X-Git-Url: http://git.megacz.com/?a=blobdiff_plain;f=Control%2FArrow.hs;h=20e367799f800033782b6043d56cc1fbde54d9ce;hb=8073392a94dc5ab198e4758d6738a0c7f5ed68cf;hp=237c37f88cc76b017300a860ba67f61911230bfb;hpb=88fcddd13e62ecf456249b535dd72a290feac0ad;p=ghc-base.git

diff --git a/Control/Arrow.hs b/Control/Arrow.hs
index 237c37f..20e3677 100644
--- a/Control/Arrow.hs
+++ b/Control/Arrow.hs
@@ -1,206 +1,278 @@
 -----------------------------------------------------------------------------
---
+-- |
 -- Module      :  Control.Arrow
 -- Copyright   :  (c) Ross Paterson 2002
 -- License     :  BSD-style (see the LICENSE file in the distribution)
 --
--- Maintainer  :  ross@soi.city.ac.uk
+-- Maintainer  :  libraries@haskell.org
 -- Stability   :  experimental
 -- Portability :  portable
 --
--- $Id: Arrow.hs,v 1.1 2002/02/26 18:19:17 ross Exp $
---
 -- Basic arrow definitions, based on
---
---	"Generalising Monads to Arrows", by John Hughes, Science of
---	Computer Programming 37, pp67-111, May 2000.
---
--- plus a couple of definitions (returnA and loop) from
---
---	"A New Notation for Arrows", by Ross Paterson, in ICFP 2001,
---	Firenze, Italy, pp229-240.
---
+--      /Generalising Monads to Arrows/, by John Hughes,
+--      /Science of Computer Programming/ 37, pp67-111, May 2000.
+-- plus a couple of definitions ('returnA' and 'loop') from
+--      /A New Notation for Arrows/, by Ross Paterson, in /ICFP 2001/,
+--      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,
+    (^>>), (>>^),
+    (>>>), (<<<), -- reexported
+    -- ** Right-to-left variants
+    (<<^), (^<<),
+    -- * Monoid operations
+    ArrowZero(..), ArrowPlus(..),
+    -- * Conditionals
+    ArrowChoice(..),
+    -- * Arrow application
+    ArrowApply(..), ArrowMonad(..), leftApp,
+    -- * Feedback
+    ArrowLoop(..)
+    ) where
+
+import Prelude hiding (id,(.))
 
 import Control.Monad
 import Control.Monad.Fix
+import Control.Category
 
 infixr 5 <+>
 infixr 3 ***
 infixr 3 &&&
 infixr 2 +++
 infixr 2 |||
-infixr 1 >>>
-infixr 1 <<<
-
------------------------------------------------------------------------------
--- Arrow classes
-
-class Arrow a where
-	arr :: (b -> c) -> a b c
-	(>>>) :: a b c -> a c d -> a b d
-	first :: a b c -> a (b,d) (c,d)
+infixr 1 ^>>, >>^
+infixr 1 ^<<, <<^
 
-	-- The following combinators are placed in the class so that they
-	-- can be overridden with more efficient versions if required.
-	-- Any replacements should satisfy these equations.
+-- | The basic arrow class.
+--
+--   Minimal complete definition: 'arr' and 'first'.
+--
+--   The other combinators have sensible default definitions,
+--   which may be overridden for efficiency.
+
+class Category a => Arrow a where
+
+    -- | Lift a function to an arrow.
+    arr :: (b -> c) -> a b c
+
+    -- | 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 mirror image of 'first'.
+    --
+    --   The default definition may be overridden with a more efficient
+    --   version if desired.
+    second :: a b c -> a (d,b) (d,c)
+    second f = arr swap >>> first f >>> arr swap
+      where
+        swap :: (x,y) -> (y,x)
+        swap ~(x,y) = (y,x)
+
+    -- | Split the input between the two argument arrows and combine
+    --   their output.  Note that this is in general not a functor.
+    --
+    --   The default definition may be overridden with a more efficient
+    --   version if desired.
+    (***) :: a b c -> a b' c' -> a (b,b') (c,c')
+    f *** g = first f >>> second g
+
+    -- | Fanout: send the input to both argument arrows and combine
+    --   their output.
+    --
+    --   The default definition may be overridden with a more efficient
+    --   version if desired.
+    (&&&) :: a b c -> a b c' -> a b (c,c')
+    f &&& g = arr (\b -> (b,b)) >>> f *** g
+
+{-# RULES
+"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)
+ #-}
 
-	second :: a b c -> a (d,b) (d,c)
-	second f = arr swap >>> first f >>> arr swap
-			where	swap ~(x,y) = (y,x)
+-- Ordinary functions are arrows.
 
