Definition exists_ (f : elt -> bool) (s : t) : bool :=
    if exists_ (P:=fun x => f x = true) (f_dec f) s
    then true
    else false.

(* Specifications: *)
Lemma exists_1 : forall (s : t) (f : elt -> bool), compat_bool E.eq f -> Exists (fun x => f x = true) s -> exists_ f s = true.
Lemma exists_2 : forall (s : t) (f : elt -> bool), compat_bool E.eq f -> exists_ f s = true -> Exists (fun x => f x = true) s.