------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties of Heyting Commutative Rings
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
open import Algebra.Apartness.Bundles using (HeytingCommutativeRing)
module Algebra.Apartness.Properties.HeytingCommutativeRing
{c ℓ₁ ℓ₂} (HCR : HeytingCommutativeRing c ℓ₁ ℓ₂) where
open import Function.Base using (_∘_)
open import Data.Product.Base using (_,_; proj₁; proj₂)
open import Algebra using (CommutativeRing; RightIdentity; Invertible; LeftInvertible; RightInvertible)
open HeytingCommutativeRing HCR
open CommutativeRing commutativeRing using (ring; *-commutativeMonoid)
open import Algebra.Properties.Ring ring
using (-0#≈0#; -‿distribˡ-*; -‿distribʳ-*; -‿anti-homo-+; -‿involutive)
open import Relation.Binary.Definitions using (Symmetric)
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
open import Algebra.Properties.CommutativeMonoid *-commutativeMonoid
private variable
x y z : Carrier
invertibleˡ⇒# : LeftInvertible _≈_ 1# _*_ (x - y) → x # y
invertibleˡ⇒# = invertible⇒# ∘ invertibleˡ⇒invertible
invertibleʳ⇒# : RightInvertible _≈_ 1# _*_ (x - y) → x # y
invertibleʳ⇒# = invertible⇒# ∘ invertibleʳ⇒invertible
x-0≈x : RightIdentity _≈_ 0# _-_
x-0≈x x = trans (+-congˡ -0#≈0#) (+-identityʳ x)
1#0 : 1# # 0#
1#0 = invertibleˡ⇒# (1# , 1*[x-0]≈x)
where
1*[x-0]≈x : 1# * (x - 0#) ≈ x
1*[x-0]≈x {x} = trans (*-identityˡ (x - 0#)) (x-0≈x x)
x#0y#0→xy#0 : x # 0# → y # 0# → x * y # 0#
x#0y#0→xy#0 {x} {y} x#0 y#0 = helper (#⇒invertible x#0) (#⇒invertible y#0)
where
helper : Invertible _≈_ 1# _*_ (x - 0#) → Invertible _≈_ 1# _*_ (y - 0#) → x * y # 0#
helper (x⁻¹ , x⁻¹*x≈1 , x*x⁻¹≈1) (y⁻¹ , y⁻¹*y≈1 , y*y⁻¹≈1)
= invertibleˡ⇒# (y⁻¹ * x⁻¹ , y⁻¹*x⁻¹*x*y≈1)
where
open ≈-Reasoning setoid
y⁻¹*x⁻¹*x*y≈1 : y⁻¹ * x⁻¹ * (x * y - 0#) ≈ 1#
y⁻¹*x⁻¹*x*y≈1 = begin
y⁻¹ * x⁻¹ * (x * y - 0#) ≈⟨ *-congˡ (x-0≈x (x * y)) ⟩
y⁻¹ * x⁻¹ * (x * y) ≈⟨ *-assoc y⁻¹ x⁻¹ (x * y) ⟩
y⁻¹ * (x⁻¹ * (x * y)) ≈⟨ *-congˡ (*-assoc x⁻¹ x y) ⟨
y⁻¹ * ((x⁻¹ * x) * y) ≈⟨ *-congˡ (*-congʳ (*-congˡ (x-0≈x x))) ⟨
y⁻¹ * ((x⁻¹ * (x - 0#)) * y) ≈⟨ *-congˡ (*-congʳ x⁻¹*x≈1) ⟩
y⁻¹ * (1# * y) ≈⟨ *-congˡ (*-identityˡ y) ⟩
y⁻¹ * y ≈⟨ *-congˡ (x-0≈x y) ⟨
y⁻¹ * (y - 0#) ≈⟨ y⁻¹*y≈1 ⟩
1# ∎
#-congʳ : x ≈ y → x # z → y # z
#-congʳ {x} {y} {z} x≈y x#z = helper (#⇒invertible x#z)
where
helper : Invertible _≈_ 1# _*_ (x - z) → y # z
helper (x-z⁻¹ , x-z⁻¹*x-z≈1# , x-z*x-z⁻¹≈1#)
= invertibleˡ⇒# (x-z⁻¹ , x-z⁻¹*y-z≈1)
where
open ≈-Reasoning setoid
x-z⁻¹*y-z≈1 : x-z⁻¹ * (y - z) ≈ 1#
x-z⁻¹*y-z≈1 = begin
x-z⁻¹ * (y - z) ≈⟨ *-congˡ (+-congʳ x≈y) ⟨
x-z⁻¹ * (x - z) ≈⟨ x-z⁻¹*x-z≈1# ⟩
1# ∎
#-congˡ : y ≈ z → x # y → x # z
#-congˡ y≈z x#y = #-sym (#-congʳ y≈z (#-sym x#y))
RetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4