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).