(* 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_age_to n {A} `{agA : ageable A} x y : level x = S n -> age x y -> age_to n x = y.