mirror of
https://github.com/Smaug123/agdaproofs
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42 lines
1.7 KiB
Agda
42 lines
1.7 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.Definition
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open import Groups.Lemmas
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open import Rings.Definition
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open import Rings.Lemmas
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open import Rings.IntegralDomains
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open import Fields.Fields
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open import Functions
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open import Setoids.Setoids
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open import Sets.EquivalenceRelations
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Fields.FieldOfFractions.Field {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) where
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open import Fields.FieldOfFractions.Setoid I
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open import Fields.FieldOfFractions.Addition I
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open import Fields.FieldOfFractions.Multiplication I
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open import Fields.FieldOfFractions.Ring I
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fieldOfFractions : Field fieldOfFractionsRing
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Field.allInvertible fieldOfFractions (fst ,, (b , _)) prA = (b ,, (fst , ans)) , need
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where
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open Setoid S
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open Equivalence eq
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need : ((b * fst) * Ring.1R R) ∼ ((fst * b) * Ring.1R R)
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need = Ring.*WellDefined R (Ring.*Commutative R) reflexive
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ans : fst ∼ Ring.0R R → False
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ans pr = prA need'
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where
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need' : (fst * Ring.1R R) ∼ (b * Ring.0R R)
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need' = transitive (Ring.*WellDefined R pr reflexive) (transitive (transitive (Ring.*Commutative R) (Ring.timesZero R)) (symmetric (Ring.timesZero R)))
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Field.nontrivial fieldOfFractions pr = IntegralDomain.nontrivial I (symmetric (transitive (symmetric (Ring.timesZero R)) (transitive (Ring.*Commutative R) (transitive pr (Ring.identIsIdent R)))))
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where
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open Setoid S
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open Equivalence eq
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pr' : (Ring.0R R) * (Ring.1R R) ∼ (Ring.1R R) * (Ring.1R R)
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pr' = pr
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