Fixpoint MapPut_behind (m:Map) : ad -> A -> Map :=
match m with
| M0 => M1
| M1 a y =>
fun (a':ad) (y':A) =>
match Nxor a a' with
| N0 => m
| Npos p => MapPut1 a y a' y' p
end
| M2 m1 m2 =>
fun (a:ad) (y:A) =>
match a with
| N0 => M2 (MapPut_behind m1 N0 y) m2
| Npos xH => M2 m1 (MapPut_behind m2 N0 y)
| Npos (xO p) => M2 (MapPut_behind m1 (Npos p) y) m2
| Npos (xI p) => M2 m1 (MapPut_behind m2 (Npos p) y)
end
end.

Lemma MapPut_behind_semantics_3_1 :
forall (m m':Map) (a:ad) (y:A),
MapPut_behind (M2 m m') a y =
(if Nbit0 a
then M2 m (MapPut_behind m' (Ndiv2 a) y)
else M2 (MapPut_behind m (Ndiv2 a) y) m').