{-# LANGUAGE CPP #-}
#if __GLASGOW_HASKELL__ >= 707
{-# LANGUAGE DeriveDataTypeable, StandaloneDeriving, Safe, DefaultSignatures #-}
#elif __GLASGOW_HASKELL__ >= 702
{-# LANGUAGE Trustworthy, DefaultSignatures #-}
#endif
#if __GLASGOW_HASKELL__ >= 706
{-# LANGUAGE PolyKinds #-}
#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 = (w a -> w a) -> w a -> w (w a)
forall (w :: * -> *) a b. Comonad w => (w a -> b) -> w a -> w b
extend w a -> w a
forall k (cat :: k -> k -> *) (a :: k). Category cat => cat a a
id

  -- |
  -- @
  -- 'extend' f = 'fmap' f . 'duplicate'
  -- @
  extend :: (w a -> b) -> w a -> w b
  extend w a -> b
f = (w a -> b) -> w (w a) -> w b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap w a -> b
f (w (w a) -> w b) -> (w a -> w (w a)) -> w a -> w b
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. w a -> w (w a)
forall (w :: * -> *) a. Comonad w => w a -> w (w a)
duplicate

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


instance Comonad ((,)e) where
  duplicate :: (e, a) -> (e, (e, a))
duplicate (e, a)
p = ((e, a) -> e
forall a b. (a, b) -> a
fst (e, a)
p, (e, a)
p)
  {-# INLINE duplicate #-}
  extract :: (e, a) -> a
extract = (e, a) -> a
forall e a. (e, a) -> a
snd
  {-# INLINE extract #-}

instance Comonad (Arg e) where
  duplicate :: Arg e a -> Arg e (Arg e a)
duplicate w :: Arg e a
w@(Arg e
a a
_) = e -> Arg e a -> Arg e (Arg e a)
forall a b. a -> b -> Arg a b
Arg e
a Arg e a
w
  {-# INLINE duplicate #-}
  extend :: (Arg e a -> b) -> Arg e a -> Arg e b
extend Arg e a -> b
f w :: Arg e a
w@(Arg e
a a
_) = e -> b -> Arg e b
forall a b. a -> b -> Arg a b
Arg e
a (Arg e a -> b
f Arg e a
w)
  {-# INLINE extend #-}
  extract :: Arg e a -> a
extract (Arg e
_ a
b) = a
b
  {-# INLINE extract #-}

instance Monoid m => Comonad ((->)m) where
  duplicate :: (m -> a) -> m -> m -> a
duplicate m -> a
f m
m = m -> a
f (m -> a) -> (m -> m) -> m -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. m -> m -> m
forall a. Monoid a => a -> a -> a
mappend m
m
  {-# INLINE duplicate #-}
  extract :: (m -> a) -> a
extract m -> a
f = m -> a
f m
forall a. Monoid a => a
mempty
  {-# INLINE extract #-}

instance Comonad Identity where
  duplicate :: Identity a -> Identity (Identity a)
duplicate = Identity a -> Identity (Identity a)
forall a. a -> Identity a
Identity
  {-# INLINE duplicate #-}
  extract :: Identity a -> a
extract = Identity a -> a
forall a. Identity a -> a
runIdentity
  {-# INLINE extract #-}

#if __GLASGOW_HASKELL__ >= 706
-- $
-- The variable `s` can have any kind.
-- For example, here it has kind `Bool`:
-- >>> :set -XDataKinds
-- >>> extract (Tagged 42 :: Tagged 'True Integer)
-- 42
#endif
instance Comonad (Tagged s) where
  duplicate :: Tagged s a -> Tagged s (Tagged s a)
duplicate = Tagged s a -> Tagged s (Tagged s a)
forall k (s :: k) b. b -> Tagged s b
Tagged
  {-# INLINE duplicate #-}
  extract :: Tagged s a -> a
extract = Tagged s a -> a
forall k (s :: k) a. Tagged s a -> a
unTagged
  {-# INLINE extract #-}

