{-# 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.Associates.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} (intDom : IntegralDomain R) where

open import Rings.Divisible.Definition R
open import Rings.IntegralDomains.Lemmas intDom
open import Rings.Associates.Definition intDom
open import Rings.Units.Definition R
open import Rings.Ideals.Definition R
open Setoid S
open Ring R
open Equivalence eq

associatesEquiv : Equivalence Associates
Equivalence.reflexive associatesEquiv {x} = 1R , ((1R , identIsIdent) ,, symmetric (transitive *Commutative identIsIdent))
Equivalence.symmetric associatesEquiv {x} {y} (a , ((invA , prInv) ,, x=ya)) = invA , ((a , transitive *Commutative prInv) ,, transitive (symmetric identIsIdent) (transitive (*WellDefined (symmetric prInv) reflexive) (transitive *Commutative (transitive *Associative (*WellDefined (symmetric x=ya) reflexive)))))
Equivalence.transitive associatesEquiv {x} {y} {z} (a , ((invA , prInvA) ,, x=ya)) (b , ((invB , prInvB) ,, y=zb)) = (a * b) , (((invA * invB) , transitive *Associative (transitive (*WellDefined (transitive *Commutative *Associative) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined (transitive *Commutative prInvA) prInvB) identIsIdent)))) ,, transitive x=ya (transitive (*WellDefined y=zb reflexive) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))))

associateImpliesMutualDivision : {a b : A} → Associates a b → a ∣ b
associateImpliesMutualDivision {a} {b} (x , ((invX , prInvX) ,, a=bx)) = invX , transitive (transitive (*WellDefined a=bx reflexive) (transitive (transitive (symmetric *Associative) (*WellDefined reflexive prInvX)) *Commutative)) identIsIdent

mutualDivisionImpliesAssociate : {a b : A} → (a ∣ b) → (b ∣ a) → ((a ∼ 0R) → False) → Associates a b
mutualDivisionImpliesAssociate {a} {b} (r , ar=b) (s , bs=a) a!=0 = s , ((r , cancelIntDom {a = a} (transitive (transitive (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined ar=b reflexive))) bs=a) (transitive (symmetric identIsIdent) *Commutative)) a!=0) ,, symmetric bs=a)

mutualDivisionImpliesAssociate' : {a b : A} → (a ∣ b) → (b ∣ a) → (a ∼ 0R) → Associates a b
mutualDivisionImpliesAssociate' {a} {b} (r , ar=b) (s , bs=a) a=0 = 1R , ((1R , identIsIdent) ,, transitive a=0 (symmetric (transitive (*WellDefined b=0 reflexive) (transitive *Commutative timesZero))))
  where
    b=0 : b ∼ 0R
    b=0 = transitive (symmetric ar=b) (transitive (transitive *Commutative (*WellDefined reflexive a=0)) (timesZero {r}))

associateImpliesGeneratedIdealsEqual : {a b : A} → Associates a b → {x : A} → generatedIdealPred a x → generatedIdealPred b x
associateImpliesGeneratedIdealsEqual {a} {b} (r , ((s , rs=1) ,, a=br)) {x} (c , ac=x) = (r * c) , transitive *Associative (transitive (*WellDefined (symmetric a=br) reflexive) ac=x)

associateImpliesGeneratedIdealsEqual' : {a b : A} → Associates a b → {x : A} → generatedIdealPred b x → generatedIdealPred a x
associateImpliesGeneratedIdealsEqual' assoc = associateImpliesGeneratedIdealsEqual (Equivalence.symmetric associatesEquiv assoc)