Section Bool.
(** Properties of [filter] *)
Variable f:elt->bool.
Variable Comp: Proper (E.eq==>Logic.eq) f.
Let Comp' : Proper (E.eq==>Logic.eq) (fun x =>negb (f x)).
(** Properties of [for_all] *)
(** Properties of [exists] *)

Lemma for_all_exists:
forall s, exists_ f s = negb (for_all (fun x =>negb (f x)) s).