{-# OPTIONS --safe --warning=error --without-K #-}

open import LogicalFormulae
open import Groups.Groups
open import Groups.Definition
open import Numbers.Naturals.Naturals
open import Orders
open import Setoids.Setoids
open import Functions
open import Rings.Definition
open import Rings.Lemmas

open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)

module Rings.IntegralDomains where
  record IntegralDomain {n m : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) : Set (lsuc m ⊔ n) where
    field
      intDom : {a b : A} → Setoid._∼_ S (a * b) (Ring.0R R) → (Setoid._∼_ S a (Ring.0R R)) || (Setoid._∼_ S b (Ring.0R R))
      nontrivial : Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False

  cancelIntDom : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) {a b c : A} → Setoid._∼_ S (a * b) (a * c) → (Setoid._∼_ S a (Ring.0R R)) || (Setoid._∼_ S b c)
  cancelIntDom {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I {a} {b} {c} ab=ac = t4
    where
      open Setoid S
      open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
      open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
      open Transitive (Equivalence.transitiveEq (Setoid.eq S))
      t1 : (a * b) + Group.inverse (Ring.additiveGroup R) (a * c) ∼ Ring.0R R
      t1 = transferToRight'' (Ring.additiveGroup R) ab=ac
      t2 : a * (b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R
      t2 = transitive (transitive (Ring.multDistributes R) (Group.wellDefined (Ring.additiveGroup R) reflexive (transferToRight' (Ring.additiveGroup R) (transitive (symmetric (Ring.multDistributes R)) (transitive (Ring.multWellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (ringTimesZero R)))))) t1
      t3 : (a ∼ Ring.0R R) || ((b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R)
      t3 = IntegralDomain.intDom I t2
      t4 : (a ∼ Ring.0R R) || (b ∼ c)
      t4 with t3
      ... | inl t = inl t
      ... | inr b = inr (transferToRight (Ring.additiveGroup R) b)