instance Comonad w => Comonad (IdentityT w) where
  extend :: (IdentityT w a -> b) -> IdentityT w a -> IdentityT w b
extend IdentityT w a -> b
f (IdentityT w a
m) = w b -> IdentityT w b
forall k (f :: k -> *) (a :: k). f a -> IdentityT f a
IdentityT ((w a -> b) -> w a -> w b
forall (w :: * -> *) a b. Comonad w => (w a -> b) -> w a -> w b
extend (IdentityT w a -> b
f (IdentityT w a -> b) -> (w a -> IdentityT w a) -> w a -> b
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. w a -> IdentityT w a
forall k (f :: k -> *) (a :: k). f a -> IdentityT f a
IdentityT) w a
m)
  extract :: IdentityT w a -> a
extract = w a -> a
forall (w :: * -> *) a. Comonad w => w a -> a
extract (w a -> a) -> (IdentityT w a -> w a) -> IdentityT w a -> a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. IdentityT w a -> w a
forall k (f :: k -> *) (a :: k). IdentityT f a -> f a
runIdentityT
  {-# INLINE extract #-}

#ifdef MIN_VERSION_containers
instance Comonad Tree where
  duplicate :: Tree a -> Tree (Tree a)
duplicate w :: Tree a
w@(Node a
_ [Tree a]
as) = Tree a -> [Tree (Tree a)] -> Tree (Tree a)
forall a. a -> [Tree a] -> Tree a
Node Tree a
w ((Tree a -> Tree (Tree a)) -> [Tree a] -> [Tree (Tree a)]
forall a b. (a -> b) -> [a] -> [b]
map Tree a -> Tree (Tree a)
forall (w :: * -> *) a. Comonad w => w a -> w (w a)
duplicate [Tree a]
as)
  extract :: Tree a -> a
extract (Node a
a [Tree a]
_) = a
a
  {-# INLINE extract #-}
#endif

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

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

instance (Comonad f, Comonad g) => Comonad (FSum.Sum f g) where
  extend :: (Sum f g a -> b) -> Sum f g a -> Sum f g b
extend Sum f g a -> b
f = (f a -> Sum f g b) -> (g a -> Sum f g b) -> Sum f g a -> Sum f g b
forall k (f :: k -> *) (a :: k) b (g :: k -> *).
(f a -> b) -> (g a -> b) -> Sum f g a -> b
coproduct
               (f b -> Sum f g b
forall k (f :: k -> *) (g :: k -> *) (a :: k). f a -> Sum f g a
FSum.InL (f b -> Sum f g b) -> (f a -> f b) -> f a -> Sum f g b
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. (f a -> b) -> f a -> f b
forall (w :: * -> *) a b. Comonad w => (w a -> b) -> w a -> w b
extend (Sum f g a -> b
f (Sum f g a -> b) -> (f a -> Sum f g a) -> f a -> b
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. f a -> Sum f g a
forall k (f :: k -> *) (g :: k -> *) (a :: k). f a -> Sum f g a
FSum.InL))
               (g b -> Sum f g b
forall k (f :: k -> *) (g :: k -> *) (a :: k). g a -> Sum f g a
FSum.InR (g b -> Sum f g b) -> (g a -> g b) -> g a -> Sum f g b
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. (g a -> b) -> g a -> g b
forall (w :: * -> *) a b. Comonad w => (w a -> b) -> w a -> w b
extend (Sum f g a -> b
f (Sum f g a -> b) -> (g a -> Sum f g a) -> g a -> b
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. g a -> Sum f g a
forall k (f :: k -> *) (g :: k -> *) (a :: k). g a -> Sum f g a
FSum.InR))
  extract :: Sum f g a -> a
extract = (f a -> a) -> (g a -> a) -> Sum f g a -> a
forall k (f :: k -> *) (a :: k) b (g :: k -> *).
(f a -> b) -> (g a -> b) -> Sum f g a -> b
coproduct f a -> a
forall (w :: * -> *) a. Comonad w => w a -> a
extract g a -> a
forall (w :: * -> *) a. Comonad w => w a -> a
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
  (<@>) = w (a -> b) -> w a -> w b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
(<*>)
#endif

