diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,30 @@ +Copyright (c) 2013, Tikhon Jelvis + +All rights reserved. + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are met: + + * Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + * Redistributions in binary form must reproduce the above + copyright notice, this list of conditions and the following + disclaimer in the documentation and/or other materials provided + with the distribution. + + * Neither the name of Tikhon Jelvis nor the names of other + contributors may be used to endorse or promote products derived + from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT +LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR +A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT +OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, +SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT +LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, +DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY +THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT +(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE +OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,2 @@ +import Distribution.Simple +main = defaultMain diff --git a/modular-arithmetic.cabal b/modular-arithmetic.cabal new file mode 100644 --- /dev/null +++ b/modular-arithmetic.cabal @@ -0,0 +1,28 @@ +-- Initial Mod.cabal generated by cabal init. For further documentation, +-- see http://haskell.org/cabal/users-guide/ + +name: modular-arithmetic +version: 1.0.0.0 +synopsis: A type for integers modulo some constant. + +description: This module provides a convenient type for working with + integers modulo some constant. It saves you from manually + wrapping numeric operations all over the place and + prevents a range of simple mistakes. + + It also provides some really cute syntax for these types + like @ℤ/7@ for integers modulo 7. + +license: BSD3 +license-file: LICENSE +author: Tikhon Jelvis +maintainer: tikhon@jelv.is +category: Math +build-type: Simple +cabal-version: >=1.8 + +library + hs-source-dirs: src + ghc-options: -Wall + exposed-modules: Data.Modular + build-depends: base ==4.6.* diff --git a/src/Data/Modular.hs b/src/Data/Modular.hs new file mode 100644 --- /dev/null +++ b/src/Data/Modular.hs @@ -0,0 +1,103 @@ +{-# LANGUAGE DataKinds #-} +{-# LANGUAGE KindSignatures #-} +{-# LANGUAGE ScopedTypeVariables #-} +{-# LANGUAGE TypeOperators #-} + +-- | +-- This module provides types for working with integers modulo some +-- constant. +-- +-- This module uses some new Haskell features introduced in 7.6. In +-- particular, it needs DataKinds and type literals +-- (GHC.TypeLits). The TypeOperators extension is needed for the nice +-- infix syntax. +-- +-- These types are created with the type constructor @Mod@ +-- (or its synonym @/@). To work with integers mod 7, you could write: +-- @ +-- Int `Mod` 7 +-- Integer `Mod` 7 +-- Integer/7 +-- ℤ/7 +-- @ +-- +-- (The last is a synonym for @Integer@ provided by this library. In +-- Emacs, you can use the Tex input mode to type it with \Bbb{Z}.) +-- +-- All the usual typeclasses are defined for these types. You can also +-- get the constant using @bound@ or extract the underlying value +-- using @unMod@. +-- +-- Here is a quick example: +-- @ +-- *Data.Modular> (10 :: ℤ/7) * (11 :: ℤ/7) +-- 5 +-- @ +-- +-- It also works correctly with negative numeric literals: +-- @ +-- *Data.Modular> (-10 :: ℤ/7) * (11 :: ℤ/7) +-- 2 +-- @ + +module Data.Modular (unMod, toMod, Mod, (/)(), ℤ) where + +import Control.Arrow (first) + +import Data.Ratio ((%)) + +import GHC.TypeLits + +-- | The actual type, wrapping an underlying @Integeral@ type @i@ in a +-- newtype annotated with the bound. +newtype i `Mod` (n :: Nat) = Mod i deriving (Eq, Ord) + +-- | Extract the underlying integral value from a modular type. +unMod :: i `Mod` n -> i +unMod (Mod i) = i + +-- | A synonym for @Mod@, inspired by the ℤ/n syntax from mathematics. +type (/) = Mod + +-- | A synonym for Integer, also inspired by the ℤ/n syntax. +type ℤ = Integer + +-- | Returns the bound of the modular type in the type itself. This +-- breaks the invariant of the type, so it shouldn't be used outside +-- this module. +_bound :: forall n i. (Integral i, SingI n) => i `Mod` n +_bound = Mod . fromInteger $ fromSing (sing :: Sing n) + +-- | Wraps the underlying type into the modular type, wrapping as +-- appropriate. +toMod :: forall n i. (Integral i, SingI n) => i -> i `Mod` n +toMod i = Mod $ i `mod` unMod (_bound :: i `Mod` n) + +instance Show i => Show (i `Mod` n) where show (Mod i) = show i +instance (Read i, Integral i, SingI n) => Read (i `Mod` n) + where readsPrec prec = map (first toMod) . readsPrec prec + +instance (Integral i, SingI n) => Num (i `Mod` n) where + fromInteger = toMod . fromInteger + + Mod i₁ + Mod i₂ = toMod $ i₁ + i₂ + Mod i₁ * Mod i₂ = toMod $ i₁ * i₂ + + abs (Mod i) = toMod $ abs i + signum (Mod i) = toMod $ signum i + negate (Mod i) = toMod $ negate i + +instance (Integral i, SingI n) => Enum (i `Mod` n) where + toEnum = fromInteger . toInteger + fromEnum = fromInteger . toInteger . unMod + +instance (Integral i, SingI n) => Bounded (i `Mod` n) where + maxBound = pred _bound + minBound = 0 + +instance (Integral i, SingI n) => Real (i `Mod` n) where + toRational (Mod i) = toInteger i % 1 + +instance (Integral i, SingI n) => Integral (i `Mod` n) where + toInteger (Mod i) = toInteger i + Mod i₁ `quotRem` Mod i₂ = let (q, r) = i₁ `quotRem` i₂ in (toMod q, toMod r)