import Mathlib set_option linter.unusedVariables.analyzeTactics true open Int Rat lemma mylemma_main_lt2 (p q r: ℤ) (hpl: 4 ≤ p) (hql: 5 ≤ q) (hrl: 6 ≤ r) : (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2 := by have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) = (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) := by norm_cast simp have hp: (↑p/↑(p-1):ℚ) ≤ ((4/3):ℚ) := by have g₁: 0 < (↑(p - 1):ℚ) := by norm_cast linarith [hpl] have g₂: ↑p * ↑(3:ℚ) ≤ ↑(4:ℚ) * (↑(p - 1):ℚ) := by norm_cast linarith refine (div_le_iff₀ g₁).mpr ?_ rw [div_mul_eq_mul_div] refine (le_div_iff₀ ?_).mpr g₂ norm_num have hq: (↑q/↑(q-1)) ≤ ((5/4):ℚ) := by have g₁: 0 < (↑(q - 1):ℚ) := by norm_cast linarith[hql] have g₂: ↑q * ↑(4:ℚ) ≤ ↑(5:ℚ) * (↑(q - 1):ℚ) := by norm_cast linarith refine (div_le_iff₀ g₁).mpr ?_ rw [div_mul_eq_mul_div] refine (le_div_iff₀ ?_).mpr g₂ norm_num have hr: (↑r/↑(r-1)) ≤ ((6/5):ℚ) := by have g₁: 0 < (↑(r - 1):ℚ) := by norm_cast linarith[hql] have g₂: ↑r * ↑(5:ℚ) ≤ ↑(6:ℚ) * (↑(r - 1):ℚ) := by norm_cast linarith refine (div_le_iff₀ g₁).mpr ?_ rw [div_mul_eq_mul_div] refine (le_div_iff₀ ?_).mpr g₂ norm_num have hub: (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (4/3:ℚ) * ((5/4):ℚ) * ((6/5):ℚ) := by have hq_nonneg: 0 ≤ (↑q:ℚ) := by norm_cast linarith have hq_1_nonneg: 0 ≤ (↑(q - 1):ℚ) := by norm_cast linarith have h₂: 0 ≤ (((q:ℚ) / ↑(q - 1)):ℚ) := by exact div_nonneg hq_nonneg hq_1_nonneg have hub1: (↑p/↑(p-1)) * (↑q/↑(q-1)) ≤ ((4/3):ℚ) * ((5/4):ℚ) := by exact mul_le_mul hp hq h₂ (by norm_num) have hr_nonneg: 0 ≤ (↑r:ℚ) := by norm_cast linarith have hr_1_nonneg: 0 ≤ (↑(r - 1):ℚ) := by norm_cast linarith have h₃: 0 ≤ (((r:ℚ) / ↑(r - 1)):ℚ) := by exact div_nonneg hr_nonneg hr_1_nonneg exact mul_le_mul hub1 hr h₃ (by norm_num) norm_num at hub rw [h₁] norm_num exact hub lemma mylemma_k_lt_2 (p q r k: ℤ) (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k) (hpl: 4 ≤ p) (hql: 5 ≤ q) (hrl: 6 ≤ r) (hden: 0 < (p - 1) * (q - 1) * (r - 1) ) : (k < 2) := by have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑2 := by exact mylemma_main_lt2 p q r hpl hql hrl have h₂: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by have g₁: ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by norm_cast linarith symm have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by norm_cast linarith[hden] exact (div_eq_iff g₂).mpr g₁ have h₃: ↑k < (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by rw [h₂] have g₁: (↑(p * q * r - 1):ℚ) < (↑(p * q * r):ℚ) := by