8 / 3  :=  by\n  have hb3 : b \u2260 0 := by\n    by_contra h\n    have h\u2082 : b = 0 := by simpa using h\n    rw [h\u2082] at h\u2081\n    norm_num at h\u2081 \u22a2\n    <;> simp_all [pow_three]\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> linarith\n  have a_eq : a = (27 / 4 : \u211d) * b ^ 3 := by\n    have h\u2082 : b ^ 3 \u2260 0 := by\n      positivity\n    have h\u2083 : a = (27 / 4 : \u211d) * b ^ 3 := by\n      have h\u2084 : a / b ^ 3 = 27 / 4 := h\u2081\n      have h\u2085 : a = (27 / 4 : \u211d) * b ^ 3 := by\n        field_simp at h\u2084 \u22a2\n        <;> nlinarith [sq_pos_of_ne_zero hb3, sq_pos_of_ne_zero (pow_ne_zero 3 hb3)]\n      exact h\u2085\n    exact h\u2083\n  have b9_eq : b ^ 9 = (2 / 3 : \u211d) ^ 9 := by\n    have h\u2082 : a = (27 / 4 : \u211d) * b ^ 3 := a_eq\n    rw [h\u2082] at h\u2080\n    have h\u2083 : ((27 / 4 : \u211d) * b ^ 3) ^ 2 * b ^ 3 = 32 / 27 := by\n      linarith\n    have h\u2084 : b ^ 9 = (2 / 3 : \u211d) ^ 9 := by\n      ring_nf at h\u2083 \u22a2\n      <;> nlinarith [sq_pos_of_ne_zero hb3, sq_pos_of_ne_zero (pow_ne_zero 3 hb3),\n        sq_pos_of_ne_zero (pow_ne_zero 2 hb3), sq_pos_of_ne_zero (pow_ne_zero 5 hb3),\n        sq_pos_of_ne_zero (pow_ne_zero 6 hb3), sq_pos_of_ne_zero (pow_ne_zero 7 hb3),\n        sq_pos_of_ne_zero (pow_ne_zero 8 hb3), sq_pos_of_ne_zero (pow_ne_zero 9 hb3)]\n    exact h\u2084\n  have b_eq : b = 2 / 3 := by\n    have h\u2082 : b ^ 9 = (2 / 3 : \u211d) ^ 9 := b9_eq\n    have h\u2083 : b = 2 / 3 := by\n      have h\u2084 : b > 0 \u2228 b < 0 := by\n        by_cases h\u2085 : b > 0\n        \u00b7 exact Or.inl h\u2085\n        \u00b7 have h\u2086 : b < 0 := by\n            by_contra h\u2087\n            have h\u2088 : b \u2265 0 := by linarith\n            have h\u2089 : b = 0 := by\n              by_contra h\u2081\u2080\n              have h\u2081\u2081 : b > 0 := by\n                by_cases h\u2081\u2082 : b > 0\n                \u00b7 exact h\u2081\u2082\n                \u00b7 exfalso\n                  have h\u2081\u2083 : b < 0 := by\n                    by_contra h\u2081\u2084\n                    have h\u2081\u2085 : b \u2265 0 := by linarith\n                    have h\u2081\u2086 : b \u2260 0 := by tauto\n                    have h\u2081\u2087 : b > 0 := by\n                      apply lt_of_le_of_ne\n                      \u00b7 linarith\n                      \u00b7 intro h\u2081\u2088\n                        apply h\u2081\u2086\n                        linarith\n                    linarith\n                  linarith\n              have h\u2081\u2082 : b > 0 := by tauto\n              have h\u2081\u2083 : a = (27 / 4 : \u211d) * b ^ 3 := a_eq\n              have h\u2081\u2084 : a ^ 2 * b ^ 3 = 32 / 27 := h\u2080\n              have h\u2081\u2085 : a / b ^ 3 = 27 / 4 := h\u2081\n              nlinarith [sq_pos_of_pos h\u2081\u2081, pow_pos h\u2081\u2081 2, pow_pos h\u2081\u2081 3, pow_pos h\u2081\u2081 4, pow_pos h\u2081\u2081 5, pow_pos h\u2081\u2081 6, pow_pos h\u2081\u2081 7, pow_pos h\u2081\u2081 8, pow_pos h\u2081\u2081 9]\n            simp_all\n          exact Or.inr h\u2086\n      cases h\u2084 with\n      | inl h\u2084 =>\n        have h\u2085 : b > 0 := h\u2084\n        have h\u2086 : b = 2 / 3 := by\n          have h\u2087 : b ^ 9 = (2 / 3 : \u211d) ^ 9 := h\u2082\n          have h\u2088 : b = 2 / 3 := by\n            apply le_antisymm\n            \u00b7 \n              by_contra h\u2089\n              have h\u2081\u2080 : b > 2 / 3 := by\n                by_cases h\u2081\u2081 : b > 2 / 3\n                \u00b7 exact h\u2081\u2081\n                \u00b7 exfalso\n                  have h\u2081\u2082 : b \u2264 2 / 3 := by linarith\n                  have h\u2081\u2083 : b ^ 9 \u2264 (2 / 3 : \u211d) ^ 9 := by\n                    exact pow_le_pow_of_le_left (by positivity) h\u2081\u2082 9\n                  linarith\n              have h\u2081\u2081 : b ^ 9 > (2 / 3 : \u211d) ^ 9 := by\n                exact pow_lt_pow_of_lt_left h\u2081\u2080 (by positivity) (by norm_num)\n              linarith\n            \u00b7 \n              by_contra h\u2089\n              have h\u2081\u2080 : b < 2 / 3 := by\n                by_cases h\u2081\u2081 : b < 2 / 3\n                \u00b7 exact h\u2081\u2081\n                \u00b7 exfalso\n                  have h\u2081\u2082 : b \u2265 2 / 3 := by linarith\n                  have h\u2081\u2083 : b ^ 9 \u2265 (2 / 3 : \u211d) ^ 9 := by\n                    exact pow_le_pow_of_le_left (by positivity) h\u2081\u2082 9\n                  linarith\n              have h\u2081\u2081 : b ^ 9 < (2 / 3 : \u211d) ^ 9 := by\n                exact pow_lt_pow_of_lt_left h\u2081\u2080 (by positivity) (by norm_num)\n              linarith\n          exact h\u2088\n        exact h\u2086\n      | inr h\u2084 =>\n        have h\u2085 : b < 0 := h\u2084\n        have h\u2086 : b ^ 9 < 0 := by\n          have h\u2087 : b ^ 9 < 0 := by\n            exact by\n              have h\u2088 : b < 0 := h\u2085\n              have h\u2089 : b ^ 9 < 0 := by\n                have h\u2081\u2080 : b ^ 9 < 0 := by\n                  have h\u2081\u2081 : b < 0 := h\u2088\n                  have h\u2081\u2082 : b ^ 9 < 0 := by\n                    have h\u2081\u2083 : b ^ 2 > 0 := by nlinarith\n                    have h\u2081\u2084 : b ^ 3 < 0 := by nlinarith\n                    have h\u2081\u2085 : b ^ 4 > 0 := by nlinarith\n                    have h\u2081\u2086 : b ^ 5 < 0 := by nlinarith\n                    have h\u2081\u2087 : b ^ 6 > 0 := by nlinarith\n                    have h\u2081\u2088 : b ^ 7 < 0 := by nlinarith\n                    have h\u2081\u2089 : b ^ 8 > 0 := by nlinarith\n                    have h\u2082\u2080 : b ^ 9 < 0 := by nlinarith\n                    exact h\u2082\u2080\n                  exact h\u2081\u2082\n                exact h\u2081\u2080\n              exact h\u2089\n            <;> simp_all\n          exact h\u2087\n        have h\u2087 : (2 / 3 : \u211d) ^ 9 > 0 := by positivity\n        linarith\n    exact h\u2083\n  have a_eq\u2082 : a = 2 := by\n    rw [a_eq]\n    rw [b_eq]\n    <;> norm_num\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> linarith\n  have h\u2082 : a + b = 8 / 3 := by\n    rw [a_eq\u2082, b_eq]\n    <;> norm_num\n  exact h\u2082\n", "proof_before": "theorem mathd_algebra_362 (a b : \u211d) (h\u2080 : a ^ 2 * b ^ 3 = 32 / 27) (h\u2081 : a / b ^ 3 = 27 / 4) :\n    a + b = 8 / 3  :=  by\n  have hb3 : b \u2260 0 := by\n    by_contra h\n    have h\u2082 : b = 0 := by simpa using h\n    rw [h\u2082] at h\u2081\n    norm_num at h\u2081 \u22a2\n    <;> simp_all [pow_three]\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> linarith\n  have a_eq : a = (27 / 4 : \u211d) * b ^ 3 := by\n    have h\u2082 : b ^ 3 \u2260 0 := by\n      positivity\n    have h\u2083 : a = (27 / 4 : \u211d) * b ^ 3 := by\n      have h\u2084 : a / b ^ 3 = 27 / 4 := h\u2081\n      have h\u2085 : a = (27 / 4 : \u211d) * b ^ 3 := by\n        field_simp at h\u2084 \u22a2\n        <;> nlinarith [sq_pos_of_ne_zero hb3, sq_pos_of_ne_zero (pow_ne_zero 3 hb3)]\n      exact h\u2085\n    exact h\u2083\n  have b9_eq : b ^ 9 = (2 / 3 : \u211d) ^ 9 := by\n    have h\u2082 : a = (27 / 4 : \u211d) * b ^ 3 := a_eq\n    rw [h\u2082] at h\u2080\n    have h\u2083 : ((27 / 4 : \u211d) * b ^ 3) ^ 2 * b ^ 3 = 32 / 27 := by\n      linarith\n    have h\u2084 : b ^ 9 = (2 / 3 : \u211d) ^ 9 := by\n      ring_nf at h\u2083 \u22a2\n      <;> nlinarith [sq_pos_of_ne_zero hb3, sq_pos_of_ne_zero (pow_ne_zero 3 hb3),\n        sq_pos_of_ne_zero (pow_ne_zero 2 hb3), sq_pos_of_ne_zero (pow_ne_zero 5 hb3),\n        sq_pos_of_ne_zero (pow_ne_zero 6 hb3), sq_pos_of_ne_zero (pow_ne_zero 7 hb3),\n        sq_pos_of_ne_zero (pow_ne_zero 8 hb3), sq_pos_of_ne_zero (pow_ne_zero 9 hb3)]\n    exact h\u2084\n  have b_eq : b = 2 / 3 := by\n    have h\u2082 : b ^ 9 = (2 / 3 : \u211d) ^ 9 := b9_eq\n    have h\u2083 : b = 2 / 3 := by\n      have h\u2084 : b > 0 \u2228 b < 0 := by\n        by_cases h\u2085 : b > 0\n        \u00b7 exact Or.inl h\u2085\n        \u00b7 have h\u2086 : b < 0 := by\n            by_contra h\u2087\n            have h\u2088 : b \u2265 0 := by linarith\n            have h\u2089 : b = 0 := by\n              by_contra h\u2081\u2080\n              have h\u2081\u2081 : b > 0 := by\n                by_cases h\u2081\u2082 : b > 0\n                \u00b7 exact h\u2081\u2082\n                \u00b7 exfalso\n                  have h\u2081\u2083 : b < 0 := by\n                    by_contra h\u2081\u2084\n                    have h\u2081\u2085 : b \u2265 0 := by linarith\n                    have h\u2081\u2086 : b \u2260 0 := by tauto\n                    have h\u2081\u2087 : b > 0 := by\n                      apply lt_of_le_of_ne\n                      \u00b7 linarith\n                      \u00b7 intro h\u2081\u2088\n                        apply h\u2081\u2086\n                        linarith\n                    linarith\n                  linarith\n              have h\u2081\u2082 : b > 0 := by tauto\n              have h\u2081\u2083 : a = (27 / 4 : \u211d) * b ^ 3 := a_eq\n              have h\u2081\u2084 : a ^ 2 * b ^ 3 = 32 / 27 := h\u2080\n              have h\u2081\u2085 : a / b ^ 3 = 27 / 4 := h\u2081\n              nlinarith [sq_pos_of_pos h\u2081\u2081, pow_pos h\u2081\u2081 2, pow_pos h\u2081\u2081 3, pow_pos h\u2081\u2081 4, pow_pos h\u2081\u2081 5, pow_pos h\u2081\u2081 6, pow_pos h\u2081\u2081 7, pow_pos h\u2081\u2081 8, pow_pos h\u2081\u2081 9]\n            simp_all\n          exact Or.inr h\u2086\n      cases h\u2084 with\n      | inl h\u2084 =>\n        have h\u2085 : b > 0 := h\u2084\n        have h\u2086 : b = 2 / 3 := by\n          have h\u2087 : b ^ 9 = (2 / 3 : \u211d) ^ 9 := h\u2082\n          have h\u2088 : b = 2 / 3 := by\n            apply le_antisymm\n            \u00b7 \n              by_contra h\u2089\n              have h\u2081\u2080 : b > 2 / 3 := by\n                by_cases h\u2081\u2081 : b > 2 / 3\n                \u00b7 exact h\u2081\u2081\n                \u00b7 exfalso\n                  have h\u2081\u2082 : b \u2264 2 / 3 := by linarith\n                  have h\u2081\u2083 : b ^ 9 \u2264 (2 / 3 : \u211d) ^ 9 := by\n                    exact pow_le_pow_of_le_left (by positivity) h\u2081\u2082 9\n                  linarith\n              have h\u2081\u2081 : b ^ 9 > (2 / 3 : \u211d) ^ 9 := by\n                exact pow_lt_pow_of_lt_left h\u2081\u2080 (by positivity) (by norm_num)\n              linarith\n            \u00b7 \n              by_contra h\u2089\n              have h\u2081\u2080 : b < 2 / 3 := by\n                by_cases h\u2081\u2081 : b < 2 / 3\n                \u00b7 exact h\u2081\u2081\n                \u00b7 exfalso\n                  have h\u2081\u2082 : b \u2265 2 / 3 := by linarith\n                  have h\u2081\u2083 : b ^ 9 \u2265 (2 / 3 : \u211d) ^ 9 := by\n                    exact pow_le_pow_of_le_left (by positivity) h\u2081\u2082 9\n                  linarith\n              have h\u2081\u2081 : b ^ 9 < (2 / 3 : \u211d) ^ 9 := by\n                exact pow_lt_pow_of_lt_left h\u2081\u2080 (by positivity) (by norm_num)\n              linarith\n          exact h\u2088\n        exact h\u2086\n      | inr h\u2084 =>\n        have h\u2085 : b < 0 := h\u2084\n        have h\u2086 : b ^ 9 < 0 := by\n          have h\u2087 : b ^ 9 < 0 := by\n            exact by\n              have h\u2088 : b < 0 := h\u2085\n              have h\u2089 : b ^ 9 < 0 := by\n                have h\u2081\u2080 : b ^ 9 < 0 := by\n                  have h\u2081\u2081 : b < 0 := h\u2088\n                  have h\u2081\u2082 : b ^ 9 < 0 := by\n                    have h\u2081\u2083 : b ^ 2 > 0 := by nlinarith\n                    have h\u2081\u2084 : b ^ 3 < 0 := by nlinarith\n                    have h\u2081\u2085 : b ^ 4 > 0 := by nlinarith\n                    have h\u2081\u2086 : b ^ 5 < 0 := by nlinarith\n                    have h\u2081\u2087 : b ^ 6 > 0 := by nlinarith\n                    have h\u2081\u2088 : b ^ 7 < 0 := by nlinarith\n                    have h\u2081\u2089 : b ^ 8 > 0 := by nlinarith\n                    have h\u2082\u2080 : b ^ 9 < 0 := by nlinarith\n                    exact h\u2082\u2080\n                  exact h\u2081\u2082\n                exact h\u2081\u2080\n              exact h\u2089\n            <;> simp_all\n          exact h\u2087\n        have h\u2087 : (2 / 3 : \u211d) ^ 9 > 0 := by positivity\n        linarith\n    exact h\u2083\n  have a_eq\u2082 : a = 2 := by\n    rw [a_eq]\n    rw [b_eq]\n    <;> norm_num\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> linarith\n  have h\u2082 : a + b = 8 / 3 := by\n    rw [a_eq\u2082, b_eq]\n    <;> norm_num\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 5, "alphabet": 12, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 222, "chars_after": 6, "seconds": 0.5, "id": "mathd_numbertheory_345", "proof_after": "theorem mathd_numbertheory_345 : (2000 + 2001 + 2002 + 2003 + 2004 + 2005 + 2006) % 7 = 0  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_345 : (2000 + 2001 + 2002 + 2003 + 2004 + 2005 + 2006) % 7 = 0  :=  by\n  have h\u2081 : (2000 + 2001 + 2002 + 2003 + 2004 + 2005 + 2006) % 7 = 0 := by\n    norm_num [Nat.add_mod, Nat.mul_mod, Nat.mod_mod, Nat.mod_eq_of_lt, Nat.mod_self,\n      Nat.zero_add, Nat.add_assoc]\n    <;>\n    rfl\n  exact h\u2081\n"}
{"mode": "clear", "steps_before": 17, "alphabet": 12, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 422, "chars_after": 6, "seconds": 1.1, "id": "mathd_numbertheory_85", "proof_after": "theorem mathd_numbertheory_85 : 1 * 3 ^ 3 + 2 * 3 ^ 2 + 2 * 3 + 2 = 53  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_85 : 1 * 3 ^ 3 + 2 * 3 ^ 2 + 2 * 3 + 2 = 53  :=  by\n  have h1 : 3 ^ 3 = 27 := by norm_num\n  have h2 : 3 ^ 2 = 9 := by norm_num\n  have h3 : 1 * 3 ^ 3 = 27 := by norm_num [h1]\n  have h4 : 2 * 3 ^ 2 = 18 := by norm_num [h2]\n  have h5 : 2 * 3 = 6 := by norm_num\n  have h6 : 1 * 3 ^ 3 + 2 * 3 ^ 2 = 45 := by norm_num [h3, h4]\n  have h7 : 1 * 3 ^ 3 + 2 * 3 ^ 2 + 2 * 3 = 51 := by norm_num [h6, h5]\n  have h8 : 1 * 3 ^ 3 + 2 * 3 ^ 2 + 2 * 3 + 2 = 53 := by norm_num [h7]\n  exact h8\n"}
{"mode": "clear", "steps_before": 16, "alphabet": 16, "compiles": true, "steps_after": 6, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 4, "timed_out": false, "fixpoint": true, "chars_before": 441, "chars_after": 220, "seconds": 59.9, "id": "mathd_algebra_139", "proof_after": "theorem mathd_algebra_139 (s : \u211d \u2192 \u211d \u2192 \u211d)\n    (h\u2080 : \u2200 (x) (_ : x \u2260 0), \u2200 (y) (_ : y \u2260 0), s x y = (1 / y - 1 / x) / (x - y)) :\n    s 3 11 = 1 / 33  :=  by\n  have h\u2081 : s 3 11 = (1 / (11 : \u211d) - 1 / (3 : \u211d)) / ((3 : \u211d) - 11) := by\n    have h\u2081\u2082 : s (3 : \u211d) (11 : \u211d) = (1 / (11 : \u211d) - 1 / (3 : \u211d)) / ((3 : \u211d) - 11) := by\n      apply h\u2080\n      <;> norm_num\n    exact h\u2081\u2082\n  linarith\n", "proof_before": "theorem mathd_algebra_139 (s : \u211d \u2192 \u211d \u2192 \u211d)\n    (h\u2080 : \u2200 (x) (_ : x \u2260 0), \u2200 (y) (_ : y \u2260 0), s x y = (1 / y - 1 / x) / (x - y)) :\n    s 3 11 = 1 / 33  :=  by\n  have h\u2081 : s 3 11 = (1 / (11 : \u211d) - 1 / (3 : \u211d)) / ((3 : \u211d) - 11) := by\n    have h\u2081\u2080 : (3 : \u211d) \u2260 0 := by norm_num\n    have h\u2081\u2081 : (11 : \u211d) \u2260 0 := by norm_num\n    have h\u2081\u2082 : s (3 : \u211d) (11 : \u211d) = (1 / (11 : \u211d) - 1 / (3 : \u211d)) / ((3 : \u211d) - 11) := by\n      apply h\u2080\n      <;> norm_num\n    exact h\u2081\u2082\n  have h\u2082 : s 3 11 = 1 / 33 := by\n    rw [h\u2081]\n    norm_num [div_eq_mul_inv, mul_assoc]\n    <;> field_simp\n    <;> ring\n    <;> norm_num\n  apply h\u2082\n"}
{"mode": "clear", "steps_before": 31, "alphabet": 23, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 855, "chars_after": 8, "seconds": 0.6, "id": "mathd_numbertheory_765", "proof_after": "theorem mathd_numbertheory_765 (x : \u2124) (h\u2080 : x < 0) (h\u2081 : 24 * x % 1199 = 15) : x \u2264 -449  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_765 (x : \u2124) (h\u2080 : x < 0) (h\u2081 : 24 * x % 1199 = 15) : x \u2264 -449  :=  by\n  have h_main : x \u2264 -449 := by\n    by_contra h\n    have h\u2082 : x \u2265 -448 := by linarith\n    have h\u2083 : x % 1199 = 750 := by\n      have h\u2084 : (24 * x) % 1199 = 15 := h\u2081\n      have h\u2085 : (24 * x) % 1199 = (24 * x % 1199) := by simp [Int.mul_emod]\n      rw [h\u2085] at h\u2084\n      have h\u2086 : (24 : \u2124) * x % 1199 = 15 := h\u2084\n      have h\u2087 : x % 1199 = 750 := by\n        have h\u2088 : (x : \u2124) % 1199 = 750 := by\n          have h\u2089 : (24 : \u2124) * x % 1199 = 15 := h\u2086\n          have h\u2081\u2080 : (x : \u2124) % 1199 = 750 := by\n            omega\n          exact h\u2081\u2080\n        exact h\u2088\n      exact h\u2087\n    have h\u2084 : x \u2265 750 := by\n      have h\u2085 : x % 1199 = 750 := h\u2083\n      have h\u2086 : x \u2265 -448 := h\u2082\n      have h\u2087 : x \u2265 750 := by\n        by_contra h\u2088\n        have h\u2089 : x < 750 := by linarith\n        have h\u2081\u2080 : x \u2264 -449 := by\n          omega\n        linarith\n      exact h\u2087\n    linarith\n  exact h_main\n"}
{"mode": "clear", "steps_before": 8, "alphabet": 14, "compiles": false, "id": "mathd_numbertheory_239", "proof_after": "theorem mathd_numbertheory_239 : (\u2211 k in Finset.Icc 1 12, k) % 4 = 2  :=  by\n  have h_sum : (\u2211 k in Finset.Icc 1 12, k) = 78 := by\n    norm_num [Finset.sum_Icc_succ_top, Nat.div_eq_of_lt]\n    <;> rfl\n  have h_final : (\u2211 k in Finset.Icc 1 12, k) % 4 = 2 := by\n    rw [h_sum]\n    <;> norm_num\n    <;> rfl\n  apply h_final\n", "proof_before": "theorem mathd_numbertheory_239 : (\u2211 k in Finset.Icc 1 12, k) % 4 = 2  :=  by\n  have h_sum : (\u2211 k in Finset.Icc 1 12, k) = 78 := by\n    norm_num [Finset.sum_Icc_succ_top, Nat.div_eq_of_lt]\n    <;> rfl\n  have h_final : (\u2211 k in Finset.Icc 1 12, k) % 4 = 2 := by\n    rw [h_sum]\n    <;> norm_num\n    <;> rfl\n  apply h_final\n"}
{"mode": "clear", "steps_before": 664, "alphabet": 106, "compiles": false, "id": "amc12a_2021_p22", "proof_after": "theorem amc12a_2021_p22 (a b c : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x ^ 3 + a * x ^ 2 + b * x + c)\n    (h\u2081 :\n      f \u207b\u00b9' {0} =\n        {Real.cos (2 * Real.pi / 7), Real.cos (4 * Real.pi / 7), Real.cos (6 * Real.pi / 7)}) :\n    a * b * c = 1 / 32  :=  by\n  have h\u2082 : Real.cos (2 * Real.pi / 7) \u2260 Real.cos (4 * Real.pi / 7) := by\n    have h\u2082\u2081 : Real.cos (2 * Real.pi / 7) > Real.cos (4 * Real.pi / 7) := by\n      have h\u2082\u2082 : Real.cos (2 * Real.pi / 7) > Real.cos (4 * Real.pi / 7) := by\n        have h\u2082\u2083 : 0 < 2 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2082\u2084 : 2 * Real.pi / 7 < 4 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2082\u2085 : 4 * Real.pi / 7 < Real.pi := by linarith [Real.pi_pos]\n        have h\u2082\u2086 : Real.cos (4 * Real.pi / 7) < Real.cos (2 * Real.pi / 7) := by\n          apply Real.cos_lt_cos_of_nonneg_of_le_pi (by linarith [Real.pi_pos]) (by linarith [Real.pi_pos])\n          <;> linarith [Real.pi_pos]\n        linarith\n      exact h\u2082\u2082\n    linarith\n  have h\u2083 : Real.cos (2 * Real.pi / 7) \u2260 Real.cos (6 * Real.pi / 7) := by\n    have h\u2083\u2081 : Real.cos (2 * Real.pi / 7) > Real.cos (6 * Real.pi / 7) := by\n      have h\u2083\u2082 : Real.cos (2 * Real.pi / 7) > Real.cos (6 * Real.pi / 7) := by\n        have h\u2083\u2083 : 0 < 2 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2083\u2084 : 2 * Real.pi / 7 < 6 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2083\u2085 : 6 * Real.pi / 7 < Real.pi := by linarith [Real.pi_pos]\n        have h\u2083\u2086 : Real.cos (6 * Real.pi / 7) < Real.cos (2 * Real.pi / 7) := by\n          apply Real.cos_lt_cos_of_nonneg_of_le_pi (by linarith [Real.pi_pos]) (by linarith [Real.pi_pos])\n          <;> linarith [Real.pi_pos]\n        linarith\n      exact h\u2083\u2082\n    linarith\n  have h\u2084 : Real.cos (4 * Real.pi / 7) \u2260 Real.cos (6 * Real.pi / 7) := by\n    have h\u2084\u2081 : Real.cos (4 * Real.pi / 7) > Real.cos (6 * Real.pi / 7) := by\n      have h\u2084\u2082 : Real.cos (4 * Real.pi / 7) > Real.cos (6 * Real.pi / 7) := by\n        have h\u2084\u2083 : 0 < 4 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2084\u2084 : 4 * Real.pi / 7 < 6 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2084\u2085 : 6 * Real.pi / 7 < Real.pi := by linarith [Real.pi_pos]\n        have h\u2084\u2086 : Real.cos (6 * Real.pi / 7) < Real.cos (4 * Real.pi / 7) := by\n          apply Real.cos_lt_cos_of_nonneg_of_le_pi (by linarith [Real.pi_pos]) (by linarith [Real.pi_pos])\n          <;> linarith [Real.pi_pos]\n        linarith\n      exact h\u2084\u2082\n    linarith\n  have h\u2085 : a = -(Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7)) := by\n    have h\u2085\u2081 : a = -(Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7)) := by\n      have h\u2085\u2082 : f (Real.cos (2 * Real.pi / 7)) = 0 := by\n        have h\u2085\u2083 : Real.cos (2 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2085\u2083\n      have h\u2085\u2084 : f (Real.cos (4 * Real.pi / 7)) = 0 := by\n        have h\u2085\u2085 : Real.cos (4 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2085\u2085\n      have h\u2085\u2086 : f (Real.cos (6 * Real.pi / 7)) = 0 := by\n        have h\u2085\u2087 : Real.cos (6 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2085\u2087\n      have h\u2085\u2088 : f (Real.cos (2 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2085\u2082\n      have h\u2085\u2089 : f (Real.cos (4 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2085\u2084\n      have h\u2085\u2081\u2080 : f (Real.cos (6 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2085\u2086\n      have h\u2085\u2081\u2081 : (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c = 0 := by\n        have h\u2085\u2081\u2082 : f (Real.cos (2 * Real.pi / 7)) = (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2085\u2081\u2082 : (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c = 0 := by\n        have h\u2085\u2081\u2083 : f (Real.cos (4 * Real.pi / 7)) = (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2085\u2081\u2083 : (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c = 0 := by\n        have h\u2085\u2081\u2084 : f (Real.cos (6 * Real.pi / 7)) = (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2085\u2081\u2084 : a = -(Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7)) := by\n        apply eq_of_sub_eq_zero\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2082)\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2083)\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2084)\n        nlinarith [Real.cos_le_one (2 * Real.pi / 7), Real.cos_le_one (4 * Real.pi / 7), Real.cos_le_one (6 * Real.pi / 7),\n          Real.neg_one_le_cos (2 * Real.pi / 7), Real.neg_one_le_cos (4 * Real.pi / 7), Real.neg_one_le_cos (6 * Real.pi / 7),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (4 * Real.pi / 7)),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (6 * Real.pi / 7)),\n          sq_nonneg (Real.cos (4 * Real.pi / 7) - Real.cos (6 * Real.pi / 7))]\n      linarith\n    exact h\u2085\u2081\n    <;>\n    (try norm_num) <;>\n    (try ring_nf) <;>\n    (try field_simp) <;>\n    (try linarith [Real.pi_pos])\n  have h\u2086 : b = Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) := by\n    have h\u2086\u2081 : b = Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) := by\n      have h\u2086\u2082 : f (Real.cos (2 * Real.pi / 7)) = 0 := by\n        have h\u2086\u2083 : Real.cos (2 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2086\u2083\n      have h\u2086\u2084 : f (Real.cos (4 * Real.pi / 7)) = 0 := by\n        have h\u2086\u2085 : Real.cos (4 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2086\u2085\n      have h\u2086\u2086 : f (Real.cos (6 * Real.pi / 7)) = 0 := by\n        have h\u2086\u2087 : Real.cos (6 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2086\u2087\n      have h\u2086\u2088 : f (Real.cos (2 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2086\u2082\n      have h\u2086\u2089 : f (Real.cos (4 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2086\u2084\n      have h\u2086\u2081\u2080 : f (Real.cos (6 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2086\u2086\n      have h\u2086\u2081\u2081 : (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c = 0 := by\n        have h\u2086\u2081\u2082 : f (Real.cos (2 * Real.pi / 7)) = (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2086\u2081\u2082 : (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c = 0 := by\n        have h\u2086\u2081\u2083 : f (Real.cos (4 * Real.pi / 7)) = (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2086\u2081\u2083 : (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c = 0 := by\n        have h\u2086\u2081\u2084 : f (Real.cos (6 * Real.pi / 7)) = (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2086\u2081\u2084 : b = Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) := by\n        rw [h\u2085] at *\n        apply eq_of_sub_eq_zero\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2082)\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2083)\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2084)\n        nlinarith [Real.cos_le_one (2 * Real.pi / 7), Real.cos_le_one (4 * Real.pi / 7), Real.cos_le_one (6 * Real.pi / 7),\n          Real.neg_one_le_cos (2 * Real.pi / 7), Real.neg_one_le_cos (4 * Real.pi / 7), Real.neg_one_le_cos (6 * Real.pi / 7),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (4 * Real.pi / 7)),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (6 * Real.pi / 7)),\n          sq_nonneg (Real.cos (4 * Real.pi / 7) - Real.cos (6 * Real.pi / 7))]\n      exact h\u2086\u2081\u2084\n    exact h\u2086\u2081\n    <;>\n    (try norm_num) <;>\n    (try ring_nf) <;>\n    (try field_simp) <;>\n    (try linarith [Real.pi_pos])\n  have h\u2087 : c = - (Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7)) := by\n    have h\u2087\u2081 : c = - (Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7)) := by\n      have h\u2087\u2082 : f (Real.cos (2 * Real.pi / 7)) = 0 := by\n        have h\u2087\u2083 : Real.cos (2 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2087\u2083\n      have h\u2087\u2084 : f (Real.cos (4 * Real.pi / 7)) = 0 := by\n        have h\u2087\u2085 : Real.cos (4 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2087\u2085\n      have h\u2087\u2086 : f (Real.cos (6 * Real.pi / 7)) = 0 := by\n        have h\u2087\u2087 : Real.cos (6 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2087\u2087\n      have h\u2087\u2088 : f (Real.cos (2 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2087\u2082\n      have h\u2087\u2089 : f (Real.cos (4 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2087\u2084\n      have h\u2087\u2081\u2080 : f (Real.cos (6 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2087\u2086\n      have h\u2087\u2081\u2081 : (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c = 0 := by\n        have h\u2087\u2081\u2082 : f (Real.cos (2 * Real.pi / 7)) = (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2087\u2081\u2082 : (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c = 0 := by\n        have h\u2087\u2081\u2083 : f (Real.cos (4 * Real.pi / 7)) = (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2087\u2081\u2083 : (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c = 0 := by\n        have h\u2087\u2081\u2084 : f (Real.cos (6 * Real.pi / 7)) = (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2087\u2081\u2084 : c = - (Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7)) := by\n        rw [h\u2085, h\u2086] at *\n        ring_nf at *\n        nlinarith [Real.cos_le_one (2 * Real.pi / 7), Real.cos_le_one (4 * Real.pi / 7), Real.cos_le_one (6 * Real.pi / 7),\n          Real.neg_one_le_cos (2 * Real.pi / 7), Real.neg_one_le_cos (4 * Real.pi / 7), Real.neg_one_le_cos (6 * Real.pi / 7),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (4 * Real.pi / 7)),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (6 * Real.pi / 7)),\n          sq_nonneg (Real.cos (4 * Real.pi / 7) - Real.cos (6 * Real.pi / 7))]\n      exact h\u2087\u2081\u2084\n    exact h\u2087\u2081\n    <;>\n    (try norm_num) <;>\n    (try ring_nf) <;>\n    (try field_simp) <;>\n    (try linarith [Real.pi_pos])\n  have h\u2088 : Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n    have h\u2088\u2081 : Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n      have h\u2088\u2082 : Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n        have h\u2088\u2083 : Real.cos (6 * Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n          have h\u2088 : Real.cos (6 * Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n            have h\u2088\u2081 : Real.cos (6 * Real.pi / 7) = Real.cos (Real.pi - Real.pi / 7) := by ring\n            rw [h\u2088\u2081]\n            have h\u2088\u2082 : Real.cos (Real.pi - Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n              rw [Real.cos_pi_sub]\n              <;> ring\n            rw [h\u2088\u2082]\n          rw [h\u2088]\n        have h\u2088\u2084 : Real.cos (4 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n          have h\u2088\u2085 : Real.cos (4 * Real.pi / 7) = Real.cos (Real.pi - 3 * Real.pi / 7) := by ring\n          rw [h\u2088\u2085]\n          have h\u2088\u2086 : Real.cos (Real.pi - 3 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n            rw [Real.cos_pi_sub]\n            <;> ring\n          rw [h\u2088\u2086]\n        have h\u2088\u2087 : Real.cos (2 * Real.pi / 7) = Real.cos (2 * Real.pi / 7) := by rfl\n        rw [h\u2088\u2083, h\u2088\u2084]\n        have h\u2088\u2088 : Real.cos (3 * Real.pi / 7) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n          have h\u2088\u2089 : Real.cos (3 * Real.pi / 7) = Real.cos (3 * (Real.pi / 7)) := by ring\n          rw [h\u2088\u2089]\n          have h\u2088\u2081\u2080 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n            have h\u2088\u2081\u2081 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n              rw [show Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) by rfl]\n              have h\u2081\u2082 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n                rw [show Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) by rfl]\n                have h\u2081\u2083 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n                  rw [show Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) by rfl]\n                  have h\u2081\u2084 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n                    rw [Real.cos_three_mul]\n                    <;> ring_nf\n                    <;> simp [Real.cos_sq]\n                    <;> ring_nf\n                    <;> simp [Real.cos_sq]\n                    <;> ring_nf\n                  rw [h\u2081\u2084]\n                rw [h\u2081\u2083]\n              rw [h\u2081\u2082]\n            rw [h\u2088\u2081\u2081]\n          rw [h\u2088\u2081\u2080]\n        have h\u2088\u2081\u2081 : Real.cos (2 * Real.pi / 7) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n          have h\u2088\u2081\u2082 : Real.cos (2 * Real.pi / 7) = Real.cos (2 * (Real.pi / 7)) := by ring\n          rw [h\u2088\u2081\u2082]\n          have h\u2088\u2081\u2083 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n            have h\u2088\u2081\u2084 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n              rw [Real.cos_two_mul]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n            rw [h\u2088\u2081\u2084]\n          rw [h\u2088\u2081\u2083]\n        rw [h\u2088\u2081\u2081, h\u2088\u2088]\n        have h\u2088\u2081\u2082 : Real.cos (Real.pi / 7) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor <;> linarith [Real.pi_gt_three]\n        have h\u2088\u2081\u2083 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n          have h\u2088\u2081\u2084 : Real.cos (Real.pi / 7) > 0 := by\n            apply Real.cos_pos_of_mem_Ioo\n            constructor <;> linarith [Real.pi_gt_three]\n          have h\u2088\u2081\u2085 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n            rw [Real.cos_three_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          have h\u2088\u2081\u2086 : Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) := by rfl\n          have h\u2088\u2081\u2087 : Real.cos (3 * (Real.pi / 7)) > 0 := by\n            apply Real.cos_pos_of_mem_Ioo\n            constructor\n            \u00b7 linarith [Real.pi_gt_three]\n            \u00b7 linarith [Real.pi_gt_three]\n          have h\u2088\u2081\u2088 : Real.cos (Real.pi / 7) > 0 := by\n            apply Real.cos_pos_of_mem_Ioo\n            constructor <;> linarith [Real.pi_gt_three]\n          have h\u2088\u2081\u2089 : Real.cos (Real.pi - 3 * (Real.pi / 7)) = -Real.cos (3 * (Real.pi / 7)) := by\n            rw [Real.cos_pi_sub]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          have h\u2088\u2082\u2080 : Real.cos (4 * (Real.pi / 7)) = Real.cos (Real.pi - 3 * (Real.pi / 7)) := by\n            ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          have h\u2088\u2082\u2081 : Real.cos (4 * (Real.pi / 7)) = -Real.cos (3 * (Real.pi / 7)) := by\n            rw [h\u2088\u2082\u2080, h\u2088\u2081\u2089]\n          have h\u2088\u2082\u2082 : Real.cos (4 * (Real.pi / 7)) = 8 * Real.cos (Real.pi / 7) ^ 4 - 8 * Real.cos (Real.pi / 7) ^ 2 + 1 := by\n            have h\u2088\u2082\u2083 : Real.cos (4 * (Real.pi / 7)) = Real.cos (2 * (2 * (Real.pi / 7))) := by ring\n            rw [h\u2088\u2082\u2083]\n            have h\u2088\u2082\u2084 : Real.cos (2 * (2 * (Real.pi / 7))) = 2 * Real.cos (2 * (Real.pi / 7)) ^ 2 - 1 := by\n              rw [Real.cos_two_mul]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n            rw [h\u2088\u2082\u2084]\n            have h\u2088\u2082\u2085 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n              rw [Real.cos_two_mul]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n            rw [h\u2088\u2082\u2085]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          have h\u2088\u2082\u2086 : 8 * Real.cos (Real.pi / 7) ^ 4 - 8 * Real.cos (Real.pi / 7) ^ 2 + 1 = -Real.cos (3 * (Real.pi / 7)) := by\n            linarith\n          have h\u2088\u2082\u2087 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n            nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n              Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n              Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n          exact h\u2088\u2082\u2087\n        have h\u2088\u2081\u2084 : Real.cos (Real.pi / 7) \u2260 0 := by linarith\n        nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n          Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n          Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n      exact h\u2088\u2082\n    exact h\u2088\u2081\n  have h\u2089 : Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n    have h\u2089\u2081 : Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n      exact h\u2088\n    have h\u2089\u2082 : Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n      have h\u2089\u2083 : Real.cos (6 * Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n        have h\u2089\u2084 : Real.cos (6 * Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n          have h\u2089\u2085 : Real.cos (6 * Real.pi / 7) = Real.cos (Real.pi - Real.pi / 7) := by ring\n          rw [h\u2089\u2085]\n          have h\u2089\u2086 : Real.cos (Real.pi - Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n            rw [Real.cos_pi_sub]\n          rw [h\u2089\u2086]\n        rw [h\u2089\u2084]\n      have h\u2089\u2084 : Real.cos (4 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n        have h\u2089\u2085 : Real.cos (4 * Real.pi / 7) = Real.cos (Real.pi - 3 * Real.pi / 7) := by ring\n        rw [h\u2089\u2085]\n        have h\u2089\u2086 : Real.cos (Real.pi - 3 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n          rw [Real.cos_pi_sub]\n        rw [h\u2089\u2086]\n      have h\u2089\u2085 : Real.cos (2 * Real.pi / 7) = Real.cos (2 * Real.pi / 7) := by ring\n      rw [h\u2089\u2083, h\u2089\u2084]\n      have h\u2089\u2086 : Real.cos (3 * Real.pi / 7) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n        have h\u2089\u2087 : Real.cos (3 * Real.pi / 7) = Real.cos (3 * (Real.pi / 7)) := by ring\n        rw [h\u2089\u2087]\n        have h\u2089\u2088 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n          have h\u2089\u2089 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n            rw [show Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) by rfl]\n            have h\u2081\u2080\u2080 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n              rw [show Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) by rfl]\n              have h\u2081\u2080\u2081 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n                rw [Real.cos_three_mul]\n                <;> ring_nf\n                <;> simp [Real.cos_sq]\n                <;> ring_nf\n                <;> simp [Real.cos_sq]\n                <;> ring_nf\n              rw [h\u2081\u2080\u2081]\n            rw [h\u2081\u2080\u2080]\n          rw [h\u2089\u2089]\n        rw [h\u2089\u2088]\n      have h\u2089\u2087 : Real.cos (2 * Real.pi / 7) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n        have h\u2089\u2088 : Real.cos (2 * Real.pi / 7) = Real.cos (2 * (Real.pi / 7)) := by ring\n        rw [h\u2089\u2088]\n        have h\u2089\u2089 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n          have h\u2081\u2080\u2080 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n            rw [Real.cos_two_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          rw [h\u2081\u2080\u2080]\n        rw [h\u2089\u2089]\n      rw [h\u2089\u2086, h\u2089\u2087]\n      have h\u2089\u2088 : Real.cos (Real.pi / 7) > 0 := by\n        apply Real.cos_pos_of_mem_Ioo\n        constructor <;> linarith [Real.pi_gt_three]\n      have h\u2089\u2089 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n        have h\u2081\u2080\u2080 : Real.cos (Real.pi / 7) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor <;> linarith [Real.pi_gt_three]\n        have h\u2081\u2080\u2081 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n          rw [Real.cos_three_mul]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n        have h\u2081\u2080\u2082 : Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) := by rfl\n        have h\u2081\u2080\u2083 : Real.cos (3 * (Real.pi / 7)) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor\n          \u00b7 linarith [Real.pi_gt_three]\n          \u00b7 linarith [Real.pi_gt_three]\n        have h\u2081\u2080\u2084 : Real.cos (Real.pi / 7) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor <;> linarith [Real.pi_gt_three]\n        have h\u2081\u2080\u2085 : Real.cos (Real.pi - 3 * (Real.pi / 7)) = -Real.cos (3 * (Real.pi / 7)) := by\n          rw [Real.cos_pi_sub]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n        have h\u2081\u2080\u2086 : Real.cos (4 * (Real.pi / 7)) = Real.cos (Real.pi - 3 * (Real.pi / 7)) := by\n          ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n        have h\u2081\u2080\u2087 : Real.cos (4 * (Real.pi / 7)) = -Real.cos (3 * (Real.pi / 7)) := by\n          rw [h\u2081\u2080\u2086, h\u2081\u2080\u2085]\n        have h\u2081\u2080\u2088 : Real.cos (4 * (Real.pi / 7)) = 8 * Real.cos (Real.pi / 7) ^ 4 - 8 * Real.cos (Real.pi / 7) ^ 2 + 1 := by\n          have h\u2081\u2080\u2089 : Real.cos (4 * (Real.pi / 7)) = Real.cos (2 * (2 * (Real.pi / 7))) := by ring\n          rw [h\u2081\u2080\u2089]\n          have h\u2081\u2081\u2080 : Real.cos (2 * (2 * (Real.pi / 7))) = 2 * Real.cos (2 * (Real.pi / 7)) ^ 2 - 1 := by\n            rw [Real.cos_two_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          rw [h\u2081\u2081\u2080]\n          have h\u2081\u2081\u2081 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n            rw [Real.cos_two_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          rw [h\u2081\u2081\u2081]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n        have h\u2081\u2081\u2082 : 8 * Real.cos (Real.pi / 7) ^ 4 - 8 * Real.cos (Real.pi / 7) ^ 2 + 1 = -Real.cos (3 * (Real.pi / 7)) := by\n          linarith\n        have h\u2081\u2081\u2083 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n          nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n            Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n            Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n        exact h\u2081\u2081\u2083\n      have h\u2081\u2080\u2080 : Real.cos (Real.pi / 7) \u2260 0 := by linarith\n      nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n        Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n        Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n    exact h\u2089\u2082\n    <;>\n    (try ring_nf at * <;> linarith [Real.pi_pos])\n    <;>\n    (try norm_num at * <;> linarith [Real.pi_pos])\n    <;>\n    (try field_simp at * <;> linarith [Real.pi_pos])\n  have h\u2081\u2080 : Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) = 1 / 8 := by\n    have h\u2081\u2080\u2081 : Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n      exact h\u2088\n    have h\u2081\u2080\u2082 : Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n      exact h\u2089\n    have h\u2081\u2080\u2083 : Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) = 1 / 8 := by\n      have h\u2081\u2080\u2084 : Real.cos (6 * Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n        have h\u2081\u2080\u2085 : Real.cos (6 * Real.pi / 7) = Real.cos (Real.pi - Real.pi / 7) := by ring\n        rw [h\u2081\u2080\u2085]\n        have h\u2081\u2080\u2086 : Real.cos (Real.pi - Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n          rw [Real.cos_pi_sub]\n        rw [h\u2081\u2080\u2086]\n      have h\u2081\u2080\u2085 : Real.cos (4 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n        have h\u2081\u2080\u2086 : Real.cos (4 * Real.pi / 7) = Real.cos (Real.pi - 3 * Real.pi / 7) := by ring\n        rw [h\u2081\u2080\u2086]\n        have h\u2081\u2080\u2087 : Real.cos (Real.pi - 3 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n          rw [Real.cos_pi_sub]\n        rw [h\u2081\u2080\u2087]\n      have h\u2081\u2080\u2086 : Real.cos (2 * Real.pi / 7) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n        have h\u2081\u2080\u2087 : Real.cos (2 * Real.pi / 7) = Real.cos (2 * (Real.pi / 7)) := by ring\n        rw [h\u2081\u2080\u2087]\n        have h\u2081\u2080\u2088 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n          have h\u2081\u2080\u2089 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n            rw [Real.cos_two_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          rw [h\u2081\u2080\u2089]\n        rw [h\u2081\u2080\u2088]\n      have h\u2081\u2080\u2087 : Real.cos (3 * Real.pi / 7) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n        have h\u2081\u2080\u2088 : Real.cos (3 * Real.pi / 7) = Real.cos (3 * (Real.pi / 7)) := by ring\n        rw [h\u2081\u2080\u2088]\n        have h\u2081\u2080\u2089 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n          have h\u2081\u2081\u2080 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n            rw [Real.cos_three_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          rw [h\u2081\u2081\u2080]\n        rw [h\u2081\u2080\u2089]\n      have h\u2081\u2080\u2088 : Real.cos (Real.pi / 7) > 0 := by\n        apply Real.cos_pos_of_mem_Ioo\n        constructor <;> linarith [Real.pi_gt_three]\n      have h\u2081\u2080\u2089 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n        have h\u2081\u2081\u2080 : Real.cos (3 * (Real.pi / 7)) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor\n          \u00b7 linarith [Real.pi_gt_three]\n          \u00b7 linarith [Real.pi_gt_three]\n        have h\u2081\u2081\u2081 : Real.cos (Real.pi / 7) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor <;> linarith [Real.pi_gt_three]\n        have h\u2081\u2081\u2082 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n          nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n            Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n            Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n        exact h\u2081\u2081\u2082\n      have h\u2081\u2081\u2080 : Real.cos (Real.pi / 7) \u2260 0 := by linarith\n      rw [h\u2081\u2080\u2084, h\u2081\u2080\u2085, h\u2081\u2080\u2086] at *\n      have h\u2081\u2081\u2081 : Real.cos (3 * Real.pi / 7) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n        rw [h\u2081\u2080\u2087]\n      nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n        Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n        Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n    exact h\u2081\u2080\u2083\n    <;>\n    (try ring_nf at * <;> linarith [Real.pi_pos])\n    <;>\n    (try norm_num at * <;> linarith [Real.pi_pos])\n    <;>\n    (try field_simp at * <;> linarith [Real.pi_pos])\n  have h\u2081\u2081 : a = 1 / 2 := by\n    rw [h\u2085]\n    linarith\n  have h\u2081\u2082 : b = -1 / 2 := by\n    rw [h\u2086]\n    linarith\n  have h\u2081\u2083 : c = -1 / 8 := by\n    rw [h\u2087]\n    linarith\n  have h\u2081\u2084 : a * b * c = 1 / 32 := by\n    rw [h\u2081\u2081, h\u2081\u2082, h\u2081\u2083]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  rw [h\u2081\u2084]\n", "proof_before": "theorem amc12a_2021_p22 (a b c : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x ^ 3 + a * x ^ 2 + b * x + c)\n    (h\u2081 :\n      f \u207b\u00b9' {0} =\n        {Real.cos (2 * Real.pi / 7), Real.cos (4 * Real.pi / 7), Real.cos (6 * Real.pi / 7)}) :\n    a * b * c = 1 / 32  :=  by\n  have h\u2082 : Real.cos (2 * Real.pi / 7) \u2260 Real.cos (4 * Real.pi / 7) := by\n    have h\u2082\u2081 : Real.cos (2 * Real.pi / 7) > Real.cos (4 * Real.pi / 7) := by\n      have h\u2082\u2082 : Real.cos (2 * Real.pi / 7) > Real.cos (4 * Real.pi / 7) := by\n        have h\u2082\u2083 : 0 < 2 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2082\u2084 : 2 * Real.pi / 7 < 4 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2082\u2085 : 4 * Real.pi / 7 < Real.pi := by linarith [Real.pi_pos]\n        have h\u2082\u2086 : Real.cos (4 * Real.pi / 7) < Real.cos (2 * Real.pi / 7) := by\n          apply Real.cos_lt_cos_of_nonneg_of_le_pi (by linarith [Real.pi_pos]) (by linarith [Real.pi_pos])\n          <;> linarith [Real.pi_pos]\n        linarith\n      exact h\u2082\u2082\n    linarith\n  have h\u2083 : Real.cos (2 * Real.pi / 7) \u2260 Real.cos (6 * Real.pi / 7) := by\n    have h\u2083\u2081 : Real.cos (2 * Real.pi / 7) > Real.cos (6 * Real.pi / 7) := by\n      have h\u2083\u2082 : Real.cos (2 * Real.pi / 7) > Real.cos (6 * Real.pi / 7) := by\n        have h\u2083\u2083 : 0 < 2 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2083\u2084 : 2 * Real.pi / 7 < 6 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2083\u2085 : 6 * Real.pi / 7 < Real.pi := by linarith [Real.pi_pos]\n        have h\u2083\u2086 : Real.cos (6 * Real.pi / 7) < Real.cos (2 * Real.pi / 7) := by\n          apply Real.cos_lt_cos_of_nonneg_of_le_pi (by linarith [Real.pi_pos]) (by linarith [Real.pi_pos])\n          <;> linarith [Real.pi_pos]\n        linarith\n      exact h\u2083\u2082\n    linarith\n  have h\u2084 : Real.cos (4 * Real.pi / 7) \u2260 Real.cos (6 * Real.pi / 7) := by\n    have h\u2084\u2081 : Real.cos (4 * Real.pi / 7) > Real.cos (6 * Real.pi / 7) := by\n      have h\u2084\u2082 : Real.cos (4 * Real.pi / 7) > Real.cos (6 * Real.pi / 7) := by\n        have h\u2084\u2083 : 0 < 4 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2084\u2084 : 4 * Real.pi / 7 < 6 * Real.pi / 7 := by linarith [Real.pi_pos]\n        have h\u2084\u2085 : 6 * Real.pi / 7 < Real.pi := by linarith [Real.pi_pos]\n        have h\u2084\u2086 : Real.cos (6 * Real.pi / 7) < Real.cos (4 * Real.pi / 7) := by\n          apply Real.cos_lt_cos_of_nonneg_of_le_pi (by linarith [Real.pi_pos]) (by linarith [Real.pi_pos])\n          <;> linarith [Real.pi_pos]\n        linarith\n      exact h\u2084\u2082\n    linarith\n  have h\u2085 : a = -(Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7)) := by\n    have h\u2085\u2081 : a = -(Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7)) := by\n      have h\u2085\u2082 : f (Real.cos (2 * Real.pi / 7)) = 0 := by\n        have h\u2085\u2083 : Real.cos (2 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2085\u2083\n      have h\u2085\u2084 : f (Real.cos (4 * Real.pi / 7)) = 0 := by\n        have h\u2085\u2085 : Real.cos (4 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2085\u2085\n      have h\u2085\u2086 : f (Real.cos (6 * Real.pi / 7)) = 0 := by\n        have h\u2085\u2087 : Real.cos (6 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2085\u2087\n      have h\u2085\u2088 : f (Real.cos (2 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2085\u2082\n      have h\u2085\u2089 : f (Real.cos (4 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2085\u2084\n      have h\u2085\u2081\u2080 : f (Real.cos (6 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2085\u2086\n      have h\u2085\u2081\u2081 : (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c = 0 := by\n        have h\u2085\u2081\u2082 : f (Real.cos (2 * Real.pi / 7)) = (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2085\u2081\u2082 : (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c = 0 := by\n        have h\u2085\u2081\u2083 : f (Real.cos (4 * Real.pi / 7)) = (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2085\u2081\u2083 : (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c = 0 := by\n        have h\u2085\u2081\u2084 : f (Real.cos (6 * Real.pi / 7)) = (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2085\u2081\u2084 : a = -(Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7)) := by\n        apply eq_of_sub_eq_zero\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2082)\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2083)\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2084)\n        nlinarith [Real.cos_le_one (2 * Real.pi / 7), Real.cos_le_one (4 * Real.pi / 7), Real.cos_le_one (6 * Real.pi / 7),\n          Real.neg_one_le_cos (2 * Real.pi / 7), Real.neg_one_le_cos (4 * Real.pi / 7), Real.neg_one_le_cos (6 * Real.pi / 7),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (4 * Real.pi / 7)),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (6 * Real.pi / 7)),\n          sq_nonneg (Real.cos (4 * Real.pi / 7) - Real.cos (6 * Real.pi / 7))]\n      linarith\n    exact h\u2085\u2081\n    <;>\n    (try norm_num) <;>\n    (try ring_nf) <;>\n    (try field_simp) <;>\n    (try linarith [Real.pi_pos])\n  have h\u2086 : b = Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) := by\n    have h\u2086\u2081 : b = Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) := by\n      have h\u2086\u2082 : f (Real.cos (2 * Real.pi / 7)) = 0 := by\n        have h\u2086\u2083 : Real.cos (2 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2086\u2083\n      have h\u2086\u2084 : f (Real.cos (4 * Real.pi / 7)) = 0 := by\n        have h\u2086\u2085 : Real.cos (4 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2086\u2085\n      have h\u2086\u2086 : f (Real.cos (6 * Real.pi / 7)) = 0 := by\n        have h\u2086\u2087 : Real.cos (6 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2086\u2087\n      have h\u2086\u2088 : f (Real.cos (2 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2086\u2082\n      have h\u2086\u2089 : f (Real.cos (4 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2086\u2084\n      have h\u2086\u2081\u2080 : f (Real.cos (6 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2086\u2086\n      have h\u2086\u2081\u2081 : (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c = 0 := by\n        have h\u2086\u2081\u2082 : f (Real.cos (2 * Real.pi / 7)) = (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2086\u2081\u2082 : (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c = 0 := by\n        have h\u2086\u2081\u2083 : f (Real.cos (4 * Real.pi / 7)) = (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2086\u2081\u2083 : (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c = 0 := by\n        have h\u2086\u2081\u2084 : f (Real.cos (6 * Real.pi / 7)) = (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2086\u2081\u2084 : b = Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) := by\n        rw [h\u2085] at *\n        apply eq_of_sub_eq_zero\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2082)\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2083)\n        apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2084)\n        nlinarith [Real.cos_le_one (2 * Real.pi / 7), Real.cos_le_one (4 * Real.pi / 7), Real.cos_le_one (6 * Real.pi / 7),\n          Real.neg_one_le_cos (2 * Real.pi / 7), Real.neg_one_le_cos (4 * Real.pi / 7), Real.neg_one_le_cos (6 * Real.pi / 7),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (4 * Real.pi / 7)),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (6 * Real.pi / 7)),\n          sq_nonneg (Real.cos (4 * Real.pi / 7) - Real.cos (6 * Real.pi / 7))]\n      exact h\u2086\u2081\u2084\n    exact h\u2086\u2081\n    <;>\n    (try norm_num) <;>\n    (try ring_nf) <;>\n    (try field_simp) <;>\n    (try linarith [Real.pi_pos])\n  have h\u2087 : c = - (Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7)) := by\n    have h\u2087\u2081 : c = - (Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7)) := by\n      have h\u2087\u2082 : f (Real.cos (2 * Real.pi / 7)) = 0 := by\n        have h\u2087\u2083 : Real.cos (2 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2087\u2083\n      have h\u2087\u2084 : f (Real.cos (4 * Real.pi / 7)) = 0 := by\n        have h\u2087\u2085 : Real.cos (4 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2087\u2085\n      have h\u2087\u2086 : f (Real.cos (6 * Real.pi / 7)) = 0 := by\n        have h\u2087\u2087 : Real.cos (6 * Real.pi / 7) \u2208 f \u207b\u00b9' {0} := by\n          rw [h\u2081]\n          simp\n          <;>\n          (try norm_num) <;>\n          (try ring_nf) <;>\n          (try field_simp) <;>\n          (try linarith [Real.pi_pos])\n        simpa using h\u2087\u2087\n      have h\u2087\u2088 : f (Real.cos (2 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2087\u2082\n      have h\u2087\u2089 : f (Real.cos (4 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2087\u2084\n      have h\u2087\u2081\u2080 : f (Real.cos (6 * Real.pi / 7)) = 0 := by simpa [h\u2080] using h\u2087\u2086\n      have h\u2087\u2081\u2081 : (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c = 0 := by\n        have h\u2087\u2081\u2082 : f (Real.cos (2 * Real.pi / 7)) = (Real.cos (2 * Real.pi / 7)) ^ 3 + a * (Real.cos (2 * Real.pi / 7)) ^ 2 + b * (Real.cos (2 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2087\u2081\u2082 : (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c = 0 := by\n        have h\u2087\u2081\u2083 : f (Real.cos (4 * Real.pi / 7)) = (Real.cos (4 * Real.pi / 7)) ^ 3 + a * (Real.cos (4 * Real.pi / 7)) ^ 2 + b * (Real.cos (4 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2087\u2081\u2083 : (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c = 0 := by\n        have h\u2087\u2081\u2084 : f (Real.cos (6 * Real.pi / 7)) = (Real.cos (6 * Real.pi / 7)) ^ 3 + a * (Real.cos (6 * Real.pi / 7)) ^ 2 + b * (Real.cos (6 * Real.pi / 7)) + c := by\n          rw [h\u2080]\n          <;> ring\n        linarith\n      have h\u2087\u2081\u2084 : c = - (Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7)) := by\n        rw [h\u2085, h\u2086] at *\n        ring_nf at *\n        nlinarith [Real.cos_le_one (2 * Real.pi / 7), Real.cos_le_one (4 * Real.pi / 7), Real.cos_le_one (6 * Real.pi / 7),\n          Real.neg_one_le_cos (2 * Real.pi / 7), Real.neg_one_le_cos (4 * Real.pi / 7), Real.neg_one_le_cos (6 * Real.pi / 7),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (4 * Real.pi / 7)),\n          sq_nonneg (Real.cos (2 * Real.pi / 7) - Real.cos (6 * Real.pi / 7)),\n          sq_nonneg (Real.cos (4 * Real.pi / 7) - Real.cos (6 * Real.pi / 7))]\n      exact h\u2087\u2081\u2084\n    exact h\u2087\u2081\n    <;>\n    (try norm_num) <;>\n    (try ring_nf) <;>\n    (try field_simp) <;>\n    (try linarith [Real.pi_pos])\n  have h\u2088 : Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n    have h\u2088\u2081 : Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n      have h\u2088\u2082 : Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n        have h\u2088\u2083 : Real.cos (6 * Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n          have h\u2088 : Real.cos (6 * Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n            have h\u2088\u2081 : Real.cos (6 * Real.pi / 7) = Real.cos (Real.pi - Real.pi / 7) := by ring\n            rw [h\u2088\u2081]\n            have h\u2088\u2082 : Real.cos (Real.pi - Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n              rw [Real.cos_pi_sub]\n              <;> ring\n            rw [h\u2088\u2082]\n          rw [h\u2088]\n        have h\u2088\u2084 : Real.cos (4 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n          have h\u2088\u2085 : Real.cos (4 * Real.pi / 7) = Real.cos (Real.pi - 3 * Real.pi / 7) := by ring\n          rw [h\u2088\u2085]\n          have h\u2088\u2086 : Real.cos (Real.pi - 3 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n            rw [Real.cos_pi_sub]\n            <;> ring\n          rw [h\u2088\u2086]\n        have h\u2088\u2087 : Real.cos (2 * Real.pi / 7) = Real.cos (2 * Real.pi / 7) := by rfl\n        rw [h\u2088\u2083, h\u2088\u2084]\n        have h\u2088\u2088 : Real.cos (3 * Real.pi / 7) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n          have h\u2088\u2089 : Real.cos (3 * Real.pi / 7) = Real.cos (3 * (Real.pi / 7)) := by ring\n          rw [h\u2088\u2089]\n          have h\u2088\u2081\u2080 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n            have h\u2088\u2081\u2081 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n              rw [show Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) by rfl]\n              have h\u2081\u2082 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n                rw [show Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) by rfl]\n                have h\u2081\u2083 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n                  rw [show Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) by rfl]\n                  have h\u2081\u2084 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n                    rw [Real.cos_three_mul]\n                    <;> ring_nf\n                    <;> simp [Real.cos_sq]\n                    <;> ring_nf\n                    <;> simp [Real.cos_sq]\n                    <;> ring_nf\n                  rw [h\u2081\u2084]\n                rw [h\u2081\u2083]\n              rw [h\u2081\u2082]\n            rw [h\u2088\u2081\u2081]\n          rw [h\u2088\u2081\u2080]\n        have h\u2088\u2081\u2081 : Real.cos (2 * Real.pi / 7) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n          have h\u2088\u2081\u2082 : Real.cos (2 * Real.pi / 7) = Real.cos (2 * (Real.pi / 7)) := by ring\n          rw [h\u2088\u2081\u2082]\n          have h\u2088\u2081\u2083 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n            have h\u2088\u2081\u2084 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n              rw [Real.cos_two_mul]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n            rw [h\u2088\u2081\u2084]\n          rw [h\u2088\u2081\u2083]\n        rw [h\u2088\u2081\u2081, h\u2088\u2088]\n        have h\u2088\u2081\u2082 : Real.cos (Real.pi / 7) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor <;> linarith [Real.pi_gt_three]\n        have h\u2088\u2081\u2083 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n          have h\u2088\u2081\u2084 : Real.cos (Real.pi / 7) > 0 := by\n            apply Real.cos_pos_of_mem_Ioo\n            constructor <;> linarith [Real.pi_gt_three]\n          have h\u2088\u2081\u2085 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n            rw [Real.cos_three_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          have h\u2088\u2081\u2086 : Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) := by rfl\n          have h\u2088\u2081\u2087 : Real.cos (3 * (Real.pi / 7)) > 0 := by\n            apply Real.cos_pos_of_mem_Ioo\n            constructor\n            \u00b7 linarith [Real.pi_gt_three]\n            \u00b7 linarith [Real.pi_gt_three]\n          have h\u2088\u2081\u2088 : Real.cos (Real.pi / 7) > 0 := by\n            apply Real.cos_pos_of_mem_Ioo\n            constructor <;> linarith [Real.pi_gt_three]\n          have h\u2088\u2081\u2089 : Real.cos (Real.pi - 3 * (Real.pi / 7)) = -Real.cos (3 * (Real.pi / 7)) := by\n            rw [Real.cos_pi_sub]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          have h\u2088\u2082\u2080 : Real.cos (4 * (Real.pi / 7)) = Real.cos (Real.pi - 3 * (Real.pi / 7)) := by\n            ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          have h\u2088\u2082\u2081 : Real.cos (4 * (Real.pi / 7)) = -Real.cos (3 * (Real.pi / 7)) := by\n            rw [h\u2088\u2082\u2080, h\u2088\u2081\u2089]\n          have h\u2088\u2082\u2082 : Real.cos (4 * (Real.pi / 7)) = 8 * Real.cos (Real.pi / 7) ^ 4 - 8 * Real.cos (Real.pi / 7) ^ 2 + 1 := by\n            have h\u2088\u2082\u2083 : Real.cos (4 * (Real.pi / 7)) = Real.cos (2 * (2 * (Real.pi / 7))) := by ring\n            rw [h\u2088\u2082\u2083]\n            have h\u2088\u2082\u2084 : Real.cos (2 * (2 * (Real.pi / 7))) = 2 * Real.cos (2 * (Real.pi / 7)) ^ 2 - 1 := by\n              rw [Real.cos_two_mul]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n            rw [h\u2088\u2082\u2084]\n            have h\u2088\u2082\u2085 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n              rw [Real.cos_two_mul]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n              <;> simp [Real.cos_sq]\n              <;> ring_nf\n            rw [h\u2088\u2082\u2085]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          have h\u2088\u2082\u2086 : 8 * Real.cos (Real.pi / 7) ^ 4 - 8 * Real.cos (Real.pi / 7) ^ 2 + 1 = -Real.cos (3 * (Real.pi / 7)) := by\n            linarith\n          have h\u2088\u2082\u2087 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n            nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n              Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n              Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n          exact h\u2088\u2082\u2087\n        have h\u2088\u2081\u2084 : Real.cos (Real.pi / 7) \u2260 0 := by linarith\n        nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n          Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n          Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n      exact h\u2088\u2082\n    exact h\u2088\u2081\n  have h\u2089 : Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n    have h\u2089\u2081 : Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n      exact h\u2088\n    have h\u2089\u2082 : Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n      have h\u2089\u2083 : Real.cos (6 * Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n        have h\u2089\u2084 : Real.cos (6 * Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n          have h\u2089\u2085 : Real.cos (6 * Real.pi / 7) = Real.cos (Real.pi - Real.pi / 7) := by ring\n          rw [h\u2089\u2085]\n          have h\u2089\u2086 : Real.cos (Real.pi - Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n            rw [Real.cos_pi_sub]\n          rw [h\u2089\u2086]\n        rw [h\u2089\u2084]\n      have h\u2089\u2084 : Real.cos (4 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n        have h\u2089\u2085 : Real.cos (4 * Real.pi / 7) = Real.cos (Real.pi - 3 * Real.pi / 7) := by ring\n        rw [h\u2089\u2085]\n        have h\u2089\u2086 : Real.cos (Real.pi - 3 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n          rw [Real.cos_pi_sub]\n        rw [h\u2089\u2086]\n      have h\u2089\u2085 : Real.cos (2 * Real.pi / 7) = Real.cos (2 * Real.pi / 7) := by ring\n      rw [h\u2089\u2083, h\u2089\u2084]\n      have h\u2089\u2086 : Real.cos (3 * Real.pi / 7) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n        have h\u2089\u2087 : Real.cos (3 * Real.pi / 7) = Real.cos (3 * (Real.pi / 7)) := by ring\n        rw [h\u2089\u2087]\n        have h\u2089\u2088 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n          have h\u2089\u2089 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n            rw [show Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) by rfl]\n            have h\u2081\u2080\u2080 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n              rw [show Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) by rfl]\n              have h\u2081\u2080\u2081 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n                rw [Real.cos_three_mul]\n                <;> ring_nf\n                <;> simp [Real.cos_sq]\n                <;> ring_nf\n                <;> simp [Real.cos_sq]\n                <;> ring_nf\n              rw [h\u2081\u2080\u2081]\n            rw [h\u2081\u2080\u2080]\n          rw [h\u2089\u2089]\n        rw [h\u2089\u2088]\n      have h\u2089\u2087 : Real.cos (2 * Real.pi / 7) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n        have h\u2089\u2088 : Real.cos (2 * Real.pi / 7) = Real.cos (2 * (Real.pi / 7)) := by ring\n        rw [h\u2089\u2088]\n        have h\u2089\u2089 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n          have h\u2081\u2080\u2080 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n            rw [Real.cos_two_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          rw [h\u2081\u2080\u2080]\n        rw [h\u2089\u2089]\n      rw [h\u2089\u2086, h\u2089\u2087]\n      have h\u2089\u2088 : Real.cos (Real.pi / 7) > 0 := by\n        apply Real.cos_pos_of_mem_Ioo\n        constructor <;> linarith [Real.pi_gt_three]\n      have h\u2089\u2089 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n        have h\u2081\u2080\u2080 : Real.cos (Real.pi / 7) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor <;> linarith [Real.pi_gt_three]\n        have h\u2081\u2080\u2081 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n          rw [Real.cos_three_mul]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n        have h\u2081\u2080\u2082 : Real.cos (3 * (Real.pi / 7)) = Real.cos (3 * (Real.pi / 7)) := by rfl\n        have h\u2081\u2080\u2083 : Real.cos (3 * (Real.pi / 7)) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor\n          \u00b7 linarith [Real.pi_gt_three]\n          \u00b7 linarith [Real.pi_gt_three]\n        have h\u2081\u2080\u2084 : Real.cos (Real.pi / 7) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor <;> linarith [Real.pi_gt_three]\n        have h\u2081\u2080\u2085 : Real.cos (Real.pi - 3 * (Real.pi / 7)) = -Real.cos (3 * (Real.pi / 7)) := by\n          rw [Real.cos_pi_sub]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n        have h\u2081\u2080\u2086 : Real.cos (4 * (Real.pi / 7)) = Real.cos (Real.pi - 3 * (Real.pi / 7)) := by\n          ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n        have h\u2081\u2080\u2087 : Real.cos (4 * (Real.pi / 7)) = -Real.cos (3 * (Real.pi / 7)) := by\n          rw [h\u2081\u2080\u2086, h\u2081\u2080\u2085]\n        have h\u2081\u2080\u2088 : Real.cos (4 * (Real.pi / 7)) = 8 * Real.cos (Real.pi / 7) ^ 4 - 8 * Real.cos (Real.pi / 7) ^ 2 + 1 := by\n          have h\u2081\u2080\u2089 : Real.cos (4 * (Real.pi / 7)) = Real.cos (2 * (2 * (Real.pi / 7))) := by ring\n          rw [h\u2081\u2080\u2089]\n          have h\u2081\u2081\u2080 : Real.cos (2 * (2 * (Real.pi / 7))) = 2 * Real.cos (2 * (Real.pi / 7)) ^ 2 - 1 := by\n            rw [Real.cos_two_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          rw [h\u2081\u2081\u2080]\n          have h\u2081\u2081\u2081 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n            rw [Real.cos_two_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          rw [h\u2081\u2081\u2081]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n          <;> simp [Real.cos_sq]\n          <;> ring_nf\n        have h\u2081\u2081\u2082 : 8 * Real.cos (Real.pi / 7) ^ 4 - 8 * Real.cos (Real.pi / 7) ^ 2 + 1 = -Real.cos (3 * (Real.pi / 7)) := by\n          linarith\n        have h\u2081\u2081\u2083 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n          nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n            Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n            Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n        exact h\u2081\u2081\u2083\n      have h\u2081\u2080\u2080 : Real.cos (Real.pi / 7) \u2260 0 := by linarith\n      nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n        Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n        Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n    exact h\u2089\u2082\n    <;>\n    (try ring_nf at * <;> linarith [Real.pi_pos])\n    <;>\n    (try norm_num at * <;> linarith [Real.pi_pos])\n    <;>\n    (try field_simp at * <;> linarith [Real.pi_pos])\n  have h\u2081\u2080 : Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) = 1 / 8 := by\n    have h\u2081\u2080\u2081 : Real.cos (2 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) + Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n      exact h\u2088\n    have h\u2081\u2080\u2082 : Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) + Real.cos (2 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) + Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) = -1 / 2 := by\n      exact h\u2089\n    have h\u2081\u2080\u2083 : Real.cos (2 * Real.pi / 7) * Real.cos (4 * Real.pi / 7) * Real.cos (6 * Real.pi / 7) = 1 / 8 := by\n      have h\u2081\u2080\u2084 : Real.cos (6 * Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n        have h\u2081\u2080\u2085 : Real.cos (6 * Real.pi / 7) = Real.cos (Real.pi - Real.pi / 7) := by ring\n        rw [h\u2081\u2080\u2085]\n        have h\u2081\u2080\u2086 : Real.cos (Real.pi - Real.pi / 7) = -Real.cos (Real.pi / 7) := by\n          rw [Real.cos_pi_sub]\n        rw [h\u2081\u2080\u2086]\n      have h\u2081\u2080\u2085 : Real.cos (4 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n        have h\u2081\u2080\u2086 : Real.cos (4 * Real.pi / 7) = Real.cos (Real.pi - 3 * Real.pi / 7) := by ring\n        rw [h\u2081\u2080\u2086]\n        have h\u2081\u2080\u2087 : Real.cos (Real.pi - 3 * Real.pi / 7) = -Real.cos (3 * Real.pi / 7) := by\n          rw [Real.cos_pi_sub]\n        rw [h\u2081\u2080\u2087]\n      have h\u2081\u2080\u2086 : Real.cos (2 * Real.pi / 7) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n        have h\u2081\u2080\u2087 : Real.cos (2 * Real.pi / 7) = Real.cos (2 * (Real.pi / 7)) := by ring\n        rw [h\u2081\u2080\u2087]\n        have h\u2081\u2080\u2088 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n          have h\u2081\u2080\u2089 : Real.cos (2 * (Real.pi / 7)) = 2 * Real.cos (Real.pi / 7) ^ 2 - 1 := by\n            rw [Real.cos_two_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          rw [h\u2081\u2080\u2089]\n        rw [h\u2081\u2080\u2088]\n      have h\u2081\u2080\u2087 : Real.cos (3 * Real.pi / 7) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n        have h\u2081\u2080\u2088 : Real.cos (3 * Real.pi / 7) = Real.cos (3 * (Real.pi / 7)) := by ring\n        rw [h\u2081\u2080\u2088]\n        have h\u2081\u2080\u2089 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n          have h\u2081\u2081\u2080 : Real.cos (3 * (Real.pi / 7)) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n            rw [Real.cos_three_mul]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n            <;> simp [Real.cos_sq]\n            <;> ring_nf\n          rw [h\u2081\u2081\u2080]\n        rw [h\u2081\u2080\u2089]\n      have h\u2081\u2080\u2088 : Real.cos (Real.pi / 7) > 0 := by\n        apply Real.cos_pos_of_mem_Ioo\n        constructor <;> linarith [Real.pi_gt_three]\n      have h\u2081\u2080\u2089 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n        have h\u2081\u2081\u2080 : Real.cos (3 * (Real.pi / 7)) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor\n          \u00b7 linarith [Real.pi_gt_three]\n          \u00b7 linarith [Real.pi_gt_three]\n        have h\u2081\u2081\u2081 : Real.cos (Real.pi / 7) > 0 := by\n          apply Real.cos_pos_of_mem_Ioo\n          constructor <;> linarith [Real.pi_gt_three]\n        have h\u2081\u2081\u2082 : 8 * Real.cos (Real.pi / 7) ^ 3 - 4 * Real.cos (Real.pi / 7) ^ 2 - 4 * Real.cos (Real.pi / 7) + 1 = 0 := by\n          nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n            Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n            Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n        exact h\u2081\u2081\u2082\n      have h\u2081\u2081\u2080 : Real.cos (Real.pi / 7) \u2260 0 := by linarith\n      rw [h\u2081\u2080\u2084, h\u2081\u2080\u2085, h\u2081\u2080\u2086] at *\n      have h\u2081\u2081\u2081 : Real.cos (3 * Real.pi / 7) = 4 * Real.cos (Real.pi / 7) ^ 3 - 3 * Real.cos (Real.pi / 7) := by\n        rw [h\u2081\u2080\u2087]\n      nlinarith [Real.cos_le_one (Real.pi / 7), Real.cos_le_one (3 * (Real.pi / 7)), Real.cos_le_one (2 * (Real.pi / 7)),\n        Real.cos_le_one (4 * (Real.pi / 7)), Real.sin_le_one (Real.pi / 7), Real.sin_le_one (3 * (Real.pi / 7)),\n        Real.sin_le_one (2 * (Real.pi / 7)), Real.sin_le_one (4 * (Real.pi / 7))]\n    exact h\u2081\u2080\u2083\n    <;>\n    (try ring_nf at * <;> linarith [Real.pi_pos])\n    <;>\n    (try norm_num at * <;> linarith [Real.pi_pos])\n    <;>\n    (try field_simp at * <;> linarith [Real.pi_pos])\n  have h\u2081\u2081 : a = 1 / 2 := by\n    rw [h\u2085]\n    linarith\n  have h\u2081\u2082 : b = -1 / 2 := by\n    rw [h\u2086]\n    linarith\n  have h\u2081\u2083 : c = -1 / 8 := by\n    rw [h\u2087]\n    linarith\n  have h\u2081\u2084 : a * b * c = 1 / 32 := by\n    rw [h\u2081\u2081, h\u2081\u2082, h\u2081\u2083]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  rw [h\u2081\u2084]\n"}
{"mode": "clear", "steps_before": 28, "alphabet": 21, "compiles": false, "id": "induction_pprime_pdvdapowpma", "proof_after": "theorem induction_pprime_pdvdapowpma (p a : \u2115) (h\u2080 : 0 < a) (h\u2081 : Nat.Prime p) : p \u2223 a ^ p - a  :=  by\n  have h\u2082 : a \u2264 a ^ p := by\n    have h\u2083 : p \u2260 0 := Nat.Prime.ne_zero h\u2081\n    have h\u2084 : a \u2260 0 := by linarith\n    have h\u2085 : a \u2264 a ^ p := by\n      apply Nat.le_self_pow\n      <;> omega\n    exact h\u2085\n  have h\u2083 : (a : ZMod p) ^ p = a := by\n    haveI : Fact p.Prime := \u27e8h\u2081\u27e9\n    have h\u2084 : (a : ZMod p) ^ p = a := by\n      rw [ZMod.pow_card]\n    exact h\u2084\n  have h\u2084 : p \u2223 a ^ p - a := by\n    haveI : Fact p.Prime := \u27e8h\u2081\u27e9\n    have h\u2085 : (p : \u2115) \u2223 a ^ p - a := by\n      have h\u2086 : (a ^ p : ZMod p) = a := by\n        simpa [ZMod.nat_cast_self] using h\u2083\n      have h\u2087 : (p : \u2115) \u2223 a ^ p - a := by\n        rw [\u2190 ZMod.nat_cast_zmod_eq_zero_iff_dvd]\n        simp_all [ZMod.nat_cast_self, pow_succ, mul_comm]\n        <;>\n        ring_nf at *\n        <;>\n        simp_all [ZMod.nat_cast_zmod_eq_zero_iff_dvd, pow_succ, mul_comm]\n      exact h\u2087\n    exact h\u2085\n  exact h\u2084\n", "proof_before": "theorem induction_pprime_pdvdapowpma (p a : \u2115) (h\u2080 : 0 < a) (h\u2081 : Nat.Prime p) : p \u2223 a ^ p - a  :=  by\n  have h\u2082 : a \u2264 a ^ p := by\n    have h\u2083 : p \u2260 0 := Nat.Prime.ne_zero h\u2081\n    have h\u2084 : a \u2260 0 := by linarith\n    have h\u2085 : a \u2264 a ^ p := by\n      apply Nat.le_self_pow\n      <;> omega\n    exact h\u2085\n  have h\u2083 : (a : ZMod p) ^ p = a := by\n    haveI : Fact p.Prime := \u27e8h\u2081\u27e9\n    have h\u2084 : (a : ZMod p) ^ p = a := by\n      rw [ZMod.pow_card]\n    exact h\u2084\n  have h\u2084 : p \u2223 a ^ p - a := by\n    haveI : Fact p.Prime := \u27e8h\u2081\u27e9\n    have h\u2085 : (p : \u2115) \u2223 a ^ p - a := by\n      have h\u2086 : (a ^ p : ZMod p) = a := by\n        simpa [ZMod.nat_cast_self] using h\u2083\n      have h\u2087 : (p : \u2115) \u2223 a ^ p - a := by\n        rw [\u2190 ZMod.nat_cast_zmod_eq_zero_iff_dvd]\n        simp_all [ZMod.nat_cast_self, pow_succ, mul_comm]\n        <;>\n        ring_nf at *\n        <;>\n        simp_all [ZMod.nat_cast_zmod_eq_zero_iff_dvd, pow_succ, mul_comm]\n      exact h\u2087\n    exact h\u2085\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 22, "alphabet": 16, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 512, "chars_after": 7, "seconds": 1.9, "id": "mathd_algebra_176", "proof_after": "theorem mathd_algebra_176 (x : \u211d) : (x + 1) ^ 2 * x = x ^ 3 + 2 * x ^ 2 + x  :=  by\n  ring\n", "proof_before": "theorem mathd_algebra_176 (x : \u211d) : (x + 1) ^ 2 * x = x ^ 3 + 2 * x ^ 2 + x  :=  by\n  have h1 : (x + 1) ^ 2 * x = (x ^ 2 + 2 * x + 1) * x := by\n    have h1\u2081 : (x + 1) ^ 2 = x ^ 2 + 2 * x + 1 := by\n      ring\n    rw [h1\u2081]\n    <;> ring\n    <;> linarith\n  have h2 : (x ^ 2 + 2 * x + 1) * x = x ^ 3 + 2 * x ^ 2 + x := by\n    have h2\u2081 : (x ^ 2 + 2 * x + 1) * x = x ^ 3 + 2 * x ^ 2 + x := by\n      ring_nf\n      <;>\n      nlinarith\n    rw [h2\u2081]\n    <;>\n    nlinarith\n  have h3 : (x + 1) ^ 2 * x = x ^ 3 + 2 * x ^ 2 + x := by\n    rw [h1]\n    rw [h2]\n    <;>\n    ring_nf\n    <;>\n    nlinarith\n  exact h3\n"}
{"mode": "clear", "steps_before": 53, "alphabet": 19, "compiles": true, "steps_after": 44, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 5, "timed_out": false, "fixpoint": true, "chars_before": 1422, "chars_after": 1246, "seconds": 237.1, "id": "algebra_absxm1pabsxpabsxp1eqxp2_0leqxleq1", "proof_after": "theorem algebra_absxm1pabsxpabsxp1eqxp2_0leqxleq1 (x : \u211d)\n    (h\u2080 : abs (x - 1) + abs x + abs (x + 1) = x + 2) : 0 \u2264 x \u2227 x \u2264 1  :=  by\n  have h\u2081 : 0 \u2264 x := by\n    by_contra h\n    by_cases h\u2083 : x < -1\n    \u00b7 \n      have h\u2084 : abs (x - 1) = -(x - 1) := by\n        rw [abs_of_neg (by linarith)]\n        <;> linarith\n      have h\u2085 : abs x = -x := by\n        rw [abs_of_neg (by linarith)]\n      have h\u2086 : abs (x + 1) = -(x + 1) := by\n        rw [abs_of_neg (by linarith)]\n        <;> linarith\n      rw [h\u2084, h\u2085, h\u2086] at h\u2080\n      have h\u2087 : - (x - 1) + -x + - (x + 1) = x + 2 := by linarith\n      have h\u2088 : x = -1 / 2 := by linarith\n      linarith\n    \u00b7 \n      have h\u2084 : x \u2265 -1 := by linarith\n      have h\u2085 : abs (x - 1) = -(x - 1) := by\n        rw [abs_of_neg (by linarith)]\n        <;> linarith\n      have h\u2086 : abs x = -x := by\n        rw [abs_of_neg (by linarith)]\n      have h\u2087 : abs (x + 1) = x + 1 := by\n        rw [abs_of_nonneg (by linarith)]\n      rw [h\u2085, h\u2086, h\u2087] at h\u2080\n      have h\u2088 : - (x - 1) + -x + (x + 1) = x + 2 := by linarith\n      have h\u2089 : x = 0 := by linarith\n      linarith\n  have h\u2082 : x \u2264 1 := by\n    by_contra h\n    have h\u2084 : abs (x - 1) = x - 1 := by\n      rw [abs_of_nonneg (by linarith)]\n    have h\u2085 : abs x = x := by\n      rw [abs_of_nonneg (by linarith)]\n    have h\u2086 : abs (x + 1) = x + 1 := by\n      rw [abs_of_nonneg (by linarith)]\n    linarith\n  exact \u27e8h\u2081, h\u2082\u27e9\n", "proof_before": "theorem algebra_absxm1pabsxpabsxp1eqxp2_0leqxleq1 (x : \u211d)\n    (h\u2080 : abs (x - 1) + abs x + abs (x + 1) = x + 2) : 0 \u2264 x \u2227 x \u2264 1  :=  by\n  have h\u2081 : 0 \u2264 x := by\n    by_contra h\n    have h\u2082 : x < 0 := by linarith\n    by_cases h\u2083 : x < -1\n    \u00b7 \n      have h\u2084 : abs (x - 1) = -(x - 1) := by\n        rw [abs_of_neg (by linarith)]\n        <;> linarith\n      have h\u2085 : abs x = -x := by\n        rw [abs_of_neg (by linarith)]\n      have h\u2086 : abs (x + 1) = -(x + 1) := by\n        rw [abs_of_neg (by linarith)]\n        <;> linarith\n      rw [h\u2084, h\u2085, h\u2086] at h\u2080\n      have h\u2087 : - (x - 1) + -x + - (x + 1) = x + 2 := by linarith\n      have h\u2088 : x = -1 / 2 := by linarith\n      linarith\n    \u00b7 \n      have h\u2084 : x \u2265 -1 := by linarith\n      have h\u2085 : abs (x - 1) = -(x - 1) := by\n        rw [abs_of_neg (by linarith)]\n        <;> linarith\n      have h\u2086 : abs x = -x := by\n        rw [abs_of_neg (by linarith)]\n      have h\u2087 : abs (x + 1) = x + 1 := by\n        rw [abs_of_nonneg (by linarith)]\n      rw [h\u2085, h\u2086, h\u2087] at h\u2080\n      have h\u2088 : - (x - 1) + -x + (x + 1) = x + 2 := by linarith\n      have h\u2089 : x = 0 := by linarith\n      linarith\n  have h\u2082 : x \u2264 1 := by\n    by_contra h\n    have h\u2083 : x > 1 := by linarith\n    have h\u2084 : abs (x - 1) = x - 1 := by\n      rw [abs_of_nonneg (by linarith)]\n    have h\u2085 : abs x = x := by\n      rw [abs_of_nonneg (by linarith)]\n    have h\u2086 : abs (x + 1) = x + 1 := by\n      rw [abs_of_nonneg (by linarith)]\n    rw [h\u2084, h\u2085, h\u2086] at h\u2080\n    have h\u2087 : x = 1 := by linarith\n    linarith\n  have h\u2083 : 0 \u2264 x \u2227 x \u2264 1 := by\n    exact \u27e8h\u2081, h\u2082\u27e9\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 698, "alphabet": 299, "compiles": false, "id": "aime_1994_p3", "proof_after": "theorem aime_1994_p3 (f : \u2124 \u2192 \u2124) (h0 : \u2200 x, f x + f (x - 1) = x ^ 2) (h1 : f 19 = 94) :\n    f 94 % 1000 = 561  :=  by\n  have h2 : f 18 = 267 := by\n    have h2\u2081 := h0 19\n    have h2\u2082 : f 19 + f 18 = 361 := by\n      norm_num at h2\u2081 \u22a2\n      <;> linarith\n    have h2\u2083 : f 18 = 267 := by\n      linarith\n    exact h2\u2083\n  have h3 : f 17 = 57 := by\n    have h3\u2081 : f 18 + f 17 = 324 := by\n      have h3\u2082 := h0 18\n      norm_num at h3\u2082 \u22a2\n      <;> linarith\n    have h3\u2083 : f 17 = 57 := by\n      linarith\n    exact h3\u2083\n  have h4 : f 16 = 232 := by\n    have h4\u2081 : f 17 + f 16 = 289 := by\n      have h4\u2082 := h0 17\n      norm_num at h4\u2082 \u22a2\n      <;> linarith\n    have h4\u2083 : f 16 = 232 := by\n      linarith\n    exact h4\u2083\n  have h5 : f 15 = 24 := by\n    have h5\u2081 : f 16 + f 15 = 256 := by\n      have h5\u2082 := h0 16\n      norm_num at h5\u2082 \u22a2\n      <;> linarith\n    have h5\u2083 : f 15 = 24 := by\n      linarith\n    exact h5\u2083\n  have h6 : f 14 = 201 := by\n    have h6\u2081 : f 15 + f 14 = 225 := by\n      have h6\u2082 := h0 15\n      norm_num at h6\u2082 \u22a2\n      <;> linarith\n    have h6\u2083 : f 14 = 201 := by\n      linarith\n    exact h6\u2083\n  have h7 : f 13 = (-5) := by\n    have h7\u2081 : f 14 + f 13 = 196 := by\n      have h7\u2082 := h0 14\n      norm_num at h7\u2082 \u22a2\n      <;> linarith\n    have h7\u2083 : f 13 = -5 := by\n      linarith\n    exact h7\u2083\n  have h8 : f 12 = 174 := by\n    have h8\u2081 : f 13 + f 12 = 169 := by\n      have h8\u2082 := h0 13\n      norm_num at h8\u2082 \u22a2\n      <;> linarith\n    have h8\u2083 : f 12 = 174 := by\n      linarith\n    exact h8\u2083\n  have h9 : f 11 = (-30) := by\n    have h9\u2081 : f 12 + f 11 = 144 := by\n      have h9\u2082 := h0 12\n      norm_num at h9\u2082 \u22a2\n      <;> linarith\n    have h9\u2083 : f 11 = -30 := by\n      linarith\n    exact h9\u2083\n  have h10 : f 10 = 151 := by\n    have h10\u2081 : f 11 + f 10 = 121 := by\n      have h10\u2082 := h0 11\n      norm_num at h10\u2082 \u22a2\n      <;> linarith\n    have h10\u2083 : f 10 = 151 := by\n      linarith\n    exact h10\u2083\n  have h11 : f 9 = (-51) := by\n    have h11\u2081 : f 10 + f 9 = 100 := by\n      have h11\u2082 := h0 10\n      norm_num at h11\u2082 \u22a2\n      <;> linarith\n    have h11\u2083 : f 9 = -51 := by\n      linarith\n    exact h11\u2083\n  have h12 : f 8 = 132 := by\n    have h12\u2081 : f 9 + f 8 = 81 := by\n      have h12\u2082 := h0 9\n      norm_num at h12\u2082 \u22a2\n      <;> linarith\n    have h12\u2083 : f 8 = 132 := by\n      linarith\n    exact h12\u2083\n  have h13 : f 7 = (-68) := by\n    have h13\u2081 : f 8 + f 7 = 64 := by\n      have h13\u2082 := h0 8\n      norm_num at h13\u2082 \u22a2\n      <;> linarith\n    have h13\u2083 : f 7 = -68 := by\n      linarith\n    exact h13\u2083\n  have h14 : f 6 = 117 := by\n    have h14\u2081 : f 7 + f 6 = 49 := by\n      have h14\u2082 := h0 7\n      norm_num at h14\u2082 \u22a2\n      <;> linarith\n    have h14\u2083 : f 6 = 117 := by\n      linarith\n    exact h14\u2083\n  have h15 : f 5 = (-81) := by\n    have h15\u2081 : f 6 + f 5 = 36 := by\n      have h15\u2082 := h0 6\n      norm_num at h15\u2082 \u22a2\n      <;> linarith\n    have h15\u2083 : f 5 = -81 := by\n      linarith\n    exact h15\u2083\n  have h16 : f 4 = 106 := by\n    have h16\u2081 : f 5 + f 4 = 25 := by\n      have h16\u2082 := h0 5\n      norm_num at h16\u2082 \u22a2\n      <;> linarith\n    have h16\u2083 : f 4 = 106 := by\n      linarith\n    exact h16\u2083\n  have h17 : f 3 = (-90) := by\n    have h17\u2081 : f 4 + f 3 = 16 := by\n      have h17\u2082 := h0 4\n      norm_num at h17\u2082 \u22a2\n      <;> linarith\n    have h17\u2083 : f 3 = -90 := by\n      linarith\n    exact h17\u2083\n  have h18 : f 2 = 99 := by\n    have h18\u2081 : f 3 + f 2 = 9 := by\n      have h18\u2082 := h0 3\n      norm_num at h18\u2082 \u22a2\n      <;> linarith\n    have h18\u2083 : f 2 = 99 := by\n      linarith\n    exact h18\u2083\n  have h19 : f 1 = (-95) := by\n    have h19\u2081 : f 2 + f 1 = 4 := by\n      have h19\u2082 := h0 2\n      norm_num at h19\u2082 \u22a2\n      <;> linarith\n    have h19\u2083 : f 1 = -95 := by\n      linarith\n    exact h19\u2083\n  have h20 : f 0 = 96 := by\n    have h20\u2081 : f 1 + f 0 = 1 := by\n      have h20\u2082 := h0 1\n      norm_num at h20\u2082 \u22a2\n      <;> linarith\n    have h20\u2083 : f 0 = 96 := by\n      linarith\n    exact h20\u2083\n  have h21 : f 20 = 306 := by\n    have h21\u2081 : f 20 + f 19 = 400 := by\n      have h21\u2082 := h0 20\n      norm_num at h21\u2082 \u22a2\n      <;> linarith\n    have h21\u2083 : f 20 = 306 := by\n      linarith\n    exact h21\u2083\n  have h22 : f 21 = 135 := by\n    have h22\u2081 : f 21 + f 20 = 441 := by\n      have h22\u2082 := h0 21\n      norm_num at h22\u2082 \u22a2\n      <;> linarith\n    have h22\u2083 : f 21 = 135 := by\n      linarith\n    exact h22\u2083\n  have h23 : f 22 = 349 := by\n    have h23\u2081 : f 22 + f 21 = 484 := by\n      have h23\u2082 := h0 22\n      norm_num at h23\u2082 \u22a2\n      <;> linarith\n    have h23\u2083 : f 22 = 349 := by\n      linarith\n    exact h23\u2083\n  have h24 : f 23 = 180 := by\n    have h24\u2081 : f 23 + f 22 = 529 := by\n      have h24\u2082 := h0 23\n      norm_num at h24\u2082 \u22a2\n      <;> linarith\n    have h24\u2083 : f 23 = 180 := by\n      linarith\n    exact h24\u2083\n  have h25 : f 24 = 396 := by\n    have h25\u2081 : f 24 + f 23 = 576 := by\n      have h25\u2082 := h0 24\n      norm_num at h25\u2082 \u22a2\n      <;> linarith\n    have h25\u2083 : f 24 = 396 := by\n      linarith\n    exact h25\u2083\n  have h26 : f 25 = 229 := by\n    have h26\u2081 : f 25 + f 24 = 625 := by\n      have h26\u2082 := h0 25\n      norm_num at h26\u2082 \u22a2\n      <;> linarith\n    have h26\u2083 : f 25 = 229 := by\n      linarith\n    exact h26\u2083\n  have h27 : f 26 = 447 := by\n    have h27\u2081 : f 26 + f 25 = 676 := by\n      have h27\u2082 := h0 26\n      norm_num at h27\u2082 \u22a2\n      <;> linarith\n    have h27\u2083 : f 26 = 447 := by\n      linarith\n    exact h27\u2083\n  have h28 : f 27 = 282 := by\n    have h28\u2081 : f 27 + f 26 = 729 := by\n      have h28\u2082 := h0 27\n      norm_num at h28\u2082 \u22a2\n      <;> linarith\n    have h28\u2083 : f 27 = 282 := by\n      linarith\n    exact h28\u2083\n  have h29 : f 28 = 502 := by\n    have h29\u2081 : f 28 + f 27 = 784 := by\n      have h29\u2082 := h0 28\n      norm_num at h29\u2082 \u22a2\n      <;> linarith\n    have h29\u2083 : f 28 = 502 := by\n      linarith\n    exact h29\u2083\n  have h30 : f 29 = 339 := by\n    have h30\u2081 : f 29 + f 28 = 841 := by\n      have h30\u2082 := h0 29\n      norm_num at h30\u2082 \u22a2\n      <;> linarith\n    have h30\u2083 : f 29 = 339 := by\n      linarith\n    exact h30\u2083\n  have h31 : f 30 = 561 := by\n    have h31\u2081 : f 30 + f 29 = 900 := by\n      have h31\u2082 := h0 30\n      norm_num at h31\u2082 \u22a2\n      <;> linarith\n    have h31\u2083 : f 30 = 561 := by\n      linarith\n    exact h31\u2083\n  have h32 : f 94 = 4561 := by\n    have h32\u2081 : f 30 = 561 := h31\n    have h32\u2082 : f 31 = 961 - 561 := by\n      have h32\u2083 := h0 31\n      norm_num at h32\u2083 \u22a2\n      <;> linarith\n    have h32\u2084 : f 31 = 400 := by\n      norm_num at h32\u2082 \u22a2\n      <;> linarith\n    have h32\u2085 : f 32 = 1024 - 400 := by\n      have h32\u2086 := h0 32\n      norm_num at h32\u2086 \u22a2\n      <;> linarith\n    have h32\u2087 : f 32 = 624 := by\n      norm_num at h32\u2085 \u22a2\n      <;> linarith\n    have h32\u2088 : f 33 = 1089 - 624 := by\n      have h32\u2089 := h0 33\n      norm_num at h32\u2089 \u22a2\n      <;> linarith\n    have h32\u2081\u2080 : f 33 = 465 := by\n      norm_num at h32\u2088 \u22a2\n      <;> linarith\n    have h32\u2081\u2081 : f 34 = 1156 - 465 := by\n      have h32\u2081\u2082 := h0 34\n      norm_num at h32\u2081\u2082 \u22a2\n      <;> linarith\n    have h32\u2081\u2083 : f 34 = 691 := by\n      norm_num at h32\u2081\u2081 \u22a2\n      <;> linarith\n    have h32\u2081\u2084 : f 35 = 1225 - 691 := by\n      have h32\u2081\u2085 := h0 35\n      norm_num at h32\u2081\u2085 \u22a2\n      <;> linarith\n    have h32\u2081\u2086 : f 35 = 534 := by\n      norm_num at h32\u2081\u2084 \u22a2\n      <;> linarith\n    have h32\u2081\u2087 : f 36 = 1296 - 534 := by\n      have h32\u2081\u2088 := h0 36\n      norm_num at h32\u2081\u2088 \u22a2\n      <;> linarith\n    have h32\u2081\u2089 : f 36 = 762 := by\n      norm_num at h32\u2081\u2087 \u22a2\n      <;> linarith\n    have h32\u2082\u2080 : f 37 = 1369 - 762 := by\n      have h32\u2082\u2081 := h0 37\n      norm_num at h32\u2082\u2081 \u22a2\n      <;> linarith\n    have h32\u2082\u2082 : f 37 = 607 := by\n      norm_num at h32\u2082\u2080 \u22a2\n      <;> linarith\n    have h32\u2082\u2083 : f 38 = 1444 - 607 := by\n      have h32\u2082\u2084 := h0 38\n      norm_num at h32\u2082\u2084 \u22a2\n      <;> linarith\n    have h32\u2082\u2085 : f 38 = 837 := by\n      norm_num at h32\u2082\u2083 \u22a2\n      <;> linarith\n    have h32\u2082\u2086 : f 39 = 1521 - 837 := by\n      have h32\u2082\u2087 := h0 39\n      norm_num at h32\u2082\u2087 \u22a2\n      <;> linarith\n    have h32\u2082\u2088 : f 39 = 684 := by\n      norm_num at h32\u2082\u2086 \u22a2\n      <;> linarith\n    have h32\u2082\u2089 : f 40 = 1600 - 684 := by\n      have h32\u2083\u2080 := h0 40\n      norm_num at h32\u2083\u2080 \u22a2\n      <;> linarith\n    have h32\u2083\u2081 : f 40 = 916 := by\n      norm_num at h32\u2082\u2089 \u22a2\n      <;> linarith\n    have h32\u2083\u2082 : f 41 = 1681 - 916 := by\n      have h32\u2083\u2083 := h0 41\n      norm_num at h32\u2083\u2083 \u22a2\n      <;> linarith\n    have h32\u2083\u2084 : f 41 = 765 := by\n      norm_num at h32\u2083\u2082 \u22a2\n      <;> linarith\n    have h32\u2083\u2085 : f 42 = 1764 - 765 := by\n      have h32\u2083\u2086 := h0 42\n      norm_num at h32\u2083\u2086 \u22a2\n      <;> linarith\n    have h32\u2083\u2087 : f 42 = 999 := by\n      norm_num at h32\u2083\u2085 \u22a2\n      <;> linarith\n    have h32\u2083\u2088 : f 43 = 1849 - 999 := by\n      have h32\u2083\u2089 := h0 43\n      norm_num at h32\u2083\u2089 \u22a2\n      <;> linarith\n    have h32\u2084\u2080 : f 43 = 850 := by\n      norm_num at h32\u2083\u2088 \u22a2\n      <;> linarith\n    have h32\u2084\u2081 : f 44 = 1936 - 850 := by\n      have h32\u2084\u2082 := h0 44\n      norm_num at h32\u2084\u2082 \u22a2\n      <;> linarith\n    have h32\u2084\u2083 : f 44 = 1086 := by\n      norm_num at h32\u2084\u2081 \u22a2\n      <;> linarith\n    have h32\u2084\u2084 : f 45 = 2025 - 1086 := by\n      have h32\u2084\u2085 := h0 45\n      norm_num at h32\u2084\u2085 \u22a2\n      <;> linarith\n    have h32\u2084\u2086 : f 45 = 939 := by\n      norm_num at h32\u2084\u2084 \u22a2\n      <;> linarith\n    have h32\u2084\u2087 : f 46 = 2116 - 939 := by\n      have h32\u2084\u2088 := h0 46\n      norm_num at h32\u2084\u2088 \u22a2\n      <;> linarith\n    have h32\u2084\u2089 : f 46 = 1177 := by\n      norm_num at h32\u2084\u2087 \u22a2\n      <;> linarith\n    have h32\u2085\u2080 : f 47 = 2209 - 1177 := by\n      have h32\u2085\u2081 := h0 47\n      norm_num at h32\u2085\u2081 \u22a2\n      <;> linarith\n    have h32\u2085\u2082 : f 47 = 1032 := by\n      norm_num at h32\u2085\u2080 \u22a2\n      <;> linarith\n    have h32\u2085\u2083 : f 48 = 2304 - 1032 := by\n      have h32\u2085\u2084 := h0 48\n      norm_num at h32\u2085\u2084 \u22a2\n      <;> linarith\n    have h32\u2085\u2085 : f 48 = 1272 := by\n      norm_num at h32\u2085\u2083 \u22a2\n      <;> linarith\n    have h32\u2085\u2086 : f 49 = 2401 - 1272 := by\n      have h32\u2085\u2087 := h0 49\n      norm_num at h32\u2085\u2087 \u22a2\n      <;> linarith\n    have h32\u2085\u2088 : f 49 = 1129 := by\n      norm_num at h32\u2085\u2086 \u22a2\n      <;> linarith\n    have h32\u2085\u2089 : f 50 = 2500 - 1129 := by\n      have h32\u2086\u2080 := h0 50\n      norm_num at h32\u2086\u2080 \u22a2\n      <;> linarith\n    have h32\u2086\u2081 : f 50 = 1371 := by\n      norm_num at h32\u2085\u2089 \u22a2\n      <;> linarith\n    have h32\u2086\u2082 : f 51 = 2601 - 1371 := by\n      have h32\u2086\u2083 := h0 51\n      norm_num at h32\u2086\u2083 \u22a2\n      <;> linarith\n    have h32\u2086\u2084 : f 51 = 1230 := by\n      norm_num at h32\u2086\u2082 \u22a2\n      <;> linarith\n    have h32\u2086\u2085 : f 52 = 2704 - 1230 := by\n      have h32\u2086\u2086 := h0 52\n      norm_num at h32\u2086\u2086 \u22a2\n      <;> linarith\n    have h32\u2086\u2087 : f 52 = 1474 := by\n      norm_num at h32\u2086\u2085 \u22a2\n      <;> linarith\n    have h32\u2086\u2088 : f 53 = 2809 - 1474 := by\n      have h32\u2086\u2089 := h0 53\n      norm_num at h32\u2086\u2089 \u22a2\n      <;> linarith\n    have h32\u2087\u2080 : f 53 = 1335 := by\n      norm_num at h32\u2086\u2088 \u22a2\n      <;> linarith\n    have h32\u2087\u2081 : f 54 = 2916 - 1335 := by\n      have h32\u2087\u2082 := h0 54\n      norm_num at h32\u2087\u2082 \u22a2\n      <;> linarith\n    have h32\u2087\u2083 : f 54 = 1581 := by\n      norm_num at h32\u2087\u2081 \u22a2\n      <;> linarith\n    have h32\u2087\u2084 : f 55 = 3025 - 1581 := by\n      have h32\u2087\u2085 := h0 55\n      norm_num at h32\u2087\u2085 \u22a2\n      <;> linarith\n    have h32\u2087\u2086 : f 55 = 1444 := by\n      norm_num at h32\u2087\u2084 \u22a2\n      <;> linarith\n    have h32\u2087\u2087 : f 56 = 3136 - 1444 := by\n      have h32\u2087\u2088 := h0 56\n      norm_num at h32\u2087\u2088 \u22a2\n      <;> linarith\n    have h32\u2087\u2089 : f 56 = 1692 := by\n      norm_num at h32\u2087\u2087 \u22a2\n      <;> linarith\n    have h32\u2088\u2080 : f 57 = 3249 - 1692 := by\n      have h32\u2088\u2081 := h0 57\n      norm_num at h32\u2088\u2081 \u22a2\n      <;> linarith\n    have h32\u2088\u2082 : f 57 = 1557 := by\n      norm_num at h32\u2088\u2080 \u22a2\n      <;> linarith\n    have h32\u2088\u2083 : f 58 = 3364 - 1557 := by\n      have h32\u2088\u2084 := h0 58\n      norm_num at h32\u2088\u2084 \u22a2\n      <;> linarith\n    have h32\u2088\u2085 : f 58 = 1807 := by\n      norm_num at h32\u2088\u2083 \u22a2\n      <;> linarith\n    have h32\u2088\u2086 : f 59 = 3481 - 1807 := by\n      have h32\u2088\u2087 := h0 59\n      norm_num at h32\u2088\u2087 \u22a2\n      <;> linarith\n    have h32\u2088\u2088 : f 59 = 1674 := by\n      norm_num at h32\u2088\u2086 \u22a2\n      <;> linarith\n    have h32\u2088\u2089 : f 60 = 3600 - 1674 := by\n      have h32\u2089\u2080 := h0 60\n      norm_num at h32\u2089\u2080 \u22a2\n      <;> linarith\n    have h32\u2089\u2081 : f 60 = 1926 := by\n      norm_num at h32\u2088\u2089 \u22a2\n      <;> linarith\n    have h32\u2089\u2082 : f 61 = 3721 - 1926 := by\n      have h32\u2089\u2083 := h0 61\n      norm_num at h32\u2089\u2083 \u22a2\n      <;> linarith\n    have h32\u2089\u2084 : f 61 = 1795 := by\n      norm_num at h32\u2089\u2082 \u22a2\n      <;> linarith\n    have h32\u2089\u2085 : f 62 = 3844 - 1795 := by\n      have h32\u2089\u2086 := h0 62\n      norm_num at h32\u2089\u2086 \u22a2\n      <;> linarith\n    have h32\u2089\u2087 : f 62 = 2049 := by\n      norm_num at h32\u2089\u2085 \u22a2\n      <;> linarith\n    have h32\u2089\u2088 : f 63 = 3969 - 2049 := by\n      have h32\u2089\u2089 := h0 63\n      norm_num at h32\u2089\u2089 \u22a2\n      <;> linarith\n    have h33\u2080 : f 63 = 1920 := by\n      norm_num at h32\u2089\u2088 \u22a2\n      <;> linarith\n    have h33\u2081 : f 64 = 4096 - 1920 := by\n      have h33\u2082 := h0 64\n      norm_num at h33\u2082 \u22a2\n      <;> linarith\n    have h33\u2083 : f 64 = 2176 := by\n      norm_num at h33\u2081 \u22a2\n      <;> linarith\n    have h33\u2084 : f 65 = 4225 - 2176 := by\n      have h33\u2085 := h0 65\n      norm_num at h33\u2085 \u22a2\n      <;> linarith\n    have h33\u2086 : f 65 = 2049 := by\n      norm_num at h33\u2084 \u22a2\n      <;> linarith\n    have h33\u2087 : f 66 = 4356 - 2049 := by\n      have h33\u2088 := h0 66\n      norm_num at h33\u2088 \u22a2\n      <;> linarith\n    have h33\u2089 : f 66 = 2307 := by\n      norm_num at h33\u2087 \u22a2\n      <;> linarith\n    have h34\u2080 : f 67 = 4489 - 2307 := by\n      have h34\u2081 := h0 67\n      norm_num at h34\u2081 \u22a2\n      <;> linarith\n    have h34\u2082 : f 67 = 2182 := by\n      norm_num at h34\u2080 \u22a2\n      <;> linarith\n    have h34\u2083 : f 68 = 4624 - 2182 := by\n      have h34\u2084 := h0 68\n      norm_num at h34\u2084 \u22a2\n      <;> linarith\n    have h34\u2085 : f 68 = 2442 := by\n      norm_num at h34\u2083 \u22a2\n      <;> linarith\n    have h34\u2086 : f 69 = 4761 - 2442 := by\n      have h34\u2087 := h0 69\n      norm_num at h34\u2087 \u22a2\n      <;> linarith\n    have h34\u2088 : f 69 = 2319 := by\n      norm_num at h34\u2086 \u22a2\n      <;> linarith\n    have h34\u2089 : f 70 = 4900 - 2319 := by\n      have h35\u2080 := h0 70\n      norm_num at h35\u2080 \u22a2\n      <;> linarith\n    have h35\u2081 : f 70 = 2581 := by\n      norm_num at h34\u2089 \u22a2\n      <;> linarith\n    have h35\u2082 : f 71 = 5041 - 2581 := by\n      have h35\u2083 := h0 71\n      norm_num at h35\u2083 \u22a2\n      <;> linarith\n    have h35\u2084 : f 71 = 2460 := by\n      norm_num at h35\u2082 \u22a2\n      <;> linarith\n    have h35\u2085 : f 72 = 5184 - 2460 := by\n      have h35\u2086 := h0 72\n      norm_num at h35\u2086 \u22a2\n      <;> linarith\n    have h35\u2087 : f 72 = 2724 := by\n      norm_num at h35\u2085 \u22a2\n      <;> linarith\n    have h35\u2088 : f 73 = 5329 - 2724 := by\n      have h35\u2089 := h0 73\n      norm_num at h35\u2089 \u22a2\n      <;> linarith\n    have h36\u2080 : f 73 = 2605 := by\n      norm_num at h35\u2088 \u22a2\n      <;> linarith\n    have h36\u2081 : f 74 = 5476 - 2605 := by\n      have h36\u2082 := h0 74\n      norm_num at h36\u2082 \u22a2\n      <;> linarith\n    have h36\u2083 : f 74 = 2871 := by\n      norm_num at h36\u2081 \u22a2\n      <;> linarith\n    have h36\u2084 : f 75 = 5625 - 2871 := by\n      have h36\u2085 := h0 75\n      norm_num at h36\u2085 \u22a2\n      <;> linarith\n    have h36\u2086 : f 75 = 2754 := by\n      norm_num at h36\u2084 \u22a2\n      <;> linarith\n    have h36\u2087 : f 76 = 5776 - 2754 := by\n      have h36\u2088 := h0 76\n      norm_num at h36\u2088 \u22a2\n      <;> linarith\n    have h36\u2089 : f 76 = 3022 := by\n      norm_num at h36\u2087 \u22a2\n      <;> linarith\n    have h37\u2080 : f 77 = 5929 - 3022 := by\n      have h37\u2081 := h0 77\n      norm_num at h37\u2081 \u22a2\n      <;> linarith\n    have h37\u2082 : f 77 = 2907 := by\n      norm_num at h37\u2080 \u22a2\n      <;> linarith\n    have h37\u2083 : f 78 = 6084 - 2907 := by\n      have h37\u2084 := h0 78\n      norm_num at h37\u2084 \u22a2\n      <;> linarith\n    have h37\u2085 : f 78 = 3177 := by\n      norm_num at h37\u2083 \u22a2\n      <;> linarith\n    have h37\u2086 : f 79 = 6241 - 3177 := by\n      have h37\u2087 := h0 79\n      norm_num at h37\u2087 \u22a2\n      <;> linarith\n    have h37\u2088 : f 79 = 3064 := by\n      norm_num at h37\u2086 \u22a2\n      <;> linarith\n    have h37\u2089 : f 80 = 6400 - 3064 := by\n      have h38\u2080 := h0 80\n      norm_num at h38\u2080 \u22a2\n      <;> linarith\n    have h38\u2081 : f 80 = 3336 := by\n      norm_num at h37\u2089 \u22a2\n      <;> linarith\n    have h38\u2082 : f 81 = 6561 - 3336 := by\n      have h38\u2083 := h0 81\n      norm_num at h38\u2083 \u22a2\n      <;> linarith\n    have h38\u2084 : f 81 = 3225 := by\n      norm_num at h38\u2082 \u22a2\n      <;> linarith\n    have h38\u2085 : f 82 = 6724 - 3225 := by\n      have h38\u2086 := h0 82\n      norm_num at h38\u2086 \u22a2\n      <;> linarith\n    have h38\u2087 : f 82 = 3499 := by\n      norm_num at h38\u2085 \u22a2\n      <;> linarith\n    have h38\u2088 : f 83 = 6889 - 3499 := by\n      have h38\u2089 := h0 83\n      norm_num at h38\u2089 \u22a2\n      <;> linarith\n    have h39\u2080 : f 83 = 3390 := by\n      norm_num at h38\u2088 \u22a2\n      <;> linarith\n    have h39\u2081 : f 84 = 7056 - 3390 := by\n      have h39\u2082 := h0 84\n      norm_num at h39\u2082 \u22a2\n      <;> linarith\n    have h39\u2083 : f 84 = 3666 := by\n      norm_num at h39\u2081 \u22a2\n      <;> linarith\n    have h39\u2084 : f 85 = 7225 - 3666 := by\n      have h39\u2085 := h0 85\n      norm_num at h39\u2085 \u22a2\n      <;> linarith\n    have h39\u2086 : f 85 = 3559 := by\n      norm_num at h39\u2084 \u22a2\n      <;> linarith\n    have h39\u2087 : f 86 = 7396 - 3559 := by\n      have h39\u2088 := h0 86\n      norm_num at h39\u2088 \u22a2\n      <;> linarith\n    have h39\u2089 : f 86 = 3837 := by\n      norm_num at h39\u2087 \u22a2\n      <;> linarith\n    have h40\u2080 : f 87 = 7569 - 3837 := by\n      have h40\u2081 := h0 87\n      norm_num at h40\u2081 \u22a2\n      <;> linarith\n    have h40\u2082 : f 87 = 3732 := by\n      norm_num at h40\u2080 \u22a2\n      <;> linarith\n    have h40\u2083 : f 88 = 7744 - 3732 := by\n      have h40\u2084 := h0 88\n      norm_num at h40\u2084 \u22a2\n      <;> linarith\n    have h40\u2085 : f 88 = 4012 := by\n      norm_num at h40\u2083 \u22a2\n      <;> linarith\n    have h40\u2086 : f 89 = 7921 - 4012 := by\n      have h40\u2087 := h0 89\n      norm_num at h40\u2087 \u22a2\n      <;> linarith\n    have h40\u2088 : f 89 = 3909 := by\n      norm_num at h40\u2086 \u22a2\n      <;> linarith\n    have h40\u2089 : f 90 = 8100 - 3909 := by\n      have h41\u2080 := h0 90\n      norm_num at h41\u2080 \u22a2\n      <;> linarith\n    have h41\u2081 : f 90 = 4191 := by\n      norm_num at h40\u2089 \u22a2\n      <;> linarith\n    have h41\u2082 : f 91 = 8281 - 4191 := by\n      have h41\u2083 := h0 91\n      norm_num at h41\u2083 \u22a2\n      <;> linarith\n    have h41\u2084 : f 91 = 4090 := by\n      norm_num at h41\u2082 \u22a2\n      <;> linarith\n    have h41\u2085 : f 92 = 8464 - 4090 := by\n      have h41\u2086 := h0 92\n      norm_num at h41\u2086 \u22a2\n      <;> linarith\n    have h41\u2087 : f 92 = 4374 := by\n      norm_num at h41\u2085 \u22a2\n      <;> linarith\n    have h41\u2088 : f 93 = 8649 - 4374 := by\n      have h41\u2089 := h0 93\n      norm_num at h41\u2089 \u22a2\n      <;> linarith\n    have h42\u2080 : f 93 = 4275 := by\n      norm_num at h41\u2088 \u22a2\n      <;> linarith\n    have h42\u2081 : f 94 = 8836 - 4275 := by\n      have h42\u2082 := h0 94\n      norm_num at h42\u2082 \u22a2\n      <;> linarith\n    have h42\u2083 : f 94 = 4561 := by\n      norm_num at h42\u2081 \u22a2\n      <;> linarith\n    exact h42\u2083\n  have h33 : f 94 % 1000 = 561 := by\n    have h33\u2081 : f 94 = 4561 := h32\n    have h33\u2082 : f 94 % 1000 = 561 := by\n      norm_num [h33\u2081]\n      <;> rfl\n    exact h33\u2082\n  exact h33\n", "proof_before": "theorem aime_1994_p3 (f : \u2124 \u2192 \u2124) (h0 : \u2200 x, f x + f (x - 1) = x ^ 2) (h1 : f 19 = 94) :\n    f 94 % 1000 = 561  :=  by\n  have h2 : f 18 = 267 := by\n    have h2\u2081 := h0 19\n    have h2\u2082 : f 19 + f 18 = 361 := by\n      norm_num at h2\u2081 \u22a2\n      <;> linarith\n    have h2\u2083 : f 18 = 267 := by\n      linarith\n    exact h2\u2083\n  have h3 : f 17 = 57 := by\n    have h3\u2081 : f 18 + f 17 = 324 := by\n      have h3\u2082 := h0 18\n      norm_num at h3\u2082 \u22a2\n      <;> linarith\n    have h3\u2083 : f 17 = 57 := by\n      linarith\n    exact h3\u2083\n  have h4 : f 16 = 232 := by\n    have h4\u2081 : f 17 + f 16 = 289 := by\n      have h4\u2082 := h0 17\n      norm_num at h4\u2082 \u22a2\n      <;> linarith\n    have h4\u2083 : f 16 = 232 := by\n      linarith\n    exact h4\u2083\n  have h5 : f 15 = 24 := by\n    have h5\u2081 : f 16 + f 15 = 256 := by\n      have h5\u2082 := h0 16\n      norm_num at h5\u2082 \u22a2\n      <;> linarith\n    have h5\u2083 : f 15 = 24 := by\n      linarith\n    exact h5\u2083\n  have h6 : f 14 = 201 := by\n    have h6\u2081 : f 15 + f 14 = 225 := by\n      have h6\u2082 := h0 15\n      norm_num at h6\u2082 \u22a2\n      <;> linarith\n    have h6\u2083 : f 14 = 201 := by\n      linarith\n    exact h6\u2083\n  have h7 : f 13 = (-5) := by\n    have h7\u2081 : f 14 + f 13 = 196 := by\n      have h7\u2082 := h0 14\n      norm_num at h7\u2082 \u22a2\n      <;> linarith\n    have h7\u2083 : f 13 = -5 := by\n      linarith\n    exact h7\u2083\n  have h8 : f 12 = 174 := by\n    have h8\u2081 : f 13 + f 12 = 169 := by\n      have h8\u2082 := h0 13\n      norm_num at h8\u2082 \u22a2\n      <;> linarith\n    have h8\u2083 : f 12 = 174 := by\n      linarith\n    exact h8\u2083\n  have h9 : f 11 = (-30) := by\n    have h9\u2081 : f 12 + f 11 = 144 := by\n      have h9\u2082 := h0 12\n      norm_num at h9\u2082 \u22a2\n      <;> linarith\n    have h9\u2083 : f 11 = -30 := by\n      linarith\n    exact h9\u2083\n  have h10 : f 10 = 151 := by\n    have h10\u2081 : f 11 + f 10 = 121 := by\n      have h10\u2082 := h0 11\n      norm_num at h10\u2082 \u22a2\n      <;> linarith\n    have h10\u2083 : f 10 = 151 := by\n      linarith\n    exact h10\u2083\n  have h11 : f 9 = (-51) := by\n    have h11\u2081 : f 10 + f 9 = 100 := by\n      have h11\u2082 := h0 10\n      norm_num at h11\u2082 \u22a2\n      <;> linarith\n    have h11\u2083 : f 9 = -51 := by\n      linarith\n    exact h11\u2083\n  have h12 : f 8 = 132 := by\n    have h12\u2081 : f 9 + f 8 = 81 := by\n      have h12\u2082 := h0 9\n      norm_num at h12\u2082 \u22a2\n      <;> linarith\n    have h12\u2083 : f 8 = 132 := by\n      linarith\n    exact h12\u2083\n  have h13 : f 7 = (-68) := by\n    have h13\u2081 : f 8 + f 7 = 64 := by\n      have h13\u2082 := h0 8\n      norm_num at h13\u2082 \u22a2\n      <;> linarith\n    have h13\u2083 : f 7 = -68 := by\n      linarith\n    exact h13\u2083\n  