| Copyright | (c) 2013 Justus Sagemüller |
|---|---|
| License | GPL v3 (see COPYING) |
| Maintainer | (@) jsag $ hvl.no |
| Safe Haskell | Trustworthy |
| Language | Haskell2010 |
Control.Category.Constrained.Prelude
Description
Synopsis
- type (×) = ProductCategory
- data ProductCategory k l p q = (k (LFactor p) (LFactor q)) :***: (l (RFactor p) (RFactor q))
- data GenericAgent k a v = GenericAgent {
- runGenericAgent :: k a v
- class Category k => HasAgent k where
- type ObjectMorphism k b c = (Object k b, Object k c, MorphObjects k b c, Object k (k b c))
- class Cartesian k => Curry k where
- type MorphObjects k b c :: Constraint
- uncurry :: (ObjectPair k a b, ObjectMorphism k b c) => k a (k b c) -> k (a, b) c
- curry :: (ObjectPair k a b, ObjectMorphism k b c) => k (a, b) c -> k a (k b c)
- apply :: (ObjectMorphism k a b, ObjectPair k (k a b) a) => k (k a b, a) b
- type CatTagged k x = Tagged (k (UnitObject k) (UnitObject k)) x
- type ObjectSum k a b = (Category k, Object k a, Object k b, SumObjects k a b, Object k (a + b))
- class (Category k, Object k (ZeroObject k)) => CoCartesian k where
- type SumObjects k a b :: Constraint
- type ZeroObject k :: *
- coSwap :: (ObjectSum k a b, ObjectSum k b a) => k (a + b) (b + a)
- attachZero :: (Object k a, zero ~ ZeroObject k, ObjectSum k a zero) => k a (a + zero)
- detachZero :: (Object k a, zero ~ ZeroObject k, ObjectSum k a zero) => k (a + zero) a
- coRegroup :: (Object k a, Object k c, ObjectSum k a b, ObjectSum k b c, ObjectSum k a (b + c), ObjectSum k (a + b) c) => k (a + (b + c)) ((a + b) + c)
- coRegroup' :: (Object k a, Object k c, ObjectSum k a b, ObjectSum k b c, ObjectSum k a (b + c), ObjectSum k (a + b) c) => k ((a + b) + c) (a + (b + c))
- maybeAsSum :: (ObjectSum k u a, u ~ UnitObject k, Object k (Maybe a)) => k (Maybe a) (u + a)
- maybeFromSum :: (ObjectSum k u a, u ~ UnitObject k, Object k (Maybe a)) => k (u + a) (Maybe a)
- boolAsSum :: (ObjectSum k u u, u ~ UnitObject k, Object k Bool) => k Bool (u + u)
- boolFromSum :: (ObjectSum k u u, u ~ UnitObject k, Object k Bool) => k (u + u) Bool
- type (+) = Either
- type ObjectPair k a b = (Category k, Object k a, Object k b, PairObjects k a b, Object k (a, b))
- class (Category k, Monoid (UnitObject k), Object k (UnitObject k)) => Cartesian k where
- type PairObjects k a b :: Constraint
- type UnitObject k :: *
- swap :: (ObjectPair k a b, ObjectPair k b a) => k (a, b) (b, a)
- attachUnit :: (unit ~ UnitObject k, ObjectPair k a unit) => k a (a, unit)
- detachUnit :: (unit ~ UnitObject k, ObjectPair k a unit) => k (a, unit) a
- regroup :: (ObjectPair k a b, ObjectPair k b c, ObjectPair k a (b, c), ObjectPair k (a, b) c) => k (a, (b, c)) ((a, b), c)
- regroup' :: (ObjectPair k a b, ObjectPair k b c, ObjectPair k a (b, c), ObjectPair k (a, b) c) => k ((a, b), c) (a, (b, c))
- class Category k => Isomorphic k a b where
- iso :: k a b
- type ConstrainedFunction isObj = ConstrainedCategory (->) isObj
- type (⊢) o k = ConstrainedCategory k o
- data ConstrainedCategory (k :: * -> * -> *) (o :: * -> Constraint) (a :: *) (b :: *)
- type Hask = Unconstrained ⊢ (->)
- class Category (k :: κ -> κ -> Type) where
- inCategoryOf :: Category k => k a b -> k c d -> k a b
- constrained :: forall o k a b. (Category k, o a, o b) => k a b -> (o ⊢ k) a b
- unconstrained :: forall o k a b. Category k => (o ⊢ k) a b -> k a b
- genericAlg :: (HasAgent k, Object k a, Object k b) => (forall q. Object k q => GenericAgent k q a -> GenericAgent k q b) -> k a b
- genericAgentMap :: (HasAgent k, Object k a, Object k b, Object k c) => k b c -> GenericAgent k a b -> GenericAgent k a c
- module Control.Functor.Constrained
- module Control.Applicative.Constrained
- module Control.Applicative.Constrained
- class (CoCartesian r, Cartesian t, Functor f r t, Object t (f (ZeroObject r))) => SumToProduct f r t where
- class (Category r, Category t) => Functor f r t | f r -> t, f t -> r where
- (<$>) :: (Functor f r (->), Object r a, Object r b) => r a b -> f a -> f b
- constrainedFmap :: (Category r, Category t, o a, o b, o (f a), o (f b)) => (r a b -> t (f a) (f b)) -> (o ⊢ r) a b -> (o ⊢ t) (f a) (f b)
- class (Monoidal f r t, Curry r, Curry t) => Applicative f r t where
- pure :: (Object r a, Object t (f a)) => a `t` f a
- (<*>) :: (ObjectMorphism r a b, ObjectMorphism t (f a) (f b), Object t (t (f a) (f b)), ObjectPair r (r a b) a, ObjectPair t (f (r a b)) (f a), Object r a, Object r b) => f (r a b) `t` t (f a) (f b)
- class (Functor f r t, Cartesian r, Cartesian t, Object t (f (UnitObject r))) => Monoidal f r t where
- pureUnit :: UnitObject t `t` f (UnitObject r)
- fzipWith :: (ObjectPair r a b, Object r c, ObjectPair t (f a) (f b), Object t (f c)) => r (a, b) c -> t (f a, f b) (f c)
- constPure :: (WellPointed r, Monoidal f r t, ObjectPoint r a, Object t (f a)) => a -> t (UnitObject t) (f a)
- fzip :: (Monoidal f r t, ObjectPair r a b, ObjectPair t (f a) (f b), Object t (f (a, b))) => t (f a, f b) (f (a, b))
- (<**>) :: (Applicative f r (->), ObjectMorphism r a b, ObjectPair r (r a b) a) => f a -> f (r a b) -> f b
- liftA :: (Applicative f r t, Object r a, Object r b, Object t (f a), Object t (f b)) => (a `r` b) -> f a `t` f b
- liftA2 :: (Applicative f r t, Object r c, ObjectMorphism r b c, Object t (f c), ObjectMorphism t (f b) (f c), ObjectPair r a b, ObjectPair t (f a) (f b)) => (a `r` (b `r` c)) -> f a `t` (f b `t` f c)
- liftA3 :: (Applicative f r t, Object r c, Object r d, ObjectMorphism r c d, ObjectMorphism r b (c `r` d), Object r (r c d), ObjectPair r a b, ObjectPair r (r c d) c, Object t (f c), Object t (f d), Object t (f a, f b), ObjectMorphism t (f c) (f d), ObjectMorphism t (f b) (t (f c) (f d)), Object t (t (f c) (f d)), ObjectPair t (f a) (f b), ObjectPair t (t (f c) (f d)) (f c), ObjectPair t (f (r c d)) (f c)) => (a `r` (b `r` (c `r` d))) -> f a `t` (f b `t` (f c `t` f d))
- constrainedFZipWith :: (Category r, Category t, o a, o b, o (a, b), o c, o (f a, f b), o (f c)) => (r (a, b) c -> t (f a, f b) (f c)) -> (o ⊢ r) (a, b) c -> (o ⊢ t) (f a, f b) (f c)
- mapM_ :: forall t k o f a b u. (Foldable t k k, WellPointed k, Monoidal f k k, u ~ UnitObject k, ObjectPair k (f u) (t a), ObjectPair k (f u) a, ObjectPair k u (t a), ObjectPair k (t a) u, ObjectPair k (f u) (f u), ObjectPair k u u, ObjectPair k b u, Object k (f b)) => (a `k` f b) -> t a `k` f u
- forM_ :: forall t k l f a b uk ul. (Foldable t k l, Monoidal f l l, Monoidal f k k, Function l, Arrow k (->), Arrow l (->), ul ~ UnitObject l, uk ~ UnitObject k, uk ~ ul, ObjectPair l ul ul, ObjectPair l (f ul) (f ul), ObjectPair l (f ul) (t a), ObjectPair l ul (t a), ObjectPair l (t a) ul, ObjectPair l (f ul) a, ObjectPair k b (f b), ObjectPair k b ul, ObjectPair k uk uk, ObjectPair k (f uk) a, ObjectPair k (f uk) (f uk)) => t a -> (a `k` f b) -> f uk
- sequence_ :: forall t k l m a b uk ul. (Foldable t k l, Arrow k (->), Arrow l (->), uk ~ UnitObject k, ul ~ UnitObject l, uk ~ ul, Monoidal m k k, Monoidal m l l, ObjectPair k a uk, ObjectPair k (t (m a)) uk, ObjectPair k uk uk, ObjectPair k (m uk) (m uk), ObjectPair k (t (m a)) ul, ObjectPair l (m ul) (t (m a)), ObjectPair l ul (t (m a)), ObjectPair l (m uk) (t (m a)), ObjectPair l (t (m a)) ul, ObjectPair k (m uk) (m a)) => t (m a) `l` m uk
- mapM :: (Traversable s t k l, k ~ l, s ~ t, Applicative m k k, Object k a, Object k (t a), ObjectPair k b (t b), ObjectPair k (m b) (m (t b)), TraversalObject k t b) => (a `k` m b) -> t a `k` m (t b)
- sequence :: (Traversable s t k l, k ~ l, s ~ t, Monoidal f k k, ObjectPair k a (t a), ObjectPair k (f a) (f (t a)), Object k (t (f a)), TraversalObject k t a) => t (f a) `k` f (t a)
- forM :: forall s t k m a b l. (Traversable s t k l, Monoidal m k l, Function l, Object k b, Object k (t b), ObjectPair k b (t b), Object l a, Object l (s a), ObjectPair l (m b) (m (t b)), TraversalObject k t b) => s a -> (a `l` m b) -> m (t b)
- newtype Kleisli m k a b = Kleisli {
- runKleisli :: k a (m b)
- class MonadPlus m k => MonadFail m k where
- class (Applicative m k k, Object k (m (UnitObject k)), Object k (m (m (UnitObject k)))) => Monad m k where
- return :: Monad m (->) => a -> m a
- (=<<) :: (Monad m k, Object k a, Object k b, Object k (m a), Object k (m b), Object k (m (m b))) => k a (m b) -> k (m a) (m b)
- (>>=) :: (Function f, Monad m f, Object f a, Object f b, Object f (m a), Object f (m b), Object f (m (m b))) => m a -> f a (m b) -> m b
- (<<) :: (Monad m k, WellPointed k, Object k a, Object k b, Object k (m a), ObjectPoint k (m b), Object k (m (m b))) => m b -> k (m a) (m b)
- (>>) :: (WellPointed k, Monad m k, ObjectPair k b (UnitObject k), ObjectPair k (m b) (UnitObject k), ObjectPair k (UnitObject k) (m b), ObjectPair k b a, ObjectPair k a b, Object k (m (a, b)), ObjectPair k (m a) (m b), ObjectPoint k (m a)) => m a -> k (m b) (m b)
- mzero :: MonadZero m (->) => m a
- mplus :: MonadPlus m (->) => m a -> m a -> m a
- when :: (Monad m k, PreArrow k, u ~ UnitObject k, ObjectPair k (m u) u) => Bool -> m u `k` m u
- unless :: (Monad m k, PreArrow k, u ~ UnitObject k, ObjectPair k (m u) u) => Bool -> m u `k` m u
- filterM :: (PreArrow k, Monad m k, SumToProduct c k k, EndoTraversable c k, ObjectPair k Bool a, Object k (c a), Object k (m (c a)), ObjectPair k (Bool, a) (c (Bool, a)), ObjectPair k (m Bool) (m a), ObjectPair k (m (Bool, a)) (m (c (Bool, a))), TraversalObject k c (Bool, a)) => (a `k` m Bool) -> c a `k` m (c a)
- type Function f = EnhancedCat (->) f
- const :: (WellPointed a, Object a b, ObjectPoint a x) => x -> a b x
- fst :: (PreArrow a, ObjectPair a x y) => a (x, y) x
- snd :: (PreArrow a, ObjectPair a x y) => a (x, y) y
- ($) :: (Function f, Object f a, Object f b) => f a b -> a -> b
- ifThenElse :: (EnhancedCat f (->), Function f, Object f Bool, Object f a, Object f (f a a), Object f (f a (f a a))) => Bool `f` (a `f` (a `f` a))
- (++) :: [a] -> [a] -> [a]
- seq :: forall {r :: RuntimeRep} a (b :: TYPE r). a -> b -> b
- zip :: [a] -> [b] -> [(a, b)]
- print :: Show a => a -> IO ()
- otherwise :: Bool
- map :: (a -> b) -> [a] -> [b]
- fromIntegral :: (Integral a, Num b) => a -> b
- realToFrac :: (Real a, Fractional b) => a -> b
- class Bounded a where
- class Enum a where
- succ :: a -> a
- pred :: a -> a
- toEnum :: Int -> a
- fromEnum :: a -> Int
- enumFrom :: a -> [a]
- enumFromThen :: a -> a -> [a]
- enumFromTo :: a -> a -> [a]
- enumFromThenTo :: a -> a -> a -> [a]
- class Eq a where
- class Fractional a => Floating a where
- class Num a => Fractional a where
- (/) :: a -> a -> a
- recip :: a -> a
- fromRational :: Rational -> a
- class (Real a, Enum a) => Integral a where
- class Num a where
- class Eq a => Ord a where
- class Read a where
- class (Num a, Ord a) => Real a where
- toRational :: a -> Rational
- class (RealFrac a, Floating a) => RealFloat a where
- floatRadix :: a -> Integer
- floatDigits :: a -> Int
- floatRange :: a -> (Int, Int)
- decodeFloat :: a -> (Integer, Int)
- encodeFloat :: Integer -> Int -> a
- exponent :: a -> Int
- significand :: a -> a
- scaleFloat :: Int -> a -> a
- isNaN :: a -> Bool
- isInfinite :: a -> Bool
- isDenormalized :: a -> Bool
- isNegativeZero :: a -> Bool
- isIEEE :: a -> Bool
- atan2 :: a -> a -> a
- class (Real a, Fractional a) => RealFrac a where
- class Show a where
- sum :: (Foldable t, Num a) => t a -> a
- product :: (Foldable t, Num a) => t a -> a
- null :: Foldable t => t a -> Bool
- minimum :: (Foldable t, Ord a) => t a -> a
- maximum :: (Foldable t, Ord a) => t a -> a
- foldr1 :: Foldable t => (a -> a -> a) -> t a -> a
- foldl1 :: Foldable t => (a -> a -> a) -> t a -> a
- elem :: (Foldable t, Eq a) => a -> t a -> Bool
- foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b
- length :: Foldable t => t a -> Int
- foldl :: Foldable t => (b -> a -> b) -> b -> t a -> b
- sequenceA :: (Traversable t, Applicative f) => t (f a) -> f (t a)
- class Semigroup a where
- (<>) :: a -> a -> a
- class Semigroup a => Monoid a where
- data Bool
- type String = [Char]
- data Char
- data Double
- data Float
- data Int
- data Integer
- data Maybe a
- data Ordering
- type Rational = Ratio Integer
- data IO a
- data Word
- data Either a b
- writeFile :: FilePath -> String -> IO ()
- readLn :: Read a => IO a
- readIO :: Read a => String -> IO a
- readFile :: FilePath -> IO String
- putStrLn :: String -> IO ()
- putStr :: String -> IO ()
- putChar :: Char -> IO ()
- interact :: (String -> String) -> IO ()
- getLine :: IO String
- getContents :: IO String
- getChar :: IO Char
- appendFile :: FilePath -> String -> IO ()
- ioError :: IOError -> IO a
- type FilePath = String
- type IOError = IOException
- userError :: String -> IOError
- or :: Foldable t => t Bool -> Bool
- notElem :: (Foldable t, Eq a) => a -> t a -> Bool
- concat :: Foldable t => t [a] -> [a]
- any :: Foldable t => (a -> Bool) -> t a -> Bool
- and :: Foldable t => t Bool -> Bool
- all :: Foldable t => (a -> Bool) -> t a -> Bool
- words :: String -> [String]
- unwords :: [String] -> String
- unlines :: [String] -> String
- lines :: String -> [String]
- reads :: Read a => ReadS a
- read :: Read a => String -> a
- either :: (a -> c) -> (b -> c) -> Either a b -> c
- readParen :: Bool -> ReadS a -> ReadS a
- lex :: ReadS String
- type ReadS a = String -> [(a, String)]
- odd :: Integral a => a -> Bool
- lcm :: Integral a => a -> a -> a
- gcd :: Integral a => a -> a -> a
- even :: Integral a => a -> Bool
- (^^) :: (Fractional a, Integral b) => a -> b -> a
- (^) :: (Num a, Integral b) => a -> b -> a
- type ShowS = String -> String
- shows :: Show a => a -> ShowS
- showString :: String -> ShowS
- showParen :: Bool -> ShowS -> ShowS
- showChar :: Char -> ShowS
- zipWith3 :: (a -> b -> c -> d) -> [a] -> [b] -> [c] -> [d]
- zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
- zip3 :: [a] -> [b] -> [c] -> [(a, b, c)]
- unzip3 :: [(a, b, c)] -> ([a], [b], [c])
- unzip :: [(a, b)] -> ([a], [b])
- takeWhile :: (a -> Bool) -> [a] -> [a]
- take :: Int -> [a] -> [a]
- tail :: [a] -> [a]
- splitAt :: Int -> [a] -> ([a], [a])
- span :: (a -> Bool) -> [a] -> ([a], [a])
- scanr1 :: (a -> a -> a) -> [a] -> [a]
- scanr :: (a -> b -> b) -> b -> [a] -> [b]
- scanl1 :: (a -> a -> a) -> [a] -> [a]
- scanl :: (b -> a -> b) -> b -> [a] -> [b]
- reverse :: [a] -> [a]
- replicate :: Int -> a -> [a]
- repeat :: a -> [a]
- lookup :: Eq a => a -> [(a, b)] -> Maybe b
- last :: [a] -> a
- iterate :: (a -> a) -> a -> [a]
- init :: [a] -> [a]
- head :: [a] -> a
- dropWhile :: (a -> Bool) -> [a] -> [a]
- drop :: Int -> [a] -> [a]
- cycle :: [a] -> [a]
- break :: (a -> Bool) -> [a] -> ([a], [a])
- (!!) :: [a] -> Int -> a
- maybe :: b -> (a -> b) -> Maybe a -> b
- subtract :: Num a => a -> a -> a
- until :: (a -> Bool) -> (a -> a) -> a -> a
- flip :: (a -> b -> c) -> b -> a -> c
- asTypeOf :: a -> a -> a
- ($!) :: forall (r :: RuntimeRep) a (b :: TYPE r). (a -> b) -> a -> b
- undefined :: forall (r :: RuntimeRep) (a :: TYPE r). HasCallStack => a
- errorWithoutStackTrace :: forall (r :: RuntimeRep) (a :: TYPE r). [Char] -> a
- error :: forall (r :: RuntimeRep) (a :: TYPE r). HasCallStack => [Char] -> a
- (&&) :: Bool -> Bool -> Bool
- not :: Bool -> Bool
- (||) :: Bool -> Bool -> Bool
The constrained-categories facilities
type (×) = ProductCategory Source #
data ProductCategory k l p q Source #
Instances
| (Category k, Category l) => Category (k × l :: Type -> Type -> Type) Source # | |
Defined in Control.Category.Constrained | |
| (Morphism k, Morphism l) => Morphism (k × l) Source # | |
Defined in Control.Arrow.Constrained Methods first :: (ObjectPair (k × l) b d, ObjectPair (k × l) c d) => (k × l) b c -> (k × l) (b, d) (c, d) Source # second :: (ObjectPair (k × l) d b, ObjectPair (k × l) d c) => (k × l) b c -> (k × l) (d, b) (d, c) Source # (***) :: (ObjectPair (k × l) b b', ObjectPair (k × l) c c') => (k × l) b c -> (k × l) b' c' -> (k × l) (b, b') (c, c') Source # | |
| (PreArrow k, PreArrow l) => PreArrow (k × l) Source # | |
| (WellPointed k, WellPointed l) => WellPointed (k × l) Source # | |
Defined in Control.Arrow.Constrained Associated Types type PointObject (k × l) x Source # Methods globalElement :: ObjectPoint (k × l) x => x -> (k × l) (UnitObject (k × l)) x Source # unit :: CatTagged (k × l) (UnitObject (k × l)) Source # const :: (Object (k × l) b, ObjectPoint (k × l) x) => x -> (k × l) b x Source # | |
| (Cartesian k, Cartesian l) => Cartesian (k × l) Source # | |
Defined in Control.Category.Constrained Methods swap :: (ObjectPair (k × l) a b, ObjectPair (k × l) b a) => (k × l) (a, b) (b, a) Source # attachUnit :: (unit ~ UnitObject (k × l), ObjectPair (k × l) a unit) => (k × l) a (a, unit) Source # detachUnit :: (unit ~ UnitObject (k × l), ObjectPair (k × l) a unit) => (k × l) (a, unit) a Source # regroup :: (ObjectPair (k × l) a b, ObjectPair (k × l) b c, ObjectPair (k × l) a (b, c), ObjectPair (k × l) (a, b) c) => (k × l) (a, (b, c)) ((a, b), c) Source # regroup' :: (ObjectPair (k × l) a b, ObjectPair (k × l) b c, ObjectPair (k × l) a (b, c), ObjectPair (k × l) (a, b) c) => (k × l) ((a, b), c) (a, (b, c)) Source # | |
| type Object (k × l :: Type -> Type -> Type) (o :: Type) Source # | |
| type UnitObject (k × l) Source # | |
Defined in Control.Category.Constrained | |
| type PointObject (k × l) o Source # | |
Defined in Control.Arrow.Constrained | |
| type PairObjects (k × l) a b Source # | |
Defined in Control.Category.Constrained type PairObjects (k × l) a b = (PairObjects k (LFactor a) (LFactor b), PairObjects l (RFactor a) (RFactor b)) | |
data GenericAgent k a v Source #
Constructors
| GenericAgent | |
Fields
| |
class Category k => HasAgent k where Source #
An agent value is a "general representation" of a category's
values, i.e. global elements. This is useful to define certain
morphisms (including ones that can't just "inherit" from ->
with arr) in ways other than point-free
composition pipelines. Instead, you can write algebraic expressions
much as if dealing with actual values of your category's objects,
but using the agent type which is restricted so any function
defined as such a lambda-expression qualifies as a morphism
of that category.
