Copyright  (c) Andrey Mokhov 20182019 

License  MIT (see the file LICENSE) 
Maintainer  andrey.mokhov@gmail.com 
Stability  experimental 
Safe Haskell  None 
Language  Haskell2010 
This is a library for selective applicative functors, or just selective functors for short, an abstraction between applicative functors and monads, introduced in this paper: https://www.staff.ncl.ac.uk/andrey.mokhov/selectivefunctors.pdf.
Synopsis
 class Applicative f => Selective f where
 (<*?) :: Selective f => f (Either a b) > f (a > b) > f b
 branch :: Selective f => f (Either a b) > f (a > c) > f (b > c) > f c
 selectA :: Applicative f => f (Either a b) > f (a > b) > f b
 apS :: Selective f => f (a > b) > f a > f b
 selectM :: Monad f => f (Either a b) > f (a > b) > f b
 ifS :: Selective f => f Bool > f a > f a > f a
 whenS :: Selective f => f Bool > f () > f ()
 fromMaybeS :: Selective f => f a > f (Maybe a) > f a
 orElse :: (Selective f, Semigroup e) => f (Either e a) > f (Either e a) > f (Either e a)
 andAlso :: (Selective f, Semigroup a) => f (Either e a) > f (Either e a) > f (Either e a)
 untilRight :: (Monoid a, Selective f) => f (Either a b) > f (a, b)
 whileS :: Selective f => f Bool > f ()
 (<>) :: Selective f => f Bool > f Bool > f Bool
 (<&&>) :: Selective f => f Bool > f Bool > f Bool
 foldS :: (Selective f, Foldable t, Monoid a) => t (f (Either e a)) > f (Either e a)
 anyS :: Selective f => (a > f Bool) > [a] > f Bool
 allS :: Selective f => (a > f Bool) > [a] > f Bool
 bindS :: (Bounded a, Enum a, Eq a, Selective f) => f a > (a > f b) > f b
 data Cases a
 casesEnum :: (Bounded a, Enum a) => Cases a
 cases :: Eq a => [a] > Cases a
 matchS :: (Eq a, Selective f) => Cases a > f a > (a > f b) > f (Either a b)
 matchM :: Monad m => Cases a > m a > (a > m b) > m (Either a b)
 newtype SelectA f a = SelectA {
 getSelectA :: f a
 newtype SelectM f a = SelectM {
 getSelectM :: f a
 newtype Over m a = Over {
 getOver :: m
 newtype Under m a = Under {
 getUnder :: m
 data Validation e a
Type class
class Applicative f => Selective f where Source #
Selective applicative functors. You can think of select
as a selective
function application: when given a value of type Left
a
, you must apply
the given function, but when given a Right
b
, you may skip the
function and associated effects, and simply return the b
.
Note that it is not a requirement for selective functors to skip unnecessary effects. It may be counterintuitive, but this makes them more useful. Why? Typically, when executing a selective computation, you would want to skip the effects (saving work); but on the other hand, if your goal is to statically analyse a given selective computation and extract the set of all possible effects (without actually executing them), then you do not want to skip any effects, because that defeats the purpose of static analysis.
The type signature of select
is reminiscent of both <*>
and >>=
, and
indeed a selective functor is in some sense a composition of an applicative
functor and the Either
monad.
Laws:
 Identity:
x <*? pure id = either id id <$> x
 Distributivity; note that
y
andz
have the same typef (a > b)
:
pure x <*? (y *> z) = (pure x <*? y) *> (pure x <*? z)
 Associativity:
x <*? (y <*? z) = (f <$> x) <*? (g <$> y) <*? (h <$> z) where f x = Right <$> x g y = a > bimap (,a) ($a) y h z = uncurry z
 Monadic
select
(for selective functors that are also monads):
select = selectM
There are also a few useful theorems:
 Apply a pure function to the result:
f <$> select x y = select (fmap f <$> x) (fmap f <$> y)
 Apply a pure function to the
Left
case of the first argument:
select (first f <$> x) y = select x ((. f) <$> y)
 Apply a pure function to the second argument:
select x (f <$> y) = select (first (flip f) <$> x) ((&) <$> y)
 Generalised identity:
x <*? pure y = either y id <$> x
 A selective functor is rigid if it satisfies
<*>
=
apS
. The following interchange law holds for rigid selective functors:
x *> (y <*? z) = (x *> y) <*? z
If f is also a Monad
, we require that select
= selectM
, from which one
can prove <*>
=
apS
.
Instances
(<*?) :: Selective f => f (Either a b) > f (a > b) > f b infixl 4 Source #
An operator alias for select
, which is sometimes convenient. It tries to
follow the notational convention for Applicative
operators. The angle
bracket pointing to the left means we always use the corresponding value.
The value on the right, however, may be skipped, hence the question mark.
branch :: Selective f => f (Either a b) > f (a > c) > f (b > c) > f c Source #
The branch
function is a natural generalisation of select
: instead of
skipping an unnecessary effect, it chooses which of the two given effectful
functions to apply to a given argument; the other effect is unnecessary. It
is possible to implement branch
in terms of select
, which is a good
puzzle (give it a try!).
We can also implement select
via branch
:
selectB :: Selective f => f (Either a b) > f (a > b) > f b selectB x y = branch x y (pure id)
selectA :: Applicative f => f (Either a b) > f (a > b) > f b Source #
We can write a function with the type signature of select
using the
Applicative
type class, but it will always execute the effects associated
with the second argument, hence being potentially less efficient.
apS :: Selective f => f (a > b) > f a > f b Source #
Recover the application operator <*>
from select
. Rigid selective
functors satisfy the law <*>
=
apS
and furthermore, the resulting
applicative functor satisfies all laws of Applicative
:
Identity:
pure id <*> v = v
Homomorphism:
pure f <*> pure x = pure (f x)
Interchange:
u <*> pure y = pure ($y) <*> u
Composition:
(.) <$> u <*> v <*> w = u <*> (v <*> w)
Conditional combinators
ifS :: Selective f => f Bool > f a > f a > f a Source #
Branch on a Boolean value, skipping unnecessary effects.
orElse :: (Selective f, Semigroup e) => f (Either e a) > f (Either e a) > f (Either e a) Source #
Return the first Right
value. If both are Left
's, accumulate errors.
andAlso :: (Selective f, Semigroup a) => f (Either e a) > f (Either e a) > f (Either e a) Source #
Accumulate the Right
values, or return the first Left
.
untilRight :: (Monoid a, Selective f) => f (Either a b) > f (a, b) Source #
Keep running an effectful computation until it returns a Right
value,
collecting the Left
's using a supplied Monoid
instance.
whileS :: Selective f => f Bool > f () Source #
Keep checking an effectful condition while it holds.
foldS :: (Selective f, Foldable t, Monoid a) => t (f (Either e a)) > f (Either e a) Source #
Generalised folding with the shortcircuiting behaviour.
anyS :: Selective f => (a > f Bool) > [a] > f Bool Source #
A lifted version of any
. Retains the shortcircuiting behaviour.
allS :: Selective f => (a > f Bool) > [a] > f Bool Source #
A lifted version of all
. Retains the shortcircuiting behaviour.
casesEnum :: (Bounded a, Enum a) => Cases a Source #
The list of all possible values of an enumerable data type.
matchS :: (Eq a, Selective f) => Cases a > f a > (a > f b) > f (Either a b) Source #
Eliminate all specified values a
from f (Either a b)
by replacing each
of them with a given f a
.
matchM :: Monad m => Cases a > m a > (a > m b) > m (Either a b) Source #
Eliminate all specified values a
from f (Either a b)
by replacing each
of them with a given f a
.
Selective functors
Any applicative functor can be given a Selective
instance by defining
select
=
selectA
. This data type captures this pattern, so you can use
it in combination with the DerivingVia
extension as follows:
newtype Over m a = Over m deriving (Functor, Applicative, Selective) via SelectA (Const m)
SelectA  