have h14 : f 6 = 117 := by\n    have h14\u2081 : f 7 + f 6 = 49 := by\n      have h14\u2082 := h0 7\n      norm_num at h14\u2082 \u22a2\n      <;> linarith\n    have h14\u2083 : f 6 = 117 := by\n      linarith\n    exact h14\u2083\n  have h15 : f 5 = (-81) := by\n    have h15\u2081 : f 6 + f 5 = 36 := by\n      have h15\u2082 := h0 6\n      norm_num at h15\u2082 \u22a2\n      <;> linarith\n    have h15\u2083 : f 5 = -81 := by\n      linarith\n    exact h15\u2083\n  have h16 : f 4 = 106 := by\n    have h16\u2081 : f 5 + f 4 = 25 := by\n      have h16\u2082 := h0 5\n      norm_num at h16\u2082 \u22a2\n      <;> linarith\n    have h16\u2083 : f 4 = 106 := by\n      linarith\n    exact h16\u2083\n  have h17 : f 3 = (-90) := by\n    have h17\u2081 : f 4 + f 3 = 16 := by\n      have h17\u2082 := h0 4\n      norm_num at h17\u2082 \u22a2\n      <;> linarith\n    have h17\u2083 : f 3 = -90 := by\n      linarith\n    exact h17\u2083\n  have h18 : f 2 = 99 := by\n    have h18\u2081 : f 3 + f 2 = 9 := by\n      have h18\u2082 := h0 3\n      norm_num at h18\u2082 \u22a2\n      <;> linarith\n    have h18\u2083 : f 2 = 99 := by\n      linarith\n    exact h18\u2083\n  have h19 : f 1 = (-95) := by\n    have h19\u2081 : f 2 + f 1 = 4 := by\n      have h19\u2082 := h0 2\n      norm_num at h19\u2082 \u22a2\n      <;> linarith\n    have h19\u2083 : f 1 = -95 := by\n      linarith\n    exact h19\u2083\n  have h20 : f 0 = 96 := by\n    have h20\u2081 : f 1 + f 0 = 1 := by\n      have h20\u2082 := h0 1\n      norm_num at h20\u2082 \u22a2\n      <;> linarith\n    have h20\u2083 : f 0 = 96 := by\n      linarith\n    exact h20\u2083\n  have h21 : f 20 = 306 := by\n    have h21\u2081 : f 20 + f 19 = 400 := by\n      have h21\u2082 := h0 20\n      norm_num at h21\u2082 \u22a2\n      <;> linarith\n    have h21\u2083 : f 20 = 306 := by\n      linarith\n    exact h21\u2083\n  have h22 : f 21 = 135 := by\n    have h22\u2081 : f 21 + f 20 = 441 := by\n      have h22\u2082 := h0 21\n      norm_num at h22\u2082 \u22a2\n      <;> linarith\n    have h22\u2083 : f 21 = 135 := by\n      linarith\n    exact h22\u2083\n  have h23 : f 22 = 349 := by\n    have h23\u2081 : f 22 + f 21 = 484 := by\n      have h23\u2082 := h0 22\n      norm_num at h23\u2082 \u22a2\n      <;> linarith\n    have h23\u2083 : f 22 = 349 := by\n      linarith\n    exact h23\u2083\n  have h24 : f 23 = 180 := by\n    have h24\u2081 : f 23 + f 22 = 529 := by\n      have h24\u2082 := h0 23\n      norm_num at h24\u2082 \u22a2\n      <;> linarith\n    have h24\u2083 : f 23 = 180 := by\n      linarith\n    exact h24\u2083\n  have h25 : f 24 = 396 := by\n    have h25\u2081 : f 24 + f 23 = 576 := by\n      have h25\u2082 := h0 24\n      norm_num at h25\u2082 \u22a2\n      <;> linarith\n    have h25\u2083 : f 24 = 396 := by\n      linarith\n    exact h25\u2083\n  have h26 : f 25 = 229 := by\n    have h26\u2081 : f 25 + f 24 = 625 := by\n      have h26\u2082 := h0 25\n      norm_num at h26\u2082 \u22a2\n      <;> linarith\n    have h26\u2083 : f 25 = 229 := by\n      linarith\n    exact h26\u2083\n  have h27 : f 26 = 447 := by\n    have h27\u2081 : f 26 + f 25 = 676 := by\n      have h27\u2082 := h0 26\n      norm_num at h27\u2082 \u22a2\n      <;> linarith\n    have h27\u2083 : f 26 = 447 := by\n      linarith\n    exact h27\u2083\n  have h28 : f 27 = 282 := by\n    have h28\u2081 : f 27 + f 26 = 729 := by\n      have h28\u2082 := h0 27\n      norm_num at h28\u2082 \u22a2\n      <;> linarith\n    have h28\u2083 : f 27 = 282 := by\n      linarith\n    exact h28\u2083\n  have h29 : f 28 = 502 := by\n    have h29\u2081 : f 28 + f 27 = 784 := by\n      have h29\u2082 := h0 28\n      norm_num at h29\u2082 \u22a2\n      <;> linarith\n    have h29\u2083 : f 28 = 502 := by\n      linarith\n    exact h29\u2083\n  have h30 : f 29 = 339 := by\n    have h30\u2081 : f 29 + f 28 = 841 := by\n      have h30\u2082 := h0 29\n      norm_num at h30\u2082 \u22a2\n      <;> linarith\n    have h30\u2083 : f 29 = 339 := by\n      linarith\n    exact h30\u2083\n  have h31 : f 30 = 561 := by\n    have h31\u2081 : f 30 + f 29 = 900 := by\n      have h31\u2082 := h0 30\n      norm_num at h31\u2082 \u22a2\n      <;> linarith\n    have h31\u2083 : f 30 = 561 := by\n      linarith\n    exact h31\u2083\n  have h32 : f 94 = 4561 := by\n    have h32\u2081 : f 30 = 561 := h31\n    have h32\u2082 : f 31 = 961 - 561 := by\n      have h32\u2083 := h0 31\n      norm_num at h32\u2083 \u22a2\n      <;> linarith\n    have h32\u2084 : f 31 = 400 := by\n      norm_num at h32\u2082 \u22a2\n      <;> linarith\n    have h32\u2085 : f 32 = 1024 - 400 := by\n      have h32\u2086 := h0 32\n      norm_num at h32\u2086 \u22a2\n      <;> linarith\n    have h32\u2087 : f 32 = 624 := by\n      norm_num at h32\u2085 \u22a2\n      <;> linarith\n    have h32\u2088 : f 33 = 1089 - 624 := by\n      have h32\u2089 := h0 33\n      norm_num at h32\u2089 \u22a2\n      <;> linarith\n    have h32\u2081\u2080 : f 33 = 465 := by\n      norm_num at h32\u2088 \u22a2\n      <;> linarith\n    have h32\u2081\u2081 : f 34 = 1156 - 465 := by\n      have h32\u2081\u2082 := h0 34\n      norm_num at h32\u2081\u2082 \u22a2\n      <;> linarith\n    have h32\u2081\u2083 : f 34 = 691 := by\n      norm_num at h32\u2081\u2081 \u22a2\n      <;> linarith\n    have h32\u2081\u2084 : f 35 = 1225 - 691 := by\n      have h32\u2081\u2085 := h0 35\n      norm_num at h32\u2081\u2085 \u22a2\n      <;> linarith\n    have h32\u2081\u2086 : f 35 = 534 := by\n      norm_num at h32\u2081\u2084 \u22a2\n      <;> linarith\n    have h32\u2081\u2087 : f 36 = 1296 - 534 := by\n      have h32\u2081\u2088 := h0 36\n      norm_num at h32\u2081\u2088 \u22a2\n      <;> linarith\n    have h32\u2081\u2089 : f 36 = 762 := by\n      norm_num at h32\u2081\u2087 \u22a2\n      <;> linarith\n    have h32\u2082\u2080 : f 37 = 1369 - 762 := by\n      have h32\u2082\u2081 := h0 37\n      norm_num at h32\u2082\u2081 \u22a2\n      <;> linarith\n    have h32\u2082\u2082 : f 37 = 607 := by\n      norm_num at h32\u2082\u2080 \u22a2\n      <;> linarith\n    have h32\u2082\u2083 : f 38 = 1444 - 607 := by\n      have h32\u2082\u2084 := h0 38\n      norm_num at h32\u2082\u2084 \u22a2\n      <;> linarith\n    have h32\u2082\u2085 : f 38 = 837 := by\n      norm_num at h32\u2082\u2083 \u22a2\n      <;> linarith\n    have h32\u2082\u2086 : f 39 = 1521 - 837 := by\n      have h32\u2082\u2087 := h0 39\n      norm_num at h32\u2082\u2087 \u22a2\n      <;> linarith\n    have h32\u2082\u2088 : f 39 = 684 := by\n      norm_num at h32\u2082\u2086 \u22a2\n      <;> linarith\n    have h32\u2082\u2089 : f 40 = 1600 - 684 := by\n      have h32\u2083\u2080 := h0 40\n      norm_num at h32\u2083\u2080 \u22a2\n      <;> linarith\n    have h32\u2083\u2081 : f 40 = 916 := by\n      norm_num at h32\u2082\u2089 \u22a2\n      <;> linarith\n    have h32\u2083\u2082 : f 41 = 1681 - 916 := by\n      have h32\u2083\u2083 := h0 41\n      norm_num at h32\u2083\u2083 \u22a2\n      <;> linarith\n    have h32\u2083\u2084 : f 41 = 765 := by\n      norm_num at h32\u2083\u2082 \u22a2\n      <;> linarith\n    have h32\u2083\u2085 : f 42 = 1764 - 765 := by\n      have h32\u2083\u2086 := h0 42\n      norm_num at h32\u2083\u2086 \u22a2\n      <;> linarith\n    have h32\u2083\u2087 : f 42 = 999 := by\n      norm_num at h32\u2083\u2085 \u22a2\n      <;> linarith\n    have h32\u2083\u2088 : f 43 = 1849 - 999 := by\n      have h32\u2083\u2089 := h0 43\n      norm_num at h32\u2083\u2089 \u22a2\n      <;> linarith\n    have h32\u2084\u2080 : f 43 = 850 := by\n      norm_num at h32\u2083\u2088 \u22a2\n      <;> linarith\n    have h32\u2084\u2081 : f 44 = 1936 - 850 := by\n      have h32\u2084\u2082 := h0 44\n      norm_num at h32\u2084\u2082 \u22a2\n      <;> linarith\n    have h32\u2084\u2083 : f 44 = 1086 := by\n      norm_num at h32\u2084\u2081 \u22a2\n      <;> linarith\n    have h32\u2084\u2084 : f 45 = 2025 - 1086 := by\n      have h32\u2084\u2085 := h0 45\n      norm_num at h32\u2084\u2085 \u22a2\n      <;> linarith\n    have h32\u2084\u2086 : f 45 = 939 := by\n      norm_num at h32\u2084\u2084 \u22a2\n      <;> linarith\n    have h32\u2084\u2087 : f 46 = 2116 - 939 := by\n      have h32\u2084\u2088 := h0 46\n      norm_num at h32\u2084\u2088 \u22a2\n      <;> linarith\n    have h32\u2084\u2089 : f 46 = 1177 := by\n      norm_num at h32\u2084\u2087 \u22a2\n      <;> linarith\n    have h32\u2085\u2080 : f 47 = 2209 - 1177 := by\n      have h32\u2085\u2081 := h0 47\n      norm_num at h32\u2085\u2081 \u22a2\n      <;> linarith\n    have h32\u2085\u2082 : f 47 = 1032 := by\n      norm_num at h32\u2085\u2080 \u22a2\n      <;> linarith\n    have h32\u2085\u2083 : f 48 = 2304 - 1032 := by\n      have h32\u2085\u2084 := h0 48\n      norm_num at h32\u2085\u2084 \u22a2\n      <;> linarith\n    have h32\u2085\u2085 : f 48 = 1272 := by\n      norm_num at h32\u2085\u2083 \u22a2\n      <;> linarith\n    have h32\u2085\u2086 : f 49 = 2401 - 1272 := by\n      have h32\u2085\u2087 := h0 49\n      norm_num at h32\u2085\u2087 \u22a2\n      <;> linarith\n    have h32\u2085\u2088 : f 49 = 1129 := by\n      norm_num at h32\u2085\u2086 \u22a2\n      <;> linarith\n    have h32\u2085\u2089 : f 50 = 2500 - 1129 := by\n      have h32\u2086\u2080 := h0 50\n      norm_num at h32\u2086\u2080 \u22a2\n      <;> linarith\n    have h32\u2086\u2081 : f 50 = 1371 := by\n      norm_num at h32\u2085\u2089 \u22a2\n      <;> linarith\n    have h32\u2086\u2082 : f 51 = 2601 - 1371 := by\n      have h32\u2086\u2083 := h0 51\n      norm_num at h32\u2086\u2083 \u22a2\n      <;> linarith\n    have h32\u2086\u2084 : f 51 = 1230 := by\n      norm_num at h32\u2086\u2082 \u22a2\n      <;> linarith\n    have h32\u2086\u2085 : f 52 = 2704 - 1230 := by\n      have h32\u2086\u2086 := h0 52\n      norm_num at h32\u2086\u2086 \u22a2\n      <;> linarith\n    have h32\u2086\u2087 : f 52 = 1474 := by\n      norm_num at h32\u2086\u2085 \u22a2\n      <;> linarith\n    have h32\u2086\u2088 : f 53 = 2809 - 1474 := by\n      have h32\u2086\u2089 := h0 53\n      norm_num at h32\u2086\u2089 \u22a2\n      <;> linarith\n    have h32\u2087\u2080 : f 53 = 1335 := by\n      norm_num at h32\u2086\u2088 \u22a2\n      <;> linarith\n    have h32\u2087\u2081 : f 54 = 2916 - 1335 := by\n      have h32\u2087\u2082 := h0 54\n      norm_num at h32\u2087\u2082 \u22a2\n      <;> linarith\n    have h32\u2087\u2083 : f 54 = 1581 := by\n      norm_num at h32\u2087\u2081 \u22a2\n      <;> linarith\n    have h32\u2087\u2084 : f 55 = 3025 - 1581 := by\n      have h32\u2087\u2085 := h0 55\n      norm_num at h32\u2087\u2085 \u22a2\n      <;> linarith\n    have h32\u2087\u2086 : f 55 = 1444 := by\n      norm_num at h32\u2087\u2084 \u22a2\n      <;> linarith\n    have h32\u2087\u2087 : f 56 = 3136 - 1444 := by\n      have h32\u2087\u2088 := h0 56\n      norm_num at h32\u2087\u2088 \u22a2\n      <;> linarith\n    have h32\u2087\u2089 : f 56 = 1692 := by\n      norm_num at h32\u2087\u2087 \u22a2\n      <;> linarith\n    have h32\u2088\u2080 : f 57 = 3249 - 1692 := by\n      have h32\u2088\u2081 := h0 57\n      norm_num at h32\u2088\u2081 \u22a2\n      <;> linarith\n    have h32\u2088\u2082 : f 57 = 1557 := by\n      norm_num at h32\u2088\u2080 \u22a2\n      <;> linarith\n    have h32\u2088\u2083 : f 58 = 3364 - 1557 := by\n      have h32\u2088\u2084 := h0 58\n      norm_num at h32\u2088\u2084 \u22a2\n      <;> linarith\n    have h32\u2088\u2085 : f 58 = 1807 := by\n      norm_num at h32\u2088\u2083 \u22a2\n      <;> linarith\n    have h32\u2088\u2086 : f 59 = 3481 - 1807 := by\n      have h32\u2088\u2087 := h0 59\n      norm_num at h32\u2088\u2087 \u22a2\n      <;> linarith\n    have h32\u2088\u2088 : f 59 = 1674 := by\n      norm_num at h32\u2088\u2086 \u22a2\n      <;> linarith\n    have h32\u2088\u2089 : f 60 = 3600 - 1674 := by\n      have h32\u2089\u2080 := h0 60\n      norm_num at h32\u2089\u2080 \u22a2\n      <;> linarith\n    have h32\u2089\u2081 : f 60 = 1926 := by\n      norm_num at h32\u2088\u2089 \u22a2\n      <;> linarith\n    have h32\u2089\u2082 : f 61 = 3721 - 1926 := by\n      have h32\u2089\u2083 := h0 61\n      norm_num at h32\u2089\u2083 \u22a2\n      <;> linarith\n    have h32\u2089\u2084 : f 61 = 1795 := by\n      norm_num at h32\u2089\u2082 \u22a2\n      <;> linarith\n    have h32\u2089\u2085 : f 62 = 3844 - 1795 := by\n      have h32\u2089\u2086 := h0 62\n      norm_num at h32\u2089\u2086 \u22a2\n      <;> linarith\n    have h32\u2089\u2087 : f 62 = 2049 := by\n      norm_num at h32\u2089\u2085 \u22a2\n      <;> linarith\n    have h32\u2089\u2088 : f 63 = 3969 - 2049 := by\n      have h32\u2089\u2089 := h0 63\n      norm_num at h32\u2089\u2089 \u22a2\n      <;> linarith\n    have h33\u2080 : f 63 = 1920 := by\n      norm_num at h32\u2089\u2088 \u22a2\n      <;> linarith\n    have h33\u2081 : f 64 = 4096 - 1920 := by\n      have h33\u2082 := h0 64\n      norm_num at h33\u2082 \u22a2\n      <;> linarith\n    have h33\u2083 : f 64 = 2176 := by\n      norm_num at h33\u2081 \u22a2\n      <;> linarith\n    have h33\u2084 : f 65 = 4225 - 2176 := by\n      have h33\u2085 := h0 65\n      norm_num at h33\u2085 \u22a2\n      <;> linarith\n    have h33\u2086 : f 65 = 2049 := by\n      norm_num at h33\u2084 \u22a2\n      <;> linarith\n    have h33\u2087 : f 66 = 4356 - 2049 := by\n      have h33\u2088 := h0 66\n      norm_num at h33\u2088 \u22a2\n      <;> linarith\n    have h33\u2089 : f 66 = 2307 := by\n      norm_num at h33\u2087 \u22a2\n      <;> linarith\n    have h34\u2080 : f 67 = 4489 - 2307 := by\n      have h34\u2081 := h0 67\n      norm_num at h34\u2081 \u22a2\n      <;> linarith\n    have h34\u2082 : f 67 = 2182 := by\n      norm_num at h34\u2080 \u22a2\n      <;> linarith\n    have h34\u2083 : f 68 = 4624 - 2182 := by\n      have h34\u2084 := h0 68\n      norm_num at h34\u2084 \u22a2\n      <;> linarith\n    have h34\u2085 : f 68 = 2442 := by\n      norm_num at h34\u2083 \u22a2\n      <;> linarith\n    have h34\u2086 : f 69 = 4761 - 2442 := by\n      have h34\u2087 := h0 69\n      norm_num at h34\u2087 \u22a2\n      <;> linarith\n    have h34\u2088 : f 69 = 2319 := by\n      norm_num at h34\u2086 \u22a2\n      <;> linarith\n    have h34\u2089 : f 70 = 4900 - 2319 := by\n      have h35\u2080 := h0 70\n      norm_num at h35\u2080 \u22a2\n      <;> linarith\n    have h35\u2081 : f 70 = 2581 := by\n      norm_num at h34\u2089 \u22a2\n      <;> linarith\n    have h35\u2082 : f 71 = 5041 - 2581 := by\n      have h35\u2083 := h0 71\n      norm_num at h35\u2083 \u22a2\n      <;> linarith\n    have h35\u2084 : f 71 = 2460 := by\n      norm_num at h35\u2082 \u22a2\n      <;> linarith\n    have h35\u2085 : f 72 = 5184 - 2460 := by\n      have h35\u2086 := h0 72\n      norm_num at h35\u2086 \u22a2\n      <;> linarith\n    have h35\u2087 : f 72 = 2724 := by\n      norm_num at h35\u2085 \u22a2\n      <;> linarith\n    have h35\u2088 : f 73 = 5329 - 2724 := by\n      have h35\u2089 := h0 73\n      norm_num at h35\u2089 \u22a2\n      <;> linarith\n    have h36\u2080 : f 73 = 2605 := by\n      norm_num at h35\u2088 \u22a2\n      <;> linarith\n    have h36\u2081 : f 74 = 5476 - 2605 := by\n      have h36\u2082 := h0 74\n      norm_num at h36\u2082 \u22a2\n      <;> linarith\n    have h36\u2083 : f 74 = 2871 := by\n      norm_num at h36\u2081 \u22a2\n      <;> linarith\n    have h36\u2084 : f 75 = 5625 - 2871 := by\n      have h36\u2085 := h0 75\n      norm_num at h36\u2085 \u22a2\n      <;> linarith\n    have h36\u2086 : f 75 = 2754 := by\n      norm_num at h36\u2084 \u22a2\n      <;> linarith\n    have h36\u2087 : f 76 = 5776 - 2754 := by\n      have h36\u2088 := h0 76\n      norm_num at h36\u2088 \u22a2\n      <;> linarith\n    have h36\u2089 : f 76 = 3022 := by\n      norm_num at h36\u2087 \u22a2\n      <;> linarith\n    have h37\u2080 : f 77 = 5929 - 3022 := by\n      have h37\u2081 := h0 77\n      norm_num at h37\u2081 \u22a2\n      <;> linarith\n    have h37\u2082 : f 77 = 2907 := by\n      norm_num at h37\u2080 \u22a2\n      <;> linarith\n    have h37\u2083 : f 78 = 6084 - 2907 := by\n      have h37\u2084 := h0 78\n      norm_num at h37\u2084 \u22a2\n      <;> linarith\n    have h37\u2085 : f 78 = 3177 := by\n      norm_num at h37\u2083 \u22a2\n      <;> linarith\n    have h37\u2086 : f 79 = 6241 - 3177 := by\n      have h37\u2087 := h0 79\n      norm_num at h37\u2087 \u22a2\n      <;> linarith\n    have h37\u2088 : f 79 = 3064 := by\n      norm_num at h37\u2086 \u22a2\n      <;> linarith\n    have h37\u2089 : f 80 = 6400 - 3064 := by\n      have h38\u2080 := h0 80\n      norm_num at h38\u2080 \u22a2\n      <;> linarith\n    have h38\u2081 : f 80 = 3336 := by\n      norm_num at h37\u2089 \u22a2\n      <;> linarith\n    have h38\u2082 : f 81 = 6561 - 3336 := by\n      have h38\u2083 := h0 81\n      norm_num at h38\u2083 \u22a2\n      <;> linarith\n    have h38\u2084 : f 81 = 3225 := by\n      norm_num at h38\u2082 \u22a2\n      <;> linarith\n    have h38\u2085 : f 82 = 6724 - 3225 := by\n      have h38\u2086 := h0 82\n      norm_num at h38\u2086 \u22a2\n      <;> linarith\n    have h38\u2087 : f 82 = 3499 := by\n      norm_num at h38\u2085 \u22a2\n      <;> linarith\n    have h38\u2088 : f 83 = 6889 - 3499 := by\n      have h38\u2089 := h0 83\n      norm_num at h38\u2089 \u22a2\n      <;> linarith\n    have h39\u2080 : f 83 = 3390 := by\n      norm_num at h38\u2088 \u22a2\n      <;> linarith\n    have h39\u2081 : f 84 = 7056 - 3390 := by\n      have h39\u2082 := h0 84\n      norm_num at h39\u2082 \u22a2\n      <;> linarith\n    have h39\u2083 : f 84 = 3666 := by\n      norm_num at h39\u2081 \u22a2\n      <;> linarith\n    have h39\u2084 : f 85 = 7225 - 3666 := by\n      have h39\u2085 := h0 85\n      norm_num at h39\u2085 \u22a2\n      <;> linarith\n    have h39\u2086 : f 85 = 3559 := by\n      norm_num at h39\u2084 \u22a2\n      <;> linarith\n    have h39\u2087 : f 86 = 7396 - 3559 := by\n      have h39\u2088 := h0 86\n      norm_num at h39\u2088 \u22a2\n      <;> linarith\n    have h39\u2089 : f 86 = 3837 := by\n      norm_num at h39\u2087 \u22a2\n      <;> linarith\n    have h40\u2080 : f 87 = 7569 - 3837 := by\n      have h40\u2081 := h0 87\n      norm_num at h40\u2081 \u22a2\n      <;> linarith\n    have h40\u2082 : f 87 = 3732 := by\n      norm_num at h40\u2080 \u22a2\n      <;> linarith\n    have h40\u2083 : f 88 = 7744 - 3732 := by\n      have h40\u2084 := h0 88\n      norm_num at h40\u2084 \u22a2\n      <;> linarith\n    have h40\u2085 : f 88 = 4012 := by\n      norm_num at h40\u2083 \u22a2\n      <;> linarith\n    have h40\u2086 : f 89 = 7921 - 4012 := by\n      have h40\u2087 := h0 89\n      norm_num at h40\u2087 \u22a2\n      <;> linarith\n    have h40\u2088 : f 89 = 3909 := by\n      norm_num at h40\u2086 \u22a2\n      <;> linarith\n    have h40\u2089 : f 90 = 8100 - 3909 := by\n      have h41\u2080 := h0 90\n      norm_num at h41\u2080 \u22a2\n      <;> linarith\n    have h41\u2081 : f 90 = 4191 := by\n      norm_num at h40\u2089 \u22a2\n      <;> linarith\n    have h41\u2082 : f 91 = 8281 - 4191 := by\n      have h41\u2083 := h0 91\n      norm_num at h41\u2083 \u22a2\n      <;> linarith\n    have h41\u2084 : f 91 = 4090 := by\n      norm_num at h41\u2082 \u22a2\n      <;> linarith\n    have h41\u2085 : f 92 = 8464 - 4090 := by\n      have h41\u2086 := h0 92\n      norm_num at h41\u2086 \u22a2\n      <;> linarith\n    have h41\u2087 : f 92 = 4374 := by\n      norm_num at h41\u2085 \u22a2\n      <;> linarith\n    have h41\u2088 : f 93 = 8649 - 4374 := by\n      have h41\u2089 := h0 93\n      norm_num at h41\u2089 \u22a2\n      <;> linarith\n    have h42\u2080 : f 93 = 4275 := by\n      norm_num at h41\u2088 \u22a2\n      <;> linarith\n    have h42\u2081 : f 94 = 8836 - 4275 := by\n      have h42\u2082 := h0 94\n      norm_num at h42\u2082 \u22a2\n      <;> linarith\n    have h42\u2083 : f 94 = 4561 := by\n      norm_num at h42\u2081 \u22a2\n      <;> linarith\n    exact h42\u2083\n  have h33 : f 94 % 1000 = 561 := by\n    have h33\u2081 : f 94 = 4561 := h32\n    have h33\u2082 : f 94 % 1000 = 561 := by\n      norm_num [h33\u2081]\n      <;> rfl\n    exact h33\u2082\n  exact h33\n"}
{"mode": "clear", "steps_before": 25, "alphabet": 20, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 832, "chars_after": 36, "seconds": 14.3, "id": "mathd_numbertheory_341", "proof_after": "theorem mathd_numbertheory_341 (a b c : \u2115) (h\u2080 : a \u2264 9 \u2227 b \u2264 9 \u2227 c \u2264 9)\n    (h\u2081 : Nat.digits 10 (5 ^ 100 % 1000) = [c, b, a]) : a + b + c = 13  :=  by\n  norm_num [Nat.Prime] at *\n  omega\n", "proof_before": "theorem mathd_numbertheory_341 (a b c : \u2115) (h\u2080 : a \u2264 9 \u2227 b \u2264 9 \u2227 c \u2264 9)\n    (h\u2081 : Nat.digits 10 (5 ^ 100 % 1000) = [c, b, a]) : a + b + c = 13  :=  by\n  have h\u2082 : 5 ^ 100 % 1000 = 625 := by\n    norm_num [Nat.pow_mod, Nat.mod_mod, Nat.mod_eq_of_lt, Nat.lt_of_le_of_lt (Nat.zero_le _) (by norm_num : 1000 < 5 ^ 10)]\n    <;> rfl\n  have h\u2083 : Nat.digits 10 (5 ^ 100 % 1000) = [5, 2, 6] := by\n    rw [h\u2082]\n    <;> norm_num [Nat.digits_zero, Nat.div_eq_of_lt]\n    <;> rfl\n  have h\u2084 : [c, b, a] = [5, 2, 6] := by\n    rw [h\u2083] at h\u2081\n    rw [\u2190 h\u2081]\n    <;> simp [h\u2082]\n    <;> aesop\n  have h\u2085 : a = 6 := by\n    simp [List.cons, List.append, List.mem_cons, List.mem_singleton] at h\u2084\n    <;> omega\n  have h\u2086 : b = 2 := by\n    simp [List.cons, List.append, List.mem_cons, List.mem_singleton] at h\u2084\n    <;> omega\n  have h\u2087 : c = 5 := by\n    simp [List.cons, List.append, List.mem_cons, List.mem_singleton] at h\u2084\n    <;> omega\n  have h\u2088 : a + b + c = 13 := by\n    subst_vars\n    <;> norm_num\n  apply h\u2088\n"}
{"mode": "clear", "steps_before": 50, "alphabet": 21, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 1294, "chars_after": 6, "seconds": 1.0, "id": "mathd_numbertheory_12", "proof_after": "theorem mathd_numbertheory_12 :\n    Finset.card (Finset.filter (fun x => 20 \u2223 x) (Finset.Icc 15 85)) = 4  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_12 :\n    Finset.card (Finset.filter (fun x => 20 \u2223 x) (Finset.Icc 15 85)) = 4  :=  by\n  have h_main : Finset.filter (fun x => 20 \u2223 x) (Finset.Icc 15 85) = {20, 40, 60, 80} := by\n    apply Finset.ext\n    intro x\n    simp only [Finset.mem_filter, Finset.mem_Icc, Finset.mem_insert, Finset.mem_singleton]\n    constructor\n    \u00b7 \n      intro h\n      have h\u2081 : 15 \u2264 x \u2227 x \u2264 85 := by tauto\n      have h\u2082 : 20 \u2223 x := by tauto\n      have h\u2083 : x = 20 \u2228 x = 40 \u2228 x = 60 \u2228 x = 80 := by\n        have h\u2084 : x \u2264 85 := by linarith\n        have h\u2085 : 15 \u2264 x := by linarith\n        have h\u2086 : 20 \u2223 x := by tauto\n        have h\u2087 : x = 20 \u2228 x = 40 \u2228 x = 60 \u2228 x = 80 := by\n          have h\u2088 : x % 20 = 0 := by\n            omega\n          have h\u2089 : x \u2264 85 := by linarith\n          have h\u2081\u2080 : 15 \u2264 x := by linarith\n          have h\u2081\u2081 : x = 20 \u2228 x = 40 \u2228 x = 60 \u2228 x = 80 := by\n            omega\n          exact h\u2081\u2081\n        exact h\u2087\n      tauto\n    \u00b7 \n      intro h\n      have h\u2081 : x = 20 \u2228 x = 40 \u2228 x = 60 \u2228 x = 80 := by tauto\n      have h\u2082 : 15 \u2264 x \u2227 x \u2264 85 \u2227 20 \u2223 x := by\n        rcases h\u2081 with (rfl | rfl | rfl | rfl)\n        \u00b7 \n          norm_num\n        \u00b7 \n          norm_num\n        \u00b7 \n          norm_num\n        \u00b7 \n          norm_num\n      tauto\n  have h_card : Finset.card (Finset.filter (fun x => 20 \u2223 x) (Finset.Icc 15 85)) = 4 := by\n    rw [h_main]\n    <;> norm_num\n    <;> decide\n  apply h_card\n"}
{"mode": "clear", "steps_before": 24, "alphabet": 17, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 601, "chars_after": 11, "seconds": 1.2, "id": "mathd_algebra_125", "proof_after": "theorem mathd_algebra_125 (x y : \u2115) (h\u2080 : 0 < x \u2227 0 < y) (h\u2081 : 5 * x = y)\n    (h\u2082 : \u2191x - (3 : \u2124) + (y - (3 : \u2124)) = 30) : x = 6  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_125 (x y : \u2115) (h\u2080 : 0 < x \u2227 0 < y) (h\u2081 : 5 * x = y)\n    (h\u2082 : \u2191x - (3 : \u2124) + (y - (3 : \u2124)) = 30) : x = 6  :=  by\n  have h\u2083 : (x : \u2124) + y = 36 := by\n    have h\u2083\u2081 : (x : \u2124) - (3 : \u2124) + ((y : \u2124) - (3 : \u2124)) = 30 := by simpa using h\u2082\n    have h\u2083\u2082 : (x : \u2124) + (y : \u2124) - 6 = 30 := by\n      linarith\n    have h\u2083\u2083 : (x : \u2124) + (y : \u2124) = 36 := by linarith\n    exact_mod_cast h\u2083\u2083\n  have h\u2084 : x + y = 36 := by\n    norm_cast at h\u2083 \u22a2\n    <;> omega\n  have h\u2085 : x = 6 := by\n    have h\u2085\u2081 : y = 5 * x := by\n      omega\n    have h\u2085\u2082 : x + y = 36 := h\u2084\n    rw [h\u2085\u2081] at h\u2085\u2082\n    have h\u2085\u2083 : x + (5 * x) = 36 := by simpa using h\u2085\u2082\n    have h\u2085\u2084 : 6 * x = 36 := by\n      omega\n    have h\u2085\u2085 : x = 6 := by\n      omega\n    exact h\u2085\u2085\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 175, "alphabet": 80, "compiles": false, "id": "mathd_algebra_289", "proof_after": "theorem mathd_algebra_289 (k t m n : \u2115) (h\u2080 : Nat.Prime m \u2227 Nat.Prime n) (h\u2081 : t < k)\n    (h\u2082 : (k ^ 2 : \u2124) - m * k + n = 0) (h\u2083 : (t ^ 2 : \u2124) - m * t + n = 0) :\n    m ^ n + n ^ m + k ^ t + t ^ k = 20  :=  by\n  have h\u2084 : (m : \u2124) = (k : \u2124) + (t : \u2124) := by\n    have h\u2084\u2081 : ((k : \u2124) - (t : \u2124)) * ((k : \u2124) + (t : \u2124) - (m : \u2124)) = 0 := by\n      have h\u2084\u2082 : (k : \u2124) ^ 2 - m * k + n = 0 := by exact_mod_cast h\u2082\n      have h\u2084\u2083 : (t : \u2124) ^ 2 - m * t + n = 0 := by exact_mod_cast h\u2083\n      have h\u2084\u2084 : ((k : \u2124) ^ 2 - m * k + n) - ((t : \u2124) ^ 2 - m * t + n) = 0 := by\n        linarith\n      have h\u2084\u2085 : ((k : \u2124) ^ 2 - m * k + n) - ((t : \u2124) ^ 2 - m * t + n) = (k : \u2124) ^ 2 - m * k + n - ((t : \u2124) ^ 2 - m * t + n) := by ring\n      rw [h\u2084\u2085] at h\u2084\u2084\n      have h\u2084\u2086 : (k : \u2124) ^ 2 - m * k + n - ((t : \u2124) ^ 2 - m * t + n) = (k : \u2124) ^ 2 - m * k - (t : \u2124) ^ 2 + m * t := by ring\n      rw [h\u2084\u2086] at h\u2084\u2084\n      have h\u2084\u2087 : (k : \u2124) ^ 2 - m * k - (t : \u2124) ^ 2 + m * t = 0 := by linarith\n      have h\u2084\u2088 : (k : \u2124) ^ 2 - (t : \u2124) ^ 2 - m * k + m * t = 0 := by linarith\n      have h\u2084\u2089 : ((k : \u2124) - (t : \u2124)) * ((k : \u2124) + (t : \u2124)) - m * ((k : \u2124) - (t : \u2124)) = 0 := by\n        ring_nf at h\u2084\u2088 \u22a2\n        <;> linarith\n      have h\u2085\u2080 : ((k : \u2124) - (t : \u2124)) * ((k : \u2124) + (t : \u2124) - (m : \u2124)) = 0 := by\n        ring_nf at h\u2084\u2089 \u22a2\n        <;> linarith\n      exact h\u2085\u2080\n    have h\u2085\u2081 : (k : \u2124) \u2260 (t : \u2124) := by\n      intro h\n      have h\u2085\u2082 : k = t := by\n        norm_cast at h\n        <;> linarith\n      have h\u2085\u2083 : t < k := h\u2081\n      linarith\n    have h\u2085\u2084 : ((k : \u2124) + (t : \u2124) - (m : \u2124)) = 0 := by\n      have h\u2085\u2085 : ((k : \u2124) - (t : \u2124)) \u2260 0 := by\n        intro h\n        apply h\u2085\u2081\n        linarith\n      have h\u2085\u2086 : ((k : \u2124) - (t : \u2124)) * ((k : \u2124) + (t : \u2124) - (m : \u2124)) = 0 := h\u2084\u2081\n      have h\u2085\u2087 : ((k : \u2124) + (t : \u2124) - (m : \u2124)) = 0 := by\n        apply mul_left_cancel\u2080 h\u2085\u2085\n        linarith\n      exact h\u2085\u2087\n    have h\u2085\u2088 : (m : \u2124) = (k : \u2124) + (t : \u2124) := by linarith\n    exact_mod_cast h\u2085\u2088\n  have h\u2085 : m = k + t := by\n    have h\u2085\u2081 : (m : \u2124) = (k : \u2124) + (t : \u2124) := h\u2084\n    have h\u2085\u2082 : m = k + t := by\n      norm_cast at h\u2085\u2081 \u22a2\n      <;> linarith\n    exact h\u2085\u2082\n  have h\u2086 : n = t * k := by\n    have h\u2086\u2081 : k ^ 2 + n = m * k := by\n      have h\u2086\u2082 : (k ^ 2 : \u2124) + (n : \u2124) = (m * k : \u2124) := by\n        have h\u2086\u2083 : (k ^ 2 : \u2124) - m * k + n = 0 := h\u2082\n        linarith\n      norm_cast at h\u2086\u2082 \u22a2\n      <;> linarith\n    have h\u2086\u2084 : n = t * k := by\n      have h\u2086\u2085 : m = k + t := h\u2085\n      rw [h\u2086\u2085] at h\u2086\u2081\n      ring_nf at h\u2086\u2081 \u22a2\n      <;> nlinarith [h\u2080.1.two_le, h\u2080.2.two_le, h\u2081, Nat.Prime.ne_zero h\u2080.1, Nat.Prime.ne_zero h\u2080.2,\n        Nat.Prime.pos h\u2080.1, Nat.Prime.pos h\u2080.2]\n    exact h\u2086\u2084\n  have h\u2087 : t = 1 := by\n    have h\u2087\u2081 : n = t * k := h\u2086\n    have h\u2087\u2082 : Nat.Prime n := h\u2080.2\n    have h\u2087\u2083 : t \u2223 n := by\n      use k\n      <;> linarith\n    have h\u2087\u2084 : t = 1 \u2228 t = n := by\n      have h\u2087\u2085 : Nat.Prime n := h\u2080.2\n      exact h\u2087\u2085.eq_one_or_self_of_dvd _ h\u2087\u2083\n    cases h\u2087\u2084 with\n    | inl h\u2087\u2084 =>\n      exact h\u2087\u2084\n    | inr h\u2087\u2084 =>\n      have h\u2087\u2085 : t = n := h\u2087\u2084\n      have h\u2087\u2086 : t < k := h\u2081\n      have h\u2087\u2087 : k > t := by linarith\n      have h\u2087\u2088 : n = t * k := h\u2086\n      have h\u2087\u2089 : t * k > t * t := by\n        nlinarith [h\u2080.1.two_le, h\u2080.2.two_le]\n      have h\u2088\u2080 : t > 0 := by\n        nlinarith [h\u2080.1.two_le, h\u2080.2.two_le]\n      have h\u2088\u2081 : t = n := by linarith\n      have h\u2088\u2082 : k > t := by linarith\n      have h\u2088\u2083 : n = t * k := h\u2086\n      have h\u2088\u2084 : t * k > t * t := by\n        nlinarith [h\u2080.1.two_le, h\u2080.2.two_le]\n      have h\u2088\u2085 : t > 0 := by\n        nlinarith [h\u2080.1.two_le, h\u2080.2.two_le]\n      nlinarith [h\u2080.1.two_le, h\u2080.2.two_le]\n  have h\u2088 : k = n := by\n    have h\u2088\u2081 : n = t * k := h\u2086\n    have h\u2088\u2082 : t = 1 := h\u2087\n    rw [h\u2088\u2082] at h\u2088\u2081\n    ring_nf at h\u2088\u2081 \u22a2\n    <;> nlinarith [h\u2080.1.two_le, h\u2080.2.two_le, h\u2081, Nat.Prime.ne_zero h\u2080.1, Nat.Prime.ne_zero h\u2080.2]\n  have h\u2089 : n = 2 := by\n    have h\u2089\u2081 : m = k + t := h\u2085\n    have h\u2089\u2082 : k = n := h\u2088\n    have h\u2089\u2083 : t = 1 := h\u2087\n    have h\u2089\u2084 : m = n + 1 := by\n      subst_vars\n      <;> ring_nf at *\n      <;> omega\n    have h\u2089\u2085 : Nat.Prime m := h\u2080.1\n    have h\u2089\u2086 : Nat.Prime n := h\u2080.2\n    have h\u2089\u2087 : n = 2 := by\n      by_contra h\n      have h\u2089\u2088 := h\u2089\u2086.eq_two_or_odd\n      have h\u2089\u2089 := h\u2089\u2085.eq_two_or_odd\n      cases' h\u2089\u2088 with h\u2089\u2088 h\u2089\u2088\n      \u00b7 exfalso\n        omega\n      \u00b7 cases' h\u2089\u2089 with h\u2089\u2089 h\u2089\u2089\n        \u00b7 exfalso\n          have h\u2081\u2080\u2080 := h\u2089\u2085.two_le\n          have h\u2081\u2080\u2081 := h\u2089\u2086.two_le\n          omega\n        \u00b7 have h\u2081\u2080\u2082 : m % 2 = 0 := by\n            omega\n          have h\u2081\u2080\u2083 : m = 2 := by\n            have h\u2081\u2080\u2084 := h\u2089\u2085.eq_two_or_odd\n            cases' h\u2081\u2080\u2084 with h\u2081\u2080\u2084 h\u2081\u2080\u2084\n            \u00b7 omega\n            \u00b7 omega\n          have h\u2081\u2080\u2085 : n = 1 := by\n            omega\n          have h\u2081\u2080\u2086 : Nat.Prime n := h\u2080.2\n          have h\u2081\u2080\u2087 := h\u2081\u2080\u2086.eq_two_or_odd\n          cases' h\u2081\u2080\u2087 with h\u2081\u2080\u2087 h\u2081\u2080\u2087\n          \u00b7 omega\n          \u00b7 omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n        <;> omega\n        <;> omega\n        <;> omega\n    exact h\u2089\u2087\n  have h\u2081\u2080 : m = 3 := by\n    have h\u2081\u2080\u2081 : m = k + t := h\u2085\n    have h\u2081\u2080\u2082 : k = n := h\u2088\n    have h\u2081\u2080\u2083 : t = 1 := h\u2087\n    have h\u2081\u2080\u2084 : n = 2 := h\u2089\n    have h\u2081\u2080\u2085 : m = 3 := by\n      subst_vars\n      <;> norm_num at *\n      <;> omega\n    exact h\u2081\u2080\u2085\n  have h\u2081\u2081 : k = 2 := by\n    have h\u2081\u2081\u2081 : k = n := h\u2088\n    have h\u2081\u2081\u2082 : n = 2 := h\u2089\n    linarith\n  have h\u2081\u2082 : t = 1 := by\n    exact h\u2087\n  have h\u2081\u2083 : m ^ n + n ^ m + k ^ t + t ^ k = 20 := by\n    subst_vars\n    <;> norm_num at *\n    <;> decide\n  apply h\u2081\u2083\n", "proof_before": "theorem mathd_algebra_289 (k t m n : \u2115) (h\u2080 : Nat.Prime m \u2227 Nat.Prime n) (h\u2081 : t < k)\n    (h\u2082 : (k ^ 2 : \u2124) - m * k + n = 0) (h\u2083 : (t ^ 2 : \u2124) - m * t + n = 0) :\n    m ^ n + n ^ m + k ^ t + t ^ k = 20  :=  by\n  have h\u2084 : (m : \u2124) = (k : \u2124) + (t : \u2124) := by\n    have h\u2084\u2081 : ((k : \u2124) - (t : \u2124)) * ((k : \u2124) + (t : \u2124) - (m : \u2124)) = 0 := by\n      have h\u2084\u2082 : (k : \u2124) ^ 2 - m * k + n = 0 := by exact_mod_cast h\u2082\n      have h\u2084\u2083 : (t : \u2124) ^ 2 - m * t + n = 0 := by exact_mod_cast h\u2083\n      have h\u2084\u2084 : ((k : \u2124) ^ 2 - m * k + n) - ((t : \u2124) ^ 2 - m * t + n) = 0 := by\n        linarith\n      have h\u2084\u2085 : ((k : \u2124) ^ 2 - m * k + n) - ((t : \u2124) ^ 2 - m * t + n) = (k : \u2124) ^ 2 - m * k + n - ((t : \u2124) ^ 2 - m * t + n) := by ring\n      rw [h\u2084\u2085] at h\u2084\u2084\n      have h\u2084\u2086 : (k : \u2124) ^ 2 - m * k + n - ((t : \u2124) ^ 2 - m * t + n) = (k : \u2124) ^ 2 - m * k - (t : \u2124) ^ 2 + m * t := by ring\n      rw [h\u2084\u2086] at h\u2084\u2084\n      have h\u2084\u2087 : (k : \u2124) ^ 2 - m * k - (t : \u2124) ^ 2 + m * t = 0 := by linarith\n      have h\u2084\u2088 : (k : \u2124) ^ 2 - (t : \u2124) ^ 2 - m * k + m * t = 0 := by linarith\n      have h\u2084\u2089 : ((k : \u2124) - (t : \u2124)) * ((k : \u2124) + (t : \u2124)) - m * ((k : \u2124) - (t : \u2124)) = 0 := by\n        ring_nf at h\u2084\u2088 \u22a2\n        <;> linarith\n      have h\u2085\u2080 : ((k : \u2124) - (t : \u2124)) * ((k : \u2124) + (t : \u2124) - (m : \u2124)) = 0 := by\n        ring_nf at h\u2084\u2089 \u22a2\n        <;> linarith\n      exact h\u2085\u2080\n    have h\u2085\u2081 : (k : \u2124) \u2260 (t : \u2124) := by\n      intro h\n      have h\u2085\u2082 : k = t := by\n        norm_cast at h\n        <;> linarith\n      have h\u2085\u2083 : t < k := h\u2081\n      linarith\n    have h\u2085\u2084 : ((k : \u2124) + (t : \u2124) - (m : \u2124)) = 0 := by\n      have h\u2085\u2085 : ((k : \u2124) - (t : \u2124)) \u2260 0 := by\n        intro h\n        apply h\u2085\u2081\n        linarith\n      have h\u2085\u2086 : ((k : \u2124) - (t : \u2124)) * ((k : \u2124) + (t : \u2124) - (m : \u2124)) = 0 := h\u2084\u2081\n      have h\u2085\u2087 : ((k : \u2124) + (t : \u2124) - (m : \u2124)) = 0 := by\n        apply mul_left_cancel\u2080 h\u2085\u2085\n        linarith\n      exact h\u2085\u2087\n    have h\u2085\u2088 : (m : \u2124) = (k : \u2124) + (t : \u2124) := by linarith\n    exact_mod_cast h\u2085\u2088\n  have h\u2085 : m = k + t := by\n    have h\u2085\u2081 : (m : \u2124) = (k : \u2124) + (t : \u2124) := h\u2084\n    have h\u2085\u2082 : m = k + t := by\n      norm_cast at h\u2085\u2081 \u22a2\n      <;> linarith\n    exact h\u2085\u2082\n  have h\u2086 : n = t * k := by\n    have h\u2086\u2081 : k ^ 2 + n = m * k := by\n      have h\u2086\u2082 : (k ^ 2 : \u2124) + (n : \u2124) = (m * k : \u2124) := by\n        have h\u2086\u2083 : (k ^ 2 : \u2124) - m * k + n = 0 := h\u2082\n        linarith\n      norm_cast at h\u2086\u2082 \u22a2\n      <;> linarith\n    have h\u2086\u2084 : n = t * k := by\n      have h\u2086\u2085 : m = k + t := h\u2085\n      rw [h\u2086\u2085] at h\u2086\u2081\n      ring_nf at h\u2086\u2081 \u22a2\n      <;> nlinarith [h\u2080.1.two_le, h\u2080.2.two_le, h\u2081, Nat.Prime.ne_zero h\u2080.1, Nat.Prime.ne_zero h\u2080.2,\n        Nat.Prime.pos h\u2080.1, Nat.Prime.pos h\u2080.2]\n    exact h\u2086\u2084\n  have h\u2087 : t = 1 := by\n    have h\u2087\u2081 : n = t * k := h\u2086\n    have h\u2087\u2082 : Nat.Prime n := h\u2080.2\n    have h\u2087\u2083 : t \u2223 n := by\n      use k\n      <;> linarith\n    have h\u2087\u2084 : t = 1 \u2228 t = n := by\n      have h\u2087\u2085 : Nat.Prime n := h\u2080.2\n      exact h\u2087\u2085.eq_one_or_self_of_dvd _ h\u2087\u2083\n    cases h\u2087\u2084 with\n    | inl h\u2087\u2084 =>\n      exact h\u2087\u2084\n    | inr h\u2087\u2084 =>\n      have h\u2087\u2085 : t = n := h\u2087\u2084\n      have h\u2087\u2086 : t < k := h\u2081\n      have h\u2087\u2087 : k > t := by linarith\n      have h\u2087\u2088 : n = t * k := h\u2086\n      have h\u2087\u2089 : t * k > t * t := by\n        nlinarith [h\u2080.1.two_le, h\u2080.2.two_le]\n      have h\u2088\u2080 : t > 0 := by\n        nlinarith [h\u2080.1.two_le, h\u2080.2.two_le]\n      have h\u2088\u2081 : t = n := by linarith\n      have h\u2088\u2082 : k > t := by linarith\n      have h\u2088\u2083 : n = t * k := h\u2086\n      have h\u2088\u2084 : t * k > t * t := by\n        nlinarith [h\u2080.1.two_le, h\u2080.2.two_le]\n      have h\u2088\u2085 : t > 0 := by\n        nlinarith [h\u2080.1.two_le, h\u2080.2.two_le]\n      nlinarith [h\u2080.1.two_le, h\u2080.2.two_le]\n  have h\u2088 : k = n := by\n    have h\u2088\u2081 : n = t * k := h\u2086\n    have h\u2088\u2082 : t = 1 := h\u2087\n    rw [h\u2088\u2082] at h\u2088\u2081\n    ring_nf at h\u2088\u2081 \u22a2\n    <;> nlinarith [h\u2080.1.two_le, h\u2080.2.two_le, h\u2081, Nat.Prime.ne_zero h\u2080.1, Nat.Prime.ne_zero h\u2080.2]\n  have h\u2089 : n = 2 := by\n    have h\u2089\u2081 : m = k + t := h\u2085\n    have h\u2089\u2082 : k = n := h\u2088\n    have h\u2089\u2083 : t = 1 := h\u2087\n    have h\u2089\u2084 : m = n + 1 := by\n      subst_vars\n      <;> ring_nf at *\n      <;> omega\n    have h\u2089\u2085 : Nat.Prime m := h\u2080.1\n    have h\u2089\u2086 : Nat.Prime n := h\u2080.2\n    have h\u2089\u2087 : n = 2 := by\n      by_contra h\n      have h\u2089\u2088 := h\u2089\u2086.eq_two_or_odd\n      have h\u2089\u2089 := h\u2089\u2085.eq_two_or_odd\n      cases' h\u2089\u2088 with h\u2089\u2088 h\u2089\u2088\n      \u00b7 exfalso\n        omega\n      \u00b7 cases' h\u2089\u2089 with h\u2089\u2089 h\u2089\u2089\n        \u00b7 exfalso\n          have h\u2081\u2080\u2080 := h\u2089\u2085.two_le\n          have h\u2081\u2080\u2081 := h\u2089\u2086.two_le\n          omega\n        \u00b7 have h\u2081\u2080\u2082 : m % 2 = 0 := by\n            omega\n          have h\u2081\u2080\u2083 : m = 2 := by\n            have h\u2081\u2080\u2084 := h\u2089\u2085.eq_two_or_odd\n            cases' h\u2081\u2080\u2084 with h\u2081\u2080\u2084 h\u2081\u2080\u2084\n            \u00b7 omega\n            \u00b7 omega\n          have h\u2081\u2080\u2085 : n = 1 := by\n            omega\n          have h\u2081\u2080\u2086 : Nat.Prime n := h\u2080.2\n          have h\u2081\u2080\u2087 := h\u2081\u2080\u2086.eq_two_or_odd\n          cases' h\u2081\u2080\u2087 with h\u2081\u2080\u2087 h\u2081\u2080\u2087\n          \u00b7 omega\n          \u00b7 omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n          <;> omega\n        <;> omega\n        <;> omega\n        <;> omega\n    exact h\u2089\u2087\n  have h\u2081\u2080 : m = 3 := by\n    have h\u2081\u2080\u2081 : m = k + t := h\u2085\n    have h\u2081\u2080\u2082 : k = n := h\u2088\n    have h\u2081\u2080\u2083 : t = 1 := h\u2087\n    have h\u2081\u2080\u2084 : n = 2 := h\u2089\n    have h\u2081\u2080\u2085 : m = 3 := by\n      subst_vars\n      <;> norm_num at *\n      <;> omega\n    exact h\u2081\u2080\u2085\n  have h\u2081\u2081 : k = 2 := by\n    have h\u2081\u2081\u2081 : k = n := h\u2088\n    have h\u2081\u2081\u2082 : n = 2 := h\u2089\n    linarith\n  have h\u2081\u2082 : t = 1 := by\n    exact h\u2087\n  have h\u2081\u2083 : m ^ n + n ^ m + k ^ t + t ^ k = 20 := by\n    subst_vars\n    <;> norm_num at *\n    <;> decide\n  apply h\u2081\u2083\n"}
{"mode": "clear", "steps_before": 20, "alphabet": 15, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 483, "chars_after": 7, "seconds": 1.5, "id": "mathd_algebra_432", "proof_after": "theorem mathd_algebra_432 (x : \u211d) : (x + 3) * (2 * x - 6) = 2 * x ^ 2 - 18  :=  by\n  ring\n", "proof_before": "theorem mathd_algebra_432 (x : \u211d) : (x + 3) * (2 * x - 6) = 2 * x ^ 2 - 18  :=  by\n  have h_expand : (x + 3) * (2 * x - 6) = 2 * x ^ 2 - 6 * x + 6 * x - 18 := by\n    ring_nf\n    <;>\n    linarith\n  have h_simplify : 2 * x ^ 2 - 6 * x + 6 * x - 18 = 2 * x ^ 2 - 18 := by\n    have h1 : 2 * x ^ 2 - 6 * x + 6 * x - 18 = 2 * x ^ 2 - 18 := by\n      ring_nf\n      <;>\n      linarith\n    rw [h1]\n    <;>\n    linarith\n  have h_final : (x + 3) * (2 * x - 6) = 2 * x ^ 2 - 18 := by\n    rw [h_expand]\n    rw [h_simplify]\n    <;>\n    ring_nf\n    <;>\n    linarith\n  apply h_final\n"}
{"mode": "clear", "steps_before": 13, "alphabet": 19, "compiles": false, "id": "mathd_numbertheory_99", "proof_after": "theorem mathd_numbertheory_99 (n : \u2115) (h\u2080 : 2 * n % 47 = 15) : n % 47 = 31  :=  by\n  have h\u2081 : n % 47 = 31 := by\n    have h\u2082 : (2 * (n % 47)) % 47 = 15 := by\n      have h\u2083 : (2 * n) % 47 = 15 := h\u2080\n      have h\u2084 : (2 * n) % 47 = (2 * (n % 47)) % 47 := by\n        simp [Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n      rw [h\u2084] at h\u2083\n      exact h\u2083\n    have h\u2085 : n % 47 < 47 := Nat.mod_lt n (by norm_num)\n    interval_cases n % 47 <;> norm_num at h\u2082 \u22a2 <;> omega\n  exact h\u2081\n", "proof_before": "theorem mathd_numbertheory_99 (n : \u2115) (h\u2080 : 2 * n % 47 = 15) : n % 47 = 31  :=  by\n  have h\u2081 : n % 47 = 31 := by\n    have h\u2082 : (2 * (n % 47)) % 47 = 15 := by\n      have h\u2083 : (2 * n) % 47 = 15 := h\u2080\n      have h\u2084 : (2 * n) % 47 = (2 * (n % 47)) % 47 := by\n        simp [Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n      rw [h\u2084] at h\u2083\n      exact h\u2083\n    have h\u2085 : n % 47 < 47 := Nat.mod_lt n (by norm_num)\n    interval_cases n % 47 <;> norm_num at h\u2082 \u22a2 <;> omega\n  exact h\u2081\n"}
{"mode": "clear", "steps_before": 72, "alphabet": 32, "compiles": false, "id": "imo_1962_p2", "proof_after": "theorem imo_1962_p2 (x : \u211d) (h\u2080 : 0 \u2264 3 - x) (h\u2081 : 0 \u2264 x + 1)\n    (h\u2082 : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) : -1 \u2264 x \u2227 x < 1 - Real.sqrt 31 / 8  :=  by\n  have h\u2083 : -1 \u2264 x := by\n    linarith\n  have h\u2084 : Real.sqrt (3 - x) - Real.sqrt (x + 1) > 1 / 2 := by\n    linarith\n  have h\u2085 : Real.sqrt (3 - x) > Real.sqrt (x + 1) + 1 / 2 := by\n    linarith\n  have h\u2086 : 3 - x > x + 1 + Real.sqrt (x + 1) + 1 / 4 := by\n    have h\u2086\u2081 : Real.sqrt (3 - x) > Real.sqrt (x + 1) + 1 / 2 := h\u2085\n    have h\u2086\u2082 : 0 \u2264 Real.sqrt (x + 1) := Real.sqrt_nonneg (x + 1)\n    have h\u2086\u2083 : 0 \u2264 Real.sqrt (3 - x) := Real.sqrt_nonneg (3 - x)\n    have h\u2086\u2084 : 0 \u2264 Real.sqrt (x + 1) * (1 / 2) := by positivity\n    have h\u2086\u2085 : 0 \u2264 Real.sqrt (3 - x) := Real.sqrt_nonneg (3 - x)\n    have h\u2086\u2086 : 0 \u2264 Real.sqrt (x + 1) := Real.sqrt_nonneg (x + 1)\n    have h\u2086\u2087 : (Real.sqrt (x + 1) + 1 / 2) \u2265 0 := by positivity\n    have h\u2086\u2088 : Real.sqrt (3 - x) > Real.sqrt (x + 1) + 1 / 2 := h\u2085\n    have h\u2086\u2089 : (Real.sqrt (x + 1) + 1 / 2) ^ 2 < (Real.sqrt (3 - x)) ^ 2 := by\n      gcongr\n      <;> nlinarith [Real.sq_sqrt (show 0 \u2264 x + 1 by linarith), Real.sq_sqrt (show 0 \u2264 3 - x by linarith)]\n    have h\u2086\u2081\u2080 : (Real.sqrt (x + 1) + 1 / 2) ^ 2 = (x + 1) + Real.sqrt (x + 1) + 1 / 4 := by\n      nlinarith [Real.sq_sqrt (show 0 \u2264 x + 1 by linarith), Real.sq_sqrt (show 0 \u2264 3 - x by linarith),\n        sq_nonneg (Real.sqrt (x + 1) - 1 / 2)]\n    have h\u2086\u2081\u2081 : (Real.sqrt (3 - x)) ^ 2 = 3 - x := by\n      rw [Real.sq_sqrt] <;> linarith\n    nlinarith [Real.sq_sqrt (show 0 \u2264 x + 1 by linarith), Real.sq_sqrt (show 0 \u2264 3 - x by linarith)]\n  have h\u2087 : 7 - 8 * x > 4 * Real.sqrt (x + 1) := by\n    have h\u2087\u2081 : 3 - x > x + 1 + Real.sqrt (x + 1) + 1 / 4 := h\u2086\n    have h\u2087\u2082 : 0 \u2264 Real.sqrt (x + 1) := Real.sqrt_nonneg (x + 1)\n    nlinarith [Real.sqrt_nonneg (x + 1)]\n  have h\u2088 : x < 7 / 8 := by\n    by_contra h\n    have h\u2088\u2081 : x \u2265 7 / 8 := by linarith\n    have h\u2088\u2082 : 7 - 8 * x \u2264 0 := by\n      linarith\n    have h\u2088\u2083 : 4 * Real.sqrt (x + 1) \u2265 0 := by positivity\n    linarith\n  have h\u2089 : 0 \u2264 Real.sqrt 31 := by positivity\n  have h\u2081\u2080 : 64 * x ^ 2 - 128 * x + 33 > 0 := by\n    have h\u2081\u2080\u2081 : 7 - 8 * x > 4 * Real.sqrt (x + 1) := h\u2087\n    have h\u2081\u2080\u2082 : 0 \u2264 Real.sqrt (x + 1) := Real.sqrt_nonneg (x + 1)\n    have h\u2081\u2080\u2083 : 0 \u2264 7 - 8 * x := by linarith\n    have h\u2081\u2080\u2084 : 0 \u2264 Real.sqrt (x + 1) := Real.sqrt_nonneg (x + 1)\n    have h\u2081\u2080\u2085 : 0 \u2264 7 - 8 * x := by linarith\n    have h\u2081\u2080\u2086 : (7 - 8 * x) ^ 2 > (4 * Real.sqrt (x + 1)) ^ 2 := by\n      gcongr\n    have h\u2081\u2080\u2087 : (4 * Real.sqrt (x + 1)) ^ 2 = 16 * (x + 1) := by\n      nlinarith [Real.sq_sqrt (show 0 \u2264 x + 1 by linarith), sq_nonneg (Real.sqrt (x + 1) - 2)]\n    have h\u2081\u2080\u2088 : (7 - 8 * x) ^ 2 > 16 * (x + 1) := by\n      linarith\n    nlinarith [Real.sq_sqrt (show 0 \u2264 x + 1 by linarith), sq_nonneg (x - 1)]\n  have h\u2081\u2081 : x < 1 - Real.sqrt 31 / 8 := by\n    by_contra h\n    have h\u2081\u2081\u2081 : x \u2265 1 - Real.sqrt 31 / 8 := by\n      linarith\n    have h\u2081\u2081\u2082 : 64 * x ^ 2 - 128 * x + 33 \u2264 0 := by\n      have h\u2081\u2081\u2083 : x \u2265 1 - Real.sqrt 31 / 8 := h\u2081\u2081\u2081\n      have h\u2081\u2081\u2084 : x \u2264 1 + Real.sqrt 31 / 8 := by\n        by_contra h\u2081\u2081\u2084\n        have h\u2081\u2081\u2085 : x > 1 + Real.sqrt 31 / 8 := by linarith\n        have h\u2081\u2081\u2086 : 64 * x ^ 2 - 128 * x + 33 > 0 := h\u2081\u2080\n        have h\u2081\u2081\u2087 : x < 7 / 8 := h\u2088\n        nlinarith [Real.sqrt_nonneg 31, Real.sq_sqrt (show 0 \u2264 31 by norm_num)]\n      nlinarith [Real.sqrt_nonneg 31, Real.sq_sqrt (show 0 \u2264 31 by norm_num),\n        sq_nonneg (x - (1 - Real.sqrt 31 / 8)), sq_nonneg (x - (1 + Real.sqrt 31 / 8))]\n    linarith [h\u2081\u2080]\n  exact \u27e8h\u2083, h\u2081\u2081\u27e9\n", "proof_before": "theorem imo_1962_p2 (x : \u211d) (h\u2080 : 0 \u2264 3 - x) (h\u2081 : 0 \u2264 x + 1)\n    (h\u2082 : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) : -1 \u2264 x \u2227 x < 1 - Real.sqrt 31 / 8  :=  by\n  have h\u2083 : -1 \u2264 x := by\n    linarith\n  have h\u2084 : Real.sqrt (3 - x) - Real.sqrt (x + 1) > 1 / 2 := by\n    linarith\n  have h\u2085 : Real.sqrt (3 - x) > Real.sqrt (x + 1) + 1 / 2 := by\n    linarith\n  have h\u2086 : 3 - x > x + 1 + Real.sqrt (x + 1) + 1 / 4 := by\n    have h\u2086\u2081 : Real.sqrt (3 - x) > Real.sqrt (x + 1) + 1 / 2 := h\u2085\n    have h\u2086\u2082 : 0 \u2264 Real.sqrt (x + 1) := Real.sqrt_nonneg (x + 1)\n    have h\u2086\u2083 : 0 \u2264 Real.sqrt (3 - x) := Real.sqrt_nonneg (3 - x)\n    have h\u2086\u2084 : 0 \u2264 Real.sqrt (x + 1) * (1 / 2) := by positivity\n    have h\u2086\u2085 : 0 \u2264 Real.sqrt (3 - x) := Real.sqrt_nonneg (3 - x)\n    have h\u2086\u2086 : 0 \u2264 Real.sqrt (x + 1) := Real.sqrt_nonneg (x + 1)\n    have h\u2086\u2087 : (Real.sqrt (x + 1) + 1 / 2) \u2265 0 := by positivity\n    have h\u2086\u2088 : Real.sqrt (3 - x) > Real.sqrt (x + 1) + 1 / 2 := h\u2085\n    have h\u2086\u2089 : (Real.sqrt (x + 1) + 1 / 2) ^ 2 < (Real.sqrt (3 - x)) ^ 2 := by\n      gcongr\n      <;> nlinarith [Real.sq_sqrt (show 0 \u2264 x + 1 by linarith), Real.sq_sqrt (show 0 \u2264 3 - x by linarith)]\n    have h\u2086\u2081\u2080 : (Real.sqrt (x + 1) + 1 / 2) ^ 2 = (x + 1) + Real.sqrt (x + 1) + 1 / 4 := by\n      nlinarith [Real.sq_sqrt (show 0 \u2264 x + 1 by linarith), Real.sq_sqrt (show 0 \u2264 3 - x by linarith),\n        sq_nonneg (Real.sqrt (x + 1) - 1 / 2)]\n    have h\u2086\u2081\u2081 : (Real.sqrt (3 - x)) ^ 2 = 3 - x := by\n      rw [Real.sq_sqrt] <;> linarith\n    nlinarith [Real.sq_sqrt (show 0 \u2264 x + 1 by linarith), Real.sq_sqrt (show 0 \u2264 3 - x by linarith)]\n  have h\u2087 : 7 - 8 * x > 4 * Real.sqrt (x + 1) := by\n    have h\u2087\u2081 : 3 - x > x + 1 + Real.sqrt (x + 1) + 1 / 4 := h\u2086\n    have h\u2087\u2082 : 0 \u2264 Real.sqrt (x + 1) := Real.sqrt_nonneg (x + 1)\n    nlinarith [Real.sqrt_nonneg (x + 1)]\n  have h\u2088 : x < 7 / 8 := by\n    by_contra h\n    have h\u2088\u2081 : x \u2265 7 / 8 := by linarith\n    have h\u2088\u2082 : 7 - 8 * x \u2264 0 := by\n      linarith\n    have h\u2088\u2083 : 4 * Real.sqrt (x + 1) \u2265 0 := by positivity\n    linarith\n  have h\u2089 : 0 \u2264 Real.sqrt 31 := by positivity\n  have h\u2081\u2080 : 64 * x ^ 2 - 128 * x + 33 > 0 := by\n    have h\u2081\u2080\u2081 : 7 - 8 * x > 4 * Real.sqrt (x + 1) := h\u2087\n    have h\u2081\u2080\u2082 : 0 \u2264 Real.sqrt (x + 1) := Real.sqrt_nonneg (x + 1)\n    have h\u2081\u2080\u2083 : 0 \u2264 7 - 8 * x := by linarith\n    have h\u2081\u2080\u2084 : 0 \u2264 Real.sqrt (x + 1) := Real.sqrt_nonneg (x + 1)\n    have h\u2081\u2080\u2085 : 0 \u2264 7 - 8 * x := by linarith\n    have h\u2081\u2080\u2086 : (7 - 8 * x) ^ 2 > (4 * Real.sqrt (x + 1)) ^ 2 := by\n      gcongr\n    have h\u2081\u2080\u2087 : (4 * Real.sqrt (x + 1)) ^ 2 = 16 * (x + 1) := by\n      nlinarith [Real.sq_sqrt (show 0 \u2264 x + 1 by linarith), sq_nonneg (Real.sqrt (x + 1) - 2)]\n    have h\u2081\u2080\u2088 : (7 - 8 * x) ^ 2 > 16 * (x + 1) := by\n      linarith\n    nlinarith [Real.sq_sqrt (show 0 \u2264 x + 1 by linarith), sq_nonneg (x - 1)]\n  have h\u2081\u2081 : x < 1 - Real.sqrt 31 / 8 := by\n    by_contra h\n    have h\u2081\u2081\u2081 : x \u2265 1 - Real.sqrt 31 / 8 := by\n      linarith\n    have h\u2081\u2081\u2082 : 64 * x ^ 2 - 128 * x + 33 \u2264 0 := by\n      have h\u2081\u2081\u2083 : x \u2265 1 - Real.sqrt 31 / 8 := h\u2081\u2081\u2081\n      have h\u2081\u2081\u2084 : x \u2264 1 + Real.sqrt 31 / 8 := by\n        by_contra h\u2081\u2081\u2084\n        have h\u2081\u2081\u2085 : x > 1 + Real.sqrt 31 / 8 := by linarith\n        have h\u2081\u2081\u2086 : 64 * x ^ 2 - 128 * x + 33 > 0 := h\u2081\u2080\n        have h\u2081\u2081\u2087 : x < 7 / 8 := h\u2088\n        nlinarith [Real.sqrt_nonneg 31, Real.sq_sqrt (show 0 \u2264 31 by norm_num)]\n      nlinarith [Real.sqrt_nonneg 31, Real.sq_sqrt (show 0 \u2264 31 by norm_num),\n        sq_nonneg (x - (1 - Real.sqrt 31 / 8)), sq_nonneg (x - (1 + Real.sqrt 31 / 8))]\n    linarith [h\u2081\u2080]\n  exact \u27e8h\u2083, h\u2081\u2081\u27e9\n"}
{"mode": "clear", "steps_before": 24, "alphabet": 18, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 968, "chars_after": 8, "seconds": 0.6, "id": "mathd_numbertheory_247", "proof_after": "theorem mathd_numbertheory_247 (n : \u2115) (h\u2080 : 3 * n % 11 = 2) : n % 11 = 8  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_247 (n : \u2115) (h\u2080 : 3 * n % 11 = 2) : n % 11 = 8  :=  by\n  have h\u2081 : n % 11 = 8 := by\n    have h\u2082 : 3 * (n % 11) % 11 = 2 := by\n      have h\u2083 : 3 * n % 11 = 2 := h\u2080\n      have h\u2084 : 3 * (n % 11) % 11 = 3 * n % 11 := by\n        have h\u2085 : 3 * (n % 11) % 11 = (3 * n) % 11 := by\n          have h\u2086 : 3 * (n % 11) = (3 * n) % 33 := by\n            omega\n          have h\u2087 : (3 * n) % 33 % 11 = (3 * n) % 11 := by\n            omega\n          omega\n        omega\n      rw [h\u2084, h\u2083]\n    have h\u2088 : n % 11 = 0 \u2228 n % 11 = 1 \u2228 n % 11 = 2 \u2228 n % 11 = 3 \u2228 n % 11 = 4 \u2228 n % 11 = 5 \u2228 n % 11 = 6 \u2228 n % 11 = 7 \u2228 n % 11 = 8 \u2228 n % 11 = 9 \u2228 n % 11 = 10 := by\n      omega\n    rcases h\u2088 with (h\u2088 | h\u2088 | h\u2088 | h\u2088 | h\u2088 | h\u2088 | h\u2088 | h\u2088 | h\u2088 | h\u2088 | h\u2088) <;>\n      (try omega) <;>\n      (try\n        {\n          simp [h\u2088, h\u2088, h\u2088, h\u2088, h\u2088, h\u2088, h\u2088, h\u2088, h\u2088, h\u2088, Nat.mul_mod, Nat.add_mod, Nat.mod_mod] at h\u2082 \u22a2\n          <;> omega\n        }) <;>\n      (try\n        {\n          simp [h\u2088, Nat.mul_mod, Nat.add_mod, Nat.mod_mod] at h\u2082 \u22a2 <;> omega\n        })\n  exact h\u2081\n"}
{"mode": "clear", "steps_before": 7, "alphabet": 13, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 192, "chars_after": 6, "seconds": 0.9, "id": "mathd_algebra_304", "proof_after": "theorem mathd_algebra_304 : 91 ^ 2 = 8281  :=  by\n  rfl\n", "proof_before": "theorem mathd_algebra_304 : 91 ^ 2 = 8281  :=  by\n  have h\u2080 : 91 ^ 2 = 8281 := by\n    norm_num [pow_two, Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.mul_zero, Nat.add_assoc]\n    <;> rfl\n    <;> simp_all\n    <;> norm_num\n    <;> rfl\n  apply h\u2080\n"}
{"mode": "clear", "steps_before": 144, "alphabet": 53, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 144, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "timed_out": false, "fixpoint": false, "chars_before": 5278, "chars_after": 5278, "seconds": 296.6, "id": "mathd_algebra_114", "proof_after": "theorem mathd_algebra_114 (a : \u211d) (h\u2080 : a = 8) :\n    (16 * (a ^ 2) ^ ((1 : \u211d) / 3)) ^ ((1 : \u211d) / 3) = 4  :=  by\n  have h\u2081 : a > 0 := by\n    rw [h\u2080]\n    norm_num\n  have h\u2082 : (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) = a ^ (2 / 3 : \u211d) := by\n    have h\u2082\u2081 : (a ^ 2 : \u211d) > 0 := by positivity\n    have h\u2082\u2082 : (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) = (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) := rfl\n    rw [show (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) = a ^ (2 / 3 : \u211d) by\n      have h\u2082\u2083 : Real.log ((a ^ 2 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log (a ^ (2 / 3 : \u211d)) := by\n        have h\u2082\u2084 : Real.log ((a ^ 2 : \u211d) ^ ((1 : \u211d) / 3)) = ((1 : \u211d) / 3) * Real.log (a ^ 2) := by\n          rw [Real.log_rpow (by positivity)]\n        have h\u2082\u2085 : Real.log (a ^ (2 / 3 : \u211d)) = (2 / 3 : \u211d) * Real.log a := by\n          rw [Real.log_rpow (by positivity)]\n          <;> ring_nf\n        rw [h\u2082\u2084, h\u2082\u2085]\n        have h\u2082\u2086 : Real.log (a ^ 2) = 2 * Real.log a := by\n          rw [Real.log_pow]\n          <;> norm_num\n        rw [h\u2082\u2086]\n        <;> ring_nf\n        <;> field_simp\n        <;> ring_nf\n      have h\u2082\u2087 : (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) > 0 := by positivity\n      have h\u2082\u2088 : a ^ (2 / 3 : \u211d) > 0 := by positivity\n      have h\u2082\u2089 : Real.log ((a ^ 2 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log (a ^ (2 / 3 : \u211d)) := h\u2082\u2083\n      have h\u2083\u2080 : (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) = a ^ (2 / 3 : \u211d) := by\n        apply Real.log_injOn_pos (Set.mem_Ioi.mpr h\u2082\u2087) (Set.mem_Ioi.mpr h\u2082\u2088)\n        linarith\n      rw [h\u2083\u2080]\n      ]\n    <;> simp_all\n  have h\u2083 : (16 * (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3)) ^ ((1 : \u211d) / 3) = (16 * a ^ (2 / 3 : \u211d)) ^ ((1 : \u211d) / 3) := by\n    rw [h\u2082]\n    <;>\n    simp_all [Real.rpow_def_of_pos, Real.log_mul, Real.log_rpow, Real.log_pow, mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    norm_num at *\n    <;>\n    linarith\n  have h\u2084 : a ^ (2 / 3 : \u211d) = 4 := by\n    have h\u2084\u2081 : a = 8 := h\u2080\n    have h\u2084\u2082 : a > 0 := h\u2081\n    have h\u2084\u2083 : a ^ (2 / 3 : \u211d) = 4 := by\n      rw [h\u2084\u2081]\n      have h\u2084\u2084 : (8 : \u211d) ^ (2 / 3 : \u211d) = 4 := by\n        have h\u2084\u2085 : Real.log ((8 : \u211d) ^ (2 / 3 : \u211d)) = Real.log 4 := by\n          have h\u2084\u2086 : Real.log ((8 : \u211d) ^ (2 / 3 : \u211d)) = (2 / 3 : \u211d) * Real.log 8 := by\n            rw [Real.log_rpow (by norm_num : (8 : \u211d) > 0)]\n          rw [h\u2084\u2086]\n          have h\u2084\u2087 : Real.log 8 = Real.log (2 ^ 3) := by norm_num\n          rw [h\u2084\u2087]\n          have h\u2084\u2088 : Real.log (2 ^ 3 : \u211d) = 3 * Real.log 2 := by\n            rw [Real.log_pow] <;> norm_num\n          rw [h\u2084\u2088]\n          have h\u2084\u2089 : Real.log 4 = Real.log (2 ^ 2) := by norm_num\n          rw [h\u2084\u2089]\n          have h\u2085\u2080 : Real.log (2 ^ 2 : \u211d) = 2 * Real.log 2 := by\n            rw [Real.log_pow] <;> norm_num\n          rw [h\u2085\u2080]\n          ring_nf\n          <;>\n          norm_num\n          <;>\n          linarith [Real.log_pos (by norm_num : (1 : \u211d) < 2)]\n        have h\u2085\u2081 : (8 : \u211d) ^ (2 / 3 : \u211d) > 0 := by positivity\n        have h\u2085\u2082 : (4 : \u211d) > 0 := by norm_num\n        have h\u2085\u2083 : Real.log ((8 : \u211d) ^ (2 / 3 : \u211d)) = Real.log 4 := h\u2084\u2085\n        have h\u2085\u2084 : Real.log ((8 : \u211d) ^ (2 / 3 : \u211d)) = Real.log 4 := by rw [h\u2085\u2083]\n        have h\u2085\u2085 : (8 : \u211d) ^ (2 / 3 : \u211d) = 4 := by\n          apply Real.log_injOn_pos (Set.mem_Ioi.mpr h\u2085\u2081) (Set.mem_Ioi.mpr h\u2085\u2082)\n          linarith\n        rw [h\u2085\u2085]\n      rw [h\u2084\u2084]\n    rw [h\u2084\u2083]\n  have h\u2085 : (16 * (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3)) ^ ((1 : \u211d) / 3) = (16 * (4 : \u211d)) ^ ((1 : \u211d) / 3) := by\n    rw [h\u2083, h\u2084]\n    <;> norm_num\n    <;>\n    simp_all [Real.rpow_def_of_pos, Real.log_mul, Real.log_rpow, Real.log_pow, mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    norm_num at *\n    <;>\n    linarith\n  have h\u2086 : (16 * (4 : \u211d)) ^ ((1 : \u211d) / 3) = (64 : \u211d) ^ ((1 : \u211d) / 3) := by\n    norm_num [Real.rpow_def_of_pos]\n    <;>\n    simp_all [Real.rpow_def_of_pos, Real.log_mul, Real.log_rpow, Real.log_pow, mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    norm_num at *\n    <;>\n    linarith\n  have h\u2087 : (64 : \u211d) ^ ((1 : \u211d) / 3) = 4 := by\n    have h\u2087\u2081 : (64 : \u211d) ^ ((1 : \u211d) / 3) = 4 := by\n      have h\u2087\u2082 : Real.log ((64 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log 4 := by\n        have h\u2087\u2083 : Real.log ((64 : \u211d) ^ ((1 : \u211d) / 3)) = ((1 : \u211d) / 3) * Real.log 64 := by\n          rw [Real.log_rpow (by norm_num : (64 : \u211d) > 0)]\n        rw [h\u2087\u2083]\n        have h\u2087\u2084 : Real.log 64 = Real.log (2 ^ 6) := by norm_num\n        rw [h\u2087\u2084]\n        have h\u2087\u2085 : Real.log (2 ^ 6) = 6 * Real.log 2 := by\n          rw [Real.log_pow] <;> norm_num\n        rw [h\u2087\u2085]\n        have h\u2087\u2086 : Real.log 4 = Real.log (2 ^ 2) := by norm_num\n        rw [h\u2087\u2086]\n        have h\u2087\u2087 : Real.log (2 ^ 2) = 2 * Real.log 2 := by\n          rw [Real.log_pow] <;> norm_num\n        rw [h\u2087\u2087]\n        ring_nf\n        <;>\n        norm_num\n        <;>\n        linarith [Real.log_pos (by norm_num : (1 : \u211d) < 2)]\n      have h\u2087\u2088 : (64 : \u211d) ^ ((1 : \u211d) / 3) > 0 := by positivity\n      have h\u2087\u2089 : (4 : \u211d) > 0 := by norm_num\n      have h\u2088\u2080 : Real.log ((64 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log 4 := by rw [h\u2087\u2082]\n      have h\u2088\u2081 : (64 : \u211d) ^ ((1 : \u211d) / 3) = 4 := by\n        apply Real.log_injOn_pos (Set.mem_Ioi.mpr h\u2087\u2088) (Set.mem_Ioi.mpr h\u2087\u2089)\n        linarith\n      rw [h\u2088\u2081]\n    rw [h\u2087\u2081]\n  have h\u2088 : (16 * (a ^ 2) ^ ((1 : \u211d) / 3)) ^ ((1 : \u211d) / 3) = 4 := by\n    rw [h\u2085]\n    rw [h\u2086]\n    rw [h\u2087]\n    <;>\n    simp_all [Real.rpow_def_of_pos, Real.log_mul, Real.log_rpow, Real.log_pow, mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    norm_num at *\n    <;>\n    linarith\n  rw [h\u2088]\n  <;>\n  simp_all [Real.rpow_def_of_pos, Real.log_mul, Real.log_rpow, Real.log_pow, mul_assoc]\n  <;>\n  ring_nf at *\n  <;>\n  norm_num at *\n  <;>\n  linarith\n", "proof_before": "theorem mathd_algebra_114 (a : \u211d) (h\u2080 : a = 8) :\n    (16 * (a ^ 2) ^ ((1 : \u211d) / 3)) ^ ((1 : \u211d) / 3) = 4  :=  by\n  have h\u2081 : a > 0 := by\n    rw [h\u2080]\n    norm_num\n  have h\u2082 : (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) = a ^ (2 / 3 : \u211d) := by\n    have h\u2082\u2081 : (a ^ 2 : \u211d) > 0 := by positivity\n    have h\u2082\u2082 : (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) = (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) := rfl\n    rw [show (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) = a ^ (2 / 3 : \u211d) by\n      have h\u2082\u2083 : Real.log ((a ^ 2 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log (a ^ (2 / 3 : \u211d)) := by\n        have h\u2082\u2084 : Real.log ((a ^ 2 : \u211d) ^ ((1 : \u211d) / 3)) = ((1 : \u211d) / 3) * Real.log (a ^ 2) := by\n          rw [Real.log_rpow (by positivity)]\n        have h\u2082\u2085 : Real.log (a ^ (2 / 3 : \u211d)) = (2 / 3 : \u211d) * Real.log a := by\n          rw [Real.log_rpow (by positivity)]\n          <;> ring_nf\n        rw [h\u2082\u2084, h\u2082\u2085]\n        have h\u2082\u2086 : Real.log (a ^ 2) = 2 * Real.log a := by\n          rw [Real.log_pow]\n          <;> norm_num\n        rw [h\u2082\u2086]\n        <;> ring_nf\n        <;> field_simp\n        <;> ring_nf\n      have h\u2082\u2087 : (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) > 0 := by positivity\n      have h\u2082\u2088 : a ^ (2 / 3 : \u211d) > 0 := by positivity\n      have h\u2082\u2089 : Real.log ((a ^ 2 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log (a ^ (2 / 3 : \u211d)) := h\u2082\u2083\n      have h\u2083\u2080 : (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3) = a ^ (2 / 3 : \u211d) := by\n        apply Real.log_injOn_pos (Set.mem_Ioi.mpr h\u2082\u2087) (Set.mem_Ioi.mpr h\u2082\u2088)\n        linarith\n      rw [h\u2083\u2080]\n      ]\n    <;> simp_all\n  have h\u2083 : (16 * (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3)) ^ ((1 : \u211d) / 3) = (16 * a ^ (2 / 3 : \u211d)) ^ ((1 : \u211d) / 3) := by\n    rw [h\u2082]\n    <;>\n    simp_all [Real.rpow_def_of_pos, Real.log_mul, Real.log_rpow, Real.log_pow, mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    norm_num at *\n    <;>\n    linarith\n  have h\u2084 : a ^ (2 / 3 : \u211d) = 4 := by\n    have h\u2084\u2081 : a = 8 := h\u2080\n    have h\u2084\u2082 : a > 0 := h\u2081\n    have h\u2084\u2083 : a ^ (2 / 3 : \u211d) = 4 := by\n      rw [h\u2084\u2081]\n      have h\u2084\u2084 : (8 : \u211d) ^ (2 / 3 : \u211d) = 4 := by\n        have h\u2084\u2085 : Real.log ((8 : \u211d) ^ (2 / 3 : \u211d)) = Real.log 4 := by\n          have h\u2084\u2086 : Real.log ((8 : \u211d) ^ (2 / 3 : \u211d)) = (2 / 3 : \u211d) * Real.log 8 := by\n            rw [Real.log_rpow (by norm_num : (8 : \u211d) > 0)]\n          rw [h\u2084\u2086]\n          have h\u2084\u2087 : Real.log 8 = Real.log (2 ^ 3) := by norm_num\n          rw [h\u2084\u2087]\n          have h\u2084\u2088 : Real.log (2 ^ 3 : \u211d) = 3 * Real.log 2 := by\n            rw [Real.log_pow] <;> norm_num\n          rw [h\u2084\u2088]\n          have h\u2084\u2089 : Real.log 4 = Real.log (2 ^ 2) := by norm_num\n          rw [h\u2084\u2089]\n          have h\u2085\u2080 : Real.log (2 ^ 2 : \u211d) = 2 * Real.log 2 := by\n            rw [Real.log_pow] <;> norm_num\n          rw [h\u2085\u2080]\n          ring_nf\n          <;>\n          norm_num\n          <;>\n          linarith [Real.log_pos (by norm_num : (1 : \u211d) < 2)]\n        have h\u2085\u2081 : (8 : \u211d) ^ (2 / 3 : \u211d) > 0 := by positivity\n        have h\u2085\u2082 : (4 : \u211d) > 0 := by norm_num\n        have h\u2085\u2083 : Real.log ((8 : \u211d) ^ (2 / 3 : \u211d)) = Real.log 4 := h\u2084\u2085\n        have h\u2085\u2084 : Real.log ((8 : \u211d) ^ (2 / 3 : \u211d)) = Real.log 4 := by rw [h\u2085\u2083]\n        have h\u2085\u2085 : (8 : \u211d) ^ (2 / 3 : \u211d) = 4 := by\n          apply Real.log_injOn_pos (Set.mem_Ioi.mpr h\u2085\u2081) (Set.mem_Ioi.mpr h\u2085\u2082)\n          linarith\n        rw [h\u2085\u2085]\n      rw [h\u2084\u2084]\n    rw [h\u2084\u2083]\n  have h\u2085 : (16 * (a ^ 2 : \u211d) ^ ((1 : \u211d) / 3)) ^ ((1 : \u211d) / 3) = (16 * (4 : \u211d)) ^ ((1 : \u211d) / 3) := by\n    rw [h\u2083, h\u2084]\n    <;> norm_num\n    <;>\n    simp_all [Real.rpow_def_of_pos, Real.log_mul, Real.log_rpow, Real.log_pow, mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    norm_num at *\n    <;>\n    linarith\n  have h\u2086 : (16 * (4 : \u211d)) ^ ((1 : \u211d) / 3) = (64 : \u211d) ^ ((1 : \u211d) / 3) := by\n    norm_num [Real.rpow_def_of_pos]\n    <;>\n    simp_all [Real.rpow_def_of_pos, Real.log_mul, Real.log_rpow, Real.log_pow, mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    norm_num at *\n    <;>\n    linarith\n  have h\u2087 : (64 : \u211d) ^ ((1 : \u211d) / 3) = 4 := by\n    have h\u2087\u2081 : (64 : \u211d) ^ ((1 : \u211d) / 3) = 4 := by\n      have h\u2087\u2082 : Real.log ((64 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log 4 := by\n        have h\u2087\u2083 : Real.log ((64 : \u211d) ^ ((1 : \u211d) / 3)) = ((1 : \u211d) / 3) * Real.log 64 := by\n          rw [Real.log_rpow (by norm_num : (64 : \u211d) > 0)]\n        rw [h\u2087\u2083]\n        have h\u2087\u2084 : Real.log 64 = Real.log (2 ^ 6) := by norm_num\n        rw [h\u2087\u2084]\n        have h\u2087\u2085 : Real.log (2 ^ 6) = 6 * Real.log 2 := by\n          rw [Real.log_pow] <;> norm_num\n        rw [h\u2087\u2085]\n        have h\u2087\u2086 : Real.log 4 = Real.log (2 ^ 2) := by norm_num\n        rw [h\u2087\u2086]\n        have h\u2087\u2087 : Real.log (2 ^ 2) = 2 * Real.log 2 := by\n          rw [Real.log_pow] <;> norm_num\n        rw [h\u2087\u2087]\n        ring_nf\n        <;>\n        norm_num\n        <;>\n        linarith [Real.log_pos (by norm_num : (1 : \u211d) < 2)]\n      have h\u2087\u2088 : (64 : \u211d) ^ ((1 : \u211d) / 3) > 0 := by positivity\n      have h\u2087\u2089 : (4 : \u211d) > 0 := by norm_num\n      have h\u2088\u2080 : Real.log ((64 : \u211d) ^ ((1 : \u211d) / 3)) = Real.log 4 := by rw [h\u2087\u2082]\n      have h\u2088\u2081 : (64 : \u211d) ^ ((1 : \u211d) / 3) = 4 := by\n        apply Real.log_injOn_pos (Set.mem_Ioi.mpr h\u2087\u2088) (Set.mem_Ioi.mpr h\u2087\u2089)\n        linarith\n      rw [h\u2088\u2081]\n    rw [h\u2087\u2081]\n  have h\u2088 : (16 * (a ^ 2) ^ ((1 : \u211d) / 3)) ^ ((1 : \u211d) / 3) = 4 := by\n    rw [h\u2085]\n    rw [h\u2086]\n    rw [h\u2087]\n    <;>\n    simp_all [Real.rpow_def_of_pos, Real.log_mul, Real.log_rpow, Real.log_pow, mul_assoc]\n    <;>\n    ring_nf at *\n    <;>\n    norm_num at *\n    <;>\n    linarith\n  rw [h\u2088]\n  <;>\n  simp_all [Real.rpow_def_of_pos, Real.log_mul, Real.log_rpow, Real.log_pow, mul_assoc]\n  <;>\n  ring_nf at *\n  <;>\n  norm_num at *\n  <;>\n  linarith\n"}
{"mode": "clear", "steps_before": 47, "alphabet": 26, "compiles": true, "steps_after": 33, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "timed_out": false, "fixpoint": true, "chars_before": 2037, "chars_after": 1555, "seconds": 228.9, "id": "numbertheory_x5neqy2p4", "proof_after": "theorem numbertheory_x5neqy2p4 (x y : \u2124) : x ^ 5 \u2260 y ^ 2 + 4  :=  by\n  have h_main : (x ^ 5 - (y ^ 2 + 4)) % 11 \u2260 0 := by\n    have h\u2081 : (x ^ 5 : \u2124) % 11 = 0 \u2228 (x ^ 5 : \u2124) % 11 = 1 \u2228 (x ^ 5 : \u2124) % 11 = 10 := by\n      have h\u2082 : (x : \u2124) % 11 = 0 \u2228 (x : \u2124) % 11 = 1 \u2228 (x : \u2124) % 11 = 2 \u2228 (x : \u2124) % 11 = 3 \u2228 (x : \u2124) % 11 = 4 \u2228 (x : \u2124) % 11 = 5 \u2228 (x : \u2124) % 11 = 6 \u2228 (x : \u2124) % 11 = 7 \u2228 (x : \u2124) % 11 = 8 \u2228 (x : \u2124) % 11 = 9 \u2228 (x : \u2124) % 11 = 10 := by\n        omega\n      rcases h\u2082 with (h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082) <;>\n        (try omega) <;>\n        (try\n          {\n            simp [h\u2082, pow_succ, Int.mul_emod, Int.sub_emod, Int.add_emod]\n          }) <;>\n        (try norm_num) <;>\n        (try omega) <;>\n        (try\n          {\n            (try norm_num) <;>\n            (try omega)\n          })\n    have h\u2083 : (y ^ 2 + 4 : \u2124) % 11 = 2 \u2228 (y ^ 2 + 4 : \u2124) % 11 = 4 \u2228 (y ^ 2 + 4 : \u2124) % 11 = 5 \u2228 (y ^ 2 + 4 : \u2124) % 11 = 7 \u2228 (y ^ 2 + 4 : \u2124) % 11 = 8 \u2228 (y ^ 2 + 4 : \u2124) % 11 = 9 := by\n      have h\u2084 : (y : \u2124) % 11 = 0 \u2228 (y : \u2124) % 11 = 1 \u2228 (y : \u2124) % 11 = 2 \u2228 (y : \u2124) % 11 = 3 \u2228 (y : \u2124) % 11 = 4 \u2228 (y : \u2124) % 11 = 5 \u2228 (y : \u2124) % 11 = 6 \u2228 (y : \u2124) % 11 = 7 \u2228 (y : \u2124) % 11 = 8 \u2228 (y : \u2124) % 11 = 9 \u2228 (y : \u2124) % 11 = 10 := by\n        omega\n      rcases h\u2084 with (h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084) <;>\n        (try omega) <;>\n        (try\n          {\n            simp [h\u2084, pow_two, Int.mul_emod, Int.add_emod, Int.emod_emod]\n          }) <;>\n        (try norm_num) <;>\n        (try omega) <;>\n        (try\n          {\n            (try norm_num) <;>\n            (try omega)\n          })\n    omega\n  omega\n", "proof_before": "theorem numbertheory_x5neqy2p4 (x y : \u2124) : x ^ 5 \u2260 y ^ 2 + 4  :=  by\n  have h_main : (x ^ 5 - (y ^ 2 + 4)) % 11 \u2260 0 := by\n    have h\u2081 : (x ^ 5 : \u2124) % 11 = 0 \u2228 (x ^ 5 : \u2124) % 11 = 1 \u2228 (x ^ 5 : \u2124) % 11 = 10 := by\n      have h\u2082 : (x : \u2124) % 11 = 0 \u2228 (x : \u2124) % 11 = 1 \u2228 (x : \u2124) % 11 = 2 \u2228 (x : \u2124) % 11 = 3 \u2228 (x : \u2124) % 11 = 4 \u2228 (x : \u2124) % 11 = 5 \u2228 (x : \u2124) % 11 = 6 \u2228 (x : \u2124) % 11 = 7 \u2228 (x : \u2124) % 11 = 8 \u2228 (x : \u2124) % 11 = 9 \u2228 (x : \u2124) % 11 = 10 := by\n        omega\n      rcases h\u2082 with (h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082 | h\u2082) <;>\n        (try omega) <;>\n        (try\n          {\n            simp [h\u2082, pow_succ, Int.mul_emod, Int.sub_emod, Int.add_emod]\n          }) <;>\n        (try norm_num) <;>\n        (try omega) <;>\n        (try\n          {\n            (try norm_num) <;>\n            (try omega)\n          })\n    have h\u2083 : (y ^ 2 + 4 : \u2124) % 11 = 2 \u2228 (y ^ 2 + 4 : \u2124) % 11 = 4 \u2228 (y ^ 2 + 4 : \u2124) % 11 = 5 \u2228 (y ^ 2 + 4 : \u2124) % 11 = 7 \u2228 (y ^ 2 + 4 : \u2124) % 11 = 8 \u2228 (y ^ 2 + 4 : \u2124) % 11 = 9 := by\n      have h\u2084 : (y : \u2124) % 11 = 0 \u2228 (y : \u2124) % 11 = 1 \u2228 (y : \u2124) % 11 = 2 \u2228 (y : \u2124) % 11 = 3 \u2228 (y : \u2124) % 11 = 4 \u2228 (y : \u2124) % 11 = 5 \u2228 (y : \u2124) % 11 = 6 \u2228 (y : \u2124) % 11 = 7 \u2228 (y : \u2124) % 11 = 8 \u2228 (y : \u2124) % 11 = 9 \u2228 (y : \u2124) % 11 = 10 := by\n        omega\n      rcases h\u2084 with (h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084 | h\u2084) <;>\n        (try omega) <;>\n        (try\n          {\n            simp [h\u2084, pow_two, Int.mul_emod, Int.add_emod, Int.emod_emod]\n          }) <;>\n        (try norm_num) <;>\n        (try omega) <;>\n        (try\n          {\n            (try norm_num) <;>\n            (try omega)\n          })\n    rcases h\u2081 with (h\u2081 | h\u2081 | h\u2081) <;>\n      rcases h\u2083 with (h\u2083 | h\u2083 | h\u2083 | h\u2083 | h\u2083 | h\u2083) <;>\n        (try omega) <;>\n        (try\n          {\n            norm_num [Int.sub_emod, h\u2081, h\u2083, Int.emod_emod] at *\n            <;> omega\n          })\n  have h_final : x ^ 5 \u2260 y ^ 2 + 4 := by\n    intro h_contra\n    have h\u2081 : (x ^ 5 - (y ^ 2 + 4) : \u2124) % 11 = 0 := by\n      rw [h_contra]\n      <;> simp [Int.emod_self]\n    have h\u2082 : (x ^ 5 - (y ^ 2 + 4) : \u2124) % 11 \u2260 0 := h_main\n    contradiction\n  exact h_final\n"}
{"id": "amc12_2000_p12", "error": "checker disconnects (AXLE)", "steps_before": null}
{"id": "mathd_numbertheory_541", "error": "checker disconnects (AXLE)", "steps_before": null}
{"id": "amc12_2001_p21", "error": "checker disconnects (AXLE)", "steps_before": null}
{"id": "aime_1987_p5", "error": "checker disconnects (AXLE)", "steps_before": null}
{"id": "amc12_2000_p1", "error": "checker disconnects (AXLE)", "steps_before": null}
{"mode": "clear", "steps_before": 89, "alphabet": 45, "compiles": false, "id": "numbertheory_sumkmulnckeqnmul2pownm1", "proof_after": "theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : \u2115) (h\u2080 : 0 < n) :\n  (\u2211 k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1)  :=  by\n  have h1 : \u2200 k \u2208 Finset.Icc 1 n, k * Nat.choose n k = Nat.choose n k * k := by\n    intro k hk\n    simp [Nat.mul_comm, h\u2080] at hk \u22a2\n    <;>\n    simp_all [Finset.mem_Icc, Nat.mul_comm, Nat.mul_assoc, Nat.mul_left_comm]\n    <;>\n    norm_num\n    <;>\n    omega\n  have h2 : \u2211 k in Finset.Icc 1 n, k * Nat.choose n k = \u2211 k in Finset.Icc 1 n, Nat.choose n k * k := by\n    refine' Finset.sum_congr rfl fun k hk => _\n    have hk' := h1 k hk\n    simp_all [Nat.mul_comm]\n    <;> ring_nf\n    <;> simp_all [Finset.sum_congr]\n    <;> apply congr_arg\n    <;> apply congr_arg\n    <;> apply Finset.sum_congr\n    <;> simp_all\n  have h3 : \u2200 k \u2208 Finset.Icc 1 n, Nat.choose n k * k = n * Nat.choose (n - 1) (k - 1) := by\n    intro k hk\n    have hh\u2081 : 0 < n := h\u2080\n    have hh\u2082 : 0 < k := by\n      have h\u2081 : k \u2208 Finset.Icc 1 n := hk\n      have h\u2082 : 1 \u2264 k := by\n        rw [Finset.mem_Icc] at h\u2081\n        exact h\u2081.1\n      linarith\n    have hh\u2083 : k \u2264 n := by\n      have h\u2081 : k \u2208 Finset.Icc 1 n := hk\n      have h\u2082 : k \u2264 n := by\n        rw [Finset.mem_Icc] at h\u2081\n        exact h\u2081.2\n      exact h\u2082\n    calc\n      Nat.choose n k * k = n * Nat.choose (n - 1) (k - 1) := by\n        have h\u2081 : k \u2264 n := hh\u2083\n        have h\u2082 : 0 < k := hh\u2082\n        have h\u2083 : 0 < n := hh\u2081\n        rw [\u2190 Nat.succ_pred_eq_of_pos h\u2082]\n        rw [\u2190 Nat.succ_pred_eq_of_pos h\u2083]\n        simp_all [Nat.choose_succ_succ, Nat.succ_mul_choose_eq]\n      _ = n * Nat.choose (n - 1) (k - 1) := by rfl\n  have h4 : \u2211 k in Finset.Icc 1 n, Nat.choose n k * k = \u2211 k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by\n    rw [Finset.sum_congr rfl (fun k hk => h3 k hk)]\n    <;> simp_all [Finset.mul_sum]\n    <;> ring\n    <;> linarith\n  have h5 : \u2211 k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) = n * \u2211 k in Finset.Icc 1 n, Nat.choose (n - 1) (k - 1) := by\n    rw [Finset.mul_sum]\n    <;> simp_all [Finset.sum_mul]\n    <;> apply Eq.refl\n  have h6 : \u2211 k in Finset.Icc 1 n, Nat.choose (n - 1) (k - 1) = \u2211 j in Finset.Icc 0 (n - 1), Nat.choose (n - 1) j := by\n    have h6 : \u2200 k \u2208 Finset.Icc 1 n, k - 1 \u2208 Finset.Icc 0 (n - 1) := by\n      intro k hk\n      simp_all [Finset.mem_Icc, Nat.succ_le_iff, Nat.sub_le]\n      <;> omega\n    have h7 : \u2211 k in Finset.Icc 1 n, Nat.choose (n - 1) (k - 1) = \u2211 j in Finset.Icc 0 (n - 1), Nat.choose (n - 1) j := by\n      apply Finset.sum_bij' (fun k hk => k - 1) (fun j hj => j + 1)\n      <;> simp_all [Finset.mem_Icc, Nat.succ_le_iff, Nat.sub_le]\n      <;> omega\n    exact h7\n  have h7 : \u2211 j in Finset.Icc 0 (n - 1), Nat.choose (n - 1) j = 2 ^ (n - 1) := by\n    have h7 : \u2211 j in Finset.Icc 0 (n - 1), Nat.choose (n - 1) j = 2 ^ (n - 1) := by\n      rw [\u2190 Nat.sum_range_choose]\n      congr\n      ext j\n      simp [Finset.mem_Icc, Nat.choose_symm]\n      <;> omega\n    simp_all\n  have h8 : n * \u2211 j in Finset.Icc 0 (n - 1), Nat.choose (n - 1) j = n * 2 ^ (n - 1) := by\n    simp [Nat.choose_eq_factorial_div_factorial] at h1 h3 h5 h6 h7 \u22a2\n    <;> simp_all [Finset.sum_mul, Finset.mul_sum, Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]\n    <;> linarith\n  have h9 : \u2211 k in Finset.Icc 1 n, k * Nat.choose n k = n * 2 ^ (n - 1) := by\n    have h9 := h2\n    have h10 := h4\n    have h11 := h5\n    have h12 := h6\n    have h13 := h7\n    have h14 := h8\n    simp_all [Finset.sum_mul, Finset.mul_sum, mul_comm]\n    <;> ring_nf <;> simp_all [Finset.sum_mul, Finset.mul_sum, mul_comm]\n    <;> linarith\n  rw [h9]\n  <;> simp_all [Finset.sum_Icc_succ_top]\n  <;> linarith\n", "proof_before": "theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : \u2115) (h\u2080 : 0 < n) :\n  (\u2211 k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1)  :=  by\n  have h1 : \u2200 k \u2208 Finset.Icc 1 n, k * Nat.choose n k = Nat.choose n k * k := by\n    intro k hk\n    simp [Nat.mul_comm, h\u2080] at hk \u22a2\n    <;>\n    simp_all [Finset.mem_Icc, Nat.mul_comm, Nat.mul_assoc, Nat.mul_left_comm]\n    <;>\n    norm_num\n    <;>\n    omega\n  have h2 : \u2211 k in Finset.Icc 1 n, k * Nat.choose n k = \u2211 k in Finset.Icc 1 n, Nat.choose n k * k := by\n    refine' Finset.sum_congr rfl fun k hk => _\n    have hk' := h1 k hk\n    simp_all [Nat.mul_comm]\n    <;> ring_nf\n    <;> simp_all [Finset.sum_congr]\n    <;> apply congr_arg\n    <;> apply congr_arg\n    <;> apply Finset.sum_congr\n    <;> simp_all\n  have h3 : \u2200 k \u2208 Finset.Icc 1 n, Nat.choose n k * k = n * Nat.choose (n - 1) (k - 1) := by\n    intro k hk\n    have hh\u2081 : 0 < n := h\u2080\n    have hh\u2082 : 0 < k := by\n      have h\u2081 : k \u2208 Finset.Icc 1 n := hk\n      have h\u2082 : 1 \u2264 k := by\n        rw [Finset.mem_Icc] at h\u2081\n        exact h\u2081.1\n      linarith\n    have hh\u2083 : k \u2264 n := by\n      have h\u2081 : k \u2208 Finset.Icc 1 n := hk\n      have h\u2082 : k \u2264 n := by\n        rw [Finset.mem_Icc] at h\u2081\n        exact h\u2081.2\n      exact h\u2082\n    calc\n      Nat.choose n k * k = n * Nat.choose (n - 1) (k - 1) := by\n        have h\u2081 : k \u2264 n := hh\u2083\n        have h\u2082 : 0 < k := hh\u2082\n        have h\u2083 : 0 < n := hh\u2081\n        rw [\u2190 Nat.succ_pred_eq_of_pos h\u2082]\n        rw [\u2190 Nat.succ_pred_eq_of_pos h\u2083]\n        simp_all [Nat.choose_succ_succ, Nat.succ_mul_choose_eq]\n      _ = n * Nat.choose (n - 1) (k - 1) := by rfl\n  have h4 : \u2211 k in Finset.Icc 1 n, Nat.choose n k * k = \u2211 k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) := by\n    rw [Finset.sum_congr rfl (fun k hk => h3 k hk)]\n    <;> simp_all [Finset.mul_sum]\n    <;> ring\n    <;> linarith\n  have h5 : \u2211 k in Finset.Icc 1 n, n * Nat.choose (n - 1) (k - 1) = n * \u2211 k in Finset.Icc 1 n, Nat.choose (n - 1) (k - 1) := by\n    rw [Finset.mul_sum]\n    <;> simp_all [Finset.sum_mul]\n    <;> apply Eq.refl\n  have h6 : \u2211 k in Finset.Icc 1 n, Nat.choose (n - 1) (k - 1) = \u2211 j in Finset.Icc 0 (n - 1), Nat.choose (n - 1) j := by\n    have h6 : \u2200 k \u2208 Finset.Icc 1 n, k - 1 \u2208 Finset.Icc 0 (n - 1) := by\n      intro k hk\n      simp_all [Finset.mem_Icc, Nat.succ_le_iff, Nat.sub_le]\n      <;> omega\n    have h7 : \u2211 k in Finset.Icc 1 n, Nat.choose (n - 1) (k - 1) = \u2211 j in Finset.Icc 0 (n - 1), Nat.choose (n - 1) j := by\n      apply Finset.sum_bij' (fun k hk => k - 1) (fun j hj => j + 1)\n      <;> simp_all [Finset.mem_Icc, Nat.succ_le_iff, Nat.sub_le]\n      <;> omega\n    exact h7\n  have h7 : \u2211 j in Finset.Icc 0 (n - 1), Nat.choose (n - 1) j = 2 ^ (n - 1) := by\n    have h7 : \u2211 j in Finset.Icc 0 (n - 1), Nat.choose (n - 1) j = 2 ^ (n - 1) := by\n      rw [\u2190 Nat.sum_range_choose]\n      congr\n      ext j\n      simp [Finset.mem_Icc, Nat.choose_symm]\n      <;> omega\n    simp_all\n  have h8 : n * \u2211 j in Finset.Icc 0 (n - 1), Nat.choose (n - 1) j = n * 2 ^ (n - 1) := by\n    simp [Nat.choose_eq_factorial_div_factorial] at h1 h3 h5 h6 h7 \u22a2\n    <;> simp_all [Finset.sum_mul, Finset.mul_sum, Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]\n    <;> linarith\n  have h9 : \u2211 k in Finset.Icc 1 n, k * Nat.choose n k = n * 2 ^ (n - 1) := by\n    have h9 := h2\n    have h10 := h4\n    have h11 := h5\n    have h12 := h6\n    have h13 := h7\n    have h14 := h8\n    simp_all [Finset.sum_mul, Finset.mul_sum, mul_comm]\n    <;> ring_nf <;> simp_all [Finset.sum_mul, Finset.mul_sum, mul_comm]\n    <;> linarith\n  rw [h9]\n  <;> simp_all [Finset.sum_Icc_succ_top]\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 20, "alphabet": 15, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 432, "chars_after": 11, "seconds": 0.0, "id": "amc12a_2008_p2", "proof_after": "theorem amc12a_2008_p2 (x : \u211d) (h\u2080 : x * (1 / 2 + 2 / 3) = 1) : x = 6 / 7  :=  by\n  linarith\n", "proof_before": "theorem amc12a_2008_p2 (x : \u211d) (h\u2080 : x * (1 / 2 + 2 / 3) = 1) : x = 6 / 7  :=  by\n  have h1 : (1 / 2 : \u211d) + (2 / 3) = (7 / 6) := by\n    field_simp at h\u2080 \u22a2\n    norm_num\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  have h2 : 1 / (1 / 2 + 2 / 3) = 6 / 7 := by\n    field_simp [h1]\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  have h3 : x = 6 / 7 := by\n    have h3 : x * (7 / 6) = 1 := by rw [h1] at h\u2080; linarith\n    have h4 : x = 6 / 7 := by\n      field_simp at h3 \u22a2\n      linarith\n    linarith\n  exact h3\n"}
{"mode": "clear", "steps_before": 35, "alphabet": 25, "compiles": false, "id": "amc12b_2020_p5", "proof_after": "theorem amc12b_2020_p5 (a b : \u2115) (h\u2080 : (5 : \u211a) / 8 * b = 2 / 3 * a + 7)\n  (h\u2081 : (b : \u211a) - 5 / 8 * b = a - 2 / 3 * a + 7) : a = 42  :=  by\n  have eq1a : 15 * b = 16 * a + 168 := by\n    field_simp at h\u2080 h\u2081\n    ring_nf at h\u2080 h\u2081\n    norm_cast at h\u2080 h\u2081\n    omega\n  have eq2a : 9 * b = 8 * a + 168 := by\n    ring_nf at h\u2080 h\u2081 \u22a2\n    field_simp at h\u2080 h\u2081 \u22a2\n    norm_cast at h\u2080 h\u2081 \u22a2\n    linarith\n  have eq3 : 6 * b = 8 * a := by\n    norm_num at *\n    field_simp at *\n    ring_nf at *\n    linarith\n  have b_eq : b = (4 * a) / 3 := by\n    field_simp at h\u2080 h\u2081 eq1a eq2a eq3 \u22a2\n    ring_nf at h\u2080 h\u2081 eq1a eq2a eq3 \u22a2\n    norm_cast at h\u2080 h\u2081 eq1a eq2a eq3 \u22a2\n    <;> omega\n  have eq4 : 9 * ((4 * a) / 3) = 8 * a + 168 := by\n    simp_all [mul_comm]\n    <;> ring_nf\n    <;> norm_cast\n    <;> omega\n  have eq5 : 12 * a = 8 * a + 168 := by\n    ring_nf at h\u2080 h\u2081 \u22a2\n    field_simp at h\u2080 h\u2081 \u22a2\n    norm_cast at h\u2080 h\u2081 \u22a2\n    linarith\n  have a_eq : a = 42 := by\n    ring_nf at eq5\n    norm_num\n    linarith\n  linarith\n", "proof_before": "theorem amc12b_2020_p5 (a b : \u2115) (h\u2080 : (5 : \u211a) / 8 * b = 2 / 3 * a + 7)\n  (h\u2081 : (b : \u211a) - 5 / 8 * b = a - 2 / 3 * a + 7) : a = 42  :=  by\n  have eq1a : 15 * b = 16 * a + 168 := by\n    field_simp at h\u2080 h\u2081\n    ring_nf at h\u2080 h\u2081\n    norm_cast at h\u2080 h\u2081\n    omega\n  have eq2a : 9 * b = 8 * a + 168 := by\n    ring_nf at h\u2080 h\u2081 \u22a2\n    field_simp at h\u2080 h\u2081 \u22a2\n    norm_cast at h\u2080 h\u2081 \u22a2\n    linarith\n  have eq3 : 6 * b = 8 * a := by\n    norm_num at *\n    field_simp at *\n    ring_nf at *\n    linarith\n  have b_eq : b = (4 * a) / 3 := by\n    field_simp at h\u2080 h\u2081 eq1a eq2a eq3 \u22a2\n    ring_nf at h\u2080 h\u2081 eq1a eq2a eq3 \u22a2\n    norm_cast at h\u2080 h\u2081 eq1a eq2a eq3 \u22a2\n    <;> omega\n  have eq4 : 9 * ((4 * a) / 3) = 8 * a + 168 := by\n    simp_all [mul_comm]\n    <;> ring_nf\n    <;> norm_cast\n    <;> omega\n  have eq5 : 12 * a = 8 * a + 168 := by\n    ring_nf at h\u2080 h\u2081 \u22a2\n    field_simp at h\u2080 h\u2081 \u22a2\n    norm_cast at h\u2080 h\u2081 \u22a2\n    linarith\n  have a_eq : a = 42 := by\n    ring_nf at eq5\n    norm_num\n    linarith\n  linarith\n"}
{"mode": "clear", "steps_before": 90, "alphabet": 33, "compiles": false, "id": "imo_1966_p5", "proof_after": "theorem imo_1966_p5 (x a : \u2115 \u2192 \u211d) (h\u2080 : a 1 \u2260 a 2) (h\u2081 : a 1 \u2260 a 3) (h\u2082 : a 1 \u2260 a 4)\n  (h\u2083 : a 2 \u2260 a 3) (h\u2084 : a 2 \u2260 a 4) (h\u2085 : a 3 \u2260 a 4) (h\u2086 : a 1 > a 2) (h\u2087 : a 2 > a 3)\n  (h\u2088 : a 3 > a 4)\n  (h\u2089 : abs (a 1 - a 2) * x 2 + abs (a 1 - a 3) * x 3 + abs (a 1 - a 4) * x 4 = 1)\n  (h\u2081\u2080 : abs (a 2 - a 1) * x 1 + abs (a 2 - a 3) * x 3 + abs (a 2 - a 4) * x 4 = 1)\n  (h\u2081\u2081 : abs (a 3 - a 1) * x 1 + abs (a 3 - a 2) * x 2 + abs (a 3 - a 4) * x 4 = 1)\n  (h\u2081\u2082 : abs (a 4 - a 1) * x 1 + abs (a 4 - a 2) * x 2 + abs (a 4 - a 3) * x 3 = 1) :\n  x 2 = 0 \u2227 x 3 = 0 \u2227 x 1 = 1 / abs (a 1 - a 4) \u2227 x 4 = 1 / abs (a 1 - a 4)  :=  by\n  have eq1 : (a 1 - a 2) * x 2 + (a 1 - a 3) * x 3 + (a 1 - a 4) * x 4 = 1 := by\n    cases' le_total 0 (a 1 - a 2) with h h <;> cases' le_total 0 (a 1 - a 3) with h' h' <;> cases' le_total 0 (a 1 - a 4) with h'' h'' <;>\n      simp_all [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos]\n    <;> linarith\n  have eq2 : (a 1 - a 2) * x 1 + (a 2 - a 3) * x 3 + (a 2 - a 4) * x 4 = 1 := by\n    cases' le_total 0 (a 1 - a 2) with h\u2081\u2083 h\u2081\u2083 <;>\n      cases' le_total 0 (a 1 - a 3) with h\u2081\u2084 h\u2081\u2084 <;>\n        cases' le_total 0 (a 1 - a 4) with h\u2081\u2085 h\u2081\u2085 <;>\n          cases' le_total 0 (a 2 - a 3) with h\u2081\u2086 h\u2081\u2086 <;>\n            cases' le_total 0 (a 2 - a 4) with h\u2081\u2087 h\u2081\u2087 <;>\n              cases' le_total 0 (a 3 - a 4) with h\u2081\u2088 h\u2081\u2088 <;>\n                simp_all only [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos] <;>\n                  linarith\n  have eq3 : (a 1 - a 3) * x 1 + (a 2 - a 3) * x 2 + (a 3 - a 4) * x 4 = 1 := by\n    cases' le_total 0 (a 1 - a 2) with h h <;>\n      cases' le_total 0 (a 1 - a 3) with h' h' <;>\n        cases' le_total 0 (a 1 - a 4) with h'' h'' <;>\n          cases' le_total 0 (a 2 - a 3) with h''' h''' <;>\n            cases' le_total 0 (a 2 - a 4) with h'''' h'''' <;>\n              cases' le_total 0 (a 3 - a 4) with h'''''' h'''''' <;>\n                simp_all only [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos] <;>\n                  linarith\n  have eq4 : (a 1 - a 4) * x 1 + (a 2 - a 4) * x 2 + (a 3 - a 4) * x 3 = 1 := by\n    have h\u2081\u2083 : (a 1 - a 4) * x 1 + (a 2 - a 4) * x 2 + (a 3 - a 4) * x 3 = 1 := by\n      cases' le_total 0 (a 1 - a 2) with h h <;>\n      cases' le_total 0 (a 1 - a 3) with h' h' <;>\n      cases' le_total 0 (a 1 - a 4) with h'' h'' <;>\n      cases' le_total 0 (a 2 - a 3) with h''' h''' <;>\n      cases' le_total 0 (a 2 - a 4) with h'''' h'''' <;>\n      cases' le_total 0 (a 3 - a 4) with h'''''' h'''''' <;>\n      simp_all only [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos] <;>\n      linarith\n    exact h\u2081\u2083\n  have x2_eq_0 : x 2 = 0 := by\n    apply Eq.symm\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2080)\n    nlinarith [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080, h\u2081\u2081, h\u2081\u2082, eq1, eq2, eq3, eq4]\n  have x3_eq_0 : x 3 = 0 := by\n    have h\u2081\u2083 : x 3 = 0 := by\n      have h\u2081\u2084 : (a 1 - a 2) * 0 + (a 1 - a 3) * x 3 + (a 1 - a 4) * x 4 = 1 := by rw [x2_eq_0] at eq1; exact eq1\n      have h\u2081\u2085 : (a 1 - a 3) * x 3 + (a 1 - a 4) * x 4 = 1 := by linarith\n      have h\u2081\u2086 : (a 1 - a 3) * x 1 + (a 2 - a 3) * 0 + (a 3 - a 4) * x 4 = 1 := by rw [x2_eq_0] at eq3; exact eq3\n      have h\u2081\u2087 : (a 1 - a 3) * x 1 + (a 3 - a 4) * x 4 = 1 := by linarith\n      have h\u2081\u2088 : (a 1 - a 3) * x 1 + (a 3 - a 4) * x 4 = 1 := by linarith\n      have h\u2081\u2089 : x 3 = 0 := by\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2081)\n        nlinarith [h\u2081\u2085, h\u2081\u2087, h\u2081\u2088]\n      exact h\u2081\u2089\n    exact h\u2081\u2083\n  have x1_val : x 1 = 1 / abs (a 1 - a 4) := by\n    have h14 : a 1 - a 4 > 0 := by linarith\n    field_simp [abs_of_pos h14]\n    nlinarith\n  have x4_val : x 4 = 1 / abs (a 1 - a 4) := by\n    simp_all [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos]\n    <;> ring_nf at *\n    <;> nlinarith\n  exact \u27e8x2_eq_0, x3_eq_0, x1_val, x4_val\u27e9\n", "proof_before": "theorem imo_1966_p5 (x a : \u2115 \u2192 \u211d) (h\u2080 : a 1 \u2260 a 2) (h\u2081 : a 1 \u2260 a 3) (h\u2082 : a 1 \u2260 a 4)\n  (h\u2083 : a 2 \u2260 a 3) (h\u2084 : a 2 \u2260 a 4) (h\u2085 : a 3 \u2260 a 4) (h\u2086 : a 1 > a 2) (h\u2087 : a 2 > a 3)\n  (h\u2088 : a 3 > a 4)\n  (h\u2089 : abs (a 1 - a 2) * x 2 + abs (a 1 - a 3) * x 3 + abs (a 1 - a 4) * x 4 = 1)\n  (h\u2081\u2080 : abs (a 2 - a 1) * x 1 + abs (a 2 - a 3) * x 3 + abs (a 2 - a 4) * x 4 = 1)\n  (h\u2081\u2081 : abs (a 3 - a 1) * x 1 + abs (a 3 - a 2) * x 2 + abs (a 3 - a 4) * x 4 = 1)\n  (h\u2081\u2082 : abs (a 4 - a 1) * x 1 + abs (a 4 - a 2) * x 2 + abs (a 4 - a 3) * x 3 = 1) :\n  x 2 = 0 \u2227 x 3 = 0 \u2227 x 1 = 1 / abs (a 1 - a 4) \u2227 x 4 = 1 / abs (a 1 - a 4)  :=  by\n  have eq1 : (a 1 - a 2) * x 2 + (a 1 - a 3) * x 3 + (a 1 - a 4) * x 4 = 1 := by\n    cases' le_total 0 (a 1 - a 2) with h h <;> cases' le_total 0 (a 1 - a 3) with h' h' <;> cases' le_total 0 (a 1 - a 4) with h'' h'' <;>\n      simp_all [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos]\n    <;> linarith\n  have eq2 : (a 1 - a 2) * x 1 + (a 2 - a 3) * x 3 + (a 2 - a 4) * x 4 = 1 := by\n    cases' le_total 0 (a 1 - a 2) with h\u2081\u2083 h\u2081\u2083 <;>\n      cases' le_total 0 (a 1 - a 3) with h\u2081\u2084 h\u2081\u2084 <;>\n        cases' le_total 0 (a 1 - a 4) with h\u2081\u2085 h\u2081\u2085 <;>\n          cases' le_total 0 (a 2 - a 3) with h\u2081\u2086 h\u2081\u2086 <;>\n            cases' le_total 0 (a 2 - a 4) with h\u2081\u2087 h\u2081\u2087 <;>\n              cases' le_total 0 (a 3 - a 4) with h\u2081\u2088 h\u2081\u2088 <;>\n                simp_all only [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos] <;>\n                  linarith\n  have eq3 : (a 1 - a 3) * x 1 + (a 2 - a 3) * x 2 + (a 3 - a 4) * x 4 = 1 := by\n    cases' le_total 0 (a 1 - a 2) with h h <;>\n      cases' le_total 0 (a 1 - a 3) with h' h' <;>\n        cases' le_total 0 (a 1 - a 4) with h'' h'' <;>\n          cases' le_total 0 (a 2 - a 3) with h''' h''' <;>\n            cases' le_total 0 (a 2 - a 4) with h'''' h'''' <;>\n              cases' le_total 0 (a 3 - a 4) with h'''''' h'''''' <;>\n                simp_all only [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos] <;>\n                  linarith\n  have eq4 : (a 1 - a 4) * x 1 + (a 2 - a 4) * x 2 + (a 3 - a 4) * x 3 = 1 := by\n    have h\u2081\u2083 : (a 1 - a 4) * x 1 + (a 2 - a 4) * x 2 + (a 3 - a 4) * x 3 = 1 := by\n      cases' le_total 0 (a 1 - a 2) with h h <;>\n      cases' le_total 0 (a 1 - a 3) with h' h' <;>\n      cases' le_total 0 (a 1 - a 4) with h'' h'' <;>\n      cases' le_total 0 (a 2 - a 3) with h''' h''' <;>\n      cases' le_total 0 (a 2 - a 4) with h'''' h'''' <;>\n      cases' le_total 0 (a 3 - a 4) with h'''''' h'''''' <;>\n      simp_all only [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos] <;>\n      linarith\n    exact h\u2081\u2083\n  have x2_eq_0 : x 2 = 0 := by\n    apply Eq.symm\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2080)\n    nlinarith [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080, h\u2081\u2081, h\u2081\u2082, eq1, eq2, eq3, eq4]\n  have x3_eq_0 : x 3 = 0 := by\n    have h\u2081\u2083 : x 3 = 0 := by\n      have h\u2081\u2084 : (a 1 - a 2) * 0 + (a 1 - a 3) * x 3 + (a 1 - a 4) * x 4 = 1 := by rw [x2_eq_0] at eq1; exact eq1\n      have h\u2081\u2085 : (a 1 - a 3) * x 3 + (a 1 - a 4) * x 4 = 1 := by linarith\n      have h\u2081\u2086 : (a 1 - a 3) * x 1 + (a 2 - a 3) * 0 + (a 3 - a 4) * x 4 = 1 := by rw [x2_eq_0] at eq3; exact eq3\n      have h\u2081\u2087 : (a 1 - a 3) * x 1 + (a 3 - a 4) * x 4 = 1 := by linarith\n      have h\u2081\u2088 : (a 1 - a 3) * x 1 + (a 3 - a 4) * x 4 = 1 := by linarith\n      have h\u2081\u2089 : x 3 = 0 := by\n        apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2081)\n        nlinarith [h\u2081\u2085, h\u2081\u2087, h\u2081\u2088]\n      exact h\u2081\u2089\n    exact h\u2081\u2083\n  have x1_val : x 1 = 1 / abs (a 1 - a 4) := by\n    have h14 : a 1 - a 4 > 0 := by linarith\n    field_simp [abs_of_pos h14]\n    nlinarith\n  have x4_val : x 4 = 1 / abs (a 1 - a 4) := by\n    simp_all [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos]\n    <;> ring_nf at *\n    <;> nlinarith\n  exact \u27e8x2_eq_0, x3_eq_0, x1_val, x4_val\u27e9\n"}
{"mode": "clear", "steps_before": 70, "alphabet": 63, "compiles": false, "id": "algebra_amgm_prod1toneq1_sum1tongeqn", "proof_after": "theorem algebra_amgm_prod1toneq1_sum1tongeqn (a : \u2115 \u2192 NNReal) (n : \u2115)\n  (h\u2080 : Finset.prod (Finset.range n) a = 1) : Finset.sum (Finset.range n) a \u2265 n  :=  by\n  have am_gm_inequality : \u2200 (x : \u2115 \u2192 NNReal) (m : \u2115), \n    Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := by\n    have step0 : \u2200 (x : \u2115 \u2192 NNReal) (m : \u2115), Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := by\n      intro x m\n      have h\u2081 : \u2200 T : Type*, T \u2192 \u2115 \u2192 NNReal \u2192 NNReal := by\n        intro T\n        intro t\n        intro m\n        intro a\n        exact a\n      have h\u2082 : Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := by\n        apply?\n      exact h\u2082\n    have step1 : \u2200 (x : \u2115 \u2192 NNReal) (m : \u2115), Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := by\n      intro x m\n      rw [ge_iff_le]\n      have h\u2081 : Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := step0 x m\n      simpa using h\u2081\n    have step2 : \u2200 (x : \u2115 \u2192 NNReal) (m : \u2115), Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := by\n      intro x m\n      have h\u2081 := step0 x m\n      have h\u2082 := step1 x m\n      simp_all\n    exact step0\n  have geometric_mean_eq_one : (Finset.prod (Finset.range n) a) ^ (1 / n) = 1 := by\n    have h\u2081 := am_gm_inequality (fun _ => 1) 0\n    have h\u2082 := am_gm_inequality (fun _ => 1) 1\n    have h\u2083 := am_gm_inequality (fun _ => 1) 2\n    have h\u2084 := am_gm_inequality (fun _ => 1) 3\n    have h\u2085 := am_gm_inequality (fun _ => 1) 4\n    simp at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085\n    <;> norm_num at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085\n    <;> linarith\n  have arithmetic_mean_ge_one : Finset.sum (Finset.range n) a / n \u2265 1 := by\n    have h\u2081 : Finset.sum (Finset.range n) a / n \u2265 1 := by\n      have h\u2082 := am_gm_inequality a n\n      have h\u2083 := am_gm_inequality (fun _ => 0) 0\n      have h\u2084 := am_gm_inequality (fun _ => 1) 0\n      have h\u2085 := am_gm_inequality (fun _ => 1) 1\n      have h\u2086 := am_gm_inequality (fun _ => 1) 2\n      have h\u2087 := am_gm_inequality (fun _ => 1) 3\n      have h\u2088 := am_gm_inequality (fun _ => 1) 4\n      have h\u2089 := am_gm_inequality (fun _ => 1) 5\n      have h\u2081\u2080 := am_gm_inequality (fun _ => 1) 6\n      have h\u2081\u2081 := am_gm_inequality (fun _ => 1) 7\n      have h\u2081\u2082 := am_gm_inequality (fun _ => 1) 8\n      have h\u2081\u2083 := am_gm_inequality (fun _ => 1) 9\n      have h\u2081\u2084 := am_gm_inequality (fun _ => 1) 10\n      norm_num at h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084\n      <;> simp_all [h\u2080, geometric_mean_eq_one]\n      <;> norm_num\n      <;> linarith\n    exact h\u2081\n  have sum_ge_n : Finset.sum (Finset.range n) a \u2265 n := by\n    have h\u2081 := am_gm_inequality (fun i => 1) 0\n    have h\u2082 := am_gm_inequality (fun i => 1) 1\n    have h\u2083 := am_gm_inequality a n\n    have h\u2084 := am_gm_inequality (fun i => 1) 2\n    have h\u2085 := am_gm_inequality (fun i => 1) 3\n    have h\u2086 := am_gm_inequality (fun i => 1) 4\n    have h\u2087 := am_gm_inequality (fun i => 1) 5\n    have h\u2088 := am_gm_inequality (fun i => 1) 6\n    have h\u2089 := am_gm_inequality (fun i => 1) 7\n    have h\u2081\u2080 := am_gm_inequality (fun i => 1) 8\n    have h\u2081\u2081 := am_gm_inequality (fun i => 1) 9\n    have h\u2081\u2082 := am_gm_inequality (fun i => 1) 10\n    simp at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n    <;> norm_num at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n    <;> nlinarith\n  simpa using sum_ge_n\n", "proof_before": "theorem algebra_amgm_prod1toneq1_sum1tongeqn (a : \u2115 \u2192 NNReal) (n : \u2115)\n  (h\u2080 : Finset.prod (Finset.range n) a = 1) : Finset.sum (Finset.range n) a \u2265 n  :=  by\n  have am_gm_inequality : \u2200 (x : \u2115 \u2192 NNReal) (m : \u2115), \n    Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := by\n    have step0 : \u2200 (x : \u2115 \u2192 NNReal) (m : \u2115), Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := by\n      intro x m\n      have h\u2081 : \u2200 T : Type*, T \u2192 \u2115 \u2192 NNReal \u2192 NNReal := by\n        intro T\n        intro t\n        intro m\n        intro a\n        exact a\n      have h\u2082 : Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := by\n        apply?\n      exact h\u2082\n    have step1 : \u2200 (x : \u2115 \u2192 NNReal) (m : \u2115), Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := by\n      intro x m\n      rw [ge_iff_le]\n      have h\u2081 : Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := step0 x m\n      simpa using h\u2081\n    have step2 : \u2200 (x : \u2115 \u2192 NNReal) (m : \u2115), Finset.sum (Finset.range m) x / m \u2265 (Finset.prod (Finset.range m) x) ^ (1 / m) := by\n      intro x m\n      have h\u2081 := step0 x m\n      have h\u2082 := step1 x m\n      simp_all\n    exact step0\n  have geometric_mean_eq_one : (Finset.prod (Finset.range n) a) ^ (1 / n) = 1 := by\n    have h\u2081 := am_gm_inequality (fun _ => 1) 0\n    have h\u2082 := am_gm_inequality (fun _ => 1) 1\n    have h\u2083 := am_gm_inequality (fun _ => 1) 2\n    have h\u2084 := am_gm_inequality (fun _ => 1) 3\n    have h\u2085 := am_gm_inequality (fun _ => 1) 4\n    simp at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085\n    <;> norm_num at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085\n    <;> linarith\n  have arithmetic_mean_ge_one : Finset.sum (Finset.range n) a / n \u2265 1 := by\n    have h\u2081 : Finset.sum (Finset.range n) a / n \u2265 1 := by\n      have h\u2082 := am_gm_inequality a n\n      have h\u2083 := am_gm_inequality (fun _ => 0) 0\n      have h\u2084 := am_gm_inequality (fun _ => 1) 0\n      have h\u2085 := am_gm_inequality (fun _ => 1) 1\n      have h\u2086 := am_gm_inequality (fun _ => 1) 2\n      have h\u2087 := am_gm_inequality (fun _ => 1) 3\n      have h\u2088 := am_gm_inequality (fun _ => 1) 4\n      have h\u2089 := am_gm_inequality (fun _ => 1) 5\n      have h\u2081\u2080 := am_gm_inequality (fun _ => 1) 6\n      have h\u2081\u2081 := am_gm_inequality (fun _ => 1) 7\n      have h\u2081\u2082 := am_gm_inequality (fun _ => 1) 8\n      have h\u2081\u2083 := am_gm_inequality (fun _ => 1) 9\n      have h\u2081\u2084 := am_gm_inequality (fun _ => 1) 10\n      norm_num at h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084\n      <;> simp_all [h\u2080, geometric_mean_eq_one]\n      <;> norm_num\n      <;> linarith\n    exact h\u2081\n  have sum_ge_n : Finset.sum (Finset.range n) a \u2265 n := by\n    have h\u2081 := am_gm_inequality (fun i => 1) 0\n    have h\u2082 := am_gm_inequality (fun i => 1) 1\n    have h\u2083 := am_gm_inequality a n\n    have h\u2084 := am_gm_inequality (fun i => 1) 2\n    have h\u2085 := am_gm_inequality (fun i => 1) 3\n    have h\u2086 := am_gm_inequality (fun i => 1) 4\n    have h\u2087 := am_gm_inequality (fun i => 1) 5\n    have h\u2088 := am_gm_inequality (fun i => 1) 6\n    have h\u2089 := am_gm_inequality (fun i => 1) 7\n    have h\u2081\u2080 := am_gm_inequality (fun i => 1) 8\n    have h\u2081\u2081 := am_gm_inequality (fun i => 1) 9\n    have h\u2081\u2082 := am_gm_inequality (fun i => 1) 10\n    simp at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n    <;> norm_num at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n    <;> nlinarith\n  simpa using sum_ge_n\n"}
{"mode": "clear", "steps_before": 2, "alphabet": 11, "compiles": false, "id": "amc12a_2003_p1", "proof_after": "theorem amc12a_2003_p1 (u v : \u2115 \u2192 \u2115) (h\u2080 : \u2200 n, u n = 2 * n + 2) (h\u2081 : \u2200 n, v n = 2 * n + 1) :\n    ((\u2211 k in Finset.range 2003, u k) - \u2211 k in Finset.range 2003, v k) = 2003  :=  by\n  simp_all only [mul_add, mul_one, mul_comm, Nat.mul_sub_left_distrib]\n  rfl\n", "proof_before": "theorem amc12a_2003_p1 (u v : \u2115 \u2192 \u2115) (h\u2080 : \u2200 n, u n = 2 * n + 2) (h\u2081 : \u2200 n, v n = 2 * n + 1) :\n    ((\u2211 k in Finset.range 2003, u k) - \u2211 k in Finset.range 2003, v k) = 2003  :=  by\n  simp_all only [mul_add, mul_one, mul_comm, Nat.mul_sub_left_distrib]\n  rfl\n"}
{"mode": "clear", "steps_before": 56, "alphabet": 31, "compiles": false, "id": "mathd_algebra_482", "proof_after": "theorem mathd_algebra_482 (m n : \u2115) (k : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : Nat.Prime m) (h\u2081 : Nat.Prime n)\n  (h\u2082 : \u2200 x, f x = x ^ 2 - 12 * x + k) (h\u2083 : f m = 0) (h\u2084 : f n = 0) (h\u2085 : m \u2260 n) : k = 35  :=  by\n  have sum_roots : m + n = 12 := by\n    simp only [h\u2082, h\u2083, h\u2084] at *\n    have h\u2086 : (m : \u211d) ^ 2 - 12 * m + k = 0 := by simp_all\n    have h\u2087 : (n : \u211d) ^ 2 - 12 * n + k = 0 := by simp_all\n    have h\u2088 : (m : \u211d) \u2260 (n : \u211d) := by simp_all\n    have h\u2089 : (m : \u211d) + (n : \u211d) = 12 := by\n      apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2088)\n      nlinarith\n    have h\u2081\u2080 : m + n = 12 := by\n      norm_cast at h\u2089\n      <;> linarith\n    exact h\u2081\u2080\n  have product_roots : m * n = k := by\n    rw [h\u2082] at h\u2083 h\u2084\n    norm_num at h\u2083 h\u2084\n    have : m = 5 \u2227 n = 7 \u2228 m = 7 \u2227 n = 5 := by\n      have : m \u2264 11 := by\n        nlinarith [Nat.Prime.two_le h\u2080, Nat.Prime.two_le h\u2081]\n      have : n \u2264 11 := by\n        nlinarith [Nat.Prime.two_le h\u2080, Nat.Prime.two_le h\u2081]\n      interval_cases m <;> interval_cases n <;> norm_num [h\u2083, h\u2084] at * <;> aesop\n    rcases this with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9) <;> norm_num [h\u2082] at h\u2083 h\u2084 \u22a2 <;> linarith\n  have prime_pair : (m = 5 \u2227 n = 7) \u2228 (m = 7 \u2227 n = 5) := by\n    have : m < 12 := by\n      linarith [Nat.Prime.two_le h\u2080, Nat.Prime.two_le h\u2081]\n    have : n < 12 := by\n      linarith [Nat.Prime.two_le h\u2080, Nat.Prime.two_le h\u2081]\n    interval_cases m <;> interval_cases n <;> norm_num [Nat.Prime] at h\u2080 h\u2081 <;> simp_all [h\u2082]\n    <;> nlinarith\n  have k_is_35 : k = 35 := by\n    have h\u2086 : k = 35 := by\n      cases' prime_pair with h\u2086 h\u2086\n      \u00b7 \n        simp_all [h\u2082, h\u2083, h\u2084]\n        linarith\n      \u00b7 \n        simp_all [h\u2082, h\u2083, h\u2084]\n        linarith\n    exact h\u2086\n  have h\u2086 := h\u2082\n  have h\u2087 := h\u2083\n  have h\u2088 := h\u2084\n  have h\u2089 := h\u2085\n  simp_all\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_482 (m n : \u2115) (k : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : Nat.Prime m) (h\u2081 : Nat.Prime n)\n  (h\u2082 : \u2200 x, f x = x ^ 2 - 12 * x + k) (h\u2083 : f m = 0) (h\u2084 : f n = 0) (h\u2085 : m \u2260 n) : k = 35  :=  by\n  have sum_roots : m + n = 12 := by\n    simp only [h\u2082, h\u2083, h\u2084] at *\n    have h\u2086 : (m : \u211d) ^ 2 - 12 * m + k = 0 := by simp_all\n    have h\u2087 : (n : \u211d) ^ 2 - 12 * n + k = 0 := by simp_all\n    have h\u2088 : (m : \u211d) \u2260 (n : \u211d) := by simp_all\n    have h\u2089 : (m : \u211d) + (n : \u211d) = 12 := by\n      apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2088)\n      nlinarith\n    have h\u2081\u2080 : m + n = 12 := by\n      norm_cast at h\u2089\n      <;> linarith\n    exact h\u2081\u2080\n  have product_roots : m * n = k := by\n    rw [h\u2082] at h\u2083 h\u2084\n    norm_num at h\u2083 h\u2084\n    have : m = 5 \u2227 n = 7 \u2228 m = 7 \u2227 n = 5 := by\n      have : m \u2264 11 := by\n        nlinarith [Nat.Prime.two_le h\u2080, Nat.Prime.two_le h\u2081]\n      have : n \u2264 11 := by\n        nlinarith [Nat.Prime.two_le h\u2080, Nat.Prime.two_le h\u2081]\n      interval_cases m <;> interval_cases n <;> norm_num [h\u2083, h\u2084] at * <;> aesop\n    rcases this with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9) <;> norm_num [h\u2082] at h\u2083 h\u2084 \u22a2 <;> linarith\n  have prime_pair : (m = 5 \u2227 n = 7) \u2228 (m = 7 \u2227 n = 5) := by\n    have : m < 12 := by\n      linarith [Nat.Prime.two_le h\u2080, Nat.Prime.two_le h\u2081]\n    have : n < 12 := by\n      linarith [Nat.Prime.two_le h\u2080, Nat.Prime.two_le h\u2081]\n    interval_cases m <;> interval_cases n <;> norm_num [Nat.Prime] at h\u2080 h\u2081 <;> simp_all [h\u2082]\n    <;> nlinarith\n  have k_is_35 : k = 35 := by\n    have h\u2086 : k = 35 := by\n      cases' prime_pair with h\u2086 h\u2086\n      \u00b7 \n        simp_all [h\u2082, h\u2083, h\u2084]\n        linarith\n      \u00b7 \n        simp_all [h\u2082, h\u2083, h\u2084]\n        linarith\n    exact h\u2086\n  have h\u2086 := h\u2082\n  have h\u2087 := h\u2083\n  have h\u2088 := h\u2084\n  have h\u2089 := h\u2085\n  simp_all\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 19, "alphabet": 18, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 653, "chars_after": 7, "seconds": 0.0, "id": "algebra_2rootsintpoly_am10tap11eqasqpam110", "proof_after": "theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : \u2102) :\n    (a - 10) * (a + 11) = a ^ 2 + a - 110  :=  by\n  ring\n", "proof_before": "theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : \u2102) :\n    (a - 10) * (a + 11) = a ^ 2 + a - 110  :=  by\n  have step1 : (a - 10) * (a + 11) = a * a + a * 11 - 10 * a - 10 * 11 := by\n    simp only [mul_add, mul_sub, mul_one, mul_neg, mul_comm]\n    ring\n    <;> ring\n    <;> ring\n    <;> ring\n  have step2 : a * a + a * 11 - 10 * a - 10 * 11 = a^2 + (11 * a - 10 * a) - 110 := by\n    rw [eq_comm] at step1\n    linear_combination step1\n  have step3 : a^2 + (11 * a - 10 * a) - 110 = a^2 + a - 110 := by\n    simp_all only [sub_mul, add_mul, mul_add, mul_sub, mul_one, mul_neg, mul_comm]\n    <;> ring\n    <;> simp_all only [mul_comm]\n    <;> ring\n  have h : (a - 10) * (a + 11) = a ^ 2 + a - 110 := by\n    rw [step1]  \n    rw [step2]  \n    rw [step3]  \n  exact h\n"}
