(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(*                      Evgeny Makarov, INRIA, 2007                     *)
(************************************************************************)
Require Export ZAddOrder.
Module Type ZMulOrderProp (Import Z : ZAxiomsMiniSig').
Include ZAddOrderProp Z.

(* Specifications: *)
Theorem mul_lt_mono_nonpos :
forall n m p q, m <= 0 -> n < m -> q <= 0 -> p < q ->  m * q < n * p.
Theorem mul_le_mono_nonpos :
forall n m p q, m <= 0 -> n <= m -> q <= 0 -> p <= q ->  m * q <= n * p.
Theorem mul_nonpos_nonpos : forall n m, n <= 0 -> m <= 0 -> 0 <= n * m.
Theorem mul_nonneg_nonpos : forall n m, 0 <= n -> m <= 0 -> n * m <= 0.
Theorem mul_nonpos_nonneg : forall n m, n <= 0 -> 0 <= m -> n * m <= 0.
Notation mul_pos := lt_0_mul (only parsing).
Theorem lt_mul_0 :
forall n m, n * m < 0 <-> n < 0 /\ m > 0 \/ n > 0 /\ m < 0.