Methods
alg :: (Object k a, Object k b) => (forall q. Object k q => AgentVal k q a -> AgentVal k q b) -> k a b Source #
($~) :: (Object k a, Object k b, Object k c) => k b c -> AgentVal k a b -> AgentVal k a c infixr 0 Source #
Instances
type ObjectMorphism k b c = (Object k b, Object k c, MorphObjects k b c, Object k (k b c)) Source #
Analogous to ObjectPair: express that k b c be an exponential object
representing the morphism.
class Cartesian k => Curry k where Source #
Methods
uncurry :: (ObjectPair k a b, ObjectMorphism k b c) => k a (k b c) -> k (a, b) c Source #
curry :: (ObjectPair k a b, ObjectMorphism k b c) => k (a, b) c -> k a (k b c) Source #
apply :: (ObjectMorphism k a b, ObjectPair k (k a b) a) => k (k a b, a) b Source #
Instances
| (Curry f, o (UnitObject f)) => Curry (o ⊢ f) Source # | |
Defined in Control.Category.Constrained Associated Types type MorphObjects (o ⊢ f) b c Source # Methods uncurry :: (ObjectPair (o ⊢ f) a b, ObjectMorphism (o ⊢ f) b c) => (o ⊢ f) a ((o ⊢ f) b c) -> (o ⊢ f) (a, b) c Source # curry :: (ObjectPair (o ⊢ f) a b, ObjectMorphism (o ⊢ f) b c) => (o ⊢ f) (a, b) c -> (o ⊢ f) a ((o ⊢ f) b c) Source # apply :: (ObjectMorphism (o ⊢ f) a b, ObjectPair (o ⊢ f) ((o ⊢ f) a b) a) => (o ⊢ f) ((o ⊢ f) a b, a) b Source # | |
| (Monad m a, Arrow a (->), Function a) => Curry (Kleisli m a) Source # | |
Defined in Control.Monad.Constrained Associated Types type MorphObjects (Kleisli m a) b c Source # Methods uncurry :: (ObjectPair (Kleisli m a) a0 b, ObjectMorphism (Kleisli m a) b c) => Kleisli m a a0 (Kleisli m a b c) -> Kleisli m a (a0, b) c Source # curry :: (ObjectPair (Kleisli m a) a0 b, ObjectMorphism (Kleisli m a) b c) => Kleisli m a (a0, b) c -> Kleisli m a a0 (Kleisli m a b c) Source # apply :: (ObjectMorphism (Kleisli m a) a0 b, ObjectPair (Kleisli m a) (Kleisli m a a0 b) a0) => Kleisli m a (Kleisli m a a0 b, a0) b Source # | |
| Curry (->) Source # | |
Defined in Control.Category.Constrained Associated Types type MorphObjects (->) b c Source # Methods uncurry :: (ObjectPair (->) a b, ObjectMorphism (->) b c) => (a -> (b -> c)) -> (a, b) -> c Source # curry :: (ObjectPair (->) a b, ObjectMorphism (->) b c) => ((a, b) -> c) -> a -> (b -> c) Source # apply :: (ObjectMorphism (->) a b, ObjectPair (->) (a -> b) a) => (a -> b, a) -> b Source # | |
type CatTagged k x = Tagged (k (UnitObject k) (UnitObject k)) x Source #
Tagged type for values that depend on some choice of category, but not on some particular object / arrow therein.
type ObjectSum k a b = (Category k, Object k a, Object k b, SumObjects k a b, Object k (a + b)) Source #
class (Category k, Object k (ZeroObject k)) => CoCartesian k where Source #
Monoidal categories need not be based on a cartesian product. The relevant alternative is coproducts.
The dual notion to Cartesian replaces such products (pairs) with
sums (Either), and unit () with void types.
Basically, the only thing that doesn't mirror Cartesian here
is that we don't require CoMonoid (. Comonoids
do in principle make sense, but not from a Haskell viewpoint
(every type is trivially a comonoid).ZeroObject k)
Haskell of course uses sum types, variants, most often without
Either appearing. But variants are generally isomorphic to sums;
the most important (sums of unit) are methods here.
Associated Types
type SumObjects k a b :: Constraint Source #
type SumObjects k a b = ()
type ZeroObject k :: * Source #
Defaults to Void.
type ZeroObject k = Void
Methods
coSwap :: (ObjectSum k a b, ObjectSum k b a) => k (a + b) (b + a) Source #
attachZero :: (Object k a, zero ~ ZeroObject k, ObjectSum k a zero) => k a (a + zero) Source #
detachZero :: (Object k a, zero ~ ZeroObject k, ObjectSum k a zero) => k (a + zero) a Source #
coRegroup :: (Object k a, Object k c, ObjectSum k a b, ObjectSum k b c, ObjectSum k a (b + c), ObjectSum k (a + b) c) => k (a + (b + c)) ((a + b) + c) Source #
coRegroup' :: (Object k a, Object k c, ObjectSum k a b, ObjectSum k b c, ObjectSum k a (b + c), ObjectSum k (a + b) c) => k ((a + b) + c) (a + (b + c)) Source #
maybeAsSum :: (ObjectSum k u a, u ~ UnitObject k, Object k (Maybe a)) => k (Maybe a) (u + a) Source #
maybeFromSum :: (ObjectSum k u a, u ~ UnitObject k, Object k (Maybe a)) => k (u + a) (Maybe a) Source #
boolAsSum :: (ObjectSum k u u, u ~ UnitObject k, Object k Bool) => k Bool (u + u) Source #
boolFromSum :: (ObjectSum k u u, u ~ UnitObject k, Object k Bool) => k (u + u) Bool Source #
Instances
| CoCartesian Op Source # | |
Defined in Control.Category.Constrained Methods coSwap :: (ObjectSum Op a b, ObjectSum Op b a) => Op (a + b) (b + a) Source # attachZero :: (Object Op a, zero ~ ZeroObject Op, ObjectSum Op a zero) => Op a (a + zero) Source # detachZero :: (Object Op a, zero ~ ZeroObject Op, ObjectSum Op a zero) => Op (a + zero) a Source # coRegroup :: (Object Op a, Object Op c, ObjectSum Op a b, ObjectSum Op b c, ObjectSum Op a (b + c), ObjectSum Op (a + b) c) => Op (a + (b + c)) ((a + b) + c) Source # coRegroup' :: (Object Op a, Object Op c, ObjectSum Op a b, ObjectSum Op b c, ObjectSum Op a (b + c), ObjectSum Op (a + b) c) => Op ((a + b) + c) (a + (b + c)) Source # maybeAsSum :: (ObjectSum Op u a, u ~ UnitObject Op, Object Op (Maybe a)) => Op (Maybe a) (u + a) Source # maybeFromSum :: (ObjectSum Op u a, u ~ UnitObject Op, Object Op (Maybe a)) => Op (u + a) (Maybe a) Source # boolAsSum :: (ObjectSum Op u u, u ~ UnitObject Op, Object Op Bool) => Op Bool (u + u) Source # boolFromSum :: (ObjectSum Op u u, u ~ UnitObject Op, Object Op Bool) => Op (u + u) Bool Source # | |
| (CoCartesian f, o (ZeroObject f)) => CoCartesian (o ⊢ f) Source # | |
Defined in Control.Category.Constrained Methods coSwap :: (ObjectSum (o ⊢ f) a b, ObjectSum (o ⊢ f) b a) => (o ⊢ f) (a + b) (b + a) Source # attachZero :: (Object (o ⊢ f) a, zero ~ ZeroObject (o ⊢ f), ObjectSum (o ⊢ f) a zero) => (o ⊢ f) a (a + zero) Source # detachZero :: (Object (o ⊢ f) a, zero ~ ZeroObject (o ⊢ f), ObjectSum (o ⊢ f) a zero) => (o ⊢ f) (a + zero) a Source # coRegroup :: (Object (o ⊢ f) a, Object (o ⊢ f) c, ObjectSum (o ⊢ f) a b, ObjectSum (o ⊢ f) b c, ObjectSum (o ⊢ f) a (b + c), ObjectSum (o ⊢ f) (a + b) c) => (o ⊢ f) (a + (b + c)) ((a + b) + c) Source # coRegroup' :: (Object (o ⊢ f) a, Object (o ⊢ f) c, ObjectSum (o ⊢ f) a b, ObjectSum (o ⊢ f) b c, ObjectSum (o ⊢ f) a (b + c), ObjectSum (o ⊢ f) (a + b) c) => (o ⊢ f) ((a + b) + c) (a + (b + c)) Source # maybeAsSum :: (ObjectSum (o ⊢ f) u a, u ~ UnitObject (o ⊢ f), Object (o ⊢ f) (Maybe a)) => (o ⊢ f) (Maybe a) (u + a) Source # maybeFromSum :: (ObjectSum (o ⊢ f) u a, u ~ UnitObject (o ⊢ f), Object (o ⊢ f) (Maybe a)) => (o ⊢ f) (u + a) (Maybe a) Source # boolAsSum :: (ObjectSum (o ⊢ f) u u, u ~ UnitObject (o ⊢ f), Object (o ⊢ f) Bool) => (o ⊢ f) Bool (u + u) Source # boolFromSum :: (ObjectSum (o ⊢ f) u u, u ~ UnitObject (o ⊢ f), Object (o ⊢ f) Bool) => (o ⊢ f) (u + u) Bool Source # | |
| (Monad m k, CoCartesian k, Object k (m (ZeroObject k)), Object k (m (m (ZeroObject k)))) => CoCartesian (Kleisli m k) Source # | |
Defined in Control.Monad.Constrained Methods coSwap :: (ObjectSum (Kleisli m k) a b, ObjectSum (Kleisli m k) b a) => Kleisli m k (a + b) (b + a) Source # attachZero :: (Object (Kleisli m k) a, zero ~ ZeroObject (Kleisli m k), ObjectSum (Kleisli m k) a zero) => Kleisli m k a (a + zero) Source # detachZero :: (Object (Kleisli m k) a, zero ~ ZeroObject (Kleisli m k), ObjectSum (Kleisli m k) a zero) => Kleisli m k (a + zero) a Source # coRegroup :: (Object (Kleisli m k) a, Object (Kleisli m k) c, ObjectSum (Kleisli m k) a b, ObjectSum (Kleisli m k) b c, ObjectSum (Kleisli m k) a (b + c), ObjectSum (Kleisli m k) (a + b) c) => Kleisli m k (a + (b + c)) ((a + b) + c) Source # coRegroup' :: (Object (Kleisli m k) a, Object (Kleisli m k) c, ObjectSum (Kleisli m k) a b, ObjectSum (Kleisli m k) b c, ObjectSum (Kleisli m k) a (b + c), ObjectSum (Kleisli m k) (a + b) c) => Kleisli m k ((a + b) + c) (a + (b + c)) Source # maybeAsSum :: (ObjectSum (Kleisli m k) u a, u ~ UnitObject (Kleisli m k), Object (Kleisli m k) (Maybe a)) => Kleisli m k (Maybe a) (u + a) Source # maybeFromSum :: (ObjectSum (Kleisli m k) u a, u ~ UnitObject (Kleisli m k), Object (Kleisli m k) (Maybe a)) => Kleisli m k (u + a) (Maybe a) Source # boolAsSum :: (ObjectSum (Kleisli m k) u u, u ~ UnitObject (Kleisli m k), Object (Kleisli m k) Bool) => Kleisli m k Bool (u + u) Source # boolFromSum :: (ObjectSum (Kleisli m k) u u, u ~ UnitObject (Kleisli m k), Object (Kleisli m k) Bool) => Kleisli m k (u + u) Bool Source # | |
| CoCartesian (->) Source # | |
Defined in Control.Category.Constrained Methods coSwap :: (ObjectSum (->) a b, ObjectSum (->) b a) => (a + b) -> (b + a) Source # attachZero :: (Object (->) a, zero ~ ZeroObject (->), ObjectSum (->) a zero) => a -> (a + zero) Source # detachZero :: (Object (->) a, zero ~ ZeroObject (->), ObjectSum (->) a zero) => (a + zero) -> a Source # coRegroup :: (Object (->) a, Object (->) c, ObjectSum (->) a b, ObjectSum (->) b c, ObjectSum (->) a (b + c), ObjectSum (->) (a + b) c) => (a + (b + c)) -> ((a + b) + c) Source # coRegroup' :: (Object (->) a, Object (->) c, ObjectSum (->) a b, ObjectSum (->) b c, ObjectSum (->) a (b + c), ObjectSum (->) (a + b) c) => ((a + b) + c) -> (a + (b + c)) Source # maybeAsSum :: (ObjectSum (->) u a, u ~ UnitObject (->), Object (->) (Maybe a)) => Maybe a -> (u + a) Source # maybeFromSum :: (ObjectSum (->) u a, u ~ UnitObject (->), Object (->) (Maybe a)) => (u + a) -> Maybe a Source # boolAsSum :: (ObjectSum (->) u u, u ~ UnitObject (->), Object (->) Bool) => Bool -> (u + u) Source # boolFromSum :: (ObjectSum (->) u u, u ~ UnitObject (->), Object (->) Bool) => (u + u) -> Bool Source # | |
type ObjectPair k a b = (Category k, Object k a, Object k b, PairObjects k a b, Object k (a, b)) Source #
Use this constraint to ensure that a, b and (a,b) are all "fully valid" objects
of your category (meaning, you can use them with the Cartesian combinators).
class (Category k, Monoid (UnitObject k), Object k (UnitObject k)) => Cartesian k where Source #
Quite a few categories (monoidal categories) will permit "products" of
objects as objects again – in the Haskell sense those are tuples – allowing
for "dyadic morphisms" (x,y) ~> r.
Together with a unique UnitObject, this makes for a monoidal
structure, with a few natural isomorphisms. Ordinary tuples may not
always be powerful enough to express the product objects; we avoid
making a dedicated associated type for the sake of simplicity,
but allow for an extra constraint to be imposed on objects prior
to consideration of pair-building.
The name Cartesian is disputable: in category theory that would rather
Imply cartesian closed category (which we represent with Curry).
Monoidal would make sense, but we reserve that to Functors.
Associated Types
type PairObjects k a b :: Constraint Source #
Extra properties two types a, b need to fulfill so (a,b) can be an
object of the category. This need not take care for a and b themselves
being objects, we do that seperately: every function that actually deals
with (a,b) objects should require the stronger ObjectPair k a b
If any two object types of your category make up a pair object, then
just leave PairObjects at the default (empty constraint).
type PairObjects k a b = ()
type UnitObject k :: * Source #
Defaults to (), and should normally be left at that.
type UnitObject k = ()
Methods
swap :: (ObjectPair k a b, ObjectPair k b a) => k (a, b) (b, a) Source #
attachUnit :: (unit ~ UnitObject k, ObjectPair k a unit) => k a (a, unit) Source #
detachUnit :: (unit ~ UnitObject k, ObjectPair k a unit) => k (a, unit) a Source #
regroup :: (ObjectPair k a b, ObjectPair k b c, ObjectPair k a (b, c), ObjectPair k (a, b) c) => k (a, (b, c)) ((a, b), c) Source #
regroup' :: (ObjectPair k a b, ObjectPair k b c, ObjectPair k a (b, c), ObjectPair k (a, b) c) => k ((a, b), c) (a, (b, c)) Source #
Instances
class Category k => Isomorphic k a b where Source #
Apart from the identity morphism, id, there are other morphisms that
can basically be considered identies. For instance, in any cartesian
category (where it makes sense to have tuples and unit () at all), it should be
possible to switch between a and the isomorphic (a, ()). iso is
the method for such "pseudo-identities", the most basic of which
are required as methods of the Cartesian class.
Why it is necessary to make these morphisms explicit: they are needed
for a couple of general-purpose category-theory methods, but even though
they're normally trivial to define there is no uniform way to do so.
For instance, for vector spaces, the baseis of (a, (b,c)) and ((a,b), c)
are sure enough structurally equivalent, but not in the same way the spaces
themselves are (sum vs. product types).
Methods
Deprecated: This generic method, while looking nicely uniform, relies on OverlappingInstances and is therefore probably a bad idea. Use the specialised methods in classes like SPDistribute instead.
Instances
type ConstrainedFunction isObj = ConstrainedCategory (->) isObj Source #
type (⊢) o k = ConstrainedCategory k o Source #
data ConstrainedCategory (k :: * -> * -> *) (o :: * -> Constraint) (a :: *) (b :: *) Source #
A given category can be specialised, by using the same morphisms but adding extra constraints to what is considered an object.
For instance, Ord⊢(->)
Instances
| Category k => Category (isObj ⊢ k :: Type -> Type -> Type) Source # | |
Defined in Control.Category.Constrained | |
| Functor [] k k => Functor [] (o ⊢ k) (o ⊢ k) Source # | |
| (o (), o Void, o [Void]) => SumToProduct [] (o ⊢ (->)) (o ⊢ (->)) Source # | |
Defined in Control.Functor.Constrained Methods sum2product :: (ObjectSum (o ⊢ (->)) a b, ObjectPair (o ⊢ (->)) [a] [b]) => (o ⊢ (->)) [a + b] ([a], [b]) Source # mapEither :: (Object (o ⊢ (->)) a, ObjectSum (o ⊢ (->)) b c, Object (o ⊢ (->)) [a], ObjectPair (o ⊢ (->)) [b] [c]) => (o ⊢ (->)) a (b + c) -> (o ⊢ (->)) [a] ([b], [c]) Source # filter :: (Object (o ⊢ (->)) a, Object (o ⊢ (->)) Bool, Object (o ⊢ (->)) [a]) => (o ⊢ (->)) a Bool -> (o ⊢ (->)) [a] [a] Source # | |
| (Foldable f s t, WellPointed s, WellPointed t, Functor f (o ⊢ s) (o ⊢ t)) => Foldable f (o ⊢ s) (o ⊢ t) Source # | |
Defined in Data.Foldable.Constrained | |
| EnhancedCat (Discrete :: Type -> Type -> Type) f => EnhancedCat (Discrete :: Type -> Type -> Type) (o ⊢ f) Source # | |
| (MorphChoice k, o (ZeroObject k)) => MorphChoice (o ⊢ k) Source # | |
Defined in Control.Arrow.Constrained Methods left :: (ObjectSum (o ⊢ k) b d, ObjectSum (o ⊢ k) c d) => (o ⊢ k) b c -> (o ⊢ k) (b + d) (c + d) Source # right :: (ObjectSum (o ⊢ k) d b, ObjectSum (o ⊢ k) d c) => (o ⊢ k) b c -> (o ⊢ k) (d + b) (d + c) Source # (+++) :: (ObjectSum (o ⊢ k) b b', ObjectSum (o ⊢ k) c c') => (o ⊢ k) b c -> (o ⊢ k) b' c' -> (o ⊢ k) (b + b') (c + c') Source # | |
| (Morphism a, o (UnitObject a)) => Morphism (o ⊢ a) Source # | |
Defined in Control.Arrow.Constrained Methods first :: (ObjectPair (o ⊢ a) b d, ObjectPair (o ⊢ a) c d) => (o ⊢ a) b c -> (o ⊢ a) (b, d) (c, d) Source # second :: (ObjectPair (o ⊢ a) d b, ObjectPair (o ⊢ a) d c) => (o ⊢ a) b c -> (o ⊢ a) (d, b) (d, c) Source # (***) :: (ObjectPair (o ⊢ a) b b', ObjectPair (o ⊢ a) c c') => (o ⊢ a) b c -> (o ⊢ a) b' c' -> (o ⊢ a) (b, b') (c, c') Source # | |
| (PreArrChoice k, o (ZeroObject k)) => PreArrChoice (o ⊢ k) Source # | |
Defined in Control.Arrow.Constrained Methods (|||) :: (ObjectSum (o ⊢ k) b b', Object (o ⊢ k) c) => (o ⊢ k) b c -> (o ⊢ k) b' c -> (o ⊢ k) (b + b') c Source # initial :: Object (o ⊢ k) b => (o ⊢ k) (ZeroObject (o ⊢ k)) b Source # coFst :: ObjectSum (o ⊢ k) a b => (o ⊢ k) a (a + b) Source # coSnd :: ObjectSum (o ⊢ k) a b => (o ⊢ k) b (a + b) Source # | |
| (PreArrow a, o (UnitObject a)) => PreArrow (o ⊢ a) Source # | |
| (SPDistribute k, o (ZeroObject k), o (UnitObject k)) => SPDistribute (o ⊢ k) Source # | |
Defined in Control.Arrow.Constrained Methods distribute :: (ObjectSum (o ⊢ k) (a, b) (a, c), ObjectPair (o ⊢ k) a (b + c), ObjectSum (o ⊢ k) b c, PairObjects (o ⊢ k) a b, PairObjects (o ⊢ k) a c) => (o ⊢ k) (a, b + c) ((a, b) + (a, c)) Source # unDistribute :: (ObjectSum (o ⊢ k) (a, b) (a, c), ObjectPair (o ⊢ k) a (b + c), ObjectSum (o ⊢ k) b c, PairObjects (o ⊢ k) a b, PairObjects (o ⊢ k) a c) => (o ⊢ k) ((a, b) + (a, c)) (a, b + c) Source # boolAsSwitch :: (ObjectSum (o ⊢ k) a a, ObjectPair (o ⊢ k) Bool a) => (o ⊢ k) (Bool, a) (a + a) Source # boolFromSwitch :: (ObjectSum (o ⊢ k) a a, ObjectPair (o ⊢ k) Bool a) => (o ⊢ k) (a + a) (Bool, a) Source # | |
| (WellPointed a, o (UnitObject a)) => WellPointed (o ⊢ a) Source # | |
Defined in Control.Arrow.Constrained Associated Types type PointObject (o ⊢ a) x Source # Methods globalElement :: ObjectPoint (o ⊢ a) x => x -> (o ⊢ a) (UnitObject (o ⊢ a)) x Source # unit :: CatTagged (o ⊢ a) (UnitObject (o ⊢ a)) Source # const :: (Object (o ⊢ a) b, ObjectPoint (o ⊢ a) x) => x -> (o ⊢ a) b x Source # | |
| (Cartesian f, o (UnitObject f)) => Cartesian (o ⊢ f) Source # | |
Defined in Control.Category.Constrained Methods swap :: (ObjectPair (o ⊢ f) a b, ObjectPair (o ⊢ f) b a) => (o ⊢ f) (a, b) (b, a) Source # attachUnit :: (unit ~ UnitObject (o ⊢ f), ObjectPair (o ⊢ f) a unit) => (o ⊢ f) a (a, unit) Source # detachUnit :: (unit ~ UnitObject (o ⊢ f), ObjectPair (o ⊢ f) a unit) => (o ⊢ f) (a, unit) a Source # regroup :: (ObjectPair (o ⊢ f) a b, ObjectPair (o ⊢ f) b c, ObjectPair (o ⊢ f) a (b, c), ObjectPair (o ⊢ f) (a, b) c) => (o ⊢ f) (a, (b, c)) ((a, b), c) Source # regroup' :: (ObjectPair (o ⊢ f) a b, ObjectPair (o ⊢ f) b c, ObjectPair (o ⊢ f) a (b, c), ObjectPair (o ⊢ f) (a, b) c) => (o ⊢ f) ((a, b), c) (a, (b, c)) Source # | |
| (CoCartesian f, o (ZeroObject f)) => CoCartesian (o ⊢ f) Source # | |
Defined in Control.Category.Constrained Methods coSwap :: (ObjectSum (o ⊢ f) a b, ObjectSum (o ⊢ f) b a) => (o ⊢ f) (a + b) (b + a) Source # attachZero :: (Object (o ⊢ f) a, zero ~ ZeroObject (o ⊢ f), ObjectSum (o ⊢ f) a zero) => (o ⊢ f) a (a + zero) Source # detachZero :: (Object (o ⊢ f) a, zero ~ ZeroObject (o ⊢ f), ObjectSum (o ⊢ f) a zero) => (o ⊢ f) (a + zero) a Source # coRegroup :: (Object (o ⊢ f) a, Object (o ⊢ f) c, ObjectSum (o ⊢ f) a b, ObjectSum (o ⊢ f) b c, ObjectSum (o ⊢ f) a (b + c), ObjectSum (o ⊢ f) (a + b) c) => (o ⊢ f) (a + (b + c)) ((a + b) + c) Source # coRegroup' :: (Object (o ⊢ f) a, Object (o ⊢ f) c, ObjectSum (o ⊢ f) a b, ObjectSum (o ⊢ f) b c, ObjectSum (o ⊢ f) a (b + c), ObjectSum (o ⊢ f) (a + b) c) => (o ⊢ f) ((a + b) + c) (a + (b + c)) Source # maybeAsSum :: (ObjectSum (o ⊢ f) u a, u ~ UnitObject (o ⊢ f), Object (o ⊢ f) (Maybe a)) => (o ⊢ f) (Maybe a) (u + a) Source # maybeFromSum :: (ObjectSum (o ⊢ f) u a, u ~ UnitObject (o ⊢ f), Object (o ⊢ f) (Maybe a)) => (o ⊢ f) (u + a) (Maybe a) Source # boolAsSum :: (ObjectSum (o ⊢ f) u u, u ~ UnitObject (o ⊢ f), Object (o ⊢ f) Bool) => (o ⊢ f) Bool (u + u) Source # boolFromSum :: (ObjectSum (o ⊢ f) u u, u ~ UnitObject (o ⊢ f), Object (o ⊢ f) Bool) => (o ⊢ f) (u + u) Bool Source # | |
| (Curry f, o (UnitObject f)) => Curry (o ⊢ f) Source # | |
Defined in Control.Category.Constrained Associated Types type MorphObjects (o ⊢ f) b c Source # Methods uncurry :: (ObjectPair (o ⊢ f) a b, ObjectMorphism (o ⊢ f) b c) => (o ⊢ f) a ((o ⊢ f) b c) -> (o ⊢ f) (a, b) c Source # curry :: (ObjectPair (o ⊢ f) a b, ObjectMorphism (o ⊢ f) b c) => (o ⊢ f) (a, b) c -> (o ⊢ f) a ((o ⊢ f) b c) Source # apply :: (ObjectMorphism (o ⊢ f) a b, ObjectPair (o ⊢ f) ((o ⊢ f) a b) a) => (o ⊢ f) ((o ⊢ f) a b, a) b Source # | |
| (EnhancedCat a k, o (UnitObject a)) => EnhancedCat (o ⊢ a) k Source # | |
| Category f => EnhancedCat (o ⊢ f) (Discrete :: Type -> Type -> Type) Source # | |
| Function f => EnhancedCat (->) (o ⊢ f) Source # | |
| type Object (isObj ⊢ k :: Type -> Type -> Type) (o :: Type) Source # | |
| type UnitObject (o ⊢ f) Source # | |
Defined in Control.Category.Constrained | |
| type ZeroObject (o ⊢ f) Source # | |
Defined in Control.Category.Constrained | |
| type PointObject (o ⊢ a) x Source # | |
Defined in Control.Arrow.Constrained | |
| type MorphObjects (o ⊢ f) a c Source # | |
Defined in Control.Category.Constrained | |
| type PairObjects (o ⊢ f) a b Source # | |
Defined in Control.Category.Constrained | |
| type SumObjects (o ⊢ f) a b Source # | |
Defined in Control.Category.Constrained | |
type Hask = Unconstrained ⊢ (->) Source #
The category of all Haskell types, with (wrapped) Haskell functions as morphisms.
