intros z1 z2 H H0; apply Zle_Zpred_inv.
apply Zabs_intro with (P := fun x => (x <= Zpred z2)%Z); auto with zarith.
Qed.

Theorem Zlt_ZERO_Zle_ONE : forall z : Z, (0 < z)%Z -> (1 <= z)%Z.
intros z H; replace 1%Z with (Zsucc 0); auto with zarith; simpl in |- *; auto.