| import Mathlib | |
| set_option linter.unusedVariables.analyzeTactics true | |
| open Nat Real | |
| lemma mylemma_xy_le_y | |
| (x y : ℕ) | |
| (h₀ : 0 < x ∧ 0 < y) | |
| -- (g : x ^ y ^ 2 = (x ^ y) ^ y) | |
| (hxy : x ≤ y) | |
| (h₁ : (x ^ y) ^ y = y ^ x) : | |
| x ^ y ≤ y := by | |
| by_contra hc | |
| push_neg at hc | |
| have h₂: y^x ≤ y^y := by | |
| { exact Nat.pow_le_pow_of_le_right h₀.2 hxy } | |
| have h₃: y^y < (x^y)^y := by | |
| refine Nat.pow_lt_pow_left hc ?_ | |
| refine Nat.pos_iff_ne_zero.mp h₀.2 | |
| rw [h₁] at h₃ | |
| linarith [h₂, h₃] | |
| lemma four_times_k_less_than_two_pow_k | |
| (k : ℕ) | |
| (hk : 5 ≤ k) : | |
| 4 * k < 2 ^ k := by | |
| -- Proceed by induction on k | |
| induction' k using Nat.case_strong_induction_on with n ih | |
| -- Base case: k = 0 is not possible since 5 ≤ k | |
| -- so we start directly with k = 5 as the base case | |
| . norm_num | |
| by_cases h₀ : n < 5 | |
| . have hn: n = 4 := by linarith | |
| rw [hn] | |
| norm_num | |
| . push_neg at h₀ | |
| have ih₁ : 4 * n < 2 ^ n := ih n (le_refl n) h₀ | |
| rw [mul_add, pow_add, mul_one, pow_one, mul_two] | |
| refine Nat.add_lt_add ih₁ ?_ | |
| refine lt_trans ?_ ih₁ | |
| refine (Nat.lt_mul_iff_one_lt_right (by norm_num)).mpr ?_ | |
| refine Nat.lt_of_lt_of_le ?_ h₀ | |
| norm_num | |
| lemma mylemma_case_xley | |
| (x y : ℕ) | |
| (h₀ : 0 < x ∧ 0 < y) | |
| (h₁ : x^(y^2) = y^x) | |
| (g₁ : x^(y^2) = (x^y)^y) | |
| (hxy : x ≤ y) : | |
| (x, y) = (1, 1) ∨ (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by | |
| rw [g₁] at h₁ | |
| have g2: x^y ≤ y := by | |
| exact mylemma_xy_le_y x y h₀ hxy h₁ | |
| have g3: x = 1 := by | |
| by_contra hc | |
| have g3: 2 ≤ x := by | |
| by_contra gc | |
| push_neg at gc | |
| interval_cases x | |
| . linarith | |
| . omega | |
| have g4: 2^y ≤ x^y := by { exact Nat.pow_le_pow_of_le_left g3 y } | |
| have g5: y < 2^y := by exact Nat.lt_two_pow_self | |
| linarith | |
| rw [g3] at h₁ | |
| simp at h₁ | |
| left | |
| norm_num | |
| exact { left := g3, right := id h₁.symm } | |
| lemma mylemma_exp_log | |
| (x: ℕ) | |
| (h₀: 0 < x): | |
| (↑x = Real.exp (Real.log ↑x)):= by | |
| have hx_pos : 0 < (↑x : ℝ) := by exact Nat.cast_pos.mpr h₀ | |
| symm | |
| exact Real.exp_log hx_pos | |
| lemma mylemma_y2_lt_x | |
| (x y : ℕ) | |