This is just a type-wrapper, morally equivalent to the (->) category itself.
The difference is that Functor instances in the (->)
category are automatically inherited from the standard Functor instances
that most packages define their type for. The benefit of that is that normal
Haskell code keeps working when the Prelude classes are replaced with the ones
from this library, but the downside is that you can't make more gradual instances
when this is desired. This is where the Hask category comes in: it only has functors
that are explicitly declared as such.
class Category (k :: κ -> κ -> Type) where Source #
In mathematics, a category is defined as a class of objects, plus a class of
morphisms between those objects. In Haskell, one traditionally works in
the category (->) (called Hask), in which any Haskell type is an object.
But of course
there are lots of useful categories where the objects are much more specific,
e.g. vector spaces with linear maps as morphisms. The obvious way to express
this in Haskell is as type class constraints, and the ConstraintKinds extension
allows quantifying over such object classes.
Like in Control.Category, "the category k" means actually k is the
morphism type constructor. From a mathematician's point of view this may
seem a bit strange way to define the category, but it just turns out to
be quite convenient for practical purposes.
Methods
id :: Object k a => k a a Source #
(.) :: (Object k a, Object k b, Object k c) => k b c -> k a b -> k a c infixr 9 Source #
Instances
inCategoryOf :: Category k => k a b -> k c d -> k a b Source #
Analogue to asTypeOf, this does not actually do anything but can
give the compiler type unification hints in a convenient manner.
constrained :: forall o k a b. (Category k, o a, o b) => k a b -> (o ⊢ k) a b Source #
Cast a morphism to its equivalent in a more constrained category, provided it connects objects that actually satisfy the extra constraint.
In practice, it is often necessary to specify to what typeclass it should be
constrained. The most convenient way of doing that is with
type-applications syntax.
E.g. constrained @Ord lengthlength function considered as a morphism in the subcategory of Hask in which all types are orderable. (Which makes it suitable for e.g. fmapping over a set.)
unconstrained :: forall o k a b. Category k => (o ⊢ k) a b -> k a b Source #
"Unpack" a constrained morphism again (forgetful functor).
Note that you may often not need to do that; in particular
morphisms that are actually Functions can just be applied
to their objects with $ right away, no need to go back to
Hask first.
genericAlg :: (HasAgent k, Object k a, Object k b) => (forall q. Object k q => GenericAgent k q a -> GenericAgent k q b) -> k a b Source #
genericAgentMap :: (HasAgent k, Object k a, Object k b, Object k c) => k b c -> GenericAgent k a b -> GenericAgent k a c Source #
module Control.Functor.Constrained
class (CoCartesian r, Cartesian t, Functor f r t, Object t (f (ZeroObject r))) => SumToProduct f r t where Source #
It is fairly common for functors (typically, container-like) to map Either
to tuples in a natural way, thus "separating the variants".
This is related to Foldable
(with list and tuple monoids), but rather more effective.
Methods
sum2product :: (ObjectSum r a b, ObjectPair t (f a) (f b)) => t (f (a + b)) (f a, f b) Source #
sum2product ≡ mapEither id
mapEither :: (Object r a, ObjectSum r b c, Object t (f a), ObjectPair t (f b) (f c)) => r a (b + c) -> t (f a) (f b, f c) Source #
mapEither f ≡ sum2product . fmap f
filter :: (Object r a, Object r Bool, Object t (f a)) => r a Bool -> t (f a) (f a) Source #
Instances
| (o (), o Void, o [Void]) => SumToProduct [] (o ⊢ (->)) (o ⊢ (->)) Source # | |
Defined in Control.Functor.Constrained Methods sum2product :: (ObjectSum (o ⊢ (->)) a b, ObjectPair (o ⊢ (->)) [a] [b]) => (o ⊢ (->)) [a + b] ([a], [b]) Source # mapEither :: (Object (o ⊢ (->)) a, ObjectSum (o ⊢ (->)) b c, Object (o ⊢ (->)) [a], ObjectPair (o ⊢ (->)) [b] [c]) => (o ⊢ (->)) a (b + c) -> (o ⊢ (->)) [a] ([b], [c]) Source # filter :: (Object (o ⊢ (->)) a, Object (o ⊢ (->)) Bool, Object (o ⊢ (->)) [a]) => (o ⊢ (->)) a Bool -> (o ⊢ (->)) [a] [a] Source # | |
| SumToProduct [] (->) (->) Source # | |
Defined in Control.Functor.Constrained Methods sum2product :: (ObjectSum (->) a b, ObjectPair (->) [a] [b]) => [a + b] -> ([a], [b]) Source # mapEither :: (Object (->) a, ObjectSum (->) b c, Object (->) [a], ObjectPair (->) [b] [c]) => (a -> (b + c)) -> [a] -> ([b], [c]) Source # filter :: (Object (->) a, Object (->) Bool, Object (->) [a]) => (a -> Bool) -> [a] -> [a] Source # | |
class (Category r, Category t) => Functor f r t | f r -> t, f t -> r where Source #
Methods
fmap :: (Object r a, Object t (f a), Object r b, Object t (f b)) => r a b -> t (f a) (f b) Source #
Instances
constrainedFmap :: (Category r, Category t, o a, o b, o (f a), o (f b)) => (r a b -> t (f a) (f b)) -> (o ⊢ r) a b -> (o ⊢ t) (f a) (f b) Source #
class (Monoidal f r t, Curry r, Curry t) => Applicative f r t where Source #
Minimal complete definition
Methods
pure :: (Object r a, Object t (f a)) => a `t` f a Source #
Note that this tends to make little sense for non-endofunctors.
Consider using constPure instead.
(<*>) :: (ObjectMorphism r a b, ObjectMorphism t (f a) (f b), Object t (t (f a) (f b)), ObjectPair r (r a b) a, ObjectPair t (f (r a b)) (f a), Object r a, Object r b) => f (r a b) `t` t (f a) (f b) infixl 4 Source #
Instances
| Applicative f => Applicative f (->) (->) Source # | |
Defined in Control.Applicative.Constrained Methods pure :: (Object (->) a, Object (->) (f a)) => a -> f a Source # (<*>) :: (ObjectMorphism (->) a b, ObjectMorphism (->) (f a) (f b), Object (->) (f a -> f b), ObjectPair (->) (a -> b) a, ObjectPair (->) (f (a -> b)) (f a), Object (->) a, Object (->) b) => f (a -> b) -> (f a -> f b) Source # | |
class (Functor f r t, Cartesian r, Cartesian t, Object t (f (UnitObject r))) => Monoidal f r t where Source #
Methods
pureUnit :: UnitObject t `t` f (UnitObject r) Source #
fzipWith :: (ObjectPair r a b, Object r c, ObjectPair t (f a) (f b), Object t (f c)) => r (a, b) c -> t (f a, f b) (f c) Source #
Instances
| Applicative f => Monoidal f (->) (->) Source # | |
Defined in Control.Applicative.Constrained Methods pureUnit :: UnitObject (->) -> f (UnitObject (->)) Source # fzipWith :: (ObjectPair (->) a b, Object (->) c, ObjectPair (->) (f a) (f b), Object (->) (f c)) => ((a, b) -> c) -> (f a, f b) -> f c Source # | |
constPure :: (WellPointed r, Monoidal f r t, ObjectPoint r a, Object t (f a)) => a -> t (UnitObject t) (f a) Source #
fzip :: (Monoidal f r t, ObjectPair r a b, ObjectPair t (f a) (f b), Object t (f (a, b))) => t (f a, f b) (f (a, b)) Source #
(<**>) :: (Applicative f r (->), ObjectMorphism r a b, ObjectPair r (r a b) a) => f a -> f (r a b) -> f b infixl 4 Source #
liftA :: (Applicative f r t, Object r a, Object r b, Object t (f a), Object t (f b)) => (a `r` b) -> f a `t` f b Source #
liftA2 :: (Applicative f r t, Object r c, ObjectMorphism r b c, Object t (f c), ObjectMorphism t (f b) (f c), ObjectPair r a b, ObjectPair t (f a) (f b)) => (a `r` (b `r` c)) -> f a `t` (f b `t` f c) Source #
liftA3 :: (Applicative f r t, Object r c, Object r d, ObjectMorphism r c d, ObjectMorphism r b (c `r` d), Object r (r c d), ObjectPair r a b, ObjectPair r (r c d) c, Object t (f c), Object t (f d), Object t (f a, f b), ObjectMorphism t (f c) (f d), ObjectMorphism t (f b) (t (f c) (f d)), Object t (t (f c) (f d)), ObjectPair t (f a) (f b), ObjectPair t (t (f c) (f d)) (f c), ObjectPair t (f (r c d)) (f c)) => (a `r` (b `r` (c `r` d))) -> f a `t` (f b `t` (f c `t` f d)) Source #
constrainedFZipWith :: (Category r, Category t, o a, o b, o (a, b), o c, o (f a, f b), o (f c)) => (r (a, b) c -> t (f a, f b) (f c)) -> (o ⊢ r) (a, b) c -> (o ⊢ t) (f a, f b) (f c) Source #
mapM_ :: forall t k o f a b u. (Foldable t k k, WellPointed k, Monoidal f k k, u ~ UnitObject k, ObjectPair k (f u) (t a), ObjectPair k (f u) a, ObjectPair k u (t a), ObjectPair k (t a) u, ObjectPair k (f u) (f u), ObjectPair k u u, ObjectPair k b u, Object k (f b)) => (a `k` f b) -> t a `k` f u Source #
The distinction between mapM_ and traverse_ doesn't really make sense
on grounds of Monoidal_ / Applicative vs Monad, but it has in fact some
benefits to restrict this to endofunctors, to make the constraint list
at least somewhat shorter.
forM_ :: forall t k l f a b uk ul. (Foldable t k l, Monoidal f l l, Monoidal f k k, Function l, Arrow k (->), Arrow l (->), ul ~ UnitObject l, uk ~ UnitObject k, uk ~ ul, ObjectPair l ul ul, ObjectPair l (f ul) (f ul), ObjectPair l (f ul) (t a), ObjectPair l ul (t a), ObjectPair l (t a) ul, ObjectPair l (f ul) a, ObjectPair k b (f b), ObjectPair k b ul, ObjectPair k uk uk, ObjectPair k (f uk) a, ObjectPair k (f uk) (f uk)) => t a -> (a `k` f b) -> f uk Source #
sequence_ :: forall t k l m a b uk ul. (Foldable t k l, Arrow k (->), Arrow l (->), uk ~ UnitObject k, ul ~ UnitObject l, uk ~ ul, Monoidal m k k, Monoidal m l l, ObjectPair k a uk, ObjectPair k (t (m a)) uk, ObjectPair k uk uk, ObjectPair k (m uk) (m uk), ObjectPair k (t (m a)) ul, ObjectPair l (m ul) (t (m a)), ObjectPair l ul (t (m a)), ObjectPair l (m uk) (t (m a)), ObjectPair l (t (m a)) ul, ObjectPair k (m uk) (m a)) => t (m a) `l` m uk Source #
mapM :: (Traversable s t k l, k ~ l, s ~ t, Applicative m k k, Object k a, Object k (t a), ObjectPair k b (t b), ObjectPair k (m b) (m (t b)), TraversalObject k t b) => (a `k` m b) -> t a `k` m (t b) Source #
traverse, restricted to endofunctors. May be more efficient to implement.
sequence :: (Traversable s t k l, k ~ l, s ~ t, Monoidal f k k, ObjectPair k a (t a), ObjectPair k (f a) (f (t a)), Object k (t (f a)), TraversalObject k t a) => t (f a) `k` f (t a) Source #
forM :: forall s t k m a b l. (Traversable s t k l, Monoidal m k l, Function l, Object k b, Object k (t b), ObjectPair k b (t b), Object l a, Object l (s a), ObjectPair l (m b) (m (t b)), TraversalObject k t b) => s a -> (a `l` m b) -> m (t b) Source #
newtype Kleisli m k a b Source #
Constructors
| Kleisli | |
Fields
| |
Instances
class (Applicative m k k, Object k (m (UnitObject k)), Object k (m (m (UnitObject k)))) => Monad m k where Source #
return :: Monad m (->) => a -> m a Source #
This is monomorphic in the category Hask, thus exactly the same as return
from the standard prelude. This allows writing expressions like
return $ case x of ...Monad m k => k a (m a).
Use pure when you want to "return" in categories other than (->); this always
works since Applicative is a superclass of Monad.
(=<<) :: (Monad m k, Object k a, Object k b, Object k (m a), Object k (m b), Object k (m (m b))) => k a (m b) -> k (m a) (m b) infixr 1 Source #
(>>=) :: (Function f, Monad m f, Object f a, Object f b, Object f (m a), Object f (m b), Object f (m (m b))) => m a -> f a (m b) -> m b infixl 1 Source #
(<<) :: (Monad m k, WellPointed k, Object k a, Object k b, Object k (m a), ObjectPoint k (m b), Object k (m (m b))) => m b -> k (m a) (m b) infixr 1 Source #
(>>) :: (WellPointed k, Monad m k, ObjectPair k b (UnitObject k), ObjectPair k (m b) (UnitObject k), ObjectPair k (UnitObject k) (m b), ObjectPair k b a, ObjectPair k a b, Object k (m (a, b)), ObjectPair k (m a) (m b), ObjectPoint k (m a)) => m a -> k (m b) (m b) infixl 1 Source #
when :: (Monad m k, PreArrow k, u ~ UnitObject k, ObjectPair k (m u) u) => Bool -> m u `k` m u Source #
unless :: (Monad m k, PreArrow k, u ~ UnitObject k, ObjectPair k (m u) u) => Bool -> m u `k` m u Source #
filterM :: (PreArrow k, Monad m k, SumToProduct c k k, EndoTraversable c k, ObjectPair k Bool a, Object k (c a), Object k (m (c a)), ObjectPair k (Bool, a) (c (Bool, a)), ObjectPair k (m Bool) (m a), ObjectPair k (m (Bool, a)) (m (c (Bool, a))), TraversalObject k c (Bool, a)) => (a `k` m Bool) -> c a `k` m (c a) Source #
type Function f = EnhancedCat (->) f Source #
Many categories have as morphisms essentially functions with extra properties: group homomorphisms, linear maps, continuous functions...
It makes sense to generalise the notion of function application to these
morphisms; we can't do that for the simple juxtaposition writing f x,
but it is possible for the function-application operator $.
This is particularly useful for ConstrainedCategory versions of Hask,
where after all the morphisms are nothing but functions.
const :: (WellPointed a, Object a b, ObjectPoint a x) => x -> a b x Source #
fst :: (PreArrow a, ObjectPair a x y) => a (x, y) x Source #
snd :: (PreArrow a, ObjectPair a x y) => a (x, y) y Source #
ifThenElse :: (EnhancedCat f (->), Function f, Object f Bool, Object f a, Object f (f a a), Object f (f a (f a a))) => Bool `f` (a `f` (a `f` a)) Source #
The compatible part of the standard Prelude
(++) :: [a] -> [a] -> [a] infixr 5 #
Append two lists, i.e.,
[x1, ..., xm] ++ [y1, ..., yn] == [x1, ..., xm, y1, ..., yn] [x1, ..., xm] ++ [y1, ...] == [x1, ..., xm, y1, ...]
If the first list is not finite, the result is the first list.
seq :: forall {r :: RuntimeRep} a (b :: TYPE r). a -> b -> b infixr 0 #
The value of seq a b is bottom if a is bottom, and
otherwise equal to b. In other words, it evaluates the first
argument a to weak head normal form (WHNF). seq is usually
introduced to improve performance by avoiding unneeded laziness.
A note on evaluation order: the expression seq a b does
not guarantee that a will be evaluated before b.
The only guarantee given by seq is that the both a
and b will be evaluated before seq returns a value.
In particular, this means that b may be evaluated before
a. If you need to guarantee a specific order of evaluation,
you must use the function pseq from the "parallel" package.
zip :: [a] -> [b] -> [(a, b)] #
\(\mathcal{O}(\min(m,n))\). zip takes two lists and returns a list of
corresponding pairs.
>>>zip [1, 2] ['a', 'b'][(1,'a'),(2,'b')]
If one input list is shorter than the other, excess elements of the longer list are discarded, even if one of the lists is infinite:
>>>zip [1] ['a', 'b'][(1,'a')]>>>zip [1, 2] ['a'][(1,'a')]>>>zip [] [1..][]>>>zip [1..] [][]
zip is right-lazy:
>>>zip [] undefined[]>>>zip undefined []*** Exception: Prelude.undefined ...
zip is capable of list fusion, but it is restricted to its
first list argument and its resulting list.
print :: Show a => a -> IO () #
The print function outputs a value of any printable type to the
standard output device.
Printable types are those that are instances of class Show; print
converts values to strings for output using the show operation and
adds a newline.
For example, a program to print the first 20 integers and their powers of 2 could be written as:
main = print ([(n, 2^n) | n <- [0..19]])
map :: (a -> b) -> [a] -> [b] #
\(\mathcal{O}(n)\). map f xs is the list obtained by applying f to
each element of xs, i.e.,
map f [x1, x2, ..., xn] == [f x1, f x2, ..., f xn] map f [x1, x2, ...] == [f x1, f x2, ...]
>>>map (+1) [1, 2, 3][2,3,4]
fromIntegral :: (Integral a, Num b) => a -> b #
general coercion from integral types
realToFrac :: (Real a, Fractional b) => a -> b #
general coercion to fractional types
The Bounded class is used to name the upper and lower limits of a
type. Ord is not a superclass of Bounded since types that are not
totally ordered may also have upper and lower bounds.
The Bounded class may be derived for any enumeration type;
minBound is the first constructor listed in the data declaration
and maxBound is the last.
Bounded may also be derived for single-constructor datatypes whose
constituent types are in Bounded.
Instances
| Bounded All | Since: base-2.1 |
| Bounded Any | Since: base-2.1 |
| Bounded Associativity | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| Bounded DecidedStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| Bounded SourceStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| Bounded SourceUnpackedness | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| Bounded Ordering | Since: base-2.1 |
| Bounded () | Since: base-2.1 |
| Bounded Bool | Since: base-2.1 |
| Bounded Char | Since: base-2.1 |
| Bounded Int | Since: base-2.1 |
| Bounded Levity | Since: base-4.16.0.0 |
| Bounded VecCount | Since: base-4.10.0.0 |
| Bounded VecElem | Since: base-4.10.0.0 |
| Bounded Word | Since: base-2.1 |
| Bounded a => Bounded (First a) | Since: base-4.9.0.0 |
| Bounded a => Bounded (Last a) | Since: base-4.9.0.0 |
| Bounded a => Bounded (Max a) | Since: base-4.9.0.0 |
| Bounded a => Bounded (Min a) | Since: base-4.9.0.0 |
| Bounded m => Bounded (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup | |
| Bounded a => Bounded (Dual a) | Since: base-2.1 |
| Bounded a => Bounded (Product a) | Since: base-2.1 |
| Bounded a => Bounded (Sum a) | Since: base-2.1 |
| Bounded a => Bounded (a) | |
| (Bounded a, Bounded b) => Bounded (a, b) | Since: base-2.1 |
| Bounded a => Bounded (Const a b) | Since: base-4.9.0.0 |
| (Applicative f, Bounded a) => Bounded (Ap f a) | Since: base-4.12.0.0 |
| Coercible a b => Bounded (Coercion a b) | Since: base-4.7.0.0 |
| Bounded b => Bounded (Tagged s b) | |
| (Bounded a, Bounded b, Bounded c) => Bounded (a, b, c) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d) => Bounded (a, b, c, d) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d, Bounded e) => Bounded (a, b, c, d, e) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d, Bounded e, Bounded f) => Bounded (a, b, c, d, e, f) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d, Bounded e, Bounded f, Bounded g) => Bounded (a, b, c, d, e, f, g) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d, Bounded e, Bounded f, Bounded g, Bounded h) => Bounded (a, b, c, d, e, f, g, h) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d, Bounded e, Bounded f, Bounded g, Bounded h, Bounded i) => Bounded (a, b, c, d, e, f, g, h, i) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d, Bounded e, Bounded f, Bounded g, Bounded h, Bounded i, Bounded j) => Bounded (a, b, c, d, e, f, g, h, i, j) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d, Bounded e, Bounded f, Bounded g, Bounded h, Bounded i, Bounded j, Bounded k) => Bounded (a, b, c, d, e, f, g, h, i, j, k) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d, Bounded e, Bounded f, Bounded g, Bounded h, Bounded i, Bounded j, Bounded k, Bounded l) => Bounded (a, b, c, d, e, f, g, h, i, j, k, l) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d, Bounded e, Bounded f, Bounded g, Bounded h, Bounded i, Bounded j, Bounded k, Bounded l, Bounded m) => Bounded (a, b, c, d, e, f, g, h, i, j, k, l, m) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d, Bounded e, Bounded f, Bounded g, Bounded h, Bounded i, Bounded j, Bounded k, Bounded l, Bounded m, Bounded n) => Bounded (a, b, c, d, e, f, g, h, i, j, k, l, m, n) | Since: base-2.1 |
| (Bounded a, Bounded b, Bounded c, Bounded d, Bounded e, Bounded f, Bounded g, Bounded h, Bounded i, Bounded j, Bounded k, Bounded l, Bounded m, Bounded n, Bounded o) => Bounded (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) | Since: base-2.1 |
Class Enum defines operations on sequentially ordered types.
The enumFrom... methods are used in Haskell's translation of
arithmetic sequences.
Instances of Enum may be derived for any enumeration type (types
whose constructors have no fields). The nullary constructors are
assumed to be numbered left-to-right by fromEnum from 0 through n-1.
See Chapter 10 of the Haskell Report for more details.
For any type that is an instance of class Bounded as well as Enum,
the following should hold:
- The calls
succmaxBoundpredminBound fromEnumandtoEnumshould give a runtime error if the result value is not representable in the result type. For example,toEnum7 ::BoolenumFromandenumFromThenshould be defined with an implicit bound, thus:
enumFrom x = enumFromTo x maxBound
enumFromThen x y = enumFromThenTo x y bound
where
bound | fromEnum y >= fromEnum x = maxBound
| otherwise = minBoundMethods
the successor of a value. For numeric types, succ adds 1.
the predecessor of a value. For numeric types, pred subtracts 1.
Convert from an Int.
Convert to an Int.
It is implementation-dependent what fromEnum returns when
applied to a value that is too large to fit in an Int.
Used in Haskell's translation of [n..] with [n..] = enumFrom n,
a possible implementation being enumFrom n = n : enumFrom (succ n).
For example:
enumFrom 4 :: [Integer] = [4,5,6,7,...]
enumFrom 6 :: [Int] = [6,7,8,9,...,maxBound :: Int]
enumFromThen :: a -> a -> [a] #
Used in Haskell's translation of [n,n'..]
with [n,n'..] = enumFromThen n n', a possible implementation being
enumFromThen n n' = n : n' : worker (f x) (f x n'),
worker s v = v : worker s (s v), x = fromEnum n' - fromEnum n and
f n y
| n > 0 = f (n - 1) (succ y)
| n < 0 = f (n + 1) (pred y)
| otherwise = y
For example:
enumFromThen 4 6 :: [Integer] = [4,6,8,10...]
enumFromThen 6 2 :: [Int] = [6,2,-2,-6,...,minBound :: Int]
enumFromTo :: a -> a -> [a] #
Used in Haskell's translation of [n..m] with
[n..m] = enumFromTo n m, a possible implementation being
enumFromTo n m
| n <= m = n : enumFromTo (succ n) m
| otherwise = [].
