{-# 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 Agda.Primitive using (Level; lzero; lsuc; _⊔_)

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

open import Setoids.Subset S
open Ring R
open Group additiveGroup
open import Groups.Subgroups.Definition additiveGroup
open Setoid S
open Equivalence eq

record Subring {c : _} (pred : A → Set c) : Set (a ⊔ b ⊔ c) where
  field
    isSubgroup : Subgroup pred
    containsOne : pred 1R
    closedUnderProduct : {x y : A} → pred x → pred y → pred (x *A y)
  isSubset = Subgroup.isSubset isSubgroup

subringMult : {c : _} {pred : A → Set c} → (s : Subring pred) → Sg A pred → Sg A pred → Sg A pred
subringMult s (a , prA) (b , prB) = (a *A b) , Subring.closedUnderProduct s prA prB

subringIsRing : {c : _} {pred : A → Set c} → (subring : Subring pred) → Ring (subsetSetoid (Subring.isSubset subring)) (subgroupOp (Subring.isSubgroup subring)) (subringMult subring)
Ring.additiveGroup (subringIsRing sub) = subgroupIsGroup (Subring.isSubgroup sub)
Ring.*WellDefined (subringIsRing sub) {r , prR} {s , prS} {t , prT} {u , prU} r=t s=u = *WellDefined r=t s=u
Ring.1R (subringIsRing sub) = (1R , Subring.containsOne sub)
Ring.groupIsAbelian (subringIsRing sub) {a , prA} {b , prB} = groupIsAbelian
Ring.*Associative (subringIsRing sub) {a , prA} {b , prB} {c , prC} = *Associative
Ring.*Commutative (subringIsRing sub) {a , prA} {b , prB} = *Commutative
Ring.*DistributesOver+ (subringIsRing sub) {a , prA} {b , prB} {c , prC} = *DistributesOver+
Ring.identIsIdent (subringIsRing sub) {a , prA} = identIsIdent