Global Instance compatible_refl : Reflexive compatible.
Global Instance compatible_comm : Symmetric compatible.

Lemma map_add_incl : forall m1 m2 m3, map_incl m1 m3 -> map_incl m2 m3 -> map_incl (map_add m1 m2) m3.
Global Instance map_join : Join (A -> option B) :=
fun a b c => forall k v, c k = Some v <-> a k = Some v \/ b k = Some v.