Safe Haskell	Safe
Language	Haskell2010

Algebra.Classes

Synopsis

type Natural = Integer
newtype Sum a = Sum {
- fromSum :: a
}
newtype Product a = Product {
- fromProduct :: a
}
newtype Exponential a = Exponential {
- fromExponential :: a
}
class Additive a where
- (+) :: a -> a -> a
- zero :: a
- times :: Natural -> a -> a
add :: (Foldable t, Additive a) => t a -> a
class Additive r => DecidableZero r where
- isZero :: r -> Bool
class Additive a => AbelianAdditive a
class Additive a => Group a where
- (-) :: a -> a -> a
- negate :: a -> a
- mult :: Integer -> a -> a
class (AbelianAdditive a, PreRing scalar) => Module scalar a where
- (*^) :: scalar -> a -> a
class Multiplicative a where
- (*) :: a -> a -> a
- one :: a
- (^) :: a -> Natural -> a
multiply :: (Multiplicative a, Foldable f) => f a -> a
type SemiRing a = (Multiplicative a, AbelianAdditive a)
type PreRing a = (SemiRing a, Group a)
fromIntegerDefault :: PreRing a => Integer -> a
class (Module a a, PreRing a) => Ring a where
- fromInteger :: Integer -> a
class Multiplicative a => Division a where
- recip :: a -> a
- (/) :: a -> a -> a
class (Ring a, Division a) => Field a where
- fromRational :: Rational -> a
type VectorSpace scalar a = (Field scalar, Module scalar a)
class Ring a => EuclideanDomain a where
- stdAssociate :: a -> a
- stdUnit :: a -> a
- normalize :: a -> (a, a)
- div, mod :: a -> a -> a
- divMod :: a -> a -> (a, a)
class (Real a, Enum a, EuclideanDomain a) => Integral a where
- quot, rem :: a -> a -> a
- quotRem :: a -> a -> (a, a)
- toInteger :: a -> Integer
data Ratio a = !a :% !a
type MyRational = Ratio Integer
gcd :: Integral a => a -> a -> a
ifThenElse :: Bool -> t -> t -> t
data InitialAdditive
- = InitialAdditive :+ InitialAdditive
- | Zero

Documentation

type Natural = Integer Source #

newtype Sum a Source #

Constructors

Sum
Fields fromSum :: a

Instances

Generic (Sum a) Source #
Instance details Defined in Algebra.Classes Associated Types type Rep (Sum a) :: Type -> Type # Methods from :: Sum a -> Rep (Sum a) x # to :: Rep (Sum a) x -> Sum a #
Additive a => Semigroup (Sum a) Source #
Instance details Defined in Algebra.Classes Methods (<>) :: Sum a -> Sum a -> Sum a # sconcat :: NonEmpty (Sum a) -> Sum a # stimes :: Integral b => b -> Sum a -> Sum a #
Additive a => Monoid (Sum a) Source #
Instance details Defined in Algebra.Classes Methods mempty :: Sum a # mappend :: Sum a -> Sum a -> Sum a # mconcat :: [Sum a] -> Sum a #
Binary a => Binary (Sum a) Source #
Instance details Defined in Algebra.Classes Methods put :: Sum a -> Put # get :: Get (Sum a) # putList :: [Sum a] -> Put #
type Rep (Sum a) Source #
Instance details Defined in Algebra.Classes type Rep (Sum a) = D1 (MetaData "Sum" "Algebra.Classes" "gasp-1.2.0.0-60xnh1c7HhuLy98Md0lIBW" True) (C1 (MetaCons "Sum" PrefixI True) (S1 (MetaSel (Just "fromSum") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)))

newtype Product a Source #

Constructors

Product
Fields fromProduct :: a

Instances

Multiplicative a => Semigroup (Product a) Source #
Instance details Defined in Algebra.Classes Methods (<>) :: Product a -> Product a -> Product a # sconcat :: NonEmpty (Product a) -> Product a # stimes :: Integral b => b -> Product a -> Product a #
Multiplicative a => Monoid (Product a) Source #
Instance details Defined in Algebra.Classes Methods mempty :: Product a # mappend :: Product a -> Product a -> Product a # mconcat :: [Product a] -> Product a #

