Definition t := Z.Definition index (z: Z): positive :=
    match z with
    | Z0 => xH
    | Zpos p => xO p
    | Zneg p => xI p
    end.

(* Specifications: *)
Lemma index_inj: forall (x y: Z), index x = index y -> x = y.
Lemma index_inj: forall (x y: Z), index x = index y -> x = y.