Section Iter.
Context {A : Type}.
Variable (f : A -> A).
Fixpoint iter_nat (n : nat) (x : A) {struct n} : A :=
match n with
| S n' => iter_nat n' (f x)
| O => x
end.

Lemma iter_nat_S :
forall (p : nat) (x : A),
iter_nat (S p) x = f (iter_nat p x).