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 joinlist_age_to {A} `{agA : ageable A} {J : Join A} {_ : Age_alg A} {_ : Perm_alg A} n (l : list A) (x : A) :
joinlist l x ->
joinlist (map (age_to n) l) (age_to n x).