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

open import LogicalFormulae
open import Numbers.Naturals.Semiring
open import Numbers.Integers.RingStructure.Ring
open import Rings.IntegralDomains.Definition

module Numbers.Integers.RingStructure.IntegralDomain where

intDom : (a b : ℤ) → a *Z b ≡ nonneg 0 → (a ≡ nonneg 0 → False) → (b ≡ nonneg 0)
intDom (nonneg zero) (nonneg b) x a!=0 = exFalso (a!=0 x)
intDom (nonneg (succ a)) (nonneg zero) a!=0 _ = refl
intDom (nonneg zero) (negSucc b) pr a!=0 = exFalso (a!=0 pr)
intDom (negSucc a) (nonneg zero) _ _ = refl

ℤIntDom : IntegralDomain ℤRing
IntegralDomain.intDom ℤIntDom {a} {b} = intDom a b
IntegralDomain.nontrivial ℤIntDom ()