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

(***************)
Lemma pred_addZ_r : forall x y : Z, addZ x (predZ y) = predZ (addZ x y).
intros; elim x.