(* 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'.
(* 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.