Data.Monoid

A class for monoids (types with an associative binary operation that has an identity) with various general-purpose instances.

Monoid typeclass

class Monoid a where

The class of monoids (types with an associative binary operation that has an identity). Instances should satisfy the following laws:

• mappend mempty x = x

• mappend x mempty = x

• mappend x (mappend y z) = mappend (mappend x y) z

• mconcat = foldr mappend mempty

The method names refer to the monoid of lists under concatenation, but there are many other instances.

Some types can be viewed as a monoid in more than one way, e.g. both addition and multiplication on numbers. In such cases we often define newtypes and make those instances of Monoid, e.g. Sum and Product.

Minimal complete definition

mempty, mappend

Methods

mempty :: a

Identity of mappend

mappend :: a -> a -> a

An associative operation

mconcat :: [a] -> a

Fold a list using the monoid. For most types, the default definition for mconcat will be used, but the function is included in the class definition so that an optimized version can be provided for specific types.

Instances

Monoid Ordering

Methods

mempty :: Ordering

mappend :: Ordering -> Ordering -> Ordering

mconcat :: [Ordering] -> Ordering

Monoid ()

Methods

mempty :: ()

mappend :: () -> () -> ()

mconcat :: [()] -> ()

Monoid Any

Methods

mempty :: Any

mappend :: Any -> Any -> Any

mconcat :: [Any] -> Any

Monoid All

Methods

mempty :: All

mappend :: All -> All -> All

mconcat :: [All] -> All

Monoid Lifetime

mappend == elSupremum

Methods

mempty :: Lifetime

mappend :: Lifetime -> Lifetime -> Lifetime

mconcat :: [Lifetime] -> Lifetime

Monoid Event

Methods

mempty :: Event

mappend :: Event -> Event -> Event

mconcat :: [Event] -> Event

Monoid [a]

Methods

mempty :: [a]

mappend :: [a] -> [a] -> [a]

mconcat :: [[a]] -> [a]

Monoid a => Monoid (Maybe a)

Lift a semigroup into Maybe forming a Monoid according to http://en.wikipedia.org/wiki/Monoid: "Any semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining e*e = e and e*s = s = s*e for all s ∈ S." Since there is no "Semigroup" typeclass providing just mappend, we use Monoid instead.

Methods

mempty :: Maybe a

mappend :: Maybe a -> Maybe a -> Maybe a

mconcat :: [Maybe a] -> Maybe a

Monoid (Last a)

Methods

mempty :: Last a

mappend :: Last a -> Last a -> Last a

mconcat :: [Last a] -> Last a

Monoid (First a)

Methods

mempty :: First a

mappend :: First a -> First a -> First a

mconcat :: [First a] -> First a

Num a => Monoid (Product a)

Methods

mempty :: Product a

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

mconcat :: [Product a] -> Product a

Num a => Monoid (Sum a)

Methods

mempty :: Sum a

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

mconcat :: [Sum a] -> Sum a

Monoid (Endo a)

Methods

mempty :: Endo a

mappend :: Endo a -> Endo a -> Endo a

mconcat :: [Endo a] -> Endo a

Monoid a => Monoid (Dual a)

Methods

mempty :: Dual a

mappend :: Dual a -> Dual a -> Dual a

mconcat :: [Dual a] -> Dual a

Monoid b => Monoid (a -> b)

Methods

mempty :: a -> b

mappend :: (a -> b) -> (a -> b) -> a -> b

mconcat :: [a -> b] -> a -> b

(Monoid a, Monoid b) => Monoid (a, b)

Methods

mempty :: (a, b)

mappend :: (a, b) -> (a, b) -> (a, b)

mconcat :: [(a, b)] -> (a, b)

Monoid (Proxy k s)

Methods

mempty :: Proxy k s

mappend :: Proxy k s -> Proxy k s -> Proxy k s

mconcat :: [Proxy k s] -> Proxy k s

Monoid a => Monoid (Const a b)

Methods

mempty :: Const a b

mappend :: Const a b -> Const a b -> Const a b

mconcat :: [Const a b] -> Const a b

(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c)

Methods

mempty :: (a, b, c)

mappend :: (a, b, c) -> (a, b, c) -> (a, b, c)

mconcat :: [(a, b, c)] -> (a, b, c)

Alternative f => Monoid (Alt * f a)

Methods

mempty :: Alt * f a

mappend :: Alt * f a -> Alt * f a -> Alt * f a

mconcat :: [Alt * f a] -> Alt * f a

(Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d)

Methods

mempty :: (a, b, c, d)

mappend :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d)

mconcat :: [(a, b, c, d)] -> (a, b, c, d)

(Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e)

Methods

mempty :: (a, b, c, d, e)

mappend :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e)

mconcat :: [(a, b, c, d, e)] -> (a, b, c, d, e)

(<>) :: Monoid m => m -> m -> m infixr 6

An infix synonym for mappend.

