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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
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(*    //   *    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)         *)
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(*                      Evgeny Makarov, INRIA, 2007                     *)
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(** * Some properties of the multiplication for modules implementing [NZBasicFunsSig']  This file defines the [NZMulProp] functor type on top of [NZAddProp]. This functor type is meant to be [Include]d in a module implementing [NZBasicFunsSig'].  This gives the following basic lemmas: - [mul_0_r], [mul_1_l], [mul_1_r] - [mul_succ_r], [mul_comm] - [mul_add_distr_r], [mul_add_distr_l] - [mul_assoc] - [mul_shuffle0] and [mul_shuffle3] to rearrange products of 3 terms - [mul_shuffle1] and [mul_shuffle2] to rearrange products of 4 terms  Notice that [NZMulProp] itself [Include]s [NZAddProp]. *)
From Stdlib.Numbers.NatInt Require Import NZAxioms NZBase NZAdd.
Module Type NZMulProp (Import NZ : NZBasicFunsSig')(Import NZBase : NZBaseProp NZ).
Include NZAddProp NZ NZBase.

(* Specifications: *)
Theorem mul_0_r : forall n, n * 0 == 0.
Theorem mul_succ_r : forall n m, n * (S m) == n * m + n.
Global Hint Rewrite mul_0_r mul_succ_r : nz.
Theorem mul_comm : forall n m, n * m == m * n.
Theorem mul_add_distr_r : forall n m p, (n + m) * p == n * p + m * p.
Theorem mul_add_distr_l : forall n m p, n * (m + p) == n * m + n * p.
Theorem mul_assoc : forall n m p, n * (m * p) == (n * m) * p.
Theorem mul_1_l : forall n, 1 * n == n.
Theorem mul_1_r : forall n, n * 1 == n.
Global Hint Rewrite mul_1_l mul_1_r : nz.
Theorem mul_shuffle0 : forall n m p, n*m*p == n*p*m.
Theorem mul_shuffle1 : forall n m p q, (n * m) * (p * q) == (n * p) * (m * q).
Theorem mul_shuffle2 : forall n m p q, (n * m) * (p * q) == (n * q) * (m * p).
Theorem mul_shuffle3 : forall n m p, n * (m * p) == m * (n * p).
End NZMulProp.