Copyright | (c) 2013 Justus Sagemüller |
---|---|
License | GPL v3 (see COPYING) |
Maintainer | (@) sagemueller $ geo.uni-koeln.de |
Safe Haskell | Trustworthy |
Language | Haskell2010 |
Synopsis
- 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 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
- 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
- newtype ConstrainedCategory (k :: * -> * -> *) (o :: * -> Constraint) (a :: *) (b :: *) = ConstrainedMorphism (k a b)
- type Hask = Unconstrained ⊢ (->)
- class Category k 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
- class (CoCartesian r, Cartesian t, Functor f r t, Object t (f (ZeroObject r))) => SumToProduct f r t where
- class (Category r, Category t, Object t (f (UnitObject r))) => 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) => 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 :: 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
- foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b
- null :: Foldable t => t a -> Bool
- length :: Foldable t => t a -> Int
- foldl :: Foldable t => (b -> a -> b) -> b -> t a -> b
- foldl1 :: Foldable t => (a -> a -> a) -> t a -> a
- sum :: (Foldable t, Num a) => t a -> a
- product :: (Foldable t, Num a) => t a -> a
- foldr1 :: Foldable t => (a -> a -> a) -> t a -> a
- maximum :: (Foldable t, Ord a) => t a -> a
- minimum :: (Foldable t, Ord a) => t a -> a
- elem :: (Foldable t, Eq a) => a -> t a -> Bool
- 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
- 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
- readIO :: Read a => String -> IO a
- readLn :: Read a => IO a
- appendFile :: FilePath -> String -> IO ()
- writeFile :: FilePath -> String -> IO ()
- readFile :: FilePath -> IO String
- interact :: (String -> String) -> IO ()
- getContents :: IO String
- getLine :: IO String
- getChar :: IO Char
- putStrLn :: String -> IO ()
- putStr :: String -> IO ()
- putChar :: Char -> IO ()
- ioError :: IOError -> IO a
- type FilePath = String
- userError :: String -> IOError
- type IOError = IOException
- notElem :: (Foldable t, Eq a) => a -> t a -> Bool
- all :: Foldable t => (a -> Bool) -> t a -> Bool
- any :: Foldable t => (a -> Bool) -> t a -> Bool
- or :: Foldable t => t Bool -> Bool
- and :: Foldable t => t Bool -> Bool
- concat :: Foldable t => t [a] -> [a]
- unwords :: [String] -> String
- words :: String -> [String]
- unlines :: [String] -> String
- lines :: String -> [String]
- read :: Read a => String -> a
- reads :: Read a => ReadS a
- either :: (a -> c) -> (b -> c) -> Either a b -> c
- lex :: ReadS String
- readParen :: Bool -> ReadS a -> ReadS a
- type ReadS a = String -> [(a, String)]
- lcm :: Integral a => a -> a -> a
- gcd :: Integral a => a -> a -> a
- (^^) :: (Fractional a, Integral b) => a -> b -> a
- (^) :: (Num a, Integral b) => a -> b -> a
- odd :: Integral a => a -> Bool
- even :: Integral a => a -> Bool
- showParen :: Bool -> ShowS -> ShowS
- showString :: String -> ShowS
- showChar :: Char -> ShowS
- shows :: Show a => a -> ShowS
- type ShowS = String -> String
- unzip3 :: [(a, b, c)] -> ([a], [b], [c])
- unzip :: [(a, b)] -> ([a], [b])
- zipWith3 :: (a -> b -> c -> d) -> [a] -> [b] -> [c] -> [d]
- zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
- zip3 :: [a] -> [b] -> [c] -> [(a, b, c)]
- (!!) :: [a] -> Int -> a
- lookup :: Eq a => a -> [(a, b)] -> Maybe b
- reverse :: [a] -> [a]
- break :: (a -> Bool) -> [a] -> ([a], [a])
- span :: (a -> Bool) -> [a] -> ([a], [a])
- splitAt :: Int -> [a] -> ([a], [a])
- drop :: Int -> [a] -> [a]
- take :: Int -> [a] -> [a]
- dropWhile :: (a -> Bool) -> [a] -> [a]
- takeWhile :: (a -> Bool) -> [a] -> [a]
- cycle :: [a] -> [a]
- replicate :: Int -> a -> [a]
- repeat :: a -> [a]
- iterate :: (a -> 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]
- init :: [a] -> [a]
- last :: [a] -> a
- tail :: [a] -> [a]
- head :: [a] -> a
- maybe :: b -> (a -> b) -> Maybe a -> b
- subtract :: Num a => a -> a -> a
- asTypeOf :: a -> a -> a
- until :: (a -> Bool) -> (a -> a) -> a -> a
- ($!) :: (a -> b) -> a -> b
- flip :: (a -> b -> c) -> b -> a -> c
- type String = [Char]
- undefined :: HasCallStack => a
- errorWithoutStackTrace :: [Char] -> a
- error :: HasCallStack => [Char] -> a
- (&&) :: Bool -> Bool -> Bool
- (||) :: Bool -> Bool -> Bool
- not :: Bool -> Bool
The constrained-categories facilities
data GenericAgent k a v Source #
GenericAgent | |
|
class Category k => HasAgent k where Source #
An agent value is a "general representation" of a category's
values, i.e. global elements. This is useful to define certain
morphisms (including ones that can't just "inherit" from '->'
with arr
) in ways other than point-free
composition pipelines. Instead, you can write algebraic expressions
much as if dealing with actual values of your category's objects,
but using the agent type which is restricted so any function
defined as such a lambda-expression qualifies as a morphism
of that category.
alg :: (Object k a, Object k b) => (forall q. Object k q => AgentVal k q a -> AgentVal k q b) -> k a b Source #
($~) :: (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 #
type MorphObjects k b c :: Constraint Source #
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 ((->) :: Type -> Type -> Type) Source # | |
Defined in Control.Category.Constrained type MorphObjects (->) b c :: Constraint Source # 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 # | |
(Curry f, o (UnitObject f)) => Curry (o ⊢ f) Source # | |
Defined in Control.Category.Constrained type MorphObjects (o ⊢ f) b c :: Constraint Source # 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 ((->) :: Type -> Type -> Type), Function a) => Curry (Kleisli m a) Source # | |
Defined in Control.Monad.Constrained type MorphObjects (Kleisli m a) b c :: Constraint Source # 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 # |
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.
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
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
.
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 UnitObject k :: * Source #
Defaults to '()', and should normally be left at that.
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).
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,
is the category of all
totally ordered data types (but with arbitrary functions; this does not require
monotonicity or anything).Ord
⊢(->)
Instances
(o (), o [()], o Void, o [Void]) => SumToProduct [] (o ⊢ ((->) :: Type -> Type -> Type)) (o ⊢ ((->) :: Type -> Type -> Type)) Source # | |
Defined in Control.Functor.Constrained 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 # | |
(Functor [] k k, o [UnitObject k]) => Functor [] (o ⊢ k) (o ⊢ k) 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 # | |
(Curry f, o (UnitObject f)) => Curry (o ⊢ f) Source # | |
Defined in Control.Category.Constrained type MorphObjects (o ⊢ f) b c :: Constraint Source # 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 # | |
(CoCartesian f, o (ZeroObject f)) => CoCartesian (o ⊢ f) Source # | |
Defined in Control.Category.Constrained type SumObjects (o ⊢ f) a b :: Constraint Source # type ZeroObject (o ⊢ f) :: Type Source # 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 # | |
(Cartesian f, o (UnitObject f)) => Cartesian (o ⊢ f) Source # | |
Defined in Control.Category.Constrained type PairObjects (o ⊢ f) a b :: Constraint Source # type UnitObject (o ⊢ f) :: Type Source # 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 # | |
Category k => Category (isObj ⊢ k) Source # | |
(WellPointed a, o (UnitObject a)) => WellPointed (o ⊢ a) Source # | |
Defined in Control.Arrow.Constrained type PointObject (o ⊢ a) x :: Constraint Source # 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 # | |
(SPDistribute k, o (ZeroObject k), o (UnitObject k)) => SPDistribute (o ⊢ k) Source # | |
Defined in Control.Arrow.Constrained 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 # | |
(PreArrChoice k, o (ZeroObject k)) => PreArrChoice (o ⊢ k) Source # | |
Defined in Control.Arrow.Constrained (|||) :: (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 # | |
(MorphChoice k, o (ZeroObject k)) => MorphChoice (o ⊢ k) Source # | |
Defined in Control.Arrow.Constrained 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 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 # | |
(EnhancedCat a k, o (UnitObject a)) => EnhancedCat (o ⊢ a) k Source # | |
Category f => EnhancedCat (o ⊢ f) (Discrete :: Type -> Type -> Type) Source # | |
Function f => EnhancedCat ((->) :: Type -> Type -> Type) (o ⊢ f) Source # | |
type ZeroObject (o ⊢ f) Source # | |
Defined in Control.Category.Constrained | |
type UnitObject (o ⊢ f) Source # | |
Defined in Control.Category.Constrained | |
type Object (isObj ⊢ k) o 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 SumObjects (o ⊢ f) a b Source # | |
Defined in Control.Category.Constrained | |
type PairObjects (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 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.
type Object k o :: Constraint Source #
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.
is the 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 Function
s 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
data GenericAgent k a v Source #
GenericAgent | |
|
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.
