(* OZ *)
reflexivity.
(* (pos n) *)
simple induction n.
(* (pos O) *)
simpl in |- *; apply succ_pred_pred_succZ.
(* (pos (S y0)) *)
intros.
do 2 rewrite (tech_add_pos_succZ n0).
rewrite H; apply succ_pred_pred_succZ.
(* neg n *)
simple induction n.
(* neg O *)
reflexivity.
(* neg (S y0) *)
intros.
do 2 rewrite (tech_add_neg_predZ n0).
elim H; reflexivity.
Qed.

(***********)
Lemma add_OZ : forall x : Z, addZ x OZ = x.
simple induction x.