newtype Exponential a Source #

Constructors

Exponential
Fields fromExponential :: a

Instances

Group a => Division (Exponential a) Source #
Instance details Defined in Algebra.Classes Methods recip :: Exponential a -> Exponential a Source # (/) :: Exponential a -> Exponential a -> Exponential a Source #
Additive a => Multiplicative (Exponential a) Source #
Instance details Defined in Algebra.Classes Methods (*) :: Exponential a -> Exponential a -> Exponential a Source # one :: Exponential a Source # (^) :: Exponential a -> Natural -> Exponential a Source #

class Additive a where Source #

Additive monoid

Minimal complete definition

(+), zero

Methods

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

zero :: a Source #

times :: Natural -> a -> a Source #

Instances

Additive Double Source #
Instance details Defined in Algebra.Classes Methods (+) :: Double -> Double -> Double Source # zero :: Double Source # times :: Natural -> Double -> Double Source #
Additive Float Source #
Instance details Defined in Algebra.Classes Methods (+) :: Float -> Float -> Float Source # zero :: Float Source # times :: Natural -> Float -> Float Source #
Additive Int Source #
Instance details Defined in Algebra.Classes Methods (+) :: Int -> Int -> Int Source # zero :: Int Source # times :: Natural -> Int -> Int Source #
Additive Integer Source #
Instance details Defined in Algebra.Classes Methods (+) :: Integer -> Integer -> Integer Source # zero :: Integer Source # times :: Natural -> Integer -> Integer Source #
Additive Word8 Source #
Instance details Defined in Algebra.Classes Methods (+) :: Word8 -> Word8 -> Word8 Source # zero :: Word8 Source # times :: Natural -> Word8 -> Word8 Source #
Additive Word16 Source #
Instance details Defined in Algebra.Classes Methods (+) :: Word16 -> Word16 -> Word16 Source # zero :: Word16 Source # times :: Natural -> Word16 -> Word16 Source #
Additive Word32 Source #
Instance details Defined in Algebra.Classes Methods (+) :: Word32 -> Word32 -> Word32 Source # zero :: Word32 Source # times :: Natural -> Word32 -> Word32 Source #
Additive CInt Source #
Instance details Defined in Algebra.Classes Methods (+) :: CInt -> CInt -> CInt Source # zero :: CInt Source # times :: Natural -> CInt -> CInt Source #
Additive InitialAdditive Source #
Instance details Defined in Algebra.Classes Methods (+) :: InitialAdditive -> InitialAdditive -> InitialAdditive Source # zero :: InitialAdditive Source # times :: Natural -> InitialAdditive -> InitialAdditive Source #
Integral a => Additive (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods (+) :: Ratio a -> Ratio a -> Ratio a Source # zero :: Ratio a Source # times :: Natural -> Ratio a -> Ratio a Source #
(Ord k, Additive v) => Additive (Map k v) Source #
Instance details Defined in Algebra.Classes Methods (+) :: Map k v -> Map k v -> Map k v Source # zero :: Map k v Source # times :: Natural -> Map k v -> Map k v Source #
(Applicative f, Additive a) => Additive (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (+) :: Euclid f a -> Euclid f a -> Euclid f a Source # zero :: Euclid f a Source # times :: Natural -> Euclid f a -> Euclid f a Source #
(Applicative f, Applicative g, Additive a) => Additive (Mat a f g) Source #
Instance details Defined in Algebra.Linear Methods (+) :: Mat a f g -> Mat a f g -> Mat a f g Source # zero :: Mat a f g Source # times :: Natural -> Mat a f g -> Mat a f g Source #

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

class Additive r => DecidableZero r where Source #

Methods

isZero :: r -> Bool Source #

Instances

DecidableZero Double Source #
Instance details Defined in Algebra.Classes Methods isZero :: Double -> Bool Source #
DecidableZero Float Source #
Instance details Defined in Algebra.Classes Methods isZero :: Float -> Bool Source #
DecidableZero Int Source #
Instance details Defined in Algebra.Classes Methods isZero :: Int -> Bool Source #
DecidableZero Integer Source #
Instance details Defined in Algebra.Classes Methods isZero :: Integer -> Bool Source #
DecidableZero Word8 Source #
Instance details Defined in Algebra.Classes Methods isZero :: Word8 -> Bool Source #
DecidableZero Word16 Source #
Instance details Defined in Algebra.Classes Methods isZero :: Word16 -> Bool Source #
DecidableZero Word32 Source #
Instance details Defined in Algebra.Classes Methods isZero :: Word32 -> Bool Source #
DecidableZero CInt Source #
Instance details Defined in Algebra.Classes Methods isZero :: CInt -> Bool Source #
(Ord k, DecidableZero v) => DecidableZero (Map k v) Source #
Instance details Defined in Algebra.Classes Methods isZero :: Map k v -> Bool Source #

