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}.