(* OZ *)
simpl in |- *; symmetry  in |- *; exact (add_OZ y).
(* pos n *)
simple induction n.
(* pos O *)
simpl in |- *; symmetry  in |- *; exact (add_IZ_succZ y).
(* pos (S y0) *)
intros; rewrite (tech_add_pos_succZ n0 y). rewrite H.
cut (succZ (pos n0) = pos (S n0)); intros. elim H0.
rewrite (succ_addZ_r y (pos n0)); reflexivity.
reflexivity.
(* neg n *)
simple induction n.
(* neg O *)
simpl in |- *; symmetry  in |- *; exact (add_mIZ_predZ y).
(* neg (S y0) *)
intros; rewrite (tech_add_neg_predZ n0 y). rewrite H.
cut (predZ (neg n0) = neg (S n0)); intros. elim H0.
rewrite (pred_addZ_r y (neg n0)); reflexivity.
reflexivity.
Qed.

(*************************)
Lemma tech_add_pos_neg_posZ :
forall n1 n2 : nat, n2 < n1 -> addZ (pos n1) (neg n2) = pos (n1 - S n2).