class Additive a => AbelianAdditive a Source #

Instances

AbelianAdditive Double Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Float Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Int Source #
Instance details Defined in Algebra.Classes
AbelianAdditive Integer Source #
Instance details Defined in Algebra.Classes
AbelianAdditive CInt Source #
Instance details Defined in Algebra.Classes
Integral a => AbelianAdditive (Ratio a) Source #
Instance details Defined in Algebra.Classes
(Ord k, AbelianAdditive v) => AbelianAdditive (Map k v) Source #
Instance details Defined in Algebra.Classes
(Applicative f, AbelianAdditive a) => AbelianAdditive (Euclid f a) Source #
Instance details Defined in Algebra.Linear
(Applicative f, Applicative g, AbelianAdditive a) => AbelianAdditive (Mat a f g) Source #
Instance details Defined in Algebra.Linear

class Additive a => Group a where Source #

Minimal complete definition

(negate | (-))

Methods

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

negate :: a -> a Source #

mult :: Integer -> a -> a Source #

Instances

Group Double Source #
Instance details Defined in Algebra.Classes Methods (-) :: Double -> Double -> Double Source # negate :: Double -> Double Source # mult :: Integer -> Double -> Double Source #
Group Float Source #
Instance details Defined in Algebra.Classes Methods (-) :: Float -> Float -> Float Source # negate :: Float -> Float Source # mult :: Integer -> Float -> Float Source #
Group Int Source #
Instance details Defined in Algebra.Classes Methods (-) :: Int -> Int -> Int Source # negate :: Int -> Int Source # mult :: Integer -> Int -> Int Source #
Group Integer Source #
Instance details Defined in Algebra.Classes Methods (-) :: Integer -> Integer -> Integer Source # negate :: Integer -> Integer Source # mult :: Integer -> Integer -> Integer Source #
Group Word8 Source #
Instance details Defined in Algebra.Classes Methods (-) :: Word8 -> Word8 -> Word8 Source # negate :: Word8 -> Word8 Source # mult :: Integer -> Word8 -> Word8 Source #
Group Word16 Source #
Instance details Defined in Algebra.Classes Methods (-) :: Word16 -> Word16 -> Word16 Source # negate :: Word16 -> Word16 Source # mult :: Integer -> Word16 -> Word16 Source #
Group Word32 Source #
Instance details Defined in Algebra.Classes Methods (-) :: Word32 -> Word32 -> Word32 Source # negate :: Word32 -> Word32 Source # mult :: Integer -> Word32 -> Word32 Source #
Group CInt Source #
Instance details Defined in Algebra.Classes Methods (-) :: CInt -> CInt -> CInt Source # negate :: CInt -> CInt Source # mult :: Integer -> CInt -> CInt Source #
Integral a => Group (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods (-) :: Ratio a -> Ratio a -> Ratio a Source # negate :: Ratio a -> Ratio a Source # mult :: Integer -> Ratio a -> Ratio a Source #
(Ord k, Group v) => Group (Map k v) Source #
Instance details Defined in Algebra.Classes Methods (-) :: Map k v -> Map k v -> Map k v Source # negate :: Map k v -> Map k v Source # mult :: Integer -> Map k v -> Map k v Source #
(Applicative f, Group a) => Group (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (-) :: Euclid f a -> Euclid f a -> Euclid f a Source # negate :: Euclid f a -> Euclid f a Source # mult :: Integer -> Euclid f a -> Euclid f a Source #
(Applicative f, Applicative g, Group a) => Group (Mat a f g) Source #
Instance details Defined in Algebra.Linear Methods (-) :: Mat a f g -> Mat a f g -> Mat a f g Source # negate :: Mat a f g -> Mat a f g Source # mult :: Integer -> Mat a f g -> Mat a f g Source #

