base-4.8.0.0: Basic libraries

Copyright (c) Andy Gill 2001, (c) Oregon Graduate Institute of Science and Technology, 2001 BSD-style (see the file libraries/base/LICENSE) libraries@haskell.org experimental portable Trustworthy Haskell2010

Data.Monoid

Description

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

Synopsis

Monoid typeclass

class Monoid a where Source

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

Methods

mempty :: a Source

Identity of mappend

mappend :: a -> a -> a Source

An associative operation

mconcat :: [a] -> a Source

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 Source Monoid () Source Monoid Any Source Monoid All Source Monoid Event Source Monoid [a] Source Monoid a => Monoid (Maybe a) Source 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. Monoid (Last a) Source Monoid (First a) Source Num a => Monoid (Product a) Source Num a => Monoid (Sum a) Source Monoid (Endo a) Source Monoid a => Monoid (Dual a) Source Monoid b => Monoid (a -> b) Source (Monoid a, Monoid b) => Monoid (a, b) Source Monoid (Proxy k s) Source Monoid a => Monoid (Const a b) Source (Monoid a, Monoid b, Monoid c) => Monoid (a, b, c) Source Alternative f => Monoid (Alt * f a) Source (Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d) Source (Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e) Source

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

An infix synonym for mappend.

Since: 4.5.0.0

newtype Dual a Source

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

Constructors

 Dual FieldsgetDual :: a

Instances

 Generic1 Dual Source Bounded a => Bounded (Dual a) Source Eq a => Eq (Dual a) Source Ord a => Ord (Dual a) Source Read a => Read (Dual a) Source Show a => Show (Dual a) Source Generic (Dual a) Source Monoid a => Monoid (Dual a) Source type Rep1 Dual Source type Rep (Dual a) Source

newtype Endo a Source

The monoid of endomorphisms under composition.

Constructors

 Endo FieldsappEndo :: a -> a

Instances

 Generic (Endo a) Source Monoid (Endo a) Source type Rep (Endo a) Source

Bool wrappers

newtype All Source

Boolean monoid under conjunction (&&).

Constructors

 All FieldsgetAll :: Bool

Instances

 Bounded All Source Eq All Source Ord All Source Read All Source Show All Source Generic All Source Monoid All Source type Rep All Source

newtype Any Source

Boolean monoid under disjunction (||).

Constructors

 Any FieldsgetAny :: Bool

Instances

 Bounded Any Source Eq Any Source Ord Any Source Read Any Source Show Any Source Generic Any Source Monoid Any Source type Rep Any Source

Num wrappers

newtype Sum a Source

Constructors

 Sum FieldsgetSum :: a

Instances

 Generic1 Sum Source Bounded a => Bounded (Sum a) Source Eq a => Eq (Sum a) Source Num a => Num (Sum a) Source Ord a => Ord (Sum a) Source Read a => Read (Sum a) Source Show a => Show (Sum a) Source Generic (Sum a) Source Num a => Monoid (Sum a) Source type Rep1 Sum Source type Rep (Sum a) Source

newtype Product a Source

Monoid under multiplication.

Constructors

 Product FieldsgetProduct :: a

Instances

 Generic1 Product Source Bounded a => Bounded (Product a) Source Eq a => Eq (Product a) Source Num a => Num (Product a) Source Ord a => Ord (Product a) Source Read a => Read (Product a) Source Show a => Show (Product a) Source Generic (Product a) Source Num a => Monoid (Product a) Source type Rep1 Product Source type Rep (Product a) Source

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 Source

Maybe monoid returning the leftmost non-Nothing value.

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

Constructors

 First FieldsgetFirst :: Maybe a

Instances

 Monad First Source Functor First Source Applicative First Source Generic1 First Source Eq a => Eq (First a) Source Ord a => Ord (First a) Source Read a => Read (First a) Source Show a => Show (First a) Source Generic (First a) Source Monoid (First a) Source type Rep1 First Source type Rep (First a) Source

newtype Last a Source

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 FieldsgetLast :: Maybe a

Instances

 Monad Last Source Functor Last Source Applicative Last Source Generic1 Last Source Eq a => Eq (Last a) Source Ord a => Ord (Last a) Source Read a => Read (Last a) Source Show a => Show (Last a) Source Generic (Last a) Source Monoid (Last a) Source type Rep1 Last Source type Rep (Last a) Source

Alternative wrapper

newtype Alt f a Source

Monoid under <|>.

Since: 4.8.0.0

Constructors

 Alt FieldsgetAlt :: f a

Instances

 Monad f => Monad (Alt * f) Source Functor f => Functor (Alt * f) Source Applicative f => Applicative (Alt * f) Source Generic1 (Alt * f) Source MonadPlus f => MonadPlus (Alt * f) Source Alternative f => Alternative (Alt * f) Source Enum (f a) => Enum (Alt k f a) Source Eq (f a) => Eq (Alt k f a) Source Num (f a) => Num (Alt k f a) Source Ord (f a) => Ord (Alt k f a) Source Read (f a) => Read (Alt k f a) Source Show (f a) => Show (Alt k f a) Source Generic (Alt k f a) Source Alternative f => Monoid (Alt * f a) Source type Rep1 (Alt k f) Source type Rep (Alt k f a) Source