mirror of
https://github.com/Smaug123/agdaproofs
synced 2025-10-17 09:28:40 +00:00
92 lines
5.1 KiB
Agda
92 lines
5.1 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.Cosets
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open import Groups.Homomorphisms.Definition
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open import Rings.Homomorphisms.Definition
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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 Setoids.Subset
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open import Setoids.Functions.Definition
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open import Setoids.Functions.Lemmas
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open import Rings.Definition
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open import Rings.Lemmas
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open import Sets.EquivalenceRelations
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open import Rings.Ideals.Definition
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open import Fields.Fields
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open import Fields.Lemmas
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open import Rings.Cosets
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open import Rings.Ideals.Maximal.Definition
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open import Rings.Ideals.Lemmas
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open import Rings.Ideals.Prime.Definition
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open import Rings.IntegralDomains.Definition
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open import Rings.Ideals.Prime.Lemmas
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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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
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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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idealMaximalImpliesQuotientField : ({d : Level} → MaximalIdeal i {d}) → Field (cosetRing R i)
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Field.allInvertible (idealMaximalImpliesQuotientField max) cosetA cosetA!=0 = ans' , ans''
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where
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gen : Ideal (cosetRing R i) (generatedIdealPred (cosetRing R i) cosetA)
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gen = generatedIdeal (cosetRing R i) cosetA
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inv : Ideal R (inverseImagePred {S = S} {T = cosetSetoid additiveGroup (Ideal.isSubgroup i)} (GroupHom.wellDefined (RingHom.groupHom (cosetRingHom R i))) (Ideal.isSubset gen))
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inv = inverseImageIsIdeal (cosetRing R i) (cosetRingHom R i) gen
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containsOnce : {a : A} → (Ideal.predicate i a) → (Ideal.predicate inv a)
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containsOnce {x} ix = x , ((x , Ideal.closedUnderPlus i (Ideal.closedUnderInverse i ix) (Ideal.isSubset i *Commutative (Ideal.accumulatesTimes i ix))) ,, Ideal.isSubset i (symmetric invLeft) (Ideal.containsIdentity i))
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notInI : A
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notInI = cosetA
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notInIIsInInv : Ideal.predicate inv notInI
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notInIIsInInv = cosetA , ((1R , Ideal.isSubset i {0R} (symmetric (transitive (+WellDefined reflexive (transitive *Commutative identIsIdent)) (invLeft {cosetA}))) (Ideal.containsIdentity i)) ,, Ideal.isSubset i (symmetric invLeft) (Ideal.containsIdentity i))
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notInIPr : (Ideal.predicate i notInI) → False
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notInIPr iInI = cosetA!=0 (Ideal.isSubset i (transitive (symmetric identLeft) (+WellDefined (symmetric (invIdent additiveGroup)) reflexive)) iInI)
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ans : {a : A} → Ideal.predicate inv a
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ans = MaximalIdeal.isMaximal max inv containsOnce (notInI , (notInIIsInInv ,, notInIPr))
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ans' : A
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ans' with ans {1R}
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... | _ , ((b , _) ,, _) = b
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ans'' : pred (inverse (Ring.1R (cosetRing R i)) + (ans' * cosetA))
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ans'' with ans {1R}
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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)
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Field.nontrivial (idealMaximalImpliesQuotientField max) 1=0 = isProper (Ideal.isSubset i (identIsIdent {proper}) (Ideal.accumulatesTimes i p))
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where
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have : pred (inverse 1R)
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have = Ideal.isSubset i identRight 1=0
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p : pred 1R
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p = Ideal.isSubset i (invTwice additiveGroup 1R) (Ideal.closedUnderInverse i have)
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quotientFieldImpliesIdealMaximal : Field (cosetRing R i) → ({d : _} → MaximalIdeal i {d})
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MaximalIdeal.notContained (quotientFieldImpliesIdealMaximal f) = proper
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MaximalIdeal.notContainedIsNotContained (quotientFieldImpliesIdealMaximal f) = isProper
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MaximalIdeal.isMaximal (quotientFieldImpliesIdealMaximal f) {bigger} biggerIdeal contained (a , (biggerA ,, notPredA)) = Ideal.isSubset biggerIdeal identIsIdent (Ideal.accumulatesTimes biggerIdeal v)
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where
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inv : Sg A (λ t → pred (inverse 1R + (t * a)))
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inv = Field.allInvertible f a λ r → notPredA (translate' R i r)
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r : A
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r = underlying inv
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s : pred (inverse 1R + (r * a))
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s with inv
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... | _ , p = p
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t : bigger (inverse 1R + (r * a))
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t = contained s
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u : bigger (inverse (r * a))
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u = Ideal.closedUnderInverse biggerIdeal (Ideal.isSubset biggerIdeal *Commutative (Ideal.accumulatesTimes biggerIdeal biggerA))
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v : bigger 1R
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v = Ideal.isSubset biggerIdeal (invTwice additiveGroup 1R) (Ideal.closedUnderInverse biggerIdeal (Ideal.isSubset biggerIdeal (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) (Ideal.closedUnderPlus biggerIdeal t u)))
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idealMaximalImpliesIdealPrime : ({d : _} → MaximalIdeal i {d}) → PrimeIdeal i
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idealMaximalImpliesIdealPrime max = quotientIntDomImpliesIdealPrime i f'
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where
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f : Field (cosetRing R i)
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f = idealMaximalImpliesQuotientField max
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f' : IntegralDomain (cosetRing R i)
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f' = fieldIsIntDom f (λ p → Field.nontrivial f (Equivalence.symmetric (Setoid.eq (cosetSetoid additiveGroup (Ideal.isSubgroup i))) p))
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