{-# OPTIONS --cubical-compatible --safe #-}
module Data.Integer.Divisibility.Signed where
open import Function.Base using (_⟨_⟩_; _$_; _$′_; _∘_; _∘′_)
open import Data.Integer.Base using (ℤ; _*_; +0; sign; _◃_; ≢-nonZero;
∣_∣; 0ℤ; +_; _+_; _-_; -_; NonZero)
open import Data.Integer.Properties
import Data.Integer.Divisibility as Unsigned
import Data.Nat.Base as ℕ
import Data.Nat.Divisibility as ℕ
import Data.Nat.Coprimality as ℕ
import Data.Nat.Properties as ℕ
import Data.Sign.Base as Sign
import Data.Sign.Properties as Sign
open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_)
open import Relation.Binary.Bundles using (Preorder)
open import Relation.Binary.Structures using (IsPreorder)
open import Relation.Binary.Definitions
using (Reflexive; Transitive; Decidable)
open import Relation.Binary.PropositionalEquality.Core
using (_≡_; trans; sym; cong; refl)
open import Relation.Binary.PropositionalEquality.Properties
using (module ≡-Reasoning; isEquivalence)
import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
open import Relation.Nullary.Decidable as Dec using (yes; no)
open import Relation.Binary.Reasoning.Syntax
infix 4 _∣_
record _∣_ (k z : ℤ) : Set where
constructor divides
field quotient : ℤ
equality : z ≡ quotient * k
open _∣_ using (quotient) public
∣ᵤ⇒∣ : ∀ {k i} → k Unsigned.∣ i → k ∣ i
∣ᵤ⇒∣ {k} {i} (Unsigned.divides 0 eq) = divides +0 (∣i∣≡0⇒i≡0 eq)
∣ᵤ⇒∣ {k} {i} (Unsigned.divides q@(ℕ.suc _) eq) with k ≟ +0
... | yes refl = divides +0 (∣i∣≡0⇒i≡0 (trans eq (ℕ.*-zeroʳ q)))
... | no neq = divides s[i*k]◃q (◃-cong sign-eq abs-eq)
where
s[i*k] = sign i Sign.* sign k
s[i*k]◃q = s[i*k] ◃ q
instance
_ = ≢-nonZero neq
_ = ◃-nonZero s[i*k] q
_ = i*j≢0 s[i*k]◃q k
sign-eq : sign i ≡ sign (s[i*k]◃q * k)
sign-eq = sym $ begin
sign (s[i*k]◃q * k) ≡⟨ sign-* s[i*k]◃q k ⟩
sign s[i*k]◃q Sign.* sign k ≡⟨ cong (Sign._* _) (sign-◃ s[i*k] q) ⟩
s[i*k] Sign.* sign k ≡⟨ Sign.*-assoc (sign i) (sign k) (sign k) ⟩
sign i Sign.* (sign k Sign.* sign k) ≡⟨ cong (sign i Sign.*_) (Sign.s*s≡+ (sign k)) ⟩
sign i Sign.* Sign.+ ≡⟨ Sign.*-identityʳ (sign i) ⟩
sign i ∎
where open ≡-Reasoning
abs-eq : ∣ i ∣ ≡ ∣ s[i*k]◃q * k ∣
abs-eq = sym $ begin
∣ s[i*k]◃q * k ∣ ≡⟨ abs-* s[i*k]◃q k ⟩
∣ s[i*k]◃q ∣ ℕ.* ∣ k ∣ ≡⟨ cong (ℕ._* ∣ k ∣) (abs-◃ s[i*k] q) ⟩
q ℕ.* ∣ k ∣ ≡⟨ eq ⟨
∣ i ∣ ∎
where open ≡-Reasoning
∣⇒∣ᵤ : ∀ {k i} → k ∣ i → k Unsigned.∣ i
∣⇒∣ᵤ {k} {i} (divides q eq) = Unsigned.divides ∣ q ∣ $′ begin
∣ i ∣ ≡⟨ cong ∣_∣ eq ⟩
