```-----------------------------------------------------------------------------
-- |
-- Module      :  Control.Arrow
-- Copyright   :  (c) Ross Paterson 2002
--
-- 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

module Control.Arrow (
-- * Arrows
Arrow(..), Kleisli(..),
-- ** Derived combinators
returnA,
(^>>), (>>^),
-- ** Right-to-left variants
(<<<), (<<^), (^<<),
-- * Monoid operations
ArrowZero(..), ArrowPlus(..),
-- * Conditionals
ArrowChoice(..),
-- * Arrow application
-- * Feedback
ArrowLoop(..)
) where

import Prelude

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 '>>>' and 'first'.  The other combinators have sensible
--   default definitions, which may be overridden for efficiency.

class 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

-- | 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 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
"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)
--	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 => 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))

-- | 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

-- | Right-to-left composition, for a better fit with arrow notation.
(<<<) :: Arrow a => a c d -> a b c -> a b d
f <<< g = g >>> 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 ArrowApply a => ArrowMonad a b = ArrowMonad (a () b)

return x = ArrowMonad (arr (\z -> x))
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)
```