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

open import LogicalFormulae
open import Groups.Groups
open import Groups.Abelian.Definition
open import Groups.Homomorphisms.Definition
open import Groups.Definition
open import Numbers.Naturals.Definition
open import Setoids.Orders
open import Setoids.Setoids
open import Functions
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Vectors
open import Lists.Lists
open import Maybe
open import Rings.Homomorphisms.Definition

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

module Rings.Polynomial.Ring {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where

open Ring R
open Setoid S
open Equivalence eq
open import Groups.Polynomials.Group additiveGroup
open import Groups.Polynomials.Definition additiveGroup
open import Rings.Polynomial.Multiplication R

polyRing : Ring naivePolySetoid _+P_ _*P_
Ring.additiveGroup polyRing = polyGroup
Ring.*WellDefined polyRing {a} {b} {c} {d} = *PwellDefined {a} {b} {c} {d}
Ring.1R polyRing = 1R :: []
Ring.groupIsAbelian polyRing {x} {y} = AbelianGroup.commutative (abelian (record { commutative = Ring.groupIsAbelian R })) {x} {y}
Ring.*Associative polyRing {a} {b} {c} = *Passoc {a} {b} {c}
Ring.*Commutative polyRing {a} {b} = p*Commutative {a} {b}
Ring.*DistributesOver+ polyRing {a} {b} {c} = *Pdistrib {a} {b} {c}
Ring.identIsIdent polyRing {a} = *Pident {a}

polyInjectionIsHom : RingHom R polyRing polyInjection
RingHom.preserves1 polyInjectionIsHom = reflexive ,, record {}
RingHom.ringHom polyInjectionIsHom = reflexive ,, (reflexive ,, record {})
GroupHom.groupHom (RingHom.groupHom polyInjectionIsHom) = reflexive ,, record {}
GroupHom.wellDefined (RingHom.groupHom polyInjectionIsHom) = SetoidInjection.wellDefined polyInjectionIsInj