    {-# LANGUAGE CPP #-}
    #if __GLASGOW_HASKELL__ >= 707
    {-# LANGUAGE DeriveDataTypeable, StandaloneDeriving, Safe, DefaultSignatures #-}
    #elif __GLASGOW_HASKELL__ >= 702
    {-# LANGUAGE Trustworthy, DefaultSignatures #-}
    #endif
     -----------------------------------------------------------------------------
    -- |
    -- Module      :  Control.Comonad
    -- Copyright   :  (C) 2008-2015 Edward Kmett,
    --                (C) 2004 Dave Menendez
    -- License     :  BSD-style (see the file LICENSE)
    --
    -- Maintainer  :  Edward Kmett <ekmett@gmail.com>
    -- Stability   :  provisional
    -- Portability :  portable
    --
    ----------------------------------------------------------------------------
    module Control.Comonad (
      -- * Comonads
        Comonad(..)
      , liftW     -- :: Comonad w => (a -> b) -> w a -> w b
      , wfix      -- :: Comonad w => w (w a -> a) -> a
      , cfix      -- :: Comonad w => (w a -> a) -> w a
      , kfix      -- :: ComonadApply w => w (w a -> a) -> w a
      , (=>=)
      , (=<=)
      , (<<=)
      , (=>>)
      -- * Combining Comonads
      , ComonadApply(..)
      , (<@@>)    -- :: ComonadApply w => w a -> w (a -> b) -> w b
      , liftW2    -- :: ComonadApply w => (a -> b -> c) -> w a -> w b -> w c
      , liftW3    -- :: ComonadApply w => (a -> b -> c -> d) -> w a -> w b -> w c -> w d
      -- * Cokleisli Arrows
      , Cokleisli(..)
      -- * Functors
      , Functor(..)
      , (<$>)     -- :: Functor f => (a -> b) -> f a -> f b
      , ($>)      -- :: Functor f => f a -> b -> f b
      ) where

    -- import _everything_
    import Data.Functor
    import Control.Applicative
    import Control.Arrow
    import Control.Category
    import Control.Monad (ap)
    #if MIN_VERSION_base(4,7,0)
    -- Control.Monad.Instances is empty
    #else
    import Control.Monad.Instances
    #endif
    import Control.Monad.Trans.Identity
    import Data.Functor.Identity
    import qualified Data.Functor.Sum as FSum
    import Data.List.NonEmpty hiding (map)
    import Data.Semigroup hiding (Product)
    import Data.Tagged
    import Prelude hiding (id, (.))
    import Control.Monad.Fix
    import Data.Typeable

    #ifdef MIN_VERSION_containers
    import Data.Tree
    #endif

    infixl 4 <@, @>, <@@>, <@>
    infixl 1 =>>
    infixr 1 <<=, =<=, =>=

    {- |

    There are two ways to define a comonad:

    I. Provide definitions for 'extract' and 'extend'
    satisfying these laws:

    @
    'extend' 'extract'      = 'id'
    'extract' . 'extend' f  = f
    'extend' f . 'extend' g = 'extend' (f . 'extend' g)
    @

    In this case, you may simply set 'fmap' = 'liftW'.

    These laws are directly analogous to the laws for monads
    and perhaps can be made clearer by viewing them as laws stating
    that Cokleisli composition must be associative, and has extract for
    a unit:

    @
    f '=>=' 'extract'   = f
    'extract' '=>=' f   = f
    (f '=>=' g) '=>=' h = f '=>=' (g '=>=' h)
    @

    II. Alternately, you may choose to provide definitions for 'fmap',
    'extract', and 'duplicate' satisfying these laws:

    @
    'extract' . 'duplicate'      = 'id'
    'fmap' 'extract' . 'duplicate' = 'id'
    'duplicate' . 'duplicate'    = 'fmap' 'duplicate' . 'duplicate'
    @

    In this case you may not rely on the ability to define 'fmap' in
    terms of 'liftW'.

