intros. elim H; intros. apply (gcd_OZ_absZ b0). unfold is_gcd in |- *.
split. elim H3; intros; elim H5; intros; exact H6.
split. elim H1; intros; elim H5; intros; elim H7; intros; rewrite H9.
apply (div_add Z IdZ addZ multZ OZ oppZ Z_ring (multZ b0 q) r d0).
elim H3; intros; elim H11; intros.
exact (div_mult Z IdZ addZ multZ OZ oppZ Z_ring b0 q d0 H12 I).
elim H3; intros; exact H10.
intros. elim H3; intros; elim H7; intros. apply (H9 q0).
cut (r = addZ a0 (oppZ (multZ b0 q))); intros. rewrite H10.
apply (div_add Z IdZ addZ multZ OZ oppZ Z_ring a0 (oppZ (multZ b0 q)) q0 H5).
apply (div_opp Z IdZ addZ multZ OZ oppZ Z_ring (multZ b0 q) q0).
exact (div_mult Z IdZ addZ multZ OZ oppZ Z_ring b0 q q0 H4 I).
elim H1; intros; elim H11; intros; elim H13; intros; rewrite H15.
elim (addZ_commutativity r (multZ b0 q)).
elim (addZ_associativity r (multZ b0 q) (oppZ (multZ b0 q))).
elim (addZ_opposite (multZ b0 q) I); intros.
elim H17; intros. elim H19; intros. rewrite H20. symmetry  in |- *. exact (add_OZ r).
exact H4.
Qed.
(*************)
Definition gcdZ (a b : Z) := pi1 Z (is_gcdZ a b) (gcdZ_exists a b).

(*********************)
Lemma positive_is_gcdZ : forall a b d : Z, is_gcdZ a b d -> leZ OZ d.
intros; elim H; intros. apply (sign_absZ b0). exact H3.