{"mode": "clear", "steps_before": 21, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 539, "chars_after": 8, "seconds": 0.7, "id": "mathd_numbertheory_458", "proof_after": "theorem mathd_numbertheory_458 (n : \u2115) (h\u2080 : n % 8 = 7) : n % 4 = 3  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_458 (n : \u2115) (h\u2080 : n % 8 = 7) : n % 4 = 3  :=  by\n  have h\u2081 : \u2203 k, n = 8 * k + 7 := by\n    use n / 8\n    have h\u2081 := Nat.div_add_mod n 8\n    simp [h\u2080, Nat.mul_comm] at h\u2081\n    omega\n  have h\u2082 : 8 % 4 = 0 := by\n    norm_num\n    <;>\n    simp_all\n    <;>\n    omega\n  have h\u2083 : 7 % 4 = 3 := by\n    simp_all [Nat.mod_eq_of_lt, show 0 < 4 by norm_num, show 0 < 8 by norm_num]\n    <;> omega\n  have h\u2084 : n % 4 = 3 := by\n    obtain \u27e8k, rfl\u27e9 := h\u2081\n    simp [h\u2082, h\u2083, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n    <;> omega\n  have h\u2085 : n % 4 = 3 := by\n    simp_all [Nat.ModEq, Nat.mod_eq_of_lt]\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 18, "alphabet": 19, "compiles": false, "id": "aime_1988_p4", "proof_after": "theorem aime_1988_p4 (n : \u2115) (a : \u2115 \u2192 \u211d) (h\u2080 : \u2200 n, abs (a n) < 1)\n  (h\u2081 : (\u2211 k in Finset.range n, abs (a k)) = 19 + abs (\u2211 k in Finset.range n, a k)) : 20 \u2264 n  :=  by\n  have h_S : \u2211 k in Finset.range n, abs (a k) \u2265 0 := by\n    nlinarith [abs_nonneg (\u2211 k in Finset.range n, a k)]\n  have h_triangle : abs (\u2211 k in Finset.range n, a k) \u2264 \u2211 k in Finset.range n, abs (a k) := by\n    cases' le_total 0 (\u2211 k in Finset.range n, a k) with h h <;> simp_all [abs_of_nonneg, abs_of_nonpos, le_refl]\n    <;> linarith\n  have h_subst : \u2211 k in Finset.range n, abs (a k) = 19 + abs (\u2211 k in Finset.range n, a k) := by\n    simpa [abs_le] using h\u2081\n  have h_sum_ge_19 : \u2211 k in Finset.range n, abs (a k) \u2265 19 := by\n    have h\u2082 := h\u2081\n    linarith [abs_nonneg (\u2211 k in Finset.range n, a k)]\n  have h_need_20 : 20 \u2264 n := by\n    contrapose! h_sum_ge_19\n    interval_cases n <;> norm_num [Finset.sum_range_succ, abs_of_nonneg, abs_of_nonpos] at h\u2081 \u22a2 <;>\n      linarith [h\u2080 0, h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, h\u2080 5, h\u2080 6, h\u2080 7, h\u2080 8, h\u2080 9, h\u2080 10, h\u2080 11, h\u2080 12, h\u2080 13,\n        h\u2080 14, h\u2080 15, h\u2080 16, h\u2080 17, h\u2080 18, h\u2080 19]\n  linarith\n", "proof_before": "theorem aime_1988_p4 (n : \u2115) (a : \u2115 \u2192 \u211d) (h\u2080 : \u2200 n, abs (a n) < 1)\n  (h\u2081 : (\u2211 k in Finset.range n, abs (a k)) = 19 + abs (\u2211 k in Finset.range n, a k)) : 20 \u2264 n  :=  by\n  have h_S : \u2211 k in Finset.range n, abs (a k) \u2265 0 := by\n    nlinarith [abs_nonneg (\u2211 k in Finset.range n, a k)]\n  have h_triangle : abs (\u2211 k in Finset.range n, a k) \u2264 \u2211 k in Finset.range n, abs (a k) := by\n    cases' le_total 0 (\u2211 k in Finset.range n, a k) with h h <;> simp_all [abs_of_nonneg, abs_of_nonpos, le_refl]\n    <;> linarith\n  have h_subst : \u2211 k in Finset.range n, abs (a k) = 19 + abs (\u2211 k in Finset.range n, a k) := by\n    simpa [abs_le] using h\u2081\n  have h_sum_ge_19 : \u2211 k in Finset.range n, abs (a k) \u2265 19 := by\n    have h\u2082 := h\u2081\n    linarith [abs_nonneg (\u2211 k in Finset.range n, a k)]\n  have h_need_20 : 20 \u2264 n := by\n    contrapose! h_sum_ge_19\n    interval_cases n <;> norm_num [Finset.sum_range_succ, abs_of_nonneg, abs_of_nonpos] at h\u2081 \u22a2 <;>\n      linarith [h\u2080 0, h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, h\u2080 5, h\u2080 6, h\u2080 7, h\u2080 8, h\u2080 9, h\u2080 10, h\u2080 11, h\u2080 12, h\u2080 13,\n        h\u2080 14, h\u2080 15, h\u2080 16, h\u2080 17, h\u2080 18, h\u2080 19]\n  linarith\n"}
{"mode": "clear", "steps_before": 27, "alphabet": 21, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1593, "chars_after": 35, "seconds": 1.0, "id": "algebra_sqineq_2at2pclta2c2p41pc", "proof_after": "theorem algebra_sqineq_2at2pclta2c2p41pc (a c : \u211d) :\n  2 * a * (2 + c) \u2264 a ^ 2 + c ^ 2 + 4 * (1 + c)  :=  by\n  linarith [sq_nonneg (a - c - 2)]\n", "proof_before": "theorem algebra_sqineq_2at2pclta2c2p41pc (a c : \u211d) :\n  2 * a * (2 + c) \u2264 a ^ 2 + c ^ 2 + 4 * (1 + c)  :=  by\n  have step1 : 2 * a * (2 + c) = 4 * a + 2 * a * c := by\n    simp only [mul_add, mul_comm, mul_left_comm, mul_assoc]\n    ring\n  have step2 : a ^ 2 + c ^ 2 + 4 * (1 + c) = a ^ 2 + c ^ 2 + 4 + 4 * c := by\n    simp_all only [mul_add, mul_one, mul_assoc, mul_comm, mul_left_comm, add_assoc, add_left_comm,\n      add_comm]\n    ring\n  have step3 : 4 * a + 2 * a * c \u2264 a ^ 2 + c ^ 2 + 4 + 4 * c := by\n    have h : 0 \u2264 (a - c) ^ 2 - 4 * (a - c) + 4 := by\n      have : 0 \u2264 (a - c - 2) ^ 2 := sq_nonneg (a - c - 2)\n      nlinarith [sq_nonneg (a - c - 2)]\n    linarith [sq_nonneg (a - c - 2)]\n  have step4 : 0 \u2264 a ^ 2 + c ^ 2 + 4 + 4 * c - 4 * a - 2 * a * c := by\n    nlinarith [sq_nonneg (a - c - 2), sq_nonneg (a - 2), sq_nonneg (c - 2),\n      sq_nonneg (a + c - 2), sq_nonneg (a + c - 1), sq_nonneg (a + c + 1)]\n  have step5 : a ^ 2 - 4 * a = (a - 2) ^ 2 - 4 := by\n    nlinarith [sq_nonneg (a - 2), sq_nonneg (a - c), sq_nonneg (c - 2)]\n  have step6 : c ^ 2 + 4 * c = (c + 2) ^ 2 - 4 := by\n    have h\u2080 : a ^ 2 - 4 * a = (a - 2) ^ 2 - 4 := step5\n    have h\u2081 : a ^ 2 - 4 * a + 4 = (a - 2) ^ 2 := by linarith\n    have h\u2082 : c ^ 2 + 4 * c = (c + 2) ^ 2 - 4 := by linarith\n    exact h\u2082\n  have step7 : 0 \u2264 (a - 2) ^ 2 - 4 + (c + 2) ^ 2 - 4 + 4 - 2 * a * c := by\n    nlinarith [step1, step2, step3, step4, step5, step6, sq_nonneg (a - 2), sq_nonneg (c + 2), sq_nonneg (a - 2 + c + 2)]\n  have step8 : 0 \u2264 (a - 2) ^ 2 + (c + 2) ^ 2 - 2 * a * c - 4 := by\n    linarith [sq_nonneg (a - 2 + c + 2), sq_nonneg (a - 2 - c - 2), step5, step6, step7]\n  linarith [step1, step2, step3, step4, step5, step6, step7, step8]\n"}
{"mode": "clear", "steps_before": 3, "alphabet": 13, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 3, "lower_bound": 3, "exhausted_depth": 2, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 125, "chars_after": 125, "seconds": 1.5, "id": "mathd_numbertheory_690", "proof_after": "theorem mathd_numbertheory_690 :\n    IsLeast { a : \u2115 | 0 < a \u2227 a \u2261 2 [MOD 3] \u2227 a \u2261 4 [MOD 5] \u2227 a \u2261 6 [MOD 7] \u2227 a \u2261 8 [MOD 9] } 314  :=  by\n  refine' \u27e8by decide, fun a \u27e8ha\u2081, ha\u2082, ha\u2083, ha\u2084, ha\u2085\u27e9 => _\u27e9\n  norm_num [ModEq, Nat.modEq_iff_dvd] at ha\u2082 ha\u2083 ha\u2084 ha\u2085\n  omega\n", "proof_before": "theorem mathd_numbertheory_690 :\n    IsLeast { a : \u2115 | 0 < a \u2227 a \u2261 2 [MOD 3] \u2227 a \u2261 4 [MOD 5] \u2227 a \u2261 6 [MOD 7] \u2227 a \u2261 8 [MOD 9] } 314  :=  by\n  refine' \u27e8by decide, fun a \u27e8ha\u2081, ha\u2082, ha\u2083, ha\u2084, ha\u2085\u27e9 => _\u27e9\n  norm_num [ModEq, Nat.modEq_iff_dvd] at ha\u2082 ha\u2083 ha\u2084 ha\u2085\n  omega\n"}
{"mode": "clear", "steps_before": 18, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1234, "chars_after": 104, "seconds": 1.6, "id": "induction_divisibility_3div2tooddnp1", "proof_after": "theorem induction_divisibility_3div2tooddnp1 (n : \u2115) : 3 \u2223 2 ^ (2 * n + 1) + 1  :=  by\n  apply Nat.dvd_of_mod_eq_zero\n  simp [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n", "proof_before": "theorem induction_divisibility_3div2tooddnp1 (n : \u2115) : 3 \u2223 2 ^ (2 * n + 1) + 1  :=  by\n  apply Nat.dvd_of_mod_eq_zero\n  apply Eq.symm\n  apply Eq.symm\n  rw [\u2190 Nat.mod_add_div (2 ^ (2 * n + 1)) 3]\n  simp [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n  <;> simp_all [Nat.pow_mod, Nat.pow_add, Nat.pow_mul, Nat.mul_mod, Nat.add_mod]\n"}
{"mode": "clear", "steps_before": 4, "alphabet": 12, "compiles": false, "id": "mathd_numbertheory_35", "proof_after": "theorem mathd_numbertheory_35 (S : Finset \u2115) (h\u2080 : \u2200 n : \u2115, n \u2223 Nat.sqrt 196) :\n    (\u2211 k in S, k) = 24  :=  by\n  simpa [Nat.sqrt_eq_zero, Nat.div_eq_of_eq_mul_left] using h\u2080 0\n  <;> decide\n  <;> simp_all [Finset.sum_eq_zero]\n  <;> decide\n", "proof_before": "theorem mathd_numbertheory_35 (S : Finset \u2115) (h\u2080 : \u2200 n : \u2115, n \u2223 Nat.sqrt 196) :\n    (\u2211 k in S, k) = 24  :=  by\n  simpa [Nat.sqrt_eq_zero, Nat.div_eq_of_eq_mul_left] using h\u2080 0\n  <;> decide\n  <;> simp_all [Finset.sum_eq_zero]\n  <;> decide\n"}
{"mode": "clear", "steps_before": 64, "alphabet": 47, "compiles": true, "steps_after": 64, "lower_bound": null, "exhausted_depth": 0, "certified": false, "rounds": 0, "fixpoint": true, "chars_before": 2808, "chars_after": 2808, "seconds": 0.2, "id": "numbertheory_sqmod4in01d", "proof_after": "theorem numbertheory_sqmod4in01d (a : \u2124) : a ^ 2 % 4 = 0 \u2228 a ^ 2 % 4 = 1  :=  by\n  have case0 : a % 4 = 0 \u2192 a ^ 2 % 4 = 0 := by\n    intro h\n    have h2 : a ^ 2 % 4 = 0 := by\n      simp [h, pow_two, Int.mul_emod, Int.add_emod]\n    exact h2\n  have case1 : a % 4 = 1 \u2192 a ^ 2 % 4 = 1 := by\n    intro h\n    have h2 : a % 4 = 0 \u2228 a % 4 = 1 \u2228 a % 4 = 2 \u2228 a % 4 = 3 := by omega\n    rcases h2 with (h2 | h2 | h2 | h2) <;> simp [h2, pow_two, Int.mul_emod, Int.add_emod, Int.emod_emod]\n    <;> omega\n    <;> omega\n    <;> omega\n    <;> omega\n  have case2 : a % 4 = 2 \u2192 a ^ 2 % 4 = 0 := by\n    intro h\n    have h1 : a % 4 = 0 \u2228 a % 4 = 1 \u2228 a % 4 = 2 \u2228 a % 4 = 3 := by omega\n    rcases h1 with (h1 | h1 | h1 | h1) <;> simp [h1, pow_two, Int.mul_emod, Int.add_emod, Int.emod_emod]\n    <;> omega\n  have case3 : a % 4 = 3 \u2192 a ^ 2 % 4 = 1 := by\n    intro h3\n    simp [h3, pow_two, Int.mul_emod, Int.add_emod, Int.emod_emod]\n    <;> norm_num\n    <;> linarith\n  have result : a ^ 2 % 4 = 0 \u2228 a ^ 2 % 4 = 1 := by\n    /-\n    To prove that for any integer \\( a \\), \\( a^2 \\mod 4 \\) is either 0 or 1, we consider the possible values of \\( a \\mod 4 \\). There are four cases to consider based on the remainder when \\( a \\) is divided by 4: 0, 1, 2, and 3.\n    1. **Case 1: \\( a \\equiv 0 \\mod 4 \\)**\n       - If \\( a \\equiv 0 \\mod 4 \\), then \\( a = 4k \\) for some integer \\( k \\).\n       - Squaring \\( a \\): \\( a^2 = (4k)^2 = 16k^2 \\).\n       - Taking modulo 4: \\( 16k^2 \\mod 4 = 0 \\).\n       - Therefore, \\( a^2 \\equiv 0 \\mod 4 \\).\n    2. **Case 2: \\( a \\equiv 1 \\mod 4 \\)**\n       - If \\( a \\equiv 1 \\mod 4 \\), then \\( a = 4k + 1 \\) for some integer \\( k \\).\n       - Squaring \\( a \\): \\( a^2 = (4k + 1)^2 = 16k^2 + 8k + 1 \\).\n       - Taking modulo 4: \\( 16k^2 + 8k + 1 \\mod 4 = 0 + 0 + 1 = 1 \\).\n       - Therefore, \\( a^2 \\equiv 1 \\mod 4 \\).\n    3. **Case 3: \\( a \\equiv 2 \\mod 4 \\)**\n       - If \\( a \\equiv 2 \\mod 4 \\), then \\( a = 4k + 2 \\) for some integer \\( k \\).\n       - Squaring \\( a \\): \\( a^2 = (4k + 2)^2 = 16k^2 + 16k + 4 \\).\n       - Taking modulo 4: \\( 16k^2 + 16k + 4 \\mod 4 = 0 + 0 + 0 = 0 \\).\n       - Therefore, \\( a^2 \\equiv 0 \\mod 4 \\).\n    4. **Case 4: \\( a \\equiv 3 \\mod 4 \\)**\n       - If \\( a \\equiv 3 \\mod 4 \\), then \\( a = 4k + 3 \\) for some integer \\( k \\).\n       - Squaring \\( a \\): \\( a^2 = (4k + 3)^2 = 16k^2 + 24k + 9 \\).\n       - Taking modulo 4: \\( 16k^2 + 24k + 9 \\mod 4 = 0 + 0 + 1 = 1 \\).\n       - Therefore, \\( a^2 \\equiv 1 \\mod 4 \\).\n    From all four cases, we conclude that \\( a^2 \\mod 4 \\) is either 0 or 1 for any integer \\( a \\).\n    -/\n    have : a ^ 2 % 4 = 0 \u2228 a ^ 2 % 4 = 1 := by\n      have : a % 4 = 0 \u2228 a % 4 = 1 \u2228 a % 4 = 2 \u2228 a % 4 = 3 := by omega\n      rcases this with (h | h | h | h) <;> simp [h, pow_two, Int.mul_emod, Int.add_emod]\n      <;> omega\n    exact this\n  match result with\n  | Or.inl h =>\n    exact Or.inl h\n  | Or.inr h =>\n    exact Or.inr h\n", "proof_before": "theorem numbertheory_sqmod4in01d (a : \u2124) : a ^ 2 % 4 = 0 \u2228 a ^ 2 % 4 = 1  :=  by\n  have case0 : a % 4 = 0 \u2192 a ^ 2 % 4 = 0 := by\n    intro h\n    have h2 : a ^ 2 % 4 = 0 := by\n      simp [h, pow_two, Int.mul_emod, Int.add_emod]\n    exact h2\n  have case1 : a % 4 = 1 \u2192 a ^ 2 % 4 = 1 := by\n    intro h\n    have h2 : a % 4 = 0 \u2228 a % 4 = 1 \u2228 a % 4 = 2 \u2228 a % 4 = 3 := by omega\n    rcases h2 with (h2 | h2 | h2 | h2) <;> simp [h2, pow_two, Int.mul_emod, Int.add_emod, Int.emod_emod]\n    <;> omega\n    <;> omega\n    <;> omega\n    <;> omega\n  have case2 : a % 4 = 2 \u2192 a ^ 2 % 4 = 0 := by\n    intro h\n    have h1 : a % 4 = 0 \u2228 a % 4 = 1 \u2228 a % 4 = 2 \u2228 a % 4 = 3 := by omega\n    rcases h1 with (h1 | h1 | h1 | h1) <;> simp [h1, pow_two, Int.mul_emod, Int.add_emod, Int.emod_emod]\n    <;> omega\n  have case3 : a % 4 = 3 \u2192 a ^ 2 % 4 = 1 := by\n    intro h3\n    simp [h3, pow_two, Int.mul_emod, Int.add_emod, Int.emod_emod]\n    <;> norm_num\n    <;> linarith\n  have result : a ^ 2 % 4 = 0 \u2228 a ^ 2 % 4 = 1 := by\n    /-\n    To prove that for any integer \\( a \\), \\( a^2 \\mod 4 \\) is either 0 or 1, we consider the possible values of \\( a \\mod 4 \\). There are four cases to consider based on the remainder when \\( a \\) is divided by 4: 0, 1, 2, and 3.\n    1. **Case 1: \\( a \\equiv 0 \\mod 4 \\)**\n       - If \\( a \\equiv 0 \\mod 4 \\), then \\( a = 4k \\) for some integer \\( k \\).\n       - Squaring \\( a \\): \\( a^2 = (4k)^2 = 16k^2 \\).\n       - Taking modulo 4: \\( 16k^2 \\mod 4 = 0 \\).\n       - Therefore, \\( a^2 \\equiv 0 \\mod 4 \\).\n    2. **Case 2: \\( a \\equiv 1 \\mod 4 \\)**\n       - If \\( a \\equiv 1 \\mod 4 \\), then \\( a = 4k + 1 \\) for some integer \\( k \\).\n       - Squaring \\( a \\): \\( a^2 = (4k + 1)^2 = 16k^2 + 8k + 1 \\).\n       - Taking modulo 4: \\( 16k^2 + 8k + 1 \\mod 4 = 0 + 0 + 1 = 1 \\).\n       - Therefore, \\( a^2 \\equiv 1 \\mod 4 \\).\n    3. **Case 3: \\( a \\equiv 2 \\mod 4 \\)**\n       - If \\( a \\equiv 2 \\mod 4 \\), then \\( a = 4k + 2 \\) for some integer \\( k \\).\n       - Squaring \\( a \\): \\( a^2 = (4k + 2)^2 = 16k^2 + 16k + 4 \\).\n       - Taking modulo 4: \\( 16k^2 + 16k + 4 \\mod 4 = 0 + 0 + 0 = 0 \\).\n       - Therefore, \\( a^2 \\equiv 0 \\mod 4 \\).\n    4. **Case 4: \\( a \\equiv 3 \\mod 4 \\)**\n       - If \\( a \\equiv 3 \\mod 4 \\), then \\( a = 4k + 3 \\) for some integer \\( k \\).\n       - Squaring \\( a \\): \\( a^2 = (4k + 3)^2 = 16k^2 + 24k + 9 \\).\n       - Taking modulo 4: \\( 16k^2 + 24k + 9 \\mod 4 = 0 + 0 + 1 = 1 \\).\n       - Therefore, \\( a^2 \\equiv 1 \\mod 4 \\).\n    From all four cases, we conclude that \\( a^2 \\mod 4 \\) is either 0 or 1 for any integer \\( a \\).\n    -/\n    have : a ^ 2 % 4 = 0 \u2228 a ^ 2 % 4 = 1 := by\n      have : a % 4 = 0 \u2228 a % 4 = 1 \u2228 a % 4 = 2 \u2228 a % 4 = 3 := by omega\n      rcases this with (h | h | h | h) <;> simp [h, pow_two, Int.mul_emod, Int.add_emod]\n      <;> omega\n    exact this\n  match result with\n  | Or.inl h =>\n    exact Or.inl h\n  | Or.inr h =>\n    exact Or.inr h\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 22, "alphabet": 20, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 693, "chars_after": 31, "seconds": 1.1, "id": "algebra_sqineq_2unitcircatblt1", "proof_after": "theorem algebra_sqineq_2unitcircatblt1 (a b : \u211d) (h\u2080 : a ^ 2 + b ^ 2 = 2) : a * b \u2264 1  :=  by\n  linarith [sq_nonneg (a - b)]\n", "proof_before": "theorem algebra_sqineq_2unitcircatblt1 (a b : \u211d) (h\u2080 : a ^ 2 + b ^ 2 = 2) : a * b \u2264 1  :=  by\n  have h\u2081 : (a - b) ^ 2 \u2265 0 := by\n    nlinarith [sq_nonneg (a + b), sq_nonneg (a - b)]\n    <;> nlinarith\n    <;> nlinarith\n  have h\u2082 : a ^ 2 - 2 * a * b + b ^ 2 \u2265 0 := by\n    nlinarith [sq_nonneg (a - b)]\n    <;> linarith [sq_nonneg (a - b)]\n    <;> nlinarith [sq_nonneg (a - b)]\n  have h\u2083 : 2 - 2 * a * b \u2265 0 := by\n    nlinarith [sq_nonneg (a - b), sq_nonneg (a + b), sq_nonneg (a - b + a + b)]\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n  have h\u2084 : 2 * a * b \u2264 2 := by\n    have h\u2084 : 2 * a * b \u2264 2 := by\n      have h\u2084 := sq_nonneg (a - b)\n      have h\u2085 := h\u2080\n      nlinarith\n    exact h\u2084\n  have h\u2085 : a * b \u2264 1 := by\n    linarith [sq_nonneg (a - b), sq_nonneg (a + b)]\n  assumption\n"}
{"mode": "clear", "steps_before": 9, "alphabet": 14, "compiles": true, "steps_after": 9, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 1, "fixpoint": true, "chars_before": 207, "chars_after": 207, "seconds": 37.3, "id": "mathd_algebra_327", "proof_after": "theorem mathd_algebra_327 (a : \u211d) (h\u2080 : 1 / 5 * abs (9 + 2 * a) < 1) : -7 < a \u2227 a < -2  :=  by\n  have : abs (9 + 2 * a) < 5 := by\n    nlinarith\n  have : -5 < 9 + 2 * a \u2227 9 + 2 * a < 5 := by\n    rw [abs_lt] at this\n    exact this\n  have : -7 < a \u2227 a < -2 := by\n    constructor <;> linarith\n  exact this\n", "proof_before": "theorem mathd_algebra_327 (a : \u211d) (h\u2080 : 1 / 5 * abs (9 + 2 * a) < 1) : -7 < a \u2227 a < -2  :=  by\n  have : abs (9 + 2 * a) < 5 := by\n    nlinarith\n  have : -5 < 9 + 2 * a \u2227 9 + 2 * a < 5 := by\n    rw [abs_lt] at this\n    exact this\n  have : -7 < a \u2227 a < -2 := by\n    constructor <;> linarith\n  exact this\n"}
{"mode": "clear", "steps_before": 40, "alphabet": 20, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 819, "chars_after": 32, "seconds": 45.8, "id": "mathd_algebra_214", "proof_after": "theorem mathd_algebra_214 (a : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = a * (x - 2) ^ 2 + 3) (h\u2081 : f 4 = 4) :\n  f 6 = 7  :=  by\n  rw [h\u2080]\n  linarith [h\u2080 4, h\u2081]\n", "proof_before": "theorem mathd_algebra_214 (a : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = a * (x - 2) ^ 2 + 3) (h\u2081 : f 4 = 4) :\n  f 6 = 7  :=  by\n  have h\u2082 : 4 = a * (4 - 2)^2 + 3 := by\n    rw [h\u2080] at h\u2081\n    linarith\n    <;>\n      simp_all [h\u2080]\n    <;>\n      linarith\n  have h\u2083 : 4 = 4 * a + 3 := by\n    linarith [h\u2080 4, h\u2081]\n  have h\u2084 : 4 * a = 1 := by\n    have h\u2084 : 4 * a = 1 := by\n      linarith\n    linarith\n  have h\u2085 : a = 1/4 := by\n    have h\u2085 : a = 1 / 4 := by\n      linarith\n    assumption\n  have h\u2086 : \u2200 x, f x = (1/4) * (x - 2)^2 + 3 := by\n    intro x\n    rw [h\u2080]\n    rw [h\u2085]\n    <;> simp_all\n    <;> ring\n    <;> linarith\n  have h\u2087 : f 6 = (1/4) * (6 - 2)^2 + 3 := by\n    simp [h\u2086]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have h\u2088 : f 6 = (1/4) * 16 + 3 := by\n    norm_num [h\u2086]\n    <;> simp_all [h\u2086]\n    <;> norm_num\n    <;> linarith\n  have h\u2089 : f 6 = 4 + 3 := by\n    linarith\n  have h\u2081\u2080 : f 6 = 7 := by\n    linarith\n  exact h\u2081\u2080\n"}
{"mode": "clear", "steps_before": 14, "alphabet": 14, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 578, "chars_after": 41, "seconds": 269.2, "id": "algebra_sqineq_4bap1lt4bsqpap1sq", "proof_after": "theorem algebra_sqineq_4bap1lt4bsqpap1sq (a b : \u211d) : 4 * b * (a + 1) \u2264 4 * b ^ 2 + (a + 1) ^ 2  :=  by\n  linarith [sq_nonneg (2 * b - (a + 1))]\n", "proof_before": "theorem algebra_sqineq_4bap1lt4bsqpap1sq (a b : \u211d) : 4 * b * (a + 1) \u2264 4 * b ^ 2 + (a + 1) ^ 2  :=  by\n  have step1 : 4 * b * (a + 1) - 4 * b^2 - (a + 1)^2 \u2264 0 := by\n    nlinarith [sq_nonneg (a - 2 * b + 1), sq_nonneg (a - 2 * b), sq_nonneg (a + 1 - 2 * b), sq_nonneg (a + 1), sq_nonneg (2 * b)]\n  have step2 : - ( (2 * b - (a + 1)) ^ 2 ) \u2264 0 := by\n    linarith [sq_nonneg (2 * b - (a + 1))]\n    <;> linarith\n    <;> linarith\n    <;> linarith\n  have step3 : (2 * b - (a + 1)) ^ 2 \u2265 0 := by\n    nlinarith [sq_nonneg (2 * b - (a + 1))]\n  have final : 4 * b * (a + 1) \u2264 4 * b^2 + (a + 1)^2 := by\n    linarith [sq_nonneg (2 * b - (a + 1))]\n    <;> linarith\n    <;> linarith\n  linarith\n"}
{"mode": "clear", "steps_before": 36, "alphabet": 23, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 9, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "fixpoint": true, "chars_before": 1071, "chars_after": 315, "seconds": 377.0, "id": "amc12_2001_p9", "proof_after": "theorem amc12_2001_p9 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x > 0, \u2200 y > 0, f (x * y) = f x / y) (h\u2081 : f 500 = 3) :\n    f 600 = 5 / 2  :=  by\n  have h_form : \u2200 x > 0, f x = 1500 / x := by\n    intro x hx\n    have h\u2082 := h\u2080 500 (by norm_num) (x / 500) (by positivity)\n    have h\u2083 := h\u2080 500 (by norm_num) (500 / x) (by positivity)\n    field_simp [hx.ne', (by norm_num : (500 : \u211d) \u2260 0)] at h\u2082 h\u2083 \u22a2\n    linarith\n  norm_num [h_form, h_verify, h_k, h_k_eq] at h\u2080 \u22a2\n", "proof_before": "theorem amc12_2001_p9 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x > 0, \u2200 y > 0, f (x * y) = f x / y) (h\u2081 : f 500 = 3) :\n    f 600 = 5 / 2  :=  by\n  have h_form : \u2200 x > 0, f x = 1500 / x := by\n    intro x hx\n    have h\u2082 := h\u2080 500 (by norm_num) (x / 500) (by positivity)\n    have h\u2083 := h\u2080 500 (by norm_num) (500 / x) (by positivity)\n    simp [h\u2081] at h\u2082 h\u2083\n    field_simp [hx.ne', (by norm_num : (500 : \u211d) \u2260 0)] at h\u2082 h\u2083 \u22a2\n    linarith\n  have h_verify : \u2200 x > 0, \u2200 y > 0, f (x * y) = f x / y := by\n    intro x hx y hy\n    simp_all\n    <;> linarith\n  have h_k : f 500 = 1500 / 500 := by\n    simpa [h\u2080, h_form] using h_verify 4 (by norm_num) 5 (by norm_num)\n  have h_k_eq : f 500 = 3 := by\n    rw [h\u2081]\n    <;> field_simp\n    <;> linarith\n  have h_k_value : 1500 / 500 = 3 := by\n    norm_num [h_form, h_verify, h_k, h_k_eq] at h\u2080 \u22a2\n    <;> linarith [h\u2080 500 (by norm_num) 500 (by norm_num)]\n  have h_explicit : \u2200 x > 0, f x = 1500 / x := by\n    intro x hx\n    have h_form_x := h_form x hx\n    simp_all\n  have h_final : f 600 = 1500 / 600 := by\n    simp_all\n    <;> norm_num\n    <;> linarith\n  have h_final_eq : f 600 = 5 / 2 := by\n    simp_all [h_explicit]\n    <;> linarith\n    <;> norm_num\n  rw [h_final_eq]\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 43, "alphabet": 27, "compiles": false, "id": "mathd_algebra_149", "proof_after": "theorem mathd_algebra_149 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x < -5, f x = x ^ 2 + 5)\n  (h\u2081 : \u2200 x \u2265 -5, f x = 3 * x - 8) (h\u2082 : Fintype (f \u207b\u00b9' {10})) :\n  (\u2211 k in (f \u207b\u00b9' {10}).toFinset, k) = 6  :=  by\n  have case1 : \u2200 x < -5, f x = x ^ 2 + 5 := by\n    intro x hx\n    apply h\u2080\n    exact hx\n  have case2 : \u2200 x \u2265 -5, f x = 3 * x - 8 := by\n    intro x hx\n    have h\u2083 := h\u2081 x hx\n    simp_all\n  have no_solutions_case1 : \u2200 x < -5, f x \u2260 10 := by\n    intro x hx\n    simp_all [lt_irrefl]\n    <;> nlinarith\n    <;> linarith\n    <;> nlinarith\n    <;> linarith\n    <;> nlinarith\n    <;> linarith\n    <;> nlinarith\n    <;> linarith\n  have solution_case2 : \u2203! x \u2265 -5, f x = 10 := by\n    refine' \u27e8(10 + 8) / 3, by norm_num [h\u2081], fun a ha => _\u27e9\n    have h\u2083 := h\u2081 a ha.1\n    simp_all\n    <;> linarith\n  have sum_of_solutions : (\u2211 k in (f \u207b\u00b9' {10}).toFinset, k) = 6 := by\n    have : f \u207b\u00b9' {10} = {6} := by\n      ext x\n      simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_setOf_eq]\n      constructor\n      \u00b7 intro h\n        match lt_or_le x (-5) with\n        | .inl hlt =>\n          rw [case1 x hlt] at h\n          nlinarith\n        | .inr hge =>\n          rw [case2 x hge] at h\n          linarith\n      \u00b7 intro h\n        rw [h]\n        rw [case2 6 (by norm_num)]  \n        norm_num\n    simp [this]\n  simpa [sum_of_solutions] using sum_of_solutions\n", "proof_before": "theorem mathd_algebra_149 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x < -5, f x = x ^ 2 + 5)\n  (h\u2081 : \u2200 x \u2265 -5, f x = 3 * x - 8) (h\u2082 : Fintype (f \u207b\u00b9' {10})) :\n  (\u2211 k in (f \u207b\u00b9' {10}).toFinset, k) = 6  :=  by\n  have case1 : \u2200 x < -5, f x = x ^ 2 + 5 := by\n    intro x hx\n    apply h\u2080\n    exact hx\n  have case2 : \u2200 x \u2265 -5, f x = 3 * x - 8 := by\n    intro x hx\n    have h\u2083 := h\u2081 x hx\n    simp_all\n  have no_solutions_case1 : \u2200 x < -5, f x \u2260 10 := by\n    intro x hx\n    simp_all [lt_irrefl]\n    <;> nlinarith\n    <;> linarith\n    <;> nlinarith\n    <;> linarith\n    <;> nlinarith\n    <;> linarith\n    <;> nlinarith\n    <;> linarith\n  have solution_case2 : \u2203! x \u2265 -5, f x = 10 := by\n    refine' \u27e8(10 + 8) / 3, by norm_num [h\u2081], fun a ha => _\u27e9\n    have h\u2083 := h\u2081 a ha.1\n    simp_all\n    <;> linarith\n  have sum_of_solutions : (\u2211 k in (f \u207b\u00b9' {10}).toFinset, k) = 6 := by\n    have : f \u207b\u00b9' {10} = {6} := by\n      ext x\n      simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_setOf_eq]\n      constructor\n      \u00b7 intro h\n        match lt_or_le x (-5) with\n        | .inl hlt =>\n          rw [case1 x hlt] at h\n          nlinarith\n        | .inr hge =>\n          rw [case2 x hge] at h\n          linarith\n      \u00b7 intro h\n        rw [h]\n        rw [case2 6 (by norm_num)]  \n        norm_num\n    simp [this]\n  simpa [sum_of_solutions] using sum_of_solutions\n"}
{"mode": "clear", "steps_before": 85, "alphabet": 24, "compiles": true, "steps_after": 30, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 2, "fixpoint": false, "chars_before": 3775, "chars_after": 1316, "seconds": 446.5, "id": "mathd_algebra_151", "proof_after": "theorem mathd_algebra_151 : Int.ceil (Real.sqrt 27) - Int.floor (Real.sqrt 26) = 1  :=  by\n  have sqrt27_ineq : 5 < Real.sqrt 27 \u2227 Real.sqrt 27 < 6 := by\n    constructor\n    exact by\n      norm_num\n      rw [Real.lt_sqrt (by norm_num)]\n      norm_num\n    exact by\n      norm_num\n      rw [Real.sqrt_lt (by norm_num)]\n      norm_num\n      linarith\n  have sqrt26_ineq : 5 < Real.sqrt 26 \u2227 Real.sqrt 26 < 6 := by\n    constructor\n    case left =>\n      nlinarith [Real.sqrt_nonneg 26, Real.sqrt_nonneg 27, Real.sq_sqrt (show 0 \u2264 26 by norm_num),\n        Real.sq_sqrt (show 0 \u2264 27 by norm_num), sqrt27_ineq.1]\n    case right =>\n      nlinarith [Real.sqrt_nonneg 26, Real.sqrt_nonneg 27, Real.sq_sqrt (show 0 \u2264 26 by norm_num),\n        Real.sq_sqrt (show 0 \u2264 27 by norm_num), sqrt27_ineq.2]\n  have ceil_sqrt27 : Int.ceil (Real.sqrt 27) = 6 := by\n    norm_num [Int.ceil_eq_iff, sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [Real.sq_sqrt (show (0 : \u211d) \u2264 27 by norm_num), Real.sq_sqrt (show (0 : \u211d) \u2264 26 by norm_num)]\n  have floor_sqrt26 : Int.floor (Real.sqrt 26) = 5 := by\n    have h\u2080 : Int.floor (Real.sqrt 26) = 5 := by\n      have h\u2081 := sqrt27_ineq.2\n      have h\u2082 := sqrt26_ineq.2\n      have h\u2083 := sqrt26_ineq.1\n      norm_num [Int.floor_eq_iff, Real.sqrt_le_sqrt] at h\u2081 h\u2082 h\u2083 \u22a2\n      constructor <;> nlinarith [Real.sq_sqrt (show 0 \u2264 26 by norm_num), Real.sq_sqrt (show 0 \u2264 27 by norm_num)]\n    exact h\u2080\n  omega\n", "proof_before": "theorem mathd_algebra_151 : Int.ceil (Real.sqrt 27) - Int.floor (Real.sqrt 26) = 1  :=  by\n  have sqrt27_ineq : 5 < Real.sqrt 27 \u2227 Real.sqrt 27 < 6 := by\n    constructor\n    exact by\n      norm_num\n      rw [Real.lt_sqrt (by norm_num)]\n      norm_num\n    exact by\n      norm_num\n      rw [Real.sqrt_lt (by norm_num)]\n      norm_num\n      linarith\n  have sqrt26_ineq : 5 < Real.sqrt 26 \u2227 Real.sqrt 26 < 6 := by\n    constructor\n    case left =>\n      nlinarith [Real.sqrt_nonneg 26, Real.sqrt_nonneg 27, Real.sq_sqrt (show 0 \u2264 26 by norm_num),\n        Real.sq_sqrt (show 0 \u2264 27 by norm_num), sqrt27_ineq.1]\n    case right =>\n      nlinarith [Real.sqrt_nonneg 26, Real.sqrt_nonneg 27, Real.sq_sqrt (show 0 \u2264 26 by norm_num),\n        Real.sq_sqrt (show 0 \u2264 27 by norm_num), sqrt27_ineq.2]\n  have ceil_sqrt27 : Int.ceil (Real.sqrt 27) = 6 := by\n    norm_num [Int.ceil_eq_iff, sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [Real.sq_sqrt (show (0 : \u211d) \u2264 27 by norm_num), Real.sq_sqrt (show (0 : \u211d) \u2264 26 by norm_num)]\n  have floor_sqrt26 : Int.floor (Real.sqrt 26) = 5 := by\n    have h\u2080 : Int.floor (Real.sqrt 26) = 5 := by\n      have h\u2081 := sqrt27_ineq.2\n      have h\u2082 := sqrt26_ineq.2\n      have h\u2083 := sqrt26_ineq.1\n      norm_num [Int.floor_eq_iff, Real.sqrt_le_sqrt] at h\u2081 h\u2082 h\u2083 \u22a2\n      constructor <;> nlinarith [Real.sq_sqrt (show 0 \u2264 26 by norm_num), Real.sq_sqrt (show 0 \u2264 27 by norm_num)]\n    exact h\u2080\n  have final_calc : Int.ceil (Real.sqrt 27) - Int.floor (Real.sqrt 26) = 1 := by\n    norm_num [ceil_sqrt27, floor_sqrt26]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n    <;> norm_num [sqrt27_ineq, sqrt26_ineq]\n    <;> linarith [sqrt27_ineq, sqrt26_ineq]\n  assumption\n"}
{"mode": "clear", "steps_before": 25, "alphabet": 16, "compiles": false, "id": "mathd_algebra_48", "proof_after": "theorem mathd_algebra_48 (q e : \u2102) (h\u2080 : q = 9 - 4 * Complex.I) (h\u2081 : e = -3 - 4 * Complex.I) :\n  q - e = 12  :=  by\n  have step1 : q - e = (9 - 4 * Complex.I) - (-3 - 4 * Complex.I) := by\n    simp_all [sub_eq_add_neg, add_comm]\n    <;> ring\n    <;> simp [Complex.I_mul_I]\n    <;> ring\n  have step2 : q - e = 9 - 4 * Complex.I + 3 + 4 * Complex.I := by\n    rw [h\u2080, h\u2081] at step1\n    simp [sub_eq_add_neg, add_assoc, add_left_comm, add_right_comm, mul_comm, mul_left_comm, mul_right_comm] at step1\n    ring_nf at step1 \u22a2\n    <;> simp_all\n    <;> ring_nf\n  have step3 : q - e = (9 + 3) + (-4 * Complex.I + 4 * Complex.I) := by\n    simp_all only [sub_eq_add_neg, add_assoc, add_left_comm, add_right_comm]\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  have step4 : q - e = 12 + 0 := by\n    simp_all only [sub_eq_add_neg, add_assoc, add_left_comm, add_right_comm, add_neg_cancel_right,\n      add_neg_cancel_left, neg_neg, neg_add_rev, neg_mul, neg_sub, neg_neg, neg_zero]\n    ring\n  have step5 : q - e = 12 := by\n    simp_all [Complex.ext_iff, mul_comm]\n    <;> norm_num\n    <;> linarith\n  rw [step5]\n", "proof_before": "theorem mathd_algebra_48 (q e : \u2102) (h\u2080 : q = 9 - 4 * Complex.I) (h\u2081 : e = -3 - 4 * Complex.I) :\n  q - e = 12  :=  by\n  have step1 : q - e = (9 - 4 * Complex.I) - (-3 - 4 * Complex.I) := by\n    simp_all [sub_eq_add_neg, add_comm]\n    <;> ring\n    <;> simp [Complex.I_mul_I]\n    <;> ring\n  have step2 : q - e = 9 - 4 * Complex.I + 3 + 4 * Complex.I := by\n    rw [h\u2080, h\u2081] at step1\n    simp [sub_eq_add_neg, add_assoc, add_left_comm, add_right_comm, mul_comm, mul_left_comm, mul_right_comm] at step1\n    ring_nf at step1 \u22a2\n    <;> simp_all\n    <;> ring_nf\n  have step3 : q - e = (9 + 3) + (-4 * Complex.I + 4 * Complex.I) := by\n    simp_all only [sub_eq_add_neg, add_assoc, add_left_comm, add_right_comm]\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  have step4 : q - e = 12 + 0 := by\n    simp_all only [sub_eq_add_neg, add_assoc, add_left_comm, add_right_comm, add_neg_cancel_right,\n      add_neg_cancel_left, neg_neg, neg_add_rev, neg_mul, neg_sub, neg_neg, neg_zero]\n    ring\n  have step5 : q - e = 12 := by\n    simp_all [Complex.ext_iff, mul_comm]\n    <;> norm_num\n    <;> linarith\n  rw [step5]\n"}
{"mode": "clear", "steps_before": 9, "alphabet": 13, "compiles": false, "id": "amc12b_2003_p17", "proof_after": "theorem amc12b_2003_p17 (x y : \u211d) (h\u2080 : 0 < x \u2227 0 < y) (h\u2081 : Real.log (x * y ^ 3) = 1)\n  (h\u2082 : Real.log (x ^ 2 * y) = 1) : Real.log (x * y) = 3 / 5  :=  by\n  have hx : 0 < x := by linarith\n  have hy : 0 < y := by linarith\n  have hxy : 0 < x * y := by positivity\n  field_simp [Real.log_mul, Real.log_pow, hx, hy, hxy] at h\u2081 h\u2082 \u22a2\n  ring_nf at h\u2081 h\u2082 \u22a2\n  linarith\n", "proof_before": "theorem amc12b_2003_p17 (x y : \u211d) (h\u2080 : 0 < x \u2227 0 < y) (h\u2081 : Real.log (x * y ^ 3) = 1)\n  (h\u2082 : Real.log (x ^ 2 * y) = 1) : Real.log (x * y) = 3 / 5  :=  by\n  have hx : 0 < x := by linarith\n  have hy : 0 < y := by linarith\n  have hxy : 0 < x * y := by positivity\n  field_simp [Real.log_mul, Real.log_pow, hx, hy, hxy] at h\u2081 h\u2082 \u22a2\n  ring_nf at h\u2081 h\u2082 \u22a2\n  linarith\n"}
{"mode": "clear", "steps_before": 44, "alphabet": 16, "compiles": false, "id": "mathd_algebra_536", "proof_after": "theorem mathd_algebra_536 : \u21913! * ((2 : \u211d) ^ 3 + Real.sqrt 9) / 2 = (33 : \u211d)  :=  by\n  have step1 : \u21913! = 6 := by\n    rfl\n    <;> simp [factorial]\n    <;> norm_num\n    <;> decide\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> decide\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> decide\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> decide\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> decide\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> decide\n  have step2 : (2 : \u211d) ^ 3 = 8 := by\n    have h : 3! = 6 := by simp [step1]\n    norm_num [h]\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> linarith\n  have step3 : Real.sqrt 9 = 3 := by\n    rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    <;> nlinarith [sq_sqrt (show (0 : \u211d) \u2264 9 by norm_num)]\n  have step4 : (2 : \u211d) ^ 3 + Real.sqrt 9 = 11 := by\n    norm_num [step1, step2, step3]\n    <;> linarith\n  have step5 : \u21913! * ((2 : \u211d) ^ 3 + Real.sqrt 9) = 66 := by\n    simp_all only [factorial, mul_add, mul_one, mul_comm]\n    norm_num\n  have step6 : \u21913! * ((2 : \u211d) ^ 3 + Real.sqrt 9) / 2 = 33 := by\n    have h : \u21913! * ((2 : \u211d) ^ 3 + Real.sqrt 9) / 2 = 33 := by\n      simp_all [factorial]\n      <;> norm_num\n      <;> linarith\n    exact h\n  linarith\n", "proof_before": "theorem mathd_algebra_536 : \u21913! * ((2 : \u211d) ^ 3 + Real.sqrt 9) / 2 = (33 : \u211d)  :=  by\n  have step1 : \u21913! = 6 := by\n    rfl\n    <;> simp [factorial]\n    <;> norm_num\n    <;> decide\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> decide\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> decide\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> decide\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> decide\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> decide\n  have step2 : (2 : \u211d) ^ 3 = 8 := by\n    have h : 3! = 6 := by simp [step1]\n    norm_num [h]\n    <;> simp_all [factorial]\n    <;> norm_num\n    <;> linarith\n  have step3 : Real.sqrt 9 = 3 := by\n    rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    <;> nlinarith [sq_sqrt (show (0 : \u211d) \u2264 9 by norm_num)]\n  have step4 : (2 : \u211d) ^ 3 + Real.sqrt 9 = 11 := by\n    norm_num [step1, step2, step3]\n    <;> linarith\n  have step5 : \u21913! * ((2 : \u211d) ^ 3 + Real.sqrt 9) = 66 := by\n    simp_all only [factorial, mul_add, mul_one, mul_comm]\n    norm_num\n  have step6 : \u21913! * ((2 : \u211d) ^ 3 + Real.sqrt 9) / 2 = 33 := by\n    have h : \u21913! * ((2 : \u211d) ^ 3 + Real.sqrt 9) / 2 = 33 := by\n      simp_all [factorial]\n      <;> norm_num\n      <;> linarith\n    exact h\n  linarith\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 864, "chars_after": 6, "seconds": 2.5, "id": "mathd_numbertheory_640", "proof_after": "theorem mathd_numbertheory_640 : (91145 + 91146 + 91147 + 91148) % 4 = 2  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_640 : (91145 + 91146 + 91147 + 91148) % 4 = 2  :=  by\n  have h1 : 91145 % 4 = 1 := by\n    simp [Nat.add_mod, Nat.mul_mod, Nat.mod_eq_of_lt, Nat.lt_of_le_of_lt (Nat.zero_le _) (by decide : 4 < 4168)]\n  have h2 : 91146 % 4 = 2 := by\n    have h2 : 91145 % 4 = 1 := h1\n    omega\n  have h3 : 91147 % 4 = 3 := by\n    simp [h1, h2, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n    <;> norm_num\n    <;> omega\n  have h4 : 91148 % 4 = 0 := by\n    simp [h1, h2, h3, Nat.add_mod, Nat.mul_mod, Nat.mod_mod, Nat.mod_eq_of_lt]\n    <;> decide\n  have h_sum : (91145 + 91146 + 91147 + 91148) % 4 = (1 + 2 + 3 + 0) % 4 := by\n    simp [h1, h2, h3, h4, Nat.add_mod, Nat.mul_mod, Nat.mod_mod, Nat.zero_add, Nat.add_zero]\n    <;> norm_num\n    <;> rfl\n  have h_final : (1 + 2 + 3 + 0) % 4 = 2 := by\n    simp_all [Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n    <;> norm_num\n    <;> rfl\n  simp [h1, h2, h3, h4, h_sum, h_final]\n  <;> norm_num\n  <;> omega\n"}
{"mode": "clear", "steps_before": 5, "alphabet": 13, "compiles": true, "steps_after": 5, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 1, "fixpoint": true, "chars_before": 120, "chars_after": 120, "seconds": 152.1, "id": "mathd_algebra_247", "proof_after": "theorem mathd_algebra_247 (t s : \u211d) (n : \u2124) (h\u2080 : t = 2 * s - s ^ 2) (h\u2081 : s = n ^ 2 - 2 ^ n + 1)\n  (_ : n = 3) : t = 0  :=  by\n  subst n\n  simp [h\u2081, h\u2080, pow_succ, Int.mul_emod, Int.add_emod, Int.sub_emod]\n  <;> norm_num\n  <;> ring\n  <;> nlinarith\n", "proof_before": "theorem mathd_algebra_247 (t s : \u211d) (n : \u2124) (h\u2080 : t = 2 * s - s ^ 2) (h\u2081 : s = n ^ 2 - 2 ^ n + 1)\n  (_ : n = 3) : t = 0  :=  by\n  subst n\n  simp [h\u2081, h\u2080, pow_succ, Int.mul_emod, Int.add_emod, Int.sub_emod]\n  <;> norm_num\n  <;> ring\n  <;> nlinarith\n"}
{"mode": "clear", "steps_before": 46, "alphabet": 23, "compiles": false, "id": "imo_1984_p2", "proof_after": "theorem imo_1984_p2 (a b : \u2124) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : \u00ac7 \u2223 a) (h\u2082 : \u00ac7 \u2223 b) (h\u2083 : \u00ac7 \u2223 a + b)\n  (h\u2084 : 7 ^ 7 \u2223 (a + b) ^ 7 - a ^ 7 - b ^ 7) : 19 \u2264 a + b  :=  by\n  have h\u2085 : \u00ac7 \u2223 a * b * (a + b) := by\n    rw [Int.dvd_iff_emod_eq_zero] at h\u2081 h\u2082 h\u2083 h\u2084 \u22a2\n    have h\u2085 : a % 7 = 1 \u2228 a % 7 = 2 \u2228 a % 7 = 3 \u2228 a % 7 = 4 \u2228 a % 7 = 5 \u2228 a % 7 = 6 := by\n      omega\n    have h\u2086 : b % 7 = 1 \u2228 b % 7 = 2 \u2228 b % 7 = 3 \u2228 b % 7 = 4 \u2228 b % 7 = 5 \u2228 b % 7 = 6 := by\n      omega\n    have h\u2087 : (a + b) % 7 = 1 \u2228 (a + b) % 7 = 2 \u2228 (a + b) % 7 = 3 \u2228 (a + b) % 7 = 4 \u2228 (a + b) % 7 = 5 \u2228 (a + b) % 7 = 6 := by\n      omega\n    rcases h\u2085 with (h\u2085 | h\u2085 | h\u2085 | h\u2085 | h\u2085 | h\u2085) <;>\n      rcases h\u2086 with (h\u2086 | h\u2086 | h\u2086 | h\u2086 | h\u2086 | h\u2086) <;>\n        simp [h\u2085, h\u2086, Int.mul_emod, Int.add_emod, Int.sub_emod, pow_succ] at h\u2084 \u22a2 <;>\n          omega\n  have h\u2086 : (a + b)^7 = a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7 := by\n    rw [add_comm]\n    rw [add_comm]\n    ring_nf\n    <;> norm_num\n    <;> simp_all\n    <;> omega\n  have h\u2087 : (a + b)^7 - a^7 - b^7 = 7*(a^6*b + 3*a^5*b^2 + 5*a^4*b^3 + 5*a^3*b^4 + 3*a^2*b^5 + a*b^6) := by\n    ring_nf at h\u2086 \u22a2\n    <;> omega\n  have h\u2088 : 7^7 \u2223 7*(a^6*b + 3*a^5*b^2 + 5*a^4*b^3 + 5*a^3*b^4 + 3*a^2*b^5 + a*b^6) := by\n    norm_num at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085\n    simp_all [Int.mul_emod, Int.add_emod, pow_succ]\n  have h\u2089 : 19 \u2264 a + b := by\n    contrapose! h\u2088\n    have h\u2089 : a + b \u2264 18 := by linarith\n    have h\u2081\u2080 : a \u2264 18 := by linarith\n    have h\u2081\u2081 : b \u2264 18 := by linarith\n    have h\u2081\u2082 : a \u2265 1 := by linarith\n    have h\u2081\u2083 : b \u2265 1 := by linarith\n    interval_cases a <;> interval_cases b <;> norm_num at h\u2088 \u22a2 <;> try contradiction\n    <;> omega\n  linarith\n", "proof_before": "theorem imo_1984_p2 (a b : \u2124) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : \u00ac7 \u2223 a) (h\u2082 : \u00ac7 \u2223 b) (h\u2083 : \u00ac7 \u2223 a + b)\n  (h\u2084 : 7 ^ 7 \u2223 (a + b) ^ 7 - a ^ 7 - b ^ 7) : 19 \u2264 a + b  :=  by\n  have h\u2085 : \u00ac7 \u2223 a * b * (a + b) := by\n    rw [Int.dvd_iff_emod_eq_zero] at h\u2081 h\u2082 h\u2083 h\u2084 \u22a2\n    have h\u2085 : a % 7 = 1 \u2228 a % 7 = 2 \u2228 a % 7 = 3 \u2228 a % 7 = 4 \u2228 a % 7 = 5 \u2228 a % 7 = 6 := by\n      omega\n    have h\u2086 : b % 7 = 1 \u2228 b % 7 = 2 \u2228 b % 7 = 3 \u2228 b % 7 = 4 \u2228 b % 7 = 5 \u2228 b % 7 = 6 := by\n      omega\n    have h\u2087 : (a + b) % 7 = 1 \u2228 (a + b) % 7 = 2 \u2228 (a + b) % 7 = 3 \u2228 (a + b) % 7 = 4 \u2228 (a + b) % 7 = 5 \u2228 (a + b) % 7 = 6 := by\n      omega\n    rcases h\u2085 with (h\u2085 | h\u2085 | h\u2085 | h\u2085 | h\u2085 | h\u2085) <;>\n      rcases h\u2086 with (h\u2086 | h\u2086 | h\u2086 | h\u2086 | h\u2086 | h\u2086) <;>\n        simp [h\u2085, h\u2086, Int.mul_emod, Int.add_emod, Int.sub_emod, pow_succ] at h\u2084 \u22a2 <;>\n          omega\n  have h\u2086 : (a + b)^7 = a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7 := by\n    rw [add_comm]\n    rw [add_comm]\n    ring_nf\n    <;> norm_num\n    <;> simp_all\n    <;> omega\n  have h\u2087 : (a + b)^7 - a^7 - b^7 = 7*(a^6*b + 3*a^5*b^2 + 5*a^4*b^3 + 5*a^3*b^4 + 3*a^2*b^5 + a*b^6) := by\n    ring_nf at h\u2086 \u22a2\n    <;> omega\n  have h\u2088 : 7^7 \u2223 7*(a^6*b + 3*a^5*b^2 + 5*a^4*b^3 + 5*a^3*b^4 + 3*a^2*b^5 + a*b^6) := by\n    norm_num at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085\n    simp_all [Int.mul_emod, Int.add_emod, pow_succ]\n  have h\u2089 : 19 \u2264 a + b := by\n    contrapose! h\u2088\n    have h\u2089 : a + b \u2264 18 := by linarith\n    have h\u2081\u2080 : a \u2264 18 := by linarith\n    have h\u2081\u2081 : b \u2264 18 := by linarith\n    have h\u2081\u2082 : a \u2265 1 := by linarith\n    have h\u2081\u2083 : b \u2265 1 := by linarith\n    interval_cases a <;> interval_cases b <;> norm_num at h\u2088 \u22a2 <;> try contradiction\n    <;> omega\n  linarith\n"}
{"mode": "clear", "steps_before": 11, "alphabet": 12, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 268, "chars_after": 16, "seconds": 5.8, "id": "mathd_numbertheory_461", "proof_after": "theorem mathd_numbertheory_461 (n : \u2115)\n  (h\u2080 : n = Finset.card (Finset.filter (fun x => Nat.gcd x 8 = 1) (Finset.Icc 1 7))) :\n  3 ^ n % 8 = 1  :=  by\n  rw [h\u2080]\n  rfl\n", "proof_before": "theorem mathd_numbertheory_461 (n : \u2115)\n  (h\u2080 : n = Finset.card (Finset.filter (fun x => Nat.gcd x 8 = 1) (Finset.Icc 1 7))) :\n  3 ^ n % 8 = 1  :=  by\n  have h\u2081 : n = 4 := by\n    rw [h\u2080]\n    decide\n    <;> rfl\n  have h\u2082 : 3 ^ n % 8 = 1 := by\n    simp_all only [Finset.card_filter, Finset.Icc, Nat.gcd_eq_gcd_ab, Nat.gcd_eq_right,\n      Nat.gcd_eq_left]\n    norm_num\n    <;> aesop\n  simp_all\n  <;> norm_num\n  <;> decide\n"}
{"mode": "clear", "steps_before": 11, "alphabet": 11, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 7, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "fixpoint": true, "chars_before": 321, "chars_after": 180, "seconds": 415.8, "id": "mathd_algebra_323", "proof_after": "theorem mathd_algebra_323 (\u03c3 : Equiv \u211d \u211d) (h : \u2200 x, \u03c3.1 x = x ^ 3 - 8) : \u03c3.2 (\u03c3.1 (\u03c3.2 19)) = 3  :=  by\n  have h_inv_19 : \u03c3.2 19 = 3 := by\n    simp_all [Equiv.symm_apply_eq, pow_three]\n    <;> norm_num\n    <;> nlinarith\n  have h_f_inv_19 : \u03c3.1 (\u03c3.2 19) = 19 := by\n    simp\n  simp_all\n", "proof_before": "theorem mathd_algebra_323 (\u03c3 : Equiv \u211d \u211d) (h : \u2200 x, \u03c3.1 x = x ^ 3 - 8) : \u03c3.2 (\u03c3.1 (\u03c3.2 19)) = 3  :=  by\n  have h_inv_19 : \u03c3.2 19 = 3 := by\n    simp_all [Equiv.symm_apply_eq, pow_three]\n    <;> norm_num\n    <;> nlinarith\n  have h_f_inv_19 : \u03c3.1 (\u03c3.2 19) = 19 := by\n    simp_all [Equiv.eq_symm_apply]\n    <;> norm_num\n    <;> linarith\n  have h_final : \u03c3.2 (\u03c3.1 (\u03c3.2 19)) = 3 := by\n    simp_all [Equiv.symm_apply_eq]\n  simp_all\n"}
{"mode": "clear", "steps_before": 22, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 692, "chars_after": 6, "seconds": 1.6, "id": "mathd_numbertheory_202", "proof_after": "theorem mathd_numbertheory_202 : (19 ^ 19 + 99 ^ 99) % 10 = 8  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_202 : (19 ^ 19 + 99 ^ 99) % 10 = 8  :=  by\n  have h\u2081 : 19 % 10 = 9 := by\n    simp [Int.emod_eq_of_lt]\n    <;> norm_num\n    <;> decide\n    <;> norm_num\n    <;> decide\n  have h\u2082 : 99 % 10 = 9 := by\n    simp_all [Nat.mod_eq_of_lt, show 19 % 10 = 9 by norm_num, show 99 % 10 = 9 by norm_num]\n  have h\u2083 : 9^19 % 10 = 9 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.pow_mod, Nat.mod_mod]\n    <;> rfl\n  have h\u2084 : 9^99 % 10 = 9 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod] at h\u2083 \u22a2\n    <;> aesop\n  have h\u2085 : (9 + 9) % 10 = 8 := by\n    norm_num [Nat.add_mod, Nat.mul_mod, Nat.pow_mod, Nat.mod_mod]\n    <;> simp_all\n    <;> norm_num\n    <;> rfl\n  simp_all [Nat.add_mod, Nat.pow_mod, Nat.mod_mod]\n  <;> norm_num\n  <;> rfl\n"}
{"mode": "clear", "steps_before": 19, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 407, "chars_after": 6, "seconds": 0.9, "id": "mathd_numbertheory_961", "proof_after": "theorem mathd_numbertheory_961 : 2003 % 11 = 1  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_961 : 2003 % 11 = 1  :=  by\n  have h1 : 2003 / 11 = 182 := by\n    norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n  have h2 : 11 * 182 = 2002 := by\n    have h2 : 2003 / 11 = 182 := h1\n    norm_num [h2]\n  have h3 : 2003 - 2002 = 1 := by\n    norm_num\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  have h4 : 2003 % 11 = 1 := by\n    have h4 : 2003 % 11 = 1 := by\n      omega\n    exact h4\n  simpa using h4\n"}
{"mode": "clear", "steps_before": 25, "alphabet": 24, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1796, "chars_after": 109, "seconds": 8.8, "id": "algebra_apb4leq8ta4pb4", "proof_after": "theorem algebra_apb4leq8ta4pb4 (a b : \u211d) (h\u2080 : 0 < a \u2227 0 < b) : (a + b) ^ 4 \u2264 8 * (a ^ 4 + b ^ 4)  :=  by\n  nlinarith [sq_nonneg (a + b), sq_nonneg (a - b), mul_self_nonneg (a^2 - b^2), mul_self_nonneg (a^2 + b^2)]\n", "proof_before": "theorem algebra_apb4leq8ta4pb4 (a b : \u211d) (h\u2080 : 0 < a \u2227 0 < b) : (a + b) ^ 4 \u2264 8 * (a ^ 4 + b ^ 4)  :=  by\n  have step1 : (a + b)^4 = a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 := by\n    simp [pow_succ, mul_add, add_mul, mul_comm, mul_left_comm, mul_assoc]\n    <;> ring\n    <;> simp [h\u2080]\n    <;> ring\n  have step2 : a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 \u2264 8*a^4 + 8*b^4 := by\n    nlinarith [sq_nonneg (a - b), sq_nonneg (a + b), sq_nonneg (a^2 - b^2), sq_nonneg (a^2 + b^2),\n      mul_nonneg (sub_nonneg.mpr h\u2080.1.le) (sub_nonneg.mpr h\u2080.2.le), sq_nonneg (a^2 - 2 * a * b + b^2),\n      sq_nonneg (a^2 + 2 * a * b + b^2), mul_nonneg (sub_nonneg.mpr h\u2080.1.le) (sub_nonneg.mpr h\u2080.2.le),\n      mul_nonneg (sub_nonneg.mpr h\u2080.1.le) (sub_nonneg.mpr h\u2080.2.le)]\n  have step3 : 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 \u2264 7*a^4 + 7*b^4 := by\n    have h\u2081 : 0 < a * b := mul_pos h\u2080.1 h\u2080.2\n    have h\u2082 : 0 < a * b ^ 2 := mul_pos h\u2080.1 (sq_pos_of_pos h\u2080.2)\n    have h\u2083 : 0 < a ^ 2 * b := mul_pos (sq_pos_of_pos h\u2080.1) h\u2080.2\n    have h\u2084 : 0 < a ^ 2 * b ^ 2 := mul_pos (sq_pos_of_pos h\u2080.1) (sq_pos_of_pos h\u2080.2)\n    nlinarith [sq_nonneg (a - b), sq_nonneg (a + b), sq_nonneg (a ^ 2 - b ^ 2), sq_nonneg (a ^ 2 + b ^ 2),\n      mul_self_nonneg (a ^ 2 - 2 * a * b + b ^ 2)]\n  have step4 : a^4 + b^4 \u2265 2*a^2*b^2 := by\n    have h\u2081 : 0 < a * b := mul_pos h\u2080.1 h\u2080.2\n    have h\u2082 : 0 < a * b ^ 2 := mul_pos h\u2080.1 (sq_pos_of_pos h\u2080.2)\n    have h\u2083 : 0 < a ^ 2 * b := mul_pos (sq_pos_of_pos h\u2080.1) h\u2080.2\n    nlinarith [sq_nonneg (a ^ 2 - b ^ 2), sq_nonneg (a ^ 2 - 2 * a * b + b ^ 2), sq_nonneg (a ^ 2 + b ^ 2 - 2 * a * b), sq_nonneg (a ^ 2 + b ^ 2 + 2 * a * b)]\n  have step5 : a^4 + b^4 \u2265 2*a^2*b^2 := by\n    linarith [step4, h\u2080.1, h\u2080.2]\n    <;> assumption\n    <;> linarith\n  have step6 : (a + b)^4 \u2264 8*(a^4 + b^4) := by\n    nlinarith [sq_nonneg (a + b), sq_nonneg (a - b), mul_self_nonneg (a^2 - b^2), mul_self_nonneg (a^2 + b^2)]\n  linarith [step6]\n"}
{"mode": "clear", "steps_before": 57, "alphabet": 32, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 10, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 6, "fixpoint": true, "chars_before": 1454, "chars_after": 251, "seconds": 469.8, "id": "numbertheory_nckeqnm1ckpnm1ckm1", "proof_after": "theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : \u2115) (h\u2080 : 0 < n \u2227 0 < k) (h\u2081 : k \u2264 n) :\n  Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1)  :=  by\n  have total : Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) := by\n    cases' n with n <;> cases' k with k <;> try { simp_all [Nat.choose_succ_succ] }\n    <;>\n    try\n      simp_all [Nat.choose_succ_succ]\n    <;>\n    ring\n  omega\n", "proof_before": "theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : \u2115) (h\u2080 : 0 < n \u2227 0 < k) (h\u2081 : k \u2264 n) :\n  Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1)  :=  by\n  have case1 : \u2200 (x : \u2115), x < n \u2192 Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) := by\n    intro x hx\n    cases n with\n    | zero =>\n      simp_all\n    | succ n =>\n      cases k with\n      | zero =>\n        simp_all\n      | succ k =>\n        cases x with\n        | zero =>\n          simp_all [Nat.choose_succ_succ, Nat.add_comm, Nat.add_assoc]\n        | succ x =>\n          simp_all [Nat.choose_succ_succ, Nat.add_comm, Nat.add_assoc]\n  have case_excludes_x : Nat.choose (n - 1) k = Nat.choose (n - 1) k := by\n    have h\u2082 := case1 0\n    have h\u2083 := case1 1\n    have h\u2084 := case1 2\n    have h\u2085 := case1 3\n    have h\u2086 := case1 4\n    have h\u2087 := case1 5\n    have h\u2088 := case1 6\n    have h\u2089 := case1 7\n    have h\u2081\u2080 := case1 8\n    have h\u2081\u2081 := case1 9\n    have h\u2081\u2082 := case1 10\n    simp_all [Nat.choose_succ_succ]\n  have case_includes_x : Nat.choose (n - 1) (k - 1) = Nat.choose (n - 1) (k - 1) := by\n    simp_all\n    <;> try simp_all [Nat.choose_succ_succ]\n    <;> linarith\n  have total : Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) := by\n    cases' n with n <;> cases' k with k <;> try { simp_all [Nat.choose_succ_succ] }\n    <;>\n    try\n      simp_all [Nat.choose_succ_succ]\n    <;>\n    try\n      linarith\n  simp_all [Nat.choose_succ_succ]\n  <;> omega\n  <;> linarith\n  <;> ring\n  <;> omega\n  <;> linarith\n  <;> ring\n  <;> omega\n  <;> linarith\n  <;> ring\n  <;> omega\n  <;> linarith\n  <;> ring\n  <;> omega\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 31, "alphabet": 23, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1037, "chars_after": 8, "seconds": 32.0, "id": "mathd_numbertheory_335", "proof_after": "theorem mathd_numbertheory_335 (n : \u2115) (h\u2080 : n % 7 = 5) : 5 * n % 7 = 4  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_335 (n : \u2115) (h\u2080 : n % 7 = 5) : 5 * n % 7 = 4  :=  by\n  have h\u2081 : \u2203 k, n = 7 * k + 5 := by\n    use n / 7\n    have h\u2081 := Nat.div_add_mod n 7\n    simp [h\u2080] at h\u2081\n    omega\n  have h\u2082 : 5 * n = 5 * (7 * Classical.choose h\u2081 + 5) := by\n    let k := Classical.choose h\u2081\n    have h\u2082 : n = 7 * k + 5 := by\n      have h\u2083 : n = 7 * k + 5 := Classical.choose_spec h\u2081\n      simpa using h\u2083\n    omega\n  have h\u2083 : 5 * n = 35 * Classical.choose h\u2081 + 25 := by\n    rw [h\u2082]\n    <;> ring\n    <;> omega\n  have h\u2084 : 35 * Classical.choose h\u2081 % 7 = 0 := by\n    cases' h\u2081 with k hk\n    simp_all [mul_comm, mul_left_comm, mul_assoc]\n    <;> omega\n  have h\u2085 : 25 % 7 = 4 := by\n    norm_num [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084]\n    <;> simp_all [Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n    <;> omega\n  have h\u2086 : 5 * n % 7 = (35 * Classical.choose h\u2081 + 25) % 7 := by\n    simp [h\u2085, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt, Nat.succ_pos]\n    <;> omega\n  have h\u2087 : 5 * n % 7 = 4 := by\n    have h\u2087 : 5 * n % 7 = (35 * Classical.choose h\u2081 + 25) % 7 := by rw [h\u2086]\n    simp [h\u2087, h\u2084, h\u2085, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n  exact h\u2087\n"}
{"mode": "clear", "steps_before": 100, "alphabet": 51, "compiles": false, "id": "mathd_numbertheory_43", "proof_after": "theorem mathd_numbertheory_43 : IsGreatest { n : \u2115 | 15 ^ n \u2223 942! } 233  :=  by\n  have h1 : \u2200 n : \u2115, 15 ^ n = 3 ^ n * 5 ^ n := by\n    intro n\n    induction n with\n    | zero => simp\n    | succ n ih =>\n      simp_all [pow_succ, mul_assoc, mul_comm, mul_left_comm]\n      <;> ring\n      <;> rfl\n  have h2 : \u2200 n : \u2115, 3 ^ n \u2223 942! \u2192 n \u2264 467 := by\n    intro n hn\n    have h1' := h1 n\n    have h2 : 3 ^ n \u2223 942! := hn\n    have h3 : n \u2264 467 := by\n      by_contra h\n      push_neg at h\n      have h4 : 3 ^ n \u2223 942! := h2\n      have h5 : 3 ^ n \u2223 942! := h4\n      have h6 : n \u2265 468 := by linarith\n      have h7 : 3 ^ 468 \u2223 942! := by\n        apply Nat.dvd_trans (Nat.pow_dvd_pow 3 (by linarith)) h5\n      have h8 : 3 ^ 468 \u2223 942! := h7\n      rw [Nat.dvd_iff_mod_eq_zero] at h8\n      have h9 : 942! % 3 ^ 468 = 0 := h8\n      have h10 : 942! % 3 ^ 468 = 3 ^ 467 := by\n        rfl\n      have h11 : 3 ^ 467 = 0 := by\n        omega\n      omega\n    exact h3\n  have h3 : \u2200 n : \u2115, 5 ^ n \u2223 942! \u2192 n \u2264 233 := by\n    intro n hn\n    have h3 : \u2200 n : \u2115, 5 ^ n \u2223 942! \u2192 n \u2264 233 := by\n      intro n hn\n      have h4 : 5 ^ n \u2223 942! := hn\n      have h5 : n \u2264 233 := by\n        by_contra h6\n        have h7 : n > 233 := by linarith\n        have h8 : 5 ^ n \u2223 942! := h4\n        have h9 : n \u2265 234 := by linarith\n        have h10 : 5 ^ 234 \u2223 942! := by\n          apply Nat.dvd_trans (Nat.pow_dvd_pow _ (by linarith)) h8\n        rw [Nat.dvd_iff_mod_eq_zero] at h10\n        have h11 : 942! % 5 ^ 234 = 0 := h10\n        have h12 : 942! % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h13 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h14 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h15 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h16 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h17 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h18 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h19 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h20 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        omega\n      exact h5\n    exact h3 n hn\n  have h4 : \u2200 n : \u2115, 15 ^ n \u2223 942! \u2194 3 ^ n \u2223 942! \u2227 5 ^ n \u2223 942! := by\n    intro n\n    have h4 : 15 ^ n = 3 ^ n * 5 ^ n := by apply h1\n    constructor\n    \u00b7 intro h\n      rw [h4] at h\n      exact \u27e8Nat.dvd_trans (by simp [h4]) h, Nat.dvd_trans (by simp [h4]) h\u27e9\n    \u00b7 rintro \u27e8h\u2081, h\u2082\u27e9\n      rw [h4]\n      exact Nat.Coprime.mul_dvd_of_dvd_of_dvd (show Nat.Coprime (3 ^ n) (5 ^ n) by\n        apply Nat.Coprime.pow\n        norm_num) h\u2081 h\u2082\n  have h5 : \u2200 n : \u2115, 15 ^ n \u2223 942! \u2192 n \u2264 233 := by\n    intro n h5\n    have h6 : 3 ^ n \u2223 942! := by\n      have h7 : 15 ^ n \u2223 942! := h5\n      rw [h4] at h7\n      exact h7.1\n    have h8 : 5 ^ n \u2223 942! := by\n      have h7 : 15 ^ n \u2223 942! := h5\n      rw [h4] at h7\n      exact h7.2\n    exact (h3 n h8)\n  have h6 : 15 ^ 233 \u2223 942! := by\n    have h6 : 15 ^ 233 \u2223 942! := by\n      apply Nat.dvd_of_mod_eq_zero\n      rfl\n    exact h6\n  refine' \u27e8by simpa using h6, fun a ha => _\u27e9\n  have h7 := h5 a\n  have h8 := h6\n  have h9 := h4 a\n  simp_all\n  <;> omega\n", "proof_before": "theorem mathd_numbertheory_43 : IsGreatest { n : \u2115 | 15 ^ n \u2223 942! } 233  :=  by\n  have h1 : \u2200 n : \u2115, 15 ^ n = 3 ^ n * 5 ^ n := by\n    intro n\n    induction n with\n    | zero => simp\n    | succ n ih =>\n      simp_all [pow_succ, mul_assoc, mul_comm, mul_left_comm]\n      <;> ring\n      <;> rfl\n  have h2 : \u2200 n : \u2115, 3 ^ n \u2223 942! \u2192 n \u2264 467 := by\n    intro n hn\n    have h1' := h1 n\n    have h2 : 3 ^ n \u2223 942! := hn\n    have h3 : n \u2264 467 := by\n      by_contra h\n      push_neg at h\n      have h4 : 3 ^ n \u2223 942! := h2\n      have h5 : 3 ^ n \u2223 942! := h4\n      have h6 : n \u2265 468 := by linarith\n      have h7 : 3 ^ 468 \u2223 942! := by\n        apply Nat.dvd_trans (Nat.pow_dvd_pow 3 (by linarith)) h5\n      have h8 : 3 ^ 468 \u2223 942! := h7\n      rw [Nat.dvd_iff_mod_eq_zero] at h8\n      have h9 : 942! % 3 ^ 468 = 0 := h8\n      have h10 : 942! % 3 ^ 468 = 3 ^ 467 := by\n        rfl\n      have h11 : 3 ^ 467 = 0 := by\n        omega\n      omega\n    exact h3\n  have h3 : \u2200 n : \u2115, 5 ^ n \u2223 942! \u2192 n \u2264 233 := by\n    intro n hn\n    have h3 : \u2200 n : \u2115, 5 ^ n \u2223 942! \u2192 n \u2264 233 := by\n      intro n hn\n      have h4 : 5 ^ n \u2223 942! := hn\n      have h5 : n \u2264 233 := by\n        by_contra h6\n        have h7 : n > 233 := by linarith\n        have h8 : 5 ^ n \u2223 942! := h4\n        have h9 : n \u2265 234 := by linarith\n        have h10 : 5 ^ 234 \u2223 942! := by\n          apply Nat.dvd_trans (Nat.pow_dvd_pow _ (by linarith)) h8\n        rw [Nat.dvd_iff_mod_eq_zero] at h10\n        have h11 : 942! % 5 ^ 234 = 0 := h10\n        have h12 : 942! % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h13 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h14 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h15 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h16 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h17 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h18 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h19 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        have h20 : 3 ^ 467 * 5 ^ 233 % 5 ^ 234 = 3 ^ 467 * 5 ^ 233 % 5 ^ 234 := by\n          rfl\n        omega\n      exact h5\n    exact h3 n hn\n  have h4 : \u2200 n : \u2115, 15 ^ n \u2223 942! \u2194 3 ^ n \u2223 942! \u2227 5 ^ n \u2223 942! := by\n    intro n\n    have h4 : 15 ^ n = 3 ^ n * 5 ^ n := by apply h1\n    constructor\n    \u00b7 intro h\n      rw [h4] at h\n      exact \u27e8Nat.dvd_trans (by simp [h4]) h, Nat.dvd_trans (by simp [h4]) h\u27e9\n    \u00b7 rintro \u27e8h\u2081, h\u2082\u27e9\n      rw [h4]\n      exact Nat.Coprime.mul_dvd_of_dvd_of_dvd (show Nat.Coprime (3 ^ n) (5 ^ n) by\n        apply Nat.Coprime.pow\n        norm_num) h\u2081 h\u2082\n  have h5 : \u2200 n : \u2115, 15 ^ n \u2223 942! \u2192 n \u2264 233 := by\n    intro n h5\n    have h6 : 3 ^ n \u2223 942! := by\n      have h7 : 15 ^ n \u2223 942! := h5\n      rw [h4] at h7\n      exact h7.1\n    have h8 : 5 ^ n \u2223 942! := by\n      have h7 : 15 ^ n \u2223 942! := h5\n      rw [h4] at h7\n      exact h7.2\n    exact (h3 n h8)\n  have h6 : 15 ^ 233 \u2223 942! := by\n    have h6 : 15 ^ 233 \u2223 942! := by\n      apply Nat.dvd_of_mod_eq_zero\n      rfl\n    exact h6\n  refine' \u27e8by simpa using h6, fun a ha => _\u27e9\n  have h7 := h5 a\n  have h8 := h6\n  have h9 := h4 a\n  simp_all\n  <;> omega\n"}
{"mode": "clear", "steps_before": 34, "alphabet": 15, "compiles": false, "id": "mathd_numbertheory_403", "proof_after": "theorem mathd_numbertheory_403 : (\u2211 k in Nat.properDivisors 198, k) = 270  :=  by\n  have prime_factors : 198 = 2^1 * 3^2 * 11^1 := by\n    norm_num [show 2^1 * 3^2 * 11^1 = 198 by ring]\n  have num_divisors : Fintype.card (Nat.divisors 198) = 12 := by\n    rw [Nat.divisors]\n    rfl\n  have all_divisors : (Nat.divisors 198) = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198} := by\n    rw [Nat.divisors]\n    rfl\n    <;> norm_num\n    <;> decide\n    <;> simp_all\n    <;> norm_num\n    <;> decide\n    <;> simp_all\n    <;> norm_num\n    <;> decide\n    <;> simp_all\n    <;> norm_num\n    <;> decide\n    <;> simp_all\n    <;> norm_num\n    <;> decide\n  have proper_divisors : (Nat.properDivisors 198) = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99} := by\n    rw [show (Nat.divisors 198) = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198} by\n      rw [all_divisors]\n    ] at num_divisors\n    simp_all [Nat.properDivisors, Finset.filter_eq]\n    <;> decide\n  have sum_proper_divisors : (\u2211 k in Nat.properDivisors 198, k) = 270 := by\n    rw [proper_divisors]\n    norm_num\n    <;> rfl\n    <;> simp_all [Nat.divisors]\n    <;> decide\n    <;> rfl\n  exact sum_proper_divisors\n", "proof_before": "theorem mathd_numbertheory_403 : (\u2211 k in Nat.properDivisors 198, k) = 270  :=  by\n  have prime_factors : 198 = 2^1 * 3^2 * 11^1 := by\n    norm_num [show 2^1 * 3^2 * 11^1 = 198 by ring]\n  have num_divisors : Fintype.card (Nat.divisors 198) = 12 := by\n    rw [Nat.divisors]\n    rfl\n  have all_divisors : (Nat.divisors 198) = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198} := by\n    rw [Nat.divisors]\n    rfl\n    <;> norm_num\n    <;> decide\n    <;> simp_all\n    <;> norm_num\n    <;> decide\n    <;> simp_all\n    <;> norm_num\n    <;> decide\n    <;> simp_all\n    <;> norm_num\n    <;> decide\n    <;> simp_all\n    <;> norm_num\n    <;> decide\n  have proper_divisors : (Nat.properDivisors 198) = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99} := by\n    rw [show (Nat.divisors 198) = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198} by\n      rw [all_divisors]\n    ] at num_divisors\n    simp_all [Nat.properDivisors, Finset.filter_eq]\n    <;> decide\n  have sum_proper_divisors : (\u2211 k in Nat.properDivisors 198, k) = 270 := by\n    rw [proper_divisors]\n    norm_num\n    <;> rfl\n    <;> simp_all [Nat.divisors]\n    <;> decide\n    <;> rfl\n  exact sum_proper_divisors\n"}
{"mode": "clear", "steps_before": 84, "alphabet": 42, "compiles": false, "id": "mathd_numbertheory_530", "proof_after": "theorem mathd_numbertheory_530 (n k : \u2115) (h\u2080 : 0 < n \u2227 0 < k) (h\u2080 : (n : \u211d) / k < 6)\n  (h\u2081 : (5 : \u211d) < n / k) : 22 \u2264 Nat.lcm n k / Nat.gcd n k  :=  by\n  have h\u2083 : \u2203 d a b, d = Nat.gcd n k \u2227 n = d * a \u2227 k = d * b \u2227 Nat.gcd a b = 1 := by\n    use Nat.gcd n k\n    use n / Nat.gcd n k\n    use k / Nat.gcd n k\n    have h\u2082 : 0 < n := by linarith\n    have h\u2083 : 0 < k := by linarith\n    have h\u2084 : 0 < Nat.gcd n k := Nat.gcd_pos_of_pos_left k h\u2082\n    have h\u2085 : n = Nat.gcd n k * (n / Nat.gcd n k) := by\n      rw [Nat.mul_div_cancel' (Nat.gcd_dvd_left n k)]\n    have h\u2086 : k = Nat.gcd n k * (k / Nat.gcd n k) := by\n      rw [Nat.mul_div_cancel' (Nat.gcd_dvd_right n k)]\n    have h\u2087 : Nat.gcd (n / Nat.gcd n k) (k / Nat.gcd n k) = 1 := by\n      rw [Nat.gcd_div (Nat.gcd_dvd_left n k) (Nat.gcd_dvd_right n k)]\n      <;> simp [h\u2084]\n    exact \u27e8rfl, h\u2085, h\u2086, h\u2087\u27e9\n  have h\u2084 : \u2200 d a b, d = Nat.gcd n k \u2192 n = d * a \u2192 k = d * b \u2192 5 * b < a \u2227 a < 6 * b := by\n    intro d a b h\u2084 h\u2085 h\u2086\n    have h\u2087 : 0 < d := by\n      have h\u2087 : 0 < Nat.gcd n k := Nat.gcd_pos_of_pos_left k (by linarith)\n      linarith\n    have h\u2088 : 0 < a := by\n      have h\u2088 : 0 < n := by linarith\n      have h\u2089 : 0 < d := by linarith\n      nlinarith\n    have h\u2089 : 0 < b := by\n      have h\u2089 : 0 < k := by linarith\n      have h\u2081\u2080 : 0 < d := by linarith\n      nlinarith\n    constructor\n    field_simp [h\u2085, h\u2086] at h\u2080 h\u2081 \u22a2\n    rw [div_lt_iff] at h\u2080 <;> rw [lt_div_iff] at h\u2081 <;> norm_cast at h\u2080 h\u2081 \u22a2 <;> nlinarith\n    field_simp [h\u2085, h\u2086] at h\u2080 h\u2081 \u22a2\n    rw [div_lt_iff] at h\u2080 <;> rw [lt_div_iff] at h\u2081 <;> norm_cast at h\u2080 h\u2081 \u22a2 <;> nlinarith\n  have h\u2085 : \u2200 a b, 5 * b < a \u2192 a < 6 * b \u2192 Nat.gcd a b = 1 \u2192 22 \u2264 a * b := by\n    have h\u2085 : \u2200 a b : \u2115, 5 * b < a \u2192 a < 6 * b \u2192 Nat.gcd a b = 1 \u2192 22 \u2264 a * b := by\n      intro a b h\u2085 h\u2086 h\u2087\n      have h\u2088 : a * b \u2265 22 := by\n        by_contra h\u2088\n        have h\u2089 : a * b \u2264 21 := by\n          linarith\n        have h\u2081\u2080 : a \u2264 21 := by\n          nlinarith\n        have h\u2081\u2081 : b \u2264 21 := by\n          nlinarith\n        interval_cases a <;> interval_cases b <;> contradiction\n      linarith\n    exact h\u2085\n  have h\u2086 : 22 \u2264 Nat.lcm n k / Nat.gcd n k := by\n    obtain \u27e8d, a, b, hd, ha, hb, hgcd\u27e9 := h\u2083\n    have h\u2086 := h\u2084 d a b hd ha hb\n    have h\u2087 := h\u2085 a b\n    have h\u2088 : 0 < d := by\n      nlinarith\n    have h\u2089 : 0 < a := by\n      nlinarith\n    have h\u2081\u2080 : 0 < b := by\n      nlinarith\n    have h\u2081\u2081 : Nat.gcd n k = d := by\n      nlinarith\n    have h\u2081\u2082 : Nat.lcm n k = d * a * b := by\n      nlinarith [Nat.gcd_mul_lcm n k, Nat.mul_comm a b, Nat.mul_comm d a, Nat.mul_comm d b]\n    have h\u2081\u2083 : d * a * b / d = a * b := by\n      rw [Nat.mul_comm]\n      exact by\n        rw [Nat.div_eq_of_eq_mul_left] <;> nlinarith\n    rw [h\u2081\u2081, h\u2081\u2082, h\u2081\u2083]\n    nlinarith [h\u2086, h\u2087]\n  exact h\u2086\n", "proof_before": "theorem mathd_numbertheory_530 (n k : \u2115) (h\u2080 : 0 < n \u2227 0 < k) (h\u2080 : (n : \u211d) / k < 6)\n  (h\u2081 : (5 : \u211d) < n / k) : 22 \u2264 Nat.lcm n k / Nat.gcd n k  :=  by\n  have h\u2083 : \u2203 d a b, d = Nat.gcd n k \u2227 n = d * a \u2227 k = d * b \u2227 Nat.gcd a b = 1 := by\n    use Nat.gcd n k\n    use n / Nat.gcd n k\n    use k / Nat.gcd n k\n    have h\u2082 : 0 < n := by linarith\n    have h\u2083 : 0 < k := by linarith\n    have h\u2084 : 0 < Nat.gcd n k := Nat.gcd_pos_of_pos_left k h\u2082\n    have h\u2085 : n = Nat.gcd n k * (n / Nat.gcd n k) := by\n      rw [Nat.mul_div_cancel' (Nat.gcd_dvd_left n k)]\n    have h\u2086 : k = Nat.gcd n k * (k / Nat.gcd n k) := by\n      rw [Nat.mul_div_cancel' (Nat.gcd_dvd_right n k)]\n    have h\u2087 : Nat.gcd (n / Nat.gcd n k) (k / Nat.gcd n k) = 1 := by\n      rw [Nat.gcd_div (Nat.gcd_dvd_left n k) (Nat.gcd_dvd_right n k)]\n      <;> simp [h\u2084]\n    exact \u27e8rfl, h\u2085, h\u2086, h\u2087\u27e9\n  have h\u2084 : \u2200 d a b, d = Nat.gcd n k \u2192 n = d * a \u2192 k = d * b \u2192 5 * b < a \u2227 a < 6 * b := by\n    intro d a b h\u2084 h\u2085 h\u2086\n    have h\u2087 : 0 < d := by\n      have h\u2087 : 0 < Nat.gcd n k := Nat.gcd_pos_of_pos_left k (by linarith)\n      linarith\n    have h\u2088 : 0 < a := by\n      have h\u2088 : 0 < n := by linarith\n      have h\u2089 : 0 < d := by linarith\n      nlinarith\n    have h\u2089 : 0 < b := by\n      have h\u2089 : 0 < k := by linarith\n      have h\u2081\u2080 : 0 < d := by linarith\n      nlinarith\n    constructor\n    field_simp [h\u2085, h\u2086] at h\u2080 h\u2081 \u22a2\n    rw [div_lt_iff] at h\u2080 <;> rw [lt_div_iff] at h\u2081 <;> norm_cast at h\u2080 h\u2081 \u22a2 <;> nlinarith\n    field_simp [h\u2085, h\u2086] at h\u2080 h\u2081 \u22a2\n    rw [div_lt_iff] at h\u2080 <;> rw [lt_div_iff] at h\u2081 <;> norm_cast at h\u2080 h\u2081 \u22a2 <;> nlinarith\n  have h\u2085 : \u2200 a b, 5 * b < a \u2192 a < 6 * b \u2192 Nat.gcd a b = 1 \u2192 22 \u2264 a * b := by\n    have h\u2085 : \u2200 a b : \u2115, 5 * b < a \u2192 a < 6 * b \u2192 Nat.gcd a b = 1 \u2192 22 \u2264 a * b := by\n      intro a b h\u2085 h\u2086 h\u2087\n      have h\u2088 : a * b \u2265 22 := by\n        by_contra h\u2088\n        have h\u2089 : a * b \u2264 21 := by\n          linarith\n        have h\u2081\u2080 : a \u2264 21 := by\n          nlinarith\n        have h\u2081\u2081 : b \u2264 21 := by\n          nlinarith\n        interval_cases a <;> interval_cases b <;> contradiction\n      linarith\n    exact h\u2085\n  have h\u2086 : 22 \u2264 Nat.lcm n k / Nat.gcd n k := by\n    obtain \u27e8d, a, b, hd, ha, hb, hgcd\u27e9 := h\u2083\n    have h\u2086 := h\u2084 d a b hd ha hb\n    have h\u2087 := h\u2085 a b\n    have h\u2088 : 0 < d := by\n      nlinarith\n    have h\u2089 : 0 < a := by\n      nlinarith\n    have h\u2081\u2080 : 0 < b := by\n      nlinarith\n    have h\u2081\u2081 : Nat.gcd n k = d := by\n      nlinarith\n    have h\u2081\u2082 : Nat.lcm n k = d * a * b := by\n      nlinarith [Nat.gcd_mul_lcm n k, Nat.mul_comm a b, Nat.mul_comm d a, Nat.mul_comm d b]\n    have h\u2081\u2083 : d * a * b / d = a * b := by\n      rw [Nat.mul_comm]\n      exact by\n        rw [Nat.div_eq_of_eq_mul_left] <;> nlinarith\n    rw [h\u2081\u2081, h\u2081\u2082, h\u2081\u2083]\n    nlinarith [h\u2086, h\u2087]\n  exact h\u2086\n"}
{"mode": "clear", "steps_before": 36, "alphabet": 28, "compiles": true, "steps_after": 33, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 2, "fixpoint": false, "chars_before": 1108, "chars_after": 1024, "seconds": 557.7, "id": "amc12a_2013_p8", "proof_after": "theorem amc12a_2013_p8 (x y : \u211d) (h\u2080 : x \u2260 0) (h\u2081 : y \u2260 0) (h\u2082 : x \u2260 y)\n  (h\u2083 : x + 2 / x = y + 2 / y) : x * y = 2  :=  by\n  have h\u2084 : x + 2 / x - y - 2 / y = 0 := by\n    linarith\n  have h\u2085 : (x - y) + (2 / x - 2 / y) = 0 := by\n    field_simp [h\u2080, h\u2081, h\u2082] at h\u2083 h\u2084 \u22a2\n    linarith\n  have h\u2086 : (x - y) + 2 * (1 / x - 1 / y) = 0 := by\n    field_simp [h\u2080, h\u2081, h\u2082] at h\u2083 h\u2084 h\u2085 \u22a2\n    linarith\n  have h\u2087 : (x - y) + 2 * ((y - x) / (x * y)) = 0 := by\n    field_simp at h\u2085 h\u2086 \u22a2\n    ring_nf at h\u2085 h\u2086 \u22a2\n    linarith\n  have h\u2088 : (x - y) - 2 * ((x - y) / (x * y)) = 0 := by\n    field_simp at h\u2085 h\u2086 h\u2087 \u22a2\n    ring_nf at h\u2085 h\u2086 h\u2087 \u22a2\n    linarith\n  have h\u2089 : (x - y) * (1 - 2 / (x * y)) = 0 := by\n    have h\u2089 : (x - y) * (1 - 2 / (x * y)) = 0 := by\n      field_simp [h\u2080, h\u2081, h\u2082] at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 \u22a2\n      linarith\n    linarith\n  have h\u2081\u2080 : 1 - 2 / (x * y) = 0 := by\n    have : x * y \u2260 0 := mul_ne_zero h\u2080 h\u2081\n    have : x - y \u2260 0 := sub_ne_zero_of_ne h\u2082\n    field_simp at h\u2089 \u22a2\n    apply mul_right_cancel\u2080 (sub_ne_zero.mpr h\u2082)\n    linarith\n  have h\u2081\u2081 : x * y = 2 := by\n    have h\u2081\u2081 : x * y = 2 := by\n      field_simp at h\u2081\u2080\n      linarith\n    exact h\u2081\u2081\n  rw [h\u2081\u2081]\n", "proof_before": "theorem amc12a_2013_p8 (x y : \u211d) (h\u2080 : x \u2260 0) (h\u2081 : y \u2260 0) (h\u2082 : x \u2260 y)\n  (h\u2083 : x + 2 / x = y + 2 / y) : x * y = 2  :=  by\n  have h\u2084 : x + 2 / x - y - 2 / y = 0 := by\n    field_simp [h\u2080, h\u2081, h\u2082] at h\u2083 \u22a2\n    ring_nf at h\u2083 \u22a2\n    apply eq_of_sub_eq_zero\n    linarith\n  have h\u2085 : (x - y) + (2 / x - 2 / y) = 0 := by\n    field_simp [h\u2080, h\u2081, h\u2082] at h\u2083 h\u2084 \u22a2\n    linarith\n  have h\u2086 : (x - y) + 2 * (1 / x - 1 / y) = 0 := by\n    field_simp [h\u2080, h\u2081, h\u2082] at h\u2083 h\u2084 h\u2085 \u22a2\n    linarith\n  have h\u2087 : (x - y) + 2 * ((y - x) / (x * y)) = 0 := by\n    field_simp at h\u2085 h\u2086 \u22a2\n    ring_nf at h\u2085 h\u2086 \u22a2\n    linarith\n  have h\u2088 : (x - y) - 2 * ((x - y) / (x * y)) = 0 := by\n    field_simp at h\u2085 h\u2086 h\u2087 \u22a2\n    ring_nf at h\u2085 h\u2086 h\u2087 \u22a2\n    linarith\n  have h\u2089 : (x - y) * (1 - 2 / (x * y)) = 0 := by\n    have h\u2089 : (x - y) * (1 - 2 / (x * y)) = 0 := by\n      field_simp [h\u2080, h\u2081, h\u2082] at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 \u22a2\n      linarith\n    linarith\n  have h\u2081\u2080 : 1 - 2 / (x * y) = 0 := by\n    have : x * y \u2260 0 := mul_ne_zero h\u2080 h\u2081\n    have : x - y \u2260 0 := sub_ne_zero_of_ne h\u2082\n    field_simp at h\u2089 \u22a2\n    apply mul_right_cancel\u2080 (sub_ne_zero.mpr h\u2082)\n    linarith\n  have h\u2081\u2081 : x * y = 2 := by\n    have h\u2081\u2081 : x * y = 2 := by\n      field_simp at h\u2081\u2080\n      linarith\n    exact h\u2081\u2081\n  rw [h\u2081\u2081]\n"}
{"mode": "clear", "steps_before": 167, "alphabet": 76, "compiles": false, "id": "numbertheory_xsqpysqintdenomeq", "proof_after": "theorem numbertheory_xsqpysqintdenomeq (x y : \u211a) (h\u2080 : (x ^ 2 + y ^ 2).den = 1) : x.den = y.den  :=  by\n  have h\u2081 : \u2203 a b, x.num = a \u2227 x.den = b \u2227 Int.gcd a b = 1 := by\n    use x.num, x.den\n    constructor\n    exact Eq.refl x.num\n    constructor\n    exact Eq.refl x.den\n    exact x.reduced\n  have h\u2082 : \u2203 c d, y.num = c \u2227 y.den = d \u2227 Int.gcd c d = 1 := by\n    refine' \u27e8y.num, y.den, by simp [Rat.num_div_den], by simp [Rat.num_div_den], _\u27e9\n    simpa [Rat.num_div_den] using y.reduced\n  have h\u2083 : x = (x.num : \u211a) / x.den := by\n    have h\u2083 : x = (x.num : \u211a) / x.den := by\n      simp_all [Rat.num_div_den]\n    exact h\u2083\n  have h\u2084 : y = (y.num : \u211a) / y.den := by\n    rw [\u2190 mul_one y]\n    field_simp [h\u2080, h\u2081, h\u2082, h\u2083]\n    <;> norm_cast\n    <;> ring_nf\n    <;> simp_all\n    <;> norm_cast\n    <;> linarith\n  have h\u2085 : x ^ 2 + y ^ 2 = (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) / (x.den ^ 2 * y.den ^ 2) := by\n    have h\u2085 : x = (x.num : \u211a) / x.den := by simpa using h\u2083\n    have h\u2086 : y = (y.num : \u211a) / y.den := by simpa using h\u2084\n    rw [h\u2085, h\u2086]\n    field_simp [mul_comm]\n    <;> norm_cast\n    <;> simp [mul_comm]\n    <;> ring_nf\n    <;> norm_cast\n    <;> simp [mul_comm]\n    <;> ring_nf\n    <;> norm_cast\n    <;> simp [mul_comm]\n    <;> ring_nf\n    <;> norm_cast\n  have h\u2086 : (x.den ^ 2 * y.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n    rw [dvd_iff_exists_eq_mul_left]\n    use (x ^ 2 + y ^ 2).num\n    rw [\u2190 Rat.num_div_den (x ^ 2 + y ^ 2)] at h\u2085\n    field_simp [h\u2080] at h\u2085 \u22a2\n    <;> norm_cast at h\u2085 \u22a2\n    <;> nlinarith\n  have h\u2087 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n    have h\u2087 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n      have h\u2088 : (x.den ^ 2 : \u2124) \u2223 (x.den ^ 2 * y.den ^ 2 : \u2124) := by\n        apply dvd_mul_of_dvd_left\n        exact dvd_refl _\n      have h\u2089 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n        exact (Int.dvd_trans h\u2088 (by simpa [h\u2085, h\u2086] using h\u2086))\n      exact h\u2089\n    exact h\u2087\n  have h\u2088 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2) := by\n    have h\u2087' : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n      exact Int.dvd_trans h\u2087 (by simpa [mul_assoc] using h\u2086)\n    have h\u2088 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2) := by\n      rw [dvd_iff_exists_eq_mul_left] at h\u2087' \u22a2\n      rcases h\u2087' with \u27e8k, hk\u27e9\n      use k - y.num ^ 2\n      linarith\n    exact h\u2088\n  have h\u2089 : Int.gcd (x.num ^ 2) (x.den ^ 2) = 1 := by\n    have h\u2089 : Int.gcd (x.num ^ 2) (x.den ^ 2) = 1 := by\n      have h\u2089\u2080 : Int.gcd (x.num) (x.den) = 1 := x.reduced\n      have h\u2089\u2081 : Int.gcd (x.num ^ 2) (x.den ^ 2) = 1 := by\n        rw [Int.gcd_eq_one_iff_coprime]\n        have h\u2089\u2082 : IsCoprime (x.num : \u2124) (x.den : \u2124) := by\n          simpa [Int.gcd_eq_one_iff_coprime] using h\u2089\u2080\n        exact h\u2089\u2082.pow\n      exact h\u2089\u2081\n    exact h\u2089\n  have h\u2081\u2080 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by\n    have h\u2081\u2080 := h\u2087\n    have h\u2081\u2081 := h\u2088\n    have h\u2081\u2082 := h\u2086\n    have h\u2081\u2083 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by\n      have h\u2081\u2084 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2) := by simpa [h\u2085, h\u2086] using h\u2088\n      have h\u2081\u2085 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by\n        have h\u2081\u2086 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2) := by simpa [h\u2085, h\u2086] using h\u2088\n        have h\u2081\u2087 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by\n          apply (Int.gcd_eq_one_iff_coprime.mp h\u2089).symm.dvd_of_dvd_mul_left\n          simpa [h\u2085, h\u2086] using h\u2088\n        exact h\u2081\u2087\n      exact h\u2081\u2085\n    exact h\u2081\u2083\n  have h\u2081\u2081 : (y.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n    have h\u2081\u2081 : (y.den : \u2124) ^ 2 \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n      have h\u2081\u2080' : (x.den : \u2124) ^ 2 \u2223 (y.den : \u2124) ^ 2 := by\n        simpa [pow_succ] using h\u2081\u2080\n      have h\u2088' : (x.den : \u2124) ^ 2 \u2223 (x.num ^ 2 * y.den ^ 2) := by\n        simpa [pow_succ] using h\u2088\n      have h\u2087' : (x.den : \u2124) ^ 2 \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n        simpa [pow_succ] using h\u2087\n      have h\u2086' : (x.den : \u2124) ^ 2 * y.den ^ 2 \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n        simpa [pow_succ] using h\u2086\n      have h\u2081\u2080'' : (y.den : \u2124) ^ 2 \u2223 (x.den : \u2124) ^ 2 * y.den ^ 2 := by\n        simpa [pow_succ] using h\u2081\u2080'\n      have h\u2088'' : (y.den : \u2124) ^ 2 \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n        refine' dvd_trans h\u2081\u2080'' h\u2086'\n      exact h\u2088''\n    exact h\u2081\u2081\n  have h\u2081\u2082 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by\n    have h\u2081\u2082 : (y.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n      exact h\u2081\u2081\n    have h\u2081\u2083 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by\n      rw [dvd_iff_exists_eq_mul_left] at h\u2081\u2082 \u22a2\n      rcases h\u2081\u2082 with \u27e8k, hk\u27e9\n      use k - x.num ^ 2\n      linarith\n    exact h\u2081\u2083\n  have h\u2081\u2083 : Int.gcd (y.num ^ 2) (y.den ^ 2) = 1 := by\n    have h\u2081\u2083 : Int.gcd (y.num ^ 2) (y.den ^ 2) = 1 := by\n      have h\u2081\u2084 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by assumption\n      have h\u2081\u2085 : Int.gcd (x.num ^ 2) (x.den ^ 2) = 1 := by assumption\n      have h\u2081\u2086 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by assumption\n      have h\u2081\u2087 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by assumption\n      have h\u2081\u2088 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by assumption\n      have h\u2081\u2089 : Int.gcd (y.num ^ 2) (y.den ^ 2) = 1 := by\n        have h\u2082\u2080 : Int.gcd (y.num) (y.den) = 1 := by\n          have h\u2082\u2081 : Int.gcd (y.num) (y.den) = 1 := by\n            exact y.reduced\n          assumption\n        have h\u2082\u2082 : Int.gcd (y.num ^ 2) (y.den ^ 2) = 1 := by\n          exact Int.gcd_eq_one_iff_coprime.mpr (IsCoprime.pow (Int.gcd_eq_one_iff_coprime.mp h\u2082\u2080))\n        assumption\n      assumption\n    assumption\n  have h\u2081\u2084 : (y.den ^ 2 : \u2124) \u2223 (x.den ^ 2) := by\n    have h\u2081\u2084 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by\n      exact Int.dvd_trans (by simpa using h\u2081\u2082) (by simp [h\u2085])\n    have h\u2081\u2085 : (y.den ^ 2 : \u2124) \u2223 (x.den ^ 2) := by\n      have h\u2081\u2086 : (y.den ^ 2 : \u2124) \u2223 (x.den ^ 2) := by\n        exact (Int.gcd_eq_one_iff_coprime.mp h\u2081\u2083).symm.dvd_of_dvd_mul_left h\u2081\u2084\n      exact h\u2081\u2086\n    exact h\u2081\u2085\n  have h\u2081\u2085 : (x.den ^ 2 : \u2124) = y.den ^ 2 := by\n    have h\u2081\u2085 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by simpa [h\u2081\u2080, h\u2081\u2081, h\u2081\u2082, h\u2081\u2083, h\u2081\u2084] using h\u2081\u2080\n    have h\u2081\u2086 : (y.den ^ 2 : \u2124) \u2223 (x.den ^ 2) := by simpa [h\u2081\u2080, h\u2081\u2081, h\u2081\u2082, h\u2081\u2083, h\u2081\u2084] using h\u2081\u2084\n    have h\u2081\u2087 : (x.den ^ 2 : \u2124) = y.den ^ 2 := by\n      apply le_antisymm\n      \u00b7 exact Int.le_of_dvd (by positivity) h\u2081\u2085\n      \u00b7 exact Int.le_of_dvd (by positivity) h\u2081\u2086\n    exact_mod_cast h\u2081\u2087\n  have h\u2081\u2086 : x.den = y.den := by\n    rw [sq] at h\u2081\u2085\n    norm_cast at h\u2081\u2085\n    apply Eq.symm\n    <;> nlinarith\n  rw [h\u2081\u2086]\n  <;>\n  ring_nf at *\n  <;>\n  simp_all\n  <;>\n  linarith\n", "proof_before": "theorem numbertheory_xsqpysqintdenomeq (x y : \u211a) (h\u2080 : (x ^ 2 + y ^ 2).den = 1) : x.den = y.den  :=  by\n  have h\u2081 : \u2203 a b, x.num = a \u2227 x.den = b \u2227 Int.gcd a b = 1 := by\n    use x.num, x.den\n    constructor\n    exact Eq.refl x.num\n    constructor\n    exact Eq.refl x.den\n    exact x.reduced\n  have h\u2082 : \u2203 c d, y.num = c \u2227 y.den = d \u2227 Int.gcd c d = 1 := by\n    refine' \u27e8y.num, y.den, by simp [Rat.num_div_den], by simp [Rat.num_div_den], _\u27e9\n    simpa [Rat.num_div_den] using y.reduced\n  have h\u2083 : x = (x.num : \u211a) / x.den := by\n    have h\u2083 : x = (x.num : \u211a) / x.den := by\n      simp_all [Rat.num_div_den]\n    exact h\u2083\n  have h\u2084 : y = (y.num : \u211a) / y.den := by\n    rw [\u2190 mul_one y]\n    field_simp [h\u2080, h\u2081, h\u2082, h\u2083]\n    <;> norm_cast\n    <;> ring_nf\n    <;> simp_all\n    <;> norm_cast\n    <;> linarith\n  have h\u2085 : x ^ 2 + y ^ 2 = (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) / (x.den ^ 2 * y.den ^ 2) := by\n    have h\u2085 : x = (x.num : \u211a) / x.den := by simpa using h\u2083\n    have h\u2086 : y = (y.num : \u211a) / y.den := by simpa using h\u2084\n    rw [h\u2085, h\u2086]\n    field_simp [mul_comm]\n    <;> norm_cast\n    <;> simp [mul_comm]\n    <;> ring_nf\n    <;> norm_cast\n    <;> simp [mul_comm]\n    <;> ring_nf\n    <;> norm_cast\n    <;> simp [mul_comm]\n    <;> ring_nf\n    <;> norm_cast\n  have h\u2086 : (x.den ^ 2 * y.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n    rw [dvd_iff_exists_eq_mul_left]\n    use (x ^ 2 + y ^ 2).num\n    rw [\u2190 Rat.num_div_den (x ^ 2 + y ^ 2)] at h\u2085\n    field_simp [h\u2080] at h\u2085 \u22a2\n    <;> norm_cast at h\u2085 \u22a2\n    <;> nlinarith\n  have h\u2087 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n    have h\u2087 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n      have h\u2088 : (x.den ^ 2 : \u2124) \u2223 (x.den ^ 2 * y.den ^ 2 : \u2124) := by\n        apply dvd_mul_of_dvd_left\n        exact dvd_refl _\n      have h\u2089 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n        exact (Int.dvd_trans h\u2088 (by simpa [h\u2085, h\u2086] using h\u2086))\n      exact h\u2089\n    exact h\u2087\n  have h\u2088 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2) := by\n    have h\u2087' : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n      exact Int.dvd_trans h\u2087 (by simpa [mul_assoc] using h\u2086)\n    have h\u2088 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2) := by\n      rw [dvd_iff_exists_eq_mul_left] at h\u2087' \u22a2\n      rcases h\u2087' with \u27e8k, hk\u27e9\n      use k - y.num ^ 2\n      linarith\n    exact h\u2088\n  have h\u2089 : Int.gcd (x.num ^ 2) (x.den ^ 2) = 1 := by\n    have h\u2089 : Int.gcd (x.num ^ 2) (x.den ^ 2) = 1 := by\n      have h\u2089\u2080 : Int.gcd (x.num) (x.den) = 1 := x.reduced\n      have h\u2089\u2081 : Int.gcd (x.num ^ 2) (x.den ^ 2) = 1 := by\n        rw [Int.gcd_eq_one_iff_coprime]\n        have h\u2089\u2082 : IsCoprime (x.num : \u2124) (x.den : \u2124) := by\n          simpa [Int.gcd_eq_one_iff_coprime] using h\u2089\u2080\n        exact h\u2089\u2082.pow\n      exact h\u2089\u2081\n    exact h\u2089\n  have h\u2081\u2080 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by\n    have h\u2081\u2080 := h\u2087\n    have h\u2081\u2081 := h\u2088\n    have h\u2081\u2082 := h\u2086\n    have h\u2081\u2083 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by\n      have h\u2081\u2084 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2) := by simpa [h\u2085, h\u2086] using h\u2088\n      have h\u2081\u2085 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by\n        have h\u2081\u2086 : (x.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2) := by simpa [h\u2085, h\u2086] using h\u2088\n        have h\u2081\u2087 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by\n          apply (Int.gcd_eq_one_iff_coprime.mp h\u2089).symm.dvd_of_dvd_mul_left\n          simpa [h\u2085, h\u2086] using h\u2088\n        exact h\u2081\u2087\n      exact h\u2081\u2085\n    exact h\u2081\u2083\n  have h\u2081\u2081 : (y.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n    have h\u2081\u2081 : (y.den : \u2124) ^ 2 \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n      have h\u2081\u2080' : (x.den : \u2124) ^ 2 \u2223 (y.den : \u2124) ^ 2 := by\n        simpa [pow_succ] using h\u2081\u2080\n      have h\u2088' : (x.den : \u2124) ^ 2 \u2223 (x.num ^ 2 * y.den ^ 2) := by\n        simpa [pow_succ] using h\u2088\n      have h\u2087' : (x.den : \u2124) ^ 2 \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n        simpa [pow_succ] using h\u2087\n      have h\u2086' : (x.den : \u2124) ^ 2 * y.den ^ 2 \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n        simpa [pow_succ] using h\u2086\n      have h\u2081\u2080'' : (y.den : \u2124) ^ 2 \u2223 (x.den : \u2124) ^ 2 * y.den ^ 2 := by\n        simpa [pow_succ] using h\u2081\u2080'\n      have h\u2088'' : (y.den : \u2124) ^ 2 \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n        refine' dvd_trans h\u2081\u2080'' h\u2086'\n      exact h\u2088''\n    exact h\u2081\u2081\n  have h\u2081\u2082 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by\n    have h\u2081\u2082 : (y.den ^ 2 : \u2124) \u2223 (x.num ^ 2 * y.den ^ 2 + y.num ^ 2 * x.den ^ 2) := by\n      exact h\u2081\u2081\n    have h\u2081\u2083 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by\n      rw [dvd_iff_exists_eq_mul_left] at h\u2081\u2082 \u22a2\n      rcases h\u2081\u2082 with \u27e8k, hk\u27e9\n      use k - x.num ^ 2\n      linarith\n    exact h\u2081\u2083\n  have h\u2081\u2083 : Int.gcd (y.num ^ 2) (y.den ^ 2) = 1 := by\n    have h\u2081\u2083 : Int.gcd (y.num ^ 2) (y.den ^ 2) = 1 := by\n      have h\u2081\u2084 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by assumption\n      have h\u2081\u2085 : Int.gcd (x.num ^ 2) (x.den ^ 2) = 1 := by assumption\n      have h\u2081\u2086 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by assumption\n      have h\u2081\u2087 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by assumption\n      have h\u2081\u2088 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by assumption\n      have h\u2081\u2089 : Int.gcd (y.num ^ 2) (y.den ^ 2) = 1 := by\n        have h\u2082\u2080 : Int.gcd (y.num) (y.den) = 1 := by\n          have h\u2082\u2081 : Int.gcd (y.num) (y.den) = 1 := by\n            exact y.reduced\n          assumption\n        have h\u2082\u2082 : Int.gcd (y.num ^ 2) (y.den ^ 2) = 1 := by\n          exact Int.gcd_eq_one_iff_coprime.mpr (IsCoprime.pow (Int.gcd_eq_one_iff_coprime.mp h\u2082\u2080))\n        assumption\n      assumption\n    assumption\n  have h\u2081\u2084 : (y.den ^ 2 : \u2124) \u2223 (x.den ^ 2) := by\n    have h\u2081\u2084 : (y.den ^ 2 : \u2124) \u2223 (y.num ^ 2 * x.den ^ 2) := by\n      exact Int.dvd_trans (by simpa using h\u2081\u2082) (by simp [h\u2085])\n    have h\u2081\u2085 : (y.den ^ 2 : \u2124) \u2223 (x.den ^ 2) := by\n      have h\u2081\u2086 : (y.den ^ 2 : \u2124) \u2223 (x.den ^ 2) := by\n        exact (Int.gcd_eq_one_iff_coprime.mp h\u2081\u2083).symm.dvd_of_dvd_mul_left h\u2081\u2084\n      exact h\u2081\u2086\n    exact h\u2081\u2085\n  have h\u2081\u2085 : (x.den ^ 2 : \u2124) = y.den ^ 2 := by\n    have h\u2081\u2085 : (x.den ^ 2 : \u2124) \u2223 (y.den ^ 2) := by simpa [h\u2081\u2080, h\u2081\u2081, h\u2081\u2082, h\u2081\u2083, h\u2081\u2084] using h\u2081\u2080\n    have h\u2081\u2086 : (y.den ^ 2 : \u2124) \u2223 (x.den ^ 2) := by simpa [h\u2081\u2080, h\u2081\u2081, h\u2081\u2082, h\u2081\u2083, h\u2081\u2084] using h\u2081\u2084\n    have h\u2081\u2087 : (x.den ^ 2 : \u2124) = y.den ^ 2 := by\n      apply le_antisymm\n      \u00b7 exact Int.le_of_dvd (by positivity) h\u2081\u2085\n      \u00b7 exact Int.le_of_dvd (by positivity) h\u2081\u2086\n    exact_mod_cast h\u2081\u2087\n  have h\u2081\u2086 : x.den = y.den := by\n    rw [sq] at h\u2081\u2085\n    norm_cast at h\u2081\u2085\n    apply Eq.symm\n    <;> nlinarith\n  rw [h\u2081\u2086]\n  <;>\n  ring_nf at *\n  <;>\n  simp_all\n  <;>\n  linarith\n"}
{"mode": "clear", "steps_before": 37, "alphabet": 26, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 10, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 5, "fixpoint": true, "chars_before": 1097, "chars_after": 293, "seconds": 295.5, "id": "numbertheory_sqmod3in01d", "proof_after": "theorem numbertheory_sqmod3in01d (a : \u2124) : a ^ 2 % 3 = 0 \u2228 a ^ 2 % 3 = 1  :=  by\n  have case1 : a % 3 = 0 \u2192 a ^ 2 % 3 = 0 := by\n    intro h\n    simp [h, pow_two, Int.mul_emod]\n  have case2 : a % 3 = 1 \u2192 a ^ 2 % 3 = 1 := by\n    intro h\n    simp [h, pow_two, Int.mul_emod]\n  have case3 : a % 3 = 2 \u2192 a ^ 2 % 3 = 1 := by\n    intro h\n    simp [h, pow_two, Int.mul_emod]\n  omega\n", "proof_before": "theorem numbertheory_sqmod3in01d (a : \u2124) : a ^ 2 % 3 = 0 \u2228 a ^ 2 % 3 = 1  :=  by\n  have case1 : a % 3 = 0 \u2192 a ^ 2 % 3 = 0 := by\n    intro h\n    simp [h, pow_two, Int.mul_emod]\n    <;> simp [h]\n    <;> simp [h]\n    <;> simp [h]\n  have case2 : a % 3 = 1 \u2192 a ^ 2 % 3 = 1 := by\n    intro h\n    have h1 : a % 3 = 0 \u2228 a % 3 = 1 \u2228 a % 3 = 2 := by omega\n    rcases h1 with (h1 | h1 | h1) <;> simp [h1, pow_two, Int.mul_emod, Int.add_emod, Int.emod_emod] at case1 \u22a2 <;> omega\n  have case3 : a % 3 = 2 \u2192 a ^ 2 % 3 = 1 := by\n    intro h\n    simp_all [pow_two, Int.mul_emod, Int.add_emod]\n    <;> omega\n  have all_cases_covered : \u2200 n : \u2124, n % 3 = 0 \u2228 n % 3 = 1 \u2228 n % 3 = 2 := by\n    intro n\n    omega\n  have final_proof : a ^ 2 % 3 = 0 \u2228 a ^ 2 % 3 = 1 := by\n    have h\u2080 : a % 3 = 0 \u2228 a % 3 = 1 \u2228 a % 3 = 2 := all_cases_covered a\n    rcases h\u2080 with (h\u2080 | h\u2080 | h\u2080) <;> simp [h\u2080, pow_two, Int.mul_emod, Int.add_emod, Int.emod_emod]\n    <;> norm_num <;> aesop\n    <;> norm_num <;> aesop\n    <;> norm_num <;> aesop\n  match all_cases_covered a with\n  | Or.inl h =>\n    exact Or.inl (case1 h)\n  | Or.inr (Or.inl h) =>\n    exact Or.inr (case2 h)\n  | Or.inr (Or.inr h) =>\n    exact Or.inr (case3 h)\n"}
{"mode": "clear", "steps_before": 20, "alphabet": 23, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 590, "chars_after": 8, "seconds": 74.2, "id": "mathd_numbertheory_110", "proof_after": "theorem mathd_numbertheory_110 (a b : \u2115) (h\u2080 : 0 < a \u2227 0 < b \u2227 b \u2264 a) (h\u2081 : (a + b) % 10 = 2)\n  (h\u2082 : (2 * a + b) % 10 = 1) : (a - b) % 10 = 6  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_110 (a b : \u2115) (h\u2080 : 0 < a \u2227 0 < b \u2227 b \u2264 a) (h\u2081 : (a + b) % 10 = 2)\n  (h\u2082 : (2 * a + b) % 10 = 1) : (a - b) % 10 = 6  :=  by\n  have h\u2083 : (a) % 10 = 9 := by\n    have h\u2083 := congr_arg (\u00b7 % 10) h\u2081\n    have h\u2084 := congr_arg (\u00b7 % 10) h\u2082\n    have h\u2085 := congr_arg (\u00b7 % 10) h\u2081\n    simp at h\u2083 h\u2084 h\u2085\n    omega\n  have h\u2084 : (b) % 10 = 3 := by\n    have h\u2084 := congr_arg (\u00b7 % 10) h\u2081\n    have h\u2085 := congr_arg (\u00b7 % 10) h\u2082\n    have h\u2086 := congr_arg (\u00b7 % 10) h\u2083\n    simp at h\u2084 h\u2085 h\u2086\n    omega\n  have h\u2085 : (a - b) % 10 = 6 := by\n    have a_mod_10 : a % 10 = 9 := h\u2083\n    have b_mod_10 : b % 10 = 3 := h\u2084\n    have a_minus_b_mod_10 : (a - b) % 10 = 6 := by\n      omega\n    exact a_minus_b_mod_10\n  have := h\u2085\n  simp_all [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084]\n"}
{"mode": "clear", "steps_before": 16, "alphabet": 27, "compiles": true, "steps_after": 16, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 573, "chars_after": 573, "seconds": 277.0, "id": "amc12a_2003_p25", "proof_after": "theorem amc12a_2003_p25 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : 0 < b)\n  (h\u2081 : \u2200 x, f x = Real.sqrt (a * x ^ 2 + b * x)) (h\u2082 : { x | 0 \u2264 f x } = f '' { x | 0 \u2264 f x }) :\n  a = 0 \u2228 a = -4  :=  by\n  have h\u2083 := h\u2082\n  simp only [Set.ext_iff, h\u2081, Set.mem_setOf_eq, Set.mem_image, Real.sqrt_nonneg] at h\u2083\n  have h\u2084 := h\u2083 0\n  have h\u2085 := h\u2083 1\n  have h\u2086 := h\u2083 (-1)\n  have h\u2087 := h\u2083 2\n  have h\u2088 := h\u2083 (-2)\n  simp at h\u2084 h\u2085 h\u2086 h\u2087 h\u2088\n  have h\u2089 := h\u2085\n  simp at h\u2089\n  have h\u2081\u2080 := h\u2086\n  simp at h\u2081\u2080\n  cases' h\u2084 with x hx\n  cases' h\u2087 with x hx\n  cases' h\u2088 with x hx\n  nlinarith [Real.sqrt_nonneg (a * x ^ 2 + b * x), Real.sqrt_nonneg (a * 2 ^ 2 + b * 2),\n    Real.sqrt_nonneg (a * (-2) ^ 2 + b * (-2)), Real.sqrt_nonneg (a * 1 ^ 2 + b * 1),\n    Real.sqrt_nonneg (a * (-1) ^ 2 + b * (-1))]\n", "proof_before": "theorem amc12a_2003_p25 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : 0 < b)\n  (h\u2081 : \u2200 x, f x = Real.sqrt (a * x ^ 2 + b * x)) (h\u2082 : { x | 0 \u2264 f x } = f '' { x | 0 \u2264 f x }) :\n  a = 0 \u2228 a = -4  :=  by\n  have h\u2083 := h\u2082\n  simp only [Set.ext_iff, h\u2081, Set.mem_setOf_eq, Set.mem_image, Real.sqrt_nonneg] at h\u2083\n  have h\u2084 := h\u2083 0\n  have h\u2085 := h\u2083 1\n  have h\u2086 := h\u2083 (-1)\n  have h\u2087 := h\u2083 2\n  have h\u2088 := h\u2083 (-2)\n  simp at h\u2084 h\u2085 h\u2086 h\u2087 h\u2088\n  have h\u2089 := h\u2085\n  simp at h\u2089\n  have h\u2081\u2080 := h\u2086\n  simp at h\u2081\u2080\n  cases' h\u2084 with x hx\n  cases' h\u2087 with x hx\n  cases' h\u2088 with x hx\n  nlinarith [Real.sqrt_nonneg (a * x ^ 2 + b * x), Real.sqrt_nonneg (a * 2 ^ 2 + b * 2),\n    Real.sqrt_nonneg (a * (-2) ^ 2 + b * (-2)), Real.sqrt_nonneg (a * 1 ^ 2 + b * 1),\n    Real.sqrt_nonneg (a * (-1) ^ 2 + b * (-1))]\n"}
{"mode": "clear", "steps_before": 43, "alphabet": 19, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 43, "lower_bound": null, "exhausted_depth": 2, "certified": false, "rounds": 6, "fixpoint": true, "chars_before": 1205, "chars_after": 1205, "seconds": 382.3, "id": "mathd_algebra_185", "proof_after": "theorem mathd_algebra_185 (s : Finset \u2124) (f : \u2124 \u2192 \u2124) (h\u2080 : \u2200 x, f x = abs (x + 4))\n  (h\u2081 : \u2200 x, x \u2208 s \u2194 f x < 9) : s.card = 17  :=  by\n  have h\u2082 : \u2200 x, |x + 4| < 9 \u2194 -9 < x + 4 \u2227 x + 4 < 9 := by\n    intro x\n    simp_all only [abs_lt]\n    <;> constructor <;> intro h\n    <;> constructor <;> linarith\n  have h\u2083 : \u2200 x, -9 < x + 4 \u2227 x + 4 < 9 \u2194 -13 < x \u2227 x < 5 := by\n    intro x\n    simp only [h\u2082, h\u2080, abs_lt] at *\n    constructor <;> intro h <;> constructor <;> linarith\n  have h\u2084 : \u2200 x : \u2124, -13 < x \u2227 x < 5 \u2194 x \u2208 Finset.Icc (-12) 4 := by\n    intro x\n    constructor\n    intro h\n    simp_all [Finset.mem_Icc]\n    omega\n    intro h\n    simp_all [Finset.mem_Icc]\n    omega\n  have h\u2085 : (Finset.Icc (-12) 4).card = 17 := by\n    simp_all only [Finset.Icc, Finset.card, Nat.cast_ofNat]\n    rfl\n    <;> decide\n  have h\u2086 : s.card = 17 := by\n    have h\u2086 : s = Finset.Icc (-12) 4 := by\n      ext x\n      simp only [Finset.mem_Icc, h\u2081, h\u2080, h\u2082, h\u2083, h\u2084, abs_lt, and_comm, lt_add_iff_pos_left,\n        and_assoc, and_iff_right_iff_imp, gt_iff_lt, lt_add_iff_pos_right, add_lt_iff_neg_right,\n        add_lt_add_iff_right, neg_lt_iff_pos_add, add_assoc]\n      constructor\n      \u00b7 intro h\n        constructor\n        \u00b7 linarith \n        \u00b7 linarith \n      \u00b7 intro h\n        constructor\n        \u00b7 linarith \n        \u00b7 linarith \n    rw [h\u2086]\n    exact h\u2085\n  rw [h\u2086]\n", "proof_before": "theorem mathd_algebra_185 (s : Finset \u2124) (f : \u2124 \u2192 \u2124) (h\u2080 : \u2200 x, f x = abs (x + 4))\n  (h\u2081 : \u2200 x, x \u2208 s \u2194 f x < 9) : s.card = 17  :=  by\n  have h\u2082 : \u2200 x, |x + 4| < 9 \u2194 -9 < x + 4 \u2227 x + 4 < 9 := by\n    intro x\n    simp_all only [abs_lt]\n    <;> constructor <;> intro h\n    <;> constructor <;> linarith\n  have h\u2083 : \u2200 x, -9 < x + 4 \u2227 x + 4 < 9 \u2194 -13 < x \u2227 x < 5 := by\n    intro x\n    simp only [h\u2082, h\u2080, abs_lt] at *\n    constructor <;> intro h <;> constructor <;> linarith\n  have h\u2084 : \u2200 x : \u2124, -13 < x \u2227 x < 5 \u2194 x \u2208 Finset.Icc (-12) 4 := by\n    intro x\n    constructor\n    intro h\n    simp_all [Finset.mem_Icc]\n    omega\n    intro h\n    simp_all [Finset.mem_Icc]\n    omega\n  have h\u2085 : (Finset.Icc (-12) 4).card = 17 := by\n    simp_all only [Finset.Icc, Finset.card, Nat.cast_ofNat]\n    rfl\n    <;> decide\n  have h\u2086 : s.card = 17 := by\n    have h\u2086 : s = Finset.Icc (-12) 4 := by\n      ext x\n      simp only [Finset.mem_Icc, h\u2081, h\u2080, h\u2082, h\u2083, h\u2084, abs_lt, and_comm, lt_add_iff_pos_left,\n        and_assoc, and_iff_right_iff_imp, gt_iff_lt, lt_add_iff_pos_right, add_lt_iff_neg_right,\n        add_lt_add_iff_right, neg_lt_iff_pos_add, add_assoc]\n      constructor\n      \u00b7 intro h\n        constructor\n        \u00b7 linarith \n        \u00b7 linarith \n      \u00b7 intro h\n        constructor\n        \u00b7 linarith \n        \u00b7 linarith \n    rw [h\u2086]\n    exact h\u2085\n  rw [h\u2086]\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 42, "alphabet": 19, "compiles": true, "steps_after": 8, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "fixpoint": true, "chars_before": 1002, "chars_after": 176, "seconds": 253.2, "id": "mathd_algebra_181", "proof_after": "theorem mathd_algebra_181 (n : \u211d) (h\u2080 : n \u2260 3) (h\u2081 : (n + 5) / (n - 3) = 2) : n = 11  :=  by\n  have cross_multiplied : n + 5 = 2 * (n - 3) := by\n    have h\u2082 : n - 3 \u2260 0 := by\n      intro H\n      apply h\u2080\n      linarith\n    field_simp [h\u2082] at h\u2081\n    linarith\n  linarith\n", "proof_before": "theorem mathd_algebra_181 (n : \u211d) (h\u2080 : n \u2260 3) (h\u2081 : (n + 5) / (n - 3) = 2) : n = 11  :=  by\n  have cross_multiplied : n + 5 = 2 * (n - 3) := by\n    have h\u2082 : n - 3 \u2260 0 := by\n      intro H\n      rw [sub_eq_zero] at H\n      apply h\u2080\n      linarith\n    field_simp [h\u2082] at h\u2081\n    linarith\n  have expanded : n + 5 = 2 * n - 6 := by\n    field_simp [h\u2080, sub_ne_zero] at h\u2081 \u22a2\n    linarith\n  have simplified : n - 2 * n = -6 - 5 := by\n    have h\u2082 : n + 5 = 2 * (n - 3) := by linarith\n    have h\u2083 : n + 5 = 2 * n - 6 := by linarith\n    have h\u2084 : n = 11 := by linarith\n    linarith\n  have isolated_n : -n = -11 := by\n    have h\u2080' : n + 5 = 2 * (n - 3) := by linarith\n    have h\u2081' : n + 5 = 2 * n - 6 := by linarith\n    have h\u2082 : n - 2 * n = -6 - 5 := by linarith\n    linarith\n  have solution : n = 11 := by\n    have h\u2082 : n - 2 * n = -6 - 5 := by linarith\n    have h\u2083 : -n = -11 := by linarith\n    have h\u2084 : n = 11 := by linarith\n    linarith\n  have verification : (11 + 5) / (11 - 3) = 2 := by\n    norm_num [h\u2080]\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  have h\u2082 : n = 11 := solution\n  exact h\u2082\n"}
{"mode": "clear", "steps_before": 6, "alphabet": 11, "compiles": true, "steps_after": 6, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 1, "fixpoint": true, "chars_before": 242, "chars_after": 242, "seconds": 35.1, "id": "mathd_algebra_22", "proof_after": "theorem mathd_algebra_22 : Real.logb (5 ^ 2) (5 ^ 4) = 2  :=  by\n  simp [Real.logb, div_eq_mul_inv, mul_assoc, mul_inv_cancel, mul_one,\n    show (5 : \u211d) \u2260 0 by norm_num, show (4 : \u211d) \u2260 0 by norm_num,\n    show (2 : \u211d) \u2260 0 by norm_num]\n  <;> norm_num\n  <;> ring\n  <;> norm_num\n  <;> field_simp\n  <;> norm_num\n", "proof_before": "theorem mathd_algebra_22 : Real.logb (5 ^ 2) (5 ^ 4) = 2  :=  by\n  simp [Real.logb, div_eq_mul_inv, mul_assoc, mul_inv_cancel, mul_one,\n    show (5 : \u211d) \u2260 0 by norm_num, show (4 : \u211d) \u2260 0 by norm_num,\n    show (2 : \u211d) \u2260 0 by norm_num]\n  <;> norm_num\n  <;> ring\n  <;> norm_num\n  <;> field_simp\n  <;> norm_num\n"}
{"mode": "clear", "steps_before": 27, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 692, "chars_after": 8, "seconds": 23.0, "id": "mathd_algebra_123", "proof_after": "theorem mathd_algebra_123 (a b : \u2115) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : a + b = 20) (h\u2082 : a = 3 * b) :\n  a - b = 10  :=  by\n  omega\n", "proof_before": "theorem mathd_algebra_123 (a b : \u2115) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : a + b = 20) (h\u2082 : a = 3 * b) :\n  a - b = 10  :=  by\n  have h\u2083 : 3 * b + b = 20 := by\n    rw [h\u2082] at h\u2081\n    simp [mul_comm, mul_one, mul_assoc, mul_left_comm] at h\u2081 \u22a2\n    omega\n  have h\u2084 : 4 * b = 20 := by\n    have h\u2084 : 4 * b = 20 := by\n      linarith\n    exact h\u2084\n  have h\u2085 : b = 5 := by\n    have h\u2085 : b = 5 := by\n      linarith\n    exact h\u2085\n  have h\u2086 : a = 3 * 5 := by\n    subst h\u2085\n    omega\n  have h\u2087 : a = 15 := by\n    simp_all only [mul_comm, mul_one, mul_add, mul_assoc, mul_left_comm]\n    <;> omega\n  have h\u2088 : a - b = 15 - 5 := by\n    simp_all\n    <;> omega\n    <;> linarith\n    <;> ring\n  have h\u2089 : a - b = 10 := by\n    simp_all only [mul_comm, mul_one, mul_assoc, mul_left_comm, mul_right_comm]\n    <;> omega\n  simpa [h\u2085, h\u2087] using h\u2089\n"}
{"mode": "clear", "steps_before": 19, "alphabet": 21, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 8, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 7, "fixpoint": true, "chars_before": 511, "chars_after": 208, "seconds": 411.5, "id": "mathd_algebra_116", "proof_after": "theorem mathd_algebra_116 (k x : \u211d) (h\u2080 : x = (13 - Real.sqrt 131) / 4)\n    (h\u2081 : 2 * x ^ 2 - 13 * x + k = 0) : k = 19 / 4  :=  by\n  have h\u2083 : 2 * ((13 - Real.sqrt 131) / 4) ^ 2 - 13 * ((13 - Real.sqrt 131) / 4) + k = 0 := by\n    rw [h\u2080] at h\u2081\n    omega\n  have h\u2084 : k = 19 / 4 := by\n    ring_nf at *\n    norm_num at *\n    linarith\n  omega\n", "proof_before": "theorem mathd_algebra_116 (k x : \u211d) (h\u2080 : x = (13 - Real.sqrt 131) / 4)\n    (h\u2081 : 2 * x ^ 2 - 13 * x + k = 0) : k = 19 / 4  :=  by\n  have h\u2082 : 2 * ((13 - Real.sqrt 131) / 4) ^ 2 - 13 * ((13 - Real.sqrt 131) / 4) + k = 0 := by\n    simp [h\u2080] at h\u2081 \u22a2\n    ring_nf at h\u2081 \u22a2\n    field_simp at h\u2081 \u22a2\n    ring_nf at h\u2081 \u22a2\n    linarith\n  have h\u2083 : 2 * ((13 - Real.sqrt 131) / 4) ^ 2 - 13 * ((13 - Real.sqrt 131) / 4) + k = 0 := by\n    norm_num at h\u2080\n    rw [h\u2080] at h\u2081\n    ring_nf at h\u2081\n    linarith\n  have h\u2084 : k = 19 / 4 := by\n    norm_num at *\n    ring_nf at *\n    field_simp at *\n    norm_num at *\n    linarith\n  have h\u2085 : k = 19 / 4 := h\u2084\n  simp_all\n"}
{"mode": "clear", "steps_before": 13, "alphabet": 16, "compiles": false, "id": "imo_1977_p5", "proof_after": "theorem imo_1977_p5 (a b q r : \u2115) (h\u2080 : r < a + b) (h\u2081 : a ^ 2 + b ^ 2 = (a + b) * q + r)\n  (h\u2082 : q ^ 2 + r = 1977) :\n  abs ((a : \u2124) - 22) = 15 \u2227 abs ((b : \u2124) - 22) = 28 \u2228\n    abs ((a : \u2124) - 22) = 28 \u2227 abs ((b : \u2124) - 22) = 15  :=  by\n  have h\u2083 : q \u2264 1977 := by\n    nlinarith [sq_nonneg (a + b : \u2124), sq_nonneg (a - b : \u2124)]\n  have h\u2084 : q = 44 := by\n    nlinarith [sq_nonneg (a + b : \u2124), sq_nonneg (a - b : \u2124)]\n  rw [h\u2084] at h\u2081 h\u2082\n  have h\u2085 : a \u2264 66 := by\n    nlinarith [sq_nonneg (a + b : \u2124), sq_nonneg (a - b : \u2124)]\n  have h\u2086 : b \u2264 66 := by\n    nlinarith [sq_nonneg (a + b : \u2124), sq_nonneg (a - b : \u2124)]\n  interval_cases a <;> interval_cases b <;> norm_num at h\u2081 h\u2082 \u22a2 <;> omega\n", "proof_before": "theorem imo_1977_p5 (a b q r : \u2115) (h\u2080 : r < a + b) (h\u2081 : a ^ 2 + b ^ 2 = (a + b) * q + r)\n  (h\u2082 : q ^ 2 + r = 1977) :\n  abs ((a : \u2124) - 22) = 15 \u2227 abs ((b : \u2124) - 22) = 28 \u2228\n    abs ((a : \u2124) - 22) = 28 \u2227 abs ((b : \u2124) - 22) = 15  :=  by\n  have h\u2083 : q \u2264 1977 := by\n    nlinarith [sq_nonneg (a + b : \u2124), sq_nonneg (a - b : \u2124)]\n  have h\u2084 : q = 44 := by\n    nlinarith [sq_nonneg (a + b : \u2124), sq_nonneg (a - b : \u2124)]\n  rw [h\u2084] at h\u2081 h\u2082\n  have h\u2085 : a \u2264 66 := by\n    nlinarith [sq_nonneg (a + b : \u2124), sq_nonneg (a - b : \u2124)]\n  have h\u2086 : b \u2264 66 := by\n    nlinarith [sq_nonneg (a + b : \u2124), sq_nonneg (a - b : \u2124)]\n  interval_cases a <;> interval_cases b <;> norm_num at h\u2081 h\u2082 \u22a2 <;> omega\n"}
{"mode": "clear", "steps_before": 7, "alphabet": 14, "compiles": false, "id": "induction_sum_1oktkp1", "proof_after": "theorem induction_sum_1oktkp1 (n : \u2115) :\n  (\u2211 k in Finset.range n, (1 : \u211d) / ((k + 1) * (k + 2))) = n / (n + 1)  :=  by\n  induction n with\n  | zero =>\n    norm_num\n  | succ n ihn =>\n    rw [Finset.range_succ, Finset.sum_insert Finset.not_mem_range_self, ihn, add_mul]\n    field_simp\n    ring_nf\n", "proof_before": "theorem induction_sum_1oktkp1 (n : \u2115) :\n  (\u2211 k in Finset.range n, (1 : \u211d) / ((k + 1) * (k + 2))) = n / (n + 1)  :=  by\n  induction n with\n  | zero =>\n    norm_num\n  | succ n ihn =>\n    rw [Finset.range_succ, Finset.sum_insert Finset.not_mem_range_self, ihn, add_mul]\n    field_simp\n    ring_nf\n"}
{"mode": "clear", "steps_before": 19, "alphabet": 16, "compiles": false, "id": "mathd_numbertheory_232", "proof_after": "theorem mathd_numbertheory_232 (x y z : ZMod 31) (h\u2080 : x = 3\u207b\u00b9) (h\u2081 : y = 5\u207b\u00b9)\n  (h\u2082 : z = (x + y)\u207b\u00b9) : z = 29  :=  by\n  have h\u2083 : x = 21 := by\n    rw [h\u2080] at *\n    rfl\n  have h\u2084 : y = 25 := by\n    rw [h\u2083] at h\u2080 h\u2082\n    norm_num [h\u2080, h\u2081, h\u2082]\n    <;> rfl\n  have h\u2085 : x + y = 15 := by\n    simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n  have h\u2086 : z = 29 := by\n    rw [h\u2082, h\u2083, h\u2084] at *\n    norm_num [h\u2083, h\u2084, h\u2085, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat]\n    <;> rfl\n  simp_all\n  <;> rfl\n  <;> norm_num\n  <;> rfl\n", "proof_before": "theorem mathd_numbertheory_232 (x y z : ZMod 31) (h\u2080 : x = 3\u207b\u00b9) (h\u2081 : y = 5\u207b\u00b9)\n  (h\u2082 : z = (x + y)\u207b\u00b9) : z = 29  :=  by\n  have h\u2083 : x = 21 := by\n    rw [h\u2080] at *\n    rfl\n  have h\u2084 : y = 25 := by\n    rw [h\u2083] at h\u2080 h\u2082\n    norm_num [h\u2080, h\u2081, h\u2082]\n    <;> rfl\n  have h\u2085 : x + y = 15 := by\n    simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n  have h\u2086 : z = 29 := by\n    rw [h\u2082, h\u2083, h\u2084] at *\n    norm_num [h\u2083, h\u2084, h\u2085, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat]\n    <;> rfl\n  simp_all\n  <;> rfl\n  <;> norm_num\n  <;> rfl\n"}
{"mode": "clear", "steps_before": 3, "alphabet": 11, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 3, "lower_bound": 3, "exhausted_depth": 2, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 84, "chars_after": 84, "seconds": 194.0, "id": "mathd_algebra_159", "proof_after": "theorem mathd_algebra_159 (b : \u211d) (f : \u211d \u2192 \u211d)\n  (h\u2080 : \u2200 x, f x = 3 * x ^ 4 - 7 * x ^ 3 + 2 * x ^ 2 - b * x + 1) (h\u2081 : f 1 = 1) : b = -2  :=  by\n  simp_all only [rpow_two, mul_one, sub_eq_add_neg, add_assoc]\n  ring_nf\n  linarith\n", "proof_before": "theorem mathd_algebra_159 (b : \u211d) (f : \u211d \u2192 \u211d)\n  (h\u2080 : \u2200 x, f x = 3 * x ^ 4 - 7 * x ^ 3 + 2 * x ^ 2 - b * x + 1) (h\u2081 : f 1 = 1) : b = -2  :=  by\n  simp_all only [rpow_two, mul_one, sub_eq_add_neg, add_assoc]\n  ring_nf\n  linarith\n"}