Instances
Functor f => Functor (SelectA f) Source #  
Applicative f => Applicative (SelectA f) Source #  
Applicative f => Selective (SelectA f) Source #  
Any monad can be given a Selective
instance by defining
select
=
selectM
. This data type captures this pattern, so you can use
it in combination with the DerivingVia
extension as follows:
newtype V1 a = V1 a deriving (Functor, Applicative, Selective, Monad) via SelectM Identity
SelectM  

Static analysis of selective functors with overapproximation.
Static analysis of selective functors with underapproximation.
data Validation e a Source #
Selective instance for the standard applicative functor Validation. This is a good example of a nontrivial selective functor which is not a monad.
Instances
Functor (Validation e) Source #  
Defined in Control.Selective fmap :: (a > b) > Validation e a > Validation e b # (<$) :: a > Validation e b > Validation e a #  
Semigroup e => Applicative (Validation e) Source #  
Defined in Control.Selective pure :: a > Validation e a # (<*>) :: Validation e (a > b) > Validation e a > Validation e b # liftA2 :: (a > b > c) > Validation e a > Validation e b > Validation e c # (*>) :: Validation e a > Validation e b > Validation e b # (<*) :: Validation e a > Validation e b > Validation e a #  
Semigroup e => Selective (Validation e) Source #  
Defined in Control.Selective select :: Validation e (Either a b) > Validation e (a > b) > Validation e b Source #  
(Show e, Show a) => Show (Validation e a) Source #  
Defined in Control.Selective showsPrec :: Int > Validation e a > ShowS # show :: Validation e a > String # showList :: [Validation e a] > ShowS # 