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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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Rings/Irreducibles/Definition.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Homomorphisms.Definition
open import Groups.Definition
open import Numbers.Naturals.Naturals
open import Setoids.Orders
open import Setoids.Setoids
open import Functions
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Rings.Homomorphisms.Definition
open import Groups.Homomorphisms.Lemmas
open import Rings.IntegralDomains.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.Irreducibles.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} (intDom : IntegralDomain R) where
open Setoid S
open Ring R
open import Rings.Units.Definition R
record Irreducible (r : A) : Set (a ⊔ b) where
field
nonzero : (r ∼ 0R) → False
nonunit : (Unit r) → False
irreducible : (x y : A) → (x * y) ∼ r → (Unit x → False) → Unit y