content: {-# LANGUAGE Trustworthy #-}
  {-# LANGUAGE NoImplicitPrelude #-}
  {-# LANGUAGE DeriveFunctor #-}
  {-# LANGUAGE DeriveGeneric #-}
  {-# LANGUAGE StandaloneDeriving #-}
  {-# OPTIONS_GHC -Wno-inline-rule-shadowing #-}
      -- The RULES for the methods of class Arrow may never fire
      -- e.g. compose/arr;  see #10528
  
  -----------------------------------------------------------------------------
  -- |
  -- Module      :  GHC.Internal.Control.Arrow
  -- Copyright   :  (c) Ross Paterson 2002
  -- License     :  BSD-style (see the LICENSE file in the distribution)
  --
  -- Maintainer  :  libraries@haskell.org
  -- Stability   :  provisional
  -- 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.
  --
  -- These papers and more information on arrows can be found at
  -- <http://www.haskell.org/arrows/>.
  
  module GHC.Internal.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 GHC.Internal.Data.Tuple ( fst, snd, uncurry )
  import GHC.Internal.Data.Either
  import GHC.Internal.Control.Monad.Fix
  import GHC.Internal.Control.Category
  import GHC.Internal.Base hiding ( (.), id )
  import GHC.Internal.Generics (Generic, Generic1)
  
  infixr 5 <+>
  infixr 3 ***
  infixr 3 &&&
  infixr 2 +++
  infixr 2 |||
  infixr 1 ^>>, >>^
  infixr 1 ^<<, <<^
  
  -- | The basic arrow class.
  --
  -- Instances should satisfy the following laws:
  --
  --  * @'arr' id = 'id'@
  --
  --  * @'arr' (f >>> g) = 'arr' f >>> 'arr' g@
  --
  --  * @'first' ('arr' f) = 'arr' ('first' f)@
  --
  --  * @'first' (f >>> g) = 'first' f >>> 'first' g@
  --
  --  * @'first' f >>> 'arr' 'fst' = 'arr' 'fst' >>> f@
  --
  --  * @'first' f >>> 'arr' ('id' *** g) = 'arr' ('id' *** g) >>> 'first' f@
  --
  --  * @'first' ('first' f) >>> 'arr' assoc = 'arr' assoc >>> 'first' f@
  --
  -- where
  --
  -- > assoc ((a,b),c) = (a,(b,c))
  --
  -- The other combinators have sensible default definitions,
  -- which may be overridden for efficiency.
  
  class Category a => Arrow a where
      {-# MINIMAL arr, (first | (***)) #-}
  
      -- | Lift a function to an arrow.
      --
      -- >   b ╭───╮ c
      -- > >───┤ f ├───>
      -- >     ╰───╯
      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.
      --
      --   The default definition may be overridden with a more efficient
      --   version if desired.
      --
      -- >   b ╭─────╮ c
      -- > >───┼─ f ─┼───>
      -- > >───┼─────┼───>
      -- >   d ╰─────╯ d
      first :: a b c -> a (b,d) (c,d)
      first = (*** id)
  
      -- | Send the second component of the input through the argument
      --   arrow, and copy the rest unchanged to the output.
      --
      --   The default definition may be overridden with a more efficient
      --   version if desired.
      --
      -- >   d ╭─────╮ d
      -- > >───┼─────┼───>
      -- > >───┼─ f ─┼───>
      -- >   b ╰─────╯ c
      second :: a b c -> a (d,b) (d,c)
      second = (id ***)
  
      -- | 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.
      --
      -- >   b ╭─────╮ c
      -- > >───┼─ f ─┼───>
      -- > >───┼─ g ─┼───>
      -- >   b'╰─────╯ c'
      (***) :: a b c -> a b' c' -> a (b,b') (c,c')
      f *** g = first f >>> arr swap >>> first g >>> arr swap
        where swap ~(x,y) = (y,x)
  
      -- | 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.
      --
      -- >     ╭───────╮ c
      -- >   b │ ┌─ f ─┼───>
      -- > >───┼─┤     │
      -- >     │ └─ g ─┼───>
      -- >     ╰───────╯ c'
      (&&&) :: 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.
  
