Program Fixpoint forall_rec (hi: Z) {wf (Zwf lo) hi}:
                 {forall x, lo <= x < hi -> P x}
                 + {exists x, lo <= x < hi /\ Q x} :=
  if zlt lo hi then
    match f (hi - 1) with
    | left _ =>
        match forall_rec (hi - 1) with
        | left _ => left _ _
        | right _ => right _ _
        end
    | right _ => right _ _
    end
  else
    left _ _
.Definition forall_dec
       (P Q: Z -> Prop) (f: forall (x: Z), {P x} + {Q x}) (i: interv) :
       {forall x, In x i -> P x} + {exists x, In x i /\ Q x} :=
 forall_rec P Q f (fst i) (snd i).Function fold_rec (hi: Z) {wf (Zwf lo) hi} : A :=
  if zlt lo hi then f (hi - 1) (fold_rec (hi - 1)) else a.

(* Specifications: *)
Lemma fold_rec_elements: forall hi, fold_rec hi = List.fold_right f a (elements_rec lo hi).