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

Data.Monoid

Contents

Monoid typeclass
Bool wrappers
Num wrappers
Maybe wrappers
Alternative wrapper

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

mempty, mappend

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 `ee = e` and `es = 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
Fields getDual :: 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
Fields appEndo :: 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
Fields getAll :: 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
Fields getAny :: 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

Monoid under addition.

Constructors

Sum
Fields getSum :: 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
Fields getProduct :: 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
Fields getFirst :: 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
Fields getLast :: 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
Fields getAlt :: 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

Monoid typeclass

Bool wrappers

Num wrappers

Maybe wrappers

Alternative wrapper

`Monoid` typeclass

`Bool` wrappers

`Num` wrappers

`Maybe` wrappers

`Alternative` wrapper