Definition in_dom (a:ad) (m:Map A) :=
    match MapGet A m a with
    | None => false
    | _ => true
    end.

Lemma in_dom_remove : forall (m:Map A) (a0 a:ad), in_dom a (MapRemove A m a0) = andb (negb (Neqb a a0)) (in_dom a m).