(* 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_ind_opp {A} `{agA : ageable A} (P : A -> Prop) : (forall x y, age x y -> P y -> P x) -> forall x n, P (age_by n x) -> P x.