Since: 4.5.0.0

newtype Dual a

The dual of a Monoid, obtained by swapping the arguments of mappend.

Constructors

Dual

Fields

• getDual :: a

Instances

Generic1 Dual

Associated Types

type Rep1 (Dual :: * -> *) :: * -> *

Methods

from1 :: Dual a -> Rep1 Dual a

to1 :: Rep1 Dual a -> Dual a

Bounded a => Bounded (Dual a)

Methods

minBound :: Dual a

maxBound :: Dual a

Eq a => Eq (Dual a)

Methods

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

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

Ord a => Ord (Dual a)

Methods

compare :: Dual a -> Dual a -> Ordering

(<) :: Dual a -> Dual a -> Bool

(<=) :: Dual a -> Dual a -> Bool

(>) :: Dual a -> Dual a -> Bool

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

max :: Dual a -> Dual a -> Dual a

min :: Dual a -> Dual a -> Dual a

Read a => Read (Dual a)

Methods

readsPrec :: Int -> ReadS (Dual a)

readList :: ReadS [Dual a]

readPrec :: ReadPrec (Dual a)

readListPrec :: ReadPrec [Dual a]

Show a => Show (Dual a)

Methods

showsPrec :: Int -> Dual a -> ShowS

show :: Dual a -> String

showList :: [Dual a] -> ShowS

Generic (Dual a)

Associated Types

type Rep (Dual a) :: * -> *

Methods

from :: Dual a -> Rep (Dual a) x

to :: Rep (Dual a) x -> Dual a

Monoid a => Monoid (Dual a)

Methods

mempty :: Dual a

mappend :: Dual a -> Dual a -> Dual a

mconcat :: [Dual a] -> Dual a

type Rep1 Dual

type Rep (Dual a)

newtype Endo a

The monoid of endomorphisms under composition.

Constructors

Endo

Fields

• appEndo :: a -> a

Instances

Generic (Endo a)

Associated Types

type Rep (Endo a) :: * -> *

Methods

from :: Endo a -> Rep (Endo a) x

to :: Rep (Endo a) x -> Endo a

Monoid (Endo a)

Methods

mempty :: Endo a

mappend :: Endo a -> Endo a -> Endo a

mconcat :: [Endo a] -> Endo a

type Rep (Endo a)

Bool wrappers

newtype All

Boolean monoid under conjunction (&&).

Constructors

All

Fields

• getAll :: Bool

Instances

Bounded All

Methods

minBound :: All

maxBound :: All

Eq All

Methods

(==) :: All -> All -> Bool

(/=) :: All -> All -> Bool

Ord All

Methods

compare :: All -> All -> Ordering

(<) :: All -> All -> Bool

(<=) :: All -> All -> Bool

(>) :: All -> All -> Bool

(>=) :: All -> All -> Bool

max :: All -> All -> All

min :: All -> All -> All

Read All

Methods

readsPrec :: Int -> ReadS All

readList :: ReadS [All]

readPrec :: ReadPrec All

readListPrec :: ReadPrec [All]

Show All

Methods

showsPrec :: Int -> All -> ShowS

show :: All -> String

showList :: [All] -> ShowS

Generic All

Associated Types

type Rep All :: * -> *

Methods

from :: All -> Rep All x

to :: Rep All x -> All

Monoid All

Methods

mempty :: All

mappend :: All -> All -> All

mconcat :: [All] -> All

type Rep All

newtype Any

Boolean monoid under disjunction (||).

Constructors

Any

Fields

• getAny :: Bool

Instances

Bounded Any

Methods

minBound :: Any

maxBound :: Any

Eq Any

Methods

(==) :: Any -> Any -> Bool

(/=) :: Any -> Any -> Bool

Ord Any

Methods

compare :: Any -> Any -> Ordering

(<) :: Any -> Any -> Bool

(<=) :: Any -> Any -> Bool

(>) :: Any -> Any -> Bool

(>=) :: Any -> Any -> Bool

max :: Any -> Any -> Any

min :: Any -> Any -> Any

Read Any

Methods

readsPrec :: Int -> ReadS Any

readList :: ReadS [Any]

readPrec :: ReadPrec Any

readListPrec :: ReadPrec [Any]

Show Any

Methods

showsPrec :: Int -> Any -> ShowS

show :: Any -> String

showList :: [Any] -> ShowS

Generic Any

Associated Types

type Rep Any :: * -> *

Methods

from :: Any -> Rep Any x

to :: Rep Any x -> Any

Monoid Any

Methods

mempty :: Any

mappend :: Any -> Any -> Any

mconcat :: [Any] -> Any

type Rep Any

Num wrappers

newtype Sum a

Monoid under addition.