For example:
enumFromTo 6 10 :: [Int] = [6,7,8,9,10]
enumFromTo 42 1 :: [Integer] = []
enumFromThenTo :: a -> a -> a -> [a] #
Used in Haskell's translation of [n,n'..m] with
[n,n'..m] = enumFromThenTo n n' m, a possible implementation
being enumFromThenTo n n' m = worker (f x) (c x) n m,
x = fromEnum n' - fromEnum n, c x = bool (>=) ((x 0)
f n y
| n > 0 = f (n - 1) (succ y)
| n < 0 = f (n + 1) (pred y)
| otherwise = y and
worker s c v m
| c v m = v : worker s c (s v) m
| otherwise = []
For example:
enumFromThenTo 4 2 -6 :: [Integer] = [4,2,0,-2,-4,-6]
enumFromThenTo 6 8 2 :: [Int] = []
Instances
The Eq class defines equality (==) and inequality (/=).
All the basic datatypes exported by the Prelude are instances of Eq,
and Eq may be derived for any datatype whose constituents are also
instances of Eq.
The Haskell Report defines no laws for Eq. However, instances are
encouraged to follow these properties:
Instances
| Eq All | Since: base-2.1 |
| Eq Any | Since: base-2.1 |
| Eq Void | Since: base-4.8.0.0 |
| Eq SpecConstrAnnotation | Since: base-4.3.0.0 |
Defined in GHC.Exts Methods (==) :: SpecConstrAnnotation -> SpecConstrAnnotation -> Bool # (/=) :: SpecConstrAnnotation -> SpecConstrAnnotation -> Bool # | |
| Eq Associativity | Since: base-4.6.0.0 |
Defined in GHC.Generics Methods (==) :: Associativity -> Associativity -> Bool # (/=) :: Associativity -> Associativity -> Bool # | |
| Eq DecidedStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods (==) :: DecidedStrictness -> DecidedStrictness -> Bool # (/=) :: DecidedStrictness -> DecidedStrictness -> Bool # | |
| Eq Fixity | Since: base-4.6.0.0 |
| Eq SourceStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods (==) :: SourceStrictness -> SourceStrictness -> Bool # (/=) :: SourceStrictness -> SourceStrictness -> Bool # | |
| Eq SourceUnpackedness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods (==) :: SourceUnpackedness -> SourceUnpackedness -> Bool # (/=) :: SourceUnpackedness -> SourceUnpackedness -> Bool # | |
| Eq MaskingState | Since: base-4.3.0.0 |
Defined in GHC.IO | |
| Eq ArrayException | Since: base-4.2.0.0 |
Defined in GHC.IO.Exception Methods (==) :: ArrayException -> ArrayException -> Bool # (/=) :: ArrayException -> ArrayException -> Bool # | |
| Eq AsyncException | Since: base-4.2.0.0 |
Defined in GHC.IO.Exception Methods (==) :: AsyncException -> AsyncException -> Bool # (/=) :: AsyncException -> AsyncException -> Bool # | |
| Eq ExitCode | |
| Eq IOErrorType | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception | |
| Eq IOException | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception | |
| Eq SrcLoc | Since: base-4.9.0.0 |
| Eq Module | |
| Eq Ordering | |
| Eq TrName | |
| Eq TyCon | |
| Eq Integer | |
| Eq () | |
| Eq Bool | |
| Eq Char | |
| Eq Double | Note that due to the presence of
Also note that
|
| Eq Float | Note that due to the presence of
Also note that
|
| Eq Int | |
| Eq Word | |
| Eq a => Eq (ZipList a) | Since: base-4.7.0.0 |
| Eq a => Eq (Complex a) | Since: base-2.1 |
| Eq a => Eq (First a) | Since: base-2.1 |
| Eq a => Eq (Last a) | Since: base-2.1 |
| Eq a => Eq (First a) | Since: base-4.9.0.0 |
| Eq a => Eq (Last a) | Since: base-4.9.0.0 |
| Eq a => Eq (Max a) | Since: base-4.9.0.0 |
| Eq a => Eq (Min a) | Since: base-4.9.0.0 |
| Eq m => Eq (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup Methods (==) :: WrappedMonoid m -> WrappedMonoid m -> Bool # (/=) :: WrappedMonoid m -> WrappedMonoid m -> Bool # | |
| Eq a => Eq (Dual a) | Since: base-2.1 |
| Eq a => Eq (Product a) | Since: base-2.1 |
| Eq a => Eq (Sum a) | Since: base-2.1 |
| Eq p => Eq (Par1 p) | Since: base-4.7.0.0 |
| Eq a => Eq (Ratio a) | Since: base-2.1 |
| Eq a => Eq (NonEmpty a) | Since: base-4.9.0.0 |
| Eq a => Eq (Maybe a) | Since: base-2.1 |
| Eq a => Eq (a) | |
| Eq a => Eq [a] | |
| (Eq a, Eq b) => Eq (Either a b) | Since: base-2.1 |
| Eq a => Eq (Arg a b) | Since: base-4.9.0.0 |
| Eq (U1 p) | Since: base-4.9.0.0 |
| Eq (V1 p) | Since: base-4.9.0.0 |
| (Eq a, Eq b) => Eq (a, b) | |
| Eq a => Eq (Const a b) | Since: base-4.9.0.0 |
| Eq (f a) => Eq (Ap f a) | Since: base-4.12.0.0 |
| Eq (f a) => Eq (Alt f a) | Since: base-4.8.0.0 |
| Eq (Coercion a b) | Since: base-4.7.0.0 |
| Eq (f p) => Eq (Rec1 f p) | Since: base-4.7.0.0 |
| Eq (URec (Ptr ()) p) | Since: base-4.9.0.0 |
| Eq (URec Char p) | Since: base-4.9.0.0 |
| Eq (URec Double p) | Since: base-4.9.0.0 |
| Eq (URec Float p) | |
| Eq (URec Int p) | Since: base-4.9.0.0 |
| Eq (URec Word p) | Since: base-4.9.0.0 |
| Eq b => Eq (Tagged s b) | |
| (Eq a, Eq b, Eq c) => Eq (a, b, c) | |
| (Eq1 f, Eq1 g, Eq a) => Eq (Product f g a) | Since: base-4.9.0.0 |
| (Eq1 f, Eq1 g, Eq a) => Eq (Sum f g a) | Since: base-4.9.0.0 |
| (Eq (f p), Eq (g p)) => Eq ((f :*: g) p) | Since: base-4.7.0.0 |
| (Eq (f p), Eq (g p)) => Eq ((f :+: g) p) | Since: base-4.7.0.0 |
| Eq c => Eq (K1 i c p) | Since: base-4.7.0.0 |
| (Eq a, Eq b, Eq c, Eq d) => Eq (a, b, c, d) | |
| (Eq1 f, Eq1 g, Eq a) => Eq (Compose f g a) | Since: base-4.9.0.0 |
| Eq (f (g p)) => Eq ((f :.: g) p) | Since: base-4.7.0.0 |
| Eq (f p) => Eq (M1 i c f p) | Since: base-4.7.0.0 |
| (Eq a, Eq b, Eq c, Eq d, Eq e) => Eq (a, b, c, d, e) | |
| (Eq a, Eq b, Eq c, Eq d, Eq e, Eq f) => Eq (a, b, c, d, e, f) | |
| (Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g) => Eq (a, b, c, d, e, f, g) | |
| (Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h) => Eq (a, b, c, d, e, f, g, h) | |
| (Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i) => Eq (a, b, c, d, e, f, g, h, i) | |
| (Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j) => Eq (a, b, c, d, e, f, g, h, i, j) | |
| (Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j, Eq k) => Eq (a, b, c, d, e, f, g, h, i, j, k) | |
| (Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j, Eq k, Eq l) => Eq (a, b, c, d, e, f, g, h, i, j, k, l) | |
| (Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j, Eq k, Eq l, Eq m) => Eq (a, b, c, d, e, f, g, h, i, j, k, l, m) | |
| (Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j, Eq k, Eq l, Eq m, Eq n) => Eq (a, b, c, d, e, f, g, h, i, j, k, l, m, n) | |
| (Eq a, Eq b, Eq c, Eq d, Eq e, Eq f, Eq g, Eq h, Eq i, Eq j, Eq k, Eq l, Eq m, Eq n, Eq o) => Eq (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) | |
class Fractional a => Floating a where #
Trigonometric and hyperbolic functions and related functions.
The Haskell Report defines no laws for Floating. However, (, +)(
and *)exp are customarily expected to define an exponential field and have
the following properties:
exp (a + b)=exp a * exp bexp (fromInteger 0)=fromInteger 1
Minimal complete definition
pi, exp, log, sin, cos, asin, acos, atan, sinh, cosh, asinh, acosh, atanh
Instances
class Num a => Fractional a where #
Fractional numbers, supporting real division.
The Haskell Report defines no laws for Fractional. However, ( and
+)( are customarily expected to define a division ring and have the
following properties:*)
recipgives the multiplicative inversex * recip x=recip x * x=fromInteger 1
Note that it isn't customarily expected that a type instance of
Fractional implement a field. However, all instances in base do.
Minimal complete definition
fromRational, (recip | (/))
Methods
Fractional division.
Reciprocal fraction.
fromRational :: Rational -> a #
Conversion from a Rational (that is Ratio IntegerfromRational
to a value of type Rational, so such literals have type
(.Fractional a) => a
Instances
| RealFloat a => Fractional (Complex a) | Since: base-2.1 |
| Integral a => Fractional (Ratio a) | Since: base-2.0.1 |
| Fractional a => Fractional (Op a b) | |
| Fractional a => Fractional (Const a b) | Since: base-4.9.0.0 |
| Fractional a => Fractional (Tagged s a) | |
class (Real a, Enum a) => Integral a where #
Integral numbers, supporting integer division.
The Haskell Report defines no laws for Integral. However, Integral
instances are customarily expected to define a Euclidean domain and have the
following properties for the div/mod and quot/rem pairs, given
suitable Euclidean functions f and g:
x=y * quot x y + rem x ywithrem x y=fromInteger 0org (rem x y)<g yx=y * div x y + mod x ywithmod x y=fromInteger 0orf (mod x y)<f y
An example of a suitable Euclidean function, for Integer's instance, is
abs.
Methods
quot :: a -> a -> a infixl 7 #
integer division truncated toward zero
integer remainder, satisfying
(x `quot` y)*y + (x `rem` y) == x
integer division truncated toward negative infinity
integer modulus, satisfying
(x `div` y)*y + (x `mod` y) == x
conversion to Integer
Instances
| Integral Integer | Since: base-2.0.1 |
Defined in GHC.Real | |
| Integral Natural | Since: base-4.8.0.0 |
Defined in GHC.Real | |
| Integral Int | Since: base-2.0.1 |
| Integral Word | Since: base-2.1 |
| Integral a => Integral (Const a b) | Since: base-4.9.0.0 |
Defined in Data.Functor.Const Methods quot :: Const a b -> Const a b -> Const a b # rem :: Const a b -> Const a b -> Const a b # div :: Const a b -> Const a b -> Const a b # mod :: Const a b -> Const a b -> Const a b # quotRem :: Const a b -> Const a b -> (Const a b, Const a b) # divMod :: Const a b -> Const a b -> (Const a b, Const a b) # | |
| Integral a => Integral (Tagged s a) | |
Defined in Data.Tagged Methods quot :: Tagged s a -> Tagged s a -> Tagged s a # rem :: Tagged s a -> Tagged s a -> Tagged s a # div :: Tagged s a -> Tagged s a -> Tagged s a # mod :: Tagged s a -> Tagged s a -> Tagged s a # quotRem :: Tagged s a -> Tagged s a -> (Tagged s a, Tagged s a) # divMod :: Tagged s a -> Tagged s a -> (Tagged s a, Tagged s a) # | |
Basic numeric class.
The Haskell Report defines no laws for Num. However, ( and +)( are
customarily expected to define a ring and have the following properties:*)
- Associativity of
(+) (x + y) + z=x + (y + z)- Commutativity of
(+) x + y=y + xfromInteger0x + fromInteger 0=xnegategives the additive inversex + negate x=fromInteger 0- Associativity of
(*) (x * y) * z=x * (y * z)fromInteger1x * fromInteger 1=xandfromInteger 1 * x=x- Distributivity of
(with respect to*)(+) a * (b + c)=(a * b) + (a * c)and(b + c) * a=(b * a) + (c * a)
Note that it isn't customarily expected that a type instance of both Num
and Ord implement an ordered ring. Indeed, in base only Integer and
Rational do.
Methods
Unary negation.
Absolute value.
Sign of a number.
The functions abs and signum should satisfy the law:
abs x * signum x == x
For real numbers, the signum is either -1 (negative), 0 (zero)
or 1 (positive).
fromInteger :: Integer -> a #
Conversion from an Integer.
An integer literal represents the application of the function
fromInteger to the appropriate value of type Integer,
so such literals have type (.Num a) => a
Instances
| Num Integer | Since: base-2.1 |
| Num Natural | Note that Since: base-4.8.0.0 |
| Num Int | Since: base-2.1 |
| Num Word | Since: base-2.1 |
| RealFloat a => Num (Complex a) | Since: base-2.1 |
| Num a => Num (Max a) | Since: base-4.9.0.0 |
| Num a => Num (Min a) | Since: base-4.9.0.0 |
| Num a => Num (Product a) | Since: base-4.7.0.0 |
Defined in Data.Semigroup.Internal | |
| Num a => Num (Sum a) | Since: base-4.7.0.0 |
| Integral a => Num (Ratio a) | Since: base-2.0.1 |
| Num a => Num (Op a b) | |
| Num a => Num (Const a b) | Since: base-4.9.0.0 |
Defined in Data.Functor.Const | |
| (Applicative f, Num a) => Num (Ap f a) | Note that even if the underlying Commutativity:
Additive inverse:
Distributivity:
Since: base-4.12.0.0 |
| Num (f a) => Num (Alt f a) | Since: base-4.8.0.0 |
| Num a => Num (Tagged s a) | |
Defined in Data.Tagged | |
The Ord class is used for totally ordered datatypes.
Instances of Ord can be derived for any user-defined datatype whose
constituent types are in Ord. The declared order of the constructors in
the data declaration determines the ordering in derived Ord instances. The
Ordering datatype allows a single comparison to determine the precise
ordering of two objects.
Ord, as defined by the Haskell report, implements a total order and has the
following properties:
- Comparability
x <= y || y <= x=True- Transitivity
- if
x <= y && y <= z=True, thenx <= z=True - Reflexivity
x <= x=True- Antisymmetry
- if
x <= y && y <= x=True, thenx == y=True
The following operator interactions are expected to hold:
x >= y=y <= xx < y=x <= y && x /= yx > y=y < xx < y=compare x y == LTx > y=compare x y == GTx == y=compare x y == EQmin x y == if x <= y then x else y=Truemax x y == if x >= y then x else y=True
Note that (7.) and (8.) do not require min and max to return either of
their arguments. The result is merely required to equal one of the
arguments in terms of (==).
Minimal complete definition: either compare or <=.
Using compare can be more efficient for complex types.
Methods
compare :: a -> a -> Ordering #
(<) :: a -> a -> Bool infix 4 #
(<=) :: a -> a -> Bool infix 4 #
(>) :: a -> a -> Bool infix 4 #
Instances
| Ord All | Since: base-2.1 |
| Ord Any | Since: base-2.1 |
| Ord Void | Since: base-4.8.0.0 |
| Ord Associativity | Since: base-4.6.0.0 |
Defined in GHC.Generics Methods compare :: Associativity -> Associativity -> Ordering # (<) :: Associativity -> Associativity -> Bool # (<=) :: Associativity -> Associativity -> Bool # (>) :: Associativity -> Associativity -> Bool # (>=) :: Associativity -> Associativity -> Bool # max :: Associativity -> Associativity -> Associativity # min :: Associativity -> Associativity -> Associativity # | |
| Ord DecidedStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods compare :: DecidedStrictness -> DecidedStrictness -> Ordering # (<) :: DecidedStrictness -> DecidedStrictness -> Bool # (<=) :: DecidedStrictness -> DecidedStrictness -> Bool # (>) :: DecidedStrictness -> DecidedStrictness -> Bool # (>=) :: DecidedStrictness -> DecidedStrictness -> Bool # max :: DecidedStrictness -> DecidedStrictness -> DecidedStrictness # min :: DecidedStrictness -> DecidedStrictness -> DecidedStrictness # | |
| Ord Fixity | Since: base-4.6.0.0 |
| Ord SourceStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods compare :: SourceStrictness -> SourceStrictness -> Ordering # (<) :: SourceStrictness -> SourceStrictness -> Bool # (<=) :: SourceStrictness -> SourceStrictness -> Bool # (>) :: SourceStrictness -> SourceStrictness -> Bool # (>=) :: SourceStrictness -> SourceStrictness -> Bool # max :: SourceStrictness -> SourceStrictness -> SourceStrictness # min :: SourceStrictness -> SourceStrictness -> SourceStrictness # | |
| Ord SourceUnpackedness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods compare :: SourceUnpackedness -> SourceUnpackedness -> Ordering # (<) :: SourceUnpackedness -> SourceUnpackedness -> Bool # (<=) :: SourceUnpackedness -> SourceUnpackedness -> Bool # (>) :: SourceUnpackedness -> SourceUnpackedness -> Bool # (>=) :: SourceUnpackedness -> SourceUnpackedness -> Bool # max :: SourceUnpackedness -> SourceUnpackedness -> SourceUnpackedness # min :: SourceUnpackedness -> SourceUnpackedness -> SourceUnpackedness # | |
| Ord ArrayException | Since: base-4.2.0.0 |
Defined in GHC.IO.Exception Methods compare :: ArrayException -> ArrayException -> Ordering # (<) :: ArrayException -> ArrayException -> Bool # (<=) :: ArrayException -> ArrayException -> Bool # (>) :: ArrayException -> ArrayException -> Bool # (>=) :: ArrayException -> ArrayException -> Bool # max :: ArrayException -> ArrayException -> ArrayException # min :: ArrayException -> ArrayException -> ArrayException # | |
| Ord AsyncException | Since: base-4.2.0.0 |
Defined in GHC.IO.Exception Methods compare :: AsyncException -> AsyncException -> Ordering # (<) :: AsyncException -> AsyncException -> Bool # (<=) :: AsyncException -> AsyncException -> Bool # (>) :: AsyncException -> AsyncException -> Bool # (>=) :: AsyncException -> AsyncException -> Bool # max :: AsyncException -> AsyncException -> AsyncException # min :: AsyncException -> AsyncException -> AsyncException # | |
| Ord ExitCode | |
Defined in GHC.IO.Exception | |
| Ord Ordering | |
Defined in GHC.Classes | |
| Ord TyCon | |
| Ord Integer | |
| Ord () | |
| Ord Bool | |
| Ord Char | |
| Ord Double | Note that due to the presence of
Also note that, due to the same,
|
| Ord Float | Note that due to the presence of
Also note that, due to the same,
|
| Ord Int | |
| Ord Word | |
| Ord a => Ord (ZipList a) | Since: base-4.7.0.0 |
| Ord a => Ord (First a) | Since: base-2.1 |
| Ord a => Ord (Last a) | Since: base-2.1 |
| Ord a => Ord (First a) | Since: base-4.9.0.0 |
| Ord a => Ord (Last a) | Since: base-4.9.0.0 |
| Ord a => Ord (Max a) | Since: base-4.9.0.0 |
| Ord a => Ord (Min a) | Since: base-4.9.0.0 |
| Ord m => Ord (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup Methods compare :: WrappedMonoid m -> WrappedMonoid m -> Ordering # (<) :: WrappedMonoid m -> WrappedMonoid m -> Bool # (<=) :: WrappedMonoid m -> WrappedMonoid m -> Bool # (>) :: WrappedMonoid m -> WrappedMonoid m -> Bool # (>=) :: WrappedMonoid m -> WrappedMonoid m -> Bool # max :: WrappedMonoid m -> WrappedMonoid m -> WrappedMonoid m # min :: WrappedMonoid m -> WrappedMonoid m -> WrappedMonoid m # | |
| Ord a => Ord (Dual a) | Since: base-2.1 |
| Ord a => Ord (Product a) | Since: base-2.1 |
| Ord a => Ord (Sum a) | Since: base-2.1 |
| Ord p => Ord (Par1 p) | Since: base-4.7.0.0 |
| Integral a => Ord (Ratio a) | Since: base-2.0.1 |
| Ord a => Ord (NonEmpty a) | Since: base-4.9.0.0 |
| Ord a => Ord (Maybe a) | Since: base-2.1 |
| Ord a => Ord (a) | |
| Ord a => Ord [a] | |
| (Ord a, Ord b) => Ord (Either a b) | Since: base-2.1 |
| Ord a => Ord (Arg a b) | Since: base-4.9.0.0 |
| Ord (U1 p) | Since: base-4.7.0.0 |
| Ord (V1 p) | Since: base-4.9.0.0 |
| (Ord a, Ord b) => Ord (a, b) | |
| Ord a => Ord (Const a b) | Since: base-4.9.0.0 |
| Ord (f a) => Ord (Ap f a) | Since: base-4.12.0.0 |
| Ord (f a) => Ord (Alt f a) | Since: base-4.8.0.0 |
Defined in Data.Semigroup.Internal | |
| Ord (Coercion a b) | Since: base-4.7.0.0 |
Defined in Data.Type.Coercion | |
| Ord (f p) => Ord (Rec1 f p) | Since: base-4.7.0.0 |
Defined in GHC.Generics | |
| Ord (URec (Ptr ()) p) | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods compare :: URec (Ptr ()) p -> URec (Ptr ()) p -> Ordering # (<) :: URec (Ptr ()) p -> URec (Ptr ()) p -> Bool # (<=) :: URec (Ptr ()) p -> URec (Ptr ()) p -> Bool # (>) :: URec (Ptr ()) p -> URec (Ptr ()) p -> Bool # (>=) :: URec (Ptr ()) p -> URec (Ptr ()) p -> Bool # max :: URec (Ptr ()) p -> URec (Ptr ()) p -> URec (Ptr ()) p # min :: URec (Ptr ()) p -> URec (Ptr ()) p -> URec (Ptr ()) p # | |
| Ord (URec Char p) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| Ord (URec Double p) | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods compare :: URec Double p -> URec Double p -> Ordering # (<) :: URec Double p -> URec Double p -> Bool # (<=) :: URec Double p -> URec Double p -> Bool # (>) :: URec Double p -> URec Double p -> Bool # (>=) :: URec Double p -> URec Double p -> Bool # | |
| Ord (URec Float p) | |
Defined in GHC.Generics | |
| Ord (URec Int p) | Since: base-4.9.0.0 |
| Ord (URec Word p) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| Ord b => Ord (Tagged s b) | |
| (Ord a, Ord b, Ord c) => Ord (a, b, c) | |
| (Ord1 f, Ord1 g, Ord a) => Ord (Product f g a) | Since: base-4.9.0.0 |
Defined in Data.Functor.Product Methods compare :: Product f g a -> Product f g a -> Ordering # (<) :: Product f g a -> Product f g a -> Bool # (<=) :: Product f g a -> Product f g a -> Bool # (>) :: Product f g a -> Product f g a -> Bool # (>=) :: Product f g a -> Product f g a -> Bool # | |
| (Ord1 f, Ord1 g, Ord a) => Ord (Sum f g a) | Since: base-4.9.0.0 |
| (Ord (f p), Ord (g p)) => Ord ((f :*: g) p) | Since: base-4.7.0.0 |
Defined in GHC.Generics | |
| (Ord (f p), Ord (g p)) => Ord ((f :+: g) p) | Since: base-4.7.0.0 |
Defined in GHC.Generics | |
| Ord c => Ord (K1 i c p) | Since: base-4.7.0.0 |
Defined in GHC.Generics | |
| (Ord a, Ord b, Ord c, Ord d) => Ord (a, b, c, d) | |
Defined in GHC.Classes | |
| (Ord1 f, Ord1 g, Ord a) => Ord (Compose f g a) | Since: base-4.9.0.0 |
Defined in Data.Functor.Compose Methods compare :: Compose f g a -> Compose f g a -> Ordering # (<) :: Compose f g a -> Compose f g a -> Bool # (<=) :: Compose f g a -> Compose f g a -> Bool # (>) :: Compose f g a -> Compose f g a -> Bool # (>=) :: Compose f g a -> Compose f g a -> Bool # | |
| Ord (f (g p)) => Ord ((f :.: g) p) | Since: base-4.7.0.0 |
Defined in GHC.Generics | |
| Ord (f p) => Ord (M1 i c f p) | Since: base-4.7.0.0 |
| (Ord a, Ord b, Ord c, Ord d, Ord e) => Ord (a, b, c, d, e) | |
Defined in GHC.Classes Methods compare :: (a, b, c, d, e) -> (a, b, c, d, e) -> Ordering # (<) :: (a, b, c, d, e) -> (a, b, c, d, e) -> Bool # (<=) :: (a, b, c, d, e) -> (a, b, c, d, e) -> Bool # (>) :: (a, b, c, d, e) -> (a, b, c, d, e) -> Bool # (>=) :: (a, b, c, d, e) -> (a, b, c, d, e) -> Bool # max :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) # min :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) # | |
| (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f) => Ord (a, b, c, d, e, f) | |
Defined in GHC.Classes Methods compare :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> Ordering # (<) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> Bool # (<=) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> Bool # (>) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> Bool # (>=) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> Bool # max :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) # min :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) # | |