| (h₀ : 0 < x ∧ 0 < y) | |
| (h₁ : x ^ y ^ 2 = y ^ x) | |
| (hxy : y < x) : | |
| y ^ 2 < x := by | |
| by_cases hy: 1 < y | |
| . have hx: 2 ≤ x := by linarith | |
| have h₂: y ^ x < x ^ x := by | |
| refine Nat.pow_lt_pow_left hxy ?_ | |
| exact Nat.ne_of_lt' h₀.1 | |
| rw [← h₁] at h₂ | |
| exact (Nat.pow_lt_pow_iff_right hx).mp h₂ | |
| . push_neg at hy | |
| interval_cases y | |
| . simp | |
| exact h₀.1 | |
| . simp at * | |
| assumption | |
| lemma mylemma_5 | |
| (x y: ℕ) | |
| (h₀: 0 < x ∧ 0 < y) | |
| (h₁: x ^ y ^ 2 = y ^ x) : | |
| (↑x / ↑y^2) ^ y ^ 2 = (↑y:ℝ)^ ((↑x:ℝ) - 2 * ↑y ^ 2) := by | |
| have g₁: (↑x:ℝ) ^ (↑y:ℝ) ^ 2 = (↑y:ℝ) ^ (↑x:ℝ) := by | |
| norm_cast | |
| have g₂: 0 < ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) := by | |
| norm_cast | |
| exact pow_pos h₀.2 _ | |
| have g₃: ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) | |
| = ((↑y:ℝ) ^ (↑x:ℝ)) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) := by | |
| refine (div_left_inj' ?_).mpr g₁ | |
| norm_cast | |
| refine pow_ne_zero _ ?_ | |
| linarith [h₀.2] | |
| have gy: 0 < (↑y:ℝ) := by | |
| norm_cast | |
| exact h₀.2 | |
| rw [← Real.rpow_sub gy (↑x) (2 * ↑y ^ 2)] at g₃ | |
| have g₄: ((↑x:ℝ) ^ (↑y:ℝ) ^ 2) / ((↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2)) | |
| = (↑x / ↑y^2) ^ y ^ 2 := by | |
| have g₅: (↑y:ℝ) ^ (2 * (↑y:ℝ) ^ 2) = ((↑y:ℝ) ^ 2) ^ ((↑y:ℝ) ^ 2) := by | |
| norm_cast | |
| refine pow_mul y 2 (y^2) | |
| rw [g₅] | |
| symm | |
| norm_cast | |
| have g₆: ((↑x:ℝ) / ↑y ^ 2) ^ y ^ 2 = ↑x ^ y ^ 2 / (↑y ^ 2) ^ y ^ 2 := by | |
| refine div_pow (↑x:ℝ) ((↑y:ℝ) ^ 2) (y^2) | |
| norm_cast at * | |
| rw [g₄] at g₃ | |
| norm_cast at * | |
| lemma mylemma_2y2_lt_x | |
| (x y : ℕ) | |
| (h₀ : 0 < x ∧ 0 < y) | |
| (h₁ : x ^ y ^ 2 = y ^ x) | |
| (hxy : y < x) : | |
| 2 * y ^ 2 < x := by | |
| by_cases hy1: y = 1 | |
| . rw [hy1] | |
| norm_num | |
| by_contra hc | |
| push_neg at hc | |
| interval_cases x | |
| . linarith | |
| . linarith | |
| . rw [hy1] at h₁ | |
| simp at h₁ | |
| . have hy: 1 < y := by | |
| contrapose! hy1 | |
| linarith | |
| clear hy1 | |
| have h₂: (↑y:ℝ) ^ 2 < ↑x := by | |
| norm_cast | |
| exact mylemma_y2_lt_x x y h₀ h₁ hxy | |
| have h₃: 1 < ↑x / (↑y:ℝ) ^ 2 := by | |
| refine (one_lt_div ?_).mpr h₂ | |
| norm_cast | |
