Definition recursion {A} : A -> (N -> A -> A) -> N -> A :=
  peano_rect (fun _ => A).#[global]
Instance recursion_wd {A} (Aeq : relation A) :
 Proper (Aeq==>(Logic.eq==>Aeq==>Aeq)==>Logic.eq==>Aeq) recursion.

Theorem iter_ind (A:Type) (f:A -> A) (a:A) (P:N -> A -> Prop) : P 0 a -> (forall n a', P n a' -> P (succ n) (f a')) -> forall n, P n (iter n f a).