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_2 : forall s s' x,
f x=false -> (Add x s s') -> filter f s [=] filter f s'.