    You may of course, choose to define both 'duplicate' /and/ 'extend'.
    In that case you must also satisfy these laws:

    @
    'extend' f  = 'fmap' f . 'duplicate'
    'duplicate' = 'extend' id
    'fmap' f    = 'extend' (f . 'extract')
    @

    These are the default definitions of 'extend' and 'duplicate' and
    the definition of 'liftW' respectively.

    -}

    class Functor w => Comonad w where
      -- |
      -- @
      -- 'extract' . 'fmap' f = f . 'extract'
      -- @
      extract :: w a -> a

      -- |
      -- @
      -- 'duplicate' = 'extend' 'id'
      -- 'fmap' ('fmap' f) . 'duplicate' = 'duplicate' . 'fmap' f
      -- @
      duplicate :: w a -> w (w a)
      duplicate = extend id

      -- |
      -- @
      -- 'extend' f = 'fmap' f . 'duplicate'
      -- @
      extend :: (w a -> b) -> w a -> w b
      extend f = fmap f . duplicate

    #if __GLASGOW_HASKELL__ >= 708
      {-# MINIMAL extract, (duplicate | extend) #-}
    #endif

    instance Comonad ((,)e) where
      duplicate p = (fst p, p)
      {-# INLINE duplicate #-}
      extract = snd
      {-# INLINE extract #-}

    #if MIN_VERSION_semigroups(0,16,2)
    instance Comonad (Arg e) where
      duplicate w@(Arg a _) = Arg a w
      {-# INLINE duplicate #-}
      extend f w@(Arg a _) = Arg a (f w)
      {-# INLINE extend #-}
      extract (Arg _ b) = b
      {-# INLINE extract #-}
    #endif

    instance Monoid m => Comonad ((->)m) where
      duplicate f m = f . mappend m
      {-# INLINE duplicate #-}
      extract f = f mempty
      {-# INLINE extract #-}

    instance Comonad Identity where
      duplicate = Identity
      {-# INLINE duplicate #-}
      extract = runIdentity
      {-# INLINE extract #-}

    instance Comonad (Tagged s) where
      duplicate = Tagged
      {-# INLINE duplicate #-}
      extract = unTagged
      {-# INLINE extract #-}

    instance Comonad w => Comonad (IdentityT w) where
      extend f (IdentityT m) = IdentityT (extend (f . IdentityT) m)
      extract = extract . runIdentityT
      {-# INLINE extract #-}

    #ifdef MIN_VERSION_containers
    instance Comonad Tree where
      duplicate w@(Node _ as) = Node w (map duplicate as)
      extract (Node a _) = a
      {-# INLINE extract #-}
    #endif

    instance Comonad NonEmpty where
      extend f w@ ~(_ :| aas) = f w :| case aas of
          []     -> []
          (a:as) -> toList (extend f (a :| as))
      extract ~(a :| _) = a
      {-# INLINE extract #-}

    coproduct :: (f a -> b) -> (g a -> b) -> FSum.Sum f g a -> b
    coproduct f _ (FSum.InL x) = f x
    coproduct _ g (FSum.InR y) = g y
    {-# INLINE coproduct #-}

    instance (Comonad f, Comonad g) => Comonad (FSum.Sum f g) where
      extend f = coproduct
                   (FSum.InL . extend (f . FSum.InL))
                   (FSum.InR . extend (f . FSum.InR))
      extract = coproduct extract extract
      {-# INLINE extract #-}

