(* The definitions and other results of age_by and age_to should be moved here from msl/ageable.v.  Alternatively, this can be moved to msl/ageable.v (or this file to msl/) eventually, but we keep it here for now to reduce compilation time. *)
(* Apply [age1] n times (meaningful when [n <= level x] *)
Definition age1' {A} `{agA : ageable A} : A -> A :=
fun x => match age1 x with Some y => y | None => x end.
Definition age_by n {A} `{agA : ageable A} : A -> A := Nat.iter n age1'.

Lemma age_age_by n {A} `{agA : ageable A} (x y : A) : age x y -> age_by (S n) x = age_by n y.