Copyright  Conor McBride and Ross Paterson 2005 

License  BSDstyle (see the LICENSE file in the distribution) 
Maintainer  [email protected] 
Stability  experimental 
Portability  portable 
Safe Haskell  Trustworthy 
Language  Haskell2010 
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
Traversable
class.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.
class Functor f => Applicative f where Source
A functor with application, providing operations to
pure
), and<*>
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 <$> x <*> y
Further, any definition must satisfy the following:
pure id <*> v = v
pure (.) <*> u <*> v <*> w = u <*> (v <*> w)
pure f <*> pure x = pure (f x)
u <*> pure y = pure ($ 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
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
(which implies that pure
and <*>
satisfy the applicative functor laws).
Lift a value.
(<*>) :: f (a > b) > f a > f b infixl 4 Source
Sequential application.
A few functors support an implementation of <*>
that is more efficient than the default one.
liftA2 :: (a > b > c) > f a > f b > f c Source
Lift a binary function to actions.
Some functors support an implementation of liftA2
that is more efficient than the default one. In particular, if fmap
is an expensive operation, it is likely better to use liftA2
than to fmap
over the structure and then use <*>
.
(*>) :: 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.
Applicative []  Since: base2.1 
Applicative Maybe  Since: base2.1 
Applicative IO  Since: base2.1 
Applicative Par1  Since: base4.9.0.0 
Applicative NonEmpty  Since: base4.9.0.0 
Defined in GHC.Base  
Applicative ReadP  Since: base4.6.0.0 
Applicative ReadPrec  Since: base4.6.0.0 
Defined in Text.ParserCombinators.ReadPrec  
Applicative Down  Since: base4.11.0.0 
Applicative Product  Since: base4.8.0.0 
Defined in Data.Semigroup.Internal  
Applicative Sum  Since: base4.8.0.0 
Applicative Dual  Since: base4.8.0.0 
Applicative Last  Since: base4.8.0.0 
Applicative First  Since: base4.8.0.0 
Applicative STM  Since: base4.8.0.0 
Applicative Identity  Since: base4.8.0.0 
Defined in Data.Functor.Identity  
Applicative ZipList 
f '<$>' 'ZipList' xs1 '<*>' ... '<*>' 'ZipList' xsN = 'ZipList' (zipWithN f xs1 ... xsN) where (\a b c > stimes c [a, b]) <$> ZipList "abcd" <*> ZipList "567" <*> ZipList [1..] = ZipList (zipWith3 (\a b c > stimes c [a, b]) "abcd" "567" [1..]) = ZipList {getZipList = ["a5","b6b6","c7c7c7"]} Since: base2.1 
Applicative Option  Since: base4.9.0.0 
Applicative Last  Since: base4.9.0.0 
Applicative First  Since: base4.9.0.0 
Applicative Max  Since: base4.9.0.0 
Applicative Min  Since: base4.9.0.0 
Applicative Complex  Since: base4.9.0.0 
Applicative (Either e)  Since: base3.0 
Defined in Data.Either  
Applicative (U1 :: Type > Type)  Since: base4.9.0.0 
Monoid a => Applicative ((,) a) 
For tuples, the ("hello ", (+15)) <*> ("world!", 2002) ("hello world!",2017) Since: base2.1 
Applicative (ST s)  Since: base4.4.0.0 
Applicative (Proxy :: Type > Type)  Since: base4.7.0.0 
Arrow a => Applicative (ArrowMonad a)  Since: base4.6.0.0 
