semirings-0.2.1.0: two monoids as one, in holy haskimony

Safe HaskellNone
LanguageHaskell98

Data.Semiring

Contents

Description

A class for semirings (types with two binary operations, one commutative and one associative, and two respective identites), with various general-purpose instances.

Synopsis

Semiring typeclass

class Semiring a where Source #

The class of semirings (types with two binary operations and two respective identities). One can think of a semiring as two monoids of the same underlying type: A commutative monoid and an associative monoid. For any type R with a Num instance, the commutative monoid is (R, '(Prelude.+)', 0) and the associative monoid is (R, '(Prelude.*)', 1).

Instances should satisfy the following laws:

additive identity
x + zero = zero + x = x
additive associativity
x + (y + z) = (x + y) + z
additive commutativity
x + y = y + x
multiplicative identity
x * one = one * x = x
multiplicative associativity
x * (y * z) = (x * y) * z
left- and right-distributivity of * over +

x * (y + z) = (x * y) + (x * z) (x + y) * z = (x * z) + (y * z)

annihilation
zero * x = x * zero = zero

Minimal complete definition

plus, zero, times, one

Methods

plus infixl 6 Source #

Arguments

:: a 
-> a 
-> a

Commutative Operation

zero Source #

Arguments

:: a

Commutative Unit

times infixl 7 Source #

Arguments

:: a 
-> a 
-> a

Associative Operation

one Source #

Arguments

:: a

Associative Unit

Instances
Semiring Bool Source # 
Instance details

Defined in Data.Semiring

Semiring Double Source # 
Instance details

Defined in Data.Semiring

Semiring Float Source # 
Instance details

Defined in Data.Semiring

Semiring Int Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Int -> Int -> Int Source #

zero :: Int Source #

times :: Int -> Int -> Int Source #

one :: Int Source #

Semiring Int8 Source # 
Instance details

Defined in Data.Semiring

Semiring Int16 Source # 
Instance details

Defined in Data.Semiring

Semiring Int32 Source # 
Instance details

Defined in Data.Semiring

Semiring Int64 Source # 
Instance details

Defined in Data.Semiring

Semiring Integer Source # 
Instance details

Defined in Data.Semiring

Semiring Natural Source # 
Instance details

Defined in Data.Semiring

Semiring Word Source # 
Instance details

Defined in Data.Semiring

Semiring Word8 Source # 
Instance details

Defined in Data.Semiring

Semiring Word16 Source # 
Instance details

Defined in Data.Semiring

Semiring Word32 Source # 
Instance details

Defined in Data.Semiring

Semiring Word64 Source # 
Instance details

Defined in Data.Semiring

Semiring () Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: () -> () -> () Source #

zero :: () Source #

times :: () -> () -> () Source #

one :: () Source #

Semiring CDev Source # 
Instance details

Defined in Data.Semiring

Semiring CIno Source # 
Instance details

Defined in Data.Semiring

Semiring CMode Source # 
Instance details

Defined in Data.Semiring

Semiring COff Source # 
Instance details

Defined in Data.Semiring

Semiring CPid Source # 
Instance details

Defined in Data.Semiring

Semiring CSsize Source # 
Instance details

Defined in Data.Semiring

Semiring CGid Source # 
Instance details

Defined in Data.Semiring

Semiring CNlink Source # 
Instance details

Defined in Data.Semiring

Semiring CUid Source # 
Instance details

Defined in Data.Semiring

Semiring CCc Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: CCc -> CCc -> CCc Source #

zero :: CCc Source #

times :: CCc -> CCc -> CCc Source #

one :: CCc Source #

Semiring CSpeed Source # 
Instance details

Defined in Data.Semiring

Semiring CTcflag Source # 
Instance details

Defined in Data.Semiring

Semiring CRLim Source # 
Instance details

Defined in Data.Semiring

Semiring Fd Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Fd -> Fd -> Fd Source #