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 #
type MorphObjects k b c :: Constraint Source #
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 ((->) :: Type -> Type -> Type) Source # | |
Defined in Control.Category.Constrained type MorphObjects (->) b c :: Constraint Source # 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 # | |
(Curry f, o (UnitObject f)) => Curry (o ⊢ f) Source # | |
Defined in Control.Category.Constrained type MorphObjects (o ⊢ f) b c :: Constraint Source # 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 ((->) :: Type -> Type -> Type), Function a) => Curry (Kleisli m a) Source # | |
Defined in Control.Monad.Constrained type MorphObjects (Kleisli m a) b c :: Constraint Source # 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 # |
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.
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
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
.
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 UnitObject k :: * Source #
Defaults to '()', and should normally be left at that.
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).
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 #
newtype 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,
is the category of all
totally ordered data types (but with arbitrary functions; this does not require
monotonicity or anything).Ord
⊢(->)
ConstrainedMorphism (k a b) |
Instances
(o (), o [()], o Void, o [Void]) => SumToProduct [] (o ⊢ ((->) :: Type -> Type -> Type)) (o ⊢ ((->) :: Type -> Type -> Type)) Source # | |
Defined in Control.Functor.Constrained 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 # | |
(Functor [] k k, o [UnitObject k]) => Functor [] (o ⊢ k) (o ⊢ k) 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 # | |
(Curry f, o (UnitObject f)) => Curry (o ⊢ f) Source # | |
Defined in Control.Category.Constrained type MorphObjects (o ⊢ f) b c :: Constraint Source # 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 # | |
(CoCartesian f, o (ZeroObject f)) => CoCartesian (o ⊢ f) Source # | |
Defined in Control.Category.Constrained type SumObjects (o ⊢ f) a b :: Constraint Source # type ZeroObject (o ⊢ f) :: Type Source # 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 # | |
(Cartesian f, o (UnitObject f)) => Cartesian (o ⊢ f) Source # | |
Defined in Control.Category.Constrained type PairObjects (o ⊢ f) a b :: Constraint Source # type UnitObject (o ⊢ f) :: Type Source # 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 # | |
Category k => Category (isObj ⊢ k) Source # | |
(WellPointed a, o (UnitObject a)) => WellPointed (o ⊢ a) Source # | |
Defined in Control.Arrow.Constrained type PointObject (o ⊢ a) x :: Constraint Source # 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 # | |
(SPDistribute k, o (ZeroObject k), o (UnitObject k)) => SPDistribute (o ⊢ k) Source # | |
Defined in Control.Arrow.Constrained 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 # | |
(PreArrChoice k, o (ZeroObject k)) => PreArrChoice (o ⊢ k) Source # | |
Defined in Control.Arrow.Constrained (|||) :: (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 # | |
(MorphChoice k, o (ZeroObject k)) => MorphChoice (o ⊢ k) Source # | |
Defined in Control.Arrow.Constrained 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 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 # | |
(EnhancedCat a k, o (UnitObject a)) => EnhancedCat (o ⊢ a) k Source # | |
Category f => EnhancedCat (o ⊢ f) (Discrete :: Type -> Type -> Type) Source # | |
Function f => EnhancedCat ((->) :: Type -> Type -> Type) (o ⊢ f) Source # | |
type ZeroObject (o ⊢ f) Source # | |
Defined in Control.Category.Constrained | |
type UnitObject (o ⊢ f) Source # | |
Defined in Control.Category.Constrained | |
type Object (isObj ⊢ k) o 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 SumObjects (o ⊢ f) a b Source # | |
Defined in Control.Category.Constrained | |
type PairObjects (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 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.
type Object k o :: Constraint Source #
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.
is the 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 Function
s 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 #
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.
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
SumToProduct [] ((->) :: Type -> Type -> Type) ((->) :: Type -> Type -> Type) Source # | |
Defined in Control.Functor.Constrained 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 # | |
(o (), o [()], o Void, o [Void]) => SumToProduct [] (o ⊢ ((->) :: Type -> Type -> Type)) (o ⊢ ((->) :: Type -> Type -> Type)) Source # | |
Defined in Control.Functor.Constrained 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 # |
class (Category r, Category t, Object t (f (UnitObject r))) => Functor f r t | f r -> t, f t -> r where 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 #
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 ((->) :: Type -> Type -> Type) ((->) :: Type -> Type -> Type) Source # | |
Defined in Control.Applicative.Constrained 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) => Monoidal f r t where Source #
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 ((->) :: Type -> Type -> Type) ((->) :: Type -> Type -> Type) Source # | |
Defined in Control.Applicative.Constrained 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 #
Kleisli | |
|
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
, which would always be ambiguous with the more general
signature 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.
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.
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] #
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, ...]
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 Bool | Since: base-2.1 |
Bounded Char | Since: base-2.1 |
Bounded Int | Since: base-2.1 |
Bounded Ordering | Since: base-2.1 |
Bounded Word | Since: base-2.1 |
Bounded VecCount | Since: base-4.10.0.0 |
Bounded VecElem | Since: base-4.10.0.0 |
Bounded () | Since: base-2.1 |
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 SourceUnpackedness | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
Bounded SourceStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
Bounded DecidedStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
Bounded a => Bounded (Min a) | Since: base-4.9.0.0 |
Bounded a => Bounded (Max a) | Since: base-4.9.0.0 |
Bounded a => Bounded (First a) | Since: base-4.9.0.0 |
Bounded a => Bounded (Last a) | Since: base-4.9.0.0 |
Bounded m => Bounded (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup minBound :: WrappedMonoid m # maxBound :: WrappedMonoid m # | |
Bounded a => Bounded (Dual a) | Since: base-2.1 |
Bounded a => Bounded (Sum a) | Since: base-2.1 |
Bounded a => Bounded (Product a) | Since: base-2.1 |
(Bounded a, Bounded b) => Bounded (a, b) | Since: base-2.1 |
(Bounded a, Bounded b, Bounded c) => Bounded (a, b, c) | 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 |
a ~ b => Bounded (a :~: b) | Since: base-4.7.0.0 |
Bounded b => Bounded (Tagged s b) | |
(Bounded a, Bounded b, Bounded c, Bounded d) => Bounded (a, b, c, d) | Since: base-2.1 |
a ~~ b => Bounded (a :~~: b) | Since: base-4.10.0.0 |
(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
andsucc
maxBound
should result in a runtime error.pred
minBound
fromEnum
andtoEnum
