[project @ 2002-02-26 18:19:17 by ross]

New Control.Arrow module, plus Arrow stub (in lang) for compatability.

diff --git a/Control/Arrow.hs b/Control/Arrow.hs
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+-----------------------------------------------------------------------------
+--
+-- Module      :  Control.Arrow
+-- Copyright   :  (c) Ross Paterson 2002
+-- License     :  BSD-style (see the LICENSE file in the distribution)
+--
+-- Maintainer  :  ross@soi.city.ac.uk
+-- 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.
+--
+-- 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
+
+import Control.Monad
+import Control.Monad.Fix
+
+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)
+
+       -- 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.
+
+       second :: a b c -> a (d,b) (d,c)
+       second f = arr swap >>> first f >>> arr swap
+                       where   swap ~(x,y) = (y,x)
+
+       (***) :: a b c -> a b' c' -> a (b,b') (c,c')
+       f *** g = first f >>> second g
+
+       (&&&) :: a b c -> a b c' -> a b (c,c')
+       f &&& g = arr (\b -> (b,b)) >>> f *** g
+
+       -- Some people prefer the name pure to arr, so both are allowed,
+       -- but you must define one of them:
+
+       pure :: (b -> c) -> a b c
+       pure = arr
+       arr = pure
+
+-----------------------------------------------------------------------------
+-- Derived combinators
+
+-- The counterpart of return in arrow notation:
+
+returnA :: Arrow a => a b b
+returnA = arr id
+
+-- Mirror image of >>>, for a better fit with arrow notation:
+
+(<<<) :: Arrow a => a c d -> a b c -> a b d
+f <<< g = g >>> f
+
+-----------------------------------------------------------------------------
+-- Monoid operations
+
+class Arrow a => ArrowZero a where
+       zeroArrow :: a b c
+
+class ArrowZero a => ArrowPlus a where
+       (<+>) :: a b c -> a b c -> a b c
+
+-----------------------------------------------------------------------------
+-- Conditionals
+
+class Arrow a => ArrowChoice a where
+       left :: a b c -> a (Either b d) (Either c d)
+
+       -- 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.
+
+       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
+
+       (+++) :: a b c -> a b' c' -> a (Either b b') (Either c c')
+       f +++ g = left f >>> right g
+
+       (|||) :: 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
+
+-----------------------------------------------------------------------------
+-- Arrow application
+
+class Arrow a => ArrowApply a where
+       app :: a (a b c, b) c
+
+-- Any instance of ArrowApply can be made into an instance if ArrowChoice
+-- by defining left = leftApp, where
+
+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
+
+-- 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:
+
+newtype ArrowApply a => 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
+
+-- 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)
+
+instance ArrowChoice (->) where
+       left f = f +++ id
+       right f = id +++ f
+       f +++ g = (Left . f) ||| (Right . g)
+       (|||) = either
+
+instance ArrowApply (->) where
+       app (f,x) = f x
+
+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)
+
+instance MonadFix m => ArrowLoop (Kleisli m) where
+       loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
+               where   f' x y = f (x, snd y)