Defined in Control.Arrow Methodspure :: a0 > ArrowMonad a a0 Source (<*>) :: ArrowMonad a (a0 > b) > ArrowMonad a a0 > ArrowMonad a b Source liftA2 :: (a0 > b > c) > ArrowMonad a a0 > ArrowMonad a b > ArrowMonad a c Source (*>) :: ArrowMonad a a0 > ArrowMonad a b > ArrowMonad a b Source (<*) :: ArrowMonad a a0 > ArrowMonad a b > ArrowMonad a a0 Source  
Monad m => Applicative (WrappedMonad m)  Since: base2.1 
Defined in Control.Applicative Methodspure :: a > WrappedMonad m a Source (<*>) :: WrappedMonad m (a > b) > WrappedMonad m a > WrappedMonad m b Source liftA2 :: (a > b > c) > WrappedMonad m a > WrappedMonad m b > WrappedMonad m c Source (*>) :: WrappedMonad m a > WrappedMonad m b > WrappedMonad m b Source (<*) :: WrappedMonad m a > WrappedMonad m b > WrappedMonad m a Source  
Applicative (ST s)  Since: base2.1 
Applicative f => Applicative (Rec1 f)  Since: base4.9.0.0 
Applicative f => Applicative (Alt f)  Since: base4.8.0.0 
Applicative f => Applicative (Ap f)  Since: base4.12.0.0 
Monoid m => Applicative (Const m :: Type > Type)  Since: base2.0.1 
Arrow a => Applicative (WrappedArrow a b)  Since: base2.1 
Defined in Control.Applicative Methodspure :: a0 > WrappedArrow a b a0 Source (<*>) :: WrappedArrow a b (a0 > b0) > WrappedArrow a b a0 > WrappedArrow a b b0 Source liftA2 :: (a0 > b0 > c) > WrappedArrow a b a0 > WrappedArrow a b b0 > WrappedArrow a b c Source (*>) :: WrappedArrow a b a0 > WrappedArrow a b b0 > WrappedArrow a b b0 Source (<*) :: WrappedArrow a b a0 > WrappedArrow a b b0 > WrappedArrow a b a0 Source  
Applicative ((>) a :: Type > Type)  Since: base2.1 
Monoid c => Applicative (K1 i c :: Type > Type)  Since: base4.12.0.0 
(Applicative f, Applicative g) => Applicative (f :*: g)  Since: base4.9.0.0 
Defined in GHC.Generics  
(Applicative f, Applicative g) => Applicative (Product f g)  Since: base4.9.0.0 
Defined in Data.Functor.Product Methodspure :: a > Product f g a Source (<*>) :: Product f g (a > b) > Product f g a > Product f g b Source liftA2 :: (a > b > c) > Product f g a > Product f g b > Product f g c Source (*>) :: Product f g a > Product f g b > Product f g b Source (<*) :: Product f g a > Product f g b > Product f g a Source  
Applicative f => Applicative (M1 i c f)  Since: base4.9.0.0 
Defined in GHC.Generics  
(Applicative f, Applicative g) => Applicative (f :.: g)  Since: base4.9.0.0 
Defined in GHC.Generics  
(Applicative f, Applicative g) => Applicative (Compose f g)  Since: base4.9.0.0 
Defined in Data.Functor.Compose Methodspure :: a > Compose f g a Source (<*>) :: Compose f g (a > b) > Compose f g a > Compose f g b Source liftA2 :: (a > b > c) > Compose f g a > Compose f g b > Compose f g c Source (*>) :: Compose f g a > Compose f g b > Compose f g b Source (<*) :: Compose f g a > Compose f g b > Compose f g a Source 
class Applicative f => Alternative f where Source
A monoid on applicative functors.
If defined, some
and many
should be the least solutions of the equations:
The identity of <>
(<>) :: f a > f a > f a infixl 3 Source
An associative binary operation
One or more.
Zero or more.
The Const
functor.