Constructors

Sum

Fields

• getSum :: a

Instances

Generic1 Sum

Associated Types

type Rep1 (Sum :: * -> *) :: * -> *

Methods

from1 :: Sum a -> Rep1 Sum a

to1 :: Rep1 Sum a -> Sum a

Bounded a => Bounded (Sum a)

Methods

minBound :: Sum a

maxBound :: Sum a

Eq a => Eq (Sum a)

Methods

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

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

Num a => Num (Sum a)

Methods

(+) :: Sum a -> Sum a -> Sum a

(-) :: Sum a -> Sum a -> Sum a

(*) :: Sum a -> Sum a -> Sum a

negate :: Sum a -> Sum a

abs :: Sum a -> Sum a

signum :: Sum a -> Sum a

fromInteger :: Integer -> Sum a

Ord a => Ord (Sum a)

Methods

compare :: Sum a -> Sum a -> Ordering

(<) :: Sum a -> Sum a -> Bool

(<=) :: Sum a -> Sum a -> Bool

(>) :: Sum a -> Sum a -> Bool

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

max :: Sum a -> Sum a -> Sum a

min :: Sum a -> Sum a -> Sum a

Read a => Read (Sum a)

Methods

readsPrec :: Int -> ReadS (Sum a)

readList :: ReadS [Sum a]

readPrec :: ReadPrec (Sum a)

readListPrec :: ReadPrec [Sum a]

Show a => Show (Sum a)

Methods

showsPrec :: Int -> Sum a -> ShowS

show :: Sum a -> String

showList :: [Sum a] -> ShowS

Generic (Sum a)

Associated Types

type Rep (Sum a) :: * -> *

Methods

from :: Sum a -> Rep (Sum a) x

to :: Rep (Sum a) x -> Sum a

Num a => Monoid (Sum a)

Methods

mempty :: Sum a

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

mconcat :: [Sum a] -> Sum a

type Rep1 Sum

type Rep (Sum a)

newtype Product a

Monoid under multiplication.

Constructors

Product

Fields

• getProduct :: a

Instances

Generic1 Product

Associated Types

type Rep1 (Product :: * -> *) :: * -> *

Methods

from1 :: Product a -> Rep1 Product a

to1 :: Rep1 Product a -> Product a

Bounded a => Bounded (Product a)

Methods

minBound :: Product a

maxBound :: Product a

Eq a => Eq (Product a)

Methods

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

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

Num a => Num (Product a)

Methods

(+) :: Product a -> Product a -> Product a

(-) :: Product a -> Product a -> Product a

(*) :: Product a -> Product a -> Product a

negate :: Product a -> Product a

abs :: Product a -> Product a

signum :: Product a -> Product a

fromInteger :: Integer -> Product a

Ord a => Ord (Product a)

Methods

compare :: Product a -> Product a -> Ordering

(<) :: Product a -> Product a -> Bool

(<=) :: Product a -> Product a -> Bool

(>) :: Product a -> Product a -> Bool

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

max :: Product a -> Product a -> Product a

min :: Product a -> Product a -> Product a

Read a => Read (Product a)

Methods

readsPrec :: Int -> ReadS (Product a)

readList :: ReadS [Product a]

readPrec :: ReadPrec (Product a)

readListPrec :: ReadPrec [Product a]

Show a => Show (Product a)

Methods

showsPrec :: Int -> Product a -> ShowS

show :: Product a -> String

showList :: [Product a] -> ShowS

Generic (Product a)

Associated Types

type Rep (Product a) :: * -> *

Methods

from :: Product a -> Rep (Product a) x

to :: Rep (Product a) x -> Product a

Num a => Monoid (Product a)

Methods

mempty :: Product a

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

mconcat :: [Product a] -> Product a

type Rep1 Product

type Rep (Product a)

Maybe wrappers

To implement find or findLast on any Foldable:

findLast :: Foldable t => (a -> Bool) -> t a -> Maybe a
findLast pred = getLast . foldMap (x -> if pred x
                                           then Last (Just x)
                                           else Last Nothing)

