cases' x with 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
case mpr.intro.intro.h.h.zero
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_ih : ∀ m < 0, f m = m
⊢ f 0 = 0

case mpr.intro.intro.h.h.succ
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 + 1, f m = m
⊢ f (x + 1) = x + 1