rw [n_ih n n.lt_succ_self, ← add_one_mul, le_iff_eq_or_lt, Nat.lt_succ_iff] at h
case h.succ
f : ℕ → ℕ
h1 : f 0 = 0
h0 : ∀ (x : ℕ), f (f x) = x
n : ℕ
n_ih : ∀ m < n + 1, f (m * f 1) = m
h : f (n * f 1 + f 1) ≤ f (n * f 1) + 1
⊢ f ((n + 1) * f 1) = n + 1
case h.succ
f : ℕ → ℕ
h1 : f 0 = 0
h0 : ∀ (x : ℕ), f (f x) = x
n : ℕ
n_ih : ∀ m < n + 1, f (m * f 1) = m
h : f ((n + 1) * f 1) = n + 1 ∨ f ((n + 1) * f 1) ≤ n
⊢ f ((n + 1) * f 1) = n + 1