-- Firenze, Italy, pp229-240.
-- See these papers for the equations these combinators are expected to
-- satisfy. These papers and more information on arrows can be found at
--- <http://www.soi.city.ac.uk/~ross/arrows/>.
-
-module Control.Arrow where
-
-import Prelude
+-- <http://www.haskell.org/arrows/>.
+
+module Control.Arrow (
+ -- * Arrows
+ Arrow(..), Kleisli(..),
+ -- ** Derived combinators
+ returnA,
+ (^>>), (>>^),
+ -- ** Right-to-left variants
+ (<<^), (^<<),
+ -- * Monoid operations
+ ArrowZero(..), ArrowPlus(..),
+ -- * Conditionals
+ ArrowChoice(..),
+ -- * Arrow application
+ ArrowApply(..), ArrowMonad(..), leftApp,
+ -- * Feedback
+ ArrowLoop(..)
+ ) where
+
+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 <<<
-
------------------------------------------------------------------------------
--- * Arrows
+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)
(&&&) :: a b c -> a b c' -> a b (c,c')
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)
+"first/arr" forall f .
+ first (arr f) = arr (first f)
+"second/arr" forall f .
+ second (arr f) = arr (second f)
+"product/arr" forall f g .
+ arr f *** arr g = arr (f *** g)
+"fanout/arr" forall f g .
+ arr f &&& arr g = arr (f &&& g)
+"compose/first" forall f g .
+ (first f) . (first g) = first (f . g)
+"compose/second" forall 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)
+-- (f *** g) ~(x,y) = (f x, g y)
+-- sorry, although the above defn is fully H'98, nhc98 can't parse it.
+ (***) 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))
------------------------------------------------------------------------------
--- ** Derived combinators
-
-- | The identity arrow, which plays the role of 'return' in arrow notation.
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
-(<<<) :: Arrow a => a c d -> a b c -> a b d
-f <<< g = g >>> f
+-- | Postcomposition with a pure function.
+(>>^) :: Arrow a => a b c -> (c -> d) -> a b d
+a >>^ f = a >>> arr f
------------------------------------------------------------------------------
--- * Monoid operations
+-- | Precomposition with a pure function (right-to-left variant).
+(<<^) :: Arrow a => a c d -> (b -> c) -> a b d
+a <<^ f = a <<< arr 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
instance MonadPlus m => ArrowPlus (Kleisli m) where
Kleisli f <+> Kleisli g = Kleisli (\x -> f x `mplus` g x)
------------------------------------------------------------------------------
--- * Conditionals
-
-- | Choice, for arrows that support it. This class underlies the
--- [if] and [case] constructs in arrow notation.
+-- @if@ and @case@ constructs in arrow notation.
-- Any instance must define 'left'. The other combinators have sensible
-- default definitions, which may be overridden for efficiency.
where untag (Left x) = x
untag (Right y) = y
+{-# RULES
+"left/arr" forall f .
+ left (arr f) = arr (left f)
+"right/arr" forall f .
+ right (arr f) = arr (right f)
+"sum/arr" forall f g .
+ arr f +++ arr g = arr (f +++ g)
+"fanin/arr" forall f g .
+ arr f ||| arr g = arr (f ||| g)
+"compose/left" forall f g .
+ left f >>> left g = left (f >>> g)
+"compose/right" forall f g .
+ right f >>> right g = right (f >>> g)
+ #-}
+
instance ArrowChoice (->) where
left f = f +++ id
right f = id +++ f
f +++ g = (f >>> arr Left) ||| (g >>> arr Right)
Kleisli f ||| Kleisli g = Kleisli (either f g)
------------------------------------------------------------------------------
--- * Arrow application
-
-- | Some arrows allow application of arrow inputs to other inputs.
class Arrow a => ArrowApply a where
leftApp f = arr ((\b -> (arr (\() -> b) >>> f >>> arr Left, ())) |||
(\d -> (arr (\() -> d) >>> arr Right, ()))) >>> app
------------------------------------------------------------------------------
--- * Feedback
-
-- | The 'loop' operator expresses computations in which an output value is
-- fed back as input, even though the computation occurs only once.
--- It underlies the [rec] value recursion construct in arrow notation.
+-- It underlies the @rec@ value recursion construct in arrow notation.
class Arrow a => ArrowLoop a where
loop :: a (b,d) (c,d) -> a b c
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