  (@>) :: w a -> w b -> w b
  w a
a @> w b
b = (b -> b) -> a -> b -> b
forall a b. a -> b -> a
const b -> b
forall k (cat :: k -> k -> *) (a :: k). Category cat => cat a a
id (a -> b -> b) -> w a -> w (b -> b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> w a
a w (b -> b) -> w b -> w b
forall (w :: * -> *) a b.
ComonadApply w =>
w (a -> b) -> w a -> w b
<@> w b
b

  (<@) :: w a -> w b -> w a
  w a
a <@ w b
b = a -> b -> a
forall a b. a -> b -> a
const (a -> b -> a) -> w a -> w (b -> a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> w a
a w (b -> a) -> w b -> w a
forall (w :: * -> *) a b.
ComonadApply w =>
w (a -> b) -> w a -> w b
<@> w b
b

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

instance ComonadApply NonEmpty where
  <@> :: NonEmpty (a -> b) -> NonEmpty a -> NonEmpty b
(<@>) = NonEmpty (a -> b) -> NonEmpty a -> NonEmpty b
forall (m :: * -> *) a b. Monad m => m (a -> b) -> m a -> m b
ap

instance Monoid m => ComonadApply ((->)m) where
  <@> :: (m -> a -> b) -> (m -> a) -> m -> b
(<@>) = (m -> a -> b) -> (m -> a) -> m -> b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
(<*>)
  (<@ ) = (<* )
  ( @>) = ( *>)

instance ComonadApply Identity where
  <@> :: Identity (a -> b) -> Identity a -> Identity b
(<@>) = Identity (a -> b) -> Identity a -> Identity b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
(<*>)
  (<@ ) = (<* )
  ( @>) = ( *>)

instance ComonadApply w => ComonadApply (IdentityT w) where
  IdentityT w (a -> b)
wa <@> :: IdentityT w (a -> b) -> IdentityT w a -> IdentityT w b
<@> IdentityT w a
wb = w b -> IdentityT w b
forall k (f :: k -> *) (a :: k). f a -> IdentityT f a
IdentityT (w (a -> b)
wa w (a -> b) -> w a -> w b
forall (w :: * -> *) a b.
ComonadApply w =>
w (a -> b) -> w a -> w b
<@> w a
wb)

#ifdef MIN_VERSION_containers
instance ComonadApply Tree where
  <@> :: Tree (a -> b) -> Tree a -> Tree b
(<@>) = Tree (a -> b) -> Tree a -> Tree b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
(<*>)
  (<@ ) = (<* )
  ( @>) = ( *>)
#endif

-- | A suitable default definition for 'fmap' for a 'Comonad'.
-- Promotes a function to a comonad.
--
-- You can only safely use 'liftW' to define 'fmap' if your 'Comonad'
-- defines '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 :: (a -> b) -> w a -> w b
liftW a -> b
f = (w a -> b) -> w a -> w b
forall (w :: * -> *) a b. Comonad w => (w a -> b) -> w a -> w b
extend (a -> b
f (a -> b) -> (w a -> a) -> w a -> b
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. w a -> a
forall (w :: * -> *) a. Comonad w => w a -> a
extract)
{-# INLINE liftW #-}

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

-- | Comonadic fixed point à la Dominic Orchard
cfix :: Comonad w => (w a -> a) -> w a
cfix :: (w a -> a) -> w a
cfix w a -> a
f = (w a -> w a) -> w a
forall a. (a -> a) -> a
fix ((w a -> a) -> w a -> w a
forall (w :: * -> *) a b. Comonad w => (w a -> b) -> w a -> w b
extend w a -> a
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 (w a -> a) -> w a
kfix w (w a -> a)
w = (w a -> w a) -> w a
forall a. (a -> a) -> a
fix ((w a -> w a) -> w a) -> (w a -> w a) -> w a
forall a b. (a -> b) -> a -> b
$ \w a
u -> w (w a -> a)
w w (w a -> a) -> w (w a) -> w a
forall (w :: * -> *) a b.
ComonadApply w =>
w (a -> b) -> w a -> w b
<@> w a -> w (w a)
forall (w :: * -> *) a. Comonad w => w a -> w (w a)
duplicate w a
u
{-# INLINE kfix #-}