should give a runtime error if the result value is not representable in the result type. For example,
is an error.toEnum
7 ::Bool
enumFrom
andenumFromThen
should 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 = minBound
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, ==
is customarily
expected to implement an equivalence relationship where two values comparing
equal are indistinguishable by "public" functions, with a "public" function
being one not allowing to see implementation details. For example, for a
type representing non-normalised natural numbers modulo 100, a "public"
function doesn't make the difference between 1 and 201. It is expected to
have the following properties:
Instances
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 Integer | |
Eq Ordering | |
Eq Word | |
Eq () | |
Eq TyCon | |
Eq Module | |
Eq TrName | |
Eq BigNat | |
Eq Void | Since: base-4.8.0.0 |
Eq SpecConstrAnnotation | Since: base-4.3.0.0 |
Defined in GHC.Exts (==) :: SpecConstrAnnotation -> SpecConstrAnnotation -> Bool # (/=) :: SpecConstrAnnotation -> SpecConstrAnnotation -> Bool # | |
Eq AsyncException | Since: base-4.2.0.0 |
Defined in GHC.IO.Exception (==) :: AsyncException -> AsyncException -> Bool # (/=) :: AsyncException -> AsyncException -> Bool # | |
Eq ArrayException | Since: base-4.2.0.0 |
Defined in GHC.IO.Exception (==) :: ArrayException -> ArrayException -> Bool # (/=) :: ArrayException -> ArrayException -> Bool # | |
Eq ExitCode | |
Eq IOErrorType | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception (==) :: IOErrorType -> IOErrorType -> Bool # (/=) :: IOErrorType -> IOErrorType -> Bool # | |
Eq MaskingState | Since: base-4.3.0.0 |
Defined in GHC.IO (==) :: MaskingState -> MaskingState -> Bool # (/=) :: MaskingState -> MaskingState -> Bool # | |
Eq IOException | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception (==) :: IOException -> IOException -> Bool # (/=) :: IOException -> IOException -> Bool # | |
Eq All | Since: base-2.1 |
Eq Any | Since: base-2.1 |
Eq Fixity | Since: base-4.6.0.0 |
Eq Associativity | Since: base-4.6.0.0 |
Defined in GHC.Generics (==) :: Associativity -> Associativity -> Bool # (/=) :: Associativity -> Associativity -> Bool # | |
Eq SourceUnpackedness | Since: base-4.9.0.0 |
Defined in GHC.Generics (==) :: SourceUnpackedness -> SourceUnpackedness -> Bool # (/=) :: SourceUnpackedness -> SourceUnpackedness -> Bool # | |
Eq SourceStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics (==) :: SourceStrictness -> SourceStrictness -> Bool # (/=) :: SourceStrictness -> SourceStrictness -> Bool # | |
Eq DecidedStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics (==) :: DecidedStrictness -> DecidedStrictness -> Bool # (/=) :: DecidedStrictness -> DecidedStrictness -> Bool # | |
Eq SrcLoc | Since: base-4.9.0.0 |
Eq a => Eq [a] | |
Eq a => Eq (Maybe a) | Since: base-2.1 |
Eq a => Eq (Ratio a) | Since: base-2.1 |
Eq p => Eq (Par1 p) | Since: base-4.7.0.0 |
Eq a => Eq (Complex a) | Since: base-2.1 |
Eq a => Eq (Min a) | Since: base-4.9.0.0 |
Eq a => Eq (Max a) | Since: base-4.9.0.0 |
Eq a => Eq (First a) | Since: base-4.9.0.0 |
Eq a => Eq (Last a) | Since: base-4.9.0.0 |
Eq m => Eq (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup (==) :: WrappedMonoid m -> WrappedMonoid m -> Bool # (/=) :: WrappedMonoid m -> WrappedMonoid m -> Bool # | |
Eq a => Eq (Option a) | Since: base-4.9.0.0 |
Eq a => Eq (ZipList a) | Since: base-4.7.0.0 |
Eq a => Eq (First a) | Since: base-2.1 |
Eq a => Eq (Last a) | Since: base-2.1 |
Eq a => Eq (Dual a) | Since: base-2.1 |
Eq a => Eq (Sum a) | Since: base-2.1 |
Eq a => Eq (Product a) | Since: base-2.1 |
Eq a => Eq (NonEmpty a) | Since: base-4.9.0.0 |
(Eq a, Eq b) => Eq (Either a b) | Since: base-2.1 |
Eq (V1 p) | Since: base-4.9.0.0 |
Eq (U1 p) | Since: base-4.9.0.0 |
(Eq a, Eq b) => Eq (a, b) | |
Eq a => Eq (Arg a b) | Since: base-4.9.0.0 |
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 a, Eq b, Eq c) => Eq (a, b, c) | |
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 (a :~: b) | Since: base-4.7.0.0 |
Eq b => Eq (Tagged s b) | |
Eq c => Eq (K1 i c p) | Since: base-4.7.0.0 |
(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 a, Eq b, Eq c, Eq d) => Eq (a, b, c, d) | |
(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 (a :~~: b) | Since: base-4.10.0.0 |
Eq (f p) => Eq (M1 i c f p) | Since: base-4.7.0.0 |
Eq (f (g p)) => Eq ((f :.: g) p) | Since: base-4.7.0.0 |
(Eq a, Eq b, Eq c, Eq d, Eq e) => Eq (a, b, c, d, e) | |
(Eq1 f, Eq1 g, Eq a) => Eq (Compose f g a) | Since: base-4.9.0.0 |
(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
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:
recip
gives 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.
fromRational, (recip | (/))
fractional division
reciprocal fraction
fromRational :: Rational -> a #
Conversion from a Rational
(that is
).
A floating literal stands for an application of Ratio
Integer
fromRational
to a value of type Rational
, so such literals have type
(
.Fractional
a) => a
Instances
Integral a => Fractional (Ratio a) | Since: base-2.0.1 |
RealFloat a => Fractional (Complex a) | Since: base-2.1 |
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 y
withrem x y
=fromInteger 0
org (rem x y)
<g y
x
=y * div x y + mod x y
withmod x y
=fromInteger 0
orf (mod x y)
<f y
An example of a suitable Euclidean function, for Integer
's instance, is
abs
.
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 Int | Since: base-2.0.1 |
Integral Integer | Since: base-2.0.1 |
Defined in GHC.Real | |
Integral Natural | Since: base-4.8.0.0 |
Defined in GHC.Real | |
Integral Word | Since: base-2.1 |
Integral a => Integral (Const a b) | Since: base-4.9.0.0 |
Defined in Data.Functor.Const | |
Integral a => Integral (Tagged s a) | |
Defined in Data.Tagged 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 + x
fromInteger 0
is the additive identityx + fromInteger 0
=x
negate
gives the additive inversex + negate x
=fromInteger 0
- Associativity of (*)
(x * y) * z
=x * (y * z)
fromInteger 1
is the multiplicative identityx * fromInteger 1
=x
andfromInteger 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.
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 Int | Since: base-2.1 |
Num Integer | Since: base-2.1 |
Num Natural | Note that Since: base-4.8.0.0 |
Num Word | Since: base-2.1 |
Integral a => Num (Ratio a) | Since: base-2.0.1 |
RealFloat a => Num (Complex a) | Since: base-2.1 |
Num a => Num (Min a) | Since: base-4.9.0.0 |
Num a => Num (Max a) | Since: base-4.9.0.0 |
Num a => Num (Sum a) | Since: base-4.7.0.0 |
Num a => Num (Product a) | Since: base-4.7.0.0 |
Defined in Data.Semigroup.Internal | |
Num a => Num (Op a b) | |
Num a => Num (Const a b) | Since: base-4.9.0.0 |
(Applicative f, Num a) => Num (Ap f a) | 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.
The Haskell Report defines no laws for Ord
. However, <=
is customarily
expected to implement a non-strict partial order and have the following
properties:
- Transitivity
- if
x <= y && y <= z
=True
, thenx <= z
=True
- Reflexivity
x <= x
=True
- Antisymmetry
- if
x <= y && y <= x
=True
, thenx == y
=True
Note that the following operator interactions are expected to hold:
x >= y
=y <= x
x < y
=x <= y && x /= y
x > y
=y < x
x < y
=compare x y == LT
x > y
=compare x y == GT
x == y
=compare x y == EQ
min x y == if x <= y then x else y
=True
max x y == if x >= y then x else y
=True
Minimal complete definition: either compare
or <=
.
