intros p1; elim p1; auto.
intros p2 Rec; pattern (Zpos (xI p2)) at 2; replace (Zpos (xI p2)) with (2 * (Zpos p2) +1)%Z; auto with zarith.
rewrite plus_IZR; rewrite mult_IZR; rewrite <- Rec.
simpl Z2R; simpl IZR; case p2; intros; simpl (P2R 1);ring.
intros p2 Rec; pattern (Zpos (xO p2)) at 2; replace (Zpos (xO p2)) with (2 * (Zpos p2))%Z; auto with zarith.
rewrite mult_IZR; rewrite <- Rec.
simpl Z2R; simpl IZR; case p2; intros; simpl (P2R 1); ring.
intros p1; elim p1; auto.
intros p2 Rec; pattern (Zneg (xI p2)) at 2; replace (Zneg (xI p2)) with ((2 * (Zneg p2) +  -1))%Z; auto with zarith.
rewrite plus_IZR; rewrite mult_IZR; rewrite <- Rec.
simpl Z2R; simpl IZR; case p2; intros; simpl (P2R 1); ring.
intros p2 Rec; pattern (Zneg (xO p2)) at 2; replace (Zneg (xO p2)) with (2 * (Zneg p2))%Z; auto with zarith.
rewrite mult_IZR; rewrite <- Rec.
simpl Z2R; simpl IZR; case p2; intros; simpl (P2R 1); ring.
Qed.

(* Specifications: *)
Theorem Z2R_le: forall p q, (p <= q)%Z -> (Z2R p <= Z2R q)%R.
intros p q; repeat rewrite Z2R_correct; intros; apply IZR_le; auto.