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https://github.com/Smaug123/agdaproofs
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51 lines
2.1 KiB
Agda
51 lines
2.1 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Groups.Groups
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open import Groups.Lemmas
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open import Groups.Definition
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open import Setoids.Setoids
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open import Rings.Definition
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open import Sets.EquivalenceRelations
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open import Rings.Ideals.Definition
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open import Rings.IntegralDomains.Definition
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open import Rings.Ideals.Prime.Definition
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open import Rings.Cosets
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Rings.Ideals.Prime.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} {c : _} {pred : A → Set c} (i : Ideal R pred) where
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open Ring R
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open Group additiveGroup
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open Setoid S
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open Equivalence eq
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open import Rings.Ideals.Lemmas R
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idealPrimeImpliesQuotientIntDom : PrimeIdeal i → IntegralDomain (cosetRing R i)
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IntegralDomain.intDom (idealPrimeImpliesQuotientIntDom isPrime) {a} {b} ab=0 a!=0 = ans
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where
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ab=0' : pred (a * b)
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ab=0' = translate' i ab=0
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a!=0' : (pred a) → False
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a!=0' prA = a!=0 (translate i prA)
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ans' : pred b
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ans' = PrimeIdeal.isPrime isPrime ab=0' a!=0'
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ans : pred (inverse (Ring.0R (cosetRing R i)) + b)
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ans = translate i ans'
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IntegralDomain.nontrivial (idealPrimeImpliesQuotientIntDom isPrime) 1=0 = PrimeIdeal.notContainedIsNotContained isPrime u
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where
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t : pred (Ring.1R (cosetRing R i))
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t = translate' i 1=0
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u : pred (PrimeIdeal.notContained isPrime)
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u = Ideal.isSubset i identIsIdent (Ideal.accumulatesTimes i {y = PrimeIdeal.notContained isPrime} t)
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quotientIntDomImpliesIdealPrime : IntegralDomain (cosetRing R i) → PrimeIdeal i
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quotientIntDomImpliesIdealPrime intDom = record { isPrime = isPrime ; notContained = Ring.1R R ; notContainedIsNotContained = notCon }
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where
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abstract
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notCon : pred 1R → False
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notCon 1=0 = IntegralDomain.nontrivial intDom (translate i 1=0)
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isPrime : {a b : A} → pred (a * b) → (pred a → False) → pred b
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isPrime {a} {b} predAB !predA = translate' i (IntegralDomain.intDom intDom (translate i predAB) λ t → !predA (translate' i t))
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