zero :: Fd Source #

times :: Fd -> Fd -> Fd Source #

one :: Fd Source #

Semiring CChar Source # 
Instance details

Defined in Data.Semiring

Semiring CSChar Source # 
Instance details

Defined in Data.Semiring

Semiring CUChar Source # 
Instance details

Defined in Data.Semiring

Semiring CShort Source # 
Instance details

Defined in Data.Semiring

Semiring CUShort Source # 
Instance details

Defined in Data.Semiring

Semiring CInt Source # 
Instance details

Defined in Data.Semiring

Semiring CUInt Source # 
Instance details

Defined in Data.Semiring

Semiring CLong Source # 
Instance details

Defined in Data.Semiring

Semiring CULong Source # 
Instance details

Defined in Data.Semiring

Semiring CLLong Source # 
Instance details

Defined in Data.Semiring

Semiring CULLong Source # 
Instance details

Defined in Data.Semiring

Semiring CFloat Source # 
Instance details

Defined in Data.Semiring

Semiring CDouble Source # 
Instance details

Defined in Data.Semiring

Semiring CPtrdiff Source # 
Instance details

Defined in Data.Semiring

Semiring CSize Source # 
Instance details

Defined in Data.Semiring

Semiring CWchar Source # 
Instance details

Defined in Data.Semiring

Semiring CSigAtomic Source # 
Instance details

Defined in Data.Semiring

Semiring CClock Source # 
Instance details

Defined in Data.Semiring

Semiring CTime Source # 
Instance details

Defined in Data.Semiring

Semiring CUSeconds Source # 
Instance details

Defined in Data.Semiring

Semiring CSUSeconds Source # 
Instance details

Defined in Data.Semiring

Semiring CIntPtr Source # 
Instance details

Defined in Data.Semiring

Semiring CUIntPtr Source # 
Instance details

Defined in Data.Semiring

Semiring CIntMax Source # 
Instance details

Defined in Data.Semiring

Semiring CUIntMax Source # 
Instance details

Defined in Data.Semiring

Semiring WordPtr Source # 
Instance details

Defined in Data.Semiring

Semiring IntPtr Source # 
Instance details

Defined in Data.Semiring

Semiring a => Semiring [a] Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: [a] -> [a] -> [a] Source #

zero :: [a] Source #

times :: [a] -> [a] -> [a] Source #

one :: [a] Source #

Semiring a => Semiring (Maybe a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Maybe a -> Maybe a -> Maybe a Source #

zero :: Maybe a Source #

times :: Maybe a -> Maybe a -> Maybe a Source #

one :: Maybe a Source #

Integral a => Semiring (Ratio a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Ratio a -> Ratio a -> Ratio a Source #

zero :: Ratio a Source #

times :: Ratio a -> Ratio a -> Ratio a Source #

one :: Ratio a Source #

Semiring a => Semiring (IO a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: IO a -> IO a -> IO a Source #

zero :: IO a Source #

times :: IO a -> IO a -> IO a Source #

one :: IO a Source #

Ring a => Semiring (Complex a) Source #

This instance can suffer due to floating point arithmetic.

Instance details

Defined in Data.Semiring

HasResolution a => Semiring (Fixed a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Fixed a -> Fixed a -> Fixed a Source #

zero :: Fixed a Source #

times :: Fixed a -> Fixed a -> Fixed a Source #

one :: Fixed a Source #

Semiring a => Semiring (Min a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Min a -> Min a -> Min a Source #

zero :: Min a Source #

times :: Min a -> Min a -> Min a Source #

one :: Min a Source #

Semiring a => Semiring (Max a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Max a -> Max a -> Max a Source #

zero :: Max a Source #

times :: Max a -> Max a -> Max a Source #

one :: Max a Source #

Semiring a => Semiring (Identity a) Source # 
Instance details

Defined in Data.Semiring

Semiring a => Semiring (Dual a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Dual a -> Dual a -> Dual a Source #

zero :: Dual a Source #

times :: Dual a -> Dual a -> Dual a Source #

one :: Dual a Source #

Semiring a => Semiring (Sum a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Sum a -> Sum a -> Sum a Source #

zero :: Sum a Source #

times :: Sum a -> Sum a -> Sum a Source #

one :: Sum a Source #

Semiring a => Semiring (Product a) Source # 
Instance details

Defined in Data.Semiring

Semiring a => Semiring (Down a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Down a -> Down a -> Down a Source #

zero :: Down a Source #

times :: Down a -> Down a -> Down a Source #

one :: Down a Source #

(Ord a, Monoid a) => Semiring (Set a) Source #

The multiplication laws are satisfied for any underlying Monoid, so we require a Monoid contraint instead of a Semiring constraint since times can use the context of either.

