Notation NoDup := (NoDupA E.eq).
Notation InA := (InA E.eq).
(** ** 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. *)

Theorem fold_rec :
forall (A:Type)(P : t -> A -> Type)(f : elt -> A -> A)(i:A)(s:t),
(forall s', Empty s' -> P s' i) ->
(forall x a s' s'', In x s -> ~In x s' -> Add x s' s'' ->
P s' a -> P s'' (f x a)) ->
P s (fold f s i).