Modular arithmetic based on Data.Nat.Bounded #2257

CHANGELOG.md

@@ -48,6 +48,13 @@ New modules
 
 * `Algebra.Module.Bundles.Raw`: raw bundles for module-like algebraic structures
 
+* Integer arithmetic modulo `n`, based on `Data.Nat.Bounded.*`:
+ ```agda
+ Data.Integer.Modulo.Base
+ Data.Integer.Modulo.Literals
+ Data.Integer.Modulo.Properties
+ ```
+
 Additions to existing modules
 -----------------------------
 

src/Data/Integer/Modulo/Base.agda

@@ -0,0 +1,118 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Integers mod n, type and basic operations
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Data.Nat.Base as ℕ
+ using (ℕ; zero; suc; NonZero; NonTrivial; nonTrivial⇒nonZero; _<_; _∸_)
+
+module Data.Integer.Modulo.Base n .{{_ : NonTrivial n}} where
+
+open import Algebra.Bundles.Raw
+ using (RawMagma; RawMonoid; RawNearSemiring; RawSemiring; RawRing)
+open import Data.Integer.Base as ℤ using (ℤ; _◂_; signAbs)
+open import Data.Nat.Bounded.Base as ℕ< hiding (fromℕ; _∼_; ≡-Modℕ)
+import Data.Nat.Properties as ℕ
+open import Data.Sign.Base as Sign using (Sign)
+open import Data.Unit.Base using (⊤)
+open import Function.Base using (id; _∘_; _$_; _on_)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+
+private
+ variable
+ m o : ℕ
+ i j : ℕ< n
+
+ instance
+ _ = nonTrivial⇒nonZero n
+
+ m∸n<m : ∀ m n → .{{NonZero m}} → .{{NonZero n}} → m ∸ n < m
+ m∸n<m m@(suc _) n@(suc o) = ℕ.s≤s (ℕ.m∸n≤m _ o)
+
+------------------------------------------------------------------------
+-- Definition
+
+-- Infixes
+infix 8 -_
+infixl 7 _*_
+infixl 6 _+_
+
+-- Type definition
+ℤmod : Set
+ℤmod = ℕ< n
+
+-- Re-export
+fromℕ : ℕ → ℤmod
+fromℕ = ℕ<.fromℕ
+
+-- Negation
+-_ : ℤmod → ℤmod
+- ⟦ m@zero ⟧< _ = ⟦0⟧<
+- ⟦ m@(suc _) ⟧< _ = ⟦ n ∸ m ⟧< m∸n<m _ _
+
+-- Addition
+_+_ : ℤmod → ℤmod → ℤmod
+i + j = fromℕ (⟦ i ⟧ ℕ.+ ⟦ j ⟧)
+
+-- Multiplication
+_*_ : ℤmod → ℤmod → ℤmod
+i * j = fromℕ (⟦ i ⟧ ℕ.* ⟦ j ⟧)
+
+------------------------------------------------------------------------
+-- Quotient map from ℤ
+
+_◃_ : Sign → ℤmod → ℤmod
+Sign.+ ◃ j = j
+Sign.- ◃ j = - j
+
+fromℤ : ℤ → ℤmod
+fromℤ i with s ◂ ∣i∣ ← signAbs i = s ◃ fromℕ ∣i∣
+
+-- the _mod_ syntax
+
+Mod : ℤ → ℤmod
+Mod = fromℤ
+
+syntax Mod {n = n} i = i mod n
+
+-- Quotient equivalence relation on ℤ induced by `fromℤ`
+
+_∼_ : Rel ℤ _
+_∼_ = _≡_ on fromℤ
+
+≡-Mod : Rel ℤ _
+≡-Mod i j = i ∼ j
+
+syntax ≡-Mod n i j = i ≡ j mod n
+
+------------------------------------------------------------------------
+-- Raw bundles
+
++-*-rawRing : RawRing _ _
++-*-rawRing = record
+ { _≈_ = _≡_
+ ; _+_ = _+_
+ ; _*_ = _*_
+ ; -_ = -_
+ ; 0# = ⟦0⟧<
+ ; 1# = ⟦1⟧<
+ }
+
+open RawRing +-*-rawRing public
+ using (+-rawMagma; *-rawMagma)
+ renaming ( +-rawMonoid to +-0-rawMonoid
+ ; *-rawMonoid to *-1-rawMonoid
+ --; rawNearSemiring to +-*-rawNearSemiring
+ ; rawSemiring to +-*-rawSemiring
+ )
+
+------------------------------------------------------------------------
+-- -- Postfix notation for when opening the module unparameterised
+
+0ℤmod = ⟦0⟧<
+1ℤmod = ⟦1⟧<
+