Using compare
can be more efficient for complex types.
compare :: a -> a -> Ordering #
(<) :: a -> a -> Bool infix 4 #
(<=) :: a -> a -> Bool infix 4 #
(>) :: a -> a -> Bool infix 4 #
Instances
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 Integer | |
Ord Ordering | |
Defined in GHC.Classes | |
Ord Word | |
Ord () | |
Ord TyCon | |
Ord BigNat | |
Ord Void | Since: base-4.8.0.0 |
Ord AsyncException | Since: base-4.2.0.0 |
Defined in GHC.IO.Exception 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 ArrayException | Since: base-4.2.0.0 |
Defined in GHC.IO.Exception 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 ExitCode | |
Defined in GHC.IO.Exception | |
Ord All | Since: base-2.1 |
Ord Any | Since: base-2.1 |
Ord Fixity | Since: base-4.6.0.0 |
Ord Associativity | Since: base-4.6.0.0 |
Defined in GHC.Generics 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 SourceUnpackedness | Since: base-4.9.0.0 |
Defined in GHC.Generics 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 SourceStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics 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 DecidedStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics 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 a => Ord [a] | |
Ord a => Ord (Maybe a) | Since: base-2.1 |
Integral a => Ord (Ratio a) | Since: base-2.0.1 |
Ord p => Ord (Par1 p) | Since: base-4.7.0.0 |
Ord a => Ord (Min a) | Since: base-4.9.0.0 |
Ord a => Ord (Max a) | Since: base-4.9.0.0 |
Ord a => Ord (First a) | Since: base-4.9.0.0 |
Ord a => Ord (Last a) | Since: base-4.9.0.0 |
Ord m => Ord (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup 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 (Option a) | Since: base-4.9.0.0 |
Defined in Data.Semigroup | |
Ord a => Ord (ZipList a) | Since: base-4.7.0.0 |
Defined in Control.Applicative | |
Ord a => Ord (First a) | Since: base-2.1 |
Ord a => Ord (Last a) | Since: base-2.1 |
Ord a => Ord (Dual a) | Since: base-2.1 |
Ord a => Ord (Sum a) | Since: base-2.1 |
Ord a => Ord (Product a) | Since: base-2.1 |
Defined in Data.Semigroup.Internal | |
Ord a => Ord (NonEmpty a) | Since: base-4.9.0.0 |
(Ord a, Ord b) => Ord (Either a b) | Since: base-2.1 |
Ord (V1 p) | Since: base-4.9.0.0 |
Ord (U1 p) | Since: base-4.7.0.0 |
(Ord a, Ord b) => Ord (a, b) | |
Ord a => Ord (Arg a b) | Since: base-4.9.0.0 |
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 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 |
Ord (URec Double p) | Since: base-4.9.0.0 |
Defined in GHC.Generics 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 |
(Ord a, Ord b, Ord c) => Ord (a, b, c) | |
Defined in GHC.Classes | |
Ord a => Ord (Const a b) | Since: base-4.9.0.0 |
Defined in Data.Functor.Const | |
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 |
Ord (Coercion a b) | Since: base-4.7.0.0 |
Defined in Data.Type.Coercion | |
Ord (a :~: b) | Since: base-4.7.0.0 |
Defined in Data.Type.Equality | |
Ord b => Ord (Tagged s b) | |
Ord c => Ord (K1 i c 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 |
(Ord (f p), Ord (g p)) => Ord ((f :*: g) p) | Since: base-4.7.0.0 |
(Ord a, Ord b, Ord c, Ord d) => Ord (a, b, c, d) | |
Defined in GHC.Classes | |
(Ord1 f, Ord1 g, Ord a) => Ord (Product f g a) | Since: base-4.9.0.0 |
Defined in Data.Functor.Product 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 |
Defined in Data.Functor.Sum | |
Ord (a :~~: b) | Since: base-4.10.0.0 |
Ord (f p) => Ord (M1 i c f p) | Since: base-4.7.0.0 |
Ord (f (g p)) => Ord ((f :.: g) 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 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) # | |
(Ord1 f, Ord1 g, Ord a) => Ord (Compose f g a) | Since: base-4.9.0.0 |
Defined in Data.Functor.Compose 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 a, Ord b, Ord c, Ord d, Ord e, Ord f) => Ord (a, b, c, d, e, f) | |
Defined in GHC.Classes 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 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 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 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 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 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 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 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 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 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 String
s, 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
Read
instance 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
Read
will parse only the record-syntax form, and furthermore, the fields must be given in the same order as the original declaration. - The derived
Read
instance 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 = 5
Note 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 = readListPrecDefault
Why 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:
instanceRead
T wherereadPrec
= ...readListPrec
=readListPrecDefault
:: 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 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 Integer | Since: base-2.1 |
Read Natural | Since: base-4.8.0.0 |
Read Ordering | Since: base-2.1 |
Read Word | Since: base-4.5.0.0 |
Read Word8 | Since: base-2.1 |
Read Word16 | Since: base-2.1 |
Read Word32 | Since: base-2.1 |
Read Word64 | Since: base-2.1 |
Read () | Since: base-2.1 |
Read Void | Reading a Since: base-4.8.0.0 |
Read ExitCode | |
Read All | Since: base-2.1 |
Read Any | Since: base-2.1 |
Read Fixity | Since: base-4.6.0.0 |
Read Associativity | Since: base-4.6.0.0 |
Defined in GHC.Generics readsPrec :: Int -> ReadS Associativity # readList :: ReadS [Associativity] # | |
Read SourceUnpackedness | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
Read SourceStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
Read DecidedStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics | |
Read Lexeme | Since: base-2.1 |
Read GeneralCategory | Since: base-2.1 |
Defined in GHC.Read | |
Read a => Read [a] | Since: base-2.1 |
Read a => Read (Maybe a) | Since: base-2.1 |
(Integral a, Read a) => Read (Ratio a) | Since: base-2.1 |
Read p => Read (Par1 p) | Since: base-4.7.0.0 |
Read a => Read (Complex a) | Since: base-2.1 |
Read a => Read (Min a) | Since: base-4.9.0.0 |
Read a => Read (Max a) | Since: base-4.9.0.0 |
Read a => Read (First a) | Since: base-4.9.0.0 |
Read a => Read (Last a) | Since: base-4.9.0.0 |
Read m => Read (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup readsPrec :: Int -> ReadS (WrappedMonoid m) # readList :: ReadS [WrappedMonoid m] # readPrec :: ReadPrec (WrappedMonoid m) # readListPrec :: ReadPrec [WrappedMonoid m] # | |
Read a => Read (Option a) | Since: base-4.9.0.0 |
Read a => Read (ZipList a) | Since: base-4.7.0.0 |
Read a => Read (First a) | Since: base-2.1 |
Read a => Read (Last a) | Since: base-2.1 |
Read a => Read (Dual a) | Since: base-2.1 |
Read a => Read (Sum a) | Since: base-2.1 |
Read a => Read (Product a) | Since: base-2.1 |
Read a => Read (NonEmpty a) | Since: base-4.11.0.0 |
(Read a, Read b) => Read (Either a b) | Since: base-3.0 |
Read (V1 p) | Since: base-4.9.0.0 |
Read (U1 p) | Since: base-4.9.0.0 |
(Read a, Read b) => Read (a, b) | Since: base-2.1 |
(Ix a, Read a, Read b) => Read (Array a b) | Since: base-2.1 |
(Read a, Read b) => Read (Arg a b) | Since: base-4.9.0.0 |
Read (f p) => Read (Rec1 f p) | Since: base-4.7.0.0 |
(Read a, Read b, Read c) => Read (a, b, c) | 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 |
a ~ b => Read (a :~: b) | Since: base-4.7.0.0 |
Read b => Read (Tagged s b) | |
Read c => Read (K1 i c p) | Since: base-4.7.0.0 |
(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 a, Read b, Read c, Read d) => Read (a, b, c, d) | 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 |
a ~~ b => Read (a :~~: b) | Since: base-4.10.0.0 |
Read (f p) => Read (M1 i c f p) | Since: base-4.7.0.0 |
Read (f (g p)) => Read ((f :.: g) 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 |
(Read1 f, Read1 g, Read a) => Read (Compose f g a) | Since: base-4.9.0.0 |
(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 #
toRational :: a -> Rational #
the rational equivalent of its real argument with full precision
Instances
Real Int | Since: base-2.0.1 |
Defined in GHC.Real toRational :: Int -> Rational # | |
Real Integer | Since: base-2.0.1 |
Defined in GHC.Real toRational :: Integer -> Rational # | |
Real Natural | Since: base-4.8.0.0 |
Defined in GHC.Real toRational :: Natural -> Rational # | |
Real Word | Since: base-2.1 |
Defined in GHC.Real toRational :: Word -> Rational # | |
Integral a => Real (Ratio a) | Since: base-2.0.1 |
Defined in GHC.Real toRational :: Ratio a -> Rational # | |
Real a => Real (Const a b) | Since: base-4.9.0.0 |
Defined in Data.Functor.Const toRational :: Const a b -> Rational # | |
Real a => Real (Tagged s a) | |
Defined in Data.Tagged 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.
floatRadix, floatDigits, floatRange, decodeFloat, encodeFloat, isNaN, isInfinite, isDenormalized, isNegativeZero, isIEEE
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
yields 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
.
