Control.Sized

A functor with application, providing operations to

• embed pure expressions (pure), and

• sequence computations and combine their results (<*> and liftA2).

A minimal complete definition must include implementations of pure and of either <*> or liftA2. If it defines both, then they must behave the same as their default definitions:

(<*>) = liftA2 id

liftA2 f x y = f Prelude.<$> x <*> y

Further, any definition must satisfy the following:

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 = (id <$ u) <*> v

• u <* v = liftA2 const u v

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

• fmap f x = pure f <*> x

It may be useful to note that supposing

forall x y. p (q x y) = f x . g y

it follows from the above that

liftA2 p (liftA2 q u v) = liftA2 f u . liftA2 g v

If f is also a Monad, it should satisfy

• pure = return

• m1 <*> m2 = m1 >>= (\x1 -> m2 >>= (\x2 -> return (x1 x2)))

• (*>) = (>>)

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

A monoid on applicative functors.

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

• some v = (:) Prelude.<$> v <*> many v

• many v = some v <|> pure []

Examples

>>> Nothing <|> Just 42Just 42

>>> [1, 2] <|> [3, 4][1,2,3,4]

>>> empty <|> print (2^15)32768

An infix synonym for fmap.

The name of this operator is an allusion to Prelude.$. Note the similarities between their types:

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

Whereas Prelude.$ is function application, <$> is function application lifted over a Functor.

Examples

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

>>> show <$> NothingNothing

>>> show <$> Just 3Just "3"

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

>>> show <$> Left 17Left 17

>>> show <$> Right 17Right "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)

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.

Examples

Perform a computation with Maybe and replace the result with a constant value if it is Just:

>>> 'a' <$ Just 2Just 'a'>>> 'a' <$ NothingNothing

One or none.

It is useful for modelling any computation that is allowed to fail.

Examples

Using the Alternative instance of Control.Monad.Except, the following functions:

>>> import Control.Monad.Except

>>> canFail = throwError "it failed" :: Except String Int>>> final = return 42                :: Except String Int

Can be combined by allowing the first function to fail:

>>> runExcept $ canFail *> finalLeft "it failed"

>>> runExcept $ optional canFail *> finalRight 42

A variant of <*> with the types of the arguments reversed. It differs from flip (<*>) in that the effects are resolved in the order the arguments are presented.

Examples

>>> (<**>) (print 1) (id <$ print 2)12

>>> flip (<*>) (print 1) (id <$ print 2)21

>>> ZipList [4, 5, 6] <**> ZipList [(+1), (*2), (/3)]ZipList {getZipList = [5.0,10.0,2.0]}

Lift a function to actions. Equivalent to Functor's fmap but implemented using only Applicative's methods: liftA f a = pure f <*> a

As such this function may be used to implement a Functor instance from an Applicative one.

Examples

Using the Applicative instance for Lists:

>>> liftA (+1) [1, 2][2,3]

Or the Applicative instance for Maybe

>>> liftA (+1) (Just 3)Just 4

Lift a ternary function to actions.

The sum of a collection of actions using (<|>), generalizing concat.

asum is just like msum, but generalised to Alternative.

Examples

Basic usage:

>>> asum [Just "Hello", Nothing, Just "World"]Just "Hello"

The Const functor.

Examples

>>> fmap (++ "World") (Const "Hello")Const "Hello"

Because we ignore the second type parameter to Const, the Applicative instance, which has (<*>) :: Monoid m => Const m (a -> b) -> Const m a -> Const m b essentially turns into Monoid m => m -> m -> m, which is (<>)

>>> Const [1, 2, 3] <*> Const [4, 5, 6]Const [1,2,3,4,5,6]

Lists, but with an Applicative functor based on zipping.

Examples

In contrast to the Applicative for GHC.List.List:

>>> (+) <$> [1, 2, 3] <*> [4, 5, 6][5,6,7,6,7,8,7,8,9]

The Applicative instance of ZipList applies the operation by pairing up the elements, analogous to zipWithN

>>> (+) <$> ZipList [1, 2, 3] <*> ZipList [4, 5, 6]ZipList {getZipList = [5,7,9]}

>>> (,,,) <$> ZipList [1, 2] <*> ZipList [3, 4] <*> ZipList [5, 6] <*> ZipList [7, 8]ZipList {getZipList = [(1,3,5,7),(2,4,6,8)]}

>>> ZipList [(+1), (^2), (/ 2)] <*> ZipList [5, 5, 5]ZipList {getZipList = [6.0,25.0,2.5]}

A sized functor is an applicative functor extended with a notion of cost/size of contained values. This is useful for any type of bounded recursion over infinite sets, most notably for various kind of enumerations.

The intention is that every sized functor definition models a (usually) infinite set (technically a bag) with a finite number of values of any given size. As long as every cyclic (recursive) definition has at least one application of pay, this invariant is guaranteed.

The module Control.Enumerable provides sized functor definitions for a lot of data types, such that the size of a value is the number of constructor applications it contains. It also allows deriving these functors for any user defined data type (using Template Haskell).