  -- | @since base-2.01
  instance Arrow (->) where
      arr f = 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 }
  
  -- | @since base-4.14.0.0
  deriving instance Generic (Kleisli m a b)
  
  -- | @since base-4.14.0.0
  deriving instance Generic1 (Kleisli m a)
  
  -- | @since base-4.14.0.0
  deriving instance Functor m => Functor (Kleisli m a)
  
  -- | @since base-4.14.0.0
  instance Applicative m => Applicative (Kleisli m a) where
    pure = Kleisli . const . pure
    {-# INLINE pure #-}
    Kleisli f <*> Kleisli g = Kleisli $ \x -> f x <*> g x
    {-# INLINE (<*>) #-}
    Kleisli f *> Kleisli g = Kleisli $ \x -> f x *> g x
    {-# INLINE (*>) #-}
    Kleisli f <* Kleisli g = Kleisli $ \x -> f x <* g x
    {-# INLINE (<*) #-}
  
  -- | @since base-4.14.0.0
  instance Alternative m => Alternative (Kleisli m a) where
    empty = Kleisli $ const empty
    {-# INLINE empty #-}
    Kleisli f <|> Kleisli g = Kleisli $ \x -> f x <|> g x
    {-# INLINE (<|>) #-}
  
  -- | @since base-4.14.0.0
  instance Monad m => Monad (Kleisli m a) where
    Kleisli f >>= k = Kleisli $ \x -> f x >>= \a -> runKleisli (k a) x
    {-# INLINE (>>=) #-}
  
  -- | @since base-4.14.0.0
  instance MonadPlus m => MonadPlus (Kleisli m a) where
    mzero = Kleisli $ const mzero
    {-# INLINE mzero #-}
    Kleisli f `mplus` Kleisli g = Kleisli $ \x -> f x `mplus` g x
    {-# INLINE mplus #-}
  
  -- | @since base-3.0
  instance Monad m => Category (Kleisli m) where
      id = Kleisli return
      (Kleisli f) . (Kleisli g) = Kleisli (\b -> g b >>= f)
  
  -- | @since base-2.01
  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.
  --
  -- >   b
  -- > >───>
  returnA :: Arrow a => a b b
  returnA = 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
  
  -- | @since base-2.01
  instance MonadPlus m => ArrowZero (Kleisli m) where
      zeroArrow = Kleisli (\_ -> mzero)
  
  -- | A monoid on arrows.
  class ArrowZero a => ArrowPlus a where
      -- | An associative operation with identity 'zeroArrow'.
      (<+>) :: a b c -> a b c -> a b c
  
  -- | @since base-2.01
  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.
  --
  -- Instances should satisfy the following laws:
  --
  --  * @'left' ('arr' f) = 'arr' ('left' f)@
  --
  --  * @'left' (f >>> g) = 'left' f >>> 'left' g@
  --
  --  * @f >>> 'arr' 'Left' = 'arr' 'Left' >>> 'left' f@
  --
  --  * @'left' f >>> 'arr' ('id' +++ g) = 'arr' ('id' +++ g) >>> 'left' f@
  --
  --  * @'left' ('left' f) >>> 'arr' assocsum = 'arr' assocsum >>> 'left' f@
  --
  -- where
  --
  -- > assocsum (Left (Left x)) = Left x
  -- > assocsum (Left (Right y)) = Right (Left y)
  -- > assocsum (Right z) = Right (Right z)
  --
  -- The other combinators have sensible default definitions, which may
  -- be overridden for efficiency.
  
  class Arrow a => ArrowChoice a where
      {-# MINIMAL (left | (+++)) #-}
  
      -- | 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)
      left = (+++ id)
  
      -- | 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 = (id +++)
  
      -- | 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 >>> arr mirror >>> left g >>> arr mirror
        where
          mirror :: Either x y -> Either y x
          mirror (Left x) = Right x
          mirror (Right y) = Left y
  
      -- | 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)
   #-}
  
  -- | @since base-2.01
  instance ArrowChoice (->) where
      left f = f +++ id
      right f = id +++ f
      f +++ g = (Left . f) ||| (Right . g)
      (|||) = either
  