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

-- | 'extend' in operator form
(<<=) :: Comonad w => (w a -> b) -> w a -> w b
<<= :: (w a -> b) -> w a -> w b
(<<=) = (w a -> b) -> w a -> w b
forall (w :: * -> *) a b. 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
w b -> c
f =<= :: (w b -> c) -> (w a -> b) -> w a -> c
=<= w a -> b
g = w b -> c
f (w b -> c) -> (w a -> w b) -> w a -> c
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. (w a -> b) -> w a -> w b
forall (w :: * -> *) a b. Comonad w => (w a -> b) -> w a -> w b
extend w a -> b
g
{-# INLINE (=<=) #-}

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

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

-- | Lift a binary function into a 'Comonad' with zipping
liftW2 :: ComonadApply w => (a -> b -> c) -> w a -> w b -> w c
liftW2 :: (a -> b -> c) -> w a -> w b -> w c
liftW2 a -> b -> c
f w a
a w b
b = a -> b -> c
f (a -> b -> c) -> w a -> w (b -> c)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> w a
a w (b -> c) -> w b -> w c
forall (w :: * -> *) a b.
ComonadApply w =>
w (a -> b) -> w a -> w b
<@> w b
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 :: (a -> b -> c -> d) -> w a -> w b -> w c -> w d
liftW3 a -> b -> c -> d
f w a
a w b
b w c
c = a -> b -> c -> d
f (a -> b -> c -> d) -> w a -> w (b -> c -> d)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> w a
a w (b -> c -> d) -> w b -> w (c -> d)
forall (w :: * -> *) a b.
ComonadApply w =>
w (a -> b) -> w a -> w b
<@> w b
b w (c -> d) -> w c -> w d
forall (w :: * -> *) a b.
ComonadApply w =>
w (a -> b) -> w a -> w b
<@> w c
c
{-# INLINE liftW3 #-}

-- | The 'Cokleisli' 'Arrow's of a given 'Comonad'
newtype Cokleisli w a b = Cokleisli { Cokleisli w a b -> w a -> b
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 w a a
id = (w a -> a) -> Cokleisli w a a
forall k (w :: k -> *) (a :: k) b. (w a -> b) -> Cokleisli w a b
Cokleisli w a -> a
forall (w :: * -> *) a. Comonad w => w a -> a
extract
  Cokleisli w b -> c
f . :: Cokleisli w b c -> Cokleisli w a b -> Cokleisli w a c
. Cokleisli w a -> b
g = (w a -> c) -> Cokleisli w a c
forall k (w :: k -> *) (a :: k) b. (w a -> b) -> Cokleisli w a b
Cokleisli (w b -> c
f (w b -> c) -> (w a -> b) -> w a -> c
forall (w :: * -> *) b c a.
Comonad w =>
(w b -> c) -> (w a -> b) -> w a -> c
=<= w a -> b
g)

