Instance map_disj_join : Join (A -> option B) :=
  fun a b c => forall k, match a k, b k with Some v, None | None, Some v => c k = Some v | None, None => c k = None | _, _ => False end.

Lemma disjoint_add : forall m1 m2 m3, disjoint m1 m2 -> disjoint m1 m3 -> disjoint m1 (map_add m2 m3).