[auto ekmett@gmail.com**20090330035856] { addfile ./doc/html/monoids/Data-Ring-ModularArithmetic.html addfile ./doc/html/monoids/src/Data-Ring-ModularArithmetic.html hunk ./doc/html/monoids/Data-Group-Combinators.html 22 ->monoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce framework(Modular s a, Integral a) => Group (Mod a s)monoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce framework(Modular s a, Integral a) => Multiplicative (Mod a s)monoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce framework + +Data.Ring.ModularArithmetic
monoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkSource codeContentsIndex
Data.Ring.ModularArithmetic
Portabilitynon-portable (MPTCs)
Stabilityexperimental
Maintainerlibraries@haskell.org
Description
Documentation
module Data.Ring
data Mod a s Source
show/hide Instances
Eq a => Eq (Mod a s)
(Modular s a, Integral a) => Num (Mod a s)
Show a => Show (Mod a s)
(Modular s a, Integral a) => Monoid (Mod a s)
(Modular s a, Integral a) => Multiplicative (Mod a s)
(Modular s a, Integral a) => LeftSemiNearRing (Mod a s)
(Modular s a, Integral a) => RightSemiNearRing (Mod a s)
(Modular s a, Integral a) => SemiRing (Mod a s)
(Modular s a, Integral a) => Group (Mod a s)
(Modular s a, Integral a) => Ring (Mod a s)
class Modular s a | s -> a whereSource
Methods
modulus :: s -> aSource
show/hide Instances
(ReflectNum s, Num a) => Modular (ModulusNum s a) a
modulus :: Modular s a => s -> aSource
withIntegralModulus :: Integral a => a -> (forall s. Modular s a => w `Mod` s) -> wSource
Produced by Haddock version 2.3.0
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Data.Monoid.Additive.Sugar, Data.Monoid.Multiplicative, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Group, Data.Group.Combinators, Data.Group.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.ModularArithmetic, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Ring.Module.AutomaticDifferentiationData.Monoid.Reducer, Data.Monoid.Reducer.Char, Data.Monoid.Lexical.UTF8.Decoder, Data.Monoid.Reducer.With, Data.Monoid.Generator, Data.Monoid.Lexical.SourcePosition, Data.Monoid.Lexical.Words, Data.Monoid.Self, Data.Monoid.Generator.LZ78, Data.Monoid.FromString, Data.Monoid.Categorical, Data.Monoid.Additive, Data.Monoid.Additive.Sugar, Data.Monoid.Multiplicative, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Group, Data.Group.Combinators, Data.Group.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Monoid.Combinators, Data.Monoid.Generator.Free, Data.Monoid.Generator.RLE, Data.Ring.Module.AutomaticDifferentiationData.Monoid.Reducer, Data.Monoid.Reducer.Char, Data.Monoid.Lexical.UTF8.Decoder, Data.Monoid.Reducer.With, Data.Monoid.Generator, Data.Monoid.Lexical.SourcePosition, Data.Monoid.Lexical.Words, Data.Monoid.Self, Data.Monoid.Generator.LZ78, Data.Monoid.FromString, Data.Monoid.Categorical, Data.Monoid.Additive, Data.Monoid.Additive.Sugar, Data.Monoid.Multiplicative, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Group, Data.Group.Combinators, Data.Group.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.ModularArithmetic, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Monoid.Combinators, Data.Monoid.Generator.Free, Data.Monoid.Generator.RLE, Data.Ring.Module.AutomaticDifferentiationData.Monoid.Reducer, Data.Monoid.Reducer.Char, Data.Monoid.Lexical.UTF8.Decoder, Data.Monoid.Reducer.With, Data.Monoid.Generator, Data.Monoid.Lexical.SourcePosition, Data.Monoid.Lexical.Words, Data.Monoid.Self, Data.Monoid.Generator.LZ78, Data.Monoid.FromString, Data.Monoid.Categorical, Data.Monoid.Additive, Data.Monoid.Additive.Sugar, Data.Monoid.Multiplicative, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Group, Data.Group.Combinators, Data.Group.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Monoid.Combinators, Data.Monoid.Generator.Free, Data.Monoid.Generator.RLE, Data.Ring.Module.AutomaticDifferentiationData.Monoid.Reducer, Data.Monoid.Reducer.Char, Data.Monoid.Lexical.UTF8.Decoder, Data.Monoid.Reducer.With, Data.Monoid.Generator, Data.Monoid.Lexical.SourcePosition, Data.Monoid.Lexical.Words, Data.Monoid.Self, Data.Monoid.Generator.LZ78, Data.Monoid.FromString, Data.Monoid.Categorical, Data.Monoid.Additive, Data.Monoid.Additive.Sugar, Data.Monoid.Multiplicative, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Group, Data.Group.Combinators, Data.Group.