intros x; elim x; auto.
unfold Zabs_nat in |- *.
intros p.
pattern p at 1 3 in |- *;
rewrite <- (pred_o_P_of_succ_nat_o_nat_of_P_eq_id p).
generalize (convert_not_O p); case (nat_of_P p); simpl in |- *;
auto with arith.
intros H'; case H'; auto.
intros n H' H'0; rewrite Ppred_succ; auto.
intros p H'; Contradict H'; auto.
Qed.

(* Specifications: *)
Theorem inject_nat_convert :
forall (p : Z) (q : positive),
p = Zpos q -> Z_of_nat (nat_of_P q) = p.