```------------------------------------------------------------------------
-- The Agda standard library
--
-- Greatest Common Divisor for integers
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Data.Integer.GCD where

open import Data.Integer.Base
open import Data.Integer.Divisibility
open import Data.Integer.Properties
open import Data.Nat.Base
import Data.Nat.GCD as ℕ
open import Data.Product
open import Relation.Binary.PropositionalEquality

open import Algebra.Definitions {A = ℤ} _≡_ as Algebra
using (Associative; Commutative; LeftIdentity; RightIdentity; LeftZero; RightZero; Zero)

------------------------------------------------------------------------
-- Definition
------------------------------------------------------------------------

gcd : ℤ → ℤ → ℤ
gcd i j = + ℕ.gcd ∣ i ∣ ∣ j ∣

------------------------------------------------------------------------
-- Properties
------------------------------------------------------------------------

gcd[i,j]∣i : ∀ i j → gcd i j ∣ i
gcd[i,j]∣i i j = ℕ.gcd[m,n]∣m ∣ i ∣ ∣ j ∣

gcd[i,j]∣j : ∀ i j → gcd i j ∣ j
gcd[i,j]∣j i j = ℕ.gcd[m,n]∣n ∣ i ∣ ∣ j ∣

gcd-greatest : ∀ {i j c} → c ∣ i → c ∣ j → c ∣ gcd i j
gcd-greatest c∣i c∣j = ℕ.gcd-greatest c∣i c∣j

gcd[0,0]≡0 : gcd 0ℤ 0ℤ ≡ 0ℤ
gcd[0,0]≡0 = cong (+_) ℕ.gcd[0,0]≡0

gcd[i,j]≡0⇒i≡0 : ∀ i j → gcd i j ≡ 0ℤ → i ≡ 0ℤ
gcd[i,j]≡0⇒i≡0 i j eq = ∣i∣≡0⇒i≡0 (ℕ.gcd[m,n]≡0⇒m≡0 (+-injective eq))

gcd[i,j]≡0⇒j≡0 : ∀ {i j} → gcd i j ≡ 0ℤ → j ≡ 0ℤ
gcd[i,j]≡0⇒j≡0 {i} eq = ∣i∣≡0⇒i≡0 (ℕ.gcd[m,n]≡0⇒n≡0 ∣ i ∣ (+-injective eq))

gcd-comm : Commutative gcd
gcd-comm i j = cong (+_) (ℕ.gcd-comm ∣ i ∣ ∣ j ∣)

gcd-assoc : Associative gcd
gcd-assoc i j k = cong (+_) (ℕ.gcd-assoc ∣ i ∣ ∣ j ∣ (∣ k ∣))

gcd-zeroˡ : LeftZero 1ℤ gcd
gcd-zeroˡ i = cong (+_) (ℕ.gcd-zeroˡ ∣ i ∣)

gcd-zeroʳ : RightZero 1ℤ gcd
gcd-zeroʳ i = cong (+_) (ℕ.gcd-zeroʳ ∣ i ∣)

gcd-zero : Zero 1ℤ gcd
gcd-zero = gcd-zeroˡ , gcd-zeroʳ
```