norm_cast exact sub_one_lt (p * q * r) have g₂: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by norm_cast exact div_lt_div_of_pos_right g₁ g₂ have h₄: (↑k:ℚ) < ↑2 := by exact lt_of_lt_of_le h₃ h₁ norm_cast at h₄ lemma mylemma_main_lt4 (p q r: ℤ) (hpl: 2 ≤ p) (hql: 3 ≤ q) (hrl: 4 ≤ r) : (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑4 := by have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) = (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) := by norm_cast simp have hp: (↑p/↑(p-1):ℚ) ≤ ↑(2:ℚ) := by have g₁: 0 < (↑(p - 1):ℚ) := by norm_cast linarith[hpl] have g₂: ↑p ≤ ↑(2:ℚ) * (↑(p - 1):ℚ) := by norm_cast linarith exact (div_le_iff₀ g₁).mpr g₂ have hq: (↑q/↑(q-1)) ≤ ((3/2):ℚ) := by have g₁: 0 < (↑(q - 1):ℚ) := by norm_cast linarith[hql] have g₂: ↑q * ↑(2:ℚ) ≤ ↑(3:ℚ) * (↑(q - 1):ℚ) := by norm_cast linarith refine (div_le_iff₀ g₁).mpr ?_ rw [div_mul_eq_mul_div] refine (le_div_iff₀ ?_).mpr g₂ norm_num have hr: (↑r/↑(r-1)) ≤ ((4/3):ℚ) := by have g₁: 0 < (↑(r - 1):ℚ) := by norm_cast linarith[hql] have g₂: ↑r * ↑(3:ℚ) ≤ ↑(4:ℚ) * (↑(r - 1):ℚ) := by norm_cast linarith refine (div_le_iff₀ g₁).mpr ?_ rw [div_mul_eq_mul_div] refine (le_div_iff₀ ?_).mpr g₂ norm_num have hub: (↑p/↑(p-1)) * (↑q/↑(q-1)) * (↑r/↑(r-1)) ≤ (2:ℚ) * ((3/2):ℚ) * ((4/3):ℚ) := by have hq_nonneg: 0 ≤ (↑q:ℚ) := by norm_cast linarith have hq_1_nonneg: 0 ≤ (↑(q - 1):ℚ) := by norm_cast linarith have h₂: 0 ≤ (((q:ℚ) / ↑(q - 1)):ℚ) := by exact div_nonneg hq_nonneg hq_1_nonneg have hub1: (↑p/↑(p-1)) * (↑q/↑(q-1)) ≤ (2:ℚ) * ((3/2):ℚ) := by exact mul_le_mul hp hq h₂ (by norm_num) have hr_nonneg: 0 ≤ (↑r:ℚ) := by norm_cast linarith have hr_1_nonneg: 0 ≤ (↑(r - 1):ℚ) := by norm_cast linarith have h₃: 0 ≤ (((r:ℚ) / ↑(r - 1)):ℚ) := by exact div_nonneg hr_nonneg hr_1_nonneg exact mul_le_mul hub1 hr h₃ (by norm_num) norm_num at hub rw [h₁] norm_num exact hub lemma mylemma_k_lt_4 (p q r k: ℤ) (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k) (hpl: 2 ≤ p) (hql: 3 ≤ q) (hrl: 4 ≤ r) (hden: 0 < (p - 1) * (q - 1) * (r - 1) ) : (k < 4) := by have h₁: (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≤ ↑4 := by exact mylemma_main_lt4 p q r hpl hql hrl have h₂: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by have g₁: ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by norm_cast linarith symm have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by norm_cast linarith [hden] exact (div_eq_iff g₂).mpr g₁ have h₃: ↑k < (↑(p * q * r) / ↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by rw [h₂] have g₁: (↑(p * q * r - 1):ℚ) < (↑(p * q * r):ℚ) := by norm_cast exact sub_one_lt (p * q * r) have g₂: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by norm_cast exact div_lt_div_of_pos_right g₁ g₂ have h₄: (↑k:ℚ) < ↑4 := by exact lt_of_lt_of_le h₃ h₁ norm_cast at h₄ lemma mylemma_k_gt_1 (p q r k: ℤ) (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k) (h₁: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ)) (hpl: 2 ≤ p) (hql: 3 ≤ q) (hrl: 4 ≤ r) (hden: 0 < (p - 1) * (q - 1) * (r - 1) ) : (1 < k) := by have hk0: 0 < (↑k:ℚ) := by have g₁: 2*3*4 ≤ p * q * r := by have g₂: 2*3 ≤ p * q := by exact mul_le_mul hpl hql (by norm_num) (by linarith[hpl]) exact mul_le_mul g₂ hrl (by norm_num) (by linarith[g₂]) have g₂: 0 < (↑(p * q * r - 1):ℚ) := by norm_cast linarith[g₁] have g₃: 0 < (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by norm_cast rw [h₁] exact div_pos g₂ g₃ norm_cast at hk0 by_contra hc push_neg at hc interval_cases k simp at hk exfalso have g₁: p*q + q*r + r*p = p+q+r := by linarith have g₂: p < p*q := by exact lt_mul_right (by linarith) (by linarith) have g₃: q < q*r := by exact lt_mul_right (by linarith) (by linarith) have g₄: r < r*p := by exact lt_mul_right (by linarith) (by linarith) have g₅: p+q+r < p*q + q*r + r*p := by linarith[g₂,g₃,g₄] linarith [g₁,g₅] lemma mylemma_p_lt_4 (p q r k: ℤ) (h₀ : 1 < p ∧ p < q ∧ q < r) (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * k) (h₁: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ)) (hpl: 2 ≤ p) (hql: 3 ≤ q) (hrl: 4 ≤ r) (hden: 0 < (p - 1) * (q - 1) * (r - 1) ) : (p < 4) := by by_contra hcp push_neg at hcp have hcq: 5 ≤ q := by linarith have hcr: 6 ≤ r := by linarith have h₃: k < 2 := by exact mylemma_k_lt_2 p q r k hk hcp hcq hcr hden have h₄: 1 < k := by exact mylemma_k_gt_1 p q r k hk h₁ hpl hql hrl hden linarith lemma q_r_divisor_of_prime (q r : ℤ) (p: ℕ) (h₀ : q * r = ↑p) (h₁: Nat.Prime p) : q = -1 ∨ q = 1 ∨ q = -p ∨ q = p := by have hq : q ≠ 0 := by intro h rw [h] at h₀ simp at h₀ symm at h₀ norm_cast at h₀ rw [h₀] at h₁ exact Nat.not_prime_zero h₁ have hr : r ≠ 0 := by intro h rw [h] at h₀ simp at h₀ norm_cast at h₀ rw [← h₀] at h₁ exact Nat.not_prime_zero h₁ have hqr : abs q * abs r = p := by have h₃: abs q = q.natAbs := by exact abs_eq_natAbs q have h₄: abs r = r.natAbs := by exact abs_eq_natAbs r rw [h₃,h₄] norm_cast exact Int.natAbs_mul_natAbs_eq h₀ have h_abs: abs (↑(q.natAbs):ℤ) = 1 ∨ abs q = p := by cases' Int.natAbs_eq q with h_1 h_2 . rw [h_1] at hqr have h₂: abs (↑(q.natAbs):ℤ) ∣ p := by exact Dvd.intro (abs r) hqr have h₃: (↑(q.natAbs):ℕ) ∣ p := by norm_cast at * have h₄: (↑(q.natAbs):ℕ) = 1 ∨ (↑(q.natAbs):ℕ) = p := by exact Nat.Prime.eq_one_or_self_of_dvd h₁ (↑(q.natAbs):ℕ) h₃ cases' h₄ with h₄₀ h₄₁ . left norm_cast at * have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q right rw [h₅] norm_cast at * . rw [h_2] at hqr rw [abs_neg _] at hqr have h₂: abs (↑(q.natAbs):ℤ) ∣ p := by exact Dvd.intro (abs r) hqr have h₃: (↑(q.natAbs):ℕ) ∣ p := by norm_cast at * have h₄: (↑(q.natAbs):ℕ) = 1 ∨ (↑(q.natAbs):ℕ) = p := by exact Nat.Prime.eq_one_or_self_of_dvd h₁ (↑(q.natAbs):ℕ) h₃ cases' h₄ with h₄₀ h₄₁ . left norm_cast at * . have h₅: abs q = q.natAbs := by exact abs_eq_natAbs q right rw [h₅] norm_cast cases' h_abs with hq_abs hq_abs . norm_cast at * have h₄: q = ↑(q.natAbs) ∨ q = -↑(q.natAbs) := by exact Int.natAbs_eq q rw [hq_abs] at h₄ norm_cast at h₄ cases' h₄ with h₄₀ h₄₁ . right left exact h₄₀ . left exact h₄₁ . right right have h₂: abs q = q.natAbs := by exact abs_eq_natAbs q rw [h₂] at hq_abs norm_cast at hq_abs refine or_comm.mp ?_ refine (Int.natAbs_eq_natAbs_iff).mp ?_ norm_cast lemma mylemma_qr_11 (q r: ℤ) (h₀: (4 - q) * (4 - r) = 11) : (4 - q = -1 ∨ 4 - q = 1 ∨ 4 - q = -11 ∨ 4 - q = 11) := by have h₁: Nat.Prime (11) := by decide exact q_r_divisor_of_prime (4-q) (4-r) 11 h₀ h₁ lemma mylemma_qr_5 (q r: ℤ) (h₀: (q - 3) * (r - 3) = 5) : (q - 3 = -1 ∨ q - 3 = 1 ∨ q - 3 = -5 ∨ q - 3 = 5) := by have h₁: Nat.Prime (5) := by decide exact q_r_divisor_of_prime (q - 3) (r - 3) 5 h₀ h₁ lemma mylemma_63qr_5 (q r: ℤ) (h₀: (6 - 3*q) * (2 - r) = 5) : (6 - 3*q = -1 ∨ 6 - 3*q = 1 ∨ 6 - 3*q = -5 ∨ 6 - 3*q = 5) := by have h₁: Nat.Prime (5) := by decide exact q_r_divisor_of_prime (6 - 3*q) (2 - r) 5 h₀ h₁ lemma mylemma_case_k_2 (p q r: ℤ) (h₀: 1 < p ∧ p < q ∧ q < r) (hpl: 2 ≤ p) (hql: 3 ≤ q) (hrl: 4 ≤ r) (hpu: p < 4) (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * 2) : (p, q, r) = (2, 4, 8) ∨ (p, q, r) = (3, 5, 15) := by interval_cases p . exfalso norm_num at * have g₁: 2*q + 2*r = 3 := by linarith linarith [g₁,hql,hrl] . right norm_num at * -- have g₁: q*r - 4*q - 4*r + 5 = 0 := by linarith have g₂: (4-q)*(4-r) = 11 := by linarith have g₃: (4-q) = -1 ∨ (4-q) = 1 ∨ (4-q) = -11 ∨ (4-q) = 11 := by exact mylemma_qr_11 q r g₂ cases' g₃ with g₃₁ g₃₂ . have hq: q = 5 := by linarith constructor . exact hq . rw [hq] at g₂ linarith[g₂] . exfalso cases' g₃₂ with g₃₂ g₃₃ . have hq: q = 3 := by linarith[g₃₂] rw [hq] at g₂ have hr: r = -7 := by