Instance details

Defined in Data.Semiring

Methods

plus :: Set a -> Set a -> Set a Source #

zero :: Set a Source #

times :: Set a -> Set a -> Set a Source #

one :: Set a Source #

(Eq a, Hashable a, Monoid a) => Semiring (HashSet a) Source #

The multiplication laws are satisfied for any underlying Monoid, so we require a Monoid contraint instead of a Semiring constraint since times can use the context of either.

Instance details

Defined in Data.Semiring

(Unbox a, Semiring a) => Semiring (Vector a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Vector a -> Vector a -> Vector a Source #

zero :: Vector a Source #

times :: Vector a -> Vector a -> Vector a Source #

one :: Vector a Source #

(Storable a, Semiring a) => Semiring (Vector a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Vector a -> Vector a -> Vector a Source #

zero :: Vector a Source #

times :: Vector a -> Vector a -> Vector a Source #

one :: Vector a Source #

Semiring a => Semiring (Vector a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Vector a -> Vector a -> Vector a Source #

zero :: Vector a Source #

times :: Vector a -> Vector a -> Vector a Source #

one :: Vector a Source #

Semiring a => Semiring (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Mul a -> Mul a -> Mul a Source #

zero :: Mul a Source #

times :: Mul a -> Mul a -> Mul a Source #

one :: Mul a Source #

Semiring a => Semiring (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Add a -> Add a -> Add a Source #

zero :: Add a Source #

times :: Add a -> Add a -> Add a Source #

one :: Add a Source #

Semiring b => Semiring (a -> b) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: (a -> b) -> (a -> b) -> a -> b Source #

zero :: a -> b Source #

times :: (a -> b) -> (a -> b) -> a -> b Source #

one :: a -> b Source #

(Semiring a, Semiring b) => Semiring (a, b) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

plus :: (a, b) -> (a, b) -> (a, b) Source #

zero :: (a, b) Source #

times :: (a, b) -> (a, b) -> (a, b) Source #

one :: (a, b) Source #

Semiring (Proxy a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Proxy a -> Proxy a -> Proxy a Source #

zero :: Proxy a Source #

times :: Proxy a -> Proxy a -> Proxy a Source #

one :: Proxy a Source #

(Ord k, Monoid k, Semiring v) => Semiring (Map k v) Source #

The multiplication laws are satisfied for any underlying Monoid as the key type, so we require a Monoid contraint instead of a Semiring constraint since times can use the context of either.

Instance details

Defined in Data.Semiring

Methods

plus :: Map k v -> Map k v -> Map k v Source #

zero :: Map k v Source #

times :: Map k v -> Map k v -> Map k v Source #

one :: Map k v Source #

(Eq k, Hashable k, Monoid k, Semiring v) => Semiring (HashMap k v) Source #

The multiplication laws are satisfied for any underlying Monoid as the key type, so we require a Monoid contraint instead of a Semiring constraint since times can use the context of either.

Instance details

Defined in Data.Semiring

Methods

plus :: HashMap k v -> HashMap k v -> HashMap k v Source #

zero :: HashMap k v Source #

times :: HashMap k v -> HashMap k v -> HashMap k v Source #

one :: HashMap k v Source #

(Semiring a, Semiring b, Semiring c) => Semiring (a, b, c) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

plus :: (a, b, c) -> (a, b, c) -> (a, b, c) Source #

zero :: (a, b, c) Source #

times :: (a, b, c) -> (a, b, c) -> (a, b, c) Source #

one :: (a, b, c) Source #

Semiring a => Semiring (Const a b) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Const a b -> Const a b -> Const a b Source #

zero :: Const a b Source #

times :: Const a b -> Const a b -> Const a b Source #

one :: Const a b Source #

(Semiring a, Semiring b, Semiring c, Semiring d) => Semiring (a, b, c, d) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

