patrick/agdaproofs
1
0
Fork 0
mirror of https://github.com/Smaug123/agdaproofs synced 2025-10-18 17:48:40 +00:00
Code Issues Packages Projects Releases Wiki Activity

Progress towards UFDs (#88)

Browse Source
This commit is contained in:
Patrick Stevens
2019-12-08 11:18:39 +00:00
committed by GitHub
parent 99c38495ce
commit 33098e94b0
20 changed files with 462 additions and 253 deletions
Download Patch File Download Diff File Expand all files Collapse all files

23
Rings/Ideals/Principal/Lemmas.agda Normal file
View File

@@ -0,0 +1,23 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Setoids.Orders
open import Setoids.Setoids
open import Functions
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.Ideals.Principal.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
open Setoid S
open Ring R
open Equivalence eq
open import Rings.Ideals.Principal.Definition R
open import Rings.Ideals.Definition R
open import Rings.Ideals.Lemmas R
open import Rings.Divisible.Definition R
generatorZeroImpliesAllZero : {c : _} {pred : A → Set c} → {i : Ideal pred} → (princ : PrincipalIdeal i) → PrincipalIdeal.generator princ ∼ 0R → {x : A} → pred x → x ∼ 0R
generatorZeroImpliesAllZero record { generator = gen ; genIsInIdeal = genIsInIdeal ; genGenerates = genGenerates } gen=0 {x} predX = generatorZeroImpliesMembersZero {x} (divisibleWellDefined gen=0 reflexive (genGenerates predX))