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