plus :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) Source #

zero :: (a, b, c, d) Source #

times :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) Source #

one :: (a, b, c, d) Source #

(Semiring a, Semiring b, Semiring c, Semiring d, Semiring e) => Semiring (a, b, c, d, e) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

plus :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) Source #

zero :: (a, b, c, d, e) Source #

times :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) Source #

one :: (a, b, c, d, e) Source #

(Semiring a, Semiring b, Semiring c, Semiring d, Semiring e, Semiring f) => Semiring (a, b, c, d, e, f) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

plus :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) Source #

zero :: (a, b, c, d, e, f) Source #

times :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) Source #

one :: (a, b, c, d, e, f) Source #

(Semiring a, Semiring b, Semiring c, Semiring d, Semiring e, Semiring f, Semiring g) => Semiring (a, b, c, d, e, f, g) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

plus :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) Source #

zero :: (a, b, c, d, e, f, g) Source #

times :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) Source #

one :: (a, b, c, d, e, f, g) Source #

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

Infix shorthand for plus.

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

Infix shorthand for times.

(^) :: (Semiring a, Integral b) => a -> b -> a infixr 8 Source #

Raise a number to a non-negative integral power. If the power is negative, this will return zero.

foldMapP :: (Foldable t, Semiring s) => (a -> s) -> t a -> s Source #

Map each element of the structure to a semiring, and combine the results using plus.

foldMapT :: (Foldable t, Semiring s) => (a -> s) -> t a -> s Source #

Map each element of the structure to a semiring, and combine the results using times.

sum :: (Foldable t, Semiring a) => t a -> a Source #

The sum function computes the additive sum of the elements in a structure. This function is lazy. For a strict version, see sum'.

product :: (Foldable t, Semiring a) => t a -> a Source #

The product function computes the product of the elements in a structure. This function is lazy. for a strict version, see product'.

sum' :: (Foldable t, Semiring a) => t a -> a Source #

The sum' function computes the additive sum of the elements in a structure. This function is strict. For a lazy version, see sum.

product' :: (Foldable t, Semiring a) => t a -> a Source #

The product' function computes the additive sum of the elements in a structure. This function is strict. For a lazy version, see product.

Types

newtype Add a Source #

Monoid under plus. Analogous to Sum, but uses the Semiring constraint rather than Num.

Constructors

Add 

Fields

Instances
Functor Add Source # 
Instance details

Defined in Data.Semiring

Methods

fmap :: (a -> b) -> Add a -> Add b #

(<$) :: a -> Add b -> Add a #

Foldable Add Source # 
Instance details

Defined in Data.Semiring

Methods

fold :: Monoid m => Add m -> m #

foldMap :: Monoid m => (a -> m) -> Add a -> m #

foldr :: (a -> b -> b) -> b -> Add a -> b #

foldr' :: (a -> b -> b) -> b -> Add a -> b #

foldl :: (b -> a -> b) -> b -> Add a -> b #

foldl' :: (b -> a -> b) -> b -> Add a -> b #

foldr1 :: (a -> a -> a) -> Add a -> a #

foldl1 :: (a -> a -> a) -> Add a -> a #

toList :: Add a -> [a] #

null :: Add a -> Bool #

length :: Add a -> Int #

elem :: Eq a => a -> Add a -> Bool #

maximum :: Ord a => Add a -> a #

minimum :: Ord a => Add a -> a #

sum :: Num a => Add a -> a #

product :: Num a => Add a -> a #

Traversable Add Source # 
Instance details

Defined in Data.Semiring

Methods

traverse :: Applicative f => (a -> f b) -> Add a -> f (Add b) #

sequenceA :: Applicative f => Add (f a) -> f (Add a) #

mapM :: Monad m => (a -> m b) -> Add a -> m (Add b) #

sequence :: Monad m => Add (m a) -> m (Add a) #

Bounded a => Bounded (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

minBound :: Add a #

maxBound :: Add a #

Enum a => Enum (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

succ :: Add a -> Add a #

pred :: Add a -> Add a #

toEnum :: Int -> Add a #

fromEnum :: Add a -> Int #

enumFrom :: Add a -> [Add a] #

enumFromThen :: Add a -> Add a -> [Add a] #

enumFromTo :: Add a -> Add a -> [Add a] #

enumFromThenTo :: Add a -> Add a -> Add a -> [Add a] #

Eq a => Eq (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

(==) :: Add a -> Add a -> Bool #

(/=) :: Add a -> Add a -> Bool #

Fractional a => Fractional (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

