induction' n with n ih
case h
x : ℕ × ℕ
n : ℕ
hlt : 0 < n
heq : (fun p => if p.1 = p.2 then (0, p.2 + 1) else (p.1 + 1, p.2))^[n] x = x
⊢ False
case h.zero
x : ℕ × ℕ
hlt : 0 < Nat.zero
heq : (fun p => if p.1 = p.2 then (0, p.2 + 1) else (p.1 + 1, p.2))^[Nat.zero] x = x
⊢ False

case h.succ
x : ℕ × ℕ
n : ℕ
ih : 0 < n → (fun p => if p.1 = p.2 then (0, p.2 + 1) else (p.1 + 1, p.2))^[n] x = x → False
hlt : 0 < Nat.succ n
heq : (fun p => if p.1 = p.2 then (0, p.2 + 1) else (p.1 + 1, p.2))^[Nat.succ n] x = x
⊢ False