  -- | @since base-2.01
  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.
  -- Instances should satisfy the following laws:
  --
  --  * @'first' ('arr' (\\x -> 'arr' (\\y -> (x,y)))) >>> 'app' = 'id'@
  --
  --  * @'first' ('arr' (g >>>)) >>> 'app' = 'second' g >>> 'app'@
  --
  --  * @'first' ('arr' (>>> h)) >>> 'app' = 'app' >>> h@
  --
  -- Such arrows are equivalent to monads (see 'ArrowMonad').
  
  class Arrow a => ArrowApply a where
      app :: a (a b c, b) c
  
  -- | @since base-2.01
  instance ArrowApply (->) where
      app (f,x) = f x
  
  -- | @since base-2.01
  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)
  
  -- | @since base-4.6.0.0
  instance Arrow a => Functor (ArrowMonad a) where
      fmap f (ArrowMonad m) = ArrowMonad $ m >>> arr f
  
  -- | @since base-4.6.0.0
  instance Arrow a => Applicative (ArrowMonad a) where
     pure x = ArrowMonad (arr (const x))
     ArrowMonad f <*> ArrowMonad x = ArrowMonad (f &&& x >>> arr (uncurry id))
  
  -- | @since base-2.01
  instance ArrowApply a => Monad (ArrowMonad a) where
      ArrowMonad m >>= f = ArrowMonad $
          m >>> arr (\x -> let ArrowMonad h = f x in (h, ())) >>> app
  
  -- | @since base-4.6.0.0
  instance ArrowPlus a => Alternative (ArrowMonad a) where
     empty = ArrowMonad zeroArrow
     ArrowMonad x <|> ArrowMonad y = ArrowMonad (x <+> y)
  
  -- | @since base-4.6.0.0
  instance (ArrowApply a, ArrowPlus a) => MonadPlus (ArrowMonad a)
  
  -- | 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, although the computation occurs only once.
  -- It underlies the @rec@ value recursion construct in arrow notation.
  -- 'loop' should satisfy the following laws:
  --
  -- [/extension/]
  --      @'loop' ('arr' f) = 'arr' (\\ b -> 'fst' ('fix' (\\ (c,d) -> f (b,d))))@
  --
  -- [/left tightening/]
  --      @'loop' ('first' h >>> f) = h >>> 'loop' f@
  --
  -- [/right tightening/]
  --      @'loop' (f >>> 'first' h) = 'loop' f >>> h@
  --
  -- [/sliding/]
  --      @'loop' (f >>> 'arr' ('id' *** k)) = 'loop' ('arr' ('id' *** k) >>> f)@
  --
  -- [/vanishing/]
  --      @'loop' ('loop' f) = 'loop' ('arr' unassoc >>> f >>> 'arr' assoc)@
  --
  -- [/superposing/]
  --      @'second' ('loop' f) = 'loop' ('arr' assoc >>> 'second' f >>> 'arr' unassoc)@
  --
  -- where
  --
  -- > assoc ((a,b),c) = (a,(b,c))
  -- > unassoc (a,(b,c)) = ((a,b),c)
  --
  class Arrow a => ArrowLoop a where
      -- |
      --
      -- >     ╭──────────────╮
      -- >   b │     ╭───╮    │ c
      -- > >───┼─────┤   ├────┼───>
      -- >     │   ┌─┤   ├─┐  │
      -- >     │ d │ ╰───╯ │  │
      -- >     │   └───<───┘  │
      -- >     ╰──────────────╯
      loop :: a (b,d) (c,d) -> a b c
  
  -- | @since base-2.01
  instance ArrowLoop (->) where
      loop f b = let (c,d) = f (b,d) in c
  
  -- | Beware that for many monads (those for which the '>>=' operation
  -- is strict) this instance will /not/ satisfy the right-tightening law
  -- required by the 'ArrowLoop' class.
  --
  -- @since base-2.01
  instance MonadFix m => ArrowLoop (Kleisli m) where
      loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
        where f' x y = f (x, snd y)
  