In particular, floatDigits
x
. If the type
contains a negative zero, also decodeFloat
0 = (0,0)
.
The result of decodeFloat
(-0.0) = (0,0)
is unspecified if either of
decodeFloat
x
or isNaN
x
is isInfinite
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) = x
is one of the two closest representable
floating-point numbers to encodeFloat
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
.
and for finite nonzero exponent
0 = 0x
,
.
If exponent
x = snd (decodeFloat
x) + floatDigits
xx
is a finite floating-point number, it is equal in value to
, where 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
,
computes the angle
(from the positive x-axis) of the vector from the origin to the
point atan2
y x(x,y)
.
returns a value in the range [atan2
y x-pi
,
pi
]. It follows the Common Lisp semantics for the origin when
signed zeroes are supported.
, with atan2
y 1y
in a type
that is RealFloat
, should return the same value as
.
A default definition of 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.
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:
n
is an integral number with the same sign asx
; andf
is 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 #
returns the integer nearest truncate
xx
between zero and x
round :: Integral b => a -> b #
returns the nearest integer to round
xx
;
the even integer if x
is equidistant between two integers
ceiling :: Integral b => a -> b #
returns the least integer not less than ceiling
xx
floor :: Integral b => a -> b #
returns the greatest integer not greater than floor
xx
Conversion of values to readable String
s.
Derived instances of Show
have the following properties, which
are compatible with derived instances of Read
:
- The result of
show
is 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
showsPrec
will produce infix applications of the constructor. - the representation will be enclosed in parentheses if the
precedence of the top-level constructor in
x
is less thand
(associativity is ignored). Thus, ifd
is0
then the result is never surrounded in parentheses; ifd
is11
it is always surrounded in parentheses, unless it is an atomic expression. - If the constructor is defined using record syntax, then
show
will 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 = 5
Note that right-associativity of :^:
is ignored. For example,
produces the stringshow
(Leaf 1 :^: Leaf 2 :^: Leaf 3)"Leaf 1 :^: (Leaf 2 :^: Leaf 3)"
.
:: 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 Bool | Since: base-2.1 |
Show Char | Since: base-2.1 |
Show Int | Since: base-2.1 |
Show Integer | Since: base-2.1 |
Show Natural | Since: base-4.8.0.0 |
Show Ordering | Since: base-2.1 |
Show Word | Since: base-2.1 |
Show RuntimeRep | Since: base-4.11.0.0 |
Defined in GHC.Show 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 CallStack | Since: base-4.9.0.0 |
Show () | Since: base-2.1 |
Show TyCon | Since: base-2.1 |
Show Module | Since: base-4.9.0.0 |
Show TrName | Since: base-4.9.0.0 |
Show KindRep | |
Show TypeLitSort | Since: base-4.11.0.0 |
Defined in GHC.Show showsPrec :: Int -> TypeLitSort -> ShowS # show :: TypeLitSort -> String # showList :: [TypeLitSort] -> ShowS # | |
Show Void | Since: base-4.8.0.0 |
Show BlockedIndefinitelyOnMVar | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception showsPrec :: Int -> BlockedIndefinitelyOnMVar -> ShowS # show :: BlockedIndefinitelyOnMVar -> String # showList :: [BlockedIndefinitelyOnMVar] -> ShowS # | |
Show BlockedIndefinitelyOnSTM | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception showsPrec :: Int -> BlockedIndefinitelyOnSTM -> ShowS # show :: BlockedIndefinitelyOnSTM -> String # showList :: [BlockedIndefinitelyOnSTM] -> ShowS # | |
Show Deadlock | Since: base-4.1.0.0 |
Show AllocationLimitExceeded | Since: base-4.7.1.0 |
Defined in GHC.IO.Exception showsPrec :: Int -> AllocationLimitExceeded -> ShowS # show :: AllocationLimitExceeded -> String # showList :: [AllocationLimitExceeded] -> ShowS # | |
Show CompactionFailed | Since: base-4.10.0.0 |
Defined in GHC.IO.Exception showsPrec :: Int -> CompactionFailed -> ShowS # show :: CompactionFailed -> String # showList :: [CompactionFailed] -> ShowS # | |
Show AssertionFailed | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception showsPrec :: Int -> AssertionFailed -> ShowS # show :: AssertionFailed -> String # showList :: [AssertionFailed] -> ShowS # | |
Show SomeAsyncException | Since: base-4.7.0.0 |
Defined in GHC.IO.Exception showsPrec :: Int -> SomeAsyncException -> ShowS # show :: SomeAsyncException -> String # showList :: [SomeAsyncException] -> ShowS # | |
Show AsyncException | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception showsPrec :: Int -> AsyncException -> ShowS # show :: AsyncException -> String # showList :: [AsyncException] -> ShowS # | |
Show ArrayException | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception showsPrec :: Int -> ArrayException -> ShowS # show :: ArrayException -> String # showList :: [ArrayException] -> ShowS # | |
Show FixIOException | Since: base-4.11.0.0 |
Defined in GHC.IO.Exception showsPrec :: Int -> FixIOException -> ShowS # show :: FixIOException -> String # showList :: [FixIOException] -> ShowS # | |
Show ExitCode | |
Show IOErrorType | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception showsPrec :: Int -> IOErrorType -> ShowS # show :: IOErrorType -> String # showList :: [IOErrorType] -> ShowS # | |
Show MaskingState | Since: base-4.3.0.0 |
Defined in GHC.IO showsPrec :: Int -> MaskingState -> ShowS # show :: MaskingState -> String # showList :: [MaskingState] -> ShowS # | |
Show IOException | Since: base-4.1.0.0 |
Defined in GHC.IO.Exception showsPrec :: Int -> IOException -> ShowS # show :: IOException -> String # showList :: [IOException] -> ShowS # | |
Show All | Since: base-2.1 |
Show Any | Since: base-2.1 |
Show Fixity | Since: base-4.6.0.0 |
Show Associativity | Since: base-4.6.0.0 |
Defined in GHC.Generics showsPrec :: Int -> Associativity -> ShowS # show :: Associativity -> String # showList :: [Associativity] -> ShowS # | |
Show SourceUnpackedness | Since: base-4.9.0.0 |
Defined in GHC.Generics showsPrec :: Int -> SourceUnpackedness -> ShowS # show :: SourceUnpackedness -> String # showList :: [SourceUnpackedness] -> ShowS # | |
Show SourceStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics showsPrec :: Int -> SourceStrictness -> ShowS # show :: SourceStrictness -> String # showList :: [SourceStrictness] -> ShowS # | |
Show DecidedStrictness | Since: base-4.9.0.0 |
Defined in GHC.Generics showsPrec :: Int -> DecidedStrictness -> ShowS # show :: DecidedStrictness -> String # showList :: [DecidedStrictness] -> ShowS # | |
Show SrcLoc | Since: base-4.9.0.0 |
Show a => Show [a] | Since: base-2.1 |
Show a => Show (Maybe a) | Since: base-2.1 |
Show a => Show (Ratio a) | Since: base-2.0.1 |
Show p => Show (Par1 p) | Since: base-4.7.0.0 |
Show a => Show (Complex a) | Since: base-2.1 |
Show a => Show (Min a) | Since: base-4.9.0.0 |
Show a => Show (Max a) | Since: base-4.9.0.0 |
Show a => Show (First a) | Since: base-4.9.0.0 |
Show a => Show (Last a) | Since: base-4.9.0.0 |
Show m => Show (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup showsPrec :: Int -> WrappedMonoid m -> ShowS # show :: WrappedMonoid m -> String # showList :: [WrappedMonoid m] -> ShowS # | |
Show a => Show (Option a) | Since: base-4.9.0.0 |
Show a => Show (ZipList a) | Since: base-4.7.0.0 |
Show a => Show (First a) | Since: base-2.1 |
Show a => Show (Last a) | Since: base-2.1 |
Show a => Show (Dual a) | Since: base-2.1 |
Show a => Show (Sum a) | Since: base-2.1 |
Show a => Show (Product a) | Since: base-2.1 |
Show a => Show (NonEmpty a) | Since: base-4.11.0.0 |
(Show a, Show b) => Show (Either a b) | Since: base-3.0 |
Show (V1 p) | Since: base-4.9.0.0 |
Show (U1 p) | Since: base-4.9.0.0 |
(Show a, Show b) => Show (a, b) | Since: base-2.1 |
(Show a, Show b) => Show (Arg a b) | Since: base-4.9.0.0 |
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 a, Show b, Show c) => Show (a, b, c) | 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 (a :~: b) | Since: base-4.7.0.0 |
Show b => Show (Tagged s b) | |
Show c => Show (K1 i c p) | Since: base-4.7.0.0 |
(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 a, Show b, Show c, Show d) => Show (a, b, c, d) | 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 (a :~~: b) | Since: base-4.10.0.0 |
Show (f p) => Show (M1 i c f p) | Since: base-4.7.0.0 |
Show (f (g p)) => Show ((f :.: g) 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 |
(Show1 f, Show1 g, Show a) => Show (Compose f g a) | Since: base-4.9.0.0 |
(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 |
foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b #
Right-associative fold of a structure.