(/) :: Add a -> Add a -> Add a #

recip :: Add a -> Add a #

fromRational :: Rational -> Add a #

Num a => Num (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

(+) :: Add a -> Add a -> Add a #

(-) :: Add a -> Add a -> Add a #

(*) :: Add a -> Add a -> Add a #

negate :: Add a -> Add a #

abs :: Add a -> Add a #

signum :: Add a -> Add a #

fromInteger :: Integer -> Add a #

Ord a => Ord (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

compare :: Add a -> Add a -> Ordering #

(<) :: Add a -> Add a -> Bool #

(<=) :: Add a -> Add a -> Bool #

(>) :: Add a -> Add a -> Bool #

(>=) :: Add a -> Add a -> Bool #

max :: Add a -> Add a -> Add a #

min :: Add a -> Add a -> Add a #

Read a => Read (Add a) Source # 
Instance details

Defined in Data.Semiring

Real a => Real (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

toRational :: Add a -> Rational #

RealFrac a => RealFrac (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

properFraction :: Integral b => Add a -> (b, Add a) #

truncate :: Integral b => Add a -> b #

round :: Integral b => Add a -> b #

ceiling :: Integral b => Add a -> b #

floor :: Integral b => Add a -> b #

Show a => Show (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

showsPrec :: Int -> Add a -> ShowS #

show :: Add a -> String #

showList :: [Add a] -> ShowS #

Generic (Add a) Source # 
Instance details

Defined in Data.Semiring

Associated Types

type Rep (Add a) :: * -> * #

Methods

from :: Add a -> Rep (Add a) x #

to :: Rep (Add a) x -> Add a #

Semiring a => Semigroup (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

(<>) :: Add a -> Add a -> Add a #

sconcat :: NonEmpty (Add a) -> Add a #

stimes :: Integral b => b -> Add a -> Add a #

Semiring a => Monoid (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

mempty :: Add a #

mappend :: Add a -> Add a -> Add a #

mconcat :: [Add a] -> Add a #

Storable a => Storable (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

sizeOf :: Add a -> Int #

alignment :: Add a -> Int #

peekElemOff :: Ptr (Add a) -> Int -> IO (Add a) #

pokeElemOff :: Ptr (Add a) -> Int -> Add a -> IO () #

peekByteOff :: Ptr b -> Int -> IO (Add a) #

pokeByteOff :: Ptr b -> Int -> Add a -> IO () #

peek :: Ptr (Add a) -> IO (Add a) #

poke :: Ptr (Add a) -> Add a -> IO () #

Semiring a => Semiring (Add a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Add a -> Add a -> Add a Source #

zero :: Add a Source #

times :: Add a -> Add a -> Add a Source #

one :: Add a Source #

Generic1 Add Source # 
Instance details

Defined in Data.Semiring

Associated Types

type Rep1 Add :: k -> * #

Methods

from1 :: Add a -> Rep1 Add a #

to1 :: Rep1 Add a -> Add a #

type Rep (Add a) Source # 
Instance details

Defined in Data.Semiring

type Rep (Add a) = D1 (MetaData "Add" "Data.Semiring" "semirings-0.2.1.0-He40r1WdcGVKOm5tgEashg" True) (C1 (MetaCons "Add" PrefixI True) (S1 (MetaSel (Just "getAdd") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)))
type Rep1 Add Source # 
Instance details

Defined in Data.Semiring

type Rep1 Add = D1 (MetaData "Add" "Data.Semiring" "semirings-0.2.1.0-He40r1WdcGVKOm5tgEashg" True) (C1 (MetaCons "Add" PrefixI True) (S1 (MetaSel (Just "getAdd") NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1))

newtype Mul a Source #

Monoid under times. Analogous to Product, but uses the Semiring constraint rather than Num.

