(* 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 pred_age1' {A} `{agA : ageable A} (R : pred A) x : app_pred R x -> app_pred R (age1' x).