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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
Rings/EuclideanDomains/Definition.agda
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@@ -22,7 +22,7 @@ module Rings.EuclideanDomains.Definition {a b : _} {A : Set a} {S : Setoid {a} {
open Setoid S
open Ring R
record DivisionAlgorithmResult {norm : {a : A} → ((a ∼ 0R) → False) → ℕ} {x y : A} (x!=0 : (x ∼ 0R) → False) (y!=0 : (y ∼ 0R) → False) : Set (a ⊔ b) where
record DivisionAlgorithmResult (norm : {a : A} → ((a ∼ 0R) → False) → ℕ) {x y : A} (x!=0 : (x ∼ 0R) → False) (y!=0 : (y ∼ 0R) → False) : Set (a ⊔ b) where
field
quotient : A
rem : A
@@ -34,4 +34,4 @@ record EuclideanDomain : Set (a ⊔ lsuc b) where
isIntegralDomain : IntegralDomain R
norm : {a : A} → ((a ∼ 0R) → False) → ℕ
normSize : {a b : A} → (a!=0 : (a ∼ 0R) → False) → (b!=0 : (b ∼ 0R) → False) → (c : A) → b ∼ (a * c) → (norm a!=0) ≤N (norm b!=0)
divisionAlg : {a b : A} → (a!=0 : (a ∼ 0R) → False) → (b!=0 : (b ∼ 0R) → False) → DivisionAlgorithmResult {norm} a!=0 b!=0
divisionAlg : {a b : A} → (a!=0 : (a ∼ 0R) → False) → (b!=0 : (b ∼ 0R) → False) → DivisionAlgorithmResult norm a!=0 b!=0