{"mode": "clear", "steps_before": 39, "alphabet": 19, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 10, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 6, "fixpoint": true, "chars_before": 1578, "chars_after": 424, "seconds": 623.5, "id": "amc12b_2003_p9", "proof_after": "theorem amc12b_2003_p9 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = a * x + b) (h\u2081 : f 6 - f 2 = 12) :\n    f 12 - f 2 = 30  :=  by\n  have h_f6 : f 6 = a * 6 + b := by\n    simp_all [mul_comm]\n    <;> linarith\n  have h_f6_minus_f2_simplified : f 6 - f 2 = 4 * a := by\n    simp_all only [mul_add, mul_sub, mul_one, mul_div_cancel_left]\n    linarith\n  have h_f12 : f 12 = 3 * 12 + b := by\n    simp_all only [mul_one, mul_zero, add_zero, mul_two, sub_eq_iff_eq_add, mul_add, mul_sub, mul_one, sub_add, sub_self, zero_add, mul_comm]\n    <;> linarith\n  linarith\n", "proof_before": "theorem amc12b_2003_p9 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = a * x + b) (h\u2081 : f 6 - f 2 = 12) :\n    f 12 - f 2 = 30  :=  by\n  have h_f_linear : \u2200 x, f x = a * x + b := by\n    intro x\n    have h\u2082 := h\u2080 x\n    have h\u2083 := h\u2080 6\n    have h\u2084 := h\u2080 2\n    simp_all\n  have h_f6 : f 6 = a * 6 + b := by\n    simp_all [mul_comm]\n    <;> linarith\n  have h_f2 : f 2 = a * 2 + b := by\n    simp_all only [h_f_linear, h_f6, sub_eq_iff_eq_add, mul_add, mul_one, mul_comm a, mul_comm b]\n    <;> linarith\n  have h_f6_minus_f2 : f 6 - f 2 = (a * 6 + b) - (a * 2 + b) := by\n    simp_all only [mul_add, mul_one, mul_sub, mul_two, add_sub_assoc, mul_left_comm]\n    <;> linarith\n  have h_f6_minus_f2_simplified : f 6 - f 2 = 4 * a := by\n    simp_all only [mul_add, mul_sub, mul_one, mul_div_cancel_left]\n    linarith\n  have h_a : a = 3 := by\n    have h\u2082 : f 6 - f 2 = 12 := h\u2081\n    have h\u2083 : f 6 - f 2 = 4 * a := h_f6_minus_f2_simplified\n    have h\u2084 : 4 * a = 12 := by linarith\n    have h\u2085 : a = 3 := by linarith\n    exact h\u2085\n  have h_f12 : f 12 = 3 * 12 + b := by\n    simp_all only [mul_one, mul_zero, add_zero, mul_two, sub_eq_iff_eq_add, mul_add, mul_sub, mul_one, sub_add, sub_self, zero_add, mul_comm]\n    <;> linarith\n  have h_f2 : f 2 = 3 * 2 + b := by\n    simp_all only [mul_one, mul_zero, sub_zero, zero_add, mul_two, mul_add, mul_sub, mul_one, sub_eq_add_neg,\n      mul_neg, mul_comm]\n    <;> linarith\n  have h_f12_minus_f2 : f 12 - f 2 = (3 * 12 + b) - (3 * 2 + b) := by\n    simp_all only [mul_one, mul_zero, mul_two, mul_add, mul_sub, mul_neg, mul_assoc]\n    <;> norm_num\n    <;> linarith\n  have h_f12_minus_f2_simplified : f 12 - f 2 = 30 := by\n    linarith\n  simpa [h_f_linear, h_a] using h_f12_minus_f2_simplified\n"}
{"mode": "clear", "steps_before": 45, "alphabet": 24, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1037, "chars_after": 11, "seconds": 188.0, "id": "mathd_algebra_144", "proof_after": "theorem mathd_algebra_144 (a b c d : \u2115) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c \u2227 0 < d) (h\u2080 : (c : \u2124) - b = d)\n    (h\u2081 : (b : \u2124) - a = d) (h\u2082 : a + b + c = 60) (h\u2083 : a + b > c) : d < 10  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_144 (a b c d : \u2115) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c \u2227 0 < d) (h\u2080 : (c : \u2124) - b = d)\n    (h\u2081 : (b : \u2124) - a = d) (h\u2082 : a + b + c = 60) (h\u2083 : a + b > c) : d < 10  :=  by\n  have b_eq : b = a + d := by\n    have h\u2084 := h\u2080\n    have h\u2085 := h\u2081\n    have h\u2086 := h\u2082\n    have h\u2087 := h\u2083\n    linarith\n  have c_eq : c = b + d := by\n    have b_eq_a_plus_d : b = a + d := by linarith\n    have c_eq_b_plus_d : c = b + d := by\n      have : (c : \u2124) - b = d := by linarith\n      have : (b : \u2124) - a = d := by linarith\n      have : a + b + c = 60 := by linarith\n      have : a + b > c := by linarith\n      have : b = a + d := by linarith\n      linarith\n    exact c_eq_b_plus_d\n  have c_eq_a : c = a + 2*d := by\n    subst_vars\n    ring_nf\n    <;> omega\n  have a_eq : a + (a + d) + (a + 2*d) = 60 := by\n    apply Eq.symm\n    ring_nf\n    linarith [h\u2080, h\u2081, h\u2082, h\u2083, b_eq, c_eq, c_eq_a]\n  have a_plus_d_eq : a + d = 20 := by\n    zify\n    linarith\n    <;> assumption\n    <;> linarith\n  have a_eq_d : a = 20 - d := by\n    omega\n  have a_gt_d : a > d := by\n    have h\u2084 := h\u2083\n    simp_all [add_assoc, add_comm, add_left_comm]\n    <;> omega\n  have d_lt_10 : d < 10 := by\n    norm_num at a_plus_d_eq a_eq_d a_gt_d\n    <;> omega\n  exact d_lt_10\n"}
{"mode": "clear", "steps_before": 19, "alphabet": 14, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 430, "chars_after": 7, "seconds": 96.9, "id": "mathd_algebra_462", "proof_after": "theorem mathd_algebra_462 : ((1 : \u211a) / 2 + 1 / 3) * (1 / 2 - 1 / 3) = 5 / 36  :=  by\n  ring\n", "proof_before": "theorem mathd_algebra_462 : ((1 : \u211a) / 2 + 1 / 3) * (1 / 2 - 1 / 3) = 5 / 36  :=  by\n  have step1 : (1 : \u211a) / 2 + 1 / 3 = 5 / 6 := by\n    norm_num [Nat.div_eq_of_lt]\n    <;> rfl\n    <;> rfl\n    <;> rfl\n    <;> rfl\n  have step2 : (1 : \u211a) / 2 - 1 / 3 = 1 / 6 := by\n    ring_nf at step1\n    linarith\n  have step3 : (5 / 6 : \u211a) * (1 / 6) = 5 / 36 := by\n    field_simp\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  nlinarith [sq_nonneg (1 / 2 + 1 / 3)]\n  <;> nlinarith\n  <;> nlinarith\n  <;> nlinarith\n  <;> nlinarith\n"}
{"mode": "clear", "steps_before": 45, "alphabet": 50, "compiles": true, "steps_after": 9, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 3, "fixpoint": true, "chars_before": 1265, "chars_after": 367, "seconds": 427.6, "id": "mathd_algebra_13", "proof_after": "theorem mathd_algebra_13 (a b : \u211d)\n  (h\u2080 : \u2200 x, x - 3 \u2260 0 \u2227 x - 5 \u2260 0 \u2192 4 * x / (x ^ 2 - 8 * x + 15) = a / (x - 3) + b / (x - 5)) :\n  a = -6 \u2227 b = 10  :=  by\n  have h\u2081 : \u2200 x, 4 * x = a * (x - 5) + b * (x - 3) := by\n    intro x\n    have h\u2081 := h\u2080 x\n    have h\u2082 := h\u2080 (-1)\n    have h\u2083 := h\u2080 1\n    norm_num [mul_comm] at h\u2081 h\u2082 h\u2083\n    nlinarith [sq_nonneg (x - 3), sq_nonneg (x - 5)]\n  refine' \u27e8_, _\u27e9 <;> linarith [h\u2081 3, h\u2081 5, h\u2081 0, h\u2081 1, h\u2081 (-1), h\u2081 (-2), h\u2081 (-3), h\u2081 (-4), h\u2081 (-5), h\u2081 (-6), h\u2081 (-7), h\u2081 (-8), h\u2081 (-9), h\u2081 (-10)]\n", "proof_before": "theorem mathd_algebra_13 (a b : \u211d)\n  (h\u2080 : \u2200 x, x - 3 \u2260 0 \u2227 x - 5 \u2260 0 \u2192 4 * x / (x ^ 2 - 8 * x + 15) = a / (x - 3) + b / (x - 5)) :\n  a = -6 \u2227 b = 10  :=  by\n  have h\u2081 : \u2200 x, 4 * x = a * (x - 5) + b * (x - 3) := by\n    intro x\n    have h\u2081 := h\u2080 x\n    have h\u2082 := h\u2080 (-1)\n    have h\u2083 := h\u2080 1\n    norm_num [mul_comm] at h\u2081 h\u2082 h\u2083\n    nlinarith [sq_nonneg (x - 3), sq_nonneg (x - 5)]\n  have h\u2082 : a = -6 := by\n    have h\u2082 := h\u2081 4\n    have h\u2083 := h\u2081 6\n    have h\u2084 := h\u2081 2\n    have h\u2085 := h\u2081 1\n    have h\u2086 := h\u2081 0\n    have h\u2087 := h\u2081 (-1)\n    have h\u2088 := h\u2081 (-2)\n    have h\u2089 := h\u2081 (-3)\n    have h\u2081\u2080 := h\u2081 (-4)\n    have h\u2081\u2081 := h\u2081 (-5)\n    have h\u2081\u2082 := h\u2081 (-6)\n    have h\u2081\u2083 := h\u2081 (-7)\n    have h\u2081\u2084 := h\u2081 (-8)\n    have h\u2081\u2085 := h\u2081 (-9)\n    have h\u2081\u2086 := h\u2081 (-10)\n    norm_num at h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085 h\u2081\u2086\n    <;> linarith\n  have h\u2083 : b = 10 := by\n    have h\u2083 := h\u2081 4\n    have h\u2084 := h\u2081 6\n    have h\u2085 := h\u2081 2\n    have h\u2086 := h\u2081 0\n    have h\u2087 := h\u2081 1\n    have h\u2088 := h\u2081 (-1)\n    have h\u2089 := h\u2081 (-2)\n    have h\u2081\u2080 := h\u2081 (-3)\n    have h\u2081\u2081 := h\u2081 (-4)\n    have h\u2081\u2082 := h\u2081 (-5)\n    have h\u2081\u2083 := h\u2081 (-6)\n    have h\u2081\u2084 := h\u2081 (-7)\n    have h\u2081\u2085 := h\u2081 (-8)\n    have h\u2081\u2086 := h\u2081 (-9)\n    have h\u2081\u2087 := h\u2081 (-10)\n    norm_num at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087\n    <;> linarith\n  refine' \u27e8_, _\u27e9 <;> linarith [h\u2081 3, h\u2081 5, h\u2081 0, h\u2081 1, h\u2081 (-1), h\u2081 (-2), h\u2081 (-3), h\u2081 (-4), h\u2081 (-5), h\u2081 (-6), h\u2081 (-7), h\u2081 (-8), h\u2081 (-9), h\u2081 (-10)]\n"}
{"mode": "clear", "steps_before": 37, "alphabet": 13, "compiles": false, "id": "mathd_numbertheory_668", "proof_after": "theorem mathd_numbertheory_668 (l r : ZMod 7) (h\u2080 : l = (2 + 3)\u207b\u00b9) (h\u2081 : r = 2\u207b\u00b9 + 3\u207b\u00b9) :\n  l - r = 1  :=  by\n  have h\u2082 : (2 + 3 : \u2124) = 5 := by\n    simp_all\n    <;> norm_num\n    <;> rfl\n    <;> simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n  have h\u2083 : (5 : ZMod 7)\u207b\u00b9 = 3 := by\n    norm_num [ZMod.nat_cast_self] at h\u2080 h\u2081 h\u2082 \u22a2\n    <;> simp_all [ZMod.nat_cast_self]\n    <;> rfl\n  have h\u2084 : (2 : ZMod 7)\u207b\u00b9 = 4 := by\n    simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n    <;> rfl\n    <;> rfl\n  have h\u2085 : (3 : ZMod 7)\u207b\u00b9 = 5 := by\n    norm_num [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, ZMod.nat_cast_self]\n    <;> rfl\n    <;> rfl\n    <;> rfl\n  have h\u2086 : (2\u207b\u00b9 + 3\u207b\u00b9 : ZMod 7) = 2 := by\n    simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n    <;> simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n  have h\u2087 : l - r = 1 := by\n    simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n    <;> aesop\n  simp_all [ZMod.int_cast_eq_int_cast_iff]\n  <;> norm_num\n  <;> rfl\n", "proof_before": "theorem mathd_numbertheory_668 (l r : ZMod 7) (h\u2080 : l = (2 + 3)\u207b\u00b9) (h\u2081 : r = 2\u207b\u00b9 + 3\u207b\u00b9) :\n  l - r = 1  :=  by\n  have h\u2082 : (2 + 3 : \u2124) = 5 := by\n    simp_all\n    <;> norm_num\n    <;> rfl\n    <;> simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n  have h\u2083 : (5 : ZMod 7)\u207b\u00b9 = 3 := by\n    norm_num [ZMod.nat_cast_self] at h\u2080 h\u2081 h\u2082 \u22a2\n    <;> simp_all [ZMod.nat_cast_self]\n    <;> rfl\n  have h\u2084 : (2 : ZMod 7)\u207b\u00b9 = 4 := by\n    simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n    <;> rfl\n    <;> rfl\n  have h\u2085 : (3 : ZMod 7)\u207b\u00b9 = 5 := by\n    norm_num [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, ZMod.nat_cast_self]\n    <;> rfl\n    <;> rfl\n    <;> rfl\n  have h\u2086 : (2\u207b\u00b9 + 3\u207b\u00b9 : ZMod 7) = 2 := by\n    simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n    <;> simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n  have h\u2087 : l - r = 1 := by\n    simp_all [ZMod.nat_cast_self]\n    <;> norm_num\n    <;> rfl\n    <;> aesop\n  simp_all [ZMod.int_cast_eq_int_cast_iff]\n  <;> norm_num\n  <;> rfl\n"}
{"mode": "clear", "steps_before": 52, "alphabet": 18, "compiles": false, "id": "mathd_algebra_493", "proof_after": "theorem mathd_algebra_493 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x ^ 2 - 4 * Real.sqrt x + 1) :\n    f (f 4) = 70  :=  by\n  have f4 : f 4 = 4 ^ 2 - 4 * Real.sqrt 4 + 1 := by\n    simp [h\u2080]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [h\u2080]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n  have f4_val : f 4 = 16 - 8 + 1 := by\n    have h\u2081 : Real.sqrt 4 = 2 := by\n      norm_num [Real.sqrt_eq_iff_sq_eq]\n    simp_all only [h\u2080, h\u2081, mul_comm, mul_one, mul_add, mul_sub, mul_assoc]\n    ring\n    <;> norm_num\n  have f4_simplified : f 4 = 9 := by\n    simp_all only [mul_comm, mul_assoc, mul_left_comm, mul_one, mul_zero]\n    <;> linarith\n  have ff4 : f (f 4) = f 9 := by\n    rw [h\u2080]\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  have f9 : f 9 = 9 ^ 2 - 4 * Real.sqrt 9 + 1 := by\n    simp_all [h\u2080, Real.sqrt_eq_iff_mul_self_eq]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have f9_val : f 9 = 81 - 12 + 1 := by\n    rw [f4_simplified] at ff4\n    simp_all only [h\u2080, ff4, f4_val, f4_simplified, f9]\n    norm_num\n    ring_nf\n    have h\u2081 : Real.sqrt 9 = 3 := by\n      rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    rw [h\u2081]\n    ring\n  have f9_simplified : f 9 = 70 := by\n    simp_all [h\u2080]\n    <;> norm_num\n    <;> ring\n    <;> rw [\u2190 Real.sqrt_sq (show (0 : \u211d) \u2264 9 by norm_num)]\n    <;> norm_num\n  have h : f (f 4) = 70 := by\n    simp_all [h\u2080, sq]\n    <;> linarith\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  simp_all [h\u2080, Real.sqrt_eq_iff_sq_eq]\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_493 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x ^ 2 - 4 * Real.sqrt x + 1) :\n    f (f 4) = 70  :=  by\n  have f4 : f 4 = 4 ^ 2 - 4 * Real.sqrt 4 + 1 := by\n    simp [h\u2080]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [h\u2080]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n  have f4_val : f 4 = 16 - 8 + 1 := by\n    have h\u2081 : Real.sqrt 4 = 2 := by\n      norm_num [Real.sqrt_eq_iff_sq_eq]\n    simp_all only [h\u2080, h\u2081, mul_comm, mul_one, mul_add, mul_sub, mul_assoc]\n    ring\n    <;> norm_num\n  have f4_simplified : f 4 = 9 := by\n    simp_all only [mul_comm, mul_assoc, mul_left_comm, mul_one, mul_zero]\n    <;> linarith\n  have ff4 : f (f 4) = f 9 := by\n    rw [h\u2080]\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  have f9 : f 9 = 9 ^ 2 - 4 * Real.sqrt 9 + 1 := by\n    simp_all [h\u2080, Real.sqrt_eq_iff_mul_self_eq]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have f9_val : f 9 = 81 - 12 + 1 := by\n    rw [f4_simplified] at ff4\n    simp_all only [h\u2080, ff4, f4_val, f4_simplified, f9]\n    norm_num\n    ring_nf\n    have h\u2081 : Real.sqrt 9 = 3 := by\n      rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    rw [h\u2081]\n    ring\n  have f9_simplified : f 9 = 70 := by\n    simp_all [h\u2080]\n    <;> norm_num\n    <;> ring\n    <;> rw [\u2190 Real.sqrt_sq (show (0 : \u211d) \u2264 9 by norm_num)]\n    <;> norm_num\n  have h : f (f 4) = 70 := by\n    simp_all [h\u2080, sq]\n    <;> linarith\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  simp_all [h\u2080, Real.sqrt_eq_iff_sq_eq]\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 26, "alphabet": 18, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 948, "chars_after": 7, "seconds": 96.6, "id": "algebra_2rootspoly_apatapbeq2asqp2ab", "proof_after": "theorem algebra_2rootspoly_apatapbeq2asqp2ab (a b : \u2102) :\n    (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b)  :=  by\n  ring\n", "proof_before": "theorem algebra_2rootspoly_apatapbeq2asqp2ab (a b : \u2102) :\n    (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b)  :=  by\n  have h1 : (a + a) * (a + b) = 2 * a * (a + b) := by\n    simp [mul_add, mul_comm, mul_left_comm, add_assoc, add_left_comm]\n    <;> ring\n    <;> ring\n    <;> ring\n  have h2 : 2 * a * (a + b) = (2 * a * a) + (2 * a * b) := by\n    simp_all only [two_mul, mul_add, add_mul]\n    <;> ring\n  have h3 : (2 * a * a) + (2 * a * b) = 2 * a^2 + 2 * (a * b) := by\n    simp_all only [mul_add, mul_comm, mul_left_comm, mul_assoc]\n    <;> ring\n  have h4 : 2 * a^2 + 2 * (a * b) = 2 * a^2 + 2 * a * b := by\n    apply Eq.symm\n    <;> ring_nf at h1 h2 h3 \u22a2\n    <;> linear_combination h3\n  have h5 : 2 * a^2 + 2 * a * b = 2 * a^2 + 2 * (a * b) := by\n    simp_all only [mul_add, mul_comm, mul_left_comm, mul_assoc, add_assoc, add_comm, add_left_comm]\n    <;> ring\n  have h6 : (a + a) * (a + b) = 2 * a^2 + 2 * (a * b) := by\n    rw [h1, h2, h3, h4, h5]\n    <;> ring\n    <;> ring_nf at h1 h2 h3 h4 h5 \u22a2\n    <;> simp_all\n    <;> ring_nf\n    <;> simp_all\n  simpa using h6\n"}
{"mode": "clear", "steps_before": 78, "alphabet": 46, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 78, "lower_bound": null, "exhausted_depth": 1, "certified": false, "rounds": 3, "fixpoint": false, "chars_before": 2000, "chars_after": 2000, "seconds": 505.0, "id": "mathd_algebra_405", "proof_after": "theorem mathd_algebra_405 (S : Finset \u2115) (h\u2080 : \u2200 x, x \u2208 S \u2194 0 < x \u2227 x ^ 2 + 4 * x + 4 < 20) :\n  S.card = 2  :=  by\n  have h\u2081 : \u2200 x : \u2115, 0 < x \u2192 x ^ 2 + 4 * x + 4 < 20 \u2192 x < 3 := by\n    intro x hx1 hx2\n    have h1 : 0 < x := hx1\n    have h2 : x ^ 2 + 4 * x + 4 < 20 := hx2\n    have h3 : x < 3 := by\n      nlinarith [sq_nonneg (x + 2)]\n    exact h3\n  have h\u2082 : Finset.Icc 1 2 = {1, 2} := by\n    ext x\n    simp only [Finset.mem_Icc, Finset.mem_insert, Finset.mem_singleton, Nat.lt_succ_iff]\n    constructor\n    \u00b7 intro hx\n      have h\u2081' : x \u2264 2 := by\n        nlinarith [h\u2080 x, h\u2081 x]\n      have h\u2082' : 1 \u2264 x := by\n        nlinarith [h\u2080 x, h\u2081 x]\n      interval_cases x <;> simp_all\n    \u00b7 intro hx\n      cases' hx with hx hx <;> simp_all [h\u2080]\n      <;> nlinarith [h\u2081 1, h\u2081 2]\n  have h\u2083 : 1 \u2208 S := by\n    have h\u2083 := h\u2080 1\n    have h\u2084 := h\u2080 2\n    have h\u2085 := h\u2081 1 (by decide)\n    have h\u2086 := h\u2081 2 (by decide)\n    simp_all [Finset.mem_Icc]\n    <;> decide\n    <;> omega\n  have h\u2084 : 2 \u2208 S := by\n    have h\u2084 := h\u2080 2\n    have h\u2085 := h\u2080 1\n    have h\u2086 := h\u2081 2 (by norm_num)\n    have h\u2087 := h\u2081 1 (by norm_num)\n    simp_all [Finset.mem_insert, Finset.mem_singleton]\n    <;> norm_num\n    <;> aesop\n  have h\u2085 : \u2200 x : \u2115, x \u2208 S \u2192 x = 1 \u2228 x = 2 := by\n    intro x hx\n    have h\u2085 : x = 1 \u2228 x = 2 := by\n      have h\u2085\u2080 : x \u2208 S \u2194 0 < x \u2227 x ^ 2 + 4 * x + 4 < 20 := h\u2080 x\n      have h\u2085\u2081 : x \u2208 S := hx\n      rw [h\u2085\u2080] at h\u2085\u2081\n      have h\u2085\u2082 : 0 < x \u2227 x ^ 2 + 4 * x + 4 < 20 := h\u2085\u2081\n      have h\u2085\u2083 : x < 3 := h\u2081 x h\u2085\u2082.1 h\u2085\u2082.2\n      have h\u2085\u2084 : x \u2264 2 := by linarith\n      have h\u2085\u2085 : x \u2265 1 := by\n        nlinarith\n      interval_cases x <;> simp_all\n    exact h\u2085\n  have h\u2086 : S.card = 2 := by\n    have h\u2086 : S = {1, 2} := by\n      ext x\n      simp only [Finset.mem_insert, Finset.mem_singleton, h\u2085, h\u2080, true_and, and_true]\n      constructor\n      \u00b7 intro h\n        match x with\n        | 0 => contradiction  \n        | 1 => simp  \n        | 2 => simp  \n        | 3 => contradiction  \n        | x + 4 =>\n          have h\u2086 := h\u2081 x\n          omega  \n      \u00b7 intro h\n        match x with\n        | 1 => simp  \n        | 2 => simp  \n    rw [h\u2086]\n    simp  \n  exact h\u2086\n", "proof_before": "theorem mathd_algebra_405 (S : Finset \u2115) (h\u2080 : \u2200 x, x \u2208 S \u2194 0 < x \u2227 x ^ 2 + 4 * x + 4 < 20) :\n  S.card = 2  :=  by\n  have h\u2081 : \u2200 x : \u2115, 0 < x \u2192 x ^ 2 + 4 * x + 4 < 20 \u2192 x < 3 := by\n    intro x hx1 hx2\n    have h1 : 0 < x := hx1\n    have h2 : x ^ 2 + 4 * x + 4 < 20 := hx2\n    have h3 : x < 3 := by\n      nlinarith [sq_nonneg (x + 2)]\n    exact h3\n  have h\u2082 : Finset.Icc 1 2 = {1, 2} := by\n    ext x\n    simp only [Finset.mem_Icc, Finset.mem_insert, Finset.mem_singleton, Nat.lt_succ_iff]\n    constructor\n    \u00b7 intro hx\n      have h\u2081' : x \u2264 2 := by\n        nlinarith [h\u2080 x, h\u2081 x]\n      have h\u2082' : 1 \u2264 x := by\n        nlinarith [h\u2080 x, h\u2081 x]\n      interval_cases x <;> simp_all\n    \u00b7 intro hx\n      cases' hx with hx hx <;> simp_all [h\u2080]\n      <;> nlinarith [h\u2081 1, h\u2081 2]\n  have h\u2083 : 1 \u2208 S := by\n    have h\u2083 := h\u2080 1\n    have h\u2084 := h\u2080 2\n    have h\u2085 := h\u2081 1 (by decide)\n    have h\u2086 := h\u2081 2 (by decide)\n    simp_all [Finset.mem_Icc]\n    <;> decide\n    <;> omega\n  have h\u2084 : 2 \u2208 S := by\n    have h\u2084 := h\u2080 2\n    have h\u2085 := h\u2080 1\n    have h\u2086 := h\u2081 2 (by norm_num)\n    have h\u2087 := h\u2081 1 (by norm_num)\n    simp_all [Finset.mem_insert, Finset.mem_singleton]\n    <;> norm_num\n    <;> aesop\n  have h\u2085 : \u2200 x : \u2115, x \u2208 S \u2192 x = 1 \u2228 x = 2 := by\n    intro x hx\n    have h\u2085 : x = 1 \u2228 x = 2 := by\n      have h\u2085\u2080 : x \u2208 S \u2194 0 < x \u2227 x ^ 2 + 4 * x + 4 < 20 := h\u2080 x\n      have h\u2085\u2081 : x \u2208 S := hx\n      rw [h\u2085\u2080] at h\u2085\u2081\n      have h\u2085\u2082 : 0 < x \u2227 x ^ 2 + 4 * x + 4 < 20 := h\u2085\u2081\n      have h\u2085\u2083 : x < 3 := h\u2081 x h\u2085\u2082.1 h\u2085\u2082.2\n      have h\u2085\u2084 : x \u2264 2 := by linarith\n      have h\u2085\u2085 : x \u2265 1 := by\n        nlinarith\n      interval_cases x <;> simp_all\n    exact h\u2085\n  have h\u2086 : S.card = 2 := by\n    have h\u2086 : S = {1, 2} := by\n      ext x\n      simp only [Finset.mem_insert, Finset.mem_singleton, h\u2085, h\u2080, true_and, and_true]\n      constructor\n      \u00b7 intro h\n        match x with\n        | 0 => contradiction  \n        | 1 => simp  \n        | 2 => simp  \n        | 3 => contradiction  \n        | x + 4 =>\n          have h\u2086 := h\u2081 x\n          omega  \n      \u00b7 intro h\n        match x with\n        | 1 => simp  \n        | 2 => simp  \n    rw [h\u2086]\n    simp  \n  exact h\u2086\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 71, "alphabet": 25, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 3305, "chars_after": 7, "seconds": 76.2, "id": "algebra_3rootspoly_amdtamctambeqnasqmbpctapcbtdpasqmbpctapcbta", "proof_after": "theorem algebra_3rootspoly_amdtamctambeqnasqmbpctapcbtdpasqmbpctapcbta (b c d a : \u2102) :\n    (a - d) * (a - c) * (a - b) =\n      -((a ^ 2 - (b + c) * a + c * b) * d) + (a ^ 2 - (b + c) * a + c * b) * a  :=  by\n  ring\n", "proof_before": "theorem algebra_3rootspoly_amdtamctambeqnasqmbpctapcbtdpasqmbpctapcbta (b c d a : \u2102) :\n    (a - d) * (a - c) * (a - b) =\n      -((a ^ 2 - (b + c) * a + c * b) * d) + (a ^ 2 - (b + c) * a + c * b) * a  :=  by\n  have h1 : (a - d) * (a - c) * (a - b) = (a - d) * ((a - c) * (a - b)) := by\n    rw [mul_assoc]\n    <;> simp [mul_comm]\n    <;> ring\n  have h2 : (a - c) * (a - b) = a^2 - (b + c)*a + b*c := by\n    simp_all [mul_assoc, mul_comm, mul_left_comm, sub_eq_add_neg, add_assoc, add_comm, add_left_comm]\n    <;> ring\n    <;> simp_all [mul_assoc, mul_comm, mul_left_comm, sub_eq_add_neg, add_assoc, add_comm, add_left_comm]\n    <;> ring\n    <;> simp_all [mul_assoc, mul_comm, mul_left_comm, sub_eq_add_neg, add_assoc, add_comm, add_left_comm]\n    <;> ring\n    <;> simp_all [mul_assoc, mul_comm, mul_left_comm, sub_eq_add_neg, add_assoc, add_comm, add_left_comm]\n    <;> ring\n  have h3 : (a - d) * (a^2 - (b + c)*a + b*c) = (a - d)*(a^2) - (a - d)*((b + c)*a) + (a - d)*(b*c) := by\n    simp_all [mul_add, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> simp_all\n  have h4 : (a - d)*(a^2) = a^3 - d*a^2 := by\n    simp_all only [mul_add, mul_comm, mul_left_comm, mul_assoc, sub_eq_add_neg, add_assoc]\n    ring\n  have h5 : (a - d)*((b + c)*a) = (b + c)*a^2 - d*(b + c)*a := by\n    simp_all only [mul_add, mul_sub, mul_comm, mul_left_comm, mul_assoc]\n    ring_nf\n    <;> simp_all only [mul_comm]\n    <;> ring_nf\n    <;> simp_all only [mul_comm]\n    <;> ring_nf\n  have h6 : (a - d)*(b*c) = a*b*c - d*b*c := by\n    simp_all only [mul_assoc, mul_comm, mul_left_comm]\n    ring_nf\n    <;> simp_all\n    <;> ring_nf\n    <;> linarith\n  have h7 : (a - d)*(a^2 - (b + c)*a + b*c) = a^3 - d*a^2 - (b + c)*a^2 + d*(b + c)*a + a*b*c - d*b*c := by\n    simp only [mul_add, mul_sub, mul_comm, mul_left_comm, mul_assoc, sub_eq_add_neg, add_assoc]\n    ring\n    <;> simp_all\n    <;> ring\n  have h8 : -((a ^ 2 - (b + c) * a + c * b) * d) + (a ^ 2 - (b + c) * a + c * b) * a = \n             - (a^2 - (b + c)*a + c*b) * d + (a^2 - (b + c)*a + c*b) * a := by\n    simp_all only [add_mul, mul_add, mul_sub, sub_mul, mul_comm, mul_left_comm, mul_assoc]\n    ring\n    <;> linarith\n  have h9 : - (a^2 - (b + c)*a + c*b) * d = -a^2*d + (b + c)*a*d - c*b*d := by\n    linear_combination h8\n  have h10 : (a^2 - (b + c)*a + c*b) * a = a^3 - (b + c)*a^2 + c*b*a := by\n    simp_all only [mul_add, mul_sub, mul_comm, mul_left_comm, mul_assoc]\n    ring_nf\n  have h11 : -a^2*d + (b + c)*a*d - c*b*d + a^3 - (b + c)*a^2 + c*b*a = \n              a^3 - (b + c)*a^2 + c*b*a - a^2*d + (b + c)*a*d - c*b*d := by\n    linear_combination h10\n    <;> simp_all only [mul_add, mul_sub, sub_mul, add_comm, add_left_comm, add_assoc]\n    <;> ring\n  have h12 : a^3 - (b + c)*a^2 + c*b*a - a^2*d + (b + c)*a*d - c*b*d = \n              a^3 - d*a^2 - (b + c)*a^2 + d*(b + c)*a + a*b*c - d*b*c := by\n    linear_combination h7\n  have h13 : (a - d) * (a - c) * (a - b) = a^3 - d*a^2 - (b + c)*a^2 + d*(b + c)*a + a*b*c - d*b*c := by\n    ring_nf\n    <;> simp_all only [mul_assoc, mul_comm, mul_left_comm, mul_one, mul_zero, sub_eq_add_neg]\n    <;> linarith\n  have h14 : -((a ^ 2 - (b + c) * a + c * b) * d) + (a ^ 2 - (b + c) * a + c * b) * a = \n              a^3 - d*a^2 - (b + c)*a^2 + d*(b + c)*a + a*b*c - d*b*c := by\n    linear_combination h13\n  have h15 : (a - d) * (a - c) * (a - b) = \n              -((a ^ 2 - (b + c) * a + c * b) * d) + (a ^ 2 - (b + c) * a + c * b) * a := by\n    linear_combination h14\n  rw [h15]\n  <;> simp_all\n  <;> ring\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 246, "alphabet": 103, "compiles": false, "id": "aimeII_2020_p6", "proof_after": "theorem aimeII_2020_p6 (t : \u2115 \u2192 \u211a) (h\u2080 : t 1 = 20) (h\u2081 : t 2 = 21)\n  (h\u2082 : \u2200 n \u2265 3, t n = (5 * t (n - 1) + 1) / (25 * t (n - 2))) :\n  \u2191(t 2020).den + (t 2020).num = 626  :=  by\n  have h3 : t 3 = (5 * t 2 + 1) / (25 * t 1) := by\n    simp [h\u2082, h\u2080, h\u2081]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n  have h4 : t 4 = (5 * t 3 + 1) / (25 * t 2) := by\n    have h4 : t 4 = (5 * t 3 + 1) / (25 * t 2) := by\n      apply h\u2082\n      all_goals norm_num\n    all_goals assumption\n  have h5 : t 5 = (5 * t 4 + 1) / (25 * t 3) := by\n    simpa [h\u2080, h\u2081, h3, h4] using h\u2082 5 (by norm_num)\n  have h6 : t 6 = (5 * t 5 + 1) / (25 * t 4) := by\n    simpa [h\u2080, h\u2081, h\u2082, h3, h4, h5] using h\u2082 6 (by decide)\n  have h7 : t 7 = (5 * t 6 + 1) / (25 * t 5) := by\n    simp_all [h\u2082]\n    <;> norm_num\n    <;> linarith\n    <;> rfl\n  have h8 : t 8 = (5 * t 7 + 1) / (25 * t 6) := by\n    norm_num [Nat.succ_eq_add_one, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] at *\n    <;> simp_all [Nat.succ_eq_add_one, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    <;> linarith\n  have h9 : t 9 = (5 * t 8 + 1) / (25 * t 7) := by\n    norm_num [h\u2082, h3, h4, h5, h6, h7, h8] at *\n    <;> simp_all [h\u2082, h3, h4, h5, h6, h7, h8] <;> norm_num\n    <;> ring\n    <;> linarith\n  have h10 : t 10 = (5 * t 9 + 1) / (25 * t 8) := by\n    norm_num [h\u2082, h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9]\n    <;> field_simp\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  have cycle : t 6 = t 1 \u2227 t 7 = t 2 \u2227 t 8 = t 3 \u2227 t 9 = t 4 \u2227 t 10 = t 5 := by\n    refine' \u27e8_, _, _, _, _\u27e9\n    <;> norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at *\n    <;> linarith\n    <;> norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at *\n    <;> linarith\n    <;> norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at *\n    <;> linarith\n    <;> norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at *\n    <;> linarith\n    <;> norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at *\n    <;> linarith\n  have mod : 2020 % 5 = 0 := by\n    have h\u2081\u2081 := cycle.1\n    have h\u2081\u2082 := cycle.2.1\n    have h\u2081\u2083 := cycle.2.2.1\n    have h\u2081\u2084 := cycle.2.2.2.1\n    have h\u2081\u2085 := cycle.2.2.2.2\n    norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10, h\u2081\u2081, h\u2081\u2082, h\u2081\u2083, h\u2081\u2084, h\u2081\u2085] at *\n    <;> ring_nf at *\n    <;> linarith\n  have t2020 : t 2020 = t 5 := by\n    have step1 : t 3 = (5 * 21 + 1) / (25 * 20) := by\n      have h\u2083 := h\u2082 3\n      have h\u2084 := h\u2082 4\n      have h\u2085 := h\u2082 5\n      have h\u2086 := h\u2082 6\n      have h\u2087 := h\u2082 7\n      have h\u2088 := h\u2082 8\n      have h\u2089 := h\u2082 9\n      have h\u2081\u2080 := h\u2082 10\n      norm_num [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080, cycle, mod] at *\n      <;> linarith\n    have step2 : t 4 = (5 * t 3 + 1) / (25 * 21) := by\n      have h\u2081\u2081 := cycle.1\n      have h\u2081\u2082 := cycle.2.1\n      have h\u2081\u2083 := cycle.2.2.1\n      have h\u2081\u2084 := cycle.2.2.2.1\n      have h\u2081\u2085 := cycle.2.2.2.2\n      have h\u2081\u2086 := h\u2082 3 (by norm_num)\n      have h\u2081\u2087 := h\u2082 4 (by norm_num)\n      have h\u2081\u2088 := h\u2082 5 (by norm_num)\n      have h\u2081\u2089 := h\u2082 6 (by norm_num)\n      have h\u2082\u2080 := h\u2082 7 (by norm_num)\n      have h\u2082\u2081 := h\u2082 8 (by norm_num)\n      have h\u2082\u2082 := h\u2082 9 (by norm_num)\n      have h\u2082\u2083 := h\u2082 10 (by norm_num)\n      norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, h\u2081\u2081, h\u2081\u2082, h\u2081\u2083, h\u2081\u2084, h\u2081\u2085, h\u2081\u2086, h\u2081\u2087, h\u2081\u2088, h\u2081\u2089, h\u2082\u2080, h\u2082\u2081, h\u2082\u2082, h\u2082\u2083] at *\n      <;> linarith\n    have step3 : t 5 = (5 * t 4 + 1) / (25 * t 3) := by\n      norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, cycle, mod, step1, step2] at *\n      <;> linarith\n      <;> norm_num\n      <;> linarith\n    have step4 : t 6 = t 1 := by\n      have h\u2081\u2081 := h\u2082 3 (by norm_num)\n      have h\u2081\u2082 := h\u2082 4 (by norm_num)\n      have h\u2081\u2083 := h\u2082 5 (by norm_num)\n      have h\u2081\u2084 := h\u2082 6 (by norm_num)\n      have h\u2081\u2085 := h\u2082 7 (by norm_num)\n      have h\u2081\u2086 := h\u2082 8 (by norm_num)\n      have h\u2081\u2087 := h\u2082 9 (by norm_num)\n      have h\u2081\u2088 := h\u2082 10 (by norm_num)\n      norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, cycle] at *\n      <;> linarith\n    have step5 : t 7 = t 2 := by\n      simp [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10, cycle, mod, step1, step2, step3, step4]\n      <;> norm_num\n      <;> linarith\n    have step6 : t 8 = t 3 := by\n      simp [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10, cycle, mod, step1, step2, step3, step4, step5] at *\n      <;> linarith\n    have step7 : t 9 = t 4 := by\n      simp [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at step1 step2 step3 cycle \u22a2\n      <;> linarith\n    have step8 : t 10 = t 5 := by\n      have h\u2080' := step4\n      have h\u2081' := step5\n      have h\u2082' := step6\n      have h\u2083' := step7\n      have h\u2084 := cycle.1\n      have h\u2085 := cycle.2.1\n      have h\u2086 := cycle.2.2.1\n      have h\u2087 := cycle.2.2.2.1\n      have h\u2088 := cycle.2.2.2.2\n      have h\u2089 := mod\n      have h\u2081\u2080 := step1\n      have h\u2081\u2081 := step2\n      have h\u2081\u2082 := step3\n      norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at h\u2080' h\u2081' h\u2082' h\u2083' h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 \u22a2\n      <;> linarith\n    have step9 : \u2200 k, t (5 * k + 1) = t 1 \u2227 t (5 * k + 2) = t 2 \u2227 t (5 * k + 3) = t 3 \u2227 t (5 * k + 4) = t 4 \u2227 t (5 * k + 5) = t 5 := by\n      intro k\n      have h11 := cycle.1\n      have h12 := cycle.2.1\n      have h13 := cycle.2.2.1\n      have h14 := cycle.2.2.2.1\n      have h15 := cycle.2.2.2.2\n      simp at h11 h12 h13 h14 h15\n      induction k <;> simp [*, Nat.mul_succ] at *\n      <;> norm_num\n      <;> linarith\n    have step10 : 2020 = 5 * 404 := by\n      have h11 := cycle.1\n      have h12 := cycle.2.1\n      have h13 := cycle.2.2.1\n      have h14 := cycle.2.2.2.1\n      have h15 := cycle.2.2.2.2\n      have h16 := step9 0\n      have h17 := h16.1\n      have h18 := h16.2.1\n      have h19 := h16.2.2.1\n      have h20 := h16.2.2.2.1\n      have h21 := h16.2.2.2.2\n      norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, h11, h12, h13, h14, h15, h17, h18, h19, h20, h21] at *\n      <;> linarith\n    have step11 : t 2020 = t (5 * 404) := by\n      norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, cycle, step1, step2, step3, step4, step5, step6, step7, step8, step9, step10] at *\n      <;> simp_all\n      <;> norm_num\n      <;> linarith\n    have step12 : t (5 * 404) = t 5 := by\n      have h\u2081\u2082 := step9 404\n      have h\u2081\u2083 := step9 403\n      have h\u2081\u2084 := step9 402\n      have h\u2081\u2085 := step9 401\n      have h\u2081\u2086 := step9 400\n      norm_num [h\u2081\u2082, h\u2081\u2083, h\u2081\u2084, h\u2081\u2085, h\u2081\u2086]\n      <;> simp_all\n      <;> ring\n      <;> linarith\n    have final : t 2020 = t 5 := by\n      simp only [step12, step11, h6, h7, h8, h9, h10, cycle, h5, h4, h3, h\u2082, h\u2080, h\u2081] at *\n      <;> norm_num\n      <;> linarith\n    simpa [final] using final\n  have simplified : t 5 = 5353 / 27825 := by\n    norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, cycle, mod, t2020] at *\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  have sum : \u2191(t 2020).den + (t 2020).num = 626 := by\n    norm_num [cycle, mod, t2020, simplified] at *\n    <;> norm_cast\n    <;> norm_num\n    <;> norm_cast\n    <;> norm_num\n    <;> norm_cast\n    <;> norm_num\n    <;> norm_cast\n    <;> norm_num\n    <;> norm_cast\n  norm_num [cycle, simplified, sum]\n  <;> simp_all [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10, t2020, simplified, sum]\n  <;> norm_cast\n  <;> rfl\n", "proof_before": "theorem aimeII_2020_p6 (t : \u2115 \u2192 \u211a) (h\u2080 : t 1 = 20) (h\u2081 : t 2 = 21)\n  (h\u2082 : \u2200 n \u2265 3, t n = (5 * t (n - 1) + 1) / (25 * t (n - 2))) :\n  \u2191(t 2020).den + (t 2020).num = 626  :=  by\n  have h3 : t 3 = (5 * t 2 + 1) / (25 * t 1) := by\n    simp [h\u2082, h\u2080, h\u2081]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n    <;> simp_all [h\u2082]\n    <;> norm_num\n  have h4 : t 4 = (5 * t 3 + 1) / (25 * t 2) := by\n    have h4 : t 4 = (5 * t 3 + 1) / (25 * t 2) := by\n      apply h\u2082\n      all_goals norm_num\n    all_goals assumption\n  have h5 : t 5 = (5 * t 4 + 1) / (25 * t 3) := by\n    simpa [h\u2080, h\u2081, h3, h4] using h\u2082 5 (by norm_num)\n  have h6 : t 6 = (5 * t 5 + 1) / (25 * t 4) := by\n    simpa [h\u2080, h\u2081, h\u2082, h3, h4, h5] using h\u2082 6 (by decide)\n  have h7 : t 7 = (5 * t 6 + 1) / (25 * t 5) := by\n    simp_all [h\u2082]\n    <;> norm_num\n    <;> linarith\n    <;> rfl\n  have h8 : t 8 = (5 * t 7 + 1) / (25 * t 6) := by\n    norm_num [Nat.succ_eq_add_one, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] at *\n    <;> simp_all [Nat.succ_eq_add_one, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    <;> linarith\n  have h9 : t 9 = (5 * t 8 + 1) / (25 * t 7) := by\n    norm_num [h\u2082, h3, h4, h5, h6, h7, h8] at *\n    <;> simp_all [h\u2082, h3, h4, h5, h6, h7, h8] <;> norm_num\n    <;> ring\n    <;> linarith\n  have h10 : t 10 = (5 * t 9 + 1) / (25 * t 8) := by\n    norm_num [h\u2082, h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9]\n    <;> field_simp\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  have cycle : t 6 = t 1 \u2227 t 7 = t 2 \u2227 t 8 = t 3 \u2227 t 9 = t 4 \u2227 t 10 = t 5 := by\n    refine' \u27e8_, _, _, _, _\u27e9\n    <;> norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at *\n    <;> linarith\n    <;> norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at *\n    <;> linarith\n    <;> norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at *\n    <;> linarith\n    <;> norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at *\n    <;> linarith\n    <;> norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at *\n    <;> linarith\n  have mod : 2020 % 5 = 0 := by\n    have h\u2081\u2081 := cycle.1\n    have h\u2081\u2082 := cycle.2.1\n    have h\u2081\u2083 := cycle.2.2.1\n    have h\u2081\u2084 := cycle.2.2.2.1\n    have h\u2081\u2085 := cycle.2.2.2.2\n    norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10, h\u2081\u2081, h\u2081\u2082, h\u2081\u2083, h\u2081\u2084, h\u2081\u2085] at *\n    <;> ring_nf at *\n    <;> linarith\n  have t2020 : t 2020 = t 5 := by\n    have step1 : t 3 = (5 * 21 + 1) / (25 * 20) := by\n      have h\u2083 := h\u2082 3\n      have h\u2084 := h\u2082 4\n      have h\u2085 := h\u2082 5\n      have h\u2086 := h\u2082 6\n      have h\u2087 := h\u2082 7\n      have h\u2088 := h\u2082 8\n      have h\u2089 := h\u2082 9\n      have h\u2081\u2080 := h\u2082 10\n      norm_num [h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080, cycle, mod] at *\n      <;> linarith\n    have step2 : t 4 = (5 * t 3 + 1) / (25 * 21) := by\n      have h\u2081\u2081 := cycle.1\n      have h\u2081\u2082 := cycle.2.1\n      have h\u2081\u2083 := cycle.2.2.1\n      have h\u2081\u2084 := cycle.2.2.2.1\n      have h\u2081\u2085 := cycle.2.2.2.2\n      have h\u2081\u2086 := h\u2082 3 (by norm_num)\n      have h\u2081\u2087 := h\u2082 4 (by norm_num)\n      have h\u2081\u2088 := h\u2082 5 (by norm_num)\n      have h\u2081\u2089 := h\u2082 6 (by norm_num)\n      have h\u2082\u2080 := h\u2082 7 (by norm_num)\n      have h\u2082\u2081 := h\u2082 8 (by norm_num)\n      have h\u2082\u2082 := h\u2082 9 (by norm_num)\n      have h\u2082\u2083 := h\u2082 10 (by norm_num)\n      norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, h\u2081\u2081, h\u2081\u2082, h\u2081\u2083, h\u2081\u2084, h\u2081\u2085, h\u2081\u2086, h\u2081\u2087, h\u2081\u2088, h\u2081\u2089, h\u2082\u2080, h\u2082\u2081, h\u2082\u2082, h\u2082\u2083] at *\n      <;> linarith\n    have step3 : t 5 = (5 * t 4 + 1) / (25 * t 3) := by\n      norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, cycle, mod, step1, step2] at *\n      <;> linarith\n      <;> norm_num\n      <;> linarith\n    have step4 : t 6 = t 1 := by\n      have h\u2081\u2081 := h\u2082 3 (by norm_num)\n      have h\u2081\u2082 := h\u2082 4 (by norm_num)\n      have h\u2081\u2083 := h\u2082 5 (by norm_num)\n      have h\u2081\u2084 := h\u2082 6 (by norm_num)\n      have h\u2081\u2085 := h\u2082 7 (by norm_num)\n      have h\u2081\u2086 := h\u2082 8 (by norm_num)\n      have h\u2081\u2087 := h\u2082 9 (by norm_num)\n      have h\u2081\u2088 := h\u2082 10 (by norm_num)\n      norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, cycle] at *\n      <;> linarith\n    have step5 : t 7 = t 2 := by\n      simp [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10, cycle, mod, step1, step2, step3, step4]\n      <;> norm_num\n      <;> linarith\n    have step6 : t 8 = t 3 := by\n      simp [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10, cycle, mod, step1, step2, step3, step4, step5] at *\n      <;> linarith\n    have step7 : t 9 = t 4 := by\n      simp [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at step1 step2 step3 cycle \u22a2\n      <;> linarith\n    have step8 : t 10 = t 5 := by\n      have h\u2080' := step4\n      have h\u2081' := step5\n      have h\u2082' := step6\n      have h\u2083' := step7\n      have h\u2084 := cycle.1\n      have h\u2085 := cycle.2.1\n      have h\u2086 := cycle.2.2.1\n      have h\u2087 := cycle.2.2.2.1\n      have h\u2088 := cycle.2.2.2.2\n      have h\u2089 := mod\n      have h\u2081\u2080 := step1\n      have h\u2081\u2081 := step2\n      have h\u2081\u2082 := step3\n      norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10] at h\u2080' h\u2081' h\u2082' h\u2083' h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 \u22a2\n      <;> linarith\n    have step9 : \u2200 k, t (5 * k + 1) = t 1 \u2227 t (5 * k + 2) = t 2 \u2227 t (5 * k + 3) = t 3 \u2227 t (5 * k + 4) = t 4 \u2227 t (5 * k + 5) = t 5 := by\n      intro k\n      have h11 := cycle.1\n      have h12 := cycle.2.1\n      have h13 := cycle.2.2.1\n      have h14 := cycle.2.2.2.1\n      have h15 := cycle.2.2.2.2\n      simp at h11 h12 h13 h14 h15\n      induction k <;> simp [*, Nat.mul_succ] at *\n      <;> norm_num\n      <;> linarith\n    have step10 : 2020 = 5 * 404 := by\n      have h11 := cycle.1\n      have h12 := cycle.2.1\n      have h13 := cycle.2.2.1\n      have h14 := cycle.2.2.2.1\n      have h15 := cycle.2.2.2.2\n      have h16 := step9 0\n      have h17 := h16.1\n      have h18 := h16.2.1\n      have h19 := h16.2.2.1\n      have h20 := h16.2.2.2.1\n      have h21 := h16.2.2.2.2\n      norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, h11, h12, h13, h14, h15, h17, h18, h19, h20, h21] at *\n      <;> linarith\n    have step11 : t 2020 = t (5 * 404) := by\n      norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, cycle, step1, step2, step3, step4, step5, step6, step7, step8, step9, step10] at *\n      <;> simp_all\n      <;> norm_num\n      <;> linarith\n    have step12 : t (5 * 404) = t 5 := by\n      have h\u2081\u2082 := step9 404\n      have h\u2081\u2083 := step9 403\n      have h\u2081\u2084 := step9 402\n      have h\u2081\u2085 := step9 401\n      have h\u2081\u2086 := step9 400\n      norm_num [h\u2081\u2082, h\u2081\u2083, h\u2081\u2084, h\u2081\u2085, h\u2081\u2086]\n      <;> simp_all\n      <;> ring\n      <;> linarith\n    have final : t 2020 = t 5 := by\n      simp only [step12, step11, h6, h7, h8, h9, h10, cycle, h5, h4, h3, h\u2082, h\u2080, h\u2081] at *\n      <;> norm_num\n      <;> linarith\n    simpa [final] using final\n  have simplified : t 5 = 5353 / 27825 := by\n    norm_num [h\u2080, h\u2081, h3, h4, h5, h6, h7, h8, h9, h10, cycle, mod, t2020] at *\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  have sum : \u2191(t 2020).den + (t 2020).num = 626 := by\n    norm_num [cycle, mod, t2020, simplified] at *\n    <;> norm_cast\n    <;> norm_num\n    <;> norm_cast\n    <;> norm_num\n    <;> norm_cast\n    <;> norm_num\n    <;> norm_cast\n    <;> norm_num\n    <;> norm_cast\n  norm_num [cycle, simplified, sum]\n  <;> simp_all [h\u2080, h\u2081, h\u2082, h3, h4, h5, h6, h7, h8, h9, h10, t2020, simplified, sum]\n  <;> norm_cast\n  <;> rfl\n"}
{"mode": "clear", "steps_before": 17, "alphabet": 15, "compiles": true, "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 391, "chars_after": 30, "seconds": 79.4, "id": "mathd_algebra_126", "proof_after": "theorem mathd_algebra_126 (x y : \u211d) (h\u2080 : 2 * 3 = x - 9) (h\u2081 : 2 * -5 = y + 1) : x = 15 \u2227 y = -11  :=  by\n  refine' \u27e8_, _\u27e9 <;> linarith\n", "proof_before": "theorem mathd_algebra_126 (x y : \u211d) (h\u2080 : 2 * 3 = x - 9) (h\u2081 : 2 * -5 = y + 1) : x = 15 \u2227 y = -11  :=  by\n  have h\u2080_simplified : 6 = x - 9 := by\n    have h\u2080' : 2 * 3 = x - 9 := h\u2080\n    have h\u2081' : 2 * -5 = y + 1 := h\u2081\n    linarith\n  have h\u2081_simplified : -10 = y + 1 := by\n    linarith\n  have x_eq_15 : x = 15 := by\n    have h\u2082 : x = 15 := by\n      linarith\n    exact h\u2082\n  have y_eq_neg_11 : y = -11 := by\n    linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n  refine' \u27e8_, _\u27e9 <;> linarith\n"}
{"mode": "clear", "steps_before": 41, "alphabet": 15, "compiles": false, "id": "mathd_numbertheory_109", "proof_after": "theorem mathd_numbertheory_109 (v : \u2115 \u2192 \u2115) (h\u2080 : \u2200 n, v n = 2 * n - 1) :\n  (\u2211 k in Finset.Icc 1 100, v k) % 7 = 4  :=  by\n  have sum_of_odds : \u2211 k in Finset.Icc 1 100, v k = 100^2 := by\n    simp_all only [Finset.sum_range_succ, h\u2080, Nat.succ_mul, Nat.zero_mul, Nat.add_succ, Nat.add_zero,\n      Nat.add_assoc]\n    rfl\n    <;> decide\n  have mod_100 : 100 % 7 = 2 := by\n    norm_num [Finset.sum_Icc_succ_top, h\u2080, Nat.mul_sub_left_distrib] at sum_of_odds\n    <;> omega\n  have mod_100_squared : (100^2) % 7 = (2^2) % 7 := by\n    simp [Finset.sum_range_succ, h\u2080, Nat.mul_mod, Nat.pow_succ, Nat.add_mod, Nat.mod_mod] at sum_of_odds \u22a2\n    <;> norm_num at sum_of_odds \u22a2\n    <;> aesop\n  have mod_4 : (2^2) % 7 = 4 := by\n    simp_all [Finset.sum_range_succ, Nat.add_mod, Nat.mul_mod, Nat.pow_mod, Nat.mod_eq_of_lt]\n    <;> norm_num\n    <;> rfl\n    <;> decide\n  have final_result : (100^2) % 7 = 4 := by\n    simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n  simp_all [Finset.sum_Icc_succ_top, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n  <;> norm_num\n  <;> rfl\n  <;> apply Eq.refl\n", "proof_before": "theorem mathd_numbertheory_109 (v : \u2115 \u2192 \u2115) (h\u2080 : \u2200 n, v n = 2 * n - 1) :\n  (\u2211 k in Finset.Icc 1 100, v k) % 7 = 4  :=  by\n  have sum_of_odds : \u2211 k in Finset.Icc 1 100, v k = 100^2 := by\n    simp_all only [Finset.sum_range_succ, h\u2080, Nat.succ_mul, Nat.zero_mul, Nat.add_succ, Nat.add_zero,\n      Nat.add_assoc]\n    rfl\n    <;> decide\n  have mod_100 : 100 % 7 = 2 := by\n    norm_num [Finset.sum_Icc_succ_top, h\u2080, Nat.mul_sub_left_distrib] at sum_of_odds\n    <;> omega\n  have mod_100_squared : (100^2) % 7 = (2^2) % 7 := by\n    simp [Finset.sum_range_succ, h\u2080, Nat.mul_mod, Nat.pow_succ, Nat.add_mod, Nat.mod_mod] at sum_of_odds \u22a2\n    <;> norm_num at sum_of_odds \u22a2\n    <;> aesop\n  have mod_4 : (2^2) % 7 = 4 := by\n    simp_all [Finset.sum_range_succ, Nat.add_mod, Nat.mul_mod, Nat.pow_mod, Nat.mod_eq_of_lt]\n    <;> norm_num\n    <;> rfl\n    <;> decide\n  have final_result : (100^2) % 7 = 4 := by\n    simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n  simp_all [Finset.sum_Icc_succ_top, Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n  <;> norm_num\n  <;> rfl\n  <;> apply Eq.refl\n"}
{"mode": "clear", "steps_before": 22, "alphabet": 20, "compiles": false, "id": "mathd_algebra_234", "proof_after": "theorem mathd_algebra_234 (d : \u211d) (h\u2080 : 27 / 125 * d = 9 / 25) : 3 / 5 * d ^ 3 = 25 / 9  :=  by\n  have r_eq : d = 5 / 3 := by\n    have h\u2081 : d = 5 / 3 := by\n      apply Eq.symm\n      field_simp at h\u2080 \u22a2\n      linarith\n    rw [h\u2081]\n  have d_cubed : d^3 = 125 / 27 := by\n    rw [r_eq]\n    field_simp\n    ring_nf\n    <;> simp_all\n    <;> linarith\n  have final_eq : 3 / 5 * d^3 = 25 / 9 := by\n    have h\u2081 : 27 / 125 * d = 9 / 25 := h\u2080\n    have h\u2082 : d = 5 / 3 := r_eq\n    have h\u2083 : d^3 = 125 / 27 := d_cubed\n    field_simp [h\u2082, h\u2083]\n    ring\n  have : d ^ 3 = 125 / 27 := d_cubed\n  have : 3 / 5 * d ^ 3 = 3 / 5 * (125 / 27) := by rw [this]\n  linarith\n", "proof_before": "theorem mathd_algebra_234 (d : \u211d) (h\u2080 : 27 / 125 * d = 9 / 25) : 3 / 5 * d ^ 3 = 25 / 9  :=  by\n  have r_eq : d = 5 / 3 := by\n    have h\u2081 : d = 5 / 3 := by\n      apply Eq.symm\n      field_simp at h\u2080 \u22a2\n      linarith\n    rw [h\u2081]\n  have d_cubed : d^3 = 125 / 27 := by\n    rw [r_eq]\n    field_simp\n    ring_nf\n    <;> simp_all\n    <;> linarith\n  have final_eq : 3 / 5 * d^3 = 25 / 9 := by\n    have h\u2081 : 27 / 125 * d = 9 / 25 := h\u2080\n    have h\u2082 : d = 5 / 3 := r_eq\n    have h\u2083 : d^3 = 125 / 27 := d_cubed\n    field_simp [h\u2082, h\u2083]\n    ring\n  have : d ^ 3 = 125 / 27 := d_cubed\n  have : 3 / 5 * d ^ 3 = 3 / 5 * (125 / 27) := by rw [this]\n  linarith\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 18, "compiles": false, "id": "mathd_numbertheory_236", "proof_after": "theorem mathd_numbertheory_236 : 1999 ^ 2000 % 5 = 1  :=  by\n  have base_simplified : 1999 % 5 = 4 := by\n    norm_num\n    <;> rfl\n    <;> rfl\n    <;> rfl\n  have euler_step : 4^4 % 5 = 1 := by\n    simp [pow_succ, pow_zero, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt, show 4 < 5 by norm_num]\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n  have exponent_simplified : 4^2000 % 5 = (4^4)^500 % 5 := by\n    norm_num [pow_succ]\n    <;> simp_all [pow_mul]\n    <;> norm_num\n    <;> rfl\n  have final_simplification : (4^4)^500 % 5 = 1 := by\n    simp [euler_step]\n  have h\u2080 : 1999 % 5 = 4 := base_simplified\n  have h\u2081 : 4 ^ 4 % 5 = 1 := euler_step\n  have h\u2082 : 4 ^ 2000 % 5 = (4 ^ 4) ^ 500 % 5 := exponent_simplified\n  have h\u2083 : (4 ^ 4) ^ 500 % 5 = 1 := final_simplification\n  simp [h\u2080, h\u2081, h\u2082, h\u2083, pow_mod, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n", "proof_before": "theorem mathd_numbertheory_236 : 1999 ^ 2000 % 5 = 1  :=  by\n  have base_simplified : 1999 % 5 = 4 := by\n    norm_num\n    <;> rfl\n    <;> rfl\n    <;> rfl\n  have euler_step : 4^4 % 5 = 1 := by\n    simp [pow_succ, pow_zero, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt, show 4 < 5 by norm_num]\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n  have exponent_simplified : 4^2000 % 5 = (4^4)^500 % 5 := by\n    norm_num [pow_succ]\n    <;> simp_all [pow_mul]\n    <;> norm_num\n    <;> rfl\n  have final_simplification : (4^4)^500 % 5 = 1 := by\n    simp [euler_step]\n  have h\u2080 : 1999 % 5 = 4 := base_simplified\n  have h\u2081 : 4 ^ 4 % 5 = 1 := euler_step\n  have h\u2082 : 4 ^ 2000 % 5 = (4 ^ 4) ^ 500 % 5 := exponent_simplified\n  have h\u2083 : (4 ^ 4) ^ 500 % 5 = 1 := final_simplification\n  simp [h\u2080, h\u2081, h\u2082, h\u2083, pow_mod, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n"}
{"mode": "clear", "steps_before": 29, "alphabet": 24, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 10, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 6, "fixpoint": true, "chars_before": 871, "chars_after": 273, "seconds": 464.3, "id": "induction_divisibility_9div10tonm1", "proof_after": "theorem induction_divisibility_9div10tonm1 (n : \u2115) (h\u2080 : 0 < n) : 9 \u2223 10 ^ n - 1  :=  by\n  have base_case : 9 \u2223 10 ^ 1 - 1 := by\n    rfl\n  have inductive_hypothesis : \u2200 k, 0 < k \u2192 9 \u2223 10 ^ k - 1 := by\n    intro k hk\n    induction' hk with k hk\n    exact base_case\n    omega\n  have conclusion : \u2200 n, 0 < n \u2192 9 \u2223 10 ^ n - 1 := by\n    omega\n  exact conclusion n h\u2080\n", "proof_before": "theorem induction_divisibility_9div10tonm1 (n : \u2115) (h\u2080 : 0 < n) : 9 \u2223 10 ^ n - 1  :=  by\n  have base_case : 9 \u2223 10 ^ 1 - 1 := by\n    apply Nat.dvd_of_mod_eq_zero\n    norm_num\n    <;> rfl\n    <;> simp_all [Nat.mul_mod, Nat.pow_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n    <;> norm_num\n    <;> omega\n  have inductive_hypothesis : \u2200 k, 0 < k \u2192 9 \u2223 10 ^ k - 1 := by\n    intro k hk\n    induction' hk with k hk\n    exact base_case\n    rw [Nat.pow_succ]\n    omega\n  have inductive_step : \u2200 k, 0 < k \u2192 9 \u2223 10 ^ (k + 1) - 1 := by\n    intro k hk\n    apply Nat.dvd_of_mod_eq_zero\n    rw [\u2190 Nat.mod_add_div (10 ^ (k + 1)) 9]\n    norm_num\n    rw [pow_succ]\n    simp [Nat.mul_mod, Nat.pow_mod, Nat.add_mod]\n    <;> simp [inductive_hypothesis]\n    <;> omega\n  have conclusion : \u2200 n, 0 < n \u2192 9 \u2223 10 ^ n - 1 := by\n    intro n hn\n    have h\u2081 := inductive_hypothesis n hn\n    have h\u2082 := inductive_step n hn\n    simp_all [Nat.dvd_iff_mod_eq_zero]\n    <;> omega\n  exact conclusion n h\u2080\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 17, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 821, "chars_after": 36, "seconds": 39.0, "id": "algebra_sqineq_36azm9asqle36zsq", "proof_after": "theorem algebra_sqineq_36azm9asqle36zsq (z a : \u211d) : 36 * (a * z) - 9 * a ^ 2 \u2264 36 * z ^ 2  :=  by\n  nlinarith [sq_nonneg (2 * z - a)]\n", "proof_before": "theorem algebra_sqineq_36azm9asqle36zsq (z a : \u211d) : 36 * (a * z) - 9 * a ^ 2 \u2264 36 * z ^ 2  :=  by\n  have step1 : 4 * (a * z) - a ^ 2 \u2264 4 * z ^ 2 := by\n    ring_nf\n    have h\u2081 : 0 \u2264 (a - 2 * z) ^ 2 := sq_nonneg _\n    linarith\n  have step2 : 4 * z ^ 2 - 4 * a * z + a ^ 2 \u2265 0 := by\n    have h1 : 0 \u2264 (2 * z - a) ^ 2 := by apply sq_nonneg\n    linarith [h1]\n  have step3 : 4 * z ^ 2 - 4 * a * z + a ^ 2 = (2 * z - a) ^ 2 := by\n    simp [sq, mul_add, mul_comm, mul_left_comm, sub_eq_add_neg, add_assoc, add_left_comm]\n    <;> ring\n  have step4 : (2 * z - a) ^ 2 \u2265 0 := by\n    have h : (2 * z - a) ^ 2 \u2265 0 := by\n      nlinarith [sq_nonneg (2 * z - a)]\n    linarith\n  have step5 : 36 * (a * z) - 9 * a ^ 2 \u2264 36 * z ^ 2 := by\n    have h1 : 36 * (a * z) - 9 * a ^ 2 \u2264 36 * z ^ 2 := by\n      linarith [step1, step2, step3, step4]\n    linarith [h1]\n  linarith [step2, step4, step5]\n  <;> nlinarith\n  <;> nlinarith\n  <;> nlinarith\n"}
{"mode": "clear", "steps_before": 21, "alphabet": 24, "compiles": true, "steps_after": 14, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 5, "fixpoint": true, "chars_before": 532, "chars_after": 396, "seconds": 286.3, "id": "numbertheory_prmdvsneqnsqmodpeq0", "proof_after": "theorem numbertheory_prmdvsneqnsqmodpeq0 (n : \u2124) (p : \u2115) (h\u2080 : Nat.Prime p) :\n  \u2191p \u2223 n \u2194 n ^ 2 % p = 0  :=  by\n  have forward : \u2191p \u2223 n \u2192 n ^ 2 % p = 0 := by\n    intro h\n    cases' h with k hk\n    rw [hk, pow_two]\n    simp [Int.mul_emod, Int.add_emod, Int.emod_emod]\n  have backward : n ^ 2 % p = 0 \u2192 \u2191p \u2223 n := by\n    intro h\u2081\n    have h\u2082 : \u2191p \u2223 n ^ 2 := by\n      rw [Int.dvd_iff_emod_eq_zero]\n      exact h\u2081\n    exact (Nat.prime_iff_prime_int.mp h\u2080).dvd_of_dvd_pow h\u2082\n  apply Iff.intro\n  assumption\n  omega\n", "proof_before": "theorem numbertheory_prmdvsneqnsqmodpeq0 (n : \u2124) (p : \u2115) (h\u2080 : Nat.Prime p) :\n  \u2191p \u2223 n \u2194 n ^ 2 % p = 0  :=  by\n  have forward : \u2191p \u2223 n \u2192 n ^ 2 % p = 0 := by\n    intro h\n    cases' h with k hk\n    rw [hk, pow_two]\n    simp [Int.mul_emod, Int.add_emod, Int.emod_emod]\n    <;> simp [h\u2080.ne_zero]\n    <;> simp [h\u2080.ne_zero]\n    <;> simp [h\u2080.ne_zero]\n  have backward : n ^ 2 % p = 0 \u2192 \u2191p \u2223 n := by\n    intro h\u2081\n    have h\u2082 : \u2191p \u2223 n ^ 2 := by\n      rw [Int.dvd_iff_emod_eq_zero]\n      exact h\u2081\n    exact (Nat.prime_iff_prime_int.mp h\u2080).dvd_of_dvd_pow h\u2082\n  apply Iff.intro\n  intro h\n  apply forward\n  assumption\n  intro h\n  apply backward\n  assumption\n"}
{"mode": "clear", "steps_before": 54, "alphabet": 23, "compiles": false, "id": "mathd_numbertheory_24", "proof_after": "theorem mathd_numbertheory_24 : (\u2211 k in Finset.Icc 1 9, 11 ^ k) % 100 = 59  :=  by\n  have h1 : 11^1 % 100 = 11 := by\n    norm_num\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n  have h2 : 11^2 % 100 = 21 := by\n    norm_num [pow_succ]\n    <;> decide\n    <;> simp_all\n    <;> decide\n  have h3 : 11^3 % 100 = 31 := by\n    have h3 : 11^2 % 100 = 21 := h2\n    have h4 : 11^3 % 100 = 31 := by\n      rw [pow_succ]\n      norm_num [Nat.mul_mod, h3, h1]\n    exact h4\n  have h4 : 11^4 % 100 = 41 := by\n    norm_num [pow_succ, mul_mod, h1, h2, h3]\n    <;> omega\n  have h5 : 11^5 % 100 = 51 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.add_mod, h1, h2, h3, h4]\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n  have h6 : 11^6 % 100 = 61 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> norm_num\n    <;> omega\n  have h7 : 11^7 % 100 = 71 := by\n    norm_num [h1, h2, h3, h4, h5, h6]\n    <;> decide\n    <;> ring_nf\n    <;> norm_num\n    <;> decide\n  have h8 : 11^8 % 100 = 81 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp [h1, h2, h3, h4, h5, h6, h7]\n    <;> rfl\n  have h9 : 11^9 % 100 = 91 := by\n    norm_num [pow_succ, h1, h2, h3, h4, h5, h6, h7, h8, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> rfl\n    <;> simp_all [Nat.pow_succ]\n    <;> norm_num\n    <;> rfl\n  have sum_terms : (11 + 21 + 31 + 41 + 51 + 61 + 71 + 81 + 91) % 100 = 59 := by\n    simp [h1, h2, h3, h4, h5, h6, h7, h8, h9, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n    <;> decide\n  simp_all [Finset.sum_range_succ, Nat.pow_mod, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n  <;> rfl\n  <;> rfl\n  <;> rfl\n", "proof_before": "theorem mathd_numbertheory_24 : (\u2211 k in Finset.Icc 1 9, 11 ^ k) % 100 = 59  :=  by\n  have h1 : 11^1 % 100 = 11 := by\n    norm_num\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n  have h2 : 11^2 % 100 = 21 := by\n    norm_num [pow_succ]\n    <;> decide\n    <;> simp_all\n    <;> decide\n  have h3 : 11^3 % 100 = 31 := by\n    have h3 : 11^2 % 100 = 21 := h2\n    have h4 : 11^3 % 100 = 31 := by\n      rw [pow_succ]\n      norm_num [Nat.mul_mod, h3, h1]\n    exact h4\n  have h4 : 11^4 % 100 = 41 := by\n    norm_num [pow_succ, mul_mod, h1, h2, h3]\n    <;> omega\n  have h5 : 11^5 % 100 = 51 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.add_mod, h1, h2, h3, h4]\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n  have h6 : 11^6 % 100 = 61 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp_all [Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> norm_num\n    <;> omega\n  have h7 : 11^7 % 100 = 71 := by\n    norm_num [h1, h2, h3, h4, h5, h6]\n    <;> decide\n    <;> ring_nf\n    <;> norm_num\n    <;> decide\n  have h8 : 11^8 % 100 = 81 := by\n    norm_num [pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> simp [h1, h2, h3, h4, h5, h6, h7]\n    <;> rfl\n  have h9 : 11^9 % 100 = 91 := by\n    norm_num [pow_succ, h1, h2, h3, h4, h5, h6, h7, h8, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> rfl\n    <;> simp_all [Nat.pow_succ]\n    <;> norm_num\n    <;> rfl\n  have sum_terms : (11 + 21 + 31 + 41 + 51 + 61 + 71 + 81 + 91) % 100 = 59 := by\n    simp [h1, h2, h3, h4, h5, h6, h7, h8, h9, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n    <;> decide\n  simp_all [Finset.sum_range_succ, Nat.pow_mod, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n  <;> rfl\n  <;> rfl\n  <;> rfl\n"}
{"mode": "clear", "steps_before": 90, "alphabet": 44, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 2672, "chars_after": 34, "seconds": 21.1, "id": "mathd_algebra_140", "proof_after": "theorem mathd_algebra_140 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c)\n  (h\u2081 : \u2200 x, 24 * x ^ 2 - 19 * x - 35 = (a * x - 5) * (2 * (b * x) + c)) : a * b - 3 * c = -9  :=  by\n  nlinarith [h\u2081 0, h\u2081 1, h\u2081 (-1)]\n", "proof_before": "theorem mathd_algebra_140 (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c)\n  (h\u2081 : \u2200 x, 24 * x ^ 2 - 19 * x - 35 = (a * x - 5) * (2 * (b * x) + c)) : a * b - 3 * c = -9  :=  by\n  have coeff_x2 : 2 * a * b = 24 := by\n    have h\u2082 := h\u2081 1\n    have h\u2083 := h\u2081 (-1)\n    have h\u2084 := h\u2081 (2)\n    have h\u2085 := h\u2081 (-2)\n    nlinarith [h\u2082, h\u2083, h\u2084, h\u2085]\n    <;> ring_nf at h\u2082 h\u2083 h\u2084 h\u2085 \u22a2\n    <;> linarith\n  have coeff_x : a * c - 10 * b = -19 := by\n    have h\u2082 := h\u2081 1\n    have h\u2083 := h\u2081 0\n    have h\u2084 := h\u2081 (-1)\n    have h\u2085 := h\u2081 (3 / 2)\n    have h\u2086 := h\u2081 (-3 / 2)\n    nlinarith\n  have coeff_const : -5 * c = -35 := by\n    have h\u2082 := h\u2081 1\n    have h\u2083 := h\u2081 0\n    have h\u2084 := h\u2081 (-1)\n    have h\u2085 := h\u2081 2\n    have h\u2086 := h\u2081 (-2)\n    have h\u2087 := h\u2081 3\n    have h\u2088 := h\u2081 (-3)\n    ring_nf at h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088\n    linarith\n  have c_eq_7 : c = 7 := by\n    have h\u2082 : c = 7 := by\n      linarith [coeff_const]\n    have h\u2083 : a * c - 10 * b = -19 := coeff_x\n    have h\u2084 : 2 * a * b = 24 := coeff_x2\n    have h\u2085 : -5 * c = -35 := coeff_const\n    nlinarith\n  have ab_eq_12 : a * b = 12 := by\n    have h\u2082 := h\u2081 1\n    have h\u2083 := h\u2081 0\n    have h\u2084 := h\u2081 (-1)\n    simp at h\u2082 h\u2083 h\u2084\n    linarith\n  have ac_minus_10b_eq_neg19 : a * c - 10 * b = -19 := by\n    have h\u2082 := h\u2081 0\n    have h\u2083 := h\u2081 1\n    have h\u2084 := h\u2081 (-1)\n    simp at h\u2082 h\u2083 h\u2084\n    linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081 2, h\u2081 3, coeff_x2, coeff_x, coeff_const, c_eq_7, ab_eq_12]\n  have a_times_7_minus_10b_eq_neg19 : a * 7 - 10 * b = -19 := by\n    have h : a * 7 - 10 * b = -19 := by\n      nlinarith [h\u2081 0, h\u2081 1, h\u2081 (-1)]\n    nlinarith\n  have a_eq_12_over_b : a = 12 / b := by\n    have h\u2082 : a = 12 / b := by\n      apply Eq.symm\n      field_simp [h\u2080.2.1.ne', h\u2080.2.2.ne', h\u2080.1.ne']\n      nlinarith [h\u2081 0, h\u2081 1, h\u2081 (-1)]\n      <;> linarith\n    exact h\u2082\n  have equation_in_b : 7 * (12 / b) - 10 * b = -19 := by\n    have h\u2082 := h\u2081 1\n    have h\u2083 := h\u2081 0\n    have h\u2084 := h\u2081 (-1)\n    simp_all [mul_comm, mul_assoc, mul_left_comm]\n    <;> nlinarith\n  have simplified_eq : 84 / b - 10 * b = -19 := by\n    field_simp [a_eq_12_over_b, c_eq_7, ab_eq_12, ac_minus_10b_eq_neg19, a_times_7_minus_10b_eq_neg19, equation_in_b] at *\n    <;> ring_nf at *\n    <;> linarith\n  have multiplied_eq : 84 - 10 * b^2 = -19 * b := by\n    have h\u2082 : 0 < b := by\n      nlinarith\n    field_simp [h\u2082.ne'] at simplified_eq\n    nlinarith\n  have quadratic_eq : 10 * b^2 - 19 * b - 84 = 0 := by\n    field_simp [h\u2080.1.ne', h\u2080.2.1.ne', h\u2080.2.2.ne'] at *\n    ring_nf at *\n    nlinarith\n  have b_eq_4 : b = 4 := by\n    have h\u2082 : b = 4 := by\n      apply mul_left_cancel\u2080 (sub_ne_zero_of_ne (show (2 : \u211d) \u2260 0 by norm_num))\n      nlinarith\n    exact h\u2082\n  have a_eq_3 : a = 3 := by\n    have h\u2082 : a = 3 := by\n      rw [a_eq_12_over_b]\n      norm_num\n      <;> linarith\n    exact h\u2082\n  have ab_minus_3c : a * b - 3 * c = -9 := by\n    subst_vars\n    norm_num\n    <;> linarith\n  linarith\n"}
{"mode": "clear", "steps_before": 52, "alphabet": 28, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 52, "lower_bound": null, "exhausted_depth": 2, "certified": false, "rounds": 4, "fixpoint": false, "chars_before": 1559, "chars_after": 1559, "seconds": 266.4, "id": "mathd_numbertheory_42", "proof_after": "theorem mathd_numbertheory_42 (S : Set \u2115) (u v : \u2115) (h\u2080 : \u2200 a : \u2115, a \u2208 S \u2194 0 < a \u2227 27 * a % 40 = 17)\n    (h\u2081 : IsLeast S u) (h\u2082 : IsLeast (S \\ {u}) v) : u + v = 62  :=  by\n  have h11_in_S : 11 \u2208 S := by\n    simp_all [h\u2080, Nat.ModEq, Nat.mod_eq_of_lt]\n    <;> norm_num\n    <;> omega\n  have h11_least : u = 11 := by\n    have h\u2081' : IsLeast S 11 := by\n      refine' \u27e8by simpa using h11_in_S, fun a h => _\u27e9\n      have h\u2080' := h\u2080 a\n      have h\u2080'' := h\u2080 11\n      simp_all\n      <;> omega\n    exact h\u2081.unique h\u2081'\n  have h51_in_S : 51 \u2208 S := by\n    have h\u2085 : 27 * 51 % 40 = 17 := by\n      norm_num\n    rw [h\u2080]\n    exact \u27e8by norm_num, h\u2085\u27e9\n  have h_lower_bound : \u2200 x \u2208 S, x \u2260 11 \u2192 x \u2265 51 := by\n    intro x hx hx_ne_11\n    have h\u2081 : IsLeast S u := h\u2081\n    have h\u2082 : IsLeast (S \\ {u}) v := h\u2082\n    have h\u2080 : \u2200 a : \u2115, a \u2208 S \u2194 0 < a \u2227 27 * a % 40 = 17 := h\u2080\n    have h11_in_S : 11 \u2208 S := h11_in_S\n    have h11_least : u = 11 := h11_least\n    have h51_in_S : 51 \u2208 S := h51_in_S\n    have h\u2081 : IsLeast S u := h\u2081\n    have h\u2082 : IsLeast (S \\ {u}) v := h\u2082\n    have h\u2080 : \u2200 a : \u2115, a \u2208 S \u2194 0 < a \u2227 27 * a % 40 = 17 := h\u2080\n    have h11_in_S : 11 \u2208 S := h11_in_S\n    have h11_least : u = 11 := h11_least\n    have h51_in_S : 51 \u2208 S := h51_in_S\n    simp_all [IsLeast, Set.mem_diff, Set.mem_singleton_iff]\n    <;> omega\n  have h51_least : v = 51 := by\n    have hv : v = 51 := by\n      have h51_least : v = 51 := by\n        apply h\u2082.unique\n        apply And.intro\n        \u00b7 \n          simp_all [h\u2080, h11_in_S, h51_in_S]\n          <;> norm_num\n        \u00b7 \n          intro x hx\n          simp_all [h\u2080, h11_in_S, h51_in_S]\n          <;> omega\n      simp_all\n    simp_all\n  subst_vars\n  norm_num\n  <;> simp_all [h\u2080, Nat.ModEq, Nat.mod_eq_of_lt]\n  <;> norm_num\n  <;> omega\n", "proof_before": "theorem mathd_numbertheory_42 (S : Set \u2115) (u v : \u2115) (h\u2080 : \u2200 a : \u2115, a \u2208 S \u2194 0 < a \u2227 27 * a % 40 = 17)\n    (h\u2081 : IsLeast S u) (h\u2082 : IsLeast (S \\ {u}) v) : u + v = 62  :=  by\n  have h11_in_S : 11 \u2208 S := by\n    simp_all [h\u2080, Nat.ModEq, Nat.mod_eq_of_lt]\n    <;> norm_num\n    <;> omega\n  have h11_least : u = 11 := by\n    have h\u2081' : IsLeast S 11 := by\n      refine' \u27e8by simpa using h11_in_S, fun a h => _\u27e9\n      have h\u2080' := h\u2080 a\n      have h\u2080'' := h\u2080 11\n      simp_all\n      <;> omega\n    exact h\u2081.unique h\u2081'\n  have h51_in_S : 51 \u2208 S := by\n    have h\u2085 : 27 * 51 % 40 = 17 := by\n      norm_num\n    rw [h\u2080]\n    exact \u27e8by norm_num, h\u2085\u27e9\n  have h_lower_bound : \u2200 x \u2208 S, x \u2260 11 \u2192 x \u2265 51 := by\n    intro x hx hx_ne_11\n    have h\u2081 : IsLeast S u := h\u2081\n    have h\u2082 : IsLeast (S \\ {u}) v := h\u2082\n    have h\u2080 : \u2200 a : \u2115, a \u2208 S \u2194 0 < a \u2227 27 * a % 40 = 17 := h\u2080\n    have h11_in_S : 11 \u2208 S := h11_in_S\n    have h11_least : u = 11 := h11_least\n    have h51_in_S : 51 \u2208 S := h51_in_S\n    have h\u2081 : IsLeast S u := h\u2081\n    have h\u2082 : IsLeast (S \\ {u}) v := h\u2082\n    have h\u2080 : \u2200 a : \u2115, a \u2208 S \u2194 0 < a \u2227 27 * a % 40 = 17 := h\u2080\n    have h11_in_S : 11 \u2208 S := h11_in_S\n    have h11_least : u = 11 := h11_least\n    have h51_in_S : 51 \u2208 S := h51_in_S\n    simp_all [IsLeast, Set.mem_diff, Set.mem_singleton_iff]\n    <;> omega\n  have h51_least : v = 51 := by\n    have hv : v = 51 := by\n      have h51_least : v = 51 := by\n        apply h\u2082.unique\n        apply And.intro\n        \u00b7 \n          simp_all [h\u2080, h11_in_S, h51_in_S]\n          <;> norm_num\n        \u00b7 \n          intro x hx\n          simp_all [h\u2080, h11_in_S, h51_in_S]\n          <;> omega\n      simp_all\n    simp_all\n  subst_vars\n  norm_num\n  <;> simp_all [h\u2080, Nat.ModEq, Nat.mod_eq_of_lt]\n  <;> norm_num\n  <;> omega\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 83, "alphabet": 51, "compiles": false, "id": "amc12a_2003_p24", "proof_after": "theorem amc12a_2003_p24 :\n  IsGreatest { y : \u211d | \u2203 a b : \u211d, 1 < b \u2227 b \u2264 a \u2227 y = Real.logb a (a / b) + Real.logb b (b / a) }\n    0  :=  by\n  have step1 : \u2200 a b : \u211d, 1 < b \u2192 b \u2264 a \u2192 \n    Real.logb a (a / b) + Real.logb b (b / a) = 2 - Real.logb a b - Real.logb b a := by\n    intro a b hb hab\n    have h\u2081 : 0 < a := by linarith\n    have h\u2082 : 0 < b := by linarith\n    have h\u2083 : 0 < Real.log a := Real.log_pos (by linarith)\n    have h\u2084 : 0 < Real.log b := Real.log_pos (by linarith)\n    field_simp [Real.logb, Real.log_mul, Real.log_div, Real.log_mul, Real.log_div, h\u2081, h\u2082]\n    ring\n    <;> simp_all [Real.log_mul, Real.log_div, h\u2081, h\u2082]\n    <;> ring\n    <;> simp_all [Real.log_mul, Real.log_div, h\u2081, h\u2082]\n    <;> ring\n  have step2 : \u2200 a b : \u211d, 1 < b \u2192 b \u2264 a \u2192 \n    Real.logb a b = (Real.log b) / (Real.log a) \u2227 Real.logb b a = (Real.log a) / (Real.log b) := by\n    intro a b h\u2081 h\u2082\n    simp_all [Real.logb, h\u2081.le, h\u2082]\n    <;> ring_nf\n    <;> simp_all [h\u2081.le, h\u2082]\n    <;> ring_nf\n    <;> linarith\n  have step3 : \u2200 a b : \u211d, 1 < b \u2192 b \u2264 a \u2192 \n    \u2203 t : \u211d, 0 < t \u2227 t \u2264 1 \u2227 Real.logb a b = t \u2227 Real.logb b a = 1 / t := by\n    intro a b hb hab\n    have h1 := step1 a b hb hab\n    have h2 := step2 a b hb hab\n    use (Real.log b) / (Real.log a)\n    have h3 : 0 < (Real.log b) / (Real.log a) := by\n      apply div_pos\n      exact Real.log_pos hb\n      exact Real.log_pos (by linarith)\n    have h4 : (Real.log b) / (Real.log a) \u2264 1 := by\n      rw [div_le_iff (Real.log_pos (by linarith))]\n      nlinarith [Real.log_le_log (by linarith) hab]\n    constructor\n    exact h3\n    constructor\n    exact h4\n    rw [h2.1]\n    rw [h2.2]\n    <;> field_simp\n  have step4 : \u2200 a b : \u211d, 1 < b \u2192 b \u2264 a \u2192 \n    \u2203 t : \u211d, 0 < t \u2227 t \u2264 1 \u2227 2 - Real.logb a b - Real.logb b a = 2 - t - 1 / t := by\n    intro a b hb h\n    obtain \u27e8t, ht1, ht2, ht3, ht4\u27e9 := step3 a b hb h\n    use t\n    constructor\n    exact ht1\n    constructor\n    exact ht2\n    simp_all [Real.logb, ht3, ht4]\n    <;> field_simp\n    <;> ring\n    <;> linarith\n  have step5 : \u2200 t : \u211d, 0 < t \u2192 t \u2264 1 \u2192 2 - t - 1 / t \u2264 0 := by\n    intro t ht ht2\n    have h : 0 < t := by linarith\n    have h2 : t \u2264 1 := by linarith\n    have h3 : 2 - t - 1 / t \u2264 0 := by\n      field_simp [h, h2]\n      rw [div_le_iff] <;> nlinarith [sq_nonneg (t - 1)]\n    exact h3\n  have step6 : IsGreatest { y : \u211d | \u2203 a b : \u211d, 1 < b \u2227 b \u2264 a \u2227 y = Real.logb a (a / b) + Real.logb b (b / a) } 0 := by\n    refine' \u27e8\u27e82, 2, by norm_num, by norm_num, by norm_num\u27e9, fun y hy => _\u27e9\n    rcases hy with \u27e8a, b, hb, hab, hy\u27e9\n    have h\u2081 := step1 a b hb hab\n    have h\u2082 := step2 a b hb hab\n    have h\u2083 := step3 a b hb hab\n    rcases h\u2083 with \u27e8t, ht, htab, hta, htb\u27e9\n    have h\u2084 := step4 a b hb hab\n    rcases h\u2084 with \u27e8t', ht', htab', hta'\u27e9\n    have h\u2085 := step5 t ht htab\n    simp_all [Real.logb, Real.log_div, Real.log_mul, mul_comm]\n    <;> norm_num\n    <;> linarith\n  simpa using step6\n", "proof_before": "theorem amc12a_2003_p24 :\n  IsGreatest { y : \u211d | \u2203 a b : \u211d, 1 < b \u2227 b \u2264 a \u2227 y = Real.logb a (a / b) + Real.logb b (b / a) }\n    0  :=  by\n  have step1 : \u2200 a b : \u211d, 1 < b \u2192 b \u2264 a \u2192 \n    Real.logb a (a / b) + Real.logb b (b / a) = 2 - Real.logb a b - Real.logb b a := by\n    intro a b hb hab\n    have h\u2081 : 0 < a := by linarith\n    have h\u2082 : 0 < b := by linarith\n    have h\u2083 : 0 < Real.log a := Real.log_pos (by linarith)\n    have h\u2084 : 0 < Real.log b := Real.log_pos (by linarith)\n    field_simp [Real.logb, Real.log_mul, Real.log_div, Real.log_mul, Real.log_div, h\u2081, h\u2082]\n    ring\n    <;> simp_all [Real.log_mul, Real.log_div, h\u2081, h\u2082]\n    <;> ring\n    <;> simp_all [Real.log_mul, Real.log_div, h\u2081, h\u2082]\n    <;> ring\n  have step2 : \u2200 a b : \u211d, 1 < b \u2192 b \u2264 a \u2192 \n    Real.logb a b = (Real.log b) / (Real.log a) \u2227 Real.logb b a = (Real.log a) / (Real.log b) := by\n    intro a b h\u2081 h\u2082\n    simp_all [Real.logb, h\u2081.le, h\u2082]\n    <;> ring_nf\n    <;> simp_all [h\u2081.le, h\u2082]\n    <;> ring_nf\n    <;> linarith\n  have step3 : \u2200 a b : \u211d, 1 < b \u2192 b \u2264 a \u2192 \n    \u2203 t : \u211d, 0 < t \u2227 t \u2264 1 \u2227 Real.logb a b = t \u2227 Real.logb b a = 1 / t := by\n    intro a b hb hab\n    have h1 := step1 a b hb hab\n    have h2 := step2 a b hb hab\n    use (Real.log b) / (Real.log a)\n    have h3 : 0 < (Real.log b) / (Real.log a) := by\n      apply div_pos\n      exact Real.log_pos hb\n      exact Real.log_pos (by linarith)\n    have h4 : (Real.log b) / (Real.log a) \u2264 1 := by\n      rw [div_le_iff (Real.log_pos (by linarith))]\n      nlinarith [Real.log_le_log (by linarith) hab]\n    constructor\n    exact h3\n    constructor\n    exact h4\n    rw [h2.1]\n    rw [h2.2]\n    <;> field_simp\n  have step4 : \u2200 a b : \u211d, 1 < b \u2192 b \u2264 a \u2192 \n    \u2203 t : \u211d, 0 < t \u2227 t \u2264 1 \u2227 2 - Real.logb a b - Real.logb b a = 2 - t - 1 / t := by\n    intro a b hb h\n    obtain \u27e8t, ht1, ht2, ht3, ht4\u27e9 := step3 a b hb h\n    use t\n    constructor\n    exact ht1\n    constructor\n    exact ht2\n    simp_all [Real.logb, ht3, ht4]\n    <;> field_simp\n    <;> ring\n    <;> linarith\n  have step5 : \u2200 t : \u211d, 0 < t \u2192 t \u2264 1 \u2192 2 - t - 1 / t \u2264 0 := by\n    intro t ht ht2\n    have h : 0 < t := by linarith\n    have h2 : t \u2264 1 := by linarith\n    have h3 : 2 - t - 1 / t \u2264 0 := by\n      field_simp [h, h2]\n      rw [div_le_iff] <;> nlinarith [sq_nonneg (t - 1)]\n    exact h3\n  have step6 : IsGreatest { y : \u211d | \u2203 a b : \u211d, 1 < b \u2227 b \u2264 a \u2227 y = Real.logb a (a / b) + Real.logb b (b / a) } 0 := by\n    refine' \u27e8\u27e82, 2, by norm_num, by norm_num, by norm_num\u27e9, fun y hy => _\u27e9\n    rcases hy with \u27e8a, b, hb, hab, hy\u27e9\n    have h\u2081 := step1 a b hb hab\n    have h\u2082 := step2 a b hb hab\n    have h\u2083 := step3 a b hb hab\n    rcases h\u2083 with \u27e8t, ht, htab, hta, htb\u27e9\n    have h\u2084 := step4 a b hb hab\n    rcases h\u2084 with \u27e8t', ht', htab', hta'\u27e9\n    have h\u2085 := step5 t ht htab\n    simp_all [Real.logb, Real.log_div, Real.log_mul, mul_comm]\n    <;> norm_num\n    <;> linarith\n  simpa using step6\n"}
{"mode": "clear", "steps_before": 41, "alphabet": 24, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1353, "chars_after": 169, "seconds": 106.3, "id": "mathd_algebra_267", "proof_after": "theorem mathd_algebra_267 (x : \u211d) (h\u2080 : x \u2260 1) (h\u2081 : x \u2260 -2)\n  (h\u2082 : (x + 1) / (x - 1) = (x - 2) / (x + 2)) : x = 0  :=  by\n  field_simp [sub_eq_zero, add_eq_zero_iff_eq_neg, h\u2080, h\u2081, sub_eq_zero, add_eq_zero_iff_eq_neg, eq_self_iff_true, not_true_eq_false, not_false_eq_true] at h\u2082\n  linarith\n", "proof_before": "theorem mathd_algebra_267 (x : \u211d) (h\u2080 : x \u2260 1) (h\u2081 : x \u2260 -2)\n  (h\u2082 : (x + 1) / (x - 1) = (x - 2) / (x + 2)) : x = 0  :=  by\n  have h\u2083 : (x + 1) * (x + 2) = (x - 2) * (x - 1) := by\n    have h\u2083 : x - 1 \u2260 0 := by\n      intro h\n      apply h\u2080\n      linarith\n    have h\u2084 : x + 2 \u2260 0 := by\n      intro h\n      apply h\u2081\n      linarith\n    field_simp [h\u2083, h\u2084, sub_eq_zero, add_eq_zero_iff_eq_neg] at h\u2082\n    linarith\n  have h\u2084 : (x + 1) * (x + 2) = x^2 + 3*x + 2 := by\n    field_simp [sub_eq_zero, add_eq_zero_iff_eq_neg, h\u2080, h\u2081, sub_eq_zero, add_eq_zero_iff_eq_neg, eq_self_iff_true, not_true_eq_false, not_false_eq_true] at h\u2082\n    ring_nf at h\u2083 \u22a2\n    <;> nlinarith\n  have h\u2085 : (x - 2) * (x - 1) = x^2 - 3*x + 2 := by\n    field_simp [h\u2080, h\u2081, sub_eq_zero, add_eq_zero_iff_eq_neg, eq_self_iff_true, not_true,\n      imp_false] at h\u2082\n    linarith\n  have h\u2086 : x^2 + 3*x + 2 = x^2 - 3*x + 2 := by\n    linarith\n  have h\u2087 : 3*x = -3*x := by\n    have h\u2087 : 3 * x = -3 * x := by\n      field_simp [h\u2080, h\u2081, sub_eq_zero, add_eq_zero_iff_eq_neg, h\u2081, h\u2080, sub_eq_zero, add_eq_zero_iff_eq_neg] at h\u2082 h\u2083 h\u2084 h\u2085 h\u2086\n      ring_nf at h\u2086 \u22a2\n      linarith\n    exact h\u2087\n  have h\u2088 : 6*x = 0 := by\n    have h\u2088 : 6 * x = 0 := by\n      nlinarith\n    exact h\u2088\n  have h\u2089 : x = 0 := by\n    have h\u2089 : x = 0 := by\n      apply Eq.symm\n      linarith\n    exact h\u2089\n  have h\u2081\u2080 : (0 + 1) / (0 - 1) = (0 - 2) / (0 + 2) := by\n    simp_all [div_eq_mul_inv]\n    <;> norm_num\n    <;> linarith\n  apply eq_of_sub_eq_zero\n  linarith\n"}
{"mode": "clear", "steps_before": 29, "alphabet": 16, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 834, "chars_after": 8, "seconds": 14.0, "id": "mathd_numbertheory_301", "proof_after": "theorem mathd_numbertheory_301 (j : \u2115) (h\u2080 : 0 < j) : 3 * (7 * \u2191j + 3) % 7 = 2  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_301 (j : \u2115) (h\u2080 : 0 < j) : 3 * (7 * \u2191j + 3) % 7 = 2  :=  by\n  have step1 : 3 * (7 * j + 3) = 21 * j + 9 := by\n    simp [mul_add, mul_comm, mul_left_comm, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    <;> linarith\n  have step2 : 21 * j % 7 = 0 := by\n    norm_num at step1\n    <;> omega\n    <;> omega\n    <;> omega\n    <;> omega\n    <;> omega\n  have step3 : 9 % 7 = 2 := by\n    norm_num\n    <;> omega\n    <;> simp_all\n    <;> omega\n  have step4 : (21 * j + 9) % 7 = (0 + 2) % 7 := by\n    simp [step1, step2, step3, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n    <;> omega\n  have step5 : (0 + 2) % 7 = 2 := by\n    simp [step2, step3, step4]\n    <;> omega\n    <;> omega\n    <;> omega\n    <;> omega\n  have final_result : 3 * (7 * j + 3) % 7 = 2 := by\n    simp_all [Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n    <;> norm_num\n    <;> omega\n  simp [step1, step2, step3, step4, step5, final_result]\n"}
{"mode": "clear", "steps_before": 33, "alphabet": 18, "compiles": false, "id": "mathd_numbertheory_739", "proof_after": "theorem mathd_numbertheory_739 : 9! % 10 = 0  :=  by\n  have h1 : 9! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 := by\n    rfl\n    <;> simp [Nat.factorial]\n    <;> rfl\n  have h2 : 2 \u2223 9! := by\n    apply Nat.dvd_of_mod_eq_zero\n    simp [h1, Nat.factorial, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt,\n      Nat.succ_pos, Nat.zero_lt_one, Nat.one_lt_two]\n  have h3 : 5 \u2223 9! := by\n    norm_num\n    simp_all [Nat.factorial]\n    <;> norm_num\n    <;> omega\n  have h4 : 10 = 2 * 5 := by\n    norm_num [factorial, h1, h2, h3]\n    <;> decide\n    <;> decide\n  have h5 : 10 \u2223 9! := by\n    rw [Nat.dvd_iff_mod_eq_zero]\n    norm_num [Nat.factorial, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n  have h6 : 9! % 10 = 0 := by\n    simp [h1, h2, h3, h4, h5, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n    <;> norm_num\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n  simp [h1, h2, h3, h4, h5, h6]\n", "proof_before": "theorem mathd_numbertheory_739 : 9! % 10 = 0  :=  by\n  have h1 : 9! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 := by\n    rfl\n    <;> simp [Nat.factorial]\n    <;> rfl\n  have h2 : 2 \u2223 9! := by\n    apply Nat.dvd_of_mod_eq_zero\n    simp [h1, Nat.factorial, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt,\n      Nat.succ_pos, Nat.zero_lt_one, Nat.one_lt_two]\n  have h3 : 5 \u2223 9! := by\n    norm_num\n    simp_all [Nat.factorial]\n    <;> norm_num\n    <;> omega\n  have h4 : 10 = 2 * 5 := by\n    norm_num [factorial, h1, h2, h3]\n    <;> decide\n    <;> decide\n  have h5 : 10 \u2223 9! := by\n    rw [Nat.dvd_iff_mod_eq_zero]\n    norm_num [Nat.factorial, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n  have h6 : 9! % 10 = 0 := by\n    simp [h1, h2, h3, h4, h5, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n    <;> norm_num\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n  simp [h1, h2, h3, h4, h5, h6]\n"}
{"mode": "clear", "steps_before": 14, "alphabet": 13, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 9, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 2, "fixpoint": true, "chars_before": 355, "chars_after": 197, "seconds": 137.4, "id": "amc12_2000_p5", "proof_after": "theorem amc12_2000_p5 (x p : \u211d) (h\u2080 : x < 2) (h\u2081 : abs (x - 2) = p) : x - p = 2 - 2 * p  :=  by\n  have h\u2082 : x = 2 - p := by\n    cases' le_total 0 (x - 2) with h h <;>\n      simp_all [abs_of_nonneg, abs_of_nonpos, sub_eq_add_neg, neg_add_rev] <;>\n      linarith\n    <;>\n    linarith\n  linarith\n", "proof_before": "theorem amc12_2000_p5 (x p : \u211d) (h\u2080 : x < 2) (h\u2081 : abs (x - 2) = p) : x - p = 2 - 2 * p  :=  by\n  have h\u2082 : x = 2 - p := by\n    cases' le_total 0 (x - 2) with h h <;>\n      simp_all [abs_of_nonneg, abs_of_nonpos, sub_eq_add_neg, neg_add_rev] <;>\n      linarith\n    <;>\n    linarith\n  have h\u2083 : x - p = 2 - 2 * p := by\n    rw [h\u2082]\n    simp_all only [mul_one, sub_sub]\n    linarith\n  simp_all only [abs_of_nonneg, sub_nonneg, sub_nonpos]\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 30, "alphabet": 21, "compiles": true, "steps_after": 17, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 8, "fixpoint": false, "chars_before": 763, "chars_after": 477, "seconds": 866.4, "id": "mathd_algebra_69", "proof_after": "theorem mathd_algebra_69 (rows seats : \u2115) (h\u2080 : rows * seats = 450)\n  (h\u2081 : (rows + 5) * (seats - 3) = 450) : rows = 25  :=  by\n  have h\u2082 : seats = 450 / rows := by\n    have h\u2082 : seats = 450 / rows := by\n      rw [Nat.div_eq_of_eq_mul_left] <;> nlinarith\n    simpa [h\u2082] using h\u2081\n  have h\u2083 : (rows + 5) * ((450 / rows) - 3) = 450 := by\n    simpa [h\u2082] using h\u2081\n  have h\u2084 : rows = 25 := by\n    have h\u2084 : rows = 25 := by\n      have h\u2088 : seats > 0 := by\n        nlinarith\n      have h\u2089 : rows \u2264 450 := by\n        nlinarith\n      interval_cases rows <;> omega\n    exact h\u2084\n  simpa [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084] using h\u2084\n", "proof_before": "theorem mathd_algebra_69 (rows seats : \u2115) (h\u2080 : rows * seats = 450)\n  (h\u2081 : (rows + 5) * (seats - 3) = 450) : rows = 25  :=  by\n  have h\u2082 : seats = 450 / rows := by\n    have h\u2082 : seats = 450 / rows := by\n      rw [Nat.div_eq_of_eq_mul_left] <;> nlinarith\n    simpa [h\u2082] using h\u2081\n  have h\u2083 : (rows + 5) * ((450 / rows) - 3) = 450 := by\n    subst h\u2082\n    ring_nf at h\u2081 \u22a2\n    <;> omega\n  have h\u2084 : rows = 25 := by\n    have h\u2084 : rows = 25 := by\n      have h\u2084 := h\u2081\n      have h\u2085 := h\u2082\n      have h\u2086 := h\u2083\n      have h\u2087 : rows > 0 := by\n        nlinarith\n      have h\u2088 : seats > 0 := by\n        nlinarith\n      have h\u2089 : rows \u2264 450 := by\n        nlinarith\n      have h\u2081\u2080 : seats \u2264 450 := by\n        nlinarith\n      have h\u2081\u2081 : rows \u2265 1 := by\n        nlinarith\n      have h\u2081\u2082 : seats \u2265 1 := by\n        nlinarith\n      interval_cases rows <;> omega\n    exact h\u2084\n  simpa [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084] using h\u2084\n"}
{"mode": "clear", "steps_before": 89, "alphabet": 30, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 3248, "chars_after": 6, "seconds": 6.8, "id": "mathd_numbertheory_30", "proof_after": "theorem mathd_numbertheory_30 :\n  (33818 ^ 2 + 33819 ^ 2 + 33820 ^ 2 + 33821 ^ 2 + 33822 ^ 2) % 17 = 0  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_30 :\n  (33818 ^ 2 + 33819 ^ 2 + 33820 ^ 2 + 33821 ^ 2 + 33822 ^ 2) % 17 = 0  :=  by\n  have h33818 : 33818 % 17 = 5 := by\n    norm_num\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n  have h33819 : 33819 % 17 = 6 := by\n    simp [Nat.add_mod, Nat.mul_mod, Nat.mod_eq_of_lt, Nat.lt_of_le_of_lt (Nat.zero_le _) (by norm_num : 17 < 33819)]\n    <;> simp_all [Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n    <;> norm_num\n    <;> omega\n  have h33820 : 33820 % 17 = 7 := by\n    simp [h33818, h33819, Nat.add_mod, Nat.mul_mod, Nat.mod_mod, Nat.mod_eq_of_lt,\n      show 2 \u2264 17 by norm_num, show 3 \u2264 17 by norm_num]\n  have h33821 : 33821 % 17 = 8 := by\n    have h33821 : 33821 % 17 = 8 := by\n      omega\n    exact h33821\n  have h33822 : 33822 % 17 = 9 := by\n    simp [h33818, h33819, h33820, h33821, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n    <;> norm_num\n    <;> rfl\n  have sq5 : 5^2 = 25 := by\n    simp [h33818, h33819, h33820, h33821, h33822, Nat.pow_succ, Nat.mul_mod, Nat.add_mod]\n    <;> decide\n  have sq6 : 6^2 = 36 := by\n    have h33818_rem : 33818 % 17 = 5 := by simpa using h33818\n    have h33819_rem : 33819 % 17 = 6 := by simpa using h33819\n    have h33820_rem : 33820 % 17 = 7 := by simpa using h33820\n    have h33821_rem : 33821 % 17 = 8 := by simpa using h33821\n    have h33822_rem : 33822 % 17 = 9 := by simpa using h33822\n    have sq5 : 5^2 = 25 := by simpa using sq5\n    have sq6 : 6^2 = 36 := by norm_num\n    exact sq6\n  have sq7 : 7^2 = 49 := by\n    have h : 7 ^ 2 = 49 := by\n      simp [h33820, h33821, h33822, sq, mul_add, mul_comm, mul_left_comm, mul_assoc]\n      <;> omega\n    simpa using h\n  have sq8 : 8^2 = 64 := by\n    have h33823 := congr_arg (\u00b7 % 17) h33820\n    have h33824 := congr_arg (\u00b7 % 17) h33821\n    have h33825 := congr_arg (\u00b7 % 17) h33822\n    simp at h33823 h33824 h33825\n    <;> norm_num at h33823 h33824 h33825\n    <;> omega\n  have sq9 : 9^2 = 81 := by\n    simp_all [sq]\n    <;> norm_num\n    <;> omega\n  have r5 : 25 % 17 = 8 := by\n    simp [h33818, h33819, h33820, h33821, h33822, sq5, sq6, sq7, sq8, sq9]\n    <;> simp_all [Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> norm_num\n    <;> omega\n  have r6 : 36 % 17 = 2 := by\n    simp_all [Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> norm_num\n    <;> omega\n  have r7 : 49 % 17 = 15 := by\n    simp_all [Nat.pow_succ, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n    <;> norm_num\n    <;> omega\n  have r8 : 64 % 17 = 13 := by\n    simp [h33818, h33819, h33820, h33821, h33822, sq5, sq6, sq7, sq8, sq9, r5, r6, r7]\n    <;> norm_num\n    <;> omega\n  have r9 : 81 % 17 = 13 := by\n    simp [h33818, h33819, h33820, h33821, h33822, sq5, sq6, sq7, sq8, sq9, r5, r6, r7, r8]\n    <;> norm_num\n    <;> aesop\n  have total_sum : 8 + 2 + 15 + 13 + 13 = 51 := by\n    simp [h33818, h33819, h33820, h33821, h33822, sq5, sq6, sq7, sq8, sq9, r5, r6, r7, r8, r9]\n    <;> norm_num\n    <;> rfl\n    <;> simp_all [Nat.pow_succ]\n    <;> norm_num\n    <;> rfl\n  have final_remainder : 51 % 17 = 0 := by\n    simp [h33818, h33819, h33820, h33821, h33822, sq5, sq6, sq7, sq8, sq9, r5, r6, r7, r8, r9, total_sum]\n    <;> decide\n  have h\u2081 : (33818 ^ 2 + 33819 ^ 2 + 33820 ^ 2 + 33821 ^ 2 + 33822 ^ 2) % 17 = 0 := by\n    norm_num [h33818, h33819, h33820, h33821, h33822, sq5, sq6, sq7, sq8, sq9, r5, r6, r7, r8, r9, total_sum, final_remainder]\n  simpa [h\u2081] using h\u2081\n"}
{"mode": "clear", "steps_before": 22, "alphabet": 18, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 526, "chars_after": 7, "seconds": 32.1, "id": "amc12a_2009_p2", "proof_after": "theorem amc12a_2009_p2 : 1 + 1 / (1 + 1 / (1 + 1)) = (5 : \u211a) / 3  :=  by\n  ring\n", "proof_before": "theorem amc12a_2009_p2 : 1 + 1 / (1 + 1 / (1 + 1)) = (5 : \u211a) / 3  :=  by\n  have step1 : 1 + 1 = 2 := by\n    norm_num\n    <;> rfl\n    <;> rfl\n    <;> rfl\n    <;> rfl\n  have step2 : 1 + 1 / (1 + 1) = (3 : \u211a) / 2 := by\n    norm_num [step1]\n    <;> field_simp [step1]\n    <;> ring_nf\n    <;> norm_cast\n  have step3 : 1 + 1 / (3 / 2) = (5 : \u211a) / 3 := by\n    norm_num [step1, step2]\n    <;> simp [div_eq_mul_inv]\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  have final : 1 + 1 / (1 + 1 / (1 + 1)) = (5 : \u211a) / 3 := by\n    norm_num [step1, step2, step3]\n    <;> field_simp\n    <;> linarith\n  rw [final]\n"}
{"mode": "clear", "steps_before": 22, "alphabet": 16, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 556, "chars_after": 20, "seconds": 32.4, "id": "mathd_algebra_55", "proof_after": "theorem mathd_algebra_55 (q p : \u211d) (h\u2080 : q = 2 - 4 + 6 - 8 + 10 - 12 + 14)\n  (h\u2081 : p = 3 - 6 + 9 - 12 + 15 - 18 + 21) : q / p = 2 / 3  :=  by\n  norm_num [h\u2080, h\u2081]\n", "proof_before": "theorem mathd_algebra_55 (q p : \u211d) (h\u2080 : q = 2 - 4 + 6 - 8 + 10 - 12 + 14)\n  (h\u2081 : p = 3 - 6 + 9 - 12 + 15 - 18 + 21) : q / p = 2 / 3  :=  by\n  have calc_q : q = 8 := by\n    norm_num [h\u2080, h\u2081]\n    <;> linarith\n  have calc_p : p = 12 := by\n    have h\u2082 : q = 8 := by\n      linarith\n    have h\u2083 : p = 12 := by\n      linarith\n    linarith\n  have fraction_result : q / p = 8 / 12 := by\n    have h\u2082 : q = 8 := calc_q\n    have h\u2083 : p = 12 := calc_p\n    rw [h\u2082, h\u2083]\n    <;> simp_all [div_eq_mul_inv, mul_inv_cancel_left\u2080]\n    <;> norm_num\n  have simplify_frac : 8 / 12 = 2 / 3 := by\n    norm_num [calc_q, calc_p, fraction_result]\n    <;> field_simp\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  linarith\n"}
{"mode": "clear", "steps_before": 35, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1063, "chars_after": 6, "seconds": 0.7, "id": "mathd_numbertheory_269", "proof_after": "theorem mathd_numbertheory_269 : (2005 ^ 2 + 2005 ^ 0 + 2005 ^ 0 + 2005 ^ 5) % 100 = 52  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_269 : (2005 ^ 2 + 2005 ^ 0 + 2005 ^ 0 + 2005 ^ 5) % 100 = 52  :=  by\n  have simplify_exp : 2005 ^ 2 + 2005 ^ 0 + 2005 ^ 0 + 2005 ^ 5 = 2005 ^ 2 + 2005 ^ 5 + 2 := by\n    norm_num\n    <;> simp [Nat.pow_succ]\n    <;> ring\n    <;> norm_num\n    <;> rfl\n    <;> rfl\n  have mod_100 : 2005 % 100 = 5 := by\n    norm_num [Nat.mod_eq_of_lt, Nat.add_mod, Nat.pow_mod]\n    <;> rfl\n    <;> simp_all\n    <;> norm_num\n    <;> rfl\n  have pow2_mod_100 : 2005 ^ 2 % 100 = 25 := by\n    have h : 2005 % 100 = 5 := by norm_num\n    norm_num [h, Nat.pow_mod, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n    <;> decide\n  have pow5_mod_100 : 2005 ^ 5 % 100 = 25 := by\n    norm_num [pow_mod, Nat.mod_eq_of_lt, Nat.succ_pos, Nat.zero_lt_one]\n    <;> simp_all [Nat.add_mod, Nat.mul_mod, Nat.pow_succ]\n    <;> norm_num\n    <;> omega\n  have final_result : (2005 ^ 2 + 2005 ^ 5 + 2) % 100 = 52 := by\n    norm_num [pow_add, pow_mul, Nat.add_mod, Nat.mul_mod, Nat.pow_mod, Nat.mod_eq_of_lt]\n    <;> simp_all\n    <;> decide\n    <;> decide\n    <;> decide\n  norm_num [simplify_exp, pow2_mod_100, pow5_mod_100, final_result]\n  <;> rfl\n  <;> rfl\n  <;> rfl\n  <;> rfl\n  <;> rfl\n"}
{"mode": "clear", "steps_before": 16, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 509, "chars_after": 8, "seconds": 0.6, "id": "mathd_numbertheory_92", "proof_after": "theorem mathd_numbertheory_92 (n : \u2115) (h\u2080 : 5 * n % 17 = 8) : n % 17 = 5  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_92 (n : \u2115) (h\u2080 : 5 * n % 17 = 8) : n % 17 = 5  :=  by\n  have h\u2081 : 5 * 7 % 17 = 1 := by\n    simp [h\u2080, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n    <;> norm_num\n    <;> omega\n  have h\u2082 : n % 17 = (7 * 8) % 17 := by\n    have h\u2082 : n % 17 = (7 * 8) % 17 := by\n      omega\n    simp [h\u2082]\n  have h\u2083 : (7 * 8) % 17 = 5 := by\n    have h\u2083 := congr_arg (\u00b7 % 17) h\u2080\n    have h\u2084 := congr_arg (\u00b7 % 17) h\u2081\n    have h\u2085 := congr_arg (\u00b7 % 17) h\u2082\n    norm_num [Nat.mul_mod, Nat.add_mod, Nat.mod_mod] at h\u2083 h\u2084 h\u2085\n    omega\n  simp_all [Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n  <;> omega\n"}
{"mode": "clear", "steps_before": 24, "alphabet": 20, "compiles": false, "id": "induction_sum2kp1npqsqm1", "proof_after": "theorem induction_sum2kp1npqsqm1 (n : \u2115) :\n  \u2211 k in Finset.range n, (2 * k + 3) = (n + 1) ^ 2 - 1  :=  by\n  have base_case : \u2211 k in Finset.range 1, (2 * k + 3) = (1 + 1) ^ 2 - 1 := by\n    simp\n    <;> rfl\n  have inductive_hypothesis : \u2200 m : \u2115, \u2211 k in Finset.range m, (2 * k + 3) = (m + 1) ^ 2 - 1 \u2192 \n    \u2211 k in Finset.range (m + 1), (2 * k + 3) = ((m + 1) + 1) ^ 2 - 1 := by\n    intro m hm\n    simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.add_mul, Nat.mul_one]\n    <;> linarith\n  have induction_step : \u2200 n : \u2115, \u2211 k in Finset.range n, (2 * k + 3) = (n + 1) ^ 2 - 1 := by\n    intro n\n    have h\u2081 := base_case\n    have h\u2082 := inductive_hypothesis\n    induction n with\n    | zero =>\n      simp_all [Finset.sum_range_zero]\n    | succ n ih =>\n      simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_succ]\n      <;> ring_nf\n      <;> omega\n  induction n with\n  | zero =>\n    simpa using base_case\n  | succ n ih =>\n    simpa [Finset.sum_range_succ, Nat.succ_eq_add_one] using inductive_hypothesis n ih\n", "proof_before": "theorem induction_sum2kp1npqsqm1 (n : \u2115) :\n  \u2211 k in Finset.range n, (2 * k + 3) = (n + 1) ^ 2 - 1  :=  by\n  have base_case : \u2211 k in Finset.range 1, (2 * k + 3) = (1 + 1) ^ 2 - 1 := by\n    simp\n    <;> rfl\n  have inductive_hypothesis : \u2200 m : \u2115, \u2211 k in Finset.range m, (2 * k + 3) = (m + 1) ^ 2 - 1 \u2192 \n    \u2211 k in Finset.range (m + 1), (2 * k + 3) = ((m + 1) + 1) ^ 2 - 1 := by\n    intro m hm\n    simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.add_mul, Nat.mul_one]\n    <;> linarith\n  have induction_step : \u2200 n : \u2115, \u2211 k in Finset.range n, (2 * k + 3) = (n + 1) ^ 2 - 1 := by\n    intro n\n    have h\u2081 := base_case\n    have h\u2082 := inductive_hypothesis\n    induction n with\n    | zero =>\n      simp_all [Finset.sum_range_zero]\n    | succ n ih =>\n      simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_succ]\n      <;> ring_nf\n      <;> omega\n  induction n with\n  | zero =>\n    simpa using base_case\n  | succ n ih =>\n    simpa [Finset.sum_range_succ, Nat.succ_eq_add_one] using inductive_hypothesis n ih\n"}
{"mode": "clear", "steps_before": 68, "alphabet": 40, "compiles": false, "id": "imo_1966_p4", "proof_after": "theorem imo_1966_p4 (n : \u2115) (x : \u211d) (h\u2080 : \u2200 k : \u2115, 0 < k \u2192 \u2200 m : \u2124, x \u2260 m * Real.pi / 2 ^ k)\n  (h\u2081 : 0 < n) :\n  (\u2211 k in Finset.Icc 1 n, 1 / Real.sin (2 ^ k * x)) = 1 / Real.tan x - 1 / Real.tan (2 ^ n * x)  :=  by\n  have base_case : (1 / Real.sin (2 * x)) = (1 / Real.tan x) - (1 / Real.tan (2 * x)) := by\n    rw [Real.tan_eq_sin_div_cos]\n    rw [Real.tan_eq_sin_div_cos]\n    by_cases hx : Real.sin x = 0 <;>\n    by_cases hx2 : Real.sin (2 * x) = 0 <;>\n    simp_all [Real.sin_two_mul, Real.cos_two_mul, mul_assoc]\n    <;>\n    field_simp\n    <;>\n    ring\n    <;>\n    simp_all [Real.sin_sq, Real.cos_sq]\n    <;>\n    ring\n    <;>\n    simp_all [Real.sin_sq, Real.cos_sq]\n    <;>\n    ring\n  have inductive_step : \u2200 m : \u2115, 0 < m \u2192 \n    (\u2211 k in Finset.Icc 1 m, 1 / Real.sin (2 ^ k * x)) = (1 / Real.tan x) - (1 / Real.tan (2 ^ m * x)) \u2192\n    (\u2211 k in Finset.Icc 1 (m + 1), 1 / Real.sin (2 ^ k * x)) = (1 / Real.tan x) - (1 / Real.tan (2 ^ (m + 1) * x)) := by\n    intro m hm h\n    have h\u2081' : m + 1 > 0 := by linarith\n    have h\u2082 : \u2211 k in Finset.Icc 1 (m + 1), 1 / Real.sin (2 ^ k * x) = \u2211 k in Finset.Icc 1 m, 1 / Real.sin (2 ^ k * x) + 1 / Real.sin (2 ^ (m + 1) * x) := by\n      rw [Finset.sum_Icc_succ_top]\n      <;> simp [hm]\n    rw [h\u2082, h]\n    have h\u2083 : 1 / Real.sin (2 ^ (m + 1) * x) = 1 / Real.tan (2 ^ m * x) - 1 / Real.tan (2 ^ (m + 1) * x) := by\n      rw [show 2 ^ (m + 1) * x = 2 * (2 ^ m * x) by ring]\n      simp [Real.tan_eq_sin_div_cos, Real.sin_two_mul, Real.cos_two_mul, mul_assoc]\n      by_cases hcos : Real.cos (2 ^ m * x) = 0 <;> by_cases hsin : Real.sin (2 ^ m * x) = 0 <;>\n        by_cases hcos' : Real.cos (2 ^ (m + 1) * x) = 0 <;> by_cases hsin' : Real.sin (2 ^ (m + 1) * x) = 0 <;>\n          field_simp [hcos, hsin, hcos', hsin']\n      <;> ring_nf\n      <;> simp_all [Real.cos_sq, Real.sin_sq]\n      <;> ring_nf\n      <;> nlinarith [Real.sin_sq_add_cos_sq (2 ^ m * x), Real.sin_sq_add_cos_sq (2 ^ (m + 1) * x)]\n    rw [h\u2083]\n    <;> nlinarith\n  have apply_induction : \u2200 n : \u2115, 0 < n \u2192 \n    (\u2211 k in Finset.Icc 1 n, 1 / Real.sin (2 ^ k * x)) = (1 / Real.tan x) - (1 / Real.tan (2 ^ n * x)) := by\n    intro n h\u2081\n    induction n with\n    | zero =>\n      cases h\u2081\n    | succ n ih =>\n      cases n with\n      | zero =>\n        simp_all [Finset.sum_Icc_succ_top, Nat.one_ne_zero, Nat.succ_pos, base_case]\n      | succ n =>\n        simp_all [Finset.sum_Icc_succ_top, Nat.succ_ne_zero, Nat.succ_pos, inductive_step]\n        <;> linarith\n  have final_conclusion : (\u2211 k in Finset.Icc 1 n, 1 / Real.sin (2 ^ k * x)) = (1 / Real.tan x) - (1 / Real.tan (2 ^ n * x)) := by\n    apply apply_induction\n    <;> simp_all\n  have h\u2082 := h\u2081\n  have h\u2083 := base_case\n  have h\u2084 := inductive_step 0\n  have h\u2085 := apply_induction 0\n  have h\u2086 := final_conclusion\n  simp_all\n", "proof_before": "theorem imo_1966_p4 (n : \u2115) (x : \u211d) (h\u2080 : \u2200 k : \u2115, 0 < k \u2192 \u2200 m : \u2124, x \u2260 m * Real.pi / 2 ^ k)\n  (h\u2081 : 0 < n) :\n  (\u2211 k in Finset.Icc 1 n, 1 / Real.sin (2 ^ k * x)) = 1 / Real.tan x - 1 / Real.tan (2 ^ n * x)  :=  by\n  have base_case : (1 / Real.sin (2 * x)) = (1 / Real.tan x) - (1 / Real.tan (2 * x)) := by\n    rw [Real.tan_eq_sin_div_cos]\n    rw [Real.tan_eq_sin_div_cos]\n    by_cases hx : Real.sin x = 0 <;>\n    by_cases hx2 : Real.sin (2 * x) = 0 <;>\n    simp_all [Real.sin_two_mul, Real.cos_two_mul, mul_assoc]\n    <;>\n    field_simp\n    <;>\n    ring\n    <;>\n    simp_all [Real.sin_sq, Real.cos_sq]\n    <;>\n    ring\n    <;>\n    simp_all [Real.sin_sq, Real.cos_sq]\n    <;>\n    ring\n  have inductive_step : \u2200 m : \u2115, 0 < m \u2192 \n    (\u2211 k in Finset.Icc 1 m, 1 / Real.sin (2 ^ k * x)) = (1 / Real.tan x) - (1 / Real.tan (2 ^ m * x)) \u2192\n    (\u2211 k in Finset.Icc 1 (m + 1), 1 / Real.sin (2 ^ k * x)) = (1 / Real.tan x) - (1 / Real.tan (2 ^ (m + 1) * x)) := by\n    intro m hm h\n    have h\u2081' : m + 1 > 0 := by linarith\n    have h\u2082 : \u2211 k in Finset.Icc 1 (m + 1), 1 / Real.sin (2 ^ k * x) = \u2211 k in Finset.Icc 1 m, 1 / Real.sin (2 ^ k * x) + 1 / Real.sin (2 ^ (m + 1) * x) := by\n      rw [Finset.sum_Icc_succ_top]\n      <;> simp [hm]\n    rw [h\u2082, h]\n    have h\u2083 : 1 / Real.sin (2 ^ (m + 1) * x) = 1 / Real.tan (2 ^ m * x) - 1 / Real.tan (2 ^ (m + 1) * x) := by\n      rw [show 2 ^ (m + 1) * x = 2 * (2 ^ m * x) by ring]\n      simp [Real.tan_eq_sin_div_cos, Real.sin_two_mul, Real.cos_two_mul, mul_assoc]\n      by_cases hcos : Real.cos (2 ^ m * x) = 0 <;> by_cases hsin : Real.sin (2 ^ m * x) = 0 <;>\n        by_cases hcos' : Real.cos (2 ^ (m + 1) * x) = 0 <;> by_cases hsin' : Real.sin (2 ^ (m + 1) * x) = 0 <;>\n          field_simp [hcos, hsin, hcos', hsin']\n      <;> ring_nf\n      <;> simp_all [Real.cos_sq, Real.sin_sq]\n      <;> ring_nf\n      <;> nlinarith [Real.sin_sq_add_cos_sq (2 ^ m * x), Real.sin_sq_add_cos_sq (2 ^ (m + 1) * x)]\n    rw [h\u2083]\n    <;> nlinarith\n  have apply_induction : \u2200 n : \u2115, 0 < n \u2192 \n    (\u2211 k in Finset.Icc 1 n, 1 / Real.sin (2 ^ k * x)) = (1 / Real.tan x) - (1 / Real.tan (2 ^ n * x)) := by\n    intro n h\u2081\n    induction n with\n    | zero =>\n      cases h\u2081\n    | succ n ih =>\n      cases n with\n      | zero =>\n        simp_all [Finset.sum_Icc_succ_top, Nat.one_ne_zero, Nat.succ_pos, base_case]\n      | succ n =>\n        simp_all [Finset.sum_Icc_succ_top, Nat.succ_ne_zero, Nat.succ_pos, inductive_step]\n        <;> linarith\n  have final_conclusion : (\u2211 k in Finset.Icc 1 n, 1 / Real.sin (2 ^ k * x)) = (1 / Real.tan x) - (1 / Real.tan (2 ^ n * x)) := by\n    apply apply_induction\n    <;> simp_all\n  have h\u2082 := h\u2081\n  have h\u2083 := base_case\n  have h\u2084 := inductive_step 0\n  have h\u2085 := apply_induction 0\n  have h\u2086 := final_conclusion\n  simp_all\n"}
{"mode": "clear", "steps_before": 56, "alphabet": 34, "compiles": false, "id": "amc12a_2002_p1", "proof_after": "theorem amc12a_2002_p1 (f : \u2102 \u2192 \u2102) (h\u2080 : \u2200 x, f x = (2 * x + 3) * (x - 4) + (2 * x + 3) * (x - 6))\n  (h\u2081 : Fintype (f \u207b\u00b9' {0})) : (\u2211 y in (f \u207b\u00b9' {0}).toFinset, y) = 7 / 2  :=  by\n  have h2 : \u2200 x, f x = (2 * x + 3) * ((x - 4) + (x - 6)) := by\n    intro x\n    rw [h\u2080]\n    ring\n    <;> simp_all\n    <;> ring\n    <;> linarith\n  have h3 : \u2200 x, f x = (2 * x + 3) * (2 * x - 10) := by\n    intro x\n    simp only [h2, mul_add, mul_sub, mul_one, mul_neg, add_mul, sub_mul, neg_mul]\n    ring\n  have h4 : f \u207b\u00b9' {0} = {-3/2, 5} := by\n    apply Set.ext\n    intro x\n    simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_insert_iff]\n    constructor\n    intro h\n    rw [h3] at h\n    have h4 : 2 * x + 3 = 0 \u2228 2 * x - 10 = 0 := by\n      apply eq_zero_or_eq_zero_of_mul_eq_zero\n      exact h\n    cases h4 with\n    | inl h4 =>\n      left\n      apply mul_left_cancel\u2080 (show (2 : \u2102) \u2260 0 by norm_num)\n      linear_combination h4\n    | inr h4 =>\n      right\n      apply mul_left_cancel\u2080 (show (2 : \u2102) \u2260 0 by norm_num)\n      linear_combination h4\n    intro h\n    cases h with\n    | inl h =>\n      rw [h3]\n      rw [h]\n      ring\n    | inr h =>\n      rw [h3]\n      rw [h]\n      ring\n  have h5 : (-3/2 + 5 : \u2102) = 7/2 := by\n    norm_num\n    <;> simp_all\n    <;> norm_num\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> linarith\n  have h6 : (\u2211 y in (f \u207b\u00b9' {0}).toFinset, y) = -3/2 + 5 := by\n    simp_all [Finset.sum_pair, h4]\n    <;> norm_num\n    <;> ring_nf at h5 \u22a2\n    <;> linear_combination h5\n  simp_all [Finset.sum_pair, h4]\n  <;> ring_nf\n  <;> norm_num\n  <;> linarith\n", "proof_before": "theorem amc12a_2002_p1 (f : \u2102 \u2192 \u2102) (h\u2080 : \u2200 x, f x = (2 * x + 3) * (x - 4) + (2 * x + 3) * (x - 6))\n  (h\u2081 : Fintype (f \u207b\u00b9' {0})) : (\u2211 y in (f \u207b\u00b9' {0}).toFinset, y) = 7 / 2  :=  by\n  have h2 : \u2200 x, f x = (2 * x + 3) * ((x - 4) + (x - 6)) := by\n    intro x\n    rw [h\u2080]\n    ring\n    <;> simp_all\n    <;> ring\n    <;> linarith\n  have h3 : \u2200 x, f x = (2 * x + 3) * (2 * x - 10) := by\n    intro x\n    simp only [h2, mul_add, mul_sub, mul_one, mul_neg, add_mul, sub_mul, neg_mul]\n    ring\n  have h4 : f \u207b\u00b9' {0} = {-3/2, 5} := by\n    apply Set.ext\n    intro x\n    simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_insert_iff]\n    constructor\n    intro h\n    rw [h3] at h\n    have h4 : 2 * x + 3 = 0 \u2228 2 * x - 10 = 0 := by\n      apply eq_zero_or_eq_zero_of_mul_eq_zero\n      exact h\n    cases h4 with\n    | inl h4 =>\n      left\n      apply mul_left_cancel\u2080 (show (2 : \u2102) \u2260 0 by norm_num)\n      linear_combination h4\n    | inr h4 =>\n      right\n      apply mul_left_cancel\u2080 (show (2 : \u2102) \u2260 0 by norm_num)\n      linear_combination h4\n    intro h\n    cases h with\n    | inl h =>\n      rw [h3]\n      rw [h]\n      ring\n    | inr h =>\n      rw [h3]\n      rw [h]\n      ring\n  have h5 : (-3/2 + 5 : \u2102) = 7/2 := by\n    norm_num\n    <;> simp_all\n    <;> norm_num\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> linarith\n  have h6 : (\u2211 y in (f \u207b\u00b9' {0}).toFinset, y) = -3/2 + 5 := by\n    simp_all [Finset.sum_pair, h4]\n    <;> norm_num\n    <;> ring_nf at h5 \u22a2\n    <;> linear_combination h5\n  simp_all [Finset.sum_pair, h4]\n  <;> ring_nf\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 4, "alphabet": 13, "compiles": true, "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 83, "chars_after": 37, "seconds": 74.9, "id": "aime_1984_p15", "proof_after": "theorem aime_1984_p15 (x y z w : \u211d)\n    (h\u2080 :\n      x ^ 2 / (2 ^ 2 - 1) + y ^ 2 / (2 ^ 2 - 3 ^ 2) + z ^ 2 / (2 ^ 2 - 5 ^ 2) +\n          w ^ 2 / (2 ^ 2 - 7 ^ 2) =\n        1)\n    (h\u2081 :\n      x ^ 2 / (4 ^ 2 - 1) + y ^ 2 / (4 ^ 2 - 3 ^ 2) + z ^ 2 / (4 ^ 2 - 5 ^ 2) +\n          w ^ 2 / (4 ^ 2 - 7 ^ 2) =\n        1)\n    (h\u2082 :\n      x ^ 2 / (6 ^ 2 - 1) + y ^ 2 / (6 ^ 2 - 3 ^ 2) + z ^ 2 / (6 ^ 2 - 5 ^ 2) +\n          w ^ 2 / (6 ^ 2 - 7 ^ 2) =\n        1)\n    (h\u2083 :\n      x ^ 2 / (8 ^ 2 - 1) + y ^ 2 / (8 ^ 2 - 3 ^ 2) + z ^ 2 / (8 ^ 2 - 5 ^ 2) +\n          w ^ 2 / (8 ^ 2 - 7 ^ 2) =\n        1) :\n    x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2 = 36  :=  by\n  norm_num at h\u2080 h\u2081 h\u2082 h\u2083\n  linarith\n", "proof_before": "theorem aime_1984_p15 (x y z w : \u211d)\n    (h\u2080 :\n      x ^ 2 / (2 ^ 2 - 1) + y ^ 2 / (2 ^ 2 - 3 ^ 2) + z ^ 2 / (2 ^ 2 - 5 ^ 2) +\n          w ^ 2 / (2 ^ 2 - 7 ^ 2) =\n        1)\n    (h\u2081 :\n      x ^ 2 / (4 ^ 2 - 1) + y ^ 2 / (4 ^ 2 - 3 ^ 2) + z ^ 2 / (4 ^ 2 - 5 ^ 2) +\n          w ^ 2 / (4 ^ 2 - 7 ^ 2) =\n        1)\n    (h\u2082 :\n      x ^ 2 / (6 ^ 2 - 1) + y ^ 2 / (6 ^ 2 - 3 ^ 2) + z ^ 2 / (6 ^ 2 - 5 ^ 2) +\n          w ^ 2 / (6 ^ 2 - 7 ^ 2) =\n        1)\n    (h\u2083 :\n      x ^ 2 / (8 ^ 2 - 1) + y ^ 2 / (8 ^ 2 - 3 ^ 2) + z ^ 2 / (8 ^ 2 - 5 ^ 2) +\n          w ^ 2 / (8 ^ 2 - 7 ^ 2) =\n        1) :\n    x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2 = 36  :=  by\n  norm_num at h\u2080 h\u2081 h\u2082 h\u2083\n  linarith\n  <;> ring_nf at h\u2080 h\u2081 h\u2082 h\u2083 \u22a2\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 17, "alphabet": 21, "compiles": false, "id": "mathd_algebra_410", "proof_after": "theorem mathd_algebra_410 (x y : \u211d) (h\u2080 : y = x ^ 2 - 6 * x + 13) : 4 \u2264 y  :=  by\n  have h\u2081 : y = (x - 3)^2 + 4 := by\n    rw [h\u2080]\n    have h\u2081 : x ^ 2 - 6 * x + 13 = (x - 3) ^ 2 + 4 := by\n      ring\n    rw [h\u2081]\n  have h\u2082 : \u2200 (x : \u211d), (x - 3)^2 + 4 \u2265 4 := by\n    intro x\n    nlinarith [sq_nonneg (x - 3)]\n  have h\u2083 : 4 \u2264 y := by\n    have h\u2083 := h\u2082 0\n    have h\u2084 := h\u2082 1\n    have h\u2085 := h\u2082 (-1)\n    have h\u2086 := h\u2082 2\n    have h\u2087 := h\u2082 (-2)\n    simp_all only [sq, mul_one, mul_neg, mul_zero, sub_zero, sub_neg_eq_add, add_zero,\n      add_right_neg, add_zero]\n    <;> nlinarith\n  linarith [h\u2082 x, h\u2082 3, h\u2082 0, h\u2082 1, h\u2082 2, h\u2082 4, h\u2082 5, h\u2082 6]\n", "proof_before": "theorem mathd_algebra_410 (x y : \u211d) (h\u2080 : y = x ^ 2 - 6 * x + 13) : 4 \u2264 y  :=  by\n  have h\u2081 : y = (x - 3)^2 + 4 := by\n    rw [h\u2080]\n    have h\u2081 : x ^ 2 - 6 * x + 13 = (x - 3) ^ 2 + 4 := by\n      ring\n    rw [h\u2081]\n  have h\u2082 : \u2200 (x : \u211d), (x - 3)^2 + 4 \u2265 4 := by\n    intro x\n    nlinarith [sq_nonneg (x - 3)]\n  have h\u2083 : 4 \u2264 y := by\n    have h\u2083 := h\u2082 0\n    have h\u2084 := h\u2082 1\n    have h\u2085 := h\u2082 (-1)\n    have h\u2086 := h\u2082 2\n    have h\u2087 := h\u2082 (-2)\n    simp_all only [sq, mul_one, mul_neg, mul_zero, sub_zero, sub_neg_eq_add, add_zero,\n      add_right_neg, add_zero]\n    <;> nlinarith\n  linarith [h\u2082 x, h\u2082 3, h\u2082 0, h\u2082 1, h\u2082 2, h\u2082 4, h\u2082 5, h\u2082 6]\n"}
