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 MapPut1_semantics' : forall (p:positive) (a a':ad) (y y':A), Nxor a a' = Npos p -> eqm (MapGet (MapPut1 a y a' y' p)) (fun a0:ad => if Neqb a' a0 then Some y' else if Neqb a a0 then Some y else None).