| (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g) => Ord (a, b, c, d, e, f, g) | |
Defined in GHC.Classes Methods compare :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> Ordering # (<) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> Bool # (<=) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> Bool # (>) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> Bool # (>=) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> Bool # max :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) # min :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) # | |
| (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h) => Ord (a, b, c, d, e, f, g, h) | |
Defined in GHC.Classes Methods compare :: (a, b, c, d, e, f, g, h) -> (a, b, c, d, e, f, g, h) -> Ordering # (<) :: (a, b, c, d, e, f, g, h) -> (a, b, c, d, e, f, g, h) -> Bool # (<=) :: (a, b, c, d, e, f, g, h) -> (a, b, c, d, e, f, g, h) -> Bool # (>) :: (a, b, c, d, e, f, g, h) -> (a, b, c, d, e, f, g, h) -> Bool # (>=) :: (a, b, c, d, e, f, g, h) -> (a, b, c, d, e, f, g, h) -> Bool # max :: (a, b, c, d, e, f, g, h) -> (a, b, c, d, e, f, g, h) -> (a, b, c, d, e, f, g, h) # min :: (a, b, c, d, e, f, g, h) -> (a, b, c, d, e, f, g, h) -> (a, b, c, d, e, f, g, h) # | |
| (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i) => Ord (a, b, c, d, e, f, g, h, i) | |
Defined in GHC.Classes Methods compare :: (a, b, c, d, e, f, g, h, i) -> (a, b, c, d, e, f, g, h, i) -> Ordering # (<) :: (a, b, c, d, e, f, g, h, i) -> (a, b, c, d, e, f, g, h, i) -> Bool # (<=) :: (a, b, c, d, e, f, g, h, i) -> (a, b, c, d, e, f, g, h, i) -> Bool # (>) :: (a, b, c, d, e, f, g, h, i) -> (a, b, c, d, e, f, g, h, i) -> Bool # (>=) :: (a, b, c, d, e, f, g, h, i) -> (a, b, c, d, e, f, g, h, i) -> Bool # max :: (a, b, c, d, e, f, g, h, i) -> (a, b, c, d, e, f, g, h, i) -> (a, b, c, d, e, f, g, h, i) # min :: (a, b, c, d, e, f, g, h, i) -> (a, b, c, d, e, f, g, h, i) -> (a, b, c, d, e, f, g, h, i) # | |
| (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j) => Ord (a, b, c, d, e, f, g, h, i, j) | |
Defined in GHC.Classes Methods compare :: (a, b, c, d, e, f, g, h, i, j) -> (a, b, c, d, e, f, g, h, i, j) -> Ordering # (<) :: (a, b, c, d, e, f, g, h, i, j) -> (a, b, c, d, e, f, g, h, i, j) -> Bool # (<=) :: (a, b, c, d, e, f, g, h, i, j) -> (a, b, c, d, e, f, g, h, i, j) -> Bool # (>) :: (a, b, c, d, e, f, g, h, i, j) -> (a, b, c, d, e, f, g, h, i, j) -> Bool # (>=) :: (a, b, c, d, e, f, g, h, i, j) -> (a, b, c, d, e, f, g, h, i, j) -> Bool # max :: (a, b, c, d, e, f, g, h, i, j) -> (a, b, c, d, e, f, g, h, i, j) -> (a, b, c, d, e, f, g, h, i, j) # min :: (a, b, c, d, e, f, g, h, i, j) -> (a, b, c, d, e, f, g, h, i, j) -> (a, b, c, d, e, f, g, h, i, j) # | |
| (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j, Ord k) => Ord (a, b, c, d, e, f, g, h, i, j, k) | |
Defined in GHC.Classes Methods compare :: (a, b, c, d, e, f, g, h, i, j, k) -> (a, b, c, d, e, f, g, h, i, j, k) -> Ordering # (<) :: (a, b, c, d, e, f, g, h, i, j, k) -> (a, b, c, d, e, f, g, h, i, j, k) -> Bool # (<=) :: (a, b, c, d, e, f, g, h, i, j, k) -> (a, b, c, d, e, f, g, h, i, j, k) -> Bool # (>) :: (a, b, c, d, e, f, g, h, i, j, k) -> (a, b, c, d, e, f, g, h, i, j, k) -> Bool # (>=) :: (a, b, c, d, e, f, g, h, i, j, k) -> (a, b, c, d, e, f, g, h, i, j, k) -> Bool # max :: (a, b, c, d, e, f, g, h, i, j, k) -> (a, b, c, d, e, f, g, h, i, j, k) -> (a, b, c, d, e, f, g, h, i, j, k) # min :: (a, b, c, d, e, f, g, h, i, j, k) -> (a, b, c, d, e, f, g, h, i, j, k) -> (a, b, c, d, e, f, g, h, i, j, k) # | |
| (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j, Ord k, Ord l) => Ord (a, b, c, d, e, f, g, h, i, j, k, l) | |
Defined in GHC.Classes Methods compare :: (a, b, c, d, e, f, g, h, i, j, k, l) -> (a, b, c, d, e, f, g, h, i, j, k, l) -> Ordering # (<) :: (a, b, c, d, e, f, g, h, i, j, k, l) -> (a, b, c, d, e, f, g, h, i, j, k, l) -> Bool # (<=) :: (a, b, c, d, e, f, g, h, i, j, k, l) -> (a, b, c, d, e, f, g, h, i, j, k, l) -> Bool # (>) :: (a, b, c, d, e, f, g, h, i, j, k, l) -> (a, b, c, d, e, f, g, h, i, j, k, l) -> Bool # (>=) :: (a, b, c, d, e, f, g, h, i, j, k, l) -> (a, b, c, d, e, f, g, h, i, j, k, l) -> Bool # max :: (a, b, c, d, e, f, g, h, i, j, k, l) -> (a, b, c, d, e, f, g, h, i, j, k, l) -> (a, b, c, d, e, f, g, h, i, j, k, l) # min :: (a, b, c, d, e, f, g, h, i, j, k, l) -> (a, b, c, d, e, f, g, h, i, j, k, l) -> (a, b, c, d, e, f, g, h, i, j, k, l) # | |
| (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j, Ord k, Ord l, Ord m) => Ord (a, b, c, d, e, f, g, h, i, j, k, l, m) | |
Defined in GHC.Classes Methods compare :: (a, b, c, d, e, f, g, h, i, j, k, l, m) -> (a, b, c, d, e, f, g, h, i, j, k, l, m) -> Ordering # (<) :: (a, b, c, d, e, f, g, h, i, j, k, l, m) -> (a, b, c, d, e, f, g, h, i, j, k, l, m) -> Bool # (<=) :: (a, b, c, d, e, f, g, h, i, j, k, l, m) -> (a, b, c, d, e, f, g, h, i, j, k, l, m) -> Bool # (>) :: (a, b, c, d, e, f, g, h, i, j, k, l, m) -> (a, b, c, d, e, f, g, h, i, j, k, l, m) -> Bool # (>=) :: (a, b, c, d, e, f, g, h, i, j, k, l, m) -> (a, b, c, d, e, f, g, h, i, j, k, l, m) -> Bool # max :: (a, b, c, d, e, f, g, h, i, j, k, l, m) -> (a, b, c, d, e, f, g, h, i, j, k, l, m) -> (a, b, c, d, e, f, g, h, i, j, k, l, m) # min :: (a, b, c, d, e, f, g, h, i, j, k, l, m) -> (a, b, c, d, e, f, g, h, i, j, k, l, m) -> (a, b, c, d, e, f, g, h, i, j, k, l, m) # | |
| (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j, Ord k, Ord l, Ord m, Ord n) => Ord (a, b, c, d, e, f, g, h, i, j, k, l, m, n) | |
Defined in GHC.Classes Methods compare :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> Ordering # (<) :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> Bool # (<=) :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> Bool # (>) :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> Bool # (>=) :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> Bool # max :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n) # min :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n) # | |
| (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f, Ord g, Ord h, Ord i, Ord j, Ord k, Ord l, Ord m, Ord n, Ord o) => Ord (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) | |
Defined in GHC.Classes Methods compare :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> Ordering # (<) :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> Bool # (<=) :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> Bool # (>) :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> Bool # (>=) :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> Bool # max :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) # min :: (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) -> (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) # | |
Parsing of Strings, producing values.
Derived instances of Read make the following assumptions, which
derived instances of Show obey:
- If the constructor is defined to be an infix operator, then the
derived
Readinstance will parse only infix applications of the constructor (not the prefix form). - Associativity is not used to reduce the occurrence of parentheses, although precedence may be.
- If the constructor is defined using record syntax, the derived
Readwill parse only the record-syntax form, and furthermore, the fields must be given in the same order as the original declaration. - The derived
Readinstance allows arbitrary Haskell whitespace between tokens of the input string. Extra parentheses are also allowed.
For example, given the declarations
infixr 5 :^: data Tree a = Leaf a | Tree a :^: Tree a
the derived instance of Read in Haskell 2010 is equivalent to
instance (Read a) => Read (Tree a) where
readsPrec d r = readParen (d > app_prec)
(\r -> [(Leaf m,t) |
("Leaf",s) <- lex r,
(m,t) <- readsPrec (app_prec+1) s]) r
++ readParen (d > up_prec)
(\r -> [(u:^:v,w) |
(u,s) <- readsPrec (up_prec+1) r,
(":^:",t) <- lex s,
(v,w) <- readsPrec (up_prec+1) t]) r
where app_prec = 10
up_prec = 5Note that right-associativity of :^: is unused.
The derived instance in GHC is equivalent to
instance (Read a) => Read (Tree a) where
readPrec = parens $ (prec app_prec $ do
Ident "Leaf" <- lexP
m <- step readPrec
return (Leaf m))
+++ (prec up_prec $ do
u <- step readPrec
Symbol ":^:" <- lexP
v <- step readPrec
return (u :^: v))
where app_prec = 10
up_prec = 5
readListPrec = readListPrecDefaultWhy do both readsPrec and readPrec exist, and why does GHC opt to
implement readPrec in derived Read instances instead of readsPrec?
The reason is that readsPrec is based on the ReadS type, and although
ReadS is mentioned in the Haskell 2010 Report, it is not a very efficient
parser data structure.
readPrec, on the other hand, is based on a much more efficient ReadPrec
datatype (a.k.a "new-style parsers"), but its definition relies on the use
of the RankNTypes language extension. Therefore, readPrec (and its
cousin, readListPrec) are marked as GHC-only. Nevertheless, it is
recommended to use readPrec instead of readsPrec whenever possible
for the efficiency improvements it brings.
As mentioned above, derived Read instances in GHC will implement
readPrec instead of readsPrec. The default implementations of
readsPrec (and its cousin, readList) will simply use readPrec under
the hood. If you are writing a Read instance by hand, it is recommended
to write it like so:
instanceReadT wherereadPrec= ...readListPrec=readListPrecDefault
Methods
Arguments
| :: Int | the operator precedence of the enclosing
context (a number from |
| -> ReadS a |
attempts to parse a value from the front of the string, returning a list of (parsed value, remaining string) pairs. If there is no successful parse, the returned list is empty.
Derived instances of Read and Show satisfy the following:
That is, readsPrec parses the string produced by
showsPrec, and delivers the value that
showsPrec started with.
Instances
| Read All | Since: base-2.1 |
| Read Any | Since: base-2.1 |
| Read Void | Reading a Since: base-4.8.0.0 |
| Read Associativity | Since: base-4.6.0.0 |
Defined in GHC.Generics Methods readsPrec :: Int -> ReadS Associativity # readList :: ReadS [Associativity] # | |
| Read DecidedStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods readsPrec :: Int -> ReadS DecidedStrictness # readList :: ReadS [DecidedStrictness] # | |
| Read Fixity | Since: base-4.6.0.0 |
| Read SourceStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods readsPrec :: Int -> ReadS SourceStrictness # readList :: ReadS [SourceStrictness] # | |
| Read SourceUnpackedness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods readsPrec :: Int -> ReadS SourceUnpackedness # readList :: ReadS [SourceUnpackedness] # | |
| Read ExitCode | |
| Read GeneralCategory | Since: base-2.1 |
Defined in GHC.Read Methods readsPrec :: Int -> ReadS GeneralCategory # readList :: ReadS [GeneralCategory] # | |
| Read Word16 | Since: base-2.1 |
| Read Word32 | Since: base-2.1 |
| Read Word64 | Since: base-2.1 |
| Read Word8 | Since: base-2.1 |
| Read Lexeme | Since: base-2.1 |
| Read Ordering | Since: base-2.1 |
| Read Integer | Since: base-2.1 |
| Read Natural | Since: base-4.8.0.0 |
| Read () | Since: base-2.1 |
| Read Bool | Since: base-2.1 |
| Read Char | Since: base-2.1 |
| Read Double | Since: base-2.1 |
| Read Float | Since: base-2.1 |
| Read Int | Since: base-2.1 |
| Read Word | Since: base-4.5.0.0 |
| Read a => Read (ZipList a) | Since: base-4.7.0.0 |
| Read a => Read (Complex a) | Since: base-2.1 |
| Read a => Read (First a) | Since: base-2.1 |
| Read a => Read (Last a) | Since: base-2.1 |
| Read a => Read (First a) | Since: base-4.9.0.0 |
| Read a => Read (Last a) | Since: base-4.9.0.0 |
| Read a => Read (Max a) | Since: base-4.9.0.0 |
| Read a => Read (Min a) | Since: base-4.9.0.0 |
| Read m => Read (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup Methods readsPrec :: Int -> ReadS (WrappedMonoid m) # readList :: ReadS [WrappedMonoid m] # readPrec :: ReadPrec (WrappedMonoid m) # readListPrec :: ReadPrec [WrappedMonoid m] # | |
| Read a => Read (Dual a) | Since: base-2.1 |
| Read a => Read (Product a) | Since: base-2.1 |
| Read a => Read (Sum a) | Since: base-2.1 |
| Read p => Read (Par1 p) | Since: base-4.7.0.0 |
| (Integral a, Read a) => Read (Ratio a) | Since: base-2.1 |
| Read a => Read (NonEmpty a) | Since: base-4.11.0.0 |
| Read a => Read (Maybe a) | Since: base-2.1 |
| Read a => Read (a) | Since: base-4.15 |
| Read a => Read [a] | Since: base-2.1 |
| (Read a, Read b) => Read (Either a b) | Since: base-3.0 |
| (Read a, Read b) => Read (Arg a b) | Since: base-4.9.0.0 |
| (Ix a, Read a, Read b) => Read (Array a b) | Since: base-2.1 |
| Read (U1 p) | Since: base-4.9.0.0 |
| Read (V1 p) | Since: base-4.9.0.0 |
| (Read a, Read b) => Read (a, b) | Since: base-2.1 |
| Read a => Read (Const a b) | This instance would be equivalent to the derived instances of the
Since: base-4.8.0.0 |
| Read (f a) => Read (Ap f a) | Since: base-4.12.0.0 |
| Read (f a) => Read (Alt f a) | Since: base-4.8.0.0 |
| Coercible a b => Read (Coercion a b) | Since: base-4.7.0.0 |
| Read (f p) => Read (Rec1 f p) | Since: base-4.7.0.0 |
| Read b => Read (Tagged s b) | |
| (Read a, Read b, Read c) => Read (a, b, c) | Since: base-2.1 |
| (Read1 f, Read1 g, Read a) => Read (Product f g a) | Since: base-4.9.0.0 |
| (Read1 f, Read1 g, Read a) => Read (Sum f g a) | Since: base-4.9.0.0 |
| (Read (f p), Read (g p)) => Read ((f :*: g) p) | Since: base-4.7.0.0 |
| (Read (f p), Read (g p)) => Read ((f :+: g) p) | Since: base-4.7.0.0 |
| Read c => Read (K1 i c p) | Since: base-4.7.0.0 |
| (Read a, Read b, Read c, Read d) => Read (a, b, c, d) | Since: base-2.1 |
| (Read1 f, Read1 g, Read a) => Read (Compose f g a) | Since: base-4.9.0.0 |
| Read (f (g p)) => Read ((f :.: g) p) | Since: base-4.7.0.0 |
| Read (f p) => Read (M1 i c f p) | Since: base-4.7.0.0 |
| (Read a, Read b, Read c, Read d, Read e) => Read (a, b, c, d, e) | Since: base-2.1 |
| (Read a, Read b, Read c, Read d, Read e, Read f) => Read (a, b, c, d, e, f) | Since: base-2.1 |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g) => Read (a, b, c, d, e, f, g) | Since: base-2.1 |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h) => Read (a, b, c, d, e, f, g, h) | Since: base-2.1 |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i) => Read (a, b, c, d, e, f, g, h, i) | Since: base-2.1 |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j) => Read (a, b, c, d, e, f, g, h, i, j) | Since: base-2.1 |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j, Read k) => Read (a, b, c, d, e, f, g, h, i, j, k) | Since: base-2.1 |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j, Read k, Read l) => Read (a, b, c, d, e, f, g, h, i, j, k, l) | Since: base-2.1 |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j, Read k, Read l, Read m) => Read (a, b, c, d, e, f, g, h, i, j, k, l, m) | Since: base-2.1 |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j, Read k, Read l, Read m, Read n) => Read (a, b, c, d, e, f, g, h, i, j, k, l, m, n) | Since: base-2.1 |
| (Read a, Read b, Read c, Read d, Read e, Read f, Read g, Read h, Read i, Read j, Read k, Read l, Read m, Read n, Read o) => Read (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) | Since: base-2.1 |
Defined in GHC.Read | |
class (Num a, Ord a) => Real a where #
Methods
toRational :: a -> Rational #
the rational equivalent of its real argument with full precision
Instances
| Real Integer | Since: base-2.0.1 |
Defined in GHC.Real Methods toRational :: Integer -> Rational # | |
| Real Natural | Since: base-4.8.0.0 |
Defined in GHC.Real Methods toRational :: Natural -> Rational # | |
| Real Int | Since: base-2.0.1 |
Defined in GHC.Real Methods toRational :: Int -> Rational # | |
| Real Word | Since: base-2.1 |
Defined in GHC.Real Methods toRational :: Word -> Rational # | |
| Integral a => Real (Ratio a) | Since: base-2.0.1 |
Defined in GHC.Real Methods toRational :: Ratio a -> Rational # | |
| Real a => Real (Const a b) | Since: base-4.9.0.0 |
Defined in Data.Functor.Const Methods toRational :: Const a b -> Rational # | |
| Real a => Real (Tagged s a) | |
Defined in Data.Tagged Methods toRational :: Tagged s a -> Rational # | |
class (RealFrac a, Floating a) => RealFloat a where #
Efficient, machine-independent access to the components of a floating-point number.
Minimal complete definition
floatRadix, floatDigits, floatRange, decodeFloat, encodeFloat, isNaN, isInfinite, isDenormalized, isNegativeZero, isIEEE
Methods
floatRadix :: a -> Integer #
a constant function, returning the radix of the representation
(often 2)
floatDigits :: a -> Int #
a constant function, returning the number of digits of
floatRadix in the significand
floatRange :: a -> (Int, Int) #
a constant function, returning the lowest and highest values the exponent may assume
decodeFloat :: a -> (Integer, Int) #
The function decodeFloat applied to a real floating-point
number returns the significand expressed as an Integer and an
appropriately scaled exponent (an Int). If decodeFloat x(m,n), then x is equal in value to m*b^^n, where b
is the floating-point radix, and furthermore, either m and n
are both zero or else b^(d-1) <= , where abs m < b^dd is
the value of floatDigits xdecodeFloat 0 = (0,0)decodeFloat (-0.0) = (0,0)decodeFloat xisNaN xisInfinite xTrue.
encodeFloat :: Integer -> Int -> a #
encodeFloat performs the inverse of decodeFloat in the
sense that for finite x with the exception of -0.0,
uncurry encodeFloat (decodeFloat x) = xencodeFloat m nm*b^^n (or ±Infinity if overflow
occurs); usually the closer, but if m contains too many bits,
the result may be rounded in the wrong direction.
exponent corresponds to the second component of decodeFloat.
exponent 0 = 0x,
exponent x = snd (decodeFloat x) + floatDigits xx is a finite floating-point number, it is equal in value to
significand x * b ^^ exponent xb is the
floating-point radix.
The behaviour is unspecified on infinite or NaN values.
significand :: a -> a #
The first component of decodeFloat, scaled to lie in the open
interval (-1,1), either 0.0 or of absolute value >= 1/b,
where b is the floating-point radix.
The behaviour is unspecified on infinite or NaN values.
scaleFloat :: Int -> a -> a #
multiplies a floating-point number by an integer power of the radix
True if the argument is an IEEE "not-a-number" (NaN) value
isInfinite :: a -> Bool #
True if the argument is an IEEE infinity or negative infinity
isDenormalized :: a -> Bool #
True if the argument is too small to be represented in
normalized format
isNegativeZero :: a -> Bool #
True if the argument is an IEEE negative zero
True if the argument is an IEEE floating point number
a version of arctangent taking two real floating-point arguments.
For real floating x and y, atan2 y x(x,y). atan2 y x-pi,
pi]. It follows the Common Lisp semantics for the origin when
signed zeroes are supported. atan2 y 1y in a type
that is RealFloat, should return the same value as atan yatan2 is provided, but implementors
can provide a more accurate implementation.
Instances
class (Real a, Fractional a) => RealFrac a where #
Extracting components of fractions.
Minimal complete definition
Methods
properFraction :: Integral b => a -> (b, a) #
The function properFraction takes a real fractional number x
and returns a pair (n,f) such that x = n+f, and:
nis an integral number with the same sign asx; andfis a fraction with the same type and sign asx, and with absolute value less than1.
The default definitions of the ceiling, floor, truncate
and round functions are in terms of properFraction.
truncate :: Integral b => a -> b #
truncate xx between zero and x
round :: Integral b => a -> b #
round xx;
the even integer if x is equidistant between two integers
ceiling :: Integral b => a -> b #
ceiling xx
floor :: Integral b => a -> b #
floor xx
Conversion of values to readable Strings.
Derived instances of Show have the following properties, which
are compatible with derived instances of Read:
- The result of
showis a syntactically correct Haskell expression containing only constants, given the fixity declarations in force at the point where the type is declared. It contains only the constructor names defined in the data type, parentheses, and spaces. When labelled constructor fields are used, braces, commas, field names, and equal signs are also used. - If the constructor is defined to be an infix operator, then
showsPrecwill produce infix applications of the constructor. - the representation will be enclosed in parentheses if the
precedence of the top-level constructor in
xis less thand(associativity is ignored). Thus, ifdis0then the result is never surrounded in parentheses; ifdis11it is always surrounded in parentheses, unless it is an atomic expression. - If the constructor is defined using record syntax, then
showwill produce the record-syntax form, with the fields given in the same order as the original declaration.
For example, given the declarations
infixr 5 :^: data Tree a = Leaf a | Tree a :^: Tree a
the derived instance of Show is equivalent to
instance (Show a) => Show (Tree a) where
showsPrec d (Leaf m) = showParen (d > app_prec) $
showString "Leaf " . showsPrec (app_prec+1) m
where app_prec = 10
showsPrec d (u :^: v) = showParen (d > up_prec) $
showsPrec (up_prec+1) u .
showString " :^: " .
showsPrec (up_prec+1) v
where up_prec = 5Note that right-associativity of :^: is ignored. For example,
show(Leaf 1 :^: Leaf 2 :^: Leaf 3)"Leaf 1 :^: (Leaf 2 :^: Leaf 3)".
