Algebra.Classes

Documentation

type Natural = Integer

newtype Sum a

Constructors

Sum

Fields

• fromSum :: a

Instances

Additive a => Monoid (Sum a)

Methods

mempty :: Sum a

mappend :: Sum a -> Sum a -> Sum a

mconcat :: [Sum a] -> Sum a

newtype Product a

Constructors

Product

Fields

• fromProduct :: a

Instances

Multiplicative a => Monoid (Product a)

Methods

mempty :: Product a

mappend :: Product a -> Product a -> Product a

mconcat :: [Product a] -> Product a

newtype Exponential a

Constructors

Exponential

Fields

• fromExponential :: a

Instances

Group a => Division (Exponential a)

Methods

recip :: Exponential a -> Exponential a

(/) :: Exponential a -> Exponential a -> Exponential a

Additive a => Multiplicative (Exponential a)

Methods

(*) :: Exponential a -> Exponential a -> Exponential a

one :: Exponential a

(^) :: Exponential a -> Natural -> Exponential a

class Additive a where

Additive monoid

Minimal complete definition

(+), zero

Methods

(+) :: a -> a -> a infixl 6

zero :: a

times :: Natural -> a -> a

Instances

Additive Double

Methods

(+) :: Double -> Double -> Double

zero :: Double

times :: Natural -> Double -> Double

Additive Float

Methods

(+) :: Float -> Float -> Float

zero :: Float

times :: Natural -> Float -> Float

Additive Int

Methods

(+) :: Int -> Int -> Int

zero :: Int

times :: Natural -> Int -> Int

Additive Integer

Methods

(+) :: Integer -> Integer -> Integer

zero :: Integer

times :: Natural -> Integer -> Integer

Additive Word8

Methods

(+) :: Word8 -> Word8 -> Word8

zero :: Word8

times :: Natural -> Word8 -> Word8

Additive Word16

Methods

(+) :: Word16 -> Word16 -> Word16

zero :: Word16

times :: Natural -> Word16 -> Word16

Additive Word32

Methods

(+) :: Word32 -> Word32 -> Word32

zero :: Word32

times :: Natural -> Word32 -> Word32

Additive CInt

Methods

(+) :: CInt -> CInt -> CInt

zero :: CInt

times :: Natural -> CInt -> CInt

Additive InitialAdditive

Methods

(+) :: InitialAdditive -> InitialAdditive -> InitialAdditive

zero :: InitialAdditive

times :: Natural -> InitialAdditive -> InitialAdditive

Integral a => Additive (Ratio a)

Methods

(+) :: Ratio a -> Ratio a -> Ratio a

zero :: Ratio a

times :: Natural -> Ratio a -> Ratio a

(Ord k, Additive v) => Additive (Map k v)

Methods

(+) :: Map k v -> Map k v -> Map k v

zero :: Map k v

times :: Natural -> Map k v -> Map k v

add :: (Foldable t, Additive a) => t a -> a

class Additive r => DecidableZero r where

Minimal complete definition

isZero

Methods

isZero :: r -> Bool

Instances

DecidableZero Double

Methods

isZero :: Double -> Bool

DecidableZero Float

Methods

isZero :: Float -> Bool

DecidableZero Int

Methods

isZero :: Int -> Bool

DecidableZero Integer

Methods

isZero :: Integer -> Bool

DecidableZero Word8

Methods

isZero :: Word8 -> Bool

DecidableZero Word16

Methods

isZero :: Word16 -> Bool

DecidableZero Word32

Methods

isZero :: Word32 -> Bool

DecidableZero CInt

Methods

isZero :: CInt -> Bool

(Ord k, DecidableZero v) => DecidableZero (Map k v)

Methods

isZero :: Map k v -> Bool

class Additive a => AbelianAdditive a

Instances

AbelianAdditive Double

AbelianAdditive Float

AbelianAdditive Int

AbelianAdditive Integer

AbelianAdditive CInt

Integral a => AbelianAdditive (Ratio a)

(Ord k, AbelianAdditive v) => AbelianAdditive (Map k v)

class Additive a => Group a where

Minimal complete definition

negate | (-)