{"mode": "clear", "steps_before": 55, "alphabet": 24, "compiles": false, "id": "mathd_algebra_421", "proof_after": "theorem mathd_algebra_421 (a b c d : \u211d) (h\u2080 : b = a ^ 2 + 4 * a + 6)\n  (h\u2081 : b = 1 / 2 * a ^ 2 + a + 6) (h\u2082 : d = c ^ 2 + 4 * c + 6) (h\u2083 : d = 1 / 2 * c ^ 2 + c + 6)\n  (h\u2084 : a < c) : c - a = 6  :=  by\n  have h\u2085 : a ^ 2 + 4 * a + 6 = 1 / 2 * a ^ 2 + a + 6 := by\n    have : a ^ 2 + 4 * a + 6 = 1 / 2 * a ^ 2 + a + 6 := by\n      rw [h\u2080] at h\u2081\n      linarith\n    nlinarith\n  have h\u2086 : a ^ 2 + 4 * a = 1 / 2 * a ^ 2 + a := by\n    have h\u2086 : a ^ 2 + 4 * a = 1 / 2 * a ^ 2 + a := by\n      linarith\n    linarith\n  have h\u2087 : 2 * a ^ 2 + 8 * a = a ^ 2 + 2 * a := by\n    have h\u2087 : a ^ 2 + 4 * a = 1 / 2 * a ^ 2 + a := by linarith\n    have h\u2088 : 2 * a ^ 2 + 8 * a = a ^ 2 + 2 * a := by\n      linarith\n    linarith\n  have h\u2088 : a ^ 2 + 6 * a = 0 := by\n    have h\u2088 : a ^ 2 + 6 * a = 0 := by\n      nlinarith\n    exact h\u2088\n  have h\u2089 : a * (a + 6) = 0 := by\n    have h\u2089 : a = 0 \u2228 a = -6 := by\n      apply or_iff_not_imp_left.mpr\n      intro h\n      apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h)\n      nlinarith\n    cases' h\u2089 with h\u2089 h\u2089\n    <;> simp_all\n    <;> nlinarith\n  have h\u2081\u2080 : a = 0 \u2228 a = -6 := by\n    apply or_iff_not_imp_right.mpr\n    intro h\n    apply Eq.symm\n    apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h)\n    nlinarith\n  have h\u2081\u2081 : a = -6 \u2227 c = 0 := by\n    have h\u2081\u2081 : a = -6 := by\n      cases' h\u2081\u2080 with h\u2081\u2080 h\u2081\u2080\n      \u00b7 \n        rw [h\u2081\u2080] at h\u2085\n        nlinarith\n      \u00b7 \n        rw [h\u2081\u2080] at h\u2085\n        nlinarith\n    have h\u2081\u2082 : c = 0 := by\n      rw [h\u2081\u2081] at h\u2084\n      nlinarith\n    exact \u27e8h\u2081\u2081, h\u2081\u2082\u27e9\n  have h\u2081\u2082 : c - a = 6 := by\n    cases' h\u2081\u2080 with h\u2081\u2080 h\u2081\u2080 <;> simp_all [h\u2081\u2081]\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n  norm_num at *\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_421 (a b c d : \u211d) (h\u2080 : b = a ^ 2 + 4 * a + 6)\n  (h\u2081 : b = 1 / 2 * a ^ 2 + a + 6) (h\u2082 : d = c ^ 2 + 4 * c + 6) (h\u2083 : d = 1 / 2 * c ^ 2 + c + 6)\n  (h\u2084 : a < c) : c - a = 6  :=  by\n  have h\u2085 : a ^ 2 + 4 * a + 6 = 1 / 2 * a ^ 2 + a + 6 := by\n    have : a ^ 2 + 4 * a + 6 = 1 / 2 * a ^ 2 + a + 6 := by\n      rw [h\u2080] at h\u2081\n      linarith\n    nlinarith\n  have h\u2086 : a ^ 2 + 4 * a = 1 / 2 * a ^ 2 + a := by\n    have h\u2086 : a ^ 2 + 4 * a = 1 / 2 * a ^ 2 + a := by\n      linarith\n    linarith\n  have h\u2087 : 2 * a ^ 2 + 8 * a = a ^ 2 + 2 * a := by\n    have h\u2087 : a ^ 2 + 4 * a = 1 / 2 * a ^ 2 + a := by linarith\n    have h\u2088 : 2 * a ^ 2 + 8 * a = a ^ 2 + 2 * a := by\n      linarith\n    linarith\n  have h\u2088 : a ^ 2 + 6 * a = 0 := by\n    have h\u2088 : a ^ 2 + 6 * a = 0 := by\n      nlinarith\n    exact h\u2088\n  have h\u2089 : a * (a + 6) = 0 := by\n    have h\u2089 : a = 0 \u2228 a = -6 := by\n      apply or_iff_not_imp_left.mpr\n      intro h\n      apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h)\n      nlinarith\n    cases' h\u2089 with h\u2089 h\u2089\n    <;> simp_all\n    <;> nlinarith\n  have h\u2081\u2080 : a = 0 \u2228 a = -6 := by\n    apply or_iff_not_imp_right.mpr\n    intro h\n    apply Eq.symm\n    apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h)\n    nlinarith\n  have h\u2081\u2081 : a = -6 \u2227 c = 0 := by\n    have h\u2081\u2081 : a = -6 := by\n      cases' h\u2081\u2080 with h\u2081\u2080 h\u2081\u2080\n      \u00b7 \n        rw [h\u2081\u2080] at h\u2085\n        nlinarith\n      \u00b7 \n        rw [h\u2081\u2080] at h\u2085\n        nlinarith\n    have h\u2081\u2082 : c = 0 := by\n      rw [h\u2081\u2081] at h\u2084\n      nlinarith\n    exact \u27e8h\u2081\u2081, h\u2081\u2082\u27e9\n  have h\u2081\u2082 : c - a = 6 := by\n    cases' h\u2081\u2080 with h\u2081\u2080 h\u2081\u2080 <;> simp_all [h\u2081\u2081]\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n  norm_num at *\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 29, "alphabet": 16, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 579, "chars_after": 12, "seconds": 11.4, "id": "mathd_algebra_37", "proof_after": "theorem mathd_algebra_37 (x y : \u211d) (h\u2080 : x + y = 7) (h\u2081 : 3 * x + y = 45) : x ^ 2 - y ^ 2 = 217  :=  by\n  nlinarith\n", "proof_before": "theorem mathd_algebra_37 (x y : \u211d) (h\u2080 : x + y = 7) (h\u2081 : 3 * x + y = 45) : x ^ 2 - y ^ 2 = 217  :=  by\n  have h\u2082 : 2 * x = 38 := by\n    have h\u2080' : y = 7 - x := by linarith\n    have h\u2081' : 3 * x + (7 - x) = 45 := by rw [h\u2080'] at h\u2081; exact h\u2081\n    have h\u2082 : 2 * x = 38 := by linarith\n    exact h\u2082\n  have h\u2083 : x = 19 := by\n    have h\u2083 : x = 19 := by\n      apply Eq.symm\n      linarith\n    simp_all\n    <;> linarith\n  have h\u2084 : y = -12 := by\n    have h\u2084 : y = -12 := by\n      linarith\n    exact h\u2084\n  have h\u2085 : x^2 - y^2 = 217 := by\n    have h\u2085 : x ^ 2 - y ^ 2 = 217 := by\n      rw [h\u2083, h\u2084]\n      norm_num\n    exact h\u2085\n  rw [h\u2083, h\u2084]\n  <;> norm_num\n  <;> linarith\n  <;> ring\n  <;> nlinarith\n"}
{"mode": "clear", "steps_before": 24, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 20, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "fixpoint": true, "chars_before": 881, "chars_after": 728, "seconds": 125.6, "id": "mathd_numbertheory_156", "proof_after": "theorem mathd_numbertheory_156 (n : \u2115) (h\u2080 : 0 < n) : Nat.gcd (n + 7) (2 * n + 1) \u2264 13  :=  by\n  have h\u2081 : Nat.gcd (n + 7) (2 * n + 1) \u2223 n + 7 := by\n    exact Nat.gcd_dvd_left (n + 7) (2 * n + 1)\n  have h\u2082 : Nat.gcd (n + 7) (2 * n + 1) \u2223 2 * n + 1 := by\n    apply Nat.gcd_dvd_right\n  have h\u2083 : Nat.gcd (n + 7) (2 * n + 1) \u2223 2 * (n + 7) - (2 * n + 1) := by\n    exact dvd_sub' (dvd_mul_of_dvd_right h\u2081 2) h\u2082\n  have h\u2084 : 2 * (n + 7) - (2 * n + 1) = 13 := by\n    omega\n  have h\u2085 : Nat.gcd (n + 7) (2 * n + 1) \u2223 13 := by\n    simp_all [Nat.dvd_iff_mod_eq_zero]\n    <;> omega\n  have h\u2086 : Nat.gcd (n + 7) (2 * n + 1) \u2264 13 := by\n    have h\u2086 : Nat.gcd (n + 7) (2 * n + 1) \u2223 13 := by assumption\n    have h\u2087 : Nat.gcd (n + 7) (2 * n + 1) \u2264 13 := by\n      apply Nat.le_of_dvd\n      \u00b7 norm_num\n      \u00b7 exact h\u2086\n    exact h\u2087\n  assumption\n", "proof_before": "theorem mathd_numbertheory_156 (n : \u2115) (h\u2080 : 0 < n) : Nat.gcd (n + 7) (2 * n + 1) \u2264 13  :=  by\n  have h\u2081 : Nat.gcd (n + 7) (2 * n + 1) \u2223 n + 7 := by\n    exact Nat.gcd_dvd_left (n + 7) (2 * n + 1)\n  have h\u2082 : Nat.gcd (n + 7) (2 * n + 1) \u2223 2 * n + 1 := by\n    apply Nat.gcd_dvd_right\n    <;> simp_all [Nat.gcd_comm]\n    <;> ring_nf\n    <;> omega\n  have h\u2083 : Nat.gcd (n + 7) (2 * n + 1) \u2223 2 * (n + 7) - (2 * n + 1) := by\n    exact dvd_sub' (dvd_mul_of_dvd_right h\u2081 2) h\u2082\n  have h\u2084 : 2 * (n + 7) - (2 * n + 1) = 13 := by\n    simp_all only [mul_add, mul_sub, mul_one, mul_neg, mul_comm, mul_left_comm, mul_assoc]\n    omega\n  have h\u2085 : Nat.gcd (n + 7) (2 * n + 1) \u2223 13 := by\n    simp_all [Nat.dvd_iff_mod_eq_zero]\n    <;> omega\n  have h\u2086 : Nat.gcd (n + 7) (2 * n + 1) \u2264 13 := by\n    have h\u2086 : Nat.gcd (n + 7) (2 * n + 1) \u2223 13 := by assumption\n    have h\u2087 : Nat.gcd (n + 7) (2 * n + 1) \u2264 13 := by\n      apply Nat.le_of_dvd\n      \u00b7 norm_num\n      \u00b7 exact h\u2086\n    exact h\u2087\n  assumption\n"}
{"mode": "clear", "steps_before": 42, "alphabet": 31, "compiles": false, "id": "mathd_algebra_224", "proof_after": "theorem mathd_algebra_224 (S : Finset \u2115)\n  (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 Real.sqrt n < 7 / 2 \u2227 2 < Real.sqrt n) : S.card = 8  :=  by\n  have h1 : \u2200 n \u2208 S, n > 4 \u2227 n < 13 := by\n    intro n h\u2081\n    simp only [h\u2080, Finset.mem_filter, Finset.mem_univ, true_and_iff] at h\u2081\n    have h\u2082 := h\u2081.1\n    have h\u2083 := h\u2081.2\n    have h\u2084 : (4 : \u211d) < n := by\n      nlinarith [sq_sqrt (by positivity : (0 : \u211d) \u2264 n)]\n    have h\u2085 : (n : \u211d) < 13 := by\n      nlinarith [sq_sqrt (by positivity : (0 : \u211d) \u2264 n)]\n    exact \u27e8by exact_mod_cast h\u2084, by exact_mod_cast h\u2085\u27e9\n  have h2 : \u2200 n : \u2115, 4 < n \u2227 n < 13 \u2192 n \u2208 S := by\n    intro n hn\n    have h2 : n \u2208 S \u2194 Real.sqrt n < 7 / 2 \u2227 2 < Real.sqrt n := h\u2080 n\n    simp_all only [Finset.mem_filter, Finset.mem_univ, true_and_iff, and_imp]\n    norm_num\n    <;> rcases hn with \u27e8h3, h4\u27e9\n    <;> interval_cases n <;> norm_num [Real.sqrt_lt, Real.lt_sqrt] at * <;> linarith\n  have h3 : S = {5, 6, 7, 8, 9, 10, 11, 12} := by\n    ext n\n    simp only [Finset.mem_insert, Finset.mem_singleton, h\u2080, true_and, and_true] at *\n    constructor\n    intro h\n    have h3 := h1 n h\n    have h4 := h2 n\n    have h5 : n < 13 := h3.2\n    have h6 : 4 < n := h3.1\n    interval_cases n <;> simp_all\n    intro h\n    have h3 : n = 5 \u2228 n = 6 \u2228 n = 7 \u2228 n = 8 \u2228 n = 9 \u2228 n = 10 \u2228 n = 11 \u2228 n = 12 := by simp_all\n    rcases h3 with (rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl) <;> simp_all\n    <;> norm_num [Real.sqrt_lt, Real.lt_sqrt]\n    <;> linarith\n  have h4 : S.card = 8 := by\n    simp_all only [Finset.card_insert_of_not_mem, Finset.mem_singleton, Finset.card_singleton,\n      Finset.card_empty, Nat.cast_ofNat]\n    <;> norm_num\n    <;> aesop\n  simpa [h3, h4] using h2 5 \u27e8by norm_num, by norm_num\u27e9\n", "proof_before": "theorem mathd_algebra_224 (S : Finset \u2115)\n  (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 Real.sqrt n < 7 / 2 \u2227 2 < Real.sqrt n) : S.card = 8  :=  by\n  have h1 : \u2200 n \u2208 S, n > 4 \u2227 n < 13 := by\n    intro n h\u2081\n    simp only [h\u2080, Finset.mem_filter, Finset.mem_univ, true_and_iff] at h\u2081\n    have h\u2082 := h\u2081.1\n    have h\u2083 := h\u2081.2\n    have h\u2084 : (4 : \u211d) < n := by\n      nlinarith [sq_sqrt (by positivity : (0 : \u211d) \u2264 n)]\n    have h\u2085 : (n : \u211d) < 13 := by\n      nlinarith [sq_sqrt (by positivity : (0 : \u211d) \u2264 n)]\n    exact \u27e8by exact_mod_cast h\u2084, by exact_mod_cast h\u2085\u27e9\n  have h2 : \u2200 n : \u2115, 4 < n \u2227 n < 13 \u2192 n \u2208 S := by\n    intro n hn\n    have h2 : n \u2208 S \u2194 Real.sqrt n < 7 / 2 \u2227 2 < Real.sqrt n := h\u2080 n\n    simp_all only [Finset.mem_filter, Finset.mem_univ, true_and_iff, and_imp]\n    norm_num\n    <;> rcases hn with \u27e8h3, h4\u27e9\n    <;> interval_cases n <;> norm_num [Real.sqrt_lt, Real.lt_sqrt] at * <;> linarith\n  have h3 : S = {5, 6, 7, 8, 9, 10, 11, 12} := by\n    ext n\n    simp only [Finset.mem_insert, Finset.mem_singleton, h\u2080, true_and, and_true] at *\n    constructor\n    intro h\n    have h3 := h1 n h\n    have h4 := h2 n\n    have h5 : n < 13 := h3.2\n    have h6 : 4 < n := h3.1\n    interval_cases n <;> simp_all\n    intro h\n    have h3 : n = 5 \u2228 n = 6 \u2228 n = 7 \u2228 n = 8 \u2228 n = 9 \u2228 n = 10 \u2228 n = 11 \u2228 n = 12 := by simp_all\n    rcases h3 with (rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl) <;> simp_all\n    <;> norm_num [Real.sqrt_lt, Real.lt_sqrt]\n    <;> linarith\n  have h4 : S.card = 8 := by\n    simp_all only [Finset.card_insert_of_not_mem, Finset.mem_singleton, Finset.card_singleton,\n      Finset.card_empty, Nat.cast_ofNat]\n    <;> norm_num\n    <;> aesop\n  simpa [h3, h4] using h2 5 \u27e8by norm_num, by norm_num\u27e9\n"}
{"mode": "clear", "steps_before": 26, "alphabet": 13, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 9, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 7, "fixpoint": true, "chars_before": 570, "chars_after": 203, "seconds": 230.2, "id": "mathd_algebra_59", "proof_after": "theorem mathd_algebra_59 (b : \u211d) (h\u2080 : (4 : \u211d) ^ b + 2 ^ 3 = 12) : b = 1  :=  by\n  have h\u2083 : (4 : \u211d) ^ b = 4 := by\n    linarith\n  have h\u2085 : b = 1 := by\n    apply_fun fun x => logb 4 x at h\u2083\n    simp_all [logb_rpow, logb_one]\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  omega\n", "proof_before": "theorem mathd_algebra_59 (b : \u211d) (h\u2080 : (4 : \u211d) ^ b + 2 ^ 3 = 12) : b = 1  :=  by\n  have h\u2081 : 2 ^ 3 = 8 := by\n    norm_num\n    <;> simp_all [pow_two]\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  have h\u2082 : (4 : \u211d) ^ b + 8 = 12 := by\n    simp_all [h\u2081]\n    <;> linarith\n  have h\u2083 : (4 : \u211d) ^ b = 4 := by\n    have h\u2083 : (4 : \u211d) ^ b = 4 := by\n      linarith\n    linarith\n  have h\u2084 : (4 : \u211d) ^ 1 = 4 := by\n    norm_num\n    <;> simp_all [pow_one]\n    <;> norm_num\n    <;> linarith\n  have h\u2085 : b = 1 := by\n    apply_fun fun x => logb 4 x at h\u2083\n    simp_all [logb_rpow, logb_one]\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  apply Eq.symm\n  simp_all\n"}
{"mode": "clear", "steps_before": 24, "alphabet": 21, "compiles": false, "id": "numbertheory_2dvd4expn", "proof_after": "theorem numbertheory_2dvd4expn (n : \u2115) (h\u2080 : n \u2260 0) : 2 \u2223 4 ^ n  :=  by\n  have step1 : 4 ^ n = (2^2) ^ n := by\n    induction n with\n    | zero =>\n      exfalso\n      simp_all\n    | succ n ih =>\n      simp_all [pow_succ]\n      <;> ring\n      <;> simp_all [pow_succ]\n      <;> ring\n  have step2 : (2^2) ^ n = 2^(2 * n) := by\n    simp_all [pow_mul]\n    <;> simp_all [pow_mul]\n    <;> omega\n  have step3 : 2 \u2223 2^(2 * n) := by\n    apply Nat.dvd_of_mod_eq_zero\n    norm_num\n    rw [\u2190 Nat.mod_add_div (2 ^ (2 * n)) 2]\n    simp [Nat.pow_mod, h\u2080, step1, step2]\n  have step4 : 2 \u2223 4 ^ n := by\n    rw [step1]\n    rw [step2]\n    exact step3\n  apply step4\n", "proof_before": "theorem numbertheory_2dvd4expn (n : \u2115) (h\u2080 : n \u2260 0) : 2 \u2223 4 ^ n  :=  by\n  have step1 : 4 ^ n = (2^2) ^ n := by\n    induction n with\n    | zero =>\n      exfalso\n      simp_all\n    | succ n ih =>\n      simp_all [pow_succ]\n      <;> ring\n      <;> simp_all [pow_succ]\n      <;> ring\n  have step2 : (2^2) ^ n = 2^(2 * n) := by\n    simp_all [pow_mul]\n    <;> simp_all [pow_mul]\n    <;> omega\n  have step3 : 2 \u2223 2^(2 * n) := by\n    apply Nat.dvd_of_mod_eq_zero\n    norm_num\n    rw [\u2190 Nat.mod_add_div (2 ^ (2 * n)) 2]\n    simp [Nat.pow_mod, h\u2080, step1, step2]\n  have step4 : 2 \u2223 4 ^ n := by\n    rw [step1]\n    rw [step2]\n    exact step3\n  apply step4\n"}
{"mode": "clear", "steps_before": 19, "alphabet": 20, "compiles": true, "steps_after": 17, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "fixpoint": true, "chars_before": 560, "chars_after": 470, "seconds": 91.5, "id": "imo_1964_p1_2", "proof_after": "theorem imo_1964_p1_2 (n : \u2115) : \u00ac7 \u2223 2 ^ n + 1  :=  by\n  have h : 2 ^ n % 7 = 1 \u2228 2 ^ n % 7 = 2 \u2228 2 ^ n % 7 = 4 := by\n    induction n with\n    | zero => simp\n    | succ n ih =>\n      cases ih with\n      | inl h =>\n        rw [Nat.pow_succ]\n        norm_num [h, Nat.mul_mod, Nat.add_mod]\n      | inr h =>\n        cases h with\n        | inl h =>\n          rw [Nat.pow_succ]\n          norm_num [h, Nat.mul_mod, Nat.add_mod]\n        | inr h =>\n          rw [Nat.pow_succ]\n          norm_num [h, Nat.mul_mod, Nat.add_mod]\n  omega\n", "proof_before": "theorem imo_1964_p1_2 (n : \u2115) : \u00ac7 \u2223 2 ^ n + 1  :=  by\n  rw [Nat.dvd_iff_mod_eq_zero]\n  have h : 2 ^ n % 7 = 1 \u2228 2 ^ n % 7 = 2 \u2228 2 ^ n % 7 = 4 := by\n    induction n with\n    | zero => simp\n    | succ n ih =>\n      cases ih with\n      | inl h =>\n        rw [Nat.pow_succ]\n        norm_num [h, Nat.mul_mod, Nat.add_mod]\n      | inr h =>\n        cases h with\n        | inl h =>\n          rw [Nat.pow_succ]\n          norm_num [h, Nat.mul_mod, Nat.add_mod]\n        | inr h =>\n          rw [Nat.pow_succ]\n          norm_num [h, Nat.mul_mod, Nat.add_mod]\n  rcases h with (h | h | h) <;> simp [h, Nat.add_mod, Nat.mul_mod]\n"}
{"mode": "clear", "steps_before": 21, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 674, "chars_after": 20, "seconds": 39.3, "id": "mathd_algebra_89", "proof_after": "theorem mathd_algebra_89 (b : \u211d) (h\u2080 : b \u2260 0) :\n  (7 * b ^ 3) ^ 2 * (4 * b ^ 2) ^ (-(3 : \u2124)) = 49 / 64  :=  by\n  field_simp\n  ring\n", "proof_before": "theorem mathd_algebra_89 (b : \u211d) (h\u2080 : b \u2260 0) :\n  (7 * b ^ 3) ^ 2 * (4 * b ^ 2) ^ (-(3 : \u2124)) = 49 / 64  :=  by\n  have step1 : (7 * b ^ 3) ^ 2 = 49 * b ^ 6 := by\n    simp [pow_mul, mul_pow, h\u2080, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  have step2 : (4 * b ^ 2) ^ (-3 : \u2124) = (1 / 64) * b ^ (-6 : \u2124) := by\n    simp_all [zpow_neg, zpow_ofNat, mul_inv_rev, inv_inv]\n    field_simp\n    ring_nf\n    <;> norm_num\n    <;> linarith\n  have step3 : 49 * b ^ 6 * ((1 / 64) * b ^ (-6 : \u2124)) = 49 / 64 := by\n    field_simp [h\u2080, zpow_neg, zpow_ofNat, mul_comm, mul_assoc, mul_left_comm]\n    <;> ring\n    <;> norm_cast\n    <;> simp_all [pow_mul]\n    <;> norm_num\n    <;> linarith\n  norm_num at *\n  simp_all [h\u2080, zpow_neg, zpow_ofNat, mul_assoc]\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 44, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 993, "chars_after": 8, "seconds": 21.0, "id": "mathd_numbertheory_284", "proof_after": "theorem mathd_numbertheory_284 (a b : \u2115) (h\u2080 : 1 \u2264 a \u2227 a \u2264 9 \u2227 b \u2264 9)\n  (h\u2081 : 10 * a + b = 2 * (a + b)) : 10 * a + b = 18  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_284 (a b : \u2115) (h\u2080 : 1 \u2264 a \u2227 a \u2264 9 \u2227 b \u2264 9)\n  (h\u2081 : 10 * a + b = 2 * (a + b)) : 10 * a + b = 18  :=  by\n  have h\u2081 : 10 * a + b = 2 * (a + b) := by\n    apply h\u2081\n  have h\u2082 : 10 * a + b = 2 * a + 2 * b := by\n    have h\u2081 := h\u2081\n    simp_all only [mul_add, mul_one, mul_comm, mul_left_comm, mul_assoc]\n    omega\n  have h\u2083 : 10 * a - 2 * a + b - 2 * b = 0 := by\n    have h\u2083 : a \u2264 9 := by linarith\n    have h\u2084 : b \u2264 9 := by linarith\n    omega\n  have h\u2084 : 8 * a - b = 0 := by\n    have h\u2084 : a \u2264 9 := by linarith\n    have h\u2085 : b \u2264 9 := by linarith\n    have h\u2086 : 8 * a - b = 0 := by omega\n    exact h\u2086\n  have h\u2085 : b = 8 * a := by\n    omega\n  have h\u2086 : 0 \u2264 8 * a \u2227 8 * a \u2264 9 := by\n    constructor <;>\n    omega\n    <;>\n    linarith\n    <;>\n    linarith\n  have h\u2087 : a = 1 := by\n    simp_all only [succ.injEq, zero_add, zero_sub, Nat.cast_zero, mul_zero, zero_mul, add_zero,\n      sub_zero]\n    omega\n  have h\u2088 : b = 8 := by\n    have h\u2088 : b = 8 := by\n      have h\u2088 : b = 8 * a := by linarith\n      rw [h\u2087] at h\u2088\n      linarith\n    exact h\u2088\n  have h\u2089 : 10 * a + b = 18 := by\n    simp_all\n    <;> omega\n  simp_all\n"}
{"mode": "clear", "steps_before": 22, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 832, "chars_after": 7, "seconds": 9.9, "id": "mathd_algebra_568", "proof_after": "theorem mathd_algebra_568 (a : \u211d) :\n    (a - 1) * (a + 1) * (a + 2) - (a - 2) * (a + 1) = a ^ 3 + a ^ 2  :=  by\n  ring\n", "proof_before": "theorem mathd_algebra_568 (a : \u211d) :\n    (a - 1) * (a + 1) * (a + 2) - (a - 2) * (a + 1) = a ^ 3 + a ^ 2  :=  by\n  have step1 : (a - 1) * (a + 1) = a^2 - 1 := by\n    ring\n    <;> ring\n    <;> ring\n    <;> ring\n  have step2 : (a^2 - 1) * (a + 2) = a^3 + 2*a^2 - a - 2 := by\n    ring_nf\n    <;> linarith [step1]\n    <;> linarith [step1]\n    <;> linarith [step1]\n  have step3 : (a - 2) * (a + 1) = a^2 - a - 2 := by\n    have h1 : (a - 2) * (a + 1) = a^2 - a - 2 := by\n      ring_nf\n    exact h1\n  have step4 : (a^3 + 2*a^2 - a - 2) - (a^2 - a - 2) = a^3 + a^2 := by\n    simp_all only [mul_add, mul_sub, mul_one, mul_neg, mul_assoc, sub_neg_eq_add, sub_eq_add_neg,\n      add_assoc, add_left_comm, add_right_comm]\n    ring\n  have step5 : (a - 1) * (a + 1) * (a + 2) - (a - 2) * (a + 1) = a^3 + a^2 := by\n    nlinarith [step1, step2, step3, step4]\n    <;> ring_nf at step1 step2 step3 step4 \u22a2\n    <;> nlinarith [step1, step2, step3, step4]\n  linarith\n"}
{"mode": "clear", "steps_before": 78, "alphabet": 35, "compiles": false, "id": "numbertheory_aneqprodakp4_anmsqrtanp1eq2", "proof_after": "theorem numbertheory_aneqprodakp4_anmsqrtanp1eq2 (a : \u2115 \u2192 \u211d) (h\u2080 : a 0 = 1)\n  (h\u2081 : \u2200 n, a (n + 1) = (\u220f k in Finset.range (n + 1), a k) + 4) :\n  \u2200 n \u2265 1, a n - Real.sqrt (a (n + 1)) = 2  :=  by\n  have a1_eq : a 1 = 5 := by\n    simp_all [Finset.prod_range_succ, Finset.prod_range_zero, Nat.cast_zero, Nat.cast_one]\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n  have a2_eq : a 2 = 9 := by\n    have h\u2082 := h\u2081 0\n    have h\u2083 := h\u2081 1\n    simp [Finset.prod_range_succ, h\u2080] at h\u2082 h\u2083 \u22a2\n    norm_num at h\u2082 h\u2083 \u22a2\n    <;> nlinarith\n  have a3_eq : a 3 = 49 := by\n    norm_num [Finset.prod_range_succ, Finset.prod_range_one] at h\u2081 \u22a2\n    have a1 := h\u2081 0\n    have a2 := h\u2081 1\n    have a3 := h\u2081 2\n    simp_all [Finset.prod_range_succ, Finset.prod_range_one]\n    <;> linarith\n  have base_case : a 1 - Real.sqrt (a 2) = 2 := by\n    simp_all [Finset.prod_range_succ, Finset.prod_range_one, mul_comm]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num [Real.sqrt_eq_iff_mul_self_eq]\n    <;> nlinarith [Real.sqrt_nonneg 9, Real.sq_sqrt (show (0 : \u211d) \u2264 9 by norm_num)]\n  have recurrence_helper : \u2200 n \u2265 1, (\u220f k in Finset.range n, a k) = a n - 4 := by\n    intro n hn\n    induction' hn with n hn\n    \u00b7 \n      simp_all [Finset.prod_range_one]\n      <;> norm_num\n    \u00b7 \n      cases n with\n      | zero =>\n        simp_all [Finset.prod_range_succ]\n      | succ n =>\n        simp_all [Finset.prod_range_succ, Nat.succ_eq_add_one]\n        <;> ring_nf\n        <;> norm_num\n        <;> linarith\n  have recurrence_pattern : \u2200 n \u2265 1, a (n + 1) = (a n - 2)^2 := by\n    intro n hn\n    have h\u2083 := recurrence_helper (n + 1) (by linarith)\n    have h\u2084 := recurrence_helper n hn\n    have h\u2085 := recurrence_helper 1 (by linarith)\n    have h\u2086 := recurrence_helper 0\n    norm_num [Finset.prod_range_succ, h\u2080] at h\u2083 h\u2084 h\u2085\n    simp_all\n    <;> nlinarith [sq_sqrt (show 0 \u2264 9 by norm_num)]\n  have sqrt_pattern : \u2200 n \u2265 1, Real.sqrt (a (n + 1)) = a n - 2 := by\n    intro n hn\n    have h_sqrt : \u2200 n \u2265 1, Real.sqrt (a (n + 1)) = a n - 2 := by\n      intro n hn\n      have h1 : a (n + 1) = (a n - 2) ^ 2 := recurrence_pattern n hn\n      have h2 : 0 \u2264 a n - 2 := by\n        have h3 : a n \u2265 4 := by\n          induction' hn with n hn\n          \u00b7 norm_num [a1_eq, a2_eq, a3_eq]\n          \u00b7 simp_all [Finset.prod_range_succ, Nat.succ_eq_add_one]\n            nlinarith\n        nlinarith\n      rw [Real.sqrt_eq_iff_sq_eq] <;> nlinarith\n    exact h_sqrt n hn\n  have main_proof : \u2200 n \u2265 1, a n - Real.sqrt (a (n + 1)) = 2 := by\n    intro n hn\n    have hn' : n \u2265 1 := by linarith\n    have h_sqrt : Real.sqrt (a (n + 1)) = a n - 2 := by\n      apply sqrt_pattern\n      linarith\n    rw [h_sqrt]\n    ring\n    <;> linarith\n  intro n hn\n  simpa [a1_eq, a2_eq, a3_eq] using main_proof n hn\n", "proof_before": "theorem numbertheory_aneqprodakp4_anmsqrtanp1eq2 (a : \u2115 \u2192 \u211d) (h\u2080 : a 0 = 1)\n  (h\u2081 : \u2200 n, a (n + 1) = (\u220f k in Finset.range (n + 1), a k) + 4) :\n  \u2200 n \u2265 1, a n - Real.sqrt (a (n + 1)) = 2  :=  by\n  have a1_eq : a 1 = 5 := by\n    simp_all [Finset.prod_range_succ, Finset.prod_range_zero, Nat.cast_zero, Nat.cast_one]\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n  have a2_eq : a 2 = 9 := by\n    have h\u2082 := h\u2081 0\n    have h\u2083 := h\u2081 1\n    simp [Finset.prod_range_succ, h\u2080] at h\u2082 h\u2083 \u22a2\n    norm_num at h\u2082 h\u2083 \u22a2\n    <;> nlinarith\n  have a3_eq : a 3 = 49 := by\n    norm_num [Finset.prod_range_succ, Finset.prod_range_one] at h\u2081 \u22a2\n    have a1 := h\u2081 0\n    have a2 := h\u2081 1\n    have a3 := h\u2081 2\n    simp_all [Finset.prod_range_succ, Finset.prod_range_one]\n    <;> linarith\n  have base_case : a 1 - Real.sqrt (a 2) = 2 := by\n    simp_all [Finset.prod_range_succ, Finset.prod_range_one, mul_comm]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num [Real.sqrt_eq_iff_mul_self_eq]\n    <;> nlinarith [Real.sqrt_nonneg 9, Real.sq_sqrt (show (0 : \u211d) \u2264 9 by norm_num)]\n  have recurrence_helper : \u2200 n \u2265 1, (\u220f k in Finset.range n, a k) = a n - 4 := by\n    intro n hn\n    induction' hn with n hn\n    \u00b7 \n      simp_all [Finset.prod_range_one]\n      <;> norm_num\n    \u00b7 \n      cases n with\n      | zero =>\n        simp_all [Finset.prod_range_succ]\n      | succ n =>\n        simp_all [Finset.prod_range_succ, Nat.succ_eq_add_one]\n        <;> ring_nf\n        <;> norm_num\n        <;> linarith\n  have recurrence_pattern : \u2200 n \u2265 1, a (n + 1) = (a n - 2)^2 := by\n    intro n hn\n    have h\u2083 := recurrence_helper (n + 1) (by linarith)\n    have h\u2084 := recurrence_helper n hn\n    have h\u2085 := recurrence_helper 1 (by linarith)\n    have h\u2086 := recurrence_helper 0\n    norm_num [Finset.prod_range_succ, h\u2080] at h\u2083 h\u2084 h\u2085\n    simp_all\n    <;> nlinarith [sq_sqrt (show 0 \u2264 9 by norm_num)]\n  have sqrt_pattern : \u2200 n \u2265 1, Real.sqrt (a (n + 1)) = a n - 2 := by\n    intro n hn\n    have h_sqrt : \u2200 n \u2265 1, Real.sqrt (a (n + 1)) = a n - 2 := by\n      intro n hn\n      have h1 : a (n + 1) = (a n - 2) ^ 2 := recurrence_pattern n hn\n      have h2 : 0 \u2264 a n - 2 := by\n        have h3 : a n \u2265 4 := by\n          induction' hn with n hn\n          \u00b7 norm_num [a1_eq, a2_eq, a3_eq]\n          \u00b7 simp_all [Finset.prod_range_succ, Nat.succ_eq_add_one]\n            nlinarith\n        nlinarith\n      rw [Real.sqrt_eq_iff_sq_eq] <;> nlinarith\n    exact h_sqrt n hn\n  have main_proof : \u2200 n \u2265 1, a n - Real.sqrt (a (n + 1)) = 2 := by\n    intro n hn\n    have hn' : n \u2265 1 := by linarith\n    have h_sqrt : Real.sqrt (a (n + 1)) = a n - 2 := by\n      apply sqrt_pattern\n      linarith\n    rw [h_sqrt]\n    ring\n    <;> linarith\n  intro n hn\n  simpa [a1_eq, a2_eq, a3_eq] using main_proof n hn\n"}
{"mode": "clear", "steps_before": 30, "alphabet": 22, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 748, "chars_after": 11, "seconds": 12.2, "id": "mathd_algebra_51", "proof_after": "theorem mathd_algebra_51 (a b : \u211d) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : a + b = 35) (h\u2082 : a = 2 / 5 * b) :\n    b - a = 15  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_51 (a b : \u211d) (h\u2080 : 0 < a \u2227 0 < b) (h\u2081 : a + b = 35) (h\u2082 : a = 2 / 5 * b) :\n    b - a = 15  :=  by\n  have h_a : a = (2 / 5) * b := by\n    linarith [h\u2081, h\u2082]\n    <;> simp_all\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  have h_sub : (2 / 5) * b + b = 35 := by\n    rw [h\u2082] at h\u2081\n    field_simp [h\u2080.1.ne', h\u2080.2.ne'] at h\u2081\n    linarith\n  have h_comb : (7 / 5) * b = 35 := by\n    field_simp [h\u2082, h_a] at h_sub \u22a2\n    linarith\n    <;> assumption\n  have h_b : b = 25 := by\n    have h\u2083 : b = 25 := by\n      field_simp at h\u2082 h_a h_sub h_comb\n      linarith\n    exact h\u2083\n  have h_a_val : a = 10 := by\n    have h\u2083 : a = 10 := by\n      rw [h\u2082] at h\u2081\n      field_simp at h\u2081\n      linarith\n    linarith\n  have h_diff : b - a = 15 := by\n    subst h_a_val\n    subst h_b\n    linarith\n  linarith [h\u2080.1, h\u2080.2, h\u2081, h\u2082, h_a, h_sub, h_comb, h_b, h_a_val, h_diff]\n"}
{"mode": "clear", "steps_before": 15, "alphabet": 12, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 451, "chars_after": 6, "seconds": 16.9, "id": "mathd_numbertheory_101", "proof_after": "theorem mathd_numbertheory_101 : 17 * 18 % 4 = 2  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_101 : 17 * 18 % 4 = 2  :=  by\n  have h17 : 17 % 4 = 1 := by\n    simp\n  have h18 : 18 % 4 = 2 := by\n    simp [*, Nat.mod_eq_of_lt, show 1 < 4 by norm_num, show 2 < 4 by norm_num]\n    <;> omega\n  have h_mul : (17 % 4) * (18 % 4) % 4 = (1 * 2) % 4 := by\n    simp [h17, h18, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt, Nat.zero_lt_two, Nat.zero_lt_one]\n    <;> decide\n  have h_final : (1 * 2) % 4 = 2 := by\n    simp_all\n  simp_all\n  <;> norm_num\n  <;> aesop\n  <;> norm_num\n  <;> aesop\n"}
{"mode": "clear", "steps_before": 26, "alphabet": 24, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 26, "lower_bound": null, "exhausted_depth": 2, "certified": false, "rounds": 7, "fixpoint": true, "chars_before": 1008, "chars_after": 1008, "seconds": 171.1, "id": "mathd_numbertheory_412", "proof_after": "theorem mathd_numbertheory_412 (x y : \u2124) (h\u2080 : x % 19 = 4) (h\u2081 : y % 19 = 7) :\n  (x + 1) ^ 2 * (y + 5) ^ 3 % 19 = 13  :=  by\n  have hx_add_1 : (x + 1) % 19 = 5 := by\n    simp [h\u2080, h\u2081, Int.add_emod, Int.mul_emod, Int.emod_emod, Int.add_assoc]\n    <;> norm_num\n    <;> rfl\n  have hx_sq : (x + 1) ^ 2 % 19 = 6 := by\n    have h\u2082 : (x + 1) ^ 2 % 19 = 6 := by\n      rw [sq]\n      norm_num [Int.mul_emod, hx_add_1, h\u2080, h\u2081, Int.add_emod, Int.mul_emod]\n    assumption\n  have hy_add_5 : (y + 5) % 19 = 12 := by\n    have h\u2082 := congr_arg (\u00b7 % 19) hx_sq\n    have h\u2083 := congr_arg (\u00b7 % 19) hx_add_1\n    have h\u2084 := congr_arg (\u00b7 % 19) h\u2080\n    have h\u2085 := congr_arg (\u00b7 % 19) h\u2081\n    norm_num at h\u2082 h\u2083 h\u2084 h\u2085\n    omega\n  have hy_cubed : (y + 5) ^ 3 % 19 = 18 := by\n    have h\u2082 := congr_arg (\u00b7 % 19) (pow_three (y + 5))\n    simp at h\u2082\n    simp_all [Int.mul_emod, Int.add_emod, pow_three]\n    <;> omega\n  have h_product : (x + 1) ^ 2 * (y + 5) ^ 3 % 19 = 13 := by\n    norm_num [Int.mul_emod, hx_sq, hy_cubed, Int.add_emod]\n    <;> decide\n  norm_num [Int.mul_emod, Int.add_emod, pow_two, pow_three] at hx_add_1 hx_sq hy_add_5 hy_cubed h_product \u22a2\n  <;> omega\n", "proof_before": "theorem mathd_numbertheory_412 (x y : \u2124) (h\u2080 : x % 19 = 4) (h\u2081 : y % 19 = 7) :\n  (x + 1) ^ 2 * (y + 5) ^ 3 % 19 = 13  :=  by\n  have hx_add_1 : (x + 1) % 19 = 5 := by\n    simp [h\u2080, h\u2081, Int.add_emod, Int.mul_emod, Int.emod_emod, Int.add_assoc]\n    <;> norm_num\n    <;> rfl\n  have hx_sq : (x + 1) ^ 2 % 19 = 6 := by\n    have h\u2082 : (x + 1) ^ 2 % 19 = 6 := by\n      rw [sq]\n      norm_num [Int.mul_emod, hx_add_1, h\u2080, h\u2081, Int.add_emod, Int.mul_emod]\n    assumption\n  have hy_add_5 : (y + 5) % 19 = 12 := by\n    have h\u2082 := congr_arg (\u00b7 % 19) hx_sq\n    have h\u2083 := congr_arg (\u00b7 % 19) hx_add_1\n    have h\u2084 := congr_arg (\u00b7 % 19) h\u2080\n    have h\u2085 := congr_arg (\u00b7 % 19) h\u2081\n    norm_num at h\u2082 h\u2083 h\u2084 h\u2085\n    omega\n  have hy_cubed : (y + 5) ^ 3 % 19 = 18 := by\n    have h\u2082 := congr_arg (\u00b7 % 19) (pow_three (y + 5))\n    simp at h\u2082\n    simp_all [Int.mul_emod, Int.add_emod, pow_three]\n    <;> omega\n  have h_product : (x + 1) ^ 2 * (y + 5) ^ 3 % 19 = 13 := by\n    norm_num [Int.mul_emod, hx_sq, hy_cubed, Int.add_emod]\n    <;> decide\n  norm_num [Int.mul_emod, Int.add_emod, pow_two, pow_three] at hx_add_1 hx_sq hy_add_5 hy_cubed h_product \u22a2\n  <;> omega\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 25, "alphabet": 17, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 617, "chars_after": 7, "seconds": 177.7, "id": "mathd_algebra_190", "proof_after": "theorem mathd_algebra_190 : ((3 : \u211d) / 8 + 7 / 8) / (4 / 5) = 25 / 16  :=  by\n  ring\n", "proof_before": "theorem mathd_algebra_190 : ((3 : \u211d) / 8 + 7 / 8) / (4 / 5) = 25 / 16  :=  by\n  have step1 : (3 / 8 : \u211d) + 7 / 8 = 10 / 8 := by\n    norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  have step2 : (10 / 8 : \u211d) = 5 / 4 := by\n    norm_num\n    <;> field_simp at step1 \u22a2\n    <;> ring_nf at step1 \u22a2\n    <;> linarith\n  have step3 : (5 / 4 : \u211d) / (4 / 5) = (5 / 4) * (5 / 4) := by\n    field_simp [step1, step2]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have step4 : (5 / 4 : \u211d) * (5 / 4) = 25 / 16 := by\n    norm_num [step1, step2, step3]\n    <;> ring_nf at *\n    <;> norm_num\n    <;> linarith\n  rw [step1, step2, step3, step4]\n  <;> norm_num\n"}
{"mode": "clear", "steps_before": 30, "alphabet": 11, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 7, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 2, "fixpoint": true, "chars_before": 769, "chars_after": 176, "seconds": 267.1, "id": "mathd_algebra_132", "proof_after": "theorem mathd_algebra_132 (x : \u211d) (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x + 2) (h\u2081 : \u2200 x, g x = x ^ 2)\n  (h\u2082 : f (g x) = g (f x)) : x = -1 / 2  :=  by\n  have f_gx : f (g x) = x^2 + 2 := by\n    simp_all [pow_two]\n    <;> linarith\n  have g_fx : g (f x) = (x + 2)^2 := by\n    simp_all [h\u2080, h\u2081, h\u2082, sq]\n    <;> linarith\n  linarith\n", "proof_before": "theorem mathd_algebra_132 (x : \u211d) (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x + 2) (h\u2081 : \u2200 x, g x = x ^ 2)\n  (h\u2082 : f (g x) = g (f x)) : x = -1 / 2  :=  by\n  have f_gx : f (g x) = x^2 + 2 := by\n    simp_all [pow_two]\n    <;> linarith\n  have g_fx : g (f x) = (x + 2)^2 := by\n    simp_all [h\u2080, h\u2081, h\u2082, sq]\n    <;> linarith\n  have eq : x^2 + 2 = (x + 2)^2 := by\n    simp_all only [rpow_two, add_right_comm]\n    <;> linarith\n  have expanded_eq : x^2 + 2 = x^2 + 4*x + 4 := by\n    have h\u2083 : x^2 + 2 = x^2 + 4*x + 4 := by\n      nlinarith\n    linarith\n  have simplified_eq1 : 2 = 4*x + 4 := by\n    have h\u2083 : 2 = 4 * x + 4 := by\n      linarith\n    linarith\n  have simplified_eq2 : -2 = 4*x := by\n    have h\u2083 : 2 = 4 * x + 4 := by linarith\n    have h\u2084 : 2 - 4 = 4 * x := by linarith\n    have h\u2085 : -2 = 4 * x := by linarith\n    have h\u2086 : -2 = 4 * x := by linarith\n    linarith\n  have final_x : x = -1 / 2 := by\n    linarith\n  linarith\n"}
{"mode": "clear", "steps_before": 26, "alphabet": 18, "compiles": false, "id": "mathd_numbertheory_543", "proof_after": "theorem mathd_numbertheory_543 : (\u2211 k in Nat.divisors (30 ^ 4), 1) - 2 = 123  :=  by\n  have prime_factors_30 : 30 = 2 * 3 * 5 := by\n    norm_num\n    <;> decide\n    <;> decide\n    <;> decide\n  have pow_30_4 : 30^4 = (2 * 3 * 5)^4 := by\n    rw [prime_factors_30]\n    <;> simp_all [pow_succ]\n    <;> norm_num\n    <;> norm_num\n    <;> norm_num\n  have expanded_pow : (2 * 3 * 5)^4 = 2^4 * 3^4 * 5^4 := by\n    simp [pow_mul, pow_succ, mul_assoc, mul_comm, mul_left_comm, prime_factors_30, pow_30_4]\n    <;> ring\n  have num_divisors : (4 + 1) * (4 + 1) * (4 + 1) = 125 := by\n    have h : (4 + 1) * (4 + 1) * (4 + 1) = 125 := by\n      simp [prime_factors_30, pow_30_4, expanded_pow]\n    exact h\n  have total_divisors : \u2211 k in Nat.divisors (30^4), 1 = 125 := by\n    simp_all only [mul_assoc, mul_comm, mul_left_comm, Nat.divisors_mul]\n    rfl\n    <;> decide\n  have result : (\u2211 k in Nat.divisors (30^4), 1) - 2 = 123 := by\n    simpa [prime_factors_30, pow_30_4, expanded_pow, num_divisors, total_divisors] using\n      total_divisors\n  simpa [prime_factors_30, pow_30_4, expanded_pow, num_divisors, total_divisors] using result\n", "proof_before": "theorem mathd_numbertheory_543 : (\u2211 k in Nat.divisors (30 ^ 4), 1) - 2 = 123  :=  by\n  have prime_factors_30 : 30 = 2 * 3 * 5 := by\n    norm_num\n    <;> decide\n    <;> decide\n    <;> decide\n  have pow_30_4 : 30^4 = (2 * 3 * 5)^4 := by\n    rw [prime_factors_30]\n    <;> simp_all [pow_succ]\n    <;> norm_num\n    <;> norm_num\n    <;> norm_num\n  have expanded_pow : (2 * 3 * 5)^4 = 2^4 * 3^4 * 5^4 := by\n    simp [pow_mul, pow_succ, mul_assoc, mul_comm, mul_left_comm, prime_factors_30, pow_30_4]\n    <;> ring\n  have num_divisors : (4 + 1) * (4 + 1) * (4 + 1) = 125 := by\n    have h : (4 + 1) * (4 + 1) * (4 + 1) = 125 := by\n      simp [prime_factors_30, pow_30_4, expanded_pow]\n    exact h\n  have total_divisors : \u2211 k in Nat.divisors (30^4), 1 = 125 := by\n    simp_all only [mul_assoc, mul_comm, mul_left_comm, Nat.divisors_mul]\n    rfl\n    <;> decide\n  have result : (\u2211 k in Nat.divisors (30^4), 1) - 2 = 123 := by\n    simpa [prime_factors_30, pow_30_4, expanded_pow, num_divisors, total_divisors] using\n      total_divisors\n  simpa [prime_factors_30, pow_30_4, expanded_pow, num_divisors, total_divisors] using result\n"}
{"mode": "clear", "steps_before": 26, "alphabet": 24, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 15, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 6, "fixpoint": true, "chars_before": 652, "chars_after": 321, "seconds": 425.0, "id": "algebra_2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4", "proof_after": "theorem algebra_2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4 (x e : \u2102) (h\u2080 : x + e = 7)\n  (h\u2081 : 2 * x + e = 3) : e = 11 \u2227 x = -4  :=  by\n  have h\u2082 : e = 7 - x := by\n    linear_combination h\u2080\n  have h\u2084 : x + 7 = 3 := by\n    linear_combination h\u2081 - h\u2080\n  have h\u2085 : x = -4 := by\n    linear_combination h\u2084\n  have h\u2086 : e = 7 - (-4) := by\n    rw [h\u2085] at h\u2082\n    linear_combination h\u2082\n  have h\u2087 : e = 11 := by\n    rw [h\u2086]\n    ring\n  constructor\n  exact h\u2087\n  exact h\u2085\n", "proof_before": "theorem algebra_2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4 (x e : \u2102) (h\u2080 : x + e = 7)\n  (h\u2081 : 2 * x + e = 3) : e = 11 \u2227 x = -4  :=  by\n  have h\u2082 : e = 7 - x := by\n    have h\u2080' : e = 7 - x := by\n      linear_combination h\u2080\n    exact h\u2080'\n  have h\u2083 : 2 * x + (7 - x) = 3 := by\n    subst h\u2082\n    linear_combination h\u2081 - h\u2080\n  have h\u2084 : x + 7 = 3 := by\n    simp_all only [mul_comm, mul_add, mul_one, sub_eq_add_neg, add_assoc, add_left_comm]\n    <;> ring_nf at *\n    <;> linear_combination h\u2081\n  have h\u2085 : x = -4 := by\n    have h\u2085 : x = -4 := by\n      linear_combination h\u2084\n    exact h\u2085\n  have h\u2086 : e = 7 - (-4) := by\n    rw [h\u2085] at h\u2082\n    linear_combination h\u2082\n  have h\u2087 : e = 11 := by\n    rw [h\u2086]\n    norm_num\n    <;> simp_all\n    <;> linear_combination h\u2080\n  constructor\n  exact h\u2087\n  exact h\u2085\n"}
{"mode": "clear", "steps_before": 7, "alphabet": 16, "compiles": false, "id": "mathd_numbertheory_22", "proof_after": "theorem mathd_numbertheory_22 (b : \u2115) (h\u2080 : b < 10)\n  (h\u2081 : Nat.sqrt (10 * b + 6) * Nat.sqrt (10 * b + 6) = 10 * b + 6) : b = 3 \u2228 b = 1  :=  by\n  have h\u2082 := congr_arg (fun x => x ^ 2) h\u2081\n  simp at h\u2082\n  interval_cases b <;> norm_num at h\u2082 \u22a2 <;> try nlinarith\n  <;> norm_num <;> nlinarith\n", "proof_before": "theorem mathd_numbertheory_22 (b : \u2115) (h\u2080 : b < 10)\n  (h\u2081 : Nat.sqrt (10 * b + 6) * Nat.sqrt (10 * b + 6) = 10 * b + 6) : b = 3 \u2228 b = 1  :=  by\n  have h\u2082 := congr_arg (fun x => x ^ 2) h\u2081\n  simp at h\u2082\n  interval_cases b <;> norm_num at h\u2082 \u22a2 <;> try nlinarith\n  <;> norm_num <;> nlinarith\n"}
{"mode": "clear", "steps_before": 75, "alphabet": 35, "compiles": false, "id": "amc12a_2008_p15", "proof_after": "theorem amc12a_2008_p15 (k : \u2115) (h\u2080 : k = 2008 ^ 2 + 2 ^ 2008) : (k ^ 2 + 2 ^ k) % 10 = 6  :=  by\n  have h\u2081 : (2008 ^ 2) % 10 = 4 := by\n    have h\u2081 : 2008 ^ 2 % 10 = 4 := by\n      have h\u2082 : 2008 % 10 = 8 := by norm_num\n      have h\u2083 : 8 ^ 2 % 10 = 4 := by norm_num\n      norm_num [h\u2082, h\u2083]\n    have h\u2082 : 2 ^ 2008 % 10 = 6 := by\n      have h\u2083 : 2 ^ 4 % 10 = 6 := by norm_num\n      have h\u2084 : 2008 % 4 = 0 := by norm_num\n      norm_num [h\u2083, h\u2084, pow_mul]\n    norm_num [h\u2080, h\u2081, h\u2082]\n  have h\u2082 : (2 ^ 2008) % 10 = 6 := by\n    have h\u2082 : (2 ^ 2008) % 10 = 6 := by\n      have h\u2083 : (2 ^ 4) % 10 = 6 := by norm_num\n      have h\u2084 : 2008 % 4 = 0 := by norm_num\n      have h\u2085 : (2 ^ 2008) % 10 = (2 ^ (4 * (2008 / 4))) % 10 := by norm_num\n      rw [pow_mul] at h\u2085\n      simp [h\u2083, h\u2084] at h\u2085\n      <;> norm_num at h\u2085\n      <;> assumption\n    simp [h\u2082, h\u2080]\n    <;> norm_num\n    <;> omega\n  have h\u2083 : k % 10 = (4 + 6) % 10 := by\n    rw [h\u2080]\n    simp [h\u2081, h\u2082, Nat.pow_mod, Nat.add_mod, Nat.mul_mod, Nat.pow_mod, Nat.add_mod, Nat.mul_mod]\n  have h\u2084 : k % 10 = 0 := by\n    simp_all [Nat.pow_mod, Nat.add_mod, Nat.mul_mod]\n    <;> norm_num\n    <;> rfl\n  have h\u2085 : (k ^ 2) % 10 = 0 := by\n    norm_num [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, Nat.pow_mod, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n  have h\u2086 : (2 ^ k) % 10 = 6 := by\n    have h\u2086 : k % 10 = 0 := by omega\n    have h\u2087 : (k ^ 2) % 10 = 0 := by omega\n    have h\u2088 : (2 ^ k) % 10 = 6 := by\n      have h\u2088 : \u2200 n : \u2115, n \u2265 1 \u2192 (2 ^ (4 * n)) % 10 = 6 := by\n        intro n hn\n        induction' hn with n hn\n        \u00b7 norm_num \n        \u00b7 simp_all [Nat.mul_succ, pow_add, pow_one, Nat.mul_mod, Nat.pow_mod, Nat.add_mod]\n      have h\u2089 : k % 4 = 0 := by omega\n      have h\u2081\u2080 : k = 4 * (k / 4) := by omega\n      rw [h\u2081\u2080]\n      apply h\u2088\n      omega\n    exact h\u2088\n  have h\u2087 : (k ^ 2 + 2 ^ k) % 10 = (0 + 6) % 10 := by\n    simp [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, Nat.add_mod, Nat.mul_mod, Nat.pow_mod, Nat.mod_mod]\n    <;> norm_num\n    <;> aesop\n  have h\u2088 : (k ^ 2 + 2 ^ k) % 10 = 6 := by\n    have h\u2081 : (2008 ^ 2) % 10 = 4 := h\u2081\n    have h\u2082 : (2 ^ 2008) % 10 = 6 := h\u2082\n    have h\u2083 : k % 10 = (4 + 6) % 10 := h\u2083\n    have h\u2084 : k % 10 = 0 := h\u2084\n    have h\u2085 : (k ^ 2) % 10 = 0 := h\u2085\n    have h\u2086 : (2 ^ k) % 10 = 6 := h\u2086\n    have h\u2087 : (k ^ 2 + 2 ^ k) % 10 = (0 + 6) % 10 := h\u2087\n    omega\n  apply h\u2088\n", "proof_before": "theorem amc12a_2008_p15 (k : \u2115) (h\u2080 : k = 2008 ^ 2 + 2 ^ 2008) : (k ^ 2 + 2 ^ k) % 10 = 6  :=  by\n  have h\u2081 : (2008 ^ 2) % 10 = 4 := by\n    have h\u2081 : 2008 ^ 2 % 10 = 4 := by\n      have h\u2082 : 2008 % 10 = 8 := by norm_num\n      have h\u2083 : 8 ^ 2 % 10 = 4 := by norm_num\n      norm_num [h\u2082, h\u2083]\n    have h\u2082 : 2 ^ 2008 % 10 = 6 := by\n      have h\u2083 : 2 ^ 4 % 10 = 6 := by norm_num\n      have h\u2084 : 2008 % 4 = 0 := by norm_num\n      norm_num [h\u2083, h\u2084, pow_mul]\n    norm_num [h\u2080, h\u2081, h\u2082]\n  have h\u2082 : (2 ^ 2008) % 10 = 6 := by\n    have h\u2082 : (2 ^ 2008) % 10 = 6 := by\n      have h\u2083 : (2 ^ 4) % 10 = 6 := by norm_num\n      have h\u2084 : 2008 % 4 = 0 := by norm_num\n      have h\u2085 : (2 ^ 2008) % 10 = (2 ^ (4 * (2008 / 4))) % 10 := by norm_num\n      rw [pow_mul] at h\u2085\n      simp [h\u2083, h\u2084] at h\u2085\n      <;> norm_num at h\u2085\n      <;> assumption\n    simp [h\u2082, h\u2080]\n    <;> norm_num\n    <;> omega\n  have h\u2083 : k % 10 = (4 + 6) % 10 := by\n    rw [h\u2080]\n    simp [h\u2081, h\u2082, Nat.pow_mod, Nat.add_mod, Nat.mul_mod, Nat.pow_mod, Nat.add_mod, Nat.mul_mod]\n  have h\u2084 : k % 10 = 0 := by\n    simp_all [Nat.pow_mod, Nat.add_mod, Nat.mul_mod]\n    <;> norm_num\n    <;> rfl\n  have h\u2085 : (k ^ 2) % 10 = 0 := by\n    norm_num [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, Nat.pow_mod, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n  have h\u2086 : (2 ^ k) % 10 = 6 := by\n    have h\u2086 : k % 10 = 0 := by omega\n    have h\u2087 : (k ^ 2) % 10 = 0 := by omega\n    have h\u2088 : (2 ^ k) % 10 = 6 := by\n      have h\u2088 : \u2200 n : \u2115, n \u2265 1 \u2192 (2 ^ (4 * n)) % 10 = 6 := by\n        intro n hn\n        induction' hn with n hn\n        \u00b7 norm_num \n        \u00b7 simp_all [Nat.mul_succ, pow_add, pow_one, Nat.mul_mod, Nat.pow_mod, Nat.add_mod]\n      have h\u2089 : k % 4 = 0 := by omega\n      have h\u2081\u2080 : k = 4 * (k / 4) := by omega\n      rw [h\u2081\u2080]\n      apply h\u2088\n      omega\n    exact h\u2088\n  have h\u2087 : (k ^ 2 + 2 ^ k) % 10 = (0 + 6) % 10 := by\n    simp [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, Nat.add_mod, Nat.mul_mod, Nat.pow_mod, Nat.mod_mod]\n    <;> norm_num\n    <;> aesop\n  have h\u2088 : (k ^ 2 + 2 ^ k) % 10 = 6 := by\n    have h\u2081 : (2008 ^ 2) % 10 = 4 := h\u2081\n    have h\u2082 : (2 ^ 2008) % 10 = 6 := h\u2082\n    have h\u2083 : k % 10 = (4 + 6) % 10 := h\u2083\n    have h\u2084 : k % 10 = 0 := h\u2084\n    have h\u2085 : (k ^ 2) % 10 = 0 := h\u2085\n    have h\u2086 : (2 ^ k) % 10 = 6 := h\u2086\n    have h\u2087 : (k ^ 2 + 2 ^ k) % 10 = (0 + 6) % 10 := h\u2087\n    omega\n  apply h\u2088\n"}
{"mode": "clear", "steps_before": 26, "alphabet": 19, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 10, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 5, "fixpoint": true, "chars_before": 622, "chars_after": 284, "seconds": 645.7, "id": "amc12b_2004_p3", "proof_after": "theorem amc12b_2004_p3 (x y : \u2115) (h\u2080 : 2 ^ x * 3 ^ y = 1296) : x + y = 8  :=  by\n  have h3 : x = 4 \u2227 y = 4 := by\n    all_goals\n      rcases x with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n        rcases y with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n          simp_all [Nat.pow_succ, Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]\n    <;>\n    omega\n  omega\n", "proof_before": "theorem amc12b_2004_p3 (x y : \u2115) (h\u2080 : 2 ^ x * 3 ^ y = 1296) : x + y = 8  :=  by\n  have h1 : 1296 = 2^4 * 3^4 := by\n    rw [show 1296 = 2 ^ 4 * 3 ^ 4 by norm_num]\n    <;> simp_all\n    <;> ring\n    <;> omega\n  have h2 : 2^x * 3^y = 2^4 * 3^4 := by\n    rw [h1] at h\u2080\n    linarith\n    <;> linarith\n  have h3 : x = 4 \u2227 y = 4 := by\n    refine' \u27e8_, _\u27e9\n    all_goals\n      rcases x with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n        rcases y with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n          simp_all [Nat.pow_succ, Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]\n    <;>\n    omega\n  have h4 : x + y = 8 := by\n    cases h3\n    subst_vars\n    norm_num\n  rw [h3.1, h3.2]\n  <;> simp\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 13, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 613, "chars_after": 12, "seconds": 120.6, "id": "amc12_2001_p2", "proof_after": "theorem amc12_2001_p2 (a b n : \u2115) (h\u2080 : 1 \u2264 a \u2227 a \u2264 9) (h\u2081 : 0 \u2264 b \u2227 b \u2264 9) (h\u2082 : n = 10 * a + b)\n  (h\u2083 : n = a * b + a + b) : b = 9  :=  by\n  nlinarith\n", "proof_before": "theorem amc12_2001_p2 (a b n : \u2115) (h\u2080 : 1 \u2264 a \u2227 a \u2264 9) (h\u2081 : 0 \u2264 b \u2227 b \u2264 9) (h\u2082 : n = 10 * a + b)\n  (h\u2083 : n = a * b + a + b) : b = 9  :=  by\n  have eq1 : 10 * a + b = a * b + a + b := by\n    simp_all only [h\u2082, h\u2083, add_comm]\n    <;> omega\n  have eq2 : 10 * a = a * b + a := by\n    have h\u2084 : 10 * a = a * b + a := by\n      linarith\n    simp_all\n  have eq3 : 10 * a = a * (b + 1) := by\n    simp_all only [Nat.mul_add, Nat.add_assoc, Nat.add_right_comm, Nat.add_comm]\n    nlinarith\n  have eq4 : 10 = b + 1 := by\n    have h\u2084 : b + 1 \u2264 10 := by\n      nlinarith\n    have h\u2085 : b \u2264 9 := by\n      nlinarith\n    have h\u2086 : b + 1 \u2265 10 := by\n      nlinarith\n    linarith\n  have eq5 : b = 9 := by\n    have h\u2085 : b = 9 := by\n      linarith\n    exact h\u2085\n  simpa using eq5\n"}
{"mode": "clear", "steps_before": 354, "alphabet": 268, "compiles": false, "id": "aime_1987_p8", "proof_after": "theorem aime_1987_p8 :\n  IsGreatest { n : \u2115 | 0 < n \u2227 \u2203! k : \u2115, (8 : \u211d) / 15 < n / (n + k) \u2227 (n : \u211d) / (n + k) < 7 / 13 } 112  :=  by\n  have h_mem : 112 \u2208 { n : \u2115 | 0 < n \u2227 \u2203! k : \u2115, (8 : \u211d) / 15 < n / (n + k) \u2227 (n : \u211d) / (n + k) < 7 / 13 } := by\n    constructor\n    \u00b7 norm_num\n    use 97\n    constructor\n    \u00b7 norm_num [div_lt_div_iff]\n    intro k hk\n    have h\u2081 := hk.1\n    have h\u2082 := hk.2\n    field_simp [add_comm] at h\u2081 h\u2082\n    rw [\u2190 sub_pos] at h\u2081 h\u2082\n    field_simp [add_comm] at h\u2081 h\u2082\n    norm_cast at h\u2081 h\u2082\n    ring_nf at h\u2081 h\u2082\n    omega\n  have h_bounds : \u2200 (n k : \u2115), (8 : \u211d) / 15 < n / (n + k) \u2227 (n : \u211d) / (n + k) < 7 / 13 \u2194 \n                   (6 : \u211d) * n / 7 < k \u2227 k < (7 : \u211d) * n / 8 := by\n    norm_num\n    have h1 : \u2200 n k : \u2115, (8 : \u211d) / 15 < n / (n + k) \u2227 (n : \u211d) / (n + k) < 7 / 13 \u2194 (6 : \u211d) * n / 7 < k \u2227 k < (7 : \u211d) * n / 8 := by\n      intro n k\n      constructor\n      \u00b7 intro h\n        have h2 : (8 : \u211d) / 15 < n / (n + k) := h.1\n        have h3 : (n : \u211d) / (n + k) < 7 / 13 := h.2\n        constructor\n        \u00b7 have h4 : (0 : \u211d) < n + k := by\n            by_contra h5\n            have h6 : (n : \u211d) \u2264 0 := by linarith\n            have h7 : (n : \u211d) / (n + k) \u2264 0 := by exact div_nonpos_of_nonneg_of_nonpos (by positivity) (by linarith)\n            have h8 : (8 : \u211d) / 15 < 0 := by linarith\n            linarith\n          field_simp at h2 h3\n          rw [\u2190 sub_pos] at h2 h3\n          field_simp at h2 h3\n          rw [\u2190 sub_pos]\n          field_simp\n          ring_nf\n          nlinarith\n        \u00b7 have h4 : (0 : \u211d) < n + k := by\n            by_contra h5\n            have h6 : (n : \u211d) \u2264 0 := by linarith\n            have h7 : (n : \u211d) / (n + k) \u2264 0 := by exact div_nonpos_of_nonneg_of_nonpos (by positivity) (by linarith)\n            have h8 : (8 : \u211d) / 15 < 0 := by linarith\n            linarith\n          field_simp at h2 h3\n          rw [\u2190 sub_pos] at h2 h3\n          field_simp at h2 h3\n          rw [\u2190 sub_pos]\n          field_simp\n          ring_nf\n          nlinarith\n      \u00b7 intro h\n        have h2 : (6 : \u211d) * n / 7 < k := h.1\n        have h3 : k < (7 : \u211d) * n / 8 := h.2\n        constructor\n        \u00b7 have h4 : (0 : \u211d) < n + k := by\n            by_contra h5\n            have h6 : (n : \u211d) \u2264 0 := by linarith\n            have h7 : (6 : \u211d) * n / 7 \u2264 0 := by exact div_nonpos_of_nonpos_of_nonneg (by linarith) (by positivity)\n            linarith\n          field_simp at h2 h3 \u22a2\n          rw [\u2190 sub_pos] at h2 h3\n          field_simp at h2 h3 \u22a2\n          rw [\u2190 sub_pos]\n          field_simp\n          ring_nf\n          nlinarith\n        \u00b7 have h4 : (0 : \u211d) < n + k := by\n            by_contra h5\n            have h6 : (n : \u211d) \u2264 0 := by linarith\n            have h7 : (7 : \u211d) * n / 8 \u2264 0 := by exact div_nonpos_of_nonpos_of_nonneg (by linarith) (by positivity)\n            linarith\n          field_simp at h2 h3 \u22a2\n          rw [\u2190 sub_pos] at h2 h3\n          field_simp at h2 h3 \u22a2\n          rw [\u2190 sub_pos]\n          field_simp\n          ring_nf\n          nlinarith\n    intro n k\n    simp_all\n  have h_interval_length : \u2200 n : \u2115, (7 : \u211d) * n / 8 - (6 : \u211d) * n / 7 = n / 56 := by\n    intro n\n    have h := h_bounds 112 0\n    have h\u2081 := h_bounds 0 0\n    have h\u2082 := h_bounds 16 0\n    have h\u2083 := h_bounds 1 0\n    have h\u2084 := h_bounds 0 1\n    have h\u2085 := h_bounds 1 1\n    have h\u2086 := h_bounds 0 1\n    have h\u2087 := h_bounds 16 1\n    norm_num at h h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 h\u2087\n    <;> linarith\n  have h_at_most_one : \u2200 n : \u2115, n \u2265 113 \u2192 \u2203 k\u2081 k\u2082 : \u2115, k\u2081 \u2260 k\u2082 \u2227 \n                       (6 : \u211d) * n / 7 < k\u2081 \u2227 k\u2081 < (7 : \u211d) * n / 8 \u2227\n                       (6 : \u211d) * n / 7 < k\u2082 \u2227 k\u2082 < (7 : \u211d) * n / 8 := by\n    intro n hn\n    norm_num [h_bounds, h_interval_length] at *\n    use 6 * n / 7 + 1, 6 * n / 7 + 2\n    constructor\n    <;> norm_num [div_lt_iff, lt_div_iff] at *\n    <;> ring_nf at *\n    <;> norm_cast at *\n    <;> omega\n  have h_n112_unique_k : \u2203! k : \u2115, (8 : \u211d) / 15 < 112 / (112 + k) \u2227 (112 : \u211d) / (112 + k) < 7 / 13 := by\n    have h\u2081 := h_bounds 112 0\n    have h\u2082 := h_bounds 112 1\n    have h\u2083 := h_bounds 112 2\n    have h\u2084 := h_bounds 112 3\n    have h\u2085 := h_bounds 112 4\n    have h\u2086 := h_bounds 112 5\n    have h\u2087 := h_bounds 112 6\n    have h\u2088 := h_bounds 112 7\n    have h\u2089 := h_bounds 112 8\n    have h\u2081\u2080 := h_bounds 112 9\n    have h\u2081\u2081 := h_bounds 112 10\n    have h\u2081\u2082 := h_bounds 112 11\n    have h\u2081\u2083 := h_bounds 112 12\n    have h\u2081\u2084 := h_bounds 112 13\n    have h\u2081\u2085 := h_bounds 112 14\n    have h\u2081\u2086 := h_bounds 112 15\n    have h\u2081\u2087 := h_bounds 112 16\n    have h\u2081\u2088 := h_bounds 112 17\n    have h\u2081\u2089 := h_bounds 112 18\n    have h\u2082\u2080 := h_bounds 112 19\n    have h\u2082\u2081 := h_bounds 112 20\n    have h\u2082\u2082 := h_bounds 112 21\n    have h\u2082\u2083 := h_bounds 112 22\n    have h\u2082\u2084 := h_bounds 112 23\n    have h\u2082\u2085 := h_bounds 112 24\n    have h\u2082\u2086 := h_bounds 112 25\n    have h\u2082\u2087 := h_bounds 112 26\n    have h\u2082\u2088 := h_bounds 112 27\n    have h\u2082\u2089 := h_bounds 112 28\n    have h\u2083\u2080 := h_bounds 112 29\n    have h\u2083\u2081 := h_bounds 112 30\n    have h\u2083\u2082 := h_bounds 112 31\n    have h\u2083\u2083 := h_bounds 112 32\n    have h\u2083\u2084 := h_bounds 112 33\n    have h\u2083\u2085 := h_bounds 112 34\n    have h\u2083\u2086 := h_bounds 112 35\n    have h\u2083\u2087 := h_bounds 112 36\n    have h\u2083\u2088 := h_bounds 112 37\n    have h\u2083\u2089 := h_bounds 112 38\n    have h\u2084\u2080 := h_bounds 112 39\n    have h\u2084\u2081 := h_bounds 112 40\n    have h\u2084\u2082 := h_bounds 112 41\n    have h\u2084\u2083 := h_bounds 112 42\n    have h\u2084\u2084 := h_bounds 112 43\n    have h\u2084\u2085 := h_bounds 112 44\n    have h\u2084\u2086 := h_bounds 112 45\n    have h\u2084\u2087 := h_bounds 112 46\n    have h\u2084\u2088 := h_bounds 112 47\n    have h\u2084\u2089 := h_bounds 112 48\n    have h\u2085\u2080 := h_bounds 112 49\n    have h\u2085\u2081 := h_bounds 112 50\n    have h\u2085\u2082 := h_bounds 112 51\n    have h\u2085\u2083 := h_bounds 112 52\n    have h\u2085\u2084 := h_bounds 112 53\n    have h\u2085\u2085 := h_bounds 112 54\n    have h\u2085\u2086 := h_bounds 112 55\n    have h\u2085\u2087 := h_bounds 112 56\n    have h\u2085\u2088 := h_bounds 112 57\n    have h\u2085\u2089 := h_bounds 112 58\n    have h\u2086\u2080 := h_bounds 112 59\n    have h\u2086\u2081 := h_bounds 112 60\n    have h\u2086\u2082 := h_bounds 112 61\n    have h\u2086\u2083 := h_bounds 112 62\n    have h\u2086\u2084 := h_bounds 112 63\n    have h\u2086\u2085 := h_bounds 112 64\n    have h\u2086\u2086 := h_bounds 112 65\n    have h\u2086\u2087 := h_bounds 112 66\n    have h\u2086\u2088 := h_bounds 112 67\n    have h\u2086\u2089 := h_bounds 112 68\n    have h\u2087\u2080 := h_bounds 112 69\n    have h\u2087\u2081 := h_bounds 112 70\n    have h\u2087\u2082 := h_bounds 112 71\n    have h\u2087\u2083 := h_bounds 112 72\n    have h\u2087\u2084 := h_bounds 112 73\n    have h\u2087\u2085 := h_bounds 112 74\n    have h\u2087\u2086 := h_bounds 112 75\n    have h\u2087\u2087 := h_bounds 112 76\n    have h\u2087\u2088 := h_bounds 112 77\n    have h\u2087\u2089 := h_bounds 112 78\n    have h\u2088\u2080 := h_bounds 112 79\n    have h\u2088\u2081 := h_bounds 112 80\n    have h\u2088\u2082 := h_bounds 112 81\n    have h\u2088\u2083 := h_bounds 112 82\n    have h\u2088\u2084 := h_bounds 112 83\n    have h\u2088\u2085 := h_bounds 112 84\n    have h\u2088\u2086 := h_bounds 112 85\n    have h\u2088\u2087 := h_bounds 112 86\n    have h\u2088\u2088 := h_bounds 112 87\n    have h\u2088\u2089 := h_bounds 112 88\n    have h\u2089\u2080 := h_bounds 112 89\n    have h\u2089\u2081 := h_bounds 112 90\n    have h\u2089\u2082 := h_bounds 112 91\n    have h\u2089\u2083 := h_bounds 112 92\n    have h\u2089\u2084 := h_bounds 112 93\n    have h\u2089\u2085 := h_bounds 112 94\n    have h\u2089\u2086 := h_bounds 112 95\n    have h\u2089\u2087 := h_bounds 112 96\n    have h\u2089\u2088 := h_bounds 112 97\n    have h\u2089\u2089 := h_bounds 112 98\n    have h\u2081\u2080\u2080 := h_bounds 112 99\n    have h\u2081\u2080\u2081 := h_bounds 112 100\n    norm_num at *\n    <;> simp_all [h_at_most_one]\n    <;> norm_num\n    <;> linarith\n  have h_n112_k_is_97 : (8 : \u211d) / 15 < 112 / (112 + 97) \u2227 (112 : \u211d) / (112 + 97) < 7 / 13 := by\n    have h\u2081 := h_bounds 112 97\n    have h\u2082 := h_interval_length 112\n    have h\u2083 := h_at_most_one 112\n    have h\u2084 := h_n112_unique_k\n    norm_num at h\u2081 h\u2082 h\u2083 h\u2084\n    <;> simp_all [div_lt_div_iff]\n    <;> norm_num\n    <;> linarith\n  have h_greatest : \u2200 n : \u2115, n \u2208 { n : \u2115 | 0 < n \u2227 \u2203! k : \u2115, (8 : \u211d) / 15 < n / (n + k) \u2227 (n : \u211d) / (n + k) < 7 / 13 } \u2192 n \u2264 112 := by\n    intro n hn\n    have hn' := hn\n    simp only [Set.mem_setOf_eq] at hn'\n    have hn'' := hn'\n    cases' hn'' with hn_pos hn_unique\n    have hn_le_112 : n \u2264 112 := by\n      by_contra h\n      have h' : n \u2265 113 := by linarith\n      have h'' := h_at_most_one n h'\n      obtain \u27e8k\u2081, k\u2082, h_k\u2081_k\u2082, h_k\u2081_ineq, h_k\u2082_ineq\u27e9 := h''\n      have h_k\u2081_eq_k\u2082 : k\u2081 = k\u2082 := by\n        apply ExistsUnique.unique hn_unique\n        <;> aesop\n      aesop\n    exact hn_le_112\n  refine' \u27e8by simpa [h_mem] using h_greatest 112, fun a ha => _\u27e9\n  have h\u2081 := h_bounds 0 0\n  have h\u2082 := h_bounds 0 1\n  have h\u2083 := h_bounds 1 0\n  have h\u2084 := h_bounds 1 1\n  have h\u2085 := h_bounds 2 1\n  have h\u2086 := h_bounds 3 1\n  have h\u2087 := h_bounds 4 1\n  have h\u2088 := h_bounds 5 2\n  have h\u2089 := h_bounds 6 2\n  have h\u2081\u2080 := h_bounds 7 3\n  have h\u2081\u2081 := h_bounds 8 3\n  have h\u2081\u2082 := h_bounds 9 4\n  have h\u2081\u2083 := h_bounds 10 4\n  have h\u2081\u2084 := h_bounds 11 5\n  have h\u2081\u2085 := h_bounds 12 6\n  have h\u2081\u2086 := h_bounds 13 7\n  have h\u2081\u2087 := h_bounds 14 8\n  have h\u2081\u2088 := h_bounds 15 9\n  have h\u2081\u2089 := h_bounds 16 10\n  have h\u2082\u2080 := h_bounds 17 11\n  have h\u2082\u2081 := h_bounds 18 12\n  have h\u2082\u2082 := h_bounds 19 13\n  have h\u2082\u2083 := h_bounds 20 14\n  have h\u2082\u2084 := h_bounds 21 15\n  have h\u2082\u2085 := h_bounds 22 16\n  have h\u2082\u2086 := h_bounds 23 17\n  have h\u2082\u2087 := h_bounds 24 19\n  have h\u2082\u2088 := h_bounds 25 20\n  have h\u2082\u2089 := h_bounds 26 21\n  have h\u2083\u2080 := h_bounds 27 22\n  have h\u2083\u2081 := h_bounds 28 23\n  have h\u2083\u2082 := h_bounds 29 24\n  have h\u2083\u2083 := h_bounds 30 26\n  have h\u2083\u2084 := h_bounds 31 27\n  have h\u2083\u2085 := h_bounds 32 29\n  have h\u2083\u2086 := h_bounds 33 30\n  have h\u2083\u2087 := h_bounds 34 31\n  have h\u2083\u2088 := h_bounds 35 33\n  have h\u2083\u2089 := h_bounds 36 34\n  have h\u2084\u2080 := h_bounds 37 35\n  have h\u2084\u2081 := h_bounds 38 37\n  have h\u2084\u2082 := h_bounds 39 38\n  have h\u2084\u2083 := h_bounds 40 40\n  have h\u2084\u2084 := h_bounds 41 41\n  have h\u2084\u2085 := h_bounds 42 43\n  have h\u2084\u2086 := h_bounds 43 44\n  have h\u2084\u2087 := h_bounds 44 46\n  have h\u2084\u2088 := h_bounds 45 47\n  have h\u2084\u2089 := h_bounds 46 49\n  have h\u2085\u2080 := h_bounds 47 51\n  have h\u2085\u2081 := h_bounds 48 52\n  have h\u2085\u2082 := h_bounds 49 54\n  have h\u2085\u2083 := h_bounds 50 56\n  have h\u2085\u2084 := h_bounds 51 58\n  have h\u2085\u2085 := h_bounds 52 60\n  have h\u2085\u2086 := h_bounds 53 62\n  have h\u2085\u2087 := h_bounds 54 64\n  have h\u2085\u2088 := h_bounds 55 66\n  have h\u2085\u2089 := h_bounds 56 68\n  have h\u2086\u2080 := h_bounds 57 71\n  have h\u2086\u2081 := h_bounds 58 73\n  have h\u2086\u2082 := h_bounds 59 75\n  have h\u2086\u2083 := h_bounds 60 78\n  have h\u2086\u2084 := h_bounds 61 80\n  have h\u2086\u2085 := h_bounds 62 83\n  have h\u2086\u2086 := h_bounds 63 85\n  have h\u2086\u2087 := h_bounds 64 88\n  have h\u2086\u2088 := h_bounds 65 91\n  have h\u2086\u2089 := h_bounds 66 94\n  have h\u2087\u2080 := h_bounds 67 97\n  have h\u2087\u2081 := h_bounds 68 100\n  have h\u2087\u2082 := h_bounds 69 103\n  have h\u2087\u2083 := h_bounds 70 107\n  have h\u2087\u2084 := h_bounds 71 110\n  have h\u2087\u2085 := h_bounds 72 114\n  have h\u2087\u2086 := h_bounds 73 118\n  have h\u2087\u2087 := h_bounds 74 122\n  have h\u2087\u2088 := h_bounds 75 126\n  have h\u2087\u2089 := h_bounds 76 131\n  have h\u2088\u2080 := h_bounds 77 135\n  have h\u2088\u2081 := h_bounds 78 140\n  have h\u2088\u2082 := h_bounds 79 145\n  have h\u2088\u2083 := h_bounds 80 150\n  have h\u2088\u2084 := h_bounds 81 155\n  have h\u2088\u2085 := h_bounds 82 161\n  have h\u2088\u2086 := h_bounds 83 166\n  have h\u2088\u2087 := h_bounds 84 172\n  have h\u2088\u2088 := h_bounds 85 178\n  have h\u2088\u2089 := h_bounds 86 184\n  have h\u2089\u2080 := h_bounds 87 190\n  have h\u2089\u2081 := h_bounds 88 197\n  have h\u2089\u2082 := h_bounds 89 203\n  have h\u2089\u2083 := h_bounds 90 210\n  have h\u2089\u2084 := h_bounds 91 217\n  have h\u2089\u2085 := h_bounds 92 224\n  have h\u2089\u2086 := h_bounds 93 232\n  have h\u2089\u2087 := h_bounds 94 239\n  have h\u2089\u2088 := h_bounds 95 247\n  have h\u2089\u2089 := h_bounds 96 255\n  have h\u2081\u2080\u2080 := h_bounds 97 263\n  have h\u2081\u2080\u2081 := h_bounds 98 271\n  have h\u2081\u2080\u2082 := h_bounds 99 280\n  have h\u2081\u2080\u2083 := h_bounds 100 288\n  norm_num at ha\n  <;> simp_all\n  <;> norm_num\n  <;> linarith\n", "proof_before": "theorem aime_1987_p8 :\n  IsGreatest { n : \u2115 | 0 < n \u2227 \u2203! k : \u2115, (8 : \u211d) / 15 < n / (n + k) \u2227 (n : \u211d) / (n + k) < 7 / 13 } 112  :=  by\n  have h_mem : 112 \u2208 { n : \u2115 | 0 < n \u2227 \u2203! k : \u2115, (8 : \u211d) / 15 < n / (n + k) \u2227 (n : \u211d) / (n + k) < 7 / 13 } := by\n    constructor\n    \u00b7 norm_num\n    use 97\n    constructor\n    \u00b7 norm_num [div_lt_div_iff]\n    intro k hk\n    have h\u2081 := hk.1\n    have h\u2082 := hk.2\n    field_simp [add_comm] at h\u2081 h\u2082\n    rw [\u2190 sub_pos] at h\u2081 h\u2082\n    field_simp [add_comm] at h\u2081 h\u2082\n    norm_cast at h\u2081 h\u2082\n    ring_nf at h\u2081 h\u2082\n    omega\n  have h_bounds : \u2200 (n k : \u2115), (8 : \u211d) / 15 < n / (n + k) \u2227 (n : \u211d) / (n + k) < 7 / 13 \u2194 \n                   (6 : \u211d) * n / 7 < k \u2227 k < (7 : \u211d) * n / 8 := by\n    norm_num\n    have h1 : \u2200 n k : \u2115, (8 : \u211d) / 15 < n / (n + k) \u2227 (n : \u211d) / (n + k) < 7 / 13 \u2194 (6 : \u211d) * n / 7 < k \u2227 k < (7 : \u211d) * n / 8 := by\n      intro n k\n      constructor\n      \u00b7 intro h\n        have h2 : (8 : \u211d) / 15 < n / (n + k) := h.1\n        have h3 : (n : \u211d) / (n + k) < 7 / 13 := h.2\n        constructor\n        \u00b7 have h4 : (0 : \u211d) < n + k := by\n            by_contra h5\n            have h6 : (n : \u211d) \u2264 0 := by linarith\n            have h7 : (n : \u211d) / (n + k) \u2264 0 := by exact div_nonpos_of_nonneg_of_nonpos (by positivity) (by linarith)\n            have h8 : (8 : \u211d) / 15 < 0 := by linarith\n            linarith\n          field_simp at h2 h3\n          rw [\u2190 sub_pos] at h2 h3\n          field_simp at h2 h3\n          rw [\u2190 sub_pos]\n          field_simp\n          ring_nf\n          nlinarith\n        \u00b7 have h4 : (0 : \u211d) < n + k := by\n            by_contra h5\n            have h6 : (n : \u211d) \u2264 0 := by linarith\n            have h7 : (n : \u211d) / (n + k) \u2264 0 := by exact div_nonpos_of_nonneg_of_nonpos (by positivity) (by linarith)\n            have h8 : (8 : \u211d) / 15 < 0 := by linarith\n            linarith\n          field_simp at h2 h3\n          rw [\u2190 sub_pos] at h2 h3\n          field_simp at h2 h3\n          rw [\u2190 sub_pos]\n          field_simp\n          ring_nf\n          nlinarith\n      \u00b7 intro h\n        have h2 : (6 : \u211d) * n / 7 < k := h.1\n        have h3 : k < (7 : \u211d) * n / 8 := h.2\n        constructor\n        \u00b7 have h4 : (0 : \u211d) < n + k := by\n            by_contra h5\n            have h6 : (n : \u211d) \u2264 0 := by linarith\n            have h7 : (6 : \u211d) * n / 7 \u2264 0 := by exact div_nonpos_of_nonpos_of_nonneg (by linarith) (by positivity)\n            linarith\n          field_simp at h2 h3 \u22a2\n          rw [\u2190 sub_pos] at h2 h3\n          field_simp at h2 h3 \u22a2\n          rw [\u2190 sub_pos]\n          field_simp\n          ring_nf\n          nlinarith\n        \u00b7 have h4 : (0 : \u211d) < n + k := by\n            by_contra h5\n            have h6 : (n : \u211d) \u2264 0 := by linarith\n            have h7 : (7 : \u211d) * n / 8 \u2264 0 := by exact div_nonpos_of_nonpos_of_nonneg (by linarith) (by positivity)\n            linarith\n          field_simp at h2 h3 \u22a2\n          rw [\u2190 sub_pos] at h2 h3\n          field_simp at h2 h3 \u22a2\n          rw [\u2190 sub_pos]\n          field_simp\n          ring_nf\n          nlinarith\n    intro n k\n    simp_all\n  have h_interval_length : \u2200 n : \u2115, (7 : \u211d) * n / 8 - (6 : \u211d) * n / 7 = n / 56 := by\n    intro n\n    have h := h_bounds 112 0\n    have h\u2081 := h_bounds 0 0\n    have h\u2082 := h_bounds 16 0\n    have h\u2083 := h_bounds 1 0\n    have h\u2084 := h_bounds 0 1\n    have h\u2085 := h_bounds 1 1\n    have h\u2086 := h_bounds 0 1\n    have h\u2087 := h_bounds 16 1\n    norm_num at h h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 h\u2087\n    <;> linarith\n  have h_at_most_one : \u2200 n : \u2115, n \u2265 113 \u2192 \u2203 k\u2081 k\u2082 : \u2115, k\u2081 \u2260 k\u2082 \u2227 \n                       (6 : \u211d) * n / 7 < k\u2081 \u2227 k\u2081 < (7 : \u211d) * n / 8 \u2227\n                       (6 : \u211d) * n / 7 < k\u2082 \u2227 k\u2082 < (7 : \u211d) * n / 8 := by\n    intro n hn\n    norm_num [h_bounds, h_interval_length] at *\n    use 6 * n / 7 + 1, 6 * n / 7 + 2\n    constructor\n    <;> norm_num [div_lt_iff, lt_div_iff] at *\n    <;> ring_nf at *\n    <;> norm_cast at *\n    <;> omega\n  have h_n112_unique_k : \u2203! k : \u2115, (8 : \u211d) / 15 < 112 / (112 + k) \u2227 (112 : \u211d) / (112 + k) < 7 / 13 := by\n    have h\u2081 := h_bounds 112 0\n    have h\u2082 := h_bounds 112 1\n    have h\u2083 := h_bounds 112 2\n    have h\u2084 := h_bounds 112 3\n    have h\u2085 := h_bounds 112 4\n    have h\u2086 := h_bounds 112 5\n    have h\u2087 := h_bounds 112 6\n    have h\u2088 := h_bounds 112 7\n    have h\u2089 := h_bounds 112 8\n    have h\u2081\u2080 := h_bounds 112 9\n    have h\u2081\u2081 := h_bounds 112 10\n    have h\u2081\u2082 := h_bounds 112 11\n    have h\u2081\u2083 := h_bounds 112 12\n    have h\u2081\u2084 := h_bounds 112 13\n    have h\u2081\u2085 := h_bounds 112 14\n    have h\u2081\u2086 := h_bounds 112 15\n    have h\u2081\u2087 := h_bounds 112 16\n    have h\u2081\u2088 := h_bounds 112 17\n    have h\u2081\u2089 := h_bounds 112 18\n    have h\u2082\u2080 := h_bounds 112 19\n    have h\u2082\u2081 := h_bounds 112 20\n    have h\u2082\u2082 := h_bounds 112 21\n    have h\u2082\u2083 := h_bounds 112 22\n    have h\u2082\u2084 := h_bounds 112 23\n    have h\u2082\u2085 := h_bounds 112 24\n    have h\u2082\u2086 := h_bounds 112 25\n    have h\u2082\u2087 := h_bounds 112 26\n    have h\u2082\u2088 := h_bounds 112 27\n    have h\u2082\u2089 := h_bounds 112 28\n    have h\u2083\u2080 := h_bounds 112 29\n    have h\u2083\u2081 := h_bounds 112 30\n    have h\u2083\u2082 := h_bounds 112 31\n    have h\u2083\u2083 := h_bounds 112 32\n    have h\u2083\u2084 := h_bounds 112 33\n    have h\u2083\u2085 := h_bounds 112 34\n    have h\u2083\u2086 := h_bounds 112 35\n    have h\u2083\u2087 := h_bounds 112 36\n    have h\u2083\u2088 := h_bounds 112 37\n    have h\u2083\u2089 := h_bounds 112 38\n    have h\u2084\u2080 := h_bounds 112 39\n    have h\u2084\u2081 := h_bounds 112 40\n    have h\u2084\u2082 := h_bounds 112 41\n    have h\u2084\u2083 := h_bounds 112 42\n    have h\u2084\u2084 := h_bounds 112 43\n    have h\u2084\u2085 := h_bounds 112 44\n    have h\u2084\u2086 := h_bounds 112 45\n    have h\u2084\u2087 := h_bounds 112 46\n    have h\u2084\u2088 := h_bounds 112 47\n    have h\u2084\u2089 := h_bounds 112 48\n    have h\u2085\u2080 := h_bounds 112 49\n    have h\u2085\u2081 := h_bounds 112 50\n    have h\u2085\u2082 := h_bounds 112 51\n    have h\u2085\u2083 := h_bounds 112 52\n    have h\u2085\u2084 := h_bounds 112 53\n    have h\u2085\u2085 := h_bounds 112 54\n    have h\u2085\u2086 := h_bounds 112 55\n    have h\u2085\u2087 := h_bounds 112 56\n    have h\u2085\u2088 := h_bounds 112 57\n    have h\u2085\u2089 := h_bounds 112 58\n    have h\u2086\u2080 := h_bounds 112 59\n    have h\u2086\u2081 := h_bounds 112 60\n    have h\u2086\u2082 := h_bounds 112 61\n    have h\u2086\u2083 := h_bounds 112 62\n    have h\u2086\u2084 := h_bounds 112 63\n    have h\u2086\u2085 := h_bounds 112 64\n    have h\u2086\u2086 := h_bounds 112 65\n    have h\u2086\u2087 := h_bounds 112 66\n    have h\u2086\u2088 := h_bounds 112 67\n    have h\u2086\u2089 := h_bounds 112 68\n    have h\u2087\u2080 := h_bounds 112 69\n    have h\u2087\u2081 := h_bounds 112 70\n    have h\u2087\u2082 := h_bounds 112 71\n    have h\u2087\u2083 := h_bounds 112 72\n    have h\u2087\u2084 := h_bounds 112 73\n    have h\u2087\u2085 := h_bounds 112 74\n    have h\u2087\u2086 := h_bounds 112 75\n    have h\u2087\u2087 := h_bounds 112 76\n    have h\u2087\u2088 := h_bounds 112 77\n    have h\u2087\u2089 := h_bounds 112 78\n    have h\u2088\u2080 := h_bounds 112 79\n    have h\u2088\u2081 := h_bounds 112 80\n    have h\u2088\u2082 := h_bounds 112 81\n    have h\u2088\u2083 := h_bounds 112 82\n    have h\u2088\u2084 := h_bounds 112 83\n    have h\u2088\u2085 := h_bounds 112 84\n    have h\u2088\u2086 := h_bounds 112 85\n    have h\u2088\u2087 := h_bounds 112 86\n    have h\u2088\u2088 := h_bounds 112 87\n    have h\u2088\u2089 := h_bounds 112 88\n    have h\u2089\u2080 := h_bounds 112 89\n    have h\u2089\u2081 := h_bounds 112 90\n    have h\u2089\u2082 := h_bounds 112 91\n    have h\u2089\u2083 := h_bounds 112 92\n    have h\u2089\u2084 := h_bounds 112 93\n    have h\u2089\u2085 := h_bounds 112 94\n    have h\u2089\u2086 := h_bounds 112 95\n    have h\u2089\u2087 := h_bounds 112 96\n    have h\u2089\u2088 := h_bounds 112 97\n    have h\u2089\u2089 := h_bounds 112 98\n    have h\u2081\u2080\u2080 := h_bounds 112 99\n    have h\u2081\u2080\u2081 := h_bounds 112 100\n    norm_num at *\n    <;> simp_all [h_at_most_one]\n    <;> norm_num\n    <;> linarith\n  have h_n112_k_is_97 : (8 : \u211d) / 15 < 112 / (112 + 97) \u2227 (112 : \u211d) / (112 + 97) < 7 / 13 := by\n    have h\u2081 := h_bounds 112 97\n    have h\u2082 := h_interval_length 112\n    have h\u2083 := h_at_most_one 112\n    have h\u2084 := h_n112_unique_k\n    norm_num at h\u2081 h\u2082 h\u2083 h\u2084\n    <;> simp_all [div_lt_div_iff]\n    <;> norm_num\n    <;> linarith\n  have h_greatest : \u2200 n : \u2115, n \u2208 { n : \u2115 | 0 < n \u2227 \u2203! k : \u2115, (8 : \u211d) / 15 < n / (n + k) \u2227 (n : \u211d) / (n + k) < 7 / 13 } \u2192 n \u2264 112 := by\n    intro n hn\n    have hn' := hn\n    simp only [Set.mem_setOf_eq] at hn'\n    have hn'' := hn'\n    cases' hn'' with hn_pos hn_unique\n    have hn_le_112 : n \u2264 112 := by\n      by_contra h\n      have h' : n \u2265 113 := by linarith\n      have h'' := h_at_most_one n h'\n      obtain \u27e8k\u2081, k\u2082, h_k\u2081_k\u2082, h_k\u2081_ineq, h_k\u2082_ineq\u27e9 := h''\n      have h_k\u2081_eq_k\u2082 : k\u2081 = k\u2082 := by\n        apply ExistsUnique.unique hn_unique\n        <;> aesop\n      aesop\n    exact hn_le_112\n  refine' \u27e8by simpa [h_mem] using h_greatest 112, fun a ha => _\u27e9\n  have h\u2081 := h_bounds 0 0\n  have h\u2082 := h_bounds 0 1\n  have h\u2083 := h_bounds 1 0\n  have h\u2084 := h_bounds 1 1\n  have h\u2085 := h_bounds 2 1\n  have h\u2086 := h_bounds 3 1\n  have h\u2087 := h_bounds 4 1\n  have h\u2088 := h_bounds 5 2\n  have h\u2089 := h_bounds 6 2\n  have h\u2081\u2080 := h_bounds 7 3\n  have h\u2081\u2081 := h_bounds 8 3\n  have h\u2081\u2082 := h_bounds 9 4\n  have h\u2081\u2083 := h_bounds 10 4\n  have h\u2081\u2084 := h_bounds 11 5\n  have h\u2081\u2085 := h_bounds 12 6\n  have h\u2081\u2086 := h_bounds 13 7\n  have h\u2081\u2087 := h_bounds 14 8\n  have h\u2081\u2088 := h_bounds 15 9\n  have h\u2081\u2089 := h_bounds 16 10\n  have h\u2082\u2080 := h_bounds 17 11\n  have h\u2082\u2081 := h_bounds 18 12\n  have h\u2082\u2082 := h_bounds 19 13\n  have h\u2082\u2083 := h_bounds 20 14\n  have h\u2082\u2084 := h_bounds 21 15\n  have h\u2082\u2085 := h_bounds 22 16\n  have h\u2082\u2086 := h_bounds 23 17\n  have h\u2082\u2087 := h_bounds 24 19\n  have h\u2082\u2088 := h_bounds 25 20\n  have h\u2082\u2089 := h_bounds 26 21\n  have h\u2083\u2080 := h_bounds 27 22\n  have h\u2083\u2081 := h_bounds 28 23\n  have h\u2083\u2082 := h_bounds 29 24\n  have h\u2083\u2083 := h_bounds 30 26\n  have h\u2083\u2084 := h_bounds 31 27\n  have h\u2083\u2085 := h_bounds 32 29\n  have h\u2083\u2086 := h_bounds 33 30\n  have h\u2083\u2087 := h_bounds 34 31\n  have h\u2083\u2088 := h_bounds 35 33\n  have h\u2083\u2089 := h_bounds 36 34\n  have h\u2084\u2080 := h_bounds 37 35\n  have h\u2084\u2081 := h_bounds 38 37\n  have h\u2084\u2082 := h_bounds 39 38\n  have h\u2084\u2083 := h_bounds 40 40\n  have h\u2084\u2084 := h_bounds 41 41\n  have h\u2084\u2085 := h_bounds 42 43\n  have h\u2084\u2086 := h_bounds 43 44\n  have h\u2084\u2087 := h_bounds 44 46\n  have h\u2084\u2088 := h_bounds 45 47\n  have h\u2084\u2089 := h_bounds 46 49\n  have h\u2085\u2080 := h_bounds 47 51\n  have h\u2085\u2081 := h_bounds 48 52\n  have h\u2085\u2082 := h_bounds 49 54\n  have h\u2085\u2083 := h_bounds 50 56\n  have h\u2085\u2084 := h_bounds 51 58\n  have h\u2085\u2085 := h_bounds 52 60\n  have h\u2085\u2086 := h_bounds 53 62\n  have h\u2085\u2087 := h_bounds 54 64\n  have h\u2085\u2088 := h_bounds 55 66\n  have h\u2085\u2089 := h_bounds 56 68\n  have h\u2086\u2080 := h_bounds 57 71\n  have h\u2086\u2081 := h_bounds 58 73\n  have h\u2086\u2082 := h_bounds 59 75\n  have h\u2086\u2083 := h_bounds 60 78\n  have h\u2086\u2084 := h_bounds 61 80\n  have h\u2086\u2085 := h_bounds 62 83\n  have h\u2086\u2086 := h_bounds 63 85\n  have h\u2086\u2087 := h_bounds 64 88\n  have h\u2086\u2088 := h_bounds 65 91\n  have h\u2086\u2089 := h_bounds 66 94\n  have h\u2087\u2080 := h_bounds 67 97\n  have h\u2087\u2081 := h_bounds 68 100\n  have h\u2087\u2082 := h_bounds 69 103\n  have h\u2087\u2083 := h_bounds 70 107\n  have h\u2087\u2084 := h_bounds 71 110\n  have h\u2087\u2085 := h_bounds 72 114\n  have h\u2087\u2086 := h_bounds 73 118\n  have h\u2087\u2087 := h_bounds 74 122\n  have h\u2087\u2088 := h_bounds 75 126\n  have h\u2087\u2089 := h_bounds 76 131\n  have h\u2088\u2080 := h_bounds 77 135\n  have h\u2088\u2081 := h_bounds 78 140\n  have h\u2088\u2082 := h_bounds 79 145\n  have h\u2088\u2083 := h_bounds 80 150\n  have h\u2088\u2084 := h_bounds 81 155\n  have h\u2088\u2085 := h_bounds 82 161\n  have h\u2088\u2086 := h_bounds 83 166\n  have h\u2088\u2087 := h_bounds 84 172\n  have h\u2088\u2088 := h_bounds 85 178\n  have h\u2088\u2089 := h_bounds 86 184\n  have h\u2089\u2080 := h_bounds 87 190\n  have h\u2089\u2081 := h_bounds 88 197\n  have h\u2089\u2082 := h_bounds 89 203\n  have h\u2089\u2083 := h_bounds 90 210\n  have h\u2089\u2084 := h_bounds 91 217\n  have h\u2089\u2085 := h_bounds 92 224\n  have h\u2089\u2086 := h_bounds 93 232\n  have h\u2089\u2087 := h_bounds 94 239\n  have h\u2089\u2088 := h_bounds 95 247\n  have h\u2089\u2089 := h_bounds 96 255\n  have h\u2081\u2080\u2080 := h_bounds 97 263\n  have h\u2081\u2080\u2081 := h_bounds 98 271\n  have h\u2081\u2080\u2082 := h_bounds 99 280\n  have h\u2081\u2080\u2083 := h_bounds 100 288\n  norm_num at ha\n  <;> simp_all\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 60, "alphabet": 35, "compiles": false, "id": "amc12_2000_p15", "proof_after": "theorem amc12_2000_p15 (f : \u2102 \u2192 \u2102) (h\u2080 : \u2200 x, f (x / 3) = x ^ 2 + x + 1)\n  (h\u2081 : Fintype (f \u207b\u00b9' {7})) : (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y / 3) = -1 / 9  :=  by\n  have h\u2082 : \u2200 y, f y = 9 * y^2 + 3 * y + 1 := by\n    intro y\n    have h\u2082 := h\u2080 (3 * y)\n    simp [mul_assoc] at h\u2082\n    have h\u2083 := h\u2080 (3 * y + 1)\n    simp [mul_assoc] at h\u2083\n    have h\u2084 := h\u2080 (3 * y - 1)\n    simp [mul_assoc] at h\u2084\n    ring_nf at h\u2082 h\u2083 h\u2084 \u22a2\n    <;> simp_all [sub_eq_add_neg, add_assoc]\n    <;> linarith\n  have h\u2083 : \u2200 y, f y = 7 \u2194 9 * y^2 + 3 * y - 6 = 0 := by\n    intro y\n    simp [h\u2082, sub_eq_zero, mul_comm]\n    <;> ring_nf <;> norm_num\n    <;> exact \u27e8fun h => by linear_combination h, fun h => by linear_combination h\u27e9\n  have h\u2084 : (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y) = -1/3 := by\n    have h\u2084 : f \u207b\u00b9' {7} = {-1, 2/3} := by\n      ext y\n      simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_insert_iff]\n      constructor\n      \u00b7 intro h\n        have h\u2085 : f y = 7 := h\n        have h\u2086 : 9 * y ^ 2 + 3 * y - 6 = 0 := by simpa [h\u2082, h\u2085] using h\u2083 y |>.mp h\u2085\n        have h\u2087 : y = -1 \u2228 y = 2 / 3 := by\n          apply or_iff_not_imp_left.mpr\n          intro h\u2088\n          apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2088)\n          linear_combination h\u2086 / 9\n        simp_all\n      \u00b7 intro h\n        have h\u2085 : y = -1 \u2228 y = 2 / 3 := h\n        have h\u2086 : f y = 7 := by\n          cases' h\u2085 with h\u2085 h\u2085\n          \u00b7 simp_all [h\u2082, h\u2085]\n            norm_num\n          \u00b7 simp_all [h\u2082, h\u2085]\n            norm_num\n        exact h\u2086\n    simp [h\u2084]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n  have h\u2085 : (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y / 3) = -1/9 := by\n    have h\u2085 : \u2200 y, f y = 7 \u2194 9 * y ^ 2 + 3 * y - 6 = 0 := by simpa using h\u2083\n    have h\u2086 : (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y) = -1/3 := by simpa using h\u2084\n    have h\u2087 : (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y / 3) = (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y) / 3 := by\n      simp [Finset.sum_div]\n    rw [h\u2087, h\u2086]\n    ring\n  norm_num at h\u2080 h\u2082 h\u2083 h\u2084 h\u2085 \u22a2\n  <;> simp_all [Complex.ext_iff]\n  <;> norm_num\n  <;> linarith\n", "proof_before": "theorem amc12_2000_p15 (f : \u2102 \u2192 \u2102) (h\u2080 : \u2200 x, f (x / 3) = x ^ 2 + x + 1)\n  (h\u2081 : Fintype (f \u207b\u00b9' {7})) : (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y / 3) = -1 / 9  :=  by\n  have h\u2082 : \u2200 y, f y = 9 * y^2 + 3 * y + 1 := by\n    intro y\n    have h\u2082 := h\u2080 (3 * y)\n    simp [mul_assoc] at h\u2082\n    have h\u2083 := h\u2080 (3 * y + 1)\n    simp [mul_assoc] at h\u2083\n    have h\u2084 := h\u2080 (3 * y - 1)\n    simp [mul_assoc] at h\u2084\n    ring_nf at h\u2082 h\u2083 h\u2084 \u22a2\n    <;> simp_all [sub_eq_add_neg, add_assoc]\n    <;> linarith\n  have h\u2083 : \u2200 y, f y = 7 \u2194 9 * y^2 + 3 * y - 6 = 0 := by\n    intro y\n    simp [h\u2082, sub_eq_zero, mul_comm]\n    <;> ring_nf <;> norm_num\n    <;> exact \u27e8fun h => by linear_combination h, fun h => by linear_combination h\u27e9\n  have h\u2084 : (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y) = -1/3 := by\n    have h\u2084 : f \u207b\u00b9' {7} = {-1, 2/3} := by\n      ext y\n      simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_insert_iff]\n      constructor\n      \u00b7 intro h\n        have h\u2085 : f y = 7 := h\n        have h\u2086 : 9 * y ^ 2 + 3 * y - 6 = 0 := by simpa [h\u2082, h\u2085] using h\u2083 y |>.mp h\u2085\n        have h\u2087 : y = -1 \u2228 y = 2 / 3 := by\n          apply or_iff_not_imp_left.mpr\n          intro h\u2088\n          apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2088)\n          linear_combination h\u2086 / 9\n        simp_all\n      \u00b7 intro h\n        have h\u2085 : y = -1 \u2228 y = 2 / 3 := h\n        have h\u2086 : f y = 7 := by\n          cases' h\u2085 with h\u2085 h\u2085\n          \u00b7 simp_all [h\u2082, h\u2085]\n            norm_num\n          \u00b7 simp_all [h\u2082, h\u2085]\n            norm_num\n        exact h\u2086\n    simp [h\u2084]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n  have h\u2085 : (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y / 3) = -1/9 := by\n    have h\u2085 : \u2200 y, f y = 7 \u2194 9 * y ^ 2 + 3 * y - 6 = 0 := by simpa using h\u2083\n    have h\u2086 : (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y) = -1/3 := by simpa using h\u2084\n    have h\u2087 : (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y / 3) = (\u2211 y in (f \u207b\u00b9' {7}).toFinset, y) / 3 := by\n      simp [Finset.sum_div]\n    rw [h\u2087, h\u2086]\n    ring\n  norm_num at h\u2080 h\u2082 h\u2083 h\u2084 h\u2085 \u22a2\n  <;> simp_all [Complex.ext_iff]\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 222, "alphabet": 70, "compiles": false, "id": "aime_1997_p11", "proof_after": "theorem aime_1997_p11 (x : \u211d)\n    (h\u2080 :\n      x =\n        (\u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180)) /\n          \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180)) :\n    Int.floor (100 * x) = 241  :=  by\n  have h\u2081 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) > 0 := by\n    have h\u2081\u2080 : \u2200 (n : \u2115), n \u2208 Finset.Icc (1 : \u2115) 44 \u2192 Real.sin (n * Real.pi / 180) > 0 := by\n      intro n hn\n      have h\u2081\u2081 : 1 \u2264 n \u2227 n \u2264 44 := Finset.mem_Icc.mp hn\n      have h\u2081\u2082 : 1 \u2264 (n : \u211d) := by exact_mod_cast h\u2081\u2081.1\n      have h\u2081\u2083 : (n : \u211d) \u2264 44 := by exact_mod_cast h\u2081\u2081.2\n      have h\u2081\u2084 : 0 < (n : \u211d) * Real.pi / 180 := by\n        have h\u2081\u2085 : 0 < Real.pi := Real.pi_pos\n        have h\u2081\u2086 : 0 < (n : \u211d) := by linarith\n        positivity\n      have h\u2081\u2085 : (n : \u211d) * Real.pi / 180 < Real.pi := by\n        have h\u2081\u2086 : (n : \u211d) \u2264 44 := by exact_mod_cast h\u2081\u2081.2\n        have h\u2081\u2087 : 0 < Real.pi := Real.pi_pos\n        nlinarith [Real.pi_gt_three]\n      have h\u2081\u2086 : Real.sin ((n : \u211d) * Real.pi / 180) > 0 := by\n        apply Real.sin_pos_of_pos_of_lt_pi\n        \u00b7 exact h\u2081\u2084\n        \u00b7 exact h\u2081\u2085\n      exact_mod_cast h\u2081\u2086\n    have h\u2081\u2087 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) > 0 := by\n      apply Finset.sum_pos\n      \u00b7 intro n hn\n        exact h\u2081\u2080 n hn\n      \u00b7 exact \u27e81, by\n          norm_num [Finset.mem_Icc]\n          <;>\n          (try norm_num) <;>\n          (try linarith [Real.pi_gt_three])\u27e9\n    exact h\u2081\u2087\n  have h\u2082 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) = (1 + Real.sqrt 2) * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n    have h\u2082\u2081 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) - \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) = Real.sqrt 2 * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n      have h\u2082\u2082 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) - \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) = \u2211 n in Finset.Icc (1 : \u2115) 44, (Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180)) := by\n        rw [\u2190 Finset.sum_sub_distrib]\n      rw [h\u2082\u2082]\n      have h\u2082\u2083 : \u2211 n in Finset.Icc (1 : \u2115) 44, (Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180)) = Real.sqrt 2 * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n        have h\u2082\u2084 : \u2211 n in Finset.Icc (1 : \u2115) 44, (Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180)) = \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sqrt 2 * Real.sin ((45 - n : \u2115) * Real.pi / 180) := by\n          apply Finset.sum_congr rfl\n          intro n hn\n          have h\u2082\u2085 : n \u2208 Finset.Icc (1 : \u2115) 44 := hn\n          have h\u2082\u2086 : 1 \u2264 n \u2227 n \u2264 44 := Finset.mem_Icc.mp h\u2082\u2085\n          have h\u2082\u2087 : Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180) = Real.sqrt 2 * Real.sin ((45 - n : \u2115) * Real.pi / 180) := by\n            have h\u2082\u2088 : (n : \u2115) \u2264 44 := by linarith\n            have h\u2082\u2089 : (n : \u2115) \u2265 1 := by linarith\n            have h\u2083\u2080 : Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180) = Real.sqrt 2 * Real.sin ((45 - n : \u2115) * Real.pi / 180) := by\n              have h\u2083\u2081 : Real.sin ((45 - n : \u2115) * Real.pi / 180) = Real.sin ((45 - n : \u2115) * Real.pi / 180) := rfl\n              rw [h\u2083\u2081]\n              have h\u2083\u2082 : Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180) = Real.sqrt 2 * Real.sin ((45 - n : \u2115) * Real.pi / 180) := by\n                have h\u2083\u2083 : Real.sin ((45 - n : \u2115) * Real.pi / 180) = Real.sin (Real.pi / 4 - (n * Real.pi / 180)) := by\n                  have h\u2083\u2084 : ((45 - n : \u2115) : \u211d) * Real.pi / 180 = Real.pi / 4 - (n * Real.pi / 180) := by\n                    have h\u2083\u2085 : n \u2264 44 := by linarith\n                    interval_cases n <;> norm_num [Nat.cast_add, Nat.cast_one, Nat.cast_sub, Nat.cast_mul] <;>\n                      (try ring_nf) <;> (try field_simp) <;> (try ring_nf) <;> (try norm_num) <;> (try linarith [Real.pi_pos])\n                    <;> (try ring_nf) <;> (try field_simp) <;> (try ring_nf) <;> (try norm_num) <;> (try linarith [Real.pi_pos])\n                  rw [h\u2083\u2084]\n                  <;> ring_nf\n                rw [h\u2083\u2083]\n                have h\u2083\u2084 : Real.sin (Real.pi / 4 - (n * Real.pi / 180)) = Real.sin (Real.pi / 4) * Real.cos (n * Real.pi / 180) - Real.cos (Real.pi / 4) * Real.sin (n * Real.pi / 180) := by\n                  rw [Real.sin_sub]\n                rw [h\u2083\u2084]\n                have h\u2083\u2085 : Real.sin (Real.pi / 4) = Real.sqrt 2 / 2 := by norm_num\n                have h\u2083\u2086 : Real.cos (Real.pi / 4) = Real.sqrt 2 / 2 := by norm_num\n                rw [h\u2083\u2085, h\u2083\u2086]\n                ring_nf\n                <;> field_simp [Real.sqrt_eq_iff_sq_eq] <;> ring_nf <;> norm_num <;>\n                  nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n              rw [h\u2083\u2082]\n            exact h\u2083\u2080\n          rw [h\u2082\u2087]\n        rw [h\u2082\u2084]\n        have h\u2082\u2085 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sqrt 2 * Real.sin ((45 - n : \u2115) * Real.pi / 180) = Real.sqrt 2 * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin ((45 - n : \u2115) * Real.pi / 180) := by\n          rw [Finset.mul_sum]\n        rw [h\u2082\u2085]\n        have h\u2082\u2086 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin ((45 - n : \u2115) * Real.pi / 180) = \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n          have h\u2082\u2087 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin ((45 - n : \u2115) * Real.pi / 180) = \u2211 k in Finset.Icc (1 : \u2115) 44, Real.sin (k * Real.pi / 180) := by\n            apply Finset.sum_bij' (fun (n : \u2115) _ \u21a6 (45 - n : \u2115)) (fun (n : \u2115) _ \u21a6 (45 - n : \u2115))\n            <;> simp_all [Finset.mem_Icc, Nat.le_of_lt_succ, Nat.succ_le_iff]\n            <;>\n              (try omega) <;>\n              (try ring_nf) <;>\n              (try simp_all [Real.sin_add, Real.sin_sub, Real.cos_add, Real.cos_sub]) <;>\n              (try field_simp) <;>\n              (try ring_nf) <;>\n              (try norm_num) <;>\n              (try linarith) <;>\n              (try omega)\n            <;>\n              (try {\n                rcases n with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n                  simp_all [Finset.mem_Icc, Nat.le_of_lt_succ, Nat.succ_le_iff] <;>\n                  (try omega) <;>\n                  (try ring_nf) <;>\n                  (try simp_all [Real.sin_add, Real.sin_sub, Real.cos_add, Real.cos_sub]) <;>\n                  (try field_simp) <;>\n                  (try ring_nf) <;>\n                  (try norm_num) <;>\n                  (try linarith) <;>\n                  (try omega)\n              })\n          rw [h\u2082\u2087]\n        rw [h\u2082\u2086]\n      rw [h\u2082\u2083]\n      <;> ring_nf\n      <;> linarith\n    have h\u2082\u2082 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) = (1 + Real.sqrt 2) * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n      have h\u2082\u2083 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) - \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) = Real.sqrt 2 * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := h\u2082\u2081\n      have h\u2082\u2084 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) = (1 + Real.sqrt 2) * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n        linarith\n      exact h\u2082\u2084\n    exact h\u2082\u2082\n  have h\u2083 : x = 1 + Real.sqrt 2 := by\n    rw [h\u2080]\n    have h\u2083\u2081 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) > 0 := h\u2081\n    have h\u2083\u2082 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) = (1 + Real.sqrt 2) * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := h\u2082\n    have h\u2083\u2083 : (\u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180)) / \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) = 1 + Real.sqrt 2 := by\n      rw [h\u2083\u2082]\n      have h\u2083\u2084 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) \u2260 0 := by linarith\n      field_simp [h\u2083\u2084]\n      <;> ring_nf\n      <;> field_simp [h\u2083\u2084]\n      <;> linarith\n    rw [h\u2083\u2083]\n    <;> ring_nf\n    <;> field_simp\n    <;> linarith\n  have h\u2084 : 100 * x = 100 + 100 * Real.sqrt 2 := by\n    rw [h\u2083]\n    <;> ring\n    <;> field_simp\n    <;> ring\n    <;> linarith [Real.sqrt_nonneg 2]\n  have h\u2085 : (241 : \u211d) \u2264 100 * x := by\n    rw [h\u2084]\n    have h\u2085\u2081 : Real.sqrt 2 \u2265 0 := Real.sqrt_nonneg _\n    have h\u2085\u2082 : (141 : \u211d) \u2264 100 * Real.sqrt 2 := by\n      nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num),\n        Real.sqrt_nonneg 2, Real.pi_gt_three]\n    linarith\n  have h\u2086 : (100 : \u211d) * x < 242 := by\n    rw [h\u2084]\n    have h\u2086\u2081 : Real.sqrt 2 \u2265 0 := Real.sqrt_nonneg _\n    have h\u2086\u2082 : (100 : \u211d) * Real.sqrt 2 < 142 := by\n      nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num), Real.sqrt_nonneg 2]\n    linarith\n  have h\u2087 : Int.floor (100 * x) = 241 := by\n    rw [Int.floor_eq_iff]\n    constructor <;> norm_num at h\u2085 h\u2086 \u22a2 <;>\n    (try linarith) <;>\n    (try\n      {\n        nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      })\n    <;>\n    (try\n      {\n        norm_num at h\u2085 h\u2086 \u22a2\n        <;>\n        (try linarith)\n        <;>\n        (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n      })\n    <;>\n    (try\n      {\n        norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n        <;>\n        (try linarith)\n        <;>\n        (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n      })\n    <;>\n    (try\n      {\n        norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n        <;>\n        (try linarith)\n        <;>\n        (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n      })\n    <;>\n    (try\n      {\n        norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n        <;>\n        (try linarith)\n        <;>\n        (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n      })\n    <;>\n    (try\n      {\n        norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n        <;>\n        (try linarith)\n        <;>\n        (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n      })\n  rw [h\u2087]\n  <;> norm_num\n  <;>\n  (try\n    {\n      norm_num at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n  <;>\n  (try\n    {\n      norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n  <;>\n  (try\n    {\n      norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n  <;>\n  (try\n    {\n      norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n  <;>\n  (try\n    {\n      norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n  <;>\n  (try\n    {\n      norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n", "proof_before": "theorem aime_1997_p11 (x : \u211d)\n    (h\u2080 :\n      x =\n        (\u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180)) /\n          \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180)) :\n    Int.floor (100 * x) = 241  :=  by\n  have h\u2081 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) > 0 := by\n    have h\u2081\u2080 : \u2200 (n : \u2115), n \u2208 Finset.Icc (1 : \u2115) 44 \u2192 Real.sin (n * Real.pi / 180) > 0 := by\n      intro n hn\n      have h\u2081\u2081 : 1 \u2264 n \u2227 n \u2264 44 := Finset.mem_Icc.mp hn\n      have h\u2081\u2082 : 1 \u2264 (n : \u211d) := by exact_mod_cast h\u2081\u2081.1\n      have h\u2081\u2083 : (n : \u211d) \u2264 44 := by exact_mod_cast h\u2081\u2081.2\n      have h\u2081\u2084 : 0 < (n : \u211d) * Real.pi / 180 := by\n        have h\u2081\u2085 : 0 < Real.pi := Real.pi_pos\n        have h\u2081\u2086 : 0 < (n : \u211d) := by linarith\n        positivity\n      have h\u2081\u2085 : (n : \u211d) * Real.pi / 180 < Real.pi := by\n        have h\u2081\u2086 : (n : \u211d) \u2264 44 := by exact_mod_cast h\u2081\u2081.2\n        have h\u2081\u2087 : 0 < Real.pi := Real.pi_pos\n        nlinarith [Real.pi_gt_three]\n      have h\u2081\u2086 : Real.sin ((n : \u211d) * Real.pi / 180) > 0 := by\n        apply Real.sin_pos_of_pos_of_lt_pi\n        \u00b7 exact h\u2081\u2084\n        \u00b7 exact h\u2081\u2085\n      exact_mod_cast h\u2081\u2086\n    have h\u2081\u2087 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) > 0 := by\n      apply Finset.sum_pos\n      \u00b7 intro n hn\n        exact h\u2081\u2080 n hn\n      \u00b7 exact \u27e81, by\n          norm_num [Finset.mem_Icc]\n          <;>\n          (try norm_num) <;>\n          (try linarith [Real.pi_gt_three])\u27e9\n    exact h\u2081\u2087\n  have h\u2082 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) = (1 + Real.sqrt 2) * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n    have h\u2082\u2081 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) - \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) = Real.sqrt 2 * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n      have h\u2082\u2082 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) - \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) = \u2211 n in Finset.Icc (1 : \u2115) 44, (Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180)) := by\n        rw [\u2190 Finset.sum_sub_distrib]\n      rw [h\u2082\u2082]\n      have h\u2082\u2083 : \u2211 n in Finset.Icc (1 : \u2115) 44, (Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180)) = Real.sqrt 2 * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n        have h\u2082\u2084 : \u2211 n in Finset.Icc (1 : \u2115) 44, (Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180)) = \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sqrt 2 * Real.sin ((45 - n : \u2115) * Real.pi / 180) := by\n          apply Finset.sum_congr rfl\n          intro n hn\n          have h\u2082\u2085 : n \u2208 Finset.Icc (1 : \u2115) 44 := hn\n          have h\u2082\u2086 : 1 \u2264 n \u2227 n \u2264 44 := Finset.mem_Icc.mp h\u2082\u2085\n          have h\u2082\u2087 : Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180) = Real.sqrt 2 * Real.sin ((45 - n : \u2115) * Real.pi / 180) := by\n            have h\u2082\u2088 : (n : \u2115) \u2264 44 := by linarith\n            have h\u2082\u2089 : (n : \u2115) \u2265 1 := by linarith\n            have h\u2083\u2080 : Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180) = Real.sqrt 2 * Real.sin ((45 - n : \u2115) * Real.pi / 180) := by\n              have h\u2083\u2081 : Real.sin ((45 - n : \u2115) * Real.pi / 180) = Real.sin ((45 - n : \u2115) * Real.pi / 180) := rfl\n              rw [h\u2083\u2081]\n              have h\u2083\u2082 : Real.cos (n * Real.pi / 180) - Real.sin (n * Real.pi / 180) = Real.sqrt 2 * Real.sin ((45 - n : \u2115) * Real.pi / 180) := by\n                have h\u2083\u2083 : Real.sin ((45 - n : \u2115) * Real.pi / 180) = Real.sin (Real.pi / 4 - (n * Real.pi / 180)) := by\n                  have h\u2083\u2084 : ((45 - n : \u2115) : \u211d) * Real.pi / 180 = Real.pi / 4 - (n * Real.pi / 180) := by\n                    have h\u2083\u2085 : n \u2264 44 := by linarith\n                    interval_cases n <;> norm_num [Nat.cast_add, Nat.cast_one, Nat.cast_sub, Nat.cast_mul] <;>\n                      (try ring_nf) <;> (try field_simp) <;> (try ring_nf) <;> (try norm_num) <;> (try linarith [Real.pi_pos])\n                    <;> (try ring_nf) <;> (try field_simp) <;> (try ring_nf) <;> (try norm_num) <;> (try linarith [Real.pi_pos])\n                  rw [h\u2083\u2084]\n                  <;> ring_nf\n                rw [h\u2083\u2083]\n                have h\u2083\u2084 : Real.sin (Real.pi / 4 - (n * Real.pi / 180)) = Real.sin (Real.pi / 4) * Real.cos (n * Real.pi / 180) - Real.cos (Real.pi / 4) * Real.sin (n * Real.pi / 180) := by\n                  rw [Real.sin_sub]\n                rw [h\u2083\u2084]\n                have h\u2083\u2085 : Real.sin (Real.pi / 4) = Real.sqrt 2 / 2 := by norm_num\n                have h\u2083\u2086 : Real.cos (Real.pi / 4) = Real.sqrt 2 / 2 := by norm_num\n                rw [h\u2083\u2085, h\u2083\u2086]\n                ring_nf\n                <;> field_simp [Real.sqrt_eq_iff_sq_eq] <;> ring_nf <;> norm_num <;>\n                  nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n              rw [h\u2083\u2082]\n            exact h\u2083\u2080\n          rw [h\u2082\u2087]\n        rw [h\u2082\u2084]\n        have h\u2082\u2085 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sqrt 2 * Real.sin ((45 - n : \u2115) * Real.pi / 180) = Real.sqrt 2 * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin ((45 - n : \u2115) * Real.pi / 180) := by\n          rw [Finset.mul_sum]\n        rw [h\u2082\u2085]\n        have h\u2082\u2086 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin ((45 - n : \u2115) * Real.pi / 180) = \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n          have h\u2082\u2087 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin ((45 - n : \u2115) * Real.pi / 180) = \u2211 k in Finset.Icc (1 : \u2115) 44, Real.sin (k * Real.pi / 180) := by\n            apply Finset.sum_bij' (fun (n : \u2115) _ \u21a6 (45 - n : \u2115)) (fun (n : \u2115) _ \u21a6 (45 - n : \u2115))\n            <;> simp_all [Finset.mem_Icc, Nat.le_of_lt_succ, Nat.succ_le_iff]\n            <;>\n              (try omega) <;>\n              (try ring_nf) <;>\n              (try simp_all [Real.sin_add, Real.sin_sub, Real.cos_add, Real.cos_sub]) <;>\n              (try field_simp) <;>\n              (try ring_nf) <;>\n              (try norm_num) <;>\n              (try linarith) <;>\n              (try omega)\n            <;>\n              (try {\n                rcases n with (_ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | _) <;>\n                  simp_all [Finset.mem_Icc, Nat.le_of_lt_succ, Nat.succ_le_iff] <;>\n                  (try omega) <;>\n                  (try ring_nf) <;>\n                  (try simp_all [Real.sin_add, Real.sin_sub, Real.cos_add, Real.cos_sub]) <;>\n                  (try field_simp) <;>\n                  (try ring_nf) <;>\n                  (try norm_num) <;>\n                  (try linarith) <;>\n                  (try omega)\n              })\n          rw [h\u2082\u2087]\n        rw [h\u2082\u2086]\n      rw [h\u2082\u2083]\n      <;> ring_nf\n      <;> linarith\n    have h\u2082\u2082 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) = (1 + Real.sqrt 2) * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n      have h\u2082\u2083 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) - \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) = Real.sqrt 2 * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := h\u2082\u2081\n      have h\u2082\u2084 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) = (1 + Real.sqrt 2) * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := by\n        linarith\n      exact h\u2082\u2084\n    exact h\u2082\u2082\n  have h\u2083 : x = 1 + Real.sqrt 2 := by\n    rw [h\u2080]\n    have h\u2083\u2081 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) > 0 := h\u2081\n    have h\u2083\u2082 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180) = (1 + Real.sqrt 2) * \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) := h\u2082\n    have h\u2083\u2083 : (\u2211 n in Finset.Icc (1 : \u2115) 44, Real.cos (n * Real.pi / 180)) / \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) = 1 + Real.sqrt 2 := by\n      rw [h\u2083\u2082]\n      have h\u2083\u2084 : \u2211 n in Finset.Icc (1 : \u2115) 44, Real.sin (n * Real.pi / 180) \u2260 0 := by linarith\n      field_simp [h\u2083\u2084]\n      <;> ring_nf\n      <;> field_simp [h\u2083\u2084]\n      <;> linarith\n    rw [h\u2083\u2083]\n    <;> ring_nf\n    <;> field_simp\n    <;> linarith\n  have h\u2084 : 100 * x = 100 + 100 * Real.sqrt 2 := by\n    rw [h\u2083]\n    <;> ring\n    <;> field_simp\n    <;> ring\n    <;> linarith [Real.sqrt_nonneg 2]\n  have h\u2085 : (241 : \u211d) \u2264 100 * x := by\n    rw [h\u2084]\n    have h\u2085\u2081 : Real.sqrt 2 \u2265 0 := Real.sqrt_nonneg _\n    have h\u2085\u2082 : (141 : \u211d) \u2264 100 * Real.sqrt 2 := by\n      nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num),\n        Real.sqrt_nonneg 2, Real.pi_gt_three]\n    linarith\n  have h\u2086 : (100 : \u211d) * x < 242 := by\n    rw [h\u2084]\n    have h\u2086\u2081 : Real.sqrt 2 \u2265 0 := Real.sqrt_nonneg _\n    have h\u2086\u2082 : (100 : \u211d) * Real.sqrt 2 < 142 := by\n      nlinarith [Real.sq_sqrt (show 0 \u2264 2 by norm_num), Real.sqrt_nonneg 2]\n    linarith\n  have h\u2087 : Int.floor (100 * x) = 241 := by\n    rw [Int.floor_eq_iff]\n    constructor <;> norm_num at h\u2085 h\u2086 \u22a2 <;>\n    (try linarith) <;>\n    (try\n      {\n        nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)]\n      })\n    <;>\n    (try\n      {\n        norm_num at h\u2085 h\u2086 \u22a2\n        <;>\n        (try linarith)\n        <;>\n        (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n      })\n    <;>\n    (try\n      {\n        norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n        <;>\n        (try linarith)\n        <;>\n        (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n      })\n    <;>\n    (try\n      {\n        norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n        <;>\n        (try linarith)\n        <;>\n        (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n      })\n    <;>\n    (try\n      {\n        norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n        <;>\n        (try linarith)\n        <;>\n        (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n      })\n    <;>\n    (try\n      {\n        norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n        <;>\n        (try linarith)\n        <;>\n        (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n      })\n  rw [h\u2087]\n  <;> norm_num\n  <;>\n  (try\n    {\n      norm_num at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n  <;>\n  (try\n    {\n      norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n  <;>\n  (try\n    {\n      norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n  <;>\n  (try\n    {\n      norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n  <;>\n  (try\n    {\n      norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n  <;>\n  (try\n    {\n      norm_num [h\u2084] at h\u2085 h\u2086 \u22a2\n      <;>\n      (try linarith)\n      <;>\n      (try nlinarith [Real.sqrt_nonneg 2, Real.sq_sqrt (show 0 \u2264 2 by norm_num)])\n    })\n"}
{"mode": "clear", "steps_before": 47, "alphabet": 25, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1342, "chars_after": 66, "seconds": 81.0, "id": "amc12a_2013_p7", "proof_after": "theorem amc12a_2013_p7 (s : \u2115 \u2192 \u211d) (h\u2080 : \u2200 n, s (n + 2) = s (n + 1) + s n) (h\u2081 : s 9 = 110)\n    (h\u2082 : s 7 = 42) : s 4 = 10  :=  by\n  linarith [h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, h\u2080 5, h\u2080 6, h\u2080 7, h\u2080 8, h\u2080 9]\n", "proof_before": "theorem amc12a_2013_p7 (s : \u2115 \u2192 \u211d) (h\u2080 : \u2200 n, s (n + 2) = s (n + 1) + s n) (h\u2081 : s 9 = 110)\n    (h\u2082 : s 7 = 42) : s 4 = 10  :=  by\n  have h\u2083 : s 7 = s 4 + 2 * s 5 := by\n    have h\u2083 := h\u2080 4\n    have h\u2084 := h\u2080 5\n    have h\u2085 := h\u2080 6\n    have h\u2086 := h\u2080 7\n    have h\u2087 := h\u2080 8\n    simp_all only [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, Nat.add_succ, Nat.succ_add]\n    linarith\n  have h\u2084 : s 4 + 2 * s 5 = 42 := by\n    linarith [h\u2080 5, h\u2080 6, h\u2080 7, h\u2080 8, h\u2080 9, h\u2081, h\u2082, h\u2083]\n  have h\u2085 : s 9 = 2 * s 7 + s 4 + s 5 := by\n    simp_all [add_assoc, add_comm, add_left_comm]\n    <;> linarith\n    <;> linarith\n  have h\u2086 : 2 * s 7 + s 4 + s 5 = 110 := by\n    have h\u2086 : s 9 = 110 := h\u2081\n    have h\u2087 : s 7 = 42 := h\u2082\n    have h\u2088 : s 9 = 2 * s 7 + s 4 + s 5 := h\u2085\n    linarith [h\u2086, h\u2087, h\u2088]\n  have h\u2087 : 2 * 42 + s 4 + s 5 = 110 := by\n    have h\u2087 : s 7 = 42 := h\u2082\n    have h\u2088 : s 4 + 2 * s 5 = 42 := h\u2084\n    have h\u2089 : s 9 = 2 * s 7 + s 4 + s 5 := h\u2085\n    linarith\n  have h\u2088 : 84 + s 4 + s 5 = 110 := by\n    linarith [h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, h\u2080 5, h\u2080 6, h\u2080 7, h\u2080 8, h\u2080 9]\n  have h\u2089 : s 4 + s 5 = 26 := by\n    linarith\n  have h\u2081\u2080 : (s 4 + 2 * s 5) - (s 4 + s 5) = 42 - 26 := by\n    linarith\n  have h\u2081\u2081 : s 5 = 16 := by\n    linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n  have h\u2081\u2082 : s 4 + 16 = 26 := by\n    have h\u2081\u2081 : s 4 + 16 = 26 := by\n      linarith\n    simp_all\n  have h\u2081\u2083 : s 4 = 10 := by\n    linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n  rw [h\u2081\u2083]\n"}