-	(***) :: a b c -> a b' c' -> a (b,b') (c,c')
-	f *** g = first f >>> second g
+instance Arrow (->) where
+    arr f = f
+    first f = f *** id
+    second f = id *** f
+--  (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)
 
-	(&&&) :: a b c -> a b c' -> a b (c,c')
-	f &&& g = arr (\b -> (b,b)) >>> f *** g
+-- | Kleisli arrows of a monad.
 
-	-- Some people prefer the name pure to arr, so both are allowed,
-	-- but you must define one of them:
+newtype Kleisli m a b = Kleisli { runKleisli :: a -> m b }
 
-	pure :: (b -> c) -> a b c
-	pure = arr
-	arr = pure
+instance Monad m => Category (Kleisli m) where
+    id = Kleisli return
+    (Kleisli f) . (Kleisli g) = Kleisli (\b -> g b >>= f)
 
------------------------------------------------------------------------------
--- Derived combinators
+instance Monad m => Arrow (Kleisli m) where
+    arr f = Kleisli (return . f)
+    first (Kleisli f) = Kleisli (\ ~(b,d) -> f b >>= \c -> return (c,d))
+    second (Kleisli f) = Kleisli (\ ~(d,b) -> f b >>= \c -> return (d,c))
 
--- The counterpart of return in arrow notation:
+-- | The identity arrow, which plays the role of 'return' in arrow notation.
 
 returnA :: Arrow a => a b b
 returnA = arr id
 
--- Mirror image of >>>, 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
+    zeroArrow :: a b c
+
+instance MonadPlus m => ArrowZero (Kleisli m) where
+    zeroArrow = Kleisli (\_ -> mzero)
 
 class ArrowZero a => ArrowPlus a where
-	(<+>) :: a b c -> a b c -> a b c
+    (<+>) :: a b c -> a b c -> a b c
 
------------------------------------------------------------------------------
--- Conditionals
+instance MonadPlus m => ArrowPlus (Kleisli m) where
+    Kleisli f <+> Kleisli g = Kleisli (\x -> f x `mplus` g x)
 
-class Arrow a => ArrowChoice a where
-	left :: a b c -> a (Either b d) (Either c d)
+-- | Choice, for arrows that support it.  This class underlies the
+--   @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.
 
-	-- The following combinators are placed in the class so that they
-	-- can be overridden with more efficient versions if required.
-	-- Any replacements should satisfy these equations.
+class Arrow a => ArrowChoice a where
 
-	right :: a b c -> a (Either d b) (Either d c)
-	right f = arr mirror >>> left f >>> arr mirror
-			where	mirror (Left x) = Right x
-				mirror (Right y) = Left y
+    -- | Feed marked inputs through the argument arrow, passing the
+    --   rest through unchanged to the output.
+    left :: a b c -> a (Either b d) (Either c d)
+
+    -- | A mirror image of 'left'.
+    --
+    --   The default definition may be overridden with a more efficient
+    --   version if desired.
+    right :: a b c -> a (Either d b) (Either d c)
+    right f = arr mirror >>> left f >>> arr mirror
+      where
+        mirror :: Either x y -> Either y x
+        mirror (Left x) = Right x
+        mirror (Right y) = Left y
+
+    -- | Split the input between the two argument arrows, retagging
+    --   and merging their outputs.
+    --   Note that this is in general not a functor.
+    --
+    --   The default definition may be overridden with a more efficient
+    --   version if desired.
+    (+++) :: a b c -> a b' c' -> a (Either b b') (Either c c')
+    f +++ g = left f >>> right g
+
+    -- | Fanin: Split the input between the two argument arrows and
+    --   merge their outputs.
+    --
+    --   The default definition may be overridden with a more efficient
+    --   version if desired.
+    (|||) :: a b d -> a c d -> a (Either b c) d
+    f ||| g = f +++ g >>> arr untag
+      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)
+ #-}
 
-	(+++) :: a b c -> a b' c' -> a (Either b b') (Either c c')
-	f +++ g = left f >>> right g
+instance ArrowChoice (->) where
+    left f = f +++ id
+    right f = id +++ f
+    f +++ g = (Left . f) ||| (Right . g)
+    (|||) = either
 