| exact pow_pos h₀.2 2 -- rw ← one_mul ((↑y:ℝ)^2) at h₂, refine lt_div_iff_mul_lt.mpr h₂, }, | |
| have h₄: 1 < (↑x / (↑y:ℝ)^2)^(y^2) := by | |
| refine one_lt_pow₀ h₃ ?_ | |
| refine Nat.ne_of_gt ?_ | |
| refine sq_pos_of_pos ?_ | |
| exact lt_of_succ_lt hy | |
| have h₅: (↑x/ (↑y:ℝ)^2)^(y^2) = (↑y:ℝ)^((↑x:ℝ) - 2*(↑y:ℝ)^2) := by | |
| exact mylemma_5 x y h₀ h₁ | |
| rw [h₅] at h₄ | |
| have h₆: 0 < (↑x:ℝ) - 2 * (↑y:ℝ) ^ 2 := by | |
| by_contra hc | |
| push_neg at hc | |
| cases' lt_or_eq_of_le hc with hlt heq | |
| . have gy: 1 < (↑y:ℝ) := by | |
| norm_cast | |
| have glt: (↑x:ℝ) - 2*(↑y:ℝ)^2 < (↑0:ℝ) := by | |
| norm_cast at * | |
| have g₁: (↑y:ℝ) ^ ((↑x:ℝ) - 2*(↑y:ℝ)^2) < (↑y:ℝ) ^ (↑0:ℝ) := by | |
| exact Real.rpow_lt_rpow_of_exponent_lt gy glt | |
| simp at g₁ | |
| linarith[ h₄,g₁] | |
| . rw [heq] at h₄ | |
| simp at h₄ | |
| simp at h₆ | |
| norm_cast at h₆ | |
| lemma mylemma_castdvd | |
| (x y: ℕ) | |
| (h₀: 0 < x ∧ 0 < y) | |
| (h₁ : x ^ y ^ 2 = y ^ x) | |
| (hyx: y < x) : | |
| (y^2 ∣ x) := by | |
| have h₂: (x ^ y ^ 2).factorization = (y^x).factorization := by | |
| exact congr_arg Nat.factorization h₁ | |
| simp at h₂ | |
| symm at h₂ | |
| have hxy1: 2 * y^2 ≤ x := by exact le_of_lt (mylemma_2y2_lt_x x y h₀ h₁ hyx) | |
| have hxy: 2 • y^2 ≤ x := by exact hxy1 | |
| have h₃: 2 • y^2 • x.factorization ≤ x • x.factorization := by | |
| rw [← smul_assoc] | |
| refine nsmul_le_nsmul_left ?_ hxy | |
| norm_num | |
| rw [← h₂] at h₃ | |
| have h₄: 2 • x • y.factorization = x • (2 • y.factorization) := by | |
| rw [← smul_assoc, ← smul_assoc] | |
| have g₄: 2 • x = x • 2 := by | |
| simp | |
| exact mul_comm 2 x | |
| rw [g₄] | |
| rw [h₄] at h₃ | |
| rw [← Nat.factorization_pow] at h₃ | |
| rw [← Nat.factorization_pow] at h₃ | |
| rw [← Nat.factorization_pow] at h₃ | |
| have h₅: (y ^ 2) ^ x ∣ x^x := by | |
| have g₁: (y ^ 2) ^ x ≠ 0 := by | |
| refine pow_ne_zero x ?_ | |
| refine pow_ne_zero 2 ?_ | |
| linarith | |
| have g₂: x ^ x ≠ 0 := by | |
| refine pow_ne_zero x ?_ | |
| linarith | |
| exact (Nat.factorization_le_iff_dvd g₁ g₂).mp h₃ | |
| refine (Nat.pow_dvd_pow_iff ?_).mp h₅ | |
| exact Nat.ne_of_gt h₀.1 | |
| lemma mylemma_xsuby_eq_2xy2_help | |
| (x y : ℕ) | |
| (h₀ : 0 < x ∧ 0 < y) | |
| (h₁ : x ^ y ^ 2 = y ^ x) | |
| (h₂ : Real.log (↑x:ℝ) = Real.log ↑y * ↑x / (↑(y ^ 2:ℕ ):ℝ) ) | |
