content: -----------------------------------------------------------------------------
  -- |
  -- 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
  --
  -- 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.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 ^<<, <<^
  
  -- | The basic arrow class.
  --   Any instance must define either 'arr' or 'pure' (which are synonyms),
  --   as well as '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: you must define either this
          --   or 'pure'.
          arr :: (b -> c) -> a b c
          arr = pure
  
          -- | A synonym for 'arr': you must define one or other of them.
          pure :: (b -> c) -> a b c
          pure = arr
  
          -- | 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)
  
          -- | 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
  "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
          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)
  
  -- | Kleisli arrows of a monad.
  
  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)
          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 identity arrow, which plays the role of 'return' in arrow notation.
  
  returnA :: Arrow a => a b b
  returnA = arr id
  
  -- | 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
  
  -- | 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 => ArrowZero (Kleisli m) where
          zeroArrow = Kleisli (\x -> mzero)
  
  class ArrowZero a => ArrowPlus a where
          (<+>) :: a b c -> a b c -> a b c
  
  instance MonadPlus m => ArrowPlus (Kleisli m) where
          Kleisli f <+> Kleisli g = Kleisli (\x -> f x `mplus` g x)
  
  -- | 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.
  
  class Arrow a => ArrowChoice a where
  
          -- | 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 (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)
   #-}
  
  instance ArrowChoice (->) where
          left f = f +++ id
          right f = id +++ f
          f +++ g = (Left . f) ||| (Right . g)
          (|||) = either
  
  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)
  
  -- | Some arrows allow application of arrow inputs to other inputs.
  
  class Arrow a => ArrowApply a where
          app :: a (a b c, b) c
  
  instance ArrowApply (->) where
          app (f,x) = f x
  
  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 'Kleisli' arrow, and any instance of 'ArrowApply' defines a monad.
  
  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)
  
  -- | Any instance of 'ArrowApply' can be made into an instance of
  --   'ArrowChoice' by defining 'left' = 'leftApp'.
  
  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 '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.
  
  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
  
  instance MonadFix m => ArrowLoop (Kleisli m) where
          loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
                  where   f' x y = f (x, snd y)
  