mirror of
https://github.com/Smaug123/agdaproofs
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40 lines
2.8 KiB
Agda
40 lines
2.8 KiB
Agda
{-# OPTIONS --warning=error --safe --without-K #-}
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open import Functions
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open import Sets.FinSet
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open import LogicalFormulae
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open import Groups.Definition
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open import Groups.Groups
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open import Groups.FiniteGroups.Definition
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open import Rings.Homomorphisms.Definition
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open import Groups.Homomorphisms.Definition
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open import Groups.Abelian.Definition
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open import Setoids.Setoids
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open import Rings.Definition
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open import Fields.FieldOfFractions.Setoid
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open import Sets.EquivalenceRelations
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open import Groups.Lemmas
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open import Groups.QuotientGroup.Definition
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module Rings.Quotients.Definition {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ _*A_ : A → A → A} {_+B_ _*B_ : B → B → B} (R : Ring S _+A_ _*A_) (R2 : Ring T _+B_ _*B_) {underf : A → B} (f : RingHom R R2 underf) where
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open import Groups.QuotientGroup.Lemmas (Ring.additiveGroup R) (Ring.additiveGroup R2) (RingHom.groupHom f)
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quotientByRingHom : Ring (quotientGroupSetoid (Ring.additiveGroup R) (RingHom.groupHom f)) _+A_ _*A_
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Ring.additiveGroup quotientByRingHom = quotientGroupByHom (Ring.additiveGroup R) (RingHom.groupHom f)
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Ring.*WellDefined quotientByRingHom fr=ft fs=fu = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (transitive (RingHom.ringHom f) (transitive (Ring.*WellDefined R2 (quotientGroupLemma' (Ring.additiveGroup R) (RingHom.groupHom f) fr=ft) (quotientGroupLemma' (Ring.additiveGroup R) (RingHom.groupHom f) fs=fu)) (symmetric (RingHom.ringHom f))))
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where
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open Setoid T
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open Equivalence eq
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Ring.1R quotientByRingHom = Ring.1R R
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Ring.groupIsAbelian quotientByRingHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.groupIsAbelian R))
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Ring.*Associative quotientByRingHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.*Associative R))
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Ring.*Commutative quotientByRingHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.*Commutative R))
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Ring.*DistributesOver+ quotientByRingHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.*DistributesOver+ R))
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Ring.identIsIdent quotientByRingHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (GroupHom.wellDefined (RingHom.groupHom f) (Ring.identIsIdent R))
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projectionMapIsHom : RingHom R quotientByRingHom id
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RingHom.preserves1 projectionMapIsHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (Equivalence.reflexive (Setoid.eq T))
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RingHom.ringHom projectionMapIsHom = quotientGroupLemma (Ring.additiveGroup R) (RingHom.groupHom f) (Equivalence.reflexive (Setoid.eq T))
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RingHom.groupHom projectionMapIsHom = projectionMapIsGroupHom
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