Fixpoint HORec' (n:nat) : X -> pred A :=
    match n with
    | S n' => f (HORec' n')
    | O    => f (fun _ => FF)
    end.

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