src/Data/Integer/Modulo/Literals.agda

@@ -0,0 +1,27 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Literals for integers mod n
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Data.Nat.Base using (ℕ; NonTrivial)
+
+module Data.Integer.Modulo.Literals n .{{_ : NonTrivial n}} where
+
+open import Agda.Builtin.FromNat using (Number)
+open import Data.Unit.Base using (⊤)
+open import Data.Integer.Modulo.Base using (ℤmod; fromℕ)
+
+------------------------------------------------------------------------
+-- Definitions
+
+Constraint : ℕ → Set
+Constraint _ = ⊤
+
+fromNat : ∀ m → {{Constraint m}} → ℤmod n
+fromNat m = fromℕ n m
+
+number : Number (ℤmod n)
+number = record { Constraint = Constraint ; fromNat = fromNat }

src/Data/Integer/Modulo/Properties.agda

@@ -0,0 +1,178 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Integers mod n, properties
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Data.Nat.Base as ℕ
+ using (ℕ; zero; suc; NonZero; NonTrivial; _<_; _∸_; _%_)
+
+module Data.Integer.Modulo.Properties n .{{_ : NonTrivial n}} where
+
+open import Algebra.Bundles.Raw
+ using (RawMagma; RawMonoid; RawNearSemiring; RawSemiring; RawRing)
+open import Algebra.Bundles
+ using (Magma; Monoid; NearSemiring; Semiring; Ring)
+import Algebra.Definitions as Definitions
+import Algebra.Structures as Structures
+open import Data.Integer.Base as ℤ using (ℤ; _◂_; signAbs)
+open import Data.Nat.Bounded.Base as ℕ< hiding (fromℕ; _∼_)
+import Data.Nat.Bounded.Properties as ℕ<
+import Data.Nat.DivMod as ℕ
+import Data.Nat.Properties as ℕ
+open import Data.Product.Base as Product using (_,_)
+open import Data.Sign.Base as Sign using (Sign)
+open import Function.Base using (_$_)
+open import Relation.Binary.PropositionalEquality
+ using (_≡_; refl; sym; trans; cong; cong₂; isEquivalence; module ≡-Reasoning)
+
+open import Data.Integer.Modulo.Base n as Modulo
+ using (ℤmod; fromℕ; fromℤ; _∼_; ≡-Mod; +-*-rawRing)
+
+open Definitions (_≡_ {A = ℤmod})
+open Structures (_≡_ {A = ℤmod})
+ using ( IsMagma; IsSemigroup; IsMonoid
