Unfolding properties for Nat.iter.

theories/numbers.v

@@ -82,7 +82,7 @@ Proof. intros. destruct (Nat_mul_split_l n x2 x1 y2 y1); auto with lia. Qed.
 Notation lcm := Nat.lcm.
 Notation divide := Nat.divide.
 Notation "( x | y )" := (divide x y) : nat_scope.
-Instance divide_dec x y : Decision (x | y).
+Instance Nat_divide_dec x y : Decision (x | y).
 Proof.
   refine (cast_if (decide (lcm x y = y))); by rewrite Nat.divide_lcm_iff.
 Defined.
@@ -94,6 +94,11 @@ Hint Extern 0 (_ | _) => reflexivity.
 Lemma Nat_divide_ne_0 x y : (x | y) → y ≠ 0 → x ≠ 0.
 Proof. intros Hxy Hy ->. by apply Hy, Nat.divide_0_l. Qed.
 
+Lemma Nat_iter_S {A} n (f: A → A) x : Nat.iter (S n) f x = f (Nat.iter n f x).
+Proof. done. Qed.
+Lemma Nat_iter_S_r {A} n (f: A → A) x : Nat.iter (S n) f x = Nat.iter n f (f x).
+Proof. induction n; f_equal/=; auto. Qed.
+
 (** * Notations and properties of [positive] *)
 Open Scope positive_scope.