base-4.8.2.0: Basic libraries

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 () mappend == elSupremum Monoid [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. Monoid (Last a) Monoid (First a) Num a => Monoid (Product a) Num a => Monoid (Sum a) Monoid (Endo a) Monoid a => Monoid (Dual a) Monoid b => Monoid (a -> b) (Monoid a, Monoid b) => Monoid (a, b) Monoid (Proxy k s) Monoid a => Monoid (Const a b) (Monoid a, Monoid b, Monoid c) => Monoid (a, b, c) Alternative f => Monoid (Alt * f a) (Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d) (Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e)

(<>) :: 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

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

newtype Endo a Source

The monoid of endomorphisms under composition.

Constructors

 Endo FieldsappEndo :: a -> a

Instances

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

Bool wrappers

newtype All Source

Boolean monoid under conjunction (&&).

Constructors

 All FieldsgetAll :: Bool

Instances

 type Rep All

newtype Any Source

Boolean monoid under disjunction (||).

Constructors

 Any FieldsgetAny :: Bool

Instances

 type Rep Any

Num wrappers

newtype Sum a Source

Constructors

 Sum FieldsgetSum :: a

Instances

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

newtype Product a Source

Monoid under multiplication.

Constructors

 Product FieldsgetProduct :: a

Instances

 Bounded a => Bounded (Product a) Eq a => Eq (Product a) Num a => Num (Product a) Ord a => Ord (Product a) Read a => Read (Product a) Show a => Show (Product a) Num a => Monoid (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 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

 Eq a => Eq (First a) Ord a => Ord (First a) Read a => Read (First a) Show a => Show (First a) Generic (First a) Monoid (First a) type Rep1 First type Rep (First a)

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

 Eq a => Eq (Last a) Ord a => Ord (Last a) Read a => Read (Last a) Show a => Show (Last a) Generic (Last a) Monoid (Last a) type Rep1 Last type Rep (Last a)

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