proposition: Theorem mod_le: forall a b, 0<=a -> 0<b -> a mod b <= a.
proof: Proof. intros a b ? ?. destruct (le_gt_cases b a). - apply le_trans with b; auto. apply lt_le_incl. destruct (mod_bound_pos a b); auto. - rewrite lt_eq_cases; right. apply mod_small; auto. Qed.
imports: NZAxioms NZMulOrder