(* 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 n m {A} `{agA : ageable A} (x : A) : age_by n (age_by m x) = age_by (n + m) x.