(** [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_rec_weak :
forall (A:Type)(P : t -> A -> Type)(f : elt -> A -> A)(i:A),
(forall s s' a, s[=]s' -> P s a -> P s' a) ->
P empty i ->
(forall x a s, ~In x s -> P s a -> P (add x s) (f x a)) ->
forall s, P s (fold f s i).