-	(|||) :: a b d -> a c d -> a (Either b c) d
-	f ||| g = f +++ g >>> arr untag
-			where	untag (Left x) = x
-				untag (Right y) = y
+instance Monad m => ArrowChoice (Kleisli m) where
+    left f = f +++ arr id
+    right f = arr 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
-	app :: a (a b c, b) c
+    app :: a (a b c, b) c
 
--- Any instance of ArrowApply can be made into an instance if ArrowChoice
--- by defining left = leftApp, where
+instance ArrowApply (->) where
+    app (f,x) = f x
 
-leftApp :: ArrowApply a => a b c -> a (Either b d) (Either c d)
-leftApp f = arr ((\b -> (arr (\() -> b) >>> f >>> arr Left, ())) |||
-		 (\d -> (arr (\() -> d) >>> arr Right, ()))) >>> app
+instance Monad m => ArrowApply (Kleisli m) where
+    app = Kleisli (\(Kleisli f, x) -> f x)
 
--- The ArrowApply class is equivalent to Monad: any monad gives rise to
--- a Kliesli arrow (see below), and any instance of ArrowApply defines
--- a monad:
+-- | The 'ArrowApply' class is equivalent to 'Monad': any monad gives rise
+--   to a 'Kleisli' arrow, and any instance of 'ArrowApply' defines a monad.
 
-newtype ArrowApply a => ArrowMonad a b = ArrowMonad (a () b)
+newtype ArrowMonad a b = ArrowMonad (a () b)
 
 instance ArrowApply a => Monad (ArrowMonad a) where
-	return x = ArrowMonad (arr (\z -> x))
-	ArrowMonad m >>= f = ArrowMonad (m >>>
-			arr (\x -> let ArrowMonad h = f x in (h, ())) >>>
-			app)
-
------------------------------------------------------------------------------
--- Feedback
-
--- The following operator expresses computations in which a value is
--- recursively defined through the computation, even though the computation
--- occurs only once:
-
-class Arrow a => ArrowLoop a where
-	loop :: a (b,d) (c,d) -> a b c
-
------------------------------------------------------------------------------
--- Arrow instances
+    return x = ArrowMonad (arr (\_ -> x))
+    ArrowMonad m >>= f = ArrowMonad $
+        m >>> arr (\x -> let ArrowMonad h = f x in (h, ())) >>> app
 
--- Ordinary functions are arrows.
+-- | Any instance of 'ArrowApply' can be made into an instance of
+--   'ArrowChoice' by defining 'left' = 'leftApp'.
 
-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)
+leftApp :: ArrowApply a => a b c -> a (Either b d) (Either c d)
+leftApp f = arr ((\b -> (arr (\() -> b) >>> f >>> arr Left, ())) |||
+             (\d -> (arr (\() -> d) >>> arr Right, ()))) >>> app
 
-instance ArrowChoice (->) where
-	left f = f +++ id
-	right f = id +++ f
-	f +++ g = (Left . f) ||| (Right . g)
-	(|||) = either
+-- | 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.
 
-instance ArrowApply (->) where
-	app (f,x) = f x
+class Arrow a => ArrowLoop a where
+    loop :: a (b,d) (c,d) -> a b c
 
 instance ArrowLoop (->) where
-	loop f b = let (c,d) = f (b,d) in c
-
------------------------------------------------------------------------------
--- Kleisli arrows of a monad
-
-newtype Kleisli m a b = Kleisli (a -> m b)
-
-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))
-
-instance MonadPlus m => ArrowZero (Kleisli m) where
-	zeroArrow = Kleisli (\x -> mzero)
-
-instance MonadPlus m => ArrowPlus (Kleisli m) where
-	Kleisli f <+> Kleisli g = Kleisli (\x -> f x `mplus` g x)
-
-instance Monad m => ArrowChoice (Kleisli m) where
-	left f = f +++ arr id
-	right f = arr id +++ f
-	f +++ g = (f >>> arr Left) ||| (g >>> arr Right)
-	Kleisli f ||| Kleisli g = Kleisli (either f g)
-
-instance Monad m => ArrowApply (Kleisli m) where
-	app = Kleisli (\(Kleisli f, x) -> f x)
+    loop f b = let (c,d) = f (b,d) in c
 
 instance MonadFix m => ArrowLoop (Kleisli m) where
-	loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
-		where	f' x y = f (x, snd y)
+    loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
+      where f' x y = f (x, snd y)