instance Comonad w => Arrow (Cokleisli w) where
  arr :: (b -> c) -> Cokleisli w b c
arr b -> c
f = (w b -> c) -> Cokleisli w b c
forall k (w :: k -> *) (a :: k) b. (w a -> b) -> Cokleisli w a b
Cokleisli (b -> c
f (b -> c) -> (w b -> b) -> w b -> c
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. w b -> b
forall (w :: * -> *) a. Comonad w => w a -> a
extract)
  first :: Cokleisli w b c -> Cokleisli w (b, d) (c, d)
first Cokleisli w b c
f = Cokleisli w b c
f Cokleisli w b c -> Cokleisli w d d -> Cokleisli w (b, d) (c, d)
forall (a :: * -> * -> *) b c b' c'.
Arrow a =>
a b c -> a b' c' -> a (b, b') (c, c')
*** Cokleisli w d d
forall k (cat :: k -> k -> *) (a :: k). Category cat => cat a a
id
  second :: Cokleisli w b c -> Cokleisli w (d, b) (d, c)
second Cokleisli w b c
f = Cokleisli w d d
forall k (cat :: k -> k -> *) (a :: k). Category cat => cat a a
id Cokleisli w d d -> Cokleisli w b c -> Cokleisli w (d, b) (d, c)
forall (a :: * -> * -> *) b c b' c'.
Arrow a =>
a b c -> a b' c' -> a (b, b') (c, c')
*** Cokleisli w b c
f
  Cokleisli w b -> c
f *** :: Cokleisli w b c -> Cokleisli w b' c' -> Cokleisli w (b, b') (c, c')
*** Cokleisli w b' -> c'
g = (w (b, b') -> (c, c')) -> Cokleisli w (b, b') (c, c')
forall k (w :: k -> *) (a :: k) b. (w a -> b) -> Cokleisli w a b
Cokleisli (w b -> c
f (w b -> c) -> (w (b, b') -> w b) -> w (b, b') -> c
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. ((b, b') -> b) -> w (b, b') -> w b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (b, b') -> b
forall a b. (a, b) -> a
fst (w (b, b') -> c) -> (w (b, b') -> c') -> w (b, b') -> (c, c')
forall (a :: * -> * -> *) b c c'.
Arrow a =>
a b c -> a b c' -> a b (c, c')
&&& w b' -> c'
g (w b' -> c') -> (w (b, b') -> w b') -> w (b, b') -> c'
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. ((b, b') -> b') -> w (b, b') -> w b'
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (b, b') -> b'
forall e a. (e, a) -> a
snd)
  Cokleisli w b -> c
f &&& :: Cokleisli w b c -> Cokleisli w b c' -> Cokleisli w b (c, c')
&&& Cokleisli w b -> c'
g = (w b -> (c, c')) -> Cokleisli w b (c, c')
forall k (w :: k -> *) (a :: k) b. (w a -> b) -> Cokleisli w a b
Cokleisli (w b -> c
f (w b -> c) -> (w b -> c') -> w b -> (c, c')
forall (a :: * -> * -> *) b c c'.
Arrow a =>
a b c -> a b c' -> a b (c, c')
&&& w b -> c'
g)

instance Comonad w => ArrowApply (Cokleisli w) where
  app :: Cokleisli w (Cokleisli w b c, b) c
app = (w (Cokleisli w b c, b) -> c) -> Cokleisli w (Cokleisli w b c, b) c
forall k (w :: k -> *) (a :: k) b. (w a -> b) -> Cokleisli w a b
Cokleisli ((w (Cokleisli w b c, b) -> c)
 -> Cokleisli w (Cokleisli w b c, b) c)
-> (w (Cokleisli w b c, b) -> c)
-> Cokleisli w (Cokleisli w b c, b) c
forall a b. (a -> b) -> a -> b
$ \w (Cokleisli w b c, b)
w -> Cokleisli w b c -> w b -> c
forall k (w :: k -> *) (a :: k) b. Cokleisli w a b -> w a -> b
runCokleisli ((Cokleisli w b c, b) -> Cokleisli w b c
forall a b. (a, b) -> a
fst (w (Cokleisli w b c, b) -> (Cokleisli w b c, b)
forall (w :: * -> *) a. Comonad w => w a -> a
extract w (Cokleisli w b c, b)
w)) ((Cokleisli w b c, b) -> b
forall e a. (e, a) -> a
snd ((Cokleisli w b c, b) -> b) -> w (Cokleisli w b c, b) -> w b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> w (Cokleisli w b c, b)
w)

instance Comonad w => ArrowChoice (Cokleisli w) where
  left :: Cokleisli w b c -> Cokleisli w (Either b d) (Either c d)
left = Cokleisli w b c -> Cokleisli w (Either b d) (Either c d)
forall (a :: * -> * -> *) b c d.
ArrowApply a =>
a b c -> a (Either b d) (Either c d)
leftApp

