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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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4
Fields/FieldOfFractions/Setoid.agda
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@@ -11,12 +11,12 @@ open import Functions
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Rings.IntegralDomains.Definition
open import Rings.IntegralDomains.Lemmas
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Fields.FieldOfFractions.Setoid {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) where
fieldOfFractionsSet : Set (a ⊔ b)
fieldOfFractionsSet = (A && (Sg A (λ a → (Setoid._∼_ S a (Ring.0R R) → False))))
@@ -38,6 +38,6 @@ Equivalence.transitive (Setoid.eq fieldOfFractionsSetoid) {a ,, (b , b!=0)} {c ,
p3 : (a * f) * d ∼ (b * e) * d
p3 = transitive p2 (transitive (*WellDefined reflexive *Commutative) *Associative)
p4 : ((d ∼ 0R) → False) → ((a * f) ∼ (b * e))
p4 = cancelIntDom R I (transitive *Commutative (transitive p3 *Commutative))
p4 = cancelIntDom I (transitive *Commutative (transitive p3 *Commutative))
p5 : (a * f) ∼ (b * e)
p5 = p4 d!=0