-----------------------------------------------------------------------
-- The Agda standard library
--
-- Unsigned divisibility
------------------------------------------------------------------------
 
-- For signed divisibility see `Data.Integer.Divisibility.Signed`
 
{-# OPTIONS --cubical-compatible --safe #-}
 
module Data.Integer.Divisibility where
 
open import Function.Base using (_on_; _$_)
open import Data.Integer.Base using (ℤ; _*_; ∣_∣; NonZero)
open import Data.Integer.Properties using (abs-*; *-comm)
import Data.Nat.Base as ℕ using (ℕ; _*_; NonZero)
import Data.Nat.Divisibility as ℕ
  using (module ∣-Reasoning; _∣_; divides; *-monoʳ-∣; *-cancelˡ-∣)
open import Level using (0ℓ)
open import Relation.Binary.Core using (Rel; _Preserves_⟶_)
 
 
------------------------------------------------------------------------
-- Divisibility
 
infix  _∣_
 
_∣_ : Rel ℤ 0ℓ
_∣_ = ℕ._∣_ on ∣_∣
 
pattern divides k eq = ℕ.divides k eq
 
------------------------------------------------------------------------
-- Properties of divisibility
 
*-monoʳ-∣ : ∀ k → (k *_) Preserves _∣_ ⟶ _∣_
*-monoʳ-∣ k {i} {j} i∣j = begin
  ∣ k * i ∣       ≡⟨ abs-* k i ⟩
  ∣ k ∣ ℕ.* ∣ i ∣ ∣⟨ ℕ.*-monoʳ-∣ ∣ k ∣ i∣j ⟩
  ∣ k ∣ ℕ.* ∣ j ∣ ≡⟨ abs-* k j ⟨
  ∣ k * j ∣       ∎
  where open ℕ.∣-Reasoning
 
*-monoˡ-∣ : ∀ k → (_* k) Preserves _∣_ ⟶ _∣_
*-monoˡ-∣ k {i} {j} rewrite *-comm i k | *-comm j k = *-monoʳ-∣ k
 
*-cancelˡ-∣ : ∀ k {i j} .{{_ : NonZero k}} → k * i ∣ k * j → i ∣ j
*-cancelˡ-∣ k {i} {j} k*i∣k*j = ℕ.*-cancelˡ-∣ ∣ k ∣ $ begin
  ∣ k ∣ ℕ.* ∣ i ∣  ≡⟨ abs-* k i ⟨
  ∣ k * i ∣        ∣⟨ k*i∣k*j ⟩
  ∣ k * j ∣        ≡⟨ abs-* k j ⟩
  ∣ k ∣ ℕ.* ∣ j ∣  ∎
  where open ℕ.∣-Reasoning
 
*-cancelʳ-∣ : ∀ k {i j} .{{_ : NonZero k}} → i * k ∣ j * k → i ∣ j
*-cancelʳ-∣ k {i} {j} rewrite *-comm i k | *-comm j k = *-cancelˡ-∣ k