Control.Applicative

This module describes a structure intermediate between a functor and a monad (technically, a strong lax monoidal functor). Compared with monads, this interface lacks the full power of the binding operation >>=, but

• it has more instances.
• it is sufficient for many uses, e.g. context-free parsing, or the Traversable class.
• instances can perform analysis of computations before they are executed, and thus produce shared optimizations.

This interface was introduced for parsers by Niklas Röjemo, because it admits more sharing than the monadic interface. The names here are mostly based on parsing work by Doaitse Swierstra.

For more details, see Applicative Programming with Effects, by Conor McBride and Ross Paterson.

Applicative functors

class Functor f => Applicative f where

A functor with application, providing operations to

• embed pure expressions (pure), and
• sequence computations and combine their results (<*>).

A minimal complete definition must include implementations of these functions satisfying the following laws:

identity

pure id <*> v = v

composition

pure (.) <*> u <*> v <*> w = u <*> (v <*> w)

homomorphism

pure f <*> pure x = pure (f x)

interchange

u <*> pure y = pure ($ y) <*> u

The other methods have the following default definitions, which may be overridden with equivalent specialized implementations:

• u *> v = pure (const id) <*> u <*> v

• u <* v = pure const <*> u <*> v

As a consequence of these laws, the Functor instance for f will satisfy

• fmap f x = pure f <*> x

If f is also a Monad, it should satisfy

• pure = return

• (<*>) = ap

(which implies that pure and <*> satisfy the applicative functor laws).

Minimal complete definition

pure, (<*>)

Methods

pure :: a -> f a

Lift a value.

(<*>) :: f (a -> b) -> f a -> f b infixl 4

Sequential application.

(*>) :: f a -> f b -> f b infixl 4

Sequence actions, discarding the value of the first argument.

(<*) :: f a -> f b -> f a infixl 4

Sequence actions, discarding the value of the second argument.

Instances

Applicative []

Applicative IO

Applicative Maybe

Applicative ReadP

Applicative ReadPrec

Applicative Last

Applicative First

Applicative STM

Applicative ZipList

Applicative Identity

Applicative ((->) a)

Applicative (Either e)

Monoid a => Applicative ((,) a)

Applicative (ST s)

Applicative (Proxy *)

Arrow a => Applicative (ArrowMonad a)

Monad m => Applicative (WrappedMonad m)

Monoid m => Applicative (Const m)

Applicative (ST s)

Applicative f => Applicative (Alt * f)

Arrow a => Applicative (WrappedArrow a b)

Alternatives

class Applicative f => Alternative f where

A monoid on applicative functors.

If defined, some and many should be the least solutions of the equations:

• some v = (:) <$> v <*> many v

• many v = some v <|> pure []

Minimal complete definition

empty, (<|>)

Methods

empty :: f a

The identity of <|>

(<|>) :: f a -> f a -> f a infixl 3

An associative binary operation

some :: f a -> f [a]

One or more.

many :: f a -> f [a]

Zero or more.

Instances

Alternative []

Alternative Maybe

Alternative ReadP

Alternative ReadPrec

Alternative STM

ArrowPlus a => Alternative (ArrowMonad a)

MonadPlus m => Alternative (WrappedMonad m)

Alternative f => Alternative (Alt * f)

(ArrowZero a, ArrowPlus a) => Alternative (WrappedArrow a b)

Instances

newtype Const a b

Constructors

Const

Fields

getConst :: a

Instances

Bifunctor Const

Functor (Const m)

Monoid m => Applicative (Const m)

Foldable (Const m)

Traversable (Const m)

Generic1 (Const a)

Eq a => Eq (Const a b)

Ord a => Ord (Const a b)

Read a => Read (Const a b)

Show a => Show (Const a b)

Generic (Const a b)

Monoid a => Monoid (Const a b)

type Rep1 (Const a)

type Rep (Const a b)

newtype WrappedMonad m a

Constructors

WrapMonad

Fields

unwrapMonad :: m a

Instances

Monad m => Monad (WrappedMonad m)

Monad m => Functor (WrappedMonad m)

Monad m => Applicative (WrappedMonad m)

Generic1 (WrappedMonad m)

MonadPlus m => Alternative (WrappedMonad m)

Generic (WrappedMonad m a)

type Rep1 (WrappedMonad m)

type Rep (WrappedMonad m a)

newtype WrappedArrow a b c

Constructors

WrapArrow

Fields

unwrapArrow :: a b c

Instances

Arrow a => Functor (WrappedArrow a b)

Arrow a => Applicative (WrappedArrow a b)

Generic1 (WrappedArrow a b)

(ArrowZero a, ArrowPlus a) => Alternative (WrappedArrow a b)

Generic (WrappedArrow a b c)

type Rep1 (WrappedArrow a b)

type Rep (WrappedArrow a b c)

newtype ZipList a

Lists, but with an Applicative functor based on zipping, so that

f <$> ZipList xs1 <*> ... <*> ZipList xsn = ZipList (zipWithn f xs1 ... xsn)

Constructors

ZipList

Fields

getZipList :: [a]

Instances

Functor ZipList

Applicative ZipList

Generic1 ZipList

Eq a => Eq (ZipList a)

Ord a => Ord (ZipList a)

Read a => Read (ZipList a)

Show a => Show (ZipList a)

Generic (ZipList a)

type Rep1 ZipList

type Rep (ZipList a)

Utility functions

(<$>) :: Functor f => (a -> b) -> f a -> f b infixl 4

An infix synonym for fmap.

Examples

Convert from a Maybe Int to a Maybe String using show:

>>> show <$> Nothing
Nothing
>>> show <$> Just 3
Just "3"

Convert from an Either Int Int to an Either Int String using show:

>>> show <$> Left 17
Left 17
>>> show <$> Right 17
Right "17"

Double each element of a list:

>>> (*2) <$> [1,2,3]
[2,4,6]

Apply even to the second element of a pair:

>>> even <$> (2,2)
(2,True)

(<$) :: Functor f => a -> f b -> f a

Replace all locations in the input with the same value. The default definition is fmap . const, but this may be overridden with a more efficient version.

(<**>) :: Applicative f => f a -> f (a -> b) -> f b infixl 4

A variant of <*> with the arguments reversed.

liftA :: Applicative f => (a -> b) -> f a -> f b

Lift a function to actions. This function may be used as a value for fmap in a Functor instance.

liftA2 :: Applicative f => (a -> b -> c) -> f a -> f b -> f c

Lift a binary function to actions.

liftA3 :: Applicative f => (a -> b -> c -> d) -> f a -> f b -> f c -> f d

Lift a ternary function to actions.

optional :: Alternative f => f a -> f (Maybe a)

One or none.