Fixpoint domain_equal (A B : Set) (m0 : Map A) (m1 : Map B) {struct m1} :
 Prop :=
  match m0, m1 with
  | M0, M0 => True
  | M0, M1 _ _ => False
  | M0, M2 _ _ => False
  | M1 _ _, M0 => False
  | M1 a _, M1 b _ => a = b
  | M1 _ _, M2 _ _ => False
  | M2 _ _, M0 => False
  | M2 _ _, M1 _ _ => False
  | M2 a b, M2 c d => domain_equal A B a c /\ domain_equal A B b d
  end.

Lemma domain_equal_mapget : forall (A B : Set) (m0 : Map A) (m1 : Map B) (a : ad) (x : A), domain_equal A B m0 m1 -> MapGet A m0 a = Some x -> exists y : B, MapGet B m1 a = Some y.