Fixpoint upd (i : nat) {A} new (l : list A) : option (list A) :=
  match i, l with
  | O, _ :: l => Some (new :: l)
  | _, nil => None
  | S i, x :: l =>
    match upd i new l with
    | Some l' => Some (x :: l')
    | None => None
    end
  end.

Lemma age_to_join_sub {A} {JA: Join A} {PA: Perm_alg A} {agA: ageable A} {AgeA: Age_alg A} : forall k x1 x2, join_sub x1 x2 -> join_sub (age_to k x1) (age_to k x2).