class (AbelianAdditive a, PreRing scalar) => Module scalar a where Source #

Module

Methods

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

Instances

Module Double Double Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Double -> Double -> Double Source #
Module Float Float Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Float -> Float -> Float Source #
Module Int Int Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Int -> Int -> Int Source #
Module Integer Integer Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Integer -> Integer -> Integer Source #
Module Rational Double Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Rational -> Double -> Double Source #
Module CInt CInt Source #
Instance details Defined in Algebra.Classes Methods (*^) :: CInt -> CInt -> CInt Source #
(Ord k, Module v v) => Module v (Map k v) Source #
Instance details Defined in Algebra.Classes Methods (*^) :: v -> Map k v -> Map k v Source #
(Applicative f, Module s a) => Module s (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (*^) :: s -> Euclid f a -> Euclid f a Source #
(Applicative f, Applicative g, Module s a) => Module s (Mat a f g) Source #
Instance details Defined in Algebra.Linear Methods (*^) :: s -> Mat a f g -> Mat a f g Source #
Integral a => Module (Ratio a) (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods (*^) :: Ratio a -> Ratio a -> Ratio a Source #

class Multiplicative a where Source #

Multiplicative monoid

Minimal complete definition

(*), one

Methods

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

one :: a Source #

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

Instances

Multiplicative Double Source #
Instance details Defined in Algebra.Classes Methods (*) :: Double -> Double -> Double Source # one :: Double Source # (^) :: Double -> Natural -> Double Source #
Multiplicative Float Source #
Instance details Defined in Algebra.Classes Methods (*) :: Float -> Float -> Float Source # one :: Float Source # (^) :: Float -> Natural -> Float Source #
Multiplicative Int Source #
Instance details Defined in Algebra.Classes Methods (*) :: Int -> Int -> Int Source # one :: Int Source # (^) :: Int -> Natural -> Int Source #
Multiplicative Integer Source #
Instance details Defined in Algebra.Classes Methods (*) :: Integer -> Integer -> Integer Source # one :: Integer Source # (^) :: Integer -> Natural -> Integer Source #
Multiplicative Word8 Source #
Instance details Defined in Algebra.Classes Methods (*) :: Word8 -> Word8 -> Word8 Source # one :: Word8 Source # (^) :: Word8 -> Natural -> Word8 Source #
Multiplicative Word16 Source #
Instance details Defined in Algebra.Classes Methods (*) :: Word16 -> Word16 -> Word16 Source # one :: Word16 Source # (^) :: Word16 -> Natural -> Word16 Source #
Multiplicative Word32 Source #
Instance details Defined in Algebra.Classes Methods (*) :: Word32 -> Word32 -> Word32 Source # one :: Word32 Source # (^) :: Word32 -> Natural -> Word32 Source #
Multiplicative CInt Source #
Instance details Defined in Algebra.Classes Methods (*) :: CInt -> CInt -> CInt Source # one :: CInt Source # (^) :: CInt -> Natural -> CInt Source #
Integral a => Multiplicative (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods (*) :: Ratio a -> Ratio a -> Ratio a Source # one :: Ratio a Source # (^) :: Ratio a -> Natural -> Ratio a Source #
Additive a => Multiplicative (Exponential a) Source #
Instance details Defined in Algebra.Classes Methods (*) :: Exponential a -> Exponential a -> Exponential a Source # one :: Exponential a Source # (^) :: Exponential a -> Natural -> Exponential a Source #
(Ring s, Applicative v, Traversable v) => Multiplicative (OrthoMat v s) Source #
Instance details Defined in Algebra.Linear Methods (*) :: OrthoMat v s -> OrthoMat v s -> OrthoMat v s Source # one :: OrthoMat v s Source # (^) :: OrthoMat v s -> Natural -> OrthoMat v s Source #

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

type SemiRing a = (Multiplicative a, AbelianAdditive a) Source #

type PreRing a = (SemiRing a, Group a) Source #

fromIntegerDefault :: PreRing a => Integer -> a Source #

class (Module a a, PreRing a) => Ring a where Source #

Minimal complete definition

Nothing

Methods

fromInteger :: Integer -> a Source #

Instances

Ring Double Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Double Source #
Ring Float Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Float Source #
Ring Int Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Int Source #
Ring Integer Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Integer Source #
Ring CInt Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> CInt Source #
Integral a => Ring (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods fromInteger :: Integer -> Ratio a Source #

class Multiplicative a => Division a where Source #

Minimal complete definition

(recip | (/))

