Fixpoint composeOptN' {A} (f : A -> option A) n x :=
  match n with
  | 0 => Some x
  | S n =>
    match composeOptN' f n x with
    | Some y => f y
    | None => None
    end
  end.

Lemma age_to_identy {A} `{asaA: Age_alg A}: forall k phi, identity phi -> identity (age_to k phi).