Theorem: Imo2007P5.generalized_imo2007_p5 (Compfiles/Imo2007P5.lean)

State before:
n : ℤ
hn : 1 < n
⊢ ∀ (a b : ℕ), 0 < a → 0 < b → n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2 → a = b
Tactic:
let P : ℕ → Prop := λ k ↦ (0 < k ∧ ∃ m : ℕ, k < m ∧ n * k * m - 1 ∣ (n * k ^ 2 - 1) ^ 2)
State after:
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
⊢ ∀ (a b : ℕ), 0 < a → 0 < b → n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2 → a = b

State before:
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
⊢ ∀ (a b : ℕ), 0 < a → 0 < b → n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2 → a = b
Tactic:
have h : ∀ k : ℕ, P k → ∃ m : ℕ, m < k ∧ P m :=
  λ k h ↦ Exists.elim (bad_exists_descent hn h)
    (λ c h0 ↦ ⟨c, h0.2.1, h0.1, k, h0.2.1, h0.2.2⟩)
State after:
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
h : ∀ (k : ℕ), P k → ∃ m < k, P m
⊢ ∀ (a b : ℕ), 0 < a → 0 < b → n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2 → a = b

State before:
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
h : ∀ (k : ℕ), P k → ∃ m < k, P m
⊢ ∀ (a b : ℕ), 0 < a → 0 < b → n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2 → a = b
Tactic:
classical
replace h := nat_pred_descent h
intro a b ha hb h0
by_contra! h1
exact (lt_or_gt_of_ne h1).elim (λ h1 ↦ h a ⟨ha, b, h1, h0⟩)
  (λ h1 ↦ h b ⟨hb, a, h1, bad_symm h0⟩)
State after:
no goals

State before:
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
h : ∀ (k : ℕ), P k → ∃ m < k, P m
⊢ ∀ (a b : ℕ), 0 < a → 0 < b → n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2 → a = b
Tactic:
replace h := nat_pred_descent h
State after:
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
h : ∀ (k : ℕ), ¬P k
⊢ ∀ (a b : ℕ), 0 < a → 0 < b → n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2 → a = b

State before:
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
h : ∀ (k : ℕ), ¬P k
⊢ ∀ (a b : ℕ), 0 < a → 0 < b → n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2 → a = b
Tactic:
intro a b ha hb h0
State after:
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
h : ∀ (k : ℕ), ¬P k
a b : ℕ
ha : 0 < a
hb : 0 < b
h0 : n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2
⊢ a = b

State before:
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
h : ∀ (k : ℕ), ¬P k
a b : ℕ
ha : 0 < a
hb : 0 < b
h0 : n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2
⊢ a = b
Tactic:
by_contra! h1
State after:
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
h : ∀ (k : ℕ), ¬P k
a b : ℕ
ha : 0 < a
hb : 0 < b
h0 : n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2
h1 : a ≠ b
⊢ False

State before:
n : ℤ
hn : 1 < n
P : ℕ → Prop := fun k => 0 < k ∧ ∃ m, k < m ∧ n * ↑k * ↑m - 1 ∣ (n * ↑k ^ 2 - 1) ^ 2
h : ∀ (k : ℕ), ¬P k
a b : ℕ
ha : 0 < a
hb : 0 < b
h0 : n * ↑a * ↑b - 1 ∣ (n * ↑a ^ 2 - 1) ^ 2
h1 : a ≠ b
⊢ False
Tactic:
exact (lt_or_gt_of_ne h1).elim (λ h1 ↦ h a ⟨ha, b, h1, h0⟩)
  (λ h1 ↦ h b ⟨hb, a, h1, bad_symm h0⟩)
State after:
no goals