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] *)

Lemma for_all_mem_4:
forall s, for_all f s=false -> {x:elt | mem x s=true /\ f x=false}.