Much of Data.Map's interface can be implemented with Data.Map.alter. Some of the rest can be implemented with a new alterA function and either First or Last:

alterA :: (Applicative f, Ord k) =>
          (Maybe a -> f (Maybe a)) -> k -> Map k a -> f (Map k a)

instance Monoid a => Applicative ((,) a)  -- from Control.Applicative

insertLookupWithKey :: Ord k => (k -> v -> v -> v) -> k -> v
                    -> Map k v -> (Maybe v, Map k v)
insertLookupWithKey combine key value =
  Arrow.first getFirst . alterA doChange key
  where
  doChange Nothing = (First Nothing, Just value)
  doChange (Just oldValue) =
    (First (Just oldValue),
     Just (combine key value oldValue))

newtype First a

Maybe monoid returning the leftmost non-Nothing value.

First a is isomorphic to Alt Maybe a, but precedes it historically.

Constructors

First

Fields

• getFirst :: Maybe a

Instances

Monad First

Methods

(>>=) :: First a -> (a -> First b) -> First b

(>>) :: First a -> First b -> First b

return :: a -> First a

fail :: String -> First a

Functor First

Methods

fmap :: (a -> b) -> First a -> First b

(<$) :: a -> First b -> First a

Applicative First

Methods

pure :: a -> First a

(<*>) :: First (a -> b) -> First a -> First b

(*>) :: First a -> First b -> First b

(<*) :: First a -> First b -> First a

Generic1 First

Associated Types

type Rep1 (First :: * -> *) :: * -> *

Methods

from1 :: First a -> Rep1 First a

to1 :: Rep1 First a -> First a

Eq a => Eq (First a)

Methods

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

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

Ord a => Ord (First a)

Methods

compare :: First a -> First a -> Ordering

(<) :: First a -> First a -> Bool

(<=) :: First a -> First a -> Bool

(>) :: First a -> First a -> Bool

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

max :: First a -> First a -> First a

min :: First a -> First a -> First a

Read a => Read (First a)

Methods

readsPrec :: Int -> ReadS (First a)

readList :: ReadS [First a]

readPrec :: ReadPrec (First a)

readListPrec :: ReadPrec [First a]

Show a => Show (First a)

Methods

showsPrec :: Int -> First a -> ShowS

show :: First a -> String

showList :: [First a] -> ShowS

Generic (First a)

Associated Types

type Rep (First a) :: * -> *

Methods

from :: First a -> Rep (First a) x

to :: Rep (First a) x -> First a

Monoid (First a)

Methods

mempty :: First a

mappend :: First a -> First a -> First a

mconcat :: [First a] -> First a

type Rep1 First

type Rep (First a)

newtype Last a

Maybe monoid returning the rightmost non-Nothing value.

Last a is isomorphic to Dual (First a), and thus to Dual (Alt Maybe a)

Constructors

Last

Fields

• getLast :: Maybe a

Instances

Monad Last

Methods

(>>=) :: Last a -> (a -> Last b) -> Last b

(>>) :: Last a -> Last b -> Last b

return :: a -> Last a

fail :: String -> Last a

Functor Last

Methods

fmap :: (a -> b) -> Last a -> Last b

(<$) :: a -> Last b -> Last a

Applicative Last

Methods

pure :: a -> Last a

(<*>) :: Last (a -> b) -> Last a -> Last b

(*>) :: Last a -> Last b -> Last b

(<*) :: Last a -> Last b -> Last a

Generic1 Last

Associated Types

type Rep1 (Last :: * -> *) :: * -> *

Methods

from1 :: Last a -> Rep1 Last a

to1 :: Rep1 Last a -> Last a

Eq a => Eq (Last a)

Methods

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

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

Ord a => Ord (Last a)

Methods

compare :: Last a -> Last a -> Ordering

(<) :: Last a -> Last a -> Bool

(<=) :: Last a -> Last a -> Bool

(>) :: Last a -> Last a -> Bool

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

max :: Last a -> Last a -> Last a

min :: Last a -> Last a -> Last a

Read a => Read (Last a)

Methods

readsPrec :: Int -> ReadS (Last a)

readList :: ReadS [Last a]

readPrec :: ReadPrec (Last a)

readListPrec :: ReadPrec [Last a]

Show a => Show (Last a)

Methods

showsPrec :: Int -> Last a -> ShowS

show :: Last a -> String

showList :: [Last a] -> ShowS

Generic (Last a)

Associated Types

type Rep (Last a) :: * -> *

Methods

from :: Last a -> Rep (Last a) x

to :: Rep (Last a) x -> Last a

Monoid (Last a)

Methods

mempty :: Last a

mappend :: Last a -> Last a -> Last a

mconcat :: [Last a] -> Last a

type Rep1 Last

type Rep (Last a)

Alternative wrapper

newtype Alt f a

Monoid under <|>.

