End Bool.
Section Bool'.
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)).

(* Specifications: *)
Lemma exists_mem_2:
forall s, exists_ f s=false -> forall x, mem x s=true -> f x=false.
Lemma exists_mem_3:
forall s x, mem x s=true -> f x=true -> exists_ f s=true.
Lemma exists_mem_4:
forall s, exists_ f s=true -> {x:elt | (mem x s)=true /\ (f x)=true}.