End HORec.
Definition HORec {A} `{ag: ageable A}  {X: Type} (f:  (X-> pred A) -> (X -> pred A)) (x: X) : pred A :=
mkPred (fun a : A => app_pred (@HORec' A ag X f (level a) x) a).

(* Specifications: *)
Lemma HORec_fold_unfold {A} `{ageable A} : forall X f (H:HOcontractive (X:=X) f),
HORec f = f (HORec f).