mirror of
https://github.com/Smaug123/agdaproofs
synced 2025-10-08 05:18:40 +00:00
24 lines
877 B
Agda
24 lines
877 B
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
|
||
|
||
open import LogicalFormulae
|
||
open import Numbers.Naturals.Semiring
|
||
open import Numbers.Naturals.Multiplication
|
||
open import Numbers.Integers.RingStructure.Ring
|
||
open import Semirings.Definition
|
||
open import Groups.Definition
|
||
open import Rings.Definition
|
||
open import Setoids.Setoids
|
||
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 ()
|