intros z1 z2 H H0; case (Zle_or_lt 0 z1); intros H1.
rewrite Zabs_eq; auto with zarith.
rewrite Zabs_eq in H; auto with zarith.
apply Zlt_trans with (2 := H).
repeat rewrite Zabs_eq_opp; auto with zarith.
Qed.

Theorem Zle_n_Zpred :
forall z1 z2 : Z, (Zpred z1 <= Zpred z2)%Z -> (z1 <= z2)%Z.