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Progress towards UFDs (#88)

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Patrick Stevens
2019-12-08 11:18:39 +00:00
committed by GitHub
parent 99c38495ce
commit 33098e94b0
20 changed files with 462 additions and 253 deletions
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15
Rings/Ideals/Maximal/Lemmas.agda
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@@ -26,7 +26,9 @@ open import Rings.Ideals.Prime.Lemmas
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.Ideals.Maximal.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) (proper : A) (isProper : pred proper → False) where
module Rings.Ideals.Maximal.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
open import Rings.Divisible.Definition R
open Ring R
open Group additiveGroup
@@ -56,7 +58,7 @@ Field.allInvertible (idealMaximalImpliesQuotientField max) cosetA cosetA!=0 = an
ans'' : pred (inverse (Ring.1R (cosetRing R i)) + (ans' * cosetA))
ans'' with ans {1R}
ans'' | a , ((b , predCAb-a) ,, pred1-a) = Ideal.isSubset i (transitive (+WellDefined (invContravariant additiveGroup) reflexive) (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)) *Commutative))) (Ideal.closedUnderPlus i (Ideal.closedUnderInverse i pred1-a) predCAb-a)
Field.nontrivial (idealMaximalImpliesQuotientField max) 1=0 = isProper (Ideal.isSubset i (identIsIdent {proper}) (Ideal.accumulatesTimes i p))
Field.nontrivial (idealMaximalImpliesQuotientField max) 1=0 = MaximalIdeal.notContainedIsNotContained max (Ideal.isSubset i (identIsIdent {MaximalIdeal.notContained (max {lzero})}) (Ideal.accumulatesTimes i p))
where
have : pred (inverse 1R)
have = Ideal.isSubset i identRight 1=0
@@ -64,8 +66,8 @@ Field.nontrivial (idealMaximalImpliesQuotientField max) 1=0 = isProper (Ideal.is
p = Ideal.isSubset i (invTwice additiveGroup 1R) (Ideal.closedUnderInverse i have)
quotientFieldImpliesIdealMaximal : Field (cosetRing R i) → ({d : _} → MaximalIdeal i {d})
MaximalIdeal.notContained (quotientFieldImpliesIdealMaximal f) = proper
MaximalIdeal.notContainedIsNotContained (quotientFieldImpliesIdealMaximal f) = isProper
MaximalIdeal.notContained (quotientFieldImpliesIdealMaximal f) = Ring.1R (cosetRing R i)
MaximalIdeal.notContainedIsNotContained (quotientFieldImpliesIdealMaximal f) p1R = Field.nontrivial f (memberDividesImpliesMember R i p1R ((inverse 1R + 0R) , identIsIdent))
MaximalIdeal.isMaximal (quotientFieldImpliesIdealMaximal f) {bigger} biggerIdeal contained (a , (biggerA ,, notPredA)) = Ideal.isSubset biggerIdeal identIsIdent (Ideal.accumulatesTimes biggerIdeal v)
where
inv : Sg A (λ t → pred (inverse 1R + (t * a)))
@@ -89,3 +91,8 @@ idealMaximalImpliesIdealPrime max = quotientIntDomImpliesIdealPrime i f'
f = idealMaximalImpliesQuotientField max
f' : IntegralDomain (cosetRing R i)
f' = fieldIsIntDom f (λ p → Field.nontrivial f (Equivalence.symmetric (Setoid.eq (cosetSetoid additiveGroup (Ideal.isSubgroup i))) p))
maximalIdealWellDefined : {d : _} {pred2 : A → Set d} (i2 : Ideal R pred2) → ({x : A} → pred x → pred2 x) → ({x : A} → pred2 x → pred x) → {e : _} → MaximalIdeal i {e} → MaximalIdeal i2 {e}
MaximalIdeal.notContained (maximalIdealWellDefined i2 pToP2 p2ToP record { notContained = notContained ; notContainedIsNotContained = notContainedIsNotContained ; isMaximal = isMaximal }) = notContained
MaximalIdeal.notContainedIsNotContained (maximalIdealWellDefined i2 pToP2 p2ToP record { notContained = notContained ; notContainedIsNotContained = notContainedIsNotContained ; isMaximal = isMaximal }) p2Not = notContainedIsNotContained (p2ToP p2Not)
MaximalIdeal.isMaximal (maximalIdealWellDefined i2 pToP2 p2ToP record { notContained = notContained ; notContainedIsNotContained = notContainedIsNotContained ; isMaximal = isMaximal }) {biggerPred} bigger pred2InBigger (r , (biggerPredR ,, notP2r)) {x} = isMaximal bigger (λ p → pred2InBigger (pToP2 p)) (r , (biggerPredR ,, λ p → notP2r (pToP2 p)))