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

Lemma map_join_spec : forall m1 m2 m3, join m1 m2 m3 <-> compatible m1 m2 /\ m3 = map_add m1 m2.
Global Program Instance map_PCM : Ghost := { valid a := True; Join_G := map_join }.