(** An alternate (and previous) specification for [fold] was based on the recursive structure of a set. It is now lemmas [fold_1] and [fold_2]. *)

Lemma fold_2 :
forall s s' x (A : Type) (eqA : A -> A -> Prop)
(st : Equivalence eqA) (i : A) (f : elt -> A -> A),
Proper (E.eq==>eqA==>eqA) f ->
transpose eqA f ->
~ In x s -> Add x s s' -> eqA (fold f s' i) (f x (fold f s i)).