∣ q * k ∣ ≡⟨ abs-* q k ⟩
∣ q ∣ ℕ.* ∣ k ∣ ∎
where open ≡-Reasoning
∣-refl : Reflexive _∣_
∣-refl = ∣ᵤ⇒∣ ℕ.∣-refl
∣-reflexive : _≡_ ⇒ _∣_
∣-reflexive refl = ∣-refl
∣-trans : Transitive _∣_
∣-trans i∣j j∣k = ∣ᵤ⇒∣ (ℕ.∣-trans (∣⇒∣ᵤ i∣j) (∣⇒∣ᵤ j∣k))
∣-isPreorder : IsPreorder _≡_ _∣_
∣-isPreorder = record
{ isEquivalence = isEquivalence
; reflexive = ∣-reflexive
; trans = ∣-trans
}
∣-preorder : Preorder _ _ _
∣-preorder = record { isPreorder = ∣-isPreorder }
module ∣-Reasoning where
private module Base = ≲-Reasoning ∣-preorder
open Base public
hiding (step-≲; step-∼; step-≈; step-≈˘)
renaming (≲-go to ∣-go)
open ∣-syntax _IsRelatedTo_ _IsRelatedTo_ ∣-go public
infix 4 _∣?_
_∣?_ : Decidable _∣_
k ∣? m = Dec.map′ ∣ᵤ⇒∣ ∣⇒∣ᵤ (∣ k ∣ ℕ.∣? ∣ m ∣)
0∣⇒≡0 : ∀ {m} → 0ℤ ∣ m → m ≡ 0ℤ
0∣⇒≡0 0|m = ∣i∣≡0⇒i≡0 (ℕ.0∣⇒≡0 (∣⇒∣ᵤ 0|m))
m∣∣m∣ : ∀ {m} → m ∣ (+ ∣ m ∣)
m∣∣m∣ = ∣ᵤ⇒∣ ℕ.∣-refl
∣m∣∣m : ∀ {m} → (+ ∣ m ∣) ∣ m
∣m∣∣m = ∣ᵤ⇒∣ ℕ.∣-refl
∣m∣n⇒∣m+n : ∀ {i m n} → i ∣ m → i ∣ n → i ∣ m + n
∣m∣n⇒∣m+n (divides q refl) (divides p refl) =
divides (q + p) (sym (*-distribʳ-+ _ q p))
∣m⇒∣-m : ∀ {i m} → i ∣ m → i ∣ - m
∣m⇒∣-m {i} {m} i∣m = ∣ᵤ⇒∣ $′ begin
∣ i ∣ ∣⟨ ∣⇒∣ᵤ i∣m ⟩
∣ m ∣ ≡⟨ ∣-i∣≡∣i∣ m ⟨
∣ - m ∣ ∎
where open ℕ.∣-Reasoning
∣m∣n⇒∣m-n : ∀ {i m n} → i ∣ m → i ∣ n → i ∣ m - n
∣m∣n⇒∣m-n i∣m i∣n = ∣m∣n⇒∣m+n i∣m (∣m⇒∣-m i∣n)
∣m+n∣m⇒∣n : ∀ {i m n} → i ∣ m + n → i ∣ m → i ∣ n
∣m+n∣m⇒∣n {i} {m} {n} i∣m+n i∣m = begin
i ∣⟨ ∣m∣n⇒∣m-n i∣m+n i∣m ⟩
m + n - m ≡⟨ +-comm (m + n) (- m) ⟩
- m + (m + n) ≡⟨ +-assoc (- m) m n ⟨
- m + m + n ≡⟨ cong (_+ n) (+-inverseˡ m) ⟩
+ 0 + n ≡⟨ +-identityˡ n ⟩
n ∎
where open ∣-Reasoning
∣m+n∣n⇒∣m : ∀ {i m n} → i ∣ m + n → i ∣ n → i ∣ m
∣m+n∣n⇒∣m {m = m} {n} i|m+n i|n rewrite +-comm m n = ∣m+n∣m⇒∣n i|m+n i|n
∣n⇒∣m*n : ∀ {i} m {n} → i ∣ n → i ∣ m * n
∣n⇒∣m*n {i} m {n} (divides q eq) = divides (m * q) $′ begin
m * n ≡⟨ cong (m *_) eq ⟩
m * (q * i) ≡⟨ *-assoc m q i ⟨
m * q * i ∎
where open ≡-Reasoning
∣m⇒∣m*n : ∀ {i m} n → i ∣ m → i ∣ m * n
∣m⇒∣m*n {m = m} n i|m rewrite *-comm m n = ∣n⇒∣m*n n i|m
*-monoʳ-∣ : ∀ k → (k *_) Preserves _∣_ ⟶ _∣_
*-monoʳ-∣ k = ∣ᵤ⇒∣ ∘ Unsigned.*-monoʳ-∣ k ∘ ∣⇒∣ᵤ
*-monoˡ-∣ : ∀ k → (_* k) Preserves _∣_ ⟶ _∣_
*-monoˡ-∣ k {i} {j} = ∣ᵤ⇒∣ ∘ Unsigned.*-monoˡ-∣ k {i} {j} ∘ ∣⇒∣ᵤ
*-cancelˡ-∣ : ∀ k {i j} .{{_ : NonZero k}} → k * i ∣ k * j → i ∣ j
*-cancelˡ-∣ k = ∣ᵤ⇒∣ ∘ Unsigned.*-cancelˡ-∣ k ∘ ∣⇒∣ᵤ
*-cancelʳ-∣ : ∀ k {i j} .{{_ : NonZero k}} → i * k ∣ j * k → i ∣ j
*-cancelʳ-∣ k {i} {j} = ∣ᵤ⇒∣ ∘′ Unsigned.*-cancelʳ-∣ k {i} {j} ∘′ ∣⇒∣ᵤ