Definition age_induction {A} `{ageable A} :=
  well_founded_induction (af_wf _).Definition fashionR {A1} `{ageable A1} {A2}`{ageable A2} (x:A1) (y: A2) : Prop :=
  level x = level y.

(* Specifications: *)
Lemma fashionR_refl {A} `{ageable A} : reflexive _ fashionR.
Lemma fashionR_trans {A} `{ageable A} {B} `{ageable B} {C} `{ageable C} : forall (x: A) (y: B) (z: C), fashionR x y -> fashionR y z -> fashionR x z.
Lemma fashionR_sym {A} `{ageable A} {B} `{ageable B}: forall (x: A) (y: B), fashionR x y -> fashionR y x.
Lemma age_level {A} `{ageable A} : forall (x y:A), age x y -> level x = S (level y).
Lemma age1_level0 {A} `{ageable A} : forall (x:A), age1 x = None <-> level x = 0.