Fixpoint MapPut1 (a:ad) (y:A) (a':ad) (y':A) (p:positive) {struct p} :
   Map :=
    match p with
    | xO p' =>
        let m := MapPut1 (Ndiv2 a) y (Ndiv2 a') y' p' in
        match Nbit0 a with
        | false => M2 m M0
        | true => M2 M0 m
        end
    | _ =>
        match Nbit0 a with
        | false => M2 (M1 (Ndiv2 a) y) (M1 (Ndiv2 a') y')
        | true => M2 (M1 (Ndiv2 a') y') (M1 (Ndiv2 a) y)
        end
    end.

Lemma MapGet_M2_bit_0_2 : forall (m m' m'':Map) (a:ad), MapGet (if Nbit0 a then M2 m m' else M2 m' m'') a = MapGet m' (Ndiv2 a).