    -- | @ComonadApply@ is to @Comonad@ like @Applicative@ is to @Monad@.
    --
    -- Mathematically, it is a strong lax symmetric semi-monoidal comonad on the
    -- category @Hask@ of Haskell types. That it to say that @w@ is a strong lax
    -- symmetric semi-monoidal functor on Hask, where both 'extract' and 'duplicate' are
    -- symmetric monoidal natural transformations.
    --
    -- Laws:
    --
    -- @
    -- ('.') '<$>' u '<@>' v '<@>' w = u '<@>' (v '<@>' w)
    -- 'extract' (p '<@>' q) = 'extract' p ('extract' q)
    -- 'duplicate' (p '<@>' q) = ('<@>') '<$>' 'duplicate' p '<@>' 'duplicate' q
    -- @
    --
    -- If our type is both a 'ComonadApply' and 'Applicative' we further require
    --
    -- @
    -- ('<*>') = ('<@>')
    -- @
    --
    -- Finally, if you choose to define ('<@') and ('@>'), the results of your
    -- definitions should match the following laws:
    --
    -- @
    -- a '@>' b = 'const' 'id' '<$>' a '<@>' b
    -- a '<@' b = 'const' '<$>' a '<@>' b
    -- @

    class Comonad w => ComonadApply w where
      (<@>) :: w (a -> b) -> w a -> w b
    #if __GLASGOW_HASKELL__ >= 702
      default (<@>) :: Applicative w => w (a -> b) -> w a -> w b
      (<@>) = (<*>)
    #endif

      (@>) :: w a -> w b -> w b
      a @> b = const id <$> a <@> b

      (<@) :: w a -> w b -> w a
      a <@ b = const <$> a <@> b

    instance Semigroup m => ComonadApply ((,)m) where
      (m, f) <@> (n, a) = (m <> n, f a)
      (m, a) <@  (n, _) = (m <> n, a)
      (m, _)  @> (n, b) = (m <> n, b)

    instance ComonadApply NonEmpty where
      (<@>) = ap

    instance Monoid m => ComonadApply ((->)m) where
      (<@>) = (<*>)
      (<@ ) = (<* )
      ( @>) = ( *>)

    instance ComonadApply Identity where
      (<@>) = (<*>)
      (<@ ) = (<* )
      ( @>) = ( *>)

    instance ComonadApply w => ComonadApply (IdentityT w) where
      IdentityT wa <@> IdentityT wb = IdentityT (wa <@> wb)

    #ifdef MIN_VERSION_containers
    instance ComonadApply Tree where
      (<@>) = (<*>)
      (<@ ) = (<* )
      ( @>) = ( *>)
    #endif

    -- | A suitable default definition for 'fmap' for a 'Comonad'.
    -- Promotes a function to a comonad.
    --
    -- You can only safely use to define 'fmap' if your 'Comonad'
    -- defined 'extend', not just 'duplicate', since defining
    -- 'extend' in terms of duplicate uses 'fmap'!
    --
    -- @
    -- 'fmap' f = 'liftW' f = 'extend' (f . 'extract')
    -- @
    liftW :: Comonad w => (a -> b) -> w a -> w b
    liftW f = extend (f . extract)
    {-# INLINE liftW #-}

    -- | Comonadic fixed point à la David Menendez
    wfix :: Comonad w => w (w a -> a) -> a
    wfix w = extract w (extend wfix w)

    -- | Comonadic fixed point à la Dominic Orchard
    cfix :: Comonad w => (w a -> a) -> w a
    cfix f = fix (extend f)
    {-# INLINE cfix #-}

    -- | Comonadic fixed point à la Kenneth Foner:
    --
    -- This is the @evaluate@ function from his <https://www.youtube.com/watch?v=F7F-BzOB670 "Getting a Quick Fix on Comonads"> talk.
    kfix :: ComonadApply w => w (w a -> a) -> w a
    kfix w = fix $ \u -> w <@> duplicate u
    {-# INLINE kfix #-}

    -- | 'extend' with the arguments swapped. Dual to '>>=' for a 'Monad'.
    (=>>) :: Comonad w => w a -> (w a -> b) -> w b
    (=>>) = flip extend
    {-# INLINE (=>>) #-}

    -- | 'extend' in operator form
    (<<=) :: Comonad w => (w a -> b) -> w a -> w b
    (<<=) = extend
    {-# INLINE (<<=) #-}