Methods
Arguments
| :: Int | the operator precedence of the enclosing
context (a number from |
| -> a | the value to be converted to a |
| -> ShowS |
Convert a value to a readable String.
showsPrec should satisfy the law
showsPrec d x r ++ s == showsPrec d x (r ++ s)
Derived instances of Read and Show satisfy the following:
That is, readsPrec parses the string produced by
showsPrec, and delivers the value that showsPrec started with.
Instances
| Show All | Since: base-2.1 |
| Show Any | Since: base-2.1 |
| Show Void | Since: base-4.8.0.0 |
| Show Associativity | Since: base-4.6.0.0 |
Defined in GHC.Generics Methods showsPrec :: Int -> Associativity -> ShowS # show :: Associativity -> String # showList :: [Associativity] -> ShowS # | |
| Show DecidedStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods showsPrec :: Int -> DecidedStrictness -> ShowS # show :: DecidedStrictness -> String # showList :: [DecidedStrictness] -> ShowS # | |
| Show Fixity | Since: base-4.6.0.0 |
| Show SourceStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods showsPrec :: Int -> SourceStrictness -> ShowS # show :: SourceStrictness -> String # showList :: [SourceStrictness] -> ShowS # | |
| Show SourceUnpackedness | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods showsPrec :: Int -> SourceUnpackedness -> ShowS # show :: SourceUnpackedness -> String # showList :: [SourceUnpackedness] -> ShowS # | |
| Show MaskingState | Since: base-4.3.0.0 |
Defined in GHC.IO Methods showsPrec :: Int -> MaskingState -> ShowS # show :: MaskingState -> String # showList :: [MaskingState] -> ShowS # | |
| Show AllocationLimitExceeded | Since: base-4.7.1.0 |
Defined in GHC.IO.Exception Methods showsPrec :: Int -> AllocationLimitExceeded -> ShowS # show :: AllocationLimitExceeded -> String # showList :: [AllocationLimitExceeded] -> ShowS # | |
| Show ArrayException | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception Methods showsPrec :: Int -> ArrayException -> ShowS # show :: ArrayException -> String # showList :: [ArrayException] -> ShowS # | |
| Show AssertionFailed | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception Methods showsPrec :: Int -> AssertionFailed -> ShowS # show :: AssertionFailed -> String # showList :: [AssertionFailed] -> ShowS # | |
| Show AsyncException | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception Methods showsPrec :: Int -> AsyncException -> ShowS # show :: AsyncException -> String # showList :: [AsyncException] -> ShowS # | |
| Show BlockedIndefinitelyOnMVar | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception Methods showsPrec :: Int -> BlockedIndefinitelyOnMVar -> ShowS # show :: BlockedIndefinitelyOnMVar -> String # showList :: [BlockedIndefinitelyOnMVar] -> ShowS # | |
| Show BlockedIndefinitelyOnSTM | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception Methods showsPrec :: Int -> BlockedIndefinitelyOnSTM -> ShowS # show :: BlockedIndefinitelyOnSTM -> String # showList :: [BlockedIndefinitelyOnSTM] -> ShowS # | |
| Show CompactionFailed | Since: base-4.10.0.0 |
Defined in GHC.IO.Exception Methods showsPrec :: Int -> CompactionFailed -> ShowS # show :: CompactionFailed -> String # showList :: [CompactionFailed] -> ShowS # | |
| Show Deadlock | Since: base-4.1.0.0 |
| Show ExitCode | |
| Show FixIOException | Since: base-4.11.0.0 |
Defined in GHC.IO.Exception Methods showsPrec :: Int -> FixIOException -> ShowS # show :: FixIOException -> String # showList :: [FixIOException] -> ShowS # | |
| Show IOErrorType | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception Methods showsPrec :: Int -> IOErrorType -> ShowS # show :: IOErrorType -> String # showList :: [IOErrorType] -> ShowS # | |
| Show IOException | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception Methods showsPrec :: Int -> IOException -> ShowS # show :: IOException -> String # showList :: [IOException] -> ShowS # | |
| Show SomeAsyncException | Since: base-4.7.0.0 |
Defined in GHC.IO.Exception Methods showsPrec :: Int -> SomeAsyncException -> ShowS # show :: SomeAsyncException -> String # showList :: [SomeAsyncException] -> ShowS # | |
| Show FractionalExponentBase | |
Defined in GHC.Real Methods showsPrec :: Int -> FractionalExponentBase -> ShowS # show :: FractionalExponentBase -> String # showList :: [FractionalExponentBase] -> ShowS # | |
| Show CallStack | Since: base-4.9.0.0 |
| Show SrcLoc | Since: base-4.9.0.0 |
| Show KindRep | |
| Show Module | Since: base-4.9.0.0 |
| Show Ordering | Since: base-2.1 |
| Show TrName | Since: base-4.9.0.0 |
| Show TyCon | Since: base-2.1 |
| Show TypeLitSort | Since: base-4.11.0.0 |
Defined in GHC.Show Methods showsPrec :: Int -> TypeLitSort -> ShowS # show :: TypeLitSort -> String # showList :: [TypeLitSort] -> ShowS # | |
| Show Integer | Since: base-2.1 |
| Show Natural | Since: base-4.8.0.0 |
| Show () | Since: base-2.1 |
| Show Bool | Since: base-2.1 |
| Show Char | Since: base-2.1 |
| Show Int | Since: base-2.1 |
| Show Levity | Since: base-4.15.0.0 |
| Show RuntimeRep | Since: base-4.11.0.0 |
Defined in GHC.Show Methods showsPrec :: Int -> RuntimeRep -> ShowS # show :: RuntimeRep -> String # showList :: [RuntimeRep] -> ShowS # | |
| Show VecCount | Since: base-4.11.0.0 |
| Show VecElem | Since: base-4.11.0.0 |
| Show Word | Since: base-2.1 |
| Show a => Show (ZipList a) | Since: base-4.7.0.0 |
| Show a => Show (Complex a) | Since: base-2.1 |
| Show a => Show (First a) | Since: base-2.1 |
| Show a => Show (Last a) | Since: base-2.1 |
| Show a => Show (First a) | Since: base-4.9.0.0 |
| Show a => Show (Last a) | Since: base-4.9.0.0 |
| Show a => Show (Max a) | Since: base-4.9.0.0 |
| Show a => Show (Min a) | Since: base-4.9.0.0 |
| Show m => Show (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup Methods showsPrec :: Int -> WrappedMonoid m -> ShowS # show :: WrappedMonoid m -> String # showList :: [WrappedMonoid m] -> ShowS # | |
| Show a => Show (Dual a) | Since: base-2.1 |
| Show a => Show (Product a) | Since: base-2.1 |
| Show a => Show (Sum a) | Since: base-2.1 |
| Show p => Show (Par1 p) | Since: base-4.7.0.0 |
| Show a => Show (Ratio a) | Since: base-2.0.1 |
| Show a => Show (NonEmpty a) | Since: base-4.11.0.0 |
| Show a => Show (Maybe a) | Since: base-2.1 |
| Show a => Show (a) | Since: base-4.15 |
| Show a => Show [a] | Since: base-2.1 |
| (Show a, Show b) => Show (Either a b) | Since: base-3.0 |
| (Show a, Show b) => Show (Arg a b) | Since: base-4.9.0.0 |
| Show (U1 p) | Since: base-4.9.0.0 |
| Show (V1 p) | Since: base-4.9.0.0 |
| (Show a, Show b) => Show (a, b) | Since: base-2.1 |
| Show a => Show (Const a b) | This instance would be equivalent to the derived instances of the
Since: base-4.8.0.0 |
| Show (f a) => Show (Ap f a) | Since: base-4.12.0.0 |
| Show (f a) => Show (Alt f a) | Since: base-4.8.0.0 |
| Show (Coercion a b) | Since: base-4.7.0.0 |
| Show (f p) => Show (Rec1 f p) | Since: base-4.7.0.0 |
| Show (URec Char p) | Since: base-4.9.0.0 |
| Show (URec Double p) | Since: base-4.9.0.0 |
| Show (URec Float p) | |
| Show (URec Int p) | Since: base-4.9.0.0 |
| Show (URec Word p) | Since: base-4.9.0.0 |
| Show b => Show (Tagged s b) | |
| (Show a, Show b, Show c) => Show (a, b, c) | Since: base-2.1 |
| (Show1 f, Show1 g, Show a) => Show (Product f g a) | Since: base-4.9.0.0 |
| (Show1 f, Show1 g, Show a) => Show (Sum f g a) | Since: base-4.9.0.0 |
| (Show (f p), Show (g p)) => Show ((f :*: g) p) | Since: base-4.7.0.0 |
| (Show (f p), Show (g p)) => Show ((f :+: g) p) | Since: base-4.7.0.0 |
| Show c => Show (K1 i c p) | Since: base-4.7.0.0 |
| (Show a, Show b, Show c, Show d) => Show (a, b, c, d) | Since: base-2.1 |
| (Show1 f, Show1 g, Show a) => Show (Compose f g a) | Since: base-4.9.0.0 |
| Show (f (g p)) => Show ((f :.: g) p) | Since: base-4.7.0.0 |
| Show (f p) => Show (M1 i c f p) | Since: base-4.7.0.0 |
| (Show a, Show b, Show c, Show d, Show e) => Show (a, b, c, d, e) | Since: base-2.1 |
| (Show a, Show b, Show c, Show d, Show e, Show f) => Show (a, b, c, d, e, f) | Since: base-2.1 |
| (Show a, Show b, Show c, Show d, Show e, Show f, Show g) => Show (a, b, c, d, e, f, g) | Since: base-2.1 |
| (Show a, Show b, Show c, Show d, Show e, Show f, Show g, Show h) => Show (a, b, c, d, e, f, g, h) | Since: base-2.1 |
| (Show a, Show b, Show c, Show d, Show e, Show f, Show g, Show h, Show i) => Show (a, b, c, d, e, f, g, h, i) | Since: base-2.1 |
| (Show a, Show b, Show c, Show d, Show e, Show f, Show g, Show h, Show i, Show j) => Show (a, b, c, d, e, f, g, h, i, j) | Since: base-2.1 |
| (Show a, Show b, Show c, Show d, Show e, Show f, Show g, Show h, Show i, Show j, Show k) => Show (a, b, c, d, e, f, g, h, i, j, k) | Since: base-2.1 |
| (Show a, Show b, Show c, Show d, Show e, Show f, Show g, Show h, Show i, Show j, Show k, Show l) => Show (a, b, c, d, e, f, g, h, i, j, k, l) | Since: base-2.1 |
| (Show a, Show b, Show c, Show d, Show e, Show f, Show g, Show h, Show i, Show j, Show k, Show l, Show m) => Show (a, b, c, d, e, f, g, h, i, j, k, l, m) | Since: base-2.1 |
| (Show a, Show b, Show c, Show d, Show e, Show f, Show g, Show h, Show i, Show j, Show k, Show l, Show m, Show n) => Show (a, b, c, d, e, f, g, h, i, j, k, l, m, n) | Since: base-2.1 |
| (Show a, Show b, Show c, Show d, Show e, Show f, Show g, Show h, Show i, Show j, Show k, Show l, Show m, Show n, Show o) => Show (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o) | Since: base-2.1 |
sum :: (Foldable t, Num a) => t a -> a #
The sum function computes the sum of the numbers of a structure.
Examples
Basic usage:
>>>sum []0
>>>sum [42]42
>>>sum [1..10]55
>>>sum [4.1, 2.0, 1.7]7.8
>>>sum [1..]* Hangs forever *
Since: base-4.8.0.0
product :: (Foldable t, Num a) => t a -> a #
The product function computes the product of the numbers of a
structure.
Examples
Basic usage:
>>>product []1
>>>product [42]42
>>>product [1..10]3628800
>>>product [4.1, 2.0, 1.7]13.939999999999998
>>>product [1..]* Hangs forever *
Since: base-4.8.0.0
null :: Foldable t => t a -> Bool #
Test whether the structure is empty. The default implementation is Left-associative and lazy in both the initial element and the accumulator. Thus optimised for structures where the first element can be accessed in constant time. Structures where this is not the case should have a non-default implementation.
Examples
Basic usage:
>>>null []True
>>>null [1]False
null is expected to terminate even for infinite structures.
The default implementation terminates provided the structure
is bounded on the left (there is a leftmost element).
>>>null [1..]False
Since: base-4.8.0.0
minimum :: (Foldable t, Ord a) => t a -> a #
The least element of a non-empty structure.
This function is non-total and will raise a runtime exception if the structure happens to be empty. A structure that supports random access and maintains its elements in order should provide a specialised implementation to return the minimum in faster than linear time.
Examples
Basic usage:
>>>minimum [1..10]1
>>>minimum []*** Exception: Prelude.minimum: empty list
>>>minimum Nothing*** Exception: minimum: empty structure
WARNING: This function is partial for possibly-empty structures like lists.
Since: base-4.8.0.0
maximum :: (Foldable t, Ord a) => t a -> a #
The largest element of a non-empty structure.
This function is non-total and will raise a runtime exception if the structure happens to be empty. A structure that supports random access and maintains its elements in order should provide a specialised implementation to return the maximum in faster than linear time.
Examples
Basic usage:
>>>maximum [1..10]10
>>>maximum []*** Exception: Prelude.maximum: empty list
>>>maximum Nothing*** Exception: maximum: empty structure
WARNING: This function is partial for possibly-empty structures like lists.
Since: base-4.8.0.0
foldr1 :: Foldable t => (a -> a -> a) -> t a -> a #
A variant of foldr that has no base case,
and thus may only be applied to non-empty structures.
This function is non-total and will raise a runtime exception if the structure happens to be empty.
Examples
Basic usage:
>>>foldr1 (+) [1..4]10
>>>foldr1 (+) []Exception: Prelude.foldr1: empty list
>>>foldr1 (+) Nothing*** Exception: foldr1: empty structure
>>>foldr1 (-) [1..4]-2
>>>foldr1 (&&) [True, False, True, True]False
>>>foldr1 (||) [False, False, True, True]True
>>>foldr1 (+) [1..]* Hangs forever *
foldl1 :: Foldable t => (a -> a -> a) -> t a -> a #
A variant of foldl that has no base case,
and thus may only be applied to non-empty structures.
This function is non-total and will raise a runtime exception if the structure happens to be empty.
foldl1f =foldl1f .toList
Examples
Basic usage:
>>>foldl1 (+) [1..4]10
>>>foldl1 (+) []*** Exception: Prelude.foldl1: empty list
>>>foldl1 (+) Nothing*** Exception: foldl1: empty structure
>>>foldl1 (-) [1..4]-8
>>>foldl1 (&&) [True, False, True, True]False
>>>foldl1 (||) [False, False, True, True]True
>>>foldl1 (+) [1..]* Hangs forever *
elem :: (Foldable t, Eq a) => a -> t a -> Bool infix 4 #
Does the element occur in the structure?
Note: elem is often used in infix form.
Examples
Basic usage:
>>>3 `elem` []False
>>>3 `elem` [1,2]False
>>>3 `elem` [1,2,3,4,5]True
For infinite structures, the default implementation of elem
terminates if the sought-after value exists at a finite distance
from the left side of the structure:
>>>3 `elem` [1..]True
>>>3 `elem` ([4..] ++ [3])* Hangs forever *
Since: base-4.8.0.0
foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b #
Right-associative fold of a structure, lazy in the accumulator.
In the case of lists, foldr, when applied to a binary operator, a
starting value (typically the right-identity of the operator), and a
list, reduces the list using the binary operator, from right to left:
foldr f z [x1, x2, ..., xn] == x1 `f` (x2 `f` ... (xn `f` z)...)
Note that since the head of the resulting expression is produced by an
application of the operator to the first element of the list, given an
operator lazy in its right argument, foldr can produce a terminating
expression from an unbounded list.
For a general Foldable structure this should be semantically identical
to,
foldr f z =foldrf z .toList
Examples
Basic usage:
>>>foldr (||) False [False, True, False]True
>>>foldr (||) False []False
>>>foldr (\c acc -> acc ++ [c]) "foo" ['a', 'b', 'c', 'd']"foodcba"
Infinite structures
⚠️ Applying foldr to infinite structures usually doesn't terminate.
It may still terminate under one of the following conditions:
- the folding function is short-circuiting
- the folding function is lazy on its second argument
Short-circuiting
( short-circuits on ||)True values, so the following terminates
because there is a True value finitely far from the left side:
>>>foldr (||) False (True : repeat False)True
But the following doesn't terminate:
>>>foldr (||) False (repeat False ++ [True])* Hangs forever *
Laziness in the second argument
Applying foldr to infinite structures terminates when the operator is
lazy in its second argument (the initial accumulator is never used in
this case, and so could be left undefined, but [] is more clear):
>>>take 5 $ foldr (\i acc -> i : fmap (+3) acc) [] (repeat 1)[1,4,7,10,13]
length :: Foldable t => t a -> Int #
Returns the size/length of a finite structure as an Int. The
default implementation just counts elements starting with the leftmost.
Instances for structures that can compute the element count faster
than via element-by-element counting, should provide a specialised
implementation.
Examples
Basic usage:
>>>length []0
>>>length ['a', 'b', 'c']3>>>length [1..]* Hangs forever *
Since: base-4.8.0.0
foldl :: Foldable t => (b -> a -> b) -> b -> t a -> b #
Left-associative fold of a structure, lazy in the accumulator. This is rarely what you want, but can work well for structures with efficient right-to-left sequencing and an operator that is lazy in its left argument.
In the case of lists, foldl, when applied to a binary operator, a
starting value (typically the left-identity of the operator), and a
list, reduces the list using the binary operator, from left to right:
foldl f z [x1, x2, ..., xn] == (...((z `f` x1) `f` x2) `f`...) `f` xn
Note that to produce the outermost application of the operator the
entire input list must be traversed. Like all left-associative folds,
foldl will diverge if given an infinite list.
If you want an efficient strict left-fold, you probably want to use
foldl' instead of foldl. The reason for this is that the latter
does not force the inner results (e.g. z `f` x1 in the above
example) before applying them to the operator (e.g. to (`f` x2)).
This results in a thunk chain \(\mathcal{O}(n)\) elements long, which
then must be evaluated from the outside-in.
For a general Foldable structure this should be semantically identical
to:
foldl f z =foldlf z .toList
Examples
The first example is a strict fold, which in practice is best performed
with foldl'.
>>>foldl (+) 42 [1,2,3,4]52
Though the result below is lazy, the input is reversed before prepending it to the initial accumulator, so corecursion begins only after traversing the entire input string.
>>>foldl (\acc c -> c : acc) "abcd" "efgh""hgfeabcd"
A left fold of a structure that is infinite on the right cannot terminate, even when for any finite input the fold just returns the initial accumulator:
>>>foldl (\a _ -> a) 0 $ repeat 1* Hangs forever *
WARNING: When it comes to lists, you always want to use either foldl' or foldr instead.
sequenceA :: (Traversable t, Applicative f) => t (f a) -> f (t a) #
Evaluate each action in the structure from left to right, and
collect the results. For a version that ignores the results
see sequenceA_.
Examples
Basic usage:
For the first two examples we show sequenceA fully evaluating a a structure and collecting the results.
>>>sequenceA [Just 1, Just 2, Just 3]Just [1,2,3]
>>>sequenceA [Right 1, Right 2, Right 3]Right [1,2,3]
The next two example show Nothing and Just will short circuit
the resulting structure if present in the input. For more context,
check the Traversable instances for Either and Maybe.
>>>sequenceA [Just 1, Just 2, Just 3, Nothing]Nothing
>>>sequenceA [Right 1, Right 2, Right 3, Left 4]Left 4
The class of semigroups (types with an associative binary operation).
Instances should satisfy the following:
Since: base-4.9.0.0
Instances
| Semigroup All | Since: base-4.9.0.0 |
| Semigroup Any | Since: base-4.9.0.0 |
| Semigroup Void | Since: base-4.9.0.0 |
| Semigroup Ordering | Since: base-4.9.0.0 |
| Semigroup () | Since: base-4.9.0.0 |
| Semigroup (Comparison a) |
(<>) :: Comparison a -> Comparison a -> Comparison a Comparison cmp <> Comparison cmp' = Comparison a a' -> cmp a a' <> cmp a a' |
Defined in Data.Functor.Contravariant Methods (<>) :: Comparison a -> Comparison a -> Comparison a # sconcat :: NonEmpty (Comparison a) -> Comparison a # stimes :: Integral b => b -> Comparison a -> Comparison a # | |
| Semigroup (Equivalence a) |
(<>) :: Equivalence a -> Equivalence a -> Equivalence a Equivalence equiv <> Equivalence equiv' = Equivalence a b -> equiv a b && equiv a b |
Defined in Data.Functor.Contravariant Methods (<>) :: Equivalence a -> Equivalence a -> Equivalence a # sconcat :: NonEmpty (Equivalence a) -> Equivalence a # stimes :: Integral b => b -> Equivalence a -> Equivalence a # | |
| Semigroup (Predicate a) |
(<>) :: Predicate a -> Predicate a -> Predicate a Predicate pred <> Predicate pred' = Predicate a -> pred a && pred' a |
| Semigroup (First a) | Since: base-4.9.0.0 |
| Semigroup (Last a) | Since: base-4.9.0.0 |
| Semigroup (First a) | Since: base-4.9.0.0 |
| Semigroup (Last a) | Since: base-4.9.0.0 |
| Ord a => Semigroup (Max a) | Since: base-4.9.0.0 |
| Ord a => Semigroup (Min a) | Since: base-4.9.0.0 |
| Monoid m => Semigroup (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup Methods (<>) :: WrappedMonoid m -> WrappedMonoid m -> WrappedMonoid m # sconcat :: NonEmpty (WrappedMonoid m) -> WrappedMonoid m # stimes :: Integral b => b -> WrappedMonoid m -> WrappedMonoid m # | |
| Semigroup a => Semigroup (Dual a) | Since: base-4.9.0.0 |
| Semigroup (Endo a) | Since: base-4.9.0.0 |
| Num a => Semigroup (Product a) | Since: base-4.9.0.0 |
| Num a => Semigroup (Sum a) | Since: base-4.9.0.0 |
| Semigroup p => Semigroup (Par1 p) | Since: base-4.12.0.0 |
| Semigroup a => Semigroup (IO a) | Since: base-4.10.0.0 |
| Semigroup (NonEmpty a) | Since: base-4.9.0.0 |
| Semigroup a => Semigroup (Maybe a) | Since: base-4.9.0.0 |
| Semigroup a => Semigroup (a) | Since: base-4.15 |
| Semigroup [a] | Since: base-4.9.0.0 |
| Semigroup (Either a b) | Since: base-4.9.0.0 |
| Semigroup a => Semigroup (Op a b) |
(<>) :: Op a b -> Op a b -> Op a b Op f <> Op g = Op a -> f a <> g a |
| Semigroup (U1 p) | Since: base-4.12.0.0 |
| Semigroup (V1 p) | Since: base-4.12.0.0 |
| (Semigroup a, Semigroup b) => Semigroup (ProductCatObj a b) Source # | |
Defined in Data.CategoryObject.Product Methods (<>) :: ProductCatObj a b -> ProductCatObj a b -> ProductCatObj a b # sconcat :: NonEmpty (ProductCatObj a b) -> ProductCatObj a b # stimes :: Integral b0 => b0 -> ProductCatObj a b -> ProductCatObj a b # | |
| Semigroup b => Semigroup (a -> b) | Since: base-4.9.0.0 |
| (Semigroup a, Semigroup b) => Semigroup (a, b) | Since: base-4.9.0.0 |
| Semigroup a => Semigroup (Const a b) | Since: base-4.9.0.0 |
| (Applicative f, Semigroup a) => Semigroup (Ap f a) | Since: base-4.12.0.0 |
| Alternative f => Semigroup (Alt f a) | Since: base-4.9.0.0 |
| Semigroup (f p) => Semigroup (Rec1 f p) | Since: base-4.12.0.0 |
| Semigroup a => Semigroup (Tagged s a) | |
| (Semigroup a, Semigroup b, Semigroup c) => Semigroup (a, b, c) | Since: base-4.9.0.0 |
| (Semigroup (f a), Semigroup (g a)) => Semigroup (Product f g a) | Since: base-4.16.0.0 |
| (Semigroup (f p), Semigroup (g p)) => Semigroup ((f :*: g) p) | Since: base-4.12.0.0 |
| Semigroup c => Semigroup (K1 i c p) | Since: base-4.12.0.0 |
| (Semigroup a, Semigroup b, Semigroup c, Semigroup d) => Semigroup (a, b, c, d) | Since: base-4.9.0.0 |
| Semigroup (f (g a)) => Semigroup (Compose f g a) | Since: base-4.16.0.0 |
| Semigroup (f (g p)) => Semigroup ((f :.: g) p) | Since: base-4.12.0.0 |
| Semigroup (f p) => Semigroup (M1 i c f p) | Since: base-4.12.0.0 |
| (Semigroup a, Semigroup b, Semigroup c, Semigroup d, Semigroup e) => Semigroup (a, b, c, d, e) | Since: base-4.9.0.0 |
class Semigroup a => Monoid a where #
The class of monoids (types with an associative binary operation that has an identity). Instances should satisfy the following:
- Right identity
x<>mempty= x- Left identity
mempty<>x = x- Associativity
x(<>(y<>z) = (x<>y)<>zSemigrouplaw)- Concatenation
mconcat=foldr(<>)mempty
The method names refer to the monoid of lists under concatenation, but there are many other instances.
