proposition: Lemma div_le_compat_l: forall p q r, 0<=p -> 0<q<=r -> p/r <= p/q.
proof: Proof. intros p q r Hp (Hq,Hqr). apply div_le_lower_bound; auto. rewrite (div_mod p r) at 2 by order. apply le_trans with (r*(p/r)). - apply mul_le_mono_nonneg_r; try order. apply div_pos; order. - rewrite <- (add_0_r (r*(p/r))) at 1. rewrite <- add_le_mono_l. destruct (mod_bound_pos p r); order. Qed.
imports: NZAxioms NZMulOrder