Constructors

Mul 

Fields

Instances
Functor Mul Source # 
Instance details

Defined in Data.Semiring

Methods

fmap :: (a -> b) -> Mul a -> Mul b #

(<$) :: a -> Mul b -> Mul a #

Foldable Mul Source # 
Instance details

Defined in Data.Semiring

Methods

fold :: Monoid m => Mul m -> m #

foldMap :: Monoid m => (a -> m) -> Mul a -> m #

foldr :: (a -> b -> b) -> b -> Mul a -> b #

foldr' :: (a -> b -> b) -> b -> Mul a -> b #

foldl :: (b -> a -> b) -> b -> Mul a -> b #

foldl' :: (b -> a -> b) -> b -> Mul a -> b #

foldr1 :: (a -> a -> a) -> Mul a -> a #

foldl1 :: (a -> a -> a) -> Mul a -> a #

toList :: Mul a -> [a] #

null :: Mul a -> Bool #

length :: Mul a -> Int #

elem :: Eq a => a -> Mul a -> Bool #

maximum :: Ord a => Mul a -> a #

minimum :: Ord a => Mul a -> a #

sum :: Num a => Mul a -> a #

product :: Num a => Mul a -> a #

Traversable Mul Source # 
Instance details

Defined in Data.Semiring

Methods

traverse :: Applicative f => (a -> f b) -> Mul a -> f (Mul b) #

sequenceA :: Applicative f => Mul (f a) -> f (Mul a) #

mapM :: Monad m => (a -> m b) -> Mul a -> m (Mul b) #

sequence :: Monad m => Mul (m a) -> m (Mul a) #

Bounded a => Bounded (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

minBound :: Mul a #

maxBound :: Mul a #

Enum a => Enum (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

succ :: Mul a -> Mul a #

pred :: Mul a -> Mul a #

toEnum :: Int -> Mul a #

fromEnum :: Mul a -> Int #

enumFrom :: Mul a -> [Mul a] #

enumFromThen :: Mul a -> Mul a -> [Mul a] #

enumFromTo :: Mul a -> Mul a -> [Mul a] #

enumFromThenTo :: Mul a -> Mul a -> Mul a -> [Mul a] #

Eq a => Eq (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

(==) :: Mul a -> Mul a -> Bool #

(/=) :: Mul a -> Mul a -> Bool #

Fractional a => Fractional (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

