Fixpoint composeOptN (A: Type) (f: A -> option A)
         (n: nat) (w: A) {struct n} : option A :=
 match n  with
 | S n' => match f w with Some w' => composeOptN A f n' w' | None => None end
 | O => Some w
 end.Definition ageN {A} `{ageable A}: nat -> A -> option A := composeOptN age1.

Lemma ageN_compose' {A} `{agA : ageable A}: forall a b phi1 phi3, ageN (a+b)%nat phi1 = Some phi3 -> exists phi2, ageN a phi1 = Some phi2 /\ ageN b phi2 = Some phi3.