(** [fold_rec_weak] is a weaker principle than [fold_rec_bis] : the step hypothesis must here be applicable anywhere. 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 : key -> elt -> A -> A)(g : key -> elt -> B -> B)(i : A)(j : B)
(m : t elt),
R i j ->
(forall k e a b, MapsTo k e m -> R a b -> R (f k e a) (g k e b)) ->
R (fold f m i) (fold g m j).