mirror of
https://github.com/Smaug123/agdaproofs
synced 2025-10-16 17:08:39 +00:00
31 lines
1.6 KiB
Agda
31 lines
1.6 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
|
||
|
||
open import LogicalFormulae
|
||
open import Groups.Definition
|
||
open import Numbers.Naturals.Definition
|
||
open import Setoids.Setoids
|
||
open import Sets.EquivalenceRelations
|
||
open import Rings.Definition
|
||
open import Rings.EuclideanDomains.Definition
|
||
open import Fields.Fields
|
||
open import Fields.Lemmas
|
||
|
||
|
||
module Rings.EuclideanDomains.Examples where
|
||
|
||
polynomialField : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) → (Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False) → EuclideanDomain R
|
||
EuclideanDomain.isIntegralDomain (polynomialField F 1!=0) = fieldIsIntDom F 1!=0
|
||
EuclideanDomain.norm (polynomialField F _) a!=0 = zero
|
||
EuclideanDomain.normSize (polynomialField F _) a!=0 b!=0 c b=ac = inr refl
|
||
DivisionAlgorithmResult.quotient (EuclideanDomain.divisionAlg (polynomialField {_*_ = _*_} F _) {a = a} {b} a!=0 b!=0) with Field.allInvertible F b b!=0
|
||
... | bInv , prB = a * bInv
|
||
DivisionAlgorithmResult.rem (EuclideanDomain.divisionAlg (polynomialField F _) a!=0 b!=0) = Field.0F F
|
||
DivisionAlgorithmResult.remSmall (EuclideanDomain.divisionAlg (polynomialField {S = S} F _) a!=0 b!=0) = inl (Equivalence.reflexive (Setoid.eq S))
|
||
DivisionAlgorithmResult.divAlg (EuclideanDomain.divisionAlg (polynomialField {S = S} {R = R} F _) {a = a} {b = b} a!=0 b!=0) with Field.allInvertible F b b!=0
|
||
... | bInv , prB = transitive (transitive (transitive (symmetric identIsIdent) (transitive *Commutative (*WellDefined reflexive (symmetric prB)))) *Associative) (symmetric identRight)
|
||
where
|
||
open Setoid S
|
||
open Equivalence eq
|
||
open Ring R
|
||
open Group additiveGroup
|