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 predicate_max:
forall (F: nat -> Prop) (Fdec: forall n, {F n}+{~ F n}) n,
F 0%nat ->
~ F n ->
exists i, F i /\ (i<n)%nat /\ ~ F (S i).