Definition Z_inj_nat (z : Z) : nat :=
  match z with
  | Z0 => 0
  | Zpos p => Pos.to_nat (p~0)
  | Zneg p => Pos.to_nat (Pos.pred_double p)
  end.Definition Z_inj_nat_rev (n : nat) : Z :=
  match n with
  | O => 0
  | S n' => match Pos.of_nat n with
            | xH =>   Zneg xH
            | xO p => Zpos p
            | xI p => Zneg (Pos.succ p)
            end
  end.

(* Specifications: *)
Lemma Pos_pred_double_inj: forall (p q : positive), Pos.pred_double p = Pos.pred_double q -> p = q.
Lemma Z_inj_nat_id: forall (z : Z), Z_inj_nat_rev (Z_inj_nat z) = z.
Lemma Z_inj_nat_inj: forall (x y : Z), Z_inj_nat x = Z_inj_nat y -> x = y.
Lemma constructive_indefinite_ground_description_Z: forall P : Z -> Prop, (forall z : Z, {P z} + {~ P z}) -> (exists z : Z, P z) -> {z : Z | P z}.