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

(** Properties of [for_all] *)
Lemma for_all_mem_1: forall s,
(forall x, (mem x s)=true->(f x)=true) -> (for_all f s)=true.