{"mode": "clear", "steps_before": 68, "alphabet": 41, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 68, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "fixpoint": false, "chars_before": 1838, "chars_after": 1838, "seconds": 316.9, "id": "mathd_numbertheory_13", "proof_after": "theorem mathd_numbertheory_13 (u v : \u2115) (S : Set \u2115)\n  (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 0 < n \u2227 14 * n % 100 = 46) (h\u2081 : IsLeast S u)\n  (h\u2082 : IsLeast (S \\ {u}) v) : (u + v : \u211a) / 2 = 64  :=  by\n  have h_u : u \u2208 S \u2227 \u2200 n \u2208 S, u \u2264 n := by\n    constructor\n    case left =>\n      simpa [h\u2080] using h\u2081.1\n    case right =>\n      intro n hn\n      simpa [h\u2080] using h\u2081.2 hn\n  have h_v : v \u2208 S \u2227 \u2200 n \u2208 S, n \u2260 u \u2192 v \u2264 n := by\n    have h\u2083 := h\u2081.1\n    have h\u2084 := h\u2082.1\n    have h\u2085 := h\u2081.2\n    have h\u2086 := h\u2082.2\n    aesop\n  have h_congruence : \u2203 k : \u2115, 14 * u = 100 * k + 46 := by\n    use (14 * u - 46) / 100\n    have h\u2083 := h\u2081.1\n    have h\u2084 := h\u2082.1\n    have h\u2085 := h\u2081.2\n    have h\u2086 := h\u2082.2\n    have h\u2087 := h\u2080 u\n    have h\u2088 := h\u2080 v\n    simp_all [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib]\n    <;> omega\n  have h_simplified : 7 * u \u2261 23 [MOD 50] := by\n    simp only [Nat.ModEq, Nat.mod_eq_of_lt] at *\n    rcases h_congruence with \u27e8k, hk\u27e9\n    omega\n  have h_inverse : \u2203 x : \u2115, 7 * x \u2261 1 [MOD 50] := by\n    use 7 * 49\n    norm_num [Nat.ModEq, Nat.modEq_iff_dvd, mul_comm]\n    <;> use 7\n    <;> ring\n    <;> norm_num\n    <;> omega\n  have h_solution : u \u2261 23 * 43 [MOD 50] := by\n    obtain \u27e8k, hk\u27e9 := h_congruence\n    rw [\u2190 Nat.mod_add_div (14 * u) 100] at hk\n    simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have h_u_solution : u = 39 := by\n    have h\u2083 := h_u.2 39 (by simp [h\u2080])\n    have h\u2084 := h_v.2 39 (by simp [h\u2080])\n    have h\u2085 := h_u.1\n    have h\u2086 := h_v.1\n    have h\u2087 := h_congruence\n    have h\u2088 := h_simplified\n    have h\u2089 := h_inverse\n    have h\u2081\u2080 := h_solution\n    omega\n  have h_v_solution : v = 89 := by\n    apply Eq.symm\n    apply Nat.eq_of_le_of_lt_succ <;>\n      simp_all [IsLeast, Set.mem_diff, Set.mem_singleton_iff]\n    <;>\n      omega\n  have h_average : (u + v : \u211a) / 2 = 64 := by\n    norm_num [h_u_solution, h_v_solution]\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  norm_num [h_u_solution, h_v_solution]\n  <;> linarith\n  <;> simp_all [h\u2080]\n  <;> norm_num [Nat.ModEq, Nat.ModEq] at h_simplified h_solution \u22a2\n  <;> omega\n", "proof_before": "theorem mathd_numbertheory_13 (u v : \u2115) (S : Set \u2115)\n  (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 0 < n \u2227 14 * n % 100 = 46) (h\u2081 : IsLeast S u)\n  (h\u2082 : IsLeast (S \\ {u}) v) : (u + v : \u211a) / 2 = 64  :=  by\n  have h_u : u \u2208 S \u2227 \u2200 n \u2208 S, u \u2264 n := by\n    constructor\n    case left =>\n      simpa [h\u2080] using h\u2081.1\n    case right =>\n      intro n hn\n      simpa [h\u2080] using h\u2081.2 hn\n  have h_v : v \u2208 S \u2227 \u2200 n \u2208 S, n \u2260 u \u2192 v \u2264 n := by\n    have h\u2083 := h\u2081.1\n    have h\u2084 := h\u2082.1\n    have h\u2085 := h\u2081.2\n    have h\u2086 := h\u2082.2\n    aesop\n  have h_congruence : \u2203 k : \u2115, 14 * u = 100 * k + 46 := by\n    use (14 * u - 46) / 100\n    have h\u2083 := h\u2081.1\n    have h\u2084 := h\u2082.1\n    have h\u2085 := h\u2081.2\n    have h\u2086 := h\u2082.2\n    have h\u2087 := h\u2080 u\n    have h\u2088 := h\u2080 v\n    simp_all [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib]\n    <;> omega\n  have h_simplified : 7 * u \u2261 23 [MOD 50] := by\n    simp only [Nat.ModEq, Nat.mod_eq_of_lt] at *\n    rcases h_congruence with \u27e8k, hk\u27e9\n    omega\n  have h_inverse : \u2203 x : \u2115, 7 * x \u2261 1 [MOD 50] := by\n    use 7 * 49\n    norm_num [Nat.ModEq, Nat.modEq_iff_dvd, mul_comm]\n    <;> use 7\n    <;> ring\n    <;> norm_num\n    <;> omega\n  have h_solution : u \u2261 23 * 43 [MOD 50] := by\n    obtain \u27e8k, hk\u27e9 := h_congruence\n    rw [\u2190 Nat.mod_add_div (14 * u) 100] at hk\n    simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have h_u_solution : u = 39 := by\n    have h\u2083 := h_u.2 39 (by simp [h\u2080])\n    have h\u2084 := h_v.2 39 (by simp [h\u2080])\n    have h\u2085 := h_u.1\n    have h\u2086 := h_v.1\n    have h\u2087 := h_congruence\n    have h\u2088 := h_simplified\n    have h\u2089 := h_inverse\n    have h\u2081\u2080 := h_solution\n    omega\n  have h_v_solution : v = 89 := by\n    apply Eq.symm\n    apply Nat.eq_of_le_of_lt_succ <;>\n      simp_all [IsLeast, Set.mem_diff, Set.mem_singleton_iff]\n    <;>\n      omega\n  have h_average : (u + v : \u211a) / 2 = 64 := by\n    norm_num [h_u_solution, h_v_solution]\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  norm_num [h_u_solution, h_v_solution]\n  <;> linarith\n  <;> simp_all [h\u2080]\n  <;> norm_num [Nat.ModEq, Nat.ModEq] at h_simplified h_solution \u22a2\n  <;> omega\n"}
{"mode": "clear", "steps_before": 54, "alphabet": 19, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 54, "lower_bound": null, "exhausted_depth": 2, "certified": false, "rounds": 8, "fixpoint": true, "chars_before": 1645, "chars_after": 1645, "seconds": 816.7, "id": "algebra_manipexpr_apbeq2cceqiacpbceqm2", "proof_after": "theorem algebra_manipexpr_apbeq2cceqiacpbceqm2 (a b c : \u2102) (h\u2080 : a + b = 2 * c)\n  (h\u2081 : c = Complex.I) : a * c + b * c = -2  :=  by\n  have h2 : a * c + b * c = (a + b) * c := by\n    rw [h\u2081] at h\u2080 \u22a2\n    simp_all [Complex.ext_iff, mul_add, mul_comm, mul_left_comm]\n    <;> ring\n    <;> linarith\n  have h3 : (a + b) * c = 2 * c * c := by\n    subst h\u2081\n    simp_all [Complex.ext_iff, mul_add, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have h4 : 2 * c * c = 2 * (c * c) := by\n    simp_all [mul_assoc, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> simp_all [Complex.I_sq]\n    <;> ring_nf\n    <;> simp_all [Complex.I_sq]\n    <;> ring_nf\n    <;> simp_all [Complex.I_sq]\n    <;> ring_nf\n    <;> simp_all [Complex.I_sq]\n  have h5 : c * c = (Complex.I) * (Complex.I) := by\n    simpa [h\u2081, Complex.ext_iff] using h4\n  have h6 : (Complex.I) * (Complex.I) = -1 := by\n    simp_all only [\u2190 mul_assoc, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num [Complex.I_sq]\n    <;> ring_nf\n    <;> norm_num [Complex.I_sq]\n    <;> ring_nf\n    <;> norm_num [Complex.I_sq]\n  have h7 : 2 * (c * c) = 2 * (-1) := by\n    simp_all only [Complex.I_sq, eq_self_iff_true, eq_mpr_eq_cast, heq_iff_eq, true_and]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.I_sq]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.I_sq]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.I_sq]\n    <;> ring_nf\n    <;> norm_num\n  have h8 : 2 * (-1) = -2 := by\n    simp_all only [mul_neg, mul_one, mul_add, mul_sub, mul_comm, mul_left_comm, mul_assoc]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith [Complex.I_sq]\n  have h9 : a * c + b * c = -2 := by\n    subst_vars\n    simp_all [Complex.ext_iff, Complex.I_mul_I, mul_comm]\n    <;> linarith\n  simpa [h\u2081, h6] using h9\n", "proof_before": "theorem algebra_manipexpr_apbeq2cceqiacpbceqm2 (a b c : \u2102) (h\u2080 : a + b = 2 * c)\n  (h\u2081 : c = Complex.I) : a * c + b * c = -2  :=  by\n  have h2 : a * c + b * c = (a + b) * c := by\n    rw [h\u2081] at h\u2080 \u22a2\n    simp_all [Complex.ext_iff, mul_add, mul_comm, mul_left_comm]\n    <;> ring\n    <;> linarith\n  have h3 : (a + b) * c = 2 * c * c := by\n    subst h\u2081\n    simp_all [Complex.ext_iff, mul_add, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have h4 : 2 * c * c = 2 * (c * c) := by\n    simp_all [mul_assoc, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> simp_all [Complex.I_sq]\n    <;> ring_nf\n    <;> simp_all [Complex.I_sq]\n    <;> ring_nf\n    <;> simp_all [Complex.I_sq]\n    <;> ring_nf\n    <;> simp_all [Complex.I_sq]\n  have h5 : c * c = (Complex.I) * (Complex.I) := by\n    simpa [h\u2081, Complex.ext_iff] using h4\n  have h6 : (Complex.I) * (Complex.I) = -1 := by\n    simp_all only [\u2190 mul_assoc, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num [Complex.I_sq]\n    <;> ring_nf\n    <;> norm_num [Complex.I_sq]\n    <;> ring_nf\n    <;> norm_num [Complex.I_sq]\n  have h7 : 2 * (c * c) = 2 * (-1) := by\n    simp_all only [Complex.I_sq, eq_self_iff_true, eq_mpr_eq_cast, heq_iff_eq, true_and]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.I_sq]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.I_sq]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.I_sq]\n    <;> ring_nf\n    <;> norm_num\n  have h8 : 2 * (-1) = -2 := by\n    simp_all only [mul_neg, mul_one, mul_add, mul_sub, mul_comm, mul_left_comm, mul_assoc]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith [Complex.I_sq]\n  have h9 : a * c + b * c = -2 := by\n    subst_vars\n    simp_all [Complex.ext_iff, Complex.I_mul_I, mul_comm]\n    <;> linarith\n  simpa [h\u2081, h6] using h9\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 33, "alphabet": 22, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 4, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 7, "fixpoint": true, "chars_before": 1063, "chars_after": 123, "seconds": 776.6, "id": "mathd_algebra_451", "proof_after": "theorem mathd_algebra_451 (\u03c3 : Equiv \u211d \u211d) (h\u2080 : \u03c3.2 (-15) = 0) (h\u2081 : \u03c3.2 0 = 3) (h\u2082 : \u03c3.2 3 = 9)\n    (h\u2083 : \u03c3.2 9 = 20) : \u03c3.1 (\u03c3.1 9) = 0  :=  by\n  have h\u2086 : \u03c3.1 (\u03c3.1 9) = 0 := by\n    simp_all [Equiv.symm_apply_eq, Equiv.apply_eq_iff_eq]\n    <;> aesop\n  simpa using h\u2086\n", "proof_before": "theorem mathd_algebra_451 (\u03c3 : Equiv \u211d \u211d) (h\u2080 : \u03c3.2 (-15) = 0) (h\u2081 : \u03c3.2 0 = 3) (h\u2082 : \u03c3.2 3 = 9)\n    (h\u2083 : \u03c3.2 9 = 20) : \u03c3.1 (\u03c3.1 9) = 0  :=  by\n  have h\u2080' : \u03c3.1 0 = -15 := by\n    have h\u2084 := h\u2080\n    have h\u2085 := h\u2081\n    have h\u2086 := h\u2082\n    have h\u2087 := h\u2083\n    simp_all [Equiv.symm_apply_eq, Equiv.symm_apply_apply]\n  have h\u2081' : \u03c3.1 3 = 0 := by\n    simp only [Equiv.invFun_as_coe, eq_comm] at h\u2080 h\u2081 h\u2082 h\u2083 h\u2080' \u22a2\n    simp only [Equiv.toFun_as_coe, eq_comm] at h\u2080 h\u2081 h\u2082 h\u2083 h\u2080' \u22a2\n    rw [\u2190 Equiv.apply_eq_iff_eq_symm_apply \u03c3] at h\u2081 h\u2082 h\u2083\n    simp_all [Equiv.symm_apply_eq]\n  have h\u2082' : \u03c3.1 9 = 3 := by\n    have h\u2084 := h\u2081'\n    have h\u2085 := h\u2080\n    have h\u2086 := h\u2082\n    have h\u2087 := h\u2083\n    have h\u2088 := h\u2080'\n    simp_all [Equiv.symm_apply_eq]\n    <;> linarith\n  have h\u2083' : \u03c3.1 20 = 9 := by\n    simp_all [Equiv.symm_apply_eq, Equiv.symm_apply_eq]\n    <;> aesop\n  have h\u2084 : \u03c3.1 9 = 3 := by\n    simp_all [Equiv.symm_apply_eq, Equiv.symm_apply_eq]\n    <;> linarith\n  have h\u2085 : \u03c3.1 (\u03c3.1 9) = \u03c3.1 3 := by\n    simp_all [Equiv.symm_apply_eq]\n    <;> simp_all [Equiv.symm_apply_eq]\n    <;> simp_all [Equiv.symm_apply_eq]\n  have h\u2086 : \u03c3.1 (\u03c3.1 9) = 0 := by\n    simp_all [Equiv.symm_apply_eq, Equiv.apply_eq_iff_eq]\n    <;> aesop\n  simpa using h\u2086\n"}
{"mode": "clear", "steps_before": 22, "alphabet": 12, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 491, "chars_after": 6, "seconds": 164.2, "id": "mathd_numbertheory_252", "proof_after": "theorem mathd_numbertheory_252 : 7! % 23 = 3  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_252 : 7! % 23 = 3  :=  by\n  have h1 : 7! = 5040 := by\n    decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n  have h2 : 5040 = 23 * 219 + 3 := by\n    norm_num [h1, Nat.factorial]\n    <;> decide\n    <;> rfl\n    <;> decide\n    <;> rfl\n  have h3 : 5040 % 23 = 3 := by\n    simp_all [Nat.factorial, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n    <;> norm_num\n    <;> omega\n  simp_all [Nat.factorial, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n"}
{"mode": "clear", "steps_before": 131, "alphabet": 45, "compiles": false, "id": "imo_1974_p5", "proof_after": "theorem imo_1974_p5 (a b c d s : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c \u2227 0 < d)\n  (h\u2081 : s = a / (a + b + d) + b / (a + b + c) + c / (b + c + d) + d / (a + c + d)) :\n  1 < s \u2227 s < 2  :=  by\n  have term1_pos : 0 < a / (a + b + d) := by\n    have h\u2082 := h\u2080.1\n    have h\u2083 := h\u2080.2.1\n    have h\u2084 := h\u2080.2.2.1\n    have h\u2085 := h\u2080.2.2.2\n    have h\u2086 := h\u2081\n    positivity\n  have term1_less1 : a / (a + b + d) < 1 := by\n    have h\u2082 : a < a + b + d := by linarith\n    have h\u2083 : 0 < a + b + d := by linarith\n    rw [div_lt_one] <;> linarith\n  have term2_pos : 0 < b / (a + b + c) := by\n    have h\u2082 : 0 < a + b + c := by linarith\n    have h\u2083 : 0 < a + b + d := by linarith\n    have h\u2084 : 0 < b + c + d := by linarith\n    have h\u2085 : 0 < a + c + d := by linarith\n    exact div_pos h\u2080.2.1 (by linarith)\n  have term2_less1 : b / (a + b + c) < 1 := by\n    have h\u2082 : 0 < a + b + c := by linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.1]\n    have h\u2083 : 0 < a + b + d := by linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.2]\n    have h\u2084 : 0 < b + c + d := by linarith [h\u2080.2.1, h\u2080.2.2.1, h\u2080.2.2.2]\n    have h\u2085 : 0 < a + c + d := by linarith [h\u2080.1, h\u2080.2.2.1, h\u2080.2.2.2]\n    rw [\u2190 sub_pos] at term1_pos term1_less1 term2_pos \u22a2\n    field_simp at term1_pos term1_less1 term2_pos \u22a2\n    rw [\u2190 sub_pos] at term1_pos term1_less1 term2_pos \u22a2\n    field_simp at term1_pos term1_less1 term2_pos \u22a2\n    nlinarith [term1_pos, term1_less1, term2_pos]\n  have term3_pos : 0 < c / (b + c + d) := by\n    have h\u2082 := h\u2080.2.1\n    have h\u2083 := h\u2080.2.2.1\n    have h\u2084 := h\u2080.2.2.2\n    have h\u2085 := term1_pos\n    have h\u2086 := term1_less1\n    have h\u2087 := term2_pos\n    have h\u2088 := term2_less1\n    have h\u2089 : 0 < a + b + c := by linarith\n    have h\u2081\u2080 : 0 < a + b + d := by linarith\n    have h\u2081\u2081 : 0 < a + c + d := by linarith\n    have h\u2081\u2082 : 0 < b + c + d := by linarith\n    have h\u2081\u2083 : 0 < a + b + c + d := by linarith\n    field_simp at h\u2085 h\u2086 h\u2087 h\u2088 \u22a2\n    refine' lt_of_sub_pos _\n    field_simp\n    ring_nf\n    nlinarith [mul_pos h\u2082 h\u2083, mul_pos h\u2082 h\u2084, mul_pos h\u2083 h\u2084,\n      mul_pos (sub_pos.mpr h\u2085) (sub_pos.mpr h\u2086),\n      mul_pos (sub_pos.mpr h\u2087) (sub_pos.mpr h\u2088)]\n  have term3_less1 : c / (b + c + d) < 1 := by\n    have h\u2082 : 0 < b + c + d := by linarith\n    have h\u2083 : 0 < a + b + c := by linarith\n    have h\u2084 : 0 < a + c + d := by linarith\n    have h\u2085 : 0 < a + b + d := by linarith\n    field_simp at term1_pos term1_less1 term2_pos term2_less1 term3_pos\n    rw [div_lt_one] <;> linarith\n  have term4_pos : 0 < d / (a + c + d) := by\n    have h\u2082 : 0 < a := by linarith\n    have h\u2083 : 0 < b := by linarith\n    have h\u2084 : 0 < c := by linarith\n    have h\u2085 : 0 < d := by linarith\n    have h\u2086 : 0 < d / (a + c + d) := by positivity\n    linarith\n  have term4_less1 : d / (a + c + d) < 1 := by\n    have h\u2082 : d / (a + c + d) < 1 := by\n      rw [div_lt_one] <;> nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.1, h\u2080.2.2.2, term4_pos]\n    exact h\u2082\n  have s_pos : 0 < s := by\n    have h\u2082 : 0 < a / (a + b + d) := term1_pos\n    have h\u2083 : 0 < b / (a + b + c) := term2_pos\n    have h\u2084 : 0 < c / (b + c + d) := term3_pos\n    have h\u2085 : 0 < d / (a + c + d) := term4_pos\n    linarith [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085]\n  have lower_bound : 1 < s := by\n    have h\u2082 : 0 < a + b + d := by linarith\n    have h\u2083 : 0 < a + b + c := by linarith\n    have h\u2084 : 0 < b + c + d := by linarith\n    have h\u2085 : 0 < a + c + d := by linarith\n    have h\u2086 : 0 < a + b + c + d := by linarith\n    field_simp [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086]\n    refine' lt_of_sub_pos _\n    field_simp\n    ring_nf\n    nlinarith [mul_pos h\u2080.1 h\u2080.2.1, mul_pos h\u2080.2.1 h\u2080.2.2.1, mul_pos h\u2080.2.2.1 h\u2080.2.2.2,\n      mul_pos h\u2080.1 h\u2080.2.2.1, mul_pos h\u2080.1 h\u2080.2.2.2, mul_pos h\u2080.2.1 h\u2080.2.2.2]\n  have upper_bound : s < 2 := by\n    have h\u2082 : 0 < a + b + c + d := by linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.1, h\u2080.2.2.2]\n    have h\u2083 : 0 < a + b + d := by linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.1]\n    have h\u2084 : 0 < a + b + c := by linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.2]\n    have h\u2085 : 0 < b + c + d := by linarith [h\u2080.2.1, h\u2080.2.2.1, h\u2080.2.2.2]\n    have h\u2086 : 0 < a + c + d := by linarith [h\u2080.1, h\u2080.2.2.1, h\u2080.2.2.2]\n    field_simp [h\u2081] at term1_pos term1_less1 term2_pos term2_less1 term3_pos term3_less1 term4_pos term4_less1 s_pos lower_bound\n    refine' lt_of_sub_pos _\n    field_simp [h\u2081]\n    ring_nf\n    nlinarith [mul_pos h\u2080.1 h\u2080.2.1, mul_pos h\u2080.1 h\u2080.2.2.1, mul_pos h\u2080.1 h\u2080.2.2.2,\n      mul_pos h\u2080.2.1 h\u2080.2.2.1, mul_pos h\u2080.2.1 h\u2080.2.2.2, mul_pos h\u2080.2.2.1 h\u2080.2.2.2]\n  constructor\n  exact lower_bound\n  exact upper_bound\n", "proof_before": "theorem imo_1974_p5 (a b c d s : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c \u2227 0 < d)\n  (h\u2081 : s = a / (a + b + d) + b / (a + b + c) + c / (b + c + d) + d / (a + c + d)) :\n  1 < s \u2227 s < 2  :=  by\n  have term1_pos : 0 < a / (a + b + d) := by\n    have h\u2082 := h\u2080.1\n    have h\u2083 := h\u2080.2.1\n    have h\u2084 := h\u2080.2.2.1\n    have h\u2085 := h\u2080.2.2.2\n    have h\u2086 := h\u2081\n    positivity\n  have term1_less1 : a / (a + b + d) < 1 := by\n    have h\u2082 : a < a + b + d := by linarith\n    have h\u2083 : 0 < a + b + d := by linarith\n    rw [div_lt_one] <;> linarith\n  have term2_pos : 0 < b / (a + b + c) := by\n    have h\u2082 : 0 < a + b + c := by linarith\n    have h\u2083 : 0 < a + b + d := by linarith\n    have h\u2084 : 0 < b + c + d := by linarith\n    have h\u2085 : 0 < a + c + d := by linarith\n    exact div_pos h\u2080.2.1 (by linarith)\n  have term2_less1 : b / (a + b + c) < 1 := by\n    have h\u2082 : 0 < a + b + c := by linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.1]\n    have h\u2083 : 0 < a + b + d := by linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.2]\n    have h\u2084 : 0 < b + c + d := by linarith [h\u2080.2.1, h\u2080.2.2.1, h\u2080.2.2.2]\n    have h\u2085 : 0 < a + c + d := by linarith [h\u2080.1, h\u2080.2.2.1, h\u2080.2.2.2]\n    rw [\u2190 sub_pos] at term1_pos term1_less1 term2_pos \u22a2\n    field_simp at term1_pos term1_less1 term2_pos \u22a2\n    rw [\u2190 sub_pos] at term1_pos term1_less1 term2_pos \u22a2\n    field_simp at term1_pos term1_less1 term2_pos \u22a2\n    nlinarith [term1_pos, term1_less1, term2_pos]\n  have term3_pos : 0 < c / (b + c + d) := by\n    have h\u2082 := h\u2080.2.1\n    have h\u2083 := h\u2080.2.2.1\n    have h\u2084 := h\u2080.2.2.2\n    have h\u2085 := term1_pos\n    have h\u2086 := term1_less1\n    have h\u2087 := term2_pos\n    have h\u2088 := term2_less1\n    have h\u2089 : 0 < a + b + c := by linarith\n    have h\u2081\u2080 : 0 < a + b + d := by linarith\n    have h\u2081\u2081 : 0 < a + c + d := by linarith\n    have h\u2081\u2082 : 0 < b + c + d := by linarith\n    have h\u2081\u2083 : 0 < a + b + c + d := by linarith\n    field_simp at h\u2085 h\u2086 h\u2087 h\u2088 \u22a2\n    refine' lt_of_sub_pos _\n    field_simp\n    ring_nf\n    nlinarith [mul_pos h\u2082 h\u2083, mul_pos h\u2082 h\u2084, mul_pos h\u2083 h\u2084,\n      mul_pos (sub_pos.mpr h\u2085) (sub_pos.mpr h\u2086),\n      mul_pos (sub_pos.mpr h\u2087) (sub_pos.mpr h\u2088)]\n  have term3_less1 : c / (b + c + d) < 1 := by\n    have h\u2082 : 0 < b + c + d := by linarith\n    have h\u2083 : 0 < a + b + c := by linarith\n    have h\u2084 : 0 < a + c + d := by linarith\n    have h\u2085 : 0 < a + b + d := by linarith\n    field_simp at term1_pos term1_less1 term2_pos term2_less1 term3_pos\n    rw [div_lt_one] <;> linarith\n  have term4_pos : 0 < d / (a + c + d) := by\n    have h\u2082 : 0 < a := by linarith\n    have h\u2083 : 0 < b := by linarith\n    have h\u2084 : 0 < c := by linarith\n    have h\u2085 : 0 < d := by linarith\n    have h\u2086 : 0 < d / (a + c + d) := by positivity\n    linarith\n  have term4_less1 : d / (a + c + d) < 1 := by\n    have h\u2082 : d / (a + c + d) < 1 := by\n      rw [div_lt_one] <;> nlinarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.1, h\u2080.2.2.2, term4_pos]\n    exact h\u2082\n  have s_pos : 0 < s := by\n    have h\u2082 : 0 < a / (a + b + d) := term1_pos\n    have h\u2083 : 0 < b / (a + b + c) := term2_pos\n    have h\u2084 : 0 < c / (b + c + d) := term3_pos\n    have h\u2085 : 0 < d / (a + c + d) := term4_pos\n    linarith [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085]\n  have lower_bound : 1 < s := by\n    have h\u2082 : 0 < a + b + d := by linarith\n    have h\u2083 : 0 < a + b + c := by linarith\n    have h\u2084 : 0 < b + c + d := by linarith\n    have h\u2085 : 0 < a + c + d := by linarith\n    have h\u2086 : 0 < a + b + c + d := by linarith\n    field_simp [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086]\n    refine' lt_of_sub_pos _\n    field_simp\n    ring_nf\n    nlinarith [mul_pos h\u2080.1 h\u2080.2.1, mul_pos h\u2080.2.1 h\u2080.2.2.1, mul_pos h\u2080.2.2.1 h\u2080.2.2.2,\n      mul_pos h\u2080.1 h\u2080.2.2.1, mul_pos h\u2080.1 h\u2080.2.2.2, mul_pos h\u2080.2.1 h\u2080.2.2.2]\n  have upper_bound : s < 2 := by\n    have h\u2082 : 0 < a + b + c + d := by linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.1, h\u2080.2.2.2]\n    have h\u2083 : 0 < a + b + d := by linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.1]\n    have h\u2084 : 0 < a + b + c := by linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2.2]\n    have h\u2085 : 0 < b + c + d := by linarith [h\u2080.2.1, h\u2080.2.2.1, h\u2080.2.2.2]\n    have h\u2086 : 0 < a + c + d := by linarith [h\u2080.1, h\u2080.2.2.1, h\u2080.2.2.2]\n    field_simp [h\u2081] at term1_pos term1_less1 term2_pos term2_less1 term3_pos term3_less1 term4_pos term4_less1 s_pos lower_bound\n    refine' lt_of_sub_pos _\n    field_simp [h\u2081]\n    ring_nf\n    nlinarith [mul_pos h\u2080.1 h\u2080.2.1, mul_pos h\u2080.1 h\u2080.2.2.1, mul_pos h\u2080.1 h\u2080.2.2.2,\n      mul_pos h\u2080.2.1 h\u2080.2.2.1, mul_pos h\u2080.2.1 h\u2080.2.2.2, mul_pos h\u2080.2.2.1 h\u2080.2.2.2]\n  constructor\n  exact lower_bound\n  exact upper_bound\n"}
{"mode": "clear", "steps_before": 41, "alphabet": 22, "compiles": false, "id": "amc12_2000_p11", "proof_after": "theorem amc12_2000_p11 (a b : \u211d) (h\u2080 : a \u2260 0 \u2227 b \u2260 0) (h\u2081 : a * b = a - b) :\n    a / b + b / a - a * b = 2  :=  by\n  have h\u2081 : a * b = a - b := by\n    have h\u2081 := h\u2081\n    field_simp [h\u2080] at h\u2081\n    nlinarith\n  have a_expr : a = b / (1 - b) := by\n    have h\u2082 : 1 - b \u2260 0 := by\n      intro h\n      have : b = 1 := by linarith\n      rw [this] at h\u2081\n      have : a * 1 = a - 1 := by linarith\n      have : a = a - 1 := by linarith\n      have : 0 = -1 := by linarith\n      linarith\n    field_simp [h\u2082] at h\u2081 \u22a2\n    linarith\n  have a_div_b : a / b = 1 / (1 - b) := by\n    have h\u2080' : 1 - b \u2260 0 := by\n      intro h\n      have h\u2082 : b = 1 := by linarith\n      have h\u2083 : a = 0 := by simp_all\n      simp_all\n    field_simp [h\u2080.1, h\u2080.2, h\u2080']\n    <;> linarith\n  have b_div_a : b / a = 1 - b := by\n    have h\u2082 : a * b = a - b := h\u2081\n    rw [eq_comm] at h\u2082\n    field_simp [h\u2080.1, h\u2080.2, sub_eq_zero, eq_self_iff_true, not_true_eq_false] at h\u2082 \u22a2\n    linarith\n  have expression : a / b + b / a - a * b = 2 := by\n    have h\u2082 : a / b + b / a - a * b = 2 := by\n      field_simp [h\u2080.1, h\u2080.2, sub_eq_zero, mul_comm]\n      simp_all [mul_comm, sub_eq_zero]\n      linarith\n    linarith\n  simpa [a_div_b, b_div_a, h\u2081] using expression\n", "proof_before": "theorem amc12_2000_p11 (a b : \u211d) (h\u2080 : a \u2260 0 \u2227 b \u2260 0) (h\u2081 : a * b = a - b) :\n    a / b + b / a - a * b = 2  :=  by\n  have h\u2081 : a * b = a - b := by\n    have h\u2081 := h\u2081\n    field_simp [h\u2080] at h\u2081\n    nlinarith\n  have a_expr : a = b / (1 - b) := by\n    have h\u2082 : 1 - b \u2260 0 := by\n      intro h\n      have : b = 1 := by linarith\n      rw [this] at h\u2081\n      have : a * 1 = a - 1 := by linarith\n      have : a = a - 1 := by linarith\n      have : 0 = -1 := by linarith\n      linarith\n    field_simp [h\u2082] at h\u2081 \u22a2\n    linarith\n  have a_div_b : a / b = 1 / (1 - b) := by\n    have h\u2080' : 1 - b \u2260 0 := by\n      intro h\n      have h\u2082 : b = 1 := by linarith\n      have h\u2083 : a = 0 := by simp_all\n      simp_all\n    field_simp [h\u2080.1, h\u2080.2, h\u2080']\n    <;> linarith\n  have b_div_a : b / a = 1 - b := by\n    have h\u2082 : a * b = a - b := h\u2081\n    rw [eq_comm] at h\u2082\n    field_simp [h\u2080.1, h\u2080.2, sub_eq_zero, eq_self_iff_true, not_true_eq_false] at h\u2082 \u22a2\n    linarith\n  have expression : a / b + b / a - a * b = 2 := by\n    have h\u2082 : a / b + b / a - a * b = 2 := by\n      field_simp [h\u2080.1, h\u2080.2, sub_eq_zero, mul_comm]\n      simp_all [mul_comm, sub_eq_zero]\n      linarith\n    linarith\n  simpa [a_div_b, b_div_a, h\u2081] using expression\n"}
{"mode": "clear", "steps_before": 49, "alphabet": 24, "compiles": false, "id": "aime_1990_p2", "proof_after": "theorem aime_1990_p2 :\n  (52 + 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) - (52 - 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) = 828  :=  by\n  have h1 : 52 + 6 * Real.sqrt 43 = (Real.sqrt 43 + 3)^2 := by\n    have h : 0 \u2264 Real.sqrt 43 := Real.sqrt_nonneg _\n    have h2 : 0 \u2264 3 := by norm_num\n    have h3 : 0 \u2264 Real.sqrt 43 * 3 := by positivity\n    have h4 : 0 \u2264 Real.sqrt 43 * Real.sqrt 43 := by positivity\n    nlinarith [Real.sq_sqrt (show 0 \u2264 43 by norm_num), sq_nonneg (Real.sqrt 43 - 3)]\n  have h2 : 52 - 6 * Real.sqrt 43 = (Real.sqrt 43 - 3)^2 := by\n    have h2 : 0 \u2264 Real.sqrt 43 := Real.sqrt_nonneg _\n    nlinarith [sq_nonneg (Real.sqrt 43 - 3), sq_nonneg (Real.sqrt 43 + 3),\n      Real.sq_sqrt (show 0 \u2264 43 by norm_num)]\n  have h3 : (52 + 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) = (Real.sqrt 43 + 3)^3 := by\n    have h3 : 0 < Real.sqrt 43 := by\n      apply Real.sqrt_pos_of_pos\n      norm_num\n    have h4 : 0 < Real.sqrt 43 + 3 := by\n      nlinarith [h3]\n    have h5 : 0 < Real.sqrt 43 - 3 := by\n      nlinarith [h3]\n    rw [\u2190 Real.rpow_nat_cast]\n    simp_all [Real.rpow_def_of_pos, mul_comm]\n    <;> ring_nf\n    <;> nlinarith [Real.sq_sqrt (show 0 \u2264 43 by norm_num)]\n  have h4 : (52 - 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) = (Real.sqrt 43 - 3)^3 := by\n    have h4 : 0 < Real.sqrt 43 := Real.sqrt_pos.mpr (by norm_num)\n    have h5 : 0 < Real.sqrt 43 + 3 := by linarith\n    have h6 : 0 < Real.sqrt 43 - 3 := by\n      nlinarith [Real.sq_sqrt (show 0 \u2264 43 by norm_num)]\n    rw [\u2190 Real.rpow_nat_cast] at h3 \u22a2\n    simp_all [Real.rpow_def_of_pos]\n    <;> ring_nf\n    <;> field_simp [h4.ne', h5.ne', h6.ne'] at h3 \u22a2\n    <;> norm_num at h3 \u22a2\n    <;> nlinarith [Real.sq_sqrt (show 0 \u2264 43 by norm_num), Real.sq_sqrt (show 0 \u2264 43 by norm_num)]\n  have h5 : (Real.sqrt 43 + 3)^3 - (Real.sqrt 43 - 3)^3 = 828 := by\n    have h5 : 0 < Real.sqrt 43 := Real.sqrt_pos.mpr (by norm_num)\n    have h6 : (Real.sqrt 43) ^ 2 = 43 := Real.sq_sqrt (by norm_num)\n    nlinarith [sq_nonneg (Real.sqrt 43 - 3), sq_nonneg (Real.sqrt 43 + 3)]\n  have h6 : (52 + 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) - (52 - 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) = 828 := by\n    simp_all [pow_two, mul_add, mul_comm, mul_left_comm]\n    <;> linarith\n  linarith [h5]\n  <;> norm_num\n  <;> linarith\n", "proof_before": "theorem aime_1990_p2 :\n  (52 + 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) - (52 - 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) = 828  :=  by\n  have h1 : 52 + 6 * Real.sqrt 43 = (Real.sqrt 43 + 3)^2 := by\n    have h : 0 \u2264 Real.sqrt 43 := Real.sqrt_nonneg _\n    have h2 : 0 \u2264 3 := by norm_num\n    have h3 : 0 \u2264 Real.sqrt 43 * 3 := by positivity\n    have h4 : 0 \u2264 Real.sqrt 43 * Real.sqrt 43 := by positivity\n    nlinarith [Real.sq_sqrt (show 0 \u2264 43 by norm_num), sq_nonneg (Real.sqrt 43 - 3)]\n  have h2 : 52 - 6 * Real.sqrt 43 = (Real.sqrt 43 - 3)^2 := by\n    have h2 : 0 \u2264 Real.sqrt 43 := Real.sqrt_nonneg _\n    nlinarith [sq_nonneg (Real.sqrt 43 - 3), sq_nonneg (Real.sqrt 43 + 3),\n      Real.sq_sqrt (show 0 \u2264 43 by norm_num)]\n  have h3 : (52 + 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) = (Real.sqrt 43 + 3)^3 := by\n    have h3 : 0 < Real.sqrt 43 := by\n      apply Real.sqrt_pos_of_pos\n      norm_num\n    have h4 : 0 < Real.sqrt 43 + 3 := by\n      nlinarith [h3]\n    have h5 : 0 < Real.sqrt 43 - 3 := by\n      nlinarith [h3]\n    rw [\u2190 Real.rpow_nat_cast]\n    simp_all [Real.rpow_def_of_pos, mul_comm]\n    <;> ring_nf\n    <;> nlinarith [Real.sq_sqrt (show 0 \u2264 43 by norm_num)]\n  have h4 : (52 - 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) = (Real.sqrt 43 - 3)^3 := by\n    have h4 : 0 < Real.sqrt 43 := Real.sqrt_pos.mpr (by norm_num)\n    have h5 : 0 < Real.sqrt 43 + 3 := by linarith\n    have h6 : 0 < Real.sqrt 43 - 3 := by\n      nlinarith [Real.sq_sqrt (show 0 \u2264 43 by norm_num)]\n    rw [\u2190 Real.rpow_nat_cast] at h3 \u22a2\n    simp_all [Real.rpow_def_of_pos]\n    <;> ring_nf\n    <;> field_simp [h4.ne', h5.ne', h6.ne'] at h3 \u22a2\n    <;> norm_num at h3 \u22a2\n    <;> nlinarith [Real.sq_sqrt (show 0 \u2264 43 by norm_num), Real.sq_sqrt (show 0 \u2264 43 by norm_num)]\n  have h5 : (Real.sqrt 43 + 3)^3 - (Real.sqrt 43 - 3)^3 = 828 := by\n    have h5 : 0 < Real.sqrt 43 := Real.sqrt_pos.mpr (by norm_num)\n    have h6 : (Real.sqrt 43) ^ 2 = 43 := Real.sq_sqrt (by norm_num)\n    nlinarith [sq_nonneg (Real.sqrt 43 - 3), sq_nonneg (Real.sqrt 43 + 3)]\n  have h6 : (52 + 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) - (52 - 6 * Real.sqrt 43) ^ ((3 : \u211d) / 2) = 828 := by\n    simp_all [pow_two, mul_add, mul_comm, mul_left_comm]\n    <;> linarith\n  linarith [h5]\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 25, "alphabet": 19, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 13, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 4, "fixpoint": true, "chars_before": 827, "chars_after": 540, "seconds": 375.8, "id": "induction_divisibility_3divnto3m2n", "proof_after": "theorem induction_divisibility_3divnto3m2n (n : \u2115) : 3 \u2223 n ^ 3 + 2 * n  :=  by\n  have inductive_step : \u2200 k, 3 \u2223 k ^ 3 + 2 * k \u2192 3 \u2223 (k + 1) ^ 3 + 2 * (k + 1) := by\n    intro k hk\n    have h1 : (k + 1) ^ 3 + 2 * (k + 1) = k ^ 3 + 2 * k + 3 * (k ^ 2 + k + 1) := by\n      ring\n    omega\n  have conclusion : 3 \u2223 n ^ 3 + 2 * n := by\n    induction n with\n    | zero =>\n      simpa using base_case\n    | succ n ihn =>\n      simpa [Nat.pow_succ, Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.mul_zero] using\n        inductive_step n (by simpa [Nat.pow_succ, Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.mul_zero] using ihn)\n  omega\n", "proof_before": "theorem induction_divisibility_3divnto3m2n (n : \u2115) : 3 \u2223 n ^ 3 + 2 * n  :=  by\n  have base_case : 3 \u2223 0 ^ 3 + 2 * 0 := by\n    norm_num\n    <;> decide\n    <;> simp_all [Nat.mul_mod, Nat.pow_succ]\n    <;> norm_num\n    <;> decide\n    <;> omega\n  have inductive_step : \u2200 k, 3 \u2223 k ^ 3 + 2 * k \u2192 3 \u2223 (k + 1) ^ 3 + 2 * (k + 1) := by\n    intro k hk\n    have h1 : (k + 1) ^ 3 + 2 * (k + 1) = k ^ 3 + 2 * k + 3 * (k ^ 2 + k + 1) := by\n      ring\n    rw [h1]\n    apply Nat.dvd_add\n    \u00b7 exact hk\n    \u00b7 simp [Nat.dvd_mul_right]\n  have conclusion : 3 \u2223 n ^ 3 + 2 * n := by\n    induction n with\n    | zero =>\n      simpa using base_case\n    | succ n ihn =>\n      simpa [Nat.pow_succ, Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.mul_zero] using\n        inductive_step n (by simpa [Nat.pow_succ, Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.mul_zero] using ihn)\n  simp_all [Nat.dvd_iff_mod_eq_zero]\n  <;> ring_nf\n  <;> omega\n"}
{"mode": "clear", "steps_before": 11, "alphabet": 12, "compiles": false, "id": "mathd_numbertheory_466", "proof_after": "theorem mathd_numbertheory_466 : (\u2211 k in Finset.range 11, k) % 9 = 1  :=  by\n  have sum_calc : \u2211 k in Finset.range 11, k = 55 := by\n    rfl\n    <;> decide\n    <;> decide\n    <;> decide\n  have remainder_calc : 55 % 9 = 1 := by\n    norm_num\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n  simpa [Finset.sum_range_succ, sum_calc, Nat.mul_mod, Nat.add_mod, Nat.mod_mod] using remainder_calc\n", "proof_before": "theorem mathd_numbertheory_466 : (\u2211 k in Finset.range 11, k) % 9 = 1  :=  by\n  have sum_calc : \u2211 k in Finset.range 11, k = 55 := by\n    rfl\n    <;> decide\n    <;> decide\n    <;> decide\n  have remainder_calc : 55 % 9 = 1 := by\n    norm_num\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n  simpa [Finset.sum_range_succ, sum_calc, Nat.mul_mod, Nat.add_mod, Nat.mod_mod] using remainder_calc\n"}
{"mode": "clear", "steps_before": 18, "alphabet": 16, "compiles": false, "id": "mathd_algebra_43", "proof_after": "theorem mathd_algebra_43 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = a * x + b) (h\u2081 : f 7 = 4)\n  (h\u2082 : f 6 = 3) : f 3 = 0  :=  by\n  have h_slope : a = 1 := by\n    simp_all only [mul_comm, mul_sub, mul_one, sub_eq_iff_eq_add, mul_add, mul_comm]\n    linarith\n  have h_intercept : b = -3 := by\n    have h\u2083 : b = -3 := by\n      rw [h\u2080] at h\u2081 h\u2082\n      linarith\n    assumption\n  have h_line_eq : \u2200 x, f x = 1 * x + (-3) := by\n    intro x\n    rw [h\u2080]\n    simp [h_slope, h_intercept]\n    <;> linarith\n  have h_x_intercept : f 3 = 0 := by\n    simp_all only [mul_one, mul_zero, add_zero, zero_add, mul_neg, mul_one]\n    <;> linarith\n  simp_all only [mul_one, mul_neg, mul_zero, add_left_neg]\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_43 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = a * x + b) (h\u2081 : f 7 = 4)\n  (h\u2082 : f 6 = 3) : f 3 = 0  :=  by\n  have h_slope : a = 1 := by\n    simp_all only [mul_comm, mul_sub, mul_one, sub_eq_iff_eq_add, mul_add, mul_comm]\n    linarith\n  have h_intercept : b = -3 := by\n    have h\u2083 : b = -3 := by\n      rw [h\u2080] at h\u2081 h\u2082\n      linarith\n    assumption\n  have h_line_eq : \u2200 x, f x = 1 * x + (-3) := by\n    intro x\n    rw [h\u2080]\n    simp [h_slope, h_intercept]\n    <;> linarith\n  have h_x_intercept : f 3 = 0 := by\n    simp_all only [mul_one, mul_zero, add_zero, zero_add, mul_neg, mul_one]\n    <;> linarith\n  simp_all only [mul_one, mul_neg, mul_zero, add_left_neg]\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 44, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 895, "chars_after": 11, "seconds": 1.8, "id": "mathd_algebra_455", "proof_after": "theorem mathd_algebra_455 (x : \u211d) (h\u2080 : 2 * (2 * (2 * (2 * x))) = 48) : x = 3  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_455 (x : \u211d) (h\u2080 : 2 * (2 * (2 * (2 * x))) = 48) : x = 3  :=  by\n  have h\u2081 : 2 * x = 2 * x := by\n    norm_num at h\u2080 \u22a2\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  have h\u2082 : 2 * (2 * x) = 4 * x := by\n    nlinarith [h\u2080]\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n  have h\u2083 : 2 * (4 * x) = 8 * x := by\n    linarith\n    <;> ring\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n  have h\u2084 : 2 * (8 * x) = 16 * x := by\n    simp_all only [mul_assoc, mul_left_comm, mul_right_comm]\n    <;> linarith\n  have h\u2085 : 16 * x = 48 := by\n    have h\u2085 : 16 * x = 48 := by\n      linarith\n    exact h\u2085\n  have h\u2086 : x = 48 / 16 := by\n    have h\u2086 : x = 48 / 16 := by\n      linarith\n    linarith\n  have h\u2087 : x = 3 := by\n    rw [h\u2086]\n    norm_num\n  linarith\n  <;> linarith\n  <;> linarith\n  <;> linarith\n  <;> linarith\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 32, "alphabet": 19, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 722, "chars_after": 8, "seconds": 0.5, "id": "mathd_numbertheory_198", "proof_after": "theorem mathd_numbertheory_198 : 5 ^ 2005 % 100 = 25  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_198 : 5 ^ 2005 % 100 = 25  :=  by\n  have base_case : 5 ^ 2 % 100 = 25 := by\n    norm_num\n    <;> rfl\n    <;> rfl\n    <;> rfl\n    <;> rfl\n    <;> rfl\n    <;> rfl\n    <;> rfl\n    <;> rfl\n    <;> rfl\n    <;> rfl\n    <;> rfl\n  have inductive_step : \u2200 k : \u2115, k \u2265 2 \u2192 5 ^ k % 100 = 25 \u2192 5 ^ (k + 1) % 100 = 25 := by\n    intro k hk h\n    rw [pow_succ]\n    norm_num [Nat.mul_mod, h]\n    <;> simp [hk]\n    <;> norm_num\n    <;> omega\n  have general_pattern : \u2200 n : \u2115, n \u2265 2 \u2192 5 ^ n % 100 = 25 := by\n    intro n hn\n    induction' hn with k hk\n    case refl => simp [base_case]\n    case step => simp_all [inductive_step]\n  have result : 5 ^ 2005 % 100 = 25 := by\n    apply general_pattern\n    norm_num\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  apply result\n"}
{"mode": "clear", "steps_before": 21, "alphabet": 13, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 509, "chars_after": 11, "seconds": 1.5, "id": "mathd_algebra_109", "proof_after": "theorem mathd_algebra_109 (a b : \u211d) (h\u2080 : 3 * a + 2 * b = 12) (h\u2081 : a = 4) : b = 0  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_109 (a b : \u211d) (h\u2080 : 3 * a + 2 * b = 12) (h\u2081 : a = 4) : b = 0  :=  by\n  have h\u2082 : 3 * 4 + 2 * b = 12 := by\n    simp_all only [mul_add, mul_comm, mul_left_comm, mul_assoc]\n    <;> linarith\n  have h\u2083 : 12 + 2 * b = 12 := by\n    simp_all only [mul_comm, mul_assoc, mul_left_comm, h\u2081]\n    linarith\n  have h\u2084 : 2 * b = 0 := by\n    have h\u2084 : 2 * b = 0 := by\n      linarith\n    linarith\n  have h\u2085 : b = 0 := by\n    have h\u2085 : b = 0 := by\n      have : 2 * b = 0 := h\u2084\n      linarith\n    simpa using h\u2085\n  linarith\n  <;> linarith\n  <;> linarith\n  <;> linarith\n  <;> linarith\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 17, "alphabet": 14, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 487, "chars_after": 7, "seconds": 2.3, "id": "mathd_numbertheory_84", "proof_after": "theorem mathd_numbertheory_84 : Int.floor ((9 : \u211d) / 160 * 100) = 5  :=  by\n  ring\n", "proof_before": "theorem mathd_numbertheory_84 : Int.floor ((9 : \u211d) / 160 * 100) = 5  :=  by\n  have decimal_eq : (9 : \u211d) / 160 = 0.05625 := by\n    norm_num\n    <;> norm_num\n    <;> norm_num\n    <;> norm_num\n    <;> norm_num\n  have shifted_decimal : (9 : \u211d) / 160 * 100 = 5.625 := by\n    norm_num [decimal_eq]\n    <;> norm_num\n    <;> norm_num\n    <;> norm_num\n    <;> norm_num\n  have floor_value : Int.floor ((9 : \u211d) / 160 * 100) = 5 := by\n    norm_num [Int.floor_eq_iff, shifted_decimal]\n    <;> norm_num\n    <;> linarith\n  simpa [shifted_decimal, floor_value] using floor_value\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 13, "compiles": false, "id": "mathd_algebra_616", "proof_after": "theorem mathd_algebra_616 (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x ^ 3 + 2 * x + 1)\n    (h\u2081 : \u2200 x, g x = x - 1) : f (g 1) = 1  :=  by\n  have g1_eq : g 1 = 1 - 1 := by\n    simp [h\u2081]\n    <;> norm_num\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  have g1_eq_zero : g 1 = 0 := by\n    simp_all only [rpow_two, mul_neg, mul_one, sub_eq_add_neg, add_assoc]\n    norm_num\n  have fg1_eq : f (g 1) = f 0 := by\n    simp_all only [rpow_two, mul_one, mul_zero, sub_zero, sub_eq_add_neg, neg_add_rev, neg_neg, neg_zero,\n      add_zero, zero_add, zero_mul, mul_zero, add_left_neg, add_right_neg, zero_sub, sub_zero, sub_neg_eq_add,\n      add_comm, add_left_comm, add_assoc]\n    <;> norm_num\n  have f0_eq : f 0 = 0 ^ 3 + 2 * 0 + 1 := by\n    simp_all only [h\u2080, h\u2081, mul_zero, mul_one, sub_zero, zero_add]\n    <;> norm_num\n    <;> linarith\n  have f0_eq_one : f 0 = 1 := by\n    simp_all [h\u2080, h\u2081, g1_eq, g1_eq_zero, fg1_eq, f0_eq]\n  have final_eq : f (g 1) = 1 := by\n    simp_all\n    <;> linarith\n  simp_all only [rpow_two, mul_one, mul_zero, add_zero, sub_self, zero_add]\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_616 (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x ^ 3 + 2 * x + 1)\n    (h\u2081 : \u2200 x, g x = x - 1) : f (g 1) = 1  :=  by\n  have g1_eq : g 1 = 1 - 1 := by\n    simp [h\u2081]\n    <;> norm_num\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  have g1_eq_zero : g 1 = 0 := by\n    simp_all only [rpow_two, mul_neg, mul_one, sub_eq_add_neg, add_assoc]\n    norm_num\n  have fg1_eq : f (g 1) = f 0 := by\n    simp_all only [rpow_two, mul_one, mul_zero, sub_zero, sub_eq_add_neg, neg_add_rev, neg_neg, neg_zero,\n      add_zero, zero_add, zero_mul, mul_zero, add_left_neg, add_right_neg, zero_sub, sub_zero, sub_neg_eq_add,\n      add_comm, add_left_comm, add_assoc]\n    <;> norm_num\n  have f0_eq : f 0 = 0 ^ 3 + 2 * 0 + 1 := by\n    simp_all only [h\u2080, h\u2081, mul_zero, mul_one, sub_zero, zero_add]\n    <;> norm_num\n    <;> linarith\n  have f0_eq_one : f 0 = 1 := by\n    simp_all [h\u2080, h\u2081, g1_eq, g1_eq_zero, fg1_eq, f0_eq]\n  have final_eq : f (g 1) = 1 := by\n    simp_all\n    <;> linarith\n  simp_all only [rpow_two, mul_one, mul_zero, add_zero, sub_self, zero_add]\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 20, "alphabet": 17, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1065, "chars_after": 143, "seconds": 13.0, "id": "mathd_algebra_437", "proof_after": "theorem mathd_algebra_437 (x y : \u211d) (n : \u2124) (h\u2080 : x ^ 3 = -45) (h\u2081 : y ^ 3 = -101) (h\u2082 : x < n)\n  (h\u2083 : \u2191n < y) : n = -4  :=  by\n  nlinarith [sq_nonneg (x - y), sq_nonneg (x + y), sq_nonneg (x + y - 1), sq_nonneg (x + y + 1), sq_nonneg (x - y + 1), sq_nonneg (x - y - 1)]\n", "proof_before": "theorem mathd_algebra_437 (x y : \u211d) (n : \u2124) (h\u2080 : x ^ 3 = -45) (h\u2081 : y ^ 3 = -101) (h\u2082 : x < n)\n  (h\u2083 : \u2191n < y) : n = -4  :=  by\n  have h4 : (-4 : \u211d) < x := by\n    nlinarith [sq_nonneg (x - y), sq_nonneg (x + y), sq_nonneg (x ^ 2 - y ^ 2), sq_nonneg (x ^ 2 + y ^ 2),\n      sq_nonneg (x - y), sq_nonneg (x + y), sq_nonneg (x ^ 2 - y ^ 2), sq_nonneg (x ^ 2 + y ^ 2)]\n  have h5 : x < (-3 : \u211d) := by\n    norm_num\n    nlinarith [h\u2080, h\u2081, h\u2082, h\u2083, h4, sq_nonneg (x + 3), sq_nonneg (x - 3), sq_nonneg (y + 3), sq_nonneg (y - 3)]\n  have h6 : (-5 : \u211d) < y := by\n    nlinarith [pow_two_nonneg (x - (-3 : \u211d)), pow_two_nonneg (y - (-5 : \u211d)),\n      pow_two_nonneg (x + 4), pow_two_nonneg (y + 5),\n      mul_self_nonneg (x - y), mul_self_nonneg (x + y),\n      mul_self_nonneg (x - y + 10), mul_self_nonneg (x + y - 10)]\n  have h7 : y < (-4 : \u211d) := by\n    nlinarith [sq_nonneg (x - y), sq_nonneg (x + y), sq_nonneg (x + y - 1), sq_nonneg (x + y + 1), sq_nonneg (x - y + 1), sq_nonneg (x - y - 1)]\n  have h8 : n = -4 := by\n    norm_num\n    <;> nlinarith [h\u2080, h\u2081, h4, h5, h6, h7]\n    <;> linarith\n    <;> nlinarith\n    <;> linarith\n    <;> nlinarith\n    <;> linarith\n    <;> nlinarith\n    <;> linarith\n  exact h8\n"}
{"mode": "clear", "steps_before": 22, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 896, "chars_after": 7, "seconds": 2.0, "id": "mathd_algebra_182", "proof_after": "theorem mathd_algebra_182 (y : \u2102) : 7 * (3 * y + 2) = 21 * y + 14  :=  by\n  ring\n", "proof_before": "theorem mathd_algebra_182 (y : \u2102) : 7 * (3 * y + 2) = 21 * y + 14  :=  by\n  have step1 : 7 * (3 * y + 2) = 7 * (3 * y) + 7 * 2 := by\n    simp [left_distrib, right_distrib, add_comm, add_left_comm, mul_comm, mul_left_comm]\n    <;> ring\n  have step2 : 7 * (3 * y) = 21 * y := by\n    have h : 7 * (3 * y + 2) = 7 * (3 * y) + 7 * 2 := step1\n    linear_combination h\n  have step3 : 7 * 2 = 14 := by\n    simp_all only [mul_add, mul_comm, mul_left_comm, mul_assoc]\n    ring_nf\n    <;> simp_all\n    <;> linarith\n  have step4 : 7 * (3 * y) + 7 * 2 = 21 * y + 14 := by\n    simp_all only [mul_add, mul_one, mul_comm, mul_left_comm, mul_assoc]\n    <;> ring\n    <;> simp_all only [mul_add, mul_one, mul_comm, mul_left_comm, mul_assoc]\n    <;> ring\n  have step5 : 7 * (3 * y + 2) = 21 * y + 14 := by\n    simp_all only [mul_add, mul_one, add_assoc, add_left_comm, add_right_comm]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  simpa [mul_add, mul_comm, mul_left_comm] using step5\n"}
{"mode": "clear", "steps_before": 27, "alphabet": 24, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 16, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "fixpoint": true, "chars_before": 849, "chars_after": 574, "seconds": 628.8, "id": "amc12a_2017_p7", "proof_after": "theorem amc12a_2017_p7 (f : \u2115 \u2192 \u211d) (h\u2080 : f 1 = 2) (h\u2081 : \u2200 n, 1 < n \u2227 Even n \u2192 f n = f (n - 1) + 1)\n  (h\u2082 : \u2200 n, 1 < n \u2227 Odd n \u2192 f n = f (n - 2) + 2) : f 2017 = 2018  :=  by\n  have inductive_step : \u2200 n, n > 1 \u2192 f n = n + 1 := by\n    intro n hn\n    cases' n with n\n    \u00b7 contradiction\n    cases' n with n\n    \u00b7 contradiction\n    induction n with\n    | zero =>\n      simp_all [Nat.succ_eq_add_one, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    | succ n ih =>\n      cases' Nat.even_or_odd n with h h <;> simp_all [Nat.succ_eq_add_one, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm, parity_simps]\n      <;> linarith\n  have f_2017 : f 2017 = 2017 + 1 := by\n    simpa [Nat.even_iff, Nat.odd_iff] using inductive_step 2017 (by norm_num)\n  linarith\n", "proof_before": "theorem amc12a_2017_p7 (f : \u2115 \u2192 \u211d) (h\u2080 : f 1 = 2) (h\u2081 : \u2200 n, 1 < n \u2227 Even n \u2192 f n = f (n - 1) + 1)\n  (h\u2082 : \u2200 n, 1 < n \u2227 Odd n \u2192 f n = f (n - 2) + 2) : f 2017 = 2018  :=  by\n  have base_case : f 1 = 2 := by\n    simp_all [Nat.lt_succ_self]\n    <;> linarith\n  have inductive_step : \u2200 n, n > 1 \u2192 f n = n + 1 := by\n    intro n hn\n    cases' n with n\n    \u00b7 contradiction\n    cases' n with n\n    \u00b7 contradiction\n    induction n with\n    | zero =>\n      simp_all [Nat.succ_eq_add_one, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    | succ n ih =>\n      cases' Nat.even_or_odd n with h h <;> simp_all [Nat.succ_eq_add_one, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm, parity_simps]\n      <;> linarith\n  have f_2017 : f 2017 = 2017 + 1 := by\n    simpa [Nat.even_iff, Nat.odd_iff] using inductive_step 2017 (by norm_num)\n  have result : f 2017 = 2018 := by\n    have h\u2083 := inductive_step 1\n    have h\u2084 := inductive_step 2\n    simp at h\u2083 h\u2084\n    have h\u2085 := h\u2082 2\n    have h\u2086 := h\u2081 2\n    simp at h\u2085 h\u2086\n    linarith\n  exact result\n"}
{"mode": "clear", "steps_before": 41, "alphabet": 34, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 29, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 2, "fixpoint": false, "chars_before": 1309, "chars_after": 1064, "seconds": 448.7, "id": "amc12b_2002_p11", "proof_after": "theorem amc12b_2002_p11 (a b : \u2115) (h\u2080 : Nat.Prime a) (h\u2081 : Nat.Prime b) (h\u2082 : Nat.Prime (a + b))\n  (h\u2083 : Nat.Prime (a - b)) : Nat.Prime (a + b + (a - b + (a + b)))  :=  by\n  have sum_expr : a + b + (a - b + (a + b)) = 3 * a + b := by\n    have h\u2084 := h\u2082.eq_one_or_self_of_dvd 2\n    have h\u2085 := h\u2083.eq_one_or_self_of_dvd 2\n    have h\u2086 := h\u2082.eq_one_or_self_of_dvd 3\n    have h\u2087 := h\u2083.eq_one_or_self_of_dvd 3\n    omega\n  have a_val : a = 5 := by\n    have h\u2084 := h\u2080.two_le\n    have h\u2085 := h\u2081.two_le\n    have h\u2086 := h\u2082.two_le\n    have h\u2087 := h\u2083.two_le\n    have h\u2088 := Nat.Prime.eq_one_or_self_of_dvd h\u2082 2\n    have h\u2089 := Nat.Prime.eq_one_or_self_of_dvd h\u2083 2\n    have h\u2081\u2080 := Nat.Prime.eq_one_or_self_of_dvd h\u2080 2\n    have h\u2081\u2081 := Nat.Prime.eq_one_or_self_of_dvd h\u2081 2\n    have h\u2081\u2082 := Nat.Prime.eq_one_or_self_of_dvd h\u2080 3\n    have h\u2081\u2083 := Nat.Prime.eq_one_or_self_of_dvd h\u2081 3\n    have h\u2081\u2084 := Nat.Prime.eq_one_or_self_of_dvd h\u2082 3\n    have h\u2081\u2085 := Nat.Prime.eq_one_or_self_of_dvd h\u2083 3\n    simp_all\n    <;> omega\n  have b_val : b = 2 := by\n    have h\u2084 := h\u2080.eq_one_or_self_of_dvd 2\n    have h\u2085 := h\u2081.eq_one_or_self_of_dvd 2\n    have h\u2086 := h\u2082.eq_one_or_self_of_dvd 2\n    have h\u2087 := h\u2083.eq_one_or_self_of_dvd 2\n    simp_all\n    <;> omega\n  norm_num [a_val, b_val]\n", "proof_before": "theorem amc12b_2002_p11 (a b : \u2115) (h\u2080 : Nat.Prime a) (h\u2081 : Nat.Prime b) (h\u2082 : Nat.Prime (a + b))\n  (h\u2083 : Nat.Prime (a - b)) : Nat.Prime (a + b + (a - b + (a + b)))  :=  by\n  have sum_expr : a + b + (a - b + (a + b)) = 3 * a + b := by\n    have h\u2084 := h\u2082.eq_one_or_self_of_dvd 2\n    have h\u2085 := h\u2083.eq_one_or_self_of_dvd 2\n    have h\u2086 := h\u2082.eq_one_or_self_of_dvd 3\n    have h\u2087 := h\u2083.eq_one_or_self_of_dvd 3\n    omega\n  have a_val : a = 5 := by\n    have h\u2084 := h\u2080.two_le\n    have h\u2085 := h\u2081.two_le\n    have h\u2086 := h\u2082.two_le\n    have h\u2087 := h\u2083.two_le\n    have h\u2088 := Nat.Prime.eq_one_or_self_of_dvd h\u2082 2\n    have h\u2089 := Nat.Prime.eq_one_or_self_of_dvd h\u2083 2\n    have h\u2081\u2080 := Nat.Prime.eq_one_or_self_of_dvd h\u2080 2\n    have h\u2081\u2081 := Nat.Prime.eq_one_or_self_of_dvd h\u2081 2\n    have h\u2081\u2082 := Nat.Prime.eq_one_or_self_of_dvd h\u2080 3\n    have h\u2081\u2083 := Nat.Prime.eq_one_or_self_of_dvd h\u2081 3\n    have h\u2081\u2084 := Nat.Prime.eq_one_or_self_of_dvd h\u2082 3\n    have h\u2081\u2085 := Nat.Prime.eq_one_or_self_of_dvd h\u2083 3\n    simp_all\n    <;> omega\n  have b_val : b = 2 := by\n    have h\u2084 := h\u2080.eq_one_or_self_of_dvd 2\n    have h\u2085 := h\u2081.eq_one_or_self_of_dvd 2\n    have h\u2086 := h\u2082.eq_one_or_self_of_dvd 2\n    have h\u2087 := h\u2083.eq_one_or_self_of_dvd 2\n    simp_all\n    <;> omega\n  have sum_val : 3 * 5 + 2 = 17 := by\n    norm_num [a_val, b_val]\n    <;> decide\n    <;> simp_all\n    <;> norm_num\n    <;> decide\n  have prime_17 : Nat.Prime 17 := by\n    norm_num [Nat.Prime]\n    <;> decide\n  subst_vars\n  simp_all [Nat.Prime]\n  <;> norm_num\n  <;> decide\n"}
{"mode": "clear", "steps_before": 25, "alphabet": 24, "compiles": false, "id": "mathd_numbertheory_303", "proof_after": "theorem mathd_numbertheory_303 (S : Finset \u2115)\n  (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 2 \u2264 n \u2227 171 \u2261 80 [MOD n] \u2227 468 \u2261 13 [MOD n]) : (\u2211 k in S, k) = 111  :=  by\n  have : S = {7, 13, 91} := by\n    ext n\n    simp only [Finset.mem_insert, Finset.mem_singleton, h\u2080, Nat.ModEq, Nat.ModEq]\n    constructor\n    \u00b7 intro h\n      match n with\n      | 0 => contradiction\n      | 1 => contradiction\n      | 2 => contradiction\n      | 7 => simp\n      | 13 => simp\n      | 91 => simp\n      | 191 => contradiction\n      | 291 => contradiction\n      | 491 => contradiction\n    \u00b7 intro h\n      match n with\n      | 7 => simp\n      | 13 => simp\n      | 91 => simp\n      | 191 => contradiction\n      | 291 => contradiction\n      | 491 => contradiction\n  rw [this]\n  norm_num\n", "proof_before": "theorem mathd_numbertheory_303 (S : Finset \u2115)\n  (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 2 \u2264 n \u2227 171 \u2261 80 [MOD n] \u2227 468 \u2261 13 [MOD n]) : (\u2211 k in S, k) = 111  :=  by\n  have : S = {7, 13, 91} := by\n    ext n\n    simp only [Finset.mem_insert, Finset.mem_singleton, h\u2080, Nat.ModEq, Nat.ModEq]\n    constructor\n    \u00b7 intro h\n      match n with\n      | 0 => contradiction\n      | 1 => contradiction\n      | 2 => contradiction\n      | 7 => simp\n      | 13 => simp\n      | 91 => simp\n      | 191 => contradiction\n      | 291 => contradiction\n      | 491 => contradiction\n    \u00b7 intro h\n      match n with\n      | 7 => simp\n      | 13 => simp\n      | 91 => simp\n      | 191 => contradiction\n      | 291 => contradiction\n      | 491 => contradiction\n  rw [this]\n  norm_num\n"}
{"mode": "clear", "steps_before": 100, "alphabet": 30, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 2435, "chars_after": 6, "seconds": 45.4, "id": "mathd_numbertheory_169", "proof_after": "theorem mathd_numbertheory_169 : Nat.gcd 20! 200000 = 40000  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_169 : Nat.gcd 20! 200000 = 40000  :=  by\n  have h1 : 20! = 2^18 * 3^8 * 5^4 * 7^2 * 11 * 13 * 17 * 19 := by\n    decide\n    <;>\n      norm_num\n    <;>\n      decide\n    <;>\n      norm_num\n    <;>\n      decide\n    <;>\n      norm_num\n    <;>\n      decide\n    <;>\n      norm_num\n    <;>\n      decide\n    <;>\n      norm_num\n    <;>\n      decide\n    <;>\n      norm_num\n    <;>\n      decide\n  have h2 : 200000 = 2^6 * 5^5 := by\n    norm_num [Nat.factorial, Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have h3 : Nat.gcd (2^18) (2^6) = 2^6 := by\n    simp_all [Nat.gcd_eq_right, Nat.pow_le_pow_of_le_right]\n    <;> norm_num\n  have h4 : Nat.gcd (5^4) (5^5) = 5^4 := by\n    simp [h3, Nat.gcd_eq_right, Nat.gcd_eq_left]\n    <;> rfl\n  have h5 : Nat.gcd (3^8) 1 = 1 := by\n    simp [Nat.gcd_eq_right]\n    <;> norm_num\n    <;> rfl\n    <;> rfl\n    <;> rfl\n  have h6 : Nat.gcd (7^2) 1 = 1 := by\n    simp [h3, h4, h5, Nat.gcd_eq_right]\n    <;> simp [h3, h4, h5, Nat.gcd_eq_right]\n    <;> simp [h3, h4, h5, Nat.gcd_eq_right]\n    <;> simp [h3, h4, h5, Nat.gcd_eq_right]\n    <;> simp [h3, h4, h5, Nat.gcd_eq_right]\n  have h7 : Nat.gcd 11 1 = 1 := by\n    simpa using h5\n  have h8 : Nat.gcd 13 1 = 1 := by\n    simp [Nat.gcd_eq_right]\n    <;> aesop\n  have h9 : Nat.gcd 17 1 = 1 := by\n    simp [Nat.gcd_eq_right]\n    <;> norm_num\n    <;> aesop\n    <;> simp_all\n    <;> norm_num\n    <;> aesop\n  have h10 : Nat.gcd 19 1 = 1 := by\n    simp_all [Nat.gcd_eq_right, Nat.gcd_eq_left]\n    <;> norm_num\n    <;> rfl\n    <;> decide\n    <;> decide\n    <;> decide\n  have h11 : Nat.gcd 20! 200000 = Nat.gcd (2^18 * 3^8 * 5^4 * 7^2 * 11 * 13 * 17 * 19) (2^6 * 5^5) := by\n    simp [h1, h2, h3, h4, h5, h6, h7, h8, h9, h10, Nat.gcd_mul_left, Nat.gcd_mul_right, Nat.gcd_assoc]\n    <;> decide\n  have h12 : Nat.gcd (2^18 * 3^8 * 5^4 * 7^2 * 11 * 13 * 17 * 19) (2^6 * 5^5) = 2^6 * 5^4 := by\n    simp_all [Nat.gcd_eq_right, Nat.gcd_eq_left]\n    <;> norm_num\n    <;> rfl\n  have h13 : 2^6 * 5^4 = 40000 := by\n    simp_all [Nat.gcd_eq_right, Nat.gcd_eq_left]\n    <;> norm_num\n    <;> ring_nf\n    <;> rfl\n    <;> simp_all [Nat.gcd_eq_right, Nat.gcd_eq_left]\n    <;> norm_num\n    <;> ring_nf\n    <;> rfl\n  simp_all [Nat.gcd_eq_right, Nat.gcd_eq_left]\n  <;> norm_num\n  <;> rfl\n  <;> apply h13\n  <;> apply h12\n  <;> apply h11\n  <;> apply h10\n  <;> apply h9\n  <;> apply h8\n  <;> apply h7\n  <;> apply h6\n  <;> apply h5\n  <;> apply h4\n  <;> apply h3\n  <;> apply h2\n  <;> apply h1\n"}
{"mode": "clear", "steps_before": 20, "alphabet": 16, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 529, "chars_after": 7, "seconds": 13.7, "id": "mathd_algebra_10", "proof_after": "theorem mathd_algebra_10 : abs ((120 : \u211d) / 100 * 30 - 130 / 100 * 20) = 10  :=  by\n  ring\n", "proof_before": "theorem mathd_algebra_10 : abs ((120 : \u211d) / 100 * 30 - 130 / 100 * 20) = 10  :=  by\n  have step1 : (120 : \u211d) / 100 * 30 = 36 := by\n    norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  have step2 : (130 : \u211d) / 100 * 20 = 26 := by\n    norm_num [mul_assoc]\n    <;> linarith\n  have step3 : (120 : \u211d) / 100 * 30 - 130 / 100 * 20 = 10 := by\n    nlinarith [step1, step2]\n    <;> norm_num\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  have step4 : |((120 : \u211d) / 100 * 30 - 130 / 100 * 20)| = 10 := by\n    rw [step3]\n    <;> simp [abs_of_pos]\n    <;> norm_num\n  simpa [abs_of_pos] using step4\n"}
{"mode": "clear", "steps_before": 69, "alphabet": 57, "compiles": false, "id": "mathd_numbertheory_32", "proof_after": "theorem mathd_numbertheory_32 (S : Finset \u2115) (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 n \u2223 36) : (\u2211 k in S, k) = 91  :=  by\n  have h\u2081 : 36 = 2^2 * 3^2 := by\n    apply Eq.symm\n    rw [show 36 = 2^2 * 3^2 by norm_num]\n    <;> rfl\n  have h\u2082 : S = {1, 2, 3, 4, 6, 9, 12, 18, 36} := by\n    have h\u2082 : S = {1, 2, 3, 4, 6, 9, 12, 18, 36} := by\n      ext n\n      simp only [h\u2080, Finset.mem_insert, Finset.mem_singleton, Nat.dvd_iff_mod_eq_zero,\n        Nat.mod_eq_of_lt]\n      constructor\n      \u00b7 intro h\n        match n with\n        | 1 => simp  \n        | 2 => simp  \n        | 3 => simp  \n        | 4 => simp  \n        | 6 => simp  \n        | 9 => simp  \n        | 12 => simp  \n        | 18 => simp  \n        | 36 => simp  \n        | 19 => contradiction  \n        | 20 => contradiction  \n        | 21 => contradiction  \n        | 22 => contradiction  \n        | 23 => contradiction  \n        | 24 => contradiction  \n        | 25 => contradiction  \n        | 26 => contradiction  \n        | 27 => contradiction  \n        | 28 => contradiction  \n        | 29 => contradiction  \n        | 30 => contradiction  \n        | 31 => contradiction  \n        | 32 => contradiction  \n        | 33 => contradiction  \n        | 34 => contradiction  \n        | 35 => contradiction  \n        | 37 => contradiction  \n        | 38 => contradiction  \n        | 39 => contradiction  \n        | 40 => contradiction  \n        | 41 => contradiction  \n        | 42 => contradiction  \n        | 43 => contradiction  \n        | 44 => contradiction  \n        | 45 => contradiction  \n        | 46 => contradiction  \n        | 47 => contradiction  \n        | 48 => contradiction  \n        | 49 => contradiction  \n        | 50 => contradiction  \n      \u00b7 intro h\n        match n with\n        | 1 => simp  \n        | 2 => simp  \n        | 3 => simp  \n        | 4 => simp  \n        | 6 => simp  \n        | 9 => simp  \n        | 12 => simp  \n        | 18 => simp  \n        | 36 => simp  \n    rw [h\u2082]\n  have h\u2083 : 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36 = 91 := by\n    simp [h\u2082, h\u2080, h\u2081, Nat.divisors]\n    <;> decide\n  have h\u2084 : (\u2211 k in S, k) = 91 := by\n    simp_all [Finset.sum_eq_multiset_sum]\n  exact h\u2084\n", "proof_before": "theorem mathd_numbertheory_32 (S : Finset \u2115) (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 n \u2223 36) : (\u2211 k in S, k) = 91  :=  by\n  have h\u2081 : 36 = 2^2 * 3^2 := by\n    apply Eq.symm\n    rw [show 36 = 2^2 * 3^2 by norm_num]\n    <;> rfl\n  have h\u2082 : S = {1, 2, 3, 4, 6, 9, 12, 18, 36} := by\n    have h\u2082 : S = {1, 2, 3, 4, 6, 9, 12, 18, 36} := by\n      ext n\n      simp only [h\u2080, Finset.mem_insert, Finset.mem_singleton, Nat.dvd_iff_mod_eq_zero,\n        Nat.mod_eq_of_lt]\n      constructor\n      \u00b7 intro h\n        match n with\n        | 1 => simp  \n        | 2 => simp  \n        | 3 => simp  \n        | 4 => simp  \n        | 6 => simp  \n        | 9 => simp  \n        | 12 => simp  \n        | 18 => simp  \n        | 36 => simp  \n        | 19 => contradiction  \n        | 20 => contradiction  \n        | 21 => contradiction  \n        | 22 => contradiction  \n        | 23 => contradiction  \n        | 24 => contradiction  \n        | 25 => contradiction  \n        | 26 => contradiction  \n        | 27 => contradiction  \n        | 28 => contradiction  \n        | 29 => contradiction  \n        | 30 => contradiction  \n        | 31 => contradiction  \n        | 32 => contradiction  \n        | 33 => contradiction  \n        | 34 => contradiction  \n        | 35 => contradiction  \n        | 37 => contradiction  \n        | 38 => contradiction  \n        | 39 => contradiction  \n        | 40 => contradiction  \n        | 41 => contradiction  \n        | 42 => contradiction  \n        | 43 => contradiction  \n        | 44 => contradiction  \n        | 45 => contradiction  \n        | 46 => contradiction  \n        | 47 => contradiction  \n        | 48 => contradiction  \n        | 49 => contradiction  \n        | 50 => contradiction  \n      \u00b7 intro h\n        match n with\n        | 1 => simp  \n        | 2 => simp  \n        | 3 => simp  \n        | 4 => simp  \n        | 6 => simp  \n        | 9 => simp  \n        | 12 => simp  \n        | 18 => simp  \n        | 36 => simp  \n    rw [h\u2082]\n  have h\u2083 : 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36 = 91 := by\n    simp [h\u2082, h\u2080, h\u2081, Nat.divisors]\n    <;> decide\n  have h\u2084 : (\u2211 k in S, k) = 91 := by\n    simp_all [Finset.sum_eq_multiset_sum]\n  exact h\u2084\n"}
{"mode": "clear", "steps_before": 36, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 4, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 8, "fixpoint": true, "chars_before": 926, "chars_after": 114, "seconds": 766.1, "id": "mathd_algebra_433", "proof_after": "theorem mathd_algebra_433 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 3 * Real.sqrt (2 * x - 7) - 8) : f 8 = 1  :=  by\n  have step6 : f 8 = 1 := by\n    simp_all only [mul_comm, mul_sub, mul_one, mul_assoc, mul_comm]\n    ring\n  omega\n", "proof_before": "theorem mathd_algebra_433 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 3 * Real.sqrt (2 * x - 7) - 8) : f 8 = 1  :=  by\n  have step1 : f 8 = 3 * Real.sqrt (2 * 8 - 7) - 8 := by\n    simp_all [mul_comm, mul_assoc]\n    <;> norm_num\n    <;> ring\n    <;> rw [\u2190 h\u2080]\n    <;> norm_num\n  have step2 : 2 * 8 - 7 = 9 := by\n    norm_num at step1\n    simp_all [mul_assoc, mul_comm, mul_left_comm, sub_eq_add_neg, add_assoc, add_left_comm]\n  have step3 : Real.sqrt 9 = 3 := by\n    rw [Real.sqrt_eq_iff_mul_self_eq] <;> norm_num\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n  have step4 : 3 * Real.sqrt 9 = 9 := by\n    rw [h\u2080] at step1\n    norm_num at step1\n    linarith\n  have step5 : 3 * Real.sqrt 9 - 8 = 1 := by\n    have h\u2081 : 3 * Real.sqrt 9 - 8 = 1 := by\n      rw [step3]\n      norm_num\n    linarith [step4]\n  have step6 : f 8 = 1 := by\n    simp_all only [mul_comm, mul_sub, mul_one, mul_assoc, mul_comm]\n    ring\n    <;> linarith\n  simp_all\n  <;> linarith\n  <;> linarith\n  <;> linarith\n  <;> linarith\n  <;> linarith\n  <;> linarith\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 3, "alphabet": 12, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 77, "chars_after": 12, "seconds": 26.7, "id": "mathd_numbertheory_48", "proof_after": "theorem mathd_numbertheory_48 (b : \u2115) (h\u2080 : 0 < b) (h\u2081 : 3 * b ^ 2 + 2 * b + 1 = 57) : b = 4  :=  by\n  nlinarith\n", "proof_before": "theorem mathd_numbertheory_48 (b : \u2115) (h\u2080 : 0 < b) (h\u2081 : 3 * b ^ 2 + 2 * b + 1 = 57) : b = 4  :=  by\n  have h\u2082 : b = 4 := by\n    nlinarith [sq_nonneg (b : \u2124), h\u2080, h\u2081]\n  simp_all\n"}
{"mode": "clear", "steps_before": 34, "alphabet": 19, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1007, "chars_after": 6, "seconds": 88.6, "id": "mathd_numbertheory_211", "proof_after": "theorem mathd_numbertheory_211 :\n  Finset.card (Finset.filter (fun n => 6 \u2223 4 * \u2191n - (2 : \u2124)) (Finset.range 60)) = 20  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_211 :\n  Finset.card (Finset.filter (fun n => 6 \u2223 4 * \u2191n - (2 : \u2124)) (Finset.range 60)) = 20  :=  by\n  have step1 : \u2200 n : \u2124, (6 \u2223 4 * n - 2) \u2194 (4 * n \u2261 2 [ZMOD 6]) := by\n    intro n\n    constructor <;> intro h\n    all_goals\n      simp_all [Int.ModEq, Int.emod_eq_of_lt]\n    <;> omega\n  have step2 : \u2200 n : \u2124, (4 * n \u2261 2 [ZMOD 6]) \u2194 (2 * n \u2261 1 [ZMOD 3]) := by\n    intro n\n    constructor\n    intro h\n    norm_num [Int.ModEq, Int.emod_eq_of_lt] at h \u22a2\n    omega\n    intro h\n    norm_num [Int.ModEq, Int.emod_eq_of_lt] at h \u22a2\n    omega\n  have step3 : \u2200 n : \u2124, (2 * n \u2261 1 [ZMOD 3]) \u2194 (n \u2261 2 [ZMOD 3]) := by\n    intro n\n    constructor\n    intro h\n    simp only [Int.ModEq, Int.ModEq] at h \u22a2\n    omega\n    intro h\n    simp only [Int.ModEq, Int.ModEq] at h \u22a2\n    omega\n  have step4 : (Finset.filter (fun n => n % 3 = 2) (Finset.range 60)).card = 20 := by\n    simp_all [Int.ModEq, Int.ModEq]\n    rfl\n    <;> decide\n    <;> decide\n    <;> decide\n  have step5 : Finset.card (Finset.filter (fun n => 6 \u2223 4 * \u2191n - (2 : \u2124)) (Finset.range 60)) = 20 := by\n    simpa [Int.ModEq, Int.emod_eq_of_lt] using step4\n  exact step5\n"}
{"mode": "clear", "steps_before": 19, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 488, "chars_after": 11, "seconds": 54.8, "id": "mathd_numbertheory_136", "proof_after": "theorem mathd_numbertheory_136 (n : \u2115) (h\u2080 : 123 * n + 17 = 39500) : n = 321  :=  by\n  linarith\n", "proof_before": "theorem mathd_numbertheory_136 (n : \u2115) (h\u2080 : 123 * n + 17 = 39500) : n = 321  :=  by\n  have h\u2081 : 123 * n = 39500 - 17 := by\n    have h\u2081 : 123 * n = 39500 - 17 := by\n      linarith\n    exact h\u2081\n  have h\u2082 : n = (39500 - 17) / 123 := by\n    apply Eq.symm\n    norm_num at h\u2081 \u22a2\n    <;> omega\n    <;> omega\n  have h\u2083 : (39500 - 17) / 123 = 321 := by\n    simp_all only [mul_comm, mul_one, mul_div_cancel_left]\n    <;> norm_num\n    <;> omega\n    <;> ring\n    <;> linarith\n  have h\u2084 : n = 321 := by\n    simp_all only [mul_comm, mul_one, mul_div_cancel_left]\n    <;> omega\n  rw [h\u2084]\n"}
{"mode": "clear", "steps_before": 112, "alphabet": 33, "compiles": false, "id": "amc12a_2019_p21", "proof_after": "theorem amc12a_2019_p21 (z : \u2102) (h\u2080 : z = (1 + Complex.I) / Real.sqrt 2) :\n  ((\u2211 k : \u2124 in Finset.Icc 1 12, z ^ k ^ 2) * (\u2211 k : \u2124 in Finset.Icc 1 12, 1 / z ^ k ^ 2)) = 36  :=  by\n  have h\u2081 : z ^ 8 = 1 := by\n    rw [h\u2080]\n    field_simp [pow_two, pow_succ, Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num <;> ring_nf <;> norm_num <;> simp [Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num <;> ring_nf <;> norm_num\n    <;> simp [Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n  have h\u2082 : (\u2211 k : \u2124 in Finset.Icc 1 12, z ^ k ^ 2) = 6 * z + 3 * z ^ 4 + 3 := by\n    have h\u2082 : z = (1 + Complex.I) / Real.sqrt 2 := h\u2080\n    have h\u2083 : z ^ 8 = 1 := h\u2081\n    have h\u2084 : \u2211 k in Finset.Icc 1 12, z ^ k ^ 2 = 6 * z + 3 * z ^ 4 + 3 := by\n      have h\u2085 : z ^ 2 = z ^ 2 := rfl\n      have h\u2086 : z ^ 4 = z ^ 4 := rfl\n      have h\u2087 : z ^ 6 = z ^ 6 := rfl\n      have h\u2088 : z ^ 8 = z ^ 8 := rfl\n      simp_all [Finset.sum_Icc_succ_top, Finset.sum_range_succ, Complex.ext_iff, pow_succ, mul_add, mul_comm, mul_left_comm]\n      <;> norm_num\n      <;> ring_nf\n      <;> norm_num\n      <;> simp_all [Complex.ext_iff, pow_succ, mul_add, mul_comm, mul_left_comm]\n      <;> norm_num\n      <;> ring_nf\n      <;> norm_num\n    exact h\u2084\n  have h\u2083 : (\u2211 k : \u2124 in Finset.Icc 1 12, 1 / z ^ k ^ 2) = 6 * z ^ 7 + 3 * z ^ 4 + 3 := by\n    simp_all [Complex.ext_iff, Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm,\n      mul_left_comm]\n    <;> field_simp [Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring\n    <;> norm_num\n    <;> simp_all [Complex.ext_iff, Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm,\n      mul_left_comm]\n    <;> field_simp [Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring\n    <;> norm_num\n    <;> simp_all [Complex.ext_iff, Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm,\n      mul_left_comm]\n    <;> field_simp [Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring\n    <;> norm_num\n  have h\u2084 : (6 * z + 3 * z ^ 4 + 3) * (6 * z ^ 7 + 3 * z ^ 4 + 3) = 36 := by\n    have h\u2084 : z = (1 + Complex.I) / Real.sqrt 2 := h\u2080\n    have h\u2085 : z ^ 8 = 1 := h\u2081\n    have h\u2086 : (\u2211 k : \u2124 in Finset.Icc 1 12, z ^ k ^ 2) = 6 * z + 3 * z ^ 4 + 3 := h\u2082\n    have h\u2087 : (\u2211 k : \u2124 in Finset.Icc 1 12, 1 / z ^ k ^ 2) = 6 * z ^ 7 + 3 * z ^ 4 + 3 := h\u2083\n    subst_vars\n    field_simp [Complex.ext_iff, pow_succ, Finset.sum_range_succ, Complex.I_sq, mul_comm, mul_assoc, mul_left_comm] at h\u2085 h\u2086 h\u2087 \u22a2\n    <;> ring_nf at h\u2085 h\u2086 h\u2087 \u22a2\n    <;> simp_all [Complex.ext_iff, pow_succ, Finset.sum_range_succ, Complex.I_sq, mul_comm, mul_assoc, mul_left_comm]\n    <;> nlinarith [sq_nonneg (z.re - z.im), sq_nonneg (z.re + z.im), sq_nonneg (z.re + z.im - 1), sq_nonneg (z.re - z.im - 1), sq_nonneg (z.re + z.im - 2), sq_nonneg (z.re - z.im - 2)]\n  have h\u2085 : ((\u2211 k : \u2124 in Finset.Icc 1 12, z ^ k ^ 2) * (\u2211 k : \u2124 in Finset.Icc 1 12, 1 / z ^ k ^ 2)) = 36 := by\n    rw [h\u2082, h\u2083]\n    <;> simp_all\n    <;> linarith\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  rw [h\u2085]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n", "proof_before": "theorem amc12a_2019_p21 (z : \u2102) (h\u2080 : z = (1 + Complex.I) / Real.sqrt 2) :\n  ((\u2211 k : \u2124 in Finset.Icc 1 12, z ^ k ^ 2) * (\u2211 k : \u2124 in Finset.Icc 1 12, 1 / z ^ k ^ 2)) = 36  :=  by\n  have h\u2081 : z ^ 8 = 1 := by\n    rw [h\u2080]\n    field_simp [pow_two, pow_succ, Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num <;> ring_nf <;> norm_num <;> simp [Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num <;> ring_nf <;> norm_num\n    <;> simp [Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> simp [Complex.ext_iff, Complex.I_mul_I, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n  have h\u2082 : (\u2211 k : \u2124 in Finset.Icc 1 12, z ^ k ^ 2) = 6 * z + 3 * z ^ 4 + 3 := by\n    have h\u2082 : z = (1 + Complex.I) / Real.sqrt 2 := h\u2080\n    have h\u2083 : z ^ 8 = 1 := h\u2081\n    have h\u2084 : \u2211 k in Finset.Icc 1 12, z ^ k ^ 2 = 6 * z + 3 * z ^ 4 + 3 := by\n      have h\u2085 : z ^ 2 = z ^ 2 := rfl\n      have h\u2086 : z ^ 4 = z ^ 4 := rfl\n      have h\u2087 : z ^ 6 = z ^ 6 := rfl\n      have h\u2088 : z ^ 8 = z ^ 8 := rfl\n      simp_all [Finset.sum_Icc_succ_top, Finset.sum_range_succ, Complex.ext_iff, pow_succ, mul_add, mul_comm, mul_left_comm]\n      <;> norm_num\n      <;> ring_nf\n      <;> norm_num\n      <;> simp_all [Complex.ext_iff, pow_succ, mul_add, mul_comm, mul_left_comm]\n      <;> norm_num\n      <;> ring_nf\n      <;> norm_num\n    exact h\u2084\n  have h\u2083 : (\u2211 k : \u2124 in Finset.Icc 1 12, 1 / z ^ k ^ 2) = 6 * z ^ 7 + 3 * z ^ 4 + 3 := by\n    simp_all [Complex.ext_iff, Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm,\n      mul_left_comm]\n    <;> field_simp [Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring\n    <;> norm_num\n    <;> simp_all [Complex.ext_iff, Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm,\n      mul_left_comm]\n    <;> field_simp [Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring\n    <;> norm_num\n    <;> simp_all [Complex.ext_iff, Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm,\n      mul_left_comm]\n    <;> field_simp [Complex.normSq, Complex.I_sq, pow_succ, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring\n    <;> norm_num\n  have h\u2084 : (6 * z + 3 * z ^ 4 + 3) * (6 * z ^ 7 + 3 * z ^ 4 + 3) = 36 := by\n    have h\u2084 : z = (1 + Complex.I) / Real.sqrt 2 := h\u2080\n    have h\u2085 : z ^ 8 = 1 := h\u2081\n    have h\u2086 : (\u2211 k : \u2124 in Finset.Icc 1 12, z ^ k ^ 2) = 6 * z + 3 * z ^ 4 + 3 := h\u2082\n    have h\u2087 : (\u2211 k : \u2124 in Finset.Icc 1 12, 1 / z ^ k ^ 2) = 6 * z ^ 7 + 3 * z ^ 4 + 3 := h\u2083\n    subst_vars\n    field_simp [Complex.ext_iff, pow_succ, Finset.sum_range_succ, Complex.I_sq, mul_comm, mul_assoc, mul_left_comm] at h\u2085 h\u2086 h\u2087 \u22a2\n    <;> ring_nf at h\u2085 h\u2086 h\u2087 \u22a2\n    <;> simp_all [Complex.ext_iff, pow_succ, Finset.sum_range_succ, Complex.I_sq, mul_comm, mul_assoc, mul_left_comm]\n    <;> nlinarith [sq_nonneg (z.re - z.im), sq_nonneg (z.re + z.im), sq_nonneg (z.re + z.im - 1), sq_nonneg (z.re - z.im - 1), sq_nonneg (z.re + z.im - 2), sq_nonneg (z.re - z.im - 2)]\n  have h\u2085 : ((\u2211 k : \u2124 in Finset.Icc 1 12, z ^ k ^ 2) * (\u2211 k : \u2124 in Finset.Icc 1 12, 1 / z ^ k ^ 2)) = 36 := by\n    rw [h\u2082, h\u2083]\n    <;> simp_all\n    <;> linarith\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  rw [h\u2085]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n  <;> ring_nf at *\n  <;> simp_all [Complex.ext_iff, Finset.sum_apply, mul_comm]\n"}
{"mode": "clear", "steps_before": 17, "alphabet": 19, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 508, "chars_after": 37, "seconds": 97.2, "id": "algebra_binomnegdiscrineq_10alt28asqp1", "proof_after": "theorem algebra_binomnegdiscrineq_10alt28asqp1 (a : \u211d) : 10 * a \u2264 28 * a ^ 2 + 1  :=  by\n  nlinarith [sq_nonneg (a - 5 / 28)]\n", "proof_before": "theorem algebra_binomnegdiscrineq_10alt28asqp1 (a : \u211d) : 10 * a \u2264 28 * a ^ 2 + 1  :=  by\n  have h1 : 28 * a ^ 2 - 10 * a + 1 \u2265 0 := by\n    have h\u2080 : 0 \u2264 (a - 5 / 28) ^ 2 := sq_nonneg _\n    have h\u2081 : 0 < 28 := by norm_num\n    nlinarith [h\u2080, h\u2081]\n  have h2 : \u2200 (a : \u211d), 28 * a ^ 2 - 10 * a + 1 \u2265 0 := by\n    intro a\n    have h2 : 28 * a ^ 2 - 10 * a + 1 \u2265 0 := by\n      have h2 := h1\n      nlinarith [sq_nonneg (a - 5 / 28)]\n    linarith\n  have h3 : \u2200 (a : \u211d), 28 * a ^ 2 - 10 * a + 1 \u2265 0 \u2192 10 * a \u2264 28 * a ^ 2 + 1 := by\n    intro a h\n    have h' := h2 a\n    linarith\n  have h4 := h3 a h1\n  linarith\n"}
{"mode": "clear", "steps_before": 43, "alphabet": 24, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 43, "lower_bound": null, "exhausted_depth": 2, "certified": false, "rounds": 6, "fixpoint": true, "chars_before": 1446, "chars_after": 1446, "seconds": 806.8, "id": "amc12a_2016_p2", "proof_after": "theorem amc12a_2016_p2 (x : \u211d) (h\u2080 : (10 : \u211d) ^ x * 100 ^ (2 * x) = 1000 ^ 5) : x = 3  :=  by\n  have h1 : 100 = (10 : \u211d)^2 := by\n    norm_num\n    <;> simp_all [pow_mul, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring_nf at h\u2080 \u22a2\n    <;> nlinarith [sq_nonneg (x - 2), sq_nonneg (x + 2)]\n  have h2 : 1000 = (10 : \u211d)^3 := by\n    rw [show (100 : \u211d) = 10 ^ 2 by norm_num] at h\u2080\n    norm_num at h\u2080\n    rw [show (1000 : \u211d) = 10 ^ 3 by norm_num]\n    <;> norm_num at h\u2080 \u22a2\n    <;> linarith\n  have h3 : (10 : \u211d)^x * ((10 : \u211d)^2)^(2*x) = ((10 : \u211d)^3)^5 := by\n    simp [h1, h2, mul_comm, mul_assoc, mul_left_comm] at h\u2080 \u22a2\n    <;> simp_all [rpow_add, rpow_mul, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> linarith\n  have h4 : (10 : \u211d)^x * ((10 : \u211d)^2)^(2*x) = (10 : \u211d)^15 := by\n    simp_all [rpow_mul, mul_assoc]\n    <;> linarith\n  have h5 : (10 : \u211d)^x * (10 : \u211d)^(4*x) = (10 : \u211d)^15 := by\n    rw [\u2190 h4]\n    <;> simp [h1, h2, h3, h4, rpow_mul]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have h6 : (10 : \u211d)^(x + 4*x) = (10 : \u211d)^15 := by\n    simp_all [rpow_add, rpow_mul, mul_add, add_mul, mul_comm, mul_left_comm]\n    <;> linarith\n    <;> assumption\n  have h7 : (10 : \u211d)^(5*x) = (10 : \u211d)^15 := by\n    have h7 : x + 4 * x = 5 * x := by ring\n    rw [h7] at h6\n    linarith\n  have h8 : 5*x = 15 := by\n    apply_fun fun a => logb 10 a at h7\n    simp_all [logb_rpow, logb_pow, mul_comm]\n    <;> ring_nf at *\n    <;> linarith\n  have h9 : x = 3 := by\n    rw [eq_comm] at h8\n    nlinarith [h\u2080, h1, h2, h3, h4, h5, h6, h7, h8]\n  linarith\n", "proof_before": "theorem amc12a_2016_p2 (x : \u211d) (h\u2080 : (10 : \u211d) ^ x * 100 ^ (2 * x) = 1000 ^ 5) : x = 3  :=  by\n  have h1 : 100 = (10 : \u211d)^2 := by\n    norm_num\n    <;> simp_all [pow_mul, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring_nf at h\u2080 \u22a2\n    <;> nlinarith [sq_nonneg (x - 2), sq_nonneg (x + 2)]\n  have h2 : 1000 = (10 : \u211d)^3 := by\n    rw [show (100 : \u211d) = 10 ^ 2 by norm_num] at h\u2080\n    norm_num at h\u2080\n    rw [show (1000 : \u211d) = 10 ^ 3 by norm_num]\n    <;> norm_num at h\u2080 \u22a2\n    <;> linarith\n  have h3 : (10 : \u211d)^x * ((10 : \u211d)^2)^(2*x) = ((10 : \u211d)^3)^5 := by\n    simp [h1, h2, mul_comm, mul_assoc, mul_left_comm] at h\u2080 \u22a2\n    <;> simp_all [rpow_add, rpow_mul, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> linarith\n  have h4 : (10 : \u211d)^x * ((10 : \u211d)^2)^(2*x) = (10 : \u211d)^15 := by\n    simp_all [rpow_mul, mul_assoc]\n    <;> linarith\n  have h5 : (10 : \u211d)^x * (10 : \u211d)^(4*x) = (10 : \u211d)^15 := by\n    rw [\u2190 h4]\n    <;> simp [h1, h2, h3, h4, rpow_mul]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have h6 : (10 : \u211d)^(x + 4*x) = (10 : \u211d)^15 := by\n    simp_all [rpow_add, rpow_mul, mul_add, add_mul, mul_comm, mul_left_comm]\n    <;> linarith\n    <;> assumption\n  have h7 : (10 : \u211d)^(5*x) = (10 : \u211d)^15 := by\n    have h7 : x + 4 * x = 5 * x := by ring\n    rw [h7] at h6\n    linarith\n  have h8 : 5*x = 15 := by\n    apply_fun fun a => logb 10 a at h7\n    simp_all [logb_rpow, logb_pow, mul_comm]\n    <;> ring_nf at *\n    <;> linarith\n  have h9 : x = 3 := by\n    rw [eq_comm] at h8\n    nlinarith [h\u2080, h1, h2, h3, h4, h5, h6, h7, h8]\n  linarith\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 50, "alphabet": 28, "compiles": false, "id": "amc12b_2003_p6", "proof_after": "theorem amc12b_2003_p6 (a r : \u211d) (u : \u2115 \u2192 \u211d) (h\u2080 : \u2200 k, u k = a * r ^ k) (h\u2081 : u 1 = 2)\n  (h\u2082 : u 3 = 6) : u 0 = 2 / Real.sqrt 3 \u2228 u 0 = -(2 / Real.sqrt 3)  :=  by\n  have h_a : a = 2 / r := by\n    have h\u2083 := h\u2080 1\n    have h\u2084 := h\u2080 3\n    simp_all [h\u2081, h\u2082]\n    have h\u2085 : r \u2260 0 := by\n      rintro rfl\n      linarith\n    field_simp [h\u2085] at *\n    <;> ring_nf at *\n    <;> nlinarith\n  have h_u3_subst : 2 / r * r^3 = 6 := by\n    have h\u2083 := h\u2080 1\n    have h\u2084 := h\u2080 3\n    rw [h_a] at h\u2083 h\u2084\n    simp_all [pow_one, pow_succ]\n    <;> ring_nf\n    <;> nlinarith\n  have h_r2 : r^2 = 3 := by\n    simp_all only [h_a, h\u2080, Nat.cast_ofNat, Nat.cast_one, Nat.cast_zero, Nat.cast_succ]\n    ring_nf\n    nlinarith\n  have h_a_pos : r = Real.sqrt 3 \u2192 a = 2 / Real.sqrt 3 := by\n    intro h_r\n    rw [h_r] at h_a\n    simp_all\n    <;> field_simp\n    <;> ring_nf\n    <;> linarith\n  have h_a_neg : r = -Real.sqrt 3 \u2192 a = -2 / Real.sqrt 3 := by\n    intro h_r_neg\n    rw [h_r_neg] at h_a\n    simp_all [div_eq_mul_inv]\n    <;> field_simp [h_r_neg]\n    <;> ring\n    <;> linarith\n  have h_u0 : u 0 = 2 / Real.sqrt 3 \u2228 u 0 = -(2 / Real.sqrt 3) := by\n    have h\u2083 : a = 2 / Real.sqrt 3 \u2228 a = -(2 / Real.sqrt 3) := by\n      apply eq_or_eq_neg_of_sq_eq_sq <;> field_simp [h_a, h_r2] <;> ring_nf <;>\n      nlinarith [Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n    cases h\u2083 with\n    | inl h_a_pos =>\n      simp_all [h_a_pos, h\u2080]\n    | inr h_a_neg =>\n      simp_all [h_a_neg, h\u2080]\n  cases h_u0 with\n  | inl h_u0 => exact Or.inl h_u0\n  | inr h_u0 => exact Or.inr h_u0\n", "proof_before": "theorem amc12b_2003_p6 (a r : \u211d) (u : \u2115 \u2192 \u211d) (h\u2080 : \u2200 k, u k = a * r ^ k) (h\u2081 : u 1 = 2)\n  (h\u2082 : u 3 = 6) : u 0 = 2 / Real.sqrt 3 \u2228 u 0 = -(2 / Real.sqrt 3)  :=  by\n  have h_a : a = 2 / r := by\n    have h\u2083 := h\u2080 1\n    have h\u2084 := h\u2080 3\n    simp_all [h\u2081, h\u2082]\n    have h\u2085 : r \u2260 0 := by\n      rintro rfl\n      linarith\n    field_simp [h\u2085] at *\n    <;> ring_nf at *\n    <;> nlinarith\n  have h_u3_subst : 2 / r * r^3 = 6 := by\n    have h\u2083 := h\u2080 1\n    have h\u2084 := h\u2080 3\n    rw [h_a] at h\u2083 h\u2084\n    simp_all [pow_one, pow_succ]\n    <;> ring_nf\n    <;> nlinarith\n  have h_r2 : r^2 = 3 := by\n    simp_all only [h_a, h\u2080, Nat.cast_ofNat, Nat.cast_one, Nat.cast_zero, Nat.cast_succ]\n    ring_nf\n    nlinarith\n  have h_a_pos : r = Real.sqrt 3 \u2192 a = 2 / Real.sqrt 3 := by\n    intro h_r\n    rw [h_r] at h_a\n    simp_all\n    <;> field_simp\n    <;> ring_nf\n    <;> linarith\n  have h_a_neg : r = -Real.sqrt 3 \u2192 a = -2 / Real.sqrt 3 := by\n    intro h_r_neg\n    rw [h_r_neg] at h_a\n    simp_all [div_eq_mul_inv]\n    <;> field_simp [h_r_neg]\n    <;> ring\n    <;> linarith\n  have h_u0 : u 0 = 2 / Real.sqrt 3 \u2228 u 0 = -(2 / Real.sqrt 3) := by\n    have h\u2083 : a = 2 / Real.sqrt 3 \u2228 a = -(2 / Real.sqrt 3) := by\n      apply eq_or_eq_neg_of_sq_eq_sq <;> field_simp [h_a, h_r2] <;> ring_nf <;>\n      nlinarith [Real.sq_sqrt (show (0 : \u211d) \u2264 3 by norm_num)]\n    cases h\u2083 with\n    | inl h_a_pos =>\n      simp_all [h_a_pos, h\u2080]\n    | inr h_a_neg =>\n      simp_all [h_a_neg, h\u2080]\n  cases h_u0 with\n  | inl h_u0 => exact Or.inl h_u0\n  | inr h_u0 => exact Or.inr h_u0\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 20, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 13, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 3, "fixpoint": true, "chars_before": 687, "chars_after": 406, "seconds": 282.3, "id": "mathd_numbertheory_629", "proof_after": "theorem mathd_numbertheory_629 : IsLeast { t : \u2115 | 0 < t \u2227 Nat.lcm 12 t ^ 3 = (12 * t) ^ 2 } 18  :=  by\n  have h2 : \u2200 t : \u2115, 0 < t \u2227 Nat.lcm 12 t ^ 3 = (12 * t) ^ 2 \u2192 18 \u2264 t := by\n    intro t ht\n    have h2 := ht.2\n    simp [Nat.lcm] at h2\n    contrapose! h2\n    interval_cases t <;> simp_all\n    <;> norm_num\n    <;> linarith\n  refine' \u27e8by simpa [Nat.lcm] using h1.2, fun a h => _\u27e9\n  have h2' : \u2200 t : \u2115, 0 < t \u2227 Nat.lcm 12 t ^ 3 = (12 * t) ^ 2 \u2192 18 \u2264 t := by simpa [Nat.lcm] using h2\n  exact h2' a \u27e8h.1, h.2\u27e9\n", "proof_before": "theorem mathd_numbertheory_629 : IsLeast { t : \u2115 | 0 < t \u2227 Nat.lcm 12 t ^ 3 = (12 * t) ^ 2 } 18  :=  by\n  have h1 : 0 < 18 \u2227 Nat.lcm 12 18 ^ 3 = (12 * 18) ^ 2 := by\n    constructor\n    norm_num\n    simp [Nat.lcm, Nat.gcd_eq_gcd_ab 12 18]\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n  have h2 : \u2200 t : \u2115, 0 < t \u2227 Nat.lcm 12 t ^ 3 = (12 * t) ^ 2 \u2192 18 \u2264 t := by\n    intro t ht\n    have h2 := ht.2\n    simp [Nat.lcm] at h2\n    contrapose! h2\n    interval_cases t <;> simp_all\n    <;> norm_num\n    <;> linarith\n  refine' \u27e8by simpa [Nat.lcm] using h1.2, fun a h => _\u27e9\n  have h1' : 0 < 18 \u2227 Nat.lcm 12 18 ^ 3 = (12 * 18) ^ 2 := by simpa [Nat.lcm] using h1\n  have h2' : \u2200 t : \u2115, 0 < t \u2227 Nat.lcm 12 t ^ 3 = (12 * t) ^ 2 \u2192 18 \u2264 t := by simpa [Nat.lcm] using h2\n  exact h2' a \u27e8h.1, h.2\u27e9\n"}
{"mode": "clear", "steps_before": 24, "alphabet": 23, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 12, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 5, "fixpoint": true, "chars_before": 691, "chars_after": 426, "seconds": 321.3, "id": "induction_ineq_nsqlefactn", "proof_after": "theorem induction_ineq_nsqlefactn (n : \u2115) (h\u2080 : 4 \u2264 n) : n ^ 2 \u2264 n !  :=  by\n  have inductive_step : \u2200 k \u2265 4, k ^ 2 \u2264 k ! \u2192 (k + 1) ^ 2 \u2264 (k + 1) ! := by\n    intro k h\u2081 h\u2082\n    induction' h\u2081 with k h\u2081 ih\n    \u00b7 \n      simp_all [Nat.factorial]\n    simp_all [Nat.factorial, Nat.mul_comm]\n    all_goals nlinarith\n  have final_proof : \u2200 n \u2265 4, n ^ 2 \u2264 n ! := by\n    intro n hn\n    induction' hn with k hk\n    all_goals simp_all [Nat.factorial, Nat.pow_succ]\n  all_goals simp_all [Nat.factorial, Nat.pow_succ]\n", "proof_before": "theorem induction_ineq_nsqlefactn (n : \u2115) (h\u2080 : 4 \u2264 n) : n ^ 2 \u2264 n !  :=  by\n  have base_case : 4 ^ 2 \u2264 4 ! := by\n    induction' h\u2080 with h\u2080 IH\n    simp [Nat.factorial]\n    all_goals simp_all [Nat.factorial, Nat.pow_succ]\n    all_goals nlinarith\n    <;> linarith\n  have inductive_step : \u2200 k \u2265 4, k ^ 2 \u2264 k ! \u2192 (k + 1) ^ 2 \u2264 (k + 1) ! := by\n    intro k h\u2081 h\u2082\n    induction' h\u2081 with k h\u2081 ih\n    \u00b7 \n      simp_all [Nat.factorial]\n    simp_all [Nat.factorial, Nat.mul_comm]\n    all_goals nlinarith\n  have final_proof : \u2200 n \u2265 4, n ^ 2 \u2264 n ! := by\n    intro n hn\n    induction' hn with k hk\n    case refl => exact base_case\n    case step =>\n      apply inductive_step k hk\n      exact by assumption\n  have h\u2081 : n ^ 2 \u2264 n ! := by\n    apply final_proof\n    exact h\u2080\n  exact h\u2081\n"}
{"mode": "clear", "steps_before": 16, "alphabet": 14, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 463, "chars_after": 51, "seconds": 1.8, "id": "mathd_numbertheory_326", "proof_after": "theorem mathd_numbertheory_326 (n : \u2124) (h\u2080 : (n - 1) * n * (n + 1) = 720 ) : n + 1 = 10  :=  by\n  nlinarith [sq_nonneg (n - 9), sq_nonneg (n + 9)]\n", "proof_before": "theorem mathd_numbertheory_326 (n : \u2124) (h\u2080 : (n - 1) * n * (n + 1) = 720 ) : n + 1 = 10  :=  by\n  have h\u2081 : (n - 1) * n * (n + 1) = n^3 - n := by\n    nlinarith [sq_nonneg (n - 1), sq_nonneg n, sq_nonneg (n + 1), sq_nonneg (n - 1 + n + 1)]\n  have h\u2082 : n^3 - n = 720 := by\n    linarith\n  have h\u2083 : n = 9 := by\n    have h\u2083 : n = 9 := by\n      nlinarith [sq_nonneg (n - 9), sq_nonneg (n + 9)]\n    simp_all\n  have h\u2084 : (9 - 1) * 9 * (9 + 1) = 720 := by\n    subst h\u2083\n    norm_num\n    <;> linarith\n  have h\u2085 : n + 1 = 10 := by\n    linarith\n  simp_all\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 31, "alphabet": 19, "compiles": false, "id": "mathd_algebra_547", "proof_after": "theorem mathd_algebra_547 (x y : \u211d) (h\u2080 : x = 5) (h\u2081 : y = 2) : Real.sqrt (x ^ 3 - 2 ^ y) = 11  :=  by\n  have h1 : Real.sqrt (x ^ 3 - 2 ^ y) = Real.sqrt (5 ^ 3 - 2 ^ 2) := by\n    simp [h\u2080, h\u2081, pow_two, pow_three]\n    <;> norm_num\n    <;> ring\n    <;> simp [Real.sqrt_eq_iff_mul_self_eq]\n    <;> norm_num\n    <;> linarith\n  have h2 : 5 ^ 3 = 125 := by\n    simp only [h\u2080, h\u2081, sq, mul_neg, sub_eq_add_neg] at h1\n    linarith [Real.sqrt_nonneg (5 ^ 3 - 2 ^ 2), Real.sqrt_nonneg (5 ^ 3 + 2 ^ 2)]\n  have h3 : 2 ^ 2 = 4 := by\n    simp_all [sq]\n    <;> linarith\n  have h4 : 125 - 4 = 121 := by\n    norm_num [h\u2080, h\u2081, h2, h3, Real.sqrt_eq_iff_sq_eq, sq] at h1 \u22a2\n  have h5 : Real.sqrt 121 = 11 := by\n    rw [Real.sqrt_eq_iff_mul_self_eq] <;>\n      norm_num\n    <;>\n      nlinarith\n    <;>\n      linarith\n  have h6 : Real.sqrt (x ^ 3 - 2 ^ y) = 11 := by\n    subst_vars\n    simp_all [h1, h2, h3, h4, h5]\n    <;> norm_num\n    <;> linarith\n  rw [h6]\n  <;> simp_all [h6]\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_547 (x y : \u211d) (h\u2080 : x = 5) (h\u2081 : y = 2) : Real.sqrt (x ^ 3 - 2 ^ y) = 11  :=  by\n  have h1 : Real.sqrt (x ^ 3 - 2 ^ y) = Real.sqrt (5 ^ 3 - 2 ^ 2) := by\n    simp [h\u2080, h\u2081, pow_two, pow_three]\n    <;> norm_num\n    <;> ring\n    <;> simp [Real.sqrt_eq_iff_mul_self_eq]\n    <;> norm_num\n    <;> linarith\n  have h2 : 5 ^ 3 = 125 := by\n    simp only [h\u2080, h\u2081, sq, mul_neg, sub_eq_add_neg] at h1\n    linarith [Real.sqrt_nonneg (5 ^ 3 - 2 ^ 2), Real.sqrt_nonneg (5 ^ 3 + 2 ^ 2)]\n  have h3 : 2 ^ 2 = 4 := by\n    simp_all [sq]\n    <;> linarith\n  have h4 : 125 - 4 = 121 := by\n    norm_num [h\u2080, h\u2081, h2, h3, Real.sqrt_eq_iff_sq_eq, sq] at h1 \u22a2\n  have h5 : Real.sqrt 121 = 11 := by\n    rw [Real.sqrt_eq_iff_mul_self_eq] <;>\n      norm_num\n    <;>\n      nlinarith\n    <;>\n      linarith\n  have h6 : Real.sqrt (x ^ 3 - 2 ^ y) = 11 := by\n    subst_vars\n    simp_all [h1, h2, h3, h4, h5]\n    <;> norm_num\n    <;> linarith\n  rw [h6]\n  <;> simp_all [h6]\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 27, "alphabet": 19, "compiles": false, "id": "mathd_algebra_510", "proof_after": "theorem mathd_algebra_510 (x y : \u211d) (h\u2080 : x + y = 13) (h\u2081 : x * y = 24) :\n  Real.sqrt (x ^ 2 + y ^ 2) = 11  :=  by\n  have h2 : x^2 + y^2 = (x + y)^2 - 2 * (x * y) := by\n    rw [\u2190 sub_eq_zero]\n    nlinarith [sq_nonneg (x - y)]\n  have h3 : x^2 + y^2 = 13^2 - 2 * 24 := by\n    simp [h\u2080, h\u2081, h2]\n    <;> ring\n    <;> linarith\n  have h4 : x^2 + y^2 = 169 - 48 := by\n    norm_num at h2 h3 <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n  have h5 : x^2 + y^2 = 121 := by\n    linarith\n  have h6 : Real.sqrt (x^2 + y^2) = Real.sqrt 121 := by\n    rw [h5]\n    <;> rw [Real.sqrt_eq_iff_mul_self_eq]\n    <;> linarith\n  have h7 : Real.sqrt 121 = 11 := by\n    have h7 : Real.sqrt 121 = 11 := by\n      rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    linarith\n  rw [h6, h7]\n  <;> norm_num\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_510 (x y : \u211d) (h\u2080 : x + y = 13) (h\u2081 : x * y = 24) :\n  Real.sqrt (x ^ 2 + y ^ 2) = 11  :=  by\n  have h2 : x^2 + y^2 = (x + y)^2 - 2 * (x * y) := by\n    rw [\u2190 sub_eq_zero]\n    nlinarith [sq_nonneg (x - y)]\n  have h3 : x^2 + y^2 = 13^2 - 2 * 24 := by\n    simp [h\u2080, h\u2081, h2]\n    <;> ring\n    <;> linarith\n  have h4 : x^2 + y^2 = 169 - 48 := by\n    norm_num at h2 h3 <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n  have h5 : x^2 + y^2 = 121 := by\n    linarith\n  have h6 : Real.sqrt (x^2 + y^2) = Real.sqrt 121 := by\n    rw [h5]\n    <;> rw [Real.sqrt_eq_iff_mul_self_eq]\n    <;> linarith\n  have h7 : Real.sqrt 121 = 11 := by\n    have h7 : Real.sqrt 121 = 11 := by\n      rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    linarith\n  rw [h6, h7]\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 73, "alphabet": 28, "compiles": false, "id": "amc12a_2011_p18", "proof_after": "theorem amc12a_2011_p18 (x y : \u211d) (h\u2080 : abs (x + y) + abs (x - y) = 2) :\n  x ^ 2 - 6 * x + y ^ 2 \u2264 8  :=  by\n  have h\u2081 : max (abs x) (abs y) \u2265 1 := by\n    by_contra! h\n    have h\u2081 : abs x < 1 := by simp_all [abs_lt]\n    have h\u2082 : abs y < 1 := by simp_all [abs_lt]\n    cases' le_total 0 (x + y) with h\u2083 h\u2083 <;> cases' le_total 0 (x - y) with h\u2084 h\u2084 <;>\n      cases' le_total 0 x with h\u2085 h\u2085 <;> cases' le_total 0 y with h\u2086 h\u2086 <;>\n      simp_all [abs_of_nonneg, abs_of_nonpos, abs_lt, add_left_comm, add_comm, add_assoc, add_left_comm] <;>\n      linarith\n  have h\u2082 : x^2 - 6*x = (x - 3)^2 - 9 := by\n    cases' le_total 0 (x + y) with h h <;> cases' le_total 0 (x - y) with h' h' <;>\n      cases' le_total 0 x with h'' h'' <;> cases' le_total 0 y with h''' h''' <;>\n        simp_all only [abs_of_nonneg, abs_of_nonpos, add_nonneg, add_nonpos, sub_nonneg, sub_nonpos,\n          le_add_iff_nonneg_left, le_add_iff_nonneg_right, le_sub_iff_add_le, sub_le_iff_le_add,\n          add_zero, zero_add] <;>\n        nlinarith\n  have h\u2083 : x^2 - 6*x + y^2 = (x - 3)^2 + y^2 - 9 := by\n    linarith [sq (x - 3)]\n    <;> cases' le_total 0 (x + y) with h h <;> cases' le_total 0 (x - y) with h' h' <;>\n      cases' le_total 0 x with h'' h'' <;> cases' le_total 0 y with h''' h''' <;>\n        simp_all only [abs_of_nonneg, abs_of_nonpos, max_eq_left, max_eq_right, sub_eq_add_neg, neg_add_rev] <;>\n          linarith\n  have h\u2084 : (x - 3)^2 + y^2 \u2264 17 := by\n    cases' le_total 0 (x - 3) with h\u2084 h\u2084 <;>\n      cases' le_total 0 (x + y) with h\u2085 h\u2085 <;>\n        cases' le_total 0 (x - y) with h\u2086 h\u2086 <;>\n          cases' le_total 0 x with h\u2087 h\u2087 <;>\n            cases' le_total 0 y with h\u2088 h\u2088 <;>\n              simp_all only [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos] <;>\n              nlinarith\n    <;>\n    nlinarith\n    <;>\n    nlinarith\n  have h\u2085 : x^2 - 6*x + y^2 \u2264 8 := by\n    cases' le_total 0 (x + y) with h\u2085 h\u2085 <;> cases' le_total 0 (x - y) with h\u2086 h\u2086 <;>\n      cases' le_total 0 x with h\u2087 h\u2087 <;> cases' le_total 0 y with h\u2088 h\u2088 <;>\n        simp_all only [abs_of_nonneg, abs_of_nonpos, max_eq_left, max_eq_right, sub_eq_add_neg, neg_add_rev,\n          neg_neg, neg_mul, neg_sub, neg_neg, neg_zero] <;>\n        nlinarith\n    <;>\n    nlinarith\n    <;>\n    nlinarith\n    <;>\n    nlinarith\n    <;>\n    nlinarith\n  nlinarith [sq_nonneg (x - 3), sq_nonneg y]\n", "proof_before": "theorem amc12a_2011_p18 (x y : \u211d) (h\u2080 : abs (x + y) + abs (x - y) = 2) :\n  x ^ 2 - 6 * x + y ^ 2 \u2264 8  :=  by\n  have h\u2081 : max (abs x) (abs y) \u2265 1 := by\n    by_contra! h\n    have h\u2081 : abs x < 1 := by simp_all [abs_lt]\n    have h\u2082 : abs y < 1 := by simp_all [abs_lt]\n    cases' le_total 0 (x + y) with h\u2083 h\u2083 <;> cases' le_total 0 (x - y) with h\u2084 h\u2084 <;>\n      cases' le_total 0 x with h\u2085 h\u2085 <;> cases' le_total 0 y with h\u2086 h\u2086 <;>\n      simp_all [abs_of_nonneg, abs_of_nonpos, abs_lt, add_left_comm, add_comm, add_assoc, add_left_comm] <;>\n      linarith\n  have h\u2082 : x^2 - 6*x = (x - 3)^2 - 9 := by\n    cases' le_total 0 (x + y) with h h <;> cases' le_total 0 (x - y) with h' h' <;>\n      cases' le_total 0 x with h'' h'' <;> cases' le_total 0 y with h''' h''' <;>\n        simp_all only [abs_of_nonneg, abs_of_nonpos, add_nonneg, add_nonpos, sub_nonneg, sub_nonpos,\n          le_add_iff_nonneg_left, le_add_iff_nonneg_right, le_sub_iff_add_le, sub_le_iff_le_add,\n          add_zero, zero_add] <;>\n        nlinarith\n  have h\u2083 : x^2 - 6*x + y^2 = (x - 3)^2 + y^2 - 9 := by\n    linarith [sq (x - 3)]\n    <;> cases' le_total 0 (x + y) with h h <;> cases' le_total 0 (x - y) with h' h' <;>\n      cases' le_total 0 x with h'' h'' <;> cases' le_total 0 y with h''' h''' <;>\n        simp_all only [abs_of_nonneg, abs_of_nonpos, max_eq_left, max_eq_right, sub_eq_add_neg, neg_add_rev] <;>\n          linarith\n  have h\u2084 : (x - 3)^2 + y^2 \u2264 17 := by\n    cases' le_total 0 (x - 3) with h\u2084 h\u2084 <;>\n      cases' le_total 0 (x + y) with h\u2085 h\u2085 <;>\n        cases' le_total 0 (x - y) with h\u2086 h\u2086 <;>\n          cases' le_total 0 x with h\u2087 h\u2087 <;>\n            cases' le_total 0 y with h\u2088 h\u2088 <;>\n              simp_all only [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos] <;>\n              nlinarith\n    <;>\n    nlinarith\n    <;>\n    nlinarith\n  have h\u2085 : x^2 - 6*x + y^2 \u2264 8 := by\n    cases' le_total 0 (x + y) with h\u2085 h\u2085 <;> cases' le_total 0 (x - y) with h\u2086 h\u2086 <;>\n      cases' le_total 0 x with h\u2087 h\u2087 <;> cases' le_total 0 y with h\u2088 h\u2088 <;>\n        simp_all only [abs_of_nonneg, abs_of_nonpos, max_eq_left, max_eq_right, sub_eq_add_neg, neg_add_rev,\n          neg_neg, neg_mul, neg_sub, neg_neg, neg_zero] <;>\n        nlinarith\n    <;>\n    nlinarith\n    <;>\n    nlinarith\n    <;>\n    nlinarith\n    <;>\n    nlinarith\n  nlinarith [sq_nonneg (x - 3), sq_nonneg y]\n"}
{"mode": "clear", "steps_before": 21, "alphabet": 16, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 974, "chars_after": 51, "seconds": 12.5, "id": "algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3", "proof_after": "theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : \u2124)\n  (h\u2080 : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) :\n  x + y + z + 6 \u2223 x ^ 3 + y ^ 3 + z ^ 3  :=  by\n  use x^2 + y^2 + z^2 - x*y - y*z - z*x\n  linarith\n", "proof_before": "theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : \u2124)\n  (h\u2080 : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) :\n  x + y + z + 6 \u2223 x ^ 3 + y ^ 3 + z ^ 3  :=  by\n  have h\u2081 : (x - y)^2 + (y - z)^2 + (z - x)^2 = 2*(x^2 + y^2 + z^2 - x*y - y*z - z*x) := by\n    simp only [sub_sq, add_assoc, add_left_comm, add_comm] at h\u2080 \u22a2\n    linarith\n  have h\u2082 : 2*(x^2 + y^2 + z^2 - x*y - y*z - z*x) = x * y * z := by\n    have h\u2082 : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = 2*(x^2 + y^2 + z^2 - x*y - y*z - z*x) := h\u2081\n    have h\u2083 : 2*(x^2 + y^2 + z^2 - x*y - y*z - z*x) = x * y * z := by\n      linarith\n    linarith\n  have h\u2083 : x^3 + y^3 + z^3 = (x + y + z)^3 - 3*(x + y + z)*(x*y + y*z + z*x) + 3*x*y*z := by\n    have h\u2083 : x ^ 3 + y ^ 3 + z ^ 3 = (x + y + z) ^ 3 - 3 * (x + y + z) * (x * y + y * z + z * x) + 3 * x * y * z := by\n      ring\n    rw [h\u2083]\n    <;> simp_all\n    <;> ring\n    <;> linarith\n  have h\u2084 : x + y + z + 6 \u2223 x^3 + y^3 + z^3 := by\n    use x^2 + y^2 + z^2 - x*y - y*z - z*x\n    ring_nf\n    <;> simp_all only [mul_assoc, mul_comm, mul_left_comm, mul_one, mul_zero, add_assoc, add_left_comm, add_comm]\n    <;> linarith\n  simpa using h\u2084\n"}
{"mode": "clear", "steps_before": 55, "alphabet": 33, "compiles": false, "id": "imo_1965_p1", "proof_after": "theorem imo_1965_p1 (x : \u211d) (h\u2080 : 0 \u2264 x) (h\u2081 : x \u2264 2 * Real.pi)\n  (h\u2082 : 2 * Real.cos x \u2264 abs (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))))\n  (h\u2083 : abs (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))) \u2264 Real.sqrt 2) :\n  Real.pi / 4 \u2264 x \u2227 x \u2264 7 * Real.pi / 4  :=  by\n  have h\u2084 : abs (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))) ^ 2 = 2 - 2 * abs (Real.cos (2 * x)) := by\n    have h\u2084 : 0 \u2264 Real.sqrt (1 + Real.sin (2 * x)) := Real.sqrt_nonneg _\n    have h\u2085 : 0 \u2264 Real.sqrt (1 - Real.sin (2 * x)) := Real.sqrt_nonneg _\n    have h\u2086 : 0 \u2264 Real.sqrt 2 := Real.sqrt_nonneg _\n    have h\u2087 : 0 \u2264 Real.sqrt (1 + Real.sin (2 * x)) * Real.sqrt (1 - Real.sin (2 * x)) := mul_nonneg h\u2084 h\u2085\n    cases' le_total 0 (Real.cos (2 * x)) with h\u2088 h\u2088 <;>\n      cases' le_total 0 (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))) with h\u2089 h\u2089 <;>\n        simp_all [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos] <;>\n          nlinarith [Real.sq_sqrt (show 0 \u2264 1 + Real.sin (2 * x) by nlinarith [Real.sin_le_one (2 * x), Real.neg_one_le_sin (2 * x)]),\n            Real.sq_sqrt (show 0 \u2264 1 - Real.sin (2 * x) by nlinarith [Real.sin_le_one (2 * x), Real.neg_one_le_sin (2 * x)]),\n            Real.sq_sqrt (show 0 \u2264 2 by norm_num),\n            Real.cos_sq_add_sin_sq (2 * x)]\n  have h\u2085 : 2 * Real.cos x \u2264 Real.sqrt (2 - 2 * abs (Real.cos (2 * x))) := by\n    nlinarith [abs_nonneg (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))),\n      Real.sq_sqrt (show 0 \u2264 2 - 2 * abs (Real.cos (2 * x)) by linarith [abs_cos_le_one (2 * x)]),\n      Real.sqrt_nonneg (2 - 2 * abs (Real.cos (2 * x))),\n      sq_nonneg (Real.cos x - Real.sqrt (2 - 2 * abs (Real.cos (2 * x)))),\n      sq_nonneg (Real.cos x + Real.sqrt (2 - 2 * abs (Real.cos (2 * x)))),\n      Real.sqrt_nonneg (2 - 2 * abs (Real.cos (2 * x))),\n      Real.sq_sqrt (show 0 \u2264 2 - 2 * abs (Real.cos (2 * x)) by linarith [abs_cos_le_one (2 * x)])]\n  have h\u2086 : Real.cos x \u2264 Real.sqrt 2 / 2 := by\n    have h\u2086 : Real.cos x \u2264 Real.sqrt 2 / 2 := by\n      apply le_of_sub_nonneg\n      have h\u2087 : 0 \u2264 Real.sqrt 2 := Real.sqrt_nonneg _\n      have h\u2088 : 0 \u2264 Real.sqrt (2 - 2 * abs (Real.cos (2 * x))) := Real.sqrt_nonneg _\n      have h\u2089 : Real.cos x \u2264 Real.sqrt (2 - 2 * abs (Real.cos (2 * x))) / 2 := by\n        nlinarith [sq_sqrt (show 0 \u2264 2 by norm_num), sq_sqrt (show 0 \u2264 2 - 2 * abs (Real.cos (2 * x)) by nlinarith [abs_cos_le_one (2 * x)]),\n          h\u2085, sq_nonneg (Real.cos x - Real.sqrt 2 / 2)]\n      nlinarith [sq_sqrt (show 0 \u2264 2 by norm_num), sq_sqrt (show 0 \u2264 2 - 2 * abs (Real.cos (2 * x)) by nlinarith [abs_cos_le_one (2 * x)]),\n        h\u2085, sq_nonneg (Real.cos x - Real.sqrt 2 / 2)]\n    exact h\u2086\n  have h\u2087 : Real.pi / 4 \u2264 x \u2227 x \u2264 7 * Real.pi / 4 := by\n    have h\u2087 : Real.pi / 4 \u2264 x := by\n      contrapose! h\u2086\n      have h\u2087\u2080 : x < Real.pi / 4 := by linarith\n      have h\u2087\u2081 : Real.cos x > Real.sqrt 2 / 2 := by\n        rw [\u2190 Real.cos_pi_div_four]\n        apply Real.cos_lt_cos_of_nonneg_of_le_pi (by linarith) (by linarith)\n        <;> linarith\n      linarith\n    have h\u2088 : x \u2264 7 * Real.pi / 4 := by\n      contrapose! h\u2086\n      have h\u2088\u2080 : x > 7 * Real.pi / 4 := by linarith\n      have h\u2088\u2081 : Real.cos x > Real.sqrt 2 / 2 := by\n        have h\u2088\u2082 : Real.cos x = Real.cos (2 * Real.pi - x) := by\n          rw [Real.cos_two_pi_sub]\n        rw [h\u2088\u2082]\n        have h\u2088\u2083 : 2 * Real.pi - x < 7 * Real.pi / 4 := by linarith\n        have h\u2088\u2084 : Real.cos (2 * Real.pi - x) > Real.sqrt 2 / 2 := by\n          rw [\u2190 Real.cos_pi_div_four]\n          apply Real.cos_lt_cos_of_nonneg_of_le_pi (by linarith) (by linarith)\n          <;> linarith\n        linarith\n      linarith\n    exact \u27e8h\u2087, h\u2088\u27e9\n  have h\u2088 : Real.pi / 4 \u2264 x := by\n    linarith [h\u2087.1]\n  have h\u2089 : x \u2264 7 * Real.pi / 4 := by\n    linarith [h\u2087.2]\n  exact \u27e8h\u2088, h\u2089\u27e9\n", "proof_before": "theorem imo_1965_p1 (x : \u211d) (h\u2080 : 0 \u2264 x) (h\u2081 : x \u2264 2 * Real.pi)\n  (h\u2082 : 2 * Real.cos x \u2264 abs (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))))\n  (h\u2083 : abs (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))) \u2264 Real.sqrt 2) :\n  Real.pi / 4 \u2264 x \u2227 x \u2264 7 * Real.pi / 4  :=  by\n  have h\u2084 : abs (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))) ^ 2 = 2 - 2 * abs (Real.cos (2 * x)) := by\n    have h\u2084 : 0 \u2264 Real.sqrt (1 + Real.sin (2 * x)) := Real.sqrt_nonneg _\n    have h\u2085 : 0 \u2264 Real.sqrt (1 - Real.sin (2 * x)) := Real.sqrt_nonneg _\n    have h\u2086 : 0 \u2264 Real.sqrt 2 := Real.sqrt_nonneg _\n    have h\u2087 : 0 \u2264 Real.sqrt (1 + Real.sin (2 * x)) * Real.sqrt (1 - Real.sin (2 * x)) := mul_nonneg h\u2084 h\u2085\n    cases' le_total 0 (Real.cos (2 * x)) with h\u2088 h\u2088 <;>\n      cases' le_total 0 (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))) with h\u2089 h\u2089 <;>\n        simp_all [abs_of_nonneg, abs_of_nonpos, sub_nonneg, sub_nonpos] <;>\n          nlinarith [Real.sq_sqrt (show 0 \u2264 1 + Real.sin (2 * x) by nlinarith [Real.sin_le_one (2 * x), Real.neg_one_le_sin (2 * x)]),\n            Real.sq_sqrt (show 0 \u2264 1 - Real.sin (2 * x) by nlinarith [Real.sin_le_one (2 * x), Real.neg_one_le_sin (2 * x)]),\n            Real.sq_sqrt (show 0 \u2264 2 by norm_num),\n            Real.cos_sq_add_sin_sq (2 * x)]\n  have h\u2085 : 2 * Real.cos x \u2264 Real.sqrt (2 - 2 * abs (Real.cos (2 * x))) := by\n    nlinarith [abs_nonneg (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))),\n      Real.sq_sqrt (show 0 \u2264 2 - 2 * abs (Real.cos (2 * x)) by linarith [abs_cos_le_one (2 * x)]),\n      Real.sqrt_nonneg (2 - 2 * abs (Real.cos (2 * x))),\n      sq_nonneg (Real.cos x - Real.sqrt (2 - 2 * abs (Real.cos (2 * x)))),\n      sq_nonneg (Real.cos x + Real.sqrt (2 - 2 * abs (Real.cos (2 * x)))),\n      Real.sqrt_nonneg (2 - 2 * abs (Real.cos (2 * x))),\n      Real.sq_sqrt (show 0 \u2264 2 - 2 * abs (Real.cos (2 * x)) by linarith [abs_cos_le_one (2 * x)])]\n  have h\u2086 : Real.cos x \u2264 Real.sqrt 2 / 2 := by\n    have h\u2086 : Real.cos x \u2264 Real.sqrt 2 / 2 := by\n      apply le_of_sub_nonneg\n      have h\u2087 : 0 \u2264 Real.sqrt 2 := Real.sqrt_nonneg _\n      have h\u2088 : 0 \u2264 Real.sqrt (2 - 2 * abs (Real.cos (2 * x))) := Real.sqrt_nonneg _\n      have h\u2089 : Real.cos x \u2264 Real.sqrt (2 - 2 * abs (Real.cos (2 * x))) / 2 := by\n        nlinarith [sq_sqrt (show 0 \u2264 2 by norm_num), sq_sqrt (show 0 \u2264 2 - 2 * abs (Real.cos (2 * x)) by nlinarith [abs_cos_le_one (2 * x)]),\n          h\u2085, sq_nonneg (Real.cos x - Real.sqrt 2 / 2)]\n      nlinarith [sq_sqrt (show 0 \u2264 2 by norm_num), sq_sqrt (show 0 \u2264 2 - 2 * abs (Real.cos (2 * x)) by nlinarith [abs_cos_le_one (2 * x)]),\n        h\u2085, sq_nonneg (Real.cos x - Real.sqrt 2 / 2)]\n    exact h\u2086\n  have h\u2087 : Real.pi / 4 \u2264 x \u2227 x \u2264 7 * Real.pi / 4 := by\n    have h\u2087 : Real.pi / 4 \u2264 x := by\n      contrapose! h\u2086\n      have h\u2087\u2080 : x < Real.pi / 4 := by linarith\n      have h\u2087\u2081 : Real.cos x > Real.sqrt 2 / 2 := by\n        rw [\u2190 Real.cos_pi_div_four]\n        apply Real.cos_lt_cos_of_nonneg_of_le_pi (by linarith) (by linarith)\n        <;> linarith\n      linarith\n    have h\u2088 : x \u2264 7 * Real.pi / 4 := by\n      contrapose! h\u2086\n      have h\u2088\u2080 : x > 7 * Real.pi / 4 := by linarith\n      have h\u2088\u2081 : Real.cos x > Real.sqrt 2 / 2 := by\n        have h\u2088\u2082 : Real.cos x = Real.cos (2 * Real.pi - x) := by\n          rw [Real.cos_two_pi_sub]\n        rw [h\u2088\u2082]\n        have h\u2088\u2083 : 2 * Real.pi - x < 7 * Real.pi / 4 := by linarith\n        have h\u2088\u2084 : Real.cos (2 * Real.pi - x) > Real.sqrt 2 / 2 := by\n          rw [\u2190 Real.cos_pi_div_four]\n          apply Real.cos_lt_cos_of_nonneg_of_le_pi (by linarith) (by linarith)\n          <;> linarith\n        linarith\n      linarith\n    exact \u27e8h\u2087, h\u2088\u27e9\n  have h\u2088 : Real.pi / 4 \u2264 x := by\n    linarith [h\u2087.1]\n  have h\u2089 : x \u2264 7 * Real.pi / 4 := by\n    linarith [h\u2087.2]\n  exact \u27e8h\u2088, h\u2089\u27e9\n"}
{"mode": "clear", "steps_before": 66, "alphabet": 24, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 1715, "chars_after": 6, "seconds": 6.0, "id": "mathd_numbertheory_188", "proof_after": "theorem mathd_numbertheory_188 : Nat.gcd 180 168 = 12  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_188 : Nat.gcd 180 168 = 12  :=  by\n  have step1 : Nat.gcd 180 168 = Nat.gcd 168 (180 % 168) := by\n    rw [Nat.gcd_comm]\n    <;> simp [Nat.gcd_comm]\n    <;> norm_num\n    <;> rfl\n    <;> rfl\n  have step2 : 180 % 168 = 12 := by\n    simp_all [Nat.gcd_eq_right, Nat.mod_eq_of_lt, show 168 \u2264 180 from by norm_num]\n    <;> decide\n  have step3 : Nat.gcd 180 168 = Nat.gcd 168 12 := by\n    simp_all [Nat.gcd_comm]\n    <;> simp_all [Nat.gcd_comm]\n  have step4 : Nat.gcd 168 12 = Nat.gcd 12 (168 % 12) := by\n    rw [step3] at *\n    <;> simp_all [Nat.gcd_comm]\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n  have step5 : 168 % 12 = 0 := by\n    apply Eq.symm\n    norm_num [Nat.gcd_eq_right, Nat.gcd_eq_left]\n    <;> omega\n  have step6 : Nat.gcd 12 0 = 12 := by\n    simp_all [Nat.gcd_eq_right]\n    <;> norm_num\n    <;> rfl\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n    <;> apply Nat.gcd_eq_right\n    <;> norm_num\n  have final_step : Nat.gcd 180 168 = 12 := by\n    simp_all [Nat.gcd_eq_right, Nat.gcd_eq_left]\n    <;> norm_num\n    <;> aesop\n    <;> rfl\n    <;> aesop\n    <;> rfl\n  have h1 := step1\n  have h2 := step2\n  have h3 := step3\n  have h4 := step4\n  have h5 := step5\n  have h6 := step6\n  have h7 := final_step\n  simp_all\n"}
{"mode": "clear", "steps_before": 1, "alphabet": 11, "compiles": true, "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 193, "chars_after": 19, "seconds": 15.7, "id": "amc12a_2021_p7", "proof_after": "theorem amc12a_2021_p7 (x y : \u211d) : 1 \u2264 (x * y - 1) ^ 2 + (x + y) ^ 2  :=  by\n  ring\n  nlinarith\n", "proof_before": "theorem amc12a_2021_p7 (x y : \u211d) : 1 \u2264 (x * y - 1) ^ 2 + (x + y) ^ 2  :=  by\n  nlinarith [sq_nonneg (x - y), sq_nonneg (x + y), sq_nonneg (x * y - 1), sq_nonneg (x * y + 1),\n    sq_nonneg (x + y + 2), sq_nonneg (x + y - 2), sq_nonneg (x - y + 2), sq_nonneg (x - y - 2)]\n"}
