Definition exists_ (m: T.t A) (f: T.elt -> A -> bool) : bool :=
  T.fold (fun b x a => b || f x a) m false.

(* Specifications: *)
Lemma exists_correct: forall m f, exists_ m f = true <-> (exists x a, T.get x m = Some a /\ f x a = true).
Lemma exists_correct: forall m f, exists_ m f = true <-> (exists x a, T.get x m = Some a /\ f x a = true).