Fixpoint MapRemove (m:Map) : ad -> Map :=
match m with
| M0 => fun _:ad => M0
| M1 a y =>
fun a':ad => match Neqb a a' with
| true => M0
| false => m
end
| M2 m1 m2 =>
fun a:ad =>
if Nbit0 a
then makeM2 m1 (MapRemove m2 (Ndiv2 a))
else makeM2 (MapRemove m1 (Ndiv2 a)) m2
end.

Lemma MapRemove_semantics :
forall (m:Map) (a:ad),
eqm (MapGet (MapRemove m a))
(fun a':ad => if Neqb a a' then None else MapGet m a').