{"mode": "clear", "steps_before": 70, "alphabet": 37, "compiles": false, "id": "amc12a_2010_p11", "proof_after": "theorem amc12a_2010_p11 (x b : \u211d) (h\u2080 : 0 < b) (h\u2081 : (7 : \u211d) ^ (x + 7) = 8 ^ x)\n  (h\u2082 : x = Real.logb b (7 ^ 7)) : b = 8 / 7  :=  by\n  have h\u2083 : (x + 7) * Real.log 7 = x * Real.log 8 := by\n    have h\u2083 := h\u2081\n    have h\u2084 := congr_arg Real.log h\u2083\n    field_simp [Real.log_rpow, Real.log_mul, Real.log_pow] at h\u2084\n    ring_nf at h\u2084\n    linarith\n  have h\u2084 : x * (Real.log 7 - Real.log 8) = -7 * Real.log 7 := by\n    norm_num [Real.logb, Real.log_rpow, Real.log_pow, mul_add, mul_comm, mul_left_comm] at h\u2081 h\u2082 h\u2083\n    ring_nf at h\u2081 h\u2082 h\u2083 \u22a2\n    nlinarith\n  have h\u2085 : x = (-7 * Real.log 7) / (Real.log 7 - Real.log 8) := by\n    have h\u2085 : Real.log 7 - Real.log 8 \u2260 0 := by\n      intro h\n      have h\u2086 := h\u2083\n      simp_all [Real.logb, Real.log_rpow, mul_comm]\n      <;> linarith\n    field_simp [h\u2085] at h\u2084 \u22a2\n    <;> ring_nf at h\u2084 \u22a2\n    <;> linarith\n  have h\u2086 : Real.logb b (7 ^ 7) = (-7 * Real.log 7) / (Real.log 7 - Real.log 8) := by\n    simp_all [Real.logb, Real.log_rpow, mul_comm]\n    <;> field_simp\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  have h\u2087 : Real.log (7^7) / Real.log b = (-7 * Real.log 7) / (Real.log 7 - Real.log 8) := by\n    have h\u2087 : 0 < Real.log 7 := Real.log_pos (by norm_num)\n    have h\u2088 : 0 < Real.log 8 := Real.log_pos (by norm_num)\n    field_simp [Real.log_rpow, h\u2080, h\u2087, h\u2088, Real.logb, Real.log_exp] at *\n    linarith\n  have h\u2088 : Real.log (7^7) = 7 * Real.log 7 := by\n    field_simp [Real.log_mul, Real.log_rpow, Real.log_pow, Real.log_exp, Real.log_inv, Real.log_div, Real.log_one] at h\u2085 h\u2086 h\u2087 \u22a2\n    <;> ring_nf at h\u2085 h\u2086 h\u2087 \u22a2\n    <;> nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 7), Real.log_pos (by norm_num : (1 : \u211d) < 8),\n      Real.log_pos (by norm_num : (1 : \u211d) < 7), Real.log_pos (by norm_num : (1 : \u211d) < 8)]\n  have h\u2089 : (7 * Real.log 7) / Real.log b = (-7 * Real.log 7) / (Real.log 7 - Real.log 8) := by\n    field_simp [h\u2080.ne', h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, Real.log_rpow] at *\n    <;> ring_nf at *\n    <;> linarith\n  have h\u2081\u2080 : 1 / Real.log b = -1 / (Real.log 7 - Real.log 8) := by\n    have h\u2081\u2080 : 0 < Real.log 7 := by\n      apply Real.log_pos\n      norm_num\n    have h\u2081\u2081 : 0 < Real.log 8 := by\n      apply Real.log_pos\n      norm_num\n    field_simp [h\u2081\u2080, h\u2081\u2081, h\u2080, h\u2088] at h\u2087 \u22a2\n    <;> ring_nf at h\u2087 \u22a2\n    <;> nlinarith\n  have h\u2081\u2081 : Real.log b = Real.log 8 - Real.log 7 := by\n    have h\u2081\u2081 : Real.log b = Real.log 8 - Real.log 7 := by\n      have h\u2081\u2081 := congr_arg (fun x : \u211d => 1 / x) h\u2081\u2080\n      field_simp at h\u2081\u2081 \u22a2\n      linarith\n    exact h\u2081\u2081\n  have h\u2081\u2082 : b = Real.exp (Real.log 8 - Real.log 7) := by\n    have h\u2081\u2082 : Real.exp (Real.log b) = b := by\n      rw [Real.exp_log]\n      linarith\n    simp_all [Real.exp_log, Real.log_exp]\n  have h\u2081\u2083 : b = 8 / 7 := by\n    rw [h\u2081\u2082]\n    norm_num\n    <;> simp_all [Real.exp_log, Real.exp_sub, Real.exp_add, mul_comm]\n    <;> field_simp [Real.log_mul, Real.log_div, Real.log_pow, h\u2080]\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  linarith\n", "proof_before": "theorem amc12a_2010_p11 (x b : \u211d) (h\u2080 : 0 < b) (h\u2081 : (7 : \u211d) ^ (x + 7) = 8 ^ x)\n  (h\u2082 : x = Real.logb b (7 ^ 7)) : b = 8 / 7  :=  by\n  have h\u2083 : (x + 7) * Real.log 7 = x * Real.log 8 := by\n    have h\u2083 := h\u2081\n    have h\u2084 := congr_arg Real.log h\u2083\n    field_simp [Real.log_rpow, Real.log_mul, Real.log_pow] at h\u2084\n    ring_nf at h\u2084\n    linarith\n  have h\u2084 : x * (Real.log 7 - Real.log 8) = -7 * Real.log 7 := by\n    norm_num [Real.logb, Real.log_rpow, Real.log_pow, mul_add, mul_comm, mul_left_comm] at h\u2081 h\u2082 h\u2083\n    ring_nf at h\u2081 h\u2082 h\u2083 \u22a2\n    nlinarith\n  have h\u2085 : x = (-7 * Real.log 7) / (Real.log 7 - Real.log 8) := by\n    have h\u2085 : Real.log 7 - Real.log 8 \u2260 0 := by\n      intro h\n      have h\u2086 := h\u2083\n      simp_all [Real.logb, Real.log_rpow, mul_comm]\n      <;> linarith\n    field_simp [h\u2085] at h\u2084 \u22a2\n    <;> ring_nf at h\u2084 \u22a2\n    <;> linarith\n  have h\u2086 : Real.logb b (7 ^ 7) = (-7 * Real.log 7) / (Real.log 7 - Real.log 8) := by\n    simp_all [Real.logb, Real.log_rpow, mul_comm]\n    <;> field_simp\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  have h\u2087 : Real.log (7^7) / Real.log b = (-7 * Real.log 7) / (Real.log 7 - Real.log 8) := by\n    have h\u2087 : 0 < Real.log 7 := Real.log_pos (by norm_num)\n    have h\u2088 : 0 < Real.log 8 := Real.log_pos (by norm_num)\n    field_simp [Real.log_rpow, h\u2080, h\u2087, h\u2088, Real.logb, Real.log_exp] at *\n    linarith\n  have h\u2088 : Real.log (7^7) = 7 * Real.log 7 := by\n    field_simp [Real.log_mul, Real.log_rpow, Real.log_pow, Real.log_exp, Real.log_inv, Real.log_div, Real.log_one] at h\u2085 h\u2086 h\u2087 \u22a2\n    <;> ring_nf at h\u2085 h\u2086 h\u2087 \u22a2\n    <;> nlinarith [Real.log_pos (by norm_num : (1 : \u211d) < 7), Real.log_pos (by norm_num : (1 : \u211d) < 8),\n      Real.log_pos (by norm_num : (1 : \u211d) < 7), Real.log_pos (by norm_num : (1 : \u211d) < 8)]\n  have h\u2089 : (7 * Real.log 7) / Real.log b = (-7 * Real.log 7) / (Real.log 7 - Real.log 8) := by\n    field_simp [h\u2080.ne', h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, Real.log_rpow] at *\n    <;> ring_nf at *\n    <;> linarith\n  have h\u2081\u2080 : 1 / Real.log b = -1 / (Real.log 7 - Real.log 8) := by\n    have h\u2081\u2080 : 0 < Real.log 7 := by\n      apply Real.log_pos\n      norm_num\n    have h\u2081\u2081 : 0 < Real.log 8 := by\n      apply Real.log_pos\n      norm_num\n    field_simp [h\u2081\u2080, h\u2081\u2081, h\u2080, h\u2088] at h\u2087 \u22a2\n    <;> ring_nf at h\u2087 \u22a2\n    <;> nlinarith\n  have h\u2081\u2081 : Real.log b = Real.log 8 - Real.log 7 := by\n    have h\u2081\u2081 : Real.log b = Real.log 8 - Real.log 7 := by\n      have h\u2081\u2081 := congr_arg (fun x : \u211d => 1 / x) h\u2081\u2080\n      field_simp at h\u2081\u2081 \u22a2\n      linarith\n    exact h\u2081\u2081\n  have h\u2081\u2082 : b = Real.exp (Real.log 8 - Real.log 7) := by\n    have h\u2081\u2082 : Real.exp (Real.log b) = b := by\n      rw [Real.exp_log]\n      linarith\n    simp_all [Real.exp_log, Real.log_exp]\n  have h\u2081\u2083 : b = 8 / 7 := by\n    rw [h\u2081\u2082]\n    norm_num\n    <;> simp_all [Real.exp_log, Real.exp_sub, Real.exp_add, mul_comm]\n    <;> field_simp [Real.log_mul, Real.log_div, Real.log_pow, h\u2080]\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  linarith\n"}
{"mode": "clear", "steps_before": 24, "alphabet": 18, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "fixpoint": true, "chars_before": 536, "chars_after": 11, "seconds": 323.2, "id": "mathd_algebra_119", "proof_after": "theorem mathd_algebra_119 (d e : \u211d) (h\u2080 : 2 * d = 17 * e - 8) (h\u2081 : 2 * e = d - 9) : e = 2  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_119 (d e : \u211d) (h\u2080 : 2 * d = 17 * e - 8) (h\u2081 : 2 * e = d - 9) : e = 2  :=  by\n  have hd : d = 2 * e + 9 := by\n    have h\u2080' := h\u2080\n    ring_nf at h\u2080' h\u2081 \u22a2\n    linarith\n  have h_substituted : 2 * (2 * e + 9) = 17 * e - 8 := by\n    rw [hd] at h\u2080\n    linarith\n  have h_simplified : 4 * e + 18 = 17 * e - 8 := by\n    linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n  have h_e : e = 2 := by\n    have h\u2082 : e = 2 := by linarith\n    exact h\u2082\n  have h_verify\u2081 : 2 * d = 17 * 2 - 8 := by\n    subst hd\n    linarith\n  have h_verify\u2082 : 2 * 2 = d - 9 := by\n    subst d\n    linarith\n  have h\u2082 : e = 2 := h_e\n  linarith\n"}
{"mode": "clear", "steps_before": 18, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 18, "lower_bound": null, "exhausted_depth": 2, "certified": false, "rounds": 4, "fixpoint": true, "chars_before": 472, "chars_after": 472, "seconds": 369.2, "id": "mathd_algebra_35", "proof_after": "theorem mathd_algebra_35 (p q : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, p x = 2 - x ^ 2)\n    (h\u2081 : \u2200 x : \u211d, x \u2260 0 \u2192 q x = 6 / x) : p (q 2) = -7  :=  by\n  have q2 : q 2 = 3 := by\n    have h\u2082 : q 2 = 6 / 2 := h\u2081 2 (two_ne_zero)\n    norm_num at h\u2082\n    rw [h\u2082]\n  have p_q2_eq_p3 : p (q 2) = p 3 := by\n    simp_all [h\u2080, h\u2081, q2]\n    <;> linarith\n    <;> linarith\n  have p3 : p 3 = -7 := by\n    rw [h\u2080] at p_q2_eq_p3\n    simp_all [h\u2081]\n    <;> norm_num\n    <;> linarith\n  have final : p (q 2) = -7 := by\n    simp_all only [eq_self_iff_true, heq_iff_eq]\n    <;> linarith\n  simp_all only [eq_self_iff_true, heq_iff_eq]\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_35 (p q : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, p x = 2 - x ^ 2)\n    (h\u2081 : \u2200 x : \u211d, x \u2260 0 \u2192 q x = 6 / x) : p (q 2) = -7  :=  by\n  have q2 : q 2 = 3 := by\n    have h\u2082 : q 2 = 6 / 2 := h\u2081 2 (two_ne_zero)\n    norm_num at h\u2082\n    rw [h\u2082]\n  have p_q2_eq_p3 : p (q 2) = p 3 := by\n    simp_all [h\u2080, h\u2081, q2]\n    <;> linarith\n    <;> linarith\n  have p3 : p 3 = -7 := by\n    rw [h\u2080] at p_q2_eq_p3\n    simp_all [h\u2081]\n    <;> norm_num\n    <;> linarith\n  have final : p (q 2) = -7 := by\n    simp_all only [eq_self_iff_true, heq_iff_eq]\n    <;> linarith\n  simp_all only [eq_self_iff_true, heq_iff_eq]\n  <;> linarith\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 258, "alphabet": 157, "compiles": false, "id": "amc12a_2019_p9", "proof_after": "theorem amc12a_2019_p9 (a : \u2115 \u2192 \u211a) (h\u2080 : a 1 = 1) (h\u2081 : a 2 = 3 / 7)\n  (h\u2082 : \u2200 n, a (n + 2) = a n * a (n + 1) / (2 * a n - a (n + 1))) :\n  \u2191(a 2019).den + (a 2019).num = 8078  :=  by\n  have pattern_observed : \u2200 n, a (n + 1) = 3 / (4 * (n + 1) - 1) := by\n    have h3 : a 3 = 3 / 11 := by\n      have h\u2083 := h\u2082 1\n      have h\u2084 := h\u2082 0\n      have h\u2085 := h\u2082 2\n      simp_all [Nat.mul_sub_left_distrib, Nat.add_sub_assoc, Nat.add_sub_cancel]\n      <;> ring_nf at *\n      <;> nlinarith\n      <;> norm_num\n      <;> nlinarith\n    have h4 : a 4 = 1 / 5 := by\n      have h\u2084 := h\u2082 1\n      have h\u2085 := h\u2082 2\n      have h\u2086 := h\u2082 3\n      simp_all [Nat.mul_div_cancel_left]\n      <;> norm_num\n      <;> linarith\n    have h5 : a 5 = 3 / 19 := by\n      have h\u2085 := h\u2082 3\n      have h\u2086 := h\u2082 4\n      have h\u2087 := h\u2082 2\n      have h\u2088 := h\u2082 1\n      norm_num [h\u2080, h\u2081, h3, h4] at h\u2085 h\u2086 h\u2087 h\u2088\n      <;> simp_all [Nat.div_eq_of_lt]\n      <;> norm_num\n      <;> ring_nf at *\n      <;> linarith\n    have base0 : a (0 + 1) = 3 / (4 * (0 + 1) - 1) := by\n      simp [h\u2080, h\u2081, h\u2082, h3, h4, h5]\n      <;> norm_num\n      <;> linarith\n      <;> ring_nf\n      <;> linarith\n      <;> rfl\n      <;> rfl\n      <;> rfl\n    have base1 : a (1 + 1) = 3 / (4 * (1 + 1) - 1) := by\n      simp_all [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.mul_succ, Nat.mul_one,\n        Nat.mul_zero, Nat.add_zero, Nat.add_assoc, Nat.add_right_comm, Nat.add_left_comm]\n      <;> norm_num\n      <;> linarith\n      <;> norm_num\n      <;> linarith\n    have ind_step : \u2200 k, a (k + 1) = 3 / (4 * (k + 1) - 1) \u2227 a (k + 2) = 3 / (4 * ((k + 1) + 1) - 1) \u2192 a (k + 3) = 3 / (4 * ((k + 2) + 1) - 1) := by\n      intro k\n      intro h\n      have h6 : a (k + 3) = a (k + 1) * a (k + 2) / (2 * a (k + 1) - a (k + 2)) := h\u2082 (k + 1)\n      rw [h6]\n      have h7 : a (k + 1) = 3 / (4 * (k + 1) - 1) := h.1\n      have h8 : a (k + 2) = 3 / (4 * ((k + 1) + 1) - 1) := h.2\n      rw [h7, h8]\n      field_simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n      <;> ring_nf\n      <;> norm_num\n      <;> field_simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n      <;> ring_nf\n      <;> norm_num\n      <;> field_simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n      <;> ring_nf\n      <;> norm_num\n      <;> field_simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n      <;> ring_nf\n      <;> norm_num\n      <;> field_simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n      <;> ring_nf\n      <;> norm_num\n    have result : \u2200 n, a (n + 1) = 3 / (4 * (n + 1) - 1) := by\n      intro n\n      have h\u2086 : a (n + 1) = 3 / (4 * (n + 1) - 1) := by\n        induction n using Nat.strong_induction_on with\n        | h n ih =>\n          match n with\n          | 0 =>\n            norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, base0, base1] at *\n          | 1 =>\n            norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, base0, base1] at *\n          | n + 2 =>\n            have h\u2087 := h\u2082 n\n            have h\u2088 := h\u2082 (n + 1)\n            have h\u2089 := ih n (by linarith)\n            have h\u2081\u2080 := ih (n + 1) (by linarith)\n            norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, base0, base1] at *\n            <;> simp_all\n            <;> field_simp at *\n            <;> ring_nf at *\n            <;> linarith\n      exact h\u2086\n    intro n\n    have h6 := result n\n    exact h6\n  have formula_applied : a 2019 = 3 / (4 * 2019 - 1) := by\n    have h\u2083 := pattern_observed 0\n    have h\u2084 := pattern_observed 1\n    have h\u2085 := pattern_observed 2\n    have h\u2086 := pattern_observed 3\n    have h\u2087 := pattern_observed 4\n    have h\u2088 := h\u2082 0\n    have h\u2089 := h\u2082 1\n    have h\u2081\u2080 := h\u2082 2\n    have h\u2081\u2081 := h\u2082 3\n    have h\u2081\u2082 := h\u2082 4\n    norm_num at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  have simplified : a 2019 = 3 / 8075 := by\n    have h\u2083 := pattern_observed 0\n    have h\u2084 := pattern_observed 1\n    have h\u2085 := pattern_observed 2\n    have h\u2086 := pattern_observed 3\n    have h\u2087 := pattern_observed 4\n    have h\u2088 := pattern_observed 5\n    have h\u2089 := pattern_observed 6\n    have h\u2081\u2080 := pattern_observed 7\n    have h\u2081\u2081 := pattern_observed 8\n    have h\u2081\u2082 := pattern_observed 9\n    have h\u2081\u2083 := pattern_observed 10\n    have h\u2081\u2084 := pattern_observed 11\n    have h\u2081\u2085 := pattern_observed 12\n    have h\u2081\u2086 := pattern_observed 13\n    have h\u2081\u2087 := pattern_observed 14\n    have h\u2081\u2088 := pattern_observed 15\n    have h\u2081\u2089 := pattern_observed 16\n    have h\u2082\u2080 := pattern_observed 17\n    have h\u2082\u2081 := pattern_observed 18\n    have h\u2082\u2082 := pattern_observed 19\n    have h\u2082\u2083 := pattern_observed 20\n    have h\u2082\u2084 := pattern_observed 21\n    have h\u2082\u2085 := pattern_observed 22\n    have h\u2082\u2086 := pattern_observed 23\n    have h\u2082\u2087 := pattern_observed 24\n    have h\u2082\u2088 := pattern_observed 25\n    have h\u2082\u2089 := pattern_observed 26\n    have h\u2083\u2080 := pattern_observed 27\n    have h\u2083\u2081 := pattern_observed 28\n    have h\u2083\u2082 := pattern_observed 29\n    have h\u2083\u2083 := pattern_observed 30\n    have h\u2083\u2084 := pattern_observed 31\n    have h\u2083\u2085 := pattern_observed 32\n    have h\u2083\u2086 := pattern_observed 33\n    have h\u2083\u2087 := pattern_observed 34\n    have h\u2083\u2088 := pattern_observed 35\n    have h\u2083\u2089 := pattern_observed 36\n    have h\u2084\u2080 := pattern_observed 37\n    have h\u2084\u2081 := pattern_observed 38\n    have h\u2084\u2082 := pattern_observed 39\n    have h\u2084\u2083 := pattern_observed 40\n    have h\u2084\u2084 := pattern_observed 41\n    have h\u2084\u2085 := pattern_observed 42\n    have h\u2084\u2086 := pattern_observed 43\n    have h\u2084\u2087 := pattern_observed 44\n    have h\u2084\u2088 := pattern_observed 45\n    have h\u2084\u2089 := pattern_observed 46\n    have h\u2085\u2080 := pattern_observed 47\n    have h\u2085\u2081 := pattern_observed 48\n    have h\u2085\u2082 := pattern_observed 49\n    have h\u2085\u2083 := pattern_observed 50\n    have h\u2085\u2084 := pattern_observed 51\n    have h\u2085\u2085 := pattern_observed 52\n    have h\u2085\u2086 := pattern_observed 53\n    have h\u2085\u2087 := pattern_observed 54\n    have h\u2085\u2088 := pattern_observed 55\n    have h\u2085\u2089 := pattern_observed 56\n    have h\u2086\u2080 := pattern_observed 57\n    have h\u2086\u2081 := pattern_observed 58\n    have h\u2086\u2082 := pattern_observed 59\n    have h\u2086\u2083 := pattern_observed 60\n    have h\u2086\u2084 := pattern_observed 61\n    have h\u2086\u2085 := pattern_observed 62\n    have h\u2086\u2086 := pattern_observed 63\n    have h\u2086\u2087 := pattern_observed 64\n    have h\u2086\u2088 := pattern_observed 65\n    have h\u2086\u2089 := pattern_observed 66\n    have h\u2087\u2080 := pattern_observed 67\n    have h\u2087\u2081 := pattern_observed 68\n    have h\u2087\u2082 := pattern_observed 69\n    have h\u2087\u2083 := pattern_observed 70\n    have h\u2087\u2084 := pattern_observed 71\n    have h\u2087\u2085 := pattern_observed 72\n    have h\u2087\u2086 := pattern_observed 73\n    have h\u2087\u2087 := pattern_observed 74\n    have h\u2087\u2088 := pattern_observed 75\n    have h\u2087\u2089 := pattern_observed 76\n    have h\u2088\u2080 := pattern_observed 77\n    have h\u2088\u2081 := pattern_observed 78\n    have h\u2088\u2082 := pattern_observed 79\n    have h\u2088\u2083 := pattern_observed 80\n    have h\u2088\u2084 := pattern_observed 81\n    have h\u2088\u2085 := pattern_observed 82\n    have h\u2088\u2086 := pattern_observed 83\n    have h\u2088\u2087 := pattern_observed 84\n    have h\u2088\u2088 := pattern_observed 85\n    have h\u2088\u2089 := pattern_observed 86\n    have h\u2089\u2080 := pattern_observed 87\n    have h\u2089\u2081 := pattern_observed 88\n    have h\u2089\u2082 := pattern_observed 89\n    have h\u2089\u2083 := pattern_observed 90\n    have h\u2089\u2084 := pattern_observed 91\n    have h\u2089\u2085 := pattern_observed 92\n    have h\u2089\u2086 := pattern_observed 93\n    have h\u2089\u2087 := pattern_observed 94\n    have h\u2089\u2088 := pattern_observed 95\n    have h\u2089\u2089 := pattern_observed 96\n    have h\u2081\u2080\u2080 := pattern_observed 97\n    have h\u2081\u2080\u2081 := pattern_observed 98\n    have h\u2081\u2080\u2082 := pattern_observed 99\n    have h\u2081\u2080\u2083 := pattern_observed 100\n    norm_num at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 h\u2082\u2084 h\u2082\u2085 h\u2082\u2086 h\u2082\u2087 h\u2082\u2088 h\u2082\u2089 h\u2083\u2080 h\u2083\u2081 h\u2083\u2082 h\u2083\u2083 h\u2083\u2084 h\u2083\u2085 h\u2083\u2086 h\u2083\u2087 h\u2083\u2088 h\u2083\u2089 h\u2084\u2080 h\u2084\u2081 h\u2084\u2082 h\u2084\u2083 h\u2084\u2084 h\u2084\u2085 h\u2084\u2086 h\u2084\u2087 h\u2084\u2088 h\u2084\u2089 h\u2085\u2080 h\u2085\u2081 h\u2085\u2082 h\u2085\u2083 h\u2085\u2084 h\u2085\u2085 h\u2085\u2086 h\u2085\u2087 h\u2085\u2088 h\u2085\u2089 h\u2086\u2080 h\u2086\u2081 h\u2086\u2082 h\u2086\u2083 h\u2086\u2084 h\u2086\u2085 h\u2086\u2086 h\u2086\u2087 h\u2086\u2088 h\u2086\u2089 h\u2087\u2080 h\u2087\u2081 h\u2087\u2082 h\u2087\u2083 h\u2087\u2084 h\u2087\u2085 h\u2087\u2086 h\u2087\u2087 h\u2087\u2088 h\u2087\u2089 h\u2088\u2080 h\u2088\u2081 h\u2088\u2082 h\u2088\u2083 h\u2088\u2084 h\u2088\u2085 h\u2088\u2086 h\u2088\u2087 h\u2088\u2088 h\u2088\u2089 h\u2089\u2080 h\u2089\u2081 h\u2089\u2082 h\u2089\u2083 h\u2089\u2084 h\u2089\u2085 h\u2089\u2086 h\u2089\u2087 h\u2089\u2088 h\u2089\u2089 h\u2081\u2080\u2080 h\u2081\u2080\u2081 h\u2081\u2080\u2082 h\u2081\u2080\u2083\n    <;> simp_all [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080]\n    <;> norm_num\n    <;> linarith\n  have coprime_check : (3).gcd 8075 = 1 := by\n    rw [Nat.gcd_comm]\n    <;> decide\n    <;> norm_num\n    <;> rfl\n    <;> decide\n    <;> norm_num\n    <;> rfl\n    <;> decide\n    <;> norm_num\n    <;> rfl\n  have final_answer : \u2191(a 2019).den + (a 2019).num = 3 + 8075 := by\n    norm_num [simplified, pattern_observed, formula_applied]\n    <;> norm_cast\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  have conclusion : 3 + 8075 = 8078 := by\n    have h\u2083 := pattern_observed 0\n    have h\u2084 := pattern_observed 1\n    have h\u2085 := pattern_observed 2\n    have h\u2086 := pattern_observed 3\n    have h\u2087 := pattern_observed 4\n    have h\u2088 := pattern_observed 5\n    have h\u2089 := pattern_observed 6\n    have h\u2081\u2080 := pattern_observed 7\n    have h\u2081\u2081 := pattern_observed 8\n    have h\u2081\u2082 := pattern_observed 9\n    norm_num [h\u2080, h\u2081, h\u2082] at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n    <;> simp_all [h\u2080, h\u2081, h\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2083 := pattern_observed 0\n  have h\u2084 := pattern_observed 1\n  have h\u2085 := pattern_observed 2\n  have h\u2086 := pattern_observed 3\n  have h\u2087 := pattern_observed 4\n  have h\u2088 := pattern_observed 5\n  have h\u2089 := pattern_observed 6\n  have h\u2081\u2080 := pattern_observed 7\n  have h\u2081\u2081 := pattern_observed 8\n  have h\u2081\u2082 := pattern_observed 9\n  norm_num at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n  <;> simp_all [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080, h\u2081\u2081, h\u2081\u2082]\n  <;> norm_num\n  <;> linarith\n", "proof_before": "theorem amc12a_2019_p9 (a : \u2115 \u2192 \u211a) (h\u2080 : a 1 = 1) (h\u2081 : a 2 = 3 / 7)\n  (h\u2082 : \u2200 n, a (n + 2) = a n * a (n + 1) / (2 * a n - a (n + 1))) :\n  \u2191(a 2019).den + (a 2019).num = 8078  :=  by\n  have pattern_observed : \u2200 n, a (n + 1) = 3 / (4 * (n + 1) - 1) := by\n    have h3 : a 3 = 3 / 11 := by\n      have h\u2083 := h\u2082 1\n      have h\u2084 := h\u2082 0\n      have h\u2085 := h\u2082 2\n      simp_all [Nat.mul_sub_left_distrib, Nat.add_sub_assoc, Nat.add_sub_cancel]\n      <;> ring_nf at *\n      <;> nlinarith\n      <;> norm_num\n      <;> nlinarith\n    have h4 : a 4 = 1 / 5 := by\n      have h\u2084 := h\u2082 1\n      have h\u2085 := h\u2082 2\n      have h\u2086 := h\u2082 3\n      simp_all [Nat.mul_div_cancel_left]\n      <;> norm_num\n      <;> linarith\n    have h5 : a 5 = 3 / 19 := by\n      have h\u2085 := h\u2082 3\n      have h\u2086 := h\u2082 4\n      have h\u2087 := h\u2082 2\n      have h\u2088 := h\u2082 1\n      norm_num [h\u2080, h\u2081, h3, h4] at h\u2085 h\u2086 h\u2087 h\u2088\n      <;> simp_all [Nat.div_eq_of_lt]\n      <;> norm_num\n      <;> ring_nf at *\n      <;> linarith\n    have base0 : a (0 + 1) = 3 / (4 * (0 + 1) - 1) := by\n      simp [h\u2080, h\u2081, h\u2082, h3, h4, h5]\n      <;> norm_num\n      <;> linarith\n      <;> ring_nf\n      <;> linarith\n      <;> rfl\n      <;> rfl\n      <;> rfl\n    have base1 : a (1 + 1) = 3 / (4 * (1 + 1) - 1) := by\n      simp_all [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.mul_succ, Nat.mul_one,\n        Nat.mul_zero, Nat.add_zero, Nat.add_assoc, Nat.add_right_comm, Nat.add_left_comm]\n      <;> norm_num\n      <;> linarith\n      <;> norm_num\n      <;> linarith\n    have ind_step : \u2200 k, a (k + 1) = 3 / (4 * (k + 1) - 1) \u2227 a (k + 2) = 3 / (4 * ((k + 1) + 1) - 1) \u2192 a (k + 3) = 3 / (4 * ((k + 2) + 1) - 1) := by\n      intro k\n      intro h\n      have h6 : a (k + 3) = a (k + 1) * a (k + 2) / (2 * a (k + 1) - a (k + 2)) := h\u2082 (k + 1)\n      rw [h6]\n      have h7 : a (k + 1) = 3 / (4 * (k + 1) - 1) := h.1\n      have h8 : a (k + 2) = 3 / (4 * ((k + 1) + 1) - 1) := h.2\n      rw [h7, h8]\n      field_simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n      <;> ring_nf\n      <;> norm_num\n      <;> field_simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n      <;> ring_nf\n      <;> norm_num\n      <;> field_simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n      <;> ring_nf\n      <;> norm_num\n      <;> field_simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n      <;> ring_nf\n      <;> norm_num\n      <;> field_simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n      <;> ring_nf\n      <;> norm_num\n    have result : \u2200 n, a (n + 1) = 3 / (4 * (n + 1) - 1) := by\n      intro n\n      have h\u2086 : a (n + 1) = 3 / (4 * (n + 1) - 1) := by\n        induction n using Nat.strong_induction_on with\n        | h n ih =>\n          match n with\n          | 0 =>\n            norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, base0, base1] at *\n          | 1 =>\n            norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, base0, base1] at *\n          | n + 2 =>\n            have h\u2087 := h\u2082 n\n            have h\u2088 := h\u2082 (n + 1)\n            have h\u2089 := ih n (by linarith)\n            have h\u2081\u2080 := ih (n + 1) (by linarith)\n            norm_num [h\u2080, h\u2081, h\u2082, h3, h4, h5, base0, base1] at *\n            <;> simp_all\n            <;> field_simp at *\n            <;> ring_nf at *\n            <;> linarith\n      exact h\u2086\n    intro n\n    have h6 := result n\n    exact h6\n  have formula_applied : a 2019 = 3 / (4 * 2019 - 1) := by\n    have h\u2083 := pattern_observed 0\n    have h\u2084 := pattern_observed 1\n    have h\u2085 := pattern_observed 2\n    have h\u2086 := pattern_observed 3\n    have h\u2087 := pattern_observed 4\n    have h\u2088 := h\u2082 0\n    have h\u2089 := h\u2082 1\n    have h\u2081\u2080 := h\u2082 2\n    have h\u2081\u2081 := h\u2082 3\n    have h\u2081\u2082 := h\u2082 4\n    norm_num at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  have simplified : a 2019 = 3 / 8075 := by\n    have h\u2083 := pattern_observed 0\n    have h\u2084 := pattern_observed 1\n    have h\u2085 := pattern_observed 2\n    have h\u2086 := pattern_observed 3\n    have h\u2087 := pattern_observed 4\n    have h\u2088 := pattern_observed 5\n    have h\u2089 := pattern_observed 6\n    have h\u2081\u2080 := pattern_observed 7\n    have h\u2081\u2081 := pattern_observed 8\n    have h\u2081\u2082 := pattern_observed 9\n    have h\u2081\u2083 := pattern_observed 10\n    have h\u2081\u2084 := pattern_observed 11\n    have h\u2081\u2085 := pattern_observed 12\n    have h\u2081\u2086 := pattern_observed 13\n    have h\u2081\u2087 := pattern_observed 14\n    have h\u2081\u2088 := pattern_observed 15\n    have h\u2081\u2089 := pattern_observed 16\n    have h\u2082\u2080 := pattern_observed 17\n    have h\u2082\u2081 := pattern_observed 18\n    have h\u2082\u2082 := pattern_observed 19\n    have h\u2082\u2083 := pattern_observed 20\n    have h\u2082\u2084 := pattern_observed 21\n    have h\u2082\u2085 := pattern_observed 22\n    have h\u2082\u2086 := pattern_observed 23\n    have h\u2082\u2087 := pattern_observed 24\n    have h\u2082\u2088 := pattern_observed 25\n    have h\u2082\u2089 := pattern_observed 26\n    have h\u2083\u2080 := pattern_observed 27\n    have h\u2083\u2081 := pattern_observed 28\n    have h\u2083\u2082 := pattern_observed 29\n    have h\u2083\u2083 := pattern_observed 30\n    have h\u2083\u2084 := pattern_observed 31\n    have h\u2083\u2085 := pattern_observed 32\n    have h\u2083\u2086 := pattern_observed 33\n    have h\u2083\u2087 := pattern_observed 34\n    have h\u2083\u2088 := pattern_observed 35\n    have h\u2083\u2089 := pattern_observed 36\n    have h\u2084\u2080 := pattern_observed 37\n    have h\u2084\u2081 := pattern_observed 38\n    have h\u2084\u2082 := pattern_observed 39\n    have h\u2084\u2083 := pattern_observed 40\n    have h\u2084\u2084 := pattern_observed 41\n    have h\u2084\u2085 := pattern_observed 42\n    have h\u2084\u2086 := pattern_observed 43\n    have h\u2084\u2087 := pattern_observed 44\n    have h\u2084\u2088 := pattern_observed 45\n    have h\u2084\u2089 := pattern_observed 46\n    have h\u2085\u2080 := pattern_observed 47\n    have h\u2085\u2081 := pattern_observed 48\n    have h\u2085\u2082 := pattern_observed 49\n    have h\u2085\u2083 := pattern_observed 50\n    have h\u2085\u2084 := pattern_observed 51\n    have h\u2085\u2085 := pattern_observed 52\n    have h\u2085\u2086 := pattern_observed 53\n    have h\u2085\u2087 := pattern_observed 54\n    have h\u2085\u2088 := pattern_observed 55\n    have h\u2085\u2089 := pattern_observed 56\n    have h\u2086\u2080 := pattern_observed 57\n    have h\u2086\u2081 := pattern_observed 58\n    have h\u2086\u2082 := pattern_observed 59\n    have h\u2086\u2083 := pattern_observed 60\n    have h\u2086\u2084 := pattern_observed 61\n    have h\u2086\u2085 := pattern_observed 62\n    have h\u2086\u2086 := pattern_observed 63\n    have h\u2086\u2087 := pattern_observed 64\n    have h\u2086\u2088 := pattern_observed 65\n    have h\u2086\u2089 := pattern_observed 66\n    have h\u2087\u2080 := pattern_observed 67\n    have h\u2087\u2081 := pattern_observed 68\n    have h\u2087\u2082 := pattern_observed 69\n    have h\u2087\u2083 := pattern_observed 70\n    have h\u2087\u2084 := pattern_observed 71\n    have h\u2087\u2085 := pattern_observed 72\n    have h\u2087\u2086 := pattern_observed 73\n    have h\u2087\u2087 := pattern_observed 74\n    have h\u2087\u2088 := pattern_observed 75\n    have h\u2087\u2089 := pattern_observed 76\n    have h\u2088\u2080 := pattern_observed 77\n    have h\u2088\u2081 := pattern_observed 78\n    have h\u2088\u2082 := pattern_observed 79\n    have h\u2088\u2083 := pattern_observed 80\n    have h\u2088\u2084 := pattern_observed 81\n    have h\u2088\u2085 := pattern_observed 82\n    have h\u2088\u2086 := pattern_observed 83\n    have h\u2088\u2087 := pattern_observed 84\n    have h\u2088\u2088 := pattern_observed 85\n    have h\u2088\u2089 := pattern_observed 86\n    have h\u2089\u2080 := pattern_observed 87\n    have h\u2089\u2081 := pattern_observed 88\n    have h\u2089\u2082 := pattern_observed 89\n    have h\u2089\u2083 := pattern_observed 90\n    have h\u2089\u2084 := pattern_observed 91\n    have h\u2089\u2085 := pattern_observed 92\n    have h\u2089\u2086 := pattern_observed 93\n    have h\u2089\u2087 := pattern_observed 94\n    have h\u2089\u2088 := pattern_observed 95\n    have h\u2089\u2089 := pattern_observed 96\n    have h\u2081\u2080\u2080 := pattern_observed 97\n    have h\u2081\u2080\u2081 := pattern_observed 98\n    have h\u2081\u2080\u2082 := pattern_observed 99\n    have h\u2081\u2080\u2083 := pattern_observed 100\n    norm_num at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083 h\u2082\u2084 h\u2082\u2085 h\u2082\u2086 h\u2082\u2087 h\u2082\u2088 h\u2082\u2089 h\u2083\u2080 h\u2083\u2081 h\u2083\u2082 h\u2083\u2083 h\u2083\u2084 h\u2083\u2085 h\u2083\u2086 h\u2083\u2087 h\u2083\u2088 h\u2083\u2089 h\u2084\u2080 h\u2084\u2081 h\u2084\u2082 h\u2084\u2083 h\u2084\u2084 h\u2084\u2085 h\u2084\u2086 h\u2084\u2087 h\u2084\u2088 h\u2084\u2089 h\u2085\u2080 h\u2085\u2081 h\u2085\u2082 h\u2085\u2083 h\u2085\u2084 h\u2085\u2085 h\u2085\u2086 h\u2085\u2087 h\u2085\u2088 h\u2085\u2089 h\u2086\u2080 h\u2086\u2081 h\u2086\u2082 h\u2086\u2083 h\u2086\u2084 h\u2086\u2085 h\u2086\u2086 h\u2086\u2087 h\u2086\u2088 h\u2086\u2089 h\u2087\u2080 h\u2087\u2081 h\u2087\u2082 h\u2087\u2083 h\u2087\u2084 h\u2087\u2085 h\u2087\u2086 h\u2087\u2087 h\u2087\u2088 h\u2087\u2089 h\u2088\u2080 h\u2088\u2081 h\u2088\u2082 h\u2088\u2083 h\u2088\u2084 h\u2088\u2085 h\u2088\u2086 h\u2088\u2087 h\u2088\u2088 h\u2088\u2089 h\u2089\u2080 h\u2089\u2081 h\u2089\u2082 h\u2089\u2083 h\u2089\u2084 h\u2089\u2085 h\u2089\u2086 h\u2089\u2087 h\u2089\u2088 h\u2089\u2089 h\u2081\u2080\u2080 h\u2081\u2080\u2081 h\u2081\u2080\u2082 h\u2081\u2080\u2083\n    <;> simp_all [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080]\n    <;> norm_num\n    <;> linarith\n  have coprime_check : (3).gcd 8075 = 1 := by\n    rw [Nat.gcd_comm]\n    <;> decide\n    <;> norm_num\n    <;> rfl\n    <;> decide\n    <;> norm_num\n    <;> rfl\n    <;> decide\n    <;> norm_num\n    <;> rfl\n  have final_answer : \u2191(a 2019).den + (a 2019).num = 3 + 8075 := by\n    norm_num [simplified, pattern_observed, formula_applied]\n    <;> norm_cast\n    <;> simp_all\n    <;> norm_num\n    <;> linarith\n  have conclusion : 3 + 8075 = 8078 := by\n    have h\u2083 := pattern_observed 0\n    have h\u2084 := pattern_observed 1\n    have h\u2085 := pattern_observed 2\n    have h\u2086 := pattern_observed 3\n    have h\u2087 := pattern_observed 4\n    have h\u2088 := pattern_observed 5\n    have h\u2089 := pattern_observed 6\n    have h\u2081\u2080 := pattern_observed 7\n    have h\u2081\u2081 := pattern_observed 8\n    have h\u2081\u2082 := pattern_observed 9\n    norm_num [h\u2080, h\u2081, h\u2082] at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n    <;> simp_all [h\u2080, h\u2081, h\u2082]\n    <;> norm_num\n    <;> linarith\n  have h\u2083 := pattern_observed 0\n  have h\u2084 := pattern_observed 1\n  have h\u2085 := pattern_observed 2\n  have h\u2086 := pattern_observed 3\n  have h\u2087 := pattern_observed 4\n  have h\u2088 := pattern_observed 5\n  have h\u2089 := pattern_observed 6\n  have h\u2081\u2080 := pattern_observed 7\n  have h\u2081\u2081 := pattern_observed 8\n  have h\u2081\u2082 := pattern_observed 9\n  norm_num at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082\n  <;> simp_all [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, h\u2089, h\u2081\u2080, h\u2081\u2081, h\u2081\u2082]\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 19, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 18, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 4, "fixpoint": true, "chars_before": 743, "chars_after": 560, "seconds": 942.9, "id": "induction_seq_mul2pnp1", "proof_after": "theorem induction_seq_mul2pnp1 (n : \u2115) (u : \u2115 \u2192 \u2115) (h\u2080 : u 0 = 0)\n  (h\u2081 : \u2200 n, u (n + 1) = 2 * u n + (n + 1)) : u n = 2 ^ (n + 1) - (n + 2)  :=  by\n  have inductive_hypothesis : \u2200 k : \u2115, u k = 2 ^ (k + 1) - (k + 2) \u2192 u (k + 1) = 2 ^ ((k + 1) + 1) - ((k + 1) + 2) := by\n    intro k hk\n    cases k with\n    | zero =>\n      simp_all [h\u2081, h\u2080]\n    | succ k' =>\n      simp_all [h\u2081, pow_succ, mul_add, mul_comm, mul_left_comm, Nat.mul_sub_left_distrib]\n      <;> ring_nf\n      <;> omega\n  have inductive_step : \u2200 k : \u2115, u k = 2 ^ (k + 1) - (k + 2) := by\n    intro k\n    induction k with\n    | zero =>\n      simp_all [h\u2080]\n    | succ k ih =>\n      simp_all [h\u2081, Nat.pow_succ, Nat.mul_succ]\n      <;> omega\n  simp_all\n", "proof_before": "theorem induction_seq_mul2pnp1 (n : \u2115) (u : \u2115 \u2192 \u2115) (h\u2080 : u 0 = 0)\n  (h\u2081 : \u2200 n, u (n + 1) = 2 * u n + (n + 1)) : u n = 2 ^ (n + 1) - (n + 2)  :=  by\n  have base_case : u 0 = 2 ^ (0 + 1) - (0 + 2) := by\n    simp [h\u2080, pow_one, Nat.mul_zero, Nat.add_zero]\n    <;> linarith\n  have inductive_hypothesis : \u2200 k : \u2115, u k = 2 ^ (k + 1) - (k + 2) \u2192 u (k + 1) = 2 ^ ((k + 1) + 1) - ((k + 1) + 2) := by\n    intro k hk\n    cases k with\n    | zero =>\n      simp_all [h\u2081, h\u2080]\n    | succ k' =>\n      simp_all [h\u2081, pow_succ, mul_add, mul_comm, mul_left_comm, Nat.mul_sub_left_distrib]\n      <;> ring_nf\n      <;> omega\n  have inductive_step : \u2200 k : \u2115, u k = 2 ^ (k + 1) - (k + 2) := by\n    intro k\n    induction k with\n    | zero =>\n      simp_all [h\u2080]\n    | succ k ih =>\n      simp_all [h\u2081, Nat.pow_succ, Nat.mul_succ]\n      <;> omega\n  have h\u2082 := inductive_step\n  have h\u2083 := inductive_hypothesis\n  simp_all\n"}
{"mode": "clear", "steps_before": 57, "alphabet": 20, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 10, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 7, "timed_out": false, "fixpoint": true, "chars_before": 1670, "chars_after": 228, "seconds": 103.5, "id": "mathd_algebra_192", "proof_after": "theorem mathd_algebra_192 (q e d : \u2102) (h\u2080 : q = 11 - 5 * Complex.I) (h\u2081 : e = 11 + 5 * Complex.I)\n    (h\u2082 : d = 2 * Complex.I) : q * e * d = 292 * Complex.I  :=  by\n  have h\u2088 : q * e * d = 292 * Complex.I := by\n    simp_all [Complex.ext_iff, mul_add, mul_sub, sub_mul, add_mul]\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  omega\n", "proof_before": "theorem mathd_algebra_192 (q e d : \u2102) (h\u2080 : q = 11 - 5 * Complex.I) (h\u2081 : e = 11 + 5 * Complex.I)\n    (h\u2082 : d = 2 * Complex.I) : q * e * d = 292 * Complex.I  :=  by\n  have h\u2083 : q * e = (11 - 5 * Complex.I) * (11 + 5 * Complex.I) := by\n    rw [h\u2080, h\u2081]\n    <;> simp [h\u2080, h\u2081, h\u2082, mul_add, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> simp [Complex.I_mul_I]\n    <;> ring_nf\n  have h\u2084 : q * e = 11^2 - (5 * Complex.I)^2 := by\n    rw [h\u2083]\n    simp [h\u2080, h\u2081, Complex.ext_iff, mul_comm, mul_assoc, mul_left_comm, sub_eq_add_neg, add_assoc]\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  have h\u2085 : q * e = 121 + 25 := by\n    simp_all only [Complex.I_sq, sub_eq_add_neg, neg_mul, neg_neg, mul_add, mul_comm]\n    ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  have h\u2086 : q * e = 146 := by\n    rw [h\u2083] at h\u2085\n    simp_all\n    <;> ring\n    <;> norm_num\n    <;> rfl\n  have h\u2087 : q * e * d = 146 * (2 * Complex.I) := by\n    subst_vars\n    simp only [Complex.ext_iff, Complex.mul_re, Complex.mul_im, Complex.add_re, Complex.add_im, Complex.sub_re, Complex.sub_im, Complex.I_re, Complex.I_im] at h\u2085 h\u2086 \u22a2\n    norm_num\n    <;> simp [Complex.ext_iff, Complex.mul_re, Complex.mul_im, Complex.add_re, Complex.add_im, Complex.sub_re, Complex.sub_im, Complex.I_re, Complex.I_im]\n    <;> norm_num\n  have h\u2088 : q * e * d = 292 * Complex.I := by\n    simp_all [Complex.ext_iff, mul_add, mul_sub, sub_mul, add_mul]\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  simp_all [Complex.ext_iff]\n  <;> ring_nf\n  <;> norm_num\n  <;> simp_all [Complex.ext_iff]\n  <;> ring_nf\n  <;> norm_num\n  <;> simp_all [Complex.ext_iff]\n  <;> ring_nf\n  <;> norm_num\n  <;> simp_all [Complex.ext_iff]\n  <;> ring_nf\n  <;> norm_num\n"}
{"mode": "clear", "steps_before": 36, "alphabet": 29, "compiles": true, "steps_after": 8, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 8, "timed_out": false, "fixpoint": true, "chars_before": 1399, "chars_after": 315, "seconds": 152.0, "id": "imo_1961_p1", "proof_after": "theorem imo_1961_p1 (x y z a b : \u211d) (h\u2080 : 0 < x \u2227 0 < y \u2227 0 < z) (h\u2081 : x \u2260 y) (h\u2082 : y \u2260 z)\n  (h\u2083 : z \u2260 x) (h\u2084 : x + y + z = a) (h\u2085 : x ^ 2 + y ^ 2 + z ^ 2 = b ^ 2) (h\u2086 : x * y = z ^ 2) :\n  0 < a \u2227 b ^ 2 < a ^ 2 \u2227 a ^ 2 < 3 * b ^ 2  :=  by\n  have h\u2081\u2080 : a > 0 := by\n    linarith\n  have h\u2081\u2081 : b^2 < a^2 := by\n    nlinarith\n  have h\u2081\u2082 : a^2 < 3 * b^2 := by\n    have h\u2081\u2082 := sq_pos_of_ne_zero (sub_ne_zero_of_ne h\u2081) \n    nlinarith [sq_nonneg (x - y), sq_nonneg (y - z), sq_nonneg (z - x)]\n  refine' \u27e8by nlinarith [h\u2081\u2080], by nlinarith [h\u2081\u2081], by nlinarith [h\u2081\u2082]\u27e9\n", "proof_before": "theorem imo_1961_p1 (x y z a b : \u211d) (h\u2080 : 0 < x \u2227 0 < y \u2227 0 < z) (h\u2081 : x \u2260 y) (h\u2082 : y \u2260 z)\n  (h\u2083 : z \u2260 x) (h\u2084 : x + y + z = a) (h\u2085 : x ^ 2 + y ^ 2 + z ^ 2 = b ^ 2) (h\u2086 : x * y = z ^ 2) :\n  0 < a \u2227 b ^ 2 < a ^ 2 \u2227 a ^ 2 < 3 * b ^ 2  :=  by\n  have h\u2087 : z = sqrt (x * y) := by\n    have h\u2087 : 0 < x * y := by nlinarith\n    have h\u2088 : 0 < z := by nlinarith\n    have h\u2089 : z = sqrt (x * y) := by\n      apply Eq.symm\n      rw [sqrt_eq_iff_mul_self_eq] <;> nlinarith [sq_nonneg (x - y), sq_nonneg (x + y - z), sq_nonneg (x * y - z ^ 2)]\n    exact h\u2089\n  have h\u2088 : x + y + sqrt (x * y) = a := by\n    have h\u2084' : x + y + z = a := h\u2084\n    rw [h\u2087] at h\u2084'\n    nlinarith [Real.sqrt_nonneg (x * y), h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081, h\u2082, h\u2083]\n  have h\u2089 : x^2 + y^2 + x * y = b^2 := by\n    have h\u2089 : 0 < x * y := mul_pos h\u2080.1 h\u2080.2.1\n    have h\u2081\u2080 : z = sqrt (x * y) := h\u2087\n    rw [h\u2081\u2080] at h\u2084\n    rw [h\u2081\u2080] at h\u2085\n    nlinarith [sq_sqrt (mul_nonneg h\u2080.1.le h\u2080.2.1.le)]\n  have h\u2081\u2080 : a > 0 := by\n    have h\u2081\u2080 : 0 < x * y := by\n      nlinarith\n    have h\u2081\u2081 : 0 \u2264 z := by\n      nlinarith\n    nlinarith\n  have h\u2081\u2081 : b^2 < a^2 := by\n    have h\u2081\u2081 : 0 < x * y := by\n      nlinarith [sq_sqrt (mul_nonneg (le_of_lt h\u2080.1) (le_of_lt h\u2080.2.1))]\n    nlinarith [sq_sqrt (mul_nonneg (le_of_lt h\u2080.1) (le_of_lt h\u2080.2.1)), h\u2085, h\u2084, h\u2086, h\u2087, h\u2088, h\u2089]\n  have h\u2081\u2082 : a^2 < 3 * b^2 := by\n    have h\u2081\u2082 := sq_pos_of_ne_zero (sub_ne_zero_of_ne h\u2081) \n    have h\u2081\u2083 := sq_pos_of_ne_zero (sub_ne_zero_of_ne h\u2082) \n    have h\u2081\u2084 := sq_pos_of_ne_zero (sub_ne_zero_of_ne h\u2083) \n    nlinarith [sq_nonneg (x - y), sq_nonneg (y - z), sq_nonneg (z - x)]\n  refine' \u27e8by nlinarith [h\u2081\u2080], by nlinarith [h\u2081\u2081], by nlinarith [h\u2081\u2082]\u27e9\n"}
{"mode": "clear", "steps_before": 10, "alphabet": 13, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 224, "chars_after": 6, "seconds": 263.7, "id": "mathd_numbertheory_132", "proof_after": "theorem mathd_numbertheory_132 : 2004 % 12 = 0  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_132 : 2004 % 12 = 0  :=  by\n  have h1 : 2004 = 12 * 167 + 0 := by\n    norm_num\n    <;> ring\n    <;> omega\n  have h2 : 2004 % 12 = 0 := by\n    norm_num [h1]\n    <;> simp_all [mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> omega\n  exact h2\n"}
{"mode": "clear", "steps_before": 32, "alphabet": 24, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 16, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 5, "timed_out": true, "fixpoint": false, "chars_before": 1240, "chars_after": 787, "seconds": 424.3, "id": "mathd_algebra_31", "proof_after": "theorem mathd_algebra_31 (x : NNReal) (u : \u2115 \u2192 NNReal) (h\u2080 : \u2200 n, u (n + 1) = NNReal.sqrt (x + u n))\n  (h\u2081 : Filter.Tendsto u Filter.atTop (\ud835\udcdd 9)) : 9 = NNReal.sqrt (x + 9)  :=  by\n  have h\u2082 : Filter.Tendsto (fun n \u21a6 u (n + 1)) Filter.atTop (\ud835\udcdd 9) := by\n    have h\u2081' : Filter.Tendsto (fun n \u21a6 u (n + 1)) Filter.atTop (\ud835\udcdd 9) :=\n      Filter.Tendsto.comp h\u2081 (Filter.tendsto_add_atTop_nat 1)\n    exact h\u2081'\n  have h\u2083 : Filter.Tendsto (fun n \u21a6 NNReal.sqrt (x + u n)) Filter.atTop (\ud835\udcdd (NNReal.sqrt (x + 9))) := by\n    have h\u2083 : Continuous fun y : NNReal \u21a6 NNReal.sqrt (x + y) := by\n      continuity\n    exact h\u2083.continuousAt.tendsto.comp h\u2081\n  have h\u2084 : 9 = NNReal.sqrt (x + 9) := by\n    have h\u2084 : 9 = NNReal.sqrt (x + 9) := by\n      have h\u2086 : Filter.Tendsto (fun n \u21a6 u (n + 1)) Filter.atTop (\ud835\udcdd 9) := h\u2082\n      have h\u2087 : Filter.Tendsto (fun n \u21a6 u (n + 1)) Filter.atTop (\ud835\udcdd (NNReal.sqrt (x + 9))) := by\n        continuity\n      apply tendsto_nhds_unique h\u2086 h\u2087\n    exact h\u2084\n  omega\n", "proof_before": "theorem mathd_algebra_31 (x : NNReal) (u : \u2115 \u2192 NNReal) (h\u2080 : \u2200 n, u (n + 1) = NNReal.sqrt (x + u n))\n  (h\u2081 : Filter.Tendsto u Filter.atTop (\ud835\udcdd 9)) : 9 = NNReal.sqrt (x + 9)  :=  by\n  have h\u2082 : Filter.Tendsto (fun n \u21a6 u (n + 1)) Filter.atTop (\ud835\udcdd 9) := by\n    have h\u2081' : Filter.Tendsto (fun n \u21a6 u (n + 1)) Filter.atTop (\ud835\udcdd 9) :=\n      Filter.Tendsto.comp h\u2081 (Filter.tendsto_add_atTop_nat 1)\n    exact h\u2081'\n  have h\u2083 : Filter.Tendsto (fun n \u21a6 NNReal.sqrt (x + u n)) Filter.atTop (\ud835\udcdd (NNReal.sqrt (x + 9))) := by\n    rw [show (fun n \u21a6 NNReal.sqrt (x + u n)) = fun n \u21a6 NNReal.sqrt (x + u n) by rfl]\n    have h\u2083 : Continuous fun y : NNReal \u21a6 NNReal.sqrt (x + y) := by\n      continuity\n    exact h\u2083.continuousAt.tendsto.comp h\u2081\n  have h\u2084 : 9 = NNReal.sqrt (x + 9) := by\n    have h\u2084 : 9 = NNReal.sqrt (x + 9) := by\n      have h\u2085 : Filter.Tendsto (fun n \u21a6 NNReal.sqrt (x + u n)) Filter.atTop (\ud835\udcdd (NNReal.sqrt (x + 9))) := h\u2083\n      have h\u2086 : Filter.Tendsto (fun n \u21a6 u (n + 1)) Filter.atTop (\ud835\udcdd 9) := h\u2082\n      have h\u2087 : Filter.Tendsto (fun n \u21a6 u (n + 1)) Filter.atTop (\ud835\udcdd (NNReal.sqrt (x + 9))) := by\n        convert h\u2085 using 1\n        ext n\n        simp [h\u2080]\n      have h\u2088 : 9 = NNReal.sqrt (x + 9) := by\n        apply tendsto_nhds_unique h\u2086 h\u2087\n      exact h\u2088\n    exact h\u2084\n  rw [h\u2084.symm]\n  <;> simp_all\n  <;> linarith\n  <;> simp_all\n  <;> linarith\n  <;> simp_all\n  <;> linarith\n  <;> simp_all\n  <;> linarith\n  <;> simp_all\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 40, "alphabet": 21, "compiles": false, "id": "mathd_numbertheory_221", "proof_after": "theorem mathd_numbertheory_221 (S : Finset \u2115)\n  (h\u2080 : \u2200 x : \u2115, x \u2208 S \u2194 0 < x \u2227 x < 1000 \u2227 x.divisors.card = 3) : S.card = 11  :=  by\n  have h\u2081 : Finset.card (Finset.filter Nat.Prime (Finset.Icc 2 31)) = 11 := by\n    rfl\n    <;> decide\n    <;> simp_all [Finset.card, Finset.filter, Nat.divisors]\n    <;> rfl\n    <;> decide\n    <;> decide\n    <;> decide\n  have h\u2082 : \u2200 p : \u2115, p \u2208 Finset.filter Nat.Prime (Finset.Icc 2 31) \u2192 p^2 < 1000 := by\n    intro p hp\n    rw [Finset.mem_filter, Finset.mem_Icc] at hp\n    have h\u2082 := h\u2081\n    have h\u2083 : p \u2264 31 := by linarith\n    have h\u2084 : p \u2265 2 := by linarith\n    have h\u2085 : p^2 < 1000 := by\n      interval_cases p <;> norm_num\n    exact h\u2085\n  have h\u2083 : \u2200 p : \u2115, p \u2208 Finset.filter Nat.Prime (Finset.Icc 2 31) \u2192 (p^2).divisors.card = 3 := by\n    intro p hp\n    have h\u2083 := h\u2082 p hp\n    simp_all [Nat.divisors_prime_pow]\n    <;> norm_num\n    <;> decide\n  have h\u2084 : S.card = Finset.card (Finset.filter Nat.Prime (Finset.Icc 2 31)) := by\n    have : S = Finset.filter (fun x => x.divisors.card = 3) (Finset.Icc 1 999) := by\n      ext x\n      simp [h\u2080, Nat.divisors]\n      <;> omega\n    simp_all only [Finset.card_filter, Finset.Icc_self, Finset.card_empty, Nat.cast_ofNat,\n      Finset.card_range, Nat.cast_one, Finset.card_filter, Finset.Icc_self, Finset.card_empty,\n      Nat.cast_ofNat, Finset.card_range, Nat.cast_one]\n    rfl\n  rw [h\u2084]\n  simp_all\n  <;> decide\n  <;> rfl\n  <;> simp_all\n  <;> decide\n  <;> rfl\n", "proof_before": "theorem mathd_numbertheory_221 (S : Finset \u2115)\n  (h\u2080 : \u2200 x : \u2115, x \u2208 S \u2194 0 < x \u2227 x < 1000 \u2227 x.divisors.card = 3) : S.card = 11  :=  by\n  have h\u2081 : Finset.card (Finset.filter Nat.Prime (Finset.Icc 2 31)) = 11 := by\n    rfl\n    <;> decide\n    <;> simp_all [Finset.card, Finset.filter, Nat.divisors]\n    <;> rfl\n    <;> decide\n    <;> decide\n    <;> decide\n  have h\u2082 : \u2200 p : \u2115, p \u2208 Finset.filter Nat.Prime (Finset.Icc 2 31) \u2192 p^2 < 1000 := by\n    intro p hp\n    rw [Finset.mem_filter, Finset.mem_Icc] at hp\n    have h\u2082 := h\u2081\n    have h\u2083 : p \u2264 31 := by linarith\n    have h\u2084 : p \u2265 2 := by linarith\n    have h\u2085 : p^2 < 1000 := by\n      interval_cases p <;> norm_num\n    exact h\u2085\n  have h\u2083 : \u2200 p : \u2115, p \u2208 Finset.filter Nat.Prime (Finset.Icc 2 31) \u2192 (p^2).divisors.card = 3 := by\n    intro p hp\n    have h\u2083 := h\u2082 p hp\n    simp_all [Nat.divisors_prime_pow]\n    <;> norm_num\n    <;> decide\n  have h\u2084 : S.card = Finset.card (Finset.filter Nat.Prime (Finset.Icc 2 31)) := by\n    have : S = Finset.filter (fun x => x.divisors.card = 3) (Finset.Icc 1 999) := by\n      ext x\n      simp [h\u2080, Nat.divisors]\n      <;> omega\n    simp_all only [Finset.card_filter, Finset.Icc_self, Finset.card_empty, Nat.cast_ofNat,\n      Finset.card_range, Nat.cast_one, Finset.card_filter, Finset.Icc_self, Finset.card_empty,\n      Nat.cast_ofNat, Finset.card_range, Nat.cast_one]\n    rfl\n  rw [h\u2084]\n  simp_all\n  <;> decide\n  <;> rfl\n  <;> simp_all\n  <;> decide\n  <;> rfl\n"}
{"mode": "clear", "steps_before": 18, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 525, "chars_after": 8, "seconds": 6.7, "id": "mathd_numbertheory_33", "proof_after": "theorem mathd_numbertheory_33 (n : \u2115) (h\u2080 : n < 398) (h\u2081 : n * 7 % 398 = 1) : n = 57  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_33 (n : \u2115) (h\u2080 : n < 398) (h\u2081 : n * 7 % 398 = 1) : n = 57  :=  by\n  have step1 : 7 * 57 % 398 = 1 := by\n    norm_num [Nat.mod_eq_of_lt (show 399 < 398 * 2 by norm_num)]\n    <;> omega\n  have step2 : \u2200 (m : \u2115), 0 \u2264 m \u2227 m < 398 \u2227 m * 7 % 398 = 1 \u2192 m = 57 := by\n    intro m hm\n    rcases hm with \u27e8hm\u2080, hm\u2081, hm\u2082\u27e9\n    omega\n  have step3 : \u2203 (n : \u2115), n < 398 \u2227 n * 7 % 398 = 1 := by\n    have h\u2081 : 7 * 57 % 398 = 1 := by simpa using step1\n    exact \u27e857, by norm_num, h\u2081\u27e9\n  have conclusion : n = 57 := by\n    have h\u2082 := step2 n\n    simp_all\n    <;> omega\n  rw [conclusion]\n  <;> simp_all\n  <;> omega\n"}
{"mode": "clear", "steps_before": 32, "alphabet": 23, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 32, "lower_bound": null, "exhausted_depth": 2, "certified": false, "rounds": 2, "timed_out": false, "fixpoint": true, "chars_before": 876, "chars_after": 876, "seconds": 229.5, "id": "amc12b_2002_p3", "proof_after": "theorem amc12b_2002_p3 (S : Finset \u2115)\n  (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 0 < n \u2227 Nat.Prime (n ^ 2 + 2 - 3 * n)) :\n  S.card = 1  :=  by\n  have h\u2081 : S = {3} := by\n    apply Finset.ext\n    intro n\n    simp only [h\u2080, Finset.mem_singleton]\n    constructor\n    \u00b7 intro h\n      have h\u2081 := h.1\n      have h\u2082 := h.2\n      have h\u2083 : n \u2264 3 := by\n        contrapose! h\u2082\n        have : n ^ 2 + 2 - 3 * n = (n - 1) * (n - 2) := by\n          cases n with\n          | zero => simp_all\n          | succ n =>\n            cases n with\n            | zero => simp_all\n            | succ n =>\n              simp_all [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.pow_succ]\n              <;> ring_nf\n              <;> omega\n        simp_all [Nat.prime_mul_iff]\n        <;> omega\n      interval_cases n <;> simp_all [Nat.Prime]\n    \u00b7 intro h\n      rw [h]\n      norm_num\n      <;> decide\n  have h\u2082 : S.card = 1 := by\n    simp_all only [Finset.card_singleton, eq_self_iff_true, and_true]\n    <;> rfl\n  simpa [h\u2081, h\u2082] using h\u2082\n", "proof_before": "theorem amc12b_2002_p3 (S : Finset \u2115)\n  (h\u2080 : \u2200 n : \u2115, n \u2208 S \u2194 0 < n \u2227 Nat.Prime (n ^ 2 + 2 - 3 * n)) :\n  S.card = 1  :=  by\n  have h\u2081 : S = {3} := by\n    apply Finset.ext\n    intro n\n    simp only [h\u2080, Finset.mem_singleton]\n    constructor\n    \u00b7 intro h\n      have h\u2081 := h.1\n      have h\u2082 := h.2\n      have h\u2083 : n \u2264 3 := by\n        contrapose! h\u2082\n        have : n ^ 2 + 2 - 3 * n = (n - 1) * (n - 2) := by\n          cases n with\n          | zero => simp_all\n          | succ n =>\n            cases n with\n            | zero => simp_all\n            | succ n =>\n              simp_all [Nat.mul_sub_left_distrib, Nat.mul_sub_right_distrib, Nat.pow_succ]\n              <;> ring_nf\n              <;> omega\n        simp_all [Nat.prime_mul_iff]\n        <;> omega\n      interval_cases n <;> simp_all [Nat.Prime]\n    \u00b7 intro h\n      rw [h]\n      norm_num\n      <;> decide\n  have h\u2082 : S.card = 1 := by\n    simp_all only [Finset.card_singleton, eq_self_iff_true, and_true]\n    <;> rfl\n  simpa [h\u2081, h\u2082] using h\u2082\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 54, "alphabet": 26, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 1571, "chars_after": 6, "seconds": 9.5, "id": "mathd_numbertheory_155", "proof_after": "theorem mathd_numbertheory_155 :\n  Finset.card (Finset.filter (fun x => x % 19 = 7) (Finset.Icc 100 999)) = 48  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_155 :\n  Finset.card (Finset.filter (fun x => x % 19 = 7) (Finset.Icc 100 999)) = 48  :=  by\n  have h1 : \u2200 (n : \u2115), n \u2208 Finset.Icc 100 999 \u2227 n % 19 = 7 \u2194 \u2203 k : \u2115, n = 19 * k + 7 \u2227 100 \u2264 n \u2227 n \u2264 999 := by\n    intro n\n    constructor\n    \u00b7 intro h\n      rw [Finset.mem_Icc] at h\n      use (n - 7) / 19\n      constructor\n      \u00b7 omega\n      \u00b7 constructor\n        \u00b7 omega\n        \u00b7 omega\n    \u00b7 rintro \u27e8k, hk1, hk2, hk3\u27e9\n      rw [Finset.mem_Icc]\n      constructor\n      \u00b7 omega\n      \u00b7 omega\n  have h2 : \u2203 k : \u2115, 19 * k + 7 \u2265 100 := by\n    use 5\n    norm_num\n    <;> aesop\n  have h3 : 5 = Nat.ceil ((100 - 7) / 19 : \u211a) := by\n    have h3 : \u2200 (n : \u2115), n \u2208 Finset.Icc 100 999 \u2227 n % 19 = 7 \u2194 \u2203 k : \u2115, n = 19 * k + 7 \u2227 100 \u2264 n \u2227 n \u2264 999 := h1\n    have h4 : \u2203 k : \u2115, 19 * k + 7 \u2265 100 := h2\n    apply Eq.symm\n    norm_num\n    <;> simp_all [Finset.mem_Icc, Nat.ceil_eq_iff]\n    <;> norm_num\n    <;> omega\n  have h4 : \u2200 k : \u2115, k \u2265 5 \u2192 19 * k + 7 \u2265 100 := by\n    intro k hk\n    norm_num at h3\n    omega\n  have h5 : \u2203 k : \u2115, 19 * k + 7 \u2264 999 := by\n    use 52\n    norm_num\n  have h6 : 52 = Nat.floor ((999 - 7) / 19 : \u211a) := by\n    apply Eq.symm\n    norm_num [Nat.floor_eq_iff, Nat.ceil_eq_iff]\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  have h7 : \u2200 k : \u2115, k \u2264 52 \u2192 19 * k + 7 \u2264 999 := by\n    intro k hk\n    norm_num [Nat.floor_le, Nat.le_floor] at h6 \u22a2\n    <;> omega\n  have h8 : 52 - 5 + 1 = 48 := by\n    norm_num\n    <;> omega\n  have h9 : Finset.card (Finset.filter (fun x => x % 19 = 7) (Finset.Icc 100 999)) = 48 := by\n    simp_all [Finset.card, Finset.filter, Finset.Icc, Nat.ceil_eq_iff, Nat.floor_eq_iff]\n    <;> aesop\n  apply h9\n"}
{"mode": "clear", "steps_before": 56, "alphabet": 36, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 56, "lower_bound": null, "exhausted_depth": 1, "certified": false, "rounds": 2, "timed_out": true, "fixpoint": false, "chars_before": 1500, "chars_after": 1500, "seconds": 582.2, "id": "mathd_algebra_101", "proof_after": "theorem mathd_algebra_101 (x : \u211d) (h\u2080 : x ^ 2 - 5 * x - 4 \u2264 10) : x \u2265 -2 \u2227 x \u2264 7  :=  by\n  have h\u2081 : x ^ 2 - 5 * x - 14 \u2264 0 := by\n    nlinarith [sq_nonneg (x - 5 / 2)]\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n  have h\u2082 : x ^ 2 - 5 * x - 14 = 0 \u2192 x = 7 \u2228 x = -2 := by\n    intro h\n    apply or_iff_not_imp_right.mpr\n    intro h\u2082\n    apply eq_of_sub_eq_zero\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2082)\n    nlinarith\n  have h\u2083 : x < -2 \u2228 -2 \u2264 x \u2227 x \u2264 7 \u2228 x > 7 := by\n    by_cases h\u2083 : x < -2 <;>\n      by_cases h\u2084 : x > 7 <;>\n        by_cases h\u2085 : x = -2 <;>\n          by_cases h\u2086 : x = 7 <;>\n            simp_all [h\u2082, le_of_lt, le_of_lt, le_of_lt, le_of_lt, le_of_lt, le_of_lt] <;>\n            nlinarith\n    <;>\n      nlinarith\n  have h\u2084 : x < -2 \u2192 x ^ 2 - 5 * x - 14 > 0 := by\n    intro h\n    have h\u2084 := h\u2082\n    have h\u2085 := h\u2083\n    cases' h\u2085 with h\u2085 h\u2085\n    . nlinarith [sq_nonneg (x + 2), sq_nonneg (x - 7)]\n    . cases' h\u2085 with h\u2085 h\u2085\n      . nlinarith [sq_nonneg (x + 2), sq_nonneg (x - 7)]\n      . nlinarith [sq_nonneg (x + 2), sq_nonneg (x - 7)]\n  have h\u2085 : -2 \u2264 x \u2227 x \u2264 7 \u2192 x ^ 2 - 5 * x - 14 \u2264 0 := by\n    rintro \u27e8h\u2085, h\u2086\u27e9\n    nlinarith [h\u2081, h\u2085, h\u2086]\n  have h\u2086 : x > 7 \u2192 x ^ 2 - 5 * x - 14 > 0 := by\n    intro hx\n    have h\u2086 : x > 7 \u2192 x ^ 2 - 5 * x - 14 > 0 := by\n      intro hx\n      have h\u2087 : x > 7 := hx\n      have h\u2088 : x ^ 2 - 5 * x - 14 > 0 := by\n        nlinarith\n      exact h\u2088\n    exact h\u2086 hx\n  have h\u2087 : x \u2265 -2 \u2227 x \u2264 7 := by\n    refine' \u27e8_, _\u27e9\n    all_goals\n      rcases h\u2083 with (h\u2083 | h\u2083 | h\u2083) <;>\n      simp_all [h\u2082, h\u2084, h\u2085, h\u2086, sq] <;>\n      nlinarith\n  exact h\u2087\n", "proof_before": "theorem mathd_algebra_101 (x : \u211d) (h\u2080 : x ^ 2 - 5 * x - 4 \u2264 10) : x \u2265 -2 \u2227 x \u2264 7  :=  by\n  have h\u2081 : x ^ 2 - 5 * x - 14 \u2264 0 := by\n    nlinarith [sq_nonneg (x - 5 / 2)]\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n  have h\u2082 : x ^ 2 - 5 * x - 14 = 0 \u2192 x = 7 \u2228 x = -2 := by\n    intro h\n    apply or_iff_not_imp_right.mpr\n    intro h\u2082\n    apply eq_of_sub_eq_zero\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2082)\n    nlinarith\n  have h\u2083 : x < -2 \u2228 -2 \u2264 x \u2227 x \u2264 7 \u2228 x > 7 := by\n    by_cases h\u2083 : x < -2 <;>\n      by_cases h\u2084 : x > 7 <;>\n        by_cases h\u2085 : x = -2 <;>\n          by_cases h\u2086 : x = 7 <;>\n            simp_all [h\u2082, le_of_lt, le_of_lt, le_of_lt, le_of_lt, le_of_lt, le_of_lt] <;>\n            nlinarith\n    <;>\n      nlinarith\n  have h\u2084 : x < -2 \u2192 x ^ 2 - 5 * x - 14 > 0 := by\n    intro h\n    have h\u2084 := h\u2082\n    have h\u2085 := h\u2083\n    cases' h\u2085 with h\u2085 h\u2085\n    . nlinarith [sq_nonneg (x + 2), sq_nonneg (x - 7)]\n    . cases' h\u2085 with h\u2085 h\u2085\n      . nlinarith [sq_nonneg (x + 2), sq_nonneg (x - 7)]\n      . nlinarith [sq_nonneg (x + 2), sq_nonneg (x - 7)]\n  have h\u2085 : -2 \u2264 x \u2227 x \u2264 7 \u2192 x ^ 2 - 5 * x - 14 \u2264 0 := by\n    rintro \u27e8h\u2085, h\u2086\u27e9\n    nlinarith [h\u2081, h\u2085, h\u2086]\n  have h\u2086 : x > 7 \u2192 x ^ 2 - 5 * x - 14 > 0 := by\n    intro hx\n    have h\u2086 : x > 7 \u2192 x ^ 2 - 5 * x - 14 > 0 := by\n      intro hx\n      have h\u2087 : x > 7 := hx\n      have h\u2088 : x ^ 2 - 5 * x - 14 > 0 := by\n        nlinarith\n      exact h\u2088\n    exact h\u2086 hx\n  have h\u2087 : x \u2265 -2 \u2227 x \u2264 7 := by\n    refine' \u27e8_, _\u27e9\n    all_goals\n      rcases h\u2083 with (h\u2083 | h\u2083 | h\u2083) <;>\n      simp_all [h\u2082, h\u2084, h\u2085, h\u2086, sq] <;>\n      nlinarith\n  exact h\u2087\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 25, "alphabet": 19, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 577, "chars_after": 11, "seconds": 158.6, "id": "mathd_algebra_104", "proof_after": "theorem mathd_algebra_104 (x : \u211d) (h\u2080 : 125 / 8 = x / 12) : x = 375 / 2  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_104 (x : \u211d) (h\u2080 : 125 / 8 = x / 12) : x = 375 / 2  :=  by\n  have h\u2081 : 125 / 8 * 12 = x := by\n    have h\u2081 : x = 125 / 8 * 12 := by\n      apply Eq.symm\n      field_simp at h\u2080\n      linarith\n    rw [h\u2081]\n  have h\u2082 : 125 * 12 = 8 * x := by\n    rw [mul_comm] at h\u2081\n    linarith\n  have h\u2083 : 1500 = 8 * x := by\n    have h\u2083 : 125 / 8 * 12 = x := h\u2081\n    norm_num at h\u2083\n    linarith\n  have h\u2084 : x = 1500 / 8 := by\n    have h\u2084 : x = 1500 / 8 := by\n      apply Eq.symm\n      field_simp at h\u2080 \u22a2\n      linarith\n    exact h\u2084\n  have h\u2085 : x = 375 / 2 := by\n    have h\u2085 : x = 375 / 2 := by\n      field_simp at h\u2080 \u22a2\n      linarith\n    linarith\n  linarith\n"}
{"mode": "clear", "steps_before": 19, "alphabet": 13, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": null, "exhausted_depth": 2, "certified": false, "rounds": 5, "timed_out": false, "fixpoint": true, "chars_before": 750, "chars_after": 93, "seconds": 415.1, "id": "mathd_algebra_67", "proof_after": "theorem mathd_algebra_67 (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 5 * x + 3) (h\u2081 : \u2200 x, g x = x ^ 2 - 2) :\n    g (f (-1)) = 2  :=  by\n  simp_all only [rpow_two, mul_neg, mul_one, mul_zero, mul_neg_one, mul_comm]\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_67 (f g : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 5 * x + 3) (h\u2081 : \u2200 x, g x = x ^ 2 - 2) :\n    g (f (-1)) = 2  :=  by\n  have hf_minus1 : f (-1) = 5 * (-1) + 3 := by\n    simp_all\n    <;> ring\n    <;> norm_num\n    <;> linarith\n  have f_minus1_value : f (-1) = -2 := by\n    simp_all only [rpow_two, mul_neg, mul_one, mul_add, mul_sub, mul_comm, mul_left_comm,\n      mul_assoc, sub_eq_add_neg, add_assoc]\n    <;> linarith\n  have hg_minus2 : g (-2) = (-2) ^ 2 - 2 := by\n    simp_all only [h\u2080, h\u2081, mul_neg, mul_one, mul_add, mul_sub, mul_comm]\n    <;> norm_num\n  have g_minus2_value : g (-2) = 2 := by\n    simp_all only [rpow_two, mul_neg, mul_one, mul_add, mul_sub, mul_comm]\n    linarith\n  have final_result : g (f (-1)) = 2 := by\n    rw [f_minus1_value]\n    rw [g_minus2_value]\n  simp_all only [rpow_two, mul_neg, mul_one, mul_zero, mul_neg_one, mul_comm]\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 9, "alphabet": 17, "compiles": false, "id": "mathd_algebra_480", "proof_after": "theorem mathd_algebra_480 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x < 0, f x = -x ^ 2 - 1)\n  (h\u2081 : \u2200 x, 0 \u2264 x \u2227 x < 4 \u2192 f x = 2) (h\u2082 : \u2200 x \u2265 4, f x = Real.sqrt x) : f Real.pi = 2  :=  by\n  have h : Real.pi \u2265 0 := Real.pi_pos.le\n  by_cases h\u2081 : Real.pi < 4\n  <;> by_cases h\u2082 : Real.pi < 0\n  <;> by_cases h\u2083 : Real.pi \u2265 4\n  <;> simp_all [h\u2080, h\u2081, h\u2082, h\u2083, Real.sqrt_eq_iff_sq_eq]\n  <;> nlinarith [Real.pi_le_four]\n  <;> nlinarith [Real.pi_pos]\n  <;> nlinarith [Real.pi_le_four]\n  <;> nlinarith [Real.pi_pos]\n", "proof_before": "theorem mathd_algebra_480 (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x < 0, f x = -x ^ 2 - 1)\n  (h\u2081 : \u2200 x, 0 \u2264 x \u2227 x < 4 \u2192 f x = 2) (h\u2082 : \u2200 x \u2265 4, f x = Real.sqrt x) : f Real.pi = 2  :=  by\n  have h : Real.pi \u2265 0 := Real.pi_pos.le\n  by_cases h\u2081 : Real.pi < 4\n  <;> by_cases h\u2082 : Real.pi < 0\n  <;> by_cases h\u2083 : Real.pi \u2265 4\n  <;> simp_all [h\u2080, h\u2081, h\u2082, h\u2083, Real.sqrt_eq_iff_sq_eq]\n  <;> nlinarith [Real.pi_le_four]\n  <;> nlinarith [Real.pi_pos]\n  <;> nlinarith [Real.pi_le_four]\n  <;> nlinarith [Real.pi_pos]\n"}
{"mode": "clear", "steps_before": 40, "alphabet": 23, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 1174, "chars_after": 11, "seconds": 9.8, "id": "mathd_algebra_96", "proof_after": "theorem mathd_algebra_96 (x y z a : \u211d) (h\u2080 : 0 < x \u2227 0 < y \u2227 0 < z)\n  (h\u2081 : Real.log x - Real.log y = a) (h\u2082 : Real.log y - Real.log z = 15)\n  (h\u2083 : Real.log z - Real.log x = -7) : a = -8  :=  by\n  linarith\n", "proof_before": "theorem mathd_algebra_96 (x y z a : \u211d) (h\u2080 : 0 < x \u2227 0 < y \u2227 0 < z)\n  (h\u2081 : Real.log x - Real.log y = a) (h\u2082 : Real.log y - Real.log z = 15)\n  (h\u2083 : Real.log z - Real.log x = -7) : a = -8  :=  by\n  have h\u2084 : Real.log z = Real.log x - 7 := by\n    ring_nf at h\u2081 h\u2082 h\u2083 \u22a2\n    have h\u2084 : Real.log z = Real.log x - 7 := by linarith\n    linarith\n  have h\u2085 : Real.log y = Real.log z + 15 := by\n    rw [\u2190 sub_eq_zero] at h\u2081 h\u2082 h\u2083 h\u2084 \u22a2\n    ring_nf at h\u2081 h\u2082 h\u2083 h\u2084 \u22a2\n    linarith [h\u2080.1, h\u2080.2.1, h\u2080.2.2, h\u2081, h\u2082, h\u2083, h\u2084]\n  have h\u2086 : Real.log y = (Real.log x - 7) + 15 := by\n    have h\u2086 : Real.log y = (Real.log x - 7) + 15 := by\n      linarith [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085]\n    linarith [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086]\n  have h\u2087 : Real.log y = Real.log x + 8 := by\n    rw [eq_comm] at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086\n    ring_nf at h\u2081 h\u2082 h\u2083 h\u2084 h\u2085 h\u2086 \u22a2\n    linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n    <;> linarith\n  have h\u2088 : Real.log x = Real.log y + a := by\n    rw [h\u2087] at h\u2081\n    linarith\n  have h\u2089 : Real.log x = (Real.log x + 8) + a := by\n    have h\u2084 : Real.log x = Real.log y + a := by linarith\n    have h\u2085 : Real.log y = Real.log z + 15 := by linarith\n    have h\u2086 : Real.log z = Real.log x - 7 := by linarith\n    linarith\n  have h\u2081\u2080 : 0 = 8 + a := by\n    ring_nf at *\n    linarith [Real.log_one]\n  have h\u2081\u2081 : a = -8 := by\n    have h\u2081\u2081 : a = -8 := by\n      linarith\n    exact h\u2081\u2081\n  linarith\n"}
{"mode": "clear", "steps_before": 55, "alphabet": 31, "compiles": false, "id": "mathd_algebra_422", "proof_after": "theorem mathd_algebra_422 (x : \u211d) (\u03c3 : Equiv \u211d \u211d) (h\u2080 : \u2200 x, \u03c3.1 x = 5 * x - 12)\n  (h\u2081 : \u03c3.1 (x + 1) = \u03c3.2 x) : x = 47 / 24  :=  by\n  have inv_def : \u2200 y, \u03c3.2 y = (y + 12) / 5 := by\n    intro y\n    have h\u2082 := h\u2080 (\u03c3.2 y)\n    have h\u2083 := h\u2080 (x + 1)\n    have h\u2084 := h\u2081\n    simp_all [Equiv.symm_apply_eq]\n    <;> ring_nf\n    <;> linarith\n  have eqn : \u03c3.2 x = \u03c3.1 (x + 1) := by\n    simp_all [Equiv.eq_symm_apply]\n    <;> norm_num\n    <;> ring\n    <;> linarith\n  have subst_eqn : (x + 12) / 5 = 5 * (x + 1) - 12 := by\n    field_simp [h\u2080, h\u2081, inv_def] at eqn \u22a2\n    ring_nf at eqn \u22a2\n    linarith\n  have simplified_eqn : (x + 12) / 5 = 5 * x + 5 - 12 := by\n    rw [h\u2080] at h\u2081; simp_all [Equiv.eq_symm_apply]\n    <;> field_simp [mul_comm] at *\n    <;> ring_nf at *\n    <;> linarith\n  have further_simplified : (x + 12) / 5 = 5 * x - 7 := by\n    field_simp [h\u2080, h\u2081, inv_def, eq_comm] at subst_eqn \u22a2\n    linarith\n  have multiplied_eqn : x + 12 = 5 * (5 * x - 7) := by\n    field_simp [h\u2080, inv_def] at eqn subst_eqn simplified_eqn further_simplified\n    linarith\n  have expanded_eqn : x + 12 = 25 * x - 35 := by\n    field_simp [h\u2080, Equiv.eq_symm_apply] at *\n    linarith\n  have rearranged_eqn : 12 + 35 = 25 * x - x := by\n    linarith [h\u2080 (x + 1), h\u2080 x, h\u2081, inv_def x, eqn, subst_eqn, simplified_eqn, further_simplified, multiplied_eqn, expanded_eqn]\n  have combined_terms : 47 = 24 * x := by\n    rw [eq_comm] at rearranged_eqn\n    linarith\n    <;> simp_all [Equiv.eq_symm_apply]\n    <;> linarith\n  have solution : x = 47 / 24 := by\n    have h\u2082 := h\u2080 (x + 1)\n    have h\u2083 := h\u2080 x\n    simp_all [Equiv.eq_symm_apply]\n    <;> ring_nf\n    <;> linarith\n    <;> assumption\n  have verification : \u03c3.2 (47 / 24) = \u03c3.1 ((47 / 24) + 1) := by\n    simp [h\u2080, h\u2081, inv_def, solution] at *\n    <;> norm_num\n    <;> ring\n    <;> linarith\n  apply Eq.symm\n  apply Eq.symm\n  field_simp [h\u2080, inv_def, Equiv.eq_symm_apply] at *\n  linarith\n", "proof_before": "theorem mathd_algebra_422 (x : \u211d) (\u03c3 : Equiv \u211d \u211d) (h\u2080 : \u2200 x, \u03c3.1 x = 5 * x - 12)\n  (h\u2081 : \u03c3.1 (x + 1) = \u03c3.2 x) : x = 47 / 24  :=  by\n  have inv_def : \u2200 y, \u03c3.2 y = (y + 12) / 5 := by\n    intro y\n    have h\u2082 := h\u2080 (\u03c3.2 y)\n    have h\u2083 := h\u2080 (x + 1)\n    have h\u2084 := h\u2081\n    simp_all [Equiv.symm_apply_eq]\n    <;> ring_nf\n    <;> linarith\n  have eqn : \u03c3.2 x = \u03c3.1 (x + 1) := by\n    simp_all [Equiv.eq_symm_apply]\n    <;> norm_num\n    <;> ring\n    <;> linarith\n  have subst_eqn : (x + 12) / 5 = 5 * (x + 1) - 12 := by\n    field_simp [h\u2080, h\u2081, inv_def] at eqn \u22a2\n    ring_nf at eqn \u22a2\n    linarith\n  have simplified_eqn : (x + 12) / 5 = 5 * x + 5 - 12 := by\n    rw [h\u2080] at h\u2081; simp_all [Equiv.eq_symm_apply]\n    <;> field_simp [mul_comm] at *\n    <;> ring_nf at *\n    <;> linarith\n  have further_simplified : (x + 12) / 5 = 5 * x - 7 := by\n    field_simp [h\u2080, h\u2081, inv_def, eq_comm] at subst_eqn \u22a2\n    linarith\n  have multiplied_eqn : x + 12 = 5 * (5 * x - 7) := by\n    field_simp [h\u2080, inv_def] at eqn subst_eqn simplified_eqn further_simplified\n    linarith\n  have expanded_eqn : x + 12 = 25 * x - 35 := by\n    field_simp [h\u2080, Equiv.eq_symm_apply] at *\n    linarith\n  have rearranged_eqn : 12 + 35 = 25 * x - x := by\n    linarith [h\u2080 (x + 1), h\u2080 x, h\u2081, inv_def x, eqn, subst_eqn, simplified_eqn, further_simplified, multiplied_eqn, expanded_eqn]\n  have combined_terms : 47 = 24 * x := by\n    rw [eq_comm] at rearranged_eqn\n    linarith\n    <;> simp_all [Equiv.eq_symm_apply]\n    <;> linarith\n  have solution : x = 47 / 24 := by\n    have h\u2082 := h\u2080 (x + 1)\n    have h\u2083 := h\u2080 x\n    simp_all [Equiv.eq_symm_apply]\n    <;> ring_nf\n    <;> linarith\n    <;> assumption\n  have verification : \u03c3.2 (47 / 24) = \u03c3.1 ((47 / 24) + 1) := by\n    simp [h\u2080, h\u2081, inv_def, solution] at *\n    <;> norm_num\n    <;> ring\n    <;> linarith\n  apply Eq.symm\n  apply Eq.symm\n  field_simp [h\u2080, inv_def, Equiv.eq_symm_apply] at *\n  linarith\n"}
{"mode": "clear", "steps_before": 15, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 484, "chars_after": 91, "seconds": 27.9, "id": "mathd_algebra_28", "proof_after": "theorem mathd_algebra_28 (c : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 2 * x ^ 2 + 5 * x + c)\n  (h\u2081 : \u2203 x, f x \u2264 0) : c \u2264 25 / 8  :=  by\n  cases' h\u2081 with x hx\n  linarith [sq_nonneg (x + (5 : \u211d) / 4), h\u2080 x, h\u2080 (x + (5 : \u211d) / 4)]\n", "proof_before": "theorem mathd_algebra_28 (c : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 2 * x ^ 2 + 5 * x + c)\n  (h\u2081 : \u2203 x, f x \u2264 0) : c \u2264 25 / 8  :=  by\n  have D_eq : 5^2 - 4 * 2 * c = 25 - 8 * c := by\n    norm_num\n    <;> ring_nf\n    <;> simp_all [mul_assoc]\n    <;> norm_num\n    <;> linarith\n  have D_nonneg : 25 - 8 * c \u2265 0 := by\n    have h\u2082 := h\u2081\n    cases' h\u2082 with x hx\n    simp only [h\u2080, mul_assoc, mul_comm, mul_left_comm] at hx\n    linarith [sq_nonneg (x + (5 / 4)), sq_nonneg (x - (5 / 4))]\n  have c_le : c \u2264 25 / 8 := by\n    cases' h\u2081 with x hx\n    linarith [sq_nonneg (x + (5 : \u211d) / 4), h\u2080 x, h\u2080 (x + (5 : \u211d) / 4)]\n  linarith\n"}
{"mode": "clear", "steps_before": 102, "alphabet": 57, "compiles": false, "id": "amc12a_2009_p15", "proof_after": "theorem amc12a_2009_p15 (n : \u2115) (h\u2080 : 0 < n)\n  (h\u2081 : (\u2211 k in Finset.Icc 1 n, \u2191k * Complex.I ^ k) = 48 + 49 * Complex.I) : n = 97  :=  by\n  have h_periodicity : \u2200 k : \u2115, Complex.I ^ (k % 4) = Complex.I ^ k := by\n    intro k\n    have h\u2082 : \u2200 n : \u2115, Complex.I ^ n = Complex.I ^ (n % 4) := by\n      intro n\n      rw [\u2190 Nat.mod_add_div n 4]\n      simp [pow_add, pow_mul, Complex.I_mul_I, mul_assoc, mul_comm, mul_left_comm]\n    rw [h\u2082 k]\n  have h_grouped_blocks : \u2200 m : \u2115, \u2211 k in Finset.Icc (4 * m + 1) (4 * m + 4), \u2191k * Complex.I ^ k = 2 - 2 * Complex.I := by\n    intro m\n    simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three, Complex.I_mul_I]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three, Complex.I_mul_I]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three, Complex.I_mul_I]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three, Complex.I_mul_I]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three, Complex.I_mul_I]\n    <;> ring_nf\n    <;> norm_num\n  have h_sum_multiple_of_4 : \u2200 m : \u2115, \u2211 k in Finset.Icc 1 (4 * m), \u2191k * Complex.I ^ k = 2 * m - 2 * m * Complex.I := by\n    intro m\n    have h_sum_group := h_grouped_blocks m\n    have h_sum_periodicity := h_periodicity\n    clear h_periodicity\n    induction m <;> simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three]\n    <;> ring_nf at * <;> norm_num <;> simp_all [Complex.ext_iff]\n    <;> norm_num <;> linarith\n  have h_contradiction_multiple_of_4 : \u00ac\u2203 m : \u2115, n = 4 * m := by\n    intro h\n    obtain \u27e8m, rfl\u27e9 := h\n    have h\u2082 := h_sum_multiple_of_4 0\n    have h\u2083 := h_sum_multiple_of_4 1\n    have h\u2084 := h_sum_multiple_of_4 2\n    have h\u2085 := h_sum_multiple_of_4 3\n    have h\u2086 := h_grouped_blocks 0\n    have h\u2087 := h_grouped_blocks 1\n    have h\u2088 := h_grouped_blocks 2\n    have h\u2089 := h_grouped_blocks 3\n    simp_all [Finset.sum_Icc_succ_top, Nat.mul_div_cancel_left]\n    <;> ring_nf at *\n    <;> simp_all [Complex.ext_iff]\n    <;> norm_num\n    <;> linarith\n  have h_n_not_multiple_of_4 : \u00ac\u2203 m : \u2115, n = 4 * m := by\n    intro h\n    obtain \u27e8m, hm\u27e9 := h\n    have h\u2082 := h_sum_multiple_of_4 m\n    have h\u2083 := h_grouped_blocks m\n    have h\u2084 := h_periodicity 0\n    have h\u2085 := h_periodicity 1\n    have h\u2086 := h_periodicity 2\n    have h\u2087 := h_periodicity 3\n    simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three]\n    all_goals\n      linarith\n  have h_form_n_plus_1 : \u2203 m : \u2115, n = 4 * m + 1 := by\n    have h\u2082 : n % 4 = 1 := by\n      by_contra! h\n      have h\u2083 : \u2203 m : \u2115, n = 4 * m \u2228 n = 4 * m + 2 \u2228 n = 4 * m + 3 := by\n        have : n % 4 = 0 \u2228 n % 4 = 1 \u2228 n % 4 = 2 \u2228 n % 4 = 3 := by omega\n        rcases this with (h\u2084 | h\u2084 | h\u2084 | h\u2084) <;> use n / 4 <;> omega\n      rcases h\u2083 with \u27e8m, h\u2083 | h\u2083 | h\u2083\u27e9 <;> simp_all [Finset.sum_Icc_succ_top, Complex.ext_iff, pow_add,\n        pow_mul, pow_two, pow_three]\n      <;> linarith [h_sum_multiple_of_4 0, h_sum_multiple_of_4 1, h_sum_multiple_of_4 2,\n        h_sum_multiple_of_4 3]\n    use n / 4\n    omega\n  have h_sum_n_plus_1 : \u2200 m : \u2115, \u2211 k in Finset.Icc 1 (4 * m + 1), \u2191k * Complex.I ^ k = 2 * m + (2 * m + 1) * Complex.I := by\n    intro m\n    have h\u2082 := h_grouped_blocks m\n    have h\u2083 := h_grouped_blocks 0\n    have h\u2084 := h_sum_multiple_of_4 m\n    have h\u2085 := h_sum_multiple_of_4 0\n    have h\u2086 := h_sum_multiple_of_4 1\n    have h\u2087 := h_sum_multiple_of_4 2\n    have h\u2088 := h_grouped_blocks 1\n    have h\u2089 := h_grouped_blocks 2\n    have h\u2081\u2080 := h_grouped_blocks 3\n    simp_all [Finset.sum_Icc_succ_top, Nat.mod_eq_of_lt, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_succ, mul_add, mul_succ, mul_one, add_assoc]\n    <;> linarith\n  have h_solve_n_plus_1 : n = 97 := by\n    cases' h_form_n_plus_1 with m hm\n    have h_sum_n_plus_1' := h_sum_n_plus_1 m\n    simp_all [Finset.sum_Icc_succ_top, Nat.succ_eq_add_one, Complex.ext_iff]\n    <;> ring_nf at * <;> norm_cast at * <;> linarith\n  obtain \u27e8m, rfl\u27e9 := h_form_n_plus_1\n  simp_all [Finset.sum_Icc_succ_top]\n  <;> ring_nf\n  <;> linarith\n", "proof_before": "theorem amc12a_2009_p15 (n : \u2115) (h\u2080 : 0 < n)\n  (h\u2081 : (\u2211 k in Finset.Icc 1 n, \u2191k * Complex.I ^ k) = 48 + 49 * Complex.I) : n = 97  :=  by\n  have h_periodicity : \u2200 k : \u2115, Complex.I ^ (k % 4) = Complex.I ^ k := by\n    intro k\n    have h\u2082 : \u2200 n : \u2115, Complex.I ^ n = Complex.I ^ (n % 4) := by\n      intro n\n      rw [\u2190 Nat.mod_add_div n 4]\n      simp [pow_add, pow_mul, Complex.I_mul_I, mul_assoc, mul_comm, mul_left_comm]\n    rw [h\u2082 k]\n  have h_grouped_blocks : \u2200 m : \u2115, \u2211 k in Finset.Icc (4 * m + 1) (4 * m + 4), \u2191k * Complex.I ^ k = 2 - 2 * Complex.I := by\n    intro m\n    simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three, Complex.I_mul_I]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three, Complex.I_mul_I]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three, Complex.I_mul_I]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three, Complex.I_mul_I]\n    <;> ring_nf\n    <;> norm_num\n    <;> simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three, Complex.I_mul_I]\n    <;> ring_nf\n    <;> norm_num\n  have h_sum_multiple_of_4 : \u2200 m : \u2115, \u2211 k in Finset.Icc 1 (4 * m), \u2191k * Complex.I ^ k = 2 * m - 2 * m * Complex.I := by\n    intro m\n    have h_sum_group := h_grouped_blocks m\n    have h_sum_periodicity := h_periodicity\n    clear h_periodicity\n    induction m <;> simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three]\n    <;> ring_nf at * <;> norm_num <;> simp_all [Complex.ext_iff]\n    <;> norm_num <;> linarith\n  have h_contradiction_multiple_of_4 : \u00ac\u2203 m : \u2115, n = 4 * m := by\n    intro h\n    obtain \u27e8m, rfl\u27e9 := h\n    have h\u2082 := h_sum_multiple_of_4 0\n    have h\u2083 := h_sum_multiple_of_4 1\n    have h\u2084 := h_sum_multiple_of_4 2\n    have h\u2085 := h_sum_multiple_of_4 3\n    have h\u2086 := h_grouped_blocks 0\n    have h\u2087 := h_grouped_blocks 1\n    have h\u2088 := h_grouped_blocks 2\n    have h\u2089 := h_grouped_blocks 3\n    simp_all [Finset.sum_Icc_succ_top, Nat.mul_div_cancel_left]\n    <;> ring_nf at *\n    <;> simp_all [Complex.ext_iff]\n    <;> norm_num\n    <;> linarith\n  have h_n_not_multiple_of_4 : \u00ac\u2203 m : \u2115, n = 4 * m := by\n    intro h\n    obtain \u27e8m, hm\u27e9 := h\n    have h\u2082 := h_sum_multiple_of_4 m\n    have h\u2083 := h_grouped_blocks m\n    have h\u2084 := h_periodicity 0\n    have h\u2085 := h_periodicity 1\n    have h\u2086 := h_periodicity 2\n    have h\u2087 := h_periodicity 3\n    simp_all [Finset.sum_Icc_succ_top, Nat.mul_succ, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_three]\n    all_goals\n      linarith\n  have h_form_n_plus_1 : \u2203 m : \u2115, n = 4 * m + 1 := by\n    have h\u2082 : n % 4 = 1 := by\n      by_contra! h\n      have h\u2083 : \u2203 m : \u2115, n = 4 * m \u2228 n = 4 * m + 2 \u2228 n = 4 * m + 3 := by\n        have : n % 4 = 0 \u2228 n % 4 = 1 \u2228 n % 4 = 2 \u2228 n % 4 = 3 := by omega\n        rcases this with (h\u2084 | h\u2084 | h\u2084 | h\u2084) <;> use n / 4 <;> omega\n      rcases h\u2083 with \u27e8m, h\u2083 | h\u2083 | h\u2083\u27e9 <;> simp_all [Finset.sum_Icc_succ_top, Complex.ext_iff, pow_add,\n        pow_mul, pow_two, pow_three]\n      <;> linarith [h_sum_multiple_of_4 0, h_sum_multiple_of_4 1, h_sum_multiple_of_4 2,\n        h_sum_multiple_of_4 3]\n    use n / 4\n    omega\n  have h_sum_n_plus_1 : \u2200 m : \u2115, \u2211 k in Finset.Icc 1 (4 * m + 1), \u2191k * Complex.I ^ k = 2 * m + (2 * m + 1) * Complex.I := by\n    intro m\n    have h\u2082 := h_grouped_blocks m\n    have h\u2083 := h_grouped_blocks 0\n    have h\u2084 := h_sum_multiple_of_4 m\n    have h\u2085 := h_sum_multiple_of_4 0\n    have h\u2086 := h_sum_multiple_of_4 1\n    have h\u2087 := h_sum_multiple_of_4 2\n    have h\u2088 := h_grouped_blocks 1\n    have h\u2089 := h_grouped_blocks 2\n    have h\u2081\u2080 := h_grouped_blocks 3\n    simp_all [Finset.sum_Icc_succ_top, Nat.mod_eq_of_lt, Complex.ext_iff, pow_add, pow_mul, pow_two, pow_succ, mul_add, mul_succ, mul_one, add_assoc]\n    <;> linarith\n  have h_solve_n_plus_1 : n = 97 := by\n    cases' h_form_n_plus_1 with m hm\n    have h_sum_n_plus_1' := h_sum_n_plus_1 m\n    simp_all [Finset.sum_Icc_succ_top, Nat.succ_eq_add_one, Complex.ext_iff]\n    <;> ring_nf at * <;> norm_cast at * <;> linarith\n  obtain \u27e8m, rfl\u27e9 := h_form_n_plus_1\n  simp_all [Finset.sum_Icc_succ_top]\n  <;> ring_nf\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 27, "alphabet": 26, "compiles": false, "id": "imo_1964_p1_1", "proof_after": "theorem imo_1964_p1_1 (n : \u2115) (h\u2080 : 7 \u2223 2 ^ n - 1) : 3 \u2223 n  :=  by\n  have h\u2081 : 2 ^ n \u2261 1 [ZMOD 7] := by\n    rw [Int.ModEq]\n    rw [\u2190 Int.coe_nat_dvd] at h\u2080\n    norm_num at h\u2080 \u22a2\n    omega\n  have h\u2082 : orderOf (2 : ZMod 7) = 3 := by\n    have h\u2082 : orderOf (2 : ZMod 7) = 3 := by\n      rw [orderOf_eq_iff] <;> decide\n    simpa using h\u2082\n  have h\u2083 : 3 \u2223 n := by\n    rw [Int.ModEq] at h\u2081\n    norm_cast at h\u2081\n    rw [Nat.dvd_iff_mod_eq_zero]\n    have h\u2083 := h\u2081\n    rw [\u2190 Nat.mod_add_div n 3] at h\u2083\n    simp [pow_add, pow_mul, Nat.pow_mod, Nat.mul_mod, Nat.mod_mod] at h\u2083\n    have h\u2084 : n % 3 = 0 := by\n      have : n % 3 = 0 \u2228 n % 3 = 1 \u2228 n % 3 = 2 := by omega\n      rcases this with (h | h | h) <;> simp [h, h\u2082, Nat.pow_mod, Nat.mul_mod, Nat.mod_mod] at h\u2083\n      <;> omega\n    exact h\u2084\n  rw [\u2190 Nat.mod_add_div n 3]\n  simp_all [Int.ModEq, Nat.ModEq]\n  <;> omega\n", "proof_before": "theorem imo_1964_p1_1 (n : \u2115) (h\u2080 : 7 \u2223 2 ^ n - 1) : 3 \u2223 n  :=  by\n  have h\u2081 : 2 ^ n \u2261 1 [ZMOD 7] := by\n    rw [Int.ModEq]\n    rw [\u2190 Int.coe_nat_dvd] at h\u2080\n    norm_num at h\u2080 \u22a2\n    omega\n  have h\u2082 : orderOf (2 : ZMod 7) = 3 := by\n    have h\u2082 : orderOf (2 : ZMod 7) = 3 := by\n      rw [orderOf_eq_iff] <;> decide\n    simpa using h\u2082\n  have h\u2083 : 3 \u2223 n := by\n    rw [Int.ModEq] at h\u2081\n    norm_cast at h\u2081\n    rw [Nat.dvd_iff_mod_eq_zero]\n    have h\u2083 := h\u2081\n    rw [\u2190 Nat.mod_add_div n 3] at h\u2083\n    simp [pow_add, pow_mul, Nat.pow_mod, Nat.mul_mod, Nat.mod_mod] at h\u2083\n    have h\u2084 : n % 3 = 0 := by\n      have : n % 3 = 0 \u2228 n % 3 = 1 \u2228 n % 3 = 2 := by omega\n      rcases this with (h | h | h) <;> simp [h, h\u2082, Nat.pow_mod, Nat.mul_mod, Nat.mod_mod] at h\u2083\n      <;> omega\n    exact h\u2084\n  rw [\u2190 Nat.mod_add_div n 3]\n  simp_all [Int.ModEq, Nat.ModEq]\n  <;> omega\n"}
{"mode": "clear", "steps_before": 56, "alphabet": 14, "compiles": false, "id": "amc12a_2008_p4", "proof_after": "theorem amc12a_2008_p4 : (\u220f k in Finset.Icc (1 : \u2115) 501, ((4 : \u211d) * k + 4) / (4 * k)) = 502  :=  by\n  norm_num [Finset.prod_range_succ]\n  <;> norm_num\n  <;> rw [show (4 : \u211d) = (4 : \u211a) by norm_num]\n  <;> norm_cast\n  <;> simp [Finset.prod_range_succ]\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n", "proof_before": "theorem amc12a_2008_p4 : (\u220f k in Finset.Icc (1 : \u2115) 501, ((4 : \u211d) * k + 4) / (4 * k)) = 502  :=  by\n  norm_num [Finset.prod_range_succ]\n  <;> norm_num\n  <;> rw [show (4 : \u211d) = (4 : \u211a) by norm_num]\n  <;> norm_cast\n  <;> simp [Finset.prod_range_succ]\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n  <;> norm_num\n  <;> ring\n  <;> simp_all\n"}
{"mode": "clear", "steps_before": 31, "alphabet": 32, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 784, "chars_after": 8, "seconds": 1.5, "id": "mathd_numbertheory_370", "proof_after": "theorem mathd_numbertheory_370 (n : \u2115) (h\u2080 : n % 7 = 3) : (2 * n + 1) % 7 = 0  :=  by\n  omega\n", "proof_before": "theorem mathd_numbertheory_370 (n : \u2115) (h\u2080 : n % 7 = 3) : (2 * n + 1) % 7 = 0  :=  by\n  have h\u2081 : \u2203 k : \u2115, n = 7 * k + 3 := by\n    use n / 7\n    have h\u2081 : n = 7 * (n / 7) + (n % 7) := by\n      rw [Nat.div_add_mod]\n    rw [h\u2080] at h\u2081\n    linarith\n  have h\u2082 : \u2200 k : \u2115, 2 * (7 * k + 3) + 1 = 14 * k + 7 := by\n    intro k\n    simp [h\u2081, mul_add, mul_comm, mul_left_comm, add_assoc, add_comm, add_left_comm]\n    <;> omega\n  have h\u2083 : \u2200 k : \u2115, 14 * k + 7 = 7 * (2 * k + 1) := by\n    intro k\n    have h\u2083 := h\u2082 k\n    linarith\n  have h\u2084 : (2 * n + 1) % 7 = 0 := by\n    obtain \u27e8k, hk\u27e9 := h\u2081\n    rw [hk]\n    simp [h\u2082, h\u2083, Nat.mul_mod, Nat.add_mod, Nat.mod_mod]\n  have h\u2085 := h\u2082 0\n  have h\u2086 := h\u2083 0\n  simp at h\u2085 h\u2086\n  have h\u2087 := h\u2082 1\n  have h\u2088 := h\u2083 1\n  simp at h\u2087 h\u2088\n  have h\u2089 := h\u2082 2\n  have h\u2081\u2080 := h\u2083 2\n  simp at h\u2089 h\u2081\u2080\n  have h\u2081\u2081 := h\u2082 3\n  have h\u2081\u2082 := h\u2083 3\n  simp at h\u2081\u2081 h\u2081\u2082\n  omega\n"}
{"mode": "clear", "steps_before": 29, "alphabet": 19, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 738, "chars_after": 64, "seconds": 46.5, "id": "amc12a_2009_p5", "proof_after": "theorem amc12a_2009_p5 (x : \u211d) (h\u2080 : x ^ 3 - (x + 1) * (x - 1) * x = 5) : x ^ 3 = 125  :=  by\n  nlinarith [sq_nonneg (x + 1), sq_nonneg (x - 1), sq_nonneg x]\n", "proof_before": "theorem amc12a_2009_p5 (x : \u211d) (h\u2080 : x ^ 3 - (x + 1) * (x - 1) * x = 5) : x ^ 3 = 125  :=  by\n  have h\u2081 : (x + 1) * (x - 1) * x = x^3 - x := by\n    ring_nf at h\u2080 \u22a2\n    <;> nlinarith [sq_nonneg (x + 1), sq_nonneg (x - 1), sq_nonneg x]\n  have h\u2082 : x^3 - x = x^3 - 5 := by\n    have h\u2082 : (x + 1) * (x - 1) * x = x^3 - x := by linarith\n    linarith\n  have h\u2083 : -x = -5 := by\n    linarith\n  have h\u2084 : x = 5 := by\n    have h\u2080 : x ^ 3 - (x + 1) * (x - 1) * x = 5 := h\u2080\n    have h\u2081 : (x + 1) * (x - 1) * x = x ^ 3 - x := by\n      ring\n    rw [h\u2081] at h\u2080\n    have h\u2082 : x ^ 3 - x = x ^ 3 - 5 := by\n      linarith\n    have h\u2083 : -x = -5 := by\n      linarith\n    have h\u2084 : x = 5 := by\n      linarith\n    exact h\u2084\n  have h\u2085 : x^3 = 125 := by\n    have h\u2085 : x = 5 := by linarith\n    subst h\u2085\n    norm_num\n  have h\u2086 : x = 5 := h\u2084\n  rw [h\u2086]\n  norm_num\n"}
{"mode": "clear", "steps_before": 40, "alphabet": 32, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 27, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 2, "timed_out": true, "fixpoint": false, "chars_before": 1027, "chars_after": 656, "seconds": 430.1, "id": "amc12b_2002_p6", "proof_after": "theorem amc12b_2002_p6 (a b : \u211d) (h\u2080 : a \u2260 0 \u2227 b \u2260 0)\n  (h\u2081 : \u2200 x, x ^ 2 + a * x + b = (x - a) * (x - b)) : a = 1 \u2227 b = -2  :=  by\n  have h\u2082 : a + b = -a := by\n    have h\u2082 := h\u2081 0\n    have h\u2083 := h\u2081 1\n    simp_all\n    <;> linarith\n  have h\u2083 : a * b = b := by\n    have h\u2083 := h\u2081 1\n    have h\u2084 := h\u2081 (-1)\n    have h\u2085 := h\u2081 0\n    simp_all\n    <;> nlinarith\n  have h\u2084 : b = -2 * a := by\n    have h\u2084 := h\u2081 1\n    have h\u2085 := h\u2081 0\n    have h\u2086 := h\u2081 a\n    have h\u2087 := h\u2081 b\n    simp_all [mul_comm, mul_assoc, mul_left_comm]\n    <;> linarith\n  have h\u2087 : b = -2 := by\n    have h\u2087 := h\u2081 0\n    have h\u2088 := h\u2081 1\n    have h\u2089 := h\u2081 (-2)\n    have h\u2081\u2080 := h\u2081 1\n    simp_all\n  constructor\n  all_goals\n    nlinarith [h\u2081 0, h\u2081 1, h\u2081 (-2), h\u2081 3, h\u2081 (-3), h\u2081 2, h\u2081 (-1), h\u2081 1, h\u2081 (-2), h\u2081 3, h\u2081 (-3), h\u2081 2, h\u2081 (-1)]\n", "proof_before": "theorem amc12b_2002_p6 (a b : \u211d) (h\u2080 : a \u2260 0 \u2227 b \u2260 0)\n  (h\u2081 : \u2200 x, x ^ 2 + a * x + b = (x - a) * (x - b)) : a = 1 \u2227 b = -2  :=  by\n  have h\u2082 : a + b = -a := by\n    have h\u2082 := h\u2081 0\n    have h\u2083 := h\u2081 1\n    simp_all\n    <;> linarith\n  have h\u2083 : a * b = b := by\n    have h\u2083 := h\u2081 1\n    have h\u2084 := h\u2081 (-1)\n    have h\u2085 := h\u2081 0\n    simp_all\n    <;> nlinarith\n  have h\u2084 : b = -2 * a := by\n    have h\u2084 := h\u2081 1\n    have h\u2085 := h\u2081 0\n    have h\u2086 := h\u2081 a\n    have h\u2087 := h\u2081 b\n    simp_all [mul_comm, mul_assoc, mul_left_comm]\n    <;> linarith\n  have h\u2085 : b * (a - 1) = 0 := by\n    rw [h\u2084] at h\u2081\n    have h\u2085 := h\u2081 0\n    have h\u2086 := h\u2081 1\n    have h\u2087 := h\u2081 (-1)\n    simp_all\n    <;> nlinarith\n  have h\u2086 : a = 1 := by\n    have : a = 1 := by\n      apply mul_left_cancel\u2080 (sub_ne_zero_of_ne h\u2080.2)\n      rw [\u2190 sub_eq_zero] at h\u2085 \u22a2\n      nlinarith [h\u2081 0, h\u2081 1, h\u2081 a, h\u2081 b, h\u2082, h\u2083, h\u2084, h\u2085]\n    simpa [h\u2080.1, h\u2080.2] using this\n  have h\u2087 : b = -2 := by\n    have h\u2087 := h\u2081 0\n    have h\u2088 := h\u2081 1\n    have h\u2089 := h\u2081 (-2)\n    have h\u2081\u2080 := h\u2081 1\n    simp_all\n  constructor\n  all_goals\n    nlinarith [h\u2081 0, h\u2081 1, h\u2081 (-2), h\u2081 3, h\u2081 (-3), h\u2081 2, h\u2081 (-1), h\u2081 1, h\u2081 (-2), h\u2081 3, h\u2081 (-3), h\u2081 2, h\u2081 (-1)]\n"}
{"mode": "clear", "steps_before": 38, "alphabet": 21, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 1057, "chars_after": 6, "seconds": 220.3, "id": "mathd_numbertheory_37", "proof_after": "theorem mathd_numbertheory_37 : Nat.lcm 9999 100001 = 90900909  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_37 : Nat.lcm 9999 100001 = 90900909  :=  by\n  have factor_9999 : 9999 = 3^2 * 11 * 101 := by\n    norm_num [pow_two, mul_assoc, mul_comm, mul_left_comm]\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n  have factor_100001 : 100001 = 11 * 9091 := by\n    have : 9999 = 3^2 * 11 * 101 := by linarith\n    norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have lcm_calculation : \n    Nat.lcm 9999 100001 = \n    Nat.lcm (3^2 * 11 * 101) (11 * 9091) := by\n    simp [factor_9999, factor_100001, Nat.lcm, Nat.gcd]\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> rfl\n  have lcm_simplified : \n    Nat.lcm (3^2 * 11 * 101) (11 * 9091) = 3^2 * 11 * 101 * 9091 := by\n    simp_all [Nat.lcm, Nat.gcd_eq_right]\n    <;> ring_nf\n    <;> norm_num\n    <;> rfl\n  have final_product : \n    3^2 * 11 * 101 * 9091 = 90900909 := by\n    linarith [factor_9999, factor_100001, lcm_calculation, lcm_simplified]\n  have final_result : \n    Nat.lcm 9999 100001 = 90900909 := by\n    rw [lcm_calculation, lcm_simplified, final_product]\n    <;> rfl\n  rw [final_result]\n"}
{"mode": "clear", "steps_before": 49, "alphabet": 30, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 49, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 0, "timed_out": true, "fixpoint": false, "chars_before": 1705, "chars_after": 1705, "seconds": 448.0, "id": "amc12a_2010_p10", "proof_after": "theorem amc12a_2010_p10 (p q : \u211d) (a : \u2115 \u2192 \u211d) (h\u2080 : \u2200 n, a (n + 2) - a (n + 1) = a (n + 1) - a n)\n  (h\u2081 : a 1 = p) (h\u2082 : a 2 = 9) (h\u2083 : a 3 = 3 * p - q) (h\u2084 : a 4 = 3 * p + q) : a 2010 = 8041  :=  by\n  have d_def : \u2200 n, a (n + 1) - a n = a 2 - a 1 := by\n    intro n\n    have h\u2085 := h\u2080 0\n    have h\u2086 := h\u2080 1\n    have h\u2087 := h\u2080 2\n    have h\u2088 := h\u2080 3\n    simp_all only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm, Nat.add_right_comm]\n    induction n <;> simp_all\n    <;> linarith\n  have d_value1 : a 2 - a 1 = 9 - p := by\n    have h\u2085 := d_def 0\n    have h\u2086 := d_def 1\n    have h\u2087 := d_def 2\n    simp_all\n    <;> linarith\n  have d_value2 : a 3 - a 2 = 3 * p - q - 9 := by\n    simp_all only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm, Nat.add_sub_cancel]\n    <;> linarith\n  have d_value3 : a 4 - a 3 = 2 * q := by\n    have d_value3 : a 4 - a 3 = 2 * q := by\n      linarith [h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, d_def 1, d_def 2, d_def 3, d_value1, d_value2, h\u2081, h\u2082, h\u2083, h\u2084]\n    exact d_value3\n  have eq1 : 9 - p = 3 * p - q - 9 := by\n    linarith [h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, h\u2081, h\u2082, h\u2083, h\u2084, d_def 1, d_def 2, d_def 3, d_value1, d_value2, d_value3]\n  have eq2 : 3 * p - q - 9 = 2 * q := by\n    linarith [h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, h\u2081, h\u2082, h\u2083, h\u2084, d_def 1, d_def 2, d_def 3, d_value1, d_value2, d_value3, eq1]\n  have d_final : a 2 - a 1 = 4 := by\n    norm_num [h\u2081, h\u2082, h\u2083, h\u2084] at *\n    linarith\n  have formula : a 2010 = p + (2009 * 4) := by\n    have h\u2085 : \u2200 n, a (n + 1) - a n = 4 := by\n      intro n\n      have h\u2085\u2080 := d_def n\n      have h\u2085\u2081 := d_final\n      linarith\n    have h\u2086 : \u2200 n, a (n + 1) = a n + 4 := by\n      intro n\n      have h\u2086\u2080 := h\u2085 n\n      have h\u2086\u2081 := h\u2085 (n + 1)\n      linarith\n    have h\u2087 : \u2200 n, a n = p + 4 * (n - 1) := by\n      intro n\n      induction n <;> simp_all [h\u2086]\n      <;> linarith\n    simp_all [h\u2087]\n    <;> linarith\n  linarith [h\u2080 0, h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, h\u2080 5, h\u2080 6, h\u2080 7, h\u2080 8, h\u2080 9, h\u2080 10]\n", "proof_before": "theorem amc12a_2010_p10 (p q : \u211d) (a : \u2115 \u2192 \u211d) (h\u2080 : \u2200 n, a (n + 2) - a (n + 1) = a (n + 1) - a n)\n  (h\u2081 : a 1 = p) (h\u2082 : a 2 = 9) (h\u2083 : a 3 = 3 * p - q) (h\u2084 : a 4 = 3 * p + q) : a 2010 = 8041  :=  by\n  have d_def : \u2200 n, a (n + 1) - a n = a 2 - a 1 := by\n    intro n\n    have h\u2085 := h\u2080 0\n    have h\u2086 := h\u2080 1\n    have h\u2087 := h\u2080 2\n    have h\u2088 := h\u2080 3\n    simp_all only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm, Nat.add_right_comm]\n    induction n <;> simp_all\n    <;> linarith\n  have d_value1 : a 2 - a 1 = 9 - p := by\n    have h\u2085 := d_def 0\n    have h\u2086 := d_def 1\n    have h\u2087 := d_def 2\n    simp_all\n    <;> linarith\n  have d_value2 : a 3 - a 2 = 3 * p - q - 9 := by\n    simp_all only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm, Nat.add_sub_cancel]\n    <;> linarith\n  have d_value3 : a 4 - a 3 = 2 * q := by\n    have d_value3 : a 4 - a 3 = 2 * q := by\n      linarith [h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, d_def 1, d_def 2, d_def 3, d_value1, d_value2, h\u2081, h\u2082, h\u2083, h\u2084]\n    exact d_value3\n  have eq1 : 9 - p = 3 * p - q - 9 := by\n    linarith [h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, h\u2081, h\u2082, h\u2083, h\u2084, d_def 1, d_def 2, d_def 3, d_value1, d_value2, d_value3]\n  have eq2 : 3 * p - q - 9 = 2 * q := by\n    linarith [h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, h\u2081, h\u2082, h\u2083, h\u2084, d_def 1, d_def 2, d_def 3, d_value1, d_value2, d_value3, eq1]\n  have d_final : a 2 - a 1 = 4 := by\n    norm_num [h\u2081, h\u2082, h\u2083, h\u2084] at *\n    linarith\n  have formula : a 2010 = p + (2009 * 4) := by\n    have h\u2085 : \u2200 n, a (n + 1) - a n = 4 := by\n      intro n\n      have h\u2085\u2080 := d_def n\n      have h\u2085\u2081 := d_final\n      linarith\n    have h\u2086 : \u2200 n, a (n + 1) = a n + 4 := by\n      intro n\n      have h\u2086\u2080 := h\u2085 n\n      have h\u2086\u2081 := h\u2085 (n + 1)\n      linarith\n    have h\u2087 : \u2200 n, a n = p + 4 * (n - 1) := by\n      intro n\n      induction n <;> simp_all [h\u2086]\n      <;> linarith\n    simp_all [h\u2087]\n    <;> linarith\n  linarith [h\u2080 0, h\u2080 1, h\u2080 2, h\u2080 3, h\u2080 4, h\u2080 5, h\u2080 6, h\u2080 7, h\u2080 8, h\u2080 9, h\u2080 10]\n"}
{"mode": "clear", "steps_before": 79, "alphabet": 19, "compiles": false, "id": "mathd_algebra_509", "proof_after": "theorem mathd_algebra_509 :\n  Real.sqrt ((5 / Real.sqrt 80 + Real.sqrt 845 / 9 + Real.sqrt 45) / Real.sqrt 5) = 13 / 6  :=  by\n  have sqrt80 : Real.sqrt 80 = 4 * Real.sqrt 5 := by\n    rw [Real.sqrt_eq_iff_sq_eq] <;> ring_nf\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n  have term1 : 5 / Real.sqrt 80 = Real.sqrt 5 / 4 := by\n    rw [sqrt80]\n    field_simp [mul_comm]\n    <;> ring\n    <;> field_simp [mul_comm]\n    <;> ring\n  have sqrt845 : Real.sqrt 845 = 13 * Real.sqrt 5 := by\n    rw [Real.sqrt_eq_iff_sq_eq] <;>\n      nlinarith [Real.sqrt_nonneg 80, Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 80 by norm_num),\n        Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num), Real.sq_sqrt (show (0 : \u211d) \u2264 845 by norm_num),\n        Real.sq_sqrt (show (0 : \u211d) \u2264 169 by norm_num), term1, sqrt80]\n  have term2 : Real.sqrt 845 / 9 = 13 * Real.sqrt 5 / 9 := by\n    simp_all [div_eq_mul_inv]\n    <;> ring_nf\n    <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have sqrt45 : Real.sqrt 45 = 3 * Real.sqrt 5 := by\n    rw [show Real.sqrt 45 = 3 * Real.sqrt 5 by\n      rw [Real.sqrt_eq_iff_sq_eq] <;> ring_nf <;> norm_num\n      ]\n  have term3 : Real.sqrt 45 = 3 * Real.sqrt 5 := by\n    rw [sqrt45]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have combined : 5 / Real.sqrt 80 + Real.sqrt 845 / 9 + Real.sqrt 45 = (169 * Real.sqrt 5) / 36 := by\n    field_simp [term1, term2, term3, sqrt80, sqrt845, sqrt45]\n    ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  have simplified : (169 * Real.sqrt 5 / 36) / Real.sqrt 5 = 169 / 36 := by\n    field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n  have sqrt169_36 : Real.sqrt (169 / 36) = 13 / 6 := by\n    rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    <;> nlinarith\n    <;> linarith\n    <;> nlinarith\n  have result : Real.sqrt ((5 / Real.sqrt 80 + Real.sqrt 845 / 9 + Real.sqrt 45) / Real.sqrt 5) = 13 / 6 := by\n    simp_all [Real.sqrt_eq_iff_sq_eq, div_eq_mul_inv, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n    <;> ring_nf\n    <;> norm_num\n  rw [result]\n  <;> norm_num\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_509 :\n  Real.sqrt ((5 / Real.sqrt 80 + Real.sqrt 845 / 9 + Real.sqrt 45) / Real.sqrt 5) = 13 / 6  :=  by\n  have sqrt80 : Real.sqrt 80 = 4 * Real.sqrt 5 := by\n    rw [Real.sqrt_eq_iff_sq_eq] <;> ring_nf\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n    <;> ring_nf\n    <;> norm_num\n  have term1 : 5 / Real.sqrt 80 = Real.sqrt 5 / 4 := by\n    rw [sqrt80]\n    field_simp [mul_comm]\n    <;> ring\n    <;> field_simp [mul_comm]\n    <;> ring\n  have sqrt845 : Real.sqrt 845 = 13 * Real.sqrt 5 := by\n    rw [Real.sqrt_eq_iff_sq_eq] <;>\n      nlinarith [Real.sqrt_nonneg 80, Real.sqrt_nonneg 5, Real.sq_sqrt (show (0 : \u211d) \u2264 80 by norm_num),\n        Real.sq_sqrt (show (0 : \u211d) \u2264 5 by norm_num), Real.sq_sqrt (show (0 : \u211d) \u2264 845 by norm_num),\n        Real.sq_sqrt (show (0 : \u211d) \u2264 169 by norm_num), term1, sqrt80]\n  have term2 : Real.sqrt 845 / 9 = 13 * Real.sqrt 5 / 9 := by\n    simp_all [div_eq_mul_inv]\n    <;> ring_nf\n    <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have sqrt45 : Real.sqrt 45 = 3 * Real.sqrt 5 := by\n    rw [show Real.sqrt 45 = 3 * Real.sqrt 5 by\n      rw [Real.sqrt_eq_iff_sq_eq] <;> ring_nf <;> norm_num\n      ]\n  have term3 : Real.sqrt 45 = 3 * Real.sqrt 5 := by\n    rw [sqrt45]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have combined : 5 / Real.sqrt 80 + Real.sqrt 845 / 9 + Real.sqrt 45 = (169 * Real.sqrt 5) / 36 := by\n    field_simp [term1, term2, term3, sqrt80, sqrt845, sqrt45]\n    ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  have simplified : (169 * Real.sqrt 5 / 36) / Real.sqrt 5 = 169 / 36 := by\n    field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n    <;> field_simp [sqrt80, sqrt845, sqrt45, term1, term2, term3, combined]\n    <;> ring_nf\n  have sqrt169_36 : Real.sqrt (169 / 36) = 13 / 6 := by\n    rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num\n    <;> nlinarith\n    <;> linarith\n    <;> nlinarith\n  have result : Real.sqrt ((5 / Real.sqrt 80 + Real.sqrt 845 / 9 + Real.sqrt 45) / Real.sqrt 5) = 13 / 6 := by\n    simp_all [Real.sqrt_eq_iff_sq_eq, div_eq_mul_inv, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [Real.sqrt_eq_iff_sq_eq]\n    <;> ring_nf\n    <;> norm_num\n  rw [result]\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 44, "alphabet": 23, "compiles": true, "steps_after": 30, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 1, "timed_out": true, "fixpoint": false, "chars_before": 1988, "chars_after": 1522, "seconds": 550.8, "id": "algebra_amgm_sqrtxymulxmyeqxpy_xpygeq4", "proof_after": "theorem algebra_amgm_sqrtxymulxmyeqxpy_xpygeq4 (x y : \u211d) (h\u2080 : 0 < x \u2227 0 < y) (h\u2081 : y \u2264 x)\n  (h\u2082 : Real.sqrt (x * y) * (x - y) = x + y) : x + y \u2265 4  :=  by\n  have h_pos : 0 < x \u2227 0 < y := by\n    exact h\u2080\n  have h_le : y \u2264 x := by\n    omega\n  have h_eq : Real.sqrt (x * y) * (x - y) = x + y := by\n    have h\u2083 : 0 < x := by linarith\n    have h\u2084 : 0 < y := by linarith\n    have h\u2085 : 0 < x * y := by positivity\n    field_simp [h\u2083.ne', h\u2084.ne', h\u2085.ne'] at h\u2082 \u22a2\n    nlinarith [sq_sqrt (show (0 : \u211d) \u2264 x * y by positivity),\n      sq_nonneg (x - y)]\n  have h_squared : (Real.sqrt (x * y) * (x - y))^2 = (x + y)^2 := by\n    have h\u2083 : 0 \u2264 (x - y) ^ 2 := sq_nonneg (x - y)\n    nlinarith [sq_sqrt (mul_nonneg (le_of_lt h_pos.1) (le_of_lt h_pos.2)),\n      sq_nonneg (x - y)]\n  have h_expanded : x * y * (x - y)^2 = (x + y)^2 := by\n    rw [\u2190 mul_one x]\n    rw [\u2190 mul_one y]\n    ring_nf at h_squared \u22a2\n    nlinarith [sq_sqrt (mul_nonneg (le_of_lt h\u2080.1) (le_of_lt h\u2080.2)),\n      mul_self_nonneg (x - y)]\n  have h_amgm : x + y \u2265 2 * Real.sqrt (x * y) := by\n    have h\u2083 : 0 \u2264 x * y := mul_nonneg h\u2080.1.le h\u2080.2.le\n    nlinarith [sq_sqrt (show 0 \u2264 x * y by linarith), sq_nonneg (x - y), h\u2081, h\u2082, h\u2083, h_expanded, h_squared,\n      h_eq, h_pos.1, h_pos.2, h_le]\n  have h_combined : x + y \u2265 4 := by\n    have h\u2083 : 0 < x * y := mul_pos h\u2080.1 h\u2080.2\n    have h\u2084 : 0 < Real.sqrt (x * y) := Real.sqrt_pos.mpr h\u2083\n    have h\u2085 : Real.sqrt (x * y) * (x - y) = x + y := h\u2082\n    nlinarith [sq_nonneg (x - y), sq_sqrt (show 0 \u2264 x * y by linarith),\n      sq_nonneg (x + y), sq_nonneg (x - y - 2 * Real.sqrt (x * y))]\n  nlinarith [h\u2080.1, h\u2080.2, h\u2081, h\u2082, h_pos.1, h_pos.2, h_le, h_eq, h_squared, h_expanded, h_amgm, h_combined]\n", "proof_before": "theorem algebra_amgm_sqrtxymulxmyeqxpy_xpygeq4 (x y : \u211d) (h\u2080 : 0 < x \u2227 0 < y) (h\u2081 : y \u2264 x)\n  (h\u2082 : Real.sqrt (x * y) * (x - y) = x + y) : x + y \u2265 4  :=  by\n  have h_pos : 0 < x \u2227 0 < y := by\n    exact h\u2080\n  have h_le : y \u2264 x := by\n    have h\u2083 : y \u2264 x := by assumption\n    have h\u2084 : Real.sqrt (x * y) * (x - y) = x + y := by assumption\n    have h\u2085 : 0 \u2264 Real.sqrt (x * y) := by apply Real.sqrt_nonneg\n    have h\u2086 : 0 \u2264 Real.sqrt (x * y) * (x - y) := by nlinarith\n    have h\u2087 : 0 \u2264 (x - y) ^ 2 := by nlinarith\n    have h\u2088 : 0 \u2264 Real.sqrt (x * y) := by apply Real.sqrt_nonneg\n    have h\u2089 : 0 \u2264 Real.sqrt (x * y) * (x - y) := by nlinarith\n    nlinarith [sq_sqrt (mul_nonneg h\u2080.1.le h\u2080.2.le), sq_nonneg (x - y)]\n  have h_eq : Real.sqrt (x * y) * (x - y) = x + y := by\n    have h\u2083 : 0 < x := by linarith\n    have h\u2084 : 0 < y := by linarith\n    have h\u2085 : 0 < x * y := by positivity\n    field_simp [h\u2083.ne', h\u2084.ne', h\u2085.ne'] at h\u2082 \u22a2\n    nlinarith [sq_sqrt (show (0 : \u211d) \u2264 x * y by positivity),\n      sq_nonneg (x - y)]\n  have h_squared : (Real.sqrt (x * y) * (x - y))^2 = (x + y)^2 := by\n    have h\u2083 : 0 \u2264 (x - y) ^ 2 := sq_nonneg (x - y)\n    nlinarith [sq_sqrt (mul_nonneg (le_of_lt h_pos.1) (le_of_lt h_pos.2)),\n      sq_nonneg (x - y)]\n  have h_expanded : x * y * (x - y)^2 = (x + y)^2 := by\n    rw [\u2190 mul_one x]\n    rw [\u2190 mul_one y]\n    ring_nf at h_squared \u22a2\n    nlinarith [sq_sqrt (mul_nonneg (le_of_lt h\u2080.1) (le_of_lt h\u2080.2)),\n      mul_self_nonneg (x - y)]\n  have h_amgm : x + y \u2265 2 * Real.sqrt (x * y) := by\n    have h\u2083 : 0 \u2264 x * y := mul_nonneg h\u2080.1.le h\u2080.2.le\n    nlinarith [sq_sqrt (show 0 \u2264 x * y by linarith), sq_nonneg (x - y), h\u2081, h\u2082, h\u2083, h_expanded, h_squared,\n      h_eq, h_pos.1, h_pos.2, h_le]\n  have h_combined : x + y \u2265 4 := by\n    have h\u2083 : 0 < x * y := mul_pos h\u2080.1 h\u2080.2\n    have h\u2084 : 0 < Real.sqrt (x * y) := Real.sqrt_pos.mpr h\u2083\n    have h\u2085 : Real.sqrt (x * y) * (x - y) = x + y := h\u2082\n    nlinarith [sq_nonneg (x - y), sq_sqrt (show 0 \u2264 x * y by linarith),\n      sq_nonneg (x + y), sq_nonneg (x - y - 2 * Real.sqrt (x * y))]\n  nlinarith [h\u2080.1, h\u2080.2, h\u2081, h\u2082, h_pos.1, h_pos.2, h_le, h_eq, h_squared, h_expanded, h_amgm, h_combined]\n"}
{"mode": "clear", "steps_before": 3, "alphabet": 11, "compiles": false, "id": "mathd_numbertheory_149", "proof_after": "theorem mathd_numbertheory_149 :\n  (\u2211 k in Finset.filter (fun x => x % 8 = 5 \u2227 x % 6 = 3) (Finset.range 50), k) = 66  :=  by\n  decide\n  <;> decide\n  <;> decide\n", "proof_before": "theorem mathd_numbertheory_149 :\n  (\u2211 k in Finset.filter (fun x => x % 8 = 5 \u2227 x % 6 = 3) (Finset.range 50), k) = 66  :=  by\n  decide\n  <;> decide\n  <;> decide\n"}
{"mode": "clear", "steps_before": 35, "alphabet": 16, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 965, "chars_after": 6, "seconds": 0.7, "id": "mathd_numbertheory_102", "proof_after": "theorem mathd_numbertheory_102 : 2 ^ 8 % 5 = 1  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_102 : 2 ^ 8 % 5 = 1  :=  by\n  have step1 : 2 ^ 1 % 5 = 2 := by\n    norm_num\n    <;> rfl\n    <;> decide\n    <;> simp_all\n    <;> decide\n    <;> decide\n  have step2 : 2 ^ 2 % 5 = 4 := by\n    norm_num\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n  have step3 : 2 ^ 3 % 5 = 3 := by\n    norm_num\n    <;> simp_all [Nat.pow_succ, Nat.mul_mod, Nat.pow_mod]\n    <;> norm_num\n    <;> omega\n  have step4 : 2 ^ 4 % 5 = 1 := by\n    norm_num [pow_succ, pow_one, Nat.mul_mod, Nat.add_mod, Nat.mod_eq_of_lt]\n    <;> decide\n    <;> norm_num\n    <;> decide\n  have step5 : 2 ^ 8 % 5 = (2 ^ 4)^2 % 5 := by\n    norm_num [pow_succ, pow_mul, pow_two]\n    <;> simp [step1, step2, step3, step4]\n    <;> omega\n  have step6 : (2 ^ 4)^2 % 5 = 1^2 % 5 := by\n    simp_all [Nat.pow_succ, Nat.mul_mod, Nat.pow_mod]\n    <;> norm_num\n    <;> rfl\n  have step7 : 1^2 % 5 = 1 := by\n    simp [step1, step2, step3, step4, step5, step6]\n    <;> simp\n    <;> norm_num\n  simpa [step1, step2, step3, step4, step5, step6, step7] using step7\n"}
{"mode": "clear", "steps_before": 23, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 893, "chars_after": 101, "seconds": 105.0, "id": "algebra_2complexrootspoly_xsqp49eqxp7itxpn7i", "proof_after": "theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i (x : \u2102) :\n    x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I)  :=  by\n  ring\n  simp [mul_add, mul_comm, mul_left_comm, sub_eq_add_neg, add_assoc, add_comm, add_left_comm]\n", "proof_before": "theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i (x : \u2102) :\n    x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I)  :=  by\n  have expand_RHS : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x * x + x * (-7 * Complex.I) + (7 * Complex.I) * x + (7 * Complex.I) * (-7 * Complex.I) := by\n    simp [mul_add, mul_comm, mul_left_comm, sub_eq_add_neg, add_assoc, add_comm, add_left_comm]\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  have simplify_RHS : x * x + x * (-7 * Complex.I) + (7 * Complex.I) * x + (7 * Complex.I) * (-7 * Complex.I) = x^2 + 49 := by\n    simp only [Complex.I_mul_I, mul_add, mul_comm, mul_left_comm, mul_assoc, add_assoc] at expand_RHS \u22a2\n    ring_nf at expand_RHS \u22a2\n    simp_all\n    <;> ring_nf\n    <;> simp_all\n    <;> ring_nf\n    <;> simp_all\n  have final_proof : x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I) := by\n    rw [eq_comm] at expand_RHS\n    rw [eq_comm] at simplify_RHS\n    simp_all\n  exact final_proof\n"}
{"mode": "clear", "steps_before": 5, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 3, "lower_bound": 3, "exhausted_depth": 2, "certified": true, "rounds": 1, "timed_out": false, "fixpoint": true, "chars_before": 122, "chars_after": 84, "seconds": 19.7, "id": "mathd_numbertheory_64", "proof_after": "theorem mathd_numbertheory_64 : IsLeast { x : \u2115 | 30 * x \u2261 42 [MOD 47] } 39  :=  by\n  refine' \u27e8by decide, fun x => _\u27e9\n  simp only [Set.mem_setOf_eq, Nat.ModEq]\n  omega\n", "proof_before": "theorem mathd_numbertheory_64 : IsLeast { x : \u2115 | 30 * x \u2261 42 [MOD 47] } 39  :=  by\n  refine' \u27e8by decide, fun x => _\u27e9\n  simp only [Set.mem_setOf_eq, Nat.ModEq]\n  intro h\n  norm_num [Nat.ModEq] at h\n  omega\n"}
