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))).

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).