+ ; IsGroup; IsAbelianGroup
+ ; IsNearSemiring; IsSemiring
+ ; IsRing)
+
+private
+ variable
+ m o : ℕ
+ i j k : ℤmod
+
+ instance
+ _ = ℕ.nonTrivial⇒nonZero n
+
+open RawRing +-*-rawRing
+open ≡-Reasoning
+
++-cong₂ : Congruent₂ _+_
++-cong₂ = cong₂ _+_
+
++-isMagma : IsMagma _+_
++-isMagma = record { isEquivalence = isEquivalence ; ∙-cong = +-cong₂ }
+
++-assoc : Associative _+_
++-assoc i j k = begin
+ i + j + k
+ ≡⟨⟩
+ fromℕ (((⟦ i ⟧ ℕ.+ ⟦ j ⟧) % n) ℕ.+ ⟦ k ⟧)
+ ≡⟨ ℕ<.≡-mod⇒fromℕ≡fromℕ ≡-mod ⟩
+ fromℕ (⟦ i ⟧ ℕ.+ ((⟦ j ⟧ ℕ.+ ⟦ k ⟧) % n))
+ ≡⟨⟩
+ i + (j + k) ∎
+ where
+ ≡-mod : (((⟦ i ⟧ ℕ.+ ⟦ j ⟧) % n) ℕ.+ ⟦ k ⟧)
+ ≡
+ (⟦ i ⟧ ℕ.+ ((⟦ j ⟧ ℕ.+ ⟦ k ⟧) % n))
+ modℕ n
+ ≡-mod = ℕ<.≡-mod-trans (ℕ<.+-distribˡ-% (⟦ i ⟧ ℕ.+ ⟦ j ⟧) ⟦ k ⟧)
+ (ℕ<.≡-mod-trans (ℕ<.≡-mod-reflexive (ℕ.+-assoc ⟦ i ⟧ ⟦ j ⟧ ⟦ k ⟧))
+ (ℕ<.≡-mod-sym (ℕ<.+-distribʳ-% ⟦ i ⟧ (⟦ j ⟧ ℕ.+ ⟦ k ⟧))))
+
++-isSemigroup : IsSemigroup _+_
++-isSemigroup = record { isMagma = +-isMagma ; assoc = +-assoc }
+
++-identityˡ : LeftIdentity 0# _+_
++-identityˡ = ℕ<.fromℕ∘toℕ≐id
+
++-identityʳ : RightIdentity 0# _+_
++-identityʳ i = trans (cong fromℕ (ℕ.+-identityʳ _)) (ℕ<.fromℕ∘toℕ≐id i)
+
++-identity : Identity 0# _+_
++-identity = +-identityˡ , +-identityʳ
+
++-isMonoid : IsMonoid _+_ 0#
++-isMonoid = record { isSemigroup = +-isSemigroup ; identity = +-identity }
+
++-inverseˡ : LeftInverse 0# -_ _+_
++-inverseˡ (⟦ zero ⟧< _) = +-identityˡ 0#
++-inverseˡ i@(⟦ suc _ ⟧< _) = ℕ<.+-inverseˡ (ℕ<.isBounded i)
+
++-inverseʳ : RightInverse 0# -_ _+_
++-inverseʳ (⟦ zero ⟧< _) = +-identityʳ 0#
++-inverseʳ i@(⟦ m@(suc _) ⟧< _) = ℕ<.+-inverseʳ (ℕ<.isBounded i)
+
++-inverse : Inverse 0# -_ _+_
++-inverse = +-inverseˡ , +-inverseʳ
+
++-0-isGroup : IsGroup _+_ 0# -_
++-0-isGroup = record { isMonoid = +-isMonoid ; inverse = +-inverse ; ⁻¹-cong = cong -_ }
+
++-comm : Commutative _+_
++-comm i j = cong fromℕ (ℕ.+-comm ⟦ i ⟧ ⟦ j ⟧)
+
++-0-isAbelianGroup : IsAbelianGroup _+_ 0# -_
++-0-isAbelianGroup = record { isGroup = +-0-isGroup ; comm = +-comm }
+
+*-cong₂ : Congruent₂ _*_