Some types can be viewed as a monoid in more than one way,
e.g. both addition and multiplication on numbers.
In such cases we often define newtypes and make those instances
of Monoid, e.g. Sum and Product.
NOTE: Semigroup is a superclass of Monoid since base-4.11.0.0.
Minimal complete definition
Methods
Identity of mappend
>>>"Hello world" <> mempty"Hello world"
An associative operation
NOTE: This method is redundant and has the default
implementation mappend = (<>)mappend is a synonym for
(<>), it is expected that the two functions are defined the same
way. In a future GHC release mappend will be removed from Monoid.
Fold a list using the monoid.
For most types, the default definition for mconcat will be
used, but the function is included in the class definition so
that an optimized version can be provided for specific types.
>>>mconcat ["Hello", " ", "Haskell", "!"]"Hello Haskell!"
Instances
| Monoid All | Since: base-2.1 |
| Monoid Any | Since: base-2.1 |
| Monoid Ordering | Since: base-2.1 |
| Monoid () | Since: base-2.1 |
| Monoid (Comparison a) |
mempty :: Comparison a mempty = Comparison _ _ -> EQ |
Defined in Data.Functor.Contravariant Methods mempty :: Comparison a # mappend :: Comparison a -> Comparison a -> Comparison a # mconcat :: [Comparison a] -> Comparison a # | |
| Monoid (Equivalence a) |
mempty :: Equivalence a mempty = Equivalence _ _ -> True |
Defined in Data.Functor.Contravariant Methods mempty :: Equivalence a # mappend :: Equivalence a -> Equivalence a -> Equivalence a # mconcat :: [Equivalence a] -> Equivalence a # | |
| Monoid (Predicate a) |
mempty :: Predicate a mempty = _ -> True |
| Monoid (First a) | Since: base-2.1 |
| Monoid (Last a) | Since: base-2.1 |
| (Ord a, Bounded a) => Monoid (Max a) | Since: base-4.9.0.0 |
| (Ord a, Bounded a) => Monoid (Min a) | Since: base-4.9.0.0 |
| Monoid m => Monoid (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup Methods mempty :: WrappedMonoid m # mappend :: WrappedMonoid m -> WrappedMonoid m -> WrappedMonoid m # mconcat :: [WrappedMonoid m] -> WrappedMonoid m # | |
| Monoid a => Monoid (Dual a) | Since: base-2.1 |
| Monoid (Endo a) | Since: base-2.1 |
| Num a => Monoid (Product a) | Since: base-2.1 |
| Num a => Monoid (Sum a) | Since: base-2.1 |
| Monoid p => Monoid (Par1 p) | Since: base-4.12.0.0 |
| Monoid a => Monoid (IO a) | Since: base-4.9.0.0 |
| Semigroup a => Monoid (Maybe a) | Lift a semigroup into Since 4.11.0: constraint on inner Since: base-2.1 |
| Monoid a => Monoid (a) | Since: base-4.15 |
| Monoid [a] | Since: base-2.1 |
| Monoid a => Monoid (Op a b) |
mempty :: Op a b mempty = Op _ -> mempty |
| Monoid (U1 p) | Since: base-4.12.0.0 |
| (Monoid a, Monoid b) => Monoid (ProductCatObj a b) Source # | |
Defined in Data.CategoryObject.Product Methods mempty :: ProductCatObj a b # mappend :: ProductCatObj a b -> ProductCatObj a b -> ProductCatObj a b # mconcat :: [ProductCatObj a b] -> ProductCatObj a b # | |
| Monoid b => Monoid (a -> b) | Since: base-2.1 |
| (Monoid a, Monoid b) => Monoid (a, b) | Since: base-2.1 |
| Monoid a => Monoid (Const a b) | Since: base-4.9.0.0 |
| (Applicative f, Monoid a) => Monoid (Ap f a) | Since: base-4.12.0.0 |
| Alternative f => Monoid (Alt f a) | Since: base-4.8.0.0 |
| Monoid (f p) => Monoid (Rec1 f p) | Since: base-4.12.0.0 |
| (Semigroup a, Monoid a) => Monoid (Tagged s a) | |
| (Monoid a, Monoid b, Monoid c) => Monoid (a, b, c) | Since: base-2.1 |
| (Monoid (f a), Monoid (g a)) => Monoid (Product f g a) | Since: base-4.16.0.0 |
| (Monoid (f p), Monoid (g p)) => Monoid ((f :*: g) p) | Since: base-4.12.0.0 |
| Monoid c => Monoid (K1 i c p) | Since: base-4.12.0.0 |
| (Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d) | Since: base-2.1 |
| Monoid (f (g a)) => Monoid (Compose f g a) | Since: base-4.16.0.0 |
| Monoid (f (g p)) => Monoid ((f :.: g) p) | Since: base-4.12.0.0 |
| Monoid (f p) => Monoid (M1 i c f p) | Since: base-4.12.0.0 |
| (Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e) | Since: base-2.1 |
Instances
| Bounded Bool | Since: base-2.1 |
| Enum Bool | Since: base-2.1 |
| Generic Bool | |
| SingKind Bool | Since: base-4.9.0.0 |
Defined in GHC.Generics Associated Types type DemoteRep Bool | |
| Read Bool | Since: base-2.1 |
| Show Bool | Since: base-2.1 |
| Eq Bool | |
| Ord Bool | |
| SingI 'False | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| SingI 'True | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| (SPDistribute k, ObjectSum k a a, ObjectPair k Bool a) => Isomorphic (k :: Type -> Type -> Type) (a + a :: Type) ((Bool, a) :: Type) Source # | |
| (SPDistribute k, ObjectSum k a a, ObjectPair k Bool a) => Isomorphic (k :: Type -> Type -> Type) ((Bool, a) :: Type) (a + a :: Type) Source # | |
| type DemoteRep Bool | |
Defined in GHC.Generics | |
| type Rep Bool | Since: base-4.6.0.0 |
| data Sing (a :: Bool) | |
The character type Char is an enumeration whose values represent
Unicode (or equivalently ISO/IEC 10646) code points (i.e. characters, see
http://www.unicode.org/ for details). This set extends the ISO 8859-1
(Latin-1) character set (the first 256 characters), which is itself an extension
of the ASCII character set (the first 128 characters). A character literal in
Haskell has type Char.
To convert a Char to or from the corresponding Int value defined
by Unicode, use toEnum and fromEnum from the
Enum class respectively (or equivalently ord and
chr).
Instances
| Bounded Char | Since: base-2.1 |
| Enum Char | Since: base-2.1 |
| Read Char | Since: base-2.1 |
| Show Char | Since: base-2.1 |
| Eq Char | |
| Ord Char | |
| Generic1 (URec Char :: k -> Type) | |
| Foldable (UChar :: TYPE LiftedRep -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable Methods fold :: Monoid m => UChar m -> m # foldMap :: Monoid m => (a -> m) -> UChar a -> m # foldMap' :: Monoid m => (a -> m) -> UChar a -> m # foldr :: (a -> b -> b) -> b -> UChar a -> b # foldr' :: (a -> b -> b) -> b -> UChar a -> b # foldl :: (b -> a -> b) -> b -> UChar a -> b # foldl' :: (b -> a -> b) -> b -> UChar a -> b # foldr1 :: (a -> a -> a) -> UChar a -> a # foldl1 :: (a -> a -> a) -> UChar a -> a # elem :: Eq a => a -> UChar a -> Bool # maximum :: Ord a => UChar a -> a # minimum :: Ord a => UChar a -> a # | |
| Traversable (UChar :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (URec Char :: TYPE LiftedRep -> Type) | Since: base-4.9.0.0 |
| Generic (URec Char p) | |
| Show (URec Char p) | Since: base-4.9.0.0 |
| Eq (URec Char p) | Since: base-4.9.0.0 |
| Ord (URec Char p) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| data URec Char (p :: k) | Used for marking occurrences of Since: base-4.9.0.0 |
| type Rep1 (URec Char :: k -> Type) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| type Rep (URec Char p) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
Double-precision floating point numbers. It is desirable that this type be at least equal in range and precision to the IEEE double-precision type.
Instances
| Floating Double | Since: base-2.1 |
| RealFloat Double | Since: base-2.1 |
Defined in GHC.Float Methods floatRadix :: Double -> Integer # floatDigits :: Double -> Int # floatRange :: Double -> (Int, Int) # decodeFloat :: Double -> (Integer, Int) # encodeFloat :: Integer -> Int -> Double # significand :: Double -> Double # scaleFloat :: Int -> Double -> Double # isInfinite :: Double -> Bool # isDenormalized :: Double -> Bool # isNegativeZero :: Double -> Bool # | |
| Read Double | Since: base-2.1 |
| Eq Double | Note that due to the presence of
Also note that
|
| Ord Double | Note that due to the presence of
Also note that, due to the same,
|
| Generic1 (URec Double :: k -> Type) | |
| Foldable (UDouble :: TYPE LiftedRep -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable Methods fold :: Monoid m => UDouble m -> m # foldMap :: Monoid m => (a -> m) -> UDouble a -> m # foldMap' :: Monoid m => (a -> m) -> UDouble a -> m # foldr :: (a -> b -> b) -> b -> UDouble a -> b # foldr' :: (a -> b -> b) -> b -> UDouble a -> b # foldl :: (b -> a -> b) -> b -> UDouble a -> b # foldl' :: (b -> a -> b) -> b -> UDouble a -> b # foldr1 :: (a -> a -> a) -> UDouble a -> a # foldl1 :: (a -> a -> a) -> UDouble a -> a # elem :: Eq a => a -> UDouble a -> Bool # maximum :: Ord a => UDouble a -> a # minimum :: Ord a => UDouble a -> a # | |
| Traversable (UDouble :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (URec Double :: TYPE LiftedRep -> Type) | Since: base-4.9.0.0 |
| Generic (URec Double p) | |
| Show (URec Double p) | Since: base-4.9.0.0 |
| Eq (URec Double p) | Since: base-4.9.0.0 |
| Ord (URec Double p) | Since: base-4.9.0.0 |
Defined in GHC.Generics Methods compare :: URec Double p -> URec Double p -> Ordering # (<) :: URec Double p -> URec Double p -> Bool # (<=) :: URec Double p -> URec Double p -> Bool # (>) :: URec Double p -> URec Double p -> Bool # (>=) :: URec Double p -> URec Double p -> Bool # | |
| data URec Double (p :: k) | Used for marking occurrences of Since: base-4.9.0.0 |
| type Rep1 (URec Double :: k -> Type) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| type Rep (URec Double p) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
Single-precision floating point numbers. It is desirable that this type be at least equal in range and precision to the IEEE single-precision type.
Instances
| Floating Float | Since: base-2.1 |
| RealFloat Float | Since: base-2.1 |
Defined in GHC.Float Methods floatRadix :: Float -> Integer # floatDigits :: Float -> Int # floatRange :: Float -> (Int, Int) # decodeFloat :: Float -> (Integer, Int) # encodeFloat :: Integer -> Int -> Float # significand :: Float -> Float # scaleFloat :: Int -> Float -> Float # isInfinite :: Float -> Bool # isDenormalized :: Float -> Bool # isNegativeZero :: Float -> Bool # | |
| Read Float | Since: base-2.1 |
| Eq Float | Note that due to the presence of
Also note that
|
| Ord Float | Note that due to the presence of
Also note that, due to the same,
|
| Generic1 (URec Float :: k -> Type) | |
| Foldable (UFloat :: TYPE LiftedRep -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable Methods fold :: Monoid m => UFloat m -> m # foldMap :: Monoid m => (a -> m) -> UFloat a -> m # foldMap' :: Monoid m => (a -> m) -> UFloat a -> m # foldr :: (a -> b -> b) -> b -> UFloat a -> b # foldr' :: (a -> b -> b) -> b -> UFloat a -> b # foldl :: (b -> a -> b) -> b -> UFloat a -> b # foldl' :: (b -> a -> b) -> b -> UFloat a -> b # foldr1 :: (a -> a -> a) -> UFloat a -> a # foldl1 :: (a -> a -> a) -> UFloat a -> a # elem :: Eq a => a -> UFloat a -> Bool # maximum :: Ord a => UFloat a -> a # minimum :: Ord a => UFloat a -> a # | |
| Traversable (UFloat :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (URec Float :: TYPE LiftedRep -> Type) | Since: base-4.9.0.0 |
| Generic (URec Float p) | |
| Show (URec Float p) | |
| Eq (URec Float p) | |
| Ord (URec Float p) | |
Defined in GHC.Generics | |
| data URec Float (p :: k) | Used for marking occurrences of Since: base-4.9.0.0 |
| type Rep1 (URec Float :: k -> Type) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| type Rep (URec Float p) | |
Defined in GHC.Generics | |
A fixed-precision integer type with at least the range [-2^29 .. 2^29-1].
The exact range for a given implementation can be determined by using
minBound and maxBound from the Bounded class.
Instances
| Bounded Int | Since: base-2.1 |
| Enum Int | Since: base-2.1 |
| Num Int | Since: base-2.1 |
| Read Int | Since: base-2.1 |
| Integral Int | Since: base-2.0.1 |
| Real Int | Since: base-2.0.1 |
Defined in GHC.Real Methods toRational :: Int -> Rational # | |
| Show Int | Since: base-2.1 |
| Eq Int | |
| Ord Int | |
| Generic1 (URec Int :: k -> Type) | |
| Foldable (UInt :: TYPE LiftedRep -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable Methods fold :: Monoid m => UInt m -> m # foldMap :: Monoid m => (a -> m) -> UInt a -> m # foldMap' :: Monoid m => (a -> m) -> UInt a -> m # foldr :: (a -> b -> b) -> b -> UInt a -> b # foldr' :: (a -> b -> b) -> b -> UInt a -> b # foldl :: (b -> a -> b) -> b -> UInt a -> b # foldl' :: (b -> a -> b) -> b -> UInt a -> b # foldr1 :: (a -> a -> a) -> UInt a -> a # foldl1 :: (a -> a -> a) -> UInt a -> a # elem :: Eq a => a -> UInt a -> Bool # maximum :: Ord a => UInt a -> a # | |
| Traversable (UInt :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (URec Int :: TYPE LiftedRep -> Type) | Since: base-4.9.0.0 |
| Generic (URec Int p) | |
| Show (URec Int p) | Since: base-4.9.0.0 |
| Eq (URec Int p) | Since: base-4.9.0.0 |
| Ord (URec Int p) | Since: base-4.9.0.0 |
| data URec Int (p :: k) | Used for marking occurrences of Since: base-4.9.0.0 |
| type Rep1 (URec Int :: k -> Type) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| type Rep (URec Int p) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
Arbitrary precision integers. In contrast with fixed-size integral types
such as Int, the Integer type represents the entire infinite range of
integers.
Integers are stored in a kind of sign-magnitude form, hence do not expect two's complement form when using bit operations.
If the value is small (fit into an Int), IS constructor is used.
Otherwise Integer and IN constructors are used to store a BigNat
representing respectively the positive or the negative value magnitude.
Invariant: Integer and IN are used iff value doesn't fit in IS
Instances
| Enum Integer | Since: base-2.1 |
| Num Integer | Since: base-2.1 |
| Read Integer | Since: base-2.1 |
| Integral Integer | Since: base-2.0.1 |
Defined in GHC.Real | |
| Real Integer | Since: base-2.0.1 |
Defined in GHC.Real Methods toRational :: Integer -> Rational # | |
| Show Integer | Since: base-2.1 |
| Eq Integer | |
| Ord Integer | |
The Maybe type encapsulates an optional value. A value of type
Maybe aa (represented as Just aNothing). Using Maybe is a good way to
deal with errors or exceptional cases without resorting to drastic
measures such as error.
The Maybe type is also a monad. It is a simple kind of error
monad, where all errors are represented by Nothing. A richer
error monad can be built using the Either type.
Instances
| MonadFail Maybe | Since: base-4.9.0.0 |
Defined in Control.Monad.Fail | |
| Foldable Maybe | Since: base-2.1 |
Defined in Data.Foldable Methods fold :: Monoid m => Maybe m -> m # foldMap :: Monoid m => (a -> m) -> Maybe a -> m # foldMap' :: Monoid m => (a -> m) -> Maybe a -> m # foldr :: (a -> b -> b) -> b -> Maybe a -> b # foldr' :: (a -> b -> b) -> b -> Maybe a -> b # foldl :: (b -> a -> b) -> b -> Maybe a -> b # foldl' :: (b -> a -> b) -> b -> Maybe a -> b # foldr1 :: (a -> a -> a) -> Maybe a -> a # foldl1 :: (a -> a -> a) -> Maybe a -> a # elem :: Eq a => a -> Maybe a -> Bool # maximum :: Ord a => Maybe a -> a # minimum :: Ord a => Maybe a -> a # | |
| Traversable Maybe | Since: base-2.1 |
| Alternative Maybe | Since: base-2.1 |
| Applicative Maybe | Since: base-2.1 |
| Functor Maybe | Since: base-2.1 |
| Monad Maybe | Since: base-2.1 |
| MonadPlus Maybe | Since: base-2.1 |
| Generic1 Maybe | |
| (Arrow k (->), WellPointed k, Function k, Functor Maybe k k) => Traversable Maybe Maybe k k Source # | |
Defined in Data.Traversable.Constrained Associated Types type TraversalObject k Maybe b Source # Methods traverse :: (Monoidal f k k, Object k a, Object k (Maybe a), ObjectPair k b (Maybe b), ObjectPair k (f b) (f (Maybe b)), TraversalObject k Maybe b) => k a (f b) -> k (Maybe a) (f (Maybe b)) Source # mapM :: (k ~ k, Maybe ~ Maybe, Applicative m k k, Object k a, Object k (Maybe a), ObjectPair k b (Maybe b), ObjectPair k (m b) (m (Maybe b)), TraversalObject k Maybe b) => k a (m b) -> k (Maybe a) (m (Maybe b)) Source # sequence :: (k ~ k, Maybe ~ Maybe, Monoidal f k k, ObjectPair k a (Maybe a), ObjectPair k (f a) (f (Maybe a)), Object k (Maybe (f a)), TraversalObject k Maybe a) => k (Maybe (f a)) (f (Maybe a)) Source # | |
| Functor Maybe (Coercion :: Type -> Type -> Type) (Coercion :: Type -> Type -> Type) Source # | |
| Functor Maybe (Discrete :: Type -> Type -> Type) (Discrete :: Type -> Type -> Type) Source # | |
| Foldable Maybe (->) (->) Source # | |
| Semigroup a => Monoid (Maybe a) | Lift a semigroup into Since 4.11.0: constraint on inner Since: base-2.1 |
| Semigroup a => Semigroup (Maybe a) | Since: base-4.9.0.0 |
| Generic (Maybe a) | |
| SingKind a => SingKind (Maybe a) | Since: base-4.9.0.0 |
Defined in GHC.Generics Associated Types type DemoteRep (Maybe a) | |
| Read a => Read (Maybe a) | Since: base-2.1 |
| Show a => Show (Maybe a) | Since: base-2.1 |
| Eq a => Eq (Maybe a) | Since: base-2.1 |
| Ord a => Ord (Maybe a) | Since: base-2.1 |
| SingI ('Nothing :: Maybe a) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| SingI a2 => SingI ('Just a2 :: Maybe a1) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| type Rep1 Maybe | Since: base-4.6.0.0 |
| type TraversalObject k Maybe b Source # | |
Defined in Data.Traversable.Constrained | |
| type DemoteRep (Maybe a) | |
Defined in GHC.Generics | |
| type Rep (Maybe a) | Since: base-4.6.0.0 |
Defined in GHC.Generics | |
| data Sing (b :: Maybe a) | |
Instances
| Monoid Ordering | Since: base-2.1 |
| Semigroup Ordering | Since: base-4.9.0.0 |
| Bounded Ordering | Since: base-2.1 |
| Enum Ordering | Since: base-2.1 |
| Generic Ordering | |
| Read Ordering | Since: base-2.1 |
| Show Ordering | Since: base-2.1 |
| Eq Ordering | |
| Ord Ordering | |
Defined in GHC.Classes | |
| type Rep Ordering | Since: base-4.6.0.0 |
A value of type IO aa.
There is really only one way to "perform" an I/O action: bind it to
Main.main in your program. When your program is run, the I/O will
be performed. It isn't possible to perform I/O from an arbitrary
function, unless that function is itself in the IO monad and called
at some point, directly or indirectly, from Main.main.
IO is a monad, so IO actions can be combined using either the do-notation
or the >> and >>= operations from the Monad
class.
Instances
| MonadFail IO | Since: base-4.9.0.0 |
Defined in Control.Monad.Fail | |
| Alternative IO | Since: base-4.9.0.0 |
| Applicative IO | Since: base-2.1 |
| Functor IO | Since: base-2.1 |
| Monad IO | Since: base-2.1 |
| MonadPlus IO | Since: base-4.9.0.0 |
| Functor IO (Coercion :: Type -> Type -> Type) (Coercion :: Type -> Type -> Type) Source # | |
| Functor IO (Discrete :: Type -> Type -> Type) (Discrete :: Type -> Type -> Type) Source # | |
| Monoid a => Monoid (IO a) | Since: base-4.9.0.0 |
| Semigroup a => Semigroup (IO a) | Since: base-4.10.0.0 |
Instances
| Bounded Word | Since: base-2.1 |
| Enum Word | Since: base-2.1 |
| Num Word | Since: base-2.1 |
| Read Word | Since: base-4.5.0.0 |
| Integral Word | Since: base-2.1 |
| Real Word | Since: base-2.1 |
Defined in GHC.Real Methods toRational :: Word -> Rational # | |
| Show Word | Since: base-2.1 |
| Eq Word | |
| Ord Word | |
| Generic1 (URec Word :: k -> Type) | |
| Foldable (UWord :: TYPE LiftedRep -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable Methods fold :: Monoid m => UWord m -> m # foldMap :: Monoid m => (a -> m) -> UWord a -> m # foldMap' :: Monoid m => (a -> m) -> UWord a -> m # foldr :: (a -> b -> b) -> b -> UWord a -> b # foldr' :: (a -> b -> b) -> b -> UWord a -> b # foldl :: (b -> a -> b) -> b -> UWord a -> b # foldl' :: (b -> a -> b) -> b -> UWord a -> b # foldr1 :: (a -> a -> a) -> UWord a -> a # foldl1 :: (a -> a -> a) -> UWord a -> a # elem :: Eq a => a -> UWord a -> Bool # maximum :: Ord a => UWord a -> a # minimum :: Ord a => UWord a -> a # | |
| Traversable (UWord :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (URec Word :: TYPE LiftedRep -> Type) | Since: base-4.9.0.0 |
| Generic (URec Word p) | |
| Show (URec Word p) | Since: base-4.9.0.0 |
| Eq (URec Word p) | Since: base-4.9.0.0 |
| Ord (URec Word p) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| data URec Word (p :: k) | Used for marking occurrences of Since: base-4.9.0.0 |
| type Rep1 (URec Word :: k -> Type) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
| type Rep (URec Word p) | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
The Either type represents values with two possibilities: a value of
type Either a bLeft aRight b
The Either type is sometimes used to represent a value which is
either correct or an error; by convention, the Left constructor is
used to hold an error value and the Right constructor is used to
hold a correct value (mnemonic: "right" also means "correct").