| (hxy : y < x) : | |
| x = y ^ (x / y ^ 2) := by | |
| have h_exp : Real.exp (Real.log ↑x) | |
| = Real.exp (Real.log ↑y * (↑x:ℝ) / ((↑y:ℝ)) ^ 2) := by | |
| rw [h₂] | |
| norm_cast | |
| rw [← mylemma_exp_log x h₀.1] at h_exp | |
| rw [← mul_div] at h_exp | |
| rw [Real.exp_mul] at h_exp | |
| rw [← mylemma_exp_log y h₀.2] at h_exp | |
| have h₃: (↑x:ℝ) / ((↑y:ℝ)^2) = (↑(x / y^2:ℕ):ℝ) := by | |
| norm_cast | |
| symm | |
| have g₂: y^2 ∣ x := by | |
| exact mylemma_castdvd x y h₀ h₁ hxy | |
| have h₃: (↑(y^(2:ℕ)):ℝ) ≠ 0 := by | |
| norm_cast | |
| exact pow_ne_zero 2 ( by linarith) | |
| exact Nat.cast_div g₂ h₃ | |
| have h₄ : (↑(y ^ (x / y ^ (2:ℕ))):ℝ) = (↑y:ℝ)^((↑x:ℝ) / ((↑y:ℝ)^2)) := by | |
| rw [Nat.cast_pow, h₃] | |
| norm_cast | |
| rw [←h₄] at h_exp | |
| exact Nat.cast_inj.mp h_exp | |
| theorem mylemma_xsuby_eq_2xy2 | |
| (x y : ℕ) | |
| (h₀ : 0 < x ∧ 0 < y) | |
| (h₁ : x ^ y ^ 2 = y ^ x) | |
| (hxy : y < x) : | |
| x = y ^ (x / y ^ 2) := by | |
| -- sketch: y^2 * log x = x * log y | |
| have h₃: Real.log (x^(y^2)) = Real.log (y^x) := by | |
| norm_cast | |
| rw [h₁] | |
| have h₄: (↑(y ^ (2:ℕ)):ℝ) * Real.log x = ↑x * Real.log y := by | |
| have h41: Real.log (y^x) = (↑x:ℝ) * Real.log (y) := by | |
| exact Real.log_pow y x | |
| have h42: Real.log (x^(y^2)) = (↑(y ^ (2:ℕ)):ℝ) * Real.log x := by | |
| exact Real.log_pow x (y^2) | |
| rw [h41,h42] at h₃ | |
| exact h₃ | |
| ring_nf at h₄ | |
| have h₅: Real.log ↑x = Real.log ↑y * ↑x / (↑(y ^ (2:ℕ)):ℝ) := by | |
| by_contra hc | |
| rw [mul_comm (Real.log ↑y) (↑x)] at hc | |
| rw [← h₄, mul_comm, ← mul_div] at hc | |
| rw [div_self, mul_one] at hc | |
| . apply hc | |
| norm_cast | |
| . norm_cast | |
| push_neg | |
| refine pow_ne_zero 2 ?_ | |
| exact Nat.ne_of_gt h₀.2 | |
| have h₆: x = y ^ (x / y ^ 2) := by | |
| exact mylemma_xsuby_eq_2xy2_help x y h₀ h₁ h₅ hxy | |
| exact h₆ | |
| theorem imo_1997_p5 | |
| (x y : ℕ) | |
| (h₀ : 0 < x ∧ 0 < y) | |
| (h₁ : x^(y^2) = y^x) : | |
| (x, y) = (1, 1) ∨ (x, y) = (16, 2) ∨ (x, y) = (27, 3) := by | |
| have g₁: x^(y^2) = (x^y)^y := by | |
| rw [Nat.pow_two] | |
| exact Nat.pow_mul x y y | |
| by_cases hxy: x ≤ y | |
| . exact mylemma_case_xley x y h₀ h₁ g₁ hxy | |
| . push_neg at hxy | |
| right | |
| have h₃: x = y ^ (x / y ^ 2) := by | |