+*-cong₂ = cong₂ _*_
+
+*-assoc : Associative _*_
+*-assoc i j k = begin
+ i * j * k
+ ≡⟨⟩
+ fromℕ (((⟦ i ⟧ ℕ.* ⟦ j ⟧) % n) ℕ.* ⟦ k ⟧)
+ ≡⟨ ℕ<.≡-mod⇒fromℕ≡fromℕ ≡-mod ⟩
+ fromℕ (⟦ i ⟧ ℕ.* ((⟦ j ⟧ ℕ.* ⟦ k ⟧) % n))
+ ≡⟨⟩
+ i * (j * k) ∎
+ where
+ ≡-mod : (((⟦ i ⟧ ℕ.* ⟦ j ⟧) % n) ℕ.* ⟦ k ⟧)
+ ≡
+ (⟦ i ⟧ ℕ.* ((⟦ j ⟧ ℕ.* ⟦ k ⟧) % n))
+ modℕ n
+ ≡-mod = ℕ<.≡-mod-trans (ℕ<.*-distribˡ-% (⟦ i ⟧ ℕ.* ⟦ j ⟧) ⟦ k ⟧)
+ (ℕ<.≡-mod-trans (ℕ<.≡-mod-reflexive (ℕ.*-assoc ⟦ i ⟧ ⟦ j ⟧ ⟦ k ⟧))
+ (ℕ<.≡-mod-sym (ℕ<.*-distribʳ-% ⟦ i ⟧ (⟦ j ⟧ ℕ.* ⟦ k ⟧))))
+
+*-identityˡ : LeftIdentity 1# _*_
+*-identityˡ i with eq ← ℕ<.⟦1⟧≡1 {n = n} rewrite eq
+ = trans (cong fromℕ (ℕ.*-identityˡ _)) (ℕ<.fromℕ≐⟦⟧< (ℕ<.isBounded i))
+
+*-identityʳ : RightIdentity 1# _*_
+*-identityʳ i with eq ← ℕ<.⟦1⟧≡1 {n = n} rewrite eq
+ = trans (cong fromℕ (ℕ.*-identityʳ _)) (ℕ<.fromℕ≐⟦⟧< (ℕ<.isBounded i))
+
+*-identity : Identity 1# _*_
+*-identity = *-identityˡ , *-identityʳ
+
+*-distribˡ-+ : _*_ DistributesOverˡ _+_
+*-distribˡ-+ i j k = ℕ<.≡-mod⇒fromℕ≡fromℕ ≡-mod
+ where
+ ≡-mod : (⟦ i ⟧ ℕ.* ⟦ j + k ⟧)
+ ≡
+ (⟦ i * j ⟧ ℕ.+ ⟦ i * k ⟧)
+ modℕ n
+ ≡-mod = ℕ<.≡-mod-trans (ℕ<.*-distribʳ-% ⟦ i ⟧ (⟦ j ⟧ ℕ.+ ⟦ k ⟧))
+ (ℕ<.≡-mod-trans (ℕ<.≡-mod-reflexive (ℕ.*-distribˡ-+ ⟦ i ⟧ ⟦ j ⟧ ⟦ k ⟧))
+ (ℕ<.≡-mod-sym (ℕ<.+-distrib-% (⟦ i ⟧ ℕ.* ⟦ j ⟧) (⟦ i ⟧ ℕ.* ⟦ k ⟧))))
+
+*-distribʳ-+ : _*_ DistributesOverʳ _+_
+*-distribʳ-+ i j k = ℕ<.≡-mod⇒fromℕ≡fromℕ ≡-mod
+ where
+ ≡-mod : (⟦ j + k ⟧ ℕ.* ⟦ i ⟧)
+ ≡
+ (⟦ j * i ⟧ ℕ.+ ⟦ k * i ⟧)
+ modℕ n
+ ≡-mod = ℕ<.≡-mod-trans (ℕ<.*-distribˡ-% (⟦ j ⟧ ℕ.+ ⟦ k ⟧) ⟦ i ⟧)
+ (ℕ<.≡-mod-trans (ℕ<.≡-mod-reflexive (ℕ.*-distribʳ-+ ⟦ i ⟧ ⟦ j ⟧ ⟦ k ⟧))
+ (ℕ<.≡-mod-sym (ℕ<.+-distrib-% (⟦ j ⟧ ℕ.* ⟦ i ⟧) (⟦ k ⟧ ℕ.* ⟦ i ⟧))))
+
+*-distrib-+ : _*_ DistributesOver _+_
+*-distrib-+ = *-distribˡ-+ , *-distribʳ-+
+
++-*-isRing : IsRing _+_ _*_ -_ 0# 1#
++-*-isRing = record
+ { +-isAbelianGroup = +-0-isAbelianGroup
+ ; *-cong = *-cong₂
+ ; *-assoc = *-assoc
+ ; *-identity = *-identity
+ ; distrib = *-distrib-+
+ }
+
+ring : Ring _ _
+ring = record { isRing = +-*-isRing }