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)).

Lemma add_filter_1 : forall s s' x,
f x=true -> (Add x s s') -> (Add x (filter f s) (filter f s')).