instance ComonadApply w => ArrowLoop (Cokleisli w) where
  loop :: Cokleisli w (b, d) (c, d) -> Cokleisli w b c
loop (Cokleisli w (b, d) -> (c, d)
f) = (w b -> c) -> Cokleisli w b c
forall k (w :: k -> *) (a :: k) b. (w a -> b) -> Cokleisli w a b
Cokleisli ((c, d) -> c
forall a b. (a, b) -> a
fst ((c, d) -> c) -> (w b -> (c, d)) -> w b -> c
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. w (w (c, d) -> (c, d)) -> (c, d)
forall (w :: * -> *) a. Comonad w => w (w a -> a) -> a
wfix (w (w (c, d) -> (c, d)) -> (c, d))
-> (w b -> w (w (c, d) -> (c, d))) -> w b -> (c, d)
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. (w b -> w (c, d) -> (c, d)) -> w b -> w (w (c, d) -> (c, d))
forall (w :: * -> *) a b. Comonad w => (w a -> b) -> w a -> w b
extend w b -> w (c, d) -> (c, d)
forall a. w b -> w (a, d) -> (c, d)
f') where
    f' :: w b -> w (a, d) -> (c, d)
f' w b
wa w (a, d)
wb = w (b, d) -> (c, d)
f ((,) (b -> d -> (b, d)) -> w b -> w (d -> (b, d))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> w b
wa w (d -> (b, d)) -> w d -> w (b, d)
forall (w :: * -> *) a b.
ComonadApply w =>
w (a -> b) -> w a -> w b
<@> ((a, d) -> d
forall e a. (e, a) -> a
snd ((a, d) -> d) -> w (a, d) -> w d
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> w (a, d)
wb))

instance Functor (Cokleisli w a) where
  fmap :: (a -> b) -> Cokleisli w a a -> Cokleisli w a b
fmap a -> b
f (Cokleisli w a -> a
g) = (w a -> b) -> Cokleisli w a b
forall k (w :: k -> *) (a :: k) b. (w a -> b) -> Cokleisli w a b
Cokleisli (a -> b
f (a -> b) -> (w a -> a) -> w a -> b
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. w a -> a
g)

instance Applicative (Cokleisli w a) where
  pure :: a -> Cokleisli w a a
pure = (w a -> a) -> Cokleisli w a a
forall k (w :: k -> *) (a :: k) b. (w a -> b) -> Cokleisli w a b
Cokleisli ((w a -> a) -> Cokleisli w a a)
-> (a -> w a -> a) -> a -> Cokleisli w a a
forall k (cat :: k -> k -> *) (b :: k) (c :: k) (a :: k).
Category cat =>
cat b c -> cat a b -> cat a c
. a -> w a -> a
forall a b. a -> b -> a
const
  Cokleisli w a -> a -> b
f <*> :: Cokleisli w a (a -> b) -> Cokleisli w a a -> Cokleisli w a b
<*> Cokleisli w a -> a
a = (w a -> b) -> Cokleisli w a b
forall k (w :: k -> *) (a :: k) b. (w a -> b) -> Cokleisli w a b
Cokleisli (\w a
w -> w a -> a -> b
f w a
w (w a -> a
a w a
w))

instance Monad (Cokleisli w a) where
  return :: a -> Cokleisli w a a
return = a -> Cokleisli w a a
forall (f :: * -> *) a. Applicative f => a -> f a
pure
  Cokleisli w a -> a
k >>= :: Cokleisli w a a -> (a -> Cokleisli w a b) -> Cokleisli w a b
>>= a -> Cokleisli w a b
f = (w a -> b) -> Cokleisli w a b
forall k (w :: k -> *) (a :: k) b. (w a -> b) -> Cokleisli w a b
Cokleisli ((w a -> b) -> Cokleisli w a b) -> (w a -> b) -> Cokleisli w a b
forall a b. (a -> b) -> a -> b
$ \w a
w -> Cokleisli w a b -> w a -> b
forall k (w :: k -> *) (a :: k) b. Cokleisli w a b -> w a -> b
runCokleisli (a -> Cokleisli w a b
f (w a -> a
k w a
w)) w a
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