-----------------------------------------------------------------------------
-- |
-- 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)
