. simp [Nat.iterate]
case zero
A✝ : Type
R : A✝ → A✝ → Prop
inhabited_A : Nonempty A✝
A : Sort u_1
f : A → A
⊢ ∀ (x : A), f^[Nat.zero + 1] x = f (f^[Nat.zero] x)

case succ
A✝ : Type
R : A✝ → A✝ → Prop
inhabited_A : Nonempty A✝
A : Sort u_1
f : A → A
n : ℕ
ihn : ∀ (x : A), f^[n + 1] x = f (f^[n] x)
⊢ ∀ (x : A), f^[Nat.succ n + 1] x = f (f^[Nat.succ n] x)
case succ
A✝ : Type
R : A✝ → A✝ → Prop
inhabited_A : Nonempty A✝
A : Sort u_1
f : A → A
n : ℕ
ihn : ∀ (x : A), f^[n + 1] x = f (f^[n] x)
⊢ ∀ (x : A), f^[Nat.succ n + 1] x = f (f^[Nat.succ n] x)