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Z is a Euclidean domain (#86)

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Patrick Stevens
2019-12-07 13:00:18 +00:00
committed by GitHub
parent cfd9787bb8
commit e192f0e1f1
38 changed files with 1018 additions and 486 deletions
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16
Rings/Ideals/Prime/Lemmas.agda
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@@ -9,6 +9,7 @@ open import Rings.Definition
open import Sets.EquivalenceRelations
open import Rings.Ideals.Definition
open import Rings.IntegralDomains.Definition
open import Rings.IntegralDomains.Lemmas
open import Rings.Ideals.Prime.Definition
open import Rings.Cosets
@@ -48,3 +49,18 @@ quotientIntDomImpliesIdealPrime intDom = record { isPrime = isPrime ; notContain
notCon 1=0 = IntegralDomain.nontrivial intDom (translate i 1=0)
isPrime : {a b : A} → pred (a * b) → (pred a → False) → pred b
isPrime {a} {b} predAB !predA = translate' i (IntegralDomain.intDom intDom (translate i predAB) λ t → !predA (translate' i t))
private
dividesZero : {a : A} → generatedIdealPred R 0R a → a ∼ 0R
dividesZero (c , pr) = symmetric (transitive (symmetric (transitive *Commutative timesZero)) pr)
zeroIdealPrimeImpliesIntDom : PrimeIdeal (generatedIdeal R 0R) → IntegralDomain R
IntegralDomain.intDom (zeroIdealPrimeImpliesIntDom record { isPrime = isPrime ; notContained = notContained ; notContainedIsNotContained = notContainedIsNotContained }) {a} {b} ab=0 a!=0 with isPrime {a} {b} (1R , transitive (transitive *Commutative timesZero) (symmetric ab=0)) λ 0|a → a!=0 (dividesZero 0|a)
... | c , 0c=b = transitive (symmetric 0c=b) (transitive *Commutative timesZero)
IntegralDomain.nontrivial (zeroIdealPrimeImpliesIntDom record { isPrime = isPrime ; notContained = notContained ; notContainedIsNotContained = notContainedIsNotContained }) 1=0 = notContainedIsNotContained (notContained , transitive (*WellDefined (symmetric 1=0) reflexive) identIsIdent)
intDomImpliesZeroIdealPrime : IntegralDomain R → PrimeIdeal (generatedIdeal R 0R)
PrimeIdeal.isPrime (intDomImpliesZeroIdealPrime intDom) (c , 0=ab) 0not|a with IntegralDomain.intDom intDom (transitive (symmetric 0=ab) (transitive *Commutative timesZero)) λ a=0 → 0not|a (0R , transitive timesZero (symmetric a=0))
... | b=0 = 0R , transitive timesZero (symmetric b=0)
PrimeIdeal.notContained (intDomImpliesZeroIdealPrime intDom) = 1R
PrimeIdeal.notContainedIsNotContained (intDomImpliesZeroIdealPrime intDom) (c , 0c=1) = IntegralDomain.nontrivial intDom (symmetric (transitive (symmetric (transitive *Commutative timesZero)) 0c=1))