End HO_REC.
Module HoRec : HO_REC.
Section HORec.
Variable A:Type.
Variable ag: ageable A.
Variable X:Type.
Variable f: (X-> pred A) -> (X -> pred A).
Fixpoint HORec' (n:nat) : X -> pred A :=
match n with
| S n' => f (HORec' n')
| O    => f (fun _ => FF)
end.
Hypothesis Hcont : HOcontractive f.

(* Specifications: *)
Lemma HORec'_unage:   forall j n x a,
(n >= level a) ->  (HORec' n x a <-> HORec' (j+n) x a).