    -- | Right-to-left 'Cokleisli' composition
    (=<=) :: Comonad w => (w b -> c) -> (w a -> b) -> w a -> c
    f =<= g = f . extend g
    {-# INLINE (=<=) #-}

    -- | Left-to-right 'Cokleisli' composition
    (=>=) :: Comonad w => (w a -> b) -> (w b -> c) -> w a -> c
    f =>= g = g . extend f
    {-# INLINE (=>=) #-}

    -- | A variant of '<@>' with the arguments reversed.
    (<@@>) :: ComonadApply w => w a -> w (a -> b) -> w b
    (<@@>) = liftW2 (flip id)
    {-# INLINE (<@@>) #-}

    -- | Lift a binary function into a 'Comonad' with zipping
    liftW2 :: ComonadApply w => (a -> b -> c) -> w a -> w b -> w c
    liftW2 f a b = f <$> a <@> b
    {-# INLINE liftW2 #-}

    -- | Lift a ternary function into a 'Comonad' with zipping
    liftW3 :: ComonadApply w => (a -> b -> c -> d) -> w a -> w b -> w c -> w d
    liftW3 f a b c = f <$> a <@> b <@> c
    {-# INLINE liftW3 #-}

    -- | The 'Cokleisli' 'Arrow's of a given 'Comonad'
    newtype Cokleisli w a b = Cokleisli { runCokleisli :: w a -> b }
    #if __GLASGOW_HASKELL__ >= 707
      deriving Typeable
    #else
    #ifdef __GLASGOW_HASKELL__
    instance Typeable1 w => Typeable2 (Cokleisli w) where
      typeOf2 twab = mkTyConApp cokleisliTyCon [typeOf1 (wa twab)]
            where wa :: Cokleisli w a b -> w a
                  wa = undefined
    #endif

    cokleisliTyCon :: TyCon
    #if MIN_VERSION_base(4,4,0)
    cokleisliTyCon = mkTyCon3 "comonad" "Control.Comonad" "Cokleisli"
    #else
    cokleisliTyCon = mkTyCon "Control.Comonad.Cokleisli"
    #endif
    {-# NOINLINE cokleisliTyCon #-}
    #endif

    instance Comonad w => Category (Cokleisli w) where
      id = Cokleisli extract
      Cokleisli f . Cokleisli g = Cokleisli (f =<= g)

    instance Comonad w => Arrow (Cokleisli w) where
      arr f = Cokleisli (f . extract)
      first f = f *** id
      second f = id *** f
      Cokleisli f *** Cokleisli g = Cokleisli (f . fmap fst &&& g . fmap snd)
      Cokleisli f &&& Cokleisli g = Cokleisli (f &&& g)

    instance Comonad w => ArrowApply (Cokleisli w) where
      app = Cokleisli $ \w -> runCokleisli (fst (extract w)) (snd <$> w)

    instance Comonad w => ArrowChoice (Cokleisli w) where
      left = leftApp

    instance ComonadApply w => ArrowLoop (Cokleisli w) where
      loop (Cokleisli f) = Cokleisli (fst . wfix . extend f') where
        f' wa wb = f ((,) <$> wa <@> (snd <$> wb))

    instance Functor (Cokleisli w a) where
      fmap f (Cokleisli g) = Cokleisli (f . g)

    instance Applicative (Cokleisli w a) where
      pure = Cokleisli . const
      Cokleisli f <*> Cokleisli a = Cokleisli (\w -> f w (a w))

    instance Monad (Cokleisli w a) where
      return = pure
      Cokleisli k >>= f = Cokleisli $ \w -> runCokleisli (f (k w)) w

    #if !(MIN_VERSION_base(4,7,0))

    infixl 4 $>

    -- | Replace the contents of a functor uniformly with a constant value.
    ($>) :: Functor f => f a -> b -> f b
    ($>) = flip (<$)

    #endif