Remark exists_for_all:
forall m f,
exists_ m f = negb (for_all m (fun x a => negb (f x a))).
Remark for_all_exists:
forall m f,
for_all m f = negb (exists_ m (fun x a => negb (f x a))).

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