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.

Lemma Z_inj_nat_id: forall (z : Z),
Z_inj_nat_rev (Z_inj_nat z) = z.