induction' x using Nat.strong_induction_on with x x_ih
case mpr.intro.intro.h
f : ℕ → ℕ
h1 : f 0 = 0
h0 : ∀ (x : ℕ), f (f x) = x
h : ∀ (y : ℕ), f (y + f 1) ≤ f y + 1
h2 : f 1 = 1
x : ℕ
⊢ f x = x
case mpr.intro.intro.h.h
f : ℕ → ℕ
h1 : f 0 = 0
h0 : ∀ (x : ℕ), f (f x) = x
h : ∀ (y : ℕ), f (y + f 1) ≤ f y + 1
h2 : f 1 = 1
x : ℕ
x_ih : ∀ m < x, f m = m
⊢ f x = x