Generic1 (Const a :: k > Type)  
Show2 (Const :: Type > Type > Type)  Since: base4.9.0.0 
Read2 (Const :: Type > Type > Type)  Since: base4.9.0.0 
Defined in Data.Functor.Classes MethodsliftReadsPrec2 :: (Int > ReadS a) > ReadS [a] > (Int > ReadS b) > ReadS [b] > Int > ReadS (Const a b) Source liftReadList2 :: (Int > ReadS a) > ReadS [a] > (Int > ReadS b) > ReadS [b] > ReadS [Const a b] Source liftReadPrec2 :: ReadPrec a > ReadPrec [a] > ReadPrec b > ReadPrec [b] > ReadPrec (Const a b) Source liftReadListPrec2 :: ReadPrec a > ReadPrec [a] > ReadPrec b > ReadPrec [b] > ReadPrec [Const a b] Source  
Ord2 (Const :: Type > Type > Type)  Since: base4.9.0.0 
Defined in Data.Functor.Classes  
Eq2 (Const :: Type > Type > Type)  Since: base4.9.0.0 
Bifunctor (Const :: Type > Type > Type)  Since: base4.8.0.0 
Bifoldable (Const :: Type > Type > Type)  Since: base4.10.0.0 
Bitraversable (Const :: Type > Type > Type)  Since: base4.10.0.0 
Defined in Data.Bitraversable Methodsbitraverse :: Applicative f => (a > f c) > (b > f d) > Const a b > f (Const c d) Source  
Functor (Const m :: Type > Type)  Since: base2.1 
Monoid m => Applicative (Const m :: Type > Type)  Since: base2.0.1 
Foldable (Const m :: Type > Type)  Since: base4.7.0.0 
Defined in Data.Functor.Const Methodsfold :: Monoid m0 => Const m m0 > m0 Source foldMap :: Monoid m0 => (a > m0) > Const m a > m0 Source foldr :: (a > b > b) > b > Const m a > b Source foldr' :: (a > b > b) > b > Const m a > b Source foldl :: (b > a > b) > b > Const m a > b Source foldl' :: (b > a > b) > b > Const m a > b Source foldr1 :: (a > a > a) > Const m a > a Source foldl1 :: (a > a > a) > Const m a > a Source toList :: Const m a > [a] Source null :: Const m a > Bool Source length :: Const m a > Int Source elem :: Eq a => a > Const m a > Bool Source maximum :: Ord a => Const m a > a Source minimum :: Ord a => Const m a > a Source  
Traversable (Const m :: Type > Type)  Since: base4.7.0.0 
Defined in Data.Traversable  
Show a => Show1 (Const a :: Type > Type)  Since: base4.9.0.0 
Read a => Read1 (Const a :: Type > Type)  Since: base4.9.0.0 
Defined in Data.Functor.Classes MethodsliftReadsPrec :: (Int > ReadS a0) > ReadS [a0] > Int > ReadS (Const a a0) Source liftReadList :: (Int > ReadS a0) > ReadS [a0] > ReadS [Const a a0] Source liftReadPrec :: ReadPrec a0 > ReadPrec [a0] > ReadPrec (Const a a0) Source liftReadListPrec :: ReadPrec a0 > ReadPrec [a0] > ReadPrec [Const a a0] Source  
Ord a => Ord1 (Const a :: Type > Type)  Since: base4.9.0.0 
Defined in Data.Functor.Classes  
Eq a => Eq1 (Const a :: Type > Type)  Since: base4.9.0.0 
Contravariant (Const a :: Type > Type)  
Bounded a => Bounded (Const a b)  Since: base4.9.0.0 
Enum a => Enum (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const Methodssucc :: Const a b > Const a b Source pred :: Const a b > Const a b Source toEnum :: Int > Const a b Source fromEnum :: Const a b > Int Source enumFrom :: Const a b > [Const a b] Source enumFromThen :: Const a b > Const a b > [Const a b] Source enumFromTo :: Const a b > Const a b > [Const a b] Source enumFromThenTo :: Const a b > Const a b > Const a b > [Const a b] Source  
Eq a => Eq (Const a b)  Since: base4.9.0.0 
Floating a => Floating (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const Methodsexp :: Const a b > Const a b Source log :: Const a b > Const a b Source sqrt :: Const a b > Const a b Source (**) :: Const a b > Const a b > Const a b Source logBase :: Const a b > Const a b > Const a b Source sin :: Const a b > Const a b Source cos :: Const a b > Const a b Source tan :: Const a b > Const a b Source asin :: Const a b > Const a b Source acos :: Const a b > Const a b Source atan :: Const a b > Const a b Source sinh :: Const a b > Const a b Source cosh :: Const a b > Const a b Source tanh :: Const a b > Const a b Source asinh :: Const a b > Const a b Source acosh :: Const a b > Const a b Source atanh :: Const a b > Const a b Source log1p :: Const a b > Const a b Source expm1 :: Const a b > Const a b Source  
