base- Basic libraries

CopyrightConor McBride and Ross Paterson 2005
LicenseBSD-style (see the LICENSE file in the distribution)
Safe HaskellTrustworthy




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 Source

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:

pure id <*> v = v
pure (.) <*> u <*> v <*> w = u <*> (v <*> w)
pure f <*> pure x = pure (f x)
u <*> pure y = pure ($ y) <*> u

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

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

If f is also a Monad, it should satisfy

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

Minimal complete definition

pure, (<*>)


pure :: a -> f a Source

Lift a value.

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

Sequential application.

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

Sequence actions, discarding the value of the first argument.

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

Sequence actions, discarding the value of the second argument.


class Applicative f => Alternative f where Source

A monoid on applicative functors.

Minimal complete definition: empty and <|>.

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

Minimal complete definition

empty, (<|>)


empty :: f a Source

The identity of <|>

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

An associative binary operation

some :: f a -> f [a] Source

One or more.

many :: f a -> f [a] Source

Zero or more.


newtype Const a b Source




getConst :: a


Functor (Const m) 
Monoid m => Applicative (Const m) 
Foldable (Const m) 
Traversable (Const m) 
Generic1 (Const a) 
Generic (Const a b) 
Monoid a => Monoid (Const a b) 
type Rep1 (Const a) 
type Rep (Const a b) 

newtype WrappedMonad m a Source




unwrapMonad :: m a

newtype WrappedArrow a b c Source




unwrapArrow :: a b c


newtype ZipList a Source

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

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




getZipList :: [a]


Utility functions

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

An infix synonym for fmap.

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

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 Source

A variant of <*> with the arguments reversed.

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

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 Source

Lift a binary function to actions.

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

Lift a ternary function to actions.

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

One or none.