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,
foldr
can produce a terminating expression from an infinite list.
For a general Foldable
structure this should be semantically identical
to,
foldr f z =foldr
f z .toList
null :: Foldable t => t a -> Bool #
Test whether the structure is empty. The default implementation is optimized for structures that are similar to cons-lists, because there is no general way to do better.
length :: Foldable t => t a -> Int #
Returns the size/length of a finite structure as an Int
. The
default implementation is optimized for structures that are similar to
cons-lists, because there is no general way to do better.
foldl :: Foldable t => (b -> a -> b) -> b -> t a -> b #
Left-associative fold of a structure.
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. This means that foldl'
will
diverge if given an infinite list.
Also note that if you want an efficient left-fold, you probably want to
use foldl'
instead of foldl
. The reason for this is that latter does
not force the "inner" results (e.g. z
in the above example)
before applying them to the operator (e.g. to f
x1(
). This results
in a thunk chain f
x2)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 =foldl
f z .toList
sum :: (Foldable t, Num a) => t a -> a #
The sum
function computes the sum of the numbers of a structure.
product :: (Foldable t, Num a) => t a -> a #
The product
function computes the product of the numbers of a
structure.
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_
.
The class of semigroups (types with an associative binary operation).
Instances should satisfy the associativity law:
Since: base-4.9.0.0
Instances
Semigroup Ordering | Since: base-4.9.0.0 |
Semigroup () | Since: base-4.9.0.0 |
Semigroup Void | Since: base-4.9.0.0 |
Semigroup All | Since: base-4.9.0.0 |
Semigroup Any | Since: base-4.9.0.0 |
Semigroup [a] | Since: base-4.9.0.0 |
Semigroup a => Semigroup (Maybe a) | Since: base-4.9.0.0 |
Semigroup a => Semigroup (IO a) | Since: base-4.10.0.0 |
Semigroup p => Semigroup (Par1 p) | Since: base-4.12.0.0 |
Semigroup (Predicate a) | |
Semigroup (Comparison a) | |
Defined in Data.Functor.Contravariant (<>) :: Comparison a -> Comparison a -> Comparison a # sconcat :: NonEmpty (Comparison a) -> Comparison a # stimes :: Integral b => b -> Comparison a -> Comparison a # | |
Semigroup (Equivalence a) | |
Defined in Data.Functor.Contravariant (<>) :: Equivalence a -> Equivalence a -> Equivalence a # sconcat :: NonEmpty (Equivalence a) -> Equivalence a # stimes :: Integral b => b -> Equivalence a -> Equivalence a # | |
Ord a => Semigroup (Min a) | Since: base-4.9.0.0 |
Ord a => Semigroup (Max 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 |
Monoid m => Semigroup (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup (<>) :: WrappedMonoid m -> WrappedMonoid m -> WrappedMonoid m # sconcat :: NonEmpty (WrappedMonoid m) -> WrappedMonoid m # stimes :: Integral b => b -> WrappedMonoid m -> WrappedMonoid m # | |
Semigroup a => Semigroup (Option 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 |
Semigroup a => Semigroup (Dual a) | Since: base-4.9.0.0 |
Semigroup (Endo a) | Since: base-4.9.0.0 |
Num a => Semigroup (Sum a) | Since: base-4.9.0.0 |
Num a => Semigroup (Product a) | Since: base-4.9.0.0 |
Semigroup (NonEmpty a) | Since: base-4.9.0.0 |
Semigroup b => Semigroup (a -> b) | Since: base-4.9.0.0 |
Semigroup (Either a b) | Since: base-4.9.0.0 |
Semigroup (V1 p) | Since: base-4.12.0.0 |
Semigroup (U1 p) | Since: base-4.12.0.0 |
(Semigroup a, Semigroup b) => Semigroup (a, b) | Since: base-4.9.0.0 |
Semigroup a => Semigroup (Op a b) | |
Semigroup (f p) => Semigroup (Rec1 f p) | Since: base-4.12.0.0 |
(Semigroup a, Semigroup b, Semigroup c) => Semigroup (a, b, c) | 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 a => Semigroup (Tagged s a) | |
Semigroup c => Semigroup (K1 i c p) | Since: base-4.12.0.0 |
(Semigroup (f p), Semigroup (g p)) => Semigroup ((f :*: g) 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 p) => Semigroup (M1 i c f p) | Since: base-4.12.0.0 |
Semigroup (f (g p)) => Semigroup ((f :.: g) 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 laws:
x
<>
mempty
= xmempty
<>
x = xx
(<>
(y<>
z) = (x<>
y)<>
zSemigroup
law)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 newtype
s and make those instances
of Monoid
, e.g. Sum
and Product
.
NOTE: Semigroup
is a superclass of Monoid
since base-4.11.0.0.
Identity of mappend
An associative operation
NOTE: This method is redundant and has the default
implementation
since base-4.11.0.0.mappend
= '(<>)'
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.