Methods

recip :: a -> a Source #

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

Instances

Division Double Source #
Instance details Defined in Algebra.Classes Methods recip :: Double -> Double Source # (/) :: Double -> Double -> Double Source #
Division Float Source #
Instance details Defined in Algebra.Classes Methods recip :: Float -> Float Source # (/) :: Float -> Float -> Float Source #
Integral a => Division (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods recip :: Ratio a -> Ratio a Source # (/) :: Ratio a -> Ratio a -> Ratio a Source #
Group a => Division (Exponential a) Source #
Instance details Defined in Algebra.Classes Methods recip :: Exponential a -> Exponential a Source # (/) :: Exponential a -> Exponential a -> Exponential a Source #
(Ring s, Applicative v, Traversable v) => Division (OrthoMat v s) Source #
Instance details Defined in Algebra.Linear Methods recip :: OrthoMat v s -> OrthoMat v s Source # (/) :: OrthoMat v s -> OrthoMat v s -> OrthoMat v s Source #

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

Minimal complete definition

Nothing

Methods

fromRational :: Rational -> a Source #

Instances

Field Double Source #
Instance details Defined in Algebra.Classes Methods fromRational :: Rational -> Double Source #
Field Float Source #
Instance details Defined in Algebra.Classes Methods fromRational :: Rational -> Float Source #
Integral a => Field (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods fromRational :: Rational -> Ratio a Source #

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

class Ring a => EuclideanDomain a where Source #

Minimal complete definition

(stdUnit | normalize), (divMod | div, mod)

Methods

stdAssociate :: a -> a Source #

stdUnit :: a -> a Source #

normalize :: a -> (a, a) Source #

div :: a -> a -> a infixl 7 Source #

mod :: a -> a -> a infixl 7 Source #

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

Instances

EuclideanDomain Int Source #
Instance details Defined in Algebra.Classes Methods stdAssociate :: Int -> Int Source # stdUnit :: Int -> Int Source # normalize :: Int -> (Int, Int) Source # div :: Int -> Int -> Int Source # mod :: Int -> Int -> Int Source # divMod :: Int -> Int -> (Int, Int) Source #
EuclideanDomain Integer Source #
Instance details Defined in Algebra.Classes Methods stdAssociate :: Integer -> Integer Source # stdUnit :: Integer -> Integer Source # normalize :: Integer -> (Integer, Integer) Source # div :: Integer -> Integer -> Integer Source # mod :: Integer -> Integer -> Integer Source # divMod :: Integer -> Integer -> (Integer, Integer) Source #
EuclideanDomain CInt Source #
Instance details Defined in Algebra.Classes Methods stdAssociate :: CInt -> CInt Source # stdUnit :: CInt -> CInt Source # normalize :: CInt -> (CInt, CInt) Source # div :: CInt -> CInt -> CInt Source # mod :: CInt -> CInt -> CInt Source # divMod :: CInt -> CInt -> (CInt, CInt) Source #

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

Minimal complete definition

toInteger

Methods

quot :: a -> a -> a Source #

rem :: a -> a -> a Source #

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

toInteger :: a -> Integer Source #

Instances

Integral Integer Source #
Instance details Defined in Algebra.Classes Methods quot :: Integer -> Integer -> Integer Source # rem :: Integer -> Integer -> Integer Source # quotRem :: Integer -> Integer -> (Integer, Integer) Source # toInteger :: Integer -> Integer Source #

data Ratio a Source #

Constructors

!a :% !a

Instances

Eq a => Eq (Ratio a) Source #
Instance details Defined in Algebra.Classes Methods (==) :: Ratio a -> Ratio a -> Bool # (/=) :: Ratio a -> Ratio a -> Bool #

type MyRational = Ratio Integer Source #

gcd :: Integral a => a -> a -> a Source #

ifThenElse :: Bool -> t -> t -> t Source #

data InitialAdditive Source #

Constructors

InitialAdditive :+ InitialAdditive
Zero

Instances

Show InitialAdditive Source #
Instance details Defined in Algebra.Classes Methods showsPrec :: Int -> InitialAdditive -> ShowS # show :: InitialAdditive -> String # showList :: [InitialAdditive] -> ShowS #
Additive InitialAdditive Source #
Instance details Defined in Algebra.Classes Methods (+) :: InitialAdditive -> InitialAdditive -> InitialAdditive Source # zero :: InitialAdditive Source # times :: Natural -> InitialAdditive -> InitialAdditive Source #