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The Coq Proof Assistant

# Library Coq.Numbers.NatInt.NZMul

# 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 Included 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 Includes NZAddProp.
From Coq.Numbers.NatInt Require Import NZAxioms NZBase NZAdd.
Module Type NZMulProp (Import NZ : NZBasicFunsSig')(Import NZBase : NZBaseProp NZ).
Include NZAddProp NZ NZBase.
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.