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https://github.com/Smaug123/agdaproofs
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24 lines
832 B
Agda
24 lines
832 B
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Numbers.Naturals.Semiring
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open import Numbers.Naturals.Multiplication
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open import Numbers.Integers.RingStructure.Ring
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open import Semirings.Definition
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open import Groups.Definition
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open import Rings.Definition
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open import Setoids.Setoids
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open import Rings.IntegralDomains
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module Numbers.Integers.RingStructure.IntegralDomain where
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intDom : (a b : ℤ) → a *Z b ≡ nonneg 0 → (a ≡ nonneg 0) || (b ≡ nonneg 0)
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intDom (nonneg zero) (nonneg b) pr = inl refl
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intDom (nonneg (succ a)) (nonneg zero) pr = inr refl
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intDom (nonneg zero) (negSucc b) pr = inl refl
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intDom (negSucc a) (nonneg zero) pr = inr refl
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ℤIntDom : IntegralDomain ℤRing
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IntegralDomain.intDom ℤIntDom {a} {b} = intDom a b
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IntegralDomain.nontrivial ℤIntDom ()
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