Methods

(-) :: a -> a -> a infixl 6

negate :: a -> a

mult :: Integer -> a -> a

Instances

Group Double

Methods

(-) :: Double -> Double -> Double

negate :: Double -> Double

mult :: Integer -> Double -> Double

Group Float

Methods

(-) :: Float -> Float -> Float

negate :: Float -> Float

mult :: Integer -> Float -> Float

Group Int

Methods

(-) :: Int -> Int -> Int

negate :: Int -> Int

mult :: Integer -> Int -> Int

Group Integer

Methods

(-) :: Integer -> Integer -> Integer

negate :: Integer -> Integer

mult :: Integer -> Integer -> Integer

Group Word8

Methods

(-) :: Word8 -> Word8 -> Word8

negate :: Word8 -> Word8

mult :: Integer -> Word8 -> Word8

Group Word16

Methods

(-) :: Word16 -> Word16 -> Word16

negate :: Word16 -> Word16

mult :: Integer -> Word16 -> Word16

Group Word32

Methods

(-) :: Word32 -> Word32 -> Word32

negate :: Word32 -> Word32

mult :: Integer -> Word32 -> Word32

Group CInt

Methods

(-) :: CInt -> CInt -> CInt

negate :: CInt -> CInt

mult :: Integer -> CInt -> CInt

Integral a => Group (Ratio a)

Methods

(-) :: Ratio a -> Ratio a -> Ratio a

negate :: Ratio a -> Ratio a

mult :: Integer -> Ratio a -> Ratio a

(Ord k, Group v) => Group (Map k v)

Methods

(-) :: Map k v -> Map k v -> Map k v

negate :: Map k v -> Map k v

mult :: Integer -> Map k v -> Map k v

class (AbelianAdditive a, Ring scalar) => Module scalar a where

Module

Minimal complete definition

(*^)

Methods

(*^) :: scalar -> a -> a infixr 7

Instances

Module Double Double

Methods

(*^) :: Double -> Double -> Double

Module Float Float

Methods

(*^) :: Float -> Float -> Float

Module Int Int

Methods

(*^) :: Int -> Int -> Int

Module Integer Integer

Methods

(*^) :: Integer -> Integer -> Integer

Module Rational Double

Methods

(*^) :: Rational -> Double -> Double

Module CInt CInt

Methods

(*^) :: CInt -> CInt -> CInt

(Ord k, Ring v) => Module v (Map k v)

Methods

(*^) :: v -> Map k v -> Map k v

class Multiplicative a where

Multiplicative monoid

Minimal complete definition

(*), one

Methods

(*) :: a -> a -> a infixl 7

one :: a

(^) :: a -> Natural -> a

Instances

Multiplicative Double

Methods

(*) :: Double -> Double -> Double

one :: Double

(^) :: Double -> Natural -> Double

Multiplicative Float

Methods

(*) :: Float -> Float -> Float

one :: Float

(^) :: Float -> Natural -> Float

Multiplicative Int

Methods

(*) :: Int -> Int -> Int

one :: Int

(^) :: Int -> Natural -> Int

Multiplicative Integer

Methods

(*) :: Integer -> Integer -> Integer

one :: Integer

(^) :: Integer -> Natural -> Integer

Multiplicative Word8

Methods

(*) :: Word8 -> Word8 -> Word8

one :: Word8

(^) :: Word8 -> Natural -> Word8

Multiplicative Word16

Methods

(*) :: Word16 -> Word16 -> Word16

one :: Word16

(^) :: Word16 -> Natural -> Word16

Multiplicative Word32

Methods

(*) :: Word32 -> Word32 -> Word32

one :: Word32

(^) :: Word32 -> Natural -> Word32

Multiplicative CInt

Methods

(*) :: CInt -> CInt -> CInt

one :: CInt

(^) :: CInt -> Natural -> CInt

Integral a => Multiplicative (Ratio a)

Methods

(*) :: Ratio a -> Ratio a -> Ratio a

one :: Ratio a

(^) :: Ratio a -> Natural -> Ratio a

Additive a => Multiplicative (Exponential a)