Instances
Monoid Ordering | Since: base-2.1 |
Monoid () | Since: base-2.1 |
Monoid All | Since: base-2.1 |
Monoid Any | Since: base-2.1 |
Monoid [a] | Since: base-2.1 |
Semigroup a => Monoid (Maybe a) | Lift a semigroup into Since 4.11.0: constraint on inner Since: base-2.1 |
Monoid a => Monoid (IO a) | Since: base-4.9.0.0 |
Monoid p => Monoid (Par1 p) | Since: base-4.12.0.0 |
Monoid (Predicate a) | |
Monoid (Comparison a) | |
Defined in Data.Functor.Contravariant mempty :: Comparison a # mappend :: Comparison a -> Comparison a -> Comparison a # mconcat :: [Comparison a] -> Comparison a # | |
Monoid (Equivalence a) | |
Defined in Data.Functor.Contravariant mempty :: Equivalence a # mappend :: Equivalence a -> Equivalence a -> Equivalence a # mconcat :: [Equivalence a] -> Equivalence a # | |
(Ord a, Bounded a) => Monoid (Min a) | Since: base-4.9.0.0 |
(Ord a, Bounded a) => Monoid (Max a) | Since: base-4.9.0.0 |
Monoid m => Monoid (WrappedMonoid m) | Since: base-4.9.0.0 |
Defined in Data.Semigroup mempty :: WrappedMonoid m # mappend :: WrappedMonoid m -> WrappedMonoid m -> WrappedMonoid m # mconcat :: [WrappedMonoid m] -> WrappedMonoid m # | |
Semigroup a => Monoid (Option a) | Since: base-4.9.0.0 |
Monoid (First a) | Since: base-2.1 |
Monoid (Last a) | Since: base-2.1 |
Monoid a => Monoid (Dual a) | Since: base-2.1 |
Monoid (Endo a) | Since: base-2.1 |
Num a => Monoid (Sum a) | Since: base-2.1 |
Num a => Monoid (Product a) | Since: base-2.1 |
Monoid b => Monoid (a -> b) | Since: base-2.1 |
Monoid (U1 p) | Since: base-4.12.0.0 |
(Monoid a, Monoid b) => Monoid (a, b) | Since: base-2.1 |
Monoid a => Monoid (Op a b) | |
Monoid (f p) => Monoid (Rec1 f p) | Since: base-4.12.0.0 |
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c) | 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 |
(Semigroup a, Monoid a) => Monoid (Tagged s a) | |
Monoid c => Monoid (K1 i c p) | Since: base-4.12.0.0 |
(Monoid (f p), Monoid (g p)) => Monoid ((f :*: g) 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 p) => Monoid (M1 i c f p) | Since: base-4.12.0.0 |
Monoid (f (g p)) => Monoid ((f :.: g) 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 |
Eq Bool | |
Ord Bool | |
Read Bool | Since: base-2.1 |
Show Bool | Since: base-2.1 |
Generic Bool | |
SingKind Bool | Since: base-4.9.0.0 |
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 (a + a) (Bool, a) Source # | |
(SPDistribute k, ObjectSum k a a, ObjectPair k Bool a) => Isomorphic k (Bool, a) (a + a) Source # | |
type Rep Bool | Since: base-4.6.0.0 |
data Sing (a :: Bool) | |
type DemoteRep Bool | |
Defined in GHC.Generics |
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 |
Eq Char | |
Ord Char | |
Read Char | Since: base-2.1 |
Show Char | Since: base-2.1 |
Generic1 (URec Char :: k -> Type) | |
Functor (URec Char :: Type -> Type) | Since: base-4.9.0.0 |
Foldable (URec Char :: Type -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable fold :: Monoid m => URec Char m -> m # foldMap :: Monoid m => (a -> m) -> URec Char a -> m # foldr :: (a -> b -> b) -> b -> URec Char a -> b # foldr' :: (a -> b -> b) -> b -> URec Char a -> b # foldl :: (b -> a -> b) -> b -> URec Char a -> b # foldl' :: (b -> a -> b) -> b -> URec Char a -> b # foldr1 :: (a -> a -> a) -> URec Char a -> a # foldl1 :: (a -> a -> a) -> URec Char a -> a # toList :: URec Char a -> [a] # length :: URec Char a -> Int # elem :: Eq a => a -> URec Char a -> Bool # maximum :: Ord a => URec Char a -> a # minimum :: Ord a => URec Char a -> a # | |
Traversable (URec Char :: Type -> Type) | 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 |
Show (URec Char p) | Since: base-4.9.0.0 |
Generic (URec Char p) | |
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
Eq Double | Note that due to the presence of
Also note that
|
Floating Double | Since: base-2.1 |
Ord Double | Note that due to the presence of
Also note that, due to the same,
|
Read Double | Since: base-2.1 |
RealFloat Double | Since: base-2.1 |
Defined in GHC.Float 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 # | |
Generic1 (URec Double :: k -> Type) | |
Functor (URec Double :: Type -> Type) | Since: base-4.9.0.0 |
Foldable (URec Double :: Type -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable fold :: Monoid m => URec Double m -> m # foldMap :: Monoid m => (a -> m) -> URec Double a -> m # foldr :: (a -> b -> b) -> b -> URec Double a -> b # foldr' :: (a -> b -> b) -> b -> URec Double a -> b # foldl :: (b -> a -> b) -> b -> URec Double a -> b # foldl' :: (b -> a -> b) -> b -> URec Double a -> b # foldr1 :: (a -> a -> a) -> URec Double a -> a # foldl1 :: (a -> a -> a) -> URec Double a -> a # toList :: URec Double a -> [a] # null :: URec Double a -> Bool # length :: URec Double a -> Int # elem :: Eq a => a -> URec Double a -> Bool # maximum :: Ord a => URec Double a -> a # minimum :: Ord a => URec Double a -> a # | |
Traversable (URec Double :: Type -> Type) | Since: base-4.9.0.0 |
Defined in Data.Traversable | |
Eq (URec Double p) | Since: base-4.9.0.0 |
Ord (URec Double p) | Since: base-4.9.0.0 |
Defined in GHC.Generics 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 # | |
Show (URec Double p) | Since: base-4.9.0.0 |
Generic (URec Double p) | |
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
Eq Float | Note that due to the presence of
Also note that
|
Floating Float | Since: base-2.1 |
Ord Float | Note that due to the presence of
Also note that, due to the same,
|
Read Float | Since: base-2.1 |
RealFloat Float | Since: base-2.1 |
Defined in GHC.Float 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 # | |
Generic1 (URec Float :: k -> Type) | |
Functor (URec Float :: Type -> Type) | Since: base-4.9.0.0 |
Foldable (URec Float :: Type -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable fold :: Monoid m => URec Float m -> m # foldMap :: Monoid m => (a -> m) -> URec Float a -> m # foldr :: (a -> b -> b) -> b -> URec Float a -> b # foldr' :: (a -> b -> b) -> b -> URec Float a -> b # foldl :: (b -> a -> b) -> b -> URec Float a -> b # foldl' :: (b -> a -> b) -> b -> URec Float a -> b # foldr1 :: (a -> a -> a) -> URec Float a -> a # foldl1 :: (a -> a -> a) -> URec Float a -> a # toList :: URec Float a -> [a] # null :: URec Float a -> Bool # length :: URec Float a -> Int # elem :: Eq a => a -> URec Float a -> Bool # maximum :: Ord a => URec Float a -> a # minimum :: Ord a => URec Float a -> a # | |
Traversable (URec Float :: Type -> Type) | Since: base-4.9.0.0 |
Defined in Data.Traversable | |
Eq (URec Float p) | |
Ord (URec Float p) | |
Defined in GHC.Generics | |
Show (URec Float p) | |
Generic (URec Float p) | |
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 |
Eq Int | |
Integral Int | Since: base-2.0.1 |
Num Int | Since: base-2.1 |
Ord Int | |
Read Int | Since: base-2.1 |
Real Int | Since: base-2.0.1 |
Defined in GHC.Real toRational :: Int -> Rational # | |
Show Int | Since: base-2.1 |
Generic1 (URec Int :: k -> Type) | |
Functor (URec Int :: Type -> Type) | Since: base-4.9.0.0 |
Foldable (URec Int :: Type -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable fold :: Monoid m => URec Int m -> m # foldMap :: Monoid m => (a -> m) -> URec Int a -> m # foldr :: (a -> b -> b) -> b -> URec Int a -> b # foldr' :: (a -> b -> b) -> b -> URec Int a -> b # foldl :: (b -> a -> b) -> b -> URec Int a -> b # foldl' :: (b -> a -> b) -> b -> URec Int a -> b # foldr1 :: (a -> a -> a) -> URec Int a -> a # foldl1 :: (a -> a -> a) -> URec Int a -> a # elem :: Eq a => a -> URec Int a -> Bool # maximum :: Ord a => URec Int a -> a # minimum :: Ord a => URec Int a -> a # | |
Traversable (URec Int :: Type -> Type) | 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 |
Show (URec Int p) | Since: base-4.9.0.0 |
Generic (URec Int p) | |
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 |
Invariant: Jn#
and Jp#
are used iff value doesn't fit in S#
Useful properties resulting from the invariants:
Instances
Enum Integer | Since: base-2.1 |
Eq Integer | |
Integral Integer | Since: base-2.0.1 |
Defined in GHC.Real | |
Num Integer | Since: base-2.1 |
Ord Integer | |
Read Integer | Since: base-2.1 |
Real Integer | Since: base-2.0.1 |
Defined in GHC.Real toRational :: Integer -> Rational # | |
Show Integer | Since: base-2.1 |
The Maybe
type encapsulates an optional value. A value of type
either contains a value of type Maybe
aa
(represented as
),
or it is empty (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
Monad Maybe | Since: base-2.1 |
Functor Maybe | Since: base-2.1 |
Applicative Maybe | Since: base-2.1 |
Foldable Maybe | Since: base-2.1 |
Defined in Data.Foldable fold :: Monoid m => Maybe m -> 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 |
MonadPlus Maybe | Since: base-2.1 |
(Arrow k ((->) :: Type -> Type -> Type), WellPointed k, Function k, Functor Maybe k k) => Traversable Maybe Maybe k k Source # | |