(/) :: Mul a -> Mul a -> Mul a #

recip :: Mul a -> Mul a #

fromRational :: Rational -> Mul a #

Num a => Num (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

(+) :: Mul a -> Mul a -> Mul a #

(-) :: Mul a -> Mul a -> Mul a #

(*) :: Mul a -> Mul a -> Mul a #

negate :: Mul a -> Mul a #

abs :: Mul a -> Mul a #

signum :: Mul a -> Mul a #

fromInteger :: Integer -> Mul a #

Ord a => Ord (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

compare :: Mul a -> Mul a -> Ordering #

(<) :: Mul a -> Mul a -> Bool #

(<=) :: Mul a -> Mul a -> Bool #

(>) :: Mul a -> Mul a -> Bool #

(>=) :: Mul a -> Mul a -> Bool #

max :: Mul a -> Mul a -> Mul a #

min :: Mul a -> Mul a -> Mul a #

Read a => Read (Mul a) Source # 
Instance details

Defined in Data.Semiring

Real a => Real (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

toRational :: Mul a -> Rational #

RealFrac a => RealFrac (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

properFraction :: Integral b => Mul a -> (b, Mul a) #

truncate :: Integral b => Mul a -> b #

round :: Integral b => Mul a -> b #

ceiling :: Integral b => Mul a -> b #

floor :: Integral b => Mul a -> b #

Show a => Show (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

showsPrec :: Int -> Mul a -> ShowS #

show :: Mul a -> String #

showList :: [Mul a] -> ShowS #

Generic (Mul a) Source # 
Instance details

Defined in Data.Semiring

Associated Types

type Rep (Mul a) :: * -> * #

Methods

from :: Mul a -> Rep (Mul a) x #

to :: Rep (Mul a) x -> Mul a #

Semiring a => Semigroup (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

(<>) :: Mul a -> Mul a -> Mul a #

sconcat :: NonEmpty (Mul a) -> Mul a #

stimes :: Integral b => b -> Mul a -> Mul a #

Semiring a => Monoid (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

mempty :: Mul a #

mappend :: Mul a -> Mul a -> Mul a #

mconcat :: [Mul a] -> Mul a #

Storable a => Storable (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

sizeOf :: Mul a -> Int #

alignment :: Mul a -> Int #

peekElemOff :: Ptr (Mul a) -> Int -> IO (Mul a) #

pokeElemOff :: Ptr (Mul a) -> Int -> Mul a -> IO () #

peekByteOff :: Ptr b -> Int -> IO (Mul a) #

pokeByteOff :: Ptr b -> Int -> Mul a -> IO () #

peek :: Ptr (Mul a) -> IO (Mul a) #

poke :: Ptr (Mul a) -> Mul a -> IO () #

Semiring a => Semiring (Mul a) Source # 
Instance details

Defined in Data.Semiring

Methods

plus :: Mul a -> Mul a -> Mul a Source #

zero :: Mul a Source #

times :: Mul a -> Mul a -> Mul a Source #

one :: Mul a Source #

Generic1 Mul Source # 
Instance details

Defined in Data.Semiring

Associated Types

type Rep1 Mul :: k -> * #

Methods

from1 :: Mul a -> Rep1 Mul a #

to1 :: Rep1 Mul a -> Mul a #

type Rep (Mul a) Source # 
Instance details

Defined in Data.Semiring

type Rep (Mul a) = D1 (MetaData "Mul" "Data.Semiring" "semirings-0.2.1.0-He40r1WdcGVKOm5tgEashg" True) (C1 (MetaCons "Mul" PrefixI True) (S1 (MetaSel (Just "getMul") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)))
type Rep1 Mul Source # 
Instance details

Defined in Data.Semiring

type Rep1 Mul = D1 (MetaData "Mul" "Data.Semiring" "semirings-0.2.1.0-He40r1WdcGVKOm5tgEashg" True) (C1 (MetaCons "Mul" PrefixI True) (S1 (MetaSel (Just "getMul") NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1))

Ring typeclass

class Semiring a => Ring a where Source #

The class of semirings with an additive inverse.

negate a + a = zero

Minimal complete definition

negate

Methods

negate :: a -> a Source #

Instances
Ring Bool Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Bool -> Bool Source #

Ring Double Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Double -> Double Source #

Ring Float Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Float -> Float Source #

Ring Int Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Int -> Int Source #

Ring Int8 Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Int8 -> Int8 Source #

Ring Int16 Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Int16 -> Int16 Source #

Ring Int32 Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Int32 -> Int32 Source #

Ring Int64 Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Int64 -> Int64 Source #

Ring Integer Source # 
Instance details

Defined in Data.Semiring

Ring Natural Source # 
Instance details

Defined in Data.Semiring

Ring Word Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Word -> Word Source #

Ring Word8 Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Word8 -> Word8 Source #

Ring Word16 Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Word16 -> Word16 Source #

Ring Word32 Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Word32 -> Word32 Source #

Ring Word64 Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Word64 -> Word64 Source #

Ring () Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: () -> () Source #

Ring CDev Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CDev -> CDev Source #

Ring CIno Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CIno -> CIno Source #

Ring CMode Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CMode -> CMode Source #

Ring COff Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: COff -> COff Source #

Ring CPid Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CPid -> CPid Source #

Ring CSsize Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CSsize -> CSsize Source #

Ring CGid Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CGid -> CGid Source #

Ring CNlink Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CNlink -> CNlink Source #

Ring CUid Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CUid -> CUid Source #

Ring CCc Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CCc -> CCc Source #

Ring CSpeed Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CSpeed -> CSpeed Source #

Ring CTcflag Source # 
Instance details

Defined in Data.Semiring

Ring CRLim Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CRLim -> CRLim Source #

Ring Fd Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Fd -> Fd Source #

Ring CChar Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CChar -> CChar Source #

Ring CSChar Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CSChar -> CSChar Source #

Ring CUChar Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CUChar -> CUChar Source #