{"mode": "clear", "steps_before": 47, "alphabet": 34, "compiles": false, "id": "algebra_amgm_faxinrrp2msqrt2geq2mxm1div2x", "proof_after": "theorem algebra_amgm_faxinrrp2msqrt2geq2mxm1div2x :\n  \u2200 x > 0, 2 - Real.sqrt 2 \u2265 2 - x - 1 / (2 * x)  :=  by\n  have step1 : \u2200 x > 0, Real.sqrt 2 \u2264 x + 1 / (2 * x) := by\n    intro x hx\n    have h\u2081 : Real.sqrt 2 \u2264 x + 1 / (2 * x) := by\n      have h\u2081 : 0 < x := hx\n      field_simp [h\u2081]\n      rw [le_div_iff (by positivity)]\n      ring_nf\n      nlinarith [sq_sqrt (show 0 \u2264 2 by norm_num),\n        sq_nonneg (x - Real.sqrt 2 / 2)]\n    exact h\u2081\n  have step2 : \u2203 x > 0, x + 1 / (2 * x) = Real.sqrt 2 := by\n    use Real.sqrt 2 / 2\n    constructor\n    exact div_pos (Real.sqrt_pos.mpr (by norm_num)) (by norm_num)\n    field_simp [mul_comm, mul_assoc, mul_left_comm]\n    <;> ring_nf\n    <;> field_simp [mul_comm, mul_assoc, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [mul_comm, mul_assoc, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [mul_comm, mul_assoc, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num\n  have step3 : \u2200 x > 0, x + 1 / (2 * x) \u2265 Real.sqrt 2 := by\n    intro x hx\n    have h1 := step1 x hx\n    have h2 := step2\n    obtain \u27e8x', hx', hx'_eq\u27e9 := h2\n    have h3 := step1 x' hx'\n    have h4 := step1 x hx\n    nlinarith [sq_sqrt (show 0 \u2264 2 by norm_num), mul_self_nonneg (x - Real.sqrt 2 / 2)]\n  have step4 : \u2200 x > 0, 2 - Real.sqrt 2 \u2265 2 - x - 1 / (2 * x) := by\n    intro x hx\n    have h := step3 x hx\n    have h\u2081 := step1 x hx\n    have h\u2082 := step2\n    cases' h\u2082 with x\u2080 hx\u2080\n    cases' hx\u2080 with hx\u2080 hx\u2080\n    have h\u2083 := step3 x\u2080 (by linarith)\n    have h\u2084 := step1 x\u2080 (by linarith)\n    linarith [h, h\u2081, h\u2083, h\u2084]\n  intro x hx\n  have := step4 x hx\n  linarith\n", "proof_before": "theorem algebra_amgm_faxinrrp2msqrt2geq2mxm1div2x :\n  \u2200 x > 0, 2 - Real.sqrt 2 \u2265 2 - x - 1 / (2 * x)  :=  by\n  have step1 : \u2200 x > 0, Real.sqrt 2 \u2264 x + 1 / (2 * x) := by\n    intro x hx\n    have h\u2081 : Real.sqrt 2 \u2264 x + 1 / (2 * x) := by\n      have h\u2081 : 0 < x := hx\n      field_simp [h\u2081]\n      rw [le_div_iff (by positivity)]\n      ring_nf\n      nlinarith [sq_sqrt (show 0 \u2264 2 by norm_num),\n        sq_nonneg (x - Real.sqrt 2 / 2)]\n    exact h\u2081\n  have step2 : \u2203 x > 0, x + 1 / (2 * x) = Real.sqrt 2 := by\n    use Real.sqrt 2 / 2\n    constructor\n    exact div_pos (Real.sqrt_pos.mpr (by norm_num)) (by norm_num)\n    field_simp [mul_comm, mul_assoc, mul_left_comm]\n    <;> ring_nf\n    <;> field_simp [mul_comm, mul_assoc, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [mul_comm, mul_assoc, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num\n    <;> field_simp [mul_comm, mul_assoc, mul_left_comm]\n    <;> ring_nf\n    <;> norm_num\n  have step3 : \u2200 x > 0, x + 1 / (2 * x) \u2265 Real.sqrt 2 := by\n    intro x hx\n    have h1 := step1 x hx\n    have h2 := step2\n    obtain \u27e8x', hx', hx'_eq\u27e9 := h2\n    have h3 := step1 x' hx'\n    have h4 := step1 x hx\n    nlinarith [sq_sqrt (show 0 \u2264 2 by norm_num), mul_self_nonneg (x - Real.sqrt 2 / 2)]\n  have step4 : \u2200 x > 0, 2 - Real.sqrt 2 \u2265 2 - x - 1 / (2 * x) := by\n    intro x hx\n    have h := step3 x hx\n    have h\u2081 := step1 x hx\n    have h\u2082 := step2\n    cases' h\u2082 with x\u2080 hx\u2080\n    cases' hx\u2080 with hx\u2080 hx\u2080\n    have h\u2083 := step3 x\u2080 (by linarith)\n    have h\u2084 := step1 x\u2080 (by linarith)\n    linarith [h, h\u2081, h\u2083, h\u2084]\n  intro x hx\n  have := step4 x hx\n  linarith\n"}
{"mode": "clear", "steps_before": 6, "alphabet": 12, "compiles": false, "id": "induction_sum_odd", "proof_after": "theorem induction_sum_odd (n : \u2115) : (\u2211 k in Finset.range n, (2 * k + 1)) = n ^ 2  :=  by\n  induction' n with n IH\n  \u00b7 \n    simp\n  \u00b7 \n    simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_succ]\n    linarith\n", "proof_before": "theorem induction_sum_odd (n : \u2115) : (\u2211 k in Finset.range n, (2 * k + 1)) = n ^ 2  :=  by\n  induction' n with n IH\n  \u00b7 \n    simp\n  \u00b7 \n    simp_all [Finset.sum_range_succ, Nat.pow_succ, Nat.mul_succ]\n    linarith\n"}
{"mode": "clear", "steps_before": 27, "alphabet": 21, "compiles": false, "id": "mathd_algebra_245", "proof_after": "theorem mathd_algebra_245 (x : \u211d) (h\u2080 : x \u2260 0) :\n  (4 / x)\u207b\u00b9 * (3 * x ^ 3 / x) ^ 2 * (1 / (2 * x))\u207b\u00b9 ^ 3 = 18 * x ^ 8  :=  by\n  have step1 : (4 / x)\u207b\u00b9 = x / 4 := by\n    field_simp [h\u2080, mul_comm]\n    <;> simp_all [h\u2080]\n    <;> linarith\n  have step2 : (3 * x ^ 3 / x) ^ 2 = 9 * x ^ 4 := by\n    have h\u2081 : x \u2260 0 := h\u2080\n    have h\u2082 : (4 / x)\u207b\u00b9 = x / 4 := step1\n    field_simp [h\u2081] at h\u2082 \u22a2\n    ring_nf at h\u2082 \u22a2\n    <;> nlinarith\n  have step3 : (1 / (2 * x))\u207b\u00b9 ^ 3 = 8 * x ^ 3 := by\n    have h1 : x \u2260 0 := by\n      intro H\n      apply h\u2080\n      linarith\n    field_simp [h1] at *\n    linarith\n  have step4 : (x / 4) * (9 * x ^ 4) * (8 * x ^ 3) = 18 * x ^ 8 := by\n    field_simp [h\u2080, step1, step2, step3]\n    <;> ring_nf\n    <;> simp_all [h\u2080, mul_assoc]\n    <;> ring_nf\n  simp_all [mul_assoc, mul_comm, mul_left_comm]\n  <;> norm_num\n  <;> field_simp [h\u2080]\n  <;> ring_nf\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_245 (x : \u211d) (h\u2080 : x \u2260 0) :\n  (4 / x)\u207b\u00b9 * (3 * x ^ 3 / x) ^ 2 * (1 / (2 * x))\u207b\u00b9 ^ 3 = 18 * x ^ 8  :=  by\n  have step1 : (4 / x)\u207b\u00b9 = x / 4 := by\n    field_simp [h\u2080, mul_comm]\n    <;> simp_all [h\u2080]\n    <;> linarith\n  have step2 : (3 * x ^ 3 / x) ^ 2 = 9 * x ^ 4 := by\n    have h\u2081 : x \u2260 0 := h\u2080\n    have h\u2082 : (4 / x)\u207b\u00b9 = x / 4 := step1\n    field_simp [h\u2081] at h\u2082 \u22a2\n    ring_nf at h\u2082 \u22a2\n    <;> nlinarith\n  have step3 : (1 / (2 * x))\u207b\u00b9 ^ 3 = 8 * x ^ 3 := by\n    have h1 : x \u2260 0 := by\n      intro H\n      apply h\u2080\n      linarith\n    field_simp [h1] at *\n    linarith\n  have step4 : (x / 4) * (9 * x ^ 4) * (8 * x ^ 3) = 18 * x ^ 8 := by\n    field_simp [h\u2080, step1, step2, step3]\n    <;> ring_nf\n    <;> simp_all [h\u2080, mul_assoc]\n    <;> ring_nf\n  simp_all [mul_assoc, mul_comm, mul_left_comm]\n  <;> norm_num\n  <;> field_simp [h\u2080]\n  <;> ring_nf\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 210, "alphabet": 60, "compiles": false, "id": "aime_1991_p6", "proof_after": "theorem aime_1991_p6 (r : \u211d) (h\u2080 : (\u2211 k in Finset.Icc (19 : \u2115) 91, Int.floor (r + k / 100)) = 546) :\n  Int.floor (100 * r) = 743  :=  by\n  have h\u2081 : r + 57 / 100 \u2265 8 := by\n    by_contra h\n    have h\u2082 : r + 57 / 100 < 8 := by linarith\n    have h\u2083 : \u2200 (k : \u2115), k \u2208 Finset.Icc 19 91 \u2192 k \u2264 57 \u2192 Int.floor (r + k / 100) \u2264 7 := by\n      intro k hk hk'\n      have h\u2084 : (k : \u211d) \u2264 57 := by exact_mod_cast hk'\n      have h\u2085 : r + k / 100 < 8 := by\n        have h\u2085\u2081 : (k : \u211d) \u2264 57 := by exact_mod_cast hk'\n        have h\u2085\u2082 : r + (k : \u211d) / 100 < 8 := by\n          have h\u2085\u2083 : r + (57 : \u211d) / 100 < 8 := by exact_mod_cast h\u2082\n          have h\u2085\u2084 : (k : \u211d) \u2264 57 := by exact_mod_cast hk'\n          have h\u2085\u2085 : (k : \u211d) / 100 \u2264 (57 : \u211d) / 100 := by\n            exact div_le_div_of_le_of_nonneg h\u2085\u2084 (by norm_num)\n          linarith\n        simpa using h\u2085\u2082\n      have h\u2086 : Int.floor (r + k / 100) \u2264 7 := by\n        have h\u2086\u2081 : Int.floor (r + k / 100) < 8 := by\n          apply Int.floor_lt.mpr\n          norm_num at h\u2085 \u22a2\n          <;> linarith\n        have h\u2086\u2082 : Int.floor (r + k / 100) \u2264 7 := by\n          omega\n        exact h\u2086\u2082\n      exact h\u2086\n    have h\u2084 : \u2200 (k : \u2115), k \u2208 Finset.Icc 19 91 \u2192 k \u2265 58 \u2192 Int.floor (r + k / 100) \u2264 8 := by\n      intro k hk hk'\n      have h\u2085 : (k : \u211d) \u2265 58 := by exact_mod_cast hk'\n      have h\u2086 : r + k / 100 < 9 := by\n        have h\u2086\u2081 : (k : \u211d) \u2264 91 := by\n          norm_num [Finset.mem_Icc] at hk\n          <;> exact_mod_cast hk.2\n        have h\u2086\u2082 : r + (k : \u211d) / 100 < 9 := by\n          have h\u2086\u2083 : r + (57 : \u211d) / 100 < 8 := by exact_mod_cast h\u2082\n          have h\u2086\u2084 : (k : \u211d) \u2265 58 := by exact_mod_cast hk'\n          have h\u2086\u2085 : (k : \u211d) / 100 \u2264 (91 : \u211d) / 100 := by\n            have h\u2086\u2085\u2081 : (k : \u211d) \u2264 91 := by exact_mod_cast (by norm_num [Finset.mem_Icc] at hk; exact hk.2)\n            have h\u2086\u2085\u2082 : (k : \u211d) / 100 \u2264 (91 : \u211d) / 100 := by\n              exact div_le_div_of_le_of_nonneg h\u2086\u2085\u2081 (by norm_num)\n            exact h\u2086\u2085\u2082\n          have h\u2086\u2086 : r + (k : \u211d) / 100 < 9 := by\n            linarith\n          simpa using h\u2086\u2086\n        simpa using h\u2086\u2082\n      have h\u2087 : Int.floor (r + k / 100) \u2264 8 := by\n        have h\u2087\u2081 : Int.floor (r + k / 100) < 9 := by\n          apply Int.floor_lt.mpr\n          norm_num at h\u2086 \u22a2\n          <;> linarith\n        have h\u2087\u2082 : Int.floor (r + k / 100) \u2264 8 := by\n          omega\n        exact h\u2087\u2082\n      exact h\u2087\n    have h\u2085 : (\u2211 k in Finset.Icc (19 : \u2115) 91, Int.floor (r + k / 100)) \u2264 545 := by\n      have h\u2085\u2081 : (\u2211 k in Finset.Icc (19 : \u2115) 91, Int.floor (r + k / 100)) \u2264 \u2211 k in Finset.Icc (19 : \u2115) 91, (if k \u2264 57 then 7 else 8 : \u2124) := by\n        apply Finset.sum_le_sum\n        intro k hk\n        have h\u2085\u2082 : k \u2208 Finset.Icc 19 91 := by simpa using hk\n        have h\u2085\u2083 : 19 \u2264 k \u2227 k \u2264 91 := by simpa using h\u2085\u2082\n        by_cases h\u2085\u2084 : k \u2264 57\n        \u00b7 have h\u2085\u2085 : Int.floor (r + k / 100) \u2264 7 := by\n            exact h\u2083 k h\u2085\u2082 (by linarith)\n          have h\u2085\u2086 : (if k \u2264 57 then 7 else 8 : \u2124) = 7 := by\n            simp [h\u2085\u2084]\n          rw [h\u2085\u2086]\n          exact h\u2085\u2085\n        \u00b7 have h\u2085\u2085 : k \u2265 58 := by\n            omega\n          have h\u2085\u2086 : Int.floor (r + k / 100) \u2264 8 := by\n            exact h\u2084 k h\u2085\u2082 (by omega)\n          have h\u2085\u2087 : (if k \u2264 57 then 7 else 8 : \u2124) = 8 := by\n            simp [h\u2085\u2084]\n          rw [h\u2085\u2087]\n          have h\u2085\u2088 : Int.floor (r + k / 100) \u2264 8 := by\n            exact h\u2084 k h\u2085\u2082 (by omega)\n          linarith\n      have h\u2085\u2082 : \u2211 k in Finset.Icc (19 : \u2115) 91, (if k \u2264 57 then 7 else 8 : \u2124) = 545 := by\n        norm_num [Finset.sum_Icc_succ_top, Finset.sum_range_succ, Finset.sum_range_zero, Finset.sum_singleton]\n        <;> rfl\n      linarith\n    linarith\n  have h\u2082 : r + 56 / 100 < 8 := by\n    by_contra h\n    have h\u2082 : r + 56 / 100 \u2265 8 := by linarith\n    have h\u2083 : \u2200 (k : \u2115), k \u2208 Finset.Icc 19 91 \u2192 k \u2265 56 \u2192 Int.floor (r + k / 100) \u2265 8 := by\n      intro k hk hk'\n      have h\u2084 : (k : \u211d) \u2265 56 := by exact_mod_cast hk'\n      have h\u2085 : r + k / 100 \u2265 8 := by\n        have h\u2085\u2081 : (k : \u211d) \u2265 56 := by exact_mod_cast hk'\n        have h\u2085\u2082 : r + (k : \u211d) / 100 \u2265 8 := by\n          have h\u2085\u2083 : r + (56 : \u211d) / 100 \u2265 8 := by exact_mod_cast h\u2082\n          have h\u2085\u2084 : (k : \u211d) \u2265 56 := by exact_mod_cast hk'\n          have h\u2085\u2085 : (k : \u211d) / 100 \u2265 (56 : \u211d) / 100 := by\n            exact div_le_div_of_le_of_nonneg h\u2085\u2084 (by norm_num)\n          linarith\n        simpa using h\u2085\u2082\n      have h\u2086 : Int.floor (r + k / 100) \u2265 8 := by\n        have h\u2086\u2081 : (8 : \u211d) \u2264 r + k / 100 := by exact_mod_cast h\u2085\n        have h\u2086\u2082 : (8 : \u2124) \u2264 Int.floor (r + k / 100) := by\n          exact Int.le_floor.mpr h\u2086\u2081\n        exact_mod_cast h\u2086\u2082\n      exact h\u2086\n    have h\u2084 : \u2200 (k : \u2115), k \u2208 Finset.Icc 19 91 \u2192 k < 56 \u2192 Int.floor (r + k / 100) \u2265 7 := by\n      intro k hk hk'\n      have h\u2085 : (k : \u211d) < 56 := by exact_mod_cast hk'\n      have h\u2086 : r + k / 100 \u2265 7 := by\n        have h\u2086\u2081 : (k : \u211d) \u2265 19 := by\n          norm_num [Finset.mem_Icc] at hk\n          <;> exact_mod_cast hk.1\n        have h\u2086\u2082 : r + (k : \u211d) / 100 \u2265 7 := by\n          have h\u2086\u2083 : r + (56 : \u211d) / 100 \u2265 8 := by exact_mod_cast h\u2082\n          have h\u2086\u2084 : (k : \u211d) < 56 := by exact_mod_cast hk'\n          have h\u2086\u2085 : (k : \u211d) / 100 \u2264 (56 : \u211d) / 100 := by\n            exact div_le_div_of_le_of_nonneg (by linarith) (by norm_num)\n          have h\u2086\u2086 : r + (k : \u211d) / 100 \u2265 7 := by\n            have h\u2086\u2087 : r + (19 : \u211d) / 100 \u2265 7 := by\n              have h\u2086\u2087\u2081 : r + (57 : \u211d) / 100 \u2265 8 := by exact_mod_cast h\u2081\n              have h\u2086\u2087\u2082 : r + (19 : \u211d) / 100 \u2265 7 := by\n                linarith\n              exact h\u2086\u2087\u2082\n            linarith\n          simpa using h\u2086\u2086\n        simpa using h\u2086\u2082\n      have h\u2087 : Int.floor (r + k / 100) \u2265 7 := by\n        have h\u2087\u2081 : (7 : \u211d) \u2264 r + k / 100 := by exact_mod_cast h\u2086\n        have h\u2087\u2082 : (7 : \u2124) \u2264 Int.floor (r + k / 100) := by\n          exact Int.le_floor.mpr h\u2087\u2081\n        exact_mod_cast h\u2087\u2082\n      exact h\u2087\n    have h\u2085 : (\u2211 k in Finset.Icc (19 : \u2115) 91, Int.floor (r + k / 100)) \u2265 547 := by\n      have h\u2085\u2081 : (\u2211 k in Finset.Icc (19 : \u2115) 91, Int.floor (r + k / 100)) \u2265 \u2211 k in Finset.Icc (19 : \u2115) 91, (if k \u2265 56 then 8 else 7 : \u2124) := by\n        apply Finset.sum_le_sum\n        intro k hk\n        have h\u2085\u2082 : k \u2208 Finset.Icc 19 91 := by simpa using hk\n        have h\u2085\u2083 : 19 \u2264 k \u2227 k \u2264 91 := by simpa using h\u2085\u2082\n        by_cases h\u2085\u2084 : k \u2265 56\n        \u00b7 have h\u2085\u2085 : Int.floor (r + k / 100) \u2265 8 := by\n            exact h\u2083 k h\u2085\u2082 (by linarith)\n          have h\u2085\u2086 : (if k \u2265 56 then 8 else 7 : \u2124) = 8 := by\n            simp [h\u2085\u2084]\n          rw [h\u2085\u2086]\n          exact h\u2085\u2085\n        \u00b7 have h\u2085\u2085 : k < 56 := by\n            omega\n          have h\u2085\u2086 : Int.floor (r + k / 100) \u2265 7 := by\n            exact h\u2084 k h\u2085\u2082 (by omega)\n          have h\u2085\u2087 : (if k \u2265 56 then 8 else 7 : \u2124) = 7 := by\n            simp [h\u2085\u2084]\n          rw [h\u2085\u2087]\n          exact h\u2085\u2086\n      have h\u2085\u2082 : \u2211 k in Finset.Icc (19 : \u2115) 91, (if k \u2265 56 then 8 else 7 : \u2124) = 547 := by\n        norm_num [Finset.sum_Icc_succ_top, Finset.sum_range_succ, Finset.sum_range_zero, Finset.sum_singleton]\n        <;> rfl\n      linarith\n    linarith\n  have h\u2083 : Int.floor (100 * r) = 743 := by\n    have h\u2084 : (743 : \u211d) \u2264 100 * r := by\n      have h\u2084\u2081 : (r + (57 : \u211d) / 100 \u2265 8) := by exact_mod_cast h\u2081\n      have h\u2084\u2082 : (100 * r \u2265 743) := by\n        norm_num at h\u2084\u2081 \u22a2\n        <;> linarith\n      exact_mod_cast h\u2084\u2082\n    have h\u2085 : (100 * r : \u211d) < 744 := by\n      have h\u2085\u2081 : (r + (56 : \u211d) / 100 < 8) := by exact_mod_cast h\u2082\n      have h\u2085\u2082 : (100 * r : \u211d) < 744 := by\n        norm_num at h\u2085\u2081 \u22a2\n        <;> linarith\n      exact h\u2085\u2082\n    have h\u2086 : Int.floor (100 * r : \u211d) = 743 := by\n      rw [Int.floor_eq_iff]\n      norm_num at h\u2084 h\u2085 \u22a2\n      constructor <;>\n      (try norm_num at h\u2081 h\u2082 \u22a2) <;>\n      (try linarith) <;>\n      (try assumption) <;>\n      (try norm_num) <;>\n      (try linarith)\n    exact_mod_cast h\u2086\n  exact h\u2083\n", "proof_before": "theorem aime_1991_p6 (r : \u211d) (h\u2080 : (\u2211 k in Finset.Icc (19 : \u2115) 91, Int.floor (r + k / 100)) = 546) :\n  Int.floor (100 * r) = 743  :=  by\n  have h\u2081 : r + 57 / 100 \u2265 8 := by\n    by_contra h\n    have h\u2082 : r + 57 / 100 < 8 := by linarith\n    have h\u2083 : \u2200 (k : \u2115), k \u2208 Finset.Icc 19 91 \u2192 k \u2264 57 \u2192 Int.floor (r + k / 100) \u2264 7 := by\n      intro k hk hk'\n      have h\u2084 : (k : \u211d) \u2264 57 := by exact_mod_cast hk'\n      have h\u2085 : r + k / 100 < 8 := by\n        have h\u2085\u2081 : (k : \u211d) \u2264 57 := by exact_mod_cast hk'\n        have h\u2085\u2082 : r + (k : \u211d) / 100 < 8 := by\n          have h\u2085\u2083 : r + (57 : \u211d) / 100 < 8 := by exact_mod_cast h\u2082\n          have h\u2085\u2084 : (k : \u211d) \u2264 57 := by exact_mod_cast hk'\n          have h\u2085\u2085 : (k : \u211d) / 100 \u2264 (57 : \u211d) / 100 := by\n            exact div_le_div_of_le_of_nonneg h\u2085\u2084 (by norm_num)\n          linarith\n        simpa using h\u2085\u2082\n      have h\u2086 : Int.floor (r + k / 100) \u2264 7 := by\n        have h\u2086\u2081 : Int.floor (r + k / 100) < 8 := by\n          apply Int.floor_lt.mpr\n          norm_num at h\u2085 \u22a2\n          <;> linarith\n        have h\u2086\u2082 : Int.floor (r + k / 100) \u2264 7 := by\n          omega\n        exact h\u2086\u2082\n      exact h\u2086\n    have h\u2084 : \u2200 (k : \u2115), k \u2208 Finset.Icc 19 91 \u2192 k \u2265 58 \u2192 Int.floor (r + k / 100) \u2264 8 := by\n      intro k hk hk'\n      have h\u2085 : (k : \u211d) \u2265 58 := by exact_mod_cast hk'\n      have h\u2086 : r + k / 100 < 9 := by\n        have h\u2086\u2081 : (k : \u211d) \u2264 91 := by\n          norm_num [Finset.mem_Icc] at hk\n          <;> exact_mod_cast hk.2\n        have h\u2086\u2082 : r + (k : \u211d) / 100 < 9 := by\n          have h\u2086\u2083 : r + (57 : \u211d) / 100 < 8 := by exact_mod_cast h\u2082\n          have h\u2086\u2084 : (k : \u211d) \u2265 58 := by exact_mod_cast hk'\n          have h\u2086\u2085 : (k : \u211d) / 100 \u2264 (91 : \u211d) / 100 := by\n            have h\u2086\u2085\u2081 : (k : \u211d) \u2264 91 := by exact_mod_cast (by norm_num [Finset.mem_Icc] at hk; exact hk.2)\n            have h\u2086\u2085\u2082 : (k : \u211d) / 100 \u2264 (91 : \u211d) / 100 := by\n              exact div_le_div_of_le_of_nonneg h\u2086\u2085\u2081 (by norm_num)\n            exact h\u2086\u2085\u2082\n          have h\u2086\u2086 : r + (k : \u211d) / 100 < 9 := by\n            linarith\n          simpa using h\u2086\u2086\n        simpa using h\u2086\u2082\n      have h\u2087 : Int.floor (r + k / 100) \u2264 8 := by\n        have h\u2087\u2081 : Int.floor (r + k / 100) < 9 := by\n          apply Int.floor_lt.mpr\n          norm_num at h\u2086 \u22a2\n          <;> linarith\n        have h\u2087\u2082 : Int.floor (r + k / 100) \u2264 8 := by\n          omega\n        exact h\u2087\u2082\n      exact h\u2087\n    have h\u2085 : (\u2211 k in Finset.Icc (19 : \u2115) 91, Int.floor (r + k / 100)) \u2264 545 := by\n      have h\u2085\u2081 : (\u2211 k in Finset.Icc (19 : \u2115) 91, Int.floor (r + k / 100)) \u2264 \u2211 k in Finset.Icc (19 : \u2115) 91, (if k \u2264 57 then 7 else 8 : \u2124) := by\n        apply Finset.sum_le_sum\n        intro k hk\n        have h\u2085\u2082 : k \u2208 Finset.Icc 19 91 := by simpa using hk\n        have h\u2085\u2083 : 19 \u2264 k \u2227 k \u2264 91 := by simpa using h\u2085\u2082\n        by_cases h\u2085\u2084 : k \u2264 57\n        \u00b7 have h\u2085\u2085 : Int.floor (r + k / 100) \u2264 7 := by\n            exact h\u2083 k h\u2085\u2082 (by linarith)\n          have h\u2085\u2086 : (if k \u2264 57 then 7 else 8 : \u2124) = 7 := by\n            simp [h\u2085\u2084]\n          rw [h\u2085\u2086]\n          exact h\u2085\u2085\n        \u00b7 have h\u2085\u2085 : k \u2265 58 := by\n            omega\n          have h\u2085\u2086 : Int.floor (r + k / 100) \u2264 8 := by\n            exact h\u2084 k h\u2085\u2082 (by omega)\n          have h\u2085\u2087 : (if k \u2264 57 then 7 else 8 : \u2124) = 8 := by\n            simp [h\u2085\u2084]\n          rw [h\u2085\u2087]\n          have h\u2085\u2088 : Int.floor (r + k / 100) \u2264 8 := by\n            exact h\u2084 k h\u2085\u2082 (by omega)\n          linarith\n      have h\u2085\u2082 : \u2211 k in Finset.Icc (19 : \u2115) 91, (if k \u2264 57 then 7 else 8 : \u2124) = 545 := by\n        norm_num [Finset.sum_Icc_succ_top, Finset.sum_range_succ, Finset.sum_range_zero, Finset.sum_singleton]\n        <;> rfl\n      linarith\n    linarith\n  have h\u2082 : r + 56 / 100 < 8 := by\n    by_contra h\n    have h\u2082 : r + 56 / 100 \u2265 8 := by linarith\n    have h\u2083 : \u2200 (k : \u2115), k \u2208 Finset.Icc 19 91 \u2192 k \u2265 56 \u2192 Int.floor (r + k / 100) \u2265 8 := by\n      intro k hk hk'\n      have h\u2084 : (k : \u211d) \u2265 56 := by exact_mod_cast hk'\n      have h\u2085 : r + k / 100 \u2265 8 := by\n        have h\u2085\u2081 : (k : \u211d) \u2265 56 := by exact_mod_cast hk'\n        have h\u2085\u2082 : r + (k : \u211d) / 100 \u2265 8 := by\n          have h\u2085\u2083 : r + (56 : \u211d) / 100 \u2265 8 := by exact_mod_cast h\u2082\n          have h\u2085\u2084 : (k : \u211d) \u2265 56 := by exact_mod_cast hk'\n          have h\u2085\u2085 : (k : \u211d) / 100 \u2265 (56 : \u211d) / 100 := by\n            exact div_le_div_of_le_of_nonneg h\u2085\u2084 (by norm_num)\n          linarith\n        simpa using h\u2085\u2082\n      have h\u2086 : Int.floor (r + k / 100) \u2265 8 := by\n        have h\u2086\u2081 : (8 : \u211d) \u2264 r + k / 100 := by exact_mod_cast h\u2085\n        have h\u2086\u2082 : (8 : \u2124) \u2264 Int.floor (r + k / 100) := by\n          exact Int.le_floor.mpr h\u2086\u2081\n        exact_mod_cast h\u2086\u2082\n      exact h\u2086\n    have h\u2084 : \u2200 (k : \u2115), k \u2208 Finset.Icc 19 91 \u2192 k < 56 \u2192 Int.floor (r + k / 100) \u2265 7 := by\n      intro k hk hk'\n      have h\u2085 : (k : \u211d) < 56 := by exact_mod_cast hk'\n      have h\u2086 : r + k / 100 \u2265 7 := by\n        have h\u2086\u2081 : (k : \u211d) \u2265 19 := by\n          norm_num [Finset.mem_Icc] at hk\n          <;> exact_mod_cast hk.1\n        have h\u2086\u2082 : r + (k : \u211d) / 100 \u2265 7 := by\n          have h\u2086\u2083 : r + (56 : \u211d) / 100 \u2265 8 := by exact_mod_cast h\u2082\n          have h\u2086\u2084 : (k : \u211d) < 56 := by exact_mod_cast hk'\n          have h\u2086\u2085 : (k : \u211d) / 100 \u2264 (56 : \u211d) / 100 := by\n            exact div_le_div_of_le_of_nonneg (by linarith) (by norm_num)\n          have h\u2086\u2086 : r + (k : \u211d) / 100 \u2265 7 := by\n            have h\u2086\u2087 : r + (19 : \u211d) / 100 \u2265 7 := by\n              have h\u2086\u2087\u2081 : r + (57 : \u211d) / 100 \u2265 8 := by exact_mod_cast h\u2081\n              have h\u2086\u2087\u2082 : r + (19 : \u211d) / 100 \u2265 7 := by\n                linarith\n              exact h\u2086\u2087\u2082\n            linarith\n          simpa using h\u2086\u2086\n        simpa using h\u2086\u2082\n      have h\u2087 : Int.floor (r + k / 100) \u2265 7 := by\n        have h\u2087\u2081 : (7 : \u211d) \u2264 r + k / 100 := by exact_mod_cast h\u2086\n        have h\u2087\u2082 : (7 : \u2124) \u2264 Int.floor (r + k / 100) := by\n          exact Int.le_floor.mpr h\u2087\u2081\n        exact_mod_cast h\u2087\u2082\n      exact h\u2087\n    have h\u2085 : (\u2211 k in Finset.Icc (19 : \u2115) 91, Int.floor (r + k / 100)) \u2265 547 := by\n      have h\u2085\u2081 : (\u2211 k in Finset.Icc (19 : \u2115) 91, Int.floor (r + k / 100)) \u2265 \u2211 k in Finset.Icc (19 : \u2115) 91, (if k \u2265 56 then 8 else 7 : \u2124) := by\n        apply Finset.sum_le_sum\n        intro k hk\n        have h\u2085\u2082 : k \u2208 Finset.Icc 19 91 := by simpa using hk\n        have h\u2085\u2083 : 19 \u2264 k \u2227 k \u2264 91 := by simpa using h\u2085\u2082\n        by_cases h\u2085\u2084 : k \u2265 56\n        \u00b7 have h\u2085\u2085 : Int.floor (r + k / 100) \u2265 8 := by\n            exact h\u2083 k h\u2085\u2082 (by linarith)\n          have h\u2085\u2086 : (if k \u2265 56 then 8 else 7 : \u2124) = 8 := by\n            simp [h\u2085\u2084]\n          rw [h\u2085\u2086]\n          exact h\u2085\u2085\n        \u00b7 have h\u2085\u2085 : k < 56 := by\n            omega\n          have h\u2085\u2086 : Int.floor (r + k / 100) \u2265 7 := by\n            exact h\u2084 k h\u2085\u2082 (by omega)\n          have h\u2085\u2087 : (if k \u2265 56 then 8 else 7 : \u2124) = 7 := by\n            simp [h\u2085\u2084]\n          rw [h\u2085\u2087]\n          exact h\u2085\u2086\n      have h\u2085\u2082 : \u2211 k in Finset.Icc (19 : \u2115) 91, (if k \u2265 56 then 8 else 7 : \u2124) = 547 := by\n        norm_num [Finset.sum_Icc_succ_top, Finset.sum_range_succ, Finset.sum_range_zero, Finset.sum_singleton]\n        <;> rfl\n      linarith\n    linarith\n  have h\u2083 : Int.floor (100 * r) = 743 := by\n    have h\u2084 : (743 : \u211d) \u2264 100 * r := by\n      have h\u2084\u2081 : (r + (57 : \u211d) / 100 \u2265 8) := by exact_mod_cast h\u2081\n      have h\u2084\u2082 : (100 * r \u2265 743) := by\n        norm_num at h\u2084\u2081 \u22a2\n        <;> linarith\n      exact_mod_cast h\u2084\u2082\n    have h\u2085 : (100 * r : \u211d) < 744 := by\n      have h\u2085\u2081 : (r + (56 : \u211d) / 100 < 8) := by exact_mod_cast h\u2082\n      have h\u2085\u2082 : (100 * r : \u211d) < 744 := by\n        norm_num at h\u2085\u2081 \u22a2\n        <;> linarith\n      exact h\u2085\u2082\n    have h\u2086 : Int.floor (100 * r : \u211d) = 743 := by\n      rw [Int.floor_eq_iff]\n      norm_num at h\u2084 h\u2085 \u22a2\n      constructor <;>\n      (try norm_num at h\u2081 h\u2082 \u22a2) <;>\n      (try linarith) <;>\n      (try assumption) <;>\n      (try norm_num) <;>\n      (try linarith)\n    exact_mod_cast h\u2086\n  exact h\u2083\n"}
{"mode": "clear", "steps_before": 7, "alphabet": 13, "compiles": true, "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 169, "chars_after": 6, "seconds": 0.9, "id": "mathd_numbertheory_200", "proof_after": "theorem mathd_numbertheory_200 : 139 % 11 = 7  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_200 : 139 % 11 = 7  :=  by\n  have remainder_calculation : 139 % 11 = 7 := by\n    norm_num\n    <;> rfl\n    <;> norm_num\n    <;> rfl\n  have h : 139 % 11 = 7 := remainder_calculation\n  simpa using h\n"}
{"mode": "clear", "steps_before": 26, "alphabet": 18, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 517, "chars_after": 6, "seconds": 1.3, "id": "mathd_numbertheory_81", "proof_after": "theorem mathd_numbertheory_81 : 71 % 3 = 2  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_81 : 71 % 3 = 2  :=  by\n  have h1 : \u2203 q r, 71 = 3 * q + r \u2227 0 \u2264 r \u2227 r < 3 := by\n    use 23, 2\n    constructor\n    norm_num\n    constructor\n    norm_num\n    norm_num\n  have h2 : \u2203 q, 71 = 3 * q + 2 := by\n    obtain \u27e8q, r, h2, h3, h4\u27e9 := h1\n    use q\n    have h5 : r = 2 := by\n      omega\n    linarith\n  have h3 : 2 < 3 := by\n    norm_num\n    <;> simp_all\n    <;> omega\n  have h4 : 71 % 3 = 2 := by\n    obtain \u27e8q, hq\u27e9 := h2\n    norm_num [Nat.mod_eq_of_lt h3, hq]\n    <;> omega\n  simp [h4]\n  <;> norm_num\n  <;> aesop\n  <;> norm_num\n  <;> aesop\n"}
{"mode": "clear", "steps_before": 35, "alphabet": 21, "compiles": false, "id": "mathd_algebra_77", "proof_after": "theorem mathd_algebra_77 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : a \u2260 0 \u2227 b \u2260 0) (h\u2081 : a \u2260 b)\n  (h\u2082 : \u2200 x, f x = x ^ 2 + a * x + b) (h\u2083 : f a = 0) (h\u2084 : f b = 0) : a = 1 \u2227 b = -2  :=  by\n  have h_sum : a + b = -a := by\n    simp_all only [h\u2082, h\u2083, h\u2084, mul_zero, mul_one, add_zero, zero_add, zero_mul, mul_one, mul_neg, neg_neg,\n      add_right_neg, add_zero]\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    ring_nf\n    <;> nlinarith\n  have h_prod : a * b = b := by\n    have h\u2085 : a * b = b := by\n      simp_all only [mul_comm, mul_one, mul_zero, add_zero, zero_add, sub_eq_add_neg, neg_mul, neg_neg]\n      apply mul_right_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n      nlinarith\n    exact h\u2085\n  have h_b_eq_neg_two_a : b = -2 * a := by\n    have h\u2085 := h\u2082 a\n    have h\u2086 := h\u2082 b\n    simp_all only [h\u2085, h\u2086, mul_comm, mul_one, add_assoc]\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    nlinarith\n  have h_a_squared_eq_neg_two_a : a * (-2 * a) = -2 * a := by\n    simp_all\n    <;> ring_nf\n    <;> nlinarith\n    <;> nlinarith\n  have h_a_eq_1 : a = 1 := by\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    ring_nf at h_a_squared_eq_neg_two_a \u22a2\n    nlinarith\n  have h_b_eq_neg_2 : b = -2 := by\n    have h\u2085 : b = -2 := by\n      apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n      linarith [h\u2082 a, h\u2082 b, h\u2083, h\u2084, h_sum, h_prod, h_b_eq_neg_two_a, h_a_squared_eq_neg_two_a, h_a_eq_1]\n    exact h\u2085\n  apply And.intro\n  all_goals\n    linarith [h\u2080.1, h\u2080.2, h\u2081, h\u2082, h\u2083, h\u2084, h_sum, h_prod, h_b_eq_neg_two_a, h_a_squared_eq_neg_two_a, h_a_eq_1, h_b_eq_neg_2]\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_77 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : a \u2260 0 \u2227 b \u2260 0) (h\u2081 : a \u2260 b)\n  (h\u2082 : \u2200 x, f x = x ^ 2 + a * x + b) (h\u2083 : f a = 0) (h\u2084 : f b = 0) : a = 1 \u2227 b = -2  :=  by\n  have h_sum : a + b = -a := by\n    simp_all only [h\u2082, h\u2083, h\u2084, mul_zero, mul_one, add_zero, zero_add, zero_mul, mul_one, mul_neg, neg_neg,\n      add_right_neg, add_zero]\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    ring_nf\n    <;> nlinarith\n  have h_prod : a * b = b := by\n    have h\u2085 : a * b = b := by\n      simp_all only [mul_comm, mul_one, mul_zero, add_zero, zero_add, sub_eq_add_neg, neg_mul, neg_neg]\n      apply mul_right_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n      nlinarith\n    exact h\u2085\n  have h_b_eq_neg_two_a : b = -2 * a := by\n    have h\u2085 := h\u2082 a\n    have h\u2086 := h\u2082 b\n    simp_all only [h\u2085, h\u2086, mul_comm, mul_one, add_assoc]\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    nlinarith\n  have h_a_squared_eq_neg_two_a : a * (-2 * a) = -2 * a := by\n    simp_all\n    <;> ring_nf\n    <;> nlinarith\n    <;> nlinarith\n  have h_a_eq_1 : a = 1 := by\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    ring_nf at h_a_squared_eq_neg_two_a \u22a2\n    nlinarith\n  have h_b_eq_neg_2 : b = -2 := by\n    have h\u2085 : b = -2 := by\n      apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n      linarith [h\u2082 a, h\u2082 b, h\u2083, h\u2084, h_sum, h_prod, h_b_eq_neg_two_a, h_a_squared_eq_neg_two_a, h_a_eq_1]\n    exact h\u2085\n  apply And.intro\n  all_goals\n    linarith [h\u2080.1, h\u2080.2, h\u2081, h\u2082, h\u2083, h\u2084, h_sum, h_prod, h_b_eq_neg_two_a, h_a_squared_eq_neg_two_a, h_a_eq_1, h_b_eq_neg_2]\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 8, "alphabet": 15, "compiles": false, "id": "aime_1983_p9", "proof_after": "theorem aime_1983_p9 (x : \u211d) (h\u2080 : 0 < x \u2227 x < Real.pi) :\n  12 \u2264 (9 * (x ^ 2 * Real.sin x ^ 2) + 4) / (x * Real.sin x)  :=  by\n  have h\u2081 : 0 < x := by linarith\n  have h\u2082 : 0 < sin x := by exact Real.sin_pos_of_pos_of_lt_pi h\u2081 h\u2080.2\n  have h\u2083 : 0 < x * sin x := mul_pos h\u2081 h\u2082\n  field_simp [h\u2081.ne', h\u2082.ne', h\u2083.ne']\n  rw [le_div_iff h\u2083]\n  nlinarith [sq_nonneg (x * sin x - 2 / 3), sq_nonneg (3 / 2 - x * sin x)]\n", "proof_before": "theorem aime_1983_p9 (x : \u211d) (h\u2080 : 0 < x \u2227 x < Real.pi) :\n  12 \u2264 (9 * (x ^ 2 * Real.sin x ^ 2) + 4) / (x * Real.sin x)  :=  by\n  have h\u2081 : 0 < x := by linarith\n  have h\u2082 : 0 < sin x := by exact Real.sin_pos_of_pos_of_lt_pi h\u2081 h\u2080.2\n  have h\u2083 : 0 < x * sin x := mul_pos h\u2081 h\u2082\n  field_simp [h\u2081.ne', h\u2082.ne', h\u2083.ne']\n  rw [le_div_iff h\u2083]\n  nlinarith [sq_nonneg (x * sin x - 2 / 3), sq_nonneg (3 / 2 - x * sin x)]\n"}
{"mode": "clear", "steps_before": 152, "alphabet": 51, "compiles": false, "id": "aimeI_2000_p7", "proof_after": "theorem aimeI_2000_p7 (x y z : \u211d) (m : \u211a) (h\u2080 : 0 < x \u2227 0 < y \u2227 0 < z) (h\u2081 : x * y * z = 1)\n  (h\u2082 : x + 1 / z = 5) (h\u2083 : y + 1 / x = 29) (h\u2084 : z + 1 / y = m) (h\u2085 : 0 < m) :\n  \u2191m.den + m.num = 5  :=  by\n  have z_expr : z = 1 / (x * y) := by\n    have h\u2086 : 0 < x * y := mul_pos h\u2080.1 h\u2080.2.1\n    have h\u2087 : 0 < x * y * z := mul_pos h\u2086 h\u2080.2.2\n    field_simp at h\u2082 h\u2083 h\u2084 \u22a2\n    nlinarith [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087]\n  have eq2 : x + 1 / (1 / (x * y)) = 5 := by\n    field_simp [h\u2080.1.ne', h\u2080.2.1.ne', h\u2080.2.2.ne', h\u2081] at *\n    nlinarith\n  have simplified_eq2 : x + x * y = 5 := by\n    have h\u2086 : 0 < x := h\u2080.1\n    have h\u2087 : 0 < y := h\u2080.2.1\n    have h\u2088 : 0 < z := h\u2080.2.2\n    field_simp [h\u2086, h\u2087, h\u2088, z_expr] at eq2 \u22a2\n    ring_nf at eq2 \u22a2\n    linarith\n  have factored_eq2 : x * (1 + y) = 5 := by\n    have : x * (1 + y) = 5 := by\n      nlinarith\n    linarith\n  have x_expr : x = 5 / (1 + y) := by\n    have h\u2086 : 0 < x := by linarith\n    have h\u2087 : 0 < y := by linarith\n    have h\u2088 : 0 < z := by linarith\n    field_simp [h\u2086, h\u2087, h\u2088] at eq2 simplified_eq2 factored_eq2 \u22a2\n    nlinarith\n  have eq3 : y + 1 / (5 / (1 + y)) = 29 := by\n    simp_all [div_eq_mul_inv]\n    <;> ring_nf\n    <;> field_simp\n    <;> ring_nf\n    <;> linarith\n  have simplified_eq3 : y + (1 + y) / 5 = 29 := by\n    have h\u2086 : y + (1 + y) / 5 = 29 := by\n      have h\u2086 := h\u2083\n      have h\u2087 := h\u2082\n      have h\u2088 := h\u2084\n      have h\u2089 := h\u2085\n      have h\u2081\u2080 := z_expr\n      have h\u2081\u2081 := eq2\n      have h\u2081\u2082 := simplified_eq2\n      have h\u2081\u2083 := factored_eq2\n      have h\u2081\u2084 := x_expr\n      have h\u2081\u2085 := eq3\n      norm_num at h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085\n      linarith\n    norm_num at h\u2086\n    linarith\n  have multiplied_eq3 : 5 * y + (1 + y) = 145 := by\n    field_simp at simplified_eq2 simplified_eq3\n    ring_nf at simplified_eq2 simplified_eq3\n    nlinarith\n  have combined_eq3 : 6 * y + 1 = 145 := by\n    ring_nf at simplified_eq2 factored_eq2 x_expr eq3 simplified_eq3 multiplied_eq3\n    linarith\n  have y_solution : y = 24 := by\n    have h\u2086 : y = 24 := by\n      linarith\n    linarith\n  have x_solution : x = 1 / 5 := by\n    have h\u2086 : x = 1 / 5 := by\n      rw [x_expr] \n      field_simp [y_solution] \n      ring \n    linarith\n  have z_solution : (1 / 5) * 24 * z = 1 := by\n    subst_vars\n    norm_num\n    <;> field_simp\n    <;> norm_num\n    <;> linarith\n  have z_val : z = 5 / 24 := by\n    have h\u2086 : z = 5 / 24 := by\n      have h\u2086 : (1 / 5) * 24 * z = 1 := z_solution\n      field_simp at h\u2086\n      linarith\n    linarith\n  have final_expr : z + 1 / y = 5 / 24 + 1 / 24 := by\n    norm_num [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, z_expr, eq2, simplified_eq2, factored_eq2, x_expr, eq3, simplified_eq3, multiplied_eq3, combined_eq3, y_solution, x_solution, z_solution, z_val]\n    <;> simp_all\n    <;> linarith\n  have simplified_final : z + 1 / y = 6 / 24 := by\n    field_simp [z_val, y_solution, x_solution] at final_expr \u22a2\n    <;> linarith\n  have reduced_final : z + 1 / y = 1 / 4 := by\n    norm_num at *\n    <;> linarith\n  have m_val : m = 1 / 4 := by\n    simp_all [div_eq_mul_inv]\n    <;> field_simp at *\n    <;> ring_nf at *\n    <;> norm_cast at *\n    <;> linarith\n    <;> nlinarith\n  have m_num : m.num = 1 := by\n    norm_num [m_val]\n    <;> simp_all\n    <;> norm_num\n    <;> field_simp\n    <;> ring\n    <;> linarith\n  have m_den : m.den = 4 := by\n    simp_all [mul_comm]\n    <;> norm_num\n    <;> field_simp [*, mul_comm] at *\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> linarith\n  have final_answer : \u2191m.den + m.num = 5 := by\n    norm_num [m_num, m_den]\n    <;> simp_all [div_eq_mul_inv]\n    <;> norm_num\n    <;> linarith\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n  simp_all [add_comm]\n  <;> norm_num\n  <;> linarith\n", "proof_before": "theorem aimeI_2000_p7 (x y z : \u211d) (m : \u211a) (h\u2080 : 0 < x \u2227 0 < y \u2227 0 < z) (h\u2081 : x * y * z = 1)\n  (h\u2082 : x + 1 / z = 5) (h\u2083 : y + 1 / x = 29) (h\u2084 : z + 1 / y = m) (h\u2085 : 0 < m) :\n  \u2191m.den + m.num = 5  :=  by\n  have z_expr : z = 1 / (x * y) := by\n    have h\u2086 : 0 < x * y := mul_pos h\u2080.1 h\u2080.2.1\n    have h\u2087 : 0 < x * y * z := mul_pos h\u2086 h\u2080.2.2\n    field_simp at h\u2082 h\u2083 h\u2084 \u22a2\n    nlinarith [h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087]\n  have eq2 : x + 1 / (1 / (x * y)) = 5 := by\n    field_simp [h\u2080.1.ne', h\u2080.2.1.ne', h\u2080.2.2.ne', h\u2081] at *\n    nlinarith\n  have simplified_eq2 : x + x * y = 5 := by\n    have h\u2086 : 0 < x := h\u2080.1\n    have h\u2087 : 0 < y := h\u2080.2.1\n    have h\u2088 : 0 < z := h\u2080.2.2\n    field_simp [h\u2086, h\u2087, h\u2088, z_expr] at eq2 \u22a2\n    ring_nf at eq2 \u22a2\n    linarith\n  have factored_eq2 : x * (1 + y) = 5 := by\n    have : x * (1 + y) = 5 := by\n      nlinarith\n    linarith\n  have x_expr : x = 5 / (1 + y) := by\n    have h\u2086 : 0 < x := by linarith\n    have h\u2087 : 0 < y := by linarith\n    have h\u2088 : 0 < z := by linarith\n    field_simp [h\u2086, h\u2087, h\u2088] at eq2 simplified_eq2 factored_eq2 \u22a2\n    nlinarith\n  have eq3 : y + 1 / (5 / (1 + y)) = 29 := by\n    simp_all [div_eq_mul_inv]\n    <;> ring_nf\n    <;> field_simp\n    <;> ring_nf\n    <;> linarith\n  have simplified_eq3 : y + (1 + y) / 5 = 29 := by\n    have h\u2086 : y + (1 + y) / 5 = 29 := by\n      have h\u2086 := h\u2083\n      have h\u2087 := h\u2082\n      have h\u2088 := h\u2084\n      have h\u2089 := h\u2085\n      have h\u2081\u2080 := z_expr\n      have h\u2081\u2081 := eq2\n      have h\u2081\u2082 := simplified_eq2\n      have h\u2081\u2083 := factored_eq2\n      have h\u2081\u2084 := x_expr\n      have h\u2081\u2085 := eq3\n      norm_num at h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085\n      linarith\n    norm_num at h\u2086\n    linarith\n  have multiplied_eq3 : 5 * y + (1 + y) = 145 := by\n    field_simp at simplified_eq2 simplified_eq3\n    ring_nf at simplified_eq2 simplified_eq3\n    nlinarith\n  have combined_eq3 : 6 * y + 1 = 145 := by\n    ring_nf at simplified_eq2 factored_eq2 x_expr eq3 simplified_eq3 multiplied_eq3\n    linarith\n  have y_solution : y = 24 := by\n    have h\u2086 : y = 24 := by\n      linarith\n    linarith\n  have x_solution : x = 1 / 5 := by\n    have h\u2086 : x = 1 / 5 := by\n      rw [x_expr] \n      field_simp [y_solution] \n      ring \n    linarith\n  have z_solution : (1 / 5) * 24 * z = 1 := by\n    subst_vars\n    norm_num\n    <;> field_simp\n    <;> norm_num\n    <;> linarith\n  have z_val : z = 5 / 24 := by\n    have h\u2086 : z = 5 / 24 := by\n      have h\u2086 : (1 / 5) * 24 * z = 1 := z_solution\n      field_simp at h\u2086\n      linarith\n    linarith\n  have final_expr : z + 1 / y = 5 / 24 + 1 / 24 := by\n    norm_num [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, z_expr, eq2, simplified_eq2, factored_eq2, x_expr, eq3, simplified_eq3, multiplied_eq3, combined_eq3, y_solution, x_solution, z_solution, z_val]\n    <;> simp_all\n    <;> linarith\n  have simplified_final : z + 1 / y = 6 / 24 := by\n    field_simp [z_val, y_solution, x_solution] at final_expr \u22a2\n    <;> linarith\n  have reduced_final : z + 1 / y = 1 / 4 := by\n    norm_num at *\n    <;> linarith\n  have m_val : m = 1 / 4 := by\n    simp_all [div_eq_mul_inv]\n    <;> field_simp at *\n    <;> ring_nf at *\n    <;> norm_cast at *\n    <;> linarith\n    <;> nlinarith\n  have m_num : m.num = 1 := by\n    norm_num [m_val]\n    <;> simp_all\n    <;> norm_num\n    <;> field_simp\n    <;> ring\n    <;> linarith\n  have m_den : m.den = 4 := by\n    simp_all [mul_comm]\n    <;> norm_num\n    <;> field_simp [*, mul_comm] at *\n    <;> ring_nf at *\n    <;> norm_num at *\n    <;> linarith\n  have final_answer : \u2191m.den + m.num = 5 := by\n    norm_num [m_num, m_den]\n    <;> simp_all [div_eq_mul_inv]\n    <;> norm_num\n    <;> linarith\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n    <;> field_simp\n    <;> norm_num\n    <;> ring\n  simp_all [add_comm]\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 30, "alphabet": 21, "compiles": false, "id": "mathd_algebra_251", "proof_after": "theorem mathd_algebra_251 (x : \u211d) (h\u2080 : x \u2260 0) (h\u2081 : 3 + 1 / x = 7 / x) : x = 2  :=  by\n  have h\u2082 : 3 + 1 / x = 7 / x := by\n    have h\u2082 : 3 + 1 / x = 7 / x := h\u2081\n    field_simp at h\u2082 \u22a2\n    linarith\n  have h\u2083 : 3 = 7 / x - 1 / x := by\n    apply Eq.symm\n    field_simp at h\u2081 h\u2082 \u22a2\n    linarith\n    <;> linarith\n  have h\u2084 : 3 = 6 / x := by\n    have h\u2084 : 3 = 6 / x := by\n      field_simp at h\u2083 \u22a2\n      linarith\n    linarith\n  have h\u2085 : 3 * x = 6 := by\n    have h\u2085 : x \u2260 0 := by\n      rintro rfl\n      simp_all\n    field_simp at h\u2081 h\u2082 h\u2083 h\u2084 \u22a2\n    linarith\n  have h\u2086 : x = 2 := by\n    have h\u2086 : x = 2 := by\n      apply Eq.symm\n      field_simp at h\u2085 h\u2084 h\u2083 h\u2082 h\u2081 h\u2080 \u22a2\n      linarith\n    apply h\u2086\n  have h\u2087 : x = 2 := h\u2086\n  have h\u2088 : x = 2 := by linarith\n  linarith\n", "proof_before": "theorem mathd_algebra_251 (x : \u211d) (h\u2080 : x \u2260 0) (h\u2081 : 3 + 1 / x = 7 / x) : x = 2  :=  by\n  have h\u2082 : 3 + 1 / x = 7 / x := by\n    have h\u2082 : 3 + 1 / x = 7 / x := h\u2081\n    field_simp at h\u2082 \u22a2\n    linarith\n  have h\u2083 : 3 = 7 / x - 1 / x := by\n    apply Eq.symm\n    field_simp at h\u2081 h\u2082 \u22a2\n    linarith\n    <;> linarith\n  have h\u2084 : 3 = 6 / x := by\n    have h\u2084 : 3 = 6 / x := by\n      field_simp at h\u2083 \u22a2\n      linarith\n    linarith\n  have h\u2085 : 3 * x = 6 := by\n    have h\u2085 : x \u2260 0 := by\n      rintro rfl\n      simp_all\n    field_simp at h\u2081 h\u2082 h\u2083 h\u2084 \u22a2\n    linarith\n  have h\u2086 : x = 2 := by\n    have h\u2086 : x = 2 := by\n      apply Eq.symm\n      field_simp at h\u2085 h\u2084 h\u2083 h\u2082 h\u2081 h\u2080 \u22a2\n      linarith\n    apply h\u2086\n  have h\u2087 : x = 2 := h\u2086\n  have h\u2088 : x = 2 := by linarith\n  linarith\n"}
{"mode": "clear", "steps_before": 54, "alphabet": 37, "compiles": false, "id": "imo_1973_p3", "proof_after": "theorem imo_1973_p3 (a b : \u211d) (h\u2080 : \u2203 x, x ^ 4 + a * x ^ 3 + b * x ^ 2 + a * x + 1 = 0) :\n  4 / 5 \u2264 a ^ 2 + b ^ 2  :=  by\n  have h\u2081 : \u2203 y, y ^ 2 + a * y + (b - 2) = 0 \u2227 (y \u2265 2 \u2228 y \u2264 -2) := by\n    obtain \u27e8x, hx\u27e9 := h\u2080\n    use (x ^ 2 + 1) / x\n    constructor\n    have hx1 : x \u2260 0 := by\n      intro h\n      rw [h] at hx\n      norm_num at hx\n    field_simp [hx1] at hx \u22a2\n    nlinarith [sq_nonneg (x ^ 2 + 1), sq_nonneg (x ^ 2 - 1), sq_nonneg (x + 1), sq_nonneg (x - 1)]\n    have hx1 : x \u2260 0 := by\n      intro h\n      rw [h] at hx\n      norm_num at hx\n    have hx2 : x > 0 \u2228 x < 0 := by\n      apply lt_or_gt_of_ne\n      intro h\n      rw [h] at hx1\n      norm_num at hx1\n    cases' hx2 with hx2 hx2\n    \u00b7 \n      apply or_iff_not_imp_left.mpr\n      intro h\n      have h1 : (x ^ 2 + 1) / x \u2265 2 := by\n        apply (le_div_iff (by positivity)).mpr\n        nlinarith [sq_nonneg (x - 1), sq_nonneg (x + 1)]\n      linarith\n    \u00b7 \n      apply or_iff_not_imp_right.mpr\n      intro h\n      have h1 : (x ^ 2 + 1) / x \u2264 -2 := by\n        apply (div_le_iff_of_neg (by linarith)).mpr\n        nlinarith [sq_nonneg (x - 1), sq_nonneg (x + 1)]\n      linarith\n  have h\u2082 : a ^ 2 - 4 * (b - 2) \u2265 0 := by\n    obtain \u27e8y, hy\u27e9 := h\u2081\n    have h\u2082 := hy.2\n    have h\u2083 := hy.1\n    have h\u2084 : a ^ 2 - 4 * (b - 2) \u2265 0 := by\n      cases' h\u2082 with h\u2082 h\u2082\n      \u00b7 nlinarith [sq_nonneg (y + 2), sq_nonneg (y - 2), sq_nonneg (a + 2 * y), sq_nonneg (a - 2 * y)]\n      \u00b7 nlinarith [sq_nonneg (y + 2), sq_nonneg (y - 2), sq_nonneg (a + 2 * y), sq_nonneg (a - 2 * y)]\n    exact h\u2084\n  have h\u2083 : a ^ 2 + b ^ 2 \u2265 4 / 5 := by\n    obtain \u27e8y, h_y, h_y_ineq\u27e9 := h\u2081\n    have h_y_ineq' := h_y_ineq\n    cases' h_y_ineq' with h_y_ge_2 h_y_le_neg_2 <;> simp_all\n    <;> nlinarith [sq_nonneg (y + 2), sq_nonneg (y - 2), sq_nonneg a, sq_nonneg (b - 2),\n      sq_nonneg (a - b * 2), sq_nonneg (a + b * 2), sq_nonneg (a ^ 2 - 4 * (b - 2))]\n  have h\u2083' : a ^ 2 + b ^ 2 \u2265 4 / 5 := by assumption\n  have h\u2082' : a ^ 2 - 4 * (b - 2) \u2265 0 := by assumption\n  nlinarith [sq_nonneg (a ^ 2 - 4 * (b - 2))]\n", "proof_before": "theorem imo_1973_p3 (a b : \u211d) (h\u2080 : \u2203 x, x ^ 4 + a * x ^ 3 + b * x ^ 2 + a * x + 1 = 0) :\n  4 / 5 \u2264 a ^ 2 + b ^ 2  :=  by\n  have h\u2081 : \u2203 y, y ^ 2 + a * y + (b - 2) = 0 \u2227 (y \u2265 2 \u2228 y \u2264 -2) := by\n    obtain \u27e8x, hx\u27e9 := h\u2080\n    use (x ^ 2 + 1) / x\n    constructor\n    have hx1 : x \u2260 0 := by\n      intro h\n      rw [h] at hx\n      norm_num at hx\n    field_simp [hx1] at hx \u22a2\n    nlinarith [sq_nonneg (x ^ 2 + 1), sq_nonneg (x ^ 2 - 1), sq_nonneg (x + 1), sq_nonneg (x - 1)]\n    have hx1 : x \u2260 0 := by\n      intro h\n      rw [h] at hx\n      norm_num at hx\n    have hx2 : x > 0 \u2228 x < 0 := by\n      apply lt_or_gt_of_ne\n      intro h\n      rw [h] at hx1\n      norm_num at hx1\n    cases' hx2 with hx2 hx2\n    \u00b7 \n      apply or_iff_not_imp_left.mpr\n      intro h\n      have h1 : (x ^ 2 + 1) / x \u2265 2 := by\n        apply (le_div_iff (by positivity)).mpr\n        nlinarith [sq_nonneg (x - 1), sq_nonneg (x + 1)]\n      linarith\n    \u00b7 \n      apply or_iff_not_imp_right.mpr\n      intro h\n      have h1 : (x ^ 2 + 1) / x \u2264 -2 := by\n        apply (div_le_iff_of_neg (by linarith)).mpr\n        nlinarith [sq_nonneg (x - 1), sq_nonneg (x + 1)]\n      linarith\n  have h\u2082 : a ^ 2 - 4 * (b - 2) \u2265 0 := by\n    obtain \u27e8y, hy\u27e9 := h\u2081\n    have h\u2082 := hy.2\n    have h\u2083 := hy.1\n    have h\u2084 : a ^ 2 - 4 * (b - 2) \u2265 0 := by\n      cases' h\u2082 with h\u2082 h\u2082\n      \u00b7 nlinarith [sq_nonneg (y + 2), sq_nonneg (y - 2), sq_nonneg (a + 2 * y), sq_nonneg (a - 2 * y)]\n      \u00b7 nlinarith [sq_nonneg (y + 2), sq_nonneg (y - 2), sq_nonneg (a + 2 * y), sq_nonneg (a - 2 * y)]\n    exact h\u2084\n  have h\u2083 : a ^ 2 + b ^ 2 \u2265 4 / 5 := by\n    obtain \u27e8y, h_y, h_y_ineq\u27e9 := h\u2081\n    have h_y_ineq' := h_y_ineq\n    cases' h_y_ineq' with h_y_ge_2 h_y_le_neg_2 <;> simp_all\n    <;> nlinarith [sq_nonneg (y + 2), sq_nonneg (y - 2), sq_nonneg a, sq_nonneg (b - 2),\n      sq_nonneg (a - b * 2), sq_nonneg (a + b * 2), sq_nonneg (a ^ 2 - 4 * (b - 2))]\n  have h\u2083' : a ^ 2 + b ^ 2 \u2265 4 / 5 := by assumption\n  have h\u2082' : a ^ 2 - 4 * (b - 2) \u2265 0 := by assumption\n  nlinarith [sq_nonneg (a ^ 2 - 4 * (b - 2))]\n"}
{"mode": "clear", "steps_before": 16, "alphabet": 18, "compiles": true, "steps_after": 9, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 4, "timed_out": false, "fixpoint": true, "chars_before": 406, "chars_after": 247, "seconds": 414.8, "id": "mathd_algebra_131", "proof_after": "theorem mathd_algebra_131 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 2 * x ^ 2 - 7 * x + 2)\n  (h\u2081 : f a = 0) (h\u2082 : f b = 0) (h\u2083 : a \u2260 b) : 1 / (a - 1) + 1 / (b - 1) = -1  :=  by\n  have h\u2085 : a * b = 1 := by\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2083)\n    nlinarith [h\u2080 a, h\u2080 b]\n  have h\u2086 : a - 1 \u2260 0 := by\n    nlinarith [h\u2080 a, h\u2080 b]\n  have h\u2087 : b - 1 \u2260 0 := by\n    nlinarith [h\u2080 a, h\u2080 b]\n  field_simp [h\u2086, h\u2087]\n  nlinarith\n", "proof_before": "theorem mathd_algebra_131 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = 2 * x ^ 2 - 7 * x + 2)\n  (h\u2081 : f a = 0) (h\u2082 : f b = 0) (h\u2083 : a \u2260 b) : 1 / (a - 1) + 1 / (b - 1) = -1  :=  by\n  have h\u2084 : a + b = 7 / 2 := by\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2083)\n    nlinarith [h\u2080 a, h\u2080 b]\n  have h\u2085 : a * b = 1 := by\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2083)\n    nlinarith [h\u2080 a, h\u2080 b]\n  have h\u2086 : a - 1 \u2260 0 := by\n    intro h\n    apply h\u2083\n    nlinarith [h\u2081, h\u2080 a, h]\n  have h\u2087 : b - 1 \u2260 0 := by\n    intro h\n    apply h\u2083\n    nlinarith [h\u2082, h\u2080 b, h]\n  field_simp [h\u2086, h\u2087]\n  nlinarith\n"}
{"mode": "clear", "steps_before": 75, "alphabet": 29, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 75, "lower_bound": null, "exhausted_depth": 2, "certified": false, "rounds": 2, "timed_out": true, "fixpoint": false, "chars_before": 2321, "chars_after": 2321, "seconds": 524.4, "id": "aimeII_2001_p3", "proof_after": "theorem aimeII_2001_p3 (x : \u2115 \u2192 \u2124) (h\u2080 : x 1 = 211) (h\u2082 : x 2 = 375) (h\u2083 : x 3 = 420)\n  (h\u2084 : x 4 = 523) (h\u2086 : \u2200 n \u2265 5, x n = x (n - 1) - x (n - 2) + x (n - 3) - x (n - 4)) :\n  x 531 + x 753 + x 975 = 898  :=  by\n  have x5 : x 5 = 267 := by\n    have h\u2085 : x 5 = 267 := by\n      rw [h\u2086]\n      norm_num [h\u2080, h\u2082, h\u2083, h\u2084]\n      linarith\n    simp_all\n  have x6 : x 6 = -211 := by\n    have h\u2085 : x 5 = 267 := x5\n    have h\u2086 : x 6 = x 5 - x 4 + x 3 - x 2 := h\u2086 6 (by norm_num)\n    simp [h\u2085, h\u2084, h\u2083, h\u2082] at h\u2086\n    linarith\n  have x7 : x 7 = -375 := by\n    simp_all [Nat.succ_eq_add_one, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    <;> linarith\n    <;> aesop\n  have x8 : x 8 = -420 := by\n    have h\u2088 := h\u2086 8 (by norm_num)\n    simp_all [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    <;> linarith\n    <;> ring\n    <;> linarith\n    <;> ring\n    <;> linarith\n  have x9 : x 9 = -523 := by\n    have h\u2089 := h\u2086 9 (by norm_num)\n    simp_all [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    <;> linarith\n  have x10 : x 10 = -267 := by\n    have h\u2081\u2080 := h\u2086 10 (by norm_num)\n    simp_all\n    <;> ring_nf\n    <;> linarith\n    <;> linarith\n  have periodicity : \u2200 n \u2265 1, x (n + 10) = x n := by\n    intro n hn\n    have h\u2081\u2081 := h\u2086 10 (by norm_num)\n    have h\u2081\u2082 := h\u2086 11 (by norm_num)\n    simp_all [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    all_goals\n      omega\n  have x531 : x 531 = 211 := by\n    norm_num [periodicity, h\u2080, h\u2082, h\u2083, h\u2084, x5, x6, x7, x8, x9, x10, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n  have x753 : x 753 = 420 := by\n    have h\u2081 : 753 % 10 = 3 := by norm_num\n    rw [\u2190 Nat.mod_add_div 753 10]\n    simp [h\u2081, periodicity, h\u2083]\n  have x975 : x 975 = 267 := by\n    norm_num [periodicity, h\u2080, h\u2082, h\u2083, h\u2084, x5, x6, x7, x8, x9, x10]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n  have final_result : x 531 + x 753 + x 975 = 898 := by\n    linarith [periodicity 1 (by norm_num), periodicity 2 (by norm_num), periodicity 3 (by norm_num), periodicity 4 (by norm_num), periodicity 5 (by norm_num), periodicity 6 (by norm_num), periodicity 7 (by norm_num), periodicity 8 (by norm_num), periodicity 9 (by norm_num), periodicity 10 (by norm_num)]\n  simpa using final_result\n", "proof_before": "theorem aimeII_2001_p3 (x : \u2115 \u2192 \u2124) (h\u2080 : x 1 = 211) (h\u2082 : x 2 = 375) (h\u2083 : x 3 = 420)\n  (h\u2084 : x 4 = 523) (h\u2086 : \u2200 n \u2265 5, x n = x (n - 1) - x (n - 2) + x (n - 3) - x (n - 4)) :\n  x 531 + x 753 + x 975 = 898  :=  by\n  have x5 : x 5 = 267 := by\n    have h\u2085 : x 5 = 267 := by\n      rw [h\u2086]\n      norm_num [h\u2080, h\u2082, h\u2083, h\u2084]\n      linarith\n    simp_all\n  have x6 : x 6 = -211 := by\n    have h\u2085 : x 5 = 267 := x5\n    have h\u2086 : x 6 = x 5 - x 4 + x 3 - x 2 := h\u2086 6 (by norm_num)\n    simp [h\u2085, h\u2084, h\u2083, h\u2082] at h\u2086\n    linarith\n  have x7 : x 7 = -375 := by\n    simp_all [Nat.succ_eq_add_one, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    <;> linarith\n    <;> aesop\n  have x8 : x 8 = -420 := by\n    have h\u2088 := h\u2086 8 (by norm_num)\n    simp_all [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    <;> linarith\n    <;> ring\n    <;> linarith\n    <;> ring\n    <;> linarith\n  have x9 : x 9 = -523 := by\n    have h\u2089 := h\u2086 9 (by norm_num)\n    simp_all [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    <;> linarith\n  have x10 : x 10 = -267 := by\n    have h\u2081\u2080 := h\u2086 10 (by norm_num)\n    simp_all\n    <;> ring_nf\n    <;> linarith\n    <;> linarith\n  have periodicity : \u2200 n \u2265 1, x (n + 10) = x n := by\n    intro n hn\n    have h\u2081\u2081 := h\u2086 10 (by norm_num)\n    have h\u2081\u2082 := h\u2086 11 (by norm_num)\n    simp_all [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n    all_goals\n      omega\n  have x531 : x 531 = 211 := by\n    norm_num [periodicity, h\u2080, h\u2082, h\u2083, h\u2084, x5, x6, x7, x8, x9, x10, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]\n  have x753 : x 753 = 420 := by\n    have h\u2081 : 753 % 10 = 3 := by norm_num\n    rw [\u2190 Nat.mod_add_div 753 10]\n    simp [h\u2081, periodicity, h\u2083]\n  have x975 : x 975 = 267 := by\n    norm_num [periodicity, h\u2080, h\u2082, h\u2083, h\u2084, x5, x6, x7, x8, x9, x10]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n    <;> simp_all [periodicity]\n    <;> linarith\n  have final_result : x 531 + x 753 + x 975 = 898 := by\n    linarith [periodicity 1 (by norm_num), periodicity 2 (by norm_num), periodicity 3 (by norm_num), periodicity 4 (by norm_num), periodicity 5 (by norm_num), periodicity 6 (by norm_num), periodicity 7 (by norm_num), periodicity 8 (by norm_num), periodicity 9 (by norm_num), periodicity 10 (by norm_num)]\n  simpa using final_result\n", "reverted": "failed independent check; original kept"}
{"mode": "clear", "steps_before": 21, "alphabet": 13, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 3, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "timed_out": false, "fixpoint": true, "chars_before": 449, "chars_after": 40, "seconds": 168.0, "id": "mathd_algebra_15", "proof_after": "theorem mathd_algebra_15 (s : \u2115 \u2192 \u2115 \u2192 \u2115)\n    (h\u2080 : \u2200 a b, 0 < a \u2227 0 < b \u2192 s a b = a ^ (b : \u2115) + b ^ (a : \u2115)) : s 2 6 = 100  :=  by\n  have h\u2081 := h\u2080 2 6 (by decide)\n  omega\n", "proof_before": "theorem mathd_algebra_15 (s : \u2115 \u2192 \u2115 \u2192 \u2115)\n    (h\u2080 : \u2200 a b, 0 < a \u2227 0 < b \u2192 s a b = a ^ (b : \u2115) + b ^ (a : \u2115)) : s 2 6 = 100  :=  by\n  have h\u2081 : s 2 6 = 2 ^ 6 + 6 ^ 2 := by\n    have h\u2081 := h\u2080 2 6 (by decide)\n    simpa using h\u2081\n  have h\u2082 : 2 ^ 6 = 64 := by\n    have h\u2082 : 0 < 2 \u2227 0 < 6 := by norm_num\n    simp_all [h\u2080, h\u2082]\n    <;> norm_num\n  have h\u2083 : 6 ^ 2 = 36 := by\n    simp_all [h\u2080]\n    <;> norm_num\n    <;> rfl\n  have h\u2084 : 64 + 36 = 100 := by\n    simp_all [h\u2080]\n    <;> norm_num\n    <;> rfl\n  simp_all [h\u2080, h\u2082, h\u2083, h\u2084]\n  <;> simp_all [h\u2080, h\u2082, h\u2083, h\u2084]\n  <;> norm_num\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 13, "alphabet": 18, "compiles": false, "id": "algebra_amgm_sumasqdivbsqgeqsumbdiva", "proof_after": "theorem algebra_amgm_sumasqdivbsqgeqsumbdiva (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) :\n  a ^ 2 / b ^ 2 + b ^ 2 / c ^ 2 + c ^ 2 / a ^ 2 \u2265 b / a + c / b + a / c  :=  by\n  have h\u2081 : 0 < a := by linarith\n  have h\u2082 : 0 < b := by linarith\n  have h\u2083 : 0 < c := by linarith\n  have h\u2084 : 0 < a * b := mul_pos h\u2081 h\u2082\n  have h\u2085 : 0 < b * c := mul_pos h\u2082 h\u2083\n  have h\u2086 : 0 < c * a := mul_pos h\u2083 h\u2081\n  field_simp [h\u2081.ne', h\u2082.ne', h\u2083.ne']\n  rw [div_le_div_iff (by positivity) (by positivity)]\n  ring_nf\n  nlinarith [sq_nonneg (a ^ 2 * c - b ^ 2 * a), sq_nonneg (b ^ 2 * a - c ^ 2 * b), sq_nonneg (c ^ 2 * b - a ^ 2 * c),\n    sq_nonneg (a ^ 2 * b - b ^ 2 * c), sq_nonneg (b ^ 2 * c - c ^ 2 * a), sq_nonneg (c ^ 2 * a - a ^ 2 * b),\n    mul_pos h\u2081 h\u2082, mul_pos h\u2082 h\u2083, mul_pos h\u2083 h\u2081, mul_pos (mul_pos h\u2081 h\u2082) (mul_pos h\u2082 h\u2083),\n    mul_pos (mul_pos h\u2082 h\u2083) (mul_pos h\u2083 h\u2081), mul_pos (mul_pos h\u2083 h\u2081) (mul_pos h\u2081 h\u2082)]\n", "proof_before": "theorem algebra_amgm_sumasqdivbsqgeqsumbdiva (a b c : \u211d) (h\u2080 : 0 < a \u2227 0 < b \u2227 0 < c) :\n  a ^ 2 / b ^ 2 + b ^ 2 / c ^ 2 + c ^ 2 / a ^ 2 \u2265 b / a + c / b + a / c  :=  by\n  have h\u2081 : 0 < a := by linarith\n  have h\u2082 : 0 < b := by linarith\n  have h\u2083 : 0 < c := by linarith\n  have h\u2084 : 0 < a * b := mul_pos h\u2081 h\u2082\n  have h\u2085 : 0 < b * c := mul_pos h\u2082 h\u2083\n  have h\u2086 : 0 < c * a := mul_pos h\u2083 h\u2081\n  field_simp [h\u2081.ne', h\u2082.ne', h\u2083.ne']\n  rw [div_le_div_iff (by positivity) (by positivity)]\n  ring_nf\n  nlinarith [sq_nonneg (a ^ 2 * c - b ^ 2 * a), sq_nonneg (b ^ 2 * a - c ^ 2 * b), sq_nonneg (c ^ 2 * b - a ^ 2 * c),\n    sq_nonneg (a ^ 2 * b - b ^ 2 * c), sq_nonneg (b ^ 2 * c - c ^ 2 * a), sq_nonneg (c ^ 2 * a - a ^ 2 * b),\n    mul_pos h\u2081 h\u2082, mul_pos h\u2082 h\u2083, mul_pos h\u2083 h\u2081, mul_pos (mul_pos h\u2081 h\u2082) (mul_pos h\u2082 h\u2083),\n    mul_pos (mul_pos h\u2082 h\u2083) (mul_pos h\u2083 h\u2081), mul_pos (mul_pos h\u2083 h\u2081) (mul_pos h\u2081 h\u2082)]\n"}
{"mode": "clear", "steps_before": 15, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 665, "chars_after": 7, "seconds": 11.6, "id": "algebra_manipexpr_2erprsqpesqeqnrpnesq", "proof_after": "theorem algebra_manipexpr_2erprsqpesqeqnrpnesq (e r : \u2102) :\n  2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2  :=  by\n  ring\n", "proof_before": "theorem algebra_manipexpr_2erprsqpesqeqnrpnesq (e r : \u2102) :\n  2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2  :=  by\n  have step1 : (-r + -e) ^ 2 = (r + e) ^ 2 := by\n    simp only [add_comm, add_left_comm]\n    ring\n    <;> simp only [sq]\n    <;> ring\n  have step2 : (r + e) ^ 2 = r ^ 2 + 2 * (e * r) + e ^ 2 := by\n    simp only [add_sq, mul_add, mul_comm, mul_left_comm, mul_assoc, neg_add_rev, neg_sq, neg_mul,\n      neg_mul, neg_neg] at step1\n    linear_combination step1\n  have step3 : r ^ 2 + 2 * (e * r) + e ^ 2 = 2 * (e * r) + (e ^ 2 + r ^ 2) := by\n    simp only [mul_comm e r] at step2\n    linear_combination step2\n  have step4 : 2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 := by\n    simp_all [pow_two, add_mul, mul_add, mul_comm, mul_left_comm]\n    <;> ring\n  apply step4\n"}
{"mode": "clear", "steps_before": 40, "alphabet": 18, "compiles": false, "id": "amc12a_2008_p8", "proof_after": "theorem amc12a_2008_p8 (x y : \u211d) (h\u2080 : 0 < x \u2227 0 < y) (h\u2081 : y ^ 3 = 1)\n  (h\u2082 : 6 * x ^ 2 = 2 * (6 * y ^ 2)) : x ^ 3 = 2 * Real.sqrt 2  :=  by\n  have y_eq_1 : y = 1 := by\n    have h\u2083 : 0 < y := by\n      linarith\n    have h\u2084 : x ^ 2 = 2 * y ^ 2 := by\n      linarith\n    have h\u2085 : 0 < x := by\n      linarith\n    nlinarith [sq_nonneg (x - y), sq_nonneg (x + y)]\n  have surface_area_eq : 6 * x^2 = 2 * (6 * 1^2) := by\n    simp_all only [y_eq_1, mul_one, mul_zero, mul_neg, mul_add, mul_sub, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> nlinarith\n  have surface_area_simplified : 6 * x^2 = 12 := by\n    simp_all only [pow_two, mul_one, mul_zero, mul_add]\n    linarith\n  have x_squared_eq_2 : x^2 = 2 := by\n    have h\u2083 : x ^ 2 = 2 := by\n      have : 6 * x ^ 2 = 12 := by linarith\n      nlinarith\n    linarith\n  have x_eq_sqrt2 : x = Real.sqrt 2 := by\n    have h\u2083 : 0 < x := by linarith\n    have h\u2084 : 0 < y := by linarith\n    have h\u2085 : x = Real.sqrt 2 := by\n      apply Eq.symm\n      rw [Real.sqrt_eq_iff_sq_eq] <;> nlinarith\n    exact h\u2085\n  have x_cubed_eq_2sqrt2 : x^3 = (Real.sqrt 2)^3 := by\n    rw [y_eq_1] at *\n    norm_num at surface_area_simplified\n    nlinarith [sq_sqrt (show 0 \u2264 2 by norm_num)]\n  have x_cubed_simplified : x^3 = 2 * Real.sqrt 2 := by\n    simp_all [pow_three, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> linarith\n  assumption\n", "proof_before": "theorem amc12a_2008_p8 (x y : \u211d) (h\u2080 : 0 < x \u2227 0 < y) (h\u2081 : y ^ 3 = 1)\n  (h\u2082 : 6 * x ^ 2 = 2 * (6 * y ^ 2)) : x ^ 3 = 2 * Real.sqrt 2  :=  by\n  have y_eq_1 : y = 1 := by\n    have h\u2083 : 0 < y := by\n      linarith\n    have h\u2084 : x ^ 2 = 2 * y ^ 2 := by\n      linarith\n    have h\u2085 : 0 < x := by\n      linarith\n    nlinarith [sq_nonneg (x - y), sq_nonneg (x + y)]\n  have surface_area_eq : 6 * x^2 = 2 * (6 * 1^2) := by\n    simp_all only [y_eq_1, mul_one, mul_zero, mul_neg, mul_add, mul_sub, mul_assoc, mul_comm, mul_left_comm]\n    <;> ring_nf\n    <;> nlinarith\n  have surface_area_simplified : 6 * x^2 = 12 := by\n    simp_all only [pow_two, mul_one, mul_zero, mul_add]\n    linarith\n  have x_squared_eq_2 : x^2 = 2 := by\n    have h\u2083 : x ^ 2 = 2 := by\n      have : 6 * x ^ 2 = 12 := by linarith\n      nlinarith\n    linarith\n  have x_eq_sqrt2 : x = Real.sqrt 2 := by\n    have h\u2083 : 0 < x := by linarith\n    have h\u2084 : 0 < y := by linarith\n    have h\u2085 : x = Real.sqrt 2 := by\n      apply Eq.symm\n      rw [Real.sqrt_eq_iff_sq_eq] <;> nlinarith\n    exact h\u2085\n  have x_cubed_eq_2sqrt2 : x^3 = (Real.sqrt 2)^3 := by\n    rw [y_eq_1] at *\n    norm_num at surface_area_simplified\n    nlinarith [sq_sqrt (show 0 \u2264 2 by norm_num)]\n  have x_cubed_simplified : x^3 = 2 * Real.sqrt 2 := by\n    simp_all [pow_three, mul_comm, mul_assoc, mul_left_comm]\n    <;> norm_num\n    <;> linarith\n  assumption\n"}
{"mode": "clear", "steps_before": 88, "alphabet": 36, "compiles": false, "id": "aime_1996_p5", "proof_after": "theorem aime_1996_p5 (a b c r s t : \u211d) (f g : \u211d \u2192 \u211d)\n  (h\u2080 : \u2200 x, f x = x ^ 3 + 3 * x ^ 2 + 4 * x - 11) (h\u2081 : \u2200 x, g x = x ^ 3 + r * x ^ 2 + s * x + t)\n  (h\u2082 : f a = 0) (h\u2083 : f b = 0) (h\u2084 : f c = 0) (h\u2085 : g (a + b) = 0) (h\u2086 : g (b + c) = 0)\n  (h\u2087 : g (c + a) = 0) (h\u2088 : List.Pairwise (\u00b7 \u2260 \u00b7) [a, b, c]) : t = 23  :=  by\n  have sum_roots : a + b + c = -3 := by\n    simp_all only [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, List.Pairwise]\n    ring_nf\n    apply mul_left_cancel\u2080 (sub_ne_zero.2 (show a \u2260 b by intro h; simp_all))\n    apply mul_left_cancel\u2080 (sub_ne_zero.2 (show a \u2260 c by intro h; simp_all))\n    nlinarith [sq_pos_of_ne_zero (sub_ne_zero.2 (show a \u2260 b by intro h; simp_all)),\n      sq_pos_of_ne_zero (sub_ne_zero.2 (show a \u2260 c by intro h; simp_all)),\n      sq_pos_of_ne_zero (sub_ne_zero.2 (show b \u2260 c by intro h; simp_all))]\n  have sum_product_roots : a*b + b*c + c*a = 4 := by\n    simp only [h\u2080, h\u2081, mul_comm] at *\n    ring_nf at *\n    have h\u2089 : a * b * c = 1 := by\n      nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n    nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n  have product_roots : a * b * c = 11 := by\n    have h\u2089 := h\u2080 a\n    have h\u2081\u2080 := h\u2080 b\n    have h\u2081\u2081 := h\u2080 c\n    simp_all\n    <;> nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (a - c), sq_nonneg (b - c)]\n  have root1 : a + b = - (r + (a + b + c)) := by\n    have h\u2089 := h\u2080 a\n    have h\u2081\u2080 := h\u2080 b\n    have h\u2081\u2081 := h\u2080 c\n    have h\u2081\u2082 := h\u2081 (a + b)\n    have h\u2081\u2083 := h\u2081 (b + c)\n    have h\u2081\u2084 := h\u2081 (c + a)\n    simp_all\n    <;> nlinarith\n  have root2 : b + c = - (r + (b + c + a)) := by\n    have h\u2089 : a ^ 2 + b ^ 2 + c ^ 2 = 18 := by\n      nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (a - c), sq_nonneg (b - c)]\n    have h\u2081\u2080 : a + b + c = -3 := sum_roots\n    have h\u2081\u2081 : a * b + b * c + c * a = 4 := sum_product_roots\n    have h\u2081\u2082 : a * b * c = 11 := product_roots\n    have h\u2081\u2083 : a + b = - (r + (a + b + c)) := root1\n    nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (a - c), sq_nonneg (b - c)]\n  have root3 : c + a = - (r + (c + a + b)) := by\n    have h\u2089 : a + b = - (r + (a + b + c)) := root1\n    have h\u2081\u2080 : b + c = - (r + (b + c + a)) := root2\n    have h\u2081\u2081 : c + a = - (r + (c + a + b)) := by\n      nlinarith [sum_roots, sum_product_roots, product_roots, h\u2089, h\u2081\u2080]\n    exact h\u2081\u2081\n  have sum_roots_second : (a + b) + (b + c) + (c + a) = 2*(a + b + c) := by\n    ring_nf at *\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n  have sum_product_roots_second : (a + b)*(b + c) + (b + c)*(c + a) + (c + a)*(a + b) = s := by\n    nlinarith [sum_roots, sum_product_roots, product_roots, root1, root2, root3, sum_roots_second, sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n  have product_roots_second : (a + b)*(b + c)*(c + a) = -t := by\n    norm_num at *\n    ring_nf at *\n    nlinarith [sq_nonneg (a + b), sq_nonneg (b + c), sq_nonneg (c + a)]\n  have expanded_product : (a + b)*(b + c)*(c + a) = a*b*c + a*b*(a + b + c) + b*c*(a + b + c) + c*a*(a + b + c) := by\n    nlinarith [product_roots, sum_product_roots, sum_roots, root1, root2, root3, sum_roots_second, sum_product_roots_second, product_roots_second]\n  have simplified_product : (a + b)*(b + c)*(c + a) = a*b*c + (a*b + b*c + c*a)*(a + b + c) := by\n    nlinarith [h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, sum_roots, sum_product_roots, product_roots, root1, root2, root3, sum_roots_second, sum_product_roots_second, product_roots_second, expanded_product]\n  have t_value : t = 23 := by\n    have h\u2089 : a * b * c = 11 := by linarith\n    have h\u2081\u2080 : a * b + b * c + c * a = 4 := by linarith\n    have h\u2081\u2081 : a + b + c = -3 := by linarith\n    have h\u2081\u2082 : a + b = - (r + (a + b + c)) := by linarith\n    have h\u2081\u2083 : b + c = - (r + (b + c + a)) := by linarith\n    have h\u2081\u2084 : c + a = - (r + (c + a + b)) := by linarith\n    have h\u2081\u2085 : (a + b) + (b + c) + (c + a) = 2 * (a + b + c) := by linarith\n    have h\u2081\u2086 : (a + b) * (b + c) + (b + c) * (c + a) + (c + a) * (a + b) = s := by linarith\n    have h\u2081\u2087 : (a + b) * (b + c) * (c + a) = -t := by linarith\n    have h\u2081\u2088 : (a + b) * (b + c) * (c + a) = a * b * c + (a * b + b * c + c * a) * (a + b + c) := by linarith\n    have h\u2081\u2089 : t = 23 := by linarith\n    exact h\u2081\u2089\n  simpa [t_value] using h\u2085\n", "proof_before": "theorem aime_1996_p5 (a b c r s t : \u211d) (f g : \u211d \u2192 \u211d)\n  (h\u2080 : \u2200 x, f x = x ^ 3 + 3 * x ^ 2 + 4 * x - 11) (h\u2081 : \u2200 x, g x = x ^ 3 + r * x ^ 2 + s * x + t)\n  (h\u2082 : f a = 0) (h\u2083 : f b = 0) (h\u2084 : f c = 0) (h\u2085 : g (a + b) = 0) (h\u2086 : g (b + c) = 0)\n  (h\u2087 : g (c + a) = 0) (h\u2088 : List.Pairwise (\u00b7 \u2260 \u00b7) [a, b, c]) : t = 23  :=  by\n  have sum_roots : a + b + c = -3 := by\n    simp_all only [h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, List.Pairwise]\n    ring_nf\n    apply mul_left_cancel\u2080 (sub_ne_zero.2 (show a \u2260 b by intro h; simp_all))\n    apply mul_left_cancel\u2080 (sub_ne_zero.2 (show a \u2260 c by intro h; simp_all))\n    nlinarith [sq_pos_of_ne_zero (sub_ne_zero.2 (show a \u2260 b by intro h; simp_all)),\n      sq_pos_of_ne_zero (sub_ne_zero.2 (show a \u2260 c by intro h; simp_all)),\n      sq_pos_of_ne_zero (sub_ne_zero.2 (show b \u2260 c by intro h; simp_all))]\n  have sum_product_roots : a*b + b*c + c*a = 4 := by\n    simp only [h\u2080, h\u2081, mul_comm] at *\n    ring_nf at *\n    have h\u2089 : a * b * c = 1 := by\n      nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n    nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n  have product_roots : a * b * c = 11 := by\n    have h\u2089 := h\u2080 a\n    have h\u2081\u2080 := h\u2080 b\n    have h\u2081\u2081 := h\u2080 c\n    simp_all\n    <;> nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (a - c), sq_nonneg (b - c)]\n  have root1 : a + b = - (r + (a + b + c)) := by\n    have h\u2089 := h\u2080 a\n    have h\u2081\u2080 := h\u2080 b\n    have h\u2081\u2081 := h\u2080 c\n    have h\u2081\u2082 := h\u2081 (a + b)\n    have h\u2081\u2083 := h\u2081 (b + c)\n    have h\u2081\u2084 := h\u2081 (c + a)\n    simp_all\n    <;> nlinarith\n  have root2 : b + c = - (r + (b + c + a)) := by\n    have h\u2089 : a ^ 2 + b ^ 2 + c ^ 2 = 18 := by\n      nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (a - c), sq_nonneg (b - c)]\n    have h\u2081\u2080 : a + b + c = -3 := sum_roots\n    have h\u2081\u2081 : a * b + b * c + c * a = 4 := sum_product_roots\n    have h\u2081\u2082 : a * b * c = 11 := product_roots\n    have h\u2081\u2083 : a + b = - (r + (a + b + c)) := root1\n    nlinarith [sq_nonneg (a + b + c), sq_nonneg (a - b), sq_nonneg (a - c), sq_nonneg (b - c)]\n  have root3 : c + a = - (r + (c + a + b)) := by\n    have h\u2089 : a + b = - (r + (a + b + c)) := root1\n    have h\u2081\u2080 : b + c = - (r + (b + c + a)) := root2\n    have h\u2081\u2081 : c + a = - (r + (c + a + b)) := by\n      nlinarith [sum_roots, sum_product_roots, product_roots, h\u2089, h\u2081\u2080]\n    exact h\u2081\u2081\n  have sum_roots_second : (a + b) + (b + c) + (c + a) = 2*(a + b + c) := by\n    ring_nf at *\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n    <;> nlinarith\n  have sum_product_roots_second : (a + b)*(b + c) + (b + c)*(c + a) + (c + a)*(a + b) = s := by\n    nlinarith [sum_roots, sum_product_roots, product_roots, root1, root2, root3, sum_roots_second, sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n  have product_roots_second : (a + b)*(b + c)*(c + a) = -t := by\n    norm_num at *\n    ring_nf at *\n    nlinarith [sq_nonneg (a + b), sq_nonneg (b + c), sq_nonneg (c + a)]\n  have expanded_product : (a + b)*(b + c)*(c + a) = a*b*c + a*b*(a + b + c) + b*c*(a + b + c) + c*a*(a + b + c) := by\n    nlinarith [product_roots, sum_product_roots, sum_roots, root1, root2, root3, sum_roots_second, sum_product_roots_second, product_roots_second]\n  have simplified_product : (a + b)*(b + c)*(c + a) = a*b*c + (a*b + b*c + c*a)*(a + b + c) := by\n    nlinarith [h\u2082, h\u2083, h\u2084, h\u2085, h\u2086, h\u2087, h\u2088, sum_roots, sum_product_roots, product_roots, root1, root2, root3, sum_roots_second, sum_product_roots_second, product_roots_second, expanded_product]\n  have t_value : t = 23 := by\n    have h\u2089 : a * b * c = 11 := by linarith\n    have h\u2081\u2080 : a * b + b * c + c * a = 4 := by linarith\n    have h\u2081\u2081 : a + b + c = -3 := by linarith\n    have h\u2081\u2082 : a + b = - (r + (a + b + c)) := by linarith\n    have h\u2081\u2083 : b + c = - (r + (b + c + a)) := by linarith\n    have h\u2081\u2084 : c + a = - (r + (c + a + b)) := by linarith\n    have h\u2081\u2085 : (a + b) + (b + c) + (c + a) = 2 * (a + b + c) := by linarith\n    have h\u2081\u2086 : (a + b) * (b + c) + (b + c) * (c + a) + (c + a) * (a + b) = s := by linarith\n    have h\u2081\u2087 : (a + b) * (b + c) * (c + a) = -t := by linarith\n    have h\u2081\u2088 : (a + b) * (b + c) * (c + a) = a * b * c + (a * b + b * c + c * a) * (a + b + c) := by linarith\n    have h\u2081\u2089 : t = 23 := by linarith\n    exact h\u2081\u2089\n  simpa [t_value] using h\u2085\n"}
{"mode": "clear", "steps_before": 35, "alphabet": 27, "compiles": false, "id": "mathd_numbertheory_780", "proof_after": "theorem mathd_numbertheory_780 (m x : \u2124) (h\u2080 : 0 \u2264 x) (h\u2081 : 10 \u2264 m \u2227 m \u2264 99) (h\u2082 : 6 * x % m = 1)\n  (h\u2083 : (x - 6 ^ 2) % m = 0) : m = 43  :=  by\n  have h_div : m \u2223 215 := by\n    have h\u2084 := h\u2081.1\n    have h\u2085 := h\u2081.2\n    have h\u2086 := h\u2082\n    have h\u2087 := h\u2083\n    interval_cases m <;> omega\n    <;> omega\n    <;> omega\n    <;> omega\n    <;> omega\n  have h_divisors : m = 5 \u2228 m = 43 := by\n    have h\u2084 := h_div\n    have h\u2085 := h\u2081.1\n    have h\u2086 := h\u2081.2\n    have h\u2087 := h\u2082\n    have h\u2088 := h\u2083\n    interval_cases m <;> norm_num at h\u2085 h\u2086 h\u2087 h\u2088 h\u2084 \u22a2 <;> try omega\n  have h_not_five : m \u2260 5 := by\n    rcases h_divisors with (rfl | rfl)\n    <;> norm_num [Int.emod_eq_of_lt] at h\u2082 h\u2083 <;> omega\n    <;> norm_num [Int.emod_eq_of_lt] at h\u2082 h\u2083 <;> omega\n  have h_final : m = 43 := by\n    have h\u2084 : m = 43 := by\n      cases' h_divisors with h\u2085 h\u2084\n      \u00b7 exfalso\n        apply h_not_five\n        linarith\n      \u00b7 linarith\n    linarith\n  simp_all\n", "proof_before": "theorem mathd_numbertheory_780 (m x : \u2124) (h\u2080 : 0 \u2264 x) (h\u2081 : 10 \u2264 m \u2227 m \u2264 99) (h\u2082 : 6 * x % m = 1)\n  (h\u2083 : (x - 6 ^ 2) % m = 0) : m = 43  :=  by\n  have h_div : m \u2223 215 := by\n    have h\u2084 := h\u2081.1\n    have h\u2085 := h\u2081.2\n    have h\u2086 := h\u2082\n    have h\u2087 := h\u2083\n    interval_cases m <;> omega\n    <;> omega\n    <;> omega\n    <;> omega\n    <;> omega\n  have h_divisors : m = 5 \u2228 m = 43 := by\n    have h\u2084 := h_div\n    have h\u2085 := h\u2081.1\n    have h\u2086 := h\u2081.2\n    have h\u2087 := h\u2082\n    have h\u2088 := h\u2083\n    interval_cases m <;> norm_num at h\u2085 h\u2086 h\u2087 h\u2088 h\u2084 \u22a2 <;> try omega\n  have h_not_five : m \u2260 5 := by\n    rcases h_divisors with (rfl | rfl)\n    <;> norm_num [Int.emod_eq_of_lt] at h\u2082 h\u2083 <;> omega\n    <;> norm_num [Int.emod_eq_of_lt] at h\u2082 h\u2083 <;> omega\n  have h_final : m = 43 := by\n    have h\u2084 : m = 43 := by\n      cases' h_divisors with h\u2085 h\u2084\n      \u00b7 exfalso\n        apply h_not_five\n        linarith\n      \u00b7 linarith\n    linarith\n  simp_all\n"}
{"mode": "clear", "steps_before": 45, "alphabet": 26, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 2, "lower_bound": 2, "exhausted_depth": 1, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 1640, "chars_after": 70, "seconds": 77.7, "id": "mathd_algebra_73", "proof_after": "theorem mathd_algebra_73 (p q r x : \u2102) (h\u2080 : (x - p) * (x - q) = (r - p) * (r - q)) (h\u2081 : x \u2260 r) :\n  x = p + q - r  :=  by\n  apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n  linear_combination h\u2080\n", "proof_before": "theorem mathd_algebra_73 (p q r x : \u2102) (h\u2080 : (x - p) * (x - q) = (r - p) * (r - q)) (h\u2081 : x \u2260 r) :\n  x = p + q - r  :=  by\n  have h2 : (x - p) * (x - q) = x^2 - (p + q) * x + p * q := by\n    ring_nf at h\u2080 \u22a2\n    <;> simp_all [sub_eq_zero, sub_ne_zero.mpr h\u2081]\n    <;> ring_nf\n    <;> simp_all [sub_eq_zero, sub_ne_zero.mpr h\u2081]\n  have h3 : (r - p) * (r - q) = r^2 - (p + q) * r + p * q := by\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    rw [\u2190 sub_eq_zero] at h\u2080 \u22a2\n    ring_nf at h\u2080 \u22a2\n    <;> simp_all\n    <;> ring\n  have h4 : x^2 - (p + q) * x + p * q = r^2 - (p + q) * r + p * q := by\n    have : x^2 - (p + q) * x + p * q = r^2 - (p + q) * r + p * q := by\n      rw [\u2190 sub_eq_zero] \n      linear_combination h\u2080 \n    exact this\n  have h5 : x^2 - (p + q) * x + p * q - (r^2 - (p + q) * r + p * q) = 0 := by\n    simp_all only [mul_add, mul_sub, mul_comm, mul_left_comm, mul_assoc]\n    ring_nf\n    <;> simp_all\n    <;> apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    <;> ring_nf\n    <;> simp_all\n  have h6 : x^2 - (p + q) * x - r^2 + (p + q) * r = 0 := by\n    rw [\u2190 sub_eq_zero] at h5\n    linear_combination h5\n  have h7 : x^2 - (p + q) * x + (p + q) * r - r^2 = 0 := by\n    linear_combination h6\n  have h8 : x^2 - (p + q) * (x - r) - r^2 = 0 := by\n    linear_combination h7\n  have h9 : x^2 - (p + q) * x + (p + q - r) * r = 0 := by\n    have h9 : x^2 - (p + q) * x + (p + q - r) * r = 0 := by\n      linear_combination h8\n    exact h9\n  have h10 : x + r = p + q := by\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    rw [\u2190 sub_eq_zero] at h9 \u22a2\n    linear_combination h9\n  have h11 : x = p + q - r := by\n    have : x = p + q - r := by\n      apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n      rw [\u2190 sub_eq_zero]\n      linear_combination h9\n    simp_all\n  rw [h11]\n"}
{"mode": "clear", "steps_before": 44, "alphabet": 23, "compiles": false, "id": "mathd_algebra_206", "proof_after": "theorem mathd_algebra_206 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x ^ 2 + a * x + b) (h\u2081 : 2 * a \u2260 b)\n  (h\u2082 : f (2 * a) = 0) (h\u2083 : f b = 0) : a + b = -1  :=  by\n  have sum_of_roots : 2 * a + b = -a := by\n    simp_all only [h\u2080, mul_add, mul_one, mul_comm, mul_left_comm, mul_assoc]\n    ring_nf\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    nlinarith\n  have product_of_roots : 2 * a * b = b := by\n    have h\u2084 := h\u2082\n    have h\u2085 := h\u2083\n    simp_all only [h\u2080, h\u2081, ne_eq, mul_eq_mul_left_iff, mul_zero, add_zero, zero_add, zero_sub, mul_neg,\n      neg_mul, neg_neg, mul_one, mul_right_comm]\n    nlinarith\n  have equation_from_sum : 3 * a + b = 0 := by\n    have h\u2084 := h\u2080 (2 * a)\n    have h\u2085 := h\u2080 b\n    simp_all\n    <;> ring_nf\n    <;> nlinarith\n  have b_in_terms_of_a : b = -3 * a := by\n    apply eq_of_sub_eq_zero\n    nlinarith\n  have equation_from_product : b * (2 * a - 1) = 0 := by\n    subst b_in_terms_of_a\n    ring_nf\n    <;> linarith\n  have a_value : a = 1 / 2 := by\n    have : b = -3 * a := by linarith\n    rw [this] at equation_from_product\n    have : 6 * a ^ 2 - 3 * a = 0 := by linarith\n    apply mul_right_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    ring_nf\n    nlinarith\n  have b_value : b = -3 / 2 := by\n    rw [b_in_terms_of_a]\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  have final_result : a + b = -1 := by\n    have h\u2084 : a + b = -1 := by\n      rw [a_value, b_value]\n      norm_num\n    simpa using h\u2084\n  simp_all only [add_left_neg, add_right_neg, mul_neg, mul_one, mul_div_cancel_left]\n  <;> linarith\n", "proof_before": "theorem mathd_algebra_206 (a b : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f x = x ^ 2 + a * x + b) (h\u2081 : 2 * a \u2260 b)\n  (h\u2082 : f (2 * a) = 0) (h\u2083 : f b = 0) : a + b = -1  :=  by\n  have sum_of_roots : 2 * a + b = -a := by\n    simp_all only [h\u2080, mul_add, mul_one, mul_comm, mul_left_comm, mul_assoc]\n    ring_nf\n    apply mul_left_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    nlinarith\n  have product_of_roots : 2 * a * b = b := by\n    have h\u2084 := h\u2082\n    have h\u2085 := h\u2083\n    simp_all only [h\u2080, h\u2081, ne_eq, mul_eq_mul_left_iff, mul_zero, add_zero, zero_add, zero_sub, mul_neg,\n      neg_mul, neg_neg, mul_one, mul_right_comm]\n    nlinarith\n  have equation_from_sum : 3 * a + b = 0 := by\n    have h\u2084 := h\u2080 (2 * a)\n    have h\u2085 := h\u2080 b\n    simp_all\n    <;> ring_nf\n    <;> nlinarith\n  have b_in_terms_of_a : b = -3 * a := by\n    apply eq_of_sub_eq_zero\n    nlinarith\n  have equation_from_product : b * (2 * a - 1) = 0 := by\n    subst b_in_terms_of_a\n    ring_nf\n    <;> linarith\n  have a_value : a = 1 / 2 := by\n    have : b = -3 * a := by linarith\n    rw [this] at equation_from_product\n    have : 6 * a ^ 2 - 3 * a = 0 := by linarith\n    apply mul_right_cancel\u2080 (sub_ne_zero.mpr h\u2081)\n    ring_nf\n    nlinarith\n  have b_value : b = -3 / 2 := by\n    rw [b_in_terms_of_a]\n    <;> linarith\n    <;> norm_num\n    <;> linarith\n  have final_result : a + b = -1 := by\n    have h\u2084 : a + b = -1 := by\n      rw [a_value, b_value]\n      norm_num\n    simpa using h\u2084\n  simp_all only [add_left_neg, add_right_neg, mul_neg, mul_one, mul_div_cancel_left]\n  <;> linarith\n"}
{"mode": "clear", "steps_before": 17, "alphabet": 15, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 1, "lower_bound": 1, "exhausted_depth": 0, "certified": true, "rounds": 0, "timed_out": false, "fixpoint": true, "chars_before": 472, "chars_after": 6, "seconds": 0.6, "id": "mathd_numbertheory_45", "proof_after": "theorem mathd_numbertheory_45 : Nat.gcd 6432 132 + 11 = 23  :=  by\n  rfl\n", "proof_before": "theorem mathd_numbertheory_45 : Nat.gcd 6432 132 + 11 = 23  :=  by\n  have h_gcd : Nat.gcd 6432 132 = 12 := by\n    apply Nat.eq_of_mul_eq_mul_right (show 0 < 12 by norm_num)\n    norm_num\n    <;> decide\n    <;> rfl\n  have h_add : Nat.gcd 6432 132 + 11 = 12 + 11 := by\n    simp [h_gcd]\n    <;> decide\n    <;> decide\n    <;> decide\n  have h_result : 12 + 11 = 23 := by\n    simp_all\n    <;> linarith\n  have h1 : Nat.gcd 6432 132 = 12 := h_gcd\n  have h2 : Nat.gcd 6432 132 + 11 = 12 + 11 := h_add\n  have h3 : 12 + 11 = 23 := h_result\n  simp_all\n"}
{"mode": "clear", "steps_before": 20, "alphabet": 18, "compiles": false, "id": "mathd_numbertheory_257", "proof_after": "theorem mathd_numbertheory_257 (x : \u2115) (h\u2080 : 1 \u2264 x \u2227 x \u2264 100)\n    (h\u2081 : 77 \u2223 (\u2211 k in Finset.range 101, k) - x) : x = 45  :=  by\n  have sum_to_100 : \u2211 k in Finset.range 101, k = 5050 := by\n    have : \u2211 k in Finset.range 101, k = 5050 := by\n      simp [Finset.sum_range_succ, mul_comm, Nat.mul_div_cancel_left, Nat.add_assoc, Nat.add_comm]\n    simpa [this] using h\u2081\n  have sum_remaining : (\u2211 k in Finset.range 101, k) - x = 5050 - x := by\n    simp [sum_to_100]\n    <;> omega\n  have h\u2082 : 77 \u2223 (5050 - x) := by\n    simpa [Finset.sum_range_succ, sum_to_100, sum_remaining] using h\u2081\n  have h\u2083 : 5050 % 77 = 45 := by\n    norm_num [Finset.sum_range_succ, Nat.dvd_iff_mod_eq_zero, Nat.mod_eq_of_lt] at h\u2081 h\u2082\n    <;> omega\n  have h\u2084 : x \u2261 45 [MOD 77] := by\n    simp_all [Nat.ModEq, Nat.modEq_iff_dvd]\n    <;> omega\n  have h\u2085 : x = 45 := by\n    have := h\u2084\n    simp only [Nat.ModEq, Nat.mod_eq_of_lt] at this\n    omega\n  simp_all\n", "proof_before": "theorem mathd_numbertheory_257 (x : \u2115) (h\u2080 : 1 \u2264 x \u2227 x \u2264 100)\n    (h\u2081 : 77 \u2223 (\u2211 k in Finset.range 101, k) - x) : x = 45  :=  by\n  have sum_to_100 : \u2211 k in Finset.range 101, k = 5050 := by\n    have : \u2211 k in Finset.range 101, k = 5050 := by\n      simp [Finset.sum_range_succ, mul_comm, Nat.mul_div_cancel_left, Nat.add_assoc, Nat.add_comm]\n    simpa [this] using h\u2081\n  have sum_remaining : (\u2211 k in Finset.range 101, k) - x = 5050 - x := by\n    simp [sum_to_100]\n    <;> omega\n  have h\u2082 : 77 \u2223 (5050 - x) := by\n    simpa [Finset.sum_range_succ, sum_to_100, sum_remaining] using h\u2081\n  have h\u2083 : 5050 % 77 = 45 := by\n    norm_num [Finset.sum_range_succ, Nat.dvd_iff_mod_eq_zero, Nat.mod_eq_of_lt] at h\u2081 h\u2082\n    <;> omega\n  have h\u2084 : x \u2261 45 [MOD 77] := by\n    simp_all [Nat.ModEq, Nat.modEq_iff_dvd]\n    <;> omega\n  have h\u2085 : x = 45 := by\n    have := h\u2084\n    simp only [Nat.ModEq, Nat.mod_eq_of_lt] at this\n    omega\n  simp_all\n"}
{"mode": "clear", "steps_before": 139, "alphabet": 89, "compiles": false, "id": "mathd_numbertheory_405", "proof_after": "theorem mathd_numbertheory_405 (a b c : \u2115) (t : \u2115 \u2192 \u2115) (h\u2080 : t 0 = 0) (h\u2081 : t 1 = 1)\n  (h\u2082 : \u2200 n > 1, t n = t (n - 2) + t (n - 1)) (h\u2083 : a \u2261 5 [MOD 16]) (h\u2084 : b \u2261 10 [MOD 16])\n  (h\u2085 : c \u2261 15 [MOD 16]) : (t a + t b + t c) % 7 = 5  :=  by\n  have def_t : \u2200 n > 1, t n = t (n - 2) + t (n - 1) := by\n    intro n hn\n    apply h\u2082\n    assumption\n  have periodicity : \u2200 n, t n \u2261 t (n % 16) [MOD 7] := by\n    intro n\n    have h\u2083' := h\u2083\n    have h\u2084' := h\u2084\n    have h\u2085' := h\u2085\n    simp only [Nat.ModEq, Nat.ModEq] at h\u2083' h\u2084' h\u2085' \u22a2\n    have h\u2086 : t n % 7 = t (n % 16) % 7 := by\n      induction n using Nat.strong_induction_on with\n      | h n ih =>\n        match n with\n        | 0 => simp [h\u2080, h\u2081, h\u2082]\n        | 1 => simp [h\u2080, h\u2081, h\u2082]\n        | 2 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 3 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 4 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 5 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 6 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 7 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 8 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 9 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 10 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 11 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 12 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 13 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 14 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 15 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | n + 16 =>\n          simp [h\u2080, h\u2081, h\u2082, Nat.add_mod] at *\n          <;> simp_all [Nat.mod_eq_of_lt]\n          <;> omega\n    simp [h\u2086]\n  have ta_mod : t a \u2261 t 5 [MOD 7] := by\n    have h\u2086 : t a \u2261 t (a % 16) [MOD 7] := periodicity a\n    have h\u2087 : a % 16 = 5 := by\n      rw [\u2190 Nat.mod_eq_of_lt (by decide : 5 < 16)]\n      simp_all [Nat.ModEq, Nat.mod_eq_of_lt]\n    rw [h\u2087] at h\u2086\n    exact h\u2086\n  have tb_mod : t b \u2261 t 10 [MOD 7] := by\n    have h\u2086 := periodicity 10\n    have h\u2087 := periodicity 16\n    have h\u2088 := periodicity 5\n    have h\u2089 := periodicity 1\n    have h\u2081\u2080 := periodicity 0\n    have h\u2081\u2081 := periodicity 2\n    have h\u2081\u2082 := periodicity 3\n    have h\u2081\u2083 := periodicity 4\n    have h\u2081\u2084 := periodicity 6\n    have h\u2081\u2085 := periodicity 7\n    have h\u2081\u2086 := periodicity 8\n    have h\u2081\u2087 := periodicity 9\n    have h\u2081\u2088 := periodicity 11\n    have h\u2081\u2089 := periodicity 12\n    have h\u2082\u2080 := periodicity 13\n    have h\u2082\u2081 := periodicity 14\n    have h\u2082\u2082 := periodicity 15\n    simp at h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082\n    <;> simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have tc_mod : t c \u2261 t 15 [MOD 7] := by\n    have h\u2086 := periodicity c\n    have h\u2087 := periodicity 15\n    simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have sum_mod : t a + t b + t c \u2261 t 5 + t 10 + t 15 [MOD 7] := by\n    have h\u2086 : t a \u2261 t 5 [MOD 7] := ta_mod\n    have h\u2087 : t b \u2261 t 10 [MOD 7] := tb_mod\n    have h\u2088 : t c \u2261 t 15 [MOD 7] := tc_mod\n    have h\u2089 : t a + t b + t c \u2261 t 5 + t 10 + t 15 [MOD 7] := by\n      simp [Nat.ModEq, Nat.ModEq] at *\n      omega\n    exact h\u2089\n  have t5_val : t 5 \u2261 5 [MOD 7] := by\n    have h\u2086 := periodicity 5\n    have h\u2087 := periodicity 10\n    have h\u2088 := periodicity 15\n    simp at h\u2086 h\u2087 h\u2088\n    simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have t10_val : t 10 \u2261 6 [MOD 7] := by\n    have h\u2086 := periodicity 10\n    have h\u2087 := periodicity 5\n    have h\u2088 := periodicity 15\n    have h\u2089 := periodicity 0\n    have h\u2081\u2080 := periodicity 1\n    have h\u2081\u2081 := periodicity 2\n    have h\u2081\u2082 := periodicity 3\n    have h\u2081\u2083 := periodicity 4\n    have h\u2081\u2084 := periodicity 6\n    have h\u2081\u2085 := periodicity 7\n    have h\u2081\u2086 := periodicity 8\n    have h\u2081\u2087 := periodicity 9\n    have h\u2081\u2088 := periodicity 11\n    have h\u2081\u2089 := periodicity 12\n    have h\u2082\u2080 := periodicity 13\n    have h\u2082\u2081 := periodicity 14\n    have h\u2082\u2082 := periodicity 15\n    simp at h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082\n    <;> simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have t15_val : t 15 \u2261 1 [MOD 7] := by\n    have h\u2086 : t 15 % 7 = 1 := by\n      rw [\u2190 Nat.mod_add_div (t 15) 7]\n      have h\u2087 := periodicity 15\n      have h\u2088 := periodicity 0\n      have h\u2089 := periodicity 1\n      have h\u2081\u2080 := periodicity 2\n      have h\u2081\u2081 := periodicity 3\n      have h\u2081\u2082 := periodicity 4\n      have h\u2081\u2083 := periodicity 5\n      have h\u2081\u2084 := periodicity 6\n      have h\u2081\u2085 := periodicity 7\n      have h\u2081\u2086 := periodicity 8\n      have h\u2081\u2087 := periodicity 9\n      have h\u2081\u2088 := periodicity 10\n      have h\u2081\u2089 := periodicity 11\n      have h\u2082\u2080 := periodicity 12\n      have h\u2082\u2081 := periodicity 13\n      have h\u2082\u2082 := periodicity 14\n      have h\u2082\u2083 := periodicity 15\n      simp at h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083\n      <;> simp_all [Nat.ModEq, Nat.ModEq]\n      <;> omega\n    exact h\u2086\n  have sum_val : t a + t b + t c \u2261 5 + 6 + 1 [MOD 7] := by\n    simp_all [Nat.ModEq, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n  have final_result : (t a + t b + t c) % 7 = 5 := by\n    simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have h\u2086 : t a + t b + t c \u2261 5 + 6 + 1 [MOD 7] := sum_val\n  have h\u2087 : 5 + 6 + 1 = 12 := by norm_num\n  have h\u2088 : 12 % 7 = 5 := by norm_num\n  simp_all [Nat.ModEq, Nat.ModEq]\n", "proof_before": "theorem mathd_numbertheory_405 (a b c : \u2115) (t : \u2115 \u2192 \u2115) (h\u2080 : t 0 = 0) (h\u2081 : t 1 = 1)\n  (h\u2082 : \u2200 n > 1, t n = t (n - 2) + t (n - 1)) (h\u2083 : a \u2261 5 [MOD 16]) (h\u2084 : b \u2261 10 [MOD 16])\n  (h\u2085 : c \u2261 15 [MOD 16]) : (t a + t b + t c) % 7 = 5  :=  by\n  have def_t : \u2200 n > 1, t n = t (n - 2) + t (n - 1) := by\n    intro n hn\n    apply h\u2082\n    assumption\n  have periodicity : \u2200 n, t n \u2261 t (n % 16) [MOD 7] := by\n    intro n\n    have h\u2083' := h\u2083\n    have h\u2084' := h\u2084\n    have h\u2085' := h\u2085\n    simp only [Nat.ModEq, Nat.ModEq] at h\u2083' h\u2084' h\u2085' \u22a2\n    have h\u2086 : t n % 7 = t (n % 16) % 7 := by\n      induction n using Nat.strong_induction_on with\n      | h n ih =>\n        match n with\n        | 0 => simp [h\u2080, h\u2081, h\u2082]\n        | 1 => simp [h\u2080, h\u2081, h\u2082]\n        | 2 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 3 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 4 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 5 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 6 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 7 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 8 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 9 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 10 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 11 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 12 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 13 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 14 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | 15 => simp [h\u2080, h\u2081, h\u2082, Nat.add_mod]\n        | n + 16 =>\n          simp [h\u2080, h\u2081, h\u2082, Nat.add_mod] at *\n          <;> simp_all [Nat.mod_eq_of_lt]\n          <;> omega\n    simp [h\u2086]\n  have ta_mod : t a \u2261 t 5 [MOD 7] := by\n    have h\u2086 : t a \u2261 t (a % 16) [MOD 7] := periodicity a\n    have h\u2087 : a % 16 = 5 := by\n      rw [\u2190 Nat.mod_eq_of_lt (by decide : 5 < 16)]\n      simp_all [Nat.ModEq, Nat.mod_eq_of_lt]\n    rw [h\u2087] at h\u2086\n    exact h\u2086\n  have tb_mod : t b \u2261 t 10 [MOD 7] := by\n    have h\u2086 := periodicity 10\n    have h\u2087 := periodicity 16\n    have h\u2088 := periodicity 5\n    have h\u2089 := periodicity 1\n    have h\u2081\u2080 := periodicity 0\n    have h\u2081\u2081 := periodicity 2\n    have h\u2081\u2082 := periodicity 3\n    have h\u2081\u2083 := periodicity 4\n    have h\u2081\u2084 := periodicity 6\n    have h\u2081\u2085 := periodicity 7\n    have h\u2081\u2086 := periodicity 8\n    have h\u2081\u2087 := periodicity 9\n    have h\u2081\u2088 := periodicity 11\n    have h\u2081\u2089 := periodicity 12\n    have h\u2082\u2080 := periodicity 13\n    have h\u2082\u2081 := periodicity 14\n    have h\u2082\u2082 := periodicity 15\n    simp at h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082\n    <;> simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have tc_mod : t c \u2261 t 15 [MOD 7] := by\n    have h\u2086 := periodicity c\n    have h\u2087 := periodicity 15\n    simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have sum_mod : t a + t b + t c \u2261 t 5 + t 10 + t 15 [MOD 7] := by\n    have h\u2086 : t a \u2261 t 5 [MOD 7] := ta_mod\n    have h\u2087 : t b \u2261 t 10 [MOD 7] := tb_mod\n    have h\u2088 : t c \u2261 t 15 [MOD 7] := tc_mod\n    have h\u2089 : t a + t b + t c \u2261 t 5 + t 10 + t 15 [MOD 7] := by\n      simp [Nat.ModEq, Nat.ModEq] at *\n      omega\n    exact h\u2089\n  have t5_val : t 5 \u2261 5 [MOD 7] := by\n    have h\u2086 := periodicity 5\n    have h\u2087 := periodicity 10\n    have h\u2088 := periodicity 15\n    simp at h\u2086 h\u2087 h\u2088\n    simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have t10_val : t 10 \u2261 6 [MOD 7] := by\n    have h\u2086 := periodicity 10\n    have h\u2087 := periodicity 5\n    have h\u2088 := periodicity 15\n    have h\u2089 := periodicity 0\n    have h\u2081\u2080 := periodicity 1\n    have h\u2081\u2081 := periodicity 2\n    have h\u2081\u2082 := periodicity 3\n    have h\u2081\u2083 := periodicity 4\n    have h\u2081\u2084 := periodicity 6\n    have h\u2081\u2085 := periodicity 7\n    have h\u2081\u2086 := periodicity 8\n    have h\u2081\u2087 := periodicity 9\n    have h\u2081\u2088 := periodicity 11\n    have h\u2081\u2089 := periodicity 12\n    have h\u2082\u2080 := periodicity 13\n    have h\u2082\u2081 := periodicity 14\n    have h\u2082\u2082 := periodicity 15\n    simp at h\u2086 h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082\n    <;> simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have t15_val : t 15 \u2261 1 [MOD 7] := by\n    have h\u2086 : t 15 % 7 = 1 := by\n      rw [\u2190 Nat.mod_add_div (t 15) 7]\n      have h\u2087 := periodicity 15\n      have h\u2088 := periodicity 0\n      have h\u2089 := periodicity 1\n      have h\u2081\u2080 := periodicity 2\n      have h\u2081\u2081 := periodicity 3\n      have h\u2081\u2082 := periodicity 4\n      have h\u2081\u2083 := periodicity 5\n      have h\u2081\u2084 := periodicity 6\n      have h\u2081\u2085 := periodicity 7\n      have h\u2081\u2086 := periodicity 8\n      have h\u2081\u2087 := periodicity 9\n      have h\u2081\u2088 := periodicity 10\n      have h\u2081\u2089 := periodicity 11\n      have h\u2082\u2080 := periodicity 12\n      have h\u2082\u2081 := periodicity 13\n      have h\u2082\u2082 := periodicity 14\n      have h\u2082\u2083 := periodicity 15\n      simp at h\u2087 h\u2088 h\u2089 h\u2081\u2080 h\u2081\u2081 h\u2081\u2082 h\u2081\u2083 h\u2081\u2084 h\u2081\u2085 h\u2081\u2086 h\u2081\u2087 h\u2081\u2088 h\u2081\u2089 h\u2082\u2080 h\u2082\u2081 h\u2082\u2082 h\u2082\u2083\n      <;> simp_all [Nat.ModEq, Nat.ModEq]\n      <;> omega\n    exact h\u2086\n  have sum_val : t a + t b + t c \u2261 5 + 6 + 1 [MOD 7] := by\n    simp_all [Nat.ModEq, Nat.add_mod, Nat.mul_mod, Nat.mod_mod]\n  have final_result : (t a + t b + t c) % 7 = 5 := by\n    simp_all [Nat.ModEq, Nat.ModEq]\n    <;> omega\n  have h\u2086 : t a + t b + t c \u2261 5 + 6 + 1 [MOD 7] := sum_val\n  have h\u2087 : 5 + 6 + 1 = 12 := by norm_num\n  have h\u2088 : 12 % 7 = 5 := by norm_num\n  simp_all [Nat.ModEq, Nat.ModEq]\n"}
{"mode": "clear", "steps_before": 20, "alphabet": 18, "compiles": false, "id": "amc12a_2017_p2", "proof_after": "theorem amc12a_2017_p2 (x y : \u211d) (h\u2080 : x \u2260 0) (h\u2081 : y \u2260 0) (h\u2082 : x + y = 4 * (x * y)) :\n  1 / x + 1 / y = 4  :=  by\n  have h\u2083 : x + y = 4 * (x * y) := by\n    rw [h\u2082]\n    <;> linarith\n  have h\u2084 : 1 / x + 1 / y = (y + x) / (x * y) := by\n    field_simp [h\u2080, h\u2081]\n    <;> ring_nf\n    <;> linarith\n  have h\u2085 : (y + x) / (x * y) = (4 * (x * y)) / (x * y) := by\n    field_simp [h\u2080, h\u2081] at h\u2084 \u22a2\n    ring_nf at h\u2084 \u22a2\n    linarith\n  have h\u2086 : (4 * (x * y)) / (x * y) = 4 := by\n    have h\u2086 : x * y \u2260 0 := mul_ne_zero h\u2080 h\u2081\n    field_simp at h\u2085 \u22a2\n    <;> simp_all\n    <;> nlinarith\n  have h\u2087 : 1 / x + 1 / y = 4 := by\n    simp_all [div_eq_mul_inv]\n    <;> linarith\n  simpa [h\u2080, h\u2081] using h\u2087\n", "proof_before": "theorem amc12a_2017_p2 (x y : \u211d) (h\u2080 : x \u2260 0) (h\u2081 : y \u2260 0) (h\u2082 : x + y = 4 * (x * y)) :\n  1 / x + 1 / y = 4  :=  by\n  have h\u2083 : x + y = 4 * (x * y) := by\n    rw [h\u2082]\n    <;> linarith\n  have h\u2084 : 1 / x + 1 / y = (y + x) / (x * y) := by\n    field_simp [h\u2080, h\u2081]\n    <;> ring_nf\n    <;> linarith\n  have h\u2085 : (y + x) / (x * y) = (4 * (x * y)) / (x * y) := by\n    field_simp [h\u2080, h\u2081] at h\u2084 \u22a2\n    ring_nf at h\u2084 \u22a2\n    linarith\n  have h\u2086 : (4 * (x * y)) / (x * y) = 4 := by\n    have h\u2086 : x * y \u2260 0 := mul_ne_zero h\u2080 h\u2081\n    field_simp at h\u2085 \u22a2\n    <;> simp_all\n    <;> nlinarith\n  have h\u2087 : 1 / x + 1 / y = 4 := by\n    simp_all [div_eq_mul_inv]\n    <;> linarith\n  simpa [h\u2080, h\u2081] using h\u2087\n"}
{"mode": "clear", "steps_before": 7, "alphabet": 14, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 3, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 1, "timed_out": false, "fixpoint": true, "chars_before": 148, "chars_after": 60, "seconds": 24.9, "id": "amc12a_2016_p3", "proof_after": "theorem amc12a_2016_p3 (f : \u211d \u2192 \u211d \u2192 \u211d)\n  (h\u2080 : \u2200 x, \u2200 (y) (_ : y \u2260 0), f x y = x - y * Int.floor (x / y)) :\n  f (3 / 8) (-(2 / 5)) = -(1 / 40)  :=  by\n  have h\u2081 := h\u2080 (3 / 8) (-(2 / 5)) (by norm_num)\n  linarith\n", "proof_before": "theorem amc12a_2016_p3 (f : \u211d \u2192 \u211d \u2192 \u211d)\n  (h\u2080 : \u2200 x, \u2200 (y) (_ : y \u2260 0), f x y = x - y * Int.floor (x / y)) :\n  f (3 / 8) (-(2 / 5)) = -(1 / 40)  :=  by\n  have h\u2081 := h\u2080 (3 / 8) (-(2 / 5)) (by norm_num)\n  norm_num at h\u2081\n  <;> simp_all [Int.floor_eq_iff]\n  <;> norm_num\n  <;> field_simp\n  <;> norm_cast\n"}
{"mode": "clear", "steps_before": 59, "alphabet": 44, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 12, "lower_bound": 2, "exhausted_depth": 1, "certified": false, "rounds": 5, "timed_out": false, "fixpoint": true, "chars_before": 1880, "chars_after": 487, "seconds": 413.6, "id": "amc12a_2009_p9", "proof_after": "theorem amc12a_2009_p9 (a b c : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f (x + 3) = 3 * x ^ 2 + 7 * x + 4)\n  (h\u2081 : \u2200 x, f x = a * x ^ 2 + b * x + c) : a + b + c = 2  :=  by\n  have h\u2082 : \u2200 x, a * (x + 3) ^ 2 + b * (x + 3) + c = 3 * x ^ 2 + 7 * x + 4 := by\n    intro x\n    simp only [h\u2081] at h\u2080 \u22a2\n    linarith [h\u2080 x, h\u2080 (x + 3), h\u2080 (x + 6)]\n  have h\u2083 : \u2200 x, a * (x ^ 2 + 6 * x + 9) + b * (x + 3) + c = 3 * x ^ 2 + 7 * x + 4 := by\n    intro x\n    have h\u2083 := h\u2082 x\n    linarith\n  have h\u2085 : \u2200 x, a * x ^ 2 + (6 * a + b) * x + (9 * a + 3 * b + c) = 3 * x ^ 2 + 7 * x + 4 := by\n    intro x\n    linarith [h\u2082 x, h\u2083 x, h\u2081 x, h\u2081 (x + 3)]\n  linarith [h\u2085 0, h\u2085 1, h\u2085 2, h\u2085 3]\n", "proof_before": "theorem amc12a_2009_p9 (a b c : \u211d) (f : \u211d \u2192 \u211d) (h\u2080 : \u2200 x, f (x + 3) = 3 * x ^ 2 + 7 * x + 4)\n  (h\u2081 : \u2200 x, f x = a * x ^ 2 + b * x + c) : a + b + c = 2  :=  by\n  have h\u2082 : \u2200 x, a * (x + 3) ^ 2 + b * (x + 3) + c = 3 * x ^ 2 + 7 * x + 4 := by\n    intro x\n    simp only [h\u2081] at h\u2080 \u22a2\n    linarith [h\u2080 x, h\u2080 (x + 3), h\u2080 (x + 6)]\n  have h\u2083 : \u2200 x, a * (x ^ 2 + 6 * x + 9) + b * (x + 3) + c = 3 * x ^ 2 + 7 * x + 4 := by\n    intro x\n    have h\u2083 := h\u2082 x\n    have h\u2084 := h\u2082 0\n    have h\u2085 := h\u2082 1\n    have h\u2086 := h\u2082 (-1)\n    have h\u2087 := h\u2080 0\n    have h\u2088 := h\u2080 1\n    have h\u2089 := h\u2080 (-1)\n    simp at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089\n    ring_nf at h\u2083 h\u2084 h\u2085 h\u2086 h\u2087 h\u2088 h\u2089 \u22a2\n    linarith\n  have h\u2084 : \u2200 x, a * x ^ 2 + 6 * a * x + 9 * a + b * x + 3 * b + c = 3 * x ^ 2 + 7 * x + 4 := by\n    intro x\n    linarith [h\u2082 x, h\u2083 x, h\u2081 x, h\u2081 (x + 3)]\n  have h\u2085 : \u2200 x, a * x ^ 2 + (6 * a + b) * x + (9 * a + 3 * b + c) = 3 * x ^ 2 + 7 * x + 4 := by\n    intro x\n    have h\u2085 := h\u2082 x\n    have h\u2086 := h\u2083 x\n    have h\u2087 := h\u2084 x\n    simp at h\u2085 h\u2086 h\u2087\n    ring_nf at h\u2085 h\u2086 h\u2087 \u22a2\n    linarith\n  have h\u2086 : a = 3 := by\n    have h\u2086 := h\u2085 0\n    have h\u2087 := h\u2085 1\n    have h\u2088 := h\u2085 (-1)\n    simp at h\u2086 h\u2087 h\u2088\n    linarith\n  have h\u2087 : 6 * a + b = 7 := by\n    simp_all only [mul_comm]\n    linarith [h\u2085 0, h\u2085 1, h\u2085 2, h\u2085 3]\n  have h\u2088 : 9 * a + 3 * b + c = 4 := by\n    have h\u2088 := h\u2085 1\n    have h\u2089 := h\u2085 0\n    have h\u2081\u2080 := h\u2085 (-1)\n    simp only [one_pow, mul_one, mul_zero, zero_add, add_zero, zero_mul, zero_sub, sub_zero,\n      mul_neg, mul_assoc] at h\u2088 h\u2089 h\u2081\u2080\n    linarith\n  have h\u2089 : a = 3 := by\n    norm_num at *\n    <;> linarith\n  have h\u2081\u2080 : b = -11 := by\n    have h\u2089 := h\u2087\n    have h\u2081\u2080 := h\u2088\n    have h\u2081\u2081 := h\u2089\n    simp_all only [mul_add, mul_one, mul_neg, mul_zero, add_zero, add_neg_cancel_left, add_zero]\n    linarith\n  have h\u2081\u2081 : c = 10 := by\n    have h\u2081\u2081 : c = 10 := by\n      linarith [h\u2088]\n    exact h\u2081\u2081\n  have h\u2081\u2082 : a + b + c = 2 := by\n    simp_all only [add_assoc, add_left_comm, add_right_comm, mul_add, mul_comm, mul_left_comm, mul_right_comm]\n    <;> linarith\n  simpa [h\u2089, h\u2081\u2080, h\u2081\u2081] using h\u2081\u2082\n"}
{"mode": "clear", "steps_before": 10, "alphabet": 15, "compiles": true, "steps_after": 10, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 1, "timed_out": false, "fixpoint": true, "chars_before": 196, "chars_after": 196, "seconds": 47.4, "id": "mathd_algebra_11", "proof_after": "theorem mathd_algebra_11 (a b : \u211d) (h\u2080 : a \u2260 b) (h\u2081 : a \u2260 2 * b)\n    (h\u2082 : (4 * a + 3 * b) / (a - 2 * b) = 5) : (a + 11 * b) / (a - b) = 2  :=  by\n  have h\u2083 : a - b \u2260 0 := by\n    intro h\n    apply h\u2080\n    linarith\n  have h\u2084 : a - 2 * b \u2260 0 := by\n    intro h\n    apply h\u2081\n    linarith\n  field_simp [h\u2083, h\u2084, sub_ne_zero, mul_sub] at *\n  linarith\n", "proof_before": "theorem mathd_algebra_11 (a b : \u211d) (h\u2080 : a \u2260 b) (h\u2081 : a \u2260 2 * b)\n    (h\u2082 : (4 * a + 3 * b) / (a - 2 * b) = 5) : (a + 11 * b) / (a - b) = 2  :=  by\n  have h\u2083 : a - b \u2260 0 := by\n    intro h\n    apply h\u2080\n    linarith\n  have h\u2084 : a - 2 * b \u2260 0 := by\n    intro h\n    apply h\u2081\n    linarith\n  field_simp [h\u2083, h\u2084, sub_ne_zero, mul_sub] at *\n  linarith\n"}
{"mode": "clear", "steps_before": 22, "alphabet": 17, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 6, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 4, "timed_out": false, "fixpoint": true, "chars_before": 631, "chars_after": 143, "seconds": 136.8, "id": "mathd_algebra_393", "proof_after": "theorem mathd_algebra_393 (\u03c3 : Equiv \u211d \u211d) (h\u2080 : \u2200 x, \u03c3.1 x = 4 * x ^ 3 + 1) : \u03c3.2 33 = 2  :=  by\n  have eq4 : \u03c3.2 33 = 2 := by\n    have h\u2081 : \u03c3.2 33 = 2 := by\n      simp_all [Equiv.symm_apply_eq]\n      <;> nlinarith\n    exact h\u2081\n  apply eq4\n", "proof_before": "theorem mathd_algebra_393 (\u03c3 : Equiv \u211d \u211d) (h\u2080 : \u2200 x, \u03c3.1 x = 4 * x ^ 3 + 1) : \u03c3.2 33 = 2  :=  by\n  have eq1 : 4 * (\u03c3.2 33) ^ 3 + 1 = 33 := by\n    have h\u2081 := \u03c3.right_inv 33\n    simp_all [Equiv.eq_symm_apply]\n    <;> ring_nf\n    <;> norm_num\n    <;> linarith\n  have eq2 : 4 * (\u03c3.2 33) ^ 3 = 32 := by\n    have h\u2081 := \u03c3.apply_symm_apply 33\n    simp_all [Equiv.eq_symm_apply]\n    <;> ring_nf\n    <;> nlinarith [sq_nonneg (\u03c3.2 33)]\n  have eq3 : (\u03c3.2 33) ^ 3 = 8 := by\n    have h\u2081 : (\u03c3.2 33) ^ 3 = 8 := by\n      apply mul_left_cancel\u2080 (show (4 : \u211d) \u2260 0 by norm_num)\n      linarith\n    rw [h\u2081]\n  have eq4 : \u03c3.2 33 = 2 := by\n    have h\u2081 : \u03c3.2 33 = 2 := by\n      simp_all [Equiv.symm_apply_eq]\n      <;> nlinarith\n    exact h\u2081\n  apply eq4\n"}
{"mode": "clear", "steps_before": 42, "alphabet": 16, "compiles": true, "note": "input uses opaque tactics; clear mode replaces them only where a clear step works", "steps_after": 8, "lower_bound": 3, "exhausted_depth": 2, "certified": false, "rounds": 6, "timed_out": false, "fixpoint": true, "chars_before": 1383, "chars_after": 355, "seconds": 167.6, "id": "mathd_algebra_110", "proof_after": "theorem mathd_algebra_110 (q e : \u2102) (h\u2080 : q = 2 - 2 * Complex.I) (h\u2081 : e = 5 + 5 * Complex.I) :\n    q * e = 20  :=  by\n  have final_simplification : q * e = 10 + 10 * Complex.I - 10 * Complex.I + 10 := by\n    simp_all only [Complex.I_sq, sub_eq_add_neg, mul_add, mul_sub, mul_one, mul_neg, mul_comm]\n    ring\n    <;> norm_num\n  have combined_terms : q * e = 20 := by\n    simp_all only [mul_comm, mul_add, mul_sub, mul_one, mul_neg, mul_assoc]\n    ring\n  rw [combined_terms]\n", "proof_before": "theorem mathd_algebra_110 (q e : \u2102) (h\u2080 : q = 2 - 2 * Complex.I) (h\u2081 : e = 5 + 5 * Complex.I) :\n    q * e = 20  :=  by\n  have expanded : q * e = (2 - 2 * Complex.I) * (5 + 5 * Complex.I) := by\n    rw [h\u2080, h\u2081]\n    <;> simp_all\n    <;> ring_nf\n    <;> norm_num\n    <;> rfl\n  have distributed : q * e = 2 * 5 + 2 * (5 * Complex.I) - 2 * Complex.I * 5 - 2 * Complex.I * (5 * Complex.I) := by\n    simp_all [mul_comm, mul_assoc, mul_left_comm, sub_eq_add_neg]\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n    <;> ring\n    <;> norm_num\n  have simplified : q * e = 10 + 10 * Complex.I - 10 * Complex.I - 10 * Complex.I^2 := by\n    subst_vars\n    simp only [Complex.ext_iff, Complex.I_mul_I, mul_neg, mul_one, mul_add, mul_sub, sub_mul, add_mul,\n      mul_comm]\n    norm_num\n  have i_squared : Complex.I^2 = -1 := by\n    have : Complex.I ^ 2 = -1 := by\n      simp [Complex.I_mul_I]\n    exact this\n  have final_simplification : q * e = 10 + 10 * Complex.I - 10 * Complex.I + 10 := by\n    simp_all only [Complex.I_sq, sub_eq_add_neg, mul_add, mul_sub, mul_one, mul_neg, mul_comm]\n    ring\n    <;> norm_num\n  have combined_terms : q * e = 20 := by\n    simp_all only [mul_comm, mul_add, mul_sub, mul_one, mul_neg, mul_assoc]\n    ring\n    <;> simp [Complex.I_mul_I]\n    <;> ring\n    <;> norm_num\n    <;> simp [Complex.I_mul_I]\n    <;> ring\n    <;> norm_num\n    <;> simp [Complex.I_mul_I]\n    <;> ring\n    <;> norm_num\n    <;> simp [Complex.I_mul_I]\n    <;> ring\n    <;> norm_num\n  rw [combined_terms]\n"}
{"id": "aime_1991_p1", "error": "checker disconnects (AXLE)", "steps_before": null}
{"id": "amc12a_2015_p10", "error": "checker disconnects (AXLE)", "steps_before": null}
{"id": "amc12a_2002_p12", "error": "checker disconnects (AXLE)", "steps_before": null}