(* OZ *)
reflexivity.
(* pos n *)
simple induction n.
(* pos O *)
symmetry  in |- *; exact (add_IZ_succZ y).
(* pos (S y0) *)
intros y0 H; do 2 rewrite (tech_mult_posZ y0).
rewrite H; elim (addZ_commutativity (pos y0) (multZ (pos y0) y)).
elim (addZ_associativity (pos y0) (multZ (pos y0) y) (succZ y)).
elim (addZ_commutativity (addZ (multZ (pos y0) y) (succZ y)) (pos y0)).
rewrite (succ_addZ_r (multZ (pos y0) y) y).
rewrite (succ_addZ_l (addZ (multZ (pos y0) y) y) (pos y0)).
elim (succ_addZ_r (addZ (multZ (pos y0) y) y) (pos y0)).
reflexivity.
(* neg n *)
simple induction n.
(* neg O *)
simpl in |- *; rewrite (add_mIZ_predZ (oppZ y)); exact (opp_succZ y).
(* neg (S y0) *)
intros y0 H; do 2 rewrite (tech_mult_negZ y0).
elim H; elim (addZ_commutativity (oppZ y) (multZ (neg y0) y)).
elim (addZ_associativity (oppZ y) (multZ (neg y0) y) (neg (S y0))).
elim (addZ_commutativity (addZ (multZ (neg y0) y) (neg (S y0))) (oppZ y)).
rewrite (opp_succZ y).
rewrite (pred_addZ_r (multZ (neg y0) (succZ y)) (oppZ y)).
rewrite H; elim (pred_addZ_l (addZ (multZ (neg y0) y) (neg y0)) (oppZ y)).
elim (pred_addZ_r (multZ (neg y0) y) (neg y0)); unfold predZ in |- *;
reflexivity.
Qed.

(*****************)
Lemma mult_predZ_r :
forall x y : Z, multZ x (predZ y) = addZ (multZ x y) (oppZ x).