-- * Arrows
Arrow(..), Kleisli(..),
-- ** Derived combinators
- returnA, (<<<),
+ returnA,
+ (^>>), (>>^),
+ -- ** Right-to-left variants
+ (<<^), (^<<),
-- * Monoid operations
ArrowZero(..), ArrowPlus(..),
-- * Conditionals
ArrowLoop(..)
) where
-import Prelude
+import Prelude hiding (id,(.))
+import qualified Prelude
import Control.Monad
import Control.Monad.Fix
+import Control.Category
infixr 5 <+>
infixr 3 ***
infixr 3 &&&
infixr 2 +++
infixr 2 |||
-infixr 1 >>>
-infixr 1 <<<
+infixr 1 ^>>, >>^
+infixr 1 ^<<, <<^
-- | The basic arrow class.
-- Any instance must define either 'arr' or 'pure' (which are synonyms),
--- as well as '>>>' and 'first'. The other combinators have sensible
+-- as well as 'first'. The other combinators have sensible
-- default definitions, which may be overridden for efficiency.
-class Arrow a where
+class Category a => Arrow a where
-- | Lift a function to an arrow: you must define either this
-- or 'pure'.
pure :: (b -> c) -> a b c
pure = arr
- -- | Left-to-right composition of arrows.
- (>>>) :: a b c -> a c d -> a b d
-
-- | Send the first component of the input through the argument
-- arrow, and copy the rest unchanged to the output.
first :: a b c -> a (b,d) (c,d)
f &&& g = arr (\b -> (b,b)) >>> f *** g
{-# RULES
+"identity"
+ arr id = id
"compose/arr" forall f g .
- arr f >>> arr g = arr (f >>> g)
+ (arr f) . (arr g) = arr (f . g)
"first/arr" forall f .
first (arr f) = arr (first f)
"second/arr" forall f .
"fanout/arr" forall f g .
arr f &&& arr g = arr (f &&& g)
"compose/first" forall f g .
- first f >>> first g = first (f >>> g)
+ (first f) . (first g) = first (f . g)
"compose/second" forall f g .
- second f >>> second g = second (f >>> g)
+ (second f) . (second g) = second (f . g)
#-}
-- Ordinary functions are arrows.
instance Arrow (->) where
arr f = f
- f >>> g = g . f
first f = f *** id
second f = id *** f
-- (f *** g) ~(x,y) = (f x, g y)
-- | Kleisli arrows of a monad.
-newtype Kleisli m a b = Kleisli (a -> m b)
+newtype Kleisli m a b = Kleisli { runKleisli :: a -> m b }
+
+instance Monad m => Category (Kleisli m) where
+ id = Kleisli return
+ (Kleisli f) . (Kleisli g) = Kleisli (\b -> g b >>= f)
instance Monad m => Arrow (Kleisli m) where
arr f = Kleisli (return . f)
- Kleisli f >>> Kleisli g = Kleisli (\b -> f b >>= g)
first (Kleisli f) = Kleisli (\ ~(b,d) -> f b >>= \c -> return (c,d))
second (Kleisli f) = Kleisli (\ ~(d,b) -> f b >>= \c -> return (d,c))
returnA :: Arrow a => a b b
returnA = arr id
--- | Right-to-left composition, for a better fit with arrow notation.
+-- | Precomposition with a pure function.
+(^>>) :: Arrow a => (b -> c) -> a c d -> a b d
+f ^>> a = arr f >>> a
+
+-- | Postcomposition with a pure function.
+(>>^) :: Arrow a => a b c -> (c -> d) -> a b d
+a >>^ f = a >>> arr f
+
+-- | Precomposition with a pure function (right-to-left variant).
+(<<^) :: Arrow a => a c d -> (b -> c) -> a b d
+a <<^ f = a <<< arr f
-(<<<) :: Arrow a => a c d -> a b c -> a b d
-f <<< g = g >>> f
+-- | Postcomposition with a pure function (right-to-left variant).
+(^<<) :: Arrow a => (c -> d) -> a b c -> a b d
+f ^<< a = arr f <<< a
class Arrow a => ArrowZero a where
zeroArrow :: a b c