Section Fold.
Notation NoDup := (NoDupA E.eq).
Notation InA := (InA E.eq).
(** Alternative specification via [fold_right] *)
(** ** Induction principles for fold (contributed by S. Lescuyer) *)
(** In the following lemma, the step hypothesis is deliberately restricted to the precise set s we are considering. *)
(** Same, with [empty] and [add] instead of [Empty] and [Add]. In this case, [P] must be compatible with equality of sets *)
(** [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).