linarith[g₂] linarith[hrl,hr] . cases' g₃₃ with g₃₃ g₃₄ . have hq: q = 15 := by linarith[g₃₃] rw [hq] at g₂ have hr: r = 5 := by linarith[g₂] linarith[hq,hr,h₀.2] . have hq: q = -7 := by linarith[g₃₄] linarith[hq,hql] lemma mylemma_case_k_3 (p q r: ℤ) (h₀: 1 < p ∧ p < q ∧ q < r) (hpl: 2 ≤ p) (hql: 3 ≤ q) (hrl: 4 ≤ r) (hpu: p < 4) (hk: p * q * r - 1 = (p - 1) * (q - 1) * (r - 1) * 3) : (p, q, r) = (2, 4, 8) ∨ (p, q, r) = (3, 5, 15) := by interval_cases p -- p = 2 . norm_num at * -- have g₁: q*r - 3*q - 3*r + 4 = 0 := by linarith have g₂: (q-3)*(r-3) = 5 := by linarith have g₃: (q-3) = -1 ∨ (q-3) = 1 ∨ (q-3) = -5 ∨ (q-3) = 5 := by exact mylemma_qr_5 q r g₂ cases' g₃ with g₃₁ g₃₂ . exfalso linarith [hql,g₃₁] . cases' g₃₂ with g₃₂ g₃₃ . have hq: q = 4 := by linarith rw [hq] at g₂ have hr: r = 8 := by linarith[g₂] exact { left := hq, right := hr } . exfalso cases' g₃₃ with g₃₃ g₃₄ . linarith[hql,g₃₃] . have hq: q = 8 := by linarith rw [hq] at g₂ norm_num at g₂ have hr: r = 4 := by linarith linarith[hrl,hr] -- p = 3 . right norm_num at * -- have g₁: 3 * q * r - 6 * q - 6 * r + 7 = 0 := by linarith have g₂: (6 - 3*q) * (2 - r) = 5 := by linarith have g₃: (6 - 3*q) = -1 ∨ (6 - 3*q) = 1 ∨ (6 - 3*q) = -5 ∨ (6 - 3*q) = 5 := by exact mylemma_63qr_5 q r g₂ exfalso cases' g₃ with g₃₁ g₃₂ . linarith[g₃₁,q] . cases' g₃₂ with g₃₂ g₃₃ . linarith[g₃₂,q] . cases' g₃₃ with g₃₃ g₃₄ . linarith[g₃₃,q] . linarith[g₃₄,q] theorem imo_1992_p1 (p q r : ℤ) (h₀ : 1 < p ∧ p < q ∧ q < r) (h₁ : (p - 1) * (q - 1) * (r - 1)∣(p * q * r - 1)) : (p, q, r) = (2, 4, 8) ∨ (p, q, r) = (3, 5, 15) := by cases' h₁ with k hk have hpl: 2 ≤ p := by linarith have hql: 3 ≤ q := by linarith have hrl: 4 ≤ r := by linarith have hden: 0 < (((p - 1) * (q - 1)) * (r - 1)) := by have gp: 0 < (p - 1) := by linarith have gq: 0 < (q - 1) := by linarith have gr: 0 < (r - 1) := by linarith exact mul_pos (mul_pos gp gq) gr have h₁: ↑k = (↑(p * q * r - 1):ℚ) / (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by have g₁: ↑(p * q * r - 1) = ↑k * (↑((p - 1) * (q - 1) * (r - 1)):ℚ) := by norm_cast linarith symm have g₂: (↑((p - 1) * (q - 1) * (r - 1)):ℚ) ≠ 0 := by norm_cast linarith[hden] exact (div_eq_iff g₂).mpr g₁ have hk4: k < 4 := by exact mylemma_k_lt_4 p q r k hk hpl hql hrl hden have hk1: 1 < k := by exact mylemma_k_gt_1 p q r k hk h₁ hpl hql hrl hden have hpu: p < 4 := by exact mylemma_p_lt_4 p q r k h₀ hk h₁ hpl hql hrl hden interval_cases k . exact mylemma_case_k_2 p q r h₀ hpl hql hrl hpu hk . exact mylemma_case_k_3 p q r h₀ hpl hql hrl hpu hk