Local Definition Comp' : Proper (E.eq==>Logic.eq) (fun x => negb (f x)).

(* Specifications: *)
Lemma exists_mem_1: forall s, (forall x, mem x s=true->f x=false) -> exists_ f s=false.
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}.