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

open import Groups.Abelian.Definition
open import Groups.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Lists.Lists


module Groups.Polynomials.Group {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) where

open import Groups.Polynomials.Definition G
open import Groups.Polynomials.Addition G public
open Setoid S
open Equivalence eq

polyGroup : Group naivePolySetoid _+P_
Group.+WellDefined polyGroup = +PwellDefined
Group.0G polyGroup = []
Group.inverse polyGroup = inverse'
Group.+Associative polyGroup {a} {b} {c} = assoc {a} {b} {c}
Group.identRight polyGroup = PidentRight
Group.identLeft polyGroup = PidentLeft
Group.invLeft polyGroup {a} = invLeft' {a}
Group.invRight polyGroup {a} = invRight' {a}

abelian : AbelianGroup G → AbelianGroup polyGroup
AbelianGroup.commutative (abelian ab) {x} {y} = comm ab {x} {y}