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

Lemma in_dom_put_behind :
forall (m:Map A) (a0:ad) (y0:A) (a:ad),
in_dom a (MapPut_behind A m a0 y0) = orb (Neqb a a0) (in_dom a m).