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_new : forall (m:Map) (a:ad) (y:A), MapGet (MapPut_behind m a y) a = match MapGet m a with | Some y' => Some y' | _ => Some y end.