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.ModularArithmetic, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Monoid.Combinators, Data.Monoid.Generator.Free, Data.Monoid.Generator.RLE, Data.Ring.Module.AutomaticDifferentiationreflectData.ReflectionReflectedData.ReflectionReflectedNumData.ReflectionReflectedNumsData.ReflectionReflectedStorableData.ReflectionreflectNumData.ReflectionreflectStorableData.ReflectionreifyData.ReflectionreifyIntegralData.ReflectionreifyIntegralsData.ReflectionreifyStorableData.Reflection, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Ring.Module.AutomaticDifferentiation, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.ModularArithmetic, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Ring.Module.AutomaticDifferentiation, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.Module, Data.Monoid.Applicative, Data.Ring.Module.AutomaticDifferentiation, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.ModularArithmetic, Data.Ring.Module, Data.Monoid.Applicative, Data.Ring.Module.AutomaticDifferentiation, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.Module, Data.Monoid.Applicative, Data.Ring.Module.AutomaticDifferentiation, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.ModularArithmetic, Data.Ring.Module, Data.Monoid.Applicative, Data.Ring.Module.AutomaticDifferentiationData.Monoid.Reducer, Data.Monoid.Reducer.Char, Data.Monoid.Lexical.UTF8.Decoder, Data.Monoid.Reducer.With, Data.Monoid.Generator, Data.Monoid.Lexical.SourcePosition, Data.Monoid.Lexical.Words, Data.Monoid.Self, Data.Monoid.Generator.LZ78, Data.Monoid.FromString, Data.Monoid.Categorical, Data.Monoid.Additive, Data.Monoid.Additive.Sugar, Data.Monoid.Multiplicative, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Group, Data.Group.Combinators, Data.Group.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Monoid.Combinators, Data.Monoid.Generator.Free, Data.Monoid.Generator.RLE, Data.Ring.Module.AutomaticDifferentiationData.Monoid.Reducer, Data.Monoid.Reducer.Char, Data.Monoid.Lexical.UTF8.Decoder, Data.Monoid.Reducer.With, Data.Monoid.Generator, Data.Monoid.Lexical.SourcePosition, Data.Monoid.Lexical.Words, Data.Monoid.Self, Data.Monoid.Generator.LZ78, Data.Monoid.FromString, Data.Monoid.Categorical, Data.Monoid.Additive, Data.Monoid.Additive.Sugar, Data.Monoid.Multiplicative, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Group, Data.Group.Combinators, Data.Group.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.ModularArithmetic, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Monoid.Combinators, Data.Monoid.Generator.Free, Data.Monoid.Generator.RLE, Data.Ring.Module.AutomaticDifferentiationData.Monoid.Reducer, Data.Monoid.Reducer.Char, Data.Monoid.Lexical.UTF8.Decoder, Data.Monoid.Reducer.With, Data.Monoid.Generator, Data.Monoid.Lexical.SourcePosition, Data.Monoid.Lexical.Words, Data.Monoid.Self, Data.Monoid.Generator.LZ78, Data.Monoid.FromString, Data.Monoid.Categorical, Data.Monoid.Additive, Data.Monoid.Additive.Sugar, Data.Monoid.Multiplicative, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Group, Data.Group.Combinators, Data.Group.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Monoid.Combinators, Data.Monoid.Generator.Free, Data.Monoid.Generator.RLE, Data.Ring.Module.AutomaticDifferentiationData.Monoid.Reducer, Data.Monoid.Reducer.Char, Data.Monoid.Lexical.UTF8.Decoder, Data.Monoid.Reducer.With, Data.Monoid.Generator, Data.Monoid.Lexical.SourcePosition, Data.Monoid.Lexical.Words, Data.Monoid.Self, Data.Monoid.Generator.LZ78, Data.Monoid.FromString, Data.Monoid.Categorical, Data.Monoid.Additive, Data.Monoid.Additive.Sugar, Data.Monoid.Multiplicative, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Group, Data.Group.Combinators, Data.Group.