Fractional a => Fractional (Const a b)  Since: base4.9.0.0 
Integral a => Integral (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const Methodsquot :: Const a b > Const a b > Const a b Source rem :: Const a b > Const a b > Const a b Source div :: Const a b > Const a b > Const a b Source mod :: Const a b > Const a b > Const a b Source quotRem :: Const a b > Const a b > (Const a b, Const a b) Source divMod :: Const a b > Const a b > (Const a b, Const a b) Source  
(Typeable k, Data a, Typeable b) => Data (Const a b)  Since: base4.10.0.0 
Defined in Data.Data Methodsgfoldl :: (forall d b0. Data d => c (d > b0) > d > c b0) > (forall g. g > c g) > Const a b > c (Const a b) Source gunfold :: (forall b0 r. Data b0 => c (b0 > r) > c r) > (forall r. r > c r) > Constr > c (Const a b) Source toConstr :: Const a b > Constr Source dataTypeOf :: Const a b > DataType Source dataCast1 :: Typeable t => (forall d. Data d => c (t d)) > Maybe (c (Const a b)) Source dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) > Maybe (c (Const a b)) Source gmapT :: (forall b0. Data b0 => b0 > b0) > Const a b > Const a b Source gmapQl :: (r > r' > r) > r > (forall d. Data d => d > r') > Const a b > r Source gmapQr :: (r' > r > r) > r > (forall d. Data d => d > r') > Const a b > r Source gmapQ :: (forall d. Data d => d > u) > Const a b > [u] Source gmapQi :: Int > (forall d. Data d => d > u) > Const a b > u Source gmapM :: Monad m => (forall d. Data d => d > m d) > Const a b > m (Const a b) Source gmapMp :: MonadPlus m => (forall d. Data d => d > m d) > Const a b > m (Const a b) Source gmapMo :: MonadPlus m => (forall d. Data d => d > m d) > Const a b > m (Const a b) Source  
Num a => Num (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const Methods(+) :: Const a b > Const a b > Const a b Source () :: Const a b > Const a b > Const a b Source (*) :: Const a b > Const a b > Const a b Source negate :: Const a b > Const a b Source abs :: Const a b > Const a b Source signum :: Const a b > Const a b Source fromInteger :: Integer > Const a b Source  
Ord a => Ord (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const  
Read a => Read (Const a b) 
This instance would be equivalent to the derived instances of the Since: base4.8.0.0 
Real a => Real (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const MethodstoRational :: Const a b > Rational Source  
RealFloat a => RealFloat (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const MethodsfloatRadix :: Const a b > Integer Source floatDigits :: Const a b > Int Source floatRange :: Const a b > (Int, Int) Source decodeFloat :: Const a b > (Integer, Int) Source encodeFloat :: Integer > Int > Const a b Source exponent :: Const a b > Int Source significand :: Const a b > Const a b Source scaleFloat :: Int > Const a b > Const a b Source isNaN :: Const a b > Bool Source isInfinite :: Const a b > Bool Source isDenormalized :: Const a b > Bool Source isNegativeZero :: Const a b > Bool Source  
RealFrac a => RealFrac (Const a b)  Since: base4.9.0.0 
Show a => Show (Const a b) 
This instance would be equivalent to the derived instances of the Since: base4.8.0.0 
Ix a => Ix (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const Methodsrange :: (Const a b, Const a b) > [Const a b] Source index :: (Const a b, Const a b) > Const a b > Int Source unsafeIndex :: (Const a b, Const a b) > Const a b > Int inRange :: (Const a b, Const a b) > Const a b > Bool Source rangeSize :: (Const a b, Const a b) > Int Source unsafeRangeSize :: (Const a b, Const a b) > Int  
IsString a => IsString (Const a b)  Since: base4.9.0.0 
Defined in Data.String MethodsfromString :: String > Const a b Source  
Generic (Const a b)  
Semigroup a => Semigroup (Const a b)  Since: base4.9.0.0 
Monoid a => Monoid (Const a b)  Since: base4.9.0.0 
FiniteBits a => FiniteBits (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const MethodsfiniteBitSize :: Const a b > Int Source countLeadingZeros :: Const a b > Int Source countTrailingZeros :: Const a b > Int Source  
