(** [exists_] **)
Definition exists_ {A : Type} (f : key -> A -> bool) dom (m : t A dom) :=
  fold (fun x v b => b || f x v) m false.

(* Specifications: *)
Lemma exists_spec_aux : forall dom A f, Proper (X.eq ==> Logic.eq ==> Logic.eq) f -> forall (m : t A dom) l b, fold_left (fun acc xv => acc || f (fst xv) (snd xv)) l b = true <-> b = true ∨ (exists x (v : A), InA (X.eq * Logic.eq)%signature (x, v) l ∧ f x v = true).
Lemma exists_spec : forall A f, Proper (X.eq ==> Logic.eq ==> Logic.eq) f -> forall dom (m : t A dom), exists_ f m = true <-> exists x v, find x m = Some v ∧ f x v = true.