Examples
The type Either String IntString or an Int. The Left constructor can be used only on
Strings, and the Right constructor can be used only on Ints:
>>>let s = Left "foo" :: Either String Int>>>sLeft "foo">>>let n = Right 3 :: Either String Int>>>nRight 3>>>:type ss :: Either String Int>>>:type nn :: Either String Int
The fmap from our Functor instance will ignore Left values, but
will apply the supplied function to values contained in a Right:
>>>let s = Left "foo" :: Either String Int>>>let n = Right 3 :: Either String Int>>>fmap (*2) sLeft "foo">>>fmap (*2) nRight 6
The Monad instance for Either allows us to chain together multiple
actions which may fail, and fail overall if any of the individual
steps failed. First we'll write a function that can either parse an
Int from a Char, or fail.
>>>import Data.Char ( digitToInt, isDigit )>>>:{let parseEither :: Char -> Either String Int parseEither c | isDigit c = Right (digitToInt c) | otherwise = Left "parse error">>>:}
The following should work, since both '1' and '2' can be
parsed as Ints.
>>>:{let parseMultiple :: Either String Int parseMultiple = do x <- parseEither '1' y <- parseEither '2' return (x + y)>>>:}
>>>parseMultipleRight 3
But the following should fail overall, since the first operation where
we attempt to parse 'm' as an Int will fail:
>>>:{let parseMultiple :: Either String Int parseMultiple = do x <- parseEither 'm' y <- parseEither '2' return (x + y)>>>:}
>>>parseMultipleLeft "parse error"
Instances
| (CoCartesian k, Object k a, u ~ ZeroObject k, ObjectSum k a u) => Isomorphic (k :: Type -> Type -> Type) (a :: Type) (a + u :: Type) Source # | |
Defined in Control.Category.Constrained | |
| (CoCartesian k, Object k a, u ~ ZeroObject k, ObjectSum k a u, ObjectSum k u a, Object k (u + a), Object k (a + u)) => Isomorphic (k :: Type -> Type -> Type) (a :: Type) (u + a :: Type) Source # | |
Defined in Control.Category.Constrained | |
| (CoCartesian k, Object k a, u ~ ZeroObject k, ObjectSum k a u) => Isomorphic (k :: Type -> Type -> Type) (a + u :: Type) (a :: Type) Source # | |
Defined in Control.Category.Constrained | |
| (CoCartesian k, Object k a, u ~ ZeroObject k, ObjectSum k a u, ObjectSum k u a, Object k (u + a), Object k (a + u)) => Isomorphic (k :: Type -> Type -> Type) (u + a :: Type) (a :: Type) Source # | |
Defined in Control.Category.Constrained | |
| (CoCartesian k, Object k a, ObjectSum k a b, ObjectSum k b c, ObjectSum k a (b + c), ObjectSum k (a + b) c, Object k c) => Isomorphic (k :: Type -> Type -> Type) ((a + b) + c :: Type) (a + (b + c) :: Type) Source # | |
| (SPDistribute k, ObjectSum k (a, b) (a, c), ObjectPair k a (b + c), ObjectSum k b c, PairObjects k a b, PairObjects k a c) => Isomorphic (k :: Type -> Type -> Type) ((a, b) + (a, c) :: Type) ((a, b + c) :: Type) Source # | |
| (CoCartesian k, Object k a, ObjectSum k a b, ObjectSum k b c, ObjectSum k a (b + c), ObjectSum k (a + b) c, Object k c) => Isomorphic (k :: Type -> Type -> Type) (a + (b + c) :: Type) ((a + b) + c :: Type) Source # | |
| (SPDistribute k, ObjectSum k a a, ObjectPair k Bool a) => Isomorphic (k :: Type -> Type -> Type) (a + a :: Type) ((Bool, a) :: Type) Source # | |
| (SPDistribute k, ObjectSum k a a, ObjectPair k Bool a) => Isomorphic (k :: Type -> Type -> Type) ((Bool, a) :: Type) (a + a :: Type) Source # | |
| (SPDistribute k, ObjectSum k (a, b) (a, c), ObjectPair k a (b + c), ObjectSum k b c, PairObjects k a b, PairObjects k a c) => Isomorphic (k :: Type -> Type -> Type) ((a, b + c) :: Type) ((a, b) + (a, c) :: Type) Source # | |
| Generic1 (Either a :: Type -> Type) | |
| Foldable (Either a) | Since: base-4.7.0.0 |
Defined in Data.Foldable Methods fold :: Monoid m => Either a m -> m # foldMap :: Monoid m => (a0 -> m) -> Either a a0 -> m # foldMap' :: Monoid m => (a0 -> m) -> Either a a0 -> m # foldr :: (a0 -> b -> b) -> b -> Either a a0 -> b # foldr' :: (a0 -> b -> b) -> b -> Either a a0 -> b # foldl :: (b -> a0 -> b) -> b -> Either a a0 -> b # foldl' :: (b -> a0 -> b) -> b -> Either a a0 -> b # foldr1 :: (a0 -> a0 -> a0) -> Either a a0 -> a0 # foldl1 :: (a0 -> a0 -> a0) -> Either a a0 -> a0 # toList :: Either a a0 -> [a0] # length :: Either a a0 -> Int # elem :: Eq a0 => a0 -> Either a a0 -> Bool # maximum :: Ord a0 => Either a a0 -> a0 # minimum :: Ord a0 => Either a a0 -> a0 # | |
| Traversable (Either a) | Since: base-4.7.0.0 |
Defined in Data.Traversable | |
| Applicative (Either e) | Since: base-3.0 |
| Functor (Either a) | Since: base-3.0 |
| Monad (Either e) | Since: base-4.4.0.0 |
| Functor (Either a) (Coercion :: Type -> Type -> Type) (Coercion :: Type -> Type -> Type) Source # | |
| Functor (Either a) (Discrete :: Type -> Type -> Type) (Discrete :: Type -> Type -> Type) Source # | |
| Semigroup (Either a b) | Since: base-4.9.0.0 |
| Generic (Either a b) | |
| (Read a, Read b) => Read (Either a b) | Since: base-3.0 |
| (Show a, Show b) => Show (Either a b) | Since: base-3.0 |
| (Eq a, Eq b) => Eq (Either a b) | Since: base-2.1 |
| (Ord a, Ord b) => Ord (Either a b) | Since: base-2.1 |
| type Rep1 (Either a :: Type -> Type) | Since: base-4.6.0.0 |
Defined in GHC.Generics type Rep1 (Either a :: Type -> Type) = D1 ('MetaData "Either" "Data.Either" "base" 'False) (C1 ('MetaCons "Left" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a)) :+: C1 ('MetaCons "Right" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) Par1)) | |
| type Rep (Either a b) | Since: base-4.6.0.0 |
Defined in GHC.Generics type Rep (Either a b) = D1 ('MetaData "Either" "Data.Either" "base" 'False) (C1 ('MetaCons "Left" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a)) :+: C1 ('MetaCons "Right" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 b))) | |
writeFile :: FilePath -> String -> IO () #
The computation writeFile file str function writes the string str,
to the file file.
readFile :: FilePath -> IO String #
The readFile function reads a file and
returns the contents of the file as a string.
The file is read lazily, on demand, as with getContents.
interact :: (String -> String) -> IO () #
The interact function takes a function of type String->String
as its argument. The entire input from the standard input device is
passed to this function as its argument, and the resulting string is
output on the standard output device.
getContents :: IO String #
The getContents operation returns all user input as a single string,
which is read lazily as it is needed
(same as hGetContents stdin).
appendFile :: FilePath -> String -> IO () #
The computation appendFile file str function appends the string str,
to the file file.
Note that writeFile and appendFile write a literal string
to a file. To write a value of any printable type, as with print,
use the show function to convert the value to a string first.
main = appendFile "squares" (show [(x,x*x) | x <- [0,0.1..2]])
File and directory names are values of type String, whose precise
meaning is operating system dependent. Files can be opened, yielding a
handle which can then be used to operate on the contents of that file.
type IOError = IOException #
The Haskell 2010 type for exceptions in the IO monad.
Any I/O operation may raise an IOException instead of returning a result.
For a more general type of exception, including also those that arise
in pure code, see Exception.
In Haskell 2010, this is an opaque type.
userError :: String -> IOError #
Construct an IOException value with a string describing the error.
The fail method of the IO instance of the Monad class raises a
userError, thus:
instance Monad IO where ... fail s = ioError (userError s)
or :: Foldable t => t Bool -> Bool #
or returns the disjunction of a container of Bools. For the
result to be False, the container must be finite; True, however,
results from a True value finitely far from the left end.
Examples
Basic usage:
>>>or []False
>>>or [True]True
>>>or [False]False
>>>or [True, True, False]True
>>>or (True : repeat False) -- Infinite list [True,False,False,False,...True
>>>or (repeat False)* Hangs forever *
notElem :: (Foldable t, Eq a) => a -> t a -> Bool infix 4 #
notElem is the negation of elem.
Examples
Basic usage:
>>>3 `notElem` []True
>>>3 `notElem` [1,2]True
>>>3 `notElem` [1,2,3,4,5]False
For infinite structures, notElem terminates if the value exists at a
finite distance from the left side of the structure:
>>>3 `notElem` [1..]False
>>>3 `notElem` ([4..] ++ [3])* Hangs forever *
concat :: Foldable t => t [a] -> [a] #
The concatenation of all the elements of a container of lists.
Examples
Basic usage:
>>>concat (Just [1, 2, 3])[1,2,3]
>>>concat (Left 42)[]
>>>concat [[1, 2, 3], [4, 5], [6], []][1,2,3,4,5,6]
any :: Foldable t => (a -> Bool) -> t a -> Bool #
Determines whether any element of the structure satisfies the predicate.
Examples
Basic usage:
>>>any (> 3) []False
>>>any (> 3) [1,2]False
>>>any (> 3) [1,2,3,4,5]True
>>>any (> 3) [1..]True
>>>any (> 3) [0, -1..]* Hangs forever *
and :: Foldable t => t Bool -> Bool #
and returns the conjunction of a container of Bools. For the
result to be True, the container must be finite; False, however,
results from a False value finitely far from the left end.
Examples
Basic usage:
>>>and []True
>>>and [True]True
>>>and [False]False
>>>and [True, True, False]False
>>>and (False : repeat True) -- Infinite list [False,True,True,True,...False
>>>and (repeat True)* Hangs forever *
all :: Foldable t => (a -> Bool) -> t a -> Bool #
Determines whether all elements of the structure satisfy the predicate.
Examples
Basic usage:
>>>all (> 3) []True
>>>all (> 3) [1,2]False
>>>all (> 3) [1,2,3,4,5]False
>>>all (> 3) [1..]False
>>>all (> 3) [4..]* Hangs forever *
words breaks a string up into a list of words, which were delimited
by white space.
>>>words "Lorem ipsum\ndolor"["Lorem","ipsum","dolor"]
lines breaks a string up into a list of strings at newline
characters. The resulting strings do not contain newlines.
Note that after splitting the string at newline characters, the last part of the string is considered a line even if it doesn't end with a newline. For example,
>>>lines ""[]
>>>lines "\n"[""]
>>>lines "one"["one"]
>>>lines "one\n"["one"]
>>>lines "one\n\n"["one",""]
>>>lines "one\ntwo"["one","two"]
>>>lines "one\ntwo\n"["one","two"]
Thus lines ss.
read :: Read a => String -> a #
The read function reads input from a string, which must be
completely consumed by the input process. read fails with an error if the
parse is unsuccessful, and it is therefore discouraged from being used in
real applications. Use readMaybe or readEither for safe alternatives.
>>>read "123" :: Int123
>>>read "hello" :: Int*** Exception: Prelude.read: no parse
either :: (a -> c) -> (b -> c) -> Either a b -> c #
Case analysis for the Either type.
If the value is Left aa;
if it is Right bb.
Examples
We create two values of type Either String IntLeft constructor and another using the Right constructor. Then
we apply "either" the length function (if we have a String)
or the "times-two" function (if we have an Int):
>>>let s = Left "foo" :: Either String Int>>>let n = Right 3 :: Either String Int>>>either length (*2) s3>>>either length (*2) n6
The lex function reads a single lexeme from the input, discarding
initial white space, and returning the characters that constitute the
lexeme. If the input string contains only white space, lex returns a
single successful `lexeme' consisting of the empty string. (Thus
lex "" = [("","")]lex fails (i.e. returns []).
This lexer is not completely faithful to the Haskell lexical syntax in the following respects:
- Qualified names are not handled properly
- Octal and hexadecimal numerics are not recognized as a single token
- Comments are not treated properly
lcm :: Integral a => a -> a -> a #
lcm x yx and y divide.
gcd :: Integral a => a -> a -> a #
gcd x yx and y of which
every common factor of x and y is also a factor; for example
gcd 4 2 = 2gcd (-4) 6 = 2gcd 0 44. gcd 0 00.
(That is, the common divisor that is "greatest" in the divisibility
preordering.)
Note: Since for signed fixed-width integer types, abs minBound < 0minBound0 or minBound
(^^) :: (Fractional a, Integral b) => a -> b -> a infixr 8 #
raise a number to an integral power
showString :: String -> ShowS #
utility function converting a String to a show function that
simply prepends the string unchanged.
utility function converting a Char to a show function that
simply prepends the character unchanged.
zipWith3 :: (a -> b -> c -> d) -> [a] -> [b] -> [c] -> [d] #
The zipWith3 function takes a function which combines three
elements, as well as three lists and returns a list of the function applied
to corresponding elements, analogous to zipWith.
It is capable of list fusion, but it is restricted to its
first list argument and its resulting list.
zipWith3 (,,) xs ys zs == zip3 xs ys zs zipWith3 f [x1,x2,x3..] [y1,y2,y3..] [z1,z2,z3..] == [f x1 y1 z1, f x2 y2 z2, f x3 y3 z3..]
zipWith :: (a -> b -> c) -> [a] -> [b] -> [c] #
\(\mathcal{O}(\min(m,n))\). zipWith generalises zip by zipping with the
function given as the first argument, instead of a tupling function.
zipWith (,) xs ys == zip xs ys zipWith f [x1,x2,x3..] [y1,y2,y3..] == [f x1 y1, f x2 y2, f x3 y3..]
For example, zipWith (+)
>>>zipWith (+) [1, 2, 3] [4, 5, 6][5,7,9]
zipWith is right-lazy:
>>>let f = undefined>>>zipWith f [] undefined[]
zipWith is capable of list fusion, but it is restricted to its
first list argument and its resulting list.
unzip :: [(a, b)] -> ([a], [b]) #
unzip transforms a list of pairs into a list of first components
and a list of second components.
>>>unzip []([],[])>>>unzip [(1, 'a'), (2, 'b')]([1,2],"ab")
takeWhile :: (a -> Bool) -> [a] -> [a] #
takeWhile, applied to a predicate p and a list xs, returns the
longest prefix (possibly empty) of xs of elements that satisfy p.
>>>takeWhile (< 3) [1,2,3,4,1,2,3,4][1,2]>>>takeWhile (< 9) [1,2,3][1,2,3]>>>takeWhile (< 0) [1,2,3][]
take n, applied to a list xs, returns the prefix of xs
of length n, or xs itself if n >= .length xs
>>>take 5 "Hello World!""Hello">>>take 3 [1,2,3,4,5][1,2,3]>>>take 3 [1,2][1,2]>>>take 3 [][]>>>take (-1) [1,2][]>>>take 0 [1,2][]
It is an instance of the more general genericTake,
in which n may be of any integral type.
\(\mathcal{O}(1)\). Extract the elements after the head of a list, which must be non-empty.
>>>tail [1, 2, 3][2,3]>>>tail [1][]>>>tail []*** Exception: Prelude.tail: empty list
splitAt :: Int -> [a] -> ([a], [a]) #
splitAt n xs returns a tuple where first element is xs prefix of
length n and second element is the remainder of the list:
>>>splitAt 6 "Hello World!"("Hello ","World!")>>>splitAt 3 [1,2,3,4,5]([1,2,3],[4,5])>>>splitAt 1 [1,2,3]([1],[2,3])>>>splitAt 3 [1,2,3]([1,2,3],[])>>>splitAt 4 [1,2,3]([1,2,3],[])>>>splitAt 0 [1,2,3]([],[1,2,3])>>>splitAt (-1) [1,2,3]([],[1,2,3])
It is equivalent to ( when take n xs, drop n xs)n is not _|_
(splitAt _|_ xs = _|_).
splitAt is an instance of the more general genericSplitAt,
in which n may be of any integral type.
span :: (a -> Bool) -> [a] -> ([a], [a]) #
span, applied to a predicate p and a list xs, returns a tuple where
first element is longest prefix (possibly empty) of xs of elements that
satisfy p and second element is the remainder of the list:
>>>span (< 3) [1,2,3,4,1,2,3,4]([1,2],[3,4,1,2,3,4])>>>span (< 9) [1,2,3]([1,2,3],[])>>>span (< 0) [1,2,3]([],[1,2,3])
scanr1 :: (a -> a -> a) -> [a] -> [a] #
\(\mathcal{O}(n)\). scanr1 is a variant of scanr that has no starting
value argument.
>>>scanr1 (+) [1..4][10,9,7,4]>>>scanr1 (+) [][]>>>scanr1 (-) [1..4][-2,3,-1,4]>>>scanr1 (&&) [True, False, True, True][False,False,True,True]>>>scanr1 (||) [True, True, False, False][True,True,False,False]>>>force $ scanr1 (+) [1..]*** Exception: stack overflow
scanr :: (a -> b -> b) -> b -> [a] -> [b] #
\(\mathcal{O}(n)\). scanr is the right-to-left dual of scanl. Note that the order of parameters on the accumulating function are reversed compared to scanl.
Also note that
head (scanr f z xs) == foldr f z xs.
>>>scanr (+) 0 [1..4][10,9,7,4,0]>>>scanr (+) 42 [][42]>>>scanr (-) 100 [1..4][98,-97,99,-96,100]>>>scanr (\nextChar reversedString -> nextChar : reversedString) "foo" ['a', 'b', 'c', 'd']["abcdfoo","bcdfoo","cdfoo","dfoo","foo"]>>>force $ scanr (+) 0 [1..]*** Exception: stack overflow
scanl1 :: (a -> a -> a) -> [a] -> [a] #
\(\mathcal{O}(n)\). scanl1 is a variant of scanl that has no starting
value argument:
scanl1 f [x1, x2, ...] == [x1, x1 `f` x2, ...]
>>>scanl1 (+) [1..4][1,3,6,10]>>>scanl1 (+) [][]>>>scanl1 (-) [1..4][1,-1,-4,-8]>>>scanl1 (&&) [True, False, True, True][True,False,False,False]>>>scanl1 (||) [False, False, True, True][False,False,True,True]>>>scanl1 (+) [1..]* Hangs forever *
scanl :: (b -> a -> b) -> b -> [a] -> [b] #
\(\mathcal{O}(n)\). scanl is similar to foldl, but returns a list of
successive reduced values from the left:
scanl f z [x1, x2, ...] == [z, z `f` x1, (z `f` x1) `f` x2, ...]
Note that
last (scanl f z xs) == foldl f z xs
>>>scanl (+) 0 [1..4][0,1,3,6,10]>>>scanl (+) 42 [][42]>>>scanl (-) 100 [1..4][100,99,97,94,90]>>>scanl (\reversedString nextChar -> nextChar : reversedString) "foo" ['a', 'b', 'c', 'd']["foo","afoo","bafoo","cbafoo","dcbafoo"]>>>scanl (+) 0 [1..]* Hangs forever *
reverse xs returns the elements of xs in reverse order.
xs must be finite.
>>>reverse [][]>>>reverse [42][42]>>>reverse [2,5,7][7,5,2]>>>reverse [1..]* Hangs forever *
replicate :: Int -> a -> [a] #
replicate n x is a list of length n with x the value of
every element.
It is an instance of the more general genericReplicate,
in which n may be of any integral type.
>>>replicate 0 True[]>>>replicate (-1) True[]>>>replicate 4 True[True,True,True,True]
repeat x is an infinite list, with x the value of every element.
>>>take 20 $ repeat 17[17,17,17,17,17,17,17,17,17...
lookup :: Eq a => a -> [(a, b)] -> Maybe b #
\(\mathcal{O}(n)\). lookup key assocs looks up a key in an association
list.
>>>lookup 2 []Nothing>>>lookup 2 [(1, "first")]Nothing>>>lookup 2 [(1, "first"), (2, "second"), (3, "third")]Just "second"
\(\mathcal{O}(n)\). Extract the last element of a list, which must be finite and non-empty.
>>>last [1, 2, 3]3>>>last [1..]* Hangs forever *>>>last []*** Exception: Prelude.last: empty list
iterate :: (a -> a) -> a -> [a] #
iterate f x returns an infinite list of repeated applications
of f to x:
iterate f x == [x, f x, f (f x), ...]
Note that iterate is lazy, potentially leading to thunk build-up if
the consumer doesn't force each iterate. See iterate' for a strict
variant of this function.
>>>take 10 $ iterate not True[True,False,True,False...>>>take 10 $ iterate (+3) 42[42,45,48,51,54,57,60,63...
\(\mathcal{O}(n)\). Return all the elements of a list except the last one. The list must be non-empty.
>>>init [1, 2, 3][1,2]>>>init [1][]>>>init []*** Exception: Prelude.init: empty list
\(\mathcal{O}(1)\). Extract the first element of a list, which must be non-empty.
>>>head [1, 2, 3]1>>>head [1..]1>>>head []*** Exception: Prelude.head: empty list
drop n xs returns the suffix of xs
after the first n elements, or [] if n >= .length xs
>>>drop 6 "Hello World!""World!">>>drop 3 [1,2,3,4,5][4,5]>>>drop 3 [1,2][]>>>drop 3 [][]>>>drop (-1) [1,2][1,2]>>>drop 0 [1,2][1,2]
It is an instance of the more general genericDrop,
in which n may be of any integral type.
cycle ties a finite list into a circular one, or equivalently,
the infinite repetition of the original list. It is the identity
on infinite lists.
>>>cycle []*** Exception: Prelude.cycle: empty list>>>take 20 $ cycle [42][42,42,42,42,42,42,42,42,42,42...>>>take 20 $ cycle [2, 5, 7][2,5,7,2,5,7,2,5,7,2,5,7...
break :: (a -> Bool) -> [a] -> ([a], [a]) #
break, applied to a predicate p and a list xs, returns a tuple where
first element is longest prefix (possibly empty) of xs of elements that
do not satisfy p and second element is the remainder of the list:
>>>break (> 3) [1,2,3,4,1,2,3,4]([1,2,3],[4,1,2,3,4])>>>break (< 9) [1,2,3]([],[1,2,3])>>>break (> 9) [1,2,3]([1,2,3],[])
(!!) :: [a] -> Int -> a infixl 9 #
List index (subscript) operator, starting from 0.
It is an instance of the more general genericIndex,
which takes an index of any integral type.
>>>['a', 'b', 'c'] !! 0'a'>>>['a', 'b', 'c'] !! 2'c'>>>['a', 'b', 'c'] !! 3*** Exception: Prelude.!!: index too large>>>['a', 'b', 'c'] !! (-1)*** Exception: Prelude.!!: negative index
maybe :: b -> (a -> b) -> Maybe a -> b #
The maybe function takes a default value, a function, and a Maybe
value. If the Maybe value is Nothing, the function returns the
default value. Otherwise, it applies the function to the value inside
the Just and returns the result.
Examples
Basic usage:
>>>maybe False odd (Just 3)True
>>>maybe False odd NothingFalse
Read an integer from a string using readMaybe. If we succeed,
return twice the integer; that is, apply (*2) to it. If instead
we fail to parse an integer, return 0 by default:
>>>import Text.Read ( readMaybe )>>>maybe 0 (*2) (readMaybe "5")10>>>maybe 0 (*2) (readMaybe "")0
Apply show to a Maybe Int. If we have Just n, we want to show
the underlying Int n. But if we have Nothing, we return the
empty string instead of (for example) "Nothing":
>>>maybe "" show (Just 5)"5">>>maybe "" show Nothing""
until :: (a -> Bool) -> (a -> a) -> a -> a #
until p ff until p holds.
flip :: (a -> b -> c) -> b -> a -> c #
flip ff.
>>>flip (++) "hello" "world""worldhello"
($!) :: forall (r :: RuntimeRep) a (b :: TYPE r). (a -> b) -> a -> b infixr 0 #
Strict (call-by-value) application operator. It takes a function and an argument, evaluates the argument to weak head normal form (WHNF), then calls the function with that value.
undefined :: forall (r :: RuntimeRep) (a :: TYPE r). HasCallStack => a #
errorWithoutStackTrace :: forall (r :: RuntimeRep) (a :: TYPE r). [Char] -> a #
A variant of error that does not produce a stack trace.
Since: base-4.9.0.0
error :: forall (r :: RuntimeRep) (a :: TYPE r). HasCallStack => [Char] -> a #
error stops execution and displays an error message.