(** Same, with [empty] and [add] instead of [Empty] and [Add]. In this case, [P] must be compatible with equality of sets *)

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