Since: 4.8.0.0

Constructors

Alt

Fields

• getAlt :: f a

Instances

Monad f => Monad (Alt * f)

Methods

(>>=) :: Alt * f a -> (a -> Alt * f b) -> Alt * f b

(>>) :: Alt * f a -> Alt * f b -> Alt * f b

return :: a -> Alt * f a

fail :: String -> Alt * f a

Functor f => Functor (Alt * f)

Methods

fmap :: (a -> b) -> Alt * f a -> Alt * f b

(<$) :: a -> Alt * f b -> Alt * f a

Applicative f => Applicative (Alt * f)

Methods

pure :: a -> Alt * f a

(<*>) :: Alt * f (a -> b) -> Alt * f a -> Alt * f b

(*>) :: Alt * f a -> Alt * f b -> Alt * f b

(<*) :: Alt * f a -> Alt * f b -> Alt * f a

Generic1 (Alt * f)

Associated Types

type Rep1 (Alt * f :: * -> *) :: * -> *

Methods

from1 :: Alt * f a -> Rep1 (Alt * f) a

to1 :: Rep1 (Alt * f) a -> Alt * f a

MonadPlus f => MonadPlus (Alt * f)

Methods

mzero :: Alt * f a

mplus :: Alt * f a -> Alt * f a -> Alt * f a

Alternative f => Alternative (Alt * f)

Methods

empty :: Alt * f a

(<|>) :: Alt * f a -> Alt * f a -> Alt * f a

some :: Alt * f a -> Alt * f [a]

many :: Alt * f a -> Alt * f [a]

Enum (f a) => Enum (Alt k f a)

Methods

succ :: Alt k f a -> Alt k f a

pred :: Alt k f a -> Alt k f a

toEnum :: Int -> Alt k f a

fromEnum :: Alt k f a -> Int

enumFrom :: Alt k f a -> [Alt k f a]

enumFromThen :: Alt k f a -> Alt k f a -> [Alt k f a]

enumFromTo :: Alt k f a -> Alt k f a -> [Alt k f a]

enumFromThenTo :: Alt k f a -> Alt k f a -> Alt k f a -> [Alt k f a]

Eq (f a) => Eq (Alt k f a)

Methods

(==) :: Alt k f a -> Alt k f a -> Bool

(/=) :: Alt k f a -> Alt k f a -> Bool

Num (f a) => Num (Alt k f a)

Methods

(+) :: Alt k f a -> Alt k f a -> Alt k f a

(-) :: Alt k f a -> Alt k f a -> Alt k f a

(*) :: Alt k f a -> Alt k f a -> Alt k f a

negate :: Alt k f a -> Alt k f a

abs :: Alt k f a -> Alt k f a

signum :: Alt k f a -> Alt k f a

fromInteger :: Integer -> Alt k f a

Ord (f a) => Ord (Alt k f a)

Methods

compare :: Alt k f a -> Alt k f a -> Ordering

(<) :: Alt k f a -> Alt k f a -> Bool

(<=) :: Alt k f a -> Alt k f a -> Bool

(>) :: Alt k f a -> Alt k f a -> Bool

(>=) :: Alt k f a -> Alt k f a -> Bool

max :: Alt k f a -> Alt k f a -> Alt k f a

min :: Alt k f a -> Alt k f a -> Alt k f a

Read (f a) => Read (Alt k f a)

Methods

readsPrec :: Int -> ReadS (Alt k f a)

readList :: ReadS [Alt k f a]

readPrec :: ReadPrec (Alt k f a)

readListPrec :: ReadPrec [Alt k f a]

Show (f a) => Show (Alt k f a)

Methods

showsPrec :: Int -> Alt k f a -> ShowS

show :: Alt k f a -> String

showList :: [Alt k f a] -> ShowS

Generic (Alt k f a)

Associated Types

type Rep (Alt k f a) :: * -> *

Methods

from :: Alt k f a -> Rep (Alt k f a) x

to :: Rep (Alt k f a) x -> Alt k f a

Alternative f => Monoid (Alt * f a)

Methods

mempty :: Alt * f a

mappend :: Alt * f a -> Alt * f a -> Alt * f a

mconcat :: [Alt * f a] -> Alt * f a

type Rep1 (Alt k f)

type Rep (Alt k f a)