Bits a => Bits (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const Methods(.&.) :: Const a b > Const a b > Const a b Source (..) :: Const a b > Const a b > Const a b Source xor :: Const a b > Const a b > Const a b Source complement :: Const a b > Const a b Source shift :: Const a b > Int > Const a b Source rotate :: Const a b > Int > Const a b Source bit :: Int > Const a b Source setBit :: Const a b > Int > Const a b Source clearBit :: Const a b > Int > Const a b Source complementBit :: Const a b > Int > Const a b Source testBit :: Const a b > Int > Bool Source bitSizeMaybe :: Const a b > Maybe Int Source bitSize :: Const a b > Int Source isSigned :: Const a b > Bool Source shiftL :: Const a b > Int > Const a b Source unsafeShiftL :: Const a b > Int > Const a b Source shiftR :: Const a b > Int > Const a b Source unsafeShiftR :: Const a b > Int > Const a b Source rotateL :: Const a b > Int > Const a b Source  
Storable a => Storable (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const MethodssizeOf :: Const a b > Int Source alignment :: Const a b > Int Source peekElemOff :: Ptr (Const a b) > Int > IO (Const a b) Source pokeElemOff :: Ptr (Const a b) > Int > Const a b > IO () Source peekByteOff :: Ptr b0 > Int > IO (Const a b) Source pokeByteOff :: Ptr b0 > Int > Const a b > IO () Source  
type Rep1 (Const a :: k > Type)  Since: base4.9.0.0 
Defined in Data.Functor.Const  
type Rep (Const a b)  Since: base4.9.0.0 
Defined in Data.Functor.Const 
newtype WrappedMonad m a Source
WrapMonad  
Fields

newtype WrappedArrow a b c Source
WrapArrow  
Fields

Lists, but with an Applicative
functor based on zipping.
ZipList  
Fields

Functor ZipList  Since: base2.1 
Applicative ZipList 
f '<$>' 'ZipList' xs1 '<*>' ... '<*>' 'ZipList' xsN = 'ZipList' (zipWithN f xs1 ... xsN) where (\a b c > stimes c [a, b]) <$> ZipList "abcd" <*> ZipList "567" <*> ZipList [1..] = ZipList (zipWith3 (\a b c > stimes c [a, b]) "abcd" "567" [1..]) = ZipList {getZipList = ["a5","b6b6","c7c7c7"]} Since: base2.1 
Foldable ZipList  Since: base4.9.0.0 
Defined in Control.Applicative Methodsfold :: Monoid m => ZipList m > m Source foldMap :: Monoid m => (a > m) > ZipList a > m Source foldr :: (a > b > b) > b > ZipList a > b Source foldr' :: (a > b > b) > b > ZipList a > b Source foldl :: (b > a > b) > b > ZipList a > b Source foldl' :: (b > a > b) > b > ZipList a > b Source foldr1 :: (a > a > a) > ZipList a > a Source foldl1 :: (a > a > a) > ZipList a > a Source toList :: ZipList a > [a] Source null :: ZipList a > Bool Source length :: ZipList a > Int Source elem :: Eq a => a > ZipList a > Bool Source maximum :: Ord a => ZipList a > a Source minimum :: Ord a => ZipList a > a Source  
Traversable ZipList  Since: base4.9.0.0 
Defined in Data.Traversable  
Alternative ZipList  Since: base4.11.0.0 
Eq a => Eq (ZipList a)  Since: base4.7.0.0 
Ord a => Ord (ZipList a)  Since: base4.7.0.0 
Defined in Control.Applicative  
Read a => Read (ZipList a)  Since: base4.7.0.0 
Show a => Show (ZipList a)  Since: base4.7.0.0 
Generic (ZipList a)  
Generic1 ZipList  
type Rep (ZipList a)  Since: base4.7.0.0 
Defined in Control.Applicative  
type Rep1 ZipList  Since: base4.7.0.0 
Defined in Control.Applicative 
(<$>) :: Functor f => (a > b) > f a > f b infixl 4 Source
An infix synonym for fmap
.
The name of this operator is an allusion to $
. Note the similarities between their types:
($) :: (a > b) > a > b (<$>) :: Functor f => (a > b) > f a > f b
Whereas $
is function application, <$>
is function application lifted over a Functor
.
Convert from a Maybe Int
to a Maybe String
using show
:
>>> show <$> 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 infixl 4 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.
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.
© The University of Glasgow and others
Licensed under a BSDstyle license (see top of the page).
https://downloads.haskell.org/~ghc/8.6.1/docs/html/libraries/base4.12.0.0/ControlApplicative.html