Ring CShort Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CShort -> CShort Source #

Ring CUShort Source # 
Instance details

Defined in Data.Semiring

Ring CInt Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CInt -> CInt Source #

Ring CUInt Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CUInt -> CUInt Source #

Ring CLong Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CLong -> CLong Source #

Ring CULong Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CULong -> CULong Source #

Ring CLLong Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CLLong -> CLLong Source #

Ring CULLong Source # 
Instance details

Defined in Data.Semiring

Ring CFloat Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CFloat -> CFloat Source #

Ring CDouble Source # 
Instance details

Defined in Data.Semiring

Ring CPtrdiff Source # 
Instance details

Defined in Data.Semiring

Ring CSize Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CSize -> CSize Source #

Ring CWchar Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CWchar -> CWchar Source #

Ring CSigAtomic Source # 
Instance details

Defined in Data.Semiring

Ring CClock Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CClock -> CClock Source #

Ring CTime Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: CTime -> CTime Source #

Ring CUSeconds Source # 
Instance details

Defined in Data.Semiring

Ring CSUSeconds Source # 
Instance details

Defined in Data.Semiring

Ring CIntPtr Source # 
Instance details

Defined in Data.Semiring

Ring CUIntPtr Source # 
Instance details

Defined in Data.Semiring

Ring CIntMax Source # 
Instance details

Defined in Data.Semiring

Ring CUIntMax Source # 
Instance details

Defined in Data.Semiring

Ring WordPtr Source # 
Instance details

Defined in Data.Semiring

Ring IntPtr Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: IntPtr -> IntPtr Source #

Ring a => Ring [a] Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: [a] -> [a] Source #

Ring a => Ring (Maybe a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Maybe a -> Maybe a Source #

Integral a => Ring (Ratio a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Ratio a -> Ratio a Source #

Ring a => Ring (IO a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: IO a -> IO a Source #

Ring a => Ring (Complex a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Complex a -> Complex a Source #

HasResolution a => Ring (Fixed a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Fixed a -> Fixed a Source #

Ring a => Ring (Min a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Min a -> Min a Source #

Ring a => Ring (Max a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Max a -> Max a Source #

Ring a => Ring (Identity a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Identity a -> Identity a Source #

Ring a => Ring (Dual a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Dual a -> Dual a Source #

Ring a => Ring (Sum a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Sum a -> Sum a Source #

Ring a => Ring (Product a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Product a -> Product a Source #

Ring a => Ring (Down a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Down a -> Down a Source #

(Unbox a, Ring a) => Ring (Vector a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Vector a -> Vector a Source #

(Storable a, Ring a) => Ring (Vector a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Vector a -> Vector a Source #

Ring a => Ring (Vector a) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Vector a -> Vector a Source #

Ring b => Ring (a -> b) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: (a -> b) -> a -> b Source #

(Ring a, Ring b) => Ring (a, b) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

negate :: (a, b) -> (a, b) Source #

(Ring a, Ring b, Ring c) => Ring (a, b, c) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

negate :: (a, b, c) -> (a, b, c) Source #

Ring a => Ring (Const a b) Source # 
Instance details

Defined in Data.Semiring

Methods

negate :: Const a b -> Const a b Source #

(Ring a, Ring b, Ring c, Ring d) => Ring (a, b, c, d) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

negate :: (a, b, c, d) -> (a, b, c, d) Source #

(Ring a, Ring b, Ring c, Ring d, Ring e) => Ring (a, b, c, d, e) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

negate :: (a, b, c, d, e) -> (a, b, c, d, e) Source #

(Ring a, Ring b, Ring c, Ring d, Ring e, Ring f) => Ring (a, b, c, d, e, f) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

negate :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) Source #

(Ring a, Ring b, Ring c, Ring d, Ring e, Ring f, Ring g) => Ring (a, b, c, d, e, f, g) Source # 
Instance details

Defined in Data.Semiring.Generic

Methods

negate :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) Source #

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

Infix shorthand for minus.

minus :: Ring a => a -> a -> a infixl 6 Source #

Substract two Ring values. For any type R with a Num instance, this is the same as '(Prelude.-)'.

x minus y = x + negate y