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.ModularArithmetic, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Monoid.Combinators, Data.Monoid.Generator.Free, Data.Monoid.Generator.RLE, Data.Ring.Module.AutomaticDifferentiation, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Ring.Module.AutomaticDifferentiation, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.ModularArithmetic, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Ring.Module.AutomaticDifferentiationwithIntegralModulusData.Ring.ModularArithmetic, Data.Monoid.Additive.Sugar, Data.Monoid.Multiplicative, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Group, Data.Group.Combinators, Data.Group.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Ring.Module.AutomaticDifferentiation, Data.Monoid.Additive.Sugar, Data.Monoid.Multiplicative, Data.Monoid.Multiplicative.Sugar, Data.Ring.Semi.Near, Data.Ring.Semi, Data.Ring.Semi.Ord, Data.Ring.Semi.Tropical, Data.Ring.Sugar, Data.Group, Data.Group.Combinators, Data.Group.Sugar, Data.Ring, Data.Ring.Boolean, Data.Ring.FromNum, Data.Ring.ModularArithmetic, Data.Ring.Module, Data.Monoid.Applicative, Data.Monoid.Monad, Data.Ring.Module.AutomaticDifferentiationmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkmonoids-0.1.15: Lots of Monoidsmonoids-0.1.17: Monoids, specialized containers and a general map/reduce frameworkLots of Monoids +>Monoids, specialized containers and a general map/reduce framework hunk ./doc/html/monoids/index.html 404 ->Data.ReflectionData.Ring.ModularArithmetic + + + +Data/Ring/ModularArithmetic.hs + + + +
{-# LANGUAGE RankNTypes, FlexibleInstances, MultiParamTypeClasses, ScopedTypeVariables, EmptyDataDecls, FunctionalDependencies, TypeOperators #-}
+-----------------------------------------------------------------------------
+-- |
+-- Module      :  Data.Ring.ModularArithmetic
+-- Copyright   :  Edward Kmett 2009, Oleg Kiselyov and Chung-chieh Shan 2004
+--                  
+-- License     :  BSD-style
+-- Maintainer  :  libraries@haskell.org
+-- Stability   :  experimental
+-- Portability :  non-portable (MPTCs)
+--
+-----------------------------------------------------------------------------
+
+module Data.Ring.ModularArithmetic
+    ( module Data.Ring
+    , Mod(getMod), Modular, modulus
+    , withIntegralModulus
+    ) where
+
+import Data.Ring
+import Data.Reflection
+
+newtype (a `Mod` s) = M { getMod :: a } 
+    deriving (Eq,Show)
+
+class Modular s a | s -> a where
+    modulus :: s -> a
+
+normalize :: (Modular s a, Integral a) => a -> (a `Mod` s)
+normalize = normalize' undefined where
+    normalize' :: (Modular s a, Integral a) => s -> a -> (a `Mod` s)
+    normalize' s a = M (a `mod` modulus s)
+
+data ModulusNum s a
+
+instance (ReflectNum s, Num a) => Modular (ModulusNum s a) a where
+    modulus _ = reflectNum (undefined :: s)
+
+withIntegralModulus :: Integral a => a -> (forall s. Modular s a => w `Mod` s) -> w
+withIntegralModulus = withIntegralModulus' undefined where
+    withIntegralModulus' :: Integral a => w -> a -> (forall s. Modular s a => w `Mod` s) -> w
+    withIntegralModulus' (_ :: w) (i :: a) k = 
+        reifyIntegral i (\(_ :: t) -> 
+        getMod (k :: w `Mod` ModulusNum t a))
+
+instance (Modular s a, Integral a) => Num (a `Mod` s) where
+    M a + M b = normalize (a + b)
+    M a - M b = normalize (a - b)
+    M a * M b = normalize (a * b)
+    negate (M a)  = normalize (negate a)
+    fromInteger i = normalize (fromInteger i)
+    signum = error "broken numerical type tower"
+    abs    = error "broken numerical type tower"
+
+instance (Modular s a, Integral a) => Monoid (a `Mod` s) where
+    mempty = 0
+    mappend = (+)
+
+instance (Modular s a, Integral a) => Multiplicative (a `Mod` s) where
+    one = 1
+    times = (*)
+
+instance (Modular s a, Integral a) => Group (a `Mod` s) where
+    gnegate = negate
+    minus = (-)
+    gsubtract = subtract
+
+instance (Modular s a, Integral a) => LeftSemiNearRing (a `Mod` s)
+instance (Modular s a, Integral a) => RightSemiNearRing (a `Mod` s)
+instance (Modular s a, Integral a) => SemiRing (a `Mod` s)
+instance (Modular s a, Integral a) => Ring (a `Mod` s)
+
+ }