(* Age [x] to the level [k] (meaningul when [k <= level x] *)
Definition age_to k {A} `{agA : ageable A} (x : A) : A := age_by (level x - k) x.

Lemma age_by_age_by_pred {A} `{agA : ageable A} (P : pred A) x n1 n2 : le n1 n2 -> app_pred P (age_by n1 x) -> app_pred P (age_by n2 x).