(** [fold_rec_weak] is a weaker principle than [fold_rec_bis] : the step hypothesis must here be applicable to any [x]. At the same time, it looks more like an induction principle, and hence can be easier to use. *)

Lemma fold_rel :
forall (A B:Type)(R : A -> B -> Type)
(f : elt -> A -> A)(g : elt -> B -> B)(i : A)(j : B)(s : t),
R i j ->
(forall x a b, In x s -> R a b -> R (f x a) (g x b)) ->
R (fold f s i) (fold g s j).