replace n_ih := congr_arg f (n_ih _ (lt_of_le_of_lt h n.lt_succ_self))
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
⊢ 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
h : f ((n + 1) * f 1) ≤ n
n_ih : f (f (f ((n + 1) * f 1) * f 1)) = f (f ((n + 1) * f 1))
⊢ f ((n + 1) * f 1) = n + 1