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).