reflexivity.
simple induction n.
reflexivity.
intros.
do 2 rewrite (tech_add_pos_succZ n0).
elim H; reflexivity.
simple induction n.
simpl in |- *; symmetry  in |- *; apply succ_pred_pred_succZ.
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.