Methods

(*) :: Exponential a -> Exponential a -> Exponential a

one :: Exponential a

(^) :: Exponential a -> Natural -> Exponential a

multiply :: (Multiplicative a, Foldable f) => f a -> a

type SemiRing a = (Multiplicative a, AbelianAdditive a)

class (SemiRing a, Group a) => Ring a where

Methods

fromInteger :: Integer -> a

Instances

Ring Double

Methods

fromInteger :: Integer -> Double

Ring Float

Methods

fromInteger :: Integer -> Float

Ring Int

Methods

fromInteger :: Integer -> Int

Ring Integer

Methods

fromInteger :: Integer -> Integer

Ring CInt

Methods

fromInteger :: Integer -> CInt

Integral a => Ring (Ratio a)

Methods

fromInteger :: Integer -> Ratio a

class Multiplicative a => Division a where

Minimal complete definition

recip | (/)

Methods

recip :: a -> a

(/) :: a -> a -> a infixl 7

Instances

Division Double

Methods

recip :: Double -> Double

(/) :: Double -> Double -> Double

Division Float

Methods

recip :: Float -> Float

(/) :: Float -> Float -> Float

Integral a => Division (Ratio a)

Methods

recip :: Ratio a -> Ratio a

(/) :: Ratio a -> Ratio a -> Ratio a

Group a => Division (Exponential a)

Methods

recip :: Exponential a -> Exponential a

(/) :: Exponential a -> Exponential a -> Exponential a

class (Ring a, Division a) => Field a where

Methods

fromRational :: Rational -> a

Instances

Field Double

Methods

fromRational :: Rational -> Double

Field Float

Methods

fromRational :: Rational -> Float

Integral a => Field (Ratio a)

Methods

fromRational :: Rational -> Ratio a

type VectorSpace scalar a = (Field scalar, Module scalar a)

class Ring a => EuclideanDomain a where

Methods

stdAssociate :: a -> a

stdUnit :: a -> a

normalize :: a -> (a, a)

div, mod :: a -> a -> a infixl 7 `div`, `mod`

divMod :: a -> a -> (a, a)

Instances

EuclideanDomain Int

Methods

stdAssociate :: Int -> Int

stdUnit :: Int -> Int

normalize :: Int -> (Int, Int)

div :: Int -> Int -> Int

mod :: Int -> Int -> Int

divMod :: Int -> Int -> (Int, Int)

EuclideanDomain Integer

Methods

stdAssociate :: Integer -> Integer

stdUnit :: Integer -> Integer

normalize :: Integer -> (Integer, Integer)

div :: Integer -> Integer -> Integer

mod :: Integer -> Integer -> Integer

divMod :: Integer -> Integer -> (Integer, Integer)

EuclideanDomain CInt

Methods

stdAssociate :: CInt -> CInt

stdUnit :: CInt -> CInt

normalize :: CInt -> (CInt, CInt)

div :: CInt -> CInt -> CInt

mod :: CInt -> CInt -> CInt

divMod :: CInt -> CInt -> (CInt, CInt)

class (Real a, Enum a, EuclideanDomain a) => Integral a where

Minimal complete definition

toInteger

Methods

quot, rem :: a -> a -> a

quotRem :: a -> a -> (a, a)

toInteger :: a -> Integer

Instances

Integral Integer

Methods

quot :: Integer -> Integer -> Integer

rem :: Integer -> Integer -> Integer

quotRem :: Integer -> Integer -> (Integer, Integer)

toInteger :: Integer -> Integer

data Ratio a

Constructors

!a :% !a

Instances

Eq a => Eq (Ratio a)

Methods

(==) :: Ratio a -> Ratio a -> Bool

(/=) :: Ratio a -> Ratio a -> Bool

type MyRational = Ratio Integer

gcd :: Integral a => a -> a -> a

ifThenElse :: Bool -> t -> t -> t

data InitialAdditive

Constructors

InitialAdditive :+ InitialAdditive

Zero

Instances

Show InitialAdditive

Methods

showsPrec :: Int -> InitialAdditive -> ShowS

show :: InitialAdditive -> String

showList :: [InitialAdditive] -> ShowS

Additive InitialAdditive

Methods

(+) :: InitialAdditive -> InitialAdditive -> InitialAdditive

zero :: InitialAdditive

times :: Natural -> InitialAdditive -> InitialAdditive