| 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 | |