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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
Require Import NZAxioms NZBase NZAdd.
Module Type NZMulProp (Import NZ : NZAxiomsSig')(Import NZBase : NZBaseProp NZ).
Include NZAddProp NZ NZBase.
Theorem mul_0_r : forall n, n * 0 == 0.
Proof.
nzinduct n; intros; now nzsimpl.
Qed.
Theorem mul_succ_r : forall n m, n * (S m) == n * m + n.
Proof.
intros n m; nzinduct n. now nzsimpl.
intro n. nzsimpl. rewrite succ_inj_wd, <- add_assoc, (add_comm m n), add_assoc.
now rewrite add_cancel_r.
Qed.
Hint Rewrite mul_0_r mul_succ_r : nz.
Theorem mul_comm : forall n m, n * m == m * n.
Proof.
intros n m; nzinduct n. now nzsimpl.
intro. nzsimpl. now rewrite add_cancel_r.
Qed.
Theorem mul_add_distr_r : forall n m p, (n + m) * p == n * p + m * p.
Proof.
intros n m p; nzinduct n. now nzsimpl.
intro n. nzsimpl. rewrite <- add_assoc, (add_comm p (m*p)), add_assoc.
now rewrite add_cancel_r.
Qed.
Theorem mul_add_distr_l : forall n m p, n * (m + p) == n * m + n * p.
Proof.
intros n m p.
rewrite (mul_comm n (m + p)), (mul_comm n m), (mul_comm n p).
apply mul_add_distr_r.
Qed.
Theorem mul_assoc : forall n m p, n * (m * p) == (n * m) * p.
Proof.
intros n m p; nzinduct n. now nzsimpl.
intro n. nzsimpl. rewrite mul_add_distr_r.
now rewrite add_cancel_r.
Qed.
Theorem mul_1_l : forall n, 1 * n == n.
Proof.
intro n. now nzsimpl'.
Qed.
Theorem mul_1_r : forall n, n * 1 == n.
Proof.
intro n. now nzsimpl'.
Qed.
Hint Rewrite mul_1_l mul_1_r : nz.
Theorem mul_shuffle0 : forall n m p, n*m*p == n*p*m.
Proof.
intros n m p. now rewrite <- 2 mul_assoc, (mul_comm m).
Qed.
Theorem mul_shuffle1 : forall n m p q, (n * m) * (p * q) == (n * p) * (m * q).
Proof.
intros n m p q. now rewrite 2 mul_assoc, (mul_shuffle0 n).
Qed.
Theorem mul_shuffle2 : forall n m p q, (n * m) * (p * q) == (n * q) * (m * p).
Proof.
intros n m p q. rewrite (mul_comm p). apply mul_shuffle1.
Qed.
Theorem mul_shuffle3 : forall n m p, n * (m * p) == m * (n * p).
Proof.
intros n m p. now rewrite mul_assoc, (mul_comm n), mul_assoc.
Qed.
End NZMulProp.
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