{-# OPTIONS --safe --warning=error --without-K #-}

open import LogicalFormulae
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Multiplication
open import Numbers.Integers.Definition
open import Numbers.Integers.Addition
open import Semirings.Definition
open import Groups.Definition
open import Rings.Definition
open import Setoids.Setoids

module Numbers.Integers.Multiplication where

infix 25 _*Z_
_*Z_ : ℤ → ℤ → ℤ
nonneg x *Z nonneg y = nonneg (x *N y)
nonneg zero *Z negSucc y = nonneg zero
nonneg (succ x) *Z negSucc y = negSucc ((succ x) *N y +N x)
negSucc x *Z nonneg zero = nonneg zero
negSucc x *Z nonneg (succ y) = negSucc ((succ y) *N x +N y)
negSucc x *Z negSucc y = nonneg ((succ x) *N (succ y))

*ZInherits : (a b : ℕ) → nonneg (a *N b) ≡ (nonneg a) *Z (nonneg b)
*ZInherits a b = refl