| exact mylemma_xsuby_eq_2xy2 x y h₀ h₁ hxy | |
| let k:ℕ := x / y^2 -- { admit }, | |
| have hk_def: k = x / y^2 := by exact rfl | |
| by_cases hk: k < 2 | |
| . rw [← hk_def] at h₃ | |
| interval_cases k | |
| . exfalso | |
| simp at h₃ | |
| linarith | |
| . exfalso | |
| simp at * | |
| linarith [hxy,h₃] --simp at h₃, rw h₃ at hxy, linarith[hxy], }, | |
| . push_neg at hk | |
| rw [← hk_def] at h₃ | |
| have h₅: k = y^(k-2) := by | |
| rw [h₃] at hk_def | |
| nth_rewrite 1 [hk_def] | |
| exact Nat.pow_div hk h₀.2 | |
| by_cases hk5: k < 5 | |
| . interval_cases k | |
| . exfalso | |
| simp at h₅ | |
| . right | |
| norm_num | |
| simp at h₅ | |
| symm at h₅ | |
| rw [h₅] at h₃ | |
| norm_num at h₃ | |
| exact { left := h₃, right := h₅ } | |
| . simp at h₅ | |
| symm at h₅ | |
| have g₂: y^4 = y^2 * y^2 := by ring_nf | |
| rw [g₂, h₅] at h₃ | |
| norm_num at h₃ | |
| left | |
| norm_num | |
| constructor | |
| . exact h₃ | |
| . have h₆ : y ^ 2 = 2 ^ 2 := by | |
| norm_num | |
| exact h₅ | |
| have h₇: 0 ≤ y := by | |
| linarith | |
| exact (sq_eq_sq₀ h₇ (by linarith)).mp (h₆) | |
| push_neg at hk5 | |
| by_cases hy: 2 ≤ y | |
| . have h₅₁: k < y^(k-2) := by | |
| have h₆: 2^(k-2) ≤ y^(k-2) := by | |
| have hk1: 3 ≤ k - 2 := by exact Nat.sub_le_sub_right hk5 2 | |
| exact (Nat.pow_le_pow_iff_left (by linarith)).mpr hy | |
| have h₇: 4*k < 2^k := by | |
| exact four_times_k_less_than_two_pow_k k hk5 | |
| have h₇: k < 2^(k-2) := by | |
| have h₈ : k < 2 ^ k / 4 := by | |
| have h81: 4 ∣ 2^k := by | |
| have h82: 2^k = 4*2^(k-2) := by | |
| have h83: k = 2 + (k -2) := by | |
| ring_nf | |
| exact (add_sub_of_le hk).symm | |
| nth_rewrite 1 [h83] | |
| rw [pow_add] | |
| norm_num | |
| rw [h82] | |
| exact Nat.dvd_mul_right 4 (2^(k-2)) | |
| exact (Nat.lt_div_iff_mul_lt' h81 k).mpr h₇ | |
| have h₉: 2 ^ k / 4 = 2 ^ (k-2) := by | |
| have g2: k = k - 2 + 2 := by | |
| exact (Nat.sub_eq_iff_eq_add hk).mp rfl | |
| have h1: 2^k = 2^(k - 2 + 2) := by | |
| exact congrArg (HPow.hPow 2) g2 | |
| have h2: 2 ^ (k - 2 + 2) = 2 ^ (k - 2) * 2 ^ 2 := by rw [pow_add] | |
| rw [h1, h2] | |
| ring_nf | |
| simp | |
| linarith | |
| linarith | |
| exfalso | |
| linarith | |
| . push_neg at hy | |
| interval_cases y | |
| . linarith | |
| . simp at h₅ | |
| simp at h₃ | |
| linarith | |