intros; elim x.
(* OZ *)
reflexivity.
(* pos n *)
simple induction n.
(* pos O *)
simpl in |- *; symmetry  in |- *; exact (add_mIZ_predZ y).
(* pos (S n0) *)
intros n0 H; unfold oppZ in |- *; do 2 rewrite (tech_mult_posZ n0).
rewrite (pred_addZ_r (multZ (pos n0) (predZ y)) y).
elim (pred_addZ_l (multZ (pos n0) (predZ y)) y).
elim (addZ_commutativity y (multZ (pos n0) y)).
elim (addZ_associativity y (multZ (pos n0) y) (neg (S n0))).
elim (addZ_commutativity (addZ (multZ (pos n0) y) (neg (S n0))) y).
rewrite H; elim (pred_addZ_r (multZ (pos n0) y) (oppZ (pos n0))).
reflexivity.
(* neg n *)
simple induction n.
(* neg O *)
simpl in |- *.
replace (pos 0) with IZ; auto.
rewrite (add_IZ_succZ (oppZ y)).
exact (opp_predZ y).
(* neg (S n0) *)
intros n0 H; do 2 rewrite (tech_mult_negZ n0).
rewrite H; rewrite (opp_predZ y).
elim (addZ_commutativity (oppZ (neg n0)) (multZ (neg n0) y)).
elim (addZ_associativity (oppZ (neg n0)) (multZ (neg n0) y) (succZ (oppZ y))).
elim
(addZ_commutativity (addZ (multZ (neg n0) y) (succZ (oppZ y)))
(oppZ (neg n0))).
rewrite (succ_addZ_r (multZ (neg n0) y) (oppZ y)).
rewrite (succ_addZ_l (addZ (multZ (neg n0) y) (oppZ y)) (oppZ (neg n0))).
elim (succ_addZ_r (addZ (multZ (neg n0) y) (oppZ y)) (oppZ (neg n0))).
reflexivity.
Qed.

(************)
Lemma mult_OZ : forall x : Z, multZ x OZ = OZ.
simple destruct x.