Defined in Data.Traversable.Constrained type TraversalObject k Maybe b :: Constraint Source # 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 ((->) :: Type -> Type -> Type) ((->) :: Type -> Type -> Type) Source # | |
Eq a => Eq (Maybe a) | Since: base-2.1 |
Ord a => Ord (Maybe a) | Since: base-2.1 |
Read a => Read (Maybe a) | Since: base-2.1 |
Show a => Show (Maybe a) | Since: base-2.1 |
Generic (Maybe a) | |
Semigroup a => Semigroup (Maybe 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 |
SingKind a => SingKind (Maybe a) | Since: base-4.9.0.0 |
Generic1 Maybe | |
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 TraversalObject k Maybe b Source # | |
Defined in Data.Traversable.Constrained | |
type Rep (Maybe a) | Since: base-4.6.0.0 |
data Sing (b :: Maybe a) | |
type DemoteRep (Maybe a) | |
Defined in GHC.Generics | |
type Rep1 Maybe | Since: base-4.6.0.0 |
Instances
Bounded Ordering | Since: base-2.1 |
Enum Ordering | Since: base-2.1 |
Eq Ordering | |
Ord Ordering | |
Defined in GHC.Classes | |
Read Ordering | Since: base-2.1 |
Show Ordering | Since: base-2.1 |
Generic Ordering | |
Semigroup Ordering | Since: base-4.9.0.0 |
Monoid Ordering | Since: base-2.1 |
type Rep Ordering | Since: base-4.6.0.0 |
A value of type
is a computation which, when performed,
does some I/O before returning 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
Monad IO | Since: base-2.1 |
Functor IO | Since: base-2.1 |
Applicative IO | Since: base-2.1 |
Alternative IO | Since: base-4.9.0.0 |
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 # | |
Semigroup a => Semigroup (IO a) | Since: base-4.10.0.0 |
Monoid a => Monoid (IO a) | Since: base-4.9.0.0 |
Instances
Bounded Word | Since: base-2.1 |
Enum Word | Since: base-2.1 |
Eq Word | |
Integral Word | Since: base-2.1 |
Num Word | Since: base-2.1 |
Ord Word | |
Read Word | Since: base-4.5.0.0 |
Real Word | Since: base-2.1 |
Defined in GHC.Real toRational :: Word -> Rational # | |
Show Word | Since: base-2.1 |
Generic1 (URec Word :: k -> Type) | |
Functor (URec Word :: Type -> Type) | Since: base-4.9.0.0 |
Foldable (URec Word :: Type -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable fold :: Monoid m => URec Word m -> m # foldMap :: Monoid m => (a -> m) -> URec Word a -> m # foldr :: (a -> b -> b) -> b -> URec Word a -> b # foldr' :: (a -> b -> b) -> b -> URec Word a -> b # foldl :: (b -> a -> b) -> b -> URec Word a -> b # foldl' :: (b -> a -> b) -> b -> URec Word a -> b # foldr1 :: (a -> a -> a) -> URec Word a -> a # foldl1 :: (a -> a -> a) -> URec Word a -> a # toList :: URec Word a -> [a] # length :: URec Word a -> Int # elem :: Eq a => a -> URec Word a -> Bool # maximum :: Ord a => URec Word a -> a # minimum :: Ord a => URec Word a -> a # | |
Traversable (URec Word :: Type -> Type) | 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 |
Show (URec Word p) | Since: base-4.9.0.0 |
Generic (URec Word p) | |
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
is either Either
a b
or Left
a
.Right
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
is the type of values which can be either
a Either
String
Int
String
or an Int
. The Left
constructor can be used only on
String
s, and the Right
constructor can be used only on Int
s:
>>>
let s = Left "foo" :: Either String Int
>>>
s
Left "foo">>>
let n = Right 3 :: Either String Int
>>>
n
Right 3>>>
:type s
s :: Either String Int>>>
:type n
n :: 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) s
Left "foo">>>
fmap (*2) n
Right 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 Int
s.
>>>
:{
let parseMultiple :: Either String Int parseMultiple = do x <- parseEither '1' y <- parseEither '2' return (x + y)>>>
:}
>>>
parseMultiple
Right 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)>>>
:}
>>>
parseMultiple
Left "parse error"
Instances
(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 a (u + a) Source # | |
Defined in Control.Category.Constrained | |
(CoCartesian k, Object k a, u ~ ZeroObject k, ObjectSum k a u) => Isomorphic k a (a + u) 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 (u + a) a Source # | |
Defined in Control.Category.Constrained | |
(CoCartesian k, Object k a, u ~ ZeroObject k, ObjectSum k a u) => Isomorphic k (a + u) a 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 ((a + b) + c) (a + (b + c)) 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 (a + (b + c)) ((a + b) + c) Source # | |
(SPDistribute k, ObjectSum k a a, ObjectPair k Bool a) => Isomorphic k (a + a) (Bool, a) Source # | |
(SPDistribute k, ObjectSum k a a, ObjectPair k Bool a) => Isomorphic k (Bool, a) (a + a) 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 ((a, b) + (a, c)) (a, b + c) 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 (a, b + c) ((a, b) + (a, c)) Source # | |
Monad (Either e) | Since: base-4.4.0.0 |
Functor (Either a) | Since: base-3.0 |
Applicative (Either e) | Since: base-3.0 |
Foldable (Either a) | Since: base-4.7.0.0 |
Defined in Data.Foldable fold :: Monoid m => Either a m -> 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 |
Generic1 (Either a :: Type -> Type) | |
Functor (Either a) (Discrete :: Type -> Type -> Type) (Discrete :: Type -> Type -> Type) Source # | |
Functor (Either a) (Coercion :: Type -> Type -> Type) (Coercion :: Type -> Type -> Type) Source # | |
(Eq a, Eq b) => Eq (Either a b) | Since: base-2.1 |
(Ord a, Ord b) => Ord (Either a b) | Since: base-2.1 |
(Read a, Read b) => Read (Either a b) | Since: base-3.0 |
(Show a, Show b) => Show (Either a b) | Since: base-3.0 |
Generic (Either a b) | |
Semigroup (Either a b) | Since: base-4.9.0.0 |
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))) |
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]])
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
).
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.
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)
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.
all :: Foldable t => (a -> Bool) -> t a -> Bool #
Determines whether all elements of the structure satisfy the predicate.
any :: Foldable t => (a -> Bool) -> t a -> Bool #
Determines whether any element of the structure satisfies the predicate.
concat :: Foldable t => t [a] -> [a] #
The concatenation of all the elements of a container of lists.
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
contains at least as many elements as newlines in 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" :: Int
123
>>>
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
, apply the first function to Left
aa
;
if it is
, apply the second function to Right
bb
.
Examples
We create two values of type
, one using the
Either
String
Int
Left
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) s
3>>>
either length (*2) n
6
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
.) If there is no legal lexeme at the
beginning of the input string, 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 #
is the smallest positive integer that both lcm
x yx
and y
divide.
gcd :: Integral a => a -> a -> a #
is the non-negative factor of both gcd
x yx
and y
of which
every common factor of x
and y
is also a factor; for example
, gcd
4 2 = 2
, gcd
(-4) 6 = 2
= gcd
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,
,
the result may be negative if one of the arguments is abs
minBound
< 0
(and
necessarily is if the other is minBound
0
or
) for such types.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.
unzip :: [(a, b)] -> ([a], [b]) #
unzip
transforms a list of pairs into a list of first components
and a list of second components.
(!!) :: [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.
lookup :: Eq a => a -> [(a, b)] -> Maybe b #
lookup
key assocs
looks up a key in an association list.
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],[])
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])
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.
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.
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.
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] == []
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.
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.
Return all the elements of a list except the last one. The list must be non-empty.
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 Nothing
False
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 #
yields the result of applying until
p ff
until p
holds.
($!) :: (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.
flip :: (a -> b -> c) -> b -> a -> c #
takes its (first) two arguments in the reverse order of flip
ff
.
>>>
flip (++) "hello" "world"
"worldhello"
undefined :: HasCallStack => a #
errorWithoutStackTrace :: [Char] -> a #
A variant of error
that does not produce a stack trace.
Since: base-4.9.0.0
error :: HasCallStack => [Char] -> a #
error
stops execution and displays an error message.