-----------------------------------------------------------------------
-- 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 4 _∣_
_∣_ : 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
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