End PROD'.
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.
Arguments composeOptN [A] _ _ _.
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.